Localized potential well vs binding site: Mapping solute dynamics in a membrane channel onto one-dimensional description

Abstract

In the one-dimensional description, the interaction of a solute molecule with the channel wall is characterized by the potential of mean force U(x), where the x-coordinate is measured along the channel axis. When the molecule can reversibly bind to certain amino acid(s) of the protein forming the channel, this results in a localized well in the potential U(x). Alternatively, this binding can be modeled by introducing a discrete localized site, in addition to the continuum of states along x. Although both models may predict identical equilibrium distributions of the coordinate x, there is a fundamental difference between the two: in the first model, the molecule passing through the channel unavoidably visits the potential well, while in the latter, it may traverse the channel without being trapped at the discrete site. Here, we show that when the two models are parameterized to have the same thermodynamic properties, they automatically yield identical translocation probabilities and mean translocation times, yet they predict qualitatively different shapes of the translocation time distribution. Specifically, the potential well model yields a narrower distribution than the model with a discrete site, a difference that can be quantified by the distribution’s coefficient of variation. This coefficient turns out to be always smaller than unity in the potential well model, whereas it may exceed unity when a discrete trapping site is present. Analysis of the translocation time distribution beyond its mean thus offers a way to differentiate between distinct translocation mechanisms.

INTRODUCTION

Channel-facilitated transport of solutes across biological membranes is a well-established and quickly developing area of science. The key transport properties of a channel, such as overall transport rates and selectivity to different solutes, are understood and quantified in terms of solute interactions with the walls of the channel water-filled nanopore.^1–18 In a one-dimensional description, the equilibrium and dynamics of a solute molecule partitioning into a membrane channel are usually modeled using two approaches:¹⁹ (i) by considering a series of binding sites located along the channel axis or (ii) by using the one-dimensional Smoluchowski equation with a potential of mean force U(x), where the x-coordinate is measured along the channel axis. In the latter model, the potential U(x) includes the interaction energy of the solute with the channel walls. Such a description is especially attractive when the potential of mean force is an experimental observable: although this potential has not yet been measured in biological channels, its shape has been quantified and even designed for model systems consisting of colloidal particles in microfluidic devices.²⁰ Likewise, potentials of mean force can be measured in other single-molecule experiments that employ optical tweezers.²¹

In the simple example of a single site of localized attraction between the channel and the molecule, such interaction can be represented by either a localized potential well of an appropriate size and depth [Fig. 1(a)] or by a reversible trapping site [Fig. 1(b)]. One can imagine that the parameters of the well and the trapping site can be chosen so as to provide the same equilibrium distribution of the molecule/solute along the channel axis for both models. Moreover, it is tempting to think that under such conditions, both models will predict not only identical thermodynamic properties of the channel but also identical transport properties.

FIG. 1.

Two models of the molecule dynamics in the channel. (a) Diffusion in a potential with a localized rectangular well. (b) Diffusion in the presence of a discrete trapping site.

In what follows, we show that such a conjecture is not true. It turns out that, although the solute translocation probabilities and its mean translocation times are identical in both cases, the distributions of the translocation time in the two models are different. Specifically, the potential well model gives a narrower distribution, while the model with a binding site predicts a broader, “heavy-tailed” distribution. The discrepancy originates from a fundamental difference between the two descriptions of the solute dynamics in the channel: the potential well model allows for only one scenario of the molecule’s passage through the channel, in which the molecule unavoidably visits the potential well. In contrast, the thermodynamically equivalent description in the framework of the trapping site model allows for two different passage scenarios: the molecule can either avoid trapping or be reversibly trapped by the site.

