aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Subdiffusion, superdiffusion and chemotaxis
Sergei Fedotov
School of Mathematics, The University of Manchester, Manchester M60 1QD, UK
We propose two nonlinear random walk models which are suitable for the analysis of both chemo-taxis and anomalous transport. We derive the balance equations for the population density for thecase when the transition rate for a random walk depends on residence time, chemotactic substanceand population density. We introduce the anomalous chemotactic sensitivity and find anomalousaggregation phenomenon. So we suggest a new explanation of the well-known effect of chemotac-tic collapse.
We develop a non-Markovian ”velocity-jump” model and obtain the superdiffusivebehavior of bacteria with power law ”run” time.
PACS numbers:
Continuous time random walks (CTRW) have beenwidely used in many fields including physics, chemistryand life sciences (see, for example, excellent reviews[1, 2]). Many biological and physical transport processesexhibit anomalous behavior for which walker mean-squared displacement increases as a fractional power µ of time: < x ( t ) > ∼ t µ (subdiffusion: µ <
1; superdif-fusion: µ >
E.coli involving the runs and tumbles. The standard as-sumption in modeling is that ”run” and ”tumble” timeintervals are exponentially distributed [5]. However ithas been found experimentally [11] that the distributionof ”run” time intervals might deviate significantly fromexponential approximation. It has a power law and leadsto superdiffusive behavior of bacteria. One of the pur-poses of this paper is to examine the effect of long-timetails on bacteria movement in terms of ”velocity-jump”model.We start with ”space-jump” random walk model in onespace dimension. The cell performs a random walk as fol-lows: it waits for a random time at each point in spacebefore making a jump to another point. The most im-portant characteristic of this walk is the transition rate γ for jumps at point x. The standard assumption in CTRW theory is that γ depends on the residence time (age) τ .This is a time interval between two successive jumps ofthe cell. The corresponding waiting time density φ ( t ) isrelated to γ ( τ ) as φ ( t ) = γ ( t ) exp (cid:16)R t γ ( τ ) dτ (cid:17) [12]. Inchemotaxis theory the jump of cells occurs in responseto a chemical signal [4]. Therefore the transition rate γ should depend on chemotactic substance (external sig-nal) S ( x, t ) and its spatial and temporal gradient. It alsodepends on macroscopic population density ρ ( x, t ). Thisdependence describes the coupling of the cells densityand chemotactic substance and crowding effects. Thus γ ( τ | x, t ) = γ ( τ | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t ) . (1)We introduce the cell density ξ ( x, t, τ ) at position x attime t with the residence time τ. The main reason forintroduction of the structured density ξ is to make a ran-dom walk Markovian. This idea has been used in [3, 12–15]. The density ξ obeys the balance equation ∂ξ∂t + ∂ξ∂τ = − γ ( τ | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t ) ξ. (2)We use the initial condition ξ ( x, , τ ) = ρ ( x ) δ ( τ ) forwhich the residence time of all cells at t = 0 equals to0; ρ ( x ) is the initial density of cells. It is clear that theresidence time τ varies from 0 to t . The condition at τ = 0 can be written as ξ ( x, t,
