Steve Smith - Bustamante Group - University of California, Berkeley

Introduction to Optical Tweezers
Steve Smith
Bustamante Group, Physics Dept.
Howard Hughes Medical Institute
University of California, Berkeley

Comet tail
Light transfers momentum to ma ter
Light exerts force on matter
James Clerk Maxwell
1831-1879

Electromagnetic waves interact
electric
with electrons in matter
DP = DU/c
V
e-
E
B
F=eBV
P = h k

Arthur Ashkin builds first optical trap
Single-beam trap
Dual-beam trap
1970
Axial
escape

Photon meets refracting object
Photon
momentum
P = h/l
Pin
F = dP/dt
q
Pout
DP
For every action there exists an equal but opposite reaction
Sir Isaac Newton
ashkin1.EXE

a stable single-beam trap
Anti-scattering force:
Forward momentum is
increased by lens -focusing
effect.
ASHKIN2.EXE

Trap live
bacteria
Infrared trap supports life !
Sort living
cells
Manipulate
organelles
inside cells
Nature, 1987

Estimating Forces
trap center
Dx
1. Assume a linear-spring restoring force
2. Determine trap stiffness k
3. Measure Dx relative to trap center
F = k Dx

Calibrating trap stiffness
(1) Stokes’ law
(2) Corner frequency
(3) Equipartition

Glass chamber
Fluid drag test force
Stokes’ law F = 6prhV but corrected for proximity to walls
Distance
detector
Data
acquisition
Motorized
Stage with
encoder
Glass
Glass
water

Brownian noise as test force
Drag force
= 6phr
for a sphere
Langevin equation:
x
F(
t)
kDx
Dx
2
(
f
Fluctuating force
= 0
< F(t) F(t’) > = 2k BTd(t-t’)
)
4k
B
T
2 2
f f
Lorentzian power spectrum
c
Trap force
Corner
frequency
f c = 2p k /

Power (nm 2 /Hz)
Power spectra
f c=k/2p
Frequency (Hz)
1/f 2
4k BT/k 2

Does not use drag coefficient, but rather ..
Equipartition method
integrate area under power curve to get
k = k BT /
However, you must have
an accurate measure of
Dx at high bandwidth.
This value is more
easily taken in AFMs
than optical traps.
½ k = ½ k BT

Position clamp avoids problems with trap linearity

Measuring forces by analyzing
momentum of the trap beam
Light ray with power W
dP/dt = nW/c
input
(nW/c) sin q
F = -D dP/dt
(nW/c) (1-cos q)
q
dP/dt
output
DdP/dt

Counter-propagating beams for
LASER
position
detector
narrow-angle trap
pbs
OBJ
DNA
pipette
OBJ
qwp
liquid
chamber
position
detector
LASER

Narrow beams stay within NA of lenses
liquid
external
force
air
detector

Light leaving trap obeys Abbe sine condition
Front focus
Focal length L
Back focal plane
X = n L sinq
Objective lens
BFP : where angle q is best
represented by offset x

How to measure light offset?
quadrant photodiode

versus PSD photodiode

+
_ _
+
_ _
_
_
QPD
PSD
+

PSD
(position sensitive detector)
Plate resistors
separated by
reverse-biased
PIN photodiode
In 1
N
Out 1
P
P
Out 2
N
In 2
opposite electrodes held at same potential
no conduction unless there is light

PSD
(position sensitive detector)
Plate resistors
separated by
reverse-biased
PIN photodiode
In 1
N
Out 1
P
P
Out 2
Suppose we shine ray of light with intensity W i in exact center of detector:
In 1 + In 2 = Out 1 + Out 2 = W i
by charge conservation
Out 1 = Out 2 = ½ W
In 1 = In 2 = ½W
by symmetry
N
In 2
(sensitivity = 1)

PSD
(position sensitive detector)
In 1
Now suppose the ray of is off center.
N
Out 1
P
P
Out 2
N
Out 1 + Out 2 = W = In 1 + In 2 still holds
In 1 > In 2 and Out 1 > Out 2 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.
In 2

PSD
(position sensitive detector)
In 1
N
Out 1
P
P
Out 2
Multiple rays add their currents linearly to the electrodes,
where each ray’s power adds W i current to the total sum.
N
In 2

PSD
(position sensitive detector)
In 1
N
Define x-y coordinates centered on detector
it can be shown
Out 1
Out 2
y
P
P
N
In 1 – In 2 = S W i x i / R D
Out 1 – Out 2 = S W i y i / R D
In 2
where R D is the half-width (or “radius”) of the detector
x

PSD
(position sensitive detector)
In 1
N
Out 1
P
P
Out 2
For arbitrary light distribution, centroid position given by difference of electrode currents
where sum = In 1 + In 2 = Out 1 + Out 2 = S W i
N
X center= R D (In 1 –In 2) / sum
Y center = R D (Out 1 – Out 2 ) / sum
Sensitivity does not depend on spot size or shape
In 2

PSD
force
sensor
samples
unfocused
beam
In 1
N
Out 1
P
P
Out 2
N
S X = In 1 – In 2 = S W i x i / R D
S Y = Out 1 – Out 2 = S W i y i / R D
In 2

