## Persistent random walk of cells involving anomalous effects and random death

Phys. Rev. E 91, 042124, April, 2015

## Abstract

The purpose of this paper is to implement a random death process into a persistent random walk model which produces sub-ballistic superdiffusion (Levy walk). We develop a stochastic two-velocity jump model of cell motility for which the switching rate depends upon the time which the cell has spent moving in one direction.It is assumed that the switching rate is a decreasing function of residence (running) time. This assumption leads to the power law for the velocity switching time distribution. This describes the anomalous persistence of cell motility: the longer the cell moves in one direction, the smaller the switching probability to another direction becomes. We derive master equations for the cell densities with the generalized switching terms involving the tempered fractional material derivatives. We show that the random death of cells has an important implication for the transport process through tempering of the superdiffusive process. In the long-time limit we write stationary master equations in terms of exponentially truncated fractional derivatives in which the rate of death plays the role of tempering of a Levy jump distribution. We find the upper and lower bounds for the stationary profiles corresponding to the ballistic transport and diffusion with the death-rate-dependent diffusion coefficient. Monte Carlo simulations confirm these bounds

Sergei Fedotov, Abby Tan, Andrey Zubarev

## Subdiffusion in a external potential: Anomalous effects hiding behind normal behavior.

Phys. Rev. E 91, 042112, April, 2015

## Abstract

We propose a model of subdiffusion in which an external force is acting on a particle at all times not only at the moment of jump. The implication of this assumption is the dependence of the random trapping time on the force with the dramatic change of particles behavior compared to the standard continuous time random walk model in the long time limit. Constant force leads to the transition from non-ergodic subdiffusion to ergodic diffusive behavior. However, we show this behavior remains anomalous in a sense that the diffusion coefficient depends on the external force and on the anomalous exponent. For quadratic potential we find that the system remains non-ergodic. The anomalous exponent in this case defines not only the speed of convergence but also the stationary distribution which is different from standard Boltzmann equilibrium.

Sergei Fedotov, Nickolay Korabel

## Transport Equations for Subdiffusion with Nonlinear Particle Interaction

J. Theor. Biology 366, 71-83 (2015)

## Abstract

We show how the nonlinear interaction effects `volume filling' and `adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with anomalous trapping and systematically derive generic non-Markovian and nonlinear governing equations for the mean concentrations of the subdiffusive cells or organisms. We uncover an interesting interaction between the nonlinearities and the non-Markovian nature of the transport. In the subdiffusive case, this interaction manifests itself in a nontrivial combination of nonlinear terms with fractional derivatives. In the long time limit, however, these equations simplify to a form without fractional operators. This provides an easy method for the study of aggregation phenomena. In particular, this enables us to show that volume filling can prevent "anomalous aggregation," which occurs in subdiffusive systems with a spatially varying anomalous exponent.

Sergei Fedotov, Peter Straka

## Nonlinear degradation-enhanced transport of morphogens performing subdiffusion

January 1, 2014. Phys. Rev. E 89, 012107 (6 pages)

## Abstract

We study a morphogen gradient formation under nonlinear degradation and subdiffusive transport. In the long-time limit, we obtain the nonlinear effect of degradation-enhanced diffusion, resulting from the interaction of non-Markovian subdiffusive transport with a nonlinear reaction. We find the stationary profile of power-law type, which has implications for robustness, with the shape of the profile being controlled by the anomalous exponent. Far away from the source of morphogens, any changes in the rate of production are not felt.

Sergei Fedotov, Steven Falconer

## Nonlinear subdiffusive fractional equations and the aggregation phenomenon

December 23, 2013. Phys. Rev. E 88, 032104 (9 pages)

## Abstract

In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on the mean density of particles. We derive a set of nonlinear subdiffusive fractional master equations and consider their diffusion approximations. We show that these equations describe the transition from an intermediate subdiffusive regime to asymptotically normal advection-diffusion transport regime. This transition is governed by nonlinear tempering parameter that generalizes the standard linear tempering. We illustrate the general results through the use of the examples from cell and population biology. We find that a nonuniform anomalous exponent has a strong influence on the aggregation phenomenon.

Sergei Fedotov, Steven Falconer

## Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

24 April, 2013. Mathematical Modelling of Natural Phenomena, Vol. 8, 28-43 (16 pages)

## Abstract

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We derive the fractional Fokker-Planck equation for the density of cells and apply this equation to the anomalous chemotaxis problem. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.

Sergei Fedotov, A O Ivanov, A Y Zubarev

## Density-dependent dispersal and population aggregation patterns

19 June, 2012. J. Theor. Biology, 309, 113-120 (8 pages)

## Abstract

We have derived reaction–dispersal–aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction–diffusion–aggregation and reaction–diffusion–advection–aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset of spatial patterns. We have analyzed the qualitative properties of the emerging spatial patterns. We have compared the conditions for the possibility of spatial instabilities for reaction–dispersal and reaction–diffusion processes with aggregation and have found that dispersal process is more stabilizing than diffusion. We have obtained a general threshold value for dispersal stability and have analyzed specific examples of biological interest.

Vicenç Méndez, Daniel Campos, Ignacio Pagonabarraga, Sergei Fedotov

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

29 May, 2012. Phys. Rev. E 85, 031132 (6 pages)

## Abstract

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.

