Updates to a ¹³C metabolic flux analysis model for evaluating energy metabolism in cultured cerebellar granule neurons from neonatal rats

Abstract

A hexose phosphate recycling model previously developed to infer fluxes through the major glucose consuming pathways in cultured cerebellar granule neurons (CGNs) from neonatal rats metabolizing [1,2-¹³C₂]glucose was revised by considering reverse flux through the non-oxidative pentose phosphate pathway (PPP) and symmetrical succinate oxidation within the tricarboxylic acid (TCA) cycle. The model adjusts three flux ratios to effect ¹³C distribution in the hexose, pentose, and triose phosphate pools, and in TCA cycle malate to minimize the error between predicted and measured ¹³C labeling in exported lactate (i.e., unlabeled, single-, double-, and triple-labeled; M, M1, M2, and M3, respectively). Inclusion of reverse non-oxidative PPP flux substantially increased the number of calculations but ultimately had relatively minor effects on the labeling of glycolytic metabolites. From the error-minimized solution in which the predicted M-M3 lactate differed by 0.49% from that measured by liquid chromatography-triple quadrupole mass spectrometry, the neurons exhibited negligible forward non-oxidative PPP flux. Thus, no glucose was used by the pentose cycle despite explicit consideration of hexose phosphate recycling. Mitochondria consumed only 16% of glucose while 45% was exported as lactate by aerobic glycolysis. The remaining 39% of glucose was shunted to pentose phosphates presumably for de novo nucleotide synthesis, but the proportion metabolized through the oxidative PPP vs. the reverse non-oxidative PPP could not be determined. The lactate exported as M1 (2.5%) and M3 (1.2%) was attributed to malic enzyme, which was responsible for 7.8% of pyruvate production (vs. 92.2% by glycolysis). The updated model is more broadly applicable to different cell types by considering bi-directional flux through the non-oxidative PPP. Its application to cultured neurons utilizing glucose as the sole exogenous substrate has demonstrated substantial oxygen-independent glucose utilization by aerobic glycolysis as well as the oxidative PPP and/or reverse non-oxidative PPP, but negligible glucose consumption by the pentose cycle.

1. Introduction

The brain’s utilization of glucose exceeds that predicted from the rate of oxygen consumption (Boyle et al., 1994; Dienel and Cruz, 2016; Madsen et al., 1995; Vaishnavi et al., 2010). Such mitochondrial-independent glucose consumption varies regionally, developmentally, and according to neuronal activity (Goyal et al., 2014; Madsen et al., 1995; Settergren et al., 1976; Vaishnavi et al., 2010), and reflects ‘aerobic glycolysis’ and pentose phosphate pathway (PPP) activities. Aerobic glycolysis involves glucose conversion to lactate despite adequate oxygen availability for its oxidation by mitochondria. Glucose consumption by the oxidative PPP or by the combined oxidative/non-oxidative pentose cycle involves its respective conversion to CO₂ and pentose phosphates, or CO2 and glyceraldehyde-3-phosphate. Glial cells may account for a considerable proportion of oxygen-independent glucose consumption in the adult brain given their high aerobic glycolytic profile and PPP activity (Amaral et al., 2011; Amaral et al., 2016; Belanger et al., 2011); however, the contribution by neurons can be significant and may profoundly influence brain development and plasticity, as well as susceptibility to neurodegeneration. Regions of higher non-mitochondrial glucose consumption correlate with greater expression of markers for synapse formation, plasticity, and remodeling (Goyal et al., 2014), although it is unknown to what extent provision of pentose phosphates, NADPH, or lactate support these communicative processes. Additionally, neuronal vulnerability to stress can be affected by oxidative PPP capacity to supply NADPH for removal of reactive oxygen species and protection of protein thiols (Herrero-Mendez et al., 2009). Assessment of glucose used by the oxidative PPP (for supplying NADPH and pentose phosphates, the latter of which is used for nucleotide synthesis) and by the pentose cycle (for supplying NADPH and ‘excess’ pentose phosphates not used for nucleotide synthesis) is challenging because of carbon recycling that occurs from the pentose cycle to the oxidative PPP. For glucose passing through the pentose cycle, two-thirds returns to glycolysis as fructose-6-phosphate/glucose-6-phosphate and is subject to recycling, while the remaining one-third is consumed by conversion to CO₂ and glyceraldehyde-3-phosphate. The goal of this study was to develop an approach to determine the proportion of glucose used by cultured neurons for the oxidative PPP, the pentose cycle, anaerobic glycolysis, and mitochondrial respiration.

Pathway fluxes can be inferred from the rates of exogenous substrate consumption, the rates of metabolic waste production, and the distribution of isotope within one or more metabolites from catabolism of an isotopically labeled substrate. Mathematical models designed to reproduce the isotope enrichment pattern in one or more metabolites can be developed to infer unknown fluxes from the measured rates and a set of constraints that effectively define the system. These constraints include (a) the number, connectedness, and compartmentation of the reactions (i.e., the system structure), (b) the carbon atom rearrangements for each substrate/product pair, (c) the stoichiometries of the reactions, and (d) any assumptions about pathway behavior (e.g., reversibility). We recently developed a model for quantifying fluxes through the major glucose consuming reaction blocks in primary cerebellar granule neurons metabolizing [1,2-¹³C₂]glucose (Gebril et al., 2016). The model quantitatively tracks ¹³C distribution in glucose-6-phosphate/fructose-6-phosphate (hexose phosphates-HxP) from the recycling of glucose carbons via the non-oxidative PPP back to the oxidative PPP, and ¹³C distribution in malate by carbon rearrangements within the tricarboxylic acid (TCA) cycle. The latter distribution was considered since ¹³C in pyruvate entering the TCA cycle can be recycled back to pyruvate via cytoplasmic and mitochondrial malic enzyme. The quantitative analysis is based on measurements of glucose consumption, lactate production, and mitochondrial respiration, and theoretical predictions on the extent of HxP recycling within the PPP (Gebril et al., 2016; WOOD and KATZ, 1958). The measured ¹³C distribution in exported lactate (i.e., the fraction having zero, one, two, or three ¹³C atoms- denoted as M, M1, M2, and M3, respectively) is reproduced from calculations (predictions) of ¹³C distributions in HxP, malate, triose phosphates (TrP; glyceraldehyde-3-phosphate and dihydroxyacetone phosphate), and pyruvate, which in turn are determined from optimizing the relative contributions of the non-oxidative PPP, glycolysis, the TCA cycle, and malic enzyme to producing these metabolites. One notable assumption made about pathway behavior was unidirectional flux through the non-oxidative PPP (Gebril et al., 2016), despite the known reversible nature of this pathway; this served to minimize the number of calculations required for assessing ¹³C distribution in HxP and TrP. This article describes revisions to the original model that improve its accuracy and applicability to different cell types by accounting for ¹³C randomization in HxP, pentose phosphates, and TrP resulting from reverse non-oxidative PPP flux, and in malate by symmetrical metabolism of succinate in the TCA cycle. The updates prompted re-analysis of ¹³C distribution in lactate from cerebellar granule neurons metabolizing [1,2-¹³C₂]glucose previously reported in (Gebril et al., 2016). The revised model and measurements indicate negligible non-oxidative PPP flux, contrary to our original conclusion of extensive pentose cycle activity.

2. Materials and Methods

Preparation of rat cerebellar granule neuron cultures, flux measurements (rates of glucose consumption, lactate production, and mitochondrial respiration), lactate labeling from [1,2-¹³C₂]glucose, and lactate derivatization with 3-nitrophenylhydrazine were detailed in (Gebril et al., 2016). Unlabeled and [3-¹³C]lactate (Cambridge Isotope Laboratories, Andover, MA, USA) standards, separately and mixed at 7.6:1 M:M1 ratio, were processed with the unknowns. The derivatized samples were diluted 100-fold in 80% methanol and analyzed on a Waters ACQUITY I-Class UPLC™ system coupled with a Waters Xevo TQ-S triple quadrupole mass spectrometer (Waters Corp, Milford, MA, USA). The separation (1 μL) was carried out on a Waters Acquity UPLC™ BEH C18 column (50mm × 2.1mm i.d., 1.7 μm). Sample and column temperatures were maintained at 10 °C and 40 °C, respectively. The mobile phase consisted of water containing 0.1% formic acid (v/v) (A) and acetonitrile with 0.1% formic acid (B). The analysis was performed using the following gradient elution at a flow rate of 0.50 mL/min: 0–2.5 min, 5% B to 18% B; 2.5–3 min, 18% B to 100% B. Each run was followed by a 2.5 min wash with 100% B and an equilibration period of 2.5 min with the initial conditions. The strong and weak solutions used to wash the auto sampler were methanol/acetonitrile/isopropanol/water (25:25:25:25, v/v/v/v/v) and methanol/water (70:30, v/v), respectively. The effluent was introduced into the TQ-S mass spectrometer equipped with electrospray ionization in negative ion mode (ESI-) for quantification of the analytes. Detection was obtained by Multiple Reaction Monitoring (MRM) mode including two MRMs for confirmation of the analytes. The quantification of analytes 224, 225, 226, and 227 (the expected masses for M, M1, M2, and M3 lactate derivatized with 3-nitrophenylhydrazine, respectively; all had 1.84 min retention times) was acquired with transitions of deprotonated ion at m/z 224.04 → 152.06 for M, 225.04 → 152.06 for M1, 226.04 → 152.06 for M2, and 227.04 → 152.06 for M3. The dwell Rme was 20 msec at cone voltage=44 V and collision energy=14 eV for each transition. The ESI–MS/MS parameters were set as follows: capillary voltage, 1.20 kV; cone voltage, 44 V; source temperature, 150 °C; desolvation temperature, 600 °C; desolvation gas flow, 600 L/h, cone gas flow, 200 L/h. Nitrogen was used as desolvation and cone gas. Argon (99.99% purity) was introduced as the collision gas into the collision cell at a flow rate of 0.15 mL/min. Data acquisition was carried out with MassLynx 4.1 software (Waters Corp., Milford, MA, USA). Monte Carlo simulations were performed using the means and standard errors of M-M3 lactate (Table 3) as previously described (Gebril et al., 2016) except that three of four labels were simulated and the fourth calculated so that M+M1+M2+M3=1.

