<?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&_udi=B6WC7-4BS4GC2-1&_user=140507&_coverDate=09%2F30%2F2004&_alid=197872630&_rdoc=1&_fmt=summary&_orig=search&_qd=1&_cdi=6731&_sort=d&_docanchor=&view=c&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&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&_udi=B6WC7-4BS4GC2-1&_coverDate=09%2F30%2F2004&_alid=197872630&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6731&_sort=d&view=c&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=b34962c344ab1a8911383073cd53016f">Full text (HTML)</ulink> and <ulink url="http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WC7-4BS4GC2-1-3Y&_cdi=6731&_orig=search&_coverDate=09%2F30%2F2004&_qd=1&_sk=999339994&view=c&wchp=dGLbVzz-zSkWz&_acct=C000011498&_version=1&_userid=140507&md5=4f701b4338556df3136f0c4815596563&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&db=PubMed&list_uids=15294427&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/>
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</bvar>
<ci> V </ci>
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<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"/>
<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/>
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<power/>
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<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
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</math>
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<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"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="dimensionless" private_interface="in" name="n"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_K_calculation">
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<ci> n </ci>
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<minus/>
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</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"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="n_diff_eq">
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<ci> n </ci>
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<ci> n </ci>
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<ci> tau_n </ci>
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<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>
<ci> V_K </ci>
</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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<cn cellml:units="dimensionless"> 5.0 </cn>
</apply>
<apply>
<plus/>
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</apply>
<apply>
<power/>
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<cn cellml:units="dimensionless"> 5.0 </cn>
</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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<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_K_ATP_calculation">
<eq/>
<ci> i_K_ATP </ci>
<apply>
<times/>
<ci> g_K_ATP </ci>
<ci> a </ci>
<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"/>
<variable units="dimensionless" name="a_infinity"/>
<variable units="micromolar" name="sa" initial_value="0.1"/>
<variable units="micromolar" name="r" initial_value="0.14"/>
<variable units="micromolar" public_interface="in" name="c"/>
<variable units="millisecond" public_interface="in" name="time"/>
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<ci> a </ci>
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<apply>
<divide/>
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</component>
<component name="ER_calcium_concentration">
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<variable units="dimensionless" name="fer" initial_value="0.01"/>
<variable units="dimensionless" name="Vcyt_Ver" initial_value="5.0"/>
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<variable units="femtoA" public_interface="in" name="i_Ca"/>
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<apply>
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<ci> c </ci>
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<component name="calcium_leak_out_of_the_ER">
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<ci> pleak </ci>
<apply>
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<ci> c </ci>
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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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<variable units="micromolar" public_interface="in" name="c_er"/>
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<ci> c </ci>
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<variable units="flux" public_interface="in" name="Jleak"/>
<variable units="flux" public_interface="in" name="JIP3"/>
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<ci> JIP3 </ci>
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<ci> J_SERCA </ci>
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</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="delayed_rectifier_potassium_current">
<component_ref component="delayed_rectifier_potassium_current_n_gate"/>
</component_ref>
<component_ref component="calcium_dependent_potassium_current">
<component_ref component="calcium_dependent_potassium_current_omega_gate"/>
</component_ref>
<component_ref component="nucleotide_sensitive_potassium_current">
<component_ref component="nucleotide_sensitive_potassium_current_a_gate"/>
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Bertram and Sherman's calcium-based phantom bursting model for pancreatic islets.
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