Ratcheted diffusion transport through crowded nanochannelsAnna Lappala, Alessio Zaccone, Eugene M. Terentjeva)

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE,U.K.

The problem of transport through nanochannels is one of the major questions in cell biology, with a widerange of applications. Brownian ratchets are fundamental in various biochemical processes, and are roughlydivided into two categories: active (usually ATP-powered) molecular motors and passive constructions with adirectional bias, where the transport is driven by thermal motion. In this paper we discuss the latter process,of spontaneous translocation of molecules (Brownian particles) by ratcheted diffusion with no external energyinput: a problem relevant for protein translocation along bacterial flagella or injectosome complex, or DNAtranslocation by bacteriophages. We use molecular dynamics simulations and statistical theory to identifytwo regimes of transport: at low rate of particles injection into the channel the process is controlled by theindividual diffusion towards the open end (the first passage problem), while at a higher rate of injectionthe crowded regime sets in. In this regime the particle density in the channel reaches a constant saturationlevel and the resistance force increases substantially, due to the osmotic pressure build-up. To achieve asteady-state transport, the apparatus that injects new particles into a crowded channel has to operate withan increasing power consumption, proportional to the length of the channel and the required rate of transport.The analysis of resistance force, and accordingly – the power required to inject the particles into a crowdedchannel to oversome its clogging, is also relevant for many microfluidics applications.

Ratcheted diffusion can be described as directionallybiased random thermal motion. Physically, this processis often described as a transport phenomenon in spa-tially periodic systems, a typical example of which is a“saw-tooth” potential that is periodic in space and hasa broken forward-backward symmetry, however, ratchetpotentials exist in various forms in terms of symmetryand periodic behavior.1 The function of biologically rel-evant “motor” enzymes, or molecular motors, is basedon non-equilibrium ratcheted diffusion, where the energyis continuously recycled in the system – typically by theATP hydrolysis. Well studied classes of these molecularmotors are myosins, kinesins and dyneins1. The physicalmechanism producing the powered linear motion alongthe filament with asymmetric (saw-tooth) potential, en-ergized by the ATP hydrolysis, is well understood af-ter the classical work of Prost et al.2,3 Myosins movealong actin filaments, and these proteins are responsiblefor muscle contraction4. Kinesin and dynein motor pro-teins move along microtubules, either towards the elon-gating/polymerising + end (which has a high concentra-tion of so called β-subunits) or the − end (which has ahigh concentration of α-subunits). Dyneins have variousadditional functions which include facilitating the move-ment of cilia and flagella.4

Our interest is different: we examine the equilibriumsituation when the directed transport is achieved sponta-neously, essentially, energized by the Brownian thermalmotion and directed by restricting the diffusion in the“wrong” direction. Examples of this equilibrium diffu-sive transport is the translocation of cargo across mem-branes through nanopores or along nanochannels.5 Es-sentially, the directed Brownian motion is achieved by

a)Electronic mail: emt1000@cam.ac.uk

simply not allowing the cargo to travel back to the sideof the channel where the process of translocation started.To achieve this, the cargo has to block the channel widthsuch that no overtaking is possible in a single-file (one-dimensional) motion.6 In this case the potential profilealong the channel is irrelevant, as the equilibrium detailedbalance would not produce any directed motion withoutan external force (in spite of any symmetry breaking).A variety of chemical asymmetries may exist inside thechannel to affect the movement of the cargo.7 However,the effect of the channel potential is merely to modify theeffective diffusion constant, which is again a well-studiedproblem.8 Biological examples of transport assumed torely on spontaneous ratcheted diffusion are found in var-ious biological systems, such as:1) flagella – hair-like protrusions from the cell body ofspecific pro- and eukaryotic cells9 grow by transport-ing flagellin protein subunits from the body of bacteriathrough a 2nm wide channel over distances of 10-20 µm,at a constant rate;10

2) the injectosome complex (type III secretion system)– a needle-like protein appendage that is found in gram-negative bacteria, the function of which is to secrete pro-teins into eukaryotic host cells to facilitate the process ofinfection11;3) the general secretory (Sec) pathway, in whichthe translocase protein complex machinery provides atranslocation pathway for proteins from the cytosolacross the cytoplasmic membrane in bacteria;12

