Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations

Fractional differentiation method: Some applications to the theory of subdiffusion-controlled reactions

The fractional differentiation method's broad possibilities are demonstrated with rather simple but important examples of the anomalous diffusion trapping problems. In particular we evaluate the reaction rate coefficients for the subdiffusion-controlled reactions and for reactions describing by a diffusion equation with a half-order time derivative as a damping term. The distinctive feature of this aproach is that the reaction rate coefficient may be obtained by means of some factorization procedure immediately, without a preliminary solution to the corresponding initial boundary value diffusion problem. The explanations given in the paper are detailed enough to provide the mathematical background for the fractional differentiation method needed to apply it to a wide range of reaction-diffusion problems with time-fractional derivatives in the Riemann-Liouville sense.

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.

Subdiffusion with particle immobilization process described by a differential equation with Riemann-Liouville-type fractional time derivative

An equation describing subdiffusion with possible immobilization of particles is derived by means of the continuous time random walk model. The equation contains a fractional time derivative of Riemann-Liouville type which is a differential-integral operator with the kernel defined by the Laplace transform; the kernel controls the immobilization process. We propose a method for calculating the inverse Laplace transform providing the kernel in the time domain. In the long time limit the subdiffusion-immobilization process reaches a stationary state in which the probability density of a particle distribution is an exponential function.

Inferences from FRAP data are model dependent: A subdiffusive analysis

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.

A New Parallelized Computation Method of HASC-N Difference Scheme for Inhomogeneous Time Fractional Fisher Equation

The fractional Fisher equation has a wide range of applications in many engineering fields. The rapid numerical methods for fractional Fisher equation have momentous scientific meaning and engineering applied value. A parallelized computation method for inhomogeneous time-fractional Fisher equation (TFFE) is proposed. The main idea is to construct the hybrid alternating segment Crank-Nicolson (HASC-N) difference scheme based on alternating segment difference technology, using the classical explicit scheme and classical implicit scheme combined with Crank-Nicolson (C-N) scheme. The unique existence, unconditional stability and convergence are proved theoretically. Numerical tests show that the HASC-N difference scheme is unconditionally stable. The HASC-N difference scheme converges to O(τ2−α+h2) under strong regularity and O(τα+h2) under weak regularity of fractional derivative discontinuity. The HASC-N difference scheme has high precision and distinct parallel computing characteristics, which is efficient for solving inhomogeneous TFFE.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

Subdiffusion-limited fractional reaction-subdiffusion equations with affine reactions: Solution, stochastic paths, and applications

In contrast to normal diffusion, there is no canonical model for reactions between chemical species which move by anomalous subdiffusion. Indeed, the type of mesoscopic equation describing reaction-subdiffusion systems depends on subtle assumptions about the microscopic behavior of individual molecules. Furthermore, the correspondence between mesoscopic and microscopic models is not well understood. In this paper, we study the subdiffusion-limited model, which is defined by mesoscopic equations with fractional derivatives applied to both the movement and the reaction terms. Assuming that the reaction terms are affine functions, we show that the solution to the fractional system is the expectation of a random time change of the solution to the corresponding integer order system. This result yields a simple and explicit algebraic relationship between the fractional and integer order solutions in Laplace space. We then find the microscopic Langevin description of individual molecules that corresponds to such mesoscopic equations and give a computer simulation method to generate their stochastic trajectories. This analysis identifies some precise microscopic conditions that dictate when this type of mesoscopic model is or is not appropriate. We apply our results to several scenarios in cell biology which, despite the ubiquity of subdiffusion in cellular environments, have been modeled almost exclusively by normal diffusion. Specifically, we consider subdiffusive models of morphogen gradient formation, fluctuating mobility, and fluorescence recovery after photobleaching (FRAP) experiments. We also apply our results to fractional ordinary differential equations.

