Exploiting Lipid Permutation Symmetry to Compute Membrane Remodeling Free Energies

A complete physical description of membrane remodeling processes, such as fusion or fission, requires knowledge of the underlying free energy landscapes, particularly in barrier regions involving collective shape changes, topological transitions, and high curvature, where Canham-Helfrich (CH) continuum descriptions may fail. To calculate these free energies using atomistic simulations, one must address not only the sampling problem due to high free energy barriers, but also an orthogonal sampling problem of combinatorial complexity stemming from the permutation symmetry of identical lipids. Here, we solve the combinatorial problem with a permutation reduction scheme to map a structural ensemble into a compact, nondegenerate subregion of configuration space, thereby permitting straightforward free energy calculations via umbrella sampling. We applied this approach, using a coarse-grained lipid model, to test the CH description of bending and found sharp increases in the bending modulus for curvature radii below 10 nm. These deviations suggest that an anharmonic bending term may be required for CH models to give quantitative energetics of highly curved states.

Figure 1

The permuted DOPC unbending trajectory $\tilde{\mathbf{x}}(t)$ at 0 ns (a) and 80 ns (b). Solvent is not shown and PO4 beads used for PR are shown as spheres. Lipid headgroups (PO4 and NC3 beads) are colored blue (dark) except for a central patch of green (light) lipids which illustrate how PR preserves lipid sorting throughout the unbending process. The red, green and blue arrows in (a) and (b) indicate the system $x$, $y$, and $z$ axes, respectively. (c) (main) The PCA derived collective coordinate $\xi$ for $\tilde{\mathbf{x}}(t)$. (inset) The leading PCA mode ${\mathbf{v}}_1$. (inset center) The average permuted structure $⟨\tilde{\mathbf{x}}(t)⟩$ between 0 and 80 ns. The 650 PO4 beads used for PCA (indices $\{\beta \}$) are colored blue (dark) and those not considered are gray (light). The linear mode ${\mathbf{v}}_1$ is illustrated for three cases as $⟨\tilde{\mathbf{x}}(t)⟩+\xi ·{\mathbf{v}}_1$ with $\xi =\{-20,0,20\}\mathrm{nm}$, from bottom to top, respectively.

Figure 2

(a) The bending free energy (PMF) $E(\xi )$ for DOPC from $16\times 450\mathrm{ns}$ umbrella sampling simulations. $E(\xi )$ was recovered using WHAM and the error bars are the standard deviations of 30 bootstrapped PMFs. (b) The bending free energy $E(\tilde{H})$ as a function of the Helfrich curvature integral $\tilde{H}$. Points lie at the umbrella window means $⟨\tilde{H}⟩$ and error bars are 95% confidence intervals. The quadratic fit is shown in green (solid line), and its linear component, the harmonic Helfrich approximation, in red (dashed line). Two snapshots (1 and 2) in (a) depict the range of bending in the ensemble and arrows in (a) and (b) show the $\xi$ and $\tilde{H}$ values. For structure 2, $R=4.6\mathrm{nm}$ at its center.

Figure 3

Midplane (curvature) radius dependent bending moduli, ${\kappa }_{\mathrm{eff}}(R)$. Points lie at the mean minimum curvature radius $⟨R_{\min }⟩$ of each umbrella window. Error bars are 95% confidence intervals. Note that ${\kappa }_{\mathrm{eff}}(R)=\kappa$ at large $R$. The legend depicts Martini lipids and their constituent coarse-grained beads. The bead types are: gray solid or open; saturated or unsaturated carbon chain (approximately 4 carbons per bead), green; glycerol, blue; PO4, orange solid or open; NC3 or NH3 (choline or amine).