<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : goforth_model_II_2002.xml
CREATED : 6th November 2002
LAST MODIFIED : 20th April 2005
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 16/01/2002 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of Goforth et al's 2002 mathematical model of Ca2+-activated K+ channels in pancreatic beta cells.
CHANGES:
09/04/2003 - AAC - Added publication date information.
20/04/2005 - PJV - Made MathML id's unique
--><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="goforth_bertram_khan_zhang_sherman_satin_2002_version02" name="goforth_bertram_khan_zhang_sherman_satin_2002_version02">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>Calcium-activated K+ Channels in Pancreatic Beta-Cells</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 is the original unchecked version of the model imported from the previous
CellML model repository, 24-Jan-2006.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
Pancreatic beta cells of the islets of Langerhans play an important role in the control of blood glucose levels by secreting insulin in response to an increase in extracellular glucose concentration. Increased glucose concentration also induces rhythmical electrical activity in beta cells. This activity consists of the periodic plateaus and spikes of Ca<superscript>2+</superscript>-dependent action potentials, which enhance Ca<superscript>2+</superscript> influx and insulin secretion. Two K<subscript>Ca</subscript> channels are present in beta cells: a voltage dependent, "fast", large conductance K<subscript>Ca</subscript>, and a slowly activating and deactivating K<subscript>Ca</subscript> current, K<subscript>slow</subscript>. K<subscript>slow</subscript> is thought to be responsible for islet pacemaking.
</para>
<para>
Experimental data suggest that both cytosolic calcium and calcium from the endoplasmic reticulum (ER) play an essential role in the activation of K<subscript>slow</subscript> channels. In their 2002 paper, P.B. Goforth, R. Bertram, F.A. Khan, M. Zhang, A. Sherman, and L.S. Satin develop two mathematical models to test this theory (see <xref linkend="fig_cell_diagram"/> below). Their first minimal model lacks a separate subspace, and they found that this two component model (cytosol and ER) was insufficient to explain Ca<superscript>2+</superscript>-induced activation of K<subscript>slow</subscript> channels. However, their second model incorporated a localised pool of Ca<superscript>2+</superscript> in a subspace located between the plasma membrane and the ER. In this model, subspace [Ca<superscript>2+</superscript>] is positively correlated with changes in cytosolic [Ca<superscript>2+</superscript>], but with a gradient that reflects Ca<superscript>2+</superscript> efflux from the ER. The simulations from this model were consistent with experimental data. </para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://www.jgp.org/cgi/content/abstract/120/3/307">Calcium-activated K<superscript>+</superscript> Channels of Mouse Beta-cells are Controlled by Both Store and Cytoplasmic Ca<superscript>2+</superscript>: Experimental and Theoretical Studies</ulink>, P.B. Goforth, R. Bertram, F.A. Khan, M. Zhang, A. Sherman, and L.S. Satin, 2002, <ulink url="http://www.jgp.org/">
<emphasis>Journal of General Physiology</emphasis>
</ulink>, 120, 307-322. (<ulink url="http://www.jgp.org/cgi/content/full/120/3/307">Full text</ulink> and <ulink url="http://www.jgp.org/cgi/reprint/120/3/307.pdf">PDF versions</ulink> of the article are available to subscribers on the Journal of General Physiology website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=12198088&dopt=Abstract">PubMed ID: 12198088</ulink>
</para>
<para>
The raw CellML descriptions of the model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>.
