INTRODUCTION

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

THE ASSEMBLY AND DISASSEMBLY of polymers synthesized from repeating monomeric units is a central theme in biology. Such polymers may be as simple as fatty acids synthesized from acetyl-CoA units or as complex as proteins synthesized from amino acids or DNA made from nucleotides. Other examples include carbohydrates (e.g., glucose from triose units, glycogen from glucose, glycoproteins), porphyrins (e.g., chlorophyll, heme), and lipids (e.g., cholesterol, triacylglycerols). Biological polymers may be homonuclear (defined as containing subunits that are identical), as in fatty acids, or heteronuclear (defined as containing more than one type of subunit), as in proteins or polynucleotides. Despite the importance of polymers in the chemistry of living systems, techniques for determining their rates of synthesis or breakdown have historically been unsatisfactory (1, 9, 18, 19). As a consequence, fields as wide ranging as lipid biosynthesis, protein metabolism, carbohydrate metabolic regulation, and control of cell proliferation have been severely constrained.

In this article, we will provide an update and eight-year perspective on a technique that provides a fundamental solution to the problem of measuring polymerization biosynthesis. Mass isotopomer distribution analysis (MIDA) is a technique based on combinatorial probabilities and the labeling patterns in intact polymers that can be said to provide a fundamental "equation for biosynthesis." Although MIDA was first presented as a systematic approach to polymerization biosynthesis only a few years ago (13-15, 20), a number of refinements, alternative calculation algorithms, and criticisms have been published since then (4, 5, 22, 31, 32). We will review here the theoretical and practical factors that must be taken into account if MIDA and related techniques are to be applied successfully.

	BIOLOGICAL BASIS OF MEASUREMENT OF POLYMERIZATION BIOSYNTHESIS BY ISOTOPE INCORPORATION

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

The principle of isotope incorporation techniques for measuring polymerization biosynthesis is, on the surface, straightforward. In a biological system, polymers that are newly synthesized mix into a pool that also contains preexisting polymer molecules. The goal of an isotope incorporation study is to quantify the fraction of molecules in the mixture that were newly synthesized during the label incorporation period (i.e., "what's new") and the rate at which the total pool of polymers is turning over. To determine the newly synthesized fraction (f) present in the mixture, one must first establish exactly how much label is contained in the population of newly synthesized polymers. Dilution of this labeled population by the population of preexisting, unlabeled molecules can then be determined, according to the precursor-product relationship (15, 22, 39, 40).

The major practical difficulty has been establishing how much label is contained in the newly synthesized population of molecules. There exists no purely physical technique for identifying in a mixed population of molecules which ones are new and which are not. No classical extraction technique can reveal where different molecules in a population came from or how long they have been present. The biochemistry of the precursor-product relationship provides a possible solution, however (Fig. 1), because the precursor pool of subunits in a cell has a physical reality and can in principle be isolated by extraction techniques.

View larger version (93K): [in this window] [in a new window]   Fig. 1.   Mass isotopomer distribution analysis (MIDA) principle. A: combinatorial probabilities determine the mass isotopomer pattern in polymers. In this simulation, natural abundance or 10% labeled pools of a subunit combine into a polymer of eight subunits. The population of each of these pools will contain a characteristic distribution of M0, M1, M2, and so on, molecular species (mass isotopomers). These proportions can be represented as a frequency histogram of the mass isotopomer pattern in the polymer and can be measured by mass spectrometry. After correction for natural abundance, degree of enrichment of precursor pool can be calculated by comparing measured patterns of mass isotopomer abundances with those predicted from theoretical precursor pool enrichments. B: simple numerical example of MIDA principle. C: three-dimensional representation of change in fractional abundance (A1) of a particular mass isotopomer (in this case, M1 of methyl palmitate) as a function of p and f. A plane is extended at values of M1 = 0.05, 0.10, 0.15, 0.20 and 0.25, demonstrating the family of solutions for all combinations of p and f that give this value of A1. Inset: a 2-dimensional projection of p vs. f in the plane A1 = 0.15. Note linearity of A1 vs. f.

Serious problems arise when investigators have tried to use surrogate monomer pools to represent the isotopic content of the true precursor pool (p) (9, 19, 34, 37, 38), however: e.g., plasma amino acids or free intracellular amino acids to represent the tRNA-amino acid precursor pool for protein synthesis, or ketone bodies to represent the acetyl-CoA pool for lipogenesis. Complicating factors deriving from subcellular or intracellular biochemical organization have been shown to affect every class of polymer so far examined in detail, including proteins (38), lipids (8, 9), carbohydrates (19, 34), and nucleic acids (18).

