Transcript Lecture 23 (4/26/10) "Optical Traps"

Light transfers momentum to matter
p = h/λ
Slides from Yann Chemla—
blame him if anything is wrong!
Optical Traps
Key points
Light generates 2 types of optical forces:
scattering, gradient
Trap strength depends on light intensity, gradient
Trap is harmonic: k ~ 0.1pN/nm
Optical scattering forces – reflection
Pf
Pi = h/λ
ΔP
θ
Pi
F = ΔP/Δt = (Pf-Pi)/Δt
Newton’s third law – for every action there is an equal and opposite reaction
Optical forces – Refraction
Pi
Pi
Pf
ΔP
Lateral gradient force
Bright ray
Dim ray
Object feels a force toward brighter light
Axial gradient force
ΔP
Pi
Focused
light
Pf
Pi
Object feels a force toward focus
Force ~ gradient intensity
IR traps and biomolecules are compatible
Neuman et al. Biophys J. 1999
Biological scales
Force: 1-100 picoNewton (pN)
Distance: <1–10 nanometer (nm)
www.scripps.edu/cb/milligan/projects.html
www.cnr.berkeley.edu/~hongwang/Project
www.alice.berkeley.edu
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
Photodetector
Condenser Objective
1) Manipulation
Want to apply forces – need ability to move stage or
trap (piezo stage, steerable mirror, AOD…)
DNA
2) Measurement
Want to measure forces, displacements – need to
detect deflection of bead from trap center
1) Video microscopy
2) Laser-based method – Back-focal plane interferometry
BFP imaged onto detector
Trap laser
∑
specimen
Relay lens
PSD
BFP
Conjugate image
planes
Position sensitive detector (PSD)
Out1
Plate resistors separated by reversebiased PIN photodiode
P
In1
N
N
In2
P
Out2
Opposite electrodes at same potential
– no conduction with no light
Multiple rays add their currents linearly to the electrodes, where each ray’s power
adds Wi current to the total sum.
Out1
SIGNAL
P
In1
N
N
P
Out2
ΔX ~ (In1-In2) / (In1 + In2)
ΔY ~ (Out1-Out2) /(Out1+Out2)
In2
POSITION
Calibration
Want to measure forces, displaces – measure voltages
from PSD – need calibration
Δx = α ΔV
F = kΔx = α kΔV
Calibrate with a
known displacement
Calibrate with a
known force
Glass
water
Glass
Glass
Move bead relative to trap
Stokes law: F = γv
Brownian motion as test force
Langevin equation:
x  kx  F (t )
Trap force
Drag force
γ = 3πηd
kBT
Fluctuating
Brownian
force
<F(t)> = 0
<F(t)F(t’)> = 2kBTγδ (t-t’)
kBT= 4.14pN-nm
Autocorrelation function x(t )x(t )
ΔtΔt Δt
x(t )x(t )
Autocorrelation function x(t )x(t )
ΔtΔt Δt
x(t )x(t )
Brownian motion as test force
Langevin equation:
x  kx  F (t )
Exponential autocorrelation f’n
k BT  k t  t 
x(t )x(t ) 
e
k

x
2
k BT

k
FT → Lorentzian power spectrum
4k BT
1
Sx  f  
2
2
k 1   f fc 
Corner
frequency
fc = k/2π
Power (V2/Hz)
4 k BT  2

2
k
fc 
k BT 2

2
2
 f
Frequency (Hz)
k
2
Reducing bandwidth reduces noise.
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
6.4 nm is a pretty large number.
[ Kinesin moves every 8.3 nm; 1 base-pair = 3.4 Å ]
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 mm bead in water).
But (<x2>BW)1/2 = [∫const*(BW)dk]1/2= [(4kBT100)/k]1/2 =
[4*4.14*10-6*100/0.1]1/2
~ 0.4 nm =
4 Angstrom!!
Basepair Resolution—Yann Chemla @ UIUC
3.40
1bp = 3.4Å
1
unpublished
2
3
1
2
2.04
4
3
5
4
1.36
5
6
6
0.68
7
7
8
UIUC - 02/11/08
0.00
0
2
Probability (a.u.)
Displacement (nm)
2.72
4
6
Time (s)
8 9
9
3.4 kb DNA
8
10
0.00
0.68
1.36
2.04
Distance (nm)
2.72
F ~ 20 pN
f = 100Hz, 10Hz
Observing individual steps
Motors move in discrete steps
Kinesin
Step size: 8nm
Asbury, et al. Science (2003)
Detailed statistics on
kinetics of stepping
& coordination
Class evaluation
1. 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.