Introduction to Optical Tweezers

Steve Smith

Bustamante Group, Physics Dept.

Howard Hughes Medical Institute

University of California, Berkeley

Light transfers momentum to matterComet tail

Light exerts force on matter

James Clerk Maxwell

1831-1879

E

V

B

F=eBVe-

Electromagnetic waves interact with electrons in matter

DP = DU/c

ele

ctr

ic

P = h k

Arthur Ashkin builds first optical trap

1970

Single-beam trap

Dual-beam trap

Axial escape

Photon meets refracting object

Pin

Pout

DP

F = dP/dt

q

P = h/l

For every action there exists an equal but opposite reactionSir Isaac Newton

Photon momentum

ashkin1.EXE

a stable single-beam trap

Anti-scattering force:Forward momentum is increased by lens -focusing effect.

ASHKIN2.EXE

Infrared trap supports life !

Trap live bacteria

Sort living cells

Manipulate organellesinside cells

Nature, 1987

Estimating Forces1. Assume a linear-spring restoring force

2. Determine trap stiffness k

3. Measure Dx relative to trap center

Dx

F = k Dx

trap center

Calibrating trap stiffness

(1) Stokes’ law

(2) Corner frequency

(3) Equipartition

Fluid drag test force

Glass chamber

Distance

detector

Motorized

Stage with

encoder

Data

acquisition

Glass

Glass

water

F = 6prhVStokes’ law but corrected for proximity to walls

xktFx D )(

Brownian noise as test forceLangevin equation:

Drag force = 6phrfor a sphere

Fluctuating force

<F(t)> = 0

< F(t) F(t’) > = 2kBTd(t-t’)

22

2 4)(

c

B

ff

Tkfx

D

Lorentzian power spectrum

Corner frequencyfc = 2p k /

Trap force

Power spectra

Pow

er (

nm2/H

z)

Frequency (Hz)

1/f2

4kBT/k2

fc=k/2p

Does not use drag coefficient, but rather ..

integrate area under power curve to get <DX2>

Equipartition method

However, you must have an accurate measure of Dx at high bandwidth. This value is more easily taken in AFMs than optical traps.

k = kBT / <Dx2>

½ k<Dx2> = ½ kBT

Position clamp avoids problems with trap linearity

Measuring forces by analyzing momentum of the trap beam

dP/dt = nW/cDdP/dt

F = -D dP/dt(nW/c) sin q

q

(nW/c) (1-cos q)Light ray with power W

input

dP/dtoutput

Counter-propagating beams for narrow-angle trap

OB

JO

BJ

LA

SE

RL

AS

ER

pipette

DNA

position detector

liquid chamber

position detector

qwppbs

external force

liquid air

detector

Narrow beams stay within NA of lenses

Fro

nt fo

cu

s

Focal length L

Ba

ck fo

ca

l p

lan

e

X = n L sinq

Light leaving trap obeys Abbe sine condition

Objective lens

BFP : where angle q is best

represented by offset x

How to measure light offset?quadrant photodiode

versus PSD photodiode

++

_ _

+

_ _

__

QPD

PSD

N

P

N

P

In1 In2

Out1

Out2

PSD (position sensitive detector)

Plate resistorsseparated byreverse-biasedPIN photodiode

opposite electrodes held at same potentialno conduction unless there is light

N

P

N

P

In1 In2

Out1

Out2

PSD (position sensitive detector)

Plate resistorsseparated byreverse-biasedPIN photodiode

In1 + In2 = Out1 + Out2 = Wi

by charge conservation

Out1 = Out2 = ½ WIn1 = In2 = ½Wby symmetry

Suppose we shine ray of light with intensity Wi in exact center of detector:

(sensitivity = 1)

N

P

N

P

In1 In2

Out1

Out2

PSD (position sensitive detector)

Now suppose the ray of is off center.

Out1 + Out2 = W = In1 + In2 still holds

In1 > In2 and Out1 > Out2 due to resistance asymmetry

Opposite electrodes held at equal potential so currents to those electrodes divide inversely to the distance of the spot from electrode.

N

P

N

P

In1 In2

Out1

Out2

PSD (position sensitive detector)

Multiple rays add their currents linearly to the electrodes,

where each ray’s power adds Wi current to the total sum.

