Model Mathematics

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Component: environment

Component: b1_AR_Gs_parameters

Component: cAMP_parameters

Component: PKA_parameters

Component: PLB_parameters

Component: LCC_parameters

Component: EC_Coupling_Parameters

Component: b1_AR_module

LR = \frac{L R}{Kl} LRG = \frac{LR Gs}{Kr} RG = \frac{R Gs}{Kc} BARK_DESENS = k_bar_kp (LR + LRG) BARK_RESENS = k_bar_km b1_AR_d PKA_DESENS = k_p_kap PKAC_I b1_AR_tot PKA_RESENS = k_p_kam b1_AR_p G_ACT = k_g_act (RG + LRG) HYD = k_hyd Gs_agtp_tot REASSOC = k_reassoc Gs_agdp Gs_bg L = L_totmax - LR - LRG R = b1_AR_tot - LR - LRG - RG Gs = Gs_tot - LRG - RG \frac{d}{d time} (b1_AR_tot) = (BARK_RESENS - BARK_DESENS) + PKA_RESENS - PKA_DESENS \frac{d}{d time} (b1_AR_d) = BARK_DESENS - BARK_RESENS \frac{d}{d time} (b1_AR_p) = PKA_DESENS - PKA_RESENS \frac{d}{d time} (Gs_agtp_tot) = G_ACT - HYD \frac{d}{d time} (Gs_agdp) = HYD - REASSOC \frac{d}{d time} (Gs_bg) = G_ACT - REASSOC

Component: cAMP_module

Gsa_GTP_AC = \frac{Gsa_GTP AC}{K_gsa} Fsk_AC = \frac{Fsk AC}{K_fsk} AC_ACT_BASAL = \frac{k_ac_basal AC ATP}{Km_basal + ATP} AC_ACT_GSA = \frac{k_ac_gsa Gsa_GTP_AC ATP}{Km_gsa + ATP} AC_ACT_FSK = \frac{k_ac_fsk Fsk_AC ATP}{Km_fsk + ATP} PDE_ACT = \frac{k_pde PDE cAMP}{Km_pde + cAMP} PDE_IBMX = \frac{PDE IBMX}{Ki_ibmx} Gsa_GTP = Gs_agtp_tot - Gsa_GTP_AC Fsk = Fsk_tot - Fsk_AC AC = AC_tot - Gsa_GTP_AC PDE = PDE_tot - PDE_IBMX IBMX = IBMX_tot - PDE_IBMX \frac{d}{d time} (cAMP_tot) = AC_ACT_BASAL + AC_ACT_GSA + AC_ACT_FSK - PDE_ACT

Component: PKA_module

PKI = \frac{PKI_tot Ki_pki}{Ki_pki + PKAC_I + PKAC_II} A2RC_I = \frac{PKAC_I}{K_d} PKAC_I (1 + \frac{PKI}{Ki_pki}) A2R_I = PKAC_I (1 + \frac{PKI}{Ki_pki}) A2RC_II = \frac{PKAC_II}{K_d} PKAC_II (1 + \frac{PKI}{Ki_pki}) A2R_II = PKAC_II (1 + \frac{PKI}{Ki_pki}) ARC_I = \frac{K_a}{cAMP} A2RC_I ARC_II = \frac{K_a}{cAMP} A2RC_II PKA_temp = \frac{K_a K_b}{K_d} + \frac{K_a cAMP}{K_d} + \frac{{cAMP}^{2}}{K_d} cAMP = cAMP_tot - (ARC_I + 2 A2RC_I + 2 A2R_I) - (ARC_II + 2 A2RC_II + 2 A2R_II) zed = 2 PKAI_tot {cAMP}^{2} - (PKAC_I (1 + \frac{PKI}{Ki_pki}) (PKA_temp PKAC_I + {cAMP}^{2})) zed1 = 2 PKAII_tot {cAMP}^{2} - (PKAC_II (1 + \frac{PKI}{Ki_pki}) (PKA_temp PKAC_II + {cAMP}^{2}))

Component: PLB_module

PLB = PLB_tot - PLB_p PLB_PHOSPH = \frac{k_pka_plb PKAC_I PLB}{Km_pka_plb + PLB} PLB_DEPHOSPH = \frac{k_pp1_plb PP1 PLB_p}{Km_pp1_plb + PLB_p} Inhib1 = Inhib1_tot - Inhib1p_tot Inhib1p_PP1 = \frac{Inhib1p PP1}{Ki_inhib1} Inhib1_PHOSPH = \frac{k_pka_i1 PKAC_I Inhib1}{Km_pka_i1 + Inhib1} Inhib1_DEPHOSPH = \frac{Vmax_pp2a_i1 Inhib1p_tot}{Km_pp2a_i1 + Inhib1p_tot} \frac{d}{d time} (PLB_p) = PLB_PHOSPH - PLB_DEPHOSPH \frac{d}{d time} (Inhib1p_tot) = Inhib1_PHOSPH - Inhib1_DEPHOSPH Inhib1p = Inhib1p_tot - Inhib1p_PP1 PP1 = PP1_tot - Inhib1p_PP1 frac_PLB_p = \frac{PLB_p}{PLB_tot} frac_PLB = \frac{PLB}{PLB_tot} frac_PLB_o = 0.9613

