Transient Subdiffusion Research Articles

Variable-order fractional diffusion: Physical interpretation and simulation within the multiple trapping model

The physical interpretation of a variable-order fractional diffusion equation within the framework of the multiple trapping model is presented. This interpretation enables the development of a numerical Monte Carlo algorithm to solve the associated subdiffusion equation. An important feature of the model is variation in energy density of localized states, when the detailed balance condition between localized and mobile particles is satisfied. The variable order anomalous diffusion equations under consideration can be applied to the description of transient subdiffusion in inhomogeneous materials, the order of which depends on the considered spatial and/or time scale. Examples of numerical solutions for different situations are demonstrated. Considering variable-order fractional drift, we calculate and analyze the transient current curves of the time-of-flight method for samples with varying density of localized states.

Hierarchically Coupled Ornstein–Uhlenbeck Processes for Transient Anomalous Diffusion

The nonlinear dependence of the mean-squared displacement (MSD) on time is a common characteristic of particle transport in complex environments. Frequently, this anomalous behavior only occurs transiently before the particle reaches a terminal Fickian diffusion. This study shows that a system of hierarchically coupled Ornstein–Uhlenbeck equations is able to describe both transient subdiffusion and transient superdiffusion dynamics, as well as their sequential combinations. To validate the model, five distinct experimental, molecular dynamics simulation, and theoretical studies are successfully described by the model. The comparison includes the transport of particles in random optical fields, supercooled liquids, bedrock, soft colloidal suspensions, and phonons in solids. The model’s broad applicability makes it a convenient tool for interpreting the MSD profiles of particles exhibiting transient anomalous diffusion.

Lateral diffusion in the presence of obstacles, reexamined.

In transient anomalous subdiffusion, diffusion is anomalous at short times, crossing over to normal at long times. This behavior is well-known for obstructed diffusion with obstacle concentrations below the percolation threshold, and is characterized by the anomalous exponent alpha, the limiting diffusion coefficient D(inf), and the crossover time t(cross), all functions of the obstacle concentration. In my work [Biophys J 66 (1994) 394], the anomalous and normal regions are chosen by eye in a log-log plot of D(t) = RSQ/t versus time t. Straight lines are found for each region by a least-squares fit, and t(cross) is defined as the intersection of these lines. Ellery et al. [J Chem Phys 144 (2016) 171104] criticized this approach and presented a method based on eigenvalues of the transition matrix. Their crossover times were significantly different from mine, and Ellery et al. claimed that their method provides a simpler and more objective evaluation of t(cross). I examine an alternative interpretation, that their method is an elegant measurement of a different characteristic time of the random walk. To test this, I compare results of the eigenvalue method with results of standard Monte Carlo calculations for various characteristic times, including distinct sites visited as a function of time, the number of visits per site (yielding the cover time and the blanket time), and the first passage time from the center of the lattice to the boundary of the region without periodic boundary conditions. The calculations are done for random walks on a square lattice with random obstacles and periodic boundary conditions. Importantly, I examine the dependence of the results on system size, and include percolation analysis so that higher obstacle concentrations can be included in some calculations. The so-called lazy random walk algorithm is also examined.

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems. In this paper, we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process. The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere. Two scenarios govern the transport process in the spherical cavity: free diffusion and diffusion under external velocity. The diffusion dynamics is described by using the concepts of probability density function (PDF) and mean square displacement (MSD) by Fokker–Planck equation in a spherical coordinate system. The effects of dead ends, sphere curvature, and velocity on PDF and MSD are analyzed numerically in detail. We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics. The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.

Anomalous diffusion in branched elliptical structure

Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues, e.g., brain, tumors, muscles, etc. is a geometrically induced complex diffusion and is relevant to different kinds of biological, physical, and chemical systems. In this paper, we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties. The ellipse domain whose boundary has the polar equation with 0 < e < 1, θ ∈ [0,2π], and b as a constant, can be obtained through stretched radius r such that Υ = r ρ (θ) with r ∈ [0,1]. We suppose that, for fixed radius r = R, an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse. The probability distribution function (PDF) in the structure and the marginal PDF and mean square displacement (MSD) along the backbone are obtained numerically. The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state. The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature.