	A SOLUTION TO THE PRECURSOR-PRODUCT PROBLEM: THE USE OF COMBINATORIAL PROBABILITIES

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

MIDA is based on a model of combinatorial probabilities. Polymerization biosynthesis can be conceptualized as a combinatorial process, with monomeric subunits from a precursor pool combining into a polymeric collection or assemblage. If the monomeric subunits are of more than one distinctive type, i.e., labeled and unlabeled, then the population of assembled polymers will not be of uniform isotopic composition. The polymers will exist as distinguishable species containing varying numbers of the different types of subunits. Some species will include no labeled subunits, some will include one labeled subunit, some will contain two labeled subunits, and so on. The relative proportion of each species of polymer is determined by and can be calculated from the binomial (or multinomial) expansion (Fig. 1A). The binomial expansion contains two variables, the number of subunits in the collection (n) and the probability (p) of each subunit being of a particular type. Because the number of subunits in a biological polymer is constant and known, the sole factor determining the relative proportions of each polymeric combination (i.e., the quantitative distribution of mass isotopomers) is p, the labeling probability in the precursor pool.

View larger version (16K): [in this window] [in a new window]   Fig. 2.   Effect of varying the natural abundance 13C value on mass isotopomer abundances and MIDA calculations for methyl palmitate. Values of f are calculated using our standard algorithm (13C = 1.09%), when sample in fact has natural abundance 13C = 1.08, 1.10, and 1.11%. Calculated values of f are shown as a function of p (calculated iteratively, at intervals of 0.001). Inset: blown-up region of graph between p = 0.05 and 0.15.

The population of intact polymeric assemblages therefore contains information about the precursor pool that is not available by analysis of the monomeric units in isolation: the combinations of labeled and unlabeled subunits in the polymer population, which are manifested for statistical or mathematical analysis as the distribution of mass isotopomers. This is the central insight on which MIDA is based. Because each isotopomeric distribution is uniquely determined by p, each distribution is characteristic of and capable of revealing the unique value of p from which it was assembled. The distribution is, moreover, immutable; it is a fingerprint that will persist throughout the lifetime of the population, as long as there is no biological discrimination (no isotope effect) between species of the polymer and no remodeling of the polymer after its original assembly.

What is the effect of mixing a population of polymers assembled from a precursor pool of labeling probability p with a population of polymers assembled from an unlabeled precursor pool? Mixing of this sort (dilution of the polymer pool) is what happens in a biological system when a labeling experiment is performed: newly synthesized polymers from the labeled pool mix with polymers that were present before the experiment began. A key mathematical feature of MIDA is that the relationships among those polymeric species that contain labeled subunits (the internal pattern among isotopomers) are unchanged by dilution from an unlabeled population of polymers (Refs. 13-15; see CENTRAL FEATURES OF MIDA SUMMARIZED).

	CENTRAL FEATURES OF MIDA SUMMARIZED

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

The first rule of MIDA is that there must be combinations possible in the molecule analyzed. At least two repeats of a probabilistically identical subunit must be present. Metabolic pathways involving other kinds of chemical transformations but no polymerization are therefore not amenable to the combinatorial approach. Polymers studied must also be analyzed intact, or with at least two subunits present, because the distribution of isotopomeric species carries the essential information. Any maneuver that reduces the population to monomeric homogeneity, such as combustion to carbon dioxide for isotope ratio measurements or hydrolysis to monomeric subunits before analysis, loses the combinatorial information and precludes application of MIDA.

The second rule of MIDA is that subpopulations of molecules must be distinguishable and quantifiable. Indeed, it is the variations within a population of assembled polymers that carry the information crucial for MIDA. The notion that there is a homogenous precursor pool and a uniform product pool is replaced by the notion of subpopulations of precursors (some A, some B) and subpopulations of products (of characteristic isotopomeric composition in quantifiable proportions). Any analytic modality must therefore be capable of discriminating among different polymeric subpopulations (species) present within the population. This is why radioisotopic methods cannot be used: specific activity is measured from the total counts and total mass of material present, treated as a uniform population; and it is why average mass measurements by electrospray ionization-mass spectrometry also cannot be used: a "centroid" average mass collapses all of the population variability in the polymer pool into a single value.

The third essential concept underlying MIDA is that dilution of the monomeric (precursor) and polymeric (product) pools affects abundance distributions differently. Both sources of dilution can alter the relative proportion of polymeric species containing no labeled subunits vs. labeled subunits, but only dilution in the precursor pool can alter the internal quantitative relationships among labeled species. It is this differential effect on "amount" (proportion of the polymer population containing any labeled subunits) vs. "pattern" (relationships within the population of labeled polymers) that allows independent calculation of p and f, respectively.

The model just described involves some simplifying assumptions, which are discussed in THEORETICAL ISSUES FOR THE USE OF MIDA.

In addition to determining p directly, MIDA offers several operational advantages over previous isotopic techniques for measuring biosynthesis rates (Table 1). One analytic consequence of the need for intact combinations is that sophisticated mass spectrometric techniques must be used in cases in which biosynthesis of high-molecular-weight polymers, such as proteins or oligonucleotides, is being measured.

Definitions

Isotopes. Atoms with the same number of protons and hence of the same element but with different numbers of neutrons (e.g., H vs. D).

Exact mass. The mass calculated by summing the exact masses of all the isotopes in the formula of a molecule (e.g., 32.04847 for CH₃NHD).

Nominal mass. The integer mass obtained by rounding the exact mass of a molecule.

