Identifying the reactions that govern a dynamical biological system is a crucial but challenging task in systems biology. In this work, we present a data-driven method to infer the underlying biochemical reaction system governing a set of observed species concentrations over time. We formulate the problem as a regression over a large, but limited, mass-action constrained reaction space and utilize sparse Bayesian inference via the regularized horseshoe prior to produce robust, interpretable biochemical reaction networks, along with uncertainty estimates of parameters. The resulting systems of chemical reactions and posteriors inform the biologist of potentially several reaction systems that can be further investigated. We demonstrate the method on two examples of recovering the dynamics of an unknown reaction system, to illustrate the benefits of improved accuracy and information obtained. Reconstructing the correlated reactions that govern a system of biochemical species from observational temporal data is an essential step in understanding many biological systems. To facilitate this process, we propose a robust, data-driven approach based on a sparse Bayesian statistical model. Our approach exploits sparse Bayesian priors and an unbiased observational model to recover a parsimonious, interpretable reaction system from mass-action relations, utilizing very little user input. On a set of simulated test problems, the method demonstrates increased robustness and decreased bias at different levels of measurement variability, while also producing interpretable reaction systems and quantifying uncertainty. As a tool, the approach can be used to flexibly interrogate biological systems while allowing incorporation of potentially uncertain domain knowledge to improve the efficiency and identifiability of the problem. PLOS Computational Biology Methods paper.This is a Developments in high-throughput experimental methodologies in biology have enabled the collection of massive amounts of time varying molecular data at small scales. This has resulted in significant advances in understanding the biochemical networks and mechanisms underlying physiological processes such as gene regulation. Indeed, greater understanding of regulatory processes at the single cell level can aid in the development of targeted therapies for diseases such as cancer –3. A majIn this work, we are primarily interested on identifying interpretable mass-action biochemical reaction networks using only the observed time series of species concentrations. Expanding upon the problem formulation first proposed as Reactive SINDy in , we autohttps://github.com/rmjiang7/bayes_reactive_sindy.The remainder of this paper proceeds as follows. In Materials and methods, we describe mass-action biochemical reaction networks and formulate the problem of inferring these networks from observational data. Next, we propose improvements to the existing methodology and describe the specifics of the proposed model applied to inference of reaction networks. In Results, we demonstrate how our methodology can be used in two different examples to retrieve interpretable networks from observational concentration data. We close in Discussion by noting a few details for usage, detailing some future directions, and mentioning the limitations of our method. All implementations and code can be found at N species at time t. The evolution of the system can be modeled using the following set of coupled ordinary differential equations (ODEs) formally known as the reaction rate equations:M reactions among N species and f(X) is the vector of all rate functions.Systems of biochemical species reacting under any number of reaction channels are commonly modeled dynamically using the framework of chemical kinetics. Specifically, denote f(X) can take the form of any nonlinear function, in this work we assume that the system follows mass-action kinetics  denote the vector of all of the reaction rates, we aim to estimate k such that, when solved, tj. Many methods exist to solve these types of problems, such as ridge, LASSO, and Elastic net regression  and estimate k separately for each using the same Sc. Trajectories of the two species are shown in τ = 1e−8 and estimate c along with the other parameters by placing the prior c ∼ Inv- Gamma. A total of 4000 samples are drawn using four MCMC chains. We note that while we use MCMC for accuracy and demonstration purposes, variational inference can also be used to obtain fast approximate solutions and is supported in our implementations. In our experiments, we found that the variational approximations were generally reliable, though this largely depends on the problem.Data is generated by solving the above system of reactions and perturbing with Log-Normal noise at fixed times to simulate a noisy measurement process. The reactions comprising the complete stoichiometric matrix In More specifically, the difference in the results demonstrates the bias introduced by estimating derivatives. At observation intervals dt = 1.0 and dt = 2.0, too much information is lost from estimating derivatives coupled with measurement noise to obtain the correct system. With the Bayesian treatment of the problem, we can also quantify uncertainty in the non-zero reaction rates. This informs us of the plausible range of reaction rates, given the observed data, and can be useful to detect which parameters the model is able to identify with evidence of correlated reactions. In Y, is observed. In this case, the statistical observational model