Model Mathematics

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Component: environment

Component: cell_geometry

Component: membrane

\frac{d}{d time} (V) = \frac{-1.0}{Cm} (i_Na + i_Ca + i_Ca_K + i_K + i_K1 + i_Kp + i_NaCa + i_NaK + i_ns_Ca + i_p_Ca + i_Ca_b + i_Na_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 = 0.32 \frac{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: time_dependent_potassium_current

g_K = 0.1128 \sqrt{\frac{Ko}{5.4}} E_K = \frac{R T}{F} \ln (\frac{Ko + PNa_K Nao}{Ki + PNa_K Nai}) i_K = g_K Xi X^{2.0} (V - E_K)

Component: time_dependent_potassium_current_X_gate

alpha_X = \frac{0.0000719 (V + 30.0)}{1.0 - (ⅇ^{(-0.148) (V + 30.0)})} beta_X = \frac{0.000131 (V + 30.0)}{(-1.0) + ⅇ^{0.0687 (V + 30.0)}} \frac{d}{d time} (X) = alpha_X (1.0 - X) - (beta_X X)

Component: time_dependent_potassium_current_Xi_gate

Xi = \frac{1.0}{1.0 + ⅇ^{\frac{V - 40.0}{40.0}}}

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 - E_K1) + 5.476)} + 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: 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) \frac{V F}{R T}}} (ⅇ^{eta \frac{V F}{R T}} {Nai}^{3.0} Cao - (ⅇ^{(eta - 1.0) \frac{V F}{R T}} {Nao}^{3.0} Cai))

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 + {(\frac{K_mNai}{Nai})}^{1.5}} \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 = Pns_Na \frac{V F^{2.0}}{R T} \frac{0.75 Nai ⅇ^{\frac{V F}{R T}} - (0.75 Nao)}{ⅇ^{\frac{V F}{R T}} - 1.0} I_ns_K = Pns_K \frac{V F^{2.0}}{R T} \frac{0.75 Ki ⅇ^{\frac{V F}{R T}} - (0.75 Ko)}{ⅇ^{\frac{V F}{R T}} - 1.0}

Component: sarcolemmal_calcium_pump

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

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_background_current

E_NaN = E_Na i_Na_b = g_Nab (V - E_NaN)

Component: L_type_Ca_channel

i_Ca = i_Ca_max y (O + O_Ca) i_Ca_max = {P_Ca}^{4.0} \frac{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} i_Ca_K = p_k 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}} alpha = 0.4 ⅇ^{\frac{V + 12.0}{10.0}} beta = 0.05 ⅇ^{\frac{- (V + 12.0)}{13.0}} alpha_a = alpha a beta_b = \frac{beta}{b} gamma = 0.5625 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_channel_y_gate

\frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y} y_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 55.0}{7.5}}} + \frac{0.1}{1.0 + ⅇ^{\frac{(- V) + 21.0}{6.0}}} tau_y = 20.0 + \frac{600.0}{1.0 + ⅇ^{\frac{V + 30.0}{9.5}}}

Component: RyR_channel_states

\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)

Component: SERCA_pump

J_up = \frac{Vmax_f fb - (Vmax_r rb)}{1.0 + fb + rb} fb = {(\frac{Cai}{k_fb})}^{n_fb} rb = {(\frac{Ca_NSR}{k_rb})}^{n_rb}

Component: intracellular_Ca_fluxes

J_rel = v1 (P_O1 + P_O2) (Ca_JSR - Ca_SS) J_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} J_xfer = \frac{Ca_SS - Cai}{tau_xfer} J_trpn = k_htrpn_plus Cai (HTRPN_tot - HTRPNCa - (k_htrpn_minus HTRPNCa)) + k_ltrpn_plus Cai (LTRPN_tot - LTRPNCa - (k_ltrpn_minus LTRPNCa))

Component: intracellular_ionic_concentrations

betai = \frac{1.0}{1.0 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Cai)}^{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} (Cai) = betai (- (J_xfer + J_up + J_trpn + (i_Ca_b + (- (2.0 i_NaCa)) + i_p_Ca) \frac{A_cap}{2.0 V_myo F})) \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}{2.0 V_SS F})) \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}) \frac{d}{d time} (Nai) = (- (i_Na + i_Na_b + i_ns_Na + i_NaCa 3.0 + i_NaK 3.0)) \frac{A_cap}{V_myo F} \frac{d}{d time} (Ki) = (- (i_Ca_K + i_K + i_K1 + i_Kp + i_ns_K + (- (i_NaK 2.0)))) \frac{A_cap}{V_myo F}

Component: troponin

\frac{d}{d time} (HTRPNCa) = k_htrpn_plus Cai (HTRPN_tot - HTRPNCa) - (k_htrpn_minus HTRPNCa) \frac{d}{d time} (LTRPNCa) = k_ltrpn_plus Cai (LTRPN_tot - LTRPNCa) - (k_ltrpn_minus LTRPNCa (0.333 + 0.667 (1.0 - Force_norm)))

Component: tropomyosin_cross_bridges

f_01 = 3.0 f_XB f_12 = 10.0 f_XB f_23 = 7.0 f_XB g_01_SL = 1.0 g_XB_SL g_12_SL = 2.0 g_XB_SL g_23_SL = 3.0 g_XB_SL g_XB_SL = g_XB (1.0 + {(1.0 - SL_norm)}^{1.6}) SL_norm = \frac{SL - 1.7}{0.7} k_trop_np = k_trop_pn {(\frac{\frac{LTRPNCa}{LTRPN_tot}}{K_trop_half})}^{N_trop} N_trop = 3.5 + 2.5 SL_norm K_trop_half = {(1.0 + \frac{K_trop_Ca}{1.7 - (0.9 SL_norm)})}^{-1.0} K_trop_Ca = \frac{k_ltrpn_minus}{k_ltrpn_plus} \frac{d}{d time} (N0) = (k_trop_pn P0 - (k_trop_np N0)) + g_01_SL N1 \frac{d}{d time} (P0) = (- ((k_trop_pn + f_01) P0)) + k_trop_np N0 + g_01_SL P1 \frac{d}{d time} (P1) = (- ((k_trop_pn + f_12 + g_01_SL) P1)) + k_trop_np N1 + k_trop_np N1 + f_01 P0 + g_12_SL P2 \frac{d}{d time} (P2) = (- ((f_23 + g_12_SL) P2)) + f_12 P1 + g_23_SL P3 \frac{d}{d time} (P3) = (- g_23_SL) P3 + f_23 P2

Component: force_computation

Force = zeta Force_norm Force_norm = phi_SL \frac{P1 + N1 + 2.0 P2 + 3.0 P3}{Force_max} Force_max = P1_max + 2.0 P2_max + 3.0 P3_max phi_SL = \{\begin{matrix} \frac{SL - 0.6}{1.4} if SL < 2.0 \land SL > 1.7 \\ 1.0 if SL < 2.2 \land SL > 2.0 \\ \frac{3.6 - SL}{1.4} if SL < 2.3 \land SL > 2.2 \end{matrix} P1_max = \frac{f_01 2.0 g_XB 3.0 g_XB}{g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23} P2_max = \frac{f_01 f_12 3.0 g_XB}{g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23} P3_max = \frac{f_01 f_12 f_23}{g_XB 2.0 g_XB 3.0 g_XB + f_01 2.0 g_XB 3.0 g_XB + f_01 f_12 3.0 g_XB + f_01 f_12 f_23}