- Author:
- pmr2.import <nobody@models.cellml.org>
- Date:
- 2009-06-17 14:14:54+12:00
- Desc:
- committing version01 of gall_susa_1999
- Permanent Source URI:
- https://models.cellml.org/workspace/gall_susa_1999/rawfile/8885c174f2a64cd670d66848368d336304a1a3f5/gall_susa_1999_b.cellml
<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : gall_model2_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 equations.
--><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_susa_1999_version02" name="gall_susa_1999_version02">
<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 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>
Pancreatic beta-cells have been the subject of both experimental and theoretical studies for several decades. One reason for this interest has been the essential role beta-cells play in glucose homeostasis - they are the only source of insulin that most cells require in order to take up and metabolise glucose, and impairment of beta-cell function contributes to diabetes. A major focus of theoretical work has been beta-cell dynamics, especially in the form of bursting electrical activity. The bursts consist of active phases of Ca<superscript>2+</superscript>-carrying action potentials alternating with silent phases of repolarisation and are accompanied by oscillations in cytosolic Ca<superscript>2+</superscript>, which drive pulses of insulin secretion. Hence [Ca<superscript>2+</superscript>]<subscript>i</subscript> regulation is of great importance due to its role as a secondary messenger in the stimulus-secretion coupling. Several transport mechanisms at the level of the plasma membrane are involved in this regulation, including L-type Ca<superscript>2+</superscript> channels, Ca<superscript>2+</superscript>-ATPases and the Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange. This antiporter couples the energy released by the influx of Na<superscript>+</superscript> ions down their electrochemical to the efflux of Ca<superscript>2+</superscript> from the cytosol. In pancreatic beta-cells the physiological role of the Na<superscript>+</superscript>/Ca<superscript>2+</superscript> remains unclear, but data suggest that it can regulate the burst duration.
</para>
<para>
In their 1999 study, David Gall and Isabella Susa use mathematical modelling to explore the impact of the activity of this transporter, both on the [Ca<superscript>2+</superscript>]<subscript>i</subscript> regulation and on the periodic bursting electrical activity. Specifically they tested the hypothesis that the Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange is able to modulate the plateau fraction, and hence the amount of insulin secreted. Gall and Susa tested this hypothesis by including I<subscript>Na/Ca</subscript> in three single cell models differing in the identity of the slow variable controlling the bursting electrical activity (see <xref linkend="fig_cell_diagram"/> below). In addition, they examined a possible link between Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange activity and glucose metabolism, finding that Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange can provide a physiological mechanism to increase the plateau fraction as the glucose concentration is raised. The mathematical models have been translated into CellML descriptions which 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.biophysj.org/cgi/content/abstract/77/1/45">Effect of Na/Ca Exchange on Plateau Fraction and [Ca<superscript>2+</superscript>]<subscript>i</subscript> in Models for Bursting in Pancreatic Beta-Cells</ulink>, David Gall and Isabella Susa, 1999, <ulink url="http://www.biophysj.org/">
<emphasis>Biophysical Journal</emphasis>
</ulink>, 77, 45-53. (<ulink url="http://www.biophysj.org/cgi/content/full/77/1/45">Full text</ulink> and <ulink url="http://www.biophysj.org/cgi/reprint/77/1/45.pdf">PDF</ulink> versions of the article are available for Journal Members on the Biophysical Journal website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=10388739&dopt=Abstract">PubMed ID: 10388739</ulink>
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell schematic for the model</title>
</objectinfo>
<imagedata fileref="cell_diagram.gif"/>
</imageobject>
</mediaobject>
<caption>Schematic diagram of the pancreatic beta-cell plasma membrane showing the ionic currents captured by the three single cell mathematical models.</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<!--
Below, we define some additional units for association with variables and
constants within the model. The identifiers are fairly self-explanatory.
