Lattice theory of trapping reactions with mobile species

Optimal searcher distribution for parallel random target searches

We consider a problem of finding a target located in a finite d-dimensional domain, using N independent random walkers, when partial information about the target location is given as a probability distribution. When N is large, the first-passage time sensitively depends on the initial searcher distribution, which invokes the question of the optimal searcher distribution that minimizes the first-passage time. Here, we analytically derive the equation for the optimal distribution and explore its limiting expressions. If the target volume can be ignored, the optimal distribution is proportional to the target distribution to the power of one third. If we consider a target of a finite volume and the probability of the initial overlapping of searchers with the target cannot be ignored in the large N limit, the optimal distribution has a weak dependence on the target distribution, with its variation being proportional to the logarithm of the target distribution. Using Langevin dynamics simulations, we numerically demonstrate our predictions in one and two dimensions.

Capture of a diffusive prey by multiple predators in confined space

The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some detail, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in one, two, and three dimensions. Due to confinement, the survival probability of the target takes a form ∼e^{-t/τ} at large times t. The characteristic capture timescale τ associated with the rare capture events are rather challenging to measure. We use a computational algorithm that allows us to estimate τ with high accuracy. We study in detail the behavior of τ as a function of the system parameters, namely, the number of searchers N, the relative diffusivity r of the target with respect to the searcher, and the system size. We find that τ deviates from the ∼1/N scaling seen in the case of a static target, and this deviation varies continuously with r and the spatial dimensions.

First-encounter time of two diffusing particles in confinement

We investigate how confinement may drastically change both the probability density of the first-encounter time and the associated survival probability in the case of two diffusing particles. To obtain analytical insights into this problem, we focus on two one-dimensional settings: a half-line and an interval. We first consider the case with equal particle diffusivities, for which exact results can be obtained for the survival probability and the associated first-encounter time density valid over the full time domain. We also evaluate the moments of the first-encounter time when they exist. We then turn to the case with unequal diffusivities and focus on the long-time behavior of the survival probability. Our results highlight the great impact of boundary effects in diffusion-controlled kinetics even for simple one-dimensional settings, as well as the difficulty of obtaining analytic results as soon as the translational invariance of such systems is broken.

Relation between structure of blocked clusters and relaxation dynamics in kinetically constrained models

We investigate the relation between the cooperative length and relaxation time, represented, respectively, by the culling time and the persistence time, in the Fredrickson-Andersen, Kob-Andersen, and spiral kinetically constrained models. By mapping the dynamics to diffusion of defects, we find a relation between the persistence time, τ_{p}, which is the time until a particle moves for the first time, and the culling time, τ_{c}, which is the minimal number of particles that need to move before a specific particle can move, τ_{p}=τ_{c}^{γ}, where γ is model- and dimension-dependent. We also show that the persistence function in the Kob-Andersen and Fredrickson-Andersen models decays subexponentially in time, P(t)=exp[-(t/τ)^{β}], but unlike previous works, we find that the exponent β appears to decay to 0 as the particle density approaches 1.

First-passage times, mobile traps, and Hopf bifurcations

For a random walk on a confined one-dimensional domain, we consider mean first-passage times (MFPT) in the presence of a mobile trap. The question we address is whether a mobile trap can improve capture times over a stationary trap. We consider two scenarios: a randomly moving trap and an oscillating trap. In both cases, we find that a stationary trap actually performs better (in terms of reducing expected capture time) than a very slowly moving trap; however, a trap moving sufficiently fast performs better than a stationary trap. We explicitly compute the thresholds that separate the two regimes. In addition, we find a surprising relation between the oscillating trap problem and a moving-sink problem that describes reduced dynamics of a single spike in a certain regime of the Gray-Scott model. Namely, the above-mentioned threshold corresponds precisely to a Hopf bifurcation that induces oscillatory motion in the location of the spike. We use this correspondence to prove the uniqueness of the Hopf bifurcation.

Group chase and escape with some fast chasers

We study group chase and escape with some fast chasers. In our model chasers look for the nearest target and move to one of the nearest sites in order to catch the target. On the other hand, targets try to escape from the nearest chaser. When a chaser catches a target, the target is removed from the system and the number of targets decreases. The lifetime of targets, at which all targets caught, decreases as t^{α} with increasing the number of chasers. When there are no fast chasers and the total number of chasers is small, the exponent α is large. When the total number of chasers is large, α becomes small. There is an optimal number of chasers to minimize the cost used in order to catch all targets. However, when we add a few fast chasers, the region with the large α vanishes. The optimal number of chasers vanishes, and the cost monotonically increases with increasing the number of chasers.

Random collisions on branched networks: how simultaneous diffusion prevents encounters in inhomogeneous structures

A huge variety of natural phenomena, including prey-predator interaction, chemical reaction kinetics, foraging, and pharmacokinetics, are mathematically described as encounters between entities performing a random motion on an appropriate structure. On homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e., a single random walker returns to its starting point with probability 1. We prove here that this property does not hold on general inhomogeneous structures, and introduce the concept of two-particle transience, providing examples of realistic recurrent structures where two particles may never meet if they both move, while an encounter is certain if either stays put. We anticipate that our results will pave the way for the study of the effects of geometry in a wide array of natural phenomena involving interaction between randomly moving agents.

Chasing and escaping by three groups of species

We study group chasing and escaping between three species. In our model, one species acts as a group of chasers for another species and acts as a group of targets for the third species. When a particle is caught by a target, the particle becomes a new chaser. Although the ratio of three species is changed, the total number of particles is conserved. When particles move randomly, the numbers of the three species change periodically but no species seems to become extinct. If particles escape from the nearest chaser and chase the nearest target, the extinction of a species occurs. The extinction induces that of the second species and finally only one species survives.

