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1985_difrancesco_noble.xml

Lloyd Catherine May c.lloyd@auckland.ac.nz The University of Auckland The Bioengineering Research Group 2001-09-28 Getting the model solving and correcting a bunch of errors and fixing up the units and moving everything to "per-unit-area" type equations. Nickerson David P 2003-10-28 Changed the model name so the model loads in the database easier. Cuellar Autumn A 2003-04-05 Added more metadata. Lloyd Catherine May 2002-07-19 Added some initial values from Penny Noble's documentation. Lloyd Catherine May 2002-05-06 Corrected several equations. Lloyd Catherine May 2002-02-25 Updated metadata to conform to the 16/1/02 CellML Metadata 1.0 Specification. Cuellar Autumn A. 2002-01-21 Created two extra components for extracellular sodium and extracellular calcium concentrations. Then changed the Nao and Cao variable public interfaces and the connections between components appropriately. Lloyd Catherine May 2002-01-03 Changed equations after using mathml validator. Lloyd Catherine May 2001-12-07 Made changes to some of the metadata, bringing them up to date with the most recent working draft (26th September) of the Metadata Specification. Lloyd Catherine May 2001-10-24 Removed document type definition as this is declared as optional according to the W3C recommendation. Lloyd Catherine May 2001-10-19 The University of Auckland, Bioengineering Research Group The Di Francesco-Noble Model of Cardiac Action Potentials in Purkinje fibres, 1985 This is the CellML description of Di Francesco and Noble's mathematical model of cardiac action potentials of Purkinje fibres. It is a significant development on the MNT model (1975), and it remains the most complete of all Purkinje fibre ionic current models to date. It is a complete replacement for the MNT model. In particular it considers changes in the interpretation of the i_K2 system, includes more accurate experimental data concerning the fast sodium current and it starts to account for fluctuations in ionic concentrations. In addition, a start is made on accounting for intracellular calcium movement between the sarcoplasmic reticulum and the myoplasm. Corrections for the model can be found in an appendix at the end of Earm & Noble, Proc. Roy. Soc. B, Vol 240(1297), 1990. Catherine Lloyd Purkinje Fibre 2578676 DiFrancesco D Noble D A model of cardiac electrical activity incorporating ionic pumps and concentration changes 1985-01-10 Philosophical Transactions of the Royal Society of London Series B 307 353 398 The membrane component is the main component for this model, containing the differential equation for the transmembrane potential (the action potential). The action potential equation consists of the sum of the trans-sarcolemmal currents plus an applied stimulus, which may be used to pace the cell model. time V I_stim i_f i_K i_K1 i_to i_Na_b i_Ca_b i_NaK i_NaCa i_Na i_si C This is a dummy equation that we simply use to make grabbing the value in CMISS much easier. IStimC I_stim The "funny" current - activated by hyperpolarisation rather than depolarisation, and consisting of sodium and potassium components. The total current is simply the sum of the two component currents. i_f i_fK i_fNa i_fK y I_fK i_fNa y I_fNa The maximal potassium current. I_fK Kc Kc Km_f g_f_K V E_K The maximal sodium current. I_fNa Kc Kc Km_f g_f_Na V E_Na The reversal potentials for the two ions. E_Na R T F Nao Nai E_K R T F Kc Ki The activation variable for the "funny" current - the y gate. The opening rate for the y gate. alpha_y 0.05e-3 -0.067 V 42.0 The closing rate for the y gate. beta_y 1.0e-3 V 42.0 1.0 -0.2 V 42.0 The Hodgkin-Huxley type kinetics for the y gate. time y alpha_y 1.0 y beta_y y The time-dependent potassium current, a current dependent on both potassium ion concentration and membrane potential. Calculation of the time-dependent potassium current. i_K x I_K I_K