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We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted L 2 norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.

Actin filaments polymerizing against membranes power endocytosis, vesicular traffic, and cell motility. In vitro reconstitution studies suggest that the structure and the dynamics of actin networks respond to mechanical forces. We demonstrate that lamellipodial actin of migrating cells responds to mechanical load when membrane tension is modulated. In a steady state, migrating cell filaments assume the canonical dendritic geometry, defined by Arp2/3-generated 70° branch points. Increased tension triggers a dense network with a broadened range of angles, whereas decreased tension causes a shift to a sparse configuration dominated by filaments growing perpendicularly to the plasma membrane. We show that these responses emerge from the geometry of branched actin: when load per filament decreases, elongation speed increases and perpendicular filaments gradually outcompete others because they polymerize the shortest distance to the membrane, where they are protected from capping. This network-intrinsic geometrical adaptation mechanism tunes protrusive force in response to mechanical load.

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SummaryUsing correlated live-cell imaging and electron tomography we found that actin branch junctions in protruding and treadmilling lamellipodia are not concentrated at the front as previously supposed, but link actin filament subsets in which there is a continuum of distances from a junction to the filament plus ends, for up to at least 1 mm. When branch sites were observed closely spaced on the same filament their separation was commonly a multiple of the actin helical repeat of 36 nm. Image averaging of branch junctions in the tomograms yielded a model for the in vivo branch at 2.9 nm resolution, which was comparable with that derived for the in vitro actinArp2/3 complex. Lamellipodium initiation was monitored in an intracellular wound-healing model and was found to involve branching from the sides of actin filaments oriented parallel to the plasmalemma. Many filament plus ends, presumably capped, terminated behind the lamellipodium tip and localized on the dorsal and ventral surfaces of the actin network. These findings reveal how branching events initiate and maintain a network of actin filaments of variable length, and provide the first structural model of the branch junction in vivo. A possible role of filament capping in generating the lamellipodium leaflet is discussed and a mathematical model of protrusion is also presented.

This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion. To cite this article: J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). RésuméHypocoercivité pour des équations cinétiques avec termes de relaxation linéaires. Cette Note est consacrée à une méthode simple pour démontrer l'hypocoercivité associée à une équation cinétique contenant un opérateur de relaxation linéaire ; il s'agit de construire une fonctionnelle de Lyapunov adaptée vérifiant une inégalité de type Gronwall. La méthode distingue clairement la coercivité au niveau microscopique, qui provient directement des propriétés de l'opérateur de relaxation, et une inégalité de trou spectral pour la densité spatiale, qui est reliée à la limite de diffusion. Elle améliore les résultats antérieurs. Notre approche est illustrée par le modèle de BGK linéaire et par un opérateur de relaxation qui correspond, au niveau macroscopique, à la diffusion rapide linéarisée. Pour citer cet article : J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).Le but de la théorie de l'hypocoercivité [10,11,8] est d'estimer des taux exponentiels de retour à un équilibre global pour des équations cinétiques dans lesquelles le terme de collision ne contrôle que le retour à un équilibre local. Nous simplifions les résultats antérieurs, [6,11,5], en distinguant l'échelle microscopique et l'échelle macroscopique, pour laquelle une propriété de trou spectral fait le lien avec les limites de diffusion, [1,2].Considérons l'équation (1) sur R d , d 1 où T := v ·∇ x −∇ x V ·∇ v , V est un potentiel extérieur et où l'opérateur de relaxation linéaire L est défini par (2). La fonction F est une mesure de probabilité strictement positive ne dépendant que de |v| 2 /2 + V (x) et on notera dμ(x, v) = F (x, v) −1 dx dv. Norme et produit scalaire sont par défaut ceux de L 2 (dμ). La donnée initiale f 0 ∈ L 2 (dμ) est normalisée par R d ×R d f 0 dx dv = 1. Théorème 1 (équilibre Maxwellien). Supposons que, alors pour tout η > 0, il existe une constante positive λ = λ(η), explicite, pour laquelle toute solution de (1) dans L 2 (dμ) vérifie :Dans le deuxième cas, on supposera que F (x, v) := ω( 1 2 |v| 2 + V (x)) −(k+1) , V (x) = (1 + |x| 2 ) β pour simplifier. Voir [3] pour des cas plus généraux. Ici, ρ F = ω 0 V d/2−k−1 , ω et ω 0 sont des constantes de normalisation.Théorème 2. Soit d 1, k > d/2 + 1. Il existe une constante β 0 > 1 tell...

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.

We develop and analyse a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The formal continuum limit of the model is a nonlinear diffusion equation with a diffusivity which can become negative if the adhesion coefficient is sufficiently large. This appears to be related to the presence of spatial oscillations and the development of plateaus (pattern formation) in numerical solutions of the discrete model. A combination of stability analysis of the discrete equations and steady-state analysis of the limiting PDE (and a higher-order correction thereof) can be used to shed light on these and other qualitative predictions of the model.