Conference Agenda

This minisymposium focuses on geometric objects arising in the study of parametrized polynomial ODEs given by biochemical reaction networks. In particular, we consider recent work that employs techniques from convex, polyhedral, and tropical geometry in order to extract properties of interest from the ODE system and to relate them to the choice of parameter values.

Specific problems covered in the minisimposium include the analysis of forward-invariant regions of the ODE system, the determination of parameter regions for multistationarity or oscillations, the performance of model reduction close to metastable regimes, and the characterization of unique existence of equilibria using oriented matroids.

Tropical prevarieties of polynomial vector fields arising from chemical reaction networks have been found useful in the analysis of such systems. For fixed parameter values it has been shown that even for systems having 30 dimensions or more tropical prevarieties can be computed. As for biochemical reaction networks there is typically high parameter uncertainty (and the qualitative dependency of the system on parameters is also of high interest) parametric computations of tropical prevarieties are very desireable. Based on experiments with the PtCut software we compare the outcome (also in terms of needed computational ressources) of simple grid sampling strategies, for which the curse of dimensionality fully applies, against computations in a polyhedral setting including parameters. The presented results are joint work with Christoph Lüders

A quite recent approach to assist in solving ODE systems (with polynomial vector fields) lies in methods of tropical geometry. Tropical geometry transforms polynomials into piece-wise linear functions and still preserves some structure of the original polynomial (like the number of roots). The polynomial is transformed into a set of polyhedra and multiple of such sets can be intersected to find common roots. Thus tropical geometry problems are combinatorial problems.

We have developed the "PtCut" program to compute the tropical prevariety resp. tropical equilibrium of a polynomial system. Details of use and implementation of PtCut are presented. Large models can cause a lot of polyhedra to be created and calculating their intersection can be very slow. Several methods of remedy are shown.

Since ODE systems often arise in biology, we wrote a free SBML-parser and used PtCut to compute tropical solutions of the curated models listed in the BioModels database. Statistics about the number of solutions, their dimension, their number of connected components and run-time are presented.

For generalized mass-action systems, uniqueness and existence of complex-balanced equilibria (in every compatibility class and for all rate constants) are equivalent to injectivity and surjectivity of a certain family of exponential maps. In this talk, we discuss when the existence of a unique solution is robust with respect to small perturbations of the exponents. In particular, we give a characterization in terms of sign vectors of the stoichiometric and kinetic-order subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. This characterization allows to formulate a robust deficiency zero theorem for generalized mass-action systems. As a corollary, we show that the classical deficiency theorem for mass-action kinetics by Horn, Jackson, and Feinberg is robust with respect to small perturbations of the kinetic orders (from the stoichiometric coefficients).


(Joint work with Josef Hofbauer and Stefan Müller)

In 2011, Deng, Feinberg, Jones, and Nachman investigated the number of positive steady states of weakly reversible mass-action systems. They proposed a proof of the existence of a positive steady state in each positive stoichiometric class. Furthermore, they claimed that there can only be finitely many positive steady states in each positive stoichiometric class. Recently, I provided a complete and clearer proof of the existence part. Moreover, together with Craciun and Yu, I constructed examples with infinitely many positive steady states within a positive stoichiometric class, thereby disproving the finiteness part. In this talk, I will sketch our method that produces such mass-action systems.