250095 VO Mathematical models of biochemical networks (2017S)

Aims

We study mathematical models of biochemical networks, in particular, of chemical reaction networks and metabolic networks.

From a reductionist point of view, cellular (sub-)systems can be seen as networks of elementary chemical reactions. Under the assumption of mass-action kinetics, statements about the system dynamics can be made without knowing the system parameters.

At a higher level of cellular organization, metabolism is modeled as a network of enzymatic reactions, often without exact knowledge of the kinetics. Linear programming is used to optimize the metabolism of microorganisms, for example, for the production of drugs or biofuels.

Contents

Dynamical systems arising from chemical reaction networks with mass-action kinetics are the subject of chemical reaction network theory (CRNT). Most notably, this theory provides statements about existence, uniqueness, and stability of positive steady states independently of the rate constants. Under certain conditions, one can characterize the qualitative dynamics of a system, based on network information only.

Recently, the applicability of CRNT has been extended to systems with generalized mass-action kinetics (power-law kinetics). Here, one strives for results that are not only independent of rate constants, but also robust with respect to kinetic orders, as determined by sign vector conditions.

Cellular organisms survive and reproduce in complex environments under permanent evolutionary pressure. As a consequence, metabolic pathways are often assumed to be highly adapted, and optimality principles are used to study the large-scale organization of metabolism. Traditionally, the analysis is based on stoichiometric information. We extend the concept of elementary flux modes (of the flux cone) to elementary vectors (of s-cones and arbitrary cones).

Recently, optimality principles have been used to study metabolic networks with kinetic information (leading to nonlinear programs). Here, one is interested in enzyme distributions that maximize a designated output flux, given a limited total enzymatic capacity. We show that, surprisingly, optimal solutions (of kinetic models) correspond to elementary flux modes (of stoichiometric models).

Methods

For the study of chemical reaction networks and metabolic networks, we combine methods from dynamical systems, graph theory, polyhedral geometry, and oriented matroids.