- Author:
- rcai987 <devnull@localhost>
- Date:
- 2015-02-05 13:17:52+13:00
- Desc:
- Added annotations to mckenzie_sneyd_1998.cellml
- Permanent Source URI:
- https://models.physiomeproject.org/workspace/mckenzie_sneyd_1998/rawfile/a5ca1bd861586641044a52eb4ec110168f7447a7/mckenzie_sneyd_1998.cellml
<?xml version='1.0'?>
<!-- FILE : mckenzie_model_1998.xml
CREATED : 21st November 2002
LAST MODIFIED : 21st November 2002
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 16/1/02 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of McKenzie and Sneyd's 1998 mathematical model of the formation and breakup of spiral waves of calcium.
CHANGES:
-->
<model name="mckenzie_1998" 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#">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Modelling the Formation and Breakup of Spiral Waves of Calcium</title>
<author>
<firstname>Catherine</firstname>
<surname>Lloyd</surname>
<affiliation>
<shortaffil>Auckland Bioengineering Institute, The 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>
In many cell types, Ca<superscript>2+</superscript> acting as a signalling agent plays an important role in the control of cellular function. It is thought that these signals are mediated through oscillations in intracellular Ca<superscript>2+</superscript> concentration ([Ca<superscript>2+</superscript>]<subscript>i</subscript>). The mechanisms underlying these oscillations are complex and they have only been partially elucidated. In addition, in larger cells such as in the <emphasis>Xenopus laevis</emphasis> oocyte, a variety of Ca<superscript>2+</superscript> wave behaviour has been seen, including plane waves, pulsating patterns and spiral waves. This has lead to the frequent use of <emphasis>Xenopus</emphasis> oocytes in experimental studies of Ca<superscript>2+</superscript> waves.
</para>
<para>
The probability of the inositol 1,4,5-triphosphate (IP<subscript>3</subscript>) receptor being in the open state is influenced by both IP<subscript>3</subscript> and Ca<superscript>2+</superscript>. This duel system of control is thought to underlie Ca<superscript>2+</superscript> waves and oscillations. IP<subscript>3</subscript> binding to the receptor activates the channel, causing a small amount of Ca<superscript>2+</superscript> to diffuse out of the ER into the cytosol. This Ca<superscript>2+</superscript> has a positive feedback effect, called calcium-induced calcium release (CICR), and it is thought to underlie Ca<superscript>2+</superscript> wave propagation. At intermediate IP<subscript>3</subscript> concentrations, a stable limit cycle appears in the kinetics, and the model becomes oscillatory. Under these conditions, wave propagation can occur even without excitability.
</para>
<para>
Several mathematical models for Ca<superscript>2+</superscript> wave propagation have been published (for some examples see <ulink url="${HTML_EXMPL_IP3_CA2+_MODEL}">DeYoung and Keizer's IP<subscript>3</subscript>-mediated Ca<superscript>2+</superscript> release model, 1992</ulink> and <ulink url="${HTML_EXMPL_LI_MODEL}">Li and Rinzel's IP<subscript>3</subscript>-mediated Ca<superscript>2+</superscript> oscillations model, 1994</ulink>), all of which are similar. The mathematical model described here was published by McKenzie and Sneyd in 1998 (see <xref linkend="fig_cell_diagram"/> below). It describes the process by which IP<subscript>3</subscript> diffuses through the cytosol and binds to receptors located on the endoplasmic reticulum (ER). These receptors also function as Ca<superscript>2+</superscript> channels. They open upon IP<subscript>3</subscript> binding, and Ca<superscript>2+</superscript> diffuses out of the ER down its electrochemical gradient into the cytosol. Ca<superscript>2+</superscript> is pumped against its electrochemical gradient out of the cytosol into the ER, and ions also leak into the cytosol from the ER and the external environment as they diffuse into the cytosol down their electrochemical gradient.
</para>
<para>
Model simulations showed that spiral waves could be induced by simulating the release of IP<subscript>3</subscript>. Increasing the size of the IP<subscript>3</subscript> additions caused a decrease in the rotation period and the breakup of the spiral wave solutions. Following this, irregular spatio-temporal patterns occurred. From their model simulations, the authors predict that a simple experimental procedure may be sufficient to reproduce unstable spiral waves.
