Topological transitions of fluid lipid membranes are fundamental processes for cell life. For example, they are required for endo- and exocytosis or to enable neurotransmitters to cross the neural synapses. Here, inspired by the idea that fusion and fission proteins could have evolved in Nature in order to carry out a minimal work expenditure, we evaluate the minimal free energy pathway for the transition between two spherical large unilamellar vesicles and a dumbbell-shaped one. To address the problem, we propose and successfully use a Ginzburg-Landau type of free energy, which allows us to uniquely describe without interruption the whole, full-scale topological change. We also compute the force fields needed to overcome the involved energy barriers. The obtained forces are in excellent agreement, in terms of intensity, scale, and spatial localization with experimental data on typical fission protein systems, whereas they suggest the presence of additional features in fusion proteins.

Lipid bilayers possess two extraordinary and apparently conflicting properties that make them essential barriers at the cellular and sub-cellular level: they are at the same time very stable and extremely plastic. Due to their plasticity, they can easily change both their shape and their topology. For example, erythrocytes are reshaped to flow into narrow capillaries, while the topological changes are required for cellular processes such as endo-and exocytosis or to enable neurotransmitters to cross the neural synapses. On the other hand, stability is a necessary prerequisite for their barrier function and for the complex cellular clockworks. It is a fact that there is currently no theoretical model able to encompass the full-scale morphological modifications of lipid bilayer vesicles. The classical description is the celebrated Canham-Helfrich model which possesses undeniable merits except the ability to deal with topological transitions. Muscular approaches such as molecular dynamics, even in its coarse-grained versions, provide invaluable microscopic insight but, for the forthcoming years, will still not be able to afford the full-scale simulation of the morphological changes in large and giant unilamellar vesicles. In this scenario, the purpose of the present paper is to produce and demonstrate an approach belonging to the class of phase-field models, which offers the potential to capture the dynamics of these full-scale lipid vesicles, and the subtle effects of global constraints induced by the Gauss-Bonnet theorem of differential geometry, and by area and volume conservation. From the technical point of view, the breakthrough consists of a novel formulation to include the contribution of the membrane Gaussian curvature in the phase-field free energy functional. It will be shown that the energy barrier that stabilizes the vesicle topology is largely due to the contribution of the Gaussian curvature, rather than to the membrane bending rigidity. The topological transition, occurring in the fusion/fission process of simple vesicles, is addressed with techniques borrowed from the statistical mechanics of rare events. The force fields capable of inducing the transition with minimal work expenditure are analyzed and found consistent with known mechanisms that operate at the biochemical level. This picture is consistent with the intuition that protein systems could have evolved in such a way as to minimize the work needed to induce the topological transition by following a minimal free energy path. The model correctly identifies the scales relevant to the two strongly asymmetric processes of vesicle fusion and fission, and provides predictions that are quantitatively consistent with experimental estimates on the energy barrier and on the strength of the forces exerted by the protein systems involved in topological transitions. Intriguingly, only quite general and macroscopic parameters are needed, namely bilayer thickness and its rigidity, suggesting that the topological change is a quite generic process.

Topological transitions of fluid lipid membranes are fundamental processes for cell life. For ex- ample, they are required for endo- and exocytosis or to enable neurotransmitters to cross the neural synapses. Inspired by the idea that fusion and fission proteins could have evolved in Nature in order to carry out a minimal work expenditure, we evaluate the minimal free energy pathway for the transition between two spherical large unilamellar vesicles and a dumbbell-shaped one. To address the problem, we propose and successfully use a Ginzburg-Landau type free energy, which allows us to uniquely describe without interruption the whole, full-scale topological change. We also compute the force fields needed to overcome the involved energy barriers. The obtained forces are in excellent agreement, in terms of intensity, scale, and spatial localization with experimental data on typical fission protein systems, whereas they suggest the presence of additional features in fusion proteins.