Detection and quantification of introgression using Bayesian inference based on conjugate priors

Detection and quantification of introgression using Bayesian inference based on conjugate priors | bioRxiv /* */ /* */ <!-- <!-- /*! * yepnope1.5.4 * (c) WTFPL, GPLv2 */ (function(a,b,c){function d(a){return"[object Function]"==o.call(a)}function e(a){return"string"==typeof a}function f(){}function g(a){return!a||"loaded"==a||"complete"==a||"uninitialized"==a}function h(){var a=p.shift();q=1,a?a.t?m(function(){("c"==a.t?B.injectCss:B.injectJs)(a.s,0,a.a,a.x,a.e,1)},0):(a(),h()):q=0}function i(a,c,d,e,f,i,j){function k(b){if(!o&&g(l.readyState)&&(u.r=o=1,!q&&h(),l.onload=l.onreadystatechange=null,b)){"img"!=a&&m(function(){t.removeChild(l)},50);for(var d in y[c])y[c].hasOwnProperty(d)&&y[c][d].onload()}}var j=j||B.errorTimeout,l=b.createElement(a),o=0,r=0,u={t:d,s:c,e:f,a:i,x:j};1===y[c]&&(r=1,y[c]=[]),"object"==a?l.data=c:(l.src=c,l.type=a),l.width=l.height="0",l.onerror=l.onload=l.onreadystatechange=function(){k.call(this,r)},p.splice(e,0,u),"img"!=a&&(r||2===y[c]?(t.insertBefore(l,s?null:n),m(k,j)):y[c].push(l))}function j(a,b,c,d,f){return q=0,b=b||"j",e(a)?i("c"==b?v:u,a,b,this.i++,c,d,f):(p.splice(this.i++,0,a),1==p.length&&h()),this}function k(){var a=B;return a.loader={load:j,i:0},a}var l=b.documentElement,m=a.setTimeout,n=b.getElementsByTagName("script")[0],o={}.toString,p=[],q=0,r="MozAppearance"in l.style,s=r&&!!b.createRange().compareNode,t=s?l:n.parentNode,l=a.opera&&"[object Opera]"==o.call(a.opera),l=!!b.attachEvent&&!l,u=r?"object":l?"script":"img",v=l?"script":u,w=Array.isArray||function(a){return"[object Array]"==o.call(a)},x=[],y={},z={timeout:function(a,b){return b.length&&(a.timeout=b[0]),a}},A,B;B=function(a){function b(a){var a=a.split("!"),b=x.length,c=a.pop(),d=a.length,c={url:c,origUrl:c,prefixes:a},e,f,g;for(f=0;f<d;f++)g=a[f].split("="),(e=z[g.shift()])&&(c=e(c,g));for(f=0;f<b;f++)c=x[f](c);return c}function g(a,e,f,g,h){var i=b(a),j=i.autoCallback;i.url.split(".").pop().split("?").shift(),i.bypass||(e&&(e=d(e)?e:e[a]||e[g]||e[a.split("/").pop().split("?")[0]]),i.instead?i.instead(a,e,f,g,h):(y[i.url]?i.noexec=!0:y[i.url]=1,f.load(i.url,i.forceCSS||!i.forceJS&&"css"==i.url.split(".").pop().split("?").shift()?"c":c,i.noexec,i.attrs,i.timeout),(d(e)||d(j))&&f.load(function(){k(),e&&e(i.origUrl,h,g),j&&j(i.origUrl,h,g),y[i.url]=2})))}function h(a,b){function c(a,c){if(a){if(e(a))c||(j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}),g(a,j,b,0,h);else if(Object(a)===a)for(n in m=function(){var b=0,c;for(c in a)a.hasOwnProperty(c)&&b++;return b}(),a)a.hasOwnProperty(n)&&(!c&&!--m&&(d(j)?j=function(){var a=[].slice.call(arguments);k.apply(this,a),l()}:j[n]=function(a){return function(){var b=[].slice.call(arguments);a&&a.apply(this,b),l()}}(k[n])),g(a[n],j,b,n,h))}else!c&&l()}var h=!!a.test,i=a.load||a.both,j=a.callback||f,k=j,l=a.complete||f,m,n;c(h?a.yep:a.nope,!!i),i&&c(i)}var i,j,l=this.yepnope.loader;if(e(a))g(a,0,l,0);else if(w(a))for(i=0;i (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='//www.googletagmanager.com/gtm.js?id='+i+dl;j.type='text/javascript';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-M677548'); Skip to main content Home About Submit ALERTS / RSS Search for this keyword Advanced Search New Results Detection and quantification of introgression using Bayesian inference based on conjugate priors Bastian Pfeifer , Durrell D. Kapan doi: https://doi.org/10.1101/2022.07.07.499145 Bastian Pfeifer 1 Institute for Medical Informatics, Statistics and Documentation. Medical University Graz , Austria Find this author on Google Scholar Find this author on PubMed Search for this author on this site For correspondence: bastian.pfeifer{at}medunigraz.at Durrell D. Kapan 2 Department of Entomology and Center for Comparative Genomics, Institute for Biodiversity Science and Sustainability. California Academy of Sciences , San Francisco, USA Find this author on Google Scholar Find this author on PubMed Search for this author on this site Abstract Full Text Info/History Metrics Preview PDF Abstract Introgression (the flow of genes between species) is a major force structuring the evolution of genomes, potentially providing raw material for adaptation. Here, we present