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  <title>Modelling a Genetic Oscillator</title>
  <author>
    <firstname>Catherine</firstname>
          <surname>Lloyd</surname>
    <affiliation>
      <shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil>
    </affiliation>
  </author>
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  <section id="sec_status">
    <title>Model Status</title>
    <para>
            This CellML modelis known to run in OpenCell and COR to reproduce the results published in the paper on which it is based (figures 2 and 3).
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  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
ABSTRACT: A wide range of organisms use circadian clocks to keep internal sense of daily time and regulate their behavior accordingly. Most of these clocks use intracellular genetic networks based on positive and negative regulatory elements. The integration of these "circuits" at the cellular level imposes strong constraints on their functioning and design. Here, we study a recently proposed model [Barkai, N. and Leibler, S. (2000) Nature (London), 403, 267-268] that incorporates just the essential elements found experimentally. We show that this type of oscillator is driven mainly by two elements: the concentration of a repressor protein and the dynamics of an activator protein forming an inactive complex with the repressor. Thus, the clock does not need to rely on mRNA dynamics to oscillate, which makes it especially resistant to fluctuations. Oscillations can be present even when the time average of the number of mRNA molecules goes below one. Under some conditions, this oscillator is not only resistant to but, paradoxically, also enhanced by the intrinsic biochemical noise.
</para>

<para>
The original paper reference is cited below:
</para>

<para>
Mechanisms of noise-resistance in genetic oscillators, Jose M. G. Vilar, Hao Yuan Kueh, Naama Barkai, and Stanislas Leibler, 2002,<emphasis>Proceedings of the National Academy of Sciences of the United States of America</emphasis>, 99, 5988-5992.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=11972055&amp;dopt=Abstract">PubMed ID: 11972055</ulink>
</para>

<informalfigure float="0" id="fig_reaction_diagram">
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    <imagedata fileref="vilar_2002.png"/>
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<caption>Biochemical network of the circadian oscillator model.</caption>
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</sect1>
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        <eq/>
        <ci>RXN10</ci>
        <apply>
          <times/>
          <ci>MA</ci>
          <ci>BetaA_1</ci>
        </apply>
      </apply>
    </math>
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    <variable name="RXN11" public_interface="out" units="flux"/>
    <variable name="A" public_interface="in" units="molecules"/>
    <variable name="DR" public_interface="in" units="molecules"/>
    <variable initial_value="1" name="Gamma_3" units="second_order_rate"/>
    <math xmlns="http://www.w3.org/1998/Math/MathML" id="RXN11_Calc">
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        <eq/>
        <ci>RXN11</ci>
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          <times/>
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          <ci>DR</ci>
          <ci>Gamma_3</ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="RXN12">
    <variable name="RXN12" public_interface="out" units="flux"/>
    <variable name="DRP" public_interface="in" units="molecules"/>
    <variable initial_value="100" name="Theta_2" units="first_order_rate"/>
    <math xmlns="http://www.w3.org/1998/Math/MathML" id="RXN12_Calc">
      <apply>
        <eq/>
        <ci>RXN12</ci>
        <apply>
          <times/>
          <ci>DRP</ci>
          <ci>Theta_2</ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="RXN13">
    <variable name="RXN13" public_interface="out" units="flux"/>
    <variable name="DR" public_interface="in" units="molecules"/>
    <variable initial_value="0.01" name="Alpha_3" units="first_order_rate"/>
    <math xmlns="http://www.w3.org/1998/Math/MathML" id="RXN13_Calc">
      <apply>
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        <ci>RXN13</ci>
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          <times/>
          <ci>DR</ci>
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        </apply>
      </apply>
    </math>
  </component>
  
  <component name="RXN14">
    <variable name="RXN14" public_interface="out" units="flux"/>
    <variable name="DRP" public_interface="in" units="molecules"/>
    <variable initial_value="50" name="Alpha_4" units="first_order_rate"/>
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        <apply>
          <times/>
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          <ci>Alpha_4</ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="RXN15">
    <variable name="RXN15" public_interface="out" units="flux"/>
    <variable name="MR" public_interface="in" units="molecules"/>
    <variable initial_value="0.5" name="Delta_5" units="first_order_rate"/>
    <math xmlns="http://www.w3.org/1998/Math/MathML" id="RXN15_Calc">
