IRISA UMR 6074, Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

CNRS, IRISA UMR 6074, Campus de Beaulieu, 35042 Rennes Cedex, France

INRIA, Campus de Beaulieu, 35042 Rennes Cedex, France

INRA, UMR1348 Pegase, F-35590 Saint-Gilles, France

Agrocampus Ouest, UMR1348 Pegase, F-35000 Rennes, France

LINA, UMR 6241, Université de Nantes, Nantes, France

Abstract

Background

When studying metabolism at the organ level, a major challenge is to understand the matter exchanges between the input and output components of the system. For example, in nutrition, biochemical models have been developed to study the metabolism of the mammary gland in relation to the synthesis of milk components. These models were designed to account for the quantitative constraints observed on inputs and outputs of the system. In these models, a compatible flux distribution is first selected. Alternatively, an infinite family of compatible set of flux rates may have to be studied when the constraints raised by observations are insufficient to identify a single flux distribution. The precursors of output nutrients are traced back with analyses similar to the computation of yield rates. However, the computation of the quantitative contributions of precursors may lack precision, mainly because some precursors are involved in the composition of several nutrients and because some metabolites are cycled in loops.

Results

We formally modeled the quantitative allocation of input nutrients among the branches of the metabolic network (AIO). It corresponds to yield information which, if standardized across all the outputs of the system, allows a precise quantitative understanding of their precursors. By solving nonlinear optimization problems, we introduced a method to study the variability of AIO coefficients when parsing the space of flux distributions that are compatible with both model stoichiometry and experimental data. Applied to a model of the metabolism of the mammary gland, our method made it possible to distinguish the effects of different nutritional treatments, although it cannot be proved that the mammary gland optimizes a specific linear combination of flux variables, including those based on energy. Altogether, our study indicated that the mammary gland possesses considerable metabolic flexibility.

Conclusion

Our method enables to study the variability of a metabolic network with respect to efficiency (i.e. yield rates). It allows a quantitative comparison of the respective contributions of precursors to the production of a set of nutrients by a metabolic network, regardless of the choice of the flux distribution within the different branches of the network.

Background

When studying metabolism, it is important to elucidate how fluxes are distributed among the different pathways of the metabolic network with respect to the available quantitative information about the system behavior. Several methods can be used to address this issue. The first approach consists of building a mechanistic description of transformations and identifying the regulations involved in the system. Continuous dynamical models are often used for this purpose, especially when time-series responses to different treatments are available to infer the dynamics of the network. Static approaches such as Petri net can also identify qualitative distributions of fluxes in a metabolic network

In this paper, our main purpose is to extend this framework to study the variability of a metabolic network at the level of

Studying the variability of AIO within a complete space of plausible flux distributions requires the solving of nonlinear optimization problems which are underdetermined in tangible applications. As a second methodological contribution, we have proposed efficient algorithms to compute lower and upper bounds for AIOs over the family of flux distributions which are compatible with both the system’s stoichiometry and the experimental datasets, regardless of the choice of a flux distribution for the internal branches of the network. An important aspect is that when the metabolic network is provided with input-output data, the complete space of plausible distributions appears to have a relatively small size, and can therefore be studied with our method. Our framework is depicted in Figure

Main functionalities of the analysis workflow.

**Main functionalities of the analysis workflow.** First, extremal vertices of the simplex polyhedron of plausible flux distributions have to be computed, including the case when this space is not bounded. Then, formal algebra is required to obtain a symbolic representation of AIO matrices, expressed as a formal function, where the variables are the coefficients of a plausible flux distribution. Finally, extrema of the AIO coefficients are computed among the complete simplex of plausible flux distributions, either with the existing optimization routine or with dedicated local-search algorithms.

Our main example of application is related to milk production. In this context, several models have been introduced, in relation with the aforementioned classification of models. One class of small-size dynamical mechanistic models predicts the blood flow and input nutrients of the metabolic system (i.e. the mammary gland)

Nonetheless, in this field of study, there has been little discussion on the impact of the choice of a single model among several (possibly infinitely many) reasonable models

To gain better understanding of the system response regardless of the choice of a flux distribution for the internal branches of the network, we applied our method to estimate the variability of AIO coefficients in our model and compared the effects of two different diets on mammary gland metabolism. Our results suggest that the bounds of AIO are sufficient to distinguish the effects of different nutritional treatments without selecting a flux distribution for the internal reactions of the metabolic network by any method - optimization of a linear combination of fluxes or a residual score. Overall, the complete study suggests considerable flexibility in mammary gland metabolism. It provides a view of the functioning of the system although its internal processes still cannot be clarified because of limitations on experimentation on large animals such as ruminants.

Results

We first investigated the set of flux distributions that are compatible with the stoichiometry of our mammary gland model (depicted in Figure

Simplified view for the stoichiometric model of ruminant mammary metabolism with no long-chain fatty acid oxidation.

**Simplified view for the stoichiometric model of ruminant mammary metabolism with no long-chain fatty acid oxidation.** The complete model is detailed in a SBML Additional file

**SBML version of the metabolic model.** The complete metabolic model of mammary gland metabolism is provided in the free and open standard SBML representation format (Systems Biology Markup Language).

Click here for file

We then successively computed the set of flux distributions compatible with the model and the real datasets of lactation metabolism in dairy cows. The datasets are given in Table _{64}), NADPH oxidation (_{19}), OAA →PYR (_{14}), OAA →G3P (_{15}) and G3P →G6P (_{8}). Therefore, by including this dataset the model became a fairly small and constrained network. Nevertheless, it was not uniquely determined since there were still several degrees of freedom.

**Input or output flux**

**(Ctrl) [**
**]**

**(CN) [**
**]**

**(HB) [**
**]**

**mmol/h/ half udder**

**mmol/h/ half udder**

**mol/d/ udder**

**mmol/h/ half udder**

Data are renormalized in mmol/h/half udder. (a) Control diet _{
I
} and _{
O
} together with additional biological linear constraints on some reaction fluxes.

^{1}Input i.e. taken up by the stoichiometric system considered (i.e.net uptake in our example for the mammary gland).

^{2}**
β
**-Hydroxybutyrate.

^{3}Total triglycerides secreted in milk considering that milk fat was composed of 100% triglycerides and that all the triglycerides were secreted in milk fat

^{4}Output i.e. leaving the system (secreted in milk).

^{5}All fatty acids synthesized within the mammary gland i.e. all C4 to C14 and 50% of C16

^{6}The balance between amino acid net uptake and amino acid net output in milk protein is calculated with established rules

^{7}Serine output corresponded to Serine synthesized minus Serine utilized in other pathways i.e. Serine required in addition to Ser uptake to synthesize milk protein.

^{8}Peptide output: number of peptide links required to synthesize the proteins exported out of the system (i.e. in milk protein).

^{9}Hanigan, 1994

^{10}Fatty acid primers were synthesized for 50% from acetate and for 50% from BHBA except C4 FA primer which was supposed to be synthetized only from BHBA

^{11}Set at zero because their inputs or outputs are set at zero (to avoid futile cycle).

