Lévy strategies in intermittent search processes are advantageous

Abstract

Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are Lévy distributed, resulting in a Lévy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting Lévy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the Lévy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher.random processes | optimization | Lévy walk | movement ecology R andom search processes occur in many areas, from chemical reactions of diffusing reactants (1) to the foraging behavior of bacteria and animals (2, 3). Of general importance is the search efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. Higher efficiency, can be achieved, for instance, by facilitated diffusion in gene regulation (4) or by controlled motion in foraging (2, 3). From theoretical and data analysis Lévy strategies, in which the searching agent performs excursions whose length is drawn from distributions with a heavy tailfor 0 < α < 2 were shown to be advantageous (5-16); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling.As an alternative to Lévy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (17-21). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation. The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (17-21).In refs. 18 and 20 relocation events were assumed to occur in a random direction for exponentially distributed time spans, giving rise to a Markovian process. We show here analytically and numerically in one dimension that this is only a partial solution to oversampling, as eventually the central limit theorem (CLT) reduces the process to a Brownian random walk with jumps on the scale of vτ 2 , where τ 2 is the typical time spent in a relocation event. In practice, revisits can be reduced by adjusting the average time spent in search and relocation phases to the density of targets. Lévy strategies, on the other hand, fundamentally circumvent the CLT, and we here demonstrate a twofold advantage of them over the exponential distribution: Lévy walk intermittent processes find the target faster than exponential strategies in the critical case of rare targets, and their performance is much less dependent on adapting to the target density. Intermittent Search with Lévy RelocationsGeneralizing the model from ref. 20, w...

Facilitated diffusion with DNA coiling

“…In 3-dimensional space both Brownian and Lévy motions are inefficient at locating a small target. Whereas typical chemical reactants in a liquid environment have no choice but to passively diffuse until mutual encounter (4) intermittent models combine different search mechanisms (28,30). An important example of intermittent search models is the facilitated diffusion of transcription factors in gene regulation.…”

“…Thus, Brownian motion, whose fractal dimension is d f = 2, in 1-dimensional and 2-dimensional geometries leads to significant oversampling in the sense that a given site is revisited many times (27,28). Lévy motion with a jump length distribution λ(x) |x| −1−α (0 < α < 2) has fractal dimension d f = α and thus a comparatively reduced degree of oversampling.…”

How DNA coiling enhances target localization by proteins

“…A protein's search for the target site is thought to be accelerated by facilitated diffusion along nonspecific DNA (3)(4)(5)(6). Recent work has yielded considerable insight in the possible search strategies of sitespecific proteins (7)(8)(9)(10)(11)(12)(13)(14)(15). Assisted by DNA looping some proteins can, for instance, intermittently bind to two DNA segments simultaneously.…”

Michaelis-Menten reaction scheme as a unified approach towards the optimal restart problem

“…That and more, the simplicity and generality of the scheme have rendered it widely applicable and it is now used to describe anything from heterogeneous catalysis [4-6] to in vivo target search kinetics [7]. As a matter of fact, one can easily convince himself that any first passage time (FPT) process [8]-be it the time to target of a simple Brownian particle, or that of more sophisticated stochastic processes [9][10][11][12] and random searchers [13][14][15]-can become subject to restart [16][17][18][19][20][21][22] and is then naturally accommodated by the MMRS. Wishing to acquire a unified view on restart phenomena we identify the MMRS as an ideal object of study.…”