This paper aims at the general mathematical framework for the equilibrium theory of two-component lipid bilayer vesicles. To take into account the influences of the local compositions together with the mean curvature and Gaussian curvature of the membrane surface, a general potential functional is constructed. We introduce two kinds of virtual displacement modes: the normal one and the tangential one. By minimizing the potential functional, the equilibrium differential equations and the boundary conditions of two-component lipid vesicles are derived. Additionally, the geometrical constraint equation and geometrically permissible condition for the two-component lipid vesicles are presented. The physical, mathematical, and biological meanings of the equilibrium differential equations and the geometrical constraint equations are discussed. The influences of physical parameters on the geometrically permissible phase diagrams are predicted. Numerical results can be used to explain recent experiments.

中文翻译：

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双组分脂质双层囊泡的平衡理论和几何约束方程

本文旨在研究双组分脂质双层囊泡平衡理论的一般数学框架。为了考虑局部成分以及膜表面的平均曲率和高斯曲率的影响，构建了一般的潜在泛函。我们介绍了两种虚拟位移模式：法向一种和切向一种。通过最小化潜在泛函，推导出平衡微分方程和双组分脂质囊泡的边界条件。此外，还提出了双组分脂质囊泡的几何约束方程和几何允许条件。讨论了平衡微分方程和几何约束方程的物理、数学和生物学意义。预测了物理参数对几何允许相图的影响。数值结果可以用来解释最近的实验。