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O. Kharchenko and R. Metzler, Phys. Chem. Chem. Phys., 2014, DOI: 10.1039/C4CP01234H.

Molecular motors pulling cargos in the viscoelastic cytosol: how powerstrokes beat subdiffusion

Igor Goychuk,∗a Vasyl O. Kharchenko,b and Ralf Metzlera,c

Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX

First published on the web Xth XXXXXXXXXX 200X

DOI: 10.1039/b000000x

The discovery of anomalous diffusion of larger biopolymersand submicron tracers such as endogenous granules, organelles, orvirus capsids in living cells, attributed to the viscoelastic nature of the cytoplasm, provokes the question whether this complexenvironment equally impacts the intracellular transport of submicron cargos by molecular motors such as kinesins: Does thepassive anomalous diffusion of free cargo always imply its anomalously slow transport by motors, the mean transport distancealong microtubule growing sublinearly rather than linearly in time? Here we analyze this question within the widely used two-state Brownian ratchet model of kinesin motors based on the continuous-state diffusion along microtubules driven by a flashingbinding potential, where the cargo particle is elasticallyattached to the motor. Depending on the cargo size, the loading force, theamplitude of the binding potential, the turnover frequencyof the molecular motor enzyme, and the linker stiffness we demonstratethat the motor transport may turn out either normal or anomalous, as indeed measured experimentally. We show how a highlyefficient normal transport mediated by motors may emerge despite the passive anomalous diffusion of the cargo, and studytheintricate effects of the elastic linker. Under different, well specified conditions the microtubule-based motor transport becomesanomalously slow and thus significantly less efficient.

1 Introduction

The thermally driven Brownian motion represents a ubiqui-tous transport mechanism in aqueous solution, especially forsmall particles such as metal and salt ions, aminoacids, sugarmolecules, and even for larger particles such as transcriptionfactors, enzymes, lipid granules, or RNAs (on scales largerthan their gyration radius). The mean squared displacement(MSD) covered by a Brownian particle without external bias(〈r〉= 0) scales linearly with time,〈(δ r)2〉= 〈r2〉−〈r〉2 ∝ Dt.If an external forcefext acts on the particle, the mean dis-tance in the direction of the bias also scales linearly withtime, 〈r〉 ∝ fextt/η . The diffusion coefficientD, the vis-cous friction coefficientη , and the temperatureT are then re-lated by the Einstein-Stokes relationD = kBT/η , wherekB isthe Boltzmann constant. This relation manifests the classicalfluctuation-dissipation theorem (FDT) at local thermal equi-librium1. The Stokes formulaη = 6πζa relates the viscousfriction with the viscosityζ of the medium and the radiusafor spherical particles in simple fluids. The mobilityµ = 1/ηand diffusion coefficient of a particle are thus inversely pro-portional to its size.

aInstitute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany. Fax: (049)331-977-1045; Tel: (049)331-977 5614; E-mail: igoychuk@uni-potsdam.deb Institute of Applied Physics, Natl. Acad. Sci. Ukraine, Sumy, Ukraine.c Department of Physics, Tampere University of Technology,Korkeakoulunkatu 3, 33101 Tampere, Finland.

Molecular transportinside living biological cells comprisesboth passive and active motion2. In complex molecularlycrowded polymeric fluids such as the cellular cytosol3, theabove simple linear dependence of the friction coefficient onthe particle size generally breaks down, and the effective fric-tion coefficientηeff may even be exponentially enhanced inthe liquid, with respect to the one in water by many orders ofmagnitude. This effect depends on the particle size, the corre-lation length of the polymeric fluid, and other parameters4,5.For instance, if for a calcium ion the viscosity in the cytosolof a cell is essentially the same as in water, a vesicle or mag-netosome with radiusa > 100 nm may rather experience aneffective viscosity comparable to the one of glycerol whichissome 1500× more viscous than the one of water at room tem-perature, or even honey (10,000× more viscous), as extrap-olated from recent systematic studies in6 for a more simplesystem. Even for particles of a typical size of a globular pro-tein (a ∼ 2.5 nm) the enhancement factor can be as large as750, as derived in7 from the experimental data in8.

Indeed, the passive diffusion of submicron particles in liv-ing cells slows down tremendously, if to consider it as anormal diffusion process on the macroscale, with respect tothe one in pure water. However, a more dramatic effect isthat the cytosol actually behaves as a viscoelastic polymericfluid and displays profound memory effects. Viscoelasticitycan cause anomalous diffusion in the subdiffusive range, dis-played by the sublinear scaling of the MSD,〈(δ r)2〉 ∝ Dα tα

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with 0 < α < 1, where the anomalous diffusion coefficientis Dα . Such subdiffusion was established in numerous ex-periments based on a large range of tracer particles both inthe cytosol of living cells and in controlled crowded environ-ments in vitro8–26. Although subdiffusion mechanisms otherthan viscoelasticity also appear relevant to explain experimen-tal results, such as continuous time random walk-type mo-tion19,21–23, and these different types of anomalous diffusionmay even mix in some systems19,22, on time scales relevant toour study the viscoelastic nature of the subdiffusion appears tobe well established both in vivo and in control experiments invitro16,19–23,25–27.

Viscoelastic subdiffusion is related to the famed fractionalBrownian motion28,29, as discussed in30. This connectioncan be derived31,32 from a Generalized Langevin Equation(GLE)1,33 with a power-law memory kernel and fractionalGaussian thermal noise30,34, see below. Dynamically theGLE governed motion with power-law correlations representsa well-founded approach with deep roots in statistical mechan-ics33,35,36. As shown in the above experiments, the highlycrowded cytosol37,38 is such a viscoelastic environment, andwe are thus going to use the GLE approach with power-law correlations in the present work. As a guiding exam-ple, we consider the subdiffusive motion of magnetosomeswith submicron radiusa = 300 nm and chains thereof con-sisting of up to 8 magnetosomes investigated experimentallyin Ref.17. These magnetosomes perform passive subdiffusionwith anomalous diffusion exponentα ≈ 0.4 in the intact cy-tosol. Their motion is characterized by subdiffusion coeffi-cients as small asD0.4 ∼ 10−16 m2/s0.4 = 100 nm2/s0.4 andeven smaller, depending on the number of magnetosomes inthe chain. Therefore, the corresponding subdiffusion spread-ing within one second is of the order of merely 10 nm.

Without help of active transport by molecular motors suchparticles would be practically localized on even appreciablylong time scales. Pulling by molecular motors under con-sumption of energy provided by ATP is thus crucial for thedelivery of such and similar cargos to more distant locationsin living cells39,40. Given the coupling of such cargos to theviscoelastic properties of their environment in the cytosol andthe resulting passive subdiffusion of cargos, the immediatequestion arises: how does this affect the active transport bymolecular motors? This is the focus of the present work.

