Model Status This model was created by Penny Noble of Oxford University. This file is known to run COR and OpenCell to reproduce the published results. The units have been checked and they are consistent.
Model Structure ABSTRACT: We hypothesize that slow inactivation of Ca2+/calmodulin-dependent kinase II (CaMKII) and its modulatory effect on sarcoplasmic reticulum (SR) Ca2+ handling are important for various interval-force (I-F) relations, in particular for the beat interval dependency in transient alternans during the decay of post-extrasystolic potentiation. We have developed a mathematical model of a single cardiomyocyte to integrate various I-F relations, including alternans, by incorporating a conceptual CaMKII kinetics model into the SR Ca2+ handling model. Our model integrates I-F relations, such as the beat interval-dependent twitch force duration, restitution and potentiation, positive staircase phenomenon and alternans. We found that CaMKII affects more or less all I-F relations, and it is a key factor for integration of the various I-F relations in our model. Alternans arises, in the model, out of a steep relation between SR Ca2+ load and release, owing to SR load-dependent changes in the releasability of Ca2+ via the ryanodine receptor. Beat interval-dependent CaMKII activity, owing to its kinetic properties and amplifying effect on SR Ca2+ load dependency of Ca2+ release, replicated the beat interval dependency of alternans, as observed experimentally. Additionally, our model enabled reproduction of the effects of various interventions on alternans, such as the slowing or accelerating of Ca2+ release and/or uptake. We conclude that a slow time-dependent factor, represented in the model by CaMKII, is important for the integration of I-F relations, including alternans, and that our model offers a useful tool for further analysis of the roles of integrative Ca2+ handling in myocardial I-F relations. The original paper reference is cited below: Modulatory effect of calmodulin-dependent kinase II (CaMKII) on sarcoplasmic reticulum Ca2+ handling and interval-force relations: a modelling study, Gentaro Iribe, Peter Kohl and Denis Noble, 2006, Philosophical Transactions of the Royal Society A , 364, (1842), 1107-1133. PubMed ID: 16608699 model diagram Schematic diagram of the Iribe et al. cell model.
i_Stim stim_amplitude time stim_start time stim_end time stim_start time stim_start stim_period stim_period stim_duration 0 time V 1 Cm i_Na i_b_Na i_K1 i_K i_to i_b_K i_Ca_L i_b_Ca i_NaCa i_NaK i_Stim E_Na R T F Na_o Na_i E_K R T F K_o K_i E_Ca 0.5 R T F Ca_o Ca_i E_mh R T F Na_o 0.12 K_o Na_i 0.12 K_i i_Na g_Na m 3 h V E_mh E0_m V 41 alpha_m 2000 E0_m delta_m 200 E0_m 1 0.1 E0_m beta_m 8000 0.056 V 66 time m alpha_m 1 m beta_m m alpha_h 20 0.125 V 75 shift_h beta_h 2000 1 320 0.1 V 75 shift_h time h alpha_h 1 h beta_h h i_K i_Kmax x K_i K_o V F R T 140 alpha_x 0.5 0.0826 V 50 1 0.057 V 50 beta_x 1.3 0.06 V 20 1 0.04 V 20 time x alpha_x 1 x beta_x x i_K1 g_K1 K_o K_o K_mk1 V E_K 1 2 F V E_K 10 R T i_to g_to s r V E_K alpha_s 0.033 V 17 beta_s 33 1 0.125 V 10 time s alpha_s 1 s beta_s s time r 333 1 1 V 4 5 r i_Ca_L_Ca 4 d f P_Ca_L_Ca V 50 F R T 1 2 V 50 F R T Ca_i 100 F R T Ca_o 2 V 50 F R T i_Ca_L_K 0.002 d f P_Ca_L_Ca V 50 F R T 1 V 50 F R T K_i 50 F R T K_o V 50 F R T i_Ca_L_Na 0.01 d f P_Ca_L_Ca V 50 F R T 1 V 50 F R T Na_i 50 F R T Na_o V 50 F R T i_Ca_L i_Ca_L_Ca i_Ca_L_K i_Ca_L_Na E0_d V 24 5 alpha_d speed_d 120 E0_d 0.00001 speed_d 30 E0_d 1 E0_d 4 beta_d speed_d 120 E0_d 0.00001 speed_d 12 E0_d 1 E0_d 10 time d alpha_d 1 d beta_d d E0_f V 34 alpha_f speed_f 25 E0_f 0.00001 speed_f 6.25 E0_f 1 E0_f 4 beta_f speed_f 50 1 E0_f 4 time f alpha_f 1 f beta_f f i_NaCa i_NaCa_max gamma V F R T Na_i 3 Ca_o gamma 1 V F R T Na_o 3 Ca_i 1 Ca_i 0.0069 i_NaK i_NaK_max K_o K_mK K_o Na_i K_mNa Na_i 1 1 0.1245 0.1 V F R T 0.0353 V F R T i_b_Ca g_b_Ca V E_Ca i_b_K g_b_K V E_K i_b_Na g_b_Na V E_Na Inf_CaMK Cmdn_Ca 0.00005 time F_CaMK Inf_CaMK F_CaMK Tau_CaMK N_CaMK F_CaMK 0.7 2 k_1 gain_k1 30625000 Ca_i 2 245 i_Ca_L k_2 gain_k2 450 1 0.36 Ca_SR k_3 gain_k3 1.885 F_SRCa_RyR 0.22 N_CaMK k_4 gain_k4 1.8 time F_1 k_3 F_3 k_4 F_1 k_1 F_1 time F_2 k_1 F_1 k_2 F_2 F_3 1 F_1 F_2 F_rel F_2 F_2 0.25 2 time F_SRCa_RyR Ca_SR F_SRCa_RyR Tau_SRCa_RyR K_rel K_rel_max F_SRCa_RyR F_SRCa_RyR 0.2 j_rel K_rel F_rel K_leak_rate Ca_SR Ca_i f_b Ca_i 0.00024 2 r_b Ca_SR 1.64 2 j_up F_CaMK V_max_f