Validity of the Electrodiffusion Model for Calculating Conductance of Simple Ion Channels

Abstract

We examine the validity and utility of the electrodiffusion (ED) equation, i.e., the generalized Nernst-Planck equation, to characterize, in combination with molecular dynamics, the electrophysiological behavior of simple ion channels. As models, we consider three systems-two naturally occurring channels formed by α-helical bundles of peptaibols, trichotoxin, and alamethicin, and a synthetic, hexameric channel, formed by a peptide that contains only leucine and serine. All these channels mediate transport of potassium and chloride ions. Starting with equilibrium properties, such as the potential of mean force experienced by an ion traversing the channel and diffusivity, obtained from molecular dynamics simulations, the ED equation can be used to determine the full current-voltage dependence with modest or no additional effort. The potential of mean force can be obtained not only from equilibrium simulations, but also, with comparable accuracy, from nonequilibrium simulations at a single voltage. The main assumptions underlying the ED equation appear to hold well for the channels and voltages studied here. To expand the utility of the ED equation, we examine what are the necessary and sufficient conditions for Ohmic and nonrectifying behavior and relate deviations from this behavior to the shape of the ionic potential of mean force.

Figure 1.

Plot of ln(1 − i/N) vs t_i to test if crossing events follow Poisson statistics for K⁺ ions (left) and Cl⁻ ions (right) for the LS3 channel (top), trichotoxin (TTX) channel (middle), and CD-Alm6 (ALM) channel (bottom).

Figure 2.

Free energy profiles for K⁺ ions (left) and Cl⁻ ions (right) for the LS3 channel (top), trichotoxin (TTX) channel (middle), and CD-Alm6 (ALM) channel (bottom). The PMF was obtained from equilibrium density profiles at 0 V (LS3 and TTX) and by way of ABF for CD-Alm6. The ED equation was used to compute the PMFs in the nonequilibrium simulations at several applied voltages.

Figure 3.

Voltage ramps computed from subtraction of the PMF from the nonequilibrium free energy profile for K⁺ ions (left) and Cl⁻ ions (right) for the LS3 channel (top), trichotoxin (TTX) channel (middle), and CD-Alm6 (ALM) channel (bottom). Values for the ramps are shown as colored dots for each applied voltage.

Figure 4.

Ion density profiles of K⁺ (left) and Cl⁻ (right) across the LS3 channel (upper) and trichotoxin (TTX) channel (lower panels) at several voltages.

Figure 5.

Magnitudes of the currents in the ED models of the LS3 channel computed as a function of the range of integration across the channel, from z_min = −16 to z = −12 to 16 Å.

Figure 6.

Relative errors in ED currents with respect to the MD values for K⁺ and Cl⁻ ions in the LS3 channel at 200, 100, and −100 mV. The x-axis is the length of integration range, z_max − z_min and the y-axis, Z, is (z_min + z_max)/2 defining the center of integration for each window.

Figure 7.

Current−Voltage curves for K⁺ ions in the LS3 channel calculated using the ED equation with the equilibrium free energy profile, A₀ (z), and the free energy profile, A₁ (z), reconstructed from the simulations at the applied voltage of 100 mV. The currents calculated from MD simulations are shown for comparison. The Ohmic current is for the equilibrium free energy profile.