Intrinsic diffusion resistance of a membrane channel, mean first-passage times between its ends, and equilibrium unidirectional fluxes

Abstract

Diffusive flux of solute molecules through a membrane channel driven by the solute concentration difference on the two sides of the membrane is inversely proportional to the channel diffusion resistance. We show that the intrinsic, channel proper, part of this resistance is the ratio of the sum of the mean first-passage times of the molecule between the channel ends and the molecule partition function in the channel. This is derived without appealing to any specific model of the channel and, therefore, is applicable to transport in channels of arbitrary shape and tortuosity and at arbitrary interaction strength of solute molecules with the channel walls.

An important characteristic of channel-facilitated membrane transport that is widely used in chemical engineering, electrochemistry, and cell biophysics, is the so-called “diffusion resistance.”^1–10 This resistance is a diffusion analog of the electrical resistance. It relates the steady-state diffusive flux of solute molecules through a membrane channel with the driving force of the transport process, the solute concentration difference in the two reservoirs separated by the membrane. In this work, we derive a general relation between the channel diffusion resistance and the mean first-passage times of the solute molecules between the channel ends, assuming perfect steering in both reservoirs. Specifically, we show that this direction-independent characteristic of transport is equal to the sum of the direction-dependent mean first-passage times of the molecule between the channel ends, divided by the molecule partition function in the channel. Our analysis is quite general in the sense that it does not appeal to any specific model of the channel and, therefore, is universally applicable to transport in channels of arbitrary shape and tortuosity, at arbitrary interaction strength of solute molecules with the channel walls.

Consider solute molecules freely diffusing in two reservoirs separated by a membrane. The molecules can pass from one reservoir to the other through a membrane channel connecting the reservoirs. In the absence of the concentration gradient, the system is in equilibrium with the molecule concentration in the reservoirs equal to c. When this concentration is low enough, the channel is mainly empty or occupied by one molecule. Under such condition, the interactions between solute molecules in the channel can be neglected, and their equilibrium concentration in the channel, c_eq(x, y, z), is given by the Boltzmann distribution,

$$c_{eq}(x,y,z)=c\exp \left[-\beta V(x,y,z)\right].$$ (1)

Here, V(x, y, z) is the potential energy of the molecule at the point (x, y, z) inside the channel, and $\beta =1/\left(k_{B}T\right)$ with k_B and T denoting the Boltzmann constant and absolute temperature, respectively. We consider molecules as point particles. This implies that we use the values of all geometric parameters of the channel corrected by the finite size of the translocating molecules.¹¹

It is assumed that the membrane is flat, and its left and right boundaries are located at x = x_L and x = x_R, respectively, where the x-axis is directed perpendicular to the membrane. It is also assumed that function V(x, y, z) is localized within the channel and vanishes at the channel ends, V(x_L, y, z) = V(x_R, y, z) = 0. The condition of low molecule concentration,

$$\iiint c_{eq}(x,y,z)dxdydz\ll 1,$$ (2)

where the integration is performed over the channel volume, implies that

$$c\ll 1/Z_{ch},Z_{ch}=\iiint \exp \left[-\beta V(x,y,z)\right]dxdydz,$$ (3)

with Z_ch denoting the molecule partition function in the channel.

Consider an ensemble of identical copies of such a system. In each copy containing a molecule in the channel, this molecule entered the channel either through its left or right opening. Thus, the equilibrium molecule concentration in the channel is the sum of two terms,

$$c_{eq}(x,y,z)=c_{eq}^{(L)}(x,y,z)+c_{eq}^{(R)}(x,y,z),$$ (4)

where the conditional equilibrium concentrations $c_{eq}^{(L)}(x,y,z)$ and $c_{eq}^{(R)}(x,y,z)$ are the contributions to c_eq(x, y, z) due to the molecules entering the channel through its left and right boundaries, respectively. At the channel ends, these concentrations (by construction) are $c_{eq}^{(L)}(x_L,y,z)=c$, $c_{eq}^{(L)}(x_R,y,z)=0$, $c_{eq}^{(R)}(x_L,y,z)=0$, and $c_{eq}^{(R)}(x_R,y,z)=c$.

