New Mathematical Models for Planar Lipid Bilyers in Bioinspired Microsystems

Lipid bilayers are currently being used for the development of many bioinspired microsystems ranging from portable and fast biosensors for detecting biological agents to biocompatible and biodegradable drug delivery carriers. Many of these microsystems work as proof-of-concept in laboratory environments but their application in real-world scenarios remains to be demonstrated due to the poor stability of lipid bilayers to mechanical stresses. An accurate characterization of the material properties of lipid bilayers, which is needed to enhance their performance, is limited by the challenges encountered in experimental in vestigations (e.g. measurement of stress and strain in the nanoscale range). Therefore, the formulation of new mathematical models is essential in making a big leap forward in the development of the next generation of bio-inspired microsystems that include lipid bilayers. We will present novel continuum models for the description of equilibrium configurations of planar lipid bilayers by accounting for their smectic A liquid crystallinity. These models represent a major improvement over existing continuum models since they incorporate significant molecular features of lipid bilayers (positional and orientational order) without requiring detailed molecular information. Unlike previous models, the proposed models capture the misalignment of the lipid molecules with the normal to the smectic layers and are derived within a new nonlinear theoretical framework for smectic A liquid crystals (IW Stewart 2007 Contin. Mech. Thermodyn. 18:343). The total energy of lipid bilayers consists of an elastic splay term, smectic layer bending and compression terms, a coupling term between the director and layer normal, a surface tension term, and a surface anchoring term. Nonlinear equilibrium equations are obtained by using variational methods and are then solved by analytical and numerical methods. The solutions illustrate the nonlinear deformations of lipid bilayers including the misalignment of lipid molecules at their interface with other media such as, for example, surface substrates.