Text S1

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

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

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 .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

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 •) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

•

◦

• 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 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 • •) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

•

◦

◦ Di ◦ ◦

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 ◦ + ◦

Di • •

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

+ 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 ◦ + ◦

Di • •

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

+ 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 ◦ + ◦

Di • •

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

+ 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 ◦ + ◦

Di • •

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

+ 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 ◦ + ◦

Di • •

◦

◦

◦

•

ν

•

•

•

•

◦

◦

◦

◦

•

+ 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 ◦

Di • •

◦

◦

◦

•

ν

•

•

•

•

◦

◦

◦

6 ◦

•

+ 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 ◦ + ◦

Di • •

◦

◦

◦

•

ν

•

•

•

•

◦

◦

◦

◦

•

+ 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 ◦ + ◦

Di • •

◦

◦

◦

•

ν

•

•

•

•

◦

◦

◦

◦

•

+ 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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

◦

ν

•

•

•

•

◦

◦

◦

◦

◦

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

7

◦• •• •• •• •• •◦ •◦ •◦ •◦ • • • • • 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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

•

•

ν

•

•

•

◦

◦

◦

◦

•

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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

•

•

ν

•

•

•

◦

◦

◦

◦

•

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 ◦ + •

Do ◦ •

•

◦

◦

◦

◦

•

•

ν

•

•

•

◦

◦

◦

◦

•

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 ◦) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

◦ Ds ◦ •

• • • • 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 •) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

◦ Ds • ◦

• • • • 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 ◦) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

◦ Ds • •

• • • • 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 •) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

• Ds ◦ ◦

• • • • 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 ◦) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

• Ds ◦ •

• • • • 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 •) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

• Ds • ◦

• • • • 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 ◦) •

◦

•

◦

•

◦

•

◦

•

◦

•

◦

• Ds • •

• • • • 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 •) •

d

◦◦ ◦ Te ◦ ◦◦

dt

◦

•

νf

◦

◦◦

•

ν

◦

•

◦

•

◦

◦

•

◦

◦

◦

◦

◦

◦

◦

•

•

◦

◦

νf

◦◦

= 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

8

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

9

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

10

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

11

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

12

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

13

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

14

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

15

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

16

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

17

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

18

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

19

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

20

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

21

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

22

Text S1 for Dynamic isotopologue model of oxygen labeling in heart

23

••

◦ ◦ ◦ • • • ◦ ◦ ◦ 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 ◦ •

◦

◦

24

◦

ν

+ 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 ◦) •

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 •) ◦

dPe ◦◦ ◦

dt ◦

De • •

d

νf

•◦

•◦

•◦

•◦

◦◦

◦◦

◦◦

◦◦

◦

ν

νf

◦

•

ν

•

•

•

= ASe((Te ••◦◦ + Te •◦◦◦ + Te ◦•◦◦ + Te ◦◦◦◦ + Te ••◦◦ + Te •◦◦◦ + Te ◦•◦◦ + Te ◦◦◦◦)We ◦) + Peor (Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + ◦

◦

◦

◦

◦

◦

+ De •◦ + De ◦• + De ◦◦)Pe ◦◦)

◦ ◦ Pe ◦ •

dt◦◦

◦

νf

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

◦•

◦◦

◦◦

◦◦

◦◦

◦◦

◦•

◦◦

◦◦

◦◦

◦◦

◦◦

◦•

= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + •◦

•◦

•◦

•◦

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ • • ◦ Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + ◦

νr Peo

◦

νf

◦

•

•

•

•

ν

◦

◦

◦

◦

(Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦◦)

◦ ◦ Pe • ◦

d

dt◦◦

•

•

•

νf

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

◦•

= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + •◦

•◦

•◦

•◦

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ • • ◦ Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + ◦

νr Peo

◦

νf

◦

•

•

•

•

ν

◦

◦

◦

◦

(Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•)

d

dt◦•

◦

◦

◦

◦ ◦ Pe • •

=

νf ASe

••

••

•◦

••

••

•◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + •◦

••

◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

•◦

••

•◦

•◦

••

◦◦

◦◦

•◦

◦• •◦ ◦◦ •◦ •• •• •◦ ◦• •◦ ◦• ◦◦ ◦• ◦◦ ◦• •• • •◦ ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

