Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = \frac{I_stim - (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K_ATP + i_KNa + i_NaCa + i_K1 + i_Kp + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca)}{C}

Component: fast_sodium_current

i_Na = g_Na P_O_Na (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

Component: Na_channel_states

\frac{d}{d time} (P_C3) = beta_11 P_C2 - (alpha_11 P_C3) \frac{d}{d time} (P_C2) = (- ((beta_11 + alpha_12) P_C2)) + alpha_11 P_C3 + beta_12 P_C1 \frac{d}{d time} (P_C1) = (- ((beta_12 + alpha_13 + beta_3) P_C1)) + alpha_12 P_C2 + beta_13 P_O_Na + alpha_3 P_IF \frac{d}{d time} (P_O_Na) = (- ((alpha_2 + beta_13) P_O_Na)) + beta_2 P_IF + alpha_13 P_C1 \frac{d}{d time} (P_IF) = (- ((beta_2 + alpha_3 + alpha_4) P_IF)) + beta_3 P_C1 + beta_4 P_IS + alpha_2 P_O_Na \frac{d}{d time} (P_IS) = alpha_4 P_IS - (beta_4 P_IF) alpha_11 = \frac{3.802}{0.1027 ⅇ^{\frac{- V}{17.0}} + 0.20 ⅇ^{\frac{- V}{150.0}}} alpha_12 = \frac{3.802}{0.1027 ⅇ^{\frac{- V}{15.0}} + 0.23 ⅇ^{\frac{- V}{150.0}}} alpha_13 = \frac{3.802}{0.1027 ⅇ^{\frac{- V}{12.0}} + 0.25 ⅇ^{\frac{- V}{150.0}}} beta_11 = 0.1917 ⅇ^{\frac{- V}{20.3}} beta_12 = 0.20 ⅇ^{\frac{- (V - 5.0)}{20.3}} beta_13 = 0.22 ⅇ^{\frac{- (V - 10.0)}{20.3}} alpha_2 = 9.178 ⅇ^{\frac{V}{29.68}} beta_2 = \frac{alpha_13 alpha_2 alpha_3}{beta_13 beta_3} alpha_3 = 0.0000000037933 ⅇ^{\frac{- V}{7.7}} beta_3 = 0.0084 + 0.00002 V alpha_4 = \frac{alpha_2}{100.0} beta_4 = alpha_3

Component: L_type_Ca_channel

i_CaCa = d f f_Ca I_CaCa i_CaNa = d f f_Ca I_CaNa i_CaK = d f f_Ca I_CaK I_CaCa = P_Ca {2.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Cai Cai ⅇ^{\frac{2.0 V F}{R T}} - (gamma_Cao Cao)}{ⅇ^{\frac{2.0 V F}{R T}} - 1.0} I_CaNa = P_Na {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Nai Nai ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Nao Nao)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0} I_CaK = P_K {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Ki Ki ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Ko Ko)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0} i_Ca_L = i_CaCa + i_CaK + i_CaNa

Component: L_type_Ca_channel_d_gate

alpha_d = \frac{d_infinity}{tau_d} d_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + 10.0}{6.24})}} tau_d = d_infinity \frac{1.0 - (ⅇ^{- (\frac{V + 10.0}{6.24})})}{0.035 (V + 10.0)} beta_d = \frac{1.0 - d_infinity}{tau_d} \frac{d}{d time} (d) = alpha_d (1.0 - d) - (beta_d d)

Component: L_type_Ca_channel_f_gate

alpha_f = \frac{f_infinity}{tau_f} f_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 35.06}{8.6}}} + \frac{0.6}{1.0 + ⅇ^{\frac{50.0 - V}{20.0}}} tau_f = \frac{1.0}{0.0197 ⅇ^{- ({(0.0337 (V + 10.0))}^{2.0})} + 0.02} beta_f = \frac{1.0 - f_infinity}{tau_f} \frac{d}{d time} (f) = alpha_f (1.0 - f) - (beta_f f)

Component: L_type_Ca_channel_f_Ca_gate

f_Ca = \frac{1.0}{1.0 + {(\frac{Cai}{Km_Ca})}^{2.0}}

Component: T_type_Ca_channel

i_Ca_T = g_Ca_T b^{2.0} g (V - E_Ca) E_Ca = \frac{R T}{2.0 F} \ln (\frac{Cao}{Cai})

