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PLoS One. 2010; 5(12): e15160.
Published online 2010 December 13. doi:  10.1371/journal.pone.0015160
PMCID: PMC3001458

Thermodynamics of Competitive Molecular Channel Transport: Application to Artificial Nuclear Pores

Laurent Kreplak, Editor

Abstract

In an analytical model channel transport is analyzed as a function of key parameters, determining efficiency and selectivity of particle transport in a competitive molecular environment. These key parameters are the concentration of particles, solvent-channel exchange dynamics, as well as particle-in-channel- and interparticle interaction. These parameters are explicitly related to translocation dynamics and channel occupation probability. Slowing down the exchange dynamics at the channel ends, or elevating the particle concentration reduces the in-channel binding strength necessary to maintain maximum transport. Optimized in-channel interaction may even shift from binding to repulsion. A simple equation gives the interrelation of access dynamics and concentration at this transition point. The model is readily transferred to competitive transport of different species, each of them having their individual in-channel affinity. Combinations of channel affinities are determined which differentially favor selectivity of certain species on the cost of others. Selectivity for a species increases if its in-channel binding enhances the species' translocation probablity when compared to that of the other species. Selectivity increases particularly for a wide binding site, long channels, and fast access dynamics. Recent experiments on competitive transport of in-channel binding and inert molecules through artificial nuclear pores serve as a paradigm for our model. It explains qualitatively and quantitatively how binding molecules are favored for transport at the cost of the transport of inert molecules.

Introduction

Understanding of molecular or particle transport through channels and pores is of paramount interest in many field, ranging from nanotechnology to life sciences [1][5]. In addition, such channel transport also serves as a paradigm for general linear transport processes like enzymatic catalysis with a 1-D reaction coordinate [6]. Optimal function of a channel in either a technical or biological setting often requires a high transport rate, which demands an adjustment of particle-channel- and interparticle interaction as well as particle concentration in the baths adjacent to the channel ends. The flow-facilitating role of a in-channel particle trapping, either by a binding site or an entropy trap, which prolongs the residence time and by this the translocation probability, has been recognized early [7][9]. When particles interact, however, this trapping hampers flow as it impedes access of other particles from the baths to the channel. This implies the existence of an optimum binding strength providing maximum flow [5], [10], [11], which depends on particle concentration, width and location, e.g. asymmetry, of the binding site [5], [12].

Despite of this previous work, many issues remain to be solved. How are the particle in-channel and interparticle interaction related to the occupation probability, i.e. a parameter observable in experiments? What is the exact mechanism responsible for an asymmetric binding site to favor transport selectively when located near the exit the flow is directed to? In which way is the optimum binding strength related to exchange dynamics at the channel ends? May also repulsive particle-channel interaction be favorable for transport? Which parameters determine the transition from a flow-facilitating binding site to a flow-facilitating repulsive interaction, and what is the mechanism behind? In a typical environment particles also compete with particles of other species for channel transport, each of them having their individual characteristics as in-channel affinity. The question arises how interspecies competition affects flow and how selectivity may be achieved e.g. by appropriate choice of in-channel interactions.

In this paper we will derive particle flow as a function of exchange dynamics and energetics at the channel ends, in-channel affinity, and interparticle interactions for single- and multi-species transport. The theory relates in-channel interaction directly to occupation probabilities of channel states, i.e. parameters accessible by experiments. A simple relation between exchange dynamics at the channel ends and particle concentration predicts whether a binding site or a repulsive force inside the channel facilitates transport. For the case that different species, each of them having its specific interaction profile, compete for channel transport, we analyze the influence of these interactions on flow of each species. Results are compared with recent experimental data [4] on transport through nuclear pores. Our model explains qualitatively and quantitatively the efficiency and selectivity of this transport process, which is of supreme importance e.g. for regulation of genomic activity.

