QUANTITATIVE LIGHT IMAGING OF

INTRACELLULAR TRANSPORT

BY

RU WANG

DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mechanical Engineering

in the Graduate College of the

University of Illinois at Urbana-Champaign, 2013

Urbana, Illinois

Doctoral Committee:

Professor Jimmy K Hsia, chair

Assistant Professor Gabriel Popescu, director of dissertation

Associate Professor Scott Paul Carney

Associate Professor Kimani C Toussaint

ii

Abstract

The interior of a living cell is a busy place. Just as understanding the flow of traffic is

essential for probing the economy of a major city, exploring the intracellular traffic patterns of

cells is fundamental to elucidating their activity. We examine intracellular traffic patterns using a

new application of spatial light interference microscopy (SLIM) (Chapter 1). We used this

quantitative phase imaging method to measure the dispersion relation, i.e. decay rate vs. spatial

mode, associated with mass transport in live cells. This approach, Dispersion-relation Phase

Spectroscopy (DPS), (Chapter 3), applies equally well to both discrete and continuous mass

distributions without the need for particle tracking. From the quadratic experimental curve specific

to diffusion, we extracted the diffusion coefficient as the only fitting parameter. The linear portion

of the dispersion relation reveals the deterministic component of the intracellular transport. Our

data show a universal behavior where the intracellular transport is diffusive at small scales and

deterministic at large scales. We further applied this method to studying transport in neurons

(Chapter 4). By modifying a traditional phase contrast microscope, we are able to use SLIM to

map the changes in index of refraction across the neuron and its extended processes. What we

found was that in dendrites and axons, the transport is mostly active, i.e., diffusion is subdominant.

Due to its ability to study specifically labeled structures, fluorescence microscopy is the

most widely used technique to study live cell dynamics and function. Fluorescence correlation

spectroscopy is an established method for studying molecular transport and diffusion coefficients

at a fixed spatial scale. We propose a new approach, dispersion-relation fluorescence spectroscopy

(DFS) (Chapter 5), to study the transport dynamics over a broad range of spatial and temporal

scales. The molecules of interest are labeled with a fluorophore whose motion gives rise to

spontaneous fluorescence intensity fluctuations that are analyzed to quantify the governing mass

iii

transport dynamics. These data are characterized by the effective dispersion relation. We report on

experiments demonstrating that DFS can distinguish diffusive from advection motion in a model

system, where we obtain quantitatively accurate values of both diffusivities and advection

velocities. Due to its spatially-resolved information, DFS can distinguish between directed and

diffusive transport in living cells. Our data indicate that the fluorescently labeled actin cytoskeleton

exhibits active transport motion along a direction parallel to the fibers and diffusive on the

perpendicular direction. And we further, for the first time, employed DFS in studying biological

structures, for example actin filaments and microtubules (Chapter 6). Our study suggested that

dispersion-relation enables to quantify both the spatial and temporal behavior of the transport

phenomenon of cytoskeleton with the aid of fluorescence tag. More importantly, in addition to

single cellular component specificity, multiple components can be studied simultaneously as long

as they are properly labeled. This ability makes the investigation of their interaction possible and

our results did show their strong interplay.

Apart from the major focus of this thesis, i.e. in-plane mass transport motion (Chapter 3-

6), we also took some efforts to study the other different motion, out-of-plane membrane

fluctuations of red blood cells (Chapter 2). We present optical measurements of nanoscale red

blood cell fluctuations obtained by highly sensitive quantitative phase imaging. These spatio-

temporal fluctuations are modeled in terms of the bulk viscoelastic response of the cell. Relating

the displacement distribution to the storage and loss moduli of the bulk has the advantage of

incorporating all geometric and cortical effects into a single effective medium behavior. The

results on normal cells indicate that the viscous modulus is much larger than the elastic one

throughout the entire frequency range covered by the measurement, indicating fluid behavior.

iv

Acknowledgements

During this long PhD journey I would sincerely thank my advisor Prof. Gabriel Popescu for his

hearted guidance and all-through support, without whom I can never have achievement today. He

is not only knowledgeable and talented but also patient and understanding. I appreciate the

opportunity he gave to allow me pursue my academic interest.

I then would like to thank all the people with whom I work, or used to work in Prof. Popescu’s

lab. They are great colleagues in work and warm friends in life.

Special thanks go to Profs. Jimmy K Hsia, Scott Paul Carney and Kimani C Toussaint for

supporting me as my Doctoral Committee and their help in many aspects to let me grow up to be

a qualified Ph.D.

v

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION OF QUANTITATIVE PHASE IMAGING (QPI) ................................................... 1

1.1 Overview of QPI ................................................................................................................................................. 1

1.2 Principle of spatial light interference microscopy (SLIM) .................................................................................. 2

CHAPTER 2. RED BLOOD CELLS VISCOELASTICITY MEASURED BY DIFFRACTION PHASE

MICROSCOPY (DPM) .............................................................................................................................................. 13

2.1 Introduction ....................................................................................................................................................... 13

2.2 Method and analysis .......................................................................................................................................... 14

2.3 Discussion ......................................................................................................................................................... 19

CHAPTER 3.DISPERSION-RELATION PHASE SPECTROSCOPY(DPS) ............................................................ 22

3.1 Introduction ....................................................................................................................................................... 22

3.2 Measurements ................................................................................................................................................... 23

3.3 Principle of dispersion-relation phase spectroscopy (DPS) ............................................................................... 25

3.4 DPS application in live cells ............................................................................................................................. 28

3.5 Discussion .......................................................................................................................................................... 33

CHAPTER 4. 1D NEURON TRANSPORT WITH DPS ............................................................................................ 35

4.1 Introduction ....................................................................................................................................................... 35

4.2 Measurements ................................................................................................................................................... 38

4.3 Analysis ............................................................................................................................................................. 41

4.4 Results ............................................................................................................................................................... 47

4.5 Discussion ......................................................................................................................................................... 53

CHAPTER 5. DISPERSION-RELATION FLUORESCENCE SPECTROSCOPY(DFS) ......................................... 55

5.1 Introduction ....................................................................................................................................................... 55

5.2 Principle and validation of DFS ........................................................................................................................ 56

5.3 Application of DFS in live cells ........................................................................................................................ 61

5.4 Discussion ......................................................................................................................................................... 65

CHAPTER 6. CYTOSKELETON FLUCTUATIONS WITH DFS ............................................................................ 67

6.1 Introduction ....................................................................................................................................................... 67

6.2 Results ............................................................................................................................................................... 69

CHAPTER 7. PHASE-RESOLVED LOW COHERENCE INTERFEROMETRY…………………………….…...82

7.1 Introduction ....................................................................................................................................................... 82

7.2 Experimental system ......................................................................................................................................... 83

7.3 Principle and Experimental validation .............................................................................................................. 86

7.4. Discussion ........................................................................................................................................................ 92

CHAPTER 8. FUTURE WORK ................................................................................................................................. 94

8.1 PHASE CORRELATION IMAGING (PCI) ..................................................................................................... 94

REFERENCE…………………………………………………………………………………………………………97

1

CHAPTER 1. Introduction of Quantitative Phase Imaging (QPI)1

1.1 Overview of QPI

Quantitative phase imaging (QPI) methods have emerged as a highly sensitive way to

quantify nanometer scale path length changes induced by a sample and thus become a growing

area to study live biology samples[1-5]. A large number of experimental setups have been

developed for QPI[6-10], however the range of QPI applications in biology has been largely

limited to red blood cell membrane fluctuations [11], or global cell parameters such as dry mass

[12], average refractive index [13], and statistical parameters of tissue slices[14]. The major reason

is the contrast in QPI images has always been limited by speckles resulting from the practice of

using highly coherent light sources such as lasers. The spatial non-uniformity caused by speckles

is due to random interference phenomenon caused by the coherent superposition of various fields

from the specimen and those scattered from, optical surfaces, imperfections or dirt [15]. Since this

superposition of fields is coherent only if the path length difference between the fields is less than

the coherence length (lc) of the light, it follows that if broadband light, with a shorter coherence

length is used the speckle will be reduced. Due to this, the image quality of laser based QPI

methods has never reached the level of white light techniques (lc ~ 1 µm) such as phase contrast

or DIC as discussed below.

To address this issue we have recently developed a new QPI method called Spatial Light

Interference Microscopy (SLIM) [16, 17]. SLIM combines two classical ideas in optics and

1 This chapter appeared in its entirety (with some modifications and additional material) in M.Mir, B. Bhaduri, R.

Wang, R. Zhu and G. Popescu,”Quantitative Phase Imaging”, Progress in Optics, Chennai, B.V.: (2012), pp. 133-217.

This material is reproduced with the permission of the publisher and is available using the ISBN: 978-0-444-59422-

8.

2

microscopy: Zernike’s phase contrast method [18] for using the intrinsic contrast of transparent

samples and Gabor’s holography to quantitatively retrieve the phase information [19]. SLIM thus

provides the spatial uniformity associated with white light methods and the stability associated

with common path interferometry. In fact, as described in greater detail below, the spatial and

temporal sensitivities of SLIM to optical path length changes have been measured to be 0.3 nm

and 0.03 nm respectively. In addition, due to the short coherence length of the illumination, SLIM

also provides excellent optical sectioning, enabling three dimensional tomography [20].

In the laser based methods the physical definition of the phase shifts that are measured is

relatively straightforward since the light source is highly monochromatic. However, for broadband

illumination the meaning of the phase that is measured must be considered carefully. It was

recently shown by Wolf [21] that if a broad band field is spatially coherent, the phase information

that is measured is that of a monochromatic field which oscillates at the average frequency of the

broadband spectrum. This concept is the key to interpreting the phase measured by SLIM. In this

section we will first discuss the physical principles behind broadband phase measurements using

SLIM, then the experimental implementation and finally various applications.

1.2 Principle of spatial light interference microscopy (SLIM)

The idea that any arbitrary image may be described as an interference phenomenon was first

proposed more than a century ago by Abbe in the context of microscopy: “The microscope image

is the interference effect of a diffraction phenomenon” [22]. This idea served as the basis for both

Zernike’s phase contrast [18] and is also the principle behind SLIM. The underlying concept here

is that under spatially coherent illumination the light passing through a sample may be thus

decomposed into its spatial average (unscattered component) and its spatially varying (scattered

component)

3

0 1

0 1

( ) ( ; )

0 1

( ; ) ( ) ( ; )

( ) ( ; )i i

U U U

U e U e

r

r r

r (1.1)

where r=(x, y). In the Fourier plane (back focal plane) of the objective lens these two components

are spatially separated, with the unscattered light being focused on-axis as shown Fig. 1.1. In the

spatial Fourier transform of the field U, ( ; )U q , it is apparent that average field U0 is proportional

to the DC component ( ; )U 0 . This is equivalent to saying that if the coherence area of the

illuminating field is larger than the field of view of the image, the average field may be written as,

0

1( , ) ( , )U U x y U x y dxdy

A (1.2)

Thus the final image may be regarded as the interference between this DC component and the

spatially varying component. Thus the final intensity that is measured may be written as:

2 2

0 1 0 1( , ) ( , ) 2 ( , ) cos ( , )I x y U U x y U U x y x y (1.3)

where Δϕ is the phase difference between the two components. Since for thin transparent samples

this phase difference is extremely small and since the Taylor expansion of the cosine term around

0 is quadratic, i.e. 2

cos( ) 12

the intensity distribution does not reveal much detail.

Zernike realized that the spatial decomposition of the field in the Fourier plane allows one to

modulate the phase and amplitude of the scattered and unscattered components relative to each

other. Thus he inserted a phase shifting material in the back focal plane that adds a π/2 shift (k=1

in Fig. 1.1) to the un-scattered light relative to the scattered light, essentially converting the cosine

to a sine which is rapidly varying around 0 ( sin( ) ). Thus Zernike coupled the phase

information into the intensity distribution and invented phase contrast microscopy. Phase contrast

(PC) has revolutionized live cell microscopy and is widely used today; however the quantitative

4

phase information is still lost in the final intensity measurement. SLIM extends Zernike’s idea to

provide this quantitative information.

As in PC microscopy, SLIM relies on the spatial decomposition of the image field into its

scattered and un-scattered components and the concept of image formation as the interference

between these two components. Thus in the space-frequency domain we may express the cross

spectral density as [23, 24]:

*

01 0 1( ; ) ( ) ( ; )W U U r r (1.4)

where the * denotes complex conjugation and the angular brackets indicate an ensemble average.

If the power spectrum 2

0( ) ( )S U has a mean frequency 0 , we may factorize the cross

spectral density as,

0[ ( ; )]

01 0 01 0( ; ) ( ; )i

W W e

r

r r (1.5)

From the Wiener–Kintchen theorem [24], the temporal cross correlation function is related to the

cross spectral density through a Fourier transform and can be expressed as

0[ ( ; )]

01 01( ; ) ( ; )i

e

r

r r

(1.6)

where 0 1( ) ( ) r r is the spatially varying phase difference. It is evident from Eq. 1.6 that

the phase may be retrieved by measuring the intensity at various time delays, . The retrieved

phase is equivalent to that of monochromatic light at frequency 0 . This can be understood by

calculating the auto-correlation function from the spectrum of the white-light source being used

(Fig 1.1c-d). It can be seen in the plot of the autocorrelation function in Fig. 1.1d that the white

light does indeed behave as a monochromatic field oscillating at a mean frequency of 0 .

5

Evidently, the coherence length is less than 2μm, which as expected is significantly shorter

compared to quasi-monochromatic light sources such as lasers and LEDs. However, as can be

seen, within this coherence length there are several full cycle modulations, in addition, the

envelope is still very flat near the central peak.

Figure 1.1. Imaging as an interference effect. (a) A simple schematic of a microscope is shown

where L1 is the objective lens which generates a Fourier transform of the image field at its back

focal plane. The un-scattered component of the field is focused on-axis and may be modulated by

the phase modulator (PM). The tube lens L2 performs an inverse Fourier transform, projecting the

image plane onto a CCD for measurement. (b) Spectrum of the white light emitted by a halogen

lamp source, with center wavelength of 531.9 nm. (c) Resampled spectrum with respect to

frequency. (d) Autocorrelation function (solid line) and its envelope (dotted line). The four circles

correspond to the phase shifts that are produced by the PM in SLIM.

6

When the delay between U0 and U1 is varied the interference is obtained simultaneously at

each pixel of the CCD, thus the CCD may be considered as an array of interferometers. The

average field, U0, is constant over the field of view and serves as a common reference for each

pixel. It is also important to note that U0 and U1 share a common optical path thus minimizing any

noise in the phase measurement due to vibrations. The intensity at the image plane may be

expressed as a function of the time delay as:

0 1 01 0( ; ) ( ) 2 ( ; ) cos ( )I I I r r r r (1.7)

In SLIM, to quantitatively retrieve the phase the time delay is varied to get phase delays of

–π, π/2, 0 and π/2 ( 0 / 2k k , k= 0, 1, 2, 3) as illustrated in Fig. 1.1d. An intensity map is

recorded at each delay and may be combined as:

( ;0) ( ; ) 2 (0) ( ) cos ( )I I r r r (1.8)

( ; ) ( ; ) 2 sin ( )2 2 2 2

I I

r r r (1.9)

For time delays around 0 that are comparable to the optical period can be assume to vary slowly

at each point as shown in Fig. 1.1d. Thus for cases where the relationship

(0) ( )2 2

holds true the spatially varying phase component may be

expressed as:

( ; / 2) ( ; / 2)( ) arg

( ;0) ( ; )

I I

I I

r rr

r r (1.10)

Letting 1 0( ) ( ) / ( )U U r r r the phase associated with the image field is determined as:

7

( )sin( ( ))( ) arg

1 ( )cos( ( ))

r rr

r r (1.11)

Thus by measuring 4 intensity maps the quantitative phase map may be uniquely determined.

Next we will discuss the experimental implementation of SLIM and its performance.

A schematic of the SLIM setup is shown in Fig. 1.2a. SLIM is designed as an add-on module to a

commercial phase contrast microscope (More details about the SLIM design and peripheral

accessories can be found in Appendix C). In order to match the illumination ring with the aperture

of the spatial light modulator (SLM), the intermediate image is relayed by a 4f system (L1 and

L2). The polarizer P ensures the SLM is operating in a phase modulation only mode. The lenseL3

and L4 form another 4f system. The SLM is placed in the Fourier plane of this system which is

conjugate to the back focal plane of the objective which contains the phase contrast ring. The active

pattern on the SLM is modulated to precisely match the size and position of the phase contrast ring

such that the phase delay between the scattered and unscattered components may be controlled as

discussed above.

To determine the relationship between the 8-bit VGA signal that is sent to the SLM and

the imparted phase delay, it is first necessary to calibrate the liquid crystal array as follows. The

SLM is first placed between two polarizers which are adjusted to be 45o to SLM axis such that it

operates in amplitude modulation mode. Once in this configuration the 8-bit grey scale signal sent

to the SLM is modulated from a value of 0 to 127 (the response from 128 to 255 is symmetric).

The intensity reflected by the SLM is then plotted vs. the grey scale value as shown in Fig. 1.2b.

The phase response is calculated from the amplitude response via a Hilbert transform (Fig. 1.2c).

From this phase response we may obtain the 3 phase shifts necessary for quantitative phase

8

reconstruction as shown in Fig. 1.2d, finally a quantitative phase map image may be determined

as described above.

Fig. 1.2e shows a quantitative phase measurement of a cultured hippocampal neuron, the

color indicates the optical path in nanometers at each pixel. The measured phase can be

approximated as

( , )

0 00

0

( , ) ( , , )

( , ) ( , )

h x y

x y k n x y z n dz

k n x y h x y

(1.12)

where, k0=2/, n(x,y,z)-n0 is the local refractive index contrast between the cell and the

surrounding culture medium, ( , )

00

1( , ) ( , , )

( , )

h x y

n x y n x y z n dzh x y

, the axially-averaged

refractive index contrast, h(x,y) the local thickness of the cell, and the mean wavelength of the

illumination light. The typical irradiance at the sample plane is ~1 nW/ m2. The exposure time is

typically 1-50 ms, which is 6-7 orders of magnitude less than that of confocal microscopy [25],

and thus there is very limited damage due to phototoxic effects. In the original SLIM system the

phase modulator has a maximum refresh rate of 60 Hz and the camera has a maximum acquisition

rate of 11 Hz, due to this the maximum rate for SLIM imaging was 2.7 Hz. Of course this is only

a practical limitation as both faster phase modulators and cameras are available commercially.

9

Figure 1.2 Experimental setup. (a) The SLIM module is attached to a commercial phase contrast

microscope (AxioObserver Z1, Zeiss). The first 4-f system (lenses L1 and L2) expands the field

of view to maintain the resolution of the microscope. The polarizer, P is used to align the

polarization of the field with the slow axis of the Spatial Light Modulator (SLM).Lens L3 projects

the back focal plane of the objective, containing the phase ring onto the SLM which is used to

impart phase shifts of 0, π/2, π and 3π/2 to the un-scattered light relative to the scattered light as

shown in the inset. Lens L4 then projects the image plane onto the CCD for measurement. (b)

Intensity modulation obtained by displaying different grayscale values on the SLM. (c) Phase

10

modulation vs. grayscale value obtained by a Hilbert transform on the data in b. (d) The 4 phase

rings and their corresponding images recorded by a CCD. (e) Reconstructed quantitative phase

image of a hippocampal neuron, the color bar indicates the optical path length in nanometers.

