Derivation of a new free energy for biological membranes

A new free energy for thin biomembranes depending on chemical composition, degree of order and membranal-bending deformations is derived in this paper. This is a result of constitutive and geometric assumptions at the three-dimensional level. The enforcement of a new symmetry group introduced in (Deseri et al., in preparation) and a 3D--2D dimension reduction procedure are among the ingredients of our methodology. Finally, the identification of the lower order term of the energy (i.e. the membranal contribution) on the basis of a bottom-up approach is performed; this relies upon standard statistical mechanics calculations. The main result is an expression of the biomembrane free energy density, whose local and non-local counterparts are weighted by different powers of the bilayer thickness. The resulting energy exhibits three striking aspects:

(i)	the local (purely membranal) energy counterpart turns out to be completely determined through the bottom-up approach mentioned above, which is based on experimentally available information on the nature of the constituents;
(ii)	the non-local energy terms, that spontaneously arise from the 3D--2D dimension reduction procedure, account for both bending and non-local membranal effects;
(iii)	the non-local energy contributions turn out to be uniquely determined by the knowledge of the membranal energy term, which in essence represents the only needed constitutive information of the model. It is worth noting that the coupling among the fields appearing as independent variables of the model is not heuristically forced, but it is rather consistently delivered through the adopted procedure.

L. Deseri gratefully acknowledges the support received by (i) the Cofin-PRIN 2005-MIUR Italian Grant Mathematical and numerical modelling and experimental investigations for advanced problems in continuum and structural mechanics, (ii) the Department of Theoretical and Applied Mechanics at Cornell University and (iii) the Center for Non-linear Analysis under the National Science Foundation Grant No. DMS 0635983 and the Department of Mathematical Sciences, Carnegie-Mellon University.