BIOLOGICAL MASS SPECTROMETRY, VOL. 20, 451458 (1991)
Mass Isotopomer Analysis :Theoretical and Practical Considerations W.-N. Paul Lee,? Lauri 0.Byerley and E. Anne Bergner Department of Pediatrics, Harbor-UCLA Medical Center, lo00 W. Carson Street, Box 16, Torrance, California 90509, USA
John Edmond Department of Biological Chemistry, UCLA School of Medicine, Center for Health Sciences, Los Angeles, California 90024, USA
A theory of mass isotopmer analysis based on the well-known principle of isotope dilution mass spectrometry is reviewed. An algorithm for the determination of isotope incorporation into a metabolic substrate from a labeled precursor using mass isotopomer analysis is presented. The step include the determination of the contribution of the derivatization reagent to the observed spectrum of the derivatized substrate and the correction of contribution from I3C natural abundance using multiple linear regression analysis. Examples of the application of this theory to determine the spectrum of the trimethylsilyl derivative of the ‘pure unlabeled’ or mononuclidic cholesterol, and the calculation of mass isotopomer distribution in cholesterol due to tracer incorporation using this ‘pure unlabeled’ spectrum, are also provided.
INTRODUCTION In the past decade, stable isotopes have been increasingly used in the fields of biochemistry and medicine as internal standards for quantitative analysis and as tracers in metabolic research. Compounds labeled with or ’H in one specific position of the molecule are commonly used. Recently, the use of labeled comin several specific or all pounds containing several positions in the same molecule have been explored in metabolic studies. The incorporation of the l3C-labeled precursor produces isomers of the compounds with different molecular weights. The modeling of distribution of these mass isotopomers (isomers with different masses) provides a novel approach to the study of complex intersecting metabolic pathways.’.’ Spectral data obtained from a mass spectrometer simply represent the distribution of ions of a compound or its fragment with different molecular weights. The observed mass :charge ratio is the sum result of experimental isotope incorporation, the presence of isotopes in heteroatoms, and the presence of natural abundance of 13C in the background. In metabolic studies, labeled substrates are often analyzed as their more volatile derivatives. The derivatizing reagents often contain isotopes which contribute to the mass isotopomer distribution of the derivatized compound as well. Correction for all such contributions is necessary before one can determine the amount of isotope incorporation and its distribution contributing to the mass isotopomers in the compound of interest. The interaction of isotopes from heteroatoms and derivatization reagent to give the final distribution of
t Author to whom correspondence should be addressed. 1052-9306/91/08045 1-08 W5.00
0 1991 by John Wiley & Sons, Ltd.
mass isotopomers can be described by the well-known principle of isotope dilution mass spectrometry. In this paper, the principle is applied to describe the formation of mass isotopomers from various functional components of a molecule. A formal relationship between mass isotopomer distribution of the derivative and of the underivatized compound is presented. The determination of mass isotopomer distribution due to isotopic tracer incorporation from spectral data is an extension of the quantitative mass spectrometric method to a multicomponent mixture. Contribution from isotopes of heteroatoms and derivatization reagent, and from backnatural abundance are corrected in sequenground tial steps. The procedure of mass isotopomer analysis as applied to tracer studies with 13C-perlabeledsubstrates involves: (i) establishing the isotopomer distribution of the derivative component in the form of a ‘derivative’or ‘pure unlabeled‘ spectrum;(ii) using the ‘derivative spectrum’ to correct for its isotopomer contribution to the observed spectrum of the derivatized compound to arrive at the mass isotopomer distribution due to in the compound of interest; and (iii) correcting for the contribution of 13C natural abundance in the labeled product to give the mass isotopomer distribution due to tracer incorporation. Examples of analysis of cholesterol will be used to illustrate the application of the theory. The prediction of mass isotopomer distribution of cholesterol from the natural abundance of ”C and the quantitative analysis of mass isotopomer distribution of a labeled cholesterol from its mass spectrum will be demonstrated. ~
EXPERIMENTAL
The cholesterol reference standard was purchased from Sigma. It was converted to its trimethylsilyl derivative using Tri-Sil (a derivatizing reagent from Pierce) prior Received I 1 November 1990 Revised 13 March 1991
452
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
to gas chromatography/mass spectrometry (GC/MS) using an HP-5985 quadrupole mass spectrometer. A 2 ft x 2 mm id. glass column packed with 1% SE-30 on 100/120 Gas Chrom Q (Alltech Associate Inc.) was used. GC conditions were: carrier gas (helium) flow rate at 23 ml min-l, injector temperature 270°C, oven temperature programmed from 220 "C to 245 "C at 10"C/ min-', jet separator and inlet line at 275"C, and the source temperature at 200 "C. Electron impact (EI) spectra obtained at -70 eV was consistent with the previously published spe~trum.~ Ion clusters around m/z 458, corresponding to the molecular ion of trimethylsilyl (TMS) cholesterol, and m/z 368, corresponding to M - 90, were monitored using selected ion monitoring (SIM) for quantitative analysis. The area under each SIM ion chromatogram representing the integrated value of multiple scans was used to calculate the normalized spectra in Tables 1 and 3. Labeled cholesterol was isolated by saponification and extraction from brain tissue samples of a newborn rat pup fed with (U-13C)acetateas 10% of dietary fat for 48 h. It was further purified by high-performance liquid chromatography (HPLC) prior to derivatization for GC/MS analysis.
dances. The isotopomer with the base mass (lowest mass) is designated as X,, and the other isotopomers are represented by XI, X 2 , X,, ..., X,, where the subscripted numbers are the incremental masses of the isotopomer over the base mass, and there are ( I + 1) isotopomers of the compound X. The relative abundance (fractional molar abundance)? of these isotopic species are represented by p , , pl, p2, ..., p,, and the sum of all the ps equal to one pi = 1).
THEORY: RELATIVE DISTRIBUTION OF MASS ISOTOPOMERS OF A MOLECULE
Rule 2. Because of the assumption of independence, the probability of detecting XiYj of a compound XY is given by the product of the individual probabilities of detecting Xi and Yj. For a compound consisting of components X and Y, the mass isotopomer distribution for the compound XY is given by the product:
Assumptions As in the theory of isotope dilution mass spectrometry, the assumptions of Pickup and McPherson are necessary for the development of a theory to predict the relative mass isotopomer distribution of a molecule.4 These assumptions were originally stated in terms of isotope, atoms and elements. For the sake of the present discussion, the terms are substituted by isotopomer, molecule and compound. Assumption 1. The relative abundance of an isotopomer of any compound is equivalent to the probability that a molecule of the compound with a mass corresponding to that of the isotopomer is drawn from or detected in the sample population. In other words, ions of different isotopomers of the same species are generated in proportion to their relative abundance. Assumption 2. The probability is independent of the location of the isotopomer in the molecule, and is independent of the presence or absence of other isotopomers in the molecule. In other words, the probability of forming or detecting an ion of a mass isotopomer is independent of the location of the isotope or the presence of other isotopes. Assumption 3. For simplicity, only integer masses are used in the discussion. However, the theory developed should be applicable to discussions where exact masses are used.
Definitions The following symbols are used to describe the different isotopomers of a compound X and their relative abun-
(1
Rules Rule 1. As a consequence of assumption 1, the descrip-
tion of a mixture of molecules of this compound X can be represented by a normalized distribution:
+
(poxo + p1X, + p2X2 ... + p,X,) and c p i = 1 Similarly, a mixture of molecules of a compound Y can be denoted by : (40'0 + 41'1 + q2Y2 + * . . + qsYJ and C qi = 1 Each formula is an algebraic statement that a compound is made up of molecules of masses X, to X, (or Yo to Y,), and the distribution of these isomers are p , to p, (or 4 , to qs),respectively.
+ PlXl + P2X2 + ...) x (40 y o + 41Yl + 42 y, + .) = (Po4ox,y, + P041XOYl + Poq2XoY2 + . . - )
(Pox,
*.
