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Polynomial dynamics of human blood genotypes frequencies. (English) Zbl 1359.92081

Summary: The frequencies of human blood genotypes in the ABO and Rh systems differ between populations. Moreover, in a given population, these frequencies typically evolve over time. The possible reasons for the existing and expected differences in these frequencies (such as disease, random genetic drift, founder effects, differences in fitness between the various blood groups etc.) are the focus of intensive research. To understand the effects of historical and evolutionary influences on the blood genotypes frequencies, it is important to know how these frequencies behave if no influences at all are present. Under this assumption the dynamics of the blood genotypes frequencies is described by a polynomial dynamical system defined by a family of quadratic forms on the 17-dimensional projective space. To describe the dynamics of such a polynomial map is a task of substantial computational complexity.
We give a complete analytic description of the evolutionary trajectory of an arbitrary distribution of human blood variations frequencies with respect to the clinically most important ABO and RhD antigens. We also show that the attracting algebraic manifold of the polynomial dynamical system in question is defined by a binomial ideal.

MSC:

92D10 Genetics and epigenetics
37N25 Dynamical systems in biology
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
92D15 Problems related to evolution

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[1] Anstee, D. J., The relationship between blood groups and disease, Blood, 115, 4635-4643 (2010)
[2] Bedford, E.; Jonsson, M., Dynamics of regular polynomial endomorphisms of \(C^k\), Am. J. Math., 122, 1, 153-212 (2000) · Zbl 0941.37027
[3] Bernstein, S. N., Principe de stationarite et generalisation de la loi de Mendel, C. R. Acad. Sci. Paris, 177, 528-531 (1923) · JFM 49.0368.03
[4] Bernstein, F., Über die Erblichkeit der Blutgruppen, Zeitschrift für induktive Abstammungs- und Vererbungslehre, 54, 1, 400-426 (1930) · JFM 57.1501.09
[5] Cantat, S.; Chambert-Loir, A.; Guedj, V., Quelques aspects des systèmes dynamiques polynomiaux, Panoramas et Synthèses, vol. 30 (2010), Société Mathématique de France: Société Mathématique de France Paris, x+341 pp. (in French) · Zbl 1237.37007
[6] Eisenbud, D.; Sturmfels, B., Binomial ideals, Duke Math. J., 84, 1, 1-45 (1996) · Zbl 0873.13021
[7] Hoffman, R., Hematology: Basic Principles and Practice (2012), Elsevier
[8] Kang, S. H., Distribution of abo genotypes and allele frequencies in a Korean population, Jpn. J. Hum. Genet., 42, 331-335 (1997)
[9] Laubenbacher, R.; Sturmfels, B., Computer algebra in systems biology, Am. Math. Mon., 116, 10, 882-891 (2009) · Zbl 1229.13027
[10] Librado, P.; Rozas, J., DnaSP v5: a software for comprehensive analysis of DNA polymorphism data, Bioinformatics, 25, 1451-1452 (2009)
[11] Lyubich, Y. I., Basic concepts and theorems on the evolutionary genetics of free populations, Russ. Math. Surv., 26, 5, 51-123 (1971)
[12] Millán, M. P.; Dickenstein, A.; Shiu, A.; Conradi, C., Chemical reaction systems with toric steady states, Bull. Math. Biol., 74, 5, 1027-1065 (2012) · Zbl 1251.92016
[13] Novitski, E., ABO blood groups and the Hardy-Weinberg equilibrium, Science, 6, 478 (1976)
[14] Ohashi, J., Polymorphisms in the ABO blood group gene in three populations in the New Georgia group of the Solomon Islands, J. Hum. Genet., 51, 407-411 (2006)
[15] Okada, Y.; Kamatani, Y., Common genetic factors for hematological traits in Humans, J. Hum. Genet., 57, 161-169 (2012)
[16] Reilly, M.; Szulkin, R., Statistical analysis of donation-transfusion data with complex correlation, Stat. Med., 26, 30, 5572-5585 (2007)
[17] Sato, T., Polymorphisms and allele frequencies of the ABO blood group gene among the Jomon, Epi-Jomon and Okhotsk people in Hokkaido, northern Japan, revealed by ancient DNA analysis, J. Hum. Genet., 55, 691-696 (2010)
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