Effects of multiple occupancy and inter-particle interactions on selective transport through narrow channels: theory versus experiment
aa r X i v : . [ q - b i o . S C ] N ov Effects of multiple occupancy and inter-particleinteractions on selective transport throughnarrow channels: theory versus experiment
Anton Zilman ∗ Theoretical Biology and Biophysics Groupand Center for Non-Linear StudiesTheoretical Division, Los Alamos National Laboratory
Abstract
Many biological and artificial transport channels function without direct input ofmetabolic energy during a transport event and without structural rearrangements in-volving transitions from a ’closed’ to an ’open’ state. Nevertheless, such channelsare able to maintain efficient and selective transport. It has been proposed that at-tractive interactions between the transported molecules and the channel can increasethe transport efficiency and that the selectivity of such channels can be based on thestrength of the interaction of the specifically transported molecules with the channel.Herein, we study the transport through narrow channels in a framework of a generalkinetic theory, which naturally incorporates multi-particle occupancy of the channeland non-single-file transport. We study how the transport efficiency and the probabil-ity of translocation through the channel are affected by inter-particle interactions inthe confined space inside the channel, and establish conditions for selective transport.We compare the predictions of the model with the available experimental data - andfind good semi-quantitative agreement. Finally, we discuss applications of the theoryto the design of artificial nano-molecular sieves.
Key words:
Transport; Channels;Selectivity; Efficiency; Diffusion; Occupancy
The proper functioning of living cells involves continuous transport of various moleculesinto and out of the cell, as well as between different cell compartments. Such transportrequires discrimination between different intra- and extra-cellular molecular signalsand demands mechanisms for efficient, selective and specific transport (1). Specifi-cally, transport devices must be able to selectively transport only certain molecularspecies while effectively filtering others, even very similar ones.In certain cases, the selectivity and efficiency of the transport is achieved throughdirect input of metabolic energy during the transport event, in the form of the hy-drolysis of ATP or GTP (1). However, in many cases, molecular transport is efficient ∗ Corresponding author. Address: LANL, POB 1663, MS B258, Los Alamos, N M 87545, U.S.A.,Tel.: (505)-667-3216, e-mail: [email protected] elective transport in nano-channels and selective without the direct input of the metabolic energy and without large scalestructural rearrangements that involve transitions from a ’closed’ to an ’open’ stateduring the transport event. Examples of transport of this type include the selectivepermeability of porins (2, 3, 4, 5, 6, 7), transport through the nuclear pore com-plex in eukaryotic cells (8, 9, 10, 11, 12), artificial nano-channels and membranes,(13, 14, 15, 16, 17, 18, 19) and other transport devices (20). Ion channels (21, 22, 23),also belong to this class of transport devices - however the selectivity of ion channelsdepends on numerous factors that set them apart (23, 24) and place them beyond thescope of the present work.Transport devices of this type commonly contain a channel or a passageway throughwhich the molecules translocate by facilitated diffusion. The selectivity and the ef-ficiency of transport are usually based not merely on the molecule size but on acombination of the size, strength of the interaction with the channel, speed of thespatial diffusion through the channel, and channel geometry (cf. Figs. 1 and 2).(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 26, 36, 37). Moreover, alarge body of experimental data shows that the specifically transported molecules inmany cases interact strongly with the channel (more strongly than the ones that arefiltered out) and can transiently bind inside the channel (2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 18, 19). Another important feature of such selective channels is thatthey are narrow, with a diameter comparable to the size of the transported molecules.Understanding mechanisms of the selectivity of transport through such channelsis an important biological question and also has important applications in nano-technology and nano-medicine. For instance, it impacts creation of artificial molecularnano-filters. In addition, it poses a fundamental physical question: how does one makea selective channel that is always open, and does not have a movable ’shutter’ specif-ically attuned to its corresponding transported molecules? Another important goal isto establish to what extent the theoretical models capture the essential properties oftransport through narrow channels by comparing the models to experimental data.The precise mechanisms and the conditions for optimal selectivity of transportthrough such channels are still unknown. These systems span a wide spectrum ofspace and time scales and biological functions. For instance, porins are involved inthe transport of small molecules into and out of the cell. They typically have channellength of several nanometers and a diameter of a couple of nanometers, tuned to thesize of their corresponding transported molecules (e.g. water or small sugars) (2, 3, 4,5, 6, 7). The transport times through porins can be shorter than one millisecond (2).In another example, the nuclear pore complex regulates the transport between the cellnucleus and the cytoplasm. It has a diameter of approximately 30 nm and a length of70 nm (8, 9, 10, 12). It can pass molecules up to 30 nm in size, within transport times ofseveral milliseconds (55, 58). Artificial selective nano-channels have been constructedseveral microns long and tens of nanometers in diameter that selectively transportmolecules of sizes in the range of tens of nanometers. Such artificial devices have beenused to selectively transport various molecules: including molecular enantiomers, shortDNA segments and synthetic polymers (13, 14, 15, 16, 17, 18, 19).Nevertheless, it has been suggested that such channels might share common mech-anisms of selectivity and efficiency. Recent theoretical works propose a mechanism ofselectivity that relies on two crucial factors, transient trapping of the cargoes insidethe pore and the resulting confinement of the cargoes in the limited space within thechannel. In particular, by modeling the transport as diffusion in an effective potential,the authors of (4, 26, 27, 29, 33, 34, 36, 37) have shown that the attractive interac- elective transport in nano-channels tions of the transported molecules with the channel, such as transient binding of themolecules to binding moieties, increase the transport efficiency. More precisely, with-out an attractive potential inside the channel, the particles entering the channel havea low probability of traversing it to the other side. Attractive interactions inside thechannel slow down the passage and increase the probability of individual molecules totranslocate through the channel (4, 26, 27, 29, 31, 34, 36, 37, 42). This mechanismof transport enhancement has also been known as ’facilitated diffusion’ in the field ofmembrane transport (36, 37, 38, 39).However, space inside the channel is limited, and if the molecules spend too muchtime inside the channel, they prevent entrance of new ones. The channel thus becomesjammed and