/home/aram148/Documents/547/ostby_oyehaug_einevoll_nagelhus_plahte_zeuthen_voipio_lloyd_ottersen_omholt_2008.cellml 0.2 19165313 <documentation xmlns="http://cellml.org/tmp-documentation"> <article> <articleinfo> <title>Astrocytic processes explaining neural activity-induced shrinkage of extraneuronal space</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 is known to run in OpenCell and COR to recreate the published results. The units have been checked and they are consistent.</para> </section> <sect1 id="sec_structure"> <title>Model Structure</title> <para>ABSTRACT: Neuronal stimulation causes approximately 30% shrinkage of the extracellular space (ECS) between neurons and surrounding astrocytes in grey and white matter under experimental conditions. Despite its possible implications for a proper understanding of basic aspects of potassium clearance and astrocyte function, the phenomenon remains unexplained. Here we present a dynamic model that accounts for current experimental data related to the shrinkage phenomenon in wild-type as well as in gene knockout individuals. We find that neuronal release of potassium and uptake of sodium during stimulation, astrocyte uptake of potassium, sodium, and chloride in passive channels, action of the Na/K/ATPase pump, and osmotically driven transport of water through the astrocyte membrane together seem sufficient for generating ECS shrinkage as such. However, when taking into account ECS and astrocyte ion concentrations observed in connection with neuronal stimulation, the actions of the Na(+)/K(+)/Cl(-) (NKCC1) and the Na(+)/HCO(3) (-) (NBC) cotransporters appear to be critical determinants for achieving observed quantitative levels of ECS shrinkage. Considering the current state of knowledge, the model framework appears sufficiently detailed and constrained to guide future key experiments and pave the way for more comprehensive astroglia-neuron interaction models for normal as well as pathophysiological situations.</para> <informalfigure float="0" id="fig_reaction_diagram"> <mediaobject> <imageobject> <objectinfo> <title>model diagram</title> </objectinfo> <imagedata fileref="ostby_2008.png" /> </imageobject> </mediaobject> <caption>Schematic diagram of the model. In the excited state there is an exchange of sodium and potassium ions across the neuron membrane and the cotransporters in the astrocyte are active.</caption> </informalfigure> <para>The original paper reference is cited below:</para> <para> Astrocytic mechanisms explaining neural-activity-induced shrinkage of extraneuronal space, Ostby I, Oyehaug L, Einevoll GT, Nagelhus EA, Plahte E, Zeuthen T, Lloyd CM, Ottersen OP, and Omholt SW, 2009, PLoS Computational Biology, 5, 1, e1000272. <ulink url="http://www.ncbi.nlm.nih.gov/pubmed/19165313">PubMed ID: 19165313</ulink> </para> </sect1> </article> </documentation> <rdf:RDF xmlns:dcterms="http://purl.org/dc/terms/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:dc="http://purl.org/dc/elements/1.1/"> <rdf:Description rdf:about="#wi" dc:title="wi"> <dcterms:alternative>ratio of astrocyte volume to astrocyte area</dcterms:alternative> </rdf:Description> </rdf:RDF> ostby_model_2008 ostby_model_2008 Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 Particle count http://bhi.washington.edu/OPB#OPB_01001 Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 Nernst potential http://bhi.washington.edu/OPB#OPB_01581 hydrogencarbonate http://purl.obolibrary.org/obo/CHEBI_17544 chloride http://purl.obolibrary.org/obo/CHEBI_17996 sodium(1+) http://purl.obolibrary.org/obo/CHEBI_29101 potassium(1+) http://purl.obolibrary.org/obo/CHEBI_29103 solute carrier family 12 member 2 http://purl.obolibrary.org/obo/PR_000014925 solute carrier family 12 member 4 http://purl.obolibrary.org/obo/PR_000014927 electrogenic sodium bicarbonate cotransporter 1 http://purl.obolibrary.org/obo/PR_000015160 Membrane http://purl.org/sig/ont/fma/fma30322 Astrocyte http://purl.org/sig/ont/fma/fma54537 Extracellular space http://purl.org/sig/ont/fma/fma70022 A physical entity that is defined by linking multiple physical entity terms with structural relations. A mathematical and/or logical expression that determines the value of a data structure. 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 discrete occurence in a simulation A computational assignment triggered by an Event 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. The output of an SBML-style Function Definition A computational or physical element of a SemSim model. A grouping of one or more data structures that comprise a computationally or physically distinct part of a model or submodel. A physical quantity chosen as a standard in terms of which other physical quantities may be expressed. hydrogencarbonate http://purl.obolibrary.org/obo/CHEBI_17544 hydrogencarbonate http://purl.obolibrary.org/obo/CHEBI_17544 chloride http://purl.obolibrary.org/obo/CHEBI_17996 chloride http://purl.obolibrary.org/obo/CHEBI_17996 sodium(1+) http://purl.obolibrary.org/obo/CHEBI_29101 sodium(1+) http://purl.obolibrary.org/obo/CHEBI_29101 potassium(1+) http://purl.obolibrary.org/obo/CHEBI_29103 potassium(1+) http://purl.obolibrary.org/obo/CHEBI_29103 Cl- flow through astrocytic membrane 1.0 1.0 ENBC ENBC.ENBC out Reversal Nernst potential of the sodium bicarbonate cotransporter Nernst potential http://bhi.washington.edu/OPB#OPB_01581 ENBC.F in ENBC.HCO3i in ENBC.HCO3o in ENBC.Nai in ENBC.Nao in ENBC.R in ENBC.T in -1 ENBC.zNBC <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>ENBC</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <ci>R</ci> <ci>T</ci> </apply> <apply> <times /> <ci>zNBC</ci> <ci>F</ci> </apply> </apply> <apply> <ln /> <apply> <divide /> <apply> <times /> <apply> <divide /> <ci>Nao</ci> <ci>Nai</ci> </apply> <apply> <power /> <ci>HCO3o</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> <apply> <power /> <ci>HCO3i</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </apply> </apply> </apply> </math> I_Cl I_Cl.ECl in Nernst potential of the Cl- ion for the astrocytic potential. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 I_Cl_I_Cl out Cloride ion flux across astrocytic membrane. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 I_Cl.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 I_Cl.elementary_charge in I_Cl.gCl in I_Cl.nA in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>I_Cl</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gCl</ci> </apply> <apply> <times /> <ci>elementary_charge</ci> <ci>nA</ci> </apply> </apply> <apply> <minus /> <ci>Vm</ci> <ci>ECl</ci> </apply> </apply> </apply> </math> I_K I_K.EK in Nernst potential of K ion for the astrosytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 I_K.F in I_K_I_K out Flux of potassium ions across astrocytic membrane. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 I_K.J_NaKATPase in I_K.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 I_K.elementary_charge in I_K.gK in I_K.nA in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>I_K</ci> <apply> <minus /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> <ci>J_NaKATPase</ci> <ci>F</ci> </apply> <apply> <times /> <ci>elementary_charge</ci> <ci>nA</ci> </apply> </apply> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gK</ci> </apply> <apply> <times /> <ci>elementary_charge</ci> <ci>nA</ci> </apply> </apply> <apply> <minus /> <ci>Vm</ci> <ci>EK</ci> </apply> </apply> </apply> </apply> </math> I_Na I_Na.ENa in Nernst potential of sodium for the astrocytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 I_Na.F in I_Na_I_Na out Na+ ion flux across astrocytic membrane. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 I_Na.J_NBC in Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 I_Na.J_NaKATPase in I_Na.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 I_Na.elementary_charge in I_Na.gNa in I_Na.nA in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>I_Na</ci> <apply> <minus /> <ci>J_NBC</ci> <apply> <plus /> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gNa</ci> </apply> <apply> <times /> <ci>elementary_charge</ci> <ci>nA</ci> </apply> </apply> <apply> <minus /> <ci>Vm</ci> <ci>ENa</ci> </apply> </apply> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">3</cn> <ci>J_NaKATPase</ci> <ci>F</ci> </apply> <apply> <times /> <ci>elementary_charge</ci> <ci>nA</ci> </apply> </apply> </apply> </apply> </apply> </math> Ion flux through sodium-potassium-chloride cotransporter 1.0 1.0 1.0 1.0 1.0 1.0 Ion flux through sodium bicarbonate cotransporter 1.0 1.0 1.0 1.0 J_KCC1 J_KCC1.Cli in J_KCC1.Clo in J_KCC1.F in J_KCC1_J_KCC1 out Ion flux through the potassium-cloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 J_KCC1.Ki in J_KCC1.Ko in J_KCC1.R in J_KCC1.T in J_KCC1.ft 1e-6 J_KCC1.gKCC1 J_KCC1.