</para>
<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>cell_diagram</title>
</objectinfo>
<imagedata fileref="goforth_2002.png"/>
</imageobject>
</mediaobject>
<caption>Schematic diagram of the model of K<subscript>slow</subscript> activation by Ca<superscript>2+</superscript>. Ca<superscript>2+</superscript> entering the beta-cell through voltage gated Ca<superscript>2+</superscript> channels (<emphasis>I<subscript>Ca</subscript>
</emphasis>) is either pumped out of the cell by the Ca<superscript>2+</superscript> ATPase (<emphasis>J<subscript>PMCA</subscript>
</emphasis>), or by Na<superscript>+</superscript>/Ca<superscript>2+</superscript> exchange, or it is pumped into the ER by SERCA (<emphasis>J<subscript>SERCA</subscript>
</emphasis>). Ca<superscript>2+</superscript> leaves the ER through specialised channels close to the K<subscript>slow</subscript> channels (<emphasis>J<subscript>RELEASE</subscript>
</emphasis> and <emphasis>J<subscript>X</subscript>
</emphasis>). This produces a local calcium concentration gradient in the subspace. Subspace calcium is passively exchanged with cytosolic calcium.</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="millisecond">
<unit units="second" prefix="milli"/>
</units>
<units name="per_millisecond">
<unit units="millisecond" exponent="-1"/>
</units>
<units name="femtoA">
<unit units="ampere" prefix="femto"/>
</units>
<units name="femtoF">
<unit units="farad" prefix="femto"/>
</units>
<units name="millivolt">
<unit units="volt" prefix="milli"/>
</units>
<units name="micromolar">
<unit units="mole" prefix="micro"/>
<unit units="litre" exponent="-1"/>
</units>
<units name="flux">
<unit units="micromolar"/>
<unit units="millisecond" exponent="-1"/>
</units>
<units name="picoS">
<unit units="siemens" prefix="pico"/>
</units>
<units name="micromolar_per_fA_ms">
<unit units="micromolar"/>
<unit units="femtoA" exponent="-1"/>
<unit units="millisecond" exponent="-1"/>
</units>
<units name="coulomb_per_mole">
<unit units="coulomb" exponent="-1"/>
<unit units="mole"/>
</units>
<units name="microlitre">
<unit units="litre" prefix="micro"/>
</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 units="millisecond" public_interface="out" name="time"/>
</component>
<component name="membrane">
<variable units="millivolt" public_interface="out" name="V" initial_value="-65.0"/>
<variable units="femtoF" name="Cm" initial_value="5300.0"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="femtoA" public_interface="in" name="i_Ca"/>
<variable units="femtoA" public_interface="in" name="i_K"/>
<variable units="femtoA" public_interface="in" name="i_KATP"/>
<variable units="femtoA" public_interface="in" name="i_KCa"/>
<variable units="femtoA" public_interface="in" name="i_Leak"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="membrane_voltage_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> V </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<plus/>
<ci> i_Ca </ci>
<ci> i_K </ci>
<ci> i_KATP </ci>
<ci> i_KCa </ci>
<ci> i_Leak </ci>
</apply>
</apply>
<ci> Cm </ci>
</apply>
</apply>
</math>
</component>
<component name="calcium_current">
<variable units="femtoA" public_interface="out" name="i_Ca"/>
<variable units="millivolt" name="V_Ca" initial_value="30.0"/>
<variable units="picoS" name="g_Ca" initial_value="1450.0"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" private_interface="in" name="m_infinity"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Ca_calculation">
<eq/>
<ci> i_Ca </ci>
<apply>
<times/>
<ci> g_Ca </ci>
<ci> m_infinity </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Ca </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_current_m_gate">
<variable units="dimensionless" public_interface="out" name="m_infinity"/>
<variable units="millivolt" name="Vm" initial_value="-13.0"/>
<variable units="millivolt" name="Sm" initial_value="8.0"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="m_infinity_calculation">
<eq/>
<ci> m_infinity </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> Vm </ci>
<ci> V </ci>
</apply>
<ci> Sm </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="rapidly_activating_K_current">
<variable units="femtoA" public_interface="out" name="i_K"/>
<variable units="millivolt" public_interface="out" name="V_K" initial_value="-70.0"/>
<variable units="picoS" name="g_K" initial_value="2500.0"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="millivolt" public_interface="in" private_interface="out" name="V"/>
<variable units="dimensionless" private_interface="in" name="n"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_K_calculation">
<eq/>
<ci> i_K </ci>
<apply>
<times/>
<ci> g_K </ci>
<ci> n </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="rapidly_activating_K_current_n_gate">
