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The geometry of recombination. (English) Zbl 1429.92107
Summary: With the tools of information geometry, we can express relations between marginals of a joint distribution in geometric terms. We develop this framework in the context of population genetics and use this to interpret the famous Ohta-Kimura formula [T. Ohta and M. Kimura, “Linkage disequilibrium due to random genetic drift”, Genet. Res. 13, No. 1, 47–55 (1969; doi:10.1017/s001667230000272x)] and discuss its generalizations for linkage equilibria in Wright-Fisher models with recombination with several loci. The state space associated with the Ohta-Kimura model is simply a Riemannian manifold of constant positive curvature. Furthermore, the equilibria states for recombination can be interpreted geometrically as a product of spheres. In the case of only 2 loci, we also derive the behavior of the mutual information between these two loci.
MSC:
92D10 Genetics and epigenetics
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[1] Akin, E.: The Geometry of Population Genetics. Springer, Berlin (1979) · Zbl 0437.92016
[2] Amari, S.I.: Information Geometry and Its Applications. Springer, Berlin (2016) · Zbl 1350.94001
[3] Amari, S.I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R.: Differential Geometry in Statistical Inference. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 10. Institute of Mathematical Statistics, Hayward (1987) · Zbl 0694.62001
[4] Amari, S.I., Nagaoka, H.: Methods of Information Geometry. Translations of Mathematical Monographs, vol. 191. American Mathematical Society/Oxford University Press, Providence/Oxford (2000). Translated from the 1993 Japanese original by Daishi Harada · Zbl 0960.62005
[5] Antonelli, PL; Strobeck, C., The geometry of random drift. I. Stochastic distance and diffusion, Adv. Appl. Prob., 9, 238-249 (1977) · Zbl 0372.60108
[6] Ay, N., Jost, J., Lê, H.V., Schwachhöfer, L.: Information Geometry. Ergebnisse der Mathematik. Springer, Berlin (2017) · Zbl 1383.53002
[7] Baake, E.; Wangenheim, U., Single-crossover recombination and ancestral recombination trees, J. Math. Biol., 68, 1371-1402 (2014) · Zbl 1284.92063
[8] Crow, J.F., Kimura, M.: Introduction to Population Genetics. Harper and Row, New York (1970) · Zbl 0246.92003
[9] Eguchi, S.; Yanagimoto, T., Asymptotical improvement of maximum likelihood estimators on Kullback-Leibler loss, J. Stat. Plan. Inference, 138, 3502-3511 (2008) · Zbl 1145.62016
[10] Ewens, W.J.: Mathematical Population Genetics I. Interdisciplinary Applied Mathematics, vol. 27, 2nd edn. Springer, New York (2004)
[11] Fisher, RA, On the dominance ratio, Proc. R. Soc. Edinb., 42, 321-341 (1922)
[12] Hofrichter, J., Jost, J., Tran, T.D.: Information Geometry and Population Genetics: The Mathematical Structure of the Wright-Fisher Model, 1st edn. Springer International Publishing, Cham (2017) · Zbl 1370.92009
[13] Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext, 7th edn. Springer, Heidelberg (2017) · Zbl 1380.53001
[14] Karlin, S.; McGregor, J., Rates and probabilities of fixation for two locus random mating finite populations without selection, Genetics, 58, 141-159 (1968)
[15] Kolmogoroff, A., Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104, 415-458 (1931) · Zbl 0001.14902
[16] Ohta, T.; Kimura, M., Linkage disequilibrium due to random genetic drift, Genet. Res., 13, 47-55 (1969)
[17] Rao, CR, Information and accuracy attainable in the estimation of statistical parameters, Bull. Calcutta. Math. Soc., 37, 81-91 (1945) · Zbl 0063.06420
[18] Tran, T.D.: Information geometry and the Wright-Fisher model of mathematical population genetics. PhD thesis, University of Leipzig (2012)
[19] Watterson, GA, On the equivalence of random mating and random union of gametes models in finite, monoecious populations, Theor. Popul. Biol., 1, 233-250 (1970) · Zbl 0244.92004
[20] Wright, S., Evolution in Mendelian populations, Genetics, 16, 97-159 (1931)
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