## The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur

Optics Express, Vol. 14, Issue 25, pp. 12517-12531 (2006)

http://dx.doi.org/10.1364/OE.14.012517

Acrobat PDF (260 KB)

### Abstract

Dynamical instrument limitations, such as finite detection bandwidth, do not simply add statistical errors to fluctuation measurements, but can create significant systematic biases that affect the measurement of steady-state properties. Such effects must be considered when calibrating ultra-sensitive force probes by analyzing the observed Brownian fluctuations. In this article, we present a novel method for extracting the true spring constant and diffusion coefficient of a harmonically confined Brownian particle that extends the standard equipartition and power spectrum techniques to account for video-image motion blur. These results are confirmed both numerically with a Brownian dynamics simulation, and experimentally with laser optical tweezers.

© 2006 Optical Society of America

## 1. Introduction

1. K. Svoboda, P.P. Mitra, and S.M. Block, “Fluctuation analysis of Motor Protein Movement and Single Enzyme Kinetics,” Proc. Natl. Acad. Sci. USA **91**, 11782–11786 (1994). [CrossRef] [PubMed]

2. T.G. Mason and D.A. Weitz, “Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids,” Phys. Rev. Lett. **74**, 1250–1253 (1995). [CrossRef] [PubMed]

3. E. Evans and K. Ritchie, “Dynamic strength of molecular adhesion bonds,” Biophys. J. **72**, 1541–1555 (1997). [CrossRef] [PubMed]

4. D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, Jr., and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature (London) **437**, 231–234 (2005). [CrossRef] [PubMed]

5. T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, “The elasticity of a single supercoiled DNA molecule.” Science **271**, 1835–1837 (1996). [CrossRef] [PubMed]

6. D.T. Chen, E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky, and A.G. Yodh, “Rheological Microscopy: Local Mechanical Properties from Microrheology,” Phys. Rev. Lett. **90**, 108301 (2003). [CrossRef] [PubMed]

7. R. Yasuda, H. Miyata, and K. Kinosita, Jr., “Direct measurement of the torsional rigidity of single actin filaments,” J. Mol. Biol. **263**, 227–236 (1996). [CrossRef] [PubMed]

8. T. Savin and P.S. Doyle, “Static and Dynamic Errors in Particle Tracking Microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

9. T. Savin and P.S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 41106 (2005). [CrossRef]

10. L.P. Ghislain and W.W Webb, “Scanning-force microscope based on an optical trap,” Opt. Lett. **18**, 1678–1680 (1993). [CrossRef] [PubMed]

11. K. Svoboda and S.M. Block, “Biological applications of optical forces.” Annu. Rev. Biophys. Biomol. Struct. **23**, 247–285 (1994). [CrossRef] [PubMed]

12. F. Gittes and C.F. Schmidt, “Signals and noise in micromechanical measurements.” Methods Cell Biol. **55**, 129–156 (1998). [CrossRef]

13. E.-L. Florin, A. Pralle, E.H.K. Stelzer, and J.K.H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A **66**, 75–78 (1998). [CrossRef]

14. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

## 2. Bias in the measured variance of a harmonically trapped Brownian particle

*X*is an average of the true position

_{m}*X*taken over a finite time interval, which we call the integration time

*W*. In the simplest model,

*t*. More complex situations can be treated by multiplying

*X*(

*t*′) by an instrument-dependent function within the integral, i.e. by using a non-rectangular integration kernel.

*U*(

*x*)/

*k*), where

_{B}T*k*is the Boltzmann constant and

_{B}*T*is the absolute temperature:

*X*. In particular, motion blur introduces a systematic bias in the measured variance,

_{m}*k*, the friction factor of the particle

*γ*, and the integration time of the imaging device

*W*[7

7. R. Yasuda, H. Miyata, and K. Kinosita, Jr., “Direct measurement of the torsional rigidity of single actin filaments,” J. Mol. Biol. **263**, 227–236 (1996). [CrossRef] [PubMed]

8. T. Savin and P.S. Doyle, “Static and Dynamic Errors in Particle Tracking Microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

9. T. Savin and P.S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 41106 (2005). [CrossRef]

*α*by expressing the exposure time

*W*in units of the trap relaxation time

*τ*=

*γ*/

*k*, i.e.

