Conservation laws and open systems on higher-dimensional networks

We discuss a framework for defining physical open systems on higher-dimensional complexes. We start with the formalization of the dynamics of open electrical circuits and the Kirchhoff behavior of the underlying open graph or 1-complex. It is discussed how the graph can be closed to an ordinary graph, and how this defines a Dirac structure on the extended graph. Then it is shown how this formalism can be extended to arbitrary k-complexes, which is illustrated by a discrete formulation of heat transfer on a two-dimensional spatial domain.

Model reduction of biochemical reactions networks by tropical analysis methods

We discuss a method of approximate model reduction for networks of biochemical reactions. This method can be applied to networks with polynomial or rational reaction rates and whose parameters are given by their orders of magnitude. In order to obtain reduced models we solve the problem of tropical equilibration that is a system of equations in max-plus algebra. In the case of networks with nonlinear fast cycles we have to solve the problem of tropical equilibration at least twice, once for the initial system and a second time for an extended system obtained by adding to the initial system the differential equations satisfied by the conservation laws of the fast subsystem. The two steps can be reiterated until the fast subsystem has no conservation laws different from the ones of the full model. Our method can be used for formal model reduction in computational systems biology

A geometric method for model reduction of biochemical networks with polynomial rate functions

Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors