In Vitro Modeling of Fatty Acid Synthesis under Conditions Simulating the Zonation of Lipogenic [13C]Acetyl-CoA Enrichment in the Liver*[boxs]

In the companion report (Bederman, I. R., Reszko, A. E., Kasumov, T., David, F., Wasserman, D. H., Kelleher, J. K., and Brunengraber, H. (2004) J. Biol. Chem. 279, 43207-43216), we demonstrated that, when the hepatic pool of lipogenic acetyl-CoA is labeled from [13C]acetate, the enrichment of this pool decreases across the liver lobule. In addition, estimates of fractional synthesis calculated by isotopomer spectral analysis (ISA), a nonlinear regression method, did not agree with a simpler algebraic two-isotopomer method. To evaluate differences between these methods, we simulated in vitro the synthesis of fatty acids under known gradients of precursor enrichment, and known values of fractional synthesis. First, we synthesized pentadecanoate from [U-13C3]propionyl-CoA and four gradients of [U-13C3]malonyl-CoA enrichment. Second, we pooled the fractions of each gradient. Third, we diluted each pool with pentadecanoate prepared from unlabeled malonyl-CoA to simulate the dilution of the newly synthesized compound by pre-existing fatty acids. This yielded a series of samples of pentadecanoate with known values of (i) lower and upper limits for the precursor enrichment, (ii) the shape of the gradient, and (iii) the fractional synthesis. At each step, the mass isotopomer distributions of the samples were analyzed by ISA and the two-isotopomer method to determine whether each method could correctly (i) detect gradients of precursor enrichment, (ii) estimate the gradient limits, and (iii) estimate the fractional synthesis. The two-isotopomer method did not identify gradients of precursor enrichment and underestimated fractional synthesis by up to 2-fold in the presence of gradients. ISA uses all mass isotopomers, correctly identified imposed gradients of precursor enrichment, and estimated the expected values of fractional synthesis within the constraints of the data.

In the companion report (1), we demonstrated that the mass isotopomer distributions (MID), 1 of fatty acids and sterols iso-lated from (i) livers of conscious dogs infused with tracer [1,2-13 C 2 ]acetate in the portal vein, and (ii) rat livers perfused with 10 mM [1,2-13 C 2 ]acetate, are not compatible with a constant enrichment of lipogenic acetyl-CoA across the liver lobule. We concluded that gradients of precursor enrichment occur even in the presence of flooding [1, 2-13 C 2 ]acetate concentrations. This probably results from the inverse zonations (2) of the activities of glycolytic (2)(3)(4)(5) and lipogenic enzymes (7-9) (perivenous Ͼ periportal) versus the activity of cytosolic acetyl-CoA synthetase (periportal Ͼ perivenous) (10). Gradients of precursor enrichment were detected using isotopomer spectral analysis (ISA) (11). In addition, we found that fractional lipogenesis calculated by the two-isotopomer method (an algebraic method similar to that described by Chinkes et al. (12)) produces lower estimates of fractional synthesis than those produced by the best fit estimates of ISA. Although the "linear gradient" ISA model (see companion report (1) for model definitions) yielded a better fit than the "Single pool" ISA model, it was not possible to evaluate the effect of gradients on estimates of fractional synthesis. Thus, we could not quantitatively evaluate the performance of the ISA in comparison to the two-isotopomer method, because the true rates of fractional synthesis in the liver dog and rat liver perfusion study were unknown.
