Model Mathematics

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Component: Environment

Component: cell

vcell = 1000 π a a l ageo = 2 π a a + 2 π a l Acap = ageo 2 vmyo = vcell 0.68 vmito = vcell 0.26 vsr = vcell 0.06 vnsr = vcell 0.0552 vjsr = vcell 0.0048 vss = vcell 0.02 AF = \frac{Acap}{F} 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) naiont = INa + 3 INaCa + 3 INaK + INal kiont = (IKr + IKs + IK1 + IKp - (2 INaK)) + Ito + 0.5 i_Stim clont = IClb + Ito2 + 0.5 i_Stim \frac{d}{d time} (V) = - (naiont + kiont + caiont + clont)

Component: INa

GNa = \{\begin{matrix} 8.25 if tissue = 0 \\ 4 8.25 otherwise \end{matrix} gNa = GNa m m m H J INa = gNa (V - ENa) ah = \{\begin{matrix} 0 if V \geq - 40 \\ 0.135 ⅇ^{\frac{80 + V}{- 6.8}} otherwise \end{matrix} aj = \{\begin{matrix} 0 if V \geq - 40 \\ \frac{((- 1.2714 e 5) ⅇ^{0.2444 V} - (3.474 e -5 ⅇ^{(- 0.04391) V})) 1 (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} otherwise \end{matrix} bh = \{\begin{matrix} \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} if V \geq - 40 \\ 3.56 ⅇ^{0.079 V} + 3.1 e 5 ⅇ^{0.35 V} otherwise \end{matrix} bj = \{\begin{matrix} \frac{0.3 ⅇ^{(- 2.535 e -7) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} if V \geq - 40 \\ \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} otherwise \end{matrix} am = \frac{0.32 1 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} bm = 0.08 ⅇ^{\frac{- V}{11}} \frac{d}{d time} (H) = ah (1 - H) - (bh H) \frac{d}{d time} (m) = am (1 - m) - (bm m) \frac{d}{d time} (J) = aj (1 - J) - (bj J)

Component: ICaL

ibarca = \frac{pca 4 (V - 15) F FonRT (gacai Ca_ss ⅇ^{2 (V - 15) FonRT} - (gacao Ca_o))}{ⅇ^{2 (V - 15) FonRT} - 1} ICaL = \{\begin{matrix} d^{dp} f f2 fca fca2 ibarca if tissue = 0 \\ d f f2 fca fca2 ibarca otherwise \end{matrix} dss = \frac{1}{1 + ⅇ^{\frac{- (V - 4)}{6.74}}} taud = 0.59 + \frac{0.8 ⅇ^{0.052 (V + 13)}}{1 + ⅇ^{0.132 (V + 13)}} fss = \frac{0.7}{1 + ⅇ^{\frac{V + 17.12}{7}}} + 0.3 f2ss = \frac{0.77}{1 + ⅇ^{\frac{V + 17.12}{7}}} + 0.23 tauf = \frac{1}{0.2411 ⅇ^{- ({(0.045 (V - 9.6914))}^{2})} + 0.0529} tauf2 = \frac{1}{0.0423 ⅇ^{- ({(0.059 (V - 18.5726))}^{2})} + 0.0054} dpss = 9 - (\frac{8}{1 + ⅇ^{\frac{- (V + 65)}{3.4}}}) fcass = \frac{0.3}{1 - (\frac{ICaL}{0.05})} + \frac{0.55}{1 + \frac{Ca_ss}{0.003}} + 0.15 fca2ss = \frac{1}{1 - (\frac{ICaL}{0.01})} taufca = \frac{10 CaMKactive}{0.15 + CaMKactive} + \frac{1}{1 + \frac{Ca_ss}{0.003}} + 0.5 taufca2 = \frac{300}{1 + ⅇ^{\frac{- ICaL - 0.175}{0.04}}} + 125 \frac{d}{d time} (d) = \frac{dss - d}{taud} \frac{d}{d time} (dp) = \frac{dpss - dp}{10} \frac{d}{d time} (f) = \frac{fss - f}{tauf} \frac{d}{d time} (f2) = \frac{f2ss - f2}{tauf2} \frac{d}{d time} (fca) = \frac{fcass - fca}{taufca} \frac{d}{d time} (fca2) = \frac{fca2ss - fca2}{taufca2}

Component: IK1

ak1 = \frac{1.02}{1 + ⅇ^{0.2385 (V - EK - 59.215)}} bk1 = \frac{0.49124 ⅇ^{0.08032 ((V - EK) + 5.476)} + 1 ⅇ^{0.06175 (V - EK - 594.31)}}{1 + ⅇ^{(- 0.5143) ((V - EK) + 4.753)}} IK1 = \frac{0.5 \sqrt{\frac{K_o}{5.4}} ak1}{ak1 + bk1} (V - EK)