the transport is diminished. To model the jamming of the channel, theauthors of (27, 30, 33, 34, 35) assumed that additional molecules cannot enter thechannel already when one molecule is present inside. They showed that particleswhose interaction with the channel is weaker than the optimal, have a low probabilityof traversing the channel, while particles that interact too strongly with the channeljam the transport. This allows discrimination between the molecules based on thestrength of their interaction with the channel, and provides a mechanism of selectivetransport; transmission efficiency is optimized for a particular interaction strength andrate of transport. Optimal trapping time, which maximizes the transmitted current,has also been demonstrated for single-file transport in (48).However, during transport, the channel can be occupied by multiple molecules,which cannot bypass each other, or do so only in the limited fashion, due to theconfinement in the limited space inside the channel (4, 45, 47, 48, 49, 50, 51). Thetransport is not necessarily single-file: the number of molecules that can be presentat a position along the channel depends on the ratio of the channel diameter to themolecule size. We must also recognize that the transport properties of narrow channelsare not dominated by the equilibrium thermodynamic channel-cargo interactions perse , but by the rates at which the cargoes enter, translocate through and exit fromthe channel with a potentially complicated geometry (1, 6, 8, 25, 27, 29, 33, 40, 48).For instance, the trapping time in the channel can be limited by the time it takesto find a narrow exit from the channel by diffusion. This phenomenon is known asentropic trapping (25, 40, 49). In the case when the rates are determined solely bythe interaction strength, stronger interactions with the channel imply slower rates andhigher trapping times (cf. Figs. 1 and 2)(4, 26, 30, 34, 35, 42).Understanding the effects of multiple channel occupancy and jamming on the trans-port selectivity is especially pertinent to the analysis of single molecule tracking ex-periments (57, 58, 59) and the design of artificial nano-molecular filters (13, 14, 15,16, 17, 18, 19).In this paper, we analyze transport through narrow channels in a framework ofa general kinetic model based on exclusion process theory as a function of the ki-netic parameters of transport Specifically, we examine the rates of entrance, hoppingthrough and exit from the channel. We extend the previous work to include multipleoccupancy and inter-particle interactions inside the channel beyond single file. Weinvestigate how the concentration of the cargoes, the channel length and radius, thedimensions of the transported molecules and the interactions between them inside thechannel influence the transport. An important goal of this paper is to explore whethera theory that has only two essential ingredients: 1) transient trapping of the moleculesinside the channel and 2) inter-particle crowding due to the confinement in the lim-ited space inside the channel, can provide an adequate explanation of the selective elective transport in nano-channels transport through narrow channels by comparing the predictions of the theory withthe available experimental data.The paper is organized as follows. We first discuss a channel that consists onlyof one ’site’ and then two-sites. Next, we discuss transport in a uniform symmetricchannel of arbitrary length, for both single-file and non-single-file transport, and es-tablish conditions for optimal transport. We then discuss the transition between twotransport regimes, jammed and un-jammed, and establish the relative contribution ofthe jamming of the channel entrance as compared to crowding inside the channel, tothe transport selectivity and efficiency. Next, we compare predictions of the theorywith the experimental data. We conclude with discussion of the results, their relationto the previous work, and consider potential applications. A transport channel can be represented as a chain of positions (sites), as illustratedin Figs. 1 and 2 (4, 27, 29, 35, 42, 47, 48, 49, 50, 51). The particles attempt toenter the channel at a given position, with an average rate J and subsequently hopback and forth between adjacent sites, if those are not fully occupied, until theyeither reach the rightmost or leftmost sites, from where they can hop out of thechannel. Hopping out from the rightmost site represents the particle reaching itsdestination compartment, while hopping out from the leftmost site channel representsan abortive transport event, where the molecule does not reach its destination (cf.Figs. 1 and 2). In the continuum limit, when the distance between the adjacentsites tends towards zero (and their number to infinity), with an appropriate choiceof the transition rates between the sites, the problem can be reduced to diffusion inan effective continuous potential (27, 29, 32, 42) (cf. also Appendix). Note that thediscrete positions (sites) do not represent the actual binding sites inside the channel.Rather, they are a convenient computational tool that allows one to explicitly takeinto account competition for space and interactions between multiple particles insidethe channel (27, 29, 42, 48, 49, 50, 51). The distance between the positions reflectsthe size of the particles.As the particles accumulate in the limited space inside the channel, they start tointerfere with the movement of the neighboring particles and prevent the entrance ofnew ones. We must differentiate between the speed, the efficiency, and the probabilityof transport. The speed is determined by the time the particles spend in the channel.The efficiency of transport is determined by the fraction of the impinging flux thatreaches the rightmost end. It depends on the kinetic parameters of the channel, suchas transition rates inside the channel and the exit rates at its ends. The selectivity oftransport is determined by the different efficiencies at different values of the kineticparameters (26, 27, 29, 30, 31, 33, 48). Transport efficiency is different from the probability that an individual particle translocates through the channel. The latter isdefined as the fraction of the particles that reach the exit after entering the channel.We discuss these issues in detail below. To get started, let us consider a ’one-site’ channel (27, 42), where all the internalspatial and energetic structure of the channel is absorbed into the forward and thebackward exit rates r → , r ← . elective transport in nano-channels Kinetic diagram of such a ’one-site’ channel is shown in Fig. 1 B . The state of thechannel is specified by the particle density (0 ≤ n ≤
