Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module

TL;DR

This paper models biological feedback loops using a type-dependent stochastic Ising model, extending the repressilator, and analyzes its dynamics, bifurcations, and oscillatory behaviors through mean-field and convergence techniques.

Contribution

It introduces a non-symmetric feedback module model based on a type-dependent Ising system, capturing gene expression variability and complex dynamics.

Findings

The model exhibits rich dynamical behaviors including oscillations.

Bifurcation analysis reveals conditions for oscillatory regimes.

Convergence from stochastic to deterministic dynamics is established.

Abstract

We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fern\'andez et. al (2009), and are useful since take account to inherent variability of gene expression. We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler (2000). We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.