Model Mathematics

Component based math viewer is available

Component: environment

Component: membrane

V_cell = π {Radius_cell}^{2} Length_cell V_myo = 0.65 V_cell V_sr = 0.035 V_cell V_sl = 0.02 V_cell V_junc = 5.39 e -4 V_cell FoRT = \frac{Frdy}{R T} F_sl = 1 - F_junc F_sl_Ca_L = 1 - F_junc_Ca_L I_Stim = \{\begin{matrix} I_Stim_Amplitude if t \geq I_Stim_Start \land t ≦ I_Stim_End \land t - I_Stim_Start - (⌊ \frac{t - I_Stim_Start}{I_Stim_Period} ⌋ I_Stim_Period) ≦ I_Stim_PulseDuration \\ 0 otherwise \end{matrix} I_Na_tot = I_Na_tot_junc + I_Na_tot_sl I_Cl_tot = I_ClCa + I_Cl_Bk I_Ca_tot = I_Ca_tot_junc + I_Ca_tot_sl I_tot = I_Na_tot + I_Cl_tot + I_Ca_tot + I_K_tot \frac{d}{d t} (V) = - (I_tot - I_Stim)

Component: Fast_Na_Current

m_ss = \frac{1}{{(1 + ⅇ^{\frac{- (56.86 + V)}{9.03}})}^{2}} tau_m = 0.1292 ⅇ^{- ({(\frac{V + 45.79}{15.54})}^{2})} + 0.06487 ⅇ^{- ({(\frac{V - 4.823}{51.12})}^{2})} a_h = \{\begin{matrix} 0 if V \geq - 40 \\ 0.057 ⅇ^{\frac{- (V + 80)}{6.8}} otherwise \end{matrix} b_h = \{\begin{matrix} \frac{5.9231}{1 + ⅇ^{\frac{- (V + 10.66)}{11.1}}} if V \geq - 40 \\ 2.7 ⅇ^{0.079 V} + 3.1 e 5 ⅇ^{0.3485 V} otherwise \end{matrix} a_j = \{\begin{matrix} 0 if V \geq - 40 \\ \frac{((- 2.5428 e 4) ⅇ^{0.2444 V} - (6.948 e -6 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} otherwise \end{matrix} b_j = \{\begin{matrix} \frac{0.6 ⅇ^{0.057 V}}{1 + ⅇ^{(- 0.1) (V + 32)}} if V \geq - 40 \\ \frac{0.02424 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} otherwise \end{matrix} tau_h = \frac{1}{a_h + b_h} h_ss = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55}{7.43}})}^{2}} tau_j = \frac{1}{a_j + b_j} j_ss = \frac{1}{{(1 + ⅇ^{\frac{V + 71.55}{7.43}})}^{2}} \frac{d}{d t} (m) = \frac{m_ss - m}{tau_m} \frac{d}{d t} (h) = \frac{h_ss - h}{tau_h} \frac{d}{d t} (j) = \frac{j_ss - j}{tau_j} I_Na_junc = F_junc G_Na m^{3} h j (V - E_Na_junc) I_Na_sl = F_sl G_Na m^{3} h j (V - E_Na_sl) I_Na = I_Na_junc + I_Na_sl

Component: Background_Na_Current

I_Na_Bk_junc = F_junc G_Na_B (V - E_Na_junc) I_Na_Bk_sl = F_sl G_Na_B (V - E_Na_sl) I_Na_Bk = I_Na_Bk_junc + I_Na_Bk_sl

Component: Na_K_Pump_Current

sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1}{7} f_NaK = \frac{1}{1 + 0.1245 ⅇ^{(- 0.1) V FoRT} + 0.0365 sigma ⅇ^{(- V) FoRT}} I_NaK_junc = \frac{F_junc Ibar_NaK f_NaK K_o}{(1 + {(\frac{Km_Naip}{Na_j})}^{4}) (K_o + Km_Ko)} I_NaK_sl = \frac{F_sl Ibar_NaK f_NaK K_o}{(1 + {(\frac{Km_Naip}{Na_sl})}^{4}) (K_o + Km_Ko)} I_NaK = I_NaK_junc + I_NaK_sl

