Cellular model#
ORdmm_Land_em_coupling#
This document is automatically generated using gotran
Model is based on the work from [OHaraViragVarroR11], [LPHS+17]
Parameters#
Name |
Value |
|---|---|
\(scale_{\mathcal{I}}\) |
\(1.02\) |
\(scale_{IK1}\) |
\(1.41\) |
\(scale_{IKr}\) |
\(1.12\) |
\(scale_{IKs}\) |
\(1.65\) |
\(scale_{INaL}\) |
\(2.27\) |
\(celltype\) |
\(0\) |
\(cao\) |
\(1.8\) |
\(ko\) |
\(5.4\) |
\(nao\) |
\(1.4\!\times\!10 ^{2}\) |
\(F\) |
\(9.65\!\times\!10 ^{4}\) |
\(R\) |
\(8.31\!\times\!10 ^{3}\) |
\(T\) |
\(3.1\!\times\!10 ^{2}\) |
\(L\) |
\(1\!\times\!10 ^{-2}\) |
\(rad\) |
\(110\!\times\!10 ^{-5}\) |
\(Ahf\) |
\(0.99\) |
\(GNa\) |
\(3.1\!\times\!10 ^{1}\) |
\(thL\) |
\(2\!\times\!10 ^{2}\) |
\(Gto\) |
\(200\!\times\!10 ^{-4}\) |
\(\delta_{epi}\) |
\(1\) |
\(Aff\) |
\(0.6\) |
\(Kmn\) |
\(200\!\times\!10 ^{-5}\) |
\(k2n\) |
\(1\!\times\!10 ^{3}\) |
\(tjca\) |
\(7.5\!\times\!10 ^{1}\) |
\(zca\) |
\(2\) |
\(bt\) |
\(4.75\) |
\(B_{0}\) |
\(2.3\) |
\(B_{1}\) |
\(-2.4\) |
\(Tot_{A}\) |
\(2.5\!\times\!10 ^{1}\) |
\(Tref\) |
\(1.2\!\times\!10 ^{2}\) |
\(Trpn_{50}\) |
\(0.35\) |
\(calib\) |
\(1\) |
\(cat_{50 ref}\) |
\(0.81\) |
\(dLambda\) |
\(0\) |
\(emcoupling\) |
\(1\) |
\(etal\) |
\(2\!\times\!10 ^{2}\) |
\(etas\) |
\(2\!\times\!10 ^{1}\) |
\(gammas\) |
\(850\!\times\!10 ^{-5}\) |
\(gammaw\) |
\(0.61\) |
\(isacs\) |
\(0\) |
\(ktrpn\) |
\(1\!\times\!10 ^{-1}\) |
\(ku\) |
\(400\!\times\!10 ^{-4}\) |
\(kuw\) |
\(0.18\) |
\(kws\) |
\(120\!\times\!10 ^{-4}\) |
\(lmbda\) |
\(1\) |
\(mode\) |
\(1\) |
\(ntm\) |
\(2.4\) |
\(ntrpn\) |
\(2\) |
\(p_{a}\) |
\(2.1\) |
\(p_{b}\) |
\(9.1\) |
\(p_{k}\) |
\(7\) |
\(\phi\) |
\(2.23\) |
\(rs\) |
\(0.25\) |
\(rw\) |
\(0.5\) |
\(CaMKo\) |
\(500\!\times\!10 ^{-4}\) |
\(KmCaM\) |
\(150\!\times\!10 ^{-5}\) |
\(KmCaMK\) |
\(0.15\) |
\(aCaMK\) |
\(500\!\times\!10 ^{-4}\) |
\(bCaMK\) |
\(680\!\times\!10 ^{-6}\) |
\(PKNa\) |
\(183\!\times\!10 ^{-4}\) |
\(Gncx\) |
\(800\!\times\!10 ^{-6}\) |
\(KmCaAct\) |
\(150\!\times\!10 ^{-6}\) |
\(kasymm\) |
\(1.25\!\times\!10 ^{1}\) |
\(kcaoff\) |
\(5\!\times\!10 ^{3}\) |
\(kcaon\) |
\(1.5\!\times\!10 ^{6}\) |
\(kna_{1}\) |
\(1.5\!\times\!10 ^{1}\) |
\(kna_{2}\) |
\(5\) |
\(kna_{3}\) |
\(8.81\!\times\!10 ^{1}\) |
\(qca\) |
\(0.17\) |
\(qna\) |
\(0.52\) |
\(wca\) |
\(6\!\times\!10 ^{4}\) |
\(wna\) |
\(6\!\times\!10 ^{4}\) |
\(wnaca\) |
\(5\!\times\!10 ^{3}\) |
\(H\) |
\(1\!\times\!10 ^{-7}\) |
\(Khp\) |
\(170\!\times\!10 ^{-9}\) |
\(Kki\) |
\(0.5\) |
\(Kko\) |
\(0.36\) |
\(Kmgatp\) |
\(170\!\times\!10 ^{-9}\) |
\(Knai_{0}\) |
\(9.07\) |
\(Knao_{0}\) |
\(2.78\!\times\!10 ^{1}\) |
\(Knap\) |
\(2.24\!\times\!10 ^{2}\) |
\(Kxkur\) |
\(2.92\!\times\!10 ^{2}\) |
\(MgADP\) |
\(500\!\times\!10 ^{-4}\) |
\(MgATP\) |
\(9.8\) |
\(Pnak\) |
\(3\!\times\!10 ^{1}\) |
\(\delta\) |
\(-0.15\) |
\(eP\) |
\(4.2\) |
\(k1m\) |
\(1.82\!\times\!10 ^{2}\) |
\(k1p\) |
\(9.49\!\times\!10 ^{2}\) |
\(k2m\) |
\(3.94\!\times\!10 ^{1}\) |
\(k2p\) |
\(6.87\!\times\!10 ^{2}\) |
\(k3m\) |
\(7.93\!\times\!10 ^{4}\) |
\(k3p\) |
\(1.9\!\times\!10 ^{3}\) |
\(k4m\) |
\(4\!\times\!10 ^{1}\) |
\(k4p\) |
\(6.39\!\times\!10 ^{2}\) |
\(zk\) |
\(1\) |
\(GKb\) |
\(300\!\times\!10 ^{-5}\) |
\(PNab\) |
\(375\!\times\!10 ^{-12}\) |
\(PCab\) |
\(250\!\times\!10 ^{-10}\) |
\(GpCa\) |
