Fisher Information matrix for single molecules with stochastic trajectories
Fisher Information matrix for single molecules with stochastic trajectories
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, that occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to equidistant time points, which have been commonly used in the modeling of dynamical systems. The measurements are distributed according to the optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a Born and Wolf profile for an out-of-focus 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.
Cramér-Rao lower bound, Fisher information matrix, Maximum likelihood estimation, Object tracking, Single molecule microscopy, Stochastic differential equation
234-264
Vahid, Milad R.
6a1a88a4-9fcc-4ac5-84a4-0d4ee1088cc6
Hanzon, Bernard
ec8a3e31-d488-4a69-8318-6ff08024dd7c
Ober, Raimund J.
31f4d47f-fb49-44f5-8ff6-87fc4aff3d36
2020
Vahid, Milad R.
6a1a88a4-9fcc-4ac5-84a4-0d4ee1088cc6
Hanzon, Bernard
ec8a3e31-d488-4a69-8318-6ff08024dd7c
Ober, Raimund J.
31f4d47f-fb49-44f5-8ff6-87fc4aff3d36
Vahid, Milad R., Hanzon, Bernard and Ober, Raimund J.
(2020)
Fisher Information matrix for single molecules with stochastic trajectories.
SIAM Journal on Imaging Sciences, 13 (1), .
(doi:10.1137/19M1242562).
Abstract
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, that occur at discrete time points, corresponding to the arrival times of a Poisson process, in contrast to equidistant time points, which have been commonly used in the modeling of dynamical systems. The measurements are distributed according to the optical diffraction theory, and therefore, they would be modeled by different distributions, e.g., an Airy profile for an in-focus and a Born and Wolf profile for an out-of-focus 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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M124256
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supplement_M124256
- Accepted Manuscript
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Accepted/In Press date: 3 January 2020
e-pub ahead of print date: 25 February 2020
Published date: 2020
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Funding Information:
∗Received by the editors February 4, 2019; accepted for publication (in revised form) January 3, 2020; published electronically February 25, 2020. https://doi.org/10.1137/19M1242562 Funding: The work of the authors was partially supported by the National Institutes of Health grant R01 GM085575. †Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843 (milad.rafiee@ tamu.edu, raimund.ober@tamu.edu). ‡Department of Biomedical Data Science, Stanford University, Stanford, CA 94305 (miladrv@stanford.edu). §Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, University College Cork, Cork, Ireland (b.hanzon@ucc.ie). ¶Department of Molecular and Cellular Medicine, Texas A&M Health Science Center, College Station, TX 77843 (milad.rafiee@tamu.edu, raimund.ober@tamu.edu). ‖Center for Cancer Immunology, Faculty of Medicine, University of Southampton, Southampton, UK (r.ober@ soton.ac.uk).
Publisher Copyright:
© 2020 Raimund Ober. Published by SIAM under the terms of the Creative Commons 4.0 license.
Keywords:
Cramér-Rao lower bound, Fisher information matrix, Maximum likelihood estimation, Object tracking, Single molecule microscopy, Stochastic differential equation
Identifiers
Local EPrints ID: 445345
URI: http://eprints.soton.ac.uk/id/eprint/445345
PURE UUID: 9e42f5fb-9838-4c3c-9dd4-6faff25493fe
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Date deposited: 02 Dec 2020 17:35
Last modified: 06 Jun 2024 04:20
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Author:
Milad R. Vahid
Author:
Bernard Hanzon
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