The goal of the present study was to evaluate the differences in estimates of precursor enrichment and fractional synthesis calculated by the two-isotopomer method and ISA. We used an experimental model where both the gradient in precursor enrichment and the fractional synthesis are known. This was accomplished by in vitro preparations that simulated the zonation of acetyl-CoA enrichment. Lipogenesis from sub-populations of hepatocytes across the liver lobule was simulated, in parallel incubations, by synthesizing a fatty acid using purified fatty acid synthase (13,14) and [U-13 C 3 ]malonyl-CoA of varying enrichment. We used gradients of malonyl-CoA enrichment, because fatty acid synthesis involves the conversion to malonyl-CoA of all acetyl units added to the primer. We used [U- 13 C 3 ]propionyl-CoA as a primer to avoid the possibility of contamination of our newly synthesized pentadecanoate with unlabeled pentadecanoate. In the presence of unlabeled malonyl-CoA, the process yields M3 2 [13,14,15-13 C 3 ]pentadecanoate. By monitoring the distribution of M3 to M15 isotopomers of pentadecanoate, we simulated in vitro the polymerization of six [ 13 C]acetyl units into a C-12 fatty acid, for multiple values of acetyl enrichment. Our goal was to simulate lipogenesis as it occurs in a real liver (i) under gradients of acetyl-CoA 13 C enrichment and (ii) in the presence of unlabeled lipids. To achieve this goal, we monitored the MID of pentadecanoate from (i) sets of incubations with progressively decreasing malonyl-CoA enrichments, (ii) pools of incubations from each set, and (iii) pools of incubations spiked with increasing amounts of "unlabeled" [13,14,15-13 C 3 ]pentadecanoate. The data were analyzed by the two-isotopomer method and by ISA (11,15 13 C 3 ]propionyl-CoA were prepared from the corresponding acids and purified as reported previously (16,17). Fatty acid synthase was isolated from livers from rats that were first starved for 2 days then re-fed with a high glucose diet for 2 days (13). The enzyme was precipitated with ammonium sulfate from the effluent of an Ultragel AcA-34 column, and the suspension was kept frozen in small aliquots at Ϫ80°C. The enzyme was used as an ammonium sulfate suspension (1 unit/ml).

In Vitro Synthesis of Pentadecanoate
Theory-The protocol was conceived to simulate decreasing gradients of 13 C enrichment of lipogenic acetyl-CoA across the liver lobule. Fatty acid synthesis involves the addition to a primer molecule (usually acetyl-CoA) of malonyl-CoA molecules formed by carboxylation of acetyl-CoA. Thus, gradients of acetyl-CoA enrichment can be reflected by gradients of malonyl-CoA enrichment. Because we wanted the acetyl units added to the primer to be labeled on both carbons, we created gradients of [U-13 C 3 ]malonyl-CoA enrichment. In the process of fatty acid synthesis, carbon 3 of [U-13 C 3 ]malonyl-CoA is lost as 13 CO 2 . Four protocols were followed to generate gradients of malonyl-CoA enrichment within four series of incubations. For three series of 15 incubations each, the gradients of M3 enrichment of malonyl-CoA decreased from 65% to 10% with the three profiles shown below in Fig. 1 (continuous lines). Note that the range of values for in vitro gradients from 65% to 10% was not chosen randomly. We observed a similar range of precursor enrichments in our in vivo models (see companion report (1)). In a fourth series of seven incubations, the M3 enrichment of malonyl-CoA decreased linearly from 10% to 1.0%. Control incubations were conducted with unlabeled malonyl-CoA and resulted in the formation of [13,14,15-13 C 3 ]pentadecanoate. The latter represents an unlabeled species, because it was prepared from unlabeled malonyl-CoA. In our simulation of liver lipogenesis, [13,14,15-13 C 3 ]pentadecanoate also represents the pre-existing, unlabeled fatty acid, which dilutes the MID of the newly synthesized labeled fatty acid. When pentadecanoate synthesis is conducted with 97% enriched [U-13 C 3 ]propionyl-CoA and [U-13 C 3 ]malonyl-CoA of various enrichments, the MID of pentadecanoate ranged from M3 up to M15 (Fig. 2).
Incubations-For each set of incubations, we prepared 15 or 7 solutions of malonyl-CoA of decreasing enrichment by mixing high-performance liquid chromatography-standardized stock solutions of unlabeled and M3 malonyl-CoA. To verify the malonyl-CoA enrichments and the shape of the gradients, aliquots of all mixed malonyl-CoA solutions were hydrolyzed in alkali, neutralized, and treated to form the tertbutyldimethylsilyl derivative of malonate, which was assayed by GC-MS (17).