Isotopomers. Isotopic isomers or species that have identical elemental compositions but are constitutionally and/or stereochemically isomeric because of isotopic substitution, as for CH₃NH₂, CH₃NHD, and CH₂DNH₂.

Isotopologues. Isotopic homologues or molecular species that have identical elemental and chemical compositions but differ in isotopic content (e.g., CH₃NH₂ vs. CH₃NHD in the example above) (36). Isotopologues are defined by their isotopic composition; therefore, each isotopologue has a unique exact mass but may not have a unique structure. An isotopologue is usually comprised of a family of isotopic isomers (isotopomers) that differ by the location of the isotopes on the molecule (e.g., CH₃NHD and CH₂DNH₂ are the same isotopologue but are different isotopomers).

Mass isotopomer. A family of isotopic isomers that is grouped on the basis of nominal mass rather than isotopic composition. A mass isotopomer may comprise molecules of different isotopic compositions, unlike an isotopologue (e.g., CH₃NHD, ¹³CH₃NH₂, CH₃¹⁵NH₂ are part of the same mass isotopomer but are different isotopologues). In operational terms, a mass isotopomer is a family of isotopologues that are not resolved by a mass spectrometer. For quadrupole mass spectrometers, this typically means that mass isotopomers are families of isotopologues that share a nominal mass. Thus the isotopologues CH₃NH₂ and CH₃NHD differ in nominal mass and are distinguished as being different mass isotopomers, but the isotopologues CH₃NHD, CH₂DNH₂, ¹³CH₃NH₂, and CH₃¹⁵NH₂ are all of the same nominal mass and hence are the same mass isotopomers. Each mass isotopomer is therefore typically composed of more than one isotopologue and has more than one exact mass. The distinction between isotopologues and mass isotopomers is useful in practice, because all individual isotopologues are not resolved using quadrupole mass spectrometers and may not be resolved even by using mass spectrometers that produce higher mass resolution, so that calculations from mass spectrometric data must be performed on the abundances of mass isotopomers rather than isotopologues. The mass isotopomer lowest in mass is represented as M₀; for most organic molecules, this is the species containing all ¹²C, ¹H, ¹⁶O, ¹⁴N, and the like. Other mass isotopomers are distinguished by their mass differences from M₀ (M₁, M₂, etc.). For a given mass isotopomer, the location or position of isotopes within the molecule is not specified and may vary (i.e., "positional isotopomers" are not distinguished).

Mass isotopomer pattern. A histogram of the abundances of the mass isotopomers of a molecule. Traditionally, the pattern is presented as percent relative abundances, where all of the abundances are normalized to that of the most abundant mass isotopomer; the most abundant isotopomer is said to be 100%. The preferred form for applications involving probability analysis, such as MIDA, however, is proportion or fractional abundance, where the fraction that each species contributes to the total abundance is used (see ). The term isotope pattern is sometimes used in place of mass isotopomer pattern, although technically the former term applies only to the abundance pattern of isotopes in an element.

Monoisotopic mass. The exact mass of the molecular species that contains all ¹H, ¹²C, ¹⁴N, ¹⁶O, ³²S, and the like. For isotopologues composed of C, H, N, O, P, S, F, Cl, Br, and I, the isotopic composition of the isotopologue with the lowest mass is unique and unambiguous, because the most abundant isotopes of these elements are also the lowest in mass (23). The monoisotopic mass is abbreviated as m₀, and the masses of other mass isotopomers are identified by their mass differences from m₀ (m₁, m₂, etc.).

Fractional abundances. The abundances of individual isotopes (for elements) or mass isotopomers (for molecules) given as the fraction of the total abundance represented by that particular isotope or mass isotopomer. This is distinguished from relative abundance, wherein the most abundant species is given the value 100 and all other species are normalized relative to 100 and expressed as percent relative abundance. For a mass isotopomer M_x where 0 to n is the range of nominal masses relative to the lowest-mass (M₀) mass isotopomer in which abundances occur. where subscript e refers to enriched and subscript b refers to baseline or natural abundance.

Isotopically perturbed. The state of an element or molecule that results from the explicit incorporation of an element or molecule with a distribution of isotopes that differs from the distribution found in nature (Table 2), whether a naturally less abundant isotope is present in excess (enriched) or in deficit (depleted).

Monomer. A chemical unit that combines during the synthesis of a polymer and that is present two or more times in the polymer.

Polymer. A molecule synthesized from and containing two or more repeats of a monomer.

	CALCULATIONS

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

A tradition in mass spectrometric applications has often been to express quantitative results as relative abundances (each species normalized to the most abundant species, which is given the value 100). In contrast, fractional abundances or analogous expressions are generally preferable for MIDA, because the method is based on combinatorial probabilities, and probabilistic events are most directly represented as fractions of the total universe of choices possible.

	THEORETICAL ISSUES FOR USE OF MIDA

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

Every model is based on assumptions, which may or may not describe real biological systems accurately. It is useful to consider and ultimately to be able to evaluate or correct for potential deviations from the simple MIDA model. For all of the simulations performed here, calculations were carried out by use of the computer algorithms described in the APPENDIX.