To quantify the spatiotemporal sensitivity of SLIM a series of 256 images with a field of

view of 10 x 10 µm2 were acquired with no sample in place. Fig. 1.3a shows the spatial and

temporal histograms associated with this data. The spatial and temporal sensitivities were

measured to be 0.28 nm and 0.029 nm respectively. Fig. 1.3b-c compares SLIM images with those

acquired using a diffraction phase microscope (DPM) [26] that was interfaced with the same

commercial microscope. The advantages provided by the broadband illumination are clear as the

SLIM background image has no structure or speckle as compared to those acquired by DPM.

11

Figure 1.3. SLIM sensitivity. (a) Spatial and temporal optical path length noise level, solid lines

indicate Gaussian fits. (b) Topographic noise in SLIM (c) Topographic noise in DPM a laser based

method. The color bar is in nanometers.

As in all QPI techniques the phase information measured by SLIM is proportional to the

refractive index times the thickness of the sample. Due to the coupling of these two variables the

natural choices for applying a QPI instrument are in situations where either the refractive index

(topography) or the thickness (refractometry) is known [27]. When these parameters are measured

dynamically, they can be used to measure membrane or density fluctuations providing mechanical

information on cellular structures [28]. Moreover, it was realized soon after the conception of

12

quantitative phase microscopy, that the integrated phase shift through a cell is proportional to its

dry mass (non-aqueous content)[29, 30], which enables studying cell mass growth [31, 32] and

mass transport [28, 33] in living cells. Furthermore, when the low-coherence illumination is

combined with a high numerical aperture objective SLIM provides excellent depth sectioning.

When this capability is combined with a linear forward model of the instrument, it can be used to

perform three-dimensional tomography on living cells [20] with sub-micron resolution. Thus the

current major applications of SLIM may be broken down into four basic categories, refractometry,

topography, dry mass measurement and tomography. In addition to basic science applications,

SLIM has also been applied to clinical applications such as blood screening [34] and cancer

diagnosis [35].

Since SLIM is coupled to a commercial phase contrast microscope that is equipped with

complete environmental control (heating, CO2, humidity), it is possible to perform long term live

cell imaging. In fact with SLIM measurements of up to a week have been performed [31]. Due to

the coupling with the commercial microscope, it is also possible to utilize all other commonly used

modalities such as fluorescence simultaneously. Fluorescence imaging can be used to add

specificity to SLIM measurements such as for identifying the stage of a cell cycle or the identity

of an observed structure. Furthermore, since it is possible to resolve sub-cellular structure with

high resolution, the inter- and intra- cellular transport of dry mass may also be quantified. Using

mosaic style imaging, it is also possible to image entire slides with sub-micron resolution imaging

by tiling and stitching adjacent fields of view. Thus SLIM may be used to study phenomenon on

time scales ranging from milliseconds to days and spatial scales ranging from sub-micron to

millimeters.

13

CHAPTER 2. Red Blood Cells Viscoelasticity Measured By Diffraction Phase Microscopy

(DPM) 2

2.1 Introduction

The red blood cell (RBC) deformability in microvasculature governs the cell’s ability to

transport oxygen in the body [36]. Interestingly, RBCs must pass periodically a deformability test

by being forced to squeeze through narrow passages (sinuses) in the spleen; upon failing this

mechanical assessment, the cell is destroyed and removed from circulation by macrophages (a type

of white cell) [37]. Thus, understanding the microrheology of RBCs is highly interesting both from

a basic science and clinical practice point of view.

However, the mechanics of this system is insufficiently understood. A number of different

techniques have been previously used to study the rheology of live cells [38]. Among these, pipette

aspiration [39], electric field deformation [40], and optical tweezers [41, 42] provide quantitative

information about the shear and bending moduli of RBC membranes. Dynamic, frequency-

dependent knowledge of the RBC mechanical response has been limited to recent developments

based on active and passive microbead rheology [43, 44].

RBC membrane thermal fluctuations have been studied intensively, as they offer a non-

perturbing window into the structure, dynamics, and function of the whole cell [45-55]. These

spontaneous motions have been modeled theoretically under both static and dynamic conditions

to connect the statistics of the membrane displacements to relevant mechanical properties of the

cell [45, 50, 51, 56-58]. With few exceptions [59, 60], all membrane models assume a flat, 2D cell

surface.

2 This chapter appeared in its entirety (with some modifications) in Ru Wang, Huafeng Ding, Mustafa Mir, Krishnarao

Tangella and Gabriel Popescu, Effective 3D viscoelasticity of red blood cells measured by diffraction phase

microscopy, Biomed. Opt. Exp., 2(3), 485 (2011)). This material is reproduced with the permission of the publisher.

14

2.2 Method and Analysis

Here we present quantitative optical measurements of RBC fluctuations obtained by highly

sensitive quantitative phase imaging [61]. These spatio-temporal fluctuations are modeled in terms

of the bulk viscoelastic response of the cell, which carries the spirit of the passive microrheology

method pioneered by Mason and Weitz [62]. Relating the displacement distribution to the storage

and loss moduli of the bulk has the advantage of incorporating all geometric and cortical effects

into a single effective medium behavior. Essentially, we treat the cell as a viscoelastic droplet that

recovers the 3D shear behavior of the cell as it exhibits itself in flow [63].

The experimental setup is a modified version of the diffraction phase microscope (DPM),

which is described in more detail elsewhere [64]. Briefly, using laser illumination and a common

path interferometer, we generate an extremely stable interferogram at the image plane of a

transmission microscope. The final quantitative phase image is obtained from this interferogram

using a Hilbert transform. Red blood cells from a healthy volunteer were imaged at a rate of 20

frames/s and the phase shift information (x, y) was translated into cell thickness distribution

h(x, y) using 0/ 2 ( )h n n , with λ=532 nm the laser wavelength, n=1.41 the refractive index

of the hemoglobin, and n0=1.34 the refractive index of plasma.

15

Figure 2.1. a) RBC topography map; color bar indicates thickness in microns. b) Instant

displacement map; color bar in nm. c) Background displacement; color bar in nm. d) Histogram

of displacements associated with the mps in b and c.

Figure 2.1a shows an RBC thickness distribution, ( , )h x y , which is the result of averaging 256

frames. A displacement map at an arbitrary moment t ( , ; ) ( , ; ) ( , )h x y t h x y t h x y is illustrated

in Fig. 2.1b and the background noise in Fig. 2.1c. Figure 2.1d shows the histogram of

displacements for both the signal (membrane) and noise, which demonstrates the ability of DPM

to retrieve membrane fluctuations with high signal to noise ratio. This noise is primarily due to

possible fluctuations from the plasma outside the RBC membrane and residual uncommon path

vibrations and air fluctuations in the interferometer. The spatially-averaged power spectra,

0

0.2x105

0.4x105

0.6x105

0.8x105

-1000 -500 0 500 1000

backgroundcell

Displacement (nm)

Co

un

ts

d

a

2μm

b

c

2µm

2µm

16

2( ) ( , ; )P h x y dxdy , for a group of normal RBCs, is illustrated in Fig. 2.2a. All measured

P() curves exhibit power law behavior

with different exponent values, 1.69 0.29 (

min 1.16 and max 2.07

) where the error captures the cell-to-cell variability (N=7). Note that

the power law predicted by 2D membrane models, assuming elastic parameters constant

frequency, predict narrow values of the exponent 4 / 3 [45, 65] or 5 / 3 [58, 66]. Of

course, our average exponent, 1.7, is compatible with the 5/3 value. The large variability measured

in the power laws,1 2 , cannot be easily explained by the 2D, frequency-independent models.

However, this power law dependence is in striking resemblance with what has been measured via

dynamic light scattering on beads embedded in complex fluids [62, 67]. There, it was found that

the power spectrum of the scattered light for a purely viscous fluid exhibits 2

dependence and

progressively broadens (i.e. exhibits lower exponents) as the elasticity increases. Our RBC

imaging measurements are indeed equivalent to scattering experiments, where the signals are due

to the interaction of light with the undulating cell surface [68]. This merger of scattering and

imaging measurements is possible due to the knowledge of phase at each point in the image plane,

which allows for numerically propagating the complex optical field at any arbitrary plane in space,

including the far-field (i.e., scattering plane) [68]. This new approach, referred to as Fourier

Transform Light Scattering, provides extremely sensitive and fast scattering measurements, which

recently allowed the non-contact probing of actin-mediated nanoscale membrane motions in live

cells [69].

We modeled the cell as a homogeneous droplet that undergoes surface fluctuations at

thermal equilibrium. In analogy to scattering-based microrheology, this viscous response, which

relates to the measured power spectrum via the fluctuation dissipation theorem, is later generalized

17

to include elasticity. The general solution of a liquid surface fluctuations under thermal equilibrium

conditions is due to Bouchiat and described in more detail in [70],

0

22 0

2

1( , ) Im( )

(1 ) 1 2

Bi

k T qh q

(2.1)

where2/ 4 q ,

2

0 / 2 q , q is the wave number (0.1-1 m-1), is the cytosol density (

31.5 /kg m ), the cytosol viscosity ( 36 10 Pa s )[71]and the surface tension (

610 /N m ) [64]. Due to the low surface tension associated with RBCs and high spatial

frequencies probed in our experiments, Eq. 2.1 reduces to

2

2

1( , )

2 ( )

Bk Th q

q

(2.2)

Note that, for a purely viscous cell, i.e. constant , the fluctuation spectrum recovers the

expected 2

behavior. Using the fluctuation dissipation theorem, the viscous response function

can be written as

2''( , ) ( , )B

q h qk T

(2.3)

The real (elastic) part of the response is obtained via the Kramers-Kronig relationship

'

'

2 ''( )'( ) 'P d

, (2.4)

In Eq.2.4, P indicates the principal value integral and the spatial dependence of '' was averaged

out, i.e. ''( ) 2 ''( , )q qdq . In order to calculate the Hilbert transform in Eq. 2.4, we used

the best fit of P() (see Fig. 2.2) with a power law function instead of the actual data. This approach

minimizes the effect of noise in evaluating the integral of Eq. 2.3, as previously described in Ref.

[62].

18

Figure 2.2. (a) Power spectra of membrane fluctuations for 7 RBCs, their average, and background,

as indicated. Dash line indicates a power law of exponent -1.7. (b). RBC viscous and elastic moduli

vs. frequency, as indicated. Dash lines show the liquid and solid behavior.

The viscoelastic modulus was obtained by using its proportionality with the complex, frequency

dependent viscosity, * *G i . Figure 2.2(b) summarizes the results in terms of the viscous

modulus, G’’(), and elastic modulus, G’(), for a set of 7 RBCs. As expected, the loss modulus

G’’ is dominant, which indicates a largely fluid behavior of the normal cell. The loss modulus G’’

has a power law dependence on frequency with and exponent between 1 and 2, as expected for a

viscoelastic fluid. This behavior agrees qualitatively with that reported by Puig-de-Morales et al.

[43], where the magnetic bead twisting method was employed to retrieve the 2D viscoelastic

moduli of the cell membrane.

10-12

10-11

10-10

10-9

10-8

10-7

0.1 1 10 100

P()averagebackground

-1.7

(rad/sec)

h

2 (m

2s)

a

0.01

0.1

1

10

100

1 10 100

G'G''

(rad/sec)

G''/

G' (P

a)

liquid

solid

b

Fitting

19

2.3 Discussion

In the following we discuss the physical interpretation of our experimental results. The

treatment of the RBC as a viscoelastic droplet originated in the experimental observation that the

power spectra measured in our experiments on RBCs have the same characteristics as those

measured by dynamic light scattering microrheology. These power spectra become continuously

broader for stiffer cells, which cannot be accounted for by previous membrane models, which

predict a fixed power law. Thus it appears that RBC flickering can be interpreted as due to the 3D

shear response of a viscoelastic medium. In the bead microrheology, the relationship between the

complex response function and shear modulus is obtained via the generalized Stokes-Einstein

equation (GSER)

1 1( )

6 ( )GSERG

a

, (2.5)

where a is the radius of the probing bead.. In the RBC case, the analog relationship is obtained, up

to a constant, by replacing a with the 1/q. We emphasize that the physical origin of this equivalence

is not entirely clear. We propose a picture that may provide a heuristic explanation for these

observations. Figure 2.3 depicts the instantaneous displacement map associated with a discocyte.

The spatial contributions from the spatial wavelength =1 m is shown in Fig. 2.3a. Our results

suggest that the surface ripples of a certain (spatial) period, 2 / q , are equivalent to

deformations produced by a spherical bead to a 3D fluid. This is illustrated in

20

Figure 2.3. Instantaneous displacement map associated with an RBC at a spatial wavelength,

=2/q, centered around 1 m. Color bar in nanometers indicates displacements in nanometers.

b) Schematics of how the membrane ripples deform the membrane material like a bead of size

comparable with . The spectrin molecules are depicted in red and connected to the lipid bilayer.

Figure 2.3b, where the 3D nature of the problem is emphasized by considering a finite thickness

of the deformed medium. This picture may explain why the complex modulus G obtained from

our data captures the 3D dynamic behavior. It is known that the cytoskeleton is solely responsible

for the finite shear resistance of the RBCs in circulation [59] . It is, therefore, not surprising that

the frequency dependence of G’ and G’’ are in qualitative agreement with passive microrheology

data obtained from polymer solutions [62]. Quantitatively, we always measured lower values for

b

a

=1 m

2 m

21

both G’ and G’’ than those reported for actin solutions [62, 72] which is expected, as the spectrin

filaments are known to be more flexible than the actin ones.

In sum, our sensitive measurements of RBC membrane displacements indicate that the

dynamics can be accounted for by applying a model of light scattering on fluid surfaces. From the

power spectrum of the fluctuations, we can extract the 3D viscoelastic response of the cells and

their effective shear moduli. The ability to account for a global descriptor of cell dynamics that

explains the measurements without relying on a priori knowledge about the bilayer and spectrin

composition may prove useful for high-throughput clinical applications such as blood screening.

22

CHAPTER 3. Dispersion-relation Phase Spectroscopy (DPS)3

3.1. Introduction

Cells have developed a complex system to govern the internal transport of materials from single

molecules to large complexes such as organelles during cell division [73]. While these processes

are essential for the maintenance of cellular functions, their physical understanding is incomplete.

It is now well documented that intracellular transport is mediated by both thermal diffusion and

molecular motors that drive the cellular material out of equilibrium [74-76]. Measurements of the

molecular motor driven transport have been made in the past via single molecule tracking (see e.g.

[77]). However, developing a more global picture of the spatial and temporal distribution of active

transport in living cells remains an unsolved problem. Approaching this question experimentally

can be reduced fundamentally to the problem of quantifying spatially heterogeneous dynamics at

the microscopic scale. In the past, cellular material has been studied successfully by both

externally-driven [78, 79] and passive particle tracking [80-82]. Recently, bending fluctuations of

microtubules have been used to probe the active dynamics in actin networks [83].

In this chapter we introduce another approach – quantitative phase imaging (QPI) – to study the

spatial and temporal distribution of active (deterministic) and passive (i.e. diffusive) transport

processes in living cells. QPI reveals the optical pathlength distribution, s(r), associated with the

transmitted field through thin specimens, including live cells, (for a review, see [84]),

( , ) ( , ) ( , )s t h t n tr r r (3.1)

3 This chapter appeared in its entirety (with some modifications) in Ru Wang, Zhuo Wang, Larry Millet, Martha

U.Gillette, A. J. Levine, and Gabriel Popescu, Optical measurement of cycle-dependent growth, Opt. Exp. 19(21),

20571 (2011). This material is reproduced with the permission of the publisher.

23

In Eq. 3.1, h is the local thickness, r=(x, y), and n the refractive index contrast with respect to that

of the surrounding medium. The pathlength fluctuations can be expressed to the first order as

,

, , , ,

, ,

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , )

s t s t s t

h t h t n t n t h t n t

n t h t h t n t

r t

r t r t r t r t

r t r t

r r r

r r r r r r

r r r r

, (3.2)

where the angular brackets indicate both temporal and spatial average. Thus, ( , )s t r , contains

information about both out-of-plane thickness fluctuations, h, and in-plane refractive index

fluctuations, n. Recently, the h-contribution to out-of-plane pathlength fluctuations has revealed

new behavior in red blood cells, where n is well approximated by a constant [55, 59, 85-87]. Here

we focus on the last term of Eq. 3.2, i.e., the in-plane mass transport revealed by n.

3.2. Measurements

Experimentally, we employed a novel method recently developed in our laboratory, referred to as

spatial light interference microscopy (SLIM) [88]. This imaging technique uses spatially coherent,

broadband light, as illustrated in Fig. 1a. SLIM operates by adding spatial modulation to the image

field of a commercial phase contrast microscope. In addition to the conventional /2 shift

introduced between the scattered and unscattered light in phase contrast microscopy [89], we

generated further phase shifts in increments of /2. This additional modulation was achieved by

using a liquid crystal phase modulator placed in the Fourier plane of the imaging field. In this

setup, four images corresponding to each phase shift (n/2, n=0, 1, 2, 3) are recorded, to produce

a quantitative phase image that is uniquely determined.

24

Fig.3.1.a) Microscope as a scattering instrument: dash line, scattered field; green, unscattered

fields. b) Quantitative phase image of a live cell. Color bar indicates phase in radians. c)

Momentum transfer in a microscope

Figure 3.1.b depicts the quantitative phase image associated with a live cell. Note that, due to

contrast changes induced by the four phase shifts, SLIM is able to minimize the effect of the well-

known halo artifact that otherwise disturbs most phase contrast images. Recently, we have shown

that the data contained in a quantitative phase image is equivalent to that of an extremely sensitive

light scattering measurement [90], as shown in Figure 3.1c. Thus, ki is the incident wave vector,

ks scattering wave vector, and q is momentum transfer (or scattering wavevector), which

corresponds to a single spatial frequency in the image. .The scattering angle (or momentum

transfer) is limited only by the numerical aperture of the objective. Thus, QPI essentially allows

measurements throughout the entire forward scattering half space from a single CCD exposure.

sk

k i

q

Sample

CCD

Objective

Tube lens

Fourier

planea

b c

k i

ein/2

10µm

25

3.3 Principle of dispersion-relation phase spectroscopy (DPS)

Due to its broadband illumination, SLIM lacks speckle-induced noise and, thus, is highly sensitive

to pathlength changes down to a fraction of nanometer [27, 88]. Throughout our measurements,

the phase image acquisition rate was around1 frame/s, which is lower than the decay rates

associated with the bending and tension modes of membrane fluctuations [85], such that we can

safely ignore these contributions. The measured pathlength fluctuations in this case report on the

dry mass transport within the cell [91]. This 2D dry mass density, n , satisfies an advection-

diffusion equation, which includes both directed and diffusive transport[92],

2 ( , ) ( , ) ( , ) 0D t t tt

r v r r , (3.3)

where D is the diffusion coefficient of the particles and v is the advection velocity due to flows in

the sample cell. The spatiotemporal autocorrelation function of is defined as

,( ', ) ( , ) ( ', )

tg t t r r

rr r , (3.4)

where the angular brackets indicate temporal and spatial average. In equation 3.4, τ denotes the

time difference. Taking the spatial Fourier transform of Eq. 3.4, we obtain the temporal

autocorrelation, g , for each spatial mode q, as

2

( , ) i Dqg e vqq (3.5)

Equation 3.5 relates the measured temporal autocorrelation function to the diffusion coefficient,

D, and velocity, v, of the matter in the cell. Note that the same autocorrelation can be measured

via dynamic light scattering at a fixed scattering angle [93] The experimental data, however, is

averaged over a broad distribution of advection velocities, so that we may write the advection-

velocity-averaged autocorrelation function as

26

2

2 20

( , )

(| |)

i Dq t

Dq t i

g e

e P e d

v

v

vv v v

q

q

q

, (3.6)

where the probability distribution P of local advection velocities remains to be determined. In order

to gain insight into the local distribution of advection speeds, we note that maximal kinesin speeds

are approximately v=0.8 m/s [94]. Given the varying loads on such motors, we expect that the

typical advection speeds will be distributed below this limit. The average advection speed may in

fact be significantly lower than this value as there must be transport in a variety of directions.