4) bacteriophage (bacterial virus) DNA injection intohost bacterial cells through the tail of the phage – a goodexample of this process is the T4 phage mechanism ofinjection.13

Independently, ratcheted transport in Brownian sys-tems is of great practical relevance for applications inlab-on-chip microfluidic devices.14 Typically, if crowdingconditions of drops/bubbles/particles are achieved at the

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inlet of the microfluidic channel, the resistance to thetransport develops, which eventually lead to microchan-nel clogging.15 The latter is a practically important phe-nomenon which limits the performance of microfluidicdevices and of industrial micromixers.16,17

One particular biological system that we shall fre-quently refer to is the bacterial flagellum, which is a tubewith a 2nm wide inner channel and the walls formed bycrystallized flagellin proteins. In order for the flagellumto grow, the protein subunits have to be translocatedto the distal (growing) end of the flagellum through thischannel, which is too narrow to accommodate folded pro-teins. Hence, subunits are transported in their unfoldedstate, after they are inserted into the channel by the ex-port apparatus.18 Flagellar export apparatus is a mul-ticomponent complex, energized by ATP hydrolysis andachieving the successive unfolding of flagellin chains andinsertion of them into the channel,19–21 thus providing abias of the subsequent 1D Brownian motion, which mo-tivates the work presented in this paper.

Very recently there has been an attempt on studyingthe diffusion of flagellin subunits along this channel,22

where many of the details and parameters of this bio-logical system are carefully presented. The shortcomingof this basic simulation is that it does not take into ac-count interactions (essentially – repulsion on contact) be-tween travelling particles, which then fails to describe thecrowding regime. Here we show molecular-dynamics sim-ulation results for translocation of Brownian particles innanochannels under conditions at the inlet, which mimiccrowded cellular environments. The effect of molecularcrowding at the channel inlet is twofold: on the one handit acts as a net drift speeding up the translocation nearthe open outlet, but on the other hand it also causes anincreasing resistance force, slowing down the transportnear the channel inlet. The range of channel length wherecrowding contributes a net resistance force increases withthe total channel length. Remarkably, this effective re-sistance force is an emergent phenomenon which in 1Ddepends entirely on the cooperative multi-body dynamicsof particles colliding in the channel. Collective diffusioncombined with a third-order virial expansion provides asimple theoretical description of the density profile in-side the channel in agreement with the simulation data.The transition from classical free diffusion to the crowd-ing regime is controlled by the competition between theparticle injection rate and the mean first passage timeof free diffusion. This fact is crucial in order to assessthe onset of crowding in both biological and engineeringsystems.

FREE SINGLE-FILE DIFFUSION

The ratcheting, or rectification of the Brownian motionalong the narrow channel is achieved by continuously in-serting new particles from one end, thus restricting thebackward motion – while the channel remains open at the

FIG. 1. A histogram representing the distribution profile ofindividual diffusion times, τdiff, along the channel of lengthL = 60σ (σ being the size of particle). The time is measures in[kts]: 1000 of simulation time steps. The solid line representsthe fit to the probability distribution discussed in the text,eq.(1), with only the normalization factor and the diffusionconstant D as fitting parameters.

distal end. The diffusion process as described in this pa-per raises three main issues: 1) how high is the resistanceforce, if the particles are inserted at a sufficiently highrate? 2) How do the two characteristic times, the meanfirst passage time τdiff and the rate of insertion 1/Tin

compete? 3) What is the crossover between the stochas-tic diffusion dominated regime (when Tin � τdiff) andthe dynamically restricted, resistance-dominated regimein a crowded channel?

In order to introduce the problem, we first look at themotion of a single particle inside a channel blocked atone end, in order to verify the properties of generalizeddiffusion, fig. 1. Since the particle essentially disappearsas soon as it reaches the open end, a distance L away, thisis a problem of first passage with a reflecting boundarycondition at x = 0 and an absorbing condition at x = L.The probability distribution of first passage times is aclassical result obtained by the method of images dueto Lord Kelvin;23 it takes the form (see “Methods” fordetail):

f(t) = const · L

2√πDt3

(e−L

2/4Dt − e−L2/Dt

). (1)

This probability distribution is used to fit the histogramof simulation results for the frequency of diffusion time ofa single particle along the channel, obtained by the Brow-nian dynamics simulation, fig. 1. In particular, the freediffusion constant in our simulation is thus measured tobe D = 0.0065 in the reduced simulation units of [σ2/τ ].An example analysis in “Methods” shows that this cor-responds to a value D = 6.5 ·10−12m2/s for 1-micron sizepolystyrene colloids in water.