Table 3.

Re-analysis of ¹³C distribution in lactate from cerebellar granule neurons by LC-TQMS.

	Area				Fractional 13C distribution
[12C Lac] (μM)	M	M1	M2	M3	M	M1	M2	M3
100	10732	221	39	0	0.9764	0.0201	0.0035	0.0000
200	21810	658	71	0	0.9677	0.0292	0.0031	0.0000
400	44146	1300	200	0	0.9671	0.0285	0.0044	0.0000
600	62110	2099	264	4	0.9633	0.0326	0.0041	0.0001
800	83522	2897	352	2	0.9625	0.0334	0.0041	0.0000
[13C1 Lac] (μM)	M	M1	M2	M3	M	M1	M2	M3
100	218	11485	292	13	0.0182	0.9564	0.0243	0.0011
200	394	23596	484	65	0.0160	0.9616	0.0197	0.0027
400	0	45987	1154	144	0.0000	0.9726	0.0244	0.0030
600	671	74063	1596	271	0.0088	0.9669	0.0208	0.0035
800	875	96330	2310	310	0.0088	0.9650	0.0231	0.0031
[12C:13C1 Lac] @ 7.6:1	M	M1	M2	M3	M	M1	M2	M3
100	10032	1339	82	0	0.8759	0.1169	0.0072	0.0000
200	19940	2989	131	5	0.8645	0.1296	0.0057	0.0002
400	40238	6144	246	13	0.8627	0.1317	0.0053	0.0003
600	57760	8990	381	29	0.8600	0.1339	0.0057	0.0004
800	75337	11686	472	13	0.8609	0.1335	0.0054	0.0001
	[lactate] (mM)				Fractional 13C distribution
samples	M	M1	M2	M3	M	M1	M2	M3
13C glc + cells	202±37	10±2	207±39	5.0±0.8	0.4751±0.0021	0.0249±0.0007	0.4881±0.0021	0.0119±0.0004
12C glc + cells	417±66	14±2	3.7±0.4	0.9±0.0	0.9556±0.0016	0.0335±0.0010	0.0088±0.0005	0.0022±0.0003

Lactate produced by cerebellar granule neurons (7×10⁶ cells cultured in 2-well Lab-Tek chambers; n=5) after 5 hr incubation with 9.6mM ¹²C or [1,2-¹³C₂] glucose were extracted with methanol and derivatized with 3-nitrophenylhydrazine. Ions 224, 225, 226, and 227 Da corresponding to derivatized lactate M, M1, M2, and M3, respectively, were separated and quantitated by liquid chromatography triple quadrupole mass spectrometry. The areas have been background subtracted using cell-free media without exogenous lactate. Lactate M and M2 concentrations were calculated from 100–800 μM M1 standards (corrected for M1 purity of 0.9625). Lactate M1 and M3 concentrations were calculated from 12–98 μM M1 standards derived from the M:M1 mixture.

3. Results and Discussion

3.1. Inferring fluxes from ¹³C distribution in lactate

3.1.1 Metabolic structure and measured fluxes

A major goal was to develop a model that accounts for ‘recycling’ processes which affect ¹³C distribution in exported lactate so that fluxes through the major glucose consuming pathways can be determined from a relatively small number of measurements. Oxidative and non-oxidative PPP activities were a particular focus because of extensive carbon reshuffling from the pentose cycle and associated HxP recycling, and reverse flux through the non-oxidative PPP. Overlooking such processes would introduce errors to the fluxes inferred from isotope enrichment of glycolytic metabolites such as lactate when these fluxes are significant. Furthermore, carbon rearrangement within the TCA cycle is extensive and can affect ¹³C deposition in lactate through reactions such as malic enzyme that recycle carbon to pyruvate. Malic enzyme has both cytoplasmic and mitochondrial isoforms, the latter of which dominates in neurons (McKenna et al., 2000; Zwingmann and Leibfritz, 2007). Pyruvate recycling may thus occur primarily in the matrix, after which transport by the pyruvate carrier would allow equilibration with the cytoplasmic pyruvate pool. Regardless of the pathway taken by malate (malate export then oxidation in the cytoplasm, or malate oxidation in the matrix followed by pyruvate export), the effect on label distribution in the cytoplasmic pyruvate pool is equivalent. The model assumed that pyruvate derived from malic enzyme mixed uniformly with that derived from glycolysis; additionally, the pyruvate derived from glucose metabolized by the pentose cycle and then glycolysis was assumed to enter the same pool, so that the ¹³C pattern in extruded lactate reflected the contribution of each pathway to labeling this pool. This assumption may not be valid, as others have concluded that pyruvate derived from the pentose cycle, glycolysis, and/or the TCA cycle may form distinct pools that are preferentially imported by mitochondria or exported as lactate, respectively (Brekke et al., 2012; McKenna et al., 2012). To evaluate the contribution of the pentose cycle, glycolysis, and malic enzyme to ¹³C enrichment in exported lactate, probability calculations were developed in Excel to quantitatively track ¹³C reshuffling in the PPP and the tricarboxylic acid cycle of cultured neurons metabolizing [1,2-¹³C₂]glucose as the only exogenous substrate (Gebril et al., 2016). The objective was to find the relative activities (i.e., their respective contributions to producing a metabolite) of the PPP and malic enzyme that yielded a predicted ¹³C distribution in lactate which reflected that measured by LC-MS/MS. Based on previous work with hepatocytes (Ainscow and Brand, 1999), twelve interconnected reaction ‘blocks’ were considered as core components of the major glucose consuming pathways (Fig. 1). Their structure is such that all fluxes could be inferred from the LC-MS/MS data provided that (a) the rates of glucose consumption, lactate production, and mitochondrial respiration were determined, (b) the system maintained an isotopic and metabolic steady-state, and (c) the reaction stoichiometries were known. This article describes revisions to the original probability calculations to account for ¹³C reshuffling by two processes overlooked in the original model: reverse flux through the non-oxidative PPP (Fig. 1; J_3f and J_3r denote the forward and reverse fluxes, respectively) and symmetrical succinate oxidation in the TCA cycle.

Figure 1.

Organization of glucose metabolism into reaction blocks for ¹³C metabolic flux analysis. Measured fluxes (red) include glucose uptake and phosphorylation (J₀), lactate production and export (J₄), and mitochondrial respiration rate (J_RR). Metabolism of [1,2-¹³C₂]glucose yields lactate (Lac) whose enrichment with one (M1), two (M2), or three (M3) ¹³C atoms depends on fluxes through the oxidative pentose phosphate pathway (PPP; J₁), the forward non-oxidative PPP (J_3f,J_3af,J_3bf) that consumes ribulose-5-phosphate (Ri5P) and produces hexose phosphates (HxP; G6P, glucose-6-phosphate; F6P, fructose-6-phosphate) and triose phosphates (TrP; dihydroxyacetone phosphate and glyceraldehyde-3-phosphate), the reverse non-oxidative PPP (J_3r,J_3ar,J_3br), glycolysis (J_2a, J_2b, which produce and consume TrP, respectively), mitochondrial pyruvate (Pyr) oxidation (J₅), and oxidation of mitochondrial malate (Mal; J₇) by malic enzyme. All fluxes can be inferred from ¹³C enrichment in lactate by predicting the relative yield of ¹³C-enriched products from each reaction. Accurately predicting M-M3 lactate is achieved by optimizing J_3af/(J₀ + J_3af), J₇/(J_2b + J₇). J_3af/J_3ar, and J₅. Consumption of NADPH (J₈), Ri5P for nucleotides (J₉), and glycolytic NADH by mitochondria (J₆) are ‘balance’ fluxes calculated from the inferred and measured fluxes that do not directly influence ¹³C enrichment in lactate.

