Joshua Lederberg School of Medicine Stanford University Stanford, California 94305 Tre calculation of molecular composi- tions consistent with a given range of mass values arises particularly in mass spectrometry. Although this can be a trivial exercise on the computer, it has been vexing to do by hand. Published tables, c.g., Beynon and Williams,! are bulky, and neverthcless cover a limited range of atom values. The values are also awkward to search, not having been sorted. The following approach was designed for a desk calculator that ought to be available to any student. As it involves only a few additions and subtractions, it ean—horribilis dictu—even be done by hand. Further- more, it lends itself to real time implementation on small computers that lack high precision ‘‘divide”’ in- structions in their hardware. The basis of the calculation is the table, which is an ordered list of the mass numbers of the formulas for H from 0 to 10, N from 0 to 5, and O from Oto 11. It contains only those compositions whose masses are an integral multiple of 12. Any number of C’s may then be added as required. The use of the table is best explained by a specific example, say m = 259.09 + 0.001. Step 1. Since 259 = 7 modulo 12, 5 H’s (5.03913) will be bor- rowed to give m’ = m+ 5H = 264.129. This is then divided into m’ = m; + my; m; = 264 (saa x 12); my = 0.129 + 0.001. Step 2. The table is searched for entries that correspond to my and whose mass does not exceed m;. (m; is expressed asm;/12 = C-equivalent.) We find none in this cycle. Step 3. We therefore remove 12 H’s (12.0989) to give m” = m’ — 12H = 252.085 + 0.001. The table now has entries at 0.034 (HgNsOs), 0.035 (HigNOs) and 0.036 (HgN;0;). These will be completed in Step 4. 12 H’s are again removed until m, falls below — 0.0498, the bottom of the table. In our example, this occurs at the next cycle. Rapid Calculation of Molecular Formulas from Mass Values Step 4. The table entries are now completed as follows Add C's Check mass to Adjust (compare make borrowed 259.0900 + up m” I's 0.0010) 34 0.034216 HisNaOs mi = Cis Cs CsthtsNsOs 259. 089 35 0.085559 HiwNOy m = Crs Cr CrH:sNOa 259.090 36 0.036895 HeNsOs mi = Cu Cs CslhisNsOs 259.092 Step &. Various criteria of chemical plausibility can be used to filter the list. Since the valence rules allow H’s to a maximum of 2+ 2C + N, none of these compositions is oversaturated. C3HisNsOg however has an odd number of H’s and may therefore represent a free radical. If wider ranges of hetero atoms are contemplated, adjustments of bloeks of 6 N (84.01844) and 12 O (191.9389) can be applied repetitively in a fashion similar to Step 3 so long as the adjusted mass allows. In fact m” = m — GN — 7H = 168.017 + 0.001 leads to Co- HyNsOu, m = 259.090. Further, m — 12N — 7H = 83.999 + 0.001. We read this as m; = 84; my; = —0.001 and find two en- tries in the table: —0.000826 (HgNO.) and 0.000510 (H2N.06¢), whose mm, however >84. The table is arranged so as to illustrate its use in a fast computer program. A linear array with 138 cells, indexed as shown, has entries that never slip more than one position away from the value of the index. The composition values can therefore be accessed by direct lookup, obviating a table search. A card deck version of the table is available on request from the author. This compilation is a greatly shortened form of some tables that were published some time ago.’ This work has been supported in part by the Advanced Re- search Projects Agency (contract SD-183), the National Aero- nautics and Space Administration (grant NGR-05-020-004), and the National Institutes of Health (grant GAM-00612-01). 1 Beynon, J. H., AnD Winutams A. E., ‘Mass and Abundance Tables for use in Mass Spectrometry,” Elsevier, Amsterdam, 1963. 2 LEDERBERG, J., “Computation of Molecular Formulas for Mass Spectrometry,” Holden-Day, San Francisco, 1964. Table of Mass Fractions for all Combinations’ of H, N,O (H S 10N S$ 60 € 11) Index ms X 108 asi N 9 = Index ms X 10° HE N oO =C Index mr X 108 i N oO =C —49 ~ 49787 0 2 11 17 0 0 0 0 0 0 31 31537 10 3 1t 9 ~45 — 45765 0 0 9 12 1 510 2 5 6 14 32 32363 4 2 1 14 —38 — 38554 0 4 10 18 2 18538 4 2 7 12 34 34216 8 4 8 16 —37 —87211 2 1 it 16 4 4532 2 3 4 9 35 35559 10 1 9 14 —34 — 34532 0 2 8 13 5 5875 4 0 5 7 36 36895 6 5 5 13 —30 —30510 0 0 6 8 6 6385 6 5 11 21 38 38238 8 2 6 1l —25 — 25978 2 3 10 17 7 7211 0 4 1 6 40 40917 6 3 3 8 — 24 — 24635 4 0 dl 15 8 8554 2 1 2 4 Al 42260 8 0 4 6 —23 — 23299 0 4 7 14 10 10407 6 3 9 16 42 42770 10 5 10 20 —21 — 21956 2 1 8 12 il 11750 8 0 10 14 43 43596 4 4 0 5 —19 — 19277 0 2 5 9 13 13086 4 4 6 13 44 44939 6 1 1 3 —15 —~ 15255 0 0 3 4 14 14429 6 1 7 1t 46 46792 10 3 8 15 —14 — 14745 2 5 9 16 15 15765 2 5 3 10 49 49471 8 4 5 12 -13 — 13402 4 2 10 18 17 17108 4 2 4 8 50 5O814 10 1 6 10 —16 — 10723 2 3 7 13 18 18961 8 4 11 20 52 52150 6 5 2 gq -9 —§380 4 0 8 11 19 19787 2 3 1 5 53 §8493 8 2 3 7 —8 — 8044 0 4 4 10 20 21130 4 0 2 3 56 56172 6 3 0 4 —6 —6701 2 1 5 8 21 21640 6 5 8 17 57 57515 8 0 1 2 —4 — 4022 0 2 2 5 22 22983 8 2 9 15 58 58025 10 5 7 16 -2 — 2169 4 4 9 17 25 25662 6 3 6 12 G2 62047 10 3 5 1t -1 — 826 6 1 10 15 27 27005 8 0 7 10 64 64726 8 4 2 8 28 28341 4 4 3 9 66 66069 10 1 3 6 29 29684 6 1 4 7 68 68748 8 2 0 3 30 31020 2 5 0 6 73 73280 10 5 4 12 77 77302 10 3 2 7 81 81324 10 1 0 2 88 88535 10 5 1 8 (~—0.049 to —0.0008) (0 to 0.03) (0,03 to 0.088) 2 Arranged so that the index for each entry agrees with 1000 xX mys + 1,9. ; _ [Reprinted from Journal of Chemical Education, Vol. 49, Page 613, September, 1972. ] Copyright 1972, by Division of Chemical Education, American Chemical Society, and reprinted by permission of the copyright owner