Assuming that a unique elemental composition could be derived from a mass spectrum, this molecular information can be furnished for metabolite annotation in either of two distinct ways: an exhaustive computation of all chemically possible isomeric structures or a query of databases for known (bio)chemical compounds.

Exhaustive methods (Figure 2) utilize either a deterministic approach 17 or a stochastic molecular isomer generator 27. For a given molecular formula, several hundred to billions of isomers can be constructed, depending on the number and nature of elements given by the chemical composition. The number of molecular formulas for the eleven most common elements at 1000 u is reported to be more than 350 millions 37. For small molecules that are analyzed by electron impact mass spectrometry, a deterministic method called MOLGEN-MS is available 20. For high molecular weight compounds, deterministic methods are quickly challenged by computing power limits due to the combinatorial explosion of isomeric structures which may render stochastic isomer generators more promising for the future 21.

For automatic annotation of metabolites in metabolomic screens, it seems today more reasonable to first search against existing chemical structures or even to limit searches for known natural product databases. A randomly chosen molecular formula of C₁₅H₁₂O₇ (304.0583 u) was taken as test case for query results, which should comprise structures like the naturally occurring pentahydroxyflavone (Figure 3). Seven repositories were compared for this exemplary case (Table 1): the life-science oriented PubChem database of the U.S. National Institutes of Health and its sub-DB ChemIDplus, the Kyoto ligand biochemical pathway database (KEGG), the CRC dictionary of natural products (DNP), a large compendium of organic chemical structures (Beilstein), a list of commercially available chemicals which could be used for confirming any given hit (MDL), a mass spectrum library used in GC/MS (NIST 5.0) and the complement of small molecules that have been described in the chemical and biochemical literature: the Chemical Abstracts Service (CAS) database.

A range of conclusions can be derived from this exercise (Table 1). Due to its limited size and its focus to consensus biochemical pathways, the KEGG database returned far fewer hits compared to more comprehensive repositories like CAS or DNP. It is important to note that therefore, automatic annotations of mass spectra must not be limited to KEGG searches alone. Conversely, however, any hit retrieved from KEGG queries might receive a higher likelihood of truly representing an identifiable metabolite due to the focus on (conserved) biochemical pathways represented in KEGG. In contrast to the small KEGG (Ligand) DB, the CAS database represents the largest database available for small molecules containing ~ 20 million organic chemicals. However, CAS cannot serve as suitable database for routine metabolite queries. On the one hand, CAS contains many compounds that have been artificially synthesized and reported by chemists, and thus are often unlikely to be present as natural compounds. On the other hand, the CAS SciFinder front end enables only a very limited and slow formula search, allowing queries of one formula at a time but not batches or series of queries. For these two reasons, CAS queries can be excluded from automated annotation efforts of complex metabolomic surveys; however, for identification purposes of selected unknown compounds in biomarker studies, the CAS database still provides the most comprehensive overview. It is interesting to note that DNP with only 170,000 entries retrieves 129 different isomeric structures of C₁₅H₁₂O₇ (among them many stereoisomers) whereas the far larger PubChem database resulted in only 19 hits. The PubChem database is a fast growing database. At the time of search it had only 800,000 entries, now it has more than 5 millions. PubChem is a freely accessible database and includes KEGG, ChemIDplus and NCBI and several other databases and should therefore be included in automatic metabolite annotations. An in-depth molecular diversity calculation could reveal any overlap 22. For an automated approach, the DNP database in SD file format (*.sdf) could be used whereas only semi-automatic procedures would be possible for the Beilstein database. Consequently, for identification routines of unknown metabolites starting from elemental compositions, DNP and PubChem search results should be combined.

