Active and Passive Transport of Cargo in a Corrugated Channel: A Lattice Model Study
AActive and Passive Transport of Cargo in a Corrugated Channel: A LatticeModel Study
Supravat Dey, a) Kevin Ching, and Moumita Das b) School of Physics and Astronomy, Rochester Institute of Technology, Rochester, New York 14623,USA. (Dated: 10 April 2018)
Inside cells, cargos such as vesicles and organelles are transported by molecular motors to their correctlocations via active motion on cytoskeletal tracks and passive, Brownian diffusion. During the transportationof cargos, motor-cargo complexes (MCC) navigate the confining and crowded environment of the cytoskeletalnetwork and other macromolecules. Motivated by this, we study a minimal two-state model of motor-drivencargo transport in confinement and predict transport properties that can be tested in experiments. Weassume that the motion of the MCC is directly affected by the entropic barrier due to confinement if it isin the passive, unbound state, but not in the active, bound state where it moves with a constant boundvelocity. We construct a lattice model based on a Fokker Planck description of the two-state system, studyit using a kinetic Monte Carlo method and compare our numerical results with analytical expressions fora mean field limit. We find that the effect of confinement strongly depends on the bound velocity and thebinding kinetics of the MCC. Confinement effectively reduces the effective diffusivity and average velocity,except when it results in an enhanced average binding rate and thereby leads to a larger average velocitythan when unconfined.PACS numbers: 05.40.Jc, 87.16.aj, 87.16.dp, 87.16.Ka
I. INTRODUCTION
Intracellular transport of cargos by molecular motorsis critical to development, maintenance, and homeosta-sis in most eukaryotic cells . There exist several typesof motors that use ATP, the energy currency of the cell,to move cargo through the cell using cytoskeletal tracks.The motors kinesin-1 and cytoplasmic dynein transportcargo using microtubules while myosin-5 and -6 do sovia actin filaments . Some motors have direction bias;some carry larger, and some smaller cargos. Examples ofintracellular cargo include organelles such as mitochon-dria, and dysfunctional or damaged protein aggregatesthat occur in disease states . While the transportof the former is essential to proper functioning of thecell, the latter need to be cleared out of the cell to pre-vent cell damage and disease progression. Understandingthe mechanistic principles underlying intracellular cargotransport will provide insights into the proper function-ing of cells, and aid in the creation of new drugs or agentsto help regain function in disease states.Microtubules and actin filaments which provide thepathway for the motors to walk during intracellulartransport, are semiflexible biopolymers that are foundthroughout the cell interior . Over the past twodecades, there have been many studies, both experimen-tal and theoretical, on cargo transport by motors onsingle microtubules in-vitro. The speed and travel dis-tance of molecular motors on surface-immobilized mi- a) Electronic mail: [email protected] b) Electronic mail: [email protected] crotubules is very well understood ; examples includethe molecular motor kinesin pulling fluid membranes ona microtubule , multiple motors transporting a singlecargo , and motors carrying cargos to multiple targetsin neurons . Within cells, however, microtubules andactin rarely exist as individual filaments. Instead, theyare found as networks of filaments and have very in-teresting mechanical structure-function properties. De-spite decades of studies of motors moving and carryingcargo on single cytoskeletal tracks, cargo transport incomplex and dynamic architectures in cells is not wellunderstood. In fact, experimental and theoreticalstudies have only recently begun to investigate howintracellular transport is affected by the physical prop-erties of the cytoskeletal network and crowded cellu-lar environments. Notable experimental studies in thisarea include in-vitro experiments that have studied howcrowding of motors and organization of microtubuleswithin bundles affect the efficiency of cargo transport.In particular, in Conway et al. found that the motionof the cargo being transported is inhibited in a bundleof randomly oriented, closely packed microtubules. Sev-eral theoretical models with active and passive transporthave also investigated the collective transport proper-ties and the spatial organizations of motors and cargoson single microtubules and inhomogeneous cytoskele-tal networks . Despite these advances, there re-main many open questions.Here we ask: How does confinement due to the cy-toskeletal network affect motor-driven cargo transport?We address this question by developing a minimal two-state model that describes cargo transport in the pres-ence of confinement. The two states are: (i) an activestate when the motor-cargo complex (MCC) is attached a r X i v : . [ c ond - m a t . s o f t ] A p r to the microtubule and moves with a constant speed, and(ii) a passive state when it is unattached and undergoesdiffusive motion. Such two-state models have been usefulin elucidating active transport of Brownian particles inconfined geometries , specifically how the cooperativerectification between geometric constraints and Brown-ian ratchets impacts net particle motion. The interplaybetween passive and active transport and confinement,as is common in intracellular transport, however, re-mains poorly understood. In this paper, we combinea Fokker Planck description with a lattice model frame-work to study how confinement, and motor dynamicsand binding kinetics interact to modify directed trans-port of cargos by motors.The paper is organized as follows. We write down theFokker Planck Equations (FPE) for the two-state modeland propose a lattice model that can capture the physicsdescribed by the FPE and reduces to the FPE in the con-tinuum limit. We simulate the lattice model using a ki-netic Monte Carlo method, and show that it reproducesknown analytical results for passive (diffusive) transportin confinement. Thereafter we investigate the full twostate problem in confinement, and calculate transportproperties such as mean squared displacement (MSD),average velocity, and effective diffusivity for the MCC,and discuss the implications of our results.We want to note that, in this paper, we use the term“active” to refer to the driven motion of cargo fueledby ATP-hydrolysis of kinesin motors via a “Brownianratchet” mechanism , and implemented as constant ve-locity motion of an MCC on a microtubule (see Sec. II).It should not be confused with self-propelled motion insoft matter literature. The confinement effect on themotion of a self-peopelled particle studied in . II. MODEL AND METHOD