0) = Z R Z t γ ( τ | x, t ) ξ ( x − z, t, τ ) w ( z | x − z, t ) dτ dz. (3)Here w ( z | x, t ) is the dispersal kernel for jumps z whichalso depends on chemotactic substance and its gradient,density ρ ( x, t ) and tw ( z | x, t ) = w ( z | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t ) . (4)It is assumed that w is independent from τ. On the lefthand side of (3) we have a density of cells just arrivingat point x at time t (zero residence time). On the righthand side of (3) we have an integration of the rate atwhich the cells with different age τ arriving at position x at time t from the different points x − z. Our purposenow is to derive the Master equation for the cell density ρ ( x, t ) = Z t ξ ( x, t, τ ) dτ (5)Using the method of characteristics, we find from (2) that ξ ( x, t, τ ) = ξ ( x, t − τ, e − R tt − τ γ ( s − ( t − τ ) | x,s ) ds . (6)Let us denote the density of cells just arriving at point x at time t by j ( x, t ) = ξ ( x, t, . We substitute (6) into (3)and take into account the initial condition for ξ. We get j ( x, t ) = Z R i ( x − z, t ) w ( z | x − z, t ) dz, (7)where the i ( x, t ) is the density of cells leaving the point x exactly at time t : i ( x, t ) = Z t j ( x, u ) φ ( x, t, u ) du + ρ ( x ) φ ( x, t, , (8) φ ( x, t, u ) = − ∂ Ψ( x, t, u ) ∂t = γ ( t − u | x, t ) e − R tu γ ( s − u | x,s ) ds (9)and Ψ( x, t, u ) is the probability that a cell is trapped atpoint x from time u to t without executing a jumpΨ( x, t, u ) = e − R tu γ ( s − u | x,s ) ds . (10)This is an extension of standard survival function fora nonlinear and nonhomogeneous case when Ψ dependson chemotactic substance S ( x, t ) and population density ρ ( x, t ). The balance equation for ρ ( x, t ) can be found bysubstitution of (6) into (5) ρ ( x, t ) = Z t j ( x, u ) Ψ( x, t, u ) du + ρ ( x ) Ψ( x, t, . (11)The system of balance equations (7), (8) and (11) is anonlinear generalization of classical CTRW renewal equa-tions [1, 15] and CTRW models for inhomogeneous andnonlinear media [16, 17]. These equations can serveas a starting point for the analysis of both chemotaxisand anomalous transport for ”space-jump” random walkmodel. If we differentiate ρ ( x, t ) in (11) with respect totime, we obtain the nonlinear Master equation ∂ρ∂t = Z R i ( x − z, t ) w ( z | x − z, t ) dz − i ( x, t ) . (12)Now we are in a position to analyze the chemotaxis andanomalous effects in more detail. First we consider thecase when a cell performs a random walk in a stationaryenvironment with the distribution of chemotactic sub-stance S ( x ). In this case γ ( τ | x, t ) = γ ( τ | S ( x )). The sur-vival probability Ψ in (10) must be a function of τ = t − u and can be written asΨ( τ | S ( x )) = e − R τ γ ( u | S ( x )) du . (13) Using the Laplace transform in (7), (8) and (11), we ob-tain i ( x, t ) = Z t K x ( t − τ ) ρ ( x, τ ) dτ, (14)where K x ( t ) is the memory kernel defined by its Laplacetransform ˆ K x ( s ) = ˆ φ ( s | S ( x ))ˆΨ ( s | S ( x )) , (15)where s is the Laplace variable. Substitution of (14) into(12) gives a generalized Master equation ∂ρ/∂t = L ρ with the operator L : L ρ = Z t Z R K x − z ( t − τ ) ρ ( x − z, τ ) w ( z | x − z, t ) dzdτ − Z t K x ( t − τ ) ρ ( x, τ ) dτ. (16)The case when the dispersal kernel w ( z | x, t ) depends onchemotactic substance S has been