Elementary vectors and autocatalytic sets for computational models of cellular growth

Traditional models of cellular growth involve an approximative biomass ''reaction'' which specifies biomass composition in terms of precursor metabolites (such as amino acids and nucleotides). On the one hand, biomass composition is often not known exactly and may vary drastically between extreme conditions; on the other hand, the predictions of computational models crucially depend on biomass. Even elementary flux modes (EFMs) depend on the biomass reaction. (To be specific: not just the numerical values of the EFMs, but also their supports and their number.) To better understand cellular phenotypes across conditions, we introduce and analyze new classes of elementary vectors for more comprehensive models of cellular growth, involving explicit synthesis reactions for all macromolecules. Growth modes (GMs) are given by stoichiometry, and elementary growth modes (EGMs) are GMs that cannot be decomposed without cancellations. Unlike EFMs, EGMs need not be support-minimal. Most importantly, every GM can be written as a sum of EGMs. In models with additional (capacity) constraints, growth vectors (GVs) and elementary growth vectors (EGVs) also depend on growth rate. In any case, EGMs/EGVs do not depend on the biomass composition. In fact, they cover all possible biomass compositions and can be seen as unbiased versions of elementary flux modes/vectors (EFMs/EFVs) used in traditional models. To relate the new concepts to other branches of theory, we define autocatalytic GMs and the corresponding autocatalytic sets of reactions. Further, we illustrate our results in a small model of a self-fabricating cell, involving glucose and ammonium uptake, amino acid and lipid synthesis, and the expression of all enzymes and the ribosome itself. In particular, we study the variation of biomass composition as a function of growth rate. In agreement with experimental data, low nitrogen uptake correlates with high carbon (lipid) storage.

Elementary growth modes/vectors and minimal autocatalytic sets for kinetic/constraint-based models of cellular growth

Elementary vectors are fundamental objects in polyhedral geometry. In metabolic pathway analysis, elementary vectors range from elementary flux modes (of the flux cone) and elementary flux vectors (of a flux polyhedron) via elementary conversion modes (of the conversion cone) to minimal cut sets (of a dual polyhedron) in computational strain design.To better understand cellular phenotypes with optimal (or suboptimal) growth rate, we introduce and analyze classes of elementary vectors for models of cellular growth. Growth modes (GMs) only depend on stoichiometry, but not on growth rate or concentrations; they are elements of the growth cone. Elementary growth modes (EGMs) are conformally nondecomposable GMs; unlike elementary flux modes, they are not support-minimal, in general. Most importantly, every GM can be written as a conformal sum of EGMs. Growth vectors (GVs) and elementary growth vectors (EGVs) also depend on growth rate, concentrations, and linear constraints; they are elements of a growth polyhedron. Again, every GV can be written as a conformal sum of EGVs. To relate the new concepts to other branches of theory, we define autocatalytic GMs and the corresponding (minimal) autocatalytic sets of reactions.As a case study, we consider whole cell models (simple kinetic models of self-fabrication). First, we use EGMs to derive an upper bound for growth rate that only depends on enzyme kinetics. Next, we study growth rate maximization (via control parameters for ribosome kinetics). In particular, we analyze growth states (GSs) and elementary growth states (EGSs) as introduced in [de Groot et al, 2020]. Unlike EGMs, EGSs depend on (metabolite) concentrations and growth rate. Most importantly, (i) we show that EGSs are support-minimal, (ii) we give a simple proof for the fact that maximum growth rate is attained at an EGS, and (iii) we show that, at every optimal EGS, the ribosome capacity constraint is active. Finally, we determine the dependence of EGSs on growth rate, and we study the relation between EGSs and minimal autocatalytic sets, EGMs, and elementary flux modes. Along the way, we point out (and resolve) mathematical issues in [de Groot et al, 2020].

