The membrane potential and its representation by a constant electric field in computer simulations

Abstract

A theoretical framework is elaborated to account for the effect of a transmembrane potential in computer simulations. It is shown that a simulation with a constant external electric field applied in the direction normal to the membrane is equivalent to the influence of surrounding infinite baths maintained to a voltage difference via ion-exchanging electrodes connected to an electromotive force. It is also shown that the linearly-weighted displacement charge within the simulation system tracks the net flow of charge through the external circuit comprising the electromotive force and the electrodes. Using a statistical mechanical reduction of the degrees of freedom of the external system, three distinct theoretical routes are formulated and examined for the purpose of characterizing the free energy of a protein embedded in a membrane that is submitted to a voltage difference. The W-route is constructed from the variations in the voltage-dependent potential of mean force along a reaction path connecting two conformations of the protein. The Q-route is based on the average displacement charge as a function of the conformation of the protein. Finally, the G-route considers the relative charging free energy of specific residues, with and without applied membrane potentials. The theoretical formulation is illustrated with a simple model of an ion crossing a vacuum slab surrounded by two aqueous bulk phases and with a fragment of the voltage-sensor of the KvAP potassium channel.

FIGURE 1

Schematic representation of the reduction to a finite system and Poisson-Boltzmann calculation. (A) A thermodynamic membrane system with a transmembrane potential is separated into a microscopic subsystem (inner region) and its surrounding baths (outer region). The inner region (delineated by the dashed box) comprises the protein as well as the membrane and bulk solution in its neighborhood, while the outer region includes the aqueous salt solutions, the electrodes, and the EMF. The membrane is centered at z = 0 and extends in the xy plane. The electrode on side II is grounded (V = 0). (B) Representation of the same system with the degrees of freedom of the outer region integrated out and their influence on the inner region approximated using a continuum electrostatic Poisson-Boltzmann theory based on Eq. 4. (C) The effect of the transmembrane potential from an infinite thermodynamic system onto the finite subsystem is illustrated using a continuum electrostatic calculation based on the modified Poisson-Boltzmann theory (3). At the top is shown the dimensionless function φ_mp acting on a cubic central inner region of 80 Å on the side (indicated by the dashed line box). The dimensionless function φ_mp varies from 0.0 of the left (white) to 1.0 on the right (red); the contour levels indicated by colors and dashed lines are set to 0.0 (white), 0.20 (yellow), 0.35 (green), 0.50 (blue), 0.65 (purple), 0.80 (violet), and 1.0 (red). The variation of φ along the z axis cutting through the center of the box is shown at the bottom. The function φ was calculated by solving Eq. 4 numerically. In the inner region, the dielectric constant is set to 1 and the ionic screening constant is set to zero. For the aqueous phase of the outer region, a dielectric constant of 80 and an ionic concentration of 150 mM were assumed. The thickness of the membrane slab in the outer region is 20 Å and its dielectric constant is set to 2. The continuum electrostatic calculations were carried out using the PBEQ module of CHARMM using a cubic grid of 201 points spaced by 1 Å. The equation was solved by using the overrelaxation method and ∼500 iterations were required to reach convergence.

FIGURE 2

(A) Simulation systems. Simulation system with a K⁺ ion solvated by water. To mimic the effect of a nonpolar membrane region, the water molecules are prevented to enter between −10 and +10 Å using a half-harmonic potential. (B) Simulation system with a 41-residue fragment of the voltage sensor of the KvAP channel, starting from Pro⁹⁹ (near the center of the S3 helix, shown in red) to Ser¹³⁹ (near the C-terminus of the S4 helix, shown in blue). To mimic the effect of a nonpolar membrane region, the water molecules are prevented to enter between −10 and +10 Å using a half-harmonic potential. The structure of the S3-S4 segment is directly taken from the x-ray structure of the voltage sensor (PDB id 1ORS). The important charged residues are displayed as sticks and the position of their center of charge are (z = 0 is at the center of the membrane): Glu¹⁰⁷ (10.8 Å), Arg¹¹⁷ (14.1 Å), Arg¹²⁰ (9.7 Å), Arg¹²³ (10.0 Å), Arg¹²⁶ (4.4 Å), Arg¹³³ (−4.2 Å), and Lys¹³⁶ (−12.7 Å).

FIGURE 3

Potential of mean force analysis. (Top) Calculated PMF W(z;V) for a K⁺ ion along the z axis of the system shown in Fig. 2 at various transmembrane potentials. The PMFs were calculated by integrating the mean force acting on the K⁺ held fixed at a series of positions along the z axis. All the PMFs were offset to be equal to zero at z = −20 Å. The results for −5.0, −1.0, and −0.5 were reconstructed from the results with 5.0, 1.0, and 0.5 V by symmetry (i.e., using the property that W(z;V) → W(−z; −V) + B, where B is an offset constant). (Bottom) Perturbative effect of the applied voltage obtained by subtracting the PMF at zero voltage from all the other PMFs, ΔW(z;V) = W(z;V) – W(z;0). The color legend is 5.0 V (orange), 1.0 V (cyan), 0.5 V (magenta), 0.0 V (black), −0.5 V (blue), −1.0 V (red), and −5.0 V (green).

FIGURE 4

The dimensionless coupling of the K⁺ to the applied transmembrane voltage deduced from three different routes is shown. The W-route: voltage-dependent PMF technique is based on Eq. 33. The Q-route: average of the displacement charge based on Eq. 34; the offset in the displacement charge is caused by the apparent capacitance of the system according to Eq. 20. The G-route: relative charging free energy based on Eq. 35 (squares). The end-point averaging technique based on Eq. 30 is also shown (solid lines). The color legend is 5.0 V (orange), 1.0 V (cyan), 0.5 V (magenta), −0.5 V (blue), −1.0 V (red), and −5.0 V (green).

FIGURE 5

KvAP voltage sensor. Illustration of the charging FEP technique based on Eq. 29 to determine the fraction of the field for specific residues for the fragment of the voltage sensor of the KvAP bacterial channel. (Top) The fraction of the field felt at each of the charged residues is shown with the position of their center of charge along the z axis: Glu¹⁰⁷ (10.8 Å); Arg¹¹⁷ (14.1 Å); Arg¹²⁰ (9.7 Å); Arg¹²³ (10.0 Å); Arg¹²⁶ (4.4 Å); Arg¹³³ (−4.2 Å); and Lys¹³⁶ (−12.7 Å). (Bottom) The variation of the fraction of the field extract at different values of the thermodynamic coupling parameter λ during the FEP calculations. The position assigned to each residue along the z axis is based on the geometric center of charge