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.

中文翻译：

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脂质膜弹性能的其他贡献：倾斜-曲率耦合和曲率梯度

在生物学相关条件下的脂质双层膜是由两个单分子层（单层）组成的柔软的横向薄流体膜，每个单分子层的厚度约为2 nm。在比双层​​厚度大得多的空间尺度上，膜的弹性由其形状很好地确定。经典的Helfrich理论将膜视为没有特定内部结构的弹性二维（2D）膜。然而，各种局部膜异质性可导致脂质相对于膜表面法线倾斜。根据3D物体的经典弹性理论，Hamm和Kozlov [ Eur。物理 J.E。 3，323（2000）]衍生的官能最一般的能量，考虑到倾斜和脂质单层曲率。最近，Terzi和Deserno [J.化学 物理 147，084702（2017）]表示哈姆和科兹洛夫的推导是不完整的，因为错过了倾斜曲率耦合项。但是，由Terzi和Deserno导出的能量函数似乎不稳定，因此对于要求最小化变形总能量的应用程序无效。在这里，我们导出了一个稳定的弹性能量函数，表明在这两个工作中都没有曲率的平方梯度。能量功能的这种变化是由于更准确地考虑了横向剪切变形项及其对膜稳定性的影响而引起的。我们还考虑了预应力项对能量功能稳定性的影响，并且我们表明，出于稳定性的考虑，应将其考虑为较小的值，而应忽略有效的高斯曲率。我们进一步概括了该理论，包括拉伸-压缩变形模式，并提供了Hamm和Kozlov先前遗漏的术语的几何解释。在位于同一单层中的两个两亲性肽的膜介导相互作用的情况下，分析了新术语的物理后果。我们还提供了导演波动的表达式，并将其与Terzi和Deserno的表达式进行了比较。在位于同一单层中的两个两亲性肽的膜介导相互作用的情况下，分析了新术语的物理后果。我们还提供了导演波动的表达式，并将其与Terzi和Deserno的表达式进行了比较。在位于同一单层中的两个两亲性肽的膜介导相互作用的情况下，分析了新术语的物理后果。我们还提供了导演波动的表达式，并将其与Terzi和Deserno的表达式进行了比较。