Answer Set Programming for Computing Constraints-Based Elementary Flux Modes: Application to Escherichia coli Core Metabolism

Elementary Flux Modes (EFMs) provide a rigorous basis to systematically characterize the steady state, cellular phenotypes, as well as metabolic network robustness and fragility. However, the number of EFMs typically grows exponentially with the size of the metabolic network, leading to excessive computational demands, and unfortunately, a large fraction of these EFMs are not biologically feasible due to system constraints. This combinatorial explosion often prevents the complete analysis of genome-scale metabolic models. Traditionally, EFMs are computed by the double description method, an efficient algorithm based on matrix calculation; however, only a few constraints can be integrated into this computation. They must be monotonic with regard to the set inclusion of the supports; otherwise, they must be treated in post-processing and thus do not save computational time. We present aspefm, a hybrid computational tool based on Answer Set Programming (ASP) and Linear Programming (LP) that permits the computation of EFMs while implementing many different types of constraints. We apply our methodology to the Escherichia coli core model, which contains 226×106 EFMs. In considering transcriptional and environmental regulation, thermodynamic constraints, and resource usage considerations, the solution space is reduced to 1118 EFMs that can be computed directly with aspefm. The solution set, for E. coli growth on O2 gradients spanning fully aerobic to anaerobic, can be further reduced to four optimal EFMs using post-processing and Pareto front analysis.

Boosting the extraction of elementary flux modes in genome-scale metabolic networks using the linear programming approach

Motivation Elementary flux modes (EFMs) are a key tool for analyzing genome-scale metabolic networks, and several methods have been proposed to compute them. Among them, those based on solving linear programming (LP) problems are known to be very efficient if the main interest lies in computing large enough sets of EFMs. Results Here, we propose a new method called EFM-Ta that boosts the efficiency rate by analyzing the information provided by the LP solver. We base our method on a further study of the final tableau of the simplex method. By performing additional elementary steps and avoiding trivial solutions consisting of two cycles, we obtain many more EFMs for each LP problem posed, improving the efficiency rate of previously proposed methods by more than one order of magnitude. Availability and implementation Software is freely available at https://github.com/biogacop/Boost_LP_EFM. Contact [email protected] Supplementary information Supplementary data are available at Bioinformatics online.

EFMviz: A COBRA Toolbox Extension to Visualize Elementary Flux Modes in Genome-Scale Metabolic Models

Elementary Flux Modes (EFMs) are a tool for constraint-based modeling and metabolic network analysis. However, systematic and automated visualization of EFMs, capable of integrating various data types is still a challenge. In this study, we developed an extension for the widely adopted COBRA Toolbox, EFMviz, for analysis and graphical visualization of EFMs as networks of reactions, metabolites and genes. The analysis workflow offers a platform for EFM visualization to improve EFM interpretability by connecting COBRA toolbox with the network analysis and visualization software Cytoscape. The biological applicability of EFMviz is demonstrated in two use cases on medium (Escherichia coli, iAF1260) and large (human, Recon 2.2) genome-scale metabolic models. EFMviz is open-source and integrated into COBRA Toolbox. The analysis workflows used for the two use cases are detailed in the two tutorials provided with EFMviz along with the data used in this study.

Finding MEMo: Minimum sets of elementary flux modes

Metabolic network reconstructions are widely used in computational systems biology for in silico studies of cellular metabolism. A common approach to analyse these models are elementary flux modes (EFMs), which correspond to minimal functional units in the network. Already for medium-sized networks, it is often impossible to compute the set of all EFMs, due to their huge number. From a practical point of view, this might also not be necessary because a subset of EFMs may already be sufficient to answer relevant biological questions. In this article, we study MEMos or minimum sets of EFMs that can generate all possible steady-state behaviours of a metabolic network. The number of EFMs in a MEMo may be by several orders of magnitude smaller than the total number of EFMs. Using MEMos, we can compute generating sets of EFMs in metabolic networks where the whole set of EFMs is too large to be enumerated.