The theory of molecular motors viewed as Brownianstochastic engines is well developed for Markovian dynam-ics in absence of any memory effects41–48. Generalizing thesemodels systematically to the motor motion in viscoelastic me-dia was started to be studied in Refs.34,49–53 by extendingseveral popular models of Brownian ratchets48 towards thedynamics with long-range viscoelastic memory. Such ratch-ets are indeed closely related to molecular motors41–45,48,54.Within the framework of such ratchet models one can think of

the motion of a compound, tightly coupled motor-cargo parti-cle as one moving in a periodic external force field providedby interaction with the polymeric fibers of the cytoskeleton,e.g., the microtubules in the case of kinesin motors consideredhere. A generalization of such a standard continuous diffu-sion ratchet model of molecular motors to include viscoelasticsubdiffusion has been put forward recently55. This model ex-plains a number of experimental facts, in particular, that thetransport by molecular motors in the viscoelastic cytosol maybe both normal and anomalously slow, depending on physi-cal parameters such as the motor’s turnover frequency and thecargo size.

In the present work, to become more realistic we establisha further generalization by considering the motor-mediatedtransport of large subdiffusive cargos, that are attached to themotor byelastic linkers. That is, we consider motor and cargoas two independent particles and thus relax assumption ofthe absolutely rigidly linked motor-cargo compound. Simi-lar elastic linker models were considered for the normal dif-fusion of both cargo and motor in dilute solvent conditions56.In our approach we assume that the passive diffusion of themotor is normal, without memory effects, when it is not at-tached to a cargo particle. We characterize the fact that themotor moves in the viscoelastic cytosol and not just water bya significantly reduced (by a factor of 10) diffusion coefficientwith respect to the one expected in water. This assumptionis justified by the relatively small size of the motor proteinitself. However, the coupling to the subdiffusive, compar-atively large cargo enforces the subdiffusion of the coupledmotor-cargo unit, as long as it is not coupled to a microtubule.Considering experimentally relevant elastic constants ofthemotor-cargo linker from Refs.18,57 and other realistic motorparameters we confirm the coexistence of normal and anoma-lous motor transport for certain parameter values similar to thefindings of Ref.55. However, as we discuss here, in this moregeneral setup of an elastic coupling between motor and cargo,the anomalous transport regime becomes in fact reinforced.Itis shown to emerge already for motor turnover frequencies ofthe order of 100 Hz for sufficiently large cargos of a typicalradiusa = 300 nm. However, even for such large cargo parti-cles the transport can become normal if the turnover frequencyis lowered to 10 Hz. Generally, the dependence of the effec-tive subdiffusive transport exponent on the cargo size and themotor turnover frequency can provide a decisive experimentalsupport for anomalous motor transport in living cells.

2 Theory and modeling

Diffusion in complex viscoelastic fluids such as the cytosoliscommonly described by the GLE58,59. We here consider over-damped dynamics of particles, neglecting inertial effects, asappropriate for the low Reynolds number limit for submicron

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particles in the cytosol. We use the single Cartesian coordi-natey for such a particle. It corresponds to the chemical coor-dinate of the motion along the microtubule when the particlebecomes attached to the motor moving on the microtubule.The GLE then reads

∫ t

−∞ηc(t − t ′)y(t ′)dt ′ = fext(t)+ ξc(t). (1)

Here the memory kernelηc(t) and the autocorrelation functionof unbiased, thermal colored Gaussian noiseξc(t) are relatedby the second FDT of Kubo,

〈ξc(t)ξc(t′)〉= kBTηc(|t − t ′|), (2)

named also the fluctuation-dissipation relation. It reflects thebalance at thermal equilibrium between the energy suppliedby thermal noise and the energy dissipated due to friction.The stochastic description in terms of the GLE (1) is not onlyconsistent with the laws of equilibrium statistical physics andthermodynamics, but it also allows one to treat strongly out-of-equilibrium transport driven, e.g., by a non-thermal fluctu-ating forcefext(t). The GLE (1) serves as a basis in passive mi-crorheology11,58,59, at thermal equilibrium, to derive the com-plex shear modulus of the medium in the frequency domain,G∗(ω) ∝ iω

∫ ∞0 exp(−iωt)η(t)dt, from measured particle tra-

jectories.The frequency dependence of the complex shear modu-

lus is commonly used to characterize viscoelastic materi-als8,11,12,58–60. In particular, the frequently observed power-law scalingG∗(ω) ∝ (iω)α with 0 < α < 1 corresponds tosubdiffusion of a free particle in this medium,〈(δx)2〉∝ Dα tα ,with fractional diffusion coefficientDα and power-law mem-ory kernel, η(t) ∝ ηα/tα , whereηα is the fractional fric-tion coefficient obeying the fractional Einstein relationDα =kBT/ηα

34. In the macroscopic theory of viscoelasticity, sim-ilar memory kernels were introduced by A. Gemant61 as ageneralization of the simplest Maxwell model with exponen-tial memory decay62. In practice, the power-law scaling ex-tends mostly over several time and frequency decades. Ahigh-frequency (short-memory) cutoff reflects the molecularnature of the condensed medium. A low-frequency, or long-memory cutoff guarantees that the macroscopic friction co-efficient ηeff =

∫ ∞0 η(t)dt remains finite, reflecting the finite

viscosity of any fluid on the macroscale30,34. The intermedi-ate power-law scaling gives rise to subdiffusion on a transienttime scale of up to several minutes, depending on the particlesize, and this can establish subdiffusion as a primary passivetransport mechanism for submicron particles on the mesoscaleinside biological cells.

The mathematical model of a strictly algebraically decayingmemory kernel corresponds to the fractional Gaussian noise(fGn) model of thermal noise. fGn presents a time-derivativeof the fractional Brownian motion (fBm). It was introduced

by Mandelbrot and van Ness29 who popularized the conceptmathematically introduced by Kolmogorov28. Both can becharacterized by the Hurst exponentH ′ = 1−α/2, whereαis the anomalous diffusion exponent defined earlier for theMSD. fGn is termed persistent for 1/2< H ′ < 1, i.e., it ex-hibits positive correlations. It corresponds to the sub-Ohmicmodel of thermal baths consisting of harmonic oscillators63.The corresponding GLE can be derived from a purely dynamichyper-dimensional Hamiltonian model assuming merely ini-tial canonical distribution of thermal bath oscillators ata giventemperature, like in a typical molecular dynamics setup33,63.The solution of GLE (1) is then also fBm, but anti-persistentand subdiffusive, with Hurst exponentH = α/2. This trans-formation occurs due to the friction with algebraically decay-ing memory32: high noise values correspond to a high friction.Importantly, this friction memory is at the heart of the veryphenomenon of viscoelasticity. Experimental values ofα varysignificantly and are in the rangeα = 0.2÷123. For example,for the intact cytoskeleton in Ref.17, α = 0.4. About the samevalue can be derived from experimental data in Ref.18, namelyfrom the power spectrumS(ω) of the transversal position fluc-tuations of the melanosome particles (sizea = 250 nm), thatare elastically attached to the motor proteins walking alongthe microtubules. For sufficiently large frequencies (exceed-ing the inverse relaxation time scale in a parabolic potentialwell), S(ω)∝ 1/ωb, with b= 1+α, and the experiment yieldsb= 1.41±0.02. We useα = 0.4 as an experimentally relevantnumerical value in what follows.

