Rendering of the source text

<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : shiferaw_model_2003.xml

CREATED : 8th January 2004

LAST MODIFIED : 8th January 2004

AUTHOR : Catherine Lloyd
         Bioengineering Instute
         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 Shiferaw et al.'s 2003 mathematical model of intracellular calcium cycling in ventricular myocytes.

CHANGES: 
  
--><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="shiferaw_watanabe_garfinkel_weiss_karma_2003_version01" name="shiferaw_watanabe_garfinkel_weiss_karma_2003_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Modelling Intracellular Calcium Cycling in Ventricular Myocytes</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>
Cardiac myocyte contraction is stimulated by a rise in the concentration of intracellular calcium ([Ca<superscript>2+</superscript>]<subscript>i</subscript>), which in turn is due to the coordinated release of calcium from the sarcoplasmic reticulum (SR) via ryanodine receptors (RyR).  These receptors are close to the L-type calcium channels in the cell surface membrane, and when the cell is depolarised, the calcium that enters the cell through the L-type channels is confined within a microdomain (the dyadic junction).  The RyR then sense this increase in [Ca<superscript>2+</superscript>]<subscript>i</subscript> and respond by opening and releasing more calcium from the SR in a process known as calcium-induced calcium releases (CICR).  The cell's contractile elements are activated by this increase in [Ca<superscript>2+</superscript>]<subscript>i</subscript> and the cell contracts.  A calcium-activated uptake pump then pumps calcium back into the SR.  This interaction between membrane potential and intracellular calcium cycling forms the basis of excitation-contraction (EC) coupling.  The calcium system is driven by an action potential waveform that is itself dependent on the dynamics of the calcium system.
</para>

<para>
There have been several efforts to mathematically model intracellular calcium dynamics, such as:
</para>
<itemizedlist>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_RJW_MODEL}">Modelling the Functional Ca<superscript>2+</superscript> Release Unit, 1999, Rice <emphasis>et al.</emphasis>, 1999</ulink>
            </para>
          </listitem>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_STERN_MODEL}">Model of Ryanodine Receptor Gating in Cardiac Muscle, 1999, Stern <emphasis>et al.</emphasis>
              </ulink>
            </para>
          </listitem>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_SOBIE_MODEL}">Sobie <emphasis>et al.</emphasis> Mechanism of Cardiac Ca<superscript>2+</superscript> Spark Termination, 2002</ulink>
            </para>
          </listitem>
</itemizedlist>
<para>
which focus on the detailed calcium dynamics within the dyadic junction.  Several authors have developed models of calcium cycling at the whole cell level, including:
</para>
<itemizedlist>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_LR_II_MODEL}">Luo-Rudy Ventricular Model II (dynamic), 1994</ulink>
            </para>
          </listitem>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_JRW_MODEL}">Jafri-Rice-Winslow Ventricular Model, 1998</ulink>
            </para>
          </listitem>
  <listitem>
            <para>
              <ulink url="${HTML_EXMPL_SNYDER_MODEL}">Snyder <emphasis>et al.</emphasis>, Cardiocyte Ca<superscript>2+</superscript> Dynamics, 2000</ulink>
            </para>
          </listitem>
</itemizedlist>  
<para>However, some of these models are lacking in physiological detail; for example, some ignore intracellular calcium compartmentation, or they overlook the fact that calcium release from the SR is in fact the summation of several discrete events.
</para>  

<para>
In the publication described here, Shiferaw <emphasis>et al.</emphasis> present a new model of EC coupling in ventricular myocytes (see <xref linkend="fig_cell_diagram"/> below).  In contrast to the previously published models this one represents calcium release from the SR as a summation of elementary release events that correspond to calcium sparks.  In addition, the model includes features that are absent in the earlier models, such as:
</para>
<itemizedlist>
  <listitem>
            <para>a description of intracellular sodium accumulation;</para>
          </listitem>
  <listitem>
            <para>compartmentation of myoplasm into a submembrane space near the sarcolemma, and the rest of the myoplasm; and</para>
          </listitem>
  <listitem>
            <para>a calcium diffusional delay between the network SR (NSR) and the junctional SR (JSR)</para>
          </listitem>
</itemizedlist>

