Periodic Oscillation in Gene Networks Modelled by Cyclic Feedback Systems

Periodic oscillations have been found in many different natural or physical systems, in particular biological organisms which have rhythmic phenomena at all levels with periods ranging from less than a second to years. From both theoretical and experiment viewpoints, it is a great challenging problem in biological science to model, analyze and further predict the periodic behaviors of living organisms. Thereby study on topics, such as, how to model, analyze, predict and even create the periodic rhythms is essential and fundamental. Generally, it is difficult to guarantee a system converging to a limit cycle or a sustained oscillation even for a simple-structured nonlinear system. Therefore, many important physiological factors such as time delays are simply ignored in order to reduce dimensionality and complexity of the systems [1, 4]. It is well known, however, that such factors may play important roles in dynamics of the biological systems. Recently, based on monotone dynamical systems, Mallet-Paret and Sell introduced successfully developed a general theory to show the existence of the omega-limit set, which opened the door to a general inquiry into not only the topological structure but also the sufficient conditions of the existence for a specific attractor, in particular periodic attractor. In this paper, we further extend the results in to more general networks and provide the sufficient conditions of limit cycles for gene regulatory systems according to both local and global analysis. Specifically, this paper aims to develop a new methodology to analyze and design biological oscillating networks with time delays, by using monotone cyclic feedback systems. We first give a local analysis of the oscillation network by local bifurcation theory. On the basis of the local result, we derive our main theorems, which ensure negative cyclic feedback systems to converge to periodic orbits globally. Furthermore, we extend our results to more general networks, which can have any types of interactions between nodes and are much easy to implement.