Mechanical and Microstructural Properties of Biological Materials

A thesis presented by

Megan Theresa Valentine

to

The Department of Physics in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in the subject of Physics.

Harvard University

Cambridge, Massachusetts

July, 2003

2003 – Megan Theresa Valentine

All rights reserved.

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Megan Theresa Valentine

Mechanical and Microstructural Properties of Biological Materials

David A. Weitz

Abstract

Biological materials, such as isolated protein networks, tissues and living cells are

extremely heterogeneous and contain structures on length scales from nanometers to

hundreds of microns. Their complex geometries and sensitivity to environmental

conditions make traditional measurements of structure and mechanics difficult,

challenging experimenters to develop new techniques. In this work, we present the

design of a novel microscope-based static light scattering instrument, and demonstrate its

usefulness by studying striated and smooth muscle and skin. We measure the two-

dimensional scattering patterns, and find that tissue structure can give rise to strong

anisotropies, which can be used to identify specific classes of tissues.

We also report the development of multiple particle tracking techniques, using the

thermal movements of colloids that have been embedded in soft complex materials to

measure local viscoelastic response. These measurements require only tens of microliters

of material, making them particularly useful for studying biological samples that are

difficult to obtain in large quantities, or inherently small like living cells. For materials

that are homogenous on the length scale of the probe particle, these thermal movements

have been previously shown to be a direct measure of the macroscopic linear frequency-

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dependent viscoelastic response1. In this work, we have extended these techniques to

probe the viscoelastic and microstructural properties of a number of heterogeneous

materials, including structured polysaccharide gels, biopolymer networks, and isolated

cytoplasm. To better understand the mechanical microenvironments of these

heterogeneous materials, we examine the effect of varying both tracer size and surface

chemistry, and present a novel protocol to render colloids protein-resistant using only

commercial reagents.

In cases where larger sample volumes are available, we also use macroscopic,

mechanical rheology methods to characterize the frequency dependent moduli and non-

linear viscoelastic response. Using a combination of microscopic and macroscopic

techniques we report the first characterization of the mechanical properties of isolated

cytoplasm, obtained from the eggs of Xenopus laevis, and discuss the role of the three

cytoskeletal filaments, actin, microtubules, and the intermediate filament cytokeratin, in

the viscoelastic response.

1 Mason, T.G., D.A. Weitz (1995) Physical Review Letters 74: 1250

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Table of Contents

1. Microscope-based light scattering instrument

1.1 Overview…………………………………………………………………… 1 1.2 Introduction………………………………………………………………... 1 1.3 Instrumental Design……………………………………………………….. 2 1.4 Scattering from dilute suspensions of colloidal spheres………………...… 6 1.5 Scattering from heterogeneous biological tissue………………………..… 6 1.6 Summary………………………………………………………………...… 11

2. The microscopic origin of light scattering in tissue

2.1 Overview……………………………………………………………..…… 12 2.2 Introduction……………………………………………………..………… 12 2.3 Materials and Methods

2.3.1 Sample processing………………………………………...……...… 20 2.3.2 Static light-scattering microscope………………………..………… 20

2.4 Results…………………………………………………………..………… 24 2.5 Discussion……………………………………………………..………..… 44

3. Microrheology

3.1 Introduction………………….…………………………..……………….. 47 3.2 Active Microrheology Methods 3.2.1 Magnetic Manipulation Techniques……………...…...…………… 50 3.2.2 Optical Tweezers Techniques…………………...……….………… 61 3.2.3 Atomic Force Microscopy Techniques…………...………...……… 72 3.3 Passive Microrheology Methods……………………………………..…… 79 3.4 Practical Applications of One-particle Microrheology…………………… 96 3.5 Two-particle Microrheology……………………………………………… 100 3.6 Summary……………………………………………………………..…… 106 4. Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking

4.1 Overview……………………………………………………………..….. 107 4.2 Introduction…………………………………………………………....… 108 4.3 Experimental Details……………………………………………..……… 115 4.4 Results and Discussion………………………………………………...… 118 4.5 Summary……………………………………………………………….… 138

5. Two-particle microrheology of inhomogeneous soft materials………..........… 141 6. Multiple particle tracking measurements of biomaterials: Effect of colloid surface chemistry

6.1 Overview…………………………………………………………...…..… 153 6.2 Introduction……………………………………………………..…...…… 154 6.3 Materials and Methods

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6.3.1 Preparation of the PEG-coated particles……………..…..………… 162 6.3.2 Preparation of the BSA-coated particles…………..……..………… 164

6.3.3 Characterization of protein adsorption using fluorescent BSA…..… 165 6.3.4 Multiple particle tracking…………………………………………... 165 6.3.5 Interpreting particle motions as mechanical response……………… 167 6.3.6 Preparation of biopolymer gels…………………………………….. 169

6.4 Results and Discussion 6.4.1 Binding capacity of CML, BSA, and PEG particles determined by adsorption of fluorescent BSA………………………………..……. 171 6.4.2 Particle mobility in a fibrin network……………………………..… 174 6.4.3 Surface chemistry effects on the microrheology of entangled F-actin networks………………………………………………..…... 179 6.4.4 One- and two-particle microrheology measurements of F-actin/ Scruin networks…………………………………………………….. 181

6.5 Summary………………………………………………………………….. 189 7. Mechanics and microstructure of Xenopus Egg cytoplasmic extracts

7.1 Overview…………………………………………………………………. 192 7.2 Introduction…………………………………………………………….… 192 7.3 Materials and Methods

7.3.1 Xenopus Egg Cytoplasmic Extracts…………………………......… 199 7.3.2 Poly(ethylene glycol) (PEG)-coated particles………………..….... 200 7.3.3 Sample preparation…………………………………………...…… 201 7.3.4 Multiple particle tracking…………………………………............. 202 7.3.5 Interpreting particle motions as mechanical response……..……… 203 7.3.6 Fixation of samples and confocal imaging………………............... 204 7.3.7 Macroscopic rheology…………………………………………..… 205 7.3.8 Contraction Assay……………………………………………..….. 206 7.4 Results and Discussion 7.4.1 Actin-based gelation causes macroscopic gel contraction….…….. 207

7.4.2 Macroscopic rheological measurements………………………...… 209 7.4.3 Imaging cytoskeletal structures…………………………………… 217 7.4.4 Measuring mechanical response on micron length scales using multiple particle tracking………………………………………….. 225 7.5 Summary…………………………………………………………………. 228

A. Experimental Techniques A1. Dynamic Light Scattering…………………………………….....……… 230

A2. Diffusing Wave Spectroscopy…………………………………..……… 237 A3. Video-based Particle Tracking………………………………..………… 246

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List of Figures 1.1 Schematic of the optical arrangement of the Static Light Scattering Microscope……………………………………………………………………... 3 1.2 I(q) measured from a dilute suspension of 2.0 µm spheres in a 1:1 mixture of H2O:D2O at φ=10-3……………………………………………………………… 8 1.3 (A-B) Real space and scattering spaces images from a monolayer of 2.0 µm particles. (C-D) Real space and scattering space images from porcine skin tissue..................................................................................................................... 9 2.1 Real space images and scattering patterns from different tissues……………… 19 2.2 Schematic of the static light scattering microscope………………………….… 23 2.3 Henyey- Greenstein fits of the form factors <I(θ)>ϕ…………………………… 27 2.4 Slice- averaged form factors of different tissue types show a power- law decay I(q)~q-z…………………………………………………………………… 30 2.5 Radial intensity distribution I(ϕ) of a single measurement for each investigated tissue type…………………………………………………………. 32 2.6 Composition of azimuthal light scattering of the investigated tissues…………. 37 2.7 Distribution of random anisotropic scattering for the two tissue types………… 38 2.8 Eigenvalues of the PCA represent the variation of scores of the PCA- analysis, performed for all tissues together…………………………………….. 40 2.9 Scatter plot of PC1 versus PC2: This plot shows the comparably strong variation in principal components of the ordered structures…………………… 41 3.1 A schematic of one experimental design of a magnetic bead microrheometer 53 3.2 A schematic view of one of the experimental designs of a magnetic bead microrheometer……………………………………………………………...…. 55 3.3 A typical creep response and recovery curve from a magnetic particle measurement…………………………………………………………...………. 58 3.4 A schematic of the experimental setup of the oscillating optical tweezers……. 71 3.5 The storage and loss moduli obtained by Dasgupta et al. (Dasgupta, et. al. 2001) for a 4% by weight 900 kDa PEO solution comparing moduli obtained with a conventional strain controlled rheometer (G’ – open square, G” – open circle) to those obtained by both DWS (G’ – solid line, G” – dash-dot) and single scattering at 20° (G’- dot, G”- dash)…………….… 97 3.6 The comparison of one- and two-particle microrheology to bulk measurements in a guar solution………………………………………………. 105 4.1 Time- and ensemble-averaged mean squared displacements as a function of time……………………………………………………………………………. 121 4.2 Individual particle mean squared displacements for several particles in Glycerol, Agarose and F-actin………………………………………………… 122 4.3 Van Hove correlation functions for particles moving in glycerol…………..… 123 4.4 Van Hove correlation functions for particles moving in agarose…………...… 128 4.5 Cluster-averaged mean squared displacement for 500 nm particles in agarose. 130 4.6 The particle trajectories for the agarose gel give a spatial map of mechanical

viii

microenvironments…………………………………………………..………… 134 4.7 Van Hove correlation functions for particles moving in actin………..……..… 135 5.1 (a) Two-point correlation function, Drr, for 0.47 µm diameter beads dispersed in a glycerol/water solution, as a function of r and τ. (b) <∆r2(τ)>D calculated from Drr (circles) overlaid on <∆r2(τ)> (line)…..……… 148 5.2 (a) Comparison of the self (triangles) and distinct (circles) displacements of 0.20 µm diameter beads in 0.25% weight guar solution. The inset shows the r dependence of rDrr for τ = 100 msec, in units of 10-3 µm3. (b) The storage (filled circles) and loss (open circles) moduli calculated using <∆r2(τ)>D , showing a crossover to elastic behavior at high frequencies, are in good agreement with rheometer measurements (solid curves). The moduli calculated using <∆r2(τ)> (triangles) do not agree……………….……. 149 5.3 Comparison of the self (triangles) and distinct (circles) displacements of 0.47 µm diameter beads in a 1 mg/ml F-actin solution………………….…..... 152

6.1 Brightfield and fluorescence images of CML, BSA, and PEG particles that have been incubated with R-BSA……………………………………………… 173 6.2 Trajectories of CML, BSA, and PEG particles in a fibrin network…………… 177 6.3 A sampling of the mean-squared displacements of individual particles moving in a fibrin network…………………………………………………..… 178 6.4 Ensemble averaged mean-squared displacements of CML, BSA, and PEG particles in an entangled actin solution……………………………… 180 6.5 The ensemble averaged MSDs for BSA and PEG particles moving in actin networks crosslinked and bundled with the actin-binding protein scruin, at various ratios of scruin:actin…………………………………………...………. 184 6.6 Two-particle mean-squared displacements for composite actin – scruin networks with β = 1/30 and β = 1/15…………………………………………. 188 7.1 The time-evolution of the viscoelastic response of the native cytoplasmic extracts, with measurements taken every 2 to 3 minutes at a fixed frequency of 1 rad/sec and strain of 0.05………………………………………………….. 210 7.2 The frequency-dependence of the viscoelastic response of the native

cytoplasmic extracts, at a strain of 0.05…………………………………...…… 211 7.3 Representative data showing the strain-dependence of bulk cytoplasm, obtained with a constant frequency of 1 rad/sec………………………..……..... 213 7.4 The storage (solid symbols) and loss (open symbols) moduli as a function of time after warming for extracts treated with A) 10 µM phalloidin, B) 10 µM taxol, C) 500 nM taxol…………………………………...……………... 215 7.5 Images of the microtubule and actin cytoskeletal filaments obtained with confocal

microscopy after incubations at room temperature of 0-, 10-, 20-, 30-, 40-minute incubations at room temperature…………………………....………………….. 220

7.6 A magnified view of the actin network after (a) a 10-minute and (b) 40-minute incubation………………………………………….………………………….... 222 7.7 Images of the actin network in an extract that has been treated with 10 µM nocodazole, after a 45 minute incubation………………….…………………… 223

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7.8 Images of the actin network in extracts that have been treated with (a) 10 µM phalloidin or (b) 10 µM taxol…………………………………...……………… 224 7.9 Ensemble-averaged MSDs of particles moving in native state extract as well as extracts that have been treated with 30 µM latrunculin B (+latB), 10 µM nocodazole (+nocod), 10 µM taxol (+Tx), 500 nM taxol (+dTx), after (A) a 10 minute or (B) 30 minute incubation at room temperature…………………… 226 A1. A standard dynamic light scattering setup…………………………………...… 231 A2. Diffusing Wave Spectroscopy setup…………………………………………… 238 List of Tables 2.1 Major azimuthal scattering components, power law coefficients and Henyey-

Greenstein factors gHG……………………………………………………….… 28 2.2 Anisotropy-based classification of tissue by cluster analysis of the scattering patterns along the dimensions of their most significant principal components.. 43

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Acknowledgements

I have been very fortunate throughout my graduate school career to be surrounded by

interesting, thoughtful, and enthusiastic individuals with whom it has been a pleasure to

work. I am most appreciative of my advisor Dave Weitz for providing a fun and exciting

place to work and for allowing me to find and follow those projects which interested me

most. I have greatly appreciated his ability to approach complex problems with simple

physical intuition and clever experimental techniques, and am grateful for the opportunity

to study in such a dynamic research group. I have been influenced by myriad post-docs

and students in his group, and am particularly grateful to Veronique Trappe, John

Crocker, Peter Kaplan, Eric Weeks, Phil Segre, and Luca Cipelletti for their dedication to

training young students to be clear thinkers and good experimentalists, and their

continued support and friendship. I enjoyed the two years I spent studying in the

Department of Physics at the University of Pennsylvania, where I received superb

training in the fundamentals of soft condensed matter and biological physics. I am

particularly thankful for excellent teaching provided by Randy Kamien, Tom Lubensky,

and Phil Nelson, and for the supportive academic community I found there. I am also

thankful for the opportunity to work and study for the past four years at Harvard

University, and appreciate the benefits of working and living in such a thriving academic

environment. I have enjoyed interacting with and learning from my colleagues and

collaborators, and have grown enormously through these experiences.

I was extremely fortunate to have the opportunity to work with Daniel Ou-Yang

for over two years as an undergraduate at Lehigh University, during which time he

xi

introduced me to the exciting world of soft matter physics. He provided me with

excellent training in good experimental procedures, and spent many hours carefully and

cheerfully answering questions, providing tangible examples of physical concepts, and

motivating my work. His influence on the scientist I am today cannot be overestimated.

The work presented in this thesis could not have been completed without the

assistance of and interactions with a number of fine collaborators and coworkers: Alois

Popp, Peter Kaplan, John Crocker, Devi Thota, Thomas Gisler, Bob Prud’homme, Martin

Beck, Eric Weeks, Margaret Gardel, Suliana Manley, Bivash Dasgupta, Jennifer Shin,

Paul Matsudaira, Vernita Gordon, Cliff Brangwynne, Andreas Bausch, Kevin Kit Parker,

Justin Jiang, Emmanuele Ostuni, Ian Wong, Hallam Stevens, and Heather Rose. I am

particularly indebted to Zach Perlman, Tim Mitchison and the members of the Mitchison

group for their interest in characterizing the mechanical properties of the cytoplasm, and

their endless enthusiasm in explaining detailed biological principles to a physicist. I am

also grateful for the expertise and reagents freely given by the laboratories of Jeff Weitz,

Erich Sackmann, Paul Matsudaira, and George Whitesides, and technical assistance given

by Pierre Tijskens, Clara Franzini-Armstrong, Marge Lehman, Geoffrey Daniels and

Jahn Hinch. I gratefully acknowledge financial support by grants from Unilever, the

NSF (DMR-0243715), the Materials Research Science and Engineering Center through

the auspices of the NSF (DMR-0213805) and NASA (NAG3-2284).

Most importantly, I thank my family and friends for their support, love,

understanding, excitement and prayers during my graduate career, particularly during the

drafting of this thesis. I especially thank my parents for their believing in me, supporting

me at any cost, and teaching me that all things are possible. Finally, I thank my partner

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and best friend Donna M. Mancusi for her endless love, comfort, patience, confidence,

and support. Without her this process would have been immeasurably more difficult, and

much less fun. I dedicate this work to her.

1

Chapter 1: Microscope-based static light scattering instrument∗

1.1 Overview

We describe a new design for a microscope-based static light scattering instrument

that provides simultaneous high-resolution images and static light scattering data. By

correlating real space images with scattering patterns, we can interpret measurements

from heterogeneous samples, which we illustrate using biological tissue.

1.2 Introduction

Static light scattering (SLS) is a well-established technique for determining size,

shape and structure by measuring the intensity, I(q), as a function of the scattering wave

vector, q. In some cases, when there is independent information about the sample, the

interpretation of I(q) is straightforward; however, when little is known about the sample,

the analysis can be more difficult. Measurement of I(q) provides an ensemble average

over the entire illumination volume, making the interpretation of scattering from highly

heterogeneous samples particularly challenging. To facilitate the interpretation of SLS in

these more complex systems, we have designed a microscope-based light scattering

instrument that allows us to illuminate selectively a volume of interest while

independently imaging the probed regions. Unlike several prior implementations, 1,2,3,4

∗ Originally published in Optics Letters, vol. 26, p. 890 (2001) by M.T. Valentine, A.K. Popp, P.D. Kaplan, and D.A. Weitz 1 Nishio I., T. Tanaka, S.-T. Sun, Y. Imanishi and S.T. Ohnishi (1983) Science 220: 1173

2

our design combines simultaneous high-resolution imaging with light scattering from

well-defined volumes at well-defined wave vectors, allowing controlled placement of the

illuminating beam, and correlation of scattering with visualized structures. With this

instrument, we probe the details of heterogeneous materials such as biological tissues.

1.3 Instrumental Design

Our design is based on a commercial inverted microscope (Leica DM-IRBE) and

is shown schematically in Figure 1.1. Illumination is from an Ar+ ion laser (Coherent

Innova 304) operating at a wavelength of λ = 514.5 nm in vacuuo. The beam is launched

from a fiber optic coupler that is mechanically mounted above the condenser lens. A

fraction of the beam is diverted onto a photodiode to monitor the intensity of the incident

beam, and a series of neutral density (ND) filters attenuate the laser intensity to typically

less than 50µW, at the condenser entrance. The laser beam is focused to a point in the

back focal plane (BFP) of a high numerical aperture (NA) oil-immersion condenser by a

lens placed conjugate to the field iris.

2 Nishizaki, T., T. Yagi, Y. Tanaka and S. Ishiwata (1993) Nature 361: 269 3 Burger, D.E., J.H. Jett and P.F. Mullaney, presented at the Los Alamos Conference on Optics, Los Alamos, NM, (1979) SPIE Proceedings, D.H. Liebenberg, ed. 190:467 4 P.S. Blank, R.B.Tishler and F.D. Carlson (1987) Applied Optics 26: 351

3

Figure 1.1 Schematic of the optical arrangement of the Static Light Scattering

Microscope.

4

With the microscope in Koehler illumination, the condenser acts as a relay lens that re-

images the collimated beam waist onto the sample, where the diameter of the excitation

beam is smaller by an amount determined by the magnification of the condenser.

Typically, a 2 mm beam at the field iris is reduced to 80 µm in the sample with a

divergence of less than 10 mrad. On the collection side of the sample, an oil-immersion

objective lens (plan-apochromatic, 100x magnification, NA=1.4) collects the scattered

light and transmitted beam. We re-image the scattered light to an intermediate plane

conjugate to the BPF of the objective, using a custom-made (Leica) projection system

positioned above the camera port of the trinocular head of the microscope. In this plane,

we insert a beam block mounted on an x-y translation stage and manually aligned for

each measurement. A final relay lens re-images the intermediate scattering plane onto a

cooled CCD detector (Princeton Instruments Model CCD-512 SF) with a 512x512 array

of 24 µm square pixels and a 16-bit dynamic range. The real space image of the sample

is simultaneously obtained by video camera (Hitachi KP-M1U) at the side port of the

microscope.

Unlike 2-dimensional spatial Fourier transforms of real-space images, the

scattering patterns measured with the static light scattering microscope are full 3-

dimensional transforms that ensure ensemble averaging over the entire illuminated

volume; this is essential to correctly capture bulk properties of localized regions of

heterogeneous samples. In addition to SLS, the collection optics can be modified to

perform dynamic light scattering measurements;5 standard microscopy techniques are

also available.

5 Kaplan, P.D., V. Trappe and D.A. Weitz (1999) Applied Optics 38: 4151

5

Quantitative measurements require several calibrations. First, we determine how

scattering wave vectors map onto the CCD detector. In the BPF of the objective, the

radial distance of the scattered light from the transmitted beam, xδ , is directly

proportional to the sine of the scattering angle,θ . However, at the detector, we measure

a non-linear relationship due to image-deforming optical aberrations caused by the relay

lens. We measure this relationship by scattering from a graticule imprinted on a glass

slide, and measuring the positions of several orders m of diffraction peaks. Using

Bragg’s law, we calculate the corresponding scattering angles: sin Gm n dθ λ= , where d

is the graticule spacing, and Gn is the index of refraction of glass. We plot sinθ as a

function of xδ , fit a third-order polynomial to the resultant data, and apply this

expression to any sample with known refractive index nS to calculate the scattering wave

vector, ( ) ( ) ( ){ }14 sin arcsin sin2s G sq n n nπ λ θ= .

There are also several calibrations for the CCD detector that must be performed to

properly account for variations in pixel sensitivity, stray light, dark counts, pixel read-out

noise and any offset in the black-level reference voltage of the camera6. To measure dark

counts caused by thermal electrons, we remove the detector from the microscope and

place it in a light-tight box to obtain a dark image at each exposure time. To perform a

flat-field correction for nonuniformities in pixel sensitivity, we uniformly illuminate the

detector and obtain a background image in which any gradients in the contrast arise only

from variations in pixel response, and normalize by dividing by the mean illumination

intensity. Corrected images are calculated by subtracting the dark image and dividing by

6 Inoue, S. and K.R. Spring, Video Microscopy: the fundamentals. New York: Plenum (1997)

6

the flat-field background image. Finally, to reduce the effects of small angle flare, the

signal obtained by scattering from a solvent-filled sample chamber is corrected as above

and subtracted from the corrected image obtained from the sample for every

measurement; in the case of the tissue samples, an empty sample chamber is used.

1.4 Scattering from dilute suspensions of colloidal spheres

To experimentally test the apparatus, we measure ( )I q for a dilute suspension of

2.0 µm diameter latex spheres (Interfacial Dynamics Corporation) in water. In the dilute

limit, the measured I(q) ~ F(q), where ( )F q is the form factor of a single sphere. As

shown in Figure 1.2; the results compare quite well to the predictions of Mie Scattering

Theory,7 as shown by the solid line. At large wave vectors, however, optical aberrations

prevent the formation of a clear image, precluding determination of the fine details in the

form factor. This is a purely optical effect that can be corrected by a more sophisticated

system of relay optics. The utility of simultaneous imaging and scattering is

demonstrated in Figure 1.3, where we show images and corresponding scattering patterns

from a partially ordered two dimensional packing of the latex spheres. Strong local

orientational ordering results in well defined Bragg spots (Figure 1.3(a)), while local

polycrystallinity results in a Bragg ring (Figure 1.3(b)); this demonstrates the sensitivity

of light scattering to even small changes in organization or symmetry.

1.5 Scattering from heterogeneous biological tissue

7 H.C.v.d. Hulst, Light scattering by small particles. New York: Dover (1981)

7

This instrument's strength is its ability to facilitate local light scattering

measurements on heterogeneous samples. This is particularly important for biological

materials that include many small structures with a wide range of shapes and orientations.

To illustrate this, we measure the scattering from thin slices of porcine skin tissue,8 where

the microscopic origins of light scattering are poorly understood. The nature of light

propagation through tissue has important implications for medical applications such as

laser surgery, optical biopsy, photodynamic therapy and laser treatment dosimetry.9,10,11

While considerable effort has focused on measuring average absorption and scattering

coefficients,12,13 little attention has been given to the effects of small yet abundant

heterogeneities on bulk scattering properties in tissue.

We use a cryo-microtome to slice porcine skin tissue into approximately 20 µm

thick slices, with total surface area of typically 1cm2, and set the laser beam diameter to

70-100 µm.

8 Popp, A.K., M.T. Valentine, P.D. Kaplan and D.A. Weitz, presented at the Optical Biopsy III, San Jose, CA (2000), SPIE Proceedings, R. R. Alfano, Ed. 3917: 22 9 Fishkin, J.B., O. Coquoz, E.R. Anderson, M. Brenner and B.J. Tromberg (1997) Applied Optics 36: 10 10 Tearney, G.J., M.E. Brezinski and J.G. Fujimoto (1997) Science 276: 2037

11 Mourant, J.R., I.J. Bigio, J. Boyer, R.L. Conn, T. Johnson and T. Shimada (1995) Lasers Surgery and Medicine 17: 350 12 Beauvoit B.and B. Chance (1998) Molecular and Cellular Biochemistry 184: 445 13 Matcher, S.J., M.Cope and D.T. Delpy (1997) Applied Optics 36: 386

8

Figure 1.2. I(q) measured from a dilute suspension of 2.0 µm spheres in a 1:1 mixture of

H2O:D2O at φ=10-3 . The solid line is the form factor predicted from Mie theory with no

fitting parameters.

0 5 10 15 20100

101

102

103

104

I (ar

bitra

ry u

nits

)

q (µm-1)

9

Figure 1.3. (A-B) Real space and scattering spaces images from a monolayer of 2.0 µm

particles. A: A region with one dominant crystal orientation resulting in pronounced

Bragg peaks; B: regions with polycrystalline domains, resulting in a Bragg ring,

indicating the average nearest neighbor distance, 2.00 ± 0.04 µm. (C-D) Real space and

scattering space images from porcine skin tissue. C: The real space image shows a region

deviod of any large heterogeneities; the resulting scattering pattern is isotropic. D: A

region from the same slice in the vicinity of a hair, resulting in an anisotropic scattering

pattern.

10

We find that in the dermis, where individual cells are separated by large regions of

extracellular matrix, large-scale structural heterogeneities, such as acellular voids, hairs

or hair follicles, and pigments can cause dramatic changes to the scattering patterns. In

Figure 1.3 (c) and (d), we show the real space and scattering space images from a typical

experiment. The upper images show a region deviod of any large heterogeneities; the

resulting scattering pattern is isotropic. The lower images show a region from the same

slice of tissue near the vicinity of a hair. The scattering pattern is anisotropic, and shows

more structure than the homogeneous case. These results suggest that models that do not

incorporate details of local tissue microstructure may not capture the anisotropic features

of the scattering patterns14,15,16. Because light can be dramatically redirected in the

vicinity of a structure, improved models are necessary to predict local radiation dosage

and to improve the precision of medical laser treatments to very small regions of

heterogeneous tissue. These improved models must correctly capture the effects of

anisotropy, demonstrated in Figure 1.3(d); this can conveniently be done through

concurrent analysis of both the radial and azimuthal dependence of the scattering. For

example, a Fourier analysis of the azimuthal dependence of the scattering will directly

identify the nature of the anisotropy, and will allow rapid and convenient identification of

specific features.

14 Hielscher, A.H. , R.E. Alcouffe and R.L. Barbour (1998) Physics in Medicine and Biology 43: 1285 15 Wilson, B.C. and G. Adam (1983) Medical Physics 10: 824 16 Saidi, I.S, S.L. Jacques and F.K. Tittel (1995) Applied Optics 34: 7410

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1.6 Summary

The static scattering microscope is sensitive to local structure in complex systems. It can

detect variations in structure and organization in thin tissue samples, making it a

potentially useful for patho-histological studies. Further development of analysis

techniques to parameterize the scattering patterns may allow the application of local light

scattering measurements to the automated classification of cellular phenotype, and the

identification of diseased tissue that is characterized by changes in size, shape or

organization.

12

Chapter 2: The microscopic origin of light scattering in tissue∗

2.1 Overview

A newly designed instrument, the static light-scattering microscope, that

combines light microscopy with static light scattering (SLS), enables us to

characterize local light scattering patterns of thin tissue sections. Each

measurement is performed with an illumination beam of 70 microns diameter. On

these length scales, tissue is not homogeneous. Both structural ordering and small

heterogeneities contribute to the scattering signal. Raw SLS data consists of a

two-dimensional intensity distribution map I(θ,ϕ), showing the dependence of the

scattered intensity I on the scattering angle θ and the azimuthal angle ϕ. In

contrast to the majority of experiments and to simulations that consider only the

scattering angle, we additionally perform an analysis of the azimuthal dependence

I(ϕ). We estimate different contributions to the azimuthal scattering variation and

show that a significant fraction of the azimuthal amplitude is the result of tissue

structure. As a demonstration of the importance of the structure-dependent part of

the azimuthal signal, we show that this fraction of the scattered light alone can be

used to classify tissue types with surprisingly high specificity and sensitivity.

2.2 Introduction

∗ Originally published in Applied Optics 42: 2871 (2003), by A.K. Popp, M. T. Valentine, P.D. Kaplan, and D.A. Weitz

13

The search for non-invasive medical techniques leads naturally to the use of light.

A growing number of optical-medical applications such as photodynamic therapy, laser-

based microsurgery, optical-coherence tomography1 and diffuse-optical tomography2,3

are based on light-tissue interactions, and thus ultimately on the single scattering event.

Our understanding of the single scattering event in tissue is limited. While a complete

spatial map of the complex index of refraction is sufficient to calculate it using

Maxwell’s equations4,5, exact calculations are computationally tedious and require an

impractical level of detail, especially knowledge of both cellular contents and exact

arrangements of scattering objects. We will show that by analyzing light scattering data

along two angular axes, we can partially account for both tissue contents and their

organization.

As most medical techniques have been developed for bulk tissue, many

experimental approaches and their analytical or numerical support rely on average tissue

scattering properties. These are expressed as scattering and absorption coefficients6,7,8,9

1 Tearney, G.J., M.E. Brezinski, B.E.Bouma, S.A. Boppard, C. Pitris, J.F. Southern, and J.G. Fujimoto (1997) Science 276: 2037 2 Siegel, A.M., J.J.A. Marotas, and D.A. Boas (1999) Optics Express 4: 287 3 Matcher, S.J., Optics Letters (1999) 24: 1729 4 Schmitt, J.M. and G. Kumar (1998) Applied Optics 37: 2788 5 Videen,G. and D. Ngo (1998) Journal of Biomedical Optics 3: 212 6 Chance, B., N.G. Wang, M. Maris, S. Nioka, and S. Sevick (1992) Advances in Experimental Medicine and Biology 317: 297 7 Fishkin, J.B., O. Coquoz, E.R. Anderson, M.Brenner, and B.J. Tromberg (1997) Applied Optics 36: 10

14

that are calculated from raw data under the assumption of a multiple scattering model.

These average scattering properties have been sufficient input to models based on the

diffusion approximation, transport theory and Monte Carlo techniques to support the

remarkable progress in developing optical tomographic techniques in recent years3,10,11,12.

None of these efforts considers cellular length scales10,13,14,15. As tissue is heterogenous

on length scales between one and one thousand microns, attempts to use continuum light-

transport models on shorter length scales may not succeed.

Unlike diffuse-optical tomography, some optical diagnostic techniques are

sensitive to cellular length scales16,17,18. Pathologists, for example, see the earliest signs

8 Cheung, W-F., S.A. Prahl, and A.J. Welch (1990) IEEE Journal of Quantum Electronics 26: 2166 9 Bevilacqua, F., P. Marquet, C. Depreursinge, and E.B. De Haller, (1995) Optics Engineering 34: 2064 10 Hielscher, A.H., R.E. Alcouffe, and R.L.Barbour (1998) Physics of Medicine and Biology 43: 1285 11 Boas, D.A., M.A. O`Leary, B. Chance, and A.G. Yodh (1994) Proceedings of the National Academy of Sciences, USA 91: 488 12 Bevilacqua, F. and C. Depreursinge (1999) Journal of the Optical Society of America A 16: 2935 13 Tuchin, V.V. (1997) Physics- Uspekhi 40: 494 14 Matcher, S.J., M. Cope and D.T. Delpy (1997) Applied Optics 36: 386 15 Smithies, D.J. and P.H. Butler (1995) Physics of Medicine and Biology 40:701 16 Luther, E. and L.A. Kamentsky (1996) Cytometry 23: 272 17 Mourant, J.R., J.P. Freyer, A.H. Hielscher, A.A. Eick, D. Shen and T.M. Johnson (1998) Applied Optics 37: 3587 18 Bevilacqua, F., P. Marquet, O. Coquoz and C. Depreursinge (1997) Applied Optics 36: 44.

15

of disease by microscopic examination of structures. The hunt for a light scattering

signature of pathological microstructure has motivated studies of tissue model systems.

These studies of tissues and cell suspensions focused on measurements of I(θ), known as

the phase function or form factor19,20,21. Most of these investigations have been

performed in backscattering geometry to mimic in vivo measurements22,22,23. The form

factor can be calculated for scatterers of various sizes and shapes. It is natural, therefore,

that static light scattering (SLS) models have focused on the content rather than the

structure of tissue. However, it is not possible to look at tissue micrographs without

noting the complex spatial arrangement of different cell types, organelles and

extracellular matrix. That the organization itself must contribute to light scattering is

well known implicitly. For example, polarized light microscopy, a useful tool for

pathologists, is sensitive to birefringence, which is a consequence of structural alignment

in tissue. Light-scattering techniques have two general advantages over direct imaging

that may lead to their preferred use in some biomedical applications. First, light

scattering captures data of direct relevance to light-transport. Second, the interpretive

strategies are more naturally quantitative than approaches based on image analysis. As

19 Jacques, S.L., C.A. Alter, and S.A. Prahl (1987) Lasers Life Science 1:309

20 Mourant, J.R., I.J. Bigio, J. Boyer, R.L. Conn, T. Johnson, and T. Shimada (1995) Laser Surgery and Medicine 17: 350 21 Perelman, L.T., V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. VanDam, J.M. Crawford, and M.S. Feld (1998) Physical Review Letters 80: 627 22 Hielscher, A.H., J.R. Mourant and I.J. Bigio (1997) Applied Optics, 36: 125

16

both light scattering and microscopy are sensitive to index of refraction contrast, they

both should be able to detect local optical property changes in tissue due to differences in

microstructure. The development of useful light-scattering approaches could lead to both

theoretical advances and practical applications in automated tissue diagnosis.

To study the role of organization at the cellular scale in light-tissue interactions,

we have developed a technique that combines SLS with microscopy. The combination

will guide our interpretation of scattering from heterogenous samples. The light

scattering apparatus is built into a fully functional microscope, which provides

simultaneous bright-field images and two-dimensional light scattering patterns24,25. We

have collected scattering patterns from several tissues. In addition to reproducing

previously published form factor results, these data point to the importance of

organization as seen by both a power law analysis of the form factor and by analyzing the

azimuthal scattering patterns for between- tissue differences. After looking at the total

azimuthally scattered power, we construct a reduced space version of the Fourier

representation of the azimuthal scattering patterns. The statistics of the patterns in this

representation show diagnostic promise and point to future applications in biomedical

optics.

As the data that follow are qualitatively different from other techniques, we

briefly preview both brightfield images and scattering patterns of different tissue types.

23 Hielscher, A.H., A.E. Eick, J.R. Mourant, D. Shen, J.P. Freyer, and I.J. Bigio (1998) Optics Express 1: 441 24 Valentine, M.T., A.K. Popp, P.D. Kaplan, and D.A. Weitz (2001) Optics Letters 26: 890

17

In Figure 2.1(a), a striated muscle has been mounted with fibers aligned parallel to the

cover slip (striated ||); the same type of muscle with fibers aligned cross-sectional to the

cover slip is shown in Figure 2.1(b). Muscle is composed of elongated fibers, which are

oriented in striated muscle and globally disoriented in the lung smooth muscle shown in

Figure 2.1(c). Skin, Figure 2.1(d), is a heterogeneous tissue containing large structures

such as hairs and sweat ducts. The influence of different organizational properties on

static light scattering patterns is displayed in the scattering patterns, which demonstrate

that directional ordering contributes to anisotropic scattering patterns. Each example has

some degree of ordering: The periodic organization of parallel aligned muscle fibers

results in highly structured azimuthal spikes (Figure 2.1(a)). The scattering from the

perpendicularly oriented striated muscle is isotropic (Figure 2.1(b)) suggesting that there

is little order in the plane perpendicular to the muscle’s contractile axis. The sarcomeres

in lung smooth muscle (Figure 2.1(c)) have local order that is difficult to detect in the

bright-field image but results in small scattering anisotropies. Rough surfaces and

interfaces such as those found on the top layer of skin, the stratum corneum, cause vague

directionality, and sharp boundaries result in spikes or lines in the scattering patterns

(Figure 2.1(d)). The scattering patterns of these four images quantitatively reflect the

qualitative features observed in the images.

25 Popp, A.K., M.T. Valentine, P.D. Kaplan, and D.A. Weitz “Light scattering microscope to investigate heterogeneities of tissues”, appearing in Optical Biopsy III, R.R. Alfano, ed., Proceedings of the SPIE (2000)

18

Figure 2.1. Real space images (left) and scattering patterns (right) from different tissues.

The real space images show a field of view of 60 µm width, displaying the surface onto

which the incident laser beam is directed. The scattering patterns are intensity

distributions and show bright spots for high scattered intensity. In (a), Rana Sartorius

muscle fibers are oriented parallel to the surface of the coverslip (‘striated muscle ||’).

Influence of orientation of the fibers on image and scattering patterns can be observed by

a comparison with (b), a striated muscle sample mostly cross-sectional oriented to the

surface of the coverslip (‘striated muscle ⊥’). (c) and (d) show real space images and

scattering patterns from smooth muscle and skin, respectively. In skin, the amount of

light scattered on the left side of the beamblock is much higher than the intensity on the

other side, another manifestation of anisotropy. The smooth muscle scatters light more

isotropically.

19

(a) Striated muscle ||

(b) Striated muscle ⊥

(c) Lung smooth muscle

(d) Skin (stratum corneum)

Figure 2.1

20

2.3 Materials and Methods

2.3.1 Sample processing

We have studied four different tissue types: Porcine smooth muscle, striated

muscle and skin, as well as Rana bifida Sartorius muscle. Porcine tissues for this study

were collected from one animal. The samples were frozen in liquid nitrogen immediately

after collection and cut into 20µm thick slices, using a standard cryo-microtome. The

slices were directly transferred from the knife to a small coverslip (18 mm x 18 mm)

which was mounted on a microslide, using ultravacuum grease to seal the chamber. The

slides were stored for up to 2 weeks at –20°C or at 4°C for less than 24 hours before

measurement.

2.3.2 Static light-scattering microscope

The physical principles as well as instrumental details of the static light scattering

microscope (Figure 2.2) have been described elsewhere in detail25-26. Briefly, a laser

beam is coupled into the illumination path of an inverted microscope. In the sample

plane, the beam is collimated with a diameter of 70 µm. Scattered light, collected by the

objective lens, forms a scattering pattern in the back focal plane. This pattern is re-

imaged onto an intermediate plane containing a beamblock to remove the transmitted

beam and finally relayed onto a cooled CCD-detector. This detector has 16 bit dynamic

range with a 512 x 512 array of 24 µm square pixels. Before a quantitative investigation

21

of the scattering function is possible, all scattering patterns are divided by the exposure

time and the input intensity and corrected for small effects including pixel differences

and flare at low angles25, thus reducing the influence of measurement errors to typically

below 1%. Flare or stray light is measured by scattering from an empty sample chamber.

The corrected intensity distribution on the detector is converted into the scattering

function I(θ,ϕ) based on considerations of optics and scattering geometry. The scattering

angle θ of a beam scattered by a sample of refractive index nS is a function of the radial

distance to the transmitted beam on the detector, δx. This relationship is calibrated by

measuring the pixel position of diffraction peaks caused by scattering from a graticule of

index of refraction nG. Using Bragg’s law to relate scattering angles to the position of

diffraction peaks on the detector δx and Snell’s law to take account of various indices of

refraction, we obtain a calibration curve with a calibration constant, C.

= xC

nn

arcsinS

G δθ (2.1)

Simultaneous bright-field imaging is performed with a CCD video camera attached to the

side port of the microscope. We average over the azimuthal angle ϕ by dividing the

pattern into concentric rings around the unscattered beam to obtain the form factor I(θ).

To obtain I(ϕ), we average over θ by binning the image into 360 wedges of 1°. To

sufficiently sample the variations of our tissue, experiments were performed for each thin

section at 25-35 randomly selected fields of view.