Methods

The model

We consider particle transport through a channel connecting two baths, labeled as (A) and (B), with respective particle concentrations An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e001.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e002.jpg. Particle motion in the channel is described as a 1-D diffusion process, and the dynamics of particle density An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e003.jpg is given by the Smoluchowski Equation [13], [14],

equation image
(1)

where An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e005.jpg is the channel coordinate, giving the position of the molecule related to the channel, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e006.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e007.jpg is the local diffusion coefficient, which is assumed to be constant, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e008.jpg. Particle-channel interaction is quantified by the force An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e009.jpg that can always be derived from a potential in the 1-D case, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e010.jpg. All energetic quantities are given in multiples of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e011.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e012.jpg the Boltzmann constant.

The exchange rates of particles, entering or leaving either channel end are An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e013.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e014.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e015.jpg (Fig. 1). So the full transport process is described by the reaction-diffusion schematic

equation image
(2)
Figure 1
Free energy profile of a channel of length An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e017.jpg with reactive ends.

The free energy levels of the baths are assumed to be equivalent, which makes flow vanish for equal concentrations An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e022.jpg. With

equation image
(3)

as the standard free energy of the reaction at the channel end An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e024.jpg, this condition is fulfilled when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e025.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e026.jpg. The more general condition for equivalent free energy levels of the baths, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e027.jpg, may always be transformed to the latter by appropriate gauging of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e028.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e029.jpg (see Appendix S1). Note that the rates An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e030.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e031.jpg in Eq. 3 describe particle exchange between a three-dimensional space (bath) and one-dimensional space (channel) with corresponding 3D and 1D particle concentrations An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e032.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e033.jpg respectively. This is accomplished by assuming that the 3D particle concentrations at the channel entrances An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e034.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e035.jpg are practically constant perpendicularly to the channel (x) axis, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e036.jpg, with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e037.jpg as the area of the channel opening. Hence, the equilibrium constant An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e038.jpg, and consequently An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e039.jpg, have units of an area, which we assume to be normalized by the channel opening area An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e040.jpg.

Results

Interacting Particles of one Species

As a simple form of particle-particle-interaction it is assumed that a particle within the channel blocks access of particles from outside, a situation which is realistic especially for transport of large long molecules. Since this ansatz depends on a reduction of state space rather than on the neglect of correlations, we do not consider it a mean field type approximation. Now particles require an empty channel to enter some end, implying that the rate of particles entering the channel from the bath An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e041.jpg is not simply An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e042.jpg, as it would be for non-interacting particles. Instead when we consider an ensemble of channels, particle transitions occur only in the fraction of empty channels. So when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e043.jpg denotes the steady state probability that a channel is empty, we obtain that the ensemble averaged transitions per unit time from the bath to the channel end An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e044.jpg is An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e045.jpg. In the steady state particle density becomes stationary An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e046.jpg and flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e047.jpg is constant throughout, i.e. reactive fluxes at the channel end and diffusive flow are equivalent,

equation image
(4)

To solve the above equations it is useful to study first particle transport in the absence of particle-particle interaction, which is realized by setting An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e049.jpg. Here flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e050.jpg is derived as a macroscopic Fick's diffusion law (see Appendix S2 and Refs. [5], [9]),

equation image
(5)

where

equation image
(6)

is the symmetrized specific particle number, which is a normalized measure of the number of particles occupying the channel, and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e053.jpg is the symmetrized first passage time, (see Appendix S2, Eqs. (S2-6)). The brackets denote the spatial average An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e054.jpg. Flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e055.jpg and its corresponding diffusive conductivity, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e056.jpg, are directly related to the translocation probability (see Appendix S2), i.e. the conditional probability that a particle starting at one end of the channel is absorbed by the bath located oppositely

equation image
(7)

It is important to stress that flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e058.jpg, and hence translocation probabilities, increase with binding strength, and that they are invariant under permutations of the potential values An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e059.jpg, since they solely depend on the mean value of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e060.jpg. In particular any asymmetry of particle in-channel interactions is not reflected in flow, as long as particles are non-interacting.