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>J_KCC1</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gKCC1</ci> <ci>ft</ci> </apply> <ci>F</ci> </apply> <ci>R</ci> <ci>T</ci> </apply> <ci>F</ci> </apply> <apply> <ln /> <apply> <divide /> <apply> <times /> <ci>Ko</ci> <ci>Clo</ci> </apply> <apply> <times /> <ci>Ki</ci> <ci>Cli</ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci>ft</ci> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <apply> <and /> <apply> <geq /> <ci>time</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second">10</cn> </apply> <apply> <lt /> <ci>time</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second">20</cn> </apply> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">0</cn> </otherwise> </piecewise> </apply> </math> J_NBC J_NBC.ENBC in Reversal Nernst potential of the sodium bicarbonate cotransporter Nernst potential http://bhi.washington.edu/OPB#OPB_01581 J_NBC.F in J_NBC_J_NBC out Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 J_NBC.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 9.03e-5 J_NBC.gNBC out <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>J_NBC</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gNBC</ci> </apply> <ci>F</ci> </apply> <apply> <minus /> <ci>Vm</ci> <ci>ENBC</ci> </apply> </apply> </apply> </math> J_NKCC1 J_NKCC1.Cli in J_NKCC1.Clo in J_NKCC1.F in J_NKCC1_J_NKCC1 out Ion flux through the sodium-potassium-chloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 J_NKCC1.Ki in J_NKCC1.Ko in J_NKCC1.Nai in J_NKCC1.Nao in J_NKCC1.R in J_NKCC1.T in J_NKCC1.ft 5.54e-6 J_NKCC1.gNKCC1 J_NKCC1.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>J_NKCC1</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gNKCC1</ci> <ci>ft</ci> </apply> <ci>F</ci> </apply> <ci>R</ci> <ci>T</ci> </apply> <ci>F</ci> </apply> <apply> <ln /> <apply> <times /> <apply> <divide /> <apply> <times /> <apply> <divide /> <ci>Nao</ci> <ci>Nai</ci> </apply> <ci>Ko</ci> </apply> <ci>Ki</ci> </apply> <apply> <power /> <apply> <divide /> <ci>Clo</ci> <ci>Cli</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci>ft</ci> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <apply> <and /> <apply> <geq /> <ci>time</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second">10</cn> </apply> <apply> <lt /> <ci>time</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="second">20</cn> </apply> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">0</cn> </otherwise> </piecewise> </apply> </math> J_NaKATPase J_NaKATPase_J_NaKATPase out 1.5 J_NaKATPase.K_Ko 10 J_NaKATPase.K_Nai J_NaKATPase.Ko in J_NaKATPase.Nai in 1.4207e-10 J_NaKATPase.Pmax <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>J_NaKATPase</ci> <apply> <divide /> <apply> <times /> <apply> <divide /> <apply> <times /> <ci>Pmax</ci> <apply> <power /> <ci>Nai</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1.5</cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci>Nai</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1.5</cn> </apply> <apply> <power /> <ci>K_Nai</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1.5</cn> </apply> </apply> </apply> <ci>Ko</ci> </apply> <apply> <plus /> <ci>Ko</ci> <ci>K_Ko</ci> </apply> </apply> </apply> </math> K+ and Cl- ion flux through the astrocytic membrane. 1.0 1.0 1.0 1.0 K+ ions flow across astrocytic membrane 1.0 1.0 N_Cli 0.32879 N_Cli.N_Cli out Time rate of change in the number of Cl- ions in the intracellular space of the astrocyte Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Cli.dN_HCO3i_dt in N_Cli.dN_Ki_dt in N_Cli.dN_Nai_dt in N_Cli.