26 Kaplan, P.D., V. Trappe, and D.A. Weitz (1999) Applied Optics 38: 4151

22

Figure 2.2. Schematic of the static light scattering microscope. A laser beam from an

Ar+- Ion laser (Coherent Innova 304, 514.5 nm), attenuated to typically 50 µW, is

coupled into the illumination path of a commercially available inverted microscope

(Leica DM – IRBE). A photodiode monitors the incoming beam intensity, which allows

us to correct for differences in input intensity. The laser beam is collimated at the sample

plane to a beam diameter of 70 µm. The conventional objective lens of the microscope

(plan-apochromatic, 100x magnification) collects both forward scattered and transmitted

light. Pixelized scattered light intensity distributions are measured at a 16 bit cooled

CCD detector (Princeton Instruments, Model CCD-512SF, 512x512 array of 24 µm

sized square pixels) located on an extension of the microscope. The scattering patterns

on the detector are enlarged images of the back focal plane of the objective (BFPO), with

the transmitted beam blocked. Additionally, the same regions of our samples are imaged

via conventional bright-field microscopy by diverting a portion of the illuminating light

to the side camera port equipped with a video camera.

23

Figure 2.2

24

The analysis below mainly concerns the light-scattering patterns. The images are

used implicitly to ensure that patterns come from the tissue of interest. The scattering

pattern and the image are not merely different representations of each other. Real space

and Fourier space results are not connected by a simple transformation due to the

different optical properties of the light sources. The coherent laser illumination in the

scattering measurement has a large effective depth of field and truly probes three-

dimensional volume25. The imaging modality does depend on all three dimensions but is

carefully designed to present planar images.

2.4 Results

We begin by comparing our results with previously published data. Typically,

scattering patterns are analyzed with respect to the scattering angle θ and compared to the

Henyey-Greenstein function, an empirical approximation for Mie-scattering from

particles with a distribution of sizes20,27:

2/3HG

2HG

2HG

0 )cosg2g1(g1

I)(Iθ

θ−+

−= . (2.2)

We have extracted the scattering angle dependent signal Iij(θ) of each local scattering

pattern i and averaged over the ensemble of fields-of view for each tissue j. In Figure 2.3

we display both the average form factors ( ) n/)(IIn

1iijj ∑=

=θθ for each tissue type and

27 Henyey, L.G. and J.L. Greenstein (1941) Astrophysics Journal 93: 70

25

fits to the Henyey-Greenstein function. The Henyey-Greenstein phase function

(Equation 2.2) contains no absolute cross section and only one parameter, the anisotropy

gHG,, a measure of forward scattering probability. In the Rayleigh scattering limit of

particles much smaller than the illumination wavelength, gHG approaches zero, while the

sharp forward scattering cone of large particles is described by a gHG close to 1. The axes

and range of angles in Figure 2.3 are chosen to match the convention of Jacques et al.20.

Both the quality of the fits and the value of gHG are in agreement with their results for

thin tissue sections (Table 2.1). In general, the match between the Henyey-Greenstein

function and light scattering data from thin tissue sections is poor at large angles18 and

only a correction to higher order terms is able to produce better agreement28,29.

As an approximation of Mie scattering, the Henyey-Greenstein function neglects

the interference resulting from organized tissue structure. If we took a different point of

view, that tissue organization is the dominant factor in light-tissue interactions, then we

would adopt a model based on the statistics of scattering from aggregates30. Aggregate

scattering models lead to power law scattering functions31 that can be used to describe

aggregation of bacterial cells. Power law functions are known to connect the theoretical

framework for phase transitions to experimental systems.

28 Flock, S.T., B.C. Wilson, and M.S. Patterson (1987) Medical Physics 14: 835 29 Marchesini, R., A. Bertoni, S. Andreola, E. Melloin, and A.E. Sichirollo(1989) Applied Optics 28: 2318 30 Dimon, P., S.K. Sinha, D.A. Weitz, C.R. Safinya, G.S. Smith, W.A. Varady, and H.M. Lindsay (1986) Physical Review Letters 57: 595 31 Nicolai, T., D. Durand, and J.-C. Gimel “Scattering properties and modelling of aggregating and gelling systems”, appearing in Light scattering: Principles and development, W. Brown, ed., New York: Oxford University Press (1996)

26

Experimentally determined power law coefficients are often used as input to the

theory of gelation processes of polymers and colloids32. To match the notation of these

models, we change axes from I(θ) to I(q). The wave vector ( ) λθπ /2/sinn4q S= is a

function of the scattering angle θ, the laser wavelength λ, and the refractive index of the

sample nS. We show in Figure 2.4 that a power law decay, I(q)~ q-z, does describe our

data well. Light scattering from many systems, such as gels and colloidal aggregates, is

described by a power law. In these systems, the power law is due to interference between

neighboring particles in an aggregate and can be derived mathematically using simple

geometric relationships and an integral over the pair correlation function33. The power

law coefficient z is a measure of the structures’ dimensionality34. Recently, it has been

stated that this concept is applicable to tissue as well4,35,36.

32 Takeda, M., T. Norisuye, and M. Shibayama (2000) Macromolecules 33: 2909 33 Teixeira, J. “Experimental methods for studying fractal aggregates”, appearing in On growth and form, H.E. Stanley and N. Ostrowski, eds, Dordrecht: Martinus Nijhoff (1986) 34 Mandelbrot, B. The fractal geometry of nature. New York: W.H. Freeman (1982) 35 Schmitt, J.M. and G. Kumar (1996) Optics Letters 21: 1310 36 Wang, R.K.K. (2000) Journal of Modern Optics 47: 103

27

5x10-3

5x10-3

4 4

3 3

2 2

1 1

0 0

[Inte

nsity

(θ)]-2

/3

0.950.900.850.800.75cos θ

smooth muscle striated muscle ⊥ skin extended fit lines

Figure 2.3. Henyey- Greenstein fits of the form factors <I(θ)>ϕ for the interval 0.995 ≥

cosθ ≥ 0.9, using the method of Jacques (1987). The systematic deviations from the small

angle Henyey-Greenstein parameters at larger angles are matched by previous studies.

This data shows the compatibility of averages of our measurements to bulk measurements

that inherently average over a large illumination area.

28

Table 2.1 Average tissue scattering properties

Skin Smooth Muscle Striated Muscle ⊥

Striated Muscle ||

Asymmetric

Anisotropy

0.48±0.18

(0.47)

0.27±0.15

(0.22)

0.24±0.1

(0.20)

0.46±0.18

(0.41)

Random scattering

component

0.48±0.16

(0.43)

0.65±0.13

(0.68)

0.68±0.09

(0.67)

0.36±0.15

(0.34)

Power law

coefficient z

3.63±0.03

2.37±0.01

2.62±0.02

2.67±0.04

Anisotropy

factor gHG

0.872

0.907

0.879

0.869

Table 2.1. Major azimuthal scattering components, power law coefficients and Henyey-

Greenstein factors gHG. The asymmetric anisotropic scattering signal can be used to

quantify differences in structural ordering between tissues. Additionally, the structure-

independent scattering fraction is given. It consists of the isotropic amplitude and the part

of the symmetric amplitude, which is linked to scatterer size. This component highlights

the difference between the less ordered tissues and the symmetrically aligned muscle

fibers of the striated muscle|| - sample (Figures 2.6 and 2.7). Here, average values are

shown with standard deviations and medians in brackets. Porcine skin is different from

muscle in that its power law coefficient is between 3 and 4, whereas the other samples

show coefficients between 2 and 3.

29

Bulk structures of dimension d have decay coefficients z=d up to 3, scatterers

organized within the surface of an object thicker than 1/q scatter light with a power law

that decays with z=6-d between 3 and 6. The best fit values of z to the form factors in

Figure 2.4 are tabulated in Table 2.1. Skin appears to be organized differently than the

other tissues. While all non-skin samples have decay coefficients between 2.3 and 2.7,

the decay coefficient for skin is 3.6. Within the framework of fractal geometry, this

difference can be explained by arguing that scatterers in skin are organized along a

macroscopic surface, while muscle scatters light from fibrous particles imbedded

throughout the bulk of the tissue, even though we have sampled only a thin slice of that

bulk-organized tissue. Thus, the plate- like cells of the stratum corneum have scattering

properties distinct from other tissues. For the fractal dimensions d, we obtained values

between 2.3 and 2.7. These values are in accordance with results of a previous study36

(d=2.6-2.72) on a number of tissues different from ours. There, the fractal dimension has

been calculated from Fourier- transformed microscope images of different resolution. In

summary, while our angular scattering data matches previously published results, a

power law analysis supports the concept that the arrangement of scattering objects

contributes qualitatively to light scattering in tissue.

30

104 104

105

105

106 106

Inte

nsity

[A.U

.]

2 3 4 5 6 7 8 910q [µm-1]

striated muscle || striated muscle ⊥ porcine skin porcine smooth muscle (lung)

Figure 2.4. Slice- averaged form factors of different tissue types show a power- law

decay I(q)~q-z. The decay law for skin (z=3.6) is substantially different than for the other

tissue types (z≈2.5).

31

The novelty of our data is the inclusion of azimuthal resolution (Figure 2.5). There is an

excellent literature about azimuthal variation in the intensity of light backscattered from

bulk tissue and tissue phantoms, in which underlying polarization effects are described as

well23,24. Our experiment is different in experimental design and investigated systems, as

we examine the azimuthal variation in forward scattering from thin tissue slices. We

observe a significant amount of light scattered into complex azimuthal patterns (Figure

2.4). In the following, we argue that both contents and structural organization contribute

to these patterns. In general, not all of the azimuthally varying signal can be assumed to

be the signature of organization, and a more detailed analysis is needed to separate

contributions of different effects. We now test the importance of azimuthal scattering in

two steps: First, by quantifying the different contributions to show that a significant part

of the light-tissue-interaction is due to structure. Second, we develop a scheme that

might be useful to identify tissue types based solely on the azimuthal scattering.

In order to separate the effects of particle shape from organization, which both

contribute to anisotropy in scattering, symmetry analysis of the single amplitudes is

crucial. In general, the polarization- dependent single scattering amplitude 'Er

in relation

to the incident beam Er

can be described by the 2 x 2 scattering matrix Sr

for any

object36.

32

Figure 2.5. Radial intensity distribution I(ϕ) of a single measurement for each

investigated tissue type. The skin sample as well as the parallel oriented muscle sample

shows a large amount of scattering dependent on the azimuthal angle. Samples showing

unordered structures have by far less anisotropic scattering.

104

104

2 2

4 4

6 68 8

105

105

2 2

4 4

6 68 8

106

106

Inte

nsity

[A.U

.]

3603303002702402101801501209060300Azimuthal angle

striated muscle⊥ smooth muscle striated muscle || skin

33

As the intensity is equal to the square of the field, the scattered intensity matrix contains

4 x 4 elements. This matrix is easily transformed into the 4 x 4 sized Mueller matrix

which completely describes the polarization state of scattered light37. For a sample of

homogenously distributed approximately spherical particles such as nuclei, which is the

usual assumption considering light- tissue interactions, the azimuthal signal can be

described applying Mie- theory36. In this case, a linear polarized incident laser beam

leads to an azimuthal signal of 180°- symmetry. Details of the pattern such as scattering

intensities depend on the state of polarization of the input beam and the size of the

scatterer. A successful and growing literature of measurements taken in backscattering

geometry are based on this scheme22-24. In these experiments, bulk tissues and tissue

phantoms produce characteristic arrays of Mueller matrices in which the 4 x 4 array is

presented as an array of images and each point on the sample contributes a pixel to each

image.

As the Mie scattering pattern is more complex in the backscattering than in the

forward scattering directions, the complexity of patterns observed in these Mueller

matrices in diffuse back-scattering techniques is higher than that of our forward scattering

data23,24,37.

On the other hand, the presence of asymmetric components in experimental data

would demonstrate the influence of local differences in particle distribution. These

microscopic differences are the building blocks of tissue structure. To effectively

estimate the relative importance of this contribution to the light- tissue interaction, we

have expanded the experimental data into a Fourier- series with complex coefficients cm.:

∑∞

−∞=

−=m

imm ec)(I ϕϕ (2.3)

34

The real amplitudes |cm| are independent of the orientation of the sample. To decrease the

systematic effects introduced by our beamblock, we replace the covered region of

typically 10° with data from its 180° symmetric counterpart as these components are a

reasonable zero order prediction for the total intensity in this small region, leading to a

conservative estimate for the asymmetric anisotropy. By back transformation, we ensure

that the resulting series represents the original function up to average differences of less

than 1 %. Using the Fourier coefficients cm, we can divide the total scattered intensity

into a symmetric component m=2k, an asymmetric component m=2k+1, and an isotropic

component m=0. Symmetric and asymmetric components together present the anisotropic

intensity.

First, we investigate the asymmetrically scattered light. Its fraction ε is:

0kc

c,m m

k2m m ≥=∑

∑

∞−∞=

≠ ε (2.4)

In a disordered sample, randomly aligned scatterers will tend to result in ε = 0. To avoid

overstating the asymmetric anisotropy factor, we calculate ε with an upper limit of m =

10 instead of ∞. This arbitrary cutoff leads to a conservative estimate that treats high

frequencies as if they are part of the isotropic signal.

Across the investigated tissues we see that a significant fraction of scattered light,

generally over 20%, scatters asymmetrically (Table 2.1). This suggests that asymmetric

scattering is a significant part of the light-tissue interaction.

Additionally, a smaller but not negligible part of the tissue structure itself might

be of symmetric shape and would thus contribute to the symmetric amplitude. Prominent

examples for symmetric tissue organization are aligned fibrous structures like muscle

35

fibers. To identify the structure dependent symmetric signal, we have resorted to an

experimental method. After taking a single measurement, the sample was rotated by at

least 90° around the center of the incident beam. Then, a second measurement of this

spot was taken and differences in the angular intensity of the symmetric signal were

estimated. The scattering signal of a homogenous particle distribution is practically

rotation- invariant, whereas the rotation of symmetrically aligned fibers leads to the

rotation of the related scattering signal. For a rough check of the presence of this effect,

distinguishable symmetric peak structures were replaced by the same angular range of the

rotated pattern. The resulting pattern was compared with the original pattern. If the

symmetric peak structures were only due to a homogenous scatterer distribution, no

change in the patterns would be observed. In fact, differences of up to 65% were

estimated in single measurements of aligned muscle. This method was extended to all

tissue types: First, the amount of anisotropic scattering was quantified using Equation

(2.4). Then, rotated and unrotated patterns were compared by calculating the minimum

of the two intensities for each angle. The resulting component contains the isotropic and

the symmetric rotation- invariant scattering signals, both effects of random scatterer

distribution. Knowing two of the three contributions, we calculate the remaining

intensity needed to reconstruct the original signal. This last component is the rotation-

variant symmetric signal due to symmetric alignment of scatterers. In general, our

estimates show that across all tissues, a significant fraction of the scattering signal can

not be explained by the assumption of homogenous scatterer distribution in tissues (Table

2.1). In Figure 2.6, we display the sample-averaged contributions to the azimuthal

scattering signal for the investigated tissues. Asymmetric and rotation- variant symmetric

36

scattering together present the signal due to the presence of irregularly shaped particles

and tissue organization. Striated muscle || shows the strongest influence of asymmetric

and symmetric rotation-variant scattering, 45% (Table 2.1) and 20%, respectively. On

average, 65% of the azimuthal signal is due to contributions that together represent tissue

organization. The highest fraction of asymmetric scattering, 48% (Table 2.1), is found in

the stratum corneum data, which additionally has the lowest fraction of rotation-variant

signal and thus the smallest amount of symmetric alignment. Perpendicularly oriented

striated muscle has the lowest amount of structure- dependent signal. But as even in this

case, tissue organization contributes about 35% to the total signal (Table 2.1), we can

assume that tissue structure and organization significantly influence the optical properties

of tissues on microscopic length scales.

Tissues are not, in general, homogenous at cellular scale. Inhomogenous and

irregular tissues present a challenge for pathological tissue typing, though extremely

different tissues should separate rather easily. To highlight the differences between

tissues, we display distributions of the rotation-invariant symmetric scattering coefficient

for the two tissue types that are most different, smooth muscle and striated muscle ||

(Table 2.1) in Figure 2.7. As these distributions do only slightly overlap, one might hope

that only one parameter alone could be used for tissue typing. While this is true for

extremely different tissues, more subtle differences will require a closer look. As an

initial attempt to apply anisotropic scattering to tissue typing, we use the simplest

multivariate method, principle component analysis (PCA), to analyze the Fourier

coefficients37,38.

37 Bickel, W.S. and W.M. Bailey (1985) American Journal of Physics 53: 468

37

1.0

0.8

0.6

0.4

0.2

0.0

frac

tion

of s

catte

red

inte

nsity

asymmetric rotation variant symmetric random

Striated muscle || Striated muscle ⊥ Lung Skin

Figure 2.6. Composition of azimuthal light scattering of the investigated tissues. Skin

(stratum corneum) shows the highest asymmetric scattering component of all investigated

tissues, striated muscle and lung the lowest. Up to 50% of the scattered light is scattered

asymmetrically. The rest is symmetric or static scattering due to a random distribution of

scatterers (`random´). All investigated tissues suggest amounts of more than 30% of the

scattering are not due to a homogenous distribution of particles. Parallel-aligned striated

muscle has the highest rotation variant symmetric scattering amplitude, skin (stratum

corneum) the lowest.

38 Liu, H., D.A Boas, Y. Zhang, A.G. Yodh, and B. Chance (1995) Physics of Medicine and Biology 40: 1983

38

30 30

25 25

20 20

15 15

10 10

5 5

0 0

Prob

abili

ty [%

]

1.00.80.60.40.2random scattering fraction

striated muscle || striated muscle ⊥

Figure 2.7. Distribution of random anisotropic scattering for the two tissue types

separable best according to the scattering component which is due to random scatterer

distribution. The probability of obtaining a certain “random” scattering fraction is shown

versus the fraction itself.

39

Our goal is to demonstrate that similar to I(θ), a high amount of physiologically relevant

information is contained in I(ϕ).

The first step in PCA is to organize the data into a mk × array, where k is the total

number of patterns, and m is the number of Fourier coefficients. To minimize sampling

errors, we reduce our representation from the 360 data points in I(ϕ) to m = 50 Fourier

amplitudes. Each of the four tissues contributed 25 patterns to our total set of k = 100

patterns. Each scattering pattern is represented by m Fourier amplitudes and can be

thought of as one point in an m dimensional data space. This space is vastly larger than

the nature of the data requires and is therefore sparsely populated. To find a simple

representation of the space of scattering patterns, we diagonalize the matrix and rotate it

so that the axes (eigenvectors) are sorted from most to least significant according to the

magnitude of their eigenvalues. This process is called principal component analysis

(PCA). The eigenvalue spectrum (Figure 2.8) indicates that over 90% of the variation

can be attributed to the first four principal components. The details of the eigenvectors

are somewhat arbitrary and should not be over-interpreted. By keeping only the

dimensions along which there is the most variation, we use PCA as a tool for data

reduction. More sophisticated multivariate analysis techniques segment data with greater

precision and produce more meaningful axes. We consider the projection of the

scattering patterns along the first five principal components as a ‘fingerprint.’ To show

how useful this representation is, we present a scatter plot of the projection of the patterns

against the first two principal components of all experiments in Figure 2.9. This plot

highlights the heterogeneity within and between tissues.

40

6

1

2

4

6

10

2

4

6

100

Perc

enta

ge o

f Eig

enva

lue

[%]

10987654321Eigenvalue No.

2

4

6

10-7

2

4

6

10-6

2

Eigenvalue

Figure 2.8. Eigenvalues of the PCA (right axis) represent the variation of scores of the

PCA- analysis, performed for all tissues together. They show the pronounced importance

of the first 4 components and can be scaled as percentage of variation (left axis). The

figure shows that more than 90% of the variation of the data is represented by the first 4

principal components.

41

Figure 2.9. Scatter plot of PC1 versus PC2: This plot shows the comparably strong

variation in principal components of the ordered structures. Much smaller variation is

observed for principal components of crossectionally oriented striated muscle and of lung

smooth muscle, which have no directional order.

20 20

15 15

10 10

5 5

PC

1

3210-1PC2

striated muscle || smooth muscle striated muscle ⊥ skin

42

The two tissue types showing repetitive micron-sized structures contain a large amount of

variation in these components, whereas the disordered striated muscle tissue and the lung

smooth muscle cluster together tightly, with the exception of a few outliers. To

qualitatively use these ‘fingerprints’, we apply k-means cluster analysis, a qualitative

measure of similarity and difference between multidimensional data. While this effort is

mainly demonstrative, we show that cluster analysis is a useful way to separate the

azimuthal light scattering signature of differently structured homogenous tissues. The

data of striated muscles, skin and lung smooth muscle are grouped into clusters using a

routine packaged with our data analysis software (IDL, Research Systems Inc., Boulder,

CO, USA). This routine divides the data into clusters with maximal distance between

their centers. As k-means clustering involves an optimization algorithm whose output

might be sensitive to initial guesses or might suffer from local minimum convergence, we

checked the algorithm shortly for its robustness by slightly varying the input data, its

order, and the number of clusters. Finding no substantial sensitivity to slight variations,

we set the number of clusters equal to the actual number of tissue types. The clustering

algorithm’s output (Table 2.2) shows the assignment of each field of view to the clusters

A, B, C and X. The distinction between ordered and disordered structures is very clear,

cluster C contains only disordered structures. Between the two disordered structures,

however, no distinction is found by cluster analysis. It is not surprising that we are

unable to distinguish cross-sectional oriented striated from smooth muscle as in some

fields of view, the smooth muscle is a very similar tissue and is locally ordered over the

diameter of our probe beam. Both sensitivity and specificity (Table 2.2) are measures

that describe the accuracy with which one tissue type is represented by a single cluster.

43

Table 2.2. Results of the cluster analysis

Classification Striated

muscle ||

Striated

muscle ⊥

Smooth

muscle

Skin Specificity

[%]

A 16 0 0 14 53

B 6 2 0 18 69

C 1 26 28 0 50

X 4 0 2 1 -

Sensitivity [%] 59 93 93 55

Table 2.2. Anisotropy-based classification of tissue by cluster analysis of the scattering

patterns along the dimensions of their most significant principal components. Ordered

samples (striated muscle ||, skin) and disordered samples (striated muscle⊥, smooth

muscle) are almost completely separated into different clusters. Sensitivity39, the

probability for measurements of one tissue type being grouped solely into one cluster, is

very high for the disordered muscles.

39 H.C. van de Hulst, Light scattering by small particles. New York: Dover (1981)

44

Sensitivity is the probability for all measurements of one tissue type to be grouped into

the same cluster. Specificity is the probability that all measurements grouped into the

same cluster belong to the same tissue type. For skin, the specificity for cluster B is 69%,

which is surprisingly high. The sensitivities obtained show that longitudinally oriented

striated muscle and skin separate from the randomly oriented samples almost completely

by azimuthal scattering. The a posteriori sensitivity and specificity of clustering is

impressive considering the limited data used for the analysis. To emphasize the

importance of I(ϕ), we have neglected all other information. Our statistical analysis of

the azimuthal angle of SLS experiments on supercellular but not macroscopic length

scales shows the influence of structure and order in light scattering from thin tissue

sections.

2.5 Discussion

We have measured static light scattering patterns from small illumination volumes

in thinly cryo-sectioned tissue in order to understand the origin of light-tissue

interactions. First, we reproduced previous results on average scattering form factors.

Then we reviewed the motivation for considering organization to be a significant

component of the light-tissue interaction. First, the influence of structure can be seen in

the large-angle power-law decay of the tissue form-factor. Second, the complex

azimuthal dependence of the scattering patterns is a consequence of tissue organization.

The common working hypothesis about the origin of light scattering from tissue is

that small, mainly spherical objects are responsible for the scattering signal. As a result,

45

many investigations focus on micron sized organelles such as vesicles, mitochondria and

nuclei18,22. Our measurements suggest that progress may be enhanced by considering

local order within the tissue and the cell. This approach will not have difficulty with

tissues mainly consisting of extracellular matrix, such as clots, connective tissues and

dermis. The static light scattering signal will capture the organizational features of mesh

and bundle size, and very likely their disturbance due to disease with higher precision

than obtained previously. We have shown that the light scattering microscope is sensitive

to heterogeneities and directionally ordered structure. The relative presence of oriented

structure and the relative contribution of heterogeneities to light scattering are optical

signatures of the state of the tissue. More challenging heterogenous tissues whose

architecture involves different cell sizes and shapes that form various structural elements,

become quantifiable, as no prediction about scatterer shape has to be made for data

analysis. Our relative success in tissue classification using a small subset of scattering

data and fairly primitive analytic methods is intriguing. It suggests a strategy to use

differences in light scattering patterns to distinguish between normal and dysplastic

tissue18,19,21. The Richards-Kortum group, for example, combines organization and

content in a fluorescence- based approach to the optical diagnosis of cervical cancer38,40.

The light scattering microscope can easily be automated to raster scan a sample or

region of interest on a sample, which might enable applications in automated quantitative

histology. As routine pathology still requires time consuming detailed visual inspection

of the whole biopsy slice, a major advance might consist of a fast automated raster

40 Ramanujam, N., M. Follen Mitchell, A. Mahadevan-Janson, S.L. Thomsen , G. Staerkel, A. Malpica, T. Wright, N. Atkinson, R. Richards-Kortum (1996) Photochemistry and Photobiology 64: 720

46

inspection using a combination of both qualitative imaging and quantitative light

scattering analysis to find small suspicious regions worth inspecting in detail.

Our report also suggests that attempts to extend models of light-transport in tissue

to shorter distances may be hindered by the existence of locally preferred directions for

light transport. Modeling the transition from photon diffusion to a few or single

scattering events is difficult as the details of tissue organization become significant. The

information about the single scattering event from the light scattering microscope might

provide important input for corrections of diffusion and transport calculations. One

application of such knowledge could be for calculations of exposure for photodynamic

therapy that take into consideration the shadows and bright spots created by local tissue

structures. In summary, researchers in the fields of optical histology and light-transport

modeling may find the light-scattering microscope to be a valuable tool.

47

Chapter 3: Microrheology∗

3.1 Introduction

Rheology is the study of the deformation and flow of a material in response to

applied stress. Simple solids store energy and provide a spring-like, elastic response,

whereas simple liquids dissipate energy through viscous flow. For more complex viscoe-

lastic materials, rheological measurements reveal both the solid- and fluid-like responses

and generally depend on the time scale at which the sample is probed1. One way to

characterize rheological response is to measure the shear modulus as a function of fre-

quency. Traditionally, these measurements have been performed on several milliliters of

material in a mechanical rheometer by applying a small amplitude oscillatory shear

strain, ( ) sin( )γ γ ω= ot t where γ o is the amplitude and ω is the frequency of oscillation, and

measuring the resultant shear stress. Typically, commercial rheometers probe frequen-

cies up to tens of Hz. The upper range is limited by the onset of inertial effects, when the

oscillatory shear wave decays appreciably before propagating throughout the entire sam-

ple. If the shear strain amplitude is small, the structure is not significantly deformed and

the material remains in equilibrium; in this case, the affine deformation of the material

controls the measured stress. The time-dependent stress is linearly proportional to the

strain, and is given by:

∗ This chapter is based on a manuscript prepared for “Microdiagnostics” edited by K.

Breuer, published by Springer-Verlag, New York (in preparations) in collaboration with M.L. Gardel and D.A. Weitz

1 Larson, R.G. The Rheology of Complex Fluids. New York: Oxford University Press

(1999)

48

[ ]( ) '( ) sin( ) "( ) cos( )ot G t G tσ γ ω ω ω ω= + (3.1)

( )G ω′ is the response in phase with the applied strain and is called the elastic or storage

modulus, a measure of the storage of elastic energy by the sample. "( )G ω is the re-

sponse out of phase with the applied strain, and in phase with the strain rate, and is called

the viscous or loss modulus, a measure of viscous dissipation of energy. The complex

shear modulus is defined as * ' "G G iG≡ + . Alternatively, it is possible to apply stress

and measure strain and obtain equivalent material properties.

Rheology measurements such as these have given valuable insight into the struc-

tural rearrangements and mechanical response of a wide range of materials. They are

particularly valuable in characterizing soft materials or complex fluids, such as colloidal

suspensions, polymer solutions and gels, emulsions, and surfactant mixtures1,2,3. How-

ever, conventional mechanical techniques are not always well-suited for all systems.

Typically, milliliter sample volumes are required, precluding the study of rare or precious

materials, including many biological samples that are difficult to obtain in large quanti-

ties. Moreover, conventional rheometers provide an average measurement of the bulk

response, and do not allow for local measurements in inhomogeneous systems. To ad-

dress these issues, a new class of microrheology measurement techniques has emerged.

To probe the material response on micrometer length scales with microliter sample vol-

umes. Microrheology methods typically use embedded micron-sized probes to locally

deform a sample. There are two broad classes of micro-rheology techniques: those in-

2 Macosko, C.W. Rheology: Principles, measurements, and applications. New York: VCH (1994) 3 Ferry, J. Viscoelastic properties of polymers. New York: Wiley (1980)

49

volving the active manipulation of probes by the local application of stress and those

measuring the passive motions of particles due to thermal or Brownian fluctuations. In

either case, when the embedded particles are much larger than any structural size of the

material, particle motions measure the macroscopic stress relaxation; smaller particles

measure the local mechanical response and also probe the effect of steric hindrances

caused by local microstructure. The use of small colloidal particles theoretically extends

the accessible frequency range by shifting the onset of inertial effects to the MHz regime;

in practice, the measurable frequency range varies with the details of the experimental

apparatus.

In this chapter, we will detail a variety of microrheology methods. In Section 3.2,

we review the active manipulation techniques in which stress is locally applied to the ma-

terial by use of electric or magnetic fields, or micromechanical forces. These active

methods often require sophisticated instrumentation and have the advantage of applying

large stresses to probe stiff materials and non-equilibrium response. Often single particle

measurements are possible, allowing measurements of local material properties in inho-

mogeneous systems. In Section 3.3, we discuss the passive measurements of thermally

excited probes, in which no external force is applied. For these methods, the mean

squared displacement of the probe particle is measured using various experimental tech-

niques and related to the macroscopic linear viscoelastic moduli of the material using a

generalized Stokes-Einstein relationship. In Section 3.4, we discuss the practical appli-

cation of microrheology techniques to number of systems, including heterogeneous mate-

rials. In heterogeneous systems, video-based multiple particle tracking methods are used

to simultaneously measure the thermally-induced motions of dozens of particles in a sin-

50

gle field of view. Single particle data is then used to map out the spatial variations in ma-

terial mechanics and microstructure. In Section 3.5, we discuss two-particle microrheol-

ogy, in which the correlated motions between pairs of particles are used to measure the

coarse-grained macroscopic material properties. This allows the characterization of bulk

material properties even in systems that are inhomogeneous on the length scale of the

probe particle. We close the chapter with a brief summary and outlook. The experimen-

tal considerations of light scattering and particle tracking techniques are discussed in de-

tail in the Appendix.

3.2 Active Microrheology Methods

One class of microrheology techniques involves the active manipulation of small

probe particles by external forces, using magnetic fields, electric fields, or microme-

chanical forces. These measurements are analogous to conventional mechanical rheology

techniques in which an external stress is applied to a sample, and the resultant strain is

measured to obtain the shear moduli; however, in this case, micron-sized probes locally

deform the material and probe the local viscoelastic response. Active measurements al-

low the possibility of applying large stresses to stiff materials in order obtain detectable

strains. They can also be used to measure non-equilibrium behavior, as sufficiently large

forces can be applied to strain the material beyond the linear regime.

3.2.1 Magnetic Manipulation Techniques

51

The oldest implementation of microrheology techniques involves the manipula-

tion of magnetic particles or iron filings, which are embedded in a material, by an exter-

nal magnetic field. This method was pioneered in the early 1920’s and has been used to

measure the mechanical properties of gelatin, cellular cytoplasm and mucus4,5,6,7,8,9. The

visual observation of the particles’ movements provided a qualitative measure of the vis-

coelastic response, but the irregularly shaped magnetic particles and simple detection

schemes did not allow a precise measure of material properties. More recently, advances

in colloidal engineering, video microscopy, and position sensitive detection have

prompted the emergence of several high precision magnetic particle micro-rheological

techniques.

One method called “magnetic bead microrheometry” or “magnetic tweezers”

combines the use of strong magnets to manipulate embedded super-paramagnetic parti-

cles, with video microscopy to measure the displacement of the particles upon application

of constant or time-dependent forces10,11,12. In this case, strong magnetic fields are re-

4 Freundlich, H. and W. Seifriz (1922) Zeitschrift fur physikalische chemie 104: 233 5 Heilbronn, A. (1922) Jahrbuch der Wissenschaftlichen Botanik 61: 284 6 Crick, F. and A . Hughes (1950) Experimental Cell Research 1: 37 7 Yagi, K. (1961) Comparative Biochemistry and Physiology 3: 73 8 Hiramoto, Y. (1969) Experimental Cell Research 56: 201 9 King, M. and P.T. Macklem (1977) Journal of Applied Physiology 42: 797 10 Ziemann, F., J. Rädler, and E. Sackmann (1994) Biophysical Journal 66: 2210 11 Amblard, F., B. Yurke, A. Pargellis, and S. Leibler (1996) Review of Scientific In-struments 67: 818

52

quired to induce a magnetic dipole in the super-paramagnetic beads, and magnetic field

gradients are applied to produce a force. The resultant particle displacements measure

the rheological response of the surrounding material. Magnetic tweezers techniques have

been used to measure the microscopic dynamics of a number of interesting materials that

are not easily probed with traditional bulk techniques, including networks of filamentous

actin10,12,13,14 living fibroblast15, macrophage16, endothelial17 and dictyostelium cells18,

and solutions of the semi-flexible filamentous bacteriophage fd19. In one experimental

design, as shown in Figure 3.1, the magnetic field is created by four pairs of soft ferro-

magnetic pole pieces arranged at right angles, where each pair is wound with a separate

field coil11.

12 Schmidt, F.G., F. Ziemann, and E. Sackmann (1996) European Biophysics Journal 24:348 13 Amblard, F., A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler (1996) Physical

Review Letters 77: 4470 14 Keller, M., J. Schilling, and E. Sackmann (2001) Review of Scientific Instruments 72:

3626 15 Bausch, A.R., F. Ziemann, A.A. Boulbitch, K. Jacobson, and E. Sackmann (1998).

Biophysical Journal 75: 2038 16 Bausch, A.R., W. Möller, and E. Sackmann (1999) Biophysical Journal 76: 573 17 Bausch, A.R., U. Hellerer, M. Essler, M. Aepfelbacher, and E. Sackmann (2001) Bio-

physical Journal 80: 2649 18 Feneberg, W., M. Westphal, and E. Sackmann (2001) European Biophysical Journal

30: 284 19 Schmidt, F.G., B. Hinner, E. Sackmann, and J.X. Tang (2000) Physical Review E 62:

5509

53

Figure 3.1. A schematic of one experimental design of a magnetic bead microrheometer.

The sample (M) is placed is the center of four coils (C) and magnetic pole pieces and

supported by a sample holder (S) on an upright research microscope. The magnetic par-

ticles in the specimen plane are imaged using the objective lens (O) and particle dis-

placements are measured with video-based detection. (Reprinted with permission11).

54

Four attached Hall probes are used to measure the local magnetic field. Each pair of

poles is precisely positioned in plane and the geometric center of the pair is at the optical

focus of the microscope, ensuring a negligible vertical component to the magnetic field.

This design creates a uniform magnetic field parallel to the focal plane and directed to

one of the poles, and allows both translation and rotation of magnetic colloidal particles,

which are typically 0.5 - 5 µm in diameter. In rotation mode, the field is uniform, with

controlled angle relative to the orientation of the pole pieces, making it a potentially use-

ful tool for measuring the local bending modulus of a material. In translation mode, a

large magnetic field is generated at one attracting pole; field lines that originate from the

attracting pole spread out to three opposing poles where smaller fields are generated. In

order to generate forces, gradient fields are required, and are created by a precise balance

of attracting flux and the total flux at the opposing poles. The force, f (t), on the particle

is given by:

( ) ( )( ) ( ) ( )xB t B tf t M t V B t

x xχ∂ ∂

= ⋅ = ⋅∂ ∂

ur uruur ur

(3.2)

where M(t) is the induced magnetic moment of the particle, B(t) is the is the imposed

magnetic field, χ is the susceptibility of the particle and V is the volume of the particle.

Typically, for 0.1-0.2 T magnetic fields and 10-20 T/m gradients, applied forces are in

the range of pN and torques are in the range of 10-14 N⋅m.

A second design, shown in Figure 3.2, uses one or two axisymmetrically arranged

magnetic coils with soft iron cores that apply a field gradient to produce a force in the fo-

cal plane10,12,14.

55

Figure 3.2. A schematic view of one of the experimental designs of a magnetic bead mi-

crorheometer. The magnet consists of two coils with cylindrical soft iron cores. The

sample and magnet are mounted on an inverted research microscope. A CCD camera is

used to capture images of the particle and image analysis routines are used to measure

particle displacements. The amplifier used to drive the magnet’s power supply also de-

livers a signal to the computer to allow measurements of phase lag. (reprinted with

permission14).

56

The force exerted depends on the details of the tip geometry, but is typically in the range

of 10 pN to 10 nN. For higher forces, only one coil is used and the tip of the pole piece is

positioned as close as 10 µm to the magnetic particle. Typically, the magnetic particles

are separated by more than 50 µm and only single particle displacements are recorded to

prevent neighboring spheres from creating induced dipolar fields. In both designs, video

microscopy is used to detect the displacements of the particles under application of force.

The instrumental precision depends on the details of the design and will be discussed in

detail in Appendix A.3; however, the spatial resolution is typically in the range of 10-20

nm and the temporal dynamic range is 0.01-100 Hz. Three modes of operation are avail-

able in any rheology measurement performed under applied force: a viscometry meas-

urement obtained by applying constant force, a creep response measurement after the ap-

plication of a pulse-like excitation, or a measurement of the frequency dependent

viscoelastic moduli in response to an oscillatory stress. In constant force mode, the vis-

cosity of a Newtonian fluid is measured by balancing the external driving force with the

viscous drag force experienced by the particle as it moves through the fluid; because the

particles are so small and light, inertial effects are typically ignored. The equation of mo-

tion is given by Stoke’s Law, 6πη=of av , where of is the constant external force, η is

the fluid viscosity, a is the particle radius and v is the velocity with which the particle

moves through the fluid. By using a fluid of known viscosity, this measurement also al-

lows for the calibration of the force exerted on the particle by a given coil current.

In addition to constant force viscometry measurements, time-dependent forces can

be applied to measure the frequency dependent viscoelastic response. In a creep-

response measurement, a force pulse is applied to the particle and the particle deflection

57

is measured as a function of time. Representative data curves are shown in Figure 3.3.

The time dependence of the bead deflection, xd (t) can be expressed as:

( )( ) ( )6df tx t J t

aπ=

(3.3)

where J(t) is the time-dependent creep compliance of the material in which the bead is

embedded. In many cases, J(t) can be interpreted with the use of mechanical equivalent

circuits of springs, which represent elastic storage, and dashpots, which represent viscous

loss, to model the local mechanical response3,15. Alternatively, an oscillatory force can

be applied to drive the particle at a controlled amplitude and frequency. The amplitude

and phase shift of the particle’s displacement with respect to the driving force are meas-

ured and are used to calculate the local frequency-dependent viscoelastic moduli. When

an oscillatory force, 0( ) exp( )f t f i tω= is applied to the particles, the displacement can be

expressed as [ ]0( ) exp ( )x t x i tω ϕ= − and the shear moduli are given by:

( ) cos ( )6

( ) sin ( )6

o

o

o

o

fG

a xf

Ga x

ω ϕ ωπ ω

ω ϕ ωπ ω

′ = ⋅

′′ = ⋅. (3.4)

If the probe particles are larger than all length scales of the material, then the

measured shear moduli report the macroscopic bulk response. If however, the particles

are embedded in an inhomogeneous material, then each particle measures the dynamic

microenvironment that surrounds it and the local material response. In materials that ac-

tively change in response to external stimuli, real-time measurements of the local dynam-

ics are possible17.

58

Figure 3.3. A typical creep response and recovery curve from a magnetic particle meas-

urement. The step pulses indicate the application of stress by turning on and off the ex-

ternal magnetic field. The resultant bead displacement consists of a fast elastic response

at the initial onset of stress, followed by a slowing down and viscous flow regime. (re-

printed with permission16)

59

Local properties can be further probed by measuring the strain field around the

each particle using mixtures of paramagnetic particles and non-magnetic latex

spheres12,15. The magnetic particles are displaced by application of constant force and the

resultant deformation field is visualized by measuring the movements of the surrounding

latex particles. For a homogeneous elastic material, the deformation field is given by:

ˆ ˆ1 (3 4 ) ( )8 (1 )

σ σπ σ

+ − + ⋅=

−

r rr f r r fu

E r (3.5)

where r is the distance from the point source, r̂ is a unit vector in the direction of r, σ is

the Poisson ratio, E is the Young’s modulus20. The elastic constants E and σ are meas-

ured, and from these the time-independent shear modulus is derived: 2(1 )Eµ σ= + . Any

deviations from the 1/r spatial decay in the strain field indicate the presence of local het-

erogeneities.

