Modeling X-linked ancestral origins in multiparental populations

56 Pages·2015·0.54 MB·English

G3: Genes|Genomes|Genetics Early Online, published on March 4, 2015 as doi:10.1534/g3.114.016154
Modeling X-linked ancestral origins in multiparental
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populations
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ChaozhiZheng∗
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Biometris,WageningenUniversityandResearchCentre,TheNetherlands
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March1,2015
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Runninghead: ModelingX-linkedancestralorigins
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Keywords: advancedintercrosslines(AIL),collaborativecross(CC),diversityoutcross(DO),
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Drosophilasyntheticpopulationresource(DSPR),mapexpansion.
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∗ Correspondingauthor:
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ChaozhiZheng
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Biometris
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WageningenUniversityandResearchCentre
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POBox16,6700AAWageningen
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TheNetherlands
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Phone: [phone]
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Email: [email protected]
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© The Author(s) 2013. Published by the Genetics Society of America.
Abstract
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The models for the mosaic structure of an individual’s genome from multiparental populations
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have been developed primarily for autosomes, whereas X chromosomes receive very little at-
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tention. In this paper, we extend our previous approach to model ancestral origin processes
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along two X chromosomes in a mapping population, which is necessary for developing hid-
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den Markov models in the reconstruction of ancestry blocks for X-linked quantitative trait lo-
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cus (QTL) mapping. The model accounts for the joint recombination pattern, the asymmetry
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between maternally and paternally derived X chromosomes, and the finiteness of population
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size. The model can be applied to various mapping populations such as the advanced inter-
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cross lines (AIL), the Collaborative Cross (CC), the heterogeneous stock (HS), the Diversity
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Outcross (DO), and the Drosophila synthetic population resource (DSPR). We further derive
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the map expansion, density (per Morgan) of recombination breakpoints, in advanced intercross
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populations with L inbred founders under the limit of an infinitely large population size. The
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analytic results show that for X chromosomes the genetic map expands linearly at a rate (per
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generation)oftwo-thirdstimes1−10/(9L)fortheAIL,andatarateoftwo-thirdstimes1−1/L
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for the DO and the HS, whereas for autosomes the map expands at a rate of 1 − 1/L for the
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AIL,theDO,andtheHS.
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Introduction
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There have been recently designed QTL mapping populations with either multiple parents to
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increase the genetic diversity of the founder population, or many intercross generations to im-
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provethemappingresolutionbyaccumulatinghistoricalrecombinationevents. Someexamples
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38 include the Collaborative Cross (CC) (CHURCHILL et al. 2004), the advanced intercross lines
39 (AIL) (DARVASI and SOLLER 1995), the heterogeneous stock (HS) (MOTT et al. 2000), the
40 diversity outcross (DO) (SVENSON et al. 2012), and the Drosophila synthetic population re-
41 source (DSPR) (KING et al. 2012). The CC can be regarded as a set of eight-way recombinant
inbredlines(RIL)bysiblingmating,whereeightfoundersofeachlinearepermuted.
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ThegenomesofindividualsinQTLmappingpopulationsarerandommosaicsofthefounders’
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genomes. The QTL mapping generally necessitates the reconstruction of these genome blocks
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along two homologous chromosomes of a sampled individual from available genotype data.
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Such reconstruction is often performed under a hidden Markov model (HMM) with the latent
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state being the pair of ancestral origins at a locus, where the transition probability of ancestral
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originsbetweentwoloci,orthetwo-locusdiplotype(two-haplotype)probabilitiesarerequired.
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Modelingancestraloriginsalongapairofautosomalchromosomeshasbeenwelldeveloped
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50 recently. BROMAN (2012a) extended the approach of HALDANE and WADDINGTON (1931)
from the two-way to four- and eight-way RIL by sibling mating, and provided recipes for cal-
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52 culating autosomal two-locus diplotype probabilities numerically. JOHANNES and COLOME-
53 TATCHE (2011) derived autosomal two-locus diplotype probabilities for the two-way RIL by
54 selfing. ZHENG et al. (2014) described a general modeling framework for ancestral origins
that can be applied to autosomes in various mapping populations such as the RIL by selfing or
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siblingmatingandtheAIL.