1) at the channel site (or, in otherwords, the probability of the channel to be occupied). It obeys the following kineticequation (27, 42): ˙ n = J (1 − n ) − ( r ← + r → ) n (1)which takes into account that the particles can enter the channel only if it is notoccupied. The average time a particle spends inside the channel is τ = 1 / ( r ← + r → )(27, 42).At steady state ( ˙ n = 0) we get for the average density and the forward flux: n = JJ + r ← + r → (2) J out = nr → = Jr → J + r ← + r → = Jr → J + 1 /τ As mentioned above, we define the transport efficiency as the ratio of the transmittedflux to the entering flux, Eff = J out /J . Thus, from the equation (2) we learn thatthe transport efficiency Eff( J, r ← , r → ) = r → J + r ← + r → is a monotonic function of boththe forward exit rate r → and the backward exit rate r ← . Therefore, for the ’one-sitechannel’ there is no optimal combination of the exit rates that would maximize thetransport. As we shall see, this is not the case for longer channels. However, even asingle site channel can have more interesting behavior, if the forward and the backwardexit rates are not independent (27, 33) (cf. Appendix). Let us consider now a longer channel consisting of two sites: 1 and 2. This is theshortest channel that explicitly takes into account the asymmetry between the channelentrance and exit, and exhibits non-trivial transport properties (4, 27, 33, 34, 49). Thekinetics of transport through such a channel is illustrated in Fig. 1 C and D . Theparticles enter the channel at the entrance site 1 with an average flux J , if it is un-occupied. The backward exit rate to the left from site 1 is r ← and the forward exit rateto the right from site 2, is r → . Once inside, a particle can hop back and forth betweensites 1 and 2 with rates r and r , respectively, if the target site is un-occupied. Theexit rates r → and r ← can be thought of as the ’off’ rates for the release of the particlesfrom the channel (27). For simplicity, in this section we assume that the channel isinternally uniform and symmetric with r = r = r and r → = r ← = r o and thateach site can be occupied only by one particle.The state of the channel is characterized by the average occupancies of the sites,0 ≤ n ≤ ≤ n ≤
1. For an internally uniform channel, these averageoccupancies can also be viewed as the probabilities that the sites 1 and 2 are occupiedby a particle (4, 45, 47). The kinetic equations describing transport through such achannel are (Fig. 1 C )˙ n = J (1 − n ) − r o n + rn (1 − n ) − rn (1 − n ) (3)˙ n = rn (1 − n ) − rn (1 − n ) − r o n and the transmitted flux is J out = r o n .The transport efficiency Eff( r o ) = J out /J is the fraction of the flux J that exitsthe channel to the right. Solving equations (3) at steady state ( ˙ n = ˙ n = 0), one gets elective transport in nano-channels for the transport efficiency:Eff( r o ) = J out /J = rr o r o (2 r + r o ) + J ( r + r o ) (4)Importantly, unlike in the one-site case, for a given entrance flux J , the transportefficiency Eff( r o ) has a maximum at a certain value of the exit rate r max o = √ Jr . Thisprovides a mechanism of selectivity; only particles whose residence time in the channel(determined by the interactions of the particles with the channel) is close to 1 /r max o are transmitted efficiently (27, 29, 30, 34, 35, 48).The total efficiency Eff = J out /J is influenced by two different effects, the jam-ming of the channel entrance and the mutual interference between the particles inside the channel. The flux that actually enters the channel is J in = J (1 − n ). The re-maining portion of the flux, Jn , does not enter the channel because the entrancesite 1 is occupied n fraction of the time. The fraction of the entering current J in that reaches the exit on the right determines the transport probability P → = J out /J in ,which characterizes transport through the channel. From the equations (3) P → = r r + r o (5)Very importantly, P → is independent of the flux J and is equal to the efficiency inthe single-particle limit, J →
0. That is, it is equivalent to the probability of a singleparticle to translocate through the channel when no other particles are present. Thus,surprisingly, the crowding of the particles inside the channel does not , on average,influence their movement through the channel. We discuss this effect at length below.To summarize this section, selective transport can arise from a balance betweentwo competing effects, enhancement of the transport by the transient trapping and theeventual jamming of the channel if the trapping times are too high(27, 29, 30, 33, 48).
In this section we study transport through a channel of arbitrary length, which ismodeled as a chain of N positions (sites):
1, 2...i...N . Particles enter at site M (notnecessarily the leftmost one) with an average flux J if the entrance site is not fullyoccupied. Once inside the channel, a particle at site i can hop to an adjacent site i ± n m , which depends on the ratio ofthe channel diameter to the size of the particles. When at an outermost site 1 or N ,a particle can leave the channel with the rate r ← and r → , respectively, or hop intothe channel with the average rate r → or r N → N − , respectively. The kinetics of thisprocess is illustrated in Fig. 2 A . The rates r i → i ± determine the speed with whichthe particles diffuse through the channel, while the exit rates r ← and r → reflect howfast the particles can leave the channel. The kinetic rates r i → i ± , r ← and r → aredetermined by the microscopic interactions of the particles with the channel, and byits geometry. As before, the exit rates r → and r ← can be thought of as the ’off’ ratesfor the release of the particles from the channel (27). In general, with a proper choiceof the transition rates, r i → i ± /r i → i ± = exp( U i +1 − U i ) /
2) in the continuum limitthe model reduces to diffusion in the potential U ( x ) (27, 32, 41, 42). Trapping of theparticles in the channel corresponds to low exit rates r → , r ← < r (27, 33, 42, 43). elective transport in nano-channels For simplicity, we assume that the channel is internally uniform, such that allthe internal transition rates are equal, r i → i ± = r for all i . At any time t , thestate of the channel is specified by the number densities of the particles at each site n , n , ..., n i , ...n N . The kinetics of transport through such a channel is described bythe following equations (4, 42, 47, 48, 49, 50)˙ n i = Jδ i,M (1 − n i n m ) + rn i − (1 − n i n m ) + rn i +1 (1 − n i n m ) − rn i (1 − n i − n m ) − rn i (1 − n i +1 n m )= Jδ i,M (1 − n i n m ) + r ( n i − + n i +1 − n i ) (6)with the boundary conditions at sites 1 and N ˙ n = Jδ ,M (1 − n n m ) − r ← n − rn (1 − n n m ) + rn (1 − n n m )= Jδ ,M (1 − n n m ) − ( r + r ← ) n + rn (7)˙ n N = − r → n N − rn N (1 − n N − n m ) + rn N − (1 − n N n m ) = − ( r + r → ) n N + rn N − where the δ -function is δ i,j = 1 if i = j and zero otherwise. The terms n i (1 − n i ± /n m ) in equations (6) and (7) reflect the fact that a particle can jump to thenext site only if it is not fully occupied, n i ± < n m . Importantly, for an internallyuniform channel, at all the internal sites the cross-terms of the form n i n i ± cancel out(47, 48). For such uniform channels, the equations (6) and (7) are exact and n i /n m is equivalent to the probability of a site i to be occupied (47, 48). Obstruction of thespace inside the channel by the particles inside of it affects only the entrance to thechannel at site M .We define the efficiency of transport as the ratio of the forward exit current J out = r → n N to the incoming flux J , Eff( r o ) = J out /J . It is the fraction of the incoming fluxthat traverses the channel. Note that the efficiency is different from the probabilityof individual particles to traverse the channel after they have entered, because someof the particles attempt to enter the channel and are rejected if the entrance site isoccupied.The linear equations (6),(7) can be solved analytically for any N (48). In the fullysymmetric case, when the forward and the backward exit rates from sites 1 and N areequal, r ← = r → = r o , the efficiency is given by:Eff( r o ) = ( r + ( M − r o ) r o r o (2 r + ( N − r o ) + Jn m ( r + ( N − r o + ( M − N − M ) r o /r ) (8)(for M < N/ J →
