Is subdiffusional transport slower than normal?

0.23 MB·English

by  Igor Goychuk

Is subdiffusional transport slower than normal?
∗
Igor Goychuk
Institute of Physics, University of Augsburg, Universita¨tstr. 1, D-86135 Augsburg, Germany
(Dated: January 26, 2012)
We consider anomalous non-Markovian transport of Brownian particles in viscoelastic fluid-like
media with very large but finite macroscopic viscosity under the influence of a constant force field
F. The viscoelastic properties of the medium are characterized by a power-law viscoelastic mem-
2 ory kernel which ultra slow decays in time on the time scale τ of strong viscoelastic correlations.
1 The subdiffusive transport regime emerges transiently for t < τ. However, the transport becomes
0 asymptotically normal for t≫τ. It is shown that even though transiently the mean displacement
2 andthevariancebothscalesublinearly,i.e. anomalouslyslow,intime,hδx(t)i∝Ftα,hδx2(t)i∝tα,
0 < α < 1, the mean displacement at each instant of time is nevertheless always larger than one
n
a obtained for normaltransport in apurelyviscous medium with thesamemacroscopic viscosity ob-
J tained in the Markovian approximation. This can have profound implications for the subdiffusive
transport in biological cells as the notion of “ultra-slowness” can be misleading in the context of
5
anomalous diffusion-limited transport and reaction processes occurring on nano- and mesoscales.
2

] PACSnumbers: 05.40.-a,05.10.Gg,87.16.Uv
h
c
e INTRODUCTION [16, 19].
m
One of approaches to anomalously slow diffusion and
- transport is traditionally based on the assumption of
t
a The widespread occurrence of anomalously slow diffu- divergent MRTs in trapping domains [1, 2, 4–6]. Of
t
s sionandtransport[1–9]inbiologicalcellsisstillnotcom- course, MRT hτi in any finite spatial domain with lin-
.
t monlyappreciatedinspiteofagrowingexperimentalevi- ear size ∆x can never diverge in real life. However, it
a
m denceandsupport[10–18]. Oneofthemainpsychological can largely exceed a characteristic diffusion time, τD ∼
obstaclesonthe waytoa widerrecognitionis thatultra- (L2/κα)1/α, required to subdiffusionally explore on av-
-
d slowness seems intuitively be rather obstructive for the erage, hδx2(t)i ∝ καtα with 0 < α < 1, a finite volume
n correspondingdiffusion-limitedbiochemicalreactions,es- with linear size L ≫ ∆x, where κα is the correspond-
o pecially if itis causedby divergentmeanresidencetimes ing subdiffusion coefficient measured in cm2/secα [23].
c
(MRTs) in trapping domains [1, 2, 4, 6] created by ran- Then,theapproximationofinfiniteMRTsbecomesphys-
[
dom meshworkofcell’s cytoskeleton. Fromthis perspec- ically justified on the relevant mesoscopic scale L. For

1 tive, the occurrence of subdiffusion might be more asso- very large times t ≫ hτi, the diffusion becomes normal,
v ciated with physics of dying, rather than with physics of hδx2(t)i ∝ t. However, the corresponding spatial scale
8
0 life. Moreover,thebulkofexperimentalbiophysicaldata can largely exceed L and therefore the normal diffusion
3 is traditionally interpreted in terms of normal diffusion (andtransport)regimecanbecomeofalittleimportance
5 andeven modernbiophysics textbooks (see e.g. [20, 21]) for certain processes in mesoscopic biochemical reactors
1. discuss only normal diffusion, despite appreciating the of living cells, such as e. g. passive transport of mRNA
0 fact of existence and importance of intracellular molecu- macromolecules or large globular proteins [15, 16], and
2 larcrowdingwhichclearlyobstructsdiffusionviaincrease in turnsubdiffusion becomes of profoundimportance for
1 of the effective medium’s viscosity [22]. The increase of such processes on mesoscopic scale. The approximation
:
v effective viscosity depends also on the size of diffusing of divergent MRTs features the continuous time random
i particles. So,“for molecules smaller than 1 nm, it’s sim- walk(CTRW)approachtosubdiffusion[1,2,4–6]. Inthis
X
ilar tothat of water; for particles of diameter 6 nm(such case,thepositionincrementscanbetotallyindependent.
