Model Mathematics

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Component: Environment

Component: cell

i_Stim = \{\begin{matrix} - 12 if time - (⌊ \frac{time}{stim_Period} ⌋ stim_Period) \geq 100 \land time - (⌊ \frac{time}{stim_Period} ⌋ stim_Period) ≦ 103 \\ 0 otherwise \end{matrix} i_tot = i_K1 + i_to + i_Kr + i_Ks + i_CaL + i_NaK + i_Na + i_b_Na + i_NaCa + i_b_Ca + i_p_K + i_p_Ca + i_Stim \frac{d}{d time} (V) = - i_tot

Component: reversal_potentials

E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) E_Ks = \frac{R T}{F} \ln (\frac{K_o + P_kna Na_o}{K_i + P_kna Na_i}) E_Ca = \frac{0.5 R T}{F} \ln (\frac{Ca_o}{Ca_i})

Component: IK1

i_K1 = g_K1_0 (\frac{T}{35} - (\frac{55}{7})) \sqrt{\frac{K_o}{5.4}} xK1_inf (V - E_K)

Component: iK1_rectification

KiMg = 2.8 ⅇ^{\frac{- (V - (fac E_K))}{180}} KbMg = 0.45 ⅇ^{\frac{- (V - (fac E_K))}{20}} Kd1SPM = 0.7 e -3 ⅇ^{\frac{- ((V - (fac E_K)) + 8 Mg_Buf)}{4.8}} Kd2SPM = 40 e -3 ⅇ^{\frac{- (V - (fac E_K))}{9.1}} temp = 1 + \frac{Mg_Buf}{KbMg} rec1 = \frac{temp temp}{\frac{SPM}{Kd1SPM} + \frac{Mg_Buf}{KiMg} + temp temp temp} rec2 = \frac{1}{1 + \frac{SPM}{Kd2SPM}} xK1_inf = phi rec1 + (1 - phi) rec2

Component: IKr

i_Kr = g_Kr_0 (\frac{T}{35} - (\frac{55}{7})) \sqrt{\frac{K_o}{5.4}} Or4 (V - E_K)

Component: iKr_Markov

alpha_xr1 = \frac{1 T}{T_Base} ⅇ^{24.335 + \frac{T_Base}{T} (0.0112 V - 25.914)} beta_xr1 = \frac{1 T}{T_Base} ⅇ^{13.688 + \frac{T_Base}{T} ((- 0.0603) V - 15.707)} alpha_xr2 = \frac{1 T}{T_Base} ⅇ^{22.746 + \frac{T_Base}{T} (0 V - 25.914)} beta_xr2 = \frac{1 T}{T_Base} ⅇ^{13.193 + \frac{T_Base}{T} (0 V - 15.707)} alpha_xr3 = \frac{1 T}{T_Base} ⅇ^{22.098 + \frac{T_Base}{T} (0.0365 V - 25.914)} beta_xr3 = \frac{1 T}{T_Base} ⅇ^{7.313 + \frac{T_Base}{T} ((- 0.0399) V - 15.707)} alpha_xr4 = \frac{1 T}{T_Base} ⅇ^{30.016 + \frac{T_Base}{T} (0.0223 V - 30.888)} {(\frac{5.4}{K_o})}^{0.4} beta_xr4 = \frac{1 T}{T_Base} ⅇ^{30.061 + \frac{T_Base}{T} ((- 0.0312) V - 33.243)} \frac{d}{d time} (Cr1) = beta_xr1 Cr2 - (alpha_xr1 Cr1) \frac{d}{d time} (Cr2) = alpha_xr1 Cr1 + beta_xr2 Cr3 - ((alpha_xr2 + beta_xr1) Cr2) \frac{d}{d time} (Cr3) = alpha_xr2 Cr2 + beta_xr3 Or4 - ((alpha_xr3 + beta_xr2) Cr3) \frac{d}{d time} (Or4) = (alpha_xr3 Cr3 + beta_xr4 Ir5 - ((alpha_xr4 + beta_xr3) Or4) - OtoB) + BtoO \frac{d}{d time} (Ir5) = alpha_xr4 Or4 - (beta_xr4 Ir5)

