Corrigendum: The Design of FluxML: A Universal Modeling Language for 13C Metabolic Flux Analysis.

[This corrects the article DOI: 10.3389/fmicb.2019.01022.].

The Design of FluxML: A Universal Modeling Language for 13C Metabolic Flux Analysis.

13C metabolic flux analysis (MFA) is the method of choice when a detailed inference of intracellular metabolic fluxes in living organisms under metabolic quasi-steady state conditions is desired. Being continuously developed since two decades, the technology made major contributions to the quantitative characterization of organisms in all fields of biotechnology and health-related research. 13C MFA, however, stands out from other "-omics sciences," in that it requires not only experimental-analytical data, but also mathematical models and a computational toolset to infer the quantities of interest, i.e., the metabolic fluxes. At present, these models cannot be conveniently exchanged between different labs. Here, we present the implementation-independent model description language FluxML for specifying 13C MFA models. The core of FluxML captures the metabolic reaction network together with atom mappings, constraints on the model parameters, and the wealth of data configurations. In particular, we describe the governing design processes that shaped the FluxML language. We demonstrate the utility of FluxML to represent many contemporary experimental-analytical requirements in the field of 13C MFA. The major aim of FluxML is to offer a sound, open, and future-proof language to unambiguously express and conserve all the necessary information for model re-use, exchange, and comparison. Along with FluxML, several powerful computational tools are supplied for easy handling, but also to maintain a maximum of flexibility. Altogether, the FluxML collection is an "all-around carefree package" for 13C MFA modelers. We believe that FluxML improves scientific productivity as well as transparency and therewith contributes to the efficiency and reproducibility of computational modeling efforts in the field of 13C MFA.

Metabolic isotopomer labeling systems. Part III: path tracing.

Isotope labeling systems (ILSs) are sets of balance equations that quantitatively describe the distribution of isotopic tracers in metabolic networks. The solution of ILSs, i.e., the calculation of isotopic labeling distributions (mostly in steady state) is the fundamental computational step of (13)C metabolic flux analysis (MFA). Aiming at a deeper analytical understanding of ILSs a new approach for solving ILSs is developed. It is based on the straightforward idea of tracing labeled molecules through the metabolic network. The new approach allows to calculate the label distribution in isotopic tracer experiments in an analytical way that directly reflects the underlying network structure. The theory of path tracing is formally developed by introducing regular expressions for representing all possible paths through the labeling network. These expressions are generated by classical path tracing algorithms, e.g. by the Kleene algorithm. As a central theoretical result, a framework for proving the correctness of such path tracing algorithms in their application to ILSs is developed. Finally, by mapping path expressions to algebraic expressions, the solution of an ILS is computed. As an offspring of the developed theory, the relation between path tracing and former approaches for ILS solution is worked out and several consequences for the numerical solution and analysis of ILSs and--more general--compartmental systems used in pharmaco-kinetic modeling will be sketched.

13CFLUX2--high-performance software suite for (13)C-metabolic flux analysis.

SUMMARY: (13)C-based metabolic flux analysis ((13)C-MFA) is the state-of-the-art method to quantitatively determine in vivo metabolic reaction rates in microorganisms. 13CFLUX2 contains all tools for composing flexible computational (13)C-MFA workflows to design and evaluate carbon labeling experiments. A specially developed XML language, FluxML, highly efficient data structures and simulation algorithms achieve a maximum of performance and effectiveness. Support of multicore CPUs, as well as compute clusters, enables scalable investigations. 13CFLUX2 outperforms existing tools in terms of universality, flexibility and built-in features. Therewith, 13CFLUX2 paves the way for next-generation high-resolution (13)C-MFA applications on the large scale. AVAILABILITY AND IMPLEMENTATION: 13CFLUX2 is implemented in C++ (ISO/IEC 14882 standard) with Java and Python add-ons to run under Linux/Unix. A demo version and binaries are available at www.13cflux.net.

The topology of metabolic isotope labeling networks.

BACKGROUND: Metabolic Flux Analysis (MFA) based on isotope labeling experiments (ILEs) is a widely established tool for determining fluxes in metabolic pathways. Isotope labeling networks (ILNs) contain all essential information required to describe the flow of labeled material in an ILE. Whereas recent experimental progress paves the way for high-throughput MFA, large network investigations and exact statistical methods, these developments are still limited by the poor performance of computational routines used for the evaluation and design of ILEs. In this context, the global analysis of ILN topology turns out to be a clue for realizing large speedup factors in all required computational procedures. RESULTS: With a strong focus on the speedup of algorithms the topology of ILNs is investigated using graph theoretic concepts and algorithms. A rigorous determination of all cyclic and isomorphic subnetworks, accompanied by the global analysis of ILN connectivity is performed. Particularly, it is proven that ILNs always brake up into a large number of small strongly connected components (SCCs) and, moreover, there are natural isomorphisms between many of these SCCs. All presented techniques are universal, i.e. they do not require special assumptions on the network structure, bidirectionality of fluxes, measurement configuration, or label input. The general results are exemplified with a practically relevant metabolic network which describes the central metabolism of E. coli comprising 10390 isotopomer pools. CONCLUSION: Exploiting the topological features of ILNs leads to a significant speedup of all universal algorithms for ILE evaluation. It is proven in theory and exemplified with the E. coli example that a speedup factor of about 1000 compared to standard algorithms is achieved. This widely opens the door for new high performance algorithms suitable for high throughput applications and large ILNs. Moreover, for the first time the global topological analysis of ILNs allows to comprehensively describe and understand the general patterns of label flow in complex networks. This is an invaluable tool for the structural design of new experiments and the interpretation of measured data.