r
a asaproteinofmass105 g/mol), it’sabout3timesthatof Within the mean-field approximation, the CTRW trans-
water. For 50-500 nm particles, it’s 30-300 times that of portiscongruent[4]withjump-like transportinrandom
water” [21](p. 571). Inthisrespect,typicalglobularpro- potentials. Moreover, the overdamped continuous space
teins are in the range of 2-20nm (diameter) [21, 22] and Markovian Langevin dynamics in spatially varying po-
mRNAmoleculesareabout400-800nmindiameter[15]. tentials can be contracted onto such a semi-Markovian
However,the traditionalthinkingandprejudgescanalso CTRWbydoingproperlyspatialcoarse-graining[24,25].
be the reasons for overlooking anomalous diffusion and Then, a potential energy disorder can result in anoma-
transport regimes (probably mostly transient) as recent lous diffusion and transport in agreementwith the semi-
experimental work uncovers [10, 14–17, 19]. Moreover, Markovian CTRW theory as recent work nicely demon-
the occurrence of subdiffusion clearly depends not only strates [26] (see also paper by Lindenberg et al. [27] in
on the size of single macromolecules, but also on their this Special Issue).
concentration, i.e. on the degree of molecular crowding Alternatively, subdiffusion can result from the
2
medium’sviscoelasticity[11,13,16,19,28–31]. Itcanbe Laplace space to
either due to viscoelasticity of the polymer actin mesh- ∞
η
work[13,16,28,29,32],orduetomacromolecularcrowd- η˜(s)= η(t)exp(−st)dt= , (2)
Z 1+(sτ)1−α
ing in complex fluids [32] as e.g. in cytoplasm of bacte- 0
rialcellswhicharelackingstaticcytoskeleton[16,22,30]. in our notations. Here, η˜(0) = η is an effective asymp-
Statisticalanalysisoftheexperimentalsingleparticledif- toticfrictioncoefficientandτ presentsalong-timemem-
fusional trajectories in bacteria [15] reveals in fact the ory cutoff. The corresponding memory kernel is approx-
primarily viscoelastic origin of subdiffusion [33]. A main imately
resultofRef. [33]isthatthefractionalBrownianmotion
η
α
scenario (see below) is more likely than one based on a η(t)≈ (3)
Γ(1−α)tα
semi-MarkovianCTRW [34]. The authors of experimen-
tal work [31] came also to a similar conclusion. Within for t ≪ τ, where η = ητα−1, and Γ(x) is the familiar
α
this alternative subdiffusional scenario all the moments gamma-function. For t ≫ τ, η(t) decays also in accor-
of random time spent in finite spatial domains remain dance with a power law, η(t) ∝ tα−2, i.e. elastic corre-
finite. The corresponding MRT is not only finite but it lations are still rather strong. However, the correspond-
scales down to zero with ∆x → 0. The physical rea- ing integral converges ensuring that the asymptotic fric-
son for subdiffusion here is very different. It occurs due tion coefficient η is finite. In the limit of infinitely large
to long-time anticorrelations in the position increments medium’s viscosity yielding η → ∞, and infinitely long
[30]. Considering Brownian particle of radius R, which memory range, τ →∞, with η =ητα−1 kept constant,
α
starts to move at t = 0 with velocity x˙(t) (we consider Eq. (3) becomes exact, η˜(s)=η sα−1, and
0 α
a one-dimensional case for simplicity), one expects it to
t η
experience a viscoelastic force Fv−el(t) = −Z Γ(1−α)α(t−t′)αx˙(t′)dt′
0
Fv−el(t)=− tη(t−t′)x˙(t′)dt′, (1) :=−ηα0D∗αx(t), (4)
Z
0
where the last equality defines fractional Caputo deriva-
tive of the order 0 < α < 1 [38] acting on x(t). η
where η(t) is a frictional memory kernel whose Laplace- α
can be named the fractional friction coefficient. Clearly,
transform η˜(s) is related to the frequency-dependent
medium’s viscosity ζ˜(iω) as η˜(s) = 6πRζ˜(s). In the Eqs. (3,4)canserveasagoodapproximationonlyforthe
times t≪τ. In the focus of this Letter is but the entire
case of purely viscous fluids, and in neglecting the hy-
time evolution,interpolating betweentransientsubdiffu-
drodynamic memory effects, ζ(t) = 2ζ δ(t), where ζ is
0 0
sion and asymptotically normal diffusion behavior. For
the fluid’s macroscopic viscosity, so that η(t) = 2η δ(t),
0
example,τ cancorrespondtothetimescaleofsecondsor
where η = 6πRζ is the Stokes viscous friction coeffi-
0 0
minutes,andthensubdiffusionemergesonthetimescale
cient. Forweaklyviscoelasticfluids,ζ(t)=ζ νexp(−νt),
0
frommicrosecondstosecondsorminutes,asinbiological
exponentiallydecaysintimewithrateν,andcorrespond-
cells [15–17, 19].
