Model Mathematics

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Component: environment

Component: membrane

i_Stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = \frac{(- 1) 1 (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 + i_Stim)}{C_sc}

Component: fast_sodium_current

E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) i_Na = g_Na m^{3} h j (V - E_Na)

Component: fast_sodium_current_m_gate

E0_m = V + 47.13 alpha_m = \{\begin{matrix} \frac{320}{0.1 - (0.005 E0_m)} if | E0_m | < 0.00001 \\ \frac{320 E0_m}{1 - (ⅇ^{(- 0.1) E0_m})} otherwise \end{matrix} beta_m = 80 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = \{\begin{matrix} alpha_m (1 - m) - (beta_m m) if V \geq - 90 \\ 0 otherwise \end{matrix}

Component: fast_sodium_current_h_gate

alpha_h = \{\begin{matrix} 135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3560 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1000}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: fast_sodium_current_j_gate

alpha_j = \{\begin{matrix} \frac{1000 (- (127140 ⅇ^{0.2444 V} + 0.00003474 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{121.2 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{300 ⅇ^{(- 0.0000002535) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} \frac{d}{d time} (j) = alpha_j (1 - j) - (beta_j j)

Component: rapid_activating_delayed_rectifiyer_K_current

E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) f_K_o = \sqrt{\frac{K_o}{4}} R_V = \frac{1}{1 + 1.4945 ⅇ^{0.0446 V}} i_Kr = g_Kr f_K_o R_V X_kr (V - E_K)

Component: rapid_activating_delayed_rectifiyer_K_current_X_kr_gate

K12 = ⅇ^{(- 5.495) + 0.1691 V} K21 = ⅇ^{- 7.677 - (0.0128 V)} X_kr_inf = \frac{K12}{K12 + K21} tau_X_kr = \frac{0.001}{K12 + K21} + tau_factor 0.027 \frac{d}{d time} (X_kr) = \frac{X_kr_inf - X_kr}{tau_X_kr}

Component: slow_activating_delayed_rectifiyer_K_current

E_Ks = \frac{R T}{F} \ln (\frac{K_o + 0.01833 Na_o}{K_i + 0.01833 Na_i}) i_Ks = g_Ks {X_ks}^{2} (V - E_Ks)

Component: slow_activating_delayed_rectifiyer_K_current_X_ks_gate

X_ks_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 24.7)}{13.6}}} tau_X_ks = \frac{0.001}{\frac{0.0000719 (V - 10)}{1 - (ⅇ^{(- 0.148) (V - 10)})} + \frac{0.000131 (V - 10)}{ⅇ^{0.0687 (V - 10)} - 1}} \frac{d}{d time} (X_ks) = \frac{X_ks_infinity - X_ks}{tau_X_ks}

Component: transient_outward_potassium_current

i_to1 = g_to1 X_to1 Y_to1 (V - E_K)

Component: transient_outward_potassium_current_X_to1_gate

alpha_X_to1 = 45.16 ⅇ^{0.03577 V} beta_X_to1 = 98.9 ⅇ^{(- 0.06237) V} \frac{d}{d time} (X_to1) = alpha_X_to1 (1 - X_to1) - (beta_X_to1 X_to1)

Component: transient_outward_potassium_current_Y_to1_gate

alpha_Y_to1 = \frac{5.415 ⅇ^{\frac{- (V + 33.5)}{5}}}{1 + 0.051335 ⅇ^{\frac{- (V + 33.5)}{5}}} beta_Y_to1 = \frac{5.415 ⅇ^{\frac{V + 33.5}{5}}}{1 + 0.051335 ⅇ^{\frac{V + 33.5}{5}}} \frac{d}{d time} (Y_to1) = alpha_Y_to1 (1 - Y_to1) - (beta_Y_to1 Y_to1)

Component: time_independent_potassium_current

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

Component: time_independent_potassium_current_K1_gate

K1_infinity_V = \frac{1}{2 + ⅇ^{\frac{1.5 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}{1 + ⅇ^{\frac{7.488 - V}{5.98}}}

