Synchrony patterns in gene regulatory networks

Abstract

Motivated by studying synchronization mechanisms in gene regulatory networks (GRNs) and their relation to evolutionary events such as genetic duplication and genetic redundancy, we consider two types of mathematical dynamical models of GRNs that depict general additive (SUM model) or multiplicative (MULT model) gene regulations. By a synchrony pattern we mean clusters of synchronized genes whose states coincide for all time. The identification of the genes in each cluster of a network synchrony pattern results into (smaller) quotient network equations representing the dynamical evolution of the original GRN equations restricted to that synchrony pattern. From the perspective of the dynamics of a GRN, in case the restricted dynamics functions as the regulatory core underlying the dynamics of the GRN, for example, if the synchrony pattern is globally attracting, the quotient network could represent a structural motif that performs a biological function, also known as a functional motif. Gene duplication in GNRs has its analog in the lifting process in the theory of coupled cell networks, which is the reverse of quotient: the unfolding of genes in a network, where each of those genes is unfolded into two or more genes, leads to a (bigger) lift network. In general, there are many lifts associated with a fixed quotient network. In this paper, we obtain results on robust synchrony patterns for SUM and MULT dynamical models inspired by the existing theoretical results in the coupled cell network formalisms. Moreover, we explore the concepts of quotient network and network lifting in the context of GRNs which are related to the process of gene duplication and the phenomenon of subfunctionalization as an outcome of functional divergence.