Possible Pitfalls of Flux Calculations Based on 13C-Labeling

Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174, available online at http:www.idealibrary.com on

Possible Pitfalls of Flux Calculations Based on

13

C-Labeling

Wouter van Winden,* , -, 1 Peter Verheijen, - and Sef Heijnen* *Bioprocestechnology Group and -Process Systems Engineering Group, Faculty of Applied Sciences, Delft University of Technology, The Netherlands Received July 11, 2000; accepted October 13, 2000; published online January 26, 2001

Metabolic engineers have enthusiastically adopted the 13C-labeling technique as a powerful tool for elucidating fluxes in metabolic networks. This tracer technique makes it possible to determine fluxes that are unobservable using only metabolite balances and allows the elimination of doubtful cofactor balances that are indispensable in flux analysis based on metabolite balancing alone. The 13C-labeling technique, however, relies on a number of assumptions that are not free from uncertainties. Two possible errors in the models that are needed to determine the metabolic fluxes from labeling data are omitted reactions and ignored occurrence of channeling. By means of two representative examples it is shown that these modeling errors may lead to serious errors in the calculated flux distributions despite the use of labeling data. A complicating fact is that the model errors are not always easily detected as poor models may still yield good fits of experimental data. Results of 13C-labeling experiments should therefore be interpreted with appropriate caution.  2001 Academic Press

INTRODUCTION Over the past two decades the 13C-labeling technique has become a well-established tool in metabolic flux analysis. The reason for this is that this 13C-tracer method adds important information to the extracellular net conversion measurements, which constitute the only available measurement data for ``classical'' flux analysis based on mass balancing alone. The 13C-tracer method does not replace, but complements the classical method. Therefore, it always offers at least as good flux estimates as the basic flux analysis. Moreover, it may, e.g., allow evaluation of separate fluxes through parallel pathways. One of the major advantages of flux analysis based on 13 C-labeling is that the extra information allows the elimination of uncertain mass balances from the set of balances. Mass balances that are often regarded as doubtful are those of metabolic cofactors ATP and NAD(P)H (van Gulik et al., 1995; Marx et al., 1996, 1999; Sauer et al., 1997; 1 To whom correspondence and reprint requests should be addressed at Kluyver Laboratory for Biotechnology, Julianalaan 67, 2628 BC Delft, The Netherlands. Fax: +31-15-278-2355. E-mail: W.A.VanWindentnw. tudelft.nl.

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Schmidt et al., 1998; Szyperski, 1998; Wiechert et al., 1997). Cofactor mass balances rely on estimated reaction stoichiometries and on controversial assumptions. Examples of the latter are a growth-independent ATP yield of the respiratory system (PO-ratio), fixed growth and non-growthassociated maintenance requirements of the cell, the presence of ATP-wasting futile cycles, NAD(P)H specificities of various enzymes, and the presence or absence of transhydrogenases interconverting NADH and NADPH. Furthermore, cofactors are involved in many reactions in the cell such that it is hardly possible to take all of them into account. As a consequence, cofactor mass balances are most probably far from complete and may lead to erroneous flux estimates when included in the mass balances. However, it should not be forgotten that the assumptions on which the 13C-labeling method is based are not free of uncertainties either. The following assumptions have been made explicit by Schmidt et al. (1997), Wiechert et al. (1997), and Szyperski (1998): (1) The network stoichiometry included in the metabolic model is complete. Any omitted reaction should be insignificant in its effect on the labeling state of the biomass; (2) Complete biochemical information on the fate of each carbon atom in the modeled reactions is available; (3) Metabolic isotope effects are absent, i.e., enzymes do not distinguish between molecules containing various numbers of labeled carbon atoms; (4) All reactions take place in compartments where metabolites are homogeneously distributed. This assumption includes: (4a) If occurring, metabolite channeling, i.e., direct transfer of metabolites from one enzyme to the next, must be taken into account; and (4b) If present, compartments in eukaryotic cells should lead to the inclusion of separate metabolite pools in the model. Although this list of assumptions has been made explicit, the full impact of some of the assumptions has not always 1096-717601 35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

been realized. An example is the error that is introduced in flux calculations based on labeling data by making assumptions about the reversibility of fluxes in a metabolic network. It was pointed out only recently that rough assumptions regarding the reversibility (e.g., assumed unidirectionality or isotopic equilibrium) are not good enough for metabolic models, since reversibility may have a severe effect on the labeling state of the metabolites and erroneous assumptions may therefore invalidate the calculated fluxes. This fact was illustrated by Follstad et al. (1998) who simulated the influence of reversible reactions in the pentose phosphate pathway (PPP from here on) on the fractional enrichments of the pathway intermediates and by Wittmann et al. (1999) who did the same for the mass distributions. Finally, to our knowledge, none of the flux analyses based on 13C-labeling data that have been published so far featured a statistical scrutiny that included testing a sensitivity analysis of the outcomes to erroneous or incomplete model assumptions. Consequently, flux analyses using the tracer method are often presented without any reservations. In this paper two possible pitfalls of the 13C-labeling method are presented. The pitfalls apply to the first and fourth assumptions in the list above. PITFALL I: INCOMPLETE METABOLIC REACTION MODELS The main assumption underlying metabolic modeling in general is that the network stoichiometry that is included in the metabolic model is complete. This assumption may not be as trivial as it seems. Although metabolic models are often based on a one-enzymeone-reaction scheme, the literature reports many enzymes that are more permissive, accepting broad ranges of substrates. Moreover, microbial genome sequencing combined with functional genomics shows that the exact amino acid sequences and thereby the specificities of enzymes may vary from species to species, which makes it dangerous to automatically assume standard textbook biochemistry when defining metabolic pathways for a given microorganism (Cordwell, 1999). Examples of broad-specificity enzymes are the f16p 2 aldolase which accepts a wide range of aldehydes instead of 2 Abbreviations used: glc, glucose; g6p, glucose 6-phosphate; f6p, fructose 6-phosphate; 6pg, 6-phosphogluconate; f16p, fructose 1,6-bisphosphate; g3p, glyceraldehyde 3-phosphate [in Figs. 1 and 2 this represents a lumped pool which includes dhap (dihydroxyacetone phosphate), 13pg (1,3-bisphosphoglycerate), 2pg and 3pg (2- and 3-phosphoglycerate), and pep ( phosphoenolpyruvate)]; pyr, pyruvate; p5p, pentose 5-phosphate [in Figs. 1 and 2 this represents a lumped pool of r5p (ribose 5-phosphate), ru5p (ribulose 5-phospate), and x5p (xylulose 5-phosphate)]; e4p, erythrose 4-phosphate; s7p, sedoheptulose 7-phosphate; tre, trehalose; man, mannitol; his, histidine; ery, erythritol; tyr, tyrosine; phe, phenylalanine.

