Operator Algebras for Dynamic Topology Models of Cytoskeleton

Abstract

Coarse-scale dynamical models of cytoskeleton can be constructed using stochastic versions of interconnected mechanical elements represented using labelled and/or parameterized graphs, with some of the parameters representing an embedding into space, provided that the dynamics of such graphs can also be represented. Since those dynamics can be stochastic and/or deterministic, and can affect both the parameters and the graph connectivity i.e. cytoskeleton topology, and can incorporate processes that happen discretely and/or continuously in time, representing such dynamics mathematically goes significantly beyond differential equations or conventional stochastic chemical reaction networks. In previous work we showed how “dynamical graph grammar” syntactic expression of such models can be systematically mapped to operator algebras as the underlying mathematical model, and gave examples involving dynamic cortical microtubules in plant (Arabidopsis) cytoskeleton. We showed that the topology-changing sector of such models can be mapped into a larger operator algebra closed under multiplication. This larger space is built out of fermion-like creation and annihilation operators as used in field theory in statistical physics. However, for designing and analyzing simulation algorithms, and for analyzing models expressed in this framework, the resulting expanded operator algebra is actually too large for human analytic work. We show that a much tighter result is possible, enabling the products and commutators of topology-changing graph rewrite rules to be re-expressed as weighted sums of other such operators. We discuss the practical benefits to cytoskeletal modeling that can be gained as a result of being able to compute directly such products and commutators.