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_Cli</ci> </apply> <apply> <plus /> <ci>dN_Ki_dt</ci> <ci>dN_Nai_dt</ci> <ci>dN_HCO3i_dt</ci> </apply> </apply> </math> N_Clo N_Clo.N_Clo out N_Clo.N_HCO3o in Rate of change in the number of HCO3 ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Clo.N_Ko in Rate of change in the number of potassium ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Clo.N_Nao in Rate of change in the number of Na ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>N_Clo</ci> <apply> <minus /> <apply> <plus /> <ci>N_Nao</ci> <ci>N_Ko</ci> </apply> <ci>N_HCO3o</ci> </apply> </apply> </math> N_HCO3i N_HCO3i.J_NBC in Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 0.58804 N_HCO3i.N_HCO3i out Rate of change in the number of HCO3 ions Particle count http://bhi.washington.edu/OPB#OPB_01001 N_HCO3i.dN_HCO3i_dt out N_HCO3i.theta_NBC in N_HCO3i.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_HCO3i</ci> </apply> <ci>dN_HCO3i_dt</ci> </apply> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>dN_HCO3i_dt</ci> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> <ci>theta_NBC</ci> <ci>J_NBC</ci> </apply> </apply> </math> N_HCO3o N_HCO3o.J_NBC in Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 0.432552 N_HCO3o.N_HCO3o out Rate of change in the number of HCO3 ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_HCO3o.theta_NBC in N_HCO3o.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_HCO3o</ci> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> <ci>theta_NBC</ci> <ci>J_NBC</ci> </apply> </apply> </math> N_Ki N_Ki.EK in Nernst potential of K ion for the astrosytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 N_Ki.F in N_Ki.J_KCC1 in Ion flux through the potassium-cloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Ki.J_NKCC1 in Ion flux through the sodium-potassium-chloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Ki.J_NaKATPase in 5.52782 N_Ki.N_Ki out Rate of change in the number of potassium ions in the intracellular space of the astrocyte Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Ki.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 N_Ki.dN_Ki_dt out N_Ki.gK in N_Ki.theta_KCC1 in N_Ki.theta_NKCC1 in N_Ki.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_Ki</ci> </apply> <ci>dN_Ki_dt</ci> </apply> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>dN_Ki_dt</ci> <apply> <minus /> <apply> <plus /> <apply> <times /> <ci>theta_NKCC1</ci> <ci>J_NKCC1</ci> </apply> <apply> <times /> <ci>theta_KCC1</ci> <ci>J_KCC1</ci> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> <ci>J_NaKATPase</ci> </apply> </apply> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gK</ci> </apply> <ci>F</ci> </apply> <apply> <minus /> <ci>Vm</ci> <ci>EK</ci> </apply> </apply> </apply> </apply> </math> N_Ko N_Ko.EK in Nernst potential of K ion for the astrosytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 N_Ko.F in N_Ko.J_KCC1 in Ion flux through the potassium-cloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Ko.J_NKCC1 in Ion flux through the sodium-potassium-chloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Ko.J_NaKATPase in 0.0807 N_Ko_N_Ko out Rate of change in the number of potassium ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Ko.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 N_Ko.gK in N_Ko.kc in N_Ko.theta_KCC1 in N_Ko.theta_NKCC1 in N_Ko.time in N_Ko.y in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_Ko</ci> </apply> <apply> <minus /> <apply> <plus /> <apply> <times /> <apply> <divide /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <ci>gK</ci> </apply> <ci>F</ci> </apply> <apply> <minus /> <ci>Vm</ci> <ci>EK</ci> </apply> </apply> <apply> <times /> <ci>kc</ci> <ci>y</ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci>theta_NKCC1</ci> <ci>J_NKCC1</ci> </apply> <apply> <times /> <ci>theta_KCC1</ci> <ci>J_KCC1</ci> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> <ci>J_NaKATPase</ci> </apply> </apply> </apply> </apply> </math> N_Nai N_Nai.ENa in Nernst potential of sodium for the astrocytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 N_Nai.F in N_Nai.J_NBC in Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Nai.J_NKCC1 in Ion flux through the sodium-potassium-chloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Nai.J_NaKATPase in 0.99796 N_Nai.N_Nai out Rate of change in the number of sodium ions in the intracellular space of the astrocyte. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Nai.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 N_Nai.dN_Nai_dt out N_Nai.gNa in N_Nai.theta_NBC in N_Nai.theta_NKCC1 in N_Nai.time in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_Nai</ci> </apply> <ci>dN_Nai_dt</ci> </apply> </math> <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>dN_Nai_dt</ci> <apply> <minus /> <apply> <plus /> <apply> <times /> <ci>theta_NKCC1</ci> <ci>J_NKCC1</ci> </apply> <apply> <times /> <ci>theta_NBC</ci> <ci>J_NBC</ci> </apply> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <apply> <plus /> <apply> <times /> <apply> <divide /> <ci>gNa</ci> <ci>F</ci> </apply> <apply> <minus /> <ci>Vm</ci> <ci>ENa</ci> </apply> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">3</cn> <ci>J_NaKATPase</ci> </apply> </apply> </apply> </apply> </apply> </math> N_Nao N_Nao.ENa in Nernst potential of sodium for the astrocytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 N_Nao.F in N_Nao.J_NBC in Ion flux through sodium bicarbonate cotransporter Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Nao.J_NKCC1 in Ion flux through the sodium-potassium-chloride cotransporter. Chemical concentration flow rate http://bhi.washington.edu/OPB#OPB_00593 N_Nao.J_NaKATPase in 4.301041 N_Nao_N_Nao out Rate of change in the number of Na ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 N_Nao.Vm in Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 N_Nao.gNa in N_Nao.kc in N_Nao.theta_NBC in N_Nao.theta_NKCC1 in N_Nao.time in N_Nao.y in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>N_Nao</ci> </apply> <apply> <minus /> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="cm2_millimolar_micrometre_per_mol"> 1 <sep /> 10 </cn> <apply> <plus /> <apply> <times /> <apply> <divide /> <ci>gNa</ci> <ci>F</ci> </apply> <apply> <minus /> <ci>Vm</ci> <ci>ENa</ci> </apply> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">3</cn> <ci>J_NaKATPase</ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci>theta_NKCC1</ci> <ci>J_NKCC1</ci> </apply> <apply> <times /> <ci>theta_NBC</ci> <ci>J_NBC</ci> </apply> <apply> <times /> <ci>kc</ci> <ci>y</ci> </apply> </apply> </apply> </apply> </math> Na+ ion flux across astrocytic membrane 1.0 1.0 Nernst_potentials Nernst_potentials.Cli in Nernst_potentials.Clo in Nernst_potentials.ECl out Nernst potential of the Cl- ion for the astrocytic potential. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 Nernst_potentials.EK out Nernst potential of K ion for the astrosytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 Nernst_potentials.ENa out Nernst potential of sodium for the astrocytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 Nernst_potentials.F in Nernst_potentials.Ki in Nernst_potentials.Ko in Nernst_potentials.Nai in Nernst_potentials.Nao in Nernst_potentials.R in Nernst_potentials.T in <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>ENa</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <ci>R</ci> <ci>T</ci> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>F</ci> </apply> </apply> <apply> <ln /> <apply> <divide /> <ci>Nao</ci> <ci>Nai</ci> </apply> </apply> </apply> </apply> <apply> <eq /> <ci>EK</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <ci>R</ci> <ci>T</ci> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>F</ci> </apply> </apply> <apply> <ln /> <apply> <divide /> <ci>Ko</ci> <ci>Ki</ci> </apply> </apply> </apply> </apply> <apply> <eq /> <ci>ECl</ci> <apply> <times /> <apply> <divide /> <apply> <times /> <ci>R</ci> <ci>T</ci> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> <ci>F</ci> </apply> </apply> <apply> <ln /> <apply> <divide /> <ci>Clo</ci> <ci>Cli</ci> </apply> </apply> </apply> </apply> </math> solute carrier family 12 member 2 http://purl.obolibrary.org/obo/PR_000014925 solute carrier family 12 member 4 http://purl.obolibrary.org/obo/PR_000014927 electrogenic sodium bicarbonate cotransporter 1 http://purl.obolibrary.org/obo/PR_000015160 C C_per_cm2 -2.0 centi C_per_cm2_s -1.0 -2.0 centi C_per_mol -1.0 J_per_mol_K -1.0 -1.0 ampere true cm2 2.0 centi cm2_millimolar_micrometre_per_mol micro -1.0 2.0 centi cm3 3.0 centi cm_per_millimolar_second centi -1.0 -1.0 coulomb true