M. Moreau

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.

Two-dimensional intermittent search processes: An alternative to Lévy flight strategies.

Lévy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of nonrevisitable targets, we propose an alternative model of two-dimensional (2D) search processes, which explicitly relies on the widely observed intermittent behavior of foraging animals. We show analytically that intermittent strategies can minimize the search time, and therefore do constitute real optimal strategies. We study two representative modes of target detection and determine which features of the search time are robust and do not depend on the specific characteristics of detection mechanisms. In particular, both modes lead to a global minimum of the search time as a function of the typical times spent in each state, for the same optimal duration of the ballistic phase. This last quantity could be a universal feature of 2D intermittent search strategies.

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.

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.

Stochastic theory of diffusion-controlled reactions

We study the kinetics of diffusion-controlled A+B→B reactions, in which both species are moving randomly on a d-dimensional lattice and react upon encounters, provided that both species are in reactive states. Particles’ reactivity fluctuates randomly between active and passive forms. We find that in low dimensions the A particle survival probability Ψ(t) is described by a stretched-exponential function of time, such that no reaction constant can be identified. In three dimensions, we recover the exponential decay law and evaluate the effective reaction constant in several particular cases. In addition, we derive some rigorous bounds on Ψ(t).

Reaction kinetics in active media

Reactants in biological cells can either freely diffuse or bind to molecular motors which perform ballistic active motion along the cytoskeletal filaments. The transport process is therefore intermittent since it alternates diffusive reactive phases and ballistic non-reactive phases. Here we present an overview of recent results (Benichou et al 2006 Phys. Rev. E 74 020102, Loverdo et al 2008 Nat. Phys. 4 134) which enables us to determine the kinetic constant of reactions limited by such intermittent transport. We address the question of optimizing the reaction rate as a function of the mean durations of each phase. To answer such questions, we calculate explicitly the mean first-passage time to the target. We conclude that intermittent transport can maximize the reaction rate, and that there are optimal durations of the two phases. We studied this model in one, two and three dimensions. All these cases are relevant to reactivity in biological cells. Indeed, structures such as dendrites can be considered as one-dimensional, membranes as two-dimensional, and bulk cytoplasm as three-dimensional. We show that the dependence on the target density is important in the one-dimensional case, weak in the two-dimensional case, and disappears in the three-dimensional case. Our results are robust, as different descriptions of the reactive phase lead to the same optimal duration of the ballistic phase.

Chance and strategy in search processes

We consider a searcher in quest of a target in two situations: in the presence of an infinite number of identical, Poisson distributed targets, and in the presence of a unique target in a finite territory. The searcher alternates intensive search phases, during which it scans the neighbouring territory but does not move, and displacement phases with no target detection. We study the problem of determining the best strategy of displacement for minimizing the mean search time: either a deterministic or a stochastic trajectory. With a reasonable simplifying hypothesis, we show that for Poisson distributed targets, deterministic, self-avoiding trajectories are more efficient than stochastic ones if the detection process involves no memory skills and can be modelled by a Markov process. In contrast, if the detection process is not Markovian, it can be better for the searcher to follow a stochastic trajectory rather than a self-avoiding trajectory, and we give an explicit example of such a memory law. In the case of a unique target, self-avoiding trajectories are always better if an infinite time is available for the search, whereas stochastic trajectories can be more efficient if the searcher has to find the target before a given deadline. Moreover, we show that the gain due to a deterministic trajectory, compared to a stochastic one, is not significant in the case of a large network containing a unique target. Additionally, for various examples of displacement trajectories, we compute the overall mean search time and study its minimization as a function of the mean duration of the detection process.

Dynamical and spatial disorder in an intermittent search process

We consider a one-dimensional model of an intermittent search process in a medium exhibiting frozen disorder. A tracer, searching for Poisson-distributed targets, alternates diffusive and ballistic motions, but can only find a target when diffusing. Preliminary theoretical results (1) are now confirmed, completed and extended, and their derivations are presented for the first time. We study the mean search timeTaccording to the laws of the searcher waiting times in the diffusive and ballistic regimes. In particular, we obtain a lower bound of � T � , which in certain circumstances is also an approximation and is valid for a very broad class of waiting time distributions. Explicit results and other approximations are presented in the case of exponential waiting times, and we study the optimization ofT � , depending on the mean durations of the diffusive and ballistic phases. Theoretical formulae are supported by numerical simulations. We show that the intermittent behaviour can allow one to minimize the search time in comparison with the purely diffusive behaviour, and that it is possible, by an adequate choice of the parameters, to increase very significantly the efficiency of the search.