## Mini-symposium 9

## Algebraic methods for biochemical reaction networks

### Tuesday, June 17, 2014

### 16.00-19.00

### Chalmers Conference Centre, Room Euler

#### Organizers

Carsten Conradi (Max Planck Institute for Dynamics of Complex Technical Systems)

Elisenda Feliu (University of Copenhagen)

Carsten Wiuf (University of Copenhagen)

#### Summary

The mini-symposium will focus on mathematical analysis of
*reaction networks* arising in systems biology.
In recent years much progress has been made towards understanding the mathematical aspects of
reaction networks using techniques from *algebraic geometry* and *computational algebra*.
The progress bypasses the need for simulation and highlights the rich mathematical structure that
underlies reaction networks.
Specifically, the mini-symposium will focus on reaction networks as classes of nonlinear dynamical systems
described by systems of ordinary differential equations (ODEs).

#### Schedule

- 16:00-16:05

**Presentation of the mini-symposium**

- 16:05-16:25

**Analysis of open chemical reaction networks governed by mass-action kinetics**

Bayu Jayawardhana, (University of Groningen)*Abstract.*In this talk, we will discuss steady-state analysis of mass-action kinetics chemical reaction networks with proportional out-flows and constant in-flows. Such class of networks has been used extensively in the simulation and analysis of bio-chemical reaction networks. Using graph of complexes, we analyze the stability and steady-state properties of such networks. The results rely on a graph extension to the framework which we have developed for closed chemical reaction networks. Hide - 16:30-16:50

**Elimination of Intermediate Species in Reaction Networks**

Daniele Cappelletti (University of Copenhagen)*Abstract.*Biochemical reactions often proceed through the formation of intermediate species. These species are transient species, such as the substrate-enzyme complex appearing in Michaelis-Menten kinetics. Suitable reduced systems with no intermediate have been defined in the deterministic setting, and the relation between the number of non-degenerate positive steady states of the full and the reduced networks have already been studied previously. We focus on stochastically modelled reaction networks and provide a rigorous asymptotic result for the elimination of the intermediate species from the model. Our key assumption is that the rates of the intermediate consumption tend to infinity fast enough compared to the rates of their production. Linear algebra techniques allow us to have a parallel result on the convergence of solutions of reaction network in the deterministic case. It is also worth noting that the rates of the reduced model we obtain for the stochastic reduced system are the same as those obtained previously. Hide - 16:55-17:15

**Generalized mass action systems and positive solutions of polynomial systems with parameters**

Georg Regensburger (RICAM, Austria)*Abstract.*Chemical reaction network theory provides statements about uniqueness, existence, and stability of positive steady states of the related dynamical systems for all rate constants. The relevant conditions depend only on the network structure and the stoichiometric subspace. In terms of polynomial equations, they guarantee existence and uniqueness of positive solutions for all parameters. We discuss an extension of several statements to generalized mass action systems where reaction rates are allowed to be power-laws in the concentrations. In this setting, uniqueness and existence additionally depend on sign vectors of the stoichiometric and kinetic-order subspaces. This is joint work with Stefan MÃ¼ller. Hide - 17:15-17:30

**Break**

- 17:30-17:50

**Ruling out Hopf bifurcations in biochemical reaction networks**

Casian Pantea (West Virginia University)*Abstract.*We describe a new graphical approach to the question of whether dynamical systems modeling networks of interacting elements admit Hopf bifurcations. The techniques make use of the spectral properties of additive compound matrices: in particular, we show that a condition on the cycles of a labelled digraph (called the DSR[2] graph) rules out the possibility of nonreal eigenvalues of the Jacobian matrix passing through the imaginary axis. Hide - 17:55-18:15

**Complex concentration coupling in chemical reaction networks**

Zoran Nikoloski/Jost Neigenfind, Max-Planck-Institute of Molecular Plant Physiology*Abstract.*Invariant flux ratios, obtained within flux coupling analysis, as well as invariant complex ratios, derived within chemical reaction network theory, can characterize robust properties of a system at steady state like absolute concentration robustness (ACR). Basically, the detection of ACR can be reduced to the problem of coarsening partitions of the set of complexes, called complex concentration coupling, independently of the deficiency of the given reaction network. Herein we describe properties of reaction networks that make the coarsening of the corresponding partition tractable. Furthermore, we describe a simple algorithm for the generation of chemical reaction networks with needed properties. Hide - 18:20-18:40

**Graph-theoretic conditions for multistationarity for mass-conserved biochemical networks**

Maya Mincheva (Northern Illinois University)*Abstract.*Understanding the dynamics of interactions in complex biochemical reaction networks is an important problem in mathematical biology. Mathematical models of biochemical networks often lead to complicated dynamical systems with many unknown parameters. Graphs associated with biochemical networks can be used to predict the existence of multistationarity in a dynamic model without knowing the parameter values. We will present a general graph-theoretic condition for multistationarity applicable to mass-conserved biochemical networks. Hide