Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level

Multifractal spectral features enhance classification of anomalous diffusion

Anomalous diffusion processes, characterized by their nonstandard scaling of the mean-squared displacement, pose a unique challenge in classification and characterization. In a previous study [Mangalam et al., Phys. Rev. Res. 5, 023144 (2023)2643-156410.1103/PhysRevResearch.5.023144], we established a comprehensive framework for understanding anomalous diffusion using multifractal formalism. The present study delves into the potential of multifractal spectral features for effectively distinguishing anomalous diffusion trajectories from five widely used models: fractional Brownian motion, scaled Brownian motion, continuous-time random walk, annealed transient time motion, and Lévy walk. We generate extensive datasets comprising 10^{6} trajectories from these five anomalous diffusion models and extract multiple multifractal spectra from each trajectory to accomplish this. Our investigation entails a thorough analysis of neural network performance, encompassing features derived from varying numbers of spectra. We also explore the integration of multifractal spectra into traditional feature datasets, enabling us to assess their impact comprehensively. To ensure a statistically meaningful comparison, we categorize features into concept groups and train neural networks using features from each designated group. Notably, several feature groups demonstrate similar levels of accuracy, with the highest performance observed in groups utilizing moving-window characteristics and p varation features. Multifractal spectral features, particularly those derived from three spectra involving different timescales and cutoffs, closely follow, highlighting their robust discriminatory potential. Remarkably, a neural network exclusively trained on features from a single multifractal spectrum exhibits commendable performance, surpassing other feature groups. In summary, our findings underscore the diverse and potent efficacy of multifractal spectral features in enhancing the predictive capacity of machine learning to classify anomalous diffusion processes.

Being Heterogeneous Is Advantageous: Extreme Brownian Non-Gaussian Searches

Redundancy in biology may be explained by the need to optimize extreme searching processes, where one or few among many particles are requested to reach the target like in human fertilization. We show that non-Gaussian rare fluctuations in Brownian diffusion dominates such searches, introducing drastic corrections to the known Gaussian behavior. Our demonstration entails different physical systems and pinpoints the relevance of diversity within redundancy to boost fast targeting. We sketch an experimental context to test our results: polydisperse systems.

Being heterogeneous is disadvantageous: Brownian non-Gaussian searches

Diffusing diffusivity models, polymers in the grand canonical ensemble and polydisperse, and continuous-time random walks all exhibit stages of non-Gaussian diffusion. Is non-Gaussian targeting more efficient than Gaussian? We address this question, central to, e.g., diffusion-limited reactions and some biological processes, through a general approach that makes use of Jensen's inequality and that encompasses all these systems. In terms of customary mean first-passage time, we show that Gaussian searches are more effective than non-Gaussian ones. A companion paper argues that non-Gaussianity becomes instead highly more efficient in applications where only a small fraction of tracers is required to reach the target.

Multiple diffusive behaviors of the random walk in inhomogeneous environments

Anomalous diffusive behaviors are observed in highly inhomogeneous but relatively stable environments such as intracellular media and are increasingly attracting attention. In this paper we develop a coupled continuous-time random walk model in which the waiting time is power-law coupled with the local environmental diffusion coefficient. We provide two forms of the waiting time density, namely, a heavy-tailed density and an exponential density. For different waiting time densities, anomalous diffusions with the diffusion exponent between 0 and 2 and Brownian yet non-Gaussian diffusion can be realized within the present model. The diffusive behaviors are analyzed and discussed by deriving the mean-squared displacement and probability density function. In addition we derive the effective jump length density corresponding to the decoupled form to help distinguish the diffusion types. Our model unifies two kinds of anomalous diffusive behavior with different characteristics in the same inhomogeneous environment into a theoretical framework. The model interprets the random motion of particles in a complex inhomogeneous environment and reproduces the experimental results of different biological and physical systems.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

Thirty years of slug control using the parasitic nematode Phasmarhabditis hermaphrodita and beyond