Component: LCC_module

LCCa = LCC_tot - LCCa_p LCCa_PHOSPH = \frac{epsilon k_pka_lcc PKAC_II LCCa}{Km_pka_lcc + epsilon LCCa} LCCa_DEPHOSPH = \frac{epsilon k_pp2a_lcc PP2A_lcc_tot LCCa_p}{Km_pp2a_lcc + epsilon LCCa_p} LCCb = LCC_tot - LCCb_p LCCb_PHOSPH = \frac{epsilon k_pka_lcc PKAC_II LCCb}{Km_pka_lcc + epsilon LCCb} LCCb_DEPHOSPH = \frac{epsilon k_pp1_lcc PP1_lcc_tot LCCb_p}{Km_pp1_lcc + epsilon LCCb_p} \frac{d}{d time} (LCCa_p) = LCCa_PHOSPH - LCCa_DEPHOSPH \frac{d}{d time} (LCCb_p) = LCCb_PHOSPH - LCCb_DEPHOSPH frac_LCCa_p = \frac{LCCa_p}{LCC_tot} frac_LCCa_po = 0.2041 frac_LCCb_p = \frac{LCCb_p}{LCC_tot} frac_LCCb_po = 0.2336

Component: Nernst_Potentials

FoRT = \frac{\frac{Frdy}{R}}{Temp} E_Na = \frac{1}{FoRT z_Na} \ln (\frac{Na_ext}{Na_i}) E_K = \frac{1}{FoRT z_K} \ln (\frac{K_ext}{K_i}) E_Ca = \frac{1}{FoRT z_Ca} \ln (\frac{Ca_ext}{Ca_i}) E_Cl = - 40

Component: Fast_Na_Current

am = \frac{0.32 (V_m + 47.13)}{1 - (ⅇ^{(- 0.1) (V_m + 47.13)})} bm = 0.08 ⅇ^{\frac{- V_m}{11}} ah = \{\begin{matrix} 0 if V_m \geq - 40 \\ 0.135 ⅇ^{\frac{80 + V_m}{- 6.8}} otherwise \end{matrix} aj = \{\begin{matrix} 0 if V_m \geq - 40 \\ \frac{((- 1.2714 e 5) ⅇ^{0.2444 V_m} - (3.474 e -5 ⅇ^{(- 0.04391) V_m})) (V_m + 37.78)}{1 + ⅇ^{0.311 (V_m + 79.23)}} otherwise \end{matrix} bh = \{\begin{matrix} \frac{1}{0.13 (1 + ⅇ^{\frac{- (V_m + 10.66)}{11.1}})} if V_m \geq - 40 \\ 3.56 ⅇ^{0.079 V_m} + 3.1 e 5 ⅇ^{0.35 V_m} otherwise \end{matrix} bj = \{\begin{matrix} \frac{0.3 ⅇ^{(- 2.575 e -7) V_m}}{1 + ⅇ^{(- 0.1) (V_m + 32)}} if V_m \geq - 40 \\ \frac{0.1212 ⅇ^{(- 0.01052) V_m}}{1 + ⅇ^{(- 0.1378) (V_m + 40.14)}} otherwise \end{matrix} \frac{d}{d time} (m) = 1 e 3 (am (1 - m) - (bm m)) \frac{d}{d time} (h) = 1 e 3 (ah (1 - h) - (bh h)) \frac{d}{d time} (j) = 1 e 3 (aj (1 - j) - (bj j)) I_Na = G_Na m^{3} h j (V_m - E_Na)

