Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary

Mean exit times as global measure of resilience of tropical forest systems under climatic disturbances-Analytical and numerical results

Both remotely sensed distribution of tree cover and models suggest three alternative stable vegetation states in the tropics: forest, savanna, and treeless states. Environmental fluctuation could cause critical transitions from the forest to the savanna state and quantifying the resilience of a given vegetation state is, therefore, crucial. While previous work has focused mostly on local stability concepts, we investigate here the mean exit time from a given basin of attraction, with partially absorbing and reflecting boundaries, as a global resilience measure. We provide detailed investigations using an established model for tropical tree cover with multistable precipitation regimes. We find that higher precipitation or weaker noise increases the mean exit time of the forest state and, thus, its resilience. Upon investigating the transition times from the forest state to other tree cover states, we find that in the bistable precipitation regime, the size of environmental fluctuations has a greater impact on the transition probabilities from the forest state compared to precipitation.

Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function g (g-subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the "ordinary" fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz-type spatial derivative. We find the function g for which the solution (Green's function, GF) to the g-subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the g-subdiffusion equation we use the g-Laplace transform method. It is shown that the scaling properties of the GF for g-subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the g-subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the g-continuous-time random walk model. The g-subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model g-subdiffusion processes, even if this process is interpreted as superdiffusion.

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach

We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

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.

Encounter-based approach to diffusion with resetting

An encounter-based approach consists in using the boundary local time as a proxy for the number of encounters between a diffusing particle and a target to implement various surface reaction mechanisms on that target. In this paper, we investigate the effects of stochastic resetting onto diffusion-controlled reactions in bounded confining domains. We first discuss the effect of position resetting onto the propagator and related quantities; in this way, we retrieve a number of earlier results but also provide complementary insights into them. Second, we introduce boundary local time resetting and investigate its impact. Curiously, we find that this type of resetting does not alter the conventional propagator governing the diffusive dynamics in the presence of a partially reactive target with a constant reactivity. In turn, the generalized propagator for other surface reaction mechanisms can be significantly affected. Our general results are illustrated for diffusion on an interval with reactive end points. Further perspectives and some open problems are discussed.

Enhancing search efficiency through diffusive echo

Despite having been studied for decades, first passage processes remain an active area of research. In this contribution we examine a particle diffusing in an annulus with an inner absorbing boundary and an outer reflective boundary. We obtain analytic expressions for the joint distribution of the hitting time and the hitting angle in two and three dimensions. For certain configurations we observe a ``diffusive echo", i.e. two well-defined maxima in the first passage time distribution to a targeted position on the absorbing boundary. This effect, which results from the interplay between the starting location and the environmental constraints, may help to significantly increase the efficiency of the random search by generating a high, sustained flux to the targeted position over a short period. Finally, we examine the corresponding one-dimensional system for which there is no well-defined echo. In a confined system, the flux integrated over all target positions always displays a shoulder. This does not, however, guarantee the presence of an echo in the joint distribution.

First-passage times to anisotropic partially reactive targets

We investigate restricted diffusion in a bounded domain towards a small partially reactive target in three- and higher-dimensional spaces. We propose a simple explicit approximation for the principal eigenvalue of the Laplace operator with mixed Robin-Neumann boundary conditions. This approximation involves the harmonic capacity and the surface area of the target, the volume of the confining domain, the diffusion coefficient, and the reactivity. The accuracy of the approximation is checked by using a finite-elements method. The proposed approximation determines also the mean first-reaction time, the long-time decay of the survival probability, and the overall reaction rate on that target. We identify the relevant lengthscale of the target, which determines its trapping capacity, and we investigate its relation to the target shape. In particular, we study the effect of target anisotropy on the principal eigenvalue by computing the harmonic capacity of prolate and oblate spheroids in various space dimensions. Some implications of these results in chemical physics and biophysics are briefly discussed.

First-encounter time of two diffusing particles in two- and three-dimensional confinement

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability S(t) and the associated first-encounter time probability density H(t) over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time 〈T〉, as well as for the decay time T characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound t_{B} for the time at which S(t) starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to T depends only on the total diffusivity D=D_{1}+D_{2}, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity D. In two dimensions, the first subleading contribution to T is found to depend weakly on the ratio D_{1}/D_{2}. We also investigate the slow-diffusion limit when D_{2}≪D_{1}, and we discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when T can be expected to be a good approximation for 〈T〉.

Subdiffusion in a system with a partially permeable partially absorbing wall

We consider subdiffusion of a particle in a one-dimensional system with a thin partially permeable and partially absorbing wall. The system with the wall can be used to filter diffusing particles. Passing through the wall, the particle can be absorbed with a certain probability. Knowing the Green's functions we derive boundary conditions at the wall. The boundary conditions take a specific form in which fractional time derivatives are involved. The temporal evolution of the probability that a diffusing particle has not been absorbed is also considered.

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.

Paradigm Shift in Diffusion-Mediated Surface Phenomena

Diffusion-mediated surface phenomena are crucial for human life and industry, with examples ranging from oxygen capture by lung alveolar surface to heterogeneous catalysis, gene regulation, membrane permeation, and filtration processes. Their current description via diffusion equations with mixed boundary conditions is limited to simple surface reactions with infinite or constant reactivity. In this Letter, we propose a probabilistic approach based on the concept of boundary local time to investigate the intricate dynamics of diffusing particles near a reactive surface. Reformulating surface-particle interactions in terms of stopping conditions, we obtain in a unified way major diffusion-reaction characteristics such as the propagator, the survival probability, the first-passage time distribution, and the reaction rate. This general formalism allows us to describe new surface reaction mechanisms such as for instance surface reactivity depending on the number of encounters with the diffusing particle that can model the effects of catalyst fooling or membrane degradation. The disentanglement of the geometric structure of the medium from surface reactivity opens far-reaching perspectives for modeling, optimization, and control of diffusion-mediated surface phenomena.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena, such as heterogeneous catalysis, permeation through semipermeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behaviors of this random variable for both bounded and unbounded domains. This analysis provides complementary insights onto the dynamics of diffusing molecules near partially reactive boundaries.

