Rendering of the source text

<?xml version='1.0' encoding='utf-8'?>
<!--  FILE :  mitochondrial_Ca_handling_1997.xml

CREATED :  11th January 2002

LAST MODIFIED : 12th August 2003

AUTHOR :  Catherine Lloyd
          Department of Engineering Science
          The University of Auckland
          
MODEL STATUS :  This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the 16/1/02 CellML Metadata 1.0 Specification.

DESCRIPTION :  This file contains a CellML description of Magnus and Keizer's
1997 minimal model of beta-call mitochondrial Ca2+ handling.

CHANGES:  
  21/01/2002 - AAC - Updated metadata to conform to the 16/1/02 CellML Metadata
                     1.0 Specification.
  26/02/2002 - CML - Corrected several equations.
  01/03/2002 - CML - Corrected units.
  22/07/2002 - CML - Added more metadata.
  09/04/2003 - AAC - Added publication date information.
  12/08/2003 - CML - Corrected equations.  
--><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="magnus_keizer_1997_version01" name="magnus_keizer_1997_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Mitochondrial Ca2+ Handling</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 is the original unchecked version of the model imported from the previous
            CellML model repository, 24-Jan-2006.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
In 1997 Gerhard Magnus and Joel Keizer published a minimal mathematical model to describe Ca<superscript>2+</superscript> handling by mitochondria in the pancreatic beta-cell.  Their kinetic model includes six transport mechanisms in the inner mitochondrial membrane, including proton pumping via respiration and proton pumping by the F0F1-ATPase, a proton leak, adenine nucleotide exchange, the Ca<superscript>2+</superscript> uniporter and Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange (see <xref linkend="fig_cell_diagram"/> below).
</para>

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

<para>
Minimal model of beta-cell mitochondrial Ca<superscript>2+</superscript> handling, Gerhard Magnus and Joel Keizer, 1997, <ulink url="http://ajpcell.physiology.org/">
            <emphasis>American Journal of Physiology</emphasis>
          </ulink>, 273, C717-C733.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=9277370&amp;dopt=Abstract">PubMed ID: 9277370</ulink>
</para>

<para>
The raw CellML description of the mitochondrial Ca<superscript>2+</superscript> handling model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.  For an example of a more complete documentation for an electrophysiological model, see <ulink url="${HTML_EXMPL_HHSA_INTRO}">The Hodgkin-Huxley Squid Axon Model, 1952</ulink>. 
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>diagram of the mitochondrial Ca2+ handling model showing ionic currents across the mitochondrial membrane</title>
    </objectinfo>
    <imagedata fileref="magnus_1997.png"/>
  </imageobject>
</mediaobject>
<caption>
A schematic representation of the ion fluxes through the mitochondrial inner membrane that are included in the Magnus-Keizer minimal model.
</caption>
</informalfigure>

<informalfigure float="0" id="fig_cellml_rendering">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>the cellml rendering of the mitochondrial Ca2+ handling model</title>
    </objectinfo>
    <imagedata fileref="cellml_rendering.gif"/>
  </imageobject>
</mediaobject>
<caption>The network defined in the CellML description of the mitochondrial Ca<superscript>2+</superscript> handling model.</caption>
</informalfigure>

</sect1>
</article>
</documentation>
  
  
  <!--
    Below, we define some additional units for association with variables and
    constants within the model. The identifiers are fairly self-explanatory.
  -->
  
  <units name="minute">
    <unit units="second" multiplier="60.0"/>
  </units>
  
  <units name="per_second">
    <unit units="second" exponent="-1"/>
  </units>
  
  <units name="millivolt">
    <unit units="volt" prefix="milli"/>
  </units>
  
  <units name="millimolar">
    <unit units="mole" prefix="milli"/>
    <unit units="litre" exponent="-1"/>
  </units>
  
  <units name="micromolar">
    <unit units="mole" prefix="micro"/>
    <unit units="litre" exponent="-1"/>
  </units>
  
  <units name="nanomolar">
    <unit units="mole" prefix="nano"/>
    <unit units="litre" exponent="-1"/>
  </units>
  
