## Optimal tracking of a Brownian particle |

Optics Express, Vol. 20, Issue 20, pp. 22585-22601 (2012)

http://dx.doi.org/10.1364/OE.20.022585

Acrobat PDF (3233 KB)

### 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.

© 2012 OSA

## 1. Introduction

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,” Cell **121**(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,” Nature **464**(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,” Science **300**(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]

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. B **71**(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]

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]

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]

20. Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano **5**(7), 5792–5799 (2011). [CrossRef] [PubMed]

## 2. General treatment

*γ*at which photons are detected is well approximated as a Gaussian function of the displacement of the fluorophore from the center of the laser spot, plus a constant background rate [21

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. Methods **7**(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]

*s*is the maximum “signal” photon rate (in photons per second) from the fluorophore,

*b*is the “background” photon rate (also in photons per second),

**x**is the particle position (written as a column vector),

**c**is the laser spot center,

**W**is the spatial covariance of the spot,

*denotes transposition, and*

^{T}^{−1}denotes matrix inversion. The probability of detecting

*n*photons during a time bin of length ∆

*t*during which the laser is stationary is the Poisson distribution

*D*. In the case of the ABEL trap, a series of electric fields

**E**

*are applied to the trap, imparting motion to the fluorophore in proportion to its electrokinetic mobility*

_{k}*μ*. The probability distribution of the molecule’s motion ∆

**x**during time step

*k*isThis equation is indifferent to the method by which the applied electric field

**E**

*is selected. In the case of the ABEL trap, the field strength is modulated so as to counteract the observed diffusion,where*

_{k}**E**is held constant or modulated independently of the molecular motion.

*μ*and

*D*reflect the charge, size, and interactions of the particle, so extracting these (possibly time-dependent) parameters is a goal of particle tracking and feedback trapping. The molecular brightness

*s*, which is sensitive to the electrochemical environment of the fluorophore, may also be of interest.

*n*,

_{j}*j*≤

*k*} denote the complete history of photon counts in discrete time bins up to and including bin

*k*, where each bin corresponds to a new laser position. We seek to construct the likelihood distribution Pr[

**x**

*|{*

_{k}*n*,

_{j}*j*≤

*k*}] for the position of the fluorophore during the

*k*

^{th}bin. The position likelihood distribution

*f*

_{a}_{|}

*≡ Pr[*

_{b}**x**

*|{*

_{a}*n*,

_{j}*j*≤

*b*}] is estimated recursively from: where the integral is over the entire two-dimensional plane and

*L*≡ Pr[

_{k}*n*|{

_{k}*n*,

_{j}*j*≤

*k*− 1}]. Equations (5) and (6) are known as “update” and “prediction” equations, respectively, and together constitute a “recursive Bayesian estimator” of the fluorophore position. To be applicable, these equations require that the molecule’s unperturbed motion be truly Brownian (e.g., no significant momentum), that the effect of the electric field

**E**be independent of the molecule’s position

**x**, and that the motion of the fluorophore during each bin be negligible.

## 3. Gaussian assumed-density filter (ADF)

^{5}multiplication operations for each convolution. It is therefore necessary to make approximations. Myriad schemes have been devised to handle nonlinear measurements (several of which are described in [23

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.

*f*

_{a}_{|}

*are approximated by simpler, mathematically tractable distributions.*

_{b}*f*

_{a}_{|}

*are normally distributed with mean*

_{b}*S*≡

*s*∆

*t*and

*B*≡

*b*∆

*t*. This distribution is no longer normally distributed, but we can approximate it as such by calculating its mean and covariance and dropping higher moments. An alternative strategy would be to project the posterior distribution onto a finite sum of Gaussian distributions, similar to a Gaussian sum filter [26

26. H. W. Sorenson and D. L. Alspach, “Recursive Bayesian estimation using Gaussian sums,” Automatica **7**(4), 465–479 (1971). [CrossRef]

*L*is simplyThe mean and covariance are These series are convergent by the alternating series test and so are well approximated by partial sums. We typically truncate each summation when subsequent terms contribute less than some fraction

_{k}*ε*<< 1 (typically 10

^{−6}) to the total.