Each incubation included 35 nmol of [U-13 C 3 ]propionyl-CoA, 100 nmol of malonyl-CoA, 600 nmol of NADPH, 0.02 unit of fatty acid synthase, in 2 ml of 0.2 M potassium phosphate buffer, pH 7.0. After 1 h of incubation at 37°C, each incubation medium in a given series was split evenly between two tubes. To one set of tubes, we added 15 nmol of [-2 H 3 ]myristate (14:0) internal standard before deproteinization with sulfosalicylic acid, extraction of fatty acids, derivatization with pentafluorobenzyl bromide, and ammonia-negative chemical ionization GC-MS assay (see companion report (1)). The analyses yielded the amount and MID of pentadecanoate synthesized in each "fraction." The other halves of all incubations of each series were pooled to simulate the extraction of a real liver and the mixing of fatty acids synthesized in all cell sub-populations. Then, the pool was redistributed into a new set of 11 tubes (1 ml/tube) to which we added increasing amounts of unlabeled [13,14,15-13 C 3 ]pentadecanoate (0 -75-nmol by 7.5-nmol increments) to simulate the dilution of newly synthesized labeled fatty acids by endogenous unlabeled fatty acids. Also, 15 nmol of [-2 H 3 ]myristate internal standard was added to each tube. Then the samples were treated for GC-MS analysis as above. GC-MS analyses of pentadecanoate, derivatized with pentafluorobenzyl bromide (m/z 244 -256), were conducted as described in the companion report (1), except that the amounts of pentadecanoate synthesized in the incubations were calculated using a standard of [-2 H 3 ]myristate (m/z 230). Derivatization of fatty acids with pentafluorobenzyl bromide (18) was selected, (i) because of the sensitivity of the negative chemical ionization assay and (ii) because the pentafluorobenzyl group splits off the fatty acyl group in the ion source and, thus, does not contribute to the MID of pentadecanoate. All analyses were run with double injection.
Calculations-Calculations of the parameters of fatty acid synthesis (precursor enrichment and fractional synthesis) were conducted using the two-isotopomer method and two variants of isotopomer spectral analysis ("Single pool" and "Gradient") that assume that the precursor enrichment is either constant, or follows a gradient of the shape defined by the model's equations (11,15).
The following four gradients were set up for in vitro simulation: Linear, Convex, Concave, and Low linear. For the large linear gradient, each synthesized sample was prepared with the precursor fractional abundance D(c) given by the following relationship, where c is an integer ranging from 0 to 14 where k specifies the degree of nonlinearity of the concave and convex gradients. k was set to 5. ISA Models for Gradients-A key feature of ISA is that it uses all measurable isotopomer data to find the best fit of model to data. As originally designed (11), ISA solves for two unknown parameters, the precursor enrichment, D, and the fraction of new synthesis at the time of sampling g(t). However, the nonlinear regression feature of ISA allows for models with additional parameters. First, gradients in precursor enrichment are modeled via ISA in discrete steps. We use 15 steps to model the gradients for ISA computed exactly as for the in vitro synthesis procedure described above. For each step of the gradient a different value is used for the precursor enrichment, D(c), as indicated by the equations above. The gradient is created by combining the values for all isotopomers for the 15 steps of the gradient and computing the fractional abundances for the combined gradient. Second, as with the conventional form of ISA, the program compares the fractional abundance values for isotopomers between data and model by calculating the weighted sum of square errors. The program searches for the best fit values of the three parameters, D min , D max , and g(t) yielding the smallest error using the Levenberg-Marquardt algorithm (11). ISA requires no correction for natural 13 C abundance, which is included in the model. A spreadsheet is included in the Supplementary Materials to demonstrate how the gradient ISA fractional abundances are created. The algebraic equations describing the steps of the gradient were developed with the assistance of the symbolic algebra facility of Mathcad (Maple) (Mathsoft, Cambridge, MA). Although the spreadsheet provides sample calculations, it does not have the capacity to perform the complete ISA calculations. The ISA program requires additional modeling that is not available in Excel; this allows finding the best-fit solution for all isotopomer equations simultaneously. For additional details about the ISA program, contact one of us (J. K. K.).
