Model Mathematics

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Component: environment

Component: membrane

\frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + I_st)

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_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 = \frac{\frac{P_Ca 2^{2} V F^{2}}{R T} (gamma_Cai Cai ⅇ^{\frac{2 V F}{R T}} - (gamma_Cao Cao))}{ⅇ^{\frac{2 V F}{R T}} - 1} I_CaNa = \frac{\frac{P_Na 1^{2} V F^{2}}{R T} (gamma_Nai Nai ⅇ^{\frac{1 V F}{R T}} - (gamma_Nao Nao))}{ⅇ^{\frac{1 V F}{R T}} - 1} I_CaK = \frac{\frac{P_K 1^{2} V F^{2}}{R T} (gamma_Ki Ki ⅇ^{\frac{1 V F}{R T}} - (gamma_Ko Ko))}{ⅇ^{\frac{1 V F}{R T}} - 1} 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}{1 + ⅇ^{\frac{- (V + 10)}{6.24}}} tau_d = \frac{d_infinity (1 - (ⅇ^{\frac{- (V + 10)}{6.24}}))}{0.035 (V + 10)} beta_d = \frac{1 - d_infinity}{tau_d} \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

Component: L_type_Ca_channel_f_gate

alpha_f = \frac{f_infinity}{tau_f} f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 35.06}{8.6}}} + \frac{0.6}{1 + ⅇ^{\frac{50 - V}{20}}} tau_f = \frac{1}{0.0197 ⅇ^{- ({(0.0337 (V + 10))}^{2})} + 0.02} beta_f = \frac{1 - f_infinity}{tau_f} \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

Component: L_type_Ca_channel_f_Ca_gate

f_Ca = \frac{1}{1 + {(\frac{Cai}{Km_Ca})}^{2}}

Component: T_type_Ca_channel

i_Ca_T = g_Ca_T b^{2} g (V - E_Ca) E_Ca = \frac{R T}{2 F} \ln (\frac{Cao}{Cai})

Component: T_type_Ca_channel_b_gate

\frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b} b_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 14)}{10.8}}} tau_b = \frac{3.7 + 6.1}{1 + ⅇ^{\frac{25 + V}{4.5}}}

Component: T_type_Ca_channel_g_gate

\frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g} g_infinity = \frac{1}{1 + ⅇ^{\frac{V + 60}{5.6}}} tau_g = \{\begin{matrix} 12 if V > 0 \\ (- 0.875) V + 12 otherwise \end{matrix}

Component: rapid_time_dependent_potassium_current

g_Kr = 0.02614 \sqrt{\frac{Ko}{5.4}} E_Kr = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_Kr = g_Kr Xr Rr (V - E_Kr)

Component: rapid_time_dependent_potassium_current_Xr_gate

\frac{d}{d time} (Xr) = \frac{Xr_infinity - Xr}{tau_Xr} Xr_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 21.5)}{7.5}}} tau_Xr = \frac{1}{\frac{0.00138 (V + 14.2)}{1 - (ⅇ^{(- 0.123) (V + 14.2)})} + \frac{0.00061 (V + 38.9)}{ⅇ^{0.145 (V + 38.9)} - 1}}

Component: rapid_time_dependent_potassium_current_Rr_gate

Rr = \frac{1}{1 + ⅇ^{\frac{V + 9}{22.4}}}

Component: slow_time_dependent_potassium_current

g_Ks = \frac{0.057 + 0.19}{1 + ⅇ^{\frac{(- 7.2) + P_Ca}{0.6}}} E_Ks = \frac{R T}{F} \ln (\frac{Ko + P_NaK Nao}{Ki + P_NaK Nai}) P_Ca = (- (\log_{10} Cai)) + 3 i_Ks = g_Ks {Xs}^{2} (V - E_Ks)

Component: slow_time_dependent_potassium_current_Xs_gate

\frac{d}{d time} (Xs) = \frac{Xs_infinity - Xs}{tau_Xs} Xs_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 1.5)}{16.7}}} tau_Xs = \frac{1}{\frac{0.0000719 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} + \frac{0.000131 (V + 30)}{ⅇ^{0.0687 (V + 30)} - 1}}

