Influence of nonlinear electrostatics on transfer energies between liquid phases: charge burial is far less expensive than Born model

Abstract

The widely used Born model describes the electrostatic response of continuous media using static dielectric constants. However, when applied to a liquid environment, a comparison of Born model predictions with experimental values (e.g., transfer free energies and pK(a) shifts) found that agreement is only achieved by using physically unrealistic dielectric constants for proteins, lipids, etc., and/or equally unrealistic atomic radii. This leads to questions concerning the physical origins for this failure of the Born model. We partially resolve this question by applying the Langevin-Debye (LD) model of a continuous distribution of point, polarizable dipoles, a model that contains an added dependence of the electrostatic response on the solvent's optical dielectric constant and both gas- and liquid-phase dipole moments, features absent in the Born model to which the LD model reduces for weak fields. The LD model is applied to simple representations of three biologically relevant systems: (i) globular proteins, (ii) lipid bilayers, and (iii) membrane proteins. The linear Born treatment greatly overestimates both the self-energy and the transfer free energy from water to hydrophobic environments (e.g., a protein interior). By using the experimental dielectric constant, the energy cost of charge burial in either globular or membrane proteins of the Born model is reduced by almost 50% with the nonlinear theory as is the pK(a) shift, and the shifts agree well with experimental trends.

Fig. 1.

Spatial dependence of ε(r). (A) The dependence of the relative permittivity ε(r) for a unit charge at a radial distance r from the ion in water (continuous line), proteins (dashed line, also see Inset), and lipids (dotted line). (B) The shift in the relative permittivity for partial charges in water. The three different partial charges are Z = 1 (solid), Z = 0.5 (dashed), and Z = 0.1 (dotted).

Fig. 2.

The geometries of the three biological systems. (A) spherical protein: R is the radius of the protein and h is the distance of the ion from the center of the protein; (B) Lipid bilayer: d is the width of the membrane and h is the relative position of the ion from the center of the membrane. (C) Cylindrical membrane protein: R is the radius of the protein, whereas d and h are as in B.

Fig. 3.

The self-energy of an ion (A) and its value relative to the counterpart in pure water (B) as a function of distance from the center of a spherical protein. The nonlinear calculation (from Eq. 9) is solid, whereas the linear (Born) energy (from Eq. 13) is dashed. The radii are protein R = 13.4 Å and ion a = 1.4 Å, and the ion charge is q = 1. The dotted vertical line represents the boundary between the protein and water.

Fig. 4.

Distance dependence of apparent dielectric constant and pK_a shift. (A) The apparent dielectric constant of a 100-residue protein as estimated from equating the Born equation (Eq. 13) with our nonlinear solvation treatment (Eq. 9). The region between the dashed vertical lines denotes the interface where ions are partly solvated by the protein and partly by water. (B) The predicted pK_a shift when transferring a charge from the surface to the interior of the protein as calculated from the nonlinear (solid) and linear (dashed) treatment, respectively. The buried depth is the distance of the ion from the protein's surface.

Fig. 5.

The self-energy of an ion (A) and its value relative to the counterpart in pure water (B) as a function of relative position from the center of a lipid bilayer. The nonlinear calculation is solid, whereas the linear (Born) energy is dashed. The membrane width is d = 30 Å, the ion radius is a = 1.4 Å, and the ion charge is q = 1. The dotted vertical lines represent the boundaries between the lipid and water.

Fig. 6.

The self-energy of an ion (A) and its value relative to the counterpart in pure water (B) as a function of relative position from the center of a lipid bilayer in which a cylindrical protein is embedded. The nonlinear calculation is solid, whereas the linear (Born) energy is dashed. The membrane width is d = 30 Å, protein radius is R = 10 Å, ion radius is a = 1.4 Å, and ion charge is q = 1. The dotted vertical lines represent the boundary between the membrane and water.