Unlike earlier work,1,8 we do not consider designed pe-riodic potentials in the channel for simplicity, and dueto the fact that this is not necessary in order to find D,the diffusion constant. The important result, reproduced

3

frequently in the literature, is that the effective diffusionconstant for traveling in a channel with a potential V (x)along its length is related to the bare diffusion constant(with no potential) as24

D =D0∫ L

0dx1

L e−V (x1)/kBT∫ L

0dx2

L eV (x2)/kBT. (2)

The product of two integrals in the denominator is alwaysgreater than unity (this is demonstrated by applying theCauchy-Schwartz inequality,24 see also25) and therefore,for any potential V (x) on the channel walls the diffusionis hindered. For our purposes of studying the effects ofcrowding in the channel, we may simply take a given fixedvalue for the effective diffusion constant, D, obtained bythe fitting of single-particle diffusion data as in fig. 1, andproceed to examine the case when many particles restricttheir motion by crowding.

CROSSOVER TO CROWDED REGIME

Having determined the average time of individual par-ticle diffusion to the far end of the channel, τdiff =∫t · f(t)dt = L2/D, we can now simulate the process

with particles injected at x = 0 with increasingly shorttime intervals Tin (i.e at an increasing rate, or the particleflux Jin = 1/Tin). As explained in “Methods” below, oursimulation algorithm only allows the insertion if a suffi-cient room (of the particle size σ) is available at x = 0;if the space is occupied, the insertion is missed and isattempted after the next Tin interval. At low rate of in-sertion this omission never happens, but at Tin � τdiff

this becomes a limiting factor. Figure 2 represents theresults in the form of the cumulative number of particleshaving passed through the channel: N(t) is essentiallythe sequential number of a particle leaving the channelat a time t. From the plot it is clear that at sufficientlylong times of simulation the progression of particles islinear on average. One may interpret this as a constantrate of delivery of cargo along the channel, however, weshall discuss below the constraints on the required powerand the force exerted by the insertion apparatus.

Figure 3 plots the average slope of the data in fig. 2on the double-logarithmic scale to allow examination ofthe two key findings. We choose to plot the inverse flux,dt/dN(t) against Tin, to have a more clear sense of theevents. The plot also shows the results for three differentchannel lengths (measured in the units of particle size σ),as labeled on the figure. In essence, every curve in fig. 2gives a single point in this graph, for a given L and Tin.Two regimes are evident: when particles are inserted ata sufficiently low rate, the insertion rate is simply equalto the constant flux J0 through the channel, with a smallcorrecting factor reflecting the channel length. However,when the attempted rate of insertion becomes high, therate of diffusive transport through the channel saturatesat a constant maximum value that depends on the chan-nel length and is determined by the increasing resistance

for the motion of the single file of particles packed in thechannel.

FIG. 2. A graph of a cumulative number of particles hav-ing passed through the channel of L = 30σ against simula-tion time-step (measured in [kts]: 1000’s of time steps, as infig. 1). The labeled values of Tin, also in [kts], represents theset interval of an attempted insertion at x = 0. The largescale of this cumulative plot makes obvious the linearity ofN(t), but masks what happens at shorter times – when theinsertion location may be occupied by other particles; fig. 3below illustrates the effect of crowding restriction.

FIG. 3. The average slope of the time-step versus the cumu-lative number of particles as a function of insertion time, Tin

for three channel lengths. For high Tin, ∆t/∆N = Tin. Theslope remains constant as a function of Tin initially due to thefact that insertion rate is too high and particles cannot be in-serted more often than Tin ≈ 1000 ts, the crossover betweenthe constant insertion rate and the linearly increasing rate athigher Tin.

