UHRMS Formula Assignment: Diophantine-Based Recalibration Yields Lorentzian Mass Error Distribution as the Limiting Factor

Ultrahigh-resolution mass spectrometry (UHRMS) is a well-established analytical method for characterizing complex molecular mixtures. It is usually performed with Fourier transform techniques, based either on ion cyclotron resonance (FTICR-MS) or mass-dependent oscillations in an ion trap (FT-Orbitrap-MS). In spite of the high technical level of these instruments, often spectral interpretation remains difficult, in particular in a nontargeted approach of complex samples. Here, we introduce a Diophantine method for molecular formula assignment. Taking the ubiquitous Gaussian distribution as an example, we first show how knowledge about random mass error can be used to assign molecular formulas in a statistically consistent way. By considering all possible attributions within a large mass error range, we show how the systematic error stemming from suboptimal calibration can be distinguished from the random mass error in peak position. Correcting for systematic mass error leaves us with a quantifiable, Lorentzian random mass error as expected for Fourier transform-based instruments with long transients. This indicates that our method is self-consistent, assigning molecular formulas close to the theoretical limit of achievable accuracy.