Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = \frac{i_Na + i_Ca_L_Ca + i_Ca_L_K + i_K + i_NaCa + i_K1 + i_Kp + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + I_stim}{Cm} 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}

Component: fast_sodium_current

i_Na = g_Na m^{3} 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 - (ⅇ^{(- 0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: fast_sodium_current_h_gate

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{80 + V}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{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{((- 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{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 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: L_type_Ca_channel

i_Ca_L_Ca = i_Ca_L_Ca_max y (O + O_Ca) i_Ca_L_K = \frac{\frac{p_k y (O + O_Ca) V F^{2}}{R T} (Ki ⅇ^{\frac{V F}{R T}} - Ko)}{ⅇ^{\frac{V F}{R T}} - 1} p_k = \frac{P_K}{1 + \frac{i_Ca_L_Ca_max}{i_Ca_L_Ca_half}} i_Ca_L_Ca_max = \frac{\frac{P_Ca 4 V F^{2}}{R T} (0.001 ⅇ^{\frac{2 V F}{R T}} - (0.341 Cao))}{ⅇ^{\frac{2 V F}{R T}} - 1} alpha = 0.4 ⅇ^{\frac{V + 12}{10}} beta = 0.05 ⅇ^{\frac{V + 12}{- 13}} alpha_a = alpha a beta_b = \frac{beta}{b} gamma = \frac{0.1875 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 + g_O_Ca + gamma a^{4} C4 - ((beta_b 4 + f_+ \frac{omega}{b^{4}}) C_Ca4) \frac{d}{d time} (O_Ca) = f_C_Ca4 - (g_O_Ca)

Component: L_type_Ca_channel_y_gate

\frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y} y_infinity = \frac{1}{1 + ⅇ^{\frac{V + 55}{7.5}}} + \frac{0.1}{1 + ⅇ^{\frac{(- V) + 21}{6}}} tau_y = 20 + \frac{600}{1 + ⅇ^{\frac{V + 30}{9.5}}}

Component: time_dependent_potassium_current

g_K = g_K_max \sqrt{\frac{Ko}{5.4}} E_K = \frac{R T}{F} \ln (\frac{Ko + P_NaK Nao}{Ki + P_NaK Nai}) i_K = g_K Xi X^{2} (V - E_K)

Component: time_dependent_potassium_current_X_gate

alpha_X = \frac{0.0000719 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} beta_X = \frac{0.000131 (V + 30)}{(- 1) + ⅇ^{0.0687 (V + 30)}} \frac{d}{d time} (X) = alpha_X (1 - X) - (beta_X X)

Component: time_dependent_potassium_current_Xi_gate

Xi = \frac{1}{1 + ⅇ^{\frac{V - 56.26}{32.1}}}

Component: time_independent_potassium_current

g_K1 = g_K1_max \sqrt{\frac{Ko}{5.4}} E_K1 = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_K1 = g_K1 K1_infinity (V - E_K1)

Component: time_independent_potassium_current_K1_gate

alpha_K1 = \frac{1.02}{1 + ⅇ^{0.2385 (V - E_K1 - 59.215)}} beta_K1 = \frac{0.49124 (ⅇ^{0.08032 (V + 5.476 - E_K1)} + ⅇ^{0.06175 (V - (E_K1 + 594.31))})}{1 + ⅇ^{(- 0.5143) ((V - E_K1) + 4.753)}} K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1}

Component: plateau_potassium_current

E_Kp = E_K1 Kp = \frac{1}{1 + ⅇ^{\frac{7.488 - V}{5.98}}} i_Kp = g_Kp Kp (V - E_Kp)

Component: Na_Ca_exchanger

i_NaCa = \frac{\frac{\frac{k_NaCa 1}{{K_mNa}^{3} + {Nao}^{3}} 1}{K_mCa + Cao} 1}{1 + k_sat ⅇ^{\frac{(eta - 1) V F}{R T}}} (ⅇ^{\frac{eta V F}{R T}} {Nai}^{3} Cao - (ⅇ^{\frac{(eta - 1) V F}{R T}} {Nao}^{3} Cai))

Component: sarcolemmal_calcium_pump

i_p_Ca = \frac{I_pCa Cai}{K_mpCa + Cai}

Component: sodium_background_current

E_NaN = E_Na i_Na_b = g_Nab (V - E_NaN)

Component: calcium_background_current

E_CaN = \frac{R T}{2 F} \ln (\frac{Cao}{Cai}) i_Ca_b = g_Cab (V - E_CaN)

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{Nao}{67.3}} - 1) i_NaK = \frac{\frac{I_NaK f_NaK 1}{1 + {(\frac{K_mNai}{Nai})}^{1.5}} Ko}{Ko + K_mKo}

Component: non_specific_calcium_activated_current

EnsCa = \frac{R T}{F} \ln (\frac{Ko + Nao}{Ki + Nai}) VnsCa = V - EnsCa i_ns_Na = \frac{I_ns_Na 1}{1 + {(\frac{K_m_ns_Ca}{Cai})}^{3}} i_ns_K = \frac{I_ns_K 1}{1 + {(\frac{K_m_ns_Ca}{Cai})}^{3}} i_ns_Ca = i_ns_Na + i_ns_K I_ns_Na = \frac{\frac{P_ns_Ca 1^{2} VnsCa F^{2}}{R T} (0.75 Nai ⅇ^{\frac{VnsCa F}{R T}} - (0.75 Nao))}{ⅇ^{\frac{VnsCa F}{R T}} - 1} I_ns_K = \frac{\frac{P_ns_Ca 1^{2} VnsCa F^{2}}{R T} (0.75 Ki ⅇ^{\frac{VnsCa F}{R T}} - (0.75 Ko))}{ⅇ^{\frac{VnsCa F}{R T}} - 1}

