The effect of network topology on the stability of discrete state models of genetic control

TL;DR

This paper develops a general method to assess the stability of large Boolean gene networks with arbitrary topologies, accounting for various gene expression biases and update schemes, by analyzing the eigenvalues of a modified adjacency matrix.

Contribution

It introduces a novel eigenvalue-based approach to determine stability in gene networks with realistic topologies and diverse gene behaviors, extending previous random-graph models.

Findings

Stability depends on the maximum eigenvalue of a modified adjacency matrix.

The method accurately predicts network responses to small perturbations.

Results are validated through numerical simulations.

Abstract

Boolean networks have been proposed as potentially useful models for genetic control. An important aspect of these networks is the stability of their dynamics in response to small perturbations. Previous approaches to stability have assumed uncorrelated random network structure. Real gene networks typically have nontrivial topology significantly different from the random network paradigm. In order to address such situations, we present a general method for determining the stability of large Boolean networks of any specified network topology and predicting their steady-state behavior in response to small perturbations. Additionally, we generalize to the case where individual genes have a distribution of `expression biases,' and we consider non-synchronous update, as well as extension of our method to non-Boolean models in which there are more than two possible gene states. We find that stability is governed by the maximum eigenvalue of a modified adjacency matrix, and we test this result by comparison with numerical simulations. We also discuss the possible application of our work to experimentally inferred gene networks.