Effect of Variations in Natural Abundance Values of the Isotopes of Elements

View larger version (19K): [in this window] [in a new window]   Fig. 3.   Loss of linearity in the relationship between measured Ax and true f, especially at high values of p, if an incomplete ion spectrum is sampled. In this example, only ions M0 to M2 are measured in palmitate-methyl-ester at various values of p. The ratio A1/A1 (measured excess M1/asymptotic M1 values) should reflect f and rise linearly as f increases from 0 to 100% (e.g., see Fig. 1C). As p increases and the proportion of unmonitored ions rises, however, linearity is lost and true f is underestimated. Simulations were shown at data points of f.

Effect of Isotope Discrimination

Effect of Incorrect Value for Expected Number of Monomeric Subunits

Calculation of p and f When the Complete Ion Spectrum is not Sampled

A numerical example follows. If M₁ theoretically represents 20% of the M₀ through M₂ ions in natural abundance molecules and 40% of the M₀ through M₂ ions at p = 0.10, but only 90% of the envelope is contained in M₀ through M₂ at p = 0.10 compared with 100% in M₀ through M₂ for the natural abundance molecules, what would be the effect of mixing these populations and monitoring only M₀ through M₂? It should be apparent that the enriched population will not contribute the theoretical 40 ions of M₁ for every 20 ions of M₁ from unlabeled molecules in an equimolar mixture when only M₀ through M₂ are measured, but will contribute only 36 (= 40% × 90% of the mole in the envelope monitored) for every 20 (= 20% × 100% of the mole of natural abundance isotopomers). It would therefore be a mistake to use percentages of the ions monitored to represent percentages of the entire population when mixing populations with different percentages of ions monitored, because f will be systematically underestimated. The solutions to this confounding factor are straightforward: either monitor an essentially complete ion spectrum for the molecules under consideration or include a mathematical correction for unequal ion spectrum sampling in the calculation algorithm.

An algebraic correction is derived and presented in the APPENDIX (Eq. A9b) for instances of significantly incomplete ion spectrum monitoring. This equation corrects for the proportion of ions monitored at the measured value of p present relative to the proportion monitored in unlabeled molecules. By use of this correction factor, mixtures of labeled and unlabeled molecules are again reduced to linear combinations of mass isotopomers, from which dilution of molecules can be calculated simply. This correction is generally extremely small and has no practical impact on most calculations (Fig. 4), because >98% of the ions within an isotopomeric envelope are typically monitored for most labeled molecules. Failure to consider the effects of incomplete ion spectrum sampling can contribute to underestimation of values of f in special cases, however, such as very high values of p, if high masses are not monitored. The impact of incomplete ion spectrum monitoring has not to our knowledge been considered previously or corrected for in other MIDA calculation algorithms.

View larger version (19K): [in this window] [in a new window]   Fig. 4.   Error in calculation of f resulting from incomplete ion spectrum monitoring of different degrees. Three different values for true f (25, 50, 75%) are shown. Note that calculated f remains 95% of actual value as long as ~85% of the ion spectrum is monitored.

Effect of Analytic ArtifactsFragment Ions

1

1

1

Effect of Isotopic Disequilibrium in the Precursor Pool

Evaluating the theoretical impact of isotopic disequilibrium. A biosynthetic system may or may not exhibit isotopic equilibrium among different pools of a monomeric subunit from which a polymer is synthesized. Accordingly, there may not always be a single value for p. A physiologically relevant example of a biosynthetic system that does not necessarily have a single value for p is gluconeogenesis. There are actually two precursors comprising the gluconeogenic triose-phosphate pool, dihydroxyacetone phosphate (DHAP) and glyceraldehyde-3-phosphate. Isotopic equilibrium between the triose-phosphates may not always be complete. One can simulate the effect of various degrees of isotopic disequilibrium between DHAP and glyceraldehyde-3-phosphate and determine the extent to which calculated values of p and f would be distorted if the standard MIDA reference table were applied to the fractional abundances generated (27). If we allow the average p to range between 0.05 and 0.15 and vary p in pool 2 from 1.0 to 2.0 times the pool 1 value, the consequences for the f can be calculated. When p in pool 1 = p in pool 2 (i.e., when isotopic equilibrium is present), the calculated value of f is of course exactly as expected from the MIDA tables (100%), whatever the value of p. When p in pool 2 is double that of pool 1, calculated f is ~12% higher than the true value for all values of p; when pool 2 is made 50% above pool 1, f is within 4% of the true value. An interesting general mathematical result to emerge from this analysis is that f is always overestimated (>100% the actual value) if isotopic disequilibrium exists within precursor subunit pools (27). The practical implication for measuring gluconeogenesis in particular is that isotopic disequilibrium in the triose-phosphate precursor pool is in principle unlikely to represent a major problem, because the MIDA calculations will work well unless DHAP and glyceraldehyde-3-phosphate are differentially enriched by a factor of >2 (27). Similar calculations can be applied to other polymers of interest.