The product of this algebraic multiplication gives the frequency distribution pi qj for each mass isotopomer Xi Y,; pi q j represents a simple multiplication product, whereas Xi Yj represents the combination of the two components Xi and Yj. The sum of the subscripts for XiYj, (i + j), gives the incremental mass of the compound Xi Y, above the base mass X, Yo. An interesting particular case of the above general expression is provided by the situation in which one of the components is mononuclidic, containing only I2C (Yo). Under such a condition, (q, = 1 and qj = 0 for j > 0), the mass isotopomer distribution of XY is the same as X except that the base mass is shifted from X, to X, Yo. Rule 3. When we assume that only integer masses are permissible, the notation adopted above allows one to identify symbols of molecules by their incremental masses, which is given by the sum of subscripts (i + j) of Xi and Yj. Since only integer masses are discernible,
t Fractional molar abundance refers to the concentration of an isomer species expressed as a fraction of the total number of molecules. It should not be confused with the abundance of an isotope. For example, if 3% of the cholesterol molecules contains 3 I3Cs (m + 3), the fractional molar abundance of this m + 3 species is 3%. The abundance of in that fraction is 3/27 of 3%. We designate fractional molar abundance of an isotopomer rn + i by m i , and the abundance of I3C is given by rn, x i/n. The former term is unique for mass isotopomer analysis and the latter term is equivalent to specific activity of radioisotopes.
MASS ISOTOPOMER ANALYSIS
453
terms of the above algebraic multiplication with the provided by the distribution of X adjusted for (shifted same incremental mass can be grouped together accordby) the corresponding incremental masses of the isotopomers of Y. The above relationship can be used to ing to the sum of their X and Y subscripts. The fractional molar abundance of any mass isotopomers of XY is predict the mass isotopomer distribution of a comgiven by the sum of coefficients of those terms, where pound XY when the mass isotopomer distribution of the components X and Y are known. the sums of the subscripts for X and Y are equal to the incremental mass. For instance, the fractional molar Corollaries abundance of the molecule with the lowest mass (m0) is given by the coefficient of the Xo Yo term which is p o g o . Several useful relationships can be derived based on the The fractional molar abundance of the molecule with above rules. one higher mass (ml) is given by poql + plqo; that of m2, Po 42 + P2 40 + P141; and so on. Corollary 1. If a molecule is made up of two components The relationship between the mass isotopomer disX and Y consisting of isotopomers ( X o , XI, X, , ...,X,) tribution (mO, m l , m2, ..., m(r + s)) and the individual and (Yo, Y,, Y,, ...,Y$, the maximum number of mass relative isotope abundance of X (p0,pl, p , , ..., pr) and isotopomers of the molecule XY is given by (r + s + 1). Y (qo, ql, q , , ..., qs) can be represented by matrix multiplication as shown in Fig. 1. The matrix which Corollaryt. For a compound of n carbon atoms and contains the contribution of each mass isotopomer of X other elements, the mass isotopomer contribution from is known in the literature as the ‘abundance m a t r i ~ ’ . ~ . ~ the presence of 13Ccarbon is given by the coeficients of The term was first used by Brauman in his paper on the the binomial expansion (poxo + plXl)”, where po and use of a least-squares approach in spectral analysis.’ p1 stand for the abundance of ”C and 13C,respectively. The ‘abundance matrix’ was used by Korzekwa et al. This is the expected consequence of applying rule 2 for quantitative analysis.6 (n - 1) times. The fractional molar abundance of mass The set of linear equations resulted from the matrix isotopomer (mi)from isotopes of carbon is given by: multiplication is shown below: mi = n ! / [ ( i ! )x (n - i ) ! ] x p$‘-i)pli Po 40 = m0 For example, ‘natural’ glucose contains six carbon =ml P140 + Po 41 atoms per molecule, and each carbon is derived from carbon with natural abundance of 1.11%. (For simpli= m2 P240 + pi41 + P o 4 2 city, we assume all hydrogen atoms are of mass 1 and P 3 q O + P 2 4 1 + PI42 + P O 4 3 = m3 oxygen atoms mass 16.) The distribution of the seven mass isotopomers (six containing ”C and one of ”C) are: m0 = 0.9358, ml = 0.0624, m2 = 0.0018, m3 = 0, ... m4 = 0, m5 = 0 and m6 = 0. This example also illustrates that in most low-molecular-weight natural organic compounds composed of carbon, oxygen, P,4s-1 + Pr-14s = m(r + s - 1) hydrogen and nitrogen, the probability of finding mass isotopomer of mass > m + 4 is rather negligible due to P r 4 , = m(r + 4 the rapid drop-off in values of the higher terms of the The similarity between the above set of linear equations binomial expansion. and those used by Lee et at.’ in their discussion of quantification of stable isotopes in mass spectrometry Corollary 3. If the mass isotopomer distribution of comshould be apparent. If we consider the mass isotopomer pound XY (mO, m l , m2, ..., m(r + s)) and the relative distribution of X to be the basis for the mass spectrum isotopomer contribution from either component X or Y of XY, , the mass spectra of XY 1, XY, , XY3, etc. are are known, the relative distribution of mass isotopomer 0,
...,
P1.
Po,
...,
Pz,
P1.
P3.
Pz. ...
..., ...,
0, 0, 0, 0,
..., ..., ...,
0, 0, 0,
..., ...)
0, 0,
Po.
...
... P . 7
4 0,
PI-I.
P,, 0,
... ...
X
... 0, 0,
0, 0,
Figure 1. Matrix multiplication of the individual relative isotopomer abundances of
X,b0,pl. pZ,...,p,) and y, (40.ql, 41 -.)4 J 1.
454
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
in the remaining component can be determined from the set of linear equations shown under rule 3. The algorithm of Biemann and the multiple linear regression method for the determination of mass isotopomer distribution of a compound using its putative ‘unlabeled spectrum’ represent two algebraic methods of solving the same problem.**9 Generally, we determine the spectrum of the compound XY to be (mO, ml, m2, . .., m(r + s)), and we also know a priori, from other determinations or from the assumption of known I3C natural abundances, the values of ps (the mass isotopomer distribution of the X portion of the molecule). Inspection of the linear equations under rule 3 will show a set of simultaneous equations with the molar fractional abundance of Y (4s) as the unknowns. The coefficients of these linear equations (ps) can be grouped to give the ‘abundance matrix’. There will be r + s + 1 number of equations for s + 1 number of unknowns. Since r + s + 1 is always greater than or equal to s + 1, the problem can be solved as an overdetermined set of linear equations by multiple linear regression analysis. In matrix notation: [Aly = m; where y is the vector of mass isotopomer distribution in Y,and m the vector of the observed mass isotopomer distribution (normalized spectral intensities) of XY. Corollary 4. In the development of the above theory, we define a compound or its components as having the properties specified in the three assumptions irrespective of their molecular compositions. Therefore, the derivatized metabolic substrate can be considered as a composite of the substrate and its derivative components. This consequence lends itself to the strategy of first establishing the mass isotopomer distribution of the ‘derivative’ component of a compound for the construction of an ‘abundance matrix’ and then applying the information to the determination of mass isotopomer analysis of many substrates of biological interest.
EXAMPLES
(a) Theoretical and observed mass isotopomer distribution in naturally occurring cholesterol. The contribution of ‘natural’ cholesterol uniformly labeled with 13C (i.e. a compound having the same proportion of I3C in every carbon) to the mass isotopomer distribution can be modeled by a simple expansion of a binomial function: (Po C plC’)”, where po and p1 are the molar fractions of ”C and I3C, and p o + p1 = 1, as stated in corollary 2. The coefficient of each term can be calculated using po = 0.9889, p1 = 0.0111 (13C natural abundance of l.ll%), and n (number of carbon for cholesterol) of 27. The coefficients from the expansion of the binomial function give the contribution of I3C in cholesterol to the observed relative mass isotopomer distribution mO, ml,m2 etc. of the derivatized cholesterol standard. Alternatively, such distribution can be experimentally determined. The EI spectrum of the TMS derivative of cholesterol showed a molecular ion at m/z 459 and an ion m/z 368 which is cholesterol without the hydroxyl group. Normalized spectral intensities of the m/z 368 ion cluster are shown in Table 1. This ion fragment has a slight tendency (about 3.9%) of losing a hydrogen atom, forming a peak at m - 1. The normalized spectral of the m/z 368 ion cluster is first adjusted for the possibility of losing a proton, and the mass isotopomer distribution is given by the fractional intensities corresponding to the different mass isotopomers of cholesterol (I/(sum of I)). The observed and expected distributions are compared in Table 2. The agreement for the isotopomers m0, ml, m2 and m3 is within the precision of the method. It is one of the assumptions in the proposed mass isotopomer analysis that the molar efficiencies of different masses are equal (i.e. the same number of ions are produced and detected for each isotope species). However, mass discrimination in ion source, the mass analyzer and the detector system is well known in mass spectrometry.” When it is possible to determine the mass isotopomer distribution directly, the problem of mass discrimination can be checked by the agreement or lack of agreement between the theoretical and observed distribution. In our case, the good agreement suggests that the assumption is realistic and practical within the range of masses monitored for cholesterol.