can be changed to,Y.In the previous example, we assumed that the species were completely observed. However, under the latent variable formulation, this is not strictly required. In this section, we demonstrate inference of the network for the identifiable Lotka-Volterra example, when only the prey species, P is lost. However, the method is still able to retrieve the correct networks, as the oscillating regime for this problem is generally unique.W = X + Y be the observed sum of X and Y, the statistical model can be stated asSimilar to above, the latent variable formulation we have presented allows for modeling of the situation where a sum of species concentrations is observed, but not any of the individual species. In this case, for the Lokta-Volterra system, letting P by a gene g in a prokaryotic cell . Furthermore, Log-Normal noise is added to the observations. At these parameter values, g and P decay rapidly, thus a small dt is required to provide sufficient information to the model. True trajectories and observations are shown in To evaluate the method on a larger reaction system with more possible reactions, we explore a simple synthetic model of auto-regulation of a protein tic cell . The modSc of 260 possible reactions. The exact reactions included can be explored in the code repository. For estimation from the regularized horseshoe model in this problem, we set τ = 1e−6 and estimate c along with the other parameters by setting c ∼ Inv- Gamma. We run several MCMC chains to obtain results however only report the best two networks obtained for each experiment.Using our library of ansatz reactions, we construct a complete stoichiometric matrix k1, k2, k3, and k4, are known with confidence and the aim is to retrieve a system of reactions which replicates the observations, given these four known reactions. Below, we present the results of this setting to demonstrate a realistic situation where partial knowledge about system. The same experiment when no reactions are known is presented in the To replicate the more common situation where the biologist has prior domain knowledge about the system under study, we explored the scenario where the first 4 reactions and rate parameters, As e−2, selected such that thresholding larger reactions changes the dynamics, the best estimated network was comprised of 24 total reactions. The full network is detailed in the Reactive SINDy is also capable of inferring a network, however it is considerably less sparse and with larger reaction rates than those from our method. Under a threshold of 1That multiple networks are obtained by different chains in this problem is largely due to the facts that our complete stoichiometric matrix constructed from the above process does not restrict many reactions. In this, an iterative procedure can be applied, where the recovered networks can be examined by the user for plausibility and implausible reactions can be excluded in future runs to converge to a different reaction system. Realistically, we expect that the complete stoichiometric matrix will often be constructed in a more careful manner so as to eliminate many of the implausible reactions before the method is used. We discuss the identifiability issue in the next section. Interestingly, the inferred networks converge largely to 2nd-order reactions to describe the system. While from a combinatorial perspective, this is not surprising considering that the ansatz library contains significantly more 2nd-order reactions than 1st-order, a possibility is to add a bias to the system for 1st-order reactions via a prior weight on certain reactions.The latent variable formulation for the observational model provides robustness when observations are noisy or observations are not made for all of the species. In these situations, it is valid and desirable to use this model as it takes into account the true measurement process as demonstrated in the Lotka-Volterra example. However, this comes at a substantial computational cost. In some situations, when all of the species are observed and the measurements are not too noisy, simply using the Bayesian Regularized Horseshoe along with estimated derivatives as an extension to Reactive SINDy is sufficient and significantly faster to identify models. The model for this is: PTLasso . We suggA major problem in the identification of reaction systems is the possibility of multiple structural networks which can produce nearly identical results, especially when data is limited and noisy. As multiple definitions of the unidentifiability problem can be found in the literature, in our work, we are referring to the case where different dynamic networks with different parameterizations and equations can yield indistinguishable outputs and thus cannot be identified from the data given. While the sparsity priors used in this paper aim to resolve this situation by biasing estimates toward systems with fewer reactions, this remains an issue as multiple structural pathways may still exist with a very similar number of reactions. Immediately, this issue can be somewhat relaxed in a few ways.First, constraining the allowed reactions will naturally bias the solutions away from certain pathways. However, this requires significant domain knowledge of the species or the system under observation. The work of Tuza et al.  presentsAlternatively, without introducing any domain knowledge, multiple MCMC chains can be used to explore all of the different networks. As each chain is trivially parallelizable, massive computational power can help explore