Abstract

We present a comprehensive theory of closed-loop particle tracking for calculating the statistics of a diffusing fluorescent particle’s motion relative to the tracking lock point. A detailed comparison is made between the theory and experimental results, with excellent quantitative agreement found in all cases. A generalization of the theory of (open-loop) fluorescence correlation spectroscopy is developed, and the relationship to previous results is discussed. Two applications of the statistical techniques are given: a method for determining a tracked particle’s localization and an algorithm for rapid particle classification based on real-time analysis of the tracking control signal.

© 2007 Optical Society of America

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  1. V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
    [Crossref] [PubMed]
  2. V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
    [Crossref] [PubMed]
  3. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
    [Crossref] [PubMed]
  4. A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. 32, 145–147 (2007).
    [Crossref]
  5. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
    [Crossref] [PubMed]
  6. H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
    [Crossref]
  7. A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
    [Crossref]
  8. A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. 94, 118102 (2005).
    [Crossref] [PubMed]
  9. A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006).
    [Crossref] [PubMed]
  10. S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. 14, 669–680 (2006).
    [Crossref]
  11. M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
    [Crossref]
  12. J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B 71, 773–777 (2000).
    [Crossref]
  13. J. Enderlein, “Positional and temporal accuracy of single molecule tracking,” Sing. Mol. 1,  225–230 (2000).
  14. A. J. Berglund and H. Mabuchi, “Feedback Controller design for tracking a single fluorescent molecule,” Appl. Phys. B 78, 653–659 (2004).
    [Crossref]
  15. S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
    [Crossref]
  16. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B 83, 127–133 (2006).
    [Crossref]
  17. D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
    [Crossref] [PubMed]
  18. O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford University Press, 1996).
  19. N. G. Van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., North-Holland, Amsterdam, 2001).
  20. C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).
  21. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed. (Springer, 1959).
  22. D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
    [Crossref]
  23. E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers 13, 1–27 (1974).
    [Crossref]
  24. D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
    [Crossref] [PubMed]
  25. O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
    [Crossref]
  26. S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. 84, 2030–2042 (2003).
    [Crossref] [PubMed]
  27. T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. 54, 983–993 (1988).
    [Crossref] [PubMed]
  28. M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
    [Crossref]
  29. A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.
  30. L. Novotny, R. D. Grover, and K. Karrai, “Reflected image of a strongly focused spot,” Opt. Lett. 26, 789–791 (2001).
    [Crossref]
  31. T. A. Laurence, S. Fore, and T. Huser, “Fast, flexible algorithm for calculating photon correlations,” Opt. Lett. 31, 829–831 (2006).
    [Crossref] [PubMed]
  32. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. 26, 373–399 (1997).
    [Crossref]
  33. S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
    [Crossref] [PubMed]
  34. E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
    [Crossref]
  35. K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
    [Crossref] [PubMed]
  36. M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).

2007 (1)

2006 (9)

T. A. Laurence, S. Fore, and T. Huser, “Fast, flexible algorithm for calculating photon correlations,” Opt. Lett. 31, 829–831 (2006).
[Crossref] [PubMed]

A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.

E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
[Crossref]

A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006).
[Crossref] [PubMed]

S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. 14, 669–680 (2006).
[Crossref]

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B 83, 127–133 (2006).
[Crossref]

D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
[Crossref] [PubMed]

2005 (8)

S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[Crossref]

A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. 94, 118102 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

2004 (2)

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
[Crossref] [PubMed]

A. J. Berglund and H. Mabuchi, “Feedback Controller design for tracking a single fluorescent molecule,” Appl. Phys. B 78, 653–659 (2004).
[Crossref]

2003 (2)

S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

2002 (1)

O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[Crossref]

2001 (1)

2000 (2)

J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B 71, 773–777 (2000).
[Crossref]

J. Enderlein, “Positional and temporal accuracy of single molecule tracking,” Sing. Mol. 1,  225–230 (2000).

1997 (1)

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. 26, 373–399 (1997).
[Crossref]

1988 (1)

T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. 54, 983–993 (1988).
[Crossref] [PubMed]

1986 (1)

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).

1974 (2)

E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers 13, 1–27 (1974).
[Crossref]

D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
[Crossref] [PubMed]

1972 (1)

D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
[Crossref]

Andersson, S. B.

S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

Armani, M. D.

