Modelisation dynamique de systemes genetiques de regulation : I. L'induction de l'operon lactose d'Escherichia coli : elaboration d'un modele

French. Dans ce premier article, on élabore un modèle mathématique pour décrire le mécanisme de l’induction de l’opéron lactose d’Escherichia coli. Les différentes interactions moléculaires sont successivement représentées par des équations différentielles qui traduisent l’évolution dans le temps : des probabilités des états possibles de la région de contrôle de l’opéron (opérateur et promoteur), de concentrations des enzymes codées par le système (perméase et (β-galactosidase), et des concentrations des substrats et produits de ces enzymes (lactose, inducteur, glucose et galactose intracellulaires). L’ensemble constitue un système de dix équations différentielles du premier ordre, présentant des non-linéarités et des arguments retardés. La cohérence de l’ensemble se justifie par des considérations sur les hiérarchies observées entre les nombres de molécules des diverses espèces moléculaires, de un gène à plusieurs millions de molécules de sucre, et entre les vitesses absolues des réactions d’association et de dissociation entre molécules. Les valeurs numériques des paramètres du modèle sont évaluées pour une bactérie de type sauvage. La plupart des paramètres peuvent être obtenus d’après des expériences, réalisées in vitro, ou in vivo sur des parties du système, mais certains paramètres inconnus doivent être estimés en utilisant le modèle. Dans ce modèle, les paramètres ont une interprétation biologique immédiate, mais l’imprécision de la détermination numérique de certains, et la complexité du modèle, empêchent de pouvoir énoncer des propriétés qualitatives générales. L’analyse de ces propriétés structurelles, de leur dépendance par rapport aux valeurs des paramètres, fait l’objet de l’article suivant, où l’on montre, notamment par des méthodes numériques, comment on peut analyser le comportement de souches mutantes, prévoir celui de génotypes inconnus, et proposer, à l’aide du modèle, de nouvelles expériences.

English. This series of papers is devoted to the building of dynamical models describing the expression of polygenic traits, when the molecular mechanisms are known from earlier biochemical and genetical analysis. The aims of these models are to provide a synthetic account of a huge body of analytical works, to make possible the simulation of further experiments, and to develop methods appropriate to the description of complex characters. This first paper is concerned with the elaboration of a dynamical system modeling the induction of the lactose operon of Escherichia coli. The different molecular mechanisms involved in the functioning of the operon are reviewed and are given a mathematical translation that takes account of the main biochemical and genetical facts, and of the achievements of previous theoretical works. The subsystems modeled is this way are the interactions between regulatory molecules (repressor, catabolite activator protein, inducers and anti-inducers, and cyclic AMP) and the control region of the operon (operator and promoter), and the biological activities of the proteins encoded by the structural genes (permease and (β-galactosidase). The biosynthesis of these proteins, which is not under the control of this operon, is modeled in a non-specific way, although it is taken advantage of the known short life-time of the messenger RNAs. Dynamical aspects of the catabolic repression by extracellular glucose are not included in the present analysis, since there are no available data that would allow to derive kinetic equations for this phenomenon. Deriving a single mathematical model that represents the whole biological system, from the juxtaposition of the preceding sub-models, is justified by some hierarchical properties. They involve the numbers of molecules in a cell (one gene, a few regulatory molecules, thousands of enzymes, millions to billions of substrates and products), and the absolute velocities of different reactions. The full model is a system of ten first-order differential equations. Five independent equations describe the states of the control region, they are linear but the coefficients depend upon the level of intracellular inducer. Two linear equations yield the rates of synthesis of the proteins, they involve delayed arguments since the arising of new active enzymes depends on the initiating of transcription a few minutes before. Two non-linear equations give the rates of change in the concentrations of intracellular lactose and inducer, they are based on the kinetic equations of the enzymes. Finally, one equation gives the rate at which glucose and galactose are produced, it is the output of the system. The parameters introduced in the model are estimated. Most of them can be derived from in vitro measurements (kinetic coefficients of association and dissociation between operator, repressor, and inducer ; kinetic parameters of the enzymes ; half life of enzymes), or from in vivo experiments dealing with sub-systems only (delays between transcription initiation and arising of enzymatic activity ; kinetic coefficients of permeation ; amplification coefficient of proteins biosynthesis). Other ones, that involve the interaction of the catabolite activator protein with the promoter, must be estimated because this interaction has not yet been quantitatively studied. The model has the advantage that its parameters have a clear meaning, each one may be related either to some biophysical property of a gene or of its product, or to some environmental characteristics. However, as it is non linear, its general properties are expected to depend on the numerical values of the parameters, which are not always known with good accuracy. The mathematical analysis of the structural properties of the model, and of their possible dependence on the parameters, will be shown in the next paper. Also, the comparative study of the wild type operon and of various mutant strains, by numerical methods, will show how the model may be used to simulate new genotypes, to predict some responses, or to estimate some parameters that are difficult to get from a direct experiment.