The Two-isotopomer Method-We used the following two-isotopomer equations to compute precursor enrichment, p, and fractional synthesis, f, using the notation of Hellerstein (19), where M i is the intensity of the signal for various isotopomers corrected for natural abundance. 2 These equations are identical to those de- scribed in the companion report (1) except that they are adjusted for pentadecanoate synthesized from unlabeled malonyl-CoA and six labeled acetyl-CoA molecules. Note that the equation for p is a function of the relative intensities of M 5 and M 7 . There is no need to divide each isotopomer intensity by that of the unlabeled M or by the sum of all isotopomers as these factors cancel out. However, the equation for f requires that the intensity at M 5 be divided by the sum of all isotopomers, ⌺M. These equations are based on probabilities and differ slightly from those proposed by Chinkes et al. (12). Note that the two-isotopomer approach requires that the data are first corrected for the natural abundance of carbon and other atoms in the mass ion analyzed. These corrections have proved to be nontrivial (21). A spreadsheet included in the Supplementary Material demonstrates the validity of the two-isotopomer method for ideal, error-free, data using any adjoining two isotopomer pair. The spreadsheet includes algebraic equations developed with Mathcad and demonstrates the agreement between the algebraic equations and idealized data.
Comparison of the Methods-To compare the two methodologies (gradient form of ISA and two-isotopomer method), we created error-free mass isotopomer spectra of pentadecanoate from given precursor enrichment and fractional synthesis parameters. Then we computed the latter parameters based on the ideal MID using ISA and the twoisotopomer method.

RESULTS AND DISCUSSION
The computations of the parameters of the synthesis of biopolymers from the analysis of mass isotopomer patterns are based on solid considerations of combinatorial analysis. This has been demonstrated in test tube experiments where polymeric compounds were synthesized from monomers labeled with heavy atoms ( 13 C, 2 H, 15 (24), acetone labeled from deuterated water by keto-enol tautomerism (25), and trimethylphosphate labeled from 18 O-labeled water (26). In all these cases, precise values of precursor enrichment were calculated. As a result, an important application of mass isotopomer analysis is the assay of the low isotopic enrichment of compounds that can be polymerized into a compound assayable by GC-MS (23, 25-28).
The usual application of mass isotopomer analysis to the synthesis of polymers in live cells assumes that the precursor enrichment is identical in all cells and does not change with time. The metabolic zonation of the liver (1), resulting from the organ's lobular architecture that functions as a plug-flow reactor (29), poses particular challenges to the measurement of fractional synthesis of biopolymers by MID analysis using this assumption. This is because each cell along the liver lobule is in contact with blood of continuously changing composition in terms of substrate concentrations and isotopic enrichment of tracers. Several studies reported that the concentration and enrichment of glycerol (30,31), NH 4 ϩ (32), and acetate (20) markedly decrease across the liver. In addition, the activities of enzymes involved in the synthesis of biopolymers also vary across the lobule. For example, there is an inverse zonation of the enzymes, which fuels lipogenesis (glucokinase (3) and ATPcitrate lyase and fatty acid synthase (7-9)) and cytosolic acetyl-CoA synthetase (which introduces label from [ 13 C]acetate in the lipogenic pathway (10)).
