A description of some computational approaches to gain new biological knowledge or to formulate hypotheses is finally provided.Cell metabolism is the biochemical machinery that provides energy and building blocks to sustain life. Understanding its fine regulation is of pivotal relevance in several fields, from metabolic engineering applications to the treatment of metabolic disorders and cancer. Sophisticated computational approaches are needed to unravel the complexity of metabolism. To this aim, a plethora of methods have been developed, yet it is generally hard to identify which computational strategy is most suited for the investigation of a specific aspect of metabolism. This review provides an up-to-date description of the computational methods available for the analysis of metabolic pathways, discussing their main advantages and drawbacks.  In particular, attention is devoted to the identification of the appropriate scale and level of accuracy in the reconstruction of metabolic networks, and to the inference of model structure and parameters, especially when dealing with a shortage of experimental measurements. The choice of the proper computational methods to derive  Catabolism and anabolism are highly interconnected and partially overlapping, forming a complex network of biochemical transformations. Moreover, they are coordinated by an elaborate regulatory structure to make the cell promptly responsive to a variety of chemical, environmental, genetic and developmental perturbations.Metabolism can be viewed as a dynamic chemical engine that converts available raw materials into energy, as well as into the building blocks needed to produce biological structures and sustain multiple cellular functions. The hydrolysis of large substrates, such as complex sugars or lipids, into smaller molecules produces energy —collectively referred to as metabolites—has been estimated to vary from 274 in the simplest bacterium M. genitalium [E. coli [metabolome [endo-metabolome, referring to metabolites present inside the cell, and the exo-metabolome, referring to metabolites excreted by the cell or present in the extracellular fluids [The number of low molecular weight metabolic intermediates and end products  or coarse-grained  models. Typically, the mechanism-based approach deals with toy or core models, while the interaction-based and constraint-based approaches are more suited for the analysis of genome-wide or core models. A schematic overview of the three main modeling approaches is given in vs. qualitative, static vs. dynamic, parameterized vs. non parameterized, single volume vs. compartmental, well-stirred vs. heterogeneous (diffusion), etc.Identify the proper mathematical formalism. A one-to-one correspondence between each modeling approach and a specific modeling purpose would facilitate the choice of the most suitable strategy to be employed. Unfortunately, a sharp-cutting separation is not always possible. In general, mechanism-based  models—which are usually defined as systems of differential equations—are considered the most likely candidates to achieve a detailed comprehension of cellular processes. Nonetheless, the usual lack of quantitative parameters represents a limit to a wide applicability of this approach for large metabolic networks. Various attempts have been proposed for the automatic estimation of missing parameters or the characterization of the parameters space [Choose the most appropriate rs space ,25,26.hubs , as well as the metabolites and reactions connecting them, which can be of particular interest within the scope of, e.g., drug target discovery [On the other side of the spectrum of modeling approaches, interaction-based models are characterized by a simplified representation of the biological process and allow to achieve qualitative knowledge only. These models can be analyzed by using, for instance, graph theory or topological analysis to investigate the “design principles” of metabolic networks, that can be considered transversal to different organisms . Moreoveiscovery .Considering the limitations of these modeling approaches, the common practice for the computational investigation of metabolism usually relies on constraint-based models. These models are based on the definition and manipulation of stoichiometric matrices, whose native application pertains to the field of metabolic engineering. In this case, the methodologies that were initially developed for the optimization of microbial strains or for the maximization of some product yields in biotechnological applications, are now widely used with different goals in the study of metabolic networks.The scale of the model is closely related to the choice of the proper modeling and computational approach, which is constrained by two key features: the nature of the biological system under examination, and the experimental data that are available or that might be measured for that system. More precisely, the modeler need to consider the following issues:in silico analysis, is presented in in silico data from metabolic networks and dynamic models, such as topological analysis and Flux Balance Analysis for large-scale metabolic networks, and simulation methods for small-scale metabolic networks. In The aim of this review is to provide a global picture of the different modeling approaches and their related computational strategies for the analysis of metabolism. We will give indications on the limits and strengths of the various methods, and present several examples on the way they have been effectively applied with different goals. However, considering the very broad scope of this work, some research on metabolic modeling had to be necessarily omitted. The general workflow of the topics discussed in this review, which range from the reconstruction of metabolic networks to their Metabolism can be described by means of mathematical models defined at different levels of detail. The scale of the model concerns both the size of the biological system that one has to analyze, defined in terms of the number of metabolites and reactions, and the chosen level of granularity. Except for very small metabolic networks, model development entails an appropriate choice of which metabolites and reactions need to be part of the model, in a tug-of-war between the desire to include as many molecular details as possible, and the opposite need to simplify the structure of the model.nodes (representing metabolites), and their respective mutual relationships, called edges (representing metabolic reactions). These formal representations are typically created in a bottom-up fashion based on genomic and bibliomic data, which can possibly be integrated with data obtained from laboratory experiments  computational analyses. In what follows, we mainly provide examples of network reconstructions that are intended for quantitative analysis, e.g., constraint-based modeling or algorithms for the simulation of the dynamics (as described in Genome-wide models. One of the main purposes in the reconstruction of genome-wide (GW) models is to summarize all the current knowledge concerning metabolic processes at the level of single gene annotation, trying to take into account every single reaction that is known to occur within an organism. These reconstructions can be exploited in a dual way: on the one hand, they represent the scaffolds for different computational analyses, such as those described in the next sections; on the other hand, they work as repositories of all the collected knowledge about metabolic pathways [pathways .etc.), by giving a general view upon the parts that compose a specific class of compounds in the cell , thus allowing the identification of all the components of the functional networks. GW models therefore represent the result of the integration of different kinds of information about metabolism. The first GW metabolic model was developed in 1999 for the bacterium H. influenzae [The curation of GW models became possible after the development of high-throughput technologies , able tofluenzae . Thanks fluenzae  to lowerfluenzae ,38 and hfluenzae ,39, inclfluenzae . These mfluenzae , which tBecause of their extension and complexity, GW metabolic network reconstructions are often the results of a community effort in which researchers are coordinated with a jamboree-approach, which may require many years of work . This meAn open challenge in Systems Biology is the reconstruction of a reliable model of human metabolism, which would allow the analysis and in-depth studies of several diseases linked to metabolic disorders. The relevance and complexity of this challenge is assessed by the presence of many databases, whose goal is to build the ultimate human metabolic pathway, and by the efforts spent to reconcile them ,51.Recon 2 [In this context, the most updated and complete human GW model reconstruction is, at present, Recon 2 . Given tRecon 1 [de novo, resulting in a total number of 7440 reactions, which is almost doubled with respect to its previous version.Recon 2 was curated starting from a previous version of the global human metabolic network, Recon 1 , a reconIt is worth mentioning here that the specific metabolic pathways that actually operate within a cell are regulated by the interaction between the genome and the environment. Therefore, from a single genomic-data driven reconstruction, many condition-specific models can arise. For instance, generic GW reconstructions of human metabolism (such as Recon 2) have been used for the generation of cell type-specific models ,52, accoad hoc versions of FBA have been developed  and cellular growth.found in ,61, wherCore models (CMs) stay in between toy and GW models. They can cover in molecular details only one or a few simple pathways  ,63, or eA representative of the former type of CMs is the metabolic reconstruction presented in , which tAn example of the latter type of CMs is instead provided in , where tCM and toy models make the analysis and interpretation of simulation results easier, but over-simplification may lead to erroneously neglect fundamental multi-factorial relationships between the state of the system and its response to different perturbations  . On the in vivo values of kinetic rates. This experimental limitation has led to the development of computational methodologies for parameter estimation (PE) [When modeling metabolic networks, or any generic biochemical system, one of the main computational problem that has to be faced consists in the definition of all model parameters, such as kinetic constants or initial molecular amounts. A proper model parameterization of the model is an indispensable requisite to perform dynamical simulations, in particular for the analysis of CMs or toy models. In a few cases, the unknown parameters can be obtained by means of laboratory experiments but, in general, they are hard or even impossible to measure, especially if one aims to determine the ion (PE) , whose gi.e., the network of reactions). Second, target time series are necessary to carry out the automatic inference of the unknown parameters. To this aim, the more precise and densely sampled the measurements are, the more reliable will be the inference of the model parameterization. In particular, time series of molecular amounts of some pivotal species, possibly obtained in different experimental conditions, can largely facilitate the assessment of the quality of the inferred set of kinetic parameters [In order to face the PE problem, two fundamental requirements