Role of the dielectric constants of membrane proteins and channel water in ion permeation

Abstract

Using both analytical solutions obtained from simplified systems and numerical results from more realistic cases, we investigate the role played by the dielectric constant of membrane proteins epsilon(p) and pore water epsilon(w) in permeation of ions across channels. We show that the boundary and its curvature are the crucial factors in determining how an ion's potential energy depends on the dielectric constants near an interface. The potential energy of an ion outside a globular protein has a dominant 1/epsilon(w) dependence, but this becomes 1/epsilon(p) for an ion inside a cavity. For channels, where the boundaries are in between these two extremes, the situation is more complex. In general, we find that variations in epsilon(w) have a much larger impact on the potential energy of an ion compared to those in epsilon(p). Therefore a better understanding of the effective epsilon(w) values employed in channel models is desirable. Although the precise value of epsilon(p) is not a crucial determinant of ion permeation properties, it still needs to be chosen carefully when quantitative comparisons with data are made.

FIGURE 1

The simplified systems used in analytical solutions of the Poisson equation: (A) plane, (B) sphere, (C) cylinder, and (D) torus.

FIGURE 2

(A) The self energy of an ion as a function of the distance d from a spherical protein. The ion has a unit charge e and the radius of the sphere is a = 20 Å. The filled circles show the exact results obtained using Eq. 5 and the solid curve shows the approximate result obtained by substituting l → l + 1 in the denominator of Eq. 5. The dotted line is the self energy of an ion at a distance d from a plane boundary. (B) The interaction energy of the same ion with a unit charge fixed at a distance d_p = 2 Å from the protein surface. The meaning of the curves are the same as in A. In both figures, dielectric constants of ɛ_p = 2 and ɛ_w = 80 are employed.

FIGURE 3

Dependence of the self (A) and Coulomb (B) energies on the dielectric constants ɛ_p and ɛ_w. Both energies are normalized with the value at ɛ_p = 2 and ɛ_w = 80. The left side shows the effect of varying ɛ_p from 2 to 20 while ɛ_w is fixed at 80, and the right side shows the variation with ɛ_w while ɛ_p = 2. The radius of the sphere is a = 20 Å, the ion is at d = 4 Å from the protein surface, and the fixed charge is at d_p = 2 Å. The solid lines show the cases when the ion is outside the sphere, which nearly overlaps with the plane results (dotted lines).

FIGURE 4

Similar to Fig. 3 but for an ion at the center of a water-filled sphere (solid line) or on the central axis of an infinite cylinder (dashed line). The results are independent of the geometrical parameters for U_s and U_c in the sphere and for U_s in the cylinder. For U_c in the cylinder, d_p/a = 0.1 is employed. For larger ratios of d_p/a (corresponding to smaller cylinder radii), the dashed lines in B move slightly toward the solid lines. Note that U_c in the sphere overlaps with the unit line.

FIGURE 5

Similar to Fig. 3 but for an ion at the center (z = 0) and mouth (z = 40 Å) of a torus-shaped channel (solid lines). On each side, the lower solid line depicts the z = 0 result. The dotted and dashed lines show the results for the plane and cylinder boundaries from Figs. 3 and 4.

FIGURE 6

Self and Coulomb energy profiles of an ion with charge e along the central axis of a cylindrical channel. A cross section of the channel along its central axis is shown in the inset. The two Coulomb energy profiles correspond to the charge configurations at the mouth (solid line) and at the center of the channel (dashed line).

FIGURE 7

Similar to Fig. 3 but for an ion at the center (z = 0) of a finite cylindrical channel. In B, the lower solid line on each side shows U_c due to the protein charges at the mouth of the channel, whereas the upper one is due to the centrally placed charges. The dotted and dashed lines show the results for the plane and infinite cylinder boundaries from Figs. 3 and 4 as before.

FIGURE 8

The Coulomb energy of a test ion along the z axis due to a fixed ion on the z axis at −12.5 Å. The self energy of the test ion is not included in U_c. The dashed line is obtained using the canonical values of ɛ_p = 2 and ɛ_w = 80. The solid lines show how U_c changes from this reference result when ɛ_p is increased to 5 while keeping ɛ_w = 80 (lower curve), and when ɛ_w is reduced to 40 while keeping ɛ_p = 2 (upper curve). The dotted line at the bottom shows U_c in bulk water.