mathematical modeling of 13c label

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Mathematical Modeling of 13C LabelIncorporation of the TCA Cycle:The Concept of Composite Precursor Function

Kai Uffmann* and Rolf Gruetter

Laboratory of Functional and Metabolic Imaging, Ecole Polytechnique Federale de Lausanne,Lausanne, Switzerland

A novel approach for the mathematical modeling of 13Clabel incorporation into amino acids via the TCA cyclethat eliminates the explicit calculation of the labeling ofthe TCA cycle intermediates is described, resulting inone differential equation per measurable time course oflabeled amino acid. The equations demonstrate that

both glutamate C4 and C3 labeling depend in a predicti-ble manner on both transmitochondrial exchange rate,VX, and TCA cycle rate, VTCA. For example, glutamate C4labeling alone does not provide any information on eitherVX or VTCA but rather a composite flux. Interestingly,glutamate C3 simultaneously receives label not only frompyruvate C3 but also from glutamate C4, described bycomposite precursor functions that depend in a proba-bilistic way on the ratio of VX to VTCA: An initial rate oflabeling of glutamate C3 (or C2) being close to zero isindicative of a high VX/VTCA. The derived analytical solu-tion of these equations shows that, when the labeling ofthe precursor pool pyruvate reaches steady state quicklycompared with the turnover rate of the measured amino

acids, instantaneous labeling can be assumed forpyruvate. The derived analytical solution has acceptableerrors compared with experimental uncertainty, thusobviating precise knowledge on the labeling kinetics ofthe precursor. In conclusion, a substantial reformulationof the modeling of label flow via the TCA cycle turnoverinto the amino acids is presented in the current study.This approach allows one to determine metabolic ratesby fitting explicit mathematical functions to measuredtime courses. VVC 2007 Wiley-Liss, Inc.

Key words: modeling; magnetic resonance spectroscopy;13C label incorporation; composite precursor function;glucose consumption

To gain insight into important cerebral metabolicreactions, 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 iso-topes, such as MR [17O (Ligeti et al., 1995), 13C (Beck-mann et al., 1991)]. Typically, in all these approaches, therate of label incorporation into a product, P, from a singleprecursor, S, is modeled by using a set of one or more or-

dinary differential equations (ODEs). The complexity ofthe used network of ODEs depends on the specific reac-tions and isotope fluxes being studied (van den Berg andGarfinkel, 1971; Cremer and Heath, 1974). Systems of upto 200 coupled ODEs have been proposed for13C NMRstudies (Chance et al., 1983) which allow the resolved

detection of multiple metabolites and resonance positions.Efficient and powerful software with a graphic user inter-face (e.g., SAAM; The SAAM Institute, Seattle, WA) hasbecome available, with contemporary computing powerallowing us to extract metabolic fluxes even from complexmulti-TCA cycle models. This classical approach has al-ready uncovered very important features of label scram-bling of glucose metabolism via the TCA cycle (Masonet al., 1992; Gruetter et al., 2001).

However, not only a thorough understanding of theinvolved biochemical processes but also knowledge of themathematics and good computer skills are required toadapt the model for a specific problem and to implementit in the chosen software. Thus, interpretation of the

underlying mathematical equations and the resultsrevealed by numerical solution are usually not intuitivelyinterpretable, as reviewed by Henry et al. (2006), and thusare 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 sensitivitylimitations, time courses of metabolite pools with lowconcentrations such as TCA cycle intermediates are diffi-cult, if not impossible, to determine. As a consequence,the number of differential equations to be solved can behigher than the number of measured time courses, whichis not a practical problem but obscures practical insight

Contract grant sponsor: Centre dImagerie BioMedical (CIBM); Contract

grant sponsor: Leenaards and Jeantet Foundations; Contract grant sponsor:

SNSF; Contract grant number: 3100A0-116220 (to R.G.).

*Correspondence to: Kai Uffmann, Ecole Polytechnique Federale de

Lausanne (EPFL), Centre dImagerie Biomedicale (CIBM), CH F1 582,

Station 6, CH-1015 Lausanne, Switzerland. E-mail: [email protected]

Received 23 November 2006; Revised 4 April 2007; Accepted 4 April

2007

Published online 28 June 2007 in Wiley InterScience (www.

interscience.wiley.com). DOI: 10.1002/jnr.21392

Journal of Neuroscience Research 85:33043317 (2007)

' 2007 Wiley-Liss, Inc.