_{2}

Glucose input ^{(1)}

237

232

12.21

254

_{95}

Glycerol input

5.84

5.74

0.033

0.69

_{96}

Acetate input

510

462

18.42

384

_{97}

BHBA input ^{(2)}

84

167

7.25

151

_{98}

Lactate input

0

0

0.023

0.48

_{62}

3C(n:m)-acycoA+glycerol-

32.96

39.11

1.52

31.67

3P →^{(3)} output ^{(4)}

**Fatty acid output (synthezized)**
^{
(5)
}

_{100}

C(4:0)

10.08

11.59

0.46

9.48

_{101}

C(6:0)

4.51

5.58

0.18

3.79

_{102}

C(8:0)

2.23

2.87

0.10

2.06

_{103}

C(10:0)

4.66

6.46

0.19

3.96

_{104}

C(12:0)

4.23

6.12

0.17

3.56

_{105}

C(14:0)

13.90

17.89

0.45

9.31

_{106}

C(16:0)

18.82

21.44

0.64

13.40

_{99}

Lactose output

73.80

83.52

3.81

79.28

**Amino acids balance**
^{
(6)
}**i.e. entry or output**

_{128}

Alanine input

3.11

0

0.105

2.19

Alanine catabolism

_{121}

Alanine output

0

3.26

0

0

Alanine synthesis

_{119}

Arginine input

4.40

4.48

0.526

10.96

Arginine catabolism

_{134}

Asparagine output

0

0

0.023

0.48

Asparagine synthesis

_{125}

Aspartate output

3.43

4.13

0.247

5.15

Aspartate synthesis

_{122}

Glutamate output

0.54

6.33

0.230

4.79

Glutamate synthesis

_{131}

Clutamine input

1.22

1.79

0.072

1.50

Glutamine catabolism

_{120}

Glycine output

4.98

3.44

0.248

5.17

Glycine synthesis

_{124}

Proline output

10.65

10.99

0.670

13.96

Proline synthesis

_{136}

Serine output ^{(7)}

7.21

7.50

0.090

1.88

Serine synthesis - Serine

used in other pathways

_{118}

Histidine input

0.23

0

0

0

Histidine catabolism

_{113}

Isoleucine input

2.19

3.57

1.518

31.63

Isoleucine catabolism

_{114}

Leucine input

2.02

3.76

0

0

Leucine catabolism

_{108}

Lysine input

2.68

3.58

0.191

3.98

Lysine catabolism

_{111}

Threonine input

0.35

0

0

0

Threonine catabolism

_{115}

Valine input

2.54

3.86

0.438

9.13

Valine catabolism

_{107}

Peptide output ^{(8)}

124.5

150.0

7.2

149.17

**Additional constraints**

_{82}

NADPH through ICDH pathways ^{(9)}

30%

30%

30%

30%

NADPH through Pentose Phosphate ^{(9)}

70%

70%

70%

70%

_{56}=3_{62}

C(n:m) →C(n:m)-acylCoA

98.87

117.32

4.56

95.00

**FA primer from Acetate**
^{
(10)
}

_{53}

C(4:0)

0

0

0

0

_{54}=_{90}

C(6:0)

2.256

2.790

0.091

1.896

_{55}=_{91}

C(8:0)

1.113

1.437

0.050

1.031

_{51}=_{52}

C(10:0)

2.331

3.230

0.095

1.979

_{86}=_{92}

C(12:0)

2.116

3.061

0.086

1.781

_{87}=_{93}

C(14:0)

6.951

8.946

0.223

4.655

_{88}=_{94}

C(16:0)

9.410

10.719

0.322

6.698

**Other constraints**
^{(11)}

_{24}

Lactate →Pyruvate

0

0

_{44}

Alanine catabolism

0

_{76}

Alanine synthesis

0

0

0

_{83}

Asparagine synthesis

0

0

_{40}

Histidine catabolism

0

0

0

_{36}

Leucine catabolism

0

0

_{33}

Threonine catabolism

0

0

0

Investigating the relevance of the optimization strategies for mammary metabolism

The balance between the ATP generated by the system and the ATP used by the system (including the ATP cost of milk component synthesis) was computed for the three datasets (Ctrl), (CN) and (HB). The results are detailed in Table

**Biological model**

**Dataset**

**ATP balance**

**Proteic**

**
Criteria of selection of a solution
**

**turnover**

Three natural assumptions are considered to model the mammary gland behavior: removal of all cycles, optimization of ATP production and study of the equilibria of a dedicated ODE-based model. All models exhibit considerable variability in their ATP balance (in mmol/h/half udder), which contradicts the assumption about the behavior of this organ. Moreover, quantitatively, the computed ATP balances are much higher than recent measurements. This suggests that ATP maximization cannot be considered as a natural objective function to model cow mammary behavior.

Mammary-gland model (Figure

(HB)

6628

0

(Ctrl)

3081

0

(CN)

2045

0

Mammary-gland model (Figure

(HB)

6628

0

(Ctrl)

3081

0

(CN)

2045

0

Model of Hannigan- Baldwin

(HB)

4375 = 6500-2125

0

(Ctrl)

(CN)

First, we considered the manual computation of fluxes with a tool named “metabolic spreadsheet”

According to this model, the ATP generated is estimated to be 6500 mmol/h/half udder (i.e. 312 mol/d/udder) while 2125 mmol/h/half udder (i.e. 102 mol/d/udder) were estimated to be used for milk component synthesis

Independently of the approach considered and the model or dataset at hand, the remaining ATP (ATP balance), after use for milk synthesis, appeared to be rather high and nonxconstant. Indeed, as shown in

In addition, both in the ODE model and the ATP-optimization approach, peptide hydrolysis was obtained at zero, implying an absence of any protein turnover. This contradicts all the observations about this pathway: considerable use of this pathway has been evidenced in several publications, although the peptide hydrolysis rates differed significantly depending on the technique used for the measurements: peptide hydrolysis (_{64} i.e. mammary protein degradation) spans from 0.25,0.23 to 0.67 of peptide synthesis (_{63} i.e. total mammary protein synthesis) in (Ctrl), (CN) and (HB), respectively

Overall, we concluded with this analysis that the energy-based optimization function may not allow an appropriate simulation of mammary gland metabolism. This was expected, considering that the system is studied at the complete organ level, involving competing processes which can rarely be modeled with a single linear objective function.

Exploring all extreme flux distributions in a refined simplex

In order to study the variability within the space of plausible flux distributions and to identify alternative relevant optimization strategies, an additional constraint was placed on the ATP balance of the system. We considered that an ATP-balance of 1250 mmol/h/half udder (60 mol/d/udder) was a relevant measure for this study _{8}) was no longer an independent variable in the system. The simplex, which was previously unbounded, appeared to be bounded with four independent variables: OAA →PYR (_{14}), OAA →G3P (_{15}), NADPH oxidation (_{19}), peptide hydrolysis (_{64}).

Applying flux variability tools _{14},_{15},_{19},_{64} was equal to zero for all treatments. The maxima of the fluxes (_{14},_{15},_{19},_{64}) were (1831,1831,669,305) for the (Crtl) treatment and (795,795,22,133) for the (CN) treatment This suggests that the (Ctrl) treatment generates a more flexible space of plausible flux distributions than the (CN) treatment.

