Autosomal Admixture Levels Are Informative About Sex Bias in Admixed Populations.  
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Sexbiased admixture has been observed in a wide variety of admixed populations. Genetic variation in sex chromosomes and functions of quantities computed from sex chromosomes and autosomes have often been examined in order to infer patterns of sexbiased admixture, typically using statistical approaches that do not mechanistically model the complexity of a sexspecific history of admixture. Here, expanding on a model of Verdu & Rosenberg (2011) that did not include sex specificity, we develop a model that mechanistically examines sexspecific admixture histories. Under the model, multiple source populations contribute to an admixed population, potentially with their male and female contributions varying over time. In an admixed population descended from two source groups, we derive the moments of the distribution of the autosomal admixture fraction from a specific source population as a function of sexspecific introgression parameters and time. Considering admixture processes that are constant in time, we demonstrate that surprisingly, although the mean autosomal admixture fraction from a specific source population does not reveal a sex bias in the admixture history, the variance of autosomal admixture is informative about sex bias. Specifically, the longterm variance decreases as the sex bias from a contributing source population increases. This result can be viewed as analogous to the reduction in effective population size for populations with an unequal number of breeding males and females. Our approach suggests that it may be possible to use the effect of sexbiased admixture on autosomal DNA to assist with methods for inference of the history of complex sexbiased admixture processes. 
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Amy Goldberg; Paul Verdu; Noah A Rosenberg 
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Type: JOURNAL ARTICLE Date: 201495 
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Title: Genetics Volume:  ISSN: 19432631 ISO Abbreviation: Genetics Publication Date: 2014 Sep 
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Journal Information Journal ID (nlmta): Genetics Journal ID (isoabbrev): Genetics Journal ID (hwp): genetics Journal ID (pmc): genetics Journal ID (publisherid): genetics ISSN: 00166731 ISSN: 19432631 Publisher: Genetics Society of America 
Article Information Download PDF Copyright © 2014 by the Genetics Society of America openaccess: Received Day: 23 Month: 6 Year: 2014 Accepted Day: 29 Month: 8 Year: 2014 Print publication date: Month: 11 Year: 2014 Electronic publication date: Day: 5 Month: 9 Year: 2014 pmcrelease publication date: Day: 5 Month: 9 Year: 2014 Volume: 198 Issue: 3 First Page: 1209 Last Page: 1229 PubMed Id: 25194159 ID: 4224161 Publisher Id: 166793 DOI: 10.1534/genetics.114.166793 
Autosomal Admixture Levels Are Informative About Sex Bias in Admixed Populations  
Amy Goldberg*1  
Paul Verdu^{†}  
Noah A. Rosenberg*  
*Department of Biology, Stanford University, Stanford, California 943055020 

†Centre National de la Recherche Scientifique–Muséum National d'Histoire Naturelle, Université Paris Diderot, Unité Mixte de Recherche 7206, Ecoanthropology and Ethnobiology, Paris, France 75005 

1Corresponding author: Stanford University, 371 Serra Mall, Gilbert Biology Bldg., Stanford, CA 94305. Email: agoldb@stanford.edu 
POPULATIONS often experience sexbiased demographic processes, in which males and females contributing to the gene pool of a population are drawn from source groups in different proportions, owing to patterns of inbreeding avoidance, dispersal, and mating practices (^{Pusey 1987}; ^{Lawson Handley and Perrin 2007}). In humans, sexbiased demography has had a particular effect on admixed populations, populations that have often been founded or influenced by periods of colonization and forced migration involving an initial or continuing admixture process (^{Mesa et al. 2000}; ^{Seielstad 2000}; ^{Wilkins and Marlowe 2006}; ^{Tremblay and Vezina 2010}; ^{Heyer et al. 2012}).
Genetic signatures of sexbiased admixture have been empirically investigated in a variety of human populations. In the Americas, these include African American, Latino, and Native American populations (^{Bolnick et al. 2006}; ^{Wang et al. 2008}; ^{Stefflova et al. 2009}; ^{Tishkoff et al. 2009}; ^{Bryc et al. 2010a},^{b}; ^{MorenoEstrada et al. 2013}; ^{Verdu et al. 2014}). Sexbiased admixture and migration have also been examined in populations throughout Asia (^{Oota et al. 2001}; ^{Wen et al. 2004}; ^{Chaix et al. 2007}; ^{Ségurel et al. 2008}; ^{Chaubey et al. 2011}; ^{Pemberton et al. 2012}; ^{Pijpe et al. 2013}), Austronesia (^{Kayser et al. 2003}, ^{2006}, ^{2008}; ^{Cox et al. 2010}; ^{Lansing et al. 2011}), and Africa (^{Wood et al. 2005}; ^{Tishkoff et al. 2007}; ^{BerniellLee et al. 2008}; ^{Beleza et al. 2013}; ^{Petersen et al. 2013}; ^{Verdu et al. 2013}).
Sexspecific admixture and migration processes have typically been studied using comparisons of the Y chromosome, which is paternally inherited, and the mitochondrial genome, inherited maternally (^{Seielstad et al. 1998}; ^{Oota et al. 2001}; ^{Wood et al. 2005}; ^{Bolnick et al. 2006}; ^{Gunnarsdóttir et al. 2011}; ^{Lacan et al. 2011}). More recently, as the Y chromosome and mitochondrial genome each represent single nonrecombining loci that provide an incomplete genomic perspective, sexbiased admixture has been examined by comparisons of autosomal DNA to the X chromosome (^{Lind et al. 2007}; ^{Wang et al. 2008}; ^{Bryc et al. 2010a},^{b}; ^{Cox et al. 2010}; ^{Beleza et al. 2013}; ^{Verdu et al. 2013}).
The Ymitochondrial and Xautosomal frameworks are both sensible, as both involve comparisons of two types of loci that follow different modes of inheritance in males and females. What has not been clear, however, is that autosomal data, which have not typically been viewed as the most informative loci for studies of sexspecific processes, can carry information about sexbiased admixture, even in the absence of a comparison with other components of the genome.
We demonstrate this surprising result through an extension of a mechanistic model for the admixture history of a hybrid population. In a diploid autosomal framework, ^{Verdu and Rosenberg (2011)} examined contributions of multiple source populations that varied through time, without considering sex specificity. Here, expanding on the model of ^{Verdu and Rosenberg (2011)}, we develop a model that mechanistically considers sexspecific admixture histories in which multiple source populations contribute to the admixed population, potentially with varying female and male contributions across generations (Figure 1). In an admixed population descended from two source populations, we derive the moments of the distribution of the fraction of autosomal admixture from a specific source population, as a function of sexspecific admixture parameters and time. We analyze the behavior of the model, considering admixture processes that are constant in time, and we show that the moments contain information about the sex bias.
Several studies have described mechanistic models of admixture (^{Chakraborty and Weiss 1988}; ^{Long 1991}; ^{Ewens and Spielman 1995}; ^{Guo et al. 2005}; ^{Verdu and Rosenberg 2011}; ^{Gravel 2012}; ^{Jin et al. 2014}). We follow the notation and style of the model of ^{Verdu and Rosenberg (2011)}, studying a hybrid population, H, which consists of immigrant individuals from M isolated source populations and hybrid individuals who have ancestors from two or more source populations. The source populations are labeled S_{α}, for α from 1 to M. We focus on the case of M = 2.
We define the parameters s_{α}_{,}_{g}_{−1} and h_{g}_{−1} as the contributions from source populations S_{α} and H, respectively, to the gene pool of the hybrid population H at the next generation, g. That is, for a random individual at generation g, the probabilities that a randomly chosen parent of the individual derives from S_{α} and H are s_{α}_{,}_{g}_{−1} and h_{g}_{−1}, respectively. We define the sexspecific parameter sα,g−1δ, for δ ∈ {f, m}, as the probability that the typeδ parent of a randomly chosen individual from the hybrid population at generation g is from source population S_{α}. Similarly, hg−1δ is the probability that the typeδ parent of a randomly chosen individual in H at generation g is from H itself. We consider a twosex model, using f for female and m for male. Because each individual has one parent of each type, female and male, we have
sα,g−1=(sα,g−1f+sα,g−1m)/2, 
hg−1=(hg−1f+hg−1m)/2. 
s1,g−1+s2,g−1+hg−1=1. 
s1,g−1f+s2,g−1f+hg−1f=s1,g−1m+s2,g−1m+hg−1m=1. 
h0=h0f=h0m=0, 
s1,0+s2,0=s1,0f+s2,0f=s1,0m+s2,0m=1. 
Our model enables us to consider complex sexbiased admixture processes by allowing uneven sexspecific contributions from each source population at each generation. It reduces to the ^{Verdu and Rosenberg (2011)} model when the sexspecific contributions are equal within source populations, that is, if for each g, s1,g−1f=s1,g−1m and s2,g−1f=s2,g−1m. We perform similar computations to those of ^{Verdu and Rosenberg (2011)}, finding that in certain cases, our results reduce to those obtained when sex specificity is not considered.
