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

TL;DR

This paper introduces density-profile jump processes to model biological signaling networks, proving their convergence to deterministic trajectories governed by differential equations, and explores bifurcations related to phase transitions.

Contribution

It presents a novel stochastic process model for biological networks and establishes almost sure convergence to deterministic paths, including analysis of bifurcations like Hopf and pitchfork.

Findings

Convergence of the empirical magnetization to deterministic trajectories.

Identification of bifurcations such as Hopf and pitchfork.

Behavior of the process varies with parameters, showing multiple stable states.

Abstract

We introduce jump processes in R^k, called density-profile process, to model biological signaling networks. They describe the macroscopic evolution of finite-size spin-flip models with k types of spins interacting through a non-reversible Glauber dynamics. We focus on the the k-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. 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 leading 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.