Department of Biochemistry and Molecular Biology, Faculty of Biology, Universitat de Barcelona, Av Diagonal 643, 08028 Barcelona, Spain

Institute of Biomedicine of Universitat de Barcelona (IBUB) and CSIC-Associated Unit, Spain

A.N.Belozersky Institute of Physico-Chemical Biology, MSU, Moscow 199899, Russia

Hospital Clínic, IDIBAPS, CIBERES; Universitat de Barcelona, Barcelona 08028, Spain

Technical Research Centre of Finland, Espoo, and Institute for Molecular Medicine, Helsinki, Finland

Institute of Cellular Medicine, The Medical School, Newcastle University, Newcastle, UK

Abstract

Background

Stable isotope tracers are used to assess metabolic flux profiles in living cells. The existing methods of measurement average out the isotopic isomer distribution in metabolites throughout the cell, whereas the knowledge of compartmental organization of analyzed pathways is crucial for the evaluation of true fluxes. That is why we accepted a challenge to create a software tool that allows deciphering the compartmentation of metabolites based on the analysis of average isotopic isomer distribution.

Results

The software Isodyn, which simulates the dynamics of isotopic isomer distribution in central metabolic pathways, was supplemented by algorithms facilitating the transition between various analyzed metabolic schemes, and by the tools for model discrimination. It simulated ^{13}C isotope distributions in glucose, lactate, glutamate and glycogen, measured by mass spectrometry after incubation of hepatocytes in the presence of only labeled glucose or glucose and lactate together (with label either in glucose or lactate). The simulations assumed either a single intracellular hexose phosphate pool, or also channeling of hexose phosphates resulting in a different isotopic composition of glycogen. Model discrimination test was applied to check the consistency of both models with experimental data. Metabolic flux profiles, evaluated with the accepted model that assumes channeling, revealed the range of changes in metabolic fluxes in liver cells.

Conclusions

The analysis of compartmentation of metabolic networks based on the measured ^{13}C distribution was included in Isodyn as a routine procedure. The advantage of this implementation is that, being a part of evaluation of metabolic fluxes, it does not require additional experiments to study metabolic compartmentation. The analysis of experimental data revealed that the distribution of measured ^{13}C-labeled glucose metabolites is inconsistent with the idea of perfect mixing of hexose phosphates in cytosol. In contrast, the observed distribution indicates the presence of a separate pool of hexose phosphates that is channeled towards glycogen synthesis.

Background

^{13}C isotope tracing, aimed in the evaluation of metabolic fluxes in living cells has been developing during last decades ^{13}C tracer fluxomics can be combined with the analysis of gene and protein expressions to provide insight into multilevel regulation of cellular processes

However, the rapidly developing experimental ^{13}C tracer metabolomics surpasses the theoretical analysis of measured data. For a long time the detailed analysis of isotopomer distribution was possible only for isotopic steady state ^{13}C tracer data could result in the discovery of unknown metabolic pathways

The topology of metabolic network could be complicated by substrate channeling

Usually, studies designed for the analysis of channeling require invasive experiments, such as permeabilization of cells and determination of diffusion of labeled metabolites from or into the presumable channel ^{13}C isotopic isomer distributions in products of metabolism of labeled substrates; i.e. in the same study, which is designed for the evaluation of metabolic flux profile, thereby, not recruiting additional experiments.

Thus, the objective of the presented work is to create and implement a tool assessing the compartmentation based on ^{13}C distribution. The challenge here is that, although the same compound, located in different subcellular spaces, likely possesses compartment-specific ^{13}C signatures, the measurements average out the compartment-specificity ^{13}C data, if the really existing compartments are taken into account. To estimate the goodness of data fit by various schemes of metabolic compartmentation we implement model discrimination analysis.

Two out of three experiments analyzed were described elsewhere

Results

Accounting for channeling in the reaction scheme of model

The dynamics of all possible isotopic isomers in glucose, lactate and glutamate from the incubation medium and glucose from glycogen in cell pellets, accumulated by two hours of incubation of liver cells with [1,2-^{13}C_{2}]D-glucose

The schemes of kinetic models used as a base for simulation of isotopologue distribution

**The schemes of kinetic models used as a base for simulation of isotopologue distribution**. Metabolites are connected by biochemical reactions represented by arrows. Various colors indicate metabolites and reactions of specific pathways: green, glycolysis/gluconeogenesis; red, pentose phosphate pathways; orange, TCA cycle. The metabolites enclosed in ellipses are considered to be in fast equilibrium. (A), the basic model that includes one pool of hexose phosphates common for glycolysis and gluconeogenesis. (B), the model that includes also the additional pool of hexose phosphates (blue) that represents channeling in gluconeogenesis. Abbreviations are explained in the list of abbreviations in the text.

Fitting the measured isotopologue distribution

The same stochastic algorithm of minimization of normalized deviations between the measured and computed data (χ^{2}) in the global space of parameters (described in Methods) was applied to each model for data fitting. The fractions of isotopologues measured in glucose, lactate, fragments of glutamate and glycogen, and their best fit using the two models are shown in Table ^{2 }is shown for each metabolite separately and Σ_{1}χ^{2 }sums all the individual χ^{2}.

