Anomalous diffusion in run-and-tumble motion

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by  Felix Thiel

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 L´evy-
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 flag-

ella 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
playedwithsolidlinesandalternateswithL´evywalkstretches
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 L´evy-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 L´evy- 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 L´evy-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 L´evy 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 propa-
gator 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 L´evy 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 L´evy 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
L´evy 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 L´evy 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 corre-
sponding 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 L´evy-counterpart. Such power-law-like 2004).
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26, 263 (1988).
occurs in several other (particularly biological) contexts
[3] A. W. Visser and U.H. Thygesen, J. Plankton Res. 25,
[10].
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We therefore expect a transition from normal to su-
[4] D.V.NicolauJr.,J.P.Armitage,andP.K.Maini,Com-
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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. Mattha¨us, M. Jagodiˇc, and J. Dobnikar, Biophys. J.
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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
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