Multistationarity and Hopf-bifurcations in families of ODE models of N-site phosphorylation


Phosphorylation is a biochemical mechanism underlying important biological processes like intracellular signalling, regulation and decision making. Biochemically a protein is altered by adding (or removing) a phosphate group at a designated binding site. Proteins often have more than one binding site, hence the term multisite phosphorylation, and N-site phosphorylation is used as a shorthand for multisite phosphorylation at N>1 binding sites. The process is well studied experimentally and through mathematical modelling. Mathematically one can distinguish three families of models parameterized by N. the number of phosphorylation sites. If, as is often the case, spatial effects are neglected, these models come in the form of systems of Ordinary Differential Equations (ODEs) with polynomial right hand side.

Measuring individual concentration variables is prohibitively expensive and technically challenging. Hence parameter values can only be given with large error bounds, if at all. In studying networks of these families therefore the following mathematical question arises naturally: are there (positive) parameter values, such that the ODEs admit a solution with property xyz - for some or all (positive) initial conditions. Here we will focus on the following three properties (i) uniqueness of steady states, (ii) existence of multiple steady states and (iii) existence of Hopf bifurcations. These are mathematically interesting and challenging and, at the same time, of interest in systems biology and medicine: multiple steady states are considered as a means of intracellular information processing and limit cycles, arising e.g. via a Hopf bifurcation are considered as clocks synchronizing intracellular processes. In essence, we propose to establish (i), (ii) or (iii) in three families of ODEs with polynomial right hand side and unknown but positive parameters and initial conditions.


01.01.2023 - 31.12.2025