Several slug species are highly pestiferous and threaten global sustainable agriculture. Current control methods rely heavily on metaldehyde pellets, which are often ineffective, harm nontarget organisms and have been banned in some countries. A viable alternative is the parasitic nematode Phasmarhabditis hermaphrodita (and recently P. californica), which has been formulated into a biological control agent (Nemaslug®) to control slugs across northern Europe. Nematodes are mixed with water and applied to soil where they seek out slugs, penetrate behind the mantle and kill them in 4-21 days. Phasmarhabditis hermaphrodita has been on the market since 1994 and since then there has been ample research on its use. Here we review the research carried out on P. hermaphrodita over the last 30 years since its development and release as a commercial product. We provide information on life cycle, worldwide distribution, history of commercialisation, gastropod immunity, host range, ecological and environmental factors that affect its success in the field, bacterial relationships, and summarise results of field trials. Finally, we suggest future directions for P. hermaphrodita research (and other Phasmarhabditis species) to enhance its use as a biological control agent to control slugs for the next 30 years. © 2023 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Universal Evolution of Fickian Non-Gaussian Diffusion in Two- and Three-Dimensional Glass-Forming Liquids

Recent works show that glass-forming liquids display Fickian non-Gaussian Diffusion, with non-Gaussian displacement distributions persisting even at very long times, when linearity in the mean square displacement (Fickianity) has already been attained. Such non-Gaussian deviations temporarily exhibit distinctive exponential tails, with a decay length λ growing in time as a power-law. We herein carefully examine data from four different glass-forming systems with isotropic interactions, both in two and three dimensions, namely, three numerical models of molecular liquids and one experimentally investigated colloidal suspension. Drawing on the identification of a proper time range for reliable exponential fits, we find that a scaling law λ(t)∝tα, with α≃1/3, holds for all considered systems, independently from dimensionality. We further show that, for each system, data at different temperatures/concentration can be collapsed onto a master-curve, identifying a characteristic time for the disappearance of exponential tails and the recovery of Gaussianity. We find that such characteristic time is always related through a power-law to the onset time of Fickianity. The present findings suggest that FnGD in glass-formers may be characterized by a "universal" evolution of the distribution tails, independent from system dimensionality, at least for liquids with isotropic potential.

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.

The Fokker-Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Krätzel function and its Fokker-Planck equation to be a fractional diffusion equation in the Erdélyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.

Time-averaging and nonergodicity of reset geometric Brownian motion with drift

How do near-bankruptcy events in the past affect the dynamics of stock-market prices in the future? Specifically, what are the long-time properties of a time-local exponential growth of stock-market prices under the influence of stochastically occurring economic crashes? Here, we derive the ensemble- and time-averaged properties of the respective "economic" or geometric Brownian motion (GBM) with a nonzero drift exposed to a Poissonian constant-rate price-restarting process of "resetting." We examine-based both on thorough analytical calculations and on findings from systematic stochastic computer simulations-the general situation of reset GBM with a nonzero [positive] drift and for all special cases emerging for varying parameters of drift, volatility, and reset rate in the model. We derive and summarize all short- and long-time dependencies for the mean-squared displacement (MSD), the variance, and the mean time-averaged MSD (TAMSD) of the process of Poisson-reset GBM under the conditions of both rare and frequent resetting. We consider three main regions of model parameters and categorize the crossovers between different functional behaviors of the statistical quantifiers of this process. The analytical relations are fully supported by the results of computer simulations. In particular, we obtain that Poisson-reset GBM is a nonergodic stochastic process, with generally MSD(Δ)≠TAMSD(Δ) and Variance(Δ)≠TAMSD(Δ) at short lag times Δ and for long trajectory lengths T. We investigate the behavior of the ergodicity-breaking parameter in each of the three regions of parameters and examine its dependence on the rate of reset at Δ/T≪1. Applications of these theoretical results to the analysis of prices of reset-containing options are pertinent.

Random diffusivity processes in an external force field

Brownian yet non-Gaussian processes have recently been observed in numerous biological systems, and corresponding theories have been constructed based on random diffusivity models. Considering the particularity of random diffusivity, this paper studies the effect of an external force acting on two kinds of random diffusivity models whose difference is embodied in whether the fluctuation-dissipation theorem is valid. Based on the two random diffusivity models, we derive the Fokker-Planck equations with an arbitrary external force, and we analyze various observables in the case with a constant force, including the Einstein relation, the moments, the kurtosis, and the asymptotic behaviors of the probability density function of particle displacement at different timescales. Both the theoretical results and numerical simulations of these observables show a significant difference between the two kinds of random diffusivity models, which implies the important role of the fluctuation-dissipation theorem in random diffusivity systems.

