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

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  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]
  15. A.V. Oppenheim, A.S. Willsky, and S.H. Nawab, Signals & systems (Prentice-Hall, Inc., Upper Saddle River, NJ, 1996).
  16. M.C. Wang and G.E. Uhlenbeck, "On the Theory of the Brownian Motion II," Rev. Mod. Phys. 17, 323-342 (1945).
    [CrossRef]
  17. D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978).
    [CrossRef]
  18. M. Doi and S.F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986).
  19. J.F. Kenney and E.S. Keeping, Mathematics of Statistics, Pt. 2, 2nd ed. (Van Nostrand, Princeton, NJ, 1951).
  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. 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]

2005

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. 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]

2004

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

2003

E.J.G. Peterman, F. Gittes, and C.F. Schmidt, "Laser-Induced Heating in Optical Traps," Biophys. J. 84, 1308-1316 (2003).
[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]

2000

1998

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]

1997

E. Evans and K. Ritchie, "Dynamic strength of molecular adhesion bonds," Biophys. J. 72, 1541-1555 (1997).
[CrossRef] [PubMed]

1996

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.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]

1995

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]

1994

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]

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

1993

1978

D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978).
[CrossRef]

1945

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

Allemand, J.F.

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]

Bensimon, A.

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]

Bensimon, D.

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]

Berg-Sørensen, K.

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

Block, S.M.

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]

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

Bustamante, C.

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]

Celliers, P.M.

Chen, D.T.

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]

Collin, D.

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]

Conia, J.

Crocker, J.C.

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]

Croquette, V.

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]

Doyle, P.S.

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]

Ermak, D.L.

D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978).
[CrossRef]

Evans, E.

E. Evans and K. Ritchie, "Dynamic strength of molecular adhesion bonds," Biophys. J. 72, 1541-1555 (1997).
[CrossRef] [PubMed]

Florin, E.-L.

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]

Flyvbjerg, H.

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

Ghislain, L.P.

Gittes, F.

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

F. Gittes and C.F. Schmidt, "Signals and noise in micromechanical measurements." Methods Cell Biol. 55, 129-156 (1998).
[CrossRef]

Gruber, J.

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]

Hörber, J.K.H.

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]

Islam, M.F.

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]

Jarzynski, C.

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]

Kinosita, K.

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]

Levine, A.J.

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]

Lubensky, T.C.

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]

Mason, T.G.

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]

McCammon, J.A.

D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978).
[CrossRef]

Mitra, P.P.

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]

Miyata, H.

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]

Peterman, E.J.G.

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

Pralle, A.

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]

Ritchie, K.

E. Evans and K. Ritchie, "Dynamic strength of molecular adhesion bonds," Biophys. J. 72, 1541-1555 (1997).
[CrossRef] [PubMed]

Ritort, F.

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]

Savin, T.

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]

T. Savin and P.S. Doyle, "Static and Dynamic Errors in Particle Tracking Microrheology," Biophys. J. 88, 623-638 (2005).
[CrossRef]

Schmidt, C.F.

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

F. Gittes and C.F. Schmidt, "Signals and noise in micromechanical measurements." Methods Cell Biol. 55, 129-156 (1998).
[CrossRef]

Smith, S.B.

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]

Stelzer, E.H.K.

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]

Strick, T.R.

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]

Svoboda, K.

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]

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

Tinoco, I.

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]

Uhlenbeck, G.E.

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

Verma, R.

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]

Wang, M.C.

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

Webb, W.W.

Weeks, E.R.

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]

Weitz, D.A.

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]

Yasuda, R.

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]

Yodh, A.G.

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]

Annu. Rev. Biophys. Biomol. Struct.

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

Appl. Opt.

Appl. Phys. A

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]

Biophys. J.

E. Evans and K. Ritchie, "Dynamic strength of molecular adhesion bonds," Biophys. J. 72, 1541-1555 (1997).
[CrossRef] [PubMed]

T. Savin and P.S. Doyle, "Static and Dynamic Errors in Particle Tracking Microrheology," Biophys. J. 88, 623-638 (2005).
[CrossRef]

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

J. Chem. Phys.

D.L. Ermak and J.A. McCammon. "Brownian dynamics with hydrodynamic interactions," J. Chem. Phys. 69, 1352-1360 (1978).
[CrossRef]

J. Mol. Biol.

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]

Methods Cell Biol.

F. Gittes and C.F. Schmidt, "Signals and noise in micromechanical measurements." Methods Cell Biol. 55, 129-156 (1998).
[CrossRef]

Nature (London)

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]

Opt. Lett.

Phys. Rev. E

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]

Phys. Rev. Lett.

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]

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]

Proc. Natl. Acad. Sci. USA

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]

Rev. Mod. Phys.

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

Rev. Sci. Instrum.

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

Science

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]

Other

A.V. Oppenheim, A.S. Willsky, and S.H. Nawab, Signals & systems (Prentice-Hall, Inc., Upper Saddle River, NJ, 1996).

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.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Brownian dynamics simulation results for measured variance as a function of exposure time. Data has been rescaled and plotted alongside S(α), the motion blur correction function of Eq. (7), showing excellent agreement within the expected error. The step size of the simulation is by 1µs, which is less than 0.01τ for all three simulations. The different simulation settings are: (i) 1.6µm bead radius, k=0.05 pN/nm, τ=0.537ms, (ii) 0.4µm bead radius, k=0.05 pN/nm, τ=0.134 ms, (iii) (1.6µm bead radius, k=0.0125 pN/nm, τ=2.148 ms) (b) Histogram of measured positions for simulation run (c) for an exposure time of 4 ms. It is a Gaussian distribution as expected [16]. The normal curve with the predicted variance is superimposed showing excellent agreement. The expected distribution for an ideal “blur-free” measurement system is superimposed as a dotted line.