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its natural substrate g3p (Toone et al., 1989), hexokinase which accepts other sugars in place of glucose (Toone et al., 1989), polyol dehydrogenase which reduces several ketones (Toone et al., 1989), alcohol dehydrogenase which reduces a variety of aldehydes and ketones (Bradshaw et al., 1992), and pyr decarboxylase, which has a side activity as carboligase linking two aldehydes (Schorken et al., 1998). Another example of a permissive enzyme, which will be more elaborately described for an illustrative purpose, is transketolase, which transfers an active glycolaldehyde group from a ketose donor to an :-hydroxyaldehyde acceptor (Schorken et al., 1998). Transketolase is widely known to catalyze the following two reactions of the nonoxidative branch of the PPP: r5p+x5p Â Ä s7p+g3p e4p+x5p Â Ä f6p+g3p. In the two above reactions, x5p, f6p, and s7p act as donors of two-carbon fragments and g3p, r5p, and e4p act as the acceptors. According to Schorken et al. (1998), transketolases of various organisms also accept glucose as an acceptor and hydroxypyruvate as a donor. Apparently, the range of the accepted substrates is relatively large. Table 1 shows an overview of the intermediates of the nonoxidative PPP and of the glycolysis which may serve as donors and acceptors of two-carbon fragments according to their chemical structure. If we combine all two-carbon fragment donors and acceptors we obtain six transketolase reactions in the nonoxidative PPP (Table 2). This table is limited to the reactions that yield reaction products with a length of at least TABLE 1 Possible C2-Donors and Acceptors of Transketolase and C3-Donors and Acceptors of Transaldolase C2C3-donor

C2C3-acceptor

D1: x5p D2: f6p D3: s7p

A1: g3p A2: e4p A3: r5p

Shared structure

Shared structure

CH 2OH | C=O | HOCH | HCOH | R

=

Transferred by transketolase

=

Transferred by transaldolase

HC=O | HCOH | R

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TABLE 2

TABLE 3

Possible Transketolase Reactions in Nonoxidative PPP

Possible Transaldolase Reactions in Nonoxidative PPP

Donor_acceptor

Resulting reaction

Donor_ acceptor

Resulting reaction

D1_A1 D1_A2 (forward) or D2_A1 (backward) D1_A3 (forward) or D3_A1 (backward) D2_A2 D3_A2 (forward) or D2_A3 (backward) D3_A3

TK1: x5p+g3p  Äg3p+x5p TK2: x5p+e4p  Äg3p+f6p TK3: x5p+r5p  Äg3p+s7p TK4: f6p+e4p  Äe4p+f6p TK5: s7p+e4p  Är5p+f6p TK6: s7p+r5p  Är5p+s7p

D2_A1 D3_A1 (forward) or D2_A2 (backward) D3_A2

TA1: f6p+g3p  Äg3p+f6p TA2: s7p+g3p  Äe4p+f6p TA2: s7p+e4p  Äe4p+s7p

three carbons and at most seven carbon atoms. The reasons for this limitation are that intermediates of less than three carbon atoms have never been reported for the nonoxidative PPP. Products of more than seven atoms, e.g., octulose, have been included in PPP models by McIntyre et al. (1989) and by Williams et al. (1987), but were only experimentally observed in small quantities in liver tissue, erythrocytes, and higher plants. Reactions TK2 and TK3 of Table 2 are the commonly applied transketolase reactions that were mentioned before. Reactions TK1, TK4, TK5, and TK6 are normally not considered in metabolic networks. Reactions TK1, TK4, and TK6 have no net stoichiometric effect, so they may be ignored in flux analysis based on mass balancing. However, these reactions do have an effect on the labeling distribution of pentose pathway intermediates so they may not be ignored when metabolic fluxes are studied using the 13 C-labeling technique. The occurrence of these additional transketolase reactions was clearly demonstrated by among others Clark et al. (1971) and Flanigan et al. (1993). Although Flanigan et al. explicitly warned for the adverse effects of the transketolase exchange reactions on isotopebased flux analyses, this seems to have been largely ignored by the metabolic engineering community. This fact may come from a confusion of words that is illustrated by a recent article by Christensen et al. (2000). In their article, Christensen et al. refer to the article of Flanigan et al. but confuse Flanigan et al.'s meaning of exchange reactions (i.e., exchange of a two-carbon fragment) with the exchange reactions that were introduced by Wiechert et al. (1997) as a suitable mathematical form to include the reversibility of reactions in metabolic models. These two meanings of the term exchange reaction are clearly different. Due to the wrong interpretation of the term, the additional model complication by reactions TK1, TK4, and TK6 was not realized. Considering the evidence for the occurrence of all other transketolase reactions, the absence of reaction TK5 in the numerous papers devoted to the PPP is surprising, especially because this reaction has a net stoichiometric effect 153