  <units name="nanomole_per_mg">
    <unit units="mole" prefix="nano"/>
    <unit units="gram" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="nanomole_per_minute_per_mg">
    <unit units="mole" prefix="nano"/>
    <unit units="minute" exponent="-1"/>
    <unit units="gram" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="capacitance_units">
    <unit units="mole" prefix="nano"/>
    <unit units="millivolt" exponent="-1"/>
    <unit units="gram" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="conductance_units">
    <unit units="mole" prefix="nano"/>
    <unit units="minute" exponent="-1"/>
    <unit units="millivolt" exponent="-1"/>
    <unit units="gram" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="millijoule_per_mole_kelvin">
    <unit units="joule" prefix="milli"/>
    <unit units="mole" exponent="-1"/>
    <unit units="kelvin" exponent="-1"/>
  </units>
  
  <units name="coulomb_per_mole">
    <unit units="coulomb"/>
    <unit units="mole" exponent="-1"/>
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  <!--
    The "environment" component is used to declare variables that are used by
    all or most of the other components, in this case just "time".
  -->
  <component name="environment">
    <variable units="minute" public_interface="out" name="time"/>
  </component>
  
  <!--
    The "membrane" component is really the `root' node of our model.
  -->
  <component name="membrane">
    <variable units="millivolt" public_interface="out" name="V"/>
    <variable units="millivolt" public_interface="out" name="V_B" initial_value="50.0"/>
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    <variable units="kelvin" public_interface="out" name="T" initial_value="310.0"/>
    <variable units="coulomb_per_mole" public_interface="out" name="F" initial_value="96845.0"/>
    <variable units="dimensionless" public_interface="out" name="delta_pH" initial_value="-0.4"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="proton_motive_force"/>
    <variable units="conductance_units" public_interface="out" name="g_H" initial_value="0.2"/>
    
    <variable units="capacitance_units" name="Cmito" initial_value="0.00145"/>
    <variable units="dimensionless" name="pH_i"/>
    <variable units="dimensionless" name="pH_m"/>
    
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="JH_res"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="JH_leak"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="JH_F1"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="J_ant"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="J_uni"/>
    <variable units="minute" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="membrane_voltage_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
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          </bvar>
          <ci> V </ci>
        </apply>
        <apply>
          <times/>
          <apply>
            <divide/>
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            <ci> Cmito </ci>
          </apply>
          <apply>
            <minus/>
            <apply>
              <plus/>
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              <ci> J_ant </ci>
              <ci> JH_leak </ci>
              <apply>
                <times/>
                <ci> J_uni </ci>
                <cn cellml:units="dimensionless"> 2.0 </cn>
              </apply>
            </apply>
            <ci> JH_res </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="delta_pH_calculation">
        <eq/>
        <ci> delta_pH </ci>
        <apply>
          <minus/>
          <ci> pH_i </ci>
          <ci> pH_m </ci>
        </apply>
      </apply>
      
      <apply id="proton_motive_force_calculation">
        <eq/>
        <ci> proton_motive_force </ci>
        <apply>
          <minus/>
          <ci> V </ci>
          <apply>
            <times/>
            <apply>
              <divide/>
              <apply>
                <times/>
                <cn cellml:units="dimensionless"> 2.303 </cn>
                <ci> R </ci>
                <ci> T </ci>
              </apply>
              <ci> F </ci>
            </apply>
            <ci> delta_pH </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="proton_pump">
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="JH_res"/>
    <variable units="nanomole_per_minute_per_mg" name="Jo"/>
    <variable units="dimensionless" name="A_res"/>
    <variable units="millimolar" name="K_res" initial_value="135000000000000000.0"/>
    <variable units="nanomolar" name="total_NAD_m" initial_value="8.0"/>
    <variable units="nanomolar" name="NAD_m"/>
    <variable units="nanomolar" name="NADH_m"/>
    <variable units="nanomole_per_mg" name="rho_res" initial_value="0.4"/>
    <variable units="dimensionless" name="r1" initial_value="0.000000000000000002077"/>
    <variable units="dimensionless" name="r2" initial_value="0.000000001728"/>
    <variable units="dimensionless" name="r3" initial_value="0.00000000000000000000000001059"/>
    <variable units="per_second" name="ra" initial_value="0.0000000006394"/>
    <variable units="per_second" name="rb" initial_value="0.0000000000001762"/>
    <variable units="per_second" name="rc1" initial_value="0.0000000000000000002656"/>
    <variable units="per_second" name="rc2" initial_value="0.000000000000000000000000008632"/>
    <variable units="dimensionless" name="g" initial_value="0.85"/>
    