This study was motivated by several recent advances in single-molecule approaches in biological and chemical physics. In particular, developments in the nanopore sensing field now allow one to advance beyond analysis of just average fluxes toward the fine details of the particle dynamics in the channel. In single-molecule experiments on channel-facilitated transport, the information comes from the analysis of transient blockages of ionic current through the channel, where a molecule entering the channel produces a measurable (but, in many practical examples, not complete) reduction in the current. The current is restored to its initial level when the molecule leaves the channel. Usually, even in the case of fully time-resolved blockages, it is not possible to assign a particular blockage event to either translocation or retraction, that is, the event when the molecule returns to the side of the membrane from which it entered the channel. One needs additional information to discriminate between the two outcomes. The possibility to distinguish between molecule translocation through the nanopore vs its return was recently demonstrated for proteins with inhomogeneous charge distribution along the peptide chain.²²

The source of the additional information in this case comes from the structure of a single blockage event. It turns out that if the charge distribution on the blocking molecule has the pattern of a block copolymer, wherein one part is mostly net-neutral and the other is highly negatively or positively charged, the blockage events acquire a sub-structure. The events are not rectangular anymore but are characterized by a more complicated time pattern, where the current in the blocked state depends on the polymer position in the nanopore, fluctuating between two levels that report on the instantaneous position of the polymer. The blockages relating to retraction start and end at the same level of the ion current through the nanopore; translocation is manifested by a difference in the current levels. This allows one to reliably separate the blockage events into a group corresponding to translocation and a group corresponding to retraction and, therefore, to analyze time distributions for each group.^22–24

Further motivation comes from single-molecule force-spectroscopy experiments²¹ and single-molecule fluorescence resonance energy transfer studies,²⁵ where improved time resolution has enabled observation of molecular transition paths of biomolecular folding, i.e., fleeting events where the molecule undergoing the folding process is caught in the act of overcoming the free energy barrier. We note that mathematical models used to describe transition paths in terms of a single reaction coordinate x (which may, for example, be the extension of a protein in a force-spectroscopy experiment) are often analogous to those used to study nanopore translocation and involve an effective potential U(x) that, in the former case, describes both the intramolecular interactions of the molecule undergoing folding and its interactions with the force probe (see, e.g., Ref. 26).

The outline of this paper is as follows. First, we formally define the two models of the solute molecule dynamics in the channel. Then, for each model, we find the molecule’s translocation probability, the first two moments of the translocation time, and the coefficient of variation C, defined as the ratio of the standard variation of the distribution to its mean, which is often used as a measure of the distribution width. We show that C is always smaller than unity for the potential well model, whereas for the trapping site model, C may exceed unity. We illustrate the difference between the translocation time distributions predicted by the two models using a numerical example.

MODELS OF A MEMBRANE CHANNEL WITH A BINDING SITE

Consider a solute molecule entering a transmembrane channel from the bulk. Two observables of interest are the translocation probability (i.e., the probability that this molecule will exit through the opposite end of the channel) and the translocation time (i.e., the molecule’s lifetime inside the channel conditional upon exiting through the opposite end of the channel). The molecule’s dynamics in the channel is frequently modeled as one-dimensional diffusion along the channel axis.^27,28 Let x be the position of the molecule in the channel, and let the channel length be L, such as we have $0⩽x⩽L$ inside the channel. The dynamics of the molecule along x is then described by the Smoluchowski equation,

$$\frac{\partial p\left(x,t\right)}{\partial t}=D\frac{\partial }{\partial x}\left(\frac{\partial p\left(x,t\right)}{\partial x}+\beta U^{\prime }\left(x\right)p\left(x,t\right)\right),$$ (1)

where $p\left(x,t\right)$ is the probability density of finding the molecule at x at time t, $U\left(x\right)$ is a potential of mean force describing the effective interaction of the molecule with the channel, $\beta ={\left(k_{B}T\right)}^{-1}$ is the inverse thermal energy, and D is the particle’s diffusivity inside the channel. Matching the three-dimensional description of the molecule dynamics outside the channel with the one-dimensional description of its dynamics in the channel is achieved by introducing radiation boundary conditions at the channel ends,^29,30

$${\left.D\frac{\partial p\left(x,t\right)}{\partial x}\right|}_{x=0}={\kappa }_{0}p\left(0,t\right);{\left.D\frac{\partial p\left(x,t\right)}{\partial x}\right|}_{x=L}=-{\kappa }_{0}p\left(L,t\right),$$ (2)

where the trapping rate ${\kappa }_0$ describes how likely the molecule is to escape from the channel into the bulk when it comes to the channel ends. This rate is equal to^29,30 ${\kappa }_0=4D_b/\pi a$, where $D_b$ is the molecule’s diffusivity in the bulk and $a$ is the channel radius.

