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On evolution operators of genetic coalgebras

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Abstract

We characterize evolutionary operators acting on coalgebras with genetic realization modeling the backwards genetic inheritance in Mendelian genetic systems. This characterization is made in terms of the different slices of the cubic stochastic matrix of type (1,2) given by the transition probabilities defining the genetic coalgebra comultiplication. We use the obtained characterization to describe all possible equilibrium states a genetic population can reach when tracing the genetic information one generation back.

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References

  • Abraham WM (1980) Linearing quadratic transformations in genetic algebras. Proc Lond Math Soc (3) 40:346–363

  • Bernstein S (1923a) Demonstration mathématique de la loi de Mendel. Comptes Rendus Acad Sci Paris 177:528–531

    MATH  Google Scholar 

  • Bernstein S (1923b) Principe de stationarité et généralisation de la loi de Mendel. Comptes Rendus Acad Sci Paris 177:581–584

    MATH  Google Scholar 

  • Bernstein S (1924) Solution of a mathematical problem connected with the theory of heredity (Russian). Ann Sci. de l’Ukraine 1:83–114

    Google Scholar 

  • Braman K (2010) Third-order tensors as linear operators on a space of matrices. Linear Algebra Appl 433:1241–1253

    Article  MathSciNet  MATH  Google Scholar 

  • Casas JM, Ladra M, Omirov BA, Rozikov UA (2014) On evolution algebras. Algebra Colloq 21:331–342

    Article  MathSciNet  MATH  Google Scholar 

  • Ching W, Ng MK (2006) Markov chains: models, algorithms and applications. International series in operations research and management science, 83. Springer, New York

    Google Scholar 

  • Etherington IHM (1939) Genetic algebras. Proc R Soc Edimburg A 59:1–99

    MathSciNet  MATH  Google Scholar 

  • Etherington IHM (1941) Non-associative algebras and the symbolism of genetics. Proc R Soc Edimburg B 61:24–42

    MathSciNet  MATH  Google Scholar 

  • Glivenkov V (1936) Algebra mendelienne. Comptes Rendus (Doklady) de l’Acad des Sci de l’URSS 13(4):385–386

  • Holgate P (1975) Genetic algebras satisfying Bernstein’s stationary principle. J Lond Math Soc 9(2):613–623

    Article  MathSciNet  MATH  Google Scholar 

  • Kolda TG, Bader BW (2009) Tensor decompositions and applications. SIAM Rev 51:455–500

    Article  MathSciNet  MATH  Google Scholar 

  • Kostitzin VA (1938) Sur les coefficients mendeliens d’heredite. Comptes Rendus de l’Acad des Sci 206:883–885

    MATH  Google Scholar 

  • Lyubich YI (1971) Fundamental concepts and theorems of the evolutional genetics of free populations. Uspehi Mat Nauk 26(5):51–116

    MathSciNet  Google Scholar 

  • Lyubich YI (1974) Two-level Bernstein populations. Math Sbornik 95(137): 606–628, 632

  • Lyubich YI (1977) Bernstein algebras. Uspehi Mat Nauk 32(6):261–263

    MathSciNet  Google Scholar 

  • Lyubich YI (1992) Mathematical structures in population genetics. Biomathematics 22, Springer-Verlag, Berlin

  • Maksimov VM (1997) Cubic stochastic matrices and their probability interpretation. Theory Prob Appl 41(1):55–69

    Article  MathSciNet  MATH  Google Scholar 

  • Paniello I (2011) Stochastic matrices arising from genetic inheritance. Linear Algebra Appl 434:791–800

    Article  MathSciNet  MATH  Google Scholar 

  • Paniello I (2014) Marginal distributions of genetic coalgebras. J Math Biol 68:1071–1087

    Article  MathSciNet  MATH  Google Scholar 

  • Reed ML (1997) Algebraic structure of genetic inheritance. Bull Am Math Soc 34(2):107–130

    Article  MathSciNet  MATH  Google Scholar 

  • Schafer RD (1949) Structure of genetic algebras. Am J Math 71:121–135

    Article  MathSciNet  MATH  Google Scholar 

  • Serebrowsky A (1934) On the properties of the Mendelian equations. Doklady A.N.SSSR 2:33–36

  • Sweedler ME (1969) Hopf algebras. W. A. Benjamin Inc., New York

    MATH  Google Scholar 

  • Tian JP, Li B-L (2004) Coalgebraic structure of genetic inheritance. Math Biosci Eng 1(2):243–266

    Article  MathSciNet  MATH  Google Scholar 

  • Tian JP (2008) Evolution algebras and their applications. Lecture notes in mathematics. Springer, Berlin

    Book  Google Scholar 

  • Wörz-Busekros A (1980) Algebras in genetics. Lecture notes in biomathematics 36, Springer-Verlag, New York

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Authors and Affiliations

  1. Department of Statistics and Operative Research, Public University of Navarre, 31006, Pamplona, Spain

    Irene Paniello

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  1. Irene Paniello
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Correspondence to Irene Paniello.

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Partially supported by the Spanish Ministerio de Ciencia y Tecnología and FEDER (MTM2013-45588-C3-2-P).

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Paniello, I. On evolution operators of genetic coalgebras. J. Math. Biol. 74, 149–168 (2017). https://doi.org/10.1007/s00285-016-1025-1

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  • DOI: https://doi.org/10.1007/s00285-016-1025-1

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