Topology-based Geometric Modelling for Biological Cellular Processes

In this paper, we present a framework to simulate biological cellular processes involving a strong topological and geometric structuration. Our framework is based on topology-based geometric modelling. The edge labels of a graph sub-class, the n-dimensional generalized maps, model the neighbouring relations between biological compartments. Thus, membranes become topological objects of the n-dimensional generalized maps representation and can be handled just as compartments. The evolutions of the underlying topology during the simulation are expressed by using graph transformation rules. These rules are conditioned by geometric properties and molecular concentrations bound to the vertices of the graph. A simplified example, inspired by the phenomenon of gap junctions, illustrates how topological and geometric parameters are involved in the simulation of biological cellular processes controlled by the concentrations of molecules in the compartments.