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Mechanism behind the Beauty: The Golden Ratio Appeared in the Shape of Red Blood Cells

Published online by Cambridge University Press:  07 February 2017

Xue-Jun Zhang  and
Zhong-Can Ou-Yang
Xue-Jun Zhang*
Affiliation:
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China
Zhong-Can Ou-Yang*
Affiliation:
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China
*
*Corresponding author.Email addresses:[email protected] (X.-J. Zhang), [email protected] (Z.-C. Ou-Yang)
*Corresponding author.Email addresses:[email protected] (X.-J. Zhang), [email protected] (Z.-C. Ou-Yang)
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Abstract

In the past two decades, a rigorous solution for the shape of human red blood cell (RBC) with a negative spontaneous curvature c0 has been derived with the Helfrich model under the condition that both the osmotic pressure Δp and tensile stress λ are equal to zero. By fitting the experimentally observed shape of RBC, c0R0 has been predicted to be –1.62, theminus golden ratio, where R0 is the radius of a sphere which has the same surface area as RBC. In this paper, we verify this prediction by comparing experimental data with an analytical equation describing the relation between volume and surface area. Furthermore, it is also found ρmax /ρB ≈ 1.6 with ρmax the maximal radius and ρB the characteristic radius of RBC, showing an approximate beautiful golden cross section of RBC. On the basis of a complete numerical calculation, we propose a mechanism behind the beauty of the minus golden ratio that c0R0 results from the balance between the minimization of the surface area and the requirement of adequate deformability of RBC to allow it passing through the spleen, the so called “physical fitness test”.

Keywords

Red blood cellgolden ratio
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 559 - 569
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Hamly, D., Hummer, G.J., Sheard, C., Fluorescence of red blood cells, Science, 106 (1947), 169–170.Google Scholar
[2] Evans, R.L., Amatuzio, D., Ebert, R., Red Blood Cell Studies: Ashby Curves, Science, 115 (1952), 572–573.Google Scholar
[3] Guest, M.M., Bond, T.P., Cooper, R.G., Derrick, J.R., Red blood cells: change in shape in capillaries, Science, 142 (1963), 1319–1321.Google Scholar
[4] Skalak, R., Branemark, P., Deformation of red blood cells in capillaries, Science, 164 (1969), 717–719.Google Scholar
[5] Ou-Yang, Z.C., Hu, J.G., Liu, J.X., Determination of spontaneous curvature and internal pressure of vesicles and red cells by their shape and the flexoelectric effect of membrane, Mod. Phys. Lett. B, 06 (1992), 1577–1582.Google Scholar
[6] Evans, E., Fung, Y.C., Improved measurements of the erythrocyte geometry, Microvac. Res, 04 (1972), 335–347.Google Scholar
[7] Livio, M., The golden ratio: The story of phi, the world's most astonishing number, Broadway Books, (2008).Google Scholar
[8] Helfrich, W., Elastic properties of lipid bilayers: theory and possible experiments, Z. Naturforsch. C, 28 (1973), 693–703.Google Scholar
[9] Kaler, E.W. et al., Spontaneous vesicle formation in aqueous mixtures of single-tailed surfactants, Science, 245 (1989), 1371–1374.Google Scholar
[10] Ou-Yang, Z.C., Helfrich, W., 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 (1989), 5280.Google Scholar
[11] Hu, J.G., Ou-Yang, Z.C., Shape equations of the axisymmetric vesicles, Phys. Rev. E, 47 (1993), 461.Google Scholar
[12] Tan, Y., Sun, D., Wang, J., and Huang, W., Mechanical Characterization of Human Red Blood Cells Under Different Osmotic Conditions by Robotic Manipulation With Optical Tweezers, Transactions on Biomedical Engineering 57, 1816 (2010).Google Scholar
[13] Naito, H., Okuda, M., Ou-Yang, Z.C., Counterexample to some shape equations for axisymmetric vesicles, Phys. Rev. E, 48 (1993), 2304.CrossRefGoogle ScholarPubMed
[14] Naito, H., Okuda, M., Zhong-can, O.Y., Polygonal shape transformation of a circular biconcave vesicle induced by osmotic pressure, Phys. Rev. E, 54 (1996), 2816.Google Scholar
[15] Deuling, H.J. and Helfrich, W., The curvature elasticity of fluid membranes : A catalogue of vesicle shapes, J. Phys. (Paris) 37 (1976), 1335–1345 CrossRefGoogle Scholar
[16] Canham, P. and Burton, A. C., Distribution of size and shape in populations of normal human red cells, Circulation Research, 22 (1968), 405–422 Google Scholar
[17] Gifford, S. C., Frank, M. G., Derganc, J., Gabel, C., Austin, R. H., Yoshida, T., and Bitensky, M.W., Parallel microchannel-based measurements of individual erythrocyte areas and volumes, Biophysical journal, 84 (2003), 623–633 CrossRefGoogle ScholarPubMed
[18] Pivkin, I.V., Peng, Z., Karniadakis, G.E., Buffet, P.A., Dao, M., Suresh, S., Biomechanics of red blood cells in human spleen and consequences for physiology and disease, Proc. Natl. Acad. Sci. U.S.A., 113 (2016), 7804–7809.Google Scholar
[19] Suzuki, Y., Tateishi, N., Soutani, M., Maeda, N., Deformation of erythrocytes in microvessels and glass capillaries: effects of erythrocyte deformability, Microcirculation, 03 (1996), 49–57.Google Scholar
[20] Tsukada, K., Sekizuka, E., Oshio, C., Minamitani, H., Direct measurement of erythrocyte deformability in diabetes mellitus with a transparent microchannel capillary model and high-speed video camera system, Microvac. Res, 61 (2001), 231–239.Google Scholar
[21] Skalak, R., Poiseuille Medal lecture. Capillary flow: past, present, and future, Biorheology, 27 (1989), 277–293.Google Scholar
[22] Secomb, T.W., Skalak, R., Ozkaya, N., Gross, J., Flow of axisymmetric red blood cells in narrow capillaries, J. Fluid. Mech, 163 (1986), 405–423.CrossRefGoogle Scholar
[23] Uzoigwe, C., The human erythrocyte has developed the biconcave disc shape to optimise the flow properties of the blood in the large vessels, Med. Hypotheses, 67 (2006), 1159–1163.Google Scholar
[24] Goodman, S.R., Kurdia, A., Ammann, L., Kakhniashvili, D., Daescu, O., The human red blood cell proteome and interactome, Exp. Biol. Med, 232 (2007), 1391–1408.Google Scholar