Measured and simulated fractions of isotopologues and total concentrations of metabolites.

**Experiment**

**Simulated**

**mean sd**

**Channeling**

**Mixed**

**Glucose**

**χ ^{2} = 0.442**

**0.406**

m0

0.512 ± 0.0069

0.511

0.511

m1

0.00913 ± 0.002

0.0084

0.00839

m2

0.478 ± 0.00652

0.481

0.481

[mM]

19.7 ± 1.92

20.4

20.2

**Lactate**

**χ ^{2} = 1.43**

**7.32**

m0

0.86 ± 0.0482

0.839

**0.81**

m1

0.0235 ± 0.00802

0.0237

**0.0178**

m2

0.0946 ± 0.0388

**0.133**

**0.17**

m3

0.022 ± 0.0438

0.00381

**0.00145**

[mM]

0.81 ± 0.51

0.959

**1.48**

**Glutamate C2-C5**

**χ ^{2} = 0.0564**

**0.0424**

m0

0.912 ± 0.0343

0.912

0.912

m1

0.0299 ± 0.0116

0.0301

0.0298

m2

0.0523 ± 0.0217

0.0574

0.0567

**Glutamate C2-C4**

**χ ^{2} = 0.0049**

**0.00437**

m0

0.919 ± 0.0339

0.919

0.919

m1

0.0365 ± 0.00175

0.0356

0.355

m2

0.0446 ± 0.0166

0.0454

0.0451

**Glycogen**

**χ ^{2} = 1.2**

**30.5**

m0

0.608 ± 0.0388

0.598

**0.658**

m1

0.0162 ± 0.0033

0.0151

**0.0271**

m2

0.362 ± 0.0351

0.375

**0.299**

m3

0.00399 ± 0.0011

0.00422

**0.00791**

m4

0.00961 ± 0.0026

0.00748

0.00749

m5

0.000464 ± 0.00016

0.000432

0.000533

mg/mL

0.355 ± 0.112

0.313

**0.232**

**Σ _{1} χ^{2}**

**3.13**

**38.28**

**Glycogen C1-C4**

**χ ^{2} = 1.42**

**6.68**

m0

0.613 ± 0.0448

0.627

**0.679**

m1

0.0224 ± 0.00834

**0.0133**

0.0297

m2

0.357 ± 0.0425

0.358

**0.289**

**Glycogen C3-C6**

**χ ^{2} = 7.97**

**30**

m0

0.952 ± 0.00767

0.952

0.951

m1

0.00743 ± 0.00211

**0.0131**

**0.018**

m2

0.0371 ± 0.00467

0.0333

**0.0279**

**Σ _{2}χ^{2}**

**9.21**

**36.68**

**Σ _{t}χ^{2} = Σ_{1}χ^{2}+Σ_{2}χ^{2}**

**12.52**

**74.95**

Isotopologues (m0, non-labeled; m1, containing one 13C isotope; m2, two 13C isotopes, etc) produced by isolated hepatocytes from glucose as the only substrate contained 50% of [1,2-13C2]D-glucose were measured in glucose from medium, glucose from glycogen and its fragments, lactate, and fragments of glutamate after two hours of incubation. The measurements are presented as mean ± standard deviation. The data were simulated using two models that either accounted for channeling or suggested a single "mixed" pool of hexose phosphates in accordance with the schemes presented in Figure 1. The fitting was performed using a stochastic algorithm described in Methods. The difference between the best fit and experimental data (χ2, see Methods) are shown for each metabolite and summarized for the whole set of data.

Model A that does not assume metabolite channeling in glycogen synthesis fitted the experimental isotopologue distribution with Σ_{1}χ^{2 }= 38.28 (Table _{1}χ^{2 }and the number of degrees of freedom defining an extremely low value of incomplete gamma function Q = 9.9·10^{-7 }unambiguously indicates that the model which does not account for channeling should be rejected

Conversely, model B that assumes channeling fits the measured isotopologue distribution much better, with Σ_{1}χ^{2 }= 3.13 (Table

Thus, the comparative study of two schemes based on the goodness of fit of the experimental data allowed rejection of the model that assumes a single common pool of hexose phosphates and acceptance of the alternative model, which accounts for channeling of intermediates in glucose metabolism.

Model validation

Electron impact ionization used in mass spectrometry often splits molecules into fragments. Since the localization of such fragments in the molecule is known, the fact that a ^{13}C atom belongs to a given fragment restricts the possible positions of this isotope in the molecule. This information can further restrict the possible set of solutions. The fractions of isotopologues from glycogen were measured not only in whole glucose molecules, but also in their fragments containing carbon atoms either 1-4 _{t}χ^{2 }= 12.52. This value and 8 degrees of freedom deduced for this model from 29 experimental points and 21 essential parameters define the value Q = 0.129. This value indicates that the model is acceptable, thus confirming the conclusion based on the simulation of ^{13}C distribution in the whole molecule of glucose from glycogen without accounting for the fragments.