Component: Rapidly_Activating_K_Current

x_r_ss = \frac{1}{1 + ⅇ^{\frac{- (V + 10)}{5}}} tau_xr = \frac{3300}{(1 + ⅇ^{\frac{- 22 - V}{9}}) (1 + ⅇ^{\frac{V + 11}{9}})} + \frac{230}{1 + ⅇ^{\frac{V + 40}{20}}} \frac{d}{d t} (x_Kr) = \frac{x_r_ss - x_Kr}{tau_xr} r_Kr = \frac{1}{1 + ⅇ^{\frac{V + 74}{24}}} I_Kr = G_Kr \sqrt{\frac{K_o}{5.4}} x_Kr r_Kr (V - E_K)

Component: Slowly_Activating_K_Current

x_s_ss = \frac{1}{1 + ⅇ^{\frac{- (V + 3.8)}{14.25}}} tau_xs = \frac{990.1}{1 + ⅇ^{\frac{- (V + 2.436)}{14.12}}} \frac{d}{d t} (x_Ks) = \frac{x_s_ss - x_Ks}{tau_xs} I_Ks_junc = F_junc G_Ks_junc {x_Ks}^{2} (V - E_Ks) I_Ks_sl = F_sl G_Ks_sl {x_Ks}^{2} (V - E_Ks) I_Ks = I_Ks_junc + I_Ks_sl

Component: Plateau_K_Current

kp_Kp = \frac{1}{1 + ⅇ^{7.488 - (\frac{V}{5.98})}} I_Kp = G_Kp kp_Kp (V - E_K)

Component: Transient_Outward_K_Current

x_to_ss = \frac{1}{1 + ⅇ^{\frac{- (V - 19)}{13}}} y_to_ss = \frac{1}{1 + ⅇ^{\frac{V + 19.5}{5}}} tau_x_tos = \frac{9}{1 + ⅇ^{\frac{V + 3}{15}}} + 0.5 tau_y_tos = \frac{800}{1 + ⅇ^{\frac{V + 60}{10}}} + 30 \frac{d}{d t} (x_to_s) = \frac{x_to_ss - x_to_s}{tau_x_tos} \frac{d}{d t} (y_to_s) = \frac{y_to_ss - y_to_s}{tau_y_tos} tau_x_tof = 8.5 ⅇ^{- ({(\frac{V + 45}{50})}^{2})} + 0.5 tau_y_tof = 85 ⅇ^{\frac{- ({(V + 40)}^{2})}{220}} + 7 \frac{d}{d t} (x_to_f) = \frac{x_to_ss - x_to_f}{tau_x_tof} \frac{d}{d t} (y_to_f) = \frac{y_to_ss - y_to_f}{tau_y_tof} I_to_s = G_to_s x_to_s y_to_s (V - E_K) I_to_f = G_to_f x_to_f y_to_f (V - E_K) I_to = I_to_s + I_to_f

Component: Inward_Rectifier_K_Current

a_K1 = \frac{4.0938}{1 + ⅇ^{0.12165 (V - E_K - 49.9344)}} b_K1 = \frac{15.7197 ⅇ^{0.06739 (V - E_K - 3.2571)} + ⅇ^{0.06175 (V - E_K - 594.31)}}{1 + ⅇ^{(- 0.16285) ((V - E_K) + 14.2067)}} K1_ss = \frac{a_K1}{a_K1 + b_K1} I_K1 = G_K1 \sqrt{\frac{K_o}{5.4}} K1_ss (V - E_K)

