Modeling Spatial Effects in Early Carcinogenesis : Stochastic Versus Deterministic Reaction-Diffusion Systems

¹ Department of Statistics, Rice University, 6100 Main Street, MS138, Houston, TX 77005, USA
² Systems Engineering Group, Silesian University of Technology, 44-100 Gliwice, Poland

^⋆ Corresponding author. E-mail: kimmel@rice.edu

Abstract

We consider the early carcinogenesis model originally proposed as a deterministic reaction-diffusion system. The model has been conceived to explore the spatial effects stemming from growth regulation of pre-cancerous cells by diffusing growth factor molecules. The model exhibited Turing instability producing transient spatial spikes in cell density, which might be considered a model counterpart of emerging foci of malignant cells. However, the process of diffusion of growth factor molecules is by its nature a stochastic random walk. An interesting question emerges to what extent the dynamics of the deterministic diffusion model approximates the stochastic process generated by the model. We address this question using simulations with a new software tool called sbioPN (spatial biological Petri Nets). The conclusion is that whereas single-realization dynamics of the stochastic process is very different from the behavior of the reaction diffusion system, it is becoming more similar when averaged over a large number of realizations. The degree of similarity depends on model parameters. Interestingly, despite the differences, typical realizations of the stochastic process include spikes of cell density, which however are spread more uniformly and are less dependent of initial conditions than those produced by the reaction-diffusion system.