As Table _{t}χ^{2 }= 74.9. With number of degrees of freedom of 11 (29 experimental points and 18 essential parameters), the value of Q was 1.42·10^{-11}. This value further indicates that the model of a homogeneous pool of hexose phosphates should be rejected.

Another validation of the channeling came from a series of two experiments where hepatocytes were incubated in the presence of glucose and lactate (as described in Methods). The conditions in the two experiments were virtually identical with the exception that glucose was labeled in one of these experiments ^{2 }and Table

Isotopologue distribution produced by isolated hepatocytes in the presence of glucose and lactate.

**Experiment 1**

**Simulations**

**Experiment 2**

**Simulations**

**label in glucose**

**B**

**A**

**label in lactate**

**B**

**A**

**glucose:**

**χ ^{2} = 7.21**

**5.72**

**χ ^{2} = 5.52**

**0.576**

m0

0.532 ± 0.0098

0.514

0.514

0.979 ± 0.01

0.976

0.984

m1

0.00846 ± 0.0022

0.0114

0.00918

0.0063 ± 0.0055

0.00484

0.00354

m2

0.459 ± 0.0103

0.473

0.475

0.0055 ± 0.0064

0.00891

0.00469

m3

-- --

--

--

0.0064 ± 0.0016

0.00996

0.00628

[mM]

20.6 ± 2.91

21

21.1

20.9 ± 2.22

20.8

20.9

**glycogen:**

**χ ^{2} = 6.46**

**27.3**

**χ ^{2} = 2.22**

**5.73**

m0

0.681 ± 0.032

0.683

0.57

0.909 ± 0.026

0.907

0.9

m1

0.0119 ± 0.031

0.00767

0.0131

0.017 ± 0.0066

0.0166

0.0225

m2

0.302 ± 0.031

0.308

0.409

0.038 ± 0.01

0.0313

0.0298

m3

0.0017 ± 0.001

0.000136

0.00208

0.0273 ± 0.0071

0.0363

0.04

m4

0.0032 ± 0.0016

0.000593

0.00565

0.0036 ± 0.0016

0.00329

0.00317

m5

-- --

--

--

0.003 ± 0.014

0.00335

0.00271

mg/mL

0.263 ± 0.084

0.256

0.196

0.262 ± 0.0691

0.256

0.196

**glgn14:**

**χ ^{2} = 5.31**

**19.2**

**χ ^{2} = 2.23**

**4.59**

m0

0.678 ± 0.032

0.692

0.588

0.93 ± 0.024

0.924

0.921

m1

0.016 ± 0.0046

0.00561

0.0112

0.033 ± 0.009

0.0343

0.0472

m2

0.3 ± 0.032

0.302

0.399

0.019 ± 0.0079

0.0178

0.0128

m3

-- --

--

--

0.014 ± 0.005

0.0209

0.0163

m4

-- --

--

--

0.004 ± 0.003

0.00267

0.0264

**glgn36:**

**χ ^{2} = 1.48**

**14.2**

**χ ^{2} = 1.97**

**4.48**

m0

0.98 ± 0.0101

0.987

0.968

0.924 ± 0.024

0.923

0.911

m1

0.00408 ± 0.0018

0.0051

0.0101

0.0265 ± 0.007

0.0332

0.0348

m2

0.0139 ± 0.0078

0.00766

0.216

0.027 ± 0.0081

0.0188

0.0219

m3

-- --

--

--

0.021 ± 0.0067

0.0221

0.0292

**lactate:**

**χ ^{2} = 2.06**

**2.73**

**χ ^{2} = 1.62**

**9.76**

m0

0.974 ± 0.026

0.991

0.985

0.636 ± 0.017

0.621

0.608

m1

0.0026 ± 0.0019

0.000974

0.00135

0.0166 ± 0.0025

0.0167

0.0172

m2

0.0094 ± 0.0037

0.00773

0.0141

0.0318 ± 0.0035

0.0302

0.0245

m3

0.00136 ± 0.023

0.00000227

0.0000436

0.316 ± 0.0213

0.332

0.35

[mM]

6.18 ± 0.75

6.8

6.73

3.18 ± 0.43

6.29

6.22

**Σχ ^{2}**

**22.52**

**69.15**

**13.56**

**25.136**

Before incubation the medium contained either 50% of [1,2-13C2]D-glucose and unlabeled lactate (experiment 1) or 50% uniformly 13C-labeled lactate and unlabeled glucose (experiment 2). The measurements are presented as mean ± standard deviation. The data were fit by two models (A and B). The conditions of incubation and measurements, and data fitting are described in Methods. The difference between the best fit and experimental data (χ2, see Methods) are shown for each metabolite and summarized for the whole set of data.

