Location: Li, Rinzel, 1994 @ f6f168138c35 / li_rinzel_1994.cellml

Author:
Hanne <Hanne@hanne-nielsens-macbook.local>
Date:
2009-12-09 12:18:12+13:00
Desc:
Added images in ai and svg format, removed non pub med references
Permanent Source URI:
http://models.cellml.org/workspace/li_rinzel_1994/rawfile/f6f168138c3524c5e8307bb541dfb882083e5ca5/li_rinzel_1994.cellml

<?xml version='1.0' encoding='utf-8'?>
<!--  FILE :  li_model_1994.xml

CREATED :  18.06.2004

LAST MODIFIED : 18th June 2004

AUTHOR :  Alice Boit
          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 Li and Rinzel's 1994 simplification of De Young and Keizer's 1992 mathematical model of the behaviour of oscillatory IP3-mediated Ca2+ release.

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="li_rinzel_1994_version02" name="li_rinzel_1994_version02">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>IP3-Mediated Ca2+ Oscillations - A Simplified Model</title>
  <author>
    <firstname>Catherine</firstname>
          <surname>Lloyd</surname>
    <firstname>Alice</firstname>
          <surname>Boit</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>
Cytosolic calcium ([Ca<superscript>2+</superscript>]<subscript>i</subscript>) shows complex spatio-temporal oscillatory profiles in a large number of cell types.  A number of mathematical models have been published that attempt to model the temporal [Ca<superscript>2+</superscript>]<subscript>i</subscript> oscillations.  Some, such as the Keizer and De Young model (see <ulink url="${HTML_EXMPL_IP3_CA2+_MODEL}"> The De Young-Keizer model, 1992</ulink>) assume that IP<subscript>3</subscript> is necessarily oscillatory.  They constructed a model of the IP<subscript>3</subscript> receptor/channel by assuming that three equivalent and independent subunits are involved in Ca<superscript>2+</superscript> conduction.  Each subunit has three binding sites: one for IP<subscript>3</subscript>, one for Ca<superscript>2+</superscript> activation, and one for Ca<superscript>2+</superscript> inactivation.  Thus, each subunit may exist in eight states with transitions governed by second-order (association) and first-order (dissociation) rate constants (see <xref linkend="fig_pathway_diagram"/> below).  All three subunits must be in the state S<subscript>110</subscript> (one IP3 and one activating Ca<superscript>2+</superscript> bound) for the channel to be open and conducting.
</para> 

<para>
In 1994, Yue-Xian Li and John Rinzel analysed the nine-variable De Young-Keizer model (1992) and reduced it to a two-variable system.  Their reduced system is analogous in form to the classic Hodgkin-Huxley equations (see <ulink url="${HTML_EXMPL_HHSA_INTRO}">The Hodgkin-Huxley Squid Axon Model, 1952</ulink>) for plasma membrane electrical excitability.  [Ca<superscript>2+</superscript>]<subscript>i</subscript> dynamics in this model thus involve endoplasmic reticulum (ER) membrane-associated excitability.  Assuming further that the binging of IP<subscript>3</subscript> does not depend on Ca<superscript>2+</superscript> occupancy at the inactivation site, Li and Rinzel obtain a "minimal" model which still retains the ability to reproduce experimental observations.
</para>

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

<para>
Equations for InsP<subscript>3</subscript> Receptor-mediated [Ca<superscript>2+</superscript>]<subscript>i</subscript> Oscillations Derived from a Detailed Kinetic Model:  A Hodgkin-Huxley Like Formalism, Yue-Xian Li and John Rinzel, 1994, <emphasis>Journal of Theoretical Biology</emphasis>, 166, 461-473.  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=8176949&amp;dopt=Abstract">PubMed ID: 8176949</ulink> 
</para>

<para>
In addition to the original version of the Li and Rinzel mathematical model, the system described in equation nine of the paper has also been implemented in CellML as a separate model (see <xref linkend="sec_download_this_model"/> for the downloads of both models in various formats).  This is a set of three coupled differential equations, namely:
</para>
<itemizedlist>
  <listitem>
            <para>One equation for describing the calcium dynamics;</para>
          </listitem>
  <listitem>
            <para>One equation which represents the gate-shutting variable h; and</para>
          </listitem>
  <listitem>
            <para>One equation which describes the influx of extracellular calcium into the cell cytosol through the plasma membrane</para>
          </listitem>
</itemizedlist> 