Component: Na_Ca_exchanger

i_NaCa = \frac{K_NaCa 5000}{({K_mNa}^{3} + {Na_o}^{3}) (K_mCa + Ca_o) (1 + K_sat ⅇ^{\frac{(eta - 1) V F}{R T}})} (ⅇ^{\frac{eta V F}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(eta - 1) V F}{R T}} {Na_o}^{3} Ca_i))

Component: sodium_potassium_pump

f_NaK = \frac{1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} sigma = \frac{1}{7} (ⅇ^{\frac{Na_o}{67.3}} - 1) i_NaK = \frac{MgATP_i}{MgATP_i0} i_NaK_winslow i_NaK_winslow = \frac{\frac{I_NaK f_NaK}{1 + {(\frac{K_mNa_i}{Na_i})}^{1.5}} K_o}{K_o + K_mK_o}

Component: sarcolemmal_calcium_pump

i_p_Ca = \frac{MgATP_i}{MgATP_i0} i_p_Ca_winslow i_p_Ca_winslow = \frac{I_pCa Ca_i}{K_mpCa + Ca_i}

Component: calcium_background_current

E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_o}{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{\frac{\frac{p_prime_k}{1 1} y (O + O_Ca) V F^{2}}{R T} (K_i ⅇ^{\frac{V F}{R T}} - K_o)}{ⅇ^{\frac{V F}{R T}} - 1} p_prime_k = \frac{P_K}{1 + \frac{i_Ca_max}{i_Ca_half}} i_Ca_max = \frac{\frac{\frac{P_Ca}{1 1} 4 V F^{2} 1000}{R T} (0.001 ⅇ^{\frac{2 V F}{R T}} - (0.341 Ca_o))}{ⅇ^{\frac{2 V F}{R T}} - 1} alpha = 400 ⅇ^{\frac{V + 2}{10}} beta = 50 ⅇ^{\frac{- (V + 2)}{13}} alpha_a = alpha a beta_b = \frac{beta}{b} gamma = \frac{103.75 Ca_ss}{1} \frac{d}{d time} (C0) = beta C1 + omega C_Ca0 - ((4 alpha + gamma) C0) \frac{d}{d time} (C1) = 4 alpha C0 + 2 beta C2 + \frac{omega}{b} C_Ca1 - ((beta + 3 alpha + gamma a) C1) \frac{d}{d time} (C2) = 3 alpha C1 + 3 beta C3 + \frac{omega}{b^{2}} C_Ca2 - ((beta 2 + 2 alpha + gamma a^{2}) C2) \frac{d}{d time} (C3) = 2 alpha C2 + 4 beta C4 + \frac{omega}{b^{3}} C_Ca3 - ((beta 3 + alpha + gamma a^{3}) C3) \frac{d}{d time} (C4) = alpha C3 + g O + \frac{omega}{b^{4}} C_Ca4 - ((beta 4 + f + gamma a^{4}) C4) \frac{d}{d time} (O) = f C4 - (g O) \frac{d}{d time} (C_Ca0) = beta_b C_Ca1 + gamma C0 - ((4 alpha_a + omega) C_Ca0) \frac{d}{d time} (C_Ca1) = 4 alpha_a C_Ca0 + 2 beta_b C_Ca2 + gamma a C1 - ((beta_b + 3 alpha_a + \frac{omega}{b}) C_Ca1) \frac{d}{d time} (C_Ca2) = 3 alpha_a C_Ca1 + 3 beta_b C_Ca3 + gamma a^{2} C2 - ((beta_b 2 + 2 alpha_a + \frac{omega}{b^{2}}) C_Ca2) \frac{d}{d time} (C_Ca3) = 2 alpha_a C_Ca2 + 4 beta_b C_Ca4 + gamma a^{3} C3 - ((beta_b 3 + alpha_a + \frac{omega}{b^{3}}) C_Ca3) \frac{d}{d time} (C_Ca4) = alpha_a C_Ca3 + gprime O_Ca + gamma a^{4} C4 - ((beta_b 4 + fprime + \frac{omega}{b^{4}}) C_Ca4) \frac{d}{d time} (O_Ca) = fprime C_Ca4 - (gprime O_Ca)