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

Belmont, A. S.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

Berglund, A. J.

A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. 32, 145–147 (2007).
[Crossref]

A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.

A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B 83, 127–133 (2006).
[Crossref]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
[Crossref] [PubMed]

A. J. Berglund and H. Mabuchi, “Feedback Controller design for tracking a single fluorescent molecule,” Appl. Phys. B 78, 653–659 (2004).
[Crossref]

Bonneau, S.

S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
[Crossref] [PubMed]

Bonnett, G.

O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[Crossref]

Brown, C. M.

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

Cang, H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
[Crossref] [PubMed]

Chaudhary, S.

S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. 14, 669–680 (2006).
[Crossref]

Chaudhary, S. V.

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

Cohen, A. E.

A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006).
[Crossref] [PubMed]

A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. 94, 118102 (2005).
[Crossref] [PubMed]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[Crossref]

Cohen, L. D.

S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
[Crossref] [PubMed]

Dahan, M.

S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
[Crossref] [PubMed]

Danuser, G.

E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
[Crossref]

DeGroot, M. H.

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).

Digman, M. A.

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

Elson, E. L.

S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
[Crossref] [PubMed]

E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers 13, 1–27 (1974).
[Crossref]

D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
[Crossref]

Enderlein, J.

J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B 71, 773–777 (2000).
[Crossref]

J. Enderlein, “Positional and temporal accuracy of single molecule tracking,” Sing. Mol. 1,  225–230 (2000).

Fore, S.

Gardiner, C. W.

C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).

Gratton, E.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

Grover, R. D.

Horwitz, A. R.

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

Huser, T.

Jacobs, O. L. R.

O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford University Press, 1996).

Jacobson, K.

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. 26, 373–399 (1997).
[Crossref]

Karrai, K.

Kis-Petikova, K.

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

Krichevsky, O.

O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[Crossref]

Laurence, T. A.

Levi, V.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

Mabuchi, H.

A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. 32, 145–147 (2007).
[Crossref]

A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B 83, 127–133 (2006).
[Crossref]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express 13, 8069–8082 (2005).
[Crossref] [PubMed]

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
[Crossref] [PubMed]

A. J. Berglund and H. Mabuchi, “Feedback Controller design for tracking a single fluorescent molecule,” Appl. Phys. B 78, 653–659 (2004).
[Crossref]

Magde, D.

D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
[Crossref] [PubMed]

E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers 13, 1–27 (1974).
[Crossref]

D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
[Crossref]

McHale, K.

A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. 32, 145–147 (2007).
[Crossref]

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
[Crossref] [PubMed]

Meijering, E.

E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
[Crossref]

Meyer, T.

T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. 54, 983–993 (1988).
[Crossref] [PubMed]

Moerner, W. E.

A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006).
[Crossref] [PubMed]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[Crossref]

Montiel, D.

D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
[Crossref] [PubMed]

Novotny, L.

Plutz, M.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

Probst, R.

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

Risken, H.

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed. (Springer, 1959).

Rizvi, A. H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

Ruan, Q.

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

Saffarian, S.

S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

Saxton, M. J.

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. 26, 373–399 (1997).
[Crossref]

Schindler, H.

T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. 54, 983–993 (1988).
[Crossref] [PubMed]

Sengupta, P.

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

Shapiro, B.

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. 14, 669–680 (2006).
[Crossref]

Smal, I.

E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
[Crossref]

Van Kampen, N. G.

N. G. Van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., North-Holland, Amsterdam, 2001).

Webb, W. W.

D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
[Crossref] [PubMed]

D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
[Crossref]

Wiseman, P. W.

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

Wong, C. M.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

Xu, C. S.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

Yang, H.

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
[Crossref] [PubMed]

Annu. Rev. Bio-phys. Biomolec. Struct. (1)

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. 26, 373–399 (1997).
[Crossref]

Appl. Phys. B (4)

J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B 71, 773–777 (2000).
[Crossref]

A. J. Berglund and H. Mabuchi, “Feedback Controller design for tracking a single fluorescent molecule,” Appl. Phys. B 78, 653–659 (2004).
[Crossref]

S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B 80, 809–816 (2005).
[Crossref]

A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B 83, 127–133 (2006).
[Crossref]

Appl. Phys. Lett. (2)

H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. 88, 223901 (2006).
[Crossref]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[Crossref]