<para>
The advantage of reducing the model to this set of three equations is that it more simple, yet it still retains the ability to explain oscillatory behaviours caused by the calcium dynamics.  All the model parameters have been set according to those which are provided in Figure 5(a) of the paper. 
</para>

<para>
The raw CellML descriptions of the Li and Rinzel 1994 model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.
</para>

<informalfigure float="0" id="fig_pathway_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>A schematic diagram of the kinetics of an IP3 receptor/channel subnuit</title>
    </objectinfo>
    <imagedata fileref="li_1994.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram of the kinetics of an IP<subscript>3</subscript> receptor/channel subunit.</caption>
</informalfigure>

</sect1>
</article>
</documentation>
  
  


  <!--#####################################################################
    Here begins the implementation
   #########################################################################
    Below, we define some additional units for association with variables and
    constants within the model.
     -->
  
  <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="flux">
    <unit units="micromolar"/>
    <unit units="second" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="first_order_rate_constant">
    <unit units="second" prefix="milli" exponent="-1"/>
  </units>

  <units name="second_order_rate_constant">
    <unit units="micromolar" exponent="-1"/>
    <unit units="second" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="ms">
    <unit units="second" prefix="milli"/>
  </units>

  <!--
    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="ms" private_interface="out" name="time"/>
  </component>
   
  <component name="C">
    <variable units="micromolar" private_interface="out" name="C" initial_value="0.2"/>
  
  
    <variable units="first_order_rate_constant" name="a" initial_value="1."/>
   
    <variable units="first_order_rate_constant" name="v1" initial_value="40."/>
    <variable units="first_order_rate_constant" name="v2" initial_value="0.02"/>
    <variable units="second_order_rate_constant" name="v3" initial_value="0.6"/>
    <variable units="second_order_rate_constant" name="v4" initial_value="1.8"/>


    <variable units="micromolar" name="d_ip3" initial_value="0.2"/>
    <variable units="micromolar" name="d_act" initial_value="0.4"/>
    <variable units="micromolar" name="d_inh" initial_value="0.4"/>
    <variable units="micromolar" name="k_er" initial_value="0.18"/>
    <variable units="micromolar" name="k_pl" initial_value="0.1"/>
    <variable units="micromolar" name="c0" initial_value="2.0"/>
    <variable units="micromolar" name="IP3" initial_value="0.6"/>

    <variable units="flux" name="j_in" initial_value="1.2"/>
    <variable units="flux" name="factor1" initial_value="0.0"/>
    <variable units="flux" name="factor2" initial_value="0.0"/>
    <variable units="flux" name="v3_summand" initial_value="0.0"/>
    <variable units="flux" name="v4_summand" initial_value="0.0"/>

    <variable units="dimensionless" name="epsilon" initial_value="0.01"/>
    <variable units="dimensionless" name="c1" initial_value="0.185"/>

    <variable units="ms" public_interface="in" name="time"/>
    
    <variable units="dimensionless" private_interface="out" name="h" initial_value="0.8"/>


    
  <!-- Just dummy ODEs to make access to the variables easier in CMISS wrapping -->


    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply id="v2_diff_eq">
        <eq/> 
       <apply>
          <diff/>
          <bvar>
            <ci>time</ci>
          </bvar>
          <ci> v2 </ci>
        </apply>
        <cn cellml:units="flux">0.0</cn>
      </apply>
    </math>
 
   <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply id="v3_diff_eq">
        <eq/> 
       <apply>
          <diff/>
          <bvar>
            <ci>time</ci>
          </bvar>
          <ci> v3 </ci>
        </apply>
        <cn cellml:units="flux">0.0</cn>
      </apply>
    </math>

   <math xmlns="http://www.w3.org/1998/Math/MathML"> 
      <apply id="j_in_diff_eq">
        <eq/> 
       <apply>
          <diff/>
          <bvar>
            <ci>time</ci>
          </bvar>
          <ci> j_in </ci>
        </apply>
        <cn cellml:units="flux">0.0</cn>
      </apply>
    </math>

