<documentation xmlns="http://cellml.org/tmp-documentation"> <articleinfo> A Bifurcation Analysis of Two Coupled Calcium Oscillators <author> <firstname>Catherine</firstname> <surname>Lloyd</surname> <affiliation> <shortaffil>Auckland Bioengineering Institute, The University of Auckland</shortaffil> </affiliation> </author> </articleinfo> Model Status This CellML model runs in both OpenCell and COR to produce an oscillating output similar to that from the original published model. The units have been checked and they are consistent. <sect1 id="sec_structure"> Model Structure ABSTRACT: In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. The original paper reference is cited below: A bifurcation analysis of two coupled calcium oscillators, Michael Bindschadler and James Sneyd, 2001, CHAOS , 11, 237-246. PubMed ID: 12779457 <informalfigure float="0" id="fig_cell_diagram"> <mediaobject> <imageobject> <objectinfo> cell diagram </objectinfo> <imagedata fileref="bindschadler_2001.png" /> </imageobject> Schematic diagram of the IP3 receptor model. The receptor with its three possible states: X, Y, and Z; representing open, shut and inactive respectively, is embedded within a model of intracellular calcium dynamics. </mediaobject> </informalfigure> </sect1> </documentation>
<rdf:RDF xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:vCard="http://www.w3.org/2001/vcard-rdf/3.0#" xmlns:bqs="http://www.cellml.org/bqs/1.0#" xmlns:j.0="http://imc.org/vCard/3.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:dcterms="http://purl.org/dc/terms/" xmlns:ns7="http://www.cellml.org/metadata/simulation/1.0#"> <rdf:Description rdf:about="http://www.cellml.org/cellml/1.0"> <cmeta:modification> <rdf:Description rdf:about="rdf:#b4886dbe-f37b-43fb-acb9-b0b8026f23fd"> <cmeta:modifier> <rdf:Description rdf:about="rdf:#c4116453-0d7f-44af-b629-9ff21e7465d6"> <vCard:N> <rdf:Description rdf:about="rdf:#928b4679-4c03-4745-be3c-b84d590156a6"> <vCard:Family>Kumar</vCard:Family> <vCard:Given>Vignesh</vCard:Given> </rdf:Description> </vCard:N> </rdf:Description> </cmeta:modifier> <rdf:value>Recoded model using parameters used by Biomodels Database as a guide</rdf:value> <dcterms:modified> <rdf:Description rdf:about="rdf:#9bcc4947-9c78-47c5-aa66-8b0418293b07"> <dcterms:W3CDTF>2007-12-06T14:07:34+13:00</dcterms:W3CDTF> </rdf:Description> </dcterms:modified> </rdf:Description> </cmeta:modification> <cmeta:modification> <rdf:Description rdf:about="rdf:#ccc38e4f-5767-4bd6-8196-a70b633111af"> <cmeta:modifier> <rdf:Description rdf:about="rdf:#e85662ee-d064-4750-93fa-ffdaa0305550"> <vCard:N> <rdf:Description rdf:about="rdf:#957cfa89-3116-450e-bbe9-7df9ee9426e6"> <vCard:Other>Richard</vCard:Other> <vCard:Family>Lawson</vCard:Family> <vCard:Given>James</vCard:Given> </rdf:Description> </vCard:N> </rdf:Description> </cmeta:modifier> <rdf:value>Fixed duplicate connections Fixed CellML 1.0/1.1 namespace mixing Updated documentation, curation status</rdf:value> <dcterms:modified> <rdf:Description rdf:about="rdf:#88f1c7e4-3b31-448c-a925-132833c561f3"> <dcterms:W3CDTF>2009-05-28T14:54:14+12:00</dcterms:W3CDTF> </rdf:Description> </dcterms:modified> </rdf:Description> </cmeta:modification> <dc:creator> <rdf:Description rdf:about="rdf:#9abd4459-6dec-4999-a3bb-3d66e8027fb7"> <vCard:N> <rdf:Description rdf:about="rdf:#e69d4683-efa8-4315-9c97-7b3f505fea2f"> <vCard:Other>May</vCard:Other> <vCard:Family>Lloyd</vCard:Family> <vCard:Given>Catherine</vCard:Given> </rdf:Description> </vCard:N> <vCard:EMAIL> <j.0:internet rdf:about="rdf:#2e4d42b3-7f5d-45ea-90bc-7ef76b410554"> <rdf:value>c.lloyd@auckland.ac.nz</rdf:value> </j.0:internet> </vCard:EMAIL> <vCard:ORG> <rdf:Description rdf:about="rdf:#309e7922-78cb-4837-8b97-05a213fe7ca0"> <vCard:Orgunit>Auckland Bioengineering Institute</vCard:Orgunit> <vCard:Orgname>The University of Auckland</vCard:Orgname> </rdf:Description> </vCard:ORG> </rdf:Description> </dc:creator> <dcterms:created> <rdf:Description rdf:about="rdf:#7a7077c2-db65-48b8-9710-5ac567ba66d9"> <dcterms:W3CDTF>2004-03-14T00:00:00+00:00</dcterms:W3CDTF> </rdf:Description> </dcterms:created> <cmeta:comment> <rdf:Description rdf:about="rdf:#f35cccde-ccfe-4150-8232-0c57ac47865f"> <rdf:value>This model has been curated and is known to run in PCEnv. The model is able to produce oscillating output similar to that shown in figures from the publication.