Model Mathematics

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Component: Ca_SR_release

\frac{d}{d time} (I) = kiSRCa Ca_sub O - (kim I) - (kom I - (koSRCa {Ca_sub}^{2.0} RI)) \frac{d}{d time} (O) = koSRCa {Ca_sub}^{2.0} R - (kom O) - (kiSRCa Ca_sub O - (kim I)) \frac{d}{d time} (R) = kim RI - (kiSRCa Ca_sub R) - (koSRCa {Ca_sub}^{2.0} R - (kom O)) \frac{d}{d time} (RI) = kom I - (koSRCa {Ca_sub}^{2.0} RI) - (kim RI - (kiSRCa Ca_sub R)) j_SRCarel = ks O (Ca_jsr - Ca_sub) kCaSR = MaxSR - (\frac{MaxSR - MinSR}{1.0 + {(\frac{EC50_SR}{Ca_jsr})}^{HSR}}) kiSRCa = kiCa kCaSR koSRCa = \frac{koCa}{kCaSR}

Component: Ca_buffering

delta_fCMi = kf_CM Cai (1.0 - fCMi) - (kb_CM fCMi) delta_fCMs = kf_CM Ca_sub (1.0 - fCMs) - (kb_CM fCMs) delta_fCQ = kf_CQ Ca_jsr (1.0 - fCQ) - (kb_CQ fCQ) delta_fTC = kf_TC Cai (1.0 - fTC) - (kb_TC fTC) delta_fTMC = kf_TMC Cai (1.0 - (fTMC + fTMM)) - (kb_TMC fTMC) delta_fTMM = kf_TMM Mgi (1.0 - (fTMC + fTMM)) - (kb_TMM fTMM) \frac{d}{d time} (fCMi) = delta_fCMi \frac{d}{d time} (fCMs) = delta_fCMs \frac{d}{d time} (fCQ) = delta_fCQ \frac{d}{d time} (fTC) = delta_fTC \frac{d}{d time} (fTMC) = delta_fTMC \frac{d}{d time} (fTMM) = delta_fTMM

Component: Ca_dynamics

\frac{d}{d time} (Ca_jsr) = j_tr - (j_SRCarel + CQ_tot delta_fCQ) \frac{d}{d time} (Ca_nsr) = j_up - (\frac{j_tr V_jsr}{V_nsr}) \frac{d}{d time} (Ca_sub) = \frac{j_SRCarel V_jsr}{V_sub} - (\frac{i_siCa + i_CaT_i_CaT - (2.0 i_NaCa_i_NaCa)}{2.0 F V_sub} + j_Ca_dif + CM_tot delta_fCMs) \frac{d}{d time} (Cai) = \frac{1.0 (j_Ca_dif V_sub - (j_up V_nsr))}{V_i} - (CM_tot delta_fCMi + TC_tot delta_fTC + TMC_tot delta_fTMC)

Component: Ca_intracellular_fluxes

P_up = P_up_basal (1.0 - b_up) b_up = \{\begin{matrix} - 0.25 if Iso_1_uM > 0.0 \\ \frac{0.7 ACh}{(9e-05) + ACh} if ACh > 0.0 \\ 0.0 otherwise \end{matrix} j_Ca_dif = \frac{Ca_sub - Cai}{tau_dif_Ca} j_tr = \frac{Ca_nsr - Ca_jsr}{tau_tr} j_up = \frac{P_up}{1.0 + ⅇ^{\frac{(- Cai) + K_up}{slope_up}}}

Component: Cell_parameters

V_cell = (1e-09) 3.14159265358979312e+00 {R_cell}^{2.0} L_cell V_i = V_i_part V_cell - V_sub V_jsr = V_jsr_part V_cell V_nsr = V_nsr_part V_cell V_sub = (1e-09) 2.0 3.14159265358979312e+00 L_sub (R_cell - (\frac{L_sub}{2.0})) L_cell

