Anomalous diffusion in run and tumble motion Felix Thiel, Lutz Schimansky-Geier, and Igor M. Sokolov Institute of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany (Dated: January 26, 2017) A random walk scheme, consisting of alternating phases of regular Brownian motion and Lévy- walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and short-time behavior of the mean squared dis- placementofthewalkerasdependingonthepropertiesofdwellingtimedistributionineachphase. Dependingonthesedistributions,normaldiffusion,superdiffusionandballisticspreadingmayarise. PACSnumbers: 05.40-a,05.45-a 7 1 0 We propose a qualitative model for the movement of 2 motilebacterialikeE.Coli. Theseorganismshaveseveral n flagella, by which they are propelled. The motion of E. a Coli can be subdivided in two different phases: tumble- J and run-mode. In the tumbling-mode the flagella rotate 5 in different directions, causing an erratic movement and 2 a fast change of orientation. In the run-mode all flagella rotate in the same direction and entangle, resulting ] h in a straight forward movement with a constant veloc- p ity. The bacteria use the tumbling phase for chemotaxis, - o swimminginthedirectionofnutrientsorawayfrompoi- i sonousareas. Thewholeprocessisdescribedforinstance b in [1] and its numerical and theoretical description is a . s subject of recent discussion, see e.g. [2–4]. FIG. 1. Sketch of the trajectory. Tumbling motion is dis- c playedwithsolidlinesandalternateswithLévywalkstretches i Random walks are the model of choice in description s (dashedlines). Afteracertaintimetheprocessisterminated, y of motion of living organisms [1, 2, 4, 5]. Run and tum- here in the run phase. h ble motion is often described either by a Lévy-walk or a p continuous time random walk (CTRW), also called “ve- [ locity” and “jumping process” [2]. CTRWs describe a of simplicity, we assume that the observation starts at a 1 random process consisting in instantaneous jumps sepa- beginningofarunorofatumblephase(inordertoavoid v ratedbyperiodsofrest(waitingtimes)followingagiven thediscussionofagingeffects). Althoughthealternating 0 probability distribution as first discussed in [6]. A Lévy- setupisnotanewprocessinthetheoryofrandomwalks, 6 3 walk is a motion at a constant speed where the direction andthegeneraldiscussionfollowsthestandardlines(see 7 ofvelocityischangedafterrandomepochs[7]. Itisessen- [9]),themodeldiscussedbelowisnewandfarfrombeing 0 tially a CTRW in which the waiting period corresponds trivial. 1. to a motion with constant velocity and jumps lead to a The displacements x in a tumbling phase are given by 0 change of the direction of motion. Although Lévy-walks the transition probability density of Brownian motion 7 are commonly used to describe the (random) motion of 1 bacteria, the origin of this kind of motion is a topic of 1 (cid:18) x2 (cid:19) v: recent studies (see e.g. [8]). λ1(x|t)= (4πDt)d2 exp −4Dt . (1) Xi Inthispaperwecombinebothprocessesandmodelthe It depends on the time t the walker has spent in this run-and-tumble motion as an alternating random walk r phase. Respectively, if t is the elapsed time in the run a consisting of phases of simple diffusion (tumble phase) phase, the transition probability density for a walk over and of Lévy walk (run phase). A sketch of the overall distance x reads: processisshowninFigure1. Thespeedofmotioninthe run phase will be denoted by c and the diffusion coeffi- 