Stability of synchronization in a multi-cellular system

Networks of biochemical reactions regulated by positiveand negative-feedback processes underlie functional dynamics in single cells. Synchronization of dynamics in the constituent cells is a hallmark of collective behavior in multi-cellular biological systems. Stability of the synchronized state is required for robust functioning of the multi-cell system in the face of noise and perturbation. Yet, the ability to respond to signals and change functional dynamics are also important features during development, disease, and evolution in living systems. In this paper, using a coupled multi-cell system model, we investigate the role of system size, coupling strength and its topology on the synchronization of the collective dynamics and its stability. Even though different coupling topologies lead to synchronization of collective dynamics, diffusive coupling through the end product of the pathway does not confer stability to the synchronized state. The results are discussed with a view to their prevalence in biological systems. Copyright c © EPLA, 2010 Introduction. – The emergence of collective behavior is an important step in the evolution of multi-cellular life from single cells. In single cells, coordinated cellular functions are performed by networks of biochemical reactions, regulated by positiveand negative-feedback processes. These cells communicate through exchange of chemicals and signals, and cooperate to work together in the multi-cellular environment (e.g., tissues and organisms). Understanding how the synchronized collective behavior of multi-cellular systems evolves depending on local intracellular properties and global features (such as, environment, system size, and interaction strength/topology), and the robustness of such behavior in response to noise in view of adaptability of biological systems are some of the most important questions in biology today [1]. Synchronization is a well-known collective phenomenon in various multi-component physical and biological systems [2]. The exchange of information (coupling) among the components can be global (all-to-all) or local (nearest neighbors). Depending on various intrinsic and extrinsic factors, the coupled dynamics can exhibit different types of synchronization such as complete synchronization, phase and lag synchronization, intermittent phase (a)Present address: Department of Mathematics and Statistics, York University Toronto, Canada. (b)E-mail: [email protected] synchronization, etc. Phase synchronization is the weakest form of synchronization and is typically observed when coupling is weak. As coupling strengths increase, more ordered stages of synchronized dynamics are obtained, culminating in the strongest form of synchronization —the complete synchronization. When phase entrainment is lost and regained intermittently in a coupled system, it is called intermittent phase synchronization [2]. An essential prerequisite for understanding the synchronization of a coupled system is to know the bounds on the coupling strengths which ensure the stability of the synchronized state. The stability of the synchronized state is hard to study mathematically for phase synchronization whereas it is relatively easy for complete synchronization. Many attempts have been made (see the references quoted in [3]) to obtain such stability conditions for physical systems —typically for small network sizes or for specific types of couplings. In order to ensure stability of the synchronized state for a general coupling scheme, analytical bounds were obtained on coupling strengths [3,4] following the notion of the master stability function [5]. Physical systems were the primary focus in these studies. For biological systems, such as tissues and organized cell assemblies, it is not clear if the stability of the synchronized state is useful or detrimental to its functions. On the one hand, synchronization is necessary for the proper functioning of the system in the face of both internal and