Since we are interested in the problem of crowding, themost valuable scenario arises when the insertion rate ishigh and so particles frequently collide in the channel.In order to understand how crowding inside the channelaffects the interactions between particles, we consideredtwo scenarios – an open channel, as described earlier, inwhich the particles essentially disappear at the far end

4

FIG. 4. The number of particles in open and closed channelsof different lengths: L = 30σ, 60σ and 90σ for the fastestattempted insertion rate (Tin = 0.001 kts); the curves arelabeled accordingly on the plot. The fluctuations in openchannels are caused by the diffusive flux of particles exitingthe channel. Due to a higher overall number of particles inlonger channels, the osmotic pressure inside is higher, whichexplains the larger gap between L and the mean plateau val-ues for longer channels.

(i.e. not able to return back into the channel once theyreach x = L) – and a closed channel with another re-flecting wall at x = L. The latter channel becomes filledwith particles at a specific rate. Because the length ofthe channel emerges as an important parameter, we haveagain compared channels of three lengths: L = 30σ, 60σand 90σ, fig. 4. As the time increases, the particles ini-tially fill the channel at a constant rate – but then theirnumber in channel saturates at a constant value: this isa lower and a highly fluctuating value if the channel isopen, and a higher and almost constant value if the chan-nel is closed and the particles pack until there is no moreinsertion possible. But even the latter value is much lessthan the channel length: 79 in the channel of 90-particleslong, 55 in 60 and 28 in 30. The difference is due tothe effective resistance to insertion of new particles whenthe density increases sufficiently; note how this differencegrows with the length of the channel, which reflects thegrowing resistance force for new particle insertion.

The growth of number of particles in the channel asa function of time was fitted with the simple exponen-tial function: n(t) = nmax(1− exp[−t/τrel]), where nmax

is the maximum number of particles in a channel at agiven time (fixed for a channel of given length): the meanplateau value in fig. 4 gives nmax = 54 for L = 90σ, 34 for60σ and 17 for 30σ. The gap between the dense-packingand the plateau value in this case (of an open channel)grows approximately linearly with the channel length, incontrast with the closed channel where this gap increasesapproximately as the square of length. Remarkably, thecharacteristic relaxation time τrel after which the dense-packing plateau is achieved was the same in all cases,τrel ≈ 40 kts.

RESISTANCE TO TRANSPORT

Here we show that the effect of crowding inside thechannel results in an effective potential within the chan-nel when the density of particles ρ is high. Let us considera steady state, at t > τrel, when the flux is constant alongthe channel, Jin = Jout. The simple one-dimensionaldiffusion with the boundary conditions (ρ0 at x = 0and ρ = 0 at r = L) gives the time-evolving profileswhich eventually would reach the steady state of linearρ(x) = [Jeq/D](L − x) and constant flux Jeq = −D∇ρ.This steady state is only possible if the density at theentrance ρ(x = 0) = [Jeq/D]L remains small, otherwisecrowding becomes relevant in the problem and the simplediffusion solutions no longer valid.

In order to examine crowding regime in more detail,fig. 5 shows the profiles of particle density within thechannel at steady-state (in other words, when the chan-nel is filled at t > τrel). The density ρ(x) here is definedas the number of particles per unit length, measuring theprobability density of finding a particle at a position x inthe channel. The deviation from the linear density profileis prominent. Generalized diffusion can be used to de-scribe transport at arbitrary particle concentrations.26,27

The relevant equation reads:

∂ρ

∂t=

∂

∂x

([D

kBT

∂Π

∂ρ

]∂ρ

∂x

), (3)

where Dcoll = D∂(βΠ)/∂ρ, with β = 1/kBT , is the col-lective diffusion coefficient, as defined in nonequilibriumthermodynamics.30 It accounts for deviations from freediffusion through the osmotic pressure Π(ρ), which for anon-ideal fluid can be written as a virial expansion in thedensity βΠ = ρ+B2ρ

2 +B3ρ3, with B2 and B3 the sec-

ond and third virial coefficients, accounting respectivelyfor two-body and three-body interactions between par-ticles in the channel. In bulk 3D systems, higher ordercoefficients are usually needed in the expansion to prop-erly describe crowding effects at high density,27 however,we shall see below that in our case of single-file 1D trans-port terms up to third order are sufficient to capture thephenomenology observed by simulations. This is simplybecause one particle in front, and one behind, determinethe progress of any given particle. The steady state so-lution of eq.(3) is easy to obtain by the chain rule:

Π = kBT [Jeq/D](L− x). (4)