In the companion report (1), we demonstrated the existence of translobular gradients of enrichment of lipogenic acetyl-CoA (labeled from [1,2-13 C 2 ]acetate) in the livers of live dogs and in perfused rat livers. In the latter animal preparations, ISA indicated the presence of a gradient of acetyl-CoA enrichment even in the absence of gradients of acetate concentration and enrichment across the liver. The MIDs of fatty acids isolated from the various livers were analyzed by the two state-of-theart computation techniques, i.e. the two-isotopomer method (modified from Chinkes (12)) and by ISA (11). The two-isotopomer method, like the more widely applied MID analysis method (19,33,34), assumes that the precursor enrichment is constant in all cells. ISA allows for either constant or variable precursor enrichment, which is calculated from a large number of isotopomer fractional abundances (versus two abundances in the two-isotopomer method). The regression-based ISA approach allows for comparing the fit of different models to the same set of data. However, a constraint of ISA is the requirement for multiple mass isotopomers of the polymer. Thus, compared with the two-isotopomer method, ISA requires either highly enriched precursors or a sufficiently long incubation so that a sufficient number of isotopomers is detected to test for the occurrence of gradients.
In this study, we modeled the zonation of enrichment of lipogenic acetyl-CoA in liver by setting up four sets of incubations in which pentadecanoate was synthesized from [U-13 C 3 ]propionyl-CoA and lots of [U-13 C 3 ]malonyl-CoA of decreasing enrichment. Each incubation simulates a population of hepatocytes, which synthesizes pentadecanoate from a pool of malonyl-CoA of defined enrichment. Each set of incubations simulates one gradient of precursor enrichment of a given shape across the liver lobule. We isolated and measured each of the four gradients of [U-13 C]malonyl-CoA enrichment used in the study. Fig. 1 (symbols) shows measured gradients of precursor enrichment. Fig. 1 (continuous lines) shows the expected shape of the corresponding gradients. It is evident that data points that fit well with the predicted curves (regression of all data has r 2 ϭ 0.99). Fig. 2 illustrates the influence of varying the enrichment of M3 malonyl-CoA on the MID of pentadecanoate synthesized from M3 propionyl-CoA. As expected, the profile of mass isotopomers of pentadecanoate shifts to the right as the enrichment of malonyl-CoA increases. The small amounts of M to M2 isotopomers correspond to traces of natural pentadecanoate present in the fatty acid synthase preparation. The priming of pentadecanoate synthesis with M3 propionyl-CoA avoids the interference of the M to M2 isotopomers of pentadecanoate with the computations. This is why, in the context of the present study, we consider M3 pentadecanoate prepared from naturally labeled malonyl-CoA as an unlabeled species.
To test whether the MIDs of newly synthesized pentadecanoate match the theoretical distributions of mass isotopomers, we plotted these theoretical distributions in Fig. 3 (continuous lines) and superimposed the measured MIDs of six samples of pentadecanoate synthesized from M3 propionyl-CoA and lots of M3 malonyl-CoA of increasing enrichments. Each symbol corresponds to a lot of pentadecanoate made from a given malonyl-CoA enrichment. The MIDs used for Fig. 3 were taken from the linear gradient experiment (from 65% to 10%). Note that the symbols of each set of data points fall on the theoretical curves. This was expected, because in each incubation, fractional pentadecanoate synthesis is 100%. Indeed, when the data of all individual incubations from the four gradients are computed using the two-isotopomer method, the average fractional synthesis is 0.96 with a coefficient of variation of 3.7%. (n ϭ 51). This reflects the precision of our measurements of isotopomer distributions.
In each of the 51 incubations, we calculated the enrichment of the malonyl-CoA precursor from the measured MIDs of pentadecanoate, using both ISA and the two-isotopomer method. In the case of ISA, we used the "Single pool" model that assumes constant precursor enrichment. Fig. 4 shows the excellent agreement between (i) the enrichments of malonyl-CoA calculated using either the two-isotopomer method (Fig. 4A) or ISA (Fig. 4B) and (ii) the actual (measured) enrichments of malonyl-CoA (96%, CV ϭ 3.7%, for both models). This confirms that the MIDs of pentadecanoate were measured under optimal conditions. In the second phase of the experiment, we simulated the extraction of a real liver by pooling equal aliquots of all incubations from each gradient. The MIDs of the four "pools" were analyzed by the two-isotopomer method and by ISA. In the latter, we used the "Gradient" model, which assumes a range of precursor enrichments. From the conditions imposed in the protocol, we knew the exact limits of the gradients of precursor enrichment (D max and D min ), as well as the average values of precursor enrichments (D average , Table I, columns 2, 4, and 6). We also knew that fractional synthesis was 100% in each pool.