    <variable units="dimensionless" public_interface="in" name="delta_pH"/>
    <variable units="millijoule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="millivolt" public_interface="in" name="V_B"/>
    <variable units="minute" public_interface="in" name="time"/>  
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> JH_res </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 360.0 </cn>
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                  <ci> ra </ci>
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                    <power/>
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                      <ci> delta_pH </ci>
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                      <apply>
                        <times/>
                        <ci> F </ci>
                        <ci> A_res </ci>
                      </apply>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
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                  <plus/>
                  <ci> ra </ci>
                  <ci> rb </ci>
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                      <times/>
                      <ci> g </ci>
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                      <ci> R </ci>
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              <plus/>
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                      <cn cellml:units="dimensionless"> 6.0 </cn>
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        </apply>
      </apply>
      
      <apply id="Jo_calculation">
        <eq/>
        <ci> Jo </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 30.0 </cn>
          <ci> rho_res </ci>
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                        <cn cellml:units="dimensionless"> 10.0 </cn>
                        <apply>
                          <times/>
                          <cn cellml:units="dimensionless"> 6.0 </cn>
                          <ci> delta_pH </ci>
                        </apply>
                      </apply>
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                      <times/>
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                      <apply>
                        <exp/>
                        <apply>
                          <divide/>
                          <apply>
                            <times/>
                            <cn cellml:units="dimensionless"> 6.0 </cn>
                            <ci> F </ci>
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              </apply>
              <apply>
                <times/>
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                <apply>
                  <exp/>
                  <apply>
                    <divide/>
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                      <ci> F </ci>
                      <ci> A_res </ci>
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                      <ci> V_B </ci>
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                    <times/>
                    <ci> r3 </ci>
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                          <ci> A_res </ci>
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      <apply id="A_res_calculation">
        <eq/>
        <ci> A_res </ci>
        <apply>
          <times/>
          <apply>
            <divide/>
            <apply>
              <times/>
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          <apply>
            <ln/>
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              <divide/>
              <apply>
                <times/>
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                  <root/>
                  <ci> NADH_m </ci>
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                <root/>
                <ci> NAD_m </ci>
              </apply>
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        <ci> total_NAD_m </ci>
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          <plus/>
          <ci> NADH_m </ci>
          <ci> NAD_m </ci>
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    </math>
  </component>
  
  <component name="proton_leak">
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="JH_leak"/>
    
    <variable units="conductance_units" public_interface="in" name="g_H"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="proton_motive_force"/>
    <variable units="minute" public_interface="in" name="time"/>  
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> JH_leak </ci>   
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          <times/>
          <ci> g_H </ci>
          <ci> proton_motive_force </ci>
        </apply>
      </apply>
    </math>
  </component>  
  
  <component name="F0_F1_ATPase">
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="JH_F1"/>
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="Jp_F1"/>
    
    <variable units="nanomole_per_mg" name="rho_F1" initial_value="0.7"/>
    <variable units="millimolar" name="K_F1" initial_value="1710000.0"/>
    <variable units="dimensionless" name="A_F1"/>
    <variable units="nanomolar" name="total_mitochondrial_adenosine_phosphate" initial_value="12.0"/>
    <variable units="nanomolar" name="ATP_m"/>
    <variable units="nanomolar" name="ADP_m_free"/>
    <variable units="millimolar" name="Pi_m" initial_value="20.0"/>
    <variable units="dimensionless" name="p1" initial_value="0.00000001346"/>
    <variable units="dimensionless" name="p2" initial_value="0.0000007739"/>
    <variable units="dimensionless" name="p3" initial_value="0.00000000000000665"/>
    <variable units="per_second" name="pa" initial_value="0.00001656"/>
    <variable units="per_second" name="pb" initial_value="0.0000003373"/>
    <variable units="per_second" name="pc1" initial_value="0.00000000000009651"/>
    <variable units="per_second" name="pc2" initial_value="0.0000000000000000004845"/>
    