Single-Particle Tracking of DNA-Binding Biomolecules in Cells: Power-Law Distributions of Dwell Times

Single-particle tracking (SPT) experiments have shown a truncated power-law distribution of dwell times for several DNA-binding species in the nucleus of living cells. Why this distribution? A standard explanation is Arrhenius escape from traps with exponentially distributed binding energies. But here the binding energies are roughly Gaussian, yielding roughly lognormal escape times. We therefore examine other mechanisms. The operational definition of SPT immobility is based on the dwell time within the volume of the effective SPT point spread function. The distribution of dwell times can be power-law at short times, but the exponent is constant and much less than observed. The dwell time problem has been treated in terms of revisits by a tracer to a trap, here recast as the number of revisits before the tracer leaves the volume of the effective SPT point spread function. The effect of revisits is limited because the sum of multiple escape times from a moderate trap is tame compared with single escapes from a deep trap, by the central limit theorem. Truncated power-law trapping yields transient anomalous subdiffusion, and obstructed diffusion includes two well-studied cases of this. In percolation the tracer concentration is low and the concentration of immobile obstacles is high. Anomalous subdiffusion is the result of static geometric trapping by dead ends, but how can these structures be made on the SPT length scale from DNA with a 50-nm persistence length? The other case is crowding, in which all species are mobile, the concentration is high, and the geometric traps are dynamic. We examine the effects of polydisperse crowders and of small concentrations of immobile species in a crowded system, to see whether crowding can bridge the gap between the SPT and DNA length scales.

An inverse problem for a time-fractional advection equation associated with a nonlinear reaction term

Fractional derivative is an important notion in the study of the contemporary mathematics not only because it is more mathematically general than the classical derivative but also it really has applications to understand many physical phenomena. In particular, fractional derivatives are related to long power-law particle jumps, which can be understood as transient anomalous sub-diffusion model (see Sabzikar F, Meerschaert M, Chen J. Tempered fractional calculus. J Comput Phys. 2015;293:14–28; Sokolov IM, Klafter J, Blumen A. Fractional kinetics. Phys Today. 2002;55:48–54; Sokolov IM, Klafter J. Anomalous diffusion spreads its wings. Phys World. 2005;18:19–22; Zhang Y, Meerschaert MM, Packman AI. Linking fluvial bed sediment transport across scales. Geophys Res Lett. 2012;39(20):20404. doi:10.1029/2012GL053476). Based on the models given in Scher H, Montroll EW. Anomalous transit-time dispersion in amorphous solids. Phys Rev B. 1975;12(6):2455–2477 and Zheng GH, Wei T. Spectral regularization method for solving a time-fractional inverse diffusion problem. Appl Math Comput. 2011;218:1972–1990, we study an inverse problem for the advection equation with a nonlinear reaction term in a two-dimensional semi-infinite domain for which we recover the initial distribution from the observation data provided at the final location x = 1. This problem is severely ill-posed in the sense of Hadamard. Thus, we propose a regularization method to construct an approximate solution for the problem. From that, convergence rate of the regularized solution is obtained under some a priori bound assumptions on the exact solution. Eventually, a numerical experiment is given to show the effectiveness of the proposed regularization methods.

Finite-range viscoelastic subdiffusion in disordered systems with inclusion of inertial effects

This work justifies further paradigmatic importance of the model of viscoelastic subdiffusion in random environments for the observed subdiffusion in cellular biological systems. Recently, we showed (2018, PCCP, 20, 24140) that this model displays several remarkable features, which makes it an attractive paradigm to explain the physical nature of subdiffusion occurring in biological cells. In particular, it combines viscoelasticity with distinct non-ergodic features. We extend this basic model to make it suitable for physical phenomena such as subdiffusion of lipids in disordered biological membranes upon including the inertial effects. For lipids, the inertial effects occur in the range of picoseconds, and a power-law decaying viscoelastic memory extends over the range of several nanoseconds. Thus, in the absence of disorder, diffusion would become normal on a time scale beyond this memory range. However, both experimentally and in some molecular-dynamical simulations, the time range of lipid subdiffusion extends far beyond the viscoelastic memory range. We study three 1d models of correlated quenched Gaussian disorder to explain the puzzle: singular short-range (exponentially correlated), smooth short-range (Gaussian-correlated), and smooth long-range (power-law correlated) disorder. For a moderate disorder strength, transient viscoelastic subdiffusion changes into the subdiffusion caused by the randomness of the environment. It is characterized by a time-dependent power-law exponent of subdiffusion α(t), which can show nonmonotonous behavior, in agreement with some recent molecular-dynamical simulations. Moreover, the spatial distribution of test particles in this disorder-dominated regime is shown to be a non-Gaussian, exponential power distribution with index χ = 1.45–2.3, which also correlates well with molecular-dynamical findings and experiments. Furthermore, this subdiffusion is nonergodic with single-trajectory averages showing a broad scatter, in agreement with experimental observations for viscoelastic subdiffusion of various particles in living cells.

Diffusion of DNA-Binding Species in the Nucleus: A Transient Anomalous Subdiffusion Model

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

Control of the transient subdiffusion exponent at short and long times

By using independent types of traps with heavy-tailed waiting times, this paper shed light on the transition between diffusive regimes, as characterized by different exponents in the mean squared displacement at short and long times. The authors show that this can be further utilized to engineer the asymptotic behavior of the mean squared displacement by changing the properties of the traps.