*L*is the likelihood of observing

_{k}*n*photons given all previous observations and the system parameters; that is,

_{k}*L*≡ Pr[

_{k}*n*|{

_{k}*n*,

_{j}*j*≤

*k*− 1};

*D*,

*μ*] (parameter conditioning previously suppressed for brevity), so the log-likelihood of the entire trajectory isMaximum likelihood parameter estimates are found by gradient ascent of this function. Simulations testing the performance of this maximum likelihood estimator are in section 6, below. Application to experimental data is in section 7 and in [13

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]

## 4. Kalman filter

*f*

_{a}_{|}

*are all taken to be Gaussian.*

_{b}*B*<<

*S*) and the scan rate is fast relative to the rate of photon detection (

*S*<< 1). In this case, the dominant term in the series is the one with

*m*=

*n*and

_{k}*i*= 0. So, the update step [Eqs. (13,14)] becomesThese equations, together with Eq. (8) allow propagation of a Gaussian likelihood function for the particle’s position using five variables (two for the position, three for the covariance), and are an instance of the well-known Kalman filter [27

27. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. Trans. ASME **82**(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.

**W**is diagonal (any beam asymmetry is along the same axes that the feedback is applied), the Kalman covariance is also diagonal and the cross-covariance term can be dropped. If the beam profile is circularly symmetric, then

**W**=

*w*

^{2}

**I**for some

*w*, and the variance in the x- and y-dimensions is always equal and can be denoted

**c**

*), the laser spot covariance (*

_{k}**W**or

*w*

^{2}), the diffusion constant (

*D*), and the electrokinetic mobility (

*μ*). The first two can be measured by scanning a point source such as a surface-immobilized fluorescent bead through the laser during a scan. The population-average diffusion constant and electrokinetic mobility can be measured independently using multi-spot fluorescence correlation spectroscopy (FCS) in the presence of an applied electric field [29

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]

20. Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano **5**(7), 5792–5799 (2011). [CrossRef] [PubMed]

*B*<<

*S*means that the Kalman filter does not distinguish between signal and background photons, so spurious counts significantly degrade the estimate. The assumption that

*S*<< 1 introduces a subtle bias, most noticeable when no photons are observed during a bin (

*n*= 0). Absence of photons suggests that the particle is not in the laser spot, and so is more likely to be elsewhere. Analogously, if one were to peer through a telescope and see darkness, one could declare that the moon was

_{k}*not*where the telescope was pointed. The ADF update step [Eq. (13)] accounts for this effect, while the Kalman update step [Eq. (16)] does not. Consequently, the Kalman estimator is slightly biased towards the most recent bin, leading to a modulation of the estimated particle position at the frequency of the laser scan.

## 5. Fundamental constraints

*a*,

*b*)

^{th}element of which iswhere angle brackets indicate the expectation (integral) over the prior probability distribution for the molecule’s position (

*f*

_{k}_{|}

_{k}_{–1}). The last term is the Fisher information of the prior distribution, which, under the Gaussian assumption of the ADF, is simply

*k*for clarity giveswhere Γ ≡ γΔ

*t*.

**c**

*. If the trapping region is well sampled by the scan pattern (the spots are spaced regularly, the distance between them is less than or comparable to the width of the beam, and the overall size of the pattern is much larger than the spread of the beam width around the particle), then the sum over the laser scan positions is well approximated by an integral. Under these conditions, the Fisher information matrix for a two-dimensional scan is approximatelywhere over-bars indicate averaging over the bins*

_{k}*k*of the scan pattern, Li

_{2}is the polylogarithm function of order 2, and

*L*is the number of spots in the scan pattern and Δ

*c*is the spacing between scan positions.) The inverse of the Fisher information is the Cramér-Rao lower bound on the measurement error covariance associated with a particular scan pattern and signal-to-background ratio. In the absence of background counts, therefore, the measurement covariance is simply the beam covariance divided by the number of detected photons, akin to the localization precision of super-resolution techniques based on Gaussian fitting [4

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,” Science **300**(5628), 2061–2065 (2003). [CrossRef] [PubMed]

*B*> 0.