The input into metabolomic queries are elemental compositions which are calculated from experimental mass spectra. Often, the performance of mass spectrometers and underlying software algorithms to calculate such molecular formulae are presented on test cases. However, molecular formulae are not uniformly distributed across the mass range. In order to exhaustively test the performance and power of algorithms calculating elemental compositions, a data set containing all chemically possible molecular formulae between a molecular mass of 20 – 500 u (using the most common elements C, H, N, O, P and S) was constructed. It is wrongly assumed by researchers outside the mass spectrometry community that within that mass range, high mass accuracy calculations of <1 ppm would result in unambiguous calculation of unique elemental compositions. We therefore have applied a number of chemical constraints to reduce the number of potential elemental compositions in the exhaustive data set to only those combinations that are allowed by chemical bonding rules. Applying constraints is the most crucial step during the whole process of formula finding and structure elucidation. Consequently, we have used the molecular weight calculator MWTWIN with a variety of restrictions: the "smart H atoms" option was used to avoid the calculation of an unreasonably high number of hydrogen atoms. This excludes species like C2₆H₂ which are chemically possibly but not relevant for metabolomics. In extremely seldom cases this can lead to an exclusion of certain formulas with multicenter bonds (C₁₀H₂₅NO₄). Secondly, metals have been excluded in our test data set because most metabolites do not contain coordinating metal atoms (although certainly a number of naturally occurring metabolites do, such as chlorophylls). However, in case trimethylsilylation was used for derivatization, search queries in GC/MS metabolite profiling data must obviously include Si which was left aside for this test data set. A third important constraint is the application of valence rules for which LEWIS and SENIOR rules were applied. These rules were found to serve as an important constraint that helped reducing an initial number of 3.5 million combinations of elemental compositions to 1.6 million for the mass range of 20–500 u (C, H, N, S O and P). Surprisingly, a number of both commercial and non-commercial formula generators are based purely on mathematical rules but do not obey the LEWIS and SENIOR chemical rules. As result, for a mass of 129.034 u species like C₉H₅O would be calculated by such formula generators which do not exist as natural compounds (however, which might exist as charged or radical species in the gas phase). Shortly, the LEWIS rule expects each compound to account for an even number of electrons with atoms that all obey the octet rule. SENIOR's theorem 2526 requires three essential conditions for the existence of molecular graphs:

A) the sum of valences is an even number, or the total number of atoms having odd valences is even;

B) the sum of valences is greater than or equal to twice the maximum valence;

C) the sum of valences is greater than or equal to twice the number of atoms minus 1.

We have written scripts that include these rules in order to reduce the number of generated formulae that are exported from current commercial or non-commercial software products. The second rule was not included because it only proofs the non-existence of very small molecules like CH₂ 26. The current script only allows atom numbers less than 100. We have not put in a further constraint that would account for the number of and double bonds (RDBE 32) or double bond equivalent (DBE) because for complex molecules with more than five atom types the calculation gets quite complicated. For example, nitrogen and phosphorous can have 3 or 5 valences, and sulphur atoms may have 2, 4 or 6 valences. For molecules that contain these three atoms in different valance states, no single solution for RDBE can be calculated but an RDBE range would result. An in-dept mathematical discussion of this problem can be found here 37. Applying the LEWIS and SENIOR check is thus much more reliable and straightforward. Our current software script obeys standard valences (ground state chemistry 17) in a conservative effort to produce an exhaustive number of formulas for ground state chemistry.

A plot of all elemental compositions between 200–300 u is given in Figure 4. It becomes immediately clear that elemental compositions are not uniformly distributed across the mass range but recurring modalities, which are due to the dependence of elemental compositions upon the chemical constraints applied. Hence, there are large areas where not a single elemental composition exists (e.g. at 297.500 u there is no formulae within +/- 0.148 u (497 ppm mass accuracy; MWTWIN smart H option). Conversely, at maximum frequency modalities, several thousand of potential formulae are chemically allowed (e.g. around 2000 elemental compositions are retrieved between 297.74–298.34 u). Mass ranges without existing molecular formulae will shift and narrow with higher mass ranges, but peak frequencies and the characteristic pattern will remain. Consequently, the performance of mass spectrometers during elemental compositions analysis tests should be shown with masses at ± 1σ around maximum peak frequencies and not with the low number of compounds that are found at the valley of the composition distributions.