*α*can also be expressed in terms of the diffusion coefficient

*D*by using the Einstein relation

*γ*=

*k*/

_{B}T*D*, i.e.

*α*=

*WDk*/(

*k*). Then as presented in appendix (A), the measured variance is given by:

_{B}T*S*(

*α*) is the motion blur correction function

## 3. Numerical studies

17. D.L. Ermak and J.A. McCammon. “Brownian dynamics with hydrodynamic interactions,” J. Chem. Phys. **69**, 1352–1360 (1978). [CrossRef]

*t*, the change in the bead position Δ

*x*is given by a discretization of the overdamped Langevin equation:

## 4. Experimental verification

### 4.1. Instrument description

_{4}laser) through a high numerical aperture oil immersion objective (Zeiss Plan Neofluar 100x/1.3) into a closed, water filled chamber. Laser power is varied with a liquid-crystal power controller (Brockton Electro-Optics). This optical tweezers system is integrated into an inverted light microscope (Zeiss Axiovert S100).

*ε*

^{2}to Eq. (6), i.e.

### 4.2. Experimental conditions

*µ*m±0.027

*µ*m). Experiments were performed by holding a bead in the optical trap and varying the power and the exposure time. The bead was held 30 µm from the closest surface, and the lamp intensity was varied with exposure time to ensure a similar intensity profile for each test. This ensured that the noise and signal quality between experimental runs was very similar, as validated by the results. For each test, both edges of the bead in one dimension were recorded and averaged to estimate the center position.

### 4.3. Experimental Results

*k*, friction factor

*γ*, and tracking error

*ε*

^{2}. Error estimates in the variance were calculated from the standard error due to the finite sample size, and variations due to vertical drift.

*γ*and

*ε*

^{2}occur at the lowest and highest powers, respectively. These estimates both agree within 2% of the error weighted average for all powers. For the nominal bead size,

*γ*agrees with the Stokes’ formula calculation to within 11%, indicating a slightly smaller bead or lower water viscosity than expected. Additionally, the estimate of tracking error

*ε*determined from the fit compares favorably with the standard deviation in position of a stuck bead, differing by about half a nanometer.

*γ*and

*ε*

^{2}are expected to remain essentially constant as the laser power is varied. Good consistency was found between the determined values of

*ε*

^{2}from different experimental runs, due to the protocol of matching the signal strength between tests. While laser heating could cause

*γ*to decrease with increasing power, this effect should be small for the < 500 mW powers used here [21

21. E.J.G. Peterman, F. Gittes, and C.F. Schmidt, “Laser-Induced Heating in Optical Traps,” Biophys. J. **84**, 1308–1316 (2003). [CrossRef] [PubMed]

22. P.M. Celliers and J. Conia, “Measurement of localized heating in the focus of an optical trap,” Appl. Opt. **39**, 3396–3407 (2000). [CrossRef]

*γ*and

*ε*

^{2}constant for all powers, the raw data was re-fit with Eq. (9) to yield

*k*. The data for all 4 powers was error-corrected by subtracting

*ε*

^{2}and was rescaled according to Eq. (6) and Eq. (7). This non-dimensionalized data is plotted alongside the motion blur correction function in Fig. 2, showing near-perfect quantitative agreement. This exceptional agreement further validates our treatment of

*γ*and

*ε*

^{2}. A plot of spring constant vs. dimensionless power is shown in Fig. 3, demonstrating the discrepancy between the blur-corrected spring constant and naïve spring constant for different integration times. Even for a modest spring constant of 0.03 pN/nm and a reasonably fast exposure time of 1 ms, the expected error is roughly 50%. We also note that the blur-corrected spring constant increases linearly with laser power as expected from optical-trapping theory. Once confirmed for a given system, this linearity can be exploited to determine not only the spring constant as a function of power but also the diffusion coefficient of the bead. This is discussed in subsection (5.3), and presented in Fig. 3.

