UC Santa Cruz

Motivated by the recent development of advanced experimental techniques in molecular biology, this paper focuses on the study of the dynamical properties of a two-gene regulatory network. A mathematical model for a two-gene regulatory network is derived and several of their properties are analyzed. Due to the presence of mixed continuous/discrete dynamics and hysteresis, we employ hybrid systems models to capture the dynamics of the system. The proposed model incorporates binary hysteresis with different thresholds capturing the interaction between the genes. We analyze properties of the solutions and asymptotic stability of equilibria in the system as a function of their parameters. As a difference to previous efforts employing piecewise-linear models, the analysis of our hybrid system model reveals the presence of limit cycles for a certain range of parameters, a behavior that is associated with the presence of hysteresis. The set of points defining the limit cycle is characterized and its asymptotic stability properties are studied. Furthermore, we determine conditions under which the stability properties of the limit cycle are robust to changes of parameters. Numerical simulations are presented to illustrate the findings. © 2014 Elsevier Inc.