considered by Lang-lands and Henry [8]. It has been pointed out by Er-ban and Othmer that movement of bacteria in favorableenvironment is determined by chemokinesis rather thanchemotaxis. In most cases the bacteria or cell ”does notfeel” a macroscopic gradient of S [6]. That is why it ismore important to study the dependence of transitionprobability γ on chemotactic substance S . To illustratethe general theory we use only a symmetrical dispersalkernel w ( z ) as a function of z .In a Markovian case, when γ does not depend on theresidence time variable τ , we have Ψ( x, t ) = e − γ ( S ( x )) t and ˆ K x ( s ) = γ ( S ( x )) . Under the diffusion approxima-tion, the Master equation (16) takes the form ∂ρ∂t = σ ∂ ∂x ( γ ( S ( x )) ρ ( x, t )) , (17)where σ = R R z w ( z ) dz . It is well known [4] that thisequation can be rewritten as ∂ρ/∂t + ∂J/∂x = 0 with theflux of cells J = χ ∂S∂x ρ − σ γ ( S ( x ))2 ∂ρ∂x , χ ( S ( x )) = − σ ∂γ ∂S , (18)where χ is the chemotactic sensitivity. When the deriva-tive ∂γ /∂S is negative, the advection (taxis) is in thedirection of increase in chemotactic substance. In gen-eral it follows from (16) that cells flux is not local intime J = − σ ∂S∂x Z t ∂K x ( t − τ ) ∂S ρ ( x, τ ) dτ − σ Z t K x ( t − τ ) ∂ρ ( x, τ ) ∂x dτ. (19)Instead of χ we have a chemotaxis memory kernel ∂K x ( t ) /∂S . Note that the memory kernel for the chemo-taxis flux is different form the memory kernel for diffusionterm (compare to [8]).Let us consider the anomalous case when the waitingtime PDF is heavy-tailed, such that the correspondingmean time is infinite. We assume that the longer cellsurvives at point x , the smaller the transition probabil-ity from x becomes. The rate γ ( τ ) is a monotonicallydecreasing function of residence time τ. For example, if γ ( τ ) = µ ( S ( x )) / ( β + τ ), it follows from (13) that thesurvival function has a power-law dependenceΨ( τ | S ( x )) = (cid:18) ββ + τ (cid:19) µ ( S ( x )) , where β is constant. Anomalous case correspondsto µ ( S ( x )) < K x ( s | S ( x )) = s − µ ( S ( x )) τ − µ ( S ( x ))0 ; τ is a parameter with units of time.The anomalous cell flux is J = − σ ∂S∂x ∂∂S τ µ ( S ( x ))0 D − µ ( S ( x )) t ρ ( x, t ) − σ τ µ ( S ( x ))0 D − µ ( S ( x )) t ∂ρ ( x, t ) ∂x , (20)where the Riemann-Liouville fractional derivative D − µ ( S ( x )) t is defined [1, 15] as D − µ ( S ( x )) t ρ ( x, t ) = 1Γ( µ ( S ( x ))) ∂∂t Z t ρ ( x, u ) du ( t − u ) − µ ( S ( x )) . (21)It should be noted that the fractional time derivative ofvariable order µ ( x ) has been considered in [16]. When µ = const , we have a classical subdiffusion transportequation for which the mean squared displacement of cellincreases with time as t µ with µ < . Let us consider the aggregation phenomenon [4]. In aMarkovian case, in a finite domain with zero flux of cellson the boundary, there exists a stationary non-uniformsolution of (17) [4]. The aggregation of cells is due to thefact that mean waiting time γ − ( S ( x ) is decreasing func-tion of the chemotactic substance S . In an anomalouscase, the system is not ergodic and there is no stationarydistribution. However, one can introduce the anomalouschemotactic sensitivity as a derivative of anomalous ex-ponent: χ µ = µ ′ ( S ( x )). When χ µ < , the cells will tendto aggregate where the exponent µ is small. The anoma-lous flux (20) leads to ρ ( x, t ) → δ ( x − x min ) as t → ∞ . Here x min is the point where the anomalous exponent µ ( S ( x )) has a minimum. It means that all cells aggregateinto a tiny region of space forming high density system atthe point x = x min . This phenomenon can be referred toas anomalous aggregation . Similar results have been ob-tained in [18] for a simple two-state system. This effectis known in a literature as chemotactic collapse [4]. Herewe suggest an explanation of this effect which is differentfrom the classical one based on Keller-Segel equations.To prevent the occurrence of delta-distribution we needto take into account the crowding effect. In what fol-lows, we consider this effect by assuming that the tran-sition rate depends on both the residence time τ and thepopulation density ρ . If the transition rate γ is independent of residencetime τ, then the system is Markovian. We assumethat γ depends on the time t and the density ρ ( x, t )or non-stationary chemotactic substance S ( x, t ) , thatis γ ( τ | x, t ) = γ ( ρ ( x, t ) , t ). Then we obtain i ( x, t ) = γ ( ρ ( x, t ) , t ) ρ ( x, t ) . The nonlinear evolution equation for ρ is ∂ρ/∂t = L ρ, where the operator L is defined as L ρ = Z R γ ( ρ ( x − z, t ) , t ) ρ ( x − z, t ) w ( z | x − z, t ) dz − γ ( ρ ( x, t ) , t ) ρ ( x, t ) . (22)Now let us consider the case when the transition proba-bility γ ( τ | x, t ) depends both on the residence time τ andthe density ρ as follows γ ( τ | x, t ) = γ ( τ | S ( x )) + γ ( ρ ( x, t ) , t ) . (23)From (7), (8) and (11), after lengthy calculations, weobtain i ( x, t ) = Z t K x ( t − τ ) e − R tτ γ ( ρ ( x,s ) ,s ) ds ρ ( x, τ ) dτ + γ ( ρ ( x, t ) , t ) . (24)It turns out that the nonlocal term in (24) involves theexponential factor with γ ( ρ ( x, t ) , t ). Although γ and γ are separable (see (23)), the corresponding terms in(24) are not separable. This is a non-Markovian memoryeffect. The generalized Master equation is ∂ρ/∂t = Lρ, where Lρ = Z t Z R K x − z ( t − τ ) ρ ( x − z, τ ) (25) × e − R tτ γ ( ρ ( x − z,s ) ,s ) ds × w ( z | x − z, t ) dzdτ − Z t K x ( t − τ ) ρ ( x, τ ) e − R tτ γ ( ρ ( x,s ) ,s ) ds dτ + L ρ. It follows from here that Lρ = L ρ + L ρ despite thefact that γ = γ + γ . Similar phenomenon related tochemical reactions has been discussed in [14, 15, 19]. Theexponential factor with γ in (25) prevents an anomalousaggregation effect in a long-time limit.Let us consider now 1-D non-Markovian ”velocity-jump” model for bacteria movement. The purpose is toget the superdiffusive behavior [11]. The bacteria movesto the right with the velocity v + and reverses the direc-tion with the rate γ + . When the bacteria moves to theleft with the velocity v − , the turning rate is γ − . In gen-eral, the turning rate depends on run time τ, on chemo-tactic substance S and macroscopic population density ρ ( x, t ) : γ ± ( τ | x, t ) = γ ± ( τ | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t ) . Let ξ + ( x, τ, t ) be the density of bacteria moving with veloc-ity v + with run time τ . The corresponding density oforganisms moving with the velocity v − is ξ − ( x, τ, t ). In-tegration of ξ ± ( x, t, τ ) over the run time variable τ givesthe mean densities ρ ± ( x, t ) = R t ξ ± ( x, t, τ ) dτ. The sys-tem of equations for ξ ± ( x, τ, t ) suggested by Alt [3] are ∂ξ ± ∂t ± v ± ∂ξ ± ∂x + ∂ξ ± ∂τ = − γ ± ( τ | x, t ) ξ ± . (26)Initial conditions are ξ ± ( x, , τ ) = ρ ± ( x ) δ ( τ ), where ρ ± ( x ) are the initial densities. Boundary conditions at τ = 0 : ξ ± ( x, t,
