First passages in bounded domains: when is the mean first passage time meaningful?

Searching for a partially absorbing target by a run-and-tumble particle in a confined space

A random search of a partially absorbing target by a run-and-tumble particle in a confined one-dimensional space is investigated. We analytically obtain the mean searching time, which shows a nonmonotonic behavior as a function of the self-propulsion speed of the active particle, indicating the existence of an optimal speed, when the absorption strength of the target is finite. In the limit of large and small absorption strengths, respectively, asymptotes of the mean searching time and the optimal speed are found. We also demonstrate that the first-passage problem of a diffusive run-and-tumble particle in high dimensions can be mapped into a one-dimensional problem with a partially absorbing target. Finally, as a practical application exploiting the existence of the optimal speed, we propose a filtering device to extract active particles with a desired speed and evaluate how the resolution of the filtering device depends on the absorption strength.

Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime

We derive general bounds on the probability that the empirical first-passage time τ[over ¯]_{n}≡∑_{i=1}^{n}τ_{i}/n of a reversible ergodic Markov process inferred from a sample of n independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct nonasymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Escape dynamics of a self-propelled nanorod from circular confinements with narrow openings

We perform computer simulations to explore the escape dynamics of a self-propelled (active) nanorod from circular confinements with narrow opening(s). Our results clearly demonstrate how the persistent and directed motion of the nanorod helps it to escape. Such escape events are absent if the nanorod is passive. To quantify the escape dynamics, we compute the radial probability density function (RPDF) and mean first escape time (MFET) and show how the activity is responsible for the bimodality of RPDF, which is clearly absent if the nanorod is passive. Broadening of displacement distributions with activity has also been observed. The computed mean first escape time decreases with activity. In contrast, the fluctuations of the first escape times vary in a non-monotonic way. This results in high values of the coefficient of variation and indicates the presence of multiple timescales in first escape time distributions and multimodality in uniformity index distributions. We hope our study will help in differentiating activity-driven escape dynamics from purely thermal passive diffusion in confinement.

Nonmonotonous Translocation Time of Polymers across Pores

Polymers confined in corrugated channels, i.e., channels of varying amplitude, display multiple local maxima and minima of the diffusion coefficient upon increasing their degree of polymerization N. We propose a theoretical effective free energy for linear polymers based on a Fick-Jacobs approach. We validate the predictions against numerical data, obtaining quantitative agreement for the effective free energy, the diffusion coefficient, and the mean first passage time. Finally, we employ the effective free energy to compute the polymer lengths N_{min} at which the diffusion coefficient presents a minimum: we find a scaling expression that we rationalize with a blob model. Our results could be useful to design porous adsorbers, that separate polymers of different sizes without the action of an external flow.

How Do Nucleosome Dynamics Regulate Protein Search on DNA?

Nearly three-fourths of all eukaryotic DNA is occupied by nucleosomes, protein-DNA complexes comprising octameric histone core proteins and ∼150 base pairs of DNA. In addition to acting as a DNA compaction vehicle, the dynamics of nucleosomes regulate the DNA site accessibility for the nonhistone proteins, thereby controlling regulatory processes involved in determining the cell identity and cell fate. Here, we propose an analytical framework to analyze the role of nucleosome dynamics on the target search process of transcription factors through a simple discrete-state stochastic description of the search process. By considering the experimentally determined kinetic rates associated with protein and nucleosome dynamics as the only inputs, we estimate the target search time of a protein via first-passage probability calculations separately during nucleosome breathing and sliding dynamics. Although both the nucleosome dynamics permit transient access to the DNA sites that are otherwise occluded by the histone proteins, our result suggests substantial differences between the protein search mechanism on a nucleosome performing breathing and sliding dynamics. Furthermore, we identify the molecular factors that influence the search efficiency and demonstrate how these factors together portray a highly dynamic landscape of gene regulation. Our analytical results are validated using extensive Monte Carlo simulations.

Encounter-based approach to the escape problem

We revise the encounter-based approach to imperfect diffusion-controlled reactions, which employs the statistics of encounters between a diffusing particle and the reactive region to implement surface reactions. We extend this approach to deal with a more general setting, in which the reactive region is surrounded by a reflecting boundary with an escape region. We derive a spectral expansion for the full propagator and investigate the behavior and probabilistic interpretations of the associated probability flux density. In particular, we obtain the joint probability density of the escape time and the number of encounters with the reactive region before escape, and the probability density of the first-crossing time of a prescribed number of encounters. We briefly discuss generalizations of the conventional Poissonian-type surface reaction mechanism described by Robin boundary condition and potential applications of this formalism in chemistry and biophysics.