We computed all extreme vertices of the simplex of plausible flux distributions for the two treatments (Ctrl), (CN). The simplex structure of the polyhedron implies that the optimum of any linear combination of metabolic fluxes involved in the model is either uniquely attained for one of these extreme points or attained by all flux distributions positioned on a face of the simplex. To gain insight on the pathways involved in the variability of our model, we also computed the linear combination of fluxes optimized for one of the extreme flux distributions. Eight extreme vertices were found for the (Ctrl) and (CN) datasets. They were obtained when optimizing the same set of linear functions^{a}. In Table _{14}), OAA →G3P (_{15}), G3P →G6P (_{8}), peptide hydrolysis (_{64}) is strongly activated whereas the three other remaining fluxes are blocked.

**Dataset**

**Model name**

**Example of**

**Combinatorics of pathways**

**Validation**

**Pathways with**

**maximized**

**nonrelevant flux**

**function**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**values**

**NADPH**

**OAA →G3P**

**OAA →PYR**

**G3P →G6P**

**Peptide**

**Peptide**

**Pyr****
→
**

**oxidation**

**hydrolysis**

**synthesis**

The qualitative properties of all vertices are shared in (Ctrl) and (CN) treatments. Both correspond to a simplex with height vertices. So are six of each of the (Ctrl) and (CN) simplex vertices. H and G vertices, in the (Ctrl) and (CN) treatments, are plausible with respect to the literature.

Extreme flux distributions within the set of plausible solutions

(Ctrl)

B

_{15}- _{19}

0

1831

0

0

0

125

1835

_{13}, _{64}

(CN)

795

150

803

(Ctrl)

F

_{14}- _{19}

0

0

1831

0

0

125

1835

_{13}, _{64}

(CN)

795

150

803

(Ctrl)

D

_{8}- _{19}

0

0

0

3662

0

125

4

_{8}, _{64}

(CN)

1590

150

8

(Ctrl)

H

_{64}- _{19}

0

0

0

0

305

430

4

(CN)

133

283

8

(Cntl)

A

_{15} + _{19}

694

1714

0

0

0

125

1718

_{13}, _{64}

(CN)

22

791

150

799

(Ctrl)

E

_{14} + _{19}

694

0

1714

0

0

125

1718

_{13}, _{64}

(CN)

22

791

150

799

(Ctrl)

C

_{8} + _{19}

694

0

0

3428

0

125

4

_{8}, _{64}

(CN)

22

1583

150

8

(Ctrl)

G

_{64} + _{19}

669

0

0

0

286

410

4

(CN)

22

132

282

8

Litterature-based upperbounds for fluxes

≤ 591

Non-zero

Lower than

≤ 266 mmol/h/half

mmol/

whole body

udder

h/half udder

protein synthesis

As discussed in a previous paragraph, flux distributions with no peptide hydrolysis cannot be considered as relevant _{63}
_{63} equals 430 or 410 mmol/h/half udder in (Ctrl) and 283 or 282 mmol/h/half udder in (CN).

Study of quantitative contributions of precursors (AIO) for plausible extreme flux distributions

As a further investigation to check the relevance of distributions G and H in the (CN) and (Ctrl) treatments, we studied the quantitative contributions of precursors of output nutrients. This study was inspired by the usual techniques in the field of nutrition - or any domain concerned with organ studies. These techniques consist in computing yield rates to elucidate how an input nutrient may contribute to the composition of an output product, for instance to clarify what proportion of glucose, acetate or alanine taken up by the mammary gland can be recovered in the milk components (lactose, fatty acids, protein) or oxidized and recovered in CO2 released in blood. To formalize this issue, we first selected carbon as the component according to which the contributions of precursors were to be computed. Then, in order to determine how much carbon introduced into the system through a given input flux can be recovered in the rate of production of an output metabolite, we introduced a precise model for the

**Origin of carbon mass in outputs for (Ctrl) treatment**

Both models have empty flux through the reactions OAA → PYR (_{14}), OAA → G3P (_{15}) and G3P → G6P (_{8}). Model (G) shows strong NADPH oxidation whereas model (H) has zero NADPH oxidation.

**Input**

**GLC**

**Glycerol**

**Acetate**

**BHBA**

**Lys**

**Threonine**

**Isoleucine**

**Leucine**

**Valine**

**Histidine**

**Arginine**

**Alanine**

**Glutamine**

**Origin of the carbon mass of each output within input (in percentage of total carbon mass of each output)**

**Output**

Model

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

Glycerol3P

87.4

95.3

12.6

4.7

0

0

0

0

0

0

0

0

0

0

0

lactose

100.0

0

0

0

0

0

0

0

0

0

0

0

0

C4

0

0

0

100.0

0

0

0

0

0

0

0

0

0

c6

0

0

66.7

33.3

0

0

0

0

0

0

0

0

0

c8

0

0

75.0

25.0

0

0

0

0

0

0

0

0

0

c10

0

0

80.0

20.0

0

0

0

0

0

0

0

0

0

c12

0

0

83.3

16.7

0

0

0

0

0

0

0

0

0

c14

0

0

85.7

14.3

0

0

0

0

0

0

0

0

0

c16

0

0

87.5

12.5

0

0

0

0

0

0

0

0

0

Glycine

73.0

78.3

9.2

3.7

9.7

9.8

4.0

4.1

0.3

1.4

0.3

0.2

0.3

0.8

0.4

0.2

0.1

Glutamate

1.3

17.2

0.2

0.8

61.5

51.2

25.5

21.3

1.4

1.1

0.1

1.7

1.4

1.6

1.3

1.3

1.1

0.2

0.1

3.4

2.8

1.0

0.7

0.9

0.8

Proline

1.3

17.2

0.2

0.8

61.5

51.2

25.5

21.3

1.4

1.1

0.1

1.7

1.4

1.6

1.3

1.3

1.1

0.2

0.1

3.4

2.8

1.0

0.7

0.9

0.8

Aspartate

1.8

17.6

0.2

0.9

60.1

50.3

25.0

20.9

1.3

1.1

0.1

2.2

1.9

1.5

1.3

2.0

1.6

0.2

0.1

3.3

2.8

1.4

0.7

0.9

0.8

Peptide

0

0

0

0

0

0

0

0

0

0

0

0

0

SERoutput

84.8

92.5

12.2

4.6

0.0

0.0

0.0

1.8

0.0

0.0

0.0

1.1

0.0

0.0

0.0

CO2Output

37.6

35.7

0.1

1.0

38.7

39.3

16.1

16.3

1.3

1.4

0.1

1.1

1.0

1.0

1.1

0.1

1.7

1.8

0.8

0.5

**Origin of carbon mass in outputs for (CN) treatment**

Both models have empty flux through the reactions OAA → PYR (_{14}), OAA → G3P (_{15}) and G3P → G6P (_{8}). Model (G) shows strong NADPH oxidation whereas model (H) has zero NADPH oxidation.

