Indentation of pore-spanning lipid membranes: Spring-stiffening or -softening responses and apparent stiffness prediction

Abstract

Atomic force microscopy (AFM)-based nanoindentation has emerged as a key nanomechanical mapping method to probe mechanical properties of two-dimensional engineering and biological materials. In comparison with extensive analytical studies on the indentation of freestanding solid thin films, surprisingly little attention has been drawn on analytical predictions of mechanical responses of freestanding lipid membranes to indentation in the framework of Helfrich membrane theory. Here we perform systematic theoretical studies with formulated analysis on indentation of nanopore-spanning lipid membranes. Two types of lipid membranes are considered. One is the membrane subject to uniform area stretching and the membrane is found to exhibit a spring-stiffening behavior during indentation, while the other is the membrane of constant tension and a spring-softening behavior at intermediate indentation is exhibited. The smaller the membrane tension is, the clearer the spring-softening feature is. For both types of membranes, analytical solutions of load–depth relationship at small indentation and apparent membrane stiffnesses are obtained at small and large tension. For the stretched lipid membranes, a cubic nonlinearity arising from membrane stretching is specified. Roles of membrane bending rigidity, (pre-)tension, or area compression modulus, and indenter size in membrane indentation are elucidated. Moreover, pressure of contact between the membrane and indenter and a line load of contact acting along the contact edge are obtained based on the shell theory. Our results provide fundamental insights on nonlinear mechanical behaviors of the lipid membrane indentation.