have to be satisfied. First, we need a precise knowledge of the physicochemical interactions between all molecular species involved in the system  [Another example of application of a PE method was presented in  for a delgorithm , a metholgorithm  42 out og (SiAn) , and refi.e., the initial guess of parameters). To avoid this drawbacks, multi-start approaches should be rather exploited in order to collect a set of putative model parameterizations.This kind of methods, also called local search algorithms, are unlikely to yield a proper set of physiological parameters, since their results strongly depend on the starting point of the optimization process , which can be recombined during the optimization process. These methods can yield, in general, better results compared to local search algorithms; however, they require a fine tuning of specific algorithmic settings, which allow to control the exploration of the search space of all model parameterizations. Indeed, PSO and DE can outperform all other algorithms only when these settings are properly identified. In [In , the autfied. In  the authfied. In  algorithA simpler methodology for PE was proposed in  to the pConcerning the PE of large-scale models, in  the CoopIt is worth mentioning that one of the main limitations of all existing PE methodologies is that they are computationally very demanding, especially in the case of models consisting in high numbers of molecular species and chemical reactions. Notwithstanding this, the calculation of the goodness of different model parameterizations can be parallelized using any parallel architecture, such as clusters of machines or Graphics Processing Units (GPUs) .The methods for PE described in The RE problem may embed the PE problem when a dynamical model needs to be inferred, since in this case both the network topology and the corresponding set of parameters have to be optimized. Therefore, as in the case of PE, RE methods require the availability of target time series of the molecular species involved in the system, so that the quality of the reverse engineered models can be evaluated.In general, the computational complexity of RE methods grows at least quadratically with the size of the biological system under investigation ; therefoindistinguishability, a problem that occurs when different network topologies exhibit the same dynamical behavior [A major drawback of RE methods is that, although they are able to generate mathematical models whose simulated dynamics can perfectly fit the experimental target data, they suffer the issue of behavior . As a coi.e., two metabolites are connected by an edge in the graph describing the network if their correlation is high). Then, the network is simplified and pruned by identifying and removing indirect interactions. Arkin et al. proposed a modified approach of correlation-based RE relying on time series data, which were used to calculate time-lagged correlations between metabolites [If simple interaction networks are to be inferred, not requiring the estimation of kinetic parameters, then correlation-based RE methods , relyingabolites . The finabolites , coupledabolites , calculadirectionality, consists in analyzing the concentration profile of metabolites, as proposed in [A way to detect indirect interactions and to perform the RE of a metabolic pathway whose edges in the interaction network have a specific posed in . This meposed in  the feasEven though RE methodologies provide scientists with mathematically sound methods for the creation or the extension of biochemical models, they are currently limited by the following intrinsic issues: (i) the aforementioned indistinguishability problem; (ii) when using correlation-based methods, the optimization strategies can be ineffective , especially if one has to deal with noisy target data ; (iii) tEnsemble modeling (EM) is an approach suitable for the reconstruction of mechanistic models of metabolism , whose aEM is more appropriate than other approaches  to direAnother advantage of EM is that it bypasses the need for a detailed characterization of kinetic parameters, and allows to define a set of models that describe relevant behaviors upon enzyme perturbations. Indeed, EM can capture phenotypes that are dependent on changes in the enzyme levels. Once the en- semble of models is produced, additional experimental data obtained in perturbation experiments—such as flux shift in response to enzyme over-expression, or knockouts on the production rate of any molecular species—are acquired and used to iteratively reduce the set of candidate models, resulting in an increasingly predictive sub-set of models. A similar approach was presented in , where ME. coli, by successively identifying single, double, and triple enzyme perturbations that cause the maximum degree of flux separation between models in the ensemble. In particular, it was found that, in the case of the analyzed metabolic network, knockouts bring about a larger scatter in the flux predictions with respect to two-fold enzyme over-expression.A different application of EM can be found in , where tAs an extension of the original EM approach, in  a strateOne of the main limits of the EM approach concerns the availability of experimental measurements or the capability of calculating the steady state fluxes, used as reference for the computational analyses. Being the main input of the models in the ensemble, the lack of these values can therefore straiten the applicability of EM.structure, given in terms of nodes and edges, disregarding any other quantitative parameter about the stoichiometry of reactions, the molecular amounts or the kinetic rates. This “network analysis” is usually applied to interaction-based models, which can represent metabolic maps collected in the KEGG [degree distributions , hubs (highly connected nodes in the network), centrality measures (stating the relative importance of nodes and edges in the network), motifs (specific subgraphs occurring with a high frequency in the network), clusters (densely connected subgraphs of nodes), etc. [The topological analysis of a metabolic network relies on the information about its the KEGG  databasethe KEGG  to systes), etc. ,110,111.s), etc. .Topological analysis can be exploited for several purposes. For example, it can serve to easily find errors or omissions in a reconstructed metabolic network, such as the dead-end metabolites and network gaps that will be discussed in scale-free networks . A fundamental consequence of the scale-freeness property is that the network is robust against random perturbations , but very fragile if hubs are disconnected [small-world  character [In this context, an interesting work appeared in , where aonnected . Anotherharacter ,117, thaharacter . Other wharacter  and moduharacter ,120. In More recently, in  the issuinvariants allow to identify conservation relations between chemical species or flux distributions at steady state, while the liveness property ensures that all reactions of a metabolic pathway can happen at any time. Further information about PN properties and their application to the study of biological systems can be found in [A special kind of bipartite graph is the Petri net (PN), a mathematical formalism extensively exploited for the modeling, simulation and analysis of the structural and behavioral properties of complex systems. According to PNs naming conventions, the graph is characterized by two kinds of nodes: the places , characterized by a state , and the transitions . PNs are directed graphs, that is, edges connecting places to transitions and transitions to places specifically describe the roles of the chemical species as either reactants or products. A PN representation of a metabolic networks can be straightforwardly converted to a reaction-based model to investigate its kinetic properties , or be efound in .Constraint-based modeling relies on the idea that all the expressed phenotypes of a given biological system must satisfy a number of constraints, which can be of physicochemical, spatial or environmental type. Accordingly, by restricting the space of all possible system’s states, it is possible to determine the functional states that a metabolic network can or cannot achieve. The fundamental assumption of constraint-based modeling is that, regardless of the environmental condition, the organism will reach a quasi-steady state that satisfies the given constraints. A plethora of methods have been developed within this framework and a comprehensive list and description can be found in ,127.The starting point of constraint-based modeling is the stoichiometric matrix see , which cobjective function (OF) that, in conventional FBA, takes the linear form z = iciv, where ic is a weight indicating how much the flux iv of reaction i contributes to the OF.FBA , which iS. cerevisiae minimizes the number of active reactions necessary to produce the required biomass.To summarize, the inputs of FBA are the stoichiometric matrix, the specification of the flux boundaries for each reaction and the OF, whereas the output is the quantification of the flux through each reaction. As long as the OF is a linear equation and all constraints are linear (dis)equations, FBA constitutes a linear programming problem; nevertheless, non-linear FBA variants were also formulated (a complete review of the optimization techniques for FBA can be found in ). An intFBA was originally developed within the scope of metabolic engineering where the objective represents some bioengineering design goal, such as maximization of the production of a biochemical compound of industrial use . HoweveGiven a certain OF, FBA tries to find a unique optimal solution; however, it is possible that, even for small networks, there exist multiple flux distributions resulting in the same optimal objective value. As different optimal flux distributions could lead to distinct biological interpretations, some strategies were proposed in order to deal with this problem. A first class of approaches tries to identify all flux distributions characterized by the same “best” objective value. To this end, Elementary Flux Modes  and ExtrAs a general remark, solutions obtained with FBA are only as good as the constraints used to identify them. Moreover, an appropriate OF is of pivotal importance in FBA studies to determine the physiological state of a cell. The identification of a plausible OF is a delicate task: knowledge about the true OF that drives the evolution is indeed limited, especially when dealing with multicellular organisms. Besides, even when the maximization of biomass yield is assumed, the stoichiometric coefficients of the biomass precursors—as well as the cofactors requirement to synthesize the monomers from the precursors—must be appropriately defined according to the requirements necessary for proliferation, using literature values of experimentally determined biomass composition . FurtherA major limit of FBA is that it is not suited for the investigation of the system dynamics, as it disregards information on metabolic concentrations and kinetic parameters. An attempt to solve the problem is provided by Dynamic Flux Balance Analysis (dFBA) methods, which are intended to partially capture metabolic dynamics by incorporating, according to experimental evidences, the modifications of flux boundaries over time due to the change in the uptake of molecules induced by nutrients availability, or genetic expression effects on transporters. Two methods within this category are: (i) Dynamic