Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics

Haas, Kevin; Yang, Haw; Chu, Jhih‐Wei

doi:10.1063/1.4820491

Cited by 14 publications

(12 citation statements)

References 59 publications

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“…For biomolecular systems and others for which the Langevin equation is employed for modeling the dynamics, the results of this work provide a theoretical foundation for quantitative analysis of data, including adopting the methods of statistical learning, 34 and for development of dynamic models. 35 …”

Section: ■ Conclusionmentioning

confidence: 99%

Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium

2014

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The analytical expression for the trajectory entropy of the overdamped Langevin equation is derived via two approaches. The first route goes through the Fokker−Planck equation that governs the propagation of the conditional probability density, while the second method goes through the path integral of the Onsager− Machlup action. The agreement of these two approaches in the continuum limit underscores the equivalence between the partial differential equation and the path integral formulations for stochastic processes in the context of trajectory entropy. The values obtained using the analytical expression are also compared with those calculated with numerical solutions for arbitrary time resolutions of the trajectory. Quantitative agreement is clearly observed consistently across different models as the time interval between snapshots in the trajectories decreases. Furthermore, analysis of different scenarios illustrates how the deterministic and stochastic forces in the Langevin equation contribute to the variation in dynamics measured by the trajectory entropy. ■ INTRODUCTIONConsider very generally a system that interacts with its environment and evolves over time. Let x be some continuous order parameter of interest (or an experimental or computational observable) that characterizes the spatial configuration and represents the state of the system. For example, x could be a molecule-scale observable in single-molecule experiments, such as the orientation-averaged dipole−dipole distance in a single-molecule Forster-type resonance energy transfer experiment, 1 the constant-force extension in a laser tweezers/atomic force pulling experiment, 2 or a collective coordinate from molecular dynamics simulations, 3 even a state variable in a quantum-control experiment. 4 In many problems of this type, including the examples given above, the time evolution of x can be described by the overdamped Langevin equation, 5where the subscript in x t is time. In the Langevin equation, D is the diffusion coefficient specifying the magnitude of the stochastic forces modeled by the Wiener process, dW t , satisfying ⟨dW t dW t′ ⟩ = δ(t − t′) dt. On the other hand, the deterministic component of the Langevin equation comes from the potential of mean force (PMF), V(x), and the mean force is F(x) = −dV(x)/dx. Here, we consider the PMF as nondimensionalized by k B T, where k B is the Boltzmann constant and T is the temperature. Suppose in one realization that the system is at an initial configuration x 0 at time zero. Then propagating the Langevin equation will trace out a trajectory X(t), a continuous but nondifferentiable function that gives a value of x t at time t, with t varying between 0 and the finite period of observation, t obs . For stationary processes, the system reaches equilibrium and the probability density of obtaining a particular value of X(t) = x along the trajectory is given by p eq (x) = exp(−V(x))/Z eq , where Z eq = ∫ dx exp(−V(x)) is the equilibrium partition function. 6 In this work, we consider an ense...

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Section: ■ Conclusionmentioning

confidence: 99%

Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium

2014

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“…In principle, the additional information for improving the inference could also be physical models. One sees such precedents in singlemolecule time series analysis: the distance and time resolutions in continuous single-molecule Förster resonance energy transfer time series analysis are drastically improved from data-driven inference (71,72) to analysis based on Langevin dynamics (73)(74)(75)(76), and the time resolution in singlemolecule on/off intensity instances could be further refined from kinetics model-free analysis (77)(78)(79) to that based on hidden Markov modeling (80).…”

Section: Discussionmentioning

confidence: 99%

Prior-Apprised Unsupervised Learning of Subpixel Curvilinear Features in Low Signal/Noise Images

Yin

Yang

2020

Biophysical Journal

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Many biophysical problems involve molecular and nanoscale targets moving next to a curvilinear track, e.g., a cytosolic cargo transported by motor proteins moving along a microtubule. For this type of problem, fluorescence imaging is usually the primary tool of choice. There is, however, an $20-fold mismatch between target-localization precision and track-imaging resolution such that questions requiring high-fidelity definition of the target's track remain inaccessible. On the other hand, if the contextual image of the tracks can be refined to a level comparable to that of the target, many intuitive yet mechanistically important issues can begin to be addressed. This work demonstrates that it is possible to statistically infer, to subpixel precision, curvilinear features in a low signal/noise image. This is achieved by a framework that consists of three stages: the Hessian-based feature enhancement, the subimage feature sampling and registration, and the statistical learning of the underlying curvilinear structure using a new, to our knowledge, method developed here for inferring the principal curves. In each stage, the descriptive prior information that the features come from curvilinear elements is explicitly taken into account. It is fully automated without user supervision, which is distinctly different from approaches that require user seeding or well-defined training data sets. Computer simulations of realistic images are used to investigate the performance of the framework and its implementation. The characterization results suggest that curvilinear features are refined to the same order of precision as that of the target and that the bootstrap confidence intervals from the analysis allow an estimate for the statistical bounds of the simulated ''true'' curve. Also shown are analyses of experimental images from three different microscopy modalities: two-photon laser-scanning microscopy, epifluorescence microscopy, and total internal reflection fluorescence microscopy. The practical application of this priorapprised unsupervised learning framework as well as its potential outlook are discussed.

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“…The errors of the parameters can be estimated from the curvature of the likelihood function at the maximum, which is related to the Fisher information. 21,28,29 This approximation of normal distribution is appropriate when the number of photons is large, so the error of the parameter estimation is small. In addition, the estimates should not be close to the boundaries (which are 0 and 1 for the FRET efficiencies and population and 0 for the relaxation rate).…”

Section: A Modelmentioning

confidence: 99%

Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET

Gopich

2015

The Journal of Chemical Physics

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Photon sequences from single-molecule Förster resonance energy transfer (FRET) experiments can be analyzed using a maximum likelihood method. Parameters of the underlying kinetic model (FRET efficiencies of the states and transition rates between conformational states) are obtained by maximizing the appropriate likelihood function. In addition, the errors (uncertainties) of the extracted parameters can be obtained from the curvature of the likelihood function at the maximum. We study the standard deviations of the parameters of a two-state model obtained from photon sequences with recorded colors and arrival times. The standard deviations can be obtained analytically in a special case when the FRET efficiencies of the states are 0 and 1 and in the limiting cases of fast and slow conformational dynamics. These results are compared with the results of numerical simulations. The accuracy and, therefore, the ability to predict model parameters depend on how fast the transition rates are compared to the photon count rate. In the limit of slow transitions, the key parameters that determine the accuracy are the number of transitions between the states and the number of independent photon sequences. In the fast transition limit, the accuracy is determined by the small fraction of photons that are correlated with their neighbors. The relative standard deviation of the relaxation rate has a "chevron" shape as a function of the transition rate in the log-log scale. The location of the minimum of this function dramatically depends on how well the FRET efficiencies of the states are separated.

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Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics

Cited by 14 publications

References 59 publications

Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium

Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium

Prior-Apprised Unsupervised Learning of Subpixel Curvilinear Features in Low Signal/Noise Images

Accuracy of maximum likelihood estimates of a two-state model in single-molecule FRET

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