Component: Ionic_values

E_K = RTONF \ln (\frac{Ko}{Ki}) E_Na = RTONF \ln (\frac{Nao}{Nai})

Component: Ki_concentration

\frac{d}{d time} (Ki) = \frac{(- 1.0) (i_Kur_i_Kur + i_to_i_to + i_Kr_i_Kr + i_Ks_i_Ks + i_fK + i_siK + i_SK_i_SK - (2.0 i_NaK_i_NaK))}{1.0 (V_i + V_sub) F}

Component: Membrane

RTONF = \frac{R T}{F} V = V_ode \frac{d}{d time} (V_ode) = \frac{- i_tot}{C} i_tot = i_f_i_f + i_Kr_i_Kr + i_Ks_i_Ks + i_to_i_to + i_NaK_i_NaK + i_NaCa_i_NaCa + i_Na_i_Na + i_CaL_i_CaL + i_CaT_i_CaT + i_KACh_i_KACh + i_Kur_i_Kur + i_SK_i_SK

Component: Nai_concentration

\frac{d}{d time} (Nai) = \frac{(- 1.0) (i_Na_i_Na + i_fNa + i_siNa + 3.0 i_NaK_i_NaK + 3.0 i_NaCa_i_NaCa)}{1.0 (V_i + V_sub) F}

Component: Rate_modulation_experiments

Component: environment

Component: i_CaL

ACh_block = \frac{0.31 ACh}{ACh + (9e-05)} Iso_increase = \{\begin{matrix} 1.23 if Iso_1_uM > 0.0 \\ 1.0 otherwise \end{matrix} i_CaL = (i_siCa + i_siK + i_siNa) (1.0 - ACh_block) 1.0 Iso_increase i_siCa = \frac{2.0 P_CaL (V - 0.0)}{RTONF (1.0 - (ⅇ^{\frac{(- 1.0) (V - 0.0) 2.0}{RTONF}}))} (Ca_sub - (Cao ⅇ^{\frac{(- 2.0) (V - 0.0)}{RTONF}})) dL fL fCa i_siK = \frac{0.000365 P_CaL (V - 0.0)}{RTONF (1.0 - (ⅇ^{\frac{(- 1.0) (V - 0.0)}{RTONF}}))} (Ki - (Ko ⅇ^{\frac{(- 1.0) (V - 0.0)}{RTONF}})) dL fL fCa i_siNa = \frac{(1.85e-05) P_CaL (V - 0.0)}{RTONF (1.0 - (ⅇ^{\frac{(- 1.0) (V - 0.0)}{RTONF}}))} (Nai - (Nao ⅇ^{\frac{(- 1.0) (V - 0.0)}{RTONF}})) dL fL fCa

Component: i_CaL_dL_gate

Iso_shift_dL = \{\begin{matrix} - 8.0 if Iso_1_uM > 0.0 \\ 0.0 otherwise \end{matrix} Iso_slope_dL = \{\begin{matrix} - 27.0 if Iso_1_uM > 0.0 \\ 0.0 otherwise \end{matrix} V_dL = - 7.7713 adVm = \{\begin{matrix} - 41.80001 if V = - 41.8 \\ 0.0 if V = 0.0 \\ - 6.80001 if V = - 6.8 \\ V otherwise \end{matrix} alpha_dL = \frac{(- 0.02839) (adVm + 41.8)}{ⅇ^{\frac{- (adVm + 41.8)}{2.5}} - 1.0} - (\frac{0.0849 (adVm + 6.8)}{ⅇ^{\frac{- (adVm + 6.8)}{4.8}} - 1.0}) bdVm = \{\begin{matrix} - 1.80001 if V = - 1.8 \\ V otherwise \end{matrix} beta_dL = \frac{0.01143 (bdVm + 1.8)}{ⅇ^{\frac{bdVm + 1.8}{2.5}} - 1.0} \frac{d}{d time} (dL) = \frac{dL_infinity - dL}{tau_dL} dL_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V - V_dL - Iso_shift_dL)}{k_dL (1.0 + \frac{Iso_slope_dL}{100.0})}}} tau_dL = \frac{0.001}{alpha_dL + beta_dL}