2.1 Earlier modeling

The simplest way to model the action of molecular motorsdriving a cargo within the GLE approach is to approximatetheir collective action in terms of a time-dependent randomforce fext(t), which itself exhibits a long range memory andis power-law correlated,〈 fext(t ′) fext(t)〉 ∝ 1/|t − t ′|γ , with0 < γ < 1. This leads to superdiffusion64, 〈δy2〉 ∝ tβ , withβ = 2α − γ > 131,64, for α > 0.5. Notice, however, thatβ inthis approach can only take the maximal value 2α for γ → 0.This corresponds to a strict 1/ f noise property of the drivingforce fext(t) generated by motors with almost non-decayingcorrelations. The origin of this limit can be easily understoodwhen we consider the transport by a time-alternating force± f0 in opposite directions. Then,〈δy(t)〉 ∝± f0tα , for a givenforce realization, and after averaging over the driving forcefluctuations〈〈δy(t)〉2〉 f0 ∝ t2α . Any decaying correlations ofthe alternating driving force render the effective diffusion ex-ponent smaller thanβmax=2α. Clearly, within such a descrip-tion superdiffusion caused by the activity of molecular motorscould only be possible forα > 0.5, at odds with experimentalresults showingβ ∼ 1.2÷1.4 for α = 0.417. Similarly, the ex-perimental results of Ref.65 cannot be explained within such a

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model, either: as shown in Fig. 3 in that reference, the passivesubdiffusion of lipid droplets withα ≈ 0.58 is changed to su-perdiffusion withβ ≈ 1.74 when they are actively transportedby molecular motors. Compared with experimental evidence,the approach in terms of a time-dependent random force istherefore too simple although it provides a number of usefulinsights.

2.2 Our model

We here consider a generalization of the viscoelastic motormodel presented in Ref.55. To start with, we assume that amotor with coordinatex along a microtubule elastically cou-ples to the cargo and exerts the forcefext = kL(x− y) on it,where the spring constant iskL. The motor itself undergoesdiffusion, most generally characterized by a friction memoryη(t), on the microtubule. The motor-microtubule binding po-tentialU(x,ζ (t)) reflects the spatial periodicityL of the mi-crotubule. The potential depends on the dynamic motor con-formationζ (t), the motor’s internal degree of freedom. Ourmodel for the coupled equations of motor and cargo coordi-natesy andx then reads

∫ t

−∞ηc(t − t ′)y(t ′)dt ′ = −kL(y− x)+ ξc(t), (3a)

∫ t

−∞η(t − t ′)x(t ′)dt ′ = kL(y− x)− ∂

∂xU(x,ζ (t))

− f0+ ξ (t), (3b)

where the memory kernelsη , ηc and the corresponding au-tocorrelation functions of the unbiased thermal colored Gaus-sian noisesξ , ξc are related by FDTs of the type (2). Fur-thermore we assume that the noise sourcesξc(t) andξ (t) areuncorrelated. The termf0 in Eq. (3b) is a constant externalloading force which can oppose the fluctuation-induced trans-port and stop it. Such a force can, for instance, be applied invivo by an optical tweezer or by using magnetic beads. Weassume further that the dynamical motor conformationsζ (t)can take two valuesζ1 andζ2. The binding potential is peri-odic,U(x+L,ζ1,2) = U(x,ζ1,2), but spatially asymmetric, asdisplayed in Fig. 1 in the lower inset. The spatial periodicityguarantees that transport is absent in the absence of confor-mational fluctuations, which are induced by ATP binding tothe molecular motor and subsequent hydrolysis. In terms ofstatistical mechanics, such transport is forbidden by the sym-metry condition of detailed balance at thermal equilibriuminthe absence of an energy source. We assume that two con-sequent conformational switches make one cycle, and thusU(x+L/2,ζ1) =U(x,ζ2). Switching between two conforma-tions is considered as symmetric two-state Markovian processwith equal transition rates 2νturn.

To be more specific, we consider the piecewise linear saw-tooth potential with amplitudeU0 and periodL. The minimum

divides the period in the ratiop : 1, and we take the particularvaluep = 3. This asymmetry defines the natural direction oftransport in direction of positivex. The maximal possible load,or stalling force for this potential at zero temperature (i.e.,without presence of thermal noise) is easy to deduce, namely,f stall0 (T = 0) = (p+1)U0/(pL). In units of thermal energy at

room temperature,kBTr = 4.1 ·10−21J = 4.1 pN×nm, L = 8nm, we find f stall

0 (T = 0) ≈ 0.6833U0/(kBTr) pN for p = 3.For T = Tr the value forf stall

0 will be lower, see below. WechooseU = 20 kBTr in our simulations. This corresponds tof stall0 (T = 0) = 13.67 pN, about twice the maximal loading

force of kinesin II at physiological temperatures. This choiceleads to a reasonable stalling force at room temperature.

In this work, we neglect memory effects for the motor par-ticle due to its relatively small size and choose theδ -formη(t) = 2ηmδ (t) with regular Stokes frictionηm = 6πamζw,wheream is the effective radius of the motor molecule andζw = 1 mPa·s. We takeam = 100 nm, about 10 times largerthan the linear geometrical size of the kinesin molecule inorder to account for the enhanced effective viscosity expe-rienced by the motor in the cytosol compared to its valuein water. Furthermore, we use the characteristic time scaleτm = L2ηm/U∗

0 to scale time in the numerical simulations,whereU∗

0 = 10kBTr. For the above parameters,τm ≈ 2.94µs.Distance is scaled in units ofL, and elastic coupling con-stants in units ofU∗

0/L2 ≈ 0.64 pN/nm. Moreover, we sin-gle out the purely viscous component in the memory ker-nel of cargo as follows:ηc(t) = 2ηcδ (t) + ηmem(t), whereηmem(t) = ηα/(Γ(1−α)tα).

It should be stressed that even for this particular choice ofthe memory kernels the studied model cannot be compareddirectly with the model in Ref.55. Indeed, if we wanted toexclude the dynamics of the cargo variable by projecting thewhole dynamics onto the subspace of the motor particle alone,such a reduced dynamics of the motor particle would be de-scribed by the Laplace transformed, reduced memory kernelηred(s) = ηm + kL/[s+ kL/(ηc + ηmem(s))], whereηmem(s) =ηα sα−1 is the Laplace transform of the kernelηmem(t) above.This can be shown for any binding potentialU(x,ζ (t)) byformally solving Eq. (3a) for the Laplace-transformed vari-abley(t) using the Laplace-transform method and substitutingthe result into the Laplace-transformed Eq. (3b). Hence, themodel of Ref.55 can only be reproduced in the formal limitof an infinitely rigid spring constant withkL → ∞, upon iden-tifying ηm + ηc with η0 in Ref.55, andηmem(t) with ηm(t)therein, as well as by adopting the same model of binding po-tentialU(x,ζ (t)). However, we are interested in realisticallysoft linkers and possible effects they can introduce, especiallywhen a fluctuating binding potential acts on the motor. In-deed, when a normally diffusing particle becomes coupled toan anomalously slow one, who will win and enforce its typeof behavior, the normal one, or the anomalous one? Or maybe

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they will find a compromise? The answer to this question isnot trivial. In the absence of the binding potential, one in-tuitively expects that the anomalously slow diffusing particlewill asymptotically enslave the normal particle. However,itis not clear what happens for a fluctuating binding potential.This circle of questions has not been studied before, neitherin the theory of anomalous diffusion and transport, nor in thefield of molecular motors. To elucidate these points is one ofthe primary goals in this work. Moreover, in this paper westudy a different model of the binding potential, and considera smaller value ofα = 0.4 in numerical simulations.