We model and study the active and passive transportof cargo by a motor moving unidirectionally on a mi-crotubule track and confined in a corrugated channelas schematically shown in Fig. 1. The motor can movemicrometer-long distances along the microtubule beforedetaching. Kinesin motors, a well-characterized family ofmotor proteins that move organelles (e.g. mitochondria)and macromolecules (e.g. RNA) in many cell types aregood examples of such motors. While the confinementfaced by an MCC in a live cell is heterogeneous and dy-namic, for simplicity, we consider an effective confiningchannel described by w ( x ) = a sin(2 πx/L ) + b , where L is the periodicity, a and b control the effective width ofthe channel, and the effective bottleneck width is givenby 2( b − a ) . The effective channel width and periodic-ity are set by the length scales associated with localizedcages and network mesh sizes.The main ingredients of the model are as follows:1. Two State Transport: Over long time scales, the off k k on w(x)−w(x) R b v FIG. 1. Schematic diagram of the model. Top: An MCC(filled circle on stick figure in green) can either bind to a mi-crotubule (straight blue line) with a rate k on and walk witha velocity v b in the + x direction or it can get detached fromthe microtubule with a rate k off and diffuse in the corrugatedchannel. The transverse position of the channel is given by ± ( R + w ( x )) (solid curved lines in red), for finite size MCCof radius R. For an equivalent point MCC, the space in thetransverse direction that is effectively available for diffusionis ± w ( x ) (dashed curved lines in red). Bottom: In the lat-tice model, an MCC either can be in the bound state (filledblue circle) or unbound state (meshed gray circle). A boundMCC only hops in the + x direction, while an unbound MCChops in both + and − x directions with appropriate rates thatdepend on the entropy barrier due to the confining wall w ( x ). MCC alternates between two states: (i) An ac-tive state which phenomenologically represents theMCC being bound to a microtubule and movingalong the microtubule in the forward direction witha speed v b , and (ii) an passive state where theMCC is detached from the microtubule and un-dergoes overdamped Brownian motion in the vis-cus medium of the cytoplasm with a free diffusionconstant D .2. Confinement: The MCC only encounters physicalconfinement due to the channel walls while under-going diffusive motion in a viscous medium, i.e.in the passive state. Our model, therefore, in-corporates an explicit, microscopic description ofthe physical confinement due to cytoskeletal net-works. This is in contrast to a more coarse-grained,continuum description where interior of the cell istreated as a viscoelastic medium . Thechannel is symmetric, and hence we do not ex-pect confinement-induced symmetry-breaking forpurely diffusive motion as in some BrownianRatchet models .3. Binding Kinetics: The binding rate, k on ( x ), of themotor can be constant or it can vary inversely withthe width of the confining channel in Fig. 1. Thelatter represents the case where tighter confine-ment leads to greater likelihood of the motor at-taching to the microtubule due to increased prox-imity to binding sites. The unbinding rate, k off ( x ),of the motor is assumed to be constant.We note that the actual width of the confining chan-nel is larger than the size of the MCC. In our formu-lation, w ( x ) is the effective channel width (depicted bythe dashed curved lines in Fig. 1), and the MCC canbe thought of as a point particle, with the actual size ofthe MCC incorporated through the free diffusivity. Withthis description, our model is valid for a wide range ofcargo sizes as long as the bound state is unaffected bythe confining channel.The model predicts three distinct regimes, a diffu-sive regime at early times, an intermediate sub-diffusiveregime, and a ballistic regime at large times, as dis-cussed in detail in the Results section. The timescalefor the crossover from diffusive to sub-diffusive motionis set by the diffusion time of the MCC in the corru-gated channel before it starts to experience the impactof confinement, while the timescale for the crossover tothe ballistic regime is set by the interplay of the bindingkinetics and the driven motion of the MCC when boundto the microtubule. A. Fokker Planck Description
In our model, the motion of the MCC is directly af-fected by the confining wall of the channel when it isundergoing Brownian motion in the unbound (passive)state. First, let us discuss the motion of an overdampedBrownian particle in a 2D confining channel, with thechannel axis along the x direction. The 2D motion in-side the channel can be described by a 1D Fokker-Planckequation, known as the Fick-Jacobs equation , ∂P ( x, t ) ∂t = D ∂∂x (cid:18) e − β A ( x ) ∂∂x e β A ( x ) P ( x, t ) (cid:19) . (1)Here, P ( x, t ) represents the probability density at a givenposition x along the direction of the channel at time t , D is the diffusion coefficient in the absence of confinement,and β = 1 /k B T , where k B is the Boltzmann constantand T is the temperature. To derive the above equation,rapid equilibration is assumed in the transverse directionof the channel. This implies that the time scale for longi-tudinal (axial) motion is very large compared to the equi-libration time scale in the transverse direction. Underthis assumption, one can successfully integrate out thetransverse variable and recast the two-dimensional mo-tion into the above Fick-Jacobs equation . The con-finement is incorporated though an effective free energy A ( x ) = V ( x ) − T S ( x ), where S ( x ) is the entropy bar-rier due to confinement and V ( x ) is an external energybarrier. The entropy due to the confining wall w ( x ) is S ( x ) = k B log(2 w ( x ) /w ave ), where w ave = 2 (cid:82) L w ( x ) dx is the average width of the channel. In the absence ofany external potential, the free energy is purely entropic, A ( x ) = − k B T log(2 w ( x ) /w ave ).Now, we return to the problem of two-state transport.The MCC walks with a velocity v b when it is bound,diffuses with a free diffusion constant D when unbound,and alternates between the two states with rates k off ( x )and k on ( x ), respectively. The Fokker-Planck equationfor the probability densities for the bound state P b ( x, t )and unbound state P ub ( x, t ) are given by: ∂P b ( x, t ) ∂t = k on ( x ) P ub ( x, t ) − k off ( x ) P b ( x, t ) − v b ∂P b ( x, t ) ∂x , (2a) ∂P ub ( x, t ) ∂t = − k on ( x ) P ub ( x, t ) + k off ( x ) P b ( x, t )+ D ∂∂x (cid:18) e − β A ( x ) ∂∂x e β A ( x ) P ub ( x, t ) (cid:19) . (2b)The first two terms in Eqs. 2a and 2b correspondto binding and unbinding transitions respectively. Thethird term in Eq. 2a represents active motion of theMCC, while the third term in Eq. 2b describes passivemotion of the MCC under confinement. Given kinesinis a highly processive motor and can take over a hun-dred steps along a microtubule before dissociating ,we neglect diffusion in the active state. However, onecan easily incorporate diffusive behavior for other motortypes by adding a diffusion term in Eq. 2a. The ana-lytical solutions of Eq. 2 are difficult, and have closedform expressions only in the passive limit and in amean field limit for two-state transport discussed laterin the paper. We, therefore, construct the correspondinglattice model which reduces to Eq. 2 in the continuumlimit and evolve the system using a kinetic Monte Carlomethod as discussed below. B. Lattice Model
We study the dynamics of a two-state MCC describedby the continuum Fokker-Plank equation (Eq. 2) usingan equivalent lattice model. The model is schematicallyshown in Fig. 1, and consists of an MCC on a one-dimensional lattice. The MCC can switch between abound and an unbound state. The bound MCC canfurther hop to its forward neighboring site while the un-bound MCC can hop to both its backward and forwardneighboring sites. The spacing between neighboring lat-tice sites is (cid:96) .Consider that the MCC is at the lattice site at position x at time t in a particular state. The transition ratesfrom the unbound state to bound state is k on ( x ) andfrom the bound state to unbound state is k off ( x ). TheMCC in the bound state can either hop to its forwardneighbor ( x + (cid:96) ) with rate λ v ( x ), or it can switch tothe unbound state with rate k off ( x ). The MCC in theunbound state can hop either to its forward neighbor ( x + (cid:96) ) with rate λ ub+ ( x ), backward neighbor ( x − (cid:96) ) with rate λ ub − ( x ), or switch to the bound state with rate k on ( x ).The master equations describing the time evolution ofthe probability densities for the bound state P b ( x, t ) andthe unbound state P ub ( x, t ) for this process are ∂P b ( x, t ) ∂t = k on ( x ) P ub ( x, t ) − k off ( x ) P b ( x, t ) + λ v ( x − (cid:96) ) P b ( x − (cid:96), t ) − λ v ( x ) P b ( x, t ) , (3a) ∂P ub ( x, t ) ∂t = − k on ( x ) P ub ( x, t ) + k off ( x ) P b ( x, t ) + λ ub+ ( x − (cid:96) ) P ub ( x − (cid:96), t ) + λ ub − ( x + (cid:96) ) P ub ( x + (cid:96), t ) − (cid:0) λ ub+ ( x ) + λ ub − ( x ) (cid:1) P ub ( x, t ) . (3b)As the bound velocity v b in our model is independentof position, the bound state hopping rate λ v ( x ) is alsoposition independent and is given by λ v ( x ) = v b /(cid:96) . Weincorporate the effect of confinement using position de-pendent hopping rates λ ub ± ( x ) for the unbound state.The hopping rates λ ub ± ( x ) depend on the free-energy A ( x ) which has a contribution from the entropic barrierdue to confinement. In the presence of an external po-tential, it also has an energy contribution. The hoppingrates are given by λ ub ± ( x ) = ( D /(cid:96) ) e − β ( A ( x ± (cid:96) ) −A ( x )) / .The factor 1 / (cid:96) → x at time t , wechoose an event out of all possible events at random witha probability proportional to its rate, and increase thetime by δt = 1 / Γ( x ), where Γ( x ) is the total rate. Forthe bound MCC, the event space consists of a forwardhopping event with probability λ v / Γ, and a transitionto the unbound state with probability k off ( x ) / Γ, wherethe total rate in the bound state Γ = k off ( x ) + λ v ( x ).For the unbound MCC, the event space consists ofa hopping event in the forward direction, a hoppingevent in the backward direction, and a transition to thebound state, with probabilities λ ub+ ( x ) / Γ, λ ub − ( x ) / Γ, and k on ( x ) / Γ respectively, where the total rate in the boundstate is Γ = k on ( x ) + λ ub+ ( x ) + λ ub − ( x ). For simplicityand efficiency, the mean of the exponential distributionΓ( x ) exp( − δt Γ( x )) is used as the time step in our sim-ulations. Although a time step drawn at random fromthe exponential distribution would have been more ap-propriate, we have checked that the choice of the meandoes not change any of our results, while it makes thesimulation more efficient. This was also verified by oneof the authors in a study of a lattice model for ballisticaggregation in .It has been shown that the introduction of a posi-tion dependent diffusivity, D ( x ) = D / (1 + w (cid:48) ( x ) ) α (with α = 1 / / w ( x ) . The qualitative behavior of the results do not change if constant diffusivity D is considered . Here,for simplicity, we study our lattice model with constantdiffusivity. However, it can be easily extended to in-corporate x-dependent diffusivity by choosing λ ub ± ( x ) =( D ( x ) /(cid:96) ) e − β ( A ( x ± (cid:96) ) −A ( x )) / . C. Simulation Details and Parameters
Throughout this study, the lengthscales associatedwith the corrugated channel are taken to be L = 1 µm , a = 1 / (2 π ) µm , and b = 1 . / (2 π ) µm . Our choiceof effective widths a and b implies fairly strong con-finement (effective bottleneck width = 2( b − a ) =0 . / (2 π ) µm ). The parameter values for the two-statemotion ( k on , k off , D , v b , and (cid:96) ) of the MCC are in-formed by experiments on kinesin motors carrying car-gos or pulling membranes . The value of thestep size or lattice spacing (cid:96) = 8 nm . The simula-tions are performed with the free diffusion constant ofthe unbound MCC D = 0 . µm s − and the off-rate k off = 0 . s − , unless otherwise specified. To exploreextended parameter space, the on-rate and bound ve-locity are varied over wide ranges, k on = 0 . − s − and v b = 0 . − . µm s − . The experimental values of k on ( ∼ . s − ) and v b ( ∼ . µm s − ) lie well within therange.For the binding (on) rate, we study two cases: (i) k on = k , and (ii) k on ( x ) ∝ k / ( w ( x )). In the lattercase, we further investigate two situations – when thespatial average of k on , in the interval L is k , to al-low for comparison with (i), and when it is greater than k . The simulations are performed with open boundarycondition, meaning that the channel can be thought ofas extending to infinity in both directions. All the datapresented in this paper are averaged over 25000 or morerealizations. III. RESULTS
We characterize the motor-driven cargo transport inour model by the mean squared displacement, (cid:104) δx ( t ) (cid:105) = (cid:104) ( x ( t + t ) − x ( t )) (cid:105) , the average velocity, (cid:104) ˙ x (cid:105) , andthe effective diffusivity, D eff of the MCC. The last twoquantities are defined in the asymptotic limit as (cid:104) ˙ x (cid:105) =lim t →∞ (cid:104) ( x ( t ) − x (0) (cid:105) t and D eff = lim t →∞ (cid:104) x ( t ) (cid:105)−(cid:104) x ( t ) (cid:105) t ,where x ( t ) is the position of the particle at time t and (cid:104)·(cid:105) represents ensemble averages . This definition of effec-tive diffusivity allows for a more accurate estimate thaninferring it from the MSD. While the MSD may or maynot grow linearly with time depending on context, thefluctuations around the mean position of the MCC inour systems grow linearly with time at large times andtherefore D eff is independent of time. A. Passive Transport in a Confining Channel
We first study the passive, diffusive transport of a par-ticle in confinement using kinetic Monte Carlo simula-tions of the lattice model. We demonstrate that the lat-tice model correctly incorporates hopping rates throughthe entropic barrier dependent free energy, and discussproperties which will be used to compare and under-stand the results of two-state transport in the next sec-tion. In this case, a particle at the lattice site x canhop to one of its neighboring sites ( x ± (cid:96) ) with rate λ ± = ( D /(cid:96) ) e − β ( A ( x ± (cid:96) ) −A ( x )) / . For Brownian mo-tion under constant force F , A ( x ) = − F x − T S ( x ),which gives λ ± ( x ) = ( D /(cid:96) ) e ± βaF/ (cid:112) w ( x ± (cid:96) ) /w ( x ).In the absence of any confinement, the rates of hoppingthen become, λ ± ( x ) = ( D /(cid:96) ) e ± β(cid:96)F/ , i.e. indepen-dent of x . Confinement makes the hopping probabili-ties x dependent, which are given by λ ± ( x ) / Γ( x ), whereΓ( x ) = λ + ( x ) + λ − ( x ).In Fig. 2(a) and (b), we present the lattice model re-sults for the scaled mobility µ eff and scaled diffusion co-efficient D eff and compare them with the correspondinganalytical predictions. The latter are obtained by solvingthe Fick-Jacobs equation (Eq. 1) for a particle undergo-ing 2D overdamped Brownian motion under an externaldriving force F in a corrugated channel w ( x ) with peri-odicity L and given by µ eff := (cid:104) ˙ x (cid:105) F = D k B T (1 − e − f ) (cid:82) L dxL I ( x, f ) f − , and (4a) D eff D = (cid:90) L dxL (cid:90) xx − L dzL e A ( x ) /k B T e A ( z ) /k B T I ( z, f ) × (cid:34)(cid:90) L I ( x, f ) dxL (cid:35) − , (4b)where I ( x, f ) := e A ( x ) /k B T (cid:82) xx − L dyL e −A ( y ) /k B T dependson the dimensionless force (or Peclet number) f := F L/k B T . Please note that the above expressions arenonlinear in f . In the absence of any geometric con-finement, the effective mobility and diffusion coefficientreduce to µ eff = D /k B T = µ and D eff = D , by substi-tuting A ( x ) = − F x in the Eq. 4. The numerical resultsare in very good agreement with the analytical predic-tions, demonstrating that our lattice model is an accu-rate representation of the Fick-Jacobs equation (Eq.1),and suggesting that this method can be used to study awide range of systems with entropic barriers. (a) µ e ff / µ f:=FL/k B T analytical (b) D e ff / D f:=FL/k B T analytical -3 -1 -2 (c) < δ x ( t ) > t CNC2D t -3 -1 -2 (d) < δ x ( t ) > t CNC2D t( β F t) FIG. 2. Lattice model results for passive transport in a chan-nel. Simulation data (solid circles) for (a) scaled mobility µ eff /µ ( µ = D /k B T ) and (b) scaled diffusion coefficient D eff /D as a function of the external driving force f showvery good agreement with the analytical predictions (solidlines) given by Eq. 4. Figure (c) shows