••

◦

νr Peo

•◦

◦•

•◦

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + ◦

νf

•

ν

•

•

•

◦

◦

◦

◦

◦

(Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•) •

◦ • Pe ◦ ◦

d

dt◦◦

•

•

νf

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

◦◦

◦◦

◦◦

◦•

= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + ◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

•◦

•◦

•◦

•◦

◦◦

• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ ◦ • Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + ◦

νr Peo

◦

νf

•

ν

•

•

•

◦

◦

◦

◦

◦

(Po •◦) − Peo(Pe •◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe •◦) ◦

◦ • Pe ◦ •

◦

d

dt◦•

=

νf ASe

◦

••

••

•◦

••

••

•◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• +

◦•

◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

•◦

•◦

••

•◦

•◦

••

◦◦

◦◦

•◦

• •◦ •• ◦• ◦◦ ◦• ◦• ◦◦ ◦• •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

◦ νr • Peo Po ◦ •

(

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + )−

◦ νf • Peo Pe ◦ •

(

ν

•

•

•

•

◦

◦

◦

) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +

◦ ◦ • De ◦ ◦ Pe ◦ •

)

)

Text S1 for Dynamic isotopologue model of oxygen labeling in heart

25

◦

dPe •• ◦

dt◦•

=

νf ASe

•◦

••

••

•◦

••

••

•◦

••

••

•◦

••

••

◦•

◦•

◦◦

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + ◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

••

•◦

•◦

••

•◦

•◦

••

◦◦

◦◦

•◦

◦• ◦◦ •◦ •• •◦ •◦ •• ◦• •◦ ◦◦ ◦• ◦• ◦◦ ◦• •• • •◦ ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

••

◦

νr Peo

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + ◦

νf

•

ν

•

•

•

◦

◦

◦

◦

◦

(Po ••) − Peo(Pe ••) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ••) ◦

◦ • Pe • •

d

dt•• • Te ◦ •◦

=

◦

νf ASe

••

dt◦◦

••

••

◦•

◦•

◦•

◦•

••

••

•◦

••

••

•◦

••

•◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

◦•

+ Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• +

◦◦

d

••

( 41 ((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We ◦ + (Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + ◦

ν

◦

νf

•

ν

•

•

•

◦

◦

◦

◦

◦

+ Peor (Po ••) − Peo(Pe ••) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ••)

• • Te ◦ ◦•)We )) • ◦ Pe ◦ ◦

◦

•

νf

•

••

•◦

•

•◦

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

◦◦

◦◦

◦◦

◦•

= ASe( 41 ((Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + ◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

•◦

•◦

•◦

•◦

◦◦

• •◦ ◦◦ ◦• ◦◦ ◦◦ ◦• ◦◦ •◦ •◦ ◦◦ ◦◦ •◦ •◦ ◦◦ ◦◦ ◦ • Te • ◦◦ + Te ◦• + Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦•)We + (Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦ + Te •◦ + Te ◦◦)We )) + •

νr Peo

•

νf

•

ν

•

•

•

◦

◦

◦

•

◦

(Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦◦) ◦

• ◦ Pe ◦ •

d

dt◦•

=

◦

νf ASe

◦

••

••

•◦

••

••

•◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• +

◦•

◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

•◦

•◦

••

•◦

•◦

••

◦◦

◦◦

•◦

• •◦ •• ◦• ◦◦ ◦• ◦• ◦◦ ◦• •◦ •• •◦ •◦ •• •◦ ◦◦ ◦• ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

••

•

νr Peo

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

••

••

•◦

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + •

νf

•

ν

•

•

•

◦

◦

◦

•

◦

(Po ◦◦) − Peo(Pe ◦◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦◦) •

• ◦ Pe • ◦

d

dt◦•

=

•

νf ASe

•

•◦

••

••

•◦

••

••

•◦

••

••

◦•

◦•

◦◦

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + ••

•◦

•◦

••

◦◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

•◦

•◦

••

◦◦

◦◦

•◦

◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦• ◦◦ ◦• ◦• ◦◦ ◦• •• •◦ • ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

••

•

νr Peo

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + •

νf

•

ν

•

•

•

◦

◦

◦

•

◦

(Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•) ◦

• ◦ Pe • •

d

dt•• • Te ◦ •◦

=

◦

νf ASe

dt◦•

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

•◦

••

••

•◦

••

◦◦

◦•

◦•

◦◦

◦•

◦•

•◦

◦•

◦•

◦◦

+ Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• +

◦◦

d

◦•

◦•

◦•

◦•

••

••

••

••

( 41 ((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We ◦ + (Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• +

