Modelling, analysis and control of biological networks

Description

The purpose of this thesis is the modeling, reduction, analysis and control of biologicalsystems. Modeling of biological networks is done by differential equations; the systemsare typically nonlinear, of large dimensions, with different time scales, and complex toanalyze. First, using techniques of monotone and compartmental systems, we study theglobal stability of the equilibrium of Michaelis-Menten enzymatic model without anyapproximation, when the system is closed or opened; we also study the general case of achain of enzymatic reactions. Biological networks are generally composed of two partsin interaction (genetic and metabolic), we therefore investigate different types of modelscoupling metabolic reactions chains with a genetic system; we reduce the full systembased on the difference in time scales (Tikhonov theorem). In the second part, we applythe same techniques of monotone systems to study a general model of gene expression.Then we consider a model of a loop where the polymerase allows the transcription ofthe gene of polymerase. This model is not monotone, but based on the parameter valuesprovided by biologists, we have reduced it to a simple and monotone model. The studyof the reduced system shows that the full system can have either a single equilibriumpoint at the origin which is globally stable or there is another one stable strictly positiveequilibrium and the origin is locally unstable. The alternative between these two casesdepends on the total amount of the concentration of ribosomes. We then study a generalmodel of the genetic machinery, taking the model studied previously for the polymeraseand coupling it with a model for the synthesis of ribosomes. We finally apply differenttypes of qualitative controls on models of small nonlinear gene networks to stabilize forexample an unstable equilibrium point or to generate a limit cycle instead of a stableequilibrium.

Abstract (French)

Le but de cette thèse est la modélisation, la réduction, l'analyse et la commande desystèmes biologiques.

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