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Model Mathematics

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Component: acacetin

fKr = 1.0 - (\frac{1.0}{1.0 + {(\frac{Aca_IC50_IKr}{conc})}^{Aca_hill_IKr}}) fKs = 1.0 - (\frac{1.0}{1.0 + {(\frac{Aca_IC50_IKs}{conc})}^{Aca_hill_IKs}}) fto = 1.0 - (\frac{1.0}{1.0 + {(\frac{Aca_IC50_Ito}{conc})}^{Aca_hill_Ito}})

Component: calcium

\frac{d}{d time} (Ca_SRi) = 1000.0 calcium_Ca_SRi_beta (DCaSR (\frac{(Ca_SRss - (2.0 Ca_SRi)) + Ca_SRi}{dx2} + \frac{Ca_SRi - Ca_SRss}{2.0 3.0 dx2}) + \frac{JSRCai}{VSRi}) \frac{d}{d time} (Ca_SRss) = 1000.0 calcium_Ca_SRss_beta (DCaSR (\frac{(Ca_SRss - (2.0 Ca_SRss)) + Ca_SRi}{dx2} + \frac{Ca_SRss - Ca_SRi}{2.0 4.0 dx2}) + \frac{JSRCaSS}{VSRss}) \frac{d}{d time} (Ca_i) = \frac{JCa}{Vnj} calcium_Ca_i_beta \frac{d}{d time} (Ca_ss) = calcium_Ca_ss_beta (\frac{JCass}{Vss} + \frac{I_tot}{2.0 Vss F}) I_tot = (- ICaL - IbCa - ICap) + 2.0 INaCa JCa = (- J_bulkSERCA) + JSRCaleaki + Jreli + Jj_nj JCass = ((- Jj_nj) + JSRCaleakss - J_bulkSERCAss) + Jrelss JSRCaSS = J_SERCASRss - JSRCaleakss - Jrelss JSRCai = J_SERCASR - JSRCaleaki - Jreli JSRCaleaki = 0.5 kSRleak (Ca_SRi - Ca_i) Vnj JSRCaleakss = 0.5 kSRleak (Ca_SRss - Ca_ss) Vss Jj_nj = \frac{2.5 DCa Aj_nj}{xj_nj} (Ca_ss - Ca_i) calcium_Ca_SRi_beta = \frac{1.0}{1.0 + \frac{CSQN KdCSQN}{{(Ca_SRi + KdCSQN)}^{2.0}}} calcium_Ca_SRss_beta = \frac{1.0}{1.0 + \frac{CSQN KdCSQN}{{(Ca_SRss + KdCSQN)}^{2.0}}} calcium_Ca_i_beta = \frac{1.0}{1.0 + \frac{BCa KdBCa}{{(Ca_i + KdBCa)}^{2.0}}} calcium_Ca_ss_beta = \frac{1.0}{1.0 + \frac{SLlow KdSLlow}{{(Ca_ss + KdSLlow)}^{2.0}} + \frac{SLhigh KdSLhigh}{{(Ca_ss + KdSLhigh)}^{2.0}} + \frac{BCa KdBCa}{{(Ca_ss + KdBCa)}^{2.0}}}

Component: cell

VSRi = 2.0 57.0 VSRss = 2.0 80.0 Vnj = 6.0 2531.0 Vss = 2.0 49.9232 dx2 = dx dx

Component: engine

Component: ibca

IbCa = 1.0 Cm gbCa (V - ECa) gbCa = 1.4 0.001131

Component: ibna

IbNa = 1.7 Cm gbNa (V - ENa) gbNa = 0.8 (6.74437500000000015e-04)

Component: ical

ICaL = 1.333333 Cm gCaL d f fca (V - ErL) \frac{d}{d time} (d) = \frac{ical_d_inf - d}{ical_d_tau} \frac{d}{d time} (f) = \frac{ical_f_inf - f}{ical_f_tau} \frac{d}{d time} (fca) = \frac{ical_fca_inf - fca}{ical_fca_tau} gCaL = 0.1294 0.75 ical_d_a = \frac{1.0}{1.0 + ⅇ^{\frac{V + 10.0}{- 6.24}}} ical_d_inf = \frac{1.0}{1.0 + ⅇ^{\frac{V + 10.0}{- 8.0}}} ical_d_tau = \{\begin{matrix} ical_d_a 4.579 if | V + 10.0 | < 1e-10 \\ \frac{ical_d_a (1.0 - (ⅇ^{\frac{V + 10.0}{- 6.24}}))}{0.035 (V + 10.0)} otherwise \end{matrix} ical_f_inf = \frac{ⅇ^{\frac{- (V + 28.0)}{6.9}}}{1.0 + ⅇ^{\frac{- (V + 28.0)}{6.9}}} ical_f_tau = \frac{1.5 2.0 3.0}{0.0197 ⅇ^{(- ({0.0337}^{2.0})) {(V + 10.0)}^{2.0}} + 0.02} ical_fca_inf = \frac{1.0}{1.0 + \frac{Ca_ss}{0.00035}}

