Conference Agenda

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Session Overview
Session
MS174, part 3: Algebraic aspects of biochemical reaction networks
Time:
Saturday, 13/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)

 

Reduction of the number of parameters

János Tóth
Budapest University of Technology and Economics

We start from a kinetic differential equation and transform it using an extended positive diagonal transformation in such a way that we prescribe the values of some of the reaction rate coefficients in the transformed system. The problems to be studied are as follows.

1. Is it possible to prescribe the value of a given set of coefficients? If yes, does the transformation obeying the prescriptions uniquely determine the values of the other coefficients? In both cases: what are the new coefficients of the transformed equation?

2. Which are all the possible subsets of new coefficients that can be independently prescribed? What is the largest number of coefficient set(s) that can be prescribed?

3. Suppose we prescribe the values of some coefficients: they should be one. In this case we usually say we only have the remaining coefficient combinations as ''independent'' coefficients. Why?

We present some statements and examples as answers to the above questions and also we show a program to carry out the corresponding transformations. Furthermore, we mention some connections to the general problem of transforming kinetic differential equations.

 

"Good children" and "bad children"

Nicola Vassena
Free University Berlin

Equilibrium bifurcations arise from sign changes of Jacobian determinants, as parameters are varied. Therefore we address here the Jacobian determinant for metabolic networks with general reaction kinetics. Our approach is based on the concept of child selections: each (mother) metabolite is mapped, injectively, to one of those (child) reactions which it drives as an input.

Our analysis distinguishes reaction network Jacobians with constant sign from the bifurcation case, where that sign depends on specific reaction rates. In particular, we distinguish "good child" selections, which do not affect the sign, from more interesting / demanding / troublesome / mischievous "bad children", which gang up towards sign changes, instability, and bifurcations.

 

Tikhonov-Fenichel parameter values for chemical reaction networks

Sebastian Walcher
RWTH Aachen

The chemical reaction networks under consideration here are described by polynomial ordinary differential equations depending on positive parameters, with the positive orthant as a positively invariant set. In order to determine parameter regions where singular perturbation reduction (in the sense of Tikhonov and Fenichel) is possible, the notion of Tikhonov-Fenichel parameter values (TFPV) was introduced some time ago in Alexandra Goeke's dissertation and subsequent publications. A TFPV is characterized by the property that small perturbations give rise to a singular perturbation reduction. It is known that the TFPV of a given system form a semi-algebraic subset, and the defining equations may be determined algorithmically by standard elimination theory.

After reviewing the above notions and results, we discuss three types of questions arising for TFPV:

(i) Singular perturbation reduction versus "classical" quasi-steady state reduction.

(ii) The unreasonable simplicity of TFPV for CRN, and the unreasonable feasibility of their computation.

(iii) Nested TFPV for multiscale systems.

The talk reports on recent joint work with Elisenda Feliu, Niclas Kruff, Christian Lax and Carsten Wiuf.

 

Parameter geography

Jeremy Gunawardena
Department of Systems Biology, Harvard Medical School

I will discuss numerical observations of the parameteric region in which a two-site, post-translational modification system exhibits bi-stationarity. Aside from general features of connectedness and near convexity, we find a substantial difference in region volume between Michaelis-Menten and realistic enzymatic assumptions, which suggests that bi-stationarity may be rare under biological conditions. We uncover a previously unsuspected parameteric relationship underlying this phenomenon. We also find that boundary parameter points move back and forth between mono-stationarity and bi-stationarity (“blinking”), as conserved quantities increase, despite apparent monotonic increase in region volume. These results rely on combining the linear framework, which allows algebraic reduction of the steady state, with numerical algebraic geometry using Bertini, which permits sampling of approximately 10^9 points in parameter space.