Detecting
external
force
from
changes
in
light
momentum
flux
liquid
n L
air
detector
external
force
External force = light force = effect from all rays:
Collector lens transforms exit angles into ray offsets
by Abbe Sine Condition: x i = L n L sin q i
PSD sums over rays to give signal S X R D= SW i x i
2L
F x = dP/dt = (n L/c) SW i sin q i
Then external transverse force is given by
F X = S XR D /cL
X

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
Light ray with power W
dP/dt = n LW/c
input
(n LW/c) sin q
F = -D dP/dt
(n LW/c) (1-cos q)
q
dP/dt
output
DdP/dt

Size of exit beam indicates axial force on
trapped object
Laser beam

Correct weighting function to extract axial
momentum flux is semi-circle
transmission
radius = n L * L
bulls-eye
optical
attenuator

Plain
Photo
diode
Placement of axial force sensors
Bulls-eye
attenuator

Path of bacterium
Flagellum wobble
10 nm
1000 samples/sec

Force (pN)
Force - Extension Behavior of dsDNA
and ssDNA
Fractional Extension

Bockelmann, Heslot, 2002
S. Koch, M. Wang, 2003
Felix Ritort et al., in preparation
Unzipping dsDNA
ssDNA
Protein
ssDNA

17 pN
16 pN
15 pN
60 nm

Motor step size:
how small can we detect?
• Effects of thermal noise and tether elasticity

Springs in series for motor
Trap center
Light
spring
k 2
Bead moves
Dx sig = Dx s k 1
k 1+k 2
Tether
spring
k 1
Motor steps
Dx s

Springs in parallel for thermal noise
Trap center
Light
spring
k 2
Combined
potential
k 1 k 2
Tether
spring
k 1

(pN 2 /Hz)
Force-noise spectral density is
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
proportional to bead size
0.27
0.54
0.82
2.03
5.10
10 100 1000 10 4
Frequency (Hz)
10 5
= 4k BT
at low frequencies

Thermal noise
Dx therm = DF therm / (k 1 + k 2)
= 2 (k BT B) 1/2 / (k 1 + k 2)
where B is bandwidth in Hz
Signal to noise ratio
SNR >1 when Dx sig > Dx therm
Dx s k 1 / (k 1 + k 2) > 2(k BT B) 1/2 / (k 1 + k 2)
Dx step > 2(k BT B) 1/2 / k 1
Tether
stiffness

Thermal limit to step detection
Dx step > 2(k BT B) 1/2 / k 1
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 k 1.
A stiff linkage (large k 1) 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.

Signal
100
50
0
-50
Averaging reduces bandwidth, suppresses noise
Noise plus Steps
-100
0 5000 10000 15000 20000 25000 30000
filtered
time
100
50
0
-50
Running Window 500
-100
0 5000 10000 15000 20000 25000 30000
time
filtered
100
50
0
-50
Running Window 10
-100
0 5000 10000 15000 20000 25000 30000
time
filtered
100
50
0
-50
filtered
100
50
0
-50
Running Window 100
-100
0 5000 10000 15000 20000 25000 30000 35000
Running Window 1000
-100
0 5000 10000 15000 20000 25000 30000
time
time

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 k 1 = dF/dx for WLC at either tension (assume P~50nm).
at 2 pN tension, k 1 = 12 pN/um
at 20 pN tension, k 1 = 170 pN/um
Then smallest resolvable step Dx s= 2(k BT B) 1/2 / k 1
Dx s= 1.5 nm @ 2 pN tension
Dx s = 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 Trap
Measures force by light momentum change
10 years gave
over 30 papers

Movable microchamber, fixed trap position
X-Y-Z piezo-flexure stage (Martoc)

Typical configuration to pull a molecule
Piezo stage
Glass
pipette

1.3nm
pipette
Methods in
Enzymology
volume 361 (2003)
Special configuration tests “drift” noise
Force (pN)
0 1 2 3
Stage position (nm)
8 0 -8 -16

Characterizing “drift” with velocity histograms
Low-pass filter position signal
(here 1 Hz)
Score average velocities
in 2 sec intervals
Fit distribution to Gaussian
Drift offset = 0.3 bp/s
Velocity SD = 2 bp/s @ 1 Hz
count
see
Neuman and Block
Cell, 2003
basepair / sec

Technical problems
Optics sensitive to:
Operator’s touch, breath, voice
Changes in room temperature, air-flow, vibration
Atmospheric fluctuations (star twinkle)
become management 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

Mini uses telecom “pump” lasers
300 mW typical
single-mode fiber
output
975 nm or 845 nm

Fixed chamber, movable traps
gives increased stability

Fiber wiggler moves trap

position
Moving the fiber is faster
than moving the chamber
Martock flexure stage Fiber wiggler
1.32 1.34 1.36 1.38 1.4 1.42 0.06 0.08 0.1 0.12 0.14 0.16
time (s) time (s)

Compact optical path avoids “twinkle” effect
10 cm from fiber to trap (3 cm air)

eam
Motorized
stage
remains
fixed
Beams move up and down
Pipette bead remains fixed
0.5 nm steps at 1 Hz

Wooden box

count
Less velocity-noise with
mini-Tweezers
basepair / sec
standard
mini
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
http:// tweezerslab.unipr.it
• Agilent Technologies Foundation