Component: IKr

gkr = gkr_const \sqrt{\frac{K_o}{5.4}} r = \frac{1}{1 + ⅇ^{\frac{V + 10}{15.4}}} IKr = gkr xr r (V - EK) xrss = \frac{1}{1 + ⅇ^{\frac{- (V + 10.085)}{4.25}}} tauxr = \frac{1}{\frac{6 e -4 (V - 1.7384)}{1 - (ⅇ^{(- 0.136) (V - 1.7384)})} + \frac{3 e -4 (V + 38.3608)}{ⅇ^{0.1522 (V + 38.3608)} - 1}} \frac{d}{d time} (xr) = \frac{xrss - xr}{tauxr}

Component: IKs

gks = 0.0248975 (1 + \frac{0.6}{1 + {(\frac{3.8 e -5}{Ca_i})}^{1.4}}) xss = \frac{1}{1 + ⅇ^{\frac{- (V - 10.5)}{24.7}}} tauxs = \frac{1}{\frac{7.61 e -5 (V + 44.6)}{1 - (ⅇ^{(- 9.97) (V + 44.6)})} + \frac{3.6 e -4 (V - 0.55)}{ⅇ^{0.128 (V - 0.55)} - 1}} IKs = gks xs1 xs2 (V - EKs) \frac{d}{d time} (xs1) = \frac{xss - xs1}{tauxs} \frac{d}{d time} (xs2) = \frac{\frac{xss - xs2}{tauxs}}{2}

Component: Ito

rv = ⅇ^{\frac{V}{300}} ay = \frac{25 ⅇ^{\frac{V - 40}{25}}}{1 + ⅇ^{\frac{V - 40}{25}}} by = \frac{25 ⅇ^{\frac{- (V + 90)}{25}}}{1 + ⅇ^{\frac{- (V + 90)}{25}}} ay2 = \frac{0.03}{1 + ⅇ^{\frac{V + 60}{5}}} by2 = \frac{0.2 ⅇ^{\frac{V + 25}{5}}}{1 + ⅇ^{\frac{V + 25}{5}}} ay3 = \frac{0.00225}{1 + ⅇ^{\frac{V + 60}{5}}} by3 = \frac{0.1 ⅇ^{\frac{V + 25}{5}}}{1 + ⅇ^{\frac{V + 25}{5}}} Ito = gitodv {ydv}^{3} ydv2 zdv rv (V - EK) \frac{d}{d time} (ydv) = ay (1 - ydv) - (by ydv) \frac{d}{d time} (ydv2) = ay2 (1 - ydv2) - (by2 ydv2) \frac{d}{d time} (zdv) = ay3 (1 - zdv) - (by3 zdv)

Component: INaK

sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1}{7} fnak = \frac{1}{1 + 0.1245 ⅇ^{(- 0.1) V FonRT} + 0.0365 sigma ⅇ^{(- V) FonRT}} INaK = \frac{\frac{ibarnak fnak 1}{1 + {(\frac{kmnai}{Na_i})}^{2}} K_o}{K_o + kmko}

Component: INaCa

ca_i_NaCa = 1.5 Ca_i allo = \frac{1}{1 + {(\frac{KmCa}{ca_i_NaCa})}^{2}} num = {Na_i}^{3} Ca_o ⅇ^{eta V FonRT} - ({Na_o}^{3} ca_i_NaCa ⅇ^{(eta - 1) V FonRT}) denom1 = 1 + ksat ⅇ^{(eta - 1) V FonRT} denom2 = KmCao {Na_i}^{3} + {KmNao}^{3} ca_i_NaCa + {KmNai}^{3} Ca_o (1 + \frac{ca_i_NaCa}{KmCai}) denom3 = KmCai {Na_o}^{3} (1 + {(\frac{Na_i}{KmNai})}^{3}) + {Na_i}^{3} Ca_o + {Na_o}^{3} ca_i_NaCa INaCa = \frac{NCXmax allo num}{denom1 (denom2 + denom3)}

Component: IKp

IKp = \frac{0.00276 (V - EK)}{1 + ⅇ^{\frac{7.488 - V}{5.98}}}

Component: IpCa

IpCa = \frac{ibarpca Ca_i}{kmpca + Ca_i}

Component: ICab

ICab = \frac{1.995084 e -7 4 V F FonRT (Ca_i ⅇ^{2 V FonRT} - (0.341 Ca_o))}{ⅇ^{2 V FonRT} - 1}

Component: Ito2

Ito2_max = \frac{PCl V F FonRT (Cl_i - (Cl_o ⅇ^{V FonRT}))}{1 - (ⅇ^{V FonRT})} AAss = \frac{1}{1 + \frac{Kmto2}{Ca_ss}} Ito2 = Ito2_max AA \frac{d}{d time} (AA) = \frac{AAss - AA}{1}

Component: IClb

IClb = GClb (V - ECl)