and thus should be included in mass balances of four of the PPP intermediates. The authors of this article see no compelling reason why this reaction would not take place as well. Transaldolase, the other enzyme of the nonoxidative PPP, has also been reported to be permissive to more substrates (Berthon et al., 1993) than those which it converts in the commonly known reaction: g3p+s7p Â Ä f6p+e4p. Therefore, similarly to transketolase, more transaldolase reactions can be hypothesized than those that are traditionally included in models of the PPP. Combining all three-carbon fragment donors and acceptors from Table 1 and selecting the combinations that yield reaction products with a length between three and seven carbon atoms, we obtain three potential transaldolase reactions. These are listed in Table 3. Reaction TA2 is the commonly applied transaldolase reaction. Reactions TA1 and TA3 do not have a net effect on the stoichiometry, but do influence the label distribution. The occurrence of reaction TA1 has been demonstrated by Ljungdahl et al. (1961). For unknown reasons, reaction TA3 has never been mentioned before. The impact of the omission of the ``forgotten'' reactions of Tables 2 and 3 on the outcome of 13C-labeling experiments will be illustrated by an example that is based on the metabolic network of the glycolysis and the PPP (Fig. 1). The network which includes the traditional reactions (Fig. 1-I) and the additional four transketolase and two transaldolase reactions (Fig. 1-II) is assumed to be the true metabolism. This extended model was used to generate a series of ``measured'' labeling data for a chosen set of fluxes and for a specific substrate labeling. The chosen flux values are shown in Fig. 1. Their values are deemed representative for those found in literature. The fluxes leading from the intermediates toward synthesis of biomass components are often small compared to the throughput of the central carbon metabolism and were set at zero in order to keep the model as simple as possible. The

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

FIG. 1. The ``true'' metabolism (Model 1) including the traditional reactions (I) and all the hypothetical transaldolase and transketolase reactions (II). The chosen flux values are shown as well. The double-headed arrows indicate bidirectional reactions and the corresponding double flux values denote the forward and backward reactions. Solid and dashed lines are only used to avoid visual confusion. The forward transketolase and transaldolase reactions are those listed in Tables 2 and 3. The corresponding backward reactions should be read from right to left in these tables. The forward reaction & 2 runs in the direction of f6p, the backward reaction toward g6p. The gray shaded boxes represent the molecules of which the NMR spectra are simulated. For abbreviations, see footnote 2.

p5p and g3p pools are lumped pools (see footnote 2) which consist of intermediates which are usually assumed to be in isotopic equilibrium due to fast exchange reactions. The fluxes of the network of Fig. 1 were chosen such that none of the intermediates accumulates. In Fig. 1-II the fluxes of reactions TK1, TK4, TK6, TA1, and TA3 could be freely chosen, since they do not have a net stoichiometric effect. They and the flux of the bidirectional reaction TK5 were chosen to be of the same order of magnitude as the traditional transketolase and transaldolase reactions TK2, TK3, and TA2 in Fig. 1-I. The applied substrate labeling was assumed to be 100 uniformly labeled glucose and 90 0 unlabeled glucose. The labeling data that were generated are 2D [ 13C, 1H] COSY NMR data (Szyperski, 1995) of a number of cell components that are derived from the intermediates. They were simulated using a model based on the method for isotopomer modeling authored by Schmidt et al. (1997). The exact nature of the simulated labeling data is explained in Appendix A. The model of Fig. 1-I represents a part of the carbon metabolism studied by many researchers who determined the fluxes for different microorganisms using 13C-labeling experiments. The split ratio between the glycolysis and the PPP has been a major target of these studies. In the reported studies, the NMR spectra of the following biomass components were measured in order to derive the labeling 154

pattern of the intermediates of the glycolysis and PPP: glucan (giving information about g6p), glycogen (g6p), trehalose (g6p), chitin (f6p), histidine (p5p), tyrosine (g3p, e4p), phenylalanine (g3p), glycine (g3p), serine (g3p), and guanosine (p5p) (Szyperski, 1995; Marx et al., 1996, 1997; Schmidt et al., 1999; Christensen et al., 2000). The biomass components of which the NMR spectra are simulated in the present study (i.e., the gray shaded boxes in Fig. 1) represent a data set that is similar to one which has actually been measured in as yet unpublished experiments that were performed in our laboratory. This extensive set of labeling data offers labeling information of all glycolysis and PPP intermediates except s7p. The set of measured labeling data that were generated using the model of Fig. 1 (Model 1) is shown in Table 4. This table also shows the set of labeling data that were generated using the model of Fig. 2 (Model 2) which only contains the glycolytic and the traditional PPP reactions. The fluxes of Model 2 were fitted in order to minimize the sum of squared residuals between the labeling data generated using this model and the measured labeling data of Model 1. Figure 2 shows that the optimally fitted Model 2 does not only fail to yield the values of the transketolase and transaldolase fluxes of Fig. 1-II; the fitted fluxes also deviate from those in Fig. 1-I. The nonoxidative PPP fluxes are clearly wrongly determined. More seriously, the amount of glucose

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

TABLE 4 Comparison of Data Simulated with the ``True Metabolism'' of Model 1 and Fitted Using Model 2

Molecule and carbon atom a

``Measured'' data (Model 1)

Fitted data (Model 2)

