Metabolite-Enzyme Coevolution: From Single Enzymes to Metabolic Pathways and Networks.

How individual enzymes evolved is relatively well understood. However, individual enzymes rarely confer a physiological advantage on their own. Judging by its current state, the emergence of metabolism seemingly demanded the simultaneous emergence of many enzymes. Indeed, how multicomponent interlocked systems, like metabolic pathways, evolved is largely an open question. This complexity can be unlocked if we assume that survival of the fittest applies not only to genes and enzymes but also to the metabolites they produce. This review develops our current knowledge of enzyme evolution into a wider hypothesis of pathway and network evolution. We describe the current models for pathway evolution and offer an integrative metabolite-enzyme coevolution hypothesis. Our hypothesis addresses the origins of new metabolites and of new enzymes and the order of their recruitment. We aim to not only survey established knowledge but also present open questions and potential ways of addressing them.

Mathematical properties of optimal fluxes in cellular reaction networks at balanced growth.

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.

BioSimulators: a central registry of simulation engines and services for recommending specific tools.

Computational models have great potential to accelerate bioscience, bioengineering, and medicine. However, it remains challenging to reproduce and reuse simulations, in part, because the numerous formats and methods for simulating various subsystems and scales remain siloed by different software tools. For example, each tool must be executed through a distinct interface. To help investigators find and use simulation tools, we developed BioSimulators (https://biosimulators.org), a central registry of the capabilities of simulation tools and consistent Python, command-line and containerized interfaces to each version of each tool. The foundation of BioSimulators is standards, such as CellML, SBML, SED-ML and the COMBINE archive format, and validation tools for simulation projects and simulation tools that ensure these standards are used consistently. To help modelers find tools for particular projects, we have also used the registry to develop recommendation services. We anticipate that BioSimulators will help modelers exchange, reproduce, and combine simulations.

SBML Level 3: an extensible format for the exchange and reuse of biological models.

Systems biology has experienced dramatic growth in the number, size, and complexity of computational models. To reproduce simulation results and reuse models, researchers must exchange unambiguous model descriptions. We review the latest edition of the Systems Biology Markup Language (SBML), a format designed for this purpose. A community of modelers and software authors developed SBML Level 3 over the past decade. Its modular form consists of a core suited to representing reaction-based models and packages that extend the core with features suited to other model types including constraint-based models, reaction-diffusion models, logical network models, and rule-based models. The format leverages two decades of SBML and a rich software ecosystem that transformed how systems biologists build and interact with models. More recently, the rise of multiscale models of whole cells and organs, and new data sources such as single-cell measurements and live imaging, has precipitated new ways of integrating data with models. We provide our perspectives on the challenges presented by these developments and how SBML Level 3 provides the foundation needed to support this evolution.

Notions of similarity for systems biology models.

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.

Parameter balancing: consistent parameter sets for kinetic metabolic models.

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.

Automated generation of bacterial resource allocation models.

Resource Balance Analysis (RBA) is a computational method based on resource allocation, which performs accurate quantitative predictions of whole-cell states (i.e. growth rate, metabolic fluxes, abundances of molecular machines including enzymes) across growth conditions. We present an integrated workflow of RBA together with the Python package RBApy. RBApy builds bacterial RBA models from annotated genome-scale metabolic models by adding descriptions of cellular processes relevant for growth and maintenance. The package includes functions for model simulation and calibration and for interfacing to Escher maps and Proteomaps for visualization. We demonstrate that RBApy faithfully reproduces results obtained by a hand-curated and experimentally validated RBA model for Bacillus subtilis. We also present a calibrated RBA model of Escherichia coli generated from scratch, which obtained excellent fits to measured flux values and enzyme abundances. RBApy makes whole-cell modelling accessible for a wide range of bacterial wild-type and engineered strains, as illustrated with a CO2-fixing Escherichia coli strain. AVAILABILITY: RBApy is available at /https://github.com/SysBioInra/RBApy, under the licence GNU GPL version 3, and runs on Linux, Mac and Windows distributions.

Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield.

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.

Global characterization of in vivo enzyme catalytic rates and their correspondence to in vitro kcat measurements.

