10:00am - 12:00pm
Algebraic aspects of biochemical reaction networks
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)
Expected number of positive real solutions to systems of polynomial equations arising from reaction networks
Given a chemical reaction network that exhibits multistationarity, a natural question is how to determine the region in the parameter space where the network is multistationary or to divide the parameter region according to the number of positive steady states. We introduce a new approach based on the Kac-Rice formula and Monte-Carlo integration to approximate the multistationarity region in the parameter space. Furthermore, we apply our method to solve two related questions. First, we provide a measure to compare two points in the multistationarity region and decide what choice is more robust to small variations of the parameters. Second, we address the problem of finding a point in a given box in the parameter space for which the network is multistationary. We apply our approach to two relevant reaction networks, namely a simple model of a hybrid histidine-kinase and the 2-site sequential distributive phosphorylation network. Finally we compare our approach with the former existing methods of studying the multistationarity region. This is a joint work with Elisenda Feliu.
Absolute concentration robustness: an algebraic perspective
How do cells maintain homeostasis in fluctuating environments? Investigations into this question led Shinar and Feinberg to introduce in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the environment changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show in this talk that they are incomplete. Accordingly, we present a new method for deciding ACR, which uses computational algebra. We illustrate our results on several biochemical signaling networks.
On the Stability of the Steady States in the n-site Futile Cycle
The multiple or n-site futile cycle is a biological process that resides in the cell. Specifically, it is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. In its standard form it has 3n+3 variables (concentrations of species) and 6n parameters. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple futile cycle (n=1) there is only one steady state which is globally stable. For the dual futile cycle (n=2) the stability of the steady states has been determined in the following sense: There exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable.
We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.
The work is joint with Alan Rendall (Mainz) and Elisenda Feliu (Copenhagen).
The DSR graph and dynamical properties of reaction networks
A significant body of recent work in mathematical biology focuses on dynamical properties of biochemical reaction networks that are a function of the network topology alone (and are therefore independent of kinetics or parameter values). One way to represent network topology is via the DSR graph, introduced by Craciun and Banaji. The structure of the DSR graph may allow conclusions on dynamical properties of the network, including multistationarity, stability of steady states, and stability under delays. In this talk we survey some old and new results on the DSR graph, and discuss connections with other graphs stemming from reaction networks.