Stochastic Gating as a Novel Mechanism for Channel Selectivity

Abstract

An ideal channel, responsible for metabolite fluxes in and out of the cells and cellular compartments, is supposed to be selective for a particular set of molecules only. However, such a channel has to be wide enough to accommodate relatively large metabolites, and, therefore, it allows passage of smaller solutes, for example, sodium, potassium, and chloride ions, thus compromising membrane’s barrier function. Here we show that stochastic gating is able to provide a mechanism for the selectivity of wide channels in favor of large metabolites. Specifically, applying our recent theory of the stochastic gating effect on channel-facilitated transport, we demonstrate that under certain conditions gating hinders translocation of fast-diffusing small solutes to a significantly higher degree than that of large solutes that diffuse much slower. We hypothesize that this can be used by Nature to minimize the shunting effect of wide channels with respect to small solutes.

Main Text

In contrast to the highly ion-selective channels studied in neurophysiology, which have narrow selectivity filters of the size of a partially dehydrated ion (1), metabolite channels are significantly wider. Indeed, they have to accommodate metabolite molecules that typically are much larger than simple mono- or divalent ions. The most studied metabolite channel, by now the paradigmatic voltage-dependent anion channel (VDAC) (2), whose dense packing in the outer mitochondrial membrane (MOM) is illustrated by Fig. 1 (3), has a β-barrel scaffold of ∼1.3-nm inner radius and 4-nm length (4, 5). Because of its relatively large radius, the VDAC pore is only slightly anion-selective (2) compared to anion channels of neuronal membranes.

Figure 1.

High-resolution AFM image shows high density of VDAC channels in the MOM (from Gonçalves et al. (3) with permission). To see this figure in color, go online.

It is clear that the large pore size of metabolite channels jeopardizes one of the main membrane functions, namely, to serve as a barrier for solutes other than the particular metabolites these channels have evolved to pass. For example, VDAC, recognized to be the major pathway for ATP and ADP exchange between mitochondria and cytosol (6), is also highly permeable for molecules smaller than these metabolites (2). VDAC is also the most abundant integral protein of the outer mitochondrial membrane (Fig. 1). However, interestingly, a number of studies have demonstrated that under physiological conditions these channels are predominantly closed (7, 8), which allows the membrane to sustain its barrier function.

In this study we propose that stochastic gating may provide a mechanism for metabolite channel selectivity in favor of slowly moving large solutes. Specifically, we show that fast gating of a predominantly closed channel leads to an increase of the channel selectivity for large, slowly diffusing molecules versus small ions by orders of magnitude.

Our analysis is based on the analytical theory of solute transport through a stochastically gated channel developed in (9). The model studied in that work assumes that transport is driven by the solute concentration difference on the two sides of the membrane: the concentration is equal to c, c > 0, on the left side of the membrane, and zero on the right side, with the gate located at the left end of the channel. The gate randomly jumps between open and closed states as shown in the inset of Fig. 2. The transitions are assumed to be Markovian and described by the gate closing and opening rate constants α and β. In the absence of gating, α = 0, the stationary flux through the channel J^ng (ng stands for nongated) is given by the product of the translocation probability, $P_{\text{tr}}^{\text{ng}}$, and the influx, k_HBPc, entering the channel from the bulk, $J^{\text{ng}}=k_{\text{HBP}}c{P}_{\text{tr}}^{\text{ng}}$. The Hill-Berg-Purcell rate constant, k_HBP, is k_HBP = 4D_bR (10, 11), with R and D_b denoting the channel radius and solute bulk diffusivity, respectively.

Figure 2.

Depending on the molecule intrachannel diffusivity (indicated by the numbers near the curves), the flux through a stochastically gated channel can differ significantly from its conventional estimate. The flux ratio, taken from Eq. 2, is the ratio of the flux at an arbitrary gating rate, J^sg, to its conventional counterpart, $J_{\text{slow}}^{\text{sg}}$, taken from Eq. 1. To see this figure in color, go online.

Gating decreases the net stationary flux because for a fraction of time, the channel is closed. Traditionally (1), this effect is described by giving the flux through a stochastically gated channel as the product of flux J^ng and the probability P_open = β/(α + β) of finding the channel open. As shown in (9), this traditional result is applicable only at slow gating, and fails when gating is fast. The reason is that gating also affects the translocation probability. This effect, neglected in the traditional theory, is taken into account in (9).

Here, the theory of (9) is used to demonstrate that stochastic gating—the phenomenon known for over a half-century and appreciated as the major means of channel regulation by the environment—is able to affect molecule translocation in a nontrivial way and provide a mechanism for selectivity in favor of slowly diffusing solutes. This occurs when the characteristic time of the gating gets comparable with or shorter than that of molecule dynamics in the channel pore. The effect is pronounced if the channel stays mostly in the closed state, i.e., when the probability of finding the channel closed, P_closed = α/(α + β), is close to unity.