simulation data forthe MSD with time with confinement (C) and without con-finement (NC) in the absence of any external driving force;Figure (d) shows the same, but with a driving force f = 25. We now discuss the results shown in Fig. 2 in more de-tail. The scaled mobility shown in Fig. 2(a) is always lessthan 1, approaching 1 asymptotically as f is increased.This suggests that a symmetric confinement without anyrectification mechanism cannot enhance the mobility ofa purely diffusive system. The behavior of the scaled ef-fective diffusivity D eff /D in Fig. 2(b) is non-monotonicwith a peak at a critical value of f , suggesting that whileat small f confinement causes the effective diffusivity todecrease, at large f the interplay of the force and con-finement leads to enhanced diffusivity. The value of thecritical force depends on the modulation of the confiningwall as discussed in ref .Confinement constrains diffusive motion of the particleby reducing the available space for movement; for f = 0,it leads to a significant decrease in the MSD comparedto the unconfined case at intermediate and large timesas seen in Fig. 2(c); the particle’s motion changes fromfree diffusion with D at early times to an intermedi-ate sub-diffusive regime, and finally to effective diffusivebehavior with D eff < D at large times. The intermedi-ate sub-diffusive regime presumably emerges due to theslowing down of motion near the neck of the channel.For large Peclet numbers f >>
1, (Fig. 2 (d)), wherethe particle motion is largely driven rather than diffu-sive, confinement, has much smaller impact compared to(c). The motion changes from free diffusion at very earlytimes to force-driven ballistic motion at large times, andas in (c) there is a visible slowing down in at intermedi-ate times for the confined case, but the gap between thetwo asymptotic MSDs is much smaller. As we will seein the next section, the intermediate slowing down dueto confinement plays a critical role in two-state cargotransport.
B. Active and Passive Transport in a Confining Channel
We now discuss the transport properties, namelyMSD, average velocity, and effective diffusivity of theMCC for the two state model with confinement, andcompare them with the results for the unconfined case.Where appropriate, we also compare our results withcorresponding steady state values in the mean field limitwithout any confinement. The following sections we referto this limit as the Mean Field No Confinement (MFNC)limit. In this limit, the probabilities of bound and un-bound states are given by ˜ P b = k on / ( k on + k off ) and˜ P ub = k off / ( k on + k off ), respectively, and the average ve-locity and effective diffusivity can be written as, V l = ˜ P b v b = k on v b k on + k off , (5) D l = ˜ P ub D + D act,ub = k off D k on + k off + k on k off v b ( k on + k off ) . (6)In Eq. 5, the velocity V l does not depend on the proba-bility of the unbound state since there is no net directedmovement during passive motion. In Eq. 6, the firstterm is due to the diffusive motion in the unbound stateand the second term accounts for an additional contribu-tion due to the stochastic transition between active andpassive motion. The analytical expression for the lattercan be found in . Maximizing this equation providesthe condition for the occurrence of the peaks in effec-tive diffusivity as well as their positions, as observed in(Fig. 5(a)).The above mean field analytical expressions are exactfor the unconfined case if the transition rates and thediffusivity of the motion are independent of position. Asthe confinement makes the diffusivity position depen-dent (through position dependent hopping rates), theseexpressions are no longer expected to hold true for theconfined case. However, comparing results with these expressions are very useful for understanding the role ofthe confinement. -3 -1 -2 (a) t c1,C t c2,C t c1,NC t c2,NC < δ x ( t ) > t CNC2D tV t -3 -1 -2 (b) < δ x ( t ) > t CNC2D tV t -3 -1 -2 (c) < δ x ( t ) > t CNC2D t -3 -1 -2 (d) < δ x ( t ) > t CNC2D tV t FIG. 3. MSD with time for k off = 0 . s − , and D =0 . µm s − , with confinement (C) and without confinement(NC). Figures (a) & (b) show data for a large binding rate( k on = 4 . s − ) of the motor to the microtubule. (a) For asmall bound velocity ( v b = 0 . µm s − ), the MSDs for bothC and NC grow as 2 D t initially, deviating at intermedi-ate times, and asymptotically converging to ballistic motion (cid:39) (cid:104) ˙ x (cid:105) t with (cid:104) ˙ x (cid:105) NC = V l > (cid:104) ˙ x (cid:105) C at large times. (b) For alarge bound velocity ( v b = 0 . µm s − ), the small t diffusivemotion and large t ballistic motion are the same for both Cand NC, but the intermediate behavior are different. Figures(c) & (d) show data for a small binding rate ( k on = 0 . s − ),for a small velocity ( v b = 0 . µm s − ) in (c) and a largevelocity ( v b = 0 . µm s − ) in (d). Mean Squared Displacements (MSD):
In Fig. 3,we present the MSD for small and large bound veloci-ties v b for two different binding rates k on . At very smalltime t , the MSD behaves as (cid:104) δx ( t ) (cid:105) (cid:39) D t . This be-havior suggests that below a crossover time scale, say t c , the effect of binding/unbinding kinetics and con-finement are negligible such that the MCC can diffusefreely. For t > t c , it shows a transition from free diffu-sion to “sub-diffusion”. In the absence of confinement,the sub-diffusive behavior is due to the time spent bythe MCC alternately transitioning between bound andunbound states, and the crossover time scale, t c , mainlydepends on the transition rates. In the confined case, inaddition to transition events, the motion of the MCCslows down even further due to the strong entropic bar-rier in the diffusive state close to a neck of the chan-nel. Consequently, the crossover time t c becomes evensmaller for the confined case than that without any con-finement. There is a second crossover from sub-diffusiveto ballistic behavior i.e., for t > t c , (cid:104) δx ( t ) (cid:105) (cid:39) (cid:104) ˙ x (cid:105) t .A representative case of the locations of t c and t c areshown in Fig. 3 (a). From the Fig. 3, it is clear that con-finement impacts the motion of the MCC more stronglyfor small v b – it spends longer times in the intermediatesub-diffusive regime leading to smaller asymptotic ve-locities at large times compared to the unconfined case.For larger v b , the sub-diffusion regime shrinks and theMCC has the same asymptotic velocities with and with-out confinement. The observed MSDs in Fig. 3 (c) and(d) are similar to (a) and (b) respectively, but with thecrossover to ballistic motion occurring at much largertimes ( t c ).The motion of the MCC and how its MSD willscale with time is determined by the interplay of fourtimescales: the inverse of the binding rate k on , theinverse of the unbinding rate k off , the diffusion time˜ L /D , and the drift timescale ˜ L/v b , where ˜ L is the char-acteristic length scale in the system and is equal to L inthe confined case. The precise dependence of t c and t c on these timescales is nontrivial and will be studied infuture work. Here we make the following qualitative ob-servations for our confining channel. For a given k off , thevalue of the first crossover timescale t c ,C seems to scaleas D /L and seems independent of v b and k on , whilethe second crossover timescale t c ,C decreases with in-creasing v b and k on . (a) < x . > / v b k v b =0.04v b =0.1v b =0.5v b =0.8v b =1.6 (b) < x . > / v b k v b =0.04v b =0.1v b =0.5v b =0.8v b =1.6 FIG. 4. Scaled average velocity (cid:104) ˙ x (cid:105) /v b of the MCC as afunction of the binding rate k on (a) without confinement and(b) with confinement for various values of v b . We have set k off = 0 . s − , and D = 0 . µm s − . Solid lines repre-sent MFNC limit. The data for v b = 0 .