We let L be a random variable indicating the source populations of the parents of a random individual from the hybrid population, H. L takes its values from the set of all possible ordered parental combinations, {S_{1}S_{1}, S_{1}H, S_{1}S_{2}, HS_{1}, HH, HS_{2}, S_{2}S_{1}, S_{2}H, S_{2}S_{2}}, listing the female parent first. We assume random mating in the hybrid population at each generation, so that the probability that an offspring has a particular pair of source populations for his or her parents is simply the product of separate probabilities associated with the female and male parents (Table 1).
We define the fraction of admixture, the random variable H_{α}_{,}_{g}_{,}_{δ}, as the probability that an autosomal genetic locus in a random individual of sex δ from the hybrid population in generation g ultimately originates from source population α. The sexspecific fractions of admixture are related to the total fraction of admixture H_{α}_{,}_{g} from source population α in generation g by H_{α}_{,}_{g} = (H_{α}_{,}_{g}_{,}_{f} + H_{α}_{,}_{g}_{,}_{m})/2.
Under the model, we derive expressions for the moments of the fraction of admixture. Autosomal DNA is inherited nonsexspecifically and from both parents; therefore, female and male offspring have identical distributions of admixture, and H_{α}_{,}_{g}_{,}_{f} and H_{α}_{,}_{g}_{,}_{m} are identically distributed. Each of these quantities depends on both the female and male fractions of admixture in the previous generation, but conditional on the previous generation (that is, on H_{α}_{,}_{g}_{−1,}_{f} and H_{α}_{,}_{g}_{−1,}_{m}), they are independent. For our twopopulation model, we consider the nonsexspecific fraction of admixture, H_{1,}_{g}_{,}_{δ}, treating δ here as representing either f or m, but retaining the same meaning through a calculation. The quantity H_{1,}_{g}_{,}_{δ} depends on both sexspecific fractions of admixture from the previous generation, H_{1,}_{g}_{−1,}_{f} and H_{1,}_{g}_{−1,}_{m}.
The definition of the model parameters and the values from Table 1 allow us to write a recursion relation for the fraction of admixture from source population 1 for a random individual of sex δ from the hybrid population at generation g, or H_{1,}_{g}_{,}_{δ}. For the first generation, g = 1, we have
H1,1,δ={1if L=S1S1,with ℙ[L=S1S1]=s1,0fs1,0m12if L=S1S2,with ℙ[L=S1S2]=s1,0fs2,0m12if L=S2S1,with ℙ[L=S2S1]=s2,0fs1,0m0if L=S2S2,with ℙ[L=S2S2]=s2,0fs2,0m. 
H1,g,δ={1if L=S1S1,with ℙ[L=S1S1]=s1,g−1fs1,g−1m1+H1,g−1,m2if L=S1H,with ℙ[L=S1H]=s1,g−1fhg−1m12if L=S1S2,with ℙ[L=S1S2]=s1,g−1fs2,g−1m1+H1,g−1,f2if L=HS1,with ℙ[L=HS1]=hg−1fs1,g−1mH1,g−1,f+H1,g−1,m2if L=HH,with ℙ[L=HH]=hg−1fhg−1mH1,g−1,f2if L=HS2,with ℙ[L=HS2]=hg−1fs2,g−1m12if L=S2S1,with ℙ[L=S2S1]=s2,g−1fs1,g−1mH1,g−1,m2if L=S2H,with ℙ[L=S2H]=s2,g−1fhg−1m0if L=S2S2,with ℙ[L=S2S2]=s2,g−1fs2,g−1m. 
ℙ(H1,1,δ=q)={s1,0fs1,0mif q=1s1,0fs2,0m+s2,0fs1,0mif q=12s2,0fs2,0mif q=0. 
ℙ(H1,g,δ=q)=hg−1fhg−1m∑r=02g−1[ℙ(H1,g−1,δ1=r2g−1)ℙ(H1,g−1,δ2=2gq−r2g−1)]+(s1,g−1fhg−1m+hg−1fs1,g−1m)ℙ(H1,g−1,δ=2q−1)+(s2,g−1fhg−1m+hg−1fs2,g−1m)ℙ(H1,g−1,δ=2q)+Ig(q). 
Ig(q)={s1,g−1fs1,g−1mif q=1s1,g−1fs2,g−1m+s2,g−1fs1,g−1mif q=12s2,g−1fs2,g−1mif q=00otherwise. 
Equations 9–11 can be used to analyze the behavior of the distribution of H_{1,}_{g}_{−1,}_{δ} over time. In Figure 2, we consider constant admixture processes after the founding of the hybrid population (sα,gδ=sαδ for each α ∈ {1, 2}, δ ∈ {f, m}, and g ≥ 1), plotting ℙ(H_{1,}_{g}_{,}_{δ}) for the first six generations, as computed recursively using Equation 10. In Figure 2, A and B, we consider a hybrid population founded with equal contributions from source populations S_{1} and S_{2}, but with no further contributions after g = 1. In both of these cases, the distribution of the autosomal admixture fraction contracts around the mean of 12. However, whereas Figure 2A has equal contributions from each sex in the founding generation, Figure 2B has a large initial sex bias. We see that the width of the distribution is smaller with the sexbiased contributions, despite equality of the total contributions s_{1,0} and s_{2,0}.
In Figures 2, C–E, we consider admixture scenarios in which the founding of the hybrid population is followed by constant contributions from the source populations over time, s_{1} = 0.1 and s_{2} = 0.3. Because the two source populations contribute after the founding, the distribution does not contract around the mean as in Figures 2, A and B. Also, because the total contributions from S_{1} and S_{2} are unequal, the distribution of H_{1,}_{g}_{,}_{δ} is no longer symmetrical. Rather, because the contribution from S_{2} is greater, the distribution is shifted toward zero.
Figures 2, C and D, have the same continuing contributions for g ≥ 2, with no sex bias in the founding generation for Figure 2C, and a large initial sex bias for Figure 2D. Despite different founding contributions, Figures 2, C and D, have similar distributions of H_{1,}_{g}_{,}_{δ} after a few generations. In Figure 2E, the hybrid population is founded without a sex bias and with equal contributions from the two source populations. The total contributions s_{1} and s_{2} are the same as in Figures 2, C and D, but unlike in Figures 2, C and D, the continuing contributions are sex biased, with s1f≠s1m and s2f≠s2m. Even with s_{1} and s_{2} held constant, the distribution of H_{1,}_{g}_{,}_{δ} depends on the sαδ. Notably, the probability of H_{1,6,}_{δ} = 0 drops from 0.157 in Figure 2C to 0.000 in Figure 2E. Similarly, ℙ(H_{1,6,}_{δ} = 1) drops to zero in Figure 2E as well. With these reductions at the extremes, we see a rise in the probability of intermediate values for H_{1,}_{g}_{,}_{δ}.
Using the law of total expectation, we write the expectation of the fraction of admixture from source population 1 for a random individual of sex δ in population H at generation g as a function of conditional expectations for all possible pairs of parents L,
E[H1,g,δ]=EL[E[H1,g,δL]]=∑ℓ∈{S1S1S1HS1S2HS1HHHS2S2S1S2HS2S2}ℙ(L=ℓ)E[H1,g,δL=ℓ]. 
E[H1,1,δ]=ℙ(L=S1S1)E[H1,1,δL=S1S1]+ℙ(L=S1S2)E[H1,1,δL=S1S2]+ℙ(L=S2S1)E[H1,1,δL=S2S1]+ℙ(L=S2S2)E[H1,1,δL=S2S2]. 
E[H1,g,δ]=ℙ(L=S1S1)E[H1,g,δL=S1S1]+ℙ(L=S1H)E[H1,g,δL=S1H]+ℙ(L=S1S2)E[H1,g,δL=S1S2]+ℙ(L=HS1)E[H1,g,δL=HS1]+ℙ(L=HH)E[H1,g,δL=HH]+ℙ(L=HS2)E[H1,g,δL=HS2]+ℙ(L=S2S1)E[H1,g,δL=S2S1]+ℙ(L=S2H)E[H1,g,δL=S2H]+ℙ(L=S2S2)E[H1,g,δL=S2S2]. 
E[H1,1,δ]=s1,0fs1,0mE[1]+(s1,0fs2,0m+s2,0fs1,0m)E[12]+s2,0fs2,0mE[0]. 
E[H1,g,δ]=s1,g−1fs1,g−1mE[1]+s1,g−1fhg−1mE[1+H1,g−1,m2]+hg−1fs1,g−1mE[1+H1,g−1,f2]+(s1,g−1fs2,g−1m+s2,g−1fs1,g−1m)E[12]+hg−1fhg−1mE[H1,g−1,f+H1,g−1,m2]+s2,g−1fhg−1mE[H1,g−1,m2]+hg−1fs2,g−1mE[H1,g−1,f2]+s2,g−1fs2,g−1mE[0]. 
