<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : keener_model_2001.xml
CREATED : 20th November 2002
LAST MODIFIED : 9th April 2003
AUTHOR : Catherine Lloyd
The 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 James Keener's 2001 mathematical model of insulin secretion oscillations in 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="keener_2001_version01" name="keener_2001_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Diffusion Induced Oscillatory Insulin Secretion</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>
Insulin is secreted from pancreatic beta cells in an oscillatory fashion. In his 2001 paper (cited below), James Keener examines a mathematical model for <emphasis>in vitro</emphasis> insulin secretion from pancreatic beta cells in a one-dimensional chemical flow reactor, where the reaction region is represented by the volume occupied by the cells (see <xref linkend="fig_cell_diagram"/> below). Glucose enters the cell through GLUT-1 and GLUT-2 transporters. Once inside the cell, intracellular glucose is metabolised, and this process activates insulin secretion via exocytosis of insulin containing granules. In the extracellular medium, insulin activates GLUT-1 transporters and inactivates GLUT-2 transporters.
</para>
<para>
In order to model the reactions, a steady flow of solution along the one-dimensional reactor is assumed, with insulin cells confined to a one-dimensional region. Keener found that the oscillations occur as a result of an important interplay between flow rate of the reactor and insulin diffusion. Without diffusion, the oscillations are eliminated.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WC7-4593YHR-R&_user=140507&_coverDate=07%2F31%2F2001&_rdoc=2&_fmt=summary&_orig=browse&_srch=%23toc%236731%232001%23999369995%23288659!&_cdi=6731&_sort=d&_docanchor=&wchp=dGLbVzz-lSztb&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=ab655a0002a58212a7c8f4893876257a">Diffusion Induced Oscillatory Insulin Secretion</ulink>, James P. Keener, 2001, <ulink url="http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=6731&_auth=y&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=2670d6bf58ba6625cf8c13344d6a4cb5">
<emphasis>Bulletin of Mathematical Biology</emphasis>
</ulink>, 63, 625-641. (A PDF version of the article is available to subscribers on the Bulletin of Mathematical Biology website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11497161&dopt=Abstract">PubMed ID: 11497161</ulink>
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell_diagram</title>
</objectinfo>
<imagedata fileref="keener_2001.png"/>
</imageobject>
</mediaobject>
<caption>A schematic diagram representing the model of glucose uptake, glucose metabolism and insulin secretion by pancreatic beta cells. Red lines represent the inactivation of GLUT-2 transporters by extracellular insulin, blue arrows represent the activation of GLUT-1 transporters by extracellular insulin and the activation of insulin secretion by glucose metabolism.</caption>
</informalfigure>
<para>
The raw CellML descriptions of the model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.
</para>
</sect1>
</article>
</documentation>
<!--
Below, we define some additional units for association with variables and
constants within the model.
-->
<units name="millimolar">
<unit units="mole" prefix="milli"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="minute">
<unit units="second" multiplier="60.0"/>
</units>
<units name="flux">
<unit units="millimolar"/>
<unit units="minute" exponent="-1"/>
</units>
<units name="cm2_per_second">
<unit units="metre" prefix="centi" exponent="2"/>
<unit units="second" exponent="-1"/>
</units>
<units name="mm">
<unit units="metre" prefix="milli"/>
</units>
<units name="mm_per_minute">
<unit units="metre" prefix="milli"/>
<unit units="minute" exponent="-1"/>
</units>
<!--
The "environment" component is used to declare variables that are used by
all or most of the other components.
