Model Mathematics

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Component: environment

Component: model_parameters

Vc = \frac{Smtot}{7850} Vmyo = Vc 0.585 VSRup = Vc 0.0315 VSRrel = Vc 0.0035 Vd = Vc 0.000075 Sms = Smtot - (0.44e4 Vc) Smt = Smtot - Sms pt = ptcm Sms Lt = \frac{Smt}{2 3.14159 radiust pt} Vt = 3.14159 {radiust}^{2} Lt pt Cmt = Smt 1 Cms = Sms 1 Rst = \frac{Rot \frac{Lt}{2}}{3.14159 {radiust}^{2} pt}

Component: common_current

i_circ = 1000 \frac{Vm_s - Vm_t}{R_st}

Component: SR_Ca_release_channel

\frac{d}{d time} (F1) = (- kap) {Ca_ss}^{4} F1 + kam F2 \frac{d}{d time} (F2) = (kap {Ca_ss}^{4} F1 - (kam F2 + kbp {Ca_ss}^{3} F2 + kcp F2)) + kbm F3 + kcm F4 \frac{d}{d time} (F3) = kbp {Ca_ss}^{3} F2 - (kbm F3) \frac{d}{d time} (F4) = kcp F2 - (kcm F4) J_CaSRrel = 1800 (F2 + F3) (CaSRrel - Ca_ss)

Component: SERCA2a_pump

fb = {(\frac{Ca_i}{K_fb})}^{N_fb} rb = {(\frac{CaSRup}{K_rb})}^{N_rb} J_CaSRup = \frac{K_SR (Vmaxf fb - (Vmaxr rb))}{1 + fb + rb}

Component: intracellular_and_SR_Ca_fluxes

J_tr = \frac{CaSRup - CaSRrel}{tau_tr} J_xfer = \frac{Ca_ss - Ca_i}{tau_xfer} J_BTRH = k_htrpn_plus Ca_i (1 - BTRH) - (k_htrpn_minus BTRH) \frac{d}{d time} (BTRH) = J_BTRH J_BTRL = k_ltrpn_plus Ca_i (1 - BTRL) - (k_ltrpn_minus BTRL) \frac{d}{d time} (BTRL) = J_BTRL J_trpn = J_BTRH + J_BTRL

Component: ion_concentrations

\frac{d}{d time} (Na_i) = \frac{- (i_Na_s + i_Na_t + i_BNa_s + i_BNa_t + i_NaCa_s 3 + i_NaCa_t 3 + i_NaK_s 3 + i_NaK_t 3)}{Vmyo F} \frac{d}{d time} (Na_t) = \frac{\frac{i_Na_t + i_BNa_t + i_NaCa_t 3 + i_NaK_t 3}{F - Jte_Na}}{Vt} i_Kext = \{\begin{matrix} 52 (Sms + Smt) if STIMULATION = 1 \land time - time0 < 0.001 \\ 0 otherwise \end{matrix} \frac{d}{d time} (K_i) = \frac{- ((- i_Kext) + i_fK_s + i_fK_t + i_Kss_s + i_Kss_t + i_K1_s + i_K1_t + i_BK_s + i_BK_t + i_NaK_s (- 2) + i_NaK_t (- 2) + i_Kto_s + i_Kto_t)}{Vmyo F} \frac{d}{d time} (K_t) = \frac{\frac{i_fK_t + i_Kss_t + i_K1_t + i_BK_t + i_NaK_t (- 2) + i_Kto_t}{F - Jte_K}}{Vt} \frac{d}{d time} (Ca_i) = \frac{1}{1 + CMDN_tot \frac{K_mCMDN}{{(K_mCMDN + Ca_i)}^{2}}} (\frac{(-2) (i_NaCa_s + i_NaCa_t) + i_BCa_s + i_BCa_t + i_CaP_s + i_CaP_t}{(-2) Vmyo F} + \frac{\frac{(Ca_ss - Ca_i) Vmyo}{tau_d} + J_CaSRrel - J_CaSRup}{Vmyo} - (J_BTRH BTRHmax + J_BTRL BTRLmax)) \frac{d}{d time} (Ca_t) = \frac{\frac{i_Ca_t + i_CaP_t + i_BCa_t + i_NaCa_t (- 2)}{2 F - Jte_Ca}}{Vt} \frac{d}{d time} (Ca_ss) = \frac{1}{1 + CMDN_tot \frac{K_mCMDN}{{(K_mCMDN + Ca_ss)}^{2}}} \frac{- (i_Ca_s + i_Ca_t)}{2 Vd F + \frac{J_CaSRrel}{Vd} - (\frac{\frac{(Ca_ss - Ca_i) Vmyo}{tau_d}}{Vd})} \frac{d}{d time} (CaSRrel) = \frac{1}{1 + \frac{CSQN_tot K_mCSQN}{{(K_mCSQN + CaSRrel)}^{2}}} (J_tr - J_CaSRrel) \frac{d}{d time} (CaSRup) = J_CaSRup - J_tr

