Model Mathematics

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Component: environment

Component: membrane

i_Stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Stim + i_K1 + i_to + i_Kr + i_Ks + i_K_ATP + i_K_ACh + i_NaK + i_Na + i_b_Na + i_p_Na + i_Ca_L_Na_cyt + i_Ca_L_Na_ds + i_NaCa_cyt + i_NaCa_ds + i_Ca_L_Ca_cyt + i_Ca_L_Ca_ds + i_Ca_L_K_cyt + i_Ca_L_K_ds + i_b_Ca + i_stretch)

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}) E_mh = \frac{R T}{F} \ln (\frac{Na_o + 0.12 K_o}{Na_i + 0.12 K_i})

Component: time_independent_potassium_current

i_K1 = \frac{\frac{g_K1 K_o}{K_o + K_mk1} (V - E_K)}{1 + ⅇ^{\frac{(V - E_K - 10) F 1.25}{R T}}}

Component: rapid_delayed_rectifier_potassium_current

i_Kr = \frac{(g_Kr1 xr1 + g_Kr2 xr2) 1}{1 + ⅇ^{\frac{V + 9}{22.4}}} (V - E_K)

Component: rapid_delayed_rectifier_potassium_current_xr1_gate

alpha_xr1 = \frac{50}{1 + ⅇ^{\frac{- (V - 5)}{9}}} beta_xr1 = 0.05 ⅇ^{\frac{- (V - 20)}{15}} \frac{d}{d time} (xr1) = alpha_xr1 (1 - xr1) - (beta_xr1 xr1)

Component: rapid_delayed_rectifier_potassium_current_xr2_gate

alpha_xr2 = \frac{50}{1 + ⅇ^{\frac{- (V - 5)}{9}}} beta_xr2 = 0.4 ⅇ^{- ({(\frac{V + 30}{30})}^{3})} \frac{d}{d time} (xr2) = alpha_xr2 (1 - xr2) - (beta_xr2 xr2)

Component: slow_delayed_rectifier_potassium_current

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

Component: slow_delayed_rectifier_potassium_current_xs_gate

alpha_xs = \frac{14}{1 + ⅇ^{\frac{- (V - 40)}{9}}} beta_xs = 1 ⅇ^{\frac{- V}{45}} \frac{d}{d time} (xs) = alpha_xs (1 - xs) - (beta_xs xs)

Component: ATP_dependent_potassium_current

i_K_ATP = \frac{g_K_ATP (V + 80)}{1 + {(\frac{ATP}{K_ATP})}^{2}}

Component: sodium_activated_potassium_current

i_KNa = \frac{g_K_Na Na_i}{Na_i + K_kna} (V - E_K)

Component: fast_sodium_current

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

Component: fast_sodium_current_m_gate

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)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

Component: fast_sodium_current_h_gate

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

Component: persistent_sodium_current

i_p_Na = \frac{g_pna 1}{1 + ⅇ^{\frac{- (V + 52)}{8}}} (V - E_Na)

Component: sodium_background_current

i_b_Na = g_bna (V - E_Na)

Component: L_type_Ca_channel

i_Ca_L_Ca_cyt = \frac{\frac{(1 - FrICa) 4 P_Ca_L d f f2 (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F 2}{R T}})} (Ca_i ⅇ^{\frac{100 F}{R T}} - (Ca_o ⅇ^{\frac{(- (V - 50)) F 2}{R T}})) i_Ca_L_K_cyt = \frac{\frac{(1 - FrICa) P_CaK P_Ca_L d f f2 (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}})} (K_i ⅇ^{\frac{50 F}{R T}} - (K_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L_Na_cyt = \frac{\frac{(1 - FrICa) P_CaNa P_Ca_L d f f2 (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}})} (Na_i ⅇ^{\frac{50 F}{R T}} - (Na_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L_Ca_ds = \frac{\frac{FrICa 4 P_Ca_L d f f2ds (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F 2}{R T}})} (Ca_i ⅇ^{\frac{100 F}{R T}} - (Ca_o ⅇ^{\frac{(- (V - 50)) F 2}{R T}})) i_Ca_L_K_ds = \frac{\frac{FrICa P_CaK P_Ca_L d f f2ds (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}})} (K_i ⅇ^{\frac{50 F}{R T}} - (K_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L_Na_ds = \frac{\frac{FrICa P_CaNa P_Ca_L d f f2ds (V - 50) F}{R T}}{1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}})} (Na_i ⅇ^{\frac{50 F}{R T}} - (Na_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L = i_Ca_L_Ca_cyt + i_Ca_L_K_cyt + i_Ca_L_Na_cyt + i_Ca_L_Ca_ds + i_Ca_L_K_ds + i_Ca_L_Na_ds