E[H1,1,δ]=s1,0, 
E[H1,g,δ]=s1,g−1+12(hg−1fE[H1,g−1,f]+hg−1mE[H1,g−1,m]). 
E[H1,g,δ]=s1,g−1+hg−1E[H1,g−1,δ]. 
We can write a general recursion for the higher moments of the admixture fraction from population S_{1} in a randomly chosen individual of sex δ from the hybrid population. For k ≥ 1, in generation g = 1,
H1,1,δk={1kif L=S1S1,with ℙ[L=S1S1]=s1,0fs1,0m(12)kif L=S1S2,with ℙ[L=S1S2]=s1,0fs2,0m(12)kif L=S2S1,with ℙ[L=S2S1]=s2,0fs1,0m0kif L=S2S2,with ℙ[L=S2S2]=s2,0fs2,0m. 
H1,g,δk={1kif L=S1S1,with ℙ[L=S1S1]=s1,g−1fs1,g−1m(1+H1,g−1,m2)kif L=S1H,with ℙ[L=S1H]=s1,g−1fhg−1m(12)kif L=S1S2,with ℙ[L=S1S2]=s1,g−1fs2,g−1m(1+H1,g−1,f2)kif L=HS1,with ℙ[L=HS1]=hg−1fs1,g−1m(H1,g−1,f+H1,g−1,m2)kif L=HH,with ℙ[L=HS1]=hg−1fhg−1m(H1,g−1,f2)kif L=HS2,with ℙ[L=HS2]=hg−1fs2,g−1m(12)kif L=S2S1,with ℙ[L=S2S1]=s2,g−1fs1,g−1m(H1,g−1,m2)kif L=S2H,with ℙ[L=S2H]=s2,g−1fhg−1m0kif L=S2S2,with ℙ[L=S2S2]=s2,g−1fs2,g−1m. 
E[H1,1,δk]=s1,0fs1,0mE[1k]+(s1,0fs2,0m+s2,0fs1,0m)E[(12)k]+s2,0fs2,0mE[0k]. 
E[H1,g,δk]=s1,g−1fs1,g−1mE[1k]+s1,g−1fhg−1mE[(1+H1,g−1,m2)k]+hg−1fs1,g−1mE[(1+H1,g−1,f2)k]+(s1,g−1fs2,g−1m+s2,g−1fs1,g−1m)E[(12)k]+hg−1fhg−1mE[(H1,g−1,f+H1,g−1,m2)k]+s2,g−1fhg−1mE[(H1,g−1,m2)k]+hg−1fs2,g−1mE[(H1,g−1,f2)k]+s2,g−1fs2,g−1mE[0k]. 
E[H1,1,δk]=s1,0fs1,0m+s1,0fs2,0m+s2,0fs1,0m2k. 
E[H1,g,δk]=s1,g−1fs1,g−1m+s1,g−1fs2,g−1m+s2,g−1fs1,g−1m2k+s1,g−1fhg−1m2k(∑i=0k(ki)E[H1,g−1,mi])+hg−1fs1,g−1m2k(∑i=0k(ki)E[H1,g−1,fi])+hg−1fhg−1m2k(∑i=0k(ki)E[H1,g−1,fk−iH1,g−1,mi])+s2,g−1fhg−1m2kE[H1,g−1,mk]+hg−1fs2,g−1m2kE[H1,g−1,fk]. 
E[H1,g,δk]=s1,g−1fs1,g−1m+s1,g−1fs2,g−1m+s2,g−1fs1,g−1m2k+s1,g−1fhg−1m+hg−1fs1,g−1m2k(∑i=0k(ki)E[H1,g−1,δi])+hg−1fhg−1m2k(∑i=0k(ki)E[H1,g−1,δk−i]E[H1,g−1,δi])+s2,g−1fhg−1m+hg−1fs2,g−1m2kE[H1,g−1,δk]. 
E[H1,1,δ]=s1,0fs1,0m+s1,0fs2,0m+s2,0fs1,0m2=s1,0, 
E[H1,g,δ]=s1,g−1fs1,g−1m+s1,g−1fs2,g−1m+s2,g−1fs1,g−1m+s1,g−1fhg−1m+hg−1fs1,g−1m2+(2hg−1fhg−1m+s1,g−1fhg−1m+hg−1fs1,g−1m+hg−1fs2,g−1m+s2,g−1fhg−1m2)E[H1,g−1,δ], 
When k = 2, Equations 24 and 26 produce a recursion for the second moment of H_{1,}_{g}_{,}_{δ}. Recalling Equations 1–6, for g = 1, we have
E[H1,1,δ2]=s1,0f(1+s1,0m)+s1,0m(1+s1,0f)4. 
E[H1,g,δ2]=s1,g−1fs1,g−1m+s1,g−1fs2,g−1m+s2,g−1fs1,g−1m4+s1,g−1fhg−1m4(1+2E[H1,g−1,m]+E[H1,g−1,m2])+hg−1fs1,g−1m4(1+2E[H1,g−1,f]+E[H1,g−1,f2])+hg−1fhg−1m4(E[H1,g−1,f2]+2E[H1,g−1,f]E[H1,g−1,m]+E[H1,g−1,m2])+s2,g−1fhg−1m4E[H1,g−1,m2]+hg−1fs2,g−1m4E[H1,g−1,f2]. 
E[H1,g,δ2]=s1,g−1f(1+s1,g−1m)+s1,g−1m(1+s1,g−1f)4+s1,g−1fhg−1m+hg−1fs1,g−1m2E[H1,g−1,δ]+hg−1fhg−1m2(E[H1,g−1,δ])2+hg−1f+hg−1m4E[H1,g−1,δ2]. 
V[H1,1,δ]=s1,0f(1−s1,0f)+s1,0m(1−s1,0m)4. 
V[H1,g,δ]=s1,g−1f(1−s1,g−1f)+s1,g−1m(1−s1,g−1m)4−s1,g−1fhg−1f+s1,g−1mhg−1m2E[H1,g−1,δ]+hg−1f(1−hg−1f)+hg−1m(1−hg−1m)4(E[H1,g−1,δ])2+hg−1f+hg−1m4V[H1,g−1,δ]. 
The recursion for the variance of the fraction of admixture of a random individual of sex δ from the hybrid population is dependent on the variance from the previous generation, the expectation from the previous generation, and its square. By contrast with the expectation, the variance of the fraction of admixture depends on the sexspecific contributions from the source populations.
Equations 32 and 33 are invariant with respect to an exchange of all variables corresponding to males (superscript m) with those corresponding with females (superscript f). Thus, although the variance is affected by the sexspecific admixture contributions, it does not identify the direction of the bias. Despite the dependence of the variance of the autosomal fraction of admixture on sexspecific contributions, under the model, the symmetry demonstrates that autosomal DNA alone does not identify which sex contributes more to the hybrid population from a given source population. This result is reasonable given the nonsexspecific inheritance pattern of autosomal DNA.
Using the recursions in Equations 17, 19, 32, and 33, we can study specific cases in which the contributions are specified. We first consider the case in which the source populations S_{1} and S_{2} do not contribute to the hybrid population after its founding: s1,gf=s1,gm=s2,gf=s2,gm=0, and h=(hgf+hgm)/2=1, for all g ≥ 1. As before, at the first generation, the hybrid population is not yet formed, and h_{0} = 0. Therefore, s1,0+s2,0=s1,0f+s2,0f=s1,0m+s2,0m=1.
Under this scenario, we can derive the exact expectation and variance of the autosomal fraction of admixture of a random individual from the hybrid population. In the case of a single admixture event, the expectation of the admixture fraction is equal to the expectation at the first generation, because the further contributions are all zero. Using Equation 19, s_{1,}_{g}_{−1} = s_{2,}_{g}_{−1} = 0 for all g ≥ 2. Therefore, from Equation 17, in the case of a single admixture event, for all g ≥ 1,
E[H1,g,δ]=s1,0. 
Using Equations 32 and 33, because s1f=s1m=s2f=s2m=0 for all g ≥ 2, the variance of the fraction of admixture follows a geometric sequence with ratio 12. For all generations g ≥ 1,
V[H1,g,δ]=s1,0f(1−s1,0f)+s1,0m(1−s1,0m)2g+1. 
With a single admixture event, the variance decreases monotonically, and its limit is zero for all parameter values. Individuals from the hybrid population mate only within the population, decreasing the variance by a factor of 2 each generation. Thus, Equation 35 predicts that the distribution of the admixture fraction for a random individual in the hybrid population contracts around the mean, converging to a constant equal to the mean admixture from the first generation.
In Equation 35, considering all possible pairs (s1,0f,s1,0m), with each entry in [0, 1], the maximal V[H1,g,δ] occurs at (s1,0f,s1,0m)=(12,12), a scenario with equal contributions from the two source populations and no sex bias. At the maximum, the variance is V[H1,g,δ]=1/2g+2. Four minima occur, at (s1,0f,s1,0m)=(0,0), (0, 1), (1, 0), and (1, 1), cases in which all individuals in generation g = 1 have the same pair of source populations for their two parents, and in later generations, all individuals continue to have the same value of H_{1,}_{g}_{,}_{δ}. In these cases, V[H1,g,δ]=0.