-->
<component name="environment">
<variable units="minute" public_interface="out" name="time"/>
<variable units="dimensionless" public_interface="out" name="rho"/>
</component>
<component name="intracellular_glucose">
<variable units="millimolar" public_interface="out" name="G_i" initial_value="0.0"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="flux" public_interface="in" name="Rm"/>
<variable units="flux" public_interface="in" name="R1"/>
<variable units="flux" public_interface="in" name="R2"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="G_i_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> G_i </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<plus/>
<ci> R1 </ci>
<ci> R2 </ci>
</apply>
<ci> Rm </ci>
</apply>
<ci> rho </ci>
</apply>
</apply>
</math>
</component>
<component name="extracellular_glucose">
<variable units="millimolar" public_interface="out" name="G" initial_value="5.0"/>
<variable units="mm" public_interface="out" name="x" initial_value="0.1"/>
<variable units="mm_per_minute" public_interface="out" name="V" initial_value="3.8"/>
<variable units="cm2_per_second" name="D_G" initial_value="6.6E-6"/>
<variable units="dimensionless" name="A"/>
<variable units="dimensionless" name="B"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="flux" public_interface="in" name="R1"/>
<variable units="flux" public_interface="in" name="R2"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="G_partialdiff_eq">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> G </ci>
</apply>
<apply>
<plus/>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<minus/>
<ci> R1 </ci>
</apply>
<apply>
<minus/>
<ci> R2 </ci>
</apply>
</apply>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> rho </ci>
</apply>
</apply>
<apply>
<times/>
<ci> D_G </ci>
<ci> A </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> G </ci>
<ci> B </ci>
</apply>
</apply>
</apply>
</apply>
<apply id="A_calculation">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
<degree>
<cn cellml:units="dimensionless"> 2.0 </cn>
</degree>
</bvar>
<degree>
<cn cellml:units="dimensionless"> 2.0 </cn>
</degree>
<ci> G </ci>
</apply>
<ci> A </ci>
</apply>
<apply id="B_calculation">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
</bvar>
<ci> G </ci>
</apply>
<ci> B </ci>
</apply>
</math>
</component>
<component name="insulin">
<variable units="millimolar" public_interface="out" name="I" initial_value="0.0"/>
<variable units="cm2_per_second" name="D_I" initial_value="2.1E-6"/>
<variable units="dimensionless" name="C"/>
<variable units="dimensionless" name="D"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="flux" public_interface="in" name="Rs"/>
<variable units="mm" public_interface="in" name="x"/>
<variable units="mm_per_minute" public_interface="in" name="V"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="I_partialdiff_eq">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> I </ci>
</apply>
<apply>
<plus/>
<ci> Rs </ci>
<apply>
<times/>
<ci> D_I </ci>
<ci> C </ci>
</apply>
<apply>
<minus/>
<apply>
<times/>
<ci> V </ci>
<ci> D </ci>
</apply>
</apply>
</apply>
</apply>
<apply id="C_calculation">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
<degree>
<cn cellml:units="dimensionless"> 2.0 </cn>
</degree>
</bvar>
<degree>
<cn cellml:units="dimensionless"> 2.0 </cn>
</degree>
<ci> I </ci>
</apply>
<ci> C </ci>
</apply>
<apply id="D_calculation">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> x </ci>
</bvar>
<ci> I </ci>
</apply>
<ci> D </ci>
</apply>
</math>
</component>
<component name="insulin_inhibition">
<variable units="dimensionless" public_interface="out" name="J"/>
<variable units="dimensionless" name="J_infinity" initial_value="2.1E-6"/>
<variable units="minute" name="J_tau" initial_value="20.0"/>
<variable units="millimolar" name="K_inh" initial_value="1.0E-6"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="millimolar" public_interface="in" name="I"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> J </ci>
</apply>
<apply>
<times/>
<ci> J_tau </ci>
<apply>
<minus/>
<ci> J_infinity </ci>
<ci> J </ci>
</apply>
</apply>
</apply>
<apply id="J_infinity_calculation">
<eq/>
<ci> J_infinity </ci>
<apply>
<divide/>
<ci> K_inh </ci>
<apply>
<plus/>
<ci> K_inh </ci>
<ci> I </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="glucose_metabolism">
<variable units="flux" public_interface="out" name="Rm"/>
<variable units="flux" name="Vm" initial_value="0.24"/>
