You are here: Home / Exposures / Cortassa, Aon, Marban, Winslow, O'Rourke, 2003 / An Integrated Model of Cardiac Mitrochondrial Energy Metabolism and Calcium Dynamics

Model Mathematics

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Component: environment

Component: ADP_m

\frac{d}{d time} (ADP_m) = V_ANT - (V_ATPase + V_SL)

Component: NADH

\frac{d}{d time} (NADH) = (- V_O2) + V_IDH + V_KGDH + V_MDH

Component: ISOC

\frac{d}{d time} (ISOC) = V_ACO - V_IDH

Component: alpha_KG

\frac{d}{d time} (alpha_KG) = V_AAT + V_IDH + (- V_KGDH)

Component: SCoA

\frac{d}{d time} (SCoA) = V_KGDH - V_SL

Component: Suc

\frac{d}{d time} (Suc) = V_SL - V_SDH

Component: FUM

\frac{d}{d time} (FUM) = V_SDH - V_FH

Component: MAL

\frac{d}{d time} (MAL) = V_FH - V_MDH

Component: OAA

\frac{d}{d time} (OAA) = V_MDH - (V_CS + V_AAT)

Component: ASP

\frac{d}{d time} (ASP) = V_AAT - V_C_ASP

Component: Ca_m

\frac{d}{d time} (Ca_m) = f 1 (V_uni - V_NaCa)

Component: Ca_i

Ca_i = \{\begin{matrix} pulse_value if time \geq stim_start \land time ≦ stim_end \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0.1 otherwise \end{matrix} stim_period = \{\begin{matrix} 0.5 if time \geq 100 \land time < 300 \\ 4 otherwise \end{matrix}

Component: Na_i

Component: ATP_i

Component: ATP_m

ATP_m = Cm - ADP_m

Component: ADP_i

ADP_i = \{\begin{matrix} pulse_value if time \geq 100 \land time < 300 \\ 0.05 otherwise \end{matrix}

Component: GLU

Component: Mg

Component: H

Component: Pi

Component: CoA

Component: AcCoA

Component: FAD

Component: FADH2

Component: NAD

NAD = C_PN - NADH

Component: CIT

CIT = C_Kint - (ISOC + alpha_KG + SCoA + Suc + FUM + MAL + OAA)

Component: mitochondrial_membrane

\frac{d}{d time} (delta_psi_m) = \frac{V_He + V_He_F + (- (V_Hu + V_ANT + V_Hleak + V_NaCa + 2.0 V_uni))}{C_mito}

Component: V_CS

V_CS = \frac{Kcat_CS ET_CS}{1.0 + \frac{Km_AcCoA}{AcCoA} + \frac{Km_OAA}{OAA} + \frac{Km_AcCoA}{AcCoA} \frac{Km_OAA}{OAA}}

Component: V_ACO

V_ACO = Kf_ACO (CIT - (\frac{ISOC}{KE_ACO}))

Component: V_IDH

V_IDH = \frac{Kcat_IDH ET_IDH}{1.0 + \frac{H}{Kh_1} + \frac{Kh_2}{H} + \frac{{(\frac{Km_ISOC}{ISOC})}^{ni}}{(1.0 + \frac{ADP_m}{Ka_ADP}) (1.0 + \frac{Ca_m}{Ka_Ca})} + \frac{Km_NAD}{NAD} (1.0 + \frac{NADH}{Ki_NADH}) + \frac{{(\frac{Km_ISOC}{ISOC})}^{ni} \frac{Km_NAD}{NAD} (1.0 + \frac{NADH}{Ki_NADH})}{(1.0 + \frac{ADP_m}{Ka_ADP}) (1.0 + \frac{Ca_m}{Ka_Ca})}}

Component: V_KGDH

V_KGDH = \frac{Kcat_KGDH ET_KGDH}{1.0 + \frac{{(\frac{Km_alpha_KG}{alpha_KG})}^{n_alpha_KG}}{(1.0 + \frac{Mg}{Kd_Mg}) (1.0 + \frac{Ca_m}{Kd_Ca})} + \frac{\frac{Km_NAD}{NAD}}{(1.0 + \frac{Mg}{Kd_Mg}) (1.0 + \frac{Ca_m}{Kd_Ca})}}

Component: V_SL

V_SL = kf_SL (SCoA ADP_m - (\frac{Suc ATP_m CoA}{Ke_SL}))

Component: V_SDH

V_SDH = \frac{Kcat_SDH ET_SDH}{1.0 + \frac{Km_Suc}{Suc} (1.0 + \frac{OAA}{Kisdh_OAA}) (1.0 + \frac{FUM}{Ki_FUM})}

Component: V_MDH

V_MDH = \frac{Kcat_MDH ET_MDH fh_a fh_i}{1.0 + \frac{Km_MAL}{MAL} (1.0 + \frac{OAA}{Ki_OAA}) + \frac{Km_NAD}{NAD} + \frac{Km_MAL}{MAL} (1.0 + \frac{OAA}{Ki_OAA}) \frac{Km_NAD}{NAD}} fh_a = \frac{1.0}{1.0 + \frac{H}{kh1} + \frac{H^{2.0}}{kh1 kh2}} + k_offset fh_i = {(\frac{1.0}{1.0 + \frac{kh3}{H} + \frac{kh3 kh4}{H^{2.0}}})}^{2.0}

