Multifractality of entangled random walks and non-uniform hyperbolic spaces

Multifractal properties of the distribution of topological invariants for a model of trajectories randomly entangled with a nonsymmetric lattice of obstacles are investigated. Using the equivalence of the model to random walks on a locally nonsymmetric tree, statistical properties of topological invariants, such as drift and return probabilities, have been studied by means of a renormalization group (RG) technique. The comparison of the analytical RG--results with numerical simulations as well as with the rigorous results of P.Gerl and W.Woess demonstrates clearly the validity of our approach. It is shown explicitly by direct counting for the discrete version of the model and by conformal methods for the continuous version that multifractality occurs when local uniformity of the phase space (which has an exponentially large number of states) has been broken.Comment: 28 pages, 11 eps-figures (enclosed

Active Gel Model of Amoeboid Cell Motility

We develop a model of amoeboid cell motility based on active gel theory. Modeling the motile apparatus of a eukaryotic cell as a confined layer of finite length of poroelastic active gel permeated by a solvent, we first show that, due to active stress and gel turnover, an initially static and homogeneous layer can undergo a contractile-type instability to a polarized moving state in which the rear is enriched in gel polymer. This agrees qualitatively with motile cells containing an actomyosin-rich uropod at their rear. We find that the gel layer settles into a steadily moving, inhomogeneous state at long times, sustained by a balance between contractility and filament turnover. In addition, our model predicts an optimal value of the gel-susbstrate adhesion leading to maximum layer speed, in agreement with cell motility assays. The model may be relevant to motility of cells translocating in complex, confining environments that can be mimicked experimentally by cell migration through microchannels.Comment: To appear in New Journal of Physic

Reactive conformations and non-Markovian reaction kinetics of a Rouse polymer searching for a target in confinement

We investigate theoretically a diffusion-limited reaction between a reactant attached to a Rouse polymer and an external fixed reactive site in confinement. The present work completes and goes beyond a previous study [T. Gu\'erin, O. B\'enichou and R. Voituriez, Nat. Chem., 4, 268 (2012)] that showed that the distribution of the polymer conformations at the very instant of reaction plays a key role in the reaction kinetics, and that its determination enables the inclusion of non-Markovian effects in the theory. Here, we describe in detail this non-Markovian theory and we compare it with numerical stochastic simulations and with a Markovian approach, in which the reactive conformations are approximated by equilibrium ones. We establish the following new results. Our analysis reveals a strongly non-Markovian regime in 1D, where the Markovian and non-Markovian dependance of the relation time on the initial distance are different. In this regime, the reactive conformations are so different from equilibrium conformations that the Markovian expressions of the reaction time can be overestimated by several orders of magnitudes for long chains. We also show how to derive qualitative scaling laws for the reaction time in a systematic way that takes into account the different behaviors of monomer motion at all time and length scales. Finally, we also give an analytical description of the average elongated shape of the polymer at the instant of the reaction and we show that its spectrum behaves a a slow power-law for large wave numbers