Homework: do Thursday the 25 th . Following Tuesday: Tom Kuhlman talking on

2. Homework: do Thursday the 25th. Following Tuesday: Tom Kuhlman talking on Following Thursday (31st): Tour of optical and magnetic traps (and maybe fluorescence) Student Presentations: November 19th (Tues.), 20th (Wed.) & 21st (Th.) Note: Wednesday will start at 6pm: get pizza! Final Exam: 7-10pm Wednesday Dec 18th.

3. Optical Traps (Tweezers) Dielectric objects are attracted to the center of the beam, slightly above the beam waist. This depends on the difference of index of refraction between the bead and the solvent (water). Vary ktrap with laser intensity such that ktrap ≈ kbio (k ≈ 0.1pN/nm) Can measure pN forces and (sub-) nm steps! http://en.wikipedia.org/wiki/Optical_tweezers

4. 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, …) [AcoustOptic Device- moveable laser pointer] 2) Measurement – collection and detection optics (BFP interferometry) 3) Calibration – convert raw data into forces (pN), displacements (nm)

5. 1) Manipulation Want to apply forces – need ability to move stage or trap (piezo stage, steerable mirror, AOD…) (Acouto Optic Device: variable placement of laser) DNA By using two beads, and taking difference, capable of removing floor movement! Get to Angstrom level!

6. 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

7. 2) Measurement: BFP imaged onto detector Trap laser ∑ specimen Relay lens PSD BFP (Image at point P will get imaged to point P’) Conjugate image planes

8. Out1 P In1 N N In2 P Out2 Position sensitive detector (PSD) Plate resistors separated by reverse-biased PIN photodiode Opposite electrodes at same potential – no conduction with no light

9. Out1 P In1 N N In2 P Out2 Multiple rays add their currents linearly to the electrodes, where each ray’s power adds Wicurrent to the total sum. SIGNAL POSITION ΔX ~ (In1-In2) / (In1 + In2) ΔY ~ (Out1-Out2) /(Out1+Out2)

10. 3) Calibration Want to measure forces, displaces – measure voltages from PSD – need calibration Δx = α ΔV F = kΔx = α kΔV a allows you to go from the photodetector voltage to distance in nm, k gets you from distance to force in pN Need to measure a, k.

11. Calibrate with a known displacement Calibrate with a known force Glass water Glass Glass Move bead relative to trap (Translation stage) Stokes law: F = γv (Water flowing)

12. Drag force γ = 6πηr Brownian motion as test force Langevin equation: ≈0 kBT Trap force Inertia term (ma) Fluctuating Brownian force <F(t)> = 0 <F(t)F(t’)> = 2kBTγδ (t-t’) Inertia term for um-sized objects is always small (…for bacteria) kBT= 4.14pN-nm δ (t-t’) = delta fcn

13. Δt Δt Δt Autocorrelation function

14. Δt Δt Δt Autocorrelation function

15. Why does tail become wider? Answer: If it’s headed in one direction, it tends to keep going in for a finite period of time. It doesn’t forget about where it is instantaneously. It has memory. <F(t)> = 0 <F(t)F(t’)> = 2kBTγδ (t-t’) This says it has no memory. Not quite correct.

16. Brownian motion as test force Langevin equation: • = Ns/m • K= N/m Exponential autocorrelation function Notice that this follows the Equilibrium Theorem FT → Lorentzian power spectrum Corner frequency fc = k/2πg kT = energy = Nm S= Nm*Ns/m/ (N/m)2 = m2sec = m2/Hz

17. As f 0, then As f fc, then As f >> fc, then

18. Langevin Equation FT: get a curve that looks like this. Determine, k, a (g = 6phr) 1. Voltages vs. time from detectors. 2. Take FT. 3. Square it to get Power spectrum. 4. Power spectrum = α2 *Sx(f). Power (V2/Hz) Note: This is Power spectrum for voltage (not Nm) kT=energy=Nm S= Nm*Ns/m/ (N/m)2 Sx(f)= m2sec = m2/Hz Power spectrum of voltage Nm V divide by a2. Frequency (Hz)

19. What is noise in measurement?. 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 ] How to decrease noise?

20. Reducing bandwidth reduces noise. If instead you collect data out to a lower bandwidth BW (100 Hz), you get a much smaller noise. Noise = integrate power spectrum over frequency. If BW < fc then it’s simple integration because power spectrum is constant, with amplitude = 4kBTg/k2 Power (V2/Hz) typical value of g (10-6 for ~1 mm bead in water). Let’s say BW = 100 Hz: But (<x2>BW)1/2 = [∫const*(BW)dk]1/2= [(4kBTg100)/k]1/2 = [4*4.14*10-6*100/0.1]1/2 ~ 0.4 nm = 4 Angstrom!! Frequency (Hz)

21. 1 2 3 1 2 4 3 5 4 5 6 6 7 7 8 9 9 8 Basepair Resolution—Yann Chemla @ UIUC unpublished 1bp = 3.4Å UIUC - 02/11/08 3.4 kb DNA F ~ 20 pN f = 100Hz, 10Hz

22. Class evaluation • 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.