Component: i_CaL_fCa_gate

\frac{d}{d time} (fCa) = \frac{fCa_infinity - fCa}{tau_fCa} fCa_infinity = \frac{Km_fCa}{Km_fCa + Ca_sub} tau_fCa = \frac{0.001 fCa_infinity}{alpha_fCa}

Component: i_CaL_fL_gate

\frac{d}{d time} (fL) = \frac{fL_infinity - fL}{tau_fL} fL_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 12.1847 + shift_fL}{}}} tau_fL = 0.001 (44.3 + 230.0 ⅇ^{- ({(\frac{V + 36.0}{10.0})}^{2.0})})

Component: i_CaT

i_CaT = \frac{2.0 P_CaT V}{RTONF (1.0 - (ⅇ^{\frac{(- 1.0) V 2.0}{RTONF}}))} (Ca_sub - (Cao ⅇ^{\frac{(- 2.0) V}{RTONF}})) dT fT

Component: i_CaT_dT_gate

\frac{d}{d time} (dT) = \frac{dT_infinity - dT}{tau_dT} dT_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 38.3)}{5.5}}} tau_dT = \frac{0.001}{1.068 ⅇ^{\frac{V + 38.3}{30.0}} + 1.068 ⅇ^{\frac{- (V + 38.3)}{30.0}}}

Component: i_CaT_fT_gate

\frac{d}{d time} (fT) = \frac{fT_infinity - fT}{tau_fT} fT_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 58.7}{3.8}}} tau_fT = \frac{1.0}{16.67 ⅇ^{\frac{- (V + 75.0)}{83.3}} + 16.67 ⅇ^{\frac{V + 75.0}{15.38}}} + offset_fT

Component: i_KACh

i_KACh = \{\begin{matrix} ACh_on g_KACh (V - E_K) (1.0 + ⅇ^{\frac{V + 20.0}{20.0}}) a if ACh > 0.0 \\ 0.0 otherwise \end{matrix}

Component: i_KACh_a_gate

\frac{d}{d time} (a) = \frac{a_infinity - a}{tau_a} a_infinity = \frac{alpha_a}{alpha_a + beta_a} alpha_a = \frac{3.5988 - 0.025641}{1.0 + \frac{1.21550000000000005e-06}{{(1.0 ACh)}^{1.6951}}} + 0.025641 beta_a = 10.0 ⅇ^{0.0133 (V + 40.0)} tau_a = \frac{1.0}{alpha_a + beta_a}

Component: i_Kr

g_Kr = (4.98909900000000031e-03) \sqrt{\frac{Ko}{5.4}} i_Kr = g_Kr (V - E_K) (0.9 paF + 0.1 paS) piy

Component: i_Kr_pa_gate

\frac{d}{d time} (paF) = \frac{pa_infinity - paF}{tau_paF} \frac{d}{d time} (paS) = \frac{pa_infinity - paS}{tau_paS} pa_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 10.0144)}{7.6607}}} tau_paF = \frac{1.0}{30.0 ⅇ^{\frac{V}{10.0}} + ⅇ^{\frac{- V}{12.0}}} tau_paS = \frac{8.46553540000000049e-01}{4.2 ⅇ^{\frac{V}{17.0}} + 0.15 ⅇ^{\frac{- V}{21.6}}}

Component: i_Kr_pi_gate

pi_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 28.6}{17.1}}} \frac{d}{d time} (piy) = \frac{pi_infinity - piy}{tau_pi} tau_pi = \frac{1.0}{100.0 ⅇ^{\frac{- V}{54.645}} + 656.0 ⅇ^{\frac{V}{106.157}}}

Component: i_Ks

E_Ks = RTONF \ln (\frac{Ko + 0.12 Nao}{Ki + 0.12 Nai}) g_Ks = \{\begin{matrix} 1.2 g_Ks_if Iso_1_uM > 0.0 \\ g_Ks_otherwise \end{matrix} i_Ks = g_Ks (V - E_Ks) n^{2.0}

