Sign vector conditions in Chemical Reaction Network Theory

Project: Research funding

Project Details

Abstract

Chemical reaction network theory (CRNT) establishes intriguing results about dynamical systems arising from networks of biochemical reactions with mass-action kinetics (MAK), in particular, about existence, uniqueness, and stability of steady states. However, the validity of the underlying mass-action law is limited to elementary reactions in homogeneous and dilute solutions. In intracellular environments, which are highly structured and characterized by macromolecular crowding, the reaction rate has to be modified.

In our previous work [MR12], we study chemical reaction networks with generalized mass-action kinetics (GMAK), where reaction rates are power-laws in the concentrations. In particular, the kinetic orders (which can be determined experimentally) can differ from the corresponding stoichiometric coefficients. A fundamental result of CRNT concerning existence and uniqueness of complex balancing equilibria can be extended in this framework, using sign vectors of the stoichiometric and kinetic-order subspaces.

A natural next step is to investigate possible generalizations of other results. For example, we want to characterize the stability of complex balancing equilibria, which is guaranteed in classical CRNT. In this and other problems, we will explore how far the kinetic orders can deviate from the stoichiometric coefficients. Our work is also motivated by the observation that results about chemical reaction networks with GMAK can be applied to dynamically equivalent networks with MAK for which classical CRNT remains silent. We want to characterize this situation in terms of network properties.

In the recent collaboration [MFR+16], we recognize our extension of Birch's theorem in [MR12] as the first partial multivariate generalization of Descartes' rule of signs. In fact, we provide sign conditions for the existence of (at most) one positive solution to a system of generalized polynomial equations. We will further investigate the implications of our results for real algebraic geometry and algebraic statistics. In [MFR+16], we characterize the injectivity of generalized polynomial maps in terms of sign vector conditions. In CRNT with GMAK, these results can be used to preclude as well as to guarantee multistationarity. A natural next step is to study the injectivity of monotone maps.

For achieving the theoretical goals, methods from dynamical systems, graph theory, polyhedral geometry, and oriented matroids will be combined in a novel way. Based on software available for oriented matroids, we will implement algorithms for the verification of sign vector conditions in computer algebra systems.

For an introduction to CRNT and an illustration of our previous results by a running example, see also [MR14].

References

[MFR+16] S. Müller, E. Feliu, G. Regensburger, C. Conradi, A. Shiu, and A. Dickenstein. Sign conditions for injectivity of generalized polynomial maps with applications to chemical reaction networks and real algebraic geometry. Foundations of Computational Mathematics, 16(1): 69–97, 2016.

[MR14] S. Müller and G. Regensburger. Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents. Proceedings of the 16th International Workshop on Computer Algebra in Scientific Computing (CASC 2014), Warsaw. Lecture Notes in Computer Science, 8660:302–323, 2014.

[MR12] S. Müller and G. Regensburger. Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces. SIAM J. Appl. Math. 72(6):1926–1947, 2012.
Short titleVorzeichenvektoren in der Theorie chemischer Reaktionsnetzwerke
StatusFinished
Effective start/end date1/01/1631/12/19

Keywords

  • chemical reaction network theory
  • generalized mass action kinetics
  • sign vectors
  • oriented matroids
  • generalized polynomial equations
  • Birch's theorem