Time Fractional Fisher-KPP and Fitzhugh-Nagumo Equations

A standard reaction-diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction-subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction-diffusion equations. In this paper, we formulate clear examples of reaction-subdiffusion systems, based on; equal birth and death rate dynamics, Fisher-Kolmogorov, Petrovsky and Piskunov (Fisher-KPP) equation dynamics, and Fitzhugh-Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction-diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

Anomalous reaction-diffusion equations for linear reactions

Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations modeling subdiffusion by merely adding reaction terms to the equations describing spatial movement. A series of previous works derived fractional reaction-diffusion equations for the spatiotemporal evolution of particles undergoing subdiffusion in one space dimension with linear reactions between a finite number of discrete states. In this paper, we first give a short and elementary proof of these previous results. We then show how this argument gives the evolution equations for more general cases, including subdiffusion following any fractional Fokker-Planck equation in an arbitrary d-dimensional spatial domain with time-dependent reactions between infinitely many discrete states. In contrast to previous works which employed a variety of technical mathematical methods, our analysis reveals that the evolution equations follow from (1) the probabilistic independence of the stochastic spatial and discrete processes describing a single particle and (2) the linearity of the integro-differential operators describing spatial movement. We also apply our results to systems combining reactions with superdiffusion.

Reaction and ultraslow diffusion on comb structures

A two-dimensional (2D) comb model is proposed to characterize reaction-ultraslow diffusion of tracers both in backbones (x direction) and side branches (y direction) of the comblike structure with two memory kernels. The memory kernels include Dirac delta, power-law, and logarithmic and inverse Mittag-Leffler (ML) functions, which can also be considered as the structural functions in the time structural derivative. Based on the comb model, ultraslow diffusion on a fractal comb structure is also investigated by considering spatial fractal geometry of the backbone volume. The mean squared displacement (MSD) and the corresponding concentration of the tracers, i.e., the solution of the comb model, are derived for reactive and conservative tracers. For a fractal structure of backbones, the derived MSDs and corresponding solutions depend on the backbone's fractal dimension. The proposed 2D comb model with different kernel functions is feasible to describe ultraslow diffusion in both the backbone and side branches of the comblike structure.

Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald–Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald–Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

Two dimensional diffusion-controlled triplet-triplet annihilation kinetics

Diffusion controlled chemical reactions are usually observed in three dimensional media. In contrast, planar bimolecular reactions taking place between reagents adsorbed at a soft interface are two-dimensional and therefore cannot be studied within the same formalism. Indeed, soft interfaces allow the adsorbed species to freely diffuse in a liquid-like manner. Here, we present the first experimental observation of a diffusion-controlled reaction in an environment that is planar at the ångström scale. By means of time-resolved surface second harmonic generation, an inherently surface sensitive technique, we observed that the kinetics of the diffusion of the reagents in the plane decreases as the surface concentration of adsorbed species increases. This is of course not the case for bulk reactions where the rates always increase with the reactant concentration. Such changes in the kinetics regime were rationalised as the evolution from a regular 2D free diffusion regime to a geometry-controlled scheme.

Modeling breast cancer progression to bone: how driver mutation order and metabolism matter

BACKGROUND

Not all the mutations are equally important for the development of metastasis. What about their order? The survival of cancer cells from the primary tumour site to the secondary seeding sites depends on the occurrence of very few driver mutations promoting oncogenic cell behaviours. Usually these driver mutations are among the most effective clinically actionable target markers. The quantitative evaluation of the effects of a mutation across primary and secondary sites is an important challenging problem that can lead to better predictability of cancer progression trajectory.

RESULTS

We introduce a quantitative model in the framework of Cellular Automata to investigate the effects of metabolic mutations and mutation order on cancer stemness and tumour cell migration from breast, blood to bone metastasised sites. Our approach models three types of mutations: driver, the order of which is relevant for the dynamics, metabolic which support cancer growth and are estimated from existing databases, and non-driver mutations. We integrate the model with bioinformatics analysis on a cancer mutation database that shows metabolism-modifying alterations constitute an important class of key cancer mutations.

CONCLUSIONS

Our work provides a quantitative basis of how the order of driver mutations and the number of mutations altering metabolic processis matter for different cancer clones through their progression in breast, blood and bone compartments. This work is innovative because of multi compartment analysis and could impact proliferation of therapy-resistant clonal populations and patient survival. Mathematical modelling of the order of mutations is presented in terms of operators in an accessible way to the broad community of researchers in cancer models so to inspire further developments of this useful (and underused in biomedical models) methodology. We believe our results and the theoretical framework could also suggest experiments to measure the overall personalised cancer mutational signature.

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.

