Calculations for Gluconeogenesis by Mass Isotopomer Distribution Analysis

Calculations relevant to the measurement of gluconeogenesis by mass isotopomer distribution analysis are presented here. These theoretical considerations are intended to complement the experimental results described in the accompanying paper. Calculations relevant to the measurement of gluconeogenesis by mass isotopomer distribution analysis are presented here. These theoretical considerations are intended to complement the experimental results described in the accompanying paper. gluconeogenesis mass isotopomer distribution analysis triose phosphate A reference table of fractional mass isotopomeric abundances can be calculated for any polymer containing repeating subunits, as a function of p (the proportion of subunits that are isotopically labeled), by application of formulae derived from the binomial or multinomial expansions. We analyze the tetraacetate fragment of glucose pentaacetate by gas chromatography/mass spectrometry, because this is the major mass spectrometric species containing all six carbons of glucose. The theoretical relationship between p and mass isotopomer fractional abundances is generated by use of a computer algorithm generated for this purpose,2 2The mass isotopomer distribution program is available from the authors on request. which takes into account the elemental composition of the molecule analyzed (C14H19O9), the composition of the constant or invariant moiety (C14H19O9), and the composition of the variable or enriched moiety (in this case, *C2, comprising the two potentially labeled monomeric subunits/glucose molecule, with one potentially enriched carbon/subunit, each with a mass increment of 1 Da). The reference table (Table I) is calculated as a function of p, shown here from p = 0.00 (natural abundance) to p = 0.30. Fractional abundances of M0, M1, and M2 only are considered here, and these abundances are “normalized” to add up to unity. Baseline fractional abundances are subtracted from the values at each p, to generate Δ fractional abundances or excesses. We have shown elsewhere (11, 12) that the relationship between excess M2 (EM2) and excess M1 (EM1), expressed as their ratio (EM2/EM1), is uniquely determined by p (over a discreet interval) and is independent of dilution from unlabeled polymer molecules. One can therefore infer p from the measured EM2/EM1 ratio. The values of Δ fractional abundances or EMx shown (Table I) represent the highest, or asymptotic, values possible at each value of p, since the calculation in the table assumes that 100% of the glucose molecules came from GNG. Dilution by unlabeled, non-GNG glucose molecules will reduce EMI and EM2 proportionately. The actual percentage GNG can therefore be calculated by comparing a measured value (e.g. EM1) to the asymptotic value (e.g. A1*).Table IReference table for gluconeogenesis from triose-P (m/z 331-333, glucose-pentaacetate derivative Open table in a new tab A sample calculation may be helpful. If we consider rat G1 at the 9-h time point for the α-anomer (Table II of accompanying paper), EM1 = 0.1723 and EM2 = 0.0442 (each calculated by subtracting base-line fractional abundances from observed fractional abundances at this time point). The ratio EM2/EM1 = 0.2566, which when checked against the reference table (Table I) or best-fit equations derived from it (legend to Table I) indicates that p = 0.1594 and A1*=0.1915. Comparing EM1 to A1* (0.1723/0.1915) reveals fractional GNG to be 89.9%. It should be noted that in practice, one need not proceed from the EM2/EM1 ratio through p to identify A1*, but one can proceed directly from the EM2/EM1 ratio to A1*, even though the table was initially generated as a function of p. Going directly from the ratio to A1* avoids concerns about compounding of error if two sequential best-fit equations had to be used. As a practical matter, disequilibrium between the triose phosphates is relatively small (<20%, see previous paper). We will show here that isotopic disequilibrium between dihydroxyacetone phosphate and glyceraldehyde 3-phosphate in theory has only a very minor impact on MIDA calculations for glucose, unless the isotopic disequilibrium is much greater (ratio ≥2:1). We will first present an intuitive explanation and then the actual calculations to prove this point. In the simplest case, where two precursor units of a dimer have the same probability of being isotopically labeled (p1 = p2 or (p1 + p2)/2 = p1 = p2) and there is no natural abundance background present, the probability of a double-labeled dimer occurring is A12. Inference of the value for p from mass isotopomer abundances in essence is performed by reversing this calculation: p12=p1=p2. Therefore, the arithmetic mean ((p1 + p2)/2) equals the geometric mean (p1·p2) when p1 = p2. If instead of two identical labeling probabilities there was a 20% lower value in one of the subunits than the other, so that p2 = 0.8(p1), the arithmetic mean of their labeling probabilities would be (0.8p1 + p1)/2 = 0.9p1, while the value calculated from probability analysis would be p1p2=0.8p12=0.894p1. For a 40% disequilibrium between the precursor subunits, the geometric mean (0.6p12=0.775p1) is still within 3.2% of the arithmetic mean (0.8p1). Although this is an oversimplified representation of the MIDA calculation, computer modeling of the actual MIDA calculations is easily made under programmed conditions of isotopic disequilibrium to test potential limitations of the method (Fig. 1). The value for fractional GNG that would be calculated is within 10% of the true value even if isotopic disequilibrium to the extent of ±40-45% exists. Only at extreme disequilibrium values (e.g. 2-fold or greater differences in isotope enrichment between the triose phosphates) are calculations substantially altered. Also, the effect is always to overestimate f, never to underestimate it. It is apparent that the effect of isotopic disequilibrium is minor for this dimer unless extreme disequilibrium conditions are present. Moreover, measured disequilibrium within the triose-P pool can be corrected for in the calculation algorithm, simply by inserting the measured ratio between the two pools (p2/p1) and revising the formulae into a quadratic rather than a binomial form. Individualized standard curves can be generated for each sample by