Modern omics analyses are able to effectively characterize the genetic, regulatory, and metabolic phenotypes of engineered microbes, yet designing genetic interventions to achieve a desired phenotype remains challenging. With recent developments in genetic engineering techniques, timelines associated with building and testing strain designs have been greatly reduced, allowing for the first time an efficient closed loop iteration between experiment and analysis. However, the scale and complexity associated with multi-omics datasets complicates manual biological reasoning about the mechanisms driving phenotypic changes. Computational techniques therefore form a critical part of the Design-Build-Test-Learn (DBTL) cycle in metabolic engineering. Traditional statistical approaches can reduce the dimensionality of these datasets and identify common motifs among high-performing strains. While successful in many studies, these methods do not take full advantage of known connections between genes, proteins, and metabolic networks. There is therefore a growing interest in model-aided design, in which modeling frameworks from systems biology are used to integrate experimental data and generate effective and non-intuitive design predictions. In this mini-review, we discuss recent progress and challenges in this field. In particular, we compare methods augmenting flux balance analysis with additional constraints from fluxomic, genomic, and metabolomic datasets and methods employing kinetic representations of individual metabolic reactions, and machine learning. We conclude with a discussion of potential future directions for improving strain design predictions in the omics era and remaining experimental and computational hurdles. The biorefinery concept involves the development of sustainable and low-impact production routes for major commodity chemicals and fuels from biomass . BiomanuA number of recent reviews have covered the growing usefulness of omics approaches in characterizing cell physiology , and theEven with access to direct measurements of activity for a wide range cellular machinery components, using these data to enhance metabolic flux for a desired pathway remains challenging. We next discuss Learn techniques that synthesize these vast data sources together with generalized knowledge of biological function.The goal of the Learn and Design steps is to use the characterization of previously engineered strains to develop improved strain designs. In its most basic form, this step can be accomplished by examining biological features  correlated with improved strain performance, and overexpressing those likely involved in the pathway of interest . DesignsConstraint-Based Reconstruction and Analysis (COBRA) methods use biological knowledge and data to place constraints on intracellular fluxes, and in recent years have expanded to consider a wide range of recent omics techniques. Here we focus on extensions of COBRA methods that pertain to guiding strain designs from omics data, while a number of recent reviews have covered COBRA methods in greater depth . A centrExtensions to the COBRA framework have therefore been proposed to impose additional constraints from experimental observations. One of the earliest such studies used transcriptomic data to block flux through reactions where gene expression for required enzymes was not observed . This meMetabolomics data are typically incorporated in constraint-based models through the explicit consideration of reaction thermodynamics. If absolute metabolite concentrations are available, thermodynamic metabolic flux analysis can provide more condition-specific information on irreversible reactions . These pde novo experimental results has not been rigorously evaluated and would serve a useful study in measuring the progress in our understanding of cellular behavior. However, even modest success rates from predictive tools are useful in guiding experimental efforts where the search space is vast. A limitation of constrained-based methods is that they are often less suitable for suggesting improvements to fine-tune the enzyme expression of an existing pathway. Such a task typically requires a kinetic description of the reactions in question, which we discuss in the next section.Constraint-Based Reconstruction and Analysis methods therefore represent an extensible and computationally efficient framework for connecting omics data of different types and have been used to successfully interpret omics data and improve strain designs in a number of studies . An advaThe goal of kinetic metabolic models is to capture the dynamic behavior of individual enzymes and integrate these expressions into the behavior of the full metabolic network. These models allow the direct prediction steady-state flux distributions as a function of enzyme expression, which typically serve as the most reliable experimental data for validation. However, models that explicitly incorporate enzyme kinetics (if parameterized correctly) are capable of predicting finer details of pathway dynamics, including the effect of slight changes in enzyme activity on metabolic flux. In constraint-based models, metabolite pools are assumed to be in a pseudo steady-state, and thus the rate rules governing flux through each reaction can be ignored. While the steady-state assumption may be justified, the specific steady-state reached inside the cell is determined, among a multitude of factors, both by external metabolite conditions as well as the kinetics and expression levels of metabolic enzymes. Kinetic modeling frameworks therefore seek to estimate these reaction rate rules from observed metabolic phenotypes to predict how enzyme perturbation will affect steady-state concentrations and fluxes.in vitro and time-course metabolite concentration measurements in fitting parameter values  become difficult to construct for reactions with many participating species. Accordingly, larger-scale kinetic models typically choose a generalizable framework for constructing rate rule expressions. These frameworks range in computational complexity and faithfulness to the underlying enzyme-substrate system, and we leave a detailed comparison of these approaches to a number of recently published reviews . SoftwarRegardless of the framework chosen, a major hurdle in using kinetic models for interpretation of omics data is the computational effort required in parameter estimation. In metabolic ensemble modeling, parameters are sampled at random and retained in the final ensemble only if they match all the considered experimental data . As a reA factor complicating the analysis of experimental data with kinetic models is the stochasticity introduced by low cell volumes and small copy numbers of several key enzymes . Cell toMachine learning methods for interpreting omics data have taken a wide range of forms, largely due to the many varied biological questions that can be asked. In this section, we focus on methods that predict future targets for strain engineering. Integrative omics analyses attempt to draw connects between disparate omics data sources, either with or without prior biological knowledge . These mSince Learn lags behind the rest of the DBTL methodology in the development of validated and standardized techniques, feasible computational techniques are still being explored and improved upon. As a result, software libraries for performing the analyses described in this minireview are relatively scarce. As the most mature method of the three, COBRA methods have relatively strong software support in both the MATLAB  and PythPSJ and YB contributed to the conception and writing of the manuscript. PSJ created the figure. YB supervised the research.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The use of mass spectrometry-based metabolomics to study human, plant and microbial biochemistry and their interactions with the environment largely depends on the ability to annotate metabolite structures by matching mass spectral features of the measured metabolites to curated spectra of reference standards. While reference databases for metabolomics now provide information for hundreds of thousands of compounds, barely 5% of these known small molecules have experimental data from pure standards. Remarkably, it is still unknown how well existing mass spectral libraries cover the biochemical landscape of prokaryotic and eukaryotic organisms. To address this issue, we have investigated the coverage of 38 genome-scale metabolic networks by public and commercial mass spectral databases, and found that on average only 40% of nodes in metabolic networks could be mapped by mass spectral information from standards. Next, we deciphered computationally which parts of the human metabolic network are poorly covered by mass spectral libraries, revealing gaps in the eicosanoids, vitamins and bile acid metabolism. Finally, our network topology analysis based on the betweenness centrality of metabolites revealed the top 20 most important metabolites that, if added to MS databases, may facilitate human metabolome characterization in the future. Metabolomics, or the comprehensive characterization and quantification of metabolites, complements upstream biochemical information obtained from genes, transcripts, and proteins, widening the current genomic reconstructions of metabolism and improving our understanding of biological and environmental processes . MetabolHere, we use genome-based reconstructions of metabolism, also called genome-scale metabolic networks ,15, to iBy analysing the coverage of metabolic networks, we have computationally deciphered which parts of the human metabolic network are poorly covered by mass spectral libraries and have identified metabolite gaps that, if added to MS databases, may enhance human metabolome characterization in the future, and consequently, provide a better understanding of cell metabolism.In silico predicted MS/MS spectra available in certain public databases [. 2016 [Only compounds with measured mass spectra were used. atabases  were not [. 2016 . This li [. 2016  yielding [. 2016  provided [. 2016 ; 1171 In [. 2016 ; 3480 In [. 