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into existing relationships. For example, it has been wellestablished that calculating the labeling of glutamate C4requires the solution of the labeling of oxoglutarate C4(Mason et al., 1992). It is, however, not discussed in theliterature that glutamate C4 labeling can be described by asingle differential equation, with a simple mathematical

solution, as will be demonstrated. The present studyhad three aims: First, to establish a mathematical for-malism that requires solving only one differential equationfor each measurable time course of13C label incorporationinto glutamate and aspartate via the TCA cycle intermedi-ates; second, to combine all terms incorporating thelabeling from precursor pools into only one expressionrepresenting a composite driving function, for whichthe concept composite precursor functions (CPF) is intro-duced; and, third, to establish analytical solutions for theamino acid labeling, allowing determination of quantita-tive criteria under which the precise measurement of therate of precursor labeling is obviated.

MATERIALS AND METHODS

Theory

The model is based on a previous model (Gruetteret al., 2001) that includes the following steady-state assump-tions: metabolite concentrations and fluxes were considered tobe constant, and the concentration of TCA cycle intermedi-ates is small compared with the concentrations of amino acids.The current paper concerns the 13C label incorporation intothe amino acids via the TCA cycle intermediates, withoutconsidering anaplerotic pathways or neuroglial compartmenta-tion; i.e., only a single TCA cycle will be considered (Fig. 1).

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

d13PCt

dt Vin

13St

S|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}influx

Vout 13PCt

P|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}efflux

Vin PF

Vout 13PCt

P: 1

This ODE describes the change of isotopic enrichment (IE) ofthe carbon of the product P at position C, 13PC, resultingfrom the difference of the sum of inflowing tracer from a sub-

strate S (first term on the right side5

influx) and the labelflowing out of the product pool (second term on the rightside 5 efflux). As the influx includes the labeling of the pre-cursor, this can be denoted by a precursor function (PF). [S]and [P] represent the total concentration of the substrate pooland the product pool, respectively.

Throughout this article, isotopic enrichment (IE) willbe used for the absolute concentration of enriched metabolite13PC(t) in lmol g

1, whereas fractional enrichment (FE)will stand for IE related to the total concentration of themetabolite pool, i.e.,

13PCt

P:

Thus the FE ranges from 0 to 1 and has no units.At this point, two notational conventions are intro-

duced. First,

d13PCt

dt;

the temporal derivative of the IE of 13PC(t) will be written asdPC (lmol g

1 min1). Second, the symbol PC denotes thedimensionless fractional enrichment (FE),

13PCt

P:

The index C refers to the position of the labeled carbon.As a result, eq. 1 is reformulated to

dPCt Vin S Vout PC V

in PF Vout PC:

2

In general, however, the term V(in) PF has to be adapted formultiple fluxes from several substrates. Accordingly, severalprecursor pools deliver label at different rates (eq. 3). Thesame applies to the effluxes.

Fig. 1. The chemical pathways involved in the label incorporationinto amino acids via the TCA cycle included in the mathematicalmodel. Metabolites that can be measured by 13C MRS are printed inboldface. The splitting of the arrows after label passed the pool of 2-oxoglutarate (OG) represents the symmetry at the succinate level.Glc, Lac, Pyr, OG, Glu, OAA, and Asp stand for glucose, lactate,pyruvate, 2-oxoglutarate, glutamate, oxaloacetate, and aspartate,respectively. The index at each metabolite name represents the posi-tion of the carbon that gets labeled in that pool. The TCA cycle fluxis represented by VTCA, whereas the transmitochondrial flux is VX.The indices denote the carbon position that becomes labeled accord-ing to the pathways of the scheme after an injection of 1-13Cglucose.

Modeling of13

C Label Incorporation Via TCA Cycle 3305

Journal of Neuroscience Research DOI 10.1002/jnr


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dPCt X

i

Vini S

i X

j

Voutj PC

Xi

Vini

8>>>:9>>>; CPF

XjV

outj PC with CPF

Pi

Vini S

i

Pi Vini8>>: 9>>;: 3

In this notation, label input comes from multiple sources andhence is termed composite precursor function (CPF). As will beshown below, reformulation of analytical equations in termsof composite precursor functions allows important insight intoisotope kinetics.