Biochem. Soc. Trans. (1)

V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, a novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. 31, 997–1000 (2003).
[Crossref] [PubMed]

Biophys. J. (6)

V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. 89, 4275–4285 (2005).
[Crossref] [PubMed]

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. 86, 3409–3422 (2004).
[Crossref] [PubMed]

S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. 84, 2030–2042 (2003).
[Crossref] [PubMed]

T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. 54, 983–993 (1988).
[Crossref] [PubMed]

M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. 90, 1317–1327 (2005).
[Crossref]

V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. 88, 2919–2928 (2005).
[Crossref] [PubMed]

Biopolymers (2)

E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers 13, 1–27 (1974).
[Crossref]

D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers 13, 29–61 (1974).
[Crossref] [PubMed]

IEEE J. Microelectromech. Syst. (1)

M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. 15, 945–956 (2006).
[Crossref]

IEEE Signal Processing Mag. (1)

E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. 23, 46–53 (2006).
[Crossref]

IEEE Trans. Contr. Syst. Technol. (1)

S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. 14, 669–680 (2006).
[Crossref]

IEEE Trans. Image Process. (1)

S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. 14, 1384–1395 (2005).
[Crossref] [PubMed]

J. Phys. Chem. B (1)

D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B 110, 19763–19770 (2006).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Ph.D. thesis, California Institute of Technology (1)

A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.

Phys. Rev. Lett. (2)

D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. 29, 705–708 (1972).
[Crossref]

A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. 94, 118102 (2005).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. 65, 251–297 (2002).
[Crossref]

Sing. Mol. 1 (1)

J. Enderlein, “Positional and temporal accuracy of single molecule tracking,” Sing. Mol. 1,  225–230 (2000).

Other (5)

O. L. R. Jacobs, Introduction to Control Theory, 2nd ed. (Oxford University Press, 1996).

N. G. Van Kampen, Stochastic processes in physics and chemistry (Elsevier Science Pub. Co., North-Holland, Amsterdam, 2001).

C. W. Gardiner, Handook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, 1985).

H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, 2nd ed. (Springer, 1959).

M. H. DeGroot, Probability and Statistics (Addison-Wesley, Reading, MA, 1986).

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Figures (8)

Fig. 1.
Fig. 1.

Block diagram of the particle tracking control system. The system is driven by the particle’s velocity U(t) and the measurement noise N(t); the system outputs are the tracking stage position X(t) and the error signal E(t). For perfect tracking control, E(t) = 0.

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Fig. 2.
Fig. 2.

Schematic diagram of the experimental apparatus for tracking freely diffusing fluorescent nanoparticles in two dimensions as described in the main text. Key: AOM, acousto-optic modulator; HWP, half-wave plate; PBS, polarizing beam splitter; QWP, quarter-wave plate; CCD, charge-coupled device camera; PD, photodiode; APD, avalanche photodiode single-photon counter; HV, high voltage.

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Fig. 3.
Fig. 3.

Tracking trajectories for the same particle at three different excitation intensities. The rate of fluorescent photon arrivals during each trajectory is shown in the upper plots, while the x and y positions of the sample stage are shown in the lower plots. A single particle was tracked for around 100 s. After each 20 s interval, data collection was paused for a few seconds while the intensity servo set point was changed manually; these pauses are indicated by vertical black lines, but are not shown on the time (t) axis.

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Fig. 4.
Fig. 4.

Mean-square deviations D̂ (Δt)/2Δt for each of the trajectories in Fig. 3, plotted together with fits to the theory developed in Sect. 2.

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Fig. 5.
Fig. 5.

Fluorescence autocorrelation functions g(τ) for each of the trajectories in Fig.3, plotted together with fits to the theory developed in Sect. 3. The curves have been offset along the vertical axis for clarity.

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Fig. 6.
Fig. 6.

Three-dimensional scatter plot of the localization L determined from D̂ (Δt), g(τ), and g 0 as described in the text. Two-dimensional scatter plots comparing each pair of methods are shown in lighter shades projected onto their respective axes. Localization values determined for the 60 nm beads are shown in blue and 210 nm beads are shown in red. Dashed lines indicate where values would lie on the projection planes if all estimates were identically equal.

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Fig. 7.
Fig. 7.

