The arithmetic behind every identity confirmation, worked in full

There is a piece of arithmetic that resolves most of this, and it takes about ten seconds. Isotope peaks in a charge-state cluster are separated by one dalton divided by the charge. A spacing of one indicates a singly charged ion, half indicates doubly charged, a third indicates triply charged. Read the spacing, assign the charge, subtract the appropriate number of proton masses, multiply, and you have the neutral molecular weight. Any spectrum with sufficient resolution to show the isotope peaks contains its own charge assignment, which is why the resolving power of the instrument matters even when nobody is looking for an impurity.

The instrument does not weigh anything

It is worth being exact about what a mass spectrometer does, because the imprecision propagates. The instrument generates ions from a sample, separates them according to the ratio of their mass to their charge, and counts them at a detector. The horizontal axis of every spectrum is mass-to-charge, conventionally written m/z and expressed in thomsons or in dimensionless units depending on the vendor’s software. Nothing is weighed. Nothing is measured against a reference mass in the sense that a balance measures against a calibration weight.

What follows from this is that every molecular weight on every certificate of analysis in this market is a calculated quantity, derived from a measured m/z by assigning a charge and subtracting the mass contribution of whatever adducted to the molecule to give it that charge — usually protons, sometimes sodium, occasionally potassium or ammonium. The assignment is normally straightforward and normally correct. It is nonetheless an assignment, and when it goes wrong it goes wrong by an integer factor, which is the kind of error that produces confident nonsense rather than a plausible discrepancy.

The practical consequence for a reader is a habit: when a mass figure appears, ask what was observed and what was inferred. A report that gives both — the m/z, the charge, and the derived neutral mass — has answered the question before it was asked.

Monoisotopic against average mass, with the numbers

The monoisotopic mass of a molecule is calculated using the exact mass of the most abundant stable isotope of each element: carbon-12 at exactly 12, hydrogen-1 at 1.00783, nitrogen-14 at 14.00307, oxygen-16 at 15.99491. The average mass uses the standard atomic weights, which are abundance-weighted means over the natural isotopic distribution: carbon at 12.011, nitrogen at 14.007, and so on. For a small molecule the two differ negligibly. For a peptide of four thousand daltons containing roughly one hundred and ninety carbon atoms, the difference is on the order of two and a half daltons.

Which one a laboratory should quote depends on what it measured. If the instrument resolved the isotopic envelope, the monoisotopic peak is identifiable and monoisotopic mass is the correct quantity to report. If the envelope was not resolved — which is the ordinary situation on a linear time-of-flight instrument at this molecular weight — the centroid of the unresolved cluster approximates the average mass, and that is what should be quoted.

The error is not in choosing one convention. It is in comparing across them. An observed monoisotopic value set against a theoretical average value will disagree by two to three daltons at incretin molecular weights, and the resulting apparent discrepancy has the size and shape of a real analytical finding. Any identity statement that does not name its convention is one step removed from being uncheckable.¹

The charge-state arithmetic, worked

For a peptide of neutral monoisotopic mass M observed as a protonated ion carrying z protons, the mass-to-charge ratio is (M + z × 1.00728) divided by z, where 1.00728 is the mass of a proton — the mass of a hydrogen atom less the mass of an electron, a distinction that matters at parts-per-million accuracy and not at all below it.

Run this for a peptide of average mass 4113.58. The singly protonated ion appears at 4114.59. The doubly protonated ion appears at 2057.80, the triply at 1372.20, the quadruply at 1029.40 and the quintuply at 823.72. All five describe the same molecule. A reader shown only the fourth of those figures, without a charge assignment, would reasonably conclude the vial contained a peptide of about a thousand daltons.

Inverting the calculation is how the neutral mass is recovered: multiply the observed m/z by the charge and subtract z proton masses. Doing this for two or three charge states from the same spectrum and finding agreement to within the instrument’s stated accuracy is the standard internal consistency check, and it is the check that catches a misassigned charge. A single m/z with a single assumed charge has no such redundancy, which is one reason electrospray with a visible charge-state envelope is more informative than a single MALDI peak even when both instruments are equally well calibrated.

The isotope pattern, and how it declares the charge

Because carbon-13 is present at roughly 1.1% natural abundance, a peptide containing one hundred and ninety carbon atoms will exist substantially as molecules containing one, two or three carbon-13 atoms. In a spectrum this appears as a series of peaks above the monoisotopic peak, separated in mass by approximately 1.00336 daltons and distributed in intensity according to the binomial statistics of the composition.

Two things follow, and both are practically useful. First, the spacing between adjacent isotope peaks in a charge-state cluster is one over the charge: a spacing of 0.5 on the m/z axis means the ion is doubly charged, 0.333 means triply, 0.25 means quadruply. This is the simplest charge assignment available and it requires no assumptions about the sample at all. Second, the relative intensities of the isotope peaks are predictable from the elemental composition, so a cluster whose shape departs markedly from the calculated envelope is evidence that two species are overlapping.

Both observations require an instrument capable of resolving the isotope peaks at the relevant m/z, which is where resolving power stops being a specification-sheet number and becomes the thing that determines whether a spectrum can be interpreted at all. Below roughly ten thousand resolving power, a multiply charged peptide envelope collapses into a single broad hump that carries neither the spacing nor the shape information.

A short glossary, because the terms are used loosely

m/z — mass-to-charge ratio, the quantity a mass spectrometer actually measures. Monoisotopic mass — mass calculated using the lightest stable isotope of each element. Average mass — mass calculated using standard atomic weights. Nominal mass — the integer sum of integer isotope masses; adequate for small molecules, useless here.

Resolving power — m divided by peak width at half height; the ability to separate nearby masses. Mass accuracy — deviation of a measurement from the true value, in parts per million. Mass defect — the difference between an exact mass and its nominal value, and the property that makes near-isobars separable.

Adduct — an ion formed by association with something other than a proton, commonly sodium or potassium. Charge-state envelope — the family of differently charged ions from one compound. Isobaric — of identical mass at the achieved precision. Isomeric — of identical composition and different structure. b and y ions — the complementary fragment series produced by amide-bond cleavage.

Precision in these terms is not decoration. Several disputes this department has been asked to adjudicate turned out, on inspection, to be disagreements about whether the word mass meant monoisotopic or average.

One correction of emphasis, since this article has spent most of its length on what cannot be seen. Mass spectrometry is a triumph of twentieth-century instrumentation and it answers the question it is asked with a precision no other technique approaches. The problem examined here is not the method. It is the practice of asking it one question and printing the answer to a larger one.