Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = - (i_Na + i_Ca + i_Ca_K + i_Kr + i_Ks + i_to1 + i_K1 + i_Kp + i_NaCa + i_NaK + i_p_Ca + i_Na_b + i_Ca_b)

Component: fast_sodium_current

i_Na = g_Na m^{3.0} h j (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

Component: fast_sodium_current_m_gate

alpha_m = \frac{0.32 (V + 47.13)}{1.0 - (ⅇ^{(-0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11.0}} \frac{d}{d time} (m) = alpha_m (1.0 - m) - (beta_m m)

Component: fast_sodium_current_h_gate

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80.0 + V}{-6.8}} if V < -40.0 \\ 0.0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000.0 ⅇ^{0.35 V} if V < -40.0 \\ \frac{1.0}{0.13 (1.0 + ⅇ^{\frac{V + 10.66}{-11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h (1.0 - h) - (beta_h h)

Component: fast_sodium_current_j_gate

alpha_j = \{\begin{matrix} ((-127140.0) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(-0.04391) V})) \frac{V + 37.78}{1.0 + ⅇ^{0.311 (V + 79.23)}} if V < -40.0 \\ 0.0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(-0.01052) V}}{1.0 + ⅇ^{(-0.1378) (V + 40.14)}} if V < -40.0 \\ \frac{0.3 ⅇ^{(-0.0000002535) V}}{1.0 + ⅇ^{(-0.1) (V + 32.0)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j (1.0 - j) - (beta_j j)

Component: rapid_activating_delayed_rectifiyer_K_current

E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_Kr = g_Kr f_Ko R_V X_kr (V - E_K) f_Ko = \sqrt{\frac{Ko}{4.0}} R_V = \frac{1.0}{1.0 + 1.4945 ⅇ^{0.0446 V}}

Component: rapid_activating_delayed_rectifiyer_K_current_X_kr_gate

\frac{d}{d time} (X_kr) = K12 (1.0 - X_kr) - (K12 X_kr) K12 = ⅇ^{(-5.495) + 0.169 V} K21 = ⅇ^{-7.677 - (0.0128 V)}

Component: slow_activating_delayed_rectifiyer_K_current

i_Ks = g_Ks {X_ks}^{2.0} (V - E_Ks) E_Ks = \frac{R T}{F} \ln (\frac{Ko + 0.01833 Nao}{Ki + 0.01833 Nai})

Component: slow_activating_delayed_rectifiyer_K_current_X_ks_gate

\frac{d}{d time} (X_ks) = \frac{X_ks_infinity - X_ks}{tau_X_ks} X_ks_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V - 24.7)}{13.6}}} tau_X_ks = \frac{1.0}{\frac{0.0000719 (V - 10.0)}{1.0 - (ⅇ^{(-0.148) (V - 10.0)})} + \frac{0.000131 (V - 10.0)}{ⅇ^{0.0687 (V - 10.0)} - 1.0}}

Component: transient_outward_potassium_current

i_to1 = g_to1 X_to1 Y_to1 (V - E_K)

Component: transient_outward_potassium_current_X_to1_gate

\frac{d}{d time} (X_to1) = alpha_X_to1 (1.0 - X_to1) - (beta_X_to1 X_to1) alpha_X_to1 = 0.04561 ⅇ^{0.03577 V} beta_X_to1 = 0.0989 ⅇ^{(-0.06237) V}

Component: transient_outward_potassium_current_Y_to1_gate

\frac{d}{d time} (Y_to1) = alpha_Y_to1 (1.0 - Y_to1) - (beta_Y_to1 Y_to1) alpha_Y_to1 = \frac{0.005415 ⅇ^{- (\frac{V + 33.5}{5.0})}}{1.0 + 0.051335 ⅇ^{- (\frac{V + 33.5}{5.0})}} beta_Y_to1 = \frac{0.005415 ⅇ^{\frac{V + 33.5}{5.0}}}{1.0 + 0.051335 ⅇ^{\frac{V + 33.5}{5.0}}}

Component: time_independent_potassium_current

i_K1 = g_K1 K1_infinity_V \frac{Ko}{Ko + K_mK1} (V - E_K)

Component: time_independent_potassium_current_K1_gate

K1_infinity_V = \frac{1.0}{2.0 + ⅇ^{1.5 \frac{F}{R T} (V - E_K)}}

Component: plateau_potassium_current

i_Kp = g_Kp Kp_V (V - E_K)