1 λ (x|t)= δ(|x|−ct) . (2) cientinthetumblingphasebyD. Thedwellingtimetin 2 S (ct)d−1 d−1 each phase is given by waiting time probability density functions (PDFs) ψ (t) and ψ (t) for the tumbling and Here S is the (hyper)surface of a d-dimensional unit 1 2 d−1 run phases, respectively. All other properties in the run ballwhichisgivenby: S =(2πd/2)/Γ(d/2),withΓ(x) d−1 andtumblingphaseswillbedenotedbysubscripts1and beingtheGammafunction. Thecorrespondingprobabil- 2 in what follows. The whole motion takes place in the itydensitiesareisotropicaslongasnochemotacticeffects space of d dimensions, with d = 2 or 3. For the sake are included. 2 We discuss the properties of the propagator yielding for walkers starting with a tumbling phase G(x ,t |x ,t ) of the complete process starting at 1 1 0 0 Λ (k,s)+σ (k,s)Λ (k,s) t=0 and stopped (observed) at time t, and concentrate G (k,s)= 1 1 2 . on the mean square displacement (MSD) 1 1−σ1(k,s)σ2(k,s) (cid:90) Since the walker could have started in both phases, we (cid:104)x2(t)(cid:105)= x2G(x,t|0,0)dx. (3) average over the two possibilities. The averaged propagator reads Typically, the MSD follows a power law pattern, (cid:104)x2(t)(cid:105) ∝ tγ; the value γ = 1 corresponds to normal G(k,s)= Λ1+Λ2+σ1Λ2+σ2Λ1 , (6) diffusion and cases with γ > 1 to superdiffusion. The 2(1−σ1σ2) special limit γ = 2 is referred to as ballistic spreading. wherewehaveomittedtheargumentsofthecorrespond- Subdiffusion, with 0 < γ < 1, is impossible in our set- ing functions. Due to its symmetry, this expression is ting. Since our situation is homogeneous both in space easiertohandleandwillbeusedforthefurtheranalysis. and in time, it is easier to use the Fourier representa- The MSD in the Laplace domain is given by tion for the coordinate and the Laplace representation for time in which G is given by an algebraic expression. (cid:104)x2(s)(cid:105)=−∆ G(k,s)| (7) k k=0 Typically, the process is stopped not at a time it changes the phase. Hence, we need two additional tran- where∆ denotestheLaplacianintheFourierspace. To k sition probability densities, which are relate this MSD to the PDFs of the waiting times and to the transition probabilities, we perform explicitly the (cid:90) ∞ Λ (x,t)=λ (x|t) ψ (t(cid:48))dt(cid:48). (4) Fourier-Laplace transforms of σ and Λ . It gives in i i i 1,2 1,2 t case of tumbling: Theydescribethedisplacementinaprematurelyfinished (cid:90) ∞ (cid:90) phase(i=1,2). Theintegralinthisexpressiongivesthe σ (k,s)= dte−st dxeikxλ (x|t)ψ (t) 1 1 1 probability that the dwelling time in phase i exceeds t. 0 (cid:90) ∞ Its Laplace transform is (1−ψi(s))/s. Thus, the Λi are = dte−st−Dk2ψ (t)=ψ (s+Dk2) 1 1 the complete analogons to the regular CTRWs waiting 0 timePDFsafterthelastjump. Eq. (4)takesintoaccount where ψ (...) is the Laplace representation of ψ (t). The the fact that the walker continuously moves during the 1 1 same procedure applied to Λ (x,t) gives waiting period. 