Location: Bertram, Sherman, 2004 @ 4e6afcf7136c / bertram_sherman_2004.cellml

Author:
pmr2.import <nobody@models.cellml.org>
Date:
2009-06-17 12:33 +1200
Desc:
committing version02 of bertram_sherman_2004
Permanent Source URI:
http://models.cellml.org/workspace/bertram_sherman_2004/rawfile/4e6afcf7136c818d12d3195e5ae936c3be1285c7/bertram_sherman_2004.cellml

<?xml version='1.0' encoding='utf-8'?>
<!--  FILE :  bertram_model_2004.xml

CREATED :  2nd September 2004

LAST MODIFIED : 2nd September 2004

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 Bertram and Sherman's
calcium-based phantom bursting model for pancreatic islets. 

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#" name="bertram_sherman_2004_version01" cmeta:id="bertram_sherman_2004_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>A Calcium-based Phantom Bursting Model for Pancreatic Islets</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 is has consistent units and has been verified as valid CellML by ValidateCellML. It is currently unsuitably constrained and can not be solved.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
Pancreatic beta-cells are located in clusters within the pancreas called the islets of Langerhans.  Beta-cells secrete the hormone insulin in response to elevated blood glucose levels, and in doing so, they play an essential role in glucose homeostasis.  When beta-cells fail to function properly, this can lead to pathologies such as type II diabetes.
</para> 

<para>
Insulin secretion is oscillatory, and it is in-phase with oscillations in the free cytosolic calcium concentration ([Ca<superscript>2+</superscript>]<subscript>i</subscript>), and theses Ca<superscript>2+</superscript> oscillations reflect a bursting pattern in the beta-cell electrical activity.  Electrical bursting consists of periodic active phases of cell firing (excitation) followed by silent phases of hyperpolarisation (rest).  These oscillations can be divided into three categories:
</para>
<itemizedlist>
  <listitem>
            <para>
              <emphasis>Fast bursting</emphasis>, which has a period between 2 and 5 seconds and which often occurs in single cells and in islets where acetylcholine is present;</para>
          </listitem>
  <listitem>
            <para>
              <emphasis>Medium bursting</emphasis>, which has a period of 10 to 60 seconds and which occurs in islets where there is a stimulatory glucose concentration; and</para>
          </listitem>
  <listitem>
            <para>
              <emphasis>Slow bursting</emphasis>, which has a period of 2 to 4 minutes and which occurs in single cells and in islets.</para>
          </listitem>
</itemizedlist>

<para>
The first mathematical models of beta-cells were developed to describe medium bursting, and the first models to address the variability in beta-cell oscillations were developed by Chay in 1995 and 1997 (see <ulink url="${HTML_EXMPL_CHAY_MODEL97}">Extracellular and Intracellular Calcium Effects on Pancreatic Beta Cells, Chay, 1997</ulink> for more details).  In these models the main mechanism for oscillations was variation in the Ca<superscript>2+</superscript> concentration in the ER, which directly or indirectly modulates one or more Ca<superscript>2+</superscript>-dependent channels.  In the Bertram and Sherman model described here the authors analyse in detail how the ER exerts its affects using a phantom bursting model (see <xref linkend="fig_cell_diagram"/> below). 
</para>

<para>
The phantom bursting model is a general paradigm for temporal plasticity in bursting in beta-cells in which bursting is driven by the interaction of two slow variables with disparate time constants (see <ulink url="${HTML_EXMPL_BERTRAM_MODEL}">The Phantom Burster Model for Pancreatic Beta-Cells, 2000</ulink> for more details).  There are three potential slow variables which could drive the phantom bursting <emphasis>in vivo</emphasis>:
</para>
<itemizedlist>
  <listitem>
            <para>cytosolic Ca<superscript>2+</superscript> concentration;</para>
          </listitem>
  <listitem>
            <para>ER Ca<superscript>2+</superscript> concentration;</para>
          </listitem>
  <listitem>
            <para>and the ADP to ATP ratio.</para>
          </listitem>
</itemizedlist>