</rdf:value> <dc:creator> <rdf:Description rdf:about="rdf:#49c5a7c9-2c51-4e11-92ea-47cd30b4e70b"> <vCard:FN>James Lawson</vCard:FN> </rdf:Description> </dc:creator> </rdf:Description> </cmeta:comment> <dc:publisher>The University of Auckland, Auckland Bioengineering Institute</dc:publisher> </rdf:Description> <rdf:Description rdf:about="http://www.cellml.org/cellml/1.0#bindschadler_sneyd_2001"> <bqs:reference> <rdf:Description rdf:about="rdf:#df80a4bf-a518-44b3-ae25-206aacbd8f83"> <bqs:JournalArticle> <rdf:Description rdf:about="rdf:#1a7bfc41-b4b7-4d7b-897e-04d16dbaaaff"> <bqs:last_page>246</bqs:last_page> <dcterms:issued> <rdf:Description rdf:about="rdf:#8b259212-c93b-4d73-880f-f4196e852e70"> <dcterms:W3CDTF>2001-03</dcterms:W3CDTF> </rdf:Description> </dcterms:issued> <bqs:Journal> <rdf:Description rdf:about="rdf:#e984bfb8-fe44-49b5-9a73-6eed0716afd8"> <dc:title>Chaos</dc:title> </rdf:Description> </bqs:Journal> <bqs:first_page>237</bqs:first_page> <bqs:volume>11</bqs:volume> <dc:title>A Bifurcation Analysis of Two Coupled Calcium Oscillators</dc:title> <dc:creator> <rdf:Seq rdf:about="rdf:#de8449ff-b4b5-4dff-87c5-eaf3c9f7cf5d"> <rdf:li> <bqs:Person rdf:about="rdf:#15671e0c-2590-4ad0-aa86-72fb2e846bb3"> <vCard:N> <rdf:Description rdf:about="rdf:#d2bcd970-a58c-4258-ac32-8937c3d4b1e3"> <vCard:Family>Bindschadler</vCard:Family> <vCard:Given>Michael</vCard:Given> </rdf:Description> </vCard:N> </bqs:Person> </rdf:li> <rdf:li> <bqs:Person rdf:about="rdf:#c95a2e51-3a66-41ef-8656-35229b6cd4d8"> <vCard:N> <rdf:Description rdf:about="rdf:#29569aeb-7683-4299-b932-8282c913a5c3"> <vCard:Family>Sneyd</vCard:Family> <vCard:Given>James</vCard:Given> </rdf:Description> </vCard:N> </bqs:Person> </rdf:li> </rdf:Seq> </dc:creator> </rdf:Description> </bqs:JournalArticle> <bqs:Pubmed_id>12779457</bqs:Pubmed_id> </rdf:Description> </bqs:reference> <bqs:reference> <rdf:Description rdf:about="rdf:#87f97fb5-0e18-4aaf-a14f-15cf01e848e6"> <dc:subject> <rdf:Description rdf:about="rdf:#42850c29-35cd-4245-8e59-1ddefdb54712"> <rdf:value> <rdf:Bag rdf:about="rdf:#0984ce3e-f973-40db-b74c-c1ae2db99b8e"> <rdf:li>bifurcation</rdf:li> <rdf:li>oscillator</rdf:li> <rdf:li>calcium dynamics</rdf:li> </rdf:Bag> </rdf:value> <bqs:subject_type>keyword</bqs:subject_type> </rdf:Description> </dc:subject> </rdf:Description> </bqs:reference> <cmeta:comment> <rdf:Description rdf:about="rdf:#d0921c45-1ac1-4a88-b7f6-37b221f9c9bd"> <rdf:value>This is the CellML description of Bindschadler and Sneyd's bifurcation analysis of two coupled calcium oscillators.</rdf:value> <dc:creator> <rdf:Description rdf:about="rdf:#f614dcd8-c814-4efb-9d7a-9b8b0327a3c1"> <vCard:FN>Catherine Lloyd</vCard:FN> </rdf:Description> </dc:creator> </rdf:Description> </cmeta:comment> <dc:title>Bindschadler and Sneyd's bifurcation analysis of two coupled calcium oscillators.</dc:title> <ns7:simulation> <rdf:Description rdf:about="rdf:#$bQgES"> <ns7:boundIntervals> <rdf:Description rdf:about="rdf:#$cQgES"> <rdf:first> <rdf:Description rdf:about="rdf:#$dQgES"> <ns7:maximumStepSize>0.001</ns7:maximumStepSize> <ns7:nonstandard-pointDensity>100000</ns7:nonstandard-pointDensity> <ns7:endingValue>50</ns7:endingValue> </rdf:Description> </rdf:first> <rdf:rest rdf:resource="http://www.w3.org/1999/02/22-rdf-syntax-ns#nil"/> </rdf:Description> </ns7:boundIntervals> </rdf:Description> </ns7:simulation> </rdf:Description> </rdf:RDF>
A bifurcation analysis of two coupled calcium oscillators
C:\Users\thomp_000\Dropbox\Work Related\Annotated Models\Bindschadler_2001\bindschadler_sneyd_2001.cellml
Chris Thompson
bindschadler_sneyd_2001
Chemical concentration
Chemical molar flow rate
calcium ion
pancreatic acinar cell
Endoplasmic reticulum
Intercellular junction
Plasma membrane
Portion of cytosol
Portion of extracellular matrix
A physical entity that is defined by linking multiple physical entity terms with structural relations.
