Homogeneous and Scalable Gene Expression Regulatory Networks with Random Layouts of Switching Parameters

LaTeX, 30 pages, 20 picturesWe consider a model of large regulatory gene expression networks where the thresholds activating the sigmoidal interactions between genes and the signs of these interactions are shuffled randomly. Such an approach allows for a qualitative understanding of network dynamics in a lack of empirical data concerning the large genomes of living organisms. Local dynamics of network nodes exhibits the multistationarity and oscillations and depends crucially upon the global topology of a "maximal" graph (comprising of all possible interactions between genes in the network). The long time behavior observed in the network defined on the homogeneous "maximal" graphs is featured by the fraction of positive interactions ($0\leq \eta\leq 1$) allowed between genes. There exists a critical value $\eta_c\eta_c,$ it tends to a fixed point (which position in the phase space is determined by the initial conditions and the certain layout of switching parameters). In networks defined on the inhomogeneous directed graphs depleted in cycles, no oscillations arise in the system even if the negative interactions in between genes present therein in abundance ($\eta_c=0$). For such networks, the bidirectional edges (if occur) influence on the dynamics essentially. In particular, if a number of edges in the "maximal" graph is bidirectional, oscillations can arise and persist in the system at any low rate of negative interactions between genes ($\eta_c=1$). Local dynamics observed in the inhomogeneous scalable regulatory networks is less sensitive to the choice of initial conditions. The scale free networks demonstrate their high error tolerance