When the cargo particle is not coupled to the motor (kL →0), its free diffusion is described by the MSD53

〈δy2(t)〉= 2DctE1−α ,2(

−[t/τin]1−α) , (4)

whereEa,b(z) = ∑∞n=0 zn/Γ(an+ b) is the generalized Mittag-

Leffler function,Dc = kBT/ηc represents the normal diffusioncoefficient, andτin = (ηc/ηα)

1/(1−α) is a time constant sepa-rating initially normal diffusion,〈δy2(t)〉 ≈ 2Dct at t ≪ τin,and subdiffusion,〈δy2(t)〉 ≈ 2Dαtα/Γ(1+ α) for t ≫ τin.Furthermore, in accordance with the methodology in30,34 weapproximate the memory kernel by a sum ofN weighted ex-ponentials

ηmem(t)≈ ηmem(t,ν0,b,N) =N

∑i=1

ki exp(−νit), (5)

with the underlying fractal scalingνi = ν0/bi−1 andki ∝ ναi .

This form ensures a power-law scaling,ηmem(ht,ν0,b,N) =h−αηmem(t,hν0,b,N), for t values well within the time-interval between two short and long time cutoffsτmin = ν−1

0and τmax = τminbN−1. The dilation parameterb > 1 con-trols the accuracy of the approximation, which exhibits smalllogarithmic oscillations with respect to the exact power law,andν0 corresponds to a high-frequency cutoff reflecting theatomic/molecular nature of any physical condensed environ-ment. In this respect, any physical fractal, either spatialor intime, has minimal and maximal ranges, defined in our case bythe ratioτmax/τmin = bN−1, and even a rough decade scalingwith b = 10 allows one to approximate the power-law with anaccuracy of several percents forα = 0.4, further increased toa one hundredth of a percent already forb = 2, see in Ref.52.Similar expansions are well known in the theory of anomalousrelaxation66.

This Markovian embedding approach provides a founda-tion for excellent numerical treatment of the GLE dynamicswith power-law memory kernel. A convenient parameteriza-tion uses

ki = ν0ηeffb1−α −1

b(i−1)α [bN(1−α)−1], (6)

where∫ ∞

0 ηmem(t)dt = ηeff characterizes a largely enhancedmacroscopic friction coefficient in the long time limitt ≫

τmax. The numberN of auxiliary modes controls the max-imal range of the subdiffusive dynamics, which eventuallyturns normal,〈δy2(t)〉 ∼ 2Defft for t ≫ τmax with Deff =kBT/(ηeff +ηc). Note that the fractional friction coefficientis related to the effective friction byηα = ηeffτα−1

max /r. Here,

r = Cα (b)Γ(1−α)

b1−α

b1−α−1[1− b−N(1−α)] is a numerical coefficient of

order of unity,r ≈ 0.93 for N ≥ 5 at α = 0.4 andb = 10,with C0.4(b)≈ 1.04. The effective relative friction coefficientηeff = ηeff/ηc, which is also effective viscosity of cytosolζeff

relative to water since we assume that the friction coefficientis proportional to the viscosity, is used as an important param-eter in our simulations, see below. It defines the time range

of subdiffusion, fromτin = τmax/η1/(1−α)eff to τmax. For exam-

ple, for ηeff = 103 andα = 0.4 one expects that subdiffusionwill extend over 5 decades in time. According to this impor-tant, nontrivial but simple relation, the relative increase of theeffective friction controls the relative time range of subdiffu-

sion,η1/(1−α)eff , independently ofb andN.

The above approximation allows one to replace the non-Markovian GLE dynamics with a higher dimensional Marko-vian dynamics by introduction of theN auxiliary Brownianquasi-particles with coordinatesyi accounting for internal vis-coelastic degrees of freedom34. In the present case,

ηmx = f (x,ζ (t))− kL(x− y)+√

2ηmkBTξm(t),

ηcy = kL(x− y)−N

∑i=1

ki(y− yi)+√

2ηckBTξ0(t),

ηiyi = ki(y− yi)+√

2ηikBTξi(t), (7)

where f (x,ζ (t)) = −∂U(x,ζ (t))/∂x − f0, and ηi = ki/νi.Furthermore, theξi(t) are uncorrelated white Gaussian noiseterms of unit intensity,〈ξi(t ′)ξ j(t)〉 = δi jδ (t − t ′), which arealso uncorrelated with the white Gaussian noise sourcesξ0(t)and ξm(t). To have a complete equivalence with the statedGLE model in Eqs. (3a) to (5), the initial positionsyi(0) aresampled from a Gaussian distribution centered aroundy(0),〈yi(0)〉= y(0) with variances〈[yi(0)− y(0)]2〉= kBT/ki. Thestochastic variableζ (t) is described by a symmetric two-stateMarkovian process with identical transition ratesν = 2νturn.This corresponds to the simplest model of molecular motorsconsidered as flashing ratchets2,41,45. Numerical solutions ofthe set of equations (7) are done with a time stepδ t ≪ ν−1.The variableζ (t) alternates its state with probabilityνδ t ateach time step, or continues to stay in the same state withprobability 1− νδ t. This is decided upon comparison of arandom number, uniformly distributed between zero and one,with νδ t.

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2.3 Thermodynamic efficiency and energetic efficiency ofthe cargo delivery

The thermodynamic efficiency of anomalous Brownian mo-tors has been addressed quite recently52,53. It turns out thatthe general approach developed for normal motors42,67 canbe almost directly applied. It yields, however, several impor-tant new results in the anomalous transport regime. Gener-ically, the work performed by the potential fluctuations in-duced by the enzyme turnover, or the input energy pumpedinto directed motion isEin(t) =

∫ t0

∂∂ t U(x, t)dt 67. This can

be evaluated within the considered model, as a sum of po-tential energy jumps∆U(x(ti)) occurring at random instantsof time ti marking the cyclic conformational transitions 1→2 → 1 → .... This input energy is invested in work againstthe external loading forcef0. The corresponding useful workis Wuse(t) = f0δx(t). The rest is used to overcome the dis-sipative influence of the environment, and thus converted toheat. The thermodynamic efficiency of isothermal motors isthus Rth(t) = 〈Wuse(t)〉/〈Ein(t)〉, after averaging over manyensemble realizations. Clearly,Rth = 0 for f0 = 0. This is alsovery clear from an energetic point of view: by relocation fromone place to another neither the potential energy of the cargo,nor that of the motor has been changed. Such behavior cor-responds to the normalmodus operandi of motors as kinesinwhich is very different from other molecular machines suchas ion pumps, which are primarily transferring ions againstanelectrochemical potential gradient, i.e., effecting an increaseof the electrochemical potential of the ions. Anomalouslyslow transport introduces principally new features forf0 > 0.Namely, for anomalous transport the useful work done againstf0 scales sublinearly with time,〈Wuse(t)〉 ∝ f0tαeff, while theinput energy scales linearly,〈Ein(t)〉 ∝ t, the latter being pro-portional to the mean number of potential fluctuations.

