Development of models of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as a case study
Alice
Boit
Auckland Bioengineering Institute, The University of Auckland
Model Status
This CellML model runs in both OpenCell and COR.
Model Structure
ABSTRACT: This study presents a method for the reduction of biophysically-based kinetic models for the active transport of ions. A lumping scheme is presented which exploits the differences in timescales associated with fast and slow transitions between model states, while maintaining the thermodynamic properties of the model. The goal of this approach is to contribute to modelling of the effects of disturbances to metabolism, associated with ischaemic heart disease, on cardiac cell function. The approach is illustrated for the sodium-potassium pump in the myocyte. The lumping scheme is applied to produce a 4-state representation from the detailed 15-state model of Lauger and Apell, Eur. Biophys. J. 13 (1986) 309, for which the principles of free energy transduction are used to link the free energy released from ATP hydrolysis (deltaGATP) to the transition rates between states of the model. An iterative minimisation algorithm is implemented to determine the transition rate parameters based on the model fit to experimental data. Finally, the relationship between deltaGATP and pump cycling direction is investigated and compared with recent experimental findings.
The original paper reference is cited below:
Development of models of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as a case study, N. P. Smith and E. J. Crampin, 2004,
Progress in Biophysics and Molecular Biology
PubMed ID: 15142754
15-state State Diagram
The 15 states of the original model.
Smith-Crampin 4-state lumping scheme
The 4 states of the Smith-Crampin model.
$\frac{d \mathrm{Vm}}{d \mathrm{time}}=1.0$
$\frac{d \mathrm{cMgADP}}{d \mathrm{time}}=0.0$
$\frac{d \mathrm{cNa\_i}}{d \mathrm{time}}=0.0$
$\mathrm{cPi}=\frac{\mathrm{cPi\_sum}}{1.0+\frac{\mathrm{cK\_i}}{\mathrm{eq\_KPi}}+\frac{\mathrm{cH}}{\mathrm{eq\_HPi}}+\frac{\mathrm{cNa\_i}}{\mathrm{eq\_NaPi}}}$
$\mathrm{dG\_Na}=\mathrm{gas\_const}\mathrm{body\_temp}\ln \left(\frac{\mathrm{cNa\_e}}{\mathrm{cNa\_i}}\right)-\mathrm{faraday\_const}\times 0.001\mathrm{Vm}$
$\mathrm{dG\_K}=\mathrm{gas\_const}\mathrm{body\_temp}\ln \left(\frac{\mathrm{cK\_i}}{\mathrm{cK\_e}}\right)+\mathrm{faraday\_const}\times 0.001\mathrm{Vm}$
$\mathrm{dG\_pump}=2.0\mathrm{dG\_K}+3.0\mathrm{dG\_Na}$
$\mathrm{dG\_ATP}=-29600.-\mathrm{gas\_const}\mathrm{body\_temp}\ln \left(\frac{\mathrm{cMgATP}}{0.001\mathrm{cMgADP}\mathrm{cPi}}\right)$
$\mathrm{net\_free\_energy}=\mathrm{dG\_ATP}+\mathrm{dG\_pump}$
$\mathrm{dimless\_Na\_i}=\frac{\mathrm{cNa\_i}}{\mathrm{eq\_Na\_base\_i}e^{\frac{\mathrm{partition\_factor}\mathrm{faraday\_const}\times 0.001\mathrm{Vm}}{3.0\mathrm{gas\_const}\mathrm{body\_temp}}}}$
$\mathrm{dimless\_Na\_e}=\frac{\mathrm{cNa\_e}}{\mathrm{eq\_Na\_base\_e}e^{\frac{(1.+\mathrm{partition\_factor})\mathrm{faraday\_const}\times 0.001\mathrm{Vm}}{3.0\mathrm{gas\_const}\mathrm{body\_temp}}}}$
$\mathrm{dimless\_K\_i}=\frac{\mathrm{cK\_i}}{\mathrm{eq\_K\_i}}$
$\mathrm{dimless\_K\_e}=\frac{\mathrm{cK\_e}}{\mathrm{eq\_K\_e}}$
$\mathrm{dimless\_MgATP}=\frac{\mathrm{cMgATP}}{\mathrm{eq\_MgATP}}$
$\mathrm{alpha1}=\frac{\mathrm{k1}\mathrm{dimless\_Na\_i}^{3}}{(1.0+\mathrm{dimless\_Na\_i})^{3}+(1.0+\mathrm{dimless\_K\_i})^{2}-1.0}$
$\mathrm{alpha2}=\mathrm{k2}$
$\mathrm{alpha3}=\frac{\mathrm{k3}\mathrm{dimless\_K\_e}^{2}}{(1.0+\mathrm{dimless\_Na\_e})^{3}+(1.0+\mathrm{dimless\_K\_e})^{2}-1.0}$
$\mathrm{alpha4}=\frac{\mathrm{k4}\mathrm{dimless\_MgATP}}{1.0+\mathrm{dimless\_MgATP}}$
$\mathrm{minus\_alpha1}=\mathrm{minus\_k1}\mathrm{cMgADP}$