3.1.2. Qualitative tracking of carbon rearrangements in the non-oxidative PPP

When NADPH demand by reactions involved in redox homeostasis and lipid biosynthesis exceeds the demand for ribulose-5-phosphate (Ri5P) by reactions involved in de novo nucleotide synthesis, then the oxidative PPP will overproduce Ri5P. For this condition, regulation of Ri5P is achieved by conversion of the excess to the glycolytic intermediates fructose-6-phosphate (F6P) and glyceraldehyde-3-phosphate (G3P) through ‘forward’ non-oxidative PPP flux. Two-thirds of the original G6P carbons deposited as excess pentose phosphates return as the hexose equivalent F6P (J_3af; and one-third enters glycolysis as TrP, J_3bf), and can either be used for glycolysis or recycled into the oxidative PPP upon conversion to G6P by hexose phosphate isomerase. Glucose utilization by the combined pentose cycle involves its conversion to CO₂ in J₁ and to TrP in J_3f (equivalent to J_3f⁄6 + J_3bf ⁄2 vs. non-cyclic utilization by the oxidative PPP and reactions synthesizing nucleotides, equivalent to (J₁ – J_3f)⁄6 + 5 ∗ J₉⁄6). For accurate flux analysis with [1,2-¹³C₂]glucose, it is important to consider the reshuffling of G6P carbons not only as they return through the non-oxidative PPP, but also as they recycle one or more times through the oxidative PPP.

The first step in tracking ¹³C reshuffling by the ‘forward’ PPP is to delineate the carbon transfers within the individual reactions. This is straightforward in the oxidative PPP, where the carbon order in G6P is retained until 6-phosphogluconate dehydrogenase, which catalyzes removal of carbon 1 as ¹³CO₂ to yield [1-¹³C]Ri5P as substrate for the non-oxidative PPP. The first two parallel reactions of the ‘forward’ non-oxidative PPP are isomerizations of Ri5P by phosphopentose isomerase and phosphopentose epimerase that do not affect the order of carbons. The isomerase converts the carbon 2 ketose to a carbon 1 aldose, yielding ribose-5-phosphate (R5P); the epimerase reverses the stereochemistry of the carbon 3 hydroxyl in Ri5P to yield the ketose xylulose-5-phosphate (X5P). Transketolase has broad substrate specificity and catalyzes the transfer of two carbons from a ketose (having specific hydroxyl configurations surrounding the keto group) to carbon 1 of an aldose (Kochetov and Solovjeva, 2014). In this case, the first two carbons of X5P are transferred to carbon 1 of R5P, forming the new ketose sedoheptulose-7-phosphate (S7P) and the three carbon aldose G3P (Fig. 2). These ‘forward’ products also satisfy the substrate requirements for transketolase, so the reverse reaction in which the first two carbons of S7P are transferred to carbon 1 of G3P also occurs. In developing the model, G3P formed by the first forward transketolase reaction was assumed a transient, PPP-specific metabolite that does not equilibrate with the glycolytic TrP pool.

Figure 2.

Carbon reshuffling in the pentose phosphate pathway. The hexose phosphate (HxP), pentose phosphate (PP), and triose phosphate (TrP) pools (boxed) are connected by the oxidative and non-oxidative PPP, and glycolysis. Carbon atoms from three glucose-6-phosphates (G6P1-3) derived from exogenous [1,2-¹³C₂]glucose via J₀ are distinguished with subscripts (C_x, C_x′, C_x″, where x=1–6) that reflect their original positions in G6P. Only C₁–C₃ of the original G6P entering the pentose cycle are reshuffled in fructose-6-phosphate (F6P) as ribose-5-phosphate (R5P) and xylulose-5-phosphate (X5P) return to the HxP pool in the forward non-oxidative PPP (J_3f and J_3af reflect the rates of pentose phosphate consumption and F6P production, respectively, by the forward non-oxidative PPP) by transketolase (TK) and transaldolase (TA). The fraction of returning F6P that equilibrates with G6P and recycles through J₁ and J_3af to return to the HxP pool at least twice is modeled as [J_3af/(J₀ + J_3af)]² while the fraction that does not recycle is (J_3af/(J₀ + J_3af))∗(1−[J_3af/(J₀ + J_3af)]). Further rounds of recycling produce a total of eight different HxP labels (unlabeled, and ¹³C at positions 1, 2, 3, 1,2, 1,3, 2,3, and 1,2,3). Phosphofructokinase, aldolase, and triose phosphate isomerase (collectively ‘upper’ glycolysis J_2a) together with non-oxidative PPP triose phosphate (TrP) production (J_3bf) dictate label distribution within the glycolytic TrP pool. In the model, glyceraldehyde-3-phosphate (G3P*) generated by the first forward transketolase reaction is assumed to not equilibrate with the glycolytic TrP pool.

Transaldolase catalyzes transfer of the first 3 carbons of S7P to carbon 1 of the aldose acceptor G3P, generating the new ketose/aldose pair F6P and erythrose-4-phosphate (E4P), both of which are adequate substrates for the reverse transaldolase reaction (Samland and Sprenger, 2009). Carbons 4–6 in F6P (F6P1 in Fig. 2) retain their original positions from G6P converted to X5P (G6P2 in Fig. 2), while carbons 2 and 3 now reside at positions 1 and 2; position 3 is occupied by carbon 2 derived from G6P converted to R5P (G6P1 in Fig. 2). The final forward reaction is a second transketolase two carbon ketose transfer from a second X5P (from G6P3 in Fig. 2) to the E4P aldose, yielding F6P2 and one G3P, the latter of which equilibrates with the glycolytic TrP pool. Positions 1 and 2 of the second F6P are from carbons 2 and 3 of G6P3, while the remaining positions 3–6 are preserved from G6P1. The TrP carbons derive from positions 4–6 of G6P3. From the forward reaction, only carbon positions 2 and 3 in the initial G6P substrates are rearranged in F6P. With [1,2-¹³C₂]glucose, the TrP produced from this pentose cycle lack ¹³C since they are derived from positions 4–6, which are not reshuffled. One of three fates was considered for F6P formed by the non-oxidative PPP: (a) it can equilibrate with G6P and recycle into the oxidative and non-oxidative PPP for additional carbon reshuffling, (b) it can equilibrate with G6P and then be processed by glycolysis to TrP, or (c) it can equilibrate with G6P, followed by partial glycolytic processing to F6P or TrP, and then return to the pentose pool through reverse flux of the non-oxidative PPP. Note that the model assumed rapid equilibration of F6P with G6P such that they form a single pool with identical ¹³C distributions.

3.1.3. Quantitative HxP recycling with a reversible non-oxidative PPP

Isotope distribution in the HxP pool was calculated from the contribution of J₀ (glucose uptake/phosphorylation) and J_3af (forward non-oxidative PPP flux, expressed as HxP formed per unit time) to HxP production (Fig. 1), together with the relative yield of each label produced by the respective blocks. Since the carbons in glucose are not reordered by J₀, the contribution of J₀, to labeling the HxP pool (as [1,2- ¹³C₂]G6P/F6P) is

$$1-\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}}$$ (1)

Note that the rate of exogenous [1,2-¹³C₂]glucose entering the HxP pool also reflects the fraction of the total pool not returning due to carbon loss as CO₂/TrP in the pentose cycle, as R5P incorporated into nucleotides, and as pyruvate formed from glycolysis. The fraction of HxP which leaves the pool and returns (as one of eight possible labels from ¹³C reshuffling one or more times in the non-oxidative PPP) (Gebril et al., 2016) is

$$\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}}$$ (2)

It follows that the fraction of HxP from J_3af that does not recycle is the product of Equations 1 and 2, while the fraction which recycles (i.e., re-enters J₁ and J_3f such that the carbons return to the HxP pool a minimum of two times) is

$${\left(\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}}\right)}^2$$ (3)

From (Gebril et al., 2016), the fractional content of a given label L returning to the HxP pool via J_3af × times is

$${}^x{F}_L^{\mathit{HxP}}={\left(\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}}\right)}^x(1-\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}})\ast \frac{{\mathrm{\sum }}_1^m{}^{x-1}{F}_a^{\mathit{PP}}{\ast }^{x-1}{F}_b^{\mathit{PP}}{\ast }^{x-1}{F}_c^{\mathit{PP}}}{{\mathrm{\sum }}_1^n{}^{x-1}{F}_{\mathit{any}}^{\mathit{PP}}{\ast }^{x-1}{F}_{\mathit{any}}^{\mathit{PP}}{\ast }^{x-1}{F}_{\mathit{any}}^{\mathit{PP}}}$$ (4)

Equation 4 is the basis for quantitatively tracking the fractional yield of each HxP label $({}^x{F}_L^{\mathit{HxP}})$ that has recycled x−1 times. The sum of m possible combinations of three pentose phosphate (PP) labels a, b, and c which yield HxP label L is normalized to the sum of all possible n combinations of three pentose phosphates recombining in J_3f. The fractional content of pentose phosphates a, b, and c from prior cycles of J₁ and $J_{3f}({}^{x-1}{F}_{a,b,c}^{\mathit{PP}})$ is determined from the previous composition of the HxP pool after taking into account the loss of carbon 1 in the oxidative PPP. Predicting the fractional M-M3 lactate is therefore partly accomplished by optimizing the flux ratio J_3af/(J₀ + J_3af), which determines the fractional abundance of the different HxP labels produced by recycling.

