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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>The gametic entropy of a population</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="2022-01-27">
        <day>27</day>
        <month>1</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>This discussion document defines the gametic entropy of a population.
It is a precise interpretation of a phenomenon occurring at the
intersection of population genetics and information theory. This
interpretation challenges minor statements in two journal articles.
Gametic entropy is the application of Shannon entropy to the genetic
information carried by gametes in the propagation of a population. This
document serves as a reference to facilitate discussion. The practical
utility of gametic entropy is not covered in this document.
Understanding of Shannon entropy and genetics is required to understand
the entire document.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="background">
      <title>Background</title>
      <p>An early application of Shannon entropy
  <xref alt="1" rid="ref-shannon_mathematical_1998" ref-type="bibr">1</xref>
  in population genetics is the highly cited article "The
  Apportionment of Human Diversity" by Lewontin in 1972
  <xref alt="2" rid="ref-lewontin_1972" ref-type="bibr">2</xref>.
  Shannon entropy is a measure of information, expressed in units of
  <italic>bits</italic> (base 2 logarithm of probability). The idea of
  genetic information being stored and quantified in bits appears in the
  context of evolutionary genetics as early as 1961
  <xref alt="3" rid="ref-kimura_natural_1961" ref-type="bibr">3</xref>.</p>
      <p>Eight years after Lewontin's article, BDH Latter wrote:</p>
      <disp-quote>
        <p>"The Shannon information measure used by Lewontin (1972) ...
    is extremely difficult to interpret genetically."
    <xref alt="4" rid="ref-latter_genetic_1980" ref-type="bibr">4</xref></p>
      </disp-quote>
      <p>This claim is about possible interpretations, in the context of
  genetics, of Shannon entropy. In contrast to the context of genetics,
  Shannon entropy was established in the context of communication
  systems.</p>
      <p>A mirror claim can be made about possible interpretations of
  genetic information in the context of communication systems. Indeed,
  CT Bergstrom &amp; M Rosvall have claimed that genetic information
  lacks an obvious interpretation in the context of communication
  theory:</p>
      <disp-quote>
        <p>"Geneticists are not so fortunate. For them, the analogy to
    communication theory is less obvious. Efforts to make this analogy
    explicit seem forced at best, ..."
    <xref alt="5" rid="ref-bergstrom_transmission_2011" ref-type="bibr">5</xref></p>
      </disp-quote>
      <p>Below we specify an interpretation and refer to it as the
  <italic>gametic entropy of a population</italic>. With an
  understanding of information theory, we propose this interpretation is
  both</p>
      <list list-type="bullet">
        <list-item>
          <p>an easy genetic interpretation of Shannon entropy, and</p>
        </list-item>
        <list-item>
          <p>a natural example of communication of genetic information.</p>
        </list-item>
      </list>
      <p>This interpretation challenges the statements in the two mentioned
  articles. It should be emphasized that the statements are peripheral
  and not core claims of their respective articles (which are
  excellent).</p>
    </sec>
    <sec id="the-interpretation">
      <title>The interpretation</title>
      <p>The <italic>gametic entropy of a population</italic> is:</p>
      <disp-quote>
        <p>The amount of information transmitted by a gamete in the
    propagation of a population.</p>
      </disp-quote>
      <p>In the specific application of the 1972 Lewontin article
  <xref alt="2" rid="ref-lewontin_1972" ref-type="bibr">2</xref>, the
  interpretation is:</p>
      <disp-quote>
        <p>The amount of information transmitted by a gamete, <bold>at a
    random locus,</bold> in the propagation of a population.</p>
      </disp-quote>
      <p>Before getting into the biology, we review the core concepts upon
  which Shannon entropy is based. We follow by observing where these
  concepts appear in the propagation of a population. Only sexually
  reproducing populations are discussed initially. Clonal populations
  will be discussed later as a degenerate special case.</p>
      <sec id="messages-senders-and-receivers">
        <title>Messages, senders, and receivers</title>
        <p>Shannon entropy is the key measurement of information theory
    <xref alt="1" rid="ref-shannon_mathematical_1998" ref-type="bibr">1</xref>.
    Entropy measures the degree of uncertainty on what messages a sender
    communicates to a receiver. Entropy measures a real tangible minimum
    capacity required of any channel or storage used to communicate
    those messages. This minimum requirement applies to all channel and
    storage mechanisms, no matter the physical mechanism.</p>
        <p>Roughly speaking, entropy is the best possible score in a game of
    <italic>20 questions</italic>. That is the minimum number of yes/no
    questions that can be asked to determine a yet to be known message.
    One bit is the quantity of information gained (uncertainty reduced)
    by one yes/no question about two equally likely possibilities.</p>
        <p>In population genetics, a key process of interest is the
    propagation of a population. In this process, what are the messages,
    senders and receivers?</p>
      </sec>
      <sec id="gametic-messages">
        <title>Gametic messages</title>
        <p>The propagation of a population requires new members to be born.