Now suppose that the molecule interacts with the channel in such a way that it can bind to a receptor site located on the channel wall. There are two ways of modeling this binding. One way is to assume that the potential $U\left(x\right)$ in Eq. (1) has an “attractive well,” whose depth and width govern the strength of the attractive interaction of the molecule with the receptor [Fig. 1(a)]. An alternative [Fig. 1(b)] is a model with a discrete trapping site coupled to one-dimensional diffusion along x. If $x=l$ is the location of the binding site along the channel and $P\left(t\right)$ is the probability that this site is occupied at time t, then the dynamics of the second model is described by the coupled equations of the form

$$\frac{\partial p\left(x,t\right)}{\partial t}=D\frac{{\partial }^{2}p\left(x,t\right)}{\partial x^2}-\delta \left(x-l\right)\left[\kappa p\left(l,t\right)-kP\left(t\right)\right],$$ (3)

$$\frac{dP\left(t\right)}{dt}=\kappa p\left(l,t\right)-kP\left(t\right),$$ (4)

where $\kappa$ and k are the trapping rate of the molecule by the binding site and the molecule dissociation rate coefficient from the site, respectively. For simplicity, we have assumed here that the molecule traveling in the channel experiences no effective force in the x-direction, except for the strongly localized interaction with the trap. Thus, Eq. (3) describes free diffusion for $x\ne l$, and the delta-function term in this equation is responsible for the exchange between the free and the trapped states of the molecule. Equations (3) and (4) must be supplemented by the boundary conditions of Eq. (2).

Here, we compare these two models and show that they predict fundamentally different dynamics even when their thermodynamic properties are identical. This difference is manifested in the distribution of the translocation time t, ${\phi }_{+}\left(t\right)$, where the plus subscript indicates a successful translocation event. Remarkably, while the two models predict identical translocation probabilities $P_{+}$ and mean translocation times [i.e., the first moments of the distribution ${\phi }_{+}\left(t\right)$],

$$⟨t_{+}⟩={\int }_0^{\infty }t{\phi }_{+}\left(t\right)dt,$$ (5)

the widths of the predicted distributions are qualitatively different. The distribution width (measured in units of its mean) is commonly quantified by the dimensionless parameter called the coefficient of variation, which is the ratio of the standard deviation of the distribution to its mean,

$$C=\frac{{\left(⟨t_{+}^2⟩-{⟨t_{+}⟩}^2\right)}^{\frac{1}{2}}}{⟨t_{+}⟩}.$$ (6)

Here,

$$⟨t_{+}^2⟩={\int }_0^{\infty }{t}^2{\phi }_{+}\left(t\right)dt$$ (7)

is the second moment of the distribution. A narrow distribution has the value of C smaller than unity, $C<1$. In contrast, a heavy-tailed distribution has $C>1$. A single-exponential distribution is a boundary between these two cases, with $C=1$.

Here, we prove that the potential well model [Eq. (1)] always predicts a narrow distribution with $C<1$. In contrast, the model with a localized trapping site [Eqs. (3) and (4)] results in values of C that can be either below or above unity. A value of C that exceeds unity is a fingerprint of essential multidimensionality of the trapping site model that highlights its fundamental difference from a purely one-dimensional diffusion model: Indeed, the inequality $C<1$ was recently proven³¹ for the distribution of the transition path time predicted by the one-dimensional diffusion model with an arbitrary potential. Formally, the transition path time in the present model can be defined as the translocation time in the limit ${\kappa }_0\to \infty$, where the radiation boundary conditions in Eq. (2) become the absorbing ones.