In addition to magnetic tweezers methods, a number of other magnetic particle

techniques have been developed to measure material response. Twisting magnetometry

measures the response of magnetic inclusions in a viscous or viscoelastic body to the

brief application of a strong external magnetic field 21,22,23,24,25. The strong field aligns

the magnetic moments of the inclusions, which can be magnetic colloidal particles or

20 Landau, L.D. and E.M Lifshitz. Theory of Elasticity. Pergamon Press, Oxford (1986) 21 Valberg, P.A. (1984) Science 224: 513 22 Valberg, P.A. and D.F. Albertini (1985) Journal of Cell Biology 101: 130 23 Valberg, P.A. and J.P. Butler (1987) Biophysical Journal 52: 537 24 Valberg, P.A. and H.A. Feldman (1987) Biophysical Journal 52: 551 25 Zaner, K.S. and Valberg P.A. (1989) Journal of Cell Biology 109: 2233

60

polycrystalline iron oxide particles or aggregates. After the field is turned off, the aligned

magnetic moments give rise to measurable magnetic field, the “remnant” field, which de-

cays as the moments become randomized. In cases where rotational diffusion is the

dominant randomizing agent, the decay can be interpreted in terms of the local viscosity.

Viscoelastic response can also be measured by incorporating a weaker twisting field, in a

direction perpendicular to the initial strong magnetic field25. When the twisting field is

first applied, it lies in a direction perpendicular to the magnetic moments of the particles

and the torque on the particles is maximal, causing the particles to rotate toward the di-

rection of the weaker twisting field. The rate at which the particles align with the twist-

ing field and the amount of recoil after the twisting field is turned off give a measure of

local viscosity and elasticity. An in-line magnetometer allows the measurement of the

angular strain and thus the local compliance, defined as a ratio of applied stress to strain.

A modification of the twisting magnetometry technique, called magnetic twisting

cytometry, has been used to apply mechanical stresses directly to cell surface receptors

using ligand coated magnetic colloidal particles that are deposited on the surface of a liv-

ing cell26,27,28. The ferromagnetic particles are magnetized in one direction, and then

twisted in a perpendicular direction to apply a controlled shear stress to the cell surface;

as the particles re-orient, a magnetometer measures the change in remnant field, which is

related to the angular strain. Alternatively, the twisting cytometry measurements can be

mounted on a research microscope, and the particle displacements measured with video

26 Wang, N., J.P. Butler, and D.E. Ingber (1993) Science 260: 1124 27 Wang, N. and D.E. Ingber (1994) Biophysical Journal 66: 1281 28 Wang, N. and D.E. Ingber (1995) Biochemistry and Cell Biology 73: 327

61

microscopy 29. When the twisting magnetic field is made to vary sinusoidally in time,

bead displacement can be interpreted in terms of the local viscoelastic response29

. The

frequency dependence of the material response can be obtained directly; however, in or-

der to calculate the stress exerted on the cell surface by the particle and the absolute vis-

coelastic moduli, the details of the contact area and amount of particle embedding in the

cell surface must be known.

Magnetic particle techniques allow measurements of bulk rheological response in

homogeneous materials or local response in heterogeneous samples. Strain-field map-

ping identifies heterogeneities and allows the measurement of local microstructure and

real-time measurements characterize dynamic changes in material response. Because

such small sample volumes are required and non-invasive magnetic fields are used, these

techniques are particularly useful for studying biological materials.

3.2.2 Optical Tweezers Techniques

Another active manipulation technique uses optical tweezers, also called optical

traps or laser tweezers, that employ highly focused beams of light to capture and manipu-

late small dielectric particles30,31,32.33. Unlike magnetic tweezers, optical tweezers apply

29 Fabry, B., G.N. Maksym, J.P. Butler, M. Glogauer, D. Navajas, and J.J. Fredberg

(2001) Physical Review Letters 87: 148102 30 Ashkin, A. (1992). Biophysical Journal 61: 569 31 Block, S.M. (1992) Nature 360:493 32 Ashkin, A. (1997) Proceedings of the National Academy of Sciences of the United

States of America 94:4853

62

force very locally and the forces are typically limited to the pN range. The two main op-

tical forces exerted on the illuminated particle are the scattering force, or radiation pres-

sure, which acts along the direction of beam propagation, and the gradient force, which

arises from induced dipole interactions with the electric field gradient and tends to pull

particles toward the focus30,32,34,35. Steep electric field gradients can be created using a

high numerical aperture objective lens to focus a laser beam onto the sample; this allows

the gradient force to dominate and forms a stable three-dimensional trap. Moving the fo-

cused laser beam forces the particle to move, apply local stress to the surrounding mate-

rial, and probe the local rheological response.

The experimental design is typically based on an inverted optical microscope with

a quality high numerical aperture oil-immersion objective lens36,37,38,39. The microscope

allows for simultaneous imaging of the sample and facilitates placement of the beam in

an inhomogeneous material. The illuminating laser beam is steered into the microscope

with an external optical train, and is commonly introduced from the epi-fluorescence port

and deflected into the optical path of the microscope by a dichroic mirror located below

33 Ashkin, A. (1998) Methods in Cell Biology 55: 1 34 Nemoto, S. and H. Togo (1998) Applied Optics 37: 6386 35 Neto, P.A.M. and H.M. Nussenzveig (2000) Europhysics Letters 50: 702 36 Fällman, E. and O. Axner (1997) Applied Optics 36: 2107 37 Visscher, K. and S.M.Block (1998) 298: 460 38 Mio, C., T. Gong, A. Terray, and D.W.M. Marr (2000) Review of Scientific Instru-

ments 71: 2196 39 Mio, C. and D.W.M. Marr (2000) Advanced Materials 12: 917

63

the objective lens. In order to achieve the most efficient and stable trapping, the beam is

collimated at the back focal plane (BFP) of the objective and the objective entrance aper-

ture is slightly overfilled. The external optical train allows the control of beam placement

and movement, and even remote control of beam movement through the use of piezo-

controlled mirror mounts or acousto-optic modulators. Steering mirrors are placed in a

plane conjugate to the BFP of the objective to ensure that the intensity and strength of the

laser trap does not fluctuate as the beam is moved36. The slight tilting of a mirror in this

plane will result in a corresponding change only in the direction of the laser beam at the

BFP of the objective, maintaining a constant degree of overfilling at the entrance aperture

and resulting only in the lateral movement of the optical trap in the specimen plane.

At the center of the optical trap, which is typically located slightly above the focal

plane, the gradient and scattering forces balance and the net optical force is zero. At the

trap center, the potential energy is given by:

2 22 2

1 /2 12 22 1

32

−− −= +

p r Ro

V n n nU I e

c n n (3.6)

where Vp is the volume of the particle, n2 is the index of refraction of the particle and n1 is

the index of refraction of the surrounding fluid, r is the radial distance from the center of

the trap and R is the 1/e width of the Gaussian laser profile at the trap40. For stable trap-

ping, n1>n2 is required. The force on the particle as it moves from the trap center is given

by:

40 Ou-Yang, H.D., “Design and applications of oscillating optical tweezers for direct measurements of colloidal forces.” Appearing in Colloid-Polymer Interactions: From Fundamentals to Practice, R.S. Farinato and P.L. Dubin, eds. Wiley, New York (1999)

64

2 2

2 2

2 21 /2 1

2 2 22 1

/

6ˆ

2

ˆ

−

−

= −∇

−= − +

≡

p r Ro

r Rot

F UrV n n n

I e rcR n n

k re r (3.7)

and for small displacements, the force is approximated by Hooke’s law with an effective

spring constant, kot40. Thus, if the laser beam center is offset from the center of the parti-

cle, the particle experiences a restoring force toward the center of the trap. By moving

the trap with respect to the position of the bead, stress can be applied locally to the sam-

ple, and the resultant particle displacement reports the strain, from which rheological in-

formation can be obtained.

In order to apply a known force, the trap spring constant, kot, must be measured.

There are several methods for calibration of kot, which is typically linearly dependent on

the intensity of the incident laser beam. The escape force method measures the amount

of force needed to remove a sphere from the trap. Although this method is simple and

doesn’t require a sophisticated detection scheme for measuring small displacements, it

depends critically on the shape of the potential at large displacements where the trapping

force response is likely non-linear. If high resolution position detection methods are

available, the trap spring constant may be measured by slowly flowing a fluid of known

viscosity past the trapped particle and measuring the displacement of the particle due to

the viscous drag force. For small displacements, the viscous drag force can be balanced

by the linear restoring force of the trap, and the trap spring constant can be determined.

Alternatively, kot can be measured from the thermal fluctuations in the position of the

trapped particle. The trap spring constant is calculated using the equipartition theorem:

21 12 2=B otk T k x

(3.8)

65

where x is the particle position and kBT is the thermal energy.

For measurements of rheological properties, particle displacements in response to

an applied force must be precisely detected. Position detection methods fall into two

broad categories: direct imaging methods and laser-based detection schemes. Direct im-

aging can be achieved using video microscopy to record images of trapped particles in

the specimen plane and centroid tracking algorithms to find the particle positions in each

frame, typically with a spatial resolution of one-tenth of a pixel41. The temporal resolu-

tion is limited by the capture rate of the camera, and is typically 30 frames per second.

Alternatively, the shadow cast by the particle can be imaged onto a quadrant photodiode

detector, which is aligned such that when the particle is in the center of the trap, the in-

tensity of light on each of the four quadrants is equal40. As the particle moves from the

trap center, the difference in light intensity on the four quadrants is recorded to measure

the particle displacement. Although the direct imaging techniques are simple to imple-

ment and interpret, they do not provide the high spatial or temporal resolution of laser

based detection schemes.

Interferometry detection is one laser-based detection scheme37,42,43. In this design,

a Wollaston prism, which is located behind the objective lens and used conventionally for

Differential Interference Microscopy (DIC), splits the laser light into two orthogonally

polarized beams. This produces two nearly overlapping, diffraction-limited spots in the

41 Crocker, J.C. and D.G. Grier (1996) Journal of Colloid and Interface Science 179: 298 42 Gittes, F., B. Schnurr, P.D. Olmsted, F.C. MacKintosh, and C.F. Schmidt (1997)

Physical Review Letters 79: 3286 43 Schnurr, B., F.Gittes, F.C. MacKintosh, and C.F. Schmidt (1997) Macromolecules 30:

7781

66

specimen plane that act as a single optical tweezers. After passing through the sample,

the beams are recombined behind the condenser lens by a second Wollaston prism; the

recombined beam passes through a quarter wave plate. If the particle is centered or the

optical trap is empty, then the recombined beam has the same linear polarization as the

incoming beam and the final beam is circularly polarized; however, if the particle is

slightly off-center there is a phase lag introduced between the two beams and final beam

is elliptically polarized. The degree of ellipticity gives a measure of the particle dis-

placement. The response is linear for displacements of up to 200 nm for 500 nm particles

and larger particles have an extended linear range; larger displacements may be detected

using a piezo-activated stage to realign the particle with the center of the laser beam.

Data is typically acquired at rates of 50-60 kHz, with subnanometer resolution42,43. This

interferometry design has the advantages of high sensitivity and inherent alignment;

however, it gives only one-dimensional data along the shear axis of the Wollaston prism.

Alternatively, the forward deflected laser light can be projected onto a photodiode

to detect the displacement of the trapped particle. A trapped bead acts as a mini-lens and

deflects the illuminating laser light forward through the condenser lens, which relays the

light onto the photodiode. In a single photodiode configuration, the detector is positioned

so that roughly 50% of the optical power from the diverging cone of light is intercepted37.

This configuration is qualitatively sensitive to both axial displacements, which change the

diameter of the intercepted light cone and the intensity of light at the detector, and lateral

displacements, which offset the cross-sectional area of the light cone from the active de-

tector area. However, quantitative measurements of particle displacements are not

straightforward with single photodiode detection since axial and lateral displacements are

67

not easily decoupled and since no information about small movements in the focal plane

is available. Alternatively, quantitative sub-nanometer displacements can be measured

with a quadrant photodiode placed in the BFP of the condenser lens. Typically, the ac-

tive area of the detector is larger than the area of the intercepted light cone, so axial dis-

placements are not detected. The forward-scattered light is detected by the photodiode,

where the photocurrent differences are amplified and converted into voltages to measure

the relative displacement of the bead with respect to the trap center. The trajectory is ob-

tained by sampling the voltages with an A/D converter at rates up to 50 kHz. The spatio-

temporal resolution is 0.01 nm/Hz1/2 above 500 Hz; at lower frequencies, the resolution

decreases and degrades to 1 nm/Hz1/2 at 20 Hz due to the mechanical resonance of the

stage44. The maximal displacement that may be detected with this technique is approxi-

mately 200 nm.

To measure rheological properties, optical tweezers are used to apply a stress lo-

cally by moving the laser beam and dragging the trapped particle through the surrounding

material; the resultant bead displacement is interpreted in terms of viscoelastic response.

Elasticity measurements are possible by applying a constant force with the optical tweez-

ers and measuring the resultant displacement of the particle. This approach has been

used to measure the time-independent shear modulus of red blood cell membranes45,46.

Alternatively, local frequency-dependent rheological properties can be measured by os-

44 Mason, T.G., T.Gisler, K. Kroy, E. Frey, and D.A. Weitz (2000) Journal of Rheology

44: 917 45 Hénon, S., G. Lenormand, A. Richert, and F. Gallet (1999). Biophysical Journal 76:

1145 46 Sleep J., D. Wilson, R. Simmons, and W. Gratzer (1999) Biophysical Journal 77: 3085

68

cillating the laser position with an external steerable mirror and measuring the amplitude

of the bead motion and the phase shift with respect to the driving force40,47. The equation

of motion for a colloidal particle in forced oscillation in a viscoelastic medium is given

by:

* *6 ( ) cos( )πη ω+ + + =&& & ot otm x ax k k x k A t (3.9)

where m* is the effective mass, η* is the effective frequency dependent viscosity of the

medium, a is the particle radius, kot is the spring constant of the trap, k is the frequency

dependent spring constant of the medium, A is the amplitude of the trap displacement and

ω is the driving frequency40. The effective mass includes the bare mass of the particle

and contribution of the inertia of the surrounding fluid entrained by the moving particle.

The effective mass is given by:

3 22 2* 33om m a aπ ηρρ π

ω= + +

(3.10)

where ρ is the mass density and η is the viscosity of the surrounding fluid. The effective

viscosity also includes an inertial correction, and is given by:

2* 1

2ρωη ηη

= +

a

. (3.11)

The equation of motion can be solved by:

( ) ( ) cos( ( ))ω ω δ ω= −x t D t . (3.12)

47 Valentine, M.T., L.E. Dewalt and H.D. Ou-Yang (1996) Journal of Physics: Condensed

Matter (U.K.) 8: 9477

69

By detecting the forward-scattered light from a second weaker probing beam and using a

lock-in amplifier, the relative displacement and phase are measured. With this dual beam

approach, ( )ωD , the amplitude of the particle displacement given by:

* 2 2 *2 2 2( )

( )ω

ω β ω=

+ − +ot

ot

k AD

k k m m (3.13)

The phase shift, ( )δ ω , is given by:

*1

* 2( ) tan βωδ ωω

−=+ −ot

mk k m (3.14)

where * *6 /a mβ πη= and kot is now given by the sum of the spring constants of the two la-

ser beams. The second probing beam is not required; the trapping beam can also be used

to measure particle displacements with modified expressions for ( )ωD and ( )δ ω . The

equations for displacement and phase are solved for the material viscosity, ( )η ω , and

elasticity, ( )ωk , from which the frequency dependent viscoelastic moduli are derived48:

( )( )2

ωωπ

′ =kG

a (3.15)

( ) ( ( ) )ω ω η ω η′′ = − solventG . (3.16)

48 Hough, L.A. and H.D. Ou-Yang (1999) Journal of Nanoparticle Research 1: 495

70

Figure 3.4. A schematic of the experimental setup of the oscillating optical tweezers. The

first polarizing beam splitter (PBS) splits the laser light into two separate beams, one

stronger trapping beam and a second weaker probing beam. Both are independently

aligned and steered by two piezoelectrically driven mirrors (PDM); rotation of the half-

wave plate (HW) changes the relative intensity of the two beams. A function generator

(FG) drives the PDM that moves the trapping beam and also delivers a signal to the lock-

in amplifier, allowing the phase lag to be measured. PD1 and PD2 are quadrant photodi-

ode detectors, and SW a switch to allow the signal from either PD1 or PD2 to reach the

lock in amplifier. PD1 is used for direct imaging of the trapped particle while PD2 is used

to measure the forward scattered laser light. The polarizer (P) placed before PD2 is used

to select either the trapping or probing beam for position detection. ND is a neutral den-

sity filter, M a mirror, PS a power supply for the piezoelectric-driver (PZT), BS a beams-

plitter, DBS a dichroic beam splitter which reflects the laser light and allows the broad-

band illumination light to pass. The sample chamber and trapped particle are located to

the right of the high-numerical-aperture objective lens (OBJ) and are not shown. (re-

printed with permission40).

71

Figure 3.4

72

The typical experimental design for frequency dependent measurements is shown in Fig.

3.4. Oscillating tweezers methods have been used to study telechelic HEUR polymers48

and collagen gels49.

Optical tweezers techniques use single particles and provide for local measure-

ments in inhomogeneous materials. Higher frequency measurements are possible with

laser detection tweezers techniques than with video-based magnetic particle manipulation

methods, and strain field mapping measurements are also possible. However, there are

disadvantages: forces are limited to the pico-Newton range, local heating can occur, as

well as local phototoxic effects in biological samples.

3.2.3 Atomic Force Microscopy Techniques

The third active manipulation technique employs the use of micromechanical

forces created by an atomic force microscope to probe local mechanical response. The

atomic force microscope (AFM), invented in 198650, has been widely used to study the

structure of soft materials, and biological materials in particular, with sub-nanometer

resolution51,52,53,54,55. In addition to imaging information about surface structure and to-

49 Velegol, D. and F. Lanni (2001) Biophysical Journal 81: 1786 50 Binning, G., C.F. Quate, and C. Gerber (1986) Physical Review Letters 56: 930 51 Drake, B., C.B. Prater, A.L. Weisenhorn, S.A.C. Gould, T.R. Albrecht, C.F. Quate,

D.S. Channell, H.G. Hansma, and P.K. Hansma (1989) Science 243: 1586 52 Henderson, E., P.G. Haydon, and D.S. Sakaguchi (1992). Science 257: 1944 53 Radmacher, M., R.W. Tillmann, M. Fritz, and H.E. Gaub (1992) Science 257: 1900 54 Bottomley, L.A., J.E. Coury, and P.N. First (1996) Biophysical Journal 74: 1564

73

pology, AFM techniques are sensitive to the force required to indent a surface and have

been used to measure local elasticity and viscoelasticity of thin samples, including bone

and bone marrow56, gelatin57,58, polyacrylamide gels59, platelets60, and living

cells59,61,62,63,64,65,66. Topographic images are obtained simultaneously with mechanical re-

sponse, allowing elasticity to be correlated with local structure67. Local elasticity meas-

urements have also provided information about the mechanical changes that accompany

55 Kasas, S., N.H. Thomson, B.L. Smith, P.K.Hansma, J. Mikossy, and H.G. Hansma

(1997) International Journal of Imaging Systems and Technology 8: 151 56 Tao, N.J., S.M. Lindsay, and S. Lees (1992) Biophysical Journal 63: 1165 57 Radmacher, M., M. Fritz, and P.K.Hansma (1995) Biophysical Journal 69: 264 58 Domke, J. and M. Radmacher (1998) Langmuir 14: 3320 59 Mahaffy, R.E., C.K. Shih, F.C. MacKintosh, and J. Käs (2000) Physical Review Letters

85: 880 60 Radmacher, M., M. Fritz, C.M. Kasher, J.P. Cleveland, and P.K. Hansma (1996) Bio-

physical Journal 70: 556 61 Hoh, J.H. and C.A. Schoenenberger (1994) Journal of Cell Science 107: 1105 62 Putman, C.A., K.O.V.D.Werf, B.G.D.Grooth, N.F.V. Hulst, and J. Greve (1994) Bio-

physical Journal 67: 1749 63 Shroff, S.G., D.R. Saner, and R. Lal (1995) American Journal of Physiology 269: C286 64 Goldman, W.H. and R.M. Ezzell (1996) Experimental Cell Research 226: 234 65 Rotsch, C., F. Braet, E. Wisse, and M. Radmacher (1997) Cell Biology International

21: 685 66 A-Hassan, E., W.F. Heinz, M.D. Antonik, N.P. D'Costa, and S. Nageswaran (1998)

Biophysical Journal 74: 1564 67 Radmacher, M., J.P.Cleveland, M. Fritz, H.G. Hansma, and P.K. Hansma (1994) Bio-

physical Journal 66: 2159

74

dynamic processes including cell division68, activation of platelets60, exocytosis69, and

drug-induced changes in the cytoskeletal structure of living cells70.

Experimentally, a commercial AFM with soft cantilever is used in constant-force

tapping mode for both imaging and elasticity measurements. The deflection of the canti-

lever is often measured with an optical “beam bounce” detection technique in which a la-

ser beam bounces off the back of the cantilever onto a position sensitive photodetector.

In constant force mode, the cantilever deflection is kept constant by a feedback circuit

that changes the height of the piezo-electrically controlled scanner head in response to the

local topography. In this mode, an image is generated from the scanner motion, and the

scan speed is limited by the response time of the feedback circuit. In tapping mode, the

tip of the cantilever is vibrated close to the sample surface, allowing the bottom of the tip

to gently touch or “tap” the surface; the oscillation amplitude changes with tip-to-sample

distance.

For each elasticity measurement, the deflection of the cantilever is measured as it

approaches the sample. For small deflections, the loading force can be calculated using

Hooke’s law once the cantilever spring constant, k, has been calibrated71. For an ex-

tremely stiff sample, the deflection of the cantilever, d, is identical to the movement of

the piezo scanner, z; however, for softer materials, the cantilever tip can indent the sam-

68 Dvorak, J.A. and E. Nagao (1998) Experimental Cell Research 242: 69 69 Schneider, S.W., S.W. Sritharan, J.P. Geibel, H. Oberleithner, and B. Jena (1997) Pro-

ceedings of the National Academy of Sciences of the United States of America 94: 316

70 Rotsch, C and M. Radmacher (2000) Biophysical Journal 78: 520 71 Butt, H-J. and M. Jaschke (1995) Nanotechnology 6: 1

75

ple. The indentation, δ, reduces the total deflection, d = z-δ, and the loading force is now

given by ( )F kd k z δ= = − . A modified Hertz model is used to describe the elastic defor-

mation of the sample by relating the indentation and loading force72,73. For a conical tip

pushing onto a flat sample, the force is given by:

22

2 tan( )1

α δπ ν

=−EF

(3.17)

where α is the half-opening angle of the AFM cone-tip, E is the Young’s modulus, and ν

is the Poisson ratio of the sample. By combining the above equations we obtain:

2

( )(2 )[ /(1 )] tan( )π ν α

−− = − +

−o

o ok d d

z z d dE (3.18)

where do is the zero deflection position and is determined from the non-contact part of the

force-curve, and zo is the contact point. Typically, ν is assumed or measured independ-

ently and the force curve is fit to determine E and zo. There are two common sources of

measurement error. For very soft samples, it is difficult to determine the exact point of

contact, leading to uncertainty in the measurement of E. Also, for very thin or soft mate-

rials, the elastic response of the underlying hard substrate that supports the soft sample

can contribute; in this case, the measured value of E should be taken as an upper bound.

The contribution of the rigid substrate is greater for higher forces and indentations, and

can be reduced by attaching a colloidal particle to the conical tip. With a spherical tip,

the contact area is increased, allowing the same stress to be applied to the sample with a

smaller force. Moreover, using a range of sizes of colloidal particles, the applied stress

72 Hertz, H. (1881) J. Reine Agnew. Mathematik 92: 156 73 Sneddon, I.N. (1965) International Journal of Engineering Science 3: 47

76

can be varied in a controlled manner; the applied stress is typically 100 – 10000 Pa for

colloids of ranging in size from less than 1 µm to 12 µm in diameter59. For a spherical tip

of radius R, the modified Hertz model predicts:

32

43(1 )

δν

=−

E RF. (3.19)

Another related technique, called Force Integration to Equal Limits (FIEL), al-

lows the mapping of relative micro-elastic response in systems with spatial inhomogenei-

ties or dynamic re-arrangements66. This technique has several advantages over traditional

analysis since the relative elastic map is independent of the tip-sample contact point and

the cantilever spring constant. In FIEL, a pair of force-curves is collected at two different

positions in the sample in constant force mode, imposing the condition that F1=F2 where

the subscript distinguishes the two measurements. For a spherical tip geometry, the force

balance gives:

3 32 2

1 21 2

4 43 3

δ δπ π

=R Rk k (3.20)

where k is the local spring constant of the sample is the defined as: 1k Eν π= − .

This equation reduces to:

32

1 1

2 2

kk

δδ

=

(3.21)

The work done by the AFM cantilever during indentation can be calculated by integrating

the force-curve over the indentation depth:

3 52 2

0

4 83 15

δ

δ δ δπ π

= =∫i

i i ii i

R Rw dk k (3.22)

77

where the index i indicates a single force-curve measurement. The relative work done is

given by the ratio of work done in each measurement:

52

1 1 2

2 2 1

δδ

=

w kw k . (3.23)

Combining the two above equations, we obtain an expression for the relative local elas-

ticity, given by:

23

1 1

2 2

=

w kw k . (3.24)

This approach works for other tip geometries as well, with a change only in the

scaling exponent; the exponent is 2/3 for a spherical or parabolic tip, 1/2 for a conical tip,

and one for a flat-end cylinder. By comparing many force curves obtained at many dif-

ferent locations in the sample, a map of relative microelasticity across the surface is ob-

tained. If the same probe is used in all measurements, the result is independent of the ex-

act probe size, cantilever spring constant, and deflection drift. This method is

independent of sample topography and does not require absolute height measurements.

FIEL mapping is most useful for measuring relative changes in elasticity in dynamic sys-

tems and is not appropriate for measuring absolute moduli. In order to obtain absolute

measures of elasticity and to compare FIEL maps obtained with different cantilevers, ad-

ditional calibration procedures are necessary.

In viscoelastic samples, AFM techniques can be modified to measure viscosity

and the frequency dependent response. Qualitative measures of the viscosity are possible

by measuring the relaxation of the tip into the sample or the hysteresis of the force-

curves. For more precise and frequency-dependent measurements, an oscillating cantile-

78

ver tip can be used59,66. A small amplitude sinusoidal signal,δ% , typically 5-20 nm, is ap-

plied normal to the surface around the initial indentation, δo, at frequencies in the range

of 20-400 Hz. A lock-in amplifier is used to measure the phase and amplitude of the re-

sponse with respect to the driving force. The modified Hertz model must now include a

frequency dependent response term, and for small oscillations the force is given by:

32 *4 3

3 2δ δ δ ≈ +

%

bead o of R E E (3.25)

where *E is analogous to the complex shear modulus, *G , and is defined as the fre-

quency-dependent part of the constant ratio E/(1-ν 2):

( )( )

*2 2 *

2 11 1

νν ν

+≡ =

− −EE

G (3.26)

and oE refers to the zero-frequency value of *E defined as:

( )( )2

2 11 (0)

νν

+≡

′−oEG (3.27)

where (0)G′ is the real part of the complex shear modulus, the elastic modulus, at zero

frequency. The second term in the modified Hertz model describes the time-dependent

response and includes contributions of both the viscous drag force on the cantilever as it

oscillates in the surrounding fluid and the viscoelastic response of the substrate. The vis-

cous force on the cantilever is dependent on the frequency of oscillation and is given by:

ωαδ= %f idrag (3.28)

where ω is oscillation frequency, and α is a constant that includes the driving amplitude,

the viscosity of the fluid, and the geometry of the cantilever. This contribution must be

subtracted from the time-dependent response in order to measure the frequency depend-

79

ent viscoelastic moduli, which depend only on the material properties of the soft sub-

strate.

AFM techniques allow measurements of elastic or viscoelastic response of thin

samples and surfaces. Images are often obtained simultaneously, allowing mechanical

properties to be correlated with local topography and microstructure. Relative elastic

mapping and fast AFM scans allow measurements of the dynamic changes in mechanics

or structure. As in all active manipulation techniques, the strain amplitude and driving

frequency can be varied and out-of-equilibrium measurements are possible.

3.3 Passive Microrheology Methods

A second class of microrheology techniques uses the Brownian dynamics of em-

bedded colloids to measure the rheology and structure of a material. Unlike active mi-

crorheology techniques that measure the response of a probe particle to an external driv-

ing force, passive measurements use only the thermal energy of embedded colloids,

determined by kBT, to measure rheological properties. For passive measurements, mate-

rials must be sufficiently soft in order for embedded colloids to move detectably with

only kBT of energy. The thermal motion of the probe in a homogeneous elastic medium

depends on the stiffness of the local microenvironment. Equating the thermal energy

density of a bead of radius a to the elastic energy needed to deform a material with an

elastic modulus 'G a length L yields:

2

3 2Bk T G La a

′=

. (3.29)

80

For most soft materials, the temperature cannot be changed significantly. Thus, the upper

limit of elastic modulus we are able to measure with passive techniques depends on both

the size of the embedded probe and on our ability to resolve small particle displacements

of order L. The resolution of detecting particle centers depends on the particular detec-

tion scheme used and typically ranges from 1 Å to 10 nm, allowing measurements with

micron-sized particles of samples with an elastic modulus up to 10 to 500 Pa. This range

is smaller than that accessible by active measurements, but is sufficient to study many

soft materials. Moreover, passive measurements share the distinct advantage that results

are always within the linear viscoelastic regime because there is no external stress ap-

plied.

To understand how the stochastic thermal energy of embedded micron-sized par-

ticles is used to probe the frequency dependent rheology of the surrounding viscoelastic

material, it is useful to first consider the motion of spheres in a purely viscous fluid then

generalize to account for elasticity. Micron-sized spheres in a purely viscous medium

undergo simple diffusion, or Brownian motion. The dynamics of particle motions are re-

vealed in the time dependent position correlation function of individual tracers. This cor-

relation function, also known as the mean squared displacement (MSD), is defined as:

22 ( ) ( ) ( )τ τ∆ = + −r r r

tx x t x t

(3.30)

where xr is the d-dimensional particle position, τ is the lag time and the brackets indicate

an average over all times t. The time-average assumes the fluid is in always in thermal

equilibrium and the material properties do not evolve in time. The diffusion coefficient,

D, of the Brownian particles is obtained from the diffusion equation:

81

2 ( ) 2τ τ∆ =rx dD

. (3.31)

From this, the viscosity η of the fluid surrounding the beads of radius a is obtained using

the Stokes-Einstein equation: / 6BD k T aπη= 74.

Many materials are more complex, exhibiting both viscous and elastic behavior.

Additionally, the responses are typically frequency dependent and depend on the time

and length scale probed by the measurement. For such materials, the thermally driven

motion of embedded spheres reflects both the viscous and elastic contributions, which are

revealed in the MSDs of the tracers75. Unlike a simple fluid where the MSDs of embed-

ded tracers evolve linearly with time, the MSDs of tracers in a complex material may

scale differently with τ,

2 ( ) ~ ατ τ∆rx , (3.32)

where α <1 and is called the diffusive exponent. The particles may exhibit subdiffusive

motion (0<α <1) or become locally constrained (α = 0) at long times.

In the case of a pure elastic homogenous material, a plateau in the MSD occurs

when the thermal energy density of the bead equals the elastic energy density of the net-

work that is deformed by the displacement of the bead:

( )2 Bk TxG a

τπ

∆ → ∞ =′ . (3.33)

74 Reif, F. Fundamentals of statistical and thermal physics. New York: McGraw-Hill, Inc. (1965) 75 Mason, T.G. and D.A. Weitz (1995) Physical Review Letters 74: 1250

82

Analogous to a harmonic oscillator, the elastic energy of deformation of a mate-

rial can be understood as the energy of a spring with a spring constant proportional to

G a′ . The energy of the spring is simply the thermal energy, kBT. A viscoelastic material

can be modeled as an elastic network that is viscously coupled to and embedded in an in-

compressible Newtonian fluid76. A natural way to incorporate the elastic response is to

generalize the standard Stokes-Einstein equation for a simple, purely viscous fluid with a

complex shear modulus G(ω)= iωη to materials that also have a real component of the

shear modulus. A generalized Langevin equation is used to describe the forces on a small

thermal particle of mass m and velocity v(t) in a complex material:

0

( ) ( ) ( ) ( )ζ τ τ τ= − −∫&t

Rmv t f t t v d (3.34)

where ( )Rf t represents all the forces acting on the particle, including both the interparticle

forces and stochastic Brownian forces. The integral represents the viscous damping of

the fluid with a time dependent memory function ζ(t) to account for the elasticity in the

network. By taking the unilateral Laplace transform of the generalized Langevin equation

and using the equipartition theorem, the viscoelastic memory function can be related to

the velocity autocorrelation function77:

( ) (0)( )ζ

=−

Bk Tv s v

ms s (3.35)

where s represents frequency in the Laplace domain. The inertial term is negligible ex-

cept at very high (~ MHz) frequencies76. When the velocity autocorrelation is written in

76 Levine A.J. and T.C. Lubensky (2000) Physical Review Letters 85: 1774 77 Mason T.G., H. Gang, and D.A. Weitz (1997). Journal of the Optical Society of Amer-

ica 14: 139

83

terms of the Laplace transform of MSD, the expression for the memory function in

Laplace space becomes

. 2 2

6( )( )

ζ =∆

%%Bk Ts

s r s (3.36)

To relate the microscopic memory function to the bulk viscoelasticity, the Stokes

law is generalized to include a frequency dependent complex viscosity75. In the Laplace

domain, this relates the complex shear modulus ( )G s% to the memory function ( )sζ% as

( )( )6ζπ

=%

% s sG sa . (3.37)

By combining these two equations, we obtain a relationship that directly relates the

mean-squared displacement of the tracers to the bulk modulus of the material:

2( )

( )π=

∆%

%Bk T

G sas r s

. (3.38)

This equation is a generalized, frequency-dependent form of the Stokes-Einstein equation

for complex fluids. In the limit of a freely diffusing particle in a purely viscous solution,

2 2( ) 6 /∆ =%r s D s , (3.39)

and the generalized Stokes-Einstein relation (GSER) recovers the frequency independent

viscosity, 0 / 6η π= Bk T aD , where D is the diffusion coefficient of the particle in the

fluid. This result of the generalized Stokes-Einstein equation is remarkable: simply by

observing the time-evolution of the MSD of thermal tracers, we obtain the linear, fre-

quency dependent viscoelastic response.

To compare with bulk rheology measurements, ( )G s% is transformed into the Fou-

rier domain to obtain *( )G ω . *( )G ω is the complex shear modulus and is the same quan-

84

tity measured with a conventional mechanical rheometer. *( )G ω can be written as the

sum of real and imaginary components: *( ) ( ) ( )G G iGω ω ω′ ′′= + . Since ( )G ω′ and ( )G ω′′ are

not independent and obey Kramers-Kronig relations, it is possible to determine both from

the single, real function ( )G s% . This can be done, in principle, by calculating the inverse

unilateral Laplace transform and then taking the Fourier transform43.

Equivalently, an alternative expression for the GSER can be written in the Fourier

domain as:

*2

( )( )

B

u

k TGai r

ωπ ω τ

=ℑ ∆

(3.40)

A unilateral Fourier transform, uℑ , is effectively a Laplace transform generalized for a

complex frequency s=iω. In practice, the numerical implementation of this process for

discretely sampled data of <∆r2(τ)> over a limited range of times can cause significant er-

rors in *( )G ω near the frequency extremes. Alternatively, a local power law expansion

of < 2 ( )∆r t > can be used to derive algebraic estimates for ( )G s% and *( )G ω 78,79. This ap-

proximation is discussed in the Appendix.

To use the generalized Stokes-Einstein relation to obtain the macroscopic viscoe-

lastic shear moduli of a material, it is necessary that the medium around the sphere may

be treated as a continuum material. This requires that the size of bead be larger than any

structural length scales of the material. For example, in a polymer network, the size of

the bead should be significantly larger than the characteristic mesh size.

78 Mason T.G., K. Ganesan, J.H. Van Zanten, D. Wirtz, S.C. Kuo (1997) Physical Review Letters 79: 3282

85

Recent theoretical work has shown that the GSER describes the thermal response

of a bead embedded in a viscoelastic medium within a certain frequency range, ωB < ω <

ω* 76. The lower limit, ωB, is the time scale at which longitudinal, or compressional,

modes become significant compared to the shear modes that are excited in the system. In

bulk rheology, the applied strain has only a shear component, but the thermally driven

probe particle responds to all of the thermally excited modes of the system including the

longitudinal modes of the elastic network. At frequencies lower than ωB, the network

compresses and the surrounding fluid drains from denser parts of the network. Above

ωB, the elastic network is coupled with the incompressible fluid and longitudinal modes

of the network are suppressed. In this regime, the probe motion is entirely due to excited

shear modes of the material43. An estimate of the lower crossover frequency, ωB, can be

determined by balancing local viscous and elastic forces. The viscous force per volume

exerted by the solvent on the network is 2~ /vη ξ , where v is the velocity of the fluid

relative to the network, η is the viscosity of the fluid and ξ is the characteristic length

scale of the elastic network. The local elastic force per volume exerted by the network is

2 2~ /G u G u a′ ′∇ at the bead surface where u is the network displacement field and a is

the radius of the bead. Viscous coupling will then dominate above a crossover frequency

2

2BG

aξω

η′

≥ . (3.41)

For a typical soft material with an elastic modulus of 1 Pa, viscosity of 0.001 Pa*sec and

a mesh size one-tenth the radius of embedded probe, this crossover frequency ωB is ap-

79 Dasgupta B.R., S.-Y. Tee , J.C. Crocker, B.J. Frisken, and D.A. Weitz (2001) Physical

Review E 65: 051505

86

proximately 10 Hz. The upper frequency limit, ω*, exists due to the onset of inertial ef-

fects of the material at the length scale of the bead. Shear waves propagated by the mo-

tion of the tracer decay exponentially from the surface of the bead through the surround-

ing medium. The characteristic length scale of decay is called the viscous penetration

depth and is proportional to 2* / ρωG where ρ is the density of the surrounding fluid

and ω is the frequency of the shear wave3. The viscous penetration depth sets a length

scale for how far information is propagated in the medium when probed with a shear

strain at a certain frequency ω. When the magnitude of the viscous penetration depth

equals the size of the bead, inertial terms cannot be neglected. For a typical soft material,

inertial effects can be expected to be significant only for frequencies larger than 1MHz43.

From these analyses, we find in typical experiments there is a very large fre-

quency range 10 Hz < ω < 100 kHz where the generalized Stokes-Einstein equation is ac-

curate and valid. Note that this is much higher than traditional mechanical measurements

where inertial effects begin to become significant around 50 Hz. Additionally, models

have been proposed for the frequency regimes where the GSER does not hold76.

To take full advantage of the range of frequencies and complex moduli accessible

in a passive microrheology experiment, it is necessary to use techniques that measure the

mean-squared displacement (MSD) of embedded spheres with excellent temporal and

spatial resolution. The MSD can be calculated from methods that directly track the parti-

cle position as a function of time or can be obtained from ensemble-averaged light scat-

tering experiments. Methods of particle detection vary significantly in temporal and spa-

tial resolution, affecting the types of measurement possible with each technique.

87

Additionally, techniques differ significantly in their ability to provide statistical accuracy

over an ensemble of probes.

Particle tracking methods have been developed to directly image the position of a

colloidal sphere as a function of time. Because the entire particle trajectory is directly

obtained, these techniques allow further analysis of individual trajectories beyond an en-

semble averaged MSD. These analyses often provide further insight into the local struc-

tures and rheology of the surrounding medium. Methods of particle tracking differ in the

spatial and temporal resolution of the particle trajectories. The temporal resolution is de-

termined by the frequency at which particle positions can be recorded. The spatial reso-

lution is determined by how precisely differences in the particle position are measured.

This, in turn, is used to determine an upper bound on the elastic modulus that can be de-

tected using thermal motion with a given experimental technique.

The motion of individual tracers can be measured with laser detection schemes

nearly identical to the optical tweezers setup described in Section 3.2.2 43,78. Unlike opti-

cal tweezers, the laser power used in laser deflection particle tracking is quite low so that

the optical forces are very small (< 5%) compared to the thermally driven forces of the

bead. The thermally driven motion will cause the bead to move off the beam’s axis and

deflection of the laser beam can be measured. From this deflection, the displacement of

the single bead is detected from which the MSD78 or the power spectral density43, the po-

sition correlation function in frequency space, can be calculated. The power spectral

density of the beads is interpreted as the viscoelastic response of the material in a similar

manner as the MSD43. This detection scheme has excellent spatiotemporal resolution;

individual particles are tracked with subnanometer precision at frequencies up to 50 kHz.

88

The enhanced frequency regime allows study of rheology of polymer networks at time

scales where single filament properties often dominate the rheological response. How-

ever, this technique is limited in its statistical accuracy in two ways. First, it is somewhat

difficult to obtain particle trajectories for a large ensemble of beads as this requires many

consecutive measurements of individual beads. Secondly, in practice, it is difficult to

track a single bead with this technique for longer than a few minutes because freely mov-

ing particles can diffuse out of the beam. While this track length is more than sufficient

for the higher end of the frequency sweep, the low frequency statistics are limited. This

strength of this technique resides in its ability to detect extremely small displacements of

individual beads at high frequencies inaccessible to other video-based methods. Laser

Deflection Particle Tracking has been used to study the rheology of F-Actin net-

works42,80,81 and living cells82.

