Rendering of the source text

<?xml version='1.0' encoding='utf-8'?>
<!--  FILE :  keener_model_2001.xml

CREATED :  20th November 2002

LAST MODIFIED : 9th April 2003

AUTHOR :  Catherine Lloyd
          The Bioengineering Institute
          The University of Auckland
          
MODEL STATUS :  This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the 16/01/2002 CellML Metadata 1.0 Specification.

DESCRIPTION :  This file contains a CellML description of James Keener's 2001 mathematical model of insulin secretion oscillations in pancreatic beta cells. 

CHANGES:  
  09/04/2003 - AAC - Added publication date information.  
 
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="keener_2001_version01" name="keener_2001_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Diffusion Induced Oscillatory Insulin Secretion</title>
  <author>
    <firstname>Catherine</firstname>
          <surname>Lloyd</surname>
    <affiliation>
      <shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
    </affiliation>
  </author>
</articleinfo>
  <section id="sec_status">
    <title>Model Status</title>
    <para>
            This model contains partial differentials and as such can not currently be solved by existing CellML tools.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
Insulin is secreted from pancreatic beta cells in an oscillatory fashion.  In his 2001 paper (cited below), James Keener examines a mathematical model for <emphasis>in vitro</emphasis> insulin secretion from pancreatic beta cells in a one-dimensional chemical flow reactor, where the reaction region is represented by the volume occupied by the cells (see <xref linkend="fig_cell_diagram"/> below).  Glucose enters the cell through GLUT-1 and GLUT-2 transporters.  Once inside the cell, intracellular glucose is metabolised, and this process activates insulin secretion via exocytosis of insulin containing granules.  In the extracellular medium, insulin activates GLUT-1 transporters and inactivates GLUT-2 transporters.   
</para>

<para>
In order to model the reactions, a steady flow of solution along the one-dimensional reactor is assumed, with insulin cells confined to a one-dimensional region.  Keener found that the oscillations occur as a result of an important interplay between flow rate of the reactor and insulin diffusion.  Without diffusion, the oscillations are eliminated.     
</para>

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

<para>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WC7-4593YHR-R&amp;_user=140507&amp;_coverDate=07%2F31%2F2001&amp;_rdoc=2&amp;_fmt=summary&amp;_orig=browse&amp;_srch=%23toc%236731%232001%23999369995%23288659!&amp;_cdi=6731&amp;_sort=d&amp;_docanchor=&amp;wchp=dGLbVzz-lSztb&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=ab655a0002a58212a7c8f4893876257a">Diffusion Induced Oscillatory Insulin Secretion</ulink>, James P. Keener, 2001, <ulink url="http://www.sciencedirect.com/science?_ob=JournalURL&amp;_cdi=6731&amp;_auth=y&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=2670d6bf58ba6625cf8c13344d6a4cb5">
            <emphasis>Bulletin of Mathematical Biology</emphasis>
          </ulink>, 63, 625-641. (A PDF version of the article is available to subscribers on the Bulletin of Mathematical Biology website.)  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=11497161&amp;dopt=Abstract">PubMed ID: 11497161</ulink> 
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>cell_diagram</title>
    </objectinfo>
    <imagedata fileref="keener_2001.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram representing the model of glucose uptake, glucose metabolism and insulin secretion by pancreatic beta cells.  Red lines represent the inactivation of GLUT-2 transporters by extracellular insulin, blue arrows represent the activation of GLUT-1 transporters by extracellular insulin and the activation of insulin secretion by glucose metabolism.</caption>
</informalfigure>

<para>
The raw CellML descriptions of the model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.  
</para>

</sect1>
</article>
</documentation>
  
  
  <!--
    Below, we define some additional units for association with variables and
    constants within the model.
  -->
  
  <units name="millimolar">
    <unit units="mole" prefix="milli"/>
    <unit units="litre" exponent="-1"/>
  </units>
  
  <units name="minute">
    <unit units="second" multiplier="60.0"/>
  </units>
  
  <units name="flux">
    <unit units="millimolar"/>
    <unit units="minute" exponent="-1"/>
  </units>
  
  <units name="cm2_per_second">
    <unit units="metre" prefix="centi" exponent="2"/>
    <unit units="second" exponent="-1"/>
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  <units name="mm">
    <unit units="metre" prefix="milli"/>
  </units>
  
  <units name="mm_per_minute">
    <unit units="metre" prefix="milli"/>
    <unit units="minute" exponent="-1"/>
  </units>
  
  <!--
    The "environment" component is used to declare variables that are used by
    all or most of the other components.
  -->
  
