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<front>
<journal-meta>
<journal-id/>
<journal-title-group>
</journal-title-group>
<issn/>
<publisher>
<publisher-name/>
</publisher>
</journal-meta>
<article-meta>
<title-group>
<article-title>Lineal admixture time: an interdisciplinary
definition</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">0000-0002-5014-4809</contrib-id>
<name>
<surname>Ellerman</surname>
<given-names>E. Castedo</given-names>
</name>
<email>castedo@castedo.com</email>
</contrib>
</contrib-group>
<pub-date date-type="eprint" publication-format="electronic" iso-8601-date="2022-04-12">
<day>12</day>
<month>4</month>
<year>2022</year>
</pub-date>
<permissions>
<copyright-statement>© 2022, Ellerman et al</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Ellerman et al</copyright-holder>
<license license-type="open-access">
<ali:license_ref xmlns:ali="http://www.niso.org/schemas/ali/1.0/">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This document is distributed under a Creative Commons
Attribution 4.0 International license.</license-p>
</license>
</permissions>
<abstract>
<p><bold>STAGE:</bold> DISCUSSION DOCUMENT</p>
<p>
<bold>OBJECTIVES:</bold>
</p>
<list list-type="bullet">
<list-item>
<p>Define <italic>lineal admixture time</italic>.</p>
</list-item>
<list-item>
<p>Invite discussion and feedback, especially regarding questions at
the end of this document.</p>
</list-item>
<list-item>
<p>Facilitate the discovery of mathematical tools, using this
definition, that are useful to downstream empirical research.</p>
</list-item>
</list>
<p><bold>SIGNIFICANCE:</bold> This definition is interpretable within
empirical research without assuming particular mathematical models of
demographic or genetic processes.</p>
<p><bold>DOWNSTREAM RESEARCH:</bold> The estimation of the timing and
extent of interbreeding between previously separated populations.</p>
</abstract>
</article-meta>
</front>
<body>
<sec id="background">
<title>Background</title>
<p>The genealogies of a population reflect population events of the
past. Lineages, at the organismal level, are lines of descent carrying
genetic information through a population's genealogies. These lineages
are a link between genetic evidence and population events of the past.
In contrast to estimates of purely genetic quantities, estimates of
quantities about lineages can have closer correspondence to
non-genetic evidence, such as archaeological, geological, and cultural
evidence.</p>
</sec>
<sec id="definition-of-lineal-admixture-time">
<title>Definition of lineal admixture time</title>
<p>We define <italic>lineal admixture time</italic> per lineage. This
measure in time duration depends on two reference time points:</p>
<list list-type="order">
<list-item>
<p>a past time horizon when all ancestors are members of separate
non-admixed subpopulations, and</p>
</list-item>
<list-item>
<p>an observation time, such as the present.</p>
</list-item>
</list>
<p>Given a lineage, the <bold>lineal admixture time</bold> is defined
as:</p>
<p><italic>the amount of time since fertilization of the first admixed
individual in the lineage (or zero if the lineage has no such
individual)</italic>.</p>
<p>Stated more compactly, lineal admixture time is time since a
lineage's first admixed fertilization. Admixed individuals are all
individual whose parents are not from the same non-admixed
subpopulations.</p>
<p>The <italic>average lineal admixture time</italic> of an individual
is an average over all lineages passing through that individual. At
every merging of lineages through offspring, equal weighting is given
to the respective sets of maternal and paternal lineages. The
<italic>average lineal admixture time</italic> of a group of
individuals is the average lineal admixture time across all those
individuals.</p>
<p>It is worth noting that non-zero admixture times do not remain
constant if the observation time changes. It follows that non-zero
average admixture times of individuals are not fixed. They will
increase if the observation time increases with age.</p>
</sec>
<sec id="an-example-with-mendels-peas">
<title>An example with Mendel's peas</title>
<p>We consider an example of lineal admixture times with Mendel's
famous peas
<xref alt="1" rid="ref-abbott_experiments_2016" ref-type="bibr">1</xref>.