f_b V_max_r r_b 1 f_b r_b dCmdn_Ca_dtime alpha_cmdn Cmdn_tot Cmdn_Ca Ca_i beta_cmdn Cmdn_Ca time Cmdn_Ca alpha_cmdn Cmdn_tot Cmdn_Ca Ca_i beta_cmdn Cmdn_Ca dTrpn_Ca_dtime alpha_trpn Trpn_tot Trpn_Ca Ca_i beta_trpn 1 2 1 Force_norm 3 Trpn_Ca time Trpn_Ca alpha_trpn Trpn_tot Trpn_Ca Ca_i beta_trpn 1 2 1 Force_norm 3 Trpn_Ca time Ca_i i_Ca_L_Ca i_b_Ca 2 i_NaCa 2 v_i F j_up j_rel v_SR v_i dCmdn_Ca_dtime dTrpn_Ca_dtime time Ca_SR j_up v_i v_SR j_rel time Na_i i_Na i_b_Na i_Ca_L_Na 3 i_NaCa 3 i_NaK v_i F time K_i i_K1 i_K i_to i_b_K i_Ca_L_K 2 i_NaK v_i F f_01 3 f_XB f_12 10 f_XB f_23 7 f_XB g_01 g_XB 2 SL_norm g_12 2 g_XB 2 SL_norm g_23 3 g_XB 2 SL_norm SL_norm SL 1.7 0.7 N_tm 3.5 2.5 SL_norm K_tm 1 1 beta_trpn alpha_trpn 0.0017 0.0009 SL_norm alpha_tm beta_tm Trpn_Ca Trpn_tot K_tm N_tm time N_0 beta_tm P_0 alpha_tm N_0 g_01 N_1 time P_0 beta_tm f_01 P_0 alpha_tm N_0 g_01 P_1 time P_1 beta_tm f_12 g_01 P_1 alpha_tm N_1 f_01 P_0 g_12 P_2 time P_2 f_23 g_12 P_2 f_12 P_1 g_23 P_3 time P_3 g_23 P_3 f_23 P_2 N_1 1 N_0 P_0 P_1 P_2 P_3 sigma_paths 1 g_XB 2 g_XB 3 g_XB 1 f_01 2 g_XB 3 g_XB 1 f_01 1 f_12 3 g_XB 1 f_01 1 f_12 1 f_23 P_1_max 1 f_01 2 g_XB 3 g_XB sigma_paths P_2_max 1 f_01 1 f_12 3 g_XB sigma_paths P_3_max 1 f_01 1 f_12 1 f_23 sigma_paths Force_max P_1_max 2 P_2_max 3 P_3_max phi_SL SL 0.6 1.4 SL 1.7 SL 2 1 SL 2 SL 2.2 3.6 SL 1.4 SL 2.2 SL 2.3 Force_norm phi_SL P_1 N_1 2 P_2 3 P_3 Force_max Force zeta Force_norm SERCA pump uptake calcium flux calcium current through L-type calcium channel background potassium current transient outward current activation variable precursor fraction of ryanodine receptors L-type calcium channel inactivation variable background sodium current concentration of calcium-bound troponin rate of change of troponin buffer concentration SERCA pump uptake calcium flux calcium reversal potential concentration of extracellular calcium sodium-potassium pump current concentration of extracellular potassium myoplasmic volume time constant factor of effect of sarcoplasmic reticulum calcium on ryanodine receptor concentration of calcium in sarcoplasmic reticulum maximum sodium-potassium pump current fast sodium current conductance concentration of calcium-bound calmodulin sodium-calcium exchanger current calcium on-rate for troponin concentration of extracellular sodium absolute temperature L-type calcium channel inactivation variable concentration of extracellular potassium L-type calcium current ryanodine receptor release flux constant SERCA pump uptake calcium flux membrane potential forward rate of tropomyosin shifting maximum force rate of change of calmodulin buffer concentration ryanodine receptor transition rate gain from precursor to product myoplasmic volume concentration of intracellular sodium forward rate of tropomyosin shifting fast sodium current inactivation variable transient outward current inactivation variable concentration of intracellular potassium potassium reversal potential concentration of intracellular calcium maximum force ryanodine receptor recovery rate gain membrane potential concentration of intracellular potassium maximum fraction of permissive tropomyosin with 3 crossbridges calcium release flux of ryanodine receptor channels membrane potential time-independent rectifier potassium current concentration of extracellular calcium concentration of intracellular potassium concentration of extracellular sodium fast sodium current M-gate reversal potential calcium-induced ryanodine receptor inactivation rate gain fast sodium current activation variable concentration of intracellular calcium sodium-potassium pump current fast sodium current concentration of extracellular sodium background sodium current fast sodium current reversal potential calcium on-rate for troponin absolute temperature time-dependent rectifier potassium current calcium-induced ryanodine receptor activation rate gain concentration of intracellular potassium background sodium current membrane potential potassium background current conductance concentration of intracellular potassium fraction of non-permissive tropomyosin with 1 crossbridge time-independent rectifier potassium current membrane potential sodium-potassium pump current concentration of intracellular potassium concentration of intracellular calcium calcium background current conductance time constant factor of effect of sarcoplasmic reticulum calcium on ryanodine receptor time-dependent