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Formation of membrane necks is crucial for fission and fusion in lipidbilayers. In this work, we seek to answer the following fundamental question:what is the relationship between protein-induced spontaneous mean curvature andthe Gaussian curvature at a membrane neck? Using an augmented Helfrich modelfor lipid bilayers to include membrane-protein interaction, we solve the shapeequation on catenoids to find the field of spontaneous curvature that satisfiesmechanical equilibrium of membrane necks. In this case, the shape equationreduces to a variable coefficient Helmholtz equation for spontaneous curvature,where the source term is proportional to the Gaussian curvature. We show howthis latter quantity is responsible for non-uniform distribution of spontaneouscurvature in minimal surfaces. We then explore the energetics of catenoids withdifferent spontaneous curvature boundary conditions and geometric asymmetriesto show how heterogeneities in spontaneous curvature distribution can couplewith Gaussian curvature to result in membrane necks of different geometries.

Gaussian curvature directs the distribution of spontaneous curvature on bilayer membrane necks