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Model Mathematics

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Component: environment

Component: ATP

\frac{\partial}{\partial time} (ATP) = (D_ATP ATP - (G_CK v_CK)) + G_AK v_AK - (G_H H_ATP)

Component: ADP

\frac{\partial}{\partial time} (ADP) = (D_ADP ADP + G_CK v_CK - (2.0 G_AK v_AK)) + G_H H_ATP

Component: AMP

\frac{\partial}{\partial time} (AMP) = D_AMP AMP + G_AK v_AK

Component: PCr

\frac{\partial}{\partial time} (PCr) = D_PCr PCr + G_CK v_CK

Component: Cr

\frac{\partial}{\partial time} (Cr) = D_Cr Cr + G_CK v_CK

Component: Pi

\frac{\partial}{\partial time} (Pi) = D_Pi Pi + G_H H_ATP

Component: v_CK

v_CK = \frac{V1 \frac{mATP Cr}{Kia Kb} - (V1_\frac{mADP PCr}{Kic Kd})}{DenCK} DenCK = 1.0 + \frac{Cr}{Kib} + \frac{PCr}{Kid} + mATP (\frac{1.0}{Kia} + \frac{Cr}{Kia Kb}) + mADP (\frac{1.0}{Kic} + \frac{PCr}{Kid Kc} + \frac{Cr}{Kic KIb}) Kc = \frac{Kic Kd}{Kid}

Component: fATP

fATP = \frac{ATP}{1.0 + \frac{Mg}{KDT}}

Component: fATPx

fATPx = \frac{ATPx}{1.0 + \frac{Mgx}{KDTx}}

Component: fATPext

fATPext = \frac{ATPext}{1.0 + \frac{Mgext}{KDText}}

Component: fATPg

fATPg = \frac{ATPg}{1.0 + \frac{Mgg}{KDTg}}

Component: fATPi

fATPi = \frac{ATPi}{1.0 + \frac{Mgi}{KDTi}}

Component: mATP

mATP = ATP - fATP

Component: mATPx

mATPx = ATPx - fATPx

Component: mATPext

mATPext = ATPext - fATPext

Component: mATPg

mATPg = ATPg - fATPg

Component: mATPi

mATPi = ATPi - fATPi

Component: fADP

fADP = \frac{ADP}{1.0 + \frac{Mg}{KDD}}

Component: fADPx

fADPx = \frac{ADPx}{1.0 + \frac{Mgx}{KDDx}}

Component: fADPext

fADPext = \frac{ADPext}{1.0 + \frac{Mgext}{KDDext}}

Component: fADPg

fADPg = \frac{ADPg}{1.0 + \frac{Mgg}{KDDg}}

Component: fADPi

fADPi = \frac{ADPi}{1.0 + \frac{Mgi}{KDDi}}

Component: mADP

mADP = ADP - fADP

Component: mADPx

mADPx = ADPx - fADPx

Component: mADPext

mADPext = ADPext - fADPext

Component: mADPg

mADPg = ADPg - fADPg

Component: mADPi

mADPi = ADPi - fADPi

Component: v_AK

v_AK = kfa fADP mADP - (kba mATP AMP)

Component: v_AKmit

v_AKmit = kfa fADP mADP - (kba mATP AMP)

Component: H_ATP

H_ATP = \{\begin{matrix} \frac{time}{30.0} H_ATPmax if time \geq 0.0 \land time < 30.0 \\ \frac{60.0 time}{30.0} H_ATPmax if time \geq 30.0 \land time < 60.0 \\ 0.0 if time \geq 60.0 \land time < 180.0 \end{matrix}

Component: G_CK

G_CK = \{\begin{matrix} 0.94 (1.0 - (\frac{x}{0.375})) if x < 0.375 \land y < 1.0 \\ 1.61 \frac{x - 0.375}{0.375} if x \geq 0.375 \land x < 0.75 \land y < 1.0 \\ 1.61 if x \geq 0.375 \lor y \geq 1.0 \end{matrix}