2016   format in Recon2  was usedAfter the creation of the undirected compound graph, we identified parts of the network that were less covered by mass spectral libraries. For this, we used the Label Propagation Algorithm (LPA) , which aTo identify key missing nodes  in mass spectral libraries, we used a network topology measure called centrality. Centrality is a very well-studied field in network science which aims at identifying important actors in a network. Among the numerous centrality indices, we chose the betweenness as the criterion for metabolite prioritisation. The betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes in the network . The betBeside topological measure, we also characterised metabolites through their prominency in scientific literature. We used the PubChem REST API  to obtaiEscherichia coli, Arabidopsis thaliana, Saccharomyces cerevisiae (yeast), Mus musculus (mouse) or Homo sapiens (human). We mapped the chemical library containing 253,927 non-redundant first block InChIKeys onto 38 different genome-scale metabolic networks, including relevant organisms such as Mus musculus, Escherichia coli and Arabidopsis thaliana, only 52–60% of their metabolomes could be potentially characterized by confronting MS data with all existing mass spectral information from pure standards (which are not currently accessible from a single resource). For human (Homo sapiens), this number drops to 42.2% and 30.5% in the case of the KEGG and Recon2 metabolic models, respectively. Second, the annotation level, i.e., specification of chemical identifiers, in genome-scale metabolic models is still very limited. Models such as Homo sapiens (Recon2 and HumanCyc) and different plants (PlantCyc) contain a large number of compounds with no compound identifier other than its name, resulting in fewer compounds than expected with associated InChIKey : 48.7% for Recon2, 48.6% for PlantCyc, and 35.7% for HumanCyc. On average, 63.2% of compounds in our metabolic models have InChIKey, which constitutes an obstacle for reliably mapping experimental metabolomics data onto metabolic models.Two significant findings can be drawn from a closer analysis of Additional to the above analysis, we have also assessed the coverage of individual mass spectral databases in metabolic models see . Figure 2 were manually removed. Out of 1597 resulting nodes in the metabolic network, 890 metabolites (55.7%) were mapped  in the poorly covered areas relative to metabolites from the well-covered areas A. The fe., systematically demonstrated that neighbouring metabolites in a metabolic network share structural similarities and have similar MS/MS spectra [Recently, Aguilar-Mogas et al spectra . On this spectra ,32 and tIn order to identify the most important metabolites currently missing in the MS databases, we performed a centrality analysis. Here we have combined cheminformatics and network analysis methods to investigate the coverage of public and commercial mass spectral databases in the metabolism of prokaryotic and eukaryotic organisms, particularly taking a closer look at human metabolism. For this, we have used genome-scale metabolic reconstructions, which are considered the most comprehensive and annotated models of metabolism in multiple organisms. Genome-scale metabolic networks contain information both on metabolites and their reactions with corresponding genes and proteins. However, most genome-scale metabolic networks are reconstructed from genomic sequences and literature, and rarely incorporate new and rapidly evolving metabolomic data. This has resulted in some of the constraints and mismatches encountered in our study.Our computational approach has revealed that many metabolites are missing from mass spectral libraries. For example, 44% of compounds with an InChIKey in Recon2 could not be matched in any mass spectral database. Our results, therefore, provide an essential resource to improve the biochemical landscape of mass spectral databases, and highlights the pressing need for standards to prioritise on to fill these gaps. However, the apparent “low metabolic content” of mass spectral libraries may also be a consequence of insufficient annotation of genome-scale metabolic models. These models (available in SBML format) were initially built for constraint-based computational studies , where the chemical structure of small molecules is not necessarily required for computation. Therefore, most of these models contain a substantial number of metabolites with only short and ambiguous names but no other standard identifiers, which represent a serious obstacle for mapping metabolomics data onto genome-scale metabolic models. Metabolites without proper identifiers can result from the lack of cross-references during their annotation by the scientific community, making computational tools unable to reach the information needed to make correspondences between mass spectral libraries and metabolic networks. One common and useful identifier in this regard is the InChI, which is directly built from the chemical structure of compounds and the hash of the structure, the InChIKey, enabling both the computational analysis performed here, as well as much broader searching of other resources. Unfortunately, we have noticed that most metabolic models often refer to classes of compounds  in order to represent the enzymatic promiscuity of substrates or to describe generic biochemical reactions. Consequently, when the metabolic networks are generated, nodes without chemical structures cannot be mapped on to the mass spectral libraries. Automated approaches to enumerate potentially matching structures to generic representations are required to capture these substances in future studies . MetabolThe difficulty of mapping metabolomics data onto metabolic networks can also stem from the different scale between models and measures: different stereoisomers may be encoded in the network but are often indistinguishable in a MS experiment . The methods proposed in this article are sufficiently generic to be applied to updated datasets and/or in-house spectral libraries. It will also be possible to apply this approach to updated versions of metabolic networks. As a matter of fact, a new version of the human metabolic network Recon has been released concurrently to our work . Our pre


Metabolic coupling of product synthesis and microbial growth is a prominent approach for maximizing production performance. Growth-coupling (GC) also helps stabilizing target production and allows the selection of superior production strains by adaptive laboratory evolution. To support the implementation of growth-coupling strain designs, we seek to identify biologically relevant, metabolic principles that enforce strong growth-coupling on the basis of reaction knockouts.We adapted an established bilevel programming framework to maximize the minimally guaranteed production rate at a fixed, medium growth rate. Using this revised formulation, we identified various GC intervention strategies for metabolites of the central carbon metabolism, which were examined for GC generating principles under diverse conditions. Curtailing the metabolism to render product formation an essential carbon drain was identified as one major strategy generating strong coupling of metabolic activity and target synthesis. Impeding the balancing of cofactors and protons in the absence of target production was the underlying principle of all other strategies and further increased the GC strength of the aforementioned strategies.Maximizing the minimally guaranteed production rate at a medium growth rate is an attractive principle for the identification of strain designs that couple growth to target metabolite production. Moreover, it allows for controlling the inevitable compromise between growth coupling strength and the retaining of microbial viability. With regard to the corresponding metabolic principles, generating a dependency between the supply of global metabolic cofactors and product synthesis appears to be advantageous in enforcing strong GC for any metabolite. Deriving such strategies manually, is a hard task, due to which we suggest incorporating computational metabolic network analyses in metabolic engineering projects seeking to determine GC strain designs.The online version of this article (10.1186/s12859-019-2946-7) contains supplementary material, which is available to authorized users. Metabolic engineering approaches strive to optimize microbial cell-factories for robust, profitable, and sustainable industrial applications . One appSaccharomyces cerevisiae and Escherichia coli, for example, such a wGC is naturally observed for fermentation products, e.g., ethanol or acetate, under anaerobic conditions or during overflow metabolism. By this means, holistic GC (hGC) is encountered if the lower production rate bound is above zero for all growth rates greater than zero  reaction is enforced. This constraint precludes the zero flux vector from the solution space and enables the identification of sGC strategies using reaction deletions only [Three distinct GC phenotypes differing in GC strength can be distinguished, which become apparent from computing and plotting so-called metabolic production envelopes . These pero Fig. b while swth Fig. c. The phons only . Hence, minimally guaranteed production and thus enforces GC at maximal growth. Complementarily to FBA, EMA utilizes the nondecomposable steady-state flux distributions, called elementary modes (EMs), of a metabolic network from which any feasible flux state can be derived by linear combinations of these EMs [Various computational algorithms exploiting the rich information content of stoichiometric metabolic models have been developed to specifically provide reaction deletion strategies leading to GC. These approaches are generally grouped into Flux Balance Analysis (FBA) and Elementary Modes Analysis (EMA) based methods. Classical FBA focuses on a particular metabolic phenotype by optimizing a biological meaningful objective function subject to steady-state mass balance constraints . Thus, Ghese EMs , 18. By hese EMs . Methodshese EMs , 21 or mhese EMs  use EMA hese EMs . Still, hese EMs , 23, 24.E. coli [Beside the in silico identification of GC intervention strategies, research on the general feasibility and driving forces of the coupling between growth and target product synthesis has been conducted. Based on the EMA approach, Klamt & Mahadevan  have buiE. coli , 20, theE. coli  proposedE. coli  in the rE. coli , 23, 26.E. coli. The resulting strategies were subsequently examined regarding the consequence of imposed growth-coupled product synthesis on metabolic network operation.In view of these disparate explanations for GC, we aimed at further unraveling key principles of reaction deletion strategies leading to GC by identifying relevant genetic intervention strategies for a set of metabolites and investigating the specific operating principle of these