Consequently, we propose that ( )P v is a Lorentzian of width v and that the mean advection

velocity averaged over the scattering volume is significantly smaller than this velocity, 0v v .

From this simple model we find that the integral in Eq. 3.6 can be evaluated as

20( , )

i q v Dqg e e

v qq . (3.7)

Thus, the mean advection velocity produces a modulation of frequency 0( ) q v q to the temporal

autocorrelation, which decays exponentially at a rate 2( )q vq Dq . This relationship between

the decay rate and its wavenumber q represents the dispersion relation associated with

intracellular transport. Thus, we refer to our method for studying transport as dispersion-relation

phase spectroscopy (DPS), not to be confused with optical dispersion. We note that here

“spectroscopy” refers to measuring spatiotemporal frequencies associated with mass transport and

should not be confused with optical spectroscopy.

In order to mimic better the viscous properties of the intracellular environment, we used SLIM to

image the Brownian motion of 1 µm polystyrene spheres in highly concentrated (99%) glycerol.

27

Fig.3.2. a) Quantitative phase image of 1μm polystyrene beads in glycerol. Colorbar indicates

pathlength in nm. b) Mean squared displacements (MSD) obtained by tracking individual beads in

a. The inset illustrates the trajectory of a single bead. c) Decay rate vs. spatial mode, (q),

associated with the beads in a. The dash ring indicates the maximum q values allowed by the

resolution limit of the microscope. d) Azimuthal average of data in c) to yield (q). The fits with

the quadratic function yields the value of the diffusion coefficient as indicated.

Figure 3.2a) shows the SLIM phase image of the sample with a field of view of 73x73 µm2. We

acquired SLIM images for 10 minutes with an acquisition rate of 1 frame/s. In principle the

expected diffusion coefficient can be calculated via the Stokes-Einstein equation, provided the

viscosity of glycerol is known. However, since the viscosity of concentrated glycerol is highly

sensitive to the absolute value of the concentration, which is prone to experimental (mixing) errors,

Г(q)Dq2, D=1.4 x 10-3µm2/s

a b

c d

MSD(τ)4Dτ, D=1.6 x 10-3µm2/s

(qx , qy)

qx

qy

28

instead we measured the diffusion coefficient by tracking individual particles in the phase image.

Figure 3.2b shows the mean-squared displacements (MSD) obtained by averaging the trajectories

of 54 particles. The linear fit yields D=(1.600.04)x10-3 m2/s. From the SLIM data, we calculated

the dispersion relation, (qx, qy), as shown in Fig. 3.2c. Thus, we first perform the spatial Fourier

transform of each frame, then calculated the temporal bandwidth, Γ, at each spatial frequency

(qx,qy) through performing the temporal Fourier transform. We then perform the azimuthal average

to obtain the radial function 2 2( ), x yq q q q . This experimental curve exhibits the expected q2

dependence, as illustrated in the log-log plot of Fig. 3.2d. The diffusion coefficient obtained is

D=(1.400.07) x10-3 m2/s and compare very well with those measured via particle tracking.

Note that measuring diffusion via DPS eliminates the need for tracking individual particles

and, more importantly, applies to particles that are not resolved in the image, i.e. are smaller than

the diffraction spot of the microscope, as long as the particles travel over distances larger than the

diffraction spot. This is largely the case in studying mass transport in live cells.

3.4 DPS application in live cells

We performed experiments on various types of live cells including glia and microglia cells, which

are common in the central nervous system. In these cells, individual, un-labeled particles cannot

be easily resolved or tracked by light microscopy. The cells were imaged in culture medium under

physiological conditions, 37 oC and 5% CO2 controls. All images were acquired with objective

40X/NA=0.75 except figure 3e) where 63x/NA= 1.4 was used. Figure 3.3 shows results obtained

on live cells. The cell-averaged ( )q curve shown in Fig. 3.3a-b exhibits a dominant quadratic

shape, which yields a diffusion coefficient D=(9.600.04) x10-4 m2/s. Further, at the low-

wavenumber end of the measurement range, we found a distinct linear dependence. From the linear

29

term in fitting the dispersion curve, we extracted the advection velocity distribution width,

obtaining a value of v=1.3 nm/s. Examining the data from live cells we note that we do not

observe sinusoidal oscillations of the autocorrelation function, which would indicate a dominant

velocity. This sets an upper limit on the mean advection velocity 0 max max 1v q . Given an

observation time of ten minutes, the upper limit on the mean advection speed is 0.1 /nm s

suggesting that, if present, a mean velocity is below this value. Thus, we can conclude that there

are no persistent currents in the cell in steady-state. Deviations from the simple quadratic

dispersion relation, however, demonstrate the presence of local directed transport, but this directed

transport does not generate coherent flows within the cell. These results are fairly insensitive to

the exact form of the advection speed distribution postulated. To obtain a part of the decay rate

that is linear in the wave number term, one needs only postulate an upper speed cutoff and a

reasonably broad distribution of speeds below this limit.

Further, we used DPS to study the mass transport in subcellular sub-domains. Figures 3.3c-h

shows the SLIM images of various cells in culture and the ( )q curves associated with the

respective regions. Note that the signal from the strips is associated with fluctuations along a single

row of pixels, which explains the relatively higher noise level. These data exhibit a diversity of

behavior, from purely diffusive in the microglia culture (Figs. 3.3c-d) to purely directed, in the

putative dendrite of a neuron (Figs. 3.3e-f) [95].

30

Fig.3.3.Quantitative phase image of a culture of glia (a, g), microglia (c) and hippocampal neurons

(e). b) Dispersion curve measured for the cell in a. The green and red lines indicate directed motion

and diffusion, respectively, with the results of the fit as indicated in the legend. Inset shows the

(qx, q

y) map. d, f, h) Dispersion curves, (q), associated with the white box regions in c),e) and

g), respectively. The corresponding fits and resulting D and v values are indicated.

~q2

Г(q)Dq2, D=7.93 x 10-4µm2/s

q

Г(q)vq, v=4.14nm/s

q

q2

Г(q)vq, v=1.3nm/s

Dq2, D=9.64 x 10-4µm2/s

a

c

e

b

d

f

g

q2

q

Г(q)vq, v=3.94nm/s

Dq2, D=1.1 x 10-3µm2/s

h

ΔΔ

Δ Δ

Δ Δ

Δv q, Δv=4.14nm/s

Δvq, Δv=1.3nm/s

Δvq, Δv = 3.94nm/s

31

The entirely directed transport measured along the dendrite is in line with what is generally

known about that ATP-consuming cargo transfer along microtubules via protein motors.

Examining a narrow strip whose long axis is oriented radially with respect to the cell nucleus (Figs.

3.3g-h), we found that the transport is diffusive at short scales (below approximately 2/q=2 m)

and directed at large scales. These findings suggest that, in general, both D and v may be

inhomogeneous and anisotropic. Remarkably, the diffusion coefficients measured in the cell have

much lower values than, for example, those associated with submicron particles in water. In order

to study further this behavior, we performed a comparison between particle tracking and DPS

measurements in both the nucleus and cytoplasm, as summarized in Fig. 3.4. Using the ImageJ

“particle tracker” plugin [96], we tracked the mass centroid of 6 microdomains in the nucleus and

the resulting averaged MSD is shown in Fig.3.4a. Interestingly, the curve indicates a strong

diffusive component that appears to have two modes, characterized by slightly different diffusion

coefficients of 0.88 m2/s and 0.55 m2/s. Remarkably, the value obtained by DPS is 0.6 m2/s,

in between the two values given by tracking (Fig. 3.4b). The directed motion component of the

MSD in Fig. 3.4a is seen more clearly for one domain in the inset. The value obtained is v=2.8

nm/s, while DPS yields v=4.1 nm/s.

32

Fig.3.4. a) and c) MSD ensemble-averaged over 7 and 6 particles in nucleus and cytoplasm regions,

respectively, as indicated in figure 3g by the red and blue boxes. Corresponding fits give diffusion

coefficients and standard deviation of drift velocity. Inset in a) shows the MSD for a particle in log

scale where directed motion is indicated by green line. b) and d) are dispersion curves for the same

area of a) and c. Inset are trajectories of two particles denoted by black arrow in figure 3.3 g). The

blue one is the particle at cytoplasm and red one is inside nucleus.

Tracking discrete particles in the cytoplasm results in higher diffusion coefficients than

obtained by DPS, 4-5 m2/s vs. 1.7 m2/s, indicating that the continuous mass included in the DPS

analysis is slower than the individual discrete particles. Interestingly, the width of the velocity

distribution from DPS,~6.9nm, as shown in figure 3.4d) is comparable with the average speed

MSD()

4D,D = 0.88x10-3 µm2/s4D,D = 0.55x10-3 µm2/s

[sec]

MSD

[µ

m2]

[sec]

MSD

[µ

m2]

MSD()4D, D = 5.0x10-3 µm2/s4D, D = 3.9x10-3 µm2/s

(v)2, v = 7.3 nm/s

q [rad/µm]

[r

ad/s

ec]

(q)Dq2, D = 0.60x10-3 µm2/s

Δvq, Δv = 4.1 nm/s

q [rad/µm]

[r

ad/s

ec]

(q)Dq2, D = 1.7x10-3 µm2/s

Δvq, Δv = 6.9 nm/s

1 2

a b

c d

x [µm]

y[µ

m]

Nucleus

Cytoplasm

Nucleustracking

Nucleusdispersion

Cytoplasmtracking

Cytoplasmdispersion

v = 2.8 nm/s

33

from particle tracking, ~7 nm/s, as shown in figure 3.4c) which is expected from a system of

particles actively transported in all directions isotropically. Our simultaneous measurements of

DPS and particle tracking confirm that transport in continuous mass distributions is generally

slower than that of discrete particles. This is the origin of the very low average values of the

diffusion coefficients reported by DPS. Previous reports, by Thompson et al. [97] and Tseng et al.

[98] reported similar values for the nuclear material. In particular, Tseng et.al. obtained values of

up to 520 Poise for the mean shear viscosity of the intranuclear region in Swiss 3T3 fibroblasts,

while the values of the diffusion coefficient obtained by particle tracking match ours very well (of

the order of 10-3 µm2/s and below). Interestingly, evidence was found for elastic behavior in nuclei

at time scales below 1 s, which is currently not covered by our current instrument. Taken together,

these results and ours indicate that the transport in live cells is far from being fully understood. We

believe our dispersion relation analysis may provide a complementary approach for studying this

phenomenon label-free, over broad spatial and temporal scales.

3.5 Discussion

In sum, we developed an approach to study the intracellular transport that is based on measuring

the dispersion curves, ( )q , from quantitative phase imaging. On average the mass transport in a

live cell is diffusive at small scales (1 m and below) and deterministic at large scales (several

microns). Our experiments show that continuous or completely transparent systems can be studied

successfully by this method, in a label-free manner. Biologically, this result is quite reasonable:

mass transport at large scales cannot be accomplished effectively by diffusion alone, as it is too

slow; thus, it requires energy consumption. For example, neurons are a particular cell type that

must accomplish transport over very large distances. Our results showing that the dendritic

transport is largely directed strongly supports this idea.

34

It is worth noting that DPS can be equally used with other quantitative phase imaging

techniques, such as these described recently in Refs. [99-101]. The temporal limitations of

studying subcellular transport are due to the acquisition rate; currently SLIM acquire 2.3 frames/s.

However, this is not a limitation of principle and higher acquisition rates can be reached by using,

for example, faster camera and spatial light modulator.

35

CHAPTER 4. 1d Neuron Transport With DPS4

4.1 Introduction

Cells rely on their ability to actively transport macromolecules and even organelles, since the

passive diffusive transport of such low mobility objects would simply be too slow. This directed

transport of intracellular components is particularly apparent during cell division, but is known to

occur during all phases of the cell cycle [73]. The necessity of active transport is especially acute

where intracellular cargos need to be carried over long distances, as in the case of the transport of

intracellular vesicles and other large objects up and down the axonal and dendritic processes of

neurons. In such narrow and elongated structures, the spatial distribution of these cellular transport

highways is particularly simple: intracellular traffic is directed along an essentially one-

dimensional, tortuous path in a three dimensional space. This bidirectional (i.e. to and from the

cell’s soma) transport is known to be mediated by some combination of thermal diffusion and

active stochastic transport driven by molecular motors (e.g. kinesin and dynein) [74-76], but these

transport phenomena could be better understood through quantitative analysis. The reduction of

complex transport networks commonly found in cell bodies to a one-dimension system provides

an opportunity for refining experimental techniques for transport measurement and for theoretical

modeling.

Using single molecule tracking, precise measurements of individual cargos that are

transported by molecular motors have been made previously, see e.g. [77]. In addition, the motion

of externally driven particles and the observation of their fluctuating position have been

4 This chapter appeared in its entirety (with some modifications) in Ru Wang, Zhuo Wang, Joe Leigh, Nahil Sobh,

Larry Millet, Martha U. Gillette, Alex J. Levine, and Gabriel Popescu, One-dimensional deterministic transport in

neurons measured by dispersion-relation phase spectroscopy, J. Phys. Cond. Matt., 23, 374107 (2011). This

material is reproduced with the permission of the publisher.

36

successfully used to monitor the viscoelastic properties of living cells [78, 79] and the “active”, or

molecular motor driven, strain fluctuations and cargo transport [80-82] in cytoskeletal networks.

The fluctuations of one- and two-dimensional objects (e.g. filaments [73] and membranes [86])

have also been studied to measure the elastic properties and motor activity in living cells. In this

article we examine a complementary technique that does not require the tracking of individual

particles and that investigates more globally the spatial and temporal nature of cargo transport over

tens of microns and thousands of seconds. While we do not resolve the motion of individual cargos,

we can quantitatively and spatially map the heterogeneous dynamics of the concentration field of

the cargos with submicron resolution. Our method relies on quantifying interferometricaly the

path-length map produced by cells [84].

Quantifying cell-induced shifts in the optical path-lengths permits nanometer scale

measurements of structures and motions in a non-contact, non-invasive manner [61]. Thus,

quantitative phase imaging (QPI) has recently become an active field of study and various

experimental approaches have been proposed [64, 102-110]. We have shown that the knowledge

of the amplitude and phase associated with the image plane is equivalent to extremely sensitive

elastic and quasi-elastic light scattering measurements [90, 111, 112]. This new approach,

represents the spatial equivalent of Fourier transform spectroscopy and is, thus, referred to as

Fourier transform light scattering (FTLS). Recently, FTLS proved sensitive enough to quantify

actin dynamics in unlabeled live cells [69].

Despite all these advances, the range of QPI applications in cellular biophysics has been

largely limited to red blood cell imaging [55, 59, 85, 86] or assessment of global cell parameters

such as dry mass [91, 103], average refractive index [113], and statistical parameters of tissue

slices [68, 112]. This limitation is mainly due to the speckle generated by the high temporal

37

coherence of the light used (typically lasers), which averages out the morphological details. Thus

the contrast in QPI images has never matched that exhibited in white light techniques such as phase

contrast and Nomarski.

Recently we developed SLIM (Spatial Light Interference Microscopy), a novel, highly

sensitive QPI method, which promises to enable unprecedented structure and dynamics studies in

biology and beyond [88]. SLIM reveals the intrinsic contrast of transparent samples like phase

contrast microscopy [114], while rendering quantitative phase information, like holography [115].

Taken together, SLIM’s features advance the field of quantitative phase imaging by several

accounts: i) provides speckle-free images, which allows for spatially sensitive optical path-length

measurement (0.3 nm); ii) uses common path interferometry, which enables temporally sensitive

optical path-length measurement (0.03nm); iii) renders 3D tomographic images of transparent

structures; iv) due to the broad band illumination, SLIM grants immediate potential for

spectroscopic (i.e. phase dispersion) imaging; v) is likely to make a broad impact by

implementation with existing phase contrast microscopes; vi) and inherently multiplexes with

fluorescence imaging for multimodal, in-depth biological studies.

The remainder of this chapter is organized as follows: In section 4.2 we describe SLIM in

more detail, focusing on its application to one-dimensional transport in neurites in section 3. In

section 4 we discuss our analysis of the data in terms of simple advection diffusion models. We

present the results of that analysis in section 5. Finally, we conclude with a discussion of the results

and our interpretation thereof in section 6.

4.2 Measurements

38

SLIM is described in more detail elsewhere [88]. In short, it is implemented as an extension to an

existing phase contrast microscope that makes the instrument capable of quantifying phase shifts

across the field of view. Both SLIM and phase contrast microscopy exploits the concept of

imaging as an interference phenomenon, which has been recognized more than a century ago by

Abbe in the context of microscopy: "The microscope image is the interference effect of a

diffraction phenomenon" [116]. Describing an image as a (complicated) interferogram later set the

basis for Gabor's development of holography [115].

Unlike the traditional phase contrast microscope, where a phase ring with a fixed phase

shift of π/2 is introduced at the back focal plane of the phase contrast objective [116], in SLIM we

map the back focal plane onto a phase-only spatial light modulator that introduces additional phase

delays of 0, π/2, π, 3π/2, as shown in Fig. 4.1a. The Fourier lenses L2 and L1 form a 4f system,

such that the spatially modulated image of the sample is recorded by the CCD. The phase shift

distribution associated with the object, (r), can thus be retrieved from a combination of 4 phase

shifted intensity images,

( )sin ( )( ) arg

1 ( )cos ( )

r rr

r r.

(4.1)

In Eq. 1, 1 0( ) ( ) /U U r r is ration between the amplitudes of the scattered (U1) and unscattered

(U0) field; is the phase difference between the scattered and unscattered light, defined as

( ;3 / 2) ( ; / 2)

( ) arg ,( ;0) ( ; )

I I

I I

r rr

r r (4.2)

where ( ; )I r is the intensity image associated with each of phase shift . Equations 1-2 show how

the quantitative phase image is retrieved via 4 successive intensity images measured for each phase

shift.

39

Compared to existing methods for quantitative phase imaging, SLIM benefits from a

number of features, such as spatial sensitivity (0.3 nm) and temporal sensitivity (0.03 nm) in

measuring optical path-lengths [88]. Since a minimal modification is brought to the existing phase

contrast microscope, SLIM can also overlay with fluorescence imaging at the same time and render

the high quality phase images as one channel for the multimodal imaging.

The ability of SLIM to render quantitative phase images of live, unlabeled cells is shown

in Fig. 4.1b. Due to the white light illumination, which alleviates the detrimental effects of speckle,

SLIM reveals the structure of the cell in great detail. The measured pathlength fluctuations report

on the dry mass transport within the cell [91]. Since most neuronal processes exhibit microtubule-

mediate transport along their elongated structures, we can assume 1D dry mass density

fluctuations, i.e., mass transport along the neurite, ( ) ( )s n s , with 2 2s x y . From the

time-lapse data, the microtubule-mediated transport along these neurites can be observed very

clearly. However, the quantitative analysis of mass transport in these 1D structures is difficult due

to their nonlinear shape. Therefore, first we numerically processed the images to “straighten” the

neurites, as described in the following section.

