Optimal tracking of a Brownian particle

doi:10.1364/OE.20.022585

Optimal tracking of a Brownian particle

Alexander P. Fields and Adam E. Cohen »View Author Affiliations

Optics Express, Vol. 20, Issue 20, pp. 22585-22601 (2012)
http://dx.doi.org/10.1364/OE.20.022585

View Full Text Article

Enhanced HTML Acrobat PDF (3233 KB)

Journal Search
Article Lookup

Article Tools

Citations

Alert me when this article is cited
Export Citation/Save

Abstract
Article Info
References (30)
Cited By
Figures (5)
Metrics
Related Content

Abstract

Optical tracking of a fluorescent particle in solution faces fundamental constraints due to Brownian motion, diffraction, and photon shot noise. Background photons and imperfect tracking apparatus further degrade tracking precision. Here we use a model of particle motion to combine information from multiple time-points to improve the localization precision. We derive successive approximations that enable real-time particle tracking with well controlled tradeoffs between precision and computational cost. We present the theory in the context of feedback electrokinetic trapping, though the results apply to optical tracking of any particle subject to diffusion and drift. We use numerical simulations and experimental data to validate the algorithms’ performance.

OCIS Codes
(180.2520) Microscopy : Fluorescence microscopy
(110.3055) Imaging systems : Information theoretical analysis
(110.4155) Imaging systems : Multiframe image processing

ToC Category:
Microscopy

History
Original Manuscript: August 2, 2012
Revised Manuscript: September 11, 2012
Manuscript Accepted: September 13, 2012
Published: September 18, 2012

Virtual Issues
Vol. 7, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Alexander P. Fields and Adam E. Cohen, "Optimal tracking of a Brownian particle," Opt. Express 20, 22585-22601 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-20-22585

Sort: Author | Year | Journal | Reset

References

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct.26(1), 373–399 (1997). [CrossRef] [PubMed]
A. D. Douglass and R. D. Vale, “Single-molecule microscopy reveals plasma membrane microdomains created by protein-protein networks that exclude or trap signaling molecules in T cells,” Cell121(6), 937–950 (2005). [CrossRef] [PubMed]
I. Chung, R. Akita, R. Vandlen, D. Toomre, J. Schlessinger, and I. Mellman, “Spatial control of EGF receptor activation by reversible dimerization on living cells,” Nature464(7289), 783–787 (2010). [CrossRef] [PubMed]
A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science300(5628), 2061–2065 (2003). [CrossRef] [PubMed]
M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A.107(42), 17864–17871 (2010). [CrossRef] [PubMed]
N. P. Wells, G. A. Lessard, P. M. Goodwin, M. E. Phipps, P. J. Cutler, D. S. Lidke, B. S. Wilson, and J. H. Werner, “Time-resolved three-dimensional molecular tracking in live cells,” Nano Lett.10(11), 4732–4737 (2010). [CrossRef] [PubMed]
A. P. Fields and A. E. Cohen, “Anti-Brownian traps for studies on single molecules,” Methods Enzymol.475, 149–174 (2010). [CrossRef] [PubMed]
A. E. Cohen and W. E. Moerner, “Principal-components analysis of shape fluctuations of single DNA molecules,” Proc. Natl. Acad. Sci. U.S.A.104(31), 12622–12627 (2007). [CrossRef] [PubMed]
Y. Jiang, Q. Wang, A. E. Cohen, N. Douglas, J. Frydman, and W. E. Moerner, “Hardware-based anti-Brownian electrokinetic trap (ABEL trap) for single molecules: control loop simulations and application to ATP binding stoichiometry in multi-subunit enzymes,” Proc. Soc. Photo Opt. Instrum. Eng.7038, 1–12 (2008). [PubMed]
R. H. Goldsmith and W. E. Moerner, “Watching conformational- and photodynamics of single fluorescent proteins in solution,” Nat. Chem.2(3), 179–186 (2010). [CrossRef] [PubMed]
H. Cang, D. Montiel, C. S. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys.129(4), 044503 (2008). [CrossRef] [PubMed]
J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B71(5), 773–777 (2000). [CrossRef]
A. P. Fields and A. E. Cohen, “Electrokinetic trapping at the one nanometer limit,” Proc. Natl. Acad. Sci. U.S.A.108(22), 8937–8942 (2011). [CrossRef] [PubMed]
A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic Press, 1970).
K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in single-molecule fluorescence microscopy,” Biophys. J.86(6), 3409–3422 (2004). [CrossRef] [PubMed]
A. J. Berglund, K. McHale, and H. Mabuchi, “Fluctuations in closed-loop fluorescent particle tracking,” Opt. Express15(12), 7752–7773 (2007). [CrossRef] [PubMed]
A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B83(1), 127–133 (2006). [CrossRef]
A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express13(20), 8069–8082 (2005). [CrossRef] [PubMed]
A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett.86(9), 093109 (2005). [CrossRef]
Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano5(7), 5792–5799 (2011). [CrossRef] [PubMed]
K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010). [CrossRef] [PubMed]
B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt.46(10), 1819–1829 (2007). [CrossRef] [PubMed]
M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process.50(2), 174–188 (2002). [CrossRef]
T. P. Minka, “A family of algorithms for approximate Bayesian inference,” Ph.D. thesis, Massachusetts Institute of Technology (2001). http://research.microsoft.com/en-us/um/people/minka/papers/ep/minka-thesis.pdf .
P. S. Maybeck, Stochastic models, estimation and control (Academic press, 1979).
H. W. Sorenson and D. L. Alspach, “Recursive Bayesian estimation using Gaussian sums,” Automatica7(4), 465–479 (1971). [CrossRef]
R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. Trans. ASME82(1), 35–45 (1960). [CrossRef]
G. Welch and G. Bishop, “An introduction to the Kalman filter,” University of North Carolina at Chapel Hill technical report TR 95–041 (2006). http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf .
M. Brinkmeier, K. Dörre, J. Stephan, and M. Eigen, “Two-beam cross-correlation: a method to characterize transport phenomena in micrometer-sized structures,” Anal. Chem.71(3), 609–616 (1999). [CrossRef] [PubMed]
P. Kapusta, “Absolute diffusion coefficients: compilation of reference data for FCS calibration,” http://www.picoquant.com/technotes/appnote_diffusion_coefficients.pdf .