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

Case 1. The equation to describe labeling incorpora-tion into the carbon at position 4 of glutamate (PC 5 Glu4)requires a second equation for the labeling of the carbon at posi-

tion 4 of oxoglutarate (PC 5 OG4). Replacing S and P in eq. 3with the abbreviations of metabolites named oxoglutarate (P 5OG) and glutamate (P 5 Glu), and considering the precursorfor these two pools, namely pyruvate labeled at carbon position3 (Pyr3) and oxoglutarate labeled at carbon position 4 (OG4),respectively, we will find as in (Mason et al., 1992)

dGlu4 VX OG4 VX Glu4

dOG4 VTCA Pyr3 VX Glu4 VX VTCA OG4:

4

Eliminating OG4 yields the following equation.

dGlu4 VX

VX VTCAdOG4 Vgt Pyr3 Glu4 with

Vgt VXVTCA

VX VTCA5

Here the term Vgt was introduced with the same definitionas in Mason et al. (1992). When assuming that the temp-oral change of the IE of pools with a much smaller concen-tration can be neglected compared with pools with higherconcentrations

e:g:dOG4

dGlu4> 1 (P

X$ 1), and

increasingly Pyr3 will serve as label precursor with decreasingVX/VTCA. Thus, the influence of both precursors can bedescribed by the composite precursor function CPF. Toemphasize the dependence of the labeling on Vgt and the ratiox 5 VX/VTCA, eq. 7 can be rewritten as

dGlu23 Vgtx 1

2x 1 CPF Glu23

Vgt f0x CPF Glu23: 8

Case 3. The model was expanded to include the

labeling of aspartate at position 2 and 3, which will bedescribed with a single equation for Asp23 as was done forGlu23 due to the symmetry of labeling at the succinate level.Using a derivation as for cases 1 and 2 the set of ODEsbecomes (for derivation see Appendix)

dAsp23

dGlu23

8>>: 9>>; fVX;VTCA 2VX VTCA VXVX 2VX VTCA

8>>: 9>>;Asp23

Glu23

8>>: 9>>;VX VTCA CPFCPFASP

8>>: 9>>; 9where

fVX;VTCA VXVTCA2VX VTCA

2 V2TCA

and

CPF PX Glu4 PTCA Pyr3

CPFAsp PX Asp23 PTCA CPF: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

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compact form as a function of Vgt and x yields

dAsp23

dGlu23

8>>: 9>>; Vgt f1x f2xf2x f1x

8>>: 9>>; Asp23Glu23

8>>: 9>>;

f3xCPF

CPFASP8>>: 9>>; 11

with

f1x 2x 1x 1

2x 12 1; f2x

xx 1

2x 12 1;

f3x x 12

2x 12 1:

Finally, to estimate the impact of each term in eq. 8 and 11on the labeling, it is very instructive to analyze the depend-ence of Glu23 and Asp23 on x 5 VX/VTCA. For x between 1and ?, 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 fi(x) are only modestly dependent on x. Thusthe label incorporation into Glu23 and Asp23 is mainly de-pendent on the CPF and Vgt, as will be shown in Results.

Analytical Solutions

In the following section, we show that for case 1 andcase 2 the ODEs for Glu4 (eq. 6) and Glu23 (eq. 8) can besolved analytically. However, it is of advantage to assume ananalytical expression for the labeling of the precursor pyruvateat position C3, Pyr3(t).

Due to the low concentration of pyruvate, it is not pos-sible to measure the IE of pyruvate at position C3 directlywith 13C MRS. In general, measurements of the plasma glu-cose or the IE of lactate at position C3 are considered todeduce the IE of pyruvate (Gruetter et al., 1992; Mason et al.,1992). To assess the effect of mildly delayed precursor enrich-ment, it was assumed that the labeling of Pyr3 follows an ex-ponential approach to steady state C:

Pyr3t C Pyr NA Pyr C Pyr ekt Pyr

n

C NA C ekto; 12

where NA is the natural abundance (NA 5 0.011) and C isthe fractional enrichment of pyruvate at steady state. The con-

stant k reflects the labeling rate of Pyr3.The solution of ODEs as used in this model can befound using the variation of constants method as shown inpart B of the Appendix. Additionally, the same method willbe used to derive the analytical expressions for the case inwhich labeling of the precursor pyruvate reaches steady stateinfinitely fast, i.e., that Pyr3 is constant for all times, whichmathematically equals performing the transition k??.