We had therefore valid reference values for the parameters to be calculated from the two-isotopomer method and the "Gradient" ISA model. The latter model yielded correct limits for all imposed gradients (Table I, compare columns 2 and 4 to columns 3 and 5). The values of precursor enrichment calculated from the gradient ISA model were then used to determine the MIDs of pentadecanoate in each pooled gradient, assuming precursor enrichments from 65% to 10% or to 1% (Low linear gradient). For each gradient, the ISA program was given the gradient shape and the value of k. The dark bars of Fig. 5 show the measured MIDs of pentadecanoate in each of the four pooled gradients. The white bars of Fig. 5 show the tight fitting of the experimental MIDs to the gradient model of ISA.
For all gradients, the two-isotopomer method computations yielded only a single value of precursor enrichment, p. In the case of the three large gradients (Concave, Convex, and Linear), p was substantially below the average value (D average ). For the Low linear gradient, p was higher than D average (Table I, last column). Table II shows the values of fractional synthesis computed by ISA (single-pool model) and the two-isotopomer method from the MIDs of each mixed pool. The expected 100% value of fractional synthesis was slightly underestimated (Ͻ3.5%) by ISA and substantially underestimated (25% to 54%) by the two-isotopomer method. ISA (D, bottom panel, B). Data from all four gradients are used to construct the graphs and compute the linear regressions.

FIG. 4. Correlations between measured enrichments of [U-13 C 3 ]malonyl-CoA used for in vitro pentadecanoate syntheses and the precursor enrichments computed from the two-isotopomer method (A) or
In the third phase of the experiment, the four pools of pentadecanoate were increasingly diluted with known amounts of unlabeled M3 pentadecanoate to simulate the dilution of newly synthesized fatty acids by the pre-existing hepatic fatty acids. Therefore, the limits of the gradient, D max and D min , were known as was the expected fractional synthesis, which equaled 1 minus the fraction of the unlabeled pentadecanoate added to the sample. To put the range of fractional synthesis into perspective, note that values of fractional synthesis reported in in vivo experiments are below 10% under normal dietary conditions (see Refs. 12 and 33 and Table I of the companion report (1)) but can exceed 40% under hypercaloric diets (35, 36). Each diluted sample was analyzed by GC-MS, and the MID was analyzed by ISA and by the two-isotopomer method. Each  panel of Fig. 6 shows (i) the values of precursor enrichment (D max and D min ) computed from the ISA gradient model (solid and open triangles), and (ii) the single value of precursor enrichment computed by the two-isotopomer method (solid squares) plotted for each of the 11 fractions. These measurements were performed using [-2 H 3 ]myristate as an internal standard. Note that fraction 1 represents the most diluted sample (ϳ20-fold), whereas fraction 11 represents the least undiluted sample ("Pool"). The dotted line represents the average precursor enrichment (D average ) for each gradient (also shown in Table I). For all gradients, the ISA program was able to identify and compute the expected gradients of precursor enrichment despite the progressive dilution. The two-isotopomer method only computed a single value of precursor enrichment that was close to the D average value. Note that the detection of a gradient with ISA requires at least three independent isotopomer fractional abundances above background or a total of four isotopomers, because the fractional abundance of one isotopomer is not independent but equals 1 minus the sum of all other fractional abundances. Three isotopomer fractional abundances are required to yield three independent equations to estimate the three parameters, D max , D min , and g(t).