The Eqs. (4) imply that switching from non-interacting to interacting particles is formally accomplished by replacing concentrations by their probability weighted values An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e061.jpg i.e. steady state flow derives formally as

equation image
(8)

For the determination of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e063.jpg, one applies conservation of probability,

equation image
(9)

which gives (see Appendix S2)

equation image
(10)

where An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e066.jpg is the asymmetric counterpart of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e067.jpg, (see Appendix S2, Eqs. (S2-7)), i.e. it is a normalized measure of asymmetric occupation capacity which vanishes for symmetric interactions.

Equation (8) relates flow of interacting particles to flow of non-interacting ones, weighted by the probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e068.jpg to find an empty channel. For low concentrations particle-particle interactions become negligible. This is reflected by An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e069.jpg approaching unity (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e070.jpg), and flow approaching that of non-interacting particles An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e071.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e072.jpg is invariant under permutations of the potential An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e073.jpg and interchange of exchange rates An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e074.jpg, the Eqs. (8, 10) make clear that any asymmetry of flow is related purely to the asymmetry of channel blocking, i.e. to the asymmetry in the probability to find an open channel An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e075.jpg. When asymmetry of flow is quantified by the difference of unidirectional flows at the same concentration An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e076.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e077.jpg, one obtains

equation image
(11)

with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e079.jpg given in Eq. (S2-32) in Appendix S2. Asymmetry of flow depends either on asymmetry of the potential An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e080.jpg, or on the difference between the exchange rates at the channel ends. We first discuss the case of equivalent exchange dynamics at both channel ends. Then a binding site located near that bath to which the flow is directed implies a higher probability to find the channel open than a binding site at the bath located oppositely, see also Appendix S2. Consequently, a binding site located in trans position of the concentration gradient implies a higher flow than in cis position. Next we consider that the potential An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e081.jpg is symmetric, but the exchange rates at the channel ends differ. In this situation flow is lower when directed to the channel end with the lower exit rate, than in reverse direction. This is not a trivial observation! One might argue that identical free energies An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e082.jpg at the channel ends imply that a lower exit rate is accompanied by a lower access rate as well. Therefore, reversing the concentration gradient should not alter flow. However, it turns out that the lower exit rate implies a higher occupation probability of the channel, which impedes flow. Asymmetry of the potential and of the exit rates may work synergistically, whenbinding site and low exit rate are located at opposite channel ends, or competitive, when both located at the same end.

Experimental Determination of Parameters

According to Eqs. (5,8,10) unidirectional flow as a function of concentration exhibits a saturation kinetics, equivalent to that obtained from the Langmuir or Michaelis-Menten Equation, in molecular adsorption or enzymatic kinetics, respectively. For facilitated carrier transport this kinetics has been suggested by Noble [15]. For channel transport it was observed in experiments on DNA transport through nanotubes [16]. The Langmuir or Michaelis-Menten constant, i.e. the concentration for which flow takes half the value of its saturation value, is

equation image
(12)

for unidirectional flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e084.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e085.jpg, respectively. So kinetic experiments should provide the symmetrized and antisymmetrized specific particle numbers An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e086.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e087.jpg respectively. With channel length An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e088.jpg, the free energy of particle channel interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e089.jpg, is then obtained from Eq. (6).

Alternatively, these parameters derive with Eq. (10) from ratios of occupation probabilities, obtained for unidirectional transport at identical concentrations

equation image
(13)
equation image
(14)

The last equations have a strong impact: Ratios of occupation probabilities are equivalent to ratios of corresponding lifetimes of channel states (see Appendix S2). The latter may be obtained experimentally by conductance measurements. From a more theoretical point of view it is of interest that symmetrized ratios of occupation probabilities, determined in the steady state, i.e. under non-equilibrium conditions, are equivalent to the Boltzmann factor corresponding to the free energy of the particle channel interaction, which, as well known, is equivalent with the ratio of the equilibrium occupation probabilities.

