Revisiting the Link between Lipid Membrane Elasticity and Microscopic Continuum Models

Abstract

Fluid lipid membranes can be described by a continuum-elastic Hamiltonian that features two central parameters mean and Gaussian curvature modulus. Of those two, the Gaussian modulus is much less understood, since it affects the membrane energetics only through topology or boundaries and is thus difficult to measure experimentally. Moreover, recent work by Hu et al. [Biophys. J. 102, 1403 (2012)] to determine this modulus from computational studies revealed discrepancies with more microscopic expectations, according to which this modulus should be given by the second moment of the lateral trans-membrane stress profile. Our goal in the present study is to revisit the arguments linking the Gaussian modulus to properties of thin sheets, using thin plate theory as a starting point, but allowing for a number of generalizations. For instance, membranes are more generally described as anisotropic continua with in-plane fluidity and a nontrivial distribution of pre-stresses. It is easy to see that in this case linear elasticity, combined with the additional but common approximation of a linear strain tensor, will predict that the Gaussian curvature modulus vanishes. We therefore need to include the possibility of nonlinear strains (while still using linear constitutive equations), but this in turn leads to some subtle inconsistencies within Monge gauge. Another difficulty is that virtually all existing theories treat Young's modulus and the Poisson ratio as constant throughout the width of the membrane. Starting from fundamental linear elasticity theory, amended by a nonlinear strain tensor, and using consistent geometrical constructions, we then investigate the elastic properties of membranes, especially the Gaussian curvature modulus.