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An element of a SemSim model that relates to the mathematical or logical representation of physical quantities or qualities.
A physical dependency defined in a specific SemSim model.
A physical entity defined in a specific SemSim model.
A physical process defined in a specific SemSim model.
A physical property defined in a specific SemSim model.
A computational element that stores numerical and/or logical information related to the model.
A data structure whose value is limited to decimals.
A data structure whose value is limited to the whole numbers.
A data structure unique to JSim's Mathematical Modeling Language (MML) used to set one or more model input values at runtime.
A quantitative relation between values of physical properties that occurs over a span of time and that determines the time-course of a physical process.
A physics analytical entity that represents the material, energy, and/or information content of material or immaterial, space-occupying continuants.
A SemSim model element used to represent, in whole or in part, the physical meaning of a computational element.
A physical processural entity that is a flow, or the control of a flow, of energy and/or information amongst physical continuants that are participants in the physical process.
A physical attribute of an instance of a physical entity, an occurrence of a physical process, or a physical dependency that has a quantitative value that is measurable by a physical device or method, or is computable from such values.
A data structure whose value is limited to the real numbers.
A physical dependency that is defined using a term in an external knowledge resource.
A physical entity that is defined using a term from an external knowledge resource.
A physical process that is defined using a term in an external knowledge resource.
A physical property that is defined using a term in an external knowledge resource.
A logical expression that constrains, but does not define, the value of a data structure. Example: x > 0.
A computational or physical element of a SemSim model.
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ATPase pumps
Ca flow from endoplasmic reticulum to cytoplasm in cell 1
Ca flow from endoplasmic reticulum to cytoplasm in cell 2
Ca leakage across cell membrane
Ca out of cell 1 cytoplasm to extracellular matrix via ATPase pumps
Ca out of cell 2 cytoplasm to extracellular matrix via ATPase pumps
First pancreatic acinar cell
IP3 Ca channel
Intercellular diffusion between two pancreatic acinar cells
Second pancreatic acinar cell
Cell 1 Ca concentration in the cytoplasm
Diffusion of Ca between coupled cells
Ca leakage across the cell membrane
Ca flow out of cell 1 via ATPase pumps
Ca flow through the IP3 receptor in cell 1
Time
<apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>c1</ci> </apply> <apply> <plus /> <apply> <minus /> <ci>j_receptor_c1</ci> <ci>j_pump_c1</ci> </apply> <ci>j_leak</ci> <ci>j_diffusion</ci> </apply> </apply>
Cell 2 Ca concentration in the cytoplasm
Ca flow out of cell 2 via ATPase pumps
Ca flow through the IP3 receptor in cell 2
<apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>c2</ci> </apply> <apply> <plus /> <apply> <minus /> <ci>j_receptor_c2</ci> <ci>j_pump_c2</ci> </apply> <ci>j_leak</ci> <ci>j_diffusion</ci> </apply> </apply>
Cell 1 active IP3 receptors on the endoplasmic reticulum in the open and shut states
IP3 receptor concentration
Rate of change in IP3 receptors from the closed state to open state in cell 1
Rate of change in IP3 receptors from the closed state to inactive state in cell 1
Rate of change in IP3 receptors from the inactive state to open state in cell 1
Rate of change in IP3 receptors from the open state to closed state in cell 1
<apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>h1</ci> </apply> <apply> <minus /> <apply> <times /> <ci>phi3_c1</ci> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>h1</ci> </apply> </apply> <apply> <divide /> <apply> <times /> <ci>phi1_c1</ci> <ci>phi2_c1</ci> <ci>h1</ci> <ci>p</ci> </apply> <apply> <plus /> <apply> <times /> <ci>phi1_c1</ci> <ci>p</ci> </apply> <ci>phi_1_c1</ci> </apply> </apply> </apply> </apply>
Cell 2 active IP3 receptors on the endoplasmic reticulum in the open and shut states
Rate of change in IP3 receptors from the closed state to open state in cell 2
Rate of change in IP3 receptors from the closed state to inactive state in cell 2
Rate of change in IP3 receptors from the inactive state to open state in cell 2
Rate of change in IP3 receptors from the open state to closed state in cell 2
<apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>h2</ci> </apply> <apply> <minus /> <apply> <times /> <ci>phi3_c2</ci> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>h2</ci> </apply> </apply> <apply> <divide /> <apply> <times /> <ci>phi1_c2</ci> <ci>phi2_c2</ci> <ci>h2</ci> <ci>p</ci> </apply> <apply> <plus /> <apply> <times /> <ci>phi1_c2</ci> <ci>p</ci> </apply> <ci>phi_1_c2</ci> </apply> </apply> </apply> </apply>
Intercellular coupling permeability rate constant.