Structural relaxation dynamics of colloidal nanotrimers

By Molecular Dynamics simulation, we investigate the dynamics of isotropic fluids of colloidal nanotrimers whose interactions are described by varying the strength of attractive and repulsive terms of the Mie potential. To provide a consistent comparison between the systems described by different force fields, we determine the phase diagram and critical points of each system, characterize the morphology of high-density liquid phases at the same reduced temperature and density, and finally investigate their long-time relaxation dynamics. In particular, we detect an especially complex dynamics that reveals the existence of slow and fast nanotrimers and the resulting occurrence of non-Gaussianity, which develops at intermediate timescales. Deviations from Gaussianity are temporary and vanish within the timescales of the system's density fluctuations decay, when a Fickian-like diffusion regime is eventually observed.

Polymers critical point originates Brownian non-Gaussian diffusion

We demonstrate that size fluctuations close to polymers critical point originate the non-Gaussian diffusion of their center of mass. Static universal exponents γ and ν-depending on the polymer topology, on the dimension of the embedding space, and on equilibrium phase-concur to determine the potential divergency of a dynamic response, epitomized by the center-of-mass kurtosis. Prospects in experiments and stochastic modeling brought about by this result are briefly outlined.

Colossal Brownian yet non-Gaussian diffusion in a periodic potential: Impact of nonequilibrium noise amplitude statistics

Last year, Białas et al. [Phys. Rev. E 102, 042121 (2020)] studied an overdamped dynamics of nonequilibrium noise driven Brownian particle dwelling in a spatially periodic potential and discovered a novel class of Brownian, yet non-Gaussian diffusion. The mean square displacement of the particle grows linearly with time and the probability density for the particle position is Gaussian; however, the corresponding distribution for the increments is non-Gaussian. The latter property induces the colossal enhancement of diffusion, significantly exceeding the well known effect of giant diffusion. Here, we considerably extend the above predictions by investigating the influence of nonequilibrium noise amplitude statistics on the colossal Brownian, yet non-Gaussian diffusion. The tail of amplitude distribution crucially impacts both the magnitude of diffusion amplification and the Gaussianity of the position and increments statistics. Our results carry profound consequences for diffusive behavior in nonequilibrium settings such as living cells in which diffusion is a central transport mechanism.

Intrinsic and environmental factors modulating autonomous robotic search under high uncertainty

Autonomous robotic search problems deal with different levels of uncertainty. When uncertainty is low, deterministic strategies employing available knowledge result in most effective searches. However, there are domains where uncertainty is always high since information about robot location, environment boundaries or precise reference points is unattainable, e.g., in cave, deep ocean, planetary exploration, or upon sensor or communications impairment. Furthermore, latency regarding when search targets move, appear or disappear add to uncertainty sources. Here we study intrinsic and environmental factors that affect low-informed robotic search based on diffusive Brownian, naive ballistic, and superdiffusive strategies (Lévy walks), and in particular, the effectiveness of their random exploration. Representative strategies were evaluated considering both intrinsic (motion drift, energy or memory limitations) and extrinsic factors (obstacles and search boundaries). Our results point towards minimum-knowledge based modulation approaches that can adjust distinct spatial and temporal aspects of random exploration to lead to effective autonomous search under uncertainty.

Integral decomposition for the solutions of the generalized Cattaneo equation

We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels t^{-α}, α∈(0,1] this equation becomes the time fractional one governed by the Caputo derivatives in which the highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with nonnegative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels t^{-α} for which the subordination based on the Brownian motion does not work if α∈(1/2,1].

Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does stochastic resetting impact nonergodicity? We examine, both analytically and by simulations, the implications of resetting on the MSD- and TAMSD-based spreading dynamics of particles executing fractional Brownian motion (FBM) with a long-time memory, heterogeneous diffusion processes (HDPs) with a power-law space-dependent diffusivity D(x)=D_{0}|x|^{γ} and their "combined" process of HDP-FBM. We find, inter alia, that the resetting dynamics of originally ergodic FBM for superdiffusive Hurst exponents develops disparities in scaling and magnitudes of the MSDs and mean TAMSDs indicating weak ergodicity breaking. For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD and observe a new trimodal form of the probability density function. For reset FBM, HDPs and HDP-FBM we compute analytically and verify by simulations the short-time MSD and TAMSD asymptotes and long-time plateaus reminiscent of those for processes under confinement. We show that certain characteristics of these reset processes are functionally similar despite a different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity-breaking parameter EB as a function of the resetting rate r. For all reset processes studied we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB∼(1/r)-decay at large r. Alongside the emerging MSD-versus-TAMSD disparity, this r-dependence of EB can be an experimentally testable prediction. We conclude by discussing some implications to experimental systems featuring resetting dynamics.

Single-Particle Tracking Reveals Anti-Persistent Subdiffusion in Cell Extracts

Single-particle tracking (SPT) has become a powerful tool to quantify transport phenomena in complex media with unprecedented detail. Based on the reconstruction of individual trajectories, a wealth of informative measures become available for each particle, allowing for a detailed comparison with theoretical predictions. While SPT has been used frequently to explore diffusive transport in artificial fluids and inside living cells, intermediate systems, i.e., biochemically active cell extracts, have been studied only sparsely. Extracts derived from the eggs of the clawfrog Xenopus laevis, for example, are known for their ability to support and mimic vital processes of cells, emphasizing the need to explore also the transport phenomena of nano-sized particles in such extracts. Here, we have performed extensive SPT on beads with 20 nm radius in native and chemically treated Xenopus extracts. By analyzing a variety of distinct measures, we show that these beads feature an anti-persistent subdiffusion that is consistent with fractional Brownian motion. Chemical treatments did not grossly alter this finding, suggesting that the high degree of macromolecular crowding in Xenopus extracts equips the fluid with a viscoelastic modulus, hence enforcing particles to perform random walks with a significant anti-persistent memory kernel.

Homeostatic swimming of zooplankton upon crowding: the case of the copepod Centropages typicus

Crowding has a major impact on the dynamics of many material and biological systems, inducing effects as diverse as glassy dynamics and swarming. While this issue has been deeply investigated for a variety of living organisms, more research remains to be done on the effect of crowding on the behaviour of copepods, the most abundant metazoans on Earth. To this aim, we experimentally investigate the swimming behaviour, used as a dynamic proxy of animal adaptations, of males and females of the calanoid copepod Centropages typicus at different densities of individuals (10, 50 and 100 ind. l^-1) by performing three-dimensional single-organism tracking. We find that the C. typicus motion is surprisingly unaffected by crowding over the investigated density range. Indeed, the mean square displacements as a function of time always show a crossover from ballistic to Fickian regime, with poor variations of the diffusion constant on increasing the density. Close to the crossover, the displacement distributions display exponential tails with a nearly density-independent decay length. The trajectory fractal dimension, D_3D ≅ 1.5, and the recently proposed 'ecological temperature' also remain stable on increasing the individual density. This suggests that, at least over the range of animal densities used, crowding does not impact on the characteristics of C. typicus swimming motion, and that a homeostatic mechanism preserves the stability of its swimming performance.

From diffusion in compartmentalized media to non-Gaussian random walks

In this work we establish a link between two different phenomena that were studied in a large and growing number of biological, composite and soft media: the diffusion in compartmentalized environment and the non-Gaussian diffusion that exhibits linear or power-law growth of the mean square displacement joined by the exponential shape of the positional probability density. We explore a microscopic model that gives rise to transient confinement, similar to the one observed for hop-diffusion on top of a cellular membrane. The compartmentalization of the media is achieved by introducing randomly placed, identical barriers. Using this model of a heterogeneous medium we derive a general class of random walks with simple jump rules that are dictated by the geometry of the compartments. Exponential decay of positional probability density is observed and we also quantify the significant decrease of the long time diffusion constant. Our results suggest that the observed exponential decay is a general feature of the transient regime in compartmentalized media.