+
(I) Determination of ‘TMSderivative’ spectrum of TMS-cholesterol
(b) Determination of mass isotopomer distributioo of TMS or ‘TMS13C(cholesterol)spectrum’ for the constructioo of the ‘abuodaoce matrix’. The existence of two ion fragments
The first step in mass isotopomer analysis is to determine the spectrum of derivative of mononuclidic cholesterol (TMS-”C(cholestero1)). The information is then used for the construction of an ‘abundance matrix’ (rule 3) for multiple linear regression analysis to correct for the contribution of the isotopes from the derivative component to the mass spectrum of the derivatized compound. Ideally, the spectra of TMS derivatives of cholesterol containing 1 to 27 I3Cs should be separately determined, and the normalized values of the experimentally determined intensities be used for the construction of the ‘abundance matrix’. Since such a set of reference standards is not available, the ‘abundance matrix’ is usually secondarily derived from a TMScholesterol spectrum of a cholesterol standard (containing 1.11% 13C).
containing the carbon skeleton of cholesterol provides us with an opportunity to demonstrate the application of corollaries 3 and 4 in mass isotopomer analysis. The m/z 458 ion of TMS-cholesterol can be viewed as our compound XY, where X is the cholesterol component and Y the TMS component. It is a common confusion as to which part of the derivatized compound should be separated as the X component. The key criteria in answering such a question is whether the group of atoms is dependent or independent of the tracer incorporation. In this analysis, the cholesterol carbons constitute OUT X component and the TMS carbons and its silicon atom form our Y.The mass isotopomer distribution of the cholesterol component (X) (accounting for all the cholesterol carbons) can be determined from the m/z 368 fragment or by simple expansion of the binomial
MASS ISOTOPOMER ANALYSIS
455
Table 1. Normalized intensities of partial spectrum of cbolesterol from six SIM analyses mlz
367
368
369
370
371
3.85 3.78 3.92 3.88 4.00 4.01
100 100 100 100 100 100
29.84 29.96 29.77 29.98 30.14 29.91
4.53 4.49 4.44 4.46 4.49 4.49
0.47 0.52 0.48 0.52 0.51 0.51
0.12 0.18 0.14 0.22 0.18 0.18
372
0.26 0.21 0.23 0.26 0.26 0.25
0.11 0.13 0.12 0.16 0.15 0.14
Average
3.91
100
29.93
4.48
0.50
0.17
0.24
0.14
Standard deviation
0.08
0
0.12
0.03
0.02
0.03
0.01
0.02
Coefficient of variation (%)
2.08
0
0.39
0.63
3.89
7.72
12.65
18.9
373
374
mass of TMS-’2C(cholesterol) (XoY), but has the mass isotopomer distribution of TMS (Y) only (see rule 2).
series as presented earlier (Table 2). The spectral data of TMS-cholesterol are presented in Table 3. The theoretical or the observed mass isotopomer distribution in cholesterol is used as the x range for this calculation. The x and y ranges of multiple linear regression analysis and the weighting factors are shown in Table 4. It should be pointed out that the x range is the ‘abundance matrix’ of Brauman.’ Since the intensities of different m/zs take on different variances (heteroscedasticity), it is necessary to normalize the variances for multiple linear regression The variances of the spectral intensities provide the proper weighting factors. The regression analysis can be performed using many microcomputer programs such as MathCad or Lotus 123. The relative concentrations of mass isotopomers are shown as the x coefficients in the regression output along with the standard errors of the estimates. Since the process of losing a proton is independent of I3C isotopes on the cholesterol, its influence on the spectral intensities is lumped into this ‘TMS”C(cholestero1) spectrum’; hence the appearance of the m - 1 ion. It should not be difficult for the reader to see that the result is the spectrum of Xo Y, which has a base
(Il) Determination of mass isotopomer distribution of a labeled cholesterol Once the putative ‘TMS-”C(cholestero1) spectrum’ is determined, the information can be applied to estimate the mass isotopomer distribution in the unknown. The spectral data of a brain cholesterol sample obtained from a newborn rat fed with perlabeled I3C(acetate)are shown in Table 3.” The ‘abundance matrix’ and regression analysis is shown in Table 5. The result of the computation is equivalent to determining the spectrum of Y from those of XY and X. In this case, we have two methods of determining the mass isotopomer distribution of cholesterol: one directly from the ions of the m/z 368 cluster and one calculated from the molecular ion of the TMS-cholesterol derivative. The agreement between the two methods of determining mass isotopomer distribution of cholesterol (Table 6) strongly suggests that conditions specified in the assumptions
Table 2. Comparison of theoretical and observed fractional molar enrichment of mass isotopomers in cbolesterol standard Theoretical Observed
mO
ml
m2
m3
m4
m5
0.7397 0.737
0.2242 0.222
0.0327 0.033
0.0030 0.004
0.0002 0.001
0.0000 0.000
Table 3. Normalized spectra of a cholesterol standard and a labeled sample mh
367
368
369
371
372
373
374
Cholesterol standard (n = 6)a Labeled cholesterolb (n = 3)
3.91
100
29.93
4.48
0.50
0.17
0.24
0.14
4.23
100
32.19
7.86
3.03
2.01
1.52
1.08
mlz
457
458
459
460
461
462
463
464
Cholesterol standard (n =6) Labeled cholesterol (n = 3)
1.57
100
38.07
10.52
2.06
0.36
0.15
0.13
1.59
100
40.35
14.35
5.02
2.44
1.71
1.26
370
‘ n = number of SIM analyses included in the normalized averaged spectrum.
Labeled cholesterol was measured as described under ExDerimental.
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
456
Table 4. Multiple linear regression analysis of cholesterol standard spectral data showing regression ranges and output mlz
0 0.740 0.224 0.033 0.003 0.0002 0 0
0.740 0.224 0.033 0.003 0.0002 0 0 0
0 0 0.740 0.224 0.033 0.003 0.0002 0
Regression output: Constant Standard error of Y estimate R2
No. of observations Degrees of freedom X coefficient@) Standard error of coefficient
mlz Normalized spectrum a
Y range'
X range
457 458 459 460 461 462 463 464
0 0 0 0.740 0.224 0.033 0.003 0.0002
0 0
0 0 0 0 0 0.740 0.224 0.033
0 0
0.740 0.224 0.033 0.003
0 0 0 0 0 0 0.740 0.224
1.57 100 38.07 10.52 2.06 0.36 0.15 0.13
Weightb
1.25 10
6.17 3.24 1.44 0.6 0.39 0.36
0 0.248 0.999 8 1
2.122 0.420 457 1.58
134.6 3.356 458 100
10.59 2.305
5.051 1.266
459 7.87
460 3.75
0.249 0.592 46 1 0.19
0.094 0.253 462 0.07
0.180 0.138 463 0.1 3
Relative intensity of mlz. Square root of Y range.
Table 5. Multiple linear regression analysis of labeled cholesterol' X range
457 458 459 460 461 462 463 464
1.577 100 7.872 3.754 0.184 0.069 0 0
0 1.577 100 7.872 3.754 0.184 0.069 0
Regression output: Constant Standard error of Y estimate
0 0 1.577 100 7.872 3.754 0.1 84 0.069
No. of observations Degrees of freedom
0 0.016 0.999 8 1
X coefficients Standard error of coefficient Coefficient of variation (%)
0.995 0.001 0.161
R2
a
0 0 0 1.577 100 7.872 3.754 0.184
0.323 0.001 0.320
Y rangeb
0 0 0 0 1.577 100 7.872 3.754
0.080 0.000 0.775
0.029 0.000 1.242
0 0 0 0 0 1.577 100 7.872
0.017 0.000 1.457
0 0 0 0 0 0 1.577 100
0.014 0.000 1.516
1.59 100 40.35 14.35 5.02 2.44 1.71 1.26
0.010 0.000 1.726
See Table 3. Relative intensity of mlz. Square root of Y range.