0) = Z t γ ∓ ( τ | S ( x, t ) , ρ ( x, t ) , t ) ξ ∓ ( x, t, τ ) dτ. (27)By using method of characteristics we solve (26) andfrom (27) after lengthy manipulations we find the non-linear system of equations for ρ ± ( x, t ) and j ± ( x, t ) = ξ ± ( x, t,
0) : ρ ± ( x, t ) = Z t j ± ( x ∓ v ± ( t − u ) , u ) Ψ ± ( x, t, u ) du + ρ ± ( x ∓ v ± t ) Ψ ± ( x, t, , (28) j ± ( x, t ) = − Z t j ∓ ( x ± v ∓ ( t − u ) , u ) ˙Ψ ∓ ( x, t, u ) du − ρ ∓ ( x ± v ∓ t ) ˙Ψ ∓ ( x, t, . (29)Here we introduce the generalized survival functionΨ ± ( x, t, u ) = e − R tu γ ± ( s − u | x ∓ v ± ( t − s ) ,s ) ds (30)and its full derivative ˙Ψ ± = ∂ Ψ ± ∂t ± v ± ∂ Ψ ± ∂x = − γ ± (cid:16) τ | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t (cid:17) Ψ ± . If we differentiate ρ ± ( x, t ) with respect to time, we obtain the system ofnonlinear equations ∂ρ ± ∂t ± v ± ∂ρ ± ∂x = j ± ( x, t ) − j ∓ ( x, t ) . (31)If the switching rates γ + and γ − are in-dependent of run time τ , then j ± ( x, t ) = γ ∓ ( τ | S ( x, t ) , ˙ S ( x, t ) , ρ ( x, t ) , t ) ρ ∓ ( x, t ) . This hyper-bolic model has been studied by Hillen et al [5].Let us illustrate the general theory by considering thecase when v ± = v and the run time PDF ψ ± ( τ ) = γ ± ( τ ) exp (cid:0)R τ γ ± ( τ ) dτ (cid:1) behaves like ψ ± ( τ ) ∼ (cid:16) τ τ (cid:17) µ , µ < , τ → ∞ . (32)The mean waiting time < τ ± > = R ∞ τ ψ ± ( τ ) dτ is in-finite. The experimental evidence of a power-law dis-tribution like (32) has been reported in [11]. We as-sume that v + = v − = v and all bacteria run in apositive direction initially. First we find the Laplacetransform of h x ( t ) i : h x ( s ) i = − i (cid:16) dρ ( k,s ) dk (cid:17) k =0 , where ρ ( k, s ) = ρ + ( k, s ) + ρ − ( k, s ) is the Fourier-Laplace trans-form of the total bacteria density of particles ρ = ρ + + ρ − . In the limit s → , we find from (28) and (29) very un-usual result that h x ( s ) i ∼ vτ µ s − µ . It means that theaverage position of bacteria is not zero as it should be inMarkovian case! It fact h x ( t ) i ∼ vτ µ t − µ as t → ∞ . Thespreading is slower than a ballistic motion ( x ( t ) = vt ) andfaster than diffusion for 0 < µ < . anomalous chemo-tactic sensitivity as a derivative of anomalous exponentwith respect to chemotaxis substance. We find the effectof anomalous aggregation when all bacteria tend to aggre-gate at the point where power-law exponent has a min-imum. So we suggest a new explanation of chemotacticcollapse which is different from the classical one based onKeller-Segel equations. Motivated by experiment on runand tumble chemotaxis [11], we set up non-Markovian”velocity-jump” model and obtain the superdiffusive be-havior of bacteria with power law ”run” time. [1] R. Metzler and J. Klafter, Phys. Rep. R161 (2004).[3] W. Alt, J. Math. Biol. , 147 (1980); H. G. Othmer, S.R. Dunbar, and W. Alt, J. Math. Biol 26, 263 (1988).[4] H. G. Othmer and A. Stevens, SIAM L. Appl. Math. 571044 (1997).[5] T. Hillen, Math. Models Meth Appl. Sci. , 1 (2002); T.Hillen and A. Stevens, Nonlinear Analysis, 1, 409 (2000).[6] R. Erban and H. G. Othmer, Multiscale Model Simul.
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