</para>
<para>
The original paper reference is cited below:
</para>
<para>
On the Formation and Breakup of Spiral Waves of Calcium, A. McKenzie and J. Sneyd, 1998, <emphasis>International Journal of Bifurcation and Chaos</emphasis>, 8, 2003-2012. (No PubMed ID)
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell_diagram</title>
</objectinfo>
<imagedata fileref="mckenzie_1998.png"/>
</imageobject>
</mediaobject>
<caption>A schematic diagram representing the model.</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="micromolar">
<unit prefix="micro" units="mole"/>
<unit exponent="-1" units="litre"/>
</units>
<units name="flux">
<unit units="micromolar"/>
<unit exponent="-1" units="second"/>
</units>
<units name="mm2_per_second">
<unit exponent="2" prefix="micro" units="metre"/>
<unit exponent="-1" units="second"/>
</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 name="time" public_interface="out" units="second"/>
</component>
<component cmeta:id="id_000000001" name="IP3">
<variable initial_value="0.0" name="P" public_interface="out" units="micromolar"/>
<variable initial_value="300.0" name="D_P" units="mm2_per_second"/>
<variable name="Pxx" units="micromolar"/>
<variable name="Pyy" units="micromolar"/>
<variable name="time" public_interface="in" units="second"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="P_partialdiff_eq">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> P </ci>
</apply>
<apply>
<times/>
<ci> D_P </ci>
<apply>
<plus/>
<ci> Pxx </ci>
<ci> Pyy </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium">
<variable initial_value="0.3" name="c" public_interface="out" units="micromolar"/>
<variable initial_value="20.0" name="D_c" units="mm2_per_second"/>
<variable name="cxx" units="micromolar"/>
<variable name="cyy" units="micromolar"/>
<variable name="time" public_interface="in" units="second"/>
<variable name="J_channel" public_interface="in" units="flux"/>
<variable name="J_pump" public_interface="in" units="flux"/>
<variable name="J_leak" public_interface="in" units="flux"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="c_partialdiff_eq">
<eq/>
<apply>
<partialdiff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> c </ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<ci> D_c </ci>
<apply>
<plus/>
<ci> cxx </ci>
<ci> cyy </ci>
</apply>
</apply>
<ci> J_channel </ci>
<ci> J_leak </ci>
<apply>
<minus/>
<ci> J_pump </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="id_000000002" name="IP3_receptor">
<variable cmeta:id="id_000000004" name="J_channel" public_interface="out" units="flux"/>
<variable cmeta:id="id_000000005" initial_value="16.0" name="k_flux" units="flux"/>
<variable cmeta:id="id_000000006" name="mu_P" units="dimensionless"/>
<variable cmeta:id="id_000000007" initial_value="0.29" name="mu_0" units="dimensionless"/>
<variable cmeta:id="id_000000008" initial_value="0.30" name="mu_1" units="dimensionless"/>
<variable cmeta:id="id_000000009" initial_value="0.7" name="k1" units="micromolar"/>
<variable cmeta:id="id_000000010" initial_value="4.0" name="k_mu" units="micromolar"/>
<variable cmeta:id="id_000000011" initial_value="0.111" name="b" units="dimensionless"/>
<variable name="time" private_interface="out" public_interface="in" units="second"/>
<variable name="c" private_interface="out" public_interface="in" units="micromolar"/>
<variable name="P" public_interface="in" units="micromolar"/>
<variable name="h" private_interface="in" units="dimensionless"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
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<eq/>
<ci> J_channel </ci>
<apply>
<times/>
<ci> k_flux </ci>
<ci> mu_P </ci>
<ci> h </ci>
<apply>
<plus/>
<ci> b </ci>
<apply>
<divide/>
<apply>
<times/>
<apply>
<minus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<ci> b </ci>
</apply>
<ci> c </ci>
</apply>
<apply>
<plus/>
<ci> k1 </ci>
<ci> c </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply id="mu_P_calculation">
<eq/>
<ci> mu_P </ci>
<apply>
<plus/>
<ci> mu_0 </ci>
<apply>
<divide/>
<apply>
<times/>
<ci> mu_1 </ci>
<ci> P </ci>
</apply>
<apply>
<plus/>
<ci> k_mu </ci>
<ci> P </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="id_000000003" name="IP3_receptor_h_gate">
<variable cmeta:id="id_000000012" name="h" public_interface="out" units="dimensionless"/>
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<variable cmeta:id="id_000000014" name="h_infinity" units="dimensionless"/>
<variable cmeta:id="id_000000015" initial_value="0.7" name="k2" units="micromolar"/>
<variable cmeta:id="id_000000016" name="c" public_interface="in" units="micromolar"/>
<variable name="time" public_interface="in" units="second"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="h_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> h </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci> h_infinity </ci>
<ci> h </ci>
</apply>
<ci> tau_h </ci>
</apply>
</apply>
<apply id="h_infinity_calculation">
<eq/>
<ci> h_infinity </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> k2 </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> k2 </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<power/>
<ci> c </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="id_000000017" name="calcium_pump">
<variable cmeta:id="id_000000018" name="J_pump" public_interface="out" units="flux"/>
<variable cmeta:id="id_000000019" initial_value="2.0" name="gamma" units="flux"/>
<variable cmeta:id="id_000000020" initial_value="0.10" name="k_gamma" units="micromolar"/>
<variable name="time" public_interface="in" units="second"/>
<variable name="c" public_interface="in" units="micromolar"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_pump_calculation">
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<apply>
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<ci> gamma </ci>
<ci> c </ci>
</apply>
<apply>
<plus/>
<ci> k_gamma </ci>
<ci> c </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component cmeta:id="id_000000021" name="calcium_leak">
<variable cmeta:id="id_000000022" name="J_leak" public_interface="out" units="flux"/>
<variable initial_value="0.0" name="beta" units="dimensionless"/>
<variable name="time" public_interface="in" units="second"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_leak_calculation">
<eq/>
<ci> J_leak </ci>
<ci> beta </ci>
</apply>
</math>
</component>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="IP3_receptor">
<component_ref component="IP3_receptor_h_gate"/>
</component_ref>
</group>
<connection>
<map_components component_1="IP3" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="calcium" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="IP3_receptor" component_2="environment"/>
<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
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</connection>
<connection>
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<map_variables variable_1="time" variable_2="time"/>
</connection>
<connection>
<map_components component_1="IP3" component_2="IP3_receptor"/>
<map_variables variable_1="P" variable_2="P"/>
</connection>
<connection>
<map_components component_1="calcium" component_2="IP3_receptor"/>
<map_variables variable_1="c" variable_2="c"/>
<map_variables variable_1="J_channel" variable_2="J_channel"/>
</connection>
<connection>
<map_components component_1="calcium" component_2="calcium_pump"/>
<map_variables variable_1="c" variable_2="c"/>
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<connection>
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