Extracting, quantifying, and comparing dynamical and biomechanical properties of living matter through single particle tracking

A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.

Theoretical insights into the full description of DNA target search by subdiffusing proteins

DNA binding proteins (DBPs) diffuse in the cytoplasm to recognise and bind with their respective target sites on the DNA to initiate several biologically important processes. The first passage time distributions (FPTDs) of DBPs are useful in quantifying the timescales of the most-probable search paths in addition to the mean value of the distribution which, strikingly, are decades of order apart in time. However, extremely crowded in vivo conditions or the viscoelasticity of the cellular medium among other factors causes biomolecules to exhibit anomalous diffusion which is usually overlooked in most theoretical studies. We have obtained approximate analytical expressions of a general FPTD and the two characteristic timescales that are valid for any single subdiffusing protein searching for its target in vivo. Our results can be applied to single-particle tracking experiments of target search.

State-Temporal Compression in Reinforcement Learning With the Reward-Restricted Geodesic Metric

It is difficult to solve complex tasks that involve large state spaces and long-term decision processes by reinforcement learning (RL) algorithms. A common and promising method to address this challenge is to compress a large RL problem into a small one. Towards this goal, the compression should be state-temporal and optimality-preserving (i.e., the optimal policy of the compressed problem should correspond to that of the uncompressed problem). In this paper, we propose a reward-restricted geodesic (RRG) metric, which can be learned by a neural network, to perform state-temporal compression in RL. We prove that compression based on the RRG metric is approximately optimality-preserving for the raw RL problem endowed with temporally abstract actions. With this compression, we design an RRG metric-based reinforcement learning (RRG-RL) algorithm to solve complex tasks. Experiments in both discrete (2D Minecraft) and continuous (Doom) environments demonstrated the superiority of our method over existing RL approaches.

Spectral fingerprints of nonequilibrium dynamics: The case of a Brownian gyrator

The same system can exhibit a completely different dynamical behavior when it evolves in equilibrium conditions or when it is driven out-of-equilibrium by, e.g., connecting some of its components to heat baths kept at different temperatures. Here we concentrate on an analytically solvable and experimentally relevant model of such a system-the so-called Brownian gyrator-a two-dimensional nanomachine that performs a systematic, on average, rotation around the origin under nonequilibrium conditions, while no net rotation takes place under equilibrium ones. On this example, we discuss a question whether it is possible to distinguish between two types of a behavior judging not upon the statistical properties of the trajectories of components but rather upon their respective spectral densities. The latter are widely used to characterize diverse dynamical systems and are routinely calculated from the data using standard built-in packages. From such a perspective, we inquire whether the power spectral densities possess some "fingerprint" properties specific to the behavior in nonequilibrium. We show that indeed one can conclusively distinguish between equilibrium and nonequilibrium dynamics by analyzing the cross-correlations between the spectral densities of both components in the short frequency limit, or from the spectral densities of both components evaluated at zero frequency. Our analytical predictions, corroborated by experimental and numerical results, open a new direction for the analysis of a nonequilibrium dynamics.

Dynamic and facilitated binding of topoisomerase accelerates topological relaxation

How type 2 Topoisomerase (TopoII) proteins relax and simplify the topology of DNA molecules is one of the most intriguing open questions in genome and DNA biophysics. Most of the existing models neglect the dynamics of TopoII which is expected of proteins searching their targets via facilitated diffusion. Here, we show that dynamic binding of TopoII speeds up the topological relaxation of knotted substrates by enhancing the search of the knotted arc. Intriguingly, this in turn implies that the timescale of topological relaxation is virtually independent of the substrate length. We then discover that considering binding biases due to facilitated diffusion on looped substrates steers the sampling of the topological space closer to the boundaries between different topoisomers yielding an optimally fast topological relaxation. We discuss our findings in the context of topological simplification in vitro and in vivo.

Reversible target-binding kinetics of multiple impatient particles

Certain biochemical reactions can only be triggered after binding a sufficient number of particles to a specific target region such as an enzyme or a protein sensor. We investigate the distribution of the reaction time, i.e., the first instance when all independently diffusing particles are bound to the target. When each particle binds irreversibly, this is equivalent to the first-passage time of the slowest (last) particle. In turn, reversible binding to the target renders the problem much more challenging and drastically changes the distribution of the reaction time. We derive the exact solution of this problem and investigate the short-time and long-time asymptotic behaviors of the reaction time probability density. We also analyze how the mean reaction time depends on the unbinding rate and the number of particles. Our exact and asymptotic solutions are compared to Monte Carlo simulations.

