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Two-boundary first exit time of Gauss-Markov processes for stochastic modeling of acto-myosin dynamics

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Abstract

We consider a stochastic differential equation in a strip, with coefficients suitably chosen to describe the acto-myosin interaction subject to time-varying forces. By simulating trajectories of the stochastic dynamics via an Euler discretization-based algorithm, we fit experimental data and determine the values of involved parameters. The steps of the myosin are represented by the exit events from the strip. Motivated by these results, we propose a specific stochastic model based on the corresponding time-inhomogeneous Gauss-Markov and diffusion process evolving between two absorbing boundaries. We specify the mean and covariance functions of the stochastic modeling process taking into account time-dependent forces including the effect of an external load. We accurately determine the probability density function (pdf) of the first exit time (FET) from the strip by solving a system of two non singular second-type Volterra integral equations via a numerical quadrature. We provide numerical estimations of the mean of FET as approximations of the dwell-time of the proteins dynamics. The percentage of backward steps is given in agreement to experimental data. Numerical and simulation results are compared and discussed.

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Notes

  1. Here, we call FET from a strip the originally two-boundary FPT random variable defined by Nobile et al. (2006).

  2. In this paper we use \((\prime )\) to indicate the derivative of the function with respect to its own argument; in particular in (1) \((\prime )\) is for the space derivative and \((\cdot )\) denotes the time derivative.

  3. The adjectives “short” and “long” used for the exit times are referred to the characteristic time \(\theta \).

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Acknowledgments

We thank the Editor and the anonymous reviewers for their constructive comments and suggestions. We also thank Angelo Pirozzi, student of Medicine at the University of Napoli Federico II, for the enlightening discussion on some biological aspects which helped us in the writing of subsection 2.1. This work was partially supported by Gruppo Nazionale per il Calcolo Scientifico (GNCS-INdAM).

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  1. Dipartimento di Matematica e Applicazioni, Univ. di Napoli FEDERICO II, Via Cintia, 80126, Napoli, Italy

    Giuseppe D’Onofrio & Enrica Pirozzi

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  1. Giuseppe D’Onofrio
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  2. Enrica Pirozzi
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Correspondence to Giuseppe D’Onofrio.

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D’Onofrio, G., Pirozzi, E. Two-boundary first exit time of Gauss-Markov processes for stochastic modeling of acto-myosin dynamics. J. Math. Biol. 74, 1511–1531 (2017). https://doi.org/10.1007/s00285-016-1061-x

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  • DOI: https://doi.org/10.1007/s00285-016-1061-x

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