Akita, R.
Alspach, D. L.
Arulampalam, M. S.
Badieirostami, M.
Berglund, A. J.
Brinkmeier, M.
Brown, P. O.
Cang, H.
Casolari, J. M.
Chung, I.
Churchman, L. S.
Clapp, T.
Cohen, A. E.
Cutler, P. J.
Dörre, K.
Douglas, N.
Douglass, A. D.
Eigen, M.
Enderlein, J.
Fields, A. P.
Flyvbjerg, H.
Forkey, J. N.
Frydman, J.
Goldman, Y. E.
Goldsmith, R. H.
Goodwin, P. M.
Gordon, N.
Ha, T.
Jacobson, K.
Jiang, Y.
Kalman, R. E.
Lessard, G. A.
Lidke, D. S.
Mabuchi, H.
Maskell, S.
McHale, K.
McKinney, S. A.
Mellman, I.
Moerner, W. E.
Montiel, D.
Mortensen, K. I.
Olivo-Marin, J. C.
Phipps, M. E.
Saxton, M. J.
Schlessinger, J.
Selvin, P. R.
Sorenson, H. W.
Spudich, J. A.
Stephan, J.
Thompson, M. A.
Toomre, D.
Vale, R. D.
Vandlen, R.
Wang, Q.
Wells, N. P.
Werner, J. H.
Wilson, B. S.
Xu, C. S.
Yang, H.
Yildiz, A.
Zerubia, J.
Zhang, B.

ACS Nano

Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano5(7), 5792–5799 (2011). [CrossRef] [PubMed]

Anal. Chem.

M. Brinkmeier, K. Dörre, J. Stephan, and M. Eigen, “Two-beam cross-correlation: a method to characterize transport phenomena in micrometer-sized structures,” Anal. Chem.71(3), 609–616 (1999). [CrossRef] [PubMed]

Annu. Rev. Biophys. Biomol. Struct.

M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct.26(1), 373–399 (1997). [CrossRef] [PubMed]

Appl. Opt.

B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt.46(10), 1819–1829 (2007). [CrossRef] [PubMed]

Appl. Phys. B

J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B71(5), 773–777 (2000). [CrossRef]
A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B83(1), 127–133 (2006). [CrossRef]

Appl. Phys. Lett.

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett.86(9), 093109 (2005). [CrossRef]

Automatica

H. W. Sorenson and D. L. Alspach, “Recursive Bayesian estimation using Gaussian sums,” Automatica7(4), 465–479 (1971). [CrossRef]

Biophys. J.

