<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : marhl_model_2000.xml
CREATED : 4th April 2002
LAST MODIFIED : 9th April 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 16/1/02 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of Marhl et al's
2000 model of Ca2+ oscillations and the role of mitochondria and cytosolic proteins.
CHANGES:
18/07/2002 - CML - Added more metadata.
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="marhl_2000" name="marhl_2000">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Complex Calcium Oscillations and the Role of Mitochondria and Cytosolic Proteins</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 CellML model runs in both OpenCell and COR to recreate the published results. The units have been checked and are consistent.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
ABSTRACT: Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority ( approximately 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.
</para>
<para>
The original paper reference is cited below:
</para>
<para>
Complex calcium oscillations and the role of mitochondria and cytosolic proteins, Marko Marhl, Thomas Haberichter, Milan Brumen and Reinhart Heinrich, 2000, <emphasis>Biosystems</emphasis>, 57, 75-86. <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11004387&dopt=Abstract">PubMed ID: 11004387</ulink>
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell schematic for the model</title>
</objectinfo>
<imagedata fileref="marhl_2000.png"/>
</imageobject>
</mediaobject>
<caption>Schematic diagram of the model sysytem.</caption>
</informalfigure>
</sect1>
</article>
</documentation>
<units name="micromolar">
<unit units="mole" prefix="micro"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="flux">
<unit units="micromolar"/>
<unit units="second" exponent="-1"/>
</units>
<units name="first_order_rate_constant">
<unit units="second" exponent="-1"/>
</units>
<units name="second_order_rate_constant">
<unit units="micromolar" exponent="-1"/>
<unit units="second" exponent="-1"/>
</units>
<component name="environment">
<variable units="second" public_interface="out" name="time"/>
</component>
<component name="total_calcium">
<variable units="micromolar" public_interface="out" name="Ca_Pr"/>
<variable units="micromolar" name="Ca_tot" initial_value="90.0"/>
<variable units="dimensionless" public_interface="in" name="rho_ER"/>
<variable units="dimensionless" public_interface="in" name="beta_ER"/>
<variable units="dimensionless" public_interface="in" name="rho_m"/>
<variable units="dimensionless" public_interface="in" name="beta_m"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<variable units="micromolar" public_interface="in" name="Ca_ER"/>
<variable units="micromolar" public_interface="in" name="Ca_m"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_Pr_calculation">
<eq/>
<ci> Ca_Pr </ci>
<apply>
<minus/>
<ci> Ca_tot </ci>
<apply>
<plus/>
<ci> Ca_cyt </ci>
<apply>
<times/>
<apply>
<divide/>
<ci> rho_ER </ci>
<ci> beta_ER </ci>
</apply>
<ci> Ca_ER </ci>
</apply>
<apply>
<times/>
<apply>
<divide/>
<ci> rho_m </ci>
<ci> beta_m </ci>
</apply>
<ci> Ca_m </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="total_protein">
<variable units="micromolar" public_interface="out" name="Pr"/>
<variable units="micromolar" name="Pr_tot" initial_value="120.0"/>
<variable units="micromolar" public_interface="in" name="Ca_Pr"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Pr_calculation">
<eq/>
<ci> Pr </ci>
<apply>
<minus/>
<ci> Pr_tot </ci>
<ci> Ca_Pr </ci>
</apply>
</apply>
</math>
</component>
<component name="cytosolic_calcium">
<variable units="micromolar" public_interface="out" initial_value="0.05" name="Ca_cyt"/>
<variable units="second_order_rate_constant" name="k_plus" initial_value="0.1"/>
<variable units="first_order_rate_constant" name="k_minus" initial_value="0.01"/>
<variable units="micromolar" public_interface="in" name="Pr"/>
<variable units="micromolar" public_interface="in" name="Ca_Pr"/>
<variable units="flux" public_interface="in" name="J_ch"/>
<variable units="flux" public_interface="in" name="J_leak"/>
<variable units="flux" public_interface="in" name="J_pump"/>
<variable units="flux" public_interface="in" name="J_out"/>
<variable units="flux" public_interface="in" name="J_in"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_cyt_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca_cyt </ci>
</apply>
<apply>
<minus/>
<apply>
<plus/>
<ci> J_ch </ci>
<ci> J_leak </ci>
<ci> J_out </ci>
<apply>
<times/>
<ci> k_minus </ci>
<ci> Ca_Pr </ci>
</apply>
</apply>
<apply>
<plus/>
<ci> J_pump </ci>
<ci> J_in </ci>
<apply>
<times/>
<ci> k_plus </ci>
<ci> Ca_cyt </ci>
<ci> Pr </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="ER_calcium">
<variable units="micromolar" public_interface="out" initial_value="1.0" name="Ca_ER"/>
<variable units="dimensionless" public_interface="out" name="rho_ER" initial_value="0.01"/>
<variable units="dimensionless" public_interface="out" name="beta_ER" initial_value="0.0025"/>
<variable units="flux" public_interface="in" name="J_ch"/>
