Mathematical modeling of ¹³C label incorporation of the TCA cycle: The concept of composite precursor function

To gain insight into important cerebral metabolic reactions, many modern methods have been established, such as radiotracer methods, e.g., PET (Raichle et al., 1975; Reivich et al., 1979), SPECT, and autoradiography (Sokoloff et al., 1977), or methods based on stable isotopes, such as MR [¹⁷O (Ligeti et al., 1995), ¹³C (Beckmann et al., 1991)]. Typically, in all these approaches, the rate of label incorporation into a product, P, from a single precursor, S, is modeled by using a set of one or more ordinary differential equations (ODEs). The complexity of the used network of ODEs depends on the specific reactions and isotope fluxes being studied (van den Berg and Garfinkel, 1971; Cremer and Heath, 1974). Systems of up to 200 coupled ODEs have been proposed for ¹³C NMR studies (Chance et al., 1983) which allow the resolved detection of multiple metabolites and resonance positions. Efficient and powerful software with a graphic user interface (e.g., SAAM; The SAAM Institute, Seattle, WA) has become available, with contemporary computing power allowing us to extract metabolic fluxes even from complex multi-TCA cycle models. This classical approach has already uncovered very important features of label scrambling of glucose metabolism via the TCA cycle (Mason et al., 1992; Gruetter et al., 2001).

However, not only a thorough understanding of the involved biochemical processes but also knowledge of the mathematics and good computer skills are required to adapt the model for a specific problem and to implement it in the chosen software. Thus, interpretation of the underlying mathematical equations and the results revealed by numerical solution are usually not intuitively interpretable, as reviewed by Henry et al. (2006), and thus are the subject of debate (Gruetter et al., 2001; Choi et al., 2002; Mangia et al., 2003; Hyder et al., 2006).

As a result of the current experimental sensitivity limitations, time courses of metabolite pools with low concentrations such as TCA cycle intermediates are difficult, if not impossible, to determine. As a consequence, the number of differential equations to be solved can be higher than the number of measured time courses, which is not a practical problem but obscures practical insight into existing relationships. For example, it has been well established that calculating the labeling of glutamate C4 requires the solution of the labeling of oxoglutarate C4 (Mason et al., 1992). It is, however, not discussed in the literature that glutamate C4 labeling can be described by a single differential equation, with a simple mathematical solution, as will be demonstrated. The present study had three aims: First, to establish a mathematical formalism that requires solving only one differential equation for each measurable time course of ¹³C label incorporation into glutamate and aspartate via the TCA cycle intermediates; second, to combine all terms incorporating the labeling from precursor pools into only one expression representing a composite “driving function,” for which the concept composite precursor functions (CPF) is introduced; and, third, to establish analytical solutions for the amino acid labeling, allowing determination of quantitative criteria under which the precise measurement of the rate of precursor labeling is obviated.

MATERIALS AND METHODS

Theory

The model is based on a previous model (Gruetter et al., 2001) that includes the following steady-state assumptions: metabolite concentrations and fluxes were considered to be constant, and the concentration of TCA cycle intermediates is small compared with the concentrations of amino acids. The current paper concerns the ¹³C label incorporation into the amino acids via the TCA cycle intermediates, without considering anaplerotic pathways or neuroglial compartmentation; i.e., only a single TCA cycle will be considered (Fig. 1).

With these assumptions, label incorporation into a carbon at position C of a product P can be written as eq. 1 (Gruetter et al., 1998), which is an ordinary differential equation (ODE):

(1)

This ODE describes the change of isotopic enrichment (IE) of the carbon of the product P at position C, ¹³P_C, resulting from the difference of the sum of inflowing tracer from a substrate S (first term on the right side = influx) and the label flowing out of the product pool (second term on the right side = efflux). As the influx includes the labeling of the precursor, this can be denoted by a precursor function (PF). [S] and [P] represent the total concentration of the substrate pool and the product pool, respectively.

Throughout this article, “isotopic enrichment (IE)” will be used for the absolute concentration of enriched metabolite ¹³P_C(t) in μmol · g^–1, whereas “fractional enrichment (FE)” will stand for IE related to the total concentration of the metabolite pool, i.e.,

Thus the FE ranges from 0 to 1 and has no units.