Figure 3 plots the variance in Equation 35 as a function of the sexspecific parameters s1,0f and s1,0m for three values of g. For g = 1, a maximum of V[H1,1,δ]=18 occurs at (s1,0f,s1,0m)=(12,12), and a minimum, V[H1,1,δ]=0, at (s1,0f,s1,0m)=(0,0),(0,1),(1,0), or (1, 1). After one generation of mixing within the hybrid population, with no further contributions from the source populations, the maximum and minima occur at the same values of (s1,0f,s1,0m), but the variance is halved (Figure 3B). That is, for a given set of values (s1,0f,s1,0m), V[H1,2,δ]=V[H1,1,δ]/2. Similarly, for g = 8 in Figure 3C, V[H1,8,δ]=V[H1,1,δ]/27. By g = 8, the hybrid population is quite homogeneous in admixture, and the variance of the admixture fraction has decreased to near zero for all sets of founding parameters. Therefore, the admixture fraction distribution is close to constant, with H_{1,8,}_{δ} ≈ s_{1,0}.
We can analyze the dependence of the variance on the sexspecific parameters by considering constant total contributions s_{1,0} and allowing the sexspecific contributions to vary, constrained by Equation 1 so that 0≤s1,0f,s1,0m≤min(1,2s1,0). Rewriting Equation 35 in terms of s_{1,0} and s1,0f,
V[H1,g,δ]=s1,0f(2s1,0−s1,0f)−s1,0(2s1,0−1)2g. 
For the specific case of s1,0=12, the total contribution for which the maximal variance occurs in Figure 3, we illustrate the variance at several locations in the allowed range for s1,0f and s1,0m (Figure 4). Four scenarios are plotted with the same total founding contribution from source population 1, s1,0=12, but with different levels of sex bias. As the female and male contributions become increasingly different, the initial variance decreases. The largest variance for s1,0=12 occurs at s1,0f=s1,0m=s1,0, with no sex bias. The minimum occurs when males all come from one source population and females all from the other. In this extreme sexbiased case, the variance is zero constantly over time, as each individual has a male parent from one population, a female parent from the other, and an admixture fraction of 12.
Next, we consider the case in which an initial admixture event founds the hybrid population and is then followed by constant nonzero contributions from the source populations. After the founding, for each g ≥ 1, all admixture parameters are constant in time: sα,gδ=sαδ for each α ∈ {1, 2} and δ ∈ {f, m}, and hgδ=hδ for each δ. Thus, we have parameter values for the founding and constant continuing admixture parameters s1f, s1m, s2f, and s2m. Each parameter takes its value in [0, 1], as do s_{1} and s_{2}. By contrast, h takes its value in (0, 1). The case of h = 1 is a single admixture event, analyzed above. The h = 0 case is trivial because the hybrid population is refounded at each generation, and the distribution of the admixture fraction thus depends only on the contribution in the previous generation. Therefore, we require s_{1} + s_{2} ≠ 0 and s_{1} + s_{2} ≠ 1. Individually, however, h^{f} and h^{m} can each vary in [0, 1], as long as they are not both zero or one.
The recursion for the expectation of the autosomal fraction of admixture, Equations 17 and 19, is equivalent to that derived by ^{Verdu and Rosenberg (2011)}. Therefore, the closed form of the expectation is equivalent as well. From ^{Verdu and Rosenberg (2011, Equation 30)} we have
E[H1,g,δ]={s1,0,g=1s1,0hg−1+s11−hg−11−h,g≥2. 
E[H1,1,δ2]=s1,0f(1+s1,0m)+s1,0m(1+s1,0f)4. 
E[H1,g,δ2]=s1f(1+s1m)+s1m(1+s1f)4+s1fhm+hfs1m2E[H1,g−1,δ]+hfhm2(E[H1,g−1,δ])2+hf+hm4E[H1,g−1,δ2]. 
E[H1,g,δ2]=ψ(g)+λE[H1,g−1,δ2], 
α0=E[H1,1,δ2]=s1,0f(1+s1,0m)+s1,0m(1+s1,0f)4. 
ψ(g)=s1f(1+s1m)+s1m(1+s1f)4+s1fhm+hfs1m2E[H1,g−1,δ]+hfhm2(E[H1,g−1,δ])2. 
ψ(g)=s1f(1+s1m)+s1m(1+s1f)4+s1fhm+hfs1m2(s1,0hg−2+s11−hg−21−h)+hfhm2(s1,0hg−2+s11−hg−21−h)2. 
E[H1,g,δ2]={α0,g=1α0(h2)g−1+∑i=2g[s1f(1+s1m)+s1m(1+s1f)4+s1fhm+hfs1m2(s1,0hi−2+s11−hi−21−h)+hfhm2(s1,0hi−2+s11−hi−21−h)2](h2)g−i,g≥2. 
E[H1,g,δ2]={α0,g=1,A1+A2hg−1+(A3+A4∑i=1g−1(2h)i)(h2)g−1,g≥2, 
A1=s1+s1fs1m(2−h)+s1(s1fhm+hfs1m)(1−h)(2−h)+s12hfhm(1−h)2(2−h), 
A2=1h(s1fhm+hfs1m+2hfhms11−h)(s1,0−s11−h), 
A3=α0+hfhms11−h[2h(s11−h−s1,0)−s1(1−h)(2−h)]+(s1fhm+hfs1m)[s1h1−h−s1,0h−s1(1−h)(2−h)]−s1+s1fs1m2−h, 
A4=hfhm2h2[s1,02+s11−h(s11−h−2s1,0)]. 
Using the relation V[H1,g,δ]=E[H1,g,δ2]−(E[H1,g,δ])2 and Equations 37 and 45, for the variance of the autosomal fraction of admixture, we have
V[H1,g,δ]={s1,0f(1−s1,0f)+s1,0m(1−s1,0m)4,g=1,A1+A2hg−1+[A3+A4∑i=1g−1(2h)i](h2)g−1−(s1,0hg−1+s11−hg−11−h)2,g≥2. 
V[H1,g,δ]={s1,0f(1−s1,0f)+s1,0m(1−s1,0m)4,g=1,A1+A2hg−1+[A3+A4(2h−(2h)g1−2h)](h2)g−1−(s1,0hg−1+s11−hg−11−h)2,g≥2. 
V[H1,g,δ]={s1,0f(1−s1,0f)+s1,0m(1−s1,0m)4,g=1,A1+A2(12)g−1+A3(14)g−1+A4(g−1)(14)g−1−[2s1+(s1,0−2s1)(12)g−1]2,g≥2. 
Figure 5 illustrates the variance of the autosomal fraction of admixture as a function of g when the contributions from the source populations are constant over time, computed using Equation 50. The figure shows that if the continuing contributions are held constant, then the longterm limiting variance does not depend on the founding parameters. Unlike in the hybrid isolation case, with constant, nonzero contributions from the source populations over time, h ≠ 0 and h ≠ 1, a nonzero limit is reached. Applying Equation 50,
limg→∞V[H1,g,δ]=A1−(s11−h)2, 
Using Equations 1, 2, 4, and 46, the limit in Equation 53 can be equivalently written in terms of the two female sexspecific contributions, s1f and s2f, and the total contributions from the two source populations, s_{1} and s_{2}. Considering admixture scenarios with constant s_{1}, s_{2}, with s_{1} + s_{2} ∈ (0, 1], but allowing s1f and s2f to range over the closed unit interval, the limiting variance depends on two independent parameters, s1f,s2f∈[0,1], subject to the constraint in Equation 1:
limg→∞V[H1,g,δ]=−(s1fs2−s2fs1)2+s1s2(s1+s2)(s1+s2)2(1+s1+s2). 
f(s1f,s2f)=−(s1fs2−s2fs1)2. 
s1fs2f=s1ms2m=s1s2. 
For fixed s_{1} and s_{2}, however, the case without sex bias is not the only maximum of the limiting variance. Figure 7 plots the variance over time for four different admixture histories, each with the same total contributions s_{1} and s_{2}, but quite different sexspecific contributions (s1f,s1m,s2f,s2m). Each of the four scenarios plotted reaches the same limit because each provides a solution to Equation 56. Because f(s1f,s2f)=0, Equation 54 depends only on the total contributions s_{1} and s_{2}. For constant s_{1} and s_{2}, any admixture history whose contributions solve f(s1f,s2f)=0 has limiting variance
limg→∞V[H1,g,δ]=s1s2(s1+s2)(1+s1+s2). 