Optimal Transport

To determine the in-channel interaction for maximum transport we restrict ourselves to interactions corresponding either to wells or barriers and do not consider potentials oscillating around zero. The probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e092.jpg to find an empty channel increases monotonically with increasing An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e093.jpg, from zero for strong binding, blocking all channels, and approaches asymptotically some value below unity. Concomitantly, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e094.jpg decreases from some value below the finite upper threshold determined by An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e095.jpg in Eq. (5), and reaches zero for infinitely high barriers. So, flow for interacting particles, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e096.jpg, vanishes for strong binding as well as for high barriers, which implies the existence of some maximum at an intermediate interaction An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e097.jpg.

The Eqs. (5,10) imply that An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e098.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e099.jpg, and, hence, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e100.jpg remain invariant under renormalization of the interaction An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e101.jpg. This property has interesting consequences for the value of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e102.jpg. With increasing activity An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e103.jpg, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e104.jpg must increase to compensate for channel blocking and may become repulsive (positive) above some threshold, see (Figs. 2, ,3).3). The value of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e105.jpg further depends on the exchange dynamics at the channel ends, i.e. on particle mobility and energetic or entropic barriers in this region (Fig. 1).

Figure 2
Flow through a channel with symmetric rectangular shaped potential (relative width An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e106.jpg, depth or height An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e107.jpg, normalized by flow at vanishing interaction.
Figure 3
Channel-particle interaction at maximum flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e110.jpg, as a function of the activity An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e111.jpg, and exchange dynamics An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e112.jpg (insert).

To analyze this in more detail we next study the variation of flow, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e113.jpg, at the transition from attractive to repulsive in-channel interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e114.jpg, for a small positive variation of in-channel interaction An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e115.jpg. A negative An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e116.jpg implies the existence of maximum of flow at an attractive interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e117.jpg. Vice versa, a positive An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e118.jpg implies some maximum for a repulsive interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e119.jpg. Since An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e120.jpg is independent of the exchange dynamics, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e121.jpg, is influenced by it only via An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e122.jpg, Eq. (5). For sufficiently slow dynamics, i.e. when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e123.jpg is very small, exchange at the channel ends becomes the time limiting step. In that case An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e124.jpg scales at An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e125.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e126.jpg, and its variation An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e127.jpg with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e128.jpg, i.e. the latter becomes negligibly small. So the variation of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e129.jpg fulfills

equation image
(15)

where we exploited the monotony of the function An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e131.jpg. Hence, in the limit of slow exchange dynamics at the channel ends the switching on of a small repulsive interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e132.jpg, leads to an increase in flow, an effect that is somewhat counterintuitive at first. However, the reason for that effect is that the channel blocking is reduced to such an extent (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e133.jpg) that it dominates the flow impeding effect of the repulsive interaction on translocation probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e134.jpg.

Explicit evaluation of the variation, Eq. (15), by its functional derivative then provides the relation between channel end activity and exchange dynamics determining the value of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e135.jpg,

equation image
(16)

Here we introduced the mean time the channel stays empty An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e137.jpg, related to the time scale of mobility within the channel, as given by the mean first passage time An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e138.jpg of a particle freely diffusing a distance An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e139.jpg,

equation image
(17)

The relation determining the exit rates at which the optimal potential switches from attractive to repulsive, resulting from Eq. (16), is given by

equation image
(18)

with An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e142.jpg leading to a repulsive optimal barrier, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e143.jpg.

As a paradigm we study a symmetric rectangular shaped potential An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e144.jpg of relative width An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e145.jpg,

equation image
(19)

which acts as a well, for An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e147.jpg, or barrier when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e148.jpg. The interaction determining maximum transport is obtained from the condition An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e149.jpg for An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e150.jpg as

equation image
(20)

with corresponding maximum flow

equation image
(21)

Increasing activity of particles An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e153.jpg, or slowing down exchange dynamics at channel ends shift An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e154.jpg toward weaker binding (Figs. 2, ,3).3). One can easily verify that the threshold determining the transition from attractive to repulsive optimal interactions in the rectangular well in Eq. (20) is given by the general results of the variational approach, Eqs. (16,18).