A major update to the model was accounting for ¹³C reshuffling by reverse flux through the nonoxidative PPP to allow greater flexibility in predicting ¹³C distribution in lactate without having to assume low nucleotide synthesis relative to oxidative PPP flux. This required introduction of a new model-optimized flux ratio-J_3af/J_3ar (along with J_3af/(J₀ + J_3af), J₇(J_2b + J₇), and J₅)- so that J_3ar and thus J_3r could be calculated. Since the model was developed to determine fluxes through reaction blocks rather than through individual reactions, ¹³C randomization by reverse (as well as forward) flux was only considered at the level of the reaction block and the explicit metabolites (Fig. 1). Thus, reverse flux was calculated as a linked, three reaction process from HxP/TrP substrates to R5P/X5P products without considering the independent reversibility of individual reactions that could return one or more implicit metabolites in the block (i.e., E4P, S7P, G3P) to the substrate pool. Reverse flux J_3r affects labeling of the pentose phosphate pool (R5P and X5P) according to its contribution to the pool (vs. that of the oxidative PPP J₁) and the fractional content of the different pentose labels produced from all combinations of HxP and TrP that enter the reverse PPP. Given that J_3r affects labeling of the R5P/X5P pool, which in turn affects label distribution in the HxP and TrP pools as the forward (recycling) reaction occurs, it follows that changes in J_3af/J_3ar will, to some degree, affect lactate labeling provided that a significant forward flux existed.

This update required compiling R5P/X5P products resulting from all combinations of labels in the HxP and TrP pools. The fractional content of pentose phosphate label L in the pool $({}^x{F}_L^{\mathit{PP}})$ is the sum of that produced by the reverse non-oxidative PPP J_3r and by the oxidative PPP J₁ following the x^th return of HxP through J_3af:

$${}^x{F}_L^{\mathit{PP}}=\frac{{\mathrm{\sum }}_1^m{}^x{F}_a^{\mathit{HxP}}\ast {}^x{F}_b^{\mathit{HxP}}\ast {}^x{F}_c^{\mathit{TrP}}}{{\mathrm{\sum }}_1^n{}^x{F}_{\mathit{any}}^{\mathit{HxP}}\ast {}^x{F}_{\mathit{any}}^{\mathit{HxP}}\ast {}^x{F}_{\mathit{any}}^{\mathit{TrP}}}\ast \left(\frac{J_{3r}}{J_1+J_{3r}}\right)+({}^x{F}_d^{\mathit{HxP}}+{}^x{F}_e^{\mathit{HxP}})\ast \frac{J_1}{J_1+J_{3r}}$$ (5)

The first term in Eq. 5 is the sum of m combinations of HxP labels a and b, and TrP label c that together yield the specific pentose phosphate label L, normalized to the sum of all n possible combinations of HxP/TrP, adjusted for the contribution of J_3r to labeling the pentose phosphate pool. The second term reflects the sum of HxP labels d and e that when metabolized through J₁ yield pentose label L, adjusted for the contribution of J₁ to labeling the pentose phosphate pool. The fractional contents of the HxP labels in Eq. 5 are calculated from Eq. 4. The fractional content of TrP label derived from glycolysis (J_2a) and the forward pentose cycle (J_3bf) after x returns of HxP through $J_{3\mathit{af}}({}^x{F}_L^{\mathit{TrP}})$ is

$$\begin{gathered} {}^x{F}_L^{\mathit{TrP}}=\frac{{\mathrm{\sum }}_1^m\frac{{}^x{F}_a^{\mathit{HxP}}}{2}}{{\mathrm{\sum }}_1^n{}^x{F}_{\mathit{all}}^{\mathit{HxP}}}\ast \frac{2J_{2a}}{2J_{2a}+J_{3bf}} \\ +{\displaystyle {\sum }_1^x[\frac{{\mathrm{\sum }}_1^m{}^x{F}_a^{\mathit{PP}}\ast {}^x{F}_b^{\mathit{PP}}\ast F_c^{\mathit{PP}}}{{\mathrm{\sum }}_1^n{}^x{F}_{\mathit{any}}^{\mathit{PP}}\ast {}^x{F}_{\mathit{any}}^{\mathit{PP}}\ast F_{\mathit{any}}^{\mathit{PP}}}\ast {(\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}})}^x(1-\frac{J_{3\mathit{af}}}{J_0+J_{3\mathit{af}}}\left)\right(\frac{J_{3bf}}{2J_{2a}+J_{3\mathit{bf}}})]} \end{gathered}$$ (6)

The first term in Eq. 6 reflects the fractional yield of TrP label L as m HxP labels are metabolized through J_2a relative to all possible n HxP labels metabolized to TrP, adjusted for the contribution of J_2a to labeling the TrP pool. The second term is the cumulative sum from x rounds of the pentose cycle J_3bf producing TrP label L from m combinations of pentose phosphate labels a, b, and c, normalized to the n possible combinations of all pentose phosphates and adjusted for the contribution of J_3bf to producing TrP.

The calculations were implemented in a step-wise, serial manner (Fig. 3). To begin, the fractional content of the pentose phosphate pool was determined from the composition of the HxP pool (100% [1,2-¹³C₂]HxP prior to operation of the pentose cycle) by summation of the fractional content of HxP labels that yield the same pentose phosphate label after decarboxylation of carbon 1. This was followed by applying Eq. 4 to update the HxP pool composition as the pentose phosphates recombine to return carbons to the HxP pool via J_3af. The TrP pool composition was then calculated with Eq. 6 from TrP generated by J_3bf and by processing of HxP through J_2a. The final step was revision of the pentose phosphate pool composition from J_3r using Eq. 5. The calculations were repeated for six rounds of HxP recycling to obtain the ‘final’ pentose phosphate, HxP, and TrP pool compositions, from which pyruvate/lactate labeling was determined. Inclusion of reverse PPP flux substantially increased the scope of calculations, as it required tracking the abundance of 32 pentose phosphate, 8 TrP, and 64 HxP labels after two rounds of recycling. All subsequent rounds of forward and reverse flux calculations required accounting for 32,768 possible combinations of substrates (32³ as three pentose phosphates recombine in J_3f; 64²×8 as two HxP and 1 TrP recombine in J_3r), which substantially lengthened the time required to complete the calculations.

Figure 3.

Reaction sequence for computations of ¹³C distribution in hexose phosphates, pentose phosphates, and triose phosphates. (A) The first forward reaction of the pentose cycle occurs. The fractional contents of R5P/X5P, F6P, and TrP are calculated from metabolism of exogenous [1,2-¹³C₂]glucose through the oxidative and forward non-oxidative PPP. Numbers below each metabolite indicate ¹³C position within the metabolite. A metabolite not enriched in ¹³C is indicated as unlabeled (UL). (B) Isomerization of F6P to G6P and glycolytic production of TrP occurs. Triose phosphate composition is updated as HxP from exogenous glucose and the non-oxidative PPP are metabolized through glycolysis and combine with TrP from the pentose cycle. (C) The first reverse non-oxidative PPP occurs. The content of R5P/X5P is updated as ten new labels enter the pentose phosphate pool from reverse non-oxidative PPP flux and combine with the existing label from the oxidative PPP. The content of each new label is calculated from the 36 possible combinations of two F6P and one TrP. (D) The second forward pentose cycle occurs. Updated calculations to the fractional contents of R5P/X5P, F6P, and TrP pools are implemented as HxP recycling begins. The R5P/X5P pool is updated as the oxidative PPP produces additional labels from the recycled HxP. This is followed by updates to F6P and TrP composition as 1000 possible combinations of pentose phosphates occur in the forward non-oxidative PPP. (E, F) Updates to the TrP (E) and pentose phosphate (F) pools are calculated as glycolysis and reverse flux through the non-oxidative PPP occur to alter their compositions. This sequential series of calculations is repeated for six rounds of forward recycling/reverse PPP flux to obtain the final composition of TrP, and thus pyruvate, derived from glycolysis.

