The material properties of lipid bilayers can affect membrane protein function whenever conformational changes in the membrane-spanning proteins perturb the structure of the surrounding bilayer. This coupling between the protein and the bilayer arises from hydrophobic interactions between the protein and the bilayer. We analyze the free energy cost associated with a hydrophobic mismatch, i.e., a difference between the length of the protein's hydrophobic exterior surface and the average thickness of the bilayer's hydrophobic core, using a (liquid-crystal) elastic model of bilayer deformations. The free energy of the deformation is described as the sum of three contributions: compression-expansion, splay-distortion, and surface tension. When evaluating the interdependence among the energy components, one modulus renormalizes the other: e.g., a change in the compression-expansion modulus affects not only the compression-expansion energy but also the splay-distortion energy. The surface tension contribution always is negligible in thin solvent-free bilayers. When evaluating the energy per unit distance (away from the inclusion), the splay-distortion component dominates close to the bilayer/inclusion boundary, whereas the compression-expansion component is more prominent further away from the boundary. Despite this complexity, the bilayer deformation energy in many cases can be described by a linear spring formalism. The results show that, for a protein embedded in a membrane with an initial hydrophobic mismatch of only 1 A, an increase in hydrophobic mismatch to 1.3 A can increase the Boltzmann factor (the equilibrium distribution for protein conformation) 10-fold due to the elastic properties of the bilayer.

The energetics of protein-induced bilayer deformation in systems with finite monolayer equilibrium curvature were investigated using an elastic membrane model. In this model the bilayer deformation energy delta G(def) has two major components: a compression-expansion component and a splay-distortion component, which includes the consequences of a bilayer curvature frustration due to a monolayer equilibrium curvature, c(0), that is different from zero. For any choice of bilayer material constants, the value of delta G(def) depends on global bilayer properties, as described by the bilayer material constants, as well as the energetics of local lipid packing adjacent to the protein. We introduce this dependence on lipid packing through the contact slope, s, at the protein-bilayer boundary. When c(0) = 0, delta G(def) can be approximated as a biquadratic function of s and the monolayer deformation at the protein/bilayer boundary, u(0): delta G(def) = a(1)u(0)(2) + a(2)u(0)s + a(3)s(2), where a(1), a(2), and a(3) are functions of the bilayer thickness, the bilayer compression-expansion and splay-distortion moduli, and the inclusion radius (this expression becomes exact when the Gaussian curvature component of delta G(def) is negligible). When c(0) not equal 0, the curvature frustration contribution is determined by the choice of boundary conditions at the protein-lipid boundary (by the value of s), and delta G(def) is the sum of the energy for c(0) = 0 plus the curvature frustration-dependent contribution. When the energetic penalty for the local lipid packing can be ignored, delta G(def) will be determined only by the global bilayer properties, and a c(0) > 0 will tend to promote a local inclusion-induced bilayer thinning. When the energetic penalty for local lipid packing is large, s will be constrained by the value of c(0). In a limiting case, where s is determined only by geometric constraints imposed by c(0), a c(0) > 0 will impede such local bilayer thinning. One cannot predict curvature effects without addressing the proper choice of boundary conditions at the protein-bilayer contact surface.

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