i_K_max Ki Kc V F R T 140.0 The voltage-dependent activation variable for the time-dependent potassium current, the x gate. The opening rate of the x gate. alpha_x 0.5e-3 V 50.0 12.1 1.0 V 50.0 17.5 The closing rate of the x gate. beta_x 1.3e-3 V 20.0 16.67 1.0 V 20.0 25.0 The kinetics of the x gate. time x alpha_x 1.0 x beta_x x The time-independent background potassium current. Calculation of the time-independent postassium current. i_K1 g_K1 Kc Kc Km_K1 V E_K 1.0 V 10.0 E_K F 2.0 R T The transient outward current (i_to) replaces the i_qr chloride-based current of the MNT model. This current is a calcium-activated, outward rectifier. It has an inactivation gate, r. Calculation of the transient outward current. i_to r g_to 0.2 Kc Km_to Kc Cai Km_Ca Cai V 10.0 1.0 -0.2 V 10.0 Ki 0.02 V Kc -0.02 V The activation variable for the transient outward current. The opening rate of the r gate. alpha_r 0.033e-3 V 17.0 The closing rate of the r gate. beta_r 33.0e-3 1.0 V 10.0 8.0 The kinetics for the r gate. time r alpha_r 1.0 r beta_r r A linear resting sodium flux. Calculation of the background sodium current. i_Na_b g_Nab V E_Na A resting background leakage calcium flux. Calculation of the calcium leakage current. i_Ca_b g_Cab V E_Ca The calcium reversal potential. E_Ca R T F 2.0 Cao Cai The Na-K exchange pump couples the free energy released by the hydrolysis of ATP to transfer sodium and potassium ions across the cell membrane against their electrochemical gradients. 3 Na ions are pumped out for every 2 K ions pumped into the cell. Calculation of the Na/K-pump current. i_NaK I_NaK Kc K_mK Kc Nai K_mNa Nai The DFN paper gives two alternative equations for the i_NaCa current. The simplest makes the current a sine function of the total energy gradient. The more realistic model uses an equation which is likely to reproduce better dependence of i_NaCa on intracellular calcium ions. We utilise the more complex version here. Calculation of the exchanger current. i_NaCa k_NaCa gamma n_NaCa 2.0 V F R T Nai n_NaCa Cao -1.0 1.0 gamma n_NaCa 2.0 V F R T Nao n_NaCa Cai 1.0 d_NaCa Cai Nao n_NaCa Cao Nai n_NaCa The DFN model retains a two-variable model of the sodium kinetics, with new equations for the gates m and h. It is acknowledged however that the model does not represent the slower components of Na inactivation and recovery. It is also assumed that the sodium channel shows a 12% permeability to K ions. Calculation of the fast sodium current. i_Na g_Na m 3.0 h V E_mh The reversal potential of the fast sodium channel, assuming a 12% permeability of potassium ions. E_mh R T F Nao 0.12 Kc Nai 0.12 Ki The voltage-dependent activation gate for the fast sodium channel - the m gate. The opening rate of the m gate. alpha_m 200.0e-3 V 41.0 1.0 -0.1 V 41.0 The closing rate of the m gate. beta_m 8000.0e-3 -0.056 V 66.0 The kinetics of the m gate. time m alpha_m 1.0 m beta_m m The voltage-dependent inactivation gate for the fast sodium current - the h gate. The opening rate of the h gate. alpha_h 20.0e-3 -0.125 V 75.0 The closing rate of the h gate. beta_h 2000.0e-3 1.0 320.0 -0.1 V 75.0 The kinetics of the h gate. time h alpha_h 1.0 h beta_h h Like the MNT model, the kinetics of the secondary inward current are still described in terms of two gate variables d and f, but the time constants for activation and inactivation processes are much shorter. The fast component, i_si of this current has been divided into the individual ion movements of Ca, K and Na. This current would later be called the L-type calcium current. The total i_si current is simply the sum of the component calcium, potassium, and sodium currents. i_si i_siCa i_siK i_siNa Calculation of the calcium component of the second inward current. i_siCa d f f2 4.0 P_Ca V 50.0 F R T 1.0 -1.0 V 50.0 F 2.0 R T Cai 50.0 F 2.0 R T Cao -2.0 V 50.0 F R T Calculation of the potassium component of the second inward current. i_siK d f f2 P_CaK P_Ca V 