Competition of Different Species: Comparison with Experimental Results in Nuclear Pore Transport

In this section we consider different species of molecules, labeled by the superscript An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e155.jpg, with concentrations An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e156.jpg in respective baths, which compete against each other for channel transport (Fig. 4). Each of the species may have its specific channel affinity. We assume the intra- and interspecies interaction of molecules as above, i.e. a channel occupied by one molecule blocks channel access of any other molecule. This implies that steady state flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e157.jpg at the channel ends is proportional to the probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e158.jpg to find the channel non-occupied as given by Eqs. (4). However, the probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e159.jpg now depends on the concentration and on the binding properties of all species. Based on the conservation of probability one derives for An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e160.jpg different species, similarly to Eq. (10) (see Appendix S2, Eq. (S2-20))

Figure 4
Two species competing for transport through channels.
equation image
(22)

Flow of the An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e162.jpg-th species is the product of the probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e163.jpg times the flow in the absence of any intra- and interspecies interaction of molecules An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e164.jpg, i.e.

equation image
(23)

Equation (22) states that all species contribute to the reduction of probability to find an empty channel proportional to their in-channel affinity and concentration. This effect uniformly hampers flow in all species, see Eq. (23). Selectivity results solely from the effects on the translocation probability of the particular species, which is proportional to An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e166.jpg, see Eq. (7). This implies that the ratio of flows of two species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e167.jpg is independent of interspecies interactions, since An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e168.jpg cancels,

equation image
(24)

Note that this ratio also does not depend on permutations of the respective interactions.

We assume in the following that the species are similar in their exchange dynamics at channel ends (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e170.jpg), and in their diffusion properties (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e171.jpg). The conductivity of unidirectional transport, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e172.jpg, of a binding species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e173.jpg always exceeds transport of a non-binding species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e174.jpg, since (see Eq. (5))

equation image
(25)

Note that we gauged for simplicity the interaction at the channel ends to zero, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e176.jpg for all species. This is not a restriction, as finite variations of the interaction at singular points do not affect the diffusive process.

Increasing the binding strength of a particular species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e177.jpg, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e178.jpg, implies that the probability to find the channel empty decreases. This has the effect that flow of all other species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e179.jpg decreases proportionally as

equation image
(26)

see also Fig. 4. The flow ratio of the species with increased binding, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e181.jpg, behaves more complicatedly. It changes to

equation image
(27)

As was the case for single species transport, the increase of binding strength has different effects on the occupation probability and on the translocation probability, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e183.jpg. While the occupation probability decreases, the translocation probability increases. Following the same arguments as for single species transport, varying the binding strength from infinitely high values to infinitely low values (corresponding to completely repulsive interaction), lets the flow vary from zero through some maximum value to zero again. Figure 5 illustrates this behavior.

Figure 5
Unidirectional flows of two competing species, transparent blue, label (1), and yellow, label(2), as a function of respective particle-channel interactions An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e184.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e185.jpg.

To summarize, the flow of a species decreases monotonically with increasing binding strength of its competitor. If the binding strength for maximal flow of this competitor is sufficiently strong, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e190.jpg, then the flow of the latter increases. In this case facilitated transport of the binding species on cost of the other species is possible.

To analyze selectivity more closely, we investigate unidirectional flow of two species of same particle concentration and initially the same symmetric particle channel interaction An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e191.jpg and reduce the binding strength of the second one, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e192.jpg increases (Fig. 6). With the free energy of particle in-channel interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e193.jpg, see Eq. (6), this implies An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e194.jpg, and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e195.jpg. So, one obtains for the flow of the species, when normalized to flow for initially equivalent interaction, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e196.jpg (see Eqs. (22–23)),

Figure 6
Unidirectional flows of two competing species, labeled (1) and (2), with symmetric rectangular shaped particle channel interactions.
equation image
(28)

Weakening the binding strength of the second species reduces the probability that the channel is blocked, and by this, facilitates transport of the first.

