Metabolic models condense biochemical knowledge about organisms in a structured and standardised way. As large-scale network reconstructions are readily available for many organisms, genome-scale models are being widely used among modellers and engineers. However, these large models can be difficult to analyse and visualise, and occasionally generate predictions that are hard to interpret or even biologically unrealistic. Of the thousands of enzymatic reactions in a typical bacterial metabolism, only a few hundred form the metabolic pathways essential to produce energy carriers and biosynthetic precursors. These pathways carry relatively high flux, are central to maintaining and reproducing the cell, and provide precursors and energy to engineered metabolic pathways. Focusing on these central metabolic subsystems, we present iCH360, a manually curated medium-scale model of energy and biosynthesis metabolism for the well-studied bacterium Escherichia coli K-12 MG1655. The model is a sub-network of the most recent genome-scale reconstruction, iML1515, and comes with an updated layer of database annotations and a range of metabolic maps for visualisation. We enriched the stoichiometric network with extensive biological information and quantitative data, including thermodynamic and kinetic constants, enhancing the scope and applicability of the model. In addition, we assess the properties of this model in comparison to its genome-scale parent and demonstrate the use of the network and supporting data in various scenarios, including enzyme-constrained flux balance analysis, elementary flux mode analysis, and thermodynamic analysis. Overall, this model holds the potential to become a reference medium-scale metabolic model for E. coli.

The physiology of biological cells evolved under physical and chemical constraints, such as mass conservation across the network of biochemical reactions, nonlinear reaction kinetics, and limits on cell density. For unicellular organisms, the fitness that governs this evolution is mainly determined by the balanced cellular growth rate. We previously introduced growth balance analysis (GBA) as a general framework to model and analyze such nonlinear systems, revealing important analytical properties of optimal balanced growth states. It has been shown that at optimality, only a minimal subset of reactions can have nonzero flux. However, no general principles have been established to determine if a specific reaction is active at optimality. Here, we extend the GBA framework to study the optimality of each biochemical reaction, and we identify the mathematical conditions determining whether a reaction is active or not at optimal growth in a given environment. We reformulate the mathematical problem in terms of a minimal number of dimensionless variables and use the Karush-Kuhn-Tucker (KKT) conditions to identify fundamental principles of optimal resource allocation in GBA models of any size and complexity. Our approach helps to identify from first principles the economic values of biochemical reactions, expressed as marginal changes in cellular growth rate; these economic values can be related to the costs and benefits of proteome allocation into the reactions' catalysts. Our formulation also generalizes the concepts of Metabolic Control Analysis to models of growing cells. We show how the extended GBA framework unifies and extends previous approaches of cellular modeling and analysis, putting forward a program to analyze cellular growth through the stationarity conditions of a Lagrangian function. GBA thereby provides a general theoretical toolbox for the study of fundamental mathematical properties of balanced cellular growth.

SUMMARY: Measured kinetic constants are key input data for metabolic models, but they are often uncertain, inconsistent and incomplete. Parameter balancing translates such data into complete and consistent parameter sets while accounting for predefined ranges and physical constraints. Based on Bayesian regression, it determines a most plausible parameter set as well as uncertainty ranges for all model parameters. Our tools for parameter balancing support standard model and data formats and enable an easy customization of prior distributions and constraints for biochemical constants. Modellers can balance kinetic constants, thermodynamic data and metabolomic data to obtain thermodynamically consistent metabolic states that comply with user-defined flux directions. AVAILABILITY AND IMPLEMENTATION: An online tool for parameter balancing, a stand-alone Python command line tool, a Python package and a Matlab toolbox (which uses the CPLEX solver) are freely available at www.parameterbalancing.net.

Systems biology models are rapidly increasing in complexity, size and numbers. When building large models, researchers rely on software tools for the retrieval, comparison, combination and merging of models, as well as for version control. These tools need to be able to quantify the differences and similarities between computational models. However, depending on the specific application, the notion of 'similarity' may greatly vary. A general notion of model similarity, applicable to various types of models, is still missing. Here we survey existing methods for the comparison of models, introduce quantitative measures for model similarity, and discuss potential applications of combined similarity measures. To frame model comparison as a general problem, we describe a theoretical approach to defining and computing similarities based on a combination of different model aspects. The six aspects that we define as potentially relevant for similarity are underlying encoding, references to biological entities, quantitative behaviour, qualitative behaviour, mathematical equations and parameters and network structure. We argue that future similarity measures will benefit from combining these model aspects in flexible, problem-specific ways to mimic users' intuition about model similarity, and to support complex model searches in databases.

Microbes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism in E. coli, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.

Briefings in Bioinformatics (2016). doi: 10.1093/bib/bbw090 The authors of the above article wish to correct the affiliation for Tim Kacprowski and the funding section for clarity. The corrections have been made online and in print. The authors apologize for this error.

