Abstract

Localizing a fluorescent particle by scanning a focused laser beam in its vicinity and analyzing the detected photon stream provides real-time information for a modern class of feedback control systems for particle tracking and trapping. We show for the full range of standard merit functions based on the Fisher information matrix (1) that the optimal path coincides with the positions of maximum slope of the square root of the beam intensity rather than with the intensity itself, (2) that this condition matches that derived from the theory describing the optimal design of experiments and (3) that in one dimension it is equivalent to maximizing the signal to noise ratio. The optimal path for a Gaussian beam scanned in two or three dimensions is presented along with the Cramér-Rao bound, which gives the ultimate localization accuracy that can be achieved by analyzing the detected photon stream. In two dimensions the optimum path is independent of the chosen merit function but this is not the case in three dimensions. Also, we show that whereas the optimum path for a Gaussian beam in two dimensions can be chosen to be continuous, it cannot be continuous in three dimensions.

© 2012 OSA

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    [CrossRef] [PubMed]
  35. M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
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2012 (1)

Z. Shen and S. Andersson, “Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[CrossRef]

2011 (2)

Z. Shen and S. Andersson, “Tracking nanometer-scale fluorescent particles in two dimensions with a confocal microscope,” IEEE Trans. Contr. Sys. Tech. 19, 1–10 (2011).

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

2010 (2)

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

Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the ABEL trap,” Appl. Phys. B 99, 23–30 (2010).
[CrossRef] [PubMed]

2009 (3)

K. McHale and H. Mabuchi, “Precise characterization of the conformation fluctuations of freely diffusing DNA: beyond Rouse and Zimm,” J. Am. Chem. Soc. 131, 17901–17907 (2009).
[CrossRef] [PubMed]

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef] [PubMed]

2008 (4)

H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. 457, 285–291 (2008).
[CrossRef]

S. Pavani and R. Piestun, “Three dimensional tracking of fluorescent microparticles using a photon-limited double-helix response system,” Opt. Express 16, 22048–22057 (2008).
[CrossRef] [PubMed]

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[CrossRef]

2007 (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]

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[CrossRef] [PubMed]

G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. 91,  224,106 (2007).
[CrossRef]

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104, 12596–12602 (2007).
[CrossRef] [PubMed]

2006 (4)

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

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,  223,901 (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]

2005 (4)

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]

A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express 13, 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, 093109 (2005).
[CrossRef]

T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

2004 (1)

R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[CrossRef] [PubMed]

2003 (3)

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (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. Technol. 31, 997–1000 (2003).
[CrossRef]

2002 (1)

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef] [PubMed]

2001 (1)

M. Cheezum, W. Walker, and W. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[CrossRef] [PubMed]

1996 (1)

J. Crocker and D. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interf. Sci. 179, 298–310 (1996).
[CrossRef]

1994 (1)

H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291–1300 (1994).
[CrossRef] [PubMed]

1986 (1)

Andersson, S.

Z. Shen and S. Andersson, “Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[CrossRef]

Z. Shen and S. Andersson, “Tracking nanometer-scale fluorescent particles in two dimensions with a confocal microscope,” IEEE Trans. Contr. Sys. Tech. 19, 1–10 (2011).

Armani, M.

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

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” 18th IEEE International Conference on MEMS855–858 (2005).

Bates, M.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef] [PubMed]

Berglund, A. J.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

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, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[CrossRef] [PubMed]

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]

Cang, H.

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[CrossRef]

H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. 457, 285–291 (2008).
[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,  223,901 (2006).
[CrossRef]

Carmichael, P.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

Chaudhary, S.

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

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” 18th IEEE International Conference on MEMS855–858 (2005).

Cheezum, M.

M. Cheezum, W. Walker, and W. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[CrossRef] [PubMed]

Cohen, A. E.

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

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

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

Crocker, J.

J. Crocker and D. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interf. Sci. 179, 298–310 (1996).
[CrossRef]

Dahan, M.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[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]

Fields, A. P.

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

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

Forkey, J. N.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Ges, I. A.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Goodman, Y. E.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Goodwin, P.

G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. 91,  224,106 (2007).
[CrossRef]

Gratton, E.

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. Technol. 31, 997–1000 (2003).
[CrossRef]

Grier, D.

J. Crocker and D. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interf. Sci. 179, 298–310 (1996).
[CrossRef]

Guilford, W.

M. Cheezum, W. Walker, and W. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[CrossRef] [PubMed]

Ha, T.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Huang, B.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef] [PubMed]

Janetopoulos, C.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Kao, H.

H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291–1300 (1994).
[CrossRef] [PubMed]

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. Technol. 31, 997–1000 (2003).
[CrossRef]

Larson, D.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef] [PubMed]

Lessard, G.

G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. 91,  224,106 (2007).
[CrossRef]

Levi, S.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

Levi, V.

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. Technol. 31, 997–1000 (2003).
[CrossRef]

Liddle, J. A.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

Luccardini, C.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

Mabuchi, H.

K. McHale and H. Mabuchi, “Precise characterization of the conformation fluctuations of freely diffusing DNA: beyond Rouse and Zimm,” J. Am. Chem. Soc. 131, 17901–17907 (2009).
[CrossRef] [PubMed]

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, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[CrossRef] [PubMed]

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]

Markov, D. A.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

McClelland, J. J.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

McHale, K.

K. McHale and H. Mabuchi, “Precise characterization of the conformation fluctuations of freely diffusing DNA: beyond Rouse and Zimm,” J. Am. Chem. Soc. 131, 17901–17907 (2009).
[CrossRef] [PubMed]

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, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[CrossRef] [PubMed]

McKinney, S. A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

McMahon, M. D.

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

Moerner, W.

Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the ABEL trap,” Appl. Phys. B 99, 23–30 (2010).
[CrossRef] [PubMed]

Moerner, W. E.

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104, 12596–12602 (2007).
[CrossRef] [PubMed]

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.

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[CrossRef]

Ober, R.

R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[CrossRef] [PubMed]

Pavani, S.

Piestun, R.

Probst, R.

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

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” 18th IEEE International Conference on MEMS855–858 (2005).

Pukelsheim, F.

F. Pukelsheim, Optimal design of experiments (Society for Industrial and Applied Mathematics, 2006).
[CrossRef]

Ram, S.

R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[CrossRef] [PubMed]

Reiserer, R. S.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Riveau, B.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

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,  223,901 (2006).
[CrossRef]

Rostaing, P.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

Ruan, Q.

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. Technol. 31, 997–1000 (2003).
[CrossRef]

Savin, T.

T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

Seale, K. T.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Selvin, P. R.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Shan Xu, C.

H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. 457, 285–291 (2008).
[CrossRef]

Shapiro, B.

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

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” 18th IEEE International Conference on MEMS855–858 (2005).

Shen, Z.

Z. Shen and S. Andersson, “Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[CrossRef]

Z. Shen and S. Andersson, “Tracking nanometer-scale fluorescent particles in two dimensions with a confocal microscope,” IEEE Trans. Contr. Sys. Tech. 19, 1–10 (2011).

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Thompson, R.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef] [PubMed]

Triller, A.

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

Verkman, A.

H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291–1300 (1994).
[CrossRef] [PubMed]

Walker, W.

M. Cheezum, W. Walker, and W. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[CrossRef] [PubMed]

Wang, Q.

Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the ABEL trap,” Appl. Phys. B 99, 23–30 (2010).
[CrossRef] [PubMed]

Ward, E.

R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[CrossRef] [PubMed]

Webb, W.

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef] [PubMed]

Werner, J.

G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. 91,  224,106 (2007).
[CrossRef]

Wikswo, J. P.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Winnick, K. A.

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,  223,901 (2006).
[CrossRef]

Wright, C.

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

Xu, C.

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[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,  223,901 (2006).
[CrossRef]

Yang, H.

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[CrossRef]

H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. 457, 285–291 (2008).
[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,  223,901 (2006).
[CrossRef]

Yildiz, A.