the space more efficiently. Starting a large number of chains at different initial points will allow the chains to converge to different modalities and present it to the user as is in the case of the Prokaryotic Auto-regulation model above. A user can prune implausible reaction networks and re-run the model to converge to better solutions. The use of ML techniques such as cross-validation on a hold-out test set to automatically rank networks based on predictive accuracy  could beL independent trajectories are observed could be stated as follows,tj for the l-th trajectory.As explored in Reactive SINDy, the incorporation of more data such as trajectories from multiple initial conditions can also aid in improving the identifiabiltiy of the process. Intuitively, this can be relevant in the case where certain dynamics are only present at certain concentration levels. In this case, a straightforward modification to the observational model where As the number of species grows, the number of possible reactions grows combinatorially. This poses a significant issue computationally, as it results in a large search space for reactions and possibly further identifiability issues as demonstrated above. The scaling issue limits the practical applicability of the method to systems with a small number of active species, which we roughly estimate to be <20 based on our experiments and the amount of time that they take. While a larger set is technically possible, the computational burden may be too great to obtain results in a reasonable amount of time. One possibility is to run the method on smaller subsets of reactions to prune reactions in a sequential procedure. However, this may lead to bias issues as combining the estimates from different subsets is a non-trivial problem, especially if dealing with partial posterior distributions. Practically, biological domain knowledge can substantially help here in limiting the allowed reactions in the system or specifying known reactions as in Example 2.Computationally, the latent variable approach with Bayesian Inference is significantly more expensive than the approach used by reactive SINDy. A large part of this is the need to compute the sensitivities of the ODE system to obtain efficient sampling. In our experiments, the auto regulatory network with 260 reactions took approximately 4 hours of time on a M1 Apple ARM processor using our approach while roughly 1.5 hours to perform a large grid search using reactive SINDy. We find that this difficulty generally scales as a function of the number of data points in addition to the number of possible reactions. A possibility on this end is to utilize Variational Inference to speed up the inference component as presented in . Furtherτ determines the level of sparsity of these networks and, in our experience, is a pivotal hyperparameter to tune when using the horseshoe prior. Generally, we find that smaller values of τ will force the near-zero reaction rates to smaller values however, this typically leads to a significant decrease in computational efficiency when estimating the networks. For this reason, through our experiments and set τ to a small enough value such that the above pruning procedure removes a large enough set of reactions while maintaining the dynamics.Selection of τ in a more interpretable way for reaction network inference problems is needed. A common strategy employed in other models is to place the prior τ ∼ Cauchy+ to allow the data to adjust τ [τ0 as,m0 can be derived as a guess for the effective non-zero coefficients and σ is the measurement noise, however our models deviate from linear regression and thus the same interpretations do not hold.Further work exploring how to properly tune and select adjust τ , howeveradjust τ  propose A common concern with Bayesian methods is whether the prior can be overcome with sufficient data. While in our experience, the utilized horseshoe priors are weakly informative, and indeed can be overcome with sufficient data to obtain the true network, however more rigorous study needs to be done for this. The particular case study demonstrated by Golchi et al.  offers gX(t) and the evolution of the system is better described using a stochastic process. While mass-action kinetics can still be applied, they now specify reaction propensities. To accommodate this, Z comes from the stochastic process as specified by [Sc remain as previously defined. However, the significant challenge here is that the posterior distribution becomes intractable due to the intractable likelihood term Many biochemical reaction systems exhibit intrinsic stochasticity. In these situations, ified by  while thequation . This prWhile there is a growing class of likelihood-free Bayesian inference methods  that canIn this work, we have presented a method to recover a parsimonious system of interpretable mass-action reactions directly from observations of species concentrations over time. Improving on the formulation presented by Reactive SINDy, we have modified the method via the Bayesian regularized horseshoe prior and by adapting the model as to not require derivative estimates. Our experiments show that, when identifiable, our modifications are able to recover the underlying system with uncertainty estimates from the Bayesian formulation even in sparse data scenarios. Alternatively, when unidentifiable, we present multiple sparse reaction networks which can reasonably explain the results and upon which a biologists can iterate.S1 Appendix(PDF)Click here for additional data file.


