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Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

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Abstract

We introduce jump processes in ℝk, called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in ℝk. We also discuss a simple signaling pathway related to cancer research, called p53 module.

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l’Université, B.P.12, 76801, St Etienne du Rouvray, France

    Roberto Fernández

  2. University of São Paulo, Rua do Matão, 1010, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brasil

    Luiz R. Fontes & E. Jordão Neves

Authors
  1. Roberto Fernández
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  2. Luiz R. Fontes
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  3. E. Jordão Neves
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Correspondence to Luiz R. Fontes.

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Fernández, R., Fontes, L.R. & Neves, E.J. Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories. J Stat Phys 136, 875–901 (2009). https://doi.org/10.1007/s10955-009-9819-9

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  • DOI: https://doi.org/10.1007/s10955-009-9819-9

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