PSD (position sensitive detector)

N

P

N

P

In1 – In2 = S Wi xi / RD

Out1 – Out2 = S Wi yi / RD

x

y

Define x-y coordinates centered on detector

it can be shown

where RD is the half-width (or “radius”) of the detector

In2In1

Out1

Out2

N

P

N

P

In1 In2

Out1

Out2

where sum = In1 + In2 = Out1 + Out2 = S Wi

Xcenter= RD (In1 –In2) / sum

Ycenter = RD (Out1 – Out2 ) / sum

PSD (position sensitive detector)

For arbitrary light distribution, centroid position given by difference of electrode currents

Sensitivity does not depend on spot size or shape

N

P

N

P

In1 In2

Out1

Out2

SX = In1 – In2 = S W i xi / RD

SY = Out1 – Out2 = S Wi yi / RD

PSD force sensor

samples

unfocused

beam

Detecting external

force from

changes in

light momentum

flux

liquid air external force

X

2L

detector

Collector lens transforms exit angles into ray offsets

by Abbe Sine Condition: xi = L nL sin qi

PSD sums over rays to give signal SX RD= SWi xi

External force = light force = effect from all rays:

Fx = dP/dt = (nL/c) SWi sin qi

Then external transverse force is given by

FX = SXRD /cL

nL

Momentum sensor calibration

Calibrate signal to power ratio for PSDs / objectives with power meter and ruler.

No test force is used.

Calibration does not change with particle size, particle shape or laser power. Particle and trap are not being calibrated (don’t matter).

Methods in Enzymology v.361 (2003)

Measuring axial forces

dP/dt = nLW/cDdP/dt

F = -D dP/dt(nLW/c) sin q

q

(nLW/c) (1-cos q)Light ray with power W

input

dP/dtoutput

Size of exit beam indicates axial force on

trapped object

Laser beam

tran

smis

sion

radius = nL * L

Correct weighting function to extract axial

momentum flux is semi-circle

bulls-eyeoptical attenuator

Placement of axial force sensors

Bulls-eyeattenuator

PlainPhotodiode

10 nm

Path of bacterium

Flagellum wobble

1000 samples/sec

Force - Extension Behavior of dsDNA and ssDNA

Fractional Extension

For

ce (

pN)

Protein

ssDNA

Unzipping dsDNA

ssDNABockelmann, Heslot, 2002

S. Koch, M. Wang, 2003

Felix Ritort et al., in preparation

15 pN

16 pN

17 pN

60 nm

Motor step size:how small can we detect?

• Effects of thermal noise and tether elasticity

Springs in series for motor

Tra

p c

ente

r

Light

springTether

spring

k1k2

Bead moves

Dxsig = Dxs k1

k1+k2

Motor steps

Dxs

Springs in parallel for thermal noise

Tra

p c

ente

r

Light

springTether

spring

Combined

potential

k2k1

k1 k2

10-8

10-7

10-6

10-5

10-4

10-3

10 100 1000 104 105

0.27

0.54

0.82

2.03

5.10

<DF

2>

(pN

2/H

z)

Frequency (Hz)

<DF2> = 4kBT at low frequencies

Force-noise spectral density is proportional to bead size

Signal to noise ratio

SNR >1 when Dxsig > Dxtherm

Dxs k1 / (k1 + k2) > 2(kBT B)1/2 / (k1 + k2)

Thermal noise

= 2 (kBT B)1/2 / (k1 + k2)

where B is bandwidth in Hz

Dxtherm = DFtherm / (k1 + k2)

Dxstep > 2(kBT B)1/2 / k1

Tether

stiffness

Dxstep > 2(kBT B)1/2 / k1

Thermal limit to step detection

Resolution depends only on tether stiffness, not trap stiffness.

Resolution degrades as (drag)1/2

Comparing AFM to laser tweezers, the force noise scales as

sqrt(cantilever length / bead diameter). Therefore a 100um cantilever has

10x more force noise than a 1 um bead, and 10x bigger distance noise

for fixed k1.

A stiff linkage (large k1) gives an AFM very good resolution when it

pushes against a hard sample. To make a DNA tether stiff requires some

tension in the tether.

-100

-50

0

50

100

0 5000 10000 15000 20000 25000 30000

Noise plus Steps

Sig

na

l

time

-100

-50

0

50

100

0 5000 10000 15000 20000 25000 30000

Running Window 10

filtere

d

time

-100

-50

0

50

100

0 5000 10000 15000 20000 25000 30000 35000

Running Window 100

filtere

d

time

-100

-50

0

50

100

0 5000 10000 15000 20000 25000 30000

Running Window 500

filtere

d

time

-100

-50

0

50

100

0 5000 10000 15000 20000 25000 30000

Running Window 1000

filtere

d

time

Averaging reduces bandwidth, suppresses noise

For example:Bead is 2 um diameter, immersed in water.