Optimal Non-Markovian Search Strategies with n-Step Memory

Stochastic search processes are ubiquitous in nature and are expected to become more efficient when equipped with a memory, where the searcher has been before. A natural realization of a search process with long-lasting memory is a migrating cell that is repelled from the diffusive chemotactic signal that it secretes on its way, denoted as an autochemotactic searcher. To analyze the efficiency of this class of non-Markovian search processes, we present a general formalism that allows one to compute the mean first-passage time (MFPT) for a given set of conditional transition probabilities for non-Markovian random walks on a lattice. We show that the optimal choice of the n-step transition probabilities decreases the MFPT systematically and substantially with an increasing number of steps. It turns out that the optimal search strategies can be reduced to simple cycles defined by a small parameter set and that mirror-asymmetric walks are more efficient. For the autochemotactic searcher, we show that an optimal coupling between the searcher and the chemical reduces the MFPT to 1/3 of the one for a Markovian random walk.

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.

First-passage fingerprints of water diffusion near glutamine surfaces

The extent to which biological interfaces affect the dynamics of water plays a key role in the exchange of matter and chemical interactions that are essential for life. The density and the mobility of water molecules depend on their proximity to biological interfaces and can play an important role in processes such as protein folding and aggregation. In this work, we study the dynamics of water near glutamine surfaces-a system of interest in studies of neurodegenerative diseases. Combining molecular-dynamics simulations and stochastic modelling, we study how the mean first-passage time and related statistics of water molecules escaping subnanometer-sized regions vary from the interface to the bulk. Our analysis reveals a dynamical complexity that reflects underlying chemical and geometrical properties of the glutamine surfaces. From the first-passage time statistics of water molecules, we infer their space-dependent diffusion coefficient in directions normal to the surfaces. Interestingly, our results suggest that the mobility of water varies over a longer length scale than the chemical potential associated with the water-protein interactions. The synergy of molecular dynamics and first-passage techniques opens the possibility for extracting space-dependent diffusion coefficients in more complex, inhomogeneous environments that are commonplace in living matter.

Translocation of a Polyelectrolyte through a Nanopore in the Presence of Trivalent Counterions: A Comparison with the Cases in Monovalent and Divalent Salt Solutions

A polyelectrolyte threading through a nanopore in a trivalent salt solution is investigated by means of molecular dynamics simulations under a reflective wall boundary. By varying the chain length N and the strength E of the driving electric field applied inside the pore, the translocation time is carefully calculated to get rid of the bouncing effect because of the boundary. The results are analyzed under the scaling form ⟨τ⟩ ∼ N ^α E ^-δ and four driving force regimes; namely, the unbiased, the weakly driven, the strongly driven trumpet, and the strongly driven isoflux regime, are distinguished. The exponents are calculated in each regime and compared with the cases in the monovalent and divalent salt solutions. Owing to strong condensation of counter ions, the changes of the exponents in the force regimes are found to be nontrivial. A large increase in translocation time can be, however, achieved as the driving field is weak. The variations of the chain size, the ion condensation, and the effective chain charge show that the process is proceeded in a quasi-equilibrium way in the unbiased regime and deviated to exhibit strong nonequilibrium characteristics as E increases. Several astonishing scaling behaviors of the waiting time function, the translocation velocity, and the diffusion properties are discovered in the study. The results provide deep insights into the phenomena of polyelectrolyte translocation in various salt solutions at different driving forces.

First passage of an active particle in the presence of passive crowders

We experimentally study the stochastic transport of a self-propelled camphor boat, driven by Marangoni forces, through a crowd of passive paper discs floating on water. We analyze the statistics of the first passage times of the active particle to travel from the center of a circular container to its boundary. While the mean times rise monotonically as a function of the covered area fraction φ of the passive paper discs, their fluctuations show a non-monotonic behavior - being higher at low and high value of φ compared to intermediate values. The reason is traced to an interplay of two distinct sources of fluctuations - one intrinsic to the dynamics, while the other due to the crowding.

