On the generic dimension and nondegeneracy of steady states of reaction networks

Abstract: In the context of (bio)chemical reaction networks, the dynamics of the concentrations of the chemical species over time are often modelled by a system of parameter-dependent ordinary differential equations, which are typically polynomial or described by rational functions. The dynamics of a reaction network are also often constrained into invariant linear subspaces called stoichiometric compatibility classes. The study of the steady states of the system translates then into the study of the positive solutions to a parametric polynomial system. The set of positive solutions lives inside a complex algebraic variety and hence tools from algebraic geometry naturally find application in this field.

In this talk I will discuss recent results addressing the following questions: What is the expected dimension of the algebraic variety of steady states? Can the dimension be "wrong" in an open set of parameters? Under what conditions is the intersection of the algebraic variety of steady states with the stoichiometric compatibility classes generically finite?

These are fundamental questions to understand the algebraic nature of the objects under study, and have been previously brought up in the study of reaction networks, for example in the context of weakly reversible reaction networks. Additionally, knowing the answer to these questions in advance is often necessary to be able to apply the mathematical machinery coming from complex algebraic geometry.

This talk is based on join work in progress with Oskar Henriksson and Beatriz Pascual-Escudero.

algebraic geometrydynamical systemsprobability

Audience: researchers in the topic

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Seminar on the Mathematics of Reaction Networks

Series comments: Subscription link: list.ku.dk/postorius/lists/morn.list.ku.dk/

This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs twice a month, typically on the 2nd and 4th Thursday of the month, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom!

The organizers.

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