On the number of genealogical ancestors tracing to the source groups of an admixed population Jazlyn A. Mooney Lily Agranat-Tamir Jonathan K. Pritchard

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On the number of genealogical ancestors

tracing to the source groups of an admixed population

Jazlyn A. Mooney∗†

, Lily Agranat-Tamir∗, Jonathan K. Pritchard∗‡

and Noah A. Rosenberg∗§

October 25, 2022

Abstract. In genetically admixed populations, admixed individuals possess ancestry from multiple

source groups. Studies of human genetic admixture frequently estimate ancestry components corre-

sponding to fractions of individual genomes that trace to speciﬁc ancestral populations. However,

the same numerical ancestry fraction can represent a wide array of admixture scenarios. Using

a mechanistic model of admixture, we characterize admixture genealogically: how many distinct

ancestors from the source populations does the admixture represent? We consider African Ameri-

cans, for whom continent-level estimates produce a 75-85% value for African ancestry on average

and 15-25% for European ancestry. Genetic studies together with key features of African-American

demographic history suggest ranges for model parameters. Using the model, we infer that if ge-

nealogical lineages of a random African American born during 1960-1965 are traced back until

they reach members of source populations, the expected number of genealogical lines terminating

with African individuals is 314, and the expected number terminating in Europeans is 51. Across

discrete generations, the peak number of African genealogical ancestors occurs for birth cohorts

from the early 1700s. The probability exceeds 50% that at least one European ancestor was born

more recently than 1835. Our genealogical perspective can contribute to further understanding the

admixture processes that underlie admixed populations. For African Americans, the results provide

insight both on how many of the ancestors of a typical African American might have been forcibly

displaced in the Transatlantic Slave Trade and on how many separate European admixture events

might exist in a typical African-American genealogy.

∗Department of Biology, Stanford, CA 94305 USA

†Department of Quantitative and Computational Biology, University of Southern California, Los Angeles, CA 90089 USA

‡Department of Genetics, Stanford University, Stanford, CA 94305 USA

§Email: noahr@stanford.edu.

arXiv:2210.12306v1 [q-bio.PE] 22 Oct 2022

Introduction

Genetically admixed populations arise when two or more source groups combine to form a new

population. After a period of multiple generations of mating among members of the incipient

admixed population and new contributors from the source groups, typical individuals in the admixed

population possess ancestry from multiple sources (Chakraborty, 1986; Korunes & Goldberg, 2021;

Gopalan et al., 2022).

The genetic history of an admixed population can be represented by a temporal sequence of

admixture contributions, starting with the founding of the new admixed group (Long, 1991; Verdu

& Rosenberg, 2011; Gravel, 2012). Among present-day members of the admixed population, genetic

patterns such as the distribution of admixture levels estimated from individual genomes can then

be used together with a model of the admixture process to uncover features such as the timing and

magnitude of the genetic contributions that characterize the admixture process (Verdu et al., 2014;

Baharian et al., 2016; Zaitlen et al., 2017).

In studies that seek to infer population parameters from genetic patterns among individuals in

the admixed population, each admixed individual is treated as a random outcome of the admixture

process. The accumulation of data on many admixed individuals then provides information about

the population history. In this perspective, for a given model of the admixture history, an individual

possesses a random genealogy conditional on the parameters of the admixture process. What

information can be obtained about a random individual genealogy under the assumptions of an

admixture model? In particular, for individual members of an admixed population, how many

distinct contributors from the source populations does their admixture represent?

In human admixed populations, questions focused on random individual genealogies can provide

information both about the population-level history of admixture and about the relationship of

individuals to that history. Consider the case of the African-American admixed population in

the United States. Living African Americans descend primarily from an admixture of African

and European source populations, much of the admixture having occurred during the period of

enslavement of most African Americans, 1619-1865. Owing to widespread patterns such as forcible

fracturing of enslaved families by enslavers, a practice of using only ﬁrst names and not surnames for

enslaved persons, lack of documentation of many of the enslaved even by ﬁrst name in the written

record, and a reticence of many formerly enslaved individuals to record genealogical information

in the period after slavery, for many African Americans, limited data are available about their

individual ancestors prior to the middle or late 1800s (Gates, 2009; Swarns, 2012; Nelson, 2016).

Thus, an admixture model has potential to recover features of African-American genealogies that

are otherwise diﬃcult to obtain.

For an African American chosen at random, how many genealogical lines traced back from

the present to a member of a source population reach an African individual? How many reach a

European or European American? The former quantity approximates the number of ancestors who

traveled from Africa to the Western Hemisphere as forced enslaved migrants in the Transatlantic

Slave Trade. The latter gives the number of occasions at which European admixture events occurred

in a random African-American genealogy. Answers to such questions are informative not only for

understanding the genealogies of individuals, but also for contributing details of the admixture

process that has given rise to the present-day population.

Model

Assumptions

We follow a mechanistic model in which admixture levels are explored in an admixed population

over time (Verdu & Rosenberg, 2011; Goldberg et al., 2014; Goldberg & Rosenberg, 2015; Goldberg

et al., 2020). Three populations are considered: source populations S1and S2, and admixed

population H. In each of a series of generations—indexed discretely with the index increasing

forward in time—an individual in the admixed population Hin generation ghas a pair of parents

probabilistically drawn from among individuals extant in generation g−1 in source populations S1

and S2and admixed population H(Figure 1).

Suppose that for an individual in generation g, the admixture contributions are s1,g−1,s2,g−1,

and hg−1, for populations S1,S2, and H, respectively. In other words, for an individual chosen

at random in admixed population H, a parent chosen at random has probability s1,g−1of having

originated from population S1,s2,g−1for population S2, and hg−1for population H. We then have

s1,g−1+hg−1+s2,g−1= 1.(1)

The sampling probabilities s1,g−1,s2,g−1, and hg−1can be interpreted as fractional contributions

from source populations S1,S2, and Hto autosomal genomes in population Hin generation g.