<apply> <eq /> <ci>j_diffusion</ci> <apply> <times /> <ci>D</ci> <apply> <minus /> <ci>c2</ci> <ci>c1</ci> </apply> </apply> </apply>
Constant for calculation of Ca flow via ATPase pumps
<apply> <eq /> <ci>j_pump_c1</ci> <apply> <divide /> <apply> <times /> <ci>Vp</ci> <apply> <power /> <ci>c1</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci>Kp</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> <apply> <power /> <ci>c1</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </apply> </apply> <apply> <eq /> <ci>j_pump_c2</ci> <apply> <divide /> <apply> <times /> <ci>Vp</ci> <apply> <power /> <ci>c2</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci>Kp</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> <apply> <power /> <ci>c2</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </apply> </apply>
Constant for calculation of Ca flow via IP3 receptor channels
<apply> <eq /> <ci>j_receptor_c1</ci> <apply> <times /> <ci>kf</ci> <apply> <power /> <apply> <divide /> <apply> <times /> <ci>p</ci> <ci>h1</ci> <ci>phi1_c1</ci> </apply> <apply> <plus /> <apply> <times /> <ci>phi1_c1</ci> <ci>p</ci> </apply> <ci>phi_1_c1</ci> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">4</cn> </apply> </apply> </apply> <apply> <eq /> <ci>j_receptor_c2</ci> <apply> <times /> <ci>kf</ci> <apply> <power /> <apply> <divide /> <apply> <times /> <ci>p</ci> <ci>h2</ci> <ci>phi1_c2</ci> </apply> <apply> <plus /> <apply> <times /> <ci>phi1_c2</ci> <ci>p</ci> </apply> <ci>phi_1_c2</ci> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">4</cn> </apply> </apply> </apply>
Constant for rate equation represented by phi 1
Constant for rate equation represented by phi 2
Constant for rate equation represented by phi 3
Constant for rate equation represented by phi -1
<apply> <eq /> <ci>phi1_c1</ci> <apply> <divide /> <apply> <times /> <ci>r2</ci> <ci>c1</ci> </apply> <apply> <plus /> <ci>R1</ci> <ci>c1</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi_1_c1</ci> <apply> <divide /> <ci>k1</ci> <apply> <plus /> <ci>R3</ci> <ci>c1</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi2_c1</ci> <apply> <divide /> <apply> <plus /> <ci>k2</ci> <apply> <times /> <ci>r4</ci> <ci>c1</ci> </apply> </apply> <apply> <plus /> <ci>R3</ci> <ci>c1</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi3_c1</ci> <apply> <divide /> <ci>k3</ci> <apply> <plus /> <ci>R5</ci> <ci>c1</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi1_c2</ci> <apply> <divide /> <apply> <times /> <ci>r2</ci> <ci>c2</ci> </apply> <apply> <plus /> <ci>R1</ci> <ci>c2</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi_1_c2</ci> <apply> <divide /> <ci>k1</ci> <apply> <plus /> <ci>R3</ci> <ci>c2</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi2_c2</ci> <apply> <divide /> <apply> <plus /> <ci>k2</ci> <apply> <times /> <ci>r4</ci> <ci>c2</ci> </apply> </apply> <apply> <plus /> <ci>R3</ci> <ci>c2</ci> </apply> </apply> </apply> <apply> <eq /> <ci>phi3_c2</ci> <apply> <divide /> <ci>k3</ci> <apply> <plus /> <ci>R5</ci> <ci>c2</ci> </apply> </apply> </apply>