0, the efficiency Eff → M/ ( N +1) without trapping ( r o → r ) and Eff → / r o → τ = N r o (27, 43, 52) to arrive atEff( r o ) = (2 τ r + ( M − N ) N ( N − N + 4 τ r + J ′ n m ( N ( τ r ) + ( N − τ r + ( M − N − M )) (9) elective transport in nano-channels where J ′ = J/r is the normalized flux. Note that the transport efficiency does notdepend on the absolute values of the transport rates r and r o , but only on the nor-malized parameters τ r and J/r . This means that the transport efficiency can be thesame for different particles, even if the kinetics of their transport through the channelis very different from each other, as long as they possess the same τ r and
J/r .As already seen in the two-site case, the transport efficiency Eff( r o ) of equation(8) has a maximum at a certain value of the exit rate (for M = 1) of r max o /r = r J/ ( rn m ) N − J max = J (( N − r max o /r + 1) + 1 (11)(cf. Appendix for M = 1).This feature provides a mechanism of selectivity; only the particles whose exit rateis close to the optimal one, r max o , are transmitted efficiently. Particles with exit rateshigher than the optimal have a higher chance of returning back because they don’tspend enough time inside the channel in order to reach the farther exit on the rightside. On the other hand, due to the limited space inside the channel, the particleswith the exit rates lower than optimal spend so much time in the channel that it getsjammed and the entrance of new particles is inhibited (4, 29, 30, 33, 34, 35).Equations (8) and (10) qualitatively agree with the results of (30, 33, 34, 35),which assumed that only one molecule can occupy the channel. Figure 3 shows howthe transport depends on the channel length N , the entrance flux J , the exit rate r o and the effective channel width n m . Note that the optimal exit rate r o decreases withthe channel length N ; for longer channels, a particle has to spend more time in thechannel in order to reach the other end. Also note that the optimal rate of equation(10), r max o /r , is less than one for J/r < N −
1; the optimal interaction is attractivefor small currents and long channels. We elaborate on this issue in Appendix.
In this section, we elaborate on why the flux through the channel decreases in thelimit of very low exit rates? Is this because new particles cannot enter or because theparticles inside the channel interfere with each other’s passage?The fraction of the incoming flux J that actually enters the channel is J in = J (1 − n /n m ). The remaining portion of the flux J n n m cannot enter because the entrancesite is occupied on average n n m fraction of the time (cf. Sec. 4 for calculation of thedensities). The total efficiency is determined by two quantities: i) the fraction of theflux that enters the channel J in and ii) the fraction of the particles that upon enteringthe channel, actually reach the rightmost end. The latter defines the probability P → = J out /J in of a particle exiting to the right after it has entered the channel and is givenby P → = J out J in = r + ( M − r o r + ( N − r o (12)Remarkably, it is independent of the flux J and is exactly equal to the efficiency inthe single particle transport limit, J →
0. This means that in unform channels the elective transport in nano-channels interactions between the particles in the channel do not affect the transport probabil-ities of the individual particles. The effect of the channel occupancy manifests only inthe jamming at the entrance site. The lower the exit rate r o , the longer the time that the particles spend inside thechannel. The trapping time varies as τ = N r o (27, 43, 52). As shown in the previoussection, at very small exit rates r o , the trapping time is so high that the channelbecomes jammed. Thus, the transport efficiency is maximized at the particular exitrate r max o . Inspection of the Fig. 3 A reveals two distinct transport regimes, roughlyseparated by the maximum of the transport efficiency at r o = r max o . At the highvalues of r o > r max o the transport of individual particles is essentially unhinderedby the presence of the others, as evidenced by the fact that the transport efficiencycurves collapse onto the dashed line, representing the zero-current, single-particle limit(Fig. 3 A ). At the low values of the exit rate where r o < r max o , the accumulatingparticles start to obstruct the entrance of the new ones. This feature provides anatural definition for the ’jamming transition’ around the r o = r max o .Solving equations (6) and (7), we get for the density profile of the particles insidethe channel, at the steady state: n i = J ( r + ( N − i ) r o ) r o (2 r + ( N − r o ) + J ( r + ( N − r o ) (13)(for M = 1, n m = 1). Note that unlike the equilibrium distribution, the maximum ofthe density profile is near the channel entrance at site 1.The total number of the particles in the channel is N tot = N X i =1 n i = N J (2 r + ( N − r o ) r o (2 r + ( N − r o ) + J ( r + ( N − r o ) (14)Note that in the limit r o → N tot → N . That is, the particles accumulate and neverleave the channel. Therefore, from equation (14) one finds that at the point of thejamming transition, r o = r max o , the number of the particles in the channel is N jam = N J/r + 2 q J/rN − (cid:18) J/r + 2( q J/rN − + N − ) (cid:19) (15)Equation (15) has important consequences (cf. Fig. 4 for illustration). It shows thatfor long channels, where N ≫ J/r , the fraction of the occupied sites at the jammingtransition tends to one half: N jam /N → / N ≫ J/r . This means thatlong channels can be filled up almost to half of their maximal capacity N before thejamming effects start to matter. For the occupancies below the jamming transition,the particles travel through the channel essentially unhindered. These effects areillustrated in Fig. 3 A and Fig. 4 A . This might explain why experiments on transportthrough narrow channels often measure apparent diffusion coefficients that are almostas large as those for the free diffusion (53, 55). elective transport in nano-channels Although the transport efficiency Eff( r o , J ) decreases with the increasing flux J , thetotal transmitted flux J out = J Eff( r o , J ) saturates at large fluxes ( J/r → ∞ ) to thelimiting value J ∞ out /r = n m r o /r N − r o /r (16)(for M = 1). This saturation of the transmitted flux at large incoming flux J isanother manifestation of the jamming of the channel entrance by the particles inside.Indeed, equation (13) shows that the density at the entrance n tends to n = 1, as J/r → ∞ . In other words, the flux saturates because no more particles can enter thechannel. This is neatly summarized by the observation that n = J out /J ∞ out .By contrast, the exit site N is not completely blocked even at high J and n N → / (1 + ( N − r o /r ) as J → ∞ . Thus, even at very large fluxes, when the entrancesite is completely blocked, the channel is not fully occupied. From equation (15), thenumber of particles in the channel is N ∞ tot = N r + ( N − r o r + ( N − r o In particular, for long channels ( N − ≫ r/r o ), the channel occupancy in the saturatedlimit is N tot /N = 1 /