The osmotic pressure has to be a linear function of coor-dinate along the channel! When the density is low andΠ = kBTρ, this reflects the classical steady-state dif-fusion with the linear density variation ρ(x). At highdensity we need to invert the dependence Π(ρ). Withinthe third-order virial approximation we obtain

ρ ≈ Jeq

D(L− x)−B2

(Jeq

D

)2

(L− x)2 (5)

+(2B22 −B3)

(Jeq

D

)3

(L− x)3,

5

FIG. 5. The steady-state local density of particles along thechannel of varying lengths (labeled on plot) when it is filledat the maximum rate Tin = 0.001 [kts]. Error bars represent1 standard deviation based on 80 snapshots at different timesduring the equilibrium stage of simulation. The solid linesrepresent theoretical fitting curves obtained with ρ = a(L −x)+b(L−x)2+c(L−x)3 (see eq.(5)), where a = (0.84/L−bL−cL2), and b = −0.0014,−0.0007,−0.0004, c = 1.8 · 10−5, 6.0 ·10−6, 2.2 · 10−6 for L = 30, 60, 90σ, respectively.

where the virial coefficients B2 and B3 are both positivefor repulsive particles used in our simulation. The coef-ficients b and c used as fitting parameters to successfullyreproduce the measured density profiles in fig. 5 are ex-plicitly expressed here, and we see that b < 0 always,while the sign of c depends on the relation between B2

and B3. For our repulsive Lennard-Jones potential thefitting suggests that 2B2

2 −B3 > 0.

The eq.(4) for the osmotic pressure illustrates the firstbarrier to transport into such a channel. The exportapparatus that injects new particles into the channel,at x = 0, has to exert a force F = −(1/ρ)[dΠ/dx] toovercome a pressure gradient at the channel entrance:Π(x = 0) over a distance comparable to the particle sizeσ. This “injection force” is Fin ≈ kBT [Jeq/D]L, linearlyincreasing with the channel length if the particle flux is tobe maintained constant, as in.10 Accordingly, the powerrequired to maintain the constant flux will also be pro-portional to the channel length: W = kBT [J2

eqσ/D]L.

The effect of crowding on the Brownian resistance forcealong the channel is described in greater detail in theAppendix. There we show that when the two aditionalvirial-coefficient terms make the steady-flux density dis-tribution nonlinear, eq.(5), the local force acting on theparticles along the channel acquires an additional contri-bution due to the collective Brownian effects:

Fdrag ≈ kBT B2

(J

D

)kBT −

1

2kBT (B2

2 −B3)

(J

D

)2

L .

(6)This equation predicts that the (negative) drag force islinear in L for suffiently long channels, and hence to thenumber of particles in the channel, n.

DISCUSSION

We have studied ratcheted diffusion transport innanochannels where translocation is achieved by meansof crowding at the channel inlet and depletion at theoutlet. This problem is paradigmatic for biomoleculetranslocation in cellular pores and flagella,9 as well asfor microfluidics where crowding leads to microchannelclogging.15,16 In a nanochannel, the cargo (protein sub-units in a flagellar channel or colloids in a microfluidicchannel) has a size comparable to the channel width: theparticles move in a single file towards the distal openend, after being injected by an appropriate export ma-chinery from the proximal end. We find many interestingaspects of such motion when the density of particles inthe channel becomes high, i.e. their mutual collisionsstart having a significant effect on the diffusion (crowdedregime). Of a particular importance is the criterion ofwhen the crowding regime sets in, which is determinedby comparing the injection period Tin = 1/Iin and thetime of diffusion along the channel, τdiff = L2/D. Whenthe insertion rate is high enough, it takes a certain timeto get into the crowded regime; this saturation (clogging)time appears to be independent of the channel length,because it is against the resistance force to the columnmotion that the clogging occurs.

But the main message and conclusion are expressed atthe end of the last section: in order to propel the longcolumn of particles along the channel, maintaining theconstant (steady-state) current, the inserting apparatushas to exert an increasing force, and expend an increas-ing power – both approximately linearly increasing withthe channel length. For long enough channels, the mo-tion will simply stop (channel clogging) until the densityslowly drops after gradual ejection of particles from theopen end and redistribution of remaining ones along thelength. For biological systems such as the flagellin trans-port, the ATP-powered export apparatus would not beable to increase its power output with the growth of theflagellar length, and instead would have to slow down therate of insertion to compensate for the growth of resis-tance. Empirical evidence so far points towards a con-stant growth rate of flagella, and further investigationsare required in the future to clarify the actual mechanism.

