Noisy attractors and ergodic sets in models of gene regulatory networks

We investigate the hypothesis that cell types are attractors. This hypothesis was criticized with the fact that real gene networks are noisy systems and, thus, do not have attractors [Kadanoff, L., Coppersmith, S., Aldana, M., 2002. Boolean Dynamics with Random Couplings. http://www.citebase.org/abstract?id=oai:arXiv.org:nlin/0204062]. Given the concept of "ergodic set" as a set of states from which the system, once entering, does not leave when subject to internal noise, first, using the Boolean network model, we show that if all nodes of states on attractors are subject to internal state change with a probability p due to noise, multiple ergodic sets are very unlikely. Thereafter, we show that if a fraction of those nodes are "locked" (not subject to state fluctuations caused by internal noise), multiple ergodic sets emerge. Finally, we present an example of a gene network, modelled with a realistic model of transcription and translation and gene-gene interaction, driven by a stochastic simulation algorithm with multiple time-delayed reactions, which has internal noise and that we also subject to external perturbations. We show that, in this case, two distinct ergodic sets exist and are stable within a wide range of parameters variations and, to some extent, to external perturbations.