The generation of a comprehensive data set of all chemically possible molecular formulae between 20–500 u enables the prediction of the upper ppm limit for unique molecular formula assignment (see Table 2). Querying masses and formulae at peak frequency distributions from Figure 4, we have determined that this mass limit is as low as 126.000 u when the most common elements (C, H, N, S, O and P) are included and a 1 ppm mass accuracy level is assumed. With these restrictions, two chemically possible formulae are generated, C₂H₈O₂P₂ and C₃H₂N₄S, both of which can be found in the CAS database and have thus indeed been reported to be existent. This level is far lower than conventionally assumed 35 and would likely be found at an even lower mass if elements like F, Cl, Br and Si were included. It is important to note that from this mass on an increasing number of formulas occur, demonstrating that <1 ppm mass accuracy alone is not sufficient for unique elemental composition assignment. Consequently, for an automatic routine, additional constraints are needed to limit the number of unique formulae from a given mass measurement.

Natural compounds on earth (such as metabolites from biological specimen) reflect the natural abundance of stable elemental isotopes, such as ¹³C (which is found at approx. 1.11% of the most frequent isotope ¹²C), ¹⁸O (0.2% of ¹⁶O), ¹⁵N (0.367 % of ¹⁴N), ²D (0.015% of ¹H) and ³³S and ³⁴ S (0.79 and 4.43 % of ³²S). The actual ratios of these stable isotopes slightly differ for each element within narrow ranges 9. Consequently, each monoisotopic pseudomolecular ion (M₀) that is used for accurate mass determinations is always accompanied by additional isotope ions. The abundance of the isotope ions (M+1, M+2, M+3) is dependent on the actual elemental composition and can therefore serve as a powerful filter in calculating unique elemental compositions from mass spectral data. In table 3 the number of calculated elemental compositions for 150.000 to 900.000 u is given at mass accuracy levels of 10-0.1 ppm without and with additional isotope abundance information. Using conventional calculations, isotope information is not included. It is clearly seen that above approx. 200 u, mass accuracies of 3–5 ppm (an error level that is usually achieved by time-of-flight mass spectrometers, TOF 11) lead to multiple chemically possible formulae, and to dozens of elemental compositions at masses above 400 u. It has therefore been argued to utilize the high resolving power and mass accuracy of Fourier transform ion cyclotron resonance mass spectrometers that achieve around 1 ppm average error in daily routines in unattended mode sometimes worse 18). However, even at 1 ppm error, ambiguity of chemical formulae increases sharply above 400 u, a range in which many secondary metabolites are detected. Use of a hypothetical mass spectrometer with only 0.1 ppm error would still not result in unique solutions above 500 u, which leads to the conclusion that improving mass accuracy is not the solution for automatic assignments of elemental compositions. In contrast, applying isotope pattern recognition greatly reduces the search space for possible elemental compositions. Today, TOF mass spectrometers are available that specify 2% absolute isotope abundance accuracy at 3 ppm mass accuracy level with a resolving power of 10,000 29. Table 3 demonstrates that such instruments may clearly outperform the 5-fold more expensive ion cyclotron resonance mass spectrometers with respect to calculation of molecular formulae. Up to 400 u, unique solutions are achieved and between 400–800 u only 2–13 possible elemental compositions are reported. A direct comparison of the list of retrieved hits at the 3 ppm level with and without exploiting the isotope abundance information confirms that applying such an orthogonal filter above 500 u removes always more than 95% of the potential formulae. It has been argued that the chemical intuition and experience of analytical chemists would sort out unlikely chemical compositions; however, such routines cannot be implemented into query algorithms and are hard to conceive even at the 1 ppm level, when hundreds of possible hits are returned at searches between 700–900 Da, the mass range of membrane lipids. The principal idea of using a combined analysis of mass spectra and isotopic distributions is known since several decades 333134. There is a further approach called MPPSIRD (mass peak profiling from selected ion recording data) 19 in which molecular formulas with non matching ion abundances are excluded. Another approach was suggested to use isotopic pattern and "virtually" enhance the resolving power of a magnetic sector instrument from 30,000 to 90,000 or that of an FT-MS from 500,000 to 1,500,000 24. It has also been argued that complementary information may be garnered from mass spectral fragmentation, sometimes including accurate mass data in an intelligent basket method 28. However such an approach is not universally applicable, and even more importantly, the interdependency of accurate mass and accurate isotope analysis for automated calculation of elemental compositions has not yet been demonstrated on a comprehensive data set of chemically possible formulae.

