article.xml
<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.2 20190208//EN"
"JATS-archivearticle1.dtd">
<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-id>
<journal-title-group>
</journal-title-group>
<issn></issn>
<publisher>
<publisher-name></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">https://orcid.org/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="pub" publication-format="electronic" iso-8601-date="2022-04-26">
<day>26</day>
<month>4</month>
<year>2022</year>
</pub-date>
<permissions>
<copyright-statement>© 2023, Ellerman et al</copyright-statement>
<copyright-year>2023</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 display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><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 display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><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 display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><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 display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><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>
</body>
<back>
<ref-list>
<title>References</title>
<ref id="ref-abbott_experiments_2016">
<element-citation publication-type="article-journal">
<person-group person-group-type="author">
<name><surname>Abbott</surname><given-names>Scott</given-names></name>
<name><surname>Fairbanks</surname><given-names>Daniel J.</given-names></name>
</person-group>
<article-title>Experiments on Plant Hybrids by Gregor Mendel</article-title>
<source>Genetics</source>
<year iso-8601-date="2016-10">2016</year><month>10</month>
<volume>204</volume>
<issue>2</issue>
<issn>1943-2631</issn>
<pub-id pub-id-type="doi">10.1534/genetics.116.195198</pub-id>
<pub-id pub-id-type="pmid">27729492</pub-id>
<fpage>407</fpage>
<lpage>422</lpage>
</element-citation>
</ref>
<ref id="ref-hartl_essential_2006">
<element-citation publication-type="book">
<person-group person-group-type="author">
<name><surname>Hartl</surname><given-names>Daniel L.</given-names></name>
<name><surname>Jones</surname><given-names>Elizabeth W.</given-names></name>
</person-group>
<source>Essential genetics: A genomics perspective</source>
<publisher-name>Jones; Bartlett Publishers</publisher-name>
<publisher-loc>Boston</publisher-loc>
<year iso-8601-date="2006">2006</year>
<edition>4th ed</edition>
<isbn>978-0-7637-3527-2</isbn>
</element-citation>
</ref>
<ref id="ref-waples_genetic_2021">
<element-citation publication-type="article-journal">
<person-group person-group-type="author">
<name><surname>Waples</surname><given-names>Ryan K.</given-names></name>
<name><surname>Hauptmann</surname><given-names>Aviaja L.</given-names></name>
<name><surname>Seiding</surname><given-names>Inge</given-names></name>
<name><surname>Jørsboe</surname><given-names>Emil</given-names></name>
<name><surname>Jørgensen</surname><given-names>Marit E.</given-names></name>
<name><surname>Grarup</surname><given-names>Niels</given-names></name>
<name><surname>Andersen</surname><given-names>Mette K.</given-names></name>
<name><surname>Larsen</surname><given-names>Christina V. L.</given-names></name>
<name><surname>Bjerregaard</surname><given-names>Peter</given-names></name>
<name><surname>Hellenthal</surname><given-names>Garrett</given-names></name>
<name><surname>Hansen</surname><given-names>Torben</given-names></name>
<name><surname>Albrechtsen</surname><given-names>Anders</given-names></name>
<name><surname>Moltke</surname><given-names>Ida</given-names></name>
</person-group>
<article-title>The genetic history of Greenlandic-European contact</article-title>
<source>Current biology: CB</source>
<year iso-8601-date="2021-05">2021</year><month>05</month>
<volume>31</volume>
<issue>10</issue>
<issn>1879-0445</issn>
<pub-id pub-id-type="doi">10.1016/j.cub.2021.02.041</pub-id>
<pub-id pub-id-type="pmid">33711251</pub-id>
<fpage>2214</fpage>
<lpage>2219.e4</lpage>
</element-citation>
</ref>
</ref-list>
</back>
</article>
