Sensitive dependence on initial conditions in gene networks

Active regulation in gene networks poses mathematical challenges that have led to conflicting approaches to analysis. Competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. An extension, due to Artstein and coauthors, of the classical singular perturbation approach was suggested as an appropriate way to handle the complex situation where non-trivial dynamics, such as a limit cycle, of fast variables occur in switching domains. This non-trivial behaviour can occur when a gene regulates multiple other genes at the same threshold. Here, it is shown that it is possible for nonuniqueness to arise in such a system in the case of limiting step-function interactions. This nonuniqueness is reminiscent of but not identical to the nonuniqueness of Filippov solutions. More realistic gene network models have sigmoidal interactions, however, and in the example considered here, it is shown numerically that the corresponding phenomenon in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different fixed point attractors.