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A special treatment is required for modeling ancestral origins along a pair of X chromo-
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58 somes. HALDANE and WADDINGTON (1931) derived the recurrence relations of the X-linked
two-locusdiplotypeprobabilitiesforthetwo-wayRILbysiblingmatingandthebi-parentalre-
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peated parent-offspring mating, and their closed form solutions for the final homozygous lines.
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61 BROMAN (2005) extended the solutions to the two- and three-locus haplotype probabilities for
62 thetwo,four,oreight-wayRILbysiblingmating. BROMAN (2012b)derivedtheX-linkedtwo-
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locushaplotypeprobabilitiesinadvancedintercrosspopulationsincludingtheAIL,theHS,and
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theDO,assuminganinfinitelylargepopulationsize.
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65 In this paper, we extend our previous work (ZHENG et al. 2014) to model the ancestral
origins along a pair of X chromosomes in a finite mapping population. This extension also
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67 builds on the theory of junctions in inbreeding (FISHER 1949, 1954). A junction is defined
as a boundary point of genome blocks on chromosomes where two distinct ancestral origins
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meet, and the boundary points that occur at the same location along multiple chromosomes are
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counted as a single junction. The map expansion is the expected junction density (per Morgan)
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on a maternally or paternally derived X chromosome, denoted by Rm or Rp, respectively. We
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denotebyρmp theoveralljunctiondensityalongtheXXchromosomesofafemale,anditcanbe
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73 usedasameasureofX-linkedQTLmappingresolution(DARVASIandSOLLER1995;WELLER
74 and SOLLER 2004).
The key feature of this extension is to account for the asymmetry between maternally and
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paternally derived X chromosomes because the latter did not experience any crossover events
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withYchromosomes. WefirstpresentamodelframeworkforX-linkedancestralorigins,where
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the recurrent relations are derived for various junction densities including the map expansions
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Rm and Rp. Then, we derive the closed form solutions for these expected densities in mapping
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populationsincludingtheRILbysiblingmating,theAIL,theHS,theDO,andtheDSPR;they
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are evaluated by forward simulation studies. Lastly, we discuss the model assumptions and the
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implicationsoftheanalyticresultsonhaplotypereconstructionsandbreedingdesigns.
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A model for X-linked ancestral origins
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Assumptions and notation
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Consideradioeciouspopulationwithtwoseparatesexes: homogameticfemaleswithsexchro-
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mosomesXXandheterogameticmaleswithsexchromosomesXY.Therearenorecombination
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events between X and Y, and thus we ignore the pseudoautosomal regions on the XY chro-
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mosomes. As in most mammals and some insects (Drosophila), some flowering species, such
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as white campion (Silene latifolia), papaya (Carica papaya), and asparagus (Asparagus offi-
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90 cianalis), have the XY sex determination system (MING and MOORE 2007). The dioecious
population was founded in generation 0, and it has non-overlapping generations. There are no
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natural or artificial selections since the founder population. The mating schemes of produc-
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ing the next generation are random, and they may vary from one generation to the next. The
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assignmentsofoffspringgendersareassumedtobeindependentofmatingschemes.
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The ancestral origins along two homologous autosomes have been modeled as a continu-
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96 ous time Markov chain (CTMC) (ZHENG et al. 2014). We extend the approach to account
for the asymmetry of XX chromosomes, using superscript m (p) for maternally (paternally)
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derived genes or chromosomes. See Table S1 for a list of symbols used in this paper. Let
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O(x) = (Om(x),Op(x))betheorderedpairoftheancestraloriginsatlocationxalongthetwo
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X chromosomes of a randomly sampled female. The ancestral origin process O(x) is assumed
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to follow a CTMC , where x is the time parameter in unit of Morgan. We assign a unique an-
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cestral origin to the X chromosomes of each inbred founder, or to each X chromosome of each
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outbred founder. Let L be the number of possible ancestral origins that Om(x) or Op(x) may
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take. L may be less than the number of inbred founders if some male founders did not produce
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daughters to pass down their X chromosomes. For example, L = 3 for the four-way RIL by
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siblingmatingsinceoneofthefoundermatingpairsproducesonlyoneson(Figure1A).