¹³C distribution in the pentose phosphates, HxP, and lactate was assessed over a broad range for J_3af/J_3ar (from 10⁵, essentially unidirectional forward non-oxidative PPP flux, to 0.55, the minimum possible assuming J_3af/(J₀ + J_3af) = 0–25) to gauge the effect of reverse PPP flux in the model. Interestingly reverse flux substantially changed R5P/X5P labeling but only modestly affected the HxP pool (Table 1). Consequently, lactate M2 and M3 were modestly enriched by a maximum of 1.55 and 0.95%, respectively at the expense of M (−1.89%) and M1 (−0.60%) when the forward non-oxidative PPP accounted for 25% labeling of the HxP pool (Fig. 4). These results are consistent with the model developed by (KATZ and Rognstad, 1967) who found that reverse flux affected labeling of the pentose phosphates more than the HxPs, and had relatively minor effects on ¹⁴C distribution in lactate. Note that the influence reverse flux has on ¹³C distribution in the HxP pool (and thus lactate) will diminish as the forward non-oxidative PPP flux decreases (i.e., decreased J_3af/(J₀ + J_3af)). It is also worth noting that because J_3af appears in two solver-optimized ratios, they are not independent of one another. This dependency was evaluated by determining the upper limit to J_3af/(J₀ + J_3af) over a range of values for J_3af/J_3ar (Fig. 5). When the latter ratio was less than 1, the upper limit of J_3af/(J₀ + J_3af) was constrained by the requirement that J₁ ≥ 0. This constraint arose to satisfy the required steady-state for the HxP pool:

$$J_0+J_{3\mathit{af}}=J_1+J_{3\mathit{ar}}+J_{2a}$$ (7)

Conversely, when J_3af/J_3ar > 1, the upper limit to J_3af/(J₀ + J_3af) was constrained by the requirement that J₉ ≥ 0 to satisfy the steady-state for the R5P/X5P pool:

$$J_1+J_{3r}=J_{3f}+J_9$$ (8)

As J_3af/J_3ar approaches unity, it is possible for J_3af/(J₀ + J_3af) to approach 0.9999, but the calculated labeling pattern in lactate will accurately reflect that produced by cells given that J_3af » J₁ and J_2a. After six rounds of recycling, less than 0.1% of the HxP that recycles would have left the HxP pool via glycolysis and the pentose cycle, and thus the final M-M3 lactate would reflect less than 0.1% of what should be produced. Practically, J_3af/(J₀ + J_3af) should be ≤0.60 to minimally account for 98% of recycled HxP. Despite the greater time necessary for computations, accounting for J_3r allows for greater applicability of the model to cells or conditions where demand for pentose phosphates for nucleotide synthesis might exceed the demand for NADPH.

Table 1.

Effect of reverse non-oxidative PPP flux on hexose and pentose phosphate pools with [1,2¹³C₂]glucose.

	Fractional 13C in Hexose Phosphates
J3af/J3ar	M	M1	M2	M3	M4	M5	M6
100000	0.0036	0.1275	0.8688	<0.0001	<0.0001	<0.0001	<0.0001
10000	0.0036	0.1275	0.8688	<0.0001	<0.0001	<0.0001	<0.0001
1000	0.0036	0.1275	0.8688	<0.0001	<0.0001	<0.0001	<0.0001
100	0.0036	0.1274	0.8688	<0.0001	<0.0001	<0.0001	<0.0001
50	0.0037	0.1272	0.8689	0.0002	<0.0001	<0.0001	<0.0001
10	0.0039	0.1258	0.8691	0.0008	0.0003	0.0002	<0.0001
5	0.0041	0.1242	0.8692	0.0017	0.0005	0.0003	<0.0001
2	0.0047	0.1197	0.8692	0.0043	0.0012	0.0009	<0.0001
1	0.0054	0.1137	0.8676	0.0091	0.0022	0.0020	<0.0001
0.55	0.0059	0.1074	0.8613	0.0181	0.0032	0.0042	<0.0001
	Fractional 13C in Pentose Phosphates
J3af/J3ar	M	M1	M2	M3	M4	M5
100000	0.1246	0.8724	0.0029	<0.0001	<0.0001	<0.0001
10000	0.1246	0.8724	0.0029	<0.0001	<0.0001	<0.0001
1000	0.1247	0.8720	0.0032	<0.0001	<0.0001	<0.0001
100	0.1254	0.8680	0.0056	0.0003	0.0006	<0.0001
50	0.1261	0.8636	0.0083	0.0006	0.0013	<0.0001
10	0.1320	0.8288	0.0296	0.0032	0.0064	<0.0001
5	0.1394	0.7859	0.0557	0.0062	0.0127	<0.0001
2	0.1617	0.6612	0.1309	0.0147	0.0313	0.0002
1	0.1988	0.4662	2465	0.0268	0.0609	0.0007
0.55	0.2574	0.1797	0.4127	0.0420	0.1056	0.0025

The measured fluxes used for the calculations were from (Gebril et al 2016): J₀ = 291, J₄ = 288, J_RR = 564 pmol glucose, lactate, or atomic oxygen used or produced per minute per 10⁶ cells, respectively. The model inputs kept constant were: J_3af/(J₀ + J_3af) = 0.25, J₇/(J_2b + J₇) = 0, J₅ = 94. The fraction of hexose (G6P, F6P) or pentose (R5P, X5P) phosphates having none (M), one (M1), or more (M2-M6) ¹³C atoms was calculated from Eq. 4 and 5, respectively.

Figure 4.

Effect of reverse non-oxidative PPP flux on ¹³C distribution in lactate. The ratio of forward (J_3af) to reverse (J_3ar) non-oxidative PPP flux was incorporated as a model-optimized parameter affecting the fraction of unlabeled (M), single- (M1), double- (M2), and triple- (M3) ¹³C labeled lactate. The remaining three optimized parameters were kept constant: J_3af/(J₀ + J_3af) = 0.25, J₇/(J_2b + J₇) = 0, J₅ = 94. The fluxes for this analysis were: J₀ = 291, J₄ = 288, J_RR = 564 pmol glucose, lactate, or atomic oxygen used or produced per minute per 10⁶ cells, respectively. Table 1 shows the corresponding changes in labeling the hexose and pentose phosphate pools for the same parameters and flux inputs used here.

Figure 5.

Maximum contribution of the forward non-oxidative PPP to labeling the HxP pool depends on the rate of reverse flux through the non-oxidative PPP. Lactate labeling is affected by the relative contributions of the forward non-oxidative PPP (J_3af) and glucose uptake/phosphorylation (J₀) to total HxP produced (J_3af/(J₀ + J_3af) and 1 − [J_3af/(J₀ + J_3af)], respectively), as well as by the ratio of forward to reverse non-oxidative PPP flux (J_3af/J_3ar). The relationship between the two ratios was assessed by determining the maximum possible value that could be assigned to J_3af/(J₀ + J_3af) for variations in J_3af/J_3ar. The maximum occurred when J₉ < 0 for J_3af/J_3ar >1 or J₁ < 0 for J_3af/J_3ar < 1. When implemented to find solutions for these variables, the model does not exceed these maximums because of explicit constraints that all fluxes must be non-negative. This relationship was assessed assuming the following conditions: $J_0=291\mathit{pmol}\frac{\mathit{glc}}{\mathit{min}}$, $J_4=288\mathit{pmol}\frac{\mathit{lac}}{\mathit{min}}$, $J_{\mathit{RR}}=567\mathit{pmol}\frac{o}{\mathit{min}}$, $J_5=97\mathit{pmol}\frac{\mathit{pyr}}{\mathit{min}}$, and $J_7/(J_{2b}+J_7)=0.05$.

3.1.4. Qualitative and quantitative ¹³C rearrangements in the TCA cycle

Malic enzyme produces pyruvate and NADPH from oxidation of malate, which is assumed to be derived exclusively from the TCA cycle. Therefore a fraction of cytoplasmic pyruvate contains ‘recycled’ ¹³C that has been subjected to reshuffling within the TCA cycle (Fig. 6). The original model quantitated malate labeling through four turns of the TCA cycle as ¹³C in pyruvate entered the cycle, but neglected to consider that succinate dehydrogenase cannot distinguish between the two carboxylic acids in succinate due to its symmetry. With the exception of unlabeled succinate, or label at positions 1,4, 2,3, or 1,2,3,4, each (of twelve possible) succinate label will be metabolized such that the ¹³C position in half of the malate will be inverted relative to the label position in succinate (e.g., for ¹³C at carbons 1 and 2 in succinate, 50% of the resulting malate will be labeled at positions 1,2 and 50% labeled at positions 3,4; Fig. 6). This doubles the number of unique malate labels in the first two turns of the cycle compared to the labels reported in Table 2 of (Gebril et al., 2016). Quantitative tracking of each label for turn x of the TCA cycle was determined essentially as previously described except that the yield of malate label L (that also has an inverse label; i.e., L= 1, 2, 3, 4, 1,2, 3,4, 1,3, 2,4, 1,2,3, 2,3,4, 1,2,4, and 1,3,4) from the sum of all m combinations of pyruvate label a and malate label b (after conversion to acetyl CoA and oxaloacetate, respectively) is half of that shown in Table 2 of Gebril et al (2016)

$${}^x{F}_L^{\mathit{Mal}}={\mathrm{\sum }}_1^m\frac{{}^{x-1}{F}_a^{\mathit{Mal}}\ast F_b^{\mathit{Pyr}}}{2}$$ (9)

By inclusion of the additional ¹³C reshuffling by succinate dehydrogenase, it was found that seven turns of the TCA cycle were necessary for malate to approach an isotopic steady state compared to the four turns in the previous model. This update substantially increased M1 and M2 pyruvate from malic enzyme while reducing M and M3 pyruvate, but the effect on exported lactate was more modest and depended on the contribution of J₇ to the pyruvate pool (Table 2).