Turnover numbers, also known as kcat values, are fundamental properties of enzymes. However, kcat data are scarce and measured in vitro, thus may not faithfully represent the in vivo situation. A basic question that awaits elucidation is: how representative are kcat values for the maximal catalytic rates of enzymes in vivo? Here, we harness omics data to calculate kmax(vivo), the observed maximal catalytic rate of an enzyme inside cells. Comparison with kcat values from Escherichia coli, yields a correlation ofr(2)= 0.62 in log scale (p < 10(-10)), with a root mean square difference of 0.54 (3.5-fold in linear scale), indicating that in vivo and in vitro maximal rates generally concur. By accounting for the degree of saturation of enzymes and the backward flux dictated by thermodynamics, we further refine the correspondence between kmax(vivo) and kcat values. The approach we present here characterizes the quantitative relationship between enzymatic catalysis in vitro and in vivo and offers a high-throughput method for extracting enzyme kinetic constants from omics data.

SBtab: a flexible table format for data exchange in systems biology.

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: : wolfram.liebermeister@gmail.com.

The Protein Cost of Metabolic Fluxes: Prediction from Enzymatic Rate Laws and Cost Minimization.

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.

Visual account of protein investment in cellular functions.

Proteomics techniques generate an avalanche of data and promise to satisfy biologists' long-held desire to measure absolute protein abundances on a genome-wide scale. However, can this knowledge be translated into a clearer picture of how cells invest their protein resources? This article aims to give a broad perspective on the composition of proteomes as gleaned from recent quantitative proteomics studies. We describe proteomaps, an approach for visualizing the composition of proteomes with a focus on protein abundances and functions. In proteomaps, each protein is shown as a polygon-shaped tile, with an area representing protein abundance. Functionally related proteins appear in adjacent regions. General trends in proteomes, such as the dominance of metabolism and protein production, become easily visible. We make interactive visualizations of published proteome datasets accessible at www.proteomaps.net. We suggest that evaluating the way protein resources are allocated by various organisms and cell types in different conditions will sharpen our understanding of how and why cells regulate the composition of their proteomes.

Glycolytic strategy as a tradeoff between energy yield and protein cost.

Contrary to the textbook portrayal of glycolysis as a single pathway conserved across all domains of life, not all sugar-consuming organisms use the canonical Embden-Meyerhoff-Parnass (EMP) glycolytic pathway. Prokaryotic glucose metabolism is particularly diverse, including several alternative glycolytic pathways, the most common of which is the Entner-Doudoroff (ED) pathway. The prevalence of the ED pathway is puzzling as it produces only one ATP per glucose--half as much as the EMP pathway. We argue that the diversity of prokaryotic glucose metabolism may reflect a tradeoff between a pathway's energy (ATP) yield and the amount of enzymatic protein required to catalyze pathway flux. We introduce methods for analyzing pathways in terms of thermodynamics and kinetics and show that the ED pathway is expected to require several-fold less enzymatic protein to achieve the same glucose conversion rate as the EMP pathway. Through genomic analysis, we further show that prokaryotes use different glycolytic pathways depending on their energy supply. Specifically, energy-deprived anaerobes overwhelmingly rely upon the higher ATP yield of the EMP pathway, whereas the ED pathway is common among facultative anaerobes and even more common among aerobes. In addition to demonstrating how protein costs can explain the use of alternative metabolic strategies, this study illustrates a direct connection between an organism's environment and the thermodynamic and biochemical properties of the metabolic pathways it employs.

Pathway thermodynamics highlights kinetic obstacles in central metabolism.

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.

Spanning high-dimensional expression space using ribosome-binding site combinatorics.

Protein levels are a dominant factor shaping natural and synthetic biological systems. Although proper functioning of metabolic pathways relies on precise control of enzyme levels, the experimental ability to balance the levels of many genes in parallel is a major outstanding challenge. Here, we introduce a rapid and modular method to span the expression space of several proteins in parallel. By combinatorially pairing genes with a compact set of ribosome-binding sites, we modulate protein abundance by several orders of magnitude. We demonstrate our strategy by using a synthetic operon containing fluorescent proteins to span a 3D color space. Using the same approach, we modulate a recombinant carotenoid biosynthesis pathway in Escherichia coli to reveal a diversity of phenotypes, each characterized by a distinct carotenoid accumulation profile. In a single combinatorial assembly, we achieve a yield of the industrially valuable compound astaxanthin 4-fold higher than previously reported. The methodology presented here provides an efficient tool for exploring a high-dimensional expression space to locate desirable phenotypes.