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The L possible ancestral origins are assumed to be exchangeable, so that we focus on the
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changesofancestralorigins. SeeFigure1BandtherelevantpartofDiscussionontheexchange-
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ability assumption. The initial distribution of O(0) at the leftmost locus x = 0 is specified by
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αmp(11),aprobabilitythatthetwoancestraloriginsarethesame(identicalbydescent,IBD)at
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alocus. Letαmp(12) = 1−αmp(11)bethenon-IBDprobability. GiveneitherIBDornon-IBD
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at the locus, the ancestral origin pair O(0) takes each of the possible combinations with equal
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probability.
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The transition rate matrix of the CTMC can be constructed from the expected densities
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Jmp(abcd) of all the junction types (abcd) along the two X chromosomes of a female. The
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junction type (abcd) denotes the four-gene IBD configuration (abcd) on both sides of a junc-
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tion,whereab(cd)isontheleft-hand(right-hand)side,haplotypeac(bd)isonthefirst(second)
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chromosome, and the same integers denote IBD. Figure 1C illustrates the seven types of junc-
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tions: (1112), (1121), (1122), (1211), (1213), (1222), and (1232) for L ≥ 3, where the two
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types (1213) and (1232) do not exist for L = 2. We do not define junction types for the eight
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two-locus configurations (1111), (1123), (1212), (1221), (1223), (1231), (1233), and (1234),
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since there are either zero or no less than two junctions between the two loci. Figure 1D shows
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thetransitionratematrixoftheCTMCinthefour-wayRILbysiblingmating. Figure1Eshows
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therelationshipsbetweentheexpecteddensitiesJmp(abcd)andthetransitionrates,andtheyare
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derived based on the interpretation that Jmp(abcd)∆d is the two-locus diplotype probability, in
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thelimitthatthegeneticdistance∆d(inMorgan)betweentwolocigoestozero.
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ThemapexpansionsRm andRp andtheoverallexpectedjunctiondensityρmp aregivenby
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Rm =Jmp(1121)+Jmp(1122)+Jmp(1222)+Jmp(1232), (1)
Rp =Jmp(1112)+Jmp(1122)+Jmp(1211)+Jmp(1213), (2)
ρmp =Rm +Rp −Jmp(1122), (3)
128 similar to those for autosomes (ZHENG et al. 2014) except that Rm (cid:54)= Rp for X chromosomes.
We have Jmp(1112) = Jmp(1211) and Jmp(1121) = Jmp(1222), since the junction densities
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do not depend on the direction of chromosomes. In contrast to the single-locus two-gene non-
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IBD probability αmp(12), the ordering of the superscripts in Jmp(abcd) generally does matter,
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that is, Jmp(abcd) (cid:54)= Jpm(abcd) except for the junction type (1122). In addition, we have
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Jmp(1213) = Jpm(1232) (see Figure 1C). Thus, the CTMC of X-linked ancestral origins can
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be described by one non-IBD probability αmp(12) and the five expected junction densities Rm,
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Rp, Jmp(1122), Jmp(1232), and Jpm(1232), under the exchangeability assumption of the L
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possibleancestralorigins.
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Single-locus non-IBD probabilities
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The calculation of the expected junction densities necessitates the introduction of the proba-
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bilities for the two- and three-gene IBD configurations at a single locus. All the following
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derivationsoftherecurrencerelationsfortheseprobabilitiesarebasedontheMendelianinher-
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itance of X-linked genes: a paternally derived gene must be a copy of the maternally derived
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gene in a male of the previous generation, and a maternally derived gene has equal probability
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ofbeingacopyofeitherthematernallyderivedgeneorthepaternallyderivedgeneinafemale
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ofthepreviousgeneration.