Figure 6.

Carbon reshuffling in the tricarboxylic acid (TCA) cycle. The carbon atom transitions are depicted for select metabolites in the TCA cycle as unlabeled oxaloacetate reacts with acetyl-CoA derived from [3-¹³C₁]pyruvate in this example for the first of seven rounds used to evaluate malate labeling. Succinate dehydrogenase does not distinguish between carboxylic acid carbons 1 and 4 in succinate since it is symmetrical, so there is a 50% chance that ¹³C in malate will reside at position 2 or 3. This aspect of carbon reshuffling in the TCA cycle was overlooked in the original model. The labeled malate from the first round is subjected to six more rounds of the cycle, each time reacting with an acetyl-CoA label derived from the pyruvate pool. From this, fractional pyruvate labeling generated by malic enzyme is determined.

Table 2.

Symmetrical vs. non-symmetrical succinate metabolism in the TCA cycle.

Updated TCA cycle: symmetrical						Original TCA cycle: non-symmetrical
	Fractional 13C in malate						Fractional 13C in malate
$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3	M4	$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3	M4
0.01	0.1864	0.3115	0.2633	0.1797	0.0590	0.01	0.1821	0.3117	0.2643	0.1810	0.0610
0.02	0.1862	0.3118	0.2641	0.1794	0.0586	0.02	0.1818	0.3120	0.2651	0.1807	0.0605
0.05	0.1855	0.3128	0.2664	0.1783	0.0571	0.05	0.1810	0.3128	0.2675	0.1797	0.0590
0.10	0.1844	0.3146	0.2702	0.1762	0.0546	0.10	0.1799	0.3144	0.2714	0.1779	0.0565
0.15	0.1836	0.3166	0.2739	0.1739	0.0520	0.15	0.1789	0.3162	0.2751	0.1758	0.0539
0.20	0.1830	0.3189	0.2774	0.1713	0.0494	0.20	0.1782	0.3183	0.2788	0.1735	0.0513
	Fractional 13C in pyruvate from ME						Fractional 13C in pyruvate from ME
$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3		$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3
0.01	0.2561	0.3614	0.2741	0.1084		0.01	0.3203	0.3052	0.2333	0.1413
0.02	0.2560	0.3620	0.2741	0.1078		0.02	0.3195	0.3062	0.2339	0.1403
0.05	0.2558	0.3641	0.2742	0.1058		0.05	0.3172	0.3095	0.2359	0.1374
0.10	0.2556	0.3677	0.2742	0.1025		0.10	0.3136	0.3149	0.2390	0.1324
0.15	0.2557	0.3714	0.2739	0.0991		0.15	0.3102	0.3206	0.2418	0.1274
0.20	0.2560	0.3752	0.2733	0.0955		0.20	0.3071	0.3264	0.2443	0.1222
	Fractional 13C in exported lactate						Fractional 13C in exported lactate
$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3		$\frac{J_7}{J_{2b}+J_7}$	M	M1	M2	M3
0.01	0.5613	0.0581	0.3791	0.0015		0.01	0.5619	0.0576	0.3787	0.0018
0.02	0.5587	0.0611	0.3776	0.0025		0.02	0.5600	0.0600	0.3768	0.0032
0.05	0.5510	0.0703	0.3731	0.0057		0.05	0.5541	0.0675	0.3711	0.0072
0.10	0.5382	0.0858	0.3655	0.0106		0.10	0.5440	0.0805	0.3620	0.0136
0.15	0.5253	0.1016	0.3579	0.0152		0.15	0.5335	0.0940	0.3531	0.0194
0.20	0.5126	0.1179	0.3502	0.0194		0.20	0.5228	0.1081	0.3444	0.0247

The measured fluxes used for the calculations were from (Gebril et al 2016): J₀ = 291, J₄ = 288, J_RR = 564 pmol glucose, lactate, or atomic oxygen used or produced per minute per 10⁶ cells, respectively. The model inputs kept constant were: J_3af/J_3ar = 10, J_3af/(J₀ + J_3af) = 0.25. The updated TCA cycle accounts for ¹³C randomization in malate as a result of succinate symmetry as it is metabolized in the TCA cycles, whereas the original model assumed carbons 1 and 2 of succinate remained exclusively at these positions in malate. The fraction of malate having none (M), one (M1), or more (M2-M4) ¹³C atoms was calculated from Eq. 9. Pyruvate labeling from malic enzyme (ME) was calculated as the sum of all malate labels that, when decarboxylated at carbon 4, yield each specific pyruvate label. Labeling in exported lactate accounts for the contribution of ME, glycolysis and pentose cycle-derived triose phosphate to the pyruvate pool.

3.2. Flux calculations

The ten fluxes not measured (Fig. 1; J₁,J_2a,J_2b,J_3f,J_3r,J₅,J₆,J₇,J₈,J₉) can be determined from J₀,J₄,J_RR, and M-M3 lactate provided the cells maintain a metabolic steady-state over the experiment. Five of the unknown fluxes (J_3f,J_3r,J₅,J₆,J₇) are determined from the measured variables in the following manner. Assume the model reproduces M-M3 lactate with J_3af(J₀ + J_3af) = X, J_3af/J_3ar = Y and J₇/(J_2b + J₇) = Z, then J_3af = X ∗ J₀/(1−X) and J_3ar =J_3af/Y (units of F6P produced or consumed/min, respectively). By stoichiometry, the remaining non-oxidative PPP fluxes are: J_3f = 1.5 ∗ J_3af (units of pentose phosphate consumed/min), J_3bf = J_3f/3 (units of TrP produced/min), J_3r = 1.5 ∗ J_3ar (units of pentose phosphate produced/min), and J_3br = J_3r/3 (units of TrP consumed/min). Concurrently, mitochondrial pyruvate oxidation J₅ is optimized within the stoichiometric constraints that J_rr = 5 ∗ J₅ + J₆ and J₅ ≥ J₆. Malic enzyme flux is determined as J₇ = Z ∗ (J₄ + J₅) since, from the steady-state pyruvate level, J_2b + J₇ = J₄ + J₅. Furthermore since pyruvate and NADH are both derived from J_2b, it is apparent from the pyruvate steady-state that production of glycolytic NADH (and pyruvate) decreases as ME flux increases, so J₆ = J₅ − J₇ given a constant NADH consumption by J₄. The remaining five fluxes can be calculated from the steady-state and reaction stoichiometries: J_2b =J₄ + J₅ − J₇, J_2a = 0.5 * (J_2b − J_3bf), J₁ =J₀ + J_3af − J_2a − J_3ar, J₈ = 2 ∗ J₁ + J₇, and J₉ = J₁ + J_3r − J_3f. In this way four measured variables can be sufficient to obtain substantial quantitative information on glucose utilization.

3.3. Application of the model

3.3.1. Re-evaluating ¹³C distribution in lactate from metabolism of [1,2-¹³C₂]glucose

As the updated model was evaluated, it became apparent that quantitation of M-M3 lactate was necessary to reach a single, ‘global’ solution, as multiple ‘local’ solutions were possible if the small contribution of M3 lactate was excluded as in (Gebril et al., 2016). This, together with the inability of the updated model to adequately reproduce M-M2 lactate measured previously by LC QTOF MS/MS, necessitated re-analysis of ¹³C distribution in lactate from CGNs. Closer inspection of the MS/MS data reported in (Gebril et al., 2016) revealed that the derivatized ions were not completely separated by QTOF prior to quantitation of the fragments. Unlabeled lactate (M=224 Da peak) contained 225 (M1) and 226 (M2) Da ions, and, similarly, the 225 Da (M1) peak contained 226 Da (M2) ions. This revelation invalidated the fractional ¹³C distribution previously reported (Gebril et al., 2016), as quantification of the 152 Da fragment from each peak was cross-contaminated with fragments from the higher molecular weight ions. Triple quadrupole mass spectrometry (LC-TQMS) was found superior to QTOF-MS/MS for both separating and detecting the ions. From analysis of M and M1 (3-¹³C lactate from Cambridge Isotope Laboratories) standards, 96.7±0.3% and 96.3±0.2% of lactate was found in their respective 224 and 225 Da peaks (Table 3). Importantly, Cambridge Isotope lot analysis reported 1.5% residual unlabeled lactate in the M1 standard (i.e., 224 Da when derivatized), which agreed well with the current LC-TQMS results of 1.3±0.3%. If the 225 Da ion had cross-contaminated the 224 Da peak, then a much greater percentage of unlabeled lactate would be expected for the M1 standard. Endogenous ¹³C present as M1 in the ¹²C lactate standards averaged 2.87±0.24%, which was similar to 3.35±0.10% endogenous M1 lactate observed in neurons metabolizing ¹²C glucose. Further evidence suggesting separation of M and M1 was from LC-TQMS analysis of 7.65:1 M:M1 mixed standards, which yielded a 6.91±0.08:1 ratio (n=5 different concentrations from 100–800 μM). With this method, M3 lactate was reliably detected above background for cells metabolizing [1,2-¹³C₂]glucose and found to constitute 1.19±0.04% of total lactate. As expected, M1 lactate was less abundant than previously measured since cross contamination with M2 was eliminated, but remarkably M1 accounted for only 2.49±0.07% of total lactate (vs. 18.6% measured previously). Thus, this more careful analysis, which supersedes that reported in (Gebril et al., 2016), yielded a substantially different lactate labeling pattern (Table 3) for use in the revised model.