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

We study a two state "jumping diffusivity" model for a Brownian process alternating between two different diffusion constants, D+>D-, with random waiting times in both states whose distribution is rather general. In the limit of long measurement times, Gaussian behavior with an effective diffusion coefficient is recovered. We show that, for equilibrium initial conditions and when the limit of the diffusion coefficient D-⟶0 is taken, the short time behavior leads to a cusp, namely a non-analytical behavior, in the distribution of the displacements P(x,t) for x⟶0. Visually this cusp, or tent-like shape, resembles similar behavior found in many experiments of diffusing particles in disordered environments, such as glassy systems and intracellular media. This general result depends only on the existence of finite mean values of the waiting times at the different states of the model. Gaussian statistics in the long time limit is achieved due to ergodicity and convergence of the distribution of the temporal occupation fraction in state D+ to a δ-function. The short time behavior of the same quantity converges to a uniform distribution, which leads to the non-analyticity in P(x,t). We demonstrate how super-statistical framework is a zeroth order short time expansion of P(x,t), in the number of transitions, that does not yield the cusp like shape. The latter, considered as the key feature of experiments in the field, is found with the first correction in perturbation theory.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

Colossal Brownian yet non-Gaussian diffusion induced by nonequilibrium noise

We report on Brownian, yet non-Gaussian diffusion, in which the mean square displacement of the particle grows linearly with time, and the probability density for the particle spreading is Gaussian like, but the probability density for its position increments possesses an exponentially decaying tail. In contrast to recent works in this area, this behavior is not a consequence of either a space- or time-dependent diffusivity, but is induced by external nonthermal noise acting on the particle dwelling in a periodic potential. The existence of the exponential tail in the increment statistics leads to colossal enhancement of diffusion, drastically surpassing the previously researched situation known as "giant" diffusion. This colossal diffusion enhancement crucially impacts a broad spectrum of the first arrival problems, such as diffusion limited reactions governing transport in living cells.

A unifying theory for two-dimensional spatial redistribution kernels with applications in population spread modelling

When building models to explain the dispersal patterns of organisms, ecologists often use an isotropic redistribution kernel to represent the distribution of movement distances based on phenomenological observations or biological considerations of the underlying physical movement mechanism. The Gaussian, two-dimensional (2D) Laplace and Bessel kernels are common choices for 2D space. All three are special (or limiting) cases of a kernel family, the Whittle-Matérn-Yasuda (WMY), first derived by Yasuda from an assumption of 2D Fickian diffusion with gamma-distributed settling times. We provide a novel derivation of this kernel family, using the simpler assumption of constant settling hazard, by means of a non-Fickian 2D diffusion equation representing movements through heterogeneous 2D media having a fractal structure. Our derivation reveals connections among a number of established redistribution kernels, unifying them under a single, flexible modelling framework. We demonstrate improvements in predictive performance in an established model for the spread of the mountain pine beetle upon replacing the Gaussian kernel by the Whittle-Matérn-Yasuda, and report similar results for a novel approximation, the product-Whittle-Matérn-Yasuda, that substantially speeds computations in applications to large datasets.

Hitchhiker model for Laplace diffusion processes

Brownian motion is a Gaussian process describing normal diffusion with a variance increasing linearly with time. Recently, intracellular single-molecule tracking experiments have recorded exponentially decaying propagators, a phenomenon called Laplace diffusion. Inspired by these developments we study a many-body approach, called the Hitchhiker model, providing a microscopic description of the widely observed behavior. Our model explains how Laplace diffusion is controlled by size fluctuations of single molecules, independently of the diffusion law which they follow. By means of numerical simulations Laplace diffusion is recovered and we show how single-molecule tracking and data analysis, in a many-body system, is highly nontrivial as tracking of a single particle or many in parallel yields vastly different estimates for the diffusivity. We quantify the differences between these two commonly used approaches, showing how the single-molecule estimate of diffusivity is larger if compared to the full tagging method.

Large Deviations for Continuous Time Random Walks

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

Packets of Diffusing Particles Exhibit Universal Exponential Tails

Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function P(X,t) of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells, and bacteria suspensions. We show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the large deviations approach for a continuous time random walk, we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to exponential decay (with logarithmic corrections) of P(X,t). This universal behavior also holds for short times, a fact that makes experimental observations readily achievable.