<para>
The model has been described here in CellML (the raw CellML description of the Bertram and Sherman 2004 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>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WC7-4BS4GC2-1&amp;_user=140507&amp;_coverDate=09%2F30%2F2004&amp;_alid=197872630&amp;_rdoc=1&amp;_fmt=summary&amp;_orig=search&amp;_qd=1&amp;_cdi=6731&amp;_sort=d&amp;_docanchor=&amp;view=c&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=e1fdd19a27b1c7938c1d568e59a560e0">A Calcium-based Phantom Bursting Model for Pancreatic Islets</ulink>, Richard Bertram and Arthur Sherman, 2004, <ulink url="http://www.molbiolcell.org/">
            <emphasis>Bulletin of Mathematical Biology</emphasis>
          </ulink>, 66, 1313-1344.  (<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WC7-4BS4GC2-1&amp;_coverDate=09%2F30%2F2004&amp;_alid=197872630&amp;_rdoc=1&amp;_fmt=&amp;_orig=search&amp;_qd=1&amp;_cdi=6731&amp;_sort=d&amp;view=c&amp;_acct=C000011498&amp;_version=1&amp;_urlVersion=0&amp;_userid=140507&amp;md5=b34962c344ab1a8911383073cd53016f">Full text (HTML)</ulink> and <ulink url="http://www.sciencedirect.com/science?_ob=MImg&amp;_imagekey=B6WC7-4BS4GC2-1-3Y&amp;_cdi=6731&amp;_orig=search&amp;_coverDate=09%2F30%2F2004&amp;_qd=1&amp;_sk=999339994&amp;view=c&amp;wchp=dGLbVzz-zSkWz&amp;_acct=C000011498&amp;_version=1&amp;_userid=140507&amp;md5=4f701b4338556df3136f0c4815596563&amp;ie=f.pdf">PDF</ulink> versions of the article are available to subscribers on the <emphasis>Bulletin of Mathematical Biology</emphasis> website.)  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=15294427&amp;dopt=Abstract">PubMed ID: 15294427</ulink>
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>cell diagram</title>
    </objectinfo>
    <imagedata fileref="bertram_2004.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram of the ionic currents and fluxes across the ER and the cell surface membranes, which are described by the mathematical model.</caption>
</informalfigure>

</sect1>
</article>
</documentation>
  
  
  
  <units name="millisecond">
    <unit units="second" prefix="milli"/>
  </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="picoS">
    <unit units="siemens" prefix="pico"/>
  </units>
  
  <units name="femtoF">
    <unit units="farad" prefix="femto"/>
  </units>
  
  <units name="femtoA">
    <unit units="ampere" prefix="femto"/>
  </units>
  
  <units name="first_order_rate_constant">
    <unit units="millisecond" exponent="-1"/>
  </units>
  
  <units name="micromolar_per_femtoA_millisecond">
    <unit units="micromolar"/>
    <unit units="femtoA" exponent="-1"/>
    <unit units="millisecond" exponent="-1"/>
  </units>
  
  <units name="flux">
    <unit units="micromolar"/>
    <unit units="millisecond" exponent="-1"/>
  </units>
  
  <component name="environment">
    <variable units="millisecond" public_interface="out" name="time"/>
  </component>
  
  <component name="membrane">
    <variable units="millivolt" public_interface="out" name="V"/>
             
    <variable units="femtoF" name="Cm" initial_value="5300.0"/>
  
    <variable units="millisecond" public_interface="in" name="time"/>
    <variable units="femtoA" public_interface="in" name="i_Ca"/>
    <variable units="femtoA" public_interface="in" name="i_K"/>
    <variable units="femtoA" public_interface="in" name="i_K_Ca"/>
    <variable units="femtoA" public_interface="in" name="i_K_ATP"/>
    
    <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_Ca </ci>
              <ci> i_K </ci>
              <ci> i_K_Ca </ci>
              <ci> i_K_ATP </ci>
            </apply>
          </apply>
          <ci> Cm </ci>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_current">
    <variable units="femtoA" public_interface="out" name="i_Ca"/>
    
    <variable units="picoS" name="g_Ca" initial_value="1200.0"/>
    <variable units="millivolt" name="V_Ca" initial_value="25.0"/>
   
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
    <variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
    
    <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>
    </math>
  </component>
  
  <component name="calcium_current_m_gate">
    <variable units="dimensionless" public_interface="out" name="m_infinity"/>
    
    <variable units="millivolt" name="vm" initial_value="-20.0"/>
    <variable units="millivolt" name="sm" initial_value="12.0"/>
  
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> m_infinity </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
              <apply>
                <divide/>
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                  <minus/>
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                <ci> sm </ci>
              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="delayed_rectifier_potassium_current">
    <variable units="femtoA" public_interface="out" name="i_K"/>
    <variable units="millivolt" public_interface="out" name="V_K" initial_value="-75.0"/>
    
    <variable units="picoS" name="g_K" initial_value="3000.0"/>
    
    <variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="i_K_calculation">
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            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="delayed_rectifier_potassium_current_n_gate">
    <variable units="dimensionless" public_interface="out" name="n"/>
    
    <variable units="millisecond" name="tau_n" initial_value="16.0"/>
    <variable units="dimensionless" name="n_infinity"/>
    <variable units="millivolt" name="vn" initial_value="-16.0"/>
    <variable units="millivolt" name="sn" initial_value="5.0"/>
  
