Computational algebraic geometry and biochemical reaction networks

A DICKENSTEIN

College Station

Conferencia; Frontiers Lectures; 2015

Texas A & M University

In recent years, techniques from computationalalgebraic geometry have been successfully used to address mathematicalchallenges in systems biology. (Bio)chemical reaction networks define systemsof ordinary differential equations with (in general, unknown) parameters. Undermass-action kinetics, these equations depend polynomially on the concentrationsof the chemical species. Biologically-relevant steady states correspond thus tothe positive real solutions of a structured system of polynomial equations.I will give a gentle introduction to the subject, based on joint work with C.Conradi, E. Feliu, S. Mueller, M. Perez Millan, G. Regensburger, and A. Shiu.One of the major questions is to preclude or to guarantee multiple positivesteady states.  I will explain this motivation and I will presentnecessary and sufficient conditions in terms of sign vectors for theinjectivity of families of polynomials maps with arbitrary real exponentsdefined on the positive orthant. I will also describe the structural propertiesof MESSI biological systems, which describe all the popular mechanisms whichpresent modifications of type enzyme-substrate or swap with intermediates.