strategies. We adapted the mixed-integer linear program formulation of OptKnock to determine GC knockout strategies for a given target compound, a specific substrate and a defined maximum number of reaction deletions. Particularly, our framework, which we termed gcOpt, maximizes the minimally guaranteed production rate at a medium, fixed growth rate and was applied to calculate GC intervention strategies for a broad range of metabolites of a core as well as a genome-scale metabolic model of fix using appropriate reaction deletions , thus pointing to an excellent combination of GC and viability of this mutant. The predicted intervention strategies at a μfix = 0.1 h− 1 a  and maximal growth rates of 0.13 h− 1. Figure The designs identified by gcOpt Fig. a-c clear− 1 Fig. a. The mah et al.  and Hädi− 1 Fig. c was int1 a Fig. b were a fix. Reducing μfix gradually favors the identification of strain designs with higher GC strengths, thus elevated guaranteed target production rates but possibly lower maximal growth rates. Moreover, the inherent approach of increasing the minimum production rate enforces the generally preferred sGC and hGC solutions, which guarantee product synthesis with growing or metabolically active organisms. This is a beneficial trait compared to alternative FBA based algorithms such as OptKnock or RobustKnock, which per se do not favor these designs over wGC solutions. In terms of computing time, these algorithms are similarly costly as compared to gcOpt due to the same mixed integer linear program (MILP) formulation  and the expected GC strength can furthermore be controlled by adapting the fixed growth rate μtion cf.  section.E. coli iAF1260 core model under aerobic as well as anaerobic conditions. The corresponding reaction and gene deletion set of each identified strategy can be found in the Additional files As mentioned in the introduction, there is a diverse discussion about possible principles and routes to enforce GC. These range between pure stoichiometric forces, such as anchor reactions, to flux-based notions which relate GC to imbalances in the households of energy and redox equivalents. As a first computational screening, we applied gcOpt to compute a comprehensive dataset of GC designs, which we analyzed in-depth to decipher general metabolic principles that trigger GC. To this end, we computed intervention strategies with one to seven reaction deletions for the 36 central carbon metabolites of the − 1 h− 1 [− 1 h− 1 and 12.2 mmol g− 1 h− 1, respectively.Under both anaerobic and aerobic condition, gcOpt simulations were additionally conducted with a decreased as well as an increased non-growth associated maintenance (NGAM) ATP requirement by changing the lower flux bound of the corresponding ATPM reaction Eq.  about 50− 1 h− 1  to 4.2 m+/H) and NADP(+/H), virtually resembling an elevated turnover of these cofactors . GCS relates the area of the accessible production envelope of the wild-type strain to the inaccessible or blocked area below the lower production rate bound of the mutant strain up to the maximum growth rate of the mutant  under aerobic and anaerobic conditions, respectively  is likely due to a global metabolic necessity rather than caused by a strict dependence of the synthesis of a particular biomass precursor on target compound production. Moreover, ATP scarcity seemed to be a metabolic trigger for GC in those sGC cases in which the synthesis of any biomass precursor was blocked by the intervention strategies. To challenge this hypothesis, the GCS of a GC strain design was investigated upon relaxing the directionality constraint of the ATPM equation cf. Eq.  thereby ons Eqs. –3 were rex and in locate the H+ protons to the extracellular and intracellular compartment, respectively. Both reactions were unbounded, i.e., allowed to carry any flux. All identified GC strategies and their GCS values under all investigated conditions are provided in the Additional files Here, the indices + conversion, the proton exchange or a combination of these strategies H/NAD(P)ies Fig. . These t+ balancing. For example, GC of lactate synthesis was abolished in most designs by relaxation of either ATP/ADP, NADH/NAD+ conversion or a free proton translocation. The basic coupling principle for this reduced metabolite is however NADH reoxidation, achieved by the reduction of pyruvate to lactate. Consequently, growth-coupling is abolished upon opening the NADH balance. Relaxation of the ATP and proton balance had the same effect as it fuels flux through the NADH transhydrogenase, which couples NADH oxidation and NADPH reduction to proton import. The formed NADPH is oxidized in biomass forming reactions making NADH re-oxidation by lactate dehydrogenase activity superfluous. Under standard conditions NADH transhydrogenase activity is limited by the cell’s potential to maintain a proton gradient over the cell membrane. In contrast, GC of ethanol was only abolished when free proton exchange was enabled. That was not expected as ethanol and lactate share almost the same synthesis pathway and as ethanol is more strongly reduced than lactate. Apparently, GC of ethanol was achieved in these designs by coupling the intracellular proton balance to the ethanol-proton symporter activity. As all intervention GC strategies for ethanol included the deletion of the ATP synthase, proton export via a reversed ATP synthase activity under relaxed ATP turnover conditions was not possible. GC of pyruvate was diminished by both free proton transport and ATP/ADP conversion. Inspection of the flux distribution under relaxed ATP/ADP conversion conditions revealed that excess ATP was used to drive the ATP synthase as proton exporter. Consequently, pyruvate secretion was enforced by the need to balance intracellular protons as was the case for ethanol. For aerobic conditions, relaxation of single or combinations of the tested constraints relieved GC for all wGC and most hGC strategies, as well. Again, formate was the only metabolite that was hard-coupled to growth by forcing flux through the pyruvate formate lyase anchor reaction. However, under aerobic conditions, this strategy is not of any relevance due to the pyruvate formate lyase’s sensitivity to oxygen [2. In vivo, this strategy might not be specific but could enforce the secretion of any upper central carbon metabolite. In our simulations, this was prohibited as only export reactions of lactate, ethanol, and formate were included in the model and the formation of these fermentation products was prevented by further reaction deletions in the GC strategies. In the remaining designs the metabolism was curtailed in a way that forced glucose oxidation through metabolic anchor reactions, here, transketolase, transaldolase or fructose bisphosphate aldolase, splitting the substrate into the target compound and an essential central carbon metabolism intermediate. As for the formate coupling strategies, the GCS of these strategies, although not completely abolished, was significantly reduced in most strategies upon relaxation of cofactor turnover and proton exchange. For the complete statistics corresponding to Figs. Disclosure of the basic coupling principles was impeded by the interrelatedness of redox cofactor, ATP and Ho oxygen . Surpris− 1 h− 1. It is furthermore practical to normalize ATPsc with the number of carbon atoms of the target product yielding the ATPsc per carbon (ATPcsc).The examination of the biomass precursor availability in the GC mutant strains and the influence of the ATP and NAD(P) H turnover and proton exchange on GC gave a first indication of the importance of the balancing of redox and energy cofactors. To further unravel the interdependency between GC and the cellular energy metabolism, the parameter ‘ATP synthesis capability’ (ATPsc) was defined. ATPsc assesses the contribution of the product synthesis to the global provision or consumption of ATP cf.  section.E. coli iAF1260 core model the ATPcsc of CO2 is the highest followed by those of fermentation and overflow metabolites such as ethanol, lactate, succinate or acetate under anaerobic as well as aerobic conditions GC strategies identified under standard conditions for which lowering the ATPM demand a posteriori weakens the GC strength of the respective mutants.Here again, this principle seems to contradict the findings in Fig. E. coli strain shifts the flux distribution towards substrate-level phosphorylation pathways, i.e., glycolysis and TCA cycle, and the secretion of correlated metabolites. We conclude that, for E. coli strains, ATP synthase deletion forms a basis for GC under aerobic conditions whereas additional knockouts enforce specificity of product secretion as can be derived from the steadily increasing mean GCS with increasing genetic interventions. Exceptions from this pattern are fermentation products or products exported via proton symporters, for which GC is induced by disrupting alternative NADH re-oxidizing or proton translocating pathways.In any way, principle 3 can be intuitively inferred from the observation that the ATP synthase is a frequent knockout target in aerobic GC strategies and more quantitatively be described with the rise of the target metabolite in the energy hierarchy. This is supported by the observation of Jensen & Michelsen  that an The formulation of a mixed integer linear program which maximizes the minimally guaranteed production rate of a target metabolite at a medium, fixed growth rate, realized in gcOpt, has been shown to yield attractive strain designs with growth-coupled target production. One advantage of the identified growth-coupling (GC) strategies, at least of the investigated test cases, is the existence of only one feasible target production rate at maximal growth rather than a range of possible production rates.Generally, metabolic network reconfigurations that render product secretion into a carbon drain necessary for metabolic activity might be the more robust GC approach as it is independent of cofactor and proton balancing that might vary under different growth conditions. However, such a coupling might not be possible for all metabolites. Our analysis revealed that coupling product formation to cofactor supply or turnover not only enhances the GCS of the former strategies but also seems to be globally applicable to any metabolite. In contrast to the more comprehendible and manually applied concept of coupling target metabolite production to biomass precursor synthesis, metabolic designs that are based on such cofactor balancing are hard to derive