    <variable units="nanomolar" public_interface="in" name="ADP_m"/> 
    <variable units="dimensionless" public_interface="in" name="delta_pH"/>    
    <variable units="millijoule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="millivolt" public_interface="in" name="V_B"/>
    <variable units="minute" public_interface="in" name="time"/>     
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="JH_F1_calculation">
        <eq/>
        <ci> JH_F1 </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> -180.0 </cn>
          <ci> rho_F1 </ci>
          <apply>
            <divide/>
            <apply>
              <minus/>
              <apply>
                <plus/>
                <apply>
                  <times/>
                  <ci> pa </ci>
                  <apply>
                    <power/>
                    <cn cellml:units="dimensionless"> 10.0 </cn>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> delta_pH </ci>
                    </apply>
                  </apply>
                  <apply>
                    <exp/>
                    <apply>
                      <divide/>
                      <apply>
                        <times/>
                        <ci> F </ci>
                        <ci> A_F1 </ci>
                      </apply>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <times/>
                  <ci> pb </ci>
                  <apply>
                    <power/>
                    <cn cellml:units="dimensionless"> 10.0 </cn>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> delta_pH </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <times/>
                <apply>
                  <plus/>
                  <ci> pa </ci>
                  <ci> pb </ci>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <plus/>
              <apply>
                <times/>
                <apply>
                  <plus/>
                  <cn cellml:units="dimensionless"> 1.0 </cn>
                  <apply>
                    <times/>
                    <ci> p1 </ci>
                    <apply>
                      <exp/>
                      <apply>
                        <divide/>
                        <apply>
                          <times/>
                          <ci> F </ci>
                          <ci> A_F1 </ci>
                        </apply>
                        <apply>
                          <times/>
                          <ci> R </ci>
                          <ci> T </ci>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V_B </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <times/>
                <apply>
                  <plus/>
                  <ci> p2 </ci>
                  <apply>
                    <times/>
                    <ci> p3 </ci>
                    <apply>
                      <exp/>
                      <apply>
                        <divide/>
                        <apply>
                          <times/>
                          <ci> F </ci>
                          <ci> A_F1 </ci>
                        </apply>
                        <apply>
                          <times/>
                          <ci> R </ci>
                          <ci> T </ci>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
      
      <apply id="Jp_F1_calculation">
        <eq/>
        <ci> Jp_F1 </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> -60.0 </cn>
          <ci> rho_F1 </ci>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <minus/>
                <apply>
                  <times/>
                  <apply>
                    <plus/>
                    <apply>
                      <times/>
                      <ci> pa </ci>
                      <apply>
                        <power/>
                        <cn cellml:units="dimensionless"> 10.0 </cn>
                        <apply>
                          <times/>
                          <cn cellml:units="dimensionless"> 3.0 </cn>
                          <ci> delta_pH </ci>
                        </apply>
                      </apply>
                    </apply>
                    <apply>
                      <times/>
                      <ci> pc1 </ci>
                      <apply>
                        <exp/>
                        <apply>
                          <divide/>
                          <apply>
                            <times/>
                            <cn cellml:units="dimensionless"> 3.0 </cn>
                            <ci> F </ci>
                            <ci> V_B </ci>
                          </apply>
                          <apply>
                            <times/>
                            <ci> R </ci>
                            <ci> T </ci>
                          </apply>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                  <apply>
                    <exp/>
                    <apply>
                      <divide/>
                      <apply>
                        <times/>
                        <ci> F </ci>
                        <ci> A_F1 </ci>
                      </apply>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <times/>
                  <ci> pa </ci>
                  <apply>
                    <exp/>
                    <apply>
                      <divide/>
                      <apply>
                        <times/>
                        <cn cellml:units="dimensionless"> 3.0 </cn>
                        <ci> F </ci>
                        <ci> V </ci>
                      </apply>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <times/>
                <ci> pc2 </ci>          
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <ci> F </ci>
                      <ci> A_F1 </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <plus/>
              <apply>
                <times/>
                <apply>
                  <plus/>
                  <cn cellml:units="dimensionless"> 1.0 </cn>
                  <apply>
                    <times/>
                    <ci> p1 </ci>
                    <apply>
                      <exp/>
                      <apply>
                        <divide/>
                        <apply>
                          <times/>
                          <ci> F </ci>
                          <ci> A_F1 </ci>
                        </apply>
                        <apply>
                          <times/>
                          <ci> R </ci>
                          <ci> T </ci>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V_B </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <times/>
                <apply>
                  <plus/>
                  <ci> p2 </ci>
                  <apply>
                    <times/>
                    <ci> p3 </ci>
                    <apply>
                      <exp/>
                      <apply>
                        <divide/>
                        <apply>
                          <times/>
                          <ci> F </ci>
                          <ci> A_F1 </ci>
                        </apply>
                        <apply>
                          <times/>
                          <ci> R </ci>
                          <ci> T </ci>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <cn cellml:units="dimensionless"> 3.0 </cn>
                      <ci> F </ci>
                      <ci> V </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>        
      