<para>
The model has been described here in CellML (the raw CellML description of the Shiferaw <emphasis>et al.</emphasis> 2003 model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>).     
</para>

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

<para>
Model of Intracellular Calcium Cycling in Ventricular Myocytes, Y. Shiferaw, M. A. Wantanabe, A. Garfinkel, J. N. Weiss, and A. Karma, 2003, <emphasis>Biophysical Journal</emphasis>, 85, 3666-3686.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=14645059&amp;dopt=Abstract">PubMed ID: 14645059</ulink>
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>cell diagram</title>
    </objectinfo>
    <imagedata fileref="shiferaw_2003.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram of the intracellular compartments relevant to calcium cycling in ventricular myocytes.</caption>
</informalfigure>

</sect1>
</article>
</documentation>
  
  
  
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    <variable units="dimensionless" name="f_infinity"/>
    <variable units="millisecond" name="tau_f" initial_value="30.0"/>
    
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
     <apply id="alpha_f_calculation">
        <eq/>
        <ci> alpha_f </ci>
        <apply>
          <divide/>
          <ci> f_infinity </ci>
          <ci> tau_f </ci>
        </apply>
      </apply>
      
      <apply id="f_infinity_calculation">
        <eq/>
        <ci> f_infinity </ci>
        <apply>
          <divide/>
          <cn cellml:units="dimensionless"> 1.0 </cn>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
              <apply>
                <divide/>
                <apply>
                  <plus/>
                  <ci> V </ci>
                  <cn cellml:units="millivolt"> 35.0 </cn>
                </apply>
                <cn cellml:units="millivolt"> 8.6 </cn>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
        
      <apply id="beta_f_calculation">
        <eq/>
        <ci> beta_f </ci>
        <apply>
          <divide/>
          <apply>
            <minus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <ci> f_infinity </ci>
          </apply>
          <ci> tau_f </ci>
        </apply>
      </apply>
      
      <apply id="df_dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> f </ci>
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> alpha_f </ci>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <ci> f </ci>
            </apply>
          </apply>
          <apply>
            <times/>
            <ci> beta_f </ci>
            <ci> f </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="Na_Ca_exchanger">
    <variable units="picoA" public_interface="out" name="I_NaCa"/>
    
    <variable units="flux" name="K_NaCa" initial_value="2E4"/>
    <variable units="millimolar" name="K_mNa" initial_value="87.5"/>
    <variable units="millimolar" name="K_mCa" initial_value="1.38"/>
    <variable units="dimensionless" name="K_sat" initial_value="0.1"/>
    <variable units="dimensionless" name="eta" initial_value="0.35"/>
    
