Model Mathematics

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Component: environment

Component: Membrane

i_tot = i_f + i_Kr + i_Ks + i_to + i_NaK + i_NaCa + i_Na + i_CaL + i_CaT + i_KACh + i_Kur \frac{d}{d time} (V_ode) = \frac{- i_tot}{C} RTONF = \frac{R T}{F} V = \{\begin{matrix} V_clamp if clamp_mode \geq 1 \\ V_ode otherwise \end{matrix}

Component: Voltage_clamp

V_clamp = \{\begin{matrix} V_test if time > t_holding \land time < t_holding + t_test \\ V_holding otherwise \end{matrix}

Component: Rate_modulation_experiments

Component: Ionic_values

E_Na = RTONF \ln (\frac{Nao}{Nai}) E_K = RTONF \ln (\frac{Ko}{Ki}) E_Ca = 0.5 RTONF \ln (\frac{Cao}{Ca_sub})

Component: Nai_concentration

\frac{d}{d time} (Nai_) = \frac{(1 - Nai_clamp) (- 1) (i_Na + i_fNa + i_siNa + 3 i_NaK + 3 i_NaCa)}{1 (V_i + V_sub) F} Nai = Nai_

Component: i_f

G_f = \frac{g_f}{\frac{Ko}{Ko + Km_f}} G_f_K = \frac{G_f}{alpha + 1} G_f_Na = alpha G_f_K g_f_Na = \frac{G_f_Na Ko}{Ko + Km_f} g_f_K = \frac{G_f_K Ko}{Ko + Km_f} i_fNa = y g_f_Na (V - E_Na) (1 - blockade) i_fK = y g_f_K (V - E_K) (1 - blockade) i_f = i_fNa + i_fK

Component: i_f_y_gate

ACh_shift = \{\begin{matrix} - 1 - (\frac{9.898 {(1 ACh)}^{0.618}}{{(1 ACh)}^{0.618} + 0.00122423}) if ACh > 0 \\ 0 otherwise \end{matrix} Iso_shift = \{\begin{matrix} 7.5 if Iso_1_uM > 0 \\ 0 otherwise \end{matrix} tau_y = \frac{1}{\frac{0.36 (V + 148.8 - ACh_shift - Iso_shift)}{ⅇ^{0.066 (V + 148.8 - ACh_shift - Iso_shift)} - 1} + \frac{0.1 (V + 87.3 - ACh_shift - Iso_shift)}{1 - (ⅇ^{(- 0.2) (V + 87.3 - ACh_shift - Iso_shift)})}} - 0.054 y_infinity = \{\begin{matrix} 0.01329 + \frac{0.99921}{1 + ⅇ^{\frac{V + 97.134 - ACh_shift - Iso_shift - y_shift}{8.1752}}} if V < - (80 - ACh_shift - Iso_shift - y_shift) \\ 0.0002501 ⅇ^{\frac{- (V - ACh_shift - Iso_shift - y_shift)}{12.861}} otherwise \end{matrix} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

Component: i_NaK

Iso_increase = \{\begin{matrix} 1.2 if Iso_1_uM > 0 \\ 1 otherwise \end{matrix} i_NaK = Iso_increase i_NaK_max {(1 + {(\frac{Km_Kp}{Ko})}^{1.2})}^{- 1} {(1 + {(\frac{Km_Nap}{Nai})}^{1.3})}^{- 1} {(1 + ⅇ^{\frac{- ((V - E_Na) + 110)}{20}})}^{- 1}

Component: i_NaCa

i_NaCa = \frac{(1 - blockade_NaCa) K_NaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4} 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) k43 = \frac{Nai}{K3ni + Nai} k12 = \frac{\frac{Ca_sub}{Kci} ⅇ^{\frac{(- Qci) V}{RTONF}}}{di} k14 = \frac{\frac{\frac{Nai}{K1ni} Nai}{K2ni} (1 + \frac{Nai}{K3ni}) ⅇ^{\frac{Qn V}{2 RTONF}}}{di} k41 = ⅇ^{\frac{(- Qn) V}{2 RTONF}} di = 1 + \frac{Ca_sub}{Kci} (1 + ⅇ^{\frac{(- Qci) V}{RTONF}} + \frac{Nai}{Kcni}) + \frac{Nai}{K1ni} (1 + \frac{Nai}{K2ni} (1 + \frac{Nai}{K3ni})) k34 = \frac{Nao}{K3no + Nao} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco V}{RTONF}}}{do} k23 = \frac{\frac{\frac{Nao}{K1no} Nao}{K2no} (1 + \frac{Nao}{K3no}) ⅇ^{\frac{(- Qn) V}{2 RTONF}}}{do} k32 = ⅇ^{\frac{Qn V}{2 RTONF}} do = 1 + \frac{Cao}{Kco} (1 + ⅇ^{\frac{Qco V}{RTONF}}) + \frac{Nao}{K1no} (1 + \frac{Nao}{K2no} (1 + \frac{Nao}{K3no}))