Table 6. Comparison of mass isotopomer distribution of a labeled cholesterol determined from two mass spectra by different methods From mlz 368" From mlz 458b
mO
ml
m2
m3
m4
m5
m6
0.677 0.677
0.218 0.220
0.053 0.054
0.021 0.020
0.014 0.012
0.01 0.009
0.007 0.007
"The individual mass isotopomer distribution is given by / i / ( ~ ~li) = ,of spectrum of mlz 368 fragment, where I= intensity and i = ions monitored from mlz 368 to m/z 374. bThe individual mass isotopomer distribution is given by the individual regression coefficient divided by the sum of all the coefficients from Table 5. It is important to note that in both instances the sum of all the ms is 1.
Weightc
1.26 10 6.35 3.79 2.24 1.56 1.31 1.12
MASS ISOTOPOMER ANALYSIS
necessary for the theory of mass isotopomer analysis have not been violated in these experimental studies. (111) Correction for incorporation of 13Cfrom natural abundance in cholesterol The mass isotopomers of the labeled cholesterol are the results of incorporation of labeled acetate and the incorporation of background acetate containing 13C at natural abundance. In order to determine tracer incorporation, the background 13C contribution to choiesterol mass isotopomers has to be subtracted. Two algorithms for background correction have been p u b l i ~ h e d . ~The * ' ~ procedure of Dunstan is a one-step calculation to correct for isotopes of the derivatization reagent and the 13C of the compound due to natural abundance. Since we have separated the isotopomer contribution of the derivatization reagent to arrive at the mass isotopomer distribution of cholesterol, we only need to consider the contribution of from natural abundance. The method for background correction is based on the recognition that as 13Cis incorporated into the cholesterol molecule from the labeled precursor, fewer carbon positions are available in the cholesterol to accept 13C from natural background. Cholesterol synthesized from natural acetate would contain 1.11% 13C. This 'unlabeled' species therefore has a mass isotopomer distribution of natural cholesterol according to the bin~ its base mass is omial distribution: (Po C + P ~ C ' ) 'and m0. When a single 13C is incorporated, the distribution of 13C from background in the remaining 26 positions would follow the binomial expansion (Po C + P ~ C ' ) ' ~ and its base mass is ml. The contribution by 13C background to cholesterol labeled with 2, 3 or 4 13C from labeled acetate precursor can be deduced similarly. In the fully labeled cholesterol, the contribution from background 13C would be zero. These distributions are used to construct another 'abundance matrix' as shown in Table 7. It should be pointed out that the subtraction
451
procedure has the same effect as the linear transformation of variables of a set of linear equations. Since there are as many equations as there are unknowns, the equations are solved exactly. The mass isotopomer distribution after background subtraction would provide the fractional enrichment of these isotopomers. Since much of the singly labeled cholesterol is due to background "C abundance, the background cholesterol fraction is 91.3%, which is made up of 67.7% of ("C)cholesterol, and approximately 20% of the singly labeled and 3% of the doubly labeled ('3C)cholesterol shown in Table 6. Examples I, I1 and 111 illustrate the main steps in the determination of mass isotopomer distribution of a labeled tracer in metabolic studies. First, the theoretical mass isotopomer distribution of the natural substance in question is constructed using the expansion of a binomial function as in example I(a). The mass spectrum of the natural analog of this compound is then used to determined the 'derivatized "C spectrum' of the compound as in example I(b). The 'derivatized "C spectrum' is then used for the determination of mass isotopomer distribution of the labeled compounds in experimental samples as in example 11. Finally, incorporation of 13Cfrom background abundance is subtracted to give the fractional molar enrichment of these isotopomers from incorporation of labeled precursors using procedure outlined in 111. DISCUSSION
Since the early experiments of Schoenheimer and Rittenberg" the application of stable isotope in metabolic research has grown e~ponentially.'~.' Concepts of stable isotope analysis using GC/MS has evolved from those of gas isotope ratio mass spectrometry. Structural information of the labeled compound is not exploited by such methods. The importance of the structural information of labeled substrates in metabolic
Table 7. Correction of 13C background from its natural abundance Theoretical distribution. m3 m4
m5
m6
Uncorrected distributionn
0 0 0 0 0 0.7840 0.1918
0 0 0 0 0 0 0.7927
0.6770 0.2190 0.0535 0.0205 0.0130 0.0095 0.0070
m0
rnl
m2
0.7418 0.2228 0.0322 0.0030 0.0002 0.0000 0.0000
0 0.7501 0.2169 0.0302 0.0027 0.0002 0.0000
0 0 0.7584 0.2109 0.0281 0.0024 0.0001
mO'
ml'
mZ'
m3'
m4'
m5
m6
0.9126
0.0209
0.0258
0.0153
0.0115
0.0086
0.0053
0 0 0 0.7669 0.2047 0.0262 0.0021
0 0 0 0 0.7754 0.1984 0.0243
Mass isotoporner distribution due to isotope incorporation
"Each column of the 'abundance matrix' contain numbers corresponding to the coefficients of a binomial expansion of b 0 C +p,C')("-'), where po = 0.989, p , = 0.011, n = 27 and i is the index for mi. The numbers are arranged in descending order such that frequency of the base mass appears at the top of the column, and the isotopomer with the largest mass appears at the bottom. Uncorrected mass isotopomer distribution from Table 6 arranged in a column from mO to m6.
458
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
studies has been well recognized, especially in the studies of gluconeogenesis and metabolism of the tricarboxylic acid (TCA) cycle^.'^.' Mass isotopomer analysis of 3C-labeled substrates provides quantitative and structural information not available to the conventional stable isotope applications. The theory behind mass isotopomer analysis is based on the same assumptions of isotope dilution mass spectrometry. The theory forms the basis for the algebraic methods such as the algorithm of Biemann* and the multiple linear regression rnethod6y9 for the determination of mass isotopomer distribution from mass spectral data. The method of Biemann is simple to apply and is often cited for mass isotopomer analysis. However, Biemann’s method does not take into account variability in counting statistics and has no mechanism for handling excessive residual. Multiple linear regression analysis has found many applications in mass spectrometry. Brauman used the method to separate the isotope contribution of heteroatoms to the molecular spectrum of an organic compound in spectral analysi~.~ Lee et al. suggested its use to exploit the information from multiple ions (m/z) for quantitative analy~is.~ Recently, the use of multiple linear regression to handle the problem of overlapping ions from components of a mixture was and its application in discussed by Korzekwa et
quantifying glucose mass isotopomers was reported by Lee.’ The statistics produced from the analysis can be used for the evaluation of propagation of errors and possible spectral contamination in a certain range of m/z monitored for quantitative analysk6 We have presented an algorithm for the determination of mass isotopomer distribution and isotope incorporation using the least-squares approach. The contributions from the derivatization reagent and from the natural background I3C abundance are corrected in separate steps. The method of correcting for TMS contribution is validated by excellent agreement of results calculated from spectral data of TMS cholesterol (m/z 458 cluster) and cholesterol (m/z 368 cluster). Data reduction can be performed using commercially available statistical software packages. The availability of statistical software for personal computers should make the determination of mass isotopomer distribution using multiple regression analysis more practical for metabolic studies.
Acknowledgements This work was supported by the UCLA Clinical Nutrition Research Unit, NIH/CNRU grant no. 6pOlCA42710, and Harbor-UCLA Clinical Research Center, NIH/GCRC grant no. 5MOlR00524.
REFERENCES 1. J. Katz, W-N. P. Lee, P. A. Wals and E. A. Bergner, J. Biol. Chem. 264.12 994 (1989). 2. B. Kalderon, A. Gopher and A. Lapidot, FEBS Lett. 204, 29 (1986). 3. C. J. W. Brooks, E. G. Horning and J. S. Young, Lipids 3, 391 (1968). 4. J. Pickup and K. McPherson, Anal. Chem. 48,1885 (1976). 5. J. I. Braurnan, Anal. Chem. 38,607 (1966). 6 . K. Korzekwa, W. N. Howald and W. F. Trager, Biomed. Environ. Mass Spectrom. 19,211 (1990). 7. W.-N. P. Lee, J. S. Whiting, A. L. Fyrnat and H. G. Boettger, Biomed. Mass Spectrom. 12,641 (1983). 8. K. Biemann, Mass Spectrometry: Organic Chemical Applications, p. 223. McGraw-Hill, New York (1962).