  <component name="environment">
    <variable units="minute" public_interface="out" name="time"/>
    <variable units="dimensionless" public_interface="out" name="rho"/>
  </component>
  
  <component name="intracellular_glucose">
    <variable units="millimolar" public_interface="out" name="G_i" initial_value="0.0"/>
    
    <variable units="minute" public_interface="in" name="time"/>
    <variable units="flux" public_interface="in" name="Rm"/>
    <variable units="flux" public_interface="in" name="R1"/>
    <variable units="flux" public_interface="in" name="R2"/>
    <variable units="dimensionless" public_interface="in" name="rho"/>
  
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        <eq/>
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            <minus/>
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            </apply>
            <ci> Rm </ci>
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        </apply>
      </apply>
    </math>
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    <variable units="mm" public_interface="out" name="x" initial_value="0.1"/>
    <variable units="mm_per_minute" public_interface="out" name="V" initial_value="3.8"/>
   
    <variable units="cm2_per_second" name="D_G" initial_value="6.6E-6"/>
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    <variable units="dimensionless" name="B"/>
   
    <variable units="minute" public_interface="in" name="time"/>
    <variable units="flux" public_interface="in" name="R1"/>
    <variable units="flux" public_interface="in" name="R2"/>
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        <eq/>
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            <ci> A </ci>
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            <minus/>
            <apply>
              <times/>
              <ci> G </ci>
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        </apply>
      </apply>
      
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          <partialdiff/>
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              <cn cellml:units="dimensionless"> 2.0 </cn>
            </degree>
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          <degree>
            <cn cellml:units="dimensionless"> 2.0 </cn>
          </degree>
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        <ci> A </ci>
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      <apply id="B_calculation">
        <eq/>
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          <bvar>
            <ci> x </ci>
          </bvar>
          <ci> G </ci>
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        <ci> B </ci>
      </apply>
    </math>
  </component>
  
  <component name="insulin">
    <variable units="millimolar" public_interface="out" name="I" initial_value="0.0"/>
   
    <variable units="cm2_per_second" name="D_I" initial_value="2.1E-6"/>
    <variable units="dimensionless" name="C"/>
    <variable units="dimensionless" name="D"/>
   
    <variable units="minute" public_interface="in" name="time"/>
    <variable units="flux" public_interface="in" name="Rs"/>
    <variable units="mm" public_interface="in" name="x"/>
    <variable units="mm_per_minute" public_interface="in" name="V"/>
   
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="I_partialdiff_eq">
        <eq/>
        <apply>
          <partialdiff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> I </ci>
        </apply>
        <apply>
          <plus/>
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            <times/>
            <ci> D_I </ci>
            <ci> C </ci>
          </apply>
          <apply>
            <minus/>
            <apply>
              <times/>
              <ci> V </ci>
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            <ci> x </ci>
          </bvar>
          <ci> I </ci>
        </apply>
        <ci> D </ci>
      </apply>
    </math>
  </component>
  
  <component name="insulin_inhibition">
    <variable units="dimensionless" public_interface="out" name="J"/>
   
    <variable units="dimensionless" name="J_infinity" initial_value="2.1E-6"/>
    <variable units="minute" name="J_tau" initial_value="20.0"/>
    <variable units="millimolar" name="K_inh" initial_value="1.0E-6"/>
   
    <variable units="minute" public_interface="in" name="time"/>
    <variable units="millimolar" public_interface="in" name="I"/>
   
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> J </ci>
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        <apply>
          <times/>
          <ci> J_tau </ci>
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            <minus/>
            <ci> J_infinity </ci>
            <ci> J </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="J_infinity_calculation">
        <eq/>
        <ci> J_infinity </ci>
        <apply>
          <divide/>
          <ci> K_inh </ci>
          <apply>
            <plus/>
            <ci> K_inh </ci>
            <ci> I </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
      
  <component name="glucose_metabolism">
    <variable units="flux" public_interface="out" name="Rm"/>
    
    <variable units="flux" name="Vm" initial_value="0.24"/>
    <variable units="millimolar" name="Km" initial_value="9.8"/>
    
    <variable units="minute" public_interface="in" name="time"/>
    <variable units="millimolar" public_interface="in" name="G_i"/>
    <variable units="dimensionless" public_interface="in" name="rho"/>
  
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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          <times/>
          <ci> rho </ci>
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              <ci> G_i </ci>
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            <apply>
              <plus/>
              <ci> Km </ci>
              <ci> G_i </ci>
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          </apply>
        </apply>
      </apply>
    </math>
  </component>
      