*k*of the scan pattern. All of the matrices commute, so we arrive at a final value for the average ADF tracking covariance ofwhere the square root is understood to indicate the unique positive definite symmetric matrix square root.

*B*= 0. In fact, Eq. (26) reproduces the background-free value of the ADF tracking error [Eq. (24)], indicating that the filters’ tracking performance differs appreciably only when the background is significant.

*t*<< (2

*w*

^{2}/

*Dγ*)

^{½}), the background-free tracking error of the ADF and the Kalman filter [Eq. (26)] simplifies toThis result can be interpreted as the product of the width of the laser spot and the mean distance the particle diffuses between photon detection events, and reproduces the tracking error formula for a continuous-time linear filtration scheme based on lock-in detection [17

17. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**(1), 127–133 (2006). [CrossRef]

*κ*that reflects the confidence that it retains validity. In the case of the Kalman filter with no background and circular symmetry [Eq. (18)], the weighting factor isUsing the approximate average tracking error from Eq. (26), the average weighting factor isThe estimate constructed

*j*bins ago will typically be weighted by

*κ*. Recasting in terms of elapsed time

^{j}*T*≡

*j*∆

*t*and setting ∆

*t*→ 0 givesThus, the timescale during which information from each photon retains utility isThe denominator in Eq. (31) represents the average velocity of the particle between photon detection events. This formula has also been derived in the continuous-time case [17

17. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**(1), 127–133 (2006). [CrossRef]

^{2}/s [30

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

*S*≈1.5 and

*B*≈0.03 for our scan pattern). Our 27-point scan pattern has spot spacing of 0.65 μm, beam width

*w*≈0.4 μm, and scan rate of 12 kHz. Equation (24) predicts a minimum tracking error of 315 nm. The timescale over which measurements retain utility [Eq. (31)] is approximately 70 μs, corresponding to a scan rate of 14 kHz. Our choice of 12 kHz approaches this limit, and trap performance shows little change when the scan rate is increased.

## 6. Simulation: trapping single fluorophores in solution

*D*= 100 μm

^{2}/s,

*μ*= −1000 μm

^{2}/(V s). The feedback electric field strength was capped at 3 V/μm. The expected photon count rate for a molecule in the center of the trap was set to 100 kHz, with a background count rate of 25 kHz. The laser scan matched our experimental setup, with 27 spots of width

*w*= 0.4 μm on a hexagonal lattice of 0.65 μm spacing, scanned with a per-spot dwell time ∆

*t*= 3.1 μs. The Kalman filter tracking and feedback algorithm also matched the experimental implementation as closely as possible. All parameters were held at these values unless otherwise noted. For each parameter set, we simulated one hundred individual molecular traces, each of length 0.1 s. Molecules were considered successfully “trapped” if their maximum displacement from the trap center never exceeded 3.4 μm.

*S*> 1 (such as when the molecule’s count rate is 100 kHz), in accordance with Eq. (26), despite the assumption

*S*<< 1 in the derivation. The primary effect of violation of the

*S*<< 1 approximation is to bias the Kalman filter towards the most recent bin (see section 4), and although this bias worsens as

*S*increases, it is more than compensated by the additional information provided by the larger number of photons.

^{2}/s); the successful cases are plotted in Fig. 3. (None of the parameter sets plotted in Fig. 2 resulted in a successful Kalman fit.) Consistent with the derivation in section 4, the Kalman filter performs well in cases where

*B*<<

*S*<< 1; indeed, its performance is nearly indistinguishable from that of the ADF in the case of zero background and

*S*< 0.9). Because the Kalman filter treats all photons as real, it vastly overestimates molecular diffusion in the presence of background and hence is extremely sensitive to SBR. Unlike the ADF, the Kalman filter fit worsens as the signal strength increases, to the point that it fails when the photon count rate is 100 kHz (

*S*≈3), even in the absence of background.

