Modelling, analysis and control of biological networks
Modélisation, analyse et commande des réseaux biologiques
Abstract
The purpose of this thesis is the modeling, reduction, analysis and control of biological
systems. Modeling of biological networks is done by differential equations; the systems
are typically nonlinear, of large dimensions, with different time scales, and complex to
analyze. First, using techniques of monotone and compartmental systems, we study the
global stability of the equilibrium of Michaelis-Menten enzymatic model without any
approximation, when the system is closed or opened; we also study the general case of a
chain of enzymatic reactions. Biological networks are generally composed of two parts
in interaction (genetic and metabolic), we therefore investigate different types of models
coupling metabolic reactions chains with a genetic system; we reduce the full system
based on the difference in time scales (Tikhonov theorem). In the second part, we apply
the 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 of
the gene of polymerase. This model is not monotone, but based on the parameter values
provided by biologists, we have reduced it to a simple and monotone model. The study
of the reduced system shows that the full system can have either a single equilibrium
point at the origin which is globally stable or there is another one stable strictly positive
equilibrium and the origin is locally unstable. The alternative between these two cases
depends on the total amount of the concentration of ribosomes. We then study a general
model of the genetic machinery, taking the model studied previously for the polymerase
and coupling it with a model for the synthesis of ribosomes. We finally apply different
types of qualitative controls on models of small nonlinear gene networks to stabilize for
example an unstable equilibrium point or to generate a limit cycle instead of a stable
equilibrium.
Le but de cette thèse est la modélisation, la réduction, l’analyse et la commande de
systèmes biologiques.
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