Component: iKr_Markov_Sotalol_block

OtoB = O Sotalol_mM kBinding BtoO = BOr4 kDiss \frac{d}{d time} (BCr1) = beta_xr1 BCr2 - (alpha_xr1 BCr1) \frac{d}{d time} (BCr2) = alpha_xr1 BCr1 + beta_xr2 BCr3 - ((alpha_xr2 + beta_xr1) BCr2) \frac{d}{d time} (BCr3) = alpha_xr2 BCr2 + beta_xr3 BOr4 - ((alpha_xr3 + beta_xr2) BCr3) \frac{d}{d time} (BOr4) = (alpha_xr3 BCr3 + beta_xr4 BIr5 - ((alpha_xr4 + beta_xr3) BOr4)) + OtoB - BtoO \frac{d}{d time} (BIr5) = alpha_xr4 BOr4 - (beta_xr4 BIr5)

Component: IKs

i_Ks = g_Ks {Xs}^{2} (V - E_Ks)

Component: iKs_Xs_gate

xs_inf = \frac{1}{1 + ⅇ^{\frac{- 5 - V}{14}}} alpha_xs = \frac{1400}{\sqrt{1 + ⅇ^{\frac{5 - V}{6}}}} beta_xs = \frac{1}{1 + ⅇ^{\frac{V - 35}{15}}} tau_xs = 1 alpha_xs beta_xs + 80 \frac{d}{d time} (Xs) = \frac{xs_inf - Xs}{tau_xs}

Component: Ito

i_to = g_to r s (V - E_K)

Component: ito_s_gate

s_inf = \frac{1}{1 + ⅇ^{\frac{V + 20}{5}}} tau_s = 85 ⅇ^{\frac{- ({(V + 45)}^{2})}{320}} + \frac{5}{1 + ⅇ^{\frac{V - 20}{5}}} + 3 \frac{d}{d time} (s) = \frac{s_inf - s}{tau_s}

Component: ito_r_gate

r_inf = \frac{1}{1 + ⅇ^{\frac{20 - V}{6}}} tau_r = 9.5 ⅇ^{\frac{- ({(V + 40)}^{2})}{1800}} + 0.8 \frac{d}{d time} (r) = \frac{r_inf - r}{tau_r}

Component: INa

i_Na = g_Na m^{3} h j (V - E_Na)

Component: iNa_m_gate

m_inf = \frac{1}{{(1 + ⅇ^{\frac{- 56.86 - V}{9.03}})}^{2}} alpha_m = \frac{1}{1 + ⅇ^{\frac{- 60 - V}{5}}} beta_m = \frac{0.1}{1 + ⅇ^{\frac{V + 35}{5}}} + \frac{0.1}{1 + ⅇ^{\frac{V - 50}{200}}} tau_m = 1 alpha_m beta_m \frac{d}{d time} (m) = \frac{m_inf - m}{tau_m}

Component: iNa_h_gate

h_inf = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55 - shift_INa_inact}{7.43}})}^{2}} alpha_h = \{\begin{matrix} 0.057 ⅇ^{\frac{- (V + 80 - shift_INa_inact)}{6.8}} if V < (- 40) + shift_INa_inact \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 2.7 ⅇ^{0.079 (V - shift_INa_inact)} + 310000 ⅇ^{0.3485 (V - shift_INa_inact)} if V < (- 40) + shift_INa_inact \\ \frac{0.77}{0.13 (1 + ⅇ^{\frac{V + 10.66 - shift_INa_inact}{- 11.1}})} otherwise \end{matrix} tau_h = \frac{1}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_inf - h}{tau_h}

