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<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 & 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 & 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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</name>
</person-group>
<article-title>Creation of a Bacterial Cell Controlled by a Chemically Synthesized Genome</article-title>
<source>Science</source>
<year iso-8601-date="2010-07">2010</year>
<month>07</month>
<date-in-citation content-type="access-date">
<year iso-8601-date="2021-12-11">2021</year>
<month>12</month>
<day>11</day>
</date-in-citation>
<volume>329</volume>
<issue>5987</issue>
<issn>0036-8075, 1095-9203</issn>
<uri>https://www.science.org/doi/10.1126/science.1190719</uri>
<pub-id pub-id-type="doi">10.1126/science.1190719</pub-id>
<fpage>52</fpage>
<lpage>56</lpage>
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</article>