Component: i_Ks_n_gate

Iso_shift = \{\begin{matrix} - 14.0 if Iso_1_uM > 0.0 \\ 0.0 otherwise \end{matrix} alpha_n = \frac{28.0}{1.0 + ⅇ^{\frac{- (V - 40.0 - Iso_shift)}{3.0}}} beta_n = 1.0 ⅇ^{\frac{- (V - Iso_shift - 5.0)}{25.0}} \frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n} n_infinity = \sqrt{\frac{1.0}{1.0 + ⅇ^{\frac{- (V + 0.6383 - Iso_shift)}{10.7071}}}} tau_n = \frac{1.0}{alpha_n + beta_n}

Component: i_Kur

i_Kur = g_Kur r_Kur s_Kur (V - E_K)

Component: i_Kur_rKur_gate

\frac{d}{d time} (r_Kur) = \frac{r_Kur_infinity - r_Kur}{tau_r_Kur} r_Kur_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 6.0}{- 8.6}}} tau_r_Kur = \frac{0.009}{1.0 + ⅇ^{\frac{V + 5.0}{12.0}}} + 0.0005

Component: i_Kur_sKur_gate

\frac{d}{d time} (s_Kur) = \frac{s_Kur_infinity - s_Kur}{tau_s_Kur} s_Kur_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 7.5}{10.0}}} tau_s_Kur = \frac{0.59}{1.0 + ⅇ^{\frac{V + 60.0}{10.0}}} + 3.05

Component: i_Na

E_mh = RTONF \ln (\frac{Nao + 0.12 Ko}{Nai + 0.12 Ki}) i_Na = i_Na_+ i_Na_L i_Na_= g_Na m^{3.0} h (V - E_mh) i_Na_L = g_Na_L m^{3.0} (V - E_mh)

Component: i_NaCa

di = 1.0 + \frac{Ca_sub}{Kci} (1.0 + ⅇ^{\frac{(- Qci) V}{RTONF}} + \frac{Nai}{Kcni}) + \frac{Nai}{K1ni} (1.0 + \frac{Nai}{K2ni} (1.0 + \frac{Nai}{K3ni})) do = 1.0 + \frac{Cao}{Kco} (1.0 + ⅇ^{\frac{Qco V}{RTONF}}) + \frac{Nao}{K1no} (1.0 + \frac{Nao}{K2no} (1.0 + \frac{Nao}{K3no})) i_NaCa = \frac{K_NaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4} k12 = \frac{\frac{Ca_sub}{Kci} ⅇ^{\frac{(- Qci) V}{RTONF}}}{di} k14 = \frac{\frac{\frac{Nai}{K1ni} Nai}{K2ni} (1.0 + \frac{Nai}{K3ni}) ⅇ^{\frac{Qn V}{2.0 RTONF}}}{di} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco V}{RTONF}}}{do} k23 = \frac{\frac{\frac{Nao}{K1no} Nao}{K2no} (1.0 + \frac{Nao}{K3no}) ⅇ^{\frac{(- Qn) V}{2.0 RTONF}}}{do} k32 = ⅇ^{\frac{Qn V}{2.0 RTONF}} k34 = \frac{Nao}{K3no + Nao} k41 = ⅇ^{\frac{(- Qn) V}{2.0 RTONF}} k43 = \frac{Nai}{K3ni + Nai} x1 = k41 k34 (k23 + k21) + k21 k32 (k43 + k41) x2 = k32 k43 (k14 + k12) + k41 k12 (k34 + k32) x3 = k14 k43 (k23 + k21) + k12 k23 (k43 + k41) x4 = k23 k34 (k14 + k12) + k14 k21 (k34 + k32)

