<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : gall_model1_1999.xml
CREATED : 10th April 2002
LAST MODIFIED : 11th August 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 Gall and Susa's
1999 Na/Ca exchange in models for pancreatic beta-cells.
CHANGES:
18/07/2002 - CML - Added more metadata.
09/04/2003 - AAC - Added publication date information.
11/08/2003 - CML - Corrected equation.
--><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="gall_1999" name="gall_1999">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Na/Ca Exchange in Models for Pancreatic Beta-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 CellML model represents Model I from the published paper. The CellML matches the published equations and the model runs in OpenCell and COR but it does not replicate the published results.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
ABSTRACT: In the presence of an insulinotropic glucose concentration, beta-cells, in intact pancreatic islets, exhibit periodic bursting electrical activity consisting of an alternation of active and silent phases. The fraction of time spent in the active phase over a period is called the plateau fraction and is correlated with the rate of insulin release. However, the mechanisms that regulate the plateau fraction remain unclear. In this paper we investigate the possible role of the plasma membrane Na+/Ca2+ exchange of the beta-cell in controlling the plateau fraction. We have extended different single-cell models to incorporate this Ca2+-activated electrogenic Ca2+ transporter. We find that the Na+/Ca2+ exchange can provide a physiological mechanism to increase the plateau fraction as the glucose concentration is raised. In addition, we show theoretically that the Na+/Ca2+ exchanger is a key regulator of the cytoplasmic calcium concentration in clusters of heterogeneous cells with gap-junctional electrical coupling.
</para>
<para>
The original paper reference is cited below:
</para>
<para>
Effect of Na/Ca Exchange on Plateau Fraction and [Ca<superscript>2+</superscript>]<subscript>i</subscript> in Models for Bursting in Pancreatic Beta-Cells, David Gall and Isabella Susa, 1999,
<emphasis>Biophysical Journal</emphasis>, 77, 45-53. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/10388739">PubMed ID: 10388739</ulink>
</para>
</sect1>
</article>
</documentation>
<units name="millisecond">
<unit units="second" prefix="milli"/>
</units>
<units name="per_millisecond">
<unit units="millisecond" exponent="-1"/>
</units>
<units name="picoA">
<unit units="ampere" prefix="pico"/>
</units>
<units name="femtoF">
<unit units="farad" prefix="femto"/>
</units>
<units name="millimolar">
<unit units="mole" prefix="milli"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="micromolar">
<unit units="mole" prefix="micro"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="millivolt">
<unit units="volt" prefix="milli"/>
</units>
<units name="picoS">
<unit units="siemens" prefix="pico"/>
</units>
<units name="mole_per_microlitre_coulomb">
<unit units="mole"/>
<unit units="litre" prefix="micro" exponent="-1"/>
<unit units="coulomb" 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" initial_value="-76.0"/>
<variable units="femtoF" name="Cm" initial_value="5310.0"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="picoA" public_interface="in" name="i_Ca"/>
<variable units="picoA" public_interface="in" name="i_K"/>
<variable units="picoA" public_interface="in" name="i_K_Ca"/>
<variable units="picoA" public_interface="in" name="i_Na_Ca"/>
<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_K_Ca </ci>
<ci> i_Na_Ca </ci>
</apply>
</apply>
<ci> Cm </ci>
</apply>
</apply>
</math>
</component>
<component name="rapidly_activating_K_current">
<variable units="picoA" 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="2500.0"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<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>
</math>
</component>
<component name="rapidly_activating_K_current_n_gate">
<variable units="dimensionless" public_interface="out" name="n" initial_value="0.1"/>
<variable units="dimensionless" name="n_infinity"/>
<variable units="dimensionless" name="lamda" initial_value="1.6"/>
<variable units="millisecond" name="tau_n"/>
<variable units="millivolt" name="V_n" initial_value="-15.0"/>
<variable units="millivolt" name="S_n" initial_value="5.6"/>
<variable units="millivolt" name="a" initial_value="65.0"/>
<variable units="millivolt" name="b" initial_value="20.0"/>
<variable units="millisecond" name="c" initial_value="6.0"/>
<variable units="millivolt" name="V_" initial_value="-75.0"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="n_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> n </ci>
</apply>
<apply>
<times/>
<ci> lamda </ci>
<apply>
<divide/>
<apply>
<minus/>
<ci> n_infinity </ci>
<ci> n </ci>
</apply>
<ci> tau_n </ci>
</apply>
</apply>
</apply>
<apply id="n_infinity_calculation">