$\mathrm{minus\_alpha2}=\frac{\mathrm{minus\_k2}\mathrm{dimless\_Na\_e}^{3}}{(1.0+\mathrm{dimless\_Na\_e})^{3}+(1.0+\mathrm{dimless\_K\_e})^{2}-1.0}$
$\mathrm{minus\_alpha3}=\frac{\mathrm{minus\_k3}\mathrm{cPi}\mathrm{cH}}{1.0+\mathrm{dimless\_MgATP}}$
$\mathrm{minus\_alpha4}=\frac{\mathrm{minus\_k4}\mathrm{dimless\_K\_i}^{2}}{(1.0+\mathrm{dimless\_Na\_i})^{3}+(1.0+\mathrm{dimless\_K\_i})^{2}-1.0}$
$\mathrm{diagram\_sum}=\mathrm{minus\_alpha3}\mathrm{minus\_alpha2}\mathrm{minus\_alpha1}+\mathrm{alpha4}\mathrm{minus\_alpha2}\mathrm{minus\_alpha1}+\mathrm{alpha4}\mathrm{alpha2}\mathrm{alpha3}+\mathrm{alpha4}\mathrm{minus\_alpha1}\mathrm{alpha3}+\mathrm{minus\_alpha3}\mathrm{minus\_alpha2}\mathrm{alpha1}+\mathrm{alpha4}\mathrm{minus\_alpha2}\mathrm{alpha1}+\mathrm{alpha4}\mathrm{alpha1}\mathrm{alpha3}+\mathrm{minus\_alpha3}\mathrm{alpha1}\mathrm{alpha2}+\mathrm{alpha4}\mathrm{alpha1}\mathrm{alpha2}+\mathrm{alpha1}\mathrm{alpha2}\mathrm{alpha3}+\mathrm{minus\_alpha4}\mathrm{minus\_alpha3}\mathrm{minus\_alpha1}+\mathrm{minus\_alpha4}\mathrm{minus\_alpha3}\mathrm{alpha2}+\mathrm{minus\_alpha4}\mathrm{minus\_alpha3}\mathrm{minus\_alpha2}+\mathrm{minus\_alpha4}\mathrm{minus\_alpha1}\mathrm{minus\_alpha2}+\mathrm{minus\_alpha4}\mathrm{alpha2}\mathrm{alpha3}+\mathrm{minus\_alpha4}\mathrm{minus\_alpha1}\mathrm{alpha3}$
$\mathrm{v\_cyc}=\frac{\mathrm{alpha1}\mathrm{alpha2}\mathrm{alpha3}\mathrm{alpha4}-\mathrm{minus\_alpha1}\mathrm{minus\_alpha2}\mathrm{minus\_alpha3}\mathrm{minus\_alpha4}}{\mathrm{diagram\_sum}}$
Forward rate constant for sodium-potassium exchanger transition from lumped states P14-P15 to lumped states P1-P6 of the Post-Albers cycle
Calculation of the diagram sum used in the denominator of the equation of cycle rate v_cyc.
Calculation of dG_pump.
Extracellular potassium ion concentration
Calculation of the foward transition rate called alpha1 from the first to the second lumped state.
The energy required to translocate a sodium ion from the intracellular cytosol to the extracellular environment
Total intracellular concentration of bound and unbound inorganic phosphate
Convenience term that is the ratio of the intracellular potassium concentration to its equilibrium constant
Magnesium-bound ADP
Intracellular concentration of magnesium-bound ADP
Convencience term for expressing equation that solves v_cyc
Equilibrium constant for dissociation of sodium ion from intracellular-facing pump when membrane potential is zero
Calculation of the dimless_Na_e parameter which is a
function of Vm because the equilibrium constant is
dependent on Vm (which occurs in the denominator's exponential). Note that the partition of the
voltage dependency in the exponential corresponds to the previous calculation of dimless_Na_i.
Intracellular concentration of inorganic phosphate
Calculation of the concentration of free inorganic phosphate cPi
related to the total measurable concentration given by cPi_sum.
The net free energy of a sodium-potassium pump cycle
Intracellular concentration of magnesium-bound ATP
Calculation of the foward transition rate called alpha2 from the second to the third state.
Note that this transition rate is identical to k2 because state P7 (see paper) is not a lumped state.
Intracellular sodium ion concentration
Convenience term that is the ratio of the extracellular sodium concentration to its equilibrium constant
Convenience term that is the ratio of the intracellular MgATP concentration to its equilibrium constant
Membrane potential
Equilibrium constant for dissociation of sodium ion from inorganic phosphate
Equilibrium constant for dissociation of proton from inorganic phosphate
Apparent first-order forward rate constant for sodium-potassium exchanger transition from lumped states P1-P6 to state P7 of the Post-Albers cycle
The Na/K-pump is an energy-consuming transporter channel within the membrane.