Component: iNa_j_gate

j_inf = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55 - shift_INa_inact}{7.43}})}^{2}} alpha_j = \{\begin{matrix} \frac{\frac{((- 25428) ⅇ^{0.2444 (V - shift_INa_inact)} - (6.948 e -6 ⅇ^{(- 0.04391) (V - shift_INa_inact)})) (V + 37.78)}{1}}{1 + ⅇ^{0.311 (V + 79.23 - shift_INa_inact)}} if V < (- 40) + shift_INa_inact \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.02424 ⅇ^{(- 0.01052) (V - shift_INa_inact)}}{1 + ⅇ^{(- 0.1378) (V + 40.14 - shift_INa_inact)}} if V < (- 40) + shift_INa_inact \\ \frac{0.6 ⅇ^{0.057 (V - shift_INa_inact)}}{1 + ⅇ^{(- 0.1) (V + 32 - shift_INa_inact)}} otherwise \end{matrix} tau_j = \frac{1}{alpha_j + beta_j} \frac{d}{d time} (j) = \frac{j_inf - j}{tau_j}

Component: INab

i_b_Na = g_bna (V - E_Na)

Component: ICaL

i_CaL = \frac{\frac{g_CaL d f f2 fCass z^{2} (V - 15) F^{2}}{R T} (0.25 Ca_ss ⅇ^{\frac{2 (V - 15) F}{R T}} - Ca_o)}{ⅇ^{\frac{2 (V - 15) F}{R T}} - 1}

Component: iCaL_d_gate

d_inf = \frac{1}{1 + ⅇ^{\frac{d_inf_shift - V}{7.5}}} alpha_d = \frac{1.4}{1 + ⅇ^{\frac{- 35 - V}{13}}} + 0.25 beta_d = \frac{1.4}{1 + ⅇ^{\frac{V + 5}{5}}} gamma_d = \frac{1}{1 + ⅇ^{\frac{50 - V}{20}}} tau_d = 1 alpha_d beta_d + gamma_d \frac{d}{d time} (d) = \frac{d_inf - d}{tau_d}

Component: iCaL_f_gate

f_inf = \frac{1}{1 + ⅇ^{\frac{V + 20}{7}}} tau_f = \frac{1102.5 ⅇ^{\frac{- ({(V + 27)}^{2})}{225}} + \frac{200}{1 + ⅇ^{\frac{13 - V}{10}}} + \frac{180}{1 + ⅇ^{\frac{V + 30}{10}}} + 20}{4} \frac{d}{d time} (f) = \frac{f_inf - f}{tau_f}

Component: iCaL_f2_gate

f2_inf = \frac{0.75}{1 + ⅇ^{\frac{V + 35}{7}}} + 0.25 tau_f2 = \frac{562 ⅇ^{\frac{- ({(V + 27)}^{2})}{240}} + \frac{31}{1 + ⅇ^{\frac{25 - V}{10}}} + \frac{80}{1 + ⅇ^{\frac{V + 30}{10}}}}{2} \frac{d}{d time} (f2) = \frac{f2_inf - f2}{tau_f2}

Component: iCaL_fCass_gate

fCass_inf = \frac{0.4}{1 + {(\frac{Ca_ss}{0.05})}^{2}} + 0.6 tau_fCass = \frac{80}{1 + {(\frac{Ca_ss}{0.05})}^{2}} + 2 \frac{d}{d time} (fCass) = \frac{fCass_inf - fCass}{tau_fCass}

Component: ICab

i_b_Ca = g_bca (V - E_Ca)

Component: IpCa

i_p_Ca = \frac{g_pCa Ca_i}{Ca_i + K_pCa}

Component: IpK

i_p_K = \frac{g_pK (V - E_K)}{1 + ⅇ^{\frac{25 - V}{5.98}}}

Component: INaK

i_NaK = \frac{\frac{\frac{P_NaK K_o}{K_o + K_mk} Na_i}{Na_i + K_mNa}}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0353 ⅇ^{\frac{(- V) F}{R T}}}