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        <ci>RXN15</ci>
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          <times/>
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          <ci>Delta_5</ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="RXN16">
    <variable name="RXN16" public_interface="out" units="flux"/>
    <variable name="MR" public_interface="in" units="molecules"/>
    <variable initial_value="5" name="BetaR_1" units="first_order_rate"/>
    <math xmlns="http://www.w3.org/1998/Math/MathML" id="RXN16_Calc">
      <apply>
        <eq/>
        <ci>RXN16</ci>
        <apply>
          <times/>
          <ci>MR</ci>
          <ci>BetaR_1</ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <connection>
    <map_components component_1="A" component_2="RXN1"/>
    <map_variables variable_1="A" variable_2="A"/>
    <map_variables variable_1="RXN1" variable_2="RXN1"/>
  </connection>
  <connection>
    <map_components component_1="A" component_2="RXN2"/>
    <map_variables variable_1="A" variable_2="A"/>
    <map_variables variable_1="RXN2" variable_2="RXN2"/>
  </connection>
  <connection>
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    <map_variables variable_1="A" variable_2="A"/>
    <map_variables variable_1="RXN5" variable_2="RXN5"/>
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  <connection>
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    <map_variables variable_1="A" variable_2="A"/>
    <map_variables variable_1="RXN11" variable_2="RXN11"/>
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    <map_variables variable_1="C" variable_2="C"/>
    <map_variables variable_1="RXN3" variable_2="RXN3"/>
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    <map_variables variable_1="RXN11" variable_2="RXN11"/>
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  <connection>
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    <map_variables variable_1="DR" variable_2="DR"/>
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    <map_variables variable_1="RXN12" variable_2="RXN12"/>
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  <connection>
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    <map_variables variable_1="DRP" variable_2="DRP"/>
    <map_variables variable_1="RXN14" variable_2="RXN14"/>
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  <connection>
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    <map_variables variable_1="MA" variable_2="MA"/>
    <map_variables variable_1="RXN9" variable_2="RXN9"/>
  </connection>
  <connection>
    <map_components component_1="MA" component_2="RXN10"/>
    <map_variables variable_1="MA" variable_2="MA"/>
    <map_variables variable_1="RXN10" variable_2="RXN10"/>
  </connection>
  <connection>
    <map_components component_1="MR" component_2="RXN15"/>
    <map_variables variable_1="MR" variable_2="MR"/>
    <map_variables variable_1="RXN15" variable_2="RXN15"/>
  </connection>
  <connection>
    <map_components component_1="MR" component_2="RXN16"/>
    <map_variables variable_1="MR" variable_2="MR"/>
    <map_variables variable_1="RXN16" variable_2="RXN16"/>
  </connection>
  <connection>
    <map_components component_1="R" component_2="RXN1"/>
    <map_variables variable_1="R" variable_2="R"/>
    <map_variables variable_1="RXN1" variable_2="RXN1"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="A"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="C"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="DA"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="DAp"/>
    <map_variables variable_1="time" variable_2="time"/>
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  <connection>
    <map_components component_1="environment" component_2="DR"/>
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  <connection>
    <map_components component_1="environment" component_2="DRP"/>
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  <connection>
    <map_components component_1="environment" component_2="MA"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="MR"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="environment" component_2="R"/>
    <map_variables variable_1="time" variable_2="time"/>
  </connection>
  <connection>
    <map_components component_1="RXN6" component_2="A"/>
    <map_variables variable_1="RXN6" variable_2="RXN6"/>
  </connection>
  <connection>
    <map_components component_1="RXN10" component_2="A"/>
    <map_variables variable_1="RXN10" variable_2="RXN10"/>
  </connection>
  <connection>
    <map_components component_1="RXN12" component_2="A"/>
    <map_variables variable_1="RXN12" variable_2="RXN12"/>
  </connection>
  <connection>
    <map_components component_1="RXN1" component_2="C"/>
    <map_variables variable_1="RXN1" variable_2="RXN1"/>
  </connection>
  <connection>
    <map_components component_1="RXN5" component_2="DAp"/>
    <map_variables variable_1="RXN5" variable_2="RXN5"/>
  </connection>
  <connection>
    <map_components component_1="RXN12" component_2="DR"/>
    <map_variables variable_1="RXN12" variable_2="RXN12"/>
  </connection>
  <connection>
    <map_components component_1="RXN11" component_2="DRP"/>
    <map_variables variable_1="RXN11" variable_2="RXN11"/>
  </connection>
  <connection>
