Model Mathematics

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Component: environment

Component: model_parameters

Component: cell_geometry

vcell = 1000 π a a l ageo = 2 π a a + 2 π a l Acap = ageo 2 AF = \frac{1 1 Acap}{F 1} Vmyo = vcell 0.678 Vsr = vcell 0.06 Vnsr = vcell 0.0552 Vjsr = vcell 0.0048 Vss_sr = vcell 0.02 Vss_CaL = vcell 0.002

Component: membrane

past = ⌊ \frac{time}{stim_period} ⌋ stim_period i_Stim = \{\begin{matrix} stim_amplitude if time - past \geq stim_offset \land time - past ≦ stim_offset + stim_duration \\ 0 otherwise \end{matrix} caiont = ICaL + ICab + IpCa - (2 (INaCa + INaCa_ss_sr)) naiont = INa + 3 (INaCa + INaCa_ss_sr) + 3 INaK + INaL + INab kiont = IKr + IKs + IK1 + IKp + (- 2) INaK + Ito1 + i_Stim clont = IClb + Ito2 \frac{d}{d time} (Vm) = - (naiont + kiont + caiont + clont)

Component: IKs

IKs = gKs OKs (Vm - EKs) gKs = gKs_max (1 + \frac{0.6}{1 + {(\frac{3.8 e -5}{Ca_i})}^{1.4}}) O2 = 1 - (C1 + C2 + C3 + C4 + C5 + C6 + C7 + C8 + C9 + C10 + C11 + C12 + C13 + C14 + C15 + O1) OKs = O1 + O2 \frac{d}{d time} (C1) = (- 4) alpha C1 + beta C2 \frac{d}{d time} (C2) = (- (3 alpha + beta + gamma)) C2 + 4 alpha C1 + 2 beta C3 + delta C6 \frac{d}{d time} (C3) = (- (2 alpha + 2 beta + 2 gamma)) C3 + 3 alpha C2 + 3 beta C4 + delta C7 \frac{d}{d time} (C4) = (- (alpha + 3 beta + 3 gamma)) C4 + 2 alpha C3 + 4 beta C5 + delta C8 \frac{d}{d time} (C5) = (- (4 beta + 4 gamma)) C5 + alpha C4 + delta C9 \frac{d}{d time} (C6) = (- (3 alpha + delta)) C6 + beta C7 + gamma C2 \frac{d}{d time} (C7) = (- (2 alpha + beta + gamma + delta)) C7 + 3 alpha C6 + 2 beta C8 + 2 gamma C3 + 2 delta C10 \frac{d}{d time} (C8) = (- (alpha + 2 beta + 2 gamma + delta)) C8 + 2 alpha C7 + 3 beta C9 + 3 gamma C4 + 2 delta C11 \frac{d}{d time} (C9) = (- (3 beta + 3 gamma + delta)) C9 + alpha C8 + 4 gamma C5 + 2 delta C12 \frac{d}{d time} (C10) = (- (2 alpha + 2 delta)) C10 + beta C11 + gamma C7 \frac{d}{d time} (C11) = (- (alpha + beta + gamma + 2 delta)) C11 + 2 alpha C10 + 2 beta C12 + 2 gamma C8 + 3 delta C13 \frac{d}{d time} (C12) = (- (2 beta + 2 gamma + 2 delta)) C12 + alpha C11 + 3 gamma C9 + 3 delta C14 \frac{d}{d time} (C13) = (- (alpha + 3 delta)) C13 + beta C14 + gamma C11 \frac{d}{d time} (C14) = (- (beta + gamma + 3 delta)) C14 + alpha C13 + 2 gamma C12 + 4 delta C15 \frac{d}{d time} (C15) = (- (4 delta + theta)) C15 + gamma C14 + eta O1 \frac{d}{d time} (O1) = (- (eta + psi)) O1 + omega O2 + theta C15 alpha = 0.01486459798086 ⅇ^{\frac{0.02987730123588 Vm F}{R T}} beta = 0.08398631219983 ⅇ^{\frac{(- 0.05546105712664) Vm F}{R T}} gamma = 0.01460066118316 ⅇ^{\frac{0.24464953099645 Vm F}{R T}} delta = 0.0031173268874 ⅇ^{\frac{(- 0.42625451944376) Vm F}{R T}} eta = 0.07731990097331 ⅇ^{\frac{(- 0.06472612248871) Vm F}{R T}} theta = 0.08953830641102 omega = 0.7940545995864 ⅇ^{\frac{(- 0.08017378192977) Vm F}{R T}} psi = 0.58638228663014 ⅇ^{\frac{0.28205554331496 Vm F}{R T}}