Fig. 2.
Fig. 2.

Fractional variance (var(Xm )/var(X)) vs. dimensionless exposure time α=W/τ for experimental optical trap data at 4 different powers. Overlaid on the data is the motion blur correction function S(α) given by Eq. (7).

Fig. 3.
Fig. 3.

Spring constant vs. power for a single bead in the optical trap. The naïve equipartition measured spring constant with 1 ms and 2 ms exposure times (red triangles and blue squares, respectively) is compared with the blur corrected spring constant (black circles). The dashed blue and red lines going through the uncorrected data represent non-linear fits to the blur model assuming a linear relationship between k and laser power, i.e. k=cP, as discussed in subsection (5.3). The values obtained from these fits for c and γ agree within error with the “black circle” values obtained by varying the exposure time.

Fig. 4.
Fig. 4.

Experimentally measured variance as a function of the high pass filter cutoff frequency shows a linear relation (line), which can be extrapolated to 0 Hz to reliably estimate the drift-free variance. The variance without filtering (cross) is 100 nm2, while the extrapolated variance (star) is 78.5nm2

Fig. 5.
Fig. 5.

A log-log plot of the one-sided power spectrum (dots) for a trapped bead, with theoretical models produced from a least squares fit to the data (blur-corrected and aliased, Eq. (23) solid line; naïve, Eq. (13) blue dashed line; naïve aliased, green dotted line). The effect of the motion blur correction function S(α) is readily apparent from the clear discrepancy between the solid red and dotted green lines.

Fig. 6.
Fig. 6.

Fractional deviation of the power spectrum data obtained by dividing the experimentally measured values (dots in Fig. 5) by the fit obtained with the blur-corrected and aliased model (solid red line in Fig. 5). Left: Scatter plot demonstrating the quality of the fit; the two dashed red lines indicate the estimated standard deviation from unity of 1 128 [14]. Right: Histogram of the fractional deviation data overlaid with a Gaussian distribution with a standard deviation of 1 128 (solid red line).

Equations (37)

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X m ( t ) = 1 W t W t X ( t ) d t
ρ X ( x ) = 1 2 π k B T k exp ( k x 2 2 k B T )
var ( X ) X 2 X 2 = k B T k
var ( X m ) var ( X )
α W τ
var ( X m ) = var ( X ) S ( α )
S ( α ) = 2 α 2 α 2 ( 1 exp ( α ) )
Δ x = D k B T f det + δ x ( Δ t )
var ( X m ) = 2 k B T k ( τ W τ 2 W 2 ( 1 exp ( W τ ) ) ) + ε 2
X m ( t ) = X ( t ) * H ( t ) X ( t ) H ( t t ) d t
H ( t ) = { 1 W 0 < t W 0 elsewhere
P m ( ω ) X ˜ m ( ω ) 2 = X ˜ ( ω ) 2 H ˜ ( ω ) 2
P ( ω ) X ˜ ( ω ) 2 = 2 γ k B T γ 2 ω 2 + k 2
H ˜ ( ω ) 2 = ( sin ( ω W 2 ) ω W 2 ) 2
var ( X ) = 1 2 π P ( ω ) d ω = k B T k
var ( X m ) = 1 2 π P m ( ω ) d ω
= 2 k B T k ( τ W τ 2 W 2 ( 1 exp ( W τ ) ) )
α W τ
var ( X m ) = var ( X ) S ( α )
S ( α ) = 2 α 2 α 2 ( 1 exp ( α ) )
P m ( ω ) = 2 γ k B T γ 2 ω 2 + k 2 ( sin ( ω W 2 ) ω W 2 ) 2
P aliased = n = + P m ( ω + n ω s )
= n = + 2 γ k B T γ 2 ( ω + n ω s ) 2 + k 2 ( sin ( ( ω + n ω s ) W 2 ) ( ω + n ω s ) W 2 ) 2
X m ( x 0 ) = 1 W 0 W X ( t x 0 ) d t
var ( X m ) X m ( X ) 2 X m ( X ) 2
var ( X m ) = ρ X ( x 0 ) X m ( X 0 ) 2 d x 0
X m ( x 0 ) 2 = 1 W 2 0 W 0 W X ( t 1 x 0 ) X ( t 2 x 0 ) d t 1 d t 2
= 2 W 2 0 W 0 t 2 X ( t 1 x 0 ) X ( t 2 x 0 ) t 2 > t 1 d t 1 d t 2
ρ t = D 2 ρ x 2 + D k B T ρ x k x + D k B T ρ k
ρ ( x , t | x 0 , t 0 ) = 1 2 π k B TV ( t t 0 ) k exp ( k ( x x 0 exp ( ( t t 0 ) τ ) ) 2 2 k B T V ( t t 0 ) )
V ( t ) = 1 exp ( 2 t τ )
X ( t 1 ) X ( t 2 ) t 2 > t 1 = x 1 x 2 ρ ( x 1 , t 1 | x 0 , 0 ) ρ ( x 2 , t 2 | x 1 , t 1 ) d x 1 d x 2
X ( t 1 ) X ( t 2 ) t 2 > t 1 = x 0 2 exp ( ( t 2 + t 1 ) τ ) + k B T V ( t 1 ) k exp ( ( t 2 t 1 ) τ )
var ( X m ) = 2 k B T k ( τ W τ 2 W 2 ( 1 exp ( W τ ) ) )
X ( t 1 ) X ( t 2 ) = k B T k exp ( t 2 t 1 τ )
S ( α ) 1 2 α 15 + α 2 60 1 + α 5
k 30 k B T 2 DW + 15 var ( X m ) + [ 225 var ( X m ) 2 + 240 DW var ( X m ) 11 D 2 W 2 ] 1 2

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