Conference Agenda

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Session Overview
Session
MS174, part 1: Algebraic aspects of biochemical reaction networks
Time:
Thursday, 11/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-105
53 seats, 70m^2

Presentations
10:00am - 12:00pm

Algebraic aspects of biochemical reaction networks

Chair(s): Alicia Dickenstein (Universidad de Buenos Aires), Georg Regensburger (Johannes Kepler University Linz)

ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Network models and polynomial positivity

Murad Banaji
Middlesex University, London

A number of problems in chemical reaction network theory - and in the study of other networks with model structure - can be restated as questions about the positivity of polynomials, or more generally the emptiness or otherwise of semialgebraic sets. In particular, positivity claims can be used to deduce the absence of particular bifurcations in networks with certain structural features. Sometimes positivity can be determined trivially, but it is not uncommon for polynomials to arise which are positive but in nontrivial ways, namely they do not belong to the smallest "natural" cone of polynomials positive on the set in question. Some problems and examples will be surveyed, and a few results using techniques from real algebraic geometry and exterior algebra will be outlined.

 

Some approaches to understand the parameter region of multistationarity

Elisenda Feliu
University of Copenhagen

In the context of chemical reaction networks, the dynamics of the concentrations in time are modelled by a system of parameter-dependent ordinary differential equations, which typically admit invariant linear subspaces, called stoichiometric compatibility classes. Multistationarity refers to the existence of two positive equilibrium points in some stoichiometric compatibility class. Numerous approaches exist to address the qualitative question of whether a network exhibits multistationarity for at least one choice of parameter values. However, tools and strategies to determine 'when' this is the case, that is, to determine for which parameter values the network is multistationary, have only recently emerged.

In this talk I will focus on conditions on the reaction rate constants that guarantee, or preclude, multistastionarity. I will discuss a result, joint with de Wolff, Kaihnsa, Sturmfels and Yürük, based on the use of Sums of Nonnegative Circuits (SONC). Our benchmark example is the n-site phosphorylation system.

 

On the bijectivity of families of exponential maps

Stefan Müller
University of Vienna

In the setting of 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 previous work, we have shown that injectivity can be characterized in terms of sign vectors of the stoichiometric and kinetic-order subspaces, that is, of the coefficient and exponent subspaces given by the family of maps. The negation of the sign-vector condition is equivalent to the existence of multiple complex-balanced equilibria (in some compatibility class and for some rate constant). In this work, we characterize the existence of a unique complex-balanced equilibrium, that is, the bijectivity of the family of exponential maps. As it turns out, the conditions for bijectivity do not only involve sign vectors, but also the exponent subspace itself. Further, we provide sufficient conditions involving only sign vectors or the Newton polytope. In terms of generalized mass-action systems, we provide an extension of the classical deficiency zero theorem.

(Joint work with Josef Hofbauer and Georg Regensburger)

 

An algebraic approach to detecting bistability in chemical reaction networks

Angélica Torres
University of Copenhagen

In recent years, algebraic parameterizations of varieties have been used to study parameter regions where a Chemical Reaction Network has multistationarity. In this work we combine these algebraic parameterizations, the Hurwitz criterion for stability and structural reduction techniques for chemical reaction networks to additionally explore the existence of bistability. The procedure can be used to find parameter regions where bistability arises. In this talk I will present our approach, how to detect bistability in a special case and some examples from cell signaling where our procedure was successfully applied.