A Jacquard

CC Cockerham

DL Harris

E Cheng

G Karigl

IJM de Boer

K Lange

KP Donnelly

L Emik

LL Lo

Luis Alberto García-Cortés

M Abney

M Gillois

R Nadot

SP Smith

Genetics Selection Evolution

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Ge n e t i c sSe lec t ionEvolut ionGarcía-Cortés Genetics Selection Evolution  (2015) 47:33 DOI 10.1186/s12711-015-0108-6SHORT COMMUNICATION Open AccessA novel recursive algorithm for the calculationof the detailed identity coefficientsLuis Alberto García-CortésAbstractBackground: A recursive algorithm to calculate the fifteen detailed coefficients of identity is introduced. Previousrecursive procedures based on the generalized coefficients of kinship provided the detailed coefficients of identityunder the assumption that the two individuals were not an ancestor of each other.Findings: By using gametic relationships to include three, four or two pairs of gametes, we can obtain thesecoefficients for any pair of individuals. We have developed a novel linear transformation that allows for the calculationof pairwise detailed identity coefficients for any pedigree given the gametic relationships. We illustrate the procedureusing the well-known pedigree of Julio and Mencha, which contains 20 Jicaque Indians of Honduras, to calculate theirdetailed coefficients.Conclusions: The proposed algorithm can be used to calculate the detailed identity coefficients of two or moreindividuals with any pedigree relationship.BackgroundThe 15 detailed states of identity were first described byHarris [1] and Gillois [2]. Throughout this paper, theywill be referred to identity coefficients as described byCockerham [3].To circumvent the absence of recurrence rules to obtainidentity coefficients, Karigl [4], following the rules forgeneralized kinship coefficients [1,3], obtained identitycoefficients using a triangular linear transformation. Thistransformation provided the 9 condensed identity coef-ficients for any pair of individuals, but the 15 detailedidentity coefficients can only be calculated using Karigl’smethod under the assumption that neither of the two indi-viduals is an ancestor of the other. To obtain generalizedkinship coefficients, Lange and Sinsheimer [5] describedan alternative way, which can calculate the detailed coef-ficients of identity without this limitation. Unfortunately,the implementation of the latter method is not a triangularlinear transformation.An alternative to using multiple kinship coefficients isthe use of multiple gametic relationships. These relation-ships, called chromosome pedigrees by Donnelly [6], havebeen succesfully used to account for dominance in linearCorrespondence: garcia@inia.esDepartamento de Mejora Genética, Instituto Nacional de InvestigaciónAgraria, Ctra. de La Coruña km. 7.5, 28040 Madrid, Spainmodels [7-9]. Here, we use these multiple gametic rela-tionships to develop a new method to calculate the 15detailed identity coefficients.Generalized gametic relationshipsThe coancestry between two individuals X and Y , φXY ,is usually calculated following simple recurrence rules.These rules can be implemented using tabular methodsor languages with recursive function support. To calculatethe whole set of coancestries for a given pedigree, only twoformulae are required [10]:φXX = (1 + φFM) /2,φXY = (φFY + φMY ) /2, (1)where F andM are the father and the mother of individualX, respectively. These equations operate successively overpairs (X,Y ) where X is assumed to be more recent thanY . Let x and x′ be the paternally and maternally inheritedcopies at a given locus carried by individual X and y andy′ the corresponding copies carried by individual Y . Thecoancestry between X and Y can then be written as:φXY = 14[P (x ≡ y) + P (x ≡ y′)+ P (x′ ≡ y)+ P (x′ ≡ y′)] ,where ≡ stands for identical by descent (IBD).