Cooperatively enhanced reactivity and "stabilitaxis" of dissociating oligomeric proteins

Many functional units in biology, such as enzymes or molecular motors, are composed of several subunits that can reversibly assemble and disassemble. This includes oligomeric proteins composed of several smaller monomers, as well as protein complexes assembled from a few proteins. By studying the generic spatial transport properties of such proteins, we investigate here whether their ability to reversibly associate and dissociate may confer on them a functional advantage with respect to nondissociating proteins. In uniform environments with position-independent association-dissociation, we find that enhanced diffusion in the monomeric state coupled to reassociation into the functional oligomeric form leads to enhanced reactivity with localized targets. In nonuniform environments with position-dependent association-dissociation, caused by, for example, spatial gradients of an inhibiting chemical, we find that dissociating proteins generically tend to accumulate in regions where they are most stable, a process that we term "stabilitaxis."

Characterizing network topology using first-passage analysis

Understanding the topological characteristics of complex networks and how they affect navigability is one of the most important goals in science today, as it plays a central role in various economic, biological, ecological, and social systems. Here we apply first-passage analysis tools to investigate the properties and characteristics of random walkers in networks with different topology. Starting with the simplest two-dimensional square lattice, we modify its topology incrementally by randomly reconnecting links between sites. We characterize these networks by first-passage time from a significant number of random walkers without interaction, varying the departure and arrival locations. We also apply the concept of first-passage simultaneity, which measures the likelihood of two walkers reaching their destination together. These measures, together with the site occupancy statistics during the processes, allowed us to differentiate the studied networks, especially the random networks from the scale-free networks, by their navigability. We also show that small-world features can also be highlighted with the proposed technique.

Field-driven tracer diffusion through curved bottlenecks: fine structure of first passage events

Using scaling arguments and extensive numerical simulations, we study the dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a hard-wall boundary.

Transport and escape in a deformable channel driven by fractional Gaussian noise

Fractional Gaussian noise (FGN) with the Hurst exponent H is an important tool to model various phenomena in biophysical systems, like subdiffusion in a single protein molecule. Considering that there also exists a confined structure which can be modeled as a channel in these systems, transport and escape driven by FGN in a deformable channel are investigated in this paper. By calculating the mean velocity, and the mean first passage time (MFPT) for crossing the nearest bottleneck and the probability distribution of the final position, effects of FGN and channel structure on the system dynamics are illustrated. Our results indicate that FGN has a complex influence mechanism under different combinations of H and the noise intensity. For a persistence case (H>0.5), the mean velocity decreases but MFPT increases with the increase of the noise intensity and H. While for an antipersistence case (H<0.5), when H is small, the relationships among the mean velocity, MFPT and the noise intensity are exactly the opposite to persistence cases. When H has a large value, the mean velocity tends to first decrease and then increase. Moreover, effects of the bottleneck and channel asymmetry are investigated. It is shown that a small H and a large channel width can lead to a large mean velocity and fast crossing. Besides, a channel asymmetry can affect the system dynamics by inducing asymmetric structure and adjusting the width of bottleneck. However, the effect of the bottleneck is the main factor. Therefore, a combination of channel with wide bottleneck and FGN in an antipersistence regime is the optimal choice to promote the transport and escape. These results provide a basis for the explanation of molecular activity in living organisms and the design of particle mixture separators.

Polymer Translocation Across a Corrugated Channel: Fick⁻Jacobs Approximation Extended Beyond the Mean First-Passage Time

Polymer translocation across a corrugated channel is a paradigmatic stochastic process encountered in diverse systems. The instance of time when a polymer first arrives to some prescribed location defines an important characteristic time-scale for various phenomena, which are triggered or controlled by such an event. Here we discuss the translocation dynamics of a Gaussian polymer in a periodically-corrugated channel using an appropriately generalized Fick⁻Jacobs approach. Our main aim is to probe an effective broadness of the first-passage time distribution (FPTD), by determining the so-called coefficient of variation γ of the FPTD, defined as the ratio of the standard deviation versus the mean first-passage time (MFPT). We present a systematic analysis of γ as a function of a variety of system's parameters. We show that γ never significantly drops below 1 and, in fact, can attain very large values, implying that the MFPT alone cannot characterize the first-passage statistics of the translocation process exhaustively well.

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.