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Zacks, S.

S. Zacks, The Theory of Statistical Inference (John Wiley & Sons, 1971).

Zhuang, X.

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef] [PubMed]

ACS Nano (1)

M. D. McMahon, A. J. Berglund, P. Carmichael, J. J. McClelland, and J. A. Liddle, “3D Particle trajectories observed by orthogonal tracking microscopy,” ACS Nano 3, 609–614 (2009).
[CrossRef] [PubMed]

Annu. Rev. Biochem. (1)

B. Huang, M. Bates, and X. Zhuang, “Super resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef] [PubMed]

Appl. Phys. B (2)

Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the ABEL trap,” Appl. Phys. B 99, 23–30 (2010).
[CrossRef] [PubMed]

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. (3)

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,  223,901 (2006).
[CrossRef]

G. Lessard, P. Goodwin, and J. Werner, “Three-dimensional tracking of individual quantum dots,” Appl. Phys. Lett. 91,  224,106 (2007).
[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. Technol. (1)

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

Biophys. J. (6)

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]

T. Savin and P. S. Doyle, “Static and dynamic errors in particle tracking microrheology,” Biophys. J. 88, 623–638 (2005).
[CrossRef]

R. Thompson, D. Larson, and W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775–2783 (2002).
[CrossRef] [PubMed]

R. Ober, S. Ram, and E. Ward, “Localization accuracy in single-molecule microscopy,” Biophys. J. 86, 1185–1200 (2004).
[CrossRef] [PubMed]

M. Cheezum, W. Walker, and W. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378–2388 (2001).
[CrossRef] [PubMed]

H. Kao and A. Verkman, “Tracking of single fluorescent particles in three dimensions: use of cylindrical optics to encode particle position,” Biophys. J. 67, 1291–1300 (1994).
[CrossRef] [PubMed]

Chem. Phys. Lett. (1)

H. Cang, C. Shan Xu, and H. Yang, “Progress in single-molecule tracking spectroscopy,” Chem. Phys. Lett. 457, 285–291 (2008).
[CrossRef]

IEEE J. Microelectromech. S. (1)

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

IEEE Trans. Contr. Sys. Tech. (1)

Z. Shen and S. Andersson, “Tracking nanometer-scale fluorescent particles in two dimensions with a confocal microscope,” IEEE Trans. Contr. Sys. Tech. 19, 1–10 (2011).

J. Am. Chem. Soc. (1)

K. McHale and H. Mabuchi, “Precise characterization of the conformation fluctuations of freely diffusing DNA: beyond Rouse and Zimm,” J. Am. Chem. Soc. 131, 17901–17907 (2009).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

H. Cang, D. Montiel, C. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. 129,  044,503 (2008).
[CrossRef]

J. Colloid Interf. Sci. (1)

J. Crocker and D. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interf. Sci. 179, 298–310 (1996).
[CrossRef]

J. Microsc. (1)

K. T. Seale, R. S. Reiserer, D. A. Markov, I. A. Ges, C. Wright, C. Janetopoulos, and J. P. Wikswo, “Mirrored pyramidal wells for simultaneous multiple vantage point microscopy,” J. Microsc. 232, 1–6 (2008).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

Mechatronics (1)

Z. Shen and S. Andersson, “Optimal measurement constellation of the fluoroBancroft localization algorithm for position estimation in tracking confocal microscopy,” Mechatronics 22, 320–326 (2012).
[CrossRef]

Method. Enzymol. (1)

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

Nano Lett. (1)

K. McHale, A. J. Berglund, and H. Mabuchi, “Quantum dot photon statistics measured by three-dimensional particle tracking,” Nano Lett. 7, 3535–3539 (2007).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Proc. Natl. Acad. Sci. USA (3)

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

W. E. Moerner, “New directions in single-molecule imaging and analysis,” Proc. Natl. Acad. Sci. USA 104, 12596–12602 (2007).
[CrossRef] [PubMed]

Science (2)

M. Dahan, S. Levi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, “Diffusion dynamics of single glycine receptors revealed by single-quantum dot tracking,” Science 302, 442–445 (2003).
[CrossRef] [PubMed]

A. Yildiz, J. N. Forkey, S. A. McKinney, T. Ha, Y. E. Goodman, and P. R. Selvin, “Myosin V walks hand-overhand: Single fluorophore imaging with 1.5-nm localization,” Science 300, 2061–2065 (2003).
[CrossRef] [PubMed]

Other (4)

M. Armani, S. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control and micro-fluidics to steer individual particles,” 18th IEEE International Conference on MEMS855–858 (2005).