Component: icap

ICap = \frac{1.26 Cm icapbar Ca_ss}{Ca_ss + kmcap}

Component: ik1

IK1 = \frac{Cm gK1 (V - EK)}{1.0 + ⅇ^{0.07 (V + 80.0)}}

Component: ikr

IKr = \frac{Cm fKr gKr xr (V - EK)}{1.0 + ⅇ^{\frac{V + 15.0}{22.4}}} gKr = 0.8 (2.94117649999999994e-02) ikr_xr_a = \{\begin{matrix} 0.0015 if | V + 14.1 | < 1e-10 \\ \frac{0.0003 (V + 14.1)}{1.0 - (ⅇ^{\frac{V + 14.1}{- 5.0}})} otherwise \end{matrix} ikr_xr_b = \{\begin{matrix} 3.78361180000000004e-04 if | V - 3.3328 | < 1e-10 \\ \frac{(7.38980000000000030e-05) (V - 3.3328)}{ⅇ^{\frac{V - 3.3328}{5.1237}} - 1.0} otherwise \end{matrix} ikr_xr_inf = \frac{1.0}{1.0 + ⅇ^{\frac{V + 14.1}{- 6.5}}} ikr_xr_tau = \frac{1.0}{ikr_xr_a + ikr_xr_b} \frac{d}{d time} (xr) = \frac{ikr_xr_inf - xr}{ikr_xr_tau}

Component: iks

IKs = Cm fKs gKs xs xs (V - EK) gKs = 0.8 (1.29411759999999987e-01) iks_xs_a = \{\begin{matrix} 0.00068 if | V - 19.9 | < 1e-10 \\ \frac{(4e-05) (V - 19.9)}{1.0 - (ⅇ^{\frac{V - 19.9}{- 17.0}})} otherwise \end{matrix} iks_xs_b = \{\begin{matrix} 0.000315 if | V - 19.9 | < 1e-10 \\ \frac{(3.5e-05) (V - 19.9)}{ⅇ^{\frac{V - 19.9}{9.0}} - 1.0} otherwise \end{matrix} iks_xs_inf = \sqrt{\frac{1.0}{1.0 + ⅇ^{\frac{V - 19.9}{- 12.7}}}} iks_xs_tau = \frac{0.5}{iks_xs_a + iks_xs_b} \frac{d}{d time} (xs) = \frac{iks_xs_inf - xs}{iks_xs_tau}

Component: ikur

\frac{d}{d time} (BC) = KC ⅇ^{(- ZKC) V FRT} conc i (1.0 - a) (1.0 - BO - BC) - (LC BC ⅇ^{(- ZLC) V FRT}) \frac{d}{d time} (BO) = KO ⅇ^{(- ZKO) V FRT} conc i a (1.0 - BO - BC) - (LO BO ⅇ^{(- ZLO) V FRT}) IKur = Cm gKur (4.5128 + \frac{1.899769}{1.0 + ⅇ^{\frac{V - 20.5232}{- 8.26597}}}) (1.0 - BO - BC) a i (V - EK) ZKC = - (3.27023573379999988e-01) ZKO = - (2.57319531729999995e-01) ZLO = - (1.28015801399999993e-02) \frac{d}{d time} (a) = \frac{ikur_a_inf - a}{ikur_a_tau} gKur = 0.006398 0.9 \frac{d}{d time} (i) = \frac{ikur_i_inf - i}{ikur_i_tau} ikur_a_inf = \frac{1.0}{1.0 + ⅇ^{\frac{- (V + 5.52)}{8.6}}} ikur_a_tau = \frac{(\frac{4.56666746825999965e+01}{1.0 + ⅇ^{\frac{V + 1.12306497072999996e+01}{1.15254705961999999e+01}}} + 4.26753514992999960e+00) (\frac{2.62186042981000011e-01}{1.0 + ⅇ^{\frac{V + 3.58658312706999993e+01}{- 3.87510627762000004e+00}}} + (2.91755017928000016e-01)) 1.0}{K_Q10} ikur_i_inf = \frac{0.52424}{1.0 + ⅇ^{\frac{V + 15.1142}{7.567021}}} + 0.4580778 ikur_i_tau = \frac{\frac{2328.0}{1.0 + ⅇ^{\frac{V - 9.435}{3.5827}}} + 1739.139}{K_Q10}