inglyη(t)=6πRζ νexp(−νt)=κexp(−νt),whereκhas
0
dimension of a linear elastic force constant. This corre-
sponds to the Maxwell theory of viscoelasticity [35] who
SIMPLE MODEL
derivedthephenomenonofviscosityfrommedium’selas-
ticity by assuming that the linear elastic force, F (t) =
el
We continue with a non-Markovian generalized
−κ[x(t)−x(0)], acting on the particle can relax in time
with rate ν, yielding a viscoelastic force, i.e. F˙v−el(t) = Langevinequation(GLE)description[39–41]foranover-
damped Brownian particle neglecting inertial effects.
−κx˙(t) − νFv−el(t). Indeed, if the force relaxation is
Then,
very fast with respect to the change of particle’s ve-
t ′ ′ ′
locity, then Fv−el(t) = − 0 η(t−t)x˙(t)dt ≈ −η0x˙(t), t ′ ′ ′
with η0 =κ/ν, whereas inRthe opposite limit, Fv−el(t)≈ Z η(t−t)x˙(t)dt =f(x,t)+ξ(t), (5)
0
−κ[x(t)−x(0)]. For strongly viscoelastic media one ex-
pects that η(t) decays in time much slower than expo- where f(x,t) is a generally nonlinear force acting on the
nential and a power law decay η(t) ∝ t−α can serve particleandξ(t)isathermalrandomforceoftheenviron-
as a better model. For a fluid-like environment the ef- ment. It is Gaussian, unbiased on average, and obeying
fective macroscopic viscosity ζ = ∞ζ(t)dt should re- the fluctuation-dissipation relation,
0 0
main,however,finite. ItcanbeveryRlarge,butyetfinite. ′ ′
hξ(t)ξ(t )i=k Tη(|t−t|), (6)
B
Therefore,along-timecutofftothepower-lawmustexist.
In 1936, A. Gemant proposed a class of power-law vis- attheenvironmentaltemperatureT. Thisisrequiredfor
coelastic models which are consistent with this demand the consistency with thermodynamics at thermal equi-
[36, 37]. Its particular representative corresponds in the librium. In the above-mentioned limit τ → ∞, η →
3
102 η ∝ ζ0 → ∞ the normally diffusing particle is get local-
ized, hδx(t)i → 0, whereas our particle still moves, but
τη/ 101 ultra slow(per definition) since hδx(t)i∝(t/τ)α. There-
F
nits of 100 nααo==r00m..75a5l fdoirseta,ntchees.“uFlutrrat-hselormw”orme,ofvoirnganpyarmtiecmleocraynkecronveelrilnartgheer
x(t), in u10-1 α=0.25 ostfutdhieedpamrtoicdleelpaonsditfioonr,ahrδbxit2r(atr)iy=cohnxs2ta(tn)ti−Fhtxh(et)vi2aroiabneyces
δ 10-2
[39]
101-30-3 10-2 10-1t/τ 100 101 102 hδx2(t)i= 2kBThδx(t)i, (8)
F
FIG.1. Meandisplacement(inunitsofFτ/η)versustime(in andthereforeitfollowstothe samepatternasinEq. (7)
unitsofτ)undertheinfluenceofconstantforceF forseveral and Fig. 1,
different values of α and the same η, τ. The limit of normal
diffusion is achieved asymptotically from above. Transiently hδx2(t)i=2κ t+2κ tα/Γ(1+α). (9)
1 α
subdiffusingparticles always coverlarger distances than nor-
mally diffusing particles with the same asymptotic frictional Here, κ is fractionaldiffusioncoefficientrelatedto tem-
α
force constant η.
perature and fractional friction coefficient by the gen-
eralized Einstein-Stokes relation, κ = k T/η , which
α B α
contains the standard one, κ = k T/η, as a particular
1 B
∞, η = const, GLE (5) is named also the fractional
α case.