It is indispensible for maintaining the electrochemical gradients of the involved ions
across the membrane which result in resting Vm. The pump splits up one ATP molecule as it undergoes a
conformational change transporting 3Na outwards and 2K inwards in each cycle.
Backward rate constant for sodium-potassium exchanger transition from lumped states P1-P6 to state P7 of the Post-Albers cycle
Calculation of the foward transition rate called alpha3 from the third to the fourth state.
Calculation of the Na contribution to dG_pump.
Calculation of the clockwise cycle rate v_cyc.
The energy required to translocate a potassium ion from the extracellular environment to the intracellular cytosol
Calculation of the dimless_K_e parameter.
Equilibrium constant for dissociation of potassium ion from extracellular-facing pump
Equilibrium constant for MgATP hydrolysis
Intracellular proton concentration
Calculation of the K contribution to dG_pump.
Calculation of the foward transition rate called alpha4 from the fourth to the first state.
Equilibrium constant for dissociation of potassium ion from inorganic phosphate
Intracellular concentration of magnesium-bound ADP
Steady-state cycle rate of sodium-potassium pump
Universal gas constant
Calculation of the dimless_MgATP parameter.
Intracellular potassium ion concentration
Backward rate constant for sodium-potassium exchanger transition from lumped states P8-P13 to lumped states P14-P15 of the Post-Albers cycle
Steady-state cycle rate of sodium-potassium pump
Membrane potential
The University of Auckland, Auckland Bioengineering Institute
The University of Auckland
Auckland Bioengineering Institute
Calculation of the reverse transition rate called minus_alpha3 from the fourth to the 3rd state.
Psuedo-first-order backward rate constant for sodium-potassium exchanger transition from lumped states P1-P6 to state P7 of the Post-Albers cycle
Convenience term that is the ratio of the extracellular potassium concentration to its equilibrium constant
Equilibrium constant for dissociation of sodium ion from extracellular-facing pump when membrane potential is zero
Apparent first-order forward rate constant for sodium-potassium exchanger transition from lumped states P8-P13 to lumped states P14-P15 of the Post-Albers cycle
The net free energy of a sodium-potassium pump cycle
Apparent first-order forward rate constant for sodium-potassium exchanger transition from state P7 to lumped states P8-P13 of the Post-Albers cycle
Calculation the net free energy of the cycle.
Calculation the energy released by ATP hydrolysis.
Forward rate constant for sodium-potassium exchanger transition from lumped states P1-P6 to state P7 of the Post-Albers cycle
Backward rate constant for sodium-potassium exchanger transition from state P7 to lumped states P8-P13 of the Post-Albers cycle
Calculation of the dimless_Na_i parameter which is a function of Vm because the equilibrium constant is
dependent on Vm (which occurs in the denominator's exponential).
Faraday constant
Convenience term that is the ratio of the intracellular sodium concentration to its equilibrium constant
Extracellular sodium ion concentration
Calculation of the reverse transition rate called minus_alpha4 from the first to the fourth lumped state.
Energy consumed for each pump cycle
Ambient temperature
Forward rate constant for sodium-potassium exchanger transition from state P7 to lumped states P8-P13 of the Post-Albers cycle
Psuedo-first-order backward rate constant for sodium-potassium exchanger transition from lumped states P14-P15 to lumped states P1-P6 of the Post-Albers cycle
Intracellular sodium ion concentration
Calculation of the dimless_K_i parameter.
Psuedo-first-order backward rate constant for sodium-potassium exchanger transition from lumped states P8-P13 to lumped states P14-P15 of the Post-Albers cycle
Calculation of the reverse transition rate called minus_alpha1 from the second to the first state.
Equilibrium constant for dissociation of potassium ion from intracellular-facing pump
The free energy of MgATP hydroysis
N. P. Smith' and E. Crampin's 2004 mathematical model of the cardiac sodium-potassium pump.
Cardiac Myocyte
Alice Boit
This is the CellML description of N. P. Smith' and E. Crampin's 2004
mathematical model of the cardiac sodium-potassium pump.
Development of models of active ion transport for whole-cell modelling: cardiac sodium-potassium pump as a case study
Progress in Biophysics & Molecular Biology
active transport
cardiac myocyte
electrophysiology
cardiac
na/k pump
Forward rate constant for sodium-potassium exchanger transition from lumped states P8-P13 to lumped states P14-P15 of the Post-Albers cycle
Backward rate constant for sodium-potassium exchanger transition from lumped states P14-P15 to lumped states P1-P6 of the Post-Albers cycle
Time domain for simulation
Psuedo-first-order backward rate constant for sodium-potassium exchanger transition from state P7 to lumped states P8-P13 of the Post-Albers cycle
Calculation of the reverse transition rate called minus_alpha2 from the 3rd to the second state.
Factor for determining how voltage dependence is partitioned between the intra- and extracellular sodium ion dissociation reactions
Apparent first-order forward rate constant for sodium-potassium exchanger transition from lumped states P14-P15 to lumped states P1-P6 of the Post-Albers cycle