3:00pm - 5:00pmPolyhedral geometry methods for biochemical reaction networks
Chair(s): Elisenda Feliu (University of Copenhagen, Denmark), Stefan Müller (University of Vienna)
This minisymposium focuses on geometric objects arising in the study of parametrized polynomial ODEs given by biochemical reaction networks. In particular, we consider recent work that employs techniques from convex, polyhedral, and tropical geometry in order to extract properties of interest from the ODE system and to relate them to the choice of parameter values.
Specific problems covered in the minisimposium include the analysis of forward-invariant regions of the ODE system, the determination of parameter regions for multistationarity or oscillations, the performance of model reduction close to metastable regimes, and the characterization of unique existence of equilibria using oriented matroids.
(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)
Endotactic Networks and Toric Differential Inclusions
Gheorghe Craciun, Abhishek Deshpande
University of Wisconsin (Madison)
An important dynamical property of biological interaction network models is persistence, which intuitively means that “no species goes extinct”. It has been conjectured that weakly reversible networks are persistent. The property of persistence is related to yet another conjecture called the Global Attractor Conjecture. Recently, Craciun has proposed a proof of the Global Attractor Conjecture. An important step in this proof is the embedding of weakly reversible dynamical systems into toric differential inclusions. We show that dynamical systems generated by the larger class of endotactic networks can be embedded into toric differential inclusions.
Approximating Convex Hulls of Curves by Polytopes
Nidhi Kaihnsa
MPI Leipzig
We study the convex hulls of trajectories of polynomial dynamical systems. Such trajec- tories also include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We approximate these convex hulls by a family of polytopes. We present numerical algorithms to identify the patches of the convex hull by classifying the facets of the polytope. An implementation based on the software Bensolve Tools is given. This is based an a joint work with Daniel Ciripoi, Andreas Lóhne and Bernd Sturmfels.
Multistationarity conditions in a network motif describing ERK activation
Carsten Conradi
HTW Berlin
ERK is an important signaling molecule that is activated by phosphorylation at two binding sites. In theory phosphorylation is either distributive or processive. It has been shown however that ERK phosphorylation is neither purely distributive nor purely processive but rather a mixture of both. While purely distributive processes are known to be multistationary, processive are not.
We study a network incorporating both mechanisms. By varying certain rate constants the contribution of the distributive mechanism can be controlled. As rate constants are hard to determine experimentally this network gives rise to a parametrized family of polynomials. In this context multistationarity refers to the existence of rate constants such that the polynomials have at least two positive solutions. Multistationarity is considered an important feature of this network and we want to understand the contribution of the distributive mechanism to the occurrence of multistationarity.
The corresponding variety admits a monomial parameterization and the family belongs to the class of systems described with Feliu, Mincheva and Wiuf. Thus multistationarity can be decided by studying the sign of the determinant of the Jacobian evaluated at this parameterization. We establish multistationarity and study whether multistationarity persists as the contribution from the distributive mechanism goes to zero.
Oscillations in a mixed phosphorylation mechanism
Maya Mincheva
Northern Illinois University
We will discuss the existence of oscillations in a phosphorylation mechanism where the phosphorylation is processive and the dephosphorylation is distributive. We show that in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface that consists of points of Hopf bifurcations. The emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system admits a Hopf bifurcation.
This is a joint work with C. Conradi and A. Shiu.
