Topological object types for morphodynamic modeling languages

Abstract

We survey useful ingredients for a new class of mathematical process-modeling languages aimed at spatial and developmental biology. Existing modeling languages for computational systems biology do not fully address the problems of spatial modeling that arise in morphodynamics (the local dynamics of form) and its applications to biological development. We seek to extend the operator algebra semantics approach from our previous "Dynamical Grammars" modeling language, whose most spatial object type is the labelled graph, to encompass more flexible topological objects. Taking clues from current developments in 3D meshing and from topological modeling for biology, illustrated by a plant tissue example, we seek language support for the approximation of lowdimensional CW complexes (which are nontrivial topological spaces, with cardinality of the continuum) and dynamic fields thereon, by finite labelled abstract complexes. Some of the proposed types would be computationally demanding, without further restriction. Restrictions and control of these approximations can be specified by use of "metricated" types. Minimally, such approximations should permit the accurate simulation of spatial diffusion processes.