  <component name="GLUT_1_transporter">
    <variable units="flux" public_interface="out" name="R1"/>
    
    <variable units="flux" name="V1" initial_value="120.0"/>
    <variable units="millimolar" name="K1" initial_value="1.4"/>
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    <variable units="minute" public_interface="in" name="time"/>
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
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    <variable units="flux" name="V2" initial_value="32.0"/>
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    <variable units="minute" public_interface="in" name="time"/>
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    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
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  <connection>
    <map_components component_2="environment" component_1="extracellular_glucose"/>
    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
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  <connection>
    <map_components component_2="environment" component_1="insulin"/>
    <map_variables variable_2="time" variable_1="time"/>
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  <connection>
    <map_components component_2="environment" component_1="insulin_inhibition"/>
    <map_variables variable_2="time" variable_1="time"/>
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  <connection>
    <map_components component_2="environment" component_1="glucose_metabolism"/>
    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
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  <connection>
    <map_components component_2="environment" component_1="GLUT_1_transporter"/>
    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
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  <connection>
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    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
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  <connection>
    <map_components component_2="environment" component_1="insulin_secretion"/>
    <map_variables variable_2="time" variable_1="time"/>
    <map_variables variable_2="rho" variable_1="rho"/>
  </connection>
  
  <connection>
    <map_components component_2="glucose_metabolism" component_1="intracellular_glucose"/>
    <map_variables variable_2="G_i" variable_1="G_i"/>
    <map_variables variable_2="Rm" variable_1="Rm"/>
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  <connection>
    <map_components component_2="GLUT_1_transporter" component_1="intracellular_glucose"/>
    <map_variables variable_2="G_i" variable_1="G_i"/>
    <map_variables variable_2="R1" variable_1="R1"/>
  </connection>
  
  <connection>
    <map_components component_2="GLUT_2_transporter" component_1="intracellular_glucose"/>
    <map_variables variable_2="G_i" variable_1="G_i"/>
    <map_variables variable_2="R2" variable_1="R2"/>
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  <connection>
    <map_components component_2="GLUT_1_transporter" component_1="extracellular_glucose"/>
    <map_variables variable_2="G" variable_1="G"/>
    <map_variables variable_2="R1" variable_1="R1"/>
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  <connection>
    <map_components component_2="GLUT_2_transporter" component_1="extracellular_glucose"/>
    <map_variables variable_2="G" variable_1="G"/>
    <map_variables variable_2="R2" variable_1="R2"/>
  </connection>
  
  <connection>
    <map_components component_2="insulin_inhibition" component_1="insulin"/>
    <map_variables variable_2="I" variable_1="I"/>
  </connection>
  
  <connection>
    <map_components component_2="insulin_secretion" component_1="insulin"/>
    <map_variables variable_2="Rs" variable_1="Rs"/>
  </connection>
  
  <connection>
    <map_components component_2="GLUT_1_transporter" component_1="insulin"/>
    <map_variables variable_2="I" variable_1="I"/>
  </connection>
  
  <connection>
    <map_components component_2="extracellular_glucose" component_1="insulin"/>
    <map_variables variable_2="x" variable_1="x"/>
    <map_variables variable_2="V" variable_1="V"/>
  </connection>
  
  <connection>
    <map_components component_2="glucose_metabolism" component_1="insulin_secretion"/>
    <map_variables variable_2="Rm" variable_1="Rm"/>
  </connection>
  

<rdf:RDF>
  <rdf:Bag rdf:about="rdf:#4387d728-b36a-44e8-a441-6a78843901ee">
    <rdf:li>oscillator</rdf:li>
    <rdf:li>beta cell</rdf:li>
    <rdf:li>calcium dynamics</rdf:li>
    <rdf:li>Pancreatic Beta-Cell</rdf:li>
    <rdf:li>insulin</rdf:li>
    <rdf:li>metabolism</rdf:li>
  </rdf:Bag>
  <rdf:Description rdf:about="">
    <dc:publisher>
        The University of Auckland, Bioengineering Institute
      </dc:publisher>
    <cmeta:modification rdf:resource="rdf:#607bdccc-bb74-4947-8ef1-3303675540e2"/>
    <dcterms:created rdf:resource="rdf:#b7878110-f9cf-4dcc-9b12-96cc7dff28f1"/>
    <dc:creator rdf:resource="rdf:#0a7a32cf-deb1-4796-ab41-0c972bd0defb"/>
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            Diffusion Induced Oscillatory Insulin Secretion
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        James Keener's 2001 mathematical model of insulin secretion oscillations         in pancreatic beta cells.
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