Location: Mosekilde, Lading, Yanchuk, Maistrenko, 2001 @ aa0e676c403f / mosekilde_lading_yanchuk_maistrenko_2001.cellml

Author:

pmr2.import <nobody@models.cellml.org>

Date:

2009-06-17 15:12:26+12:00

Desc:

committing version01 of mosekilde_lading_yanchuk_maistrenko_2001

Permanent Source URI:

http://models.cellml.org/workspace/mosekilde_lading_yanchuk_maistrenko_2001/rawfile/aa0e676c403f6edc747b9a481fcfbb039baab391/mosekilde_lading_yanchuk_maistrenko_2001.cellml

<?xml version='1.0' encoding='utf-8'?>
<!--  FILE :  mosekilde_model_2001.xml

CREATED :  14th January 2003

LAST MODIFIED : 9th April 2003

AUTHOR :  Catherine Lloyd
          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 Mosekilde et al's
2001 mathematical model of bursting pancreatic beta cells.

CHANGES:  
  09/04/2003 - AAC - Added publication date information.  
    
--><model xmlns="http://www.cellml.org/cellml/1.0#" xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="http://www.cellml.org/cellml/1.0#" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" cmeta:id="mosekilde_lading_yanchuk_maistrenko_2001_version01" name="mosekilde_lading_yanchuk_maistrenko_2001_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>Bifurcation Structure of a Model of Bursting Pancreatic Cells</title>
  <author>
    <firstname>Catherine</firstname>
          <surname>Lloyd</surname>
    <affiliation>
      <shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
    </affiliation>
  </author>
</articleinfo>
  <section id="sec_status">
    <title>Model Status</title>
    <para>
            This model contains partial differentials and as such can not currently be solved by existing CellML tools.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
Through the continuous activity of ion pumps in the plasma membrane, relatively steep transmembrane electrochemical gradients are maintained.  Because the transmembrane distributions of many ions such as Na<superscript>+</superscript>, K<superscript>+</superscript> and Ca<superscript>2+</superscript> are far from equilibrium, cells are inclined towards excitable electrical activity, or the membrane potential, determined by relative charge distribution, exhibits complicated patterns of slow and fast oscillations associated with variations in the transmembrane ionic currents.  These dynamics play an essential role in intracellular function and intercellular communication.  
</para>

<para>
When exposed to a threshold concentration of glucose, pancreatic beta-cells exhibit a complicated pattern of electrical activity.  Bursts of action potential spikes (the "active" phase) are observed, separated by a "silent" phase of membrane repolarisation.  Insulin secretion is dependent on the relative amount of time the cells spend in the active state, and this time increases with the concentration of extracellular glucose.  Bursting dynamics control the influx of Ca<superscript>2+</superscript> into the cell, and in turn, Ca<superscript>2+</superscript> triggers the release of insulin.  So, via this mechanism, the bursting dynamics of the pancreatic beta-cells regulate their response to varying glucose concentrations.  Below a threshold glucose concentration (about 5mM), the cells do not burst.  Above a certain concentration (&gt; 22 mM), the cells display continuous spiking, and insulin secretion becomes saturated. 
</para>

<para>
Mathematical models of pancreatic beta-cells are usually based on the Hodgkin-Huxley formalism (for the original model see <ulink url="${HTML_EXMPL_HHSA_INTRO}">The Hodgkin-Huxley Squid Axon Model, 1952</ulink>), with elaborations to account for Ca<superscript>2+</superscript> storage, glucose metabolism, and regulation by ATP and various hormones.  Several models have been published (see <ulink url="${HTML_REPOSITORY_INTRODUCTION}">the CellML repository</ulink> for more pancreatic beta-cell and calcium oscillation models), but although some have been established for some years, their bifurcation structure is so complicated it is not yet fully understood.  The goal of the 2001 Mosekilde <emphasis>et al.</emphasis> paper analysed here was to give an accurate, but simplified account of the bifurcation structure of the individual beta-cell (see <xref linkend="fig_cell_diagram"/> below).
</para>