-->
<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>
<!--
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="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="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_slow"/>
<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_slow </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="2700.0"/>
<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"/>
<variable units="dimensionless" name="n_infinity"/>
<variable units="dimensionless" name="lamda" initial_value="1.0"/>
<variable units="millisecond" name="tau_n" initial_value="20.0"/>
<variable units="millivolt" name="V_n" initial_value="-16.0"/>
<variable units="millivolt" name="S_n" initial_value="5.6"/>
<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>
</math>
</component>
<component name="calcium_current">
<variable units="picoA" public_interface="out" name="i_Ca"/>
<variable units="millivolt" name="V_Ca" initial_value="25.0"/>
<variable units="picoS" name="g_Ca" initial_value="1000.0"/>
<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"/>
<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="V_m" initial_value="-20.0"/>
<variable units="millivolt" name="S_m" initial_value="12.0"/>
<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="slow_K_current">
<variable units="picoA" public_interface="out" name="i_slow"/>
<variable units="picoS" name="g_s" initial_value="200.0"/>
<variable units="millivolt" public_interface="in" name="V_K"/>
<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="s"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_slow_calculation">
<eq/>
<ci> i_slow </ci>
<apply>
<times/>
<ci> g_s </ci>
<ci> s </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="slow_K_current_s_gate">
<variable units="dimensionless" public_interface="out" name="s"/>
<variable units="dimensionless" name="s_infinity"/>
<variable units="millisecond" name="tau_s" initial_value="12000.0"/>
<variable units="millivolt" name="V_s" initial_value="-52.0"/>
<variable units="millivolt" name="S_s" initial_value="10.0"/>
<variable units="dimensionless" name="R_s" initial_value="0.58"/>
<variable units="dimensionless" name="S_V_R_s"/>
<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="s_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> s </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci> S_V_R_s </ci>
<ci> s </ci>
</apply>
<ci> tau_s </ci>
</apply>
</apply>
<apply id="S_V_R_s_calculation">
<eq/>
<ci> S_V_R_s </ci>
<apply>
<plus/>
<ci> s_infinity </ci>
<ci> R_s </ci>
</apply>
</apply>
<apply id="s_infinity_calculation">
<eq/>
<ci> s_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_s </ci>
<ci> V </ci>
</apply>
<ci> S_s </ci>
</apply>
</apply>
</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="350.0"/>
<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>
<times/>
<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>
<minus/>
<ci> V </ci>
<ci> V_Na_Ca </ci>
</apply>
</apply>
</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"/>
<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="micromolar" name="Ca_ret"/>
<variable units="dimensionless" name="f" initial_value="0.02"/>
<variable units="per_millisecond" name="k_Ca" initial_value="0.64"/>
<variable units="per_millisecond" name="k_rel" initial_value="0.0006"/>
<variable units="per_millisecond" name="k_pump" initial_value="0.2"/>
<variable units="mole_per_microlitre_coulomb" name="alpha" initial_value="0.00006"/>
<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>
<minus/>
<apply>
<plus/>
<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>
<times/>
<ci> k_rel </ci>
<apply>
<minus/>
<ci> Ca_ret </ci>
<ci> Ca_i </ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci> k_pump </ci>
<ci> Ca_i </ci>
</apply>
</apply>
</apply>
<apply id="Ca_ret_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca_ret </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<apply>
<minus/>
<ci> k_rel </ci>
</apply>
<apply>
<minus/>
<ci> Ca_ret </ci>
<ci> Ca_i </ci>
</apply>
</apply>
<apply>
<times/>
<ci> k_pump </ci>
<ci> Ca_i </ci>
</apply>
</apply>
</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="rapidly_activating_K_current">
<component_ref component="rapidly_activating_K_current_n_gate"/>
</component_ref>
<component_ref component="slow_K_current">
<component_ref component="slow_K_current_s_gate"/>
</component_ref>
<component_ref component="Na_Ca_exchanger_current"/>
<component_ref component="ionic_concentrations"/>
</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="rapidly_activating_K_current">
<component_ref component="rapidly_activating_K_current_n_gate"/>
</component_ref>
<component_ref component="slow_K_current">
<component_ref component="slow_K_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="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="slow_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="slow_K_current" component_1="membrane"/>
<map_variables variable_2="V" variable_1="V"/>
<map_variables variable_2="i_slow" variable_1="i_slow"/>
</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="slow_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_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="rapidly_activating_K_current_n_gate" component_1="rapidly_activating_K_current"/>
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The Gall and Susa 1999 model of Na/Ca exchange in Model II for pancreatic beta-cells.
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