9. W.- N. P. Lee, J. Biol. Chem. 264, 13002 (1989). 10. D. Hachey, W. W. Wong, T. W. Boutton and P. D. Klein, Mass Spectrom. Rev. 8, 289 (1987). 11. J. Edmond, R. A. Korsak, E. A. Bergner and W. P. Lee, FASEB J. 4, A1 932 (1990). 12. R. H. Dunstan, Biomed. Environ. Mass Spectrom. 15, 473 (1988). 13. R. Schoenheimer and D. Rittenberg, J. Biol. Chem. 127, 285 (1939). 14. D. M . Bier and D. E. Matthews, Fed. Prm. 41,2679 (1982). 15. D. Halliday and M . J. Rennie, Clin. Sci. 83,485 (1982). 16. G. Hetenyi Jr, Fed. Proc. 41, 104 (1982). 17. R. Geobel, M. Berman and D. Foster, Fed. Prm. 41, 96 (1982).
Mass Isotopomer Analysis :Theoretical and Practical Considerations W.-N. Paul Lee,? Lauri 0.Byerley and E. Anne Bergner Department of Pediatrics, Harbor-UCLA Medical Center, lo00 W. Carson Street, Box 16, Torrance, California 90509, USA
John Edmond Department of Biological Chemistry, UCLA School of Medicine, Center for Health Sciences, Los Angeles, California 90024, USA
A theory of mass isotopmer analysis based on the well-known principle of isotope dilution mass spectrometry is reviewed. An algorithm for the determination of isotope incorporation into a metabolic substrate from a labeled precursor using mass isotopomer analysis is presented. The step include the determination of the contribution of the derivatization reagent to the observed spectrum of the derivatized substrate and the correction of contribution from I3C natural abundance using multiple linear regression analysis. Examples of the application of this theory to determine the spectrum of the trimethylsilyl derivative of the ‘pure unlabeled’ or mononuclidic cholesterol, and the calculation of mass isotopomer distribution in cholesterol due to tracer incorporation using this ‘pure unlabeled’ spectrum, are also provided.
INTRODUCTION In the past decade, stable isotopes have been increasingly used in the fields of biochemistry and medicine as internal standards for quantitative analysis and as tracers in metabolic research. Compounds labeled with or ’H in one specific position of the molecule are commonly used. Recently, the use of labeled comin several specific or all pounds containing several positions in the same molecule have been explored in metabolic studies. The incorporation of the l3C-labeled precursor produces isomers of the compounds with different molecular weights. The modeling of distribution of these mass isotopomers (isomers with different masses) provides a novel approach to the study of complex intersecting metabolic pathways.’.’ Spectral data obtained from a mass spectrometer simply represent the distribution of ions of a compound or its fragment with different molecular weights. The observed mass :charge ratio is the sum result of experimental isotope incorporation, the presence of isotopes in heteroatoms, and the presence of natural abundance of 13C in the background. In metabolic studies, labeled substrates are often analyzed as their more volatile derivatives. The derivatizing reagents often contain isotopes which contribute to the mass isotopomer distribution of the derivatized compound as well. Correction for all such contributions is necessary before one can determine the amount of isotope incorporation and its distribution contributing to the mass isotopomers in the compound of interest. The interaction of isotopes from heteroatoms and derivatization reagent to give the final distribution of
t Author to whom correspondence should be addressed. 1052-9306/91/08045 1-08 W5.00
0 1991 by John Wiley & Sons, Ltd.
mass isotopomers can be described by the well-known principle of isotope dilution mass spectrometry. In this paper, the principle is applied to describe the formation of mass isotopomers from various functional components of a molecule. A formal relationship between mass isotopomer distribution of the derivative and of the underivatized compound is presented. The determination of mass isotopomer distribution due to isotopic tracer incorporation from spectral data is an extension of the quantitative mass spectrometric method to a multicomponent mixture. Contribution from isotopes of heteroatoms and derivatization reagent, and from backnatural abundance are corrected in sequenground tial steps. The procedure of mass isotopomer analysis as applied to tracer studies with 13C-perlabeledsubstrates involves: (i) establishing the isotopomer distribution of the derivative component in the form of a ‘derivative’or ‘pure unlabeled‘ spectrum;(ii) using the ‘derivative spectrum’ to correct for its isotopomer contribution to the observed spectrum of the derivatized compound to arrive at the mass isotopomer distribution due to in the compound of interest; and (iii) correcting for the contribution of 13C natural abundance in the labeled product to give the mass isotopomer distribution due to tracer incorporation. Examples of analysis of cholesterol will be used to illustrate the application of the theory. The prediction of mass isotopomer distribution of cholesterol from the natural abundance of ”C and the quantitative analysis of mass isotopomer distribution of a labeled cholesterol from its mass spectrum will be demonstrated. ~
EXPERIMENTAL
The cholesterol reference standard was purchased from Sigma. It was converted to its trimethylsilyl derivative using Tri-Sil (a derivatizing reagent from Pierce) prior Received I 1 November 1990 Revised 13 March 1991
452
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
to gas chromatography/mass spectrometry (GC/MS) using an HP-5985 quadrupole mass spectrometer. A 2 ft x 2 mm id. glass column packed with 1% SE-30 on 100/120 Gas Chrom Q (Alltech Associate Inc.) was used. GC conditions were: carrier gas (helium) flow rate at 23 ml min-l, injector temperature 270°C, oven temperature programmed from 220 "C to 245 "C at 10"C/ min-', jet separator and inlet line at 275"C, and the source temperature at 200 "C. Electron impact (EI) spectra obtained at -70 eV was consistent with the previously published spe~trum.~ Ion clusters around m/z 458, corresponding to the molecular ion of trimethylsilyl (TMS) cholesterol, and m/z 368, corresponding to M - 90, were monitored using selected ion monitoring (SIM) for quantitative analysis. The area under each SIM ion chromatogram representing the integrated value of multiple scans was used to calculate the normalized spectra in Tables 1 and 3. Labeled cholesterol was isolated by saponification and extraction from brain tissue samples of a newborn rat pup fed with (U-13C)acetateas 10% of dietary fat for 48 h. It was further purified by high-performance liquid chromatography (HPLC) prior to derivatization for GC/MS analysis.
dances. The isotopomer with the base mass (lowest mass) is designated as X,, and the other isotopomers are represented by XI, X 2 , X,, ..., X,, where the subscripted numbers are the incremental masses of the isotopomer over the base mass, and there are ( I + 1) isotopomers of the compound X. The relative abundance (fractional molar abundance)? of these isotopic species are represented by p , , pl, p2, ..., p,, and the sum of all the ps equal to one pi = 1).
THEORY: RELATIVE DISTRIBUTION OF MASS ISOTOPOMERS OF A MOLECULE
Rule 2. Because of the assumption of independence, the probability of detecting XiYj of a compound XY is given by the product of the individual probabilities of detecting Xi and Yj. For a compound consisting of components X and Y, the mass isotopomer distribution for the compound XY is given by the product:
Assumptions As in the theory of isotope dilution mass spectrometry, the assumptions of Pickup and McPherson are necessary for the development of a theory to predict the relative mass isotopomer distribution of a molecule.4 These assumptions were originally stated in terms of isotope, atoms and elements. For the sake of the present discussion, the terms are substituted by isotopomer, molecule and compound. Assumption 1. The relative abundance of an isotopomer of any compound is equivalent to the probability that a molecule of the compound with a mass corresponding to that of the isotopomer is drawn from or detected in the sample population. In other words, ions of different isotopomers of the same species are generated in proportion to their relative abundance. Assumption 2. The probability is independent of the location of the isotopomer in the molecule, and is independent of the presence or absence of other isotopomers in the molecule. In other words, the probability of forming or detecting an ion of a mass isotopomer is independent of the location of the isotope or the presence of other isotopes. Assumption 3. For simplicity, only integer masses are used in the discussion. However, the theory developed should be applicable to discussions where exact masses are used.
Definitions The following symbols are used to describe the different isotopomers of a compound X and their relative abun-
(1
Rules Rule 1. As a consequence of assumption 1, the descrip-
tion of a mixture of molecules of this compound X can be represented by a normalized distribution:
+
(poxo + p1X, + p2X2 ... + p,X,) and c p i = 1 Similarly, a mixture of molecules of a compound Y can be denoted by : (40'0 + 41'1 + q2Y2 + * . . + qsYJ and C qi = 1 Each formula is an algebraic statement that a compound is made up of molecules of masses X, to X, (or Yo to Y,), and the distribution of these isomers are p , to p, (or 4 , to qs),respectively.