*τ*=

*k*/

*γ*, it is feasible to calibrate the trap using the bead position power spectrum, allowing comparisons to the previous results at low laser power. Power spectrum fitting with the blur-corrected and aliased expression (Eq. (23)) at the lowest power yielded both a spring constant and friction factor that agree with the blur-corrected equipartition values to within 1%. Fits of the same data using the naïve expression (Eq. (13), not corrected for exposure time or aliasing) provided slightly worse results, overestimating the spring constant by 3% and the friction factor by 7%. (See Appendix C for procedural details.)

## 5. Discussion: Practical suggestions for calibrating an optical trap

*k*and diffusion coefficient

*D*of a harmonically confined Brownian particle. We will assume that the temperature

*T*is known. The approaches here are generic, and can be used even if the confining potential is not an optical trap (e.g. beads embedded in a gel, etc.) We will continue to treat the measured position as an unweighted time average of the true position over the integration time

*W*(Eq. (1)), which is consistent with the experimental results for our detection system. In other situations, e.g. if the rise and fall time are not negligible relative to the exposure time, these equations and ideas can be readily generalized as noted in section (2).

### 5.1. Determining k from D andW.

*D*of the confined particle and the integration time

*W*of the instrument is known, the correct spring constant

*k*can be directly obtained from the measured variance var(

*X*) by using equation 6. If the tracking error

_{m}*ε*is significant,

*ε*

^{2}should first be subtracted from the measured variance as in equation 9. While we cannot in general isolate

*k*in this transcendental equation, it can easily be found numerically by utilizing a standard rootfinding method. Alternatively, an approximate closed form solution for k is derived in Appendix D.

### 5.2. Determining k and D by varying W

*k*and

*D*can still be determined by measuring the variance at different shutter speeds and fitting to the blur-corrected variance function. This technique is demonstrated in the experimental results section, and yields accurate measurements provided the integration time is not too much larger than the trap relaxation time (

*α*is not much larger than 1). Practically speaking, this is a useful technique, as the maximum shutter speed of a camera is often much faster than the maximum data acquisition speed (e.g. it is much easier to obtain a video camera with a 0.1 ms shutter speed than a camera with a frame rate of 10 kHz). Furthermore, this approach for quantifying the power spectrum from the blur is quite general, and could be used in other systems. As long as the form of the power spectrum is known, the model parameters could be determined by measuring the total variance over a suitable spectrum of shutter speeds.

### 5.3. Determining k and D by varying k

*k*and

*D*by measuring the variance of a trapped bead at different laser powers but with the same shutter speed. Such data can be fit to the blur model (equation 6, recalling that

*α*=

*WDk*/(

*k*)) by introducing an additional fitting parameter

_{B}T*c*that relates the laser power

*P*to the spring constant, i.e. we make the substitution

*k*=

*cP*, and perform a non-linear fit to var(

*X*) vs. power data in order to determine

_{m}*c*and

*D*. Equivalently, we can express the naïve spring constant

*k*=

_{m}*k*/var(

_{B}T*X*) as a function of

_{m}*c*and

*P*, and perform a fit to

*k*vs.

_{m}*P*data as shown in Fig. 3 of the experimental results subsection (4.3), where the viability of this method is demonstrated.

### 5.4. Design strategies for using the blur technique

## 6. Conclusions

### A. Calculation of the measured variance of a harmonically trapped Brownian particle

*k*, the diffusion coefficient of the particle

*D*, and the integration time of the imaging device

*W*(notation is as introduced in section (2)). The derivation follows standard techniques (e.g. [15]) and is similar to calculations presented in references [7

7. R. Yasuda, H. Miyata, and K. Kinosita, Jr., “Direct measurement of the torsional rigidity of single actin filaments,” J. Mol. Biol. **263**, 227–236 (1996). [CrossRef] [PubMed]

8. T. Savin and P.S. Doyle, “Static and Dynamic Errors in Particle Tracking Microrheology,” Biophys. J. **88**, 623–638 (2005). [CrossRef]

9. T. Savin and P.S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E **71**, 41106 (2005). [CrossRef]

16. M.C. Wang and G.E. Uhlenbeck, “On the Theory of the Brownian Motion II,” Rev. Mod. Phys. **17**, 323–342 (1945). [CrossRef]

#### A.1. Frequency-space calculation

*X*(

_{m}*t*) can be calculated by convolving the true trajectory

*X*(

*t*) with a rectangular function,

*H*(

*t*) is defined by:

*t*′ is integrated from -∞ to +∞), which is our convention whenever limits are not explicitly written. The width of the rectangle

*W*is simply the integration time as previously defined. This convolution acts as an ideal moving average filter in time, and is consistent with the integral expression for

*X*(

_{m}*t*) given in Eq. (1).

