Formation of membrane necks is crucial for fission and fusion in lipid bilayers. In this work, we seek to answer the following fundamental question: what is the relationship between protein-induced spontaneous mean curvature and the Gaussian curvature at a membrane neck? Using an augmented Helfrich model for lipid bilayers to include membrane–protein interaction, we solve the shape equation on catenoids to find the field of spontaneous curvature that satisfies mechanical equilibrium of membrane necks. In this case, the shape equation reduces to a variable coefficient Helmholtz equation for spontaneous curvature, where the source term is proportional to the Gaussian curvature. We show how this latter quantity is responsible for non-uniform distribution of spontaneous curvature in minimal surfaces. We then explore the energetics of catenoids with different spontaneous curvature boundary conditions and geometric asymmetries to show how heterogeneities in spontaneous curvature distribution can couple with Gaussian curvature to result in membrane necks of different geometries.

中文翻译：

---

高斯曲率指导自发曲率在双层膜颈上的分布†

膜颈的形成对于脂质双层中的裂变和融合至关重要。在这项工作中，我们试图回答以下基本问题：蛋白质诱导的自发平均曲率与膜颈高斯曲率之间的关系是什么？使用针对脂质双层的增强型Helfrich模型来包括膜-蛋白质相互作用，我们求解了链状体的形状方程，以找到满足膜颈机械平衡的自发曲率场。在这种情况下，形状方程可简化为自发曲率的可变系数亥姆霍兹方程，其中源项与高斯曲率成比例。我们显示了后一个量如何导致自发曲率在最小表面上的不均匀分布。