40

Figure 4.1. a) Microscope as a scattering instrument: dash line, scattered field; solid green line,

unscattered field; L1,

L2 lenses; CCD, charged coupled device; ki wave vector associated with

the incident plane wave. b) Quantitative phase images of live supraoptic magnocellular neurons

in culture. The objective was 40x, 0.65NA for b. Color bar indicates phase in radians. Scale bar:

10 μm.

S

CCD

L1

L2

Fourier

planea

b

k i

1, ei/2, ei

, ei3/2

[rad]

41

4.3 Analysis

For image processing, we used ImageJ, a Java-based program developed at the National Institutes

of Health [117]. This platform allows the implementation of a plugin, referred to as NeuronJ, which

was originally developed by Meijering et al. to numerically trace neurites in fluorescence images

[118]. We used this algorithm for SLIM images to delineate and segment neurites of arbitrary

trajectories. Further, we used a cubic-spline interpolation method to straighten the neurites from a

2D path, 2 2s x y , to a 1D trajectory. This computational tool was developed by Kocsis et al.

in the context of electron microscopy (for more details, see Ref. [118]). Figures 4.2-4.3 illustrate

how the pathlength information along individual neurites is extracted and represented along single

lines. This numerical procedure was repeated for all the frames in the time-resolved data. The

resulting x-t phase images were analyzed in terms of the dispersion relation, as detailed below.

The data from the phase images of many neurities are combined to compute the two-point

correlation function of the mass density fluctuations

(x,t) (assumed to be proportional to the

fluctuations in the local index of refraction) in them:

',( , ) ( ' , ) ( ', )

x tg x x x t x t , (4.3)

where the angled brackets here denote a spatial and temporal average. The observed temporal

decay of this correlation function allows us to interpret the experimental data in terms of the

stochastic dynamics of mass transport in the system. Fourier transforming the data

( , ) ( , ) iqxg q dx g x e , (4.4)

we find that the decay of

˜ g (q,) is well fit by a single exponential so that we can determine the q-

dependence of that rate constant

(q) , characterizing stochastic dynamics on length scales of

42

2 / q . Representative examples of these extracted decay rates are shown in Fig. 3 as a function

of wavenumber q.

The decay of correlations reflects the net effect of the motion of material having various

indices of refraction and velocities. We cannot distinguish a priori the independent contributions

to the signal coming from various equilibrium (e.g. diffusion) and nonequilibrium processes such

as the molecular-motor-driven motion of vesicles or the dynamics of polymerization and

depolymerization of e.g. microtubules. Examining the data – see Fig. 4.2 (d-g) – one notes that the

observable dynamic heterogeneities appear to be localized or point-like objects, consistent with

vesicles or groups of vesicles moving within the neurites. For brevity we refer to mass transport

dynamics generating the decay of correlations as “vesicle motion” hereafter, although this cannot

be concluded from the phase images alone. Independent of the ultimate identification of these

mobile structures, we propose to characterize their dynamics in terms of an advection diffusion

model in one dimension

2 ( , ) v ( , ) ( , ) 0D x t x t x tt

(4.5)

describing the dynamics of the mass density field in terms of the combination of diffusion with

collective diffusion constant D and advection or active transport with speed v. We note that the

effective diffusion constant is not expected to be coincide with the equilibrium one as active

stochastic active transport processes contribute to the effective diffusion of mass density in the

system. We then assume that the observed decay of correlations results from the incoherent motion

of many compact bodies having a distribution of advection speeds P(v). We discuss the suitability

of these assumptions below.

There is a long history of studying transport in one-dimensional (1D) or nearly (to be

defined below) 1D structures, which has implications for any analysis of mass transport in neurites.

43

The long length scale properties of 1D transport is strongly influenced by two features. First the

effect of steric interactions between the moving elements can be significant, an effect commonly

encountered in traffic jams [119]. Alexander and Pincus showed that the non-driven or equilibrium

dynamics particles on a 1D track is actually subdiffusive, due to such steric effects [120]. The

dynamics of the transport system of current interest is, however, expected to be dominated by the

non-equilibrium advective effects of molecular motors at long length scales. Two classes of

models have been studied that generate such steady-state currents, i.e. net mass transport that are

distinguished by how they break the break the left/right symmetry of the problem in order to

generated such currents. In the first class the symmetry breaking term is introduced via boundary

conditions by attaching the 1D track to a particle source and sink on opposite sides, as done in,

e.g., Ref [52]. In the second class one postulates a nonzero mean force acting on the particles. We

expect this latter class of models to be appropriate for system of current interest since it is known

that molecular motors actively transport cargos in fixed directions relative to polarized tracks

within neurites [119, 121]. More generally, both inherently symmetric models driven by boundary

conditions, known as symmetric exclusion processes (SEP), and symmetry breaking systems or

asymmetric exclusion processes (ASEP) have been explored in a variety of contexts including

models of proton transfer in water channels [122], mRNA translation [123], in addition to the

problem under consideration, the motion of cargos driven by molecular motors along microtubules

[119, 121, 124]. The more idealized system of perfectly unidirectional stochastic motion (i.e. no

backwards steps allowed) motion, known as totally asymmetric exclusion processes (TASEP) has

also generated intense interest in part from its role as a tractable model system with which to

explore nontrivial non-equilibrium steady-states [125].

44

Our approach to the dynamics is in the spirit of various hydrodynamic approximations to

the ASEP models used to treat the long wavelength and low frequency dynamics of such systems.

These models, often including binding/unbinding or Langmuir kinetics in addition to the 1D

motion, result in complex dynamics for the particle density field [126-128]. These dynamics can

be described by a Burgers equation for the density field, which is well-known to exhibit moving

shock fronts [126] or rapid jumps in particle density. Of course, transport in the neurite takes place

along a bundle of 1D tracks with driven transport in both directions. Such generalizations having

multiple tracks [129, 130] have been explored for both ASEP [131-133] and SEP systems.

Our analysis of the dynamics as the sum of a variety of independent advection/diffusion

processes neglects large correlated motions resulting from the excluded volume interaction. We

believe that this linearized model is a reasonable starting point for data analysis for two reasons.

There is evidence that multiple tracks or allowing for slippage of cargos past each other (i.e.

“nearly 1D models”) minimizes the role of large correlated motions [52]. In the presence of such

strong interactions, however, we expect the collective excitations of the particle density (e.g.

Burgers shocks) to move roughly independently so that even in the case of strong correlations in

the density of moving particles, the index of refraction changes, to which we are sensitive, should

reflect the dynamics of weakly interacting collective phenomena when viewed at long length

scales.

Returning to our advection diffusion model, we determine from Eq. 4.5 the temporal decay

of the autocorrelation function of the mass density g (defined by Eqs. [4.3,4.4]). It is

straightforward to show that at wavenumber q, this function takes the form

˜ g (q,) g0e[iqvDq 2 ] . (4.6)

45

Equation 4.6 relates the measured temporal autocorrelation function to the diffusion coefficient,

D, and advection velocity, v, of the matter in the system. The experimental data, however, reflects

the net effect of the motion of many vesicles, each having its own local transport velocity

depending on e.g. the size of the vesicle and number of molecular motors attached to it. The

observed autocorrelation function then should be the sum or average over a broad distribution of

advection velocities

P(v) . The advection-velocity-averaged autocorrelation function is

proportional to the Fourier transform of the advection velocity distribution

2

0[ ( , )] ( )Dq ivq

Pg q g e dvP v e , (4.7)

where the square brackets P denote an average over the distribution of advection velocities P(v).

This distribution remains to be determined. The integral in Eq. 4.7 is the moment generating

function of P(v) with

iq as the independent variable. Thus the natural logarithm of Eq.4.7,

2

0

1ln , ln ( )

Pg q g Dq K iq

, (4.8)

can be written in terms of the cumulant generating function ( ) ln ( ) ixvK x dvP v e . For any

distribution having a finite second moment we find from Eq.4.8 that

22 2 2 2

0

1ln , ln

2Pg q g Dq iq v q v v , (4.9)

where the higher order terms are proportional to the product of the nth cumulant of the advection

velocity distribution and

(q)n

. Examining Eq. 4.9 we note that the dominant terms controlling

the time evolution of the correlation function at small wavenumber consist of an oscillatory term

linear in q and proportional to the mean advection velocity plus a term quadratic in q that depends

on a combination the diffusion constant D and the variance of the advection velocity distribution.

Comparing this result to our data, we observe two things. First this result is not consistent with the

46

observed decay rate that is linear in wavenumber q and does not exhibit oscillations. Second we

do not observe a term proportional to

2

. We conclude the mean velocity must be sufficiently

small that the oscillatory contribution to the correlation function is a frequencies so low as to be

not observable over the time of observation. This is not surprising in that we expect to observe the

net effect of mass transport along both directions in each neurite. Such bi-directional transport is

in itself not surprising. Since there is little or no net transport of mass down these structures over

the period of observation, the observed mean velocity should vanish. Moreover, we note that

advection velocity distribution must be sufficiently broad as to have a divergent second moment.

Based on this analysis we attempt to fit the data by choosing a Lorentzian distribution with

width

v and small but nonzero mean 0v

v . This leads, by direct integration of Eq.4.7, to a

linear dependence of the decay rate on wavenumber

[ ˜ g (q,)]P eDq 2ei v qv q

. (4.10)

Of course is not possible for the distribution of motor velocities to be truly Lorentzian since the

motors’ maximal speed is finite. For kinesin motors, this maximum is know to be approximately

vmax=0.8 m/s [94]. Accordingly, we take a P(v) to be Lorentzian with a both a mean

vm and

width

vmuch smaller than the cutoff velocity,

vm,v vmax :

P(v) P0

(v vm)2 v2(vmax |v |) (4.11)

where

x is the Heaviside step function and the normalization constant

P0 v

2arctanvmax

v

for a small mean velocity. The normalization is exact in the limit that

vm 0. Since we expect to observe transport in both directions with essentially equal probability,

we expect the mean advection speed to be significantly lower than the maximum motor speed. Of

47

course the assumption that both the width and mean of the advection velocity distribution are small

compared to the maximum velocity can be checked a posteriori.

Applying this cutoff Lorentzian model to the integral in Eq.4.7 one can show that

2 max

0 max

max

2 cos 2( , ) 1 Si

2m

q v q viq v Dq q v

ve v q q veg q g e e v q

v

(4.12)

where Si(x) is the sine integral. The final term in the product may be neglected in the limit that

vvmax

1 and max 1v q , both of which are valid for the data in question. Thus, the effect of

the large velocity cutoff is subdominant and the wavenumber dependent decay rate of the

autocorrelation function is a sum of linear and quadratic terms in q, representing the effect of

motor-generated advection and diffusion, respectively. We may write this decay rate as

2( )q vq Dq . (4.13)

This relationship between the decay rate and its wavenumber q represents the dispersion

relationship associated with intracellular transport and provides insight into the diffusion

coefficient and velocity distribution. We refer to this method as dispersion-relation phase

spectroscopy (DPS).

4. 4 Results

We applied DPS to various neuronal structures in different cultures and over a broad range of

temporal and spatial scales. Figure 4.2 summarizes the results obtained by imaging a field of

neurites for 18 minutes at a rate of 1 frame per 15 seconds. A quantitative phase map measured by

SLIM is shown in Fig. 4.2a, where the grayscale bar indicates path-length in nm and the neuritic

structures of interest are highlighted. Figure 4.2b exemplifies the pathlength fluctuation around a

single neurite and indicates the highly inhomogeneous nature of the neurites structure. Further,

48

this signal, which is, up to a constant, the mass density, ( )x , is characterized by high signal to

noise, which translates into high spatial sensitivity to density fluctuations. Temporally, the density

fluctuations associated with two arbitrary points are shown in Fig. 4.2c. These signals show that

SLIM can detect nanometer scale path-length changes, which translate into minute changes in

local mass density.

Figure 4.2 a) Phase image of neuron at 40X with frame rate of about 15sec/frame to determine

mass transport in neurites labeled with a pink (highlighted) line. b) Path length static distribution

along an individual neurite(d). c) Path length dynamic fluctuations over time at two locations

(labeled blue circle(point1) and red square (point2)) of a neurite (d); d-g) Neurites of image (a)

were traced and straightened with Image J plugins - Neuron J and Straighten.

ab) c)

d)

e)

f)

g)

10µm

Point 1Point 2

2

1

s [nm]

49

We applied the algorithm described in Section 4.3 to numerically “straighten” the neurites

(Fig. 4.2d-g) such that the 1D diffusion-advection equation can be employed in a straight forward

manner. For each neurite, the x-t data was Fourier transformed both spatially and temporally and

the decay rate, ( )q , was retrieved. Remarkably, as shown in Fig. 4.3, all the ( )q plots exhibit

linear behavior, which is consistent with a signal resulting from a broad distribution of advection

velocities. It is not consistent with an advection velocity distribution that has a finite second

moment. We do not observe a signal consistent with diffusive transport. This is reasonable since

it is known that diffusion is not an efficient mechanism for this type of long-range transport

associated with neurites [134] compared to active, or motor-driven mass transport. Interestingly,

we found that the bandwidth of the speed distribution for the three neurites,

v 1.86 nm/s, 1.48

nm/s, 1.78 nm/s are very close in value, which may indicate a universal mechanism for this directed

stochastic motion.

In order to confirm these findings on a different cell culture, with different imaging

resolution and acquisition rates, we applied DPS to a neurite system where the acquisition rate was

7.5 times faster and the spatial resolution higher by a factor of 2 (Figure 4.4). The analysis

procedure is analogous to that shown in Figs. 4.2-4.3. Notably, the results of these measurements

at different spatio-temporal resolutions match closely those obtained before. Thus, for all neurites

the results indicate deterministic transport, as shown in Fig. 4.5. The speed bandwidth values,

v 2.08 nm/s, 3.94 nm/s, 1.92 nm/s, are somewhat higher that in the previous case, but still in

the same range. This similarity in the measured values suggests a commonality of transport

mechanisms in the dendritic arbor of neurons.

The mean advection velocity extracted from the data is consistent with zero. From the

resolution limit (

1m) and observation time (

1000s), we may set an upper limit on the net drift

50

velocity of 1 nm/s. We do not observe net mass transport down the neurites. Finally, we note that

the assumptions made in the use of the cutoff Lorentzian appear to be justified

Figure 4.3. a-c) Dispersion relation curves(blue) for straightened neurites d-f) of figure 4.2. Solid

lines indicate linear fit. The fit parameter, v, represents the bandwidth of directed motion speed

distribution.

a)

b)

c)

(q)vq, v=1.86nm/s

(q)

(q)

vq, v=1.78nm/s

vq, v=1.48nm/s

51

Figure 4.4. a) Phase image of mass transport in neurites of hippocampal neurons. The image was

acquired at 63X with a frame rate of 2 sec/frame, the pink lines (highlighted) represent the data

subsets analyzed in b-g. b) Path length static distribution along an individual neurite(g). c) Path

length dynamic fluctuations over time at two locations (labeled blue circle (point1) and red square

(point2)) of a neurite (d) ;d-g) Neurite of image (a) were traced and straightened with Image J

plugins - Neuron J and Straighten.

10µm

ab) c)

d)

e)

f)

g)

Point 1Point 2

1

2

s [nm]

52

Figure 4.5. a-c) Dispersion relation curves (circles) for straightened neurites d-f) of figure 4.4.

Note the broad q-coverage due to the high NA objective. Solid lines indicate linear fit. The fit

parameter, v, represents the bandwidth of directed motion speed distribution.

a)

b)

c)(q)

(q)

(q)

vq, v=1.92nm/s

vq, v=3.94nm/s

vq, v=2.08nm/s

53

4.5 Discussion

We have found that the autocorrelation function in the neurites is inconsistent with purely

passive diffusive motion of cargos, but can be interpreted using a simple model of advective

transport in which individual cargos are moved in both directions and with a Lorentzian range of

speeds having nearly zero mean (due to equal rates of mass transport towards and away from the

cell’s soma) and a fairly narrow distribution about that vanishing mean, at least compared to the

maximum velocity associated with active motors. This result is puzzling in that given the extracted

distribution of advection velocities, one cannot expect to observe cargos transported at speeds

comparable to the reported maximum motor velocities. We may speculate as the cause of the

absence of fast movers. Assuming that the observed mass transport reflects the motion of

individual motors, there are two possible interpretations of this result. First, the motors may, in

fact, be operating near the stall force, where mean velocities are low and velocity fluctuations

become significant. This proposed scenario seems to be physiologically inefficient and therefore

unlikely. Alternatively, the motors are operating at forces where cargo detachment becomes

frequent. If the cargos detach from the motors sufficiently frequently in the crowded viscoelastic

environment of the neurites, then although the motors are still moving rapidly, the fraction of the

time in which the cargos are being actively transported is small and consequently their observed

advection speeds may be much smaller than that of the motors. Finally, it is possible that the mass

transport to which we are sensitive reflect slow collective motions of many cargos, e.g. shock

fronts in the density field, that may not move with the same velocity as the underlying motor-

driven cargos. This appears to be the most reasonable explanation since it is known that the shock

fronts can be significantly slower than the transport speeds of the individual cargos [135]. Clearly,

simultaneous tracking of the cargos, motors, and mass density is necessary resolve this issue.

54

Similarly, theoretical advances are necessary to better understand the collective behavior of the

mass density field in multi-track driven systems with steric interactions. We note that our results,

i.e. the linear in wavenumber decay rate of the autocorrelation function is not consistent with a

velocity distribution that decays sufficiently rapidly at large velocities to have a finite second

moment. The proposed Lorentzian, albeit with a narrow width compared to the maximum motor

velocity, reflects a broad probability distribution.

55

CHAPTER 5. Dispersion-relation fluorescence spectroscopy(DFS) 5

5.1 Introduction

Trafficking inside live cells is the result of both passive diffusion and active or molecular-motor-

driven processes (see, e.g., Ref. [136]). In order to understand intracellular trafficking, one must

be able to distinguish these types of dynamics with sufficient spatial and temporal resolution so as

to identify the structure of intracellular transport networks and monitor their changes over the

course of the cell cycle. Experimentally, this task is challenging due to the multitude of temporal

and spatial scales involved. Diffusion of fluorescently-tagged molecules has been studied

successfully by fluorescence correlation spectroscopy (FCS) [137-142] and fluorescence recovery

after photobleaching (FRAP) [143-146]. In this case, the temporal scales are in the range of s-ms

and the spatial scale is fixed by the excitation beam size. Image correlation spectroscopy (ICS)

[147], spatiotemporal image correlation spectroscopy (STICS) [148], and raster image correlation

spectroscopy (RICS) [149] have been successfully developed to extract information about

fluorophore transport. STICS complements ICS in the sense that it allows measuring the direction

of the velocity, in addition to its magnitude. RICS extends ICS to faster diffusion temporal scales.

Note that all these methods use confocal scanning imaging.

For studying the transport of larger objects in the cell, e.g. organelles and vesicles, particle

tracking has been used successfully [150-152]. However, the cell contains many extended objects

or continuous media, such as actin cytoskeleton, which, when viewed on scales large compared to

its mesh size, cannot be resolved into separately traceable objects. For this reason, the

spatiotemporal fluctuations of such continuous media cannot be investigated by particle tracking.

5 This chapter appeared in its entirety (with some modifications) in Ru Wang, Lei Lei, Yingxiao Wang, A. J. Levine,

and Gabriel Popescu, Dispersion-relation fluorescence spectroscopy, Phys. Rev. Lett., 109 (18), 188104 (2012).This

material is reproduced with the permission of the publisher.