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="joule_per_mole_kelvin" public_interface="in" name="R"/>
    <variable units="kelvin" public_interface="in" name="T"/>
    <variable units="coulomb_per_mole" public_interface="in" name="F"/>
    <variable units="millimolar" public_interface="in" name="Nai"/>
    <variable units="millimolar" public_interface="in" name="Nao"/>
    <variable units="micromolar" public_interface="in" name="cs"/>
    <variable units="millimolar" public_interface="in" name="Cao"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="Na_Ca_exchanger">
        <eq/>
        <ci> I_NaCa </ci>
        <apply>
          <times/> 
          <ci> K_NaCa </ci>
          <apply>
            <divide/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <plus/>
              <apply>
                <power/>
                <ci> K_mNa </ci>
                <cn cellml:units="dimensionless"> 3.0 </cn>
              </apply>
              <apply>
                <power/>
                <ci> Nao </ci>
                <cn cellml:units="dimensionless"> 3.0 </cn>
              </apply>
            </apply>
          </apply>
          <apply>
            <divide/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <plus/>
              <ci> K_mCa </ci>
              <ci> Cao </ci>
            </apply>
          </apply>
          <apply>
            <divide/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <plus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <apply>
                <times/>
                <ci> K_sat </ci>
                <apply>
                  <exp/>
                  <apply>
                    <times/>
                    <apply>
                      <minus/>
                      <ci> eta </ci>
                      <cn cellml:units="dimensionless"> 1.0 </cn>
                    </apply>
                    <ci> V </ci>
                    <apply>
                      <divide/>
                      <ci> F </ci>
                      <apply>
                        <times/>
                        <ci> R </ci>
                        <ci> T </ci>
                      </apply>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <minus/>
            <apply>
              <times/>
              <apply>
                <exp/>
                <apply>
                  <times/>
                  <ci> eta </ci>
                  <ci> V </ci>
                  <apply>
                    <divide/>
                    <ci> F </ci>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <power/>
                <ci> Nai </ci>
                <cn cellml:units="dimensionless"> 3.0 </cn>
              </apply>
              <ci> Cao </ci>
            </apply>
            <apply>
              <times/>
              <apply>
                <exp/>
                <apply>
                  <times/>
                  <apply>
                    <minus/>
                    <ci> eta </ci>
                    <cn cellml:units="dimensionless"> 1.0 </cn>
                  </apply>
                  <ci> V </ci>
                  <apply>
                    <divide/>
                    <ci> F </ci>
                    <apply>
                      <times/>
                      <ci> R </ci>
                      <ci> T </ci>
                    </apply>
                  </apply>
                </apply>
              </apply>
              <apply>
                <power/>
                <ci> Nao </ci>
                <cn cellml:units="dimensionless"> 3.0 </cn>
              </apply>
              <ci> cs </ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="uptake_current">
    <variable units="picoA" public_interface="out" name="I_up"/>
    
    <variable units="flux" name="vup" initial_value="250.0"/>
    <variable units="micromolar" name="cup" initial_value="0.5"/>
    
    <variable units="micromolar" public_interface="in" name="ci"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="I_up_calculation">
        <eq/>
        <ci> I_up </ci>
        <apply>
          <divide/>
          <apply>
            <times/>
            <ci> vup </ci>
            <apply>
              <power/>
              <ci> ci </ci>
              <cn cellml:units="dimensionless"> 2.0 </cn>
            </apply>
          </apply>
          <apply>
            <plus/>
            <apply>
              <power/>
              <ci> ci </ci>
              <cn cellml:units="dimensionless"> 2.0 </cn>
            </apply>
            <apply>
              <power/>
              <ci> cup </ci>
              <cn cellml:units="dimensionless"> 2.0 </cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_buffering">
    <variable units="dimensionless" public_interface="out" name="beta_ci"/>
    <variable units="dimensionless" public_interface="out" name="beta_cs"/>
    <variable units="picoA" public_interface="out" name="I_trpn_i"/>
    <variable units="picoA" public_interface="out" name="I_trpn_s"/>
    
    <variable units="micromolar" name="CaTi"/>
    <variable units="micromolar" name="CaTs"/>
    <variable units="second_order_rate_constant" name="kT_on" initial_value="32.7"/>
    <variable units="first_order_rate_constant" name="kT_off" initial_value="19.6"/>
    <variable units="micromolar" name="BT" initial_value="70.0"/>
    <variable units="micromolar" name="BSR" initial_value="47.0"/>
    <variable units="micromolar" name="BCd" initial_value="24.0"/>
    <variable units="micromolar" name="KSR" initial_value="0.6"/>
    <variable units="micromolar" name="KCd" initial_value="7.0"/>
     
    <variable units="micromolar" public_interface="in" name="ci"/>
    <variable units="micromolar" public_interface="in" name="cs"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="CaTi_calculation">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> CaTi </ci>
        </apply>
        <ci> I_trpn_i </ci>
      </apply>
      