2. Also note that the saturated flux is proportional to n m , andthat it decreases with r o /r .The results of this section closely parallel Michaelis-Maenten kinetics of multi-stepenzymatic reactions (46) and are important for the estimation of binding affinitiesfrom the channel transport experiments (53, 54, 56), as well as for comparison withexperiments on flux through artificial nano-channels ( - Sec 4). 4. In experimental systems, the exit rates and the rates of transport through the channelare determined by a potentially complicated kinetics of binding and unbinding in-side the channel. Can the theory adequately describe these experiments? Facilitateddiffusion theories produced results consistent with the experimental observations ofthe transport of gases through functionalized membranes (36, 37, 39) enhancement oftransport of oxygen by myoglobin (38) and the transport through bacterial porins(4).In this section, we compare the theoretical predictions of this paper with the experi-ments of Ref. (16).Briefly, in the experiments of Ref.(16) that we chose for comparison with theo-retical predictions, transport of short DNA segments through artificial nano-channelswas studied. The flux of the DNA segments through these channels was measured intwo cases: 1) empty channels and 2) channels were lined with single-stranded DNAhairpins, grafted to the walls. Each hairpin has a stretch of 18 un-paired bases inthe middle. The transported particles were 18 base ssDNA segments with the se-quence complementary to the un-paired regions of the ssDNA hairpins inside thechannels. Thus, the transported DNA segments can transiently hybridize with theDNA grafted inside the channel. That investigation found that the flux through theDNA-containing channels is higher than through the channels without DNA hair-pins inside, providing evidence that the transient trapping indeed facilitates transportthrough nano-channels. However, eventually the interactions between the particles in elective transport in nano-channels the limited space inside the nano-channel block the passage, causing the transmittedflux to saturate with the increase in the incoming flux. This is another signature ofthe transient trapping discussed above in Sec. 3 (4, 14, 15, 16).The radius of an empty channel is R ≃ L = 6 µ (16).The grafted ssDNA hairpins reduce the passageway radius, which, for the purposesof comparison with the theory, we roughly estimate as R ≃ S ≃ n m = 6 for the empty channels and n m = 3 for the channels with the grafted DNA hairpins inside. Furthermore, weestimate the incoming flux as 4 D out cR (41), where D out = k B T πηS H is the diffusioncoefficient of the transported DNA coils outside the channel (60); η is the viscosity ofthe solvent, S H ≃ . S is the hydrodynamic radius of the coils (60) and c is the outsideconcentration of the transported DNA (41, 42). To model the finite capacity of thechannel, we estimate the number of available positions in the channel as N = L/ (2 S ),where L = 6 µ (4, 48, 49, 51)(cf. also Appendix).Finally, r o /r = π D out D in LNR Z , where Z is the reduction in the exit rate due to thetrapping inside the channel (27)(cf. also Appendix). We return to the question ofhow Z is related to the actual binding energy below. The ratio D out /D in and Z arethe two independent fitting parameters of the model (note that the r and r o appearas independent parameters in eq.(8).We first tested the model for the case without DNA segments attached insidethe channel. The data (black dots) and the fit (black line) with D in /D out = 0 .
42 and Z = 1 are shown in black dots in Fig. 5 A . Analogously, for the channels with the DNAhairpins inside, the fit of the equation (8) to the data (red dots) is shown in the redline in Fig. 5 A , with the best fitting parameters D in /D out = 0 .
004 and Z = 0 . r o by a factor Z = 0 . ǫ , then the function Z ∼ exp( − ǫ/kT ) shoulddescribe the trend in the dependence of Z on ǫ (4, 27, 35, 42, 48). The authorsof (16) measured fluxes through the channel for DNA segments possessing differentnumbers of mismatches to the DNA grafted inside and found that the flux decreaseswith the number of mismatches. Thus, assuming as a first approximation that thebinding energy ǫ decreases linearly with the number of mismatches n , so that ǫ ( n ) = ǫ n =0 (18 − n ) /
18 we get Z = exp(ln(0 . n − / Z is compared with the data in Fig. 5 B . It shows that thissimple estimate correctly reproduces the trend in reduction of the transmitted fluxwith the number of mismatches. Note that there are no additional fitting parametersused in this figure.That the simplified theory developed in this paper can correctly reproduce thetrends in the observed fluxes, and even gives semi-quantitative fit of the data forreasonable values of the parameters, is encouraging. This demonstrates that a theorythat is built upon only two essential assumptions, 1) facilitation of diffusion by thetransient trapping inside the channel and 2) mutual interference between the particlescrowded in the confined space inside the channel, does provide an adequate explanation elective transport in nano-channels of the experimental data. Moreover, the theory provides verifiable predictions abouthow the flux should change with the channel diameter and length, as well as the particlesize and concentration, as described in section 3. Comparison of these theoreticalpredictions with future quantitative experiments will lead to further ramification of thetheoretical approach and will facilitate the design of artificial selective nano-channelswith desired properties.Discussion of other effects observed in (16), which are attributable to a confor-mational transition of the hairpin layer during transport, is outside the scope of thepresent work. Proper functioning of living cells requires constant transport of different molecularsignals into and out of the cell, as well as between different cell compartments. Tocarry out this task, the living cells have evolved various mechanisms for efficient andselective transport.One class of transport devices comprises narrow channels whose diameter is com-parable to the size of the molecules transported through it. Examples include selectivetransport through the nuclear pore complex, bacterial porins, (1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11, 12) and other non-biological transport systems such as zeolites (25, 40, 49). Acrucial feature of such transport channels is their ability to selectively transport theirspecific signalling molecules while efficiently blocking the passage of all others.Driven by the notion that natural evolution has optimized the function of suchdevices, large effort is being currently invested into the creation of artificial nano-molecular sorting devices that mimic the function of biological channels (13, 14, 15,16, 17, 18, 19, 55). The design of such devices requires detailed understanding of theprinciples of selective transport through narrow channels.The precise conditions for the optimal transport selectivity through narrow chan-nels still elude our understanding. A large body of experimental work indicatesthat the selectivity is often based on the differential interactions of the transportedmolecules with their corresponding transport channels. Moreover, interaction of thetransported molecules with their corresponding transport channels is strong, exceed-ing the interaction of the non-specific competitors. Another salient feature of suchchannels is that they are narrow, so that the particles cannot freely bypass each other(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16). .Recent theoretical works have shown that selective transport through narrow chan-nels can arise from a balance between efficiency and speed; transient trapping insidethe channel increases the probability of a molecule to pass through the channel, butleads to jamming at too high trapping times (4, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 48).Extending previous work, in this paper we have analyzed transport through narrowchannels in the framework of generalized kinetic theory. We