Protein-induced bilayer deformations: the lipid tilt degree of freedom

The theory of hydrophobic interaction between a transmembrane protein and a lipid bilayer is reinvestigated. The protein is modeled as a cylindrically symmetric rigid inclusion, residing in a symmetric, tension-free lipid bilayer. The hydrophobic coupling between the inclusion and the lipids may induce an elastic bilayer deformation, which is commonly described in terms of stretching (or compressing) the hydrocarbon chains of the lipids. In the present work, we additionally include the possibility of the average lipid director to tilt with respect to the normal direction of the hydrocarbon-water interface. The corresponding membrane deformation energy is expressed using both a phenomenological description of elastic lipid layer perturbations and employing a specific molecular lipid model. The molecular lipid model accounts for head group repulsions, interfacial tension, and the chain conformational free energy. Assuming incompressibility of the hydrocarbon chains, we estimate and compare typical membrane deformation energies induced by single gramicidin A channels, with and without the lipid tilt degree of freedom taken into account. The membrane deformation energies are conveniently expressed using a spring constant. We argue that the consideration of the lipid tilt degree of freedom leads to a severalfold reduction of the spring constant and should thus not be excluded from the description of protein-induced membrane deformations. Possible limits of membrane elasticity-based theories for lipid-protein interactions are discussed. Finally, we calculate inclusion-induced deformations of electrostatically charged bilayers, illuminating the coupling between electrostatic and elastic energies in charged membranes.