Optimization Approach , which eDespite all the aforementioned challenges, FBA has significant potential to support the interpretation of increasingly available, yet non-comprehensive, metabolic data, and there have been many innovative developments to improve its predictive capabilities . In partAn emerging research front that exploits FBA and that deserves a final remark has the purpose of identifying the topological features of a metabolic network. For instance, groups of reactions in a network that always appear together in functional states were identified by means of flux coupling finder  and Extretc.), and (iii) the initial state of the system . As a consequence, the simulation of the dynamics can be generally done for  mechanism-based model.The simulation of a biological phenomenon through a mathematical model allows to obtain the (quantitative) temporal evolution for each molecular species occurring in the model itself see . Simulatn-fold over-expression of one or more chemical species. In general, it is possible to automatically perturb each model parameter to systematically analyze the corresponding system response, in order to gain insights into the underlying biological process in a very reduced time with respect to laboratory experiments.A main advantage of simulation algorithms regards the opportunity of easily testing the behavior of the system under different perturbations. For instance, by increasing (decreasing) the value of a kinetic constant, the system can be simulated in the condition corresponding to an enhanced (reduced) activity of that chemical reaction. Moreover, the initial state of the system can be varied as well, hence mimicking the biological conditions which range from the deletion to an arbitrary Simulations can be realized by exploiting different computational tools, which can be chosen according to the scale of the modeled system, to the nature of its components and to any possible role played by biological noise. In what follows, we provide a brief description of the most used simulation methods for metabolic processes modeled as systems of differential equations.Ordinary differential equations. When the molecular species involved in a system are present in large amounts, as it is usually assumed in metabolic networks, the effect of noise can be neglected. In these case, it is therefore convenient to apply a deterministic modeling approach, where each molecular species is described by means of a continuous variable (representing its cellular concentration) and its variation in time is formally described through ODEs. ODEs represent the standard mathematical framework for the simulation of the dynamics of biological systems: given the initial state of the system , together with the kinetic parameters, the dynamics of the system is usually obtained by numerically solving the set of differential equations. Being one of the most applied approaches in almost any area of the science, there exist many efficient numerical integrators to rely on [ rely on ,151.bc1 complex , leading to the production of reactive oxygen species (ROS) involved in the pathophysiology of several diseases. This small-scale mathematical model was able to reproduce the experimental data of the activity of the mitochondrial electron transfer chain, and allowed to investigate the response of complex III in the presence of the inhibitor antimycin A, as well as the effect of different values of the membrane potential on ROS generation. The behavior of the electron transfer chain was also analyzed with a novel composable kinetic rate equation  in [Mechanistic models based on ODEs can give a detailed description of the metabolic pathway under investigation, or even of the functioning of some specific molecular components, thus facilitating the investigation of the relationships between the modeled system and various related diseases. For instance, in  the func law) in , solving law) in ,156, theOn a larger scale, a comprehensive kinetic model of human hepatic glucose metabolism was presented in . This ODIn , the autTrypanosoma brucei was presented in [A model of glycolysis in the parasitic protozoan ented in . The autReaction-Diffusion models. The classic modeling approaches for the description of a biochemical system do not usually take into account the diffusion of molecules. However, if the effect of chemical species localization plays a relevant role on the system dynamics , then te, e.g., ,162,163.Through the development of a RD model, in  the authIn , a RD moDifferent simulation and computational methods can be integrated each other, as proposed in ,167, wheThe simulation methods described above share a common limitation related to the high level of details of the information required to properly set up the model, posing serious constraints on the size (in terms of metabolites and reactions) of the biological system that can be investigated. However, many efforts were focused on the acceleration of these algorithms, for instance by using special hardware like field-programmable gate arrays , computead hoc model-driven experiments, along with the possible definition of new laboratory protocols, in order to validate the computational hypotheses.In this section we present different computational strategies to perform thorough analyses of metabolic models. The outcome of these methodologies can be exploited to gain new insights about the functioning of a metabolic pathway, and to formulate novel hypotheses about its behavior in different conditions. This usually leads to the design of in silico metabolic model must undergo a validation process  qualitatively, e.g., to assess their capability to generate all the precursor metabolites, with particular regard to biomass components, and/or all the metabolites that the organism is known to produce or degrade; (ii) quantitatively, by comparing the model behavior with various experimental observation such as secretion products and gene essentiality. For example, in  the resue supply ,174, is gene deletion analysis deserves particular attention as it might be exploited either to validate a model, or to infer novel experimental hypotheses after model validation. This method consists in simulating the inhibition of a metabolic gene by excluding from the model the reactions associated to that gene. For example, in [E. coli. The model was able to qualitatively predict the growth potential of mutant strains in 86% of the examined cases. Along similar lines, the effect on cell viability of 87.8% of 599 single gene deletion experiments in a genome scale model of S. cerevisiae resulted in agreement with experimental observations when growth on synthetic complete medium was simulated [The strategy of mple, in , FBA wasimulated .The behavior of mathematical models of biological systems, especially in the case of complex metabolic pathways, highly depends on the values of the parameters  that characterize them. Model outcomes may therefore present a substantial variability due to (i) a large uncertainty, associated to some input parameters, that can propagate throughout the model; (ii) a possible intrinsic sensitivity of the model, such that small changes in the input result in large changes in the output.maxV of 10-formyltetrahydrofolate synthase varies between 100 and 486,000 µM/h.The source of uncertainty in the definition of the model inputs can derive from errors in the experimental measurements or, more often, from lacking or contradicting information present in literature. A clear example is given in , where tparameter sweep analysis, whose outcome is a qualitative visualization of the relationship between model input and output. In fact, the outcome of SA are sensitivity indexes (or coefficients) that represent the “importance” of each parameter and allow to rank them, with higher values indicating those parameters the model is most sensitive to. Remarkably, a ranking of the parameters can bring about useful insights at different stages of the modeling process and for several applications [In the situations described above, a computational analysis aimed at characterizing the relationships between model input and model output might be fundamental: sensitivity analysis (SA) represents a methodology well suited for this purpose . Given tications .E. coli from 122 to 22 reactions without significant discrepancies in the model predictions.At the stage of model calibration, SA can provide indications on which parameters require a precise knowledge in order to reduce the uncertainty in the model output. On the contrary, the least sensitive parameters could be disregarded from the model: indeed, SA can support model reduction, an important step to eliminate redundancy in the models and to uncover the most important control mechanisms of biological systems. For instance, in , the ideSA allows to investigate the robustness of the model and of its predictions, but perhaps its most important application is related to MCA , a theorSensitive parameters identified by means of SA may indeed represent either control points of the system behavior, or spots susceptible to failure. Identification of these parameters can therefore guide the design or facilitate the choice of which are the most suitable control experiments to carry out, thereby reducing laboratory costs and efforts. For example, in  a SA of local SA methods have been mainly applied [baseline  value) of the model parameters. Typically, they exploit partial derivative coefficients, which can be estimated either numerically, by finite difference approximation .ad hoc experiments, and the uncertainty is thus related to the experimental error, or when the model is already validated, and there is a good confidence about the parameter values. However, parameter uncertainty is often uncontrolled, especially when parameters values are taken from literature, or when they are inferred via PE methods , which found in , due to found in ). It is found in , or the found in ).Although the computation costs associated to the number of possible parameterizations can become very high, it is worth noting that the simulations needed for the computation of the sensitivity indexes are independent from one another and can thus be performed in a parallel fashion. Therefore, parallel architectures such as computer clusters or GPUs can be exploited to drastically reduce the overall running time.etc.