manually. This is mainly due to the complex interconnectedness of energy and redox cofactors within metabolic networks. Accordingly, we argue that computer-aided network analysis can accelerate the development of strain designs strictly coupling production to microbial growth by predicting effective GC strategies with a reasonable number of gene deletions.vt of a target compound t for a fixed growth rate μfix. The corresponding bi-level optimization problem is formulated as follows:gcOpt is geared to existing multi-level optimization frameworks and their lower-level formulations optimizing an engineering objective by searching for appropriate sets of reaction deletions , 16. gcOi ∈ R within the reversible metabolic model if i is inactive (0) or active (1). With constraint (6), the size of the reaction deletion set is limited to K interventions. Note that the gcOpt formulation requires the splitting of all reversible reaction of the original model into irreversible forward and backward reactions. This results in an irreversible metabolic model containing N reactions with strictly positive fluxes (Eq. 10). The flux value of each irreversible reaction is contained in the vector S ∈ ℝM| × |N|| is the stoichiometric matrix where each non-zero value Sj, i denotes the stoichiometric coefficient of metabolite j ∈ M participating in reaction i ∈ N. Each flux vi is constrained by lower and upper bounds k indicates a knockout (Eq. 9). The connection between a reversible reaction k and one of its irreversible counterparts i is kept in the mapping matrix B ∈ {0, 1}N| × |R|| by Bi, k = 1. Since the biomass reaction vbm is fixed to μfix by Eq. 11, μfix needs to be lower than the maximally achievable biomass formation rate where Solving the nested mixed-integer optimization problem (Eq. 5-11) using linear programming solvers is intractable . By virtiJO1366 of E. coli K-12 MG1655 [iJO1366  was reduced following a preprocessing routine similar to a protocol of Feist et al. (2010), which was integrated into the gcOpt framework. More specifically, essential as well as exchange, diffusion, transport and spontaneous reactions were excluded from the set of possible target reactions for deletion. Furthermore, reactions contained in the subsystems cell envelope biosynthesis, membrane lipid metabolism, murein biosynthesis, tRNA charging and glycerophospholipid metabolism were also not regarded as deletion targets. In addition to reducing the solution space of the problem posed by gcOpt, the actual number of reactions within the considered metabolic model was trimmed by entirely removing all reactions unable to carry any flux. These so called blocked reactions were identified by flux variability analysis [In this work, core models of the central carbon metabolism introduced by Trinh et al.  and Orth2 MG1655  were useanalysis  by meanshttps://github.com/Spherotob/gcOpt). For solving the single-level MILP derived from problem (1), the Gurobi Optimizer  was utilized. All computations in this work were conducted on a Windows 7 machine with a maximum configuration of 16 GB of RAM and an AMD FX-8350 Eight-Core (à 4.00 GHz) processor.The gcOpt framework was implemented in MATLAB 2016b and is freely available on GitHub . This expresses the principle that the flux modes with the lowest yields are sequentially made inaccessible the stronger the coupling between growth and product synthesis becomes. To account for point (2), the minimally guaranteed target product yield at maximal growth Ymax is considered as a factor in the formula for the GCS (Eq. 6.1-6.3). For strain designs with similar GC levels according to an evaluation of the production envelope areas, this factor promotes those that predicts high yields at physiologically relevant growth rates. To be able to directly distinguish the GC types sGC, hGC and wGC as stated in requirement (3), the intersection of the lower production rate boundary and the growth axis was further integrated. Following this, the GCS is finally calculated as follows:To quantify and compare the GC level or strength of microbial strain designs we sought for a distinct measure. We required this measure to simultaneously (1) reflect the actual coupling strength in terms of the position of the lower production rate bound, (2) the yield at physiologically relevant growth conditions, as well as (3) the qualitative coupling type. Hence, we defined GCS, a novel measure for the growth coupling strength based on the production envelope representation. As visualized in Fig. Here, m in the biomass equation a new, unbounded reaction of the form υmM → was defined. Similarly, for each product N and stoichiometric coefficient υn a reaction → υnN was added to the model. Likewise, if the consumption of a precursor was coupled to the production of a certain compound, e.g. ATP and ADP, both were linked in a mass-balanced reaction of the form υmM → υnN. The original biomass equation was erased from the model. The availability of a biomass precursor of this original equation was then tested by setting up a linear program that maximized the related singularized reaction subject to all mass balance, substrate uptake and thermodynamic constraints of the original model. A maximal achievable flux of zero, implies loss of the metabolic capacity to synthesize the respective precursor and hence, this precursor is inaccessible. For cases with positive maximal fluxes, the synthesis of the precursor is not impaired.To evaluate the capability of a metabolic network to synthesize a particular biomass precursor the biomass synthesis equation was singularized into separate, independent reactions, i.e., for each single reactant M with stoichiometric coefficient υATPM max/dvtarget exchange with vATPM max being the maximal flux through the ATPM reaction  parameter was created to deduce the influence of byproduct secretion on the synthesis and provision of the cellular energy equivalent. For a given metabolic network and production rate of a target compound, the ATPsc describes the change of the maximal flux through the reaction ATPM in response to a change of the production rate of the target chemical. Mathematically, the ATPsc is defined by the derivative dvtion Eq.  computedAdditional file 1:E. coli iAF1260 core model. GCS values are additionally provided for an unconstrained ATP/ADP and NADH/NAD conversion, an unhindered proton translocation reaction, as well as combinations of these relaxations. The dataset includes the respective reaction and gene deletions for each identified strategy. (XLS 87 kb)GCS values for the identified GC strategies of all metabolites of the central carbon metabolism under anaerobic conditions. GC strategies with different numbers of maximally allowable reaction deletions were computed using gcOpt with the Additional file 2:E. coli iAF1260 core model. GCS values are additionally provided for an unconstrained ATP/ADP and NADH/NAD conversion, an unhindered proton translocation reaction, as well as combinations of these relaxations. The dataset includes the respective reaction and gene deletions for each identified strategy. (XLS 107 kb)GCS values for the identified GC strategies of all metabolites of the central carbon metabolism under aerobic conditions. GC strategies with different numbers of maximally allowable reaction deletions were computed using gcOpt with the Additional file 3:Figure S1. Succinate production envelopes of strain designs identified by gcOpt (A) and OptKnock (B) under aerobic conditions. Figure S2. Lactate production envelopes of strain designs identified by gcOpt (A) and OptKnock (B) under anaerobic conditions. Figure S3. Ethanol yield spaces of GC strain designs identified by gcOpt in comparison to designs taken from literature. Figure S4. Relation between maximal ATP maintenance flux (ATPM) and the production rate of several metabolites under anaerobe and aerobe conditions. Figure S5. ATP synthesis capability values normalized by the number of carbon atoms (ATPcsc) for several metabolites of the central carbon metabolism. (DOCX 1805 kb)Additional file 4:Table S1. Reaction deletions of GC strain designs identified by gcOpt in comparison to designs taken from literature. Table S2. Reaction deletions of GC strain designs identified by gcOpt and OptKnock for the production of succinate under aerobic conditions. Table S3. Reaction deletions of GC strain designs identified by gcOpt and OptKnock for the production of lactate under anaerobic conditions. Table S4. Effect of the relaxation of ATP/ADP and NADH/NAD+ conversion as well as the proton translocation on the GCS of the identified GC strain designs for anaerobic conditions. Table S5. Effect of the relaxation of ATP/ADP and NADH/NAD+ conversion as well as the proton translocation on the GCS of the identified GC strain designs for aerobic conditions. (DOCX 59 kb)


E. coli model. We show that ETFL is able to reproduce proteome-limited growth. We also subject it to several analyses, including the prediction of feasible mRNA and enzyme concentrations and gene essentiality.Systems biology has long been interested in models capturing both metabolism and expression in a cell. We propose here an implementation of the metabolism and expression model formalism (ME-models), which we call ETFL, for Expression and Thermodynamics Flux models. ETFL is a hierarchical model formulation, from metabolism to RNA synthesis, that allows simulating thermodynamics-compliant intracellular fluxes as well as enzyme and mRNA concentration levels. ETFL formulates a mixed-integer linear problem (MILP) that enables both relative and absolute metabolite, protein, and mRNA concentration integration. ETFL is compatible with standard MILP solvers and does not require a non-linear solver, unlike the previous state of the art. It also accounts for growth-dependent parameters, such as relative protein or mRNA content. We present ETFL along with its validation using results obtained from a well-characterized  Accounting for the effects of genetic expression in genome-scale metabolic models is challenging. Here, the authors introduce a model formulation that efficiently simulates thermodynamic-compliant fluxes, enzyme and mRNA concentration levels, allowing omics integration and broad analysis of in silico cellular physiology. Metabolic modeling, which helps making sense of the metabolism in a biological network, is an important tool for engineering biocatalysts, with applications in biofuels, drug design, microbial community analysis, and personalized medicine. Model accuracy is instrumental to the success of these