Component: plateau_potassium_current_Kp_gate

Kp_V = \frac{1.0}{1.0 + ⅇ^{\frac{7.488 - V}{5.98}}}

Component: Na_Ca_exchanger

i_NaCa = K_NaCa \frac{5000.0}{{K_mNa}^{3.0} + {Nao}^{3.0}} \frac{1.0}{K_mCa + Cao} \frac{1.0}{1.0 + K_sat ⅇ^{(eta - 1.0) V \frac{F}{R T}}} (ⅇ^{eta V \frac{F}{R T}} {Nai}^{3.0} Cao - (ⅇ^{(eta - 1.0) V \frac{F}{R T}} {Nao}^{3.0} Ca_i))

Component: sodium_potassium_pump

i_NaK = S_MgATP I_NaK

Component: sarcolemmal_calcium_pump

i_p_Ca = S_MgATP I_p_Ca

Component: SERCA_pump

J_up = S_MgATP j_up

Component: S_MgATP

S_MgATP = \frac{MgATP_i}{MgATP_i0}

Component: calcium_background_current

E_Ca = \frac{R T}{2.0 F} \ln (\frac{Cao}{Ca_i}) i_Ca_b = g_Cab (V - E_Ca)

Component: sodium_background_current

i_Na_b = g_Nab (V - E_Na)

Component: L_type_Ca_current

i_Ca = i_Ca_max y (O + O_Ca) i_Ca_K = \frac{p_k}{C_sc} y (O + O_Ca) \frac{V F^{2.0}}{R T} \frac{Ki ⅇ^{\frac{V F}{R T}} - Ko}{ⅇ^{\frac{V F}{R T}} - 1.0} p_k = \frac{P_K}{1.0 + \frac{i_Ca_max}{i_Ca_half}} i_Ca_max = \frac{P_Ca}{C_sc} \frac{4.0 V F^{2.0}}{R T} \frac{0.001 ⅇ^{\frac{2.0 V F}{R T}} - (0.341 Cao)}{ⅇ^{\frac{2.0 V F}{R T}} - 1.0} alpha = 0.4 ⅇ^{\frac{V + 2.0}{10.0}} beta = 0.05 ⅇ^{\frac{- (V + 2.0)}{13.0}} alpha_a = alpha a beta_b = \frac{beta}{b} gamma = 0.10375 Ca_ss \frac{d}{d time} (C0) = beta C1 + omega C_Ca0 - ((4.0 alpha + gamma) C0) \frac{d}{d time} (C1) = 4.0 alpha C0 + 2.0 beta C2 + \frac{omega}{b} C_Ca1 - ((beta + 3.0 alpha + gamma a) C1) \frac{d}{d time} (C2) = 3.0 alpha C1 + 3.0 beta C3 + \frac{omega}{b^{2.0}} C_Ca2 - ((beta 2.0 + 2.0 alpha + gamma a^{2.0}) C2) \frac{d}{d time} (C3) = 2.0 alpha C2 + 4.0 beta C4 + \frac{omega}{b^{3.0}} C_Ca3 - ((beta 3.0 + alpha + gamma a^{3.0}) C3) \frac{d}{d time} (C4) = alpha C3 + g O + \frac{omega}{b^{4.0}} C_Ca4 - ((beta 4.0 + f + gamma a^{4.0}) C4) \frac{d}{d time} (O) = f C4 - (g O) \frac{d}{d time} (C_Ca0) = beta_b C_Ca1 + gamma C_Ca0 - ((4.0 alpha_a + omega) C_Ca0) \frac{d}{d time} (C_Ca1) = 4.0 alpha_a C_Ca0 + 2.0 beta_b C_Ca2 + gamma a C1 - ((beta_b + 3.0 alpha_a + \frac{omega}{b}) C_Ca1) \frac{d}{d time} (C_Ca2) = 3.0 alpha_a C_Ca1 + 3.0 beta_b C_Ca3 + gamma a^{2.0} C2 - ((beta_b 2.0 + 2.0 alpha_a + \frac{omega}{b^{2.0}}) C_Ca2) \frac{d}{d time} (C_Ca3) = 2.0 alpha_a C_Ca2 + 4.0 beta_b C_Ca4 + gamma a^{3.0} C3 - ((beta_b 3.0 + alpha_a + \frac{omega}{b^{3.0}}) C_Ca3) \frac{d}{d time} (C_Ca4) = alpha_a C_Ca3 + g_O_Ca + gamma a^{4.0} C4 - ((beta_b 4.0 + f_+ \frac{omega}{b^{4.0}}) C_Ca4) \frac{d}{d time} (O_Ca) = f_C_Ca4 - (g_O_Ca)