When the four gradient pools were diluted by increasing amounts of unlabeled pentadecanoate, the expected fractional synthesis value for each set of diluted samples was known. The MIDs of pentadecanoate in each dilution were analyzed by both ISA and the two-isotopomer method. We plotted the values of fractional synthesis calculated by either the two-isotopomer method or ISA against the expected values of fractional synthesis and performed linear regressions to evaluate the goodness of the correlation (Fig. 7). The expected slope in each of these correlations equals 1.0. In each panel of Fig. 7, the points of both data sets lie precisely on the regression line as shown by the values of r 2 ϭ 0.99. However, Fig. 7 shows that, in the case of the two-isotopomer method, the slope is less than expected (slopes range from 0.74 to 0.46). This demonstrates that the two-isotopomer method consistently underestimated fractional synthesis up to 2-fold. The r 2 value of 0.99 indicates that the underestimate occurs across the whole range of fractional synthesis. In contrast, the ISA method correctly determined the expected values of fractional synthesis (slopes ϭ 0.96 -0.99).
The ISA method was further tested to determine if it could detect the gradient shape. Given the choice of a strongly convex or concave gradient (k ϭ 5) or a linear gradient, ISA found the best fit with the correct gradient shape when the fractional synthesis was greater than 10% and D min Ͼ 0.1. However, in the case where both the precursor enrichment and fractional synthesis rate are low, the number of detectable isotopomers declines, the background noise in the mass spectra becomes substantial, and gradients shapes cannot be distinguished. Our test of gradient detection cannot cover the many possible gra-dients shapes that may exist in vivo. However, we present this comparison of gradients to demonstrate the unique capabilities of ISA that allow the detection of gradients.
To illustrate how the two-isotopomer method differs from ISA, theoretical data for the fractional abundance of isotopomers of pentadecanoate synthesized from propionyl-CoA [U-13 C]acetyl-CoA were generated using ISA probability-based equations. First, consider ideal, error-free data, generated mathematically from probability equations with pre-set parameters (D ϭ 0.5 and g(t) ϭ 0.7; Fig. 8A). The ISA model requires an equation for the fractional abundance of each isotopomer as a function of the parameters D and g(t) (11). If T n and N n indicate the fractional abundance of tracer and natural acetate of mass, n, the equation for the M0 isotopomer of a polymer of six acetate units given in Equation 6.
This equation contains two unknowns, D and g(t). Plotting all pairs of D and g(t) values that yield a specific value or M0 produces a line. Each of the continuous lines of Fig. 8A represents a mass isotopomer of pentadecanoate generated from ideal, error-free, data where the MIDs of pentadecanoate were calculated with D ϭ 0.5 and g(t) ϭ 0.7. Because the curves were generated from the ideal data, it is not surprising to find that all continuous lines intersect at a sharp point defined by the imposed parameters D ϭ 0.5 and g(t) ϭ 0.7. Both ISA and the two-isotopomer method described above will find this solution. Thus, for ideal data, there is no difference between the two methods.
Next, consider mathematically generated data of fractional abundances assuming a linear gradient of precursor enrichments ranging from D max ϭ 0.65 to D min ϭ 0.1 with g(t) ϭ 0.7. The range of possible solutions for each isotopomer is plotted for a model assuming a single value for D (Fig. 8B). Note that the continuous isotopomer lines no longer intersect at a single point. This is because the data are not consistent with a model that assumes a single value of D. The two-isotopomer method can find a solution by choosing any of the intersection of pairs of isotopomer lines. However, the "solution" for each pair of adjoining isotopomer lines is different and none are the statistical "best fit." For example, the M5/M7 pair of lines cross at p ϭ 0.297 and f ϭ 0.495, which is identical to the solution found using the two-isotopomer equations for pentadecanoate presented above. However, these values for p and f do not produce a good fit to the entire isotopomer spectrum as shown in Fig. 8C. Although they fit M5 and M7 perfectly, this fit overestimates M0 and underestimates masses greater than M7. The effect of this forced fit to M5 and M7 is an underestimation of fractional synthesis computed by the twoisotopomer method. The ISA approach to data is