\(500\!\times\!10 ^{-6}\) |
\(Esac_{ns}\) |
\(-1\!\times\!10 ^{1}\) |
\(Gsac_{k}\) |
\(1.1\) |
\(Gsac_{ns}\) |
\(600\!\times\!10 ^{-5}\) |
\(\lambda_{max}\) |
\(1.1\) |
\(amp\) |
\(-8\!\times\!10 ^{1}\) |
\(duration\) |
\(0.5\) |
\(BSLmax\) |
\(1.12\) |
\(BSRmax\) |
\(470\!\times\!10 ^{-4}\) |
\(KmBSL\) |
\(870\!\times\!10 ^{-5}\) |
\(KmBSR\) |
\(870\!\times\!10 ^{-6}\) |
\(cmdnmax\) |
\(500\!\times\!10 ^{-4}\) |
\(csqnmax\) |
\(1\!\times\!10 ^{1}\) |
\(kmcmdn\) |
\(238\!\times\!10 ^{-5}\) |
\(kmcsqn\) |
\(0.8\) |
\(kmtrpn\) |
\(500\!\times\!10 ^{-6}\) |
\(trpnmax\) |
\(700\!\times\!10 ^{-4}\) |
States#
Name |
Value |
|---|---|
\(CaMKt\) |
\(0\) |
\(hf\) |
\(1\) |
\(hs\) |
\(1\) |
\(hsp\) |
\(1\) |
\(j\) |
\(1\) |
\(jp\) |
\(1\) |
\(m\) |
\(0\) |
\(hL\) |
\(1\) |
\(hLp\) |
\(1\) |
\(mL\) |
\(0\) |
\(a\) |
\(0\) |
\(ap\) |
\(0\) |
\(iF\) |
\(1\) |
\(iFp\) |
\(1\) |
\(iS\) |
\(1\) |
\(iSp\) |
\(1\) |
\(d\) |
\(0\) |
\(fcaf\) |
\(1\) |
\(fcafp\) |
\(1\) |
\(fcas\) |
\(1\) |
\(\operatorname{FallingFactorial}\) |
\(1\) |
\(ffp\) |
\(1\) |
\(fs\) |
\(1\) |
\(jca\) |
\(1\) |
\(nca\) |
\(0\) |
\(xrf\) |
\(0\) |
\(xrs\) |
\(0\) |
\(xk_{1}\) |
\(1\) |
\(xs_{1}\) |
\(0\) |
\(xs_{2}\) |
\(0\) |
\(v\) |
\(-8.7\!\times\!10 ^{1}\) |
\(Jrelnp\) |
\(0\) |
\(Jrelp\) |
\(0\) |
\(cajsr\) |
\(1.2\) |
\(cansr\) |
\(1.2\) |
\(cass\) |
\(1\!\times\!10 ^{-4}\) |
\(ki\) |
\(1.45\!\times\!10 ^{2}\) |
\(kss\) |
\(1.45\!\times\!10 ^{2}\) |
\(nai\) |
\(7\) |
\(nass\) |
\(7\) |
\(CaTrpn\) |
\(0\) |
\(Cd\) |
\(0\) |
\(TmB\) |
\(1\) |
\(XS\) |
\(0\) |
\(XW\) |
\(0\) |
\(Zetas\) |
\(0\) |
\(Zetaw\) |
\(0\) |
\(cai\) |
\(1\!\times\!10 ^{-4}\) |
State expressions#
\[
\frac{dCaMKt}{dt} = - bCaMK CaMKt + aCaMK \left(CaMKb + CaMKt\right) CaMKb
\]
\[
\frac{dm}{dt} = \frac{1}{tm} \left(- m + mss\right)
\]
\[
\frac{dhf}{dt} = \frac{1}{thf} \left(- hf + hss\right)
\]
\[
\frac{dhs}{dt} = \frac{1}{ths} \left(- hs + hss\right)
\]
\[
\frac{dj}{dt} = \frac{1}{tj} \left(- j + jss\right)
\]
\[
\frac{dhsp}{dt} = \frac{1}{thsp} \left(- hsp + hssp\right)
\]
\[
\frac{djp}{dt} = \frac{1}{tjp} \left(- jp + jss\right)
\]
\[
\frac{dmL}{dt} = \frac{1}{tmL} \left(- mL + mLss\right)
\]
\[
\frac{dhL}{dt} = \frac{1}{thL} \left(- hL + hLss\right)
\]
\[
\frac{dhLp}{dt} = \frac{1}{thLp} \left(- hLp + hLssp\right)
\]
\[
\frac{da}{dt} = \frac{1}{ta} \left(- a + ass\right)
\]
\[
\frac{diF}{dt} = \frac{1}{tiF} \left(- iF + iss\right)
\]
\[
\frac{diS}{dt} = \frac{1}{tiS} \left(- iS + iss\right)
\]
\[
\frac{dap}{dt} = \frac{1}{ta} \left(- ap + assp\right)
\]
\[
\frac{diFp}{dt} = \frac{1}{tiFp} \left(- iFp + iss\right)
\]
\[
\frac{diSp}{dt} = \frac{1}{tiSp} \left(- iSp + iss\right)
\]
\[
\frac{dd}{dt} = \frac{1}{td} \left(- d + dss\right)
\]
\[
\frac{dff}{dt} = \frac{1}{tff} \left(- ff + fss\right)
\]
\[
\frac{dfs}{dt} = \frac{1}{tfs} \left(- fs + fss\right)
\]
\[
\frac{dfcaf}{dt} = \frac{1}{tfcaf} \left(- fcaf + fcass\right)
\]
\[
\frac{dfcas}{dt} = \frac{1}{tfcas} \left(- fcas + fcass\right)
\]
\[
\frac{djca}{dt} = \frac{1}{tjca} \left(- jca + fcass\right)
\]
\[
\frac{dffp}{dt} = \frac{1}{tffp} \left(- ffp + fss\right)
\]
\[
\frac{dfcafp}{dt} = \frac{1}{tfcafp} \left(- fcafp + fcass\right)
\]
\[
\frac{dnca}{dt} = k2n anca - km2n nca
\]
\[
\frac{dxrf}{dt} = \frac{1}{txrf} \left(- xrf + xrss\right)