© 2015 García-Cortés; licensee BioMed Central. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedicationwaiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwisestated.García-Cortés Genetics Selection Evolution  (2015) 47:33 Page 2 of 4Analogous relationships have been described forgametes [6] as ψab = P (a ≡ b) and recurrence rules havebeen developed for their pairwise relationships [6,7]:ψaa = 1,ψab = 1/2(ψgb + ψhb), (2)where a and b denote two gametes in the pedigree. Bothg and h are the direct ancestral gametes of a, that is, thegametes of the father or the mother if a is a paternalor maternal gamete, respectively. Although Equations (1)and (2) are closely related, they have different inter-pretations.We use the three-way (ψabc), the four-way (ψabcd) andthe two-pair (ψab,cd) gametic relationships as counter-parts of the conventional generalized kinship coefficients[4]. These generalized gametic relationships correspondto the probability of three or four gametes to be IBD. Notethat these multiple gametic relationships correspond tomultiple gametic identities, regardless of the identity bydescent with other gametes. For instance, for individu-als X and Y , whose paternally and maternally inheritedgametes are described above,ψxx′y = P(x ≡ x′ ≡ y) = P (x ≡ x′ ≡ y ≡ y′)+ P (x ≡ x′ ≡ y ≡ y′) .ψab,cd is the probability that gametes a and b are IBDand simultaneously c and d are also IBD. For instance, forindividuals X and Y ,ψxx′,yy′ = P(x ≡ x′ ≡ y ≡ y′)+ P (x ≡ x′ ≡ y ≡ y′) .The recursive formulae for the whole set of multiplegametic relationships areψaa = 1,ψab = 12(ψgb + ψhb),ψaaa = 1,ψaab = ψab,ψabc = 12(ψgbc + ψhbc),ψaaaa = 1,ψaaab = ψab,ψaabc = ψabc,ψabcd = 12(ψgbcd + ψhbcd),ψaa,aa = 1,ψaa,ab = ψab,ψab,ac = ψabc,ψaa,bc = ψbc,ψab,cd = 12(ψgb,cd + ψhb,cd).(3)For an easier implementation, Equation (3) can besummarized in a simple set of rules1. In any n-way relationship, merge groups sharing agiven gamete, that is, ψab,ac = ψabac2. In any n-way relationship, discard repeated gametes,i.e. ψaabc = ψabc or ψaa,bc = ψa,bc.3. Given that the probability of a gamete to be IBD toitself is 1, discard groups of identity including a singlegamete, i.e. ψa,bc = ψbc.4. Identities with a single gamete are 1 and identitieswith two or more founder gametes at the same groupare 0.5. Calculate ψaθ = 12(ψgθ + ψhθ), where θ stands forany identity pattern and a for a gamete of theyoungest individual. For instanceψab,cd = 12(ψgb,cd + ψhb,cd)orψabc = 12(ψgbc + ψhbc).As long as these rules are correct and regardless of thenumber of gametes involved, they can be used to calculateidentities involving more than two individuals.The detailed identity coefficientsThe 15 detailed coefficients of identity for individuals Xand Y can be calculated from the generalized gameticrelationships by a simple linear transformation. We willconsider the 15 partitions or identity states as describedby Gillois [2] and Jacquard [11], i.e.,S1 : x ≡ x′ ≡ y ≡ y′ S2 : x ≡ x′ ≡ y ≡ y′ S3 : x ≡ x′ ≡ y′ ≡ yS4 : x ≡ y ≡ y′ ≡ x′ S5 : x′ ≡ y ≡ y′ ≡ x S6 : x ≡ x′ ≡ y ≡ y′S7 : x ≡ x′ ≡ y ≡ y′ S8 : x ≡ x′ ≡ y ≡ y′ S9 : x ≡ y ≡ x′ ≡ y′S10 : x ≡ y ≡ x′ ≡ y′ S11 : x ≡ y ≡ x′ ≡ y′ S12 : x ≡ y′ ≡ x′ ≡ yS13 : x ≡ y′ ≡ x′ ≡ y S14 : x ≡ y′ ≡ x′ ≡ y S15 : x ≡ y′ ≡ x′ ≡ yThe detailed identity coefficients δi are the probabilitiesof each state Si for a given pair of individuals.Generalized kinships [4] are a linear transformation ofthe pairwise detailed identity coefficients. For instance,ψxx′yy′ requires the state S1 to be true, then we can writeψxx′yy′ = δ1. Similarly, ψxx′y requires either S1 or S2 to betrue and therefore ψxx′y = δ1 + δ2, and so on for all ψ . Aset of 15 equalities is defined to set up the following linearsystem of equations.⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 0 0 0 0 0 0 0 0 0 0 0 0 01 0 1 0 0 0 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 0 0 01 0 0 0 1 0 0 0 0 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0 0 0 0 0 01 1 1 0 0 1 1 0 0 0 0 0 0 0 01 0 0 1 1 1 0 1 0 0 0 0 0 0 01 0 0 0 0 0 0 0 1 0 0 0 0 0 01 1 0 1 0 0 0 0 1 1 0 0 0 0 01 0 1 0 1 0 0 0 1 0 1 0 0 0 01 0 0 0 0 0 0 0 0 0 0 1 0 0 01 0 1 1 0 0 0 0 0 0 0 1 1 0 01 1 0 0 1 0 0 0 0 0 0 1 0 1 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣δ1δ2δ3δ4δ5δ6δ7δ8δ9δ10δ11δ12δ13δ14δ15⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ψxx′yy′ψxx′yψxx′y′ψxyy′ψx′yy′ψxx′ ,yy′ψxx′ψyy′ψxy,x′y′ψxyψx′y′ψxy′ ,x′yψxy′ψx′y1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(4)García-Cortés Genetics Selection Evolution  (2015) 47:33 Page 3 of 4Figure 1 The pedigree of the Jicaque Indians Julio and Mencha.The coefficient matrix in Equation (4) is triangular andis equivalent to formula (8) in [4]. Nevertheless, the righthand side here includes the generalized gametic relation-ships instead of their kinship counterparts.An example: the detailed identity coefficients ofthe Jicaque IndiansWe reanalyzed the pedigree of the Jicaque indians Julioand Mencha [12] presented in Figure 1 and previouslyanalyzed in [4,11,13]. Previous analyses have focused onthe identity coefficients between Julio and Mencha andbetween two of their progeny. Table 1 shows these resultsand those of two other pairs: Julio vs. one of their progenyand Mencha vs. one of their progeny.Program source codes written in Fortran 90 andexamples are available at https://github.com/agarcor21/IdentityCoefficients.DiscussionThe identity by descent of two gametes [6-9] is conceptu-ally simpler than the kinship between individuals becauseit avoids any random sampling of genes. Although thenumber of gametes is larger than the number of individu-als, its simple definition results in more intuitive recursiveformulae and easier calculations.Here, we have provided a novel method to derive thedetailed identity coefficients [1,2]. Karigl derived a sim-ple transformation to calculate these coefficients from themultiple kinship coefficients, which provided the 15 iden-tity coefficients when the individuals were not ancestor-descendant related. Lange and Sinsheimer [5] first pro-vided exact equations for these 15 coefficients withoutany assumption. Depending on the purpose of the study,researchers have used 7 or 9 condensed identity coeffi-cients. For instance, in order to apply the identity coeffi-cients to dominance models, 7 condensed coefficients canbe used [8,9]. To our knowledge, the algorithm proposedin [9] has never been used to calculate the 15 detailed coef-ficients, but the formulas presented in their paper couldbe used to obtain them in a direct way.We have proposed using multiple gametic relationshipsto calculate the 15 detailed identity coefficients. The lin-ear transformation in this paper is similar to equation 8 ofKarigl’s approach [4]. Reordering the rows in Equation (4)to follow Karigl’s pattern [4], both formulae only differon the right hand side. Note that the meaning of bothformulae is rather different.The procedure presented here has been successfullyimplemented on small pedigrees. In large data sets, thenumber of calls to the recursive function will dependon the number of generations and the structure of thepedigree. Implementing the procedure in such scenariosTable 1 Detailed coefficients of identity for four pairs involving Julio, Mencha and their progenyJulio-Mencha Progeny-progeny Julio-progeny Mencha-progenyδ1 0.01025 0.06580 0.03467 0.03467δ2 0.02393 0.04333 0.08252 0.00000δ3 0.02490 0.04333 