To gain some feeling for the magnitude of the effect, let us examine the limiting cases of slow and fast gating. The gating dynamics are considered to be slow when its characteristic time, (α + β)⁻¹, significantly exceeds the characteristic time required for the solute to traverse the channel, (α + β)⁻¹ ≫ L²/D_ch, where L is the channel length and D_ch is the molecule intrachannel diffusivity. In the case of slow gating, the channel has enough time to establish the stationary flux, J^ng, when it is open between consecutive gating events. As mentioned above, in the traditional theory (1) that holds true only for slow gate dynamics, the flux through the stochastically gated (sg) channel, $J_{\text{slow}}^{\text{sg}}$, is given by

$$J_{\text{slow}}^{\text{sg}}=J^{\text{ng}}{P}_{\text{open}}.$$ (1)

The situation is different when the characteristic gating time is comparable or shorter than its diffusion counterpart. In this case, molecular flux can be significantly higher than the conventional estimate taken from Eq. 1. Introducing notations $\lambda =L\sqrt{\left(\alpha +\beta \right)/D_{\text{ch}}}$ and κ = 4D_b/(πR), we can write the ratio of the flux J^sg at an arbitrary gating rate to the conventional estimate, $J_{\text{slow}}^{\text{sg}}$, as (see Eq. 3.12 in (9)):

$$\frac{J^{\text{sg}}}{J_{\text{slow}}^{\text{sg}}}=\frac{J^{\text{sg}}}{J^{\text{ng}}{P}_{\text{open}}}=\frac{2+\kappa L/D_{\text{ch}}}{\left(2+\kappa L/D_{\text{ch}}\right)\left(1-P_{\text{closed}}\right)+P_{\text{closed}}F\left(\lambda \right)},$$ (2)

where function F(λ) is given by

$$F\left(\lambda \right)=\frac{2\left(\kappa L/D_{\text{ch}}\right)\lambda +\left[{\left(\kappa L/D_{\text{ch}}\right)}^2+{\lambda }^2\right]\tanh \left(\lambda \right)}{\lambda \left(\kappa L/D_{\text{ch}}+\lambda \tanh \left(\lambda \right)\right)}.$$ (3)

The selectivity enhancement due to stochastic gating is illustrated in Fig. 2, which gives the dependence of the flux ratio on the probability of finding the channel in the closed state for different values of molecule diffusivity inside the channel. For this particular illustration, we took channel radius and length using the VDAC-like geometric parameters given above. (It has to be noted that although the effective channel radius is smaller than the pure geometrical radius of the water-filled VDAC pore (2, 4, 5), nevertheless, for simplicity, we use the crystallographic value.) Other parameters are D_b = 2⋅10⁻⁹ m²/s, and gating rates (Fig. 2, inset) are varied as α = 5⋅10⁷ P_closed s⁻¹ and β = 5⋅10⁷ (1 –P_closed) s⁻¹, with the value of P_closed given at the x axis. This way the characteristic time of the gating, (α + β)⁻¹, is kept constant for all curves in Fig. 2, whereas the diffusion time, L²/D_ch, increases as the values of D_ch (shown at the curves) decrease from the bottom to the top curve. The effect of slowing down of solute diffusion by the channel may be strong (12). For example, using noise analysis for one of the main VDAC’s substrates, ATP, it was inferred (13) that its diffusivity inside the VDAC pore is nearly two orders-of-magnitude lower than in free solution.

The message from the figure is twofold. First, the gating-induced increase in the flux ratio depends on the molecule intrachannel diffusivity: for the fast-moving molecules and ions with diffusivity of 10⁻⁹ m²/s, the effect of gating is small (the bottom curve), whereas, for slowly moving solutes, it can reach several orders of magnitude (the top curve). Second, and, probably, most important in this context, the effect is greatest when the channel is predominantly closed, i.e., when P_closed → 1. Because fast gating of a predominantly closed channel reduces the flux of more mobile molecules much more strongly than the flux of less mobile ones, it provides a mechanism of selectivity in favor of slow solutes.

One of the possibilities of an intuitive interpretation of the above flux ratio dependence on P_closed is as follows (9). The fate of a molecule entering the channel depends on the state of the gate: when the gate is open, the molecule translocation probability is well below unity (14). However, if the gate closes just after the molecule enters the channel and stays closed, the translocation probability is unity (see the inset in Fig. 2). At slow gating, the fraction of molecules entering the channel, which feel the change in the gate state, is negligibly small. The overwhelming majority of the entering molecules is “unaware” of gating, because they leave the channel before the gate changes its state. This is why the conventional approximation works well in this case. In contrast, when gating is fast, all molecules entering the channel feel the gate opening and closing that results in the increase in their translocation probabilities. Consequently, the flux under the fast gating conditions is higher than its conventional estimate.

As was indicated by one of the reviewers, the intuitive interpretation above sounds like Maxwell’s demon because it seems to imply that gating would increase the flux only in one direction by blocking the pore entrance and preventing the molecule from going back. In fact, it is not really the case: it can be shown that our result for the flux through a stochastically gated channel remains unchanged when the gate is located at the opposite channel end, i.e., at the channel exit (unpublished data).

We hypothesize that this, to our knowledge, newly described mechanism of the selectivity due to stochastic gating provides an explanation for the long-known but puzzling observation that VDAC in intact mitochondria is mostly closed (7, 8). Our conjecture is that the large number of predominantly closed VDAC channels is necessary to keep ATP/ADP transport at a sufficiently high level while effectively suppressing small-ion leakage through the MOM.

To conclude, it is worth mentioning that the stochastic gating effect considered here might not be restricted to the VDAC case. It could shed light on some aspects of functioning of other predominantly plugged β-barrel channels, such as FhuA (15, 16, 17) and PapC (18, 19) of the Escherichia coli outer membrane. It may also be important in other biological processes, which are controlled by passage through fluctuating bottlenecks.

Author Contributions

A.M.B. and S.M.B. equally contributed to research design, analytical tools, data analysis, and writing the article.

Editor: Anatoly Kolomeisky.