*X̃*(

*ω*)=∫

*X*(

*t*)exp(

*iωt*)d

*t*, and

*ω*is the frequency in radians/second. The theoretical power spectrum

*P*(

*ω*) is given by:

*P*(

*ω*) yields the true variance of

*X*(

*t*),

*X*) as a function of the exposure time

_{m}*W*and the friction factor

*γ*by integrating the power spectrum of the measured position (Eq. (12)):

*τ*=

*γ*/

*k*, the trap relaxation time. Writing this formula in terms of the dimensionless exposure time,

*X*)=

*k*/

_{B}T*k*yields:

*S*(

*α*) is the motion blur correction function:

#### A.2. Blur-corrected filtered power spectrum

*P*(

*ω*) that accounts for both exposure time effects and aliasing. Combining expressions 12, 13 and 14, we can see the effect of exposure time on the measured power spectrum:

*ω*is the angular sampling frequency (i.e. the data acquisition rate times 2

_{s}*π*). Aliasing changes the shape of the power spectrum, so neglecting it when fitting can cause errors. The sum in Eq. (23) can be calculated numerically and fit to experimental data. It is typically sufficient to calculate only the first few terms.

*X*) is unchanged. A detailed discussion of power spectrum calibration with an emphasis on photodiode detection systems is given in reference [14

_{m}14. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

#### A.3. Real-space calculation

*X*(

*t*), the measured position

*X*is a random function of the true position of the particle at the start of the integration time, i.e.

_{m}*X*(

*t*|

*x*

_{0}) is the actual position of the bead at time

*t*given that it is at position

*x*

_{0}at time zero, andW is the integration time as defined previously. In other words, even with knowledge of the initial particle position, it is not possible to predict what the measured position will be. However, the distribution of

*X*is well-defined, and one can determine its moments.

_{m}*X*, and the measured position for a given initial position

*X*(

_{m}*x*). For the harmonic potential

*X*(

_{m}*X*)〉=0 by symmetry, so the variance reduces to

*ρX*(

*x*

_{0}) is the probability density of the initial position, and the integral is taken over all space (consistent with our previously stated convention). In equilibrium,

*ρX*(

*x*

_{0}) is simply the Boltzmann distribution given in Eq. (2).

*t*

_{2}>

*t*

_{1}is added, which changes the limits of integration.

*ρ*(

*x*,

*t*|

*x*

_{0},

*t*

_{0}). The Green’s function represents the probability density for finding the particle at position

*x*at time

*t*given that it is at

*x*

_{0}at time

*t*

_{0}. It can be found by solving the diffusion equation

*ρ*(

*x*,

*t*

_{0})=

*δ*(

*x*-

*x*

_{0}). The solution to this problem is well-known [18, 16

16. M.C. Wang and G.E. Uhlenbeck, “On the Theory of the Brownian Motion II,” Rev. Mod. Phys. **17**, 323–342 (1945). [CrossRef]

*τ*=

*k*/(

_{B}T*kD*)=

*γ*/

*k*. Notice that this is simply a spreading Gaussian distribution with the mean given by the deterministic (non-Brownian) position of a particle connected to a spring in an overdamped environment, and with a variance that looks like free diffusion at short time scales (i.e. initially increasing as 2

*D*(

*t*-

*t*

_{0})), but exponentially approaching the equilibrium value of

*k*/

_{B}T*k*on longer time scales.

*V*(

*t*) and

*τ*are as defined above.

*t*

_{1}≫

*τ*, which yields the simplified equation:

*t*

_{2}-

*t*

_{1}) we obtain the standard result of Eq. (13).