      <apply id="A_F1_calculation">
        <eq/>
        <ci> A_F1 </ci>
        <apply>
          <times/>
          <apply>
            <divide/>
            <apply>
              <times/>
              <ci> R </ci>
              <ci> T </ci>
            </apply>
            <ci> F </ci>
          </apply>
          <apply>
            <ln/>
            <apply>
              <divide/>
              <apply>
                <times/>
                <ci> K_F1 </ci>
                <ci> ATP_m </ci>
              </apply>
              <apply>
                <times/>
                <ci> ADP_m_free </ci>
                <ci> Pi_m </ci>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
      
      <apply id="total_mitochondrial_adenosine_phosphate_calculation">
        <eq/>
        <ci> total_mitochondrial_adenosine_phosphate </ci>
        <apply>
          <plus/>
          <ci> ATP_m </ci>
          <ci> ADP_m </ci>
        </apply>
      </apply>
      
      <apply id="ADP_m_free_calculation">
        <eq/>
        <ci> ADP_m_free </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 0.8 </cn>
          <ci> ADP_m </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="ATP_ADP_exchange">
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="J_ant"/>
    <variable units="nanomolar" public_interface="out" name="ADP_m"/>
    
    <variable units="nanomole_per_minute_per_mg" name="Jmax_ant" initial_value="1000.0"/>
    <variable units="nanomolar" name="ATP_i"/>
    <variable units="nanomolar" name="ATP_m"/>
    <variable units="nanomolar" name="ADP_i"/>
    <variable units="nanomolar" name="ATP4_i"/>
    <variable units="nanomolar" name="ATP4_m"/>
    <variable units="nanomolar" name="ADP3_i"/>
    <variable units="nanomolar" name="ADP3_m"/>
    <variable units="dimensionless" name="f" initial_value="0.5"/>
   
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="Jp_F1"/>
    <variable units="millijoule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="minute" public_interface="in" name="time"/>  
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_ant_calculation">
        <eq/>
        <ci> J_ant </ci>
        <apply>
          <times/>
          <ci> Jmax_ant </ci>
          <apply>
            <divide/>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <apply>
                <times/>
                <apply>
                  <divide/>
                  <apply>
                    <times/>
                    <ci> ATP4_i </ci>
                    <ci> ADP3_m </ci>
                  </apply>
                  <apply>
                    <times/>
                    <ci> ADP3_i </ci>
                    <ci> ATP4_m </ci>
                  </apply>
                </apply>
                <apply>
                  <exp/>
                  <apply>
                    <divide/>
                    <apply>
                      <times/>
                      <apply>
                        <minus/>
                        <ci> F </ci>
                      </apply>
                      <ci> V </ci>
                    </apply>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <times/>
              <apply>
                <plus/>
                <cn cellml:units="dimensionless"> 1.0 </cn>
                <apply>
                  <times/>
                  <apply>
                    <divide/>
                    <ci> ATP4_i </ci>
                    <ci> ADP3_i </ci>
                  </apply>
                  <apply>
                    <exp/>
                    <apply>
                      <divide/>
                      <apply>
                        <times/>
                        <apply>
                          <minus/>
                          <ci> f </ci>
                        </apply>
                        <ci> F </ci>
                        <ci> V </ci>
                      </apply>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <plus/>
                <cn cellml:units="dimensionless"> 1.0 </cn>
                <apply>
                  <divide/>
                  <ci> ADP3_m </ci>
                  <ci> ATP4_m </ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>    
        