+ PlXl + P2X2 + ...) x (40 y o + 41Yl + 42 y, + .) = (Po4ox,y, + P041XOYl + Poq2XoY2 + . . - )
(Pox,
*.
The product of this algebraic multiplication gives the frequency distribution pi qj for each mass isotopomer Xi Y,; pi q j represents a simple multiplication product, whereas Xi Yj represents the combination of the two components Xi and Yj. The sum of the subscripts for XiYj, (i + j), gives the incremental mass of the compound Xi Y, above the base mass X, Yo. An interesting particular case of the above general expression is provided by the situation in which one of the components is mononuclidic, containing only I2C (Yo). Under such a condition, (q, = 1 and qj = 0 for j > 0), the mass isotopomer distribution of XY is the same as X except that the base mass is shifted from X, to X, Yo. Rule 3. When we assume that only integer masses are permissible, the notation adopted above allows one to identify symbols of molecules by their incremental masses, which is given by the sum of subscripts (i + j) of Xi and Yj. Since only integer masses are discernible,
t Fractional molar abundance refers to the concentration of an isomer species expressed as a fraction of the total number of molecules. It should not be confused with the abundance of an isotope. For example, if 3% of the cholesterol molecules contains 3 I3Cs (m + 3), the fractional molar abundance of this m + 3 species is 3%. The abundance of in that fraction is 3/27 of 3%. We designate fractional molar abundance of an isotopomer rn + i by m i , and the abundance of I3C is given by rn, x i/n. The former term is unique for mass isotopomer analysis and the latter term is equivalent to specific activity of radioisotopes.
MASS ISOTOPOMER ANALYSIS
453
terms of the above algebraic multiplication with the provided by the distribution of X adjusted for (shifted same incremental mass can be grouped together accordby) the corresponding incremental masses of the isotopomers of Y. The above relationship can be used to ing to the sum of their X and Y subscripts. The fractional molar abundance of any mass isotopomers of XY is predict the mass isotopomer distribution of a comgiven by the sum of coefficients of those terms, where pound XY when the mass isotopomer distribution of the components X and Y are known. the sums of the subscripts for X and Y are equal to the incremental mass. For instance, the fractional molar Corollaries abundance of the molecule with the lowest mass (m0) is given by the coefficient of the Xo Yo term which is p o g o . Several useful relationships can be derived based on the The fractional molar abundance of the molecule with above rules. one higher mass (ml) is given by poql + plqo; that of m2, Po 42 + P2 40 + P141; and so on. Corollary 1. If a molecule is made up of two components The relationship between the mass isotopomer disX and Y consisting of isotopomers ( X o , XI, X, , ...,X,) tribution (mO, m l , m2, ..., m(r + s)) and the individual and (Yo, Y,, Y,, ...,Y$, the maximum number of mass relative isotope abundance of X (p0,pl, p , , ..., pr) and isotopomers of the molecule XY is given by (r + s + 1). Y (qo, ql, q , , ..., qs) can be represented by matrix multiplication as shown in Fig. 1. The matrix which Corollaryt. For a compound of n carbon atoms and contains the contribution of each mass isotopomer of X other elements, the mass isotopomer contribution from is known in the literature as the ‘abundance m a t r i ~ ’ . ~ . ~ the presence of 13Ccarbon is given by the coeficients of The term was first used by Brauman in his paper on the the binomial expansion (poxo + plXl)”, where po and use of a least-squares approach in spectral analysis.’ p1 stand for the abundance of ”C and 13C,respectively. The ‘abundance matrix’ was used by Korzekwa et al. This is the expected consequence of applying rule 2 for quantitative analysis.6 (n - 1) times. The fractional molar abundance of mass The set of linear equations resulted from the matrix isotopomer (mi)from isotopes of carbon is given by: multiplication is shown below: mi = n ! / [ ( i ! )x (n - i ) ! ] x p$‘-i)pli Po 40 = m0 For example, ‘natural’ glucose contains six carbon =ml P140 + Po 41 atoms per molecule, and each carbon is derived from carbon with natural abundance of 1.11%. (For simpli= m2 P240 + pi41 + P o 4 2 city, we assume all hydrogen atoms are of mass 1 and P 3 q O + P 2 4 1 + PI42 + P O 4 3 = m3 oxygen atoms mass 16.) The distribution of the seven mass isotopomers (six containing ”C and one of ”C) are: m0 = 0.9358, ml = 0.0624, m2 = 0.0018, m3 = 0, ... m4 = 0, m5 = 0 and m6 = 0. This example also illustrates that in most low-molecular-weight natural organic compounds composed of carbon, oxygen, P,4s-1 + Pr-14s = m(r + s - 1) hydrogen and nitrogen, the probability of finding mass isotopomer of mass > m + 4 is rather negligible due to P r 4 , = m(r + 4 the rapid drop-off in values of the higher terms of the The similarity between the above set of linear equations binomial expansion. and those used by Lee et at.’ in their discussion of quantification of stable isotopes in mass spectrometry Corollary 3. If the mass isotopomer distribution of comshould be apparent. If we consider the mass isotopomer pound XY (mO, m l , m2, ..., m(r + s)) and the relative distribution of X to be the basis for the mass spectrum isotopomer contribution from either component X or Y of XY, , the mass spectra of XY 1, XY, , XY3, etc. are are known, the relative distribution of mass isotopomer 0,
...,
P1.
Po,
...,
Pz,
P1.
P3.
Pz. ...
..., ...,
0, 0, 0, 0,
..., ..., ...,
0, 0, 0,
..., ...)
0, 0,
Po.
...
... P . 7
4 0,
PI-I.
P,, 0,
... ...
X
... 0, 0,
0, 0,
Figure 1. Matrix multiplication of the individual relative isotopomer abundances of
X,b0,pl. pZ,...,p,) and y, (40.ql, 41 -.)4 J 1.
454
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
in the remaining component can be determined from the set of linear equations shown under rule 3. The algorithm of Biemann and the multiple linear regression method for the determination of mass isotopomer distribution of a compound using its putative ‘unlabeled spectrum’ represent two algebraic methods of solving the same problem.**9 Generally, we determine the spectrum of the compound XY to be (mO, ml, m2, . .., m(r + s)), and we also know a priori, from other determinations or from the assumption of known I3C natural abundances, the values of ps (the mass isotopomer distribution of the X portion of the molecule). Inspection of the linear equations under rule 3 will show a set of simultaneous equations with the molar fractional abundance of Y (4s) as the unknowns. The coefficients of these linear equations (ps) can be grouped to give the ‘abundance matrix’. There will be r + s + 1 number of equations for s + 1 number of unknowns. Since r + s + 1 is always greater than or equal to s + 1, the problem can be solved as an overdetermined set of linear equations by multiple linear regression analysis. In matrix notation: [Aly = m; where y is the vector of mass isotopomer distribution in Y,and m the vector of the observed mass isotopomer distribution (normalized spectral intensities) of XY. Corollary 4. In the development of the above theory, we define a compound or its components as having the properties specified in the three assumptions irrespective of their molecular compositions. Therefore, the derivatized metabolic substrate can be considered as a composite of the substrate and its derivative components. This consequence lends itself to the strategy of first establishing the mass isotopomer distribution of the ‘derivative’ component of a compound for the construction of an ‘abundance matrix’ and then applying the information to the determination of mass isotopomer analysis of many substrates of biological interest.