Thus far, we have considered the maximal limiting variance as a function of the sexspecific parameters given constant total contributions s_{1} and s_{2}. We can also identify the values of s_{1} and s_{2} that maximize the limiting variance, considering all s_{1}, s_{2} ∈ [0, 1]. For each choice of s_{1} and s_{2}, the maximal variance over values of s1f and s2f is given by Equation 57. We can therefore find the s_{1} and s_{2} that maximize Equation 57. As shown by ^{Verdu and Rosenberg (2011)}, given s_{1} + s_{2}, the maximal limiting variance occurs when s_{1} = s_{2}. Over the range of possible choices for s_{1} + s_{2} ∈ (0, 1), the maximum occurs when s1=s2=12. Unlike in ^{Verdu and Rosenberg (2011)}, however, this maximum requires the sexspecific contributions to solve f(s1f,s2f)=0.
Interestingly, one of the minima of the limiting variance occurs when s1=s2=12, but with f(s1f,s2f)≠0. Specifically, when s1=s2=12, but all males come from one source population and all females from the other, (s1f,s1m,s2f,s2m)=(1,0,0,1) or (0, 1, 1, 0), the limiting variance in Equation 54 is zero. In this case, L=S1S2 or L=S2S1 for every individual in the hybrid population. By Equation 8, the hybrid population is founded anew at each generation, each individual having admixture fraction H1,g,δ=12.
More generally, given s1 and s2, the minimum occurs when (s1f,s2f)=(2s1,0) or (s1f,s2f)=(0,2s2). Given s1 and s2, the limiting variance is minimized with respect to s1f and s2f when f(s1f,s2f) is smallest (Equation 54). Because f is the negative of the square of a difference of products, it is smallest when one term is zero and the other is at its maximum, as at (s1f,s2f)=(2s1,0) or (s1f,s2f)=(0,2s2). These points represent the maximal sex bias for fixed (s_{1}, s_{2}).
If we allow s_{1} and s_{2} to vary, because a variance is bounded below by zero, any set of parameters that produces zero variance is a minimum. In Equation 54, if either s_{1} = 0 or s_{2} = 0, then the limiting variance of the admixture fraction is zero. When only one population contributes after the founding, in the limit, all ancestry in the hybrid population traces to that population.
The limiting variance of the fraction of admixture over time in Equation 53 is a function of the sexspecific contributions from the hybrid population, h^{f} and h^{m}, and source population 1, s1f and s1m. Recalling Equation 4, the limiting variance is equivalently written as a function of the sexspecific contributions from source population 2, s2f and s2m, and either source population 1 (Equation 54), or the hybrid population. It can be viewed as a function of all six sexspecific parameters (s1f,s1m,s2f,s2m,hf,hm), four of which can be selected while assigning the other two by the constraint from Equation 4.
We can therefore analyze the behavior of the limiting variance as a function of two of the sexspecific parameters by specifying two other parameters and allowing the final two parameters, one female and one male, to vary according to Equation 4. Of the four parameters we consider, using the constraint from Equation 4 separately in males and females, two must be male and two must be female. Because the variance is invariant with respect to exchanging the source populations or the sexes, the sixdimensional parameter space has a number of symmetries. Figure 8, Figure 9, Figure 10, Figure 11, and Figure 12 examine the five possible, nonredundant ways of choosing two populations and the corresponding male and female parameters from those populations and holding two corresponding parameters fixed (either from the same sex in the two populations, or for males and females from one population) while allowing the other two to vary. Figure 13 then highlights an informative case that considers the limiting variance in terms of a male and a female parameter from different populations.
Each figure shows multiple contour plots of the limiting variance as a function of two sexspecific parameters, for fixed values of two other parameters. Three cases plot the limiting variance as a function of the female and male parameters from a given population, with the female and male contributions of another population specified. In two other cases, parameters for a single sex from two populations are plotted, specifying the contributions from the other sex for those populations.
By considering these parameter combinations, we can examine the dependence of the variance on sexspecific parameters and parameter interactions, as well as potential bounds on both the parameters and the variance. We highlight a number of symmetries in the limiting variance. The plots also illustrate the maxima and minima found in the previous section.
In Figure 8, we consider the variance of the fraction of admixture as a function of s1f, the female contribution from S_{1}, on the xaxis, and s1m, the male contribution from S_{1}, on the yaxis, computed using Equation 53. We plot the variance for fixed h^{f}, the female contribution from H, and h^{m}, the male contribution from H. The domain for s1f and s1m is constrained by Equation 4, with s1f taking values in [0, 1 − h^{f}], and s1m taking values in [0, 1 − h^{m}].
Figure 8, top left, shows the variance as a function of s1f and s1m, with h^{f} = h^{m} = h = 0. Here, the hybrid population is founded anew by the source populations each generation, and s1f and s1m both take values from the full domain [0, 1]. For h^{f} = h^{m} = h = 0, the maximal limiting variance is limg→∞V[H1,g,δ]=18, occurring when s1f=s1m=s1=0.5. At this maximum, given Equation 4 and h^{f} = h^{m} = 0, we have s2f=s2m=s2=0.5. As in Equation 54, the maximum occurs when female and male contributions from the source populations are equal and the total contributions from the source populations are equal.
The minima of limg→∞V[H1,g,δ]=0 occur at the four corners of the plot. At the origin, when s1f=s1m=s1=0, the limiting variance is zero because only S_{2} contributes to the hybrid population. Individuals in the hybrid population all have parents L = S_{2}S_{2} and admixture fraction zero (Equation 8). By exchanging S_{1} for S_{2}, the case of s1f=s1m=s1=1 is similar. Additional minima occur at (s1f,s1m)=(1,0) or (0, 1), where all males come from one source population and all females from the other, and all individuals at the next generation of the hybrid population have admixture fraction 12 (Equation 8).
For h^{f} = h^{m} = h = 0, the limiting variance is symmetrical over the line s1m=s1f, as a result of the symmetry between males and female in the variance (Equation 33). Because the hybrid population provides no contribution and the variance of the fraction of admixture is symmetric with respect to source population, the variance is also symmetric over the lines s1f=0.5 and s1m=0.5.
The columns of Figure 8 consider increasing, fixed values for h^{f}, and the rows consider increasing, fixed values for h^{m}, both from {0, 0.25, 0.5, 0.75, 0.95}. All the cells maintain the general shape of the limiting variance as a function of s1f and s1m seen for h^{f} = h^{m} = 0. However, as the domain for s1f and s1m shrinks with increasing h^{f} and h^{m}, the location of the maximal variance changes across cells. In all cases, the maximum of the limiting variance occurs when s1f and s1m each lie at the midpoints of their respective domains, s1f=(1−hf)/2 and s1m=(1−hm)/2. The magnitude of the limiting variance at each maximum decreases as its location moves away from s1f=s1m=0.5.
In all the cells, the minimum limg→∞V[H1,g,δ]=0 occurs when s1f and s1m are either both zero or they lie at the maxima of their respective domains. In these cases, only one source population contributes to the hybrid population, and all individuals in the hybrid population have an admixture fraction from S_{1} of either 0, when s1f=s1m=0, or 1, when s1f=1−hf and s1m=1−hm. The limiting variance is no longer zero at the two corners of each plot where only one of {s1f,s1m} is at the maximum of its domain; these corners, however, are minima of the variance given the values of s_{1} and s_{2}. In these cases, males all come from one source population and females from the other, producing a minimum of Equation 54 for fixed s_{1} and s_{2}.
As in the case of h^{f} = h^{m} = 0, each cell is symmetrical in reflecting over both the midpoint of the xaxis, s1f=(1−hf)/2, and that of the yaxis, s1m=(1−hm)/2. The limiting variance is symmetrical with respect to source population (Equation 54), and this pair of reflections corresponds to an exchange of source populations. For h^{f} = h^{m} = 0, the line s1f=s1m generates circular contours, but as the contributions from the hybrid population increase, the contours become elliptical.
In Figure 8, cells on the diagonal have equal contributions from males and females in the hybrid population, h^{f} = h^{m} = h. For h^{f} ≠ h^{m}, cells above the diagonal are equivalent to those below the diagonal with an exchange of female for male contributions, for both s_{1} and h. For example, the cell with h^{f} = 0.25 and h^{m} = 0.5 is equivalent to the cell with h^{f} = 0.5 and h^{m} = 0.25 if the axes are also switched so that s1m appears along the xaxis and s1f is on the yaxis.
Figure 9 plots the limiting variance as a function of s1f on the xaxis and s1m on the yaxis, but we now fix values of s2f by column and s2m by row using Equations 54 and 1. The maxima and minima occur at the same parameter values found in Figure 8, but they appear in different locations on the plots. For example, in Figure 9, the global maximum across cells occurs in the plot with s2f=s2m=0.5 specified, and s1f=s2m=0.5. By Equation 4, this location has h^{f} = h^{m} = 0, the cell with the maximal variance in Figure 8. In Figure 9, top left, the variance is zero for all s1f and s1m, because s_{2} = 0 (Equation 54).
Whereas all cells in Figure 8 are symmetric in reflecting over the midpoints of both domains, in Figure 9, only the cells with s2f=s2m are symmetric over the line s1m=s1f. However, the symmetry corresponding to transposing males and females is visible in that a cell above the diagonal and its corresponding cell below the diagonal are equivalent if the axes for s1f and s1m are switched.