—have been applied for the investigation of biological systems e, e.g., ,198.etc., do not hamper or impair the cell functioning. On the contrary, positive feedback (feedforward) deliberately destabilizes the system equilibrium by increasing the difference between the current and the previous system’s response, as in (second messenger-mediated) signal amplification in transduction pathways.In particular, the goal of control theory is to identify the nature of the control structures that regulate the behavior of cellular systems, such as feedback, feedforward and cascades  Figure . FeedbacE. coli. Interestingly, the analysis was based on ChIP-chip and gene expression data, interfaced with a curated GW metabolic model of E. coli. Thanks to this approach, the authors showed that the discovered feedback-feedforward architecture between ArcA and Fnr is able to control the metabolism of E. coli at the genome level.In this context, some recent works highlighted the feasibility of control theory for the analysis of different aspects of metabolism. These works usually rely on small-scale mechanistic models of metabolic pathways, described by means of ODEs. In , for insIt is now generally accepted that many problems in Systems Biology have analogies with problems in control theory. Though, biological systems are much different from engineering systems and require an accurate application of standard analysis methods. For instance, cellular processes are far from being linear systems, they are not continually at steady state conditions, and there is usually a lack of adequate or sufficient information about their kinetic parameterization. Therefore, classical methods of control theory still need to be challenged by novel questions in order to be fully applicable and predictive for biological systems ,198,203.The methods and procedures described above offer to computational and systems biologists a vast array of tools to investigate metabolism. Consistently, different questions may be asked. In this section we outline some relevant questions that different metabolic models may help to answer, in both basic and applied research.For a biologist, the first aim of model building is to rationalize and give structure to existing knowledge that may be scattered through a large number of papers and databases, which are difficult to integrate and often contain disparate and/or contradictory data sets. A typical example of this type of application is the construction of GW models, that allow to refine metabolic maps, to resolve contradictory data, to fill empty spaces in the reconstructed networks and iteratively build more and more precise maps.M. genitalium integrating 28 submodels of cellular processes, ranging from metabolism and transport to gene expression, DNA replication and cell cycle control. Each of these processes was simulated independently on a short time scale using an adequate computational strategy , and subsequently integrating the results with those obtained on longer time scales with a numerical integration algorithm.An outstanding case of a model increasing and validating biological knowledge can be found in . In thisM. genitalium. In particular, in the domain of metabolism, the model correctly predicted that flux through glycolysis is more than hundred times higher than the flux through pentose phosphates and lipid biosynthesis. Moreover, the authors identified the availability of dNTPs as a cell cycle regulator, highlighting a strong connection between cell cycle and metabolism. Finally, from the energetic point of view, the model suggested that the production of mRNA and proteins are the most consuming ATP and GTP processes, and it also evidenced a relevant discrepancy in energy production and consumption due to cellular maintenance costs and futile cycles.On the whole, this model was efficiently employed to improve the biological knowledge on S. cerevisiae and tumor cells share several features, since both yeast and cancer cells preferentially use fermentation rather than respiration to generate ATP. These metabolic behaviors—referred to as Crabtree and Warburg effect in yeast and cancer cell, respectively [et al. developed a toy model of ATP production that includes stoichiometric reactions for glycolysis, oxidative phosphorylation and biosynthetic reactions [The comprehension of the complex regulatory properties of metabolic networks would be of great benefit in many fields, including the ability to properly fine tune and/or control fermentation of industrially relevant yeasts, or the clarification of metabolic alterations in multifactorial diseases, such as cancer. From a metabolic point of view, the budding yeast ectively ,205—are eactions . The reset al. developed a different toy model that integrates several subsystems within a complete self-replicating system [Molenar g system . They shg system . These ret al. [i.e., strictly stoichiometric analysis), but a constraint on enzyme solvent capacity was required. The authors reported that their analysis match the metabolic behavior observed experimentally during oncogenic progression of ras-transformed cells. Extension of this work allowed to detect the Warburg effect using GW metabolic models of each of the NCI-60 cell lines and to identify its association with cell migration [Despite extensive experimental investigation, the molecular bases of the Warburg effect still remain elusive. Resendis-Antonio and coworkers designed a CM, encompassing glycolysis, TCA cycle, pentose phosphate, glutaminolysis and oxidative phosphorylation, and analyzed its behavior using FBA . Based oet al.  used theigration . These ret al. constructed a generic GW cancer model, which was validated by correctly identifying genes essential for cellular proliferation in cancer cell lines [By coupling transcriptomic data to Recon 1, Folger ll lines . The avall lines .et al. predicted putative oncometabolites that could result from the loss-of-function or gain-of-function mutations of metabolic enzymes in cancer cells by constraining Recon 2 GW models with large-scale mutational and gene expression data, indicating that constraint-based GW models have the potential for discovering novel oncometabolites in different cancers [Nam  cancers . Constra cancers , as well cancers .As in the case of GW models, also CMs can be effectively exploited in the identification of potential drugs to target cancer metabolism. In , the stuin silico strategies able to guide strain development [S. cerevisiae  cell factory over-producing succinic acid [E. coli strain that produces 1,4-butanediol (BDO) at high yields was designed. BDO is an important commodity chemical, that is massively produced to satisfy global markets but that is not occurring naturally in any known organism. Putative biochemical pathways able to convert an endogenous E. coli metabolite to BDO were identified, and the final pathway was chosen on thermodynamic grounds. A GW metabolic model guided E. coli engineering, so that it could produce enough reducing power to drive the BDO pathway starting from either pure sugars or biomass-derived mixed sugars. These works show that flux optimization (via FBA) of GW models plays an important role in strain design and development, allows to identify novel and non-intuitive metabolic engineering strategies, and enables new bioprocesses even for commodity chemicals not naturally produced by living cells.Metabolic engineering is the directed genetic modification of cellular reactions. It aims to alter the metabolic architecture of microorganisms to efficiently produce target chemicals . The sucelopment . GW metaelopment ,217,218,nic acid ,58. Anotnic acid , where aClostridium saccharoperbutylacetonicum N1-4 was proposed in [i.e., acidogenesis and solventogenesis), and allowed to achieve a correct dynamics of metabolites even under perturbed initial conditions, proving the validity of the model for further analyses and its potentiality to formulate predictions about the existence of metabolic bottlenecks for high production of butanol. Similarly, in [Lactococcus lactis IO-1, in which the competitive/uncompetitive inhibition by lactate and the substrate activation/inhibition by xylose was integrated in the rate equations. In both cases, the CMs allowed to correctly simulate the metabolites dynamics, and were used to gain biological insights by means of additional analysis based on sensitivity methods.In the context of metabolic engineering, an ODE-based dynamic model describing the metabolic pathways involved in acetone-butanol-ethanol production by posed in . The CM arly, in  the authFinally, also SA methods can be applied within the field of metabolic engineering. For instance, in ,184, theIn the last decades, mathematical modeling and computational methods have become an indispensable mean to achieve a thorough understanding of the functioning and for the identification of the design principles of complex systems. In general, the mechanism-based modeling approach is characterized by the greatest predictive capability concerning the functioning of biological systems at the molecular level. Anyway, in the context of metabolism the applicability of this approach is limited because of its unsuitability for the analysis of systems with a large size, such as metabolic networks, which are characterized by a high number of metabolites and reactions. On the contrary, interaction-based models and, especially, constraint-based models are the most widely used for the study of metabolism, although they neglect most of the quantitative and kinetic information. To provide a bird’s eye view of mathematical models and the respective computational methods applied for the study of metabolism, we provide in Irrespective of the chosen modeling approach and of the scale of the model, different computational methods are usually exploited together for the analysis of the system. For instance, reverse engineering techniques usually require parameter estimation, which both rely on algorithms for the simulation of the dynamics of the system .ad hoc comprehensive quantitative measure. As a matter of fact, the performance of each methodology is strongly affected by the size and the complexity of the model under investigation, as well as by the quality of the available experimental data. Therefore, general quantitative discussions concerning the goodness of a method with respect to similar methods cannot be provided, unless all these methods are applied to the study of the same model.Unfortunately, given a specific computational task and the available methods to solve it, it is not feasible to provide a realistic comparison of the various techniques by defining some ad hoc experimental measurements during the computational phases of model construction and validation, as well as to test its predictive capability. In a Systems Biology workflow, the interplay between experimental biologists and modelers is indeed indispensable to understand each other’s requirements and to take advantage of the respective expertise. As an example, experimental biologists would be required to design appropriate laboratory experiments, while modelers should be prepared to develop appropriate modeling strategies and simulation tools. This interplay is then expected to increase the efficacy and broaden the scope of mathematical models in the study of metabolism, and increase our knowledge of metabolism and its regulatory properties.In order for a model to be useful to experimental biologists, it has to strongly interconnect with biological data and All in all, it is an exciting moment for metabolism modeling. The availability of vastly improved platforms allowing shorter and shorter computation times on the one hand, and increasingly accurate, sensitive and quantitative metabolomics techniques on the other hand, will allow the development of complex, mechanism-based metabolic models and their interconnection with models of other cellular functions, paving the way to the development of whole cell models .