1 Letusassumethatwalksstartwiththebeginningofa 1−ψ (s+Dk2) tumble phase. To find the propagator, we note that the Λ1(k,s)= s1+Dk2 . (8) corresponding process may have completed some num- ber m ≥ 0 of full tumble and run cycles before the Let us now turn to the Lévy walk phase. It reads observation at time t took place, and that the time is σ (k,s)= (9) stopped either during the tumble or during the following 2 run phase of the last, incomplete, cycle. A propagator = (cid:90) de(cid:90) ∞drrd−1(cid:90) ∞dtδ(r−ct)ψ2(t)e−st+irke. corresponding to a full cycle is simply a convolution of S (ct)d−1 Ωd−1 0 0 d−1 σ (x,t) = λ (x|t)ψ (t) and σ (x,t) = λ (x|t)ψ (t). In 1 1 1 2 2 2 a Fourier-Laplace domain σ (x,t) transforms to σ (k,s). Performing integration in r we obtain i i TheFourier-Laplacerepresentationofthepropagatorfor 1 (cid:90) m completed cycles is given by [σ1(k,s)σ2(k,s)]m. The σ2(k,s)= S deψ2(s−icke) (10) remaining last cycle can be stopped either in the tumble d−1 Ωd−1 phase or in the run phase. In the first case, the joint Here Ω denotes the hypersurface of a d-dimensional d−1 PDF of displacement and time in this last phase will be unitball,andeisanunitvectordefiningthepointonthis given by Λ1(k,s), or, in the Fourier-Laplace represen- surface. Integration in e corresponds in d = 2,3 to the tation, by Λ1(k,s). If the incomplete cycle finishes in angle-integration or over the solid angle. Analogously, the run phase, this PDF is given by the convolution of σ1(x,t)andΛ2(x,t),correspondingtoσ1(k,s)Λ2(k,s)in Λ (k,s)= 1 (cid:90) de1−ψ2(s−icke). (11) the Fourier-Laplace representation. Summing up all the 2 S s−icke d−1 Ωd−1 possibilities, we get Evaluating the Laplacian of the propagator one ob- G (k,s)= (5) tains a lengthy expression involving σ , Λ and their 1 1,2 1,2 ∞ first and second partial derivatives taken at k = 0. All (cid:88) = [σ (k,s)σ (k,s)]m[Λ (k,s)+σ (k,s)Λ (k,s)], items are expressed through ψ and their first and sec- 1 2 1 1 2 1,2 m=0 ond derivatives with respect to their argument. For the 3 running phase, they have additionally to be integrated with α,β < 1. In this case Eq.(12) gives in the leading over the surface of the unit ball. All first derivatives of order σ and Λ vanish due to spacial symmetry. Terms co1n,2taining 1a,2second derivative of ψ1,2 enter expressions (cid:10)x2(s)(cid:11)≈ 2A2c2(1−β)sβ−2 . (18) A sα+1+A sβ+1 for σ and for Λ with opposite signs and cancel. The 1 2 1,2 1,2 compact final result consists of two parts corresponding Remarkably, this asymptotic expression does not explic- to the running and the tumbling phase. It reads: itlydependonthediffusioncoefficientD ofthetumbling phase. Transformed back to the time domain, the diffu- dD(1+ψ ) 1−ψ (cid:104)x2(s)(cid:105)= 2 1 (12) sive scaling depends on the relation between α and β: s(1−ψ ψ ) s 1 2 c2(1+ψ ) (cid:18)ψ(cid:48) 1−ψ (cid:19) t2 forβ <α + 1 2 + 2 ,  s(1−ψ1ψ2) s s2 (cid:104)x2(t)(cid:105)(cid:39)c2(1−β) A1A+2A2t2 forβ =α  2 A2t2−(β−α) forβ >α where the derivative ψ(cid:48) is taken with respect to the Γ(3+α−β)A1 (19) Laplace-variable s. Hence, we obtain superdiffusion in the long-time limit, Exponential Waiting Time PDFs. We now pro- which is always ballistic for β ≤ α. It is independent ceed by making specific assumptions about the waiting on the properties of the tumbling phase for β < α, and time PDFs ψ . We first assume that all waiting time i depends on a prefactors A and A if α = β. The first PDFs take exponential forms so that ψ (s)=1/(sτ +1) 1 2 i i two regimes are dominated by Lévy walks. The third with τ being mean dwelling times in the corresponding i superdiffusive one with β > α is close to a sequence of phases. By plugging these expressions into (12), we can Lévy walks interrupted by rests. The tumbling periods get an expression for the second moment in the spectral hardly contribute to the displacement, and, therefore, a domain dependence on D is still absent. (cid:104)x2(s)(cid:105)= (13) As one can see, the asymptotic expressions (19) fail for β ≥ 1. In this case, subleading orders in Eqs.