Non-Gaussian subdiffusion of single-molecule tracers in a hydrated polymer network

Single molecule tracking experiments inside a hydrated polymer network have shown that the tracer motion is subdiffusive due to the viscoelastic environment inside the gel-like network. This property can be related to the negative autocorrelation of the instantaneous displacements at short times. Although the displacements of the individual tracers exhibit Gaussian statistics, the displacement distribution of all the trajectories combined from different spatial locations of the polymer network exhibits a non-Gaussian distribution. Here, we analyze many individual tracer trajectories to show that the central portion of the non-Gaussian distribution can be well approximated by an exponential distribution that spreads sublinearly with time. We explain all these features seen in the experiment by a generalized Langevin model for an overdamped particle with algebraically decaying correlations. We show that the degree of non-Gaussianity can change with the extent of heterogeneity, which is controlled in our model by the experimentally observed distributions of the motion parameters.

Robust random search with scale-free stochastic resetting

A new model of search based on stochastic resetting is introduced, wherein rate of resets depends explicitly on time elapsed since the beginning of the process. It is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self-similarity and linear growth of the mean-square displacement with nonlocality and non-Gaussian propagator. It is argued that such resetting protocol offers a general and efficient search-boosting method that does not need to be optimized with respect to the scale of the underlying search problem (e.g., distance to the goal) and is not very sensitive to other search parameters. Both subdiffusive and superdiffusive regimes of the mean-squared displacement scaling are demonstrated with more general rate functions.

Transport dynamics of complex fluids

Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.

Anomalous Dynamics of in Vivo Cargo Delivery by Motor Protein Multiplexes

Vesicle transport conducted by motor protein multiplexes (MPMs), which is ubiquitous among eukaryotes, shows anomalous and stochastic dynamics qualitatively different from the dynamics of thermal motion and artificial active matter; the relationship between in vivo vesicle-delivery dynamics and the underlying physicochemical processes is not yet quantitatively understood. Addressing this issue, we perform accurate tracking of individual vesicles, containing upconverting nanoparticles, transported by kinesin-dynein-multiplexes along axonal microtubules. The mean-square-displacement of vesicles along the microtubule exhibits unusual dynamic phase transitions that are seemingly inconsistent with the scaling behavior of the mean-first-passage time over the travel length. These paradoxical results and the vesicle displacement distribution are quantitatively explained and predicted by a multimode MPM model, developed in the current work, where ATP-hydrolysis-coupled motion of MPM has both unidirectional and bidirectional modes.

Diffusive ferromagnetic roller gas

An ensemble of actively rotating ferromagnetic particles is used to realize an active roller gas. Here, we investigate the diffusive properties of such a gas in experiments and simulations. We reveal that ferromagnetic rollers demonstrate a normal (Fickian) diffusion with a characteristic linear growth of the mean-squared displacement, while statistics of displacements stay non-Gaussian. At short times the system has a bimodal distribution of the displacements that transitions with time to a quasi-Gaussian distribution (Gaussian core with overpopulated tails) for a range of studied particle number densities. Inert particles introduced into the active roller gas exhibit similar diffusive behavior. The results provide insights into diffusive properties of active colloidal systems with activity originating from spinning degrees of freedom.

Non-Gaussian, non-ergodic, and non-Fickian diffusion of tracers in mucin hydrogels

Native mucus is polymer-based soft-matter material of paramount biological importance. How non-Gaussian and non-ergodic is the diffusive spreading of pathogens in mucus? We study the passive, thermally driven motion of micron-sized tracers in hydrogels of mucins, the main polymeric component of mucus. We report the results of the Bayesian analysis for ranking several diffusion models for a set of tracer trajectories [C. E. Wagner et al., Biomacromolecules, 2017, 18, 3654]. The models with "diffusing diffusivity", fractional and standard Brownian motion are used. The likelihood functions and evidences of each model are computed, ranking the significance of each model for individual traces. We find that viscoelastic anomalous diffusion is often most probable, followed by Brownian motion, while the model with a diffusing diffusion coefficient is only realised rarely. Our analysis also clarifies the distribution of time-averaged displacements, correlations of scaling exponents and diffusion coefficients, and the degree of non-Gaussianity of displacements at varying pH levels. Weak ergodicity breaking is also quantified. We conclude that-consistent with the original study-diffusion of tracers in the mucin gels is most non-Gaussian and non-ergodic at low pH that corresponds to the most heterogeneous networks. Using the Bayesian approach with the nested-sampling algorithm, together with the quantitative analysis of multiple statistical measures, we report new insights into possible physical mechanisms of diffusion in mucin gels.