The equilibrium in the system is dynamic. There are equilibrium unidirectional fluxes of molecules between the reservoirs flowing through the channel in opposite directions. Molecules entering the channel from the left reservoir and exiting to the right one form the left-to-right equilibrium unidirectional flux. Correspondingly, molecules passing through the channel in the opposite direction form the right-to-left equilibrium unidirectional flux. These fluxes, $J_{L\to R}^{eq}$ and $-J_{R\to L}^{eq}$ (both $J_{L\to R}^{eq}$ and $J_{R\to L}^{eq}$ are positive), are equal in magnitude,

$$J_{L\to R}^{eq}=J_{R\to L}^{eq},$$ (5)

so that they compensate each other, and the net flux through the channel is zero.

There are non-trivial relationships between the conditional equilibrium concentrations $c_{eq}^{(L)}(x,y,z)$ and $c_{eq}^{(R)}(x,y,z)$ and the magnitude of the equilibrium unidirectional fluxes $J_{L\to R}^{eq}$ and $J_{R\to L}^{eq}$. Namely, the integral of $c_{eq}^{(L)}(x,y,z)$ over the channel volume divided by the flux $J_{L\to R}^{eq}$ is equal to the molecule mean first-passage time from the left channel boundary to the right one, conditional on the left boundary being reflecting. Denoting this mean first-passage time by ${\tau }_{FP}^{ref}\left(L\to R\right)$, we can write

$$\frac{1}{J_{L\to R}^{eq}}\iiint c_{eq}^{(L)}(x,y,z)dxdydz={\tau }_{FP}^{ref}\left(L\to R\right).$$ (6)

Similarly, for molecule transitions through the channel in the opposite direction, we have

$$\frac{1}{J_{R\to L}^{eq}}\iiint c_{eq}^{(R)}(x,y,z)dxdydz={\tau }_{FP}^{ref}\left(R\to L\right),$$ (7)

where ${\tau }_{FP}^{ref}\left(R\to L\right)$ is the mean first-passage time from the reflecting right channel boundary to the left one.

To prove the identity in Eq. (6) consider a hypothetical situation where the left and right channel openings are reflecting and absorbing boundaries for diffusing molecules, respectively. Suppose that the flux $J_{L\to R}^{eq}$ is constantly injected into the channel near its reflecting left boundary. Then, the steady-state concentration of the molecules in the channel is equal to the conditional equilibrium concentration $c_{eq}^{(L)}(x,y,z)$. Keeping this in mind, one can see that the relation in Eq. (6) is a special case of the general expression for the mean first-passage time obtained by Reimann et al.¹²

The sum of the mean first-passage times ${\tau }_{FP}^{ref}\left(L\to R\right)$ and ${\tau }_{FP}^{ref}\left(R\to L\right)$ is

$${\tau }_{FP}^{ref}\left(x_L\to x_R\right)+{\tau }_{FP}^{ref}\left(x_R\to x_L\right)=\frac{1}{J_{L\to R}^{eq}}\iiint c_{eq}(x,y,z)dxdydz,$$ (8)

where we have taken advantage of the fact that the equilibrium unidirectional fluxes are equal in magnitude, Eq. (5), and that the sum of the conditional equilibrium concentrations $c_{eq}^{(L)}(x,y,z)$ and $c_{eq}^{(R)}(x,y,z)$ is the equilibrium molecule concentration in the channel c_eq(x, y, z), Eq. (4). Using Eq. (1), one can see that the integral entering Eq. (8) is the product of the equilibrium bulk concentration c and the molecule partition function in the channel Z_ch, Eq. (3). Thus, we can write Eq. (8) as

$${\tau }_{FP}^{ref}\left(x_L\to x_R\right)+{\tau }_{FP}^{ref}\left(x_R\to x_L\right)=\frac{c}{J_{L\to R}^{eq}}{Z}_{ch}.$$ (9)