applications through an efficient engineering of the host organisms. However, incorporating expression information into metabolic networks poses a significant challenge, and most current models do not even attempt it—effectively excluding an important network in biological systems that can drastically affect results. In metabolic engineering, strains are modified and controlled at the genome level through the transcriptome, and the effects are observed at the fluxome level, which accounts for the range of metabolic reactions in an organism. In between these two levels is the proteome that performs the biochemical transformations according to the genetic template, though it is this middle step in the process that cannot yet be robustly and efficiently incorporated into models of metabolic systems. Because of the complex interplay between these different layers of control, understanding expression and incorporating this into future models is key for improving metabolic engineering.1, it is possible to generate GEMs for hundreds of different species at a time. GEMs are particularly amenable to flux balance analysis (FBA), which models metabolism at the fluxome level using linear optimization techniques. However, plain FBA has been known to predict biochemically unrealistic solutions like free high-flux cycles or thermodynamically infeasible pathways. It also scales growth linearly with carbon uptake, which is not observed at high-uptake fluxes. FBA also fails to capture growth-dependent and protein-level effects, such as enzyme saturation or proteome-related limitations. Hence, several efforts have been made to supplement FBA with additional constraints to improve its predictive power. For example, thermodynamics-based flux analysis (TFA)3 uses thermodynamic constraints to enforce thermodynamically consistent reaction directionalities and to allow the integration of metabolomics. Resource balance models add a total proteome capacity constraint, as formulated in Beg et al.4, to model the proteome-related limitations of the cell, as enzymes have to compete for the constrained total amount of cellular proteins. Frameworks like GECKO5 further build on this resource balance idea and include flux constraints based on proteomics, such as 7 were the first to integrate the entirety of the expression mechanisms of the cell from the bottom-up, including mRNA and protein synthesis.Classically, model-based strain design has relied on tools that use the DNA sequence of an organism and homology with well-studied organisms to infer a network of metabolic reactions that happen inside a cell of that organism, which is called a genome-scale model (GEM). With current technologies and tools like metagenome sequencing10. Mixing these methods would require the inclusion of integers in ME-models, which is not straightforward and would lead to more complex mixed-integer nonlinear programs (MINLP) that are computationally intensive to solve. In addition, the amount of RNA and protein, the RNA and protein expression rates, and their stabilities are all growth dependent11, and including accurate representations of these variables leads to even more complex, nonlinear models. Meanwhile, although resource balance models such as GECKO could theoretically be integrated into TFA or ME-models in the current formulations, to the best of our knowledge, no link with TFA or ME-models has been proposed. Therefore, the metabolic engineering community needs a common formulation for these methodologies to build the most accurate models.However, simultaneously accounting for all of these constraints is challenging because of the formulation of each method, as TFA models involve integer variables that yield a mixed-integer linear program (MILP), whereas ME-models involve bilinear constraints that require special optimization procedures and a high-precision (quad-precision) solverE. coli model, iJO136612.We investigated the development of such a framework and propose herein a unified formulation for Expression and Thermodynamics-enabled FLux models (ETFL) that can account for the above integration issues. To our knowledge, ETFL is the first formulation that can account at the same time for expression, thermodynamics, and growth-dependant variables. It is also the first to do so using common double-precision MILP solvers. In ETFL, we address the compatibility of the formulations by expressing the growth rate variable in bilinear products as a piecewise constant function. We also address the issue of solver precision by performing a scaling that reduces the range of orders of magnitude of the variables. This reformulation allows us to transform the problem into a MILP, which we can solve efficiently using common open source or commercial solvers. The resulting model is then effectively able to directly integrate thermodynamic constraints as well as expression constraints and growth-dependent parameters. In this model, metabolite, enzyme, and mRNA concentration levels are explicitly defined to enable fast and easy omics integration: metabolites through their log-concentration variables in thermodynamics constraints, and enzymes and mRNA through their total concentration variables in the expression constraint. Finally, we show an application of this framework to a well-characterized Important assumptions are made to derive this formulation. The two most notables ones are (i) we can neglect the dilution rate of metabolites, and (ii) the steady-state approximation holds. While these assumptions are commonly made in FBA, we discuss them in details in the Supplementary Note ETFL is an ME-model implementation because it proposes a formulation that both accounts for metabolism and expression constraints. ME-models do not aim to replace kinetic models, but to account for the expression cost of making the enzymes that are necessary to carry a biochemical flux. In ETFL, this includes the cost of peptide and mRNA synthesis, as well as the competition for ribosomes and RNA polymerase in a limited proteome.5 and ME-model8.To transparently account for expression mechanisms and increase the predictive power of our models, we needed to derive the equations that could bridge the biochemistry with the optimization problem that is ETFL. Here, we present a summary of these equations, and detail their derivation in the Methods section. We derived these equations using assumptions similar to those used in the formulation of the GECKOThis formulation relies on derivations rooted in the biological mechanism of expression and depends on a number of biochemical parameters related to the cell. In particular, the mass balances of the macromolecules are expressed using concentration variables. Each mass balance will yield an equation where the concentrations of the macromolecules will be variables, thus effectively formulating a new constraint of the model and allowing us to calculate concentration values by solving the model.We can write the quasi-steady-state mass balance for macromolecules as follows:Using this formalism, for each macromolecule we can define and link together a synthesis flux, a degradation flux, and the macromolecule’s concentration. Knowing enzyme concentrations allows us to bound the variables representing metabolic reaction fluxes with their maximum catalytic rate according to the classical equation:The part of the matrix that has been added to the FBA problem to account for expression has been termed the expression problem (EP). Although this initial formulation is bilinear, we detail in the Methods section how we cast it to a MILP.11 report for instance that mRNA and protein mass ratios in the cell change with growth rate.In FBA, the biomass reaction is an artificial, lumped reaction that represents the consumption of metabolites in proportion to the cell growth rate. This consumption reflects nucleoside triphosphate (NTP) requirements for mRNAs, amino acid requirements for proteins, lipid requirements for the cell wall, or metal ion needs. Biomass reaction inclusiveness depends on the modeling assumptions made during the model curation process and can vary significantly among models of the same species. The consumed amount of each metabolite is usually estimated experimentally by measuring the the amounts of these metabolites in dried cell mass. Because the stoichiometric ratios of metabolites in the biomass reaction are fixed, the abundance of metabolites is the same for all growth rates. This simplifying assumption, necessary in FBA, goes against experimental evidence. Neidhardt and Curtis11. In this context, the approximation made in FBA can be written using ETFL terms:Because ETFL has explicit expression requirements through transcription, translation, and tRNA-charging reactions, it is possible to account for varying ratios of NTPs and amino acids as the growth rate changes, an effect that is captured in experimentsHere we show the formulation of the constraints of ETFL. For clarity, we use different indexing sets, each referring to a specific object in the model. The definition of these, as well as that of the variables and the parameters, are detailed in Table Metabolite mass balanceSetting Constraining Constraining If applicable, relaxing the allocation constraints,If applicable, relaxing the thermodynamic coupling constraints.In the ETFL formulation, enzyme synthesis is driven by the coupling between FBA and EP through the catalytic constraints. To carry flux, the cell needs to produce enzymes which will also use the metabolic resources of the cell. If allocation constraints are enforced, the amount of protein and mRNA synthesized must meet predefined mass ratios for the problem to be feasible. Hence, the metabolic requirement terms for the expression machinery (amino acids and NTP) have been removed from the biomass reaction and are accounted for in the tRNA charging and transcription reactions. Thus, the FBA solutions can be recovered from the ETFL formulation by the following routine:12 is a well-curated and well-studied GEM of E. coli that is closely related to the GEM used in developing both ME-models iOL1650-ME7 and iJL1678b-ME8. In addition, this model has been extensively applied in the literature, and is aligned with a variety of data sets that can be used for data integration. We wanted to subject the model to classical studies that would highlight the power of ETFL, particularly as pertains to proteome-limited growth, macromolecule concentration variability analysis, and gene knockout studies. We also wanted to assess the sensitivity of the model with respect