    <map_components component_1="RXN7" component_2="MA"/>
    <map_variables variable_1="RXN7" variable_2="RXN7"/>
  </connection>
  <connection>
    <map_components component_1="RXN8" component_2="MA"/>
    <map_variables variable_1="RXN8" variable_2="RXN8"/>
  </connection>
  <connection>
    <map_components component_1="RXN13" component_2="MR"/>
    <map_variables variable_1="RXN13" variable_2="RXN13"/>
  </connection>
  <connection>
    <map_components component_1="RXN14" component_2="MR"/>
    <map_variables variable_1="RXN14" variable_2="RXN14"/>
  </connection>
  <connection>
    <map_components component_1="RXN3" component_2="R"/>
    <map_variables variable_1="RXN3" variable_2="RXN3"/>
  </connection>
  <connection>
    <map_components component_1="RXN4" component_2="R"/>
    <map_variables variable_1="RXN4" variable_2="RXN4"/>
    <map_variables variable_1="R" variable_2="R"/>
  </connection>
  <connection>
    <map_components component_1="RXN16" component_2="R"/>
    <map_variables variable_1="RXN16" variable_2="RXN16"/>
  </connection>
  <connection>
    <map_components component_1="RXN6" component_2="DA"/>
    <map_variables variable_1="RXN6" variable_2="RXN6"/>
  </connection>
  



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    <rdf:li>circadian rhythms</rdf:li>
    <rdf:li>noise-resistance</rdf:li>
    <rdf:li>signal transduction</rdf:li>
    <rdf:li>gene regulation</rdf:li>
    <rdf:li>genetic oscillator</rdf:li>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">James</vCard:Given>
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    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Richard</vCard:Other>
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    <ns7:endingValue xmlns:ns7="http://www.cellml.org/metadata/simulation/1.0#">400</ns7:endingValue>
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    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2008-09-03T14:33:04+12:00</dcterms:W3CDTF>
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    <vCard:FN xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">James Lawson</vCard:FN>
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  <rdf:Description rdf:about="rdf:#092ccb9f-e343-4f58-b48d-ebbe3172ed40">
    <bqs:subject_type xmlns:bqs="http://www.cellml.org/bqs/1.0#">keyword</bqs:subject_type>
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    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2003-11-03T00:00:00+00:00</dcterms:W3CDTF>
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    <dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">PNAS</dc:title>
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    <dcterms:modified xmlns:dcterms="http://purl.org/dc/terms/" rdf:resource="rdf:#6b31cce9-847c-4f65-904d-2276d8008efb"/>
    <rdf:value>added cmeta:id's to some variables</rdf:value>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">James</vCard:Given>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Jose</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Vilar</vCard:Family>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">G.</vCard:Other>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">M.</vCard:Other>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Catherine</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Lloyd</vCard:Family>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">May</vCard:Other>
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    <vCard:FN xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Catherine Lloyd</vCard:FN>
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    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2008-09-01T13:12:26+12:00</dcterms:W3CDTF>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Hao</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Kueh</vCard:Family>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Yuan</vCard:Other>
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    <dcterms:modified xmlns:dcterms="http://purl.org/dc/terms/" rdf:resource="rdf:#4a858c75-c353-402e-9b38-9cde2c76eaca"/>
    <rdf:value>This model has been recoded using the Biomodels Database model as a template</rdf:value>
    <cmeta:modifier rdf:resource="rdf:#4e2213ae-fdc0-4cc8-9347-12ff246912c9"/>
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    <dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:resource="rdf:#6e776b10-823c-4701-a469-3bd7d4e49f98"/>
    <rdf:value>This model is known to run in PCEnv and COR reproduce the results published in the paper on which it is based (figures 2 and 3)</rdf:value>
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  <rdf:Description rdf:about="rdf:#4e2213ae-fdc0-4cc8-9347-12ff246912c9">
    <vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:resource="rdf:#ef5c8ff3-9197-4151-951e-9745118b88bc"/>
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  <rdf:Description rdf:about="#vilar_kueh_barkai_leibler_2002">
    <ns7:simulation xmlns:ns7="http://www.cellml.org/metadata/simulation/1.0#" rdf:resource="rdf:#$dmNO53"/>
    <dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Vilar et al.'s 2002 mathematical model of a genetic oscillator.</dc:title>