Absolute difference

Standard deviation b for ;=0.015

His :-s His :-d1 His :-d2 His :-dd His ;-s His ;-d His ;-t His $-s His $-d Tyr :-s Tyr :-d1 Tyr :-d2 Tyr :-dd Tyr ;-s Tyr ;-s Tyr ;-d Tyr ;-t Tyr 26-s Tyr 26-d Tyr 26-t Tyr 35-s Tyr 35-d Tyr 35-t Phe :-s Phe :-d1 Phe :-d2 Phe :-dd Phe ;-s Phe ;-d Phe ;-t Man 16-s Man 16-d Man 25-s Man 25-d Man 25-t Tre 1-s Tre 1-d Tre 2-s Tre 2-d Tre 2-t Tre 3-s Tre 3-d Tre 3-t Tre 4-s Tre 4-d Tre 4-t Tre 5-s Tre 5-d Tre 5-t Tre 6-s Tre 6-d Ery 14-s Ery 14-d

0.107 0.097 0.004 0.792 0.185 0.469 0.346 0.287 0.713 0.113 0.005 0.143 0.739 0.105 0.105 0.798 0.097 0.098 0.804 0.098 0.191 0.343 0.466 0.113 0.005 0.143 0.739 0.105 0.798 0.097 0.121 0.879 0.126 0.216 0.658 0.127 0.873 0.120 0.211 0.669 0.175 0.166 0.659 0.111 0.087 0.802 0.097 0.020 0.883 0.099 0.901 0.210 0.790

0.111 0.091 0.004 0.793 0.183 0.470 0.347 0.246 0.754 0.119 0.006 0.134 0.741 0.111 0.111 0.792 0.096 0.102 0.800 0.097 0.202 0.335 0.463 0.119 0.006 0.134 0.741 0.111 0.792 0.096 0.130 0.870 0.133 0.216 0.651 0.139 0.861 0.130 0.207 0.662 0.184 0.163 0.653 0.112 0.094 0.794 0.098 0.0200.882 0.100 0.900 0.222 0.778

0.004 0.006 0.000 0.001 0.002 0.001 0.001 0.041 0.041 0.006 0.001 0.009 0.002 0.006 0.006 0.005 0.001 0.004 0.004 0.000 0.011 0.008 0.003 0.006 0.001 0.009 0.002 0.006 0.005 0.001 0.009 0.009 0.007 0.000 0.007 0.011 0.011 0.010 0.003 0.007 0.008 0.003 0.006 0.001 0.007 0.008 0.001 0.000 0.001 0.001 0.001 0.012 0.012

0.013 0.019 0.021 0.026 0.023 0.022 0.022 0.012 0.012 0.015 0.022 0.018 0.027 0.024 0.024 0.031 0.026 0.023 0.032 0.027 0.023 0.020 0.024 0.014 0.020 0.019 0.026 0.024 0.033 0.027 0.013 0.013 0.024 0.019 0.024 0.013 0.013 0.023 0.019 0.025 0.023 0.018 0.025 0.022 0.019 0.028 0.024 0.021 0.029 0.014 0.014 0.012 0.012

FIG. 2. The ``traditional'' metabolism (Model 2) showing the optimally fitted flux values for noise-free ``measurement'' data.

entering the PPP is seriously overestimated (by more than 500). This latter overestimation leads to erroneous estimations of amounts of NAD(P)H and ATP that are involved in this part of the carbon metabolism. It should be noted that the deviations between the fluxes (Model 2) and the ``true'' fluxes (Model 1) depend on the sizes of the new transaldolase and transketose reactions that were assumed (Fig. 1-II). When the sizes of TK1, TK4, TK6, TA1, and TA3 were reduced to only 20 0 of the values in Fig. 1-II and the sizes of the forward and backward reactions TK5 were reduced to 8 and 2, respectively, the optimally fitted flux v 5 reduced from 34 (see Fig. 2) to 26. This means that the amount of glucose entering the PPP is still overestimated by almost 25 0. From Table 4 it is clear that the optimally fitted Model 2 yields labeling data that do not deviate much from those of Model 1 when compared to fits of similar data sets in literature [e.g., (Schmidt et al., 1999)]. This is a rather surprising result if one considers that Model 1 contains seven more reactions than Model 2 (the bidirectional reaction TK5 consisting of a separate forward and backward flux). The only data in Table 4 that show any difference are the relative peak areas of the $-carbon of histidine. It must be emphasized that Model 1 was used to generate error-free 2D [ 13C, 1H] COSY NMR data and that the fluxes of Model 2 were fitted to these error-free data. Due to the absence of any measurement error, the lack of fit between the two sets of relative intensities is caused by a 155

a

The names and symbols are explained in Appendix A. The standard deviation of the noisy ``measurement'' data that are generated for the level of noise where the modeling error is just detectable at a 5 0 significance level (see text). b

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FIG. 3. The minimized covariance weighted sum of squared residuals (SS min, weighted ) of the noise-free data of Model 1 and the fitted data of Model 2 (g, performed fits, , approximation of Eq. (1)) and the onetailed probability of Eq. (2) () versus level of measurement noise (;).

modeling error. Therefore, Table 4 theoretically indicates the presence of a modeling error. In practice, however, measurement errors will be present, making it more difficult to detect a modeling error. The maximal level of measurement noise at which the above modeling error is still detectable can be determined using statistical testing. This is discussed in the following section.