**Input**

**GLC**

**Glycerol**

**Acetate**

**BHBA**

**Lys**

**Isoleucine**

**Leucine**

**Valine**

**Arginine**

**Glutamine**

**Origin of the carbon mass of each output within input**

**(in percentage of total carbon mass of each output)**

**Output**

Model

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

(G)

(H)

Glycerol3P

90.3

90.7

9.7

9.3

0

0

0

0

0

0

0

0

0

Lactose

100.0

0

0

0

0

0

0

0

0

0

0

C4

0

0

0

100.0

0

0

0

0

0

0

c6

0

0

66.7

33.3

0

0

0

0

0

0

c8

0

0

75.0

25.0

0

0

0

0

0

0

c10

0

0

80.0

20.0

0

0

0

0

0

0

c12

0

0

83.3

16.7

0

0

0

0

0

0

c14

0

0

85.7

14.3

0

0

0

0

0

0

c16

0

0

87.5

12.5

0

0

0

0

0

0

Glycine

73.8

74.1

7.3

7.0

5.6

10.8

0.5

0.5

0.5

0.4

0.5

0.2

Alanine

90.3

90.7

9.7

9.3

0

0

0

0

0

0

0

0

Glutamate

2.5

3.0

0.2

0.3

28.7

28.6

56.1

55.8

1.6

2.4

2.5

1.7

3.0

1.2

Proline

2.5

3.0

0.2

0.3

28.7

28.6

56.1

55.8

1.6

2.4

2.5

1.7

3.0

1.2

Aspartate

3.8

4.3

0.4

27.8

27.6

54.2

53.9

1.6

1.5

3.2

2.4

2.6

2.9

1.2

Peptide

0

0

0

0

0

0

0

0

0

0

SERoutput

90.3

90.7

9.7

9.3

0

0

0

0

0

0

0

0

CO2Output

24.2

24.1

0.2

22.2

43.4

1.9

1.8

2.0

1.7

1.9

0.8

This precise analysis of the origin of carbon in lactose synthesis with AIO suggests that extreme flux distributions G and H have to be rejected, although both flux distributions are consistent with flux variability criteria. We concluded that none of the extreme vertices of the set of plausible flux distributions could be considered biologically relevant with respect to the model and data at hand. Notice that the optimization of any linear combination of metabolic fluxes is either reached by these extreme distributions or reached by the infinite number of flux distributions lying in a face of the simplex. This suggests that the functioning of the mammary gland cannot be uniquely modeled by the optimization of any linear combination of metabolic fluxes involved in the current model.

Discriminate treatments despite mammary gland flexibility: maxima of AIO on the complete polyhedron of flux distributions

Previous studies suggest that the response of the mammary gland cannot be modeled uniquely by the optimization of a linear objective function of fluxes. However, there are several nonoptimal flux distributions that satisfy the literature-based information that we have used so far^{b}. More generally, many flux distributions compatible with the additional constraints, including the condition on carbon precursors for lactose can be shown for both (CN) and (Ctrl) treatments. Topological arguments prove that an infinite set of flux distributions exist. Nonetheless, without an idea about the exact shape of the space of feasible fluxes (because of the nonlinear nature of the condition of carbon precursors), we cannot select a plausible point within this space. In other words, the available knowledge appears insufficient to determine uniquely a flux distribution of nutrients among the different branches of the proposed model.

To understand the functioning of the mammary gland and despite this difficulty, we introduced a method to estimate the variability of nutrient allocation among pathways on a carbon basis (see Eq. (3)) by computing the range (min-max) of AIO coefficients. As these coefficients are nonlinear functions of flux variables, computing these min and max over the complete space of plausible flux distribution requires solving nonlinear optimization problems. Our dedicated algorithm detailed in the method section scaled properly to the real case that we are studying^{c}. Interestingly, with this approach, we did not favor any internal functioning of the system since we parsed all objective functions (linear combinations of flux variables), in order to have a complete description of the space of plausible flux distributions.

The min-max tables for allocations of nutrients in the different pathways provided a clearer view of nutrient utilization within the mammary gland. Unlike the functioning of the extremal distributions (G) and (H) shown in Tables

**(Ctrl) treatment**

A local-search algorithm allowed us to compute the minima and maxima of each AIO coefficient for the two treatments (Ctrl), (CN) (in mmol/h/half udder of Carbon). These tables allow discriminating the response of the mammary gland to the two treatments without requiring selection of a flux distribution for reactions in the metabolic network. (CN) treatment (protein intake by food) is characterized by a lower proportion of glucose which is oxidized in CO2 than in (Ctrl).

^{(1)}Amino acid input corresponded to positive balances between amino acid net uptake and amino acid and utilization in milk protein (i.e. peptide output).

^{(2)}Amino acid Output corresponded to negative balance between, amino acid net uptake and utilization in milk protein (i.e. peptide output).