UNLABELLED: : SBtab is a table-based data format for Systems Biology, designed to support automated data integration and model building. It uses the structure of spreadsheets and defines conventions for table structure, controlled vocabularies and semantic annotations. The format comes with predefined table types for experimental data and SBML-compliant model structures and can easily be customized to cover new types of data. AVAILABILITY AND IMPLEMENTATION: SBtab documents can be created and edited with any text editor or spreadsheet tool. The website www.sbtab.net provides online tools for syntax validation and conversion to SBML and HTML, as well as software for using SBtab in MS Excel, MATLAB and R. The stand-alone Python code contains functions for file parsing, validation, conversion to SBML and HTML and an interface to SQLite databases, to be integrated into Systems Biology workflows. A detailed specification of SBtab, including examples and descriptions of table types and available tools, can be found at www.sbtab.net CONTACT: : [email protected].

Bacterial growth depends crucially on metabolic fluxes, which are limited by the cell's capacity to maintain metabolic enzymes. The necessary enzyme amount per unit flux is a major determinant of metabolic strategies both in evolution and bioengineering. It depends on enzyme parameters (such as kcat and KM constants), but also on metabolite concentrations. Moreover, similar amounts of different enzymes might incur different costs for the cell, depending on enzyme-specific properties such as protein size and half-life. Here, we developed enzyme cost minimization (ECM), a scalable method for computing enzyme amounts that support a given metabolic flux at a minimal protein cost. The complex interplay of enzyme and metabolite concentrations, e.g. through thermodynamic driving forces and enzyme saturation, would make it hard to solve this optimization problem directly. By treating enzyme cost as a function of metabolite levels, we formulated ECM as a numerically tractable, convex optimization problem. Its tiered approach allows for building models at different levels of detail, depending on the amount of available data. Validating our method with measured metabolite and protein levels in E. coli central metabolism, we found typical prediction fold errors of 4.1 and 2.6, respectively, for the two kinds of data. This result from the cost-optimized metabolic state is significantly better than randomly sampled metabolite profiles, supporting the hypothesis that enzyme cost is important for the fitness of E. coli. ECM can be used to predict enzyme levels and protein cost in natural and engineered pathways, and could be a valuable computational tool to assist metabolic engineering projects. Furthermore, it establishes a direct connection between protein cost and thermodynamics, and provides a physically plausible and computationally tractable way to include enzyme kinetics into constraint-based metabolic models, where kinetics have usually been ignored or oversimplified.

In metabolism research, thermodynamics is usually used to determine the directionality of a reaction or the feasibility of a pathway. However, the relationship between thermodynamic potentials and fluxes is not limited to questions of directionality: thermodynamics also affects the kinetics of reactions through the flux-force relationship, which states that the logarithm of the ratio between the forward and reverse fluxes is directly proportional to the change in Gibbs energy due to a reaction (ΔrG'). Accordingly, if an enzyme catalyzes a reaction with a ΔrG' of -5.7 kJ/mol then the forward flux will be roughly ten times the reverse flux. As ΔrG' approaches equilibrium (ΔrG' = 0 kJ/mol), exponentially more enzyme counterproductively catalyzes the reverse reaction, reducing the net rate at which the reaction proceeds. Thus, the enzyme level required to achieve a given flux increases dramatically near equilibrium. Here, we develop a framework for quantifying the degree to which pathways suffer these thermodynamic limitations on flux. For each pathway, we calculate a single thermodynamically-derived metric (the Max-min Driving Force, MDF), which enables objective ranking of pathways by the degree to which their flux is constrained by low thermodynamic driving force. Our framework accounts for the effect of pH, ionic strength and metabolite concentration ranges and allows us to quantify how alterations to the pathway structure affect the pathway's thermodynamics. Applying this methodology to pathways of central metabolism sheds light on some of their features, including metabolic bypasses (e.g., fermentation pathways bypassing substrate-level phosphorylation), substrate channeling (e.g., of oxaloacetate from malate dehydrogenase to citrate synthase), and use of alternative cofactors (e.g., quinone as an electron acceptor instead of NAD). The methods presented here place another arrow in metabolic engineers' quiver, providing a simple means of evaluating the thermodynamic and kinetic quality of different pathway chemistries that produce the same molecules.

BACKGROUND: To enable automatic searches, alignments, and model combination, the elements of systems biology models need to be compared and matched across models. Elements can be identified by machine-readable biological annotations, but assigning such annotations and matching non-annotated elements is tedious work and calls for automation. RESULTS: A new method called "semantic propagation" allows the comparison of model elements based not only on their own annotations, but also on annotations of surrounding elements in the network. One may either propagate feature vectors, describing the annotations of individual elements, or quantitative similarities between elements from different models. Based on semantic propagation, we align partially annotated models and find annotations for non-annotated model elements. CONCLUSIONS: Semantic propagation and model alignment are included in the open-source library semanticSBML, available on sourceforge. Online services for model alignment and for annotation prediction can be used at http://www.semanticsbml.org.