Component: L_type_Ca_current_y_gate

\frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y} y_infinity = \frac{0.8}{1.0 + ⅇ^{\frac{V + 12.5}{5.0}}} + 0.2 tau_y = 20.0 + \frac{600.0}{1.0 + ⅇ^{\frac{V + 20.0}{9.5}}}

Component: RyR_channel

\frac{d}{d time} (P_C1) = (- k_a_plus) {Ca_ss}^{n} P_C1 + k_a_minus P_O1 \frac{d}{d time} (P_O1) = (k_a_plus {Ca_ss}^{n} P_C1 - (k_a_minus P_O1 + k_b_plus {Ca_ss}^{m} P_O1 + k_c_plus P_O1)) + k_b_minus P_O2 + k_c_minus P_C2 \frac{d}{d time} (P_O2) = k_b_plus {Ca_ss}^{m} P_O1 - (k_b_minus P_O2) \frac{d}{d time} (P_C2) = k_c_plus P_O1 - (k_c_minus P_C2) J_rel = v1 (P_O1 + P_O2) (Ca_JSR - Ca_ss)

Component: intracellular_Ca_fluxes

J_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} J_xfer = \frac{Ca_ss - Ca_i}{tau_xfer} J_trpn = J_HTRPNCa + J_LTRPNCa J_HTRPNCa = \frac{d}{d time} (HTRPNCa) \frac{d}{d time} (HTRPNCa) = k_htrpn_plus Ca_i (HTRPN_tot - HTRPNCa) - (k_htrpn_minus HTRPNCa) J_LTRPNCa = \frac{d}{d time} (LTRPNCa) \frac{d}{d time} (LTRPNCa) = k_ltrpn_plus Ca_i (LTRPN_tot - LTRPNCa) - (k_ltrpn_minus LTRPNCa)

Component: intracellular_ion_concentrations

\frac{d}{d time} (Ca_i) = beta_i (J_xfer - (J_up + J_trpn + ((i_Ca_b - (2.0 i_NaCa)) + i_p_Ca) \frac{A_cap C_sc}{2.0 V_myo F})) + k_minus_CaATP CaATP_i + k_minus_CaADP CaADP_i - (k_plus_CaATP Ca_i ATP_i + k_plus_CaADP Ca_i ADP_i) \frac{d}{d time} (Nai) = (- (i_Na + i_Na_b + i_NaCa 3.0 + i_NaK 3.0)) \frac{A_cap C_sc}{V_myo F} \frac{d}{d time} (Ki) = (- (i_Ca_K + i_Kr + i_Ks + i_K1 + i_Kp + i_to1 + i_NaK (-2.0))) \frac{A_cap C_sc}{V_myo F} beta_i = \frac{1.0}{1.0 + \frac{CMDN_tot K_mCMDN}{K_mCMDN + {Ca_i}^{2.0}}} beta_SS = \frac{1.0}{1.0 + \frac{CMDN_tot K_mCMDN}{K_mCMDN + {Ca_ss}^{2.0}}} beta_JSR = \frac{1.0}{1.0 + \frac{CSQN_tot K_mCSQN}{K_mCSQN + {Ca_JSR}^{2.0}}} \frac{d}{d time} (Ca_ss) = beta_SS (J_rel \frac{V_JSR}{V_ss} - (J_xfer \frac{V_myo}{V_ss}) - (i_Ca \frac{A_cap C_sc}{2.0 V_ss F})) + k_minus_CaATP CaATP_ss - (k_plus_CaATP Ca_ss ATP_ss + k_plus_CaADP Ca_ss (ADP_ss + k_minus_CaADP CaADP_ss)) \frac{d}{d time} (Ca_JSR) = beta_JSR (J_tr - J_rel) \frac{d}{d time} (Ca_NSR) = J_up \frac{V_myo}{V_NSR} - (J_tr \frac{V_JSR}{V_NSR})