The SS min, weighted of Eq. (1) represents the modeling error, since it results from a fit based on the noise-free measurement data of Model 1. When noise is added to the measurement data, it is expected that the SS min, weighted will increase due to a contribution of the pure measurement error. This contribution has a / 2 distribution with a number of degrees of freedom that equals the number of independent data minus the number of parameters of Model 2. Table 4 shows that the number of simulation data equals 52 of which 18 are dependent, since the relative intensities of the 18 simulated spectra sum up to 1. Model 2 has 5 degrees of freedom, leading to a total degree of freedom of the a / 2 distribution of 52&18&5=29. The expected mean outcome of a / 2 distribution with 29 degrees of freedom equals 29. Therefore, it is expected that when Model 2 is used to fit the measurement data of Model 1 to which noise is added, SS min, weighted will be 29 higher than those shown in Fig. 3. This was verified by means of Monte-Carlo simulations at various levels of measurement noise. The various levels of noise that were added to the data of Model 1 were generated from the corresponding covariance matrices as described by Johnson (1987). To test whether the model error is detectable, the SS min, weighted of the noisy data was tested for the null hypothesis that Model 2 is correct. Using Eq. (1), the test can be formulated as follows:

Lack of Fit Detection Threshold To determine how accurate the NMR measurements should be in order to make the modeling error detectable, it was determined how much the modeling error and pure measurement error contribute to the lack of fit for several levels of measurement noise. For this purpose, five covariance matrices of the measurement data of Model 1 were generated at increasing levels of measurement noise (see Appendix B). The magnitude of the error is expressed by the variable ;, which is a measure for the relative error. The covariance matrices were used to weigh the sum of squared residuals in five fits of the simulated data of Model 2 to those of Model 1. The resulting minimized weighted sums of squared residuals are shown in Fig. 3. It can be clearly seen that at increasing levels of noise (i.e., increasing values of ;), the weighted sum of squares decreases. This follows from the fact that the unweighted sum of squared residuals remains constant, whereas the covariance matrix by which the sum is weighted increases with ;. From the data in Fig. 3 it can be derived that the minimized covariance weighted sum of squared residuals (SS min, weighted ) equals SS min, weighted =

1 . ; 2 } 316

(1) 156

\

SS min, weighted

1

\; } 316+29+ } t/ (29) <:. +

P SS min, weighted 

2

2

(2)

The probability P of Eq. (2) is plotted versus ; in Fig. 3. According to Eq. (2), the null hypothesis must be rejected when the probability of the observed deviation is smaller than :. When : is chosen to be 0.05, the null hypothesis is rejected (i.e., the modeling error is detected) for values of ; smaller than 0.015. The covariance matrix for ;=0.015 was used to generate a large number of noisy measurement data sets. The standard deviations of the separate relative intensities in these sets were determined and are shown in Table 4. The allowed errors vary from a relative error of over 500 0 for small relative intensities (e.g., his :-d2) to a relative error of 1.5 0 (e.g., tre 1-d). This example gives an impression of the accuracy of the NMR measurements that is required to detect the omission of the additional transaldolase and transketolase reactions. Note that the accuracy of the NMR measurements that is required to detect the omission in the model depends on the model error and thus on the sizes of the new transaldolase and transketose reactions that were assumed in Model 1.

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It was already pointed out before that the $-carbon of histidine largely contributes to the modeling error part of the sum of squared residuals (see Table 4). A consequence of this fact is that the maximally allowed level of noise at which the modeling error is detectable drastically decreases if the 13C-labeling data of this carbon atom are not included in the fit. This illustrates the importance of measurement data that are sensitive toward modeling errors. It is therefore recommended to look for these model-sensitive measurement data in the design of future labeling experiments. Additional

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C-Labeling Data

To check whether a more extensive set of 13C-labeling data would increase the lack of fit between Models 1 and 2, the dataset of Table 4 was extended with a set of MS data including all the mass fractions of the intermediates of the PPP and of the branch point intermediates of the glycolysis and the PPP. These 37 mass fractions (7 for both g6p and f6p, 6 for p5p, 5 for e4p, 8 for s7p, and 4 for g3p) were also simulated with the model of Fig. 1 and fitted by varying the fluxes of the model of Fig. 2. The resulting fitted flux set was only marginally different from the set shown in Fig. 2 and fitted both the NMR and MS data very well (results not shown). In other words, in this specific example even the extensive dataset of combined MS and NMR data does not reveal the serious modeling errors, unless the measurement errors are accurately known and small. Note that a general conclusion about the identifiability of the modeling error cannot be given here, because the sensitivities of the simulated data to the modeling error depend on the actual values of the metabolic fluxes. PITFALL II: MICROCOMPARTMENTATION DUE TO METABOLITE CHANNELING Besides the assumption of a correct metabolic network that was discussed in the previous section, another important assumption of the 13C-labeling method is that metabolite channeling must be taken into account when it is proven to occur in cells. Metabolic channeling describes the mechanism whereby the product of an enzymatic reaction is transferred to the next enzyme without mixing with the bulk-phase metabolite pool (Kholodenko et al., 1996). In their review of metabolic network analysis, Christensen et al. (1999) mention the influence of channeling on the outcome of 13C-labeling experiments. Several researchers have included the possibility of channeling in the metabolic models they used for the simulation of 13C-labeling (Portais et al., 1993; Schmidt et al., 1999). In each of these papers channeling is 157

