- Author:
- rcai987 <devnull@localhost>
- Date:
- 2015-02-02 13:17:02+13:00
- Desc:
- Added annotations to keener_2001.cellml model
- Permanent Source URI:
- http://models.cellml.org/workspace/keener_2001/rawfile/587dbd7c8f9572fd20b10f74edcba99fd036335f/keener_2001.cellml
<?xml version="1.0"?>
<!-- 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:bqs="http://www.cellml.org/bqs/1.0#" xmlns:cellml="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:dcterms="http://purl.org/dc/terms/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" xml:base="file:///H:/keener_2001/keener_2001.cellml" 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>
Diffusion Induced Oscillatory Insulin Secretion, James P. Keener, 2001, <emphasis>Bulletin of Mathematical Biology</emphasis>
, 63, 625-641. <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 prefix="milli" units="mole"/>
<unit exponent="-1" units="litre"/>
</units>
<units name="minute">
<unit multiplier="60.0" units="second"/>
</units>
<units name="flux">
<unit units="millimolar"/>
<unit exponent="-1" units="minute"/>
</units>
<units name="cm2_per_second">
<unit exponent="2" prefix="centi" units="metre"/>
<unit exponent="-1" units="second"/>
</units>
<units name="mm">
<unit prefix="milli" units="metre"/>
</units>
<units name="mm_per_minute">
<unit prefix="milli" units="metre"/>
<unit exponent="-1" units="minute"/>
</units>
<!--
The "environment" component is used to declare variables that are used by
all or most of the other components.
-->
<component name="environment">
<variable name="time" public_interface="out" units="minute"/>
<variable name="rho" public_interface="out" units="dimensionless"/>
</component>
<component cmeta:id="id_00001" name="intracellular_glucose">
<variable cmeta:id="id_00002" initial_value="0.0" name="G_i" public_interface="out" units="millimolar"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="Rm" public_interface="in" units="flux"/>
<variable name="R1" public_interface="in" units="flux"/>
<variable name="R2" public_interface="in" units="flux"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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 cmeta:id="id_00003" name="extracellular_glucose">
<variable cmeta:id="id_00004" initial_value="5.0" name="G" public_interface="out" units="millimolar"/>
<variable initial_value="0.1" name="x" public_interface="out" units="mm"/>
<variable initial_value="3.8" name="V" public_interface="out" units="mm_per_minute"/>
<variable initial_value="6.6E-6" name="D_G" units="cm2_per_second"/>
<variable name="A" units="dimensionless"/>
<variable name="B" units="dimensionless"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="R1" public_interface="in" units="flux"/>
<variable name="R2" public_interface="in" units="flux"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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 cmeta:id="id_00004" name="insulin">
<variable cmeta:id="id_00005" initial_value="0.0" name="I" public_interface="out" units="millimolar"/>
<variable initial_value="2.1E-6" name="D_I" units="cm2_per_second"/>
<variable name="C" units="dimensionless"/>
<variable name="D" units="dimensionless"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="Rs" public_interface="in" units="flux"/>
<variable name="x" public_interface="in" units="mm"/>
<variable name="V" public_interface="in" units="mm_per_minute"/>
<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 cmeta:id="id_00006" name="insulin_inhibition">
<variable name="J" public_interface="out" units="dimensionless"/>
<variable initial_value="2.1E-6" name="J_infinity" units="dimensionless"/>
<variable initial_value="20.0" name="J_tau" units="minute"/>
<variable initial_value="1.0E-6" name="K_inh" units="millimolar"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="I" public_interface="in" units="millimolar"/>
<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 cmeta:id="id_00007" name="glucose_metabolism">
<variable cmeta:id="id_00008" name="Rm" public_interface="out" units="flux"/>
<variable initial_value="0.24" name="Vm" units="flux"/>
<variable initial_value="9.8" name="Km" units="millimolar"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="G_i" public_interface="in" units="millimolar"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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 cmeta:id="id_00009" name="GLUT_1_transporter">
<variable name="R1" public_interface="out" units="flux"/>
<variable initial_value="120.0" name="V1" units="flux"/>
<variable initial_value="1.4" name="K1" units="millimolar"/>
<variable initial_value="4.0E-5" name="Ki" units="millimolar"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="I" public_interface="in" units="millimolar"/>
<variable name="G" public_interface="in" units="millimolar"/>
<variable name="G_i" public_interface="in" units="millimolar"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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 cmeta:id="id_00010" name="GLUT_2_transporter">
<variable name="R2" public_interface="out" units="flux"/>
<variable initial_value="32.0" name="V2" units="flux"/>
<variable initial_value="17.0" name="K2" units="millimolar"/>
<variable name="Jm" units="dimensionless"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="G" public_interface="in" units="millimolar"/>
<variable name="G_i" public_interface="in" units="millimolar"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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 cmeta:id="id_00011" name="insulin_secretion">
<variable name="Rs" public_interface="out" units="flux"/>
<variable initial_value="0.034" name="Vs" units="flux"/>
<variable initial_value="0.13" name="Ks" units="millimolar"/>
<variable initial_value="0.01" name="L" units="flux"/>
<variable name="time" public_interface="in" units="minute"/>
<variable name="Rm" public_interface="in" units="flux"/>
<variable name="rho" public_interface="in" units="dimensionless"/>
<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_1="intracellular_glucose" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="extracellular_glucose" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="insulin" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="insulin_inhibition" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="glucose_metabolism" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="GLUT_1_transporter" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="GLUT_2_transporter" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="insulin_secretion" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
<map_variables variable_1="rho" variable_2="rho"/>
</connection>
<connection>
<map_components component_1="intracellular_glucose" component_2="glucose_metabolism"/>
<map_variables variable_1="G_i" variable_2="G_i"/>
<map_variables variable_1="Rm" variable_2="Rm"/>
</connection>
<connection>
<map_components component_1="intracellular_glucose" component_2="GLUT_1_transporter"/>
<map_variables variable_1="G_i" variable_2="G_i"/>
<map_variables variable_1="R1" variable_2="R1"/>
</connection>
<connection>
<map_components component_1="intracellular_glucose" component_2="GLUT_2_transporter"/>
<map_variables variable_1="G_i" variable_2="G_i"/>
<map_variables variable_1="R2" variable_2="R2"/>
</connection>
<connection>
<map_components component_1="extracellular_glucose" component_2="GLUT_1_transporter"/>
<map_variables variable_1="G" variable_2="G"/>
<map_variables variable_1="R1" variable_2="R1"/>
</connection>
<connection>
<map_components component_1="extracellular_glucose" component_2="GLUT_2_transporter"/>
<map_variables variable_1="G" variable_2="G"/>
<map_variables variable_1="R2" variable_2="R2"/>
</connection>
<connection>
<map_components component_1="insulin" component_2="insulin_inhibition"/>
<map_variables variable_1="I" variable_2="I"/>
</connection>
<connection>
<map_components component_1="insulin" component_2="insulin_secretion"/>
<map_variables variable_1="Rs" variable_2="Rs"/>
</connection>
<connection>
<map_components component_1="insulin" component_2="GLUT_1_transporter"/>
<map_variables variable_1="I" variable_2="I"/>
</connection>
<connection>
<map_components component_1="insulin" component_2="extracellular_glucose"/>
<map_variables variable_1="x" variable_2="x"/>
<map_variables variable_1="V" variable_2="V"/>
</connection>
<connection>
<map_components component_1="insulin_secretion" component_2="glucose_metabolism"/>
<map_variables variable_1="Rm" variable_2="Rm"/>
</connection>
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