K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in single-molecule fluorescence microscopy,” Biophys. J.86(6), 3409–3422 (2004). [CrossRef] [PubMed]

Cell

A. D. Douglass and R. D. Vale, “Single-molecule microscopy reveals plasma membrane microdomains created by protein-protein networks that exclude or trap signaling molecules in T cells,” Cell121(6), 937–950 (2005). [CrossRef] [PubMed]

IEEE Trans. Signal Process.

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process.50(2), 174–188 (2002). [CrossRef]

J. Basic Eng. Trans. ASME

R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. Trans. ASME82(1), 35–45 (1960). [CrossRef]

J. Chem. Phys.

H. Cang, D. Montiel, C. S. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys.129(4), 044503 (2008). [CrossRef] [PubMed]

Methods Enzymol.

A. P. Fields and A. E. Cohen, “Anti-Brownian traps for studies on single molecules,” Methods Enzymol.475, 149–174 (2010). [CrossRef] [PubMed]

Nano Lett.

N. P. Wells, G. A. Lessard, P. M. Goodwin, M. E. Phipps, P. J. Cutler, D. S. Lidke, B. S. Wilson, and J. H. Werner, “Time-resolved three-dimensional molecular tracking in live cells,” Nano Lett.10(11), 4732–4737 (2010). [CrossRef] [PubMed]

Nat. Chem.

R. H. Goldsmith and W. E. Moerner, “Watching conformational- and photodynamics of single fluorescent proteins in solution,” Nat. Chem.2(3), 179–186 (2010). [CrossRef] [PubMed]

Nat. Methods

K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010). [CrossRef] [PubMed]

Nature

I. Chung, R. Akita, R. Vandlen, D. Toomre, J. Schlessinger, and I. Mellman, “Spatial control of EGF receptor activation by reversible dimerization on living cells,” Nature464(7289), 783–787 (2010). [CrossRef] [PubMed]

Opt. Express

Proc. Natl. Acad. Sci. U.S.A.

A. P. Fields and A. E. Cohen, “Electrokinetic trapping at the one nanometer limit,” Proc. Natl. Acad. Sci. U.S.A.108(22), 8937–8942 (2011). [CrossRef] [PubMed]
M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A.107(42), 17864–17871 (2010). [CrossRef] [PubMed]
A. E. Cohen and W. E. Moerner, “Principal-components analysis of shape fluctuations of single DNA molecules,” Proc. Natl. Acad. Sci. U.S.A.104(31), 12622–12627 (2007). [CrossRef] [PubMed]

Proc. Soc. Photo Opt. Instrum. Eng.

Y. Jiang, Q. Wang, A. E. Cohen, N. Douglas, J. Frydman, and W. E. Moerner, “Hardware-based anti-Brownian electrokinetic trap (ABEL trap) for single molecules: control loop simulations and application to ATP binding stoichiometry in multi-subunit enzymes,” Proc. Soc. Photo Opt. Instrum. Eng.7038, 1–12 (2008). [PubMed]

Science

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science300(5628), 2061–2065 (2003). [CrossRef] [PubMed]

Other

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic Press, 1970).
G. Welch and G. Bishop, “An introduction to the Kalman filter,” University of North Carolina at Chapel Hill technical report TR 95–041 (2006). http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf .
T. P. Minka, “A family of algorithms for approximate Bayesian inference,” Ph.D. thesis, Massachusetts Institute of Technology (2001). http://research.microsoft.com/en-us/um/people/minka/papers/ep/minka-thesis.pdf .
P. S. Maybeck, Stochastic models, estimation and control (Academic press, 1979).
P. Kapusta, “Absolute diffusion coefficients: compilation of reference data for FCS calibration,” http://www.picoquant.com/technotes/appnote_diffusion_coefficients.pdf .

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Click here to see a list of articles that cite this paper

Figures


Fig. 1	Fig. 2	Fig. 3


Fig. 4	Fig. 5

« Previous Article | Next Article »

OSA is a member of CrossRef.

[1] M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct.26(1), 373–399 (1997). [CrossRef] [PubMed]

[2] A. D. Douglass and R. D. Vale, “Single-molecule microscopy reveals plasma membrane microdomains created by protein-protein networks that exclude or trap signaling molecules in T cells,” Cell121(6), 937–950 (2005). [CrossRef] [PubMed]

[3] I. Chung, R. Akita, R. Vandlen, D. Toomre, J. Schlessinger, and I. Mellman, “Spatial control of EGF receptor activation by reversible dimerization on living cells,” Nature464(7289), 783–787 (2010). [CrossRef] [PubMed]

[4] A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goldman, and P. R. Selvin, “Myosin V walks hand-over-hand: single fluorophore imaging with 1.5-nm localization,” Science300(5628), 2061–2065 (2003). [CrossRef] [PubMed]