56

In response to the challenge presented by having to track intracellular dynamics of a broad range

of spatial and temporal scales, we have recently developed a label-free method that can be used to

study transport in live cells over a broad range of spatiotemporal scales [153, 154]. This technique

uses quantitative phase imaging [155], works with intrinsic contrast and, thus, can be used

indefinitely, without restrictions due to photobleaching or photoxicity. However, it lacks

specificity, i.e., it cannot report on the transport of a specific chemical species or structures.

5.2 Principle and validation of DFS

In this chapter, we propose a fluorescence imaging approach for studying the dynamics of

specific structures, both continuous and discrete. This method, referred to as dispersion-relation

fluorescence spectroscopy (DFS), uses data collected via time-resolved fluorescence full-field

imaging. Therefore, unlike FCS or FRAP, DFS renders temporal as well as spatial information.

Because it operates in the frequency domain, DFS is complementary to other measurements

developed for the same purpose, such as ICS [147], STICS [148], RICS [149]. Analyzing the

frequency (k-) domain, i.e., the dispersion relation of the dynamic transport, is ideal for

understanding the physical phenomenon because it directly relates to the governing equation of

motion. This allows studying the dynamic properties of intracellular transport in a scale dependent

manner. For example, as detailed below, we are able to identify directed and diffusive transport at

different spatial scales in living cells.

From the 3D fluorescence data (x, y, t), we calculate (q), the wavenumber-dependent

decay of temporal correlations in the fluorescence signal. The (q) curve is a fundamental

characteristic of the transport phenomenon under investigation. We fit this dispersion relation

using a model based on the diffusion-advection equation, but one in which we assume that the

57

decay of correlations results from incoherent sum of the advective (i.e., actively driven) motion of

many different intracellular elements, such as vesicles or organelles with a distribution of

advection velocities. This leads to an expression for the normalized density autocorrelation

function of the form

2

( , )i vq Dqg q e e

0v q , (5.1a)

and the decay rate

2( )q vq Dq . (5.1b)

In Eqs. 5.1a-b, q q

is the wavenumber, D is the diffusion constant of cargoes in the intracellular

medium. In order to generate the observed linear-in-q decay rate of correlations, we assume a

Lorentzian distribution of advection velocities with a mean v0 and a width Dv

.

In order to further validate this interpretation of the data, we demonstrate DFS on standard

samples. We recorded time lapses of fluorescently labeled polystyrene beads both under Brownian

motion (Figs. 5.1a-b) and drift (Figs. 5.1c-d). These 1.2 µm diameter fluorescent beads were

suspended in water between two coverslips and imaged under fluorescence microscopy at

acquisition rates of 23 Hz (Fig. 1a) and 0.2 Hz (Fig.1c), up to 256 frames and 128 frames,

respectively. Figures 5.1b and 5.1d show the trajectories of individual beads corresponding to

Figures 5.1a and 5.1c, respectively.

The mean-squared displacements (MSD) and DFS signals associated with the particles in

Fig. 5.1 are shown in Fig. 5.2. The MSD for the Brownian particles (Fig. 5.1a-b) were obtained by

averaging the trajectories of 229 particles. Figure 5.2a shows a linear fit of this MSD, which yields

a diffusion coefficient 0.27 0.05D µm2/s. In order to compute the dispersion relation, (qx, qy),

from the fluorescence movie (Fig. 5.1a), we first performed the spatial Fourier transform of each

58

frame, then calculated the temporal bandwidth, Γ, at each spatial frequency (qx, qy). The (qx, qy)

map is shown in Fig. 5.2b. We then performed the azimuthal average to obtain the radial profile

2 2( ), x yq q q q . This experimental curve exhibits the expected q2 dependence associated

with diffusion (Fig. 5.2c). The resulting diffusion coefficient is 0.21 0.04D µm2/s and

compares very well the value obtained via particle tracking.

For the case of the drift motion shown in Fig. 5.1c, the squared displacements along x direction,

2( )x , and y direction, 2( )y , are shown in Fig. 5.2d. The resulting components of the

drift velocity are 32.5xv µm/s, 8.8yv µm/s, obtained by fitting a quadratic curve. This drift

velocity produces a shift in the power spectrum at frequency ( ) x x y yv q v q q , which is

essentially a q-dependent Doppler frequency shift. We projected angular frequency versus wave

vector components qx (Fig. 5.2e) and qy (Fig. 5.2f). The corresponding slope yields the drift

velocity 33.3xv µm/s and 9.1yv µm/s, which compare very well with the results of particle

tracking.

59

Fig.5 1a) Fluorescence images of 1.2µm polystyrene fluorescence beads in water under Brownian

motion. b) Trajectories of individual beads in 1a). c) Fluorescence images of 1.2µm polystyrene

fluorescence beads drifting from left to the right. d) Trajectories of individual beads in 1c).

a)

b)

c)

d)

60

Fig.5.2a Mean squared displacement (MSD) obtained by tracking individual beads in figure 5.1a).

b) Decay rate vs. spatial mode, (qx, qy), associated with the beads in a. The dash ring indicates

the maximum q values allowed by the resolution limit of the microscope. c) Azimuthal average of

data in b) to yield (q). The fits with the quadratic function yields the value of the diffusion

coefficient as indicated. d) Mean squared displacement along x direction, , and y direction of

beads in Fig. 5.1c). e) Temporal frequency vs spatial frequency. The slope yields the drifting

1 2 3 4 5 6 70

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10

1

2

3

4

5

6

7

8

9

10

MSD

[µ

m2]

MSD(τ )

4Dτ,D=0.27 0.05 µm2/s

τ[s]

q [rad/µm]

Γ[r

ad/s

]

Γ(q)

Dq2,D= 0.21 0.04 µm2/s

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0

5000

10000

< x2

>/<

y2>[

µm

2]

τ[s]

<x2>(τ ),2 2

x xv v = 32.5μm / s

<y2>(τ ),2 2

y yv v = 8.8 μm / s

78.42o

53.13o

, 33.3 /m sx x x

v q v

, 9.01 /m sy y y

v q v

a)

b)

c)

d)

e)

f)

(qx, qy) x

y

xq

yq

61

velocity along x direction. f) Temporal frequency vs spatial frequency. The slope yields the drifting

velocity along y direction.

5.3 Application of DFS in live cells

To prove that DFS works equally well for live cells, we performed experiments in S2

drosophila cells, whose peroxisomes were labeled with green fluorescent protein. The images

were acquired at rate 1 sec/frame up to 2 minutes with epi-fluoresce microscopy, as shown in Fig.

5.3a. MSD of peroxisomes were calculated after their trajectories were recorded with the ImageJ

software, as shown in Fig. 5.3b. The linear fit of MSD yields the diffusion coefficients

3(7 6) 10D µm2/s (Fig. 5.3d). We applied DFS to the acquired images and found that the

motion of peroxisome were diffusive at time scales of the order of seconds, with 3(9 1) 10D

µm2/s (Fig. 5.3b), which agrees with that obtained from particle tracking. Thus, DFS can be easily

implemented to study motions of discrete particles in live cells, without the need for particle

tracking.

62

Fig. 5.3a) Fluorescence images of culture of S2 drosophila cells whose peroxisomes were labeled

with GFP. b) Dispersion curve measured for the cell in a. The quadratic curve fitting indicates

dominant diffusion motion. And the fitting coefficients yields the diffusion coefficient. Inset

shows the (qx, q

y) map. c) Trajectories of peroxisomes in a). d) Mean squared displacement

(MSD) obtained tracking the trajectories of peroxisomes. The linear relationship indicates

diffusive motion and fitting coefficient gives rise to the corresponding diffusion coefficient.

q [rad/µm]

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Γ(q)

Dq2,D= 0.009 0.001µm2/s

Γ[r

ad/s

]

τ[s]

0 20 40 60 800

1

2

3

4

MSD

[µ

m2]

MSD(τ )

4Dτ,D=0.007 0.006µm2/s

a)

b) d)

c)

63

Note that, since DFS relies on analyzing fluctuations in the density of fluorophores rather

than particle tracking, it applies equally well to continuous mass distributions, such as cytoskeletal

structures. To explore such a physical problem, we studied mouse embryonic fibroblasts (MEFs)

with their actin cytoskeleton labeled with GFP, as shown in Fig. 5.4a. The actin bundles are clearly

visible, aligned along the line 1. We acquired 256 fluorescence images at a rate of 1 frame/s. The

inset of Fig. 5.4b shows the dispersion map, ( , )x yq q , which is calculated from those acquired

images. Clearly, this dispersion map is no longer isotropic, which is due to the fact that the

transport in directions parallel and perpendicular to the actin bundles differs. We identify from the

images the directions parallel1 ,and perpendicular

2 to the actin bundles and determine the

dispersion curves for motion projected onto these axes. These dispersion relations are plotted on a

logarithmic scale in Fig. 5.4b. As might be expected, the results indicate that the transport along

the actin fibers is directed, as the dispersion curve ( )q displays q1 dependence. By contrast, the

transport perpendicular to the actin fibers is dominated by diffusion, as indicated by the q2

dependence. These findings are consistent with the well-known phenomenon of “tread milling” in

which seemingly stable actin filaments are actually continually polymerizing at one end and

depolymerizing at the other [156-158]. This active process leads to directed 1D transport of actin

monomers down the filaments. Transport of actin monomers perpendicular to fiber direction,

however, does not result from this driven process so that mass transport along 2 must be

controlled by thermal diffusion.

The reason why q and q2 curves flatten at the far right part is due to the limited acquisition

rate (the Γ range is limited at high-values). At that small scale (high q part), the fast dynamics is

not sampled fast enough due to finite frame rate. We kept the acquisition rate low purposely to

reduce the amount of exposure and the probability of photobleaching. By doing this, we can

64

distinguish two modes of transport mechanism in the slow dynamics/large scale (low q part). While

photobleaching is an important photochemical phenomenon which usually complicates the

biological observation and is especially problematic in time-lapse microscopy, it can also be

exploited to study the diffusion of molecules in approaches such as FRAP. In this letter, we

simplified the investigation by keeping low exposure, to the point where photobleaching effects

are negligible. Thus the signals we observe fully represent the movement of fluorophores and

cellular components that they attach to.

Fig. 5.4a) Fluorescence images of culture of mouse embryonic fibroblast (MEF) whose actin were

labeled with GFP. b) Dispersion curve measured for the cell in a. The black and red lines indicate

directed motion and diffusion along and direction respectively. Inset shows the (qx, q

y) map

associated with a).

1 2 3 4 5 6 7 8 9100.01

0.1

1

q [rad/µm]

Γ[r

ad/s

]

θ1

θ2

θ1

θ2

θ1

θ2

q

q2

a) b)

65

5.4 Discussion

In summary, in this chapter we proposed dispersion-relation fluorescence spectroscopy

(DFS) to study full-field fluorescence signal fluctuations caused by mass transport. The specific

structures of interest are labeled with a fluorophore whose motions give rise to fluorescence

intensity fluctuations that are further analyzed to quantify the governing molecular mass transport

dynamics. These data are characterized by the effective dispersion relation in the form of a power

law, qq ~ , which describes the relaxation rate of the spatial mode q (1/Γ describes the

characteristic time of moving particles to travel an 1/q mean distance). This dispersion should not

be confused with optical dispersion. “Spectroscopy” refers to measuring spatiotemporal

frequencies associated with mass transport and should not be confused with optical spectroscopy.

We have demonstrated that DFS can distinguish diffusive from directed (or advective)

motion both in model system where we were able to independently confirm the values of the

diffusion constant and advection velocity extracted via DFS. Turning to cellular systems, we found

that DFS can be used to make similar distinctions in intracellular mass transport. By studying the

transport of monomeric actin in an oriented filament network, we were able to use DFS to observe

directed transport along the filaments, consistent with tread milling, and diffusive transport in the

direction perpendicular to them.

Measuring the dispersion relation itself informs about the spatial scales at which each of

the two types of transport, diffusion and advection, is dominant. This description comes naturally

because each spatial frequency is characterized by a specific temporal frequency. In the k-

representation, the two regimes of transport are directly separated even though their respective

time constants may differ by orders of magnitude (see, e.g., Fig. 5.4b). On the other hand, the

space-time analysis (correlation calculations) of ICS, RICS, and STICS gives an average effect

66

over all frequencies. In this case, mathematically the fitting function is a temporal convolution

between the diffusive and advective terms. Thus, the two contributions overlap in this

representation. Unless they have comparable correlation times, the diffusive and directed transport

cannot be readily distinguished (Ref. [148] discusses extensively this issue). Furthermore, the

frequency domain representation automatically excludes the effects of the static structures in the

specimen (=0 contributions), which was discussed in detail by Brown et al. [159]. Finally, DFS

works with full field fluorescence imaging (not confocal), which allows collecting data from all

points simultaneously and simplifies the analysis.

DFS can be used with particles that cannot be individually resolved, i.e. that are smaller

than the diffraction spot of the microscope, as long as the particles travel over distances larger than

the diffraction spot. This is typically the case of interest when studying mass transport in live cells.

The capability of performing these measurements in a live system provides a new window into the

physics of out of equilibrium systems. In the chapter 6, we are using DFS to study the cytoskeleton

dynamics under the influence of various protein-motor inhibitors. Besides quantifying transport

characteristics of molecules, in principle DFS is also capable to estimate the number of fluorescent

molecules that contribute to each motion (spatial) component. In order to estimate quantitatively

the concentration of fluorescent molecules we need to perform a series of calibrations, which is

subject to our future work.

67

CHAPTER 6. Cytoskeleton Fluctuations with DFS

6.1 Introduction

Mass transport along actin and microtubules within the live cell plays a crucial role in

regulating cell function, from cell differentiation and motility, to mitosis and apoptosis [160].

Studying cytoskeletal dynamics is extremely interesting not only at the basic science but also

clinical level, as it may open new avenues to treatment of important disease [161]. For example,

disruption of cargo transport along neural processes can lead to neurodegenerative diseases such

Alzheimer’s [162, 163].

Despite great progress on the subject, the role of cytoskeleton in intracellular transport

remains insufficiently understood [164-166]. Trafficking at intracellular level results from thermal

diffusion as well as active motion driven by molecular motors [167]. This process has been

examined through techniques that generally rely on fluorescence imaging [168], e.g., Fluorescent

Speckle Microcopy (FSM) [169], Fluorescence correlation spectroscopy (FCS) [170],

Fluorescence resonance energy transfer (FRET) [171], Fluorescence recovery after

photobleaching (FREP) [172], Fluorescence Lifetime Imaging Microscopy (FLIM) [173].

Recently, there has been significant progress in superlocalization microscopy such as STORM

[174], PALM [175, 176] and their precursor, FIONA [77], which are closing the gap between the

cellular and molecular scale biology. In combination with fluorescence microscopy, particle

tracking is a widely used approach for studying the transport of discrete objects inside the cell,

such as embedded particles, organelles, and vesicles [78, 81, 177, 178]. Analyzing the trajectories

of these particles in terms of mean squared displacements over a significant ensemble provides

valuable information about the nature of transport and even the rheology of the surrounding

medium. However, this approach has limited applicability when the system of interest is

68

continuous, such as actin and microtubule cytoskeleton at spatial scales larger that the mesh size.

Furthermore, particle tracking does not contain explicit spatial information.

In order to meet this experimental challenge of investigating mass transport in live cells

over broad spatiotemporal scales, in the chapter 5 we have proposed a new approach, referred to

as dispersion-relation fluorescence spectroscopy (DFS) [179]. This method analyzes the time-

lapse, full-field fluorescence images to obtain the dispersion relation associated with the dynamic

transport. This dispersion relation is in the form of a power law, ~q q , which describes the

relaxation rate (in rad/s) of the spatial mode q (in m/s). Physically, 1/Γ describes the correlation

time associated with a spatial disturbance of wavelength 1/q. Importantly, the dispersion curve

allows us to distinguish between diffusion and directed transport and the specific spatiotemporal

scales at which each is dominant.

In our studies, we used time-lapse imaging of green fluorescence protein (GFP) fused to

actin and microtubules to study their spatiotemporal fluctuations. We found that DFS is capable

of revealing subtle changes in the cytoskeleton dynamics associated with the integrity of the

polymerization process. Specifically, we demonstrate that DFS can quantitatively monitor the

transition from deterministic to random transport when the polymerization of actin and

microtubule filaments was blocked. Our studies indicate that cells have an interesting mechanism

for active transport of actin along microtubules.

69

6.2 RESULTS

We used DFS to study actin dynamics under the effect of cyto-D. GFP-fused actin of

mouse embryonic fibroblast (MEF) cells was imaged continuously using epi-fluorescence (see

Materials and Methods for details on GFP transfection and imaging). Time lapse images were

acquired over 4 min at 1 frames/s. Individual images of normal cells revealed the actin bundles

oriented along certain directions, as illustrated in Fig. 6.1a. Consequently, the dispersion map (Fig.

6.1d) is also anisotropic, indicating that transport along the bundles is significantly different than

that across them, as expected. Interestingly, along the filament bundles, DFS reveals deterministic

transport, as indicated by the linear dependence of the vs. q curve (black squares in Fig. 6.1g).

By contrast, the dynamics across the actin filaments is diffusive, q2. This result suggests that

the continuous polymerization process at the end of the actin filaments is an active process

governed by ATP consumption rather than thermal noise. In order to test this hypothesis, we

imaged the same cells after adding cyto-D, which blocks the polymerization process. Clearly,

Cyto-D destroys the actin filamentous structure, as evidenced by Fig. 6.1b. As a result of crumbling

the filaments, the dispersion map becomes isotropic (Fig. 6.1e). More importantly, the dispersion

relation indicates a clear q2 behavior and, thus, Brownian transport (blue circles in Fig. 6.1g).

Further, as a control experiment, we washed out the cyto-D using culture medium and incubated

the cells at 37o and 5% CO2 for one hour. After the wash out, the actin structure recovered (Fig.

6.1c) and, remarkably, the transport along the filaments returned to deterministic behavior (red

triangles in Fig. 6.1g).

These experiments demonstrate that the active transport is strongly dependent on the

integrity of the actin filaments. At the same time, DFS appears to be a valuable tool for studying

such dynamics, where particle tracking methods are limited. Note that, at high-q values, the curves

70

in Fig. 6.1g deviate from the power laws of q1, q2 and level off. This is because of the finite

acquisition rate of our measurement, which limits the values of maximum . The q1 curves level

at lower q-values due to the faster transport process.

71

Figure 6.1. a) Actin-GFP image of a transfected MEF cell. b) The same cell as in a imaged after

Cyto-D was added into the cell dish. c) The same cell as in a and b imaged after the cyto-D was

θ1θ2

Normal actin Cyto-D Wash-out

1 2 3 4 5 6 7 8 9100.01

0.1

1

q

q2

Normal

Cyto-D

Wash-out

Γ[r

ad/s

ec]

q [rad/µm]

a b c

d e f

g

50µm 50µm 50µm

72

washed out. After 2 hours, the actin filament structure gradually recovered. d)-f) The dispersion

map, Γ(qx, qy), associated with the time lapse images a)-c); g) The dispersion curves along fiber

directions (red line in a and c) were plotted on the same graph. The black squares represent the

normal MEF shown in a) and exhibits linear dependence (directed transport). The blue circles

correspond to b where the cells were treated with cyto-D and exhibit a quadratic dependence

(diffusion). The red triangles represent the cell in c, recovered after washing out cyto-D. Note that

the normal and recovered cell exhibit identical dispersion relation, which indicates that the

dynamics is fully recovered as a result of wash-out. All time lapse images were acquired at 1

frame/s for 4.25 minutes using epi-fluorescence microscopy.