*B*<<

*S*<< 1 cause the Kalman filter to report that the particle is moving more than it actually is, due to the treatment of background photons as real when

*B*> 0, and due to the bias towards the most recent bin when

*S*> 0. When the Kalman filter is supplied with a diffusion coefficient not too far from the true value (e.g. when it is used for tracking), this extraneous motion is largely suppressed by the filter and the tracking error remains low. However, when the Kalman filter is used for parameter fitting, the model inflates the diffusion coefficient to explain the excess motion. The ADF does not encounter this problem because it can increase the parameters

*B*or

*S*to account for the detected photons.

*B*> 0, the Kalman filter’s tracking error is minimized when the model diffusion coefficient is set below its true value (or, equivalently, when the beam width is set to a larger value); this is because the introduction of background broadens the effective beam width, an effect that is compensated in the ADF but not in the Kalman filter.

## 7. Experiment: trapping single molecules of Alexa 647

**108**(22), 8937–8942 (2011). [CrossRef] [PubMed]

*n*= 137 trapping events was

*D =*348 ± 2 μm

^{2}/s (s.e.m.), in reasonable agreement with the value 325 μm

^{2}/s we obtained from a different data set [13

**108**(22), 8937–8942 (2011). [CrossRef] [PubMed]

^{2}/s measured by FCS [30

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

*μ*= −5.0 × 10

^{3}μm

^{2}/(V s). For a free particle of size much smaller than the Debye screening length, the Einstein-Smoluchowski relation predicts an electrophoretic mobility

*μ*=

*qD*/

*k*, where

_{B}T*q*is the charge of the particle. From our measurement of

*D*, this relation predicts

*μ*= −1.4 × 10

^{4}μm

^{2}/(V s) assuming each molecule bears a single negative charge, significantly different from our data. However, the Einstein-Smoluchowski formula does not take into account electroosmotic flow, the fluid motion induced by application of an electric field to water in a small capillary. This flow velocity depends on details of chemical state of the channel walls, and is expected to be counter to the direction of the electrophoretic motion in our experiments.

**108**(22), 8937–8942 (2011). [CrossRef] [PubMed]

## 8. Conclusion

**108**(22), 8937–8942 (2011). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Biophys. Biomol. Struct. |

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,” Cell |

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,” Nature |

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,” Science |

5. | M. A. Thompson, J. M. Casolari, M. Badieirostami, P. O. Brown, and W. E. Moerner, “Three-dimensional tracking of single mRNA particles in |

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. |

7. | A. P. Fields and A. E. Cohen, “Anti-Brownian traps for studies on single molecules,” Methods Enzymol. |

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. |

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. |

10. | R. H. Goldsmith and W. E. Moerner, “Watching conformational- and photodynamics of single fluorescent proteins in solution,” Nat. Chem. |

11. | H. Cang, D. Montiel, C. S. Xu, and H. Yang, “Observation of spectral anisotropy of gold nanoparticles,” J. Chem. Phys. |

12. | J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B |

13. | A. P. Fields and A. E. Cohen, “Electrokinetic trapping at the one nanometer limit,” Proc. Natl. Acad. Sci. U.S.A. |

14. | A. H. Jazwinski, |

15. | K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in single-molecule fluorescence microscopy,” Biophys. J. |

16. | A. J. Berglund, K. McHale, and H. Mabuchi, “Fluctuations in closed-loop fluorescent particle tracking,” Opt. Express |

17. | A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B |

18. | A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence correlation spectroscopy of individual particles,” Opt. Express |

19. | A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. |

20. | Q. Wang and W. E. Moerner, “An adaptive anti-Brownian electrokinetic trap with real-time information on single-molecule diffusivity and mobility,” ACS Nano |

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. Methods |

22. | B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. |

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. |

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, |

26. | H. W. Sorenson and D. L. Alspach, “Recursive Bayesian estimation using Gaussian sums,” Automatica |

27. | R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. Trans. ASME |

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. |

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

**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/oe/abstract.cfm?URI=oe-20-20-22585

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### 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]
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