EXAMPLES
(a) Theoretical and observed mass isotopomer distribution in naturally occurring cholesterol. The contribution of ‘natural’ cholesterol uniformly labeled with 13C (i.e. a compound having the same proportion of I3C in every carbon) to the mass isotopomer distribution can be modeled by a simple expansion of a binomial function: (Po C plC’)”, where po and p1 are the molar fractions of ”C and I3C, and p o + p1 = 1, as stated in corollary 2. The coefficient of each term can be calculated using po = 0.9889, p1 = 0.0111 (13C natural abundance of l.ll%), and n (number of carbon for cholesterol) of 27. The coefficients from the expansion of the binomial function give the contribution of I3C in cholesterol to the observed relative mass isotopomer distribution mO, ml,m2 etc. of the derivatized cholesterol standard. Alternatively, such distribution can be experimentally determined. The EI spectrum of the TMS derivative of cholesterol showed a molecular ion at m/z 459 and an ion m/z 368 which is cholesterol without the hydroxyl group. Normalized spectral intensities of the m/z 368 ion cluster are shown in Table 1. This ion fragment has a slight tendency (about 3.9%) of losing a hydrogen atom, forming a peak at m - 1. The normalized spectral of the m/z 368 ion cluster is first adjusted for the possibility of losing a proton, and the mass isotopomer distribution is given by the fractional intensities corresponding to the different mass isotopomers of cholesterol (I/(sum of I)). The observed and expected distributions are compared in Table 2. The agreement for the isotopomers m0, ml, m2 and m3 is within the precision of the method. It is one of the assumptions in the proposed mass isotopomer analysis that the molar efficiencies of different masses are equal (i.e. the same number of ions are produced and detected for each isotope species). However, mass discrimination in ion source, the mass analyzer and the detector system is well known in mass spectrometry.” When it is possible to determine the mass isotopomer distribution directly, the problem of mass discrimination can be checked by the agreement or lack of agreement between the theoretical and observed distribution. In our case, the good agreement suggests that the assumption is realistic and practical within the range of masses monitored for cholesterol.
+
(I) Determination of ‘TMSderivative’ spectrum of TMS-cholesterol
(b) Determination of mass isotopomer distributioo of TMS or ‘TMS13C(cholesterol)spectrum’ for the constructioo of the ‘abuodaoce matrix’. The existence of two ion fragments
The first step in mass isotopomer analysis is to determine the spectrum of derivative of mononuclidic cholesterol (TMS-”C(cholestero1)). The information is then used for the construction of an ‘abundance matrix’ (rule 3) for multiple linear regression analysis to correct for the contribution of the isotopes from the derivative component to the mass spectrum of the derivatized compound. Ideally, the spectra of TMS derivatives of cholesterol containing 1 to 27 I3Cs should be separately determined, and the normalized values of the experimentally determined intensities be used for the construction of the ‘abundance matrix’. Since such a set of reference standards is not available, the ‘abundance matrix’ is usually secondarily derived from a TMScholesterol spectrum of a cholesterol standard (containing 1.11% 13C).
containing the carbon skeleton of cholesterol provides us with an opportunity to demonstrate the application of corollaries 3 and 4 in mass isotopomer analysis. The m/z 458 ion of TMS-cholesterol can be viewed as our compound XY, where X is the cholesterol component and Y the TMS component. It is a common confusion as to which part of the derivatized compound should be separated as the X component. The key criteria in answering such a question is whether the group of atoms is dependent or independent of the tracer incorporation. In this analysis, the cholesterol carbons constitute OUT X component and the TMS carbons and its silicon atom form our Y.The mass isotopomer distribution of the cholesterol component (X) (accounting for all the cholesterol carbons) can be determined from the m/z 368 fragment or by simple expansion of the binomial
MASS ISOTOPOMER ANALYSIS
455
Table 1. Normalized intensities of partial spectrum of cbolesterol from six SIM analyses mlz
367
368
369
370
371
3.85 3.78 3.92 3.88 4.00 4.01
100 100 100 100 100 100
29.84 29.96 29.77 29.98 30.14 29.91
4.53 4.49 4.44 4.46 4.49 4.49
0.47 0.52 0.48 0.52 0.51 0.51
0.12 0.18 0.14 0.22 0.18 0.18
372
0.26 0.21 0.23 0.26 0.26 0.25
0.11 0.13 0.12 0.16 0.15 0.14
Average
3.91
100
29.93
4.48
0.50
0.17
0.24
0.14
Standard deviation
0.08
0
0.12
0.03
0.02
0.03
0.01
0.02
Coefficient of variation (%)
2.08
0
0.39
0.63
3.89
7.72
12.65
18.9
373
374
mass of TMS-’2C(cholesterol) (XoY), but has the mass isotopomer distribution of TMS (Y) only (see rule 2).
series as presented earlier (Table 2). The spectral data of TMS-cholesterol are presented in Table 3. The theoretical or the observed mass isotopomer distribution in cholesterol is used as the x range for this calculation. The x and y ranges of multiple linear regression analysis and the weighting factors are shown in Table 4. It should be pointed out that the x range is the ‘abundance matrix’ of Brauman.’ Since the intensities of different m/zs take on different variances (heteroscedasticity), it is necessary to normalize the variances for multiple linear regression The variances of the spectral intensities provide the proper weighting factors. The regression analysis can be performed using many microcomputer programs such as MathCad or Lotus 123. The relative concentrations of mass isotopomers are shown as the x coefficients in the regression output along with the standard errors of the estimates. Since the process of losing a proton is independent of I3C isotopes on the cholesterol, its influence on the spectral intensities is lumped into this ‘TMS”C(cholestero1) spectrum’; hence the appearance of the m - 1 ion. It should not be difficult for the reader to see that the result is the spectrum of Xo Y, which has a base
(Il) Determination of mass isotopomer distribution of a labeled cholesterol Once the putative ‘TMS-”C(cholestero1) spectrum’ is determined, the information can be applied to estimate the mass isotopomer distribution in the unknown. The spectral data of a brain cholesterol sample obtained from a newborn rat fed with perlabeled I3C(acetate)are shown in Table 3.” The ‘abundance matrix’ and regression analysis is shown in Table 5. The result of the computation is equivalent to determining the spectrum of Y from those of XY and X. In this case, we have two methods of determining the mass isotopomer distribution of cholesterol: one directly from the ions of the m/z 368 cluster and one calculated from the molecular ion of the TMS-cholesterol derivative. The agreement between the two methods of determining mass isotopomer distribution of cholesterol (Table 6) strongly suggests that conditions specified in the assumptions
Table 2. Comparison of theoretical and observed fractional molar enrichment of mass isotopomers in cbolesterol standard Theoretical Observed
mO
ml
m2
m3
m4
m5
0.7397 0.737
0.2242 0.222
0.0327 0.033
0.0030 0.004
0.0002 0.001
0.0000 0.000
Table 3. Normalized spectra of a cholesterol standard and a labeled sample mh
367
368
369
371
372
373
374
Cholesterol standard (n = 6)a Labeled cholesterolb (n = 3)
3.91
100
29.93
4.48
0.50
0.17
0.24
0.14
4.23
100
32.19
7.86
3.03
2.01
1.52
1.08
mlz
457
458
459
460
461
462
463
464
Cholesterol standard (n =6) Labeled cholesterol (n = 3)
1.57
100
38.07
10.52
2.06
0.36
0.15
0.13
1.59
100
40.35
14.35
5.02
2.44
1.71
1.26
370
‘ n = number of SIM analyses included in the normalized averaged spectrum.
Labeled cholesterol was measured as described under ExDerimental.
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
456
Table 4. Multiple linear regression analysis of cholesterol standard spectral data showing regression ranges and output mlz
0 0.740 0.224 0.033 0.003 0.0002 0 0
0.740 0.224 0.033 0.003 0.0002 0 0 0
0 0 0.740 0.224 0.033 0.003 0.0002 0
Regression output: Constant Standard error of Y estimate R2
No. of observations Degrees of freedom X coefficient@) Standard error of coefficient
mlz Normalized spectrum a
Y range'
X range
457 458 459 460 461 462 463 464
0 0 0 0.740 0.224 0.033 0.003 0.0002
0 0
0 0 0 0 0 0.740 0.224 0.033
0 0
0.740 0.224 0.033 0.003
0 0 0 0 0 0 0.740 0.224
1.57 100 38.07 10.52 2.06 0.36 0.15 0.13
Weightb
1.25 10
6.17 3.24 1.44 0.6 0.39 0.36
0 0.248 0.999 8 1
2.122 0.420 457 1.58
134.6 3.356 458 100
10.59 2.305
5.051 1.266
459 7.87
460 3.75
0.249 0.592 46 1 0.19
0.094 0.253 462 0.07
0.180 0.138 463 0.1 3
Relative intensity of mlz. Square root of Y range.