Tether is 10 kbp of dsDNA and tether tension is either 2 pN or 20 pN.

Signal of interest is at 1 Hz, so that much bandwidth is required.

Tether stiffness k1 = dF/dx for WLC at either tension (assume P~50nm).

at 2 pN tension, k1 = 12 pN/um

at 20 pN tension, k1 = 170 pN/um

Then smallest resolvable step Dxs= 2(kBT B)1/2 / k1

Dxs= 1.5 nm @ 2 pN tension

Dxs = 0.1 nm @ 20 pN tension (in 1 Hz bandwidth)

Averaging for infinite time will reduce B to zero and resolve infinitely small steps

[ but completely lose temporal resolution]

Slow down the process?

[ now limited by position drift of instrument ]

Work-Horse Optical TrapMeasures force by light momentum change

10 years gaveover 30 papers

Movable microchamber, fixed trap position

X-Y-Z piezo-flexure stage (Martoc)

Glasspipette

Piezo stage

Typical configuration to pull a molecule

Methods in Enzymology

volume 361 (2003)

pipette

1.3nmFor

ce (pN

)

0

1

2

3

Sta

ge position (nm

) 8 0

-8 -

16

Special configuration tests “drift” noise

basepair / sec

count

Drift offset = 0.3 bp/sVelocity SD = 2 bp/s @ 1 Hz

see Neuman and BlockCell, 2003

Characterizing “drift” with velocity histograms

Low-pass filter position signal

(here 1 Hz)

Score average velocities

in 2 sec intervals

Fit distribution to Gaussian

Optics sensitive to:Operator’s touch, breath, voiceChanges in room temperature, air-flow, vibrationAtmospheric fluctuations (star twinkle)

become management problems

Technical problems

Only works in special room in basement Needs operator training for good data Competition for machine time Expensive to build extra machines

Our solution: “Mini-Tweezers”

Table-top

instrument

Optics head

hangs from

bungee cord

Works OK on

upper floors

300 mW typical

single-mode fiber

output

975 nm or 845 nm

Mini uses telecom “pump” lasers

Fixed chamber, movable traps

gives increased stability

Fiber wiggler moves trap

1.32 1.34 1.36 1.38 1.4 1.42 0.06 0.08 0.1 0.12 0.14 0.16

Martock flexure stage Fiber wiggler

posi

tion

time (s) time (s)

Moving the fiber is faster

than moving the chamber

Compact optical path avoids “twinkle” effect

10 cm from fiber to trap (3 cm air)

0.5 nm steps at 1 Hz

beam

Motorized stage remains fixed

Beams move up and downPipette bead remains fixed

Wooden box

Less velocity-noise with

mini-Tweezers

basepair / sec

count

mini

standard

Velocity noise = 0.4 bp/s @ 1Hz BW

Analytical

optical

traps

can do:

RNA hairpins assay helicase activity

RNA secondary structure, folding and refolding

Phage packaging motors

Polymer entropic elasticity

DNA mechanics (torsional rigidity, phase transitions)

DNA condensation phase transitions

DNA thermodynamics, base-pair energies

Force-melting DNA shows sequence (unzipping)

Molecular motors in muscle (myosin, actin)

Cell transport: kinesin on tubulin, dynein on tubulin

Cell import: endosome degradation

Protein folding and refolding (RnaseH, T4 Lysozyme)

Protein folding multimers (Titin)

Enzyme movements, kinetics: topoisomerase, gyrase

Polymerases (DNA, RNA)

Affinity studies: antibody, ligand

DNA/protein binding, e.g. recA,

Chromatin structure and remodeling

Combinatorial chemistry, bead sorting

Cell sorting by drag coefficients

Rheology of polymers

Reptation studies

Electrophoresis forces

Cell wall deformability

Statistical mechanics (Jarzynski, Crooks theorems)

Bacterial motility (swimming force) in 3 dimensions

Education / training in biophysics

Thanks:

• Carlos Bustamante and lab members

• Howard Hughes Medical Institute

• Claudio Rivetti, University of Parma

• Agilent Technologies Foundation

http:// tweezerslab.unipr.it