Model B fits both the experiments with Σχ^{2 }= 36.1 (22.52 when label is in glucose and 13.56 when label is in lactate). Both the experiments together provided 46 points. With 24 essential parameters, this problem has 22 degrees of freedom that corresponds to Q = 0.09, thus indicating that the model is acceptable. If this set of parameters which gives the best fit for model B is used for model A (where the same flux of glycogen synthesis is directed from the common pool of hexoses), Σχ^{2 }increases up to 331, where glycogen gave the greatest Σχ^{2}. The fitting procedure reduced Σχ^{2 }for model A down to 94.286 (69.15 when label was in glucose and 25.13 when label was in lactate) with the isotopologue distribution shown in Table ^{-11}, which indicates that model A is incorrect.

Metabolic flux distribution

If model discrimination analysis indicates that a model should be rejected, as in the case of model A, the distribution of metabolic fluxes obtained with such a model cannot be reliable. In contrary, if the analysis suggests accepting a model, as in the case of model B, there is much more confidence that it evaluates true metabolic fluxes. Therefore we analyzed the distribution of metabolite fluxes computed with model B. Table

Metabolic fluxes corresponding to the best fit of experimental data and their 99% confidence intervals

**Glucose as the only substrate**

**Glucose with lactate**

**99% confidence interval**

**99% confidence interval**

**bestfit (min - max)**

**bestfit (min - max)**

**Model A**

**hk1**

0.0026894(0.002150-0.003080)

0.152266(0.074815-0.377413)

0.01981

**hk2**

0.0021436(0.001710-0.002480)

0.0056939(0.0041563-0.0076078)

**g6pase1**

5.50E-05(2.70E-5-7.80E-5)

0.150611(0.073125-0.375565)

0.01809

**g6pase2**

2.13E-05(0.0-0.000066)

0.005166(0.0029064-0.0071982)

**pfk1**

0.003048(0.002370-0.003530)

0.0023558(0.0015063-0.0046942)

0.00345

**pfk2**

0.0004816(0.0-0.000950)

0.0007975(0.0006197-0.0012778)

**fbpase1**

0.0004446(0.000270-0.000560)

0.0013469(0.0005781-0.0023709)

0.00363

**fbpase2**

0.0005496(0.000330-0.000690)

0.0023329(0.00161-0.0032427)

**gp**

2.56E-06(1.70E-6-5.10E-6)

0.000143(5.12E-05-0.0001883)

0.00000

**gs**

0.0021929(0.001660-0.002680)

0.0022087(0.0018593-0.0027097)

0.00188

**aldf**

0.0192413(0.007480-0.022580)

0.0159083(0.0153142-0.0163931)

0.01608

**aldr**

0.016706(0.004480-0.020410)

0.0164349(0.0153054-0.016708)

0.01625

**aldex**

0.0424386(0.010420-0.056150)

0.0137425(0.0103922-0.0195718)

0.01565

**g3pep**

0.0064103(0.004490-0.007700)

0.258278(0.1677705-0.3394195)

0.12224

**pepg3**

0.0013391(0.000140-0.002370)

0.258701(0.166264-0.339572)

0.12257

**pk**

0.0050704(0.003910-0.006230)

0.0199774(0.0157079-0.0239355)

0.01497

**lacin**

1.71E-07(1.20E-7-2.60E-7)

0.231716(0.1714965-0.2686475)

0.12266

**lacout**

0.00507(0.003910-0.006230)

0.222326(0.1621185-0.2590095)

0.11074

**pc**

1.20E-07(5.70E-8-1.90E-7)

0.0204457(0.0155337-0.0233495)

0.01542

**pepck**

1.16E-08(5.30E-9-2.80E-8)

0.020401(0.0154933-0.0232863)

0.01530

**maloa**

1.85E-07(9.60E-8-3.00E-7)

0.0835934(0.048937-0.111892)

0.03396

**oamal**

3.80E-08(1.30E-8-8.10E-8)

0.0747156(0.0368199-0.0997713)

0.02261

**cs**

2.55E-07(1.30E-7-4.00E-7)

0.008922(0.0069631-0.0147641)

0.01147

**citmal**

1.47E-07(8.40E-8-2.30E-7)

0.0088778(0.0069227-0.0146991)

0.01135

**pdh**

2.75E-07(7.70E-8-5.00E-7)

0.0089212(0.0069623-0.0147633)

0.01147

**g6pdh**

3.87E-06(2.80E-6-7.50E-6)

0.0018986(0.0013506-0.0021492)

0.00000

**p5p > s7p**

0.0011849(0.000880-0.002270)

0.0006229(0.0004459-0.0007215)

0.00051

**s7p > r5p**

0.0011925(0.000890-0.002280)

4.35E-06(2.56E-06-2.98E-05)

0.00047

**f6p > p5p**

9.59E-06(4.20E-6-3.40E-5)

4.02E-05(1.17E-05-9.84E-05)

0.00000

**p5p > f6p**

4.93E-06(2.80E-6-2.80E-5)

0.0006793(0.0004937-0.000751)

0.00000

**f6p > s7p**

1.85E-05(9.20E-6-7.30E-5)

1.40E-05(4.56E-06-2.45E-05)