Cancer is considered a high‐risk condition for severe illness resulting from COVID‐19. The interaction between severe acute respiratory syndrome coronavirus‐2 (SARS‐CoV‐2) and human metabolism is key to elucidating the risk posed by COVID‐19 for cancer patients and identifying effective treatments, yet it is largely uncharacterised on a mechanistic level. We present a genome‐scale map of short‐term metabolic alterations triggered by SARS‐CoV‐2 infection of cancer cells. Through transcriptomic‐ and proteomic‐informed genome‐scale metabolic modelling, we characterise the role of RNA and fatty acid biosynthesis in conjunction with a rewiring in energy production pathways and enhanced cytokine secretion. These findings link together complementary aspects of viral invasion of cancer cells, while providing mechanistic insights that can inform the development of treatment strategies.  AGP, alpha‐1 acid glycoproteinCOVID‐19, coronavirus disease 2019DARs, differentially active reactionsDEGs, differentially expressed genesDEPs, differentially expressed proteinsdUMP, deoxyuridine monophosphateFBA, flux balance analysisFVA, flux variability analysisGO, gene ontologyGSMMs, genome‐scale metabolic modelshpi, hours post‐infectionIP10, interferon γ‐induced protein‐10MIP‐1β, macrophage inflammatory protein‐1βPPARalpha, peroxisome proliferator‐activated receptor alphaSARS‐CoV‐2, severe acute respiratory syndrome coronavirus‐2SDH, succinate dehydrogenaseVEGFA, vascular endothelial growth factor AThe rapid spread of severe acute respiratory syndrome coronavirus‐2 (SARS‐CoV‐2) has created a global public health emergency that is currently affecting countries across the globe , 2. The Cancer is considered by the Centers for Disease Control and Prevention a high‐risk condition for severe illness resulting from COVID‐19. However, it is not clear to what extent this comorbidity affects the cellular mechanisms observed in both diseases. While cancer patients infected with SARS‐CoV‐2 are known to be at increased risk, the human metabolic response to SARS‐CoV‐2 infection in cancer cells remains largely uncharacterised to date , 4, 5, 6et al. [Further, liver dysfunction has been reported to increase with increasing severity of COVID‐19 infection , 11, 12.et al.  demonstret al. , 17.Recently, time‐course transcriptomic measurements of SARS‐CoV‐2‐infected Huh7 cells have been reported . In thisSimilarly, studies of circulating blood metabolites in COVID‐19 patients have yielded interesting biomarkers of infection, including hijacking of nucleic acid intermediates , 22, 23,To unify and interpret heterogeneous omics data, systems biology offers powerful tools such as genome‐scale metabolic modelling . This apTo address these gaps, we apply a systems biology approach to explore how SARS‐CoV‐2 infection impacts human cancer cell metabolism and how distinct pathways are affected over time. To this end, we build genome‐scale metabolic models (GSMMs) that incorporate multiple transcriptional and protein expression states after infection and that reflect consecutive stages of immune response to viral invasion. Thanks to a flux balance analysis (FBA) framework, such models allowed us to explore in detail how the disease insurgence impacts single metabolic reactions. Through this approach, we identify altered pathway capabilities and reconstruct a global overview of metabolic rewiring under SARS‐CoV‐2 infection , which is associated with the cytokine storm, was highly upregulated. Finally, we suggest some biological processes that could be targeted by therapeutic treatments based on our findings.et al. [RNA‐seq data were taken from the SRA database (accession PRJNA627100). Briefly, Huh7 cells were used to perform transcriptomic analysis in uninfected and 24, 48 and 72 hours post‐infection (hpi) by the SARS‐CoV‐2 virus. Three specimens for each time point were collected. Complete data production is extensively reported in the original paper of Appelberg et al. . Preprocfastp v0.20.0 [star v2.7.0 [rsem v1.3.3 [rsem command (rsem‐generate‐data‐matrix). Genes with less than 10 counts in all samples were then filtered out. Gene differential expression (pairwise comparisons) was computed by edger [P‐value ≤ 0.05 were considered differentially expressed.RNA read preprocessing was performed by using  v0.20.0 , applyinr v2.7.0  