Leon Avery , Applied Mathematics, University of Waterloo
Models of social behavior in C. elegans larvae
Newly hatched larvae (L1s) of the nematode Caenorhabditis elegans can survive for more than two weeks without food. Alex Artyukhin has shown that when placed on a petri plate after two days of starvation, these L1s spontaneously come together into roughly circular aggregates. We neither know how they aggregate, nor why. Artyukhin proposes that the worms release a diffusible, unstable attractant into solution. This attractant diffuses away from the source and decomposes, forming a gradient. The worms are then proposed to move up the gradient -- this leads to the formation of aggregates.
This "Active Walker" model is described mathematically by a reaction-diffusion partial differential equation (PDE) governing the diffusion and decomposition of the attractant, and a series of stochastic different equations, one for each worm. Since the number of worms is large (typically a million in a single experiment), the worms may be approximated by a continuous density function, whose behavior is governed by an advection-diffusion PDE. Together the attractant PDE and density PDE form a nonlinear coupled system. Keller and Segel (1970) showed that this coupled PDE system becomes unstable above a threshold density, leading to spontaneous aggregation. Numerical solution of the PDEs will produce results that can be compared to experiments in animals, testing and refining the model.
An equally interesting question, but a more difficult one, is why the worms aggregate? That is, what purpose (if any) does aggregation serve, or why would such a behavior have evolved? We speculate that it improves the ability of the worms to survive starvation, perhaps by allowing them to pool information about the environment. This speculation is supported by the observation that the survival of C elegans L1 larvae is indeed improved when they aggregate. Furthermore, of four Caenorhabditis species tested, two show both density-dependent survival and L1 aggregation, and two show neither behavior. We hope to test the feasibility of these explanations in future modeling work.