Correcting for documented isotopic disequilibrium. If one is not satisfied with theoretical arguments discounting the importance of isotopic disequilibrium within a precursor pool, it is possible to modify the calculation algorithm to incorporate deviations from isotopic equilibrium within precursor pools. Theoretical tables can be generated by modifying the algorithm described above, if the degree of isotopic disequilibrium is measurable. In the case of gluconeogenesis, for example, mass spectrometric fragmentation of the molecule into "top" (C-1 to C-3) and "bottom" (C-4 to C-6) halves can reveal whether labeling was equal in DHAP and glyceraldehyde-3-phosphate, respectively (26). If enrichments differ by a certain proportion, then this value can be incorporated into the probability calculations to generate an appropriate, individualized standard curve that adjusts asymptotic values appropriately. Thus individualized standard curves can be generated for each experiment on the basis of the observed degree of isotopic disequilibrium within precursor pools, if the latter is significant and can be determined experimentally. Because the mathematical approach that we have described (see APPENDIX and Refs. 13-15) is based on empirical relationships between derived values (fractional abundances) rather than expressions of the pure binomial or multinomial expansion, deviations from the simple combinatorial probability model can be accounted for relatively easily and without compromising mathematical rigor.

Effect of Isotopic Inhomogeneity in the Precursor Pool (e.g., More Than One Biosynthetic Site, an Isotopic Gradient Across a Tissue, or Time Variations in p)

The consequence of these scenarios is that, instead of a single binomial distribution, there will be a mixture of distributions from the different values of p in newly synthesized polymers. This mixture of distributions itself mixes with, and needs to be distinguished from, the natural abundance distribution that represents old molecules. Any attempt to model biosynthesis as two populations or two distributions, enriched and unenriched, will not be rigorously correct mathematically but will instead be an approximation, because binomial or multinomial expansions cannot themselves be averaged (i.e., are not linear) (21). The M₂ isotopomer changes approximately as the power of 2 for changes in p, the M₃ as the power of 3, etc., so that different expansions cannot be combined and averaged as if they were linear.

The practical questions are, how much does this matter? what impact on estimated parameters will there be if p is inconstant in time or space? and can it be identified or corrected for when present? A simulation for gluconeogenesis in two tissues or at two time points with different p values has been presented elsewhere (26): f remains 0.8-0.85, even when pool 2 enrichment is 2-3 times pool 1. Another example is a gradient in precursor pool enrichment across a tissue (Fig. 5A). If one models 10 pools contributing equally to gluconeogenesis with a labeling gradient that spans an approximately twofold range [e.g., 0.105-0.195 molar excess (ME)], the consequence is minor (underestimation of true values of f by a factor of <5%, Fig. 5A). Even for a fourfold gradient (0.06-0.24 ME), f is only underestimated by ~15%; i.e., if the actual value of f = 0.50, measured f will be 0.425. Only at very large gradients (e.g., 15-fold, from 0.02 to 0.29 ME) is even a 25% underestimation observed. An analogous situation can be simulated for lipogenesis, with an isotopic gradient of acetyl-CoA across a lipogenic tissue such as liver. If 10 pools contribute equally to lipogenesis, with a gradient from 0.03 to 0.30 ME (at intervals of 0.03), f is 87.3% instead of 100% (i.e., underestimated by 12.7%). If a gradient from 0.02 to 0.10 ME is simulated, with 100 pools contributing equally to lipogenesis, the value of f calculated by MIDA by use of M₁ and M₂ isotopomers is 90.1%. Thus inconstancy in p over time or space means that the simple binomial model becomes an approximation rather than an exact description of biosynthesis, but the practical impact varies according to physiological conditions and typically is fairly small. An investigator is able to evaluate the likelihood of significant error and the practical acceptability of this degree of error by performing a simulation of this type.

View larger version (22K): [in this window] [in a new window]   Fig. 5.   Effects of isotopic inhomogeneity in the precursor pool on calculated values of p and f. A: effect on gluconeogenesis of an isotopic gradient within a tissue. Ten sites are simulated to contribute equally to gluconeogenesis, with different degrees of isotopic gradient in the region around p = 0.15 (from 0.125-0.175 up to 0.01-0.29, at the intervals stated). B: use of high-mass isotopomer analysis to identify inhomogeneity in the precursor pool, or in this case, two precursor pools. Isotopomeric envelopes resulting in palmitate-methylester if two biosynthetic sites contribute equally, one at p = 0.05 and one at p = 0.30, are shown (left). The envelope from the summed distribution is then compared with the envelope expected for p = 0.175 (the average value) (right). C: use of high-mass isotopomer analysis to identify inhomogeneity in the precursor pool, in this case, an isotopic gradient in the precursor pool. A labeling gradient from 0.03 to 0.30 for p is compared with a homogenous pool at the value of p (0.166) calculated from the observed M2-to-M1 ratio.

