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 P i = h/λ PfPf ΔPΔP Newton’s third law – for every action there is an equal and opposite reaction F = ΔP/Δt = (P f -P i )/Δt PiPi θ

Optical forces – Refraction PiPi PfPf PiPi ΔPΔP

Lateral gradient force Bright ray Dim ray Object feels a force toward brighter light

Axial gradient force PiPi PfPf PiPi ΔPΔP 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 Force: 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 microscopy 2) 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 In 1 In 2 Out 1 Out 2 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 In 1 In 2 Out 1 Out 2 ΔX ~ (In 1 -In 2 ) / (In 1 + In 2 ) ΔY ~ (Out 1 -Out 2 ) /(Out 1 +Out 2 ) POSITION SIGNAL Multiple rays add their currents linearly to the electrodes, where each ray’s power adds W i current to the total sum.

Calibration Want to measure forces, displaces – measure voltages from PSD – need calibration Δx = α ΔV F = kΔx = α kΔV

Glass water Glass Calibrate with a known displacement Calibrate with a known force Move bead relative to trap Stokes law: F = γv

Brownian motion as test force Langevin equation: Drag force γ = 3πηd Fluctuating Brownian force Trap force = 0 = 2k B Tγδ (t-t’) kBTkBT k B T= 4.14pN-nm

ΔtΔt ΔtΔtΔtΔt Autocorrelation function

ΔtΔt ΔtΔtΔtΔt

Brownian motion as test force Langevin equation: FT → Lorentzian power spectrum Corner frequency f c = k/2π  Exponential autocorrelation f’n

Power (V 2 /Hz) Frequency (Hz)

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. 1/2 = (k B T/k) 1/2 = (4.14/0.1) 1/2 = (41.4) 1/2 ~ 6.4 nm Reducing bandwidth reduces noise. But ( BW ) 1/2 = [∫const*(BW)dk] 1/2 = [(4k B T  100)/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 ]

3.4 kb DNA F ~ 20 pN f = 100Hz, 10Hz 1bp = 3.4Å UIUC - 02/11/08 Basepair Resolution—Yann UIUC unpublished

Kinesin Asbury, et al. Science (2003) Step size: 8nm Observing individual steps Motors move in discrete stepsDetailed 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.