By the same token as fractional transport cannot be charac-terized by a constant mean velocity, it also cannot be char-acterized by any mean power or useful work in a station-ary regime. One way around this is to define a fractionalpower and fractional efficiency52,53. Or, one interprets phys-ically the non-constant thermodynamic efficiencyRth. In thepresent case it decays algebraically in time,Rth(t) ∝ 1/t1−αeff

and can still be comparatively high even for large times, asshown below. In the anomalous transport regime, asymptoti-cally Rth(t)→ 0 independently off0. Even though the usefulwork performed againstf0 is always finite, a benchmark ofany genuine Brownian motor, it becomes a negligible portionof the input energy in the course of time. The input energy isspent mostly to overcome the dissipative influence of the en-vironment, that effects a hugely enhanced effective viscosity.It is dissipated as heat,Q(t) = 〈Ein(t)〉− 〈Wuse(t)〉.

The primary task of molecular motors such as kinesin con-sists in the delivery of various cargos to certain destinations,

and not in the increase of their own potential energy. Forthis reason, numerous Stokes efficiencies have been definedin addition toRth

68–70. Which of them is most appropriateremains somewhat vague, especially with respect to anoma-lous transport53. We proposed a different measure to quantifythe motor performance atRth = 0, namely, the energetic de-livery performance55. It reflects the optimization of the meandelivery velocity per energy spent. Quantifying the net in-put energy by the number of enzyme turnovers, a natural def-inition is D = d/(t〈Nturn〉), whered is the delivery distancereached during timet, after 〈Nturn〉 cyclic turnovers on aver-age. Clearly,〈Nturn〉 = νturnt, and for an ideal motor in thetight coupling power stroke regime, whose processive motionis perfectly synchronized with the turnovers of the ‘catalyticwheel’71, d = Lνturnt, and therefore Dideal = L2νturn/d. Forany givend, the increase ofνturn leads to a linearly increaseddelivery performance. However, in reality there will alwaysexist deviations from this idealization.

3 Results and Discussion

In our simulations we useN = 10, b = 10, andν0 = 100for the Markovian embedding, yieldingτmax ≈ 29.4 s forτm = 2.94 µs. The numerical solution of the stochastic dif-ferential equations (7) was achieved via the stochastic Heunmethod72 in CUDA on NVIDIA Kepler graphical processorunits. The time step of the integration wasδ t = 5 · 10−3 inscaled units, and the terminal time was 106 (2.94 s) in mostsimulations. We considered a cargo with radiusa = 300 nm(ηc/ηm = 3) and two different values ofηeff = 3 · 104, the‘larger’ effective viscosity, andηeff = 3 · 103, the ‘smaller’one. This corresponds to two different values of the subdif-

fusion coefficientD(1)α ≈ 171 nm2/s0.4, similar to typical val-

ues measured for magnetosomes in Ref.17 and the ten times

largerD(2)α ≈ 1710 nm2/s0.4, corresponding to a smaller par-

ticle. Moreover, two different values for the elastic spring

constant were used,k(1)L = 0.32 pN/nm (‘strong’), which cor-responds to measurements in vitro57, and a ten times softer

springk(2)L = 0.032 pN/nm (‘weak’), in accordance with re-

cent results in Ref.18 in living cells. Finally, ν(1)turn = 85 Hz

andν(2)turn = 17 Hz denote the ‘fast’ and ‘slow’ turnover fre-

quencies. According to these parameters we consider the sixdifferent parameter sets listed in Table 1.

In Fig. 1a we illustrate both the active and passive motionof the cargo and motor particles for single trajectories. Con-sider first the case that the motor particle is detached from themicrotubule. If motor and cargo are unconnected, the mo-tor particle according to our model assumption would diffusenormally at all times,〈(δ r)2〉 ∝ D1t, while the larger cargoparticle would exhibit anomalous diffusion,〈(δ r)2〉 ∝ Dα tα ,

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1time [s]

0

100

200

300

400

500

600

700

800

900

[nm

]

Lνturntcount. processmotorcargo

10-7

10-6

10-5

10-4

10-3

10-2

10-1

time [s]

10-1

100

101

102

motorcargo

8 160

20

δx2 (

t)__

_

x

[nm]x

U [k

Tr]

B

δ

t0.4

a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1time [s]

0

100

200

300

400

500

600

700

800

900

[nm

]

Lνturntcount. processmotorcargo

0.50 0.52 0.54300

310

320

330

340

350

8 160

20

x

[nm]x

U [k

Tr]

B

δ

b)

Fig. 1 Positions of the motor (black line) and cargo (blue line)versus time for asingle trajectory realization in the case ofanomalous transport (parta, setS1, see Table 1 for the parameterscorresponding to various setsSi) and normal transport (partb, setS2. Turnover frequencyνturn = 85 Hz. The realizations of countingprocess (number of potential switches multiplied with potentialhalf-periodL/2) are depicted by red lines (difficult to detect in partb because of a perfect synchronization). The broken black linesdepict the dependence of the averaged (over many trajectoryrealizations) position of motor on time in the case of a perfectsynchronization (ideal power stroke like mechanism). Upper inset inthe parta, shows diffusion of coupled motor and cargo in theabsence of binding potential. The position variances have beenobtained using a corresponding single-trajectory time averaging, asdescribed in the text. The upper inset in the partb magnifies a partof motor and cargo trajectories making the step-wise motionofmotor obvious. It is perfectly synchronized with the potentialswitches. Cargo randomly fluctuates around the motor position. Thelower insets show two conformations of binding potential.

Table 1Parameter sets

Set ηeff kL,pN/nm

νturn,Hz

S1 3 ·104 0.320 85S2 3 ·103 0.320 85S3 3 ·103 0.032 85S4 3 ·104 0.032 85S5 3 ·104 0.320 17S6 3 ·104 0.032 17

over a large intermediate time range. Initially, both particlesdiffuse normally as the viscoelastic friction is not relevant atsuch early times. Interestingly, a similar feature, i.e., initiallynormal diffusion without any persistence or anti-persistencein the motion was found experimentally for lipid droplets inRef.65. When both motor and cargo particles are harmoni-cally coupled, the motor particle initially diffuses normally buteventually becomes enslaved by the cargo particle, and bothsubdiffuse. This is shown in the top inset of Fig. 1a, where weplot the single trajectory MSD in terms of the sliding average

δx2(t) =1

T − t

∫

T −t

0δx2(t|t ′)dt ′ (8)

over the particle position varianceδx2(t|t ′) = [x(t + t ′) −x(t ′)]2. In the limit t ≪ T this quantity provides the sameinformation as the ensemble average〈δx2(t)〉, since the con-sidered viscoelastic diffusion is ergodic30,73,74. We note that,on average, no mean displacementδx(t) of the motor-cargocomplex occurs. This is in accordance with our intuition: thesubdiffusing particle eventually enslaves the normal one,inthe absence of external forces acting on the latter.