to the presence of thermodynamic constraints, as well as growth-dependent parameters.iJO36611. The following Table Thus, we first experimented with four different models using ETFL with or without thermodynamic constraints and growth-dependent protein/RNA/DNA allocation following Table 2 as reported by Neidhardt et al.To study the behavior of the model at different carbon uptake rates, we simulated growth on a minimal medium with only glucose as a carbon source, unlimited oxygen, and some essential inorganic compounds. This would allow us to show that at a higher carbon uptake, the model would predict a limited growth—unlike FBA that would predict an unlimited linear increase.7 and GECKO5 formulations, within the context of models accounting for protein usage, this is caused by (i) the protein burden necessary to metabolize higher fluxes; (ii) the increased demand in protein synthesis at higher growth rates; and (iii) for the models with allocation constraints, the allowed protein and RNA mass ratio. We can see that models featuring protein, RNA, and DNA allocation constraints (vE[T]FL) consistently predict a lower growth rate than models without allocation constraints. This is expected, as the data we input requires additional proteins and mRNA to account for non-metabolism-related macromolecules. Models featuring thermodynamic constraints ([v]ETFL) also predict a lower growth rate, consistent with the fact that thermodynamics constrain the model to valid solutions whose flux is in the subspace of the FBA feasible space. The most constrained model (vETFL) consequently has the lowest growth rate at any glucose uptake. This is in accordance with published TFA results that eliminated biologically infeasible flux profiles yielding non-realistic higher growth rates2. Figure We summarize the constraint matrix of the EP of vETFL in Supplementary Table Although we initially focused on including only enzymes for which we had all the necessary information , we wanted to assess the robustness of our model when the missing enzymes were modeled as well as check our model’s sensitivity to changes in the catalytic rate constants. Thus, we additionally built three more models, based on vETFL, with the following properties: (i) all the missing enzymes were estimated by averaging the properties of the known enzymes based on the curation for the vETFL iJO1366  that is done by performing a variability analysis on the enzyme (mRNA) concentration variables. This type of study can, for instance, be compared with transcriptomics to check if the expression profile of an engineered strain corresponds to what is expected in its corresponding model. A way to visualize the average allowed proteome (transcriptome) is to use the average value of the variability of each enzyme (mRNA) concentration as a feasible observation. Due to the convexity of the solution space, it is a solution to the problem. This observation is then plotted on a finite area, which can be done using the online software Proteomaps Figure We used the mean of the variability analysis as the observation rather than a single optimal solution because the optimality principle in LP only guarantees a unique global optimum value and not a unique optimal solution. Moreover, solver heuristics give sparse and extreme results (corners of the explored simplex), which do not accurately represent the full extent of the considered solution space.The ETFL framework can also analyze the essentiality of specific genes by performing single gene knockouts. The growth of models with knocked-out genes can then be compared with the experimental data to assess the quality of the model as a validation.12. We use the Matthew’s correlation coefficient (MCC) as a metric for the quality of the prediction, which is preferred over accuracy as it is not sensitive to the imbalance between the number of essential genes and non-essential genes. The MCC reads like a usual correlation coefficient, with 1 being a perfect correlation, −1 perfect anti-correlation, and 0 no correlation. We used the essentiality data and conventions given in the Supplementary Material of Orth et al.12, as explained in Fig. We performed a gene essentiality analysis using in ETFL and compared it with the results reported in the publication of iJO1366 by Orth et al.Compared with iJO1366, we observe that ETFL predicts fewer false negatives , However, ETFL also presents more false positives . This indicates that ETFL is less constrained than iJO1366—the cell has more genetic alternatives for growth. This is an artificial effect that stems from the missing enzyme data.As we add more enzyme information to the model, the false positive rate decreases. This is verified by Fig. A detailed interpretation of the differences between gene knockout in ETFL and FBA is discussed in the Methods section. The 16, gpSampler17, or optGpSampler18. Once it has converged, a sampling should provide a better representation of the center of the solution space than the mean of the variability analysis.Sampling the feasible solution space of FBA is a common way to study solution robustness and variability. Since there are often multiple FBA solutions at the optimal objective value, representative solutions are often sought, and sampling is one way to obtain them. However, because ETFL contains integer variables, it is not compatible with traditional sampling methods in its current formulation. It is possible, though, to make the model convex, and hence amenable to sampling, by fixing the integers to their values at a given growth rate and, if applicable, TFA directionality. This will block the flux directions (if TFA is performed) as well as the growth-dependent parameters. The resulting model is then solely linear, and sampling can be performed with traditional techniques, such as artificially centered hit and run (ACHR)20, as it is more robust to outliers than the arithmetic mean and more sensitive to unevenness than the median.For robustly reporting solution times of ETFL, we logged solving times each time a model was optimized during the redaction of this article. In that respect, some observations are the result of iterated optimizations, others from different optimization problems. In particular, variability and gene essentiality analyses require thousands of optimizations. We aggregated the solution times report the corresponding histograms, by model type, in Fig. 21, Gurobi22), we report a geometric mean solution time of 7.47 s for vETFL, with 95% of the problems solved in <100 s on the test hardware. This is three orders of magnitude better than the reported solution time for O’Brien et al.7  part of the solution space that might contain physiological phenotypes. In particular, catalytic constraints become equalities, and the flux carried by reactions is set to be proportional to the amount of available enzyme instead of being upper-bounded by it. This gives less flexibility to the cell and prevents the representation of transient phenotypes. As an example, a cell that has been growing on a carbon source (e.g. glucose) will have a proteome suited to utilize this carbon source. However, once exhausted, it will need to reallocate its proteome to a new carbon source (e.g. lactose). In this transient state, some enzymes related to the first carbon source metabolism (e.g. glucose transporters) will carry no flux. In this case, cobraME would predict no flux, and also no enzyme concentration. In constrast, ETFL would allow for non-utilized enzymes and avoids such trade-offs, which is also crucial for accurately integrating proteomics data.Such performance enhancements allow studies that would have been excessively time consuming using prior ME-model formulations. We show in Table Finally, ETFL relies on solver-specific MILP algorithms and heuristics, which also means that great variability in performances can be observed depending on the solver parameters. We provide tuned presets for different tasks  with the package, and recommend that users run their own solver tuning if long run times are observed. We witnessed an up to The ETFL formulation is amenable to further kinds of analyses. Leveraging both the explicit expression constraints and the MILP nature of the problem, we present several possibilities for future studies using ETFL:11. Although such dependency is not taken into account in the presented results, it is possible to account for this by (i) discretizing It has been reported that several other parameters, such as the ribosome transcription rate constant 23, iMAT24, GIMME25, or MINEA26, can also be adequately reformulated for ETFL. Metabolomics can still be integrated using TFA3.Explicit mRNA and enzyme concentrations allow the direct integration of absolute or relative proteomics and transcriptomics by changing the bounds of the corresponding variables in the EP. An additional gauge constraint will be needed for the relative data. Previous transcriptomic integration methods, such as REMI27. The underlying hypothesis is that the enzyme distribution and assignments remain the same, except for the knocked-out gene. With ETFL, it is possible to directly compute a Minimization of Protein Adjustment (MOPA) by reformulating the objective function as such:In the original paper, the hypothesis behind the Minimization of Metabolic Adjustment (MOMA) method is that the metabolic fluxes of an organism subject to a gene knockout show a minimal change compared with the metabolic fluxes of the wild-type organism17 was developed to address the high fluxes of some of the solutions given by FBA. Although this concern is addressed in ETFL by the combined actions of the EP and thermodynamics, pFBA can be adapted to ETFL to study an organism under parsimonious constraints. For example, it is possible to reformulate it into a parsimonious expression problem to find the minimal expression level required to meet a growth target using objective functions similar to (MOPA), (MORA), and (MOXA). It is also possible to turn the problem around to consider the allowed enzyme amounts under minimal flux constraint obtained by pFBA to assess the metabolic flexibility of an organism.Parsimonious FBA (pFBA)28 is a method that uses FBA to predict the dynamics of a biological system represented with a stoichiometric model. In its original static optimization approach (SOA) formulation, a FBA problem is solved at each time step. The value of boundary fluxes of the FBA problem are updated at each iteration with values produced with a kinetic law, such as Michaelis–Menten glucose uptake and oxygen