Component: ICaL

OI_star = 1 - (C + O + C_star + O_star + CI + OI + CI_star) ICaL = ICaL_max (O + O_star) ICaL_max = \frac{\frac{PCa 4 Vm F^{2}}{R T} (gamma_Cai Ca_ss_CaL ⅇ^{\frac{2 Vm F}{R T}} - (gamma_Cao Ca_o))}{ⅇ^{\frac{2 Vm F}{R T}} - 1} \frac{d}{d time} (C) = (- (alpha + delta + y)) C + beta O + theta C_star + x CI \frac{d}{d time} (O) = (- (beta + delta + y)) O + alpha C + theta O_star + x OI \frac{d}{d time} (C_star) = (- (alpha + theta + y_star)) C_star + delta C + beta O_star + x_star CI_star \frac{d}{d time} (O_star) = (- (beta + theta + y_star)) O_star + delta O + alpha C_star + x_star OI_star \frac{d}{d time} (CI) = (- (alpha + delta_I + x)) CI + y C + theta_I CI_star + beta OI \frac{d}{d time} (OI) = (- (beta + delta_I + x)) OI + y O + theta_I OI_star + alpha CI \frac{d}{d time} (CI_star) = (- (alpha + theta_I + x_star)) CI_star + delta_I CI + y_star C_star + beta OI_star ACT_tau = 0.59 + \frac{0.8 ⅇ^{0.052 (Vm + 13)}}{1 + ⅇ^{0.132 (Vm + 13)}} ACT_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm - 13.56)}{9.45}}} alpha = \frac{ACT_infinity}{ACT_tau} beta = \frac{1 - ACT_infinity}{ACT_tau} IV_infinity = \frac{\frac{1}{1 + ⅇ^{\frac{Vm + 17.5}{3}}} + 0.25}{1.25} IV_tau = \frac{1}{\frac{1}{24.828 (1 + ⅇ^{\frac{Vm + 49.1}{10.349}})} + \frac{1}{30.553 (1 + ⅇ^{\frac{- (Vm + 0.213)}{10.807}})}} x = \frac{IV_infinity}{IV_tau} y = \frac{1 - IV_infinity}{IV_tau} IV_infinity_star = \frac{\frac{1}{1 + ⅇ^{\frac{Vm + 17.5}{3}}} + 0.0001}{1.0001} IV_tau_star = \frac{1}{\frac{1}{24.828 (1 + ⅇ^{\frac{Vm + 49.1}{10.349}})} + IV_beta_star} IV_beta_star = \frac{1}{IV_beta_infinity_star (1 + ⅇ^{\frac{- (Vm + 0.213)}{10.807}})} IV_beta_infinity_star = 25 - (\frac{17.5}{1 + {(\frac{0.003}{Ca_ss_CaL})}^{4}}) x_star = \frac{IV_infinity_star}{IV_tau_star} y_star = \frac{1 - IV_infinity_star}{IV_tau_star} delta = \frac{3}{1 + {(\frac{0.003}{Ca_ss_CaL})}^{4}} delta_I = \frac{theta_I x y_star delta}{y x_star theta}

Component: INa

INa = g_Na m^{3} h j (Vm - ENa)

Component: INa_m_gate

am = \frac{0.32 (Vm + 47.13)}{1 - (ⅇ^{(- 0.1) (Vm + 47.13)})} bm = 0.08 ⅇ^{\frac{- Vm}{11}} \frac{d}{d time} (m) = am (1 - m) - (bm m)

Component: INa_h_gate

ah = \{\begin{matrix} 0 if Vm \geq - 40 \\ 0.135 ⅇ^{\frac{80 + Vm}{- 6.8}} otherwise \end{matrix} bh = \{\begin{matrix} \frac{1}{0.13 (1 + ⅇ^{\frac{Vm + 10.66}{- 11.1}})} if Vm \geq - 40 \\ 3.56 ⅇ^{0.079 Vm} + 3.1 e 5 ⅇ^{0.35 Vm} otherwise \end{matrix} \frac{d}{d time} (h) = ah (1 - h) - (bh h)