Continuous time random walk with local particle-particle interaction

The continuous time random walk (CTRW) is often applied to the study of particle motion in disordered media. Yet most such applications do not allow for particle-particle (walker-walker) interaction. In this paper, we consider a CTRW with particle-particle interaction; however, for simplicity, we restrain the interaction to be local. The generalized Chapman-Kolmogorov equation is modified by introducing a perturbation function that fluctuates around 1, which models the effect of interaction. Subsequently, a time-fractional nonlinear advection-diffusion equation is derived from this walking system. Under the initial condition of condensed particles at the origin and the free-boundary condition, we numerically solve this equation with both attractive and repulsive particle-particle interactions. Moreover, a Monte Carlo simulation is devised to verify the results of the above numerical work. The equation and the simulation unanimously predict that this walking system converges to the conventional one in the long-time limit. However, for systems where the free-boundary condition and long-time limit are not simultaneously satisfied, this convergence does not hold.

Time-fractional characterization of brine reaction and precipitation in porous media

Brine reaction and precipitation in porous media sometimes occur in the presence of a strong fluid flowing field, which induces the mobilization of the precipitated salts and distorts their spatial distribution. It is interesting to investigate how the distribution responds to such mobilization. We view these precipitates as random walkers in the complex inner space of the porous media, where they make stochastic jumps among locations and possibly wait between successive transitions. In consideration of related experimental results, the waiting time of the precipitates at a particular position is allowed to range widely from short sojourn to permanent residence. Through the model of a continuous-time random walk, a class of time-fractional equations for the precipitate's concentration profile is derived, including that in the Riemann-Liouville formalism and the Prabhakar formalism. The solutions to these equations show the general pattern of the precipitate's spatiotemporal evolution: a coupling of mass accumulation and mass transport. And the degree to which the mass is mobilized turns out to be monotonically correlated to the fractional exponent α. Moreover, to keep the completeness of the model, we further discuss how the interaction among the precipitates influences the precipitation process. In doing so, a time-fractional non-linear Fokker-Planck equation with source term is introduced and solved. It is shown that the interaction among the precipitates slightly perturbs their spatial distribution. This distribution is largely dominated by the brine reaction itself and the interaction between the precipitates and the porous media.

Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. These models often involve fractional derivatives. The important physical extension of this work to processes occurring in growing materials has proven highly nontrivial. Here we derive evolution equations for modeling subdiffusive transport in a growing medium. The derivation is based on a continuous-time random walk. The concise formulation of these evolution equations requires the introduction of a new, comoving, fractional derivative. The implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.

Continuous-time random-walk model for anomalous diffusion in expanding media

Expanding media are typical in many different fields, e.g., in biology and cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties such as the set of positional moments and the Green's function. Here, we focus on the characterization of such effects when the diffusion process is described by the continuous-time random-walk (CTRW) model. As is well known, when the medium is static this model yields anomalous diffusion for a proper choice of the probability density function (pdf) for the jump length and the waiting time, but the behavior may change drastically if a medium expansion is superimposed on the intrinsic random motion of the diffusing particle. For the case where the jump length and the waiting time pdfs are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Lévy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Green's function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. In the specific case of a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This "big crunch" effect, totally absent in the case of normal diffusion, stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model in an expanding medium. From this hierarchy, the full time evolution of the second-order moment is obtained for some specific types of expansion. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained, whence the long-time behavior of moments of arbitrary order is subsequently inferred. Our analytical and numerical results for both Lévy flights and subdiffusive CTRWs confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Lévy flights, we quantify this effect by means of the so-called "Lévy horizon."

Reaction–diffusion with stochastic decay rates

Microscopic physical and chemical fluctuations in a reaction–diffusion system lead to anomalous chemical kinetics and transport on the mesoscopic scale. Emergent non-Markovian effects lead to power-law reaction times and localization of reacting species.

Anomalous dielectric relaxation with linear reaction dynamics in space-dependent force fields

The anomalous dielectric relaxation of disordered reaction with linear reaction dynamics is studied via the continuous time random walk model in the presence of space-dependent electric field. Two kinds of modified reaction-subdiffusion equations are derived for different linear reaction processes by the master equation, including the instantaneous annihilation reaction and the noninstantaneous annihilation reaction. If a constant proportion of walkers is added or removed instantaneously at the end of each step, there will be a modified reaction-subdiffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps, there will be a standard linear reaction kinetics term but a fractional order temporal derivative operating on an anomalous diffusion term. The dielectric polarization is analyzed based on the Legendre polynomials and the dielectric properties of both reactions can be expressed by the effective rotational diffusion function and component concentration function, which is similar to the standard reaction-diffusion process. The results show that the effective permittivity can be used to describe the dielectric properties in these reactions if the chemical reaction time is much longer than the relaxation time.