different, because it uses all of the isotopomer data and a regression approach that provides an estimate of how well the model fits the data. In the example of a gradient of precursor enrichment illustrated in Fig. 8B, the ISA model uses a constant value for D to estimate for D ϭ 0.376 and g(t) ϭ 0.628. Also, ISA produces a statistical evaluation of the computation, indicating that this solution is a poor fit of model to data characterized by a large sum of squares error, 0.005. If a linear gradient ISA model is used instead, this model finds a solution (D max ϭ 0.65, D min ϭ 0.1, and g(t) ϭ 0.7) with a very small error, less than 10 Ϫ14 . Thus, the linear gradient model is a better choice for these data. Fig. 8B also demonstrates that all two-isotopomer equations using adjoining pairs of isotopomers, such as M7/M9 and other pairs, underestimate fractional synthesis when a gradient in precursor enrichment is present. Thus, the poor outcome of the two-isotopomer  (34) acknowledged that gradients in precursor enrichment can be detected by variations in the estimated precursor enrichment as shown in Fig. 8B. Lee et al. (37) also noted that the gradients of precursor enrichment lead to incorrect estimates of fractional synthesis. Here, we show that ISA can both detect gradients and correctly estimate precursor enrichment and frac- tional synthesis when the data are of high quality, i.e. being close to the theoretical values and providing sufficient number of detectable mass isotopomers. However, note that all reported in vivo measurements of fatty acid and sterol synthesis labeled from [ 13 C]acetate (12, 19, 33, 36, 38 -41) were conducted under conditions that yield low precursor enrichment, and thus low abundances of heavy mass isotopomers of the biopolymers. These conditions are similar to the in vitro synthesis conditions simulated by our Low linear gradient (D max ϭ 0.1 (Fig. 1)). Indeed, the MID of pentadecanoate synthesized in this gradient includes only low abundances of light mass isotopomers with two or four 13 C atoms derived from [U-13 C 3 ]malonyl-CoA (bottom panel of Fig. 5). In most in vivo studies of fatty acid and sterol synthesis, polymers are labeled from plasma [ 13 C]acetate of low enrichment and are diluted by the abundant endogenous species. Under these conditions, the sampled polymers, i.e. fatty acids and sterols, do not include sufficient amounts of labeled mass isotopomers. Thus, one is unable to evaluate the possibility of gradients (33,36,39). However, if gradients are present, our analysis indicates that applying methods that assume constant precursor enrichment results in underestimated fractional lipogenesis.
An unresolved question is the magnitude of the gradient in vivo when the enrichment is low. Assuming that supplying the tracer at high enrichment and concentration yields a linear gradient and 5-fold change in enrichment across the liver, D max ϭ 0.5 and D min ϭ 0.1. This does not imply that tracer supplied at low enrichment to yield a D max of 0.05 would also produce a linear gradient with D min ϭ 0.01. If the liver extracts acetate vigorously, at low enrichment, the precursor labeling may decrease to undetectable level at some point across the lobule (20). Then, the newly synthesized polymer in the downstream area of the lobule would be unlabeled and would not be distinguishable from the pre-existing polymer. This scenario would result in large underestimates of fractional synthesis. Additional studies are required to determine whether gradients at low enrichment are simply scaled versions of high enrichment gradients and whether their shape and endpoints are affected by the concentration of acetate.
What are the implications of the above findings on estimates of hepatic fractional lipogenesis by the two-isotopomer and the ISA models in the companion report (1)? In the companion study we imposed high levels of precursor enrichment in dog livers and in perfused rat livers, to ensure the presence of multiple mass isotopomers. Thus, the conditions were similar to those of the in vitro constructed gradients using D max ϭ 0.65. Also, in the companion report we provide convincing evidence that, in the presence of [ 13 C]acetate, the enrichment of lipogenic acetyl-CoA follows sharp descending gradients across the liver. The present report validates the ability of ISA to detect gradients of precursor enrichment, to estimate the limits of these gradients (Fig. 6), and to generate reliable values of fractional synthesis (Fig. 7) under gradient conditions. These validations support the conclusions of the companion report.