a versatile Bayesian model selection approach for the detection and quantification of introgression. The proposed d f - BF approach builds upon the recently published distance-based d f statistic. Unlike d f , d f - BF takes into account the number of variant sites within a genomic region. The d f - BF method quantifies introgression with the inferred θ parameter, and at the same time enables weighing the strength of evidence for introgression based on Bayes Factors. To ensure fast computation we make use of conjugate priors with no need for computational demanding MCMC iterations. We compare our method with other approaches including d f , f d , and Patterson’s D using a wide range of coalescent simulations. Furthermore, we showcase the applicability of the d f - BF approach using whole genome mosquito data. Finally, we integrate the new method into the powerful genomics R-package PopGenome. I. B ackground Our methodology builds upon the recently published d f statistic, we introduced as an estimator of the proportion of introgression [ 1 ]. It is formulated as where ABBA k , BABA k , and BBAA k represent SNP sharing patterns on a four-taxon tree, which we show can be expressed in terms of genetic distance: where p xk refers to the mutant allele frequency in population x at variant site k . Here d xyk is the average pairwise nucleotide difference between population x and population y at variant site k. L is the total number of bi-allelic sites in a genomic region. The first two taxa are closely related species the third taxon is a potential donor of mutant allele B at variable sites, and the fourth taxon refers to the outgroup as in the original work by Patterson [ 2 ]. Note, the d f statistic calculates the fraction of introgression based on variant sites where the outgroup (taxon 4) is monomorphic for allele A . From equation 1 it can be seen that when either p 2 k · d 13 k ( ABBA k + BBAA k ) or p 1 k · d 23 k ( BABA k + BBAA k ) is zero, the d f statistic estimate is 1 or -1, respectively. This can generate false positives in low diversity regions, e.g in low recombining regions comprising only a few bi-allelic markers. This issue is not unique to the d f statistic and applies to other ABBA - BABA methods, such as f d [ 3 ], and Patterson’s D since they do not explicitly account for the number of bi-allelic sites which provide import evidence of introgression. II. N ew A pproach To tackle this problem, we transform the d f statistic into a Bayesian model selection problem. We define two competing models of introgression. Model M 1 and M 2 are represented by the following binomial likelihood functions: The parameter θ 1 in model M 1 includes information about the fraction of the data explained by the ABBA+BBAA ( p 2 d 13 ) patterns. In model M 2 , θ 1 captures the BABA+BBAA ( p 1 · d 23 ) signals. The parameter θ 2 includes the species tree pattern BBAA ( p 1 · p 2 · (1− p 3 )), it is used as an approximate measure of the neutral (non-introgressed) signal within the data. The proposed Bayesian model assumes that the observed data D can be approximately explained by the species tree pattern (BBAA) plus the corresponding introgression frequency patterns (ABBA and BABA). We use the conjugate Beta distribution as a prior where λ is the average population size of P 1 , P 2 and P 3 . In order to form the posterior we propose the following updating scheme of the Beta distribution per variant site k The corresponding posterior density distributions of the models M 1 and M 2 are where d fθ are the inferred Beta model parameter to quantify the gene-flow between P 3 ↔ P 2 (model M 1 ) and P 3 ↔ P 1 (model M 2 ). Finally, evidences of introgression are calculated using Bayes Factors as allowing researchers to judge the relative merit of the two competing introgression models. The resulting Bayes Factors are interpreted according to Jeffrey’s Table; d f - BF = 1 (no evidence), d f - BF = 1 − 3 (anecdotal evidence), d f - BF = 3 − 10 (moderate evidence), d f - BF = 10 − 30 (strong evidence), d f - BF = 30 − 100 (very strong evidence), and d f - BF > = 100 (extreme evidence). III. R esults To validate the d f - BF approach we generated topologies with different levels of introgression using Hudson’s ms program [ 4 ]. The sequence alignments were produced by the seqgen program [ 5 ]. We generated 5kb sequence with split times t 12 = 1 × 4 N, t 123 = 2 × 4 N and t 123 O = 3 × 4 N generations ago. The time of gene-flow from P 3 to P 2 was set to t GF = 0.1 × 4 N generations ago with a fraction of introgression of f = 0.1. The recombination rate was set to r = 0.01, and a Hasegawa-Kishino-Yano substitution model was applied with a branch scaling factor of s = 0.01. We varied the fraction of introgression and the time of gene-flow and compared d f - BF with Patterson’s D ( D ), f d and d f . Figure 1a shows the results when varying the fraction of introgression from population P 3 to P 2 . The d f - BF model parameter θ (denoted as d fθ ) precisely quantifies the fraction of introgression and produces almost identical results as the d f statistic. The corresponding Bayes Factors for each introgression level are shown in Figure 1b . With the current setting strong evidence of introgression is reported when the fraction of introgression is greater than 0.8. We also varied the time of gene-flow. We confirm the results reported in [ 1 ], d f is almost not affected by the time of gene flow, and quantifies the fraction of introgression more accurate compared to Patterson’s D and f d . We report the same properties for d fθ (not shown). Download figure Open in new tab Fig. 1. Simulation results. (a) Shown are the results of d f , f d , D , and d fθ on simulated data with varying levels of introgression (100 iterations each). The horizontal lines refer to the real fraction of introgression. (b) The d f - BF Bayes Factor values of the corresponding levels of introgression shown in (a). We fully integrated the d f - BF approach into the R-package PopGenome [ 6 ]. C onclusion In summary, our flexible Bayesian model selection frame-work quantifies introgression, is equally or more accurate than the d f statistic upon which it based, and at the same time enables quantification of the strength of evidence for introgression based on Bayes Factors. R eferences [1]. ↵ B. Pfeifer and D. D. Kapan , “ Estimates of introgression as a function of pairwise distances ,” BMC bioinformatics , vol. 20 , no. 1 , pp. 1 – 11 , 2019 . OpenUrl CrossRef [2]. ↵ R. E. Green , J. Krause , A. W. Briggs , T. Maricic , U. Stenzel , M. Kircher , N. Patterson , H. Li , W. Zhai , M. H.-Y. Fritz , et al. , “ A draft sequence of the neandertal genome ,” science , vol. 328 , no. 5979 , pp. 710 – 722 , 2010 . OpenUrl Abstract / FREE Full Text [3]. ↵ S. H. Martin , J. W. Davey , and C. D. Jiggins , “ Evaluating the use of abba–baba statistics to locate introgressed loci ,” Molecular biology and evolution , vol. 32 , no. 1 , pp. 244 – 257 , 2015 . OpenUrl CrossRef PubMed [4]. ↵ R. R. Hudson , “ Generating samples under a wright–fisher neutral model of genetic variation ,” Bioinformatics , vol. 18 , no. 2 , pp. 337 – 338 , 2002 . OpenUrl CrossRef PubMed Web of Science [5]. ↵ A. Rambaut and N. C. Grass , “ Seq-gen: an application for the monte carlo simulation of dna sequence evolution along phylogenetic trees ,” Bioinformatics , vol. 13 , no. 3 , pp. 235 – 238 , 1997 . OpenUrl CrossRef PubMed [6]. ↵ B. Pfeifer , U. Wittelsbürger , S. E. Ramos-Onsins , and M. J. Lercher , “ Popgenome: an efficient swiss army knife for population genomic analyses in r ,” Molecular biology and evolution , vol. 31 , no. 7 , pp. 1929 – 1936 , 2014 . OpenUrl CrossRef PubMed Web of Science Back to top Previous Next Posted July 10, 2022. Download PDF Email Thank you for your interest in spreading the word about bioRxiv. NOTE: Your email address is requested solely to identify you as the sender of this article. Your Email * Your Name * Send To * Enter multiple addresses on separate lines or separate them with commas. You are going to email the following Detection and quantification of introgression using Bayesian inference based on conjugate priors Message Subject (Your Name) has forwarded a page to you from bioRxiv Message Body (Your Name) thought you would like to see this page from the bioRxiv website. Your Personal Message CAPTCHA This question is for testing whether or not you are a human visitor and to prevent automated spam submissions. 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