A fluctuating elastic plate and a cell model for lipid membranes

The thermal fluctuations of lipid bi-layer membranes are key to their interaction with cellular components as well as the measurement of their mechanical properties. Typically, membrane fluctuations are analyzed by decomposing into normal modes or by molecular simulations. Here we propose two new approaches to calculate the partition function of a membrane. In the first approach we view the membrane as a fluctuating von Karman plate and discretize it into triangular elements. We express its energy as a function of nodal displacements, and then compute the partition function and co-variance matrix using Gaussian integrals. We recover well-known results for the dependence of the projected area of the membrane on the applied tension and recent simulation results on the dependence of membrane free energy on geometry, spontaneous curvature and tension. As new applications we compute the fluctuations of the membrane of a malaria infected cell and analyze the effects of boundary conditions on fluctuations. Our second approach is based on the cell model of Lennard-Jones and Devonshire. This model, which was developed for liquids, assumes that each molecule fluctuates within a cell on which a potential is imposed by all the surrounding molecules. We adapt the cell model to a lipid membrane by recognizing that it is a 2D liquid with the ability to deform out of plane whose energetic penalty must be factored into the partition function of a cell. We show, once again, that some results on membrane fluctuations can be recovered using this new cell model. However, unlike some well established results, our cell model gives an entropy that scales with the number of molecules in a membrane. Our model makes predictions about the heat capacity of the membrane that can be tested in experiments.