rectifier potassium current activation variable membrane potential sodium-potassium pump current background potassium current myoplasmic volume concentration of intracellular calcium sodium current through L-type calcium channel calcium reversal potential L-type calcium channel activation variable saturation potential of fast sodium current inactivation gate fast sodium current reversal potential fraction of non-permissive tropomyosin with 0 crossbridges potassium reversal potential time-dependent rectifier potassium current fast sodium current time-dependent rectifier potassium current maximum time-dependent rectifier potassium current membrane potential time-dependent rectifier potassium current rate of change of troponin buffer concentration calcium release flux of ryanodine receptor channels calcium on-rate for troponin minimum transition rate from strong to weak crossbridge transient outward current activation variable transient outward potassium current sodium-calcium exchanger current rate of change of troponin buffer concentration absolute temperature absolute temperature calcium on-rate for calmodulin sarcoplasmic reticulum volume membrane potential concentration of extracellular potassium fast sodium current reversal potential calmodulin-dependent kinase II activity factor L-type calcium channel activation variable fast sodium current conductance myoplasmic volume concentration of extracellular potassium concentration of extracellular potassium maximum fraction of permissive tropomyosin with 1 crossbridge concentration of calcium-bound calmodulin calcium on-rate for calmodulin membrane capacitance sarcoplasmic reticulum volume SERCA pump uptake calcium flux conversion factor for normalisation to physiological force membrane potential L-type calcium chanel F-gate reversal potential absolute temperature time-dependent rectifier potassium current concentration of intracellular calcium extracellular potassium half-saturation constant of sodium-potassium pump current externally applied stimulus current membrane potential time-dependent rectifier potassium current background potassium current membrane capacitance calcium release flux of ryanodine receptor channels SERCA pump reverse rate parameter fraction of permissive tropomyosin with 2 crossbridges concentration of calcium-bound calmodulin sodium-potassium pump current concentration of calcium in sarcoplasmic reticulum calcium on-rate for troponin L-type calcium channel inactivation variable intracellular sodium half-saturation constant of sodium-potassium pump current concentration of intracellular calcium membrane potential sodium-calcium exchanger current calmodulin-dependent kinase II activity factor saturation potential of fast sodium current activation gate sodium current through L-type calcium channel basic transition rate from weak to strong crossbridge transient outward current conductance fraction of permissive tropomyosin with 3 crossbridges membrane potential concentration of extracellular calcium L-type calcium current membrane potential concentration of extracellular calcium concentration of intracellular sodium potassium background current conductance concentration of intracellular potassium activator fraction of ryanodine receptors concentration of extracellular potassium background sodium current concentration of total calmodulin calcium current through L-type calcium channel sarcoplasmic reticulum volume membrane potential membrane capacitance background sodium current maximum L-type calcium current conductance concentration of calcium-bound troponin myoplasmic volume transient outward potassium current rate of change of calmodulin buffer concentration product fraction of ryanodine receptors SERCA pump reverse rate parameter myoplasmic volume concentration of extracellular potassium concentration of extracellular calcium potassium reversal potential absolute temperature product fraction of ryanodine receptors concentration of extracellular potassium time-independent potassium current conductance concentration of extracellular calcium sarcoplasmic reticulum volume transient outward current inactivation variable membrane potential concentration of extracellular calcium ryanodine receptor release flux constant activator fraction of ryanodine receptors minimum transition rate from strong to weak crossbridge membrane potential precursor fraction of ryanodine receptors time-independent potassium