04, and 0 . µm s − do not follow MFNC limit. The dashed line is a fit of0 . µm s − data to Eq. 5 which yields k effoff = 0 . s − and v eff b = 0 . µm s − . The dotted line is a fit of 0 . µm s − data with k effoff = 0 . s − and v eff b = 0 . µm s − . Average velocity:
In Fig. 4(a) and (b), we show theresults for the scaled average velocity for various val-ues of v b , without and with confinement, respectively.In the absence of confinement, for large separation be-tween the times spent by the MCC in the unbound andbound states (Eq. 5), the average velocity approachesconstant values– for k on (cid:29) k off , (cid:104) ˙ x (cid:105) approaches v b andmotion of the MCC is predominantly ballistic, while forfor k on (cid:28) k off , it approaches zero suggesting diffusive(or no) motion. In the intermediate regime, their is an interplay between ballistic and diffusive motion, and thescaled average velocities grow monotonically with k on .Fig. 4 (a) captures this behavior, with the data for all v b following Eq. 5, as expected.Confinement reduces the average velocity of the MCCfor small bound velocities v b , while for large v b the resultsfollow the MFNC limit (Fig. 4(b)). For small v b (0 . . µm s − ), confinement leads to a reduction in thescaled average velocity and it stays below the MFNClimit described by Eq. 5. However, the data can be fitto Eq. 5 with an effective k effoff , and an effective v eff b . Wefind that k effoff > k off and v eff b < v b suggesting that con-finement renormalizes the bound velocity and the un-binding rate to values lower than without confinement,and therefore effectively reduces the processivity of themotor. For large v b (0 . , . , and , . µm s − ), the scaledaverage velocities are unaffected by confinement. Theseresults are consistent with MSDs discussed earlier. (a) D e ff / D k v b =0.04v b =0.1v b =0.5v b =0.8v b =1.6 (b) D e ff / D k v b =0.04v b =0.1v b =0.5v b =0.8v b =1.6 FIG. 5. Scaled effective diffusivity D eff /D as a function ofthe binding rate k on (a) without confinement and (b) withconfinement for various values of v b . We have set k off =0 . s − , and D = 0 . µm s − . Dashed lines represent theMFNC limit. The data for unconfined case follow the MFNClimit, while for confined case data stay below their MFNClimit. Effective Diffusivity :
Confinement has a muchmore striking impact on the effective diffusivity of theMCC. Unlike for the average velocity, we do not find adata collapse in this case. We show the scaled diffusioncoefficient as a function of k on for the unconfined casein Fig. 5(a). For small k on , the motion of the MCC isdominated by passive diffusion and D eff is large. Whilefor large k on , as the motion is largely ballistic, D eff be-come smaller. As expected, the data for the unconfinedcase follow the analytical prediction given by Eq. 6. Thecondition for a maximum to exist is predicted by Eq.6: v b > (cid:112) k off /D (= 0 .
81 for parameters used here).We observed that the scaled diffusion coefficient showsa maximum for v b = 1 . µm s − at k on (cid:39) . s − . Inthe case of confinement (Fig. 5(b)), the effective diffu-sion coefficient becomes smaller than for the case withoutconfinement, but they are well separated with prominentpeaks, and the presence of a maximum at k on (cid:39) . s − is now seen for small v b . For v b = 0 . µm s − we ob-serve a peak here which is not possible for the uncon-fined case. The lowering of effective diffusivity suggeststhat the confinement reduces the noise in the motion bylimiting the availability of space for the motion.The observed maximum in diffusivity is reminiscentof similar behavior for a particle undergoing diffusion ina tiled washboard potential , or in a periodic confinedprofile . There the appearance of a peak in the diffusiv-ity is associated with a “locked-to-running” transition .In the locked state, the particle shows no net movementover a significant amount of time, while in the runningstate the particle has a net drift velocity. Transitions be-tween the two states can be induced via occasional largekicks due to noise. We observe peaks in the two-statemodel even without any confinement, with the passive(unbound) and active (bound) states corresponding tothe locked and running states respectively. The transi-tion between the two states are induced by the bindingkinetics of the motors to the microtubule. Confinementmakes the peaks much more pronounced, and leads togreater separation between the scaled effective diffusiv-ity curves for different v b .The qualitative behavior of the average velocity andeffective diffusivity does not depend on the free diffu-sion constant D and the effective widths of the chan-nel. We have checked this by studying systems with D = 0 . µm s − and b = 1 . / (2 π ) µm (results notpresented here). C. Active and Passive Transport in a Confining Channelwith Spatially Varying Binding Rates
Next we study cargo transport by motors with a spa-tially varying binding rate that depends on the localwidth of the confining channel. We have studied thefollowing two cases: (i) k on ( x ) = k √ b − a / ( L w ( x )),where the binding rate is normalized to ensure thatthe spatial average (cid:104) k on ( x ) (cid:105) = k , and (ii) k on ( x ) = k /w ( x ), where the binding rate is not normalized, andits spatial average (cid:104) k on ( x ) (cid:105) = k L/ √ b − a . For both(i) and (ii), the unbinding rate k off is assumed to be in-dependent of the spatial variation of the channel width.Let us first consider the case (i) with the normalizedspatially varying binding rate. We present the scaled ve-locities against k for two different values of unbindingrates k off = 0 .