Component: G_AK

G_AK = G_CK

Component: G_H

G_H = \{\begin{matrix} \frac{4.0}{3.0} \frac{x}{0.125} if x < 0.125 \land y < 1.0 \\ \frac{4.0}{3.0} if x \geq 0.125 \land x < 0.75 \land y < 1.0 \\ \frac{4.0}{3.0} (1.0 - (\frac{x - 0.75}{0.125})) if x \geq 0.75 \land x < 0.875 \land y < 1.0 \\ 0.0 if x \geq 0.875 \lor y \geq 1.0 \end{matrix}

Component: diffusion_through_mitochondrial_membrane

\frac{\partial}{\partial n} (ATP_diff) = \frac{R_ATP (ATPi - ATP)}{D_ATP} \frac{\partial}{\partial n} (ADP_diff) = \frac{R_ADP (ADPi - ADP)}{D_ADP} \frac{\partial}{\partial n} (AMP_diff) = \frac{R_AMP (AMPi - AMP)}{D_AMP} \frac{\partial}{\partial n} (PCr_diff) = \frac{R_PCr (PCri - PCr)}{D_PCr} \frac{\partial}{\partial n} (Cr_diff) = \frac{R_Cr (Cri - Cr)}{D_Cr} \frac{\partial}{\partial n} (Pi_diff) = \frac{R_Pi (Pii - Pi)}{D_Pi}

Component: ATPi

\frac{d}{d time} (ATPi) = (\frac{R_ATP}{Li} (ATP - ATPi) + \frac{v_ANT}{FVi} - (\frac{v_MiCK}{FVi})) + \frac{v_AKmit}{FVi}

Component: ADPi

\frac{d}{d time} (ADPi) = (\frac{R_ADP}{Li} (ADP - ADPi) - (\frac{2.0 v_AKmit}{FVi} + \frac{v_ANT}{FVi})) + \frac{v_MiCK}{FVi}

Component: AMPi

\frac{d}{d time} (AMPi) = \frac{R_AMP}{Li} (AMP - AMPi) + \frac{v_AKmit}{FVi}

Component: PCri

\frac{d}{d time} (PCri) = \frac{R_PCr}{Li} (PCr - PCri) + \frac{v_AKmit}{FVi}

Component: Cri

\frac{d}{d time} (Cri) = \frac{R_Cr}{Li} (Cr - Cri) - (\frac{v_MiCK}{FVi})

Component: Pii

\frac{d}{d time} (Pii) = \frac{R_Pi}{Li} (Pi - Pii) - (\frac{v_PI}{FVi})

Component: v_MiCK

v_MiCK = v_MiCK_G + v_MiCK_I v_MiCK_G = \frac{V1 \frac{mATPg Cri}{Kia_G Kb} - (V1_\frac{mADPg PCri}{Kic_G Kd})}{DenMiCK} v_MiCK_I = \frac{V1 \frac{mATPi Cri}{Kia Kb} - (V1_\frac{mADPi PCri}{Kic Kd})}{DenMiCK} DenMiCK = 1.0 + \frac{Cri}{Kib} + \frac{PCri}{Kid} + (\frac{mATPg}{Kia_G} + \frac{mATPi}{Kia}) (1.0 + \frac{Cri}{Kb}) + (\frac{mADPg}{Kic_G} + \frac{mADPi}{Kic}) (1.0 + \frac{PCri}{Kd} + \frac{Cri}{KIb})

Component: v_exch

v_exch_ATP = R_exch_ATP (ATPi - ATPg) v_exch_ADP = R_exch_ADP (ADPi - ADPg)

Component: UQ

\frac{d}{d time} (UQ) = \frac{v_C3 - v_C1}{FVx}

Component: c3plus

\frac{d}{d time} (c3plus) = \frac{2.0 (2.0 v_O2 - v_C3)}{FVx}

Component: NAD

\frac{d}{d time} (NAD) = \frac{v_C1 - v_dh}{FVx}

Component: Hx

\frac{d}{d time} (Hx) = delta_Hx delta_Hx = \frac{- (2.0 (2.0 + u) v_O2 + 4.0 v_C1 + (4.0 - (2.0 u)) v_C3 - (na v_sn + v_ANT u + (1.0 - u) v_PI + v_leak))}{FVx r_buff}