At this point, two notational conventions are introduced. First,

the temporal derivative of the IE of ¹³P_C(t) will be written as dP_C (μmol · g^–1 · min^–1). Second, the symbol P_C denotes the dimensionless fractional enrichment (FE),

The index C refers to the position of the labeled carbon.

As a result, eq. 1 is reformulated to

(2)

In general, however, the term V⁽ⁱⁿ⁾ · PF has to be adapted for multiple fluxes from several substrates. Accordingly, several precursor pools deliver label at different rates (eq. 3). The same applies to the effluxes.

(3)

In this notation, label input comes from multiple sources and hence is termed composite precursor function (CPF). As will be shown below, reformulation of analytical equations in terms of composite precursor functions allows important insight into isotope kinetics.

In this paper, three cases were examined: case 1 deals with the label incorporation into Glu₄, case 2 with incorporation into Glu₄ and Glu₂ or Glu₃, and case 3 with incorporation into Glu₄, Glu₂, or Glu₃ and Asp₂ or Asp₃.

Case 1

The equation to describe labeling incorporation into the carbon at position 4 of glutamate (P_C = Glu₄) requires a second equation for the labeling of the carbon at position 4 of oxoglutarate (P_C = OG₄). Replacing S and P in eq. 3 with the abbreviations of metabolites named oxoglutarate (P = OG) and glutamate (P = Glu), and considering the precursor for these two pools, namely pyruvate labeled at carbon position 3 (Pyr₃) and oxoglutarate labeled at carbon position 4 (OG₄), respectively, we will find as in (Mason et al., 1992)

(4)

Eliminating OG₄ yields the following equation.

(5)

Here the term V_gt was introduced with the same definition as in Mason et al. (1992). When assuming that the temporal change of the IE of pools with a much smaller concentration can be neglected compared with pools with higher concentrations

leads to the simplified ODE

(6)

The PF introduced here is analogous to eq. 2 or 3, reflecting the role of Pyr₃ as the precursor of Glu₄. The equation will not differ when considering label flow from [2-¹³C]acetyl-CoA or [1-¹³C]Glc or [6-¹³C]Glc, provided that labeling delays in glycolytic intermediates can be neglected.

Case 2

The symmetry of the succinate molecule implies that the labeling time course of Glu₂ and Glu₃ can be represented by the same equation (Yu et al., 1997), where Glu₂₃ represents label in Glu₂ or Glu₃, respectively. Using the same mathematical assumption as for the derivation of eq. 6 the following expression can be derived (for the derivation see Appendix)

(7)

with CPF = P_X · Glu₄ + P_TCA · Pyr₃ and

It is of interest to note that, despite a single label precursor, Pyr₃, Glu₂₃ receives label input from two sources, namely, Glu₄ and Pyr₃. For example, Glu₂₃ will receive all label input from Glu₄ when V_X/V_TCA >> 1 (P_X ∼ 1), and increasingly Pyr₃ will serve as label precursor with decreasing V_X/V_TCA. Thus, the influence of both precursors can be described by the composite precursor function CPF. To emphasize the dependence of the labeling on V_gt and the ratio x = V_X/V_TCA, eq. 7 can be rewritten as

(8)

Case 3

The model was expanded to include the labeling of aspartate at position 2 and 3, which will be described with a single equation for Asp₂₃ as was done for Glu₂₃ due to the symmetry of labeling at the succinate level. Using a derivation as for cases 1 and 2 the set of ODEs becomes (for derivation see Appendix)

(9)

where

and

(10)

Eq. 6 holds for all cases in addition to the case-specific ODEs, eq. 8 for case 2 and eq. 9 for case 3. Writing eq. 9 in a more compact form as a function of V_gt and x yields

(11)

with

Finally, to estimate the impact of each term in eq. 8 and 11 on the labeling, it is very instructive to analyze the dependence of Glu₂₃ and Asp₂₃ on x = V_X/V_TCA. For x between 1 and ∞, which represent the complete range of reported x, f₀(x) ranges from 0.67 to 0.5, f₁(x) from –0.83 to –1, f₂(x) from 0.29 to 0.5, and f₃(x) from 0.57 to 0.5. In other words, the functions f_i(x) are only modestly dependent on x. Thus the label incorporation into Glu₂₃ and Asp₂₃ is mainly dependent on the CPF and V_gt, as will be shown in Results.

