Figure 1

For each leaflet, $z^{(\alpha )}$ represents the smooth (protrusionless) surface running though the hydrocarbon water surfaces. $z^{(m)}$ is the smooth surface which separates the top and bottom leaflets, so that the top monolayer is bounded by $z^{(1)}$ and $z^{(m)}$, while the bottom monolayer is bounded by $z^{(m)}$ and $z^{(2)}$. The coarse-grained shape of the membrane averaged over the top and bottom monolayers is given by $z^{+}\equiv \frac{1}{2}[z^{(1)}+z^{(2)}]$. The unit vectors ${\mathbb{N}}^{(\alpha )}$ are normal to $z^{(\alpha )}$ and point toward the interior of the bilayer. Assuming the membrane is nearly flat, the normals may be approximated as ${\mathbb{N}}^{(\alpha )}=(-1{)}^{\alpha }[-\nabla z^{(\alpha )},1]$. Molecular orientation of the lipids is described by the unit vector field ${\mathbb{n}}^{(\alpha )}$, which points from $z^{(\alpha )}$ to $z^{(m)}$ along a vector connecting the two ends of the lipid hydrocarbon chain(s). The $xy$ components of the dashed vectors denote the tilt vector ${\mathbf{m}}^{(\alpha )}={\mathbf{n}}^{(\alpha )}-{\mathbf{N}}^{(\alpha )}$.Reuse & Permissions

Figure 2

Two modes of membrane bending (each box represents a lipid and $\epsilon =0$ for simplicity). Top: Membrane bending associated with splay in lipid orientation ($\nabla ·\widehat{\mathbf{n}}$), in the absence of any lipid tilting ($\widehat{\mathbf{m}}$). Bottom: Membrane bending associated with lipid tilt, in the absence of splay. The top mode represents the dominant contribution to bending at long wavelengths and the source of $q^{-4}$ scaling in $⟨|h_{\mathbf{q}}{|}^2⟩$. The bottom mode involves an increase in the average area per lipid exposed to solvent and is associated with a $q^{-2}$ scaling characteristic shape fluctuations damped by surface tension.Reuse & Permissions

Figure 3

The spectrum of longitudinal molecular orientation fluctuations $⟨|{\widehat{n}}_{\mathbf{q}}^{\parallel }{|}^2⟩$ multipled by $q^2$. Simulation data are shown for an implicit solvent model (CG) (◊), the MARTINI model for DPPC (□), and a united-atom force field for DMPC (◯). Open symbols denote wavelengths greater than twice the bilayer thickness, used in determination of $K_c$. For clarity, the data sets for MARTINI and CG were vertically shifted by 0.1 and 0.2, respectively.Reuse & Permissions

Figure 4

The transverse orientation spectrum $⟨|{\widehat{n}}_{\mathbf{q}}^{\perp }{|}^2⟩$ (see Fig. 3 for details). In contrast to $⟨|{\widehat{n}}_{\mathbf{q}}^{\parallel }{|}^2⟩$, the theory (solid lines) and simulation data agree down to wavelengths of a few nanometers [Eq. (5)]. While different definitions of the molecular orientation lead to quantitative changes in the $⟨|{\widehat{n}}_{\mathbf{q}}^{\perp }{|}^2⟩$ data for MARTINI and UA, the resulting spectra (not shown) retain excellent agreement with Eq. (5).Reuse & Permissions

Figure 5

The spectrum of height fluctuations $⟨|h_{\mathbf{q}}{|}^2⟩$ multiplied by $q^4$ for the same simulations as in Fig. 3. The gray symbols represent “correcting” these results, as suggested by Eq. (3), to account for lipid tilt: $[⟨|h_{\mathbf{q}}{|}^2⟩-\frac{k_{B}T}{K_{\theta }{q}^2}]q^4$. For clarity, the MARTINI and CG data sets were vertically shifted by 0.1 and 0.2, respectively.Reuse & Permissions