Bayesian analysis of single-particle tracking data using the nested-sampling algorithm: maximum-likelihood model selection applied to stochastic-diffusivity data

We employ Bayesian statistics using the nested-sampling algorithm to compare and rank multiple models of ergodic diffusion (including anomalous diffusion) as well as to assess their optimal parameters for in silico-generated and real time-series. We focus on the recently-introduced model of Brownian motion with "diffusing diffusivity"-giving rise to widely-observed non-Gaussian displacement statistics-and its comparison to Brownian and fractional Brownian motion, also for the time-series with some measurement noise. We conduct this model-assessment analysis using Bayesian statistics and the nested-sampling algorithm on the level of individual particle trajectories. We evaluate relative model probabilities and compute best-parameter sets for each diffusion model, comparing the estimated parameters to the true ones. We test the performance of the nested-sampling algorithm and its predictive power both for computer-generated (idealised) trajectories as well as for real single-particle-tracking trajectories. Our approach delivers new important insight into the objective selection of the most suitable stochastic model for a given time-series. We also present first model-ranking results in application to experimental data of tracer diffusion in polymer-based hydrogels.

Non-Gaussianity, population heterogeneity, and transient superdiffusion in the spreading dynamics of amoeboid cells

What is the underlying diffusion process governing the spreading dynamics and search strategies employed by amoeboid cells?

A random acceleration model of individual animal movement allowing for diffusive, superdiffusive and superballistic regimes

Patterns of individual animal movement attracted considerable attention over the last two decades. In particular, question as to whether animal movement is predominantly diffusive or superdiffusive has been a focus of discussion and controversy. We consider this problem using a theory of stochastic motion based on the Langevin equation with non-Wiener stochastic forcing that originates in animal’s response to environmental noise. We show that diffusive and superdiffusive types of motion are inherent parts of the same general movement process that arises as interplay between the force exerted by animals (essentially, by animal’s muscles) and the environmental drag. The movement is superballistic with the mean square displacement growing with time as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle {x}^{2}(t)\rangle \sim {t}^{4}$$\end{document}〈x2(t)〉∼t4 at the beginning and eventually slowing down to the diffusive spread \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle {x}^{2}(t)\rangle \sim t$$\end{document}〈x2(t)〉∼t. We show that the duration of the superballistic and superdiffusive stages can be long depending on the properties of the environmental noise and the intensity of drag. Our findings demonstrate theoretically how the movement pattern that includes diffusive and superdiffusive/superballistic motion arises naturally as a result of the interplay between the dissipative properties of the environment and the animal’s biological traits such as the body mass, typical movement velocity and the typical duration of uninterrupted movement.

Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics

The mechanism of diffusing diffusivity predicts that, in environments where the diffusivity changes gradually, the displacement distribution becomes non-Gaussian, even though the mean-square displacement grows linearly with time. Here, we report single-particle tracking measurements of the diffusion of colloidal spheres near a planar substrate. Because the local effective diffusivity is known, we have been able to carry out a direct test of this mechanism for diffusion in inhomogeneous media.

Lévy flight with absorption: A model for diffusing diffusivity with long tails

We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of "diffusing diffusivity" [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)JPCBFK1520-610610.1021/acs.jpcb.6b01527], it was shown that the mean square displacement of particle remains Fickian, i.e., 〈x^{2}(T)〉∝T at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of D in that model has a steady state limit which is exponential, i.e., π_{e}(D)∼e^{-D/D_{0}}. In the present study, we model the diffusivity of a particle as a Lévy flight process so that D has a power-law tailed distribution, viz., π_{e}(D)∼D^{-1-α} with 0<α<1. We find that in the short time limit, the width of displacement distribution is proportional to sqrt[T], implying that the diffusion is Fickian. But for long times, the width is proportional to T^{1/2α} which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index 2α which preserves its shape at all times.

