article: The functioning of living cells requires efficient and selective transport of materials into and out of the cell, and between different cellular compartments. Much of this transport occurs through nano-scale channels that do not require large scale molecular re-arrangements (such as transition from a ‘closed’ to an ‘open’ state) and do not require a direct input of metabolic energy during transport. Nevertheless, these ‘always open’ channels are highly selective and pass only their cognate molecules, while efficiently excluding all others; indeed, these channels can efficiently transport specific molecules even in the presence of a vast excess of non-specific molecules. Such biological transporters have inspired the creation of artificial nano-channels. These channels can be used as nano-molecular sorters, and can also serve as testbeds for examining modes of biological transport. In this paper, we propose a simple kinetic mechanism that explains how the selectivity of such ‘always open’ channels can be based on the exclusion of non-specific molecules by specific ones, due to the competition for limited space inside the channel. The predictions of the theory account for the behavior of the nuclear pore complex and of artificial nanopores that mimic its function. This theory provides the basis for future work aimed at understanding the selectivity of various biological transport phenomena.
  Living cells require the efficient and selective trafficking of molecules through various transport channels [1], [2]. Some transporters require large conformational changes, involving transitions from ‘closed’ to ‘open’ states and a direct input of metabolic energy during transport [1]. However, many other transporters provide efficient and selective transport without large conformational changes and without a direct input of metabolic energy during transport. Examples of the latter transport mechanisms include selective permeability of porins [3]–[8], transport through the nuclear pore complex (NPC) [9]–[13], and the access of ligands to the active sites of certain enzymes [14]. In the context of transport through the NPC, such a mode of transport has been termed ‘virtual gating’ [12], [15]. Ion channels also belong to this class of transporters, although factors specific to ion channels set them beyond the scope of the present work [16]. Recently, artificial molecular nano-channel devices have been built that mimic and utilize the principles upon which the function of natural transporters is based [17]–[24]. In this paper, we focus on an artificial nano-molecular channel that mimics the functioning of the NPC [23], as the mimic provides important insights into the function of the underlying biological channel. Despite their variety, such natural and artificial transporters appear to share common mechanisms of transport selectivity and efficiency. They commonly include a channel or a passageway, through which molecules translocate by diffusion [2]–[24]. Often, selective transport involves transient interactions of the transported molecules with corresponding receptors inside the channel [2]–[24], which leads to transient trapping of the transported molecules in the channel. The selectivity mechanisms of such channels are still a matter of debate. A crucial insight is that the channel geometry, even in the absence of any physical barrier for particle entrance, the probability of a particle to transolcate through a channel is low [25], [26]. Transient trapping increases the probability of transport of individual molecules and thus enhances the transport. Related effects arise in selective membrane transport, known as ‘facilitated diffusion’ in that context [2], [15], [25]–[32]. However, if molecules spend too much time in the channel, the rate at which they leave the channel is lower than the rate at which they attempt to enter - which leads to jamming and a decrease of transport. Hence, transport efficiency can be optimized by tuning the interaction strength of the transported molecules with the channel. The selectivity of such channels can thus be based on the differences in the trapping times of the optimally interacting molecules compared to others [5], [15], [31], [33]–[37]. It is important to emphasize that the efficiency and selectivity of transport are determined not by the equilibrium interaction strength of the molecules with the channel per se, but by the rates at which the molecules enter, translocate through, and exit from the transport channel [15], [34], [36], [37]. These rates are in many cases determined by the strength of the interactions with the transport device, but can be also determined by its geometry [2]–[23]. For instance, the trapping times inside a channel can be limited by diffusion through convoluted passages inside the channel (e. g. in zeolites), known as ‘entropic trapping’ [38]–[41]. Theories based on these ideas provide an adequate explanation of transport selectivity of artificial nano-channels for single species transport (for instance, [42]). However, in nature (and in order to be useful in many technological applications such as molecular sorters) the selected molecules have to be transported through a channel in a vast background of other molecules, many of which can interact weakly and non-specifically with the transport channel. Thus, transport channels have to be able to constantly select their cognate molecules from such a background. It is still not clear precisely how biological and artificial channels can perform selective transport under such conditions, but any useful theoretical description must take into account this non- specific competition. It is likely that various mechanisms can contribute to selectivity. For instance, in some cases, the selectivity arises from the presense of a physical or energetic barrier for the entrance of non-specific molecules into the channel [27]–[29]. In this paper we focus on the universal selectivity properties of channels, which do not depend on the specific molecular details pertinent to each specific transporter. We show that highly selective transport is possible in the presence of non-specific competition even when the non-specific molecules are free to interact with and enter into the channel. We study the case of a mixture of two molecular species of different trapping strenghts attempting to traverse the channel. Our model relies on only two essential ingredients: transient trapping of the molecules in the channel and inter-molecular competition for the limited space inside the channel. Analysis of the model reveals a novel kinetic mechanism of the enhancement of transport selectivity through narrow channels, which relies on the sequential exclusion of weakly trapped (low affinity) non-specific molecules from the channel due to competition with strongly trapped (high affinity) cognate molecules that spend a longer time in the channel. Comparison of the theoretical predictions with experimental data shows that the predicted mechanism accounts for the transport selectivity observed in an artificial nano-channel that mimics the NPC. Due to its generality, the proposed mechanism of selectivity is expected to play a role in various biological and artificial nano-channels.