=

νf ASe

•

ν

•

νf

•

ν

•

◦

•

•

◦

•

◦

◦

+ Peor (Po ◦•) − Peo(Pe ◦•) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe ◦•)

• • Te ◦ ◦•)We )) • • Pe ◦ ◦

◦

•

•

•

•◦

••

••

••

••

••

•◦

••

••

•◦

••

( 16 ((Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• + Te ◦••◦ + Te ◦•◦• + Te ◦••• + Te ◦◦•◦ + Te ◦◦◦• + Te ◦◦•• + Te •••◦ + Te ••◦• + Te •••• +

◦•

◦•

◦◦

◦◦

◦•

◦•

◦•

••

◦◦

•◦

•◦

••

•◦

•◦

••

◦◦

◦◦

•◦

◦• ◦◦ •◦ •• •◦ •◦ •• •◦ ◦• ◦◦ ◦• ◦• ◦◦ ◦• •• •◦ • ◦ Te • ◦◦ + Te ◦• + Te ◦• + Te •◦ + Te •• + Te •• + Te ◦◦ + Te ◦• + Te ◦•)We + (Te •◦ + Te •◦ + Te •• + Te ◦◦ + Te ◦◦ + Te ◦• + Te •◦ + Te •◦ + •◦ ◦ Te ◦ ••

••

•

νr Peo

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

+ Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦• + Te ••◦◦ + Te •••◦ + Te ••◦• + Te •◦◦◦ + Te •◦•◦ + Te •◦◦• + Te ◦•◦◦ + Te ◦••◦ + Te ◦•◦• + Te ◦◦◦◦ + Te ◦◦•◦ + Te ◦◦◦•)We •)) + νf

•

•

ν

•

•

•

◦

◦

◦

•

◦

(Po •◦) − Peo(Pe •◦) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• + De ◦◦)Pe •◦) ◦

• • Pe ◦ •

d

dt•• • Te ◦ •◦

=

◦

νf ASe

◦

••

••

••

••

◦•

◦•

◦•

◦•

••

••

•◦

••

••

•◦

◦◦

◦•

◦•

◦◦

◦•

◦•

•◦

••

••

•◦

( 41 ((Te •••• + Te •◦•• + Te ◦••• + Te ◦◦•• + Te •••• + Te •◦•• + Te ◦••• + Te ◦◦••)We ◦ + (Te •••◦ + Te ••◦• + Te •••• + Te •◦•◦ + Te •◦◦• + Te •◦•• +

••

•◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

+ 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

•

•

•

•

◦

◦

◦

) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +

• ◦ • De ◦ ◦ Pe ◦ •

)

)

•

dPe •• ◦

dt

=

νf ASe

••

••

••

••

◦•

◦•

◦•

◦•

••

••

( 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 ••

••

•◦

••

••

•◦

◦•

◦•

◦◦

26

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

◦•

+ Te ◦•◦• + 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

•

•

•

•

◦

◦

◦

) − ASer ((De •• + De •◦ + De ◦• + De ◦◦ + De •• + De •◦ + De ◦• +

• ◦ • De ◦ ◦ Pe • ◦

)

)

•

dPe •• •

dt ◦

De • •

d

νf

••

◦

◦

◦

•

+ De •◦ + De ◦• + De ◦◦)Pe ••) •

dt

νf

◦

ν

d

dt d

dt d

dt

νf

◦

ν

d

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

•◦

•◦

••

•◦

•◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

•◦

•◦

•◦

•◦

◦◦

◦◦

◦◦

•◦

••

••

•◦

◦•

◦•

•◦

•◦

••

•◦

•◦

••

+ 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

••

•◦

••

••

•◦

◦

••

••

= 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 ◦•)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 • ••

•

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 ◦•)Ws )) − • • • • • • ◦ ◦ ◦ ◦ νf νr ◦ ◦ • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) ◦

