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

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 Rk. We also discuss a simple signaling pathway related to cancer research, called p53 module.

Universidade de São Paulo USP-COFECUB

Universidade de São Paulo USP-COFECUB

Antonio Prudente Cancer Research (FAPESP-CEPID)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

CNRS-FAPESP agreement

Centre National de la Recherche Scientifique (CNRS)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

CNPq[307978/2004-4]

CNPq[484351/2006-0]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

FAPESP[2004/07276-2]

FAPESP[06/03227-2]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)