The effect on the second species is more complex. A reduction of its binding strength reduces its translocation probability, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e203.jpg, see Eq. (5,7). However the flow hampering effect of blocking is reduced as well. Following the same arguments as in the previous section there exists a maximum of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e204.jpg at some value An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e205.jpg, when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e206.jpg is varied from infinitely high to low (repulsive) binding strengths. Hence, the behavior of the flow An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e207.jpg for increasing values of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e208.jpg depends on the location of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e209.jpg. Flow of the second species decreases with decreasing binding strength when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e210.jpg, see the left panel of Fig. 6, i.e. small variations of the binding strength have a strong effect on selectivity. Vice versa, when An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e211.jpg, a decreasing binding strength makes flow first run through some maximum, before it decreases (right panel Fig. 6). Hence, larger variations of binding strength are necessary to achieve selectivity. The first scenario,in which small variations of binding strength of the second species resulted in transport selectivity, demands that maximal flow of this species occurs at a sufficient strong binding strength, so that An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e212.jpg. This condition is best fulfilled when the width of the binding region is large, and channel-solvent access dynamics is fast, see Eq. (20).

When we quantify selectivity as the ratio of relative flows of the two species we get

equation image
(29)

i.e. it is identical to the ratio of the translocation probabilities, see Eqs. (5–7). Figure 7 demonstrates that this selectivity increases with the width of the binding site, as does the translocation probability for a binding site. Selectivity works better the faster the access dynamics is in relation to the time scale of channel crossing, measured by the corresponding time scales An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e214.jpg and An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e215.jpg, respectively. In other words, selectivity works better the more the species differ in their binding strength, the longer the channel is, the slower diffusion is within, and the faster the particles enter the channel from outside.

Figure 7
Selectivity, defined as the ratio of relative flows for two species An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e216.jpg, as a function of the difference of free binding energy An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e217.jpg, and relative width of the rectangular potential is shown in the above panel.

This theory explains experimental results on selectivity and competition in artificial nanopores mimicking the nuclear pore complex [4]. These pores contain nucleoporins which transiently bind to transport factors plus cargo, and by this control flow of the latter through the nuclear envelope. The authors investigated competitive transport of the human nuclear transport factor 2-gluthatione S-transferase, NTF2-GST (NTF), and of bovine serum albumin (BSA), which is similar in size and diffusion properties to NTF. The pores were functionalized either with nucleoporins (NSP1) or PEG-thiol, which are comparable in size and polymer properties. However, the NTF binds solely to the NSP1 pore, but not to the PEG-thiol one. The inert BSA binds to neither of the functionalized pores. After replacement of the PEG-thiol by the NTF binding NSP1 pore, BSA flux decreased, whereas that of the competing NTF increased. This is shown in Fig. 8, where the binding strength of the NTF is varied. An increasing binding strength (decrease of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e222.jpg) monotonously decreases the flow of the inert molecule (BSA), whereas that of the binding NTF runs through some maximum. An increasing binding strength of NTF increases its translocation probability, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e223.jpg. Since exchange dynamics at the channel ends is sufficiently fast (see Appendix S3), this effect of NTF binding dominates that of reducing An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e224.jpg, implying the existence of some optimal binding strength, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e225.jpg, at which maximum NTF flux occurs. Conversely, the translocation probability of the inert BSA, An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e226.jpg, is not affected, and BSA flow is reduced due to the NTF-binding related decrease of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e227.jpg, Eq. (26).

Figure 8
Competitive transport of bovine serum albumin (BSA, blue line) and nuclear transport factor (NTF, black line) through a nuclear pore as a function of the NTF in-channel interaction.