3.3.2. Updated assessment of metabolic fluxes and pentose cycle activity in cultured neurons

Cerebellar granule neuron fluxes J₀, J₄, and J_RR (where RR denotes mitochondrial respiration rate) measured previously with [¹²C]glucose as the sole substrate (n=6) (Gebril et al., 2016) were used with the revised model to solve for the unknown fluxes by optimizing J_3af/J_3ar, J_3af/(J₀ +J_3af), J₇/(2J_2a + J_3bf + J₇), and J₅ to reproduce the re-measured fractional M-M3 lactate (n=5 for neurons using [1,2-¹³C₂]glucose as sole substrate) reported in Table 3. However, the model was unable to reproduce the average M-M3 lactate pattern (Table 4). As a result, the unknown fluxes were estimated by minimizing the summed difference (i.e., error) between the predicted M-M3 pattern $(M_i^{\mathit{Predict}})$ and the measured M-M3 pattern $(M_i^{\mathit{Measured}})$:

$${\mathrm{\sum }}_{i=0}^3|M_i^{\mathit{Predict}}-M_i^{\mathit{Measured}}|=\mathit{minimum}$$ (10)

It was not surprising that, for all experiments, the model was constrained (by the requirement that J_3af/(J₀ + J_3af)≥ 0.00001) to the same solution given that M-M3 lactate was measured from separate experiments (Table 4). Monte Carlo simulations conducted as in (Gebril et al., 2016) provided an estimate of the additional variation in fluxes to be expected for simultaneous measurements of M-M3 lactate and J₀, J₄, and J_RR (Table 4, minimum and maximum from the simulations designated in parentheses below each mean). Most of the error was associated with underestimating M2 lactate and correspondingly overestimating M3 lactate.

Table 4.

Cerebellar granule neuron fluxes estimated from the updated model.

	Optimized Flux Ratio
	${}^{J_{3\mathrm{af}}}/{}_{J_{3\mathit{ar}}}$	${}^{J_{3\mathrm{af}}}/{}_{(J_0+J_{3\mathit{ar}})}$	${}^{J_7}/{}_{(2J_{2a}+J_{3\mathit{bf}}+J_7)}$
Model Result	nd	<0.0001±0.0000 (<0.0001–0.0023)	0.0782±0.0000 (0.0691–0.0803)
	Fractional 13C distribution in lactate
	M	M1	M2	M3
Model result	0.4755±0.0000 (0.4749–0.4784)	0.0249±0.0000 (0.0220–0.0256)	0.4857±0.0000 (0.4850–0.4872)	0.0140±0.0000 (0.0124–0.0143)
Measured	0.4751±0.0021	0.0249±0.0007	0.4881±0.0021	0.0119±0.0004
error	0.0004 (0.0002–0.0030)	<0.0001 (<0.0001–0.0013)	0.0024 (0.0006–0.0040)	0.0021 (0.0011–0.0034)
Reaction block	Flux, set 1 (J3af/J3ar = max) (pmol/min × 106 cells)		Flux, set 2 (J3af/J3ar = min) (pmol/min × 106 cells)
Glucose uptake (J0)	291±23		291±23
Oxidative PPP (J1)	112±24 (111–113)		0±0
Glycolysis, upper (J2a)	178±23 (178–180)		201±21
Glycolysis, lower (J2b)	357±45 (356–361)		357±45
Non-oxidative PPP, for. (J3f)	0±0 (0–1.1)		0±0
Non-oxidative PPP, rev. (J3r)	0±0 (0–1.1)		135±30
Lactate production (J4)	288±46		288±46
Pyruvate oxidation (J5)	99±5 (98–99)		99±5
NADH oxidation (J6)	67±4 (67–72)		67±4
Cyto. malic enzyme (J7)	30±4 (26–31)		30±4
NADPH oxidation (J8)	255±47 (247–257)		30±4
Nucleotide synthesis (J9)	112±24 (110–113)		135±30
Mito. respiration (JRR)	564±29		564±29

Measured fluxes (J₀,J₄,J_RR; mean±standard error of the mean, n=6) from (Gebril et al., 2016) and re-measured fractional ¹³C distribution in lactate (M-M3; mean±standard error of the mean, n=5) were used to solve for the remaining fluxes (expressed as pmol substrate consumed, except for J_3r which is pmol product formed) by optimizing the three flux ratios and J₅. The model results reflect minimization of the total error between the predicted and measured values. J_3af/J_3ar could not be determined (nd) from the data given J_3af was negligible (J_3af = 0.0029 ± 0.0002 pmol F6P/min × 10⁶ cells from the constraint that J_3af/(J₀ + J_3af) ≥ 0.00001). Set 1 fluxes assumed the maximum possible ratio (i.e., the same negligible reverse as forward non-oxidative PPP flux so that J_3af/J_3ar = 1) while set 2 fluxes assumed a minimum possible ratio (which averaged 4.29±1.02 × 10⁻⁵; maximal J_3ar was reached when oxidative PPP flux was < 0.01 pmol G6P/min × 10⁶ cells). The maximum and minimum ratios therefore provide a range of uncertainties for the five fluxes (J₁,J_2a,J_3r,J₈,J₉) dependent on J_3af/J_3ar. Numbers in parentheses reflect the highest and lowest means from 100 Monte Carlo simulated data sets for J_3ar = J_3af. The simulations were not performed for J_3af/J_3ar = mín as the variation was expected to be similar given virtually no contribution of J_3af to labeling the hexose phosphate pool.

The most important result from this new analysis was negligible forward non-oxidative PPP flux, and thus no apparent pentose cycle activity or recycling of hexose phosphates from the non-oxidative PPP into the oxidative PPP. The forward flux was designated negligible rather than zero for validity of the calculations (hence the constraint J_3af/(J₀ + J_3af) ≥ 0.00001). Thus, in cultured CGNs from newborn rats with depolarizing levels of extracellular K⁺ (25mM), effectively none of the glucose taken up was consumed by the pentose cycle (although up to 39% of glucose was consumed by the oxidative PPP and reactions synthesizing nucleotides- see below). This result contrasts with our previous data suggesting the pentose cycle consumed 19.5% of glucose, and is consistent with other reports also evaluating isotope distribution in lactate (Ben-Yoseph et al., 1996; Brekke et al., 2012) that have demonstrated very low basal pentose cycle activity (0–4% of total glucose consumption) in cultured neurons. In such cases of minimal pentose cycle activity, the difference in flux estimation should be small for HxP recycling and non-recycling models. The differences between models are expected to be more substantial for in vivo brain metabolism experiments, glial cell enriched cultures, or oxidatively stressed neuronal cultures where apparent pentose cycle activity consumes from 4–15% (unstressed cultured glial cells; neonatal or adult brain) to 25–40% (stressed neurons) of total glucose (Amaral et al., 2011; Amaral et al., 2016; Ben-Yoseph et al., 1995; Ben-Yoseph et al., 1996; Brekke et al., 2014; Dusick et al., 2007; Morken et al., 2014), in which case accounting for HxP recycling becomes more important. Since the unstressed CGNs in this study exhibited negligible pentose cycle flux, ¹³C reshuffling by the TCA cycle and malic enzyme largely explained the deviation in lactate labeling from 50% M/M2 expected by glycolysis. The results suggest approximately 7.8% of pyruvate was derived from malic enzyme, which is consistent with what we reported previously (Gebril et al., 2016) despite incomplete assessment of TCA cycle ¹³C randomization in the original model. This is also similar to malic enzyme in cultured astrocytes which was responsible for producing 5% of total pyruvate (Amaral et al., 2011). The fact that the model did not exactly reproduce M-M3 lactate could indicate that (1) isotopic steady-states may not have been reached in the glutamate and/or aspartate pools that are subject to transaminase exchange reactions with TCA cycle α-ketoglutarate and oxaloacetate pools, respectively, (2) contaminating glial cells may have supplied alternatively labeled substrates (e.g., glutamine, pyruvate) that affected ¹³C distribution in the relevant neuronal metabolite pools (e.g., α-ketoglutarate, pyruvate), (3) differentially labeled pyruvate pools (from incomplete mixing of pyruvate from glycolysis, malic enzyme, and/or the pentose cycle) may have been preferentially channeled to either mitochondria or lactate dehydrogenase, and/or (4) significant reverse flux through malic enzyme may have affected TCA cycle malate labeling by fixation of unlabeled CO₂. Nonetheless, the small error between measured and predicted M-M3 lactate suggests that the underlying assumptions, constraints, and system structure reasonably reflect the metabolic behavior of neurons.