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In a dioecious mapping population, the single-locus two-gene probabilities of IBD con-
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figuration (ab) depend on whether the two homologous genes are in a single individual or
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not. Thus, we denote by βmm(ab), βmp(ab), and βpp(ab) the two-gene probability of IBD
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configuration (ab), given that the two homologous genes are in two distinct individuals in gen-
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eration t and have parental origins mm, mp, and pp, respectively (Figure 2A); it holds that
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βpm(ab) = βmp(ab).
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The recurrence relations of the two-gene non-IBD probabilities are derived by tracing the
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parental origins of two homologous genes from generation t ≥ 1 into the previous generation,
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andtheyaregivenby
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(cid:20) (cid:21)
1 1 1 1
βmm(12) =sm αmp(12)+(1−sm) βmm(12)+ βmp + βpp (4a)
t t 2 t−1 t 4 t−1 2 t−1 4 t−1
1 1
βmp(12) = βmm(12)+ βmp(12) (4b)
t 2 t−1 2 t−1
βpp(12) =(1−sp)βmm(12) (4c)
t t t−1
αmp(12) =βmp(12) (4d)
t t
whereequation(4d)holdsimmediatelyafteronegenerationofrandommating,althoughitmay
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not hold in the founder population at t = 0. In equation (4a), the first term on the right-hand
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sidereferstothescenariothatthetwogeneswithparentoriginsmmingenerationtcomefrom
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asinglefemaleofthepreviousgenerationwiththeprobabilitysm,andwithprobability1/2that
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they come from different genes of the female. In equation (4b), the two genes with parental
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origins mp cannot merge since they must come from one male and one female of the previous
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generation. In equation (4c), the two genes with parental origins pp in generation t come from
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a single male of the previous generation with the probability sp; if so, they must merge since
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thereisonlyoneXchromosomeinamale.
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We introduce the single-locus three-gene probabilities of IBD configuration (abc). Let
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βmmm(abc), βmmp(abc), βmpp(abc), and βppp(abc) be the probabilities of IBD configuration
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(abc), given that the three homologous genes are in three distinct individuals in generation t
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and have parental origins mmm, mmp, mpp, and ppp, respectively (Figure 2B). Similarly, we
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define αmmp(abc) and αmpp(abc) for three homologous genes in two distinct individuals. The
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ordering of the superscripts does not matter for these three-gene probabilities, for example,
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βmmp(abc) = βmpm(abc) = βpmm(abc).
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The recurrence relations of the three-gene non-IBD probabilities are derived by tracing the
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parentaloriginsofthreehomologousgenesfromgenerationt ≥ 1intothepreviousgeneration,
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andtheyaregivenby
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(cid:20) (cid:21)
1 1 1
βmmm(123) =3qm αmmp(123)+ αmpp(123) +(1−sm −2qm)
t t 2 2 t−1 2 t−1 t t
(5a)
(cid:20) (cid:21)
1 3 3 1
× βmmm(123)+ βmmp(123)+ βmpp(123)+ βppp(123)
8 t−1 8 t−1 8 t−1 8 t−1
1
βmmp(123) =sm αmmp(123)+(1−sm)
t t 2 t−1 t
(cid:20) (cid:21) (5b)
1 1 1
× βmmm(123)+ βmmp(123)+ βmpp(123)
4 t−1 2 t−1 4 t−1
(cid:20) (cid:21)
1 1
βmpp(123) =(1−sp) βmmm(123)+ βmmp(123) (5c)
t t 2 t−1 2 t−1
βppp(123) =(1−sp −2qp)βmmm(123) (5d)
t t t t−1
αmmp(123) =βmmp(123) (5e)
t t
αmpp(123) =βmpp(123) (5f)
t t
where qm is the coalescence probability of three maternally derived genes in generation t that
173 t
a particular pair of genes come from a single female of the previous generation and the third
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comes from another female of the previous generation, and similarly qp for three paternally
175 t
derivedgenes. Theequations(5e,5f)holdimmediatelyafteronegenerationofrandommating,
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althoughtheymaynotholdinthefounderpopulationatt = 0.