Langevinequation [9] upon the use of the corresponding
Wesupposethatourobservationisrathergeneral. For
abbreviation (4) for its lhs. In this limit, ξ(t) is noth-
example, the results in Ref. [15] seem to agree with our
ing elsebut the fractionalGaussiannoise byMandelbrot
line of reasoning. Indeed, mRNA macromolecules have
andvanNess[42]whichpresentsaninstanceof1/f noise
in the related experiments radii in the range of 200-500
with the spectral power density S(ω) ∝ 1/ω1−α. Notice
nm. Furthermore,the normaldiffusion coefficient in wa-
that generally the lower integration limit in Eq. (5) is
ter was found to be κ =1µm2/sec (see Supplementary
1
t →−∞. Itcanbe replaced,however,with t =0 since
0 0 Material in [15]). From this, given the water viscosity
we assume that the particle starts to move at this time ζ =0.9·10−3Pa·sec,onecanestimatethecorresponding
w
being initially localized, i.e. v(t)=x˙(t)=0 for t<t .
0 radiusasR=k T/(6πζ κ )whichgivesR≈242nmfor
B w 1
Let us consider the transport under the influence of
T = 300 K. Let us assume that R ≈ 250 nm. Then, the
constant force F. Then, the above GLE can be easily
correspondingmacroscopicnormaldiffusioncoefficientin
solvedformally using the Laplace-transformmethod, for
cytosol should be by the factor of r ≈ 300 smaller than
any memory kernel η(t). Transforming back to the time
one in water [21] (see the above quotation in Introduc-
domain for the memory kernel (2), one obtains for the
tion). This yields κ(cyt) ≈ κ /r ≈ 3.33·10−3 µm2/sec.
averagedmean displacement the simple result, 1 1
However, the experiment yields not normal but subdif-
Ft Ftα fusion with α ≈ 0.7 (see Fig. 2(a) in [15]) and κα in
hδx(t)i= η + η Γ(1+α) the range from 10−3 to 10−2 µm2/sec0.7 [34]. Assuming
Fτ t α 1 t α κα =10−2 µm2/sec0.7 for this value of R (smaller parti-
= + . (7) cles in experiment should also subdiffuse faster) one can
η (cid:20)τ Γ(1+α)(cid:18)τ(cid:19) (cid:21)
conclude that subdiffusion can indeed cover larger dis-
ThisexactsolutioniscomparedinFig.1withthesolution tancesthannormaldiffusionwithκ(1cyt) ∼κ1/r. Further-
of the ordinary Langevinequation with memoryless fric- more, one can estimate the transition time τ. Given the
tionfortheparticlewhichexperiencesthesamefrictional relation τ = (η/ηα)1/(1−α) = (κα/κ(1cyt))1/(1−α) which
force for the whole time span as our particle asymptoti- follows within our model one obtains for it τ ≈ 55 sec.
cally, or,saiddifferently, the resultof the Markovianap- This is a rather reasonable estimate since subdiffusion
proximation to the considered dynamics. Clearly, for all regime lasts in those experiments up to 30 sec, cf. Fig.
times our particle moves in fact faster, covering larger 2(a) in [15].
distances and approaching gradually the limit of nor-
mal diffusion from above. The instant time-dependent
CONCLUSIONS
ensemble-averaged velocity hv(t)i := hδx˙(t)i = F/η +
F/[η Γ(α)t1−α]is alsoalwayslargerthan its asymptotic
α
value F/η (a spurious singularity at t = 0 can be elimi- The discussed phenomenon might seem paradoxical,
nated, if to take the initial inertial effects into account). even though its explanation is almost trivial. Neverthe-
Mathematically,thisissimplybecausexα ≫x,forx≪1 less, it has profound implications for subdiffusion in bi-
and 0 < α < 1. The physics is also clear. In the limit ological cells. First of all, the occurrence of subdiffusion
4
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ACKNOWLEDGMENTS
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Cell Biology, 2nd. ed. (Sounders Elsevier, Philadelphia,
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