   <!-- end of dummy ODEs -->  
 
   <!-- c0 differential equation -->  
     
   <math xmlns="http://www.w3.org/1998/Math/MathML"> 
    <apply id="c0_diff_eq">
        <eq/>
     <apply>
          <diff/>
      <bvar>
            <ci> time </ci>
          </bvar>
       <ci> c0 </ci>
      </apply>

      <apply>
          <times/>
       <ci> epsilon </ci>
       <cn cellml:units="dimensionless">0.001</cn>
          <!--apply sec2msec factor-->
      <apply>
            <minus/>
       <ci> j_in </ci>
      <apply>
              <divide/>
       <apply>
                <times/>
        <ci> v4 </ci>
        <apply>
                  <power/>
         <ci> C </ci>
         <cn cellml:units="dimensionless">2.0</cn> 
        </apply>
       </apply>
              <!--end times nominator-->
       <apply>
                <plus/>
                <!--start denominator-->
        <apply>
                  <power/>
         <ci> k_pl </ci>
         <cn cellml:units="dimensionless">2.0</cn> 
        </apply>
        <apply>
                  <power/>
         <ci> C </ci>
         <cn cellml:units="dimensionless">2.0</cn> 
        </apply>
       </apply>
              <!--end plus-->
      </apply>
            <!--end divide-->
     </apply>
          <!--end minus-->
    </apply>
        <!--end times epsilon-->
   </apply>
  </math>

 <!-- differential equation for gating variable h (shutting)-->  

    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
     <apply id="h_diff_eq">
        <eq/>
      <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> h </ci>
      </apply>
     <apply>
          <times/>
      <cn cellml:units="dimensionless">0.001</cn> 
      <apply>
            <minus/>
       <apply>
              <times/> 
        <ci> a </ci>
        <ci> d_inh </ci>
       </apply>
       <apply>
              <times/>
        <ci> h </ci>
        <ci> a </ci>
        <apply>
                <plus/>
         <ci> C </ci>
         <ci> d_inh </ci>
        </apply>
       </apply>
      </apply>
     </apply>    
    </apply>
   </math>

  <!-- calculation of factor 1 for the [Ca] differential equation-->  

    <math xmlns="http://www.w3.org/1998/Math/MathML"> 
     <apply id="factor1_calc">
        <eq/> 
      <ci> factor1 </ci>
     <apply>
          <times/>
      <cn cellml:units="dimensionless">0.001</cn> 
      <apply>
            <plus/>
       <apply>
              <times/>
        <ci> v1 </ci>
        <apply>
                <power/>
                <!--begin power IP3-->
         <apply>
                  <divide/>
          <ci> IP3 </ci>
          <apply>
                    <plus/>
           <ci> IP3 </ci>
	   <ci> d_ip3 </ci>
          </apply>
                  <!--end plus-->
         </apply>
                <!--end divide in power IP3-->
         <cn cellml:units="dimensionless">3.0</cn> 
        </apply>
              <!--end power IP3-->
       <apply>
                <power/>
                <!--begin power Ca-->
        <apply>
                  <divide/>
         <ci> C </ci>
         <apply>
                    <plus/>
          <ci> C </ci>
	  <ci> d_act </ci>
         </apply>
                  <!--end plus-->
        </apply>
                <!--end divide in power Ca-->
        <cn cellml:units="dimensionless">3.0</cn> 
       </apply>
              <!-- end power Ca -->
        <apply>
                <power/>
         <ci> h </ci>
	 <cn cellml:units="dimensionless">3.0</cn> 
        </apply>
              <!--end power h^3-->
       </apply>
            <!--end times v1-->
       <ci> v2 </ci>
      </apply>
          <!--end plus v2-->
     </apply>
        <!--end times sec2msec factor -->
    </apply>
   </math>

  <!-- calculation of factor 2 for the [Ca] differential equation--> 

  <math xmlns="http://www.w3.org/1998/Math/MathML">     
   <apply id="factor2_calc">
        <eq/> 
    <ci> factor2 </ci>
    <apply>
          <minus/>
     <ci> c0 </ci>
     <apply>
            <times/>
      <ci> C </ci>
      <apply>
              <plus/>   
       <cn cellml:units="dimensionless">1.0</cn> 
       <ci> c1 </ci>
      </apply>
     </apply>
    </apply>
   </apply>
  </math>

