Gillespie-Lindenmayer systems for stochastic simulation of morphogenesis

Abstract

Lindenmayer systems (L-systems) provide a useful framework for modelling the development of multicellular structures and organisms. The parametric extension of L-systems allows for incorporating molecular-level processes into the models. Until now, the dynamics of these processes has been expressed using differential equations, implying continuously valued concentrations of the substances involved. This assumption is not satisfied, however, when the numbers of molecules are small. A further extension that accounts for the stochastic effects arising in this case is thus needed. We integrate L-systems and the Gillespie’s Stochastic Simulation Algorithm to simulate stochastic processes in fixed and developing linear structures. We illustrate the resulting formalism with stochastic implementations of diffusion-decay, reaction-diffusion and auxin-transport-driven morphogenetic processes. Our method and software can be used to simulate molecular and higher-level spatially explicit stochastic processes in static and developing structures, and study their behaviour in the presence of stochastic perturbations.