Component: i_NaK

Iso_increase = \{\begin{matrix} 1.2 if Iso_1_uM > 0.0 \\ 1.0 otherwise \end{matrix} i_NaK = Iso_increase i_NaK_max {(1.0 + {(\frac{Km_Kp}{Ko})}^{1.2})}^{- 1.0} {(1.0 + {(\frac{Km_Nap}{Nai})}^{1.3})}^{- 1.0} {(1.0 + ⅇ^{\frac{- ((V - E_Na) + 110.0)}{20.0}})}^{- 1.0}

Component: i_Na_h_gate

alpha_h = 20.0 ⅇ^{(- 0.125) (V + 75.0)} beta_h = \frac{2000.0}{320.0 ⅇ^{(- 0.1) (V + 75.0)} + 1.0} \frac{d}{d time} (h) = \frac{h_infinity - h}{tau_h} h_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 69.804}{4.4565}}} tau_h = \frac{1.0}{alpha_h + beta_h}

Component: i_Na_m_gate

E0_m = V + 41.0 alpha_m = \{\begin{matrix} 2000.0 if | E0_m | < delta_m \\ \frac{200.0 E0_m}{1.0 - (ⅇ^{(- 0.1) E0_m})} otherwise \end{matrix} beta_m = 8000.0 ⅇ^{(- 0.056) (V + 66.0)} \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m} m_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 42.0504)}{8.3106}}} tau_m = \frac{1.0}{alpha_m + beta_m}

Component: i_SK

i_SK = g_SK (V - E_K) x

Component: i_SK_x_gate

tau_x = \frac{0.001}{0.047 \frac{1000.0 Ca_sub}{1.0} + \frac{1.0}{76.0}} \frac{d}{d time} (x) = \frac{x_infinity - x}{tau_x} x_infinity = \frac{0.81 {(Ca_sub (1.0))}^{n_SK}}{{(\frac{Ca_sub}{1.0})}^{n_SK} + {(\frac{EC50_SK}{1.0})}^{n_SK}}

Component: i_f

i_f = i_fNa + i_fK i_fK = y g_f_K (V - E_K) i_fNa = y g_f_Na (V - E_Na)

Component: i_f_y_gate

ACh_shift = \{\begin{matrix} - 1.0 - (\frac{9.898 {(1.0 ACh)}^{0.618}}{{(1.0 ACh)}^{0.618} + (1.22422999999999998e-03)}) if ACh > 0.0 \\ 0.0 otherwise \end{matrix} Iso_shift = \{\begin{matrix} 7.5 if Iso_1_uM > 0.0 \\ 0.0 otherwise \end{matrix} tau_y = \frac{1.0}{\frac{0.36 (V + 148.8 - ACh_shift - Iso_shift)}{ⅇ^{0.066 (V + 148.8 - ACh_shift - Iso_shift)} - 1.0} + \frac{0.1 (V + 87.3 - ACh_shift - Iso_shift)}{1.0 - (ⅇ^{(- 0.2) (V + 87.3 - ACh_shift - Iso_shift)})}} - 0.054 \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y} y_infinity = \{\begin{matrix} 0.01329 + \frac{0.99921}{1.0 + ⅇ^{\frac{V + 97.134 - ACh_shift - Iso_shift - y_shift}{8.1752}}} if V < - (80.0 - ACh_shift - Iso_shift - y_shift) \\ 0.0002501 ⅇ^{\frac{- (V - ACh_shift - Iso_shift - y_shift)}{12.861}} otherwise \end{matrix}

Component: i_to

i_to = g_to (V - E_K) q r

Component: i_to_q_gate

\frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q} q_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{V + 49.0}{13.0}}} tau_q = 0.001 0.6 (\frac{65.17}{0.57 ⅇ^{(- 0.08) (V + 44.0)} + 0.065 ⅇ^{0.1 (V + 45.93)}} + 10.1)

Component: i_to_r_gate

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1.0}{1.0 + ⅇ^{\frac{- (V - 19.3)}{15.0}}} tau_r = 0.001 0.66 1.4 (\frac{15.59}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.369 ⅇ^{(- 0.12) (V + 23.84)}} + 2.98)