High-dimensional isotomics, part 1: A mathematical framework for isotomics

Abstract

Molecules can exist in a variety of isotopic forms, called isotopologues, with varying numbers of isotopic substitutions at symmetrically nonequivalent atomic positions. The concentrations of these isotopologues in a sample, referred to here as the sample's isotome, encodes information about that sample's physical and chemical history. While much of this information remains inaccessible due to experimental challenges, recent advances have enabled the measurement of many new constraints on a sample's isotome. These constraints, which may be obtained from several different technologies, currently consist of ratios of subsets of the isotome and in almost all cases fail to directly observe most isotopologues. Thus, it is challenging to relate the set of all measured constraints to the abundances of all possible isotopologues. We here develop a mathematical framework for understanding how various measurements of a sample's isotome relate to one another. We first show a method for tracking isotopologues through complicated experimental designs, to rigorously and precisely state what subsets of the isotome are being measured. We then propose the generalization of the so-called ‘clumped’ isotope ratios to a new ratio type, the “U” value, which gives the concentration of any set of isotopologues relative to the unsubstituted isotopologue; this is a more appropriate way to report many isotopologue measurements. The U value can be used to compare and combine clumped, molecular-average, and site-specific measures of isotopic content; we demonstrate that for molecules with near-stochastic distributions of isotopes (and thus for many cases of interest), the molecular-average or site-specific U values are approximately equal to the corresponding molecular-average or site-specific isotope ratio. The U values therefore provide a convenient framework for comparing and manipulating many different types of observations. To demonstrate our work in practice, we apply it to a MS/MS experiment in which a subset of isotopologues with a given cardinal mass is selected, subjected to collisional fragmentation, and then observed in an Orbitrap mass spectrometer. This design, which is now practical, offers many constraints on a sample's isotome that are conceptually difficult to relate to concentrations of individual isotopologues. We analyze a simulated MS/MS experiment offering over 100 constraints on a methionine isotome (we plan to present our experimental results from this experiment in a companion publication). Our framework enables us to report conventional data products, such as overall molecular δPDB13C values, as well as measurements of various singly and multiply-substituted (including triply-substituted) isotopologues, demonstrating the efficacy and generalizability of our mathematical methods.