Component: L_Type_Calcium_Current

a_lcc = 400 ⅇ^{\frac{V_m + 2}{10}} b_lcc = 50 ⅇ^{\frac{(- 1) (V_m + 2)}{13}} f_lcc = f (\frac{0.375 frac_LCCa_p}{frac_LCCa_po} + 0.625) y_lcc_inf = \frac{1}{1 + ⅇ^{\frac{V_m + 55}{7.5}}} + \frac{0.1}{1 + ⅇ^{\frac{(- V_m) + 21}{6}}} tau_y_lcc = 0.02 + \frac{0.3}{1 + ⅇ^{\frac{V_m + 30}{9.5}}} gamma = gamma_o Ca_i v_gamma = gamma ({(1 - v)}^{4} + 2 v {(1 - v)}^{3} + 4 v^{2} {(1 - v)}^{2} + 8 v^{3} (1 - v) + 16 v^{4} (1 - (\frac{f_lcc}{g}))) v_omega = omega ({(1 - w)}^{4} + 0.5 w {(1 - w)}^{3} + 0.25 w^{2} {(1 - w)}^{2} + 0.125 w^{3} (1 - w) + 0.0625 w^{4}) \frac{d}{d time} (v) = a_lcc (1 - v) - (b_lcc v) \frac{d}{d time} (w) = 2 a_lcc (1 - w) - (\frac{b_lcc}{2} w) \frac{d}{d time} (x) = f_lcc (1 - x) - (g x) \frac{d}{d time} (y) = \frac{y_lcc_inf - y}{tau_y_lcc} \frac{d}{d time} (z) = v_omega (1 - z) - (v_gamma z) i_bar_Ca = \frac{alpha p_Ca 4 V_m Frdy FoRT (1 e -3 ⅇ^{2 V_m FoRT} - (0.341 Ca_ext))}{ⅇ^{2 V_m FoRT} - 1} i_bar_K = \frac{alpha p_K V_m Frdy FoRT (K_i ⅇ^{V_m FoRT} - K_ext)}{ⅇ^{V_m FoRT} - 1} f_avail = 0.5 (\frac{0.4 frac_LCCb_p}{frac_LCCb_po} + 0.6) I_Ca = i_bar_Ca N_lcc f_avail v^{4} x y z I_CaK = \frac{i_bar_K}{1 + \frac{I_Ca}{I_Ca05}} N_lcc f_avail v^{4} x y z I_Ca_tot = I_Ca + I_CaK

Component: Transient_Outward_K_Current

r_toss = \frac{1}{1 + ⅇ^{\frac{V_m + 10.6}{- 11.42}}} s_toss = \frac{1}{1 + ⅇ^{\frac{V_m + 43.5}{6.8841}}} tau_r_to = \frac{1}{45.16 ⅇ^{0.03577 (V_m + 50)} + 98.9 ⅇ^{(- 0.1) (V_m + 38)}} tau_s_to = 0.35 ⅇ^{(- 1) {(\frac{V_m + 70}{15})}^{2}} + 0.035 tau_ss_to = 3.7 ⅇ^{(- 1) {(\frac{V_m + 70}{30})}^{2}} + 0.035 \frac{d}{d time} (r_to) = \frac{r_toss - r_to}{tau_r_to} \frac{d}{d time} (s_to) = \frac{s_toss - s_to}{tau_s_to} \frac{d}{d time} (ss_to) = \frac{s_toss - ss_to}{tau_ss_to} I_to = G_to r_to (0.886 s_to + 0.114 ss_to) (V_m - E_K)

Component: Steady_State_K_Current

r_ss_inf = \frac{1}{1 + ⅇ^{\frac{V_m + 11.5}{- 11.82}}} tau_r_ss = \frac{10}{45.16 ⅇ^{0.03577 (V_m + 50)} + 98.9 ⅇ^{(- 0.1) (V_m + 38)}} s_ss_inf = \frac{1}{1 + ⅇ^{\frac{V_m + 87.5}{10.3}}} tau_s_ss = 2.1 \frac{d}{d time} (r_ss) = \frac{r_ss_inf - r_ss}{tau_r_ss} \frac{d}{d time} (s_ss) = \frac{s_ss_inf - s_ss}{tau_s_ss} I_ss = G_ss r_ss s_ss (V_m - E_K)

Component: Time_Independent_K_Current

a_Ki = \frac{1.02}{1 + ⅇ^{0.2385 (V_m - E_K - 59.215)}} b_Ki = \frac{0.49124 ⅇ^{0.08032 (V_m + 5.476 - E_K)} + ⅇ^{0.06175 (V_m - E_K - 594.31)}}{1 + ⅇ^{(- 0.5143) ((V_m - E_K) + 4.753)}} Ki_ss = \frac{a_Ki}{a_Ki + b_Ki} I_Ki = G_Ki_bar \sqrt{\frac{K_ext}{5.4}} Ki_ss (V_m - E_K)

Component: Plateau_K_Current

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

Component: Na_Ca_Exchanger_Current

s4 = 1 ⅇ^{eta V_m FoRT} {Na_i}^{3} Ca_ext s5 = 1 ⅇ^{(eta - 1) V_m FoRT} {Na_ext}^{3} Ca_i I_NCX = \frac{k_NaCa}{({Km_Na}^{3} + {Na_ext}^{3}) (Km_Ca + Ca_ext) (1 + k_sat ⅇ^{(eta - 1) V_m FoRT})} (s4 - s5)