It is also possible to identify, and even correct for, isotopic gradients or variations in precursor pool enrichments by application of combinatorial principles. If a large isotopic gradient is present or two precursors of different enrichment contributed to biosynthesis, there will be a divergence between the isotopomer pattern in high-mass (multiply labeled) vs. low-mass (single- and double-labeled) polymers (Fig. 5, B and C). If a gradient exists within the precursor pool, high-mass species will be produced that would otherwise never be observed if the average value of p were uniformly present. The pattern of higher-mass isotopomers predicted by analyzing lower masses will not be observed; similarly, the pattern of low-mass isotopomers predicted by the higher masses will not be met (Fig. 5, B and C).

The occurrence of "inappropriate" multiply labeled species relative to the pattern among the less-labeled species (Fig. 5, B and C) can therefore be used diagnostically to confirm or exclude the existence of an isotopic gradient. One simple approach is to monitor higher-mass isotopomers and compare calculated values of p from higher- vs. lower-mass relationships. In the case of a labeling gradient across a lipogenic tissue spanning 0.03 to 0.30 (Fig. 5C), for example, the pattern of excess M₃/M₂ isotopomers in palmitate indicates p = 0.213; the ratio in M₄/M₃ indicates p = 0.226, whereas M₂/M₁ isotopomers reveal a lower p (0.166). The divergence is even greater for simple two-precursor-pool systems (Fig. 5B). If there were an equal mixture of a palmitate population derived from 0.30 and a 0.50 value of p, analysis of M₁ and M₂ would reveal p = 0.173, whereas analysis of M₃ and M₄ would indicate p = 0.29. In contrast, if a single, homogenous precursor pool is present, calculated values of p are identical whatever masses are used for its calculation (15). The finding of different p values by analysis of high- vs. low-mass isotopomer patterns represents a technique for identifying the presence of an isotopic gradient.

Conversion of Fractional Synthesis Values into Chemical Fluxes: Combining MIDA Calculations with Administration of Exogenous Stable Isotope-Labeled Polymers

For example, it is useful to measure the plasma glucose turnover concurrently with fractional gluconeogenesis to determine the absolute rate of gluconeogenesis. If [1-²H]glucose or [6,6-²H₂]glucose is used for turnover, labeled species have the same nominal mass as the isotopomers analyzed for MIDA calculations (M₁ and M₂). Because quadrupole mass spectrometers do not have sufficient resolving power to distinguish between ²H and ¹³C exact masses, a technique for determining the contributions from [²H]glucose vs. [¹³C]triose-phosphate is required. This can be achieved by analyzing a derivative that is stripped of the labeled hydrogen (e.g., aldonitrile-pentaacetate or saccharic acid derivatives from which position 1 and 1,6 hydrogens are removed, respectively) in addition to a derivative that contains both inputs (e.g., pentaacetate). The MIDA calculation on the derivative stripped of ²H is routine. The problem, however, is how to "subtract" or correct the ¹³C contribution from the combined spectrum to establish the ²H labeling, by difference.

A calculation algorithm can be used to correct for the underlying isotopomeric distribution from incorporation of ¹³C gluconeogenic precursors, for example, to measure [²H]glucose enrichment (7). The glucose molecules present during a simultaneous measurement of fractional gluconeogenesis and glucose turnover consist of a mixture of three populations: gluconeogenic product molecules arising from the labeled triose-phosphate precursor pool, labeled glucose molecules infused exogenously as tracer, and unlabeled molecules with a natural abundance distribution. The key is that the isotopomeric distribution of each of these components is known. The infused tracer ([6,6-²H₂]- or [1-²H]glucose) has an isotopomer distribution that is easy to calculate; natural abundance glucose also has a known distribution; and the gluconeogenic population has a distribution of isotopomers that is a function of p and that is measurable from the deuterium-stripped derivative. The distributions from each of these three components of the mixture can therefore simply be added to construct a theoretical standard curve, simulating the effect of adding ²H-labeled glucose to mixtures of the other two populations at the measured f. If gluconeogenesis f = 0.33 and p = 0.15, for example, the calculation consists of adding 0.33 times the abundance of each isotopomer from gluconeogenesis at p = 0.15, then (0.67  z) times the abundance of each isotopomer from natural abundance glucose, and z times the abundance of each isotopomer from the [²H]glucose, where z is the fraction of ²H-labeled glucose added to generate the theoretical standard curve. Two or more values of z are simulated to construct a linear standard curve, wherein the isotopomers of interest are plotted against z to generate a slope and intercept for calculation of ²H enrichment and dilution in the intact molecule (Fig. 6). This algorithm must be applied separately for each time point sampled, because each sample will have a unique p and f (8, 36), and thus a unique slope and intercept for the standard curve.

View larger version (16K): [in this window] [in a new window]   Fig. 6.   Standard curve for calculation of [6,6-2H2]glucose enrichment in the presence of an underlying 13C combinatorial distribution from gluconeogenesis. In this example, the value for p is 0.15.

