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

## TL;DR

This paper introduces a new Diophantine method for assigning molecular formulas in ultrahigh-resolution mass spectrometry, improving accuracy by distinguishing systematic and random mass errors.

## Contribution

A novel Diophantine-based recalibration method is introduced, achieving Lorentzian mass error distribution as a limiting factor for UHRMS.

## Key findings

- The Diophantine method distinguishes systematic and random mass errors in UHRMS.
- Correcting systematic errors results in a Lorentzian random mass error distribution.
- The method achieves molecular formula assignments near the theoretical accuracy limit.

## Abstract

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.