Theoretical analysis of hydrophobic matching and membrane-mediated interactions in lipid bilayers containing gramicidin

We present a quantitative analysis of the effects of hydrophobic matching and membrane-mediated protein-protein interactions exhibited by gramicidin embedded in dimyristoylphosphatidylcholine (DMPC) and dilauroylphosphatidylcholine (DLPC) bilayers (Harroun et al., 1999. Biophys. J. 76:937-945). Incorporating gramicidin, at 1:10 peptide/lipid molar ratio, decreases the phosphate-to-phosphate (PtP) peak separation in the DMPC bilayer from 35.3 Å without gramicidin to 32.7 Å. In contrast, the same molar ratio of gramicidin in DLPC increases the PtP from 30.8 Å to 32.1 Å. Concurrently, x-ray in-plane scattering showed that the most probable nearest-neighbor separation between gramicidin channels was 26.8 Å in DLPC, but reduced to 23.3 Å in DMPC. In this paper we review the idea of hydrophobic matching in which the lipid bilayer deforms to match the hydrophobic surface of the embedded proteins. We use a simple elasticity theory, including thickness compression, tension, and splay terms to describe the membrane deformation. The energy of membrane deformation is compared with the energy cost of hydrophobic mismatch. We discuss the boundary conditions between a gramicidin channel and the lipid bilayer. We used a numerical method to solve the problem of membrane deformation profile in the presence of a high density of gramicidin channels and ran computer simulations of 81 gramicidin channels to find the equilibrium distributions of the channels in the plane of the bilayer: The simulations contain four parameters: bilayer thickness compressibility 1/B, bilayer bending rigidity K(c), the channel- bilayer mismatch D(o), and the slope of the interface at the lipid-protein boundary s. B, K(c), and D(o) were experimentally measured; the only free parameter is s. The value of s is determined by the requirement that the theory produces the experimental values of bilayer thinning by gramicidin and the shift in the peak position of the in-plane scattering due to membrane- mediated channel-channel interactions. We show that both hydrophobic matching and membrane-mediated interactions can be understood by the simple elasticity theory.