Abstract: Modeling the dynamics of a chemical reaction systems is an interesting topic where the application of convex and toric geometry can be of great advantage. As an illustrative example a model for the electrocatalytic oxidation of formic acid will be presented. It will be shown how a realistic system involving different rate laws like mass action kinetics and Butler-Volmer kinetics can be described by a set of purely polynomial ordinary differential equations. The property of the set of positive stationary solutions of such systems as an intersection set of a convex cone and the variety of a toric ideal will be explained in details. It allows to reduce this high dimensional set to a curve and leads to a new parameterization of the steady state. The stability analysis can then be executed with the usual theorems stemming from the theory of dynamical systems. Of special interest are bistability and oscillatory instabilities. The consideration of this problem under the light of toric geometry allows to make justified conclusions about the occurrence of such dynamical phenomena and about the parameter range in which they are to be expected.
Abstract: Several different approaches to modeling of gene regulatory networks have been used in recent years, in particular ODE-based models and Boolean networks. In this talk we propose a modeling approach that can be thought of interpolating between these two frameworks. It is based on the concept of a dynamic network represented by polynomial functions on finite, but arbitrarily large state sets. Applications contain the reverse-engineering of gene regulatory networks from data, as well as the reverse-engineering of dynamics. The polynomial framework makes it possible to bring to bear the powerful machinery of algorithmic polynomial algebra.:
Abstract: The behavior of the concentrations of chemical species in a system of chemical reactions is given by a polynomial differential system. We are interested in the stationary solutions and thus the positive solutions of a system of sparse polynomial equations. The system comes with a rich structure given by two graphs, a directed graph describing the chemical reactions and a bipartite graph for the stoichiometric coefficients. The polynomials depend on many unknown parameter. We show the existence of special solutions of type complex balancing with the Cayley trick. The existence of several positive solutions is shown by a variation of the Viro method. This is based on the approach to intersect a deformed toric variety with a convex polyhedral cone. The generators of this cone are known as extreme fluxes or extreme currents. The so-called stoichoimetric generators are central for multistationarity. Our result gives the parameter region where multistationarity is expected to occur.
Abstract: Model discrimination is concerned with the ability to make distinctions between different reaction networks (i.e. directed graphs) based on available experimental data and qualitative properties of the reaction networks and their associated dynamical models. Whereas it is a straithforward procedure to translate a network structure into a set of ordinary differential equations (ODEs) with yet unknown parameters (i.e. a dynamical model), it is not a-priori clear, which network structure will yield a dynamical model that is compatatible with the available experimental data. We suggest to use qualitative properties, such as the number of steady states, to discriminate between different network structures. The solution set of a system of sparse polynomial equations is identical to the staedy states of a system of ODEs derived from a (bio)chemical reaction network. Since the unknowns represent chemcial concentrations we are only interested in real nonnegative solutions. Our contribution show how results from [1] can be applied to model discrimination.
Two simplified, yet realistic network structures describing the transition between two phases in the cell of Saccharomyces cerevisiae (budding yeast) are examined: Under the assumed conditions the qualitative behavior of the system is primarily characterized by the existence of two staedy states, one corresponding to the G1-phase and one to S-pahse and Mitasis. For model discrimination it is sufficient to show that the ODE system corresponding to one of those network structures will never exhibit two steady states, regardless of the parameter values, whereas for some parameter values the system associated with the second network structure indeed admits two steady states.
Thus two sets of sparse polynomial equations are investigated by using certain triangulations of the Newton polytopes associated with those sets. Each triangulation defines a subnetwork of the original reaction network and thus a set of polynomial equations involving less terms. Based on results from [1] the existence of two nonnegative solutions for these sets of polynomial equations is explored. Extension to the complete set of polynomial equations and thus to the complete network is then possible by again using results from [1].
[1] Karin Gatermann and Matthias Wolfrum. Bernstein's 2. theorem and Viro's method for sparse polynomial systems in chemistry.
Abstract: The set of systems of differential equations that are in normal form with respect to a particular linear part has the structure of a module of equivariants, and it is best described by giving a Stanley decomposition of that module. Groeber basis methods(implementable in a computer algebra system) are used to determine the Stanley decomposition of the ring of invariants, that arise in normal forms for systems with nilpotent linear part consisting of repeated 2x2 Jordan blocks. An efficient algorithm developed by Murdock is then used to produce a Stanley decomposition of the module of the equivariants from the Stanley decomposition of the ring of invariants.
Abstract: Consider a deuteron colliding with another deuteron ignoring charge and spin. In the case where after the collision two deuterons are returned, there are only two possible reactions. Either nothing happened or a particle was exchanged. Generalizing this simple problem from scattering theory results in an excursion into Polya's theory of counting and the theory of double cosets. Until the advent of computer algebra, the theory of double cosets has been restricted to a few elegant but computationally impossible theorems. Impossible in the sense that in principle the calculation can be done but it will take ten thousand years. Today, using Computer Algebra much can be calculated quickly. Using Macsyma and Maple in the special case of Young group generated double cosets, we will see just how valuable Computer Algebra can be. Some surprising and stimulating patterns emerge after a just few computer algebra experiments. The results presented last year will be extended and expanded. Some proofs will be sketched.