# Networks Seminar

# Introduction and Overview

Networks arise in many scientific applications, from resource allocation/transportation problems, to electrical engineering networks, to the study of biochemical reactions systems. While the intended applications and analytic techniques vary significantly in these disciplines, the core goal remains the same: to extract as much information as possible about the dynamical behaviors of the resulting systems from the structure of the network itself.

# Organization and Contacts

The Networks Seminar is currently organized by Matthew D. Johnston, David F. Anderson, and Gheorghe Craciun. Contact information, including how to join the mailing list, is contained below:

James Brunner

**Email:** jdbrunner[at]math.wisc.edu

**Webpage:** www.math.wisc.edu/~jdbrunner

Jinsu Kim

**Email:** jskim[at]math.wisc.edu

**Webpage:** www.math.wisc.edu/~jskim

David F. Anderson

**Tel:** 608-263-4943

**Email:** anderson[at]math.wisc.edu

**Webpage:** www.math.wisc.edu/~anderson

Gheorghe Craciun

**Tel:** 608-265-3391

**Email:** craciun[at]math.wisc.edu

**Webpage:** www.math.wisc.edu/~craciun

**Join Mailing List:** join-math-networks-seminar[at]lists.wisc.edu

**Email Mailing List:** networksem[at]math.wisc.edu

# Spring 2015

The seminar will take place on **Wednesdays in 901 Van Vleck Hall at 2:25 PM**, unless otherwise noted.

## January 30, February 4, and February 11 (2:00 p.m. in Van Vleck 901), Gheorghe Craciun (UW-Madison)

**Title:** Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

**Abstract:** The global attractor conjecture says that toric dynamical systems (i.e., a class of polynomial dynamical systems on the positive orthant) have a globally attracting point within each positive linear invariant subspace -- or, equivalently, complex balanced mass-action systems have a globally attracting point within each positive stoichiometric compatibility class. We introduce toric differential inclusions, and we show that each positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the origin. In particular, we show that similar invariant regions prevent positive solutions of weakly reversible k-variable polynomial dynamical systems from approaching the origin. We use this result to prove the global attractor conjecture.

## Wednesday, March 11, Claudia Solis-Lemus (UW-Madison statistics)

**Title:** Statistical inference of phylogenetic networks

**Abstract:** Bacteria and other organisms do not follow the paradigm of vertical inheritance of genetic material. Human beings, for example, inherit their DNA from their parents only (vertical transfer), but bacteria can share DNA between different species (horizontal transfer). Therefore, their evolution cannot be modeled by a tree. To incorporate these organisms to the tree of life, we need methods to infer phylogenetic networks. In this talk, I will present a statistical method to infer phylogenetic networks from DNA sequences. I will discuss the challenges and results on assessing the identifiability of the model. Our techniques to learn phylogenetic networks will enable scientists to incorporate organisms to the tree of life in parts that are more net-like than tree-like, and thus, complete a broader picture of evolution.

## Wednesday, March 18, James Brunner (UW-Madison)

**Title:** Predator-prey systems: non-robust permanence

**Abstract:** The classic Lotka-Volterra predator-prey system can be thought of as a chemical reaction network which is persistent, but not k-variable persistent or permanent. A geometric argument for this inspires a small change that makes the system k-variable permanent. However, the resulting system is not robustly permanent. In fact, an infinitesimal perturbation will cause the system to lose even the property of persistence. Similar situations are very common in CRNT, arising whenever a reaction in a network does not involve one or more species. Determining permanence of such systems remains a challenge.

## Wednesday, April 15, Murad Banaji, University of Portsmouth

**Title:** Logarithmic norms, compound matrices, and applications to chemical reaction networks

**Abstract:** I'll discuss some background from analysis and exterior algebra and how it might be applied to chemical reaction networks or other models in biology. Starting with basic calculus, we can write down differential/integral inequalities bounding the growth rates of (generalised) lengths of curves, areas of surfaces, etc. under the action of a semiflow. Local information - encoded in Jacobian matrices - can then be used to derive global conclusions about the nature of limit-sets, and the possibility of oscillation or chaos.

