Transition path from two apposed membranes to a stalk obtained by a combination of particle simulations and string method

The formation of an hourglass-shaped passage (stalk) connecting two apposed membranes is an essential initial step in membrane fusion. The most probable transition path from two separate membranes to a stalk, i.e., the minimum free-energy path (MFEP), is constructed using a combination of particle simulations and string method. For the reversible transition path in the coarse-grained membrane model, a collective order parameter, m, can be identified as the local difference of hydrophilic and hydrophobic densities. In particle simulations, the free energy F[m] as a functional of m is not readily available. This difficulty is overcome by an equation-free approach, where the morphology and the excess free energy along the MFEP are obtained by an on-the-fly string method. The transition state is confirmed by diagonalization of order-parameter fluctuations and by the probability of reaching either stalk or bilayer morphology from different positions along the MFEP.