S. Zacks, The Theory of Statistical Inference (John Wiley & Sons, 1971).

F. Pukelsheim, Optimal design of experiments (Society for Industrial and Applied Mathematics, 2006).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of a laser-scanning particle localization experiment. A Gaussian beam is scanned along a time-dependent (continuous or discrete) path rL(t) and a modulated stream of photons is detected. The optimal design problem is to determine which scan path encodes maximal information about a particles location in the detected photon stream.

Fig. 2
Fig. 2

Plot of f (r) as defined in Eq. (14). The scan path given by Eqs. (13a)(13c) are proved to be optimal by observing that f (r) is less than1 for all other values of r/w0 or z/zR. Note that because the two optimal points, i.e., the peaks in the graph in Fig. 2, are separated by a valley (f < 1) there is no smoothly varying continuous path in 3D that is optimal for a Gaussian beam. This is in contrast to the 2D case with a Gaussian beam where all the points on the circle r = w 0 / 2 have f = 1 and hence a continuous path can be used if desired. [26].

Equations (46)

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

σ x 2 + σ y 2 w 0 2 2 N p h
σ x 2 + σ y 2 + σ z 2 w 0 2 N p h ( 0.5 + 4.92 w 0 λ + 11.10 w 0 2 λ 2 ) .
σ x 2 + σ y 2 + σ z 2 λ 2 N p h 2 NA ( 0.08 + 0.31 NA + 0.28 NA 2 ) .
p ( t 1 , , t K | r ) = 1 K ! k = 1 K Γ ( r r L ( t k ) ) exp [ 0 τ d t Γ ( r r L ( t ) ) ]
[ F ] j k F j k = K = 0 0 τ d t 1 d t K p ( t 1 , , t K | r ) ( j ln [ p ( t 1 , t K | x ) ] ) ( k ln [ p ( t 1 , , t K | r ) ] )
ϕ p [ F ] = ( 1 d Tr [ F p ] ) 1 / p = ( 1 d Tr [ F F F p times ] ) 1 / p
F j k = 0 τ d t 1 Γ ( r r L ( t ) ) ( j Γ ( r r L ( t ) ) ) ( k Γ ( r r L ( t ) ) ) = 4 0 T d t ( j Γ ( r r L ( t ) ) ) ( k Γ ( r r L ( t ) ) )
g = 2 Γ ( r r L ( t ) ) τ = 2 ξ τ a ( r r L ( t ) )
F = 1 τ 0 τ d t g g T .
F = n = 1 N c n g n g n T , c n = Δ t n / τ = 4 ξ n = 1 N Δ t n a ( r r L ( n ) ) ) a ( r r L ( n ) ) ) T
g T F * p 1 g Tr [ F * p ]
( a ( r ) ) T F * p 1 ( a ( r ) ) 1 4 ξ τ Tr [ F * p ]
0 = ( 1 d Tr [ F p ] ) 1 / p 1 [ F p 1 ] k j ( k a ( r r L ( t ) ) ) ( i j a ( r r L ( t ) ) )