Component: Ca_Activated_Cl_Current

I_ClCa_junc = \frac{F_junc G_ClCa}{1 + \frac{Kd_ClCa}{Ca_j}} (V - E_Cl) I_ClCa_sl = \frac{F_sl G_ClCa}{1 + \frac{Kd_ClCa}{Ca_sl}} (V - E_Cl) I_ClCa = I_ClCa_junc + I_ClCa_sl

Component: Background_Cl_Current

I_Cl_Bk = G_Cl_B (V - E_Cl)

Component: L_Type_Calcium_Current

d_ss = \frac{1}{1 + ⅇ^{\frac{- (V + 5)}{6}}} 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 = alpha_d beta_d + gamma_d f_ss = \frac{1}{1 + ⅇ^{\frac{V + 20}{7}}} alpha_f = 1102.5 ⅇ^{- ({(\frac{V + 27}{15})}^{2})} beta_f = \frac{200}{1 + ⅇ^{\frac{13 - V}{10}}} gamma_f = \frac{180}{1 + ⅇ^{\frac{V + 30}{10}}} + 20 tau_f = alpha_f + beta_f + gamma_f f_2_ss = \frac{0.67}{1 + ⅇ^{\frac{V + 35}{7}}} + 0.33 alpha_f_2 = 300 ⅇ^{\frac{- ({(V + 25)}^{2})}{170}} beta_f_2 = \frac{31}{1 + ⅇ^{\frac{25 - V}{10}}} gamma_f_2 = \frac{16}{1 + ⅇ^{\frac{V + 30}{10}}} tau_f_2 = alpha_f_2 + beta_f_2 + gamma_f_2 \frac{d}{d t} (d) = \frac{d_ss - d}{tau_d} \frac{d}{d t} (f) = \frac{f_ss - f}{tau_f} \frac{d}{d t} (f_2) = \frac{f_2_ss - f_2}{tau_f_2} \frac{d}{d t} (f_Ca_B_j) = 1.7 Ca_j (1 - f_Ca_B_j) - (11.9 e -3 f_Ca_B_j) \frac{d}{d t} (f_Ca_B_sl) = 1.7 Ca_sl (1 - f_Ca_B_sl) - (11.9 e -3 f_Ca_B_sl) Ibar_Ca_j = \frac{p_Ca V Frdy FoRT (Ca_j ⅇ^{2 V FoRT} - Ca_o)}{ⅇ^{2 V FoRT} - 1} Ibar_Ca_sl = \frac{p_Ca V Frdy FoRT (Ca_sl ⅇ^{2 V FoRT} - Ca_o)}{ⅇ^{2 V FoRT} - 1} Ibar_K = \frac{p_K V Frdy FoRT (K_i ⅇ^{V FoRT} - K_o)}{ⅇ^{V FoRT} - 1} Ibar_Na_j = \frac{p_Na V Frdy FoRT (Na_j ⅇ^{V FoRT} - Na_o)}{ⅇ^{V FoRT} - 1} Ibar_Na_sl = \frac{p_Na V Frdy FoRT (Na_sl ⅇ^{V FoRT} - Na_o)}{ⅇ^{V FoRT} - 1} I_Ca_junc = F_junc_Ca_L Ibar_Ca_j d f f_2 (1 - f_Ca_B_j) I_Ca_sl = F_sl_Ca_L Ibar_Ca_sl d f f_2 (1 - f_Ca_B_sl) I_Ca = I_Ca_junc + I_Ca_sl I_Ca_K = Ibar_K d f f_2 (F_junc_Ca_L (1 - f_Ca_B_j) + F_sl_Ca_L (1 - f_Ca_B_sl)) I_Ca_Na_junc = F_junc_Ca_L Ibar_Na_j d f f_2 (1 - f_Ca_B_j) I_Ca_Na_sl = F_sl_Ca_L Ibar_Na_sl d f f_2 (1 - f_Ca_B_sl) I_Ca_Na = I_Ca_Na_junc + I_Ca_Na_sl I_Ca_L = I_Ca + I_Ca_K + I_Ca_Na

