Topological Approach to Measuring the Gaussian Curvature Modulus of Lipid Membranes in Simulation

Fission and fusion of lipid membranes are ubiquitous shape transformations in living cells, necessary for maintaining the shape of organelles and enabling a host of transport processes between them. These topology-changing events involve curvature-elastic free energy changes proportional to the Gaussian curvature modulus, κ[over ¯]. However, precisely because of its topological nature, the value of this modulus is exceptionally hard to determine. Here we propose a novel method to computationally determine κ[over ¯] that involves simulating triply periodic minimal surfaces. We measure their excess curvature energy E and infer the associated free energy F via a thermodynamic argument that rests on the strong temperature dependence of E and the relatively weak temperature dependence of E/F. Our approach is unique in that it circumvents the complications arising in alternative procedures that require an open membrane edge to break the constraints of the Gauss-Bonnet theorem. As an illustration, we determine κ[over ¯] for a coarse-grained lipid model over a range of temperatures and lipid shapes. We show that the observed changes of both the Gaussian curvature modulus as well as its ratio to the ordinary bending modulus follow widely held expectations.