Component: T_type_Ca_channel_b_gate

b_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + 14.0}{10.8})}} tau_b = \frac{3.7 + 6.1}{1.0 + ⅇ^{\frac{25.0 + V}{4.5}}} \frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b}

Component: T_type_Ca_channel_g_gate

g_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 60.0}{5.6}}} tau_g = \{\begin{matrix} 12.0 if V > 0.0 \\ (-0.875) V + 12.0 otherwise \end{matrix} \frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g}

Component: rapid_time_dependent_potassium_current

i_Kr = g_Kr P_O (V - E_Kr) g_Kr = 0.0135 {Ko}^{0.59} E_Kr = \frac{R T}{F} \ln (\frac{Ko}{Ki})

Component: Kr_channel_states

\frac{d}{d time} (P_C3) = beta P_C2 - (alpha P_C3) \frac{d}{d time} (P_C2) = (- ((beta + alpha_in) P_C2)) + alpha P_C3 + beta_in P_C1 \frac{d}{d time} (P_C1) = (- ((beta_in + alpha_alpha + alpha_alpha) P_C1)) + alpha_in P_C2 + beta_beta P_O \frac{d}{d time} (P_O) = (- ((beta_beta + beta_i) P_O)) + alpha_alpha P_C1 + alpha_i P_I \frac{d}{d time} (P_I) = (- ((mu + alpha_i) P_I)) + alpha_alpha P_C1 + beta_i P_O alpha = 0.00555 ⅇ^{0.05547153 (V - 12.0)} beta = 0.002357 ⅇ^{(-0.036588) V} alpha_alpha = 0.00655 ⅇ^{0.05547153 (V - 36.0)} beta_beta = 0.0029357 ⅇ^{(-0.02158) V} alpha_i = 0.439 ⅇ^{(-0.02352) (V + 25.0)} \frac{4.5}{Ko} beta_i = 0.656 ⅇ^{0.000942 V} \frac{{4.5}^{0.3}}{{Ko}^{0.3}} mu = \frac{alpha_i beta_beta alpha_alpha}{alpha_alpha beta_i}

Component: slow_time_dependent_potassium_current

g_Ks = 0.433 \frac{1.0 + 0.6}{1.0 + {(\frac{0.000038}{Cai})}^{1.4}} E_Ks = \frac{R T}{F} \ln (\frac{Ko + P_NaK Nao}{Ki + P_NaK Nai}) i_Ks = g_Ks Xs1 Xs2 (V - E_Ks)

Component: slow_time_dependent_potassium_current_Xs1_gate

Xs_infinity = \frac{1.0}{1.0 + ⅇ^{- (\frac{V + (-1.5)}{16.7})}} tau_Xs1 = \frac{1.0}{\frac{0.0000719 (V + 30.0)}{1.0 - (ⅇ^{(-0.148) (V + 30.0)})} + \frac{0.000131 (V + 30.0)}{ⅇ^{0.0687 (V + 30.0)} - 1.0}} \frac{d}{d time} (Xs1) = \frac{Xs_infinity - Xs1}{tau_Xs1}

Component: slow_time_dependent_potassium_current_Xs2_gate

tau_Xs2 = 4.0 tau_Xs1 \frac{d}{d time} (Xs2) = \frac{Xs_infinity - Xs2}{tau_Xs2}

Component: sodium_activated_potassium_current

i_KNa = g_KNa P_oNai P_oV (V - E_K) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) P_oV = 0.8 - (\frac{0.65}{1.0 + ⅇ^{\frac{V + 125.0}{15.0}}}) P_oNai = \frac{0.85}{1.0 + {(\frac{K_D}{Nai})}^{n}}

Component: ATP_dependent_potassium_current

i_K_ATP = g_K_ATP (V - E_K) g_K_ATP = G_K_ATP P_ATP {(\frac{Ko}{Ko_normal})}^{n} E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) G_K_ATP = \frac{0.00000195}{Nichols_area}

Component: Na_Ca_exchanger

i_NaCa = K_NaCa \frac{1.0}{{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} Cai))

Component: time_independent_potassium_current

g_K1 = 0.75 \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}{1.0 + 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))}}{1.0 + 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{1.0}{1.0 + ⅇ^{\frac{7.488 - V}{5.98}}} i_Kp = g_Kp Kp (V - E_Kp)