\]
\[
\frac{dxrs}{dt} = \frac{1}{txrs} \left(- xrs + xrss\right)
\]
\[
\frac{dxs_{1}}{dt} = \frac{1}{txs_{1}} \left(- xs_{1} + xs1ss\right)
\]
\[
\frac{dxs_{2}}{dt} = \frac{1}{txs_{2}} \left(- xs_{2} + xs2ss\right)
\]
\[
\frac{dxk_{1}}{dt} = \frac{1}{txk_{1}} \left(- xk_{1} + xk1ss\right)
\]
\[
\frac{dv}{dt} = - Isac_{P k} - Isac_{P ns} - ICaK - \mathcal{I} - ICaNa - ICab - IK_{1} - IKb - IKr - IKs - INa - INaCa_{i} - INaCa_{ss} - INaK - INaL - INab - IpCa - Istim - Ito
\]
\[
\frac{dJrelnp}{dt} = \frac{1}{\tau_{rel}} \left(- Jrelnp + Jrel_{inf}\right)
\]
\[
\frac{dJrelp}{dt} = \frac{1}{\tau_{relp}} \left(- Jrelp + Jrel_{infp}\right)
\]
\[
\frac{dnai}{dt} = \frac{JdiffNa vss}{vmyo} + \frac{Acap}{F vmyo} \left(- INa - INaL - INab - \frac{Isac_{P ns}}{3} - 3 INaCa_{i} - 3 INaK\right)
\]
\[
\frac{dnass}{dt} = - JdiffNa + \frac{Acap}{F vss} \left(- ICaNa - 3 INaCa_{ss}\right)
\]
\[
\frac{dki}{dt} = \frac{JdiffK vss}{vmyo} + \frac{Acap}{F vmyo} \left(- Isac_{P k} - IK_{1} - IKb - IKr - IKs - Istim - Ito - \frac{Isac_{P ns}}{3} + 2 INaK\right)
\]
\[
\frac{dkss}{dt} = - JdiffK - \frac{Acap ICaK}{F vss}
\]
\[
\frac{dcass}{dt} = \left(- Jdiff + \frac{Jrel vjsr}{vss} + \frac{0.5 Acap}{F vss} \left(- \mathcal{I} + 2 INaCa_{ss}\right)\right) Bcass
\]
\[
\frac{dcansr}{dt} = - \frac{Jtr vjsr}{vnsr} + Jup
\]
\[
\frac{dcajsr}{dt} = \left(- Jrel + Jtr\right) Bcajsr
\]
\[
\frac{dXS}{dt} = kws XW - XS gammasu - XS ksu
\]
\[
\frac{dXW}{dt} = kuw XU - kws XW - XW gammawu - XW kwu
\]
\[
\frac{dCaTrpn}{dt} = ktrpn \left(- CaTrpn + \left(\frac{1\!\times\!10 ^{3}}{cat_{50}} cai\right)^{ntrpn} \left(1 - CaTrpn\right)\right)
\]
\[\begin{split}
\frac{dTmB}{dt} = \left(\begin{cases} CaTrpn^{- \frac{ntm}{2}} & \text{for}\: CaTrpn^{- \frac{ntm}{2}} < 1\!\times\!10 ^{2} \\1\!\times\!10 ^{2} & \text{otherwise} \end{cases}\right) XU kb - ku CaTrpn^{\frac{ntm}{2}} TmB
\end{split}\]
\[
\frac{dZetas}{dt} = dLambda As - Zetas cs
\]
\[
\frac{dZetaw}{dt} = dLambda Aw - Zetaw cw
\]
\[
\frac{dCd}{dt} = \frac{p_{k}}{\eta} \left(- Cd + C\right)
\]
\[
\frac{dcai}{dt} = \left(- J_{TRPN} + \frac{Jdiff vss}{vmyo} - \frac{Jup vnsr}{vmyo} + \frac{0.5 Acap}{F vmyo} \left(- ICab - IpCa - \frac{Isac_{P ns}}{3} + 2 INaCa_{i}\right)\right) Bcai
\]
Expressions#
\[
vcell = 3.14\!\times\!10 ^{3} L rad^{2}
\]
\[
Ageo = 6.28 rad^{2} + 6.28 L rad
\]
\[
Acap = 2 Ageo
\]
\[
vmyo = 0.68 vcell
\]
\[
vnsr = 552\!\times\!10 ^{-4} vcell
\]
\[
vjsr = 480\!\times\!10 ^{-5} vcell
\]
\[
vss = 200\!\times\!10 ^{-4} vcell
\]
\[
CaMKb = \frac{CaMKo \left(1 - CaMKt\right)}{1 + \frac{KmCaM}{cass}}
\]
\[
CaMKa = CaMKb + CaMKt
\]
\[
mss = \frac{1}{1 + 146\!\times\!10 ^{-5} e^{- 0.13 v}}
\]
\[
tm = \frac{1}{9.45 e^{288\!\times\!10 ^{-4} v} + 193\!\times\!10 ^{-7} e^{- 0.17 v}}
\]
\[
hss = \frac{1}{1 + 3.03\!\times\!10 ^{5} e^{0.16 v}}
\]
\[
thf = \frac{1}{118\!\times\!10 ^{-7} e^{- 0.16 v} + 6.31 e^{493\!\times\!10 ^{-4} v}}
\]
\[
ths = \frac{1}{516\!\times\!10 ^{-5} e^{- 357\!\times\!10 ^{-4} v} + 0.37 e^{176\!\times\!10 ^{-4} v}}
\]
\[
Ahs = 1 - Ahf
\]
\[
h = Ahf hf + Ahs hs
\]
\[
jss = hss
\]
\[
tj = 2.04 + \frac{1}{0.31 e^{260\!\times\!10 ^{-4} v} + 113\!\times\!10 ^{-9} e^{- 0.12 v}}
\]
\[
hssp = \frac{1}{1 + 8.2\!\times\!10 ^{5} e^{0.16 v}}
\]
\[
thsp = 3 ths
