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.

What is the fastest way of finding a randomly hidden target? This question of general relevance is of vital importance for foraging animals. Experimental observations reveal that the search behaviour of foragers is generally intermittent: active search phases randomly alternate with phases of fast ballistic motion. In this letter, we study the efficiency of this type of two states search strategies, by calculating analytically the mean first passage time at the target. We model the perception mecanism involved in the active search phase by a diffusive process. In this framework, we show that the search strategy is optimal when the average duration of "motion phases" varies like the power either 3/5 or 2/3 of the average duration of "search phases", depending on the regime. This scaling accounts for experimental data over a wide range of species, which suggests that the kinetics of search trajectories is a determining factor optimized by foragers and that the perception activity is adequately described by a diffusion process. PACS numbers:Searching for a randomly located object is one of the most frequent tasks of living organisms, be it for obtaining food, a sexual partner or a shelter [1]. In these examples, the search time is generally a limiting factor which has to be optimized for the survival of the species. The question of determining the efficiency of a search behaviour is thus a crucial problem of behavioral ecology, which has inspired numerous experimental [1,2,3,4,5] and theoretical [6,7,8,9,10] works . It is also relevant to broader domains such as stochastic processes theory [11,12], applied mathematics [13] and molecular biology [14,15].Anyone who has ever lost his keys knows that instinctively we adopt an intermittent behaviour combining local scanning phases and relocating phases. Indeed, numerous studies of foraging behaviour of a broad range of animal species show that such an intermittent behaviour is commonly observed and that the durations of search and displacement phases vary widely[1, 2, 3]. The spectrum, which goes from cruise strategy (ex. for large fishes that swim continuously such as tuna), to ambush or sitand-wait search, where the forager remains stationary for long periods (such as rattlesnake), has never been interpreted quantitatively. The intermittent strategy, often referred to as "saltatory" [2,3], can be understood intuitively when the targets are "difficult" to detect and sparsely distributed, as it is the case for many foragers (such as ground foraging birds, lizards, planktivorous fish...): since a fast movement is known to significantly degrade perception abilities [2,3], the forager must search slowly. Then, it has to relocate as fast as possible in order to explore a previously unscanned space, and search slowly again.Even though numerous models based on optimization of the net energy gain [4,5,6] predict an optimal strategy for foragers, the large number of unknown parameters used to model the complexity of the energetic constraint, renders a quantitative comparison wi...

It is widely recognized that the cleaving rate of a restriction enzyme on target DNA sequences is several orders-of-magnitude faster than the maximal one calculated from the diffusion-limited theory. It was therefore commonly assumed that the target site interaction of a restriction enzyme with DNA has to occur via two steps: one-dimensional diffusion along a DNA segment, and long-range jumps coming from association-dissociation events. We propose here a stochastic model for this reaction which comprises a series of one-dimensional diffusions of a restriction enzyme on nonspecific DNA sequences interrupted by three-dimensional excursions in the solution until the target sequence is reached. This model provides an optimal finding strategy which explains the fast association rate. Modeling the excursions by uncorrelated random jumps, we recover the expression of the mean time required for target site association to occur given by Berg et al. in 1981, and we explicitly give several physical quantities describing the stochastic pathway of the enzyme. For competitive target sites we calculate two quantities: processivity and preference. By comparing these theoretical expressions to recent experimental data obtained for EcoRV-DNA interaction, we quantify: 1), the mean residence time per binding event of EcoRV on DNA for a representative one-dimensional diffusion coefficient; 2), the average lengths of DNA scanned during the one-dimensional diffusion (during one binding event and during the overall process); and 3), the mean time and the mean number of visits needed to go from one target site to the other. Further, we evaluate the dynamics of DNA cleavage with regard to the probability for the restriction enzyme to perform another one-dimensional diffusion on the same DNA substrate following a three-dimensional excursion.

The cell cytoskeleton is a striking example of an 'active' medium driven out-of-equilibrium by ATP hydrolysis 1 . Such activity has been shown to have a spectacular impact on the mechanical and rheological properties of the cellular medium 2-10 , as well as on its transport properties 11-14 : a generic tracer particle freely diffuses as in a standard equilibrium medium, but also intermittently binds with random interaction times to motor proteins, which perform active ballistic excursions along cytoskeletal filaments. Here, we propose an analytical model of transport-limited reactions in active media, and show quantitatively how active transport can enhance reactivity for large enough tracers such as vesicles. We derive analytically the average interaction time with motor proteins that optimizes the reaction rate, and reveal remarkable universal features of the optimal configuration. We discuss why active transport may be beneficial in various biological examples: cell cytoskeleton, membranes and lamellipodia, and tubular structures such as axons 1 . Various motor proteins such as kinesins or myosins are able to convert the chemical fuel provided by ATP into mechanical work by interacting with the semiflexible oriented filaments (mainly F-actin and microtubules) of the cytoskeleton 1 . As many molecules or larger cellular organelles such as vesicles, lysosomes or mitochondria, hereafter referred to as tracer particles, can randomly bind and unbind to motors, the overall transport of a tracer in the cell can be described as alternating phases of standard diffusive transport and phases of active directed transport powered by motor proteins 1,15,16 . Active transport in cells has been extensively studied both experimentally, for instance by single-particle tracking methods 11,12 , and theoretically by evaluating the mean displacement of a tracer 13,17 , or stationary concentration profiles 14 . On the other hand, most cell functions are regulated by coordinated chemical reactions that involve low concentrations of reactants (such as ribosomes or vesicles carrying targeted proteins), and are therefore limited by transport. However, up to now a general quantitative analysis of the impact of active transport on reaction kinetics in cells, and more generally in generic active media, is still missing, even though a few specific examples have been tackled 18 . Here, we propose an analytical model that enables us to determine for the first time the kinetic constant of transportlimited reactions in active media.The model relies on the idea of intermittent dynamics introduced in the context of search processes [19][20][21][22][23][24][25][26][27] . We consider a tracer particle evolving in a d-dimensional space (in practice d = 1, 2, 3) that exhibits thermal diffusion phases of diffusion coefficient D (denoted phases 1), randomly interrupted by ballistic excursions bound to motors (referred to as phases 2) of constant velocity v and direction pointing in the solid angle ω v (Fig. 1a). The distribution of the filaments' orie...

We present a novel computational method of first-passage times between a starting site and a target site of regular bounded lattices. We derive accurate expressions for all the moments of this first-passage time, validated by numerical simulations. Their range of validity is discussed. We also consider the case of a starting site and two targets. In addition, we present the extension to continuous Brownian motion. These results are of great relevance to any system involving diffusion in confined media.

Lévy flights are known to be optimal search strategies in the particular case of revisitable targets. In the relevant situation of non revisitable targets, we propose an alternative model of bidimensional 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 bidimensional intermittent search strategies.

Dynamics of a tracer particle subject to a constant driving force E in a onedimensional lattice gas of hard-core particles whose transition rates are symmetric. We show that the mean displacement of the driven tracer, X T (E, t),

No abstract