Randomly distributed Dictyostelium discoideum cells form cooperative territories by signaling to each other with cAMP. Cells initiate the process by sending out pulsatile signals, which propagate as waves. With time, circular and spiral patterns form. We show that by adding spatial and temporal noise to the levels of an important regulator of external cAMP levels, the cAMP phosphodiesterase inhibitor, we can explain the natural progression of the system from randomly firing cells to circular waves whose symmetries break to form double-and single-or multi-armed spirals. When phosphodiesterase inhibitor is increased with time, mimicking experimental data, the wavelength of the spirals shortens, and a proportion of them evolve into pairs of connected spirals. We compare these results to recent experiments, finding that the temporal and spatial correspondence between experiment and model is very close.Perhaps the most striking single feature of the Dictyostelium discoideum life cycle is the ability of the amoebae to selforganize and produce well-spaced territories that will later differentiate into highly structured fruiting bodies (1, 2). Self-organization begins when a few scattered cells spontaneously secrete a single pulse of cAMP (3)(4)(5). Cells in the surround respond by secreting more cAMP (2, 6) and begin to move toward the cAMP source (7,8). cAMP diffuses to the next layer of cells, inducing a similar response, while the remaining cAMP is degraded by an external phosphodiesterase (PDE) (9) regulated by its inhibitor (PDI) (10-12); both proteins are secreted by the cells. The net result is a cAMP wave spreading outward from the excited cells.Cell signaling produces characteristic bands of light and dark rings and spirals as territories begin to form (7,(13)(14)(15). It is known that the light bands correspond approximately to high cAMP concentrations (16). An example of the evolution of these patterns is shown in Fig. 1.Although the evolution of spirals from preexisting spirals or broken wave segments has been modeled by Tyson and coworkers (18) and others (19), very little is known about how they emerge from biologically plausible initial conditions, why there are target patterns in some areas of a field of excitable cells and spirals in others, why some laboratory conditions appear to favor one over the other, and why for a particular strain and circumstance one pattern tends to dominate (20).These are important questions because they bear on early events governing the organization of the slime mold fruiting body. In addition, circular and spiral waves are generic to excitable systems, and thus an understanding of their origins in slime molds has wider applicability-for example, understanding spiral and planar waves in cardiac muscle (21,22), the Belousov-Zhabotinskii reaction (23), Ca2+ waves in Medaka eggs (24), and mitochondrial activity in Xenopus oocytes (25).The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "adver...