MAPPING BETWEEN THE TWO MODELS

To compare the two models, we require that their equilibrium properties are the same. In particular, the two models should predict the same probability for the solute molecule to be trapped when the system is in equilibrium and the net flux of solute molecules through the channel is zero. To analyze the equilibrium case, it is expedient (but not necessary—see below) to imagine the situation where a single molecule is trapped inside the channel whose entrance and exit are closed (i.e., reflecting boundary conditions are imposed at $x=0,L$) and demand that the equilibrium probability distributions for both models are the same. For mathematical convenience, for the diffusion model of the channel, we choose a rectangular shape for the trapping potential [Fig. 1(a)], with

$$U\left(x\right)=0,x\notin \left(a,b\right),$$ (8)

and

$$U\left(x\right)=-U_0,x\in \left(a,b\right),$$ (9)

where

$$a=l-\frac{\mathrm{\Delta }}{2},b=l+\frac{\mathrm{\Delta }}{2},$$ (10)

and the limit $\mathrm{\Delta }\ll L$ will be assumed. Then, the equilibrium distribution of the coordinate x is given by a piecewise function,

$$p_{eq}\left(x\right)=\frac{1}{\mathrm{\Delta }{e}^{\beta U_0}+L-\mathrm{\Delta }}\left\{\begin{array}{cc}1,\hfill & x\notin \left(a,b\right)\hfill \\ e^{\beta U_0},\hfill & x\in \left(a,b\right).\hfill \end{array}\right.$$ (11)

For the trapping-site model [Fig. 1(b)], the equilibrium distribution can be written as

$$p_{eq}\left(x\right)=P_t\delta \left(x-l\right)+\frac{1-P_t}{L},$$ (12)

where the equilibrium population of the trap satisfies the condition $P_{t}k=\frac{1-P_t}{L}\kappa$. This leads to

$$P_t=\frac{1}{1+kL/\kappa }.$$ (13)

We compare Eqs. (11) and (12) and demand that the probabilities to be trapped are the same in both models,

$$P_t=\frac{1}{1+kL/\kappa }=\frac{\mathrm{\Delta }{e}^{\beta U_0}}{\mathrm{\Delta }{e}^{\beta U_0}+L-\mathrm{\Delta }},$$ (14)

which gives

$$e^{\beta U_0}=\frac{P_t}{1-P_t}\left(\frac{L}{\mathrm{\Delta }}-1\right)=\frac{\kappa }{kL}\left(\frac{L}{\mathrm{\Delta }}-1\right).$$ (15)

Here, we are interested in the limiting case of an infinitely narrow well, $\frac{\mathrm{\Delta }}{L}\to 0$, where we have

$$\underset{\mathrm{\Delta }\to 0}{lim}\mathrm{\Delta }{e}^{\beta U_0}=\frac{\kappa }{k}.$$ (16)

In the absence of reflecting boundary conditions at the channel’s entrance and exit, Eqs. (11)–(13) are still correct, provided that $p_{eq}\left(x\right)$ is interpreted as the equilibrium probability conditional upon finding a solute molecule inside the channel, and thus, introduction of these boundary conditions does not affect the validity of Eq. (16). In fact, Eq. (16) can be derived from the following general argument. Let $p_o$ be the probability density of finding the molecule outside the trap. Then, for the potential well model, the probability density $p_w$ for points belonging to the interval $\left(a,b\right)$ is obtained by using the Boltzmann distribution, $p_w=p_o{e}^{\beta U_0}$, and thus, the probability of finding the molecule trapped inside the well of length $\mathrm{\Delta }$ is $P_t=\mathrm{\Delta }{p}_w=\mathrm{\Delta }$ $e^{\beta U_0}{p}_o$. On the other hand, for the trapping site model, equilibrium requires $P_{t}k=p_o\kappa$, and thus, $P_t=\frac{p_o\kappa }{k}.$ Equating the expressions for the trapping probability $P_t$ from both models leads immediately to Eq. (16).

OUTLINE OF THE CALCULATION OF THE TRANSLOCATION PROBABILITY P₊ AND OF THE DISTRIBUTION OF THE TRANSLOCATION TIME φ₊(t)

We envision that at $t=0$ the molecule enters the channel through the left entrance ($x=0$). For the first model, we then solve Eq. (1) with the initial condition $p\left(x,0\right)=\delta \left(x\right)$ and the boundary conditions of Eq. (2). For the second model, we solve Eqs. (3) and (4) with the initial conditions $p\left(x,0\right)=\delta \left(x\right),P\left(0\right)=0$ and the same boundary conditions of Eq. (2). Once we know $p\left(x,t\right)$, we can calculate the flux escaping through the right boundary at $x=L$,

$$f_{+}\left(t\right)={\kappa }_{0}p\left(L,t\right).$$ (17)