Component: INa_j_gate

aj = \{\begin{matrix} 0 if Vm \geq - 40 \\ \frac{((- 1.2714 e 5) ⅇ^{0.2444 Vm} - (6.948 e -5 ⅇ^{(- 0.04391) Vm})) (Vm + 37.78)}{1 + ⅇ^{0.311 (Vm + 79.23)}} otherwise \end{matrix} bj = \{\begin{matrix} \frac{0.3 ⅇ^{(- 2.535 e -7) Vm}}{1 + ⅇ^{(- 0.1) (Vm + 32)}} if Vm \geq - 40 \\ \frac{0.1212 ⅇ^{(- 0.01052) Vm}}{1 + ⅇ^{(- 0.1378) (Vm + 40.14)}} otherwise \end{matrix} \frac{d}{d time} (j) = aj (1 - j) - (bj j)

Component: INaK

INaK = ibarnak fv PK PNa phi = \frac{sigma (Vm - V_half) F}{R T} fv = \frac{1}{1 + ⅇ^{- phi}} PK = \frac{K_o}{K_o + kmko} PNa = {(\frac{Na_i}{Na_i + kmnai})}^{3}

Component: INaCa

numerator = 0.8 Vmax ({Na_i}^{3} Ca_o ⅇ^{\frac{eta Vm F}{R T}} - ({Na_o}^{3} Ca_i ⅇ^{\frac{(eta - 1) Vm F}{R T}})) denom_1 = 1 + {(\frac{KmCa_act}{Ca_i})}^{2} denom_2 = 1 + ksat ⅇ^{\frac{(eta - 1) Vm F}{R T}} denom_3 = KmCao {Na_i}^{3} + {KmNao}^{3} Ca_i + {KmNai}^{3} Ca_o (1 + \frac{Ca_i}{KmCai}) denom_4 = KmCai {Na_o}^{3} (1 + {(\frac{Na_i}{KmNai})}^{3}) + {Na_i}^{3} Ca_o + {Na_o}^{3} Ca_i num_ss = 0.2 Vmax ({Na_ss_sr}^{3} Ca_o ⅇ^{\frac{eta Vm F}{R T}} - ({Na_o}^{3} Ca_ss_sr ⅇ^{\frac{(eta - 1) Vm F}{R T}})) denom_ss_1 = 1 + {(\frac{KmCa_act}{Ca_ss_sr})}^{2} denom_ss_2 = 1 + ksat ⅇ^{\frac{(eta - 1) Vm F}{R T}} denom_ss_3 = KmCao {Na_ss_sr}^{3} + {KmNao}^{3} Ca_ss_sr + {KmNai}^{3} Ca_o (1 + \frac{Ca_ss_sr}{KmCai}) denom_ss_4 = KmCai {Na_o}^{3} (1 + {(\frac{Na_ss_sr}{KmNai})}^{3}) + {Na_ss_sr}^{3} Ca_o + {Na_o}^{3} Ca_ss_sr INaCa_ss_sr = \frac{num_ss}{denom_ss_1 denom_ss_2 (denom_ss_3 + denom_ss_4)} INaCa_cai = \frac{numerator}{denom_1 denom_2 (denom_3 + denom_4)} INaCa = INaCa_cai + INaCa_ss_sr

Component: IKp

IKp = gKp Kp (Vm - EK) Kp = \frac{1}{1 + ⅇ^{\frac{7.488 - Vm}{5.98}}}

Component: IpCa

IpCa = \frac{gpCa Ca_i}{KmpCa + Ca_i}

Component: ICab

ICab = \frac{\frac{PCab 4 Vm F^{2}}{R T} (gamma_Ca_i Ca_i ⅇ^{\frac{2 Vm F}{R T}} - (gamma_Ca_o Ca_o))}{ⅇ^{\frac{2 Vm F}{R T}} - 1}

Component: INab

phi = \frac{F Vm}{R T} INab = \frac{F PNab phi (Na_i ⅇ^{phi} - Na_o)}{ⅇ^{phi} - 1}

Component: IClb

IClb = gClb (Vm - ECl)

Component: INaL

INaL = gNaL {mL}^{3} hL (Vm - ENa)

Component: INaL_mL_gate

amL = \frac{0.32 (Vm + 47.13)}{1 - (ⅇ^{(- 0.1) (Vm + 47.13)})} bmL = 0.08 ⅇ^{\frac{- Vm}{11}} \frac{d}{d time} (mL) = amL (1 - mL) - (bmL mL)

Component: INaL_hL_gate

hL_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 91}{6.1}}} \frac{d}{d time} (hL) = \frac{hL_infinity - hL}{tau_hL}