**Input**

**Glucose**

**Glycerol**

**Acetate**

**BHBA**

**Lysine**

**Threonine**
^{
1
}

**Isoleucine**
^{
1
}

**Leucine**
^{
1
}

**Valine**
^{
1
}

**Histidine**
^{
1
}

**Arginine**
^{
1
}

**Alanine**
^{
1
}

**Glutamine**
^{
1
}

**1422**

**17,5**

**1020**

**336**

**16,1**

**1,40**

**13,1**

**12,1**

**12,7**

**1,38**

**26,4**

**9,33**

**6,10**

Output

**Minimum and maximum utilisation of Input in each output (in mmol/h/half udder of Carbon)**

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Glycerol3P

98.9

32.9

97.0

1.24

12.5

0

38.4

0

16.0

0

1.1

0

0.1

0

1.3

0

1.0

0

1.2

0

0.1

0

2.0

0

1.0

0

0.6

Lactose

886

751

886

0

7.7

0

79.8

0

33.2

0

2.3

0

0.1

0

2.6

0

2.0

0

2.4

0

0.2

0

4.0

0

2.0

0

1.1

C4

40.3

0

0

0

40.3

0

0

0

0

0

0

0

0

0

c6

27.1

0

0

18.0

9.0

0

0

0

0

0

0

0

0

0

c8

17.8

0

0

13.4

4.5

0

0

0

0

0

0

0

0

0

c10

46.6

0

0

37.3

9.3

0

0

0

0

0

0

0

0

0

c12

50.8

0

0

42.3

8.5

0

0

0

0

0

0

0

0

0

c14

195

0

0

167

27.8

0

0

0

0

0

0

0

0

0

c16

301

0

0

263

37.6

0

0

0

0

0

0

0

0

0

Glycine ^{2}

10.0

3.5

8.0

0.1

0.9

1.0

3.7

0.4

1.5

0.0

0.1

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.2

0.0

0.1

0.0

0.1

Glutamate ^{2}

2.7

0.0

1.1

0.0

0.1

1.0

1.6

0.4

0.7

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

Proline ^{2}

53.3

0.7

20.8

0.0

0.7

20.0

32.4

8.3

13.5

0.5

0.7

0.0

0.1

0.4

0.9

0.5

0.8

0.3

0.7

0.0

0.1

1.0

1.8

0.2

0.6

0.3

0.5

Aspartate ^{2}

13.7

0.2

7.4

0.0

0.3

3.8

8.2

1.6

3.4

0.1

0.2

0.0

0.1

0.1

0.3

0.1

0.2

0.1

0.3

0.0

0.1

0.2

0.4

0.0

0.2

0.0

0.1

Serine ^{2}

21.6

7.0

20.6

0.3

2.6

0.0

8.2

0.0

3.4

0.0

0.2

0.4

0.4

0.0

0.3

0.0

0.2

0.0

0.3

0.2

0.3

0.0

0.4

0.0

0.2

0.0

0.1

CO2

1126

396

565

1.3

14.5

343

446

142

185

12.1

15.3

0.7

0.8

9.1

12.4

8.7

11.3

9.1

12.1

0.7

1.0

15.1

20.3

6.3

8.8

4.2

5.6

**(CN) treatment**

A local-search algorithm allowed us to compute the minima and maxima of each AIO coefficient for the two treatments (Ctrl), (CN) (in mmol/h/half udder of Carbon). These tables allow discriminating the response of the mammary gland to the two treatments without requiring selection of a flux distribution for reactions in the metabolic network. (CN) treatment (protein intake by food) is characterized by a lower proportion of glucose which is oxidized in CO2 than in (Ctrl).

^{(1)}Amino acid input corresponded to positive balances between amino acid net uptake and amino acid and utilization in milk protein (i.e. peptide output).

^{(2)}Amino acid Output corresponded to negative balance between, amino acid net uptake and utilization in milk protein (i.e. peptide output).

**Input**

**Glucose**

**Glycerol**

**Acetate**

**BHBA**

**Lysine**
^{
1
}

**Isoleucine**
^{
1
}

**Leucine**
^{
1
}

**Valine**
^{
1
}

**Arginine**
^{
1
}

**Glutamine**
^{
1
}

**1392**

**17.2**

**924**

**668**

**21.5**

**21.4**

**22.6**

**19.3**

**26.9**

**8.95**

Output

**Minimum and maximum utilisation of input in each output (in mmol/h/half udder of Carbon)**

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Min

Max

Glycerol3P

117

36.1

115

1.8

11.3

0

22.8

0

44.5

0

1.6

0

2.3

0.0

2.0

0.0

2.0

0.0

2.2

0.0

0.9

Lactose

1002

866

1002

0

9.2

0

37.9

0

74.0

0

2.6

0

3.9

0.0

3.3

0.0

3.3

0.0

3.7

0.0

1.5

C4

46.4

0

0

0

46.4

0

0

0

0

0

0

C6

33.5

0

0

22.3

11.2

0

0

0

0

0

0

C8

23.0

0

0

17.2

5.7

0

0

0

0

0

0

C10

64.6

0

0

51.7

12.9

0

0

0

0

0

0

C12

73.4

0

0

61.2

12.2

0

0

0

0

0

0

C14

250

0

0

215

35.8

0

0

0

0

0

0

C16

343

0

0

300

42.9

0

0

0

0

0

0

glycine ^{2}

6.9

2.1

5.5

0.0

0.5

0.4

1.3

0.7

2.6

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

0.0

0.1

Alanine ^{2}

9.8

1.4

9.6

0.0

0.9

0

2.4

0

4.7

0

0.2

0

0.3

0

0.2

0

0.2

0

0.2

0

0.1

glutamate ^{2}

31.7

0.8

8.8

0.0

0.4

6.7

9.0

13.0

17.6

0.4

0.5

0.4

0.7

0.6

0.8

0.3

0.5

0.7

0.9

0.3

0.4

proline ^{2}

55.0

1.4

15.3

0.0

0.6

11.6

15.6

22.6

30.5

0.7

0.9

0.8

1.3

1.0

1.4

0.5

0.9

1.2

1.6

0.5

0.7

aspartate ^{2}

16.5

0.6

7.0

0.0

0.3

2.7

4.5

5.3

8.9

0.2

0.3

0.3

0.5

0.2

0.4

0.2

0.4

0.3

0.5

0.1

0.2

serine ^{2}

22.5

6.9

22.0

0.3

2.2

0

4.4

0

8.5

0

0.3

0

0.4

0

0.4

0

0.4

0

0.4

0

0.2

CO2

1021

244

402

2.0

12.5

180

227

351

444

16.3

19.8

14.2

19.1

15.8

20.0

13.4

17.4

14.8

19.2

5.9

7.8

More generally, the intervals of distributions of nutrients in different pathways were used to compare the effects of the (Ctrl) and (CN) treatments. Biologically, in comparison to the (Ctrl) treatment, the (CN) treatment was characterized by a lower proportion of glucose (on a carbon basis) which is oxidized in CO2, and a larger ratio used for lactose synthesis. This hypothesis can be sustained since the intervals of distribution of carbon from glucose in CO2 and lactose almost did not overlap in the (Ctrl) and (CN) treatments. More precisely, in the (Ctrl) treatment, from

Altogether, this analysis allowed us to discriminate the effects of the different treatments whatever the internal functioning of the system may be: the (CN) treatment (increase in protein supply to cows) was characterized by a lower proportion of glucose oxidized in CO2 than in (Ctrl). It appears to be a suitable strategy to analyze the metabolism flexibility without selecting a precise flux distribution or making any assumption on the internal metabolic fluxes.

As a final study, we used an interior point exploration method to estimate the AIO variability over the boundary of the simplex, rather than the complete space that was explored previously. As both tables appeared to be equal, we concluded that optimized AIO are reached on the boundary of the simplex. However, not all extreme vertices of the simplex are relevant biologically and they do not optimize AIO. This suggests that the flux distributions that optimize an AIO coefficient are placed at the interior of the simplex faces. In the future, it may be interesting to study the biological significance with multi-objective approaches

Impact of long-chain fatty acids oxidation over the model predictions

As a last study, we studied the robustness of our conclusion with respect to changes in the modeling of input (net uptake) of long-chain fatty acids (LCFA), C16 and C18, in the tricarboxylic acid (TCA) cycle. In the study of the (Ctrl) and (CN) treatments, such an input of LCFA was not considered in the system since they are not synthesized by the mammary gland and we hypothesized that they are not oxidized within the mammary gland, based on isotope measurements in studies in fed lactating goats and in nonruminants

**Main characteristics of models with different ratios of LCFA oxidized in TCA**

**Model**

**CO2 prediction**

**ATP**

**ATP = 1250 mmol/h/half udder**

**optimum**

**Extreme flux distribution (see Table 9)**

**Variability of AIO coefficients**

**Ratio of**

**Dataset**

**Predicted**

**Ratio of**

**Total number**

**Nonplausible**

**Nonplausible**

**Glucose carbon**

**Glucose carbon**

**long-chain FA**

**CO2**

**predicted CO2**

**flux values**

**AIO**

**required to**

**oxidized**

**oxidated in TCA**

**and measured CO2**

**produce lactose**

To study the impact of the variability of FA oxidation, a ratio of long-chain FA (10%, 20%, 25%) was introduced in the TCA cycle. For datasets (CN), (Ctrl) and (HB) which were compatible with a given ratio of LCFA oxidation, extreme flux distributions and AIO coefficients variability were studied. Our main conclusions are robust to the introduction of LCFA oxidation (Table _{15} is highly constrained by measurements, so that this flux cannot vanish when _{14} is optimized or when _{19} is maximized. All the vertices for the (HB)-simplex contradict knowledge-based literature.