0.00000

**s7p > f6p**

9.56E-06(4.80E-6-6.60E-5)

1.65E-06(9.03E-07-8.56E-06)

0.00000

**p5p-g3p**

0.0006152(2.70E-4-2.83E-3)

0.0017868(0.0007706-0.0033249)

0.00036

**f6p-s7p**

7.69E-08(2.10E-8-7.70E-7)

1.53E-05(4.62E-06-2.78E-05)

0.00000

**p5p-s7p**

0.0022967(1.37E-3-7.06E-3)

1.51E-06(8.37E-07-8.16E-06)

0.00067

**f6p > s7p**

0.0015144(0.000850-0.001940)

0.000862(0.0002796-0.0021263)

0.00117

**s7p-f6p**

0.0015015(0.000850-0.001920)

0.0014894(0.0008194-0.0026708)

0.00119

**f6p-g3p**

0.0075338(0.002170-0.013330)

0.0164459(0.0085357-0.0352201)

0.00230

**s7p-e4p**

0.0003018(0.000140-0.000550)

7.81E-05(2.01E-05-0.0002102)

0.00061

The names of fluxes are given in the list of abbreviations in the text.

At first glance, a notable difference is seen between metabolic fluxes under the two different conditions. The fluxes for the best fit indicate that the presence of lactate had perturbed the entire central carbohydrate metabolism of hepatocytes. Without lactate, almost half of the glucose consumed(hk) was used to replenish the glycogen store (glgsn) that was exhausted during starvation, and the rest was mainly converted to lactate except a small part that was burned in TCA cycle. Although net consumption of glucose did not change much by the addition of lactate, the fluxes of glucose input (hk) and output (g6ph) taken separately are increased by almost two orders of magnitude. Thus, recycling of metabolites increased without affecting the net influx of glucose. The addition of lactate increased recycling in many other points downstream of glucose entrance. This refers to the flux through fructose bisphosphatase (fbpase), which forms a futile cycle with phosphofructokinase (pfk). The increase of flux transforming glyceraldehyde 3-phosphate into phosphoenolpyruvate (g3pep, it lumped a set of reactions) is accompanied by the increase of reactions in the reverse direction (pepg3). Essentially there is an increased futile recycling through pyruvate kinase (pk), pyruvate carboxylase (pc) and phosphoenolpyruvate carboxykinase (pepck). This recycling is accompanied by an increase of flux through the TCA cycle (pdh, cs, citmal) linked with enhancing of energy production. Some changes took place in the pentose phosphate pathway, but they were not as impressive as in glycolysis and TCA cycle.

The confidence intervals for some of the fluxes (e.g. hkI) were large. However, many intervals for the two studied conditions do not overlap, and so the changes described above for the best fit remain qualitatively the same for the whole intervals.

For each fit (as it can be seen in best fit fluxes presented in Table

Among the fluxes of pentose phosphate pathway, the most essential are the exchange between triose and pentose phosphates, and fructose and sedoheptulose phosphates. These exchanges also contribute to the difference between isotopomer content of glycogen and hexose phosphates fueling glycolysis.

Although model A was rejected, the fluxes corresponding to the best fit in the presence of lactate are shown in the last column of Table

Discussion

Possible sources of errors and the implemented way of avoiding them

Stable isotope tracing is a promising sensitive technique for the study of metabolism in living cells. However, it is sensitive to various flaws and incompleteness in the data analysis. That is why tools for tracer data analysis must take into account the possible sources of errors. In particular, omitting some isotope exchange reactions may lead to significant errors in the calculated flux distributions

Not accounting for compartmentation due to metabolic channeling is another pitfall that can result in incorrect estimation of metabolic fluxes ^{13}C distribution in metabolites does not require specific experiments. Instead, it requires a specific analysis related with the implementation of various schemes and application of model discrimination analysis to define the compatibility of the schemes with the data.

Experimental design facilitating the analysis

To restrict the possible ways of label propagation, the experimental system was simplified to a maximum by excluding the other sources of carbon except glucose or lactate. This permitted us to find that measured isotopologue distribution even in a small number of metabolites limits the possible solutions sufficiently to reject the hypothesis about unique well mixed pool of hexose phosphates.

The more metabolites analyzed, the more information about the topology of metabolic network they can potentially bring. To extract such information much more various hypothetic topologies must be analyzed. As an introduction to the isotopomer-based analysis of network topology, we presented a simple case of a few metabolites. However, the commencement from small dataset facilitates further extension of this method to larger datasets, including those obtained with NMR or MS

Channeling from the point of view of limited diffusion

Although it is known that from the point of view of ^{13}C distribution channeling can be simulated as an additional compartment _{ATP }channels

Matching schemes to the types of isotopomer distribution

The acceptance of model B does not mean that the set of parameters and respective fluxes can be defined unambiguously. The large confidence intervals for some fluxes (shown in Table

Change of hepatocyte metabolism in the presence of lactate

Despite the large confidence intervals, the change of metabolic state in the presence of lactate is evident. Most of the metabolic fluxes increased so much that confidence intervals for them do not overlap with those found for glucose as the only substrate. Lactate induced the substantial increase of metabolite recycling. This result of modeling, in principle, agrees with the direct observation, that an essential amount of label from lactate ascends up to medium glucose, and an essential amount of label from glucose descends down to lactate. It is in accordance with the function of the liver, which can utilize lactate to synthesize glucose.

