Dynamical properties of a gene-protein model

A major limitation of the classical random Boolean network model of gene regulatory networks is its synchronous updating, which implies that all the proteins decay at the same rate. Here a model is discussed, where the network is composed of two different sets of nodes, labelled G and P with reference to “genes” and “proteins”. Each gene corresponds to a protein (the one it codes for), while several proteins can simultaneously affect the expression of a gene. Both kinds of nodes take Boolean values. If we look at the genes only, it is like adding some memory terms, so the new state of the gene subnetwork network does no longer depend upon its previous state only. In general, these terms tend to make the dynamics of the network more ordered than that of the corresponding memoryless network. The analysis is focused here mostly on dynamical critical states. It has been shown elsewhere that the usual way of computing the Derrida parameter, starting from purely random initial conditions, can be misleading in strongly non-ergodic systems. So here the effects of perturbations on both genes’ and proteins’ levels is analysed, using both the canonical Derrida procedure and an “extended” one. The results are discussed. Moreover, the stability of attractors is also analysed, measured by counting the fraction of perturbations where the system eventually falls back onto the initial attractor.