  Let us first consider a ‘one-site’ channel model (Fig. 1). All the details of the potentially complicated kinetics of transport through the channel are absorbed into the forward and backward exit rates and. These exit rates can be thought of as ‘off’ rates for the release of the particles from the channel. Particles of two different species (denoted as n and m) attempt to enter the channel from the left (Fig. 1). Particles of species enter the channel with the rate if the channel is unoccupied, exit at the right end with the rate, or return to the left side with the rate. The respective rates for the other species, particles of type, are, and (Fig. 1). The channel can be in three states: occupied by an -species particle, occupied by an -species particle, or un-occupied, with the respective probabilities, , and. This scheme explicitly allows only one particle of any type to be present in the channel at any time. In other words, if the channel is occupied by a particle of either species, other particles cannot enter until the residing particle hops out. Note the parallel between transport through such one-site channel and the Michaelis-Menten kinetics of enzymatic reactions – the channel is analogous to the enzyme molecule, while the transported particles are analogous to the substrates. The master equation describing the kinetics of transport through the channel is [2], [43]: (1) Note that, (so that) because the channel has to be in some state. Transmitted fluxes to the right of the particles of each type are and, respectively. Solving equations (1), we get: (2) We define the efficiency of transport as the ratio of the transmitted flux to the impinging flux, . However, not all the particles that attempt to enter the channel succeed, because the channel is occupied with the probability. The transport efficiency is thus different from the translocation probability of a particle that has entered the channel to exit on the right - a fact that will become important below. Mathematically, the translocation probability is defined as. From eq. (2), in the absence of competition, when particles of only one type are present (say), in the limit of small currents (when), the efficiency and the probability are identical and equal to. In the case when both particle species are competing for space in the channel, from equation (2), the ratio of transport efficiencies of m-species and n-species is (3) Thus, the transport efficiency of the particles of each type through a single-site channel is not influenced by the presence of particles of the other type. As we show below this is not so for channels that can accommodate more than one particle. Selectivity conditions change when one considers transport in a mixture of two different species of molecules in longer channels, where the molecules can interfere with each other' s passage through the channel. The main result is that in the presence of more strongly trapped species, the transport of more weakly trapped species is strongly inhibited, compared to the case when they are present alone. We now turn to comparison of the theoretical predictions with recent experiments on transport through artificial nano-channels that mimic NPC function [23], where many of the parameters discussed above can be varied experimentally. In these experiments, the channels were functionalized with natively unfolded proteins that naturally line the passageway of the NPC (commonly known as the FG-nups). These proteins bind strongly (although transiently) and specifically to nuclear transport factors, but weakly and non-specifically (or not at all) to other proteins (see Fig. 8A). Here we confine ourselves to qualitative comparison with the experiments, to establish the basic mechanisms of selectivity that operate in such channels (Fig. 8B). More quantitative comparisons require more detailed understanding of the local binding-unbinding kinetics of the multiple binding sites on the transport factors to unfolded filamentous proteins within the NPC, as well as realistic modeling of the dynamics of the filaments themselves [59]–[62]. At this stage, the understanding of the mechanistic details of the interactions of the transport factors with the FG-nups and of the movement of the transport factors from one FG-nup to the next is lacking. Jovanovic-Talisman et al. [23] investigated the transport of various nuclear transport factors and of non-specific “control” proteins such as bovine serum albumin (BSA) through the artificial channels described above, and compared the fluxes when they are present either separately or in mixtures. A subset of the experimental results of [23], where the GST tagged nuclear transport factor 2 (NTF2-GST) and BSA were compared, issummarized in Fig. 8C. It was observed that the transport of non-binding control protein (BSA) was inhibited by the presence of NTF2-GST, in accord with the theoretical predictions (above). Likewise, the magnitude of the inhibition increased with the length of the trapping region and decreased with the channel width, in accord with the theortical predictions. Thus, the mechanism proposed in this paper account for the experimental results and indicates that selective nano-filters can be built relatively simply, using just the basic stochastic kinetics of the transport process and competition for space inside the channel.