In a recent study of denitrification dynamics in hyporheic zone sediments, we observed a significant time lag  in enzymatic response to the changes in substrate concentration. To explore an underlying mechanism and understand the interactive dynamics between enzymes and nutrients, we developed a trait-based model that associates a community's traits with functional enzymes, instead of typically used species guilds . This enzyme-based formulation allows to collectively describe biogeochemical functions of microbial communities without directly parameterizing the dynamics of species guilds, therefore being scalable to complex communities. As a key component of modeling, we accounted for microbial regulation occurring through transcriptional and translational processes, the dynamics of which was parameterized based on the temporal profiles of enzyme concentrations measured using a new signature peptide-based method. The simulation results using the resulting model showed several days of a time lag in enzymatic responses as observed in experiments. Further, the model showed that the delayed enzymatic reactions could be primarily controlled by transcriptional responses and that the dynamics of transcripts and enzymes are closely correlated. The developed model can serve as a useful tool for predicting biogeochemical processes in natural environments, either independently or through integration with hydrologic flow simulators. Microbes in terrestrial and aquatic ecosystems are primary drivers of biogeochemical processes, including carbon and nitrogen cycling , but also suggested a complex regulatory machinery in microbes as a plausible mechanism for the observed time lags in enzyme responses. The model also provided novel insights into the biological roles of transcripts as a control point of dynamic metabolic regulation.The microbial community model developed in this article provides several new conceptual components for microbial community modeling, which we summarize in the following section as an essential pre-requisite for understanding the idea of the proposed model framework and the implications of the simulation results.Modeling environmental microbial communities is challenging, primarily due to immense complexity in their compositional diversity , therefore requiring no a priori knowledge of the functional roles of individual organisms.As an alternative, the complexity of microbial communities can be reduced by associating a community's metabolic traits with Figure 2. The functionally versatile organisms  can contribute to more than one catalytic step as implied by the multiple arrows. Since the enzyme functions are directly modeled, the enzyme-based formulation does not have to track which organisms are involved in each step; thereby reducing the number of parameters to describe the community dynamics in comparison to the guild-based formulation.Figure The enzyme-based formulation describes an entire microbial community as a single unit that responds to the environment by regulating its metabolism through expression of genes and enzymes. As a key advantage, this approach facilitates the application of existing dynamic modeling platforms that have been developed to predict metabolic behaviors of single organisms . This postulate removes the need to account for detailed mechanistic information on regulation, which is generally unknown for environmental organisms. The term “cybernetics” denotes a goal-seeking aspect of metabolism Wiener, . The presingle organisms , dissolved nitrate (2O reductase (NosZ), which represent the first and last steps in denitrification. Other enzymes associated with intermediate steps (such as NirS/NirK and Nor) were not measured. Nutrient concentrations were determined by analyzing three replicates. The enzyme  concentrations were obtained as the average of their two signature peptide concentrations: each signature peptide was determined from two replicates. Dissimilatory nitrate reduction to ammonium (DNRA) is another potential reaction pathway for the anaerobic reduction of nitrate, but the experimental data by Li et al. (2O (an intermediate during denitrification) was detected. While it is known that N2O could also be released as a byproduct of DNRA 's subsurface biogeochemistry research group. As all experimental details on data collection and enzyme assay are published elsewhere  , so that the coefficients will represent the stoichiometric relationships between reactants and products for a unit mole of consumed organic carbon, as follows:We normalized the above reactions with respect to one mole of dissolved organic carbon , indicating its consumption is faster than generation, and (2) no NO measurement was available. Therefore, we simplified the denitrification process into two step reactions by assuming relatively fast dynamics of NO and N2O, i.e.,Accurate parameterization of some of the intermediate reactions was difficult because (1) the concentration of NC5H7O2N) as follows:We also consider the synthesis of biomass  or (4) or both, depending on which electron acceptors (f1 and f2 denote the fractions of energy production associated with CH2O), but involve different electron acceptors . In the above reactions, we excluded H+ and H2O to focus on carbon and nitrogen balances. For simplicity, hereafter we use DOC (dissolved organic carbon), DIC (dissolved inorganic carbon) and BM (biomass) to denote CH2O, CO2 and C5H7O2N.where In a homogeneous batch reactor, mass balances of substrates and biomass can be written based on stoichiometric equations (6) and (7) as follows:x denotes concentrations of reactants and products, and r1 and r2 are reaction rates associated with BM and DOC to account for biomass degradation and the impact on the formation of organic carbon as follows:In the above equation, kdeg denotes the rate of biomass degradation. The coefficient factor “5” on the right hand side of Equation (10) denotes the carbon-based stoichiometric relationship between BM  degradation and the resulting increase of DOC .where DOC uptake rate through the reaction i (ri) as being regulated by the enzyme concentration (ei), i.e.,It is often that simple Monod-type kinetic equations do not properly describe biogeochemical reactions  and translation . With a focus on the dynamics of transcripts and enzymes, we treated the genes as part of the resource pool to minimize the number of parameters. The resulting model contains mass balance equations for the three major components associated with regulation: the resource pool (denoted by R), transcripts (T), and enzymes (E). The equations are given as follows:i and τi denote the concentrations of the resource pool and transcripts associated with the reaction i; rR,i, rT,i, and rE,i represent the kinetics of synthesis of resource pools, transcripts and enzymes, respectively; αR,i is the constitutive synthesis rate of resource pool i; βR,i, βT,i, and βE,i denote the parameters of degradation rates of resource pool Ri, transcript Ti, and enzyme Ei, respectively. The three terms on the right hand side of Equation (12), respectively, denote the constitutive synthesis rate, inductive synthesis rate, and the degradation rate of resource pool associated with the reaction i, where the variable ui denotes the control of inductive synthesis of resource. Similarly, the two terms on the right hand side of Equations (13) and (14) denote the synthesis and degradation rates of transcripts and enzymes. Note that the resource allocation is determined by the cybernetic variable ui in Equation (12), which subsequently dictates the syntheses of transcripts and enzymes as in Equations (13) and (14).where ρui in Equation is determined by the cybernetic control law termed the matching law , which reflects the expected metabolic benefit derived from the allocation of internal resources for the syntheses of transcripts and enzymes and for the subsequent catalysis of specific biogeochemical reactions  in this work. The functional form of ui in Equation (15) was heuristically determined in earlier cybernetic models, but was later theoretically derived from an optimal control theory  any alternative forms of kinetic models can be used in the cybernetic modeling formulation, and (2) even in the case where simple Monod-type kinetics were used for where the 2 is originally composed of three reduction steps catalyzed by NirS/NirK, Nor, and NosZ1/NosZ2, respectively , the rate constants of the syntheses of internal resources , transcripts , and enzymes , the biomass degradation rate (kdeg), and the fraction of energy production associated with f1). Due to the limited availability of experimental data, the parameters associated with the dynamics of internal resources and transcripts  were indirectly identified by fitting the profiles of nutrients and enzymes. Other parameters were fixed based on the literature values  or determined by manual adjustment . For fitted parameters, we used the bootstrapping technique to quantify the uncertainty in the estimates based on 95% confidence intervals. Bootstrapping was performed by estimating the best-fit parameter values for 500 experimental data sets, where each data set was created by randomly sampling a uniform distribution of the measured values at each time point. The sampled values varied within ±standard deviation from the mean experimental value. The resulting parameter sets containing values below the 2.5th and above the 97.5th quantile for at least one parameter were removed. The lowest and highest values of each parameter in the remaining parameter sets were taken as the lower and upper 95% confidence limits, respectively. The resulting parameter values (with lower and upper bounds) are summarized in Table We determined key parameters of the denitrification model through the data fit such that the sum of the squared errors between simulations and measured data is minimized. Experimental data used for this purpose included temporal profiles of substrates . Figure dynamic momentum in metabolic control. In addition to delayed enzymatic responses, the data showed that enzyme concentrations  uptake rate, i.e.,f1 < f2; see section Methods); gradually decreases for the first 3 days; maintains an almost constant level until day 5.7 where the relative ratio of u1 and u2. The f2 is close to 1  in Equation (7).Figure 2, but the level of time delay is less pronounced, particularly when considering the fact that the In Figure For the reduction of This result also reveals the complex interplay between resources, transcripts, and enzymes in denitrification. The dynamic profiles of enzymes  for modeling complex microbial communities. There are several unique features of the RSCM as highlighted in the following. First, the RSCM is built upon the functional enzyme-based approach, which enables the consistent model formulation of complex microbial communities. This enzyme-based parameterization is particularly useful in circumstances where metabolic functions of individual organisms are difficult to uniquely characterize (and therefore difficult to group). Second, the RSCM elaborates regulatory control by accounting for transcription and translation, the products of which can be quantified. Considering the difficulty in obtaining reliable estimates of guild-specific biomass for the function guild-based approach, the ability to directly quantify transcripts and enzyme concentrations is likely an advantage of the enzyme-based approach. As another key distinction, the RSCM describes the dynamic microbial regulation based on the cybernetic approach. In the current practices of biogeochemical modeling, metabolic regulation is often accounted for based on empirical inhibition kinetics, which not only increases the number of parameters to identify, but also may lead to less accurate predictions  (Stephanopoulos et al., All authors contributed to the design of the research. HS developed the modeling concept and performed simulations. DT, XS, and XC contributed to the parameter identification. HS and DT drafted out the article, which was edited by JS and JF. All authors read and approved the final manuscript.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.