Component: i_Na

E_mh = RTONF \ln (\frac{Nao + 0.12 Ko}{Nai + 0.12 Ki}) i_Na_= g_Na m^{3} h (V - E_mh) i_Na_L = g_Na_L m^{3} (V - E_mh) i_Na = i_Na_+ i_Na_L

Component: i_Na_m_gate

m_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 42.0504)}{8.3106}}} E0_m = V + 41 alpha_m = \{\begin{matrix} 2000 if | E0_m | < delta_m \\ \frac{200 E0_m}{1 - (ⅇ^{(- 0.1) E0_m})} otherwise \end{matrix} beta_m = 8000 ⅇ^{(- 0.056) (V + 66)} tau_m = \frac{1}{alpha_m + beta_m} \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m}

Component: i_Na_h_gate

h_infinity = \frac{1}{1 + ⅇ^{\frac{V + 69.804}{4.4565}}} alpha_h = 20 ⅇ^{(- 0.125) (V + 75)} beta_h = \frac{2000}{320 ⅇ^{(- 0.1) (V + 75)} + 1} tau_h = \frac{1}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_infinity - h}{tau_h}

Component: i_CaL

Iso_increase = \{\begin{matrix} 1.23 if Iso_1_uM > 0 \\ 1 otherwise \end{matrix} i_siCa = \frac{2 P_CaL (V - 0)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 0) 2}{RTONF}}))} (Ca_sub - (Cao ⅇ^{\frac{(- 2) (V - 0)}{RTONF}})) dL fL fCa i_siK = \frac{0.000365 P_CaL (V - 0)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 0)}{RTONF}}))} (Ki - (Ko ⅇ^{\frac{(- 1) (V - 0)}{RTONF}})) dL fL fCa i_siNa = \frac{0.0000185 P_CaL (V - 0)}{RTONF (1 - (ⅇ^{\frac{(- 1) (V - 0)}{RTONF}}))} (Nai - (Nao ⅇ^{\frac{(- 1) (V - 0)}{RTONF}})) dL fL fCa ACh_block = \frac{0.31 ACh}{ACh + 0.00009} i_CaL = (i_siCa + i_siK + i_siNa) (1 - ACh_block) 1 Iso_increase

Component: i_CaL_dL_gate

Iso_shift_dL = \{\begin{matrix} - 8 if Iso_1_uM > 0 \\ 0 otherwise \end{matrix} Iso_slope_dL = \{\begin{matrix} - 27 if Iso_1_uM > 0 \\ 0 otherwise \end{matrix} dL_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - V_dL - Iso_shift_dL)}{k_dL (1 + \frac{Iso_slope_dL}{100})}}} tau_dL = \frac{0.001}{alpha_dL + beta_dL} alpha_dL = \frac{(- 0.02839) (adVm + 41.8)}{ⅇ^{\frac{- (adVm + 41.8)}{2.5}} - 1} - (\frac{0.0849 (adVm + 6.8)}{ⅇ^{\frac{- (adVm + 6.8)}{4.8}} - 1}) adVm = \{\begin{matrix} - 41.80001 if V = - 41.8 \\ 0 if V = 0 \\ - 6.80001 if V = - 6.8 \\ V otherwise \end{matrix} beta_dL = \frac{0.01143 (bdVm + 1.8)}{ⅇ^{\frac{bdVm + 1.8}{2.5}} - 1} bdVm = \{\begin{matrix} - 1.80001 if V = - 1.8 \\ V otherwise \end{matrix} \frac{d}{d time} (dL) = \frac{dL_infinity - dL}{tau_dL}

Component: i_CaL_fL_gate

fL_infinity = \frac{1}{1 + ⅇ^{\frac{V + 37.4 + shift_fL}{5.3 + k_fL}}} tau_fL = 0.001 (44.3 + 230 ⅇ^{- ({(\frac{V + 36}{10})}^{2})}) \frac{d}{d time} (fL) = \frac{fL_infinity - fL}{tau_fL}