Spatial Cytoskeleton Organization Supports Targeted Intracellular Transport

The efficiency of intracellular cargo transport from specific sources to target locations is strongly dependent upon molecular motor-assisted motion along the cytoskeleton. Radial transport along microtubules and lateral transport along the filaments of the actin cortex underneath the cell membrane are characteristic for cells with a centrosome. The interplay between the specific cytoskeleton organization and the motor performance results in a spatially inhomogeneous intermittent search strategy. To analyze the efficiency of such intracellular search strategies, we formulate a random velocity model with intermittent arrest states. We evaluate efficiency in terms of mean first passage times for three different, frequently encountered intracellular transport tasks: 1) the narrow escape problem, which emerges during cargo transport to a synapse or other specific region of the cell membrane; 2) the reaction problem, which considers the binding time of two particles within the cell; and 3) the reaction-escape problem, which arises when cargo must be released at a synapse only after pairing with another particle. Our results indicate that cells are able to realize efficient search strategies for various intracellular transport tasks economically through a spatial cytoskeleton organization that involves only a narrow actin cortex rather than a cell body filled with randomly oriented actin filaments.

Theory for controlling individual self-propelled micro-swimmers by photon nudging I: directed transport

Photon nudging is a new experimental method which enables the force-free manipulation and localization of individual self-propelled artificial micro-swimmers in fluidic environments.

Acceleration of bursty multiprotein target search kinetics on DNA by colocalisation

Proteins are capable of locating specific targets on DNA by employing a facilitated diffusion process with intermittent 1D and 3D search steps. We here uncover the implications of colocalisation of protein production and DNA binding sites via computer simulations.

Multitarget search on complex networks: A logarithmic growth of global mean random cover time

We investigate multitarget search on complex networks and derive an exact expression for the mean random cover time that quantifies the expected time a walker needs to visit multiple targets. Based on this, we recover and extend some interesting results of multitarget search on networks. Specifically, we observe the logarithmic increase of the global mean random cover time with the target number for a broad range of random search processes, including generic random walks, biased random walks, and maximal entropy random walks. We show that the logarithmic growth pattern is a universal feature of multi-target search on networks by using the annealed network approach and the Sherman-Morrison formula. Moreover, we find that for biased random walks, the global mean random cover time can be minimized, and that the corresponding optimal parameter also minimizes the global mean first passage time, pointing towards its robustness. Our findings further confirm that the logarithmic growth pattern is a universal law governing multitarget search in confined media.

Quantifying non-ergodicity of anomalous diffusion with higher order moments

Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.

Noise-Driven Return Statistics: Scaling and Truncation in Stochastic Storage Processes

In countless systems, subjected to variable forcing, a key question arises: how much time will a state variable spend away from a given threshold? When forcing is treated as a stochastic process, this can be addressed with first return time distributions. While many studies suggest exponential, double exponential or power laws as empirical forms, we contend that truncated power laws are natural candidates. To this end, we consider a minimal stochastic mass balance model and identify a parsimonious mechanism for the emergence of truncated power law return times. We derive boundary-independent scaling and truncation properties, which are consistent with numerical simulations, and discuss the implications and applicability of our findings.

Spatial heterogeneity can facilitate the target search of self-propelled particles

A numerical investigation of the target search dynamics of self-propelled particles (SPPs) in heterogeneous environments is presented in this work. We show that the spatial heterogeneity has a dramatic effect on the target search dynamics of SPPs. The relative magnitude of the self-propulsion length l_p and the radius of the circular domain R_c determines how the mean search time of SPPs τ depends on the area fraction of fixed obstacles ϕ_ob. For l_p < R_c, the target search process is diffusion-dominated so that a monotonic increase in τ with increasing ϕ_ob is observed. For l_p > R_c, τ is shown to be a non-monotonic convex function as a function of ϕ_ob due to the interplay of the distribution-dominated and diffusion-dominated dynamic regimes. Furthermore, at fixed ϕ_ob, τ shows a minimum upon increasing the self-propulsion velocity v₀ of a SPP of a slow rotational diffusion when it searches for a target at low ϕ_ob, while it decreases monotonically at high ϕ_ob. The present work highlights that the introduction of spatial heterogeneity causes rich dynamic behaviors of a SPP searching for a target, and deepens our understanding of the transport of active matter in heterogeneous media.

