Abstract

We study the statistical properties of recordings that contain time-dependent positions of a bead trapped in optical tweezers. Analysis of such a time series indicates that the commonly accepted model, i.e., the autoregressive process of first-order, is not sufficient to fit the data. We show the presence of a first-order moving average part in the dynamical model of the system. We explain the origin of this part as an influence of the high-frequency CCD camera on the measurements. We show that this influence evidently depends on the applied exposure time. The proposed autoregressive moving average model appears to reflect perfectly all statistical features of the high-frequency recording data.

© 2014 Optical Society of America

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References

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  1. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
    [CrossRef]
  2. A. Ranaweera, Investigations with Optical Tweezers: Construction, Identification, and Control (University of California, 2004).
  3. L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
    [CrossRef]
  4. M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
    [CrossRef]
  5. S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
    [CrossRef]
  6. M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
    [CrossRef]
  7. A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
    [CrossRef]
  8. D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
    [CrossRef]
  9. W. Coffey, Yu. Kalmykov, and J. Waldron, The Langevin Equation (World Scientific, 2005).
  10. K. Sobczyk, Stochastic Differential Equations (Kluwer Academic, 1991).
  11. D. Grier and Y. Roichman, “Holographic optical trapping,” Appl. Opt. 45, 880–887 (2006).
    [CrossRef]
  12. M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
    [CrossRef]
  13. A. Horst and N. Forde, “Power spectral analysis for optical trap stiffness calibration from high-speed camera position detection with limited bandwidth,” Opt. Express 18, 7670–7677 (2010).
    [CrossRef]
  14. P. Brockwell and R. Davis, Time Series: Theory and Methods (Springer-Verlag, 2006).
  15. G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, 1994).
  16. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–613 (2004).
    [CrossRef]
  17. S. J. Orfanidis, Introduction to Signal Processing (Prentice-Hall, 1995).

2010 (1)

2009 (1)

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

2006 (1)

2005 (1)

2004 (2)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–613 (2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

1999 (1)

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

1997 (2)

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

1996 (1)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef]

1994 (1)

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Arsenault, M. E.

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

Baua, H. H.

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–613 (2004).
[CrossRef]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Box, G.

G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, 1994).

Brockwell, P.

P. Brockwell and R. Davis, Time Series: Theory and Methods (Springer-Verlag, 2006).

Bustamante, C.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef]

Coffey, W.

W. Coffey, Yu. Kalmykov, and J. Waldron, The Langevin Equation (World Scientific, 2005).

Cui, Y.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef]

Davis, R.

P. Brockwell and R. Davis, Time Series: Theory and Methods (Springer-Verlag, 2006).

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–613 (2004).
[CrossRef]

Forde, N.

Gelles, J.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Ghislain, L. P.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Goldman, Y. E.

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

Grier, D.

Grier, D. G.

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
[CrossRef]

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

Horst, A.

Jenkins, G.

G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, 1994).

Kalmykov, Yu.

W. Coffey, Yu. Kalmykov, and J. Waldron, The Langevin Equation (World Scientific, 2005).

Ladavac, K.

Landick, R.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Lee, S.-H.

Mehta, A. D.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Orfanidis, S. J.

S. J. Orfanidis, Introduction to Signal Processing (Prentice-Hall, 1995).

Polin, M.

Ranaweera, A.

A. Ranaweera, Investigations with Optical Tweezers: Construction, Identification, and Control (University of California, 2004).

Reinsel, G.

G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, 1994).

Rief, M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Roichman, Y.

Simmons, R. M.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Smith, D. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Smith, S. B.

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef]

Sobczyk, K.

K. Sobczyk, Stochastic Differential Equations (Kluwer Academic, 1991).

Spudich, J. A.

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Sun, Y.

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

Switz, N. A.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Waldron, J.

W. Coffey, Yu. Kalmykov, and J. Waldron, The Langevin Equation (World Scientific, 2005).

Wang, M. D.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Webb, W. W.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Yin, H.

