Streaming instability of slime mold amoebae: An analytical model

Abstract

waves of the signaling molecule cAMP with cAMP-directed cell movement causes the breakup of a uniform cell layer into branching patterns of cell streams. Recent numerical and experimental investigations emphasize the pivotal role of the cell-density dependence of the chemical wave speed for the occurrence of the streaming instability. A simple, analytically tractable, model of Dictyostelium aggregation is developed to test this idea. The interaction of cAMP waves with cAMP-directed cell movement is studied in the form of coupled dynamics of wave front geometries and cell density. Comparing the resulting explicit instability criterion and dispersion relation for cell streaming with the previous findings of model simulations and numerical stability analyses, a unifying interpretation of the streaming instability as a cAMP wave-driven chemotactic instability is proposed. @S1063-651X~97!14408-7# PACS number~s!: 87.22.2q I. INTRODUCTION Spontaneous symmetry breaking under nonequilibrium conditions is characteristic of a wide variety of physical and chemical systems; it is found in areas as diverse as fluid flow, nonlinear optics, and oscillations and waves in chemical reactions @1#. On the other hand, conclusive evidence for such processes to underly spatial patterning in biological systems is relatively rare ~cf. @2#!. The amoeboid microorganism Dictyostelium discoideum has long been considered a paradigm for the study of biological pattern formation, and recently a mechanism of self-organized patterning akin to those in inanimate systems has been implicated in a morphological transition in its life cycle. When switching from unicellular to a multicellular mode of existence, cell aggregates emerge from an initially uniform layer of single cells, forming a pattern of dense cell streams which coalesce into aggregation centers @Fig. 1~a!#. A range of mathematical models based on experimentally established single-cell properties has been employed to investigate the mechanism of cell streaming @3‐8#. Numerical simulations of these models demonstrate that aggregation via cell streaming is the result of an interaction of reaction-diffusion waves of an intercel