+ Wr")) +Abro (Do:) â xbibu. (De!) â K&:(PbH +Pep +. F .00. E. E ..... dom = abbau () +aÄrno (ba:) â (Dw:) â abra(Dwó). = xém (Dr) + abus ( Ds:) â spéus (Duo) ...
Text S1 Text S1 for the paper entitled: Sensitivity analysis of flux determination in heart by H218 O-provided labeling using a dynamic isotopologue model of energy transfer pathways David W. Schryer, Pearu Peterson, Ardo Illaste, and Marko Vendelin Laboratory of Systems Biology, Institute of Cybernetics, Tallinn University of Technology, Estonia
This document presents the definition of the phosphotransfer network studied in the main text and the derivation of the model equations. All intermediate derivation steps are included as well as the full set of model equations.
Contents 1 Definition of phosphotransfer network 1.1 Compartments included in the model . 1.2 Species included in the model . . . . . 1.3 Phosphotransfer network reactions . . 1.4 Oxygen atom mappings . . . . . . . .
1 1 1 2 3
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2 Model derivation 2.1 Individual isotopic transformations . . . 2.2 Kinetic equations for isotopomers . . . . 2.3 Pool definitions . . . . . . . . . . . . . . 2.4 Kinetic equations for mass isotopomers .
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3 . 4 . 4 . 30 . 34
1
DEFINITION OF PHOSPHOTRANSFER NETWORK
The phosphotransfer network under consideration is compartmentalized such that each of the three types of enzymatic reactions occurs in several compartments (see Figure S7 below). The species that participate in these reactions move between compartments via transport reactions. 1.1
Compartments included in the model
The reactions and species that form this network are located in three compartments: cytosol (o), intermitochondrial membrane space (i), and the mitochondrial matrix (m). In addition, enzyme bound compartments are included for both ATP synthase (s) and the ATPase reactions (e). The names of all species, reactions and fluxes include one or two of these compartmental tags. 1.2
Species included in the model
All species that become labeled with
18
O are considered. We introduce a compact notation for these species:
• five species of ADP (D):
Dm , Di , Do , De , Ds ;
• five species of ATP (T):
Tm , Ti , To , Te , Ts ;
• four species of inorganic phosphate (P): • two species of phosphocreatine (C):
Pm , Po , Pe , Ps ;
Ci , Co ;
1
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
Wo
ATPoe 1
Weo
ASe
Peo
Cytosol
To
We Te
AdKo
ADPeo
2
Co
CKo
ATPio
2
Pe De
Po
Do
IMS
Pom
ADPoi
Matrix Pm
Ti
Dm
Ps Ds
1 ATPmi
Pms
ADPms
ASs
Wos
ATPsm
Wo Ws Ts
Tm
Cio
AdKi
CKi
2 ADPim
Di
Ci
Figure S7: Network diagram with flux names. Note that this is the same as Figure 1 in the main text with flux values replaced with the names of each reaction used in this document, and the metabolite species names that are used in the equations in this document. The species names include subscripts that indicate compartmental location. • two species of water (W):
We , Ws ;
Adenosine monophosphate is not included since it does not become isotopically labeled with labeling state of intercellular water (Wo ) is specified as a function for model simulation. 1.3
18
O. The isotopic
Phosphotransfer network reactions
The compartmentalized model of the phosphotransfer network is defined by six bidirectional enzymatic reactions (AdKi, AdKo, ASe, ASs, CKi, CKo), two unidirectional substrate exchange reactions (ATPoe, ATPsm), six bidirectional substrate exchange reactions (Peo, Pms, Weo, Wos, ADPeo, ADPms), five bidirectional transport reactions (ADPim, ADPoi, ATPio, ATPmi, Cio), and one unidirectional transport reaction (Pom). The network of these fluxes is presented in Figure S7, and their reaction definitions are given below.
νf AdKi
AdK reaction in the IMS:
ATPase reactions:
Ti
− − * ) −− − − Di + Di
− * + We − ) − − De + Pe νf CKi
−− * ) − − Di + Ci
Transport of ATP (cytosol, ATPase):
To
−−−→ Te
Transport of P (cytosol, ATPase):
Pe
− − * ) − − Po
Transport of water (cytosol, ATPase):
Transport of ADP (cytosol, ATPase):
ATP synthase:
νr ASe
Ti
CK reaction in the IMS:
AdK reaction in the cytosol:
νf ATPoe
− )− −* − Wo
De
− − * ) −− −− − − Do νr ADPeo
− * + Ps − ) − − Ts + Ws νr ASs
νf CKo
Transport of ATP (matrix, ATP syn.):
Ts
−−−→ Tm
Transport of P (matrix, ATP syn.):
Pm
− − * ) − − Ps
Wo
− − * ) − − Ws
Dm
− − * ) −− −− − − Ds
Transport of water (cytosol, ATP syn.):
νr Weo
νf ADPeo
νf ASs
Ds
−− ) −* − Co + Do
νr Peo
νf Weo
νr AdKo
To
CK reaction in the cytosol:
νr CKi
νf Peo
We
To
νr AdKi νf ASe
Te
νf AdKo
− − * ) −− − − Do + Do
Transport of ADP (matrix, ATP syn.):
νr CKo
νf ATPsm
νf Pms νr Pms
νf Wos νr Wos
νf ADPms νr ADPms
Text S1 for Dynamic isotopologue model of oxygen labeling in heart νf ADPim
Transport of ADP (IMS, matrix):
− − * ) −− −− − − Dm
Ti
−− * ) −− − − To
Ci
−− * ) − − Co
Transport of ATP (IMS, matrix):
νr ATPio νf Cio
Transport of CK (cytosol, IMS):
Transport of ADP (cytosol, IMS):
νr ADPim νf ATPio
Transport of ATP (cytosol, IMS):
1.4
Di
νr Cio
Transport of P (cytosol, matrix):
3 νf ADPoi
Do
−− ) −− −* − Di
Tm
− )− −− −* − Ti
Po
−−→ Pm
νr ADPoi νf ATPmi
νr ATPmi
νf Pom
Oxygen atom mappings
All three oxygen atoms in every phosphoryl group of all species have an equal probability of being isotopically labeled (see main text). 2
MODEL DERIVATION
The derivation of the mass balance equations used to calculate the dynamic change in labeling state for all oxygen atoms in this system is discussed in the main text. In short, (I) the full set of individual isotopic transformations is generated (Section 2.1), (II) these transformations are combined into mass balances around each isotopologue in the system (Section 2.2), (III) mass isotopologue pool relations are composed taking into account oxygen atom mappings (Section 2.3), and (IV) mass isotopologue balances are composed by collecting the isotopologue balances according to the pool relations (Section 2.4). A program was written in Python to generate these equations. This program implements symbolic manipulation tools specifically designed to carry out steps (I) through (IV), and is available upon request. Note that a number of subexpressions, such as the sum of all inorganic phosphate isotopologues in the first four mass isotopologue balances (and others), always equal one. These subexpressions were not simplified since these terms act as stabilizing attractors during integration. A succinct notation was devised to make the isotopologue equations more readable. The labeling state of the oxygen atoms attached to the mobile phosphorus atom are written in a column of filled or unfilled circles corresponding to 18O and 16O, respectively. If a species has more than one phosphoryl group, for example, the two phosphoryl groups in T each is written in a separate columns of three oxygen atoms with the outermost row corresponding to •• the outermost phosphoryl group (i.e. γ-ATPin the case of T). T is written as Te ••••, and the four oxygen atoms of P
are written in one column (i.e.
• •
Po •). •
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
2.1
4
Individual isotopic transformations
The isotopic transformations for one reaction are provided as an example. The kinetic equations in Section 2.2 are constructed from the set of all transformations from all reactions. •• • Ti • ••
•◦ • Ti • ••
••
•
•
νf ,νr AKi
−− ) −− −* −
• Di • •
+
νf ,νr AKi
•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
◦
•◦ • Ti • ◦•
− )− −− −* − Di •◦ + Di ••
•◦
•• • Ti ◦ ◦•
− )− −− −* − Di ◦◦ + Di ••
◦• • Ti • ••
◦◦ • Ti • ••
◦•
νf ,νr AKi
−− ) −− −* −
◦ Di • •
+
νf ,νr AKi
◦
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
−− ) −− −* − Di •◦ + Di ••
◦◦ • Ti • ◦•
− )− −− −* − Di •◦ + Di ••
◦• • Ti ◦ ••
◦◦
− )− −− −* − Di ◦• + Di ••
◦• • Ti ◦ ◦•
− )− −− −* − Di ◦◦ + Di ••
◦◦ • Ti ◦ ◦•
◦•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
◦
νf ,νr AKi
−− ) −− −* −
◦ Di • •
+
νf ,νr AKi
◦
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
− )− −− −* − Di ◦• + Di •◦
◦• • Ti ◦ ◦◦
− )− −− −* − Di ◦◦ + Di •◦
◦◦
◦•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
◦
•
−− ) −− −* − Di ◦• + Di ◦• ◦
−− ) −− −* − Di ◦◦ + Di ◦• •
− )− −− −* − Di •• + Di ◦• νf ,νr AKi
−− ) −− −* −
◦ Di • •
+
◦ Di ◦ •
νf ,νr AKi
◦
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
•
◦ Ti • ◦•
−− ) −− −* − Di •◦ + Di ◦•
◦◦ ◦ Ti • ◦•
− )− −− −* − Di •◦ + Di ◦•
◦• ◦ Ti ◦ ••
◦◦
−− ) −− −* − Di ◦• + Di ◦• ◦
◦ Ti ◦ ••
− )− −− −* − Di ◦• + Di ◦•
◦• ◦ Ti ◦ ◦•
− )− −− −* − Di ◦◦ + Di ◦•
◦◦
−− ) −− −* − Di ◦◦ + Di •◦
•
− )− −− −* − Di ◦◦ + Di ◦•
◦◦ ◦ Ti • ••
◦
• Ti ◦ •◦
νf ,νr AKi
•• ◦ Ti ◦ ◦•
◦•
−− ) −− −* − Di ◦• + Di •◦
+
◦ Di ◦ •
− )− −− −* − Di ◦• + Di ◦•
◦ Ti • ••
•
−− ) −− −* −
• Di • •
◦ Ti ◦ ••
•◦
◦ Di • ◦
νf ,νr AKi
− )− −− −* − Di •◦ + Di ◦•
◦ Ti ◦ ◦•
•
•
•◦ ◦ Ti • ◦•
•◦
−− ) −− −* − Di •• + Di •◦
•
−− ) −− −* − Di •◦ + Di ◦•
••
−− ) −− −* − Di ◦◦ + Di •◦
νf ,νr AKi
− )− −− −* − Di •• + Di ◦•
◦ Ti • ◦•
◦ Ti ◦ ••
◦
− )− −− −* − Di •◦ + Di •◦
• Ti ◦ ◦◦
••
−− ) −− −* − Di ◦• + Di •◦
◦◦ • Ti • ◦◦
◦◦
•◦ ◦ Ti • ••
•
−− ) −− −* − Di •◦ + Di •◦
◦•
−− ) −− −* − Di ◦◦ + Di ••
•
• Ti • ◦◦
• Ti ◦ •◦
◦
• Ti ◦ ••
νf ,νr AKi
− )− −− −* − Di ◦◦ + Di •◦
◦◦ • Ti • •◦
−− ) −− −* − Di ◦• + Di ••
+
•• • Ti ◦ ◦◦
◦•
•• ◦ Ti • ••
◦ Di • ◦
− )− −− −* − Di ◦• + Di •◦
• Ti • •◦
•
• Ti • ◦•
−− ) −− −* −
• Di • •
• Ti ◦ •◦
•◦
◦ Di • •
νf ,νr AKi
− )− −− −* − Di •◦ + Di •◦
• Ti ◦ ◦◦
•
•
•◦ • Ti • ◦◦
•◦
− )− −− −* − Di •• + Di ••
•
−− ) −− −* − Di •◦ + Di •◦
••
−− ) −− −* − Di ◦◦ + Di ••
νf ,νr AKi
− )− −− −* − Di •• + Di •◦
• Ti • ◦◦
• Ti ◦ •◦
◦
− )− −− −* − Di ◦• + Di ••
•◦
••
−− ) −− −* − Di ◦• + Di ••
• Ti ◦ ••
• Ti ◦ ◦•
•◦ • Ti • •◦
•
−− ) −− −* − Di •◦ + Di ••
••
•• • Ti • •◦
◦ Di • •
• Ti • ◦•
• Ti ◦ ••
2.2
νf ,νr AKi
− )− −− −* − Di •• + Di ••
◦ Ti ◦ ◦•
−− ) −− −* − Di ◦◦ + Di ◦•
•• ◦ Ti • •◦
•◦ ◦ Ti • •◦
••
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
•
•
νf ,νr AKi
•
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
νf ,νr AKi
◦
•
νf ,νr AKi
◦
◦
− )− −− −* − Di •• + Di ◦◦ −− ) −− −* − Di •• + Di ◦◦
◦ Ti • ◦◦
−− ) −− −* − Di •◦ + Di ◦◦
•◦ ◦ Ti • ◦◦
− )− −− −* − Di •◦ + Di ◦◦
•• ◦ Ti ◦ •◦
•◦
−− ) −− −* − Di ◦• + Di ◦◦
◦ Ti ◦ •◦
− )− −− −* − Di ◦• + Di ◦◦
•• ◦ Ti ◦ ◦◦
− )− −− −* − Di ◦◦ + Di ◦◦
•◦ ◦ Ti ◦ ◦◦
◦• ◦ Ti • •◦
◦◦ ◦ Ti • •◦
◦•
−− ) −− −* − Di ◦◦ + Di ◦◦ − )− −− −* − Di •• + Di ◦◦ −− ) −− −* − Di •• + Di ◦◦
◦ Ti • ◦◦
−− ) −− −* − Di •◦ + Di ◦◦
◦◦ ◦ Ti • ◦◦
− )− −− −* − Di •◦ + Di ◦◦
◦• ◦ Ti ◦ •◦
◦◦
−− ) −− −* − Di ◦• + Di ◦◦
◦ Ti ◦ •◦
− )− −− −* − Di ◦• + Di ◦◦
◦• ◦ Ti ◦ ◦◦
− )− −− −* − Di ◦◦ + Di ◦◦
◦◦ ◦ Ti ◦ ◦◦
−− ) −− −* − Di ◦◦ + Di ◦◦
Kinetic equations for isotopomers ◦
• • dDe ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ νf νf νr ν = ASe((Te ◦◦•• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦◦◦ + Te ◦◦•• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦◦◦)(We • + We ◦)) + ADPeo (Do ◦◦) − ADPeo(De ◦◦) − ASer ((Pe •• + Pe •• + • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De ◦ ◦) •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
◦
◦ De ◦ •
• • d ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ νf νf νr ν = ASe((Te ◦••• + Te ◦••◦ + Te ◦•◦• + Te ◦•◦◦ + Te ◦••• + Te ◦••◦ + Te ◦•◦• + Te ◦•◦◦)(We • + We ◦)) + ADPeo (Do ◦•) − ADPeo(De ◦•) − ASer ((Pe •• + Pe •• + • ◦ dt • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • • ◦ ◦ • • ◦ ◦ ◦ ◦ • ◦ ◦ ◦ ◦ • • • Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De ◦ •) •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
◦
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
5
◦
• • dDe •◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ νf νf νr ν (Do •◦) − ADPeo(De •◦) − ASer ((Pe •• + Pe •• + = ASe((Te •◦•• + Te •◦•◦ + Te •◦◦• + Te •◦◦◦ + Te •◦•• + Te •◦•◦ + Te •◦◦• + Te •◦◦◦)(We • + We ◦)) + ADPeo • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De • ) ◦ •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
◦
◦ De • •
• • d ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ νf νf νr ν (Do ••) − ADPeo(De ••) − ASer ((Pe •• + Pe •• + = ASe((Te •••• + Te •••◦ + Te ••◦• + Te ••◦◦ + Te •••• + Te •••◦ + Te ••◦• + Te ••◦◦)(We • + We ◦)) + ADPeo • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ • Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De •) •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
◦
• De ◦ ◦
• • d •• •• •• •• •◦ •◦ •◦ •◦ • • νf νf νr ν (Do ◦◦) − ADPeo(De ◦◦) − ASer ((Pe •• + Pe •• + = ASe((Te ◦◦•• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦◦◦ + Te ◦◦•• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦◦◦)(We • + We ◦)) + ADPeo • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ ◦ Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De ◦) •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
◦
• De ◦ •
• • d •• •• •• •• •◦ •◦ •◦ •◦ • • νf νf νr ν (Do ◦•) − ADPeo(De ◦•) − ASer ((Pe •• + Pe •• + = ASe((Te ◦••• + Te ◦••◦ + Te ◦•◦• + Te ◦•◦◦ + Te ◦••• + Te ◦••◦ + Te ◦•◦• + Te ◦•◦◦)(We • + We ◦)) + ADPeo • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ ◦ Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De •) •
◦
•
◦
•
◦
•
◦
•
◦
•
◦
•
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• • d •• •• •• •• •◦ •◦ •◦ •◦ • • νf νf νr ν (Do •◦) − ADPeo(De •◦) − ASer ((Pe •• + Pe •• + = ASe((Te •◦•• + Te •◦•◦ + Te •◦◦• + Te •◦◦◦ + Te •◦•• + Te •◦•◦ + Te •◦◦• + Te •◦◦◦)(We • + We ◦)) + ADPeo • ◦ dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ • Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De ◦) •
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• • d • • •◦ •◦ •◦ •◦ •• •• •• •• νf νf ν νr (Do ••) − ADPeo(De ••) − ASer ((Pe •• + Pe •• + = ASe((Te •••• + Te •••◦ + Te ••◦• + Te ••◦◦ + Te •••• + Te •••◦ + Te ••◦• + Te ••◦◦)(We • + We ◦)) + ADPeo ◦ • dt • • • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦ + Pe • + Pe • + Pe ◦ + Pe ◦)De • •) •
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d ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ •◦ •◦ •◦ ◦ •◦ ◦◦ νf νf = ADPoi(Do ◦◦) + AKi(2Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦•• + Ti ◦◦•◦ + Ti ◦◦◦• + Ti ◦◦◦◦ + Ti ◦◦•• + Ti ◦◦•◦ + Ti ◦◦◦•) + dt ◦• • • • • ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• νf νf νr νr νr ◦ ◦ • • ◦ ◦ ◦ ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• • CKi(Ti ◦ ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦) + ADPim(Dm ◦) − ADPim(Di ◦) − ADPoi(Di ◦) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ◦◦) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ◦◦) ◦
dDi ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ •◦ •◦ •◦ •◦ ◦ ◦◦ νf νf = ADPoi(Do ◦•) + AKi(2Ti ◦•◦• + Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦• + Ti ••◦• + Ti •◦◦• + Ti ◦••• + Ti ◦••◦ + Ti ◦•◦• + Ti ◦•◦◦ + Ti ◦••• + Ti ◦••◦ + Ti ◦•◦◦ + Ti ◦◦◦•) + dt ◦• • • • • ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• νf νf νr νr νr ◦ ◦ • • ◦ ◦ ◦ ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• • CKi(Ti ◦ •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦) + ADPim(Dm •) − ADPim(Di •) − ADPoi(Di •) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ◦•) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ◦•) ◦
dDi •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ •◦ •◦ •◦ •◦ ◦◦ ◦ νf νf = ADPoi(Do •◦) + AKi(2Ti •◦•◦ + Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦ + Ti •••◦ + Ti •◦•• + Ti •◦•◦ + Ti •◦◦• + Ti •◦◦◦ + Ti •◦•• + Ti •◦◦• + Ti •◦◦◦ + Ti ◦••◦ + Ti ◦◦•◦) + dt ◦• • • • ◦ ◦ • ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• νf νf νr νr νr ◦ ◦ • • • • • •◦ •◦ •• •• •◦ •◦ • •• CKi(Ti • ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦) + ADPim(Dm ◦) − ADPim(Di ◦) − ADPoi(Di ◦) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di •◦) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di •◦) ◦
dDi •• ◦ ◦◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• •◦ •◦ •◦ νf νf = ADPoi(Do ••) + AKi(2Ti •••• + Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦•• + Ti •••• + Ti •••◦ + Ti ••◦• + Ti ••◦◦ + Ti •••◦ + Ti ••◦• + Ti ••◦◦ + Ti •◦•• + Ti ◦••• + Ti ◦◦••) + dt ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ • • • • νf νf νr νr νr • •• •◦ •◦ •• •• •◦ •◦ • • • • • ◦ ◦ CKi(Ti • •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦) + ADPim(Dm •) − ADPim(Di •) − ADPoi(Di •) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ••) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ••) •