Next, we tested the sensitivity of DFS to microtubule dynamics. We performed

experiments on GFP-labeled microtubules of MEFs under normal and nocodazole-treatment

conditions (Figure 6.2). The fluorescence images were taken with the same acquisition rate, 1

frame/s, for 4 minutes. Figures 6.2a and 6.2d show the image and DFS map, respectively, of a

normal MEF with its microtubules labeled by GFP. The dispersion curve (Figure 6.2g) indicates

that the transport along microtubules filaments is directed (q). The same cell is shown in Fig.

6.2b after nocodazole was added into the cell petri dish to depolymerize the microtubules.

Interestingly, while the image itself is not informative about the microtubule polymerization state,

the DFS map, i.e., Fig. 6.2e, reveals significant changes from its normal counterpart, shown in Fig.

6.2d. Due to the nocodazole, the DFS map becomes narrower and isotropic. This transition from

anisotropy to isotropy indicates that the microtubules are characterized by polarity, which plays a

role in intracellular transport. The effect is further illustrated in Fig. 6.2g, where profiles along the

direction indicated in Fig. 6.2a are presented. Note that, after adding the drug, the dispersion curve

73

flattened (blue-circle curve). Again, we performed a control experiment where we washed out the

drug with fresh medium and kept the cells at 37 degree temperature and 5% CO2 for approximately

an hour. This washout procedure recovers the filament structure. After the washout, the dispersion

curve (red triangles in Fig. 6.2g) overlapped perfectly with the one before treatment (black square),

indicating that the microtubule active dynamics also recovered.

74

Figure 6.2. a) Tubulin-GFP image of a transfected MEF cell. b) The same cell as in a imaged after

Nocodazole was added into the cell dish. c) The same cell as in a and b imaged after the nocodazole

θ1

Normal tubulin Nocodazole Wash-outa b c

d e f

1 2 3 4 5 6 7 8 9100.01

0.1

1

before

noco

washout

Normal

Nocodazle

Wash-out

q

q [rad/µm]

Γ[r

ad/s

ec]

g

50µm 50µm 50µm

75

was washed out. d)-f) The dispersion map, Γ(qx, qy), associated with the time lapse images a)-c);

g) The dispersion curves along fiber directions (red line in a ) were plotted on the same graph. The

black squares represent the normal MEF shown in a) and exhibits linear dependence (directed

transport). The blue circles correspond to b where the cells were treated with nocodazole, causing

the cure to flaten out. The red triangles represent the cell in c, recovered after washing out the

nocodazole. Note that the normal and recovered cell exhibit identical dispersion relation, which

indicates that the dynamics is fully recovered as a result of wash-out. All time lapse images were

acquired at 1 frame/s for 4.25 minutes using epi-fluorescence microscopy.

The normal-block-recover experiments demonstrated the sensitivity of DFS to cytoskeleton

dynamics. Next, we studied in more detail the effect of the two drugs as it develops in time. Figure

6.3 illustrates the gradual effect of nocodazole used to block the microtubule polymerization. In

order to monitor this highly dynamic process, we recorded time-lapse image sets at intervals of 20

minutes. Each fluorescence data set (Fig. 6.3 a-d) was taken at the same acquisition rate of 1

frame/s over 4 minutes. It can be seen that, due to nocodazole, the DFS maps gradually became

isotropic (Fig.6.3 e-h). The dispersion relation was analyzed along two particular directions,

marked by the black and red line in Fig.6.3 e-h; the azimuthal average is also shown. These

dispersion curves consistently flattened out after the polymerization of tubulin was disrupted by

nocodazole, as shown in Fig. 6.3 i-l. As the microtubule filaments are broken down, the resulting

monomers are smaller in size and, thus, move faster in the solution. Since our acquisition rate and,

76

therefore, maximum , is constant, the effect of the drug is to gradually level off the DFS curves.

Figure 6.3. a) Tubulin-GFP image of a transfected MEF cell. b-d) The same cell at different points

in time, as indicated, after adding nocodazole. e)-h) Dispersion maps Γ(qx, qy) associated with

images a)-d) with two axes of symmetry shown; i)-l) Dispersion relation curves plotted along the

two axes (1, 2), and azimuthally-averaged, as indicated. Before treatment linear dependence

(deterministic motion) was dominant. Under nocodazole treatment, the curves flatten out

progressively. All time lapse images were acquired at 1 frame/s for 4.25 minutes using epi-

fluorescence microscopy.

1 2 3 4 5 6 789100.01

0.1

1

1 2 3 4 5 6 789100.01

0.1

1

1 2 3 4 5 6 789100.01

0.1

1

1 2 3 4 5 6 789100.01

0.1

1

q [rad/µm] q [rad/µm] q [rad/µm] q [rad/µm]

Γ[r

ad/s

ec]

Normal MEF Nocodazole (T=0)

a) b) c) d)

~ q1

a b c d

T = 20min T = 40min

e f g h

θ1

θ2θ1 θ1 θ1

θ2 θ2θ2

θ1

i j k l

θ2averaged

θ2averaged

θ1θ2averaged

θ2averaged

θ1 θ1

~ q1

50µm 50µm 50µm 50µm

77

We performed an analog experiment to study the progression of the cyto-D effect on actin.

The results are summarized in Fig. 6.4 and reveal an unexpected behavior, as follows. Immediately

after adding cyto-D (Fig 6.4b), not surprisingly, the actin filament transport transitioned from a

combination of directed and diffusive motions (Fig 6.4i) to a largely diffusive motion (Fig 6.4j).

However, after approximately 20 minutes, the data indicated directed transport again, as illustrated

by the q1 dependence in Fig. 6.4k. Imaging the cell over a longer period of time did not reveal a

tendency to return to diffusion. This result was unexpected. We hypothesized that this effect was

due to the actin fragments that became attached to the microtubules and, thus, were subjected to

active transport. In order to test this hypothesis, we ran a double-treatment test, i.e., after the actin

polymerization was blocked with cyto-D (Figs. 6.5b and c), the microtubule polymerization was

further disrupted with nocodazole (fig 6.5d, e and f). Thus, following cyto-D- induced transition

from directed motion, (q1 in Fig 6.5n) to diffusive transport (q2 on Fig 6.5o) and directed again (q1

on Fig 6.5p), the actin filament transport eventually became purely diffusive (q2 Fig 6.5q and 6.5r)

after adding nocodazole. These results indicate that along with the microtubule structures

dissociation, the actin fragments eventually dissolved in the cytoplasm and underwent random

motion, thus proving our hypothesis.

78

Figure 6.4. a) Actin-GFP image of a transfected MEF cell. b-d) The same cell at different point in

time, as indicated, after adding cyto-D. e)-h) Dispersion maps Γ(qx, qy) associated with images

a)-d) with two axes of symmetry shown; i)-l) Dispersion relation curves plotted along the two axes

(1, 2), and azimuthally-averaged, as indicated. Immediately after cyto-D was applied, the

transport becomes diffusive (q2). However, with time, the deterministic trend (q1) is observed again

(k, l). All time lapse images were acquired at 1 frame/s for 4.25 minutes using epi-fluorescence

microscopy.

Normal MEF Cyto-D (T=0) T = 20 minutes T = 40 minutes

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-4

1E-3

0.01

0.1

1

q [rad/µm] q [rad/µm] q [rad/µm] q [rad/µm]

[r

ad/s

ec]

~ q1

~ q1

~ q2

~ q2

~ q1

a b c d

e f g h

θ1 θ2

θ1

0.1 1 101E-3

0.01

0.1

1

θ1

θ1 θ1

θ2θ2

θ2 θ2

averagedaveraged

averaged averaged

i j k l

50µm 50µm 50µm 50µm

79

Figure 6.5. a) Actin-GFP image of a transfected MEF cell. b-c) The same cell at different points

in time, as indicated, after adding cyto-D. e-f) The same cell at different points in time after adding

nocodazole. g-l) Dispersion maps, Γ(qx, qy), associated with images a)-f), with two axes of

symmetry shown in g; m-r) Dispersion relation curves plotted along the two axes (1, 2), and

azimuthally-averaged, as indicated. Immediately after cyto-D was applied, the transport becomes

diffusive (o) and, later, deterministic trend (p). However, after the nocodazole treatment, diffusion

becoems dominant (q, r). All time lapse images were acquired at 1 frame/s for 4.25 minutes using

epi-fluorescence microscopy.

In summary, we used the dispersion relation to study actin and microtubule polymerization

and depolymerization dynamics in live cells. DFS is a full-field and tracking-free method, which

makes it ideally suitable for studying any compartment’s dynamics inside living cell, whether

continuous and discrete. The ability to distinguish between diffusion and deterministic transport

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

0.1 1 101E-3

0.01

0.1

1

q [rad/µm] q [rad/µm] q [rad/µm] q [rad/µm] q [rad/µm] q [rad/µm]

[r

ad/s

ec]

Normal MEF Cyto-D (T=0) T=20minutes Nocodazole(T=23 minutes) T = 53 minutes T = 83 minutes

~ q1

~ q2

~ q1

~ q2 ~ q1

~ q1

~ q2

a b c d e f

g h i j k l

m n o p q r

θ1

θ2

θ1

θ2

averaged

50µm 50µm 50µm 50µm 50µm 50µm

80

over broad spatiotemporal scales promises to shed new light on understanding many cell functions.

The specific study presented here on cytoskeleton dynamics under the influence of various protein-

motor inhibitors suggests the cross-talk between actin and microtubule transport. Since many basic

and clinical questions related to actin and microtubule-mediated transport are insufficiently

understood, we anticipate that DFS will contribute meaningfully to such studies in the future.

Materials and Methods

Cell culture

Mouse embryonic fibroblasts (MEFs) were obtained from American Type Culture Collection (ATCC).

These cells were cultured in a 37 C incubator with humid atmosphere, 5% CO2 and 95% air and were

grown in DMEM media supplemented with 100 g/ml ampicillin, 100 g/ml streptomycin, 2 g/L sodium

bicarbonate, plus 10% fetal bovine serum (FBS).

Transfection, cell pretreatment

MEFs were transfected with lipofectamine 2000 (Invitrogen USA) and plated on glass bottom dish and

cultured for 36-48 h in DMEM medium with 10% FBS. During the imaging process, the cells were

maintained in CO2-independent medium (Gibco BRL) with 10 % FBS. 1 M Cytochalasin D, Nocodazole

were added during imaging.

Time-lapse epi-fluorescence imaging

To perform time-lapse epi-fluorescence imaging, we used a motorized Zeiss microscope (Z1

model) equipped with 63x/1.4 oil-immersion objective. During time-lapse imaging the cells were

81

maintained in an incubator with both heat and CO2 control. The camera is PhotonMAX-EMCCD

from Princeton Instruments and the illumination source in fluoresce microscopy is mercury arc

lamp HBO 100. The filter set used for imaging MEF cells is GFP/FITC with excitation wavelength

of 495 nm and emission wavelength 517nm, respectively.

82

CHAPTER 7. Phase-resolved low coherence interferometry6

7.1 Introduction

Soft materials, such z polymer solutions, colloids, and biological matter, are characterized

by multiple characteristic spatiotemporal scales and, as a consequence, their dynamic response to

external strains is non-trivial [180]. Microrheology studies the mechanical response of such

materials at the microscopic scale. This type of investigation is relevant to a number of fields,

especially biology, where cell mechanics is essential for the proper functioning of a living system

[181]. It has been shown that optical methods (e.g., dynamic light scattering, fluorescence

correlation spectroscopy, optical trapping) are uniquely suitable for studying dynamic properties

of soft-matter [182]. Recent microrheology techniques rely on applying strain to the fluid through

embedded ‘probe’ particles, either undergoing thermal (Brownian) or externally driven motion

[78, 183]. In such techniques, the statistics of particle motion is described via the ensemble-

averaged mean squared displacements, which requires the recording of multiple trajectories.

Here we present measurements of the nanoscale surface fluctuations associated with small

fluid droplets at thermal equilibrium. Using the fluctuation-dissipation theorem [184], these

motions are used to infer the mechanical properties of the fluid. Our experimental setup is

implemented with optical fibers and relies on phase-resolved interferometry with broadband fields.

Amplitude-based low-coherence interferometry (LCI) has been used in the past for performing

dynamic light scattering and microrheology measurement [67, 185-188]. However, such

amplitude-based techniques are not sensitive enough to detect quantitatively nanometer scale

6 This chapter appeared in its entirety (with some modifications and additional material) in Ru Wang, Taewoo Kim,

Mustafa Mir, and Gabriel Popescu, “Nanoscale Fluctuations and Surface Tension Measurements in droplets using

Phase-resolved Low-coherence Interferometry”, Appl. Opt. (Special Issue on Holography), 52 (1), A177-A181

(2013). This material is reproduced with the permission of the publisher and is available using the ISBN: 978-0-444-

59422-8.

83

motions; quantitative phase measurements are necessary instead [189, 190]. We show that

resolving the phase of the field allows us to quantify these fluctuations and extract the respective

surface tension coefficient of the liquid.

7.2 Experimental system

Our experimental system is described in Fig. 7.1. Light from a swept source, i.e., a laser

whose emission wavelength is sweeping a 100 nm spectral range around 0=1325 nm at frequency

of 8kHz, is coupled into a single mode fiber coupler. A droplet of interest is in contact with the

end of the single-mode optical fiber such that it is suspended in air. The optical field emerging

from the fiber core is partially specularly reflected at the fiber/fluid interface, while the transmitted

component propagates through the droplet, reflects at the fluid-air interface, and propagates back

into the fiber. Thus, the specular field, U1, can be considered as a reference for the sample field,

U2, which carries information about the position of the droplet surface. The two fields propagate

back through the output of the 1x2 fiber coupler and to the detector, which is a conventional OCT

balanced detector. The intensity measured by the detector at each optical frequency has the form

2

1 2 1 2 12( ) ( ) ( ) 2ReI U U I I W (7.1)

In Eq. 7.1, I1,2 are the intensities of the two fields and 12W is the cross-spectral density

associated with the two fields, which is defined as [24]

0

12 1 2 1 2( ) *( ) ( ) ( )i

W U U U U e

(7.2)

In Eq. 7.2, 0 2 /nd c , where n is the refractive index of droplet and d is the thickness of

the droplet, for the case of water n= 1.33 . τ0 indicates the average time delay associated with U2

due to propagation through the droplet and is the phase shift due to the droplet fluctuations

84

around the average thickness. Note that the two fields are identical in terms of frequency content,

except for the phase term and irrelevant scaling of the amplitude. Thus, the modulus of the cross

spectral density is the power spectrum of the original field, 2

1( ) ( )S U , and Eq.7.2 may be

written as

0

12 ( )i

W S e

. (7.3)

From Eq. 7.1 it is clear that experimentally we only have access to the real part of the cross-

spectral density. However, the imaginary part of W is obtained using a Hilbert transform as

12

12

Re '1Im '

'

WW P d

, (7.4)

where P indicates a principal value integral. From W12, we can now express the cross-

correlation function, 12 , via a Fourier transform

12 12

iW e d

. (7.5)

Combining Eqs. 7.3 and 7.5, we have

12 0 0( ) ie , (7.6)

Where 0 is the autocorrelation function of either field, the Fourier transform of which is

the spectrum, S. Thus, Eq. 7.6 indicates that the thickness of the droplet can be obtained from this

shift as 0 / 2d c n . The resolution in measuring this thickness is given by the modulus of 0 , i.e.,

the coherence length of the field, which is 17.6 µm in our case. However, the fluctuations of the

free surface can be measured with much better accuracy, as this information is carried by the phase

of the cross-correlation. Here, it is important to note that because the two interfering fields are

traveling along a common path, the phase relationship between them is extremely stable. The

standard deviation of the phase noise was measured by assessing the phase noise in the absence of

85

a sample. The resulting pathlength noise level was measured as 0.62 nm, which sets the limit on

the lowest pathlength fluctuation that the instrument can detect. This value translates to a

0.62/(2x1.33)=0.23 nm change in thickness associated with the droplet.

Figure 7.1 Phase-resolved Low-coherence Interferometry (pr-LCI) setup and the expanded view

of the fiber tip. The Gaussian field propagation inside the droplet is shown.

-

Swept

source

d

U2U1

Detector

86

7.3 Principle and Experimental validation

Our system performs 8,000 scans per second. Figure 7.2a shows an example of time-resolved

pathlength distributions obtained by measuring a water droplet. In Fig. 7.2b, we show a pathlength

profile at approximately t=30 ms, indicated by an arrow in Fig. 7.2a. The peak associated with the

reflection from the droplet is clearly observed at 656.6 m. The respective phase measurement is

shown in Fig. 7.2c. Outside the peak of interest, there are areas where the phase map suffers from

phase wrapping due to lack of signal. However, in order to resolve the droplet fluctuations, only

the phase information around the reflection peak is necessary. Such time-resolved phase traces are

shown in Fig. 7.2d for alcohol, water and oil droplets. Interestingly, the decay in time of the

pathlength signal is due to the evaporation of the droplets. As expected, the alcohol evaporates at

the fastest rate, i.e., 8.5 m/s, and the oil evaporation was measured to be 100 times slower.

To study the droplet fluctuations, we first fit the linear slope and subtract it from the data.

The inset of Fig. 7.2c. shows the fluctuation track after subtracting the evaporation contribution.

The power spectra of these fluctuations for water, alcohol, and oil droplets are shown in Figs. 7.3a-

c. Interestingly, all the power spectra for these liquids exhibit a power law behavior, with an

exponent of =-2. We fit these data with a model borrowed from light scattering by liquid surfaces

[70]. For a flat surface of a liquid at thermal equilibrium, the displacement, h¸ has the following

dependence on the in-plane wavenumber, 2 2

x yk k k , and vibrating frequency

2

0 0

2

0 0

(1 ) 1 2( , )

(1 ) 1 2

i t i th k

i y i t i t

, (7.7)

where 2/ 4y k , 2

0 / 2t k , and , , and represent the density, viscosity and surface

tension, respectively. Note that the Gaussian field incident on the droplet surface has its own spatial

dependence due to the original field at the end of the fiber and the propagation through the fiber.

87

Thus, the calculation of the field of interest U2 (Eq. 7.1) is performed in several steps. First, the

Gaussian field exiting the fiber, 2 2

0/

0

r wU r e

, (where w0 is the beam waist and 2 2r x y is the

radial coordinate) propagates through the droplet to yield field 0 ' 'U at the surface. Upon

reflection off the fluctuating surface, the field becomes 02 ( ', )

0 0'' ', ' 'i k h

U U e

, where k0

is the light mean wavenumber and’ is the conjugate variable to k. After Gaussian propagation

back to the fiber, we obtain the final field U2. Of course, the calculation is performed more

efficiently in the k variable, as in this case the Fresnel propagation of Gaussian beams reduces to

a product with a quadratic phase term. The final field U2 can be expressed symbolically as a

function of the initial field U0 and displacement distribution h:

2 2

2 0 0

0 0

( , ) exp( 2 ( , )** ( )exp( ) exp( )2 2

z dU k i k h k U k i k i k

k k

. (7.8)

In Eq. 7.8, symbol ** indicates a 2D convolution integral with respect to variable (kx, ky)

and d is the thickness of the droplet. Physically, this convolution indicates averaging of the high

frequency ripples in the surface (of high k values). The two products by 2

0

exp2

di k

k

indicate

Gaussian propagation back and forth through the droplet; plane wave terms, 0exp ik d , were

ignored as they do not contribute to the fluctuation spectrum. Finally, since the optical fiber

integrates the information spatially, the field U2 is integrated with respect to k from 0 to 10,000

rad/m which corresponds to about 1µm wavelength. The numbers used for this calculation for each

liquid is shown in Table 7.1. All the physical constants are given at room temperature 25oC and

the test oil sample was transparent oil which is widely used for immersion oil objective to increase

the resolution of a microscope.