We imagine one of his classic experiments starting in 1860 and ending
in 1862. Because the pea plant is an annual plant, each generation is
separated by one year. During the lifetime of one generation,
fertilization of the next generation occurs, in the form of seeds
(illustrated with a <monospace>.</monospace> symbol).</p>
<preformat>1860 A-.-Y B-.-Z
| |
1861 M---.---N
|
1862 Q</preformat>
<p>In this example we will treat the non-admixed ancestral
subpopulations to be true-breeding pea plant varieties in 1860. Plants
<monospace>A</monospace> and <monospace>B</monospace> in the diagram
are non-admixed plants with round peas. Plants
<monospace>Y</monospace> and <monospace>Z</monospace> are non-admixed
plants with wrinkled peas. These two subpopulations grow and are
crossed in 1860 producing seeds that same year. Because they were
crossed, the offspring <monospace>M</monospace> and
<monospace>N</monospace> are both admixed (hybrid). This first admixed
(hybrid) generation will produce seeds in 1861 for the second admixed
generation, plant <monospace>Q</monospace>.</p>
<p>Plant <monospace>Q</monospace> has four lineages:</p>
<list list-type="bullet">
<list-item>
<p>
<monospace>Q-M-A</monospace>
</p>
</list-item>
<list-item>
<p>
<monospace>Q-M-Y</monospace>
</p>
</list-item>
<list-item>
<p>
<monospace>Q-N-B</monospace>
</p>
</list-item>
<list-item>
<p>
<monospace>Q-N-Z</monospace>
</p>
</list-item>
</list>
<p>In these lineages the first admixed individuals are
<monospace>M</monospace> and <monospace>N</monospace>, both of which
were fertilized in 1860. The first admixed fertilizations occur in
1860. Thus the lineal admixture time for all lineages of
<monospace>Q</monospace>, observed in 1862, is <bold>2
years</bold>.</p>
</sec>
<sec id="hybrid-generation-numbers">
<title>Hybrid generation numbers</title>
<p>A related definition measures generation number rather than time.
Given a lineage, the <italic>lineal hybrid generation number</italic>
is defined as:</p>
<p><italic>the number of admixed individual in the
lineage</italic>.</p>
<p>Because peas are annual, the lineal admixture times in years
coincide exactly with the lineal hybrid generation numbers. Today, the
first and second hybrid generations are often referred to as
<inline-formula><alternatives><tex-math><![CDATA[F_1]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>
and <inline-formula><alternatives><tex-math><![CDATA[F_2]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>
<xref alt="2" rid="ref-hartl_essential_2006" ref-type="bibr">2</xref>.
Lineal average hybird generation numbers are a generalization of
indexes given to hybrid generations <inline-formula><alternatives><tex-math><![CDATA[F_1]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>,
<inline-formula><alternatives><tex-math><![CDATA[F_2]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>,
etc...</p>
</sec>
<sec id="an-example-with-humans">
<title>An example with humans</title>
<p>These definitions are inspired by the use of generation numbers to
infer admixture timing in the study of admixture in Greenland
<xref alt="3" rid="ref-waples_genetic_2021" ref-type="bibr">3</xref>.</p>
<p>We give a hypothetical example in Greenland at an observation time
of 1600 CE. We categorize the ancestors at a time horizon of 1000 CE
into separate non-interbreeding Inuit and European subpopulations. We
denote the fertilization and death of an individual with symbols
<monospace>.</monospace> and <monospace>x</monospace>, respectively,
and write names at the time an individual is a 3-month-old baby.</p>
<preformat>1500 Erik
1510 | Aaju
1520 | | Natar Atuat
1530 |----.----| | |
1540 | Tagak | |----.----|
1550 | | x x Mikak |
1560 x | | |
1570 |-------.-------| x
1580 x Kiviaq |
1590 | x
1600 | </preformat>
<p>We imagine Erik as only having European ancestors and Aaju, Natar
and Atuat only having Intuit ancestors. As of 1600 CE, the lineages
are:</p>
<list list-type="bullet">
<list-item>
<p>Kiviaq-Tagak-Erik</p>
</list-item>
<list-item>
<p>Kiviaq-Tagak-Aaju</p>
</list-item>
<list-item>
<p>Kiviaq-Mikak-Natar</p>
</list-item>
<list-item>
<p>Kiviaq-Mikak-Atuat</p>
</list-item>
</list>
<p>For the Tagak lineages, the first admixed fertilization is in 1530.
