Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T17:44:43.663Z Has data issue: false hasContentIssue false
  • English
  • Français

An Exactly Solved Model for Mutation, Recombination and Selection

Published online by Cambridge University Press:  20 November 2018

Michael Baake  and
Ellen Baake
Michael Baake
Affiliation:
Institut für Mathematik und Informatik Universität Greifswald Jahnstr. 15a 17487 Greifswald Germany, email: mbaake@uni-greifswald.de
Ellen Baake
Affiliation:
Institut für Mathematik und Informatik Universität Greifswald Jahnstr. 15a 17487 Greifswald Germany, email: ellen.baake@uni-greifswald.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.

Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Keywords

92D1034L3037N3006A0760J25population geneticsrecombinationnonlinear ODEsmeasure-valued dynamical systemsMöbius inversion
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 1 , 01 February 2003 , pp. 3 - 41
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Aigner, M., Combinatorial Theory. Springer, Berlin, 1979, reprint, 1997.Google Scholar
[2] Akin, E., The Geometry of Population Genetics. Lecture Notes in Biomathematics 31, Springer, Berlin, 1979.Google Scholar
[3] Akin, E., Cycling in simple genetic systems. J. Math. Biol. 13 (1982), 305–324.Google Scholar
[4] Amann, H., Gewöhnliche Differentialgleichungen. 2nd ed., de Gruyter, Berlin, 1995, (older) English edition Ordinary Differential Equations. de Gruyter, Berlin, 1990.Google Scholar
[5] Arendt, W., Characterization of positive semigroups on Banach lattices. In: One-parameter Semigroups of Positive Operators, (ed. R. Nagel), Lecture Notes in Math. 1184, Springer, Berlin, 1986, 247–291.Google Scholar
[6] Baake, E., Mutation and recombination with tight linkage. J. Math. Biol. 42 (2001), 455–488.Google Scholar
[7] Baake, E. and Gabriel, W., Biological evolution through mutation, selection and drift: An introductory review. In: Annual Review of Computational Physics, VII, (ed. D. Stauffer), World Scientific, Singapore, 2000, 203–264; cond-mat/9907372.Google Scholar
[8] Ben Arous, G. and Zeitouni, O., Increasing propagation of chaos for mean field models. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), 85–102.Google Scholar
[9] Berberian, S. K., Measure and Integration. Macmillan, New York, 1965.Google Scholar
[10] Berge, C., Principles of Combinatorics. Academic Press, New York, 1971.Google Scholar
[11] Bürger, R., The Mathematical Theory of Selection, Recombination, and Mutation, Wiley, Chichester, 2000.Google Scholar
[12] Bürger, R. and Bomze, I., Stationary distributions under mutation-selection balance: structure and properties. Adv. in Appl. Probab. 28 (1996), 227–251.Google Scholar
[13] Christiansen, F. B., The effect of population subdivision on multiple loci without selection. In: Mathematical evolutionary theory, (ed. M.W. Feldman), Princeton University Press, Princeton, 1989, 71–85.Google Scholar
[14] Christiansen, F. B., Population Genetics of Multiple Loci. Wiley, Chichester, 2000.Google Scholar
[15] Clark, A. et al., Haplotype structure and population genetic inferences from nucleotide-sequence variation in human lipoprotein lipase. Am. J. Hum. Genet. 63 (1998), 595–612.Google Scholar
[16] Dawson, K. J., The decay of linkage disequilibria under random union of gametes: how to calculate Bennett's principal components. Theor. Popul. Biol. 58 (2000), 1–20.Google Scholar
[17] Dawson, K. J., The evolution of a population under recombination: How to linearise the dynamics. Linear Algebra Appl. 348 (2002), 115–137.Google Scholar
[18] Dudley, R. M., Real Analysis and Probability. Chapman and Hall, New York, 1989.Google Scholar