Component: reversal_potentials

ENa = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) EK = \frac{R T}{F} \ln (\frac{K_o}{K_i}) EKs = \frac{R T}{F} \ln (\frac{K_o + prnak Na_o}{K_i + prnak Na_i}) ECl = \frac{(- R) T}{F} \ln (\frac{Cl_o}{Cl_i})

Component: IK1

IK1 = gK1 K1 (Vm - EK) gK1 = g_K1_max \sqrt{\frac{K_o}{5.4}}

Component: IK1_K1_gate

alpha_k1 = \frac{1.02}{1 + ⅇ^{0.2385 (Vm - EK - 59.215)}} beta_k1 = \frac{0.49124 ⅇ^{0.08032 ((Vm - EK) + 5.476)} + ⅇ^{0.06175 (Vm - EK - 594.31)}}{1 + ⅇ^{(- 0.5143) ((Vm - EK) + 4.753)}} K1 = \frac{alpha_k1}{alpha_k1 + beta_k1}

Component: CT_Na_Cl

CT_Na_Cl = \frac{CT_Na_Cl_max {(ENa - ECl)}^{4}}{{(ENa - ECl)}^{4} + {87.8251}^{4}}

Component: CT_K_Cl

CT_K_Cl = \frac{CT_K_Cl_max (EK - ECl)}{EK + 87.8251 - ECl}

Component: IKr

gKr = gKr_max \sqrt{\frac{K_o}{5.4}} IKr = gKr xr r (Vm - EK)

Component: IKr_xr_gate

xr_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm + 10.085)}{4.25}}} tau_xr = \frac{1}{\frac{0.0006 (Vm - 1.7384)}{1 - (ⅇ^{(- 0.136) (Vm - 1.7384)})} + \frac{0.0003 (Vm + 38.3608)}{ⅇ^{0.1522 (Vm + 38.3608)} - 1}} \frac{d}{d time} (xr) = \frac{xr_infinity - xr}{tau_xr}

Component: IKr_r_gate

r = \frac{1}{1 + ⅇ^{\frac{Vm + 10}{15.4}}}

Component: Ito1

Ito1 = gto1 a^{3} i1f i1s rto1 (Vm - EK) rto1 = ⅇ^{\frac{Vm}{550}}

Component: Ito1_a_gate

alpha_a = \frac{1}{1.2089 (1 + ⅇ^{\frac{Vm - 18.4099}{- 29.3814}})} beta_a = \frac{3.5}{1 + ⅇ^{\frac{Vm + 100}{29.3814}}} tau_a = \frac{1}{alpha_a + beta_a} a_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 9.437}{- 7.133}}} \frac{d}{d time} (a) = \frac{a_infinity - a}{tau_a}

Component: Ito1_i1f_gate

beta_i1f = \frac{1}{9.7953 (1 + ⅇ^{\frac{Vm + 19}{- 9}})} alpha_i1f = \frac{0.025}{1 + ⅇ^{\frac{Vm + 58}{5}}} \frac{d}{d time} (i1f) = alpha_i1f (1 - i1f) - (beta_i1f i1f)

Component: Ito1_i1s_gate

beta_i1s = \frac{1}{9.7953 (1 + ⅇ^{\frac{Vm + 19}{- 9}})} alpha_i1s = \frac{1}{250 (1 + ⅇ^{\frac{Vm + 60}{5}})} \frac{d}{d time} (i1s) = alpha_i1s (1 - i1s) - (beta_i1s i1s)

Component: Ito2

Ito2 = Ito2_max Ito2_max_scaling_factor i2f KCa_ito2 Ito2_max = \frac{\frac{PCl {zCl}^{2} Vm F^{2}}{R T} (Cl_i - (Cl_o ⅇ^{\frac{Vm F}{R T}}))}{1 - (ⅇ^{\frac{Vm F}{R T}})} KCa_ito2 = 1 - (\frac{1}{1 + {(\frac{Irel}{kCa_ito2})}^{2}})

Component: Ito2_i2f_gate

\frac{d}{d time} (i2f) = \frac{i2f_infinity - i2f}{tau_i2f} i2f_infinity = \frac{alpha_i2f}{alpha_i2f + beta_i2f} beta_i2f = \frac{1}{5 (1 + ⅇ^{\frac{Vm + 19}{- 9}})} alpha_i2f = \frac{0.025}{1 + ⅇ^{\frac{Vm + 58}{5}}}