      <apply id="I_trpn_i_calculation">
        <eq/>
        <ci> I_trpn_i </ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> kT_on </ci>
            <ci> ci </ci>
            <apply>
              <minus/>
              <ci> BT </ci>
              <ci> CaTi </ci>
            </apply>
          </apply>
          <apply>
            <times/>
            <ci> kT_off </ci>
            <ci> CaTi </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="beta_ci_calculation">
        <eq/>
        <ci> beta_ci </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <divide/>
              <apply>
                <times/>
                <ci> BSR </ci>
                <ci> KSR </ci>
              </apply>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <ci> ci </ci>
                  <ci> KSR </ci>
                </apply>
                <cn cellml:units="dimensionless"> 2.0 </cn>
              </apply>
            </apply>
            <apply>
              <divide/>
              <apply>
                <times/>
                <ci> BCd </ci>
                <ci> KCd </ci>
              </apply>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <ci> ci </ci>
                  <ci> KCd </ci>
                </apply>
                <cn cellml:units="dimensionless"> 2.0 </cn>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
      
      <apply id="CaTs_calculation">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> CaTs </ci>
        </apply>
        <ci> I_trpn_s </ci>
      </apply>
      
      <apply id="I_trpn_s_calculation">
        <eq/>
        <ci> I_trpn_s </ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> kT_on </ci>
            <ci> cs </ci>
            <apply>
              <minus/>
              <ci> BT </ci>
              <ci> CaTs </ci>
            </apply>
          </apply>
          <apply>
            <times/>
            <ci> kT_off </ci>
            <ci> CaTs </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="beta_cs_calculation">
        <eq/>
        <ci> beta_cs </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <divide/>
              <apply>
                <times/>
                <ci> BSR </ci>
                <ci> KSR </ci>
              </apply>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <ci> cs </ci>
                  <ci> KSR </ci>
                </apply>
                <cn cellml:units="dimensionless"> 2.0 </cn>
              </apply>
            </apply>
            <apply>
              <divide/>
              <apply>
                <times/>
                <ci> BCd </ci>
                <ci> KCd </ci>
              </apply>
              <apply>
                <power/>
                <apply>
                  <plus/>
                  <ci> cs </ci>
                  <ci> KCd </ci>
                </apply>
                <cn cellml:units="dimensionless"> 2.0 </cn>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="Ca_concentration_in_the_submembrane_space">
    <variable units="micromolar" public_interface="out" name="cs"/>
    
    <variable units="microlitre" name="vi" initial_value="0.0001"/>
    <variable units="microlitre" name="vs" initial_value="0.00001"/>
    
    <variable units="millisecond" public_interface="in" name="taus"/>
    <variable units="dimensionless" public_interface="in" name="beta_cs"/>
    <variable units="micromolar" public_interface="in" name="ci"/>
    <variable units="picoA" public_interface="in" name="I_Ca"/>
    <variable units="picoA" public_interface="in" name="I_NaCa"/>
    <variable units="picoA" public_interface="in" name="I_r"/>
    <variable units="picoA" public_interface="in" name="I_trpn_s"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dcs_dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> cs </ci>
        </apply>
        <apply>
          <times/>
          <ci> beta_cs </ci>
          <apply>
            <minus/>
            <apply>
              <times/>
              <apply>
                <divide/>
                <ci> vi </ci>
                <ci> vs </ci>
              </apply>
              <apply>
                <plus/>
                <apply>
                  <minus/>
                  <ci> I_r </ci>
                  <apply>
                    <plus/>
                    <apply>
                      <divide/>
                      <apply>
                        <minus/> 
                        <ci> cs </ci>
                        <ci> ci </ci>
                      </apply>
                      <ci> taus </ci>
                    </apply>
                    <ci> I_Ca </ci>
                  </apply>
                </apply>
                <ci> I_NaCa </ci>
              </apply>
            </apply>
            <ci> I_trpn_s </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="Ca_concentration_in_the_cytosol">
    <variable units="micromolar" public_interface="out" name="ci"/>
    