Alternatively, it is possible to directly image the embedded beads using a simple

video microscopy setup. Video microscopy of single beads is often used in active meas-

urements as well, in particular with the magnetic bead microrheology experiments dis-

cussed in Section 3.2.1. Techniques in image processing have been developed to auto-

mate the process of accurate particle center location to simultaneously track hundreds of

embedded probes in a single field of view of the microscope with submicron precision.

While video microscopy is limited to frequencies available to the camera, the strength of

the technique is in its ability to obtain good statistics on ensembles of beads.

80 McGrath J.L., J.H. Hartwig, and S.C.Kuo (2000) Biophysical Journal 79: 3258 81 Tseng Y. and D. Wirtz (2001) Biophysical Journal 81: 1643 82 Yamada, S., D. Wirtz, and S.C. Kuo (2000) Biophysical Journal 78: 1736

89

Embedded spheres are imaged with a conventional light microscope using either fluores-

cence or bright field microscopy. Using bright field microscopy, spheres larger than a

few hundred nanometers can be observed but the diffraction limited resolving power of

the microscope precludes the study of smaller probes. Fluorescent labeling offers the

ability to observe smaller probes, which now act as point sources of light, as well as the

opportunity to perform colocalization studies with differently dyed beads. Fluorescently

labeled colloidal spheres are commercially available from 20 nm up to several microns.

In a typical video based experiment, a time series of microscope images is ob-

tained with a CCD camera and recorded in analog format onto a S-VHS cassette using

commercially available video tape recorders. Video images are digitized using a com-

puter equipped with a frame grabber card. Until recently, it was not possible to write a

full frame image of 480 x 640 pixels directly to the hard drive at the standard video rates

of 30 Hz due to limitations in the speed at which information can be transferred. With

advances in computer technology, it is now possible to write full frame images directly to

the hard drive at 30 Hz 14; nonetheless, capturing movies to videotape affords the ability

to store large amounts of data conveniently.

Of some concern is that the frequency ranges for video microscopy are around 10

Hz, earlier estimated as the lower frequency limit of the generalized Stokes-Einstein rela-

tion (GSER). This frequency limit is simply an estimate for when compressibility effects

may become significant in a viscoelastic medium and preclude the use of the GSER.

However, this lower frequency limit varies widely in different samples. Some materials,

like simple fluids, are known to be incompressible at all frequencies. Additionally, in

90

Section 3.5, formalism will be introduced to test for and even quantify compressibility ef-

fects using particle tracking with a video microscopy apparatus.

In a homogeneous, isotropic material, it is sufficient to examine the projection of

the particle trajectory along a single axis. In heterogeneous materials, it may be useful to

be able to obtain two- or three-dimensional particle trajectory. In video microscopy, the

motion of the particle is projected into the plane of the focus and a two dimensional tra-

jectory is obtained for further analysis. Techniques have been developed to track parti-

cles in the direction perpendicular to the plane of focus either by modifying the optics in

a conventional microscope83 or by carefully examining the structure of the Airy disk cre-

ated by circular particles in bright field microscopy83,84,85. However, confocal micros-

copy is currently most widely used to follow the three dimensional motion of fluores-

cently tagged colloids86,87.

The power in using video microscopy for microrheology lies in the potential of

following the motions of roughly a hundred colloidal particles simultaneously and the

ability to obtain the ensemble averaged mean-squared displacement (MSD) while still re-

taining each of the individual particle trajectories. The ability to accurately and precisely

find the centers of the two-dimensional colloidal images in each frame of video is crucial.

83 Kao, H.P. and A.S. Verkman (1994) Biophysical Journal 67: 1291 84 Ovryn, B (2000) Experiments in Fluids 29: S175 85 Ovryn, B. and S.H. Izen (2000) Journal of the Optical Society of America, A 17: 1202 86 Weeks, E.R., J.C.Crocker, A.C. Levitt, A. Schofield and D.A. Weitz (2000) Science

287: 627 87 Dinsmore, A.D., E.R. Weeks, V. Prasad, A.C. Levitt, and D.A. Weitz (2001) Applied

Optics 40: 4152

91

Algorithms have been developed to automate the process of finding particle centers and

accurately find particles to roughly 1/10 of a pixel, and are described in the Appendix41.

For a typical magnification, this is a resolution of 10 nm. Once colloidal particles are lo-

cated in a sequence of video images, particle positions in each image are correlated with

positions in later images to produce trajectories. To track more than one particle, care is

required to uniquely identify each particle in each frame41.

Multiparticle tracking is particularly well suited to study materials that are hetero-

geneous at the length scales of the bead; for these systems, single bead measurements are

not sufficient to describe a bulk response, but particle movements do reveal details of the

local mechanics and microstructure. Measurements on heterogeneous materials will be

discussed in Section 3.4. In the case of homogeneous materials, each thermally activated

particle measures the same continuum viscoelastic response; as a result, measurements on

ensembles of particles are often preferable to single bead measurements by providing bet-

ter statistical accuracy in calculating the MSD and moduli. Ensemble averaged behaviors

can be obtained by averaging many single particle motions that are simultaneously ob-

tained with video microscopy; however, light scattering techniques are often preferable.

Light scattering methods inherently average over a large ensemble of particles, and are

not appropriate for samples that may exhibit local heterogeneity; however, for homoge-

neous samples, light scattering has the advantage of better averaging and statistical accu-

racy and a larger accessible frequency range than any macroscopic measurement or

video-based microrheology technique.

In a typical dynamic light scattering measurement, a laser beam impinges on a

sample and is scattered by the particles into a detector placed at an angle, θ, with respect

92

to the incoming beam88. As the particles diffuse and rearrange in the sample, the inten-

sity of light that reaches the detector fluctuates in time. In the simplest case, each photon

is scattered only once within the illumination volume directly into the detector. The in-

tensity fluctuations are measured as a function of time, I (t), and the normalized intensity

correlation function, g2 (τ), is calculated as a function of lag time τ:

2 2

( ) ( )( )

( )

I t I tg

I t

ττ

+=

(3.42)

where the brackets indicate an average over time. The measured g2, can be related to the

calculated field correlation function, g1, which is given by:

1 2

( ) *( )( )

( )

E t E tg

E t

ττ

+=

(3.43)

where E is the scattered electric field, using the Siegert relation:

22 1( ) 1τ β= +g g (3.44)

where β is determined by the coherence of the detection scheme. For measurements of a

single coherence area, or speckle, 1β ≈ . If all particles are statistically independent, and

moving randomly due to thermal impulses only, then:

2 21( ) exp[ ( ) / 6]τ τ= − ∆g q r (3.45)

where 2 ( )r τ∆ is the ensemble averaged three-dimensional MSD, and q is the scattering

wave vector given by:

4 sin2

nq π θλ

= (3.46)

88 Berne, B.J. and R. Pecora. Dynamic Light Scattering with applications to chemistry,

biology, and physics. Mineola: Dover (2000)

93

where n is the index of refraction of the sample and λ is the wavelength of the laser in

vacuuo. The correlation function decays as the scatters move a distance 1/q. For some

elastic samples, particles may be locally constrained and unable to move 1/q during the

measurement, leading to “non-ergodic” behavior. In order to extract the ensemble aver-

age of the field correlation function from the measured time-averaged intensity fluctua-

tions, a different method of analysis is required89,90,91,92. Once the MSD is obtained, the

generalized Stokes-Einstein formalism can be applied to extract the frequency dependent

viscoelastic moduli. Single light scattering techniques are typically sensitive to frequen-

cies in the range of 0.01 – 10 Hz, similar to the frequency range available with a conven-

tional macroscopic rheometer.

A second light scattering technique, Diffusing Wave Spectroscopy (DWS), allows

measurements of multiple scattering media and extends the accessible frequency range to

much higher frequencies93,94. The experimental set-up is similar to that of the single-

scattering experiment; however, in this case, a laser beam impinges on an opaque sample

89 Pusey, P.N. and W. van Megen (1989) Physica A 157: 705 90 Joosten, J.G.H., E.T.F. Gelade, and P.N. Pusey (1990) Physical Review A 42: 2161 91 van Megen, W., S.M. Underwood, and P.N. Pusey (1991) Physical Review Letters 67:

1586 92 Xue, J.Z., D.J. Pine, S.T. Milner, X.L.Wu, and P.M. Chaikin (1992) Physical Review A

46:6550 93 Pine D.J., D.A. Weitz, P.M. Chaikin, and E. Herbolzheimer (1988) Physical Review

Letters 60: 1134 94 Weitz D.A. and D.J. Pine, “Diffusing-wave spectroscopy,” appearing in Dynamic

Light Scattering, W. Brown (ed). Oxford: Oxford University Press (1993)

94

and the light is scattered multiple times before exiting. The diffusion equation is used to

describe the propagation of light through the sample. All q-dependent information is lost

as the photons average over all possible angles, resulting in only two experimental ge-

ometries: transmission and backscattering. Like single scattering experiments, the inten-

sity of a single coherence area is detected; fluctuations in intensity reflect the dynamics of

the scattering medium and the mean squared displacement of the particles can be ob-

tained. The field correlation function can be expressed as:

( )2 2

1 *0

( )( ) exp

3

ok s rg P s ds

l

ττ

∞ ∆ ∝ −

∫ (3.47)

where P(s) is the probability of light traveling a path length s, and is determined by solv-

ing the diffusion equation with the experimental boundary conditions, ko = 2πn/λ where n

is the index of refraction of the solution, λ is the wavelength of light in vacuuo, and l* is

transport mean free path and is defined as the distance the light must travel before its di-

rection is completely randomized. The mean free path is determined in an independent

measurement of the transmitted intensity and is typically much smaller than the thickness

of the sample chamber. The correlation function decays when the total path length of the

light through the sample changes by roughly the wavelength of the incident beam. To

achieve this, each particle in the path of the light needs to move only a fraction of a

wavelength; as a result, DWS is sensitive to motions on much smaller length scales and

faster time scales than single scattering measurements. Analytic inversion is used to ob-

tain the MSD from 1( )g τ . The typical frequency range in a DWS measurement is 10 – 105

Hz, allowing direct measurements of the high-frequency response of polymer solutions

and other materials that are impossible with traditional mechanical measurements. The

95

microrheology of flexible and semi-flexible polymer solutions has been measured using

light scattering techniques78,79,95,96,97,98,99. As seen in Figure 3.5, Dasgupta et al. have

demonstrated excellent agreement between moduli obtained for a flexible polymer solu-

tion, polyethylene oxide (PEO), using DWS and single light scattering microrheology

and mechanical rheometry.

3.4 Practical Applications of One-Particle Microrheology

Because the techniques discussed in Sections 3.2 and 3.3 interpret the dynamics

of individual probe particles as a viscoelastic response, these techniques have come to be

known as one-particle microrheology. One-particle microrheology is a powerful tool to

study the rheological properties of samples with extremely small sample volumes at fre-

quencies inaccessible to bulk measurements. The development of microrheological tech-

niques is currently an active field and the full range of possibilities, and limitations, of

microrheology tools has yet to be completely understood. Here we discuss important is-

sues to consider before interpreting the individual bead motion as a bulk rheological re-

sponse of the material.

95 Mason, T.G., H.Gang, and D.A. Weitz (1996) Journal of Molecular Structure 383: 81 96 Gisler, T. and D.A. Weitz (1998) Current Opinion in Colloid and Interface Science 3:

586 97 Gisler T. and D.A.Weitz (1999) Physical Review Letters 82: 1606 98 Palmer, A., T.G. Mason, J. Xu, S.C. Kuo, and D. Wirtz (1999) Biophysical Journal 76:

1063 99 Mason, T.G. (2000) Rheologica Acta 39: 371

96

One-particle microrheology assumes the local environment of the bead re-

flects that of the bulk. If the surface chemistry of the bead modifies the structure of the

material around the bead, the one-particle response will be a reflection of the local micro-

environment rather than bulk rheology. Interactions between the embedded probe and

sample are of much interest and are highly system dependent. In studies with polyethyl-

ene oxide, an uncross-linked flexible polymer solution, there is no effect of bead chemis-

try79. However, surface chemistry seems to have significant effects with biopolymer net-

works80.

97

Figure 3.5. The storage and loss moduli obtained by Dasgupta et al. (Dasgupta, et.

al. 2001) for a 4% by weight 900 kDa PEO solution comparing moduli obtained with a

conventional strain controlled rheometer (G’ – open square, G” – open circle) to those

obtained by both DWS (G’ – solid line, G” – dash-dot) and single scattering at 20o (G’-

dot, G” – dash). The beads used in both light scattering techniques are 0.65 micron

spheres. Using the three techniques, it is possible to obtain data over ~6 decades in fre-

quency. Additionally, the moduli obtained via light scattering are in excellent agreement

with bulk measurement. Similar agreement between DWS and bulk rheology was found

for other concentrations of PEO as well. (reprinted with permission79)

98

The charge groups used to stabilize commercially available colloids are reactive with

many proteins, leading to unspecific binding. In a careful study with F-actin networks,

the bead chemistry was changed to either inhibit or to encourage binding of actin to the

bead surface80. The beads that prevented actin binding were insensitive to changes in the

mechanical properties of the network. By contrast, beads that bound to actin filaments

reflected the bulk properties of the networks more accurately. However, it is often diffi-

cult to precisely control protein adsorption onto the beads. Moreover, although in some

cases binding helps one-particle microrheology probe bulk properties, there may be other

consequences. In the worst-case scenario, there is significant aggregation of the probes

and the macroscopic gel-like structures are significantly altered. In less extreme cases,

the presence of the bead affects only the surrounding local network but the bulk proper-

ties are unchanged. In this case, it is possible to obtain a modulus from one-particle mi-

crorheology but it is unclear whether the measured local modulus reports the bulk re-

sponse.

In general, it is advisable to test for probe surface chemistry effects in a new sys-

tem. In Section 3.5, two-particle microrheology will be discussed. Unlike one-particle

microrheology that measures individual bead response to the surrounding microenviron-

ment, two-particle microrheology is independent of bead surface chemistry.

One-particle microrheology has the additional feature that bulk response is meas-

ured only if the probe size is larger than the length scale of heterogeneity in the sample.

These length scales are often unknown prior to a microrheology experiment. When the

particle diameter is comparable to or smaller than the length scale of structures in the

medium, the tracers can move within small cavities and their motions are not only a

99

measure of the viscoelastic response, but also of the effect of steric hindrances caused by

the cavity walls100. A material like agarose is a prime example. Agarose is known to be

structurally heterogeneous and consists of a network of fibrous molecules.

The elastic modulus of agarose, as measured in a bulk rheometer, is roughly 2700

Pa and too high to measure with a passive microrheological technique. However, while

agarose contains large elastic structures that span the sample and bear macroscopic stress,

these structures do not form a homogeneous elastic medium. Instead, agarose is charac-

terized by many smaller voids, or pores, through which smaller particles may move. By

observing the dynamics of smaller particles within the pores, one is actually characteriz-

ing the structural and mechanical properties of the weaker pores rather than the bulk con-

tinuum properties.

In a particle tracking experiment, a plateau in the ensemble averaged mean-

squared displacement at long lag times, < 2 ( )τ∆ → ∞x >, indicates that the particles are

constrained by the material, but it is necessary to examine how this plateau varies with

particle radius, a, to determine the nature of the constraint. If the plateau is a measure of

local elasticity G then < 2 ( )τ∆ → ∞x >= /πkT Ga , so < 2 ( )τ∆ → ∞x > scales with 1/a. If the

plateau is a measure of a pore size d then d=a + h where h is the size of the fluid filled

gap between the particle and the pore wall and can be approximated by

h=< 2 ( )τ∆ → ∞x >1/2. Thus, if the particle is measuring a purely steric boundary then

+a < 2 ( )τ∆ → ∞x >1/2 remains roughly constant. It is important to test trends in mean-

squared displacement as a function of bead size to determine the nature of the constraints

100 Valentine, M.T., P.D. Kaplan, D. Thota, J.C. Crocker, T. Gisler, R.K. Prud'homme,

M. Beck, and D.A. Weitz (2001) Physical Review E 64: 061506

100

felt by the bead. The technique of two-particle microrheology is able to distinguish be-

tween measurement of a steric boundary and a local modulus.

In heterogeneous materials, individual particles movements must be considered

since different particles may be exploring different microenvironments. Statistical tech-

niques have been developed to compare the individual particles and map spatial and tem-

poral variation in mechanical response100. Furthermore, particles in similar microenvi-

ronments can be grouped together into a meaningful ensemble and average rheological

and structural properties can be obtained.

3.5 Two-particle Microrheology

One-particle microrheology is very sensitive to the local environment of the em-

bedded bead. If the tracers locally modify the structure of the medium or sample only the

weak pores in a heterogeneous material, then one-particle microrheology determines the

structure and mechanics of the local microenvironment rather than bulk rheology. The

recently developed technique of two-particle microrheology101 eliminates motion due to

purely local structure and mechanics by measuring the cross-correlated motion of pairs of

tracer particles within the sample. The correlated motion of the particles is not affected

by the size, or even shape, of the tracer particles and is independent of the specific cou-

pling between the probe and the medium. Furthermore, the length scale being probed is

not the individual bead radius, a, but is the distance, r, between the tracers which is typi-

101 Crocker, J.C., M.T. Valentine, E.R. Weeks, T. Gisler, P.D. Kaplan, A.G. Yodh, and

D.A.Weitz (2000) Physical Review Letters 85: 888

101

cally 10-100 microns rather than 1 micron. This increase in length scale means that the

technique is insensitive to short wavelength heterogeneities in the sample smaller than the

bead separation distance and thus may probe bulk rheology even if individual particles do

not. Additionally, probing longer length scales lowers the frequency limit, Bω , of the

generalized Stokes-Einstein equation by replacing a by r in Equation 3.40.

Two-particle microrheology directly maps the long-range deformation or flow of

the material due to a single tracer’s motion. Since one tracer’s strain field will entrain a

second particle, the cross-correlated motion of two tracers’ movements is a direct map of

the strain field in the material. In a medium that is homogeneous at long length scales

and characterized by a bulk viscoelasticity, the strain field is proportional to the tracer

motion and decays like a/r where r is the distance from the tracer. Local heterogeneities

either intrinsic to the material or created by the presence of the probe will affect individ-

ual particle motions but the movements will be uncorrelated at large distances.

With modifications, a two-particle measurement is possible with most of the ac-

tive and passive techniques previously discussed. Video based multiparticle tracking is

particularly well suited to examine cross-correlated motion because several hundred trac-

ers can be imaged simultaneously101. In a typical two-particle measurement, roughly one

hundred tracers are observed for several hundred seconds at video rate to gather sufficient

statistics. Vector displacements of individual tracers are calculated as a function of lag

time, τ, and absolute time, t: ( , ) ( ) ( )α α ατ τ∆ = + −r t r t r t . Then the ensemble averaged

tensor product of the vector displacements is calculated:

,( , ) ( , ) ( , ) ( )ji ij

i j tD r r t r t r R tαβ α βτ τ τ δ

≠ = ∆ ∆ − (3.48)

102

where i and j label two particles, α and β are coordinate axes and Rij is the distance be-

tween particles i and j. The average is taken over the only the distinct terms ( i j≠ ); the

“self” term yields the one-particle mean-squared displacement, < 2 ( )τ∆r >.

The two-particle correlation for particles in an incompressible continuum is calcu-

lated by treating each thermal particle as a point stress source and mapping its expected

strain field20. In the limit where the particle separation, r, is much greater than the parti-

cle radius a (r >> a), this is calculated by multiplying the one-particle mean-squared dis-

placement predicted by conventional generalized Stokes-Einstein relation in (3.38) by

a/r, to obtain:

( , )2 ( )

12

Brr

rr

k TD r srsG s

D D Dθθ φφ

π=

= =

%%

(3.49)

where ( , )rrD r s% is the Laplace transform of ( , )rrD r t and the off-diagonal terms vanish.

While this result has been obtained for an incompressible medium, compressible materi-

als can be treated by using a modifed Stokes-Einstein relation and strain field20. The

Brownian motion of a single probe is the superposition of all modes with wavelengths

greater than the particle radius, a. The correlated motion of two tracers a separation r

apart is driven only by modes with wavelengths greater than the separation distance.

Therefore, two tracers that are separated by more than the coarse-grained length scale in

an inhomogeneous medium will depend on the coarse-grained, macroscopic complex

modulus. In an experiment, it is necessary to confirm that ( , ) ~ 1/%rrD r s r by examining

the correlated motion of at least several pairs of probes with different pairwise separa-

103

tions. If the strain field follows 1/r within a certain range of interparticle distances, the

material can be treated as a homogeneous continuum at those length scales.

Comparing the longitudinal two-point correlation to the generalized Stokes-

Einstein equation used in one-particle microrheology suggests defining a new quantity:

the distinct mean-squared displacement, < 2 ( )τ∆r >D, as

2 2( ) ( , )rrD

rr D r sa

τ∆ = %

(3.50)

This is the thermal motion obtained by extrapolating the long-wavelength thermal undu-

lations of the medium to the bead radius. In a homogenous material where the GSER is

valid, the distinct mean-squared displacement matches the conventional one-particle

mean-squared displacement. In inhomogeneous materials, differences between

< 2 ( )τ∆r >D and < 2 ( )τ∆r > provide insight into the local microenvironment experienced by

the tracers. In this case, < 2 ( )τ∆r > may be understood as a superposition of a long-

wavelength motion described by < 2 ( )τ∆r >D plus a local motion in a cavity. Figure 3.6a

shows the comparison between the self and distinct mean squared displacements of 0.20

µm beads in a 0.25% guar solution. Guar is a naturally occurring neutral polysaccharide

extracted from guar gum bean. A small concentration of guar in water dramatically

changes the viscoelastic properties because of the formation of high-molecular weight,

mesoscopic aggregates102. Two-particle microrheology results are obtained by substitut-

ing < 2 ( )τ∆r >D into the generalized Stokes-Einstein equation in place of < 2 ( )τ∆r >. The

self and distinct mean squared displacements results for the guar solution do not corre-

102 Gittings M.R., L. Cipelletti, V. Trappe, and D.A. Weitz, In M, Marques C (2000)

Journal of Physical Chemistry B 104: 4381

104

spond, but disagree by a factor of two. In Figure 3.6b, the data are converted into

'( )G ω and "( )G ω , and compared to results obtained from a macroscopic strain-controlled

mechanical rheometer. As shown, the moduli calculated from the two-point correlation

function are in good agreement with the results obtained with the rheometer.

Single-particle microrheology provides qualitatively different moduli and com-

pletely fails to detect the crossover frequency. Unlike one-particle microrheology, two-

particle microrheology is successful in determining the bulk rheological behavior of an

inhomogeneous medium. This allows measurements in a larger range of materials than

previously accessible with one-particle microrheology.

105

Figure 3.6. The comparison of one- and two-particle microrheology to bulk measure-

ments in a guar solution seen by Crocker et al. (Crocker, et al. 2000) (a) Comparison of

the one-particle MSD (triangles) and distinct MSD (circles) of 0.20 µm diameter beads in

0.25% weight guar solution. Due to the heterogeneous nature of the medium, the curves

differ by a factor of two. The solid line is a smooth fit to the data used for calculating

rheology. (b) The elastic (filled circles) and loss (open circles) moduli calculated using

the distinct MSD showing a crossover at high frequencies are in good agreement with the

mechanical bulk measurement (solid curves). The moduli calculated using the one-

particle MSD (triangles) do not agree. (reprinted with permission101)

106

3.6 Summary

Microrheological techniques are powerful methods to characterize the mechanics and

structure of novel and complex materials on length scales much shorter than those meas-

ured with bulk techniques. In an incompressible, homogeneous material the response of

an individual probe due to external forcing or thermal fluctuations is a reflection of the

bulk, viscoelastic properties of the surrounding medium. In heterogeneous materials, the

motion of individual beads allows characterization of local mechanics while macroscopic

rheological response is obtained by calculating the cross-correlated motion of the tracers

from the same data. By combining the local and macroscopic measurements, a new un-

derstanding of how structure and mechanical response at the micron length scale relates

to the bulk material properties emerges. Only microliter sample volumes are required, al-

lowing the application of rheological techniques to materials too costly or difficult to syn-

thesize in large quantities, or systems that are inherently small like living cells. Further-

more, viscoelastic response can be measured at frequencies ranging from 0.01 Hz to 105

Hz, much larger than the range of traditional mechanical measurements, allowing direct

measurements of high frequency response in a wide range of soft materials. These tech-

niques allow the study of both new systems and different material properties than con-

ventional methods, and open new possibilities for understanding the microscopic proper-

ties of complex materials.

107

Chapter 4: Investigating the microenvironments of inhomogeneous soft materials with

multiple particle tracking∗

4.1 Overview

We develop a new multiple particle tracking technique for making precise,

localized measurements of the mechanical microenvironments of inhomogeneous

materials. Using video microscopy, we simultaneously measure the Brownian dynamics

of roughly one hundred fluorescent tracer particles embedded in a complex medium and

interpret their motions in terms of local viscoelastic response. To help overcome the

inherent statistical limitations due to the finite imaging volume and limited imaging

times, we develop new statistical techniques and analyze the distribution of particle

displacements in order to make meaningful comparisons of individual particles and thus

characterize the diversity and properties of the microenvironments. The ability to

perform many local measurements simultaneously allows more precise measurements

even in systems that evolve in time. We show several examples of inhomogeneous

materials to demonstrate the flexibility of the technique and learn new details of the

mechanics of the microenvironments that small particles explore. This technique extends

other microrheological methods to allow simultaneous measurements of large numbers of

probe particles, enabling heterogeneous samples to be studied more effectively.

∗ Originally Published in Physical Review E vol. 64, p. 061506 (2001) , by M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud'homme, M. Beck, and D. A. Weitz. Copyright(2001) by the American Physical Society.

108

4.2 Introduction

Little is known of the mechanical microenvironments of complex materials such

as polymer gels and the cytoplasm, yet understanding these local environments is critical

for characterizing the microscopic processes that take place in within them. Local

measurements are necessary to explore the microscopic dynamics and mechanical

properties that control many chemical and physical processes, such as the dynamics of

colloidal particles in confined geometries, the diffusion of reactants for the chemical

modification of gels, and the separation and transport of macromolecules and proteins in

structured gels1,2,3,4. In biological materials, inter- and intra-cellular transport, activities

within cellular compartments, and the resistance encountered by molecular motors

depend on local, not bulk, mechanical response to locally generated forces5,6,7,8. Often,

the details of this microscopic response cannot be inferred from in vitro or bulk

1 Tamai, N. ,M. Ishikawa, N. Kitamura and H. Masuhara (1991) Chemical Physics Letters 184: 398 2 Greiss, G., K.B. Guiseley and P. Serwer (1993) Biophysical Journal 65: 138 3 Evans, D.F. and H. Wennerström, The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet. New York: VCH Publishers (1994) 4 Ciszkowska, M. and M.D. Guillaume (1999) Journal Physical Chemistry A 103: 607 5 Luby-Phelps, K., F. Lanni and D.L. Taylor (1988) Annual Review of Biophysical and Biophysical Chemistry 17: 369 6 Luby-Phelps, K. and D.L. Taylor (1988) Cell Motility Cytoskeleton 10: 28 7 Provance, D.W. ,A. McDowall, M. Marko and K. Luby-Phelps (1993) Journal of Cell Science 106: 565 8 Janson, L.W., K. Ragsdale and K. Luby-Phelps (1996) Biophysical Journal 71: 1228

109

measurements. For example, intracellular biochemical reaction rates may differ

significantly from those measured in dilute solution because macromolecular crowding

and the physical constraints imposed by the cytoskeletal network dramatically alter the

mobility of reactants9,10,11,12,13,14. In this paper, we present a technique to study local

mechanical properties by simultaneously tracking many Brownian tracer particles and

using a combination of concepts of microrheology and careful statistical tools to describe

the range of microenvironments encountered in a single system.

Macroscopically, rheological measurements completely describe the mechanical

response of a material. Simple fluids are characterized by their viscosity and ideal elastic

solids by their elastic modulus. Although more complex, viscoelastic materials that

display properties of both solids and liquids can also be fully characterized by their

frequency-dependent complex shear modulus. Microscopically, we can characterize the

local mechanical response by a local viscosity and elasticity. However, we must also

consider the ability of the large structures found in many complex materials to constrain

small particles or macromolecules. At length scales where the constraining effects of

9 Kramers, H.A. (1940) Physica (Utrecht) 7: 284 10 Zimmerman, S.B. and A.P. Minton (1993) Annual Review Biophysics and Biomathematics 22: 27 11 Arrio-Dupont, M., S. Cribier, G. Foucault, P.F. Devaux and A. dAlbis (1996) Biophysical Journal 70: 2327 12 Arrio-Dupont, M., G. Foucault, M. Vacher, P.F. Devaux and S. Cribier (2000) Biophysical Journal 78: 901 13 Odde, D.J. (1997) Biophysical Journal 73: 88 14 Srivastava, A. and G. Krishnamoorthy (1997) Archives of Biochemistry and Biophysics 340: 159

110

these structures become important, the microscopic response is no longer connected to

the bulk response by simple scaling laws, and direct local measurements are essential. In

order to be interpreted in terms of mechanical microenvironments, these measurements

should be local, should not depend on any assumptions about homogeneity and should be

sensitive to variations in local response and the presence of steric constraints.

A number of techniques have been employed to measure structure and dynamics

in complex fluids. Static measurements, such as freeze fracture electron microscopy, x-

ray diffraction and fluorescence studies, give detailed structural information, but do not

connect structure with dynamic mechanical response 2,15,16,17. Fluorescence recovery

after photobleaching (FRAP) experiments probe dynamics and measure the average long-

time collective diffusion coefficient, but are insensitive to very local variations in

response or constraining volumes18,19,20; thus, local measurements are often preferable.

For example, particle tracking experiments of individually fluorescently tagged lipids in

the plasma cell membrane directly measure the diffusion of single lipids and reveal the

15 Attwood, T.K. and D.B. Sellen (1990) Biopolymers 29:1325 16 Attwood, T.K., B.J. Nelmes and D.B. Sellen (1988) Biopolymers 27: 201 17 Pernodet, N., M. Maaloum and B. Tinland, (1997) Electrophoresis 18: 55

18 Hou, L., F. Lanni and K. Luby-Phelps (1990) Biophysical Journal 58: 31

19 Jones, J.D. and K. Luby-Phelps (1996) Biophysical Journal 71: 2742 20 Johnson, E.M., D.A. Berk, R.K. Jain and W.M. Deen (1996) Biophysical Journal 70: 1017

111

localized interactions that give rise to the average trends measured with FRAP21.

Recently, a similar technique has been used to measure the diffusion of proteins inside

living cells22.

Microrheology is another dynamic technique that can be used to probe very

localized mechanical properties using tracer particles embedded in a complex fluid23,24,25.

By using spherical tracer particles of a known size, particle motion can be interpreted

quantitatively in terms of the local viscoelastic properties of the surrounding medium.

While often interpreted in terms of the macroscopic bulk modulus, the real power of

microrheology lies in the fact that individual tracer beads probe microenvironments and

the motion of the individual tracer particles reflects the local mechanical response of the

surrounding material. Typically in microrheology measurements, the tracer particles are

much larger than the characteristic structures of the medium and particle motions are

interpreted in terms of the linear frequency-dependent viscoelastic moduli. However,

when the particle diameter is comparable to or smaller than the length scale of structures

in the medium, the tracers can move within small cavities and their motions are not only

a measure of the local viscoelastic response, but also of the effect of steric hindrances

21 Saxton, M.J. and K. Jacobson (1997) Annual Review of Biophysical and Biomolecular Structure 26: 373 22 Goulian, M. and S.M. Simon (2000) Biophyiscal Journal 79: 2188 23 MacKintosh, F.C. and C.F. Schmidt (1999) Current Opinion in Colloid Interface Science 4:300 24 Mason, T.G. and D.A. Weitz (1995) Physical Review Letters 74: 1250

25 Levine, A.J. and T.C. Lubensky (2000) Physical Review Letters 85: 1774

112

caused by the cavity walls. Thus, traditional microrheology techniques can be used to

measure both local rheology and local constraining volumes in order to characterize fully

the mechanical response of the microenvironments that small particles explore.

There are several different implementations of microrheology. One technique

involves the active manipulation of embedded probe particles using optical or magnetic

forces to move small probe particles and apply local stress to complex

materials26,27,28,29,30,31. Although active measurements can be extremely useful, especially

in stiff materials where large stresses are necessary to attain a measurable strain, applying

a well-controlled force to the probe particle requires sophisticated instrumentation and

calibration procedures. Additionally, large particles are often required to apply

sufficiently well-controlled forces to the material, preventing the application of this

technique to very small length scales.

Other microrheology techniques involve the passive observation of the thermal

fluctuations of the probe particles. In this case, there are no limits on particle size, but

the material must be sufficiently soft to allow detectable motion of particles moving with

26 Ziemann, F., J. Rädler and E. Sackmann (1994) Biophysical Journal 66: 2210 27 Amblard, F., A.C. Maggs, B. Yurke, A.N. Pargellis and S. Leibler (1996) Physical Review Letters 77: 4470 28 Schmidt, F.G., F. Ziemann and E. Sackmann (1996) European Biophysical Journal 24: 348 29 Valentine, M.T. , L.E. Dewalt and H.D. Ou-Yang (1996) Journal of Physics: Condensed Matter (U.K.) 8: 9477 30 Hough, L.A. and H.D. Ou-Yang (1999) Journal of Nanoparticle Research 1: 494 31 Bausch, A.R., W. Möller and E. Sackmann (1999) Biophysical Journal 76: 573

113

only kBT of energy. In passive measurements, the key parameter is the mean squared

displacement of the tracer particles, and this can be measured by various methods.

Dynamic light scattering techniques work well and report the mean squared displacement

averaged over an ensemble of tracer particles24,32,33. By contrast, single particle tracking

microrheology uses a single probe particle to measure highly localized mechanical

response34,35,36,37 with high spatial and temporal sensitivity. Because this method uses a

single probe instead of averaging over an ensemble, it is well suited to studying local

microenvironments. However, in heterogeneous systems many repeated experiments

with different beads in different parts of the sample are required to completely

characterize the material response. Another passive technique, two-particle

microrheology, measures macroscopic properties in inhomogeneous samples by

correlating the displacements of separated particles to measure the long wavelength

thermal fluctuations of the material, but does not characterize different

32 Gisler, T. and D.A. Weitz (1999) Physical Review Letters 82: 1606 33Palmer, A., T.G. Mason, J. Xu, S.C. Kuo and D. Wirtz (1999) Biophysical Journal 76: 1063 34 Gittes, F., B. Schnurr, P.D. Olmsted, F.C. MacKintosh and C.F. Schmidt (1997) Physical Review Letters 79: 3286 35 Mason, T.G., K. Ganesan, J.H. vanZanten, D. Wirtz and S.C. Kuo (1997) Physical Review Letters 79: 3282 36 Schnurr, B., F. Gittes, F.C. MacKintosh and C.F. Schmidt (1997) Macromolecules 30: 7781 37 Yamada, S., D. Wirtz and S.C. Kuo (2000) Biophysical Journal 78: 1736

114

microenvironments directly38. None of the techniques allow for simultaneous local

measurements in very heterogeneous samples, yet this is imperative for understanding

microscopic processes in materials that evolve in time.

In this paper, we report a new approach that uses video microscopy to

simultaneously track roughly one hundred fluorescently labeled thermally activated

particles in a single field of view. By analyzing the motions of the particles individually,

we are able to perform many experiments in parallel. The strength of this technique lies

in its experimental simplicity and our ability to collect large amounts of data in relatively

short times. This allows us to directly probe and characterize heterogeneous

microenvironments even in samples that are dynamically changing in time so that

consecutive experiments are not possible.

There is, however, an intrinsic limitation in this technique: because of our finite

imaging volume, the amount of data for any given particle is limited, reducing the

statistical accuracy with which the mean squared displacement can be determined. If we

could collect sufficiently long trajectories, then the differences in the mean squared

displacements of different particles would be a direct measure of the variation in

microenvironments found in the sample. However, our trajectories are not long enough

to make precise individual measurements; thus, we are forced to use a more statistical

approach. Rather than relying exclusively on the mean squared displacement, we also

develop new statistical tools that allow us to use higher order statistics and compare

distributions of displacements for individual particles. Using a formal statistical test to

38 Crocker, J.C., M.T. Valentine, E.R. Weeks, T. Gisler, P.D. Kaplan, A.G. Yodh and D.A. Weitz (2000) Physical Review Letters 85: 888

115

account for limited sampling, we determine which particles are measuring detectable

differences in the local microenvironments. We group together indistinguishable

particles and average over the mean squared displacements of particles within the group

to obtain more accurate measures of the local response for each grouping; we can thereby

determine statistically meaningful variations throughout the sample. In samples where

we detect spatial heterogeneity, we can measure the range of variation in local response

and can physically map out the variations across the sample. We are thus able to identify

and characterize different types of heterogeneity in samples with complex structures and

dynamics.

To illustrate the flexibility and range of applicability of this technique, we

examine three systems that display different types and ranges of heterogeneity. The first,

a glycerol/water solution is purely viscous and homogeneous and is included as a

benchmark to compare with more complex samples. The second sample is agarose, a

fibrous polysaccharide gel that is characterized by many small pores that determine its

utility as a common separation medium16,17. Agarose displays spatial heterogeneity and

shows the importance of differentiating between length scales when measuring

mechanical response. Finally, a solution of the semi-flexible biopolymer F-actin, an

important structural component of the cellular cytoplasm, is chosen as an example that

illustrates the consequences of time-dependent fluctuations and their effect on small

probe particles.

4.3 Experimental Details

116

Fluorescent tracer particles were imaged with video microscopy epi-fluorescence

(Leica DM-IRB/E) using a 100x, oil-immersion objective with a numerical aperture of

1.4 at a magnification of 129 nm/CCD pixel. Video half frames were acquired to obtain

tracer positions with 60 Hz temporal resolution. In each frame, the positions of the

fluorescent particles were identified by finding the brightness averaged centroid position

with a subpixel accuracy of approximately 10 nm39. The fluorescent beads appeared

bright even as the particles moved slightly in and out of focus, allowing a more precise

determination of the centroid position than is possible with non-fluorescently tagged

particles. Samples were prepared with roughly one hundred particles visible in each field

of view. Note that although better temporal and spatial resolution can be obtained with

laser deflection particle tracking (LDPT) measurements of single particles35, the

resolution given by our simple video detection scheme is sufficient to study the dynamics

of a wide range of soft materials. Furthermore, the ability to measure many particles

simultaneously allows the determination of the dynamic correlations between

neighboring particles and the local distribution of mechanical properties without

numerous repeated experiments. This is particularly important for the study of samples

that can evolve with time, where measurements must be done in a short period of time so

that the aging has not modified the sample or its properties.

Ten minutes of video were recorded on S-VHS tape and digitized and analyzed

offline, yielding a few million particle positions for each sample. We considered only

particles that remain in focus for at least five seconds, to ensure minimal statistical

39 Crocker, J.C. and D.G. Grier (1996) Journal of Colloid Interface Science 179: 298

117

accuracy for individual particles. This condition has the potential to bias our observed

distributions by under-weighting the fraction of fast-moving particles. However, for a

typical solution with viscosity of 5 cp, the distance that a 0.5µm particle diffuses in 5

seconds is smaller than the depth of field, which is roughly 2µm; thus we expect the

effects of the biasing to be very small. In more viscous media, or in materials that

constrain particles within local structures, the probability of a particle making a long

excursion from the focal plane is even further reduced. Here, we consider only one-

dimensional particle motion to maximize the accuracy of our tracking algorithms; since

each video half-frame consists of half as many rows as the full image, the accuracy is

degraded in the direction perpendicular to the row direction39. Two dimensional analysis

of the particle tracks can also be done in cases when the anisotropy of the individual

microenvironments is of importance.

Agarose (Seakem LE; FMC) was dissolved in 45 mM Tris buffer at pH=8 with

0.01% Triton X-100 surfactant at a concentration of 0.36% w/v and heated to boiling for

approximately 1 minute. The solution was cooled to 80°C, above the gelation

temperature, and 500 nm diameter rhodamine-labeled carboxylate-modified latex

polystyrene spheres (Molecular Probes) were added at a volume fraction of 0.2 %. The

solution was then injected into a heated chamber, consisting of a #1.5 cover-slip and

microscope slide. The chamber was hermetically sealed using UV-cured epoxy (Norland

#81) and quench-cooled to room temperature, triggering gelation of the agarose.

118

Actin prepared from rabbit muscle40 and stored in G-buffer (2mM Tris-Cl, 0.2

mM ATP, 0.5 mM dithiothreitol (DTT), 0.2 mM CaCl2) was mixed with rhodamine-

labeled 470 nm diameter tracers and polymerized in the sample chamber at a

concentration of 1 mg/mL with the addition of KCl to 75 mM, MgCl2 to 2.5 mM and

HEPES to 10 mM. Phalloidin was added in a 1.2:1 molar ratio to G-actin to stabilize the

actin filaments.