Our model does not only describe qualitatively the experiments, but also provides some quantitative insights into the binding energetics. From the data of Jovanovic-Talisman et al. one can determine the ratio of diffusive conductivities of NTF and the inert BSA, Eq. (25), for a pore functionalized with the nucleoporin NSP1, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e237.jpg (see Appendix S3). The access of transport factors to the channel ends, and hence exchange dynamics here, may be estimated to be very fast when compared to transport inside the channel (see Appendix S3). This implies that the ratio of conductivities of binding NTF (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e238.jpg) and non-binding BSA (An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e239.jpg) in Eq. (25) simplifies to

equation image
(30)

The observed value of approximately 50% for the reduction of flow of the inert BSA molecule competing with NTF when switching from the non-binding PEG-thiol pore to the NTF binding NSP1 pore determines with Eq. (26) the ratios of probabilities,

equation image
(31)

where we assumed a symmetric channel. The above equation reveals after insertion of activities and structural data (see Appendix S3), the free energy An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e242.jpg of in-channel interaction of NTF is

equation image
(32)

The average of the Boltzmann factor and its inverse in Eqs. (30,32), determined from experimental data, correctly reflect the Cauchy-Schwartz inequality

equation image
(33)

This product is unity if, and only if the interaction is constant throughout the channel, i.e. An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e245.jpg. Hence, a product close to one, as it is the case above, implies that An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e246.jpg is still approximately constant over a significant domain of the channel. In fact, approximating particle channel interaction by a rectangular potential (Eq. (19)), which selfconsistently reproduces the average Boltzmann factor and its inverse (Eqs. (30,32)), provides a relative width of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e247.jpg, i.e. close to unity, and a binding energy of An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e248.jpg, which is close to the free energy of in-channel interaction.

Discussion

The model presented here allows us to analyze how interparticle interactions, particle in-channel energetics, as well as exchange dynamics and energetics at the interface of channel and solvent affect particle transport. Exchange at the interface was simplified by a two-side exchange process between solvent and channel end, where the corresponding rates comprise the dynamics of energetic or entropic barrier crossing. Inside the channel, diffusive particle dynamics is subject to forces which derive from an in-channel potential. Our model may also be extended to include entropic forces/barriers within the channel as presented by Reguera and Rubi [17]; however the detailed analysis would be beyond the scope of this article.

Interparticle interaction was approximated by the assumption that a molecule inside the channel completely blocks the access of others, i.e. interactions of several particles within the channel were excluded. This single occupancy condition has been described very early in literature for discrete and continuous models in the limit of fast solvent-channel exchange [18], [19] and is meanwhile often applied in models of channel transport [5], [10][12], [20], [21]. Discrete models which considered multiple occupancies for single file [22][24], or non-single file transport [25][27] were suggested in the past. However, analytical solutions for these models require that the discrete model is restricted to very few sites, or that the interaction force inside the channel takes a simple form, e.g. it is constant or even vanishes. For the latter case, which in the continuum limit corresponds to a constant potential, the effect of interparticle interaction on flow cancels on average for single species transport within the channel, but is present at the channel ends [26]. As long as single-file transport is a valid approximation, our model can be extended to include the interaction of several particles within the channel by adapting the condition of conserved probability in Eq. (9) for the maximum number of particles occupying the channel.

We derived flow explicitly in terms of occupation probabilities, free energy of particle in-channel binding and exchange dynamics. This allowed us to determine the free energy of particle-channel interaction, i.e. a measure of binding strength, from key parameters of the Michaelis-Menten kinetics, Eq. (12, as well as to determine the occupation probabilities, Eq. (14). Both quantities are accessible by experiments.

Flow with interparticle interaction could be factorized into a term An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e249.jpg that describes only non-interacting particles, and into the probability An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e250.jpg to find a non-occupied channel. Since An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e251.jpg is proportional to the translocation probability and independent of the actual form of the in-channel interaction An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e252.jpg, the only influence of the actual form of that interaction potential on flow is through its effect on An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e253.jpg. An important application of this results is the asymmetry of transport: When the direction of the concentration gradient is reversed, flow is higher when the binding site is located near the channel end of lower concentration. This result was derived in the past by us [5] and others [12] from respective models. However, it is now clear that it this result is related mainly to the asymmetry in the occupation probability.