<variable units="millimolar" name="Km" initial_value="9.8"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="millimolar" public_interface="in" name="G_i"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Rm_calculation">
<eq/>
<ci> Rm </ci>
<apply>
<times/>
<ci> rho </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> Vm </ci>
<ci> G_i </ci>
</apply>
<apply>
<plus/>
<ci> Km </ci>
<ci> G_i </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="GLUT_1_transporter">
<variable units="flux" public_interface="out" name="R1"/>
<variable units="flux" name="V1" initial_value="120.0"/>
<variable units="millimolar" name="K1" initial_value="1.4"/>
<variable units="millimolar" name="Ki" initial_value="4.0E-5"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="millimolar" public_interface="in" name="I"/>
<variable units="millimolar" public_interface="in" name="G"/>
<variable units="millimolar" public_interface="in" name="G_i"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="R1_calculation">
<eq/>
<ci> R1 </ci>
<apply>
<times/>
<ci> rho </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> V1 </ci>
<apply>
<minus/>
<ci> G </ci>
<ci> G_i </ci>
</apply>
</apply>
<apply>
<times/>
<apply>
<plus/>
<ci> K1 </ci>
<ci> G_i </ci>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<divide/>
<ci> G </ci>
<ci> K1 </ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<divide/>
<ci> I </ci>
<apply>
<plus/>
<ci> Ki </ci>
<ci> I </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="GLUT_2_transporter">
<variable units="flux" public_interface="out" name="R2"/>
<variable units="flux" name="V2" initial_value="32.0"/>
<variable units="millimolar" name="K2" initial_value="17.0"/>
<variable units="dimensionless" name="Jm"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="millimolar" public_interface="in" name="G"/>
<variable units="millimolar" public_interface="in" name="G_i"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="R2_calculation">
<eq/>
<ci> R2 </ci>
<apply>
<times/>
<ci> rho </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> V2 </ci>
<apply>
<minus/>
<apply>
<times/>
<ci> G </ci>
<ci> Jm </ci>
</apply>
<ci> G_i </ci>
</apply>
</apply>
<apply>
<times/>
<apply>
<plus/>
<ci> K2 </ci>
<ci> G_i </ci>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<divide/>
<ci> G </ci>
<ci> K2 </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="insulin_secretion">
<variable units="flux" public_interface="out" name="Rs"/>
<variable units="flux" name="Vs" initial_value="0.034"/>
<variable units="millimolar" name="Ks" initial_value="0.13"/>
<variable units="flux" name="L" initial_value="0.01"/>
<variable units="minute" public_interface="in" name="time"/>
<variable units="flux" public_interface="in" name="Rm"/>
<variable units="dimensionless" public_interface="in" name="rho"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Rs_calculation">
<eq/>
<ci> Rs </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> Vs </ci>
<apply>
<plus/>
<apply>
<power/>
<ci> Rm </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<times/>
<apply>
<power/>
<ci> rho </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<power/>
<ci> L </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> Rm </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<times/>
<apply>
<power/>
<ci> rho </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<power/>
<ci> Ks </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
</apply>
<apply>
<times/>
<apply>
<power/>
<ci> rho </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
<apply>
<power/>
<ci> L </ci>
<cn cellml:units="dimensionless"> 4.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<connection>
<map_components component_2="environment" component_1="intracellular_glucose"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="environment" component_1="extracellular_glucose"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="environment" component_1="insulin"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="insulin_inhibition"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="glucose_metabolism"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="environment" component_1="GLUT_1_transporter"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="environment" component_1="GLUT_2_transporter"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="environment" component_1="insulin_secretion"/>
<map_variables variable_2="time" variable_1="time"/>