only mentioned as a mechanism that causes orientationconserved transfer of symmetric molecules which would otherwise cause scrambling of the labeling pattern. The occurrence of orientation-conserved transfer is controversial, as Christensen et al. (1999) show in their review of the scientific debate regarding the channeling of the TCA cycle intermediates succinate and fumarate. The symmetry of these molecules causes randomization of the C 1 C 4 and C 2 C 3 labeling, unless the molecules are passed from succinate thiokinase to succinate dehydrogenase and fumarase without being released. In that case their orientation is conserved and scrambling does not occur. In fact, orientation-conserved transfer is only one of the effects that channeling may have on the label distribution in a metabolic network. Channeling between two enzymes may also lead to microcompartmentation of the reaction intermediate. In contrast to orientation-conserved transfer, this effect of channeling has never been taken into account in 13C labeling studies. Still, it may be a common phenomenon in many major pathways of both eukaryotic and prokaryotic cells (Ovadi et al., 1988; Mathews, 1993; Kholodenko et al., 1996). In glycolysis direct enzyme-toenzyme transfer of metabolites has been claimed to occur between aldolase and g3p dehydrogenase (Ovadi et al., 1978a,b, 1983, 1990). This claim is not uncontroversial as is evidenced by the paper of Kvassman et al. (1988). Other glycolytic enzymeenzyme interactions have been claimed for g3p dehydrogenase and phosphoglycerate kinase (Srivastava et al., 1986). Interactions between glycolytic enzymes and enzymes of the PPP have been studied by Wood et al. (1985). They found evidence of complexes between g3p dehydrogenase, transketolase, and transaldolase. Debnam et al. (1997) concluded from their research that intermediates of the oxidative branch of the PPP are channeled between hexokinase, g6p dehydrogenase, and 6pg dehydrogenase. Further channeling complexes have been suggested between aldolase and glycerol phosphate dehydrogenase, between aspartate aminotransferase and glutamate dehydrogenase (Ovadi, 1990), and between the enzymes of the tricarboxylic acid cycle (Haggie et al., 1999). It is important to realize that microcompartmentation due to channeling clearly conflicts with the assumption of homogeneous metabolite pools unless the microcompartments are considered as separate pools. When this fact is neglected, the 13C-labeling data of the pools will give inaccurate information about ratios of the fluxes entering the pool, since the ratios only reflect those of the nonchanneled flux fractions. To our knowledge the impact of this fact has never been acknowledged. The importance of the fact will be illustrated by means of an example system shown in Fig. 4.

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If we define v i as the flux catalyzed by enzyme e i , irrespective of the state of complexation of e i , and if we assume that in Fig. 4-II : 0 of the flux v 2 is channeled, then we can derive what percentage ; of flux v 3 must be channeled in order to fulfil the constraint that the absolute amount of C that is channeled by e 2 must equal the amount channeled by e3 : v2 v 1 +v 2 =v 3 . ;=: } : } v 2 =; } v 3 (v 1 +v 2 )

=

(3)

Assume that in Fig. 4 the fluxes are v 1 =100 mols, v 2 =200 mols, and by consequence v 3 =300 mols. If : equals 750, then Eq. (3) tells that b must equal 50 0. Furthermore, assume that the label distribution of metabolites A and B is known. Written in the form of isotopomer distribution vectors (Schmidt et al., 1997), the labeling shown in Fig. 4 is a=(1,0) T and b=(0,1) T. From these isotopomer distribution vectors, the labeling of C can be calculated as follows for cases I and II in Fig. 4: O I: v 1 } a +v 2 } b =v 3 } c =(v 1 +v 2 ) } c o     v1 v2 c= } a+ } bo O  (v 1 +v 2 )  (v 1 +v 2 ) 

FIG. 4. A metabolic node with and without channeling. A black sphere denotes a 13C-atom and a white sphere a 12C-atom. (I) Metabolites A and B are converted to C by enzymes e 1 and e 2 . C is converted to D by enzyme e 3 . (II) A fraction of e 2 forms an enzyme complex with enzyme e 3 . This e 2e 3 complex converts B to D and channels the intermediate C without releasing it. (III) All of the enzyme e 2 complexes with enzyme e 3 and forms a ``leaky channel'' from which some of the intermediate C leaks to the bulk pool.

Figure 4-I represents the standard situation around a metabolic branch point. In Fig. 4-II, enzymes e 2 and e 3 are assumed to be present both in complexed (allowing channeling) and in uncomplexed forms, which leads to a certain fraction of a one-carbon molecule C that is channeled and does not mix with the bulk pool of C. Alternatively, as shown in Fig. 4-III, it may be assumed that enzyme e 2 is only present in the complexed state but forms a so-called ``leaky channel'' (Ovadi, 1990). In this case the nonleaked fraction of C does not mix with the bulk pool of C. In both cases II and III, the label distribution of the free metabolite C does not only depend on the fluxes through enzymes e 1 and e 2 but also on the complexed fraction of enzyme e 2 (case II) or on the leaked fraction of the channeled metabolite (case III). 158

13 200 0 100 1 } } + = c= 300 1 23  300 0

\+

\+ \ +

(4)

II: v 1 } a +(1&:) } v 2 } b =(1&;) } v 3 } c    O =(v 1 +(1&:) } v 2 ) } c o v1 (1&:) } v 2 c= } a+ } bo O  (v 1 +(1&:) } v 2 )  (v 1 +(1&:) } v 2 )  0 23 50 100 1 + = . } } c= 13 150 1  150 0

\+

\+ \ +

(5)

The outcomes of Eqs. (4) and (5) show that the fraction of the flux v 2 that is channeled influences the labeling state of C. In case I, measuring the labeling state of C by means of NMR or MS yields one independent measurement that fixes the flux ratio v 1 v 2 . This solves the flux analysis if only one of the fluxes is known. In this case, measuring the label distribution of metabolite D will not yield any extra information. In case II, however, measuring only the labeling state of C does not suffice to solve the two degrees of freedom formed by the flux ratio v 1 v 2 and by the parameter :. In this case, the measurement of the labeling state of D is required too:

Pitfalls of the

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

I: v 3 } c =v 3 } d o O   c =d o O   13 d= 23 

of several intermediates of a linear pathway differ, then channeling effects or unknown inputs in the intermediate pools must be expected.