dDi ◦◦ • •• •• •• •• •• •• •• •◦ •◦ •◦ •◦ ◦• ◦• ◦• ◦• νf νf = ADPoi(Do ◦◦) + AKi(2Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦•• + Ti ◦◦•◦ + Ti ◦◦◦• + Ti ◦◦•• + Ti ◦◦•◦ + Ti ◦◦◦• + Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦) + dt •• •• •• •• •◦ •◦ •◦ •◦ • • • • • • • νf νf νr νr νr • ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦ ◦ ◦ • • ◦ ◦ CKi(Ti ◦ ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦) + ADPim(Dm ◦) − ADPim(Di ◦) − ADPoi(Di ◦) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ◦◦) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ◦◦) •
dDi ◦• • •• •• •• •• •• •• •◦ •◦ •◦ •◦ •• ◦• ◦• ◦• ◦• νf νf = ADPoi(Do ◦•) + AKi(2Ti ◦•◦• + Ti ••◦• + Ti •◦◦• + Ti ◦••• + Ti ◦••◦ + Ti ◦•◦◦ + Ti ◦••• + Ti ◦••◦ + Ti ◦•◦• + Ti ◦•◦◦ + Ti ◦◦◦• + Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦•) + dt •• •◦ •◦ •◦ • • • • • • • •• •• •• •◦ νf νf νr νr νr ◦◦ ◦◦ ◦ ◦ ◦ • • ◦ ◦ • ◦• ◦◦ ◦◦ ◦• ◦• CKi(Ti ◦ •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦) + ADPim(Dm •) − ADPim(Di •) − ADPoi(Di •) − AKi(2(Di • + Di ◦ + Di • + Di ◦ +
Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ◦•) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ◦•) •
dDi •◦ •• • •• •• •• •• •◦ •◦ •◦ •◦ •• •• ◦• ◦• ◦• ◦• νf νf = ADPoi(Do •◦) + AKi(2Ti •◦•◦ + Ti •••◦ + Ti •◦•• + Ti •◦◦• + Ti •◦◦◦ + Ti •◦•• + Ti •◦•◦ + Ti •◦◦• + Ti •◦◦◦ + Ti ◦••◦ + Ti ◦◦•◦ + Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦) + dt •• •• •• •• •◦ •◦ •◦ •◦ • • • • • • • νf νf νr νr νr • •• •◦ •◦ •• •• •◦ •◦ • • • • • ◦ ◦ CKi(Ti • ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦ + Ti ◦• + Ti ◦◦) + ADPim(Dm ◦) − ADPim(Di ◦) − ADPoi(Di ◦) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di •◦) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di •◦) •
dDi •• • •• •• •• •• •◦ •◦ •◦ •◦ •• •• •• ◦• ◦• ◦• ◦• νf νf = ADPoi(Do ••) + AKi(2Ti •••• + Ti •••◦ + Ti ••◦• + Ti ••◦◦ + Ti •••• + Ti •••◦ + Ti ••◦• + Ti ••◦◦ + Ti •◦•• + Ti ◦••• + Ti ◦◦•• + Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦••) + dt •• •• •• •• •◦ •◦ •◦ •◦ • • • • • • • νf νf νr νr νr • •• •◦ •◦ •• •• •◦ •◦ • • • • • ◦ ◦ CKi(Ti • •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦ + Ti •• + Ti •◦) + ADPim(Dm •) − ADPim(Di •) − ADPoi(Di •) − AKi(2(Di • + Di ◦ + Di • + Di ◦ + ◦
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+ Di •◦ + Di ◦• + Di ◦◦)Di ••) − CKir ((Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦ + Ci •• + Ci •◦ + Ci ◦• + Ci ◦◦)Di ••) ◦
dDm ◦◦ ◦ ◦ ◦ ◦ νf νf νr νr = ADPim(Di ◦◦) + ADPms (Ds ◦◦) − ADPms(Dm ◦◦) − ADPim (Dm ◦◦) dt ◦
dDm ◦• ◦ ◦ ◦ ◦ νf νf νr νr = ADPim(Di ◦•) + ADPms (Ds ◦•) − ADPms(Dm ◦•) − ADPim (Dm ◦•) dt ◦
dDm •◦ ◦ ◦ ◦ ◦ νf νf νr νr = ADPim(Di •◦) + ADPms (Ds •◦) − ADPms(Dm •◦) − ADPim (Dm •◦) dt ◦
dDm •• ◦ ◦ ◦ ◦ νf νf νr νr = ADPim(Di ••) + ADPms (Ds ••) − ADPms(Dm ••) − ADPim (Dm ••) dt •
dDm ◦◦ • • • • νf νf νr νr (Dm ◦◦) (Ds ◦◦) − ADPms(Dm ◦◦) − ADPim = ADPim(Di ◦◦) + ADPms dt •
dDm ◦• • • • • νf νf νr νr = ADPim(Di ◦•) + ADPms (Ds ◦•) − ADPms(Dm ◦•) − ADPim (Dm ◦•) dt •
dDm •◦ • • • • νf νf νr νr = ADPim(Di •◦) + ADPms (Ds •◦) − ADPms(Dm •◦) − ADPim (Dm •◦) dt •
dDm •• • • • • νf νf νr νr (Dm ••) (Ds ••) − ADPms(Dm ••) − ADPim = ADPim(Di ••) + ADPms dt ◦
dDo ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ •◦ •◦ •◦ ◦ •◦ ◦◦ νf νf = ADPeo(De ◦◦) + AKo(2To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦•• + To ◦◦•◦ + To ◦◦◦• + To ◦◦◦◦ + To ◦◦•• + To ◦◦•◦ + dt◦◦ • • ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• νf νf νr νr νr • • ◦ ◦ ◦ ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦ To ◦ ◦•) + CKo(To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦) + ADPoi(Di ◦) − ADPoi(Do ◦) − ADPeo(Do ◦) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do ◦◦) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do ◦◦) ◦
dDo ◦• ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ •◦ •◦ ◦◦ •◦ ◦ ◦◦ •◦ νf νf = ADPeo(De ◦•) + AKo(2To ◦•◦• + To ••◦• + To •◦◦• + To ◦•◦• + To ◦◦◦• + To ••◦• + To •◦◦• + To ◦••• + To ◦••◦ + To ◦•◦• + To ◦•◦◦ + To ◦••• + To ◦••◦ + To ◦•◦◦ + dt◦◦ • ◦ ◦ • ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• νf νf νr νr νr • ◦ ◦ • ◦ ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦ To ◦ ◦•) + CKo(To •• + To •◦ + To •• + To •◦ + To •• + To •◦ + To •• + To •◦) + ADPoi(Di •) − ADPoi(Do •) − ADPeo(Do •) − AKo(2(Do • + Do ◦ + •
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dDo •◦ ◦ ◦◦ •◦ •◦ •◦ •◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ νf νf = ADPeo(De •◦) + AKo(2To •◦•◦ + To •••◦ + To •◦•◦ + To ◦••◦ + To ◦◦•◦ + To •••◦ + To •◦•• + To •◦•◦ + To •◦◦• + To •◦◦◦ + To •◦•• + To •◦◦• + To •◦◦◦ + To ◦••◦ + dt◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ • • νf νf νr νr νr • •• •• •◦ •◦ •• •• •◦ •◦ • • • • • To ◦ ◦◦) + CKo(To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦) + ADPoi(Di ◦) − ADPoi(Do ◦) − ADPeo(Do ◦) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do •◦) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do •◦) ◦
dDo •• ◦ ◦◦ •◦ •◦ •◦ •◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf = ADPeo(De ••) + AKo(2To •••• + To •••• + To •◦•• + To ◦••• + To ◦◦•• + To •••• + To •••◦ + To ••◦• + To ••◦◦ + To •••◦ + To ••◦• + To ••◦◦ + To •◦•• + To ◦••• + dt◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ • • νf νf νr νr νr •• • •• •◦ •◦ •• •• •◦ •◦ • • • • • To ◦ ◦•) + CKo(To •• + To •◦ + To •• + To •◦ + To •• + To •◦ + To •• + To •◦) + ADPoi(Di •) − ADPoi(Do •) − ADPeo(Do •) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do ••) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do ••) •
dDo ◦◦ •◦ ◦• ◦• ◦• •• •• •• •• •• •• •◦ •◦ •◦ • •• νf νf = ADPeo(De ◦◦) + AKo(2To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦•• + To ◦◦•◦ + To ◦◦◦• + To ◦◦•• + To ◦◦•◦ + To ◦◦◦• + To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + dt
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
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◦• •• •• •• •• •◦ •◦ •◦ •◦ • • • • • νf νf νr νr νr ◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦ ◦ ◦ • • To ◦ ◦◦) + CKo(To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦) + ADPoi(Di ◦) − ADPoi(Do ◦) − ADPeo(Do ◦) − AKo(2(Do • + Do ◦ + • • • • ◦ ◦ ◦ ◦ • • • ◦ ◦ ◦ ◦ • ν r • • ◦ ◦ • • ◦ ◦ ◦ ◦ • • ◦ ◦ ◦ Do ◦ • + Do ◦ + Do • + Do ◦ + Do • + Do ◦)Do ◦) − CKo((Co • + Co ◦ + Co • + Co ◦ + Co • + Co ◦ + Co • + Co ◦)Do ◦) •
dDo ◦• • •• •• •• •• •• •• •◦ •◦ •◦ •◦ •• ◦• ◦• ◦• νf νf = ADPeo(De ◦•) + AKo(2To ◦•◦• + To ••◦• + To •◦◦• + To ◦••• + To ◦••◦ + To ◦•◦◦ + To ◦••• + To ◦••◦ + To ◦•◦• + To ◦•◦◦ + To ◦◦◦• + To ••◦• + To •◦◦• + To ◦•◦• + dt◦• •• •• •• •• •◦ •◦ •◦ •◦ • • • • • νf νf νr νr νr ◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦ ◦ ◦ • • To ◦ ◦•) + CKo(To •• + To •◦ + To •• + To •◦ + To •• + To •◦ + To •• + To •◦) + ADPoi(Di •) − ADPoi(Do •) − ADPeo(Do •) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do ◦•) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do ◦•) •
dDo •◦ ◦• ◦• ◦• •• •• •◦ •◦ •◦ •◦ •• •• •• •• • •• νf νf = ADPeo(De •◦) + AKo(2To •◦•◦ + To •••◦ + To •◦•• + To •◦◦• + To •◦◦◦ + To •◦•• + To •◦•◦ + To •◦◦• + To •◦◦◦ + To ◦••◦ + To ◦◦•◦ + To •••◦ + To •◦•◦ + To ◦••◦ + dt◦• •• •• •• •• •◦ •◦ •◦ •◦ • • • • • νf νf νr νr νr • •• •• •◦ •◦ •• •• •◦ •◦ • • • • • To ◦ ◦◦) + CKo(To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦ + To ◦• + To ◦◦) + ADPoi(Di ◦) − ADPoi(Do ◦) − ADPeo(Do ◦) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do •◦) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do •◦) •
dDo •• • •• •• •• •• •◦ •◦ •◦ •◦ •• •• •• ◦• ◦• ◦• νf νf = ADPeo(De ••) + AKo(2To •••• + To •••◦ + To ••◦• + To ••◦◦ + To •••• + To •••◦ + To ••◦• + To ••◦◦ + To •◦•• + To ◦••• + To ◦◦•• + To •••• + To •◦•• + To ◦••• + dt◦• •• •• •• •• •◦ •◦ •◦ •◦ • • • • • νf νf νr νr νr • •• •• •◦ •◦ •• •• •◦ •◦ • • • • • To ◦ ◦•) + CKo(To •• + To •◦ + To •• + To •◦ + To •• + To •◦ + To •• + To •◦) + ADPoi(Di •) − ADPoi(Do •) − ADPeo(Do •) − AKo(2(Do • + Do ◦ + •
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r + Do ◦◦ + Do •• + Do •◦ + Do ◦• + Do ◦◦)Do ••) − CKo ((Co •• + Co •◦ + Co ◦• + Co ◦◦ + Co •• + Co •◦ + Co ◦• + Co ◦◦)Do ••) ◦
• • • • dDs ◦◦ ◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ νf νf ν = ADPms(Dm ◦◦) + ASsr ((Ts ◦◦•• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦◦◦ + Ts ◦◦•• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + • ◦ • ◦ dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ν ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ r ◦ Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds ◦ ◦) − ADPms(Ds ◦) •
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• • • • d ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦ ◦• νf νf ν = ADPms(Dm ◦•) + ASsr ((Ts ◦••• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦•◦◦ + Ts ◦••• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦•◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + ◦ • ◦ • dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ν ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ r ◦ Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds ◦ •) − ADPms(Ds •) •
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• • • • d ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦ ◦• ◦• νf νf ν = ADPms(Dm •◦) + ASsr ((Ts •◦•• + Ts •◦•◦ + Ts •◦◦• + Ts •◦◦◦ + Ts •◦•• + Ts •◦•◦ + Ts •◦◦• + Ts •◦◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + ◦ • ◦ • dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ νr ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ • • Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds ◦) − ADPms(Ds ◦) •
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• • • • d ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦ ◦• ◦• νf νf ν = ADPms(Dm ••) + ASsr ((Ts •••• + Ts •••◦ + Ts ••◦• + Ts ••◦◦ + Ts •••• + Ts •••◦ + Ts ••◦• + Ts ••◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + ◦ • ◦ • dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ νr ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ • • Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds •) − ADPms(Ds •) •
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• • • • d •◦ •◦ •◦ •◦ •• •• • •• •• νf νf ν = ADPms(Dm ◦◦) + ASsr ((Ts ◦◦•• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦◦◦ + Ts ◦◦•• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + • ◦ ◦ • dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • νr ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ ◦ ◦ Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds ◦) − ADPms(Ds ◦) •
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• • • • d •◦ •◦ •◦ •◦ •• •• • •• •• νf νf ν = ADPms(Dm ◦•) + ASsr ((Ts ◦••• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦•◦◦ + Ts ◦••• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦•◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + ◦ • • ◦ dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • νr ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ ◦ ◦ Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds •) − ADPms(Ds •) •
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• • • • d • •• •• •• •• •◦ •◦ •◦ •◦ νf νf ν = ADPms(Dm •◦) + ASsr ((Ts •◦•• + Ts •◦•◦ + Ts •◦◦• + Ts •◦◦◦ + Ts •◦•• + Ts •◦•◦ + Ts •◦◦• + Ts •◦◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + • ◦ • ◦ dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ν ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ r • Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds • ◦) − ADPms(Ds ◦) •
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• • • • d • •• •• •• •• •◦ •◦ •◦ •◦ νf νf ν = ADPms(Dm ••) + ASsr ((Ts •••• + Ts •••◦ + Ts ••◦• + Ts ••◦◦ + Ts •••• + Ts •••◦ + Ts ••◦• + Ts ••◦◦)(Ws • + Ws ◦)) − ASs((Ps •• + Ps •• + Ps •◦ + Ps •◦ + • ◦ • ◦ dt • • • • ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ • • ν ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ r • Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦ + Ps • + Ps • + Ps ◦ + Ps ◦)Ds • •) − ADPms(Ds •) •
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= ATPoe(To ◦◦◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦◦)
Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦◦• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To ◦◦◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦•◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ◦◦•◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • • ◦ • • • ◦ ◦ ◦ dTe ◦◦•• ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ◦◦••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦◦◦ ◦• ◦ ◦• νf νf ν = ATPoe(To ◦◦◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe ◦◦◦• ◦ ◦• ◦• νf νf ν = ATPoe(To ◦◦◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦•) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe ◦◦•◦ ◦• ◦ ◦• νf νf ν = ATPoe(To ◦◦•◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦•◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • • ◦ dTe ◦◦•• ◦ ◦• ◦• νf νf ν = ATPoe(To ◦◦••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ◦◦) − ASe((We • + We ◦)Te ◦◦••) • ◦ • • • dt ◦◦
• ◦ ◦ ◦ ◦ dTe ◦•◦◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ◦•◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ◦•) − ASe((We • + We ◦)Te ◦•◦◦) ◦ ◦ ◦ • ◦ dt ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦•◦• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To ◦•◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦••◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ◦••◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦••◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
◦ ◦ ◦ ◦ • • • • • • dTe ◦••• ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ◦•••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•••) • ◦ • • • • • ◦ ◦ ◦ dt ◦•
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦•◦◦ ◦ ◦• ◦• νf νf ν = ATPoe(To ◦•◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe ◦•◦• ◦ ◦• ◦• νf νf ν = ATPoe(To ◦•◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦•) • • • ◦ • ◦ • • ◦ ◦ dt ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe ◦••◦ ◦• ◦ ◦• νf νf ν = ATPoe(To ◦••◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦••◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • dTe ◦••• ◦• ◦ ◦• νf νf ν = ATPoe(To ◦•••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ◦•) − ASe((We • + We ◦)Te ◦•••) ◦ • • • • dt ◦◦
◦ ◦ ◦ ◦ • dTe •◦◦◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To •◦◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De •◦) − ASe((We • + We ◦)Te •◦◦◦) • ◦ ◦ ◦ ◦ dt ◦◦
• • ◦ • ◦ ◦ • ◦ ◦ ◦ dTe •◦◦• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To •◦◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦•) ◦ ◦ ◦ ◦ ◦ • • ◦ • • dt ◦◦
• ◦ ◦ • • • ◦ ◦ ◦ ◦ dTe •◦•◦ ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To •◦•◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • • • • ◦ • ◦ ◦ ◦ dTe •◦•• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To •◦••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦••) ◦ ◦ • • • ◦ • ◦ • • dt ◦•
• ◦ • • ◦ ◦ • ◦ ◦ ◦ dTe •◦◦◦ ◦• ◦ ◦• νf νf ν = ATPoe(To •◦◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦◦) ◦ ◦ ◦ ◦ ◦ • • ◦ • • dt ◦•
◦ ◦ ◦ ◦ • • • • • • dTe •◦◦• ◦ ◦• ◦• νf νf ν = ATPoe(To •◦◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦•) • • • • ◦ • ◦ • ◦ ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe •◦•◦ ◦• ◦ ◦• νf νf ν = ATPoe(To •◦•◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦•◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • dTe •◦•• ◦ ◦• ◦• νf νf ν = ATPoe(To •◦••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De •◦) − ASe((We • + We ◦)Te •◦••) ◦ • • • • dt ◦◦