Global mean first-passage times of random walks on complex networks

We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position, the so-called global mean first-passage time (GMFPT). This bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the GMFPT with the network size. We show that this minimal scaling, which can be arbitrarily slow, is realized under the simple condition that the random walk is transient at the target site and independently of the small-world, scale-free, or fractal properties of the network. Last, we put forward that the GMFPT to a specific target is not a representative property of the network since the target averaged GMFPT satisfies much more restrictive bounds.

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.

Survival of an evasive prey

We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a predator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln P(ev)(t) approximately (N/V)(2)ln P(imm)(t) between the survival probabilities of an evasive and an immobile prey. Hence, when the density rho = N/V of the predators is low, rho << 1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.

First passage time analysis of animal movement and insights into the functional response

Movement plays a role in structuring the interactions between individuals, their environment, and other species. Although movement models coupled with empirical data are widely used to study animal distribution, they have seldom been used to study search time. This paper proposes first passage time as a novel approach for understanding the effect of the landscape on animal movement and search time. In the context of animal movement, first passage time is the time taken for an animal to reach a specified site for the first time. We synthesize current first passage time theory and derive a general first passage time equation for animal movement. This equation is related to the Fokker-Planck equation, which is used to describe the distribution of animals in the landscape. We illustrate the first passage time method by analyzing the effect of territorial behavior on the time required for a red fox to locate prey throughout its home range. Using first passage time to compute search times, we consider the effect of two different searching modes on a functional response. We show that random searching leads to a Holling type III functional response. First passage time analysis provides a new tool for studying how animal movement may influence ecological processes.

Exact asymptotics for nonradiative migration-accelerated energy transfer in one-dimensional systems

We study direct energy transfer by multipolar or exchange interactions between diffusive excited donor and diffusive unexcited acceptors. Extending over the case of long-range transfer of an excitation energy a nonperturbative approach by Bray and Blythe [Phys. Rev. Lett. 89, 150601 (2002)], originally developed for contact diffusion-controlled reactions, we determine exactly long-time asymptotics of the donor decay function in one-dimensional systems.

Survival probability of a particle in a sea of mobile traps: a tale of tails

We study the long-time tails of the survival probability P(t) of an A particle diffusing in d-dimensional media in the presence of a concentration rho of traps B that move subdiffusively, such that the mean square displacement of each trap grows as tgamma with 0 < or = gamma < or =1. Starting from a continuous time random walk description of the motion of the particle and of the traps, we derive lower and upper bounds for P(t) and show that for gamma < or =2/(d+2) these bounds coincide asymptotically, thus determining asymptotically exact results. The asymptotic decay law in this regime is exactly that obtained for immobile traps. This means that for sufficiently subdiffusive traps, the moving A particle sees the traps as essentially immobile, and Lifshitz or trapping tails remain unchanged. For gamma >2/(d+2) and d< or =2 the upper and lower bounds again coincide, leading to a decay law equal to that of a stationary particle. Thus, in this regime the moving traps see the particle as essentially immobile. For d>2 , however, the upper and lower bounds in this gamma regime no longer coincide, and the decay law for the survival probability of the A particle remains ambiguous.

Mobile trap algorithm for zinc detection using protein sensors

We present a mobile trap algorithm to sense zinc ions using protein-based sensors such as carbonic anhydrase (CA). Zinc is an essential biometal required for mammalian cellular functions although its intracellular concentration is reported to be very low. Protein-based sensors like CA molecules are employed to sense rare species like zinc ions. In this study, the zinc ions are mobile targets, which are sought by the mobile traps in the form of sensors. Particle motions are modeled using random walk along with the first passage technique for efficient simulations. The association reaction between sensors and ions is incorporated using a probability (p1) upon an ion-sensor collision. The dissociation reaction of an ion-bound CA molecule is modeled using a second, independent probability (p2). The results of the algorithm are verified against the traditional simulation techniques (e.g., Gillespie's algorithm). This study demonstrates that individual sensor molecules can be characterized using the probability pair (p1,p2), which, in turn, is linked to the system level chemical kinetic constants, kon and koff. Further investigations of CA-Zn reaction using the mobile trap algorithm show that when the diffusivity of zinc ions approaches that of sensor molecules, the reaction data obtained using the static trap assumption differ from the reaction data obtained using the mobile trap formulation. This study also reveals similar behavior when the sensor molecule has higher dissociation constant. In both the cases, the reaction data obtained using the static trap formulation reach equilibrium at a higher number of complex molecules (ion-bound sensor molecules) compared to the reaction data from the mobile trap formulation. With practical limitations on the number sensors that can be inserted/expressed in a cell and stochastic nature of the intracellular ionic concentrations, fluorescence from the number of complex sensor molecules at equilibrium will be the measure of the intracellular ion concentration. For reliable detection of zinc ions, it is desirable that the sensors must not bind all the zinc ions tightly, but should rather bind and unbind. Thus for a given fluorescence and with association-dissociation reactions between ions and sensors, the static trap approach will underestimate the number of zinc ions present in the system.

Intermittent search process and teleportation

The authors study an intermittent search process combining diffusion and "teleportation" phases in a d-dimensional spherical continuous system and in a regular lattice. The searcher alternates diffusive phases, during which targets can be discovered, and fast phases (teleportation) which randomly relocate the searcher, but do not allow for target detection. The authors show that this alternation can be favorable for minimizing the time of first discovery, and that this time can be optimized by a convenient choice of the mean waiting times of each motion phase. The optimal search strategy is explicitly derived in the continuous case and in the lattice case. Arguments are given to show that much more general intermittent motions do provide optimal search strategies in d dimensions. These results can be useful in the context of heterogeneous catalysis or in various biological examples of transport through membrane pores.