Assessing the Sensitivity of MIDA-Calculated Biosynthetic Parameters to Measurement Error

	ALTERNATIVE CALCULATION ALGORITHMS

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

MIDA as we now understand it was developed from 1990 to 1992 by Hellerstein and co-workers (13-15, 17) and Kelleher et al. (20), working independently. MIDA is defined by its purpose, its method, and its calculations. The purpose or field of MIDA is to measure the synthesis of biological polymers, the isotopic labeling of the monomeric precursor pool from which the polymers were assembled, and related kinetic parameters. The method involves measurement and analysis of mass isotopomer abundance distributions in intact polymers according to a combinatorial probability model, after introduction of a stable isotopically labeled monomeric subunit. The calculation approach uses the binomial or multinomial expansion as a basis for interpreting incorporation of isotopically labeled repeating subunits in the intact polymer and for inferring dilution in the monomeric and polymeric (precursor and product) pools.

It is in the area of a calculation algorithm that modifications have been presented (1, 5, 22, 32) since the original approaches described by Hellerstein and co-workers (13-15, 17) and Kelleher et al. (21). All the calculation algorithms presented so far, however, share fundamental assumptions and postulate a common model of biosynthesis. All postulate a combinatorial (binomial/multinomial) precursor-product biosynthetic model, and all postulate two confounding factors that may then modify the simple binomial/multinomial distributions: first, the natural abundance isotopes in the molecule, which interact with label-derived isotopomers; and second, the fact that two sources of dilution exist in biosynthetic systems (in the product pool as well as the precursor pool) and that these two sources of dilution influence isotopomer abundances differently in the product.

Only two general solutions have been proposed for each of the problems just noted. The influence of natural abundance isotopes on label distributions is a strictly computational problem. The solution has been either to create a computational model that incorporates all sources of isotope from both labeled precursor and natural abundance isotopes, and thus does not conform to a simple binomial distribution (Hellerstein and Neese, Ref. 15, and Kelleher et al., Ref. 20), or to transform the data in a way that removes the influence of natural abundance isotopes and restores a pure binomial distribution from the labeled precursor (Lee et al., Ref. 22, and Chinkes et al., Ref. 5). The problem of two sources of dilution, in contrast, reflects a biological issue. Again, there have been two computational solutions proposed. Most methods (Hellerstein, Lee, and Chinkes) have used a step-wise approach. Ratios among natural abundance-corrected terms are first computed. Use of internal ratios of isotopomers in the polymer sample analyzed removes the effect of varying product dilution, because all isotopomers sampled in the labeled molecules are equally diluted by natural abundance molecules, so that precursor pool dilution can be calculated independently. Once this unknown (p) is solved, the second unknown (f) can be solved algebraically (Fig. 1C). Alternatively, the two sources of dilution (Fig. 1C) can be solved simultaneously, by best fit (nonlinear regression analysis) of multiple solution sets for the two unknowns taken together (21).

The most important point is that all the calculation algorithms presented to date give essentially identical results. Thus data of Byerley et al. (4), from ²H₂O incorporation into cellular cholesterol in cultured cells, give identical values for p and f whether calculated by the method of Lee (22) or by our approach (unpublished results). Identical results are obtained when data are analyzed by the approach of Kelleher et al. (21) and ours (unpublished observations; and T. Masterson, personal communication, April 1994). Chinkes et al. (5) also compared calculation algorithms using simulated data and concluded that results were essentially identical.

It should be noted that all these approaches are of roughly equal computational complexity. They all require a computer program¹, to calculate abundances and to generate either a model or algebraic correction factors, and a specific software package that has to be applied separately for each molecule analyzed.

	ANALYTIC AND EXPERIMENTAL DESIGN CONSIDERATIONS

TOP ABSTRACT INTRODUCTION BIOLOGICAL BASIS OF MEASUREMENT... A SOLUTION TO THE... CENTRAL FEATURES OF MIDA... CALCULATIONS THEORETICAL ISSUES FOR USE... ALTERNATIVE CALCULATION... ANALYTIC AND EXPERIMENTAL... SOME FUTURE DIRECTIONS FOR... REFERENCES APPENDIX

The single most difficult problem facing the use of MIDA at present, both in theory and in practice, relates to quantitative accuracy of measurements, i.e., the analytic performance of mass spectrometers. This problem is widely recognized by workers in the field but has only rarely been noted in the literature (3, 10, 11, 23, 24, 26, 30). MIDA is based on analysis of numerical distributions in the context of a model of combinatorial probabilities. If the instrument generates inaccurate numbers, measured distributions will no longer reflect the actual isotopomeric distributions present. The actual effect on kinetic estimates (p, f) will depend on the nature and extent of the experimental inaccuracy. How then does one apply equations that are based on a model of combinatorial probabilities if the numbers do not fit the actual distributions generated? The best solution would be if the experimenter understood the cause and exact nature of instrumental inaccuracy: if one knew that the explanation was, for example, inadequate resolution of adjacent masses by the mass analyzer, a suitable correction algorithm might be applied. This is in essence what is done with liquid scintillation counting of radioisotopes, wherein ³H and ¹⁴C spillover is accounted for, so that each isotope can be independently measured. Unfortunately, the analytic basis of quantitative inaccuracy by current mass spectrometers is not understood in a sufficiently definitive way at present to allow simple correction (see Strategies for Evaluating Quantitative Instrument Performance and Data Acceptability). Another reasonable approach is to use standard curves, as one does when measuring dilution of an exogenous labeled product. There are problems also that make this difficult for biosynthetic MIDA methods, however. These we will discuss.