Component: V_FH

V_FH = kf_FH (FUM - (\frac{MAL}{Ke_FH}))

Component: V_AAT

V_AAT = kf_AAT (OAA GLU - (\frac{alpha_KG ASP}{Ke_AAT}))

Component: V_C_ASP

V_C_ASP = k_C_ASP ASP

Component: oxidative_phosphorylation

V_O2 = 0.5 rho_res \frac{((ra + rc1 ⅇ^{\frac{6.0 F delta_psi_B}{R T}}) ⅇ^{\frac{Ares F}{R T}} - (ra ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}})) + rc2 ⅇ^{\frac{Ares F}{R T}} ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}}}{(1.0 + r1 ⅇ^{\frac{F Ares}{R T}}) ⅇ^{\frac{6.0 F delta_psi_B}{R T}} + (r2 + r3 ⅇ^{\frac{F Ares}{R T}}) ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}}} V_He = 6.0 rho_res \frac{ra ⅇ^{\frac{F Ares}{R T}} - ((ra + rb) ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}})}{(1.0 + r1 ⅇ^{\frac{F Ares}{R T}}) ⅇ^{\frac{6.0 F delta_psi_B}{R T}} + (r2 + r3 ⅇ^{\frac{F Ares}{R T}}) ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}}} Ares = \frac{R T}{F} \ln (Kres {(\frac{NADH}{NAD})}^{0.5}) V_He_F = 6.0 rho_res_F \frac{ra ⅇ^{\frac{F Ares_F}{R T}} - ((ra + rb) ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}})}{(1.0 + r1 ⅇ^{\frac{F Ares_F}{R T}}) ⅇ^{\frac{6.0 F delta_psi_B}{R T}} + (r2 + r3 ⅇ^{\frac{F Ares_F}{R T}}) ⅇ^{\frac{g 6.0 F delta_mu_H}{R T}}} Ares_F = \frac{R T}{F} \ln (Kres_F {(\frac{FADH2}{FAD})}^{0.5}) V_Hleak = gH delta_mu_H delta_mu_H = \frac{R T}{F} delta_pH + delta_psi_m V_ATPase = (- rho_F1) \frac{(100.0 pa + pc1 ⅇ^{\frac{3.0 F delta_psi_B}{R T}}) ⅇ^{\frac{AF1 F}{R T}} - (pa ⅇ^{\frac{3.0 F delta_mu_H}{R T}} + pc2 ⅇ^{\frac{AF1 F}{R T}} ⅇ^{\frac{3.0 F delta_mu_H}{R T}})}{(1.0 + p1 ⅇ^{\frac{F AF1}{R T}}) ⅇ^{\frac{3.0 F delta_psi_B}{R T}} + (p2 + p3 ⅇ^{\frac{F AF1}{R T}}) ⅇ^{\frac{3.0 F delta_mu_H}{R T}}} V_Hu = (-3.0) rho_F1 \frac{100.0 pa (1.0 + ⅇ^{\frac{F AF1}{R T}}) - ((pa + pb) ⅇ^{\frac{3.0 F delta_mu_H}{R T}})}{(1.0 + p1 ⅇ^{\frac{F AF1}{R T}}) ⅇ^{\frac{3.0 F delta_psi_B}{R T}} + (p2 + p3 ⅇ^{\frac{F AF1}{R T}}) ⅇ^{\frac{3.0 F delta_mu_H}{R T}}} AF1 = \frac{R T}{F} \ln (KF1 \frac{ATP_m}{ADP_m Pi})

Component: calcium_dynamics

V_ANT = Vmax_ANT \frac{1.0 - (\frac{0.05 ATP_i 0.45 0.8 ADP_m}{0.45 ADP_i 0.05 ATP_m})}{(1.0 + \frac{0.05 ATP_i}{0.45 ADP_i} ⅇ^{\frac{(- h) F delta_psi_0}{R T}}) (1.0 + \frac{0.45 0.8 ADP_m}{0.05 ATP_m})} V_uni = Vmax_uni \frac{\frac{Ca_i}{K_trans} {(1.0 + \frac{Ca_i}{K_trans})}^{3.0} \frac{2.0 F (delta_psi_m - delta_psi_0)}{R T}}{{(1.0 + \frac{Ca_i}{K_trans})}^{4.0} + \frac{L}{{(1.0 + \frac{Ca_i}{K_act})}^{na}} (1.0 - (ⅇ^{\frac{(-2.0) F (delta_psi_m - delta_psi_0)}{R T}}))} V_NaCa = Vmax_NaCa \frac{ⅇ^{\frac{b F (delta_psi_m - delta_psi_0)}{R T}} ⅇ^{\ln (\frac{Ca_i}{Ca_m})}}{{(1.0 + \frac{KNa}{Na_i})}^{n} (1.0 + \frac{KCa}{Ca_m})}

Source: Derived from workspace Cortassa, Aon, Marban, Winslow, Orourke, 2003 at changeset a126ab075023.

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An Integrated Model of Cardiac Mitrochondrial Energy Metabolism and Calcium Dynamics