0%

(HB)

1546

Non available

6628

Nonrelevant hypothesis

(CN)

1021

99%

2045

8

6

2

[62.2 ; 72.0]

[17.5 ; 28.9]

(Ctrl)

1126

121%

3081

8

6

2

[52.8 ; 62.3]

[27.8 ; 39.7]

10%

(HB)

1640

Non available

7395

13

13

0

[47.7 ; 62.4]

[28.8 ; 45.9]

(CN)

1107

107%

2739

8

6

2

[59.4 ; 72.0]

[17.5 ; 32.0]

(Ctrl)

1202

129%

3701

8

6

2

[51.6 ; 62.3]

[27.9 ; 41.2]

20%

(HB)

1756

Non available

8336

13

13

0

[46.9 ; 62.4]

[29.0; 47.0]

(CN)

1214

118%

3607

8

6

2

[57.3 ; 72.0]

[17.5 ; 34.6]

(Ctrl)

1298

140%

4476

Nonrelevant hypothesis: predicted CO2 is not compatible with measured CO2

25%

(HB)

1826

Non available

8396

13

13

0

[46.5; 62.4]

[29.0; 47.5]

(CN)

1279

124%

4128

8

6

2

[56.3 ; 72.0]

[17.6 ; 35.8]

(Ctrl)

1355

146%

4938

Nonrelevant hypothesis: predicted CO2 is not compatible with measured CO2

ATP balance is too high

Extreme distributions are not biologically relevant

For plausible ratios of long-chain FA in TCA,(CN) treatment is characterized by a lower proportion of glucose (on a carbon basis) which is oxidized in CO2, and a larger ratio used for lactose synthesis.

**Main properties of the simplex vertices, assuming constant ATP-production,**

**with different ratios of LCFA oxidized in TCA**

**Dataset**

**% of FA**

**Model**

**Example of**

**Combinatorics**

**Validation**

**oxidated**

**name**

**maximized**

**of pathways**

**in TCA**

**function**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**
R
**

**NADPH**

**OAA →G3P**

**OAA →PYR**

**G3P →G6P**

**Peptide**

**Peptide**

**Pyr****
→
**

**oxidation**

**hydrolysis**

**synthesis**

To study the impact of the variability of FA oxidation, a ratio of long-chain FA (10%, 20%, 25%) was introduced in the TCA cycle. For datasets (CN), (Ctrl) and (HB) which were compatible with a given ratio of LCFA oxidation, extreme flux distributions and AIO coefficients variability were studied. Our main conclusions are robust to the introduction of LCFA oxidation (Table _{15} is highly constrained by measurements, so that this flux cannot vanish when _{14} is optimized or when _{19} is maximized. All the vertices for the (HB)-simplex contradict knowledge-based literature.

(Ctrl)

0%

1831

125

1835

Non relevant flux values for _{13}, _{64}

10%

2451

2455

(CN)

0%

B

_{15}- _{19}

0

795

0

0

0

150

803

10%

1489

1497

20%

2357

2365

25%

2878

2886

(HB)

10%

6173

150

6145

20%

7115

7086

25%

7675

7646

(Ctrl)

0%

1831

125

1835

Non relevant flux values for _{13}, _{64}

10%

2451

2455

(CN)

0%

F

_{14}- _{19}

0

0

795

0

0

150

803

10%

1489

1497

20%

2357

2365

25%

2878

2886

(HB)

10%

6173

150

6145

20%

7115

7086

25%

7675

7646

(Ctrl)

0%

3662

125

4

Non relevant flux values for _{8}, _{64}

10%

4902

(CN)

0%

D

_{8}- _{19}

0

0

0

1590

0

150

8

10%

2978

20%

4714

25%

5756

(HB)

10%

12289

20%

D1

_{8}- _{19}- _{14}

29

0

14172

25%

0

15292

0

150

0

10%

12289

20%

D2

_{8}- _{19}- _{15}

0

29

14172

25%

15292

(Ctrl)

0%

305

430

4

Glucose is the unique precursor of lactose synthesis (AIO)

10%

409

533

(CN)

0%

H

_{64}- _{19}

0

0

0

0

133

283

8

10%

248

398

20%

393

543

25%

480

630

(HB)

10%

H1

_{64}- _{19}- _{14}

0

29

0

0

1024

1174

0

Non relevant flux values for _{63}

20%

1181

1331

25%

1274

1424

10%

H2

_{64}- _{19}- _{15}

0

29

1024

1174

20%

1181

1331

25%

1274

1424

(Ctrl)

0%

669

1714

125

1718

Non relevant flux values for _{13}, _{64}

10%

694

2330

2334

(CN)

0%

A

_{15} + _{19}

791

0

0

0

150

799

10%

22

1485

1493

20%

2353

2361

(HB)

10%

1216

5961

150

5932

20%

6902

6873

Extreme flux distributions within the set of plausible solutions

25%

7462

7433

(Ctrl)

0%

694

1714

125

1718

10%

2330

2334

(CN)

0%

E

_{14} + _{19}

22

0

791

0

0

150

799

10%

1485

1493

20%

2353

2361

25%

2874

2882

Non relevant flux values for _{13}, _{64}

(HB)

10%

E1

_{14} + _{19}

1216

32

5929

0

0

150

5932

20%

1216

32

6902

6873

25%

1216

32

7462

6920

10%

6008

5979

20%

E2

_{14} + _{19}- 50_{15}

946

0

6949

6920

25%

7509

7481

(Ctrl)

0%

694

3428

125

4

10%

4659

(CN)

0%

C

_{8} + _{19}

22

0

0

1583

0

150

8

10%

2971

15%

4706

20%

5749

(HB)

10%

C1

_{8} + _{19}

1216

32

0

11858

3

20%

13840

25%

14961

0

150

10%

C2

_{8} + _{19}- 50_{15}

946

0

29

11958

0

Non relevant flux values for _{8}, _{64}

20%

13840

25%

14961

(Ctrl)

0%

694

286

410

4

Glucose is the unique precursor of lactose synthesis (AIO)

10%

388

513

(CN)

0%

G

_{64} + _{19}

22

0

0

0

132

282

8

10%

248

398

20%

392

542

25%

479

629

(HB)

10%

G1

_{64} + _{19}

1216

32

0

0

989

1139

3

Non relevant flux values for _{63}, _{64}

20%

1145

1295

25%

1238

1388

10%

G2

_{64} + _{19}- 50_{15}

946

0

29

996

1146

0

20%

1146

1296

25%

1247

1397

Litterature-based upperbounds for fluxes

≤ 591

Non-zero

Lower than

≤ 266 mmol/h/half

mmol/

whole body

udder

h/half udder

protein synthesis

From this literature study, it appears that LCFA oxidation may depend on both the environmental and experimental contexts, and no single model can be favored yet. To study the impact of LCFA oxidation, we successively introduced a ratio of LCFA in the TCA cycle (10%, 20%, 25%), and assumed that 50% of C16 output are synthesized within the mammary gland

Modeling long-chain fatty acids oxidation (LCFA) in tricarboxylic acid (TCA) cycle.