4.4 Results and Discussion

Although we are ultimately interested in local microenvironments, the ensemble-

averaged mechanical response gives valuable information about the dominant physics of

the material. Using multiparticle tracking, millions of positions are assigned to thousands

of particle trajectories, from which we calculate the ensemble-averaged mean squared

tracer displacement, 22 ( ) ( )x x t x tτ∆ = + − , as a function of lag time, τ , where the

angled brackets indicate an average over many starting times, t, as shown in Figure 4.1.

The beads in glycerol are purely diffusive, as expected for a simple fluid, and 2x∆ is

linear in τ , as shown in Figure 4.1(a). These data can be interpreted as a measure of

viscosity, η, using 2 ( ) 2x Dτ τ∆ = in one dimension where 6BD k T aπη= and a is the

tracer particle radius. The beads in agarose move significantly differently, as shown in

Figure 4.1(b). The particles show approximately diffusive behavior at short times;

40 Pardee, J.D. and J.A. Spudich (1982) Methods of Enzymology 85: 164

119

however, at the longest times, the mean squared displacement appears to be approaching

a plateau, indicating that the particles are becoming constrained. The F-actin sample

shows no true plateau, as shown in Figure 4.1(c) and is subdiffusive for all accessible

times, with power-law behavior having slopes well below one; this suggests that, while

the beads are not permanently caged, their motions are affected by the surrounding

network.

Although the ensemble response gives valuable insight into average mechanics, in

order to characterize microenvironments, it is necessary to examine the individual

particles directly. In Figure 4.2 we show the time evolution of the mean squared

displacement of several individual particles for each system. There are very large

variations from particle to particle. While it is tempting to explain different

displacements by variations in the local microenvironments, it is not correct to do so.

This is most clearly evidenced by the results for the glycerol solution; although the

microenvironments are strictly homogeneous, the time-averaged mean squared

displacements show large deviations, as shown in Figure 4.2(a). This reflects the fact

that these data are not an appropriate measure of heterogeneity since individual particles

are not sufficiently time-averaged to determine a statistically accurate mean squared

displacement; thus, at least part of the particle to particle variation is a result of poor

statistics41. In the agarose sample, as shown in Figure 4.2(b), the mean squared

displacements of the individual particles appear to vary much more than the particles in

the glycerol sample, suggesting there may be by a range of underlying

microenvironments. The actin sample, as shown in Figure 4.2(c), also displays variations

41 Qian, H., M.P. Sheetz and E.L. Elson (1991) Biophysical Journal 60: 910

120

in mean squared displacements that are larger than those of the glycerol but not quite as

large as those of the agarose. However, in order to make any quantitative comparisons,

we must be careful to isolate the variations due to different local mechanical

environments from the variations due to limited statistical sampling.

As evidenced most strikingly by the glycerol data, we do not have sufficient

statistics to accurately determine the mean squared displacements of the particles directly.

Instead we use higher order statistics to better distinguish the effects of poor statistics

from true variations in the sample. We examine the distribution of displacements at lag

time τ, ( ),P x τ∆ , known as the van Hove correlation function. The ensemble-averaged

van Hove correlation function for the glycerol/water solution is shown for two different

lag times, 0.033 and 0.1 seconds, in Figure 4.3(a). The lines are Gaussian fits to the data,

and there is good agreement as expected for a purely diffusive system where the particle

motion is described by a simple random walk, with

( ) ( ) ( )1/ 2 2, 4 exp / 4P x D x Dτ π τ τ−∆ = − ∆ . The variance of the distribution provides a

measure of the diffusion coefficient, and hence, the viscosity. A sampling of van Hove

correlation functions for individual particles at a lag time of 0.1 seconds is shown in

Figure 4.3(b). The solid lines indicate the Gaussian fit to the distribution. The particles

each follow Gaussian statistics and appear to be indistinguishable, indicating that each

tracer bead experiences the same local diffusion coefficient; thus the solution is

homogeneous. In fact, the Gaussian behavior of the ensemble-averaged van Hove

correlation function is a strong indicator of the homogeneity of the solution, since the

sum of the individual particle data will be Gaussian only if each particle displays

Gaussian statistics, and each has the same diffusion coefficient.

121

Figure 4.1. Time- and ensemble-averaged mean squared displacements as a function of

time. The solid line in each figure indicates a slope of 1. (a) Glycerol: 2x∆ is linear in

time, as expected for a simple fluid. (b). Agarose: Tracer particles diffuse at short times

and 2x∆ is linear in time. At longer times, the particles are caged, and we observe the

onset of a plateau: 2 ( )x τ∆ → ∞ =0.2 µm2 suggesting a plateau modulus of

2/ ( )BG k T a xπ τ= ∆ → ∞ ≈0.1 Pa. (c) F-actin: The motion is sub-diffusive at short

times, and even further constrained at longer times.

122

Figure 4.2. Individual particle mean squared displacements for several particles in (a)

Glycerol, (b) Agarose and (c) F-actin. The spread of 2x∆ reflects the uncertainties due

to statistical fluctuations which result from short data collection times.

123

Figure4.3. (a) The ensemble-averaged data is fit by a Gaussian distribution over several

orders of magnitude, as expected for a purely viscous fluid. (b) Individual particles,

shown at a time lag of 0.1 seconds, also display Gaussian statistics, and measure the same

diffusion coefficient, indicating the homogeneity of the solution. The small differences

between the fits are due to limited statistics in the evaluation of ( , )P x t∆ .

124

Although qualitative comparisons are possible by observing the shapes of the van

Hove correlation functions, for statistically accurate comparisons of the individual

particle data, we use a more formal test. One useful parameter is the F-statistic,

( ) ( )2 2/ / /k k l lf n nσ σ= , which compares the variances of any two independent, random,

normally distributed quantities, distinguished by labels k and l, with variance 2σ and n

degrees of freedom42,43. For particle trajectories, n is the number of statistically

independent time steps that build up the van Hove correlation function. For time series

( )x t of duration T mδ= , sampled to ( )ix t , it iδ= , assembling ( )2 ,P x jτ δ∆ = from

/m j non-overlapping segments will give /n m j= degrees of freedom. However, this

sampling procedure discards a large fraction of our data. Alternatively, we can assemble

( ),P x jτ δ∆ = from every pair ( ) ( )( ),i i jx t x t + to produce better statistics but such

sampling does not produce n m j= − degrees of freedom because sequential pairs are not

statistically independent. In order to determine the number of independent degrees of

freedom in such over-sampled data, we perform Monte Carlo simulations of an ensemble

of particles, where each experiences the same local diffusion coefficient. We calculate

the van Hove correlation functions for each particle, and calculate the F-statistic for each

pair. Empirically, we find that the F-statistic derived from over-sampled data is very

nearly F-distributed with 1.5 /n m j= , and we use this relationship to determine the

number of independent degrees of freedom in our data.

42 Martin, B.R. Statistics for Physicists. London: Academic Press (1971) 43 Roe, B.P. Probability and Statistics in Experimental Physics. New York: Springer-Verlag (1992)

125

To make statistically accurate comparisons of individual particles, we use the F-

test, which compares the variances of two distributions using the F-statistic and returns

the confidence level with which we can determine the two distributions to be statistically

different. In some soft materials, neighboring particles show correlated motion due to

long-range hydrodynamic or viscoelastic effects38; however, because every particle is

influenced by many neighboring particles, we expect any correlations to cancel out in the

distribution of displacements for a single particle. Thus we expect the individual van

Hove correlation functions to be statistically independent, and the variances of these

distributions to be F-distributed. Our ability to make meaningful distinctions using the F-

test depends upon the number of independent events that build up the individual

distributions. For the materials studied here, we found that a particle volume fraction of

roughly 0.001 and observation times of 5-20 minutes were sufficient to gather enough

data to make meaningful comparisons. At these volume fractions we do not expect the

presence of the particles to alter network formation; consistent with this, our results do

not depend on particle volume fraction. For softer or less viscous materials, we expect

increased particle motion and require a lower volume fraction of particles in order to

ensure that the inter-particle spacing is greater than the distance a particle moves between

subsequent frames, thus allowing a unique identification of each particle at each time

step. Lower magnification optics will capture a larger field of view, and may be useful

for measurements in such materials; however, the apparent size of individual particles

must be at least five pixels to ensure accurate tracking of the center of brightness.

Additionally, longer observation times may be required to gather sufficient statistics since

particles in softer materials are more likely to move out of the field of view.

126

Using a 95% certainty of difference, we compare particles pairwise to identify

those particles that are statistically distinguishable. Unlike other methods, which present

only distributions of statistical behaviors, using the F-test we are able to make actual

comparisons of individual particle tracks. We then group the particles into “clusters”

where all the particles in a given cluster are statistically indistinguishable. The beads in

the glycerol/water solution group into only one cluster, as expected for a homogeneous

solution where all particles measure the same local viscosity. Thus, despite the large

apparent variation in the mean squared displacements, the statistics are nevertheless

sufficiently good to show that there is in fact only a single microenvironment; however,

we must perform this more careful statistical analysis to determine this.

By contrast, in agarose, the beads display markedly different behavior, reflecting

the complex nature of the medium. Agarose is known to be structurally heterogeneous,

consisting of a complex network of fibrous molecules16. We measure the macroscopic

elastic plateau modulus to be roughly 2700 Pa, using a strain-controlled rheometer (Ares;

Rheometrics) for frequencies from 0.01 to 100 rad/s. In a homogeneous material, this

shear modulus would be too large to produce a detectable amount of motion with

thermally activated tracer particles. The calculated mean squared displacements for a 1

µm-diameter particle in the agarose would be 1 nm2, which we cannot resolve with our

particle tracking method; consistent with this, we observe that beads that are 1 µm in

diameter do not move detectably, indicating that the particles are larger than the

characteristic elastic structures and probe the bulk elastic response. However, while

agarose does contain large elastic structures that span the sample and bear stress

macroscopically, at short length scales, these structures do not form a homogeneous

127

elastic continuum; instead, agarose is characterized by many smaller voids, or pores,

through which smaller particles may move17. By observing the dynamics of the smaller

particles, we can characterize the mechanical microenvironment of the pores themselves.

Although constrained, the smaller particles move within the irregularly shaped

pores, and do move in and out of focus, limiting our ability to track an individual particle

for long times. Because we are limited by trajectories of finite length, we again use

higher order statistics to make quantitative comparisons. The ensemble-averaged van

Hove correlation function is shown in Figure 4.4(a) at a time lag of 0.033 seconds and 0.1

seconds. The lines indicate the Gaussian fit to each distribution; however, unlike the

homogeneous glycerol solution, the agarose data deviates from the Gaussian fit at all

displacements at both time lags. These deviations from Gaussian behavior indicate a

detectable heterogeneity in mechanical response. A sampling of the individual van Hove

correlation functions at a lag time of 0.1 seconds are shown in Figure 4.4(b), and the

Gaussian fit for each distribution is indicated by a solid line. Within our limited

statistical accuracy, the distribution of particle displacements for each bead appears to be

Gaussian; however, the widths of the distributions are different, reflecting different local

diffusion coefficients.

At short lag times, where the statistical accuracy is best, the F-test identifies

several statistically different groups of particles, indicating microscopic differences in the

local properties. However, at longer lag times, where the particles are more constrained,

we are unable to distinguish any differences in the distributions with the F-test.

128

Figure 4.4 Van Hove correlation functions for particles moving in agarose. (a) The

ensemble-averaged distribution is non-Gaussian for even small displacements at all time

lags, and is approximately described by an exponential distribution. (b) Individual

particle distributions, shown at a time lag of 0.1 seconds, are each fit by a Gaussian of a

different width, indicating a different local diffusion coefficient.

129

In this long time limit, the number of independent time steps is considerably lower,

decreasing the statistical accuracy of the comparison. However, there are still wide

variations in the mean squared displacements; thus, in order to better characterize the

long time behavior, we average the data in a statistically meaningful way. We use the F-

test results to group the particles into clusters according to their local viscosity based on

the distributions at short times and average the mean squared displacements over the

particles in each cluster. Because all particles in a given cluster are statistically

indistinguishable and measure the same microscopic response, they form a meaningful

ensemble and it is appropriate to consider the behavior averaged over particles in the

cluster. The cluster-averaged mean squared displacements for the six most populated

clusters are shown in Figure 4.5. At short times, the motion is very nearly diffusive and

we measure a range of local viscosities from 2.3 cp to 8.6 cp. At long times the cluster-

averaged mean-squared displacements do not converge, suggesting that particles in

different clusters do experience different microenvironments. The plateau in the mean

squared displacement at long times indicates that the particles are constrained by the

material but we require further tests to reveal the nature of this constraint. In a

homogeneous medium, the plateau is interpreted in terms of the elastic modulus of the

material; however, in a heterogeneous microenvironment, the plateau may be a measure

of local elasticity, or a measure of the size of the local constraining volume, or pore. To

differentiate between these two possibilities, we use probe particles of different sizes and

determine how 2 ( )x< ∆ ∞ > scales with particle radius, a. If the plateau is a measure of

local elasticity, G, then 2 ( ) /x kT Gaπ< ∆ ∞ >= , so 2 ( )x< ∆ ∞ > scales with 1/a.

130

Figure 4.5. Cluster-averaged mean squared displacement for 500 nm particles in agarose.

The local viscosity ranges from 2.3 cp to 8.6 cp and the pore sizes from 700 nm to 1200

nm. There is a correlation between short time viscosity and the long time plateau in the

mean squared displacement due to increased hydrodynamic resistance in the smallest

pores.

131

If, however, the plateau is a measure of pore size, d, then d a h= + , where h is the size of

the fluid filled gap between the particle surface and the pore wall and is approximated by

2 1/ 2( ( ) )h x= < ∆ ∞ > . We have repeated our measurements with 200 nm particles, and

compared the results with those obtained with 500 nm particles. We find that

2 ( )x< ∆ ∞ > does not scale with particle size; instead 2 1/ 2( ( ) )a x+ < ∆ ∞ > remains roughly

constant. This implies that in agarose the plateau in the mean squared displacement does

not provide a measurement of local elasticity, instead it probes the average pore size,

which is found to be approximately 1µm.

Once we have determined the nature of the constraint that gives rise to the long

time plateau, we can begin to characterize and compare the microenvironments in

agarose. The cluster-averaged mean squared displacements suggest that the particles that

are the most tightly constrained at longer times experience the highest local viscosity at

earlier times. The plateau in mean squared displacement is a measure of an effective pore

size, and our data suggest that particles in smaller pores are diffusing more slowly; this

may be due to hydrodynamic effects. Particles moving in fluid-filled pores will

experience an apparent increase in viscosity since in order to move, the particle must

squeeze fluid out of a thin gap between the particle surface and the pore wall44. A

particle of size a moving in a fluid-filled pore of size a h+ , where h a<< , will

experience a drag force 4 33 / 2F a hπη= and an effective diffusion coefficient of

3 42 / 3BD k Th aπη= . If we assume spherical, impermeable pores and again estimate the

44 Acheson, D.J. Elementary Fluid Dynamics. Oxford: Oxford University Press (1990)

132

pore size by assuming ( )1

2 2( )h x≈ ∆ ∞ , we find that for the smallest pores, where the

approximation h a<< is most valid, the measured increase in viscosity is consistent with

increased hydrodynamic resistance. In reality, the pore walls are likely irregular, fibrous

and semi-permeable, giving rise to complex flow fields and long-range hydrodynamic

effects. Nonetheless, our data suggest that the walls do decrease the mobility of the

particles they constrain and further this simple scaling relation describes the general trend

we observe.

To verify the effects of pore size on the diffusivity of small particles, we examine

the behavior of the 200 nm particles, which should be less sensitive to hydrodynamic

effects. In this case, we again distinguish several clusters of particles that experience

different microenvironments; however, we measure a smaller range of local viscosities,

from 2.4 cP to 3.7 cP, in agreement with the viscosities measured by the least constrained

500 nm particles. This suggests that the pores are filled with a soft polymer that raises

the viscosity to 2-3 times that of water; however, hydrodynamics do play a role in further

decreasing the diffusivity of the 500 nm particles in the smallest pores.

To better characterize the range of microenvironments present in the sample, we

do Monte Carlo simulations of an ensemble of beads performing random walks with a

known distribution of local diffusion coefficients. We use the same particle trajectory

lengths that we measure experimentally to ensure the same statistical limitations and vary

the width of the distribution of local diffusion coefficients until we recover the same

clustering behavior found experimentally. Our data is consistent with a distribution of

diffusion coefficients that varies by 20% +/- 5%. Both the cluster analysis and the

Monte Carlo simulations show that despite the large contrast between macroscopic and

133

microscopic environments, the range of mechanical microenvironments within the pore

structure is quite small. This result could not have been predicted from bulk

measurements alone and emphasizes the importance of direct local measurements.

We can further investigate the nature of the microenvironments by plotting the

trajectories of all particles and color-coding the individual tracks according to cluster

membership as shown in Figure 4.6. This allows us to physically map the variation in

local mechanical response across the sample and to look for spatial correlations of similar

microenvironments. We observe no long-range spatial correlations indicating that, in

agarose, the length scale of heterogeneity is small, and nearby pores have different sizes

and are characterized by different local viscosities.

To further test the flexibility of this technique, and to explore the effect of time

dependent fluctuations on small probe particles, we measure individual particles moving

in a solution of F-actin. The embedded tracer particles are 470 nm in diameter, slightly

larger than the average mesh size of approximately 300 nm expected for a 1mg/mL

solution of F-actin45. Unlike the beads in agarose, however, the tracer particles in actin

are constrained by the semi-flexible polymer network but are not permanently caged or

confined in pores. The ensemble-averaged van Hove correlation function shows roughly

Gaussian behavior at shortest lag time of 0.033 seconds, but deviates at slightly longer

time lags, above 0.1 seconds, to give a broader tail as shown in Figure 4.7(a).

45 Schmidt, C.F., M. Bärmann, G. Isenberg, and E. Sackmann (1989) Macromolecules 22: 3638

134

Figure 4.6. The particle trajectories for the agarose gel give a spatial map of mechanical

microenvironments. The trajectories are color-coded according to cluster analysis; beads

of the same color experience the same local viscosity. There is no long-range spatial

correlation, suggesting that the length scale of heterogeneity is small – of order a particle

diameter.

135

Figure 4.7. Van Hove correlation functions for particles moving in actin. (a) The

ensemble-averaged distributions do not compare well to a Gaussian fit. There are far too

many large displacement events for a Gaussian displacement model, a deviation that

grows with time. (b) Individual particle distributions, shown at a time lag of 0.1 seconds,

are similar, although there are differences for larger displacements.

136

Such non-Gaussian behavior indicates that the system is not a simple homogeneous

solution at the length scale of the probe particles. The van Hove correlation functions of

individual particles at a time lag of 0.1 seconds are shown in Figure 4.7(b). The

distributions look similar for small displacements, although there is deviation at larger

displacements and some indication of non-Gaussian behavior; however, this must be

viewed with caution given the poor statistics.

Our attempts to use the F-test to classify particles by statistically different

diffusivities failed to identify any distinguishable clusters. Our inability to distinguish

between individual particles despite the non-Gaussian behavior of the ensemble-averaged

distribution suggests the possibility of a temporal heterogeneity. In this case, the time-

averaged response of even an individual particle may be a superposition of data from

several microenvironments. Such a superposition would result in a more complicated

form for the van Hove correlation function of an individual particle; however, our limited

statistical accuracy does not allow us to distinguish any difference. Preliminary work in

our laboratory suggests that a single particle that is roughly equal in size to the average

mesh size can in fact sample several different cages formed by the F-actin network, and

can even “jump” repeatedly between two neighboring cages46. This suggests that the

temporal heterogeneity may be due to the actin network re-arranging in time, or the beads

exploring different static microenvironments over the course of the measurement, or

both, resulting in a time-averaged response that is not necessarily a simple representation

of any of the instantaneous microenvironments. The characteristic cross-over frequency

from viscous to elastic behavior for similar F-actin solutions has been reported in the

46 Gardel, M.L., I. Wong, A. Bausch, and D. A. Weitz, (unpublished)

137

range of 0.1 to 1 Hz 47, corresponding to approximately the same time lags we are

investigating and indicating the ability of the network to rearrange on this time scale.

The ability of probe particles to move between different microenvironments in a

fluctuating network is consistent with previous measurements in actin, where small

particles have reportedly failed to measure bulk mechanical response38,47,48,49. One

explanation for this failure is that probe particles that are much smaller than the

persistence length of the actin filaments, typically 10-20 µm, introduce non-affine

deformations of the network as filaments are bent around the probes with a characteristic

radius comparable to the bead diameter47,48. This non-affine excitation results in a

different elastic response than is measured in macroscopic rheology. This effect is very

sensitive to filament length, and the wide distribution of filament lengths that our samples

likely possess could cause mechanical heterogeneity if the small particles are able to

move through different microenvironments characterized by different average filament

lengths during the measurement. Furthermore, although reptation of filaments typically

occurs on time scales of 1000 seconds, much longer than the time scales of our

measurements, the relaxation time might be considerably shorter if short filaments

dominate the local response. The resultant distribution of relaxation times could also

give rise to temporal heterogeneity.

A recent comparison of one-particle and two-particle microrheology in actin

samples also showed that the mean squared displacement calculated from the “self-

47 Schmidt, F.G., B. Hinner, and E. Sackmann (2000) Physical Review E 61: 5646 48 Maggs, A.C. (1998) Physical Review E 57: 2091 49 Morse, D.C. (1998)Macromolecules 31: 7044

138

diffusion” of a single probe particle is probably not solely a measure of the fluctuation

spectrum of the actin network38. Rather, the mean squared displacement likely reflects a

superposition of the long wavelength thermal undulations of the polymer network and the

local motion of the particle inside a transient “cage” formed by the surrounding actin

filaments. This local motion could give rise to temporal heterogeneity if the average

spacing between filaments fluctuates in time, or if a particle samples several different

cages during the course of the measurement. These results may be important to help

rationalize previous measurements in actin where particles had been assumed to measure

a homogeneous network response.

4.5 Summary

In this paper, we have described a new multiparticle tracking technique that

investigates mechanical microenvironments of complex systems at the micron– scale.

Such local measurements are essential to fully characterizing inhomogeneous systems,

and are imperative to understanding the microscopic processes that occur in structured

materials. By simultaneously tracking multiple particles moving in complex solutions,

we can perform measurements even in time-evolving samples, such as gels undergoing

polymerization and the cytoplasm of living cells. Although the finite imaging volume

limits the statistical accuracy with which we can calculate the mean squared displacement

of individual particles, by adopting a more statistical approach, we can compare

distributions of particle displacements to identify quantitative differences in local

microenvironments.

139

In agarose, large fibrous polymers form heterogeneous elastic structures that bear

macroscopic stress but allow the motion of particles in confined pores. By examining

statistics of individual particles and clustering those particles that are statistically

indistinguishable, we measure variations in the properties of the pores and physically

map those variations across the sample. Despite the significant differences between

macroscopic rheological properties and microscopic environment explored by small

particles, the variation within local microenvironments is quite small, of order 20%. We

do not observe spatial correlation of similar microenvironments, indicating that

neighboring pores have different sizes and local viscosities. Furthermore, we detect a

decrease in the mobility of particles in the smallest pores due to increased hydrodynamic

drag. These results highlight the importance of measuring mechanical response at

different length scales in highly structured heterogeneous materials. For instance, small

macromolecules can easily move through the pores of the gel but large complexes of

several macromolecules may be dramatically slowed or even trapped.

In actin, we find signatures of heterogeneity in the ensemble-averaged

distributions of particle displacements, but are unable to statistically distinguish any

individual particles. We interpret this result as evidence of temporal heterogeneity, and

suggest that individual particles may explore several different microenvironments during

the course of the measurement. The ability of particles to move through different

mechanical environments over short times may impact the previous interpretation of actin

microrheology experiments that have been interpreted in terms of bulk rheological

response. Furthermore, the ability of the multiparticle tracking technique to identify

140

temporal heterogeneity may allow the measurement of characteristic time scales in

structurally complex systems that dynamically change in time.

141

Chapter 5: Two-point microrheology of inhomogeneous soft materials∗

Many interesting and important materials such as polymers, gels and biomaterials

are viscoelastic; when responding to an external stress, they both store and dissipate

energy. This behavior is quantified by the complex shear modulus, G*(ω), which provides

insight about the material's microscopic dynamics. Typically, G*(ω) is measured by

applying oscillatory strain to a sample and measuring the resulting stress. Recently a new

method, called microrheology, has been developed which determines G*(ω) from the

thermal motion of microscopic tracer particles embedded in the material1,2.

Microrheology offers significant potential advantages: it provides a local probe of G*(ω)

in miniscule sample volumes and can do so at very high frequencies.

While microrheology provides an accurate measure of G*(ω) for simple systems,

its validity in common complex systems is far from certain. If the tracers locally modify

the structure of the medium, or sample only pores in an inhomogeneous matrix, then bulk

rheological properties will not be determined. Such subtle effects currently limit many

interesting applications of microrheology.

∗ Originally published in Physical Review Letters (2000) vol 85, p.888, by John C. Crocker, M.T. Valentine, Eric R. Weeks, T. Gisler, P.D. Kaplan, A.G. Yodh, and D.A. Weitz. Copyright (2000) by theAmerican Physical Society. 1 Mason, T.G. and D.A.Weitz (1995) Physical Review Letters 74:1250; Mason, T.G., K.Ganesan, J.H. vanZanten, D.Wirtz and S.C.Kuo (1997) Physical Review Letters 79: 3282 2 Gittes, F., B.Schnurr, P.D.Olmstead, F.C.MacKintosh and C.F.Schmidt (1997) Physical Review Letters 79: 3286; Schnurr, B., F.Gittes, F.C.MacKintosh and C.F.Schmidt (1997) Macromolecules 30: 7781

142

In this Letter, we introduce a new formalism, which we term `two-point

microrheology', based on measuring the cross-correlated thermal motion of pairs of tracer

particles to determine G*(ω). This new technique overcomes the limitations of single

particle microrheology. It does not depend on the size or shape of the tracer particle;

moreover it is independent of the coupling between the tracer and the medium. We

demonstrate the validity of this approach with measurements on a highly inhomogeneous

material, a solution of the polysaccharide guar. Two-point microrheology correctly

reproduces results obtained with a mechanical rheometer, whereas single particle

microrheology gives erroneous results. We also compare ordinary and two-point

microrheology of F-actin2,3,4, a semiflexible biopolymer constituent of the cytoskeleton.

Different results are obtained with the two techniques, suggesting that earlier

interpretations of F-actin microrheology should be reexamined.

Conventional microrheology1,2 uses the equation:

2 ( )( )

Bk Tr ssaG sπ

=%% (5.1)

where 2 ( )r s% is the Laplace transform of the tracers' mean squared displacement, <∆r2(τ)>

as a function of Laplace frequency s, and a is their radius. Equation 5.1 is the Stokes-

Einstein equation generalized to a frequency-dependent viscosity 1[ ( )]s G s− % that accounts

for elasticity1. Equation 5.1 is subject to the same conditions as the Stokes calculation:

overdamped spherical tracer particles in a homogeneous, incompressible continuum with

3 Müller, O., H.E.Gaub, M.Bärmann and E.Sackmann (1991) Macromolecules 24: 3111; Janmey, P.A., S. Hvidt, J. Kas, D. Lerche, A. Maggs, E. Sackmann, M. Schliwa, and T.P. Stossel (1994) Journal of Biological Chemistry 269: 32503 4 Gisler, T. and D.A.Weitz (1999) Physical Review Letters 82: 1606

143

no-slip boundaries. If the tracers inhabit cavities in a porous medium, or create their own

cavities by steric or chemical interactions with the material itself, their mobilities may be

much greater than predicted by Equation 5.15.

Since the effect of such inhomogeneities is difficult to quantify, we seek a means

to discriminate between a tracer moving in a soft pore in an otherwise very rigid matrix

from another tracer moving with the same amplitude in an homogeneous soft matrix. One

difference between these cases arises from the long-range deformation or flow in the

matrix due to the tracer's motion. In the homogeneous case, this strain field is

proportional to the tracer motion and decays ~a/r, where r is the distance from the tracer.

For the hypothetical tracer in a soft cavity, the strain field is localized to the cavity itself.

Since one tracer's strain field will entrain a second particle, we can measure the

strain field by cross-correlating two tracers' motion. Recent experiments have probed

such correlated motion in viscous6 and elastic7 materials. We use multi-particle video

tracking8, to measure the vector displacements of the tracers

( , ) ( ) ( )r t r t r tα α ατ τ∆ = + − where t is the absolute time and t is the lag time. We then

calculate the ensemble averaged tensor product of the tracer displacements:

, ,

( , ) ( , ) ( , ) ( )i j ij

i j tD r r t r t r R tαβ α βτ τ τ δ

≠ = ∆ ∆ − (5.2)

5 Morse, D.C. (1998) Macromolecules 31: 7044 6 Crocker, J.C. (1997) Journal of Chemical Physics 106: 2837; Meiners, J.C. and S.R.Quake (1999) Physical Review Letters 82: 2211 7 Schmidt, F.G., F.Ziemann and E.Sackmann (1996) European Biophysics Journal 24: 348; Bausch, A.R., W.Moller and E.Sackmann (1999) Biophysical Journal 76: 573 8 Crocker, J.C. and D.G.Grier (1996) Journal of Colloid and Interface Science 179: 298

144

where i and j label different particles, α and β label different coordinates, and Rij is the

distance between particles i and j. The average is taken over the `distinct' terms (i ≠ j);

the `self' terms yield <∆r2(τ)> × δ(r).

For an incompressible continuum, the expected 2-point correlation is computed

by multiplying the displacement predicted in Equation 5.1 by the strain field of a point

stress9. The result, in the limit r >> a, is:

( , ) ,2 ( )

Brr

k TD r srsG sπ

=%% 1

2 rrD D Dθθ φφ= = (5.3)

where ( , )rrD r s% is the Laplace transform of ( , )rrD r τ and the off-diagonal tensor

elements vanish. Significantly, Equation 5.3 has no dependence on a, suggesting that

Dαβ (r,τ) is independent of the tracer's size, shape and boundary conditions with the

medium, in the limit r >> a.

While we will treat our samples as incompressible, Equation 5.3 can be

generalized to compressible media by using a different Stokes-Einstein relation and strain

field9. The compressibility changes the anisotropy of the strain field, and modifies the

relative amplitude of the tensor elements. Thus, measuring the different tensor elements

should enable measurements of compressibility7.

We use Equation 5.3 as the basis for the microrheology of inhomogeneous media

which cannot be modeled by Equation 5.1. We demonstrate the effectiveness of this

approach empirically, and present a simple argument in its favor. The thermal motion of

a soft

9 Landau, L.D. and E.M.Lifshitz. Theory of Elasticity: Third Edition. Oxford: Pergamon Press (1986)

145

viscoelastic medium can be described as a stochastic, time-fluctuating strain field

characterized by a spectral density that depends on frequency and spatial wavelength10.

The Brownian motion of a single tracer is the superposition of such modes with

wavelengths greater than the particle radius, a. The correlated motion of two tracers is

driven by those modes with wavelengths greater than their separation r rather than a,

since shorter wavelength modes do not move the tracers in phase. Thus, the correlated

motion of two tracers separated by more than the coarse-graining lengthscale in an

inhomogeneous medium will depend only on the coarse-grained, macroscopic, modulus.

Comparing Equation 5.3 and Equation 5.1 suggests that we define a `distinct' mean

squared displacement, <∆r2(τ)>D as:

2 2( ) ( , ).rrD

rr D ra

τ τ∆ = (5.4)

This quantity is just the thermal motion obtained by extrapolating the long-wavelength

thermal undulations of the medium down to the bead size. If and only if the assumptions

implicit in Equation 5.1 are valid will <∆r2(τ)>=<∆r2(τ)>D. Any difference in the

displacements provides insight into the local micro-environment experienced by the

tracers. We can then understand <∆r2(τ)> as a superposition of a long-wavelength motion

described by <∆r2(τ)>D plus a local motion in a boundary layer or cavity.

In practice, we first confirm that Drr ~ 1/r, which indicates that the medium can be

treated as a (coarse-grained) homogeneous continuum. This was the case for all our

samples over the length scales we studied, 3 < r < 30 µm. We then use the average value

of rDrr over that range to calculate <∆r2(τ)>D from Equation 5.4. Finally, we calculate

10 Chaikin, P.M. and T. Lubensky, Principles of Condensed Matter Physics. Cambridge: Cambridge University Press (1994)

146

the two-point microrheology result by substituting <∆r2(τ)>D into Equation (5.1) in place

of <∆r2(τ)>.

Several numerical procedures for calculating the shear modulus from

<∆r2(τ)> have appeared in the literature1,2,4. We first approximate <∆r2(τ)> locally by a

second-order polynomial (spline) in the logarithmic plane to obtain a smoothed estimate

of the function and its first logarithmic derivative. We then use approximate, local

algebraic expressions to calculate the storage, G′(ω), and loss, G″(ω), moduli (defined by

G* = G′+ i G″), from the smoothed value and derivative1. This method approaches the

exact result as <∆r2(τ)>approaches a power law, and has the advantage that it does not

require the experimental data to be fit to an analytic model, nor does it suffer from the

truncation errors of numerical integral transforms. On the basis of numerical tests, we

expect all the moduli presented here to have systematic errors smaller than 5% of |G*|,

although larger errors are possible near the frequency extrema.

We used sub-micron fluorescent beads as tracers (Molecular Probes, Rhodamine

Red-X labeled carboxylate-modified latex/polystyrene). We sealed the samples between a

#1.5 glass cover-slip and microscope slide with UV-curing epoxy (Norland #81). The

tracers were imaged with epi-fluorescence (Leica, inverted microscope, DM-IRB/E)

using a 100 ×, NA=1.4 oil-immersion objective at a magnification of 129nm/CCD pixel

and a video shutter time of 2msec. To minimize wall effects, we focused at least 25 µm

into the 150 µm thick sample chambers. A few hundred particles were located within the

field of view and 2 µm depth of focus. For each sample, 10 minutes of video were

recorded, which yields a few million tracer positions with 60 Hz temporal resolution and

roughly 20 nm spatial resolution8.

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As a control, we measured the two-point correlation, Dαβ, of 0.47 µm diameter tracers in

a glycerol/water mixture. The results confirm our expectations for a simple fluid, as

shown in Figure 5.1(a). The functional form is Drr, Dφφ ∝ τ/r to within statistical error,

with Dφφ /Drr = 0.43 ± 0.06, in agreement with Equation (5.3) for an incompressible

medium. As shown in Figure 5.1(b), we find that <∆r2(τ)>D is equal to <∆r2(τ)> and is

linear to within statistical errors, at least when a small constant is subtracted from the

latter. This small constant added onto <∆r2(τ)> is simply the squared measurement error

of the tracer positions. Since the errors for two tracers are uncorrelated, <∆r2(τ)>D is

unaffected.

To demonstrate the effectiveness of two-point microrheology in inhomogeneous

media, we compared measurements of a guar solution with those from a mechanical

rheometer. Guar is a naturally occurring neutral polysaccharide (MW ≈ 106) extracted

from the guar gum bean. A small concentration of guar in water dramatically changes the

viscoelastic properties, because of the formation of high-molecular weight, mesoscopic

aggregates11 resulting from random associations of the guar molecules. This presents a

highly inhomogeneous medium, with a very large range of characteristic length scales,

ideally suited to testing our technique.

Our results for 0.20 µm diameter tracers in a 0.25%-by-weight guar solution are

shown in Figure 5.2.

11 Gittings, M.R., L.Cipelletti, V.Trappe, D.A.Weitz, M.In and C.Marques (preprint); P.Molyneux, in Water Soluble Polymers: Synthesis, Solution Properties and Applications, S.W.Shalaby, C.L.McCormick and G.B.Butler, eds (ACS Symposium Series 467, 1991). pp. 232-248.

148

Figure 5.1. (a) Two-point correlation function, Drr, for 0.47 µm diameter beads

dispersed in a glycerol/water solution, as a function of r and τ. In a triple-log plot, the

surface is a plane, and is ∝ τ / r. (b) <∆r2(τ)>D calculated from Drr (circles) overlaid on

<∆r2(τ)> (line). The agreement between the two indicates that the fluid satisfies

Equation 5.1.

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Figure 5.2. (a) Comparison of the self (triangles) and distinct (circles) displacements of

0.20 µm diameter beads in 0.25% weight guar solution. The solid line is a smooth fit to

the data, used for calculating the rheology. The inset shows the r dependence of rDrr for

τ = 100 msec, in unitsof 10-3 µm3. (b) The storage (filled circles) and loss (open circles)

moduli calculated using <∆r2(τ)>D , showing a crossover to elastic behavior at high

frequencies, are in good agreement with rheometer measurements (solid curves). The

moduli calculated using <∆r2(τ)> (triangles) do not agree.

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Unlike the simple fluid case, the two mean squared displacements do not correspond, but

disagree by roughly a factor of two and have a somewhat different functional form, as

shown in Figure 5.2(a). We converted both to their corresponding G′(ω) and G″(ω), and

compared the results to directly measured moduli from a controlled-strain rheometer

(Rheometric Scientific, Ares F). As shown in Figure 5.2(b) the moduli calculated from

the two-point correlation function are in good agreement with the results obtained with

the rheometer, correctly capturing the crossover of G′(ω) and G″(ω) at ~ 40 rad/s. Single

particle microrheology provides qualitatively different moduli and completely fails to

detect the crossover. This confirms the underlying concepts of the two-point method, and

verifies its accuracy in determining the bulk rheological behavior of an inhomogeneous

medium.

As an application of the two-point method, we measured polymerized F-actin,

obtained from purified rabbit muscle4,12. The sample was prepared at 1 mg/ml in G-

buffer (2mM tris-Cl, 0.2 mM ATP, 0.5 mM DTT and 0.1 mM CaCl2), mixed with 0.47

µm diameter tracers and polymerized in the sample chamber by the addition of MgCl2 to

75 mM. Phalloidin was added in a 1.2:1 molar ratio to G-actin to stabilize the actin

filaments.

The measured <∆r2(τ)> and <∆r2(τ)>D are shown in Figure 5.3(a); <∆r2(τ)>

resembles that reported in other microrheology experiments2,4, increasing

subdiffusively with a weak turnover at ≈ 50 msec. <∆r2(τ)>D shows no turnover, scaling

as τ1/2. Significantly, <∆r2(τ)> is up to five times larger than <∆r2(τ)>D , suggesting that

12 Pardee, J.D. and J.A.Spudich (1982) Methods in Enzymology 85: 164

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most of the motion of the beads is inside a `cage' formed by the actin filaments5. Since it

is not due to the fluctuation spectrum of the actin solution, the turnover in <∆r2(τ)> may

result from collisions of particles with the cage. This highlights the important distinction

between effects due to continuum mechanics and those due to the tracers'

micro-environments. Some of the discrepancy may also be due to compressibility of the

network at the small spatial wavelengths probed in the single-particle data.

The corresponding moduli for F-actin are shown in Figure 5.3(b). While <∆r2(τ)> gives

moduli similar to other microrheology results2,4, <∆r2(τ)>D yields a very simple

rheological spectrum varying as ω1/2, which, interestingly, corresponds with the exponent

seen at lower frequencies in macroscopic measurements3. The significant difference

between the two-point and single-particle measurements suggests that previous

microrheology results must be reexamined. More work, in both theory and experiment is

required to connect the internal dynamics of the actin filaments, shown recently to scale

as τ3/4, to the G* determined by microrheology5,13,14. Two-point microrheology has

several advantages, in addition to its ability to probe inhomogeneous media. Its

robustness should enable accurate microrheology with polydisperse, non-spherical or

unknown size tracers, as in studies of the cytoskeleton using organelles. Its ability to

simultaneously probe both bulk rheology and the tracers' micro-environments will likely

be key to understanding such complex media as biopolymers and cells.

13 Gittes, F. and F.C.MacKintosh (1998) Physical Review E 58: R1241 D.C.Morse (1998) Physical Review E 58: R1237 14 Caspi, A., M.Elbaum, R.Granek, A.Lachish and D.Zbaida (1998) Physical Review Letters 80: 1106

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Figure 5.3. Comparison of the self (triangles) and distinct (circles) displacements of 0.47

µm diameter beads in a 1 mg/ml F-actin solution. (inset) Shows the r dependence of rDrr

for τ = 67 msec, in units of 10-3 µm3. (b) The storage (filled circles) and loss (open

circles) moduli calculated using <∆r2(τ)>D vary as ω1/2. The moduli calculated with

<∆r2(τ)> (triangles) show a different form approaching ω3/4 The short lines indicate

slopes of 1/2 and 3/4.

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Chapter 6: Multiple particle tracking measurements of biomaterials: Effects of colloid

surface chemistry∗

6.1 Overview:

We present a simple protocol to render colloids protein-resistant by grafting short

methoxy-terminated poly(ethylene glycol) (PEG) to the surface of carboxylate-modified

colloids using commercially available reagents, and demonstrate the usefulness of PEG-

coated particles for microrheology and multiple particle tracking measurements of

inhomogeneous materials. We compare the protein binding capacity and mobility of the

PEG-coated particles to particles coated with physisorbed BSA and bare carboxylate-

modified latex spheres. We demonstrate that these surfaces adsorb differing amounts of

protein, and using video-based particle tracking techniques, show that the different

particles measure different physical properties of the biomaterials they probe. To

illustrate the range of behavior, we investigate the effect of particle surface chemistry on

measurements of networks of fibrin, entangled F-actin, and composite F-actin solutions

that are crosslinked and bundled with the actin-binding protein scruin.