Impact of anomalous transport kinetics on the progress of wound healing

This work focuses on the transport kinetics of chemical and cellular species during wound healing. Anomalous transport kinetics, coupling sub- and superdiffusion with chemotaxis, and fractional viscoelasticity of soft tissues are analyzed from a modeling point of view. The paper presents a generalization of well stablished mechano-chemical models of wound contraction (Murphy et al., 2012; Valero et al., 2014) to include the previously mentioned anomalous effects by means of partial differential equations of fractional order. Results show the effect that anomalous dynamics have on the contraction rate and extension and on the distribution of biological species, and indicators of fibroproliferative disorders are identified.

A model of space-fractional-order diffusion in the glial scar

Implantation of neuroprosthetic electrodes induces a stereotypical state of neuroinflammation, which is thought to be detrimental for the neurons surrounding the electrode. Mechanisms of this type of neuroinflammation are still poorly understood. Recent experimental and theoretical results point to a possible role of the diffusing species in this process. The paper considers a model of anomalous diffusion occurring in the glial scar around a chronic implant in two simple geometries - a separable rectilinear electrode and a cylindrical electrode, which are solvable exactly. We describe a hypothetical extended source of diffusing species and study its concentration profile in steady-state conditions. Diffusion transport is assumed to obey a fractional-order Fick law, derivable from physically realistic assumptions using a fractional calculus approach. Presented fractional-order distribution morphs into integer-order diffusion in the case of integral fractional exponents. The model demonstrates that accumulation of diffusing species can occur and the scar properties (i.e. tortuosity, fractional order, scar thickness) and boundary conditions can influence such accumulation. The observed shape of the concentration profile corresponds qualitatively with GFAP profiles reported in the literature. The main difference with respect to the previous studies is the explicit incorporation of the apparatus of fractional calculus without assumption of an ad hoc tortuosity parameter. The approach can be adapted to other studies of diffusion in biological tissues, for example of biomolecules or small drug molecules.

MESOSCOPIC MODELING OF STOCHASTIC REACTION-DIFFUSION KINETICS IN THE SUBDIFFUSIVE REGIME

Subdiffusion has been proposed as an explanation of various kinetic phenomena inside living cells. In order to fascilitate large-scale computational studies of subdiffusive chemical processes, we extend a recently suggested mesoscopic model of subdiffusion into an accurate and consistent reaction-subdiffusion computational framework. Two different possible models of chemical reaction are revealed and some basic dynamic properties are derived. In certain cases those mesoscopic models have a direct interpretation at the macroscopic level as fractional partial differential equations in a bounded time interval. Through analysis and numerical experiments we estimate the macroscopic effects of reactions under subdiffusive mixing. The models display properties observed also in experiments: for a short time interval the behavior of the diffusion and the reaction is ordinary, in an intermediate interval the behavior is anomalous, and at long times the behavior is ordinary again.

A Fractional Order Recovery SIR Model from a Stochastic Process

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

Distributed-order diffusion equations and multifractality: Models and solutions

We study distributed-order time fractional diffusion equations characterized by multifractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed-order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed-order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multifractal properties of distributed-order diffusion equations is provided.

Integrodifferential formulations of the continuous-time random walk for solute transport subject to bimolecular A+B→0 reactions: From micro- to mesoscopic

We develop continuous-time random walk (CTRW) equations governing the transport of two species that annihilate when in proximity to one another. In comparison with catalytic or spontaneous transformation reactions that have been previously considered in concert with CTRW, both species have spatially variant concentrations that require consideration. We develop two distinct formulations. The first treats transport and reaction microscopically, potentially capturing behavior at sharp fronts, but at the cost of being strongly nonlinear. The second, mesoscopic, formulation relies on a separation-of-scales technique we develop to separate microscopic-scale reaction and upscaled transport. This simplifies the governing equations and allows treatment of more general reaction dynamics, but requires stronger smoothness assumptions of the solution. The mesoscopic formulation is easily tractable using an existing solution from the literature (we also provide an alternative derivation), and the generalized master equation (GME) for particles undergoing A+B→0 reactions is presented. We show that this GME simplifies, under appropriate circumstances, to both the GME for the unreactive CTRW and to the advection-dispersion-reaction equation. An additional major contribution of this work is on the numerical side: to corroborate our development, we develop an indirect particle-tracking-partial-integro-differential-equation (PIDE) hybrid verification technique which could be applicable widely in reactive anomalous transport. Numerical simulations support the mesoscopic analysis.