<eq/>
<ci> n_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>
<minus/>
<ci> V_n </ci>
<ci> V </ci>
</apply>
<ci> S_n </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="tau_n_calculation">
<eq/>
<ci> tau_n </ci>
<apply>
<divide/>
<ci> c </ci>
<apply>
<plus/>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V </ci>
<ci> V_ </ci>
</apply>
<ci> a </ci>
</apply>
</apply>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V_ </ci>
<ci> V </ci>
</apply>
<ci> b </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_current">
<variable units="picoA" public_interface="out" name="i_Ca"/>
<variable units="millivolt" name="V_Ca" initial_value="110.0"/>
<variable units="picoS" name="g_Ca" initial_value="1400.0"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" private_interface="in" name="m_infinity"/>
<variable units="dimensionless" private_interface="in" name="h"/>
<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>
<ci> h </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="V_m" initial_value="4.0"/>
<variable units="millivolt" name="S_m" initial_value="14.0"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="m_infinity_calculation">
<eq/>
<ci> m_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>
<minus/>
<ci> V_m </ci>
<ci> V </ci>
</apply>
<ci> S_m </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_current_h_gate">
<variable units="dimensionless" public_interface="out" name="h"/>
<variable units="millivolt" name="V_h" initial_value="-10.0"/>
<variable units="millivolt" name="S_h" initial_value="-10.0"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="h_calculation">
<eq/>
<ci> h </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> V_h </ci>
<ci> V </ci>
</apply>
<ci> S_h </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_activated_K_current">
<variable units="picoA" public_interface="out" name="i_K_Ca"/>
<variable units="picoS" name="g_K_Ca" initial_value="30000.0"/>
<variable units="micromolar" name="K_d" initial_value="100.0"/>
<variable units="millivolt" public_interface="in" name="V_K"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="micromolar" public_interface="in" name="Ca_i"/>
<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>
<apply>
<divide/>
<ci> Ca_i </ci>
<apply>
<plus/>
<ci> K_d </ci>
<ci> Ca_i </ci>
</apply>
</apply>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Na_Ca_exchanger_current">
<variable units="picoA" public_interface="out" name="i_Na_Ca"/>
<variable units="picoS" name="g_Na_Ca" initial_value="234.0"/>
<variable units="micromolar" name="K_1_2" initial_value="1.5"/>
<variable units="millivolt" name="V_Na_Ca"/>
<variable units="millivolt" name="RT_F" initial_value="26.54"/>
<variable units="dimensionless" name="nH" initial_value="5.0"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="micromolar" public_interface="in" name="Ca_i"/>
<variable units="micromolar" public_interface="in" name="Ca_o"/>
<variable units="millimolar" public_interface="in" name="Na_i"/>
<variable units="millimolar" public_interface="in" name="Na_o"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Na_Ca_calculation">
<eq/>
<ci> i_Na_Ca </ci>
<apply>
<times/>
<ci> g_Na_Ca </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> Ca_i </ci>
<ci> nH </ci>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> K_1_2 </ci>
<ci> nH </ci>
</apply>
<apply>
<power/>
<ci> Ca_i </ci>
<ci> nH </ci>
</apply>
</apply>
</apply>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Na_Ca </ci>
</apply>
</apply>
</apply>
<apply id="V_Na_Ca_calculation">
<eq/>
<ci> V_Na_Ca </ci>
<apply>
<times/>
<ci> RT_F </ci>
<apply>
<times/>
<cn cellml:units="dimensionless"> 3.0 </cn>
<apply>
<ln/>
<apply>
<minus/>
<apply>
<divide/>
<ci> Na_o </ci>
<ci> Na_i </ci>
</apply>
<apply>
<ln/>
<apply>
<divide/>
<ci> Ca_o </ci>
<ci> Ca_i </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="ionic_concentrations">
<variable units="micromolar" public_interface="out" name="Ca_i" initial_value="0.52"/>
<variable units="micromolar" public_interface="out" name="Ca_o" initial_value="2600.0"/>
<variable units="millimolar" public_interface="out" name="Na_i" initial_value="10.0"/>
<variable units="millimolar" public_interface="out" name="Na_o" initial_value="140.0"/>
<variable units="dimensionless" name="f" initial_value="0.001"/>
<variable units="per_millisecond" name="k_Ca" initial_value="0.03"/>
<variable units="mole_per_microlitre_coulomb" name="alpha" initial_value="0.0000045055"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="picoA" public_interface="in" name="i_Ca"/>
<variable units="picoA" public_interface="in" name="i_Na_Ca"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_i_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca_i </ci>
</apply>
<apply>
<times/>
<ci> f </ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<minus/>
<ci> alpha </ci>
</apply>
<apply>
<minus/>
<ci> i_Ca </ci>
<apply>
<times/>
<cn cellml:units="dimensionless"> 2.0 </cn>