Histogram of estimates D̂ (Δt = 10ms) calculated for a single trajectory of each type of particle (60 and 210 nm) versus the number of samples N used for the estimate. The solid black curves show the expected distribution with mean value set equal to the mean value from each data set.

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Fig. 8.
Fig. 8.

Measured probability of correct classification Pcorr as the estimation time T and the sample time Δt are varied. The dashed curves show the expected success probability calculated from χ2 statistics.

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Tables (2)

Tables Icon

Table 3. Table of fit parameters for the mean-square deviation curves of Fig. 4 and fluorescence autocorrelation functions of Fig. 5. Each set of fit parameters is labeled above by the photon count rate in kHz. For the fits to g(τ), the beam waist and diffusion coefficient were constrained to w = 1 μm (determined by scanning the excitation laser over an immobilized fluorescent nanoparticle) and D = 5.1 μm2/s (the average value determined from the mean-square deviation curves); constrained parameters are indicated by *. Aside from the constraint on D in g(τ), the two parameter sets are otherwise independent.

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Tables Icon

Table 4. Table comparing the localization L and measurement noise n determined from D̂ (Δt), g(τ), and g 0 for both 60 and 210 nm diameter particles. Shot-noise limited, theoretical optimum values are denoted by “Opt.” The error values are the observed standard deviation.

View Table | View all tables in this article

Equations (87)

