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 experimental 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 corresponds 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 expects 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 localized, 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 normally diffusing particles with the same asymptotic frictional Here, κ is fractionaldiffusioncoefficientrelatedto tem- α force constant η. perature and fractional friction coefficient by the generalized 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

Is subdiffusional transport slower than normal?

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