Case 1. The ODE for the labeling of glutamate atposition 4 (eq. 6) in detailed notation to emphasize the timedependence of respective terms is given by

dGlu4t

dt Vgt

Glu4t

Glu Vgt

Pyr3t

Pyr: 13

Utilizing the initial condition Glu4(t 5 0) 5 NA [Glu], thetime course of Glu4 is given by

Glu4tVgt Ck

NACk

ekt NAC kkkk

ekt ;14

with k 5 Vgt/[Glu]. When NA is negligible (NA/C >: 9>>;with s

Vgt

Glu

x 1

2x 1 k

x 1

2x 1; 18

with the initial condition Glu23 5 NA [Glu], the analyticalequation is given by

Glu23 NA Glu est

s Glu PX k C

k s 1 est

NA C

k ks k

8>>:

ekt est

NA C k

kk ksk

ekt est9>>;

PTCA C

s 1 est

NA C

s k

ekt est8>>: 9>>;;

19

Modeling of13




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with k and s as defined in eq. 18. When assuming fast label-ing of Pyr3 (k/k >> 1 and k/s >> 1) and NA/C ? were calcu-

lated. As defined in eq. 12, the range of k values represents possi-ble labeling velocities, so that 95% steady-state FE of Pyr3 isreached after 30, 12, 6, 3, 1.5, 0.6, 0.3, 0.2, 0.1, > 1 reveals

that the numerical values of fi(x) do not vary strongly (seeMaterials and Methods), whereas the CPF change theircomplete shape, because Glu4 labels with a time constantk1 5 (Vgt/[Glu])

1 when pyruvate is labeled fastest. Withincreasing VX/VTCA, the probability that label is comingdirectly from a preceding labeled amino acid is increas-ingly dominating (PX approaching 1) such that Pyr3 losesits functionality as an isotopic label precursor.

Numerical Evaluation

Elimination of TCA Cycle Intermediates. Thesolutions of the exact and the simplified ODEs (Fig. 2)show the typical evolution of13C labeling time courses aspresented previously (Gruetter et al., 2001). In both cases2 and 3, when increasing the ratio VX/VTCA Glu4approaches steady state faster, and subsequent labeling ofGlu23 starts more slowly but later also increases morestrongly, thus describing an increasingly sigmoid curve. Incase 3, increasing VX/VTCA also emphasize the increasingsigmoid shape of the FE of Asp23. In comparison with




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case 2, the sigmoid shape for the time course of Glu23

becomes slightly more intense in case 3 because of the fur-ther 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 deriva-tion of the present model yields a negligible deviationfrom the enrichment curves calculated using the full set ofODEs (Fig. 2). The deviation was maximally below$0.6%, which is far below the experimental error. Evalu-ating the dependence of the error on the two cases, theerror of Glu23 increases when Asp23 was included, whereasthe error of Glu4 remained constant. The error was lowerwith faster Pyr3 labeling, i.e., larger k (data not shown).For Figure 2, k 5 4 min1 was used to yield an estimationof the upper bound of the error.

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

Assumption of a Constant Precursor Label-ing. When assuming that pyruvate labeling is rapid, i.e.,k/k >> 1 and k/s >> 1, and comparing this with the so-lution assuming a constant instantaneous Pyr3 labeling Cat t > 0, a small dependence on the ratio VX/VTCA wasobserved (Fig. 3), which overall was negligible. Asexpected, as k was increased, the deviations decreased

exponentially. The time course of Glu23 showed less devi-

ation 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 reachedafter 30 min, results in a deviation below 16% and 8% forthe FE of Glu4 and Glu23, respectively (Fig. 4). If such alabel concentration can be reached after only 12 min, thisalready reduces the deviation to 8% and 3%, respectively.These errors can be assumed to be negligible in relation toexperimental error.

DISCUSSION

The current paper describes a novel mathematical

framework for simplifying the mathematical modeling oftracer incorporation, illustrated with the example of TCAcycle rate determination by 13C NMR. We show that, ifchanges in the labeling of TCA cycle intermediates can beneglected compared with that in amino acids, the numberof differential equations required to describe the labelincorporation can be halved. Interestingly, the flux rateinto Glu4 calculated thus is identical to the Vgt derived byMason et al. (1992), who used probability arguments.Here we demonstrate that this derivation is valid only ifthe rate of labeling of OG4 is negligible compared with

Fig. 2. The fractional enrichment of Glu4, Glu23, and Asp23 calcu-lated with the set of ODEs describing the labeling pathway eitherincluding the TCA cycle intermediates (exact 5 sold lines) or with-out them (simple 5 dots) and calculated from the analytical solutionsof the simplified ODE (analyt. 5 crosses). This calculation was per-formed for different ratios of VX/VTCA 5 1, 2, 8 (columns from left

to right). The FEs in the upper row of graphs are derived from themodel neglecting the labeling of oxaloacetate and aspartate butinclude the analytical solutions (case 2), whereas the lower row showsFEs resulting from the model including labeling of aspartate (case 3).With increasing VX, the sigmoid character of the time coursesbecomes more intense.