• ◦ ◦ ◦ ◦ dTe ••◦◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To ••◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ••) − ASe((We • + We ◦)Te ••◦◦) ◦ ◦ ◦ • ◦ dt ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ••◦• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To ••◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe •••◦ ◦ ◦◦ ◦◦ νf νf ν = ATPoe(To •••◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te •••◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • • ◦ • • • ◦ ◦ ◦ dTe •••• ◦◦ ◦ ◦◦ νf νf ν = ATPoe(To ••••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ••◦◦ ◦ ◦• ◦• νf νf ν = ATPoe(To ••◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦•
• • • • ◦ • • ◦ ◦ ◦ dTe ••◦• ◦ ◦• ◦• νf νf ν = ATPoe(To ••◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦•) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • ◦ • • • ◦ ◦ ◦ dTe •••◦ ◦• ◦ ◦• νf νf ν = ATPoe(To •••◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te •••◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • • ◦ dTe •••• ◦ ◦• ◦• νf νf ν = ATPoe(To ••••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ••) − ASe((We • + We ◦)Te ••••) • ◦ • • • dt •◦
◦ ◦ ◦ ◦ • dTe ◦◦◦◦ • •◦ •◦ νf νf ν = ATPoe(To ◦◦◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦◦) ◦ • ◦ ◦ ◦ dt •◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦◦• • •◦ •◦ νf νf ν = ATPoe(To ◦◦◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦•◦ • •◦ •◦ νf νf ν = ATPoe(To ◦◦•◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦•◦) ◦ ◦ • ◦ • ◦ • • ◦ ◦ dt •◦
• • • ◦ • • • ◦ ◦ ◦ dTe ◦◦•• •◦ • •◦ νf νf ν = ATPoe(To ◦◦••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦••) ◦ • • • ◦ ◦ • ◦ • • dt ••
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦◦◦◦ •• • •• νf νf ν = ATPoe(To ◦◦◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
◦ • ◦ • ◦ ◦ • • • • dTe ◦◦◦• •• • •• νf νf ν = ATPoe(To ◦◦◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦◦•) • ◦ • • • • ◦ • ◦ ◦ dt ••
• • • • • ◦ • ◦ ◦ ◦ dTe ◦◦•◦ •• • •• νf νf ν = ATPoe(To ◦◦•◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦◦) − ASe((We • + We ◦)Te ◦◦•◦) ◦ ◦ • • ◦ • • ◦ • • dt ••
• • • • ◦ dTe ◦◦•• •• • •• νf νf ν = ATPoe(To ◦◦••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ◦◦) − ASe((We • + We ◦)Te ◦◦••) • ◦ • • • dt •◦
◦ • ◦ ◦ ◦ dTe ◦•◦◦ •◦ • •◦ νf νf ν = ATPoe(To ◦•◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ◦•) − ASe((We • + We ◦)Te ◦•◦◦) ◦ ◦ ◦ ◦ • dt •◦
• ◦ • • ◦ ◦ • ◦ ◦ ◦ dTe ◦•◦• •◦ • •◦ νf νf ν = ATPoe(To ◦•◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦•) ◦ ◦ ◦ ◦ ◦ • • ◦ • • dt •◦
◦ ◦ ◦ ◦ • ◦ ◦ • • • dTe ◦••◦ • •◦ •◦ νf νf ν = ATPoe(To ◦••◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦••◦) • ◦ ◦ • ◦ • ◦ • ◦ ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart •◦
• • • ◦ • • • ◦ ◦ ◦ dTe ◦••• •◦ • •◦ νf νf ν = ATPoe(To ◦•••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•••) ◦ • • • ◦ ◦ • ◦ • • dt ••
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ◦•◦◦ • •• •• νf νf ν = ATPoe(To ◦•◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
• • • ◦ • • • ◦ ◦ ◦ dTe ◦•◦• • •• •• νf νf ν = ATPoe(To ◦•◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦•◦•) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • ◦ • • • ◦ ◦ ◦ dTe ◦••◦ •• • •• νf νf ν = ATPoe(To ◦••◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ◦•) − ASe((We • + We ◦)Te ◦••◦) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • ◦ • dTe ◦••• • •• •• νf νf ν = ATPoe(To ◦•••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ◦•) − ASe((We • + We ◦)Te ◦•••) ◦ • • • • dt •◦
◦ • ◦ ◦ ◦ dTe •◦◦◦ •◦ • •◦ νf νf ν = ATPoe(To •◦◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De •◦) − ASe((We • + We ◦)Te •◦◦◦) ◦ ◦ ◦ ◦ • dt •◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe •◦◦• • •◦ •◦ νf νf ν = ATPoe(To •◦◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
• • ◦ ◦ ◦ • • ◦ ◦ ◦ dTe •◦•◦ • •◦ •◦ νf νf ν = ATPoe(To •◦•◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
• • • ◦ • • • ◦ ◦ ◦ dTe •◦•• •◦ • •◦ νf νf ν = ATPoe(To •◦••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦••) ◦ • • • ◦ ◦ • ◦ • • dt ••
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe •◦◦◦ • •• •• νf νf ν = ATPoe(To •◦◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
◦ ◦ ◦ ◦ • • • • • • dTe •◦◦• • •• •• νf νf ν = ATPoe(To •◦◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦◦•) • ◦ • • • • • ◦ ◦ ◦ dt ••
• • • ◦ • • • ◦ ◦ ◦ dTe •◦•◦ • •• •• νf νf ν = ATPoe(To •◦•◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De •◦) − ASe((We • + We ◦)Te •◦•◦) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • • ◦ dTe •◦•• •• • •• νf νf ν = ATPoe(To •◦••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De •◦) − ASe((We • + We ◦)Te •◦••) • • • • ◦ dt •◦
◦ • ◦ ◦ ◦ dTe ••◦◦ •◦ • •◦ νf νf ν = ATPoe(To ••◦◦) + ASer ((Pe ◦◦ + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦))De ••) − ASe((We • + We ◦)Te ••◦◦) ◦ ◦ ◦ ◦ • dt •◦
• ◦ ◦ ◦ • • • ◦ ◦ ◦ dTe ••◦• •◦ • •◦ νf νf ν = ATPoe(To ••◦•) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
◦ ◦ ◦ • ◦ ◦ ◦ • • • dTe •••◦ •◦ • •◦ νf νf ν = ATPoe(To •••◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te •••◦) • ◦ • • ◦ • ◦ ◦ ◦ ◦ dt •◦
• • • • • ◦ • ◦ ◦ ◦ dTe •••• •◦ • •◦ νf νf ν = ATPoe(To ••••) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••••) ◦ ◦ • • ◦ • • ◦ • • dt ••
• ◦ ◦ • • • ◦ ◦ ◦ ◦ dTe ••◦◦ •• • •• νf νf ν = ATPoe(To ••◦◦) + ASer (( 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
• • • • • ◦ • ◦ ◦ ◦ dTe ••◦• •• • •• νf νf ν = ATPoe(To ••◦•) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te ••◦•) ◦ ◦ • • • ◦ • ◦ • • dt ••
• • • • • ◦ • ◦ ◦ ◦ dTe •••◦ •• • •• νf νf ν = ATPoe(To •••◦) + ASer (( 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••) + 61 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•))De ••) − ASe((We • + We ◦)Te •••◦) ◦ • ◦ ◦ • • • ◦ • • dt ••
◦ • • • • dTe •••• • •• •• νf νf ν = ATPoe(To ••••) + ASer ((Pe •• + 14 (Pe •• + Pe •◦ + Pe ◦• + Pe ••))De ••) − ASe((We • + We ◦)Te ••••) • • • • ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦◦
dTi ◦◦◦◦ ◦◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr ν νr ν (Ti ◦◦◦◦) = ATPmi(Tm ◦◦◦◦) + AKir (Di ◦◦Di ◦◦) + ATPio (To ◦◦◦◦) + CKir (Di ◦◦Ci ◦◦) − AKi(Ti ◦◦◦◦) − ATPio(Ti ◦◦◦◦) − CKi(Ti ◦◦◦◦) − ATPmi dt ◦◦
dTi ◦◦◦• ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦◦•) + AKir (Di ◦•Di ◦◦) + ATPio (To ◦◦◦•) + CKir (Di ◦◦Ci ◦•) − AKi(Ti ◦◦◦•) − ATPio(Ti ◦◦◦•) − CKi(Ti ◦◦◦•) − ATPmi (Ti ◦◦◦•) dt ◦◦
dTi ◦◦•◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦•◦) + AKir (Di •◦Di ◦◦) + ATPio (To ◦◦•◦) + CKir (Di ◦◦Ci •◦) − AKi(Ti ◦◦•◦) − ATPio(Ti ◦◦•◦) − CKi(Ti ◦◦•◦) − ATPmi (Ti ◦◦•◦) dt ◦◦
dTi ◦◦•• ◦ ◦ ◦◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr (Ti ◦◦••) (To ◦◦••) + CKir (Di ◦◦Ci ••) − AKi(Ti ◦◦••) − ATPio(Ti ◦◦••) − CKi(Ti ◦◦••) − ATPmi = ATPmi(Tm ◦◦••) + AKir (Di ••Di ◦◦) + ATPio dt ◦•
dTi ◦◦◦◦ ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦◦◦) + AKir (Di ◦◦Di ◦◦) + ATPio (To ◦◦◦◦) + CKir (Di ◦◦Ci ◦◦) − AKi(Ti ◦◦◦◦) − ATPio(Ti ◦◦◦◦) − CKi(Ti ◦◦◦◦) − ATPmi (Ti ◦◦◦◦) dt ◦•
dTi ◦◦◦• ◦• ◦ • • ◦ ◦• ◦• ◦• ◦• ◦• νf νf νf νf νr ν ν νr (Ti ◦◦◦•) (To ◦◦◦•) + CKir (Di ◦◦Ci ◦•) − AKi(Ti ◦◦◦•) − ATPio(Ti ◦◦◦•) − CKi(Ti ◦◦◦•) − ATPmi = ATPmi(Tm ◦◦◦•) + AKir (Di ◦•Di ◦◦) + ATPio dt ◦•
dTi ◦◦•◦ ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦•◦) + AKir (Di •◦Di ◦◦) + ATPio (To ◦◦•◦) + CKir (Di ◦◦Ci •◦) − AKi(Ti ◦◦•◦) − ATPio(Ti ◦◦•◦) − CKi(Ti ◦◦•◦) − ATPmi (Ti ◦◦•◦) dt ◦•
dTi ◦◦•• ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦••) + AKir (Di ••Di ◦◦) + ATPio (To ◦◦••) + CKir (Di ◦◦Ci ••) − AKi(Ti ◦◦••) − ATPio(Ti ◦◦••) − CKi(Ti ◦◦••) − ATPmi (Ti ◦◦••) dt ◦◦
dTi ◦•◦◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr (Ti ◦•◦◦) = ATPmi(Tm ◦•◦◦) + AKir (Di ◦•Di ◦◦) + ATPio (To ◦•◦◦) + CKir (Di ◦•Ci ◦◦) − AKi(Ti ◦•◦◦) − ATPio(Ti ◦•◦◦) − CKi(Ti ◦•◦◦) − ATPmi dt ◦◦
dTi ◦•◦• ◦◦ ◦◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ νf νf νf νf νr ν ν νr (Ti ◦•◦•) (To ◦•◦•) + CKir (Di ◦•Ci ◦•) − AKi(Ti ◦•◦•) − ATPio(Ti ◦•◦•) − CKi(Ti ◦•◦•) − ATPmi = ATPmi(Tm ◦•◦•) + AKir (Di ◦•Di ◦•) + ATPio dt ◦◦
dTi ◦••◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ νf νf νf νf νr ν νr ν (Ti ◦••◦) (To ◦••◦) + CKir (Di ◦•Ci •◦) − AKi(Ti ◦••◦) − ATPio(Ti ◦••◦) − CKi(Ti ◦••◦) − ATPmi = ATPmi(Tm ◦••◦) + AKir (Di •◦Di ◦•) + ATPio dt ◦◦
dTi ◦••• ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr ν ν νr = ATPmi(Tm ◦•••) + AKir (Di ••Di ◦•) + ATPio (To ◦•••) + CKir (Di ◦•Ci ••) − AKi(Ti ◦•••) − ATPio(Ti ◦•••) − CKi(Ti ◦•••) − ATPmi (Ti ◦•••) dt ◦•
dTi ◦•◦◦ ◦• ◦• ◦• ◦ • ◦• • ◦ ◦• ◦• νf νf νf νf νr ν ν νr (Ti ◦•◦◦) (To ◦•◦◦) + CKir (Di ◦•Ci ◦◦) − AKi(Ti ◦•◦◦) − ATPio(Ti ◦•◦◦) − CKi(Ti ◦•◦◦) − ATPmi = ATPmi(Tm ◦•◦◦) + AKir (Di ◦◦Di ◦•) + ATPio dt ◦•
dTi ◦•◦• ◦• ◦• ◦• ◦• • ◦ ◦• ◦ • ◦• νf νf νf νf νr ν νr ν (Ti ◦•◦•) (To ◦•◦•) + CKir (Di ◦•Ci ◦•) − AKi(Ti ◦•◦•) − ATPio(Ti ◦•◦•) − CKi(Ti ◦•◦•) − ATPmi = ATPmi(Tm ◦•◦•) + AKir (Di ◦•Di ◦•) + ATPio dt ◦•
dTi ◦••◦ ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr ν νr ν (Ti ◦••◦) = ATPmi(Tm ◦••◦) + AKir (Di •◦Di ◦•) + ATPio (To ◦••◦) + CKir (Di ◦•Ci •◦) − AKi(Ti ◦••◦) − ATPio(Ti ◦••◦) − CKi(Ti ◦••◦) − ATPmi dt ◦•
dTi ◦••• ◦• ◦• ◦• ◦• ◦ • ◦• • ◦ ◦• νf νf νf νf νr ν νr ν (Ti ◦•••) (To ◦•••) + CKir (Di ◦•Ci ••) − AKi(Ti ◦•••) − ATPio(Ti ◦•••) − CKi(Ti ◦•••) − ATPmi = ATPmi(Tm ◦•••) + AKir (Di ••Di ◦•) + ATPio dt ◦◦
dTi •◦◦◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr (To •◦◦◦) + CKir (Di •◦Ci ◦◦) − AKi(Ti •◦◦◦) − ATPio(Ti •◦◦◦) − CKi(Ti •◦◦◦) − ATPmi (Ti •◦◦◦) = ATPmi(Tm •◦◦◦) + AKir (Di •◦Di ◦◦) + ATPio dt ◦◦
dTi •◦◦• ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm •◦◦•) + AKir (Di •◦Di ◦•) + ATPio (To •◦◦•) + CKir (Di •◦Ci ◦•) − AKi(Ti •◦◦•) − ATPio(Ti •◦◦•) − CKi(Ti •◦◦•) − ATPmi (Ti •◦◦•) dt ◦◦
dTi •◦•◦ ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm •◦•◦) + AKir (Di •◦Di •◦) + ATPio (To •◦•◦) + CKir (Di •◦Ci •◦) − AKi(Ti •◦•◦) − ATPio(Ti •◦•◦) − CKi(Ti •◦•◦) − ATPmi (Ti •◦•◦) dt ◦◦
dTi •◦•• ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr (Ti •◦••) = ATPmi(Tm •◦••) + AKir (Di ••Di •◦) + ATPio (To •◦••) + CKir (Di •◦Ci ••) − AKi(Ti •◦••) − ATPio(Ti •◦••) − CKi(Ti •◦••) − ATPmi dt ◦•
dTi •◦◦◦ ◦• ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• νf νf νf νf νr ν νr ν (To •◦◦◦) + CKir (Di •◦Ci ◦◦) − AKi(Ti •◦◦◦) − ATPio(Ti •◦◦◦) − CKi(Ti •◦◦◦) − ATPmi (Ti •◦◦◦) = ATPmi(Tm •◦◦◦) + AKir (Di ◦◦Di •◦) + ATPio dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦•
dTi •◦◦• ◦• ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• νf νf νf νf νr ν νr ν (Ti •◦◦•) = ATPmi(Tm •◦◦•) + AKir (Di ◦•Di •◦) + ATPio (To •◦◦•) + CKir (Di •◦Ci ◦•) − AKi(Ti •◦◦•) − ATPio(Ti •◦◦•) − CKi(Ti •◦◦•) − ATPmi dt ◦•
dTi •◦•◦ ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm •◦•◦) + AKir (Di •◦Di •◦) + ATPio (To •◦•◦) + CKir (Di •◦Ci •◦) − AKi(Ti •◦•◦) − ATPio(Ti •◦•◦) − CKi(Ti •◦•◦) − ATPmi (Ti •◦•◦) dt ◦•
dTi •◦•• ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm •◦••) + AKir (Di ••Di •◦) + ATPio (To •◦••) + CKir (Di •◦Ci ••) − AKi(Ti •◦••) − ATPio(Ti •◦••) − CKi(Ti •◦••) − ATPmi (Ti •◦••) dt ◦◦
dTi ••◦◦ ◦ ◦ ◦◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr (Ti ••◦◦) (To ••◦◦) + CKir (Di ••Ci ◦◦) − AKi(Ti ••◦◦) − ATPio(Ti ••◦◦) − CKi(Ti ••◦◦) − ATPmi = ATPmi(Tm ••◦◦) + AKir (Di ••Di ◦◦) + ATPio dt ◦◦
dTi ••◦• ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ••◦•) + AKir (Di ••Di ◦•) + ATPio (To ••◦•) + CKir (Di ••Ci ◦•) − AKi(Ti ••◦•) − ATPio(Ti ••◦•) − CKi(Ti ••◦•) − ATPmi (Ti ••◦•) dt ◦◦
dTi •••◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr ν ν νr (Ti •••◦) (To •••◦) + CKir (Di ••Ci •◦) − AKi(Ti •••◦) − ATPio(Ti •••◦) − CKi(Ti •••◦) − ATPmi = ATPmi(Tm •••◦) + AKir (Di ••Di •◦) + ATPio dt ◦◦
dTi •••• ◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ••••) + AKir (Di ••Di ••) + ATPio (To ••••) + CKir (Di ••Ci ••) − AKi(Ti ••••) − ATPio(Ti ••••) − CKi(Ti ••••) − ATPmi (Ti ••••) dt ◦•
dTi ••◦◦ ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr = ATPmi(Tm ••◦◦) + AKir (Di ◦◦Di ••) + ATPio (To ••◦◦) + CKir (Di ••Ci ◦◦) − AKi(Ti ••◦◦) − ATPio(Ti ••◦◦) − CKi(Ti ••◦◦) − ATPmi (Ti ••◦◦) dt ◦•
dTi ••◦• ◦• • ◦ ◦• ◦ • ◦• ◦• ◦• ◦• νf νf νf νf ν νr ν νr (Ti ••◦•) = ATPmi(Tm ••◦•) + AKir (Di ◦•Di ••) + ATPio (To ••◦•) + CKir (Di ••Ci ◦•) − AKi(Ti ••◦•) − ATPio(Ti ••◦•) − CKi(Ti ••◦•) − ATPmi dt ◦•
dTi •••◦ ◦• ◦• ◦• ◦ • ◦• • ◦ ◦• ◦• νf νf νf νf νr ν ν νr (Ti •••◦) (To •••◦) + CKir (Di ••Ci •◦) − AKi(Ti •••◦) − ATPio(Ti •••◦) − CKi(Ti •••◦) − ATPmi = ATPmi(Tm •••◦) + AKir (Di •◦Di ••) + ATPio dt ◦•
dTi •••• ◦• ◦• ◦• ◦• • ◦ ◦• ◦ • ◦• νf νf νf νf νr ν νr ν (Ti ••••) (To ••••) + CKir (Di ••Ci ••) − AKi(Ti ••••) − ATPio(Ti ••••) − CKi(Ti ••••) − ATPmi = ATPmi(Tm ••••) + AKir (Di ••Di ••) + ATPio dt •◦
dTi ◦◦◦◦ • ◦ • ◦ •◦ •◦ •◦ •◦ •◦ •◦ νf νf νf νf νr ν ν νr = ATPmi(Tm ◦◦◦◦) + AKir (Di ◦◦Di ◦◦) + ATPio (To ◦◦◦◦) + CKir (Di ◦◦Ci ◦◦) − AKi(Ti ◦◦◦◦) − ATPio(Ti ◦◦◦◦) − CKi(Ti ◦◦◦◦) − ATPmi (Ti ◦◦◦◦) dt •◦
dTi ◦◦◦• •◦ •◦ •◦ • ◦ •◦ • ◦ •◦ •◦ νf νf νf νf νr ν ν νr (Ti ◦◦◦•) (To ◦◦◦•) + CKir (Di ◦◦Ci ◦•) − AKi(Ti ◦◦◦•) − ATPio(Ti ◦◦◦•) − CKi(Ti ◦◦◦•) − ATPmi = ATPmi(Tm ◦◦◦•) + AKir (Di ◦◦Di ◦•) + ATPio dt •◦
dTi ◦◦•◦ •◦ •◦ •◦ •◦ ◦ • •◦ ◦ • •◦ νf νf νf νf νr ν νr ν (Ti ◦◦•◦) (To ◦◦•◦) + CKir (Di ◦◦Ci •◦) − AKi(Ti ◦◦•◦) − ATPio(Ti ◦◦•◦) − CKi(Ti ◦◦•◦) − ATPmi = ATPmi(Tm ◦◦•◦) + AKir (Di ◦◦Di •◦) + ATPio dt •◦
dTi ◦◦•• •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr ν νr ν (Ti ◦◦••) = ATPmi(Tm ◦◦••) + AKir (Di ◦◦Di ••) + ATPio (To ◦◦••) + CKir (Di ◦◦Ci ••) − AKi(Ti ◦◦••) − ATPio(Ti ◦◦••) − CKi(Ti ◦◦••) − ATPmi dt ••
dTi ◦◦◦◦ •• •• •• •• • • •• • • •• νf νf νf νf νr ν νr ν (Ti ◦◦◦◦) (To ◦◦◦◦) + CKir (Di ◦◦Ci ◦◦) − AKi(Ti ◦◦◦◦) − ATPio(Ti ◦◦◦◦) − CKi(Ti ◦◦◦◦) − ATPmi = ATPmi(Tm ◦◦◦◦) + AKir (Di ◦◦Di ◦◦) + ATPio dt ••
dTi ◦◦◦• •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr (To ◦◦◦•) + CKir (Di ◦◦Ci ◦•) − AKi(Ti ◦◦◦•) − ATPio(Ti ◦◦◦•) − CKi(Ti ◦◦◦•) − ATPmi (Ti ◦◦◦•) = ATPmi(Tm ◦◦◦•) + AKir (Di ◦•Di ◦◦) + ATPio dt ••
dTi ◦◦•◦ •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦•◦) + AKir (Di •◦Di ◦◦) + ATPio (To ◦◦•◦) + CKir (Di ◦◦Ci •◦) − AKi(Ti ◦◦•◦) − ATPio(Ti ◦◦•◦) − CKi(Ti ◦◦•◦) − ATPmi (Ti ◦◦•◦) dt ••
dTi ◦◦•• •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦◦••) + AKir (Di ••Di ◦◦) + ATPio (To ◦◦••) + CKir (Di ◦◦Ci ••) − AKi(Ti ◦◦••) − ATPio(Ti ◦◦••) − CKi(Ti ◦◦••) − ATPmi (Ti ◦◦••) dt •◦
dTi ◦•◦◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr (Ti ◦•◦◦) = ATPmi(Tm ◦•◦◦) + AKir (Di ◦•Di ◦◦) + ATPio (To ◦•◦◦) + CKir (Di ◦•Ci ◦◦) − AKi(Ti ◦•◦◦) − ATPio(Ti ◦•◦◦) − CKi(Ti ◦•◦◦) − ATPmi dt •◦
dTi ◦•◦• •◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ νf νf νf νf νr ν νr ν (To ◦•◦•) + CKir (Di ◦•Ci ◦•) − AKi(Ti ◦•◦•) − ATPio(Ti ◦•◦•) − CKi(Ti ◦•◦•) − ATPmi (Ti ◦•◦•) = ATPmi(Tm ◦•◦•) + AKir (Di ◦•Di ◦•) + ATPio dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart •◦
dTi ◦••◦ •◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ νf νf νf νf νr ν νr ν (Ti ◦••◦) = ATPmi(Tm ◦••◦) + AKir (Di ◦•Di •◦) + ATPio (To ◦••◦) + CKir (Di ◦•Ci •◦) − AKi(Ti ◦••◦) − ATPio(Ti ◦••◦) − CKi(Ti ◦••◦) − ATPmi dt •◦
dTi ◦••• •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ◦•••) + AKir (Di ◦•Di ••) + ATPio (To ◦•••) + CKir (Di ◦•Ci ••) − AKi(Ti ◦•••) − ATPio(Ti ◦•••) − CKi(Ti ◦•••) − ATPmi (Ti ◦•••) dt ••
dTi ◦•◦◦ •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦•◦◦) + AKir (Di ◦•Di ◦◦) + ATPio (To ◦•◦◦) + CKir (Di ◦•Ci ◦◦) − AKi(Ti ◦•◦◦) − ATPio(Ti ◦•◦◦) − CKi(Ti ◦•◦◦) − ATPmi (Ti ◦•◦◦) dt ••
dTi ◦•◦• • • •• •• • • •• •• •• •• νf νf νf νf ν νr ν νr (Ti ◦•◦•) (To ◦•◦•) + CKir (Di ◦•Ci ◦•) − AKi(Ti ◦•◦•) − ATPio(Ti ◦•◦•) − CKi(Ti ◦•◦•) − ATPmi = ATPmi(Tm ◦•◦•) + AKir (Di ◦•Di ◦•) + ATPio dt ••
dTi ◦••◦ •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr = ATPmi(Tm ◦••◦) + AKir (Di •◦Di ◦•) + ATPio (To ◦••◦) + CKir (Di ◦•Ci •◦) − AKi(Ti ◦••◦) − ATPio(Ti ◦••◦) − CKi(Ti ◦••◦) − ATPmi (Ti ◦••◦) dt ••
dTi ◦••• •• • • • • •• •• •• •• •• νf νf νf νf νr ν ν νr (Ti ◦•••) (To ◦•••) + CKir (Di ◦•Ci ••) − AKi(Ti ◦•••) − ATPio(Ti ◦•••) − CKi(Ti ◦•••) − ATPmi = ATPmi(Tm ◦•••) + AKir (Di ••Di ◦•) + ATPio dt •◦
