  Lipid bilayer membranes are flexible thin laterally fluid films consisting of
two unimolecular layers of lipids. On spatial scales much larger than the
bilayer thickness, the membrane elasticity is well determined by its shape and
adequately described by the classical Helfrich Hamiltonian. However, various
local membrane heterogeneities can result in a lipids tilt relative to the
membrane surface normal. On the basis of the classical elasticity theory of 3D
bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] derived the most
general energy functional, taking into account the tilt and bending. Recently,
Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)] showed that Hamm and
Kozlov's derivation was incomplete because the tilt-curvature coupling term had
been missed. However, the energy functional derived by Terzi and Deserno
appeared to be unstable, thereby being invalid for applications. Here, we
derive a stable elastic energy functional, showing that the squared gradient of
the curvature was missed in both of these works. This change in the energy
functional arises from a more accurate consideration of the transverse shear
deformation terms and their influence on the membrane stability. We also
consider the influence of the prestress terms on the stability of the energy
functional, and we show that the effective Gaussian curvature should be
neglected because of the stability requirements. We further generalize the
theory, including the stretching-compressing deformation modes, and we provide
the geometrical interpretation of the terms that were previously missed by Hamm
and Kozlov. The physical consequences of the new terms are analyzed in the case
of a membrane-mediated interaction of two amphipathic peptides located in the
same monolayer. We also provide the expression for director fluctuations,
comparing it with that obtained by Terzi and Deserno.