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U ( t ) = 2 D dW p ( t ) d t ,
X p ( t ) = ∫ 0 t U ( t ′ ) dt ′ = 2 D ∫ 0 t d W p ( t ′ )
E ( t ) = X ( t ) − X p ( t )
N ( t ) = n dW n ( t ) d t
( X ˜ ( s ) E ˜ ( s ) ) = ( T X U ( s ) T X N ( s ) T E U ( s ) T E N ( s ) ) ( U ˜ ( s ) N ˜ ( s ) ) .
T X U ( s ) = L ( s ) s [ 1 + L ( s ) ]
T X N ( s ) = L ( s ) 1 + L ( s )
T E U ( s ) = 1 s [ 1 + L ( s ) ]
T E N ( s ) = L ( s ) 1 + L ( s ) .
X ˜ ( s ) = X ˜ U ( s ) + X ˜ N ( s )
E ˜ ( s ) = E ˜ U ( s ) + E ˜ N ( s )
T ( s ) = C ( s I − A ) − 1 B .
𝔼 [ X ( t ) ] = 0
𝔼 [ X ( t ) 2 ] = C ∑ ∞ C T
𝔼 [ X ( t + τ ) X ( t ) ] = C e Aτ ∑ ∞ C T
𝔼 { X ( t + τ ) − X ( t ) 2 } = 2 C ( I − e Aτ ) ∑ ∞ C T
𝔼 [ Δ X Δ t ( t + τ ) Δ X Δ t ( t ) ] = − C [ 2 e Aτ − e A ( τ − Δ t ) − e A ( τ + Δ t ) ] ∑ ∞ C T
≈ ( Δ t ) 2 CA 2 e Aτ ∑ ∞ C T ( Δ t small )
d q ( t ) = Aq ( t ) d t + B U ( t ) d t
X ( t ) = Cq ( t ) .
d q ( t ) = Aq ( t ) d t + α B d W ( t ) .
∂ ∂ t p q t = ∑ j k A j k ∂ ∂ q j [ q k p q t ] + α 2 ∑ j k ( BB T ) j k ∂ 2 ∂ q j ∂ q k p q t
p t ( q | q 0 ) = 𝓝 [ q ; e A t q 0 ∑ t ]
𝓝 [ q ; m , ∑ ] = ∫ d ( m ) k ( 2 π ) m exp [ − i k T ( q − m ) − 1 2 k T ∑ k ]
d ∑ t d t = A ∑ t + ∑ t A T + α BB T , ∑ t = 0 = 0 .
A ∑ ∞ + ∑ ∞ A T + α BB T = 0 .
∑ t = α ∫ 0 t e A ( t − t ′ ) BB T e A T ( t − t ′ ) d t ′
= α ( ∑ ∞ − e A t ∑ ∞ e A T t ) , ( A < 0 ) .
pτ q 2 q 1 = 𝓝 [ ( q 2 q 1 ) ; 0 , ( ∑ ∞ e A τ ∑ ∞ ∑ ∞ e A T τ ∑ ∞ ) ]
p τ X 2 X 1 = 𝓝 [ ( X 2 X 1 ) ; 0 , ( C ∑ ∞ C T C e Aτ ∑ ∞ C T C e A τ ∑ ∞ C T C ∑ ∞ C T ) ]
X ˜ ( s ) = 1 s T ̅ ( s ) U ˜ ( s )
X ˙ ( t ) = d d t X ( t ) ↔ X ˜ ˙ ( s ) = s X ˜ ( s )
𝔼 [ X ( t ) ] = 0
𝔼 [ X ( t ) 2 ] = 2 C ̅ A ̅ − 2 ( e A ̅ t − A ̅ t − I ) ∑ ̅ ∞ C ̅ T
𝔼 [ X ( t ) X ( t + τ ) ] = C ̅ A ̅ − 2 ( e A ̅ ( t + τ ) + e A ̅ t − e A ̅ τ − 2 A ̅ t − I ) ∑ ̅ ∞ C ̅ T
𝔼 { X ( t + τ ) − X ( t ) 2 } = 2 C ̅ A ̅ − 2 [ e A ̅ τ − A ̅ τ − I ] ∑ ̅ ∞ C ̅ T
𝔼 [ Δ X Δ t ( t + τ ) Δ X Δ t ( t ) ] = − C ̅ A ̅ − 2 [ 2 e A ̅ τ − e A ̅ ( τ − Δ t ) − e A ̅ ( τ + Δ t ) ] ∑ ̅ ∞ C ̅ T
≈ ( Δ t ) 2 C ̅ e A ̅ τ ∑ ̅ ∞ C ̅ T ( Δ t small )
X ˜ ˙ ( s ) = s X ˜ ( s ) = T ̅ ( s ) U ˜ ( s )
X ( t ) = ∫ 0 t X ˙ ( t ′ ) d t ′
𝔼 [ X ( t ) ] = 0 .
𝔼 [ X ( t ) X ( t + τ ) ] = ∫ 0 t d t ′ ∫ 0 t + τ d t ″ 𝔼 [ X ˙ ( t ′ ) X ˙ ( t ″ ) ]
= C ¯ [ ∫ 0 t d t ′ ∫ 0 t + τ d t ″ e A ¯ ∣ t ′ − t ″ ∣ ] ∑ ¯ ∞ C ¯ T .
C ( s ) = γ c s , P ( s ) = 1
C ( s ) = γ c s , P ( s ) = 1 1 + s γ p .
𝔼 [ E U ( t ) E U ( t + τ ) ] = C e A τ ∑ ∞ C T = D γ c e − γ c τ .
𝔼 [ E ( t ) E ( t + τ ) ] = 𝔼 [ E U ( t ) E U ( t + τ ) ] + 𝔼 [ E N ( t ) E N ( t + τ ) ] = D ¯ γ c e − γ c τ