\]
\[
hp = Ahf hf + Ahs hsp
\]
\[
tjp = 1.46 tj
\]
\[
fINap = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
INa = GNa m^{3} \left(- ENa + v\right) \left(\left(1 - fINap\right) h j + fINap hp jp\right)
\]
\[
mLss = \frac{1}{1 + 292\!\times\!10 ^{-6} e^{- 0.19 v}}
\]
\[
tmL = tm
\]
\[
hLss = \frac{1}{1 + 1.21\!\times\!10 ^{5} e^{0.13 v}}
\]
\[
hLssp = \frac{1}{1 + 2.76\!\times\!10 ^{5} e^{0.13 v}}
\]
\[
thLp = 3 thL
\]
\[
GNaL = 750\!\times\!10 ^{-5} scale_{INaL}
\]
\[
fINaLp = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
INaL = \left(- ENa + v\right) \left(\left(1 - fINaLp\right) hL + fINaLp hLp\right) GNaL mL
\]
\[
ass = \frac{1}{1 + 2.63 e^{- 675\!\times\!10 ^{-4} v}}
\]
\[
ta = \frac{1.05}{\frac{1}{1.21 + 2.26 e^{- 340\!\times\!10 ^{-4} v}} + \frac{3.5}{1 + 3.01\!\times\!10 ^{1} e^{340\!\times\!10 ^{-4} v}}}
\]
\[
iss = \frac{1}{1 + 2.19\!\times\!10 ^{3} e^{0.17 v}}
\]
\[
tiF = 4.56 + \frac{\delta_{epi}}{0.14 e^{- 1\!\times\!10 ^{-2} v} + 1.63 e^{603\!\times\!10 ^{-4} v}}
\]
\[
tiS = 2.36\!\times\!10 ^{1} + \frac{\delta_{epi}}{276\!\times\!10 ^{-6} e^{- 169\!\times\!10 ^{-4} v} + 242\!\times\!10 ^{-4} e^{0.12 v}}
\]
\[
AiF = \frac{1}{1 + 0.24 e^{661\!\times\!10 ^{-5} v}}
\]
\[
AiS = 1 - AiF
\]
\[
i = AiF iF + AiS iS
\]
\[
assp = \frac{1}{1 + 5.17 e^{- 675\!\times\!10 ^{-4} v}}
\]
\[
dti_{develop} = 1.35 + \frac{1\!\times\!10 ^{-4}}{266\!\times\!10 ^{-7} e^{629\!\times\!10 ^{-4} v} + 4.55\!\times\!10 ^{24} e^{- 4.64 v}}
\]
\[
dti_{recover} = 1 - \frac{0.5}{1 + 3.31\!\times\!10 ^{1} e^{500\!\times\!10 ^{-4} v}}
\]
\[
tiFp = dti_{develop} dti_{recover} tiF
\]
\[
tiSp = dti_{develop} dti_{recover} tiS
\]
\[
ip = AiF iFp + AiS iSp
\]
\[
fItop = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
Ito = Gto \left(- EK + v\right) \left(\left(1 - fItop\right) a i + ap fItop ip\right)
\]
\[
dss = \frac{1}{1 + 0.39 e^{- 0.24 v}}
\]
\[
td = 0.6 + \frac{1}{3.53 e^{900\!\times\!10 ^{-4} v} + 0.74 e^{- 500\!\times\!10 ^{-4} v}}
\]
\[
fss = \frac{1}{1 + 2\!\times\!10 ^{2} e^{0.27 v}}
\]
\[
tff = 7 + \frac{1}{333\!\times\!10 ^{-4} e^{1\!\times\!10 ^{-1} v} + 609\!\times\!10 ^{-6} e^{- 1\!\times\!10 ^{-1} v}}
\]
\[
tfs = 1\!\times\!10 ^{3} + \frac{1}{100\!\times\!10 ^{-7} e^{- 0.25 v} + 805\!\times\!10 ^{-7} e^{0.17 v}}
\]
\[
Afs = 1 - Aff
\]
\[
f = Aff ff + Afs fs
\]
\[
fcass = fss
\]
\[
tfcaf = 7 + \frac{1}{708\!\times\!10 ^{-4} e^{- 0.14 v} + 226\!\times\!10 ^{-4} e^{0.14 v}}
\]
\[
tfcas = 1\!\times\!10 ^{2} + \frac{1}{120\!\times\!10 ^{-6} e^{0.14 v} + 120\!\times\!10 ^{-6} e^{- 0.33 v}}
\]
\[
Afcaf = 0.3 + \frac{0.6}{1 + 0.37 e^{1\!\times\!10 ^{-1} v}}
\]
\[
Afcas = 1 - Afcaf
\]
\[
fca = Afcaf fcaf + Afcas fcas
\]
\[
tffp = 2.5 tff
\]
\[
fp = Aff ffp + Afs fs
\]
\[
tfcafp = 2.5 tfcaf
\]
\[
fcap = Afcaf fcafp + Afcas fcas
\]
\[
km2n = 1 jca
\]
\[
anca = \frac{1}{\left(1 + \frac{Kmn}{cass}\right)^{4} + \frac{k2n}{km2n}}
\]
\[
\mathcal{\Phi} = \frac{4 vffrt}{-1 + e^{2 vfrt}} \left(- 0.34 cao + cass e^{2 vfrt}\right)
\]
\[
PhiCaNa = \frac{1 vffrt}{-1 + e^{1 vfrt}} \left(- 0.75 nao + 0.75 e^{1 vfrt} nass\right)
\]
\[
PhiCaK = \frac{1 vffrt}{-1 + e^{1 vfrt}} \left(- 0.75 ko + 0.75 e^{1 vfrt} kss\right)
\]
\[
PCa = 1\!\times\!10 ^{-4} scale_{\mathcal{I}}
\]
\[
PCap = 1.1 PCa
\]
\[
PCaNa = 125\!\times\!10 ^{-5} PCa