The translocation probability is the integral of this flux,

$$P_{+}={\int }_0^{\infty }dt{f}_{+}\left(t\right),$$ (18)

and the distribution of the translocation time is the normalized flux,

$${\phi }_{+}\left(t\right)=f_{+}\left(t\right)/P_{+}.$$ (19)

We find the solutions for $p\left(x,t\right)$ and $P\left(t\right)$ in the Laplace space, where Eqs. (1)–(4) take the form

$$s\widehat{p}\left(x,s\right)-1=D\frac{\partial }{\partial x}\left(\frac{\partial \widehat{p}\left(x,s\right)}{\partial x}+\beta U^{\prime }\left(x\right)\widehat{p}\left(x,s\right)\right),$$ (1’)

$${\left.D\frac{\partial \widehat{p}\left(x,s\right)}{\partial x}\right|}_{x=0}={\kappa }_0\widehat{p}\left(0,s\right);{\left.D\frac{\partial \widehat{p}\left(x,s\right)}{\partial x}\right|}_{x=L}=-{\kappa }_0\widehat{p}\left(L,s\right),$$ (2’)

$$s\widehat{p}\left(x,s\right)-1=D\frac{{\partial }^2\widehat{p}\left(x,s\right)}{\partial x^2}-\delta \left(x-l\right)\left[\kappa \widehat{p}\left(l,s\right)-k\widehat{P}\left(s\right)\right],$$ (3’)

$$s\widehat{P}\left(s\right)=\kappa \widehat{p}\left(l,s\right)-k\widehat{P}\left(s\right).$$ (4’)

Here, the hat denotes the Laplace transform of the function [i.e., $ĝ\left(s\right)={\int }_0^{\infty }dt{e}^{-st}g\left(t\right)$ for an arbitrary function g$\left(t\right)$]. Using Eqs. (17)–(19), we now obtain the following expressions for the translocation probability and the Laplace-transformed translocation time distribution:

$$P_{+}={\widehat{f}}_{+}\left(0\right)={\kappa }_0\widehat{p}\left(L,0\right),{\widehat{\phi }}_{+}\left(s\right)=\frac{{\widehat{f}}_{+}\left(s\right)}{P_{+}}=\frac{{\kappa }_0\widehat{p}\left(L,s\right)}{P_{+}}.$$ (20)

The mean translocation time [Eq. (5)] and the second moment of the translocation time distribution [Eq. (7)] are now readily found as

$$⟨t_{+}⟩=-{\left.\frac{d{\widehat{\phi }}_{+}\left(s\right)}{ds}\right|}_{s=0}\equiv -{\widehat{\phi }}_{+}^{\prime }\left(0\right),$$ (21)

$$⟨t_{+}^2⟩={\left.\frac{d^2{\widehat{\phi }}_{+}\left(s\right)}{d{s}^2}\right|}_{s=0}\equiv {\widehat{\phi }}_{+}^{″}\left(0\right).$$ (22)

According to Eq. (6), the conditions that the distribution is broad ($C>1$) or narrow ($C<1$) are, respectively, equivalent to ${\widehat{\phi }}_{+}^{″}\left(0\right)>2{\left[{{\widehat{\phi }}^{\prime }}_{+}\left(0\right)\right]}^2$ or ${\widehat{\phi }}_{+}^{″}\left(0\right)<2{\left[{{\widehat{\phi }}^{\prime }}_{+}\left(0\right)\right]}^2$.

RESULTS: COMPARISON BETWEEN THE TWO MODELS

To establish the correspondence between the two models, we consider the limit of an infinitely narrow potential well, $\mathrm{\Delta }\to 0$, and demand that the depth and the width of the potential well satisfy Eq. (16). This equation provides a mapping between the parameters $\left(U_0,\mathrm{\Delta }\right)$ of the first model and $\left(k,\kappa \right)$ of the second model, and thus allows us to write the solutions of both models in terms of the rate coefficient $k$ and the trapping rate $\kappa$ that are defined in the model with the trapping site. We assume that both the potential well and the trapping site are located in the middle of the channel, i.e., $l=L/2$. Superscripts “1” and “2” are used here to denote the results obtained for the first (the potential well) and the second (trapping site) models.