diffusion. Because ETFL allows direct access to enzyme concentrations, it is possible to use the latter to reformulate dFBA in its SOA. The original SOA approach uses ad hoc constraints on the absolute flux change at each time step. However, in ETFL, it is possible to bound flux changes indirectly by bounding enzyme and mRNA concentration changes in the EP. Effectively, this approach allows the movement from ad hoc constraints to physiological constraints.Dynamic FBA (dFBA)Often, kinetic frameworks require a reference flux distribution as an input. ETFL can provide such a distribution, with an increased accuracy as compared with FBA.Building an ETFL model from a genome-scale model follows a detailed procedure, for which a SOP is provided in the Supplementary Note 7 used In ETFL, and ME-models in general, catalytic constraints are what links the metabolism to the expression problem. Because of this, the accuracy of the ETFL reconstruction is also heavily dependent on the quality of the catalytic rate constants 29, and literature. As explained in the previous paragraph, special attention should be given to the ATP synthase, the RNA polymerase, the ribosome, and the enzymes of the main carbon pathway. Macromolecule degradation rates are less critical and can be averaged. Growth-dependent protein, RNA, and DNA ratios drastically improve the quality of the model, as they allow to account for the expression activity that is related to non-metabolic processes.Another key component for catalytically constraining the model is to have quality enzyme composition information. Indeed, marking an enzyme as a monomer instead of a dimer halves its synthesis cost. A good source for this information is MetaCycE. coli model should be done with care. Similarly to gap filling and the use of template reactions, conserving parameters across close species is helpful; however, conserving parameters across a large phylogenetic distance is erroneous. An example is the ribosome translation rate, which can vary by one order of magnitude between S. cerevisiae and E. coli.In the construction of a model for another organism, approximating parameters based on values from an Finally, great care should be taken with respect to the units. Different conventions are used across sources. Parameters for which this has been observed include catalytic rate constants, molecular weights, and concentrations.1 can be directly fed to the framework to generate models for each of the 773 bacteria they identified. Integration with platforms like KBase30 can also be envisioned to automatically draft ETFL reconstructions parametrized by curated organism-specific data. In a more general way, ETFL can assess the allowed expression profiles of any biological system amenable to genome-scale modeling, such as in the metabolic engineering of biocatalysts, microbial communities, drug design, or personalized medicine.ETFL is a framework which implements expression and thermodynamic formalism using mainstream double-precision MILP solvers. This could not be previously accomplished using state-of-the-art ME-models, which use specialized quad-precision solvers and do not support integer variables. The formalism itself is based on the explicit and direct relationship with the underlying biochemistry and provides a way to incorporate growth-dependent variables using MILP linearization techniques. These new growth-dependent variables provide a finer modeling of expression because they consider phenotypic differences in different growth regimens, which are key for accurate modeling. ETFL can also compute explicit mRNA and enzyme concentrations as well as perform direct -omics data integration. In this, ETFL complements and extends FBA capabilities by using explicit relationships in lieu of the typical assumptions on the relationships between the transcriptome, proteome, and fluxome. This explicit accounting of expression mechanisms provides a finer level of control and a more relevant prediction of gene-editing outcomes. ETFL is robust to missing data, as missing enzymes and their composition can be approximated using average enzyme characteristics. Because of this and its operational similarity with classic FBA-related analyses, ETFL can be efficiently integrated in standard model-based pipelines. For this intent, we provide in the Supplementary Note The mass balances of the macromolecules in ME-models are written with respect to their concentration variables. If we assume the cell is growing at a specific growth rate The mass balance of a macromolecule We next combine Eqs.  and 6)  6) and d31. It is a common assumption that the concentrations inside the cell remain time invariant (quasi-steady-state assumption), effectively yielding the constraint:By definition, n reactants to m products is trivial.It is also understood from the formulation of the FBA that adding a new reaction to the system, such as:32. When used, we specify their identification number as well as the original source from which the value was reported. Finally, we will represent products between a parameter value and a variable by the symbol “Several parameter values are taken from the BioNumbers databaseHereafter, we propose a detailed top–down approach to formulate the constraints being built for ETFL, starting from the metabolite network and moving down to RNA synthesis. The general organization for each macromolecule is to write down its mass balance, apply assumptions, and then detail its synthesis and consumption mechanisms.From FBA, the mass–balance relationship for metabolites can be written as:For the rest of the formulation, it is necessary to split the net flux The distinction between the bounds of the forward and backward net fluxes is important, as some enzymes have different catalytic activities, depending on the direction of the flux.Each enzyme We model the degradation reaction of the enzyme in the following manner:Like any other enzyme, ribosomes verify the mass balance:The ribosome differs from other enzymes in that it takes ribosomal peptides As explained earlier, the stoichiometric coefficients When ribosomes are degraded, their constituting amino acids and ribonucleotides are recovered:The degradation rate is constrained in a manner similar to the constraint Finally, we can then write the total ribosome capacity constraint:If we know the ratio RNAP is an enzyme, and hence it also satisfies mass balance:In addition, the total capacity of RNAP follows a capacity constraint similar to that of ribosomes:The peptide concentrations obey the mass–balance equation:We assume in the current model that the protein assembly rates are much faster than dilution and degradation, and thus simplify this mass balance to:The synthesis of peptides consumes charged tRNAs, which are subsequently uncharged during the current peptide synthesis by a ribosome. The process consumes two GTPs and releases two GDPs and two Pi per amino acid:As explained in the section Preliminaries, conventions, and notations, this reaction adds a supplementary term in the mass balances of the metabolites , The peptides are the product of a translation reaction that is catalyzed by a ribosome. As we did with the catalytic constraints for general biochemistry reactions, we can apply the ribosome maximum catalytic rate as an upper bound to its translation rate During the translation, an mRNA is read to produce a peptide. mRNAs are subject the same mass–balance constraints:E. coli, BNID 10006033), and Again, the stoichiometric coefficients will appear in the mass balances of each of the metabolites and macromolecules involved. The transcription process is catalyzed by RNA polymerase (RNAP). For each transcription of mRNA, we can put an upper bound on the transcription rate We must also take into account the relationship between ribosome assignment and mRNA concentration. On each strand of E. coli, 34, 10012135), which amounts to ~60 base pairs . From there we can get the additional constraint:For We consider the following degradation reaction for mRNAs:And, again, we know the degradation rates:37. Thus, their mass balance in the model reads:rRNAs are used in the ribosome assembly reaction. According to the definition of The degradation reaction is the same as for mRNA, and is part of the total degradation of the ribosome.37, we neglect their degradation. Let Since tRNAs are relatively stable moleculestRNAs are produced with a charging reaction and consumed by peptide synthesis. We use the following charging reaction:By default, this reaction is assumed to happen spontaneously, but catalytic constraints can be applied if the adequate catalytic rate constants and enzyme compositions are known. Once again, the stoichiometric coefficients of each reactant will appear in the stoichiometric matrix in the column corresponding to this reaction.The main issue with the EP formulation presented previously lies in the continuous bilinear terms that describe the dilution of the macromolecules, 7. As such, its recent adaptation in Lloyd et al.8 uses the two-level iterative algorithm SolveME9 that requires a dedicated nonlinear solver. In this fashion, iterative approaches which try to sequentially improve a value of the growth are a way to deal with the bilinearity. We present instead a MILP approximation of the problem that makes it compatible and solvable with mainstream MILP solvers in a single optimization formulation. We achieve this through the discretization and linearization of the bilinear products. This operation can be understood as locally approximating the bilinear problem by several linear subproblems and choosing the best approximation.In this state, the dilution term is bilinear, and the formulation requires a bilinear solver or potentially a mixed-integer bilinear solver if thermodynamics are to be added. The original ME-model formulation has similar terms as we are presenting here38. However, in the case of non-monotonically increasing or decreasing parameters with respect to the growth rate, such guarantees are harder to prove in general QCP cases, and thus MILP provides a strong framework, with global optimality guarantees and enumeration of alternative solutions. Second, by displacing the solving complexity to the solver, it also allows us to rely on the latest advances in MILP solving, which is a very dynamic field, with new solver releases every 6–12 months.Using a MILP approximation rather than an iterative scheme has two clear advantages. First, it allows to simulate growth-dependent parameters (such as RNA/protein mass ratios) with guarantees on