Component: L_type_Ca_channel_d_gate

E0_d = V + 24 - 5 alpha_d = \{\begin{matrix} 120 if | E0_d | < 0.0001 \\ \frac{30 E0_d}{1 - (ⅇ^{\frac{- E0_d}{4}})} otherwise \end{matrix} beta_d = \{\begin{matrix} 120 if | E0_d | < 0.0001 \\ \frac{12 E0_d}{ⅇ^{\frac{E0_d}{10}} - 1} otherwise \end{matrix} \frac{d}{d time} (d) = speed_d (alpha_d (1 - d) - (beta_d d))

Component: L_type_Ca_channel_f_gate

E0_f = V + 34 alpha_f = \{\begin{matrix} 25 if | E0_f | < delta_f \\ \frac{6.25 E0_f}{ⅇ^{\frac{E0_f}{4}} - 1} otherwise \end{matrix} beta_f = \frac{12}{1 + ⅇ^{\frac{(- 1) (V + 34)}{4}}} \frac{d}{d time} (f) = speed_f (alpha_f (1 - f) - (beta_f f))

Component: L_type_Ca_channel_f2_gate

\frac{d}{d time} (f2) = 1 - (1 (\frac{Ca_i}{Km_f2 + Ca_i} + f2))

Component: L_type_Ca_channel_f2ds_gate

\frac{d}{d time} (f2ds) = R_decay (1 - (\frac{Ca_ds}{Km_f2ds + Ca_ds} + f2ds))

Component: calcium_background_current

i_b_Ca = g_bca (V - E_Ca)

Component: transient_outward_current

i_to = g_to (g_tos + s (1 - g_tos)) r (V - E_K)

Component: transient_outward_current_s_gate

alpha_s = 0.033 ⅇ^{\frac{- V}{17}} beta_s = \frac{33}{1 + ⅇ^{(- 0.125) (V + 10)}} \frac{d}{d time} (s) = alpha_s (1 - s) - (beta_s s)

Component: transient_outward_current_r_gate

\frac{d}{d time} (r) = 333 (\frac{1}{1 + ⅇ^{\frac{- (V + 4)}{5}}} - r)

Component: ACh_dependent_potassium_current

i_K_ACh = \frac{g_KACh \frac{K_o}{K_o + K_mk1} x_ACh \frac{1}{1 + {(\frac{K_D}{ACh})}^{2}} (V - E_K)}{1 + ⅇ^{\frac{2 F (V - (E_K + 10))}{R T}}}

Component: ACh_dependent_potassium_current_xACh_gate

\frac{d}{d time} (x_ACh) = alpha_ACh (1 - x_ACh) - (beta_ACh x_ACh)

Component: sodium_potassium_pump

i_NaK = \frac{\frac{i_NaK_max K_o}{K_mK + K_o} Na_i}{K_mNa + Na_i}

Component: sodium_calcium_exchanger

i_NaCa_cyt = \frac{(1 - FRiNaCa) k_NaCa (ⅇ^{\frac{gamma (n_NaCa - 2) V F}{R T}} {Na_i}^{n_NaCa} Ca_o - (ⅇ^{\frac{(gamma - 1) (n_NaCa - 2) V F}{R T}} {Na_o}^{n_NaCa} Ca_i))}{(1 + d_NaCa (Ca_i {Na_o}^{n_NaCa} + Ca_o {Na_i}^{n_NaCa})) (1 + \frac{Ca_i}{0.0069})} i_NaCa_ds = \frac{FRiNaCa k_NaCa (ⅇ^{\frac{gamma (n_NaCa - 2) V F}{R T}} {Na_i}^{n_NaCa} Ca_o - (ⅇ^{\frac{(gamma - 1) (n_NaCa - 2) V F}{R T}} {Na_o}^{n_NaCa} Ca_ds))}{(1 + d_NaCa (Ca_ds {Na_o}^{n_NaCa} + Ca_o {Na_i}^{n_NaCa})) (1 + \frac{Ca_ds}{0.0069})} i_NaCa = i_NaCa_cyt + i_NaCa_ds

Component: sarcoplasmic_reticulum_calcium_pump

K_1 = \frac{K_cyca K_xcs}{K_srca} K_2 = Ca_i + Ca_up K_1 + K_cyca K_xcs + K_cyca i_up = \frac{Ca_i}{K_2} alpha_up - (\frac{Ca_up K_1}{K_2} beta_up)

Component: calcium_translocation

i_trans = 50 (Ca_up - Ca_rel)

Component: calcium_release

VoltDep = ⅇ^{0.08 (V - 40)} CaiReg = \frac{Ca_i}{Ca_i + K_m_Ca_cyt} CadsReg = \frac{Ca_ds}{Ca_ds + K_m_Ca_ds} RegBindSite = CaiReg + (1 - CaiReg) CadsReg ActRate = 0 VoltDep + 500 {RegBindSite}^{2} InactRate = 60 + 500 {RegBindSite}^{2} SpeedRel = \{\begin{matrix} 5 if V < - 50 \\ 1 otherwise \end{matrix} SRLeak = \{\begin{matrix} K_leak_rate ⅇ^{gama_SR_IT isometric_tension} if isometric_tension > 0 \\ K_leak_rate ⅇ^{gama_SR_SL sarcomere_length} otherwise \end{matrix} PrecFrac = 1 - ActFrac - ProdFrac \frac{d}{d time} (ActFrac) = PrecFrac SpeedRel ActRate - (ActFrac SpeedRel InactRate) \frac{d}{d time} (ProdFrac) = ActFrac SpeedRel InactRate - (SpeedRel 1 ProdFrac) i_rel = ({(\frac{ActFrac}{ActFrac + 0.25})}^{2} K_m_rel + SRLeak) Ca_rel