Component: i_CaL_fCa_gate

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

Component: i_CaT

i_CaT = \frac{2 P_CaT V}{RTONF (1 - (ⅇ^{\frac{(- 1) V 2}{RTONF}}))} (Ca_sub - (Cao ⅇ^{\frac{(- 2) V}{RTONF}})) dT fT

Component: i_CaT_dT_gate

dT_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 38.3)}{5.5}}} tau_dT = \frac{0.001}{1.068 ⅇ^{\frac{V + 38.3}{30}} + 1.068 ⅇ^{\frac{- (V + 38.3)}{30}}} \frac{d}{d time} (dT) = \frac{dT_infinity - dT}{tau_dT}

Component: i_CaT_fT_gate

fT_infinity = \frac{1}{1 + ⅇ^{\frac{V + 58.7}{3.8}}} tau_fT = \frac{1}{16.67 ⅇ^{\frac{- (V + 75)}{83.3}} + 16.67 ⅇ^{\frac{V + 75}{15.38}}} + offset_fT \frac{d}{d time} (fT) = \frac{fT_infinity - fT}{tau_fT}

Component: Ca_SR_release

j_SRCarel = ks O (Ca_jsr - Ca_sub) diff = Ca_jsr - Ca_sub kCaSR = MaxSR - (\frac{MaxSR - MinSR}{1 + {(\frac{EC50_SR}{Ca_jsr})}^{HSR}}) koSRCa = \frac{koCa}{kCaSR} kiSRCa = kiCa kCaSR \frac{d}{d time} (R) = kim RI - (kiSRCa Ca_sub R) - (koSRCa {Ca_sub}^{2} R - (kom O)) \frac{d}{d time} (O) = koSRCa {Ca_sub}^{2} R - (kom O) - (kiSRCa Ca_sub O - (kim I)) \frac{d}{d time} (I) = kiSRCa Ca_sub O - (kim I) - (kom I - (koSRCa {Ca_sub}^{2} RI)) \frac{d}{d time} (RI) = kom I - (koSRCa {Ca_sub}^{2} RI) - (kim RI - (kiSRCa Ca_sub R)) P_tot = R + O + I + RI

Component: Ca_intracellular_fluxes

b_up = \{\begin{matrix} - 0.25 if Iso_1_uM > 0 \\ \frac{0.7 ACh}{0.00009 + ACh} if ACh > 0 \\ 0 otherwise \end{matrix} P_up = P_up_basal (1 - b_up) j_Ca_dif = \frac{Ca_sub - Cai}{tau_dif_Ca} j_up = \frac{P_up}{1 + ⅇ^{\frac{(- Cai) + K_up}{slope_up}}} j_tr = \frac{Ca_nsr - Ca_jsr}{tau_tr}

Component: Ca_buffering

\frac{d}{d time} (fTC) = delta_fTC delta_fTC = kf_TC Cai (1 - fTC) - (kb_TC fTC) \frac{d}{d time} (fTMC) = delta_fTMC delta_fTMC = kf_TMC Cai (1 - (fTMC + fTMM)) - (kb_TMC fTMC) \frac{d}{d time} (fTMM) = delta_fTMM delta_fTMM = kf_TMM Mgi (1 - (fTMC + fTMM)) - (kb_TMM fTMM) \frac{d}{d time} (fCMi) = delta_fCMi delta_fCMi = kf_CM Cai (1 - fCMi) - (kb_CM fCMi) \frac{d}{d time} (fCMs) = delta_fCMs delta_fCMs = kf_CM Ca_sub (1 - fCMs) - (kb_CM fCMs) \frac{d}{d time} (fCQ) = delta_fCQ delta_fCQ = kf_CQ Ca_jsr (1 - fCQ) - (kb_CQ fCQ)

Component: Ca_dynamics

\frac{d}{d time} (Cai) = \frac{1 (j_Ca_dif V_sub - (j_up V_nsr))}{V_i} - (CM_tot delta_fCMi + TC_tot delta_fTC + TMC_tot delta_fTMC) \frac{d}{d time} (Ca_sub) = \frac{j_SRCarel V_jsr}{V_sub} - (\frac{i_siCa + i_CaT - (2 i_NaCa)}{2 F V_sub} + j_Ca_dif + CM_tot delta_fCMs) \frac{d}{d time} (Ca_nsr) = j_up - (\frac{j_tr V_jsr}{V_nsr}) \frac{d}{d time} (Ca_jsr) = j_tr - (j_SRCarel + CQ_tot delta_fCQ)