In contrast, the Mikak lineages have a first admixed fertilization in
1570. Thus the lineal admixture times are equal parts 70 years and 30
years, for an average lineal admixture time of 50 years. The
respective lineal hybrid generation numbers are 1 and 2 resulting in
an average lineal hybrid generation number of 1.5.</p>
</sec>
<sec id="advantages-of-lineal-admixture-time">
<title>Advantages of lineal admixture time</title>
<p>Lineal admixture time has a number of advantages as a quantity for
research.</p>
<p>Firstly, it is independent of any particular model of an admixture
process. With enough genealogical information one could calculate
lineal admixture times of a real group of individuals. There is one
true set of lineal admixture times for a real population. But
realistically, we can only hope to estimate those numbers based on
evidence and models. But the definition exists independent of any
particular model used for estimation.</p>
<p>Secondly, the interpretation of lineal admixture time does not
require fluency in probability theory. Researchers with interests in
empirical evidence and not mathematics can make use of estimates of
lineal admixture times.</p>
<p>Thirdly, we conjecture that <italic>distributions</italic> of
lineal admixture times across individuals, and groups of individuals
will prove to be a useful mathematical tool in the timing of
admixture. This conjecture is based on not-yet-documented mathematical
work by ECE. Lineal admixture time is conveniently representable as a
random variable from a stochastic process in which lineages are random
objects.</p>
<p>The same populations events that affect admixture also affect
hybridization, introgression, and gene flow. We conjecture that
statistical tools for estimating distributions of lineal admixture
time will also be useful to these related topics in addition to
admixture timing.</p>
<p>The Greenland example illustrates how average lineal admixture time
is easily testable against non-genetic lines of evidence. If a genetic
model estimates the average lineal admixture time of present day
Greenlanders to be 500 years, we can consider historical and
archaeological evidence to falsify that estimate.</p>
</sec>
<sec id="end">
<title>End</title>
<p>This document presents a definition of <italic>lineal admixture
time</italic>. This quantity is the basis for current mathematical
studies of ECE. Feedback and input is greatly appreciated, especially
regarding the relevance to downstream empirical research. In
particular, the following questions are of particular interest:</p>
<list list-type="order">
<list-item>
<p>Is <italic>lineal admixture time</italic> easily understood and
interpretable without fluency in probability theory?</p>
</list-item>
<list-item>
<p>Does <italic>lineal admixture time</italic> benefit from being
comparable to non-genetic lines of evidence (i.e archaeological,
geological, linguistic, historical, cultural)?</p>
</list-item>
<list-item>
<p>What are existing terms or literature for this specific
definition of admixture time?</p>
</list-item>
<list-item>
<p>Are the following terms:</p>
<list list-type="bullet">
<list-item>
<p>lineage</p>
</list-item>
<list-item>
<p>genealogy</p>
</list-item>
<list-item>
<p>lineal admixture time</p>
</list-item>
<list-item>
<p>lineal hybrid generation number</p>
</list-item>
</list>
<p>potentially confusing in the way they are used in this
document?</p>
</list-item>
</list>
</sec>
<sec id="references">
<title>References</title>
</sec>
</body>
<back>
<ref-list>
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<element-citation publication-type="article-journal">
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<given-names>Daniel J.</given-names>
</name>
</person-group>
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<volume>204</volume>
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<ref id="ref-hartl_essential_2006">
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<surname>Hartl</surname>
<given-names>Daniel L.</given-names>
</name>
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<surname>Jones</surname>
<given-names>Elizabeth W.</given-names>
</name>
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</back>
</article>
