Marc Lefranc

The Topology of Chaos

Intro.-01 Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 Embed-02 Embed-03 Embed-04a Embed-04b Topology of Orbits-01 Topology of Orbits-02 Topology of Orbits-03a Topology of Orbits-03b Topology of Orbits-04 Topology of Orbits-04a Topology of Orbits-04b Topology of Orbits-05a Topology of Orbits-05b Topology of Orbits-06 Topology of Orbits-07 Topology of Orbits-08 Topology of Orbits-09 Topology of Orbits-10 Topology of Orbits-11 Topology of Orbits-12 Topology of Orbits-13 Topology of Orbits-14 Topology of Orbits-15 Topology of Orbits-16 Topology of Orbits-17 Program-01 Program-02 Program-03 Program-04 Program-05 Program-06 Program-07 Program-08 Program-09a Program-09b Program-10a Program-10b Program-11 Program-12 Reps-01 Reps-02 Reps-03 Reps-04 Reps-05 Reps-06 Reps-07 Reps-08 Reps-09 Class-01 Class-02 Class-03 Class-04 Class-05 Bases-01 Bases-02 Bases-03 Bases-04 Tori-01 Tori-02 Tori-03 Tori-04 Tori-05 Tori-06 Tori-07 Tori-08 Tori-09 Tori-10 Tori-11 Tori-12 Tori-13 Tori-14 Tori-15 Tori-16a Tori-16b Tori-16c Tori-17 Tori-18 Tori-19 Tori-20 Summary-01 Summary-02 Summary-03 Summary-04 Summary-05 Summary-06 Summary-07 Summary-08 Summary-09 Summary-10 The Topology of Chaos

Homoclinic chaos in a laser containing a saturable absorber

Checks of homoclinic chaos made with nonlinear analysis techniques have been performed on the signals coming from a CO2 laser containing CH3I as a saturable absorber. The one-dimensional return maps of the regimes appearing inside the alternating periodic chaotic sequence are typical of homoclinic chaos. Numerical simulations give results in good agreement with the experimental observations. In the case of a fast absorber, a homoclinic tangency to a cycle created in a suberitical Hopf bifurcation is seen to be responsible for the homo-clinic behavior observed in the model.

Robustness of circadian clocks to daylight fluctuations: hints from the picoeucaryote Ostreococcus tauri.

The development of systemic approaches in biology has put emphasis on identifying genetic modules whose behavior can be modeled accurately so as to gain insight into their structure and function. However, most gene circuits in a cell are under control of external signals and thus, quantitative agreement between experimental data and a mathematical model is difficult. Circadian biology has been one notable exception: quantitative models of the internal clock that orchestrates biological processes over the 24-hour diurnal cycle have been constructed for a few organisms, from cyanobacteria to plants and mammals. In most cases, a complex architecture with interlocked feedback loops has been evidenced. Here we present the first modeling results for the circadian clock of the green unicellular alga Ostreococcus tauri. Two plant-like clock genes have been shown to play a central role in the Ostreococcus clock. We find that their expression time profiles can be accurately reproduced by a minimal model of a two-gene transcriptional feedback loop. Remarkably, best adjustment of data recorded under light/dark alternation is obtained when assuming that the oscillator is not coupled to the diurnal cycle. This suggests that coupling to light is confined to specific time intervals and has no dynamical effect when the oscillator is entrained by the diurnal cycle. This intringuing property may reflect a strategy to minimize the impact of fluctuations in daylight intensity on the core circadian oscillator, a type of perturbation that has been rarely considered when assessing the robustness of circadian clocks.

Robust entrainment of circadian oscillators requires specific phase response curves.

The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earths rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.

On proving the absence of oscillations in models of genetic circuits

Using computer algebra methods to prove that a gene regulatory network cannot oscillate appears to be easier than expected. We illustrate this claim with a family of models related to historical examples.

From template analysis to generating partitions I: periodic orbits, knots and symbolic encodings

Abstract We present a detailed algorithm to construct symbolic encodings for chaotic attractors of three-dimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by Lefranc et al. [Phys. Rev. Lett. 73 (1994) 1364]. For each orbit, the symbolic names that are consistent with its knot-theoretic invariants and with the topological structure of the attractor are first obtained using template analysis. This information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations. We provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations, and obtain partitions whose border is localized with a precision of 0.01%. A distinctive advantage of this approach is that the solution is progressively refined using higher-period orbits, which makes it robust to noise, and suitable for analyzing experimental time series. Furthermore, the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one-dimensional and hyperbolic maps.

Applying a Rigorous Quasi-Steady State Approximation Method for Proving the Absence of Oscillations in Models of Genetic Circuits

In this paper, we apply a rigorous quasi-steady state approximation method on a family of models describing a gene regulated by a polymer of its own protein. We study the absence of oscillations for this family of models and prove that Poincare-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result presented in a former paper at Algebraic Biology 2007is thereby generalized. The rigorous method is illustrated over the basic enzymatic reaction.

A Mathematical Model of the Liver Circadian Clock Linking Feeding and Fasting Cycles to Clock Function

To maintain energy homeostasis despite variable energy supply and consumption along the diurnal cycle, the liver relies on a circadian clock synchronized to food timing. Perturbed feeding and fasting cycles have been associated with clock disruption and metabolic diseases; however, the mechanisms are unclear. To address this question, we have constructed a mathematical model of the mammalian circadian clock, incorporating the metabolic sensors SIRT1 and AMPK. The clock response to various temporal patterns of AMPK activation was simulated numerically, mimicking the effects of a normal diet, fasting, and a high-fat diet. The model reproduces the dampened clock gene expression and NAD+ rhythms reported for mice on a high-fat diet and predicts that this effect may be pharmacologically rescued by timed REV-ERB agonist administration. Our model thus identifies altered AMPK signaling as a mechanism leading to clock disruption and its associated metabolic effects and suggests a pharmacological approach to resetting the clock in obesity.

A robust two-gene oscillator at the core of Ostreococcus tauri circadian clock

The microscopic green alga Ostreococcus tauri is rapidly emerging as a promising model organism in the green lineage. In particular, recent results by Corellou et al. [Plant Cell 21, 3436 (2009)] and Thommen et al. [PLOS Comput. Biol. 6, e1000990 (2010)] strongly suggest that its circadian clock is a simplified version of Arabidopsis thaliana clock, and that it is architectured so as to be robust to natural daylight fluctuations. In this work, we analyze the time series data from luminescent reporters for the two central clock genes TOC1 and CCA1 and correlate them with microarray data previously analyzed. Our mathematical analysis strongly supports both the existence of a simple two-gene oscillator at the core of Ostreococcus tauri clock and the fact that its dynamics is not affected by light in normal entrainment conditions, a signature of its robustness.

Model Reduction of Chemical Reaction Systems using Elimination

There exist different schemes of model reduction for parametric ordinary differential systems arising from chemical reaction systems. In this paper, we focus on some schemes which rely on quasi-steady states approximations. We show that these schemes can be formulated by means of differential and algebraic elimination. Our formulation is simpler than the classical ones. It permitted us to obtain an approximation of the basic enzymatic reaction system which is different from those of Henri–Michaëlis–Menten and Briggs–Haldane.