with star v2.7.0 , annotatr v2.7.0 ). Alignmm v1.3.3  to estimby edger  from rawlimma r package [P‐values after a Benjamini–Hochberg (BH) adjustment. Proteins with an alpha value less than 0.05 were considered significant. For both proteomic and transcriptomic results, no threshold was applied on the log2 fold change.Starting from quantile‐normalised protein abundance, proteomic profiles were processed as described in the original paper . Proteom package . In the biomart package [r 3.6, querying available Ensembl Gene IDs and retrieving Gene Names and HGNC gene IDs to allow mapping of genes onto the metabolic model. Then, GO (gene ontology) and pathway enrichment analyses were performed on the KEGG [Re‐annotation of differentially expressed genes (DEGs) and proteins was performed using the  package  in r 3.6the KEGG  and Reacthe KEGG  public pWe modelled the hepatocyte‐derived cellular carcinoma cell line Huh7 metabolism based on the Recon 2.2 genome‐scale reconstruction . This moS where the rows correspond to metabolites and the columns represent reactions. Under a steady‐state assumption, there is no net change in mass in the system and the mass is conserved. Therefore, the rate of production of each internal metabolite equals its rate of consumption. A column vector v represents the flux through the system (reaction rate of each reaction). Under the steady‐state assumption, the matrix multiplication of the stoichiometric matrix S and column vector v provides the linear equations representing the constraints (vS = 0).Flux balance analysis is a mathematical approach for analysing the flux of biochemical reactions through a genome‐scale reconstruction of metabolic networks , 29. FBAv ≤ ub. These are vectors representing the lowest and highest reaction rate possible for each reaction. These constraints reduce the possible solution space and can be set for each condition, therefore creating condition‐specific metabolic models [Further constraints are added through RNA‐seq expression profiles, and specifically the lower and upper bounds of each metabolic flux lb ≤ c models .metrade [Temporal progression was modelled through the creation of condition‐specific models, based on the gene and protein expression fold change profiles over the three time points. Gene expression for the unaffected cells was used as a control when calculating the fold changes. Condition‐specific models were generated through a modified version of metrade  where flIn addition to the transcriptional constraints described above, we utilised additional constraints devised based on the literature and our RNA‐seq analysis results. Regulation of lipid metabolism by peroxisome proliferator‐activated receptor alpha , which increases the size and number of peroxisomes, was identified from the pathway analysis. PPARalpha is known to be involved in regulating fatty acid metabolism and the immune response in hepatocytes . Acyl‐CoAfter having obtained a constrained model, flux variability analysis (FVA) was used to evaluate the minimum and maximum value of the flux that each reaction can carry, while still satisfying the given constraints. FVA was run by solving a double linear programming problem on each condition‐specific metabolic model. Estimation of metabolic potential on each time point was performed by FVA as implemented in the COBRA toolbox . A metabFor each time point, we determined the most perturbed metabolic pathways by a one‐sided hypergeometric test with false discovery rate correction for multiple testing on the differentially active reactions (DARs). Overactive DARs were defined as those reactions whose maximal flux fold change was 1.5 or greater and above the 95th percentile of the fold change distribution. For underactivity, we considered reactions whose maximal flux fold change was 0.8 or lower and fell in the 5th fold change percentile. In this way, we could select metabolic reactions characterised by activity changes with both statistical relevance and meaningful effect size.et al. [To evaluate how SARS‐CoV‐2 infection affects metabolic activity along short‐term post‐infection time points , we applied our condition‐specific modelling approach, as described in the et al. .As a result of transcriptomic analysis, we obtained a high percentage of uniquely mapped reads, ranging from 88.02 and 89.13, with