Component: Cell_parameters

V_cell = 0.000000001 π {R_cell}^{2} L_cell V_sub = 0.000000001 2 π L_sub (R_cell - (\frac{L_sub}{2})) L_cell V_jsr = V_jsr_part V_cell V_i = V_i_part V_cell - V_sub V_nsr = V_nsr_part V_cell

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}{1 + ⅇ^{\frac{V + 6}{- 8.6}}} tau_r_Kur = \frac{0.009}{1 + ⅇ^{\frac{V + 5}{12}}} + 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}{1 + ⅇ^{\frac{V + 7.5}{10}}} tau_s_Kur = \frac{0.59}{1 + ⅇ^{\frac{V + 60}{10}}} + 3.05

Component: i_to

i_to = g_to (V - E_K) q r

Component: i_to_q_gate

q_infinity = \frac{1}{1 + ⅇ^{\frac{V + 49}{13}}} tau_q = 0.001 0.6 (\frac{65.17}{0.57 ⅇ^{(- 0.08) (V + 44)} + 0.065 ⅇ^{0.1 (V + 45.93)}} + 10.1) \frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q}

Component: i_to_r_gate

r_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 19.3)}{15}}} 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) \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

Component: i_Kr

i_Kr = g_Kr (V - E_K) (0.9 paF + 0.1 paS) piy

Component: i_Kr_pa_gate

alfapaF = \frac{\frac{1}{1 + ⅇ^{\frac{- (V + 23.2)}{6.6}}}}{\frac{0.84655354}{37.2 ⅇ^{\frac{V}{11.9}} + 0.96 ⅇ^{\frac{- V}{18.5}}}} betapaF = 4 (\frac{37.2 ⅇ^{\frac{V}{15.9}} + 0.96 ⅇ^{\frac{- V}{22.5}}}{0.84655354} - (\frac{\frac{1}{1 + ⅇ^{\frac{- (V + 23.2)}{10.6}}}}{\frac{0.84655354}{37.2 ⅇ^{\frac{V}{15.9}} + 0.96 ⅇ^{\frac{- V}{22.5}}}})) pa_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 10.0144)}{7.6607}}} tau_paS = \frac{0.84655354}{4.2 ⅇ^{\frac{V}{17}} + 0.15 ⅇ^{\frac{- V}{21.6}}} tau_paF = \frac{1}{30 ⅇ^{\frac{V}{10}} + ⅇ^{\frac{- V}{12}}} \frac{d}{d time} (paS) = \frac{pa_infinity - paS}{tau_paS} \frac{d}{d time} (paF) = \frac{pa_infinity - paF}{tau_paF}

Component: i_Kr_pi_gate

tau_pi = \frac{1}{100 ⅇ^{\frac{- V}{54.645}} + 656 ⅇ^{\frac{V}{106.157}}} pi_infinity = \frac{1}{1 + ⅇ^{\frac{V + 28.6}{17.1}}} \frac{d}{d time} (piy) = \frac{pi_infinity - piy}{tau_pi}

Component: i_Ks

g_Ks = \{\begin{matrix} 1.2 g_Ks_if Iso_1_uM > 0 \\ g_Ks_otherwise \end{matrix} E_Ks = RTONF \ln (\frac{Ko + 0.12 Nao}{Ki + 0.12 Nai}) i_Ks = g_Ks (V - E_Ks) n^{2}

Component: i_Ks_n_gate

Iso_shift = \{\begin{matrix} - 14 if Iso_1_uM > 0 \\ 0 otherwise \end{matrix} n_infinity = \sqrt{\frac{1}{1 + ⅇ^{\frac{- (V + 0.6383 - Iso_shift)}{10.7071}}}} tau_n = \frac{1}{alpha_n + beta_n} alpha_n = \frac{28}{1 + ⅇ^{\frac{- (V - 40 - Iso_shift)}{3}}} beta_n = 1 ⅇ^{\frac{- (V - Iso_shift - 5)}{25}} \frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n}

Component: i_KACh

i_KACh = \{\begin{matrix} ACh_on g_KACh (V - E_K) (1 + ⅇ^{\frac{V + 20}{20}}) a if ACh > 0 \\ 0 otherwise \end{matrix}

Component: i_KACh_a_gate

alpha_a = \frac{3.5988 - 0.025641}{1 + \frac{0.0000012155}{{(1 ACh)}^{1.6951}}} + 0.025641 beta_a = 10 ⅇ^{0.0133 (V + 40)} a_infinity = \frac{alpha_a}{alpha_a + beta_a} tau_a = \frac{1}{alpha_a + beta_a} \frac{d}{d time} (a) = \frac{a_infinity - a}{tau_a}