      <apply id="ADP_m_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> ADP_m </ci>
        </apply>
        <apply>
          <minus/>
          <ci> J_ant </ci>
          <ci> Jp_F1 </ci>
        </apply>
      </apply>

      <apply id="ADP3_m_calculation">
        <eq/>
        <ci> ADP3_m </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 0.45 </cn>
          <cn cellml:units="dimensionless"> 0.8 </cn>
          <ci> ADP_m </ci>
        </apply>
      </apply>

      <apply id="ADP3_i_calculation">
        <eq/>
        <ci> ADP3_i </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 0.45 </cn>
          <cn cellml:units="dimensionless"> 1.0 </cn>
          <ci> ADP_i </ci>
        </apply>
      </apply>

      <apply id="ATP4_m_calculation">
        <eq/>
        <ci> ATP4_m </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 0.05 </cn>
          <ci> ATP_m </ci>
        </apply>
      </apply>

      <apply id="ATP4_i_calculation">
        <eq/>
        <ci> ATP4_i </ci>
        <apply>
          <times/>
          <cn cellml:units="dimensionless"> 0.05 </cn>
          <ci> ATP_i </ci>
        </apply>
      </apply>
    </math>
  </component>
        
  <component name="Ca_uniporter">
    <variable units="nanomole_per_minute_per_mg" public_interface="out" name="J_uni"/>
    <variable units="millivolt" public_interface="out" name="V_offset" initial_value="91.0"/>
    
    <variable units="nanomole_per_minute_per_mg" name="Jmax_uni" initial_value="300.0"/>
    <variable units="nanomolar" name="Ca_i"/>
    <variable units="micromolar" name="K_trans" initial_value="19.0"/>
    <variable units="micromolar" name="K_act" initial_value="0.38"/>
    <variable units="dimensionless" name="L" initial_value="110.0"/>
    <variable units="dimensionless" name="na" initial_value="2.8"/>
    
    <variable units="millijoule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="minute" public_interface="in" name="time"/>      
  
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_uni_calculation">
        <eq/>
        <ci> J_uni </ci>
        <apply>
          <times/>
          <ci> Jmax_uni </ci>
          <apply>
            <divide/>
            <apply>
              <times/>
              <apply>
                <divide/>
                <ci> Ca_i </ci>
                <ci> K_trans </ci>
              </apply>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <cn cellml:units="dimensionless"> 1.0 </cn>
                  <apply>
                    <divide/>
                    <ci> Ca_i </ci>
                    <ci> K_trans </ci>
                  </apply>
                </apply>
                <cn cellml:units="dimensionless"> 3.0 </cn>
              </apply>
            </apply>
            <apply>
              <plus/>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <cn cellml:units="dimensionless"> 1.0 </cn>
                  <apply>
                    <divide/>
                    <ci> Ca_i </ci>
                    <ci> K_trans </ci>
                  </apply>
                </apply>
                <cn cellml:units="dimensionless"> 4.0 </cn>
              </apply>
              <apply>
                <divide/>
                <ci> L </ci>
                <apply>
                  <power/>
                  <apply>
                    <plus/>
                    <cn cellml:units="dimensionless"> 1.0 </cn>
                    <apply>
                      <divide/>
                      <ci> Ca_i </ci>
                      <ci> K_act </ci>
                    </apply>
                  </apply>
                  <ci> na </ci>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <divide/>
            <apply>
              <divide/>
              <apply>
                <times/>
                <cn cellml:units="dimensionless"> 2.0 </cn>
                <ci> F </ci>
                <apply>
                  <minus/>
                  <ci> V </ci>
                  <ci> V_offset </ci>
                </apply>
              </apply>
              <apply>
                <times/>
                <ci> R </ci>
                <ci> T </ci>
              </apply>
            </apply>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <apply>
                <exp/>
                <apply>
                  <divide/>
                  <apply>
                    <times/>
                    <cn cellml:units="dimensionless"> -2.0 </cn>
                    <ci> F </ci>
                    <apply>
                      <minus/>
                      <ci> V </ci>
                      <ci> V_offset </ci>
                    </apply>
                  </apply>
                  <apply>
                    <times/>
                    <ci> R </ci>
                    <ci> T </ci>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="Na_Ca_exchanger">
    <variable units="nanomole_per_minute_per_mg" name="J_Na_Ca"/>
    <variable units="nanomole_per_minute_per_mg" name="Jmax_Na_Ca" initial_value="25.0"/>
    <variable units="nanomolar" name="Ca_m"/>
    <variable units="millimolar" name="Na_i" initial_value="30.0"/>
    <variable units="millimolar" name="K_Na" initial_value="9.4"/>
    <variable units="nanomole_per_mg" name="K_Ca" initial_value="0.003"/>
    <variable units="dimensionless" name="b" initial_value="0.0"/>
    <variable units="dimensionless" name="n" initial_value="2.0"/>
    <variable units="dimensionless" name="fm" initial_value="0.0003"/>  
   