Transient and persistent particle subdiffusion in a disordered chain coupled to bosons

We consider the propagation of a single particle in a random chain, assisted by the coupling to dispersive bosons. Time evolution treated with rate equations for hopping between localized states reveals a qualitative difference between dynamics due to noninteracting bosons and hard-core bosons. In the first case the transient dynamics is subdiffusive, but multi-boson processes allow for long-time normal diffusion, while hard-core effects suppress multi-boson processes leading to persistent subdiffusive transport, consistent with numerical results for a full many-body evolution. In contrast, analogous study for a quasiperiodic potential reveals a stable long-time diffusion.

Transient subdiffusion from an Ising environment.

We introduce a model in which a particle performs a continuous-time random walk (CTRW) coupled to an environment with Ising dynamics. The particle shows locally varying diffusivity determined by the geometrical properties of the underlying Ising environment, that is, the diffusivity depends on the size of the connected area of spins pointing in the same direction. The model shows anomalous diffusion when the Ising environment is at critical temperature. We show that any finite scale introduced by a temperature different from the critical one, or a finite size of the environment, cause subdiffusion only during a transient time. The characteristic time, at which the system returns to normal diffusion after the subdiffusive plateau depends on the limiting scale and on how close the temperature is to criticality. The system also displays apparent ergodicity breaking at intermediate time, while ergodicity breaking at longer time occurs only under the idealized infinite environment at the critical temperature.

Effects of soft interactions and bound mobility on diffusion in crowded environments: a model of sticky and slippery obstacles

Crowded environments modify the diffusion of macromolecules, generally slowing their movement and inducing transient anomalous subdiffusion. The presence of obstacles also modifies the kinetics and equilibrium behavior of tracers. While previous theoretical studies of particle diffusion have typically assumed either impenetrable obstacles or binding interactions that immobilize the particle, in many cellular contexts bound particles remain mobile. Examples include membrane proteins or lipids with some entry and diffusion within lipid domains and proteins that can enter into membraneless organelles or compartments such as the nucleolus. Using a lattice model, we studied the diffusive movement of tracer particles which bind to soft obstacles, allowing tracers and obstacles to occupy the same lattice site. For sticky obstacles, bound tracer particles are immobile, while for slippery obstacles, bound tracers can hop without penalty to adjacent obstacles. In both models, binding significantly alters tracer motion. The type and degree of motion while bound is a key determinant of the tracer mobility: slippery obstacles can allow nearly unhindered diffusion, even at high obstacle filling fraction. To mimic compartmentalization in a cell, we examined how obstacle size and a range of bound diffusion coefficients affect tracer dynamics. The behavior of the model is similar in two and three spatial dimensions. Our work has implications for protein movement and interactions within cells.

Density Propagator for Many-Body Localization: Finite-Size Effects, Transient Subdiffusion, and Exponential Decay.

We investigate charge relaxation in quantum wires of spinless disordered fermions (t-V model). Our observable is the time-dependent density propagator Π_{ϵ}(x,t), calculated in windows of different energy density ϵ of the many-body Hamiltonian and at different disorder strengths W, not exceeding the critical value W_{c}. The width Δx_{ϵ}(t) of Π_{ϵ}(x,t) exhibits a behavior dlnΔx_{ϵ}(t)/dlnt=β_{ϵ}(t), where the exponent function β_{ϵ}(t)≲1/2 is seen to depend strongly on L at all investigated parameter combinations. (i)We confirm the existence of a region in phase space that exhibits subdiffusive dynamics in the sense that β_{ϵ}(t)<1/2 in a large window of times. However, subdiffusion might possibly be transient, only, finally giving way to a conventional diffusive behavior with β_{ϵ}=1/2. (ii)We cannot confirm the existence of many-body mobility edges even in regions of the phase diagram that have been reported to be deep in the delocalized phase. (iii)(Transient) subdiffusion 0<β_{ϵ}(t)≲1/2 coexists with an enhanced probability for returning to the origin Π_{ϵ}(0,t), decaying much slower than 1/Δx_{ϵ}(t). Correspondingly, the spatial decay of Π_{ϵ}(x,t) is far from Gaussian, being exponential or even slower. On a phenomenological level, our findings are broadly consistent with the effects of strong disorder and (fractal) Griffiths regions.

Fractional and fractal derivative models for transient anomalous diffusion: Model comparison

Transient anomalous diffusion characterized by transition between diffusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian diffusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional diffusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), breakthrough curves (exhibiting nuance in diffusive state transition), and possible hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures transient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to capture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications.

