Inflation of the Edge of Chaos in a Simple Model of Gene Interaction Networks

We study a set of linearized catalytic reactions to model gene and protein interactions. The model is based on experimentally motivated interaction network topologies and is designed to capture some key properties of gene expression statistics. We impose a non-linearity to the system by enforcing a boundary condition which guarantees non-negative concentrations of chemical substances. System stability is quantified by maximum Lyapunov exponents. We find that the non-negativity constraint leads to a drastic inflation of those regions in parameter space where the Lyapunov exponent exactly vanishes. Within the model this finding can be fully explained as a result of a symmetry breaking mechanism induced by the positivity constraint. The robustness of this finding with respect to network topologies and the role of intrinsic molecular- and external noise is discussed. We argue that systems with inflated ’edges of chaos’ could be much more easily favored by natural selection than systems where the Lyapunov exponent vanishes only on a parameter set of measure zero.