Analytical Solutions

In the following section, we show that for case 1 and case 2 the ODEs for Glu₄ (eq. 6) and Glu₂₃ (eq. 8) can be solved analytically. However, it is of advantage to assume an analytical expression for the labeling of the precursor pyruvate at position C3, Pyr₃(t).

Due to the low concentration of pyruvate, it is not possible to measure the IE of pyruvate at position C3 directly with ¹³C MRS. In general, measurements of the plasma glucose or the IE of lactate at position C3 are considered to deduce the IE of pyruvate (Gruetter et al., 1992; Mason et al., 1992). To assess the effect of mildly delayed precursor enrichment, it was assumed that the labeling of Pyr₃ follows an exponential approach to steady state C:

(12)

where NA is the natural abundance (NA = 0.011) and C is the fractional enrichment of pyruvate at steady state. The constant k reflects the labeling rate of Pyr₃.

The solution of ODEs as used in this model can be found using the variation of constants method as shown in part B of the Appendix. Additionally, the same method will be used to derive the analytical expressions for the case in which labeling of the precursor pyruvate reaches steady state infinitely fast, i.e., that Pyr₃ is constant for all times, which mathematically equals performing the transition k→∞.

Case 1

The ODE for the labeling of glutamate at position 4 (eq. 6) in detailed notation to emphasize the time dependence of respective terms is given by

(13)

Utilizing the initial condition Glu₄(t = 0) = NA · [Glu], the time course of Glu₄ is given by

(14)

with λ = V_gt/[Glu]. When NA is negligible (NA/C << 1), eq. 14 can be rewritten as

(15)

Assuming that the labeling of the precursor Pyr₃ reaches steady state quickly compared with the turnover of Glu₄ and thus is constant for all times, and using the same method to solve

(16)

yields

(17)

In other words, if it can be assumed that Pyr₃ reaches steady-state fast compared with Glu₄ turnover, a step function can be assumed for Pyr₃ turnover and its explicit measurement is not necessary.

Case 2

The labeling of glutamate at position C3 (eq. 8) will be solved in detailed notation explicitly representing the time-dependent terms and is given by:

(18)

with the initial condition Glu₂₃ = NA · [Glu], the analytical equation is given by

(19)

with λ and τ as defined in eq. 18. When assuming fast labeling of Pyr₃ (k/λ >> 1 and k/τ >> 1) and NA/C << 1, this equation will be simplified to

(20)

In summary, we derived simplified analytical expressions that describe the time course of Glu₄ (eq. 17) and Glu₂₃ (eq. 20), assuming a constant labeling of Pyr₃ for all times as well as a negligible natural abundance.

RESULTS

Numerical Evaluation

Elimination of TCA Cycle Intermediates

The solutions of the exact and the simplified ODEs (Fig. 2) show the typical evolution of ¹³C labeling time courses as presented previously (Gruetter et al., 2001). In both cases 2 and 3, when increasing the ratio V_X/V_TCA Glu₄ approaches steady state faster, and subsequent labeling of Glu₂₃ starts more slowly but later also increases more strongly, thus describing an increasingly sigmoid curve. In case 3, increasing V_X/V_TCA also emphasize the increasing sigmoid shape of the FE of Asp₂₃. In comparison with case 2, the sigmoid shape for the time course of Glu₂₃ becomes slightly more intense in case 3 because of the further delay caused by preceding labeling of Asp₂₃.