\]
\[
PCaK = 357\!\times\!10 ^{-6} PCa
\]
\[
PCaNap = 125\!\times\!10 ^{-5} PCap
\]
\[
PCaKp = 357\!\times\!10 ^{-6} PCap
\]
\[
fICaLp = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
\mathcal{I} = \left(1 - fICaLp\right) \left(\left(1 - nca\right) f + fca jca nca\right) PCa \mathcal{\Phi} d + \left(\left(1 - nca\right) fp + fcap jca nca\right) PCap \mathcal{\Phi} d fICaLp
\]
\[
ICaNa = \left(1 - fICaLp\right) \left(\left(1 - nca\right) f + fca jca nca\right) PCaNa PhiCaNa d + \left(\left(1 - nca\right) fp + fcap jca nca\right) PCaNap PhiCaNa d fICaLp
\]
\[
ICaK = \left(1 - fICaLp\right) \left(\left(1 - nca\right) f + fca jca nca\right) PCaK PhiCaK d + \left(\left(1 - nca\right) fp + fcap jca nca\right) PCaKp PhiCaK d fICaLp
\]
\[
xrss = \frac{1}{1 + 0.29 e^{- 0.15 v}}
\]
\[
txrf = 1.3\!\times\!10 ^{1} + \frac{1}{102\!\times\!10 ^{-6} e^{0.26 v} + 430\!\times\!10 ^{-6} e^{- 491\!\times\!10 ^{-4} v}}
\]
\[
txrs = 1.86 + \frac{1}{592\!\times\!10 ^{-6} e^{0.14 v} + 355\!\times\!10 ^{-7} e^{- 386\!\times\!10 ^{-4} v}}
\]
\[
Axrf = \frac{1}{1 + 4.2 e^{262\!\times\!10 ^{-4} v}}
\]
\[
Axrs = 1 - Axrf
\]
\[
xr = Axrf xrf + Axrs xrs
\]
\[
rkr = \frac{1}{\left(1 + 2.08 e^{133\!\times\!10 ^{-4} v}\right) \left(1 + 0.72 e^{333\!\times\!10 ^{-4} v}\right)}
\]
\[
GKr = 460\!\times\!10 ^{-4} scale_{IKr}
\]
\[
IKr = 0.43 \sqrt{ko} \left(- EK + v\right) GKr rkr xr
\]
\[
xs1ss = \frac{1}{1 + 0.27 e^{- 0.11 v}}
\]
\[
txs_{1} = 8.17\!\times\!10 ^{2} + \frac{1}{350\!\times\!10 ^{-5} e^{562\!\times\!10 ^{-4} v} + 518\!\times\!10 ^{-6} e^{- 435\!\times\!10 ^{-5} v}}
\]
\[
xs2ss = xs1ss
\]
\[
txs_{2} = \frac{1}{226\!\times\!10 ^{-5} e^{- 323\!\times\!10 ^{-4} v} + 821\!\times\!10 ^{-6} e^{500\!\times\!10 ^{-4} v}}
\]
\[
KsCa = 1 + \frac{0.6}{1 + 648\!\times\!10 ^{-9} \left(\frac{1}{cai}\right)^{1.4}}
\]
\[
GKs = 340\!\times\!10 ^{-5} scale_{IKs}
\]
\[
IKs = \left(- EKs + v\right) GKs KsCa xs_{1} xs_{2}
\]
\[
xk1ss = \frac{1}{1 + e^{\frac{-1.45\!\times\!10 ^{2} - v - 2.55 ko}{3.81 + 1.57 ko}}}
\]
\[
txk_{1} = \frac{1.22\!\times\!10 ^{2}}{194\!\times\!10 ^{-5} e^{- 491\!\times\!10 ^{-4} v} + 3.04\!\times\!10 ^{1} e^{144\!\times\!10 ^{-4} v}}
\]
\[
rk_{1} = \frac{1}{1 + 6.92\!\times\!10 ^{4} e^{0.10 v - 0.27 ko}}
\]
\[
GK_{1} = 0.19 scale_{IK1}
\]
\[
IK_{1} = \sqrt{ko} \left(- EK + v\right) GK_{1} rk_{1} xk_{1}
\]
\[
a_{rel} = 0.5 bt
\]
\[
Jrel_{inf} = - \frac{\mathcal{I} a_{rel}}{1 + 2.56\!\times\!10 ^{1} \left(\frac{1}{cajsr}\right)^{8}}
\]
\[
\tau_{rel tmp} = \frac{bt}{1 + \frac{123\!\times\!10 ^{-4}}{cajsr}}
\]
\[\begin{split}
\tau_{rel} = \begin{cases} 0.00 & \text{for}\: \tau_{rel tmp} < 0.00 \\\tau_{rel tmp} & \text{otherwise} \end{cases}
\end{split}\]
\[
btp = 1.25 bt
\]
\[
a_{relp} = 0.5 btp
\]
\[
Jrel_{infp} = - \frac{\mathcal{I} a_{relp}}{1 + 2.56\!\times\!10 ^{1} \left(\frac{1}{cajsr}\right)^{8}}
\]
\[
\tau_{relp tmp} = \frac{btp}{1 + \frac{123\!\times\!10 ^{-4}}{cajsr}}
\]
\[\begin{split}
\tau_{relp} = \begin{cases} 0.00 & \text{for}\: \tau_{relp tmp} < 0.00 \\\tau_{relp tmp} & \text{otherwise} \end{cases}
\end{split}\]
\[
fJrelp = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
Jrel = \left(1 - fJrelp\right) Jrelnp + Jrelp fJrelp
\]
\[