[5] M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in Saccharomyces cerevisiae using a double-helix point spread function,” Proc. Natl. Acad. Sci. U.S.A.107(42), 17864–17871 (2010). [CrossRef] [PubMed]

[6] N. P. Wells, G. A. Lessard, P. M. Goodwin, M. E. Phipps, P. J. Cutler, D. S. Lidke, B. S. Wilson, and J. H. Werner, “Time-resolved three-dimensional molecular tracking in live cells,” Nano Lett.10(11), 4732–4737 (2010). [CrossRef] [PubMed]

[7] A. P. Fields and A. E. Cohen, “Anti-Brownian traps for studies on single molecules,” Methods Enzymol.475, 149–174 (2010). [CrossRef] [PubMed]

[8] A. E. Cohen and W. E. Moerner, “Principal-components analysis of shape fluctuations of single DNA molecules,” Proc. Natl. Acad. Sci. U.S.A.104(31), 12622–12627 (2007). [CrossRef] [PubMed]

[9] Y. Jiang, Q. Wang, A. E. Cohen, N. Douglas, J. Frydman, and W. E. Moerner, “Hardware-based anti-Brownian electrokinetic trap (ABEL trap) for single molecules: control loop simulations and application to ATP binding stoichiometry in multi-subunit enzymes,” Proc. Soc. Photo Opt. Instrum. Eng.7038, 1–12 (2008). [PubMed]

[10] R. H. Goldsmith and W. E. Moerner, “Watching conformational- and photodynamics of single fluorescent proteins in solution,” Nat. Chem.2(3), 179–186 (2010). [CrossRef] [PubMed]

[11] H. Cang, D. Montiel, C. S. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys.129(4), 044503 (2008). [CrossRef] [PubMed]

[12] J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B71(5), 773–777 (2000). [CrossRef]

[13] A. P. Fields and A. E. Cohen, “Electrokinetic trapping at the one nanometer limit,” Proc. Natl. Acad. Sci. U.S.A.108(22), 8937–8942 (2011). [CrossRef] [PubMed]

[14] A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic Press, 1970).

[15] K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in single-molecule fluorescence microscopy,” Biophys. J.86(6), 3409–3422 (2004). [CrossRef] [PubMed]

[16] A. J. Berglund, K. McHale, and H. Mabuchi, “Fluctuations in closed-loop fluorescent particle tracking,” Opt. Express15(12), 7752–7773 (2007). [CrossRef] [PubMed]

[17] A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B83(1), 127–133 (2006). [CrossRef]

[18] A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express13(20), 8069–8082 (2005). [CrossRef] [PubMed]

[19] A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett.86(9), 093109 (2005). [CrossRef]

[20] Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano5(7), 5792–5799 (2011). [CrossRef] [PubMed]

[21] K. I. Mortensen, L. S. Churchman, J. A. Spudich, and H. Flyvbjerg, “Optimized localization analysis for single-molecule tracking and super-resolution microscopy,” Nat. Methods7(5), 377–381 (2010). [CrossRef] [PubMed]

[22] B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt.46(10), 1819–1829 (2007). [CrossRef] [PubMed]

[23] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process.50(2), 174–188 (2002). [CrossRef]

[24] T. P. Minka, “A family of algorithms for approximate Bayesian inference,” Ph.D. thesis, Massachusetts Institute of Technology (2001). http://research.microsoft.com/en-us/um/people/minka/papers/ep/minka-thesis.pdf .

[25] P. S. Maybeck, Stochastic models, estimation and control (Academic press, 1979).

[26] H. W. Sorenson and D. L. Alspach, “Recursive Bayesian estimation using Gaussian sums,” Automatica7(4), 465–479 (1971). [CrossRef]

[27] R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. Trans. ASME82(1), 35–45 (1960). [CrossRef]

[28] G. Welch and G. Bishop, “An introduction to the Kalman filter,” University of North Carolina at Chapel Hill technical report TR 95–041 (2006). http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf .

[29] M. Brinkmeier, K. Dörre, J. Stephan, and M. Eigen, “Two-beam cross-correlation: a method to characterize transport phenomena in micrometer-sized structures,” Anal. Chem.71(3), 609–616 (1999). [CrossRef] [PubMed]

[30] P. Kapusta, “Absolute diffusion coefficients: compilation of reference data for FCS calibration,” http://www.picoquant.com/technotes/appnote_diffusion_coefficients.pdf .

OpticsInfoBase

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

Optimal tracking of a Brownian particle