Component: ina

\frac{d}{d time} (BA) = drug_Ka drug_concen m^{3.0} h j (1.0 - BA - BI) - (drug_La BA) \frac{d}{d time} (BI) = drug_Ki drug_concen (1.0 - h) (1.0 - BA - BI) - (drug_Li BI) INa = gNa m^{3.0} h j (V - ENa) Cm (1.0 - BA - BI) drug_Ka = \frac{0.1}{1000.0} drug_Ki = \frac{0.1}{1000.0} \frac{d}{d time} (h) = ina_h_alpha (1.0 - h) - (ina_h_beta h) ina_h_alpha = \{\begin{matrix} 0.0 if V \geq - 40.0 \\ 0.135 ⅇ^{\frac{V + 80.0}{- 6.8}} otherwise \end{matrix} ina_h_beta = \{\begin{matrix} \frac{1.0}{0.13 (1.0 + ⅇ^{\frac{V + 10.66}{- 11.1}})} if V \geq - 40.0 \\ 3.56 ⅇ^{0.079 V} + 310000.0 ⅇ^{0.35 V} otherwise \end{matrix} ina_j_alpha = \{\begin{matrix} 0.0 if V \geq - 40.0 \\ \frac{((- 127140.0) ⅇ^{0.2444 V} - ((3.474e-05) ⅇ^{(- 0.04391) V})) (V + 37.78)}{1.0 + ⅇ^{0.311 (V + 79.23)}} otherwise \end{matrix} ina_j_beta = \{\begin{matrix} \frac{0.3 ⅇ^{(- (2.535e-07)) V}}{1.0 + ⅇ^{(- 0.1) (V + 32.0)}} if V \geq - 40.0 \\ \frac{0.1212 ⅇ^{(- 0.01052) V}}{1.0 + ⅇ^{(- 0.1378) (V + 40.14)}} otherwise \end{matrix} ina_m_alpha = \{\begin{matrix} 3.2 if | V + 47.13 | < 1e-10 \\ \frac{0.32 (V + 47.13)}{1.0 - (ⅇ^{(- 0.1) (V + 47.13)})} otherwise \end{matrix} ina_m_beta = 0.08 ⅇ^{\frac{- V}{11.0}} \frac{d}{d time} (j) = ina_j_alpha (1.0 - j) - (ina_j_beta j) \frac{d}{d time} (m) = ina_m_alpha (1.0 - m) - (ina_m_beta m)

Component: inaca

INaCa = \frac{\frac{\frac{1.4 Cm knacalr}{{kmnalr}^{3.0} + {Na_o}^{3.0}}}{kmcalr + Ca_o}}{1.0 + ksatlr ⅇ^{(gammalr - 1.0) V FRT}} ({Na_i}^{3.0} Ca_o ⅇ^{V gammalr FRT} - ({Na_o}^{3.0} Ca_ss ⅇ^{V (gammalr - 1.0) FRT}))

Component: inak

INaK = \frac{\frac{1.28 Cm I_bar fnak K_o}{K_o + kmko}}{1.0 + {(\frac{kmnai}{Na_i})}^{4.0}} I_bar = 1.4 (5.99338739999999981e-01) fnak = \frac{1.0}{1.0 + 0.1245 ⅇ^{(- 0.1) V FRT} + 0.0365 sigma ⅇ^{(- V) FRT}} sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1.0}{7.0}

Component: ion

Component: ito

Ito = 1.05 fto Cm gto r s (V - EK) gto = 0.75471 0.1962 ito_r_inf = \frac{1.0}{1.0 + ⅇ^{\frac{V - 1.0}{- 11.0}}} ito_r_tau = 3.5 ⅇ^{(- (\frac{V}{30.0})) 2.0} + 1.5 ito_s_inf = \frac{1.0}{1.0 + ⅇ^{\frac{V + 40.5}{11.5}}} ito_s_tau = 25.635 ⅇ^{(- (\frac{V + 52.45}{15.8827})) 2.0} + 14.14 \frac{d}{d time} (r) = \frac{ito_r_inf - r}{ito_r_tau} \frac{d}{d time} (s) = \frac{ito_s_inf - s}{ito_s_tau}