Similarly to Figure 8, Figure 10 considers the limiting variance of the admixture fraction over time as a function of the four variables s1f, s1m, h^{f}, and h^{m} using Equation 53. In Figure 10, each cell shows h^{f} on the xaxis and h^{m} on the yaxis, for s1f and s1m specified, with the domains of h^{f} and h^{m} constrained by Equation 4. The cells on the diagonal have s1f=s1m, and there is a symmetry over this line of cells in that if values of s1f and s1m are switched, then the cells will be equivalent with a transposition of the axes.
In Figure 10, top left, s1f=s1m=s1=0, and the limiting variance is a constant zero. In Figure 10, the maximal variance occurs at the origin (h^{f} = h^{m} = h = 0) of the cell with s1f=s1m=s1=0.5. As in Figure 8, at the maximum, by Equation 4, s2f=s2m=s2=0.5. In this case, females and males contribute equally. Both source populations contribute maximally to pull the distribution of the fraction of admixture toward the extremes of zero and one.
Because the limiting variance is symmetrical with respect to source population, and recalling Equation 4, each cell in Figure 10 is equivalent to a corresponding cell in Figure 9 reflected along both the xaxis and yaxis. For example, the cell in Figure 10 with s1f=0.5 and s1m=0.25 is equivalent to the Figure 9 cell with s2f=0.5 and s2m=0.25 if reflected on both the x and yaxes.
Figure 8, Figure 9, and Figure 10 illustrate that the global maximum of the limiting variance occurs when the two source populations contribute equally, the contributions from the two sexes are equal, and the hybrid population does not contribute to the next generation. As the parameters move from the location of the maximal limiting variance to the minimum, the variance monotonically decreases.
Next we plot on the x and yaxes two parameters of the same sex from different populations. Because the variance is invariant with respect to transposition of females and males, we consider only females without loss of generality. Figure 11 plots the limiting variance as a function of s1f on the xaxis and h^{f} on the yaxis, for fixed values of s1m and h^{m}. Figure 12 plots the limiting variance as a function of s1f and s2f, for fixed s1m and s2m. For Figure 11 and Figure 12, the domains of s1f,s2f, and h^{f} are constrained by Equation 4.
For Figure 11, the maximal limiting variance occurs in the cell with s1m=0.5 and h^{m} = 0, at (s1f,hf)=(0.5,0). By Equation 4, this location is the same parameter set for the maximum in Figures 8, Figure 9, and Figure 10. The maximum in each cell occurs when (s1f,hf)=(0.5,0), but the magnitude of the variance decreases with increased distance from the cell with fixed s1m=0.5 and h^{m} = 0. Similarly, within each cell, the limiting variance decreases with distance from (s1f,hf)=(0.5,0).
In the first column of Figure 11, where s1m=0, the line s1f=0 produces zero variance because the hybrid population is homogenous, with only one source population contributing. Similarly, in cells with s1m+hm=1, as s2f=s2m=0 by Equation 4, the line s1f+hf=1 has minimal variance.
In Figure 12, because of the symmetry in sex in Equation 54, the cells above and below those where s1m=s2m are equivalent with a transposition of axes. As in Figure 8, Figure 9, Figure 10, and Figure 11, the maximal variance occurs in the cell with s1m=s2m=0.5 at s1f=s2f=0.5. Also, similar to Figure 11, in the first column, when s1m=0, the line s1f=0 is a minimum; in the first row, when s2m=0, the line s2f=0 is a minimum.
Analogous to the similarity between Figure 9 and Figure 10, by Equation 4, each cell in Figure 12 is a transformation of a cell in Figure 11. For example, for the cell with s1m=0.25 and h^{m} = 0.5 specified in Figure 11, because the male contributions sum to one, this cell also specifies s2m=0.25. Therefore, we can compare this cell to the cell with s1m=s2m=0.25 in Figure 12. Both show s1f on the xaxis, and using Equation 4, we can rewrite the yaxis in Figure 12 as s2f=1−hf.
Finally, we consider a case in which males from one population in (S_{1}, S_{2}, H) are compared to females from a different population. Although multiple parameter configurations are possible, we plot one that is particularly informative, providing a perspective on Equation 54 beyond the observations visible in Figure 8, Figure 9, Figure 10, Figure 11, and Figure 12. Figure 13 plots the limit of the variance of the admixture fraction as a function of s1f on the xaxis and s2m on the yaxis, for fixed s2f and s1m. We rewrite Equation 54 as a function of s1f,s1m,s2f, and s2m using Equation 1:
limg→∞V[H1,g,δ]=−2(s1fs2m−s2fs1m)2+(s1f+s1m)(s2f+s2m)(s1f+s1m+s2f+s2m)(s1f+s1m+s2f+s2m)2(2+s1f+s1m+s2f+s2m). 
Our model demonstrates the potential informativeness of autosomal DNA in the study of sexbiased admixture histories. Under a framework in which admixture occurs over time, potentially with different male and female contributions from the source populations, we have derived recursive expressions for the expectation, variance, and higher moments of the fraction of autosomal admixture. For the special case of constant admixture over time, we have analyzed the behavior of the variance of the admixture fraction. Although the expectation of the autosomal admixture fraction depends only on the total contributions from the source populations, we found that the variance of the autosomal admixture can contain a signature of sexspecific contributions. In particular, for constant admixture over time, the variance of the autosomal admixture fraction decreases as the male and female contributions become increasingly unequal.
That autosomal DNA possesses a signature of sexbiased admixture might at first appear counterintuitive, as unlike the sex chromosomes, autosomes are carried equally in both sexes. The phenomenon can, however, be understood by analogy with the wellknown result that increasing sex bias decreases the effective size of populations (^{Wright 1931}; ^{Crow and Dennison 1988}; ^{Caballero 1994}; ^{Hartl and Clark 2007}). In a computation of effective size using the coalescent, for example (^{Nordborg and Krone 2002}; ^{Ramachandran et al. 2008}), the sex bias causes pairs of genetic lineages to be likely to find common ancestors more recently than in a nonsexbiased population, as the reduced chance of a coalescence in the sex that represents a larger fraction of the breeding population is outweighed by the greater chance of a coalescence in the less populous sex. In a similar manner, if admixture is sexbiased, because lineages are more likely to travel along paths through populations with the larger sexspecific contributions, the variability of genealogical paths—and hence, the variance of the admixture fraction—is reduced compared to the nonsexbiased case.
Autosomal DNA, with its multitude of independent loci, potentially provides more information about the complex histories of hybrid populations, and the autosomal genome might be less susceptible to locusspecific selective pressures than the sex chromosomes. To take advantage of autosomal information, many recent efforts to study sexbiased demography have compared autosomal DNA with the X chromosome (^{Ramachandran et al. 2004}, ^{2008}; ^{Wilkins and Marlowe 2006}; ^{Hammer et al. 2008}, ^{2010}; ^{Bustamante and Ramachandran 2009}; ^{Keinan et al. 2009}; ^{Casto et al. 2010}; ^{Emery et al. 2010}; ^{Keinan and Reich 2010}; ^{Labuda et al. 2010}; ^{Lambert et al. 2010}; ^{Gottipati et al. 2011}; ^{Heyer et al. 2012}; ^{Arbiza et al. 2014}). Our study enhances the set of frameworks available for considering effects of admixture and sex bias on autosomal variation. Further, our theoretical results are potentially important to the interpretation of existing methods that utilize admixture fractions. In particular, a decreased variance, often interpreted as older admixture timing, can instead be a consequence of sex bias.
For a single admixture event, the expectation of the autosomal admixture fraction is constant in time and not dependent on sexspecific contributions. Unlike in the case of hybrid isolation, if constant nonzero contributions from the source populations occur over time, then the variance of the fraction of autosomal admixture reaches a nonzero limit, dependent on these continuing sexspecific admixture rates, but not on the founding contributions. In both scenarios, the variance contains information about the magnitude of a sex bias in the admixture history of a hybrid population. For an arbitrary constant total contribution from a source population, the maximal variance occurs when there is no sex bias. The maximal variance across allowable parameter values of the constant admixture model is seen when there is no sex bias and equal contributions from both source populations, that is, s1f=s1m=s2f=s2m=0.5. Two types of admixture history minimize the variance of the autosomal admixture fraction. First, the variance is zero when only one source population contributes to the hybrid population. Second, the variance is zero if all males come from one source population and all females from the other. In this scenario, all individuals in the hybrid population have admixture fraction 12.
Although the variance of the autosomal admixture fraction suggests that autosomal DNA is informative about sexbiased admixture, the relationship between the variance and the sexspecific parameters is complex. We uncovered an interesting case in which quite different sexspecific histories can lead to the same variance over time (Figure 7). The variance is in fact dependent on the product of multiple sexspecific parameters, not on each parameter separately (Figure 13). In particular, when s1fs2m=s2fs1m, the variance is maximized (Equations 54–56), and depends only on the total contributions from the source populations, s_{1} and s_{2} (Equation 57). The symmetry arises from the nonsexspecific inheritance of autosomal DNA.
