Modeling framework for isotopic labeling of heteronuclear moieties

Borkum et al. J Cheminform (2017) 9:14 DOI 10.1186/s13321-017-0201-7 Open Access RESEARCH ARTICLE Modeling framework for isotopic labeling of heteronuclear moieties Mark I. Borkum1*, Patrick N. Reardon1, Ronald C. Taylor2 and Nancy G. Isern1 Abstract Background: Isotopic labeling is an analytic technique that is used to track the movement of isotopes through reaction networks. In general, the applicability of isotopic labeling techniques is limited to the investigation of reaction networks that consider homonuclear moieties, whose atoms are of one tracer element with two isotopes, distinguished by the presence of one additional neutron. Results: This article presents a reformulation of the modeling framework for isotopic labeling, generalized to arbitrarily large, heteronuclear moieties, arbitrary numbers of isotopic tracer elements, and arbitrary numbers of isotopes per element, distinguished by arbitrary numbers of additional neutrons. Conclusions: With this work, it is now possible to simulate the isotopic labeling states of metabolites in completely arbitrary biochemical reaction networks. Keywords: Isotopic labeling, Isotopomers, Cumomers, Elementary metabolite units, Metabolic engineering Introduction Isotopic labeling is an analytic technique that is used to track the movement of isotopes through reaction networks. First, specific atoms of the reagent moieties are replaced with detectable, “labeled” isotopic variants. Then, after the reactions have been allowed to proceed, the position and relative abundance of labeled isotopic atoms are determined by experiment. Subsequent analysis of the measured quantities elucidates the characteristics of the reaction network, e.g., the rate constants of the reactions. In general, the applicability of isotopic labeling techniques is limited to the investigation of reaction networks that consider homonuclear moieties, whose atoms are of one isotopic tracer element with two isotopes, distinguished by the presence of one additional neutron. The contribution of this article is a reformulation of the modeling framework for isotopic labeling, generalized to arbitrarily large, heteronuclear moieties, arbitrary numbers of isotopic tracer elements, and arbitrary numbers of *Correspondence: 1 Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, 3335 Innovation Boulevard, Richland, WA 99354, USA Full list of author information is available at the end of the article isotopes per element, distinguished by arbitrary numbers of additional neutrons. Background The first group to give a partial solution to the problem of the representation of isotopic labeling states of arbitrary moieties was Malloy et al. [1]. Representing isotopic labeling states of carbon atoms in backbones of metabolites as vectors of Boolean truth values, which they referred to as “isotopomers” (a contraction of the term “isotopic isomers”), using 0 and 1 to denote, respectively, 12C-unlabeled and 13C-labeled atoms, they showed that relative abundances of metabolites in biological systems could be calculated using nonlinear functions of relative abundances of isotopomers, which they referred to as “isotopomer fractions.” For example, a metabolite of two carbon atoms has 22 = 4 isotopomers, 00, 01, 10 and 11; and hence, the isotopic labeling state of the metabolite is given by 4 isotopomer fractions. Aside from being nonlinear, Malloy et al.’s construction suffers from the fact that an exponential number of isotopomer fractions are required in order to determine the isotopic labeling state of any given metabolite. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/ publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Borkum et al. J Cheminform (2017) 9:14 Wiechert et al. [2] noted that, by the nature of the problem, isotopic labeling states of biochemical reaction network substrates are wholly determined by specific subsets of carbon atoms; and therefore, that isotopic labeling states of the complement of each subset can be omitted, incurring no loss of information. Representing isotopic labeling states of carbon atoms in backbones of metabolites as vectors of placeholder variables, which they referred to as “cumomers” (a contraction of the term “cumulative isotopomers”), using 1 and x to denote, respectively, determinate (13C-labeled) and indeterminate (12C-unlabeled or 13C-labeled) isotopic labeling states, with a total ordering given by x < 1, they showed that Malloy et al.’s construction could be reformulated as a cascade system of linear functions of relative abundances of cumomers, which they referred to as “cumomer fractions,” with the original construction being recovered via a “suitable variable transformation” [2, Eq. 7] in the form of an invertible square matrix. For example, a metabolite of two carbon atoms has 22 = 4 cumomers, xx, x1, 1x and 11; and hence, the isotopic labeling state of the metabolite is given by 4 cumomer fractions. Antoniewicz et al. [3] shed new light on this subject. Manipulating isotopic labeling states of specific subsets of carbon atoms as aggregations, rather than as singletons, they showed that, under certain conditions, Wiechert et al.’s cascade system could be optimized using graph-theoretic methods, e.g., vertex reachability analysis, edge smoothing and Dulmage–Mendelsohn decomposition, thereby significantly reducing the total number of system variables. Representing isotopic labeling states of carbon atoms in backbones of metabolites as mass distributions, which they referred to as “Elementary Metabolite Units (EMU),” they showed that every EMU has a unique factorization, and that the mass distribution of a given EMU is equal to the vector convolution of the mass distributions of its proper factors. Moreover, they showed that, in a given mass distribution, the mass fraction corresponding to the greatest number of mass shifts is equivalent to a specific cumomer fraction. Antoniewicz et al.’s work provided the foundation for a whole host of important biological investigations [4–6], and inspired many software implementations [7– 10]. Even so, the cases not solved by Antoniewicz et al. merit attention for four reasons. First, the subject is very closely tied to the concept of isotopic labeling, and can thus serve to bring greater clarity and determinacy to its mathematical formulation. In this respect, treatment of the subject possesses an immediate interest. Second, the EMU method suffers from the same sy (...truncated)