50.0 F R T 1.0 -1.0 V 50.0 F R T Ki 50.0 F R T Kc -1.0 V 50.0 F R T Calculation of the sodium component of the second inward current. i_siNa d f f2 P_CaNa P_Ca V 50.0 F R T 1.0 -1.0 V 50.0 F R T Nai 50.0 F R T Nao -1.0 V 50.0 F R T The voltage-dependent activation gate for the second inward current - the d gate. The opening rate of the d gate. alpha_d 30.0e-3 V 19.0 1.0 -1.0 V 19.0 4.0 The closing rate of the d gate. beta_d 12.0e-3 V 19.0 V 19.0 10.0 1.0 The kinetics of the d gate. time d alpha_d 1.0 d beta_d d The voltage-dependent inactivation gate for the second inward current - the f gate. The opening rate of the f gate. alpha_f 6.25e-3 V 34.0 V 34.0 4.0 1.0 The closing rate of the f gate. beta_f 50.0e-3 1.0 -1.0 V 34.0 4.0 The kinetics of the f gate. time f alpha_f 1.0 f beta_f f The DFN model also includes a description of Ca-dependent inactivation. When calcium ions bind to a regulatory site on the channel protein, they induce a conformational change such that the channel no longer conducts, and the secondary current slows or ceases. The kinetics of the f2 gate. time f2 alpha_f2 1.0 f2 beta_f2 f2 beta_f2 Cai alpha_f2 K_mf2 A representation of extracellular sodium ion concentration, held constant in this model. This component contains the description of intracellular sodium concentration change. The rate of change of intracellular sodium concentration is the sum of the ions entering the intracellular volume via the sodium transmembrane currents. time Nai Am i_Na i_Na_b i_fNa i_siNa i_NaK 3.0 i_NaCa n_NaCa n_NaCa 2.0 V_i F A representation of extracellular calcium ion concentration, held constant in this model. Changes in [Ca]i were first modelled in the BR model and has only been slightly developed in the DFN model. Calcium is sequestered in the sarcoplasmic reticulum ([Ca]up). A fraction is transferred to a release store in the junctional SR ([Ca]rel) before being released into the intracellular space. The Ca concentrations in each of these various stores is modelled together with the transfer between the calcium sites and the Ca transfer across the cell membrane via the other ionic currents. The calcium current into the uptake store. i_up alpha_up Cai Ca_up_max Ca_up alpha_up 1.0 tau_up Ca_up_max The calcium current between the uptake and release stores, which uses a voltage-dependent activation variable - the p gate. i_tr alpha_tr p Ca_up Ca_rel alpha_tr 1.0 tau_rep The opening rate of the p gate. alpha_p 0.625e-3 V 34.0 V 34.0 4.0 1.0 The closing rate of the p gate. beta_p 5.0e-3 1.0 -1.0 V 34.0 4.0 The kinetics of the p gate. time p alpha_p 1.0 p beta_p p The calcium release current from the SR into the cytosol. i_rel alpha_rel Ca_rel Cai 2.0 Cai 2.0 K_mCa alpha_rel 1.0 tau_rel The rate of change of calcium concentration in the uptake store. time Ca_up i_up V_i V_up i_tr V_rel V_up The rate of change of calcium concentration in the release store. time Ca_rel i_tr i_rel The rate of change of intracellular calcium concentration. time Cai i_rel V_rel V_i i_up Am i_siCa i_Ca_b 2.0 i_NaCa n_NaCa 2.0 2.0 V_i F A representation of extracellular potassium ion concentration based on a homogeneous three-compartment model. The rate of change in the cleft potassium concentration is the combination of potassium ions crossing the cellular membrane and ions diffusing into the volume from the bulk extracellular stores. time Kc P Kc Kb Am i_mK V_i F The total transmembrane potassium current. i_mK i_K1 i_K i_fK i_siK i_to 2.0 i_NaK The change in intracellular potassium ion concentration. The rate of change of intracellular potassium concentration is the sum of the ions crossing the membrane. time Ki Am i_mK V_i F A component used to group the geometry parameters and calculations. The intracellular volume fraction. V_i 1.0 V_ecs The extracellular volume fraction. V_e V_ecs The calcium uptake store volume fraction. V_up 0.05 V_i The calcium release store volume fraction. V_rel 0.02 V_i