∗ This work was performed in collaboration with Z.E. Perlman and TJ. Mitchison (Department of Cell Biology and Institute of Chemistry and Cell Biology, Harvard Medical School), J.H. Shin (Department of Mechanical Engineering, Massachusetts Institute of Technology), and P.T. Matsudaira (Department of Biology, Massachusetts Institute of Technology and the Whitehead Institute for Biomedical Research), M.L. Gardel (Department of Physics, Harvard).

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6.2 Introduction

Microrheology techniques allow sensitive measurements of the mechanical

properties of soft materials, and have been used to measure the microscopic dynamics of

a number of interesting biomaterials that are not easily probed with traditional bulk

techniques1. These include networks of filamentous actin2,3,4,5,6, solutions of the semi-

flexible filamentous bacteriophage fd7, solutions of the intermediate filament desmin8,

and living fibroblast9, macrophage10, endothelial11, COS-712, and dictyostelium cells13.

1 Gardel, M.L., M.T. Valentine, and D.A. Weitz. “Microrheology” appearing in Microdiagnostics, K. Breuer, ed. New York: Springer (In preparation) 2 Ziemann, F., J. Rädler, and E. Sackmann (1994) Biophysical Journal 66: 2210 3 Amblard, F., A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler (1996) Physical Review Letters 77: 4470 4 Schmidt, F.G., F. Ziemann, and E. Sackmann (1996) European Biophysics Journal 24: 348 5 Keller, M., J. Schilling, and E. Sackmann (2001) Review of Scientific Instruments 72: 3626 6 Gardel, M.L., M.T. Valentine, J.C. Crocker, A.R. Bausch and D.A. Weitz. (Submitted) 7 Schmidt, F.G., B. Hinner, E. Sackmann, and J.X. Tang (2000) Physical Review E 62: 5509 8 Hohenadl, M. T. Storz, H. Kirpal, K. Kroy, and R. Merkel (1999) Biophysical Journal 77: 2199 9 Bausch, A.R., F. Ziemann, A.A. Boulbitch, K. Jacobson, and E. Sackmann (1998) Biophysical Journal 75: 2038 10 Bausch, A.R., W. Möller, and E. Sackmann (1999) Biophysical Journal 76: 573 11 Bausch, A.R., U. Hellerer, M. Essler, M. Aepfelbacher, and E. Sackmann (2001). Biophysical Journal 80: 2649

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There are several implementations of microrheology, each using the motions of

small tracer particles that are embedded in a complex material to measure the local

viscoelastic response. One method employs the active manipulation of probe particles by

optical or magnetic forces to locally apply stress to a complex material 2,4,14,15,16. Active

measurements are extremely useful in probing the mechanical properties of stiff

biological materials, where large stresses are required to attain detectable strains. These

methods require sophisticated instrumentation and calibration procedures and often

require large particles in order to apply sufficiently large controlled stresses. An

alternative approach uses only the passive Brownian motion of embedded spheres to

probe the material response; in this case, samples must be sufficiently soft to allow

particles moving with only kBT of energy to locally deform the material 17,18,19,20.

12 Yamada, S., D. Wirtz, and S.C. Kuo (2000) Biophysical Journal 78: 1736 13 Feneberg, W., M. Westphal, and E. Sackmann (2001) European Biophysical Journal 30: 284 14 Amblard, F., B. Yurke, A. Pargellis, S. Leibler (1996) Review of Scientific Instruments 67: 818 15 Valentine, M.T., L.E. Dewalt, and H.D. Ou-Yang (1996) Journal of Physics: Condensed Matter (U.K.) 8: 9477 16 Hough, L.A. and H.D. Ou-Yang (1999) Journal of Nanoparticle Research 1: 495 17 Mason, T.G., and D.A. Weitz (1995) Physical Review Letters 74: 1250 18 Mason, T.G., H. Gang, and D.A. Weitz (1997) Journal of the Optical Society of America 14:139 19 Mason, T.G., K. Ganesan, J.H. Van Zanten, D. Wirtz, and S.C. Kuo (1997) Physical Review Letters 79:3282 20 Gittes, F., B. Schnurr, P.D. Olmsted, F.C. MacKintosh, and C.F. Schmidt (1997) Physical Review Letters 79: 3286

156

We employ a video-based multiple particle tracking technique to measure the

passive thermal fluctuations of dozens of micron-sized colloidal particles simultaneously

and use the dynamics of single particles, as well as the correlated movements of

separated tracers to probe the mechanical response of soft materials21,22,23 (Crocker, et al.,

2000; Levine and Lubensky, 2000; Valentine, et al., 2001). The combination of small

sample volumes, in the range of 20 microliters, and non-invasive measurement

techniques is particularly attractive for the study of biological samples. By varying the

particle size with respect to the native structural length scales of the material, we are able

to perform three distinct measurements. When embedded probe particles are large

compared to all structural sizes of the material, we determine the mean-squared

displacement (MSD) of the individual particles and average the MSDs of the ensemble of

particles in our imaging volume to improve our statistical accuracy. We use

microrheology formalism to relate the ensemble-averaged MSD to the linear frequency-

dependent viscous and elastic moduli using a generalized Stokes-Einstein relation; this

allows an accurate measurement of the macroscopic rheological response17,18,19,24.

When the embedded particles are approximately equal to or smaller than the

structural length scales of the material, particles move within small cavities and their

dynamics are no longer directly related to the bulk viscoelastic response. Rather, their

21 Crocker J.C., M.T. Valentine, E.R. Weeks, T. Gisler, P.D. Kaplan, A.G. Yodh, and D.A. Weitz (2000) Physical Review Letters 85: 888 22 Levine A.J. and T.C. Lubensky (2000) Physical Review Letters 85: 1774 23 Valentine M.T., P.D. Kaplan, D. Thota, J.C. Crocker, T. Gisler, R.K. Prud'homme, M. Beck, and D.A. Weitz (2001) Physical Review E 64: 061506 24 Dasgupta B.R., S.-Y. Tee, J.C. Crocker, B.J. Frisken, and D.A. Weitz (2001) Physical Review E 65: 051505

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Brownian movements allow us to characterize the physical properties of the

heterogeneous microenvironments they explore. In many cases individual particles

probe mechanically distinct microenvironments, preventing meaningful ensemble-

averaging; however, from the MSDs of individual particles it is possible to measure the

distribution of such important physical parameters as pore size or viscosity within the

small cavities23.

In a third measurement, called two-particle microrheology, the correlated

movements of separated pairs of particles are analyzed to measure the long-range

deformation of a material21,22. For homogeneous materials, individual particle dynamics

reflect the bulk viscoelastic response, and using two-particle analysis to measure

correlations on larger length scales provides the same information as obtained with

individual tracers. For locally inhomogeneous materials, particle dynamics are a

superposition of local fluctuations, which are experienced by particles in soft cavities and

are unrelated to polymer dynamics, and long wavelength modes resulting from the

thermal fluctuations of the network itself. Correlated motions arise only from network

fluctuations on length scales greater than the interparticle separation distance. We

assume that on these large length scales, the material is homogenous, and use two-

particle analysis to eliminate the uncorrelated local fluctuations, thus allowing

measurements of coarse-grained macroscopic rheological response, even in materials that

are inhomogeneous on the length scale of a single probe.

In each case, in order to properly interpret tracer motions, assumptions must be

made about the nature of the coupling of the particle to the medium. The ability to

characterize and modify the surface interactions, as well as tracer size, is crucial to

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properly interpreting multiple particle tracking measurements of complex biomaterials.

The ideal interactions between the particle and surrounding material vary depending on

which of the three measurements is being performed. For microrheology measurements,

in which the particle size is large in comparison to all structural length scales, particles

must be sufficiently resistant to protein adsorption to prevent local modification of

network architecture and introduction of small heterogeneities. For measurements of

microenvironments using tracers that are smaller than the structural length scales of a

heterogeneous material, protein adsorption can cause particles to adhere to cavity walls,

preventing them from fully exploring small pores and leading to uncertainties in the

interpretation of particle dynamics. When local variations in microenvironments or

changes in particle mobility due to protein adsorption are uncorrelated over large

distances, two-particle microrheology is insensitive to surface chemistry effects;

however, we find that the signal-to-noise ratio of these measurements remains sensitive

to the coupling of the tracer to the medium.

Understanding the detailed microscopic surface interactions between colloids and

proteins is critical for many other applications; however, these interactions are extremely

complex and remain poorly understood25. For example, the commercial colloids that are

typically used in microrheology experiments interact with proteins in myriad ways.

These colloids are typically charged, often with surface-bound carboxyl groups, to cause

repulsion between particles and prevent aggregation. Proteins often interact with and

adsorb to these charged colloids through non-specific electrostatic and hydrophobic

interactions. One common approach to reduce the non-specific adsorption of a particular

25 Haynes, C.A. and W. Norde (1994) Colloids and Surfaces B: Biointerfaces 2: 517

159

protein is to pre-incubate the colloids with another protein solution, known to have a high

surface affinity, such as bovine serum albumin (BSA), in order to block any available

protein binding sites. Such coatings have been shown to reduce the adsorption of some

proteins to colloidal surfaces and in many cases BSA-coated particles provide an

excellent alternative to bare tracer particles; however, the adsorbed BSA monolayer is

patchy and does not render the particles completely inert26. Moreover, because the

protein layer is simply adsorbed, and not covalently bound, desorption of the BSA from

the surface is possible.

A useful alternative to charge stabilization is the steric stabilization of colloids

achieved by the adsorption or grafting of polymer chains to the particle surface27,28. In

this case, the presence of a dense polymer brush prevents aggregation, and may

additionally shield the bare particle surface, thus blocking potential protein binding

sites29. Poly(ethylene glycol) (PEG) is an uncharged hydrophilic polymer that is used

extensively to improve the biocompatibility of materials, as PEG-coated surfaces are

resistant to the adsorption of proteins, and are nontoxic and nonimmunogenic30. The

origin of protein resistance in PEG-coated surfaces is generally attributed to steric

repulsion effects arising from both the loss of conformational entropy of the polymer as 26 McGrath, J.L., J.H. Hartwig, and S.C. Kuo (2000) Biophysical Journal 79: 3258 27 de Gennes, P.G. (1987) Advanced Colloid Interface Science 27: 189 28 Jeon, S.I., J.H. Lee, J.D. Andrade, and P.G. de Gennes (1991) Journal of Colloid and Interface Science 142: 149 29 Prime, K.L. and G.M. Whitesides (1993) Journal of the American Chemical Society 115: 10714 30 Harris, J. M., ed. Poly(ethylene glycol) chemistry: biotechnical and biomedical applications. New York : Plenum Press (1992)

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the brush is compressed and the disfavorable desolvation of the chains as water

molecules are expelled from the polymer layer; however, a detailed understanding of the

microscopic mechanism remains elusive29,31,32,33. Empirically, PEG has been shown to

reduce the adsorption of a wide variety of proteins when grafted onto flat surfaces29,31.

However, the adsorption properties of alternative geometries have not been fully

explored, as PEG-coated colloidal particles are not readily available commercially, and

are not widely used34,35,36,37,38.

In this paper, we present a simple and robust protocol to graft short methoxy-

terminated poly(ethylene glycol) to the surface of carboxylate-modified colloids using

commercially available reagents. We use these particles to investigate the interactions

between colloids and protein solutions using three different particle surface chemistries:

PEG-coated particles, particles coated with physisorbed BSA and bare carboxylate-

modified latex (CML) spheres. We demonstrate that these surfaces adsorb differing

31 Prime, K.L. and G.M Whitesides (1991) Science 252: 1164 32 Ostuni, E., R.G. Chapman, R. E. Holmlin, S. Takayama, and G.M. Whitesides (2001) Langmuir 17: 5605 33 Harder, P., M. Grunze, R. Dahint, G. M. Whitesides, and P.E. Laibinis (1998) Journal of Physical Chemistry B 102: 426 34 Shay, J.S., R.J. English, R.J. Spontak, C.M. Balik, and S.A. Khan (2000) Macromolecules 33: 6664 35 Weisbecker, C.S., M.V. Merritt, and G.M. Whitesides (1996) Langmuir 12: 3763 36 Shay, J.S., S.R. Raghavan, and S.A. Khan (2001) Journal of Rheology: 45: 913 37 Liu, J., L.M. Gan, C.H. Chew, C.H. Quek, H. Gong, and L.H. Gan (1997) Journal of Polymer Science A: Polymer Chemistry 35: 3575 38 De Sousa Delgado, A., M. Léonard, and E. Dellacherie (2001) Langmuir 17: 4386

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amounts of protein. Using multiple particle tracking techniques, we show that the

different particles measure different physical properties of the biomaterials they probe.

To illustrate the effect of surface chemistry on particle dynamics, we choose three protein

systems that display different adsorption characteristics. The first is fibrinogen, a

globular protein found in blood plasma that polymerizes into the filamentous protein

fibrin and is involved in the formation of clots in vivo; in vitro, fibrinogen self-assembles

into a crosslinked and branched semi-flexible fibrin network upon activation by

thrombin39. The adsorption properties of fibrinogen have been extensively studied due to

their strong surface affinity and structural similarity to the extracellular matrix protein

fibronectin, which is known to promote cell adhesion; moreover, because fibrinogen is

found in blood plasma, characterizing and controlling its ability to interact with foreign

surfaces is important to the successful biomedical implantation of therapeutic devices and

prostheses29,32,33,40,41,42,43. The second sample is filamentous actin (F-actin), a well-

studied and important biopolymer that is found in cellular cytoplasm and plays a role in

determining the strength, motility, and shape of living cells44,45,46. F-actin has a

39 Doolittle, R.F. (1981) Scientific American 245: 126 40 Lahiri, J., L. Issacs, J. Tien, and G.M. Whitesides (1999) Analytical Chemistry 71: 777 41 Alcantar, N. A., E.S. Aydil, and J.N. Israelachvili (2000) Journal of Biocompatible Materials Research 51: 343 42 Chapman, R.G., E. Ostuni, L. Yan, and G.M. Whitesides (2000) Langmuir 16: 6927 43 Chirakul, P. V.H. Perez-Luna, H. Owen, G.P. Lopez, and P.D. Hampton (2002) Langmuir 18: 4324 44 Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. Molecular Biology of the Cell. NewYork: Garland (2002)

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persistence length that is orders of magnitude larger than its diameter, and is an excellent

model system with which to explore the rheology and dynamics of semi-flexible polymer

networks47,48,49. F-actin networks have been studied extensively with particle tracking

and microrheology techniques, and understanding and controlling the surface interactions

of F-actin with colloids is imperative for properly interpreting those data 2,3,4,5,6,26. The

final sample, a composite protein network composed of F-actin and the actin-bundling

protein scruin, is chosen to investigate the effect of surface interactions with embedded

tracer particles in networks that more closely approximate physiologically relevant actin

gels, which are often crosslinked and bundled by actin-binding proteins in vivo.

6.3 Materials and Methods

6.3.1 Preparing PEG-coated Particles:

To create the PEG-coated particles, we attach amine-terminated methoxy-PEG,

(mPEG-NH2), NH2-(CH2-CH2-O)n-OCH3, where, on average, n=16, resulting in an

average molecular weight of 750 Da (Rapp Polymere, Tübingen, Germany), to

carboxylate-modified latex (CML) particles (Molecular Probes, Eugene, OR or 45 Boal, D. Mechanics of the Cell. Cambridge: Cambridge University Press (2002) 46 Howard, J. Mechanics of motor proteins and the cytoskeleton. Sunderland: Sinauer (2001) 47 Maggs, A.C. (1997) Physical Review E 57: 2091 48 Morse, D.C. (1998) Macromolecules 31: 7030 49 Gittes, F., and F.C. MacKintosh (1998) Physical Review E 58: R1241

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Interfacial Dynamics Corporation, Portland OR) using standard carbodiimide coupling

chemistry. In this reaction, the carboxylic acids are activated by the formation at low pH

of a reactive N-hydroxysuccinimide (NHS) ester, proceeding through the formation of a

less stable ester with 1-[3-(Dimethylamino)propyl]-3-ethylcarbodiimide (EDC) (Sigma-

Aldrich, St. Louis, MO). The activated esters are then mixed with mPEG-NH2 at higher

pH in order to accelerate deprotonation of the amine, which reacts with the NHS-ester to

yield a stable amide bond.

In order to minimize the probability of colloidal aggregation, we perform all

buffer exchanges slowly using dialysis tubing, rather then the more traditional

centrifugation or filtration techniques. Also, we foreshorten the timing of the low-pH

ester formation step, since we observe significant aggregation when this reaction is

allowed to proceed for more than 30 minutes, which we attribute to the loss of

electrostatic repulsion. All reactions and washes are performed under constant slow

stirring, and with the buffer volume exceeding the particle solution volume at least 100-

fold.

Spheres are loaded into dialysis tubing (SpectraPor, 10 kD cutoff) at number

densities of approximately 1011 - 1013 particles per mL; we observe that higher number

densities result in aggregation and poor coupling efficiency. The bags are submerged in

MES buffer (100 mM 2-(N-morpholino)ethanesulfonic acid) at pH 6.0 for two hours.

Bags are then rinsed with deionized (DI) water and submerged in a solution containing 15

mM EDC, 5 mM NHS, and a 10-fold excess of mPEG-NH2 in MES buffer for 30

minutes. Both the EDC and NHS are in vast excess, so the concentrations may be varied

slightly without loss of coupling efficiency; however, using significantly higher

164

concentrations of either reagent may accelerate the loss of charge stability and promote

aggregation. The mPEG-NH2 is added at this step primarily to ensure its immediate

presence in the dialysis bag after the pH is raised; a 100-fold increase in concentration

has no effect on reaction efficiency or protein resistivity of the particles.

The bags are then submerged into borate buffer (50 mM boric acid, 36 mM

sodium tetraborate) at pH 8.5, with NHS and mPEG-NH2 at the same concentrations as

above. The reaction is allowed to proceed for at least 8 hours under constant gentle

stirring, and is repeated twice with fresh buffer and reagents. After the third reaction, the

particles are washed in pure borate buffer for at least two hours to remove any unreacted

reagents and polymer. The particles are then recovered and stored at 4 ºC; they remain

stable against aggregation and protein-resistant for at least several months.

6.3.2 Preparation of BSA-coated particles:

Bovine Serum Albumin (Sigma-Aldrich, St. Louis, MO) is slowly dissolved in

Phosphate Buffered Saline (PBS) (Gibco, Invitrogen Life Technologies, Carlsbad CA) at

room temperature at concentrations of 5 mg/mL. CML particles, with diameters of either

1.0 or 0.84 µm, are incubated with the BSA solutions for one hour at room temperature,

or overnight at 4 ºC. The suspensions are centrifuged at low speed for 30 minutes, the

supernatant is discarded, and the particles are resuspended in fresh PBS. This washing is

repeated 3 or more times to remove any unbound BSA. Particles are stored at 4 ºC and

used within 48 hours of preparation, in order to reduce the likelihood of the BSA

desorbing from the particle surface.

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6.3.3 Characterization of protein adsorption using fluorescent BSA:

Undyed particles of each surface chemistry are incubated with bovine-derived

albumin, labeled with tetramethylrhodamine isothiocyanate (R-BSA) (Sigma-Aldrich, St.

Louis, MO), then observed with fluorescence microscopy to determine the amount of

protein adsorption. Experimentally, the R-BSA is dissolved in PBS at a concentration of

1 mg/mL and stored at 4 ºC for up to 48 hours. For each chemistry, we add 50 µL of 0.84

µm particles at a volume fraction of approximately 0.3% to 500 µL of the R-BSA

solution and incubate the solution overnight at 4 ºC under constant slow rotation.

Volume fractions of the BSA- and PEG-coated particles are determined by visually

comparing the diluted particle suspensions with optical microscopy to solutions of CML

particles at known volume fractions. The suspensions are then centrifuged at low speed

for 30 minutes, the supernatant is discarded, and the particles are resuspended in fresh

PBS. This washing is repeated three or more times to remove any unbound R-BSA.

Particles are observed with a Leica DM-IRB inverted microscope with a 100x

magnification, oil-immersion objective, with numerical aperture 1.4; using Metamorph

acquisition software, brightfield images are obtained with a Hamamatsu C2400 CCD

camera, and fluorescence images are obtained using a Hamamatsu EB-CCD intensified

camera. Images are analyzed using Adobe Photoshop.

6.3.4 Multiple particle tracking:

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For visualization of particle dynamics, particles are loaded into microscope

observation chambers that consist of three 1-mm spacers that are positioned in a “U-

shape” and sandwiched between a glass slide and a No. 1.5 glass coverslip. The slide and

coverglass are rinsed with water and methanol, and air dried before use. All pieces are

held in place with UV-cured optical glue (#81 or # 61 Norland, Cranbury NJ). To reduce

the amount of non-specific protein adsorption onto the glass surfaces, a 5 mg/mL BSA

solution is loaded into the chamber and incubated at room temperature for at least 1 hour

prior to use. After incubation, the chambers are rinsed exhaustively with DI water to

remove any remaining unbound BSA and the chambers are air-dried before use. Sample

volumes of 30 to 50-µL are loaded into the observation chambers, and the open side is

sealed with high vacuum grease to prevent evaporation. Care is taken to prevent any air

bubbles from contacting the sample as such bubbles can lead to slow leakage and

contribute to macroscopic drift of the embedded particles. Once loaded, the chamber is

left undisturbed for at least 30 minutes at room temperature to allow the protein networks

to form. The slides are then gently transferred to an inverted research microscope (Leica

DM-IRB) for observation. For two-particle microrheology measurements, we image at

least 100 microns into the sample to minimize hydrodynamic interactions with the walls.

Particles are imaged with brightfield or epi-fluorescence microscopy, and the

particle movements are recorded with a CCD camera (Cohu) onto S-VHS video tape or

are directly digitized in real-time using custom-written image analysis software5. Video

frames are acquired to obtain tracer positions with 30 Hz temporal resolution. In each

frame, the positions of the particles are identified by finding the brightness-averaged

167

centroid position with a subpixel accuracy of approximately 10 - 20 nm50. Positions are

then linked in time to create two-dimensional particle trajectories.

6.3.5 Interpreting particle motions as mechanical response:

With passive microrheology methods, small colloidal particles are embedded into

a complex fluid, and their thermally-activated Brownian motions are used to measure

local viscoelastic response17,18,19. The ensemble averaged mean-squared displacement

(MSD), 22 ( ) ( ) ( )x x t x tτ τ∆ = + − , is calculated as a function of lag time, τ, where

the angled brackets indicate an average over many starting times t and the ensemble of

particles in the field of view. For spherical tracers that are embedded in a homogeneous

and incompressible medium, the MSD is directly related to the viscoelastic response of

the surrounding material17,18,19. Physically, this can be understood by considering by two

limiting cases: a purely viscous fluid and a completely elastic solid. For a purely viscous

fluid, the d-dimensional MSD will increase linearly with lag time 2 ( ) 2x dDτ τ∆ = . The

viscosity / 6Bk T Daη π= may be calculated from the diffusion coefficient D, where a is

the particle radius. For a purely elastic material, the MSD will reach an average plateau

value 2px∆ that is independent of lag time and is determined by the elastic modulus of

the material. Equating the thermal energy to the elastic deformation energy, we estimate

the elastic plateau modulus 2~ /p B pG k T x a∆ . In general, the full frequency

dependence of the viscoelastic moduli is obtained from the MSD using the generalized 50 Crocker, J.C. and D.G. Grier (1996) Science 179: 298

168

Stokes-Einstein equation: 2 ( ) / ( )Bx s k T saG sπ= %% where 2 ( )x s% is the Laplace transform

of 2 ( )x τ∆ , and ( )G s% is the viscoelastic response as a function of the Laplace frequency s

17,18,19.

For materials that are inhomogeneous on the length scale of the tracers, the MSD

is more difficult to interpret, as particles are sensitive to variations in local viscoelastic

response as well as changes in local microstructure, which may consist of pores found in

the native material, or cavities that are sterically-induced by the presence of the tracers51.

In many cases, careful measurements of the time-averaged MSDs of individual particles

by multiple particle tracking allow characterization of the spatial distribution of the

physical properties of these pores23. However, these analyses rely on assumptions about

the coupling of the particle to the medium, and uncontrolled protein adsorption can result

in errant interpretations. By contrast, two-particle microrheology uses the correlated

movements of pairs of separated particles to measure the long-wavelength macroscopic

response of materials that are inhomogeneous on the length scale of the tracer but

homogeneous on the length scale of several particles21,22. Because very local differences

in material properties or protein adsorption are uncorrelated over distances larger than a

few particle diameters, two-particle microrheology is typically insensitive to these

effects. In this case, we calculate the ensemble-averaged tensor product of the tracer

displacements: ,

( , ) ( , ) ( , ) [ ( )]i j ij

i j tD r r t r t r R tαβ α βτ τ τ δ

≠= ∆ ∆ − where i and j label different

particles, α and β label different coordinates, and Rij is the distance between particles i

51 Chen, D.T., E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky, and A.G. Yodh (2003) Physical Review Letters 90:108301/1

169

and j. In the limit, r >> a, for an incompressible medium, ( , ) / 2 ( )rr BD r s k T rsG sπ= % ,

with no dependence on the particle size, shape or boundary conditions. Using only the Rij

where we observe a 1/r decay, to ensure that on these length scales the material is

homogeneous, we extrapolate the correlated motion to the length scale a and define the

two-particle mean-squared displacement 2 ( ) 2 / ( , )rrDr r aD rτ τ∆ = .

6.3.6 Preparation of biopolymer gels:

To demonstrate the impact of protein adsorption on microrheology experiments of

biopolymers, we measure particle movements in fibrin networks, entangled actin gels,

and a composite actin network, crosslinked and bundled with the actin-binding protein

scruin.

To form the fibrin network, fibrinogen monomers, at a concentration of 0.44

mg/mL, are rapidly thawed from -70 ºC, placed at room temperature and used within four

hours. Concentrated thrombin is rapidly thawed from -70 ºC, stored on ice for up to two

hours, and diluted to 0.6 µM immediately before use. The network is formed by mixing

50 µL 0.44 mg/mL fibrinogen in Tris-buffered saline (145 mM NaCl, 20 mM Trizma

Base, 0.3 mM Sodium Azide, 0.01% Tween 20; pH = 7.4), 0.5 µL 1M CaCl2, 0.5 µL of

the particle solution, and 0.5 µL of 0.6 µM thrombin in a small plastic microcentrifuge

tube. Once the thrombin is added, the fibrinogen solution quickly polymerizes to form

the fibrin network, and must be transferred immediately into the measurement chamber.

To form the entangled filamentous actin (F-actin) networks, lyophilized globular

actin (G-actin) is thawed from -20 ºC, dissolved in DI water and dialyzed against fresh G-

170

Buffer (2 mM Tris HCl, 0.2 mM ATP, 0.2 mM CaCl2, 0.2 mM dithiothreitol (DTT),

0.005% Sodium Azide) at pH 8.0 and 4 ºC for 24 hours, during which time G-Buffer is

replaced every eight hours. Solutions of G-Actin are kept at 4 ºC and used within seven

days of preparation. G-actin is mixed with the colloidal particle suspensions and fresh G-

buffer to adjust final concentration. Polymerization is initiated by addition of 1/10 of the

final sample volume of 10x F-buffer (20 mM Tris HCl, 20 mM MgCl2, 1 M KCl, 2 mM

DTT, 2 mM CaCl2, 5 mM ATP, pH 7.5). Samples are mixed gently for 10 seconds, then

loaded into observation chambers and allowed to equilibrate for 60 minutes at room

temperature before particle movements are recorded.

To form composite F-actin gels, we use scruin, an actin-bundling protein found

uniquely in the sperm of horseshoe crabs, where it bundles F- actin in the acrosome52,53.

Scruin purification is performed as previously described with minor modification54,55.

The integrity of the protein is checked by SDS-PAGE gel electrophoresis before each

experiment to ensure low level of proteolysis or degradation. The concentrations are

determined either by the Bradford assay, using BSA as a standard, or by absorbance at

280 nm56. Crosslinked and bundled actin networks are prepared by adding G-actin to the

mixture of scruin, particles, and 10x F-buffer. Samples are allowed to equilibrate for one

hour before observation. 52 Schmid, M.F., J.M. Agris, J. Jakana, P. Matsudaira, and J.W. Chiu (1995) Journal of Cell Biology 124: 341 53 Tilney, L.G. M.S. Tilney, and G.M. Guild (1996) Journal of Cell Biology 133: 61 54 Sun, S., M. Footer, and P. Matsudaira (1997) Molecular Biology of the Cell 8: 421 55 Shin, J.H., M.L. Gardel, A.R. Bausch, L. Mahadevan, P. Matsudaira, and D.A. Weitz. (in preparation) 56 Bradford, M.M. (1976) Analytical Chemistry 72: 248

171

6.4 Results and Discussion

6.4.1 Binding Capacity of CML, BSA, and PEG-coated particles determined by

adsorption of fluorescent BSA

In order to characterize the amount of protein adsorbed onto particles with

different surface modifications, we incubate 0.84-µm undyed CML and BSA-, and PEG-

coated particles with bovine-derived albumin, labeled with tetramethylrhodamine

isothiocyanate (R-BSA) and observe the particles with brightfield and fluorescence

microscopy to detect the amount of protein adsorbed. Brightfield images of particles of

the three surface chemistries are shown in Figure 6.1(a). Since the particles are not

inherently fluorescent, only particles that have adsorbed a considerable amount of

fluorescently-labeled protein will be visible with fluorescence imaging. As shown in

Figure 6.1(b), the CML and BSA-coated particles do fluoresce, indicating that both

surface chemistries allow significant protein adsorption. By contrast, the PEG-coated

particles show negligible fluorescence, indicating that very little protein has adsorbed.

To quantify the amount of adsorption, we calculate the relative fluorescence intensities of

the particles by measuring the total intensity of the image, subtracting the average

background intensity, which is measured in the absence of the particles, and then

normalizing by the number of particles in the field of view. As shown in Figure 6.1(c),

the intensity of the BSA-coated particles is only 60 percent of the intensity of the CML

particles, indicating that the BSA coating does prevent some amount of additional protein

172

adsorption, but does not render the particles completely inert. A partial reduction in

adsorption is consistent with previous measurements of BSA coating on CML spheres,

which indicate that even with coating solutions of up to 400 mg/mL BSA, the surface

coverage of physisorbed BSA never exceeds 50% of the total particle surface area, and

thus cannot completely prevent protein adsorption26. By contrast, the PEG-coated

particles are significantly more protein resistant than either the CML or BSA-coated

particles; the intensity of the PEG-coated particles is only two percent of that of the

untreated CML particles.

173

Figure 6.1. Brightfield (A) and fluorescence (B) images of CML, BSA, and PEG-coated

particles that have been incubated with R-BSA. The fluorescence intensity indicates the

binding capacity of each particle. The CML and BSA-coated particles adsorb a

significant amount of protein, while the adsorption of the PEG-coated particles is very

small. There is a slight shift in the field view between the brightfield and fluorescence

images due to slightly different optics along the two paths; in some cases particles have

diffused slightly between the image acquisitions. (C) Normalized fluorescence intensity

of the CML and BSA- and PEG-coated particles incubated with R-BSA. The intensity of

the BSA-coated particles is 60% of the CML particles, indicating that the BSA coating

does prevent some additional protein adsorption, but does not render the particles

completely inert. The intensity of the PEG-coated particles is only 2% of the CML

particles, indicating a significant improvement in protein resistivity

CML

BSA

PEG

Inte

nsity

(arb

itrar

y un

its)

A CB

174

6.4.2 Particle mobility in a fibrin network

Another measure of protein adsorption is the degree of incorporation of colloidal

particles into a protein network. Even a very small amount of adsorption can cause

particles to adhere to the protein network, dramatically affecting their mobility and

complicating the interpretation of particle tracking measurements. To explore the amount

of protein adsorption for particles of each surface chemistry, we use a crosslinked and

heterogeneous network of the semiflexible biopolymer fibrin, and record the movements

of 1-µm particles embedded in the gel. The mesh size of the 0.43 mg/mL fibrin network

is roughly 5 to 10-µm, as observed with fluorescence microscopy of networks formed

with rhodamine-dyed fibrinogen (data not shown). Because the mesh size is significantly

larger than the diameter of the tracers, we expect that particles that are resistant to the

adsorption of proteins will freely diffuse through the network. Particles with some

degree of adsorption, on the other hand, may adhere to the gel, and move only due to the

thermal undulations of the network. Thus, we expect that the degree of particle mobility

will qualitatively indicate the amount of protein adsorption onto the surface.

Experimentally, we record movies of particle dynamics, and identify the position

of each particle in each image; by comparing subsequent images, we connect these

positions in time to construct particle trajectories. In Figure 6.2 we show representative

trajectories for 1-µm particles moving in a 0.43 mg/mL fibrin network. The CML and

BSA-coated particles are constrained and barely move, while the PEG-coated particles

are much more mobile and the shape of their trajectories suggests a random walk. To

better quantify the particle motions, we calculate the one-dimensional MSD. In no cases

175

do the CML and BSA-coated tracers move detectably, and the value of the MSD is a

reflection only of our error in identifying particle centers. This suggests that the CML

and BSA-coated particles are adhering to an extremely stiff elastic material, whose

modulus is too large to allow detectable thermal movements of the tracers. Although we

are unable to accurately determine the MSD of either the CML and BSA-coated particles,

we are able to place an upper bound on the MSD; from the squared measurement error of

the particle positions, we estimate 2px∆ ≤ 4 x 10-4 µm2. For particles moving in a

heterogeneous network, characterized by an average mesh size ξ, we estimate the elastic

plateau modulus to be 2~ /p B pG k T x ξ∆ .57 Using our upper bound for the MSD and ξ

~ 5 µm, we obtain a lower bound for the elastic modulus G′ ≥ 5 Pa.

By contrast, the PEG-coated particles do move detectably, showing a variety of

different behaviors. Unlike the MSDs of the CML and BSA-coated particles, the MSD of

each PEG-coated particle is well above our resolution limit, and thus accurately reflects

the local microenvironment surrounding the tracer. In Figure 6.3 we display the one-

dimensional MSDs of 46 individual PEG-coated particles. Of the 46 particles, 38 move

diffusively for lag times up to one second, with MSDs increasing linearly in τ. To

improve our statistical accuracy, we calculate the ensemble-averaged MSD of all 38

mobile PEG-coated particles. Using the ensemble averaged MSD, we determine the local

viscosity to be 1.7 mPa-sec, consistent with the viscosity of the buffer, which we

independently measure to be roughly 1 mPa-sec; the small increase may be due to

57 Krall, A.H. and D.A. Weitz (1998) Physical Review Letters 80: 778

176

increased hydrodynamic drag as the particles move through the polymer network23,58.

This suggests that most of the PEG-coated particles completely resist protein adsorption,

and are able to freely diffuse nearly unperturbed by the polymer strands at lag times up to

one second. The remaining 8 particles, 17 % of the total, move subdiffusively with some

reaching a saturated plateau value at long lag times.

Using the individual PEG-coated particle trajectories and MSDs, we are unable to

distinguish experimentally between particles that have adhered to the network due to a

small amount of adsorption of fibrin, and particles that are not bound, but merely reside

in a dense region of the gel where the local mesh size is approximately equal to or

smaller than the tracer. Because the network is heterogeneous on the length scale of the

particle, in no case do we measure the bulk rheological response.

Measurements of particle mobility in fibrin networks highlight the importance of

understanding and controlling the surface interactions between colloids and protein gels.

Both the CML and BSA-coated particles adsorb proteins, and adhere to the network; as a

result, their movements are too small to be detected by our video-based particle tracking

method. By contrast, the PEG-coated spheres are much less likely to adsorb protein, and

most of the particles freely diffuse through the stiff elastic meshwork; these “inert”

particles are sensitive to local microstructure, and the viscosity of the background fluid.

This illustrates that by simply altering the protein binding capacity of the colloidal

tracers, we can tune the sensitivity of our tracers to their local microenvironment.

58 Jones, J.D. and K. Luby-Phelps (1996) Biophysical Journal 71: 2742

177

Figure 6.2. Trajectories of CML and BSA- and PEG-coated particles in a fibrin network.

The BSA-coated and CML particles are incorporated into the stiff fibrin network and are

constrained; circles indicate their static positions. By contrast, the PEG-coated particles

are resistant to the non-specific protein adsorption and thus are mobile with trajectories

that resemble random walks. The scale bar indicates 20 µm.

178

Figure 6.3. A sampling of the mean-squared displacements of individual particles

moving in a fibrin network. Most diffuse, but some particles are locally constrained,

leading to a plateau in their MSDs at long lag times.

<∆x2 >

(µm

2 )

τ (s)

179

6.4.3 Surface chemistry effects on the microrheology of entangled F-actin networks

In order to further explore the effect of surface chemistry on the protein

adsorption and mobility of colloids in biomaterials, we measure the movements of 1.0-

µm CML, and BSA- and PEG-coated particles in an 11.9 µM entangled filamentous actin

(F-actin) network. F-actin networks have been studied extensively with microrheology

techniques2,3,4,5,6, and the surface chemistry effects of CML and BSA-coated particles in

actin networks have been previously reported in detail26. Here, we calculate the

ensemble-averaged MSD for the PEG-coated particles, and compare to those of the CML

and BSA-coated particles, as shown in Figure 6.4. There is a measurable difference

between the plateau value of the MSD, 2px∆ , of the untreated CML particles, and that of

the BSA- and PEG-coated particles. There is a reduction in 2px∆ of the CML tracers

by a factor of roughly two in comparison to that of the BSA- and PEG-coated particles, in

agreement with previous work26. This reduction may indicate that the CML particles are

more tightly bound to the polymer network, or that by binding to actin filaments, the

CML particles introduce local crosslinking that slightly increases the local elastic

modulus. There is no measurable difference between the value of 2px∆ for the BSA-

and PEG-coated particles, suggesting that the binding affinity of actin to the particles is

weak in comparison to fibrin. In each case, the overall effect of varying surface

chemistry for entangled actin solutions is small.

180

Figure 6.4. Ensemble averaged mean-squared displacements of CML ( ), BSA-coated

( ), and PEG-coated ( ) particles in an entangled actin solution. There is measurable

decrease in the MSD of the CML particles as compared to the BSA- or PEG-coated

particles; however, the overall effect is small, suggesting that the affinity of actin to bare

CML particle surfaces is weak.

10-2 10-1 100 10110-4

10-3

10-2

10-1

100<∆

x2 (τ)>

(µm

2 )

τ (s)

181

6.4.4 One- and two-particle microrheology measurements of F-actin/Scruin networks

We measure only weak effects of particle surface modifications in pure actin

solutions; however, physiological actin networks are stiffened by actin-binding proteins

that crosslink, bundle, and nucleate filaments, and in cells these networks form in the

presence of numerous other globular and filamentous proteins44. Actin-binding proteins

may display more striking differences in binding capacity that are important to particle

mobility and to the interpretation of microrheology measurements in physiologically

relevant actin gels26. To further investigate these effects, we compare the movements of

1.0-µm CML, and BSA- and PEG-coated particles in F-actin networks that are

crosslinked and bundled by the actin-bundling protein scruin. Scruin is found in the

acrosomal bundle of the horseshoe crab sperm, where it decorates individual actin

filaments at a 1:1 stoichiometric ratio52. In vivo, scruin-scruin interactions align F-actin

in a tight parallel array of neighboring filaments. In vitro, actin-scruin composite

networks are used as model systems to investigate the mechanics and microstructure of

crosslinked F-actin gels; in this case, the ratio of scruin to G-actin, R, is varied from 1/30

to 1 55. Over these ratios, scruin serves to both crosslink and bundle actin filaments into a

three dimensional network. For small R, infrequent scruin-scruin interactions crosslink

neighboring actin filaments that remain randomly oriented without promoting bundle

formation. The network structure is only slightly perturbed from that of a purely

entangled actin network, and is characterized by a mesh size in microns 0.3 / Acξ =

182

where cA is the concentration of actin in mg/mL 59. At larger R, bundles form and as R is

increased for a constant actin concentration, the bundle thickness increases, leading to a

concomitant increase in the mesh size of the network.

To investigate the role of scruin in the surface interactions between the tracer

colloids and the gel, and to study the transition from a weakly-crosslinked to a highly-

bundled network, we embed CML, and BSA- and PEG-coated particles into the

composite actin-scruin networks and observe the resultant particle displacements. In

contrast to the solutions of purely entangled F-actin, we see dramatic differences in both

the magnitude and time dependence of the MSDs of particles with different surface

modifications. At the lowest concentration of crosslinkers, R = 1/30, the CML particles

aggregate and no single particles are observed, precluding measurements of particle

dynamics. At higher crosslinker densities, this aggregation worsens and the CML

particles form large clumps that sediment out of solution. By contrast, the BSA- and

PEG-coated particles remain, consistent with their ability to resist protein adsorption.