Component: Na_K_Pump_Current

sigma = \frac{ⅇ^{\frac{Na_ext}{67.3}} - 1}{7} f_NaK = \frac{1}{1 + 0.1245 ⅇ^{(- 0.1) V_m FoRT} + 0.0365 sigma ⅇ^{(- V_m) FoRT}} I_NaK = \frac{\frac{I_bar_NaK f_NaK}{1 + {(\frac{Km_Nai}{Na_i})}^{1.5}} K_ext}{K_ext + Km_Ko}

Component: Sarcolemmal_Ca_Pump_Current

I_PCa = \frac{I_bar_PCa Ca_i}{Km_PCa + Ca_i}

Component: Ca_Background_Current

I_CaB = G_CaB (V_m - E_Ca)

Component: Na_Background_Current

I_NaB = G_NaB (V_m - E_Na)

Component: Total_Membrane_Currents

I_Na_tot = I_Na + I_NaB + 3 I_NCX + 3 I_NaK I_K_tot = (I_to + I_ss + I_Ki + I_Kp - (2 I_NaK)) + I_CaK I_Ca_tot = I_Ca + I_CaB + I_PCa - (2 I_NCX)

Component: Calcium_Induced_Calcium_Release

t_rel_sub = trel + 2 e -3 ryr_on = 1 - (ⅇ^{\frac{- t_rel_sub}{tau_on}}) ryr_off = ⅇ^{\frac{- t_rel_sub}{tau_off}} g_rel = \frac{G_max_rel}{1 (1 + ⅇ^{\frac{I_Ca_tot + 5}{0.9}})} I_rel = g_rel ryr_on ryr_off (Ca_jsr - Ca_i)

Component: Other_SR_Fluxes_and_Concentrations

Km_up = \frac{Km_up0 (1 + 2 frac_PLB)}{1 + 2 frac_PLB_o} I_up = \frac{I_up_bar {Ca_i}^{2}}{{Km_up}^{2} + {Ca_i}^{2}} I_leak = \frac{I_up_bar Ca_nsr}{NSR_bar} I_tr = \frac{Ca_nsr - Ca_jsr}{tau_tr} B_jsr = \frac{1}{1 + \frac{CSQN_bar Km_CSQN}{{(Km_CSQN + Ca_jsr)}^{2}}} \frac{d}{d time} (Ca_nsr) = I_up - I_leak - (\frac{I_tr V_jsr}{V_nsr}) \frac{d}{d time} (Ca_jsr) = B_jsr (I_tr - I_rel) SR_content = 1 e 3 (\frac{(Ca_jsr + \frac{Ca_jsr}{B_jsr}) V_jsr}{V_myo} + \frac{Ca_nsr V_nsr}{V_myo})

Component: Cytoplasmic_Calcium_Buffering

b_trpn = \frac{TRPN_bar Km_TRPN}{{(Km_TRPN + Ca_i)}^{2}} b_cmdn = \frac{CMDN_bar Km_CMDN}{{(Km_CMDN + Ca_i)}^{2}} b_indo = \frac{INDO_bar Km_INDO}{{(Km_INDO + Ca_i)}^{2}} B_myo = \frac{1}{1 + b_cmdn + 2 b_trpn + b_indo}

Component: Ion_Concentrations_and_Membrane_Potential

V_clamp = \{\begin{matrix} V_test if time > 59.1 \land time < 59.5 \\ V_hold otherwise \end{matrix} I_app = \{\begin{matrix} 0 if protocol = 0 \\ I_pace if protocol = 1 \\ \frac{V_clamp - V_m}{R_clamp} otherwise \end{matrix} \frac{d}{d time} (Na_i) = \frac{lambda I_Na_tot A_Cap}{V_myo z_Na Frdy} \frac{d}{d time} (K_i) = \frac{lambda I_K_tot A_Cap}{V_myo z_K Frdy} \frac{d}{d time} (Ca_i) = B_myo ((\frac{lambda I_Ca_tot A_Cap}{V_myo z_Ca Frdy} - (\frac{(I_up - I_leak) V_nsr}{V_myo})) + \frac{I_rel V_jsr}{V_myo}) \frac{d}{d time} (V_m) = (- 1 e 3) (I_Ca_tot + I_K_tot + I_Na_tot - I_app) \frac{d}{d time} (trel) = \{\begin{matrix} 1 - (1 e 4 trel) if (- 1 e 3) (I_Ca_tot + I_K_tot + I_Na_tot - I_app) > 30 e 3 \\ 1 otherwise \end{matrix} I_pace = \{\begin{matrix} 10 if \sin (2 π time) > 0.9999 \\ 0 otherwise \end{matrix}