42 and 0 . s − in Fig. 6(a). For eachvalue of the unbinding rate, we consider three differentbound velocities v b . For each unbinding rate, the aver-age velocity data collapse onto a single curve, suggestingthat unlike for constant binding rates the scaled veloci-ties do not depend on v b . We further find that for bothunbinding rates, the average velocities of the MCC staybelow the corresponding velocities V l in the MFNC limitgiven by Eq. 5. The decrease in the scaled velocity sug-gests that the effective unbinding rate k effoff has increased,and the collapse to a single curve suggests that k effoff is in-dependent of bound velocity v b (see Fig. 6(c)), unlike thecase with constant binding rate. (a) < x . > / v b k v b =0.04v b =0.1v b =0.8 (b) < x . > / v b k v b =0.04v b =0.8 (c) < x . > / v b γ k v b =0.04v b =0.8 FIG. 6. Scaled average velocity as a function of k for motorswith spatially varying binding rate k on ( x ) = k /w ( x ). (a)Shows data for case (i) with k off = 0 . s − (solid symbols)and k off = 0 . s − (open symbols) for v b = 0 . , . , and0 . µm s − . The corresponding MFNC predictions are shownwith dotted and dashed lines respectively. (b) Shows datafor case (i) (solid symbols) and case (ii) (open symbols) for k off = 0 . s − , and the MFNC prediction (solid line). (c)Shows the data collapse as a function of γk , where γ = 1for case (i) and γ = L/ √ b − a for case (ii). The solidline represents the MFNC limit for k off = 0 . s − , and thedashed line represents the fit to MFNC limit with an effectiveunbinding rate k effoff = 2 . s − . Note that k effoff is very largecompared to the actual unbinding rate. Next we compare case (i) with case (ii), where thebinding rate is not normalized so that the average (cid:104) k on ( x ) (cid:105) is not equal to k and is, in fact, larger than k . In Fig. 6(b), we show the scaled velocities against k for case (i) and case (ii), for a given unbinding rate k off = 0 . s − along with the corresponding MFNC pre-diction (Eq. 5). We observe that for both cases, thedata for all bound velocities show a good collapse as inFig. 6(a). More interestingly, the average velocity of theMCC for case (ii) is greater than the MFNC velocity V l ,unlike case (i) where it is always less than V l . This en-hancement in the average velocity is because of (cid:104) k on ( x ) (cid:105) being larger than k by a factor of by L/ √ b − a ; mul-tiplying k by this factor can collapse both data ontoa single curve which stays below V l (Fig. 6(c)). Never-theless this suggests that if confinement were to causean enhancement of the average binding rate (cid:104) k on ( x ) (cid:105) , itwould lead to larger average velocities of the MCC. Infact, in a study similarly to (ii) but for Brownian ratchetsin confined media, the authors found an enhancement ofthe net particle for non-processive motors with a con-finement dependent binding rate . The qualitativebehavior of the effective diffusion coefficient for an MCCwith spatially varying binding rates was observed to besimilar to that for an MCC with constant binding rates(results not shown here). D. Reduced Probability Density for Bound and UnboundStates
To understand the behavior of the scaled average ve-locity, in particular the observed data collapsed, we ex-amine the probability densities of the bound and un-bound states. Given the periodic nature of the corru-gated confining channel , we study the reduced prob-ability densities for the bound state ˆ P b ( x, t ) and un-bound state ˆ P b ( x, t ) defined as follows:ˆ P b ( x, t ) = n = ∞ (cid:88) n = −∞ P b ( x + nL, t ) , (7a)ˆ P ub ( x, t ) = n = ∞ (cid:88) n = −∞ P ub ( x + nL, t ) , (7b)with (cid:82) L dx [ ˆ P b ( x, t ) + ˆ P ub ( x, t )] = 1. We have checkednumerically that these probability densities reach theirsteady state values ˆ P st b ( x ) and ˆ P st ub ( x ) at large times.In Fig. 7, we have shown ˆ P st b ( x ) and ˆ P st ub ( x ) for threecases: (i) unconfined (Fig. 7(b)), (ii) confinement withconstant binding rate with k on = k (Fig. 7(c)), and(iii) confinement with spatially varying binding ratewith k on ( x ) = k (cid:112) ( b − a ) / ( L w ( x )) (Fig. 7(d)). InFig. 7(a), we plot the spatial profile of the confiningwall w ( x ) and its entropic barrier. For case (i), ˆ P st b ( x )and ˆ P st ub ( x ) are uniform in x , independent of v b , and fol-low the corresponding analytical predictions as expected.Interestingly, for case (ii), the ˆ P st b ( x ) and ˆ P st ub ( x ) arenot uniform but modulate with the same wavelength asthat associated with the spatial variation of the confin-ing channel. The probability density ˆ P st ub ( x ) is indepen-dent of v b , while ˆ P st b ( x ) vary with v b , approaching toa constant value at large v b . For case (iii), ˆ P st ub followsthe spatial variation of the channel in x , while ˆ P st b isuniform– this is because the normalization of the bind-ing rate involves scaling by w ( x ); both these reducedprobability densities are independent of v b . It is impor-tant to note that for the latter case the value of ˆ P st ub ( ˆ P st b ) has significantly increased (decreased) (Fig. 7(c))compared to that for the confined MCC with constantbinding rate (fig. 7(b)). This enhancement of ˆ P st ub anddecrease in ˆ P st b leads to smaller scaled velocity in thecase of spatially varying (normalized) binding rate.In summary, when the reduced probability densitiesare independent of v b , the corresponding scaled velocitydata will collapse onto a single curve, and vice versa. (a) x w(x)-log(w(x)) (b) x Pˆ stb , v b =0.04Pˆ stub , v b =0.04Pˆ stb , v b =0.8Pˆ stub , v b =0.8k on /(k on + k off )k off /(k on + k off ) (c) x Pˆ stb , v b =0.04Pˆ stub , v b =0.04Pˆ stb , v b =0.8Pˆ stub , v b =0.8Pˆ stb , v b =1.6Pˆ stub , v b =1.6 (d) x Pˆ stb , v b =0.04Pˆ stub , v b =0.04Pˆ stb , v b =0.8Pˆ stub , v b =0.8 FIG. 7. (a) The wall function w ( x ) with L = 1 and the en-tropic barrier due to w ( x ). Figures (b), (c), and (d) show thesteady state reduced probability densities ˆ P st b ( x ) and ˆ P st ub ( x ).Data is shown for small and large v b (0 .
04, and 0 . µm s − ),while D = 0 . µm s − , k = 0 . s − , and k off = 0 . s − .Figure (b) shows data for the unconfined case, figure (c) fora confined MCC with constant binding rates, and figure (d)for a confined MCC with that has spatially varying bindingrates scaled by the confinement w ( x ). This also explains why for the case of confinement withconstant binding rate, scaled velocity data collapse wasonly observed for large v b , but not for small v b . Fur-thermore, the increase in unbinding probability densityin the case of spatially varying binding rate explains thedecrease in the scaled velocity (in Fig. 6). IV. SUMMARY AND DISCUSSION
