Assessing smectic liquid-crystal continuum models for elastic bilayer deformations

Abstract

For four decades, since W. Helfrich's pioneering study of smectic A liquid crystals in 1973, continuum elastic models (CEMs) have been employed as tools to understand the energetics of protein-induced lipid bilayer deformations. Among the assumptions underlying this use is that all relevant protein-lipid interactions can be included in the continuum representation of the protein-bilayer interactions through the physical parameters determined for protein-free bilayers and the choice of boundary conditions at the protein/bilayer interface. To better understand this assumption, we review the general structure of CEMs, examine how different choices of boundary conditions and physical moduli profiles alter the predicted bilayer thickness profiles around gramicidin A (gA) and mitochondrial voltage-dependent anion channels (VDAC), respectively, and compare these profiles with those obtained from all-atom molecular dynamics simulations. We find that the profiles differ qualitatively in the first lipid shell around the channels, indicating that the CEMs do not capture accurately the consequences of the protein-induced local changes in lipid bilayer dynamics. Therefore, one needs to be careful when interpreting the results of CEM-based analyses of lipid bilayer-membrane protein interactions.

Fig. 1

Bilayer deformation profiles upon inclusion of a cylindrically symmetric protein. (A) Positive hydrophobic mismatch (l > d₀) and (B) negative hydrophobic mismatch (l < d₀), where l represents the hydrophobic length of the transmembrane protein and d₀ represents that of the unperturbed lipid bilayer. The hydrophobic match condition is used at the protein-lipid contact (r = r₀) and the unperturbed membrane condition is assumed at r = r_∞. Also, symmetry of the upper and lower leaflets is assumed.

Fig. 2

The membrane (hydrophobic thickness) profiles obtained and smoothed from the MD simulations of (A) gA and (B) VDAC systems. The dotted cylinder represents the cylindrical approximation of the protein structure in the CEM and the radii are 7.5 A̋ (gA) and 21.5 A̋ (VDAC), respectively.

Fig. 3

Space-dependent modulus profiles used for the gA system. We used three different choices of moduli: first, the uniform profile, which assumes that the bulk values apply throughout the system (not shown); second, the space-dependent modulus profile from Partenskii and Jordan (PJ) (Partenskii and Jordan, 2002); as well as four profiles determined from MD-based areal compressibility profiles.

Fig. 4

Comparison of the CEM-derived thickness profiles with the MD-derived profiles for the gA systems in membranes composed of (A) DLPC, (B), DMPC, (C) DOPC, and (D) POPC.

Fig. 5

Comparison of the thickness profiles around VDAC from MD and CEM. The black lines are for the MD-derived profile (dotted) and its smoothed profile (solid). The boundary conditions for the CEMs were from the MD profiles (s₀ = s_MD) with uniform modulus (red) and the PJ model (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)