Component: Na_Ca_Exchanger_Current

Ka_junc = \frac{1}{1 + {(\frac{Kd_act}{Ca_j})}^{2}} Ka_sl = \frac{1}{1 + {(\frac{Kd_act}{Ca_sl})}^{2}} s1_junc = ⅇ^{nu V FoRT} {Na_j}^{3} Ca_o s1_sl = ⅇ^{nu V FoRT} {Na_sl}^{3} Ca_o s2_junc = ⅇ^{(nu - 1) V FoRT} {Na_o}^{3} Ca_j s2_sl = ⅇ^{(nu - 1) V FoRT} {Na_o}^{3} Ca_sl s3_junc = Km_Ca_i {Na_o}^{3} (1 + {(\frac{Na_j}{Km_Na_i})}^{3}) + {Km_Na_o}^{3} Ca_j (1 + \frac{Ca_j}{Km_Ca_i}) + Km_Ca_o {Na_j}^{3} + {Na_j}^{3} Ca_o + {Na_o}^{3} Ca_j s3_sl = Km_Ca_i {Na_o}^{3} (1 + {(\frac{Na_sl}{Km_Na_i})}^{3}) + {Km_Na_o}^{3} Ca_sl (1 + \frac{Ca_sl}{Km_Ca_i}) + Km_Ca_o {Na_sl}^{3} + {Na_sl}^{3} Ca_o + {Na_o}^{3} Ca_sl I_ncx_junc = \frac{F_junc Ibar_NCX Ka_junc (s1_junc - s2_junc)}{s3_junc (1 + k_sat ⅇ^{(nu - 1) V FoRT})} I_ncx_sl = \frac{F_sl Ibar_NCX Ka_sl (s1_sl - s2_sl)}{s3_sl (1 + k_sat ⅇ^{(nu - 1) V FoRT})} I_ncx = I_ncx_junc + I_ncx_sl

Component: Sarcolemmal_Ca_Pump_Current

I_pCa_junc = \frac{F_junc Ibar_PMCA {Ca_j}^{1.6}}{{Km_P_Ca}^{1.6} + {Ca_j}^{1.6}} I_pCa_sl = \frac{F_sl Ibar_PMCA {Ca_sl}^{1.6}}{{Km_P_Ca}^{1.6} + {Ca_sl}^{1.6}} I_pCa = I_pCa_junc + I_pCa_sl

Component: Background_Ca_Current

I_Ca_Bk_junc = F_junc G_Ca_B (V - E_Ca_junc) I_Ca_Bk_sl = F_sl G_Ca_B (V - E_Ca_sl) I_Ca_Bk = I_Ca_Bk_junc + I_Ca_Bk_sl

Component: SR_Fluxes

k_Ca_SR = Max_SR - (\frac{Max_SR - Min_SR}{1 + {(\frac{ec50_SR}{Ca_SR})}^{2.5}}) ko_SR_Ca = \frac{ko_Ca}{k_Ca_SR} ki_SR_Ca = ki_Ca k_Ca_SR RI = 1 - Ry_Rr - Ry_Ro - Ry_Ri \frac{d}{d t} (Ry_Rr) = ki_m RI - (ki_SR_Ca Ca_j Ry_Rr) - (ko_SR_Ca {Ca_j}^{2} Ry_Rr - (ko_m Ry_Ro)) \frac{d}{d t} (Ry_Ro) = ko_SR_Ca {Ca_j}^{2} Ry_Rr - (ko_m Ry_Ro) - (ki_SR_Ca Ca_j Ry_Ro - (ki_m Ry_Ri)) \frac{d}{d t} (Ry_Ri) = ki_SR_Ca Ca_j Ry_Ro - (ki_m Ry_Ri) - (ko_m Ry_Ri - (ko_SR_Ca {Ca_j}^{2} RI)) J_SR_Ca_rel = ks Ry_Ro (Ca_SR - Ca_j) J_ser_Ca = \frac{V_max_SR_CaP ({(\frac{Ca_i}{Km_f})}^{hill_SR_CaP} - ({(\frac{Ca_SR}{Km_r})}^{hill_SR_CaP}))}{1 + {(\frac{Ca_i}{Km_f})}^{hill_SR_CaP} + {(\frac{Ca_SR}{Km_r})}^{hill_SR_CaP}} J_SR_leak = 5.348 e -6 (Ca_SR - Ca_j)