We have studied the role of confinement in two-statecargo transport in a two-dimensional corrugated channelusing the Fick-Jacobs formalism, and an equivalent one-dimensional lattice model. The effect of confinement isincorporated through a position dependent entropic bar-rier. At any given time, the MCC can be in one of twostates: an active state where it moves on a microtubuletrack with a constant speed, and a passive state whenit is detached from the microtubule and undergoes dif-fusive motion. We assumed small cargo sizes such thatwhile the diffusive motion is impaired by confinement,the bound state directed motion is not. The results fromthe lattice model exactly match known analytical re-sults for purely diffusive motion in confinement, demon-strating that the Arrhenius description for hopping ratesworks for our system and other similar systems with en-0tropic barriers. Moreover, the lattice based approachand simple evolution rules make our model computation-ally more efficient for simulating two state transport incomplex confinement profiles than numerical simulationsof the corresponding 2D Langevin equations.In order to understand and quantify how confinementimpacts transport properties, we computed and com-pared the MSD, as well as the average velocity and ef-fective diffusivity of the MCC with and without con-finement. The MSD of the confined MCC shows threedistinct dynamical regimes corresponding to diffusivemotion at small times, ballistic motion at large times,and sub-diffusive motion at intermediate times. Thecrossover timescale ( ∼ − s ) from diffusive to sub-diffusive motion is determined by the interplay betweenpassive or diffusive motion and confinement, and sug-gests a mesh size ∼ nm for the parameters used inour study if the confinement were due to a cytoskeletalnetwork. The crossover from sub-diffusive to the ballisticmotion is dictated by the motor properties, specificallythe binding kinetics and the speed of the motor whenbound. Confinement significantly reduces the crossovertime from diffusive to sub-diffusive behavior, and alsoleads to a significant intermediate sub-diffusive regime.For unconfined MCCs, this intermediate regime is eitherabsent or much smaller than for confined MCCs.We also found that confinement effectively enhancesthe motor unbinding rate and thus reduces the aver-age velocity when the bound velocity is small, but hasa negligible effect otherwise. The impact of confine-ment on the effective diffusivity is more remarkable. Inthe absence of any confinement, for less active MCCs( v b (cid:46) . µm s − ), an increase in the binding rateleads to a decrease in the effective diffusivity becauseof the comparatively less time spent in the unboundstate; for more active MCCs ( v b (cid:38) . µm s − ), how-ever, the diffusivity initially increases with the bindingrate reaching a peak, and then decreases. This can be at-tributed to locked-to-running transitions in the two statemodel. While confinement leads to smaller diffusivities,the peaks now start appearing at smaller v b ∼ . µm s − and are more prominent. Since kinesin-1 motors havean in vitro speed of 0 . µm s − and an in vivo speed of2 . µm s − , the peaks should be readily observed inexperiments in live cells. In vitro, the predictions of ourmodel can be tested in experiments on kinesin-based mi-crotubule transport in enclosed microfluidic channels .An exception to the above confinement induced slow-ing down of the MCC is observed when confinement en-hanced the average rate of binding of the MCC to themicrotubule, thereby leading to an enhancement in theaverage velocity. This suggests that the impact of con-finement on cargo transport strongly depends on if andhow it modulates the binding kinetics of the motors. Itsimpact on binding kinetics can be obtained in enclosedmicrochannel experiments by measuring the MCC res- idence times in the bound and unbound states for differ-ent channel widths.The same experimental set up can be used to obtainthe scaled average velocity, and thus test the predictionsof our study. In addition to studying motor driven cargotransport in confinement, such microfabricated enclosedchannels can be potentially used to deliver specific pro-teins or to separate DNA or RNA strands from a complexmixture by binding them to microtubules and transport-ing them to desired locations. Our results therefore maynot only be useful in understanding cargo transport incells, but also may help in advancing the nanoscale drugdelivery system within cells and sequencing techniquesfor DNA and RNA. Our model can also be easily ex-tended to study bidirectional cargo transport .Finally, for completeness, we comment on the effect ofhydrodynamic coupling between the wall and the cargoin the light of a recent experimental study . In ourwork, we study only the effect of entropic barrier ignor-ing the hydrodynamic coupling between the confiningchannel and the diffusing particle, as the latter is nottaken into account in the Fick-Jacobs approach .A recent experiment on colloidal diffusion in corrugatedmicro-channels found that confinement can increase thehydrodynamic drag which is not captured by the Fick-Jacobs theory using free diffusivities . However, theauthors have demonstrated that this theory can be usedto explain their results if it is reformulated in terms ofthe experimentally measured diffusion coefficients. Itmay be interesting to study how hydrodynamic effectsimpact our system; while it is outside the scope of ourcurrent study, we will pursue this in future work. Sucheffects have been found to be important for microswim-mers which, unlike diffusive particles, create and use hy-drodynamic flow fields for their propulsion . ACKNOWLEDGMENTS
The authors would like to thank Jennifer Ross, MeganValentine, and Ajay Gopinathan for illuminating andhelpful discussions, and acknowledge helpful suggestionsfrom anonymous reviewers. SD would also like to thankDibyendu Das for useful discussions. This research isfunded in part by the Gordon and Betty Moore Foun-dation through Grant GBMF5263.02 to MD. MD andKC were also partially supported by a Cottrell CollegeScience Award from Research Corporation for ScienceAdvancement.
Appendix A: Continuous limit of the Lattice model
As discussed in the main text, the master equationsdescribing the time evolution of the probability densitiesof the bound (active) and unbound (passive) state aregiven by,1 ∂P b ( x, t ) ∂t = k on ( x ) P ub ( x, t ) − k off ( x ) P b ( x, t ) + λ v P b ( x − (cid:96), t ) − λ v P b ( x, t ) (A1) ∂P ub ( x, t ) ∂t = − k on ( x ) P ub ( x, t ) + k off ( x ) P b ( x, t ) + λ ub+ ( x − (cid:96) ) P ub ( x − (cid:96), t ) + λ ub − ( x + (cid:96) ) P ub ( x + (cid:96), t ) − (cid:0) λ ub+ ( x ) + λ ub − ( x ) (cid:1) P ub ( x, t ) , (A2)where k on ( x ) and k off ( x ) are the transition rates for the bound and unbound state respectively, λ v = v b /(cid:96) isthe hopping rate in the forward direction when the motor-cargo complex is in the bound state, and λ ub ± ( x ) =( D /(cid:96) )e − β ( A ( x ± (cid:96) ) −A ( x )) / is the unbound state hopping rates for the forward and backward direction respectively.Using Taylor’s expansion for P b , ub ( x ± (cid:96), t ) and λ ub+ ( x ± (cid:96) ) around x and keeping the terms up to 2 nd order in (cid:96) , weget ∂P b ( x, t ) ∂t = k on P ub − k off P b − v b ∂P b ∂x + D v ∂ P b ∂x + O ( (cid:96) ) , (A3) ∂P ub ( x, t ) ∂t = − k on P ub + k off P b + (cid:96) P ub (cid:20) − dλ ub+ dx + dλ ub − dx + (cid:96) (cid:18) dλ ub+ dx − dλ ub − dx (cid:19)(cid:21) + (cid:96) ∂P ub ∂x (cid:20) − λ ub+ + λ ub − + (cid:96) ( dλ ub+ dx + dλ ub − dx ) (cid:21) + (cid:96) ∂ P ub ∂x (cid:0) λ ub+ + λ ub − (cid:1) + O ( (cid:96) ) , (A4)where D v ≡ (cid:96) λ v / (cid:96) v b /
