The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a spontaneous curvature parameter c_0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes.

中文翻译：

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Helfrich 模型中球体的刚性和稳定性

由 H^{c_0} 表示的 Helfrich 泛函是 Helfrich (1973) 提出的一个数学表达式，用于表示弹性磷脂双层携带的自然自由能。Helfrich 的理论是理想化的弹性磷脂双层在所有可能的配置中最小化 H^{c_0}。该函数将自发曲率参数 c_0 与双层的平均曲率以及对面积和体积的约束整合在一起，通过包含渗透压差和拉伸应力或其他方式。使用嵌入的可定向表面的数学概念来表示双层的配置，人们可能期望能够采用微分几何和变分计算的方法，根据它所依赖的参数对双层配置进行精细分析. 在本文中，我们重点讨论球形红细胞的情况，以期更好地了解球形红细胞和球形红细胞增多症。我们根据 Helfrich 模型中的参数提供了球形解的完整分类。我们还对球形细胞的刚性和稳定性进行了一些进一步的分析。