represent a transportchannel as a sequence of positions (sites) and the transport through the channel isdetermined by the hopping rates from one position to another inside the channel, aswell as by the exit (’off’) rates from the channel at its ends. To take into accountthe limited space inside the channel, and the finite size of the transported particles,we allow only a limited occupancy at each position, n m . Thus, a particle presentat a given position along the channel, can hop to an adjacent position only if thelatter is occupied by less than n m particles. Our model allows one to naturally treatchannel occupancy by multiple particles and extends the treatment beyond the single- elective transport in nano-channels file transport. The main determinant of the transport properties of the channel isnot the interaction strength of the particles with the channel per se , but the kineticproperties of the channel, which determine the trapping time τ and also depend onthe geometrical properties of diffusion in the confined space inside the channel. Thesepossibilities are illustrated in the Fig. 2.We briefly summarize our major findings below. In qualitative agreement withprevious work, we find that the transient trapping of the particles in the channelincreases the transport probability; particles that have high exit rates do not stay in thechannel long enough to reach the exit into the destination compartment on the rightside (cf. Fig. 2), and have a high probability to return back (26, 27, 29, 31, 36, 37, 42).Essentially, transient trapping increases the time that the particles spend inside thechannel to be long enough in order to reach the exit on the right side. Thus, althougheach individual particle spends more time in the channel, the transmitted flux ishigher. If the only measured quantity is the flux through the channel, experimentscan not easily distinguish between the probability of transport and transport speed.(3, 53, 55, 56). However, they can be distinguished in the experiments that followtransport of individual molecules (57, 58, 59).When the exit rate is too slow or the incoming flux is too high, the rate of particles’entrance to the channel becomes higher than the rate of exit and the particles startto accumulate inside the channel, because the space inside is limited. This leads totwo distinct effects. First, the particles inside the channel start to interfere with thepassage of each other. Second, they block the entrance site and inhibit the entranceof new particles. The channel thus becomes jammed. We must distinguish betweentranslocation probability and transport efficiency . Transport efficiency is the fractionof the total incoming flux that reaches the exit. Only a certain fraction of the in-coming flux can enter the channel because the entrance site can be occupied whenparticles attempt to enter. This effect decreases the capability to enter the channeland, as a consequence, decreases the transport efficiency. Interactions between theparticles inside the channel can also influence the probability of individual particles totranslocate through the channel upon entering compared to the single particle case.However, we found that for internally uniform channels the crowding of the particlesinside the channel does not affect the probability of individual particles to translocatethrough the pore. Thus, the effect of particle accumulation in the channel manifestsonly in the blocking of the entrance to the channel, which leads to the decrease in thetotal transport efficiency (and transmitted flux) at low exit rate or high incoming flux(Fig 4 B ). Thus, we predict that the kinetic profile near the entrance is an importantfactor in determining the selectivity of transport.For symmetric channels, this balance between the transport probability and theobstruction of the particle entrance to the channel, determines the optimal exit rate r max o , (cf. Fig. 4) which maximizes the transport. This provides a basis for selectivity,whereby different molecules can be selected by the kinetics of their transport throughthe channel (27, 29, 30, 33, 34, 35, 48). In the case discussed in this paper, when manyparticles can be present in the channel simultaneously, the optimal exit rate and theoptimal flux depend on the length of the channel (cf. Section 3). Notably, this is apurely kinetic selectivity mechanism: although a low exit rate can be due to energeticinteractions between the transported particles and the channel; the transport efficiencyis not determined by the equilibrium occupancy considerations; the selectivity can gobeyond the difference in the equilibrium binding affinities between different molecules.The fact that the transport efficiency has a maximum at a certain value of the exit elective transport in nano-channels rate r o = r max o provides a natural definition for the ’jamming transition’. Particleswith the exit rates faster than r max o pass through the channel essentially unhinderedby the interactions with other particles because they do not stay in the channel longenough(Fig. 3 A .) On the other hand, particles with exit rates slower than r max compete with each other for entrance into the limited space inside the channel andthe channel becomes jammed. Importantly, we found that the interactions betweenthe particles, and the competition for the limited space inside the channel do notplay an important role until quite a few of them accumulate in the channel. Forlong channels, approximately half of the available channel sites are occupied at thejamming transition ( Fig. 4). This implies that in many experimental situations theinteractions between the transported particles do not play a significant role, and mayexplain why the apparent diffusion coefficient in many flux measurement experimentsis found to be almost as high as for free diffusion (8, 53, 55, 58).Although many particles can be crowded inside the channel, and the entrance tothe channel is blocked, transmitted flux does not disappear even at high fluxes anddensities, but rather saturates to the limiting value determined by the trapping timeand the channel length (cf. Fig. 3 and Fig. 5.) This closely parallels Michaelis-Maenten kinetics of enzymatic reactions (1, 46) and might be relevant to estimationof binding strengths from flux experiments (53, 54, 56). Notably,the selective andefficient transport persists beyond the single file transport, even when the ratio of thechannel diameter to the particle size is large. In this case, the optimal exit rate r max o is simply shifted to lower values.In order to determine whether the theory developed in this paper can providean adequate description of experiments, we compared predictions of the theory tothe experiments reported in (16). That work found that at low concentrations ofthe transported particles the flux through artificial nano-channels increases if theparticles can transiently bind inside the channel. Moreover, as the binding energy ofthe particles was decreased, the enhancement of the flux was lower. However, as theconcentration of the particles in the origin compartment increases, the flux saturatesfor the channels with transient binding, while the saturation if not observed for non-binding channels, at the experimental range of concentrations. Both these resultsare in agreement with the theory and can be semi-quantitatively described by thetheoretical predictions, as shown in section 4.Thus, we find that the theory based on only two main ingredients: 1) transienttrapping of the molecules inside the channel and 2) crowding