<variable units="dimensionless" public_interface="out" name="n" initial_value="0.0001"/>
<variable units="dimensionless" name="n_infinity"/>
<variable units="millisecond" name="tau_n" initial_value="10.8"/>
<variable units="millivolt" name="Vn" initial_value="-15.0"/>
<variable units="millivolt" name="Sn" initial_value="5.6"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="second" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="n_diff_eq">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> n </ci>
</apply>
<apply>
<divide/>
<apply>
<minus/>
<ci> n_infinity </ci>
<ci> n </ci>
</apply>
<ci> tau_n </ci>
</apply>
</apply>
<apply id="n_infinity_calculation">
<eq/>
<ci> n_infinity </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<plus/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<exp/>
<apply>
<divide/>
<apply>
<minus/>
<ci> Vn </ci>
<ci> V </ci>
</apply>
<ci> Sn </ci>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Ca_activated_K_current">
<variable units="femtoA" public_interface="out" name="i_KCa"/>
<variable units="picoS" name="g_KCa" initial_value="1200.0"/>
<variable units="millivolt" public_interface="in" name="V_K"/>
<variable units="millisecond" public_interface="in" private_interface="out" name="time"/>
<variable units="micromolar" public_interface="in" private_interface="out" name="c_SS"/>
<variable units="millivolt" public_interface="in" name="V"/>
<variable units="dimensionless" private_interface="in" name="w"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="Ca_activated_K_current_i_KCa_calculation">
<eq/>
<ci> i_KCa </ci>
<apply>
<times/>
<ci> g_KCa </ci>
<ci> w </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="Ca_activated_K_current_w_gate">
<variable units="dimensionless" public_interface="out" name="w"/>
<variable units="micromolar" name="Kd" initial_value="0.7"/>
<variable units="dimensionless" name="q" initial_value="8.0"/>
<variable units="micromolar" public_interface="in" name="c_SS"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="w_calculation">
<eq/>
<ci> w </ci>
<apply>
<divide/>
<apply>
<power/>
<ci> c_SS </ci>
<ci> q </ci>
</apply>
<apply>
<plus/>
<apply>
<power/>
<ci> c_SS </ci>
<ci> q </ci>
</apply>
<apply>
<power/>
<ci> Kd </ci>
<ci> q </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="ATP_dependent_K_current">
<variable units="femtoA" public_interface="out" name="i_KATP"/>
<variable units="picoS" name="g_KATP" initial_value="63.0"/>
<variable units="millivolt" public_interface="in" name="V_K"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="ATP_dependent_K_current_i_KCa_calculation">
<eq/>
<ci> i_KATP </ci>
<apply>
<times/>
<ci> g_KATP </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_K </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="leak_current">
<variable units="femtoA" public_interface="out" name="i_Leak"/>
<variable units="picoS" name="g_Leak" initial_value="14.0"/>
<variable units="millivolt" name="V_Leak" initial_value="-30.0"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="millivolt" public_interface="in" name="V"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="i_Leak_calculation">
<eq/>
<ci> i_Leak </ci>
<apply>
<times/>
<ci> g_Leak </ci>
<apply>
<minus/>
<ci> V </ci>
<ci> V_Leak </ci>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_concentrations">
<variable units="micromolar" public_interface="out" name="c" initial_value="0.027"/>
<variable units="micromolar" public_interface="out" name="c_ER" initial_value="111.17"/>
<variable units="micromolar" public_interface="out" name="c_SS" initial_value="0.29"/>
<variable units="dimensionless" name="f_CYT" initial_value="0.01"/>
<variable units="dimensionless" name="f_ER" initial_value="0.005"/>
<variable units="dimensionless" name="f_SS" initial_value="0.04"/>
<variable units="micromolar_per_fA_ms" name="alpha" initial_value="4.5E-6"/>
<variable units="coulomb_per_mole" name="F" initial_value="96845.0"/>
<variable units="microlitre" name="V_CYT"/>
<variable units="microlitre" name="V_ER"/>
<variable units="microlitre" name="V_SS"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="femtoA" public_interface="in" name="i_Ca"/>
<variable units="flux" public_interface="in" name="J_PMCA"/>
<variable units="flux" public_interface="in" name="J_SERCA"/>
<variable units="flux" public_interface="in" name="J_RELEASE"/>
<variable units="flux" public_interface="in" name="J_X"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="c_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> c </ci>
</apply>
<apply>
<times/>
<ci> f_CYT </ci>
<apply>
<plus/>
<apply>
<minus/>
<apply>
<plus/>
<apply>
<times/>
<ci> alpha </ci>
<ci> i_Ca </ci>
</apply>
<ci> J_PMCA </ci>
