The window at the edge of chaos in a simple model of gene interaction networks

As a model for gene and protein interactions we study a set for molecular catalytic reactions. The model is based on experimentally motivated interaction network topologies, and is designed to capture some key statistics of gene expression statistics. We impose a non-linearity to the system by a boundary condition which guarantees non-negative concentrations of chemical concentrations and study the system stability 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. We explain the finding as a self-organized critical phenomenon. The robustness of this finding with respect to different 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.