Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories |
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Authors: | Roberto?Fernández Luiz?R?Fontes E?Jord?o?Neves |
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Institution: | 1.Laboratoire de Mathématiques Rapha?l Salem,UMR 6085 CNRS-Université de Rouen,St Etienne du Rouvray,France;2.University of S?o Paulo,S?o Paulo,Brasil |
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Abstract: | We introduce jump processes in ℝ
k
, 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 ℝ
k
. We also discuss a simple signaling pathway related to cancer research, called p53 module. |
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Keywords: | |
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