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that of Glu4, which is a condition that has not been estab-lished to date.Specifically, the presented model yielded a simplified

set of a reduced number of ODEs describing the labelincorporation into the amino acids resulting from meta-bolic reactions ascribed to the TCA cycle. The labeling ofglutamate at positions C2, C3, C4, and of aspartate atpositions C2, C3 can be described with only five ODEs,namely, eq. 6 and 9.

Label flows into Asp23 from either Pyr3 or Glu4 andsimilarly for Glu23, and this flow can be described by com-posite precursor functions. We are not aware of anydescription of the labeling of Asp23 and Glu23 in such amanner. When interpreting the driving input functions of

the ODEs (composite precursor function) as terms repre-senting the probability that label comes from the directlypreviously labeled amino acid or from preceding labeledpools, for example, for a very high VX/VTCA, the label ismostly received from the latest labeled amino acid poolinstead of any other pool labeled earlier in the metabolicpathway.

From eq. 6, it is clear that, with a step function inPyr3, Glu4 will follow an exponential time course (1 e

kt)with k 5 Vgt/Glu, which is confirmed by the analyticalexpression for Glu4 (eq. 17). As has been shown previ-ously (Choi and Gruetter, 2004), with such a significantdelay, when Glu4 is the precursor, (VX/VTCA >> 1) willlead to a sigmoidal shape of the label curve for Glu23. For

example, from eq. 20, VX/VTCA >> 1 implies PX $ 1and PTCA $ 0 and hence Glu23/[Glu] 5 C(1 2est1

ekt) 5 C(1 est)2. Clearly, the first derivative of thisfunction with time is initially zero; hence, with VX/VTCA>> 1, it is shown analytically that Glu23 labeling at t 5 0should be very slow. The approximations for this deriva-tion require fast precursor labeling for Glu23 (k/k >> 1)and a high enrichment (NA/C


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tion of the CPFs, the sensitivity of the labeling timecourses to VX/VTCA can be evaluated. The increasing sig-moidal shape further implies that the ratio of VX/VTCA isbest determined at the initial points of the labeling timecourses 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 experi-mental data with the help of mathematical modeling(Mason et al., 1992). Thus this interpretation of the math-ematical expressions reflects only the intuitive character ofthe concept of CPFs. Furthermore, eq. 6 explicitly statesthat it is not possible to determine either the TCA cycleflux VTCA or the mitochondrial flux VX from measuringthe labeling time course of glutamate in position C4.Instead, from the definition of Vgt (eq. 6), it is apparentthat Vgt is always smaller than VTCA. Thus, assuming thatVgt 5 VTCA (VX >> VTCA) leads to an underestimationof VTCA and consequently of the total glucose consump-tion (CMRglc) rate of the brain. In this context, it is of in-terest to note that some previously published values of

VTCA, ranging from $0.47 to $0.53 lmol/g, obtainedwith the assumption that VX >> VTCA (Hyder et al.,1996, 1997; Sibson et al., 1998; Patel et al., 2004), arelower than results found with 13C NMR (Henry et al.,2002) as well as with other tracer methods, such as 17ONMR (Zhu et al., 2002) or autoradiography (Nakaoet al., 2001), which ranged from $0.71 to $0.83 lmol/gunder similar anesthesia conditions (a-chloralose).

Analytical solutions of the ODEs could also bederived. However, not using any further simplification,unlike the resulting expressions for Glu4 (eq. 14), which isfairly simple, the function Glu23 (eq. 19) already has afairly 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 andwill be presented elsewhere.However, the fact that now one analytical solution is

available for either fitting the time course of Glu4 to deter-mine Vgt or Glu3 to determine the ratio VX/VTCA revealsa unique tool for analyzing the data of a 1-13C-glucoseMRS experiment. Consequently, measuring both labelingcurves allows for derivation of both fluxes, VX and VTCA.