Bcass = \frac{1}{1 + \frac{BSLmax KmBSL}{\left(KmBSL + cass\right)^{2}} + \frac{BSRmax KmBSR}{\left(KmBSR + cass\right)^{2}}}
\]
\[
Bcajsr = \frac{1}{1 + \frac{csqnmax kmcsqn}{\left(kmcsqn + cajsr\right)^{2}}}
\]
\[\begin{split}
XS_{max} = \begin{cases} XS & \text{for}\: XS > 0 \\0 & \text{otherwise} \end{cases}
\end{split}\]
\[\begin{split}
XW_{max} = \begin{cases} XW & \text{for}\: XW > 0 \\0 & \text{otherwise} \end{cases}
\end{split}\]
\[\begin{split}
CaTrpn_{max} = \begin{cases} CaTrpn & \text{for}\: CaTrpn > 0 \\0 & \text{otherwise} \end{cases}
\end{split}\]
\[
kwu = - kws + kuw \left(-1 + \frac{1}{rw}\right)
\]
\[
ksu = kws rw \left(-1 + \frac{1}{rs}\right)
\]
\[
Aw = \frac{Tot_{A} rs}{rs + rw \left(1 - rs\right)}
\]
\[
As = Aw
\]
\[
cw = \frac{kuw \phi}{rw} \left(1 - rw\right)
\]
\[
cs = \frac{kws \phi}{rs} rw \left(1 - rs\right)
\]
\[\begin{split}
\lambda_{min12} = \begin{cases} lmbda & \text{for}\: lmbda < 1.2 \\1.2 & \text{otherwise} \end{cases}
\end{split}\]
\[\begin{split}
\lambda_{min087} = \begin{cases} \lambda_{min12} & \text{for}\: \lambda_{min12} < 0.87 \\0.87 & \text{otherwise} \end{cases}
\end{split}\]
\[
h_{\lambda prima} = 1 + B_{0} \left(-1.87 + \lambda_{min087} + \lambda_{min12}\right)
\]
\[\begin{split}
h_{\lambda} = \begin{cases} h_{\lambda prima} & \text{for}\: h_{\lambda prima} > 0 \\0 & \text{otherwise} \end{cases}
\end{split}\]
\[
XU = 1 - TmB - XS - XW
\]
\[
gammawu = gammaw \left|{Zetaw}\right|
\]
\[\begin{split}
gammasu = gammas \left(\begin{cases} Zetas \left(Zetas > 0\right) & \text{for}\: Zetas \left(Zetas > 0\right) > \left(-1 - Zetas\right) \left(Zetas < -1\right) \\\left(-1 - Zetas\right) \left(Zetas < -1\right) & \text{otherwise} \end{cases}\right)
\end{split}\]
\[
cat_{50} = cat_{50 ref} + B_{1} \left(-1 + \lambda_{min12}\right)
\]
\[
kb = \frac{ku Trpn_{50}^{ntm}}{1 - rs - rw \left(1 - rs\right)}
\]
\[
Ta = \frac{Tref h_{\lambda}}{rs} \left(\left(1 + Zetas\right) XS + XW Zetaw\right)
\]
\[
C = -1 + \lambda_{min12}
\]
\[
dCd = - Cd + C
\]
\[\begin{split}
\eta = \begin{cases} etas & \text{for}\: dCd < 0 \\etal & \text{otherwise} \end{cases}
\end{split}\]
\[
Fd = dCd \eta
\]
\[
F_{1} = -1 + e^{p_{b} C}
\]
\[
Tp = p_{a} \left(F_{1} + Fd\right)
\]
\[
Ttot = Ta + Tp
\]
\[
Bcai = \frac{1}{1 + \frac{cmdnmax kmcmdn}{\left(kmcmdn + cai\right)^{2}}}
\]
\[
J_{TRPN} = trpnmax \frac{d}{d t} CaTrpn
\]
\[
ENa = \frac{R T}{F} \log{\left (\frac{nao}{nai} \right )}
\]
\[
EK = \frac{R T}{F} \log{\left (\frac{ko}{ki} \right )}
\]
\[
EKs = \frac{R T}{F} \log{\left (\frac{ko + PKNa nao}{PKNa nai + ki} \right )}
\]
\[
vffrt = \frac{F^{2} v}{R T}
\]
\[
vfrt = \frac{F v}{R T}
\]
\[
hca = e^{\frac{F qca v}{R T}}
\]
\[
hna = e^{\frac{F qna v}{R T}}
\]
\[
h_{1 i} = 1 + \frac{nai}{kna_{3}} \left(1 + hna\right)
\]
\[
h_{2 i} = \frac{hna nai}{kna_{3} h_{1 i}}
\]
\[
h_{3 i} = \frac{1}{h_{1 i}}
\]
\[
h_{4 i} = 1 + \frac{nai}{kna_{1}} \left(1 + \frac{nai}{kna_{2}}\right)
\]
\[
h_{5 i} = \frac{nai^{2}}{kna_{1} kna_{2} h_{4 i}}
\]
\[
h_{6 i} = \frac{1}{h_{4 i}}
\]
\[
h_{7 i} = 1 + \frac{nao}{kna_{3}} \left(1 + \frac{1}{hna}\right)
\]
\[
h_{8 i} = \frac{nao}{kna_{3} h_{7 i} hna}
\]
\[
h_{9 i} = \frac{1}{h_{7 i}}
\]
\[
h_{10 i} = 1 + kasymm + \frac{nao}{kna_{1}} \left(1 + \frac{nao}{kna_{2}}\right)
\]
\[
h_{11 i} = \frac{nao^{2}}{kna_{1} kna_{2} h_{10 i}}