In crowded environments, non-specific interactions be-tween molecules lead to deviations from free diffusion.Here we have shown that repulsive interactions betweenparticles lead to an effective drag force, slowing down dif-fusive transport, which is an emergent cooperative phe-nomenon. This is an important effect which has to betaken into account in the analysis of many both naturaland engineered systems: from biomolecule translocationin membrane pores to the design of microfluidic lab-on-chip devices.

6

Methods

We approach the problem of ratcheted diffusion byBrownian dynamics simulations. Our setup consists ofa cylindrical channel into which particles are added at acertain rate. The force field of the model is rather simple– there is only hard core repulsion between particles aswell as between particles and the wall of the cylindricalchannel, which is represented by the repulsive part of theLennard-Jones potential. The channel is blocked fromone side, which only allows particles to translocate to theother side of the channel. As soon as particles reach theend of the channel they disappear from the system. Thedynamics of the process of translocation as well as par-ticle’s positions and forces acting and each particle wererecorded at every time-step. The number of injected - aswell as ejected particles was also monitored. Based onthis information, the diffusion profile inside the channelcan be defined as soon as the injection/ejection equilib-rium state is reached in the channel, meaning that at thispoint, the channel is maximally occupied, and the rateof injection is equal to the rate of ejection. New particlesare only added when the position at the beginning of thechannel is not occupied by other particles, and thereforeeven if the particles are set to be added every time-step,the addition will not happen if another particle is foundless than σ distance away from the addition site. Ad-ditionally, we studied how the system behavior changesif the channel is closed, as opposed to deleting particleswhen they reach the end of the channel.

In order to be able to convert the results of simulations,Lennard-Jones (reduced) units were used. In moleculardynamics algorithms, physical quantities are expressed asdimensionless units, which results in all physical quanti-ties being around unity, hence reducing the problems ofworking with numbers that are very small or very large,making errors more visible and also simplifying the out-put produced and increasing computational efficiency asthe equations of motion become simplified due to the ab-sorption of model-defining parameters into reduced units.Reduced units also allow one to work on different prob-lems with a single model. The following reduced unitsare used for a Lennard-Jones system: length - σ, en-ergy - ε and mass m. From these, one can express unitsof all other physical quantities as follows: time (τ) as

τ = σ√m/ε, temperature kBT in units of ε and pressure

P in units of ε/σ3. The simulation time step was takenas ts= 0.01τ .

For instance, a ‘typical’ colloid particle of size σ =1µm and mass density 1000 kg/m3 would have a massm ≈ 4.2 · 10−15kg (staying strictly in SI units here).Taking kBT = ε ≈ 4.2 · 10−21J (for room temperature),we obtain the time scale τ = 10−3s. This means thatthe diffusion constant that we obtained from the fit infig. 1, D ≈ 0.0065 in units [σ2/τ ] would have the valueD ≈ 6.5 · 10−12m2/s, which is almost exactly the magni-tude of a widely measured diffusion constant for micron-size sterically stabilized polystyrene particles in water.

First passage time. Here we summarize the stan-dard facts about the first passage problem. This is theclassical problem addressing the question: if a diffusingparticle starts at, e.g. x = 0 – on average how long wouldit take to reach a given point, x = L, for the first time(hence the name of “first passage”). Obviously in thestochastic problem one can only have the answer for theaverage time, and so the aim is to find the probabilitydistribution of such times (and then evaluate the averageif required). In our case we remain in the 1-dimensionallimit and the generic diffusion equation for the concen-tration, or the probability to find a particle at (x, t), hasthe form of continuity relation:

∂p(x, t)

∂t= − ∂

∂xJ(x, t), (7)

where J(x, t) is the generalized diffusion flux. For thefree diffusion, the Green’s function of this equation withJ = −D∇p(x, t) is the Gaussian

G(x, x′; t, 0) =1√

4πDte−(x−x′)2/4Dt. (8)

The “first passage” condition is implemented by apply-ing the absorbing boundary condition, that is, once theparticle first reaches x = L, it disappears from the analy-sis: p(L, t) = 0. Such a condition is traditionally treatedby the method of images, that is, imposing two oppositevalues – one starting from x = 0, the other from x = 2L:

pabs(x, t) =1√

4πDt

(e−x

2/4Dt − e−(2L−x)2/4Dt). (9)

Figure 6 plots several curves of this probability distribu-tion, for D = 1 and the increasing time (the particlesstart from x = 0 and x = 2L at t = 0), to show howthe method of images helps to maintain the constraint ofp(L) = 0 at all times.