Component: Irel

alpha_rel = beta_tau kappa beta_tau = beta_0 (1 + delta_beta_CaMK) delta_beta_CaMK = \frac{delta_beta_0}{1 + {(\frac{K_beta}{1 CaMK_active})}^{h_beta}} tau_rel = \frac{beta_tau}{1 + \frac{Krel_tau}{Ca_JSR}} rel_infinity = \frac{ICaL alpha_rel}{1 + {(\frac{Krel_infinity}{Ca_JSR})}^{h_rel}} \frac{d}{d time} (Irel) = \frac{- (rel_infinity + Irel)}{tau_rel}

Component: Iup

Iup = \frac{(delta_iupCaMK + 1) iupbar Ca_i}{Ca_i + kmup - delta_kmPLB} delta_iupCaMK = \frac{delta_iupCaMK_bar CaMK_active 1}{kmCaMK + CaMK_active 1} delta_kmPLB = \frac{delta_kmPLB_bar CaMK_active 1}{kmCaMK + CaMK_active 1}

Component: Ileak

Ileak = \frac{0.004375}{nsrbar} Ca_NSR

Component: Itr

Itr = \frac{Ca_NSR - Ca_JSR}{tautr}

Component: Ca

Idiff = \frac{Ca_ss_sr - Ca_i}{tau_diff} Idiff_ss = \frac{Ca_ss_sr - Ca_ss_CaL}{tau_diff_ss} bmyo = \frac{1}{1 + \frac{cmdn_bar km_cmdn}{{(Ca_i + km_cmdn)}^{2}} + \frac{km_trpn trpn_bar}{{(Ca_i + km_trpn)}^{2}}} \frac{d}{d time} (Ca_i) = bmyo (\frac{(- (ICab + IpCa - (2 INaCa))) AF}{Vmyo 2} + \frac{(Ileak - Iup) Vnsr}{Vmyo} + \frac{Idiff Vss_sr}{Vmyo}) \frac{d}{d time} (Ca_ss_sr) = (- bss_sr) (Idiff + Idiff_ss - (\frac{2 INaCa_ss_sr AF}{2 Vss_sr} + \frac{Irel Vjsr}{Vss_sr})) bss_sr = \frac{1}{1 + \frac{BSRmax KmBSR}{{(KmBSR + Ca_ss_sr)}^{2}} + \frac{BSLmax KmBSL}{{(KmBSL + Ca_ss_sr)}^{2}}} \frac{d}{d time} (Ca_ss_CaL) = (- bss_cal) (\frac{ICaL AF}{2 Vss_CaL} - (\frac{Idiff_ss Vss_sr}{Vss_CaL})) bss_cal = \frac{1}{1 + \frac{BSRmax KmBSR}{{(KmBSR + Ca_ss_CaL)}^{2}} + \frac{BSLmax KmBSL}{{(KmBSL + Ca_ss_CaL)}^{2}}} \frac{d}{d time} (Ca_NSR) = Iup - (Ileak + \frac{Itr Vjsr}{Vnsr}) bcsqn = \frac{1}{1 + \frac{kmcsqn csqnbar}{{(Ca_JSR + kmcsqn)}^{2}}} \frac{d}{d time} (Ca_JSR) = bcsqn (Itr - Irel)

Component: Na

Idiff_Na = \frac{Na_ss_sr - Na_i}{tau_diff} \frac{d}{d time} (Na_i) = - (\frac{(3 INaCa + 3 INaK + INa + INaL + INab) AF}{Vmyo} - (CT_Na_Cl + \frac{Idiff_Na Vss_sr}{Vmyo})) \frac{d}{d time} (Na_ss_sr) = - (\frac{3 INaCa_ss_sr AF}{Vss_sr} + Idiff_Na)

Component: Cl

Idiff_Cl = \frac{Cl_ss - Cl_i}{tau_diff} \frac{d}{d time} (Cl_i) = - (\frac{IClb AF}{(- 1) Vmyo} - (CT_Na_Cl + CT_K_Cl + \frac{Idiff_Cl Vss_sr}{Vmyo})) \frac{d}{d time} (Cl_ss) = - (\frac{Ito2 AF}{(- 1) Vss_sr} + Idiff_Cl)

Component: K

\frac{d}{d time} (K_i) = - (\frac{(IKs + IKr + IK1 + Ito1 + IKp + I_stim - (2 INaK)) AF}{Vmyo} - CT_K_Cl)

Component: CaMK_active

\frac{d}{d time} (CaMK_trap) = alpha_CaMK CaMK_active (CaMK_active - CaMK_trap) - (beta_CaMK CaMK_trap) CaMK_active = \frac{CaMK_0 (1 - CaMK_trap)}{1 + \frac{Km}{Ca_ss_sr}} + CaMK_trap