R. Voituriez

Intermittent search strategies

This review examines intermittent target search strategies, which combine phases of slow motion, allowing the searcher to detect the target, and phases of fast motion during which targets cannot be detected. We first show that intermittent search strategies are actually widely observed at various scales. At the macroscopic scale, this is for example the case of animals looking for food ; at the microscopic scale, intermittent transport patterns are involved in reaction pathway of DNA binding proteins as well as in intracellular transport. Second, we introduce generic stochastic models, which show that intermittent strategies are efficient strategies, which enable to minimize the search time. This suggests that the intrinsic efficiency of intermittent search strategies could justify their frequent observation in nature. Last, beyond these modeling aspects, we propose that intermittent strategies could be used also in a broader context to design and accelerate search processes.

Spontaneous flow transition in active polar gels

We study theoretically the effects of confinement on active polar gels such as the actin network of eukaryotic cells. Using generalized hydrodynamics equations derived for active gels, we predict, in the case of quasi–one-dimensional geometry, a spontaneous flow transition from a homogeneously polarized immobile state for small thicknesses, to a perturbed flowing state for larger thicknesses. The transition is not driven by an external field but by the activity of the system. We suggest several possible experimental realizations.

Generic Phase Diagram of Active Polar Films

We study theoretically the phase diagram of compressible active polar gels such as the actin network of eukaryotic cells. Using generalized hydrodynamics equations, we perform a linear stability analysis of the uniform states in the case of an infinite bidimensional active gel to obtain the dynamic phase diagram of active polar films. We predict, in particular, modulated flowing phases and a macroscopic phase separation at high activity. This qualitatively accounts for experimental observations of various active systems, such as actomyosin gels, microtubules and kinesins in vitro solutions, or swimming bacterial colonies.

A minimal model of intermittent search in dimension two

We propose and analyse a model of bidimensional search processes, explicitly relying on the widely observed intermittent behaviour of foraging animals, which involves a searcher enjoying minimal orientational and temporal memory skills. We show analytically that, in the case of non-revisitable targets, intermittent strategies can minimize the search time, and therefore constitute real optimal strategies, as opposed to Levy flights strategy which are optimal only in the particular case of revisitable targets. Two representative modes of target detection are presented, and they allow us to determine which characteristics of the optimal strategy are robust and do not depend on the specific characteristics of detection mechanisms. In particular, our study tends to show that the optimal duration of the ballistic phase is a universal feature of bidimensional intermittent search strategies. Last, by comparing the results of our minimal model to systematic search strategies, we show that if temporal and orientational memory skills speed up the search, they do not change the order of magnitude of the search time.

Optimizing intermittent reaction paths

Various examples of biochemical reactions in cells, such as DNA/protein interactions, reveal that in extremely diluted regimes reaction paths are not always simple brownian trajectories. They can rather be qualified as intermittent, since they combine slow diffusion phases on one hand and a second mode of faster transport on the other hand, which can be either a faster diffusion mode, as in the case of DNA-binding proteins, or a ballistic mode powered by molecular motors in the case of intracellular transport. In this article, we introduce simple theoretical models which permit to calculate explicitly the reaction rates for reactions limited by intermittent transport. This approach shows quantitatively that intermittent reaction pathways are actually very efficient, since they permit to significantly increase the reaction rates, which could explain why they are observed so often. Moreover, we give theoretical arguments which suggest that intermittent transport could also be useful for in vitro chemistry. Indeed, we show that intermittent transport naturally pops up in the context of reaction at interfaces, where reactants combine surface diffusion phases and bulk excursions, and could permit to enhance reactivity. In this case, adjusting chemically the affinity of reactants with the interface makes possible to optimize the reaction rate.

Averaged residence times of stochastic motions in bounded domains

Two years ago, Blanco and Fournier (Blanco S. and Fournier R., Europhys. Lett. 61 (2003) 168) calculated the mean first exit time of a domain of a particle undergoing a randomly reoriented ballistic motion which starts from the boundary. They showed that it is simply related to the ratio of the volumes domain over its surface. This work was extended by Mazzolo (Mazzolo A., Europhys. Lett. 68 (2004) 350), who studied the case of trajectories which start inside the volume. In this letter, we propose an alternative formulation of the problem which allows us to calculate not only the mean exit time, but also the mean residence time inside a sub-domain. The cases of any combinations of reflecting and absorbing boundary conditions are considered. Lastly, we generalize our results for a wide class of stochastic motions.

A stochastic model for intermittent search strategies

It is often necessary, in scientific or everyday life problems, to find a randomly hidden target. What is then the optimal strategy to reach it as rapidly as possible? In this article, we develop a stochastic theory for intermittent search behaviours, which are often observed: the searcher alternates phases of intensive search and slow motion with fast displacements. The first results of this theory have already been announced recently. Here we provide a detailed presentation of the theory, as well as the full derivation of the results. Furthermore, we explicitly discuss the minimization of the time needed to find the target.

Robustness of optimal intermittent search strategies in one, two, and three dimensions

Search problems at various scales involve a searcher, be it a molecule before reaction or a foraging animal, which performs an intermittent motion. Here we analyze a generic model based on such type of intermittent motion, in which the searcher alternates phases of slow motion allowing detection and phases of fast motion without detection. We present full and systematic results for different modeling hypotheses of the detection mechanism in space in one, two, and three dimensions. Our study completes and extends the results of our recent letter [Loverdo, Nat. Phys. 4, 134 (2008)] and gives the necessary calculation details. In addition, another modeling of the detection case is presented. We show that the mean target detection time can be minimized as a function of the mean duration of each phase in one, two, and three dimensions. Importantly, this optimal strategy does not depend on the details of the modeling of the slow detection phase, which shows the robustness of our results. We believe that this systematic analysis can be used as a basis to study quantitatively various real search problems involving intermittent behaviors.