Anomalous diffusion in time-fluctuating non-stationary diffusivity landscapes

We investigate the diffusive and ergodic properties of massive and confined particles in a model disordered medium, in which the local diffusivity fluctuates in time while its mean has a power law dependence on the diffusion time.

Ultimate failure of the Lévy Foraging Hypothesis: Two-scale searching strategies outperform scale-free ones even when prey are scarce and cryptic

The "Lévy Foraging Hypothesis" promotes Lévy walk (LW) as the best strategy to forage for patchily but unpredictably located prey. This strategy mixes extensive and intensive searching phases in a mostly cue-free way through strange, scale-free kinetics. It is however less efficient than a cue-driven two-scale Composite Brownian walk (CBW) when the resources encountered are systematically detected. Nevertheless, it could be assumed that the intrinsic capacity of LW to trigger cue-free intensive searching at random locations might be advantageous when resources are not only scarcely encountered but also so cryptic that the probability to detect those encountered during movement is low. Surprisingly, this situation, which should be quite common in natural environments, has almost never been studied. Only a few studies have considered "saltatory" foragers, which are fully "blind" while moving and thus detect prey only during scanning pauses, but none of them compared the efficiency of LW vs. CBW in this context or in less extreme contexts where the detection probability during movement is not null but very low. In a study based on computer simulations, we filled the bridge between the concepts of "pure continuous" and "pure saltatory" foraging by considering that the probability to detect resources encountered while moving may range from 0 to 1. We showed that regularly stopping to scan the environment can indeed improve efficiency, but only at very low detection probabilities. Furthermore, the LW is then systematically outperformed by a mixed cue-driven/internally-driven CBW. It is thus more likely that evolution tends to favour strategies that rely on environmental feedbacks rather than on strange kinetics.

Apparent power-law distributions in animal movements can arise from intraspecific interactions

Lévy flights have gained prominence for analysis of animal movement. In a Lévy flight, step-lengths are drawn from a heavy-tailed distribution such as a power law (PL), and a large number of empirical demonstrations have been published. Others, however, have suggested that animal movement is ill fit by PL distributions or contend a state-switching process better explains apparent Lévy flight movement patterns. We used a mix of direct behavioural observations and GPS tracking to understand step-length patterns in females of two related butterflies. We initially found movement in one species (Euphydryas editha taylori) was best fit by a bounded PL, evidence of a Lévy flight, while the other (Euphydryas phaeton) was best fit by an exponential distribution. Subsequent analyses introduced additional candidate models and used behavioural observations to sort steps based on intraspecific interactions (interactions were rare in E. phaeton but common in E. e. taylori). These analyses showed a mixed-exponential is favoured over the bounded PL for E. e. taylori and that when step-lengths were sorted into states based on the influence of harassing conspecific males, both states were best fit by simple exponential distributions. The direct behavioural observations allowed us to infer the underlying behavioural mechanism is a state-switching process driven by intraspecific interactions rather than a Lévy flight.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

Diffusing diffusivity: a model for anomalous, yet Brownian, diffusion

Wang et al. [Proc. Natl. Acad. Sci. U.S.A. 106, 15160 (2009)] have found that in several systems the linear time dependence of the mean-square displacement (MSD) of diffusing colloidal particles, typical of normal diffusion, is accompanied by a non-Gaussian displacement distribution G(x,t), with roughly exponential tails at short times, a situation they termed “anomalous yet Brownian” diffusion. The diversity of systems in which this is observed calls for a generic model. We present such a model where there is diffusivity memory but no direction memory in the particle trajectory, and we show that it leads to both a linear MSD and a non-Gaussian G(x,t) at short times. In our model, the diffusivity is undergoing a (perhaps biased) random walk, hence the expression “diffusing diffusivity”. G(x,t) is predicted to be exactly exponential at short times if the distribution of diffusivities is itself exponential, but an exponential remains a good fit for a variety of diffusivity distributions. Moreover, our generic model can be modified to produce subdiffusion.