250075 VO Mathematics of Reaction Networks (2025W)

Aims

Fundamental cellular functions including signaling, gene regulation, and metabolism involve numerous molecular species interacting via chemical reactions. More than one century of biochemistry and several decades of molecular biology have provided an unprecedented window into the complexity of such chemical reaction networks in living cells. Recent experimental techniques even allow real-time observations of complex dynamical behaviour such as hysteresis, oscillations, and stochastic fluctuations.

Mathematics has played a pivotal role in coping with the complexity of chemical reaction networks and is a cornerstone of current systems biology. In this lecture, we will consider two modeling frameworks in more detail.

Contents

Chemical networks: Many cellular systems can be modeled as networks of chemical reactions, often with mass-action kinetics (leading to ordinary differential equations with polynomial right-hand sides). Interestingly, for large classes of networks, the qualitative behaviour of the dynamical systems is independent of the system parameters.

In this lecture, we will prove a classical result that guarantees existence, uniqueness, and stability of positive equilibria independently of the rate constants (for networks with deficiency zero). Moreover, we will study extensions of the theory to systems with generalized mass-action kinetics.

Metabolic networks: As a particular cellular system, metabolism is modeled as a network of enzymatic reactions, often without exact knowledge of the kinetics. Since cellular organisms survive and reproduce in complex environments under permanent evolutionary pressure, metabolic pathways are assumed to be highly adapted, and optimality principles are used to study the organization of metabolism. Traditionally, the analysis of genome-scale metabolic models is based on stoichiometric data, leading to linear programs for fluxes (steady-state reaction rates).

In this lecture, we will also consider kinetic data and study optimal enzyme allocation, leading to nonlinear problems. Importantly, optimal solutions are (combinations of) elementary flux modes (elementary vectors of the flux cone), representing minimal metabolic pathways.

Methods

For the study of chemical and metabolic networks, we combine concepts and methods from graph theory, dynamical systems, polyhedral geometry, and oriented matroids (such as Laplacian matrices, Lyapunov functions, polyhedral cones, and elementary vectors).