Component: calcium_subsystem

V_SS = 5.828 e -5 V_myo V_NSR = 0.081 V_myo V_JSR = 0.00464 V_myo J_rel = v1 RyR_open (Ca_JSR - Ca_SS) RyR_open = P_O1 + P_O2 \frac{d}{d time} (P_C1) = (- k_a_plus) {Ca_SS}^{nCa} P_C1 + k_a_minus P_O1 \frac{d}{d time} (P_O1) = (k_a_plus {Ca_SS}^{nCa} P_C1 - (k_a_minus P_O1 + k_b_plus {Ca_SS}^{mCa} 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}^{mCa} P_O1 - (k_b_minus P_O2) \frac{d}{d time} (P_C2) = k_c_plus P_O1 - (k_c_minus P_C2) J_leak = v2 (Ca_NSR - Cai) J_up = \frac{v3 {Cai}^{2}}{{K_mup}^{2} + {Cai}^{2}} J_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} J_xfer = \frac{Ca_SS - Cai}{tau_xfer} J_htrpn = k_htrpn_plus Cai (HTRPN_tot - HTRPNCa) - (k_htrpn_minus HTRPNCa) J_ltrpn = k_ltrpn_plus Cai (LTRPN_tot - LTRPNCa) - (k_ltrpn_minus LTRPNCa) J_trpn = J_htrpn + J_ltrpn \frac{d}{d time} (HTRPNCa) = J_htrpn \frac{d}{d time} (LTRPNCa) = J_ltrpn Bi = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Cai)}^{2}}} B_SS = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_SS)}^{2}}} B_JSR = \frac{1}{1 + \frac{CSQN_tot K_mCSQN}{{(K_mCSQN + Ca_JSR)}^{2}}} \frac{d}{d time} (Cai) = Bi (J_leak + J_xfer - (J_up + J_trpn + \frac{((i_Ca_b - (2 i_NaCa)) + i_p_Ca) Am}{2 V_myo F})) \frac{d}{d time} (Ca_SS) = B_SS (\frac{J_rel V_JSR}{V_SS} - (\frac{J_xfer V_myo}{V_SS}) - (\frac{i_Ca_L_Ca Am}{2 V_SS F})) \frac{d}{d time} (Ca_JSR) = B_JSR (J_tr - J_rel) \frac{d}{d time} (Ca_NSR) = \frac{(J_up - J_leak) V_myo}{V_NSR} - (\frac{J_tr V_JSR}{V_NSR})

Component: ionic_concentrations

\frac{d}{d time} (Nai) = \frac{(- (i_Na + i_Na_b + i_ns_Na + i_NaCa 3 + i_NaK 3)) Am}{V_myo F} \frac{d}{d time} (Ki) = \frac{(- (i_Ca_L_K + i_K + i_K1 + i_Kp + i_ns_K + (- i_NaK) 2)) Am}{V_myo F} \frac{d}{d time} (Ko) = \frac{(i_Ca_L_K + i_K + i_K1 + i_Kp + i_ns_K + (- i_NaK) 2) Am}{V_myo F}

Component: Myofilaments

\frac{d}{d time} (Cab) = kon Cai (TRPN_tot - Cab) - (koff (1 - (\frac{Tension}{gamma_trpn Tref})) Cab) K_2 = \frac{1 alphar2 {zp}^{nRel}}{{zp}^{nRel} + {Kz}^{nRel}} (1 - (\frac{nRel {Kz}^{nRel}}{{zp}^{nRel} + {Kz}^{nRel}})) K_1 = \frac{1 alphar2 {zp}^{nRel - 1} nRel {Kz}^{nRel}}{{({zp}^{nRel} + {Kz}^{nRel})}^{2}} z_max = \frac{alpha0 1}{{(\frac{CaTRPN50}{TRPN_tot})}^{nHill}} - (\frac{K_2 1}{alphar1 + K_1 1 + \frac{alpha0}{{(\frac{CaTRPN50}{TRPN_tot})}^{nHill}}}) Ca50 = Ca50ref (1 + beta1 (lambda - 1)) CaTRPN50 = \frac{Ca50 TRPN_tot}{Ca50 + \frac{koff}{kon} (1 - (\frac{(1 + beta0 (lambda - 1)) 0.5}{gamma_trpn}))} \frac{d}{d time} (z) = alpha0 {(\frac{Cab}{CaTRPN50})}^{nHill} (1 - z) + (- z) alphar1 + \frac{(- alphar2) z^{nRel}}{z^{nRel} + {Kz}^{nRel}} T0max = Tref (1 + beta0 (lambda - 1)) T0 = \frac{T0max z}{z_max} Tension = \{\begin{matrix} \frac{T0 (a Q + 1)}{1 - Q} if Q < 0 \\ \frac{T0 (1 - ((a + 2) Q))}{1 + Q} otherwise \end{matrix} \frac{d}{d time} (Q1) = A1 dlambdadt - (1 alpha1 Q1) \frac{d}{d time} (Q2) = A2 dlambdadt - (1 alpha2 Q2) \frac{d}{d time} (Q3) = A3 dlambdadt - (1 alpha3 Q3) Q = Q1 + Q2 + Q3