# Past Seminars

# Fall 2014

## Wednesday, September 17, Matthew D. Johnston (UW-Madison)

**Title:** An Introduction to Chemical Reaction Network Theory

**Abstract:** There has been significant interest recently in the relationship between the kinetic of models of biochemical reaction networks and the underlying network structure. In this introductory talk, we will discuss the basic features of so-called Chemical Reaction Network Theory which has been an active area of research since the 1970s. In particular, graph-theoretic notions such as weak reversibility and network deficiency will be introduced. We will also discuss reasonable modeling choices---including deterministic and stochastic formulations---and summarize a few surprising results which hold for deterministically modeled mass action systems.

## Monday, September 22 (2:25 p.m. in Van Vleck B105) Casian Pantea (WVU)

**Title:** Injectivity and multistationarity in chemical reaction networks

**Abstract:** Much attention has been paid recently to bistability and switch-like behavior that might be resident in important biochemical reaction networks. It turns out that large classes of extremely complex networks cannot give rise to multistationarity, no matter what their reaction rates might be. In turn, absence of multistationarity in a reaction network is often a consequence of the corresponding vector field being injective. In this talk I will give an overview of both older and newer injectivity results for vector fields associated with a biochemical reaction network. As much as possible, the matrix-theoretic techniques behind these results will also be discussed.

## Wednesday, October 1 David F. Anderson (UW-Madison)

**Title:** Deficiency and stochastic models of biochemical reaction networks

**Abstract:** The deficiency of a reaction network is central to many results from chemical reaction network theory. In this talk, I will explain what the deficiency of a reaction network is, and how it can be used to shed light on the behavior of the associated mathematical models. I will try to discuss results for both deterministic and stochastic models.

## Wednesday, October 15, Daniele Cappelletti (University of Copenhagen)

**Title:** Elimination of intermediate species in biochemical reaction networks

**Abstract:** Biochemical reactions often proceed through the formation of transient intermediate species. These species are usually more unstable than the other species and degraded at a faster rate. Due to the complexity and intractability of many reaction networks, intermediate species are therefore often ignored in the models. In this talk I will formally introduce stochastic reaction networks and unveil a connection with the deterministic ones. Further, I will show an asymptotic result giving some condition on when it is possible to safely ignore intermediate species, both in the stochastic and deterministic frameworks. I will finally show some further issues on intermediate species I am currently working on.

## Wednesday, October 22; October 29; November 5, Gheorghe Craciun (UW-Madison)

**Title:** Persistence properties of mass action systems

**Abstract:** A positive trajectory of a dynamical system is called persistent if, in the long run, it does not approach the boundary of the positive orthant. In biological applications, the persistence property is critical in deciding if a species in an ecosystem will become extinct, an infection will die off, or a chemical species will be completely consumed by a reaction network. We describe some classes of dynamical systems for which all positive trajectories are persistent. We also describe connections to the Global Attractor Conjecture, which says that a large class of mass-action systems (called complex balanced or toric dynamical systems) have a global attractor within any invariant subspace.

## Wednesday, November 19 Matthew D. Johnston (UW-Madison)

**Title:** Recent Results in Chemical Reaction Network Theory

**Abstract:** In this talk, I will present some recent results in two related areas of Chemical Reaction Network Theory. (1) I will present a method called network translation which is often capable of determinining characteristics of the steady state sets of deterministically modeled mass action systems. The method associates the original reaction network to a related non-physical network which is more strongly connected than the original one. (2) I will investigate differences in long-term behavior between traditional deterministic models of chemical reaction systems and the more physically realistic stochastic models. I will present a classification of systems for which the deterministic model often predicts positivity, but for which an extinction event necessarily occurs when modeled stochastically.

## Friday, December 5 (11:00 a.m. in Van Vleck 901), Nikki Meshkat (NCSU)

**Title:** Finding identifiable functions of linear and nonlinear biological models

**Abstract:** Identifiability concerns finding which unknown parameters of a model can be quantified from given input-output data. Many linear ODE models, used primarily in Systems Biology, are unidentifiable, which means that parameters can take on an infinite number of values and yet yield the same input-output data. For a given unidentifiable model, the goal is then to find a set of identifiable functions, i.e. parameter combinations, and then attempt to find a scaling reparametrization so that the resulting model is identifiable. We examine this problem of finding a simple set of identifiable parameter combinations for both linear and nonlinear models. In particular, we use a graph-theoretic approach to find identifiable parameter combinations for linear compartment models and also demonstrate an algorithm using Grobner bases to find identifiable parameter combinations for nonlinear models. We demonstrate these results using our web implementation, COMBOS.