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

<para>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6T2K-4447MJD-J&amp;_user=140507&amp;_coverDate=12%2F31%2F2001&amp;_alid=70440630&amp;_rdoc=1&amp;_fmt=summary&amp;_orig=search&amp;_qd=1&amp;_cdi=4921&amp;_sort=d&amp;_docanchor=&amp;wchp=dGLbVlz-lSzBk&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=dc19bf053e438dae9918fa3c279f2f7e">Bifurcation structure of a model of bursting pancreatic cells</ulink>, Erik Mosekilde, Brian Lading, Sergiy Yanchuk and Yuri Maistrenko, 2001, <ulink url="http://www.sciencedirect.com/science?_ob=JournalURL&amp;_cdi=4921&amp;_auth=y&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=511bbadc6ebfd16e10115a252026e228">
            <emphasis>Biosystems</emphasis>
          </ulink>, 63, 3-13.  (PDF and <ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6T2K-4447MJD-J&amp;_coverDate=12%2F31%2F2001&amp;_alid=70440630&amp;_rdoc=1&amp;_fmt=&amp;_orig=search&amp;_qd=1&amp;_cdi=4921&amp;_sort=d&amp;wchp=dGLbVlz-lSzBk&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=63f03834b470d0be17a65ca137c26f84">full text</ulink> versions of the article are available to subscribers on the journal website.)  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=11595325&amp;dopt=Abstract">PubMed ID: 11595325</ulink>
</para>

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

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>diagram of the model</title>
    </objectinfo>
    <imagedata fileref="mosekilde_2001.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic representation of the ionic currents described by the Mosekilde <emphasis>et al.</emphasis> 2001 pancreatic beta-cell model.  The model includes a calcium current, I <subscript>Ca<superscript>2+</superscript>
            </subscript>; a potassium current, I<subscript>K</subscript>; and a slow current, I<subscript>S</subscript>.</caption>
</informalfigure>

</sect1>
</article>
</documentation>
  
  
  <!--
    Below, we define some additional units for association with variables and
    constants within the model. 
  -->
  
  <units name="per_second">
    <unit units="second" exponent="-1"/>
  </units>
  
  <units name="millivolt">
    <unit units="volt" prefix="milli"/>
  </units>
  
  <units name="micromolar">
    <unit units="mole" prefix="micro"/>
    <unit units="litre" exponent="-1"/>
  </units>
  
  <units name="microS_per_cm2">
    <unit units="siemens" prefix="micro"/>
    <unit units="metre" prefix="centi" exponent="-2"/>
  </units>
  
  <units name="nanoA_per_cm2">
    <unit units="ampere" prefix="nano"/>
    <unit units="metre" prefix="centi" exponent="-2"/>
  </units>
  
  <component name="environment">
    <variable units="second" public_interface="out" name="time"/>
  </component>
  
  <component name="membrane">
    <variable units="millivolt" public_interface="out" name="V"/>
    <variable units="second" public_interface="out" name="tau" initial_value="0.02"/>
             
    <variable units="second" public_interface="in" name="time"/>
    <variable units="nanoA_per_cm2" public_interface="in" name="i_K"/>
    <variable units="nanoA_per_cm2" public_interface="in" name="i_Ca"/>
    <variable units="nanoA_per_cm2" public_interface="in" name="i_S"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="membrane_voltage_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> V </ci>
        </apply>
        <apply>
          <divide/>
          <apply>
            <minus/>
            <apply>
              <plus/>
              <ci> i_K </ci>
              <ci> i_Ca </ci>
              <ci> i_S </ci>
            </apply>
          </apply>
          <ci> tau </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_current">
    <variable units="nanoA_per_cm2" public_interface="out" name="i_Ca"/>
    <variable units="dimensionless" public_interface="out" name="J11"/>
   
    <variable units="millivolt" name="V_Ca" initial_value="25.0"/>
    <variable units="microS_per_cm2" name="g_Ca" initial_value="3.6"/>
    
    <variable units="second" public_interface="in" name="time"/>
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
    <variable units="millivolt" public_interface="in" name="V_K"/>
    <variable units="dimensionless" private_interface="in" name="m_infinity"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="i_Ca_calculation">
        <eq/>
        <ci> i_Ca </ci>
        <apply>
          <times/>
          <ci> g_Ca </ci>
          <ci> m_infinity </ci>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_Ca </ci>
          </apply>
        </apply>
      </apply>

      <apply id="J11_calculation">
        <eq/>
        <ci> J11 </ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <apply>
              <minus/>
              <ci> g_Ca </ci>
            </apply>
            <apply>
              <partialdiff/>
              <bvar>
                <ci> V </ci>
              </bvar>
              <ci> m_infinity </ci>
            </apply>
            <apply>
              <minus/>
              <ci> V </ci>
              <ci> V_Ca </ci>
            </apply>
          </apply>
          <apply>
            <times/>
            <ci> g_Ca </ci>
            <ci> m_infinity </ci>
            <apply>
              <divide/>
              <apply>
                <minus/>
                <ci> V_Ca </ci>
                <ci> V_K </ci>
              </apply>
              <apply>
                <minus/>
                <ci> V </ci>
                <ci> V_K </ci>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_current_m_gate">
    <variable units="dimensionless" public_interface="out" name="m_infinity"/>
    
