## Geometry Seminar

**Date:** November 4, 2019

**Time:** 3:00PM - 4:00PM

**Location:** BLOC 628

**Speaker:** Nida Obatake, Texas A&M University

**Title:** *Polyhedral methods for chemical reaction networks*

**Abstract:** Chemical Reaction Network theory is an area of mathematics that analyzes the behaviors of chemical processes. A major problem asks about the stability of steady states of these networks. Rubinstein et al. (2016) showed that the ERK network exhibits multiple steady states, bistability, and undergoes periodic oscillations for some choice of rate constants and total species concentrations. The ERK network reduces to the processive dual-site phosphorylation network when certain reactions are omitted, and this network is known to have a unique, stable steady state (Conradi and Shiu, 2015). To investigate how the dynamics change as reactions are removed from the ERK network, we analyze subnetworks of the ERK network. In particular, we prove that oscillations persist even after we greatly simplify the model by making all reactions irreversible and removing intermediates. To prove this, we introduce the Newton-polytope Method: an algorithmic procedure that uses techniques from polyhedral geometry to construct a positive point where a pair of polynomials achieve certain desired sign conditions. We then use our algorithm to apply an algebraic criterion for Hopf bifurcations that relies on analyzing polynomials (Yang, 2002). Additionally, we investigate the maximum number of steady states of a system by defining a notion of a mixed volume of a chemical reaction network. In general, the mixed volume is an upper bound on the number of complex-number steady states, but we show that this bound is tight for ERK networks. Joint work with Anne Shiu, Xiaoxian Tang and Angelica Torres.