a mean value of 88.5. Then, we calculated DEGs among all the possible time point combinations, focusing on the differences between uninfected cells and the most advanced stage of virus incubation (72 hpi), which is expected to disclose the most valuable and significant results. Analogously, we obtained DEPs for the same contrasts. At 72 hpi, we identified 3464 DEGs and 883 DEPs Table , associaTo obtain a mechanistic understanding of the alterations observed on a transcriptional level, we integrated the gene and protein expression profiles described above within Recon 2.2, a GSMM of human cells. The selected GSMM is among the most complete templates for the human metabolic network and was chosen based on its manually curated pathways and the validated representation of energy generation across various carbon sources . As RecoOur modelling approach assumes a metabolic steady state, which can be assumed by considering different stages of infection that are distant in time. This approach has previously been used to detect biomarkers and drug targets for a range of disorders , 31, 73.To obtain a global overview of how SARS‐CoV‐2 affects cancer cell metabolism, we closely inspected altered pathways and individual reactions. In the following sections, we describe the main patterns identified and discuss their potential role in SARS‐CoV‐2 invasion inside host cells.de novo dTTP (thymidine triphosphate) synthesis. Thymidylate synthase converts dUMP to TMP and dTMP to dTDP. NDP (nucleoside diphosphate) kinase then converts dTDP into dTTP. This last reaction is among the second most highly upregulated reactions  [see transcriptomic‐informed results in Fig.  CI, CII,In parallel, the majority of the reactions of glycolysis in both sets of flux data are upregulated or unchanged through all time points, with a dip in some upregulated reactions at 48 hpi. The main rate‐limiting enzyme of glycolysis, phosphofructokinase, is progressively upregulated through all time points in the transcriptomic‐informed flux data  were found to be upregulated in the proteomic‐informed data, while in the transcriptomics‐informed data upregulation was found for IL‐6, IL‐8, IL‐12A, vascular endothelial growth factor A (VEGFA), chemokine (C‐C motif) ligand 20 (CCL20) and alpha‐1 acid glycoprotein (AGP). Analysis of the metabolic alterations further revealed high upregulation of IL‐17A, IL‐15 and IL‐7.As expected, protein secretion exhibited a widespread upregulation, though visible especially at 24 and 72 hpi, with 10 upregulated cytokine/chemokine secretory pathways. Macrophage inflammatory protein‐1β (MIP‐1β/CCL4), IP10/d‐alanine metabolism also detected in the metabolic fluxes may be a mechanism for producing cytotoxic oxidative stress as part of the innate immune response [As shown in Fig. response  (Fig. 3AFigure It should be noted that individuals suffering from cancer may already be taking medication. While the treatments suggested here do not interact with common liver cancer drugs according to the interactions listed in the Joint Formulary Committee  Table S, cautionWhile transcriptomic‐only approaches can provide insights into changes in gene expression caused by viral infection, translating these data into metabolic data can give deeper and mechanistic insights into the effects of viral infection on the host cell. Here, we use genome‐scale‐metabolic modelling of Huh7 cells with transcriptomic‐ and proteomic‐informed FVA to explore the metabolic effects of altered gene expression from COVID‐19‐infected cancer cells on cellular phenotype. This method allows the identification of dysregulation at the enzyme/reaction level, in contrast to previous studies of metabolism. Our models were constrained to represent the viral‐infected cell using evidence from literature and the pathway analysis of DEGs.CXCL10, eotaxin, MIP‐1β and IL‐6. Finally, we suggest therapeutic treatments based on mediating the inflammatory response and metabolic key changes that enable the virus to replicate in the host cancer cell. Future studies could enhance our understanding by investigating multicellular metabolic changes as well as how individual data such as gender, age and ethnicity affect the metabolic changes caused by COVID‐19 infection. This will facilitate the investigation of more personalised therapeutic interventions for cancer