Sergei Fedotov, Steven Falconer

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

19 Dec, 2011. Phys. Rev. E 84, 061131 (8 pages)

## Abstract

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 switchingcase we estimate the average size of the near-core-outer region that corresponds to sublinear growth ⟨r(t)⟩∼tμ 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.

Sergei Fedotov, Alexander Iomin, Lev Ryashko

## Subdiffusion, chemotaxis, and anomalous aggregation

18 February, 2011. Phys. Rev. E 83, 021110 (5 pages)

## Abstract

Abstract 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.

Sergei Fedotov

## Anomalous transport and nonlinear reactions in spiny dendrites

6 October, 2010. Phys. Rev. E 82, 041103 (9 pages)

## Abstract

We present a mesoscopic description of the anomalous transport and reactions of particles in spiny dendrites. As a starting point we use two-state Markovian model with the transition probabilities depending on residence time variable. The main assumption is that the longer a particle survives inside spine, the smaller becomes the transition probability from spine to dendrite. We extend a linear model presented in Fedotov et al. [Phys. Rev. Lett. 101, 218102 (2008)] and derive the nonlinear Master equations for the average densities of particles inside spines and parent dendrite by eliminating residence time variable. We show that the flux of particles between spines and parent dendrite is not local in time and space. In particular the average flux of particles from a population of spines through spines necks into parent dendrite depends on chemical reactions in spines. This memory effect means that one cannot separate the exchange flux of particles and the chemical reactions inside spines. This phenomenon does not exist in the Markovian case. The flux of particles from dendrite to spines is found to depend on the transport process inside dendrite. We show that if the particles inside a dendrite have constant velocity, the mean particle’s position ⟨x(t)⟩ increases as tμ with μ<1 (anomalous advection). We derive a fractional advection-diffusion equation for the total density of particles.

Sergei Fedotov, Hamed Al-Shamsi, Alexey Ivanov, Andrey Zubarev

## Stochastic model for population migration and the growth of human, settlements during the Neolithic

12 August, 2008. Phys. Rev. E 78, 026107 (7 pages)

## Abstract

We present a stochastic two-population model that describes the migration and growth of semisedentary foragers and sedentary farmers along a river valley during the Neolithic transition. The main idea is that random migration and transition from a sedentary to a foraging way of life, and backwards, is strongly coupled with the local crop production and associated degradation of land. We derive a nonlinear integral equation for the population density coupled with the equations for the density of soil nutrients and crop production. Our model provides a description of the formation of human settlements along the river valley. The numerical results show that the individual farmers have a tendency for aggregation and clustering. We show that the large-scale pattern is a transient phenomenon which eventually disappears due to land degradation.

Sergei Fedotov, David Moss, Daniel Campos

## Anomalous reaction transport processes: The dynamics beyond the law of mass action

20 June, 2008. Phys. Rev. E 77, 061130 (5 pages)

## Abstract

In this paper we reconsider the mass action law (MAL) for the anomalous reversible reaction A⇄B with diffusion. We provide a mesoscopic description of this reaction when the transitions between two states A and B are governed by anomalous (heavy-tailed) waiting-time distributions. We derive the set of mesoscopic integro-differential equations for the mean densities of reacting and diffusing particles in both states. We show that the effective reaction rate memory kernels in these equations and the uniform asymptotic states depend on transport characteristics such as jumping rates. This is in contradiction with the classical picture of MAL. We find that transport can even induce an extinction of the particles such that the density of particles A or B tends asymptotically to zero. We verify analytical results by Monte Carlo simulations and show that the mesoscopic densities exhibit a transient growth before decay.

Daniel Campos, Sergei Fedotov, Vicenç Méndez

## Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion

12 March, 2008. Phys. Rev. E 77, 031911 (10 pages)

## Abstract

The proliferation and migration dichotomy of the tumor cell invasion is examined within a two-component continuous time random walk (CTRW) model. The balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived. The transport of tumor cells is formulated in terms of the CTRW with an arbitrary waiting time distribution law, while proliferation is modeled by a logistic growth. The overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined.

Sergei Fedotov, Alexander Iomin

## The effects of distributed life cycles on the dynamics of viral infections

21 September, 2008. J. Theor. Biology, 254, 430– 438 (10 pages)

## Abstract

We explore the role of cellular life cycles for viruses and host cells in an infection process. For this purpose, we derive a generalized version of the basic model of virus dynamics (Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74–79) from a mesoscopic description. In its final form the model can be written as a set of Volterra integrodifferential equations. We consider the role of distributed lifespans and a intracellular (eclipse) phase. These processes are implemented by means of probability distribution functions. The basic reproductive ratio R0 of the infection is properly defined in terms of such distributions by using an analysis of the equilibrium states and their stability. It is concluded that the introduction of distributed delays can strongly modify both the value of R0 and the predictions for the virus loads, so the effects on the infection dynamics are of major importance. We also show how the model presented here can be applied to some simple situations where direct comparison with experiments is possible. Specifically, phage–bacteria interactions are analyzed. The dynamics of the eclipse phase for phages is characterized analytically, which allows us to compare the performance of three different fittings proposed before for the one-step growth curve.

Daniel Campos, Sergei Fedotov, Vicenç Méndez