Synchronization in coupled cells with activator-inhibitor pathways S. Rajesh, 1 Sudeshna Sinha, 2 and Somdata Sinha 1, * 1 Centre for Cellular and Molecular Biology, Uppal Road, Hyderabad 500007, India 2 The Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India Received 5 July 2006; revised manuscript received 16 October 2006; published 9 January 2007 The functional dynamics exhibited by cell collectives are fascinating examples of robust, synchronized, collective behavior in spatially extended biological systems. To investigate the roles of local cellular dynamics and interaction strength in the spatiotemporal dynamics of cell collectives of different sizes, we study a model system consisting of a ring of coupled cells incorporating a three-step biochemical pathway of regulated activator-inhibitor reactions. The isolated individual cells display very complex dynamics as a result of the nonlinear interactions common in cellular processes. On coupling the cells to nearest neighbors, through diffusion of the pathway end product, the ring of cells yields a host of interesting and unusual dynamical features such as, suppression of chaos, phase synchronization, traveling waves, and intermittency, for varying interaction strengths and system sizes. But robust complete synchronization can be induced in these coupled cells with a small degree of random coupling among them even where regular coupling yielded only intermit- tent synchronization. Our studies indicate that robustness in synchronized functional dynamics in tissues and cell populations in nature can be ensured by a few transient random connections among the cells. Such connections are being discovered only recently in real cellular systems. DOI: 10.1103/PhysRevE.75.011906 PACS numbers: 87.18.Hf, 47.54.-r, 05.45.-a I. INTRODUCTION A single cell, which is the basic building block of all living organisms, performs its functions through various sub- strates that are produced by the intracellular network of regu- lated biochemical reaction pathways. Even though end- product inhibition is the single most common motif of regulation in biochemical pathways, as it ensures homeosta- sis, a large number of biochemical pathways consist of mul- tiple regulatory loops through positive and negative feedback processes such as, enzyme activation-inhibition, gene induction-repression, etc. 1–3. The chemical kinetics of these feedback reactions and other intracellular processes in- volve high order of nonlinearity, and, therefore, these path- ways in the cells often show a variety of nonlinear phenom- ena such as self-sustained oscillations, birhythmicity and chaos 4 –6. In a population and in the multicellular state e.g., tis- sues, cells interact with each other directly or indirectly. Hence, the dynamics of an individual cell may be influenced by the interaction or coupling with other cells. Living sys- tems use such interactions to coordinate and control many biological functions 7–11. There is a diversity of coupling mechanisms that nature uses to enforce communication among cells in a cellular ensemble. In biological tissues, the arrangement and types of contacts complement their specific functions. Such intercellular signaling couples the biochemi- cal reaction pathways within each cell through diffusion of the products of these reactions. Such diffusive coupling oc- curs in metabolically coupled cells, which leads to robust synchrony among cells and spatial patterns in cellular ensembles 12–14. In reality, a small degree of randomness in spatial cou- pling can be expected to exist along with the strict nearest neighbor scenarios discussed above. Indeed, many systems of biological, technological, and physical significance are better described by randomizing some fraction of the regular links 15, as it allows information to be transferred at longer distances in lesser time. Recently a diversity of interactions have been shown to enforce communication among spatially non-neighboring cells. Recent experimental demonstrations of mechanisms of transient long distance interactions through substrates or cellular processes “nanotubes”, are shown to regulate multicellular functions 16. It is not clearly understood how the local functional dynamics of each cell, the features of intercellular signaling, and the sys- tem size interact to ensure that robustness and regulative capacity emerges at the tissue or population level. The most interesting feature of the coupled system is its global behavior under different dynamic conditions of its constituent cells. The two most important emergent behav- iors in coupled systems are—synchrony and spatiotemporal patterns 17. Synchronization is a phenomena that widely occurs in coupled nonlinear systems. Natural systems as di- verse as clocks, flashing fireflies, cardiac pacemakers and firing neurons exhibit a tendency to operate in synchrony. One can have synchronization of a periodic oscillator by ex- ternal force, or the well-known phenomena of phase locking and frequency entrainment of periodic oscillators. Interest- ingly, chaotic systems, though much more complex, also synchronize in varying degrees, such as i complete syn- chronization CS where the difference between signals vir- tually disappears 18; ii lag synchronization LS where the subsystems are synchronized with a delay or time shift 19; iii generalized synchronization GS where the instan- taneous states of subsystems are interrelated by a functional dependence 20; iv phase synchronization PS where the systems remain largely uncorrelated, but the mean time scales of their oscillations coincide or become commensurate 21, i.e., the phases of the systems are locked even though the amplitudes may be uncorrelated; and v intermittent *Electronic address: sinha@ccmb.res.in PHYSICAL REVIEW E 75, 011906 2007 1539-3755/2007/751/01190611 ©2007 The American Physical Society 011906-1