𝔼 [ E U ( t ) E U ( t + τ ) ] = D e − γ p τ 2 [ ( 1 γ c + 1 γ p ) cosh ( ν τ 2 ) + γ p ν ( 1 γ c − 1 γ p ) sinh ( ν τ 2 ) ]
𝔼 [ E N ( t ) E N ( t + τ ) ] = n 2 γ c 2 e − γ c τ 2 [ cosh ( ν τ 2 ) + γ p ν sinh ( ν τ 2 ) ] .
Γ t = Φ [ X p ( t ) − X ( t ) ] = Φ [ E ( t ) ] .
Φ ( x ) = Γ 0 exp ( − 2 w 2 x 2 ) .
Γ t = Φ [ E ( t ) − x L ( t ) ]
G ( t ; τ ) = 𝔼 [ Γ t Γ t + τ ] .
E = ( E ( t ) E ( t + τ ) ) , x L = ( x L ( t ) x L ( t + τ ) ) .
𝔼 [ E ] = 0 , 𝔼 [ EE T ] = ( σ 0 2 σ τ 2 σ τ 2 σ 0 2 )
σ 0 2 = C ∑ ∞ C T , σ τ 2 = C e Aτ ∑ ∞ C T .
G ( t ; τ ) = ∫ ∫ d 2 E p ( E ) Φ [ E ( t ) − x L ( t ) ] Φ [ E ( t + τ ) − x L ( t + τ ) ]
= ( Γ 0 2 w 2 4 det M τ ) exp ( − 1 2 x L T M τ − 1 x L )
M τ = ( σ 0 2 + w 2 4 σ τ 2 σ τ 2 σ 0 2 + w 2 4 ) .
g ( t ; τ ) = 𝔼 [ Γ t Γ t + τ ] 𝔼 [ Γ t ] 𝔼 [ Γ t + τ ] − 1
= σ ̅ 0 2 det M τ exp [ − 1 2 ( x L T M τ − 1 x L − 1 σ ¯ 0 2 x L T x L ) ] − 1 .
G ( t ; τ ) = Γ 0 2 ( G x ( t ; τ ) Γ 0 2 ) ( G y ( t ; τ ) Γ 0 2 ) ( G z ( t ; τ ) Γ 0 2 )
g ( t ; τ ) = [ g x ( t ; τ ) + 1 ] [ g y ( t ; τ ) + 1 ] [ g z ( t ; τ ) + 1 ] − 1 .
σ τ 2 = C e Aτ ∑ ∞ C T
≈ C ( I + A τ ) ∑ ∞ C T + O ( A τ ) 2
= σ 0 2 + τCA ∑ ∞ C T
= σ 0 2 + τ 2 C ( A ∑ ∞ + ∑ ∞ A T ) C T
= σ 0 2 − τ 2 CBB T C T
= σ 0 2 − D τ
G ( t ; τ ) ( Γ 0 2 w 2 4 det M 0 ) = det M 0 det M τ exp ( − 1 2 x L T M τ − 1 x L )
≈ σ ¯ 0 4 − σ 0 4 σ ¯ 0 4 − ( σ 0 2 − Dτ ) 2 exp [ − 1 2 x L T ( σ ¯ 0 2 σ 0 2 − Dτ σ 0 2 − Dτ σ ¯ 0 2 ) − 1 x L ]
→ 1 1 + τ τ D exp [ − ∣ x L ( t ) − x L ( t + τ ) ∣ 2 w 2 ( 1 + τ τ D ) ] ,
x L ( t ) = r cos ω 0 t , y L ( t ) = r sin ω 0 t .
G ( τ ) = ( Γ 0 2 w 2 4 det M τ ) 2 exp [ − 1 2 ( x L T M τ − 1 x L + y L T M τ − 1 y L ) ]
= ( Γ 0 w 2 4 σ ̅ 0 4 − σ τ 4 ) 2 exp [ − r 2 ( σ ̅ 0 2 − σ τ 2 cos ω 0 τ σ ̅ 0 4 − σ 0 4 ) ] ,
g ( τ ) = σ ̅ 0 4 σ ̅ 0 4 − σ 0 4 exp [ − r 2 ( σ ¯ 0 2 − σ τ 2 cos ω 0 τ σ ¯ 0 4 − σ 0 4 ) + r 2 σ ̅ 0 2 ] − 1 .
g ̅ ( τ ) = ω 0 2 π ∫ τ τ + 2 π ω 0 g ( τ ′ ) dτ ′ ≈ σ ̅ 0 4 σ ̅ 0 4 − σ 0 4 exp [ − r 2 ( σ ̅ 0 2 σ ̅ 2 4 − σ 0 4 − 1 σ ̅ 0 2 ) ] I 0 [ r 2 σ τ 2 σ ̅ 0 4 − σ 0 4 ] ,
g ( τ ) = 1 N ̅ 1 + τ τ D ,
σ 0 2 w 2 = 1 4 ( g 0 + g 0 ( 1 + g 0 ) ) .
g 0 + = g ( τ = 0 ) , g 0 − = g ( τ = π ω 0 )
1 ζ = 1 4 log ( 1 + g 0 + 1 + g 0 − ) .
σ 0 2 w 2 = 1 8 ( ζ r 2 w 2 − 1 ) − 1 .
D ̂ ( Δ t ) = Var [ X ( t + Δ t ) − X ( t ) ] 2 Δ t
L = 〈 E ( t ) 2 〉 t = σ 0 .
L = 𝔼 [ E U ( t ) 2 ] + 𝔼 [ E N ( t ) 2 ] = D ( 1 γ c + 1 γ p ) + n 2 γ c 2 ,
D t h = D t h * = D 1 D 2 D 2 − D 1 ( log D 2 D 1 + 2 N − 1 log λ 1 1 − λ 1 ) .
D t h * : λ 1 p N ( D t h * ; D 1 ) = λ 2 p N ( D t h * ; D 2 ) ,

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