    <variable units="millisecond" public_interface="in" name="taus"/>
    <variable units="dimensionless" public_interface="in" name="beta_ci"/>
    <variable units="micromolar" public_interface="in" name="cs"/>
    <variable units="picoA" public_interface="in" name="I_up"/>
    <variable units="picoA" public_interface="in" name="I_trpn_i"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dci_dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> ci </ci>
        </apply>
        <apply>
          <times/>
          <ci> beta_ci </ci>
          <apply>
            <minus/>
            <apply>
              <divide/>
              <apply>
                <minus/> 
                <ci> cs </ci>
                <ci> ci </ci>
              </apply>
              <ci> taus </ci>
            </apply>
            <apply>
              <plus/>
              <ci> I_up </ci>
              <ci> I_trpn_i </ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="total_Ca_concentration_in_the_SR">
    <variable units="micromolar" public_interface="out" name="cj"/>
    
    <variable units="picoA" public_interface="in" name="I_up"/>
    <variable units="picoA" public_interface="in" name="I_r"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dcj_dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> cj </ci>
        </apply>
        <apply>
          <minus/>
          <ci> I_up </ci>
          <ci> I_r </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="average_Ca_concentration_in_unrecruited_JSR_compartments">
    <variable units="micromolar" public_interface="out" name="cj_"/>
    
    <variable units="millisecond" name="taua" initial_value="50.0"/>
    
    <variable units="micromolar" public_interface="in" name="cj"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dcj__dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> cj_ </ci>
        </apply>
        <apply>
          <divide/>
          <apply>
            <minus/> 
            <ci> cj </ci>
            <ci> cj_ </ci>
          </apply>
          <ci> taua </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="Ca_flux_out_of_the_SR">
    <variable units="picoA" public_interface="out" name="I_r"/>
   
    <variable units="sparks" name="N"/>
    <variable units="millisecond" name="taur" initial_value="20.0"/>
    <variable units="sparks_per_micromolar" name="g" initial_value="1.5E4"/>
    <variable units="micromolar" name="Q"/>
    <variable units="micromolar" name="s"/>
    <variable units="dimensionless" name="A"/>
    <variable units="first_order_rate_constant" name="u" initial_value="11.3"/>
    
    <variable units="picoA" public_interface="in" name="I_Ca"/>
    <variable units="micromolar" public_interface="in" name="cj_"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="I_r_calculation">
        <eq/>
        <ci> I_r </ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> g </ci>
            <ci> I_Ca </ci>
            <ci> Q </ci>
            <apply>
              <divide/>
              <ci> I_r </ci>
              <ci> taur </ci>
            </apply>
          </apply>
        </apply>
      </apply>
     
      <apply id="N_calculation">
        <eq/>
        <ci> N </ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci> g </ci>
            <ci> I_Ca </ci>
            <ci> A </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="Q_calculation">
        <eq/>
        <ci> Q </ci>
        <piecewise>
          <piece>
            <cn cellml:units="micromolar"> 0.0 </cn>
            <apply>
              <and/>
              <apply>
                <lt/>
                <ci> cj_ </ci> 
                <cn cellml:units="micromolar"> 50.0 </cn>
              </apply>
              <apply>
                <gt/>
                <ci> cj_ </ci> 
                <cn cellml:units="micromolar"> 0.0 </cn>
              </apply>
            </apply>
          </piece>
          <piece>
            <apply>
              <minus/>
              <ci> cj_ </ci>
              <cn cellml:units="micromolar"> 50.0 </cn>
            </apply>
            <apply>
              <and/>
              <apply>
                <lt/>
                <ci> cj_ </ci> 
                <cn cellml:units="micromolar"> 115.0 </cn>
              </apply>
              <apply>
                <geq/>
                <ci> cj_ </ci> 
                <cn cellml:units="micromolar"> 50.0 </cn>
              </apply>
            </apply>
          </piece> 
          <piece>
            <apply>
              <plus/>
              <apply>
                <times/>
                <ci> u </ci>
                <ci> cj_ </ci>
              </apply>
              <ci> s </ci>
            </apply>
            <apply>
              <geq/>
              <ci> cj_ </ci> 
              <cn cellml:units="micromolar"> 115.0 </cn>
            </apply>
          </piece>
        </piecewise>
      </apply>
    </math>
  </component>
  