2. In (cid:96) → D v in Eq. A5 we recover the continuumFokker-Planck equation for the bound state (Eq. (2a) in the main text) ∂P b ( x, t ) ∂t = k on P ub − k off P b − v b ∂P b ∂x . (A5)Considering the leading order contributions for the coefficients of P ub and dP ub dx in Eq. A6 we get ∂P ub ( x, t ) ∂t = − k on P ub + k off P b + (cid:96) P ub (cid:20) − dλ ub+ dx + dλ ub − dx (cid:21) + (cid:96) ∂P ub ∂x (cid:2) − λ ub+ + λ ub − (cid:3) + (cid:96) ∂ P ub ∂x (cid:0) λ ub+ + λ ub − (cid:1) . (A6)For (cid:96) → λ ub ± ( x ) = ( D /(cid:96) )e ∓ β(cid:96) d A dx . In this limit, the coefficients of P ub , ∂P ub ∂x , and ∂ P ub ∂x are given by, (cid:96) (cid:16) − dλ ub+ dx + dλ ub − dx (cid:17) (cid:39) D β d A dx , (cid:96) (cid:0) − λ ub+ + λ ub − (cid:1) (cid:39) D β d A dx , and (cid:96) (cid:0) λ ub+ + λ ub − (cid:1) / (cid:39) D respectively. Using theseexpressions in Eq. A6 we recover the continuum Fokker-Planck equation for the unbound state (Eq. (2b) in the maintext), ∂P ub ( x, t ) ∂t = − k on P ub + k off P b + D β ∂∂x (cid:18) P ub d A dx (cid:19) + D ∂ P ub ∂x = − k on P ub + k off P b + D ∂∂x (cid:18) e − β A ( x ) ∂∂x e β A ( x ) P ub ( x, t ) (cid:19) . (A7) B. Alberts, A. Johnson, J. H. Lewis, and D. Morgan,
Molecularbiology of the cell (Garland Science, 2015). R. D. Vale, Cell , 467 (2003). J. L. Ross, M. Y. Ali, and D. M. Warshaw, Current Opinion inCell Biology , 41 (2008). J. L. Ross, H. Shuman, E. L. Holzbaur, and Y. E. Goldman,Biophysical Journal , 3115 (2008). S. Gunawardena and L. S. B. Goldstein, Journal of Neurobiology , 258 (2003). N. Hirokawa and R. Takemura, Nature Reviews Neuroscience ,201 (2005). F. J¨ulicher, A. Ajdari, and J. Prost, Rev. Mod. Phys. , 1269(1997). C. Appert-Rolland, M. Ebbinghaus, and L. Santen, PhysicsReports , 1 (2015). M. Schuh, Nat. Cell Biol. , 1431 (2013). E. L. F. Holzbaur, Intracellular Traffic and Neurodegenera-tive Disorders Research and Perspectives in Alzheimers Disease , 27 (2006). S. Millecamps and J.-P. Julien, Nature Reviews Neuroscience , 161 (2013). A. L. Zajac, Y. E. Goldman, E. L. Holzbaur, and E. M. Ostap,Current Biology , 1173 (2013). P. C. Bressloff and J. M. Newby, Rev. Mod. Phys. , 135 (2013). M. Pilhofer, M. S. Ladinsky, A. W. McDowall, G. Petroni, andG. J. Jensen, PLOS Biology , 1 (2011). M. T. Valentine, P. M. Fordyce, T. C. Krzysiak, S. P. Gilbert,and S. M. Block, Nature Cell Biology , 470 (2006). O. Campas, C. Leduc, P. Bassereau, J. Casademunt, J.-F.Joanny, and J. Prost, Biophysical journal , 5009 (2008). S. Klumpp and R. Lipowsky, Proceedings of the NationalAcademy of Sciences of the United States of America , 17284 (2005). P. Bressloff and J. Newby, New Journal of Physics , 023033(2009). L. Conway, D. Wood, E. T¨uzel, and J. L. Ross, Proceedings ofthe National Academy of Sciences , 20814 (2012). L. Conway, M. W. Gramlich, S. M. Ali Tabei, and J. L. Ross,Cytoskeleton , 595 (2014). W. W. Ahmed and T. A. Saif, Scientific reports (2014). N. Fakhri, A. D. Wessel, C. Willms, M. Pasquali, D. R. Klopfen-stein, F. C. MacKintosh, and C. F. Schmidt, Science , 1031(2014). M. Guo, A. J. Ehrlicher, M. H. Jensen, M. Renz, J. R. Moore,R. D. Goldman, J. Lippincott-Schwartz, F. C. Mackintosh, andD. A. Weitz, Cell , 822 (2014). C. P. Brangwynne, G. H. Koenderink, F. C. MacKintosh, andD. A. Weitz, Trends in Cell Biology , 423 (2009). P. Greulich and L. Santen, The European Physical Journal E , 191 (2010). I. Neri, N. Kern, and A. Parmeggiani, Phys. Rev. Lett. ,098102 (2013). L. Ciandrini, I. Neri, J. C. Walter, O. Dauloudet, andA. Parmeggiani, Physical Biology , 056006 (2014). D. Ando, N. Korabel, K. Huang, and A. Gopinathan, Biophys-ical Journal , 1574 (2015). F. H¨ofling and T. Franosch, Rep. Prog. Phys. , 046602 (2013). I. Goychuk, V. O. Kharchenko, and R. Metzler, PLoS ONE ,e91700 (2014). I. Goychuk, Phys. Biol. , 016013 (2015). P. Malgaretti, I. Pagonabarraga, and J. M. Rub´ı, Phys. Rev. E , 010105 (2012). P. Malgaretti, I. Pagonabarraga, and J. M. Rubi, The Journalof Chemical Physics , 194906 (2013). C. Bustamante, D. Keller, and G. Oster, Accountsof Chemical Research , 412 (2001), pMID: 11412078,https://doi.org/10.1021/ar0001719. P. K. Ghosh, V. R. Misko, F. Marchesoni, and F. Nori, Phys.Rev. Lett. , 268301 (2013). N. Suetsugu, N. Yamada, T. Kagawa, H. Yonekura, T. Q. P.Uyeda, A. Kadota, and M. Wada, Proceedings of the NationalAcademy of Sciences , 8860 (2010). D. Reguera, G. Schmid, P. S. Burada, J. M. Rub´ı, P. Reimann,and P. H¨anggi, Phys. Rev. Lett. , 130603 (2006). I. Goychuk, V. O. Kharchenko, and R. Metzler, Phys. Chem.Chem. Phys. , 16524 (2014). S. Bouzat, Phys. Rev. E , 062707 (2014). P. Reimann, Physics Reports , 57 (2002). P. H¨anggi and F. Marchesoni, Reviews of Modern Physics ,387 (2009). R. Zwanzig, The Journal of Physical Chemistry , 3926 (1992). D. Reguera and J. M. Rub´ı, Phys. Rev. E , 061106 (2001). S. M. Block, L. S. B. Goldstein, and B. J. Schnapp, Nature ,348 (1990). E. Taylor and G. Borisy, Journal of Cell Biology , F27 (2000). S. Dey, D. Das, and R. Rajesh, EPL (Europhysics Letters) ,44001 (2011). K. Visscher, M. J. Schnitzer, and S. M. Block, Nature CellBiology , 718 (2000). C. B. Korn, S. Klumpp, R. Lipowsky, and U. S. Schwarz, TheJournal of Chemical Physics , 245107 (2009). P. Burada, G. Schmid, D. Reguera, J. Rub´ı, and P. H¨anggi,EPL (Europhysics Letters) , 50003 (2009). P. S. Burada, P. H¨anggi, F. Marchesoni, G. Schmid, andP. Talkner, ChemPhysChem , 45 (2009). P. Burada, G. Schmid, P. Talkner, P. H¨anggi, D. Reguera, andJ. Rubi, Biosystems , 16 (2008). M. Dogterom and S. Leibler, Phys. Rev. Lett. , 1347 (1993). G. Costantini and F. Marchesoni, EPL (Europhysics Letters) , 491 (1999). Because of the channel periodicity and open boundary conditionsin our model, P b ( x, t ) and P ub ( x, t ) → R. P. Ron Milo,
Cell Biology by the Numbers (Garland Science,2015). Y. Huang, M. Uppalapati, W. Hancock, and T. Jackson, IEEETransactions on Advanced Packaging , 564 (2005). M. Welte, Current Biology , R525 (2004). X. Yang, C. Liu, Y. Li, F. Marchesoni, P. H¨anggi, and H. P.Zhang, Proceedings of the National Academy of Sciences ,9564 (2017). P. Malgaretti and H. Stark, The Journal of Chemical Physics146