**Modeling long-chain fatty acids oxidation (LCFA) in tricarboxylic acid (TCA) cycle.** Introducing LCFA (C(16:0), C(18:0)) oxidation in the TCA cycle may be required to consistently model the response to several treatments, such as the (HB) dataset (Table _{141}) and C(18:0) (_{142}), and output of C(18:0) (_{143}). C(16:0) oxidation (_{20}) and C(18:0) oxidation (_{21}) are modified accordingly.

For the (Ctrl) treatment, by comparing the predicted and measured CO2 quantities, we concluded that the hypotheses of 20% and 25% of FA oxidized in the TCA cycle were not in agreement with our experimental data: the increase of CO2 prediction was too large both when compared to measured CO2 (see Table

For all remaining compatible pairs of model and datasets, we first studied the ATP maximization hypothesis. In all cases, our results, shown in Table

Then we enumerated extreme flux distributions and studied their biological relevance. As shown in Table ^{d}. More precisely, for the (HB) treatment, we still obtain a division according to the activation of the four fluxes OAA →PYR (_{14}), OAA →G3P (_{15}), G3P →G6P (_{8}), peptide hydrolysis (_{64}). Nonetheless, an intricate phenomenon appears. Indeed, measurements imply that OAA →G3P (_{15}) is very constrained. It cannot vanish at the same time as OAA →PYR (_{14}) or when NADPH oxidation (_{19}) is at its maximal value. Therefore, for several optimal conditions related to NADPH oxidation (_{19}), OAA →PYR (_{14}), G3P →G6P (_{8}), peptide hydrolysis (_{64}), we need to decide whether OAA →G3P (_{15}) is slightly non-zero, or whether OAA →G3P is assumed to vanish although the optimal is not exactly reached for the initial flux considered. Despite this difference of structure between the (HB) dataset and the (Ctrl) and (CN) datasets, our analysis suggested that extreme distributions were not biologically relevant, independently from the ratio of LCFA oxidation or the dataset under study.

Finally, comparing the variability of the AIO coefficients for the (Ctrl) and (CN) treatments when oxidizing 10% of LCFA in the TCA cycle still suggested that the (CN) treatment is characterized by a lower proportion of glucose (on a carbon basis) which is oxidized in CO2, and a larger ratio used for lactose synthesis.

Altogether, this study suggests that the main characteristics of the (Ctrl) and the (CN) treatments are robust and could be elucidated despite lacking information on the precise internal behavior of LCFA oxidation.

Discussions

Analyzing the distribution of nutrients in a metabolic network to study the flexibility of a metabolism at the organ level

An important challenge in applicational fields of metabolism studies at the organ level is to understand how the components of inputs are transformed into some expected outputs, under some assumptions about the functioning of a system. To that end, great use is made of comparisons between yield rates describing the allocation of input nutrients within the set of outputs. Nonetheless, to allow a precise comparison of nutrients, these studies require insights on the distribution of matter components across the output of

To elucidate how input nutrients are allocated among the output nutrients of a metabolic system despite experimental limitations, we have introduced novel methods which refine the flux balance analysis of a metabolic system related to an organ of a large animal. Our method can be seen as an extension of Flux Variability Analysis

As an example of application, we have studied the mammary metabolism in ruminants (dairy cows)

As a methodological innovation, we introduced a method to estimate the

The latter point is where our main algorithmic innovation lies. Indeed, AIO coefficients are not linear with respect to the flux variables

This approach applied to a stoichiometry model taking ATP variations into account permitted a better understanding of ruminant mammary gland metabolism in comparison to previous studies based on similar models and datasets

Optimization strategies within tissue or organs

Our study allowed us to revisit optimization-based hypotheses on the functioning of the mammary metabolism. We have provided evidence that flux distributions corresponding to an optimal production of energy (ATP) cannot describe appropriately the metabolism of the mammary gland, as would be the case for bacterial metabolisms, which tend to optimize biomass-related functions

As an alternative, we have hypothesized that ATP balance remains nonoptimal and almost constant in response to several treatments, and we have introduced in the model a recent estimation for this quantity

Therefore, no optimization of a linear combination of flux variables could be found to uniquely describe the metabolism of the mammary gland, a multicellular organ, as an extreme flux distribution of our model. To overcome this limitation of FBA-inspired analyses at the mammalian tissue or organ level, we studied the variability of AIO coefficients by introducing the computation of min-max ranges of AIO. This led to a general overview of the effects of treatments without precluding any steady state internal behavior.

Benefits of studying the variability of the allocation of input in output (AIO) and future model refinements

The study of the variability of AIO, that is, intervals of allocation of nutrients, made it possible to distinguish the metabolism of the mammary gland in two different nutritional conditions corresponding to an increase in protein supply (Ctrl and CN). Notably, the intervals of allocation of glucose in CO2 and lactose were different in the two nutritional conditions, reflecting the mammary gland’s metabolic flexibility. Nonetheless, several improvements can be made to the model.

First, the simplex of plausible distributions could be reduced by introducing knowledge currently available on the kinetic bounds for enzymatic activities, introduced in previous numerical models _{6} which transforms G3P to PYR under the regulation of five different enzymes, together with NAD+ and ATP availability. Therefore, proposing relevant maximal values requires the use of advanced methods for model reductions.

A second improvement of the model is dependent on the production of additional observations to clarify the set of possible behaviors of the system. Particularly, the contribution of acetate and

The last improvement of the model consists of including constraints that do not correspond to rule-based metabolic equations. More precisely, in several numerical models, the effects of external fluxes were introduced to predict the response of the mammary gland in a consistent way

Conclusion

We have introduced a method in the framework of flux-balance analysis of a metabolic network. As a main novelty, our approach allows studying the variability of efficiencies (or yield rates) of a metabolic model provided with input-output measurements. More precisely, our approach allows a quantitative estimation of the minimum and maximum proportions of the carbon quantity of each input nutrient which is recovered in each output component of the system. The main innovation is to propose a method which does not require determining the quantitative distribution of nutrients between the branches of the system. To that end, we have performed a parsing of the space of flux distributions which are compatible with both the model stoichiometry and input-output measurements.

This method was applied to study the response of the mammary gland to several treatments. It allowed us to distinguish two different metabolic responses of the system, corresponding to two nutritional situations and accurately reflecting metabolic flexibility. Overall, our method appears to be configured to study the variability of the yield rates of a metabolic system at the multicellular or organ level without making any hypothesis on the internal behavior of the system.

Method

Flux modeling based on Flux Balance Analysis

Metabolic models are described according to the generic framework of Flux Balance Analysis

Let _{
j
} has the form _{
m,j
} and _{
i,j
})_{
i≤p+n+q,j≤r
}, where

As a usual assumption, intermediary metabolites cannot accumulate in the cell: the consumption and production flux rates of intermediary metabolites are balanced. A linear constraint is derived on the rate vector

Assuming that reactions are irreversible and fluxes have physical upper-bounds, the set of solutions to Eq.(1) is a convex polyhedron, called _{
I
} and _{
O
}.

