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 $\mathcal{F}[m]$ 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.

Figure 1

Contour plot of the string $m_s(\mathbf{r})$ in the midplane of the system; every second configuration is depicted. Hydrophobic regions are colored gray (red), and hydrophilic regions are shown in dark gray (blue).

Figure 2

(Right axis) Free energy along the MFEP (black, left axis) and the probability that configurations generated with the restrained Hamiltonian ${\mathcal{H}}_c$ at $m_c=m_s$ decay into two apposed bilayers at a time $1.41{\tau }_0$ after the restrained has been removed [gray (blue), right axis]. Results have been obtained for 256 independent configurations at each value of $s$. The thermal energy scale, $k_{B}T$, is indicated for our coarse-grained membrane model (amphiphiles) and biological lipids. The inset presents a configuration snapshot at the saddle point, $s^{*}=0.532$. Hydrophilic beads are colored light gray (yellow), hydrophobic beads are shown in dark gray (red), and solvent particles are not shown. Only every 10th linear amphiphile is depicted. When we identify one double-tailed lipid with two linear amphiphiles, the molecular density in the snapshot matches that of a lipid bilayer.

Figure 3

Left panel: Saddle point by string method $s=0.532$ (left half) and by diagonalization of fluctuation spectrum according to Eq. (8) (right half). Right panel: Contour plot of the difference, $m^{*}-m_{0.532}$ between the estimate of the fluctuation analysis and the saddle-point estimate of the string method. The range of contour levels is reduced by a factor of 10 compared with that of the left panel.