Component: t_tubular_ion_fluxes

Jte_Na = \frac{1 Vt}{tau_Na (Na_t - Na_e)} Jte_Ca = \frac{1 Vt}{tau_Ca (Ca_t - Ca_e)} Jte_K = \frac{1 Vt}{tau_K (K_t - K_e)}

Component: surface_membrane

Component: membrane

\frac{d}{d time} (Vm_s) = 1000 \frac{i_Kext + i_circ - (i_Na_s + i_Ca_s + i_Kto_s + i_Kss_s + i_f_s + i_K1_s + i_B_s + i_NaK_s + i_NaCa_s + i_CaP_s)}{Cms}

Component: sodium_current

E_Na = \frac{R T}{F} \ln (\frac{Na_e}{Na_i}) i_Na_s = g_Nas m^{3} h (Vm_s - (\log_{10} (\frac{Na_e + 0.12 K_e}{Na_i + 0.12 K_i}) \frac{R T}{F})) g_Nas = g_Na (Sms + Smt) (1 - fNat)

Component: sodium_current_m_gate

alpha_m = \frac{117.26 (Vm_s + 59.3)}{1 - (ⅇ^{(- 0.55) (Vm_s + 59.3)})} beta_m = 3800 ⅇ^{(- 0.072) (Vm_s + 61)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: sodium_current_h_gate

alpha_h = \frac{284.4}{1 + ⅇ^{0.0812 (Vm_s + 115.9)}} beta_h = \frac{18.77 (Vm_s + 64.4)}{1 - (ⅇ^{(- 0.22) (Vm_s + 64.4)})} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: calcium_current

Co_inf = \frac{1}{1 + ⅇ^{\frac{Vm_s + 15.3}{- 5}}} tauCo = 0.00305 ⅇ^{(- 0.0045) {(Vm_s + 7)}^{2}} + 0.00105 ⅇ^{(- 0.002) {(Vm_s - 18)}^{2}} + 0.00025 \frac{d}{d time} (Co) = \frac{Co_inf - Co}{tauCo} C1_inf = \frac{1}{1 + ⅇ^{\frac{Vm_s + 26.7}{5.4}}} tauC1 = 0.105 ⅇ^{- ({(\frac{Vm_s + 45}{12})}^{2})} + \frac{0.04}{1 + ⅇ^{\frac{(- Vm_s) + 25}{25}}} + \frac{0.015}{1 + ⅇ^{\frac{Vm_s + 75}{25}}} + 0.0017 \frac{d}{d time} (C1) = \frac{C1_inf - C1}{tauC1} C2_inf = \frac{1}{1 + ⅇ^{\frac{Vm_s + 26.7}{5.4}}} tauC2 = 0.041 ⅇ^{- ({(\frac{Vm_s + 47}{12})}^{2})} + \frac{0.08}{1 + ⅇ^{\frac{Vm_s + 55}{- 5}}} + \frac{0.015}{1 + ⅇ^{\frac{Vm_s + 75}{25}}} + 0.0017 \frac{d}{d time} (C2) = \frac{C2_inf - C2}{tauC2} C_cast_inf = \frac{1}{1 + \frac{Ca_ss}{0.01}} \frac{d}{d time} (C_cast) = \frac{C_cast_inf - C_cast}{tauC_cast} g_Cas = g_Ca (Sms + Smt) (1 - fCat) i_Ca_s = g_Cas 4 \frac{Vm_s F^{2}}{R T} \frac{Ca_ss ⅇ^{\frac{2 Vm_s F^{2}}{R T}} - (0.341 Ca_e)}{ⅇ^{\frac{2 Vm_s F^{2}}{R T}} - 1} Co ((0.9 + \frac{C_cast}{10.0}) C1 + (0.1 + \frac{C_cast}{10.0}) C2)

Component: Ca_independent_transient_outward_K_current

E_K = \frac{R T}{F} \ln (\frac{K_e}{K_i}) i_Kto_s = g_Ktos r (a s + b s_slow) (Vm_s - E_K) g_Ktos = g_Kto (Sms + Smt) (1 - fKtot)