We analyzed in detail the binding strength for maximal flow. In the past, scenarios had been discussed in which attractive interactions favored transport [5], [10], [12], [26], [28], explaining experiments e.g. for DNA [1] and nuclear pore transport [4]. Here we demonstrated that for high chemical activity of particles, or a slow exchange dynamics at channel ends, maximal transport can occur also for repulsive interactions. The effect of high activity An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e254.jpg was also reported by Kolomeisky [20]. The effect of slow exchange dynamics, which, for example, may be due to energetic or entropic barriers a molecule has to pass at the channel entrance, is new, and required a kinetic explanation. The flow enhancing effect of repulsive interactions was not observed in our previous work [5], where exchange dynamics at the channel ends was assumed to be much faster than the first passage time to pass the channel. In that limit only attractive interactions can optimize transport.

We could extend our model straightforwardly to describe particles of different competing species, each having its own specific channel affinity. Binding favors flow of one species, if its effect on increasing the translocation probability exceeds its flow hampering effect due to increased channel blocking.

Note that only the latter effect, increased channel-blocking, affects the non-binding species, i.e. its flow is reduced when compared to vanishing binding. So binding of a species may enhance its flow on cost of the non-binding species.

We demonstrated for two species, both having initially equivalent particle-channel interaction, that a reduction of the binding strength of one species leads to an increased flow of the other. Flow of the species with reduced binding strength exhibits a more complex behavior. If the binding strength for maximal flow of this species is lower than the initially equivalent binding strength, flow for this species goes through a maximum before it declines with decreasing binding, see Fig. 5. Otherwise, flow of the species with reduced binding strength declines promptly and continuously, which is more favorable to achieve selectivity, see Fig. 6. This behavior of the flow was also observed in simulations of a multi-occupancy model for two competing species in [25], [27]. Interestingly, that multi-occupancy model revealed some moderate cross-dependence of the translocation probability of one species on the binding strength of the other. So the translocation probability of the species with conserved binding strength went through some maximum while reducing the binding strength of its competitor [27]. This feature is related to the fact, that the channel allowed multi-occupancy. A reduction of binding strength implies less particles of this species in the channel, i.e intra-channel interactions with the competing species, having conserved binding strength, decrease, which results in an increased translocation probability. In the single occupancy model, i.e. the model described here, the translocated particle blocks the channel during the whole process. So translocation of one species is independent of interparticle interactions, i.e. in particular independent of interactions with the other species. So, as discussed above, cross interactions derive solely from cross dependencies of occupation probabilities.

Our model also explains the experimental data for competitive transport of nuclear transport factors through artificial nuclear pores described in Ref. [4]. There flow of two competing species through pores was investigated. The two transported species were inert or potentially binding, respectively, while the pores were functionalized either with binding or inert sites. As derived from our model, the flow of the binding (non-binding) species within the pore functionalized with binding sites increased (decreased) when related to that of the inert pore. These ratios of flow also allowed a quantitative estimation of the strength and extent of binding.

Supporting Information

Appendix S1

Herein the interdependence of the particle-channel interaction potential An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e255.jpg at the channel ends and the exit rates An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e256.jpg is analyzed, which allows appropriate gauging of both.

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Appendix S2

This Appendix derives in detail the dependence of channel flow on first passage time and channel occupation number and probability. These parameters are related to the translocation probability and the lifetime of channel states. In this context the effect of asymmetry of in-channel binding site on flow is derived.

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Appendix S3

The access dynamics of the nuclear transport factor (NTF) from outside to the nuclear pore is estimated. Furthermore the diffusive conductivities of bovine serum albumin (BSA) and nuclear transport factor (NTF) through the nuclear pores and their activities An external file that holds a picture, illustration, etc.
Object name is pone.0015160.e257.jpg are computed from experimental data.

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Acknowledgments

The authors are grateful to Philipp Schön and Jens Ulmer (BIOLAB technology AG, Zürich, Switzerland) for fruitful discussions.

Footnotes

Competing Interests: The authors have declared that no competing interests exist.

Funding: This work was partially supported by the Collaborative Research Centre 688 of the German Research Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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