Additional contributions to elastic energy of lipid membranes: Tilt-curvature coupling and curvature gradient

Lipid bilayer membranes under biologically relevant conditions are flexible thin laterally fluid films consisting of two unimolecular layers (monolayers) each about 2 nm thick. On spatial scales much larger than the bilayer thickness, the membrane elasticity is well determined by its shape. The classical Helfrich theory considers the membrane as an elastic two-dimensional (2D) film, which has no particular internal structure. However, various local membrane heterogeneities can result in a lipids tilt relative to the membrane surface normal. On the basis of the classical elasticity theory of 3D bodies, Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] derived the most general energy functional, taking into account the tilt and lipid monolayer curvature. Recently, Terzi and Deserno [J. Chem. Phys. 147, 084702 (2017)] showed that Hamm and Kozlov's derivation was incomplete because the tilt-curvature coupling term had been missed. However, the energy functional derived by Terzi and Deserno appeared to be unstable, thereby being invalid for applications that require minimizations of the overall energy of deformations. Here, we derive a stable elastic energy functional, showing that the squared gradient of the curvature was missed in both of these works. This change in the energy functional arises from a more accurate consideration of the transverse shear deformation terms and their influence on the membrane stability. We also consider the influence of the prestress terms on the stability of the energy functional, and we show that it should be considered small and the effective Gaussian curvature should be neglected because of the stability requirements. We further generalize the theory, including the stretching-compressing deformation modes, and we provide the geometrical interpretation of the terms that were previously missed by Hamm and Kozlov. The physical consequences of the new terms are analyzed in the case of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer. We also provide the expression for director fluctuations, comparing it with that obtained by Terzi and Deserno.

Figure 1

The illustration of the additional contributions to the elastic energy of lipid membranes. For the sake of simplicity, all deformations are presented in the one-dimensional case. All shapes are exactly calculated under the incompressibility constraint. (a) The undeformed lipid monolayer in the reference configuration. The hydrophobic thickness of the monolayer is chosen to be 1.5 nm; the thickness of the lipid heads region is set to 0.4 nm for illustrative purposes. (b) The pure tilt deformation at an angle of 20°. (c) The constant stretching gradient deformation with $\frac{d\alpha }{dx}=0.1\mathrm{n}{\mathrm{m}}^{-1}$. (d) The pure bending deformation of the dividing surface (no stretching and zero tilt with respect to the dividing surface) to a sine wave 0.05sin($2x$). (e) The deformation of zero curvature and varying tilt divergence: $T_x=0.05\sin \left(2x\right)$, where $T_x$ is the $x$-component of the tilt field. Panels (c), (d), and (e) show that a local tilt, which is responsible for the additional contributions, can emerge inside the lipid monolayer even if there is no tilt with respect to the dividing surface, or it is small as in panel (e). Such a local tilt emerges due to thickness variations that result in transverse shear deformations inside a lipid monolayer. For illustrative purposes, the local tilt in panels (c), (d), and (e) is shown for surfaces that in the reference configuration correspond to planes parallel to the dividing surface at a distance equal to the hydrophobic thickness of the monolayer.

Figure 2

Physical consequences of the additional contributions, $k_c^{}\mathbf{T}·\left(\mathbf{\nabla }\tilde{K}\right)$ and $\frac{k_{\mathrm{gr}}}{2}{\left(\mathbf{\nabla }\tilde{K}\right)}_{}^2$, to the elastic energy of lipid membranes, illustrated by an example of a membrane-mediated interaction of two amphipathic peptides located in the same monolayer of a lipid membrane. The figures show the dependence of the elastic energy of the bilayer on the distance $d$ between the peptides. (a) Schematic representation of two amphipathic peptides of diameter $D=1.3\mathrm{nm}$, inducing a jump in the boundary directors, $n_{2x}-n_{1x}=n_{4x}-n_{3x}=-0.8$. (b) The energy profile obtained within the framework of the HK Hamiltonian (red curve) in comparison with the profiles obtained using the Eq. (16) Hamiltonian with different values of $k_{\mathrm{gr}}$ and fixed $k_c=0$. (c) The energy profile obtained within the framework of the HK Hamiltonian (red curve) in comparison with the profile obtained using the Eq. (16) Hamiltonian with theoretically estimated values of $k_c=–5k_{B}T$ and $k_{\mathrm{gr}}=3k_{B}T\mathrm{n}{\mathrm{m}}^2$ (green curve). (d) The energy profile obtained within the framework of the HK Hamiltonian (red curve) in comparison with the profiles obtained using the Eq. (16) Hamiltonian with different values of $k_c$ and fixed $k_{\mathrm{gr}}=12k_{B}T\mathrm{n}{\mathrm{m}}^2$.

Figure 3

The lipid membrane is unstable if the tilt-curvature coupling modulus $k_c$ is nonzero, while the curvature gradient modulus $k_{\mathrm{gr}}$ is zero. The instability is caused by membrane oscillations that are energetically unbounded from below. The figure shows two amphipathic peptides (black ellipses) embedded in a box of length 30 nm with periodic boundary conditions. The distance between the peptides is 14.2 nm, and the analytically calculated shapes of the dividing surfaces of the upper and lower monolayers are shown. (a) $k_c=-5k_{B}T, k_{\mathrm{gr}}=3k_{B}T\mathrm{n}{\mathrm{m}}^2$ (as estimated in Sec. 4b). (b) $k_c=-5k_{B}T, k_{\mathrm{gr}}=0k_{B}T\mathrm{n}{\mathrm{m}}^2$, amplitude $A_0$ of the oscillations is 0.5 nm. The rest of the membrane elastic parameters are the same as in Fig. 2. In panel (a) the membrane is stable, while in panel (b) the membrane is unstable due to the arising oscillations: energy $w$ of the membrane inside the box scales as $w=a{A}_0^2+b{A}_0+c$, where $a\approx -140k_{B}T/\mathrm{n}{\mathrm{m}}^3, b\approx 6.7\times {10}^{-3}{k}_{B}T/\mathrm{n}{\mathrm{m}}^2, c\approx 1.4k_{B}T/\mathrm{nm}$ (the energy is considered per unit length along the axis of symmetry), implying that when the oscillation amplitude $A_0$ increases, the energy of the membrane inside the box approaches minus infinity.

Figure 4

The fluctuation spectrum of $q^2\langle {|n_q^{\parallel }|}^2\rangle$ as predicted by the HK ($k_c=k_{\mathrm{gr}}=0$, red curve), TD ($k_c\ne 0, k_{\mathrm{gr}}=0$, cyan curve), and Eq. (14) ($k_c\ne 0, k_{\mathrm{gr}}\ne 0$, black curve) Hamiltonians, where $k_c$ and $k_{\mathrm{gr}}$ are the tilt-curvature coupling and curvature gradient modulus, respectively. The spectrum is constant in the case of the HK Hamiltonian, monotonically increasing and monotonically decreasing in the case of the TD and Eq. (14) Hamiltonians, respectively. Although the TD Hamiltonian rightly reflects an upward trend observed in simulations at high $q$-values, it is divergent (the corresponding asymptote is depicted as a dashed line). Fixing the divergence leads to the Eq. (14) Hamiltonian, which is monotonically decreasing.