current conductance time-independent rectifier potassium current background potassium current concentration of calcium-bound calmodulin membrane potential saturation potential of fast sodium current activation gate potassium reversal potential membrane capacitance sodium reversal potential maximum fraction of permissive tropomyosin with 3 crossbridges voltage dependence parameter of sodium-calcium exchanger maximum fraction of permissive tropomyosin with 1 crossbridge time-dependent rectifier potassium current activation variable background calcium current sarcoplasmic reticulum volume membrane potential effect factor of sarcoplasmic reticulum calcium on ryanodine receptors sarcomere length extracellular potassium half-saturation constant of time-independent rectifier potassium current recovery rate constant of ryanodine receptor calcium background current conductance transient outward potassium current sarcomere length concentration of intracellular calcium concentration of extracellular sodium membrane potential potassium current through L-type calcium channel myoplasmic volume transient outward current inactivation variable absolute temperature concentration of extracellular sodium calcium-induced ryanodine receptor activation rate constant background calcium current membrane potential myoplasmic volume calcium-induced ryanodine receptor inactivation rate constant time-independent rectifier potassium current membrane potential time-dependent rectifier potassium current activation variable calcium reversal potential concentration of extracellular potassium maximum fraction of permissive tropomyosin with 2 crossbridges transient outward potassium current extracellular potassium half-saturation constant of sodium-potassium pump current concentration of intracellular calcium sodium reversal potential calcium-induced ryanodine receptor activation rate constant fraction of non-permissive tropomyosin with 0 crossbridges potassium current through L-type calcium channel externally applied stimulus current reverse rate of tropomyosin shifting membrane potential concentration of extracellular sodium potassium current through L-type calcium channel L-type calcium chanel D-gate reversal potential fast sodium current inactivation variable sodium reversal potential transient outward current inactivation variable calcium off-rate for calmodulin fast sodium current calcium off-rate for calmodulin concentration of calcium in sarcoplasmic reticulum membrane potential fraction of non-permissive tropomyosin with 1 crossbridge sodium-calcium exchanger current concentration of extracellular potassium L-type calcium chanel F-gate reversal potential calcium release flux of ryanodine receptor channels calcium current through L-type calcium channel concentration of intracellular calcium sarcoplasmic reticulum volume maximum calcium off-rate for troponin background potassium current membrane potential potassium reversal potential calmodulin-dependent kinase II activity factor concentration of extracellular calcium membrane potential fraction of permissive tropomyosin with 0 crossbridges background calcium current concentration of intracellular sodium membrane potential concentration of total calmodulin concentration of calcium-bound troponin maximum sodium-calcium exchanger current background potassium current effect factor of sarcoplasmic reticulum calcium on ryanodine receptors concentration of total troponin background sodium current intracellular sodium half-saturation constant of sodium-potassium pump current concentration of intracellular calcium sodium-potassium pump current sodium-calcium exchanger current membrane potential calcium current through L-type calcium channel sodium-potassium pump current SERCA pump uptake calcium flux fraction of permissive tropomyosin with 1 crossbridge SERCA pump forward rate parameter maximum time-dependent rectifier potassium current conversion factor for normalisation to physiological force transient outward current conductance time-dependent rectifier potassium current activation variable background calcium current transient outward current activation variable maximum sodium-calcium exchanger current rate of change of troponin buffer concentration time constant for calmodulin-dependent kinase II kinetics L-type calcium current maximum calcium off-rate for troponin membrane potential absolute temperature sodium-calcium exchanger current membrane