of the molecules inthe limited space inside the channel, captures the essential features of the selectivetransport through nano-channels. Moreover, the theory provides verifiable predictionsregarding how the flux and selectivity of such channels depend on the channel length,channel radius, the size of the transported molecules and the strength of the interac-tions of the molecules with the channel. In particular, we predict that the flux throughsuch channels can be optimized by varying the interaction parameters and the chan-nel dimensions. Such predictions are useful for the design of artificial nano-sortingdevices. Further quantitative experiments and comparison with the theory are neededin order to test the theory and for its further refinement.We expect that the effects described in this paper should play a role in selectivetransport through any narrow channel. For instance, the effects described in thispaper might be relevant in determining the selectivity of the ion channels, althoughother factors might be dominant (21, 22, 23, 24). In each particular system othereffects related to molecular details might be dominant determinants of selectivity. elective transport in nano-channels Such effects might include the long range electrostatics and channel fluctuations inthe ion channels, the details of the transfer of the transported molecules from onebinding moiety to another, and conformational changes of the filaments that carry thebinding moieties (as in the nuclear pore complex and other polymer-based systems).Finally, we note that the theory developed in this paper can also be applied toother signal-transducing schemes, such as signalling cascades and multi-step enzymaticreactions (46, 61, 62, 63).The author is thankful to C. Connaughton, B. Chait, I. Nemenman, J. Pearson, A.Perelson, Y. Rabin, K. Rasmussen, M. Rout, N. Sinitsyn, T. Talisman, Z. Schuss forstimulating discussions, P. Welch for comments on the manuscript, and anonymousreviewers for helpful suggestions. This research was performed under the auspices ofthe U.S. Department of Energy under contract DE-AC52-06NA25396.
Appendix
Single particle occupancy: connection to previous work
In this section we show that the model of this paper can be reduced to previousmodels, in a proper limit. Let us assume, following (27, 30, 33, 34, 35) that alreadywhen the channel is occupied only by one particle, it prevents the entrance of others.The channel, however, is long, and the particle can obey complicated kinetics inside,which determines its probability to traverse the channel, and the time it spends inside.Physically, such situation can arise, for instance, due to strong long-range repulsionbetween the particles.In this case, the problem reduces to a ’single-site’ channel of Sec. 2.1 but withforward exit rate r → , backward exit rate r ← that are not independent, but are de-termined by the internal kinetics of the channel, and are related through the single-particle dwelling time τ and transport probability P tr . As in Sec. 2.1, the transmittedflux is J out = Jr → J + r → + r ← (17)From equation (8), the probability of a single particle to traverse the channel oflength N (for J →
0) is P tr = 1 / (2 + ( N − N/ ( τ r )) = r → / ( r → + r ← ), and theresidence time is τ = N/ (2 r o ) = 1 / ( r → + r ← ) (52). Thus, we get for the transmittedflux: J out = J Jτ ) (cid:16) ( N − N τr (cid:17) (18)which is identical to expressions obtained in Ref.(27), if one bears in mind that theflux is J = k on c , where c is the concentration of the particles outside the channel.In is important to note that the optimal exit rate in this case is r max o = q JrNN − ,that is almost independent of N for long channels. This is in contrast to the modelof Sec. 2.4, which takes into account multiple occupancy of the channel by manyparticles - where the optimal exit rate decreases with N . The optimal current is, bycontrast, higher for multiple-occupance channels. This is natural - if more particles arecan occupy the channel before it becomes jammed, the channel can sustain a highercurrent. elective transport in nano-channels Connection between continuum and discrete models.
Discrete model of equation (6) reduces to a continuum description of transport insidethe channel, if one defines the one-dimensional particle density c ( x ) = n i /a where a is the distance between the ’sites’, so that x = ai , with a diffusion coefficient D in = a r (27, 41, 42). For comparison with real systems, one-dimensional diffusioninside the channel must be matched to the three-dimensional diffusion outside thechannel, through the choice of r o (see e.g.(27, 28, 41, 55)). For clarity, we re-derivethis connection here without the inter-particle interactions inside the channel - seeFig. 6 for illustration.We denote the three-dimensional concentration of particles at the left side far awayfrom the channel as c ∞ L ; we assume that concentration on the right side far away fromthe channel is zero. At steady state, a density profile will be established such thatthe flux through the pore is F , the (three-dimensional) density at the pore entranceon the left is c L and the density at the exit on the right is c R ; the correspondingone-dimensional densities are c L = c L βR , and c R = c R βR , where R is the channelradius, and β is a geometrical pre-factor that depends on the shape of the channelopening ( β = π for circular opening).At steady state, the flux that enters the channel from the left is (41): J = α ( c ∞ L − c L ) RD out = F (19)where α is a geometrical pre-factor that depends on the shape of the channel opening; α = 4 for a circular opening(41). Note that if all the impinging particles would gothrough the channel, the entering flux would be J = αc ∞ L RD out - the flux to a fullyabsorbing patch of radius R (41). However, even in the absence of jamming, not allparticles go through - some of them return back, after hopping back and forth insidethe channel, as reflected in the returned portion of the flux − αc L RD out (41).The flux that exits the channel to the right is (27, 41): J out = αc R RD out = F (20)The flux inside the channel, for a flat potential profile, is: (26, 28, 29, 42) F = c L − c R LZ D in (21)where Z = h e E i is the average inverse Boltzmann factor of the attractive energy insidethe channel, E <
0. Solving the above equations, we get: F = J αβ LR D out D in Z (22)And thus the fraction of the transmitted flux is P tr = 12 + αβ LR D out D in Z (23)On the other hand, equation (8) gives without jamming ( J → P tr = 12 + ( N − r o /r = 12 + La r o r (24) elective transport in nano-channels Finally, choosing r o /r = J out Z/n N a /D in = αβ D o D in aR Z , the discrete and continuous formu-lations become equivalent as long as N = L/a ≫ and overall conclusions are not sensitive to small variations in theestimates of the parameters of the model (data not shown). Expressions for M = 1 For completeness, we present here the expressions for the general case 1 ≤ M < N/ n m = 1 The optimal exit rate (for the values of J, M and N when the optimum exists): r max o /r = J/r (1 − M ) − p J/r ( − M + N + 1) J/r ( M − + (2 M − N −
1) (25)The channel occupancy
J/r (2 + ( N − r o /r )( N + ( M − N − M ) r o /r )2( r o /r (2 + ( N − r o /r ) + J/r (1 + ( M − r o /r )(1 + ( N − M ) r o /r )) (26)and the saturation current in the J/r → ∞ limit: r o (1 + ( M − r o /r )( M − N − M )( r o /r ) + ( N − r o /r + 1 (27) References
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Biopolymers elective transport in nano-channels Figure Legends
Figure 1.
