IsoSwap: A Toeplitz Operator Framework for Isotope Transformations in Mass Spectrometry

Abstract Accurate prediction of isotope distributions is essential for biomolecule identification and quantitation in mass spectrometry. Existing algorithms, such as recursive expansions or generating-function methods, achieve high accuracy but suffer from limited scalability, specialized handling of polyisotopic elements such as sulfur, and poor adaptability to chemical modifications. We present IsoSwap, a convolution-based framework that reformulates isotope distribution calculations as structured linear algebra operations. By representing isotopic profiles as discrete probability vectors indexed by neutron count, we express convolution as outer-product expansions and introduce Hankel and Toeplitz matrices. This formulation directly yields both aggregated isotope probabilities and centroid masses, and naturally extends to deconvolution for modeling leaving groups. IsoSwap computes complete isotopic envelopes in milliseconds, supports GPU acceleration, and generalizes to higher-order convolutions without element-specific adaptations. On simulated phosphorothioate-modified oligonucleotides, IsoSwap reproduces distributions computed directly from the modified elemental formula to floating-point precision, at approximately 5.6 times the speed of direct recomputation. Applied to an experimental mass spectrum of myoglobin, the operators interconvert the apo and heme-bound holo forms in both directions, with centroid-mass accuracy below 1 ppm. This provides a scalable, algebraically transparent method for predicting and transforming isotope distributions.