Application of the Helfrich elasticity theory to the morphology of red blood cells

Abstract

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.