    <variable units="nanomole_per_minute_per_mg" public_interface="in" name="J_uni"/>
    <variable units="millivolt" public_interface="in" name="V_offset"/>
    <variable units="millijoule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="minute" public_interface="in" name="time"/>      
  
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_Na_Ca_calculation">
        <eq/>
        <ci> J_Na_Ca </ci>
        <apply>
          <times/>
          <ci> Jmax_Na_Ca </ci>
          <apply>
            <divide/>
            <apply>
              <exp/>
              <apply>
                <divide/>
                <apply>
                  <times/>
                  <ci> b </ci>
                  <ci> F </ci>
                  <apply>
                    <minus/>
                    <ci> V </ci>
                    <ci> V_offset </ci>
                  </apply>
                </apply>
                <apply>
                  <times/>
                  <ci> R </ci>
                  <ci> T </ci>
                </apply>
              </apply>
            </apply>
            <apply>
              <times/>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <cn cellml:units="dimensionless"> 1.0 </cn>
                  <apply>
                    <divide/>
                    <ci> K_Na </ci>
                    <ci> Na_i </ci>
                  </apply>
                </apply>
                <ci> n </ci>
              </apply>
              <apply>
                <plus/>
                <cn cellml:units="dimensionless"> 1.0 </cn>
                <apply>
                  <divide/>
                  <ci> K_Ca </ci>
                  <ci> Ca_m </ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
      
      <apply id="Ca_m_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> Ca_m </ci>
        </apply>
        <apply>
          <times/>
          <ci> fm </ci>
          <apply>
            <minus/>
            <ci> J_uni </ci>
            <ci> J_Na_Ca </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
                    
  <group>
    <relationship_ref relationship="containment"/>
    <component_ref component="membrane">
      <component_ref component="proton_pump"/>
      <component_ref component="proton_leak"/>
      <component_ref component="F0_F1_ATPase"/>
      <component_ref component="ATP_ADP_exchange"/>
      <component_ref component="Ca_uniporter"/>
      <component_ref component="Na_Ca_exchanger"/>                    
    </component_ref>
  </group>
  
  <!--
    "Time" is passed from the "environment" component into all other components.
  -->
  
  <connection>
    <map_components component_2="environment" component_1="membrane"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="proton_pump" component_1="environment"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="proton_leak" component_1="environment"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection> 
  
  <connection>
    <map_components component_2="F0_F1_ATPase" component_1="environment"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="ATP_ADP_exchange" component_1="environment"/>
    <map_variables variable_2="time" variable_1="time"/>
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  <connection>
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    <map_variables variable_2="time" variable_1="time"/>
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  <connection>
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  <!--
    Several variables are passed between the "membrane" and its sub-components.
  -->
  
  <connection>
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  <connection>
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    <map_variables variable_2="JH_leak" variable_1="JH_leak"/>
    <map_variables variable_2="g_H" variable_1="g_H"/>
    <map_variables variable_2="proton_motive_force" variable_1="proton_motive_force"/>
  </connection>
  