However, once the motor is attached to the microtubulartrack, it processively steps in the direction defined by theasymmetry of the ratchet potential, and now the cargo hasto move along with the motor. This is seen in the main partof Fig. 1a. Initially, the motor steps are perfectly coupledto the potential fluctuations, where it should be noted that aparticular realization of the counting process in units ofL/2in this figure is faster than the average value corresponding

to the optimal transport distanced = Lν(1)turnt, cf. the broken

line. Indeed, different realizations of single enzyme dynam-ics yield different realizations of the potential fluctuations andthe above optimal transport distance is the ensemble averageover many trajectories. The particular realization can be bothfaster and slower, with a typical fluctuation in the number ofsteps of the order of

√νturnt. After some transient time one

can clearly see the motor backstepping. Moreover, the cargoparticle always fluctuates around the motor position, but on

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average it lags somewhat behind the motor. It is not obvi-ous from this figure that the transport is anomalously slow andnot just corresponds to some suboptimal mean motor velocityv< vopt= Lνturn. However, the transport is indeed anomalous,〈δx(t)〉∝ tαeff , after an initially normal transport regime as canbe deduced after averaging over 1000 different trajectory re-alizations, as shown in Fig. 2. This allows us to extract theanomalous transport exponentαeff. As already verified in ourprevious studies of viscoelastic subdiffusive dynamics inpe-riodic nonlinear potentials, the value of thisαeff is generallytime-dependent. However, it relaxes to a long-time limitingvalue in the course of time, which is displayed in Fig. 3 asfunction of the loadf0 for different sets of parameters.

It is very important for physical reasons that the cargo lagsbehind the motor, following at a certain mean distance withfluctuations around it. Therefore, it makes sense to considertwo effective exponentsαeff for the motor and cargo sepa-rately. For a stronger linker, both exponents are very simi-lar because the average distance between the motor and thecargo is relatively small, see the inset in Fig. 2a. Even in thestrongly anomalous case S1, the average distance saturates ataroundδx = 17 nm, which largely exceeds the correspond-ing amplitude of the thermal fluctuations, given by the rootmean square (rms)δxT =

√

kBTr/kL ∼ 3.58 nm. The elas-tic spring energy corresponding to the above average distance

is EL = k(1)L (δx)2/2∼ 46.24pN·nm∼ 11kBTr. It can be eas-ily sustained by a covalent bond. However, for a weaker linkerthe scenario becomes highly nontrivial and in fact stronglyde-pends on the character of the transport regime. In this case,thedifference between two exponentsαeff for motor and cargo canbe essential, see in Fig. 3 for larger cargo. For a smaller cargoin the normal transport case S3, the distance is about 29 nmat the end of the simulation, see the inset in Fig. 2a, and thespring energy is about 3.3 kBTr only. However, in the stronglyanomalous regime S4, the mean distance increases to 170 nmand the spring energy rises accordingly until about 111kBTr.This is already a typical energy of a covalent bond. Therefore,such a linker, considered as globular entropic spring, willbedefinitely unfolded to reach its maximal length, which can in-deed be around 150 nm when it is fully stretched and is suf-ficiently long75. However, the most probable scenario is thatsuch a linker will simply break or, alternatively, the cargowilldissociate from the linker and will be lost by the motor. Thisis the first important result of this work. Namely, to trans-fer a large cargo in a strongly anomalous transport regime thelinker must be sufficiently strong. With these caveats in mind,we consider the case S4 in more detail, as it also reveals an-other remarkable feature, which is expected to be present forstronger linkers, as well, albeit to a lesser degree. Namely,in the case S4 the motor moves faster than in the case of astronger linker for S1, see Fig. 2a. This is a remarkable obser-vation, which was not expecteda priori. Given this feature,

0 1 2 3time [s]

0

500

1000

1500

2000

<δx

>[n

m]

S1 motorS1 cargoS4 motorS4 cargoS2 motorS2 cargoS3 motorS3 cargo

0 1 2 30

50

100

150

200

S4

S1

S3a)

mot

or-c

argo

0 1 2 3time [s]

0

50

100

150

200

250

300

350

400

<δx

>[n

m]

S5 motorS5 cargoS6 motorS6 cargofit ~ t1.081

0 1 2 30

1020304050

S5S6b)

mot

or-c

argo

Fig. 2 Ensemble-averaged trajectories (103 different realizations) ofmotor and cargo particles for different sets of parameters at twovalues of turnover frequencies, 85 Hz (a), and 17 Hz (b). Opposingforce is absent,f0 = 0, in all cases. Insets show the differencesbetween the motor and cargo positions for the chosen sets. Transportof smaller particles is normal. (a) Within anomalous transportregime of larger particles, the motor-cargo distance increases andsaturates at about 17 nm for S1 and about 170 nm for S4. Thesevalues largely exceed the corresponding thermal rms distances of3.58 nm and 11.32 nm, respectively. This means that weaker linkercan be unfolded during the transport of larger particle, or even breakdown. (b) For a smaller turnover frequency, anomalous transportbecomes normal both for motor and cargo in the case of stronglinker, S5. Surprisingly, for a weak linker, S6, the motor operatesnormally, while cargo enters a paradoxical super-transport regimewith exponentαeff ≈ 1.081, even if moves slower than motor.

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the motor can be expected to perform a larger work against anexternal loadf0 for a weaker linker. Moreover, on decreas-ing the turnover frequency down to 17 Hz (see the case S6 inFig. 2b) the motor operates normally, with the same mean ve-locity as in caseS5 corresponding to the optimal motor regimeat this frequency. However, the cargo paradoxically entersasuper-transport regime with a scaling exponentαeff ≈ 1.081,see also Fig. 3, while movingslower than the motor. Thismeans that the transport exponentβ = 2αeff can, in principle,become larger than two, i.e. the observed motor-mediated dif-fusion of the cargo can be transiently super-ballistic. This isour second nontrivial and surprising result. The explanation ofthis seemingly paradoxical behavior is quite simple. Namely,it occurs because the distance between the motor and cargoincreases sublinearly in time, before entering the saturationregime. To realize this, let us assume first that this mutual dis-tance grows linearly in time. Then, the cargo distance wouldalso grow linearly with an effective velocity equal to the mo-tor velocity minus the velocity of the mutual distance growth.However, if the mutual distance increases only sublinearly, i.e.with a velocity which drops in time, see inset in Fig. 2b, it ap-pears that the cargo moves faster than in the normal transportregime characterized by the effective velocity. Paradoxically,an ever increasing delay of the cargo motion past the opti-mally operating motor due to the weakness of the motor-cargolinker can be interpreted as a super-transport of cargo. How-ever, upon a saturation of the motor-cargo distance this tran-sient regime will be over. This example shows that it can bevery misleading to associate the value of anomalous transportexponentαeff with the speed of transport.

