Francine Diener

Simplified Models for the Mammalian Circadian Clock

Numerous biological mechanisms are synchronized by the circadian rhythm in species so diverse as mushrooms, drosophiles or mammals. Because of its ubiquity and of its implication in numerous cellular functions, we believe important to be able to extract the main “coarse-grain” mechanisms common to a majority of organisms. In this article we consider both differential equation models and discrete models and we deliberately choose to push the simplicity of the model as far as possible, focusing only on a few biological behaviours of interest. The hope is to get the essential abstract causalities that govern these behaviours.

Optimal phenotypic plasticity in a stochastic environment minimises the cost/benefit ratio

This paper addresses the question of optimal phenotypic plasticity as a response to environmental fluctuations while optimising the cost/benefit ratio, where the cost is energetic expense of plasticity, and benefit is fitness. The dispersion matrix Σ of the genes’ response (H=ln|Σ|) is used: (i) in a numerical model as a metric of the phenotypic variance reduction in the course of fitness optimisation, then (ii) in an analytical model, in order to optimise parameters under the constraint of limited energy availability. Results lead to speculate that such optimised organisms should maximise their exergy and thus the direct/indirect work they exert on the habitat. It is shown that the optimal cost/benefit ratio belongs to an interval in which differences between individuals should not substantially modify their fitness. Consequently, even in the case of an ideal population, close to the optimal plasticity, a certain level of genetic diversity should be long conserved, and a part, still to be determined, of intra-populations genetic diversity probably stem from environment fluctuations. Species confronted to monotonous factors should be less plastic than vicariant species experiencing heterogeneous environments. Analogies with the MaxEnt algorithm of Jaynes (1957a,b) are discussed, leading to the conjecture that this method may be applied even in case of multivariate but non multinormal distributions of the responses.

Jumps of solutions of the equations εx″ = f ( t,x, .- x -)

One of the main difficulties in the study of the equations

The steady-state phase distribution of the motor switch complex model of Halobacterium salinarum.

\varepsilon \ddot x = f(t,x,\dot x)

Ducks and rivers: three existence results

for small

DERIVATIVES: A SCIENTIFIC REVOLUTION OF THE SEVENTIES

\varepsilon

A Pedestrian Proof of the Hopf-Bifurcation Theorem

is created by the existence of parts of solutions, called “jumps” for which the velocity is large. We give here precise definitions of such jumps, their extremities, and their thickness, and we show that, for most equations of the type considered, it is easy to compute, up to an infinitesimal, the position of jumps, extremities and thickness for each solution. Our methods use both nonstandard analysis and a geometrical approach consisting of, among other things, observing solutions in some convenient planes, (plans d’observabilite) of the phase space.

Analyse non standard

Steady-state analysis is performed on the kinetic model for the switch complex of the flagellar motor of Halobacterium salinarum (Nutsch et al.). The existence and uniqueness of a positive steady-state of the system is established and it is demonstrated why the steady-state is centered around the competent phase, a state of the motor in which it is able to respond to light stimuli. It is also demonstrated why the steady-state shifts to the refractory phase when the steady-state value of the response regulator CheYP increases. This work is one aspect of modeling in systems biology wherein the mathematical properties of a model are established.

Nonstandard analysis in practice

The beginnings of nonstandard analysis in Prance are strongly related to the study of singular perturbations of the Van der Pol equation [110] and its ducks [29, 12]. This equation is an interesting and simple example of a slow-fast equation, in other words, an equation of the form: where e > 0 is a fixed i-small number, f a near-standard function, and f0 := 0f. We will study here a typical equation of such a kind and this will provide us with the opportunity to look at some of the tools that have been developed for their study.

Chasse au canard (première partie)

AbstractThe paper presents the main ideas of the Black, Scholes and Merton theory as a simplified introduction for non specialist. One explains what kind of financial product are options, how to price them and how to build a hedging portfolio that allows, according to the theory, to get rid of the risk.