<ci> J_SERCA </ci>
</apply>
</apply>
<ci> J_X </ci>
</apply>
</apply>
</apply>
<apply id="alpha_calculation">
<eq/>
<ci> alpha </ci>
<apply>
<divide/>
<cn cellml:units="dimensionless"> 1.0 </cn>
<apply>
<times/>
<cn cellml:units="dimensionless"> 2.0 </cn>
<ci> F </ci>
<ci> V_CYT </ci>
</apply>
</apply>
</apply>
<apply id="V_CYT_calculation1">
<eq/>
<ci> V_CYT </ci>
<apply>
<times/>
<cn cellml:units="dimensionless"> 25.0 </cn>
<ci> V_ER </ci>
</apply>
</apply>
<apply id="V_SS_calculation">
<eq/>
<ci> V_SS </ci>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.4 </cn>
<ci> V_CYT </ci>
</apply>
</apply>
<apply id="V_ER_calculation">
<eq/>
<ci> V_ER </ci>
<apply>
<times/>
<cn cellml:units="dimensionless"> 0.1 </cn>
<ci> V_SS </ci>
</apply>
</apply>
<apply id="c_ER_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> c_ER </ci>
</apply>
<apply>
<times/>
<ci> f_ER </ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<divide/>
<ci> V_CYT </ci>
<ci> V_ER </ci>
</apply>
<ci> J_SERCA </ci>
</apply>
<ci> J_RELEASE </ci>
</apply>
</apply>
</apply>
<apply id="c_SS_dt">
<eq/>
<apply>
<diff/>
<bvar>
<ci> time </ci>
</bvar>
<ci> c_SS </ci>
</apply>
<apply>
<times/>
<ci> f_SS </ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<divide/>
<ci> V_ER </ci>
<ci> V_SS </ci>
</apply>
<ci> J_RELEASE </ci>
</apply>
<apply>
<times/>
<apply>
<divide/>
<ci> V_CYT </ci>
<ci> V_SS </ci>
</apply>
<ci> J_X </ci>
</apply>
</apply>
</apply>
</apply>
</math>
</component>
<component name="calcium_fluxes">
<variable units="flux" public_interface="out" name="J_PMCA"/>
<variable units="flux" public_interface="out" name="J_SERCA"/>
<variable units="flux" public_interface="out" name="J_RELEASE"/>
<variable units="flux" public_interface="out" name="J_X"/>
<variable units="per_millisecond" name="k_PMCA" initial_value="0.18"/>
<variable units="per_millisecond" name="k_SERCA" initial_value="0.1"/>
<variable units="per_millisecond" name="p_ER" initial_value="0.0015"/>
<variable units="per_millisecond" name="p_X" initial_value="0.025"/>
<variable units="micromolar" public_interface="in" name="c"/>
<variable units="micromolar" public_interface="in" name="c_ER"/>
<variable units="micromolar" public_interface="in" name="c_SS"/>
<variable units="millisecond" public_interface="in" name="time"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply id="J_PMCA_calculation">
<eq/>
<ci> J_PMCA </ci>
<apply>
<times/>
<ci> k_PMCA </ci>
<ci> c </ci>
</apply>
</apply>
<apply id="J_SERCA_calculation">
<eq/>
<ci> J_SERCA </ci>
<apply>
<times/>
<ci> k_SERCA </ci>
<ci> c </ci>
</apply>
</apply>
<apply id="J_RELEASE_calculation">
<eq/>
<ci> J_RELEASE </ci>
<apply>
<times/>
<ci> p_ER </ci>
<apply>
<minus/>
<ci> c_ER </ci>
<ci> c_SS </ci>
</apply>
</apply>
</apply>
<apply id="J_X_calculation">
<eq/>
<ci> J_X </ci>
<apply>
<times/>
<ci> p_X </ci>
<apply>
<minus/>
<ci> c_SS </ci>
<ci> c </ci>
</apply>
</apply>
</apply>
</math>
</component>
<group>
<relationship_ref relationship="containment"/>
<component_ref component="membrane">
<component_ref component="calcium_current">
<component_ref component="calcium_current_m_gate"/>
</component_ref>
<component_ref component="rapidly_activating_K_current">
<component_ref component="rapidly_activating_K_current_n_gate"/>
</component_ref>
<component_ref component="Ca_activated_K_current">
<component_ref component="Ca_activated_K_current_w_gate"/>
</component_ref>
<component_ref component="ATP_dependent_K_current"/>
<component_ref component="leak_current"/>
<component_ref component="calcium_concentrations"/>
<component_ref component="calcium_fluxes"/>
</component_ref>
</group>
<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="calcium_current">
<component_ref component="calcium_current_m_gate"/>
</component_ref>
<component_ref component="rapidly_activating_K_current">
<component_ref component="rapidly_activating_K_current_n_gate"/>
</component_ref>
<component_ref component="Ca_activated_K_current">
<component_ref component="Ca_activated_K_current_w_gate"/>
</component_ref>
</group>
<connection>
<map_components component_2="environment" component_1="membrane"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="calcium_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="rapidly_activating_K_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="Ca_activated_K_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="ATP_dependent_K_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="leak_current"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
<map_components component_2="environment" component_1="calcium_concentrations"/>
<map_variables variable_2="time" variable_1="time"/>
</connection>
<connection>
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