With negligible forward non-oxidative PPP flux, J_3af/J_3ar became irrelevant to influencing labeling of the HxP and TrP pools and thus exported lactate, so this ratio could not be optimized in these experiments. As a result, the input data were insufficient to conclusively infer J_3ar, J₁, J_2a, J₈, and J₉ given that the proportion of glucose consumed by the oxidative PPP vs. the reverse non-oxidative PPP could not be distinguished. If J_3r is negligible (i.e., assume J_3af/J_3ar= 1; flux set 1 Table 4), then all glucose not exported as lactate nor oxidized by mitochondria (39%), would be consumed by the oxidative PPP and used subsequently for nucleotide synthesis. In this case, the upper limit to J₁ would be 112±24 pmol G6P consumed/min × 10⁶ cells, which would correspond to lower limits for J₉ (112±24 pmol R5P consumed/min × 10⁶ cells) and J_2a (178±23 pmol G6P consumed/min × 10⁶ cells). At the other extreme, if the extra glucose not exported as lactate nor oxidized by mitochondria is consumed by the reverse non-oxidative PPP (flux set 2, Table 4), then J₄ would be negligible and the upper limit to J_3r = 135±30 pmol pentose phosphates produced/min × 10⁶ cells, which in turn yields upper limits to J_2a (201±21 pmol G6P/min × 10⁶ cells) and J₉ (135±30 pmol R5P/min × 10⁶ cells). For maximal J₁ and negligible J_3r, the pentose phosphate pool should be 100% M1; in contrast, the predicted ¹³C distribution with negligible J₁ and maximal J_3r is M=0.3333, M2=0.5000, and M4=0.1667. Thus, with negligible forward non-oxidative PPP flux, measurement of fractional M-M5 content of the pentose phosphate pool would provide the constraints necessary to optimize J_3af/J_3ar to accurately assess J₁ and J_3r. In either case, the substantial percentage of glucose used for nucleotide synthesis is surprising and may be related to the young age of these neurons (isolated from P7 neonatal rats; see below).

It is likely that the true CGN fluxes are closer to set 1 than set 2 given the critical role of the oxidative PPP in NADPH synthesis/antioxidant defenses and neuronal viability (Chinta et al., 2007; Dunn et al., 2014; Fernandez-Fernandez et al., 2012; Filosa et al., 2003; Herrero-Mendez et al., 2009; Levonen et al., 2014; Li et al., 2014). Assuming negligible reverse non-oxidative PPP flux, then 39% of glucose consumption could be attributed to the oxidative PPP and the downstream reactions involved in de novo nucleotide synthesis (vs 61% to glycolysis) despite negligible pentose cycle activity. In this state, 7% of glucose would be lost as CO₂ in the oxidative PPP, and 32% converted to pentose phosphates for use in nucleotide synthesis. The upper limit to oxidative PPP flux is less than half of that reported previously (Gebril et al., 2016), but nonetheless suggests that up to 88% of NADPH is from J₁ vs. 12% from malic enzyme. Of course since J_3r might consume some of the glucose, it will be necessary to test this ‘upper limit’ of oxidative PPP glucose consumption by analysis of ¹³C distribution in the pentose phosphates.

For glucose consumed by glycolysis, 45% was exported as lactate while the remaining 16% was oxidized by mitochondria. Hence 84% of glucose consumption was independent of mitochondrial respiration, with aerobic glycolysis accounting for the majority (54%) of this. As discussed previously (Gebril et al., 2016), the high non-mitochondrial glucose use (particularly by aerobic glycolysis) may partly be attributable to the in vitro culture conditions (10 mM glucose, no exogenous lactate or amino acids). However, some of the oxygen-independent glucose use may be physiologically relevant and associated with the early age of neurons in this study (isolated from 7 day old Wistar rats). A meta-analysis demonstrated that human whole brain oxygen-independent glucose consumption peaked in early childhood (approximately 5 years old) and accounted for about one-third of total glucose consumption (Goyal et al., 2014). Such use was correlated with high rates of neuronal growth, maturation, synapse formation, and/or plasticity and thus may be important for brain development. In this study, the extent to which aerobic glycolysis (45% of total glucose) vs. the oxidative PPP (up to 39% of total glucose) may be important for these processes is unknown. Nonetheless, the high rate of non-mitochondrial glucose consumption of these relatively immature neurons may partly reflect ongoing in vitro growth/maturation.

4. Summary and conclusions

Fluxes through the major glucose consuming reactions have been re-assessed in cerebellar granule neurons metabolizing [1,2-¹³C₂]glucose using an updated model originally described in (Gebril et al., 2016) and new measurements of ¹³C lactate labeling by triple quadrupole mass spectrometry. The updates increased the applicability of the model to different cell types by considering the reversible nature of the non-oxidative PPP, and increased the reliability of quantitatively tracking ¹³C distribution in malate by accounting for symmetrical succinate oxidation within the TCA cycle. Reverse flux through the non-oxidative PPP had relatively minor effects on ¹³C distribution in the hexose phosphate pool and by extension in lactate, but this depended on the contribution of the forward non-oxidative PPP flux to labeling the HxP pool. The effect of symmetrical succinate metabolism was such that pyruvate produced by malic enzyme was predicted to have substantially greater M1/M2 and less M/M3 content, which in turn affected ¹³C distribution in lactate according to the contribution of malic enzyme vs. glycolysis to labeling the pyruvate pool. When applied to ¹³C lactate from cultured CGNs, the error-minimized solution indicated negligible pentose cycle activity but highly active oxidative PPP and/or reverse non-oxidative PPP flux that consumed 39% of glucose. Malic enzyme and glycolysis were responsible for 7.8 and 92.2% of pyruvate production, respectively, and largely accounted for ¹³C labeling in exported lactate. The majority (74%) of glucose used by glycolysis was exported as lactate as opposed to oxidized by mitochondria. The high mitochondrial-independent glucose utilization may be a consequence of the young age of neurons as well as the in vitro conditions used. The current approach offers a few advantages compared to other models that evaluate fluxes from ¹³C labeling in glycolytic and/or TCA cycle metabolites by (1) providing more accurate estimations of pentose cycle activity for redox homeostasis by accounting for HxP recycling and reverse non-oxidative PPP flux, particularly in situations where the cycle becomes a significant glucose consumer (e.g., in stressed neurons, glial cells, or the intact brain), (2) quantifying glucose utilization by the oxidative PPP for both redox homeostasis and provision of pentose phosphates used for nucleotide synthesis, a flux not typically evaluated with the pentose cycle, (3) considering the affect that malic enzyme has on lactate labeling, and (4) requiring relatively few experiments for quantifying fluxes through all major glucose consuming reactions.

Highlights.

• Reverse non-oxidative PPP flux was incorporated in a ¹³C metabolic flux model.

• The model was applied to cerebellar granule neurons from neonatal rat brains.

• The oxidative and reverse non-oxidative PPP consumed 39% of the glucose.

• Glucose consumption by the pentose cycle was negligible.

• The updated model offers advantages to assessing pathway fluxes.

Abbreviations list

PPP

pentose phosphate pathway

LC-MS/MS

liquid chromatography tandem mass spectrometry

LC-TQ-MS

liquid chromatography triple quadrupole mass spectrometry

LC-QTOF

liquid chromatography quadrupole time of flight mass spectrometry

HxP

hexose phosphate (referring to both glucose-6-phosphate and fructose-6-phosphate)

G6P

glucose-6-phosphate

F6P

fructose-6-phosphate

TrP

triose phosphates (referring to both dihydroxyacetone phosphate and glyceraldehyde-3-phosphate)

G3P

glyceraldehyde-3-phosphate

R5P

ribose-5-phosphate

X5P

xyulose-5-phosphate

Ri5P

ribulose-5-phosphate

TCA

tricarboxylic acid

M

a metabolite having no ¹³C atoms derived from [1,2-¹³C₂]glucose

M1

a metabolite having one ¹³C atom derived from [1,2-¹³C₂]glucose

M2

a metabolite having two ¹³C atoms derived from [1,2-¹³C₂]glucose

M3

a metabolite having three ¹³C atoms derived from [1,2-¹³C₂]glucose

J_x

flux through reaction x