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Analytical solutions of the membrane shape equation
1Institute of Theoretical Physics, Chinese Academy of Science, Beijing, China.
Abstract:
Helfrich's liquid crystal membrane theory successfully establishes a quantitative framework for describing biomembrane morphology by combining surface differential geometry with membrane elasticity mechanics, laying the foundation for the physics of biomembranes. The Zhong-Can-Helfrich equation provides a central mathematical tool for this theory, enabling the analytical solution of complex biological shapes such as the biconcave disk of red blood cells and promoting theoretical predictions and experimental validations of various membrane structures, including toroidal vesicles. This review article focuses on analytical solutions of the Zhong-Can-Helfrich shape equation for fluid membranes. We also review applications of this equation to membranes with open boundaries and to multisphere solutions of the equation relevant to myelin formation in red blood cells. At the end of this paper, the membrane shape of red blood cells in a vessel is studied for slow blood velocity, and it is found that conical red blood cells can exist in flowing blood.
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