(17) dDτ (sτ +2) c2τ2(sτ +2) = 1 2 + 2 1 , must be taken into account. With β > 1, the Laplace s2(sτ1τ2+τ1+τ2) s2(sτ2+1)(sτ1τ2+τ1+τ2) transform of the waiting time density in the run phase reads andfromthat,theasymptoticscalinginthetimedomain ψ =1−A s+B sβ +···. (20) (cid:104)x2(t)(cid:105)(cid:39)dDt for t→0, (14) 2 2 2 (cid:104)x2(t)(cid:105)(cid:39)2dDτ1+c2τ22t for t→∞. (15) A2 stands for the finite mean waiting time in this phase. τ +τ Thecaseβ =1leadstologarithmiccorrections,whichdo 1 2 not change the qualitative behaviour. We now consider In order to obtain this result, we used Tauberian theo- thecase1<β ≤2whenthefirstmomentofthedwelling rems, which state that the small s limit corresponds to a time in the run phase does exist, but the second one large t limit in original domain and vice versa. In both diverges except for the limiting value β = 2. Again we limits t → 0 and t → ∞, one observes normal diffusion, let 0 < α ≤ 1 for the tumble phase. Repeating the albeit with different diffusion coefficients. procedure, we get Power Law PDFs. We will now consider waiting time distributions asymptotically following power laws dDA sα−1+B c2βsβ−2 for t → ∞: ψ1(t) ∝ t−(1+α) and ψ2(t) ∝ t−(1+β). Inter- (cid:104)x2(s)(cid:105)≈2 A1 sα+1+A2 s2 (21) 1 2 estingnewfeaturesariseifα≤1andβ ≤2. Iftheexpo- nentsareequalorlargerthanunity,thefirstmomentsof wherein an explicit dependence on D appears. The be- the waiting times are finite but the second ones diverge. havior in the time domain depends on relation between If the exponents are less than unity, i.e. α,β < 1, even α and β: the means do not exist. As we proceed to show, the dif- fusivebehaviorisgovernedbythefirstnon-analyticterm  Γ2((cid:16)(3β+−α1−)cβ2)BA12t2−(β−α)(cid:17) forβ−1<α<1 of ψ1(s), resp. ψ2(s). (cid:104)x2(t)(cid:105)(cid:39) 2 dD+c2(β−1)B2 t forβ−1=α<1 Let us first discuss the case when the first moments  A1 2dDt forα<β−1<1 of both dwelling time distributions diverge. The cor- (22) responding expansions of the Laplace-transforms in the If β−1<α, the behavior of the walker still corresponds limit s→0 read to superdiffusion, and the D-dependence is suppressed. ψ (s)=1−A sα+··· (16) This case again is similar to Lévy walks interrupted by 1 1 rests where the tumbling does not contribute to the dis- ψ (s)=1−A sβ +··· (17) 2 2 placement. Inthetwofollowingsituationswithβ−1≥α, 4 the walker exhibits a normal diffusive behavior. In the model may prove to be useful to model situations that lastonewithβ >1+α,theMSDisdominatedbytheran- exhibitsuchtransitions,suchastheonedescribedin[11]. dom motion in the tumbling phase. The running phase hasnoeffectonthevalueofthediffusioncoefficient. For short times t → 0 the motion is always dominated by normal diffusion. In the limit α → 1 the PDF of waiting times in the tumbling phase gets a finite first moment. The corresponding scalings are obtained from Eqs. (19) and (22). Ifβ <1, i.e. ifthewaitingtimesintherunphasedonot possess neither a first nor a second moment, the motion is ballistic. With growing β > 1 the motion tends to a superdiffusive one. The special limit is when α = 1 and β =2 corresponds to the purely diffusive case. Then ψ 1 does possess the first moment τ corresponding to A in 1 1 the expansion (17) and ψ does possess the second mo- 2 ment with value 2τ2 expressed by 2B in the Laplace FIG. 2. Phase diagram of the diffusive regimes. The figure 2 2 showswhichdiffusivebehaviorhastobeexpectedinthelong- transform (17). The MSD can be obtained from the ex- timelimit. αistheexponentcorrespondingtothetumbling- pression(21)bytakingnowintoaccountthecontribution phase, and β is the exponent of the run phase. of the second term in the denominator. This result coin- cides in the long time asymptotics with the one for the exponentially distributed waiting times Eq.