Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks

We generalize random Boolean networks by softening the hard binary discretization into multiple discrete states. These multistate networks are generic models of gene regulatory networks, where each gene is known to assume a finite number of functionally different expression levels. We analytically determine the critical connectivity that separates the biologically unfavorable frozen and chaotic regimes. This connectivity is inversely proportional to a parameter which measures the heterogeneity of the update rules. Interestingly, the latter does not necessarily increase with the mean number of discrete states per node. Still, allowing for multiple states decreases the critical connectivity as compared to random Boolean networks, and thus leads to biologically unrealistic situations. Therefore, we study two approaches to increase the critical connectivity. First, we demonstrate that each network can be kept in its frozen regime by sufficiently biasing the update rules. Second, we restrict the randomly chosen update rules to a subclass of biologically more meaningful functions. These functions are characterized based on a thermodynamic model of gene regulation. We analytically show that their usage indeed increases the critical connectivity. From a general point of view, our thermodynamic considerations link discrete and continuous models of gene regulatory networks.