dTi •◦◦◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr = ATPmi(Tm •◦◦◦) + AKir (Di •◦Di ◦◦) + ATPio (To •◦◦◦) + CKir (Di •◦Ci ◦◦) − AKi(Ti •◦◦◦) − ATPio(Ti •◦◦◦) − CKi(Ti •◦◦◦) − ATPmi (Ti •◦◦◦) dt •◦
dTi •◦◦• •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr = ATPmi(Tm •◦◦•) + AKir (Di •◦Di ◦•) + ATPio (To •◦◦•) + CKir (Di •◦Ci ◦•) − AKi(Ti •◦◦•) − ATPio(Ti •◦◦•) − CKi(Ti •◦◦•) − ATPmi (Ti •◦◦•) dt •◦
dTi •◦•◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr (Ti •◦•◦) = ATPmi(Tm •◦•◦) + AKir (Di •◦Di •◦) + ATPio (To •◦•◦) + CKir (Di •◦Ci •◦) − AKi(Ti •◦•◦) − ATPio(Ti •◦•◦) − CKi(Ti •◦•◦) − ATPmi dt •◦
dTi •◦•• •◦ •◦ •◦ • ◦ •◦ • ◦ •◦ •◦ νf νf νf νf νr ν ν νr (Ti •◦••) (To •◦••) + CKir (Di •◦Ci ••) − AKi(Ti •◦••) − ATPio(Ti •◦••) − CKi(Ti •◦••) − ATPmi = ATPmi(Tm •◦••) + AKir (Di •◦Di ••) + ATPio dt ••
dTi •◦◦◦ •• •• •• •• • • •• • • •• νf νf νf νf νr ν νr ν (Ti •◦◦◦) (To •◦◦◦) + CKir (Di •◦Ci ◦◦) − AKi(Ti •◦◦◦) − ATPio(Ti •◦◦◦) − CKi(Ti •◦◦◦) − ATPmi = ATPmi(Tm •◦◦◦) + AKir (Di •◦Di ◦◦) + ATPio dt ••
dTi •◦◦• • • • • •• •• •• •• •• •• νf νf νf νf νr ν ν νr = ATPmi(Tm •◦◦•) + AKir (Di •◦Di ◦•) + ATPio (To •◦◦•) + CKir (Di •◦Ci ◦•) − AKi(Ti •◦◦•) − ATPio(Ti •◦◦•) − CKi(Ti •◦◦•) − ATPmi (Ti •◦◦•) dt ••
dTi •◦•◦ •• •• •• • • •• • • •• •• νf νf νf νf νr ν ν νr (Ti •◦•◦) (To •◦•◦) + CKir (Di •◦Ci •◦) − AKi(Ti •◦•◦) − ATPio(Ti •◦•◦) − CKi(Ti •◦•◦) − ATPmi = ATPmi(Tm •◦•◦) + AKir (Di •◦Di •◦) + ATPio dt ••
dTi •◦•• •• •• •• •• • • •• • • •• νf νf νf νf νr ν νr ν (Ti •◦••) (To •◦••) + CKir (Di •◦Ci ••) − AKi(Ti •◦••) − ATPio(Ti •◦••) − CKi(Ti •◦••) − ATPmi = ATPmi(Tm •◦••) + AKir (Di ••Di •◦) + ATPio dt •◦
dTi ••◦◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr ν νr ν (Ti ••◦◦) = ATPmi(Tm ••◦◦) + AKir (Di ••Di ◦◦) + ATPio (To ••◦◦) + CKir (Di ••Ci ◦◦) − AKi(Ti ••◦◦) − ATPio(Ti ••◦◦) − CKi(Ti ••◦◦) − ATPmi dt •◦
dTi ••◦• •◦ •◦ •◦ •◦ • ◦ •◦ • ◦ •◦ νf νf νf νf νr ν νr ν (Ti ••◦•) (To ••◦•) + CKir (Di ••Ci ◦•) − AKi(Ti ••◦•) − ATPio(Ti ••◦•) − CKi(Ti ••◦•) − ATPmi = ATPmi(Tm ••◦•) + AKir (Di ••Di ◦•) + ATPio dt •◦
dTi •••◦ •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr (To •••◦) + CKir (Di ••Ci •◦) − AKi(Ti •••◦) − ATPio(Ti •••◦) − CKi(Ti •••◦) − ATPmi (Ti •••◦) = ATPmi(Tm •••◦) + AKir (Di ••Di •◦) + ATPio dt •◦
dTi •••• •◦ • ◦ •◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf ν νr ν νr = ATPmi(Tm ••••) + AKir (Di ••Di ••) + ATPio (To ••••) + CKir (Di ••Ci ••) − AKi(Ti ••••) − ATPio(Ti ••••) − CKi(Ti ••••) − ATPmi (Ti ••••) dt ••
dTi ••◦◦ •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr = ATPmi(Tm ••◦◦) + AKir (Di ••Di ◦◦) + ATPio (To ••◦◦) + CKir (Di ••Ci ◦◦) − AKi(Ti ••◦◦) − ATPio(Ti ••◦◦) − CKi(Ti ••◦◦) − ATPmi (Ti ••◦◦) dt ••
dTi ••◦• •• • • •• • • •• •• •• •• νf νf νf νf ν νr ν νr (Ti ••◦•) = ATPmi(Tm ••◦•) + AKir (Di ••Di ◦•) + ATPio (To ••◦•) + CKir (Di ••Ci ◦•) − AKi(Ti ••◦•) − ATPio(Ti ••◦•) − CKi(Ti ••◦•) − ATPmi dt ••
dTi •••◦ •• •• • • •• • • •• •• •• νf νf νf νf νr ν νr ν (To •••◦) + CKir (Di ••Ci •◦) − AKi(Ti •••◦) − ATPio(Ti •••◦) − CKi(Ti •••◦) − ATPmi (Ti •••◦) = ATPmi(Tm •••◦) + AKir (Di ••Di •◦) + ATPio dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
dTi •••• •• •• • • •• • • •• •• •• νf νf νf νf νr ν νr ν (Ti ••••) = ATPmi(Tm ••••) + AKir (Di ••Di ••) + ATPio (To ••••) + CKir (Di ••Ci ••) − AKi(Ti ••••) − ATPio(Ti ••••) − CKi(Ti ••••) − ATPmi dt ◦◦
dTm ◦◦◦◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ◦◦◦◦) + ATPmi (Ti ◦◦◦◦) − ATPmi(Tm ◦◦◦◦) dt ◦◦
dTm ◦◦◦• ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ◦◦◦•) + ATPmi (Ti ◦◦◦•) − ATPmi(Tm ◦◦◦•) dt ◦◦
dTm ◦◦•◦ ◦◦ ◦◦ ◦◦ νf νf νr (Ti ◦◦•◦) − ATPmi(Tm ◦◦•◦) = ATPsm(Ts ◦◦•◦) + ATPmi dt ◦◦
dTm ◦◦•• ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ◦◦••) + ATPmi (Ti ◦◦••) − ATPmi(Tm ◦◦••) dt ◦•
dTm ◦◦◦◦ ◦• ◦• ◦• νf νf νr (Ti ◦◦◦◦) − ATPmi(Tm ◦◦◦◦) = ATPsm(Ts ◦◦◦◦) + ATPmi dt ◦•
dTm ◦◦◦• ◦• ◦• ◦• νf νf νr = ATPsm(Ts ◦◦◦•) + ATPmi (Ti ◦◦◦•) − ATPmi(Tm ◦◦◦•) dt ◦•
dTm ◦◦•◦ ◦• ◦• ◦• νf νf νr = ATPsm(Ts ◦◦•◦) + ATPmi (Ti ◦◦•◦) − ATPmi(Tm ◦◦•◦) dt ◦•
dTm ◦◦•• ◦• ◦• ◦• νf νf νr = ATPsm(Ts ◦◦••) + ATPmi (Ti ◦◦••) − ATPmi(Tm ◦◦••) dt ◦◦
dTm ◦•◦◦ ◦◦ ◦◦ ◦◦ νf νf νr (Ti ◦•◦◦) − ATPmi(Tm ◦•◦◦) = ATPsm(Ts ◦•◦◦) + ATPmi dt ◦◦
dTm ◦•◦• ◦◦ ◦◦ ◦◦ νf νf νr (Ti ◦•◦•) − ATPmi(Tm ◦•◦•) = ATPsm(Ts ◦•◦•) + ATPmi dt ◦◦
dTm ◦••◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ◦••◦) + ATPmi (Ti ◦••◦) − ATPmi(Tm ◦••◦) dt ◦◦
dTm ◦••• ◦◦ ◦◦ ◦◦ νf νf νr (Ti ◦•••) − ATPmi(Tm ◦•••) = ATPsm(Ts ◦•••) + ATPmi dt ◦•
dTm ◦•◦◦ ◦• ◦• ◦• νf νf νr (Ti ◦•◦◦) − ATPmi(Tm ◦•◦◦) = ATPsm(Ts ◦•◦◦) + ATPmi dt ◦•
dTm ◦•◦• ◦• ◦• ◦• νf νf νr = ATPsm(Ts ◦•◦•) + ATPmi (Ti ◦•◦•) − ATPmi(Tm ◦•◦•) dt ◦•
dTm ◦••◦ ◦• ◦• ◦• νf νf νr = ATPsm(Ts ◦••◦) + ATPmi (Ti ◦••◦) − ATPmi(Tm ◦••◦) dt ◦•
dTm ◦••• ◦• ◦• ◦• νf νf νr (Ti ◦•••) − ATPmi(Tm ◦•••) = ATPsm(Ts ◦•••) + ATPmi dt ◦◦
dTm •◦◦◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts •◦◦◦) + ATPmi (Ti •◦◦◦) − ATPmi(Tm •◦◦◦) dt ◦◦
dTm •◦◦• ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts •◦◦•) + ATPmi (Ti •◦◦•) − ATPmi(Tm •◦◦•) dt ◦◦
dTm •◦•◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts •◦•◦) + ATPmi (Ti •◦•◦) − ATPmi(Tm •◦•◦) dt ◦◦
dTm •◦•• ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts •◦••) + ATPmi (Ti •◦••) − ATPmi(Tm •◦••) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦•
dTm •◦◦◦ ◦• ◦• ◦• νf νf νr = ATPsm(Ts •◦◦◦) + ATPmi (Ti •◦◦◦) − ATPmi(Tm •◦◦◦) dt ◦•
dTm •◦◦• ◦• ◦• ◦• νf νf νr = ATPsm(Ts •◦◦•) + ATPmi (Ti •◦◦•) − ATPmi(Tm •◦◦•) dt ◦•
dTm •◦•◦ ◦• ◦• ◦• νf νf νr = ATPsm(Ts •◦•◦) + ATPmi (Ti •◦•◦) − ATPmi(Tm •◦•◦) dt ◦•
dTm •◦•• ◦• ◦• ◦• νf νf νr (Ti •◦••) − ATPmi(Tm •◦••) = ATPsm(Ts •◦••) + ATPmi dt ◦◦
dTm ••◦◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ••◦◦) + ATPmi (Ti ••◦◦) − ATPmi(Tm ••◦◦) dt ◦◦
dTm ••◦• ◦◦ ◦◦ ◦◦ νf νf νr (Ti ••◦•) − ATPmi(Tm ••◦•) = ATPsm(Ts ••◦•) + ATPmi dt ◦◦
dTm •••◦ ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts •••◦) + ATPmi (Ti •••◦) − ATPmi(Tm •••◦) dt ◦◦
dTm •••• ◦◦ ◦◦ ◦◦ νf νf νr = ATPsm(Ts ••••) + ATPmi (Ti ••••) − ATPmi(Tm ••••) dt ◦•
dTm ••◦◦ ◦• ◦• ◦• νf νf νr = ATPsm(Ts ••◦◦) + ATPmi (Ti ••◦◦) − ATPmi(Tm ••◦◦) dt ◦•
dTm ••◦• ◦• ◦• ◦• νf νf νr (Ti ••◦•) − ATPmi(Tm ••◦•) = ATPsm(Ts ••◦•) + ATPmi dt ◦•
dTm •••◦ ◦• ◦• ◦• νf νf νr (Ti •••◦) − ATPmi(Tm •••◦) = ATPsm(Ts •••◦) + ATPmi dt ◦•
dTm •••• ◦• ◦• ◦• νf νf νr = ATPsm(Ts ••••) + ATPmi (Ti ••••) − ATPmi(Tm ••••) dt •◦
dTm ◦◦◦◦ •◦ •◦ •◦ νf νf νr (Ti ◦◦◦◦) − ATPmi(Tm ◦◦◦◦) = ATPsm(Ts ◦◦◦◦) + ATPmi dt •◦
dTm ◦◦◦• •◦ •◦ •◦ νf νf νr (Ti ◦◦◦•) − ATPmi(Tm ◦◦◦•) = ATPsm(Ts ◦◦◦•) + ATPmi dt •◦
dTm ◦◦•◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦◦•◦) + ATPmi (Ti ◦◦•◦) − ATPmi(Tm ◦◦•◦) dt •◦
dTm ◦◦•• •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦◦••) + ATPmi (Ti ◦◦••) − ATPmi(Tm ◦◦••) dt ••
dTm ◦◦◦◦ •• •• •• νf νf νr (Ti ◦◦◦◦) − ATPmi(Tm ◦◦◦◦) = ATPsm(Ts ◦◦◦◦) + ATPmi dt ••
dTm ◦◦◦• •• •• •• νf νf νr = ATPsm(Ts ◦◦◦•) + ATPmi (Ti ◦◦◦•) − ATPmi(Tm ◦◦◦•) dt ••
dTm ◦◦•◦ •• •• •• νf νf νr = ATPsm(Ts ◦◦•◦) + ATPmi (Ti ◦◦•◦) − ATPmi(Tm ◦◦•◦) dt ••
dTm ◦◦•• •• •• •• νf νf νr = ATPsm(Ts ◦◦••) + ATPmi (Ti ◦◦••) − ATPmi(Tm ◦◦••) dt •◦
dTm ◦•◦◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦•◦◦) + ATPmi (Ti ◦•◦◦) − ATPmi(Tm ◦•◦◦) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart •◦
dTm ◦•◦• •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦•◦•) + ATPmi (Ti ◦•◦•) − ATPmi(Tm ◦•◦•) dt •◦
dTm ◦••◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦••◦) + ATPmi (Ti ◦••◦) − ATPmi(Tm ◦••◦) dt •◦
dTm ◦••• •◦ •◦ •◦ νf νf νr = ATPsm(Ts ◦•••) + ATPmi (Ti ◦•••) − ATPmi(Tm ◦•••) dt ••
dTm ◦•◦◦ •• •• •• νf νf νr (Ti ◦•◦◦) − ATPmi(Tm ◦•◦◦) = ATPsm(Ts ◦•◦◦) + ATPmi dt ••
dTm ◦•◦• •• •• •• νf νf νr = ATPsm(Ts ◦•◦•) + ATPmi (Ti ◦•◦•) − ATPmi(Tm ◦•◦•) dt ••
dTm ◦••◦ •• •• •• νf νf νr (Ti ◦••◦) − ATPmi(Tm ◦••◦) = ATPsm(Ts ◦••◦) + ATPmi dt ••
dTm ◦••• •• •• •• νf νf νr = ATPsm(Ts ◦•••) + ATPmi (Ti ◦•••) − ATPmi(Tm ◦•••) dt •◦
dTm •◦◦◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts •◦◦◦) + ATPmi (Ti •◦◦◦) − ATPmi(Tm •◦◦◦) dt •◦
dTm •◦◦• •◦ •◦ •◦ νf νf νr = ATPsm(Ts •◦◦•) + ATPmi (Ti •◦◦•) − ATPmi(Tm •◦◦•) dt •◦
dTm •◦•◦ •◦ •◦ •◦ νf νf νr (Ti •◦•◦) − ATPmi(Tm •◦•◦) = ATPsm(Ts •◦•◦) + ATPmi dt •◦
dTm •◦•• •◦ •◦ •◦ νf νf νr (Ti •◦••) − ATPmi(Tm •◦••) = ATPsm(Ts •◦••) + ATPmi dt ••
dTm •◦◦◦ •• •• •• νf νf νr = ATPsm(Ts •◦◦◦) + ATPmi (Ti •◦◦◦) − ATPmi(Tm •◦◦◦) dt ••
dTm •◦◦• •• •• •• νf νf νr (Ti •◦◦•) − ATPmi(Tm •◦◦•) = ATPsm(Ts •◦◦•) + ATPmi dt ••
dTm •◦•◦ •• •• •• νf νf νr (Ti •◦•◦) − ATPmi(Tm •◦•◦) = ATPsm(Ts •◦•◦) + ATPmi dt ••
dTm •◦•• •• •• •• νf νf νr = ATPsm(Ts •◦••) + ATPmi (Ti •◦••) − ATPmi(Tm •◦••) dt •◦
dTm ••◦◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts ••◦◦) + ATPmi (Ti ••◦◦) − ATPmi(Tm ••◦◦) dt •◦
dTm ••◦• •◦ •◦ •◦ νf νf νr (Ti ••◦•) − ATPmi(Tm ••◦•) = ATPsm(Ts ••◦•) + ATPmi dt •◦
dTm •••◦ •◦ •◦ •◦ νf νf νr = ATPsm(Ts •••◦) + ATPmi (Ti •••◦) − ATPmi(Tm •••◦) dt •◦
dTm •••• •◦ •◦ •◦ νf νf νr = ATPsm(Ts ••••) + ATPmi (Ti ••••) − ATPmi(Tm ••••) dt ••
dTm ••◦◦ •• •• •• νf νf νr = ATPsm(Ts ••◦◦) + ATPmi (Ti ••◦◦) − ATPmi(Tm ••◦◦) dt ••
dTm ••◦• •• •• •• νf νf νr = ATPsm(Ts ••◦•) + ATPmi (Ti ••◦•) − ATPmi(Tm ••◦•) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
dTm •••◦ •• •• •• νf νf νr = ATPsm(Ts •••◦) + ATPmi (Ti •••◦) − ATPmi(Tm •••◦) dt ••
dTm •••• •• •• •• νf νf νr = ATPsm(Ts ••••) + ATPmi (Ti ••••) − ATPmi(Tm ••••) dt ◦◦
dTo ◦◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦◦) + AKo (Do ◦◦Do ◦◦) + CKo (Do ◦◦Co ◦◦) − AKo(To ◦◦◦◦) − ATPoe(To ◦◦◦◦) − CKo(To ◦◦◦◦) − ATPio (To ◦◦◦◦) dt ◦◦
dTo ◦◦◦• ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (Do ◦•Do ◦◦) + CKo (Do ◦◦Co ◦•) − AKo(To ◦◦◦•) − ATPoe(To ◦◦◦•) − CKo(To ◦◦◦•) − ATPio (To ◦◦◦•) = ATPio(Ti ◦◦◦•) + AKo dt ◦◦
dTo ◦◦•◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti ◦◦•◦) + AKo (Do •◦Do ◦◦) + CKo (Do ◦◦Co •◦) − AKo(To ◦◦•◦) − ATPoe(To ◦◦•◦) − CKo(To ◦◦•◦) − ATPio (To ◦◦•◦) dt ◦◦
dTo ◦◦•• ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To ◦◦••) (Do ◦◦Co ••) − AKo(To ◦◦••) − ATPoe(To ◦◦••) − CKo(To ◦◦••) − ATPio (Do ••Do ◦◦) + CKo = ATPio(Ti ◦◦••) + AKo dt ◦•
dTo ◦◦◦◦ ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦◦) + AKo (Do ◦◦Do ◦◦) + CKo (Do ◦◦Co ◦◦) − AKo(To ◦◦◦◦) − ATPoe(To ◦◦◦◦) − CKo(To ◦◦◦◦) − ATPio (To ◦◦◦◦) dt ◦•
dTo ◦◦◦• ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦•) + AKo (Do ◦•Do ◦◦) + CKo (Do ◦◦Co ◦•) − AKo(To ◦◦◦•) − ATPoe(To ◦◦◦•) − CKo(To ◦◦◦•) − ATPio (To ◦◦◦•) dt ◦•
dTo ◦◦•◦ ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr (To ◦◦•◦) = ATPio(Ti ◦◦•◦) + AKo (Do •◦Do ◦◦) + CKo (Do ◦◦Co •◦) − AKo(To ◦◦•◦) − ATPoe(To ◦◦•◦) − CKo(To ◦◦•◦) − ATPio dt ◦•
dTo ◦◦•• ◦• ◦• ◦• • ◦ ◦ • ◦• ◦• νf νf νf νf νr νr νr (To ◦◦••) (Do ••Do ◦◦) + CKo (Do ◦◦Co ••) − AKo(To ◦◦••) − ATPoe(To ◦◦••) − CKo(To ◦◦••) − ATPio = ATPio(Ti ◦◦••) + AKo dt ◦◦
dTo ◦•◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ νf νf νf νf νr νr νr (To ◦•◦◦) (Do ◦•Co ◦◦) − AKo(To ◦•◦◦) − ATPoe(To ◦•◦◦) − CKo(To ◦•◦◦) − ATPio (Do ◦•Do ◦◦) + CKo = ATPio(Ti ◦•◦◦) + AKo dt ◦◦
dTo ◦•◦• ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti ◦•◦•) + AKo (Do ◦•Do ◦•) + CKo (Do ◦•Co ◦•) − AKo(To ◦•◦•) − ATPoe(To ◦•◦•) − CKo(To ◦•◦•) − ATPio (To ◦•◦•) dt ◦◦
dTo ◦••◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To ◦••◦) (Do •◦Do ◦•) + CKo (Do ◦•Co •◦) − AKo(To ◦••◦) − ATPoe(To ◦••◦) − CKo(To ◦••◦) − ATPio = ATPio(Ti ◦••◦) + AKo dt ◦◦
dTo ◦••• ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ νf νf νf νf νr νr νr (To ◦•••) (Do ◦•Co ••) − AKo(To ◦•••) − ATPoe(To ◦•••) − CKo(To ◦•••) − ATPio (Do ••Do ◦•) + CKo = ATPio(Ti ◦•••) + AKo dt ◦•
dTo ◦•◦◦ ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti ◦•◦◦) + AKo (Do ◦◦Do ◦•) + CKo (Do ◦•Co ◦◦) − AKo(To ◦•◦◦) − ATPoe(To ◦•◦◦) − CKo(To ◦•◦◦) − ATPio (To ◦•◦◦) dt ◦•
dTo ◦•◦• ◦• ◦• ◦• • ◦• ◦ • ◦ ◦• νf νf νf νf νr νr νr (To ◦•◦•) (Do ◦•Co ◦•) − AKo(To ◦•◦•) − ATPoe(To ◦•◦•) − CKo(To ◦•◦•) − ATPio = ATPio(Ti ◦•◦•) + AKo (Do ◦•Do ◦•) + CKo dt ◦•
dTo ◦••◦ ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr (Do •◦Do ◦•) + CKo (Do ◦•Co •◦) − AKo(To ◦••◦) − ATPoe(To ◦••◦) − CKo(To ◦••◦) − ATPio (To ◦••◦) = ATPio(Ti ◦••◦) + AKo dt ◦•
dTo ◦••• ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti ◦•••) + AKo (Do ••Do ◦•) + CKo (Do ◦•Co ••) − AKo(To ◦•••) − ATPoe(To ◦•••) − CKo(To ◦•••) − ATPio (To ◦•••) dt ◦◦
dTo •◦◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti •◦◦◦) + AKo (Do •◦Do ◦◦) + CKo (Do •◦Co ◦◦) − AKo(To •◦◦◦) − ATPoe(To •◦◦◦) − CKo(To •◦◦◦) − ATPio (To •◦◦◦) dt ◦◦
dTo •◦◦• ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To •◦◦•) = ATPio(Ti •◦◦•) + AKo (Do •◦Do ◦•) + CKo (Do •◦Co ◦•) − AKo(To •◦◦•) − ATPoe(To •◦◦•) − CKo(To •◦◦•) − ATPio dt ◦◦
dTo •◦•◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (Do •◦Co •◦) − AKo(To •◦•◦) − ATPoe(To •◦•◦) − CKo(To •◦•◦) − ATPio (To •◦•◦) = ATPio(Ti •◦•◦) + AKo (Do •◦Do •◦) + CKo dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦◦
dTo •◦•• ◦◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To •◦••) = ATPio(Ti •◦••) + AKo (Do ••Do •◦) + CKo (Do •◦Co ••) − AKo(To •◦••) − ATPoe(To •◦••) − CKo(To •◦••) − ATPio dt ◦•
dTo •◦◦◦ ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti •◦◦◦) + AKo (Do ◦◦Do •◦) + CKo (Do •◦Co ◦◦) − AKo(To •◦◦◦) − ATPoe(To •◦◦◦) − CKo(To •◦◦◦) − ATPio (To •◦◦◦) dt ◦•
dTo •◦◦• ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti •◦◦•) + AKo (Do ◦•Do •◦) + CKo (Do •◦Co ◦•) − AKo(To •◦◦•) − ATPoe(To •◦◦•) − CKo(To •◦◦•) − ATPio (To •◦◦•) dt ◦•
dTo •◦•◦ • ◦ ◦ • ◦• ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr (Do •◦Do •◦) + CKo (Do •◦Co •◦) − AKo(To •◦•◦) − ATPoe(To •◦•◦) − CKo(To •◦•◦) − ATPio (To •◦•◦) = ATPio(Ti •◦•◦) + AKo dt ◦•
dTo •◦•• ◦• • ◦ ◦ • ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti •◦••) + AKo (Do ••Do •◦) + CKo (Do •◦Co ••) − AKo(To •◦••) − ATPoe(To •◦••) − CKo(To •◦••) − ATPio (To •◦••) dt ◦◦
dTo ••◦◦ ◦ ◦ ◦◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To ••◦◦) (Do ••Co ◦◦) − AKo(To ••◦◦) − ATPoe(To ••◦◦) − CKo(To ••◦◦) − ATPio (Do ••Do ◦◦) + CKo = ATPio(Ti ••◦◦) + AKo dt ◦◦
dTo ••◦• ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti ••◦•) + AKo (Do ••Do ◦•) + CKo (Do ••Co ◦•) − AKo(To ••◦•) − ATPoe(To ••◦•) − CKo(To ••◦•) − ATPio (To ••◦•) dt ◦◦
dTo •••◦ ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr = ATPio(Ti •••◦) + AKo (Do ••Do •◦) + CKo (Do ••Co •◦) − AKo(To •••◦) − ATPoe(To •••◦) − CKo(To •••◦) − ATPio (To •••◦) dt ◦◦
dTo •••• ◦◦ ◦ ◦ ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ νf νf νf νf νr νr νr (To ••••) = ATPio(Ti ••••) + AKo (Do ••Do ••) + CKo (Do ••Co ••) − AKo(To ••••) − ATPoe(To ••••) − CKo(To ••••) − ATPio dt ◦•
dTo ••◦◦ ◦• ◦• ◦• • ◦ ◦ • ◦• ◦• νf νf νf νf νr νr νr (To ••◦◦) (Do ◦◦Do ••) + CKo (Do ••Co ◦◦) − AKo(To ••◦◦) − ATPoe(To ••◦◦) − CKo(To ••◦◦) − ATPio = ATPio(Ti ••◦◦) + AKo dt ◦•
dTo ••◦• ◦• ◦• ◦• ◦• • ◦ ◦ • ◦• νf νf νf νf νr νr νr (To ••◦•) (Do ••Co ◦•) − AKo(To ••◦•) − ATPoe(To ••◦•) − CKo(To ••◦•) − ATPio (Do ◦•Do ••) + CKo = ATPio(Ti ••◦•) + AKo dt ◦•
dTo •••◦ • ◦ ◦ • ◦• ◦• ◦• ◦• ◦• νf νf νf νf νr νr νr = ATPio(Ti •••◦) + AKo (Do •◦Do ••) + CKo (Do ••Co •◦) − AKo(To •••◦) − ATPoe(To •••◦) − CKo(To •••◦) − ATPio (To •••◦) dt ◦•
dTo •••• ◦• ◦• ◦• • ◦ ◦ • ◦• ◦• νf νf νf νf νr νr νr (To ••••) (Do ••Do ••) + CKo (Do ••Co ••) − AKo(To ••••) − ATPoe(To ••••) − CKo(To ••••) − ATPio = ATPio(Ti ••••) + AKo dt •◦
dTo ◦◦◦◦ •◦ •◦ •◦ •◦ ◦ • ◦ • •◦ νf νf νf νf νr νr νr (To ◦◦◦◦) (Do ◦◦Co ◦◦) − AKo(To ◦◦◦◦) − ATPoe(To ◦◦◦◦) − CKo(To ◦◦◦◦) − ATPio (Do ◦◦Do ◦◦) + CKo = ATPio(Ti ◦◦◦◦) + AKo dt •◦
dTo ◦◦◦• •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦•) + AKo (Do ◦◦Do ◦•) + CKo (Do ◦◦Co ◦•) − AKo(To ◦◦◦•) − ATPoe(To ◦◦◦•) − CKo(To ◦◦◦•) − ATPio (To ◦◦◦•) dt •◦
dTo ◦◦•◦ •◦ •◦ •◦ ◦ •◦ • • ◦ •◦ νf νf νf νf νr νr νr (To ◦◦•◦) (Do ◦◦Co •◦) − AKo(To ◦◦•◦) − ATPoe(To ◦◦•◦) − CKo(To ◦◦•◦) − ATPio = ATPio(Ti ◦◦•◦) + AKo (Do ◦◦Do •◦) + CKo dt •◦
dTo ◦◦•• •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (Do ◦◦Do ••) + CKo (Do ◦◦Co ••) − AKo(To ◦◦••) − ATPoe(To ◦◦••) − CKo(To ◦◦••) − ATPio (To ◦◦••) = ATPio(Ti ◦◦••) + AKo dt ••
dTo ◦◦◦◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦◦) + AKo (Do ◦◦Do ◦◦) + CKo (Do ◦◦Co ◦◦) − AKo(To ◦◦◦◦) − ATPoe(To ◦◦◦◦) − CKo(To ◦◦◦◦) − ATPio (To ◦◦◦◦) dt ••