  <component name="membrane">
    <variable units="millivolt" public_interface="out" name="V"/>
    
    <variable units="millivolt" name="Vmin" initial_value="-80.0"/>
    <variable units="millivolt" name="Vmax" initial_value="30.0"/>
    <variable units="dimensionless" name="m"/>
    <variable units="dimensionless" name="x"/>
    <variable units="dimensionless" name="a" initial_value="0.666667"/>
    <variable units="second" name="T"/>
    
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="V_calculation">
        <eq/>
        <ci> V </ci>
        <piecewise>
          <piece>
            <apply>
              <plus/>
              <ci> Vmin </ci>
              <apply>
                <times/>
                <apply>
                  <minus/>
                  <ci> Vmax </ci>
                  <ci> Vmin </ci>
                </apply>
                <apply>
                  <power/>
                  <apply>
                    <minus/>
                    <cn cellml:units="dimensionless"> 1.0 </cn>
                    <apply>
                      <power/>
                      <apply>
                        <divide/>
                        <apply>
                          <minus/>
                          <ci> time </ci>
                          <apply>
                            <times/>
                            <ci> m </ci>
                            <ci> T </ci>
                          </apply>
                        </apply>
                        <apply>
                          <times/>
                          <ci> m </ci>
                          <ci> T </ci>
                        </apply>
                      </apply>
                      <cn cellml:units="dimensionless"> 2.0 </cn>
                    </apply>
                  </apply>
                  <cn cellml:units="dimensionless"> 0.5 </cn>
                </apply>
              </apply>
            </apply>
            <apply>
              <and/>
              <apply>
                <leq/>
                <ci> time </ci>
                <apply>
                  <plus/>
                  <apply>
                    <times/>
                    <ci> m </ci>
                    <ci> T </ci>
                  </apply>
                  <apply>
                    <times/>
                    <ci> x </ci>
                    <ci> T </ci>
                  </apply>
                </apply>
              </apply>
              <apply>
                <geq/>
                <ci> time </ci>
                <apply>
                  <times/>
                  <ci> m </ci>
                  <ci> T </ci>
                </apply>
              </apply>
            </apply>
          </piece>
          <piece>
            <ci> Vmin </ci>
            <apply>
              <and/>
              <apply>
                <lt/>
                <ci> time </ci>
                <apply>
                  <times/>
                  <apply>
                    <plus/>
                    <ci> m </ci>
                    <cn cellml:units="dimensionless"> 1.0 </cn>
                  </apply>
                  <ci> T </ci>
                </apply>
              </apply>
              <apply>
                <gt/>
                <ci> time </ci>
                <apply>
                  <plus/>
                  <apply>
                    <times/>
                    <ci> m </ci>
                    <ci> T </ci>
                  </apply>
                  <apply>
                    <times/>
                    <ci> x </ci>
                    <ci> T </ci>
                  </apply>
                </apply>
              </apply>
            </apply>
          </piece>
        </piecewise>
      </apply>
      
      <apply id="x_calculation">
        <eq/>
        <ci> x </ci>
        <apply>
          <divide/>
          <ci> a </ci>
          <apply>
            <plus/>
            <ci> a </ci>
            <ci> T </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="intracellular_Na_concentration">
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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  <connection>
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            Model of Intracellular Calcium Cycling in Ventricular Myocytes
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