Component: Ca_independent_transient_outward_K_current_r_gate

r_max = \frac{1}{1 + ⅇ^{\frac{Vm_s + 10.6}{- 11.42}}} tau_r = \frac{1}{45.16 ⅇ^{0.03577 (Vm_s + 50)} + 98.9 ⅇ^{(- 0.1) (Vm_s + 38)}} \frac{d}{d time} (r) = \frac{r_max - r}{tau_r}

Component: Ca_independent_transient_outward_K_current_s_gate

s_max = \frac{1}{1 + ⅇ^{\frac{Vm_s + 45.3}{6.8841}}} tau_s = 0.35 ⅇ^{- ({(0.0666 (Vm_s + 70))}^{2})} + 0.035 \frac{d}{d time} (s) = \frac{s_max - s}{tau_s}

Component: Ca_independent_transient_outward_K_current_s_slow_gate

s_slow_max = \frac{1}{1 + ⅇ^{\frac{Vm_s + 45.3}{6.8841}}} tau_s_slow = 3.7 ⅇ^{- ({(0.0333 (Vm_s + 70))}^{2})} + 0.035 \frac{d}{d time} (s_slow) = \frac{s_slow_max - s_slow}{tau_s_slow}

Component: steady_state_outward_K_current

i_Kss_s = g_Ksss r_ss s_ss (Vm_s - E_K) g_Ksss = g_Kss (Sms + Smt) (1 - fKsst)

Component: steady_state_outward_K_current_r_ss_gate

r_ss_max = \frac{1}{1 + ⅇ^{\frac{- (Vm_s + 11.5)}{11.82}}} tau_r_ss = \frac{10}{45.16 ⅇ^{0.03577 (Vm_s + 50)} + 98.9 ⅇ^{(- 0.1) (Vm_s + 38)}} \frac{d}{d time} (r_ss) = \frac{r_ss_max - r_ss}{tau_r_ss}

Component: steady_state_outward_K_current_s_ss_gate

s_ss_max = \frac{1}{1 + ⅇ^{\frac{Vm_s + 87.5}{10.3}}} tau_s_ss = 2.1 \frac{d}{d time} (s_ss) = \frac{s_ss_max - s_ss}{tau_s_ss}

Component: inward_rectifier

i_K1_s = \frac{g_K1s (Vm_s - (E_K + 1.73))}{1 + ⅇ^{\frac{1.613 F}{R T} (Vm_s - (E_K + 1.73))}} (1 + ⅇ^{\frac{- (K_e - 0.9988)}{0.124}}) g_K1s = g_K1 (Sms + Smt) (1 - fK1t)

Component: hyperpolarisation_activated_current

f_K = 1 - f_Na i_fNa_s = g_fs y f_Na (Vm_s - E_Na) i_fK_s = g_fs y f_K (Vm_s - E_K) i_f_s = g_fs y (0.2 (Vm_s - E_Na) + 0.8 (Vm_s - E_K)) g_fs = g_f (Sms + Smt) (1 - fft)

Component: hyperpolarisation_activated_current_y_gate

y_infinity = \frac{1}{1 + ⅇ^{\frac{Vm_s + 138.6}{10.48}}} tau_y = \frac{1}{0.11885 ⅇ^{\frac{Vm_s + 80}{28.37}} + 0.5623 ⅇ^{\frac{Vm_s + 80}{- 14.19}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

Component: background_currents

i_BNa_s = g_B_Nas (Vm_s - E_Na) i_BCa_s = g_B_Cas (Vm_s - E_Ca) i_BK_s = g_B_Ks (Vm_s - E_K) i_B_s = i_BNa_s + i_BCa_s + i_BK_s g_B_Cas = g_B_Ca (Sms + Smt) (1 - fCabt) g_B_Nas = g_B_Na (Sms + Smt) (1 - fNabt) g_B_Ks = g_B_K (Sms + Smt) (1 - fKbt)

Component: sodium_potassium_pump

sigma = \frac{ⅇ^{\frac{Na_e}{67.3}} - 1}{7} i_NaK_s = \frac{\frac{\frac{i_NaK_max_s}{1 + 0.1245 ⅇ^{\frac{(- 0.1) Vm_s F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- Vm_s) F}{R T}}} K_e}{K_e + K_m_K}}{1 + {(\frac{K_m_Na}{Na_i})}^{1.5}} i_NaK_max_s = i_NaK_max (Sms + Smt) (1 - fNaKt)

Component: sarcolemmal_calcium_pump_current

i_CaP_s = \frac{i_CaP_max_s Ca_i}{Ca_i + 0.0004} i_CaP_max_s = i_CaP_max (Sms + Smt) (1 - fCaPt)

