Department of Mathematics, UCI

Biochemical reaction networks provide a paradigm for many dynamical systems in biology. The paradigm can be generalized to describe "variable-structure systems" in which objects larger than molecules (such as cells) also change in number and in their relationships over time. In the course of building mathematical and software tools for understanding such networks and dynamical systems, we have identified some interesting applied mathematical problems whose reformulation and solution would be very useful in current computational biology. For example, we can identify partial differential equations whose solution would be especially instructive for enzyme kinetics. These problems arise at the level of small reaction networks, multimolecular complexes, and the development of multicellular tissues. Developmental examples include modeling the shoot meristem of a plant.

A common mathematical framework for models at these different spatial scales can be given in terms of "dynamical grammars". In a dynamical grammar, an input/output syntax for an elementary chemical or biological processes is mapped to an operator algebra expression for the generator of the temporal dynamics associated with that process. Many processes act simultaneously (in parallel) if their generators are summed. Contingent spatial relationships are expressed in terms of dynamical graph grammars, whose formulation could perhaps be improved by use of ideas from topology and differential geometry. By solving such problems, we may hope to construct a useful modeling language of sufficient generality to describe multiscale, variable-structure dynamical systems that arise naturally in biology.