(15). Wehaveseenthatbychoosingproperwaitingtimedis- It also may be suitable to describe non-biological situa- tributions of an alternating CTRW process as described tionslikethemotionofnanorods,whichmayalsoexhibit above one can describe different propagation regimes in superdiffusion [12, 13]. the long-time limit ranging from normal diffusion up to The authors acknowledge financial support by DFG ballisticspreading. Inthecaseofpower-lawwaitingtime within IRTG 1740 research and training group project. PDFs normal diffusive behavior can only be achieved if the distribution for the Brownian part has a heavier tail than the one for runs (α ≤ β −1 ≤ 1), which has to possess a first moment, i.e. the PDF of the Brownian tumbling phase has to have significantly more mass in [1] H. C. Berg, E. Coli in Motion (Springer, Heidelberg, the tail than its Lévy-counterpart. Such power-law-like 2004). behavior of a waiting time distribution is no rarity and [2] H. G. Othmer, S.R. Dunbar, and W. Alt, J. Math. Biol. 26, 263 (1988). occurs in several other (particularly biological) contexts [3] A. W. Visser and U.H. Thygesen, J. Plankton Res. 25, [10]. 1157 (2003). We therefore expect a transition from normal to su- [4] D.V.NicolauJr.,J.P.Armitage,andP.K.Maini,Com- perdiffusion in cases with α > β−1 > 0 or with β < 1. put. Biol. Chem. 33, 269 (2009). Fig. 2 shows what diffusive regime may be expected for [5] F. J. R. Meysman, V. S. Malyuga, B. P. Boudreau, and different values of α and β. J. J. Middelburg, Geochim. Cosmochim. Ac. 72, 3460 We note that if the observation does not start at time (2008). [6] E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167 of a phase change, the model with the considered power- (1965). law distributed waiting times shows aging effects [9]. It [7] M. F. Shlesinger, B. J. West, and J. Klafter, Phys. Rev. may result in changes of prefactors in the corresponding Lett. 58, 1100 (1987). asymptoticexpressionsbutwon’tleadtodifferentasymp- [8] F. Matthaüs, M. Jagodiˇc, and J. Dobnikar, Biophys. J. totics, so that the classification given in Fig. 2 still will 97, 946 (2009). hold true. [9] J.KlafterandI.M.Sokolov,Firststepsinrandomwalks, Let us summarize our findings. We proposed a phe- 1st ed. (Oxford UP, 2011). [10] J.Mejias,H.J.Kappen,andJ.J.Torres,PLoSONE5, nomenological model for bacteria performing a run-and- e13651 (2010). tumble motion in the absence of chemotaxis. We have [11] C.Metzner,C.Raupach,D.ParanhosZitterbart,andB. determinedthepossiblediffusiveregimesofsuchmotion. Fabry, Phys. Rev. E 76, 021925 (2007). Dependingonthedistributionoftimesinthetwophases [12] D.CamposandV.Méndez,J.Chem.Phys.130,134711 of motion, the model shows transitions from normal dif- (2009). fusive to superdiffusive and to ballistic behavior. Our [13] R. Golestanian, Phys. Rev. Lett. 102, 188305 (2009).

Anomalous diffusion in run-and-tumble motion

Similar Anomalous diffusion in run-and-tumble motion