Abstract
My PhD is about formal methods for analysing biochemical reaction network models. My focus is on methods coping with parametric uncertainty, whose parameters aregiven as intervals or orders of magnitudes.The thesis splits in two parts: one deals with interval methods, the other is about tropical methods.In the first part of my thesis, I introduce a new algorithm based on interval arithmetics, constraint methods, and optimization, that allows to test homeostasis, defined as dependence of steady states on the parameters. This concept includes absolute concentration robustness that has been introduced elsewhere.I also show how to use the same kind of methods in order to test if biochemical network models have a single or multiple steady states.In the second part of my thesis, I present two novel contributions to model reduction of biochemical reaction networks whose parameters are given by their orders of magnitude.The first contribution concerns the concept of approximated conservation laws. A method for model reduction combining tropical geometry and singular singular perturbation results have been recently proposed by our team and others, but there are some cases when this method fails. One cause of failure is when the fast subsystem defined by the tropical method has conservation laws, that are not conserved by the full system. This case covers the "quasi-equilibrium" situation, well known in biochemistry. We prove that approximated conservation laws are slower than species involved in it and can be used as supplementary slow variables.By Gaussian elimination of some fast variables, we transform the system into a system without approximated conservation laws, that can be reduced using previously developed methods.We also provide algorithmic methods for finding approximated conservation laws (linear, monomial or polynomial) and for testing the hyperbolicity conditions needed for the validity of the model reduction techniques.Another direction is to generalise the full tropical equilibrations, previously introduced by our team and others, into partial tropical equilibrations. The concept of partial equilibrations allows to identify reduced models valid in regions of the phase and parameter space where there are no full equilibrations. Heuristically, the concept is justified by the fact that slow species of multiple time scales models don't need to be equilibrated in the model reduction process. I provide algorithmic methods for computing the polyhedral complex of partial equilibrations and discuss how these can be used for rescaling and reducing biochemical reaction models.