Optimization and universality of Brownian search in a basic model of quenched heterogeneous media

Random search with resetting in heterogeneous environments

We investigate random searches under stochastic position resetting at rate r, in a bounded 1D environment with space-dependent diffusivity D(x). For arbitrary shapes of D(x) and prescriptions of the associated multiplicative stochastic process, we obtain analytical expressions for the average time T for reaching the target (mean first-passage time), given the initial and reset positions, in good agreement with stochastic simulations. For arbitrary D(x), we obtain an exact closed-form expression for T, within a Stratonovich scenario, while for other prescriptions, like Itô and anti-Itô, we derive asymptotic approximations for small and large rates r. Exact results are also obtained for particular forms of D(x), such as the linear one, with arbitrary prescriptions, allowing to outline and discuss the main effects introduced by diffusive heterogeneity on a random search with resetting. We explore how the effectiveness of resetting varies with different types of heterogeneity.

Nature of barriers determines first passage times in heterogeneous media

Intuition suggests that passage times across a region increase with the number of barriers along the path. Can this fail depending on the nature of the barrier? To probe this fundamental question, we exactly solve for the first passage time in general d-dimensions for diffusive transport through a spatially patterned array of obstacles - either entropic or energetic, depending on the nature of the obstacles. For energetic barriers, we show that first passage times vary non-monotonically with the number of barriers, while for entropic barriers it increases monotonically. This non-monotonicity for energetic barriers is further reflected in the behaviour of effective diffusivity as well. We then design a simple experiment where a robotic bug navigates in a heterogeneous environment through a spatially patterned array of obstacles to validate our predictions. Finally, using numerical simulations, we show that this non-monotonic behaviour for energetic barriers is general and extends to even super-diffusive transport.

Particle-environment interactions in arbitrary dimensions: A unifying analytic framework to model diffusion with inert spatial heterogeneities

Inert interactions between randomly moving entities and spatial disorder play a crucial role in quantifying the diffusive properties of a system, with examples ranging from molecules advancing along dendritic spines to antipredator displacements of animals due to sparse vegetation. Despite the ubiquity of such phenomena, a general framework to model the movement explicitly in the presence of spatial heterogeneities is missing. Here, we tackle this challenge and develop an analytic theory to model inert particle-environment interactions in domains of arbitrary shape and dimensions. We use a discrete space formulation, which allows us to model the interactions between an agent and the environment as perturbed dynamics between lattice sites. Interactions from spatial disorder, such as impenetrable and permeable obstacles or regions of increased or decreased diffusivity, as well as many others, can be modelled using our framework. We provide exact expressions for the generating function of the occupation probability of the diffusing particle and related transport quantities such as first-passage, return, and exit probabilities and their respective means. We uncover a surprising property, the disorder indifference phenomenon of the mean first-passage time in the presence of a permeable barrier in quasi-1D systems. We demonstrate the widespread applicability of our formalism by considering three examples that span across scales and disciplines. (1) We explore an enhancement strategy of transdermal drug delivery. (2) We represent the movement decisions of an animal undergoing thigomotaxis, the tendency to remain at the peripheries of its enclosure, using a spatially disordered environment. (3) We illustrate the use of spatial heterogeneities to model inert interactions between particles by modeling the search for a promoter region on the DNA by transcription factors during gene transcription.

First-passage time analysis of diffusion-controlled reactions in single-molecule detection

Single-molecule detection (SMD) aims to achieve the ultimate limit-of-detection (LOD) in biosensing. This method detects a countable number of targeted analyte molecules in solution, where the dynamics of molecule diffusion, capturing, identification and delivery greatly impact the SMD's efficiency and accuracy. In this study, we adopt the first-passage time method to investigate the diffusion-controlled reaction process in SMD. We analyze the influence of detection conditions on incubation time and the expected coefficient of variation (CV) under three SMD molecule capturing strategies, including solid-phase capturing (one-dimensional solid-liquid interface fixation), liquid-phase magnetic bead (MB) capturing, and liquid-phase direct fluorescence pair labeling. We find that inside a finite-sized reaction chamber, a finite average reaction time exists in all three capturing strategies, while the liquid-phase strategies are in general more efficient than the solid-phase approaches. CV can be estimated by averaging first-passage time solely in all three strategies, and the CV reduction is achievable given an extended reaction time. To further enable zeptomolar detection, extra treatments, such as adopting liquid-phase fluorescence pairs with high diffusion rates to label the molecule, or designing specific sensing devices with large effective sensing areas would be required. This framework provides solid theoretical support to guide the design of SMD sensing strategies and sensor structures to achieve desired measurement time and CV.

Universal cover-time distribution of heterogeneous random walks

The cover-time problem, i.e., the time to visit every site in a system, is one of the key issues of random walks with wide applications in natural, social, and engineered systems. Addressing the full distribution of cover times for random walk on complex structures has been a long-standing challenge and has attracted persistent efforts. Usually it is assumed that the random walk is noncompact, to facilitate theoretical treatments by neglecting the correlations between visits. The known results are essentially limited to noncompact and homogeneous systems, where different sites are on an equal footing and have identical or close mean first-passage times, such as random walks on a torus. In contrast, realistic random walks are prevailingly heterogeneous with diversified mean first-passage times. Does a universal distribution still exist? Here, by considering the most general situations of noncompact random walks, we uncover a generalized rescaling relation for the cover time, exploiting the diversified mean first-passage times that have not been accounted for before. This allows us to concretely establish a universal distribution of the rescaled cover times for heterogeneous noncompact random walks, which turns out to be the Gumbel universality class that is ubiquitous for a large family of extreme value statistics. Our analysis is based on the transfer matrix framework, which is generic in that, besides heterogeneity, it is also robust against biased protocols, directed links, and self-connecting loops. The finding is corroborated with extensive numerical simulations of diverse heterogeneous noncompact random walks on both model and realistic topological structures. Our technical ingredient may be exploited for other extreme value or ergodicity problems with nonidentical distributions.