Schematic diagram of transport through a channel A.
Schematic illustrationof the transport through a narrow channel. B. Kinetic diagram of a one-site channel. C. Kinetic diagram of a two-site channel.
Figure 2.
Kinetic diagrams of transport through a channel of an arbitrary length A.
Symmetric channel consisting of N positions (sites). The particles enter the channel ata site M with an average rate J . B. Equivalent energetic diagram in the case when theexit rates are determined by the interaction (binding energy) with the channel. Theexit rates at the channel ends are given by Arrhenius-Boltzmann factors of the energybarriers at the exits, E → and E ← : r → ∼ exp( − E → /kT ) and r ← ∼ exp( − E ← /kT ) C. Equivalent geometry of the channel in the case when the exit rates are due to spatialbottlenecks at the channel ends.
Figure 3.
Efficiency of transport through a channel of an arbitrary length A.
Transportefficiency as a function of the exit rate for
J/r = 0 . n m = 1 for different entrancesites M . Black line: M = 1 , N = 10, gray line: M = 4 , N = 40; corresponding dashedlines show the probability of a particle to traverse the channel; it is identical to a singleparticle transport efficiency in the limit J → B. Transport efficiency as afunction of channel length N , for the optimal value of exit rate r o = ( Jr/ ( N − / , M = 1, J/r = 0 . n m = 1. Black line: J/r = 0 .
01, dashed line
J/r = 0 . C. Transmitted flux J out /J ∞ out - cf. equations (8) and (16), as a function of the normalizedincoming flux J/r ; black line: r o /r = 0 .
01, dashed line r o /r = 1; M = 1; n m = 1. Notethat the transmitted flux saturates to a constant value J ∞ out in the jammed regime. D. Optimal exit rate as a function of the channel length N for M = 1, J/r = 0 . n m = 1, (black line). Dashed line: same for J/r = 0 . Figure 4.
Occupancy of the channel at the jamming transition A.
Occupied fractionof the channel at the jamming transition, r o = r max , as a function of the channellength N , for different values of the incoming flux J/r . It shows that the channel canbe occupied to a considerable degree - up to half of the available sites - before thejamming becomes significant. B. Densities at the entrance site 1 (black line) and exitsite N (dashed line) as a function of the incoming flux J/r for r o /r = 0 . N = 5, n m = 1. Density at the entrance site saturates to 1, which causes the saturation ofthe transmitted flux. Density at the exit site stays low even in the regime when thetransmitted flux through the pore saturates. Figure 5.
Flux through nano-channels: comparison with experiment A.
Flux throughthe nano-channel as a function of the outside concentration of the transported ssDNA.Black dots - experimental data from Ref.(16) for a nano-channel without trapping elective transport in nano-channels inside. Corresponding black line - theoretical fit from eq. (8) with n m = 6, Z=1, D in /D out = 0 . N = L/ (2 S ). Red dots - experimental data from Ref.(16) for a nano-channel with ssDNA hairpins grafted inside the channel, which are complementary tothe transported ssDNA. Corresponding red line is the theoretical prediction of eq.(8) with n m = 3, D in /D out = 0 . Z = 0 . N = L/ (2 S ) B. Reduction ofthe flux through the channel as a function of the number of mismatches betweentransported ssDNA and the ssDNA hairpins grafted inside, relative to the flux of theperfect complement ssDNA measured at the feed ssDNA concentration 9 µ M. Dots -experimental data from Ref.(16) for a single mismatch at the edge of the transportedDNA segment; square - single mismatch in the middle of the transported ssDNAsegment; line - theoretical model; same parameter values as used in panel A - cf. text. Figure 6.
Three-dimensional diffusion outside the channel
Schematic illustration of thethree-dimensional diffusion outside the channel and one-dimensional diffusion inside.See text in Appendix. elective transport in nano-channels Jr r rrJ rr
One site Two sites J r r Schematic
AB C
Kinetic diagrams
Figure 1: elective transport in nano-channels i-1 Ni+1 i 2 N-1 rr rr rr rr J A Channel is represented by N positions (sites) r B Equivalent potential C Equivalent geometry r E E r ~ exp( / kT) E r ~ exp( / kT) E Figure 2: elective transport in nano-channels J/r O p t i m a l ex i t r a t e T r a n s po r t e ff i c i e n cy T r a n s m i tt e d f l u x Incoming fluxExit rate r o Channel length N O p t i m a l t r a n s po r t e ff i c i e n cy A BC D
10 20 30 40 500.10.20.30. 4 10 20 30 40 50
Channel length N/r o r m ax Figure 3: elective transport in nano-channels C h a nn e l l e ng t h N O cc up i e d f r ac t i on I n c o m i ng f l u x J / r D e n s i t y a t t h e e n t r a n ce s i t e D e n s i t y a t t h e ex i t s i t e J / r =0 . J / r =0 . J / r =1
20 40 60 80 1000.20.30.40.50.1
Figure 4: elective transport in nano-channels Feed concentration of the transported ssDNA [ µ M] E x i t f l u x [ m o l e c u l e s / s e c / po r e ] Number of mismatches R e du c t i on i n t h e f l u x Figure 5: R J = α D (c -c ) out LL c L J = α D c out out R c L c R L