  <!-- calculation of the v3-related summand for the [Ca] differential equation--> 

  <math xmlns="http://www.w3.org/1998/Math/MathML"> 
   <apply id="v3_summand_calc">
        <eq/> 
    <ci> v3_summand </ci>
    <apply>
          <divide/>
     <apply>
            <times/>
      <ci> v3 </ci>
      <cn cellml:units="dimensionless">0.001</cn> 
      <apply>
              <power/>
       <ci> C </ci>
       <cn cellml:units="dimensionless">2.0</cn> 
      </apply>
     </apply>
     <apply>
            <plus/>
      <apply>
              <power/>
       <ci> k_er </ci>
       <cn cellml:units="dimensionless">2.0</cn> 
      </apply>
      <apply>
              <power/>
       <ci> C </ci>
       <cn cellml:units="dimensionless">2.0</cn> 
      </apply>
     </apply>
    </apply>
   </apply>
  </math>


  <!-- calculation of the v4-related summand for the [Ca] differential equation--> 

   <math xmlns="http://www.w3.org/1998/Math/MathML"> 
    <apply id="v4_summand_calc">
        <eq/> 
     <ci> v4_summand </ci>
     <apply>
          <times/>
      <cn cellml:units="dimensionless">0.001</cn> 	
      <ci> epsilon </ci>
      <apply>
            <minus/>
       <ci> j_in </ci>
       <apply>
              <divide/>
        <apply>
                <times/>
        <ci> v4 </ci>
                <!--start nom-->
         <apply>
                  <power/>
          <ci> C </ci>
          <cn cellml:units="dimensionless">2.0</cn> 
         </apply>
        </apply>
              <!--end times zaehler-->
        <apply>
                <plus/>
                <!--start denom-->
         <apply>
                  <power/>
          <ci> k_pl </ci>
          <cn cellml:units="dimensionless">2.0</cn> 
         </apply>
         <apply>
                  <power/>
          <ci> C </ci>
          <cn cellml:units="dimensionless">2.0</cn> 
         </apply>
        </apply>
              <!--end denom-->
       </apply>
            <!--end divide-->
      </apply>
          <!--end minus-->
     </apply>
        <!--end times epsilon-->
    </apply>
   </math>

  <!-- [Ca] differential equation--> 

   <math xmlns="http://www.w3.org/1998/Math/MathML"> 
    <apply id="C_diff_eq">
        <eq/> 
     <apply>
          <diff/>
      <bvar>
            <ci>time</ci>
          </bvar>
       <ci> C </ci>
     </apply>
     <apply>
          <plus/>  
      <apply>
            <minus/> 
       <apply>
              <times/>
        <ci> factor1 </ci>
        <ci> factor2 </ci>
       </apply>
       <ci> v3_summand </ci>
      </apply>
          <!--end minus v3-related summand -->
       <ci> v4_summand </ci>     
     </apply>
        <!--end plus v4-related summand -->
    </apply>
      <!--end quality-->
   </math>

  </component>
  
  
  
  <group>
    <relationship_ref relationship="encapsulation"/>
    <component_ref component="environment">
      <component_ref component="C"/>
    </component_ref>
  </group>
  
  <connection>
    <map_components component_2="environment" component_1="C"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  





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    <rdf:li>calcium dynamics</rdf:li>
    <rdf:li>IP3</rdf:li>
    <rdf:li>electrophysiology</rdf:li>
    <rdf:li>ip3r</rdf:li>
    <rdf:li>hodgkin-huxley</rdf:li>
  </rdf:Bag>
  <rdf:Seq rdf:about="rdf:#citationAuthorsSeq">
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    <vCard:Given>John</vCard:Given>
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    <dc:title>Equations for InsP3 Receptor-mediated [Ca2+]i Oscillations Derived from a Detailed Kinetic Model: A Hodgkin-Huxley Like Formalism</dc:title>
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    <dc:title>
        The Li and Rinzel 1994 simplification of the De Young-Keizer 1992 model 
        of oscillatory calcium release through the IP3 stimulated channel
      </dc:title>
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    <dc:title>Journal of theoretical Biology</dc:title>
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