\ +

(6)

CONCLUSIONS

O II: (1&;) } v 3 } c +; } v 3 } b =v 3 } d o    (1&;) } v 3 ; } v3 d= } c+ } bo O v3 v3    150 23 150 0 13 d= + = . } } 300 1 23  300 13

\ +

\+ \ +

(7)

Equation (6) shows that in case I, measuring D indeed does not give any additional information. This is obvious, since the labeling state of D always equals that of C. In case II, however, the labeling state of D forms an additional independent measurement. Combining the measured labeling states of C and D and Eq. (3), the flux ratio v 1 v 2 and the parameters : and ; can be determined. Note that Eq. (7) might have been derived in an alternative way by considering that the net result of the fluxes in case II is a flux v 1 from A to D and a flux v 2 from B to D: II: v 1 } a +v 2 } b =v 3 } d =(v 1 +v 2 ) } d o O     v2 v1 } a+ } bo O d=  (v 1 +v 2 )  (v 1 +v 2 )  13 200 0 100 1 } } + = . d= 300 1 23  300 0

\+

\+ \ +

(8)

Case III is completely analogous to case II when the fraction a of flux v 2 that is channeled in case II is read as the fraction a of flux v 2 that is not leaked from the leaky channel of case III. It will be clear that for large metabolic networks considering all the possible complexes of neighboring enzymes and introducing additional parameters for fractions of fluxes that are channeled and thus bypass metabolite pools will lead to many extra degrees of freedom in the network. These degrees of freedom may cause inobservabilities of parts of the network, even when large amounts of high-quality 13 C-labeling data are available. Therefore, the implications of channeling on the outcome of 13C-labeling studies will have to be seriously considered. To detect the occurrence of channeling in a metabolic network, one will have to measure the labeling of all the intermediates of the pathways in the network. If this is not done, one may not automatically assume that the labeling data of one of the intermediates of a linear pathway are representative for all other intermediates. If labeling results 159

The two pitfalls of the 13C-labeling technique that were described and illustrated in this paper show some of the uncertainties that still surround this technique. Both incomplete metabolic models and the occurrence of channeling between enzymes of the network may cause serious errors in the determined fluxes. If one does not take these possible sources of errors into account, the iterative numerical procedure that is commonly used to find fluxes that fit simulated to experimental data still yields an answer. At first sight, this answer may seem acceptable, because the labeling data are well fitted. It has been shown, however, that a satisfactory fit is no guarantee of a correct set of fluxes. Therefore, publications of flux analysis based on 13Clabeling studies should pay due attention to the sensitivity of the results to these modeling uncertainties. To make modeling errors detectable, as many labeled metabolites as possible should be measured. Besides a reasonably large quantity of measurement data, a thorough statistical data evaluation is required to enable the rejection of models that do not yield an acceptable fit between measured and simulated labeling data. Assuming values for the measurement errors will not do; errors will have to be firmly established based on repeated independent experiments or signalto-noise analysis of NMR or MS spectra. This allows one to check a posteriori whether modeling assumptions such as complete metabolic models and homogeneous distributions of intermediates were justifiable. APPENDIX A: EXPLANATION OF SIMULATED LABELING DATA In 2D [ 13C, 1H] COSY NMR spectra the following multiplets may be discerned: a singlet peak (denoted as ``s''), a doublet peak (``d''), a double doublet (``dd''), and a triplet (``t'') (Szyperski, 1995). Singlets and doublets are found in spectra of a two-carbon fragment within a molecule. Singlets, doublets, and double doublets are observed in spectra of three-carbon fragments. In the case that the two 13 C 13C scalar coupling constants in a three-carbon fragment are identical, only one doublet is observed and the two middle double doublet peaks overlap, leading to a triplet. Alternatively, if the 13C 13C scalar coupling constants differ, two doublets are observed (``d1'' having a larger coupling constant than ``d2'') and a double doublet is found. This appendix shows the chemical structures of the molecules of which NMR spectra were assumed to be

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

measured in the calculations in ``Pitfall I.'' The numbers between brackets indicate the numbering of the carbon atoms. For each of the spectra in Table 4 it is indicated which carbon atoms give rise to the concerning spectrum and whether the 13C 13C scalar coupling constants are identical or not. Histidine C(5)HN N C(1)O(OH)C(2)H(NH2)C(3)H2C(4)

|

n

TYR :: observed carbon atom: 2 and its (different) C 13C scalar couplings with atoms 1 and 3; TYR ;: observed carbon atom: 3 and its (identical) 13 C 13C scalar couplings with atoms 2 and 4; TYR C26: observed carbon atoms: 5 and 9 and their (identical) 13C 13C scalar couplings with atoms 4 and 6, respectively, 4 and 8. TYR C35: observed carbon atoms: 6 and 8 and their (identical) 13C 13C scalar couplings with atoms 5 and 7, respectively, 7 and 9. 13

Phenylalanine C(9)HC(8)H N N n n

N=CH

C(1)O(OH)C(2)H(NH2)C(3)H2C(4) C(7)H n N C(5)H=C(6)H

HIS :: observed carbon atom: 2 and its (different) 13 C 13C scalar couplings with atoms 1 and 3; HIS ;: observed carbon atom: 3 and its (identical) 13 C 13C scalar couplings with atoms 2 and 4; HIS $: observed carbon atom: 5 and its 13C 13C scalar coupling with atom 4.

