Advances in deterministic models of biochemical interaction networks

Anne Shiu

(Texas A&M University, USA)

"Absolute concentration robustness in networks with many conservation laws"

The concept of absolute concentration robustness (ACR) was introduced by Shinar and Feinberg in their investigations into how biochemical systems maintain their function despite changes in the environment. A reaction system exhibits ACR in some species if the positive steady-state value of that species does not depend on initial conditions. Mathematically, this means that the positive part of the variety of the steady-state ideal lies entirely in a hyperplane of the form x_i=c, for some c>0. Deciding whether a given reaction system -- or those arising from some reaction network -- exhibits ACR is difficult in general, but this talk describes how for many simple networks, assessing ACR is straightforward. Indeed, our criteria for ACR can be performed by simply inspecting a network or its standard embedding into Euclidean space. Our main results pertain to networks with many conservation laws, so that all reactions are parallel to one other. Such 'one-dimensional' networks include those networks having only one species. We also consider networks with only two reactions, and show that ACR is characterized by a well-known criterion of Shinar and Feinberg. Finally, up to some natural ACR-preserving operations -- relabeling species, lengthening a reaction, and so on -- only three families of networks with two reactions and two species have ACR.

Stefan Mueller

(University of Vienna, Austria)

"Monomial parametrizations of positive equilibria"

We consider positive steady states of chemical reaction networks with (generalized) mass-action kinetics that allow a monomial parametrization. The latter is often a prerequisite in applications where one studies phenomena such as multistationarity and absolute concentration robustness. In particular, we review results on complex-balanced equilibria (special equilibria given by binomial equations) and toric steady states (where all steady states are binomial). For example, a recent result states that a network with mass-action kinetics has toric steady states if it is dynamically equivalent to a network with generalized mass-action kinetics that has zero effective and kinetic-order deficiencies and hence complex-balanced (and no other) equilibria. Finally, we discuss steps towards a characterization of networks with monomial parametrizations.

Badal Joshi

(California State University San Marcos, USA)

"Preserving Robust Output despite highly variable reactant supplies"

A cell/biochemical network must produce a consistently robust, easily readable output when interacting with its environment. However, the internal conditions of the cell and the available supplies of reactants are highly variable. To overcome this, the biochemical network must have architecture which is capable of producing the same output despite variations in reactant supplies, a property we will refer to as output robustness. As a possible means of achieving a robust system output, Shinar and Feinberg suggested the property of ACR (absolute concentration robustness), which requires that all steady states be in a hyperplane parallel to a coordinate hyperplane. However, ACR is neither necessary nor sufficient for output robustness, a fact that can be noticed in simple biochemical systems. To develop a stronger connection with output robustness, we define dynamic ACR, a property related to dynamics, rather than only to equilibrium behavior, and one that ensures convergence to a robust value. We illustrate the definition, and certain natural sub-types of dynamic ACR, with a rich body of examples of reaction networks. Towards the end, we will give a brief description of certain minimal motifs of dynamic ACR networks.

Jiaxin Jin

(University of Wisconsin, Madison, USA)

"Uniqueness of weakly reversible and deficiency zero realization"

Weakly reversible, deficiency zero mass-action systems, being complex-balanced for any choice of rate constants, are remarkably stable. Here we show that if a dynamical system is generated by a weakly reversible network that has deficiency equal to zero, then this network must be unique. Moreover, we show that both weak reversibility and deficiency zero are necessary for uniqueness. We also describe an algorithm that can determine whether or not a system of differential equations can admit a weakly reversible, deficiency zero realization.