potential calmodulin-dependent kinase II activity factor concentration of intracellular potassium transient outward potassium current rate of change of calmodulin buffer concentration concentration of total troponin L-type calcium current transient outward current activation variable concentration of total troponin calmodulin-dependent kinase II activity factor membrane potential transient outward potassium current concentration of intracellular calcium concentration of intracellular sodium sodium-calcium exchanger current time constant for calmodulin-dependent kinase II kinetics L-type calcium current sodium background current conductance background calcium current time-independent rectifier potassium current concentration of intracellular calcium concentration of intracellular calcium fast sodium current inactivation variable concentration of calcium-bound calmodulin SERCA pump uptake calcium flux membrane potential absolute temperature iribe_model_2006 L-type calcium channel inactivation variable SERCA pump forward rate parameter extracellular potassium half-saturation constant of time-independent rectifier potassium current concentration of intracellular sodium concentration of intracellular sodium absolute temperature fast sodium current basic transition rate from weak to strong crossbridge saturation potential of fast sodium current inactivation gate concentration of calcium-bound troponin concentration of calcium-bound troponin concentration of total troponin potassium reversal potential absolute temperature absolute temperature concentration of intracellular sodium potassium current through L-type calcium channel time-independent rectifier potassium current background calcium current absolute temperature voltage dependence parameter of sodium-calcium exchanger maximum calcium off-rate for troponin concentration of intracellular sodium fast sodium current activation variable sodium background current conductance sodium current through L-type calcium channel calcium reversal potential sodium current through L-type calcium channel fast sodium current L-type calcium channel activation variable concentration of intracellular calcium L-type calcium channel activation variable ryanodine receptor transition rate gain from precursor to product fast sodium current activation variable fraction of permissive tropomyosin with 1 crossbridge membrane potential calmodulin-dependent kinase II activity factor membrane potential fast sodium current M-gate reversal potential myoplasmic volume L-type calcium chanel D-gate reversal potential ryanodine receptor transition rate from precursor to product sodium-calcium exchanger current concentration of calcium in sarcoplasmic reticulum fast sodium current reversal potential potassium reversal potential concentration of extracellular sodium myoplasmic volume fast sodium current activation variable maximum fraction of permissive tropomyosin with 2 crossbridges concentration of calcium-bound calmodulin calcium release flux of ryanodine receptor channels concentration of intracellular calcium fraction of permissive tropomyosin with 0 crossbridges membrane potential concentration of intracellular calcium maximum calcium off-rate for troponin concentration of intracellular sodium potassium reversal potential membrane potential membrane potential sodium-potassium pump current concentration of calcium-bound troponin concentration of extracellular sodium reverse rate of tropomyosin shifting concentration of extracellular potassium calcium-induced ryanodine receptor inactivation rate gain absolute temperature fraction of permissive tropomyosin with 3 crossbridges concentration of calcium in sarcoplasmic reticulum sodium reversal potential maximum sodium-potassium pump current calcium-induced ryanodine receptor inactivation rate constant L-type calcium current calcium-induced ryanodine receptor activation rate gain fast sodium current inactivation variable fast sodium current ryanodine receptor recovery rate gain fraction of permissive tropomyosin with 2 crossbridges calcium release flux of ryanodine receptor channels ryanodine receptor transition rate from precursor to product concentration of calcium in sarcoplasmic reticulum maximum L-type calcium current conductance recovery rate constant of ryanodine receptor concentration of intracellular sodium rate of change of calmodulin buffer concentration concentration of extracellular potassium