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Exact simulation of coupled Wright–Fisher diffusions

Part of: Markov processes Stochastic analysis Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  22 November 2021

Celia García-Pareja Open the ORCID record for Celia García-Pareja [Opens in a new window] ,
Henrik Hult  and
Timo Koski
Celia García-Pareja*
Affiliation:
KTH Royal Institute of Technology
Henrik Hult*
Affiliation:
KTH Royal Institute of Technology
Timo Koski*
Affiliation:
KTH Royal Institute of Technology and University of Helsinki
*
*Postal address: Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.
*Postal address: Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.
*Postal address: Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.
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Abstract

In this paper an exact rejection algorithm for simulating paths of the coupled Wright–Fisher diffusion is introduced. The coupled Wright–Fisher diffusion is a family of multivariate Wright–Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright–Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright–Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm’s complexity is derived and its performance demonstrated in a simulation study.

Keywords

Exact simulationrejection algorithmmultivariate diffusionspopulation geneticscoupled Wright–Fisher modelepistasis

MSC classification

Primary: 60J60: Diffusion processes 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations 65C05: Monte Carlo methods

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 923 - 950
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Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Aurell, E., Ekeberg, M. and Koski, T. (2019). On a multilocus Wright–Fisher model with mutation and a Svirezhev–Shahshahani gradient-like selection dynamics. Preprint. Available at https://arxiv.org/abs/1906.00716.Google Scholar
Beaumont, M. A., Zhang, W. and Balding, D. J. (2002). Approximate Bayesian computation in population genetics. Genetics 162, 2025–2035.10.1093/genetics/162.4.2025CrossRefGoogle ScholarPubMed
Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098.CrossRefGoogle Scholar
Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2008). A factorisation of diffusion measure and finite sample path constructions. Methodology Comput. Appl. Prob. 10, 85–104.10.1007/s11009-007-9060-4CrossRefGoogle Scholar
Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Prob. 15, 2422–2444.10.1214/105051605000000485CrossRefGoogle Scholar
Blanchet, J. and Zhang, F. (2020). Exact simulation for multivariate Itô diffusions. Advances in Applied Probability, 52(4), 1003–1034.CrossRefGoogle Scholar
Bürger, R. (2000). The Mathematical Theory of Selection, Recombination, and Mutation. John Wiley, Chichester.Google Scholar
Buzbas, E. O., Joyce, P. and Rosenberg, N. A. (2011). Inference on the strength of balancing selection for epistatically interacting loci. Theoret. Pop. Biol. 79, 102–113.10.1016/j.tpb.2011.01.002CrossRefGoogle ScholarPubMed
Casella, B. and Roberts, G. O. (2008). Exact Monte Carlo simulation of killed diffusions. Adv. Appl. Prob. 40, 273–291.10.1239/aap/1208358896CrossRefGoogle Scholar
Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Operat. Res. 38, 591–616.CrossRefGoogle Scholar
Corander, J. et al. (2017). Frequency-dependent selection in vaccine-associated pneumococcal population dynamics. Nature Ecol. Evol. 1, 1950–1960.CrossRefGoogle Scholar
Crow, J. F. and Kimura, M. (1970). An Introduction to Population Genetics Theory. Harper & Row, New York, Evanston, London.Google Scholar
Devroye, L. (2006). Nonuniform random variate generation. Handbooks Operat. Res. Manag. Sci. 13, 83–121.10.1016/S0927-0507(06)13004-2CrossRefGoogle Scholar
Ekeberg, M. et al. (2013). Improved contact prediction in proteins: using pseudolikelihoods to infer Potts models. Phys. Rev. E 87, 012707.CrossRefGoogle ScholarPubMed
Etheridge, A. M. and Griffiths, R. C. (2009). A coalescent dual process in a Moran model with genic selection. Theoret. Pop. Biol. 75, 320–330.10.1016/j.tpb.2009.03.004CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (2009). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Ethier, S. N. and Nagylaki, T. (1989). Diffusion approximations of the two-locus Wright–Fisher model. J. Math. Biol. 27, 17–28.CrossRefGoogle ScholarPubMed
Ewens, W. J. (2004). Mathematical Population Genetics, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Favero, M., Hult, H. and Koski, T. (2019). A dual process for the coupled Wright–Fisher diffusion. To appear in J. Math. Biol.Google Scholar