To further investigate the effect of surface chemistry on particle dynamics and to

investigate the transition from an entangled to a bundled network, we examine the MSDs

of the BSA- and PEG-coated particles at different scruin:actin ratios. The concentration

of actin is fixed at 11.9 µM, and the amount of scruin is varied to achieve the desired

stoichiometric ratio, as shown in Figure 6.5(a). For each R, the MSD of the BSA-coated

particles (solid symbols) increases at short lag times, and saturates at a plateau value

2px∆ at approximately τ = 0.3 seconds; the value of 2

px∆ decreases with increasing

59 Schmidt, C.F., M. Baermann, G. Isenberg, and E. Sackmann (1989) Macromolecules 22: 3638

183

crosslinker density. By contrast, the PEG-coated particles (open symbols) show a

completely different trend, as shown in Figure 6.5(b). At the lowest crosslinker density,

R = 1/30 ( ), the MSD saturate to a plateau value by a lag time of roughly 0.5 seconds;

however, this value is more than five times larger than the plateau observed with the

BSA-coated particles at the same R. When the amount of scruin is increased slightly to R

= 1/15 ( ) and R = 1/10 ( ), there is no change in the MSD of the PEG-coated particles

despite the three-fold change in crosslinker concentration. At higher crosslinker densities

of R = 1/5 ( ), or greater, the ensemble-averaged MSD does not reach a true plateau and

the value of the MSD at long times is larger than that of particles moving in less

crosslinked samples, or the purely entangled F-actin solutions. At the highest crosslinker

density of R = 1 ( ), the MSD is nearly linear at short lag times and never reaches a

plateau value within our experimental time range. The viscosity calculated from the

short time data is roughly 3 mPa-sec, similar to the viscosity of the background aqueous

fluid, which we independently measure to be approximately 1 mPa-sec.

184

Figure 6.5. The ensemble averaged MSDs for BSA-coated particles (solid symbols) and

PEG-coated particles (open symbols) moving in actin networks crosslinked and bundled

with the actin-binding protein scruin, at various ratios of scruin:actin: R = 1/30 ( ),

1/15( ), 1/10 ( ), 1/5( ), 1/2.5( ) and 1( ). A representative error bar is shown for

one point for each surface modification. The BSA-coated particles are constrained for

each scruin:actin ratio, reaching a plateau <∆xp2> that decreases with increasing amounts

of scruin. The ensemble averaged MSDs for PEG-coated particles moving in actin-scruin

networks, with R ranging from 1/30 to 1. At the lower concentration of crosslinkers, the

particles are constrained; however, the particle mobility increases with increasing

amounts of scruin.

10-2 10-1 100 10110-4

10-3

10-2

10-1

100

∆x 2 (µ

m2 )

τ (s)

185

The differences in the movements of the BSA-coated and PEG-coated particles

are consistent with the differences in their protein-binding capacities. The BSA-coated

particles, although better able to resist protein adsorption than the bare CML spheres, are

still somewhat “sticky” and adhere to the network; as a result, their movements reflect the

thermal fluctuations of the polymer strands. As we increase the amount of scruin, the

average bundle diameter d increases, causing an increase in the bending modulus of the

bundled actin strands; for thin, isotropic rods, B ~ d 4. Because the BSA-coated particles

adhere to these bundles, they are sensitive to changes in local stiffness and move less

with increasing amounts of scruin. By contrast, the PEG-coated particles are more

protein-resistant, and do not adhere to the network, rather they are merely constrained by

the elastic cage formed by the surrounding polymer strands. At low and intermediate

scruin concentrations we observe no change in the MSD measured by the PEG-coated

particles with increasing amounts of scruin, indicating little change in the local

microenvironment; however, at larger concentrations of scruin, ξ increases beyond the

fixed particle radius, allowing the PEG-coated particles to move more.

Because we measure no difference in the movements of BSA- and PEG-coated

particles in pure entangled actin networks we surmise that in the actin-scruin networks,

the BSA-coated particles preferentially bind to scruin. This raises the possibility that the

BSA-coated particles may be preferentially distributed to the scruin-rich regions of the

gel, or alternatively that the BSA-coated particles recruit scruin to their surfaces, thereby

increasing the effective concentration of scruin around the particles. It is therefore

possible that the BSA-coated particles change the distribution of crosslinkers in the

sample and create locally stiffer, more crosslinked and bundled regions of the gel near the

186

particle surface; this may lead to a decrease in the amplitude of their thermal fluctuations.

By contrast, the PEG-coated particles do not recruit scruin or modify the spatial

distribution of crosslinkers, and are this more randomly distributed among the native

mechanical microenvironments of the gel. Consistent with this, we measure differences

in 2px∆ for the BSA- and PEG-coated particles at low and intermediate scruin

concentrations, a regime in which particles with both surface modifications are

constrained by the network. The plateau values of the MSD for the two surface

modifications differ by a factor or five to ten, with the PEG-coated particles moving

more. Alternatively, it has been suggested that during the initial stages of gelation before

a percolated network has formed, non-binding particles will migrate to the weaker

regions of heterogeneous gels, as this separation maximizes particle motions and thus is

entropically favored26. We are unable to experimentally distinguish among these

possibilities, or precisely interpret 2px∆ for the constrained particles in these composite

networks.

To further explore these differences, and reveal the underlying polymer dynamics,

we use two-particle microrheology to measure the long wavelength fluctuations of the

actin-scruin networks to determine the bulk viscoelastic response. In this analysis, the

correlated motions of pairs of separated particles, which are insensitive to local changes

in the coupling of the probe particle to the network, are used to measure the macroscopic

rheology. The two-particle MSD for the PEG- coated particles (open symbols) and

BSA-coated particles (solid symbols) in a network with R = 1/30 is shown in Figure

6.6(a). There is a slight difference between the two-particle MSD measured with the

BSA- and PEG-coated particles; however, this difference is significantly smaller than that

187

measured with the one-particle MSD, and is within the sample to sample variation. The

two-particle MSDs for a more strongly crosslinked network with R = 1/15 are shown in

Figure 6.6(b); there is a small decrease in the two-particle MSD for the more highly

crosslinked and bundled sample as compared to the data shown in Figure 6.6(a), and

good agreement between the data for the BSA- and PEG-coated particles. In all cases the

two-particle MSDs plateau at long lag times, and the plateau values for <∆x2(τ)>D are

lower than those measured by the one-particle techniques by either the BSA- or PEG-

coated particles. Furthermore, we find that the two-particle analysis eliminates the

differences between the BSA- and PEG-coated particles. This suggests that although

there are local differences in the coupling of the BSA- and PEG-coated particles to the

network that give rise to differences in the one-particle MSDs, the particles do not induce

large length scale inhomogeneities; moreover, the macroscopic stress relaxation

measured by the two particle types is similar. To our knowledge, this is the first

demonstration that two-particle microrheology methods successfully eliminate the local

variations in mechanical response caused by differences in tracer surface chemistry.

Interestingly, for both samples, the two-particle MSD calculated from the data for

the PEG-coated particles is noisier than that calculated using the data from the BSA-

coated particles. In both cases, we suspect that the particle dynamics are a superposition

of local fluctuations, which are uncorrelated and unrelated to the long wavelength

polymer fluctuations, and long wavelength modes due to the thermal fluctuations of the

network itself, which lead to the correlated motions measured by the two-particle

dynamics.

188

Figure 6.6. Two-particle mean-squared displacements for composite actin – scruin

networks with (a) β = 1/30 and (b) β = 1/15. The long wavelength elastic modes are

unaffected by local coupling of the particles to the network, and differences between the

BSA-coated particles (solid symbols) and PEG-coated particles (open-symbols) measured

with one-particle techniques are eliminated. In all cases, the MSDs show a plateau at

long lag times, and the plateau values of <xp2> are significantly smaller than those

measured by one-particle techniques.

10-3

10-2

0.01 0.1 1 1010-4

10-3

<∆x D

2 > (µ

m2 )

(A)<∆x D

2 > (µ

m2 )

(B)

t (s)

189

Because the BSA-coated particles are somewhat sticky, they are physically coupled to the

polymer filaments; thus, a greater proportion of their Brownian dynamics reflects the

fluctuations of the network. By contrast, the PEG-coated particles are less likely to bind

to the network and a smaller proportion of their Brownian motion reflects the correlated

network dynamics. This leads to a much poorer signal-to-noise ratio for the PEG-coated

particles than for the BSA-coated particles, leading to noisier data in the two-particle

MSD.

6.5 Summary

The examples of the fibrin, F-actin and F-actin-scruin composite networks

illustrate the rich information about local viscosity, elasticity, and microstructure

available at a range of length scales with multiple particle tracking measurements and

demonstrate the challenges of interpreting particle movements in complex, heterogeneous

samples. In order to achieve the full potential of these methods, the delicate interactions

between the embedded probe colloids and the biopolymer networks must be understood

and controlled. In particular, to understand the interplay between mechanics and

structure in complex biomaterials, and to address important biological questions in vivo,

it is critical to identify which aspect of the multi-component system is being probed.

In this paper, we employ two types of particles with commonly-used surface

modifications, CML particles, and CML particles that have been coated with BSA, and

compare their adsorption properties and mobility to those of PEG-coated tracers. The

binding capacity of the particles varies greatly depending on the surface modifications

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and the particular protein of interest; however, in all cases, we find the CML particles

adsorb the most protein, while the PEG-coated particles adsorb the least. Moreover, we

find a correlation between increased binding capacity and decreased particle mobility.

Thus differences in particle dynamics measured with multiple particle tracking

techniques may reflect differences in the particles’ adsorption characteristics, as well as

differences in the local viscoelastic modulus or changes in local microstructure. When

encountering a new material, careful studies of the effect of both particle size and surface

chemistry are absolutely required for the proper interpretation of microrheology and

multiple particle tracking experiments.

We have further demonstrated that binding and non-binding particles are sensitive

to different physical properties of the heterogeneous networks they probe. Our

measurements of particle displacements in fibrin networks and crosslinked and bundled

actin gels suggest that particles that are weakly bound to the network are sensitive to

changes in local bending stiffness, but not microstructure, while the more inert PEG-

coated particles are sensitive to changes in local mesh size and viscosity, but not elastic

modulus. Thus, the ideal particle-polymer interactions vary depending on the particular

physical property of interest, and the ability to characterize and tune the adsorption

characteristics of the embedded particles, is critical to fully explore the microscopic

material response.

Additionally, using two-particle microrheology techniques, we measure the

macroscopic viscoelastic response of locally inhomogeneous material by measuring the

correlated movements of separated particles. Although the interpretation of these

measurements does not rely on understanding the nature of the particle coupling to the

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medium, the signal-to-noise ratio does depend on the ability of the particles to couple to

the long wavelength viscoelastic modes of the material, suggesting that some amount of

binding is desirable. In cases where a small amount of adsorption is desired, the

uncontrolled synthesis, ill-defined characterization, and short shelf-life of the BSA-

coated particles may be impractical. Similarly, non-specific adsorption onto bare CML

particles is often unacceptable; for example, in the actin-scruin networks, the CML

particles formed large aggregates that sedimented out of solution, preventing

measurements of particle dynamics. In these cases, it may be possible, with slight

modifications to the protocol described here, to create designer particles with dense

methoxy-terminated PEG brushes that prevent non-specific adsorption, but bind to a

particular target protein by the addition of a small number of reactive PEG chains that

display terminal groups that may be anchored to a particular protein of interest by an

antibody or other specific linkage40. Such designer particles may allow mechanical

measurements of a single component of a complex composite structure, such as in the

cytoplasm of living cells.

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Chapter 7: Mechanical Properties of Xenopus Egg Cytoplasmic Extracts∗

7.1 Overview

We report the initial characterization of the mechanical properties of Xenopus egg

cytoplasmic extracts. To measure response at micron length scales, we use multiple

particle tracking methods to measure the thermal fluctuations of embedded colloidal

probes, and find a mostly viscous response with viscosity η ~ 20 mPa-sec. At larger

length scales, the extract forms a soft viscoelastic solid, with elastic modulus G′ ~ 2 – 10

Pa, and loss modulus, G″ ~ 0.5 - 5 Pa. To reveal the molecular basis for gel elasticity, we

use drug treatments to independently remove F-actin, microtubules, and cytokeratin and

use confocal microscopy to observe the structural changes that accompany gelation. We

find that the actin and microtubule filaments cooperate to organize the cytoskeleton and

give mechanical strength. We also observe an actin-mediated gel contraction and find

that is resisted by microtubule filaments in vitro.

7.2 Introduction

Cellular cytoplasm consists of the cytosol, a background fluid of globular

proteins, and the cytoskeleton, a complex filamentous network consisting of F-actin,

microtubules, and intermediate filaments. Cytoskeletal interactions drive a number of

∗ In collaboration with Z.E. Perlman and TJ. Mitchison, Harvard Medical School

193

vital biological processes including locomotion, adhesion, and division1,2,3. By forming a

continuous three-dimensional network, the cytoskeleton provides a scaffolding upon

which motor proteins move and a large surface area for localization and immobilization

of specific cytoplasmic molecules4. The cytoskeleton is also implicated in signal

transduction, and can remodel, deform, and flow in response to external stress4. Yet,

despite great interest in cellular mechanics and detailed structural information about the

component systems, an understanding of integrated cytoplasmic mechanics has remained

elusive.

Because living cells are small, dynamic, and fragile with complicated geometries,

these systems challenge experimenters to develop new techniques to measure

micromechanical response. A number of promising methods have emerged, including

measurements of the diffusion of small macromolecules in cytosol5,6,7,8 and time-

dependent viscoelastic properties using magnetic9,10,11,12,13,14,15,16, optical17,18 or

1 Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. Molecular Biology of the Cell. NewYork: Garland (2002). 2 Howard, J. Mechanics of motor proteins and the cytoskeleton. Sunderland: Sinauer (2001). 3 Boal, D. Mechanics of the Cell. Cambridge: Cambridge University Press (2002) 4 Janmey, P.A. (1998) Physiological Reviews 78: 763 5 Luby-Phelps, K., D. Lansing Taylor, and F. Lanni (1986) Journal of Cell Biology 102: 2015 6 Wojcieszyn, J.W., R.A. Schlegel, E.-S. Wu, and K.A. Jacobson (1981) Proceedings of the National Academy of Science, USA 75: 4407 7 Luby-Phelps, K. (1994) Current Opinion in Cell Biology 6: 3 8 Goulian, M. and S.M. Simon (2000) Biophysical Journal 79: 2188 9 Wang N., J.P. Butler, and D.E. Ingber (1993) Science 260:1124

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mechanical forces19,20,21,22,23,24,25 to locally apply stress to the cell surface or interior.

Other techniques track the thermal movements of organelles and endogenous lipid

10 Wang, N., and D.E. Ingber (1994) Biophysical Journal 66: 1281 11 Wang, N., and D.E. Ingber (1995) Biochemistry and Cell Biology 73: 327 12 Bausch, A.R., F. Ziemann, A.A. Boulbitch, K. Jacobson, and E. Sackmann (1998) Biophysical Journal 75: 2038 13 Bausch, A.R., W. Möller, and E. Sackmann (1999) Biophysical Journal 76: 573 14 Bausch, A.R., U. Hellerer, M. Essler, M. Aepfelbacher, and E. Sackmann (2001) Biophysical Journal 80: 2649 15 Feneberg, W., M. Westphal, and E. Sackmann (2001) European Biophysical Journal 30:284 16 Fabr,. B., G.N. Maksym, J.P.Butler, M. Glogauer, D. Navajas, and J.J. Fredberg (2001) Physical Review Letters 87: 148102 17 Hénon, S., G. Lenormand, A. Richert, and F. Gallet (1999) Biophysical Journal 76: 1145 18 Sleep, J., D. Wilson, R. Simmons, and W. Gratzer (1999) Biophysical Journal 77: 3085 19 Mahaffy, R.E., C.K. Shih, F.C. MacKintosh, and J. Käs (2000) Physical Review Letters 85: 880 20 Hoh, J.H. and C.A. Schoenenberger (1994) Journal of Cell Science 107: 1105 21 Putman, C.A., K.O.V.D. Werf, B.G.D. Grooth, N.F.V. Hulst, and J. Greve (1994) Biophysical Journal 67: 1749 22 Shroff, S.G., D.R. Saner, and R. Lal (1995) American Journal of Physiology 269: C286 23 Goldman, W.H., and R.M. Ezzell (1996). Experimental Cell Research 226: 234 24 Rotsch, C, F.Braet , E. Wisse, and M. Radmacher (1997). Cell Biology International 21: 685 25 A-Hassan, E., W.F. Heinz, M.D. Antonik, N.P. D'Costa, and S. Nageswaran (1998) Biophysical Journal 74:1564

195

granules26, or the active transport of endosomes27,28 to measure the physical properties of

cellular microenvironments. These experiments have provided vital information about

the local elastic and viscous responses in the cytoplasm, but have not provided a clear

understanding of the mechanisms that govern this response.

Much theoretical work to characterize the mechanical properties of the cytoplasm

has focused on networks of isolated protein filaments, such as actin. Cytoskeletal

filaments have a persistence length that is orders of magnitude larger than their diameter,

and are thus excellent model systems for the study of semiflexible polymers29,30,31,32.

Experimentally, actin networks have been studied extensively with microrheology

techniques, in which small colloids are embedded in a material, and their thermal or

externally forced motions are used to measure rheological response; in some cases,

samples can be obtained in large quantities, allowing conventional macroscopic

tests33,34,35,36,37,38. These measurements provide important information about the

26 Yamada, S., D. Wirtz, and S.C. Kuo (2000) Biophysical Journal 78: 1736 27 Capsi, A., R. Granek, and M. Elbaum (2000) Physical Review Letters 85:5655 28 Capsi, A., R. Granek, and M. Elbaum (2002) Diffusion and directed motion in cellular transport. Physical Review E 66: 011916 29 Maggs, A.C. (1997). Physical Review E 57: 2091 30 Maggs, A.C. (1997). Physical Review E 55: 7396 31 Gittes, F. and F.C. MacKintosh (1998) Physical Review E 58: R1241 32 Morse, D.C. (1998) Macromolecules 31: 7030 33 Ziemann, F., J. Rädler, and E. Sackmann (1994) Biophysical Journal 66: 2210

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dynamics of single filaments, and the viscoelasticity of entangled polymer networks;

however, isolated networks may not have physiologically relevant architecture, making

comparisons to cellular cytoplasm difficult. Moreover, single component protein

networks cannot be used to investigate the interactions between the different cytoskeletal

systems nor the role of numerous cytosolic binding proteins found in vivo.

To address these issues, and provide an intermediate system between isolated

protein networks, and living cells, we have developed interphase Xenopus egg extracts as

a model system to study cytoplasmic mechanics. Extracts are obtained from meiotically

mature eggs that have been crushed by high speed centrifugation to isolate the globular

cytoplasmic proteins. Upon warming to room temperature, F-actin, microtubules, and

cytokeratin, the only intermediate filament present in the extracts39,40, polymerize into

filaments, forming a soft viscoelastic gel. Xenopus egg extracts have unique technical

advantages that make them especially useful for our study. Because the eggs are

deformable, and are physically packed by gentle centrifugation prior to disruption by

34 Amblard, F., A.C. Maggs, B. Yurke, A.N. Pargellis, and S. Leibler (1996) Physical Review Letters 77: 4470 35 Schmidt, F.G., F. Ziemann, and E. Sackmann (1996) European Biophysics Journal 24: 348 36 McGrath, J.L., J.H. Hartwig, and S.C. Kuo (2000) Biophysical Journal 79: 3258 37 Keller, M., J. Schilling, and E. Sackmann (2001) Review of Scientific Instruments 72: 3626 38 Gardel, M.L., M.T. Valentine, J.C. Crocker, A.R. Bausch and D.A. Weitz (Submitted) 39 Franz, J.K., L. Gall, M.A. Williams, B. Picheral, W.W. Franke (1983) Proceedings of the National Academy of Sciences USA 80: 6254 40 Franz, J.K., W.W. Franke (1986) Proceedings of the National Academy of Sciences USA 83: 6475

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faster centrifugation, the cytosol is diluted very little with extract buffer during

homogenization, with a dilution of only 10-20% 41. Extracts also remain metabolically

active, with energy in the form of ATP supplied by the metabolism of endogenous

glycogen and added phosphocreatine; this prevents myosin motors from forming rigor

bonds onto actin filaments, thus artificially stiffening the gel. Additionally, the actin,

microtubule and cytokeratin filaments may each be removed or stabilized by the addition

of pharmacological or immunological agents, allowing us to probe each system

independently. Moreover, it is possible to harvest relatively large amounts of cytoplasm

in order to perform macroscopic mechanical tests.

Xenopus egg extracts have previously been shown to be excellent model systems

for cytoplasmic structure 42,43,44 and have been used extensively to study mitotic spindle

assembly, and to biochemically purify proteins that influence microtubule dynamics

45,46,47. Such extracts have also been used as a model system to study polymerization-

41 Murray, A. (1991) “Cell cycle extracts” In Methods in Cell Biology, B.K. Kay and B. Peng, eds. San Diego: Academic Press p. 508-605. 42 Sider, J.R., C.A. Mandato, K.L. Weber, A.J. Zandy, D. Beach, R.J. Finst, J. Skoble, and W.M. Bement (1999) Journal of Cell Science: 112: 1947 43 Waterman-Storer, C., D.Y. Duey, K.L. Weber, J. Keech, R.E. Cheney, E.D. Salmon, and W.M. Bement (2000) Journal of Cell Biology 150: 361 44 Weber, K.L., and W.M. Bement (2002) Journal of Cell Science 115: 1373 45 Desai, A. and T.J. Mitchison (1997) Annual Review of Cell and Developmental Biology 113: 83 46 Desai, A., A. Murray, T.J. Mitchison, and C.E. Walczak (1999) Methods of Cell Biology 61: 385 47 Shirasu, M., A. Yonetani, and C.E. Walczak (1999) Microscopy Research and Technique 44: 435

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based motility 48,49 and the physiological sol-gel transitions that may drive such cellular

processes as cell spreading and crawling50. However, few experiments have focused on

quantitative measurements of the important viscoelastic properties of this material.

In this paper, we present initial characterization of the mechanical properties of

Xenopus-derived bulk cytoplasm. Because the F-actin, microtubules, and cytokeratin

form a network with a large mesh size of several microns, the viscoelastic response varies

greatly depending on what length scales we probe the material. We use a multiple

particle tracking technique in which the thermal motions of small embedded colloidal

particles are used to measure the local mechanical properties of the extract. At length

scales of a micron, the sample is mostly viscous with viscosity η ~ 20 mPa-sec. To probe

the material at larger length scales, we use a macroscopic mechanical rheometer, and find

that the composite network forms a soft viscoelastic solid. We investigate the role of

each of the cytoskeletal filaments by selectively removing or stabilizing the F-actin,

microtubules, or cytokeratin independently using pharmacological and immunological

disruption techniques. We find that the F-actin and microtubule networks cooperate to

organize the extract and give mechanical strength. We also find that microtubules resist

the actomyosin-dependent contraction of the gel in vitro.

48 Theriot, J.A. J. Rosenblatt, D.A. Portnoy, P.J. Goldschmidt-Clermont, and T.J. Mitchison (1994) Cell 76: 505 49 Cameron, L.A., M.J. Footer, A. van Oudenaarden, and J.A. Theriot (1999) Proceedings of the National Academy of Sciences USA 96: 4908 50 Clark, T.G. and R.W. Merriam (1978) Journal of Cell Biology 77: 427

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7.3 Materials and Methods

7.3.1 Xenopus Egg Cytoplasmic Extracts

Cytoplasmic extracts are prepared from Xenopus laevis as previously described,

with minor modifications41,46. Briefly, laid eggs are harvested from adult Xenopus

females and washed with MMR (5 mM HEPES, pH 7.8; 0.1 mM EDTA, 100 mM NaCl,

2 mM KCl, 1 mM MgCl2, 2 mM CaCl2) and all lysed, puffy or irregular eggs are

discarded. In order to maximize yield, frogs are returned to MMR at 16 ºC for a second

preparation in which both laid and squeezed eggs are used. Eggs are dejellied in 2%

cysteine in water, with KOH added to pH 7.8, then washed twice with cold XB

containing cycloheximide at a concentration of 1 µg/mL. Eggs are cycled into interphase

by incubating with a calcium ionophore (Sigma- Aldrich; A23187) at a concentration of

200 ng/mL and cycloheximide at 1 µg/mL in XB (10 mM HEPES, pH 7.7; 1 mM

MgCL2, 0.1 mM CaCl2, 100 mM KCl, 50 mM sucrose) for 5 minutes at 4 ºC.

Cycloheximide is added at each successive buffer exchange to prevent production of

cyclin and arrest the cell cycle state. In order to have sufficient quantities of extract for

bulk rheology measurements, large-scale preparations using 20 to 30 frogs are necessary.

The large quantity of eggs requires extended preparation time; to reduce protein

degradation during the preparation, we perform all steps following the addition of the

ionophore at 4 ºC. In order to preserve the filamentous actin in the final extract, we omit

cytochalsin D. Eggs are gently transferred into 50 mL centrifuge tubes with a 25 mL

plastic pipette that has been cutoff to enlarge the opening and fire-polished to remove

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sharp edges. The packing spin consists of 1 minute at 700 rpm and 30 sec at 1000 rpm in

a clinical centrifuge, after which time all excess buffer is removed; the crushing spin is

performed at 12.5 krpm for 15 minutes at 4 ºC to separate cytoplasmic proteins from

pigment, organelles, lipids and fats. Crude extract is recovered, supplemented with

protease inhibitors (10 µg/mL final concentration each of leupeptin, pepstatin, and

chymostatin), energy mix (7.5 mM creatine phosphate, 1 mM ATP, 1 mM MgCl2, final

concentrations), and 50 mM sucrose. Aliquots of 500 µL for use in rheology experiments

and 100 µL for use in particle tracking and imaging experiments are flash frozen in

liquid nitrogen and stored at -70 ºC until use.

7.3.2 Poly(ethylene glycol) (PEG)-coated Particles:

To create protein resistant colloidal particles, we attach amine-terminated

methoxy-poly(ethylene glycol), (mPEG-NH2), NH2-(CH2-CH2-O)n-OCH3, with <n> = 16

for an average molecular weight of 750 Da, (Rapp Polymere, Tübingen, Germany), to

carboxylate-modified latex (CML) particles (Molecular Probes, Eugene, OR or

Interfacial Dynamics Corporation, Portland OR) using standard carbodiimide coupling

chemistry51. Spheres are loaded into dialysis tubing (SpectraPor, 10 kD cutoff) at

number densities of approximately 1011 - 1013 particles per mL. The bags are submerged

in MES buffer (100 mM 2-(N-morpholino)ethanesulfonic acid) at pH 6.0 for two hours.

Bags are then rinsed with deionized (DI) water and submerged in a solution containing 15

51 Valentine, M.T., Z.E. Perlman, M..L. Gardel, J.H. Shin, P.T. Matsudaira, T.J. Mitchison, and D.A. Weitz (in preparation)

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mM 1-[3-(Dimethylamino)propyl]-3-ethylcarbodiimide (EDC), 5 mM N-

hydroxysuccinimide (NHS), and a 10-fold excess of mPEG-NH2 in MES buffer for 30

minutes. All reactions and washes are performed under constant slow stirring, and with

the buffer volume exceeding the particle solution volume at least 100-fold. The bags are

then submerged into borate buffer (50 mM boric acid, 36 mM sodium tetraborate) at pH

8.5, with NHS and mPEG-NH2 at the same concentrations as above. The reaction is

allowed to proceed for at least 8 hours under constant gentle stirring, and is repeated

twice with fresh buffer and reagents. After the third reaction, the particles are washed in

pure borate buffer for at least two hours to remove any unreacted reagents and polymer.

The particles are then recovered and stored at 4 ºC.

7.3.3 Sample Preparation:

In all cases, extracts are kept cold until just before the measurement in order to

prevent the onset of the temperature-dependent polymerization and cross linking of the

cytoskeletal filaments. To isolate the molecular basis for gel elasticity, we use a variety

of pharmacological and immunological techniques to selectively disrupt or stabilize a

single component of the cytoplasm. Actin is disrupted by addition of 30 µM latrunculin B

(Calbiochem) and microtubules are disrupted by addition of 10 µM nocodazole (Sigma-

Aldrich). Phalloidin (Cytoskeleton) and taxol (Cytoskeleton) are each added at a

concentration of 10 µM to stabilize actin and microtubule networks respectively.

Nocodazole, latrunculin B, phalloidin and taxol are all dissolved in DMSO; the final

concentration of DMSO in the extracts is less than 1%. Cytokeratin assembly is blocked

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by the addition of 1 mg/mL cytokeratin antibody C11 (Sigma-Aldrich). Control

experiments with the addition of 1 mg/mL of the non-specific antibody IgG and an

equivalent amount of pure PBS are also performed. Due to difficulties in obtaining

highly concentrated IgG samples, the addition of antibodies to the extract introduces a 1:8

dilution.

7.3.4 Multiple particle tracking:

In order to characterize the mechanical response on micron length scales, small

PEG-coated colloidal particles are added to the extract and the particle movements

recorded. For visualization of particle dynamics, particles are loaded into microscope

observation chambers that consist of three 170-µm glass spacers that are positioned in a

“U-shape” and sandwiched between a glass slide and a No. 1.5 glass coverglass. The

slide and coverglass are rinsed with water and methanol, and air-dried before use. All

pieces are held in place with UV-cured optical glue (#81 or # 61 Norland, Cranbury NJ).

To reduce the amount of non-specific protein adsorption onto the glass surfaces, a 5

mg/mL BSA solution is loaded into the chamber and incubated at room temperature for at

least 30 minutes prior to use. After incubation, the chambers are rinsed exhaustively with

DI water to remove any remaining unbound BSA and the chambers are air-dried before

use. Sample volumes of 30 to 50-µL are loaded into the observation chambers, and the

open side is sealed with high vacuum grease to prevent evaporation. Care is taken to

prevent any air bubbles from contacting the sample as such bubbles can lead to slow

leakage and contribute to macroscopic drift of the embedded particles. The slides are

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then gently transferred to an inverted research microscope (Leica DM-IRB) for

observation with a 40x oil-immersion lens. Typically, we image at least 25 microns into

the sample to minimize hydrodynamic interactions between the particles and walls.

Particles are imaged with brightfield microscopy, and images of the particles are recorded

with a CCD camera (Cohu) onto S-VHS video tape or are directly digitized in real-time

using custom-written image analysis software37; the magnification is 250 nm per pixel.

Video frames are analyzed to obtain tracer positions with 30 Hz temporal resolution. In

each frame, the positions of the particles are identified by finding the brightness-averaged

centroid position with a subpixel accuracy of approximately 10 - 20 nm1. Positions are

then linked in time to create two-dimensional particle trajectories.

7.3.5 Interpreting particle motions as mechanical response:

With passive microrheology methods, small colloidal particles are embedded into

a complex fluid, and their thermally-activated Brownian motions are used to measure

local mechanical response at the length scale of the sphere2,3,4. The ensemble averaged

mean-squared displacement (MSD), 22 ( ) ( ) ( )r r t r tτ τ∆ = + − , is calculated as a

function of lag time, τ, where the angled brackets indicate an average over many starting 1 Crocker, J.C. and D.G. Grier (1996) Journal of Colloid and Interface Science 179: 298 2 Mason, T.G. and D.A. Weitz (1995) Physical Review Letters 74: 1250 3 Mason, T.G., K. Ganesan, J.H. Van Zanten, D. Wirtz, S.C. Kuo (1997) Physical Review Letters 79: 3282 4 Mason, T.G., H. Gang, and D.A. Weitz (1997) Journal of the Optical Society of America 14: 139

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times t and the ensemble of particles in the field of view. In a homogeneous material, the

MSD is directly related to the viscoelastic response of the surrounding material. In the

case of a purely viscous fluid, the d-dimensional MSD will increase linearly with lag

time 2 ( ) 2r dDτ τ∆ = . The viscosity / 6Bk T Daη π= may be calculated from the

diffusion coefficient D, where a is the particle radius. For materials that are

inhomogeneous on the length scale of the tracers, the MSD is more difficult to interpret,

as particles are sensitive to variations in local viscoelastic response as well as changes in

local microstructure, which may consist of pores found in the native material, or cavities

that are sterically-induced by the presence of the tracers5.

7.3.6 Fixation of Samples and Confocal Imaging:

In order to visualize the cytoskeletal networks, we prepare rapidly frozen extract

samples for confocal microscopy, as previously described, with minor modifications44,6.

Briefly, clean 22 x 22 mm2 glass coverslips are coated with Rain X, and rinsed with

ethanol. One the Rain X- treated coverslips are dry,15 µL extract, supplemented with 3

µM rhodamine-actin (0.3 labeling stiochiometry) and Alexa-488 tubulin (0.6-0.7 labeling

stiochiometry) is pipetted onto them, and an 18 x 18 mm2 untreated glass coverslip is

inverted on top. The small coverslips are scored with a diamond scribe to identify each

sample. The samples are incubated at room temperature for 0, 10, 20, and 40 minutes for 5 Chen, D.T., E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky and A.G. Yodh (2003) Physical Review Letters 90: 108301/1 6 Mandato, C.A., K.L. Weber, A.J. Zandy, T.J. Keating, and W.M. Bement (2000) Methods of Molecular Biology 161: 229

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the time series, and 45 minutes for the pharmacological disruption series before being

plunged into liquid nitrogen for 10 to 30 minutes. The samples are recovered, and placed

on a glass Petri dish chilled on dry ice, and the small coverslips are pried off and placed

immediately in methanol, chilled to -20 ºC, to fix the samples. After 10 minutes the

samples are rinsed with Tris-buffered saline (TBS), pH 7.5 to wash out any remaining

methanol. The coverslips are inverted onto a drop of mounting media (containing

phenylalamine and glycerol) placed on clean glass slides and the chambers are sealed

with nail polish. Images are obtained using a Nikon spinning disk confocal microscope,

using a 100x oil-objective at a magnification of 0.15 µm per pixel. Final sample

thicknesses ranged from 15 to 40 µm; the images presented here are taken at the mid-

plane.

7.3.7 Macroscopic Rheology:

To measure the bulk viscoelastic response, we use a mechanical strain-controlled

rheometer (ARES) with a cone and plate geometry, with a radius of 25 mm, cone angle of

0.02 rad, gap of approximately 0.8 mm; the required sample volume is 500 µL. We load

all samples cold in order to prevent distortions or rupture of the networks that form at

room temperature. The lower plate fixture is chilled to 5 °C, and the upper cone fixture is

chilled on ice prior to mounting. Individual aliquots of extract are thawed within one

hour of use and stored on ice until they are loaded. To reduce the likelihood of the

sample drying during the course of the measurement, we use a room humidifier, and

surround the cone and plate with wet paper towels and a solvent trap. Once the sample is

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loaded, we raise the temperature of the lower tool to 22 °C, and allow the sample to

equilibrate. To probe the rheological response of the isolated cytoplasm, we apply a

small amplitude oscillatory shear strain, γ(t) = γo sin(ωt) where ω is the frequency of

oscillation and γ o is the strain amplitude and measure the resultant shear stress. For

small strains, the time-dependent stress σ is linearly proportional to γo,

[ ]( ) ( )sin( ) ( ) cos( )ot G t G tσ γ ω ω ω ω′ ′′= + , where G′ is the storage or elastic modulus and

G″ loss or viscous modulus. To study the onset of gelation, we perform measurements

with ω = 1 rad/s and γo = 0.05 every several minutes for roughly one hour after warming

and monitor the change in G′ and G″ as a function of time. For times significantly

beyond one hour, our measurement often becomes unreliable. Drying and denaturing of

the sample sometimes occur, forming a thick crust at the edge of the cone and plate and

causing a large and sudden increase in the modulus; we also sometimes observe

monotonic decreases in modulus at long times. We suspect that this decrease in modulus

is an artifact of gel shrinkage or contraction, which allows slip at the interface of the

sample and tool. For those samples that exhibit steady state behavior, we perform

frequency-dependent measurements, with γ = 0.01 or 0.05, and ω ranging from 1 – 100

rad/s. We also probe the non-linear viscoelastic regime by measuring the stress σ, as

well as G′ and G″ as a function of γ for ω = 1 rad/s and γo ranging from 0.005 to 25.

7.3.8 Contraction Assay:

To investigate the role of actomyosin gelation in the contraction of interphase

Xenopus egg extracts, we observe samples with various pharmocological treatments as a

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function of time. Chambers are prepared as described for microscopy, using 1 mm

spacers, and 100 µL of sample is loaded into each chamber. Images are obtained with a

Canon Powershot S-200 digital ELPH camera.

7.4 Results and Discussion

7.4.1 Actin-based gelation causes macroscopic gel contraction

Mitotic Xenopus egg extracts undergo a fast actin-based contraction within 15-30

minutes of warming to room temperature, making mechanical measurements of mitotic

bulk cytoplasm difficult. Interphase extracts also exhibit an actin-mediated gelation and

contraction on longer time scales. At t = 0 the extracts are loaded into the observation

chambers and warmed to room temperature, initiating temperature-dependent

polymerization and crosslinking of the cytoskeletal filaments; samples are observed at 10,

30, 60, and 100 minutes after warming. For the native extracts, there is little difference

in the samples for the first three time points, indicating that for waiting times up to one

hour, there is little macroscopic contraction. However, for a waiting time of 100 minutes,

the sample has phase separated into a dark actin-dense gel located in the center of the

sample chamber, which is surrounded by a lighter viscous fluid. To investigate the

molecular basis of this contraction, and to explore the experimental challenges of

handling these gels, we compare the macroscopic contraction of native interphase

extracts to extracts that have been treated with various pharmacological treatments to

dissemble or stabilize of the various cytoskeletal filaments. We use 30 µM latrunculin B

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(+latB) to remove the filamentous actin network, 10 µM nocodazole (+nocod) to

depolymerize the microtubule network, 10 µM taxol (+Tx)or 500 nM taxol (+dTx) to

promote formation of stable microtubules, and 10 µM phalloidin (+Ph) to stabilize F-

actin. At waiting times of 10 and 30 minutes, none of the samples shows macroscopic

contraction. By 60 minutes waiting time, the extract that has been treated with 10 µM

phalloidin shows significant contraction, while the remaining samples are unchanged.

At 100 minutes waiting time, all samples, with the exception of the extract treated with

latrunculin B, have contracted. Disassembling the actin network by addition of

latrunculin B prevents any contraction, while the addition of phalloidin to stabilize actin

filaments appears to accelerate the effect. Although we do observe significant

macroscopic differences between in the extract that has been treated with nocodazole and

the native extract, we do see evidence of an accelerated contraction on very small length

scales using video microscopy. At waiting times of roughly 30 minutes, we observe an

increase in the non-uniform cytoplasmic flow that is characteristic of the onset of

contraction. We observe similar flow patterns in the extract that has been treated with

phalloidin at waiting times of approximately 10 minutes, while the remaining samples are

unaffected for waiting times up to one hour. This suggests that the microtubule network

may serve to resist contraction either by locally withstanding the stress exerted on the

cytoplasm by the actin network, or because actin-microtubule interactions compete with

the actomyosin interactions that drive contraction. In mitotic extracts, the microtubule

networks are locally organized into asters and spindles, and do not form the space-

spanning three-dimensional networks found in interphase gels; this local rearrangement

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of the microtubules may prevent them from resisting the actin-based contraction, leading

to the fast contractions that are typically observed.

7.4.2 Macroscopic Rheological Measurements

To further investigate the temperature-induced gelation of the cytoplasmic

extracts, we use a strain-controlled macroscopic rheometer to probe the linear and non-

linear viscoelastic response. The time-evolution of the viscoelastic response of the native

cytoplasmic extracts is shown in Figure 7.1, with measurements taken every 2 to 3

minutes at a fixed frequency ω = 1 rad/sec and strain γ = 0.05. Each curve represents an

independent measurement, and is shown to demonstrate the variation in response. In all

cases, the extracts form weak viscoelastic solids, with elastic moduli G′ in the range of 2

to 10 Pa, and viscous moduli G″ in the range of 1 to 5 Pa. By comparison, for a purified

sample of 25 µM entangled actin, the frequency plateau value of the elastic modulus is

approximately 0.1 Pa38. For extracts that are harvested on the same day and are pooled

before being aliquotted and frozen, we measure smaller differences in the modulus, with

variations in both G′ and G″ of roughly a factor of two. This suggests that some of the

variation in mechanical response is due to differences in composition from harvest to

harvest. The onset of gelation, defined as the time at which the sample produces a

measurable stress with γ = 0.05, occurs at 15 to 50 minutes.

For long waiting times, the moduli reach an approximate steady-state value, and

we are able to make frequency-dependent measurements at a fixed strain of 0.05, as

shown in Figure 7.2. Each curve represents an independent measurement, and shows the

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variation in response. In all cases, the elastic modulus dominates the loss modulus by a

factor of roughly two, and both G′ and G″ show a similar weak dependence on frequency.

Unlike strongly crosslinked gels, there is a significant loss modulus at all frequencies,

suggesting a broad spectrum of relaxation processes.

Figure 7.1. The time-evolution of the viscoelastic response of the native cytoplasmic

extracts, with measurements taken every 2 to 3 minutes at a fixed frequency of 1 rad/sec

and strain of 0.05. Each curve represents an independent measurement, and shows the

variation in response. In all cases, the extracts form weak viscoelastic solids, with elastic

moduli G′ (left, solid symbols) in the range of 1-10 Pa, and viscous moduli G″ (right,

open symbols) in the range of 0.5 – 5 Pa.