For all three cases and any choice of parameters, the time courses verified that the simplification in the derivation of the present model yields a negligible deviation from the enrichment curves calculated using the full set of ODEs (Fig. 2). The deviation was maximally below ∼0.6%, which is far below the experimental error. Evaluating the dependence of the error on the two cases, the error of Glu₂₃ increases when Asp₂₃ was included, whereas the error of Glu₄ remained constant. The error was lower with faster Pyr₃ labeling, i.e., larger k (data not shown). For Figure 2, k = 4 min^–1 was used to yield an estimation of the upper bound of the error.

The upper row of graphs in Figure 2 also contains a plot of the analytical solutions for Glu₄ (eq. 14) and Glu₂₃ (eq. 19; crosses). As expected, the solutions are identical to those of the simplified ODEs.

Assumption of a Constant Precursor Labeling

When assuming that pyruvate labeling is rapid, i.e., k/λ >> 1 and k/τ >> 1, and comparing this with the solution assuming a constant instantaneous Pyr₃ labeling C at t > 0, a small dependence on the ratio V_X/V_TCA was observed (Fig. 3), which overall was negligible. As expected, as k was increased, the deviations decreased exponentially. The time course of Glu₂₃ showed less deviation for low values of k in comparison with Glu₄ (Fig. 3). Adapting the infusion protocol of the labeled glucose, such that 95% of the steady-state concentration is reached after 30 min, results in a deviation below 16% and 8% for the FE of Glu₄ and Glu₂₃, respectively (Fig. 4). If such a label concentration can be reached after only 12 min, this already reduces the deviation to 8% and 3%, respectively. These errors can be assumed to be negligible in relation to experimental error.

DISCUSSION

The current paper describes a novel mathematical framework for simplifying the mathematical modeling of tracer incorporation, illustrated with the example of TCA cycle rate determination by ¹³C NMR. We show that, if changes in the labeling of TCA cycle intermediates can be neglected compared with that in amino acids, the number of differential equations required to describe the label incorporation can be halved. Interestingly, the flux rate into Glu₄ calculated thus is identical to the V_gt derived by Mason et al. (1992), who used probability arguments. Here we demonstrate that this derivation is valid only if the rate of labeling of OG₄ is negligible compared with that of Glu₄, which is a condition that has not been established to date.

Specifically, the presented model yielded a simplified set of a reduced number of ODEs describing the label incorporation into the amino acids resulting from metabolic reactions ascribed to the TCA cycle. The labeling of glutamate at positions C2, C3, C4, and of aspartate at positions C2, C3 can be described with only five ODEs, namely, eq. 6 and 9.

Label flows into Asp₂₃ from either Pyr₃ or Glu₄ and similarly for Glu₂₃, and this flow can be described by composite precursor functions. We are not aware of any description of the labeling of Asp₂₃ and Glu₂₃ in such a manner. When interpreting the driving input functions of the ODEs (composite precursor function) as terms representing the probability that label comes from the directly previously labeled amino acid or from preceding labeled pools, for example, for a very high V_X/V_TCA, the label is mostly received from the latest labeled amino acid pool instead of any other pool labeled earlier in the metabolic pathway.

From eq. 6, it is clear that, with a step function in Pyr₃, Glu₄ will follow an exponential time course (1 – e^–kt) with k = V_gt/Glu, which is confirmed by the analytical expression for Glu₄ (eq. 17). As has been shown previously (Choi and Gruetter, 2004), with such a significant delay, when Glu₄ is the precursor, (V_X/V_TCA >> 1) will lead to a sigmoidal shape of the label curve for Glu₂₃. For example, from eq. 20, V_X/V_TCA >> 1 implies P_X ∼ 1 and P_TCA ∼ 0 and hence Glu₂₃/[Glu] = C(1 – 2e^–τt + e^–λt) = C(1 – e^–τt)². Clearly, the first derivative of this function with time is initially zero; hence, with V_X/V_TCA >> 1, it is shown analytically that Glu₂₃ labeling at t = 0 should be very slow. The approximations for this derivation require fast precursor labeling for Glu₂₃ (k/λ >> 1) and a high enrichment (NA/C << 1), so they can be easily verified from experimental data fulfilling these conditions. As expected from eq. 8 or analytically from eq. 20, it was found that the sigmoid nature of the labeling curves increases with V_X. If V_X is higher, more label enters into the amino acids and thus arrives later in the next pool of amino acids on the pathway. This delay causes a less steep slope at the beginning of the labeling time course of later-labeled amino acids and thus a more curved sigmoid, implying that the CPF is dominated by the direct precursor, e.g., Glu₄ for Asp₂₃, Asp₂₃ for Glu₂₃, or Glu₄ for Glu₂₃, if Asp₂₃ is not included. Hence, with the formulation of the CPFs, the sensitivity of the labeling time courses to V_X/V_TCA can be evaluated. The increasing sigmoidal shape further implies that the ratio of V_X/V_TCA is best determined at the initial points of the labeling time courses and not in the data acquired at steady state.