  In nature, transport channels have to select for their cognate cargoes over a vast background of other species that might interact with the channel non-specifically. How can they maintain selective transport in such conditions? It is likely that many different mechanisms of selectivity may be operational in such channels [24], [61]–[64]. Here, we have studied a minimal kinetic mechanism of selectivity enhancement, which relies only on the inherent properties of stochastic transport through narrow channels. The model includes only two essential ingredients: transient trapping of the particles inside the channel, and the competition for the limited space inside the channel. The model predicts that weakly trapped (non-specific) species are effectively excluded from transport through the channel by competition with strongly trapped cognate cargoes that spend more time in the channel. In a mixture of two different species - one that is transiently trapped in the channel longer than the other - the transport of the particles of the more weakly trapped species is strongly inhibited compared to the case when they are present alone. Moreover, the theory predicts that transport of the more strongly trapped species is enhanced by the presence of the non-specific competitors. These effects are described in Figs. 4 and 7. In the main, inhibition of non-specific competitor transport is not due to prevention of entrance into the channel. Rather, this inhibition is largely due to the diminished probability of translocating through the channel (and so the increased probability of returning to the entrance compartment) after the particle has entered the channel. Remarkably, the transport of non-specific particles is inhibited even if their flux greatly exceeds that of the specifically binding particles. This selectivity enhancement is a purely kinetic, non-equilibrium mechanism. Notably, it does not require input of metabolic energy [1], [65]–[67], but rather stems from the inherent properties of the stochastic transport process. Thus, this effect is expected to hold for various molecular mechanisms of transport through the channels, channel widths, and particle sizes. Even in channels where other effects may be dominant, the effect described here is likely to play a role. It is important to emphasize that for the purposes of the present theory, it is immaterial as to which physical mechanism determines the rate of “hopping” through the channel and the escape rates – i. e. , whether they are determined by the binding energies of the particles inside the channel (as in ion channels or porins) [33], [36], [37], [43], geometrical effects such as entropic trapping [38]–[41], or a mixture of the two (e. g. during transport through the nuclear pore complex and artificial nano-channels [8]–[12], [17]–[24]). Predictions of our theory are in agreement with recent experiments on transport through artificial nano-channels that mimic the nuclear pore complex function [23] - Fig. 8. Thus, both theory and experiment emphasize the need to always consider non-specific competition when studying transport selectivity of both biological and artificial nano-channels. They also highlight the role that the specifc molecules play in the selectivity – they can be viewed as an essential part of the selectivity mechanism. In their absence, it is possible that the channel can be essentially non-selective and can pass various non-specific molecules; it is the presence of the specific molecules that makes the transport selective. The theory also makes verifiable predictions on how the addition of non-specific molecules affects the transport of the specific ones. We expect that future comparison of the theory with experimental data will lead to further refinements of the theory and elucidation of additional selectivity mechanisms, thus allowing the design of more selective artificial nano-channels. Future questions include the mutual influence between the fluxes of particle species in multi-species case, as well as more detailed modeling of the diffusion of the transport factors through the layer of the FG-nups (in the context of the NPC transport) and the analysis of single molecule tracking experiments [57], [68]–[71]. Finally, our theory can be generalized to describe mechanisms of selectivity in arbitrary signal transduction schemes [67], [72]–[74].
  The analytical calculations were perfromed by pencil and paper with the help of Mathematica 5. 2 package. Simulations were implemented in C language and run on a cluster of opteron processors under UNIX. The simulation code is presented in Text S1.
summary: Various channels and transporters shuttle molecules into and out of the cell, as well as between different cell compartments. Such channels have be selective, i. e. to pass only certain molecular species in a given direction, while efficiently blocking the passage of all others. Transport properties of some channels (e. g. ion channels), have been extensively studied. However, the mechanisms of channels that conduct larger molecules, such as the nuclear pore complex, which gates all transport between the cell nucleus and the cytoplasm, are less understood. In particular, it is still not clear how such channels can efficiently transport their specific molecules even in the presence of a vast excess of non-specific molecules that potentially could clog the channel. Understanding how such channels work is also important for technological applications, such as design of artificial nano-filters. In this paper, we propose a mechanism of selectivity of such channels in the presence of vast amounts of background molecular noise. The predictions of the theory account for the behavior of the nuclear pore complex and of artificial nanochannels that mimic its function. The theory provides the basis for future work aimed at understanding the selectivity of transport through various biological and artificial channels.
keywords: biophysics/theory and simulation
  computational biology/systems biology
title: Enhancement of Transport Selectivity through Nano-Channels by Non-Specific Competition