Solving Eqs. (1’)–(4’) and using Eq. (20), we obtain

$$P_{+}^{\left(1\right)}=P_{+}^{\left(2\right)}=\frac{1}{2+{\kappa }_{0}D/L}.$$ (23)

Thus, the translocation probabilities predicted by both models are identical, provided that the equilibrium properties of the models are the same [i.e., when Eq. (16) is satisfied].

The general expressions for the distributions of the translocation time and their moments are rather cumbersome; to somewhat simplify these expressions, we are focusing here on the limit ${\kappa }_{0}D/L\to \infty$. According to Eq. (23), the translocation probability vanishes in this limit since the radiation boundary condition reduces to the absorbing one, and a molecule starting from the absorbing boundary has no chance to pass through the channel. Nevertheless, the distribution of the translocation time remains finite and becomes independent of ${\kappa }_0$ for ${\kappa }_{0}D/L\gg 1$, i.e., reaching a well-defined limit as ${\kappa }_{0}D/L\to \infty$.

In this limit, for the first model, we find

$${\widehat{\phi }}_{+}^{\left(1\right)}\left(s\right)=\frac{qL/2}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{qL}{2}\right)\left[\cosh \left(\frac{qL}{2}\right)+\frac{q\kappa }{2k}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{qL}{2}\right)\right]},$$ (24)

where we have defined

$$q=\sqrt{s/D},$$ (25)

and for the second model, we have

$${\widehat{\phi }}_{+}^{\left(2\right)}\left(s\right)=\frac{qL/2}{\sinh \left(\frac{qL}{2}\right)\left[\cosh \left(\frac{qL}{2}\right)+\frac{q\kappa }{2\left(s+k\right)}\sinh \left(\frac{qL}{2}\right)\right]}.$$ (26)

The two expressions, Eqs. (24) and (26), are strikingly similar. In fact, using Eq. (21), one can verify that they predict identical mean translocation times, $⟨t_{+}^{\left(1\right)}⟩=⟨t_{+}^{\left(2\right)}⟩$. However, when it comes to the second moments of the distribution and, consequently, to the values of the coefficient of variation, the predictions of the two models are drastically different. Specifically, the first model always predicts $C<1$. Indeed, using Eq. (24), we find

$${\left({\widehat{\phi }}_{+}^{\left(1\right)}\right)}^{\prime \prime }\left(0\right)-2{\left[{\left({\widehat{\phi }}_{+}^{\left(1\right)}\right)}^{\prime }\left(0\right)\right]}^2=-\left(\frac{2}{5}+\frac{\kappa }{kL}\right)\frac{L^4}{24D^2}<0.$$ (27)

The second model, in contrast, can result in either $C<1$ or $C>1$, depending on the parameters of the system. Indeed, from Eq. (26), we obtain

$${\left({\widehat{\phi }}_{+}^{\left(2\right)}\right)}^{\prime \prime }\left(0\right)-2{\left[{\left({\widehat{\phi }}_{+}^{\left(2\right)}\right)}^{\prime }\left(0\right)\right]}^2=-\left[\frac{2}{5}+\frac{\kappa }{kL}\left(1-12\frac{D}{L^{2}k}\right)\right]\frac{L^4}{24D^2}.$$ (28)

This expression can have either a positive or negative sign. Importantly, in the limit $L\to 0$, the leading term of Eq. (28) is positive, and thus, for a sufficiently short channel, we universally have $C>1$, a behavior opposite to the prediction of the one-dimensional model with a potential well. [Note that, for L = 0, Eqs. (6), (21), (22), and (28) predict C = 1, that is, a purely exponential distribution, as should be expected.] Distributions with $C>1$ also arise when the dissociation rate constant k tends to zero, and a trapped molecule spends long time in the bound state. A common distinctive feature of both cases where $C>1$ is that the characteristic passage time through the channel of a finite length, $L^2/D$, and the molecule’s lifetime at the binding site, $1/k$, are very different.

