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

We introduce jump processes in R, 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 kdimensional 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, associated to a biological 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. 1 Motivation and introduction The interest in the analysis of dynamical processes going on within a biological system and the corresponding signal-processing mechanisms, together with recent successes in molecular biology and advances in computer technology, spurred a revival of systems biology ideas [1]. This approach [2, 3], which exploits ideas from dynamical systems and control theory [28], can be traced back at least sixty five years ago to Erwin Schrodinger’s question: What is life? [5]. ∗[email protected] †[email protected] ‡[email protected]