dimensionless true first_order_rate_constant -1.0 joule true kelvin true litre true metre true micrometre micro micrometre_millimolar_per_second -1.0 micro micrometre_per_centimetre -1.0 centi micro micrometre_per_millimolar micro -1.0 micrometre_per_millimolar_second micro -1.0 -1.0 millimolar milli -1.0 millimolar_micrometre micro millimolar_micrometre_per_second micro -1.0 millimole milli millimole_per_cm2 -2.0 centi millimole_per_cm2_millisecond -1.0 milli milli -2.0 centi millimole_per_cm2_second -2.0 centi milli -1.0 millisecond milli mole true mole_per_cm2_second -1.0 -2.0 centi ohm true per_millimolar -1.0 per_mole -1.0 per_ohm_cm2 -2.0 centi -1.0 picoampere pico second true true volt environment environment_time out true Membrane http://purl.org/sig/ont/fma/fma30322 Astrocyte http://purl.org/sig/ont/fma/fma54537 Extracellular space http://purl.org/sig/ont/fma/fma70022 Extracellular space http://purl.org/sig/ont/fma/fma70022 membrane in membrane.ECl Nernst potential of the Cl- ion for the astrocytic potential. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 in membrane.EK Nernst potential of K ion for the astrosytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 in membrane.ENBC Reversal Nernst potential of the sodium bicarbonate cotransporter Nernst potential http://bhi.washington.edu/OPB#OPB_01581 in membrane.ENa Nernst potential of sodium for the astrocytic membrane. Nernst potential http://bhi.washington.edu/OPB#OPB_01581 in membrane.F in membrane.J_NaKATPase membrane_Vm out Astrocytic membrane potential. Electrochemical potential http://bhi.washington.edu/OPB#OPB_01364 in membrane.gCl in membrane.gK in membrane.gNBC in membrane.gNa in membrane.theta_NBC in membrane.time <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>Vm</ci> <apply> <divide /> <apply> <minus /> <apply> <plus /> <apply> <times /> <ci>gNa</ci> <ci>ENa</ci> </apply> <apply> <times /> <ci>gK</ci> <ci>EK</ci> </apply> <apply> <times /> <ci>gCl</ci> <ci>ECl</ci> </apply> <apply> <times /> <ci>theta_NBC</ci> <ci>gNBC</ci> <ci>ENBC</ci> </apply> </apply> <apply> <times /> <ci>J_NaKATPase</ci> <ci>F</ci> </apply> </apply> <apply> <plus /> <ci>gNa</ci> <ci>gK</ci> <ci>gCl</ci> <apply> <times /> <ci>theta_NBC</ci> <ci>gNBC</ci> </apply> </apply> </apply> </apply> </math> model_parameters model_parameters.Cli out model_parameters.Clo out 9.649e4 model_parameters.F out model_parameters.HCO3i out model_parameters.HCO3o out model_parameters.Ki out model_parameters.Ko out 2.1e-4 model_parameters.Lp out in model_parameters.N_Cli Time rate of change in the number of Cl- ions in the intracellular space of the astrocyte Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_Clo in model_parameters.N_HCO3i Rate of change in the number of HCO3 ions Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_HCO3o Rate of change in the number of HCO3 ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_Ki Rate of change in the number of potassium ions in the intracellular space of the astrocyte Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_Ko Rate of change in the number of potassium ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_Nai Rate of change in the number of sodium ions in the intracellular space of the astrocyte. Particle count http://bhi.washington.edu/OPB#OPB_01001 in model_parameters.N_Nao Rate of change in the number of Na ions in the extracellular space. Particle count http://bhi.washington.edu/OPB#OPB_01001 model_parameters.Nai out model_parameters.Nao out 8.315 model_parameters.R out 300 model_parameters.T out 12.41 model_parameters.Xi out 2 model_parameters.alpha 14 model_parameters.beta model_parameters.deltaT 1.6e-19 model_parameters.elementary_charge out 8.797e-5 model_parameters.gCl out 0.004 model_parameters.gK out 1.314e-4 model_parameters.gNa out model_parameters.gamma_alpha model_parameters.gamma_alpha_beta model_parameters.gamma_beta 7.35e-2 model_parameters.kc out 6.0221415e23 model_parameters.nA out 10 model_parameters.t0 20 model_parameters.t1 30 model_parameters.t2 1 model_parameters.theta_KCC1 out 1 model_parameters.theta_NBC out 1 model_parameters.theta_NKCC1 out in model_parameters.time in model_parameters.wi in model_parameters.wo