However, quantitatively, some of the results are not so evident. Table ^{2 }criterion is very sensitive to the value of such recycling: its two-fold decrease leads to the χ^{2 }increase from 36.1 to 46.6 (data not shown), which indicates that the flux that decreased twice was out of 99.9% confidence interval for this recycling. The high velocity of this futile cycle results in the high rate of ATP consumption. However, it agrees in the order of magnitude with ATP production, taking into account that the flux through TCA cycle produces five folds more NADH (15 folds more ATP).

The net glucose consumed as well as lactate produced are burned through the TCA cycle thus producing energy necessary for the recycling. Thus, in the presence of only glucose, its essential part is used to replenish glycogen, whereas in the presence of both glucose and lactate the cultured liver cells apparently burns these substrates (preferentially lactate).

The presented analysis of the entire set of experiments characterized the capacity of hepatocytes to modify metabolic state under extreme conditions. The characterization of metabolism of hepatocytes is inseparable from the detection of the real compartmentation of considered pathways. Application of this methodology to larger datasets will reveal new information about the network topology. It opens a perspective to examine the compartmentation and metabolic flux profile in various cells under physiological and patho-physiological conditions.

Conclusions

Compartmentation of intracellular metabolism, appeared as a general phenomenon, results that the analysis of metabolic flux distribution should be inseparable from the analysis of compartmental structure of studied pathways. Here we proposed a methodology implemented in our software to reveal compartmental structure and metabolic flux distribution from the distribution of ^{13}C isotopomers measured in the products of cells incubated with ^{13}C labeled substrates. This methodology is based on varying the schemes for simulation of measured data and applying the model discrimination analysis. The application of this methodology to the analysis of ^{13}C isotopomer distributions measured in metabolites of isolated liver cells revealed a separate compartment of hexose phosphates related with substrate channeling in glycogen metabolism. This analysis provided the distribution of metabolic fluxes in central carbohydrate metabolism of the cells incubated with ^{13}C labeled glucose, and revealed the changes of fluxes that were induced by addition of lactate in the incubation media.

Methods

To analyze cellular metabolic flux profiles for specific conditions

Models and data fitting

The systems of differential equations corresponding to the schemes presented in Figure

**Differential equations of the used kinetic models**. Kinetic models were used to simulate the total fluxes and concentrations of metabolites. Based on these calculated total values Isodyn further simulates the distribution of isotopic isomers.

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The concentrations of isotopologues needed to be compared with experimental data were calculated as a sum of the respective isotopomer concentrations computed by Isodyn. Fitting of the experimental data was performed by minimizing χ^{2}, the square of deviations between measured isotopologue fractions (y_{i}) and values (y(x_{i}, a)) computed for the set of parameters, a, as fractions of isotopologues x_{i}, normalized by experimental standard deviations (σ_{i}):

The minimization was performed in the global space of parameters using our implementation of simulated annealing algorithm supplemented by coordinate descent in local area ^{2}. Multiple application of the optimization resulted in multiple sets of fluxes characterized by different values of ^{2}. Application of ^{2} threshold to the obtained sets of fluxes

Modification of reaction scheme

A change of equations of the basic kinetic model could be usually performed easily, even graphically: there are algorithms that transform a drawn scheme into a system of ordinary differential equations (ODE)

Aldolase reaction

An essential restriction, which helps to distinguish between models, is including the interdependency of fluxes catalyzed by the same enzyme, as we have shown for transketolase and transaldolase

The scheme in Figure _{3}, because another part of v_{3} produces dhap originated not from fbp, but from the same pool of trioses, bound to the enzyme through the reaction v_{-3}. The steady state fraction of v_{3} that produces dhap originated from fbp (that equals to the fraction of bound dhap originated from fbp (PfE-dhap)) can be expressed as the ratio of input of molecules originated from fbp to the total input in E-dhap:

Isotope-exchange fluxes in the aldolase-catalysed reaction

**Isotope-exchange fluxes in the aldolase-catalysed reaction**. **(A) **shows the whole reaction cycle when the enzyme (E) forms a complex with fbp (E-fbp), releases g3p keeping dhap (E-dhap), and finally releases dhap returning to (E). Forward flux (green lines) through the whole reaction cycle brings isotopes originated from the fbp pool into the pools of dhap and g3p, and reverse flux (red lines) brings isotopes originated from the pools of dhap and g3p into fbp pool. **(B) **shows forward (green lines) and reverse (red lines) fluxes that only exchange isotopes of upper part of fbp molecule with g3p pool without releasing dhap. v_{i }designate the respective rates of elementary steps of reaction mechanism.