  <connection>
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    <map_variables variable_2="JH_F1" variable_1="JH_F1"/>
    <map_variables variable_2="delta_pH" variable_1="delta_pH"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="V_B" variable_1="V_B"/>
    <map_variables variable_2="R" variable_1="R"/>
    <map_variables variable_2="T" variable_1="T"/>
    <map_variables variable_2="F" variable_1="F"/>
  </connection> 
  
  <connection>
    <map_components component_2="ATP_ADP_exchange" component_1="membrane"/>
    <map_variables variable_2="J_ant" variable_1="J_ant"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="R" variable_1="R"/>
    <map_variables variable_2="T" variable_1="T"/>
    <map_variables variable_2="F" variable_1="F"/>
  </connection> 
  
  <connection>
    <map_components component_2="Ca_uniporter" component_1="membrane"/>
    <map_variables variable_2="J_uni" variable_1="J_uni"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="R" variable_1="R"/>
    <map_variables variable_2="T" variable_1="T"/>
    <map_variables variable_2="F" variable_1="F"/>
  </connection> 
  
  <connection>
    <map_components component_2="Na_Ca_exchanger" component_1="membrane"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="R" variable_1="R"/>
    <map_variables variable_2="T" variable_1="T"/>
    <map_variables variable_2="F" variable_1="F"/>
  </connection>

  <!--  A few variables are passed between the sibling components. -->  
  
  <connection>
    <map_components component_2="F0_F1_ATPase" component_1="ATP_ADP_exchange"/>
    <map_variables variable_2="Jp_F1" variable_1="Jp_F1"/>
    <map_variables variable_2="ADP_m" variable_1="ADP_m"/>
  </connection> 
  
  <connection>
    <map_components component_2="Ca_uniporter" component_1="Na_Ca_exchanger"/>
    <map_variables variable_2="J_uni" variable_1="J_uni"/>
    <map_variables variable_2="V_offset" variable_1="V_offset"/>
  </connection>

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    <rdf:li>calcium dynamics</rdf:li>
    <rdf:li>electrophysiology</rdf:li>
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    <rdf:li>mitochondria</rdf:li>
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          Corrected units.
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    <vCard:Given>Catherine</vCard:Given>
    <vCard:Family>Lloyd</vCard:Family>
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    <vCard:Given>Catherine</vCard:Given>
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            Minimal Model of beta-cell mitochondrial Ca2+ handling
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    <dcterms:W3CDTF>2003-04-09</dcterms:W3CDTF>
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    <dcterms:W3CDTF>2003-08-12</dcterms:W3CDTF>
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          Corrected equations: JH_F1 in F0_F1_ATPase component, J_uni in 
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          This is the CellML description of Gerhard Magnus and Joel Keizer's 
          1997 minimal model of beta-call mitochondrial Ca2+ handling.  They 
          developed a simplified mathematical model to describe Ca2+ handling by
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          Added publication date information.
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    <dcterms:W3CDTF>1997-08</dcterms:W3CDTF>
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    <dc:title>
        Minimal model of beta-cell mitochondrial Ca2+ handling
      </dc:title>
    <cmeta:bio_entity>Pancreatic Beta-cell</cmeta:bio_entity>
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    <vCard:FN>Catherine Lloyd</vCard:FN>
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    <dcterms:W3CDTF>2002-07-22</dcterms:W3CDTF>
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          Corrected several equations.
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          Added more metadata.
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          Updated metadata to conform to the 16/1/02 CellML Metadata 1.0 
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        The University of Auckland, Bioengineering Research Group
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    <vCard:Given>Autumn</vCard:Given>
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    <vCard:Other>A</vCard:Other>
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    <vCard:Given>Catherine</vCard:Given>
    <vCard:Family>Lloyd</vCard:Family>
    <vCard:Other>May</vCard:Other>
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    <dcterms:W3CDTF>2002-01-11</dcterms:W3CDTF>
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  <rdf:Description rdf:about="rdf:#67d0541f-e12b-4598-9015-c25e1c578c67">
    <dc:title>American Journal of Physoiology</dc:title>
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    <bqs:subject_type>keyword</bqs:subject_type>
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    <dcterms:W3CDTF>2002-02-26</dcterms:W3CDTF>
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