Quantum subdiffusion with two- and three-body interactions

We study the dynamics of a few-quantum-particle cloud in the presence of two- and three-body interactions in weakly disordered one-dimensional lattices. The interaction is dramatically enhancing the Anderson localization length $\xi_1$ of noninteracting particles. We launch compact wave packets and show that few-body interactions lead to transient subdiffusion of wave packets, $m_2 \sim t^{\alpha}$, $\alpha<1$, on length scales beyond $\xi_1$. The subdiffusion exponent is independent of the number of particles. Two-body interactions yield $\alpha\approx0.5$ for two and three particles, while three-body interactions decrease it to $\alpha\approx0.2$. The tails of expanding wave packets exhibit exponential localization with a slowly decreasing exponent. We relate our results to subdiffusion in nonlinear random lattices, and to results on restricted diffusion in high-dimensional spaces like e.g. on comb lattices.

Computing the non-Markovian coarse-grained interactions derived from the Mori-Zwanzig formalism in molecular systems: Application to polymer melts.

Memory effects are often introduced during coarse-graining of a complex dynamical system. In particular, a generalized Langevin equation (GLE) for the coarse-grained (CG) system arises in the context of Mori-Zwanzig formalism. Upon a pairwise decomposition, GLE can be reformulated into its pairwise version, i.e., non-Markovian dissipative particle dynamics (DPD). GLE models the dynamics of a single coarse particle, while DPD considers the dynamics of many interacting CG particles, with both CG systems governed by non-Markovian interactions. We compare two different methods for the practical implementation of the non-Markovian interactions in GLE and DPD systems. More specifically, a direct evaluation of the non-Markovian (NM) terms is performed in LE-NM and DPD-NM models, which requires the storage of historical information that significantly increases computational complexity. Alternatively, we use a few auxiliary variables in LE-AUX and DPD-AUX models to replace the non-Markovian dynamics with a Markovian dynamics in a higher dimensional space, leading to a much reduced memory footprint and computational cost. In our numerical benchmarks, the GLE and non-Markovian DPD models are constructed from molecular dynamics (MD) simulations of star-polymer melts. Results show that a Markovian dynamics with auxiliary variables successfully generates equivalent non-Markovian dynamics consistent with the reference MD system, while maintaining a tractable computational cost. Also, transient subdiffusion of the star-polymers observed in the MD system can be reproduced by the coarse-grained models. The non-interacting particle models, LE-NM/AUX, are computationally much cheaper than the interacting particle models, DPD-NM/AUX. However, the pairwise models with momentum conservation are more appropriate for correctly reproducing the long-time hydrodynamics characterised by an algebraic decay in the velocity autocorrelation function.

Langevin equation with fluctuating diffusivity: A two-state model.

Recently, anomalous subdiffusion, aging, and scatter of the diffusion coefficient have been reported in many single-particle-tracking experiments, though the origins of these behaviors are still elusive. Here, as a model to describe such phenomena, we investigate a Langevin equation with diffusivity fluctuating between a fast and a slow state. Namely, the diffusivity follows a dichotomous stochastic process. We assume that the sojourn time distributions of these two states are given by power laws. It is shown that, for a nonequilibrium ensemble, the ensemble-averaged mean-square displacement (MSD) shows transient subdiffusion. In contrast, the time-averaged MSD shows normal diffusion, but an effective diffusion coefficient transiently shows aging behavior. The propagator is non-Gaussian for short time and converges to a Gaussian distribution in a long-time limit; this convergence to Gaussian is extremely slow for some parameter values. For equilibrium ensembles, both ensemble-averaged and time-averaged MSDs show only normal diffusion and thus we cannot detect any traces of the fluctuating diffusivity with these MSDs. Therefore, as an alternative approach to characterizing the fluctuating diffusivity, the relative standard deviation (RSD) of the time-averaged MSD is utilized and it is shown that the RSD exhibits slow relaxation as a signature of the long-time correlation in the fluctuating diffusivity. Furthermore, it is shown that the RSD is related to a non-Gaussian parameter of the propagator. To obtain these theoretical results, we develop a two-state renewal theory as an analytical tool.

Editorial: Quantitative Biology: Dynamics of Living Systems.

Editorial: Quantitative Biology: Dynamics of Living Systems.

Effects of Obstacle Binding and Bound Mobility in a Lattice Model of Protein Diffusion

Crowded environments modify the diffusion of macromolecules, generally slowing their movement and inducing transient anomalous subdiffusion. Obstacles also modify the kinetics and equilibrium behavior of interaction between mobile proteins. In some biophysical contexts, bound molecules can still experience mobility, for example transcription factors sliding along DNA, membrane proteins with some entry and diffusion within lipid domains, or proteins that can enter into non-membrane bound compartments such as the nucleolus. We used a lattice model to study the diffusive behavior of tracer particles which bind to obstacles and can diffuse within them. We show that binding significantly alters the motion of tracers. The type and degree of motion while bound is a key determinant of the tracer mobility. Our work has implications for protein-protein movement and interactions within living cells.