Component: INal

INal = GNaL {mL}^{3} hL (V - ENa) amL = \frac{0.32 1 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} bmL = 0.08 ⅇ^{\frac{- V}{11}} hLss = \frac{1}{1 + ⅇ^{\frac{V + 91}{6.1}}} \frac{d}{d time} (mL) = amL (1 - mL) - (bmL mL) \frac{d}{d time} (hL) = \frac{hLss - hL}{600}

Component: reversal_potentials

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

Component: Irel

CaMKbound = \frac{CaMK0 (1 - CaMKtrap)}{1 + \frac{Km}{Ca_ss}} CaMKactive = CaMKbound + CaMKtrap Grel = 3000 vg vg = \{\begin{matrix} \frac{1}{1 + ⅇ^{\frac{ibarca + 13}{5}}} if tissue = 0 \\ 1 otherwise \end{matrix} cafac = \frac{1}{1 + ⅇ^{\frac{ICaL + 0.05}{0.015}}} dro_inf = \frac{{Ca_jsr}^{1.9}}{{Ca_jsr}^{1.9} + {(\frac{49.28 Ca_ss}{Ca_ss + 0.0028})}^{1.9}} dtau_rel = \frac{dtau_rel_max CaMKactive}{KmCaMK + CaMKactive} ross = \frac{dro_inf}{{(\frac{1}{ICaL})}^{2} + 1} riss = \frac{1}{1 + ⅇ^{\frac{(Ca_ss - 4 e -4) + 0.002 cafac}{2.5 e -5}}} tauri = 3 + dtau_rel + \frac{350 - dtau_rel}{1 + ⅇ^{\frac{(Ca_ss - 0.003) + 0.003 cafac}{2 e -4}}} irelcicr = Grel ro ri (Ca_jsr - Ca_ss) \frac{d}{d time} (ro) = \frac{ross - ro}{3} \frac{d}{d time} (ri) = \frac{riss - ri}{tauri} \frac{d}{d time} (CaMKtrap) = 0.05 CaMKactive (CaMKactive - CaMKtrap) - (6.8 e -4 CaMKtrap)

Component: Iup_Ileak

dKmPLB = \frac{dKmPLBmax CaMKactive}{KmCaMK + CaMKactive} dJup = \frac{dJupmax CaMKactive}{KmCaMK + CaMKactive} iup = \frac{(dJup + 1) iupmax Ca_i}{Ca_i + Kmup - dKmPLB} ileak = \frac{iupmax Ca_nsr}{nsrmax}

Component: Idiff_Itr

idiff = \frac{Ca_ss - Ca_i}{0.2} itr = \frac{Ca_nsr - Ca_jsr}{120}

Component: Na

CTNaCl = \frac{CTNaClmax {(ENa - ECl)}^{4}}{{(ENa - ECl)}^{4} + {87.8251}^{4}} \frac{d}{d time} (Na_i) = \frac{(- naiont) AF}{vmyo} + CTNaCl

Component: K

CTKCl = \frac{CTKClmax (EK - ECl)}{(EK - ECl) + 87.8251} \frac{d}{d time} (K_i) = \frac{(- kiont) AF}{vmyo} + CTKCl

Component: Cl

\frac{d}{d time} (Cl_i) = \frac{clont AF}{vmyo} + CTNaCl + CTKCl

Component: Ca

\frac{d}{d time} (Ca_i) = bmyo (\frac{(- (ICab + IpCa - (2 INaCa))) AF}{vmyo 2} + \frac{(ileak - iup) vnsr}{vmyo} + \frac{idiff vss}{vmyo}) \frac{d}{d time} (Ca_ss) = bss (\frac{(- ICaL) AF}{vss 2} + \frac{irelcicr vjsr}{vss} - idiff) \frac{d}{d time} (Ca_nsr) = iup - (\frac{itr vjsr}{vnsr}) - ileak \frac{d}{d time} (Ca_jsr) = bcsqn (itr - irelcicr) bcsqn = \frac{1}{1 + \frac{kmcsqn csqnbar}{{(Ca_jsr + kmcsqn)}^{2}}} bmyo = \frac{1}{1 + \frac{cbar kmc}{{(Ca_i + kmc)}^{2}} + \frac{kmt tbar}{{(Ca_i + kmt)}^{2}}} bss = \frac{1}{1 + \frac{BSRmax KmBSR}{{(KmBSR + Ca_ss)}^{2}} + \frac{BSLmax KmBSL}{{(KmBSL + Ca_ss)}^{2}}}

Model Mathematics

Component: Environment

Component: cell

Component: INa

Component: ICaL

Component: IK1

Component: IKr

Component: IKs

Component: Ito

Component: INaK

Component: INaCa

Component: IKp

Component: IpCa

Component: ICab

Component: Ito2

Component: IClb

Component: INal

Component: reversal_potentials

Component: Irel

Component: Iup_Ileak

Component: Idiff_Itr

Component: Na

Component: K

Component: Cl

Component: Ca

Component: intracellular_currents

Component: transmembrane_currents

Component: Ions_n_reversal_potentials