Gene regulation models and evolution

We examine the dynamics of gene regulation for a system with K environments with each environment favoring a different pattern of gene transcription and regulation in a prokaryotic organism. We recast the model of the system as discrete-time K-periodic. This allows us to exploit the well known properties of periodic systems to characterize the steady-state behavior of wild type versus mutant populations of the organism: We also obtain conditions on the parameters of the state matrix using the properties of positive linear systems. We examine in more detail the example of the lac operon of E. Coli which has two environments: a high demand environment H and a low demand environment L. The gene regulation system alternates between the two environments as the organism completes one life cycle. We derive conditions under which the system reaches steady growth alternating between a state x_H and x_L as each life cycle is completed. Our results show the dependence of the mutant to wild type ratio on the ratio of high demand to low demand duration.