Advances in deterministic models of biochemical interaction networks

Balazs Boros

(University of Vienna, Austria)

"Oscillations in deficiency-one mass-action systems"

Whereas the positive equilibrium of a mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present examples with centers or multiple limit cycles.

Beatriz Pascual Escudero

(University of Copenhagen, Denmark)

"Detecting concentration robustness in Reaction Networks"

A biological system has absolute concentration robustness (ACR) for some species if the concentration of this species is identical at any possible equilibrium that the network admits. In particular, this concentration must be independent of the initial conditions. While some classes of networks with ACR have been described, as well as some techniques to check ACR for a given network, finding networks with this property is a difficult task in general. The connection of this global version of robustness with other local notions leads to a practical test on necessary conditions for ACR, by means of algebraic-geometric techniques. This test allows to analyze networks in the search for the possibility of ACR or local ACR for some values of the reaction rates, or discard it for all values. This is based on joint work with E. Feliu.

Alan Rendall

(Johannes Gutenberg University, Mainz, Germany)

"Global convergence to steady states in a model for the in-host dynamics of hepatitis C"

We consider a model for the concentration of hepatitis C virus particles in a host which includes a simple description of the virus replication. This model has two virus-free steady states and two corresponding basic reproduction numbers. It has at most three positive steady states. Although it is not known whether there can be more than one steady state we prove that for certain ranges of the parameters every solution converges to a steady state. This is accomplished by applying a method of Li and Muldowney which uses the Lozinskii measure corresponding to a certain norm. An estimate for this Lozinskii measure of the second additive compound of the Jacobian matrix is the key condition which is required. The central idea of the method is to exclude all other kinds of asymptotic behaviour, such as convergence to a periodic solution.

Murad Banaji

(Middlesex University London, UK)

"Building Reaction Networks with Prescribed Properties"

In general, the problem of identifying reaction networks with some prescribed dynamical property is challenging. As an example of a dynamical property, let's consider stable oscillation. The question then becomes: does a given network allow stable oscillation for some choice of parameters (e.g., rate constants if the reaction network has mass action kinetics)? As networks grow in size, this question becomes harder and harder to check numerically. One way of making progress is via theorems which tell us how, given an oscillatory network, we can build a larger oscillatory network with more species or reactions. I'll give an overview of such theorems, focussing mainly on oscillation.