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Published in: BIT Numerical Mathematics 3/2019

06-04-2019

Behavior of different numerical schemes for random genetic drift

Authors: Shixin Xu, Minxin Chen, Chun Liu, Ran Zhang, Xingye Yue

Published in: BIT Numerical Mathematics | Issue 3/2019

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Abstract

In the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.

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Literature
1.
go back to reference Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)MATH Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)MATH
2.
go back to reference Der, R., Epstein, C.L., Plotkin, J.B.: Generalized population models and the nature of genetic drift. Theor. Popul. Biol. 80(2), 80–99 (2011)CrossRefMATH Der, R., Epstein, C.L., Plotkin, J.B.: Generalized population models and the nature of genetic drift. Theor. Popul. Biol. 80(2), 80–99 (2011)CrossRefMATH
4.
go back to reference Eymard, R., Gallouët, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook for Numerical Analysis, pp. 715–1022. North Holland, Amsterdam (2000) Eymard, R., Gallouët, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook for Numerical Analysis, pp. 715–1022. North Holland, Amsterdam (2000)
5.
go back to reference Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–431 (1922)CrossRef Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–431 (1922)CrossRef
6.
go back to reference Hössjer, O., Tyvand, P.A., Miloh, T.: Exact Markov chain and approximate diffusion solution for hapliod genetic drift with one-way mutation. Math. Biosci. 272, 100–112 (2016)MathSciNetCrossRefMATH Hössjer, O., Tyvand, P.A., Miloh, T.: Exact Markov chain and approximate diffusion solution for hapliod genetic drift with one-way mutation. Math. Biosci. 272, 100–112 (2016)MathSciNetCrossRefMATH
7.
go back to reference Kimura, M.: Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harb. Symp. Quant. Biol. 20, 33–53 (1955)CrossRef Kimura, M.: Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harb. Symp. Quant. Biol. 20, 33–53 (1955)CrossRef
8.
go back to reference Kimura, M.: Random genetic drift in multi-allelic locus. Evolution 9(4), 419–435 (1955)CrossRef Kimura, M.: Random genetic drift in multi-allelic locus. Evolution 9(4), 419–435 (1955)CrossRef
9.
go back to reference Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47(6), 713–719 (1962) Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47(6), 713–719 (1962)
11.
go back to reference Kimura, M.: The Neutral Theory of Molecular Evolution: A Review of Recent Evidence. Cambridge University Press, Cambridge (1983)CrossRef Kimura, M.: The Neutral Theory of Molecular Evolution: A Review of Recent Evidence. Cambridge University Press, Cambridge (1983)CrossRef
12.
go back to reference LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefMATH LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefMATH
13.
go back to reference McKane, A.J., Waxman, D.: Sigular solution of the diffusion equation of population genetics. J. Theor. Biol. 247, 849–858 (2007)CrossRef McKane, A.J., Waxman, D.: Sigular solution of the diffusion equation of population genetics. J. Theor. Biol. 247, 849–858 (2007)CrossRef
15.
go back to reference Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996)CrossRefMATH Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996)CrossRefMATH
16.
go back to reference Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, New York (2006)MATH Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, New York (2006)MATH
18.
go back to reference Tran, T.D., Hofrichter, J., Jost, J.: An introduction to the mathematical structure of the Wright–Fisher model of population genetics. Theory Biosci. 132, 73–82 (2013)CrossRef Tran, T.D., Hofrichter, J., Jost, J.: An introduction to the mathematical structure of the Wright–Fisher model of population genetics. Theory Biosci. 132, 73–82 (2013)CrossRef
19.
go back to reference Waxman, D.: Fixation at a locus with multiple alleles: structure and solution of the Wright Fisher model. J. Theor. Biol. 257, 245–251 (2009)MathSciNetCrossRefMATH Waxman, D.: Fixation at a locus with multiple alleles: structure and solution of the Wright Fisher model. J. Theor. Biol. 257, 245–251 (2009)MathSciNetCrossRefMATH
20.
21.
22.
go back to reference Zhao, L., Yue, X., Waxman, D.: Complete numerical solution of the diffusion equation of random genetic drift. Genetics 194(4), 973–985 (2013)CrossRef Zhao, L., Yue, X., Waxman, D.: Complete numerical solution of the diffusion equation of random genetic drift. Genetics 194(4), 973–985 (2013)CrossRef
Metadata
Title
Behavior of different numerical schemes for random genetic drift
Authors
Shixin Xu
Minxin Chen
Chun Liu
Ran Zhang
Xingye Yue
Publication date
06-04-2019
Publisher
Springer Netherlands
Published in
BIT Numerical Mathematics / Issue 3/2019
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-019-00749-4

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