PHE : and PHE ;: see TYR : and TYR ;. Mannitol C(1)H2(OH)C(2)H(OH)C(3)H(OH)

Tyrosine

C(4)H(OH)C(5)H(OH)C(6)H2(OH)

C(9)HC(8)H N N n n

MAN C16: observed carbon atoms: 1 and 6 and their C 13C scalar couplings with atom 2, respectively, 5; MAN C25: observed carbon atoms: 2 and 5 and their (identical) 13C 13C scalar couplings with atoms 1 and 3, respectively, 4 and 6. 13

C(1)O(OH)C(2)H(NH2)C(3)H2C(4) C(7)H n N C(5)H=C(6)H Trehalose C(6)H(OH)

C(6)OH

| C(5)H wwww O N n C(4)H(OH) n

| w C(5)H O www N n

C(1)H(OH)C(1)H(OH) N n

C(3)H(OH)C(2)H(OH)

C(4)H(OH) N

C(2)H(OH)C(3)H(OH)

TRE C1: observed carbon atom: 1 and its 13C 13C scalar coupling with atom 2; TRE C2: observed carbon atom: 2 and its (identical) 13 C 13C scalar couplings with atoms 1 and 3; TRE C3: observed carbon atom: 3 and its (identical) 13 C 13C scalar couplings with atoms 2 and 4; TRE C4: observed carbon atom: 4 and its (identical) 13 C 13C scalar couplings with atoms 3 and 5;

TRE C5: observed carbon atom: 5 and its (identical) C 13C scalar couplings with atoms 4 and 6; TRE C6: observed carbon atom: 6 and its 13C 13C scalar coupling with atom 5. 13

Erythritol C(1)H2(OH)C(2)H(OH)C(3)H(OH)C(4)H2(OH) 160

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13

13

C-Labeling Method

Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

ERY C14: observed carbon atoms: 1 and 4 and their C 13C scalar coupling with atom 2, respectively, 3. APPENDIX B: GENERATION OF COVARIANCE MATRIX

The data that are obtained from 2D [ 13C, 1H] COSY NMR experiments are the relative areas of the spectral peaks. These areas will contain measurement errors due to the spectral noise. On the basis of observations of experimental spectra, we hereby postulate an assumption regarding the experimental error which we will use to generate covariance matrices for the simulated 2D [ 13C, 1 H] COSY NMR data: The width of a spectral peak is independent of its height. As the error in the calculated peak area is a function of the variance of the spectral noise (_ 2noise ) and of the width of the peak, the absolute error of each peak in a spectrum is identical. In other words, the variance of the error in the area of a multiplet consisting of m peaks is given by

where : is a constant. Based on this assumption, we can generate covariance matrices for the multiplet areas of the spectra of the measured carbon atoms: (B1)

\ +

0 2 0 0

0 0 2 0

0 0 0 4

\ +

3 , MC= 0 = 0

0 0 2 0

\ +

Aj A tot

.

(B2)

In this equation, A j represents the summed area of the peaks of multiplet j and A tot is the total area of all the peaks of the spectrum of the observed carbon atom. The covariance matrix of the relative intensities can be derived from the covariance matrix of the multiplet areas: Cov RI =J } Cov A } J t =: } _ 2noise } J } M } J t. = = = = =

1&RI j

\A + = \ $ A + &RI \A + $ RI j k

In Eq. (B1), Cov A is the covariance matrix, and M is a = diagonal matrix that has the numbers of the peaks that constitute the multiplet as its diagonal elements. There are three options for M: = 1 0 1 0 , MB= M = =A 0 0 2 = 0

RI j =

(B3)

In Eq. (B-3), Cov RI is the covariance matrix of the relative intensities and J is the Jacobian containing the partial = derivatives of the relative intensities of each multiplet to the areas. The diagonal and off-diagonal elements of J are = calculated from Eq. (B2):

(: } _ 2noise } m),

Cov A =: } _ 2noise } M. =

the two peaks that do not overlap with the singlet by a factor of two. Furthermore, the pure singlet area is calculated by subtracting the total area of the two aforementioned triplet peaks from the observed singlet area. The simulated data presented in Table 4 are not the multiplet areas, but so called ``relative intensities'' of the various multiplets. The relative intensity of a multiplet j (RI j ) is the normalized area, which is calculated as follows:

{

for

j=k,

(B4)

tot

j

for

j{k.

(B5)

tot

Filling in these equations in Eq. (B3) yields Cov RI =; 2 } [= I &RI } i T ] } = M } [= I &RI } i T ] T,  

.

0 4

M is valid for the simplest possible spectrum (arising from =A a single 13C 13C scalar coupling with a neighboring carbon atom), consisting of a singlet (one peak) and a doublet (two peaks) (Szyperski, 1995, 1998). For spectra with nine peaks (arising from two different 13C 13C scalar couplings) M B = should be used in Eq. (B1). Finally, M C is applicable for = 13 spectra with five peaks (arising from two identical C 13C scalar couplings). In this case, it is taken into account that the triplet area is calculated by multiplying the total area of 161

(B6)

where RI is the vector containing the relative intensities of a given carbon atom, I is an identity matrix of appropriate = size, and i is a vector of the size of RI and contains only ones.  2 Finally, ; has been substituted for ((: } _ 2noise )A tot ). Equation (B6) yields the covariance matrix of the relative intensities of a spectrum. Several of these small covariance matrices relating to the various measured carbon atoms can be combined to one large block-diagonal covariance matrix. By doing so we ignore the fact that the subspectra of several carbon atoms in the 2D [ 13C, 1H] COSY NMR spectrum will have varying signal-to-noise ratios (i.e., various values of ;). Furthermore, we ignore possible overlapping of one or more peaks in a spectrum, which certainly does occur in experimental spectra. Still, the obtained covariance matrix

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Metabolic Engineering 3, 151162 (2001) doi:10.1006mben.2000.0174

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will yield more realistic estimated errors than simply assuming constant absolute or relative errors of the relative intensities, which is usually done in 13C-labeling literature.

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