    For sexually reproducing species, a new birth requires genetic
    information to be passed down from a mother and a father in the
    population. Each half of this genetic information is stored and
    transmitted by a gamete. It is at the unit of a gamete that we
    clearly see a complete message, sent by each of the two parents to
    new offspring, the receiver.</p>
        <p>For a next possible birth in a population, there is a
    probabilistic distribution of possible genetic messages carried by
    each gamete. The Shannon entropy of the distribution of possible
    messages is the gametic entropy of the population. One can roughly
    think of gametic entropy as the number of yes/no questions offspring
    need to ask per parent to find out what alleles they are to inherit:
    "Hey parent, do I get an Rh+ or an Rh- allele from
    you?"</p>
        <p>When only looking at the entropy of autosomal genetic
    information, the distinction between maternal and paternal does not
    matter, but when considering the information in sex chromosomes, the
    gametic entropy can be specifically maternal or paternal. The
    unqualified gametic entropy is a per gamete entropy and thus the
    average of both maternal and paternal gametic entropies, since every
    birth requires one of each kind of gametic message.</p>
      </sec>
    </sec>
    <sec id="toy-illustration">
      <title>Toy illustration</title>
      <p>We consider a toy example of a sexually reproducing unicorn
  population with biallelic chromosomes. Imagine a population of
  unicorns with 10 autosomal chromosomes of which the entire lengths of
  each chromosome consists of only one of two equally frequent
  haplotypes. In contrast, the X and Y chromosomes of these unicorns are
  completely fixed with only one X haplotype and only one Y haplotype in
  the population. The gametic entropy across the 10 autosomal
  chromosomes is exactly 10 bits. The maternal gametic entropy for the
  sex chromosome is zero because there is no uncertainty about which sex
  chromosome (or its haplotype) is transmitted. The total maternal
  gametic entropy is thus 10 bits. In contrast, the paternal gametic
  entropy for the sex chromosome is 1 bit since there is an equal chance
  of an X or a Y chromosome being communicated (and no additional
  uncertainty regarding each sex chromosome's respective haplotype).
  Thus the total paternal gametic entropy is 11 bits. The overall
  gametic entropy of this unicorn population is
  <inline-formula><alternatives><tex-math><![CDATA[10.5]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>10.5</mml:mn></mml:math></alternatives></inline-formula>
  bits, the average of the parent-specific entropies.</p>
    </sec>
    <sec id="decoupling-medium-of-transmission">
      <title>Decoupling medium of transmission</title>
      <p>One of the key insights from information (communication) theory is
  that fundamental properties of information, such as entropy, exist
  regardless of the forms of storage or transmission.</p>
      <p>At current levels of technology, DNA is the only medium of
  transmission of interest for the genetic information carried by
  gametes. Nonetheless, a thought experiment of hypothetical but almost
  plausible transmission mediums helps elucidate the independence of the
  "pure" information transmitted from the medium of
  transmission.</p>
      <sec id="a-thought-experiment">
        <title>A thought experiment</title>
        <p>Reflecting on DNA sequencing technology, maternal spindle
    transfer in "three-parent" IVF
    <xref alt="6" rid="ref-amato_three-parent_2014" ref-type="bibr">6</xref>
    <xref alt="7" rid="ref-zhang_live_2017" ref-type="bibr">7</xref> and
    artificial synthesis of an entire (bacterial) genome
    <xref alt="8" rid="ref-gibson_creation_2010" ref-type="bibr">8</xref>,
    it is not hard to imagine a possibility of completely dematerialized
    transmission of the genetic information carried by gametes. This
    transmission can be over any of the channels described by
    information theory, including telegraph schemes.</p>
        <p>We now can propose an answer to the following question posed by
    CT Bergstrom &amp; M Rosvall:</p>
        <disp-quote>
          <p>"Is there a clean mapping from informational processes in
      biology onto the telegraph schema?"
      <xref alt="9" rid="ref-bergstrom_response_2011" ref-type="bibr">9</xref></p>
        </disp-quote>
        <p>The practical utility of such telegraphy, if any, is hard to
    imagine today. But as an entertaining thought experiment in science
    fiction, we can image the utility of interplanetary transmission of
    the genetic information in donor gametes for IVF. Rather than
    physically transporting gametes between planets, which could take
    months to years, the pure data can be transmitted in minutes or
    hours. All of the facts about transmission rates in telegraphy apply
    in a hypothetical interplanetary IVF system. For a given population
    of equally likely possible donors, the maternal (paternal) gametic
    entropy, is the best possible rate, measures in bits, for
    transmitting the genetic information of a egg (sperm) donor.</p>
      </sec>
    </sec>
    <sec id="degenerate-case-of-clonal-populations">
      <title>Degenerate case of clonal populations</title>
      <p>In the case of asexually reproducing clonal populations, there is
  no distinction between the genetic information in a gamete vs a
  parent. The "gametic message" is simply the entire genome of
  a parent, which is sent in its entirety to its single-parent
  offspring. The gametic entropy of a clonal population is the same as
  the entropy of the distribution of distinct genomes in the
  population.</p>
    </sec>
    <sec id="concluding-questions">
      <title>Concluding questions</title>
      <p>This document challenges claims in two previous journal articles
  regarding two respective questions:</p>
      <list list-type="order">
        <list-item>
          <p>Is the gametic entropy of a population an extremely difficult
      genetic interpretation of Shannon entropy?</p>
        </list-item>
        <list-item>
          <p>Is the gametic entropy of a population a forced analogy in
      communcation theory for geneticists?</p>
        </list-item>
      </list>
    </sec>
    <sec id="acknowledgements">
      <title>Acknowledgements</title>
      <p>ECE thanks Steven Orzack and John Novembre for fruitful discussions
  about the Lewontin 1972 paper.</p>
    </sec>
    <sec id="references">
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    </sec>
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</article>
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