ForS1 and f0 = 0 shown in Fig. 1a,αeff ≈ 0.63, both for themotor and cargo. The same value also characterizes the motorin the caseS4. However, for the cargo the transport exponenthere is larger,αeff ≈ 0.70, even if it moves absolutely slower,an interesting feature by itself. This once again indicatesthatthere is no simple general relation between the value of thetransport exponent and the transport speed, contrary to com-mon belief. The observed features explain the origin of thesuperdiffusive exponent in the rangeβ = 1.2− 1.4, in spiteof the low value of the free subdiffusion exponentα = 0.4in17. However, for a smaller particle (or rather for a smallerηα in our model) corresponding to the caseS2, the transportbecomes normal at the same flashing frequency and withoutbiasing back load,f0 = 0, as demonstrated in Fig. 1b. In-terestingly, the corresponding single trajectory realization inFig. 1b does not correspond to a larger transport distance att = 1 s as compared with the anomalous transport in Fig. 1a.This is simply because the number of turnovers performed un-til this time is larger in Fig. 1a than in Fig. 1b, for the partic-ular realizations presented. This reflects statistical variationsof the corresponding counting process, or, in physical terms,the stochastic nature of single motor proteins. The upper in-

0 1 2 3 4 5 6

0.4

0.6

0.8

1.0

S1

S2

S3

S4 motorS4 cargoS5

S6 motorS6 cargo

α

f0

eff

[pN]

Fig. 3 Effective transport exponentsαeff of the motor and cargo asfunctions of loading forcef0 for various sets of parameters, asdeduced from the ensemble-average at the end point of simulations(2.94 sec). These two values ofαeff differ significantly for weaklinker and large cargo, the casesS4 andS6 shown in this plot. Forother cases, the differences are much less pronounced. For thisreason, the onlyαeff for the motor is displayed for other sets. Noticethatαeff of the cargo can be larger than one for a normally operatingmotor in the case of a weak linker,S6.

set in Fig. 1b reveals a perfect synchronization between thepotential switches and the motor steps. It renders the red linecorresponding to the potential switches barely distinguishablein this plot. One can also see in this upper inset that the cargoparticle fluctuates symmetrically around the motor. It doesnotlag behind the motor on average, in contrast to Fig. 1a. Fur-thermore, within a transport regime closer to normal transporta much softer linker does not significantly change the results,as demonstrated by Fig. 3 for a smaller cargo. For a largercargo, though, anomalous transport becomes clearly affectedin a very nontrivial way, as discussed above.

Notably, the motor can operate normally if the turnover fre-quency is reduced, see the results forS5 and S6 at f0 = 0in Fig. 3. However, the transport of cargo can become bothnormal (strong linker,S5), or even, seemingly paradoxical,anomalously fast, if to judge from the transport exponentαeff

for a weak linker,S6. This super-transport is in fact abso-lutely slower than the normal one, see Fig. 2b and the relateddiscussion above. These results show that motors can real-ize both normal and anomalous transport in the viscoelasticcytosol of living cells, where large submicron cargos subdif-fuse on the time scale from milliseconds to tens of seconds.Which transport regime will be realized depends specificallyon the particle size (or fractional friction coefficientηα ), theenzyme turnover frequencyνturn, and on the linker rigidity.This presents one of the important results of our work.

Furthermore, if a counter-forcef0 > 0 is applied, the

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101

102

103

100

101

102

S , d = 400 nmS , d = 800 nmS , d = 800 nmS , d = 800 nmS , d = 8 m

ν [Hz]

4

D [n

m/s

]

1

1

2

2 µ

turn

Fig. 4 Cargo transport delivery efficiency D as function of turnoverfrequencyνturn for different sets of parameters and different deliverydistancesd. Broken lines correspond to ideal power-strokedependence Dideal= L2νturn/d, to compare with.

anomalous transport regime becomes promoted, as shown inFig. 3. This effect is primarily due to the reduction of the bind-ing potential amplitudeU0. Anomalous transport will alsoemerge immediately for the studied parameters ifU0 is de-creased, for instance, to the lower value 10kBTr.

The transport efficiency of cargo delivery in the viscoelas-tic cytosol can be of the same order as that for the motion ina dilute solution, despite the passive subdiffusion. This effi-ciency is demonstrated in Fig. 4 for realistic turnover frequen-cies (lower than 200 Hz). It persists even for large distances ofsome 8µm. The discussed power stroke mechanism can thusperfectly overcome the subdiffusive resistance of the medium,if the cargo size is not too large. In the anomalous transportregime with strongly decreased transport efficiency there stillexists an optimal turnover frequency, which depends on thedelivery distance. Indeed, in the anomalous transport regimewith increasing turnover frequency or load even normally op-erating motors do not have sufficient time to relax down to thepotential minimum after each potential jump, or escape eventsbecome important. This results in backsteps, and after reach-ing a maximum versusνturn the delivery efficiency will nec-essarily drop. In the anomalous transport regime, the optimalνopt

turn(d) necessarily decays with increasing delivery distanced. Energetically, it makes then less sense to hydrolyze moreATP molecules for efficient cargo delivery. A correspondingoptimization can be important for the cell economy. As ex-pected, anomalous transport on a stronger linker is more effi-cient for realistic turnover frequencies, a fact that agrees withintuition. This is because the cargo follows at a larger distancefrom the motor in case of a weaker linker.

The thermodynamic efficiency becomes strongly affectedby the fact that an increase off0 promotes the onset of the

1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

S1

S2

S3

S4

S5

S6

R

f0

th

[pN]

Fig. 5 Motor thermodynamic efficiency as function of loading forcef0 at the end of simulations (corresponding totmax= 2.94 s) fordifferent sets.

anomalous transport regime, as observed in Fig. 3. Even if thetransport was normal forf0 = 0 it may become anomalous un-der a sufficiently strong external loadf0. Hence, whenever thetransport is anomalous, the thermodynamic efficiency startsto depend on time and vanishes asymptotically. However, itcan be quite large even at long times such astmax ∼ 3 s, seeFig. 5, where it still amounts to some 23% at the maximum,see the results for casesS2 andS3, where the thermodynamicefficiency practically does not depend on the linker rigidity.This is an important results of the present work, which isapriori counterintuitive. Namely, in a highly efficient transportregime, which is closer to normal transport, neither transportexponentαeff (Fig. 3) nor the thermodynamic efficiency arepractically affected by the rigidity of the linker. However, inthe low efficiency transport regime, the influence of the linkerrigidity becomes evident in Figs. 3 and 5. Remarkably, thethermodynamic efficiency of the motor is slightly larger foraweaker linker. This interesting feature is due to the fact that themotor covers larger distances in this case, as shown in Fig. 2.Interestingly, the stalling force does not depend onkL at allfor the studied parameters. The dependence ofRth(t, f0) onthe load f0 is very illustrative:Rth(t, f0) vanishes not only atf0 = 0, but also at somef stall

0 (T,U0,νturn). This defines thestalling force, which is time-independent but strongly dependson potential amplitude, temperature, and driving frequency.The thermodynamic efficiency has thus a maximum at someoptimal loading forcef opt

0 (t), andRth(t, f opt0 ) can be relatively

high in the anomalous transport regime, for a sufficiently highpotential amplitudeU0, comparable with the maximal thermo-dynamic efficiency of kinesins in the normal regime of about50%. The dependence ofRth(t, f0) on f0 is strongly asymmet-ric in the thermodynamically highly efficient regime. How-ever, it becomes more symmetric in a low efficient anomalous

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5 10 15 20 250

4

8

12

16 T = 0T = 0.5 Tr

T = Tr

fit

U0

f 0

stal

l[p

N]

[k Tr]Β

Fig. 6 Stalling force as function of barrier heightU0 at differenttemperatures and fixedνturn = 85 Hz. Numerical results arecompared with an analytical fit by Eq. 9 forU0 >UmT/Tr .

regime, where it is described approximately by the parabolicdependenceRth ∝ f0(1− f0/ f stall

0 )/ f stall0 with f opt

0 = f stall0 /2,

and a proportionality coefficient which slowly drops in time.This phenomenon of a time-dependence ofRth(t) can be ver-balized as fatigue of molecular motors caused by the vis-coelasticity of the cytosol. It can be used as direct means toprobe our model experimentally.