We have considered two scenarios, isolation of a hybrid population after its founding, and constant contributions from source populations to the hybrid population over time. Although the admixture history of real hybrid populations is likely more complex than these, jointly considering the mean, variance, and potentially higher moments of the admixture fractions, our models can provide a starting point for statistical frameworks to estimate parameters of mechanistic admixture models. We have not numerically analyzed complex timevarying admixture histories, but our recursive expressions flexibly accommodate a range of population histories, especially if simplifying assumptions are employed to reduce the number of parameters.
Our model omits a number of potentially important phenomena. First, assortative mating by ancestry, preferential mating of individuals with those with similar admixture fractions, has been empirically observed in admixed populations (^{Risch et al. 2009}), and may have sexspecific patterns. Second, our focus on a randomly chosen locus in a deterministic model amounts to a potentially unrealistic assumption of an infinite chromosome with infinitely many independent segments. ^{Gravel (2012)}, however, calculated the variance of the admixture fraction including both finite chromosomes and finite population sizes, for a model similar to the one presented here, albeit without sex bias. ^{Gravel (2012)} found that the genealogy of individuals in the hybrid population—which our model explicitly examines—is the main factor affecting the variance when admixture is recent, showing that the ^{Verdu and Rosenberg (2011)} variance provides a good fit to the finitepopulation finitechromosome result in that context. We expect that this suitability to conditions of recent admixture applies similarly for our sexbiased version of the ^{Verdu and Rosenberg (2011)} model.
Finally, although sex bias does influence autosomal variation, because autosomal DNA is not inherited sexspecifically, the sex that contributes more from a given source population is nonidentifiable with autosomal DNA alone. Because the X chromosome has a sexspecific mode of inheritance, consideration of the X chromosome alongside autosomal data under the mechanistic model may assist in differentiating between scenarios that produce the same variance with different choices of the sex with a greater contribution.
Notes
Available freely online through the authorsupported open access option.
fn1Communicating editor: B. A. Payseur
We thank Ethan Jewett and Michael D. Edge for useful discussions. We acknowledge support from an National Science Foundation (NSF) Graduate Research Fellowship and from NSF grant BCS1147534.
Literature Cited
Arbiza L.,Gottipati S.,Siepel A.,Keinan A.. , Year: 2014 Contrasting Xlinked and autosomal diversity across 14 human populations.Am. J. Hum. Genet.94: 827–844.24836452  
Beleza S.,Campos J.,Lopes J.,Araújo I. I.,Almada A. H.,et al. , Year: 2013 The admixture structure and genetic variation of the archipelago of Cape Verde and its implications for admixture mapping studies.PLoS ONE7: e51103.23226471  
BerniellLee G.,Plaza S.,Bosch E.,Calafell F.,Jourdan E.,et al. , Year: 2008 Admixture and sexual bias in the population settlement of La Réunion Island (Indian Ocean).Am. J. Phys. Anthropol.136: 100–107.18186507  
Bolnick D. A.,Bolnick D. I.,Smith D. G.. , Year: 2006 Asymmetric male and female genetic histories among Native Americans from eastern North America.Mol. Biol. Evol.23: 2161–2174.16916941  
Bryc K.,Auton A.,Nelson M. R.,Oksenberg J. R.,Hauser S. L.,et al. , Year: 2010 a. Genomewide patterns of population structure and admixture in West Africans and African Americans.Proc. Natl. Acad. Sci. USA107: 786–791.20080753  
Bryc K.,Velez C.,Karafet T.,MorenoEstrada A.,Reynolds A.,et al. , Year: 2010 b. Genomewide patterns of population structure and admixture among Hispanic/Latino populations.Proc. Natl. Acad. Sci. USA107: 8954–8961.20445096  
Bustamante C. D.,Ramachandran S.. , Year: 2009 Evaluating signatures of sexspecific processes in the human genome.Nat. Genet.41: 8–10.19112457  
Caballero A.. , Year: 1994 Developments in the prediction of effective population size.Heredity73: 657–679.7814264  
Casto A. M.,Li J. Z.,Absher D.,Myers R.,Ramachandran S.,et al. , Year: 2010 Characterization of Xlinked SNP genotypic variation in globally distributed populations.Genome Biol.11: R10.20109212  
Chaix R.,QuintanaMurci L.,Hegay T.,Hammer M. F.,Mobasher Z.,et al. , Year: 2007 From social to genetic structures in central Asia.Curr. Biol.17: 43–48.17208185  
Chakraborty R.,Weiss K. M.. , Year: 1988 Admixture as a tool for finding linked genes and detecting that difference from allelic association between loci.Proc. Natl. Acad. Sci. USA85: 9119–9123.3194414  
Chaubey G.,Metspalu M.,Choi Y.,Magi R.,Romero I. G.,et al. , Year: 2011 Population genetic structure in Indian Austroasiatic speakers: the role of landscape barriers and sexspecific admixture.Mol. Biol. Evol.28: 1013–1024.20978040  
Cox M. P.,Karafet T. M.,Lansing J. S.,Sudoyo H.,Hammer M. F.. , Year: 2010 Autosomal and Xlinked single nucleotide polymorphisms reveal a steep Asian–Melanesian ancestry cline in eastern Indonesia and a sex bias in admixture rates.Proc. R. Soc. Lond. B Biol. Sci.277: 1589–1596.  
Crow J. F.,Dennison C.. , Year: 1988 Inbreeding and variance effective population numbers.Evolution42: 482–495.  
Cull P.,Flahive M.,Robson R.. , Year: 2005 Difference Equations: From Rabbits to Chaos. SpringerVerlag, New York.  
Emery L. S.,Felsenstein J.,Akey J. M.. , Year: 2010 Estimators of the human effective sex ratio detect sex biases on different timescales.Am. J. Hum. Genet.87: 848–856.21109223  
Ewens W. J.,Spielman R. J.. , Year: 1995 The transmission/disequilibrium test: history, subdivision, and admixture.Am. J. Hum. Genet.57: 455–464.7668272  
Gottipati S.,Arbiza L.,Siepel A.,Clark A. G.,Keinan A.. , Year: 2011 Analyses of Xlinked and autosomal genetic variation in populationscale whole genome sequencing.Nat. Genet.43: 741–743.21775991  
Gravel S.. , Year: 2012 Population genetics models of local ancestry.Genetics191: 607–619.22491189  
Gunnarsdóttir E. D.,Nandineni M. R.,Li M.,Myles S.,Gil D.,et al. , Year: 2011 Larger mitochondrial DNA than Ychromosome differences between matrilocal and patrilocal groups from Sumatra.Nat. Commun.2: 228.21407194  
Guo W.,Fung W. K.,Shi N.,Guo J.. , Year: 2005 On the formula for admixture linkage disequilibrium.Hum. Hered.60: 177–180.16352907  
Hammer M. F.,Mendez F. L.,Cox M. P.,Woerner A. E.,Wall J. D.. , Year: 2008 Sexbiased evolutionary forces shape genomic patterns of human diversity.PLoS Genet.4: e1000202.18818765  
Hammer M. F.,Woerner A. E.,Mendez F. L.,Watkins J. C.,Cox M. P.,et al. , Year: 2010 The ratio of human X chromosome to autosome diversity is positively correlated with genetic distance from genes.Nat. Genet.42: 830–831.20802480  
Hartl D. L.,Clark A. G.. , Year: 2007 Principles of Population Genetics, Ed. 4Sinauer, Sunderland, MA.  
Heyer E.,Chaix R.,Pavard S.,Austerlitz F.. , Year: 2012 Sexspecific demographic behaviors that shape human genomic variation.Mol. Ecol.21: 597–612.22211311  
Jin W.,Li R.,Zhou Y.,Xu S.. , Year: 2014 Distribution of ancestral chromosomal segments in admixed genomes and its implications for inferring population history and admixture mapping.Eur. J. Hum. Genet.22: 930–937.24253859  
Kayser M.,Brauer S.,Weiss G.,Schiefenhövel W.,Underhill P.,et al. , Year: 2003 Reduced Ychromosome, but not mitochondrial DNA, diversity in human populations from West New Guinea.Am. J. Hum. Genet.72: 281–302.12532283  
Kayser M.,Brauer S.,Cordaux R.,Casto A.,Lao O.,et al. , Year: 2006 Melanesian and Asian origins of Polynesians: mtDNA and Y chromosome gradients across the Pacific.Mol. Biol. Evol.23: 2234–2244.16923821  
Kayser M.,Lao O.,Saar K.,Brauer S.,Wang X.,et al. , Year: 2008 Genomewide analysis indicates more Asian than Melanesian ancestry of Polynesians.Am. J. Hum. Genet.82: 194–198.18179899  
Keinan A.,Reich D.. , Year: 2010 Can a sexbiased human demography account for the reduced effective population size of chromosome X in nonAfricans?Mol. Biol. Evol.27: 2312–2321.20453016  
Keinan A.,Mullikin J. C.,Patterson N.,Reich D.. , Year: 2009 Accelerated genetic drift on chromosome X during the human dispersal out of Africa.Nat. Genet.41: 66–70.19098910  
Labuda D.,Lefebvre J. F.,Nadeau P.,RoyGagnon M. H.. , Year: 2010 Femaletomale breeding ratio in modern humans—an analysis based on historical recombinations.Am. J. Hum. Genet.86: 353–363.20188344  
Lacan M.,Keyser C.,Ricaut F. X.,Brucato N.,Duranthon F.,et al. , Year: 2011 Ancient DNA reveals male diffusion through the Neolithic Mediterranean route.Proc. Natl. Acad. Sci. USA108: 9788–9791.21628562  
Lambert C. A.,Connelly C. F.,Madeoy J.,Qiu R.,Olson M. V.,et al. , Year: 2010 Highly punctuated patterns of population structure on the X chromosome and implications for African evolutionary history.Am. J. Hum. Genet.86: 34–44.20085712  
Lansing J. S.,Cox M. P.,de Vet T. A.,Downey S. S.,Hallmark B.,et al. , Year: 2011 An ongoing Austronesian expansion in Island Southeast Asia.J. Anthropol. Archaeol.30: 262–272.  