Escherichia coli proteome by mass. Genome-scale models of Metabolism and macromolecular Expression (ME) compute proteome allocation linked to metabolism and fitness. Using proteomics data, we formulated allocation constraints for key proteome sectors in the ME model. The resulting calibrated model effectively computed the “generalist” (wild-type) E. coli proteome and phenotype across diverse growth environments. Across 15 growth conditions, prediction errors for growth rate and metabolic fluxes were 69% and 14% lower, respectively. The sector-constrained ME model thus represents a generalist ME model reflecting both growth rate maximization and “hedging” against uncertain environments and stresses, as indicated by significant enrichment of these sectors for the general stress response sigma factor σS. Finally, the sector constraints represent a general formalism for integrating omics data from any experimental condition into constraint-based ME models. The constraints can be fine-grained  or coarse-grained  as demonstrated here. This flexible formalism provides an accessible approach for narrowing the gap between the complexity captured by omics data and governing principles of proteome allocation described by systems-level models.Integrating omics data to refine or make context-specific models is an active field of constraint-based modeling. Proteomics now cover over 95% of the  Genome-scale models have been used to conduct systems-level studies of cellular metabolism for over 15 years348910et al. recently published a proteomics data ResourceE. coli genes (>95% by mass) under 22 experimental conditions. Using genome-scale models, we show how such proteomics resources can be used to reveal principles underlying proteome allocation. ME models compute growth optimal proteomes consistent with laboratory evolved strains accuratelyE. coli allocate a fraction of the proteome to non-growth related functions. Collectively, such allocation can be viewed as “hedging” against unknown enviromental challengesSchmidt E. coli in a variety of different growth environments. We then compare the “optimal” versus the “generalist” proteomes to reveal principles underlying proteome allocation.Elucidating trade-offs between multiple cellular objectives is an active field of systems biology research16E. coli proteome has been shown to exhibit relatively simple relationships when proteins were grouped into meaningful “sectors” We first computed optimal proteome allocation that maximized growth rate. Measured proteome allocation differed notably from these computed optimal proteomes. While the interactions between thousands of individual proteins is complex, the We then identified proteome sectors whose measured mass fractions were greater  and smaller  compared to the optimal proteomes across growth conditions . For ourE. coli. To this end, we added new “sector constraints” to the ME model (constraint (5) in Methods). Here we constrained the sum of protein mass fractions within each of the six sectors; however, the formulation is general in that any individual protein or different sector definition (coarser or finer-grained) can be used. It is important to note that while our sector constraints involved 966 of the 1678 genes in the iJL1678 ME model, only six actual constraints were added to the model–only the sum of mass fraction of each sector was constrained. Therefore, the individual protein mass fractions were still computed by the ME model. Because our objective was to develop a generalist ME model, we used these coarse-grained sector constraints to prevent over-fitting the proteome to specific conditions.We hypothesized that by constraining the ME model to more accurately allocate proteome to these large sectors, we could better predict growth rate and metabolism of the wild-type, generalist The addition of the six sector constraints led to markedly improved agreement in growth rate predictions across all conditions, with 69% lower sum of squared error (SSE), overall . We alsoσS  and macromolecular expression (E) model subsystem . Note thmeasured .P < 0.05) fluxes in 5 of 8 of the subsystems for glucose , and all 8 metabolic subsystems for acetate . Interestingly, the Generalist model showed a clear and significant negative correlation between total proteome size and growth rate, whereas the Optimal model did not  data by Gerosa keff). To this end, we randomly perturbed effective rate constants by ±50% of their nominal values and maximized growth rate. Due to the considerable computational burden of many ME simulations, we limited the sensitivity analysis to glucose and acetate conditions. We ended up with 80 simulations that were still feasible after perturbations. We found that on average, the sensitivity to keff uncertainty was greatest at the proteome mass fraction level, decreased for metabolism, and was smallest for the growth rate. For protein mass fractions, the coefficient of variation (CV) of proteins across randomly perturbed simulations ranged between 0.073 to 3.9 (median of 0.31) and 0.086 to 8.9 (median of 0.62) for the Optimal and Generalist models, respectively . Metabolic fluxes showed similar variability with CVs ranging from 0.29 to 4.8 (median of 0.38) and 0.28 to 9.7 (median of 0.61) for the Optimal and Generalist models, respectively . Finally, growth rates varied with median CVs of 0.28 and 0.31 for Optimal and Generalist models, respectively (P = 1). Therefore, we conclude that while uncertainty in keff can lead to wide variability in the expression of individual proteins, the effects are attenuated for metabolic fluxes and eventually growth rate.Due to proteome constraints, ME models exhibit essentially no flux variability at the proteome level at maximum growth rateectively . The Optectively . Again, ectively , which wE. coli strains17E. coli.One of the primary uses of ME models in previous studies has been to model optimal −1, up to 95% of the generalist proteome was not beneficial for growth. Much of the apparently wasted proteome was related to general stress response and “hedging” against environmental uncertainty. By integrating proteomics data into a ME model, we revealed the cellular cost of dedicating resources to maintaining this generalist proteome. We showed that proteomics data can be used to identify key proteome sector constraints and calibrate ME models. This approach can be extended in future work using sectors other than COGs, or determining novel sectors using the ME model itself. In particular, here we defined sector constraints capturing broad trends across many conditions, rather than fitting individual conditions. Yet, nearly all of the 15 minimal media examined showed improved predictions in terms of proteome allocation, growth rate, and metabolic fluxes. In future work, the sector constraints may be extended to include known regulation. For example, transcriptional regulation of carbon transport and utilization pathways has been well-studied for phosphotransferase system (PTS) sugarsThe ME model indicated that at growth rates as low as 0.1 het al.Sσ. Thus, new data sets that describe other experimental conditions or environmental pressures could be used to place additional constraints on the proteome using a ME model. Such experiments might measure the proteome across multiple nutrients under various stresses such as low or high pHFurther, we believe that the efforts here provide a framework moving forward for using novel data sets to tailor models to represent cellular objectives other than maximum growth rate. For example, with the various growth conditions in the data provided by Schmidt One of the biggest challenges facing systems biology is to integrate new data types into genome-scale mathematical models to provide biologically meaningful phenotypic predictions. ME models provide mechanistic understanding of how metabolic flux states are linked to protein expression. Integrating data into a structured framework thus leads to an improved understanding of systems-level properties of organisms, suggesting a combined experimental and modeling approach to meet the “Big Data to Knowledge” challenge.E. coli (iJL1678)102635−20. These tight tolerances were necessary because ME fluxes can vary by 15 orders of magnitudeWe used the latest published ME model of li iJL16781034, weThe generalist ME model includes “sector constraints” in addition to the standard ME model formulation. The complete formulation of the optimization problem associated with the “generalist” ME model is the following:g(μ) is a function of growth rate μ, p is the total simulated proteome mass, Translation is the set of translation fluxes, nsector is the number of constrained sectors, Sector(k) is the index set of translation reactions in sector k, wj is the molecular weight for protein (in g/mmol) j, vj is the translation flux for protein j (in mmol/gDW/h), ϕk is the mass fraction of sector k, and v, l and u are the vectors of reaction fluxes, lower and upper flux bounds, respectively. “Optimal” ME models are formulated similarly, except without constraints (4) and (5).where Constraint (5) is the “sector constraint”, which constrains the summed mass fraction of a proteome sector to reach the specified amount. In this case, we constrained each sector by an inequality so that allocation to the constrained sector was greater than or equal to the measured mass fraction. The constraint is derived from the relation,v are the rates of reactions that synthesize protein. The macromolecule Expression (E) matrix in the ME model enables the explicit computation of protein synthesis rate.noting that the translation fluxes et al.Our formulation can also be considered a more generalized extension of work by O’Brien keff) were kept identical for both the Optimal and Generalist ME models, and were the same values as the original iJL1678 ME modelkeff and fluxes v are related as v = keffE, where E is the enzyme concentration. Therefore, the maximum flux of a reaction is affected by proteome allocation across conditions.Effective catalytic rate constants  of each protein by its molecular weight (mi): fi = mivi/∑j∈Translationmjvj, where Translation is the index set of translation reactions.Computed growth rates P (grams protein/gram dry weight). Because total protein synthesized was diluted by cell division, we haveME models compute the protein fraction of dry cell weight, wj is the molecular weight of protein j, vjtrsl, is the translation flux of protein j, and μ is the growth rate (h−1). Therefore, P = ∑jwjvjtrsl,/μ.where keff). We (uniformly) randomly perturbed these keff by ±50% of their original values in the published iJL1678 modelWe analyzed the sensitivity of proteome mass fraction, metabolic flux, and growth rate predictions to uncertainty in 1322 effective rate constants , and 11 minimal media . We chose the six sectors having at least 5% mass fraction across all conditions. The constrained mass fraction for each sector (ϕi in constraint (5)) was the minimum mass fraction across all conditions.In this study, we used proteomics data from 15 minimal media conditionsHow to cite this article: Yang, L. et al. Principles of proteome allocation are revealed using proteomic data and genome-scale models. Sci. Rep.6, 36734; doi: 10.1038/srep36734 (2016).Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