Table 5. Multiple linear regression analysis of labeled cholesterol' X range
457 458 459 460 461 462 463 464
1.577 100 7.872 3.754 0.184 0.069 0 0
0 1.577 100 7.872 3.754 0.184 0.069 0
Regression output: Constant Standard error of Y estimate
0 0 1.577 100 7.872 3.754 0.1 84 0.069
No. of observations Degrees of freedom
0 0.016 0.999 8 1
X coefficients Standard error of coefficient Coefficient of variation (%)
0.995 0.001 0.161
R2
a
0 0 0 1.577 100 7.872 3.754 0.184
0.323 0.001 0.320
Y rangeb
0 0 0 0 1.577 100 7.872 3.754
0.080 0.000 0.775
0.029 0.000 1.242
0 0 0 0 0 1.577 100 7.872
0.017 0.000 1.457
0 0 0 0 0 0 1.577 100
0.014 0.000 1.516
1.59 100 40.35 14.35 5.02 2.44 1.71 1.26
0.010 0.000 1.726
See Table 3. Relative intensity of mlz. Square root of Y range.
Table 6. Comparison of mass isotopomer distribution of a labeled cholesterol determined from two mass spectra by different methods From mlz 368" From mlz 458b
mO
ml
m2
m3
m4
m5
m6
0.677 0.677
0.218 0.220
0.053 0.054
0.021 0.020
0.014 0.012
0.01 0.009
0.007 0.007
"The individual mass isotopomer distribution is given by / i / ( ~ ~li) = ,of spectrum of mlz 368 fragment, where I= intensity and i = ions monitored from mlz 368 to m/z 374. bThe individual mass isotopomer distribution is given by the individual regression coefficient divided by the sum of all the coefficients from Table 5. It is important to note that in both instances the sum of all the ms is 1.
Weightc
1.26 10 6.35 3.79 2.24 1.56 1.31 1.12
MASS ISOTOPOMER ANALYSIS
necessary for the theory of mass isotopomer analysis have not been violated in these experimental studies. (111) Correction for incorporation of 13Cfrom natural abundance in cholesterol The mass isotopomers of the labeled cholesterol are the results of incorporation of labeled acetate and the incorporation of background acetate containing 13C at natural abundance. In order to determine tracer incorporation, the background 13C contribution to choiesterol mass isotopomers has to be subtracted. Two algorithms for background correction have been p u b l i ~ h e d . ~The * ' ~ procedure of Dunstan is a one-step calculation to correct for isotopes of the derivatization reagent and the 13C of the compound due to natural abundance. Since we have separated the isotopomer contribution of the derivatization reagent to arrive at the mass isotopomer distribution of cholesterol, we only need to consider the contribution of from natural abundance. The method for background correction is based on the recognition that as 13Cis incorporated into the cholesterol molecule from the labeled precursor, fewer carbon positions are available in the cholesterol to accept 13C from natural background. Cholesterol synthesized from natural acetate would contain 1.11% 13C. This 'unlabeled' species therefore has a mass isotopomer distribution of natural cholesterol according to the bin~ its base mass is omial distribution: (Po C + P ~ C ' ) 'and m0. When a single 13C is incorporated, the distribution of 13C from background in the remaining 26 positions would follow the binomial expansion (Po C + P ~ C ' ) ' ~ and its base mass is ml. The contribution by 13C background to cholesterol labeled with 2, 3 or 4 13C from labeled acetate precursor can be deduced similarly. In the fully labeled cholesterol, the contribution from background 13C would be zero. These distributions are used to construct another 'abundance matrix' as shown in Table 7. It should be pointed out that the subtraction
451
procedure has the same effect as the linear transformation of variables of a set of linear equations. Since there are as many equations as there are unknowns, the equations are solved exactly. The mass isotopomer distribution after background subtraction would provide the fractional enrichment of these isotopomers. Since much of the singly labeled cholesterol is due to background "C abundance, the background cholesterol fraction is 91.3%, which is made up of 67.7% of ("C)cholesterol, and approximately 20% of the singly labeled and 3% of the doubly labeled ('3C)cholesterol shown in Table 6. Examples I, I1 and 111 illustrate the main steps in the determination of mass isotopomer distribution of a labeled tracer in metabolic studies. First, the theoretical mass isotopomer distribution of the natural substance in question is constructed using the expansion of a binomial function as in example I(a). The mass spectrum of the natural analog of this compound is then used to determined the 'derivatized "C spectrum' of the compound as in example I(b). The 'derivatized "C spectrum' is then used for the determination of mass isotopomer distribution of the labeled compounds in experimental samples as in example 11. Finally, incorporation of 13Cfrom background abundance is subtracted to give the fractional molar enrichment of these isotopomers from incorporation of labeled precursors using procedure outlined in 111. DISCUSSION
Since the early experiments of Schoenheimer and Rittenberg" the application of stable isotope in metabolic research has grown e~ponentially.'~.' Concepts of stable isotope analysis using GC/MS has evolved from those of gas isotope ratio mass spectrometry. Structural information of the labeled compound is not exploited by such methods. The importance of the structural information of labeled substrates in metabolic
Table 7. Correction of 13C background from its natural abundance Theoretical distribution. m3 m4
m5
m6
Uncorrected distributionn
0 0 0 0 0 0.7840 0.1918
0 0 0 0 0 0 0.7927
0.6770 0.2190 0.0535 0.0205 0.0130 0.0095 0.0070
m0
rnl
m2
0.7418 0.2228 0.0322 0.0030 0.0002 0.0000 0.0000
0 0.7501 0.2169 0.0302 0.0027 0.0002 0.0000
0 0 0.7584 0.2109 0.0281 0.0024 0.0001
mO'
ml'
mZ'
m3'
m4'
m5
m6
0.9126
0.0209
0.0258
0.0153
0.0115
0.0086
0.0053
0 0 0 0.7669 0.2047 0.0262 0.0021
0 0 0 0 0.7754 0.1984 0.0243
Mass isotoporner distribution due to isotope incorporation
"Each column of the 'abundance matrix' contain numbers corresponding to the coefficients of a binomial expansion of b 0 C +p,C')("-'), where po = 0.989, p , = 0.011, n = 27 and i is the index for mi. The numbers are arranged in descending order such that frequency of the base mass appears at the top of the column, and the isotopomer with the largest mass appears at the bottom. Uncorrected mass isotopomer distribution from Table 6 arranged in a column from mO to m6.
458
W.-N. P. LEE, J. EDMOND, L. 0. BYERLEY AND E. A. BERGNER
studies has been well recognized, especially in the studies of gluconeogenesis and metabolism of the tricarboxylic acid (TCA) cycle^.'^.' Mass isotopomer analysis of 3C-labeled substrates provides quantitative and structural information not available to the conventional stable isotope applications. The theory behind mass isotopomer analysis is based on the same assumptions of isotope dilution mass spectrometry. The theory forms the basis for the algebraic methods such as the algorithm of Biemann* and the multiple linear regression rnethod6y9 for the determination of mass isotopomer distribution from mass spectral data. The method of Biemann is simple to apply and is often cited for mass isotopomer analysis. However, Biemann’s method does not take into account variability in counting statistics and has no mechanism for handling excessive residual. Multiple linear regression analysis has found many applications in mass spectrometry. Brauman used the method to separate the isotope contribution of heteroatoms to the molecular spectrum of an organic compound in spectral analysi~.~ Lee et al. suggested its use to exploit the information from multiple ions (m/z) for quantitative analy~is.~ Recently, the use of multiple linear regression to handle the problem of overlapping ions from components of a mixture was and its application in discussed by Korzekwa et
quantifying glucose mass isotopomers was reported by Lee.’ The statistics produced from the analysis can be used for the evaluation of propagation of errors and possible spectral contamination in a certain range of m/z monitored for quantitative analysk6 We have presented an algorithm for the determination of mass isotopomer distribution and isotope incorporation using the least-squares approach. The contributions from the derivatization reagent and from the natural background I3C abundance are corrected in separate steps. The method of correcting for TMS contribution is validated by excellent agreement of results calculated from spectral data of TMS cholesterol (m/z 458 cluster) and cholesterol (m/z 368 cluster). Data reduction can be performed using commercially available statistical software packages. The availability of statistical software for personal computers should make the determination of mass isotopomer distribution using multiple regression analysis more practical for metabolic studies.
Acknowledgements This work was supported by the UCLA Clinical Nutrition Research Unit, NIH/CNRU grant no. 6pOlCA42710, and Harbor-UCLA Clinical Research Center, NIH/GCRC grant no. 5MOlR00524.
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