Component: sarcolemmal_calcium_pump

i_p_Ca = I_pCa \frac{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}{2.0 F} \ln (\frac{Cao}{Cai}) i_Ca_b = g_Cab (V - E_CaN)

Component: sodium_potassium_pump

f_NaK = \frac{1.0}{1.0 + 0.1245 ⅇ^{(-0.1) \frac{V F}{R T}} + 0.0365 sigma ⅇ^{- (\frac{V F}{R T})}} sigma = \frac{1.0}{7.0} (ⅇ^{\frac{Nao}{67.3}} - 1.0) i_NaK = I_NaK f_NaK \frac{1.0}{1.0 + \sqrt{{(\frac{K_mNai}{Nai})}^{3.0}}} \frac{Ko}{Ko + K_mKo}

Component: non_specific_calcium_activated_current

i_ns_Na = I_ns_Na \frac{1.0}{1.0 + {(\frac{K_m_ns_Ca}{Cai})}^{3.0}} i_ns_K = I_ns_K \frac{1.0}{1.0 + {(\frac{K_m_ns_Ca}{Cai})}^{3.0}} i_ns_Ca = i_ns_Na + i_ns_K I_ns_Na = P_ns_Ca {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Nai Nai ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Nao Nao)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0} I_ns_K = P_ns_Ca {1.0}^{2.0} \frac{V F^{2.0}}{R T} \frac{gamma_Ki Ki ⅇ^{\frac{1.0 V F}{R T}} - (gamma_Ko Ko)}{ⅇ^{\frac{1.0 V F}{R T}} - 1.0}

Component: ionic_concentrations

\frac{d}{d time} (Nai) = (- (i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa 3.0 + i_NaK 3.0)) \frac{A_cap}{V_myo F} \frac{d}{d time} (Cai) = (i_CaCa + i_p_Ca + i_Ca_b + i_Ca_T - i_NaCa) \frac{A_cap}{2.0 V_myo F} + i_rel \frac{V_JSR}{V_myo} + (i_leak - i_up) \frac{V_NSR}{V_myo} \frac{d}{d time} (Ki) = (- (i_CaK + i_Kr + i_Ks + i_K_ATP + i_KNa + i_K1 + i_Kp + i_ns_K + (- (i_NaK 2.0)))) \frac{A_cap}{V_myo F} \frac{d}{d time} (Ko) = (i_CaK + i_Kr + i_Ks + i_K_ATP + i_KNa + i_Kp + i_ns_K + (- (i_NaK 2.0))) \frac{A_cap}{V_cleft F} \frac{d}{d time} (Ca_JSR) = - ((i_rel - i_tr) \frac{V_NSR}{V_JSR}) \frac{d}{d time} (Ca_NSR) = - (i_leak + i_tr - i_up) \frac{d}{d time} (Ca_foot) = (- i_CaCa) \frac{A_cap}{2.0 V_myo F} R_A_V

Component: calcium_buffers_in_the_myoplasm

Tn_buff = Tn_max \frac{Cai}{Cai + K_mTn} CMDN_buff = CMDN_max \frac{Cai}{Cai + K_mCMDN}

Component: calcium_fluxes_in_the_SR

i_rel = G_rel (Ca_JSR - Cai) G_rel = \{\begin{matrix} G_rel_max \frac{delta_Ca_i2 - delta_Ca_ith}{K_mrel + delta_Ca_i2 - delta_Ca_ith} (1.0 - (ⅇ^{- (\frac{t}{tau_on})})) ⅇ^{- (\frac{t}{tau_off})} if calcium_overload = 0.0 \\ G_rel_max (1.0 - (ⅇ^{- (\frac{t}{tau_on})})) ⅇ^{- (\frac{t}{tau_off})} otherwise \end{matrix} G_rel_max = \{\begin{matrix} 0.0 if CSQN_buff < CSQN_th \\ 4.0 otherwise \end{matrix} CSQN_buff = CSQN_max \frac{Ca_JSR}{Ca_JSR + K_mCSQN} i_up = I_up \frac{Cai}{Cai + K_mup} i_leak = K_leak Ca_NSR K_leak = \frac{I_up}{Ca_NSR_max} i_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr}