{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:26:48Z","timestamp":1760243208288,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2014,11,14]],"date-time":"2014-11-14T00:00:00Z","timestamp":1415923200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called \u201callele-frequency spectrum\u201d or \u201csite-frequency spectrum\u201d (SFS). Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.<\/jats:p>","DOI":"10.3390\/computation2040199","type":"journal-article","created":{"date-parts":[[2014,11,17]],"date-time":"2014-11-17T03:15:21Z","timestamp":1416194121000},"page":"199-220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials"],"prefix":"10.3390","volume":"2","author":[{"given":"Claus","family":"Vogl","sequence":"first","affiliation":[{"name":"Institute of Animal Breeding and Genetics, University of Veterinary Medicine, Vienna, Veterinarplatz 1, 1210 Vienna, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2014,11,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1226","DOI":"10.1093\/molbev\/msq046","article-title":"On the utility of short intron sequences as a reference for the detection of positive and negative selection in Drosophila","volume":"27","author":"Parsch","year":"2010","journal-title":"Mol. Biol. Evol."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fisher, R. (1930). The Genetical Theory of Natural Selection, Clarendon Press.","DOI":"10.5962\/bhl.title.27468"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/genetics\/16.2.97","article-title":"Evolution in Mendelian populations","volume":"16","author":"Wright","year":"1931","journal-title":"Genetics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1017\/S0305004100033193","article-title":"Random processes in genetics","volume":"54","author":"Moran","year":"1958","journal-title":"Proc. Camb. Philos. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kimura, M. (1983). The Neutral Theory of Molecular Evolution, Cambridge University Press.","DOI":"10.1017\/CBO9780511623486"},{"key":"ref_6","unstructured":"Kimura, M. (1994). Population Genetics, Molecular Evolution, and the Neutral Theory: Selected Papers, University of Chicago Press."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.tpb.2006.06.005","article-title":"Non-equilibrium theory of the allele frequency spectrum","volume":"71","author":"Evans","year":"2007","journal-title":"Theor. Popul. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.tpb.2011.03.003","article-title":"Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory","volume":"79","author":"Zivkovic","year":"2011","journal-title":"Theor. Popul. Biol."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ewens, W. (2004). Mathematical Population Genetics, Springer. [2nd ed.].","DOI":"10.1007\/978-0-387-21822-9"},{"key":"ref_10","unstructured":"Wolfram Research, Inc. Mathematica, Version 10.0. Available online: http:\/\/wolfram.com\/."},{"key":"ref_11","unstructured":"Matlab 8.4. Available online: http:\/\/www.mathworks.de\/."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1534\/genetics.111.136929","article-title":"A simple method for finding explicit analytic transition densities of diffusion processes with general diploid selection","volume":"190","author":"Song","year":"2012","journal-title":"Genetics"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Baake, E., and Bialowons, R. (2008). Ancestral Processes with Selection: Branching and Moran Models, Banach Center Publications.","DOI":"10.4064\/bc80-0-2"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/j.tpb.2009.03.004","article-title":"A coalescent dual process in a Moran model with genic selectio","volume":"75","author":"Etheridge","year":"2009","journal-title":"Theor. Popul. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.tpb.2012.01.001","article-title":"The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates","volume":"81","author":"Vogl","year":"2012","journal-title":"Theor. Popul. Genet."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Hein, J., Schierup, M., and Wiuf, C. (2005). Gene Genealogies, Variation, and Evolution: A Primer in Coalescent Theory, Oxford University Press.","DOI":"10.1093\/oso\/9780198529958.001.0001"},{"key":"ref_17","unstructured":"Hazewinkel, M. (2001). Encyclopedia of Mathematics, Springer."},{"key":"ref_18","unstructured":"Griffiths, R., and Span\u00f2, D. (2010). Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman, Cambridge University Press."},{"key":"ref_19","unstructured":"Abramowitz, M., and Stegun, I. (1970). Handbook of Mathematical Functions, Dover. [9th ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1073\/pnas.41.3.144","article-title":"Solution of a process of random genetic drift with a continuous model","volume":"41","author":"Kimura","year":"1955","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Vogl, C. (2014). Estimating the Scaled Mutation Rate and Mutation Bias with Site Frequency Data. Theor. Popul. Biol., in press.","DOI":"10.1016\/j.tpb.2014.10.002"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/j.jtbi.2007.04.016","article-title":"Singular solutions of the diffusion equation of population genetics","volume":"247","author":"McKane","year":"2007","journal-title":"J. Theor. Biol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s12064-012-0170-3","article-title":"An introduction to the mathematical structure of the Wright-Fisher model of population genetics","volume":"132","author":"Tran","year":"2013","journal-title":"Theory Biosci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1111\/j.1420-9101.2012.02580.x","article-title":"Unconstrained evolution in short introns?\u2014An analysis of genome-wide polymorphism and divergence data from Drosophila","volume":"25","author":"Clemente","year":"2012","journal-title":"J. Evol. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2582","DOI":"10.1111\/jeb.12003","article-title":"Evidence for complex selection on four-fold degenerate sites in Drosophila melanogaster","volume":"25","author":"Clemente","year":"2012","journal-title":"J. Evol. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/0040-5809(74)90020-3","article-title":"A note on the sampling theory for infinite alleles and infinite sites models","volume":"6","author":"Ewens","year":"1974","journal-title":"Theor. Popul. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1016\/0040-5809(75)90020-9","article-title":"On the number of segregating sites in genetical models without recombination","volume":"7","author":"Watterson","year":"1975","journal-title":"Theor. Popul. Biol."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1093\/genetics\/132.4.1161","article-title":"Population genetics of polymorphism and divergence","volume":"132","author":"Sawyer","year":"1992","journal-title":"Genetics"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.tpb.2010.05.003","article-title":"Sufficiency of the number of segregating sites in the limit under finite-sites mutation","volume":"78","author":"RoyChoudhury","year":"2010","journal-title":"Theor. Popul. Biol."},{"key":"ref_30","unstructured":"Vogl, C. (2014). Biallelic Mutation-Drift Diffusion in the Limit of Small Scaled Mutation Rates. ArXiv E-Prints."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"e1000695","DOI":"10.1371\/journal.pgen.1000695","article-title":"Inferring the Joint Demographic History of Multiple Populations from Multidimensional SNP Frequency Data","volume":"5","author":"Gutenkunst","year":"2009","journal-title":"PLoS Genet."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1093\/molbev\/mst205","article-title":"Massive habitat-specific genomic response in D. melanogaster populations during experimental evolution in hot and cold environments","volume":"31","author":"Tobler","year":"2014","journal-title":"Mol. Biol. Evol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1093\/genetics\/152.2.755","article-title":"Using maximum likelihood to estimate population size from temporal changes in allele frequencies","volume":"152","author":"Williamson","year":"1999","journal-title":"Genetics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1093\/genetics\/156.4.2109","article-title":"Monte Carlo evaluation of the likelihood for Ne from temporally spaced samples","volume":"156","author":"Anderson","year":"2000","journal-title":"Genetics"},{"key":"ref_35","unstructured":"R Core Team (2013). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5477","DOI":"10.1088\/0305-4470\/36\/20\/309","article-title":"Theory and computation of spheroidal wave functions","volume":"36","author":"Falloon","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1093\/genetics\/162.4.2025","article-title":"Approximate Bayesian Computation in Population Genetic","volume":"162","author":"Beaumont","year":"2002","journal-title":"Genetics"},{"key":"ref_38","unstructured":"Stratton, J. (1954). Spheroidal Wave Functions, The Technology Press of the Massachusetts Institute of Technology."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Meixner, J., and Sch\u00e4fke, F. (1954). Mathieusche Funktionen und Sph\u00e4roidfunktionen, Springer. (In German).","DOI":"10.1007\/978-3-662-00941-3"},{"key":"ref_40","unstructured":"Flammer, C. (1957). Spheroidal Wave Functions, Stanford University Press."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"6792","DOI":"10.1103\/PhysRevE.58.6792","article-title":"Computations of spheroidal harmonics with complex arguments: A review with an algorithm","volume":"58","author":"Li","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_42","unstructured":"Falloon, P.E. (2001). Theory and Computation of Spheroidal Harmonics with General Arguments. [Master\u2019s Thesis, Department of Physics, The University of Western Australia]."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/2\/4\/199\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:09:27Z","timestamp":1760216967000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/2\/4\/199"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,14]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,12]]}},"alternative-id":["computation2040199"],"URL":"https:\/\/doi.org\/10.3390\/computation2040199","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2014,11,14]]}}}