Marc Lefranc

University of Lille (ULille) · Physics

How nuclear proteins diffuse and find their targets remains a key question in the transcription field. Dynamic proteins in the nucleus are classically subdiffusive and undergo anomalous diffusion, yet the underlying physical mechanisms are still debated. In this study, we explore the contribution of interactions to the generation of anomalous diffu...

Two studies show that noise is a key ingredient of new mechanisms for entraining the NF-κB system.

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 const...

We investigate the dynamics of the heterodimer autorepression loop (HAL), a small genetic module in which a protein A acts as an autorepressor and binds to a second protein B to form an AB dimer. For suitable values of the rate constants, the HAL produces pulses of A alternating with pulses of B. By means of analytical and numerical calculations, w...

Most organisms anticipate daily environmental variations and orchestrate cellular functions thanks to a circadian clock which entrains robustly to the day/night cycle, despite fluctuations in light intensity due to weather or seasonal variations. Marine organisms are also subjected to fluctuations in light spectral composition as their depth varies...

We characterize the systematic changes in the topological structure of chaotic attractors that occur as spike-adding and homoclinic bifurcations are encountered in the slow-fast dynamics of neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose neuron model, where we show that the unstable periodic orbits appearing after each spike...

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that wh...

Fundamental biological processes such as transcription and translation, where a genetic sequence is sequentially read by a macromolecule, have been well described by a classical model of nonequilibrium statistical physics, the totally asymmetric exclusion principle (TASEP). This model describes particles hopping between sites of a one-dimensional l...

Circadian rhythms are ubiquitous on earth from cyanobacteria to land plants and animals. Circadian clocks are synchronized to the day/night cycle by environmental factors such as light and temperature. In eukaryotes, clocks rely on complex gene regulatory networks involving transcriptional regulation but also post-transcriptional and post-translati...

The knot-theoretic characterization of three-dimensional strange attractors has proved an invaluable tool for comparing models to experiments, understanding the structure of bifurcation diagrams, constructing symbolic encodings or obtaining signatures of chaos. In four dimensions and above, however, all closed curves can be deformed into each other...

Daylight is the primary cue used by circadian clocks to entrain to the day/night cycle so as to synchronize physiological processes with periodic environmental changes induced by Earth rotation. However, the temporal daylight pattern is not the same every day due to erratic weather fluctuations or regular seasonal changes. Then, how do circadian cl...

Local bathymetric quasi-periodic patterns of oscillation are identified from monthly profile surveys taken at two shore-perpendicular transects at the USACE field research facility in Duck, North Carolina, USA, spanning 24.5 years and covering the swash and surf zones. The chosen transects are the two furthest (north and south) from the pier locate...

Description of Methods. (DOCX)

The green microscopic alga Ostreococcus tauri has recently emerged as a promising model for understanding how circadian clocks, which drive the daily biological rhythms of many organisms, synchronize to the day/night cycle in changing weather and seasons. Here, we analyze translational reporter time series data for the central clock genes CCA1 and...

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Most theoretical approaches describe them as purely deterministic or stochastic dynamical systems, depending on which point of view is favored. Here, we investigate the dynamics of a self-repressing g...

Author Summary In the life cycle of sexual organisms, a specialized cell division -meiosis- reduces the number of chromosomes in gametes or spores while fertilization or mating restores the original number. The essential feature that distinguishes meiosis from mitosis (the usual division) is the succession of two rounds of division following a sing...

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 th...

Definition of Dynamical SystemsExistence and Uniqueness TheoremExamples of Dynamical SystemsChange of VariablesFixed PointsPeriodic OrbitsFlows Near Nonsingular PointsVolume Expansion and ContractionStretching and SqueezingThe Fundamental IdeaSummary

The Fundamental ProblemFrom Template Matrices to Topological InvariantsIdentifying Templates from InvariantsConstructing Generating PartitionsSummary

Is Template Analysis Valid for Non-Hyperbolic Systems?Can Template Analysis Be Applied to Weakly Dissipative Systems?What About Higher-Dimensional Systems?

Lorenz EquationsOptically Pumped Molecular LaserFluid ExperimentsWhy A3?Summary

Stretching & Folding vs. Tearing & SqueezingInflationBoundary of InflationIndexProjectionNature of SingularitiesTrinionsPoincar? Surface of SectionConstruction of Canonical FormsPerestroikasSummary

Closed LoopsWhat Does This Have to Do with Dynamical Systems?General Properties of Branched ManifoldsBirman?Williams TheoremRelaxation of RestrictionsExamples of Branched ManifoldsUniqueness and NonuniquenessStandard FormTopological InvariantsAdditional PropertiesSubtemplatesSummary

DiffeomorphismsMappings of DataTests for EmbeddingsTests of Embedding TestsGeometric Tests for EmbeddingsDynamical Tests for EmbeddingsTopological Test for EmbeddingsPostmortem on Embedding TestsStationarityBeyond EmbeddingsSummary