Moreover, the model presented shows that a preciseknowledge of the precursor labeling of pyruvate is nolonger required, as long as the infusion protocol guaran-tees a sufficiently fast labeling compared with the aminoacid label turnover rates. Even a less perfectly adaptedinfusion protocol that reaches a steady state of pyruvate

labeling after about 30 min results in negligibly low devi-ations from the correct expression of the time courses ofGlu4 and Glu23 when related to experimental accuracy.Henry et al. (2002) established an infusion protocol of la-beled glucose that fulfills this criterion. With this addi-tional simplification, Pyr3 5 const, the analytical expres-sions of Glu4 and Glu23 fall into a straightforward formsuch that the fitting process can be done with even com-mon software, such as Excel or Origin. With this step, asignificant practical problem of in vivo 13C MRS studiesis solved.

CONCLUSIONS

We conclude that the explicit solution of the label-ing of the TCA cycle intermediates can be eliminatedfrom the mathematical model, resulting in a reduced num-ber of necessary ODEs to one equation per measurablemetabolite pool. Formulation of the composite precursor

functions showed that measuring time points at the begin-ning of the labeling yields valuable information for thedetermination of VX. The model thus offers a formalmethod to investigate the principles of label kinetics insuch 13C turnover experiments leading to analyticalexpressions. We further conclude that, provided that theprecursor, e.g., Pyr, approaches steady-state quickly com-pared with the rate of labeling of the amino acid investi-gated, the analytical solution assuming instantaneous label-ing can be used.

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method for the measurement of local cerebral glucose utilization:

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field. Proc Natl Acad Sci U S A 99:1319413199.

APPENDIX

Derivation of the Simplified ODE

With the use of eq. 1, the following set of 10 ODEs(eq. A1) describes the label kinetics related to the chemicalpathways represented schematically in Figure 1. Commas inthe index refer to differential equations of the same form butdescribing label incorporation at different carbon positions.

dGlu2;3;4 VX OG2;3;4 VX Glu2;3;4 ) OG2;3;4 1

VXdGlu2;3;4 Glu2;3;4

dAsp2;3 VX OAA2;3 VX Asp2;3 ) OAA2;3 1

VXdAsp2;3 Asp2;3

dOG2;3 VTCA OAA2;3 VX Glu2;3 VX VTCA OG2;3dOG4 VTCA Pyr3 VX Glu4 VX VTCA OG4

dOAA2;3 VTCA

2 OG3

VTCA

2 OG4 VX Asp2;3 VX VTCA OAA2;3:

A1

Due to the symmetry of succinate, labeling of gluta-mate and aspartate at carbon position 2 and 3 can bedescribed by the same equation in our model (Yuet al., 1997), which will be denoted below by theindex 23.

Case 2. To filter out the information hidden inthe labeling time courses of Glu4 and Glu23 only, we willignore the labeling incorporation into the pools of aspar-tate, i.e., oxaloacetate, for the time being. Thus, the set ofequations (eq. A1) reduces to

dGlu23;4 VX OG23;4 VX Glu23;4 ) OG23;4 1

VXdGlu23;4 Glu23;4

dOG23 VTCA

2 OG23

VTCA

2 OG4 VX Glu23 VX VTCA OG23

dOG4 VTCA Pyr3 VX Glu4 VX VTCA OG4

A2




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Aiming for elimination of the expressions for theTCA cycle intermediates, we replace the terms of OAAand OG in the last two equations with the expressionsfrom the first equation. This gives eq. 5 and eq. A3.

dGlu4 VX

VX VTCA dOG4

VX VTCAVX VTCA

Pyr3 Glu4

5

dGlu23 2VX

2VX VTCA dOG23

VXVTCA

2VX VTCAVX VTCA dOG4

VXVTCA

2VX VTCAVX VTCAfVX VTCA Glu23 VX Glu4 VTCA Pyr3g:

A3

In view of the fact that the concentration of the TCAintermediates is much lower than the pool size of theamino acids, the enrichment of the OAA2,3 reaches steady

state much faster than Asp2,3. Because the labeling followsan exponential evolution in reaching a steady state,dOAA2,3 approaches zero, whereas the labeling velocityof Asp2,3 (dAsp2,3) is still of significant amplitude. Thisapplies in the same way for OG2,3,4 and Glu2,3,4, respec-tively. Additionally,

VX

VX VTCA: 9>;: A6By using the same arguments as above, the terms relatedto the TCA cycle intermediates can again be eliminated.