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 + 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: 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 + \sqrt{{(\frac{K_mNai}{Nai})}^{3}}} Ko}{Ko + K_mKo}

Component: non_specific_calcium_activated_current

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} V F^{2}}{R T} (gamma_Nai Nai ⅇ^{\frac{1 V F}{R T}} - (gamma_Nao Nao))}{ⅇ^{\frac{1 V F}{R T}} - 1} I_ns_K = \frac{\frac{P_ns_Ca 1^{2} V F^{2}}{R T} (gamma_Ki Ki ⅇ^{\frac{1 V F}{R T}} - (gamma_Ko Ko))}{ⅇ^{\frac{1 V F}{R T}} - 1}

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: calcium_buffers_in_the_myoplasm

TRPN_buff = \frac{TRPN_max Cai}{Cai + K_mTRPN} CMDN_buff = \frac{CMDN_max Cai}{Cai + K_mCMDN} Ca_JSR_new = \frac{\sqrt{{b1}^{2} + 4 c1} - b1}{2} delta_Ca_JSR = Ca_JSR_old - Ca_JSR_new delta_Cai = Cai_old - Cai b1 = K_mCSQN - (CSQN_max + CSQN_buff + delta_Ca_JSR + Ca_JSR_old) c1 = K_mCSQN (CSQN_buff + delta_Ca_JSR + Ca_JSR_old) Cai = \frac{2}{3} \sqrt{b^{2} - (3 c)} \cos (\frac{\arccos (\frac{9 b c - (2 b^{3} + 27 d)}{2 {(b^{2} - (3 c))}^{\frac{3}{2}}})}{3}) - (\frac{b}{3}) b = CMDN_max + TRPN_max + K_mTRPN + K_mCMDN - Ca_total c = K_mCMDN K_mTRPN + TRPN_max K_mCMDN + CMDN_max K_mTRPN - (Ca_total (K_mTRPN + K_mCMDN)) d = (- K_mTRPN) K_mCMDN Ca_total Ca_total = TRPN_buff + CMDN_buff + delta_Cai + Cai_old

Component: calcium_fluxes_in_the_SR

i_rel = G_rel (Ca_JSR_new - Cai) G_rel = \{\begin{matrix} \frac{G_rel_max (delta_Ca_i2 - delta_Ca_ith)}{K_mrel + delta_Ca_i2 - delta_Ca_ith} (1 - (ⅇ^{\frac{- t}{tau_on}})) ⅇ^{\frac{- t}{tau_off}} if calcium_overload = 0 \\ G_rel_max (1 - (ⅇ^{\frac{- t}{tau_on}})) ⅇ^{\frac{- t}{tau_off}} otherwise \end{matrix} G_rel_max = \{\begin{matrix} 0 if calcium_overload = 0 \land delta_Ca_i2 < delta_Ca_ith \\ 60 if calcium_overload = 0 \land delta_Ca_i2 \geq delta_Ca_ith \\ 0 if CSQN_buff < CSQN_th \\ 4 otherwise \end{matrix} CSQN_buff = \frac{CSQN_max Ca_JSR_new}{Ca_JSR_new + K_mCSQN} i_up = \frac{I_up 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_new}{tau_tr}

Component: ionic_concentrations

\frac{d}{d time} (Nai) = \frac{(- (i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa 3 + i_NaK 3)) A_cap}{V_myo F} \frac{d}{d time} (Ki) = \frac{(- (i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_ns_K + (- i_NaK) 2)) A_cap}{V_myo F} \frac{d}{d time} (Ko) = \frac{(i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_ns_K + (- i_NaK) 2) A_cap}{V_cleft F} \frac{d}{d time} (Ca_NSR) = - (i_leak + i_tr - i_up) \frac{d}{d time} (Ca_foot) = \frac{(- i_CaCa) A_cap}{2 V_myo F} R_A_V