The ratio $c/J_{L\to R}^{eq}$ in the above equation is the intrinsic diffusion resistance of the membrane channel. When the solute concentrations c_L and c_R on the two sides of the membrane are not equal, c_L ≠ c_R, there is a steady-state flux J of the solute molecules through the channel. This flux is proportional to the concentration difference,

$$J=\frac{c_L-c_R}{R_{dif}},$$ (10)

where R_dif is the channel diffusion resistance. At low solute concentrations, R_dif is independent of c_L and c_R and depends only on the channel geometry and the solute interaction strength with the channel walls. Total diffusion resistance of a membrane channel consists of three parts: left and right access resistances associated with the molecule entrance/exit transitions between the channel and the left and right reservoirs, respectively, and the intrinsic diffusion resistance of the channel proper. The access resistances are inversely proportional to the solute diffusivity in the bulk, whereas the intrinsic diffusion resistance is a function of the solute diffusivity in the channel.

When the solutions on both sides of the membrane are well stirred (ws), or the bulk diffusivity of the solute molecules, denoted by D_b, tends to infinity, the access resistances vanish since under such conditions, the solute concentrations near the channel openings are equal to their bulk values c_L and c_R. As a consequence, in this case, the channel diffusion resistance is equal to its intrinsic part, which we denote by R_ws, ${\left.R_{ws}=R_{dif}\right|}_{D_b\to \infty }$, and Eq. (10) reduces to

$$J_{ws}={\left.J\right|}_{D_b\to \infty }=\frac{c_L-c_R}{{\left.R\right|}_{D_b\to \infty }}=\frac{c_L-c_R}{R_{ws}}.$$ (11)

Thus, we arrive at the following expression for the intrinsic diffusion resistance

$$R_{ws}=\frac{c_L-c_R}{J_{ws}}.$$ (12)

One can see that the ratio $c/J_{L\to R}^{eq}$ in Eq. (9) is just the intrinsic diffusion resistance of the channel,

$$R_{ws}=\frac{c}{J_{L\to R}^{eq}}=\frac{c}{J_{R\to L}^{eq}}.$$ (13)

This allows us to write Eq. (9) as

$${\tau }_{FP}^{ref}\left(x_L\to x_R\right)+{\tau }_{FP}^{ref}\left(x_R\to x_L\right)=R_{ws}{Z}_{ch}$$ (14)

and to express the channel intrinsic resistance in terms of the mean first-passage times of the molecule between the channel ends and the molecule partition function in the channel,

$$R_{ws}=\frac{{\tau }_{FP}^{ref}\left(x_L\to x_R\right)+{\tau }_{FP}^{ref}\left(x_R\to x_L\right)}{Z_{ch}}.$$ (15)

This and the expressions for the mean first-passage times between the channel ends, Eqs. (6) and (7), are main results of the present work.

As an illustration of the general relation in Eq. (15), consider the case of free diffusion, V(x, y, z) = 0, through a straight cylindrical channel of smoothly varying radius R(x), $\left|dR(x)/dx\right|<1$,¹³ schematically shown in Fig. 1. Here, explicit expressions for the diffusion resistance and the mean first-passage times between the channel ends can be obtained since the three-dimensional molecule diffusion dynamics in the channel can be approximated as effectively one-dimensional and described in terms of the generalized Fick–Jacobs equation.^14–20

FIG. 1.

Channel of a smoothly varying radius.

Let the concentrations of the solute molecules in the left and right reservoirs, respectively, be c_L and c_R. Denoting the position-dependent molecule diffusion coefficient in the channel by D(x) and the channel cross section area by A(x) = πR²(x), we can write the steady-state flux J through the channel as

$$J=-D(x)A(x)\frac{d}{dx}\left[\frac{c_1(x)}{A(x)}\right],x_L\le x\le x_R,$$ (16)

where c₁(x) is the one-dimensional steady-state concentration of the solute molecules in the channel, measured in units of molecules/length. The above equation should be solved subject to the boundary conditions c₁(x_L) = c_LA(x_L) and c₁(x_R) = c_RA(x_R), which assume perfect stirring in both reservoirs (or D_b → ∞). Dividing both sides of Eq. (16) by the product D(x)A(x), we obtain

$$\frac{J}{D(x)A(x)}=-\frac{d}{dx}\left[\frac{c_1(x)}{A(x)}\right].$$ (17)

Next, we integrate both sides of this equality over x from x_L to x_R. The result is

$$J\underset{x_L}{\overset{x_R}{\int }}\frac{dx}{D(x)A(x)}=-{\left.\frac{c_1(x)}{A(x)}\right|}_{x_L}^{x_R}=c_L-c_R,$$ (18)

where we have used the boundary conditions at the channel ends mentioned above. Thus, the steady-state flux through the channel is given by

$$J=\frac{c_L-c_R}{\underset{x_L}{\overset{x_R}{\int }}\frac{dx}{D(x)A(x)}},$$ (19)

and hence, the intrinsic diffusion resistance of the channel is

$$R_{ws}=\frac{c_L-c_R}{J}=\underset{x_L}{\overset{x_R}{\int }}\frac{dx}{D(x)A(x)}.$$ (20)

Now, we compare the obtained expression for R_ws with the sum of the mean first-passage times ${\tau }_{FP}^{ref}\left(x_L\to x_R\right)$ and ${\tau }_{FP}^{ref}\left(x_R\to x_L\right)$. For the case under consideration, these mean first-passage times are given by (see, e.g., Chap. XII of Ref. 21)

$${\tau }_{FP}^{ref}\left(x_L\to x_R\right)=\underset{x_L}{\overset{x_R}{\int }}\frac{dy}{D(y)A(y)}\underset{x_L}{\overset{y}{\int }}A(x)dx$$ (21)

and

$${\tau }_{FP}^{ref}\left(x_R\to x_L\right)=\underset{x_L}{\overset{x_R}{\int }}\frac{dy}{D(y)A(y)}\underset{y}{\overset{x_R}{\int }}A(x)dx.$$ (22)

The sum of these mean first-passage times is

$${\tau }_{FP}^{ref}\left(x_L\to x_R\right)+{\tau }_{FP}^{ref}\left(x_R\to x_L\right)=\left(\underset{x_L}{\overset{x_R}{\int }}\frac{dy}{D(y)A(y)}\right)\left(\underset{x_L}{\overset{x_R}{\int }}A(x)dx\right).$$ (23)

The term in the first parentheses on the right-hand side of the above equality is the intrinsic diffusion resistance in Eq. (20). The term in the second parentheses is the channel volume, which is the partition function of the channel when V(x, y, z) = 0, $Z_{ch}=\underset{x_L}{\overset{x_R}{\int }}A(x)dx$ [see Eq. (3)]. Thus, the equality in Eq. (23) is a special case of the general relation between the channel intrinsic diffusion resistance and the sum of the mean first-passage times between the channel ends, Eq. (14), for the case of free diffusion through a channel of smoothly varying radius.

To summarize, main results of this work are the expressions for the mean first-passage time between the ends of the membrane channel, Eqs. (6), (7), and (15), which establishes the relation between these mean first-passage times and the intrinsic diffusion resistance of the channel. The diffusion resistance is equal to the sum of the forward and backward mean first-passage times divided by the molecule partition function in the channel. In deriving the relation in Eq. (15), we have not made any assumptions about the channel geometry and/or molecule interaction strength with the channel walls. As a consequence, this relation is quite general and applicable to any membrane channel. It is interesting that a similar relation was obtained in the theory of activated rate processes [see Eqs. (17), (20), and (21) in Ref. 22].

Our analysis is based on the consideration of the equilibrium unidirectional fluxes flowing through the channel in the opposite directions. These fluxes are formed by the molecules entering the channel from the left and right reservoirs. Therefore, the key step in our approach is the splitting of the equilibrium molecule concentration in the channel into two conditional equilibrium concentrations, Eq. (4). Thus, our analysis provides a new insight into the relation between equilibrium and transport properties of membrane channels.

It is worth indicating that the relation in Eq. (15) can be instrumental in computing the intrinsic diffusion resistance of the channel numerically. The point is that it allows one to avoid dealing with transport under non-equilibrium conditions, when the molecule concentrations on the two sides of the membrane are different. Instead, one can determine the mean first-passage times of the molecule between the channel ends and the molecule partition function in the channel that can be done with relative ease. Substituting these quantities into Eq. (15), one finds the diffusion resistance.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts of interest to disclose.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.