Component: ATPx

\frac{d}{d time} (ATPx) = \frac{v_sn - v_ANT}{FVx}

Component: Pix

\frac{d}{d time} (Pix) = \frac{v_PI - v_sn}{FVx}

Component: oxidative_phosphorylation

v_dh = \frac{k_dh}{{(1.0 + \frac{k_mN}{\frac{NAD}{NADH}})}^{pD}} v_C1 = k_C1 (Eu - (En + 2.0 delta_p)) v_C3 = k_C3 (Ec - (Eu + (2.0 - u) delta_p)) v_O2 = 0.5 k_C4 a2plus c2plus \frac{O2}{O2 + k_mO} v_sn = k_sn \frac{{10.0}^{\frac{delta_Gsn}{Z}} - 1.0}{{10.0}^{\frac{delta_Gsn}{Z}} + 1.0} v_PI = vfPI Hext \frac{Pii}{1.0 + {10.0}^{pHext - pKa}} - (vbPI Hx \frac{Pix}{1.0 + {10.0}^{pHx - pKa}}) v_leak = k_L1 ({10.0}^{k_L2 delta_p} - 1.0)

Component: NADH

NADH = NAD_tot - NAD

Component: UQH2

UQH2 = UQ_tot - UQ

Component: c2plus

c2plus = ctot - c3plus

Component: a2plus

a2plus = atot - a3_2

Component: ADPx

ADPx = Ax - ATPx

Component: pHx

pHx = - (\log_{10} (eta Hx))

Component: delta_pH

delta_pH = Z (pHext - pHx)

Component: delta_p

delta_p = \frac{1.0}{1.0 - u} delta_pH

Component: delta_psi

delta_psi = - (delta_p - delta_pH)

Component: Hext

Hext = eta {1.0}^{- pHext}

Component: r_buff

r_buff = r_buff0 eta \frac{delta_pH}{delta_Hx}

Component: Z

Z = \ln 10.0 \frac{R T}{F}

Component: v_ANT

v_ANT = v_ANT_XG + v_ANT_XI v_ANT_XG = v_ANT \frac{fADPg}{Kg \frac{10.0 + fADPg}{Kg + \frac{fADPi}{Ki}}} (\frac{fADPg}{fATPg {10.0}^{\frac{(-0.35) delta_psi}{Z}} + fADPg} - (\frac{fADPx}{fATPx {10.0}^{\frac{(-0.65) delta_psi}{Z}} + fADPx})) v_ANT_XI = v_ANT \frac{fADPi}{Ki \frac{10.0 + fADPg}{Kg + \frac{fADPi}{Ki}}} (\frac{fADPi}{fATPi {10.0}^{\frac{(-0.35) delta_psi}{Z}} + fADPi} - (\frac{fADPx}{fATPx {10.0}^{\frac{(-0.65) delta_psi}{Z}} + fADPx}))

Component: delta_Gsn

delta_Gsn = (na delta_p - (\frac{delta_Gsn0}{F})) + Z \log_{10} (\frac{eta ATPx}{ADPx Pix})

Component: redox_potentials

En = En0 + \frac{Z}{2.0} \log_{10} (\frac{NAD}{NADH}) Eu = Eu0 + \frac{Z}{2.0} \log_{10} (\frac{UQ}{UQH2}) Ec = Ec0 + \frac{Z}{2.0} \log_{10} (\frac{c3plus}{c2plus}) Ea = Ec + (1.0 + u) delta_p a3_2 = {10.0}^{(Ea - Ea0) Z}

Component: diffusion_constants

Component: Mg_buffering

Component: H_buffering

Component: diffusion_through_mitochondrial_membrane_parameters

Component: general_constants

Component: fractional_volumes

Source: Derived from workspace Vendelin, Kongas, Saks, 2000 at changeset 54152205044e.

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Regulation of mitochondrial respiration in heart cells analyzed by reaction-diffusion model of energy transfer