Indeed, for low-efficient anomalous transport, the depen-dence ofRth on the load in Fig. 5 is approximately parabolic,with a maximum atf opt

0 = f stall0 /2. Counterintuitively, it is

slightly larger for a softer linker. This is because the usefulwork was defined as the work performed by motors againstf0, and not by the cargo, and that the motors step ahead oftheir cargos on slightly larger distances for a softer linker. Themaximal loading, or stalling forcef stall

0 is time-independent.It also does not depend on the cargo size. However, it stronglydepends on the flashing frequency (Fig. 5), as well as on thepotential amplitude and temperature (Fig. 6). Numerical re-sults show that forU0 >Um(νturn)T/Tr,

f stall0 (T,U0,νturn)≈

43L

(

U0−Um(νturn)TTr

)

, (9)

and a fit to numerical data yieldsUm ≈ 11.2 kBTr at νturn = 85Hz. The corresponding stalling force becomesf stall

0 ≈ 6 pNfor U0 = 20 kBTr, but is smaller forνturn = 17 Hz (Fig. 5).From this we conclude that a reasonably strong motor requiresbinding potential amplitudes larger than some tenkBT . Wemay interpret the result in Eq. 9 in terms of an effective freeenergy barrierF0(T ) = U0− TS0 with an effective ‘entropy’S = Um(νturn)/Tr. Then, f stall

0 (T,U0,νturn) ≈ 4F0(T )/3L forpositiveF0.

4 Conclusions

In this work, we presented a model for active transport of sub-micron cargo particles by molecular motors in the viscoelasticcytosol of living cells. This model goes significantly beyondour previous discussion in Ref.55 by the inclusion of a realisticelastic linker between the motor and its cargo, giving rise to anumber of surprising effects. When decoupled from the mo-tor, the cargo particles subdiffuse and the motor operates nor-mally. As shown here, the effective transport of the coupledmotor-cargo complex may be both normal and anomalous, de-pending on the physical parameters of the system.

The viscoelasticity of the cellular cytosol is taken into ac-count by the formulation of the cargo dynamics in terms of aGeneralized Langevin Equation with a memory kernel, whichscales in accordance with a power law between two cutoffs.Passive subdiffusion according to our model occurs on time

scales betweenτmin = τmax/ζ 1/(1−α)rel andτmax, whereζrel is

the effective cytosol viscosity relative to the water viscosity.It is distinctly enhanced,ζrel ≫ 1, for submicron particles.Asymptotically, normal diffusion of the free particle is re-stored, but with this increased value ofζrel. For α = 0.4 andtypical valuesζrel ∼ 104 or ζrel ∼ 103, subdiffusion then per-sists over about 6.5 or 5 decades in time, respectively. Suchtransient subdiffusion is described by the multidimensionalMarkovian embedding with a well-controlled accuracy of theapproximation. The molecular motor is described by a variantof the standard Markovian model of flashing Brownian mo-tors with a periodic saw-tooth potential, that randomly fluc-tuates between two realizations differing by phase, such thattwo potential flashes correspond to one complete enzyme cy-cle, that can advance the motor by one spatial period. Thismodel describes, in particular, a perfect power stroke ratchettransport in the case of highly processive molecular motors.It explains how a perfect ratchet mechanism can win oversubdiffusional restrictions in living cells. A perfect transportregime with maximal transport efficiency emerges when themotion of the motor is locked onto the potential fluctuationseffected by the internal motor turnover. As an important newresult, we clearly showed that in order to operate efficientlywith a non-vanishingly small stalling force in the viscoelas-tic environment of living cells the amplitude of the bindingpotentialU0 should be no less than around tenkBTr, or 0.25eV. For a realistic valueU0 ∼ 20kBTr ∼ 0.5 eV, with a stallingforce about 5-6 pN within the present model, depending onthe motor operating frequency it turns out that variation ofthe spring constant of the linker in the experimentally rele-vant rangekL ∼ 0.03− 0.3 pN/nm does not change signifi-cantly the major results in the efficient transport regime forsufficiently small cargos. However, in a strongly anomalousregime emerging for large cargos the weakness of the linkercan distinctly influence the transport. In fact, our resultsshow

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that a weak linker withkL ∼ 0.03 pN/nm might simply not beable to sustain anomalous transport of sufficiently large car-gos with a = 300 nm in such a regime. It will be stronglystretched and eventually break, or the cargo will dissociatefrom it and become lost (other scenarios are also possible).However, if the motor operates slowly and enters the optimalnormal regime, such a weak linker can yet support transport ofsuch large cargos on a smaller spatial scale with cargo enter-ing a paradoxical regime of transient super-transport withαeff

larger than one, the origin of which is rationalized within ourmodel. Furthermore, the cargo delivery efficiency is generallylarger for a stronger linker, but the thermodynamic efficiencyof the motor is somewhat smaller. In our view, these conclu-sions are interesting and important, and will prompt furtherstudies in this direction.

The crucial point which our model explains is how one andthe same motor in the same cell can realize both normal andanomalous transport. The occurrence of a particular trans-port regime depends on the binding potential amplitude, frac-tional frictional strengthηα of the cargo particle (dependingon its size), loading external forcef0, and enzyme turnoverfrequency. The effective transport exponentαeff can vary fromα to one, and this can easily explain the emergence of anoma-lously fast diffusion withβ = 2αeff mediated by motors in thecells with α ≤ 0.5. Strikingly enough, the transport can beperfectly normal (in agreement with most experiments), re-flecting an almost perfect synchronization between the enzy-matic turnovers and the motor’s stepping along microtubule.This can be rationalized within a power-stroke mechanism andexplains how a power-stroke like operation can beat subdiffu-sion. However, subdiffusion can also tremendously slow downtransport by making it anomalously slow. It depends in partic-ular on the cargo size.

As a final conclusion, we reaffirmed in a more realistic set-ting the major qualitative results obtained in Ref.55 based on amore simplistic approach. Thus, the occurrence of anomalousversus normal transport depending on the cargo size and themotor turnover frequencies now appears to be well rational-ized. In particular, in order to beat subdiffusion with a powerstroke one needs an appreciably strong linker between the mo-tor and its cargo.

We hope that these new results will inspire new experi-ments, as they should be present in more complex models ofmolecular motors operating in viscoelastic cytosol. We hopethat this can be done in a nearest future.

Acknowledgments

Support of this research by the German Research Foundation,Grant GO 2052/1-2, as well as funding from the Academy ofFinland (FiDiPro scheme) are gratefully acknowledged.

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