<variable units="flux" public_interface="in" name="J_leak"/>
<variable units="flux" public_interface="in" name="J_pump"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_ER_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca_ER </ci>
</apply>
<apply>
<times/>
<apply>
<divide/>
<ci> beta_ER </ci>
<ci> rho_ER </ci>
</apply>
<apply>
<minus/>
<ci> J_pump </ci>
<apply>
<plus/>
<ci> J_ch </ci>
<ci> J_leak </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="mitochondrial_calcium">
<variable units="micromolar" public_interface="out" initial_value="0.4" name="Ca_m"/>
<variable units="dimensionless" public_interface="out" name="rho_m" initial_value="0.01"/>
<variable units="dimensionless" public_interface="out" name="beta_m" initial_value="0.0025"/>
<variable units="flux" public_interface="in" name="J_in"/>
<variable units="flux" public_interface="in" name="J_out"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_m_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> Ca_m </ci>
</apply>
<apply>
<times/>
<apply>
<divide/>
<ci> beta_m </ci>
<ci> rho_m </ci>
</apply>
<apply>
<minus/>
<ci> J_in </ci>
<ci> J_out </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="ATP_dependent_Ca_uptake_into_the_ER">
<variable units="flux" public_interface="out" name="J_pump"/>
<variable units="first_order_rate_constant" name="k_pump" initial_value="20.0"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_pump_calculation">
<eq/>
<ci> J_pump </ci>
<apply>
<times/>
<ci> k_pump </ci>
<ci> Ca_cyt </ci>
</apply>
</apply>
</math>
</component>
<component name="Ca_efflux_from_the_ER">
<variable units="flux" public_interface="out" name="J_ch"/>
<variable units="first_order_rate_constant" name="k_ch" initial_value="4100.0"/>
<variable units="micromolar" name="K1" initial_value="5.0"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<variable units="micromolar" public_interface="in" name="Ca_ER"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_ch_calculation">
<eq/>
<ci> J_ch </ci>
<apply>
<times/>
<ci> k_ch </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> K1 </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
</apply>
</apply>
<apply>
<minus/>
<ci> Ca_ER </ci>
<ci> Ca_cyt </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Ca_leak_flux_from_the_ER">
<variable units="flux" public_interface="out" name="J_leak"/>
<variable units="first_order_rate_constant" name="k_leak" initial_value="0.05"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<variable units="micromolar" public_interface="in" name="Ca_ER"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_leak_calculation">
<eq/>
<ci> J_leak </ci>
<apply>
<times/>
<ci> k_leak </ci>
<apply>
<minus/>
<ci> Ca_ER </ci>
<ci> Ca_cyt </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="mitochondrial_Ca_uptake">
<variable units="flux" public_interface="out" name="J_in"/>
<variable units="flux" name="k_in" initial_value="300.0"/>
<variable units="micromolar" name="K2" initial_value="0.8"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_in_calculation">
<eq/>
<ci> J_in </ci>
<apply>
<times/>
<ci> k_in </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 8.0 </cn>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> K2 </ci>
<cn cellml:units="dimensionless"> 8.0 </cn>
</apply>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 8.0 </cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="mitochondrial_Ca_release">
<variable units="flux" public_interface="out" name="J_out"/>
<variable units="first_order_rate_constant" name="k_out" initial_value="125.0"/>
<variable units="first_order_rate_constant" name="k_m" initial_value="0.00625"/>
<variable units="micromolar" name="K3" initial_value="5.0"/>
<variable units="micromolar" public_interface="in" name="Ca_cyt"/>
<variable units="micromolar" public_interface="in" name="Ca_m"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_out_calculation">
<eq/>
<ci> J_out </ci>
<apply>
<times/>
<apply>
<plus/>
<apply>
<times/>
<ci> k_out </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> K3 </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
<apply>
<power/>
<ci> Ca_cyt </ci>
<cn cellml:units="dimensionless"> 2.0 </cn>
</apply>
</apply>
</apply>
</apply>
<ci> k_m </ci>
</apply>
<ci> Ca_m </ci>
</apply>
</apply>
</math>
</component>
<connection>
<map_components component_2="environment" component_1="cytosolic_calcium"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="ER_calcium"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="mitochondrial_calcium"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="ER_calcium" component_1="total_calcium"/>
<map_variables variable_2="rho_ER" variable_1="rho_ER"/>
<map_variables variable_2="beta_ER" variable_1="beta_ER"/>
<map_variables variable_2="Ca_ER" variable_1="Ca_ER"/>
</connection>
<connection>
<map_components component_2="mitochondrial_calcium" component_1="total_calcium"/>
<map_variables variable_2="rho_m" variable_1="rho_m"/>
<map_variables variable_2="beta_m" variable_1="beta_m"/>
<map_variables variable_2="Ca_m" variable_1="Ca_m"/>
</connection>
<connection>
