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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="1.2" article-type="other">
  <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="2022-04-26">
        <day>26</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 ending at 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 the non-zero
  average admixture time of an individual is not fixed. It 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. 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 population 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 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 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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              <surname>Jones</surname>
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            </name>
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  </back>
</article>
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