[19] Engel, K.-J. and Nagel, R., One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Math. 194, Springer, New York, 2000.Google Scholar
[20] Eshel, I., Evolution processes with continuity of types. Adv. in Appl. Probab. 4 (1972), 475–507.Google Scholar
[21] Ethier, S. N. and Kurtz, T. G., Markov Processes: Characterization and Convergence. Wiley, New York, 1986.Google Scholar
[22] Ewens, W. J., A generalized fundamental theorem of natural selection. Genetics 63 (1969), 531–537.Google Scholar
[23] Ewens, W. J., Mean fitness increases when fitnesses are additive. Nature 221(1969), 1076.Google Scholar
[24] Freidlin, M. I. and Wentzell, A. D., Random Perturbations of Dynamical Systems. 2nd ed., Springer, New York, 1998.Google Scholar
[25] Greiner, G., Spectral theory of positive semigroups on Banach lattices. In: One-parameter Semigroups of Positive Operators, (ed. R. Nagel), Lecture Notes in Math. 1184, Springer, Berlin, 1986, 292–332.Google Scholar
[26] Guckenheimer, J. and Holmes, Ph., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. corr. 3rd printing, Springer, New York, 1990.Google Scholar
[27] Hofbauer, J., The selection-mutation equation. J. Math. Biol. 23 (1985), 41–53.Google Scholar
[28] Jones, B. L., Some principles governing selection in self-reproducing macromolecular systems.an analog of Fisher's fundamental theorem. J. Math. Biol. 6 (1978), 169–175.Google Scholar
[29] van Kampen, N. G., Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam, 1981.Google Scholar
[30] Karlin, S., General two-locus selection models: Some objectives, results and interpretation. Theor. Popul. Biol. 7 (1975), 364–398.Google Scholar
[31] Karlin, S. and Liberman, U., Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes. Genetics 91 (1979), 777–798.Google Scholar
[32] Kimura, M., A stochastic model concerning the maintenance of genetic variability in quantitative characters. Proc. Natl. Acad. Sci. 54 (1965), 731–736.Google Scholar
[33] Kingman, J. F. C., Markov population processes. J. Appl. Prob. 6 (1969), 1–18.Google Scholar
[34] Kirzhner, V. and Lyubich, Yu., Multilocus dynamics under haploid selection. J. Math. Biol. 35 (1997), 391–408.Google Scholar
[35] Lang, S., Real and Functional Analysis. 3rd ed., Springer, New York, 1993.Google Scholar
[36] Lyubich, Yu. I., Mathematical Structures in Population Genetics. Springer, Berlin, 1992.Google Scholar
[37] Manos, H. and Liberman, U., Discrete chiasma formation models and their associated high order interference. J. Math. Biol. 36 (1998), 448–468.Google Scholar
[38] McHale, D. and Ringwood, G. A., Haldane linearisation of baric algebras. J. LondonMath. Soc. (2) 28 (1983), 17–26.Google Scholar
[39] Reed, M. and Simon, B., Functional Analysis. 2nd ed., Academic Press, San Diego, California, 1980.Google Scholar
[40] Ringwood, G. A., Hypergeometric algebras and Mendelian genetics. Niew Archief voor Wiskunde (4) 3 (1985), 69–83.Google Scholar
[41] Rudin, W., Real and Complex Analysis. 3rd ed., McGraw-Hill, New York, 1987.Google Scholar
[42] Schaefer, H. H., Banach Lattices and Positive Operators. Springer, Berlin, 1974.Google Scholar
[43] Schaeffer, S. and Miller, E. L., Estimates of linkage disequilibrium and the recombination parameter determined from segregating nucleotide sites in the alcohol dehydrogenase region of Drosophila pseudoobscura. Genetics 135 (1993), 541–552.Google Scholar
[44] Thompson, C. J. and McBride, J. L., On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules. Math. Biosci. 21 (1974), 127–142.Google Scholar
[45] Werner, D., Funktionalanalysis. 3rd ed., Springer, Berlin, 2000.Google Scholar