Efficiency of random search with space-dependent diffusivity

We address the problem of random search for a target in an environment with a space-dependent diffusion coefficient D(x). Considering a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain and analyze the first-passage-time distribution and use it to compute the search efficiency E=〈1/t〉. For the paradigmatic power-law diffusion coefficient D(x)=D_{0}|x|^{α}, where x is the distance from the target and α<2, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed-form expression for E, valid for arbitrary diffusion coefficient D(x). This result depends only on the distribution of diffusivity values and not on its spatial organization. Furthermore, the analytical expression predicts that a heterogeneous diffusivity profile leads to a lower efficiency than the homogeneous one with the same average level within the space between the target and the searcher initial position, but this efficiency can be exceeded for other interpretations.

Towards a full quantitative description of single-molecule reaction kinetics in biological cells

The first-passage time (FPT), i.e., the moment when a stochastic process reaches a given threshold value for the first time, is a fundamental mathematical concept with immediate applications. In particular, it quantifies the statistics of instances when biomolecules in a biological cell reach their specific binding sites and trigger cellular regulation. Typically, the first-passage properties are given in terms of mean first-passage times. However, modern experiments now monitor single-molecular binding-processes in living cells and thus provide access to the full statistics of the underlying first-passage events, in particular, inherent cell-to-cell fluctuations. We here present a robust explicit approach for obtaining the distribution of FPTs to a small partially reactive target in cylindrical-annulus domains, which represent typical bacterial and neuronal cell shapes. We investigate various asymptotic behaviours of this FPT distribution and show that it is typically very broad in many biological situations, thus, the mean FPT can differ from the most probable FPT by orders of magnitude. The most probable FPT is shown to strongly depend only on the starting position within the geometry and to be almost independent of the target size and reactivity. These findings demonstrate the dramatic relevance of knowing the full distribution of FPTs and thus open new perspectives for a more reliable description of many intracellular processes initiated by the arrival of one or few biomolecules to a small, spatially localised region inside the cell.

Diffusive escape through a narrow opening: new insights into a classic problem

We study the mean first exit time (T_ε) of a particle diffusing in a circular or a spherical micro-domain with an impenetrable confining boundary containing a small escape window (EW) of an angular size ε.

Effect of crowding and confinement on first-passage times: A model study

We study the "color dynamics" of a hard-disk fluid confined in an annulus, as well as the corresponding hard-sphere system in three dimensions, using event-driven simulation in order to explore the effect of confinement and self-crowding on the search for targets. We compute the mean first-passage times (MFPTs) of red particles transiting from the outer to the inner boundary as well as those of blue particles passing from the inner to the outer boundary for different packing fractions and geometries. In the steady state the reaction rate, defined as the rate of collision of red particles with the inner boundary, is inversely proportional to the sum of the MFPTs. The reaction rate is wall mediated (ballistic) at low densities and diffusion controlled at higher densities and displays a maximum at intermediate densities. At moderate to high densities, the presence of layering has a strong influence on the search process. The numerical results for the reaction rate and MFPTs are compared with a ballistic model at low densities and a Smoluchowski approach with uniform diffusivities at higher densities. We discuss the reasons for the limited validity of the theoretical approaches. The maximum in the reaction rate is qualitatively well rendered by a Bosanquet-like approach that interpolates between the two regimes. Finally, we compute the position-dependent diffusivity from the MFPTs and observe that it is out of phase with the radial density.

First passage time distribution in heterogeneity controlled kinetics: going beyond the mean first passage time

The first passage is a generic concept for quantifying when a random quantity such as the position of a diffusing molecule or the value of a stock crosses a preset threshold (target) for the first time. The last decade saw an enlightening series of new results focusing mostly on the so-called mean and global first passage time (MFPT and GFPT, respectively) of such processes. Here we push the understanding of first passage processes one step further. For a simple heterogeneous system we derive rigorously the complete distribution of first passage times (FPTs). Our results demonstrate that the typical FPT significantly differs from the MFPT, which corresponds to the long time behaviour of the FPT distribution. Conversely, the short time behaviour is shown to correspond to trajectories connecting directly from the initial value to the target. Remarkably, we reveal a previously overlooked third characteristic time scale of the first passage dynamics mirroring brief excursion away from the target.

First-Passage Times in d-Dimensional Heterogeneous Media

Although there are many theoretical studies of the mean first-passage time (MFPT), most neglect the diffusive heterogeneity of real systems. We present exact analytical expressions for the MFPT and residence times of a pointlike particle diffusing in a spherically symmetric d-dimensional heterogeneous system composed of two concentric media with different diffusion coefficients with an absorbing inner boundary (target) and a reflecting outer boundary. By varying the convention, e.g., Itō, Stratonovich, or isothermal, chosen to interpret the overdamped Langevin equation with multiplicative noise describing the diffusion process, we find different predictions and counterintuitive results for the residence time in the outer region and hence for the MFPT, while the residence time in the inner region is independent of the convention. This convention dependence of residence times and the MFPT could provide insights about the heterogeneous diffusion in a cell or in a tumor, or for animal and insect searches inside their home range.