Model of anomalous diffusion-absorption process in a system consisting of two different media separated by a thin membrane

We present the model of a diffusion-absorption process in a system which consists of two media separated by a thin partially permeable membrane. The kind of diffusion as well as the parameters of the process may be different in both media. Based on a simple model of a particle's random walk in a membrane system we derive the Green's functions, then we find the boundary conditions at the membrane. One of the boundary conditions is rather complicated and takes a relatively simple form in terms of the Laplace transform. Assuming that particles diffuse independently of one another, the obtained boundary conditions can be used to solve differential or differential-integral equations describing the processes in multilayered systems for any initial condition. We consider normal diffusion, subdiffusion, and slow subdiffusion processes, and we also suggest how superdiffusion could be included in this model. The presented method provides the functions in terms of the Laplace transform and some useful methods of calculation of the inverse Laplace transform are shown.

Diffusion-limited reactions in dynamic heterogeneous media

Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule.

“Diffusing diffusivity” concept has been recently put forward to account for rapid structural rearrangements in soft matter and biological systems. Here the authors propose a general mathematical framework to compute the distribution of first-passage times in a dynamically heterogeneous medium.

Heterogeneous continuous-time random walks

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

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.

The escape problem for mortal walkers

We introduce and investigate the escape problem for random walkers that may eventually die, decay, bleach, or lose activity during their diffusion towards an escape or reactive region on the boundary of a confining domain. In the case of a first-order kinetics (i.e., exponentially distributed lifetimes), we study the effect of the associated death rate onto the survival probability, the exit probability, and the mean first passage time. We derive the upper and lower bounds and some approximations for these quantities. We reveal three asymptotic regimes of small, intermediate, and large death rates. General estimates and asymptotics are compared to several explicit solutions for simple domains and to numerical simulations. These results allow one to account for stochastic photobleaching of fluorescent tracers in bio-imaging, degradation of mRNA molecules in genetic translation mechanisms, or high mortality rates of spermatozoa in the fertilization process. Our findings provide a mathematical ground for optimizing storage containers and materials to reduce the risk of leakage of dangerous chemicals or nuclear wastes.

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

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

Analytical representations of the spread harmonic measure density

We study the spread harmonic measure that characterizes the spatial distribution of reaction events on a partially reactive surface. For Euclidean domains in which Brownian motion can be split into independent lateral and transverse displacements, we derive analytical formulas for the spread harmonic measure density and analyze its asymptotic behavior. This analysis is applicable to slab domains, general cylindrical domains, and a half-space. We investigate the spreading effect due to multiple reflections on the surface, and the underlying role of finite reactivity. We discuss further extensions and applications of analytical results to describe Laplacian transfer phenomena such as permeation through semipermeable membranes, secondary current distribution on partially blocking electrodes, and surface relaxation in nuclear magnetic resonance.

Moving boundary problems governed by anomalous diffusion

Anomalous diffusion can be characterized by a mean-squared displacement 〈x(2)(t)〉 that is proportional to t(α) where α≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (α<1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Subdiffusion in a system with thin membranes

We study both theoretically and experimentally a process of subdiffusion in a system with two thin membranes. The theoretical model uses Green's functions obtained for the membrane system by means of the generalized method of images. These Green's functions are combinations of the fundamental solutions to a fractional subdiffusion equation describing subdiffusion in a homogenous, unbounded system. Using Green's functions we find analytical formulas describing the time evolution of concentration profiles and the time evolution of the amount of substance that remains in the region between the membranes. The concentration profiles fulfill a new boundary condition at the membrane, in which the membrane permeability is assumed to change over time according to the special formula presented in the paper. These concentration profiles fulfill a standard subdiffusion equation with fractional Riemann-Liouville time derivative only approximately, but they coincide very well with the experimental data. Fitting the theoretical functions in with the experimental results, we also estimate the subdiffusion coefficient of polyethylene glycol 2000 in agarose hydrogel.

Temporal scaling characteristics of diffusion as a new MRI contrast: findings in rat hippocampus

Features of the diffusion-time dependence of the diffusion-weighted magnetic resonance imaging (MRI) signal provide a new contrast that could be altered by numerous biological processes and pathologies in tissue at microscopic length scales. An anomalous diffusion model, based on the theory of Brownian motion in fractal and disordered media, is used to characterize the temporal scaling (TS) characteristics of diffusion-related quantities, such as moments of the displacement and zero-displacement probabilities, in excised rat hippocampus specimens. To reduce the effect of noise in magnitude-valued MRI data, a novel numerical procedure was employed to yield accurate estimation of these quantities even when the signal falls below the noise floor. The power-law dependencies characterize the TS behavior in all regions of the rat hippocampus, providing unique information about its microscopic architecture. The relationship between the TS characteristics and diffusion anisotropy is investigated by examining the anisotropy of TS, and conversely, the TS of anisotropy. The findings suggest the robustness of the technique as well as the reproducibility of estimates. TS characteristics of the diffusion-weighted signals could be used as a new and useful marker of tissue microstructure.

Subdiffusive master equation with space-dependent anomalous exponent and structural instability

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent μ in a bounded domain [0,L] are not structurally stable with respect to the nonhomogeneous variations of parameter μ. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of μ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a "Black Swan," the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.