In this work, we present in detail, in an accessible manner for undergraduate and graduate physics students, the model of spontaneous curvature, due to Helfrich, that quantitatively explains why the red blood cells in their natural state adopt a biconcave shape. The main hypothesis is that the equilibrium cell shape satisfies the principle of minimum free energy. Therefore, in the model, an expression for the membrane free energy is postulated based on the Helfrich theory. In that approximation, the membrane is modelled as a two-dimensional surface and the energy is written as a function of the surface principal curvatures and three parameters, including the spontaneous curvature, c₀, which is associated with the chemical composition of the membrane. The negative values for c₀ induce invaginations in the cell membrane. The model predicts the discocyte-spherocyte transition for the red blood cell. In the article, the concepts involved in the theory are developed in detail, and an algorithm that allows obtaining the contour of the cell is presented in detail as supplementary material.

中文翻译：

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Helfrich弹性理论在红细胞形态学中的应用

在这项工作中，我们以通俗易懂的方式为本科生和研究生的物理学生呈现了由赫尔夫里希（Helfrich）引起的自发曲率模型，该模型定量地解释了自然状态下的红细胞为何采用双凹形。主要假设是，平衡孔的形状满足最小自由能的原理。因此，在该模型中，基于Helfrich理论假定了膜自由能的表达式。在这种近似中，将膜建模为二维表面，并根据表面主曲率和三个参数（包括与膜的化学成分相关的自发曲率c _0）来写入能量。c的负值₀诱导细胞膜内陷。该模型预测了红细胞的盘状细胞-球状细胞过渡。在本文中，详细介绍了该理论中涉及的概念，并详细介绍了一种可获取细胞轮廓的算法，作为补充材料。