Chaos in a ring circuit

A ring-shaped logic circuit is proposed here as a robust design for a True Random Number Generator (TRNG). Most existing TRNGs rely on physical noise as a source of randomness, where the underlying idealized deterministic system is simply oscillatory. The design proposed here is based on chaotic dynamics and therefore intrinsically displays random behavior, even in the ideal noise-free situation. The paper presents several mathematical models for the circuit having different levels of detail. They take the form of differential equations using steep sigmoid terms for the transfer functions of logic gates. A large part of the analysis is concerned with the hard step-function limit, leading to a model known in mathematical biology as a Glass network. In this framework, an underlying discrete structure (a state space diagram) is used to describe the likely structure of the global attractor for this system. The latter takes the form of intertwined periodic paths, along which trajectories alternate unpredictably. It is also invariant under the action of the cyclic group. A combination of analytical results and numerical investigations confirms the occurrence of symmetric chaos in this system, which when implemented in (noisy) hardware, should therefore serve as a robust TRNG

Global stability of enzymatic chain of full reversible Michaelis-Menten reactions

International audienceWe consider a chain of metabolic reactions catalyzed by enzymes, of reversible Michaelis-Menten type with full dynamics, i.e. not reduced with any quasi- steady state approximations. We study the corresponding dynamical system and show its global stability if the equilibrium exists. If the system is open, the equilibrium may not exist. The main tool is monotone systems theory. Finally we study the implications of these results for the study of coupled genetic-metabolic systems

Les réseaux de régulation génétique : controle d'un oscillateur

National audienceno abstrac

Global stability of full open reversible Michaelis-Menten reactions

International audienceno abstrac

Modélisation, analyse et commande des réseaux biologiques

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

Mathematical study of the global dynamics of a concave gene expression model

International audienceWe describe in this paper the global dynamical behavior of a mathematical model of expression of polymerase in bacteria. This model is given by a differential system and algebraic equations. We use some tools from monotone systems theory with concavity of nonlinearities to obtain a global qualitative result: either the trivial equilibrium is globally stable, either there exists a unique positive equilibrium which is globally stable in the positive orthant. The same result holds for a class of qualitatively defined functions. Some generalizations of this result are given

Stability analysis and reduction of gene transcription models

International audienceThe aim of this paper is to analyze the dynamical behaviour of models of gene transcription in a cell, developed in Kremling (2007), which are a detailed representation of transcription process from DNA to RNA with action of polymerase. Using monotone system theory and compartmental systems theory, we study several versions of these models. We use our results of global stability to show that, under assumptions of time scales separation, it is possible to reduce the models to much smaller ones

Global Stability of Full Open Reversible Michaelis-Menten Reactions

International audienceWe consider the full closed and open Michaelis-Menten enzymatic reactions. We study the corresponding dynamical system and show its global stability if the equilibrium exists. If the system is open, the equilibrium may not exist. Then we consider an open chain of reversible metabolic reactions and also prove global stability. Our mathematical tools are monotone systems theory and compartmental systems theory

Analysis and reduction of transcription translation coupled models for gene expression

International audienc