SUMMARY: Systems Biology Markup Language (SBML) is the leading exchange format for mathematical models in Systems Biology. Semantic annotations link model elements with external knowledge via unique database identifiers and ontology terms, enabling software to check and process models by their biochemical meaning. Such information is essential for model merging, one of the key steps towards the construction of large kinetic models. SemanticSBML is a tool that helps users to check and edit MIRIAM annotations and SBO terms in SBML models. Using a large collection of biochemical names and database identifiers, it supports modellers in finding the right annotations and in merging existing models. Initially, an element matching is derived from the MIRIAM annotations and conflicting element attributes are categorized and highlighted. Conflicts can then be resolved automatically or manually, allowing the user to control the merging process in detail. AVAILABILITY: SemanticSBML comes as a free software written in Python and released under the GPL 3. A Debian package, a source package for other Linux distributions, a Windows installer and an online version of semanticSBML with limited functionality are available at http://www.semanticsbml.org. A preinstalled version can be found on the Linux live DVD SB.OS, available at http://www.sbos.eu. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: Standard rate laws are a key requisite for systematically turning metabolic networks into kinetic models. They should provide simple, general and biochemically plausible formulae for reaction velocities and reaction elasticities. At the same time, they need to respect thermodynamic relations between the kinetic constants and the metabolic fluxes and concentrations. RESULTS: We present a family of reversible rate laws for reactions with arbitrary stoichiometries and various types of regulation, including mass-action, Michaelis-Menten and uni-uni reversible Hill kinetics as special cases. With a thermodynamically safe parameterization of these rate laws, parameter sets obtained by model fitting, sampling or optimization are guaranteed to lead to consistent chemical equilibrium states. A reformulation using saturation values yields simple formulae for rates and elasticities, which can be easily adjusted to the given stationary flux distributions. Furthermore, this formulation highlights the role of chemical potential differences as thermodynamic driving forces. We compare the modular rate laws to the thermodynamic-kinetic modelling formalism and discuss a simplified rate law in which the reaction rate directly depends on the reaction affinity. For automatic handling of modular rate laws, we propose a standard syntax and semantic annotations for the Systems Biology Markup Language. AVAILABILITY: An online tool for inserting the rate laws into SBML models is freely available at www.semanticsbml.org. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

MOTIVATION: The availability of modern sequencing techniques has led to a rapid increase in the amount of reconstructed metabolic networks. Using these models as a platform for the analysis of high throughput transcriptomic, proteomic and metabolomic data can provide valuable insight into conditional changes in the metabolic activity of an organism. While transcriptomics and proteomics provide important insights into the hierarchical regulation of metabolic flux, metabolomics shed light on the actual enzyme activity through metabolic regulation and mass action effects. Here we introduce a new method, termed integrative omics-metabolic analysis (IOMA) that quantitatively integrates proteomic and metabolomic data with genome-scale metabolic models, to more accurately predict metabolic flux distributions. The method is formulated as a quadratic programming (QP) problem that seeks a steady-state flux distribution in which flux through reactions with measured proteomic and metabolomic data, is as consistent as possible with kinetically derived flux estimations. RESULTS: IOMA is shown to successfully predict the metabolic state of human erythrocytes (compared to kinetic model simulations), showing a significant advantage over the commonly used methods flux balance analysis and minimization of metabolic adjustment. Thereafter, IOMA is shown to correctly predict metabolic fluxes in Escherichia coli under different gene knockouts for which both metabolomic and proteomic data is available, achieving higher prediction accuracy over the extant methods. Considering the lack of high-throughput flux measurements, while high-throughput metabolomic and proteomic data are becoming readily available, we expect IOMA to significantly contribute to future research of cellular metabolism.

Abstract Motivation: The expression of genes is controlled by specific combinations of cellular variables. We applied Independent Component Analysis (ICA) to gene expression data, deriving a linear model based on hidden variables, which we term ‘expression modes’. The expression of each gene is a linear function of the expression modes, where, according to the ICA model, the linear influences of different modes show a minimal statistical dependence, and their distributions deviate sharply from the normal distribution. Results: Studying cell cycle-related gene expression in yeast, we found that the dominant expression modes could be related to distinct biological functions, such as phases of the cell cycle or the mating response. Analysis of human lymphocytes revealed modes that were related to characteristic differences between cell types. With both data sets, the linear influences of the dominant modes showed distributions with large tails, indicating the existence of specifically up- and downregulated target genes. The expression modes and their influences can be used to visualize the samples and genes in low-dimensional spaces. A projection to expression modes helps to highlight particular biological functions, to reduce noise, and to compress the data in a biologically sensible way. Availability: The FastICA algorithm (Hyvärinen, IEEE Trans. Neural Netw. , 10, 626–634, 1999) is freely available at http://www.cis.hut.fi/projects/ica/fastica/. Additional matlab scripts and detailed results can be downloaded from http://www.molgen.mpg.de/research/lehrach/projects/genica/ Contact: [email protected]