On the Effects of Firing Memory in the Dynamics of Conjunctive Networks

Abstract

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several update schemes that range from parallel or synchronous to asynchronous. However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called firing memory in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is PSPACE-complete.

Non-deterministic updates of Boolean networks

“…Until now, all the updating modes that have been discussed depend on deterministic updates that are context free, which leads to deal with memoryless dynamical systems. In [14,15] have been introduced another model of BNs, called Memory Boolean networks (MBNs). The first objective of MBNs is to capture the biologically relevant gene-protein BN model introduced in [18], that builds on the following principles: automata are split in two types: a half models genes, the other half models their associated one-to-one proteins; each protein has its own decay time: the number of time steps during which it remains present in the cell after having been produced by the punctual expression of its associated gene.…”

“…Memory Boolean networks (MBNs) [14,15] take into account some kind of delay for the decrease of automata. They have been introduced by the means of a deterministic dynamical system with non-binary configurations, whose updates are computed deterministically from the BN and a memory vector, specifying the delay for each automaton.…”