Detecting multistationarity in mass-action networks.

• Applicants: Professor Dr.-Ing. Carsten Conradi; Professor Dr. Thomas Kahle
• Subject Area: Mathematics
• Term: from 2015 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 284057449

Biochemical reaction networks with mass-action kinetics (mass-action networks) are widely used in systems biology. Every mass-action network defines a system of ordinary differential equations (ODEs) with polynomial right hand sides. Parameter values in these ODEs can only be specified within large intervals (due to high measurement uncertainty and difficult experimental conditions). Every mass-action network therefore defines a parametrized family of ODEs, where determining steady states numerically can be challenging already. Hence there is a growing interest of mathematicians in the underlying structures that determine dynamical or stationary behavior of a chemical reaction network.In this proposal we focus on structural conditions for the existence of multiple positive steady states (multistationarity), which is a desired property in many biological applications (e.g. modeling of cell division or programmed cell death). Deciding whether or not multistationarity is possible is mathematically equivalent to deciding whether or not a polynomial system with unknown coefficients has at least two positive solutions and hence a challenging question. Despite this being a question of real algebraic origin, most of the work in the area has been carried out in chemical engineering and mathematical biology. In this proposal we want to join our complementary experience in mathematical biology and algebraic geometry to make progress on understanding multistationarity in mass-action networks.Specific goals are (i) An understanding of when the steady state ideal is binomial. (ii) A description of boundary steady states and in particular their dependence on parameters and the network. (iii) An implementation of an automatic method for the detection of multistationarity..(iv) A better understanding of the geometry of parameter regions associated to multistationarity for selected classes of polynomial models from systems biology. Completion of these goals will be beneficial to mathematicians and biologists alike. This will open the field to scientists working in (real) algebraic geometry, providing many new and challenging problems and it will yield valuable tools for scientists working in systems biology.

• DFG Programme: Research Grants