The shape and free energy of a lipid bilayer surrounding a membrane inclusion

Membrane inclusion interactions are studied within the scope of continuum theory. We show that the free energy functional for the membrane thickness can be rewritten as a constant times a dimensionless integral. For cylindrical inclusions, the resulting differential equation gives a thickness profile that depends on the radius of the cylinder and one single lipid property, a correlation length that is determined by the ratio of the thickness compressibility and bending moduli. The solutions decay in a non-monotonic fashion with one single observable minimum. A solution for planar geometry may either be explicitly constructed or obtained by letting the radius of the cylinder go to infinity. In dimensionless units the initial derivative of the thickness profile is universal and equal to -1/√2. In physical units, the derivative depends on the size of the hydrophobic mismatch as well as the membrane correlation length and will usually be fairly small but clearly non-zero. The line tension between the protein inclusion and a fluid phase membrane will depend on the hydrophobic mismatch and be of the order of 10 pN (larger for the gel phase). This results in free energy costs for the inclusion that will be up to tens of kJ/mol (in the fluid phase).