Component: Ca_and_Mg_buffering_by_ATP

ATP_ss = ATP_tot - (CaATP_ss + MgATP_ss) \frac{d}{d time} (CaATP_ss) = k_plus_CaATP Ca_ss ATP_ss - (Jxfer_CaATP \frac{V_myo}{V_ss} + k_minus_CaATP CaATP_ss) \frac{d}{d time} (MgATP_ss) = k_plus_MgATP Mg_ss ATP_ss - (Jxfer_MgATP \frac{V_myo}{V_ss} + k_minus_MgATP MgATP_ss) ATP_i = ATP_tot - (CaATP_i + MgATP_i) \frac{d}{d time} (CaATP_i) = Jxfer_CaATP + k_plus_CaATP Ca_i ATP_i - (k_minus_CaATP CaATP_i) \frac{d}{d time} (MgATP_i) = Jxfer_MgATP + k_plus_MgATP Mg_i ATP_i - (k_minus_MgATP MgATP_i) Jxfer_CaATP = \frac{CaATP_ss - CaATP_i}{tau_xfer_CaATP} Jxfer_MgATP = \frac{MgATP_ss - MgATP_i}{tau_xfer_MgATP} ADP_ss = ADP_tot - (CaADP_ss + MgADP_ss) \frac{d}{d time} (CaADP_ss) = k_plus_CaADP Ca_ss ADP_ss - (Jxfer_CaADP \frac{V_myo}{V_ss} + k_minus_CaADP CaADP_ss) \frac{d}{d time} (MgADP_ss) = k_plus_MgADP Mg_ss ADP_ss - (Jxfer_MgADP \frac{V_myo}{V_ss} + k_minus_MgADP MgADP_ss) ADP_i = ADP_tot - (CaADP_i + MgADP_i) \frac{d}{d time} (CaADP_i) = Jxfer_CaADP + k_plus_CaADP Ca_i ADP_i - (k_minus_CaADP CaADP_i) \frac{d}{d time} (MgADP_i) = Jxfer_MgADP + k_plus_MgADP Mg_i ADP_i - (k_minus_MgADP MgADP_i) Jxfer_CaADP = \frac{CaADP_ss - CaADP_i}{tau_xfer_CaADP} Jxfer_MgADP = \frac{MgADP_ss - MgADP_i}{tau_xfer_MgADP} Mg_ss = k_minus_MgATP MgATP_ss + k_minus_MgADP MgADP_ss - (k_plus_MgATP Mg_ss ATP_ss + k_plus_MgADP Mg_ss ADP_ss + Jxfer_Mg \frac{V_myo}{V_ss}) Mg_i = Jxfer_Mg + k_minus_MgATP MgATP_i + k_minus_MgADP MgADP_i - (k_plus_MgATP Mg_i ATP_i + k_plus_MgADP Mg_i ADP_i) Jxfer_Mg = \frac{Mg_ss - Mg_i}{tau_xfer_Mg}

Model Mathematics

Component: environment

Component: membrane

Component: fast_sodium_current

Component: fast_sodium_current_m_gate

Component: fast_sodium_current_h_gate

Component: fast_sodium_current_j_gate

Component: rapid_activating_delayed_rectifiyer_K_current

Component: rapid_activating_delayed_rectifiyer_K_current_X_kr_gate

Component: slow_activating_delayed_rectifiyer_K_current

Component: slow_activating_delayed_rectifiyer_K_current_X_ks_gate

Component: transient_outward_potassium_current

Component: transient_outward_potassium_current_X_to1_gate

Component: transient_outward_potassium_current_Y_to1_gate

Component: time_independent_potassium_current

Component: time_independent_potassium_current_K1_gate

Component: plateau_potassium_current

Component: plateau_potassium_current_Kp_gate

Component: Na_Ca_exchanger

Component: sodium_potassium_pump

Component: sarcolemmal_calcium_pump

Component: SERCA_pump

Component: S_MgATP

Component: calcium_background_current

Component: sodium_background_current

Component: L_type_Ca_current

Component: L_type_Ca_current_y_gate

Component: RyR_channel

Component: intracellular_Ca_fluxes

Component: intracellular_ion_concentrations

Component: Ca_and_Mg_buffering_by_ATP

Component: standard_ionic_concentrations

Component: model_parameters