Optical Tweezers and their calibration

How Trap works 1

http://en.wikipedia.org/wiki/File:Optical_Trap_As_a_Spring.jpg

How Trap works 2

http://en.wikipedia.org/wiki/File:Optical_Trap_Ray_Optics_Explanation.jpg

Calibration Methods

1. Histogram of Bead Positions

2. Power Spectrum of Bead Positions

3. Stokes Flow or Stokes Friction

Histogram of Bead Positions 1

a. Motivation and preliminary calculations.

Idea : If the bead in trap makes a Brownian motion,The histogram of positions should be a gaussian !!!

Application : The width of the gaussian is related with the stiffness of trap.

Math : Compare, gaussian function, probability density, energy stored in a spring and Equipartition Theorem.

Tk

xE

BCep)(

2

2

1)( xxE 2

2

1

2

1xTkB

2

20

2xx

Cep

Leads to:

TkB2

E.-L. Florin, A. Pralle, E.H.K. Stelzer, J.K.H. Hörber, Photonic force microscope calibration by thermal noise analysisApplied Physics A 66, S75-S78 (1998)

Histogram of Bead Positions 2

b. Position Sensitive Detector voltages corresponding to bead positions in trap.

Histogram of Bead Positions 3

c. Histogram of centralized data and fitting to a gaussian

Histogram of Bead Positions 4

d. Function to fit, parameter values, trap stiffness

22

23

21

b

bx

ebp

432

31 1041909.1,0024789.0,1017798.5 bbb

CTK

NmkB

00

23 25,1038.1

21610695.6V

Nm Units are important !!!

Power Spectrum of Bead Positions 1

a. Motivation and preliminary calculations 1.

Idea : If the motion of a bead in trap obey the Langevin equation,the solution of differential equation can be utilized to find stiffness !!!

Application : In the absolute value square of fourier transform of differential equation (power spectrum), corner frequency is related with stiffness.

Math : Start with the Langevin equation, add a time dependent noise,find the fourier transform, calculate magnitude of fourier transform.

)()( tvxFma Langevin Equation :

Power Spectrum of Bead Positions 1

a. Motivation and preliminary calculations 2.

Ignore inertial terms (Low Reynolds number), take the force term as spring force Also take the noise as Brownian noise. Then…

)(0 txx 0t Tkf B 4~ 2 with &

Take the fourier transform of equation gives…

)(~)(~)2(2 ffxif

Magnitude square of the fourier transform of positions (power spectrum) is…

)()(~

222

2

ff

Tkfx

c

B

with

2

cf

Joshua W. Shaevitz, A Practical Guide to Optical Trapping, ??? (2006)

Power Spectrum of Bead Positions 1

a. Motivation and preliminary calculations 3.

Having only PSD voltage values corresponding to actual position valuesLeads us to change equation for power spectrum correspondingly…

)()(

~2222

2

ff

TkfV

c

B

Vx with

G. Romano, L. Sacconi, M. Capitanio, F.S. Pavone, Force and Torque Measurements using magnetic micro beads forSingle molecule biopysics, Optics Communications 215, 323-331(2003)

Power Spectrum of Bead Positions 2

b. Power Spectrum of centralized data and fitting to a lorenzian…

Power Spectrum of Bead Positions 3

c. Function to fit, parameter values, trap stiffness 1

22

21

1)(

~

fbbfV

32

31 10445.9,10725.4 bb

With…

2

21

b

bfc

22 Tkb B

V

mD 5106146.1 Hzfc 6.48

281051.1s

kg

Power Spectrum of Bead Positions 3

c. Function to fit, parameter values, trap stiffness 2

21510203.1V

Nm Units are important !!!

Using the scaling factor for detector, stiffness coefficients are…

nm

pNPS 310611.4

nm

pNH 310567.2

Stokes Flow or Stokes Friction 1

a. Motivation and preliminary calculations 1.

Idea : If we move our trap with constant velocity while we there is a bead in it, equilibrium position of the trapped bead should change !!!

Application : The new equilibrium position caused from constant velocity should be related with stiffness of the trap !!!

Math : Start with the Langevin equation, add a constant velocity term,rearrange terms to observe shift in equilibrium position .

)(0 txx xvx with

)()(0 txv

x with

v

xeq

Stokes Flow or Stokes Friction 1

a. Motivation and preliminary calculations 2.

Approximated stage position in timev = constant !!!

Approximated force on bead in timehence the equilibrium position

v

xvF eq

Stokes Flow or Stokes Friction 2

b. Centralized bead position voltages …

Stokes Flow or Stokes Friction 2

b. Histogram of centralized data and fitting to a double gaussian…

Stokes Flow or Stokes Friction 3

c. Fitting function, parameter values, trap stiffness 1

V

mD 5106146.1 V

mS 610575.7

Now, we utilize the scaling factor belongs detector (from power spectrum calculation) and another scaling factor belongs stage (from company manual) for calculating xeq and v.

25

26

22

23

24

21

b

bx

b

bx

ebebp

0199.0,00368.0,10097.1 323

1 bbb

0197.0,0030.0,105.1 653

4 bbb

s

mv 510575.7 mxeq

71018.3

Stokes Flow or Stokes Friction 3

c. Fitting function, parameter values, trap stiffness 2

nm

pNSS 310597.3

v

xeq Using the relation

Results & Discussion

nm

pNSS 310597.3 nm

pNPS 310611.4 nm

pNH 310567.2

Calculated trap stiffness values are;

• Stiffness values are similar to the values are calculated in article below …• They are (same order of magnitude) close but not so close to tell a certainstiffness value...• The scale factor for detector may need a check for its accuracy. This is postponed to another presentation… • Fitting the histogram of positions to double gaussian gives similar width valuesso that similar stiffness values. But there are almost 20% difference among them…• If the response of the detector to a single axis movement on the stage is afunction of all four detector voltages, all of above analysis will need refinement…• If the azimuthal symmetry of the incoming gaussian beam somehow disturbed,again all of analysis will need refinement…

L. Oddershede, S. Grego, S.F. Norrelykke, K. Berg-Sorensen , Optical Tweezers: Probing Biological SurfacesProbe Microscopy 2, 129-137(2000)