### B. High-pass filtering in variance measurements

*f*=k/2

_{c}*πγ*). However, as the filtering frequencies approach zero, drift causes the measured variance to increase beyond its expected value. By applying a linear fit and extrapolating to the 0 Hz cutoff, we can reliably estimate the “drift-free” variance of bead position.

### C. Experimental power spectrum calibration

12. F. Gittes and C.F. Schmidt, “Signals and noise in micromechanical measurements.” Methods Cell Biol. **55**, 129–156 (1998). [CrossRef]

14. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. **75**, 594–612 (2004). [CrossRef]

**75**, 594–612 (2004). [CrossRef]

**75**, 594–612 (2004). [CrossRef]

### D. Approximate analytical expression for k

*W*is not significantly larger than the trap relaxation time, i.e.

*α*=

*W*/

*τ*=

*Wk*/

*γ*is not much larger than 1, an approximate version of equation 6 can be inverted to give a closed form solution for

*k*. First, we use a Padé approximation to express the motion blur correction function as:

*k*. This results in the following approximation for the true spring constant:

*α*<3, which corresponds to a blur correction factor of

*S*(3)≐0.46. In other words, if the uncorrected equipartition method gives a spring constant which is within a factor of 2 of the true value, this approximation formula should be accurate to within 3%, as we have tested numerically.

## Acknowledgments

## References and links

1. | K. Svoboda, P.P. Mitra, and S.M. Block, “Fluctuation analysis of Motor Protein Movement and Single Enzyme Kinetics,” Proc. Natl. Acad. Sci. USA |

2. | T.G. Mason and D.A. Weitz, “Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids,” Phys. Rev. Lett. |

3. | E. Evans and K. Ritchie, “Dynamic strength of molecular adhesion bonds,” Biophys. J. |

4. | D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, Jr., and C. Bustamante, “Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies,” Nature (London) |

5. | T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, “The elasticity of a single supercoiled DNA molecule.” Science |

6. | D.T. Chen, E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky, and A.G. Yodh, “Rheological Microscopy: Local Mechanical Properties from Microrheology,” Phys. Rev. Lett. |

7. | R. Yasuda, H. Miyata, and K. Kinosita, Jr., “Direct measurement of the torsional rigidity of single actin filaments,” J. Mol. Biol. |

8. | T. Savin and P.S. Doyle, “Static and Dynamic Errors in Particle Tracking Microrheology,” Biophys. J. |

9. | T. Savin and P.S. Doyle, “Role of a finite exposure time on measuring an elastic modulus using microrheology,” Phys. Rev. E |

10. | L.P. Ghislain and W.W Webb, “Scanning-force microscope based on an optical trap,” Opt. Lett. |

11. | K. Svoboda and S.M. Block, “Biological applications of optical forces.” Annu. Rev. Biophys. Biomol. Struct. |

12. | F. Gittes and C.F. Schmidt, “Signals and noise in micromechanical measurements.” Methods Cell Biol. |

13. | E.-L. Florin, A. Pralle, E.H.K. Stelzer, and J.K.H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. A |

14. | K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. |

15. | A.V. Oppenheim, A.S. Willsky, and S.H. Nawab, |

16. | M.C. Wang and G.E. Uhlenbeck, “On the Theory of the Brownian Motion II,” Rev. Mod. Phys. |

17. | D.L. Ermak and J.A. McCammon. “Brownian dynamics with hydrodynamic interactions,” J. Chem. Phys. |

18. | M. Doi and S.F. Edwards, |

19. | J.F. Kenney and E.S. Keeping, |

20. | Data aquisition software was written by Volkmar Heinrich. |

21. | E.J.G. Peterman, F. Gittes, and C.F. Schmidt, “Laser-Induced Heating in Optical Traps,” Biophys. J. |

22. | P.M. Celliers and J. Conia, “Measurement of localized heating in the focus of an optical trap,” Appl. Opt. |

