A state space based approach to localizing single molecules from multi-emitter images

Vahid, Milad R.; Chao, Jerry; Ober, Raimund J.

doi:10.1117/12.2253175

Cited by 3 publications

(5 citation statements)

References 17 publications

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“…(2.4) and (2.5), respectively. We next in Corollary 5.5 to Theorem 5.2 calculate the Fisher information matrix for these special cases, and show that the obtained results are consistent with the results presented in [22,37,35,36].…”

Section: )supporting

confidence: 83%

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Vahid¹,

Hanzon²,

Ober³

2020

SIAM J. Imaging Sci.

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Tracking of objects in cellular environments has become a vital tool in molecular cell biology. A particularly important example is single molecule tracking which enables the study of the motion of a molecule in cellular environments by locating the molecule over time and provides quantitative information on the behavior of individual molecules in cellular environments, which were not available before through bulk studies. Here, we consider a dynamical system where the motion of an object is modeled by stochastic differential equations (SDEs), and measurements are the detected photons emitted by the moving fluorescently labeled object, which occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to uniform time points which have been commonly used in similar dynamical systems. The measurements are distributed according to optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a three dimensional profile, such as Born and Wolf, for an out-offocus molecule with respect to the detector. For some special circumstances, Gaussian image models have been proposed. In this paper, we introduce a stochastic framework in which we calculate the maximum likelihood estimates of the biophysical parameters of the molecular interactions, e.g., diffusion and drift coefficients. More importantly, we develop a general framework to calculate the Cramér-Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, for the estimation of unknown parameters and use it as a benchmark in the evaluation of the standard deviation of the estimates. There exists no established method, even for Gaussian measurements, to systematically calculate the CRLB for the general motion model that we consider in this paper. We apply the developed methodology to simulated data of a molecule with linear trajectories and show that the standard deviation of the estimates matches well with the square root of the CRLB. We also show that equally sampled and Poisson distributed time points lead to significantly different Fisher information matrices.

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Section: )supporting

confidence: 83%

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Vahid¹,

Hanzon²,

Ober³

2020

SIAM J. Imaging Sci.

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“…We also calculated the standard deviations of the estimates. These standard deviations, which are computed as 57.4 nm and 59.6 nm for the x 0 -and y 0 -locations of the molecule, respectively, are close to the localization accuracy, i.e., the positive definite square root of the CRLB, which is given as 58.37 nm for both xand y-directions, reported in [18], [28], [29], [30].…”

Section: Example 53supporting

confidence: 64%

“…For the infinite practical data model, as calculated in Theorem 6.2, the above expression can be rewritten as Eq. (30).…”

Section: Effect Of Noisementioning

confidence: 99%

“…We also calculated the standard deviations of the estimates. These standard deviations, which are computed as 57.4 nm and 59.6 nm for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$x_0$\end{document} - and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$y_0$\end{document} -locations of the molecule, respectively, are close to the localization accuracy, i.e., the positive definite square root of the CRLB, which is given as 58.37 nm for both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$x$\end{document} - and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$y$\end{document} -directions, reported in [18] , [28] – [30] .

Fig.…”

Section: Maximum Likelihood Estimationmentioning

confidence: 99%

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Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

Vahid

Hanzon

Ober

2021

IEEE Trans. Comput. Imaging

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The advent of single molecule microscopy has revolutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions.

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“…These photons are detected using a pixelated detector of pixel size and image size of 6.5 × 6.5 µm and 60 × 60 pixels, respectively. the positive definite square root of the CRLB, which is given as 58.37 nm for both xand y-directions, reported in [18], [26], [27], [28].…”

Section: B Maximum Likelihood Estimation For Practical Data Modelmentioning

confidence: 99%

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

Vahid,

Hanzon,

Ober

2019

Preprint

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The advent of single molecule microscopy has revolutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which were not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel. Therefore, the precise temporal information of detection of the photons is not available. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories from pixelated images a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Cramér-Rao lower bound (CRLB), given by the inverse of the Fisher information matrix, on the variance of the parameter estimates. Even in cases that we have a small number of photons, the obtained results show that we are able to estimate the parameters of the molecule trajectory from simulated fluorescence microscopy images using our proposed method.

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A state space based approach to localizing single molecules from multi-emitter images

Cited by 3 publications

References 17 publications

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Fisher Information Matrix for Single Molecules with Stochastic Trajectories

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

Effect of Pixelation on the Parameter Estimation of Single Molecule Trajectories

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