Several questions concerning mass spectrometric quantitative inaccuracy need to be addressed. We will not review these issues extensively here but will note some practical implications of each.

The Nature of Quantitative Inaccuracy in Mass Spectrometric Measurement of Isotope Ratios

One potential cause of inaccuracy is incomplete resolution of adjacent ions in the ion envelope (peak tailing), resulting in contamination of adjacent mass channels. If the mass analyzer is operating at the limit of its mass-resolving capacity, the degree of misidentification of ions due to ion scattering and peak tailing (24) could vary from run to run. According to this explanation, mass analyzers that achieve significantly better resolution (e.g., magnetic sector compared with quadrupole mass analyzers) would be predicted to exhibit better accuracy. This prediction has not yet been systematically tested. Another prediction is that higher abundances in adjacent mass isotopomers should worsen resolution (increase peak tailing), resulting in worse concentration sensitivity of fractional isotopomer abundances. Abundance sensitivity (24), the observation that the most abundant ion in an envelope tends to be underestimated quantitatively, has been identified as a problem in the field of isotopic analysis of elements by use of isotope ratio mass spectrometers. The physical explanation and methods for instrumental correction continue to be debated, however. A second possible cause of inaccuracy is nonlinearity of the detector response at different abundances or for different ions. If the detector output at each mass does not faithfully and consistently reflect the number of ions that reach it, numerical distributions will be skewed (10). Detector nonlinearity might be correctable by use of suitable standards, if these could be synthesized, or by use of improved multiplier-detectors. Detector nonlinearity would also predispose to abundance-sensitivity effects: if abundances of different isotopomers span a large dynamic range, then the slope of detector output vs. injected material will be different for each. A third possibility is the occurrence of chemistry in the ion source (10, 11, 30). Chemical reactions such as hydrogen abstraction or addition could alter mass isotopomer abundances. This problem could be addressed by using derivatives that are unlikely to undergo these reactions (such as fluorinated molecules lacking exchangeable hydrogens) or by use of different ionization conditions (e.g., metastable atom bombardment). This problem might also explain concentration effects on isotope ratios (increased concentrations in the ion source lead to more chemical interactions, as postulated in Refs. 10, 11, and 30).

Regardless of the physical mechanisms involved, some empirical observations may be helpful operationally. First, concentration effects (relative abundances of isotopomers varying as a function of the total amount of material injected onto the mass spectrometer) are a problem for most molecules and tend to be more pronounced the greater the dynamic range among the masses monitored. With glucose-pentaacetate, for example, in natural abundance samples the ratio of the M₀ (0.8396) isotopomer to the M₁ (0.1348) and M₂ (0.0256) isotopomer abundances spans almost two orders of magnitude. In contrast, at higher values of precursor enrichment (e.g., p = 0.24), the ion envelope is more evenly spread out (M₀ = 0.3060, M₁ = 0.4400 and M₂ = 0.2550, for a span of less than twofold), and observed concentration sensitivity is in fact less of a problem (Neese, R. A., R. Bandsma, and M. K. Hellerstein, unpublished observations). Second, the highest abundance isotopomers tend to be relatively underestimated as total ion abundance increases (as has also been observed with inorganic elemental analyses using isotope ratio mass spectrometers, Ref. 24). Third, and counterintuitively, masses with very low baseline isotope abundances (i.e., higher masses with few or no natural isotope abundances) can be analytically undesirable when they have to be compared quantitatively with higher abundance masses. Although it may seem attractive to avoid having to subtract baseline values, the extremely large dynamic range maximizes the relative concentration sensitivity of different isotopomers and may thereby lead to concentration-sensitivity for isotopomer abundances.

In summary, a number of analytic factors influence the relative abundances of mass isotopomers measured in an envelope (Table 2). These factors range from chemical events in the ion source (abstraction of H) to performance of the mass analyzer (ion transmission efficiency, mass resolution, peak tailing), characteristics of the ion detector (velocity dependence of multiplier, nonlinearity due to threshold or saturation effects), or accuracy of the integration software (baseline value that is subtracted, peak-fitting algorithm used). Until basic mass spectrometry research identifies the causes of quantitative inaccuracy, it will not be possible to correct post hoc in a definitive manner for substantial deviations from expected isotopomer abundances. Other strategies are therefore required. The most important of these in practice involve assessment of instrument performance and analytic techniques for prevention of inaccuracy.

Strategies for Evaluating Quantitative Instrument Performance and Data Acceptability

Measurements on natural abundance standards. The theoretically expected mass isotopomer abundances can be calculated for any natural abundance molecule or ion fragment of known chemical composition (see APPENDIX). The impact of variations in natural isotope abundances is extremely small (Fig. 2), so that natural abund