Proportion Regulation in Globally Coupled Nonlinear Systems

As a model of proportion regulation in differentiation process of biological system, globally coupled activator-inhibitor systems are studied. Formation and destabilization of one and two cluster state are predicted analytically. Numerical simulations show that the proportion of units of clusters is chosen within a finite range and it is selected depend on the initial condition. 87.10.+e,87.22.-q,87.22.As Typeset using REVTEX e-mail address: [email protected] e-mail adderss: [email protected] 1 The regulation of proportion among different cell types in a tissue is a general and important aspect of biological development. It is well known that the proportion between the two different cell types is roughly constant irrespective of the slug size of cellular slime mold Dictyostelium discoideum (Dd) amoebae [1–4]. Initially the same type of aggregative cells, when dissociated, randomly mixed, and reaggregated, differentiate into two types cells (prespore and prestalk cells) without pattern formation. It is known now that cell differentiation starts independently on the cell position, and later cell sorting forms the twozoned prestalk-prespore pattern in slug of Dd [3–5]. Similar regulation mechanism can be observed in caste populations of social insects such as ants and bees [6,7]. In the division of work, proportion is regulated irrespective of the size of society nor the artificial partial extinction by the experimenter. No theoretical model exists to describe the proportion regulation. Any pattern formation model such as Turing type instability with diffusive coupling [8] is incompatible with the observation that the Dd cells start to differentiate independent to their positions. A large population of identical units interacting equally to the other units (globally coupled nonlinear system) is a good candidate to describe the phenomena. It is an idealized model of the cases, when the diffusion length of chemical factor, e.g. differentiation inducing factor (DIF) or pheromone, is large enough compared to the cell size, or when the individual units moves around to interact with others. Recently, globally coupled chaotic map [9,10] and globally coupled oscillators [11–16] are studied and interesting phenomena including clustering and their destabilization are observed. However, analysis of cluster state is difficult for these systems because the unit itself is complex enough. It is also doubtful that chaos or oscillation is playing essential roles in proportion regulation of biological system such as Dd. In this respect, a minimum model of clustering is preferable. Our model of globally coupled system is composed of N activator-inhibitor type units which have two variables u and v. The dynamics of each unit is modeled as