Belousov?Zhabotinskii Chemical ReactionLaser with Saturable AbsorberStringed InstrumentLasers with Low-Intensity SignalsThe Lasers in LilleThe Laser in ZaragozaNeuron with Subthreshold OscillationsSummary

Embeddings, Representations, EquivalenceSimplest Class of Strange AttractorsRepresentation LabelsEquivalence of Representations with Increasing DimensionGenus-g AttractorsRepresentation LabelsEquivalence in Increasing DimensionSummary

Stretching and Squeezing MechanismsLinking NumbersRelative Rotation RatesRelation between Linking Numbers and Relative Rotation RatesAdditional Uses of Topological InvariantsSummary

Reduction of DimensionEquivalenceStructure TheoryGermsUnfoldingPathsRankComplex ExtensionsCoxeter–Dynkin DiagramsReal FormsLocal vs. Global ClassificationCover–Image RelationsSymmetry Breaking and RestorationSummary

IntroductionLogistic MapBifurcation DiagramsElementary Bifurcations in the Logistic MapMap ConjugacyFully Developed Chaos in the Logistic MapOne-Dimensional Symbolic DynamicsShift Dynamical Systems, Markov Partitions, and EntropyFingerprints of Periodic Orbits and Orbit ForcingTwo-Dimensional Dynamics: Smale's HorseshoeHénon MapCircle MapsAnnulus M...

Information Loss and GainCover and Image RelationsRotation Symmetry 1: ImagesRotation Symmetry 2: CoversPeeling: a New Global BifurcationInversion Symmetry: Driven OscillatorsDuffing OscillatorVan der Pol OscillatorSummary

Catastrophe Theory as a ModelUnfolding of Branched Manifolds: Branched Manifolds as GermsUnfolding within Branched Manifolds: Unfolding of the HorseshoeMissing OrbitsRoutes to ChaosOrbit Forcing and Topological Entropy: Mathematical AspectsTopological Measures of Chaos in ExperimentsSummary

Brief Summary of the Topological Analysis ProgramOverview of the Topological Analysis ProgramDataEmbeddingsPeriodic OrbitsComputation of Topological InvariantsIdentify TemplateValidate TemplateModel DynamicsValidate ModelSummary

Review of Classification Theory in R3General SetupFlows in R4Cusps in Weakly Coupled, Strongly Dissipative Chaotic SystemsCusp Bifurcation DiagramsNonlocal SingularitiesGlobal Boundary ConditionsFrom Braids to Triangulations: toward a Kinematics in Higher DimensionsSummary

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 Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A temptin...

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...

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...

Les fonctions cellulaires reposent en grande partie sur des réseaux de gènes et de protéines en interaction. Ces réseaux peuvent présenter des comportements dynamiques complexes, en particulier des oscillations. Nousétudions ici l'apparition d'oscillations dans un modèle minimal de circuit génétique, où un gène est réprimé par sa propre protéine. U...

Les fonctions cellulaires reposent en grande partie sur des réseaux de gènes et de protéines en interaction. Ces réseaux peuvent présenter des comportements dynamiques complexes, en particulier des oscillations. Nousétudions ici l'apparition d'oscillations dans un modèle minimal de circuit génétique, où un gène est réprimé par sa propre protéine. U...

We show experimentally that parametric interaction can induce a cooperative oscillation of non-simultaneously resonant transverse modes in an optical parametric oscillator. More generally, this effect is expected to occur in any spatially extended system subjected to boundary conditions where nonlinear wave mixing of two nonresonant spatial modes c...

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinit...

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 Poincaré-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result...

Poincaré developed topology and exploited this new branch of mathematics in ingenious ways to study the properties of differential equations. Ideas and tools from this branch of mathematics are particularly well suited to describe and to classify a restricted but enormously rich class of chaotic dynamical systems, and thus the term chaos topology r...

IntroductionLogistic MapBifurcation DiagramsElementary Bifurcations in the Logistic MapMap ConjugacyFully Developed Chaos in the Logistic MapOne-Dimensional Symbolic DynamicsShift Dynamical Systems, Markov Partitions, and EntropyFingerprints of Periodic Orbits and Orbit ForcingTwo-Dimensional Dynamics: Smale's HorseshoeH?non MapCircle MapsSummary

Half TitleTitleCopyrightPrefaceContents

Review of Classification Theory in R3General SetupFlows in R4Cusp Bifurcation DiagramsNonlocal SingularitiesGlobal Boundary ConditionsSummary

Reduction of DimensionEquivalenceStructure TheoryGermsUnfoldingPathsRankComplex ExtensionsCoxeter-Dynkin DiagramsReal FormsLocal vs. Global ClassificationCover-Image RelationsSymmetry Breaking and RestorationSummary

