Relations between the conditions of admitting cycles in Boolean and ODE network systems

Abstract

Boolean (BL) systems and coupled ordinary differential equations (ODEs) are popular models for studying biological networks. BL systems can be set up without detailed reaction mechanisms and rate constants and provide qualitatively useful information, but they cannot capture the continuous dynamics of biological systems. On the other hand, ODEs are able to capture the continuous dynamic features of biological networks and provide more information on how the activities of components depend on other components and parameter values. However, a useful coupled ODE model requires details about interactions and parameter values. The introduction of the relationships between the two types of models will enable us to leverage their advantages and better understand the target network systems. In this paper, we investigate the relations between the conditions of the existence of limit cycles in ODE networks and their homologous discrete systems. We prove that for a single feedback loop, as long as the corresponding governing functions of the homologous continuous and discrete systems have the same upper and lower asymptotes, the limit cycle borne via Hopf bifurcation corresponds to the cycle of the discrete system. However, for some coupled feedback loops, besides having the same upper and lower asymptotes, parameters such as the decay rates also play crucial roles.