Component: Cytosolic_Ca_Buffers

\frac{d}{d t} (TnC_l) = k_on_TnC_l Ca_i (B_max_TnC_low - TnC_l) - (k_off_TnC_l TnC_l) \frac{d}{d t} (TnC_h_c) = k_on_TnC_h_Ca Ca_i (B_max_TnC_high - TnC_h_c - TnC_h_m) - (k_off_TnC_h_Ca TnC_h_c) \frac{d}{d t} (TnC_h_m) = k_on_TnC_h_Mg Mg_i (B_max_TnC_high - TnC_h_c - TnC_h_m) - (k_off_TnC_h_Mg TnC_h_m) \frac{d}{d t} (CaM) = k_on_CaM Ca_i (B_max_CaM - CaM) - (k_off_CaM CaM) \frac{d}{d t} (Myo_c) = k_on_myo_Ca Ca_i (B_max_myosin - Myo_c - Myo_m) - (k_off_myo_Ca Myo_c) \frac{d}{d t} (Myo_m) = k_on_myo_Mg Mg_i (B_max_myosin - Myo_c - Myo_m) - (k_off_myo_Mg Myo_m) \frac{d}{d t} (SRB) = k_on_SR Ca_i (B_max_SR - SRB) - (k_off_SR SRB) J_Ca_B_cytosol = \frac{d}{d t} (TnC_l) + \frac{d}{d t} (TnC_h_c) + \frac{d}{d t} (TnC_h_m) + \frac{d}{d t} (CaM) + \frac{d}{d t} (Myo_c) + \frac{d}{d t} (Myo_m) + \frac{d}{d t} (SRB)

Component: Junctional_and_SL_Ca_Buffers

B_max_SL_low_sl = \frac{37.4 e -3 V_myo}{V_sl} B_max_SL_low_j = \frac{4.6 e -4 V_myo}{V_junc} B_max_SL_high_sl = \frac{13.4 e -3 V_myo}{V_sl} B_max_SL_high_j = \frac{1.65 e -4 V_myo}{V_junc} \frac{d}{d t} (SLL_j) = k_on_sl_l Ca_j (B_max_SL_low_j - SLL_j) - (k_off_sl_l SLL_j) \frac{d}{d t} (SLL_sl) = k_on_sl_l Ca_sl (B_max_SL_low_sl - SLL_sl) - (k_off_sl_l SLL_sl) \frac{d}{d t} (SLH_j) = k_on_sl_h Ca_j (B_max_SL_high_j - SLH_j) - (k_off_sl_h SLH_j) \frac{d}{d t} (SLH_sl) = k_on_sl_h Ca_sl (B_max_SL_high_sl - SLH_sl) - (k_off_sl_h SLH_sl) J_Ca_B_junction = \frac{d}{d t} (SLL_j) + \frac{d}{d t} (SLH_j) J_Ca_B_sl = \frac{d}{d t} (SLL_sl) + \frac{d}{d t} (SLH_sl)