\]
\[
h_{12 i} = \frac{1}{h_{10 i}}
\]
\[
k_{1 i} = cao kcaon h_{12 i}
\]
\[
k_{2 i} = kcaoff
\]
\[
k3p_{i} = wca h_{9 i}
\]
\[
k3pp_{i} = wnaca h_{8 i}
\]
\[
k_{3 i} = k3p_{i} + k3pp_{i}
\]
\[
k4p_{i} = \frac{wca}{hca} h_{3 i}
\]
\[
k4pp_{i} = wnaca h_{2 i}
\]
\[
k_{4 i} = k4p_{i} + k4pp_{i}
\]
\[
k_{5 i} = kcaoff
\]
\[
k_{6 i} = kcaon cai h_{6 i}
\]
\[
k_{7 i} = wna h_{2 i} h_{5 i}
\]
\[
k_{8 i} = wna h_{11 i} h_{8 i}
\]
\[
x_{1 i} = \left(k_{2 i} + k_{3 i}\right) k_{5 i} k_{7 i} + \left(k_{6 i} + k_{7 i}\right) k_{2 i} k_{4 i}
\]
\[
x_{2 i} = \left(k_{1 i} + k_{8 i}\right) k_{4 i} k_{6 i} + \left(k_{4 i} + k_{5 i}\right) k_{1 i} k_{7 i}
\]
\[
x_{3 i} = \left(k_{2 i} + k_{3 i}\right) k_{6 i} k_{8 i} + \left(k_{6 i} + k_{7 i}\right) k_{1 i} k_{3 i}
\]
\[
x_{4 i} = \left(k_{1 i} + k_{8 i}\right) k_{3 i} k_{5 i} + \left(k_{4 i} + k_{5 i}\right) k_{2 i} k_{8 i}
\]
\[
E_{1 i} = \frac{x_{1 i}}{x_{1 i} + x_{2 i} + x_{3 i} + x_{4 i}}
\]
\[
E_{2 i} = \frac{x_{2 i}}{x_{1 i} + x_{2 i} + x_{3 i} + x_{4 i}}
\]
\[
E_{3 i} = \frac{x_{3 i}}{x_{1 i} + x_{2 i} + x_{3 i} + x_{4 i}}
\]
\[
E_{4 i} = \frac{x_{4 i}}{x_{1 i} + x_{2 i} + x_{3 i} + x_{4 i}}
\]
\[
allo_{i} = \frac{1}{1 + \left(\frac{KmCaAct}{cai}\right)^{2}}
\]
\[
zna = 1
\]
\[
JncxNa_{i} = E_{3 i} k4pp_{i} - E_{2 i} k3pp_{i} + 3 E_{4 i} k_{7 i} - 3 E_{1 i} k_{8 i}
\]
\[
JncxCa_{i} = E_{2 i} k_{2 i} - E_{1 i} k_{1 i}
\]
\[
INaCa_{i} = 0.8 Gncx \left(zca JncxCa_{i} + zna JncxNa_{i}\right) allo_{i}
\]
\[
h_{1} = 1 + \frac{nass}{kna_{3}} \left(1 + hna\right)
\]
\[
h_{2} = \frac{hna nass}{kna_{3} h_{1}}
\]
\[
h_{3} = \frac{1}{h_{1}}
\]
\[
h_{4} = 1 + \frac{nass}{kna_{1}} \left(1 + \frac{nass}{kna_{2}}\right)
\]
\[
h_{5} = \frac{nass^{2}}{kna_{1} kna_{2} h_{4}}
\]
\[
h_{6} = \frac{1}{h_{4}}
\]
\[
h_{7} = 1 + \frac{nao}{kna_{3}} \left(1 + \frac{1}{hna}\right)
\]
\[
h_{8} = \frac{nao}{kna_{3} h_{7} hna}
\]
\[
h_{9} = \frac{1}{h_{7}}
\]
\[
h_{10} = 1 + kasymm + \frac{nao}{kna_{1}} \left(1 + \frac{nao}{kna_{2}}\right)
\]
\[
h_{11} = \frac{nao^{2}}{kna_{1} kna_{2} h_{10}}
\]
\[
h_{12} = \frac{1}{h_{10}}
\]
\[
k_{1} = cao kcaon h_{12}
\]
\[
k_{2} = kcaoff
\]
\[
k3p_{ss} = wca h_{9}
\]
\[
k3pp = wnaca h_{8}
\]
\[
k_{3} = k3p_{ss} + k3pp
\]
\[
k4p_{ss} = \frac{wca h_{3}}{hca}
\]
\[
k4pp = wnaca h_{2}
\]
\[
k_{4} = k4p_{ss} + k4pp
\]
\[
k_{5} = kcaoff
\]
\[
k_{6} = kcaon cass h_{6}
\]
\[
k_{7} = wna h_{2} h_{5}
\]
\[
k_{8} = wna h_{11} h_{8}
\]
\[
x_{1 ss} = \left(k_{2} + k_{3}\right) k_{5} k_{7} + \left(k_{6} + k_{7}\right) k_{2} k_{4}
\]
\[
x_{2 ss} = \left(k_{1} + k_{8}\right) k_{4} k_{6} + \left(k_{4} + k_{5}\right) k_{1} k_{7}
\]
\[
x_{3 ss} = \left(k_{2} + k_{3}\right) k_{6} k_{8} + \left(k_{6} + k_{7}\right) k_{1} k_{3}
\]
\[
x_{4 ss} = \left(k_{1} + k_{8}\right) k_{3} k_{5} + \left(k_{4} + k_{5}\right) k_{2} k_{8}
\]
\[
E_{1 ss} = \frac{x_{1 ss}}{x_{1 ss} + x_{2 ss} + x_{3 ss} + x_{4 ss}}
\]
\[
E_{2 ss} = \frac{x_{2 ss}}{x_{1 ss} + x_{2 ss} + x_{3 ss} + x_{4 ss}}
\]
\[
E_{3 ss} = \frac{x_{3 ss}}{x_{1 ss} + x_{2 ss} + x_{3 ss} + x_{4 ss}}
\]
\[
E_{4 ss} = \frac{x_{4 ss}}{x_{1 ss} + x_{2 ss} + x_{3 ss} + x_{4 ss}}
\]
\[
allo_{ss} = \frac{1}{1 + \left(\frac{KmCaAct}{cass}\right)^{2}}
\]
\[
JncxNa_{ss} = E_{3 ss} k4pp - E_{2 ss} k3pp + 3 E_{4 ss} k_{7} - 3 E_{1 ss} k_{8}