Modeling the quantitative contribution of input metabolites to output nutrients: AIO

In studies at the organ level such as in nutrition, the choice of a plausible flux distribution

We introduce the following formalism. Let ^{
r
}(

Let _{
I
}(_{
O
}[

To determine these ratios, we introduce _{
P
}[_{
O
}[_{
O
}[_{1} and _{2} are defined in Table _{
O
}[_{1} is a square matrix such that all diagonal coefficients are equal to 1 and the others are nonpositive. This family of matrices is named the _{1} is invertible so that the vectors of allocations _{
O
}[_{
P
}[_{1} can be used.

^{
r
}(_{
k
})

=

Ratio of the flux of component _{
j
} recovered in the composition of the product _{
k
}. It is the sum of individual substrate contributions.

We are given a stoichiometric matrix _{
i
} which is recovered in the flux of the output

_{
i
})

=

Total metabolite rate involved in the production of an intermediary metabolite _{
i
}, before its degradation by other reactions

_{
k,i
}

=

Ratio of a product flux (

_{1}[

=

(_{
k,i
})_{
p<k,i≤p+n+q
}

Linear transformation of matter components contained in intermediary or output metabolites

_{2}[

=

(_{
k,i
})_{
p<k≤p+n+q,1≤i≤p
}

Linear transformation of matter component contained in input metabolites

=

Rates of fluxes of component brought by the

=

Constraints on component fluxes deduced from thematter-invariance law, derived from Eq.(2) below.

Overall, the _{
O
}[_{1}], …, _{
O
}[_{
p
}], as follows:

With these formulas at hand, as soon as the flux distribution _{1} can be used. Altogether, the computation of a complete AIO table takes ^{3}+

Computing extrema of the AIO coefficients among the complete polyhedron of plausible flux distributions: solving nonlinear optimization problems

Computing the extrema of the AIO coefficients among the convex polyhedron of plausible flux distributions

These problems are nonlinear programming (NLP) problems

When both the simplex search space was bounded and an analytical expression for

When an analytical expression was not available, we implemented an alternative strategy making use of some local search routines together with the fact that _{0},_{1},…,_{
n
}) with _{
i
}∈[0,2_{
n
}∈[0,_{0},_{0},_{1},…,_{
n
}) is a parameterization of the boundary of the simplex. This parameterization is used to determine a discretization of the simplex boundary. Given an integer _{
i
} is assumed to be of the form 2_{
n
} rewrites as _{1},…,2_{
n
}) of _{1},…,2_{
n
}) and (_{1},…,2_{
n
}). We can finally apply a local search algorithm based on a dichotomy principle to improve the discretization level and reach the solution to the optimization problem on the simplex boundary.

The same algorithm is extended to the full space by introducing a supplementary coordinate standing for the distance between _{0} of the Chebyshev ball. This makes it possible to check whether the optimum identified previously lies on the boundary of the simplex.

Analysis workflow

A web application dedicated to the computation of AIO for metabolic networks is freely available ^{e}.

The complete workflow of analysis requires the online webpage together with the use of several environments. First,

Mammary gland stoichiometry model

As the basis of the ruminant mammary gland metabolism that is studied in this paper, we used a generic model of mammary metabolism

The model was first extended and detailed in order to include the reactions included in other models of mammary metabolism

Protein synthesis was summarized by the number of peptide links synthesized (peptide synthesis) and the ATP required for each peptide synthesis was fixed at 5 in the present model

Finally, four output reactions were included in the model in order to calculate the overall allocations in carbon (_{137}), nitrogen (_{138}), _{139}) and Serine (_{135}).

As a last step, this model was restricted to the case of ruminants. Some reactions, such as fatty acid oxidations (_{20} to _{23}) and pyruvate synthesis through malic enzyme _{66}) and some input and output fluxes present in the generic stoichiometric model _{25}); glucose production from _{10}), acetone synthesis (_{69}), _{70}), acetoacetate synthesis (_{71}), and ureogenesis (_{50}).

Altogether, this mammary gland model contained 140 reactions, involving eleven intermediary metabolites, four cofactors and ATP, CO2, O2 and NH3. The stoichiometry of the system implies that the balance (production minus consumption) of these four last outputs can be uniquely deduced from the choice of a plausible flux distribution. The full model is shown in Figure

Additional information: datasets, inputs, outputs, additional knowledge, component

Two datasets

Fluxes of nutrients taken up on a net basis by the mammary gland were considered as inputs whereas fluxes of nutrients produced in milk, such as lactose, milk protein, milk fatty acid arranged in triglycerides or CO2 produced and released in blood, were considered as outputs. Special attention was paid to amino acid inputs and outputs, milk protein output, fatty acids synthesized within the mammary gland and total triglyceride output according to established calculation rules _{28}) and triglyceride hydrolysis (_{65}) in

Eight additional linear constraints were introduced to the fluxes to ensure that the model was relevant biologically ^{f}. The rate of fatty-acid acylCoA (

Availability of supporting data

The datasets supporting the results of this article are included within the article (Table

Endnotes

^{a} These functions are: _{8}+_{19}, _{8}-_{19}, _{14}+_{19}, _{14}-_{19}, _{15}+_{19}, _{15}-_{19}, _{64}+_{19}, _{64}-_{19}

^{b} For instance, for the (Crtl) treatment, there is a distribution such that _{8}<591 and _{13}<266 are lower than the bounds introduced in _{64}/_{63}=0.62<0.67 and glucose is the precursor for 90.79% of the lactose carbon

^{c} All the computations were performed by using

^{d} These functions are: _{8}+_{19}, _{8}+_{19}-50 _{15}, _{8}-_{19}-_{14}, _{8}-_{19}-_{15}, _{14}+_{19}, _{14}+_{19}-50 _{15}, _{14}-_{19}, _{15}+_{19}, _{15}-_{19}, _{64}+_{19}, _{64}+_{19}-50 _{15}, _{64}-_{19}-_{14}, _{64}-_{19}-_{15}.

^{e}

^{f} Mathematically, these constraints can be derived as: 0.7_{82}-0.3_{58}=0; _{51}-_{52}=0; _{54}-_{90}=0; _{55}-_{91}=0; _{86}-_{92}=0; _{87}-_{93}=0; _{88}-_{94}=0; _{56}-3_{62}=0.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

OA-A, JB and AS formalized the computation AIO as an optimization issue. OA-A and JB conceived the algorithms to compute AIO. OA-A performed experimentations. OA-A, SL and JV-M built the stoichiometric model. SL prepared experimental datasets and performed the biological interpretation of the results. AS and SL studied the structure on the simplex of plausible flux distributions. All authors contributed to the redaction of the paper. All authors read and approved the final manuscript.

Acknowledgements

Part of this program was supported by the French National Agency for Research (ANR-10-BLANC- 0218). The authors wish to thank Damien Eveillard for fruitful discussions.