model_parameters.y out <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>y</ci> <piecewise> <piece> <apply> <times /> <apply> <divide /> <ci>gamma_alpha_beta</ci> <apply> <times /> <ci>gamma_alpha</ci> <ci>gamma_beta</ci> </apply> </apply> <apply> <power /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <apply> <divide /> <apply> <minus /> <ci>time</ci> <ci>t0</ci> </apply> <ci>deltaT</ci> </apply> </apply> <apply> <minus /> <ci>beta</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> <apply> <power /> <apply> <divide /> <apply> <minus /> <ci>time</ci> <ci>t0</ci> </apply> <ci>deltaT</ci> </apply> <apply> <minus /> <ci>alpha</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <and /> <apply> <geq /> <ci>time</ci> <ci>t0</ci> </apply> <apply> <leq /> <ci>time</ci> <ci>t1</ci> </apply> </apply> </piece> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">-1</cn> <apply> <and /> <apply> <gt /> <ci>time</ci> <ci>t1</ci> </apply> <apply> <lt /> <ci>time</ci> <ci>t2</ci> </apply> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">0</cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>deltaT</ci> <apply> <minus /> <ci>t1</ci> <ci>t0</ci> </apply> </apply> <apply> <eq /> <ci>gamma_alpha_beta</ci> <apply> <factorial /> <apply> <minus /> <apply> <plus /> <ci>alpha</ci> <ci>beta</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <eq /> <ci>gamma_alpha</ci> <apply> <factorial /> <apply> <minus /> <ci>alpha</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <eq /> <ci>gamma_beta</ci> <apply> <factorial /> <apply> <minus /> <ci>beta</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <eq /> <ci>Ki</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Ki</ci> <ci>wi</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Ki</ci> <ci>wi</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>Ko</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Ko</ci> <ci>wo</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Ko</ci> <ci>wo</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>Nai</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Nai</ci> <ci>wi</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Nai</ci> <ci>wi</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>Nao</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Nao</ci> <ci>wo</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Nao</ci> <ci>wo</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>Cli</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Cli</ci> <ci>wi</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Cli</ci> <ci>wi</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>Clo</ci> <piecewise> <piece> <apply> <divide /> <ci>N_Clo</ci> <ci>wo</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_Clo</ci> <ci>wo</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>HCO3i</ci> <piecewise> <piece> <apply> <divide /> <ci>N_HCO3i</ci> <ci>wi</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_HCO3i</ci> <ci>wi</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> <apply> <eq /> <ci>HCO3o</ci> <piecewise> <piece> <apply> <divide /> <ci>N_HCO3o</ci> <ci>wo</ci> </apply> <apply> <gt /> <apply> <divide /> <ci>N_HCO3o</ci> <ci>wo</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="millimolar">0</cn> </apply> </piece> <otherwise> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" type="e-notation" cellml:units="millimolar"> 1 <sep /> -180 </cn> </otherwise> </piecewise> </apply> </math> wi in wi.Cli in wi.Clo in wi.HCO3i in wi.HCO3o in wi.Ki in wi.Ko in wi.Lp in wi.Nai in wi.Nao in wi.Xi in wi.time 0.061 out wi.wi <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>wi</ci> </apply> <apply> <times /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="micrometre_per_centimetre">1</cn> <ci>Lp</ci> <apply> <minus /> <apply> <plus /> <ci>Nai</ci> <ci>Ki</ci> <ci>Cli</ci> <ci>HCO3i</ci> <apply> <divide /> <ci>Xi</ci> <ci>wi</ci> </apply> </apply> <apply> <plus /> <ci>Nao</ci> <ci>Ko</ci> <ci>Clo</ci> <ci>HCO3o</ci> </apply> </apply> </apply> </apply> </math> wo 0.0879 wo.w_tot in wo.wi out wo.wo <math xmlns="http://www.w3.org/1998/Math/MathML"> <apply> <eq /> <ci>wo</ci> <apply> <minus /> <ci>w_tot</ci> <ci>wi</ci> </apply> </apply> </math>