Light transfers momentum to matter

p = h/λ

Slides from Yann Chemla—blame him if anything is wrong!

Optical Traps

Key points

Trap strength depends on light intensity, gradient

Trap is harmonic: k ~ 0.1pN/nm

Light generates 2 types of optical forces: scattering, gradient

Optical scattering forces – reflection

Pi = h/λ

Pf

ΔP

Newton’s third law – for every action there is an equal and opposite reaction

F = ΔP/Δt = (Pf-Pi)/Δt

Pi

θ

Optical forces – Refraction

Pi

Pf

Pi

ΔP

Lateral gradient force

Bright ray

Dim ray

Object feels a force toward brighter light

Axial gradient force

Pi

Pf

PiΔP

Force ~ gradient intensity

Object feels a force toward focus

Focused light

IR traps and biomolecules are compatible

Neuman et al. Biophys J. 1999

Biological scales

www.alice.berkeley.eduwww.cnr.berkeley.edu/~hongwang/Project

www.scripps.edu/cb/milligan/projects.html

Force: 1-100 picoNewton (pN)Distance: <1–10 nanometer (nm)

Requirements for a quantitative optical trap:

1) Manipulation – intense light (laser), large gradient (high NA objective), moveable stage (piezo stage) or trap (piezo mirror, AOD, …)

2) Measurement – collection and detection optics (BFP interferometry)

3) Calibration – convert raw data into forces (pN), displacements (nm)

Laser Beam expander

ObjectiveCondenser

Photodetector

1) Manipulation

DNA

Want to apply forces – need ability to move stage or trap (piezo stage, steerable mirror, AOD…)

2) Measurement

Want to measure forces, displacements – need to detect deflection of bead from trap center

1) Video microscopy2) Laser-based method – Back-focal plane interferometry

BFP imaged onto detector

Trap laser

BFP

specimen

PSD

Relay lens

Conjugate image planes

∑

N

P

N

P

In1 In2

Out1

Out2

Position sensitive detector (PSD)

Plate resistors separated by reverse-biased PIN photodiode

Opposite electrodes at same potential – no conduction with no light

N

P

N

P

In1 In2

Out1

Out2

ΔX ~ (In1-In2) / (In1 + In2)

ΔY ~ (Out1-Out2) /(Out1+Out2)

POSITION

SIGNAL

Multiple rays add their currents linearly to the electrodes, where each ray’s power

adds Wi current to the total sum.

Calibration

Want to measure forces, displaces – measure voltages from PSD – need calibration

Δx = α ΔVF = kΔx = α kΔV

Glass

Glass

water

Glass

Calibrate with a known displacement

Calibrate with a known force

Move bead relative to trap Stokes law: F = γv

)(tFkxx

Brownian motion as test force

Langevin equation:

Drag forceγ = 3πηd

Fluctuating Brownian force

Trap force

<F(t)> = 0<F(t)F(t’)> = 2kBTγδ (t-t’)

kBT

kBT= 4.14pN-nm

ΔtΔtΔt

)()( txtx

Autocorrelation function )()( txtx

)()( txtx ΔtΔtΔt

Autocorrelation function )()( txtx

)(tFkxx

Brownian motion as test forceLangevin equation:

22 1

14

c

Bx

ffk

TkfS

FT → Lorentzian power spectrum

Corner frequencyfc = k/2π

k

Tkx B 2ttkB e

k

Tktxtx )()(

Exponential autocorrelation f’n

Po

we

r (V

2/H

z)

Frequency (Hz)

22

4 k

TkB

222

f

TkB

2

kfc

The noise in position using equipartition theorem you calculate for noise at all frequencies (infinite bandwidth).

For a typical value of stiffness (k) = 0.1 pN/nm.

<x2>1/2 = (kBT/k)1/2 = (4.14/0.1)1/2 = (41.4)1/2 ~ 6.4 nm

Reducing bandwidth reduces noise.

But (<x2>BW)1/2 = [∫const*(BW)dk]1/2= [(4kBT100)/k]1/2 = [4*4.14*10-6*100/0.1]1/2

~ 0.4 nm = 4 Angstrom!!

If instead you collect data out to a lower bandwidth BW (100 Hz), you get a much smaller noise.

(Ex: typical value of (10-6 for ~1 m bead in water).

6.4 nm is a pretty large number.

[ Kinesin moves every 8.3 nm; 1 base-pair = 3.4 Å ]

0.00 0.68 1.36 2.04 2.72

Probability (a.u.)

Distance (nm)0 2 4 6 8 10

0.00

0.68

1.36

2.04

2.72

3.40

Dis

plac

emen

t (n

m)

Time (s)

3.4 kb DNA

F ~ 20 pN

f = 100Hz, 10Hz

1bp = 3.4Å1

2

3

4

5

6

7

8 9

12

34

5

6

78

9UIUC - 02/11/08

Basepair Resolution—Yann Chemla @ UIUC

unpublished

Kinesin

Asbury, et al. Science (2003)

Step size: 8nm

Observing individual steps

Motors move in discrete steps Detailed statistics on kinetics of stepping & coordination

Class evaluation1. What was the most interesting thing you learned in class today?

2. What are you confused about?

3. Related to today’s subject, what would you like to know more about?

4. Any helpful comments.

Answer, and turn in at the end of class.