Here Pf_{E-fbp} is the fraction of v_{2}, which brings the carbons originated from fbp to E-dhap (or the fraction of bound fbp originated from fbp). Another fraction of v_{2}, (1-P^{f}_{E-fbp}) brings carbons originated from triose pool, which were bound through reactions v_{-3 }and v_{-2. }Pf_{E-fbp}, in turn, can be expressed as the ratio of input of molecules originated from fbp to total input in E-fbp:

The solution of equations (1) and (2) is

where the rates vi could be expressed through the rate constants and substrate concentrations. The forward flux through the whole cycle (indicated by green lines in Figure

The reverse flux of fbp formation (red lines in Figure ^{t}_{E-fbp }) could be described similar to (4):

Figure ^{fg}_{E-dhap}) in (E-fbp) is:

and the forward (thin black in Figure

The flux in the opposite direction (indicated by thick gray lines) is described likewise.

Thus, the model accounts for three isotope-exchange fluxes related with aldolase activity: forward and reverse flux through the whole cycle of enzyme reaction, and pure isotope exchange flux between f6p and g3p, without the change of total concentrations of these metabolites. They are not used in classical kinetic simulations, where only the net flux is important, but they are necessary for the subsequent simulation of isotopologue distribution. The isotope-exchange fluxes of transketolase and transaldolase were implemented similarly as described elsewhere

χ^{2 }criterion for the acceptance or rejection of model

To analyze the structure of metabolic networks, model discrimination analysis was used to test various kinetic models of the same pathways and reject the ones inconsistent with experimental data. Isodyn implements criteria for acceptance or rejection of a model based on the values of normalized square of difference between experimental data and simulation (χ^{2}) and numbers of degrees of freedom

The fitting algorithm implemented in Isodyn identifies the global minimum for the function χ^{2 }and the respective set of parameters and fluxes. If the model is acceptable, the estimated fluxes are also acceptable as a model prediction. The value of χ^{2 }is used in Isodyn as a criterion for acceptance or rejection of model as it is described in ^{2 }by chance could exceed a determined value, is given as an incomplete gamma function (Q(a,x), where a = F/2 (F is the number of degrees of freedom), and x = χ^{2}/2):

Here Γ(a) is gamma function:

The model is acceptable if Q value is larger than 0.05. It can be acceptable even with Q value larger than 0.001, if the errors are not normal or have been moderately underestimated. But if the Q value is lower than 0.001, the model must be rejected as inconsistent with experimental data.

Estimation of number of degrees of freedom

Formally, the number of degrees of freedom (F) is calculated as the difference between number of data points (N, which in our case is the number of fractions of isotopologues for all measured metabolites and total metabolite concentrations) and parameters (P) in the model:

However, in the case if the model is underdetermined, it could happen that the fit of the given data is insensitive to some parameters or there are ambiguous combinations equally affecting the fit, so that the parameters could not be distinguished. The presence of such parameters does not improve the fit and thus do not decrease the number of degrees of freedom. Both situations result in the fact that the Hessian matrix (the matrix of second derivatives of objective function ^{2} with respect to

This matrix is calculated as follows. First derivative of χ^{2} with respect to the parameters is:

where k = 1,2,...M (number of parameters)

Differentiation of these functions gives

Where:

The second term under the sum, which contains the second derivative of fitting function y(xi,a) is usually ignored

As it is indicated above, the singularity of this matrix means that some parameters of the model could not be defined in principle, given the specific dataset analyzed. To find out what characteristics of the studied system (parameters of the model) could be revealed, given the model and a specific dataset, the standard procedure of singular value decomposition of Hessian matrix was used

Following this standard routine the Hessian matrix A is decomposed to the product of orthogonal matrix U, the vector (or diagonal matrix) of singular values W, and the orthogonal matrix V. The ratio of maximal and minimal values in the vector W, called condition number, characterizes the singularity of A. If some values of W are zeros, matrix A is strictly singular, but it could be numerically close to singularity, or ill-conditioned, if the condition number is close to machine precision. The covariance matrix C = A-1 was found from singular value decomposition as C = (V·W-1·UT), where W-1 = [diag(1/wi)]. Diagonal elements of C are the variances of parameters and the other elements are covariances. If matrix A is ill-conditioned, its inverse C cannot be defined and the failure of finding the inverse indicates that the number of parameters is excessive.

Isodyn finds the maximal set of parameters of the model, which, being considered as a subject for fitting, give a non-singular Hessian matrix. The size of this set could be considered as the number of parameters, which affects the number of degrees of freedom in the model with regard to given experimental data. If in fact the model has more parameters, the other parameters are not distinguishable by the given experimental data and must be considered as constants. The number of degrees of freedom (F) is defined as a difference between the numbers of experimental data and effective model parameters, and the value of incomplete gamma function Q(F/2, χ2/2), defined as described in

Experimental methods

Materials

[1,2-^{13}C_{2}]D-glucose (> 99% enriched) and [U-^{13}C_{3}]L-lactate (> 99% enriched) were purchased from Isotec (Miamisburg, OH), and other reagents from Sigma-Aldrich Company (St. Louis, MO).