dTo ◦◦◦• •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ◦◦◦•) + AKo (Do ◦•Do ◦◦) + CKo (Do ◦◦Co ◦•) − AKo(To ◦◦◦•) − ATPoe(To ◦◦◦•) − CKo(To ◦◦◦•) − ATPio (To ◦◦◦•) dt ••
dTo ◦◦•◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr (To ◦◦•◦) = ATPio(Ti ◦◦•◦) + AKo (Do •◦Do ◦◦) + CKo (Do ◦◦Co •◦) − AKo(To ◦◦•◦) − ATPoe(To ◦◦•◦) − CKo(To ◦◦•◦) − ATPio dt ••
dTo ◦◦•• •• • • • • •• •• •• •• νf νf νf νf νr νr νr (Do ◦◦Co ••) − AKo(To ◦◦••) − ATPoe(To ◦◦••) − CKo(To ◦◦••) − ATPio (To ◦◦••) = ATPio(Ti ◦◦••) + AKo (Do ••Do ◦◦) + CKo dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart •◦
dTo ◦•◦◦ •◦ •◦ • ◦ • ◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (To ◦•◦◦) = ATPio(Ti ◦•◦◦) + AKo (Do ◦•Do ◦◦) + CKo (Do ◦•Co ◦◦) − AKo(To ◦•◦◦) − ATPoe(To ◦•◦◦) − CKo(To ◦•◦◦) − ATPio dt •◦
dTo ◦•◦• •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti ◦•◦•) + AKo (Do ◦•Do ◦•) + CKo (Do ◦•Co ◦•) − AKo(To ◦•◦•) − ATPoe(To ◦•◦•) − CKo(To ◦•◦•) − ATPio (To ◦•◦•) dt •◦
dTo ◦••◦ •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti ◦••◦) + AKo (Do ◦•Do •◦) + CKo (Do ◦•Co •◦) − AKo(To ◦••◦) − ATPoe(To ◦••◦) − CKo(To ◦••◦) − ATPio (To ◦••◦) dt •◦
dTo ◦••• • ◦ • ◦ •◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (Do ◦•Do ••) + CKo (Do ◦•Co ••) − AKo(To ◦•••) − ATPoe(To ◦•••) − CKo(To ◦•••) − ATPio (To ◦•••) = ATPio(Ti ◦•••) + AKo dt ••
dTo ◦•◦◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ◦•◦◦) + AKo (Do ◦•Do ◦◦) + CKo (Do ◦•Co ◦◦) − AKo(To ◦•◦◦) − ATPoe(To ◦•◦◦) − CKo(To ◦•◦◦) − ATPio (To ◦•◦◦) dt ••
dTo ◦•◦• • • •• • • •• •• •• •• νf νf νf νf νr νr νr (To ◦•◦•) (Do ◦•Co ◦•) − AKo(To ◦•◦•) − ATPoe(To ◦•◦•) − CKo(To ◦•◦•) − ATPio (Do ◦•Do ◦•) + CKo = ATPio(Ti ◦•◦•) + AKo dt ••
dTo ◦••◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ◦••◦) + AKo (Do •◦Do ◦•) + CKo (Do ◦•Co •◦) − AKo(To ◦••◦) − ATPoe(To ◦••◦) − CKo(To ◦••◦) − ATPio (To ◦••◦) dt ••
dTo ◦••• •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ◦•••) + AKo (Do ••Do ◦•) + CKo (Do ◦•Co ••) − AKo(To ◦•••) − ATPoe(To ◦•••) − CKo(To ◦•••) − ATPio (To ◦•••) dt •◦
dTo •◦◦◦ •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (To •◦◦◦) = ATPio(Ti •◦◦◦) + AKo (Do •◦Do ◦◦) + CKo (Do •◦Co ◦◦) − AKo(To •◦◦◦) − ATPoe(To •◦◦◦) − CKo(To •◦◦◦) − ATPio dt •◦
dTo •◦◦• •◦ •◦ •◦ • ◦ • ◦ •◦ •◦ νf νf νf νf νr νr νr (To •◦◦•) (Do •◦Do ◦•) + CKo (Do •◦Co ◦•) − AKo(To •◦◦•) − ATPoe(To •◦◦•) − CKo(To •◦◦•) − ATPio = ATPio(Ti •◦◦•) + AKo dt •◦
dTo •◦•◦ •◦ •◦ •◦ •◦ ◦ • ◦ • •◦ νf νf νf νf νr νr νr (To •◦•◦) (Do •◦Co •◦) − AKo(To •◦•◦) − ATPoe(To •◦•◦) − CKo(To •◦•◦) − ATPio (Do •◦Do •◦) + CKo = ATPio(Ti •◦•◦) + AKo dt •◦
dTo •◦•• • ◦ • ◦ •◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti •◦••) + AKo (Do •◦Do ••) + CKo (Do •◦Co ••) − AKo(To •◦••) − ATPoe(To •◦••) − CKo(To •◦••) − ATPio (To •◦••) dt ••
dTo •◦◦◦ •• •• •• • • • • •• •• νf νf νf νf νr νr νr (To •◦◦◦) (Do •◦Do ◦◦) + CKo (Do •◦Co ◦◦) − AKo(To •◦◦◦) − ATPoe(To •◦◦◦) − CKo(To •◦◦◦) − ATPio = ATPio(Ti •◦◦◦) + AKo dt ••
dTo •◦◦• •• •• •• •• • • • • •• νf νf νf νf νr νr νr (To •◦◦•) (Do •◦Co ◦•) − AKo(To •◦◦•) − ATPoe(To •◦◦•) − CKo(To •◦◦•) − ATPio (Do •◦Do ◦•) + CKo = ATPio(Ti •◦◦•) + AKo dt ••
dTo •◦•◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti •◦•◦) + AKo (Do •◦Do •◦) + CKo (Do •◦Co •◦) − AKo(To •◦•◦) − ATPoe(To •◦•◦) − CKo(To •◦•◦) − ATPio (To •◦•◦) dt ••
dTo •◦•• •• •• •• • •• • • • •• νf νf νf νf νr νr νr (To •◦••) (Do •◦Co ••) − AKo(To •◦••) − ATPoe(To •◦••) − CKo(To •◦••) − ATPio = ATPio(Ti •◦••) + AKo (Do ••Do •◦) + CKo dt •◦
dTo ••◦◦ •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (Do ••Do ◦◦) + CKo (Do ••Co ◦◦) − AKo(To ••◦◦) − ATPoe(To ••◦◦) − CKo(To ••◦◦) − ATPio (To ••◦◦) = ATPio(Ti ••◦◦) + AKo dt •◦
dTo ••◦• •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti ••◦•) + AKo (Do ••Do ◦•) + CKo (Do ••Co ◦•) − AKo(To ••◦•) − ATPoe(To ••◦•) − CKo(To ••◦•) − ATPio (To ••◦•) dt •◦
dTo •••◦ •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr = ATPio(Ti •••◦) + AKo (Do ••Do •◦) + CKo (Do ••Co •◦) − AKo(To •••◦) − ATPoe(To •••◦) − CKo(To •••◦) − ATPio (To •••◦) dt •◦
dTo •••• •◦ • ◦ • ◦ •◦ •◦ •◦ •◦ νf νf νf νf νr νr νr (To ••••) = ATPio(Ti ••••) + AKo (Do ••Do ••) + CKo (Do ••Co ••) − AKo(To ••••) − ATPoe(To ••••) − CKo(To ••••) − ATPio dt ••
dTo ••◦◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr (Do ••Co ◦◦) − AKo(To ••◦◦) − ATPoe(To ••◦◦) − CKo(To ••◦◦) − ATPio (To ••◦◦) = ATPio(Ti ••◦◦) + AKo (Do ••Do ◦◦) + CKo dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
dTo ••◦• •• •• • • • • •• •• •• νf νf νf νf νr νr νr (To ••◦•) = ATPio(Ti ••◦•) + AKo (Do ••Do ◦•) + CKo (Do ••Co ◦•) − AKo(To ••◦•) − ATPoe(To ••◦•) − CKo(To ••◦•) − ATPio dt ••
dTo •••◦ •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti •••◦) + AKo (Do ••Do •◦) + CKo (Do ••Co •◦) − AKo(To •••◦) − ATPoe(To •••◦) − CKo(To •••◦) − ATPio (To •••◦) dt ••
dTo •••• •• • • • • •• •• •• •• νf νf νf νf νr νr νr = ATPio(Ti ••••) + AKo (Do ••Do ••) + CKo (Do ••Co ••) − AKo(To ••••) − ATPoe(To ••••) − CKo(To ••••) − ATPio (To ••••) dt ◦◦
◦ • ◦ ◦ ◦ dTs ◦◦◦◦ νf ◦ ◦◦ ◦◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ◦◦) − ATPsm(Ts ◦◦◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦◦) ◦ ◦ ◦ ◦ • dt ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦◦◦• νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦•) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦◦•◦ νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦•◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • ◦ • • • ◦ ◦ ◦ dTs ◦◦•• νf 1 •• ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦••) − ASsr ((Ws • + Ws ◦)Ts ◦◦••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• ◦ ◦ ◦ • • ◦ ◦ ◦ dTs ◦◦◦◦ νf 1 •◦ ◦ ◦• ◦• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs ◦◦◦• νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦•) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦•) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs ◦◦•◦ νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦•◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦•◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
◦ • • • • dTs ◦◦•• νf ◦• ◦ ◦• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ◦◦) − ATPsm(Ts ◦◦••) − ASsr ((Ws • + Ws ◦)Ts ◦◦••) • • • • ◦ dt ◦◦
• ◦ ◦ ◦ ◦ dTs ◦•◦◦ νf ◦ ◦◦ ◦◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ◦•) − ATPsm(Ts ◦•◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦•◦◦) ◦ ◦ ◦ • ◦ dt ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦•◦• νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦•) − ASsr ((Ws • + Ws ◦)Ts ◦•◦•) ◦ ◦ • ◦ • ◦ • • ◦ ◦ dt ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦••◦ νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦••◦) − ASsr ((Ws • + Ws ◦)Ts ◦••◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • ◦ • • • ◦ ◦ ◦ dTs ◦••• νf 1 •• ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•••) − ASsr ((Ws • + Ws ◦)Ts ◦•••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
◦ ◦ ◦ • ◦ ◦ ◦ • • dTs ◦•◦◦ νf 1 •◦ ◦• ◦• ◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦•◦◦) • ◦ • • ◦ • ◦ ◦ ◦ ◦ dt ◦•
• • • • ◦ • ◦ ◦ ◦ dTs ◦•◦• νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦•) − ASsr ((Ws • + Ws ◦)Ts ◦•◦•) ◦ ◦ • • ◦ • • ◦ • • dt ◦•
• • • • • ◦ ◦ ◦ ◦ dTs ◦••◦ νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦••◦) − ASsr ((Ws • + Ws ◦)Ts ◦••◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • • ◦ dTs ◦••• νf ◦ ◦• ◦• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ◦•) − ATPsm(Ts ◦•••) − ASsr ((Ws • + Ws ◦)Ts ◦•••) • ◦ • • • dt ◦◦
◦ • ◦ ◦ ◦ dTs •◦◦◦ νf ◦ ◦◦ ◦◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds •◦) − ATPsm(Ts •◦◦◦) − ASsr ((Ws • + Ws ◦)Ts •◦◦◦) ◦ ◦ ◦ ◦ • dt ◦◦
◦ ◦ ◦ ◦ ◦ ◦ • • • dTs •◦◦• νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦•) − ASsr ((Ws • + Ws ◦)Ts •◦◦•) • ◦ ◦ • ◦ • ◦ • ◦ ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs •◦•◦ νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦•◦) − ASsr ((Ws • + Ws ◦)Ts •◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • ◦ • • • ◦ ◦ ◦ dTs •◦•• νf 1 •• ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦••) − ASsr ((Ws • + Ws ◦)Ts •◦••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs •◦◦◦ νf 1 •◦ ◦ ◦• ◦• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦◦) − ASsr ((Ws • + Ws ◦)Ts •◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs •◦◦• νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦•) − ASsr ((Ws • + Ws ◦)Ts •◦◦•) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs •◦•◦ νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦•◦) − ASsr ((Ws • + Ws ◦)Ts •◦•◦) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • • • ◦ dTs •◦•• νf ◦ ◦• ◦• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds •◦) − ATPsm(Ts •◦••) − ASsr ((Ws • + Ws ◦)Ts •◦••) • ◦ • • • dt ◦◦
◦ • ◦ ◦ ◦ dTs ••◦◦ νf ◦ ◦◦ ◦◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ••) − ATPsm(Ts ••◦◦) − ASsr ((Ws • + Ws ◦)Ts ••◦◦) ◦ ◦ ◦ ◦ • dt ◦◦
• ◦ ◦ ◦ • • ◦ ◦ ◦ dTs ••◦• νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦•) − ASsr ((Ws • + Ws ◦)Ts ••◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs •••◦ νf 1 •◦ ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts •••◦) − ASsr ((Ws • + Ws ◦)Ts •••◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ◦◦
• • ◦ • • • ◦ ◦ ◦ dTs •••• νf 1 •• ◦ ◦◦ ◦◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••••) − ASsr ((Ws • + Ws ◦)Ts ••••) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
◦ ◦ ◦ ◦ ◦ • ◦ • • dTs ••◦◦ νf 1 •◦ ◦ ◦• ◦• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦◦) − ASsr ((Ws • + Ws ◦)Ts ••◦◦) • ◦ • • ◦ • ◦ ◦ ◦ ◦ dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs ••◦• νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦•) − ASsr ((Ws • + Ws ◦)Ts ••◦•) ◦ • • • ◦ ◦ • ◦ • • dt ◦•
• • ◦ • • • ◦ ◦ ◦ dTs •••◦ νf 1 •• ◦ ◦• ◦• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts •••◦) − ASsr ((Ws • + Ws ◦)Ts •••◦) • • • ◦ • ◦ • • ◦ ◦ dt ◦•
• • • • ◦ dTs •••• νf ◦ ◦• ◦• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ••) − ATPsm(Ts ••••) − ASsr ((Ws • + Ws ◦)Ts ••••) • ◦ • • • dt •◦
• ◦ ◦ ◦ ◦ dTs ◦◦◦◦ νf • •◦ •◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ◦◦) − ATPsm(Ts ◦◦◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦◦) ◦ ◦ ◦ • ◦ dt •◦
◦ ◦ ◦ • ◦ ◦ ◦ • • dTs ◦◦◦• νf 1 •◦ •◦ •◦ • νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦•) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦•) • ◦ • • ◦ • ◦ ◦ ◦ ◦ dt •◦
• ◦ • ◦ ◦ • ◦ ◦ ◦ dTs ◦◦•◦ νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦•◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦•◦) ◦ ◦ ◦ ◦ ◦ • • ◦ • • dt •◦
• • • • • ◦ ◦ ◦ ◦ dTs ◦◦•• νf 1 •• • •◦ •◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦••) − ASsr ((Ws • + Ws ◦)Ts ◦◦••) ◦ • • • ◦ ◦ • ◦ • • dt ••
◦ • • ◦ ◦ • ◦ ◦ ◦ dTs ◦◦◦◦ νf 1 •◦ • •• •• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦◦) ◦ ◦ ◦ ◦ • ◦ • ◦ • • dt ••
• • • • ◦ • ◦ ◦ ◦ dTs ◦◦◦• νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦◦•) − ASsr ((Ws • + Ws ◦)Ts ◦◦◦•) ◦ • ◦ ◦ • • • ◦ • • dt ••
◦ ◦ ◦ ◦ • • • • • dTs ◦◦•◦ νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦◦) − ATPsm(Ts ◦◦•◦) − ASsr ((Ws • + Ws ◦)Ts ◦◦•◦) • • • • ◦ • ◦ • ◦ ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
• • • • ◦ dTs ◦◦•• νf • •• •• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ◦◦) − ATPsm(Ts ◦◦••) − ASsr ((Ws • + Ws ◦)Ts ◦◦••) • ◦ • • • dt •◦
• ◦ ◦ ◦ ◦ dTs ◦•◦◦ νf • •◦ •◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ◦•) − ATPsm(Ts ◦•◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦•◦◦) ◦ ◦ ◦ • ◦ dt •◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦•◦• νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦•) − ASsr ((Ws • + Ws ◦)Ts ◦•◦•) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦••◦ νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦••◦) − ASsr ((Ws • + Ws ◦)Ts ◦••◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
• • ◦ • • • ◦ ◦ ◦ dTs ◦••• νf 1 •• • •◦ •◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•••) − ASsr ((Ws • + Ws ◦)Ts ◦•••) ◦ • • • ◦ ◦ • ◦ • • dt ••
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ◦•◦◦ νf 1 •◦ • •• •• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦◦) − ASsr ((Ws • + Ws ◦)Ts ◦•◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
• • ◦ • • • ◦ ◦ ◦ dTs ◦•◦• νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦•◦•) − ASsr ((Ws • + Ws ◦)Ts ◦•◦•) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • ◦ • • ◦ ◦ ◦ dTs ◦••◦ νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ◦•) − ATPsm(Ts ◦••◦) − ASsr ((Ws • + Ws ◦)Ts ◦••◦) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • • ◦ dTs ◦••• νf • •• •• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ◦•) − ATPsm(Ts ◦•••) − ASsr ((Ws • + Ws ◦)Ts ◦•••) • ◦ • • • dt •◦
◦ • ◦ ◦ ◦ dTs •◦◦◦ νf • •◦ •◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds •◦) − ATPsm(Ts •◦◦◦) − ASsr ((Ws • + Ws ◦)Ts •◦◦◦) ◦ ◦ ◦ ◦ • dt •◦
◦ ◦ ◦ ◦ ◦ • ◦ • • dTs •◦◦• νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦•) − ASsr ((Ws • + Ws ◦)Ts •◦◦•) • ◦ • • ◦ • ◦ ◦ ◦ ◦ dt •◦
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs •◦•◦ νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦•◦) − ASsr ((Ws • + Ws ◦)Ts •◦•◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt •◦
• • ◦ • • • ◦ ◦ ◦ dTs •◦•• νf 1 •• • •◦ •◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦••) − ASsr ((Ws • + Ws ◦)Ts •◦••) • • • ◦ • ◦ • • ◦ ◦ dt ••
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs •◦◦◦ νf 1 •◦ • •• •• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦◦) − ASsr ((Ws • + Ws ◦)Ts •◦◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
• • ◦ • • • ◦ ◦ ◦ dTs •◦◦• νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦◦•) − ASsr ((Ws • + Ws ◦)Ts •◦◦•) ◦ • • • ◦ ◦ • ◦ • • dt ••
◦ • ◦ • ◦ ◦ • • • dTs •◦•◦ νf 1 •• •• •• • νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds •◦) − ATPsm(Ts •◦•◦) − ASsr ((Ws • + Ws ◦)Ts •◦•◦) • ◦ • • • • ◦ • ◦ ◦ dt ••
• • • • ◦ dTs •◦•• νf • •• •• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds •◦) − ATPsm(Ts •◦••) − ASsr ((Ws • + Ws ◦)Ts •◦••) • ◦ • • • dt •◦
◦ • ◦ ◦ ◦ dTs ••◦◦ νf • •◦ •◦ νf ν = ASs((Ps ◦◦ + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦))Ds ••) − ATPsm(Ts ••◦◦) − ASsr ((Ws • + Ws ◦)Ts ••◦◦) ◦ ◦ ◦ ◦ • dt •◦
◦ • • ◦ ◦ • ◦ ◦ ◦ dTs ••◦• νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦•) − ASsr ((Ws • + Ws ◦)Ts ••◦•) ◦ ◦ ◦ ◦ • ◦ • ◦ • • dt •◦
◦ • • ◦ ◦ • ◦ ◦ ◦ dTs •••◦ νf 1 •◦ • •◦ •◦ νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts •••◦) − ASsr ((Ws • + Ws ◦)Ts •••◦) ◦ ◦ ◦ ◦ ◦ • • ◦ • • dt •◦
◦ ◦ ◦ ◦ • • • • • dTs •••• νf 1 •• • •◦ •◦ νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••••) − ASsr ((Ws • + Ws ◦)Ts ••••) • • • • ◦ • ◦ • ◦ ◦ dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart
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••
◦ ◦ ◦ • • • ◦ ◦ ◦ dTs ••◦◦ νf 1 •◦ • •• •• νf ν = ASs(( 4 (Ps ◦ + Ps •◦ + Ps ◦• + Ps ◦◦) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦◦) − ASsr ((Ws • + Ws ◦)Ts ••◦◦) ◦ ◦ ◦ • ◦ ◦ • ◦ • • dt ••
• • ◦ • • • ◦ ◦ ◦ dTs ••◦• νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts ••◦•) − ASsr ((Ws • + Ws ◦)Ts ••◦•) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • ◦ • • • ◦ ◦ ◦ dTs •••◦ νf 1 •• • •• •• νf ν = ASs(( 4 (Ps • + Ps •◦ + Ps ◦• + Ps ••) + 16 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•))Ds ••) − ATPsm(Ts •••◦) − ASsr ((Ws • + Ws ◦)Ts •••◦) ◦ • • • ◦ ◦ • ◦ • • dt ••
• • • • ◦ dTs •••• νf • •• •• νf ν = ASs((Ps •• + 14 (Ps •• + Ps •◦ + Ps ◦• + Ps ••))Ds ••) − ATPsm(Ts ••••) − ASsr ((Ws • + Ws ◦)Ts ••••) • ◦ • • • dt ◦
dCi ◦◦ •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ • • • • ◦ ◦ νf νf ν ν = CKi(Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦) + Cior (Co ◦◦) − Cio(Ci ◦◦) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt ◦ ◦ ◦ ◦ ◦ Di ◦ • + Di ◦)Ci ◦) ◦
dCi ◦• •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ • • • • ◦ ◦ νf νf ν ν = CKi(Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦• + Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦•) + Cior (Co ◦•) − Cio(Ci ◦•) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt ◦ ◦ ◦ ◦ ◦ Di ◦ • + Di ◦)Ci •) ◦