The explosive growth of microbiome research has yielded great quantities of data. These data provide us with many answers, but raise just as many questions. 16S rDNA—the backbone of microbiome analyses—allows us to assess α-diversity, β-diversity, and microbe-microbe associations, which characterize the overall properties of an ecosystem. However, we are still unable to use 16S rDNA data to directly assess the microbe-microbe and microbe-environment interactions that determine the broader ecology of that system. Thus, properties such as competition, cooperation, and nutrient conditions remain insufficiently analyzed. Here, we apply predictive community metabolic models of microbes identified with 16S rDNA data to probe the ecology of microbial communities.Desulfovibrio piger is affected by, and affects the growth of, other members in a simplified gut community under metabolic conditions suggested to be determinant for their dynamics.We developed a methodology for the large-scale assessment of microbial metabolic interactions (MMinte) from 16S rDNA data. MMinte assesses the relative growth rates of interacting pairs of organisms within a community metabolic network and whether that interaction has a positive or negative effect. Moreover, MMinte’s simulations take into account the nutritional environment, which plays a strong role in determining the metabolism of individual microbes. We present two case studies that demonstrate the utility of this software. In the first, we show how diet influences the nature of the microbe-microbe interactions. In the second, we use MMinte’s modular feature set to better understand how the growth of By applying metabolic models to commonly available sequence data, MMinte grants the user insight into the metabolic relationships between microbes, highlighting important features that may relate to ecological stability, susceptibility, and cross-feeding. These relationships are at the foundation of a wide range of ecological questions that impact our ability to understand problems such as microbially-derived toxicity in colon cancer.The online version of this article (doi:10.1186/s12859-016-1230-3) contains supplementary material, which is available to authorized users. Advances in sequencing technology have culminated in an explosion of 16S rDNA–based microbiome projects, both small  and largEfforts to understand complex microbial communities range from inference based on 16S rDNA sequences  to the uBridging the gap between association and mechanism in microbe-microbe interactions requires an approach centered on mechanistic principles. One avenue to deciphering the role of a microbe in a community is through the use of a predictive modeling approach , 23. MetThe use of metabolic modeling to understand community dynamics is a thriving area of study, as demonstrated by the variety of tools being developed by different groups –37. NetCMMinte (pronounced /‘minti/) is an integrated pipeline that allows users to explore the pairwise interactions (positive or negative) that occur in a microbial network. From an association network and 16S rDNA sequence data, MMinte identifies corresponding genomes, reconstructs metabolic models, estimates growth under specific metabolic conditions, analyzes pairwise interactions, assigns interaction types  to netwoMMinte consists of minimally overlapping functions that come together to perform a single task. In designing it, our goal was to facilitate code re-use by focusing on modularity, allowing the user to streamline the parts presented here for other purposes. Indeed, we do not view MMinte as a single-purpose code, but as a set of widgets that can be repurposed for multiple queries, ranging from testing interactions between a set pair of species to reconstructing a community metabolic network. The user is required to provide a file with a measure of association between OTUs in Widget 1 that will define the pairwise interaction that will be analyzed. However, if the user wants to simulate all pairwise analyses between a list of sequences independently of a pre-assumed association, the user can start such an analysis in Widget 2.www.github.com/mendessoares/MMinte) for use by the community, and the authors welcome contributions to further its development.The web browser interface creates a point-and-click experience that allows the user to perform complex analysis on large data sets without programming expertise. For those seeking more control or to implement their own pipelines using MMinte widgets, MMinte functions can also be run in a command-line environment. Because all of the code is provided to the user, it can be changed to fit a particular need. Finally, MMinte is under continuous development, it is publicly available on Github  that will not be used in future analyses, based on an existing list of OTU associations. For plotting of the network in Widget 7, MMinte first creates a network based on the information on the file listing the associations between pairs of OTUs to be analyzed. If pairs of OTUs are not listed in the file as having some level of association, no edges between them are represented. This network is then used as the basis for the depiction of the kind of interactions that are predicted to occur between the OTUs (see description of Widget 7).(1) 16S rDNA sequences of all representative OTUs .Using BLAST, the 16S rDNA sequences are matched with 16S rDNA sequences from publicly available, complete genomes in NCBI . Importa16S rDNA sequences of OTUs.(1) Genome IDs and (2) percent similarity table.This widget uses the web based ModelSEED  frameworGenome IDs.Single-species metabolic models.Using COBRApy , this fu(1) Species-species associations  and (2) single-species metabolic models in the Systems Biology Markup Language (SBML) format.Two-species metabolic models.This step estimates the growth rates for each species under defined nutrient conditions, in isolation and in the presence of another species, by running a flux balance analysis in COBRApy , 47 and (1) Two-species metabolic models and (2) choice of metabolic conditions to be used from media file .Growth-rate predictions.Using previously calculated growth rates, this function quantifies the effect of pairwise interactions and assigns an interaction type to each pair, following Heinken and Thiele . The intGrowth-rate predictions.Quantitative effect of interaction and interaction type predicted.Widget 6). The shading of a node reflects the percent similarity between OTUs and genomes (Widget 2).This function generates a color-coded interaction network using the D3.js  visualiz(1) Association table between pairs of OTUs, (2) percent similarity table, and (3) quantitative effect of interaction and interaction type predicted.Metabolic interaction network see Fig. .Fig. 2NeBelow we present two case studies that exemplify how MMinte can be used to predict microbial interactions under user defined metabolic conditions. The user can choose to run the default example representing a mock community created by the authors as a test for the performance of the software. All files used in these examples, and a full tutorial on how to perform these analyses, can be found in the Additional file With just two user-provided files, one containing correlations between pairs of OTUs and the other representative sequences from a microbiome study, MMinte creates an association network in which the color of the links represents the types of predicted interactions between pairs of OTUs in a microbial community.http://hmpdacc.org/HMQCP/, uncompressed files from rep_set_v13.fna.gz and otu_table_v13.txt.gz) [In Case Study 1, we use data from the Human Microbiome Project (HMP) to demonstrate MMinte’s potential for exploratory research studies that focus on the dynamics of host-associated microbial communities. The HMP is a multi-institutional collaboration that allowed the description of the microbial diversity of several sites of the human body in healthy individuals. The study was reviewed by each participating institution’s Institutional Review Boards. Full information can be found in . To demo.txt.gz) , 5, allo.txt.gz) . The redThe number of positive and negative interactions in a community influences its level of stability and consequently its resistance to invasion by pathogens . With MMleft panel). We then reran Widget 5 with different metabolite availability conditions. Figure right panel) shows the same correlation network with a metabolite availability that is 10 times lower, resulting in different predicted interactions (58 positive and 651 negative interactions between OTUs and 45 pairs lacking any interaction). This result is consistent with the prediction that lower nutrient availability will favor more competition between organisms.We ran the full MMinte pipeline by clicking the “run all” tab on MMinte’s introductory page and providing two files, corrs.txt (which contains the associations between OTUs) and seqs.txt (which contains representative sequences) using the default setting of “complete” for the metabolite availability condition, which represents a condition with 380 metabolites available in large amounts. MMinte predicted 252 positive and 323 negative interactions between pairs of OTUs in addition to 179 pairs lacking any type of of interaction  are likely to promote more stability. In addition, if we assume that stronger positive correlation values between pairs of species indicate positive interactions [The results of this analysis highlight some possible characteristics of the community that could not be inferred solely from association data. For instance, assuming stability, and thus protection against pathogen invasion, is greater in communities with more competitive interactions , then thractions , the netDesulfovibrio piger in the guts of gnotobiotic mice, in the presence of eight other bacterial species and under different nutritional conditions. D. piger is the most commonly found sulfate-reducing bacterium in healthy adults, and is thought to shape the responses of the gut microbiota to dietary changes [In the following example, we use data from Rey et al. , who inv changes . However changes .D. piger and other members of the gut microbial community have been shown to influence the level of H2S in the gut. However, D. piger has a variety of potential metabolic pathways, only some of which will lead to the production of H2S. The role of interactions in determining the metabolic niche of D. piger in the gut is both important and not fully understood. Using MMinte, we explored the types of interactions that are predicted to occur between nine different species of microbes that co-occur in the human gut and whose interactions are believed to be metabolically based [The interactions between ly based , 53. We ly based .D. piger: model 411464.8, Bacteroides thetaiotaomicron: model 226186.12, Bacteroides caccae: model 411901.7, Bacteroides ovatus: model 28116.7, Eubacterium rectale: model 657318.4, Marvinbryantia formatexigens: model 478749.5, Collinsella aerofaciens: model 411903.6, Escherichia coli: model 83333.113, and Clostridium symbiosium: model 742740.3). After reconstructing the individual species metabolic models and creating two-species communities (Widget 4), we predicted species growth rates in the presence and absence of another species in the community by running Widget 5 under 17 different metabolic conditions, listed in Additional file We started by providing a list of species IDs to Widget 3 (D. piger in the presence and absence of other species in the community shows that this species is likely to benefit from the presence of each of the other species in the community under “Complete” metabolic conditions. D. piger is consistently predicted to grow under aerobic conditions, but under anaerobic conditions, growth is only predicted to occur if either B. ovatus, B. thetatiotaomicron, B. caccae, C. symbiosium, or E. coli are present. Thus, using the models reconstructed using ModelSEED, MMinte predicts an obligate association between D. piger and these species in anaerobic environments (Additional file D. piger impaired the growth of most species it was paired with under all conditions except “Complete” (Additional file E. coli and E. rectale; the magnitude of the effect of D. piger on their growth depended on the flux conditions for oxygen, chondroitin sulfate, and sulfate (Additional file A look at the predicted growth rate of Overall, the number of each type of interaction changed depending on metabolite availability, but not linearly Fig. . For insMMinte bridges the gap between association and mechanism in microbe-microbe interactions by assessing the metabolic influence that two microbes have on each other. Our predictive modeling approach involves reconstructing the pairwise metabolic community models that make up the basic unit of interaction within a community. More specifically, MMinte advances microbiome research by assigning functional interactions instead of simply calculating associations or correlations based on abundance . This alThe metabolic interactions that MMinte identifies can be used to understand the broader ecological features of a biological system. Dynamic ecological features such as stability and robustness are linked to competitive-cooperative interactions and the nature of the positive-negative feedback loops they engender . For exaD. piger and other organisms commonly found in the gut. In vivo experiments have shown that mice colonized solely with D. piger have significantly increased levels of H2S, which is a genotoxic metabolite that may be involved in the development of colorectal cancer [D. piger can help inform dietary interventions aimed at reducing the abundance of this species in the gastrointestinal tract.Case Study 2 highlights the modular nature of MMinte and the ability of the user to explore the effect that changes in the availability of a particular metabolite may have on the interactions between organisms. The results give us an important window into the role of the environment, specifically the presence or absence of oxygen, chondroitin sulfate, and sulfate, on the interactions between l cancer , 60, comDesulfovibrio vulgaris and Methanococcus maripaludis. The model accurately predicted the ratio of the two species during cell growth and the flux of several metabolites, thus demonstrating the utility of this modeling approach to understand the mechanistic bases of interactions [Constraint-based metabolic models were first used to explore the mechanistic bases of interactions between a sulfate-reduced and a methanogen , 62. In ractions . Furtherractions  were ablLike all algorithms, MMinte has potential limitations. For example, the predictions created by MMinte are only as accurate as the metabolic models used. These metabolic models are linked from 16S in a multi-step process that involves identification of genomes and metabolic network reconstruction using ModelSEED . This auMMinte is a tool that predicts the type of interactions occurring between organisms in a complex microbial community under defined metabolite conditions based on the metabolic models of each species. A full run takes data of an association network and 16S rDNA sequences, identifies the genomes, reconstructs metabolic models, and estimates the effect of being in a two-species community for each species under user defined metabolic conditions. The predicted interactions are then plotted in an interaction network. Additionally, the widgets that make up MMinte can be run independently allowing the user to perform specific tasks and bypass some of the steps of the analysis. We have incorporated the design principles of clear modularity, usability, and open access into the development of MMinte. In our view, part of the value of MMinte to the development of predictive community metabolic modeling is the potential for integration into other analytical platforms. We view the ability to build on existing development efforts as critical to expanding systems biology tools to wider and broader scales of ecology and data . MMinte