    <variable units="millivolt" name="theta_m" initial_value="12.0"/>
    <variable units="millivolt" name="V_m" initial_value="-20.0"/>
    
    <variable units="millivolt" public_interface="in" name="V"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="m_infinity_calculation">
        <eq/>
        <ci> m_infinity </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
              <apply>
                <divide/>
                <apply>
                  <minus/>
                  <ci> V_m </ci>
                  <ci> V </ci>
                </apply>
                <ci> theta_m </ci>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="potassium_current">
    <variable units="nanoA_per_cm2" public_interface="out" name="i_K"/>
    <variable units="millivolt" public_interface="out" name="V_K" initial_value="-75.0"/>
    <variable units="dimensionless" public_interface="out" name="J12"/>
    <variable units="dimensionless" public_interface="out" private_interface="in" name="J21"/>
    <variable units="dimensionless" public_interface="out" private_interface="in" name="J22"/>
    
    <variable units="microS_per_cm2" name="g_K" initial_value="10.0"/>
    
    <variable units="second" public_interface="in" private_interface="out" name="time"/>
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
    <variable units="second" public_interface="in" private_interface="out" name="tau"/>
    
    <variable units="dimensionless" private_interface="in" name="n"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="i_K_calculation">
        <eq/>
        <ci> i_K </ci>
        <apply>
          <times/>
          <ci> g_K </ci>
          <ci> n </ci>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="J12_calculation">
        <eq/>
        <ci> J12 </ci>
        <apply>
          <times/>
          <apply>
            <minus/>
            <ci> g_K </ci>
          </apply>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="potassium_current_n_gate">
    <variable units="dimensionless" public_interface="out" name="n"/>
    <variable units="dimensionless" public_interface="out" name="J21"/>
    <variable units="dimensionless" public_interface="out" name="J22"/>
    
    <variable units="dimensionless" name="n_infinity"/>
    <variable units="millivolt" name="theta_n" initial_value="5.6"/>
    <variable units="dimensionless" name="delta" initial_value="0.85"/>
    <variable units="millivolt" name="V_n" initial_value="-16.0"/>
    
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="second" public_interface="in" name="tau"/>
    <variable units="second" public_interface="in" name="time"/>
   
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="n_infinity_calculation">
        <eq/>
        <ci> n_infinity </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
              <apply>
                <divide/>
                <apply>
                  <minus/>
                  <ci> V_n </ci>
                  <ci> V </ci>
                </apply>
                <ci> theta_n </ci>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
      
      <apply id="n_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> n </ci>
        </apply>
        <apply>
          <divide/>
          <apply>
            <times/>
            <ci> delta </ci>
            <apply>
              <minus/>
              <ci> n_infinity </ci>
              <ci> n </ci>
            </apply>
          </apply>
          <ci> tau </ci>
        </apply>
      </apply>
      
      <apply id="J21_calculation">
        <eq/>
        <ci> J21 </ci>
        <apply>
          <times/>
          <ci> delta </ci>
          <apply>
            <partialdiff/>
            <bvar>
              <ci> V </ci>
            </bvar>
            <ci> n_infinity </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="J22_calculation">
        <eq/>
        <ci> J22 </ci>
        <apply>
          <minus/>
          <ci> delta </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="slow_current">
    <variable units="nanoA_per_cm2" public_interface="out" name="i_S"/> 
    <variable units="second" private_interface="out" name="tau_S" initial_value="35.0"/>
    <variable units="dimensionless" public_interface="out" name="J13"/>
    <variable units="dimensionless" public_interface="out" name="J33"/>
    <variable units="dimensionless" public_interface="out" private_interface="in" name="J31"/>
    
    <variable units="microS_per_cm2" name="g_S" initial_value="4.0"/>
    <variable units="dimensionless" name="k_S"/>
    
    <variable units="second" public_interface="in" private_interface="out" name="time"/>
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
    <variable units="millivolt" public_interface="in" name="V_K"/>
    <variable units="dimensionless" private_interface="in" name="S"/>
    <variable units="second" public_interface="in" name="tau"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="i_S_calculation">
        <eq/>
        <ci> i_S </ci>
        <apply>
          <times/>
          <ci> g_S </ci>
          <ci> S </ci>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
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        </apply>
      </apply>
      