dCi •◦ •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ ◦ • • • • ◦ ◦ νf νf ν ν = CKi(Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦ + Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦) + Cior (Co •◦) − Cio(Ci •◦) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt ◦ ◦ ◦ ◦ • Di ◦ • + Di ◦)Ci ◦) ◦
dCi •• •◦ •◦ ◦ ◦ • • • • ◦ ◦ ◦◦ ◦◦ ◦◦ ◦◦ •◦ •◦ νf νf ν ν = CKi(Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦•• + Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦••) + Cior (Co ••) − Cio(Ci ••) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt ◦ ◦ ◦ ◦ • Di ◦ • + Di ◦)Ci •) •
dCi ◦◦ ◦ ◦ • • • • • • ◦• ◦• ◦• ◦• •• •• •• •• νf νf ν ν = CKi(Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦ + Ti ••◦◦ + Ti •◦◦◦ + Ti ◦•◦◦ + Ti ◦◦◦◦) + Cior (Co ◦◦) − Cio(Ci ◦◦) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt • ◦ ◦ ◦ ◦ Di ◦ • + Di ◦)Ci ◦) •
dCi ◦• ◦ ◦ • • • • • • ◦• ◦• ◦• ◦• •• •• •• •• νf νf ν ν = CKi(Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦• + Ti ••◦• + Ti •◦◦• + Ti ◦•◦• + Ti ◦◦◦•) + Cior (Co ◦•) − Cio(Ci ◦•) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt • ◦ ◦ ◦ ◦ Di ◦ • + Di ◦)Ci •) •
dCi •◦ ◦ ◦ • • • • • • ◦• ◦• ◦• ◦• •• •• •• •• νf νf ν ν = CKi(Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦ + Ti •••◦ + Ti •◦•◦ + Ti ◦••◦ + Ti ◦◦•◦) + Cior (Co •◦) − Cio(Ci •◦) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt • ◦ ◦ • ◦ Di ◦ • + Di ◦)Ci ◦) •
dCi •• ◦ • • ◦ • • • • ◦• ◦• ◦• ◦• •• •• •• •• νf νf ν ν = CKi(Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦•• + Ti •••• + Ti •◦•• + Ti ◦••• + Ti ◦◦••) + Cior (Co ••) − Cio(Ci ••) − CKir ((Di •• + Di •◦ + Di ◦• + Di ◦◦ + Di •• + Di •◦ + dt ◦ • ◦ ◦ • Di ◦ • + Di ◦)Ci •) ◦
dCo ◦◦ •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ • • • • ◦ ◦ νf νf νr = CKo(To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦◦◦) + Cio(Ci ◦◦) − CKo ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + dt◦ ◦ ◦ ◦ νr ◦ ◦ ◦ Do ◦ • + Do ◦)Co ◦) − Cio(Co ◦) ◦
dCo ◦• •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ • • • • ◦ ◦ νf νf νr = CKo(To ••◦• + To •◦◦• + To ◦•◦• + To ◦◦◦• + To ••◦• + To •◦◦• + To ◦•◦• + To ◦◦◦•) + Cio(Ci ◦•) − CKo ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + dt◦ ◦ ◦ ◦ νr ◦ ◦ ◦ Do ◦ • + Do ◦)Co •) − Cio(Co •) ◦
dCo •◦ •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ • • • • ◦ ◦ νf νf νr = CKo(To •••◦ + To •◦•◦ + To ◦••◦ + To ◦◦•◦ + To •••◦ + To •◦•◦ + To ◦••◦ + To ◦◦•◦) + Cio(Ci •◦) − CKo ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + dt◦ ◦ ◦ ◦ νr ◦ • • Do ◦ • + Do ◦)Co ◦) − Cio(Co ◦) ◦
dCo •• • • • • ◦ ◦ •◦ •◦ •◦ •◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦ νf νf νr ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + = CKo(To •••• + To •◦•• + To ◦••• + To ◦◦•• + To •••• + To •◦•• + To ◦••• + To ◦◦••) + Cio(Ci ••) − CKo dt
Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦
Do ◦ •
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24
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+ Do ◦◦)Co ••) − Cior (Co ••) •
dCo ◦◦ •• •• •• •• ◦• ◦• ◦• ◦• • • • • • ◦ ◦ νf νf νr ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + = CKo(To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦◦◦ + To ••◦◦ + To •◦◦◦ + To ◦•◦◦ + To ◦◦◦◦) + Cio(Ci ◦◦) − CKo dt◦ ◦ • • νr ◦ ◦ ◦ Do ◦ • + Do ◦)Co ◦) − Cio(Co ◦) •
dCo ◦• •• •• •• •• ◦• ◦• ◦• ◦• • • • • • ◦ ◦ νf νf νr ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + = CKo(To ••◦• + To •◦◦• + To ◦•◦• + To ◦◦◦• + To ••◦• + To •◦◦• + To ◦•◦• + To ◦◦◦•) + Cio(Ci ◦•) − CKo dt◦ ◦ • • νr ◦ ◦ ◦ Do ◦ • + Do ◦)Co •) − Cio(Co •) •
dCo •◦ •• •• •• •• ◦• ◦• ◦• ◦• • • • • • ◦ ◦ νf νf νr ((Do •• + Do •◦ + Do ◦• + Do ◦◦ + Do •• + Do •◦ + = CKo(To •••◦ + To •◦•◦ + To ◦••◦ + To ◦◦•◦ + To •••◦ + To •◦•◦ + To ◦••◦ + To ◦◦•◦) + Cio(Ci •◦) − CKo dt◦ ◦ • • νr ◦ • • Do ◦ • + Do ◦)Co ◦) − Cio(Co ◦) •
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+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + ◦
νf
•
ν
•
•
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(Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•) •
◦ • Pe ◦ ◦
d
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= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + ◦◦
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• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ ◦ • Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + ◦
νr Peo
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νf
•
ν
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(Po •◦) − Peo(Pe •◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe •◦) ◦
◦ • Pe ◦ •
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( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• +
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• •◦ •• ◦• ◦◦ ◦• ◦• ◦◦ ◦• •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••
◦ νr • Peo Po ◦ •
(
••
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+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + )−
◦ νf • Peo Pe ◦ •
(
ν
•
•
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•
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) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +
◦ ◦ • De ◦ ◦ Pe ◦ •
)
)
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
25
◦
dPe •• ◦
dt◦•
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( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + ◦◦
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◦• ◦◦ •◦ •• •◦ •◦ •• ◦• •◦ ◦◦ ◦• ◦• ◦◦ ◦• •• • •◦ ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••
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+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + ◦
νf
•
ν
•
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(Po ••) − Peo(Pe ••) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ••) ◦
◦ • Pe • •
d
dt•• • Te ◦ •◦
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+ Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• +
◦◦
d
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( 41 ((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We ◦ + (Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + ◦
ν
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νf
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ν
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+ Peor (Po ••) − Peo(Pe ••) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ••)
• • Te ◦ ◦•)We )) • ◦ Pe ◦ ◦
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νf
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= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + ◦◦
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• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ ◦ • Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + •
νr Peo
•
νf
•
ν
•
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•
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•
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(Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦◦) ◦
• ◦ Pe ◦ •
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( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• +
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νf
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(Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦◦) •
• ◦ Pe • ◦
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( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + ••
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• ◦ Pe • •
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=
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+ Peor (Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•)
• • Te ◦ ◦•)We )) • • Pe ◦ ◦
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◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦• ◦◦ ◦• ◦• ◦◦ ◦• •• •◦ • ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••
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• • Pe ◦ •
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+ Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• +
◦◦
• • Te ◦ ◦•)We ))
+
• νr • Peo Po ◦ •
(
)−
• νf • Peo Pe ◦ •
(
ν
•
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) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +
• ◦ • De ◦ ◦ Pe ◦ •
)
)
•
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( 41 ((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We ◦ + (Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• +
Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
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26
◦•
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• Te ◦ •◦ ◦◦
• • Te ◦ ◦•)We ))
+
• νr • Peo Po • ◦
(
)−
• νf • Peo Pe • ◦
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ν
•
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) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +
• ◦ • De ◦ ◦ Pe • ◦
)
)
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dt d
dt
νf
◦
ν
d
dt d
dt
◦
νf
◦
νf
◦
ν
◦
◦
νf
◦
= Pom(Po ◦•) + Pmsr (Ps ◦•) − Pms(Pm ◦•) νf
◦
•
ν
◦
•
νf
◦
= Pom(Po •◦) + Pmsr (Ps •◦) − Pms(Pm •◦) νf
◦
◦
ν
◦
◦
νf
◦
= Pom(Po •◦) + Pmsr (Ps •◦) − Pms(Pm •◦) νf
◦
•
ν
◦
•
νf
◦
= Pom(Po ••) + Pmsr (Ps ••) − Pms(Pm ••) νf
◦
◦
◦
◦
◦ • Pm • •
◦ •
◦
•
◦ • Pm • ◦
νf
= Pom(Po ◦•) + Pmsr (Ps ◦•) − Pms(Pm ◦•)
◦
◦ • Pm ◦ •
◦
◦
•
•
◦ • Pm ◦ ◦
◦
= Pom(Po ◦◦) + Pmsr (Ps ◦◦) − Pms(Pm ◦◦)
◦
◦ ◦ Pm • •
νf
◦
•
◦ ◦ Pm • ◦
◦
= Pom(Po ◦◦) + Pmsr (Ps ◦◦) − Pms(Pm ◦◦) ◦
◦ ◦ Pm ◦ •
ν
◦
νf
◦
= Pom(Po ••) + Pmsr (Ps ••) − Pms(Pm ••) •
•
•
•
dPm ◦◦ ◦
dt • ◦ Pm ◦ •
d
dt • ◦ Pm • ◦
d
dt • ◦ Pm • •
d
dt • • Pm ◦ ◦
d
dt • • Pm ◦ •
d
dt • • Pm • ◦
d
dt • • Pm • •
dt
◦•
◦•
◦•
ν
• •
◦ ◦ Pm ◦ ◦
d
◦•
νf
•
ν
•
•
•
•
= ASe((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We •) + Peor (Po ••) − Peo(Pe ••) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ +
νf
•
ν
•
νf
•
= Pom(Po ◦◦) + Pmsr (Ps ◦◦) − Pms(Pm ◦◦) νf
•
◦
◦
◦
ν
•
νf
•
= Pom(Po ◦◦) + Pmsr (Ps ◦◦) − Pms(Pm ◦◦) •
νf
•
•
ν
•
•
νf
•
= Pom(Po ◦•) + Pmsr (Ps ◦•) − Pms(Pm ◦•) ◦
νf
•
◦
ν
•
◦
νf
•
= Pom(Po ◦•) + Pmsr (Ps ◦•) − Pms(Pm ◦•) •
νf
•
•
ν
•
•
νf
•
= Pom(Po •◦) + Pmsr (Ps •◦) − Pms(Pm •◦) ◦
νf
•
◦
ν
•
◦
νf
•
= Pom(Po •◦) + Pmsr (Ps •◦) − Pms(Pm •◦) •
νf
•
•
ν
•
•
νf
•
= Pom(Po ••) + Pmsr (Ps ••) − Pms(Pm ••) ◦
νf
•
◦
ν
•
◦
νf
•
= Pom(Po ••) + Pmsr (Ps ••) − Pms(Pm ••) •
•
•
•
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
27
◦
dPo ◦◦ ◦
dt
◦
νf
◦
νf
◦
◦ ◦ Po ◦ •
d
dt
◦
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
νf
d
dt
◦
◦
νf
◦
ν
= Peo(Pe •◦) − Pom(Po •◦) − Peor (Po •◦) •
◦
νf
•
◦
νf
◦
ν
= Peo(Pe ••) − Pom(Po ••) − Peor (Po ••) ◦
νf
◦
◦
◦
νf
◦
ν
= Peo(Pe ••) − Pom(Po ••) − Peor (Po ••) •
•
νf
•
•
νf
•
ν
= Peo(Pe ◦◦) − Pom(Po ◦◦) − Peor (Po ◦◦) •
νf
◦
◦
•
νf
•
ν
= Peo(Pe ◦◦) − Pom(Po ◦◦) − Peor (Po ◦◦) •
• ◦ Po • ◦
◦
ν
◦
◦
• ◦ Po ◦ •
•
= Peo(Pe •◦) − Pom(Po •◦) − Peor (Po •◦)
•
• ◦ Po ◦ ◦
◦
ν
•
◦
◦ • Po • •
◦
= Peo(Pe ◦•) − Pom(Po ◦•) − Peor (Po ◦•)
•
◦ • Po • ◦
◦
ν
◦
◦
◦ • Po ◦ •
•
= Peo(Pe ◦•) − Pom(Po ◦•) − Peor (Po ◦•)
•
◦ • Po ◦ ◦
◦
ν
•
◦
◦ ◦ Po • •
◦
= Peo(Pe ◦◦) − Pom(Po ◦◦) − Peor (Po ◦◦) •
◦ ◦ Po • ◦
◦
ν
= Peo(Pe ◦◦) − Pom(Po ◦◦) − Peor (Po ◦◦)
•
•
νf
•
•
νf
•
ν
= Peo(Pe ◦•) − Pom(Po ◦•) − Peor (Po ◦•) ◦
◦
◦
•
dPo ◦• •
dt d
dt d
dt d
dt d
dt d
dt
•
ν
•
νf
◦
◦
•
νf
•
ν
= Peo(Pe •◦) − Pom(Po •◦) − Peor (Po •◦) •
•
νf
•
•
νf
•
ν
= Peo(Pe ••) − Pom(Po ••) − Peor (Po ••) ◦
•
νf
◦
•
νf
•
ν
= Peo(Pe ••) − Pom(Po ••) − Peor (Po ••) •
◦ ◦ Ps ◦ ◦
•
◦
νf
•
•◦
ν
•◦
•◦
•◦
◦◦
◦◦
◦◦
νf
◦◦
•
•
•
•
◦
◦
= Pms(Pm ◦◦) + ASsr ((Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦ + Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦)Ws ◦) − ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + ◦
◦
◦
◦
ν
+ Ds ◦◦)Ps ◦◦) − Pmsr (Ps ◦◦)
◦ ◦ Ps ◦ •
dt
•
νf
◦
• • Po • •
d
•
νf
•
= Peo(Pe •◦) − Pom(Po •◦) − Peor (Po •◦)
•
• • Po • ◦
•
ν
•
◦
• • Po ◦ •
◦
•
νf
•
• • Po ◦ ◦
Ds ◦ •
•
νf
= Peo(Pe ◦•) − Pom(Po ◦•) − Peor (Po ◦•)
◦
νf
◦
◦
ν
••
•◦
•◦
••
•◦
•◦
••
•◦
•◦
••
•◦
•◦
◦•
◦◦
= Pms(Pm ◦◦) + ASsr ( 14 ((Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ••◦◦ + Ts •••◦ + •
Text S1 for Dynamic isotopologue model of oxygen labeling in heart ◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
28
◦◦
•◦
•◦
•◦
•◦
◦◦
◦◦
◦◦
•◦
••
•◦
•◦
◦•
◦◦
•◦
•◦
•◦
◦◦
◦◦
◦◦
•◦
••
••
•◦
◦•
◦•
•◦
•◦
••
•◦
•◦
••
+ Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦•)Ws ◦ + (Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦ + Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ +
◦ Ts • ••
νf
◦◦
•
•
•
•
◦
◦
◦
− ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• +
◦ • Ts ◦ ◦◦)Ws ))
◦ ◦ ◦ Ds ◦ ◦ Ps ◦ •
)
)−
◦ νr ◦ Pms Ps ◦ •
••
•◦
(
)
◦
dPs ◦• ◦
dt ◦◦
◦
νf
••
ν
•◦
•◦
••
•◦
•◦
= Pms(Pm ◦•) + ASsr ( 14 ((Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ••◦◦ + Ts •••◦ + ◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
•◦
+ Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦•)Ws ◦ + (Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦ + Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ +
◦ Ts • ••
νf
◦◦
◦ ◦ Ps • •
d
dt ◦◦
•
•
•
•
◦
◦
◦
◦
◦
◦
ν
− ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦)Ps ◦•) − Pmsr (Ps ◦•)
◦ • Ts ◦ ◦◦)Ws ))
◦
◦
νf
••
ν
••
•◦
••
••
•◦
◦
••
••
= Pms(Pm ◦•) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + •
◦•
◦•
◦◦
◦•
◦•
◦◦
•◦
••
•◦
•◦
◦•
◦◦
◦•
◦•
◦◦
••
+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ +
• Ts • •• •◦
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦◦ • ◦◦ • Ts ◦ •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦•)Ws )) − ◦ ◦ • • • • ◦ ◦ ◦ ◦ νf νr ◦ ◦ • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) •
◦ • Ps ◦ ◦
d
dt ◦◦
◦
νf
••
ν
•◦
•◦
••
•
•◦
•◦
••
•◦
•◦
••
•◦
•◦
◦•
◦◦
•◦
•◦
•◦
◦◦
◦◦
◦◦
•◦
••
••
•◦
◦•
◦•
•◦
•◦
••
•◦
•◦
••
= Pms(Pm •◦) + ASsr ( 14 ((Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ••◦◦ + Ts •••◦ + ◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
•◦
+ Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦•)Ws ◦ + (Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦ + Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ +
◦ Ts • ••
νf
◦◦
◦ • Ps ◦ •
d
dt ◦◦
•
•
•
•
◦
◦
◦
◦
◦
◦
ν
− ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦)Ps •◦) − Pmsr (Ps •◦)
◦ • Ts ◦ ◦◦)Ws ))
◦
◦
νf
••
ν
••
••
••
•◦
••
••
•◦
◦
= Pms(Pm •◦) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + •
◦•
◦•
◦◦
◦•
◦•
◦◦
•◦
••
•◦
•◦
◦•
◦◦
••
◦◦
◦•
◦•
+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ +
• Ts • •• •◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
• ◦◦ ◦◦ ◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ • Ts ◦ •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦•)Ws )) − ◦ ◦ • • • • ◦ ◦ ◦ ◦ νf ν • • r • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps ◦) − Pms(Ps ◦) •
◦ • Ps • ◦
d
dt ◦◦
•◦
••
••
•◦
◦•
◦•
◦•
◦•
◦◦
◦•
◦•
◦◦
•◦ ◦ Ts ◦ ••
•• ◦ Ts ◦ ◦◦
•◦ • Ts ◦ ◦◦
•◦ ◦ Ts ◦ ◦•
◦• ◦ Ts • •◦
+
◦◦ • Ts • •◦
+
◦ Ds ◦ •
••
◦◦
◦•
◦•
•◦
•◦
••
•◦
•◦
••
+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ +
•◦ • Ts ◦ •◦ ASs
••
••
•◦
••
••
•◦
••
••
ν
= Pms(Pm ••) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + ◦
• Ts • ••
νf
◦
νf
•
+
• Ds • •
((
+
+
• Ds • ◦
+
+
• Ds ◦ •
+
+
• Ds ◦ ◦
+
+
◦ Ds • •
+
◦ Ds • ◦
+
+
◦◦ ◦ Ts • ••
◦ Ds ◦ ◦
)
◦• ◦ Ts • ◦◦
+
◦ • Ps • ◦
)−
◦◦ • Ts • ◦◦ ◦ νr • Pms Ps • ◦
+
(
+
◦◦ ◦ Ts • ◦•
+
◦• ◦ Ts ◦ •◦
+
◦◦ • Ts ◦ •◦
+
◦◦ ◦ Ts ◦ ••
+
◦• ◦ Ts ◦ ◦◦
+
◦◦ • Ts ◦ ◦◦
+
◦◦ ◦ Ts ◦ ◦•
)Ws •)) −
)
◦
dPs •• •
dt •◦
=
νf Pms
◦
••
••
ν
••
••
◦•
◦•
◦•
◦•
◦•
◦•
••
••
••
••
◦◦
◦•
◦•
•◦
(Pm ••) + ASsr ( 14 ((Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦•• + Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦••)Ws ◦ + (Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + •
•◦
••
••
•◦
••
••
◦•
◦•
◦◦
◦◦
◦•
◦•
◦◦ ◦• ◦• ◦• ◦◦ •◦ •• •• •• •• •◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ • Ts • ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + Ts ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + Ts ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + ◦ ◦ ◦ ◦ ◦ ◦ • • • • ◦◦ νf νr • • • ◦ ◦ • • ◦ ◦ • • • Ts ◦ ◦•)Ws )) − ASs((Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) •