A further example supports this notion of a high impact of an orthogonal isotopic pattern filter. Actual measurement data were taken from analysis of trimethylsilylated (TMS) sorbitol, which was calculated as a pseudomolecular ion with a mass/charge 615.324 u at 5 ppm error under chemical ionization using a gas chromatography – time of flight mass spectrometer (GC-TOF, 36). In Figure 5 all 370 possible elemental compositions are plotted that are calculated from this mass including elements C, H, N, O, S, Si and P using MWTWIN with smart hydrogen option, without a restriction on the number of elements. When LEWIS and SENIOR checks were applied together with a 5% isotope abundance error, 12 result possible elemental compositions were obtained. In comparison, at 1 ppm mass accuracy still 56 formulae were calculated without orthogonal filter applied. For trimethylsilylated compounds in GC-TOF analysis, actually further constraints can be applied. After subtraction of elements counting for the trimethylsilyl group, the correct formula of the non-derivatized molecule is obtained.

Exhaustive calculation of formulae from 20–500 u using C, H, N, O, S and P was performed using the Molecular Weight Calculator MWTWIN 23 on a 1.7 GHz Pentium M with 1 GByte RAM. Calculation time and data cleaning with Textpad 40 was about 24 h. As valence values and molecular masses for each of the elements are constant, the resulting patterns of these calculations are also applicable to higher mass ranges. It is feasible to calculate molecular formulas in much higher range using CHEFOEG 30. LEWIS and SENIOR rules were checked using self-written scripts in Visual Basic which were implemented into Statistica Dataminer v7 23 and Microsoft Excel 2003. A demo version of Molgen 3.5 42 was downloaded and used for the calculation of the number of structural isomers of some formulae given as examples.

Isotopic pattern were calculated with a modified Mercury6 version 13. This version takes the molecular formula as input and writes the isotopic abundances with the according masses to a log file. It can process 1 million formulas in 3 hours on a Pentium M 1.7 GHz. The resulting formulae and isotopic patterns of a single example were transferred to an MS Excel sheet where a simple matching function was implemented. Isotopic abundances are normalized to 100. The root mean square error (RMS) of the isotopic abundances is given in percent. This Excel function adds the differences between the calculated and target intensities for each of the M+1, M+2 and M+3 peaks and matches the sum of these differences against the target intensities. Furthermore an MS Excel array formula was implemented to report the number of remaining formulae when manually entering the isotope abundance accuracy in percent (according to the mass spectrometer specifications).

Mass spectrometry always reports charged species. For the correct use of the software, the neutral form of the molecule is required. In this case the charge of molecular ion can be removed and hydrogen is added or subtracted to retrieve the neutral form of the molecule (mass of proton and electron = 1.007825 u). Any other adduct must be removed in the same manner.

In table 3, isotope abundance examples were taken from individual compounds that were randomly selected from 48,000 example formulae in the range of 150–900 u, each of which had to pass LEWIS and SENIOR checks and an inclusion of C and H out of the list of C, H, N, S, O and P. Accordingly, selection of another compound for each mass example would change the single result given in the 'isotope abundance accuracy' columns, but not the overall conclusions. For all cases, the MWTWIN smart H option was applied, excluding potential formulae with a high combination of elements (e.g. C₂₆H₄) 39 that are inexistent in metabolome compositions. A complete matrix containing all results for 10, 5, 3, 1 and 0.1 ppm and 20, 10, 5, 2 and 1% isotopic abundance accuracy for 150–900 ppm can be found at http://fiehnlab.ucdavis.edu.