Animals

180-200 g male Wistar rats were used. They were maintained in a 12h:12h light-dark cycle with free access to standard laboratory rat chow pellets (Panlab) and water. Animals were deprived of food 24 hours prior to hepatocyte isolation. Experiments were conducted according to guidelines accepted by the University Animal Care and Use Committee. Appropiate measures were taken to minimize pain or discomfort of animals.

Preparation of cells and incubation

Suspensions of isolated parenchymal liver cells were prepared from 24-h starved animals as previously described ^{6 }cells/ml, were incubated at 37°C with gassing and continuous shaking (160 strokes/min, which is the minimum shaking that assures total suspension of cells) for 2 h, as it is the optimum time to ensure maximum glycogen synthesis without diminishing cell viability. Conditions for cell incubation were: a) 20 mM glucose, with glucose enriched 50% in [1,2-^{13}C_{2}]-glucose, b) 20 mM glucose + 9 mM lactate + 1 mM pyruvate (10 mM lactate/pyruvate (9:1)), with glucose enriched 50% in [1,2-^{13}C_{2}]-glucose and c) 20 mM glucose + 10 mM lactate/pyruvate (9:1), with lactate enriched 50% in [U-^{13}C_{3}]-lactate.

Measurement of metabolites

At the beginning and end of incubations, cells were centrifuged (3000 g, 20 s), and incubation medium and cell pellets were obtained. For glycogen determination, cell pellets were immediately homogenized with 30% (w/v) KOH using a modification of Chan et al. methodology

Gas Chromatography/Mass Spectrometry sample processing and analysis

At the end of incubations, cells were centrifuged so that the incubation medium and cell pellet are separated and everything was frozen in liquid nitrogen and stored at -80°C until processing for GC/MS analysis. Incubation media were processed for isolation of lactate, glucose, and glutamate using previously established methods

Results of the mass isotopologues in glucose, lactate and glutamate are reported as molar fractions of m0, m1, m2, etc, where m0, m1, m2... indicate the number of ^{13}C atoms in the molecule ^{13}C isotope enrichment are presented in Additional File

**Measured distributions of ^{13}C isotopomers in metabolites**. The table presents the data for each independent experiment obtained as described in Methods after subtraction natural

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List of abbreviations

Metabolites: glc: glucose; glu: glutamate; lac: lactate; glgn: glycogen; g6p: glucose 6 phosphate; f6p: fructose 6-phosphate; fbp: fructose 1,6-bisphosphate; dhap: dihydroxyacetone phosphate; g3p: glyceraldehyde 3-phosphate; pep: phosphoenolpyruvate; pyr: pyruvate; accoa: acetyl coenzyme A; e4p: erythrose 4-phosphate; s7p: sedoheptulose 7-phosphate; r5p: ribose 5-phosphate; xu5p: xylulose 5-phosphate; mal: malate; oaa: oxaloacetic acid; cit: citrate. Enzymes: g6pase: glucose 6-phosphatase; gs: glycogen synthase; gp: glycogen phosphorylase; hk1 & hk2: hexokinase; fbp1 and fbp2: fructose 1,6-bisphosphatase; pfk1 and pfk2: phosphofructokinase; ald: aldolase; tk: transketolase; ta: transaldolase; g6pdh: glucose 6-phosphate dehydrogenase; pck: phosphoenolpyruvate carboxykinase; pk: pyruvate kinase; pdh: pyruvate dehydrogenase complex; cs: citrate synthase; pc: pyruvate carboxylase.

Authors' contributions

IMM performed the analysis and wrote the paper, VAS developed the algorithms and wrote the paper, SM made the experiments and analyzed the data, JR analyzed the data, MO analyzed the data, LA analyzed the data and wrote the paper, MC analyzed the data and wrote the paper. All authors read and approved the final manuscript.

Acknowledgements and Funding

This work was supported by the European Commission Seventh Framework Programme FP7 (Diaprepp Health-F2-2008-202013, Etherpaths KBBE-grant n°222639, Synergy-COPD project grant agreement n° 270086); the Spanish Government and the European Union FEDER funds (SAF2011-25726) and Instituto de Salud Carlos III, Ministerio de Ciencia e Innovación of Spanish Government & European Regional Development Fund (ERDF) "Una manera de hacer Europa" ISCIII-RTICC (RD06/0020/0046); Generalitat de Catalunya-AGAUR, (2009SGR1308 and 2009 CTP 00026); Foundation Marato TV3-042010.

MC acknowledges the support received through the prize "ICREA Academia" for excellence in research, funded by ICREA foundation-Generalitat de Catalunya.

Authors thank Dr J.J. Guinovart and A. Adrover (Universitat de Barcelona and Institute for Research of Barcelona, Spain) for their financial and technical support in hepatocyte incubation, Dr W.N. P. Lee (Harbor-UCLA Medical Center, CA, USA) for his help in GC/MS analysis, and Anusha Jayaraman for the important assistance in the preparation of manuscript.