Surgery in membrane fusion

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Abstract

The Gaussian curvature term of the Helfrich energy is topological, so a theory posed at fixed topological type discards it. Membrane fusion is a sequence of topology changes, and we determine what that term carries along one. The two leaflets divide a pathway’s surfaces into two classes, and the class imbalance $\Delta$, the difference of their director degrees, is insensitive to lipid tilt of any magnitude, equals the difference of the two classes’ surgery numbers, and has the parity of their sum, so hemifusion is an odd surgery number. A pore rim has $\int K \,\mathrm{d}A = -4\pi$ whatever its profile and its radii. Under a single saddle-splay modulus the energy depends on the surgery number alone, and one surgery is $91$ to $119\,k_BT$ at the measured modulus; that quantity enters no line tension, and the rim whose energy it is exists only after the nucleation barrier is crossed, so neither quantity reported for a pore bears on it. A bilayer of half-thickness $\ell$ has $\sup|K| < \ell^{-2}$ and bending energy nonincreasing along the constrained gradient flow, every topology change requires $\mathcal{W} \ge 8\pi$, and a merge dissipates at least $4\pi - \int_N (H^2 - K)\,\mathrm{d}A$, so the whole pathway performs at most $\mathcal{W}(0)/2\pi - \chi(0)/2 - 1$ events, which $k$ round vesicles attain.