Global warming precipitation accumulation increases above the current-climate cutoff scale

Precipitation accumulations, integrated over rainfall events, can be affected by both intensity and duration of the storm event. Thus, although precipitation intensity is widely projected to increase under global warming, a clear framework for predicting accumulation changes has been lacking, despite the importance of accumulations for societal impacts. Theory for changes in the probability density function (pdf) of precipitation accumulations is presented with an evaluation of these changes in global climate model simulations. We show that a simple set of conditions implies roughly exponential increases in the frequency of the very largest accumulations above a physical cutoff scale, increasing with event size. The pdf exhibits an approximately power-law range where probability density drops slowly with each order of magnitude size increase, up to a cutoff at large accumulations that limits the largest events experienced in current climate. The theory predicts that the cutoff scale, controlled by the interplay of moisture convergence variance and precipitation loss, tends to increase under global warming. Thus, precisely the large accumulations above the cutoff that are currently rare will exhibit increases in the warmer climate as this cutoff is extended. This indeed occurs in the full climate model, with a 3 °C end-of-century global-average warming yielding regional increases of hundreds of percent to >1,000% in the probability density of the largest accumulations that have historical precedents. The probabilities of unprecedented accumulations are also consistent with the extension of the cutoff.

Numerical analysis of homogeneous and inhomogeneous intermittent search strategies

Random search processes for targets that are inhomogeneously distributed in a search domain require spatially inhomogeneous search strategies to find the target as fast as possible. Here, we compare systematically the efficiency of homogeneous and inhomogeneous strategies for intermittent search, which alternates stochastically between slow, diffusive motion in which the target can be detected and fast ballistic motion during which targets cannot be detected. We analyze the mean first-passage time of homogeneous and inhomogeneous strategies for three paradigmatic search problems: (1) the narrow escape problem, i.e., the searcher looks for a small area on the boundary of the search domain, (2) reaction kinetics, i.e., the detection of an immobile target in the interior of a search domain, and (3) the reaction-escape problem, i.e., the searcher first needs to find a mobile target before it can escape through a narrow area on the boundary. Using families of inhomogeneous strategies, partially motivated by the organization of the cytoskeleton in cells with a centrosome, we show that they are almost always more efficient than homogeneous strategies.

Uniform asymptotic approximation of diffusion to a small target: Generalized reaction models

The diffusion of a reactant to a binding target plays a key role in many biological processes. The reaction radius at which the reactant and target may interact is often a small parameter relative to the diameter of the domain in which the reactant diffuses. We develop uniform in time asymptotic expansions in the reaction radius of the full solution to the corresponding diffusion equations for two separate reactant-target interaction mechanisms: the Doi or volume reactivity model and the Smoluchowski-Collins-Kimball partial-absorption surface reactivity model. In the former, the reactant and target react with a fixed probability per unit time when within a specified separation. In the latter, upon reaching a fixed separation, they probabilistically react or the reactant reflects away from the target. Expansions of the solution to each model are constructed by projecting out the contribution of the first eigenvalue and eigenfunction to the solution of the diffusion equation and then developing matched asymptotic expansions in Laplace-transform space. Our approach offers an equivalent, but alternative, method to the pseudopotential approach we previously employed [Isaacson and Newby, Phys. Rev. E 88, 012820 (2013)PLEEE81539-375510.1103/PhysRevE.88.012820] for the simpler Smoluchowski pure-absorption reaction mechanism. We find that the resulting asymptotic expansions of the diffusion equation solutions are identical with the exception of one parameter: the diffusion-limited reaction rates of the Doi and partial-absorption models. This demonstrates that for biological systems in which the reaction radius is a small parameter, properly calibrated Doi and partial-absorption models may be functionally equivalent.

Hybrid asymptotic-numerical approach for estimating first-passage-time densities of the two-dimensional narrow capture problem

A hybrid asymptotic-numerical method is presented for obtaining an asymptotic estimate for the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with a reflecting boundary. As motivation for this study, we calculate the variance in the capture time of a random walker by a single interior trap and determine this quantity to be comparable in magnitude to the mean. This implies that the mean is not necessarily reflective of typical capture times and that the full density must be determined. To solve the underlying diffusion equation, the method of Laplace transforms is used to obtain an elliptic problem of modified Helmholtz type. In the limit of vanishing trap sizes, each trap is represented as a Dirac point source that permits the solution of the transform equation to be represented as a superposition of Helmholtz Green's functions. Using this solution, we construct asymptotic short-time solutions of the first-passage-time density, which captures peaks associated with rapid capture by the absorbing traps. When numerical evaluation of the Helmholtz Green's function is employed followed by numerical inversion of the Laplace transform, the method reproduces the density for larger times. We demonstrate the accuracy of our solution technique with a comparison to statistics obtained from a time-dependent solution of the diffusion equation and discrete particle simulations. In particular, we demonstrate that the method is capable of capturing the multimodal behavior in the capture time density that arises when the traps are strategically arranged. The hybrid method presented can be applied to scenarios involving both arbitrary domains and trap shapes.