Cattaneo-type subdiffusion-reaction equation

Subdiffusion in a system in which mobile particles A can chemically react with static particles B according to the rule A+B→B is considered within a persistent random-walk model. This model, which assumes a correlation between successive steps of particles, provides hyperbolic Cattaneo normal diffusion or fractional subdiffusion equations. Starting with the difference equation, which describes a persistent random walk in a system with chemical reactions, using the generating function method and the continuous-time random-walk formalism, we will derive the Cattaneo-type subdiffusion differential equation with fractional time derivatives in which the chemical reactions mentioned above are taken into account. We will also find its solution over a long time limit. Based on the obtained results, we will find the Cattaneo-type subdiffusion-reaction equation in the case in which mobile particles of species A and B can chemically react according to a more complicated rule.

Subdiffusion-reaction processes with A→B reactions versus subdiffusion-reaction processes with A+B→B reactions

We consider the subdiffusion-reaction process with reactions of a type A+B→B (in which particles A are assumed to be mobile, whereas B are assumed to be static) in comparison to the subdiffusion-reaction process with A→B reactions which was studied by Sokolov, Schmidt, and Sagués [Phys. Rev. E 73, 031102 (2006)]. In both processes a rule that reactions can only occur between particles which continue to exist is taken into account. Although in both processes a probability of the vanishing of particle A due to a reaction is independent of both time and space variables (assuming that in the system with the A+B→B reactions, particles B are distributed homogeneously), we show that subdiffusion-reaction equations describing these processes as well as their Green's functions are qualitatively different. The reason for this difference is as follows. In the case of the former reaction, particles A and B have to meet with some probability before the reaction occurs in contradiction with the case of the latter reaction. For the subdiffusion process with the A+B→B reactions we consider three models which differ in some details concerning a description of the reactions. We base the method considered in this paper on a random walk model in a system with both discrete time and discrete space variables. Then the system with discrete variables is transformed into a system with both continuous time and continuous space variables. Such a method seems to be convenient in analyzing subdiffusion-reaction processes with partially absorbing or partially reflecting walls. The reason is that within this method we can determine Green's functions without a necessity of solving a fractional differential subdiffusion-reaction equation with boundary conditions at the walls. As an example, we use the model to find the Green's functions for a subdiffusive reaction system (with the reactions mentioned above), which is bounded by a partially absorbing wall. This example shows how the model can be used to analyze the subdiffusion-reaction process in a system with partially absorbing or reflecting thin membranes. Employing a simple phenomenological model, we also derive equations related to the reaction parameters used in the considered models.

Memory-induced anomalous dynamics in a minimal random walk model

A discrete-time dynamics of a non-Markovian random walker is analyzed using a minimal model where memory of the past drives the present dynamics. In recent work [N. Kumar et al., Phys. Rev. E 82, 021101 (2010)] we proposed a model that exhibits asymptotic superdiffusion, normal diffusion, and subdiffusion with the sweep of a single parameter. Here we propose an even simpler model, with minimal options for the walker: either move forward or stay at rest. We show that this model can also give rise to diffusive, subdiffusive, and superdiffusive dynamics at long times as a single parameter is varied. We show that in order to have subdiffusive dynamics, the memory of the rest states must be perfectly correlated with the present dynamics. We show explicitly that if this condition is not satisfied in a unidirectional walk, the dynamics is only either diffusive or superdiffusive (but not subdiffusive) at long times.