88

The fluctuating phase of U2 is extracted as shown in Eq. 7.6 and its power spectrum is used to fit

the experimental data. As indicated by the high quality fit (r2=0.995), this theoretical model

describes the data very well. From the integral of the power spectrum, which is essentially the

variance of the fluctuation, the surface tension of ethanol aqueous solutions in different

concentrations is measured. As shown in Fig 7.4, the measured data agree with the expected values

within the experimental errors [191]. The error bars indicate variability from measurement to

measurement (N=10 for each concentration). Our errors are surprisingly high despite the

sensitivity of our measurement and their understanding requires further investigation. The major

possibility could be the surface fluctuation of droplet is very small (nm scale), so the environmental

air flow around it could greatly change its profile which might contribute large errors. And the

laser may increase the local temperature which might result in some variation of physical constants

we used to calculate the expected value.

Table 7.1. Physical constants of different liquids

n ρ

(kg/m3)

η

(mNs/m2)

σ

(mN/m)

Ethanol 1.36 789 1.2 22.39

Water 1.33 1000 1.0 72.8

Oil 1.518 1093 0.486 106.7

89

Figure 7.2. Result of a pr-LCI measurement on a liquid droplet. a) Modulus of the cross-correlation

function measured over 100ms. b) Vertical profile taken from a). The arrow indicates the droplet

thickness of 656.6µm. c) Phase of the cross-correlation function over 100 ms; inset: Horizontal

profile at the arrow after removing the slope. d) Horizontal profile taken from c) for ethanol, water,

and the immersion oil. The evaporation speed calculated from the slope of each profile is shown.

0 500 1000 1500 2000 2500 3000 3500 400010

-5

10-4

10-3

10-2

10-1

100

0.0 0.1 0.2 0.3 0.40

1000

2000

3000

4000

Pa

thle

ng

th (

nm

)

Time (sec)

0 20 40 60 80 100

3000

2500

2000

1500

1000

500

0

Path

len

gth

(m

)

Time (ms)

Power

0 20 40 60 80 100

3000

2500

2000

1500

1000

500

0

Path

len

gth

(m

)

Time (ms)

Phase

656.6µm

P(s

)

Pathlength (µm)

P(s, t)

a)

c)

b)

d)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-300

-200

-100

0

100

200

300

time (s)

path

length

(nm

)

Alcohol:α=8.5x103nm/sWater: α =7.3x102nm/sOil: α =89nm/s

90

Figure 7.3. Power spectra of the surface fluctuation (left) and the histogram of the fluctuation

amplitude distribution (right) for a) Ethanol b) Water c) Immersion oil. The red lines in the power

91

spectra show the calculated result from our model and the red lines in the histogram plots show

Gaussian fits (full width half maxima as indicated).

92

Figure 7.4. Estimated surface tension of water-ethanol mixture at various mole concentrations (red)

and expected value of water-ethanol mixture surface tension (black). Error bars were obtained

from 10 different measurements made for each concentration.

7.4 Discussion

In summary, we have introduced a measurement method for nanoscale surface thermal fluctuations

of liquid droplets using phase-resolved low coherence interferometry. With the aid of low

coherence fields, common-path interferometry and a fast sampling rate, the sensitivity to thickness

0 20 40 60 80 100

10

100

Expected

Measured

Su

rfa

ce

Te

nsio

n (

mN

/m)

Alcohol Mole Fraction (%)

93

fluctuations is a fraction of a nanometer. The data is modeled based on the fluctuations-dissipation

theorem to yield the surface tension of liquid droplets. The measurement of ethanol aqueous

solutions at different concentrations shows good agreement with the expected physical values. We

anticipate that such a fiber optic sensor will find important applications in studying the mechanical

properties of small quantities of fluids, e.g. cytoskeleton solutions or blood plasma, without the

need for probing beads.

94

CHAPTER 8. Future Work

8.1Phase Correlation Imaging (PCI)

The family of image correlation spectroscopy (ICS), such as spatiotemporal image

correlation spectroscopy (STICS), and raster image correlation spectroscopy (RICS), Dynamic

image correlation spectroscopy (DICS) provides powerful tools to extract information about cell

molecular transport. Note that all these methods use intensity-base confocal scanning imaging. As

introduced in Chapter 1, quantitative phase imaging directly measures cellular components’

density/refraction index without adding contrast agents, for example fluorophores, and thus its

correlation time at each spatial scale actually represents a full-field cell activity. Here we propose

for the first time a phase correlation method to analyze cell dynamics and functions.

When a spatial and temporal resolved phase ( , , )x y t data was acquired, at each spatial

coordinate r ( , )i ix y , correlation function ( )rG can be calculated as 2

( ) ( )( )

( )

r rr

r

t tG

t

.

Then correlation time 0 ( , )i ix y is defined as correlation function, ( )rG , 1/e decay time. And the

larger 0 indicated slower dynamics, while smaller 0 faster dynamics. Then Fig 8.1 b) suggests

that at celluar dynamics were faster at the cell outside membrane as well as nucleus membrane

region, which can be intuitively understood. In comparison with the Fig 8.1 d), whose actin

polymerization was disrupted with cyto-D, the dynamics of periphery cell membrane where

enriched with actin filaments slowed down a lot, which also can easily observed from its movie

Fig. 8.1-b.

Correlation time analysis were performed in space-time domain, and each spatial scale

convolved with all spatial wave vector values. To separate all wave vectors values, we carried on

a high-pass filter and low-pass filter on the correlation time images Fig. 8.1c) and d) and obtained

95

Fig 8.2, which resulted in high q/local scale and low q/large scale cell activity map. And compared

to the major difference between Fig. 8.2 a) and b), it looked very similar on Fig. 8.2 c) and d). This

results suggested that actin govern large scale cell dynamics. This case also demonstrates that

phase correlation image analysis provides a powerful tool to study cell functionality.

Figure 8.1 a) Normal glia cell movie imaged with SLIM using 40x/0.75 objective at 1sec/frame.

96

b) Add cyto-D to block the actin polymerization in a) and acquire a movie with SLIM with the

same objective at the same acquisition rate. c) And d) calculate the correlation time of the time

lapse phase fluctuations at each spatial pixels.

8

Fig 8.2 a) and b) are the low-pass filter results of Fig. 8.1 c) and d) while c) and d) are the high-

pass filter results of Fig. 8.1c) and b)

97

Reference

[1] D. Paganin and K. A. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light,"

Physical Review Letters, vol. 80, pp. 2586-2589, 1998. [2] D. Zicha and G. A. Dunn, "An Image-Processing System for Cell Behavior Studies in Subconfluent

Cultures," Journal of Microscopy-Oxford, vol. 179, pp. 11-21, Jul 1995. [3] C. H. Yang, et al., "Phase-dispersion optical tomography," Optics Letters, vol. 26, pp. 686-688, May

15 2001. [4] M. A. Choma, et al., "Spectral-domain phase microscopy," Optics Letters, vol. 30, pp. 1162-1164,

May 15 2005. [5] C. Joo, et al., "Spectral-domain optical coherence phase microscopy for quantitative phase-

contrast imaging," Optics Letters, vol. 30, pp. 2131-2133, Aug 15 2005. [6] V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of

diffraction tomography and a novel tomographic microscope," Journal of Microscopy-Oxford, vol. 205, pp. 165-176, Feb 2002.

[7] F. Charriere, et al., "Living specimen tomography by digital holographic microscopy: morphometry of testate amoeba," Optics Express, vol. 14, pp. 7005-7013, Aug 7 2006.

[8] F. Charriere, et al., "Cell refractive index tomography by digital holographic microscopy," Optics Letters, vol. 31, pp. 178-180, Jan 15 2006.

[9] W. Choi, et al., "Tomographic phase microscopy," Nature Methods, vol. 4, pp. 717-719, Sep 2007. [10] W. S. Choi, et al., "Extended depth of focus in tomographic phase microscopy using a propagation

algorithm," Optics Letters, vol. 33, pp. 171-173, Jan 15 2008. [11] G. Popescu, et al., "Optical measurement of cell membrane tension," Physical Review Letters, vol.

97, Nov 24 2006. [12] G. Popescu, et al., "Optical imaging of cell mass and growth dynamics," American Journal of

Physiology-Cell Physiology, vol. 295, pp. C538-C544, Aug 2008. [13] N. Lue, et al., "Live cell refractometry using microfluidic devices," Optics Letters, vol. 31, pp. 2759-

2761, Sep 15 2006. [14] H. F. Ding, et al., "Optical properties of tissues quantified by Fourier-transform light scattering,"

Optics Letters, vol. 34, pp. 1372-1374, May 1 2009. [15] J. W. Goodman, Statistical optics, Wiley classics library ed. New York: Wiley, 2000. [16] Z. Wang, et al., "Spatial light interference microscopy (SLIM)," Optics Express, vol. 19, pp. 1016-

1026, Jan 17 2011. [17] Z. Wang and G. Popescu, "Quantitative phase imaging with broadband fields," Applied Physics

Letters, vol. 96, pp. -, Feb 1 2010. [18] F. Zernike, "How I Discovered Phase Contrast," Science, vol. 121, pp. 345-349, 1955. [19] D. Gabor, "A new microscopic principle," Nature, vol. 161, p. 777, May 15 1948. [20] Z. Wang, et al., "Spatial light interference tomography (SLIT)," Optics Express, vol. 19, pp. 19907-

19918, Oct 10 2011. [21] E. Wolf, "Solution of the Phase Problem in the Theory of Structure Determination of Crystals from

X-Ray Diffraction Experiments," Physical Review Letters, vol. 103, Aug 14 2009. [22] E. Abbe, "Beitrage zur Theorie des Mikroscps und der mikrskopischen Wahrnehmung," Arch.

Mikrosk. Anat., vol. 9, 1873. [23] Z. Wang, et al., "Spatial light interference microscopy (SLIM)," Optics Express, vol. 19, pp. 1016-

26, Jan 17 2011. [24] L. Mandel and E. Wolf, Optical coherence and quantum optics. Cambridge ; New York: Cambridge

University Press, 1995.

98

[25] S. J. Wright and D. J. Wright, "Introduction to confocal microscopy," Cell Biological Applications of Confocal Microscopy, Second Edition, vol. 70, pp. 1-+, 2002.

[26] G. Popescu, et al., "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt. Lett., vol. 31, pp. 775-777, 2006.

[27] Z. Wang, et al., "Topography and refractometry of nanostructures using spatial light interference microscopy," Optics Letters, vol. 35, pp. 208-210, Jan 15 2010.

[28] Z. Wang, et al., "Label-free intracellular transport measured by spatial light interference microscopy," Journal of Biomedical Optics, vol. 16, Feb 2011.

[29] R. Barer, "Interference microscopy and mass determination," Nature, vol. 169, pp. 366-7, Mar 1 1952.

[30] R. Barer, "Determination of dry mass, thickness, solid and water concentration in living cells," Nature, vol. 172, pp. 1097-8, Dec 12 1953.

[31] M. Mir, et al., "Optical measurement of cycle-dependent cell growth," Proc Natl Acad Sci U S A, vol. 108, pp. 13124-9, Aug 9 2011.

[32] G. Popescu, et al., "Optical imaging of cell mass and growth dynamics," Am J Physiol Cell Physiol, vol. 295, pp. C538–C544, 2008.

[33] R. Wang, et al., "One-dimensional deterministic transport in neurons measured by dispersion-relation phase spectroscopy," Journal of Physics-Condensed Matter, vol. 23, Sep 21 2011.

[34] M. Mir, et al., "Blood testing at the single cel level using quantitative phase and amplitude microscopy," Biomed. Opt. Exp., vol. 2, pp. 3259-3266, 2011.

[35] Z. Wang, et al., "Tissue refractive index as marker of disease," Journal of Biomedical Optics, vol. 16, pp. 116017-1-5, 2011.

[36] N. Mohandas and P. G. Gallagher, "Red cell membrane: past, present, and future," Blood, vol. 112, pp. 3939-48, Nov 15 2008.

[37] R. Cotran, et al., Robbins pathologic basis of disease, 7 ed.: WB Saunders Company, 2004. [38] G. Bao and S. Suresh, "Cell and molecular mechanics of biological materials," Nature Mat., vol. 2,

pp. 715-725, 2003. [39] D. E. Discher, et al., "Molecular maps of red cell deformation: hidden elasticity and in situ

connectivity," Science, vol. 266, pp. 1032-5, Nov 11 1994. [40] H. Engelhardt, et al., "Viscoelastic properties of erythrocyte membranes in high-frequency electric

fields," Nature, vol. 307, pp. 378-80, Jan 26-Feb 1 1984. [41] M. Dao, et al., "Mechanics of the human red blood cell deformed by optical tweezers," Journal of

the Mechanics and Physics of Solids, vol. 51, pp. 2259-2280, Nov-Dec 2003. [42] J. Sleep, et al., "Elasticity of the red cell membrane and its relation to hemolytic disorders: an

optical tweezers study," Biophys J, vol. 77, pp. 3085-95, Dec 1999. [43] M. Puig-de-Morales, et al., "Viscoelasticity of the human red blood cell," J. Appl. Physiol., vol. 293,

pp. 597-605, 2007. [44] M. S. Amin, et al., "Microrheology of red blood cell membranes using dynamic scattering

microscopy," Opt. Express., vol. 15, p. 17001, 2007. [45] F. Brochard and J. F. Lennon, "Frequency spectrum of the flicker phenomenon in erythrocytes," J.

Physique, vol. 36, pp. 1035-1047, 1975. [46] S. Levin and R. Korenstein, "Membrane Fluctuations In Erythrocytes Are Linked To Mgatp-

Dependent Dynamic Assembly Of The Membrane Skeleton," Biophys. J., vol. 60, pp. 733-737, Sep 1991.

[47] D. H. Boal, et al., "Dual Network Model For Red-Blood-Cell Membranes," Phys. Rev. Lett., vol. 69, pp. 3405-3408, Dec 7 1992.

[48] S. Tuvia, et al., "Correlation Between Local Cell-Membrane Displacements And Filterability Of Human Red-Blood-Cells," FEBS Lett., vol. 304, pp. 32-36, Jun 8 1992.

99

[49] S. Tuvia, et al., "Cell membrane fluctuations are regulated by medium macroviscosity: Evidence for a metabolic driving force," Proc. Natl. Acad. Sci. U. S. A., vol. 94, pp. 5045-5049, May 13 1997.

[50] N. Gov, et al., "Cytoskeleton confinement and tension of red blood cell membranes," Phys. Rev. Lett., vol. 90, p. 228101, Jun 6 2003.

[51] N. Gov, "Membrane undulations driven by force fluctuations of active proteins," Phys. Rev. Lett., vol. 93, p. 268104, Dec 31 2004.

[52] L. C. L. Lin and F. L. H. Brown, "Brownian dynamics in Fourier space: Membrane simulations over long length and time scales," Phys. Rev. Lett., vol. 93, Dec 17 2004.

[53] G. Popescu, et al., "Optical measurement of cell membrane tension," Phys Rev Lett, vol. 97, p. 218101, Nov 24 2006.

[54] L. C. L. Lin, et al., "Nonequilibrium membrane fluctuations driven by active proteins," Journal of Chemical Physics, vol. 124, pp. -, Feb 21 2006.

[55] Y. K. Park, et al., "Refractive index maps and membrane dynamics of human red blood cells parasitized by Plasmodium falciparum," Proc Natl Acad Sci U S A, vol. 105, p. 13730, 2008.

[56] N. S. Gov and S. A. Safran, "Red blood cell membrane fluctuations and shape controlled by ATP-induced cytoskeletal defects," Biophys. J., vol. 88, pp. 1859-1874, Mar 2005.

[57] R. Lipowski and M. Girardet, "Shape fluctuations of polymerized or solidlike membranes," Phys. Rev. Lett., vol. 65, pp. 2893-2896, 1990.

[58] A. J. Levine and F. C. MacKintosh, "Dynamics of viscoelastic membranes," Phys. Rev. E, vol. 66, Dec 2002.

[59] Y. K. Park, et al., "Metabolic remodeling of the human red blood cell membran," Proc. Nat. Acad. Sci., vol. 107, p. 1289, 2010.

[60] M. A. Peterson, "Geometrical Methods For The Elasticity Theory Of Membranes," Journal Of Mathematical Phys., vol. 26, pp. 711-717, 1985.

[61] G. Popescu, "Quantitative phase imaging of nanoscale cell structure and dynamics," in Methods in Cell Biology. vol. 90, P. J. Bhanu, Ed., ed: Elsevier, 2008, p. 87.

[62] T. G. Mason and D. A. Weitz, "Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids," Phys. Rev. Lett., vol. 74, pp. 1250-1253, Feb 13 1995.

[63] J. D. Wan, et al., "Dynamics of shear-induced ATP release from red blood cells," Proc Natl Acad Sci U S A, vol. 105, pp. 16432-16437, Oct 28 2008.

[64] G. Popescu, et al., "Diffraction phase microscopy for quantifying cell structure and dynamics," Opt Lett, vol. 31, pp. 775-777, 2006.

[65] N. Gov, et al., "Cytoskeleton confinement of red blood cell membrane fluctuations," Biophys. J., vol. 84, pp. 486A-486A, Feb 2003.

[66] A. G. Zilman and R. Granek, "Undulations and dynamic structure factor of membranes," Phys. Rev. Lett., vol. 77, pp. 4788-4791, Dec 2 1996.

[67] G. Popescu, et al., "Spatially resolved microrheology using localized coherence volumes," Phys. Rev. E, vol. 65, p. 041504, Apr 2002.

[68] H. Ding, et al., "Fourier transform light scattering of inhomogeneous and dynamic structures," Phys. Rev. Lett., vol. 101, p. 238102, 2008.

[69] H. Ding, et al., "Actin-driven cell dynamics probed by Fourier transform light scattering," Biomed. Opt. Express, vol. 1, p. 260, 2010.

[70] D. Langevin, Light scattering by liquid surfaces and complementary techniques. New York: M. Dekker, 1992.

[71] Y. C. Fung, Biomechanics : mechanical properties of living tissues, 2nd ed. New York: Springer-Verlag, 1993.

[72] F. Gittes and F. C. MacKintosh, "Dynamic shear modulus of a semiflexible polymer network," Physical Review E, vol. 58, pp. R1241-R1244, Aug 1998.

100

[73] R. B. Vallee and M. P. Sheetz, "Targeting of motor proteins," Science, vol. 271, pp. 1539-1544, Mar 15 1996.

[74] F. C. MacKintosh and C. F. Schmidt, "Active cellular materials," Current Opinion in Cell Biology, vol. 22, pp. 29-35, Feb 2010.

[75] C. P. Brangwynne, et al., "Cytoplasmic diffusion: molecular motors mix it up," Journal of Cell Biology, vol. 183, pp. 583-587, Nov 17 2008.

[76] C. P. Brangwynne, et al., "Intracellular transport by active diffusion," Trends Cell Biol, vol. 19, pp. 423-7, Sep 2009.

[77] A. Yildiz, et al., "Myosin V walks hand-over-hand: Single fluorophore imaging with 1.5-nm localization," Science, vol. 300, pp. 2061-2065, Jun 27 2003.

[78] X. Trepat, et al., "Universal physical responses to stretch in the living cell," Nature, vol. 447, pp. 592-+, May 31 2007.

[79] B. Fabry, et al., "Scaling the microrheology of living cells," Phys. Rev. Lett., vol. 8714, pp. art. no.-148102, Oct 1 2001.

[80] A. Caspi, et al., "Enhanced diffusion in active intracellular transport," Phys. Rev. Lett., vol. 85, pp. 5655-5658, Dec 25 2000.