Component: INaCa

i_NaCa = \frac{K_NaCa (ⅇ^{\frac{gamma V F}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(gamma - 1) V F}{R T}} {Na_o}^{3} Ca_i alpha))}{({Km_Nai}^{3} + {Na_o}^{3}) (Km_Ca + Ca_o) (1 + K_sat ⅇ^{\frac{(gamma - 1) V F}{R T}})}

Component: Ca

\frac{d}{d time} (Ca_i) = Ca_i_bufc (\frac{(i_leak - i_up) V_sr}{Vol_c} + i_xfer - (\frac{(i_b_Ca + i_p_Ca - (2 i_NaCa)) Cm}{2 Vol_c F})) \frac{d}{d time} (Ca_SR) = Ca_sr_bufsr (i_up - (i_rel + i_leak)) \frac{d}{d time} (Ca_ss) = Ca_ss_bufss (\frac{(- i_CaL) Cm}{2 V_ss F} + \frac{i_rel V_sr}{V_ss} - (\frac{i_xfer Vol_c}{V_ss}))

Component: Ileak_Iup_Ixfer

i_up = \frac{Vmax_up}{1 + \frac{{K_up}^{2}}{{Ca_i}^{2}}} i_leak = Vol_leak (Ca_SR - Ca_i) i_xfer = Vol_xfer (Ca_ss - Ca_i)

Component: Irel

kcasr = max_sr - (\frac{max_sr - min_sr}{1 + {(\frac{EC}{Ca_SR})}^{2}}) k1 = \frac{k1_prime}{kcasr} k2 = k2_prime kcasr O = \frac{k1 {Ca_ss}^{2} R_prime}{k3 + k1 {Ca_ss}^{2}} i_rel = Vol_rel O (Ca_SR - Ca_ss) \frac{d}{d time} (R_prime) = (- k2) Ca_ss R_prime + k4 (1 - R_prime)

Component: Ca_buffer

Ca_i_bufc = \frac{1}{1 + \frac{Buf_c K_buf_c}{{(Ca_i + K_buf_c)}^{2}}} Ca_sr_bufsr = \frac{1}{1 + \frac{Buf_sr K_buf_sr}{{(Ca_SR + K_buf_sr)}^{2}}} Ca_ss_bufss = \frac{1}{1 + \frac{Buf_ss K_buf_ss}{{(Ca_ss + K_buf_ss)}^{2}}}

Component: Na

\frac{d}{d time} (Na_i) = \frac{(- (i_Na + i_b_Na + 3 i_NaK + 3 i_NaCa)) Cm}{Vol_c F}

Component: K

\frac{d}{d time} (K_i) = \frac{- (i_K1 + i_to + i_Kr + i_Ks + i_p_K + i_Stim - (2 i_NaK))}{Vol_c F} Cm

Model Mathematics

Component: Environment

Component: cell

Component: reversal_potentials

Component: IK1

Component: iK1_rectification

Component: IKr

Component: iKr_Markov

Component: iKr_Markov_Sotalol_block

Component: IKs

Component: iKs_Xs_gate

Component: Ito

Component: ito_s_gate

Component: ito_r_gate

Component: INa

Component: iNa_m_gate

Component: iNa_h_gate

Component: iNa_j_gate

Component: INab

Component: ICaL

Component: iCaL_d_gate

Component: iCaL_f_gate

Component: iCaL_f2_gate

Component: iCaL_fCass_gate

Component: ICab

Component: IpCa

Component: IpK

Component: INaK

Component: INaCa

Component: Ca

Component: Ileak_Iup_Ixfer

Component: Irel

Component: Ca_buffer

Component: Na

Component: K

Component: transmembrane_currents

Component: intracellular_currents

Component: Ions_n_reversal_potentials