This interpretation of the ODEs as presented above confirms former findings based on an analysis of experimental data with the help of mathematical modeling (Mason et al., 1992). Thus this interpretation of the mathematical expressions reflects only the intuitive character of the concept of CPFs. Furthermore, eq. 6 explicitly states that it is not possible to determine either the TCA cycle flux V_TCA or the mitochondrial flux V_X from measuring the labeling time course of glutamate in position C4. Instead, from the definition of V_gt (eq. 6), it is apparent that V_gt is always smaller than V_TCA. Thus, assuming that V_gt = V_TCA (V_X >> V_TCA) leads to an underestimation of V_TCA and consequently of the total glucose consumption (CMR_glc) rate of the brain. In this context, it is of interest to note that some previously published values of V_TCA, ranging from ∼0.47 to ∼0.53 μmol/g, obtained with the assumption that V_X >> V_TCA (Hyder et al., 1996, 1997; Sibson et al., 1998; Patel et al., 2004), are lower than results found with ¹³C NMR (Henry et al., 2002) as well as with other tracer methods, such as ¹⁷O NMR (Zhu et al., 2002) or autoradiography (Nakao et al., 2001), which ranged from ∼0.71 to ∼0.83 μmol/g under similar anesthesia conditions (α-chloralose).

Analytical solutions of the ODEs could also be derived. However, not using any further simplification, unlike the resulting expressions for Glu₄ (eq. 14), which is fairly simple, the function Glu₂₃ (eq. 19) already has a fairly complex structure, which is difficult to interpret. Solving the coupled ODEs of Glu₂₃ and Asp₂₃ yields terms that are beyond the scope of this manuscript and will be presented elsewhere.

However, the fact that now one analytical solution is available for either fitting the time course of Glu₄ to determine V_gt or Glu₃ to determine the ratio V_X/V_TCA reveals a unique tool for analyzing the data of a 1-¹³C-glucose MRS experiment. Consequently, measuring both labeling curves allows for derivation of both fluxes, V_X and V_TCA.

Moreover, the model presented shows that a precise knowledge of the precursor labeling of pyruvate is no longer required, as long as the infusion protocol guarantees a sufficiently fast labeling compared with the amino acid label turnover rates. Even a less perfectly adapted infusion protocol that reaches a steady state of pyruvate labeling after about 30 min results in negligibly low deviations from the correct expression of the time courses of Glu₄ and Glu₂₃ when related to experimental accuracy. Henry et al. (2002) established an infusion protocol of labeled glucose that fulfills this criterion. With this additional simplification, Pyr₃ = const, the analytical expressions of Glu₄ and Glu₂₃ fall into a straightforward form such that the fitting process can be done with even common software, such as Excel or Origin. With this step, a significant practical problem of in vivo ¹³C MRS studies is solved.

CONCLUSIONS

We conclude that the explicit solution of the labeling of the TCA cycle intermediates can be eliminated from the mathematical model, resulting in a reduced number of necessary ODEs to one equation per measurable metabolite pool. Formulation of the composite precursor functions showed that measuring time points at the beginning of the labeling yields valuable information for the determination of V_X. The model thus offers a formal method to investigate the principles of label kinetics in such ¹³C turnover experiments leading to analytical expressions. We further conclude that, provided that the precursor, e.g., Pyr, approaches steady-state quickly compared with the rate of labeling of the amino acid investigated, the analytical solution assuming instantaneous labeling can be used.