<map_components component_2="cytosolic_calcium" component_1="total_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="Ca_Pr" variable_1="Ca_Pr"/>
</connection>
<connection>
<map_components component_2="total_protein" component_1="total_calcium"/>
<map_variables variable_2="Ca_Pr" variable_1="Ca_Pr"/>
</connection>
<connection>
<map_components component_2="total_protein" component_1="cytosolic_calcium"/>
<map_variables variable_2="Pr" variable_1="Pr"/>
</connection>
<connection>
<map_components component_2="ATP_dependent_Ca_uptake_into_the_ER" component_1="cytosolic_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="J_pump" variable_1="J_pump"/>
</connection>
<connection>
<map_components component_2="Ca_efflux_from_the_ER" component_1="cytosolic_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="J_ch" variable_1="J_ch"/>
</connection>
<connection>
<map_components component_2="Ca_leak_flux_from_the_ER" component_1="cytosolic_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="J_leak" variable_1="J_leak"/>
</connection>
<connection>
<map_components component_2="mitochondrial_Ca_release" component_1="cytosolic_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="J_out" variable_1="J_out"/>
</connection>
<connection>
<map_components component_2="mitochondrial_Ca_uptake" component_1="cytosolic_calcium"/>
<map_variables variable_2="Ca_cyt" variable_1="Ca_cyt"/>
<map_variables variable_2="J_in" variable_1="J_in"/>
</connection>
<connection>
<map_components component_2="ATP_dependent_Ca_uptake_into_the_ER" component_1="ER_calcium"/>
<map_variables variable_2="J_pump" variable_1="J_pump"/>
</connection>
<connection>
<map_components component_2="Ca_efflux_from_the_ER" component_1="ER_calcium"/>
<map_variables variable_2="Ca_ER" variable_1="Ca_ER"/>
<map_variables variable_2="J_ch" variable_1="J_ch"/>
</connection>
<connection>
<map_components component_2="Ca_leak_flux_from_the_ER" component_1="ER_calcium"/>
<map_variables variable_2="Ca_ER" variable_1="Ca_ER"/>
<map_variables variable_2="J_leak" variable_1="J_leak"/>
</connection>
<connection>
<map_components component_2="mitochondrial_Ca_release" component_1="mitochondrial_calcium"/>
<map_variables variable_2="Ca_m" variable_1="Ca_m"/>
<map_variables variable_2="J_out" variable_1="J_out"/>
</connection>
<connection>
<map_components component_2="mitochondrial_Ca_uptake" component_1="mitochondrial_calcium"/>
<map_variables variable_2="J_in" variable_1="J_in"/>
</connection>
<rdf:RDF>
<rdf:Bag rdf:about="rdf:#a3cd0a8d-7520-490f-a0de-85204d064f73">
<rdf:li>calcium dynamics</rdf:li>
<rdf:li>electrophysiology</rdf:li>
<rdf:li>mitochondria</rdf:li>
<rdf:li>oscillator</rdf:li>
</rdf:Bag>
<rdf:Seq rdf:about="rdf:#918348d1-2e8a-44ff-a480-fea01f35c99f">
<rdf:li rdf:resource="rdf:#13a4596c-df9b-4b0f-bf61-16b26cb9b805"/>
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<rdf:li rdf:resource="rdf:#62fdc3aa-0065-4775-8a7a-64d80e71325b"/>
</rdf:Seq>
<rdf:Description rdf:about="">
<dc:publisher>
The University of Auckland, Auckland Bioengineering Institute
</dc:publisher>
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<rdf:Description rdf:about="rdf:#e7678806-38af-4487-8ea9-db9fbd85d250">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
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<bqs:subject_type>keyword</bqs:subject_type>
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<rdf:value>
Added more metadata.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#7bde58e6-98fb-4f43-b526-3d0901db4ac8"/>
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<rdf:Description rdf:about="rdf:#9c5d8eef-f870-4a3e-97c5-778e1199cb20">
<vCard:Given>Reinhart</vCard:Given>
<vCard:Family>Heinrich</vCard:Family>
</rdf:Description>
<rdf:Description rdf:about="rdf:#7ab1934e-2d33-458a-94dc-07c8ae55359a">
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<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
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<rdf:Description rdf:about="rdf:#0d73188d-74e2-442f-a701-9f606389317e">
<vCard:Given>Marko</vCard:Given>
<vCard:Family>Marhl</vCard:Family>
</rdf:Description>
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<dcterms:modified rdf:resource="rdf:#2000f7b1-550d-437b-99d8-2c65dfc18115"/>
<rdf:value>
Added publication date information.
</rdf:value>
<cmeta:modifier rdf:resource="rdf:#8f42ee6a-af98-4e4c-9851-ed08e4fcd7bc"/>
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<rdf:Description rdf:about="rdf:#1537ac2f-3559-416d-96c2-a8c989066679">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>May</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#bc3dac2b-36a6-46d4-8361-89ec946e2b07">
<dcterms:W3CDTF>2000-07</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#928f627c-922a-449b-afd8-47f0b58d7518">
<vCard:FN>Catherine Lloyd</vCard:FN>
</rdf:Description>
<rdf:Description rdf:about="#marhl_2000">
<dc:title>
A mathematical model of Ca2+ oscillations and the role of mitochondria
and cytosolic proteins.
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Complex calcium oscillations and the role of mitochondria and cytosolic proteins
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