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    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">James</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Lawson</vCard:Family>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Richard</vCard:Other>
  </rdf:Description>
  <rdf:Description rdf:about="rdf:#488414d1-491e-4e40-99e7-94213eab43f0">
    <dc:creator xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:resource="rdf:#d2b09214-9391-49d2-add4-e6c8db79b79b"/>
    <rdf:value>This is the CellML description of Vilar et al.'s 2002 mathematical model of a genetic oscillator.</rdf:value>
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  <rdf:Description rdf:about="rdf:#4a858c75-c353-402e-9b38-9cde2c76eaca">
    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2008-02-05T11:39:57+13:00</dcterms:W3CDTF>
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  <rdf:Description rdf:about="rdf:#fa2451b2-21c1-44d8-9e14-e716863955a1">
    <rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
    <vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:resource="rdf:#b167a7db-a498-4547-9a62-82b4b7baa854"/>
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    <dcterms:modified xmlns:dcterms="http://purl.org/dc/terms/" rdf:resource="rdf:#5ba0e82b-e1ba-461d-b725-d1e1cdeeed90"/>
    <rdf:value>Added some cmeta:ids to varibles to allow referencing by a PCEnv session</rdf:value>
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    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2008-02-20T14:54:14+13:00</dcterms:W3CDTF>
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    <vCard:Orgname xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">The University of Auckland</vCard:Orgname>
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    <vCard:EMAIL xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:resource="rdf:#d2fceb36-78cd-492d-9226-b5a863ab0ff7"/>
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    <rdf:type rdf:resource="http://imc.org/vCard/3.0#internet"/>
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    <rdf:value>Added CellML 1.0 namespace

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    <dc:title xmlns:dc="http://purl.org/dc/elements/1.1/">Mechanisms of noise-resistance in genetic oscillators</dc:title>
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    <bqs:last_page xmlns:bqs="http://www.cellml.org/bqs/1.0#">5992</bqs:last_page>
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  <rdf:Description rdf:about="rdf:#e52579ff-7412-4023-a63f-4d9540642421">
    <rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
    <vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:resource="rdf:#e95de634-eb12-4283-b9c7-791c675a8038"/>
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    <dcterms:W3CDTF xmlns:dcterms="http://purl.org/dc/terms/">2002-04-30</dcterms:W3CDTF>
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    <vCard:N xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" rdf:resource="rdf:#10cbbde9-48b9-457e-b929-269fb4394a69"/>
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  <rdf:Description rdf:about="">
    <dc:publisher xmlns:dc="http://purl.org/dc/elements/1.1/">The University of Auckland, Auckland Bioengineering Institute</dc:publisher>
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    <dcterms:created xmlns:dcterms="http://purl.org/dc/terms/" rdf:resource="rdf:#f9a23f78-da42-4635-998f-9f21e3c14a88"/>
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    <cmeta:modification rdf:resource="rdf:#695fc7df-e54d-47fc-832b-50c8e9ac9da2"/>
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  <rdf:Description rdf:about="rdf:#b167a7db-a498-4547-9a62-82b4b7baa854">
    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Naama</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Barkai</vCard:Family>
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  <rdf:Description rdf:about="rdf:#$dmNO53">
    <ns7:boundIntervals xmlns:ns7="http://www.cellml.org/metadata/simulation/1.0#" rdf:resource="rdf:#$emNO53"/>
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  <rdf:Description rdf:about="rdf:#1b4f8009-249a-454f-8826-add810038fd5">
    <bqs:Pubmed_id xmlns:bqs="http://www.cellml.org/bqs/1.0#">11972055</bqs:Pubmed_id>
    <bqs:JournalArticle xmlns:bqs="http://www.cellml.org/bqs/1.0#" rdf:resource="rdf:#14119b20-ee64-49a1-b3f9-52383fc35b3d"/>
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  <rdf:Description rdf:about="rdf:#6fcccf65-6d32-4645-b5ea-19ca8765eead">
    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Stanislas</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Leibler</vCard:Family>
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  <rdf:Description rdf:about="rdf:#62c278f7-6a0a-43b8-878a-dd73a10f995e">
    <dc:subject xmlns:dc="http://purl.org/dc/elements/1.1/" rdf:resource="rdf:#092ccb9f-e343-4f58-b48d-ebbe3172ed40"/>
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  <rdf:Description rdf:about="rdf:#ef5c8ff3-9197-4151-951e-9745118b88bc">
    <vCard:Given xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Vignesh</vCard:Given>
    <vCard:Family xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">Kumar</vCard:Family>
    <vCard:Other xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#">B</vCard:Other>
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</model>