Component: Sodium_Concentrations

I_Na_tot_junc = I_Na_junc + I_Na_Bk_junc + 3 I_ncx_junc + 3 I_NaK_junc + I_Ca_Na_junc I_Na_tot_sl = I_Na_sl + I_Na_Bk_sl + 3 I_ncx_sl + 3 I_NaK_sl + I_Ca_Na_sl \frac{d}{d t} (Na_j) = \frac{(- I_Na_tot_junc) C_mem}{V_junc Frdy} + \frac{J_Na_juncsl}{V_junc} (Na_sl - Na_j) - (\frac{d}{d t} (Na_B_j)) \frac{d}{d t} (Na_sl) = \frac{(- I_Na_tot_sl) C_mem}{V_sl Frdy} + \frac{J_Na_juncsl}{V_sl} (Na_j - Na_sl) + \frac{J_Na_slmyo}{V_sl} (Na_i - Na_sl) - (\frac{d}{d t} (Na_B_sl)) \frac{d}{d t} (Na_i) = \frac{J_Na_slmyo}{V_myo} (Na_sl - Na_i) \frac{d}{d t} (Na_B_j) = k_on_Na Na_j (B_max_Na_j - Na_B_j) - (k_off_Na Na_B_j) \frac{d}{d t} (Na_B_sl) = k_on_Na Na_sl (B_max_Na_sl - Na_B_sl) - (k_off_Na Na_B_sl)

Component: Potassium_Concentrations

I_K_tot = (I_to + I_Kr + I_Ks + I_K1 - (2 I_NaK)) + I_Ca_K + I_Kp

Component: Calcium_Concentrations

I_Ca_tot_junc = I_Ca_junc + I_Ca_Bk_junc + I_pCa_junc - (2 I_ncx_junc) I_Ca_tot_sl = I_Ca_sl + I_Ca_Bk_sl + I_pCa_sl - (2 I_ncx_sl) B_max_csqn = \frac{140 e -3 V_myo}{V_sr} \frac{d}{d t} (Csqn_b) = k_on_csqn Ca_SR (B_max_csqn - Csqn_b) - (k_off_csqn Csqn_b) \frac{d}{d t} (Ca_j) = (\frac{(- I_Ca_tot_junc) C_mem}{V_junc 2 Frdy} + \frac{J_Ca_juncsl}{V_junc} (Ca_sl - Ca_j) - J_Ca_B_junction) + \frac{J_SR_Ca_rel V_sr}{V_junc} + \frac{J_SR_leak V_myo}{V_junc} \frac{d}{d t} (Ca_sl) = \frac{(- I_Ca_tot_sl) C_mem}{V_sl 2 Frdy} + \frac{J_Ca_juncsl}{V_sl} (Ca_j - Ca_sl) + \frac{J_Ca_slmyo}{V_sl} (Ca_i - Ca_sl) - J_Ca_B_sl \frac{d}{d t} (Ca_i) = (\frac{(- J_ser_Ca) V_sr}{V_myo} - J_Ca_B_cytosol) + \frac{J_Ca_slmyo}{V_myo} (Ca_sl - Ca_i) \frac{d}{d t} (Ca_SR) = J_ser_Ca - (\frac{J_SR_leak V_myo}{V_sr} + J_SR_Ca_rel) - (\frac{d}{d t} (Csqn_b))

Component: Magnesium_Concentrations

Component: Chlorine_Concentrations

Component: Nerst_Potentials

E_Na_junc = \frac{1}{FoRT} \ln (\frac{Na_o}{Na_j}) E_Na_sl = \frac{1}{FoRT} \ln (\frac{Na_o}{Na_sl}) E_Ks = \frac{1}{FoRT} \ln (\frac{K_o + p_Na_K Na_o}{K_i + p_Na_K Na_i}) E_K = \frac{1}{FoRT} \ln (\frac{K_o}{K_i}) E_Ca_junc = \frac{1}{2 FoRT} \ln (\frac{Ca_o}{Ca_j}) E_Ca_sl = \frac{1}{2 FoRT} \ln (\frac{Ca_o}{Ca_sl}) E_Cl = \frac{1}{FoRT} \ln (\frac{Cl_i}{Cl_o})