use of the revised algorithm (not shown). Thus, by measuring the degree of isotopic disequilibrium within the triose-P pool (see accompanying paper) and making corrections if differences are documented, the MIDA approach can be modified to remain applicable with equal theoretical rigor even if the assumption of isotopic equilibrium is not met. Several strategies might be pursued for testing whether 13CO2 reincorporation is a practical problem. The presence of excess triple-labeled glucose molecules would be supportive evidence (representing combination of an M2-triose with an M1-triose), but the probability of this event is small, and analytic imprecision might prevent its accurate quantification. Another approach would be to compare results from [1-13C]lactate to [3-13C]lactate (see previous paper), since double-labeled phosphoenolpyruvate cannot result from 13CO2 addition to [1-13C]pyruvate. This is an indirect approach, however, and would not answer the question definitively for [3-13]lactate or [1-13C]glycerol. The most direct approach to this question is to look for double-labeling in the triose portion of the glucose molecules analyzed. The observation of a higher than expected abundance of M2 isotopomers in either half of the glucose molecule would suggest that 13CO2 reincorporation into already labeled triose units is an important problem for the measurement of GNG by MIDA. This last strategy can be achieved by first reducing glucose to sorbitol, to remove the necessity of analyzing separate fragments for carbons 1-3 and 4-6 of glucose, and then calculating the abundance of M2-isotopomers that is expected due to combination of a labeled 13C atom with a natural abundance 13C or 2H atom. Any difference between this expected value for M2 in the triose fragment and the observed M2 for the triose fragment reveals excess (double-labeled) triose-P subunits. We calculated the fractional abundances expected for M0, M1, and M2 in the triose fragment of sorbitol due to two molecular species: unlabeled units (all atoms in the triose unit at natural isotope abundances only) and single-labeled units (one labeled 13Catom present, with the remainder of the atoms in the triose unit at natural abundances). If M2-isotopomers remained that were not accounted for by these two molecular species, it would imply that a third species (double 13C-Iabeled triose units) must also be present. The M0-M2 isotopomers attributable to unlabeled (all natural abundance) units are easily calculated, by starting with the measured abundance of the M0 isotopomer, since an M0 can only be present if the molecule contains no labeled 13C atoms. Each M0 isotopomer implies a corresponding abundance of M1 and M2 isotopomers on the basis of natural abundance distributions alone, which can be calculated. This M1 and M2 represents the values from unlabeled (natural abundance) units. For animal G1 (see Table II and legend for Table VI of accompanying paper), for example, measured M0 was = 0.8011, which implies that natural abundance M1 = 0.8011 × 0.1034 = 0.0828 and natural abundance M2 = 0.8011 × 0.0171 = 0.0137. Subtracting these unlabeled (natural abundance) M1 and M2 values from measured values of M1 and M2 reveals M1 and M2 from all 13C-labeled species. In the example, the difference between measured and unlabeled M1 = 0.1775 - 0.0828 = 0.0947 and for M2 = 0.0214 - 0.0137 = 0.0077. This observed distribution of M1 and M2 fragments is then compared with the expected distribution if labeling were due only to the presence of a single 13C atom. The expected ratio of M1/M2 is 1:0.0911, if due only to the presence of one 13C atom with all the other atoms at their natural isotope abundances. In the example, expected M1 = 0.0938; M2 = 0.0087. When these values are compared with observed values, it is apparent that the measured EM2 attributable to label incorporation (0.0077) is not higher than the expected EM2 due to the presence of a single 13C atom (0.0087). In fact, in this example, the measured EM2 value was slightly (0.0010) lower than the expected value. The presence of double-labeled triose subunits can be evaluated in this manner. An important practical issue to address is the sensitivity of derived parameters to experimental error, in particular, the question of whether analytic error is amplified in the derived parameters. The effect of measurement error on calculated parameters was evaluated at different levels of precursor pool enrichments (Fig. 2). This was done by modeling the effect of an analytic coefficient of variation (cv)for the most sensitive measurements, in this case EM2 or the EM2/EM1 ratio. The key question is whether propagation of analytic error causes significant error amplification in the final parameters. Fig. 2 demonstrates that the degree of error amplification is dependent upon the precursor enrichment achieved. Thus, at p = 0.20, a 1.7% error in the EM2/EM1 ratio results in a 2.6% change in p and in a 1.9% change in f, only 1.54- and 1.14-fold error amplifications, respectively. At p = 0.15, which was the precursor enrichment achieved from [2-13C]glycerol in these studies (see Tables I and II and Fig. 3 of accompanying article), error amplification was 1.95-fold for p and 1.58-fold for f. At p = 0.08, which was the precursor enrichment achieved from [3-13C]-lactate (Table I, Fig. 3 of accompanying article), error amplification increased to 3.38-fold for p and 3.05 for f. At p = 0.05, error amplification is 5.21-fold for p and 4.90-fold for f. What does this analysis mean in practical terms? Since cv values of <1% for EM2 and ≪1% for EM1 were routinely achieved in these studies by application of the analytic principles noted (Table II, of previous paper), a 1.58-fold error amplification (at p = 0.15) will result in a cv in f < 2%, due to analytic error. A cv of 2-3% would be observed at p = 0.08. Thus, if true f is 80%, measured values might vary between 79 and 81% (at p = 0.15) or 78-82% (at p = 0.08) due to analytic imprecision. The finding that [2-13C]glycerol was a much more effective way to label the triose-P pool (see previous paper) is relevant in this context, since error sensitivity falls as p increases. It is also clear that analytic precision is best kept high (i.e. cv in the most sensitive measurement kept within a maximum of 3-5%) in order to reduce uncertainty in derived parameters.