DeconVolution of Mass Spectrosopic Patterns
SCHEME 1. Hydrogen-Deuterium Exchange in the
r-Positions of 3-Octanone
columns). The pattern results on the one hand from the natural
distribution of 13C and other atoms showing significant isotope
distributions (e.g., Cl) as well as from fragmentation. For these
theoretical considerations we have chosen a starting material
that can be labeled only in two positions. Since we deal with
low-resolution MS, the abundance pattern of the single deute-
rium labeled derivative d1 has to be the same as the unlabeled
one but shifted one mass unit upward. The d2-derivative pattern
is shifted two mass units to higher mass and so on. The pattern
of a mixture (the analyte) results from the sum of the fraction
of each derivative multiplied by the abundance at the corre-
sponding mass. For instance, in Table 1 at m/z of M + 2 the
abundance of the analyte (here z2) is proportional to c‚x(d0) +
b‚x(d1) + a‚x(d2), where x(di) is the fraction of the corresponding
derivative. Therefore, for the simple example in Table 1 five
equations (for each m/z M to M + 4 one equation) are obtained.
In general, considering the number of equations that are
obtained and the number of derivatives, the number of equations
is higher than the number of derivatives. Taking the example
of Table 1, we have three derivatives (d0-d2) but five equations
(M until M + 4, or z0 to z4). Therefore we deal with an
overdetermined linear equation system, which cannot be exactly
solved (for real data). The equations can be written in a
mathematical way, by using matrixes (eq 1), whereby A is the
matrix (for the example in Table 1, it is the one in the box, the
places without a value are set to 0), x is a vector containing the
fraction of each derivative {for the example [x(d0), x(d1), x(d2)]T,
T means transposed}, and b are the abundances of the analyte,
e.g., (z0, z1, z2, z3, z4)T.
of the labeling degree was required to be able to properly
optimize the synthesis of the desired labeled ketones. We
herewith present a novel algorithm for the exact determination
of the degree of labeling that is based on the evaluation of mass
spectroscopic data by computerized methods.
Results and Discussion
Generation of Deuterium-Labeled Ketones. Our studies
commenced with the investigation of the hydrogen-deuterium
exchange reaction in simple ketones using deuterium oxide
(D2O) as the deuterium source and CD3COOD as a catalyst.
Deuterium exchange in D2O/CD3COOD6 of a rather simple
ketone like 3-octanone 1a employing controlled microwave
heating in sealed vessels or conventional reflux conditions
proceeded via various deuterated intermediates. Considering the
four acidic R-protons of 1a, five derivatives are possible, namely
d1, d2, d3, d4, and the unlabeled starting material d0.
In order to follow the degree of deuterium labeling in
3-octanone 1a during the reaction and to characterize the
obtained products, low-resolution MS analysis may be consid-
ered as an appropriate method. However, the mixture of
compounds labeled at the two different R-positions to varying
degrees led to a rather complex mass spectrum (Figure 1b).
We expected that the deconvolution of such a pattern is well
described in the literature. To our astonishment we found a rather
unsatisfying procedure in two standard reference books.8 Es-
sentially, this procedure proposes to take the abundance of the
lowest mass peak in the area of interest that can only be
attributed to one derivative and subtract the corresponding
abundance of this derivative from the abundance at higher mass
and then continue with the next higher substituted derivative,
and by using this iterative procedure the fraction of each
derivative is obtained. It is evident that this method is rather
inaccurate, since it relies completely on the accuracy of a single
abundance at a single mass (the first one), which leads to a
high inaccuracy. An additionally performed thorough search in
the primary literature did not lead to any more sophisitcated
algorithms for this specific deconvolution problem.
A‚x ) b
(1)
where A is the m × n matrix, m * n, m is the number of
columns, and n is the number of rows; x is a vector (1 × m),
containing the fraction of each derivative, i.e., [x(d0), x(d1), ...,
x(dm)]T; and b is a vector (1 × n), representing the abundances
of the analyte, i.e., (z0, z1, ..., zn)T.
Since we have an overdetermined system and we deal with
data of measurements which have a certain error, an algorithm
to minimize the error will definitely be superior to the rather
cumbersome literature procedure.8 Applying the least-squares
method to the overdetermined linear equation system, the
solution is found by applying the pseudoinverse matrix of A
which is (AT‚A)-1‚A.9 From a mathematical point of view, it
is required that the column vectors of A have to be linear
independent, which is fulfilled for our problem.
Therefore x of eq 1 can easily be obtained by the expression
given in eq 2.
Analyzing the problem in more detail, we listed the abundance
pattern of a hypothetical organic (carbon atom-based) starting
material at the different m/z ratios of interest (Table 1, two left
x ) (AT‚A)-1‚A‚b
(2)
Although this expression looks rather short, the determination
of x requires a number of mathematical operations. Standard
programs like Microsoft Excel can deal with these expressions.
However, to circumvent problems associated with Excel soft-
ware we prepared an Excel sheet for both PC and Macintosh
computer platforms.10 Since the same question arises when
dealing with 13C or 17O labeling, the identical sheet is also
applicable for these derivatives. In the case of tritium, 18O, or
(6) (a) Fodor-Csorba, K.; Galli, G.; Holly, S.; Gacs-Baitz, E. Tetrahedron
Lett. 2002, 43, 3789. (b) Jones, J. R.; Lockley, W. J. S.; Lu, S. Y.;
Thompson, S. P. Tetrahedron Lett. 2001, 42, 331. (c) Loupy, A.; Petit, H.;
Hamelin, J.; Texier Boullet, F.; Jacquault, P.; Mathe, D. Synthesis 1998,
1213. (d) Anto, S.; Getvoldsen, G. S.; Harding, J. R.; Jones, J. R.; Lu, S.
Y.; Russel, J. C. J. Chem. Soc., Perkin Trans. 2 2000, 2208.
(7) For a general review on controlled microwave synthesis, see: (a)
Kappe, C. O. Angew. Chem., Int. Ed. 2004, 43, 6250. (b) For a review on
microwave-assisted radiochemistry, see: Jones, J. R.; Lu, S.-Y. In
MicrowaVes in Organic Synthesis, 2nd ed.; Loupy, A., Ed.; Wiley-VCH:
Weinheim, Germany, 2006; Chapter 13, pp 435-462.
(8) (a) Hesse, M.; Meier, H.; Zeeh, B. Spectroscopic Methods in Organic
Chemistry; Thieme: New York, 1997; pp 256-267. (b) Budzikiewicz, H.
Massenspektrometrie; Wiley-VCH: Weinheim, Germany, 1998; pp 70-
71.
(9) Poole, D. Linear Algebra; Thomson Brooks/Cole: New York, 2006.
(10) The Excel sheets can be downloaded free of charge from: ftp://
biocatalysis.uni-graz/pub/IsoPat2/. When opening, the Excel-file macros
must be activated.
J. Org. Chem, Vol. 72, No. 15, 2007 5779