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Appl. Opt. (1)

Biophys J. (1)

M. D. Wang, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Stretching DNA with optical tweezers,” Biophys J. 72, 1335–1346 (1997).
[CrossRef]

Curr. Opin. Colloid Interface Sci. (1)

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

Opt. Express (2)

Phys. Chem. Chem. Phys. (1)

M. E. Arsenault, Y. Sun, H. H. Baua, and Y. E. Goldman, “Using electrical and optical tweezers to facilitate studies of molecular motors,” Phys. Chem. Chem. Phys. 11, 4834–4839 (2009).
[CrossRef]

Rev. Sci. Instrum. (3)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–613 (2004).
[CrossRef]

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Science (2)

S. B. Smith, Y. Cui, and C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef]

A. D. Mehta, M. Rief, J. A. Spudich, D. A. Smith, and R. M. Simmons, “Single-molecule biomechanics with optical methods,” Science 283, 1689–1695 (1999).
[CrossRef]

Other (6)

S. J. Orfanidis, Introduction to Signal Processing (Prentice-Hall, 1995).

A. Ranaweera, Investigations with Optical Tweezers: Construction, Identification, and Control (University of California, 2004).

W. Coffey, Yu. Kalmykov, and J. Waldron, The Langevin Equation (World Scientific, 2005).

K. Sobczyk, Stochastic Differential Equations (Kluwer Academic, 1991).

P. Brockwell and R. Davis, Time Series: Theory and Methods (Springer-Verlag, 2006).

G. Box, G. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control (Prentice-Hall, 1994).

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Figures (6)

Fig. 1.
Fig. 1.

Plot of an exemplary trajectory recorded with sampling frequency 5000 fps (1000 measurements).

Fig. 2.
Fig. 2.

Estimate of the psd (gray line) and the fitted AR(1) psd (dashed black line) for recording made with sampling frequency 104fps.

Fig. 3.
Fig. 3.

Partial autocorrelation estimated from the bead’s trajectory for low-frequency sampling at 103fps.

Fig. 4.
Fig. 4.

Partial autocorrelation estimated from the bead’s trajectory for high-frequency sampling at 104fps.

Fig. 5.
Fig. 5.

Estimate of the psd (gray line) and the fitted ARMA(1,1) psd (dashed black line).

Fig. 6.
Fig. 6.

Estimated pacf of trajectory taken with frequency of sampling 103fps, but exposure time increased to 895 μs.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

0=FS+FO+FT,
dX=dt·λX+DdB,
X(t)=DtdB(s)e(ts)λ.
Xn:=X(nΔt),n{0,1,2,},Δt=const.
Xn=aXn1+ξn,
ξn=DnΔtnΔt+ΔtdB(s)eλ(nΔts).
a^=iXiXi1iXi2.
Xn=a2Xn2+ξn+aξn1=ak+1Xnk1+ξn+aξn1++akξnk.
Xn=k=0akξnk.
acovX(i,j):=cov(Xi,Xj).
acovX(k)=σ21a2a|k|.
psdX(ω):=k=acovX(k)eiωk.
psdX(ω)=σ21+a22acos(ω).
pacfX(k):=corr(XiPXi+1,,Xi+k1(Xi),Xi+kPXi+1,,Xi+k1(Xi+k)),
pacfX(k)={1,k=0;a,k=1;0,k>1.
Xn=aXn1+ξn+bξn1.
Xn=(a+b)Xn1abXn2b2ξn2.
Xn=(a+b)k=1(b)k1Xnk.
Xn=Xn+bXn1=a(Xn1+bXn2)+ξn+bξn1=aXn1+ξn+bξn1,
acovX(k)=cov(Xn+bXn1,Xn+k+bXn1+k)=(1+b2)acovX(k)+bacovX(k1)+bacovX(k+1).
psdX(ω)=σ21+b2+2bcos(ω)1+a22acos(ω).

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