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Figure 7.2. The frequency-dependence of the viscoelastic response of the native

cytoplasmic extracts, at a strain of 0.05. Each curve represents an independent

measurement, and shows the variation in response. In all cases, the elastic modulus

dominates the loss modulus by a factor of roughly two. Both G′ (left, solid symbols) and

G″ (right, open symbols) show a weak dependence on frequency. The dotted line has a

slope of 0.15.

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We investigate the non-linear rheological response by measuring stress, G′ and G″

as a function of strain γ at a fixed frequency ω = 1 rad/sec, as shown with a representative

data set in Figure 7.3. The linear regime, in which stress is linearly dependent on strain,

is small, with strain softening occurring for γ > 0.01, and G′ dominates until very large

strains. At the highest strains, G″ dominates, and we estimate the viscosity η = ω−1G″ ~

40 mPa-sec.

In order to reveal the microscopic origins of this viscoelastic response, we use a

variety of pharmacological and immunological disruption techniques to alter the

cytoplasmic structure, and measure the change in rheological response. To probe the role

of actin, we use extracts treated with 30 µM latrunculin B. These samples are extremely

weak and never develop a measurable elastic modulus, up to γ ~ 1. To study the

contribution of the microtubule network, we use extracts treated with 10 µM nocodazole.

These exhibit weak viscoelastic behavior, with moduli at the limits of our ability to

measure with this experimental apparatus, in the range of 0.1 – 0.5 Pa or smaller. To

explore the role of the intermediate filament cytokeratin, we add the anti-cytokeratin

antibody C11 to a final concentration of 1 mg/mL. We were unable to obtain highly

concentrated antibody solutions, so we introduce a rather large dilution (1:8) with

addition of C11. We compare the rheological response of extracts containing 1 mg/mL

C11 with extracts containing the non-specific antibody IgG at 1 mg/mL, or extracts

containing an equivalent amount of pure PBS. The dilution is responsible for an overall

reduction in elastic response, but we measure no specific reduction due to the removal of

the cytokeratin network within the repeatability of the experiment.

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Figure 7.3. (left) Representative data showing the strain-dependence of bulk cytoplasm,

obtained with a constant frequency of 1 rad/sec. The linear regime is small, with strain

softening occurring for γ > 0.01. G′ dominates until very large strains. At the highest

strains, G″ dominates, with a viscosity η = ωG″ ~ 40 mPa-sec.

214

In addition to adding drugs to disassemble the cytoskeleton, we can also

selectively stabilize the actin or microtubule networks with phalloidin or taxol

respectively. Without taxol, microtubules undergo dynamic instability, an unusual

polymerization mechanism where microtubules alternate between periods of prolonged

elongation and rapid catastrophic disassembly45. Taxol prevents this catastrophe, but

allows additional monomers to be added, stimulating the growth of stable microtubules.

At low concentrations of taxol, small numbers of long microtubules form, while higher

concentrations of taxol stimulate the growth of many shorter filaments. We have used

concentrations of 10 mM and 500 nM to explore the effect of taxol concentration on

microtubule length and on the rheological response. The time-evolution of the elastic

(solid symbols) and viscous (open symbols) moduli for phalloidin- and taxol-stabilized

extracts are shown in Figure 7.4, with several independent measurements shown for each

condition to indicate the variability in the response. We observe similar responses for

extracts stabilized with either phalloidin (Figure 7.4(a)) or 10 mM taxol (Figure 7.4(b)).

Both form a soft viscoelastic solid with moduli similar to what was measured for the

untreated native extracts. The onset of gelation occurs at 20-35 minutes in both cases,

and the moduli fail to reach a steady-state value within the experimental observation

time. In both cases we observe a decrease in the modulus at long waiting times, which

perhaps signals the onset of contraction which may cause the gel to disengage from the

cone and plate fixture used to measure the rheological response. In all cases the critical

strain γc, the largest strain in the linear regime, is roughly 0.01. For the extracts treated

with 500 nM taxol (Figure 7.4(c)), the onset of gelation is more rapid, occurring within

10-20 minutes and in each case, the moduli do reach steady state.

215

Figure 7.4 The storage (solid symbols) and loss (open symbols) moduli as a function of

time after warming for extracts treated with A) 10 µM phalloidin, B) 10 µM taxol, C) 500

nM taxol. In each case, the extracts form weak viscoelastic solids, similar to the

untreated native cytoplasmic gels. The time dependence of G′ and G″ is similar, and

G″/ G′ is roughly 0.5.

A

C

B

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The overall value of the modulus, as well as the frequency- and strain-dependence of the

viscoelastic response is similar to the native state.

Our mechanical measurements demonstrate that F-actin plays a crucial role in the

mechanical response of the cytoplasm. When the actin network is disassembled by

addition of Latrunculin B, we observe no measurable elastic response. In addition to its

important role in the organization and active transport mechanisms in the cell, the

microtubule lattice also plays an important mechanical role. When the microtubule

network is disrupted by addition of nocodazole, the resultant cytoplasm is substantially

weaker than the native state. Previous measurements of F-actin-microtubule interactions

in Xenopus egg extracts have shown that microtubules, in conjunction with the motor

protein cytoplasmic dynein, form physical contact with and exert force on F-actin,

promoting the assembly of actin filaments into larger bundles42,43. As a result, the

microtubule lattice may impact mechanical response directly by forming an independent

stress-bearing network, with large bending modulus due to the extremely long persistence

length of microtubules, of order 1 mm, or may affect the mechanical properties of the

cytoplasm indirectly by remodeling the actin filaments into bundles that are better able to

resist deformation43.

Our macroscopic rheological measurements indicate that both the actin and

microtubule networks make important contributions to the elastic response of the

cytoplasm; however, we observe no change in viscoelastic response upon the addition of

the monoclonal anti-cytokeratin antibody C11, suggesting a minor role for the cytokeratin

filament system in the mechanical properties of isolated cytoplasm. Intermediate

filaments have been reported to contribute mechanical strength to living cells and are

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present in significant quantities in cells that undergo large stresses, including nerve cell

axons, muscle cells, and epithelial cells1. In vivo, intermediate filaments are often

anchored to the plasma membrane at cell-cell junctions and are found in the other

membrane-associated structures such as the nuclear lamina1,1 ;our measurements suggest

that their mechanical strength may be compromised in bulk cytoplasm.

7.4.3 Imaging Cytoskeletal Structures:

Our macroscopic rheology measurements indicate that both the actin and

microtubule filament systems play an important role in mechanical strength of the

cytoplasm. To further investigate their interactions and structure, we image fluorescently

labeled actin and tubulin in rapidly frozen extract samples using confocal microscopy.

Representative images are shown in Figure 7.5 with images acquired after 0-, 10-, 20-,

30-, and 40-minute incubation periods at room temperature; the magnification is the same

for each image, and the scale bar in the uppermost left image represents 25 µm. After a

10-minute incubation, a dense, heterogeneous actin network has formed, but there is little

visible structure among the microtubules. For times greater than 10 minutes, the network

undergoes dramatic remodeling as the actin filaments are bundled into thick fibers. By t

= 40 minutes, the mesh size of the actin network has increased significantly and large

bundles are visible, while the microtubules network is now also clearly visible and is

closely associated with actin. In Figure 7.6, we show enlarged images of the actin

1 Herrmann, H., U. Aebi (2000) Current Opinion in Cell Biology 12: 79

218

network after 10- and 40-minute incubations; the scale bar represents 5 µm. There is a

significant increase in the mesh size at longer times.

Our rheological measurements indicate that in the absence of microtubules, the

cytoplasm is quite weak, with an elastic modulus of 0.5 Pa or smaller. To investigate the

structural changes that cause this mechanical difference, we image the actin network of

cytoplasmic extracts that have been treated with 10 µM nocodazole and incubated at

room temperature for 45 minutes, as shown in Figure 7.7. Unlike the native extracts,

which are remodeled into bundled networks by t = 40 minutes, the actin filaments in the

nocodazole-treated samples have not been remodeled and instead resemble the thin and

dispersed native-state networks observed immediately after the untreated gels are warmed

to room temperature, as shown in Figure 8a. The disruption of the regular mesh work

observed in the native interphase gels is consistent with the notion of microtubule-

dependent remodeling of the actin lattice43. Additionally, in some areas of the sample,

local clusters of actin with tenuous connections between aggregates are observed, as

shown in Figure 7.7b. The inability of the actin network to form stiff bundles, and the

formation of disconnected aggregates is consistent with the reduced elastic response in

the nocodazole-treated egg extracts.

To better understand the role of filament stabilizers on mechanical response, we

investigate the changes in cytoskeletal organization upon addition of 10 µM phalloidin or

10 µM taxol, which stabilize actin filaments or microtubules respectively. In both cases,

actin-remodeling occurs and the local mesh size of the actin network increases, as shown

in the fluorescent images of the actin cytoskeleton in Figure 7.8. We observe subtle

differences in the long-range organization of the taxol-stabilized extracts (Figure 7.8b),

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with the formation of long continuous bundles of actin that extend for 40 µm or more.

However, in both cases, the gross network morphology is similar to that of the native

state gels. Small changes in network structure as compared to the native extracts are

consistent with the negligible changes in elastic response upon addition of stabilizing

proteins as measured with macroscopic rheology.

220

Figure 7.5. Images of the microtubule and actin cytoskeletal filaments obtained with

confocal microscopy after incubations at room temperature of 0-, 10-, 20-, 30-, 40-minute

incubations at room temperature. By t = 10 minutes, a dense, heterogeneous actin

network has formed. By t = 40 minutes, the actin filaments have bundled into thick

fibers and the average mesh size has increased significantly. The microtubules are

unstructured at early times, but at later times, a network is visible and is closely

associated with actin.

221

222

Figure 7.6 A magnified view of the actin network after (a) a 10-minute and (b) 40-

minute incubation. After 10 minutes, dense and heterogeneous network has formed.

After 40 minutes, the acrin filaments have been remodeled into a bundled network with

mesh size of approximately 5 µm. In each image the scale bar is 5 µm long.

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Figure 7.7. Images of the actin network in an extract that has been treated with 10 µM

nocodazole, after a 45 minute incubation. The bundles observed in the native state gels

are not present, and the network consists of a dense heterogeneous network with mesh

size of roughly 1 micron, much smaller than observed in the native gels, with some

disconnected local aggregates of actin.

224

Figure 7.8. Images of the actin network in extracts that have been treated with (a) 10

µM phalloidin or (b) 10 µM taxol. In both cases the gross network morphology closely

resembles that of the untreated native extracts.

225

7.4.4 Measuring mechanical response on micron length scales using multiple particle

tracking

In order to further explore the mechanical microenvironments of the cytoplasmic

extracts, we embedded 1-µm colloidal particles that have been coated with poly(ethylene

glycol) to reduce the non-specific adsorption of proteins into the extracts, and record their

thermal motions with video microscopy. We measure the mean-squared displacement

(MSD) as a function of lag time, and interpret this quantity in terms of local viscoelastic

response. We record particle dynamics for native interphase extracts, as well as for

particles that are embedded in extracts that have been treated with 30 µM latrunculin B

(+latB), 10 µM nocodazole (+nocod), 10 µM taxol (+Tx), 500 nM taxol (+dTx), and 10

µM phalloidin (+ph), and calculate the two-dimensional ensemble-averaged MSD of

particles after a 10-minute incubation and a 30-minute incubation, as shown in Figure

7.9. For purely diffusive motion, <∆r2(τ)> should evolve linearly with lag time τ, and

with a slope of one on a log-log plot (solid line). After 10 minutes, our data evolve with

a slightly smaller power law dependence on lag time, with a slope closer to 0.85 on this

log-log plot (dotted line), indicating that the material is not a pure fluid on these length

scales. However, the slope is only slightly reduced from one and the response is still

dominated by the local viscosity. If we use the simple Stokes-Einstein relation

22 3 ( )Bk T r aη τ π τ= ∆ , where η is the viscosity, kBT is the thermal energy, and a is

the particle radius, to get a rough estimate of the viscosity , we measure the η ~ 10-30

mPa-sec.

226

Figure 7.9. Ensemble-averaged MSDs of particles moving in native state extract as well

as extracts that have been treated with 30 µM latrunculin B (+latB), 10 µM nocodazole

(+nocod), 10 µM taxol (+Tx), 500 nM taxol (+dTx), after (A) a 10 minute or (B) 30

minute incubation at room temperature. (A) After 10 minutes, our data evolve slightly

subdiffusively with lag time. The solid line represents a slope of 1, the dotted line a slope

of 0.85. The response is dominated by the local viscosity, in the range of 10-30 mPa-sec.

We measure no significant change in particle dynamics upon disassembly of either the

actin or microtubule networks, or the addition of taxol to stabilize the microtubule.

Microscopic contraction of the actin network prevents measurements of particle

dynamics upon addition of phalloidin, and causes a slight upturn in the data for the

+nocod case at the longest lag times. (B) After 30 minutes, the extracts are still

dominated by viscous relaxation, with viscosity in the range of 10-30 mPa-sec.

Microscopic contraction of the actin network prevents measurements of particle

dynamics upon addition of phalloidin, or nocodazole.

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We measure no significant change in particle dynamics upon disassembly of either the

actin or microtubule networks, or the addition of taxol to stabilize the microtubules,

suggesting that the filament structures are not playing a large role in the viscous response

at micron length scales. Microscopic contraction of the actin network prevents

measurements of particle dynamics upon addition of phalloidin, and causes a slight

upturn in the data for the extracts treated with nocodazole at the longest lag times. After

30 minutes, the extracts are still dominated by viscous relaxation, with viscosity in the

same range of 10-30 mPa-sec. Microscopic contraction of the actin network prevents

measurements of particle dynamics upon addition of phalloidin, or nocodazole.

The viscosity we measure is similar to, although slightly larger than previous

measurements of the cytosol, the background fluid of globular proteins that surrounds the

cytoskeleton1. Our confocal images indicate that a dense, but heterogeneous actin

network forms within the first 10 minutes of warming to room temperature, and is then

remodeled into thick filaments within 30 to 40 minutes of warming. Surprisingly, there is

no significant change between the particle dynamics recorded after a 10-minute

incubation and a 30-minute incubation, and no significant difference in particle

movements with either the actin or microtubule networks removed. We do measure

slightly subdiffusive particle dynamics, indicating the presence of an elastic material or

local microstructure that provide local resistance for tracer movement, but we are unable

to identify what structures give rise to this resistance. Previous measurements of living

cells also indicated the heterogeneous nature of cellular cytoplasm and the presence of

1 Luby-Phelps, K. (2000) International Review of Cytology 192: 189

228

local clusters of densely packed or strongly crosslinked filaments, separated by very soft

sol-gel-like region13.

7.5 Summary

The results of this study provide an initial characterization of isolated bulk

cytoplasm using the model system of Xenopus-derived cytoplasmic extracts. At

millimeter length scales, we have demonstrated the mechanical interplay between the

actin and microtubule filament systems by measuring the time-evolution of the

viscoelastic response as well as the frequency- and strain-dependence of the elastic and

viscous moduli. Native interphase cytoplasm is a soft viscoelastic solid with elastic

modulus in the range or 2-10 Pa, and a viscous modulus of 1-5 Pa. The disassembly of

actin filaments prevents the formation of an elastic gel, and the disruption of the

microtubule lattice significantly weakens the elastic response, and accelerates the

contraction that accompanies actin-based gelation. We do not measure a contribution

from the cytokeratin network to cell viscoelasticity. At micron sized length scales we

measure a mostly viscous response with a local viscosity in the range of 10-30 mPa-sec,

consistent with measurements of bulk cytoplasm under high strain. Within an hour of

warming to room temperature, the actin network is remodeled from a dense network into

a tightly bundled array with mesh size greater than several microns, dramatically

changing the long wavelength elastic properties of the material without modifying the

ability of small macromolecules to diffuse.

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This Xenopus-based cell extract system holds promise as an excellent model

system for the study not only of cytoskeletal structure 39,40,41, but cytoplasmic mechanics.

In addition to investigations of bulk properties of isolated cytoplasm, it is also possible to

investigate the physical mechanisms that guide such cellular processes as active transport

or cell division in physiologically relevant conditions.

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Appendix: Experimental Techniques

Here we discuss in detail the various experimental techniques used in our lab:

Dynamic Light Scattering, Diffusing Wave Spectroscopy, and Video-based particle

tracking methods. For each technique, we use embedded colloidal particles that are

purchased commercially from Bangs Laboratories Inc. (Fishers, IN), Molecular Probes

(Eugene, OR), or IDC, Interfacial Dynamics Corporation (Portland, OR). A variety of

surface chemistries that include surface-bound carboxyl, amine, or sulfate groups, sizes

that range from approximately 20 nm to several microns, and different fluorescent dyes

are available.

A.1 Dynamic Light Scattering

The typical dynamic light scattering apparatus includes a light source, a goniometer that

contains the sample and defines the scattering geometry, a detector, and a digital

correlator that calculates the intensity autocorrelation function in real time, as shown in

Figure A.11,2. The common light source is a continuous wave laser, a coherent,

monochromatic source of light with an emission wavelength in the range of 400-700 nm.

For single light scattering measurements, our lab employs a Coherent Innova 300-series

Argon-Ion Laser (Santa Clara, CA), which is operated at a wavelength of 514.5 nm; the

laser power at the sample is typically in the range of 50-300 mW. The operational

1 Berne B.J., Pecora R. Dynamic Light Scattering with applications to chemistry, biology, and physics. Mineola: Dover (2000) 2 Johnson C.S., Gabriel DA Laser Light Scattering. New York: Dover (1995)

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wavelength is not critical, although the absorbance lines for a sample are avoided to

prevent local heating, convection, and thermal lensing effects.

Figure A.1. A standard dynamic light scattering setup

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The excitation wavelengths for fluorescent samples are also rejected. In cases where the

excitation spectrum is wide, and it is not possible to eliminate fluorescence, filters are

used to prevent any small emitted signal from reaching the detector. Plane-polarized

light is required, as fluctuations in polarization give rise to undesired fluctuations in the

intensity of the scattered light. To reduce mechanical vibrations, the experiment is often

placed on an air-cushioned optical table, and to prevent relative movement of the laser

and the sample, all optical elements are placed on the same plate.

The goniometer holds the sample and defines the scattering geometry. Several

commercial goniometers are available; our lab uses models manufactured by Brookhaven

Instruments (Holtzville, NY) and ALV (Langen, Germany). An entrance lens is used to

focus the laser to a beam diameter of 50-100 microns, which increases the intensity of

each coherence area, or speckle. Sample cells made from optical quality glass are used to

reduce scattering and unwanted reflections from irregularities on the surface. Cylindrical

vials are used for measurements at multiple scattering angles and are positioned such that

the laser beam passes through the center of the vial; square cells are also useful for

measurements at a fixed scattering angle of 90 degrees. Often the vial is placed in a vat

filled with a fluid such as decalin or toluene with an index of refraction nearly equal to

that of glass, in order to reduce the refraction and scattering at the vial surface. A

window along the circumference of the vat allows the scattered light to reach the

detector, which is placed on an arm that rotates about the center of the vial and collects

the light at various angles. Typically, scattering angles ranging from 15 to 150 degrees

are accessible, corresponding to q-vectors in the range of approximately 4-30 µm-1 (using

Eqn. 3.46). At smaller angles, flare or scattering from large dust particles can dominate

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the signal and prevent good measurements of the particle mean-squared displacements

(MSD). Using multiply filtered solutions may reduce the amount of dust present in the

sample.

Detection optics, which include a series of lenses and pinholes or a fiber optic

cable, define the scattering volume and collect the scattered light. A bright speckle

occurs when light interferes constructively at the detector and the angular extent of a each

speckle is given by: ∆θ = λ/l, where λ is the wavelength of the incoming light and l is the

size of the beam in the scattering volume. For a beam of 50 microns in diameter that

passes through a vial of 5 mm in diameter, the speckle size will be roughly 25 microns in

width and 2.5 mm in height at a distance of 0.5 m from the center of the beam. The

detector is typically positioned at a specific distance from the scattering volume to ensure

that the collection area is limited to 1-2 speckles. When larger numbers of speckles are

collected with traditional pinhole optics systems, their fluctuations tend to cancel and

reduce the overall signal; however, this effect is not observed in fiber coupled systems3.

The detector typically consists of a photomultiplier tube (PMT), which is mounted on the

goniometer arm and collects the incoming photons at a specific angle. Within the PMT,

each photon strikes the cathode and emits an electron that is accelerated and collides with

a dynode, which in turn emits several electrons. The current passes through a series of

additional dynodes, amplifying a single electron roughly 106 times1. Once one photon

strikes the cathode of the PMT, there is a time delay before another photon may be

detected. This “dead time” is due to the electron transit time and is typically in the range

3 Gisler T., Ruger H., Egelhaaf S.U., Tschumi J., Schurtenberger P., Ricka J. (1995) Applied Optics 34:3546

234

of tens of nanoseconds, setting a lower limit on the temporal resolution of this device.

For most DLS measurements, this transit time is much shorter than any time scale of

interest.

The amplified signal from the PMT is passed to a pulse amplifier discriminator

circuit (PAD) that converts the small incoming voltage pulses into standard logic pulses

of defined duration and height, which are then passed to the correlator. Typically PADs

that are designed specifically for spectroscopy applications are used. The amplifier

discriminator sets a threshold level for the incoming pulses in order to reject the low

voltage signals that arise from spurious electrons, which are inadvertently amplified

through the cascade circuit. In addition to these rogue electrons, positive ions are

occasionally generated at some point in the electron cascade and return to the cathode or

an early dynode to initiate a second cascade. In this case, a second pulse follows the first

by several hundred nanoseconds and results in a peak in the correlation function at short

delay times. The shortest delay times used experimentally are usually in the microsecond

range, and this “afterpulsing” peak is not observed.

In cases where the short time dynamics are of interest, the scattered light is directed by a

beam splitter onto two independent PMTs, which are processed by two amplifier

discriminator circuits and the output of both is passed to the correlator. A pseudo-cross-

correlated signal is then obtained, <IA(t)IB(t+τ)> where the indices A and B distinguish the

signals received by the two PMTs. The cross-correlation also reduces dead time effects4.

4 Weitz D.A, D.J. Pine. Diffusing-wave spectroscopy. Appearing in Dynamic Light Scattering. W. Brown, ed .Oxford: Oxford University Press (1993)

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After processing by the amplifier-discriminator, the signal is then transmitted to a

digital correlator that calculates the intensity autocorrelation function in real time. The

correlator is typically available as a PCI or ISA compatible card with a software

interface. A range of delay times is available, and the shortest time is approximately 10 -

25 nanoseconds. Through the software interface, the number of channels, or delay times

that are calculated, is selected and the spacing of the channels is also specified. Often the

last few channels are shifted out to very long times to allow an independent measure of

the baseline intensity, which is used to normalize the correlation function.

In order to calculate the mean squared displacement of the particles reliably, the intensity

auto correlation function must be calculated with good statistical accuracy. There are two

issues that reduce the signal to noise ratio: the finite intensity of the incoming light, and

the limited duration of the experiment. The output signal from the PAD is typically in

the range of one hundred thousand of photons per second, expressed as a count rate of

100 kHz. The photon counting statistics are Poisson whereby the errors are given

roughly by N1/2, where N is the number of independent measurements. For a count rate

of 100 kHz, the counting error would be roughly 0.3%; however, for a count rate of only

100 Hz, the counting error would be much larger, of order 10%. Additionally, for such

low count rates the amount of noise present in the data becomes significant. The noise

level is primarily due to the “dark current” of the PMT and is roughly 50 Hz, although the

level varies depending on the details of the detector. This dark current arises from

thermally excited electrons that produce an anode current even in the absence of any

incoming light. In rare cases where the signal is so small that the dark current noise is

appreciable, the PMT may be cooled to reduce the contribution of these thermal

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electrons. Alternatively, when the scattering signal is very large, some attenuation may

be required; in general, count rates of greater than 1 MHz cause a nonlinear response, and

count rates greater than 20 MHz damage the photosensitive PMT.

The second consideration in the statistical accuracy of the correlation function is

the duration of the experiment. For simple fluids, the field correlation function, given in

Equation. 3.45 can be re-written as g1(τ)=exp[-q2Dτ] by substituting <∆r2(τ)>=6Dτ. The

decay time is thus given by 1/q2D. Although viscoelastic materials will display a

different dependence on τ, an estimate of the decay time can similarly obtained. The

square root of the ratio of the decay time to the duration of the experiment gives a rough

estimate of the statistical uncertainty in the MSD. For errors to be of order 3%, the

duration must be is at least one thousand times longer than any decay time, a criterion

that is easy to meet at large angles but more challenging at smaller angles due to the q-2

dependence on wave-vector. We collect data for roughly an hour at each angle to ensure

sufficient statistical accuracy. Long experiments are not possible in time-evolving

systems that may sediment or undergo chemical reactions during the course of the

measurement. In this case, many consecutive short runs may be averaged together to

obtain the MSD. Another disadvantage of long collection times is the increased

likelihood of measuring contributions from stray dust particles or other contaminants that

are present in solution; careful sample preparation will reduce this possibility.

For DLS measurements, only samples that are nearly transparent may be used in

order to insure that each incoming photon is scattered only once before reaching the

detector. For fluids or low contrast polymer solutions, a small amount of colloidal

particles are added to a final volume fraction of 10-5 – 10-6. Typically, the scattering

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signal from the colloidal particles is significantly larger than the scattering from the

solvent. For cloudy and opaque samples, single light scattering measurements are not

possible, and the Diffusing Wave Spectroscopy (DWS) method should be used.

A.2 Diffusing Wave Spectroscopy

The Diffusing Wave Spectroscopy (DWS) apparatus is similar to that of the single

light scattering experiment, and consists of a laser source, a simple optical train and

sample holder, a detector, and digital correlator5. The optical train allows for two

boundary conditions, in which either a collimated or focused beam impinges on the

sample cell. There are two experimental geometries: forward and backscattering. Our lab

typically employs the forward scattering geometry, as shown in Figure A.2. Higher laser

powers are required for DWS measurements than for single scattering experiments and

the laser power at the sample is roughly 100mW. Additionally, the coherence length of

the laser is critical in DWS and must be larger than the longest paths of the scattered light

through the sample, to insure that these paths are not discarded. The coherence length is

increased by the addition of an intercavity etalon in our Coherent Innova 300 series

Argon-ion laser (Santa Clara, CA) that insures operation in a single longitudinal mode.

Photons that are multiply scattered report an angular average and lose their q-

dependence; thus, a goniometer is not required. Multiply scattered light is depolarized

and signals of equal intensity are found with polarization parallel or perpendicular to the

incident beam. However, because each polarization is independent, this reduces the 5 Pine DJ, Weitz DA, Chaikin PM, Herbolzheimer E (1988) Physical Review Letters 60: 1134

238

signal to noise ratio and causes the intercept of the intensity correlation function to fall to

0.5. To prevent this, a polarization analyzer is placed before the detector to restrict the

scattered light to a single polarization.

Figure A.2. Diffusing Wave Spectroscopy setup.

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Often in DWS experiments, the timescales of interest are extremely short and are

comparable to the dead time of the PMT. In order to measure the dynamics of the sample

on these timescales, a pseudo-cross-correlated signal is used. Cross-correlation also

reduces after-pulsing effects, which occur at time lags of roughly 100 nanoseconds. For

DWS experiments, the volume fraction of the embedded colloidal spheres is large,

roughly 10-2, in order to insure that the transport mean path, l* is a small fraction of the

length of the sample chamber, l. The ratio of l/l* is typically greater than five. The mean

free path is determined by comparing the transmitted intensity of a reference sample,

whose transport mean path is known, to the experimental sample6.

A.3 Video Microscopy

Several of the microrheological techniques rely on accurately tracking the time

evolved positions of individual probe particles. Laser Deflection Particle Tracking 7,8

measures the motion of individual beads with subnanometer precision at frequencies up

to 50 kHz. This technique has excellent spatiotemporal resolution but is not designed to

observe the dynamics of large ensembles of beads easily. Our lab uses video microscopy

to follow the dynamics of ~100 beads simultaneously with a spatial resolution of 10 nm

and temporal resolution of ~30 Hz. The time and spatial resolution of a particle tracking

experiment with video microscopy relies on both the hardware used to capture images 6 Dasgupta BR, Tee S-Y, Crocker JC, Frisken BJ, Weitz DA (2001) Physical Review E 65:051505 7 Mason TG, Ganesan K, Van Zanten JH, Wirtz D, Kuo SC (1997) Physical Review Letters 79:3282 8 Schnurr B, Gittes F, MacKintosh FC, Schmidt CF (1997) Macromolecules 30: 7781

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and the software algorithms used to detect particle centers. In the simplest setup an

analog CCD camera (COHU, San Diego, CA) is attached to the side port of an optical

microscope (Leica, Bannockburn, IL) and 480 x 640 pixel images are captured with a S-

VHS video cassette recorder at a rate of 30 images per second. For most experiments, an

eight bit analog, uncooled monochrome CCD camera with a variable shutter speed is

sufficient. Monochrome CCD cameras are preferable to color cameras because they are

more sensitive to subtle brightness variations, and are significantly cheaper. A larger

dynamic range may be necessary for experiments with very dim or low contrast particles.

The videotape of particle dynamics is then digitized by a computer equipped with a frame

grabber card; digital images are analyzed to determine the particle positions in each

frame.

For one micron spheres, images are typically obtained with bright field

microscopy using a 40x air objective at a magnification of 250 nm/CCD pixel. In order

for particle tracking algorithms to accurately determine particle centers, the number of

pixels subtended by the image of each particle must be four or more CCD pixels. The

image magnifications necessary to achieve this diameter will change based on the size of

the embedded probe and if particles are imaged with bright field or fluorescence

microscopy.

Colloidal particles are commercially available in sizes between 20 nm and several

microns in diameter. The diffraction limited resolution, d, of a microscope is

0.61 . .λ= ∗d N A where N.A. is the numerical aperture of the objective used and λ is the

wavelength of light. For light microscopy, this resolution limit is a few hundred

nanometers. Colloidal spheres larger than this resolution limit can be observed directly

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with bright field microscopy. For smaller sizes, it is possible to use fluorescence

microscopy to image dyed colloids. In either method, the colloidal particle position is

observed as a circularly symmetric Gaussian image intensity profile centered at its

geometrical center. In general, particle tracking routines are able to locate particle

positions with subpixel accuracy by iterating an initial position, which is determined as

the brightest pixel in a local region, with the offsets in position, determined by calculating

the brightness-weighted centroid of the surrounding region, to refine the location of the

peak in brightness within a single pixel. At a magnification of 100x (10 pixels/micron),

subpixel accuracy corresponds to a spatial resolution of 10 nm, an order of magnitude

better than diffraction limited resolution! Because these particle tracking routines depend

on variations of brightness over distances larger than the diffraction limited resolution,

the numerical aperture of the objective does not affect the accuracy of detecting centers.

However, a lower numerical aperture increases the depth of focus and projection error of

particle motion perpendicular to the field of view increases.

The shutter speed of the CCD camera sets the exposure time, τe, of a single

image. If the exposure time is long enough to allow significant particle motion, the

microscope image of the particle will not be circularly symmetric which results in a

decreased ability of finding an accurate center. The shutter speed is set so a probe of

radius a embedded in a fluid of viscosity η diffuses less than the spatial resolution, 10

nm, in τe. We calculate the one dimensional root mean-squared displacement,

( )1/ 22 1/ 2( ) 2 10e ex Dτ τ< ∆ > = ≤ nm, where the diffusion constant, D, is determined by the

Einstein relation, / 6BD k T aπη= . For a typical experiment with a one micron probe

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embedded in a medium with a viscosity of 0.001 Pa*sec, a shutter speed of 1 msec is

sufficient.

Typically, a single CCD image is captured with alternate rows of pixels captured

every 1/60 of a second. This allows the temporal resolution of video microscopy to be

extended to 60 Hz by extracting the even and odd rows of the image to give two 240 x

640 images captured every 1/60 of a second9. Without extracting the separate fields, the

full 480 x 640 image is a superposition of two vertically interlaced half frame images, or

fields, taken 1/60 second apart. If the particle moves significantly during that time, it is

impossible to resolve an accurate particle center. A 100 nm particle in water (η=0.001

Pa*sec) imaged at a magnification of 10 pixels/µm (100x oil immersion objective) moves

100 nm, or one pixel, in 1/60 sec. Without a field analysis, the particle tracking

resolution decreases by an order of magnitude and the particle dynamics at the shortest

lag times near 1/30 sec, are seriously affected. Unless the particle motion between

captured fields is beneath the noise floor of the particle tracking routines, the short time

dynamics analyzed with an interlaced image will be affected by image averaging error.

The resolution with a field analysis is significantly improved for the horizontal direction,

but in the vertical direction the resolution is often adversely affected because of the

number of pixels subtended by the image in that direction is halved.

The CCD camera BNC output is directly connected to a television for viewing

and a high quality S-VHS cassette recorder (Sanyo, Chatsworth, CA) to record time

series of images at video rate. A computer equipped with a frame grabber card is

interfaced with the video cassette recorder to digitize images at rates accessible to the

9 Crocker JC, Grier DG (1996) Journal of Colloid and Interface Science 179:298

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available RAM of the computer by capturing images for a short time and then pausing the

movie while images are written to a file. In most cases, the limitation to writing images

directly to the hard drive at video rates is that it takes longer than 1/30 second to write an

image from RAM to the disk. With a special configuration of the hardware of a personal

computer and custom software, it is now possible to write full frame images directly to

the hard drive at video rates in standard movie file format10. Commercially available

software (Universal Imaging Corporation, Downingtown, PA; Scion, Frederick, MA) to

control frame grabbing hardware is also available.

Digital cameras (Hamamatsu, Bridgewater, NJ) now offer an alternative to traditional

CCD cameras and allow control over pixel resolution, integration time, size and capture

rate of the images. This enables us to optimize image quality for very dim particles or

achieve capture rates faster than a traditional CCD camera by decreasing resolution and

frame size. Digital movies are converted to a three-dimensional tiff stack of images and

software is used to locate colloidal features in each image. We use particle tracking

routines developed by John Crocker and David Grier9. An online tutorial of this method

with software routines written in IDL is maintained by John Crocker and Eric Weeks and

is available at http://glinda.lrsm.upenn.edu/~weeks/idl/. The particle tracking algorithms

are comprised of three steps: identification of appropriate parameters to detect the

position of desired particles appearing in a single image, using a macro to automate the

identification for every tiff image in a 3D tiff array and linking the positions found in

each frame to form particle trajectories. The particle identification software detects bright

‘spots’ against a dark, zero intensity background pixel map. A ‘spot’, or feature, is a

10 Keller M, Schilling J, Sackmann E (2001) Review of Scientific Instruments 72: 3626

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region of locally higher intensity with a circularly symmetric Gaussian brightness profile

across its diameter. Because our tracking programs are dependent on variations in

brightness; the image contrast should be maximized without saturating the image so there

is a unique maximum in the brightness profile across the bead image.

To identify features, the tiff image is smoothed using a spatial bandpass filter with

the lower bound of allowed wavelengths set to retain subtle variations in brightness

(usually one pixel) and the upper bound set to the average diameter of the features. After

smoothing the image, the average background intensity is subtracted so the final image is

a low intensity background with sharply peaked circular ‘spots’ where the original

particle images are. After locating the brightest pixels in a given region, a circular

Gaussian mask with a diameter greater than or equal to the upper bound of the spatial

bandpass is iterated around this guess to refine the located center of these circular peaks

in the image brightness. Given a smoothed image and diameter of a Gaussian mask, our

feature finding program will return a five dimensional array with the x-centriod, y-

centroid, integrated brightness, radius of gyration and eccentricity for each feature found.

To obtain subpixel accuracy, we choose the diameter of the mask to be slightly larger

than that subtended by the particle image. This ensures the algorithm is able to find the

unique maximum of the feature upon iteration.

This feature finding routine is very sensitive and will detect many possible

particles. To separate the actual particles from false identifications, we further

discriminate the found features based on their integrated brightness, eccentricity and

radius of gyration. Colloidal features will fall into a broad cluster around certain values

of radius of gyration and brightness. For any given image, false particle identifications

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will tend to lie outside this target cluster and can be clipped. Eccentricity, a geometrical

measure of how circular an ellipse is, ranges from zero for circles and one for lines.

Since we track circular features, we can set an upper bound on the eccentricity of 0.2 to

allow for slight deviations from circular symmetry. We find the parameters that

successfully eliminate rogue particle identifications for a single image and use identical

bounds for subsequent images. It is often useful to overlay the located particle positions

with the original microscope image to visually check particle identifications.

Once colloidal particles are located in a sequence of video images, particle

positions in each image are correlated with positions in later images to produce

trajectories. To track more than one particle, care is required to uniquely identify each

particle in each frame9. In practice, the typical distance a particle moves between images

must be significantly smaller than the typical interparticle spacing. Otherwise, particle

positions will be confused between snapshots. Particles that are a few hundred

nanometers in diameter typically diffuse a distance smaller than their diameters between

frames and, therefore, relatively high concentrations of particles can be used.

Because particles can move in and out of the focused field of view, it is useful to

include a memory function in the tracking algorithm that allows gaps in a trajectory

where a feature is not found. The last known locations of missing particles are retained

in case unassigned particles reappear sufficiently nearby to resume the trajectory. Using

a memory function allows the length of individual tracks to be maximized, therefore

greatly increasing the statistics for the mean-squared displacement (MSD). Trajectories

are identified for every feature in the field of view and saved as an array. In a typical

multiparticle tracking experiment, millions of positions are assigned to thousands of

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particle trajectories. We usually require 1000 independent events to calculate the MSD at

a given lag time. The number of events, or independent particle displacements, increases

with both the number of particles in a field of view and the length of particle tracks11.

For an experiment tracking forty beads with a diameter of one micron in a viscous

medium of 0.001 Pa*sec, around ten minutes of video is required to calculate the MSD to

lag times of one minute.

A.4 Obtaining G*(ω) from 2 ( )r t∆r

In all the methods discussed, we desire to obtain knowledge of the complex shear

modulus from the bead dynamics. In order to relate the mean-squared displacement of

embedded Brownian probes, 2 ( )r t∆r , to a frequency dependent complex modulus,

*( )G ω , using the generalized Stokes-Einstein relation, it is necessary to calculate the

numerical Laplace transform of 2 ( )r t∆r to obtain ( )G s% using Equation 3.37. We then

fit ( )G s% to a continuous functional form in the real variable s and substitute s=iω into

( )G s% to obtain *( ) '( ) "( )G G iGω ω ω= + . While direct, this method of transforming the

mean-squared displacement can introduce significant errors in the moduli obtained. The

numerical Laplace transform is typically implemented by selecting a frequency s,

multiplying 2 ( )r t∆r by a decaying exponential and integrating using the trapezoid rule.

While this method is very accurate within frequency extremes, it introduces errors near

11 Valentine MT, Kaplan PD, Thota D, Crocker JC, Gisler T, Prud'homme RK, Beck M, Weitz DA (2001) Physical Review E 64: 061506

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frequency extremes due to truncation of the data set. These errors are usually significant

for data within a decade of either extrema. Additionally, the analytic continuation of

( )G s% requires finding an appropriate functional form of discrete data. Without an accurate

functional form of the discrete data, significant errors can result in the elastic and loss

complex moduli.

To overcome these errors, Mason, et. al. developed a method to estimate the

transforms algebraically by using a local power law around a frequency of interest, ω,

and retaining the leading term7:

2 2 ( )( ) (1/ ) ( )r t r t α ωω ω∆ ≈ ∆r r

(A.1)

where 2 (1/ )r ω∆r is the magnitude of at t=1/ω and

2

1/

ln ( )( )

lnt

d r t

d tω

α ω

=

∆≡

r

(A.2)

is the power law exponent describing the logarithmic slope of 2 ( )r t∆r at t=1/ω. The

Fourier transform, 2 ( )r tℑ ∆r , of the power law is directly evaluated:

[ ]2 2 ( )( ) (1/ ) 1 ( )i r t r i α ωω ω α ω −ℑ ∆ ≈ ∆ Γ + (A.3)

where Γ is the gamma function. Using a local power law approximation implicitly

assumes that contributions to the transform integral from the behavior of < 2 ( )r t∆ > at

times different than 1/ω can be neglected. By substitution into the Fourier representation

of the GSER and the use of Euler’s equation, we obtain:

'( ) *( ) cos( ( ) / 2)

"( ) *( ) sin( ( ) / 2)

G G

G G

ω ω πα ω

ω ω πα ω

=

= (A.4)

where

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[ ]2*( )

(1/ ) 1 ( )Bk TG

a rω

π ω α ω≈

∆ Γ + (A.5)

These relations provide a direct physical intuition of how the elastic and loss moduli

depend on < 2 ( )r t∆ >. In a pure viscous medium, the sphere diffuses and 1α ≈ resulting in

a dominant loss modulus. Constrained in a pure elastic medium, the α approaches zero

and the elastic modulus dominates.

While this estimation method is convenient and intuitive to use, it can fail to give

an accurate estimation of the moduli when the mean-squared displacement is sharply

curved with a rapidly changing slope. These regions can be of particular interest because

such changes reflect significant relaxation times of the sample. To account for these

issues, modifications have been made to Equation 3.56 to better account for curvature in

the mean-squared displacement and give improved estimates of the moduli in these

regions6.