d
• ◦ Ps ◦ ◦
dt ◦◦ ◦ Ts • ••
νf
•◦
•◦
••
•◦
•◦
••
•◦
•◦
••
•◦
•◦
◦•
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
◦•
◦◦
◦◦
•◦
•◦
•◦
•◦
◦◦
◦◦
◦◦
•◦
••
••
•◦
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+ Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦•)Ws ◦ + (Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ + Ts ◦◦◦◦ + Ts ••◦◦ + Ts •◦◦◦ + Ts ◦•◦◦ +
• ◦ Ps ◦ •
• Ts • ••
•◦
◦
◦◦
dt ◦◦
••
ν
= Pms(Pm ◦◦) + ASsr ( 14 ((Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ••◦◦ + Ts •••◦ +
◦ • Ts ◦ ◦◦)Ws ))
d
•
•
νf
νf
•
•
•
•
◦
◦
◦
•
◦
ν
•
− ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦)Ps ◦◦) − Pmsr (Ps ◦◦) ◦
•
••
ν
••
•◦
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•◦
◦
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= Pms(Pm ◦◦) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + •
◦•
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+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + ◦◦
◦•
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• ◦◦ ◦◦ ◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ • Ts ◦ •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦•)Ws )) − • • • • • ◦ ◦ ◦ ◦ • νf νr ◦ ◦ ◦ ◦ • • ◦ ◦ • ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps ◦) − Pms(Ps ◦) •
•
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
29
•
dPs ◦• ◦
dt ◦◦ • Ts • ••
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νf
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ν
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= Pms(Pm ◦•) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + ◦
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+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ +
•◦
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• ◦◦ ◦◦ ◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ • Ts ◦ •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦•)Ws )) − • • • • • • ◦ ◦ ◦ ◦ νf νr ◦ ◦ • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) ◦
• ◦ Ps • •
d
dt •◦
=
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νf Pms
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ν
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(Pm ◦•) + ASsr ( 14 ((Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦•• + Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦••)Ws ◦ + (Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + •
••
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• ◦• ◦◦ ◦• ◦• ◦◦ ◦• •• •◦ •• •• •◦ •• ◦• ◦◦ ◦• ◦• ◦◦ Ts • ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + Ts ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + Ts ◦• + Ts •◦ + Ts •• + Ts •• + Ts ◦◦ + Ts ◦• + • • ◦◦ • • • • ◦ ◦ ◦ ◦ νf ν ◦ ◦ r • • • ◦ ◦ • • ◦ ◦ • Ts ◦ ◦•)Ws )) − ASs((Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) •
d
• • Ps ◦ ◦
dt ◦◦ • Ts • ••
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νf
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ν
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= Pms(Pm •◦) + ASsr ( 16 ((Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •••◦ + Ts ••◦• + ◦
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+ Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••)Ws ◦ + (Ts ••◦◦ + Ts •••◦ + Ts ••◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ +
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◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦◦ • ◦◦ • Ts ◦ •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦• + Ts •◦ + Ts •◦ + Ts •• + Ts ◦◦ + Ts ◦◦ + Ts ◦•)Ws )) − • • • • • • ◦ ◦ ◦ ◦ νf ν • • r • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps ◦) − Pms(Ps ◦) ◦
• • Ps ◦ •
d
dt •◦
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(Pm •◦) + ASsr ( 14 ((Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦•• + Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦••)Ws ◦ + (Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + •
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•
d
• • Ps • ◦
dt •◦
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νf Pms
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(Pm ••) + ASsr ( 14 ((Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦•• + Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦••)Ws ◦ + (Ts •••◦ + Ts ••◦• + Ts •••• + Ts •◦•◦ + Ts •◦◦• + ◦
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◦
d
• • Ps • •
dt ◦
Ds ◦ •
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ν
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νf
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= Pms(Pm ••) + ASsr ((Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦•• + Ts •••• + Ts •◦•• + Ts ◦••• + Ts ◦◦••)Ws •) − ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + •
•
◦
ν
•
+ Ds ◦◦)Ps ••) − Pmsr (Ps ••) •
•
• • • ◦ ◦ ◦ • • • ◦ • • • • ◦ ◦ ◦ ◦ dWe ν = ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)(Pe ◦◦ + 12 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•) + 14 (Pe •• + Pe •◦ + Pe ◦• + ◦ ◦ • ◦ • • ◦ • • ◦ dt ◦ • ◦ ◦ ◦ •• •• •• •◦ •◦ •◦ •• •• •◦ •• •• νf νr • ◦ • ◦ ◦ 3 • • • • • ◦ • • ◦ • • • • • ◦ • ◦ • • • • ◦ Pe •) + 4 (Pe ◦ + Pe ◦ + Pe • + Pe ◦))) + Weo(Wo ) − ASe((Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦◦ •+ ◦
• ••
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◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• •◦ •◦ ◦ •• •• Te • ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + •◦ ◦• •• •◦ •◦ •• •• •◦ •• •• •◦ •◦ •• •• •◦ •◦ ◦ ◦◦ •• Te ◦ ◦• + Te ◦◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te •• + ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ νf • ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦ ◦ Te ◦ •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦)We ) − Weo(We ) • • • • ◦ • ◦ ◦ ◦ ◦ • • • • ◦ ◦ ◦ ◦ dWe • ν = ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)(Pe •• + 12 (Pe •◦ + Pe ◦• + Pe ◦◦ + Pe •• + Pe •◦ + Pe ◦•) + 14 (Pe ◦◦ + Pe •◦ + Pe ◦• + • ◦ ◦ • ◦ ◦ ◦ ◦ • • dt • • ◦ • ◦ •• •• •• •• •◦ •◦ •◦ •◦ •• •• •• νf νr • ◦ • ◦ • 3 • •• •◦ •◦ •• •• •◦ •◦ •• •• •◦ Pe ◦) + 4 (Pe • + Pe ◦ + Pe • + Pe •))) + Weo(Wo •) − ASe((Te • •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + • ••
•◦
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◦ •• •• •◦ •◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• Te • ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + ◦ ◦◦ •• •• •◦ •◦ •• •• •◦ •◦ •• •• •◦ •◦ •• •• •◦ •◦ ◦• Te ◦ ◦• + Te ◦◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te •• + ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ νf • ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ • • Te ◦ •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te •• + Te •◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦ + Te ◦• + Te ◦◦)We ) − Weo(We ) ◦ • • • ◦ ◦ ◦ • • • • • • • ◦ ◦ ◦ ◦ dWs ◦ νf = ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦)(Ps ◦◦ + 12 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•) + 14 (Ps •• + Ps •◦ + Ps ◦• + ◦ ◦ ◦ • ◦ • • ◦ • • dt ◦ • ◦ ◦ ◦ •◦ •◦ •• •• •• •• •• •• •• •◦ •◦ νf νr • ◦ • ◦ ◦ 3 ◦ • ◦ • • • • • • • • • • ◦ • ◦ • • • • • ◦ Ps •) + 4 (Ps ◦ + Ps ◦ + Ps • + Ps ◦))) + Wos(Wo ) − ASs((Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦◦ •+ •
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••
•◦
•◦ ◦ Ts ◦ ◦• ◦• • Ts ◦ •◦
•◦ ◦ Ts ◦ ◦◦ ◦• ◦ Ts ◦ ••
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◦ •• •• •◦ •◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• Ts • ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ +
+ +
+
◦• • Ts • •• ◦• ◦ Ts ◦ •◦
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◦• • Ts • •◦ ◦◦ • Ts ◦ ••
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◦• ◦ Ts • •• ◦◦ • Ts ◦ •◦
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◦• ◦ Ts • •◦ ◦◦ ◦ Ts ◦ ••
+
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+ ◦◦
+ ◦•
+ Ts ◦•◦◦ + Ts ◦◦•• +
◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦ ◦ • • ◦ ◦ • • ◦ Ts • Ts • Ts • Ts • Ts • Ts • Ts • Ts • Ts • ◦◦ ◦• ◦◦ ◦• ◦◦ ◦• ◦◦ ◦• •◦ ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ν r • ◦ ◦ • • ◦ ◦ ◦ Ts ◦ Ts ◦ Ts ◦ Ts ◦ Ts ◦ Ts ◦ Ts ◦ Wos Ws ◦ ◦◦ ◦• ◦◦ ◦• ◦◦ ◦• ◦◦ W s
+
+
+
+
+
+
)
)−
◦•
+ Ts ◦••• +
+
+
+
+
+
+
+
+
+
(
)
• • • • ◦ ◦ ◦ • ◦ ◦ • • • • ◦ ◦ ◦ ◦ dWs • νf = ASs((Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦ + Ds •• + Ds •◦ + Ds ◦• + Ds ◦◦)(Ps •• + 12 (Ps •◦ + Ps ◦• + Ps ◦◦ + Ps •• + Ps •◦ + Ps ◦•) + 14 (Ps ◦◦ + Ps •◦ + Ps ◦• + • ◦ ◦ • ◦ • • ◦ ◦ ◦ dt ◦ • • • ◦ •• •• •• •◦ •◦ •◦ •◦ •• •• •• •• νf ν ◦ • • ◦ • r 3 •◦ •• •• •◦ •◦ •• •• •◦ •◦ • •• Ps ◦) + 4 (Ps • + Ps ◦ + Ps • + Ps •))) + Wos(Wo •) − ASs((Ts • •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + • ••
•◦
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◦◦
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◦•
De 0
= De ◦◦ •
◦
◦
De 1
= De ◦◦ + De •◦ + De ◦• •
•
◦
De 2
= De •◦ + De ◦• + De ••
De 3
= De ••
Di 0
= Di ◦◦
Di 1
= Di ◦◦ + Di •◦ + Di ◦•
Di 2
= Di •◦ + Di ◦• + Di ••
Di 3
= Di ••
Dm 0
= Dm ◦◦
Dm 1
= Dm ◦◦ + Dm •◦ + Dm ◦•
Dm 2
= Dm •◦ + Dm ◦• + Dm ••
Dm 3
= Dm ••
Do 0
= Do ◦◦
Do 1
= Do ◦◦ + Do •◦ + Do ◦•
Do 2
= Do •◦ + Do ◦• + Do ••
Do 3
= Do ••
Ds 0
= Ds ◦◦
Ds 1
= Ds ◦◦ + Ds •◦ + Ds ◦•
Ds 2
= Ds •◦ + Ds ◦• + Ds ••
Ds 3
= Ds ••
Te 00
= Te ◦◦◦◦
Te 01
= Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•
Te 02
= Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦••
Te 03
= Te ◦◦••
Te 10
= Te ◦◦◦◦ + Te •◦◦◦ + Te ◦•◦◦
Te 11
= Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦•
Te 12
= Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦•••
◦ •• •• •◦ •◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• Ts • ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + ◦• •◦ •◦ •• •• •◦ •◦ •• •• •◦ •◦ •• •• •◦ •◦ •• •• ◦ ◦◦ Ts ◦ ◦• + Ts ◦◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts •• + ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ◦• ◦• ◦• ◦• ◦◦ ◦◦ ◦◦ ◦◦ ν r • ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ • • Ts ◦ •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts •• + Ts •◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦ + Ts ◦• + Ts ◦◦)Ws ) − Wos(Ws )
2.3
Pool definitions ◦
•
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•
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart
31
••
◦•
◦•
•◦
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◦◦
••
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◦•
••
•◦
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••
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•◦
Te 13
= Te ◦◦•• + Te •◦•• + Te ◦•••
Te 20
= Te •◦◦◦ + Te ◦•◦◦ + Te ••◦◦
Te 21
= Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ••◦◦ + Te •••◦ + Te ••◦•
Te 22
= Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te •••◦ + Te ••◦• + Te ••••
Te 23
= Te •◦•• + Te ◦••• + Te ••••
Te 30
= Te ••◦◦
Te 31
= Te ••◦◦ + Te •••◦ + Te ••◦•
Te 32
= Te •••◦ + Te ••◦• + Te ••••
Te 33
= Te ••••
Ti 00
= Ti ◦◦◦◦
Ti 01
= Ti ◦◦◦◦ + Ti ◦◦•◦ + Ti ◦◦◦•
Ti 02
= Ti ◦◦•◦ + Ti ◦◦◦• + Ti ◦◦••
Ti 03
= Ti ◦◦••
Ti 10
= Ti ◦◦◦◦ + Ti •◦◦◦ + Ti ◦•◦◦
Ti 11
= Ti ◦◦◦◦ + Ti ◦◦•◦ + Ti ◦◦◦• + Ti •◦◦◦ + Ti •◦•◦ + Ti •◦◦• + Ti ◦•◦◦ + Ti ◦••◦ + Ti ◦•◦•
Ti 12
= Ti ◦◦•◦ + Ti ◦◦◦• + Ti ◦◦•• + Ti •◦•◦ + Ti •◦◦• + Ti •◦•• + Ti ◦••◦ + Ti ◦•◦• + Ti ◦•••
Ti 13
= Ti ◦◦•• + Ti •◦•• + Ti ◦•••
Ti 20
= Ti •◦◦◦ + Ti ◦•◦◦ + Ti ••◦◦
Ti 21
= Ti •◦◦◦ + Ti •◦•◦ + Ti •◦◦• + Ti ◦•◦◦ + Ti ◦••◦ + Ti ◦•◦• + Ti ••◦◦ + Ti •••◦ + Ti ••◦•
Ti 22
= Ti •◦•◦ + Ti •◦◦• + Ti •◦•• + Ti ◦••◦ + Ti ◦•◦• + Ti ◦••• + Ti •••◦ + Ti ••◦• + Ti ••••
Ti 23
= Ti •◦•• + Ti ◦••• + Ti ••••
Ti 30
= Ti ••◦◦
Ti 31
= Ti ••◦◦ + Ti •••◦ + Ti ••◦•
Ti 32
= Ti •••◦ + Ti ••◦• + Ti ••••
Ti 33
= Ti ••••
Tm 00
= Tm ◦◦◦◦
Tm 01
= Tm ◦◦◦◦ + Tm ◦◦•◦ + Tm ◦◦◦•
Tm 02
= Tm ◦◦•◦ + Tm ◦◦◦• + Tm ◦◦••
Tm 03
= Tm ◦◦••
Tm 10
= Tm ◦◦◦◦ + Tm •◦◦◦ + Tm ◦•◦◦
Tm 11
= Tm ◦◦◦◦ + Tm ◦◦•◦ + Tm ◦◦◦• + Tm •◦◦◦ + Tm •◦•◦ + Tm •◦◦• + Tm ◦•◦◦ + Tm ◦••◦ + Tm ◦•◦•
Tm 12
= Tm ◦◦•◦ + Tm ◦◦◦• + Tm ◦◦•• + Tm •◦•◦ + Tm •◦◦• + Tm •◦•• + Tm ◦••◦ + Tm ◦•◦• + Tm ◦•••
Tm 13
= Tm ◦◦•• + Tm •◦•• + Tm ◦•••
Tm 20
= Tm •◦◦◦ + Tm ◦•◦◦ + Tm ••◦◦
Tm 21
= Tm •◦◦◦ + Tm •◦•◦ + Tm •◦◦• + Tm ◦•◦◦ + Tm ◦••◦ + Tm ◦•◦• + Tm ••◦◦ + Tm •••◦ + Tm ••◦•
Tm 22
= Tm •◦•◦ + Tm •◦◦• + Tm •◦•• + Tm ◦••◦ + Tm ◦•◦• + Tm ◦••• + Tm •••◦ + Tm ••◦• + Tm ••••
Tm 23
= Tm •◦•• + Tm ◦••• + Tm ••••
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart
32
•◦
Tm 30
= Tm ••◦◦
Tm 31
= Tm ••◦◦ + Tm •••◦ + Tm ••◦•
Tm 32
= Tm •••◦ + Tm ••◦• + Tm ••••
Tm 33
= Tm ••••
To 00
= To ◦◦◦◦
To 01
= To ◦◦◦◦ + To ◦◦•◦ + To ◦◦◦•
To 02
= To ◦◦•◦ + To ◦◦◦• + To ◦◦••
To 03
= To ◦◦••
To 10
= To ◦◦◦◦ + To •◦◦◦ + To ◦•◦◦
To 11
= To ◦◦◦◦ + To ◦◦•◦ + To ◦◦◦• + To •◦◦◦ + To •◦•◦ + To •◦◦• + To ◦•◦◦ + To ◦••◦ + To ◦•◦•
To 12
= To ◦◦•◦ + To ◦◦◦• + To ◦◦•• + To •◦•◦ + To •◦◦• + To •◦•• + To ◦••◦ + To ◦•◦• + To ◦•••
To 13
= To ◦◦•• + To •◦•• + To ◦•••
To 20
= To •◦◦◦ + To ◦•◦◦ + To ••◦◦
To 21
= To •◦◦◦ + To •◦•◦ + To •◦◦• + To ◦•◦◦ + To ◦••◦ + To ◦•◦• + To ••◦◦ + To •••◦ + To ••◦•
To 22
= To •◦•◦ + To •◦◦• + To •◦•• + To ◦••◦ + To ◦•◦• + To ◦••• + To •••◦ + To ••◦• + To ••••
To 23
= To •◦•• + To ◦••• + To ••••
To 30
= To ••◦◦
To 31
= To ••◦◦ + To •••◦ + To ••◦•
To 32
= To •••◦ + To ••◦• + To ••••
To 33
= To ••••
Ts 00
= Ts ◦◦◦◦
Ts 01
= Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦•
Ts 02
= Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦••
Ts 03
= Ts ◦◦••
Ts 10
= Ts ◦◦◦◦ + Ts •◦◦◦ + Ts ◦•◦◦
Ts 11
= Ts ◦◦◦◦ + Ts ◦◦•◦ + Ts ◦◦◦• + Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦•
Ts 12
= Ts ◦◦•◦ + Ts ◦◦◦• + Ts ◦◦•• + Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦•••
Ts 13
= Ts ◦◦•• + Ts •◦•• + Ts ◦•••
Ts 20
= Ts •◦◦◦ + Ts ◦•◦◦ + Ts ••◦◦
Ts 21
= Ts •◦◦◦ + Ts •◦•◦ + Ts •◦◦• + Ts ◦•◦◦ + Ts ◦••◦ + Ts ◦•◦• + Ts ••◦◦ + Ts •••◦ + Ts ••◦•
Ts 22
= Ts •◦•◦ + Ts •◦◦• + Ts •◦•• + Ts ◦••◦ + Ts ◦•◦• + Ts ◦••• + Ts •••◦ + Ts ••◦• + Ts ••••
Ts 23
= Ts •◦•• + Ts ◦••• + Ts ••••
Ts 30
= Ts ••◦◦
Ts 31
= Ts ••◦◦ + Ts •••◦ + Ts ••◦•
Ts 32
= Ts •••◦ + Ts ••◦• + Ts ••••
Ts 33
= Ts ••••
••
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Ci 0
= Ci ◦◦
Text S1 for Dynamic isotopologue model of oxygen labeling in heart •
◦
◦
•
•
◦
Ci 1
= Ci ◦◦ + Ci •◦ + Ci ◦•
Ci 2
= Ci •◦ + Ci ◦• + Ci ••
Ci 3
= Ci ••
Co 0
= Co ◦◦
Co 1
= Co ◦◦ + Co •◦ + Co ◦•
Co 2
= Co •◦ + Co ◦• + Co ••
Co 3
= Co ••
Pe 0
= Pe ◦◦
Pe 1
= Pe ◦◦ + Pe •◦ + Pe ◦• + Pe ◦◦
•
◦
•
◦
◦
•
•
◦
•
◦ ◦ •
◦
◦
◦
◦
◦
◦
=
• • Pe ◦ ◦
+
• ◦ Pe • ◦
+
• ◦ Pe ◦ •
=
• • Pe • ◦
+
• • Pe ◦ •
+
• ◦ Pe • •
Pe 4
=
• • Pe • •
Pm 0
= Pm ◦◦
Pm 1
= Pm ◦◦ + Pm •◦ + Pm ◦• + Pm ◦◦
Pe 2 Pe 3
• ◦
◦
◦
+ Pe •• + Pe •◦ + Pe ◦• ◦
•
•
◦ • Pe • •
+
◦ ◦ •
◦
◦
◦
◦
◦
◦
=
• • Pm ◦ ◦
+
• ◦ Pm • ◦
+
• ◦ Pm ◦ •
Pm 3
=
• • Pm • ◦
+
• • Pm ◦ •
+
• ◦ Pm • •
Pm 4
= Pm ••
Po 0
=
◦ ◦ Po ◦ ◦
Po 1
= Po ◦◦ + Po •◦ + Po ◦• + Po ◦◦
Pm 2
• ◦
◦
◦
+ Pm •• + Pm •◦ + Pm ◦• +
•
•
◦
◦ • Pm • •
• •
•
◦
◦
◦
◦
◦
◦
=
• • Po ◦ ◦
+
• ◦ Po • ◦
+
• ◦ Po ◦ •
Po 3
=
• • Po • ◦
+
• • Po ◦ •
+
• ◦ Po • •
Po 4
= Po ••
Ps 0
=
◦ ◦ Ps ◦ ◦
Ps 1
= Ps ◦◦ + Ps •◦ + Ps ◦• + Ps ◦◦
Po 2
• ◦
◦
◦
+ Po •• + Po •◦ + Po ◦• +
•
•
◦
◦ • Po • •
• •
•
Ps 2 Ps 3 Ps 4
◦
◦
◦
◦
◦
•
=
• • Ps ◦ ◦
+
• ◦ Ps • ◦
+
• ◦ Ps ◦ •
+
◦ • Ps • ◦
=
• • Ps • ◦
+
• • Ps ◦ •
+
• ◦ Ps • •
+
◦ • Ps • •
=
• • Ps • •
We 0
= We ◦
We 1
= We •
Wo 0
= Wo ◦
Wo 1
= Wo •
0
= Ws ◦
Ws 1
= Ws •
Ws
◦
◦
◦
+ Ps •◦ + Ps ◦• •
•
33
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
2.4
34
Kinetic equations for mass isotopomers