**OCIS Codes**

(110.4280) Imaging systems : Noise in imaging systems

(140.7010) Lasers and laser optics : Laser trapping

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

**ToC Category:**

Trapping

**History**

Original Manuscript: July 21, 2006

Revised Manuscript: October 24, 2006

Manuscript Accepted: November 24, 2006

Published: December 11, 2006

**Virtual Issues**

Vol. 2, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Wesley P. Wong and Ken Halvorsen, "The effect of integration time on fluctuation measurements: calibrating an optical trap in the presence of motion blur," Opt. Express **14**, 12517-12531 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-12517

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### References

- K. Svoboda, P.P. Mitra, and S.M. Block, "Fluctuation analysis of Motor Protein Movement and Single Enzyme Kinetics," Proc. Natl. Acad. Sci. USA 91, 11782-11786 (1994). [CrossRef] [PubMed]
- T.G. Mason and D.A. Weitz, "Optical Measurements of Frequency-Dependent Linear Viscoelastic Moduli of Complex Fluids," Phys. Rev. Lett. 74, 1250-1253 (1995). [CrossRef] [PubMed]
- E. Evans and K. Ritchie, "Dynamic strength of molecular adhesion bonds," Biophys. J. 72, 1541-1555 (1997). [CrossRef] [PubMed]
- D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, Jr., and C. Bustamante, "Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies," Nature (London) 437, 231-234 (2005). [CrossRef] [PubMed]
- T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, "The elasticity of a single supercoiled DNA molecule." Science 271, 1835-1837 (1996). [CrossRef] [PubMed]
- D.T. Chen, E.R. Weeks, J.C. Crocker, M.F. Islam, R. Verma, J. Gruber, A.J. Levine, T.C. Lubensky, and A.G. Yodh, "Rheological Microscopy: Local Mechanical Properties from Microrheology," Phys. Rev. Lett. 90, 108301 (2003). [CrossRef] [PubMed]
- R. Yasuda, H. Miyata, and K. Kinosita, Jr., "Direct measurement of the torsional rigidity of single actin filaments," J. Mol. Biol. 263, 227-236 (1996). [CrossRef] [PubMed]
- T. Savin and P.S. Doyle, "Static and Dynamic Errors in Particle Tracking Microrheology," Biophys. J. 88, 623-638 (2005). [CrossRef]
- T. Savin and P.S. Doyle, "Role of a finite exposure time on measuring an elastic modulus using microrheology," Phys. Rev. E 71, 41106 (2005). [CrossRef]
- L.P. Ghislain and W.W. Webb, "Scanning-force microscope based on an optical trap," Opt. Lett. 18, 1678-1680 (1993). [CrossRef] [PubMed]
- K. Svoboda and S.M. Block, "Biological applications of optical forces." Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994). [CrossRef] [PubMed]
- F. Gittes and C.F. Schmidt, "Signals and noise in micromechanical measurements." Methods Cell Biol. 55, 129-156 (1998). [CrossRef]
- E.-L. Florin, A. Pralle, E.H.K. Stelzer, and J.K.H. Hörber, "Photonic force microscope calibration by thermal noise analysis," Appl. Phys. A 66, 75-78 (1998). [CrossRef]
- K. Berg-Sørensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum. 75, 594-612 (2004). [CrossRef]
- A.V. Oppenheim, A.S. Willsky, and S.H. Nawab, Signals & systems (Prentice-Hall, Inc., Upper Saddle River, NJ, 1996).
- M.C. Wang and G.E. Uhlenbeck, "On the Theory of the Brownian Motion II," Rev. Mod. Phys. 17, 323-342 (1945). [CrossRef]
- D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978). [CrossRef]
- M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).
- J.F. Kenney and E.S. Keeping, Mathematics of Statistics, Pt. 2, 2nd ed. (Van Nostrand, Princeton, NJ, 1951).
- Data aquisition software was written by Volkmar Heinrich.
- E.J.G. Peterman, F. Gittes, and C.F. Schmidt, "Laser-Induced Heating in Optical Traps," Biophys. J. 84, 1308-1316 (2003). [CrossRef] [PubMed]
- P.M. Celliers and J. Conia, "Measurement of localized heating in the focus of an optical trap," Appl. Opt. 39, 3396-3407 (2000). [CrossRef]

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