Leaflet invariants and topology change in membrane fusion

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Abstract

The Gaussian curvature term of the Helfrich energy is topological, and a theory posed at fixed topological type therefore discards it. Membrane fusion is a sequence of topology changes, and we study what that term records along one. The two leaflets divide such 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. For a surface of revolution the integrated Gaussian curvature depends only on the profile’s radial direction at its endpoints, so a pore rim carries $-4\pi$ whatever its shape or its radii. Under a single saddle-splay modulus the energy depends on the surgery number alone, and at the measured modulus one surgery is $91$ to $119\,k_BT$. That quantity is the whole discontinuity of the Canham–Helfrich energy at a surgery and is independent of the pore radius, so it enters no line tension, and the nucleation free energy reported for a pore is recorded before any rim exists; neither measurement bears on it. Restoring the tension term gives the open pore a barrier and a lysis tension. With the tube radius pinned the barrier is a function of $\kappa$, $\ell$ and $\sigma$ alone and the saddle-splay modulus cancels from it; with the tube radius relaxed the transition state sits at the aspect ratio $\tfrac12(\pi + \sqrt{\pi^2-8})$, where a pore trades no area, its energy is independent of the tension, and three fifths of it is saddle-splay. Both readings put the barrier in the hundreds of $k_BT$, as does every line tension in the measured range, so the barrier a membrane crosses is reached before any rim exists and the rim is a post-nucleation object. A bilayer of half-thickness $\ell$ has $\sup|K| < \ell^{-2}$ and $|2H(H^2-K)| \le 8/27\ell^{3}$, its bending energy is nonincreasing along the constrained gradient flow, and every topology change requires $\mathcal{W} \ge 8\pi$, so all of them occur in an initial interval. A disc of the surface carries at least $2\pi$ of bending energy less the total geodesic curvature of its rim, and the two rims of a merge carry exactly minus the inserted neck’s Gaussian curvature, so a merge dissipates at least $4\pi - \int_N (H^2-K)\,\mathrm{d}A$, which a pore attains. That floor is $4\pi$ for a neck of vanishing area, $4\pi(1 - \tanh(w/a))$ for a catenoidal one of waist $a$ and half-height $w$, and positive for any neck below $4\pi\ell^2$ in area. At the floor the whole pathway performs at most $\mathcal{W}(0)/2\pi - \chi(0)/2 - 1$ events, which is $k-1$ for $k$ round vesicles and is attained.