The survival probability for the particle to remain any-where between 0 and L, having started at x = 0 and t = 0

is obtained by integration: Q(t) =∫ L

0pabs(x, t)dx. The

result of this integration is the difference of two errorfunctions:

Q(t) = erf

[L√4Dt

]− 1

2erf

[L√Dt

]. (10)

Since x = L is an absorbing boundary, the domain of0 − L is independent from the domain L − 2L. Giventhe definition of the survival probability, the fraction−∂Q(t)/∂t is absorbed between t and t+dt. This meansthat f(t) = −∂Q(t)/∂t is actually the probability densityof the time t that takes the particle to reach x = L forthe first time, given in the main text as eq.(1). Figure 7plots a collection of these distributions for different val-ues of the diffusion constant and fixed L = 1. In bothplots we did not normalize the probability density func-tions or their parameters, just assuming all values arenon-dimensional for qualitative illustration. To find the

7

FIG. 6. The probability distribution Eq.(9) for particles withthe absorbing boundary enforced at x = L by the method ofimages.

FIG. 7. The probability distributions eq.(1) of the firstpassage times in the channel of 0−L, for several values of thediffusion constant D.

average value of the first passage time we need to inte-grate, obtaining the very much expected result:

τdiff =

∫ ∞0

t f(t) dt =L2

D. (11)

Forces in crowded channel. The effect of crowdingon the ratcheted transport can be understood via theBrownian force acting on a particle in the system is givenin general terms as F = −(1/ρ)dΠ

dx . Since we work in 1-dimensional geometry, with particles unable to overtakeeach other, only the horizontal projections of all forcesmatter. We can take as positive all forces parallel to the xaxis along the translocation direction (down the densitygradient), and with the minus sign we take the resistanceforces which oppose the translocation, i.e. drag forces.Using the chain rule,

F = −1

ρ

(dΠ

dρ

)dρ

dx,

and employing the virial expansion for Π(ρ) and expres-sion for ρ(x) given by the eq.(5) of the main text, af-ter minor manipulation we obtain the additional effectiveforce determined by the collective effects in the crowdedregime:

Fcrowd ≈kBTB2J

D− kBT (B2

2 −B3)J2

D2(L− x), (12)

where the virial coefficients B2 and B3 were inroduced inthe main text. The last term can be negative, i.e. con-tribute a net drag force or resistance to diffusive motion.It is easy to carry out the explicit calculation of the totaldrag force per particle due to crowding in the channel,by taking the integral over the channel length, with theresult given in eq.(6).

It is also interesting to observe that the effect of crowd-ing on diffusive transport can be mapped onto a classi-cal diffusion equation in the effective force-field given byFcrowd, in the spirit of an approximation introduced in,26

∂

∂x

[(D∂ρ

∂x

)− ρFcrowd

kBT

]= 0 (13)

This is a classical (steady-state) diffusion equation in theforce-field Fcrowd which takes care of the additional ρ-dependent crowding effect. Hence, the diffusive transportin a crowded environment can be effectively described bymeans of classical diffusion in the additional force-fielddue to crowding. The latter can be calculated analyt-ically using (12) whenever the inter-particle interactionand the corresponding virial coefficients are known. Thesolution to this equation is given by the expression for ρthat we presented as eq.(5)

References

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Acknowledgments

This work was funded by the Osk. HuttunenFoundation (Finland) and the Ernest OppenheimerFund, Cambridge. The simulations were performedusing the Darwin Supercomputer of the Universityof Cambridge High Performance Computing Service(http://www.hpc.cam.ac.uk/), provided by Dell Inc. us-ing Strategic Research Infrastructure Funding from theHigher Education Funding Council for England.