    <variable units="millivolt" public_interface="in" name="V"/>
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <apply>
          <diff/>
          <bvar>
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          </bvar>
          <ci> n </ci>
        </apply>
        <apply>
          <divide/>
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            <minus/>
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            <ci> n </ci>
          </apply>
          <ci> tau_n </ci>
        </apply>
      </apply>
      
      <apply id="n_infinity_calculation">
        <eq/>
        <ci> n_infinity </ci>
        <apply>
          <power/>
          <apply>
            <plus/>
            <cn cellml:units="dimensionless"> 1.0 </cn>
            <apply>
              <exp/>
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              </apply>
            </apply>
          </apply>
          <cn cellml:units="dimensionless"> -1.0 </cn>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_dependent_potassium_current">
    <variable units="femtoA" public_interface="out" name="i_K_Ca"/>
    
    <variable units="picoS" name="g_K_Ca" initial_value="300.0"/>
    
    <variable units="millivolt" public_interface="in" name="V_K"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="micromolar" public_interface="in" private_interface="out" name="c"/>
    <variable units="millisecond" public_interface="in" name="time"/>
    
    <variable units="dimensionless" private_interface="in" name="omega"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="i_K_Ca_calculation">
        <eq/>
        <ci> i_K_Ca </ci>
        <apply>
          <times/>
          <ci> g_K_Ca </ci>
          <ci> omega </ci>
          <apply>
            <minus/>
            <ci> V </ci>
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          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="calcium_dependent_potassium_current_omega_gate">
    <variable units="dimensionless" public_interface="out" name="omega"/>
    
    <variable units="micromolar" name="kD" initial_value="0.3"/>
    
    <variable units="micromolar" public_interface="in" name="c"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="omega_eq">
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        <ci> omega </ci>
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          <divide/>
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          </apply>
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            </apply>
            <apply>
              <power/>
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            </apply>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="nucleotide_sensitive_potassium_current">
    <variable units="femtoA" public_interface="out" name="i_K_ATP"/>
    
    <variable units="picoS" name="g_K_ATP" initial_value="500.0"/>
    
    <variable units="millivolt" public_interface="in" name="V_K"/>
    <variable units="millivolt" public_interface="in" name="V"/>
    <variable units="micromolar" public_interface="in" private_interface="out" name="c"/>
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    <variable units="dimensionless" private_interface="in" name="a"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> i_K_ATP </ci>
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          <times/>
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          <apply>
            <minus/>
            <ci> V </ci>
            <ci> V_K </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="nucleotide_sensitive_potassium_current_a_gate">
    <variable units="dimensionless" public_interface="out" name="a"/>
    
    <variable units="millisecond" name="tau_a" initial_value="300000.0"/>
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    <variable units="micromolar" name="sa" initial_value="0.1"/>
    <variable units="micromolar" name="r" initial_value="0.14"/>
  
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    <variable units="dimensionless" name="fcyt" initial_value="0.01"/>
    
    <variable units="flux" public_interface="in" name="Jmem"/>
    <variable units="flux" public_interface="in" name="Jer"/>
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  <component name="ER_calcium_concentration">
    <variable units="micromolar" public_interface="out" name="c_er"/>
    
    <variable units="dimensionless" name="fer" initial_value="0.01"/>
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    <variable units="flux" public_interface="in" name="Jer"/>
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    <math xmlns="http://www.w3.org/1998/Math/MathML">
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    <variable units="flux" public_interface="out" name="Jmem"/>
    
    <variable units="micromolar_per_femtoA_millisecond" name="alpha" initial_value="4.5E-6"/>
    <variable units="first_order_rate_constant" name="kPMCA" initial_value="0.2"/>
    
    <variable units="micromolar" public_interface="in" name="c"/>
    <variable units="femtoA" public_interface="in" name="i_Ca"/>
   
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <eq/>
        <ci> Jmem </ci>
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            <apply>
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    <variable units="first_order_rate_constant" name="kSERCA" initial_value="0.4"/>
    
    <variable units="micromolar" public_interface="in" name="c"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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      </apply>
    </math>
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  <component name="calcium_leak_out_of_the_ER">
    <variable units="flux" public_interface="out" name="Jleak"/>
    
    <variable units="first_order_rate_constant" name="pleak" initial_value="0.0005"/>
    
    <variable units="micromolar" public_interface="in" name="c"/>
    <variable units="micromolar" public_interface="in" name="c_er"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
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        <ci> Jleak </ci>
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            <ci> c </ci>
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  <component name="calcium_efflux_through_the_IP3R">
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    <variable units="first_order_rate_constant" name="O_infinity"/>
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    <variable units="micromolar" name="d_IP3" initial_value="0.5"/>
    <variable units="micromolar" name="d_inact" initial_value="0.4"/>
    <variable units="micromolar" name="IP3"/>
    
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