0 = ( x a ( x x L ( t ) ) ) ( x 2 a ( x x L ( t ) ) )
ϕ p [ F ] = ( 1 D s = 1 D Δ t s p | s a ( r r L ( s ) ) | 2 p ) 1 / p
ϕ p [ F ] ( Δ t s ) = 0 for s = 1 to D 1
a ( x , y , z ) = a 0 1 + z 2 / z R 2 exp [ 1 w 0 2 ( x 2 + y 2 1 + z 2 / z R 2 ) ]
Δ t 1 = Δ t 2 = w 0 τ 2 w 0 + 9 z R / 6 e Δ t 3 = ( 9 z R / 6 e ) τ 2 w 0 + 9 z R / 6 e
g = 4 r w 0 2 Γ ( r ) τ .
F * = 4 w 0 2 Γ 0 τ e ( 1 0 0 1 ) = 4 N p h w 0 2 ( 1 0 0 1 )
( x 2 + y 2 ) e 2 w 0 2 ( x 2 + y 2 ) w 0 2 2 e ,
σ x 2 + σ y 2 w 0 2 2 N p h
Γ ( r ) = Γ ( x , y , z ) = Γ 0 1 + z 2 / z R 2 exp [ 2 w 0 2 ( x 2 + y 2 1 + z 2 / z R 2 ) ] .
r L ( 1 ) = ( ± w 0 2 , 0 , 0 ) Δ t 1 = w 0 τ 2 w 0 + 9 z R / 6 e
r L ( 2 ) = ( 0 , ± w 0 2 , 0 ) Δ t 1 = w 0 τ 2 w 0 + 9 z R / 6 e
r L ( 3 ) = ( 0 , 0 , ± z R 2 ) Δ t 3 = ( 9 z R / 6 e ) τ 2 w 0 + 9 z R / 6 e .
F * 11 = F * 22 = 8 Γ 0 τ / e 2 w 0 2 + 9 z R / 6 e Γ 0 τ 0.68 w 0 2 + 0.76 w 0 z R F * 33 = 16 Γ 0 τ / 6 e 6 w 0 z R + 27 z R 2 / 6 e Γ 0 τ 1.51 w 0 z R + 1.69 z R 2
f ( r ) = g T F * 2 g Tr [ F * 1 ] = 4 ξ τ ( a ( r ) ) T F * 2 ( a ( r ) ) Tr [ F * 1 ] .
f ( r ) = 8 e r ¯ 2 ( 1 + z ¯ 2 ) 2 + 27 z ¯ 2 ( 1 2 r ¯ 2 + z ¯ 2 ) 2 4 ( 1 + z ¯ 2 ) 5 exp [ 2 r ¯ 2 1 + z ¯ 2 ] .
σ x 2 + σ y 2 + σ z 2 Tr [ F * 1 ] .
σ x 2 + σ y 2 + σ z 2 w 0 2 2 N p h [ 1 + 3 2 ( e + 3 2 e ) z R w 0 + 9 4 z R 2 w 0 2 ] .
ϕ p [ F ] ϕ p [ F * ]
Tr [ F p ] Tr [ F * p ]
Tr [ F * p ] = n = 1 N g * n i [ F * p 1 ] i j g * n j = n = 1 N g * n T F * p 1 g * n
v T F * v = n = 1 N ( v i g * n i ) 2 > 0
g * n = i = 1 D g ¯ * n i e i
g ¯ * n i = g ¯ * i δ n , i
n = 1 N g ¯ * n i g ¯ * n j = g ¯ * i g ¯ * j δ i j = f * i δ i j
n = 1 N g * n T f * p 1 g * n = g * T f * p 1 g *
g T F * p 1 g Tr [ F * p ]
Γ [ r r L ( t ) ] Γ [ r L ( t ) ] + r T Γ | r L ( t ) + 1 2 r T H ( Γ ) | r L ( t ) r + O ( r 3 )
A r MLE + b = 0
A = 0 τ H ( Γ ) | r L ( t ) d t + k = 1 K H ( log Γ ) | r L ( t k )
b = 0 τ ( Γ ) | r L ( t ) d t + k = 1 K ( log Γ ) | r L ( t k ) .
A = K ( 1 0 0 1 ) , b = w 2 k = 1 K r L ( t k )
r MLE = { 0 , K = 0 w 2 ( 1 K k = 1 K cos 2 π t k τ , 1 K k = 1 K sin 2 π t k τ ) T , K > 0

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