      <apply id="J13_calculation">
        <eq/>
        <ci> J13 </ci>
        <apply>
          <times/>
          <apply>
            <minus/>
            <ci> g_S </ci>
          </apply>
          <ci> k_S </ci>
          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="k_S_calculation">
        <eq/>
        <ci> k_S </ci>
        <apply>
          <divide/>
          <ci> tau </ci>
          <ci> tau_S </ci>
        </apply>
      </apply>
      
      <apply id="J33_calculation">
        <eq/>
        <ci> J33 </ci>
        <apply>
          <minus/>
          <ci> k_S </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="slow_current_S_gate">
    <variable units="dimensionless" public_interface="out" name="S"/>
    <variable units="dimensionless" public_interface="out" name="J31"/>
    
    <variable units="dimensionless" name="S_infinity"/>
    <variable units="millivolt" name="theta_S" initial_value="10.0"/>
    <variable units="millivolt" name="V_S" initial_value="-38.34"/>
    
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="second" public_interface="in" name="tau_S"/>
    <variable units="second" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="S_infinity_calculation">
        <eq/>
        <ci> S_infinity </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
              <apply>
                <divide/>
                <apply>
                  <minus/>
                  <ci> V_S </ci>
                  <ci> V </ci>
                </apply>
                <ci> theta_S </ci>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
      
      <apply id="S_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> S </ci>
        </apply>
        <apply>
          <divide/>
          <apply>
            <minus/>
            <ci> S_infinity </ci>
            <ci> S </ci>
          </apply>
          <ci> tau_S </ci>
        </apply>
      </apply>
      
      <apply id="J31_calculation">
        <eq/>
        <ci> J31 </ci>
        <apply>
          <partialdiff/>
          <bvar>
            <ci> V </ci>
          </bvar>
          <ci> S_infinity </ci>
        </apply>
      </apply>
    </math>
  </component> 
  
  <component name="jacobian_matrix">
    <variable units="dimensionless" name="J"/>
    
    <variable units="dimensionless" public_interface="in" name="J11"/>
    <variable units="dimensionless" public_interface="in" name="J12"/>
    <variable units="dimensionless" public_interface="in" name="J13"/>
    <variable units="dimensionless" public_interface="in" name="J21"/>
    <variable units="dimensionless" public_interface="in" name="J22"/>
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    <variable units="dimensionless" public_interface="in" name="J33"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_calculation">
        <eq/>
        <ci> J </ci>
        <matrix>
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            <ci> J11 </ci>
            <ci> J12 </ci>
            <ci> J13 </ci>
          </matrixrow>
          <matrixrow>
            <ci> J21 </ci>
            <ci> J22 </ci>
            <cn cellml:units="dimensionless"> 0 </cn>
          </matrixrow>
          <matrixrow>
            <ci> J31 </ci>
            <cn cellml:units="dimensionless"> 0 </cn>
            <ci> J33 </ci> 
          </matrixrow>
        </matrix>
      </apply>
    </math>
  </component>
  
  <group>
    <relationship_ref relationship="containment"/>
    <component_ref component="membrane">
      <component_ref component="calcium_current">
        <component_ref component="calcium_current_m_gate"/>
      </component_ref>
      <component_ref component="potassium_current">
        <component_ref component="potassium_current_n_gate"/>
      </component_ref>
      <component_ref component="slow_current">
        <component_ref component="slow_current_S_gate"/>
      </component_ref>
      <component_ref component="jacobian_matrix"/>
    </component_ref>
  </group>
  
  <group>
    <relationship_ref relationship="encapsulation"/> 
    <component_ref component="calcium_current">
      <component_ref component="calcium_current_m_gate"/>
    </component_ref>
    <component_ref component="potassium_current">
      <component_ref component="potassium_current_n_gate"/>
    </component_ref>
    <component_ref component="slow_current">
      <component_ref component="slow_current_S_gate"/>
    </component_ref>
  </group>
      
  <connection>
    <map_components component_2="environment" component_1="membrane"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="calcium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="potassium_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="slow_current"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="calcium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_Ca" variable_1="i_Ca"/>
  </connection>
  
  <connection>
    <map_components component_2="membrane" component_1="potassium_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_K" variable_1="i_K"/>
    <map_variables variable_2="tau" variable_1="tau"/>
  </connection>  
  
  <connection>
    <map_components component_2="membrane" component_1="slow_current"/>
    <map_variables variable_2="V" variable_1="V"/>
    <map_variables variable_2="i_S" variable_1="i_S"/>
    <map_variables variable_2="tau" variable_1="tau"/>
  </connection>
  
  <connection>
    <map_components component_2="jacobian_matrix" component_1="calcium_current"/>
    <map_variables variable_2="J11" variable_1="J11"/>
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  <connection>
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