### Focus Group Meeting: Mathematical and Computational Models in Biological Networks: Abstracts and Lecture Materials

The Deficiency Zero Theorem, Global Stability, and Stationary Distributions
David Anderson, Department of Mathematics, University of Wisconsin-Madison

The dynamics of biochemical reaction systems can be modeled either deterministically or stochastically. Typically, the equations governing the dynamics of these models are quite complex. Further, there is oftentimes little knowledge about the exact values of the different system parameters, and, worse still, these system parameter values may vary from cell to cell. However, the network structure of a given system induces the corresponding equations (up to parameter values) governing its dynamics. I will show in this talk how this fact may be exploited to infer qualitative properties of large classes of biochemical systems and, most importantly, to learn which properties are independent of the details of the system parameters. I will give results for both stochastically and deterministically modeled systems. For deterministically modeled systems I will focus on persistence of trajectories, which in some important cases is sufficient to guarantee global asymptotic stability of equilibria. For stochastically modeled systems I will focus on the existence, and form, of stationary distributions.

Multistationarity in biochemical reaction networks with mass action kinetics
Carsten Conradi, Max-Planck-Institut Dynamik komplexer technischer Systeme

The standard model of biochemical reaction networks with mass action kinetics is a first order ODE in the concentration vector x of the involved species. The right hand side of this vector ODE is given by the product of the so-called stoichiometric matrix N and a vector of reaction rates. If the matrix N does not have full row rank, the evolution of states obeys a number of conservation laws and thus is algebraically confined by a system Zx(t)=Zx(0) of affine equations. Multistationarity is then equivalent to a positive answer to the following question: Is there a vector k of (positive) rate constants such that two different positive steady state solutions a and b exist that satisfy the same algebraic constraints. i.e. Za=Zb?

This question has been analyzed in chemical engineering in the 1980s and 1990s and a partial answer is given by Feinberg's Chemical Reaction Network Theory (CRNT). Only recently we proposed an approach that was motivated by an in-depth study of CRNT. Based on a result from CRNT, the algebraic constraints are disregarded at first. The solution set (a,b,k) is parameterized via solutions of a new system of equations obtained by transforming the equations defining stationarity (of a and b). This first step is based on the idea of generators for pointed polyhedral cones, in particular of generators for the intersection of the kernel of N and the nonnegative orthant (of the n-dimensional Euclidean space). A second step then allows the formulation of necessary and sufficient conditions for multistationarity.

Here two points are highlighted: (1) For many reaction networks of biological relevance the transformed equations can be reduced to a system of linear equations, however the precise connection to the network structure is still open. As a proof-of-concept, an analysis of the ERK-cascade is discussed. (2) The transformed equations contain many degrees of freedom. There are various ways to formulate and to interpret the arising conditions. Each way is emphasizing different aspects of multistationarity and thereby offering new insights.

Multiple equilibria in biochemical reaction networks
Gheorghe Craciun, Department of Mathematics, University of Wisconsin-Madison

We describe the Jacobian criterion and the Species-Reactions Graph approach to discriminating between biochemical reaction networks that can admit multiple equilibria and others that cannot.

Persistence in biochemical reaction networks
Patrick De Leenheer, Department of Mathematics, University of Florida

This talk considers the notion of persistence in BRN's. Roughly speaking, we say that a BRN is persistent if all chemical species remain present asymptotically, when they are present initially. We will provide graphical conditions which, combined with simple algebraic conditions, guarantee persistence of a given BRN. We will also consider a few examples of fairly complex models taken from the recent molecular biology literature to illustrate the theory. Time permitting, we will sketch how persistence can play a role in establishing convergence of (almost all) solutions of BRN's to equilibria, based on a strong monotonicity approach. Finally, I will present an open problem to the audience.

Studying switches and oscillations using I/O monotone dynamical systems
German Enciso, Systems Biology Department, Harvard Medical School

Determining the long-term behavior of large biochemical models has proved to be a remarkably difficult problem. Yet these models exhibit several characteristics that might make them amenable to study under the right perspective. One possible approach (first suggested by Sontag and Angeli) is their decomposition in terms of monotone systems, which can be thought of as systems with exclusively positive feedback. In this talk I discuss some general properties of monotone dynamical systems with inputs and outputs, and how they can be used to model dynamical behaviors such as global attractivity to an equilibrium, switches and oscillations.

Phenotypes in mathematics: multistability, long transients, or "fuzzy math"
Adam Halasz, GRASP Laboratory, University of Pennsylvania

We investigate the relationship between biological phenotype and dynamical equilibria in the context of noisy dynamical systems representing gene regulatory networks. The lac operon is a well known example of the paradigm of the on/off gene. The two states of the operon correspond dynamical equilibria and to biologically distinct metabolic states. We will discuss two other examples, where the biological distinction is well established, the case of competence in B.subtilis and the still open question of persistence in E.coli. In the case of competence, it has been recently proposed that this alternative phenotype does not correspond to a dynamical steady state of the genetic regulatory network, rather, it represents a long excursion, triggered by a subtle, controlled stochastic molecular mechanism. A similar mechanism seems to apply to persistence in E.coli. The general question that arises is that it seems that the biological notion of phenotype, which is traditionally identified with steady states of the gene regulatory network, may include examples where a phenotype corresponds to subsets of state space which do not contain equilibria. It is not clear how a mathematical notion of phenotype can be constructed in order to include such examples along with the traditional [stable] equilibrium.

Monotone Dynamics I
Morris Hirsch, Department of Mathematics, UC, Berkeley

I will review the basic results of monotone dynamical systems: most trajectories go to equilibrium, stable equilibria exist, and attractors are not chaotic.

Monotone Dynamics II
Morris Hirsch, Department of Mathematics, UC, Berkeley

Recent work with David Angeli and Eduardo Sontag showing that certain biologically relevant systems that are not monotonic can be expressed as feed-forward nets of monotone control systems. This permits the results of my first talk to rule out chaotic attractors. Some other applications of monotonicity will also be discussed.

The smallest multistationary mass-preserving chemical reaction network
Anne Shiu, Department of Mathematics, University of California, Berkeley

Chemical reaction network theory can provide sufficient conditions for the existence of bistability, and on the other hand can rule out the possibility of multiple steady states. Understanding small networks is important because the existence of multiple steady states in a subnetwork of a biochemical model can sometimes be lifted to establish multistationarity in the larger network. We establish the smallest reversible, mass-preserving network that admits bistability and determine the semi-algebraic set of parameters for which more than one steady state exists.