Conceived and designed the experiments: WRB WN. Performed the experiments: WRB WN. Analyzed the data: WRB. Contributed reagents/materials/analysis tools: WRB WN. Wrote the paper: WRB.
The authors have declared that no competing interests exist.
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.
Understanding of molecular or particle transport through channels and pores is of paramount interest in many field, ranging from nanotechnology to life sciences
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
We consider particle transport through a channel connecting two baths, labeled as (A) and (B), with respective particle concentrations
where
The exchange rates of particles, entering or leaving either channel end are
The particle in-channel interaction
The free energy levels of the baths are assumed to be equivalent, which makes flow vanish for equal concentrations
as the standard free energy of the reaction at the channel end
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
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
where
is the symmetrized specific particle number, which is a normalized measure of the number of particles occupying the channel, and
It is important to stress that flow
The Eqs. (4) imply that switching from non-interacting to interacting particles is formally accomplished by replacing concentrations by their probability weighted values
For the determination of
which gives (see
where
Equation (8) relates flow of interacting particles to flow of non-interacting ones, weighted by the probability
with
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
for unidirectional flow
Alternatively, these parameters derive with Eq. (10) from ratios of occupation probabilities, obtained for unidirectional transport at identical concentrations
The last equations have a strong impact: Ratios of occupation probabilities are equivalent to ratios of corresponding lifetimes of channel states (see
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
The Eqs. (5,10) imply that
Different chemical activities and exchange dynamics at channel ends are studied
Other parameters are as in
To analyze this in more detail we next study the variation of flow,
where we exploited the monotony of the function
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
Here we introduced the mean time the channel stays empty
The relation determining the exit rates at which the optimal potential switches from attractive to repulsive, resulting from Eq. (16), is given by
with
As a paradigm we study a symmetric rectangular shaped potential
which acts as a well, for
with corresponding maximum flow
Increasing activity of particles
In this section we consider different species of molecules, labeled by the superscript
One has the capability of in-channel binding (green-red binding sites), the other is inert. The binding species dominates in-channel sojourn, and by this increases translocation probabilit. Hence, transport of the binding species is increased on the cost of the inert species, the channel access of which is hampered.
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
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 (
Note that we gauged for simplicity the interaction at the channel ends to zero, i.e.
Increasing the binding strength of a particular species
see also
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,
Flows are normalized to that in the absence of particle-channel interactions. A symmetric rectangular shaped potential with relative width
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.
To analyze selectivity more closely, we investigate unidirectional flow of two species of same particle concentration and initially the same symmetric particle channel interaction
The free energy of binding
The effect on the second species is more complex. A reduction of its binding strength reduces its translocation probability,
When we quantify selectivity as the ratio of relative flows of the two species we get
i.e. it is identical to the ratio of the translocation probabilities, see Eqs. (5–7).
The access dynamics is here assumed to be very fast
This theory explains experimental results on selectivity and competition in artificial nanopores mimicking the nuclear pore complex
The interaction potential is assumed to have a symmetric rectangular shape, with height/depth
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.
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,
where we assumed a symmetric channel. The above equation reveals after insertion of activities and structural data (see
The average of the Boltzmann factor and its inverse in Eqs. (30,32), determined from experimental data, correctly reflect the Cauchy-Schwartz inequality
This product is unity if, and only if the interaction is constant throughout the channel, i.e.
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
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
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
We analyzed in detail the binding strength for maximal flow. In the past, scenarios had been discussed in which attractive interactions favored 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
Our model also explains the experimental data for competitive transport of nuclear transport factors through artificial nuclear pores described in Ref.
Herein the interdependence of the particle-channel interaction potential
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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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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
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The authors are grateful to Philipp Schön and Jens Ulmer (BIOLAB technology AG, Zürich, Switzerland) for fruitful discussions.