Component: L_type_Ca_current_y_gate

y_infinity = \frac{0.8}{1 + ⅇ^{\frac{V + 12.5}{5}}} + 0.2 tau_y = \frac{20 + \frac{600}{1 + ⅇ^{\frac{V + 20}{9.5}}}}{1000} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

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: SERCA2a_pump

fb = {(\frac{Ca_i}{K_fb})}^{N_fb} rb = {(\frac{Ca_NSR}{K_rb})}^{N_rb} J_up = \frac{MgATP_i}{MgATP_i0} J_up_winslow J_up_winslow = \frac{K_SR (Vmaxf fb - (Vmaxr rb))}{1 + fb + rb}

Component: intracellular_Ca_fluxes

J_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} J_xfer = \frac{Ca_ss - Ca_i}{tau_xfer} J_trpn = HTRPN_tot J_HTRPNCa + LTRPN_tot J_LTRPNCa J_HTRPNCa = \frac{d}{d time} (HTRPNCa) \frac{d}{d time} (HTRPNCa) = k_htrpn_plus Ca_i (1 - HTRPNCa) - (k_htrpn_minus HTRPNCa) J_LTRPNCa = \frac{d}{d time} (LTRPNCa) \frac{d}{d time} (LTRPNCa) = k_ltrpn_plus Ca_i (1 - LTRPNCa) - (k_ltrpn_minus LTRPNCa)

Component: intracellular_ion_concentrations

\frac{d}{d time} (Ca_i) = beta_i ((J_xfer - (J_up + J_trpn)) + (- ((i_p_Ca + i_Ca_b - (2 i_NaCa)) \frac{A_cap C_sc}{2 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} (Na_i) = \frac{(- 0) (i_Na + i_Na_b + i_NaCa 3 + i_NaK 3) A_cap 1}{V_myo F} \frac{d}{d time} (K_i) = \frac{(- 0) (i_Ca_K + i_Kr + i_Ks + i_K1 + i_Kp + i_to1 + i_NaK (- 2)) A_cap 1}{V_myo F} beta_i = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_i)}^{2}} + \frac{EGTA_tot K_mEGTA}{{(K_mEGTA + Ca_i)}^{2}}} beta_SS = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_ss)}^{2}} + \frac{EGTA_tot K_mEGTA}{{(K_mEGTA + Ca_ss)}^{2}}} beta_JSR = \frac{1}{1 + \frac{CSQN_tot K_mCSQN}{{(K_mCSQN + Ca_JSR)}^{2}}} \frac{d}{d time} (Ca_ss) = beta_SS (\frac{J_rel V_JSR}{V_ss} + k_minus_CaATP CaATP_ss + k_minus_CaADP CaADP_ss - (\frac{J_xfer V_myo}{V_ss} + i_Ca \frac{A_cap 1}{2 V_ss F} + k_plus_CaATP Ca_ss ATP_ss + k_plus_CaADP Ca_ss ADP_ss)) \frac{d}{d time} (Ca_JSR) = beta_JSR (J_tr - J_rel) \frac{d}{d time} (Ca_NSR) = \frac{J_up V_myo}{V_NSR} - (\frac{J_tr 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} \frac{d}{d time} (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}) \frac{d}{d time} (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: calcium_background_current

Component: sodium_background_current

Component: L_type_Ca_current

Component: L_type_Ca_current_y_gate

Component: RyR_channel

Component: SERCA2a_pump

Component: intracellular_Ca_fluxes

Component: intracellular_ion_concentrations

Component: Ca_and_Mg_buffering_by_ATP

Component: extracellular_ion_concentrations

Component: model_parameters