• ◦ Ps • •

d

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 ν ◦ ◦ r • • • ◦ ◦ • • ◦ ◦ • Ts ◦ ◦•)Ws )) − ASs((Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps •) − Pms(Ps •) •

d

• • Ps ◦ ◦

dt ◦◦ • Ts • ••

•

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 ◦•)Ws )) − • • • • • • ◦ ◦ ◦ ◦ νf ν • • r • ◦ ◦ • • ◦ ◦ ASs((Ds • • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦ + Ds • + Ds ◦)Ps ◦) − Pms(Ps ◦) ◦

• • Ps ◦ •

d

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 •◦

=

•

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 ◦

Ds ◦ •

•

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 ••) •

•

• • • ◦ ◦ ◦ • • • ◦ • • • • ◦ ◦ ◦ ◦ 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 ◦◦ •+ ◦

• ••

•◦

◦

◦ •◦

◦ •◦

• •◦

••

••

••

••

•◦

•◦

•◦

•◦

••

••

••

••

•◦

•◦

•◦

•◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• •◦ •◦ ◦ •• •• 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 ◦• + • ••

•◦

• •◦

• •◦

• •◦

••

••

••

••

•◦

•◦

•◦

•◦

••

••

••

••

•◦

•◦

•◦

•◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

◦

◦ •• •• •◦ •◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• 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 ◦◦ •+ •

◦

◦

◦

•

Text S1 for Dynamic isotopologue model of oxygen labeling in heart ••

•◦

•◦ ◦ Ts ◦ ◦• ◦• • Ts ◦ •◦

•◦ ◦ Ts ◦ ◦◦ ◦• ◦ Ts ◦ ••

•◦

•◦

•◦

••

••

••

••

◦◦ • Ts • ••

◦◦ • Ts • •◦

◦◦ ◦ Ts • ••

•◦

•◦

30 •◦

•◦

••

••

••

••

•◦

•◦

◦ •• •• •◦ •◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• ◦◦ ◦◦ ◦• ◦• 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 ◦ 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 ◦• + • ••

•◦

◦

• •◦

• •◦

• •◦

••

••

••

••

•◦

•◦

•◦

•◦

••

••

••

••

•◦

•◦

•◦

•◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

◦•

◦•

◦•

◦◦

◦◦

◦◦

◦◦

◦•

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 ◦

•

◦

•

◦

◦

•

•

◦

•

◦

•

◦

◦

•

•

◦

•

◦

•

◦

◦

•

•

◦

•

◦

•

◦

◦

•

•

◦

•

◦◦

◦•

◦◦

◦◦

◦•

◦•

◦◦

•◦

◦◦

◦◦

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

◦•

Text S1 for Dynamic isotopologue model of oxygen labeling in heart

31

••

◦•

◦•

•◦

•◦

◦◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

••

••

◦•

••

•◦

•◦

••

••

•◦

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 ••••

•◦

••

◦◦

◦•

◦◦

◦◦

◦•

◦•

◦◦

•◦

◦◦

◦◦

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

◦•

◦•

•◦

•◦

◦◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

••

••

◦•

••

•◦

•◦

••

••

•◦

◦•

•◦

••

◦◦

◦•

◦◦

◦◦

◦•

◦•

◦◦

•◦

◦◦

◦◦

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

◦•

◦•

•◦

•◦

◦◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

••

••

◦•

◦•

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 ••••

••

•◦

•◦

••

••

•◦

••

◦◦

◦•

◦◦

◦◦

◦•

◦•

◦◦

•◦

◦◦

◦◦

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

◦•

◦•

•◦

•◦

◦◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

••

••

◦•

••

•◦

•◦

••

••

•◦

◦•

•◦

••

◦◦

◦•

◦◦

◦◦

◦•

◦•

◦◦

•◦

◦◦

◦◦

••

•◦

•◦

◦•

◦◦

◦◦

◦•

◦◦

◦◦

••

••

•◦

◦•

◦•

◦◦

◦•

◦•

◦◦

••

◦•

◦•

•◦

•◦

◦◦

••

•◦

•◦

••

•◦

•◦

◦•

◦◦

◦◦

••

••

•◦

••

••

•◦

◦•

◦•

◦◦

••

••

◦•

••

•◦

•◦

••

••

•◦

◦•

•◦

••

◦

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

νf

ν

νf

ν

νf

ν

νf

ν

νf

ν

νf

ν

νf

ν

νf

ν

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

35

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

36

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

37

Text S1 for Dynamic isotopologue model of oxygen labeling in heart

38

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

39

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)