Lawson Handley L. J.,Perrin N.. , Year: 2007 Advances in our understanding of mammalian sexbiased dispersal.Mol. Ecol.16: 1559–1578.17402974  
Lind J. M.,HutchesonDilks H. B.,Williams J. H.,Moore J. H.,Essex M.,et al. , Year: 2007 Elevated male European and female African contributions to the genomes of African American individuals.Hum. Genet.120: 713–722.17006671  
Long J. C.. , Year: 1991 The genetic structure of admixed populations.Genetics127: 417–428.2004712  
Mesa N. R.,Mondragón M. C.,Soto I. D.,Parra M. V.,Duque C.,et al. , Year: 2000 Autosomal, mtDNA, and Ychromosome diversity in Amerinds: pre and postColumbian patterns of gene flow in South America.Am. J. Hum. Genet.67: 1277–1286.11032789  
MorenoEstrada A.,Gravel S.,Zakharia F.,McCauley J. L.,Byrnes J. K.,et al. , Year: 2013 Reconstructing the population genetic history of the Caribbean.PLoS Genet.9: e1003925.24244192  
Nordborg M.,Krone S. M.. , Year: 2002 Separation of time scales and convergence to the coalescent in structured populations, pp. 194–232 in Modern Developments in Theoretical Population Genetics: The Legacy of Gustave Malécot, edited by Slatkin M.,Veuille M.Oxford University Press, Oxford.  
Oota H.,SettheethamIshida W.,Tiwawech D.,Ishida T.,Stoneking M.. , Year: 2001 Human mtDNA and Ychromosome variation is correlated with matrilocal versus patrilocal residence.Nat. Genet.29: 20–21.11528385  
Pemberton T. J.,Li F. Y.,Hanson E. K.,Mehta N. U.,Choi S.,et al. , Year: 2012 Impact of restricted marital practices on genetic variation in an endogamous Gujarati group.Am. J. Phys. Anthropol.149: 92–103.22729696  
Petersen D. C.,Libiger O.,Tindall E. A.,Hardie R. A.,Hannick L. I.,et al. , Year: 2013 Complex patterns of genomic admixture within Southern Africa.PLoS Genet.9: e1003309.23516368  
Pijpe J.,de Voogt A.,van Oven M.,Henneman P.,van der Gaag K. J.,et al. , Year: 2013 Indian Ocean crossroads: human genetic origin and population structure in the Maldives.Am. J. Phys. Anthropol.151: 58–67.23526367  
Pusey A. E.. , Year: 1987 Sexbiased dispersal and inbreeding avoidance in birds and mammals.Trends Ecol. Evol.2: 295–299.21227869  
Ramachandran S.,Rosenberg N. A.,Zhivotovsky L. A.,Feldman M. W.. , Year: 2004 Robustness of the inference of human population structure: a comparison of Xchromosomal and autosomal microsatellites.Hum. Genomics1: 87–97.15601537  
Ramachandran S.,Rosenberg N. A.,Feldman M. W.,Wakeley J.. , Year: 2008 Population differentiation and migration: coalescence times in a twosex island model for autosomal and Xlinked loci.Theor. Popul. Biol.74: 291–301.18817799  
Risch N.,Choudhry S.,Via M.,Basu A.,Sebro R.,et al. , Year: 2009 Ancestryrelated assortative mating in Latino populations.Genome Biol.10: R132.19930545  
Ségurel L.,MartínezCruz B.,QuintanaMurci L.,Balaresque P.,Georges M.,et al. , Year: 2008 Sexspecific genetic structure and social organization in Central Asia: insights from a multilocus study.PLoS Genet.4: e1000200.18818760  
Seielstad M. T.. , Year: 2000 Asymmetries in the maternal and paternal genetic histories of Colombian populations.Am. J. Hum. Genet.67: 1062–1066.11023812  
Seielstad M. T.,Minch E.,CavalliSforza L. L.. , Year: 1998 Genetic evidence for a higher female migration rate in humans.Nat. Genet.20: 278–280.9806547  
Stefflova K.,Dulik M. C.,Pai A. A.,Walker A. H.,ZeiglerJohnson C. M.,et al. , Year: 2009 Evaluation of group genetic ancestry of populations from Philadelphia and Dakar in the context of sexbiased admixture in the Americas.PLoS ONE4: e7842.19946364  
Tishkoff S. A.,Gonder M. K.,Henn B. M.,Mortensen H.,Knight A.,et al. , Year: 2007 History of clickspeaking populations of Africa inferred from mtDNA and Y chromosome genetic variation.Mol. Biol. Evol.24: 2180–2195.17656633  
Tishkoff S. A.,Reed F. A.,Friedlaender F. R.,Ehret C.,Ranciaro A.,et al. , Year: 2009 The genetic structure and history of Africans and African Americans.Science324: 1035–1044.19407144  
Tremblay M.,Vezina H.. , Year: 2010 Genealogical analysis of maternal and paternal lineages in the Quebec population.Hum. Biol.82: 179–198.20649399  
Verdu P.,Rosenberg N. A.. , Year: 2011 A general mechanistic model for admixture histories of hybrid populations.Genetics189: 1413–1426.21968194  
Verdu P.,Becker N. S.,Froment A.,Georges M.,Grugni V.,et al. , Year: 2013 Sociocultural behavior, sexbiased admixture, and effective population sizes in Central African Pygmies and nonPygmies.Mol. Biol. Evol.30: 918–937.23300254  
Verdu P.,Pemberton T. J.,Laurent R.,Kemp B. M.,GonzalezOliver A.,et al. , Year: 2014 Patterns of admixture and population structure in native populations of northwest North America.PLoS Genet.10: e1004530.25122539  
Wang S.,Ray N.,Rojas W.,Parra M. V.,Bedoya G.,et al. , Year: 2008 Geographic patterns of genome admixture in Latin American mestizos.PLoS Genet.4: e1000037.18369456  
Wen B.,Xie X.,Gao S.,Li H.,Shi H.,et al. , Year: 2004 Analyses of genetic structure of TibetoBurman populations reveals sexbiased admixture in southern TibetoBurmans.Am. J. Hum. Genet.74: 856–865.15042512  
Wilkins J. F.,Marlowe F. W.. , Year: 2006 Sexbiased migration in humans: What should we expect from genetic data?BioEssays28: 290–300.16479583  
Wood E. T.,Stover D. A.,Ehret C.,DestroBisol G.,Spedini G.,et al. , Year: 2005 Contrasting patterns of Y chromosome and mtDNA variation in Africa: evidence for sexbiased demographic processes.Eur. J. Hum. Genet.13: 867–876.15856073  
Wright S.. , Year: 1931 Evolution in Mendelian populations.Genetics16: 97–159.17246615 
Figures
Tables
Case  Female parent’s population  Male parent’s population  Probability 

1  S_{1}  S_{1}  s1,g−1fs1,g−1m 
2  S_{1}  H  s1,g−1fhg−1m 
3  S_{1}  S_{2}  s1,g−1fs2,g−1m 
4  H  S_{1}  hg−1fs1,g−1m 
5  H  H  hg−1fsg−1m 
6  H  S_{2}  hg−1fs2,g−1m 
7  S_{2}  S_{1}  s2,g−1fs1,g−1m 
8  S_{2}  H  s2,g−1fhg−1m 
9  S_{2}  S_{2}  s2,g−1fs2,g−1m 
The parameter sα,gδ is the probability that the parent of sex δ for a randomly chosen individual from the hybrid population, at generation g, is from the source population α. Similarly, the probability that this parent is from H is hgδ.
Article Categories:
Keywords: admixture, demographic inference, model, population history, sex bias. 
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