Leon Avery | Applied Math, 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 difusion 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. Equilibria of this system can be analyzed by setting the time derivative to zero. At equilibrium, the PDEs are uncoupled and linear. Under certain assumptions, in the presence of a given attractant gradient, the density has a unique stable equilibrium -- a Boltzmann distribution. The coupled system shows a density threshold for the formation of aggregates. Below a certain mean density, the uniform distribution of worms is stable. Above the threshold, aggregates form spontaneously. This matches the actual observed behavior of the worms.
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 at high density. 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.