<ci> i_Na_Ca </ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci> k_Ca </ci>
<ci> Ca_i </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="calcium_current">
<component_ref component="calcium_current_m_gate"/>
<component_ref component="calcium_current_h_gate"/>
</component_ref>
<component_ref component="rapidly_activating_K_current">
<component_ref component="rapidly_activating_K_current_n_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="rapidly_activating_K_current"/>
<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="calcium_activated_K_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Na_Ca_exchanger_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="ionic_concentrations"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="calcium_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_Ca" variable_1="i_Ca"/>
</connection>
<connection>
<map_components component_2="rapidly_activating_K_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_K" variable_1="i_K"/>
</connection>
<connection>
<map_components component_2="calcium_activated_K_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_K_Ca" variable_1="i_K_Ca"/>
</connection>
<connection>
<map_components component_2="Na_Ca_exchanger_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_Na_Ca" variable_1="i_Na_Ca"/>
</connection>
<connection>
<map_components component_2="calcium_activated_K_current" component_1="rapidly_activating_K_current"/>
<map_variables variable_2="V_K" variable_1="V_K"/>
</connection>
<connection>
<map_components component_2="calcium_current" component_1="ionic_concentrations"/>
<map_variables variable_2="i_Ca" variable_1="i_Ca"/>
</connection>
<connection>
<map_components component_2="Na_Ca_exchanger_current" component_1="ionic_concentrations"/>
<map_variables variable_2="i_Na_Ca" variable_1="i_Na_Ca"/>
<map_variables variable_2="Ca_i" variable_1="Ca_i"/>
<map_variables variable_2="Na_i" variable_1="Na_i"/>
<map_variables variable_2="Ca_o" variable_1="Ca_o"/>
<map_variables variable_2="Na_o" variable_1="Na_o"/>
</connection>
<connection>
<map_components component_2="calcium_activated_K_current" component_1="ionic_concentrations"/>
<map_variables variable_2="Ca_i" variable_1="Ca_i"/>
</connection>
<connection>
<map_components component_2="calcium_current_m_gate" component_1="calcium_current"/>
<map_variables variable_2="m_infinity" variable_1="m_infinity"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="calcium_current_h_gate" component_1="calcium_current"/>
<map_variables variable_2="h" variable_1="h"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="rapidly_activating_K_current_n_gate" component_1="rapidly_activating_K_current"/>
<map_variables variable_2="n" variable_1="n"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<rdf:RDF>
<rdf:Bag rdf:about="rdf:#5192e075-a20c-42cf-9a51-0cc968b59ced">
<rdf:li>Pancreatic Beta-Cell</rdf:li>
<rdf:li>calcium dynamics</rdf:li>
<rdf:li>electrophysiology</rdf:li>
</rdf:Bag>
<rdf:Seq rdf:about="rdf:#citationAuthorsSeq">
<rdf:li rdf:resource="rdf:#author1Vcard"/>
<rdf:li rdf:resource="rdf:#author2Vcard"/>
</rdf:Seq>
<rdf:Description rdf:about="rdf:#927afa79-bd81-45e4-919a-ab64e76006a1">
<vCard:FN>Catherine Lloyd</vCard:FN>
</rdf:Description>
<rdf:Description rdf:about="rdf:#21a2c39d-abcc-41b9-a872-145ad5c59c25">
<dcterms:W3CDTF>2002-07-18</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author1Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author1VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#6c77af87-b5ad-4b35-ba4a-db9e62da6608">
<vCard:Given>Autumn</vCard:Given>
<vCard:Family>Cuellar</vCard:Family>
<vCard:Other>A</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#4f02b49e-8563-4e8f-82ac-d8277588dbab">
<bqs:Pubmed_id>10388739</bqs:Pubmed_id>
<bqs:JournalArticle rdf:resource="rdf:#5b63df64-d157-4674-8408-6757b62985e7"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#3fa2328c-1067-4e5a-81bc-9fca1f1c9f3b">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value rdf:resource="rdf:#5192e075-a20c-42cf-9a51-0cc968b59ced"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2VcardN">
<vCard:Given>Isabella</vCard:Given>
<vCard:Family>Susa</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="rdf:#1a977171-ef8f-4b68-a4d9-ec762db9ea1e">
<vCard:N rdf:resource="rdf:#2be9d1b3-ccb6-4a53-ae84-4b3a6d3c9d12"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author2VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#0990d294-6776-415c-9fef-95716c7543b5">
<dcterms:W3CDTF>1999-07-01</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="">
<dc:title>Effect of Na/Ca Exchange on Plateau Fraction and [Ca]i in Models for Bursting in Pancreatic Beta-Cells (Model A)</dc:title>
<dc:publisher>
The University of Auckland, Bioengineering Institute
</dc:publisher>
<cmeta:modification rdf:resource="rdf:#b9893046-8a83-45ab-9b92-42602cacb1b7"/>
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