Temporal Correlations of the Running Maximum of a Brownian Trajectory

We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0,t_{1}] and [0,t_{2}], with t_{2}>t_{1}. We determine the exact forms of the distribution functions P(m,M) and P(G=M-m), and calculate the moments E{(M-m)^{k}} and the cross-moments E{m^{l}M^{k}} with arbitrary integers l and k. We show that correlations between m and M decay as sqrt[t_{1}/t_{2}] when t_{2}/t_{1}→∞, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient ρ(m,M) and the power spectrum of M_{t}, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.

Random pure states: Quantifying bipartite entanglement beyond the linear statistics

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

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.

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

The kinetics of a variety of transport-controlled processes can be reduced to the problem of determining the mean time needed to arrive at a given location for the first time, the so-called mean first-passage time (MFPT) problem. The occurrence of occasional large jumps or intermittent patterns combining various types of motion are known to outperform the standard random walk with respect to the MFPT, by reducing oversampling of space. Here we show that a regular but spatially heterogeneous random walk can significantly and universally enhance the search in any spatial dimension. In a generic minimal model we consider a spherically symmetric system comprising two concentric regions with piecewise constant diffusivity. The MFPT is analyzed under the constraint of conserved average dynamics, that is, the spatially averaged diffusivity is kept constant. Our analytical calculations and extensive numerical simulations demonstrate the existence of an optimal heterogeneity minimizing the MFPT to the target. We prove that the MFPT for a random walk is completely dominated by what we term direct trajectories towards the target and reveal a remarkable universality of the spatially heterogeneous search with respect to target size and system dimensionality. In contrast to intermittent strategies, which are most profitable in low spatial dimensions, the spatially inhomogeneous search performs best in higher dimensions. Discussing our results alongside recent experiments on single-particle tracking in living cells, we argue that the observed spatial heterogeneity may be beneficial for cellular signaling processes.

First-passage time of run-and-tumble particles

We solve the problem of first-passage time for run-and-tumble particles in one dimension. Exact expression is derived for the mean first-passage time in the general case, considering external force fields and chemotactic fields, giving rise to space-dependent swim speed and tumble rate. Agreement between theoretical formulae and numerical simulations is obtained in the analyzed case studies --constant and sinusoidal force fields, constant gradient chemotactic field. Reported findings can be useful to get insights into very different phenomena involving active particles, such as bacterial motion in external fields, intracellular transport, cell migration, animal foraging.

Geometry-induced fluctuations of olfactory searches in bounded domains

In olfactory search an immobile target emits chemical molecules at constant rate. The molecules are transported by the medium, which is assumed to be turbulent. Considering a searcher able to detect such chemical signals and whose motion follows the infotaxis strategy, we study the statistics of the first-passage time to the target when the searcher moves on a finite two-dimensional lattice of different geometries. Far from the target, where the concentration of chemicals is low, the direction of the searcher's first movement is determined by the geometry of the domain and the topology of the lattice, inducing strong fluctuations on the average search time with respect to the initial position of the searcher. The domain is partitioned in well-defined regions characterized by the direction of the first movement. If the search starts over the interface between two different regions, large fluctuations in the search time are observed.

Particle invasion, survival, and non-ergodicity in 2D diffusion processes with space-dependent diffusivity

We study the thermal Markovian diffusion of tracer particles in a 2D medium with spatially varying diffusivity D(r), mimicking recently measured, heterogeneous maps of the apparent diffusion coefficient in biological cells. For this heterogeneous diffusion process (HDP) we analyse the mean squared displacement (MSD) of the tracer particles, the time averaged MSD, the spatial probability density function, and the first passage time dynamics from the cell boundary to the nucleus. Moreover we examine the non-ergodic properties of this process which are important for the correct physical interpretation of time averages of observables obtained from single particle tracking experiments. From extensive computer simulations of the 2D stochastic Langevin equation we present an in-depth study of this HDP. In particular, we find that the MSDs along the radial and azimuthal directions in a circular domain obey anomalous and Brownian scaling, respectively. We demonstrate that the time averaged MSD stays linear as a function of the lag time and the system thus reveals a weak ergodicity breaking. Our results will enable one to rationalise the diffusive motion of larger tracer particles such as viruses or submicron beads in biological cells.