\]
\[
JncxCa_{ss} = E_{2 ss} k_{2} - E_{1 ss} k_{1}
\]
\[
INaCa_{ss} = 0.2 Gncx \left(zca JncxCa_{ss} + zna JncxNa_{ss}\right) allo_{ss}
\]
\[
Knai = Knai_{0} e^{\frac{0.33 F \delta v}{R T}}
\]
\[
Knao = Knao_{0} e^{\frac{0.33 F v}{R T} \left(1 - \delta\right)}
\]
\[
P = \frac{eP}{1 + \frac{H}{Khp} + \frac{nai}{Knap} + \frac{ki}{Kxkur}}
\]
\[
a_{1} = \frac{k1p \left(\frac{nai}{Knai}\right)^{3}}{-1 + \left(1 + \frac{ki}{Kki}\right)^{2} + \left(1 + \frac{nai}{Knai}\right)^{3}}
\]
\[
b_{1} = MgADP k1m
\]
\[
a_{2} = k2p
\]
\[
b_{2} = \frac{k2m \left(\frac{nao}{Knao}\right)^{3}}{-1 + \left(1 + \frac{ko}{Kko}\right)^{2} + \left(1 + \frac{nao}{Knao}\right)^{3}}
\]
\[
a_{3} = \frac{k3p \left(\frac{ko}{Kko}\right)^{2}}{-1 + \left(1 + \frac{ko}{Kko}\right)^{2} + \left(1 + \frac{nao}{Knao}\right)^{3}}
\]
\[
b_{3} = \frac{H k3m P}{1 + \frac{MgATP}{Kmgatp}}
\]
\[
a_{4} = \frac{MgATP k4p}{Kmgatp \left(1 + \frac{MgATP}{Kmgatp}\right)}
\]
\[
b_{4} = \frac{k4m \left(\frac{ki}{Kki}\right)^{2}}{-1 + \left(1 + \frac{ki}{Kki}\right)^{2} + \left(1 + \frac{nai}{Knai}\right)^{3}}
\]
\[
x_{1} = a_{1} a_{2} a_{4} + a_{1} a_{2} b_{3} + a_{2} b_{3} b_{4} + b_{2} b_{3} b_{4}
\]
\[
x_{2} = a_{1} a_{2} a_{3} + a_{2} a_{3} b_{4} + a_{3} b_{1} b_{4} + b_{1} b_{2} b_{4}
\]
\[
x_{3} = a_{2} a_{3} a_{4} + a_{3} a_{4} b_{1} + a_{4} b_{1} b_{2} + b_{1} b_{2} b_{3}
\]
\[
x_{4} = a_{1} a_{3} a_{4} + a_{1} a_{4} b_{2} + a_{1} b_{2} b_{3} + b_{2} b_{3} b_{4}
\]
\[
E1 = \frac{x_{1}}{x_{1} + x_{2} + x_{3} + x_{4}}
\]
\[
E_{2} = \frac{x_{2}}{x_{1} + x_{2} + x_{3} + x_{4}}
\]
\[
E_{3} = \frac{x_{3}}{x_{1} + x_{2} + x_{3} + x_{4}}
\]
\[
E_{4} = \frac{x_{4}}{x_{1} + x_{2} + x_{3} + x_{4}}
\]
\[
JnakNa = 3 E_{1} a_{3} - 3 E_{2} b_{3}
\]
\[
JnakK = 2 E_{4} b_{1} - 2 E_{3} a_{1}
\]
\[
INaK = Pnak \left(zk JnakK + zna JnakNa\right)
\]
\[
xkb = \frac{1}{1 + 2.2 e^{- 545\!\times\!10 ^{-4} v}}
\]
\[
IKb = GKb \left(- EK + v\right) xkb
\]
\[
INab = \frac{PNab vffrt}{-1 + e^{vfrt}} \left(- nao + e^{vfrt} nai\right)
\]
\[
ICab = \frac{4 PCab vffrt}{-1 + e^{2 vfrt}} \left(- 0.34 cao + cai e^{2 vfrt}\right)
\]
\[
IpCa = \frac{GpCa cai}{500\!\times\!10 ^{-6} + cai}
\]
\[
Isac_{P ns} = 0
\]
\[
Isac_{P k} = 0
\]
\[
Istim = amp \left(t \leq duration\right)
\]
\[
JdiffNa = 0.5 nass - 0.5 nai
\]
\[
JdiffK = 0.5 kss - 0.5 ki
\]
\[
Jdiff = 5 cass - 5 cai
\]
\[
Jupnp = \frac{438\!\times\!10 ^{-5} cai}{920\!\times\!10 ^{-6} + cai}
\]
\[
Jupp = \frac{120\!\times\!10 ^{-4} cai}{750\!\times\!10 ^{-6} + cai}
\]
\[
fJupp = \frac{1}{1 + \frac{KmCaMK}{CaMKa}}
\]
\[
Jleak = 262\!\times\!10 ^{-6} cansr
\]
\[
Jup = - Jleak + \left(1 - fJupp\right) Jupnp + Jupp fJupp
\]
\[
Jtr = 1\!\times\!10 ^{-2} cansr - 1\!\times\!10 ^{-2} cajsr
\]
References#
[LPHS+17]
Sander Land, So-Jin Park-Holohan, Nicolas P Smith, Cristobal G Dos Remedios, Jonathan C Kentish, and Steven A Niederer. A model of cardiac contraction based on novel measurements of tension development in human cardiomyocytes. Journal of Molecular and Cellular Cardiology, 106:68–83, 2017.
[OHaraViragVarroR11]
Thomas O'Hara, László Virág, András Varró, and Yoram Rudy. Simulation of the undiseased human cardiac ventricular action potential: model formulation and experimental validation. PLoS computational biology, 7(5):e1002061, 2011.