Component: Na_Ca_ion_exchanger_current

i_NaCa_s = \frac{i_NaCa_max_s (ⅇ^{0.0187 Vm_s} {Na_i}^{3} Ca_e - (ⅇ^{(-0.0187) Vm_s} {Na_e}^{3} Ca_i))}{1 + 0.0001 (Ca_i {Na_e}^{3} + Ca_e {Na_i}^{3})} i_NaCa_max_s = i_NaCa_max (Sms + Smt) (1 - fNaCat)

Component: tubular_membrane

Component: membrane

\frac{d}{d time} (Vm_t) = 1000 \frac{- (i_circ + i_Na_t + i_Ca_t + i_Kto_t + i_Kss_t + i_f_t + i_K1_t + i_B_t + i_NaK_t + i_NaCa_t + i_CaP_t)}{Cmt}

Component: sodium_current

E_Na = \frac{R T}{F} \ln (\frac{Na_t}{Na_i}) i_Na_t = g_Nat m^{3} h (Vm_t - (\log_{10} (\frac{Na_t + 0.12 K_t}{Na_i + 0.12 K_i}) \frac{R T}{F})) g_Nat = g_Na (Sms + Smt) fNat

Component: sodium_current_m_gate

alpha_m = \frac{117.26 (Vm_t + 59.3)}{1 - (ⅇ^{(- 0.55) (Vm_t + 59.3)})} beta_m = 3800 ⅇ^{(- 0.072) (Vm_t + 61)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: sodium_current_h_gate

alpha_h = \frac{284.4}{1 + ⅇ^{0.0812 (Vm_t + 115.9)}} beta_h = \frac{18.77 (Vm_t + 64.4)}{1 - (ⅇ^{(- 0.22) (Vm_t + 64.4)})} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

Component: calcium_current

Co_inf = \frac{1}{1 + ⅇ^{\frac{Vm_t + 15.3}{- 5}}} tauCo = 0.00305 ⅇ^{(- 0.0045) {(Vm_t + 7)}^{2}} + 0.00105 ⅇ^{(- 0.002) {(Vm_t - 18)}^{2}} + 0.00025 \frac{d}{d time} (Co) = \frac{Co_inf - Co}{tauCo} C1_inf = \frac{1}{1 + ⅇ^{\frac{Vm_t + 26.7}{5.4}}} tauC1 = 0.105 ⅇ^{- ({(\frac{Vm_t + 45}{12})}^{2})} + \frac{0.04}{1 + ⅇ^{\frac{(- Vm_t) + 25}{25}}} + \frac{0.015}{1 + ⅇ^{\frac{Vm_t + 75}{25}}} + 0.0017 \frac{d}{d time} (C1) = \frac{C1_inf - C1}{tauC1} C2_inf = \frac{1}{1 + ⅇ^{\frac{Vm_t + 26.7}{5.4}}} tauC2 = 0.041 ⅇ^{- ({(\frac{Vm_t + 47}{12})}^{2})} + \frac{0.08}{1 + ⅇ^{\frac{Vm_t + 55}{- 5}}} + \frac{0.015}{1 + ⅇ^{\frac{Vm_t + 75}{25}}} + 0.0017 \frac{d}{d time} (C2) = \frac{C2_inf - C2}{tauC2} C_cast_inf = \frac{1}{1 + \frac{Ca_ss}{0.01}} \frac{d}{d time} (C_cast) = \frac{C_cast_inf - C_cast}{tauC_cast} g_Cat = g_Ca (Sms + Smt) fCat i_Ca_t = g_Cat 4 \frac{Vm_t F^{2}}{R T} \frac{Ca_ss ⅇ^{\frac{2 Vm_t F^{2}}{R T}} - (0.341 Ca_t)}{ⅇ^{\frac{2 Vm_t F^{2}}{R T}} - 1} Co ((0.9 + \frac{C_cast}{10.0}) C1 + (0.1 + \frac{C_cast}{10.0}) C2)

Component: Ca_independent_transient_outward_K_current

E_K = \frac{R T}{F} \ln (\frac{K_t}{K_i}) i_Kto_t = g_Ktot r (a s + b s_slow) (Vm_t - E_K) g_Ktot = g_Kto (Sms + Smt) fKtot

Component: Ca_independent_transient_outward_K_current_r_gate

r_max = \frac{1}{1 + ⅇ^{\frac{Vm_t + 10.6}{- 11.42}}} tau_r = \frac{1}{45.16 ⅇ^{0.03577 (Vm_t + 50)} + 98.9 ⅇ^{(- 0.1) (Vm_t + 38)}} \frac{d}{d time} (r) = \frac{r_max - r}{tau_r}