Analytical model for macromolecular partitioning during yeast cell division

BACKGROUND

Asymmetric cell division, whereby a parent cell generates two sibling cells with unequal content and thereby distinct fates, is central to cell differentiation, organism development and ageing. Unequal partitioning of the macromolecular content of the parent cell - which includes proteins, DNA, RNA, large proteinaceous assemblies and organelles - can be achieved by both passive (e.g. diffusion, localized retention sites) and active (e.g. motor-driven transport) processes operating in the presence of external polarity cues, internal asymmetries, spontaneous symmetry breaking, or stochastic effects. However, the quantitative contribution of different processes to the partitioning of macromolecular content is difficult to evaluate.

RESULTS

Here we developed an analytical model that allows rapid quantitative assessment of partitioning as a function of various parameters in the budding yeast Saccharomyces cerevisiae. This model exposes quantitative degeneracies among the physical parameters that govern macromolecular partitioning, and reveals regions of the solution space where diffusion is sufficient to drive asymmetric partitioning and regions where asymmetric partitioning can only be achieved through additional processes such as motor-driven transport. Application of the model to different macromolecular assemblies suggests that partitioning of protein aggregates and episomes, but not prions, is diffusion-limited in yeast, consistent with previous reports.

CONCLUSIONS

In contrast to computationally intensive stochastic simulations of particular scenarios, our analytical model provides an efficient and comprehensive overview of partitioning as a function of global and macromolecule-specific parameters. Identification of quantitative degeneracies among these parameters highlights the importance of their careful measurement for a given macromolecular species in order to understand the dominant processes responsible for its observed partitioning.

Occupation times on a comb with ramified teeth

We investigate occupation time statistics for random walks on a comb with ramified teeth. This is achieved through the relation between the occupation time and the first passage times. Statistics of occupation times in half space follows Lamperti's distribution, i.e., the generalized arcsine law holds. Transitions between different behaviors are observed, which are controlled by the size of the backbone and teeth of the comb, as well as bias. Occupation time on a nonsimply connected domain is analyzed with a mean-field theory and numerical simulations. In that case, the generalized arcsine law is not valid.

Topography of chance

We present a model of multiplicative Langevin dynamics that is based on two foundations: the Langevin equation and the notion of multiplicative evolution. The model is a nonlinear mechanism transforming a white-noise input to a dynamic-equilibrium output, using a single control: an underlying convex U-shaped potential function. The output is quantified by a stationary density which can attain a given number of shapes and a given number of randomness categories. The model generates each admissible combination of the output's shape and randomness in a universal and robust fashion. Moreover, practically all the probability distributions that are supported on the positive half-line, and that are commonly encountered and applied across the sciences, can be reverse engineered by this model. Hence, this model is a universal equilibrium mechanism, in the context of multiplicative dynamics, for the robust generation of "chance": the model's output. In turn, the properties of the produced "chance," the output's shape and randomness, are determined with mathematical precision by the control's landscape, its topography. Thus, a topographic map of chance is established. As a particular application, probability distributions with power-law tails are shown to be universally and robustly generated by controls on the "edge of convexity": convex U-shaped potential functions with asymptotically linear wings.

Search times with arbitrary detection constraints

Random encounters in space are central to describing diffusion-limited reactions, animal foraging, search processes, and many other situations in nature. These encounters, however, are often constrained by the capacity of the searcher to detect and/or recognize its target. This can be due to limited binding and perception abilities of the searcher or hiding and avoiding mechanisms used by the target. Hence detection failure upon passage over the target location turns the process into an n-passage problem, with n being random. Here we provide a general description of this detection problem for arbitrary dimensions and arbitrary detection constraints. The mean detection time (MDT) for a random searcher embedded in a sea of homogeneously distributed targets is obtained as a function of the target density ρ, the size domain L, and the effective detection distance a. While the scaling with ρ and L is found to be universal and equivalent to that found for the corresponding first-passage problem, the scaling of the MDT on a depends on the specific detection mechanism considered.

First-passage properties of molecular spiders

Molecular spiders are synthetic catalytic DNA-based nanoscale walkers. We study the mean first-passage time for abstract models of spiders moving on a finite two-dimensional lattice with various boundary conditions and compare it with the mean first-passage time of spiders moving on a one-dimensional track. We evaluate by how much the slowdown on newly visited sites, owing to catalysis, can improve the mean first-passage time of spiders and show that in one dimension, when both ends of the track are an absorbing boundary, the performance gain is lower than in two dimensions, when the absorbing boundary is a circle; this persists even when the absorbing boundary is a single site.