Anomalous versus slowed-down Brownian diffusion in the ligand-binding equilibrium

Measurements of protein motion in living cells and membranes consistently report transient anomalous diffusion (subdiffusion) that converges back to a Brownian motion with reduced diffusion coefficient at long times after the anomalous diffusion regime. Therefore, slowed-down Brownian motion could be considered the macroscopic limit of transient anomalous diffusion. On the other hand, membranes are also heterogeneous media in which Brownian motion may be locally slowed down due to variations in lipid composition. Here, we investigate whether both situations lead to a similar behavior for the reversible ligand-binding reaction in two dimensions. We compare the (long-time) equilibrium properties obtained with transient anomalous diffusion due to obstacle hindrance or power-law-distributed residence times (continuous-time random walks) to those obtained with space-dependent slowed-down Brownian motion. Using theoretical arguments and Monte Carlo simulations, we show that these three scenarios have distinctive effects on the apparent affinity of the reaction. Whereas continuous-time random walks decrease the apparent affinity of the reaction, locally slowed-down Brownian motion and local hindrance by obstacles both improve it. However, only in the case of slowed-down Brownian motion is the affinity maximal when the slowdown is restricted to a subregion of the available space. Hence, even at long times (equilibrium), these processes are different and exhibit irreconcilable behaviors when the area fraction of reduced mobility changes.

Propagation failure for a front between stable states in a system with subdiffusion

The propagation of subdiffusion-reaction fronts is studied in the framework of a model recently suggested by Fedotov [ Phys. Rev. E 81 011117 (2010)]. An exactly solvable model with a piecewise linear reaction function is considered. A drastic difference between the cases of normal diffusion and subdiffusion has been revealed. While in the case of normal diffusion, a traveling wave solution between two locally stable phases always exists, and is unique, in the case of the subdiffusion such solutions do not exist. The numerical simulation shows that the velocity of the front decreases with time according to a power law. The only kind of fronts moving with a constant velocity are waves which propagate solely due to the reaction, with a vanishing subdiffusive flux.

Evanescent continuous-time random walks

We study how an evanescence process affects the number of distinct sites visited by a continuous-time random walker in one dimension. We distinguish two very different cases, namely, when evanescence can only occur concurrently with a jump, and when evanescence can occur at any time. The first is characteristic of trapping processes on a lattice, whereas the second is associated with spontaneous death processes such as radioactive decay. In both of these situations we consider three different forms of the waiting time distribution between jumps, namely, exponential, long tailed, and ultraslow.

Reaction-subdiffusion front propagation in a comblike model of spiny dendrites

Fractional reaction-diffusion equations are derived by exploiting the geometrical similarities between a comb structure and a spiny dendrite. In the framework of the obtained equations, two scenarios of reaction transport in spiny dendrites are explored, where both a linear reaction in spines and nonlinear Fisher-Kolmogorov-Petrovskii-Piskunov reactions along dendrites are considered. In the framework of fractional subdiffusive comb model, we develop a Hamilton-Jacobi approach to estimate the overall velocity of the reaction front propagation. One of the main effects observed is the failure of the front propagation for both scenarios due to either the reaction inside the spines or the interaction of the reaction with the spines. In the first case the spines are the source of reactions, while in the latter case, the spines are a source of a damping mechanism.

Mixing-Driven Equilibrium Reactions in Multidimensional Fractional Advection Dispersion Systems

We study instantaneous, mixing-driven, bimolecular equilibrium reactions in a system where transport is governed by a multidimensional space fractional dispersion equation. The superdiffusive, nonlocal nature of the system causes the location and magnitude of reactions that take place to change significantly from a classical Fickian diffusion model. In particular, regions where reaction rates would be zero for the Fickian case become regions where the maximum reaction rate occurs when anomalous dispersion operates. We also study a global metric of mixing in the system, the scalar dissipation rate and compute its asymptotic scaling rates analytically. The scalar dissipation rate scales asymptotically as t^-(d+α)/α , where d is the number of spatial dimensions and α is the fractional derivative exponent.

Random death process for the regularization of subdiffusive fractional equations

The description of subdiffusive transport in complex media by fractional equations with a constant anomalous exponent is not robust where the stationary distribution is concerned. The Gibbs-Boltzmann distribution is radically changed by even small spatial perturbations to the anomalous exponent [S. Fedotov and S. Falconer, Phys. Rev. E 85, 031132 (2012)]. To rectify this problem we propose the inclusion of the random death process in the random walk scheme, which is quite natural for biological applications including morphogen gradient formation. From this, we arrive at the modified fractional master equation and analyze its asymptotic behavior, both analytically and by Monte Carlo simulation. We show that this equation is structurally stable against spatial variations of the anomalous exponent. We find that the stationary flux of the particles has a Markovian form with rate functions depending on the anomalous rate functions, the death rate, and the anomalous exponent. Additionally, in the continuous limit we arrive at an advection-diffusion equation where advection and diffusion coefficients depend on both the death rate and anomalous exponent.

Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach

We calculate the survival probability of an immobile target surrounded by a sea of uncorrelated diffusive or subdiffusive evanescent traps (i.e., traps that disappear in the course of their motion). Our calculation is based on a fractional reaction-subdiffusion equation derived from a continuous time random walk model of the system. Contrary to an earlier method valid only in one dimension (d=1), the equation is applicable in any Euclidean dimension d and elucidates the interplay between anomalous subdiffusive transport, the irreversible evanescence reaction, and the dimension in which both the traps and the target are embedded. Explicit results for the survival probability of the target are obtained for a density ρ(t) of traps which decays (i) exponentially and (ii) as a power law. In the former case, the target has a finite asymptotic survival probability in all integer dimensions, whereas in the latter case there are several regimes where the values of the decay exponent for ρ(t) and the anomalous diffusion exponent of the traps determine whether or not the target has a chance of eternal survival in one, two, and three dimensions.

Application of hyperbolic scaling for calculation of reaction-subdiffusion front propagation

A technique of hyperbolic scaling is applied to calculate a reaction front velocity in an irreversible autocatalytic conversion reaction A+B → 2A under subdiffusion. The method, based on the geometric optics approach is a technically elegant observation of the propagation front failure obtained in Phys. Rev. E 78, 011128 (2008).

Delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

It has been alleged in several papers that the so-called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate the accuracy of the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in L(2) norm to the DCTRWs than the telegraph equation. We conclude, therefore, that first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.

Non-Markovian models for migration-proliferation dichotomy of cancer cells: anomalous switching and spreading rate

Proliferation and migration dichotomy of the tumor cell invasion is examined within two non-Markovian models. We consider the tumor spheroid, which consists of the tumor core with a high density of cells and the outer invasive zone. We distinguish two different regions of the outer invasive zone and develop models for both zones. In model I we analyze the near-core-outer region, where biased migration away from the tumor spheroid core takes place. We suggest non-Markovian switching between the migrating and proliferating phenotypes of tumor cells. Nonlinear master equations for mean densities of cancer cells of both phenotypes are derived. In anomalous switching case we estimate the average size of the near-core-outer region that corresponds to sublinear growth (r(t)) ~ t(μ) for 0 < μ < 1. In model II we consider the outer zone, where the density of cancer cells is very low. We suggest an integrodifferential equation for the total density of cancer cells. For proliferation rate we use the classical logistic growth, while the migration of cells is subdiffusive. The exact formulas for the overall spreading rate of cancer cells are obtained by a hyperbolic scaling and Hamilton-Jacobi techniques.

Particle-tracking simulation of fractional diffusion-reaction processes

Computer simulation of reactive transport in heterogeneous systems remains a challenge due to the multiscale nature of reactive dynamics and the non-Fickian behavior of transport. This study develops a fully Lagrangian approach via particle tracking to describe the reactive transport controlled by the tempered super- or subdiffusion. In the particle-tracking algorithm, the local-scale reaction is affected by the interaction radius between adjacent reactants, whose motion can be simulated by the Langevin equations corresponding to the tempered stable models. Lagrangian simulation results show that the transient superdiffusion enhances the reaction by enhancing the degree of mixing of the reactants. The proposed particle-tracking scheme can also be extended conveniently to multiscale superdiffusion. For the case of transient subdiffusion, the trapping of solutes in the immobile phase can either decrease or accelerate the reaction rate, depending on the initial condition of the reactant particles. Further practical applications show that the new solver efficiently captures bimolecular reactions observed in laboratories.

Subdiffusion, chemotaxis, and anomalous aggregation

We propose a nonlinear random walk model which is suitable for the analysis of both chemotaxis and anomalous subdiffusive transport. We derive the master equations for the population density for the case when the transition rate for a random walk depends on residence time, chemotactic substance, and population density. We introduce the anomalous chemotactic sensitivity and find an anomalous aggregation phenomenon. So we suggest a different explanation of the well-known effect of chemotactic collapse.