Definition of Dynamical SystemsExistence and Uniqueness TheoremExamples of Dynamical SystemsChange of VariablesFixed PointsPeriodic OrbitsFlows near Nonsingular PointsVolume Expansion and ContractionStretching and SqueezingThe Fundamental IdeaSummary

Information Loss and GainCover and Image RelationsRotation Symmetry 1: ImagesRotation Symmetry 2: CoversPeeling: A New Global BifurcationInversion Symmetry: Driven OscillatorsDuffing Oscillatorvan der Pol OscillatorSummary

The Fundamental ProblemFrom Template Matrices to Topological InvariantsIdentifying Templates from InvariantsConstructing Generating PartitionsSummary

Stretching and Squeezing MechanismsLinking NumbersRelative Rotation RatesRelation between Linking Numbers and Relative Rotation RatesAdditional Uses of Topological InvariantsSummary

Lorenz EquationsOptically Pumped Molecular LaserFluid ExperimentsWhy A3?Summary

Catastrophe Theory as a ModelUnfolding of Branched Manifolds: Branched Manifolds as GermsUnfolding within Branched Manifolds: Unfolding of the Horseshoe35Missing OrbitsRoutes to ChaosSummary

Belousov-Zhabotinskii Chemical ReactionLaser with Saturable AbsorberStringed InstrumentLasers with Low-Intensity SignalsThe Lasers in LilleNeuron with Subthreshold OscillationsSummary

Brief Summary of the Topological Analysis ProgramOverview of the Topological Analysis ProgramDataEmbeddingsPeriodic OrbitsComputation of Topological InvariantsIdentify TemplateValidate TemplateModel DynamicsValidate ModelSummary

Closed LoopsWhat Has This Got to Do with Dynamical Systems?General Properties of Branched ManifoldsBirman-Williams TheoremRelaxation of RestrictionsExamples of Branched ManifoldsUniqueness and NonuniquenessStandard FormTopological InvariantsAdditional PropertiesSubtemplatesSummary

Theoretical investigations of dynamical behavior in optical parametric oscillators (OPO) have generally assumed that the cavity detunings of the interacting fields are controllable parameters. However, OPOs are known to experience mode hops, where the system jumps to the mode of lowest cavity detuning. We note that this phenomenon significantly lim...

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orb...

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.

When a low-dimensional chaotic attractor is embedded in a three-dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: Parity, global torsion, and knot type. We discuss how they can change with the embedding. Finally, we show that the mechanism tha...

Chaotic behavior in a deterministic dynamical system results from the interplay in state space of two geometric mechanisms, stretching and squeezing, which conspire to separate arbitrarily close trajectories while confining the dynam-ics to a bounded subset of state space, a strange attractor. A topological method has been designed to classify the...

When a low-dimensional chaotic attractor is embedded in a three-dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: Parity, global torsion, and knot type. We discuss how they can change with the embedding. Finally, we show that the mechanism tha...

The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built--determinism and continuity--apply in any dimension. We propose an alternative...

The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built\char22{}determinism and continuity\char22{}apply in any dimension. We propose...

Nous discutons quelques mécanismes d'instabilité récemment observés dans un oscillateur paramétrique optique (OPO) : d'une part des instabilités opto-thermiques où le système oscille autour des courbes de résonance d'un ou plusieurs modes, d'autre part des oscillations rapides résultant de l'interaction de plusieurs modes transverses. La première o...

The topological methods for obtaining first experimental signature of deterministic chaos, in an optical parametric oscillator using a short segment of a nonstationary time series were analyzed. It was observed that if a positive-entropy closed orbit did not change its braid type along the homotopy path, then it provided a signature of chaos. The t...

We investigate theoretically and experimentally the interplay between cavity and double-refraction in continuous-wave optical parametric oscillators. We show that very basic geometrical effects can prevent transverse wavevector matching for the TEM₀₀ modes, and thus increase the threshold and change dramatically the beam structures when...

Optical parametric oscillators (OPOs) are promising tunable sources of coherent light, in particular for generating nonclassical states of light. As many nonlinear optical devices (e.g., lasers), OPOs can exhibit instabilities. However, their dynamics has so far been relatively little studied: although periodic and chaotic behavior have been predic...

Different forms of bursting oscillations with frequencies from a few MHz to hundreds of MHz separated by intervals of no activity have been observed experimentally for an optical parametric oscillator (OPO) system subject to thermal effects. These oscillations have been simulated numerically using previously derived equations for two interacting tr...

This paper concentrates on spatiotemporally chaotic regimes. The investigation shows different types of spatiotemporal chaos (extensive and non-extensive) stemming from at least three different origins. The first mechanism is linked to spatial nonuniformities (here the Gaussian lineshape of the laser) that can induce low dimensional spatiotemporal...