Modeling of13




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dGlu23 VTCA

VX VTCA dAsp23

VX VTCAVX VTCA

Asp23 Glu23

dGlu4 VX VTCA

VX VTCA Pyr3 Glu4

dAsp23 VTCA

2 VX VTCA dGlu23 dGlu4

VX VTCAVX VTCA

1

2Glu23

1

2Glu4 Asp23

8>: 9>;:A7

Further on, this set of ODEs can be rephrased to give sep-arate equations for each labeled carbon position of all

involved amino acids (eq. A8).

dGlu4 Vgt fGlu4 Pyr3g with Vgt VXVTCA

VXVTCA

dAsp23 fVX=VTCAf2VX VTCA Asp3 VX Glu23 Glu4 VTCA Pyr3g

dGlu23 fVX=VTCA 2VX Asp3 2VX VTCA Glu23 VTCA

VX VTCA VX Glu4 VTCA Pyr3

:

A8

The function

f VX

VTCA

8: 9; VXVTCA2VX VTCA

2 V2TCA

x

2x 12 1with x VX

VTCA

in eq. A8 is dimensionless.

Applying the concept of composite precursor func-tions again leads to a comprehensive form of the set of

ODEs (eq. A8) for labeling of aspartate and glutamate atcarbon positions 2 and 3.

dAsp23dGlu23

8>: 9>; fVX;VTCA 2VX VTCA VXVX 2VX VTCA8>: 9>; Asp23

Glu23

8>: 9>; VX VTCA CPFCPFASP8>: 9>;

9

with f as above and

CPF PX Glu4 PTCA Pyr3

CPFAsp PX Asp23 PTCA CPF:10

Also in this case, labeling of glutamate at carbon position 4still follows

dGlu4 Vgt fGlu4 Pyr3g with Vgt VXVTCA

VX VTCA:

6

Analytical Solutions of the Simplified ODE

In the following, according to case 1 and case 2, the

ODEs for Glu4 and Glu23 will be solved analytically.Case 1. The ODE for the labeling of glutamate atposition 4 is

dGlu4 Vgt Pyr3 Glu4 Vgt PF Vgt Glu4

with Vgt VXVTCA

VX VTCA: 6

For the following calculation, we will again use the moredetailed notation to emphasize the time dependence of




12/14

the terms in eq. 6:

dGlu4t

dt Vgt

Glu4t

Glu Vgt

Pyr3t

Pyr: 13

Labeling of Pyr3 is assumed to follow

Pyr3t C Pyr NA Pyr C Pyr ekt

Pyr fC NA C ektg; 12

where NA is the natural abundance (NA 5 0.011) and C isthe fraction of the concentration of labeled molecules reachedat steady state. The constant k reflects the labeling velocityof Pyr3. Since this time course is not constant and nonzero,eq. 13 is a nonhomogeneous ODE of the first order.

The solution of such an ODE consists of the addi-tion of a solution, yH, of the homogeneous equation

dGlu4

dt k Glu4 0 with k

Vgt

Glu B1

and one specific solution of the inhomogeneous ODE,yS. With the usual ansatz, the latter equation yieldsyH5 c e

kt.Using the variation of constants method with the

ansatz yS 5 c(t) ekt the product rule yields y0S 5 c

0(t) ekt c(t) k ekt, where the apostrophe designates thetemporal derivative. Inserting this and the expression forPyr3 (eq. 12) into the nonhomogeneous ODE (eq. 13)gives

c0t ekt ct k ekt k ct ekt

Vgt fC NA C ektg: B2

Integration of the resulting equation

c0t Vgt fC ekt NA C ekktg; B3

yields

ct Vgt C

k

NA C

k k ekt

ekt: B4

Thus one specific solution of the ODE (eq. 13) is

yS Vgt C

k

NA Ck k

ekt

: B5

In this manner, an explicit function describing thetime course of Glu4 was found:

Glu4t yS yH Vgt C

k

NA C

k k ekt

c ekt:

B6

Finally, the integration constant c remains to bedetermined by using the initial condition Glu4(t 5 0) 5NA [Glu]. With

