Abstract
The energy of the bending of the membranes within the Canham model is directly justified as an extension of the case of bending of the elastic strips known as Euler’s Elasticas. Later, this model was elaborated upon by Helfrich & Deuling in the form of a non-linear system of two equations for the curvatures of the axially-symmetric membranes. Finally, Ou-Yang and Helfrich introduced the model which is currently seen to be most adequate for the description of membrane configurations. Since we are primarily interested in analytical solutions, it is quite natural that the equation by Ou-Yang and Helfrich be examined for the presence of symmetries, because they are in direct connection with solutions. When this is done, it becomes clear that the most general group of symmetries of the shape equation of the form coincides with the group of Euclidean motions in the real three-dimensional space. Among the generators of this group are those of rotations and translations, which hint about the existence of the analytical solutions discussed in the present and the subsequent chapters.
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References
G. Bluman, S. Kumei, Symmetries and Differential Equations (Springer, New York, 1989)
B. Canham, The minimum energy of bending as a possible explanation of biconcave shape of the human red blood cell. J. Theoret. Biol. 26, 61–81 (1970)
H. Davson, J. Danielli, The Permeability of Natural Membranes, 2nd edn. (Cambridge University Press, Cambridge, 1952)
G. de Matteis, Group analysis of the membrane shape equation, in Nonlinear Physics: Theory and Experiment II, ed. by M. Ablowitz, M. Boiti, F. Pempinelli, B. Prinari (World Scientific, Singapore, 2003), pp. 221–226
H. Deuling, W. Helfrich, Red blood cell shapes as explained on the basis of curvature elasticity. Biophys. J. 16, 861–868 (1976)
Y.-C. Fung, P. Tong, Theory of the sphering of red blood cell. Biophys. J. 8, 175–198 (1968)
W. Helfrich, Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch c 28, 693–703 (1973)
W. Helfrich, J. Deuling, Some theoretical shapes of red blood cells. J. Phys. (Colloq.) 36, 327–329 (1975)
N. Ibragimov, Transformation Groups Applied to Mathematical Physics (Riedel, Boston, 1985)
T. Jenkins, Static equilibrium configurations of a model red blood cell. J. Math. Biol. 4, 149–169 (1977a)
T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32, 755–764 (1977b)
J. Murphy, Erythrocyte metabolism VI., Cell shape and the location of cholesterol in the erythrocyte membrane. J. Lab. Clin. Med. 65, 756–774 (1965)
P. Olver, Applications of Lie Groups to Differential Equations, 2nd edn. (Springer, New York, 1993)
Z.-C. OuYang, W. Helfrich, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288 (1989)
L. Ovsiannikov, Group Analysis of Differential Equations (Academic, New York, 1982)
V. Pulov, M. Hadzhilazova, V. Lyutskanov, I. Mladenov, Helfrich’s shape model of biological membranes: Group analysis, determining system and symmetries, in Proceedings of the Third International Scientific Congress 7 (TU Varna, 2012), pp. 271–275
V. Pulov, E. Chacarov, I. Uzunov, A computer algebra application to determination of Lie symmetries of partial differential equations. Serdica J. Comput. 1, 241–251 (2007)
R. Rand, Mechanical properties of the red cell membrane, II. Viscoelastic breakdown of the membrane. Biophys. J. 4, 303–316 (1964)
R. Rand, A. Burton, Mechanical properties of the red cell membrane, I. Membrane stiffness and intracellular pressure. Biophys. J. 4, 115–135 (1964)
M. Schneider, J. Jenkins, W. Webb, Thermal fluctuations of large cylindrical phospholipid vesicles. Biophys. J. 45, 891–899 (1984)
V. Vassilev, P. Djonjorov, I. Mladenov, Symmetry groups, conservation laws and group-invariant solutions of the membrane shape equation. Geom. Integrab. Quant. 7, 265–279 (2006)
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Mladenov, I.M., Hadzhilazova, M. (2017). Equations of Equilibrium States of Membranes. In: The Many Faces of Elastica. Forum for Interdisciplinary Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-61244-7_5
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DOI: https://doi.org/10.1007/978-3-319-61244-7_5
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