Component: extracellular_sodium_concentration

Component: intracellular_sodium_concentration

\frac{d}{d time} (Na_i) = \frac{- 1}{1 V_i F} (i_Na + i_p_Na + i_b_Na + 3 i_NaK + 3 i_NaCa_cyt + i_Ca_L_Na_cyt + i_Ca_L_Na_ds)

Component: extracellular_calcium_concentration

Component: extracellular_potassium_concentration

Component: intracellular_potassium_concentration

\frac{d}{d time} (K_i) = \frac{- 1}{1 V_i F} (i_K1 + i_Kr + i_Ks + i_Ca_L_K_cyt + i_Ca_L_K_ds + i_to - (2 i_NaK))

Component: intracellular_calcium_concentration

V_Cell = \frac{3.141592654 {(\frac{radius}{1000})}^{2} length}{1000} V_i_ratio = 1 - V_e_ratio - V_up_ratio - V_rel_ratio V_i = V_Cell V_i_ratio \frac{d}{d time} (Ca_i) = \frac{- 1}{2 1 V_i F} (i_Ca_L_Ca_cyt + i_b_Ca - (2 i_NaCa_cyt) - (2 i_NaCa_ds)) + Ca_ds V_ds_ratio Kdecay + \frac{i_rel V_rel_ratio}{V_i_ratio} - (\frac{d}{d time} (Ca_Calmod)) - (\frac{d}{d time} (Ca_Trop)) - i_up \frac{d}{d time} (Ca_ds) = \frac{(- 1) i_Ca_L_Ca_ds}{2 1 V_ds_ratio V_i F} - (Ca_ds Kdecay) \frac{d}{d time} (Ca_up) = \frac{V_i_ratio}{V_up_ratio} i_up - i_trans \frac{d}{d time} (Ca_rel) = \frac{V_up_ratio}{V_rel_ratio} i_trans - i_rel \frac{d}{d time} (Ca_Calmod) = alpha_Calmod Ca_i (Calmod - Ca_Calmod) - (beta_Calmod Ca_Calmod) alpha_Trop = KTrop ⅇ^{gama_Trop_SL sarcomere_length} \frac{d}{d time} (Ca_Trop) = alpha_Trop Ca_i (Trop - Ca_Trop) - (beta_Trop Ca_Trop)

Component: contraction

XCont2 = \frac{Ca_Calmod}{Calmod} XCont1 = \frac{Ca_Trop}{Trop} \frac{d}{d time} (light_chain) = KCont1 {XCont1}^{2} XCont2 (1 - light_chain) - (KCont2 light_chain) \frac{d}{d time} (cross_bridge) = KCont3 light_chain (1 - cross_bridge) - (KCont4 cross_bridge) tension_rest = (2e-4) ⅇ^{2 sarcomere_length} tension_active = \{\begin{matrix} 1 - (ⅇ^{(- 3) (sarcomere_length - 1)}) if sarcomere_length > 1 \\ 0 otherwise \end{matrix} overlap = \{\begin{matrix} 1 - (0.625 (sarcomere_length - 2)) if sarcomere_length > 2 \\ 1 otherwise \end{matrix} cross_bridge_availability = tension_active overlap cross_bridge_density isometric_tension = cross_bridge cross_bridge_availability + tension_rest

Component: stretch_current

f_stretch = \{\begin{matrix} \frac{1}{1 + ⅇ^{(- 2) gama_SAC_IT (isometric_tension - ITHST)}} if isometric_tension > 0 \\ \frac{1}{1 + ⅇ^{(- 2) gama_SAC_SL (sarcomere_length - SLHST)}} otherwise \end{matrix} i_Ca_stretch = g_Ca_stretch f_stretch (V - E_Ca) i_Na_stretch = g_Na_stretch f_stretch (V - E_Na) i_K_stretch = g_K_stretch f_stretch (V - E_K) i_An_stretch = g_An_stretch f_stretch (V - E_An_stretch) i_stretch = i_Ca_stretch + i_Na_stretch + i_K_stretch + i_An_stretch

Component: fibroblast

i_fibro = g_fibro (V_fibro + 20) + g_fibro_stretch (V_fibro - E_fibro_stretch) i_fibro_junct = (- g_fibro_junct) (V - V_fibro) \frac{d}{d time} (V_fibro) = \frac{- (i_fibro + i_fibro_junct)}{c_fibro}