convergence and global optimality directly inherited from the MILP nature of the problem. In the case of parameters that are monotonically increasing or decreasing with respect to growth rate, guarantees exist in quadratically-constrained programming (QCP), such as showed in SteadyCom39, a particular case of the Glover linearization scheme40 that was previously used in metabolic engineering by Hatzimanikatis et al.42.In ETFL, we approximate the growth rate Let With this notation, The model needs two more constraint to ensure that As an example, let us consider modeling an organism whose growth rate does not exceed The values of Solve the FBA for Select the corresponding, closest value of Apply it to compute dilution valuesSolve the EP with fixed dilutionApply the catalytic constraints to the FBARecalculate the FBA under catalytic constraintsIf MILP solvers use a variety of algorithms and heuristics to solve MILP problems. In this case, the difficulty lies in the fact that the EP and the FBA are almost independent and linked through a limited number of equations and variables. Even though the automated solving methods of the solver might seem obscure to a human, we thought useful to provide a human-understandable heuristic for solving a formulation such as ETFL. It might prove useful in the case where one needs to find an initial non-optimal solution, which sometimes greatly improve solver performances. Thus, conceptually, a heuristic for solving an ETFL problem would be:40 to replace the product In the previous derivation, we replaced the growth rate variable by a discrete number of acceptable values. We can approximate the continuous product Because of the binary expansion, the complexity of the model grows only as 39 to transform this product into an equivalent system of one new variable and three new constraints:After discretization of the growth rate, the dilution term for the macromolecule With this method, we can directly reformulate generalized mass balances as described in Eq.  for mRNAAnd we get the additional linearization constraints:11. We thus aim to approximate the nonlinear function Since the growth has been discretized, it is now possible to also directly discretize other growth-dependent parameters of the problem, regardless of whether they are in a linear or nonlinear relationship with growth. This is a direct consequence of the formulation of ETFL, which allows some flexibility in the modeling assumptions of the user. As an example, we described the relationship between growth and protein and mRNA mass ratios, In addition, it is necessary to have the integer index of The first term represents the growth integer index  are interpolated using a piecewise linear function (dashed line), which is then discretized (full line). Using the integer constraints described above, the model can be forced to display a protein content that corresponds to its growth. We apply the same techniques to mRNA and DNA content.Figure 11. We model the DNA reaction synthesis as follows:To further increase the scope of macromolecules covered by the model, it is also possible to add growth-dependent DNA content. DNA mass ratios at specific growth rates are reported in Neidhardt et al.We consider that the DNA does not degrade, meaning the only source of DNA consumption is dilution caused by the growth of the cell and A critical issue in the formulation of this problem is that the variables are different orders of magnitude. Fluxes are typically between To circumvent these limitations, we scale the EP, which will reduce the numerical difficulty of the problem, using nondimensionalization. We create nondimensionalized variables by dividing the variables of the initial problem by an estimated upper bound. For example, by definition, macromolecule concentrations cannot exceed In this scheme, mentclass2pt{minimFor the sake of clarity, all problem formulations will be kept in their dimensionalized form in the subsequent equations although the implementation is in fact nondimensionalized. The nondimensionalized problem is described further in Supplementary Note ETFL is amenable to modeling more intricate expression processes. A short selection of these is detailed below.Enzyme-mediated complex assembly: By default, all the peptides are assumed to assemble spontaneously, without an enzyme. However, in the case of an enzyme-mediated assembly, it is possible to limit the assembly rate by a catalytic constraint if needed, in a fashion similar to Eq. . If we dEnzyme activation and posttranslational modifications: Some enzymes require to be modified in order to be active, and sometimes by metabolites of the cell. This can be captured by adding a new species representing the active enzyme, and an activation reaction transforming the inactive enzyme to the active form. If the metabolite The mass balances of This activation reaction can itself be catalytically limited if needed (see previous paragraph), and require the participation of metabolites. Thus, ETFL allows to capture protein-metabolite interactions.Enzyme association: It is also possible to model the partition between free enzymes and associated enzymes. In that case, we simply need to operate the following adaptations: (i) replace the 44.Dilution and degradation assumptions: In the current formulation, some species have their dilution or degradation neglected because of high reactivity or slow degradation rate constants. This can be relaxed by simply editing the mass balance reaction according to the assumption to be relaxed. In particular, enzyme-mediated degradation can be modeled by adding suitable catalytic constraints on the degradation reactions. In addition, the dilution term for metabolites can be taken into account if needed, in a manner similar to what Benyamini et al. describe in their method for FBA accounting for dilutionMILP-based gene knockout strategies for strain design: The ETFL formulation of gene knockout using an upper bound on the translation rate allows to directly formulate MILP-based gene knockout strategies for strain design. Indeed, for each 3 imposes constraints on a FBA problem to couple reaction directionality to the standard free energy of reactions and metabolite concentrations. We also introduce constraints that couple the sign of the Gibbs energy of a reaction to its directionality through the use of integer variables and a mixed-integer linear coupling formulation. This framework reduces the feasible flux space and improves the predictive power of FBA by removing thermodynamically invalid flux profiles.Thermodynamics flux analysis (TFA)Considering 45 to directly calculate We use the group contribution methodSome metabolites are not fully characterized, e.g. metabolites with -R groups such as fatty acids, or metabolites attached to a Coenzyme A or acyl-carrier protein. In these cases, the group contribution method allows to directly calculate the net change in the standard Gibbs energy. Since these -R groups are often conserved in the reaction, their contribution terms cancel out when calculating the Gibbs energy of the reaction.2:The concentration variables are bounded by experimental measurements or physiological assumptions, and the standard Gibbs energies are bounded by the measurement or estimation error. Since the net flux of each reaction has already been split between forward flux (mentclasspt{minimaEquation  defines Equation  and 57)\documentx in Eqs  59) if  if \docux in Eqs  is added46. We converted the reported half lives into rate constants using the classical relationship 47).mRNA degradation rates constants 48 for homomeric enzymes. Complex formation reactions for non-homomer enzymes were taken from the supplementary information of O’Brien et al.7 and Lloyd et al.8. EC numbers were obtained from BiGG49 and the iJO1366 publication12. Their corresponding 50.Catalytic rate constants 48. Other peptide compositions of enzymes were taken from the Supplementary Information of O’Brien et al.7 and Lloyd et al.8. Additional information was obtained from the Metacyc/Biocyc database51 using specialized SmartTables queries52.Homomer compositions were obtained from Davidi et al.Selenocysteine as a metabolite.Cysteine to selenocysteine conversion as a pseudo reaction.Replacements for the tRNA metabolites and their charging reaction, as dilution has to be considered.The initial model was subjected to minor changes to accommodate for ETFL modeling. In particular, we added:We also modified the biomass reaction by removing its nucleotide and amino acid components, since they are already taken into account by the expression problem as explained in the section Biomass reaction synthesis and mass balance.Given a reaction in the model, if no enzyme is supplied but the reaction possesses a gene reaction rule, it is possible to infer an enzyme from it. The rule expression is expanded, and each term separated by an OR boolean operator is interpreted as an isozyme, while terms separated by an AND boolean operator are interpreted as unit peptide stoichiometric requirements. The enzyme is then assigned an average catalytic rate constant and degradation rate constant.The method for testing gene essentiality in FBA is to evaluate for each reaction the gene-protein-reaction association rules (GPRs) containing the gene of interest. The GPR is a boolean expression where the symbols represent whether a gene is expressed. OR operators represent isozymes, and AND operators the assembly of several peptides in a complex. To knock a gene out, its symbol in each GPR is simply assigned the value False. The GPR of all reactions is subsequently evaluated, and the reactions whose GPR evaluates to False are set to have a net flux of 0. Knocking a gene out in ETFL works differently: we replaces GPRs with mass balances, and the direct interaction between gene transcription, peptide translation, enzyme assembly, and metabolism. In this context, knocking-out a gene is done by forcing its transcription rate to 0. Indeed, gene-reactions relationships are conveyed directly through the direct contribution of the relevant peptides either as components of the enzyme complex (AND operator in GPRs) or as isozymes (OR operator). An advantage of this formulation is that it can be used in strain design strategies to optimize directly for knockouts in a single optimization problem.If a knocked-out gene does not have enzyme associated with it ; 2 Further information on research design is available in the Supplementary InformationDescription of Additional Supplementary FilesSupplementary Data 1Reporting Summary