[81] D. Mizuno, et al., "Nonequilibrium mechanics of active cytoskeletal networks," Science, vol. 315, pp. 370-373, Jan 19 2007.

[82] B. Wang, et al., "Anomalous yet Brownian," Proceedings of the National Academy of Sciences of the United States of America, vol. 106, pp. 15160-15164, Sep 8 2009.

[83] C. P. Brangwynne, et al., "Nonequilibrium microtubule fluctuations in a model cytoskeleton," Physical Review Letters, vol. 100, pp. -, Mar 21 2008.

[84] G. Popescu, "Quantitative Phase Imaging of Nanoscale Cell Structure and Dynamics," in Methods in Cell Biology. vol. Methods in Nano Cell Biology, B. P. Jena, Ed., ed San Diego: Academic Press, 2008, pp. 87-115.

[85] G. Popescu, et al., "Optical measurement of cell membrane tension," Phys. Rev. Lett., vol. 97, p. 218101, 2006.

[86] Y. K. Park, et al., "Measurement of red blood cell mechanics during morphological changes," Proc. Nat. Acad. Sci., vol. 107, p. 6731, 2010.

[87] G. Popescu, et al., "Imaging red blood cell dynamics by quantitative phase microscopy," Blood Cells Molecules and Diseases, vol. 41, pp. 10-16, Jul-Aug 2008.

[88] Z. Wang, et al., "Spatial light interference microscopy (SLIM)," Optics express, vol. 19, p. 1016, 2011.

[89] F. Zernike, "How I discovered phase contrast," Science, vol. 121, p. 345, 1955. [90] H. F. Ding, et al., "Fourier Transform Light Scattering of Inhomogeneous and Dynamic Structures,"

Physical Review Letters, vol. 101, p. 238102, Dec 5 2008. [91] G. Popescu, et al., "Optical imaging of cell mass and growth dynamics," Am J Physiol Cell Physiol,

vol. 295, pp. C538-44, Aug 2008. [92] R. B. Bird, Stewart, W.E. and Lightfoot, E.N., Transport Phenomena John Wiley & Sons ed.: John

Wiley & Sons, 1960. [93] B. J. Berne and R. Pecora, Dynamic light scattering with applications to chemistry, biology and

Physics. New York: Wiley, 1976. [94] D. L. Coy, et al., "Kinesin takes one 8-nm step for each ATP that it hydrolyzes," J Biol Chem, vol.

274, pp. 3667-71, Feb 5 1999. [95] R. Wang, et al., "One-dimensional deterministic transport in neurons measured by dispersion-

relation phase spectroscopy," Journal of Physics:Condensed Matter, vol. 23, 2011. [96] I. F. Sbalzarini and P. Koumoutsakos, "Feature point tracking and trajectory analysis for video

imaging in cell biology," Journal of Structural Biology, vol. 151, pp. 182-195, Aug 2005.

101

[97] M. A. Thompson, et al., "Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function," Proceedings of the National Academy of Sciences of the United States of America, vol. 107, pp. 17864-17871, Oct 19 2010.

[98] Y. Tseng, et al., "Micro-organization and visco-elasticity of the interphase nucleus revealed by particle nanotracking," Journal of Cell Science, vol. 117, pp. 2159-67, Apr 15 2004.

[99] P. Marquet, et al., "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Optics Letters, vol. 30, pp. 468-470, Mar 1 2005.

[100] J. Frank, et al., "Non-interferometric, non-iterative phase retrieval by Green's functions," Journal of the Optical Society of America. A, Optics, image science, and vision, vol. 27, pp. 2244-51, Oct 1 2010.

[101] P. Bon, et al., "Quadriwave lateral shearing interferometry for quantitative phase microscopy of living cells," Optics express, vol. 17, pp. 13080-94, Jul 20 2009.

[102] D. Paganin and K. A. Nugent, "Noninterferometric phase imaging with partially coherent light," Phys. Rev. Lett. , vol. 80, pp. 2586-2589, Mar 23 1998.

[103] D. Zicha and G. A. Dunn, "An Image-Processing System For Cell Behavior Studies In Subconfluent Cultures," J. Microscopy, vol. 179, pp. 11-21, Jul 1995.

[104] C. H. Yang, et al., "Phase-dispersion optical tomography," Opt Lett, vol. 26, pp. 686-688, May 15 2001.

[105] M. A. Choma, et al., "Spectral-domain phase microscopy," Opt. Lett., vol. 30, pp. 1162-1164, May 15 2005.

[106] C. Fang-Yen, et al., "Imaging voltage-dependent cell motions with heterodyne Mach-Zehnder phase microscopy," Optics Letters, vol. 32, pp. 1572-1574, Jun 1 2007.

[107] C. Joo, et al., "Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging," Opt. Lett., vol. 30, pp. 2131-2133, Aug 15 2005.

[108] W. S. Rockward, et al., "Quantitative phase measurements using optical quadrature microscopy," Applied Optics, vol. 47, pp. 1684-1696, Apr 1 2008.

[109] T. Ikeda, et al., "Hilbert phase microscopy for investigating fast dynamics in transparent systems," Opt. Lett., vol. 30, pp. 1165-1168, 2005.

[110] G. Popescu, et al., "Fourier phase microscopy for investigation of biological structures and dynamics," Opt. Lett., vol. 29, pp. 2503-2505, Nov 1 2004.

[111] H. F. Ding, et al., "Fourier Transform Light Scattering (FTLS) of Cells and Tissues," Journal of Computational and Theoretical Nanoscience, vol. 7, pp. 2501-2511, Dec 2010.

[112] H. Ding, et al., "Optical properties of tissues quantified by Fourier transform light scattering " Opt. Lett., vol. 34, p. 1372, 2009.

[113] N. Lue, et al., "Live cell refractometry using microfluidic devices," Opt. Lett., vol. 31, p. 2759, 2006. [114] F. Zernike, "How I discovered phase contrast," Science vol. 121, p. 345, 1955. [115] D. Gabor, "A new microscopic principle," Nature, vol. 161, p. 777, 1948. [116] F. Zernike, "Phase contrast, a new method for the microscopic observation of transparent objects,

Part 1," Physica, vol. 9, pp. 686-698, 1942. [117] W. S. Rasband, "ImageJ," U.S. National Institutes of Health, Bethesda, Maryland, USA,

http://rsb.info.nih.gov/ij/, 1997-2009. [118] E. Meijering, et al., "Design and validation of a tool for neurite tracing and analysis in fluorescence

microscopy images," Cytometry Part A, vol. 58A, pp. 167-176, Apr 2004. [119] D. Chowdhury, et al., "Physics of transport and traffic phenomena in biology: from molecular

motors and cells to organisms," Physics of Life Reviews, vol. 2, pp. 318-352, Dec 2005. [120] S. Alexander and P. Pincus, "Diffusion of Labeled Particles on One-Dimensional Chains," Physical

Review B, vol. 18, pp. 2011-2012, 1978.

102

[121] D. Chowdhury, et al., "Intra-cellular traffic: bio-molecular motors on filamentary tracks," European Physical Journal B, vol. 64, pp. 593-600, Aug 2008.

[122] T. Chou, "Water alignment, dipolar interactions, and multiple proton occupancy during water-wire proton transport," Biophysical Journal, vol. 86, pp. 2827-2836, May 2004.

[123] S. L. Shaw, et al., "Sustained microtubule treadmilling in Arabidopsis cortical arrays," Science, vol. 300, pp. 1715-1718, Jun 13 2003.

[124] P. Pierobon, "Traffic of molecular motors: from theory to experiments," Traffic and Granular Flow’07, pp. 679-688, 2009.

[125] F. Spitzer, "Interaction of Markov processes," Advances in Mathematics, vol. 5, pp. 246-290, 1970. [126] M. R. Evans, et al., "Shock formation in an exclusion process with creation and annihilation,"

Physical Review E, vol. 68, pp. -, Aug 2003. [127] A. Parmeggiani, et al., "Phase coexistence in driven one-dimensional transport," Physical Review

Letters, vol. 90, pp. -, Feb 28 2003. [128] A. Parmeggiani, et al., "Totally asymmetric simple exclusion process with Langmuir kinetics,"

Physical Review E, vol. 70, pp. -, Oct 2004. [129] K. H. Kim and M. den Nijs, "Dynamic screening in a two-species asymmetric exclusion process,"

Physical Review E, vol. 76, pp. -, Aug 2007. [130] A. J. Wood, "A totally asymmetric exclusion process with stochastically mediated entrance and

exit," Journal of Physics a-Mathematical and Theoretical, vol. 42, pp. -, Nov 6 2009. [131] T. Chou, "Kinetics and thermodynamics across single-file pores: Solute permeability and rectified

osmosis," Journal of Chemical Physics, vol. 110, pp. 606-615, Jan 1 1999. [132] A. B. Kolomeisky, "Exact solutions for a partially asymmetric exclusion model with two species,"

Physica A, vol. 245, pp. 523-533, Nov 1 1997. [133] E. Pronina and A. B. Kolomeisky, "Two-channel totally asymmetric simple exclusion processes,"

Journal of Physics a-Mathematical and General, vol. 37, pp. 9907-9918, Oct 22 2004. [134] R. Wang, et al., "Spatiotemporal mass transport in living cells," Phys. Rev. Lett., under review. [135] E. Frey, "Personal communication," ed, 2011. [136] N. Segev, Trafficking inside cells : pathways, mechanisms, and regulation. Austin, Tex., New York,

N.Y.: Landes Bioscience ; Springer Science+Business Media, 2009. [137] D. Magde, et al., "Thermodynamic Fluctuations in a Reacting System - Measurement by

Fluorescence Correlation Spectroscopy," Physical Review Letters, vol. 29, pp. 705-&, 1972. [138] M. A. Digman and E. Gratton, "Fluorescence correlation spectroscopy and fluorescence cross-

correlation spectroscopy," Wiley Interdisciplinary Reviews-Systems Biology and Medicine, vol. 1, pp. 273-282, Sep-Oct 2009.

[139] H. M. Chen, et al., "Dynamics of equilibrium structural fluctuations of apomyoglobin measured by fluorescence correlation spectroscopy," Proceedings of the National Academy of Sciences of the United States of America, vol. 104, pp. 10459-10464, Jun 19 2007.

[140] U. Haupts, et al., "Dynamics of fluorescence fluctuations in green fluorescent protein observed by fluorescence correlation spectroscopy," Proceedings of the National Academy of Sciences of the United States of America, vol. 95, pp. 13573-13578, Nov 10 1998.

[141] D. Lumma, et al., "Dynamics of large semiflexible chains probed by fluorescence correlation spectroscopy," Physical Review Letters, vol. 90, May 30 2003.

[142] K. Bacia, et al., "Fluorescence cross-correlation spectroscopy in living cells," Nature Methods, vol. 3, pp. 83-89, Feb 2006.

[143] D. Axelrod, et al., "Mobility measurement by analysis of fluorescence photobleaching recovery kinetics," Biophysical Journal, vol. 16, pp. 1055-1069, 1976.

[144] J. Yao, et al., "Dynamics of heat shock factor association with native gene loci in living cells," Nature, vol. 442, pp. 1050-1053, Aug 31 2006.

103

[145] L. Y. Wang, et al., "In situ measurement of solute transport in the bone lacunar-canalicular system," Proceedings of the National Academy of Sciences of the United States of America, vol. 102, pp. 11911-11916, Aug 16 2005.

[146] J. C. Politz, et al., "Intranuclear diffusion and hybridization state of oligonucleotides measured by fluorescence correlation spectroscopy in living cells," Proceedings of the National Academy of Sciences of the United States of America, vol. 95, pp. 6043-6048, May 26 1998.

[147] N. O. Petersen, et al., "Quantitation of Membrane-Receptor Distributions by Image Correlation Spectroscopy - Concept and Application," Biophysical Journal, vol. 65, pp. 1135-1146, Sep 1993.

[148] B. Hebert, et al., "Spatiotemporal image correlation Spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells," Biophysical Journal, vol. 88, pp. 3601-3614, May 2005.

[149] M. A. Digman, et al., "Measuring Fast Dynamics in Solutions and Cells with a Laser Scanning Microscope," Biophysical Journal, vol. 89, pp. 1317-1327, 2005.

[150] M. T. Valentine, et al., "Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking," Physical Review E, vol. 6406, pp. -, Dec 2001.

[151] J. H. Shin, et al., "Relating microstructure to rheology of a bundled and cross-linked F-actin network in vitro," Proceedings of the National Academy of Sciences of the United States of America, vol. 101, pp. 9636-9641, Jun 29 2004.

[152] T. A. Waigh, "Microrheology of complex fluids," Rep. Prog. Phys., vol. 68, pp. 685-742, Mar 2005. [153] R. Wang, et al., "Dispersion-relation phase spectroscopy of intracellular transport," Opt. Express,

vol. 19, pp. 20571-20579, 2011. [154] R. Wang, et al., "One-dimensional deterministic transport in neurons measured by dispersion-

relation phase spectroscopy," J. Phys.: Cond. Matter, vol. 23, p. 374107, 2011. [155] G. Popescu, Quantitative phase imaging of cells and tissues. New York: McGraw-Hill, 2011. [156] Y. Aratyn-Schaus and M. L. Gardel, "BIOPHYSICS Clutch Dynamics," Science, vol. 322, pp. 1646-

1647, Dec 12 2008. [157] C. E. Chan and D. J. Odde, "Traction Dynamics of Filopodia on Compliant Substrates," Science, vol.

322, pp. 1687-1691, Dec 12 2008. [158] T. D. Pollard and J. A. Cooper, "Actin, a Central Player in Cell Shape and Movement," Science, vol.

326, pp. 1208-1212, Nov 27 2009. [159] C. M. Brown, et al., "Raster image correlation spectroscopy (RICS) for measuring fast protein

dynamics and concentrations with a commercial laser scanning confocal microscope," Journal of Microscopy-Oxford, vol. 229, pp. 78-91, Jan 2008.

[160] N. Segev, "Trafficking Inside Cells: Pathways, Mechanisms and Regulation," Trafficking inside Cells: Pathways, Mechanisms and Regulation, pp. 1-445, 2009.

[161] T. Ohba, et al., "Self-organization of microtubule asters induced in Xenopus egg extracts by GTP-bound Ran," Science, vol. 284, pp. 1356-1358, May 21 1999.

[162] N. Hirokawa and R. Takemura, "Molecular motors in neuronal development, intracellular transport and diseases," Current Opinion in Neurobiology, vol. 14, pp. 564-73, Oct 2004.

[163] E. Mandelkow and E. M. Mandelkow, "Kinesin motors and disease," Trends in Cell Biology, vol. 12, pp. 585-91, Dec 2002.

[164] M. Schuh, "An actin-dependent mechanism for long-range vesicle transport," Nature Cell Biology, vol. 13, pp. 1431-U115, Dec 2011.

[165] M. Often, et al., "Local Motion Analysis Reveals Impact of the Dynamic Cytoskeleton on Intracellular Subdiffusion," Biophysical Journal, vol. 102, pp. 758-767, Feb 22 2012.

[166] J. M. Trifaro, et al., "Cytoskeletal control of vesicle transport and exocytosis in chromaffin cells," Acta Physiologica, vol. 192, pp. 165-172, Feb 2008.

104

[167] R. Wang, et al., "Dispersion-relation phase spectroscopy of intracellular transport," Optics Express, vol. 19, pp. 20571-20579, Oct 10 2011.

[168] J. L. Ross, et al., "Cargo transport: molecular motors navigate a complex cytoskeleton," Current Opinion in Cell Biology, vol. 20, pp. 41-47, Feb 2008.

[169] G. Danuser and C. M. Waterman-Storer, "Quantitative fluorescent speckle Microscopy of cytoskeleton dynamics," Annual Review Of Biophysics And Biomolecular Structure, vol. 35, pp. 361-387, 2006.

[170] O. Krichevsky and G. Bonnet, "Fluorescence correlation spectroscopy: the technique and its applications," Reports on Progress in Physics, vol. 65, pp. 251-297, Feb 2002.

[171] B. N. G. Giepmans, et al., "Review - The fluorescent toolbox for assessing protein location and function," Science, vol. 312, pp. 217-224, Apr 14 2006.

[172] E. A. J. Reits and J. J. Neefjes, "From fixed to FRAP: measuring protein mobility and activity in living cells," Nature Cell Biology, vol. 3, pp. E145-E147, Jun 2001.

[173] T. W. J. Gadella, et al., "Fluorescence Lifetime Imaging Microscopy (Flim) - Spatial-Resolution of Microstructures on the Nanosecond Time-Scale," Biophysical Chemistry, vol. 48, pp. 221-239, Dec 1993.

[174] M. J. Rust, et al., "Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM)," Nature Methods, vol. 3, pp. 793-795, Oct 2006.

[175] E. Betzig, et al., "Imaging intracellular fluorescent proteins at nanometer resolution," Science, vol. 313, pp. 1642-1645, Sep 15 2006.

[176] S. T. Hess, et al., "Ultra-high resolution imaging by fluorescence photoactivation localization microscopy," Biophysical Journal, vol. 91, pp. 4258-4272, Dec 2006.

[177] B. Fabry, et al., "Scaling the microrheology of living cells," Physical Review Letters, vol. 87, Oct 1 2001.

[178] A. Caspi, et al., "Enhanced diffusion in active intracellular transport," Physical Review Letters, vol. 85, pp. 5655-5658, Dec 25 2000.

[179] R. Wang, et al., "Dispersion relation flourescence spectroscopy," Phys. Rev. Lett, vol. 109, pp. 188104-1-188104-5, 2012.

[180] P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics. Cambridge ; New York, NY, USA: Cambridge University Press, 1995.

[181] D. H. Boal, Mechanics of the cell. Cambridge, UK ; New York: Cambridge University Press, 2002. [182] R. Borsali and R. Pecora, Soft-matter characterization, 1st ed. New York: Springer, 2008. [183] A. J. Levine and T. C. Lubensky, "One- and two-particle microrheology," Phys. Rev. Lett., vol. 85,

pp. 1774-1777, Aug 21 2000. [184] R. Kubo, "The fluctuation-dissipation theorem," Rep. Prog. Phys., vol. 29, pp. 255-284, 1966. [185] G. Popescu and A. Dogariu, "Dynamic light scattering in localized coherence volumes," Opt. Lett.,

vol. 26, pp. 551-553, Apr 15 2001. [186] K. K. Bizheva, et al., "Path-length-resolved dynamic light scattering in highly scattering random

media: The transition to diffusing wave spectroscopy," Physical Review E, vol. 58, pp. 7664-7667, Dec 1998.

[187] A. L. Oldenburg, et al., "Phase-resolved magnetomotive OCT for imaging nanomolar concentrations of magnetic nanoparticles in tissues," Optics Express, vol. 16, pp. 11525-11539, Jul 21 2008.

[188] A. Wax, et al., "Measurement of angular distributions by use of low-coherence interferometry for light-scattering spectroscopy," Optics Letters, vol. 26, pp. 322-324, Mar 15 2001.

[189] G. Popescu, Quantitative phase imaging of cells and tissues. New York: McGraw-Hill, 2011.

105

[190] B. Varghese, et al., "Quantification of optical Doppler broadening and optical path lengths of multiply scattered light by phase modulated low coherence interferometry," Optics Express, vol. 15, pp. 9157-9165, Jul 23 2007.

[191] M. Aratono, et al., "Thermodynamic study on the surface formation of the mixture of water and ethanol," Journal of Colloid and Interface Science, vol. 191, pp. 146-153, 1997.