<map_variables variable_2="rho" variable_1="rho"/>
</connection>
<connection>
<map_components component_2="glucose_metabolism" component_1="intracellular_glucose"/>
<map_variables variable_2="G_i" variable_1="G_i"/>
<map_variables variable_2="Rm" variable_1="Rm"/>
</connection>
<connection>
<map_components component_2="GLUT_1_transporter" component_1="intracellular_glucose"/>
<map_variables variable_2="G_i" variable_1="G_i"/>
<map_variables variable_2="R1" variable_1="R1"/>
</connection>
<connection>
<map_components component_2="GLUT_2_transporter" component_1="intracellular_glucose"/>
<map_variables variable_2="G_i" variable_1="G_i"/>
<map_variables variable_2="R2" variable_1="R2"/>
</connection>
<connection>
<map_components component_2="GLUT_1_transporter" component_1="extracellular_glucose"/>
<map_variables variable_2="G" variable_1="G"/>
<map_variables variable_2="R1" variable_1="R1"/>
</connection>
<connection>
<map_components component_2="GLUT_2_transporter" component_1="extracellular_glucose"/>
<map_variables variable_2="G" variable_1="G"/>
<map_variables variable_2="R2" variable_1="R2"/>
</connection>
<connection>
<map_components component_2="insulin_inhibition" component_1="insulin"/>
<map_variables variable_2="I" variable_1="I"/>
</connection>
<connection>
<map_components component_2="insulin_secretion" component_1="insulin"/>
<map_variables variable_2="Rs" variable_1="Rs"/>
</connection>
<connection>
<map_components component_2="GLUT_1_transporter" component_1="insulin"/>
<map_variables variable_2="I" variable_1="I"/>
</connection>
<connection>
<map_components component_2="extracellular_glucose" component_1="insulin"/>
<map_variables variable_2="x" variable_1="x"/>
<map_variables variable_2="V" variable_1="V"/>
</connection>
<connection>
<map_components component_2="glucose_metabolism" component_1="insulin_secretion"/>
<map_variables variable_2="Rm" variable_1="Rm"/>
</connection>
<rdf:RDF>
<rdf:Bag rdf:about="rdf:#4387d728-b36a-44e8-a441-6a78843901ee">
<rdf:li>oscillator</rdf:li>
<rdf:li>beta cell</rdf:li>
<rdf:li>calcium dynamics</rdf:li>
<rdf:li>Pancreatic Beta-Cell</rdf:li>
<rdf:li>insulin</rdf:li>
<rdf:li>metabolism</rdf:li>
</rdf:Bag>
<rdf:Description rdf:about="">
<dc:publisher>
The University of Auckland, Bioengineering Institute
</dc:publisher>
<cmeta:modification rdf:resource="rdf:#607bdccc-bb74-4947-8ef1-3303675540e2"/>
<dcterms:created rdf:resource="rdf:#b7878110-f9cf-4dcc-9b12-96cc7dff28f1"/>
<dc:creator rdf:resource="rdf:#0a7a32cf-deb1-4796-ab41-0c972bd0defb"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#ba8ca845-6b1f-4515-b083-bd9ce2c13320">
<vCard:FN>Catherine Lloyd</vCard:FN>
</rdf:Description>
<rdf:Description rdf:about="rdf:#140e36d2-77be-46ac-9d8c-e10a9bd2e75e">
<dc:subject rdf:resource="rdf:#15ae8bab-e3fa-4b1c-9f95-cbf6857fde89"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#7e4528c0-a2dd-4383-be66-40361909ae08">
<vCard:Given>Autumn</vCard:Given>
<vCard:Family>Cuellar</vCard:Family>
<vCard:Other>A</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#0a7a32cf-deb1-4796-ab41-0c972bd0defb">
<vCard:ORG rdf:resource="rdf:#7bbfc7ba-aded-44a8-acd9-a0447c4ea144"/>
<vCard:EMAIL rdf:resource="rdf:#1036b57a-114c-44c7-bef0-ec95fdfb46a2"/>
<vCard:N rdf:resource="rdf:#b4b93091-2370-40c7-b529-96846bc87729"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#b4b93091-2370-40c7-b529-96846bc87729">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#d96130ee-4877-44fb-a82d-498e7161d2b9">
<vCard:Given>James</vCard:Given>
<vCard:Family>Keener</vCard:Family>
<vCard:Other>P</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#7bbfc7ba-aded-44a8-acd9-a0447c4ea144">
<vCard:Orgname>The University of Auckland</vCard:Orgname>
<vCard:Orgunit>The Bioengineering Institute</vCard:Orgunit>
</rdf:Description>
<rdf:Description rdf:about="rdf:#15ae8bab-e3fa-4b1c-9f95-cbf6857fde89">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value rdf:resource="rdf:#4387d728-b36a-44e8-a441-6a78843901ee"/>
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This is the CellML description of James Keener's 2001 mathematical
model of insulin secretion oscillations in pancreatic beta cells.
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Diffusion Induced Oscillatory Insulin Secretion
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James Keener's 2001 mathematical model of insulin secretion oscillations in pancreatic beta cells.
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