Our results for the translocation time distributions ${\phi }_{+}\left(t\right)$ are illustrated in Fig. 2, where the inverse Laplace transforms of Eqs. (24) and (26) were computed numerically and plotted for both models. The distributions ${\phi }_{+}\left(t\right)$ predicted by each model have exactly the same mean, but their shapes are different. The model with a discrete trapping site, in particular, shows a shaper peak at short times combined with a long exponential tail [as readily seen on a logarithmic scale, Fig. 2(b)], resulting in a coefficient of variation exceeding 1. We note that a perfunctory inspection of Fig. 2(a) may suggest that the model with a trapping site predicts a narrower distribution as the slow-decaying distribution tail is not evident until the distribution is plotted on a log scale. This highlights the importance of a careful analysis of the tails of experimental distributions for distinguishing between different dynamical models. While this may impose an extra demand on the statistical quality of the experimental data, we remark that distribution tails obviously require lower time resolution than that required to analyze a distribution’s short-time behavior.

FIG. 2.

Distributions of the translocation time for the diffusion model with a potential well (model 1) and for the model with a discrete trapping site (model 2). (b) presents the same data as (a) but on logarithmic scale to highlight the long-time behavior of the distributions. The time is reported in dimensionless units with $L^2/D$ being the time unit. Other parameters are $k=2D/L^2$ and $\kappa =0.5kL$. The coefficient of variation is $C\approx 0.74<1$ for the one-dimensional diffusion model and $C\approx 1.26>1$ for the model with a trapping site.

DISCUSSION AND CONCLUSIONS

The main result of this study is that even in the case when both the equilibrium distribution of solutes partitioning into a membrane channel and the net rates of channel-facilitated transport are perfectly described by a one-dimensional diffusion model that utilizes the potential of mean force, such a description is not necessarily correct and may miss essential physics. It turns out that there are physical situations where modeling involving a localized binding site is necessary for the comprehensive description of solute dynamics in the channel.

An atomistic rationalization behind the two descriptions discussed above is that the solute molecules may interact with the channel in two different ways. For charged molecules and narrow channels, one can think about Coulombic interactions acting on every solute molecule that approaches the region of the channel containing charged amino acids.^12,32 In this case, every molecule passing through the channel interacts with these amino acids, and the description involving the potential of mean force is appropriate. Alternatively, for wide channels and neutral or zwitterionic solutes, interactions can be mostly limited to a site formed by a special arrangement of the residues of the channel protein.^1,33 In this case, solutes reversibly bind only upon direct collision with this arrangement and bypass it otherwise—the scenario described above by the trapping site model.

The difference in the translocation time distributions arising from the distinction between the two translocation mechanisms, one of which can be crudely thought of as involving an on-pathway intermediate (attractive potential well) and the other involving an off-pathway intermediate (a distinct trapping site), is in agreement with the results of a recent study,³¹ which showed that any 1D diffusion model leads to a coefficient variation below unity for the distribution of the transition path time. Thus, a multidimensional model is required to explain any distribution with C > 1. The trapping site model studied here is an example of such a multidimensional model as it allows for two parallel pathways, one bypassing the trapping site and the other involving trapping. Remarkably, mapping of this model onto 1D (i.e., the potential well model) results in correct prediction of kinetic properties such as the mean translocation time and the translocation probability; yet, a more careful analysis of the translocation time distribution and, particularly, its second moment shows that the 1D model misses essential physics. The increased experimental availability of the distributions of translocation times^20,22 (in the case of channel transport studies) and transition path times^21,34–38 (in the case of biomolecular folding and binding studies) thus offers an opportunity to zoom in on the physical mechanisms of the corresponding processes.

Finally, we note that, although this paper focuses on the case where the potential of mean force $U\left(x\right)$ is flat except at the location where binding takes place, both models can be easily extended to allow for arbitrary potential of mean force and/or multiple binding sites (although numerical solution of the Smoluchowski equation may be required). In particular, many experimental studies involve translocation driven by an electric field, which may be described by introducing an additional linear potential into the Smoluchowski equation. As long as the driving field is weak enough, the qualitative difference between the two types of models observed here should pertain.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.