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Thederivationsoftherecurrenceequations(5a–5d)forthethree-genenon-IBDprobabilities
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are similar to equations (4a–4c) for the two-gene non-IBD probabilities. In equation (5a), the
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pre-factor3denotesthateachofthethreepossiblepairsofgenesmaycomefromasinglefemale
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ofthepreviousgeneration;theterm(1−sm −2qm)istheprobabilitythatthethreematernally
181 t t
derivedgenesingenerationtcomefromthreedistinctfemalesofthepreviousgeneration,andit
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isobtainedbytheprobability1−smthatonepairofgenescomefromtwodistinctfemalesminus
183 t
theprobability2qm thatthethirdgeneandeithergeneofthepaircomefromasinglefemaleof
184 t
the previous generation. Similarly, the term (1−sp −2qp) in equation (5d) is the probability
185 t t
that the three paternally derived genes in generation t come from three distinct males of the
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previousgeneration.
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Expected junction densities
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We derive the recurrence relations for Rm, Rp, Jmp(1122), Jmp(1232), and Jpm(1232). The
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190 recurrencerelationforRm followsfromthetheoryofjunctions(FISHER 1954): anewjunction
is formed whenever a recombination event occurs between two X chromosomes that are non-
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IBDatthelocationofacrossover. TherecurrencerelationsforthemapexpansionsRm andRp
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aregivenby
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1 1
Rm = Rm + Rp +αmp(12), (6a)
t 2 t−1 2 t−1 t−1
Rp = Rm , (6b)
t t−1
where equation (6b) follows directly from no recombination events occurring between the XY
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chromosomesinamaleofthepreviousgeneration.
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Tomeasuredifferentialmapexpansionsbetweenmaternallyandpaternallyderivedchromo-
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somes,wedefineRX = (2Rm+Rp)/3andR− = (Rm−Rp)/2,andtheirrecurrencerelations
197 t t t t t t
aregivenby
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2
RX =RX + αmp(12), (7a)
t t−1 3 t−1
1 1
R− =− R− + αmp(12), (7b)
t 2 t−1 2 t−1
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accordingtotherecurrenceequations(6a,6b). Ifthereareequalnumbersofmalesandfemales
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in the population, a randomly chosen X chromosome is maternally derived with probability
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2/3, and it is paternally derived with probability 1/3. Thus RX can be interpreted as the map
201 t
expansiononarandomlychosenXchromosome.
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For comparisons, we denote by RA the map expansion on a random chosen autosome, and
203 t
204 anditsrecurrencerelationisgivenby(MACLEOD etal.2005; ZHENG etal.2014)
RA = RA +αAA(12), (8)
t t−1 t−1
whereαAA(12)referstothenon-IBDprobabilitybetweentwohomologousautosomalgenesin
205 t
an individual. The equations (7a, 8) show that the map expansion RX for an X chromosome is
206 t
two-thirds RA for an autosome if the non-IBD probability αAA(12) for autosomes is the same
207 t t
asαmp(12)forXXchromosomes,andthesexratiois1.
208 t
In addition to Jmp(abcd) and Jpm(abcd), we define Kmm(abcd), Kmp(abcd) , Kpm(abcd),
209 t t t t t
and Kpp(abcd) for haplotypes ac and bd that are in two distinct individuals and have parental
210 t
originsmm,mp,pm,andpp,respectively(Figure2C).Thecontributionstothejunctionsinthe
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current generation come from either the existing junctions at the previous generation, or a new
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junction via a crossover event. In the following, we focus on the formation of a new junction,
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since the contributions of the existing junctions in the previous generation are similar to those
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forthetwo-genenon-IBDprobabilitiesintherecurrenceequations(4a–4c).
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Theschematicsoftherecurrencerelationsforjunctiontypes(1232)and(1122)areshownin
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FigureS1. Theancestrytransitionsoftype(1122)occuronbothhaplotypesacandbdatexactly
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the same location, and thus a new junction of type (1122) can be formed only by duplicating a
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chromosome segment. It holds that Jmp(1122) = Jpm(1122) and Kmp(1122) = Kpm(1122)
219 t t t t
becauseofthesymmetryoftype(1122). Wehave
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10