Component: Ca_independent_transient_outward_K_current_s_gate

s_max = \frac{1}{1 + ⅇ^{\frac{Vm_t + 45.3}{6.8841}}} tau_t = 0.35 ⅇ^{- ({(0.0666 (Vm_t + 70))}^{2})} + 0.035 \frac{d}{d time} (s) = \frac{s_max - s}{tau_t}

Component: Ca_independent_transient_outward_K_current_s_slow_gate

s_slow_max = \frac{1}{1 + ⅇ^{\frac{Vm_t + 45.3}{6.8841}}} tau_s_slow = 3.7 ⅇ^{- ({(0.0333 (Vm_t + 70))}^{2})} + 0.035 \frac{d}{d time} (s_slow) = \frac{s_slow_max - s_slow}{tau_s_slow}

Component: steady_state_outward_K_current

i_Kss_t = g_Ksst r_ss s_ss (Vm_t - E_K) g_Ksst = g_Kss (Sms + Smt) fKsst

Component: steady_state_outward_K_current_r_ss_gate

r_ss_max = \frac{1}{1 + ⅇ^{\frac{- (Vm_t + 11.5)}{11.82}}} tau_r_ss = \frac{10}{45.16 ⅇ^{0.03577 (Vm_t + 50)} + 98.9 ⅇ^{(- 0.1) (Vm_t + 38)}} \frac{d}{d time} (r_ss) = \frac{r_ss_max - r_ss}{tau_r_ss}

Component: steady_state_outward_K_current_s_ss_gate

s_ss_max = \frac{1}{1 + ⅇ^{\frac{Vm_t + 87.5}{10.3}}} tau_s_ss = 2.1 \frac{d}{d time} (s_ss) = \frac{s_ss_max - s_ss}{tau_s_ss}

Component: inward_rectifier

i_K1_t = \frac{g_K1t (Vm_t - (E_K + 1.73))}{1 + ⅇ^{\frac{1.613 F}{R T} (Vm_t - (E_K + 1.73))}} (1 + ⅇ^{\frac{- (K_t - 0.9988)}{0.124}}) g_K1t = g_K1 (Sms + Smt) fK1t

Component: hyperpolarisation_activated_current

f_K = 1 - f_Na i_fNa_t = g_ft y f_Na (Vm_t - E_Na) i_fK_t = g_ft y f_K (Vm_t - E_K) i_f_t = g_ft y (0.2 (Vm_t - E_Na) + 0.8 (Vm_t - E_K)) g_ft = g_f (Sms + Smt) fft

Component: hyperpolarisation_activated_current_y_gate

y_infinity = \frac{1}{1 + ⅇ^{\frac{Vm_t + 138.6}{10.48}}} tau_y = \frac{1}{0.11885 ⅇ^{\frac{Vm_t + 80}{28.37}} + 0.5623 ⅇ^{\frac{Vm_t + 80}{- 14.19}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

Component: background_currents

i_BNa_t = g_B_Nat (Vm_t - E_Na) i_BCa_t = g_B_Cat (Vm_t - E_Ca) i_BK_t = g_B_Kt (Vm_t - E_K) i_B_t = i_BNa_t + i_BCa_t + i_BK_t g_B_Cat = g_B_Ca (Sms + Smt) fCabt g_B_Nat = g_B_Na (Sms + Smt) fNabt g_B_Kt = g_B_K (Sms + Smt) fKbt

Component: sodium_potassium_pump

sigma = \frac{ⅇ^{\frac{Na_t}{67.3}} - 1}{7} i_NaK_t = \frac{\frac{\frac{i_NaK_max_t}{1 + 0.1245 ⅇ^{\frac{(- 0.1) Vm_t F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- Vm_t) F}{R T}}} K_t}{K_t + K_m_K}}{1 + {(\frac{K_m_Na}{Na_i})}^{1.5}} i_NaK_max_t = i_NaK_max (Sms + Smt) fNaKt

Component: sarcolemmal_calcium_pump_current

i_CaP_t = \frac{i_CaP_max_t Ca_i}{Ca_i + 0.0004} i_CaP_max_t = i_CaP_max (Sms + Smt) fCaPt

Component: Na_Ca_ion_exchanger_current

i_NaCa_t = \frac{i_NaCa_max_t (ⅇ^{0.0187 Vm_t} {Na_i}^{3} Ca_t - (ⅇ^{(-0.0187) Vm_t} {Na_t}^{3} Ca_i))}{1 + 0.0001 (Ca_i {Na_t}^{3} + Ca_t {Na_i}^{3})} i_NaCa_max_t = i_NaCa_max (Sms + Smt) fNaCat