Glu40 Vgt C

k

NA C

k k c NA Glu;B7

we find

Glu4t Vgt C

k

NA C

k k ekt NA C

k

k k k ekt

: 14

Assuming that the time course of the precursor Pyr3 isconstant, i.e., is a step function, and using the same

method to solve the following equation

dGlu4t

dt Vgt

Glu4t

Glu Vgt

Pyr3

Pyrwith

Pyr3 const: C Pyr 16

yields

Glu4t Glu fC NA C ektg: B8

This equals the assumption that Pyr3 reaches steady stateinfinitely fast; i.e., k is infinitely high. Thus analyzing

eq. 14 in the limit k?? and replacing

k Vgt

Glu

leads to the same expression (eq. B8).Furthermore, we can assume that the natural

abundance, NA, is very small compared with the steadyconcentration C and set NA 5 0. This yields a verycompact form for the function describing the labelingof Glu4:

Glu4t C Glu 1 ekt: 17

Case 2. By analogy to the derivation above, it isalso possible to solve equation eq. 8:

dGlu23 Vgtx 1

2x 1 CPF Glu23

Vgt f0x CPF Glu23; 8

where x5VX/VTCA and CPF PX Glu4 PTCA Pyr3:

Modeling of13




13/14

The respective nonhomogeneous equation with the com-mon extended notation reads as follows:

dGlu23t

dt

s Glu PX

Glu4t

Glu P

TCA8>>: Pyr3tPyr

Glu23t

Glu 9>>;with s

Vgt

Glu

x 1

2x 1 k

x 1

2x 1; 18

i.e.,

dGlu23

dt s Glu23 s Glu PX

Glu4

Glu PTCA

Pyr3Pyr

8>>: 9>>;or

dGlu23

dt

s Glu23 s Glu (PX kC

k

NA C

k k ekt NA C

k

kk k ekt

8>>: 9>>; PTCA C NA C e

kt

): B9

Solving the homogeneous equation

dGlu23

dt s Glu23 0

yields yH5

c est

.As above, using the ansatz yS 5 c(t) est and y0S 5

c0(t) est c(t) s est for the specific solution, we haveto integrate

c0t s Glu n

PX k

C

k est

NA C

k k eskt NA C

k

kk k eskt

8>>: 9>>;PTCA C e

st NA C eskt o

; B10

which results in

ct s Glu

(PX k

C

k s est

NA C

k ks k

8>>: eskt NA C

k

kk ksk eskt

9>>; PTCA C

s est NA C

s k eskt8>>: 9>>;) B11

and finally yields

yS ctest sGlu PX k

C

ks

NAC

kkskekt

8>>:NAC

k

kkkskekt

9>>;PTCA

C

s

NAC

skekt

8>>:

9>>;

: B12

Thus, the complete solution is

Glu23 yS yH c est s Glu

PX k C

k s

NA C

k ks k ekt

8>>:NA C

k

kk ks k ekt

9>>;PTCA

C

s

NA C

s k ekt

8>>: 9>>;: B13To determine the constant of integration, we use the

initial condition Glu23 (0)5

NA [Glu] and find

c NA Glu s Glu

PX k C

k s

NA C

k ks k

8>>: 9>>; NA C

k

kk ks k

9>>; PTCA Cs

NA C

s k

8>>: 9>>;:B14

Inserting this result into the equation above, thetime course of Glu23 follows

Glu23 NA Glu est s Glu PX k

C

k s est

NAC

kkskest NAC

k

kkkskest

8>>: 9>>;

PTCA C

s

NAC

sk

8>>: 9>>; ests Glu PX k Ck s

NAC

kksk ekt NAC

k

kkksk ekt

8>>: 9>>;

PTCA C

s

NAC

sk ekt

8>>: 9>>; B15




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C

k s

1 est

NA C

k ks k

ekt est

NA C k

kk ksk

8>>:

ekt est

PTCA C

s

1 est

NA C

s k

ekt est8>

>:

9>>;

9>>;

: 19

As before, we want to analyze the assumption of aconstant precursor function, i.e., k?? and find


C

k s 1 est

NA C

ksk ekt est

8>>: 9>>; PTCA Cs

1 est

8>: 9>;

NA Glu est s Glu PX C

s 1 est

NA C

sk ekt est

8>: 9>; PTCA Cs

1 est

8>: 9>; ;B16

which is exactly what would be found if the nonhomoge-neous differential equation would be solved with theassumption the Pyr35 C [Pyr] 5 const.

Under the assumption, again, that NA is negligiblecompared with C, it can be eliminated from the equationabove, which results in

Glu23 C Glu PX 1 est

s

sk ekt est

8: 9;n PTCA 1 e

sto: 20

Modeling of13



mathematical modeling of 13c label

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