0.00000 0.08252δ4 0.02393 0.04333 0.06665 0.06665δ5 0.02490 0.04333 0.08228 0.08228δ6 0.00708 0.00729 0.00000 0.00000δ7 0.05103 0.02383 0.00000 0.00000δ8 0.05103 0.02383 0.00000 0.00000δ9 0.05127 0.24713 0.08228 0.06665δ10 0.10937 0.15900 0.29248 0.00000δ11 0.16992 0.15900 0.00000 0.29248δ12 0.00708 0.00729 0.06665 0.08228δ13 0.05103 0.02383 0.00000 0.29248δ14 0.05103 0.02383 0.29248 0.00000δ15 0.34326 0.08582 0.00000 0.00000García-Cortés Genetics Selection Evolution  (2015) 47:33 Page 4 of 4is beyond our goal, but path counting or graph theory-based methods [14,15] have been developed to improvethe computing efficiency of this calculation.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsLAGC conceived the study, designed and performed all the analyses anddrafted the manuscript. The author read and approved the final manuscript.AcknowledgementsThis work was funded by grant CGL2012-39861-C02-02 from Ministerio deEconomía y Competitividad. We are most grateful to Ángeles de Cara, LuisGómez-Raya and Andrés Legarra for comments on the manuscript. We arealso grateful to Rohan Fernando for providing the source code of one of thereferred algorithms.Received: 8 March 2014 Accepted: 12 February 2015References1. Harris DL. Genotypic covariances between inbred relatives. Genetics.1964;50:1319–48.2. Gillois M. La relation d’identité en génétique. Ann Inst Henri Poincaré.1964;B2:1–94.3. Cockerham CC. Higher order probability functions of identity of alleles bydescent. Genetics. 1971;69:235–46.4. Karigl G. A recursive algorithm for the calculation of identity coefficients.Ann Hum Genet. 1981;45:299–305.5. Lange K, Sinsheimer JS. Calculation of genetic identity coefficients.Ann Hum Genet. 1992;56:339–46.6. Donnelly KP. The probability that related individuals share some sectionof genome identical by descent. Theor Popul Biol. 1983;23:34–63.7. Smith SP, Allaire FR. Efficient selection rules to increase non-linear merit:application to selection. Genet Sel Evol. 1985;17:387–406.8. de Boer IJM, Hoeschele I. Genetic evaluation methods for populationswith dominance and inbreeding. Theor Appl Genet. 1993;86:245–258.9. Lo LL, Fernando RL, Cantet RJC, Grossman M. Theory for modellingmeans and covariances in a two-breed population with dominanceinheritance. Theor Appl Genet. 1995;90:49–62.10. Emik L, Terrill CE. Systematic procedures for calculating inbreedingcoefficients. J Hered. 1949;40:51–55.11. Jacquard A. The genetic structure of populations. New York: Springer;1973.12. Chapman AM, Jacquard AM. Un isolat d’Amérique Centrale: Les indiensJicaques du Honduras. I.N.E.P., editor. Génétique et Populations. PressesUniversitaires de France; 1971.13. Nadot R, Vaysseix G. Apparentement et identité, algorithme de calcul descoefficients d’identité. Biometrics. 1973;29:347–59.14. Cheng E, Elliott B, Ozsoyoglu ZM. Efficient computation of kinship andidentity coefficients on large pedigrees. J Bioinform Comput Biol.2009;7:429–53.15. Abney M. A graphical algorithm for fast computation of identitycoefficients and generalized kinship coefficients. Bioinformatics.2009;25:1561–3. Submit your next manuscript to BioMed Centraland take full advantage of: • Convenient online submission• Thorough peer review• No space constraints or color ﬁgure charges• Immediate publication on acceptance• Inclusion in PubMed, CAS, Scopus and Google Scholar• Research which is freely available for redistributionSubmit your manuscript at www.biomedcentral.com/submit

A novel recursive algorithm for the calculation of the detailed identity coefficients

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A novel recursive algorithm for the calculation of the detailed identity coefficients

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