Simultaneous Emergence of Cooperative Response and Mutational Robustness in Gene Regulatory Networks

Gene regulatory networks (GRN) are complex systems in which many genes mutually regulate the expression by the transcription factors for changing the cell state adaptively to the change of environmental conditions. Living systems have gained such adaptive responses through the Darwinian evolution and thus the interactions between genes are not at all random interactions. In this respect, GRNs utilized by life are very rare among “all the possible GRNs”. In the present study, we consider the following two problems: (1) How rare are GRNs that have been produced by the evolution (2) Relationship between the robustness against mutations and the function of GRNs.
We use a simplified mathematical model and consider GRNs consisting of N genes with one input gene and one output gene. The difference in the expression level of the output gene between the input states “on” and “off” is taken as the fitness. That is, the GRNs exhibiting more sensitive responses are regarded as “more evolved”. A number of GRNs with a broad range of fitness from the minimum to the maximum are generated by means of the Multi-Canonical Monte Carlo method, and the appearance probability of each value of the fitness in the random GRNs is estimated. The figure shows the appearance probability for the case that each gene interacts with 5 other genes on average. This figure is regarded to express the “fitness landscape”. The “fittest ensemble”, which consists of GRNs with the highest fitness, is of particular interest, because GRNs produced by the evolution should be included in it.
We obtained the following results: (1) Almost all GRNs generated randomly are either non-functional or having a very low fitness and the GRNs in the fittest ensemble are very rare. (2) GRNs in the fittest ensemble respond step-function-like to the input level. This fact suggests that the GRNs respond by switching the fixed-point of the dynamical system. (3) Responses are quite robust against the fluctuations of the input level, as a result of the fixed-point switching. (4) Majority of the GRNs in the fittest ensemble are robust against the mutation of the single-interaction deletion. These results suggest that while producing highly sensitive GRNs is difficult, once they are produced, the robustness against the mutations and that against the input fluctuations are readily achieved. In other words, these two robustnesses are the universal properties regardless to the evolutionary processes.