Component: membrane

\frac{d}{d time} (V) = \frac{- (i_ion + i_stim)}{Cm} i_ion = + INa + Ito + IKur + IKr + IKs + ICaL + IK1 + IbNa + IbCa + INaCa + INaK + ICap

Component: nernst

ECa = 13.35 \ln (\frac{Ca_o}{Ca_i}) EK = 26.71 \ln (\frac{K_o}{K_i}) ENa = 26.71 \ln (\frac{Na_o}{Na_i})

Component: phys

FRT = \frac{\frac{F}{R}}{T}

Component: potassium

\frac{d}{d time} (K_i) = 0.0

Component: ryr

Jreli = nui o_i c_i RyRSRCai (Ca_SRi - Ca_i) Jrelss = nuss o_ss c_ss RyRSRCass (Ca_SRss - Ca_ss) RyRSRCai = 1.0 - (\frac{1.0}{1.0 + ⅇ^{\frac{Ca_SRi - 0.3}{0.1}}}) RyRSRCass = 1.0 - (\frac{1.0}{1.0 + ⅇ^{\frac{Ca_SRss - 0.3}{0.1}}}) \frac{d}{d time} (a_i) = \frac{ryr_a_i_inf - a_i}{ryr_a_i_tau} \frac{d}{d time} (a_ss) = \frac{ryr_a_ss_inf - a_ss}{ryr_a_ss_tau} \frac{d}{d time} (c_i) = \frac{ryr_c_i_inf - c_i}{ryr_c_i_tau} \frac{d}{d time} (c_ss) = \frac{ryr_c_ss_inf - c_ss}{ryr_c_ss_tau} nui = 0.001 Vnj nuss = 0.625 Vss \frac{d}{d time} (o_i) = \frac{ryr_o_i_inf - o_i}{ryr_o_i_tau} \frac{d}{d time} (o_ss) = \frac{ryr_o_ss_inf - o_ss}{ryr_o_ss_tau} ryr_a_i_inf = 0.505 - (\frac{0.427}{1.0 + ⅇ^{\frac{2000.0 Ca_i - 0.29}{0.082}}}) ryr_a_ss_inf = 0.505 - (\frac{0.427}{1.0 + ⅇ^{\frac{1000.0 Ca_ss - 0.29}{0.082}}}) ryr_c_i_inf = \frac{1.0}{1.0 + ⅇ^{\frac{2000.0 Ca_i - (a_i + 0.02)}{0.01}}} ryr_c_i_tau = 2.0 15.0 ryr_c_ss_inf = \frac{1.0}{1.0 + ⅇ^{\frac{1000.0 Ca_ss - (a_ss + 0.02)}{0.01}}} ryr_o_i_inf = 1.0 - (\frac{1.0}{1.0 + ⅇ^{\frac{2000.0 Ca_i - (a_i + 0.22)}{0.03}}}) ryr_o_ss_inf = 1.0 - (\frac{1.0}{1.0 + ⅇ^{\frac{1000.0 Ca_ss - (a_ss + 0.22)}{0.03}}})

Component: serca

J_SERCASR = 0.75 ((- k3) {Ca_SRi}^{2.0} (cpumps - SERCACa) + k4 SERCACa) Vnj 2.0 J_SERCASRss = 0.75 ((- k3) {Ca_SRss}^{2.0} (cpumps - SERCACass) + k4 SERCACass) Vss 2.0 J_bulkSERCA = 0.75 (k1 {Ca_i}^{2.0} (cpumps - SERCACa) - (k2 SERCACa)) Vnj 2.0 J_bulkSERCAss = 0.75 (k1 {Ca_ss}^{2.0} (cpumps - SERCACass) - (k2 SERCACass)) Vss 2.0 \frac{d}{d time} (SERCACa) = \frac{(- J_SERCASR) + J_bulkSERCA}{Vnj} \frac{d}{d time} (SERCACass) = \frac{(- J_SERCASRss) + J_bulkSERCAss}{Vss}

Component: sodium

\frac{d}{d time} (Na_i) = \frac{(- 3.0) INaK - (3.0 INaCa) - IbNa - INa}{F Vi}

Component: stimulus

amplitude = - 8000.0 i_stim = amplitude pace pace = \{\begin{matrix} 1.0 if time - offset - (period ⌊ \frac{time - offset}{period} ⌋) < duration \\ 0.0 otherwise \end{matrix}

Source: Derived from workspace Ni et al. 2017 Human atrial model at changeset 28bb4238f19c.

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