dDe 0 νf νf νr ν = ASe((Te 00 + Te 01 + Te 02 + Te 03)(We 0 + We 1)) + ADPeo (Do 0) − ADPeo(De 0) − ASer ((Pe 0 + Pe 1 + Pe 2 + Pe 3 + Pe 4)De 0) dt dDe 1 νf νf νr ν = ASe((Te 10 + Te 11 + Te 12 + Te 13)(We 0 + We 1)) + ADPeo (Do 1) − ADPeo(De 1) − ASer ((Pe 0 + Pe 1 + Pe 2 + Pe 3 + Pe 4)De 1) dt dDe 2 νf νf νr ν = ASe((Te 20 + Te 21 + Te 22 + Te 23)(We 0 + We 1)) + ADPeo (Do 2) − ADPeo(De 2) − ASer ((Pe 0 + Pe 1 + Pe 2 + Pe 3 + Pe 4)De 2) dt dDe 3 νf νf νr ν = ASe((Te 30 + Te 31 + Te 32 + Te 33)(We 0 + We 1)) + ADPeo (Do 3) − ADPeo(De 3) − ASer ((Pe 0 + Pe 1 + Pe 2 + Pe 3 + Pe 4)De 3) dt dDi 0 νf νf νf νf νr = ADPoi(Do 0) + AKi(2Ti 00 + Ti 01 + Ti 02 + Ti 03 + Ti 10 + Ti 20 + Ti 30) + CKi(Ti 00 + Ti 01 + Ti 02 + Ti 03) + ADPim (Dm 0) − ADPim(Di 0) − dt νr νr νr ADPoi(Di 0) − AKi(2(Di 0 + Di 1 + Di 2 + Di 3)Di 0) − CKi((Ci 0 + Ci 1 + Ci 2 + Ci 3)Di 0) dDi 1 νf νf νf νf νr = ADPoi(Do 1) + AKi(2Ti 11 + Ti 01 + Ti 10 + Ti 12 + Ti 13 + Ti 21 + Ti 31) + CKi(Ti 10 + Ti 11 + Ti 12 + Ti 13) + ADPim (Dm 1) − ADPim(Di 1) − dt νr νr νr ADPoi(Di 1) − AKi(2(Di 0 + Di 1 + Di 2 + Di 3)Di 1) − CKi((Ci 0 + Ci 1 + Ci 2 + Ci 3)Di 1) dDi 2 νf νf νf νf νr = ADPoi(Do 2) + AKi(2Ti 22 + Ti 02 + Ti 12 + Ti 20 + Ti 21 + Ti 23 + Ti 32) + CKi(Ti 20 + Ti 21 + Ti 22 + Ti 23) + ADPim (Dm 2) − ADPim(Di 2) − dt νr νr νr ADPoi(Di 2) − AKi(2(Di 0 + Di 1 + Di 2 + Di 3)Di 2) − CKi((Ci 0 + Ci 1 + Ci 2 + Ci 3)Di 2) dDi 3 νf νf νf νf νr = ADPoi(Do 3) + AKi(2Ti 33 + Ti 03 + Ti 13 + Ti 23 + Ti 30 + Ti 31 + Ti 32) + CKi(Ti 30 + Ti 31 + Ti 32 + Ti 33) + ADPim (Dm 3) − ADPim(Di 3) − dt νr νr νr ADPoi(Di 3) − AKi(2(Di 0 + Di 1 + Di 2 + Di 3)Di 3) − CKi((Ci 0 + Ci 1 + Ci 2 + Ci 3)Di 3) dDm 0 νf νf νr νr = ADPim(Di 0) + ADPms (Ds 0) − ADPms(Dm 0) − ADPim (Dm 0) dt dDm 1 νf νf νr νr = ADPim(Di 1) + ADPms (Ds 1) − ADPms(Dm 1) − ADPim (Dm 1) dt dDm 2 νf νf νr νr = ADPim(Di 2) + ADPms (Ds 2) − ADPms(Dm 2) − ADPim (Dm 2) dt dDm 3 νf νf νr νr = ADPim(Di 3) + ADPms (Ds 3) − ADPms(Dm 3) − ADPim (Dm 3) dt dDo 0 νf νf νf νr = ADPeo(De 0) + AKo(2To 00 + To 01 + To 02 + To 03 + To 10 + To 20 + To 30) + CKo(To 00 + To 01 + To 02 + To 03) + ADPoi (Di 0) − dt νf νr νr νr ADPoi(Do 0) − ADPeo(Do 0) − AKo(2(Do 0 + Do 1 + Do 2 + Do 3)Do 0) − CKo((Co 0 + Co 1 + Co 2 + Co 3)Do 0) dDo 1 νf νf νf νr = ADPeo(De 1) + AKo(2To 11 + To 01 + To 10 + To 12 + To 13 + To 21 + To 31) + CKo(To 10 + To 11 + To 12 + To 13) + ADPoi (Di 1) − dt νf ν ν ν r r r ADPoi(Do 1) − ADPeo(Do 1) − AKo(2(Do 0 + Do 1 + Do 2 + Do 3)Do 1) − CKo((Co 0 + Co 1 + Co 2 + Co 3)Do 1) dDo 2 νf νf νf νr = ADPeo(De 2) + AKo(2To 22 + To 02 + To 12 + To 20 + To 21 + To 23 + To 32) + CKo(To 20 + To 21 + To 22 + To 23) + ADPoi (Di 2) − dt νf νr νr νr ADPoi(Do 2) − ADPeo(Do 2) − AKo(2(Do 0 + Do 1 + Do 2 + Do 3)Do 2) − CKo((Co 0 + Co 1 + Co 2 + Co 3)Do 2) dDo 3 νf νf νf νr (Di 3) − = ADPeo(De 3) + AKo(2To 33 + To 03 + To 13 + To 23 + To 30 + To 31 + To 32) + CKo(To 30 + To 31 + To 32 + To 33) + ADPoi dt νf νr νr νr ADPoi(Do 3) − ADPeo(Do 3) − AKo(2(Do 0 + Do 1 + Do 2 + Do 3)Do 3) − CKo((Co 0 + Co 1 + Co 2 + Co 3)Do 3) dDs 0 dt dDs 1 dt dDs 2 dt dDs 3 dt
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r = ADPms(Dm 0) + ASsr ((Ts 00 + Ts 01 + Ts 02 + Ts 03)(Ws 0 + Ws 1)) − ASs((Ps 0 + Ps 1 + Ps 2 + Ps 3 + Ps 4)Ds 0) − ADPms (Ds 0)
r = ADPms(Dm 1) + ASsr ((Ts 10 + Ts 11 + Ts 12 + Ts 13)(Ws 0 + Ws 1)) − ASs((Ps 0 + Ps 1 + Ps 2 + Ps 3 + Ps 4)Ds 1) − ADPms (Ds 1)
r = ADPms(Dm 2) + ASsr ((Ts 20 + Ts 21 + Ts 22 + Ts 23)(Ws 0 + Ws 1)) − ASs((Ps 0 + Ps 1 + Ps 2 + Ps 3 + Ps 4)Ds 2) − ADPms (Ds 2)
r = ADPms(Dm 3) + ASsr ((Ts 30 + Ts 31 + Ts 32 + Ts 33)(Ws 0 + Ws 1)) − ASs((Ps 0 + Ps 1 + Ps 2 + Ps 3 + Ps 4)Ds 3) − ADPms (Ds 3)
Text S1 for Dynamic isotopologue model of oxygen labeling in heart dTe 00 νf νf ν = ATPoe(To 00) + ASer ((Pe 0 + 14 Pe 1)De 0) − ASe((We 0 + We 1)Te 00) dt dTe 01 νf νf ν = ATPoe(To 01) + ASer (( 12 Pe 2 + 34 Pe 1)De 0) − ASe((We 0 + We 1)Te 01) dt dTe 02 νf νf ν = ATPoe(To 02) + ASer (( 12 Pe 2 + 34 Pe 3)De 0) − ASe((We 0 + We 1)Te 02) dt dTe 03 νf νf ν = ATPoe(To 03) + ASer ((Pe 4 + 14 Pe 3)De 0) − ASe((We 0 + We 1)Te 03) dt dTe 10 νf νf ν = ATPoe(To 10) + ASer ((Pe 0 + 14 Pe 1)De 1) − ASe((We 0 + We 1)Te 10) dt dTe 11 νf νf ν = ATPoe(To 11) + ASer (( 12 Pe 2 + 34 Pe 1)De 1) − ASe((We 0 + We 1)Te 11) dt dTe 12 νf νf ν = ATPoe(To 12) + ASer (( 12 Pe 2 + 34 Pe 3)De 1) − ASe((We 0 + We 1)Te 12) dt dTe 13 νf νf ν = ATPoe(To 13) + ASer ((Pe 4 + 14 Pe 3)De 1) − ASe((We 0 + We 1)Te 13) dt dTe 20 νf νf ν = ATPoe(To 20) + ASer ((Pe 0 + 14 Pe 1)De 2) − ASe((We 0 + We 1)Te 20) dt dTe 21 νf νf ν = ATPoe(To 21) + ASer (( 12 Pe 2 + 34 Pe 1)De 2) − ASe((We 0 + We 1)Te 21) dt dTe 22 νf νf ν = ATPoe(To 22) + ASer (( 12 Pe 2 + 34 Pe 3)De 2) − ASe((We 0 + We 1)Te 22) dt dTe 23 νf νf ν = ATPoe(To 23) + ASer ((Pe 4 + 14 Pe 3)De 2) − ASe((We 0 + We 1)Te 23) dt dTe 30 νf νf ν = ATPoe(To 30) + ASer ((Pe 0 + 14 Pe 1)De 3) − ASe((We 0 + We 1)Te 30) dt dTe 31 νf νf ν = ATPoe(To 31) + ASer (( 12 Pe 2 + 34 Pe 1)De 3) − ASe((We 0 + We 1)Te 31) dt dTe 32 νf νf ν = ATPoe(To 32) + ASer (( 12 Pe 2 + 34 Pe 3)De 3) − ASe((We 0 + We 1)Te 32) dt dTe 33 νf νf ν = ATPoe(To 33) + ASer ((Pe 4 + 14 Pe 3)De 3) − ASe((We 0 + We 1)Te 33) dt dTi 00 νf νf νf νf ν νr ν νr = ATPmi(Tm 00) + AKir (Di 0Di 0) + ATPio (To 00) + CKir (Di 0Ci 0) − AKi(Ti 00) − ATPio(Ti 00) − CKi(Ti 00) − ATPmi (Ti 00) dt dTi 01 νf νf νf νf ν νr ν νr = ATPmi(Tm 01) + AKir (Di 0Di 1) + ATPio (To 01) + CKir (Di 0Ci 1) − AKi(Ti 01) − ATPio(Ti 01) − CKi(Ti 01) − ATPmi (Ti 01) dt dTi 02 νf νf νf νf ν νr ν νr = ATPmi(Tm 02) + AKir (Di 0Di 2) + ATPio (To 02) + CKir (Di 0Ci 2) − AKi(Ti 02) − ATPio(Ti 02) − CKi(Ti 02) − ATPmi (Ti 02) dt dTi 03 νf νf νf νf ν νr ν νr = ATPmi(Tm 03) + AKir (Di 0Di 3) + ATPio (To 03) + CKir (Di 0Ci 3) − AKi(Ti 03) − ATPio(Ti 03) − CKi(Ti 03) − ATPmi (Ti 03) dt dTi 10 νf νf νf νf ν νr ν νr = ATPmi(Tm 10) + AKir (Di 0Di 1) + ATPio (To 10) + CKir (Di 1Ci 0) − AKi(Ti 10) − ATPio(Ti 10) − CKi(Ti 10) − ATPmi (Ti 10) dt dTi 11 νf νf νf νf ν νr ν νr = ATPmi(Tm 11) + AKir (Di 1Di 1) + ATPio (To 11) + CKir (Di 1Ci 1) − AKi(Ti 11) − ATPio(Ti 11) − CKi(Ti 11) − ATPmi (Ti 11) dt dTi 12 νf νf νf νf ν νr ν νr = ATPmi(Tm 12) + AKir (Di 1Di 2) + ATPio (To 12) + CKir (Di 1Ci 2) − AKi(Ti 12) − ATPio(Ti 12) − CKi(Ti 12) − ATPmi (Ti 12) dt dTi 13 νf νf νf νf ν νr ν νr = ATPmi(Tm 13) + AKir (Di 1Di 3) + ATPio (To 13) + CKir (Di 1Ci 3) − AKi(Ti 13) − ATPio(Ti 13) − CKi(Ti 13) − ATPmi (Ti 13) dt dTi 20 νf νf νf νf ν νr ν νr = ATPmi(Tm 20) + AKir (Di 0Di 2) + ATPio (To 20) + CKir (Di 2Ci 0) − AKi(Ti 20) − ATPio(Ti 20) − CKi(Ti 20) − ATPmi (Ti 20) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart dTi 21 νf νf νf νf ν νr ν νr = ATPmi(Tm 21) + AKir (Di 1Di 2) + ATPio (To 21) + CKir (Di 2Ci 1) − AKi(Ti 21) − ATPio(Ti 21) − CKi(Ti 21) − ATPmi (Ti 21) dt dTi 22 νf νf νf νf ν νr ν νr = ATPmi(Tm 22) + AKir (Di 2Di 2) + ATPio (To 22) + CKir (Di 2Ci 2) − AKi(Ti 22) − ATPio(Ti 22) − CKi(Ti 22) − ATPmi (Ti 22) dt dTi 23 νf νf νf νf ν νr ν νr = ATPmi(Tm 23) + AKir (Di 2Di 3) + ATPio (To 23) + CKir (Di 2Ci 3) − AKi(Ti 23) − ATPio(Ti 23) − CKi(Ti 23) − ATPmi (Ti 23) dt dTi 30 νf νf νf νf ν νr ν νr = ATPmi(Tm 30) + AKir (Di 0Di 3) + ATPio (To 30) + CKir (Di 3Ci 0) − AKi(Ti 30) − ATPio(Ti 30) − CKi(Ti 30) − ATPmi (Ti 30) dt dTi 31 νf νf νf νf ν νr ν νr = ATPmi(Tm 31) + AKir (Di 1Di 3) + ATPio (To 31) + CKir (Di 3Ci 1) − AKi(Ti 31) − ATPio(Ti 31) − CKi(Ti 31) − ATPmi (Ti 31) dt dTi 32 νf νf νf νf ν νr ν νr = ATPmi(Tm 32) + AKir (Di 2Di 3) + ATPio (To 32) + CKir (Di 3Ci 2) − AKi(Ti 32) − ATPio(Ti 32) − CKi(Ti 32) − ATPmi (Ti 32) dt dTi 33 νf νf νf νf ν νr ν νr = ATPmi(Tm 33) + AKir (Di 3Di 3) + ATPio (To 33) + CKir (Di 3Ci 3) − AKi(Ti 33) − ATPio(Ti 33) − CKi(Ti 33) − ATPmi (Ti 33) dt dTm 00 νf νf νr = ATPsm(Ts 00) + ATPmi (Ti 00) − ATPmi(Tm 00) dt dTm 01 νf νf νr = ATPsm(Ts 01) + ATPmi (Ti 01) − ATPmi(Tm 01) dt dTm 02 νf νf νr = ATPsm(Ts 02) + ATPmi (Ti 02) − ATPmi(Tm 02) dt dTm 03 νf νf νr = ATPsm(Ts 03) + ATPmi (Ti 03) − ATPmi(Tm 03) dt dTm 10 νf νf νr = ATPsm(Ts 10) + ATPmi (Ti 10) − ATPmi(Tm 10) dt dTm 11 νf νf νr = ATPsm(Ts 11) + ATPmi (Ti 11) − ATPmi(Tm 11) dt dTm 12 νf νf νr = ATPsm(Ts 12) + ATPmi (Ti 12) − ATPmi(Tm 12) dt dTm 13 νf νf νr = ATPsm(Ts 13) + ATPmi (Ti 13) − ATPmi(Tm 13) dt dTm 20 νf νf νr = ATPsm(Ts 20) + ATPmi (Ti 20) − ATPmi(Tm 20) dt dTm 21 νf νf νr = ATPsm(Ts 21) + ATPmi (Ti 21) − ATPmi(Tm 21) dt dTm 22 νf νf νr = ATPsm(Ts 22) + ATPmi (Ti 22) − ATPmi(Tm 22) dt dTm 23 νf νf νr = ATPsm(Ts 23) + ATPmi (Ti 23) − ATPmi(Tm 23) dt dTm 30 νf νf νr = ATPsm(Ts 30) + ATPmi (Ti 30) − ATPmi(Tm 30) dt dTm 31 νf νf νr = ATPsm(Ts 31) + ATPmi (Ti 31) − ATPmi(Tm 31) dt dTm 32 νf νf νr = ATPsm(Ts 32) + ATPmi (Ti 32) − ATPmi(Tm 32) dt dTm 33 νf νf νr = ATPsm(Ts 33) + ATPmi (Ti 33) − ATPmi(Tm 33) dt dTo 00 νf νf νf νf νr νr νr = ATPio(Ti 00) + AKo (Do 0Do 0) + CKo (Do 0Co 0) − AKo(To 00) − ATPoe(To 00) − CKo(To 00) − ATPio (To 00) dt dTo 01 νf νf νf νf νr νr νr = ATPio(Ti 01) + AKo (Do 0Do 1) + CKo (Do 0Co 1) − AKo(To 01) − ATPoe(To 01) − CKo(To 01) − ATPio (To 01) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart dTo 02 νf νf νf νf νr νr νr = ATPio(Ti 02) + AKo (Do 0Do 2) + CKo (Do 0Co 2) − AKo(To 02) − ATPoe(To 02) − CKo(To 02) − ATPio (To 02) dt dTo 03 νf νf νf νf νr νr νr = ATPio(Ti 03) + AKo (Do 0Do 3) + CKo (Do 0Co 3) − AKo(To 03) − ATPoe(To 03) − CKo(To 03) − ATPio (To 03) dt dTo 10 νf νf νf νf νr νr νr = ATPio(Ti 10) + AKo (Do 0Do 1) + CKo (Do 1Co 0) − AKo(To 10) − ATPoe(To 10) − CKo(To 10) − ATPio (To 10) dt dTo 11 νf νf νf νf νr νr νr = ATPio(Ti 11) + AKo (Do 1Do 1) + CKo (Do 1Co 1) − AKo(To 11) − ATPoe(To 11) − CKo(To 11) − ATPio (To 11) dt dTo 12 νf νf νf νf νr νr νr = ATPio(Ti 12) + AKo (Do 1Do 2) + CKo (Do 1Co 2) − AKo(To 12) − ATPoe(To 12) − CKo(To 12) − ATPio (To 12) dt dTo 13 νf νf νf νf νr νr νr = ATPio(Ti 13) + AKo (Do 1Do 3) + CKo (Do 1Co 3) − AKo(To 13) − ATPoe(To 13) − CKo(To 13) − ATPio (To 13) dt dTo 20 νf νf νf νf νr νr νr = ATPio(Ti 20) + AKo (Do 0Do 2) + CKo (Do 2Co 0) − AKo(To 20) − ATPoe(To 20) − CKo(To 20) − ATPio (To 20) dt dTo 21 νf νf νf νf νr νr νr = ATPio(Ti 21) + AKo (Do 1Do 2) + CKo (Do 2Co 1) − AKo(To 21) − ATPoe(To 21) − CKo(To 21) − ATPio (To 21) dt dTo 22 νf νf νf νf νr νr νr = ATPio(Ti 22) + AKo (Do 2Do 2) + CKo (Do 2Co 2) − AKo(To 22) − ATPoe(To 22) − CKo(To 22) − ATPio (To 22) dt dTo 23 νf νf νf νf νr νr νr = ATPio(Ti 23) + AKo (Do 2Do 3) + CKo (Do 2Co 3) − AKo(To 23) − ATPoe(To 23) − CKo(To 23) − ATPio (To 23) dt dTo 30 νf νf νf νf νr νr νr = ATPio(Ti 30) + AKo (Do 0Do 3) + CKo (Do 3Co 0) − AKo(To 30) − ATPoe(To 30) − CKo(To 30) − ATPio (To 30) dt dTo 31 νf νf νf νf νr νr νr = ATPio(Ti 31) + AKo (Do 1Do 3) + CKo (Do 3Co 1) − AKo(To 31) − ATPoe(To 31) − CKo(To 31) − ATPio (To 31) dt dTo 32 νf νf νf νf νr νr νr = ATPio(Ti 32) + AKo (Do 2Do 3) + CKo (Do 3Co 2) − AKo(To 32) − ATPoe(To 32) − CKo(To 32) − ATPio (To 32) dt dTo 33 νf νf νf νf νr νr νr = ATPio(Ti 33) + AKo (Do 3Do 3) + CKo (Do 3Co 3) − AKo(To 33) − ATPoe(To 33) − CKo(To 33) − ATPio (To 33) dt dTs 00 νf νf ν = ASs((Ps 0 + 41 Ps 1)Ds 0) − ATPsm(Ts 00) − ASsr ((Ws 0 + Ws 1)Ts 00) dt dTs 01 νf 1 2 3 1 0 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 01) − ASsr ((Ws 0 + Ws 1)Ts 01) dt dTs 02 νf 1 2 3 3 0 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 02) − ASsr ((Ws 0 + Ws 1)Ts 02) dt dTs 03 νf νf ν = ASs((Ps 4 + 41 Ps 3)Ds 0) − ATPsm(Ts 03) − ASsr ((Ws 0 + Ws 1)Ts 03) dt dTs 10 νf νf ν = ASs((Ps 0 + 41 Ps 1)Ds 1) − ATPsm(Ts 10) − ASsr ((Ws 0 + Ws 1)Ts 10) dt dTs 11 νf 1 2 3 1 1 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 11) − ASsr ((Ws 0 + Ws 1)Ts 11) dt dTs 12 νf 1 2 3 3 1 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 12) − ASsr ((Ws 0 + Ws 1)Ts 12) dt dTs 13 νf νf ν = ASs((Ps 4 + 41 Ps 3)Ds 1) − ATPsm(Ts 13) − ASsr ((Ws 0 + Ws 1)Ts 13) dt dTs 20 νf νf ν = ASs((Ps 0 + 41 Ps 1)Ds 2) − ATPsm(Ts 20) − ASsr ((Ws 0 + Ws 1)Ts 20) dt dTs 21 νf 1 2 3 1 2 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 21) − ASsr ((Ws 0 + Ws 1)Ts 21) dt dTs 22 νf 1 2 3 3 2 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 22) − ASsr ((Ws 0 + Ws 1)Ts 22) dt
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Text S1 for Dynamic isotopologue model of oxygen labeling in heart
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dTs 23 νf νf ν = ASs((Ps 4 + 41 Ps 3)Ds 2) − ATPsm(Ts 23) − ASsr ((Ws 0 + Ws 1)Ts 23) dt dTs 30 νf νf ν = ASs((Ps 0 + 41 Ps 1)Ds 3) − ATPsm(Ts 30) − ASsr ((Ws 0 + Ws 1)Ts 30) dt dTs 31 νf 1 2 3 1 3 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 31) − ASsr ((Ws 0 + Ws 1)Ts 31) dt dTs 32 νf 1 2 3 3 3 νf ν = ASs(( 2 Ps + 4 Ps )Ds ) − ATPsm(Ts 32) − ASsr ((Ws 0 + Ws 1)Ts 32) dt dTs 33 νf νf ν = ASs((Ps 4 + 41 Ps 3)Ds 3) − ATPsm(Ts 33) − ASsr ((Ws 0 + Ws 1)Ts 33) dt dCi 0 νf νf ν ν = CKi(Ti 00 + Ti 10 + Ti 20 + Ti 30) + Cior (Co 0) − Cio(Ci 0) − CKir ((Di 0 + Di 1 + Di 2 + Di 3)Ci 0) dt dCi 1 νf νf ν ν = CKi(Ti 01 + Ti 11 + Ti 21 + Ti 31) + Cior (Co 1) − Cio(Ci 1) − CKir ((Di 0 + Di 1 + Di 2 + Di 3)Ci 1) dt dCi 2 νf νf ν ν = CKi(Ti 02 + Ti 12 + Ti 22 + Ti 32) + Cior (Co 2) − Cio(Ci 2) − CKir ((Di 0 + Di 1 + Di 2 + Di 3)Ci 2) dt dCi 3 νf νf ν ν = CKi(Ti 03 + Ti 13 + Ti 23 + Ti 33) + Cior (Co 3) − Cio(Ci 3) − CKir ((Di 0 + Di 1 + Di 2 + Di 3)Ci 3) dt dCo 0 νf νf νr ν = CKo(To 00 + To 10 + To 20 + To 30) + Cio(Ci 0) − CKo ((Do 0 + Do 1 + Do 2 + Do 3)Co 0) − Cior (Co 0) dt dCo 1 νf νf νr ν = CKo(To 01 + To 11 + To 21 + To 31) + Cio(Ci 1) − CKo ((Do 0 + Do 1 + Do 2 + Do 3)Co 1) − Cior (Co 1) dt dCo 2 νf νf νr ν = CKo(To 02 + To 12 + To 22 + To 32) + Cio(Ci 2) − CKo ((Do 0 + Do 1 + Do 2 + Do 3)Co 2) − Cior (Co 2) dt dCo 3 νf νf νr ν = CKo(To 03 + To 13 + To 23 + To 33) + Cio(Ci 3) − CKo ((Do 0 + Do 1 + Do 2 + Do 3)Co 3) − Cior (Co 3) dt dPe 0 νf νf ν ν = ASe((Te 00 + Te 10 + Te 20 + Te 30)We 0) + Peor (Po 0) − Peo(Pe 0) − ASer ((De 0 + De 1 + De 2 + De 3)Pe 0) dt dPe 1 νf νf ν ν = ASe((Te 00 + Te 10 + Te 20 + Te 30)We 1 + (Te 01 + Te 11 + Te 21 + Te 31)We 0) + Peor (Po 1) − Peo(Pe 1) − ASer ((De 0 + De 1 + De 2 + dt De 3)Pe 1) dPe 2 νf νf ν ν = ASe((Te 01 + Te 11 + Te 21 + Te 31)We 1 + (Te 02 + Te 12 + Te 22 + Te 32)We 0) + Peor (Po 2) − Peo(Pe 2) − ASer ((De 0 + De 1 + De 2 + dt De 3)Pe 2) dPe 3 νf νf ν ν = ASe((Te 02 + Te 12 + Te 22 + Te 32)We 1 + (Te 03 + Te 13 + Te 23 + Te 33)We 0) + Peor (Po 3) − Peo(Pe 3) − ASer ((De 0 + De 1 + De 2 + dt De 3)Pe 3) dPe 4 νf νf ν ν = ASe((Te 03 + Te 13 + Te 23 + Te 33)We 1) + Peor (Po 4) − Peo(Pe 4) − ASer ((De 0 + De 1 + De 2 + De 3)Pe 4) dt dPm 0 νf νf ν = Pom(Po 0) + Pmsr (Ps 0) − Pms(Pm 0) dt dPm 1 νf νf ν = Pom(Po 1) + Pmsr (Ps 1) − Pms(Pm 1) dt dPm 2 νf νf ν = Pom(Po 2) + Pmsr (Ps 2) − Pms(Pm 2) dt dPm 3 νf νf ν = Pom(Po 3) + Pmsr (Ps 3) − Pms(Pm 3) dt dPm 4 νf νf ν = Pom(Po 4) + Pmsr (Ps 4) − Pms(Pm 4) dt dPo 0 νf νf ν = Peo(Pe 0) − Pom(Po 0) − Peor (Po 0) dt
Text S1 for Dynamic isotopologue model of oxygen labeling in heart
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dPo 1 νf νf ν = Peo(Pe 1) − Pom(Po 1) − Peor (Po 1) dt dPo 2 νf νf ν = Peo(Pe 2) − Pom(Po 2) − Peor (Po 2) dt dPo 3 νf νf ν = Peo(Pe 3) − Pom(Po 3) − Peor (Po 3) dt dPo 4 νf νf ν = Peo(Pe 4) − Pom(Po 4) − Peor (Po 4) dt dPs 0 νf νf ν ν = Pms(Pm 0) + ASsr ((Ts 00 + Ts 10 + Ts 20 + Ts 30)Ws 0) − ASs((Ds 0 + Ds 1 + Ds 2 + Ds 3)Ps 0) − Pmsr (Ps 0) dt dPs 1 νf νf ν ν = Pms(Pm 1)+ASsr ((Ts 00+Ts 10+Ts 20+Ts 30)Ws 1+(Ts 01+Ts 11+Ts 21+Ts 31)Ws 0)−ASs((Ds 0+Ds 1+Ds 2+Ds 3)Ps 1)−Pmsr (Ps 1) dt dPs 2 νf νf ν ν = Pms(Pm 2)+ASsr ((Ts 01+Ts 11+Ts 21+Ts 31)Ws 1+(Ts 02+Ts 12+Ts 22+Ts 32)Ws 0)−ASs((Ds 0+Ds 1+Ds 2+Ds 3)Ps 2)−Pmsr (Ps 2) dt dPs 3 νf νf ν ν = Pms(Pm 3)+ASsr ((Ts 02+Ts 12+Ts 22+Ts 32)Ws 1+(Ts 03+Ts 13+Ts 23+Ts 33)Ws 0)−ASs((Ds 0+Ds 1+Ds 2+Ds 3)Ps 3)−Pmsr (Ps 3) dt dPs 4 νf νf ν ν = Pms(Pm 4) + ASsr ((Ts 03 + Ts 13 + Ts 23 + Ts 33)Ws 1) − ASs((Ds 0 + Ds 1 + Ds 2 + Ds 3)Ps 4) − Pmsr (Ps 4) dt dWe 0 νf ν νr = ASer ((De 0 + De 1 + De 2 + De 3)(Pe 0 + 12 Pe 2 + 14 Pe 3 + 43 Pe 1)) + Weo (Wo 0) − ASe((Te 00 + Te 01 + Te 02 + Te 03 + Te 10 + Te 11 + dt νf Te 12 + Te 13 + Te 20 + Te 21 + Te 22 + Te 23 + Te 30 + Te 31 + Te 32 + Te 33)We 0) − Weo(We 0) dWe 1 νf ν νr = ASer ((De 0 + De 1 + De 2 + De 3)(Pe 4 + 12 Pe 2 + 14 Pe 1 + 43 Pe 3)) + Weo (Wo 1) − ASe((Te 00 + Te 01 + Te 02 + Te 03 + Te 10 + Te 11 + dt νf Te 12 + Te 13 + Te 20 + Te 21 + Te 22 + Te 23 + Te 30 + Te 31 + Te 32 + Te 33)We 1) − Weo(We 1) dWs 0 νf νf ν = ASs((Ds 0 + Ds 1 + Ds 2 + Ds 3)(Ps 0 + 12 Ps 2 + 14 Ps 3 + 34 Ps 1)) + Wos(Wo 0) − ASsr ((Ts 00 + Ts 01 + Ts 02 + Ts 03 + Ts 10 + Ts 11 + dt νr Ts 12 + Ts 13 + Ts 20 + Ts 21 + Ts 22 + Ts 23 + Ts 30 + Ts 31 + Ts 32 + Ts 33)Ws 0) − Wos(Ws 0) dWs 1 νf νf ν = ASs((Ds 0 + Ds 1 + Ds 2 + Ds 3)(Ps 4 + 12 Ps 2 + 14 Ps 1 + 34 Ps 3)) + Wos(Wo 1) − ASsr ((Ts 00 + Ts 01 + Ts 02 + Ts 03 + Ts 10 + Ts 11 + dt νr Ts 12 + Ts 13 + Ts 20 + Ts 21 + Ts 22 + Ts 23 + Ts 30 + Ts 31 + Ts 32 + Ts 33)Ws 1) − Wos(Ws 1)