Generation g= 1 represents the founding of the admixed population from members of the source

population from generation g= 0. The admixed population does not exist in generation g= 0, so

that h0= 0, and s1,0+s2,0= 1.

Previous studies with these modeling assumptions have tracked properties of random variables

that describe admixture proportions in the source populations S1and S2at generation g. In partic-

ular, Verdu & Rosenberg (2011) studied recursions for the probability distribution and moments of

a random variable H1,g , representing the autosomal fraction of admixture from source population 1

for an individual in the admixed population at generation g. We instead study the random variable

Z1,g, the number of genealogical ancestors from source population 1 for an individual in the admixed

population at generation g, and Z2,g, the number of genealogical ancestors from source population

2. In the sense in which we consider genealogical ancestors, once a source population is reached

along a genealogical line in a speciﬁc ancestor, that ancestor is tabulated as a genealogical ancestor

from the associated source population, and the line is not traced any farther back (Figure 2).

Recursion for the number of genealogical ancestors

We review expressions that we will need for the mean and variance of autosomal admixture under

the model (Verdu & Rosenberg, 2011). The mean ancestry fraction from population 1 in generation

gis (Verdu & Rosenberg, 2011, eqs. 10 and 11):

E[H1,g] = 









s1,0, g = 1,

s1,g−1+hg−1E[H1,g−1], g ≥2.

(2)

The variance of the ancestry fraction from population 1 in generation gis (Verdu & Rosenberg,

2011, eqs. 22 and 23)

V[H1,g] =











s1,0(1−s1,0)

2, g = 1,

s1,g−1(1−s1,g−1)

2−s1,g−1hg−1E[H1,g−1] + hg−1(1−hg−1)

2E[H1,g−1]2

+hg−1

2V[H1,g−1], g ≥2.

(3)

Note that the mean ancestry fraction from population 2 is one minus the mean ancestry fraction

from population 1, and the variances of the two ancestry fractions are equal.

A recursion describing the admixture fraction H1,g (Verdu & Rosenberg, 2011) can be modiﬁed

to obtain a recursion for Z1,g. Whereas the random autosomal admixture fraction H1,g of an

individual is the mean of the corresponding admixture fractions of the parents of the individual,

the random number of ancestors Z1,g is the sum of the numbers of ancestors of the parents (from

population 1).

Let Lbe a random variable that gives the source populations of the parents of a random

individual from the admixed population. Listing the mother ﬁrst, Ltakes a value in the set L=

{S1S1, S1H, S1S2, HS1, HH, HS2, S2S1, S2H, S2S2}. Based on eqs. 1 and 2 of Verdu & Rosenberg

(2011), for generation g= 1, we have

Z1,1=











2 if L=S1S1,with P[L=S1S1] = s1,0s1,0

1 if L=S1S2,with P[L=S1S2] = s1,0s2,0

1 if L=S2S1,with P[L=S2S1] = s2,0s1,0

0 if L=S2S2,with P[L=S2S2] = s2,0s2,0.

(4)

For subsequent generations, g≥2,

Z1,g =











2 if L=S1S1,with P[L=S1S1] = s1,g−1s1,g−1

1 + Z1,g−1if L=S1H, with P[L=S1H] = s1,g−1hg−1

1 if L=S1S2,with P[L=S1S2] = s1,g−1s2,g−1

Z1,g−1+ 1 if L=HS1,with P[L=HS1] = hg−1s1,g−1

Z1,g−1+Z0

1,g−1if L=HH, with P[L=HS1] = hg−1hg−1

Z1,g−1if L=HS2,with P[L=HS2] = hg−1s2,g−1

1 if L=S2S1,with P[L=S2S1] = s2,g−1s1,g−1

Z1,g−1if L=S2H, with P[L=S2H] = s2,g−1hg−1

0 if L=S2S2,with P[L=S2S2] = s2,g−1s2,g−1.

(5)

For L=HH,Z1,g−1and Z0

1,g−1are independent and identically distributed copies of the same

random variable. Eqs. 4 and 5 enable us to compute the probability distribution of Z1,g , the number

of population-1 ancestors of an individual in the admixed population in generation g.Z1,g and Z2,g

range in Qg={0,1,...,2g}. For qin Qg, we compute the probability P[Z1,g =q] that a random

individual from population Hat generation ghas qgenealogical ancestors from population 1.

Analogously to eqs. 3-5 of Verdu & Rosenberg (2011), we have for g≥1

P[Z1,1=q] =











1,0, q = 2,

2s1,0s2,0, q = 1,

2,0, q = 0.

(6)

For g≥2 and qin Qg,

P[Z1,g =q] = h2

g−1

2g−1

r=0 P[Z1,g−1=r]P[Z1,g−1=q−r]

+(2s1,g−1hg−1)P[Z1,g−1=q−1] + (2s2,g−1hg−1)P[Z1,g−1=q] + Ig(q).(7)

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OnthenumberofgenealogicalancestorstracingtothesourcegroupsofanadmixedpopulationJazlynA.Mooney*,LilyAgranat-Tamir,JonathanK.Pritchard,andNoahA.Rosenberg§October25,2022Abstract.Ingeneticallyadmixedpopulations,admixedindividualspossessancestryfrommultiplesourcegroups.Studiesofhumangeneticadmixture...

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On the number of genealogical ancestors tracing to the source groups of an admixed population Jazlyn A. Mooney Lily Agranat-Tamir Jonathan K. Pritchard

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