Optimal enzyme profiles in unbranched metabolic pathways.

How to optimize the allocation of enzymes in metabolic pathways has been a topic of study for many decades. Although the general problem is complex and nonlinear, we have previously shown that it can be solved by convex optimization. In this paper, we focus on unbranched metabolic pathways with simplified enzymatic rate laws and derive analytic solutions to the optimization problem. We revisit existing solutions based on the limit of mass-action rate laws and present new solutions for other rate laws. Furthermore, we revisit a known relationship between flux control coefficients and enzyme abundances in optimal metabolic states. We generalize this relationship to models with density constraints on enzymes and metabolites, and present a new local relationship between optimal reaction elasticities and enzyme amounts. Finally, we apply our theory to derive simple kinetics-based formulae for protein allocation during bacterial growth.

A neural-mechanistic hybrid approach improving the predictive power of genome-scale metabolic models.

Constraint-based metabolic models have been used for decades to predict the phenotype of microorganisms in different environments. However, quantitative predictions are limited unless labor-intensive measurements of media uptake fluxes are performed. We show how hybrid neural-mechanistic models can serve as an architecture for machine learning providing a way to improve phenotype predictions. We illustrate our hybrid models with growth rate predictions of Escherichia coli and Pseudomonas putida grown in different media and with phenotype predictions of gene knocked-out Escherichia coli mutants. Our neural-mechanistic models systematically outperform constraint-based models and require training set sizes orders of magnitude smaller than classical machine learning methods. Our hybrid approach opens a doorway to enhancing constraint-based modeling: instead of constraining mechanistic models with additional experimental measurements, our hybrid models grasp the power of machine learning while fulfilling mechanistic constrains, thus saving time and resources in typical systems biology or biological engineering projects.

RBAtools: a programming interface for Resource Balance Analysis models.

Motivation: Efficient resource allocation can contribute to an organism's fitness and can improve evolutionary success. Resource Balance Analysis (RBA) is a computational framework that models an organism's growth-optimal proteome configurations in various environments. RBA software enables the construction of RBA models on genome scale and the calculation of medium-specific, growth-optimal cell states including metabolic fluxes and the abundance of macromolecular machines. However, existing software lacks a simple programming interface for non-expert users, easy to use and interoperable with other software. Results: The python package RBAtools provides convenient access to RBA models. As a flexible programming interface, it enables the implementation of custom workflows and the modification of existing genome-scale RBA models. Its high-level functions comprise simulation, model fitting, parameter screens, sensitivity analysis, variability analysis and the construction of Pareto fronts. Models and data are represented as structured tables and can be exported to common data formats for fluxomics and proteomics visualization. Availability and implementation: RBAtools documentation, installation instructions and tutorials are available at https://sysbioinra.github.io/rbatools/. General information about RBA and related software can be found at rba.inrae.fr.

Fibration symmetry uncovers minimal regulatory networks for logical computation in bacteria.

Symmetry principles have proven important in physics, deep learning and geometry, allowing for the reduction of complicated systems to simpler, more comprehensible models that preserve the system's features of interest. Biological systems often show a high level of complexity and consist of a high number of interacting parts. Using symmetry fibrations, the relevant symmetries for biological 'message-passing' networks, we reduced the gene regulatory networks of E. coli and B. subtilis bacteria in a way that preserves information flow and highlights the computational capabilities of the network. Nodes that share isomorphic input trees are grouped into equivalence classes called fibers, whereby genes that receive signals with the same 'history' belong to one fiber and synchronize. We further reduce the networks to its computational core by removing "dangling ends" via k-core decomposition. The computational core of the network consists of a few strongly connected components in which signals can cycle while signals are transmitted between these "information vortices" in a linear feed-forward manner. These components are in charge of decision making in the bacterial cell by employing a series of genetic toggle-switch circuits that store memory, and oscillator circuits. These circuits act as the central computation machine of the network, whose output signals then spread to the rest of the network.