Fearnhead, P. (2006). The stationary distribution of allele frequencies when selection acts at unlinked loci. Theoret. Pop. Biol. 70, 376–386.10.1016/j.tpb.2006.02.001CrossRefGoogle ScholarPubMed
Fearnhead, P. and Prangle, D. (2012). Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation. J. R. Statist. Soc. B [Statist. Methodology] 74, 419–474.CrossRefGoogle Scholar
Fitzsimmons, P., Pitman, J. and Yor, M. (1993). Markovian bridges: construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, 1992, Birkhäuser, Boston, pp. 101–134.10.1007/978-1-4612-0339-1_5CrossRefGoogle Scholar
Gao, C.-Y., Zhou, H.-J. and Aurell, E. (2018). Correlation-compressed direct-coupling analysis. Phys. Rev. E 98, 032407.10.1103/PhysRevE.98.032407CrossRefGoogle Scholar
Griffiths, R. C. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Pop. Biol. 17, 37–50.CrossRefGoogle ScholarPubMed
Griffiths, R. C. (1984). Asymptotic line-of-descent distributions. J. Math. Biol. 21, 67–75.CrossRefGoogle Scholar
Griffiths, R. C. (2006). Coalescent lineage distributions. Adv. Appl. Prob. 38, 405–429.10.1239/aap/1151337077CrossRefGoogle Scholar
Griffiths, R. C., Jenkins, P. A. and Spanò, D. (2018). Wright–Fisher diffusion bridges. Theoret. Pop. Biol. 122, 67–77.10.1016/j.tpb.2017.09.005CrossRefGoogle ScholarPubMed
Griffiths, R. C. and Li, W.-H. (1983). Simulating allele frequencies in a population and the genetic differentiation of populations under mutation pressure. Theoret. Pop. Biol. 23, 19–33.10.1016/0040-5809(83)90003-5CrossRefGoogle Scholar
Griffiths, R. C. and Spanò, D. (2010). Diffusion processes and coalescent trees. In Probability and Mathematical Genetics, Papers in Honour of Sir John Kingman, eds Bingham, N. H. and Goldie, C. M., Cambridge University Press, pp. 358–375.CrossRefGoogle Scholar
Hössjer, O., Tyvand, P. A. and Miloh, T. (2016). Exact Markov chain and approximate diffusion solution for haploid genetic drift with one-way mutation. Math. Biosci. 272, 100–112.CrossRefGoogle ScholarPubMed
Jenkins, P. A. (2013). Exact simulation of the sample paths of a diffusion with a finite entrance boundary. Preprint. Available at https://arxiv.org/abs/1311.5777.Google Scholar
Jenkins, P. A. and Spanò, D. (2017). Exact simulation of the Wright–Fisher diffusion. Ann. Appl. Prob. 27, 1478–1509.CrossRefGoogle Scholar
Jewett, E. M. and Rosenberg, N. A. (2014). Theory and applications of a deterministic approximation to the coalescent model. Theoret. Pop. Biol. 93, 14–29.10.1016/j.tpb.2013.12.007CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1998). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Kimura, M. (1955). Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harb. Symp. Quant. Biol. 20, 33–53.10.1101/SQB.1955.020.01.006CrossRefGoogle ScholarPubMed
Kimura, M. (1964). Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.10.2307/3211856CrossRefGoogle Scholar
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, New York.CrossRefGoogle Scholar
Neher, R. A. and Shraiman, B. I. (2011). Statistical genetics and evolution of quantitative traits. Rev. Mod. Phys. 83, 1283–1300.CrossRefGoogle Scholar
Nené, N. R., Mustonen, V. and Illingworth, C. J. R. (2018). Evaluating genetic drift in time-series evolutionary analysis. J. Theoret. Biol. 437, 51–57.CrossRefGoogle Scholar
Pollock, M., Johansen, A. M. and Roberts, G. O. (2016). On the exact and $\varepsilon$-strong simulation of (jump) diffusions. Bernoulli 22, 794–856.Google Scholar
Rohlfs, R. V., Swanson, W. J. and Weir, B. S. (2010). Detecting coevolution through allelic association between physically unlinked loci. Amer. J. Human Genet. 86, 674–685.CrossRefGoogle ScholarPubMed
Schubert, B., Maddamsetti, R., Nyman, J., Farhat, M. R. and Marks, D. S. (2019). Genome-wide discovery of epistatic loci affecting antibiotic resistance in Neisseria gonorrhoeae using evolutionary couplings. Nature Microbiol. 4, 328–338.10.1038/s41564-018-0309-1CrossRefGoogle Scholar
Skwark, M. J. et al. (2017). Interacting networks of resistance, virulence and core machinery genes identified by genome-wide epistasis analysis. PLoS Genet. 13, e1006508.10.1371/journal.pgen.1006508CrossRefGoogle ScholarPubMed
Steinrücken, M., Bhaskar, A. and Song, Y. S. (2014). A novel spectral method for inferring general diploid selection from time series genetic data. Ann. Appl. Statist. 8, 2203–2222.CrossRefGoogle ScholarPubMed
Tataru, P., Simonsen, M., Bataillon, T. and Hobolth, A. (2017). Statistical inference in the Wright–Fisher model using allele frequency data. Systematic Biol. 66, e30–e46.Google ScholarPubMed
Tavaré, S., Balding, D. J., Griffiths, R. C. and Donnelly, P. (1997). Inferring coalescence times from DNA sequence data. Genetics 145, 505–518.10.1093/genetics/145.2.505CrossRefGoogle ScholarPubMed