An important hallmark of the human gut microbiota is its species diversity and complexity. Various diseases have been associated with a decreased diversity leading to reduced metabolic functionalities. An important hallmark of the human gut microbiota is its species diversity and complexity. Various diseases have been associated with a decreased diversity leading to reduced metabolic functionalities. Common approaches to investigate the human microbiota include high-throughput sequencing with subsequent correlative analyses. However, to understand the ecology of the human gut microbiota and consequently design novel treatments for diseases, it is important to represent the different interactions between microbes with their associated metabolites. Computational systems biology approaches can give further mechanistic insights by constructing data- or knowledge-driven networks that represent microbe interactions. In this minireview, we will discuss current approaches in systems biology to analyze the human gut microbiota, with a particular focus on constraint-based modeling. We will discuss various community modeling techniques with their advantages and differences, as well as their application to predict the metabolic mechanisms of intestinal microbial communities. Finally, we will discuss future perspectives and current challenges of simulating realistic and comprehensive models of the human gut microbiota. Microbial communities are abundantly present throughout nature and form many beneficial symbiotic interactions with different eukaryotic hosts. In many cases, the host acquires novel functionalities with mutualists that can contribute to host fitness and well-being  by, e.g.While the symbiosis between the human gut ecosystem and its host is less well understood than other symbioses, intestinal microbial communities have been associated with human well-being . In partA loss of microbial and metabolic diversity can lead to various microbiota-associated diseases, such as obesity , type 2 With the advent of high-throughput sequencing technologies, we can now enumerate and describe the microbial and functional diversity of the human gut with respect to different diseases and conditions. These analyses have unraveled a complex microbial ecosystem that is influenced by diet and environmental factors . FurtherTo understand the ecological mechanisms that drive the human gut microbiota better, it is important to go beyond the descriptive nature of high-throughput data analysis. This can be achieved by formulating knowledge- or data-driven models, which can describe the underlying ecosystem and give mechanistic insights for hypothesis generation and designing subsequent experiments. A recent review described the relevance of such models in giving novel causal relationships in a field that is dominated by data and correlation . Here, wNetwork-based approaches are used to identify relevant microbes or metabolites of the human gut microbiota. Networks are usually represented by interactions in the form of edges that connect biological components in the form of nodes. With respect to the human gut microbiota, these components can represent species that interact  or metabSpecies cooccurrence networks can be used to investigate ecological interactions between microbes. Based on high-throughput data analyses, simple networks can be constructed with the species abundance information for different patients . Such neTo overcome these limitations, metabolism-based networks have been used to assess the functional aspect of the human gut microbiota. Therefore, metabolic functions or enzymes annotated from high-throughput sequencing of different human individuals can be used to identify pathway or metabolite differences . By connIn the bottom-up network approach, biochemical knowledge is used to generate specific networks, e.g., biochemical networks, from information provide in databases, such as the Kyoto Encyclopedia of Genes and Genomes (KEGG) , MetaCycIn another study, the metabolic exchange between microbes has been represented in a global interaction map, which helped the identification of metabolic deficiencies in certain diseases . On the In a complementary approach, genome-scale metabolic reconstructions of organisms are constructed based on genomic and biochemical data to represent comprehensively their metabolism to investigate the metabolic potential. To facilitate the reconstruction process, available automatic reconstruction pipelines, such as ModelSEED , can be While the automatic reconstruction process provides a good approximation of a species’ metabolism, the manual curation of reactions is essential for metabolic modeling. On the basis of biochemical knowledge of reaction directionality and substrate uptake, reactions are refined within the metabolic reconstruction to be more congruent with experiments . This mavB, which is subject to S × v = 0, and vi,min ≤ vi ≤ vi,max, where v represents the flux values  through all model reactions i while maximizing the flux vBthrough the metabolic objective reaction B. The steady-state assumption implies that metabolites cannot accumulate and the total flux into the network must equal the total outflux, and it is represented by S × v = 0, where S is the aforementioned stoichiometric matrix. The constraints, represented as vi,min ≤ vi ≤ vi,max, can reflect medium conditions, in which the uptake of metabolites via exchange reactions is limited or limitations of internal reaction fluxes that come from experimental data  is based on genome-scale metabolic reconstructions, which are formalized as metabolic models to simulate biological relevant physiological states. By applying condition-specific constraints, reconstructions are converted into metabolic models . Often, tal data . The biotal data  and can tal data . Constratal data –45 or coal data ––50. Compl data ––, facilitAn advantage of the COBRA approach is that it can assess the spectrum of metabolism for an organism . The steCompartmentalized microbial community models can be constructed by integrating individual microbial metabolic reconstructions via their S-matrices . In thisPairwise models of microbial communities can give new insights in terms of the metabolic interactions that occur in the human gut. In a recent study, we applied pairwise community models to the human gut microbiota by modeling all possible pairs of more than 700 microbes and found strikingly mostly negative metabolic interactions, e.g., competition for resources, of different microbes , which uPopulation-based COBRA modeling can be used to analyze temporal dynamics of metabolic interactions within microbial communities. With the optimization of biomass growth, FBA intrinsically models growth of microbe populations. The combined S-matrix approach can also be used to model population dynamics with respect to a temporal scale . EssentiPopulation-based COBRA modeling can give new insights in the dynamic change of metabolic interactions of human gut microbial communities. A recent approach applied population-based metabolic models to a simple community of six microbes . The resThe metabolic interactions between distinct microbial organisms can be modeled by combining the COBRA approach with individual-based modeling (IBM). IBM is a method also used in classical ecology  to modelIndividual-based COBRA modeling can give insights into the spatial and temporal community structure of the human gut microbiota. For instance, a recent study has explored the effects of antibiotics on a two-species community model of human gut microbes, represented by a combination of individual-based and kinetic modeling . SimilarThe COBRA approaches for analyzing microbial communities provide promising tools for investigating the metabolic effect of the human gut microbiota. Several studies have been conducted in this context, mostly focusing on small communities representative of the intestinal microbiota . These sA challenging aspect in community modeling of the human gut microbiota is horizontal gene transfer (HGT). HGT seems to occur frequently within the human gut microbiota , 79 and in vitro (One of the most striking hallmarks of the human gut microbiota is its species diversity, which poses challenges to the model simulations that need to be addressed. Simplified microbiota models of less than 10 species are relevant for studying metabolic interactions in general and can be used to simulate experiments that are conducted with small microbial communities in gnotobiotic animals  or in viin vitro . Howeverin vitro , we creain vitro , 83. By in vitro , 81, demin vitro  and are Model complexity also poses the problem of subsequent data analysis and visualization. By simulating large-scale microbiota models with a high number of variables, it becomes difficult to identify the explaining pathways. The large amount of simulations and the complexity of the models further require data analysis approaches to find the most relevant parameters influencing the modeled system with the aim to propose an experimentally testable hypothesis. State-of-the art machine learning approaches have already been used to aid the analyses of COBRA simulation results , 85 but High-throughput omics data can be used to contextualize genome-scale metabolic models , 87. SimExperimental validation of community simulation results plays an important role to assess the predictive potential of the microbial community models. While community models can give interesting novel hypotheses with their simulation, particular attention should be paid to the biological relevance of the predictions. It is therefore important to relate and validate at least part of the simulations with experimental values that come from existing knowledge or direct experiments. Existing knowledge can be used to validate the predictions of community models, e.g., by comparing simulated with measured metabolite concentrations . This wiThe introduced COBRA community modeling approaches are promising tools to give novel insights and hypotheses of microbial consortia in the human gut. The models allow for the integration of multidimensional omics data and go beyond the descriptive nature of more traditional high-throughput data analysis approaches with the potential to unravel novel mechanisms. The different discussed modeling paradigms rely on various assumptions that need to be assessed before starting an analysis. On the basis of this assessment, the model that is least complicated and most explanatory for a specific research question should be chosen. It could also be fruitful to combine different approaches to see their consistency or potential differences for a specific problem. For example, it would be interesting to model single isolated species and communities separately to identify beneficial or detrimental effects on the single species level. Further comparisons can be made between network-based and modeling approaches. Can, for example, the negative and positive interactions in a coexistence correlation network be explained by the underlying metabolic interactions? Comparisons between algorithms could give further hints on candidate experiments that can be performed for validation. This can drive further research on the human microbiota, mechanistically predicting different treatments for various diseases.