patients.A pathway analysis revealed an upregulation of PPARalpha, known to be involved in fatty acid metabolism and regulation of the immune system . The anaCA conceived and supervised the study; LB and GB performed the RNA‐seq data analyses; EY and GZ developed the software and performed the simulations; EY and GZ analysed the data; all authors contributed to drafting the manuscript; EY, GZ and CA produced the final version of the manuscript.Table S1. Differentially expressed genes and proteins.Click here for additional data file.Table S2. Enrichment analysis of differentially expressed genes.Click here for additional data file.Table S3. Enrichment analysis of differentially expressed proteins.Click here for additional data file.Table S4. Transcriptomic‐informed flux rates for reactions and pathways.Click here for additional data file.Table S5. Proteomic‐informed flux rates for reactions and pathways.Click here for additional data file.Table S6. Common liver cancer drugs and their interactions.Click here for additional data file.


E. coli BL21(DE3) using pET based expression vectors leads to metabolic stress comparable to a carbon overfeeding response. Opposite to original expectations generation of energy as well as catabolic provision of precursor metabolites were excluded as limiting factors for growth and protein production. On the contrary, accumulation of ATP and precursor metabolites revealed their ample formation but insufficient withdrawal as a result of protein production mediated constraints in anabolic pathways. Thus, not limitation but excess of energy and precursor metabolites were identified as being connected to the protein production associated metabolic burden.Recently it was shown that production of recombinant proteins in E. coli BL21(DE3) but also occurs in E. coli K12. Most notably, it was demonstrated that the IPTG-induced production of hFGF-2 using a tac-promoter based expression vector in the E. coli K12 strain TG1 was leading to persistent accumulation of key regulatory molecules such as ATP, fructose-1,6-bisphosphate and pyruvate.Here we show that the protein production associated accumulation of energy and catabolic precursor metabolites is not unique to E. coli K12 strain TG1 using another promoter/vector combination. These findings confirm that energy is not a limiting factor for recombinant protein production. Moreover, the data also show that an accelerated glycolytic pathway flux aggravates the protein production associated “metabolic burden”. Under conditions of compromised anabolic capacities cells are not able to reorganize their metabolic enzyme repertoire as required for reduced carbon processing.Excessive energy generation, respectively, accumulation of ATP during recombinant protein production is not unique to the BL21(DE3)/T7 promoter based expression system but also observed in the  Escherichia coli is frequently associated with an impact on the host cell metabolism known as “metabolic burden”. The extent of the metabolic burden depends on the properties of the recombinant gene, in particular on the properties of the encoded transcript and protein product [Recombinant protein production in  product , the env product , 3.E. coli BL21(DE3)/T7 promoter expression vector combination that the growth inhibitory effect of recombinant protein production is not elevated energy and precursor withdrawal for plasmid encoded functions but, on the opposite, ample catabolic generation and insufficient withdrawal of energy and precursor for anabolic purposes [Until lately it was common understanding that the recombinant protein production associated metabolic burden is the result of redirection of metabolites and energy away from host cell towards plasmid encoded functions. More recently, it was shown for the purposes . The gropurposes , 6. TherE. coli BL21(DE3)/T7 promoter expression vector combination it has been shown that induced recombinant gene expression leads to ATP accumulation which is being further subjected to degradation pathways finally leading to the extracellular accumulation of hypoxanthine [E. coli K12 strain TG1 where it was shown that the ATP level as well as the adenylate energy charge decreased in response to temperature-induced (shift to 42 °C) production [For the xanthine . This haoduction .E. coli K12 strain TG1 using a temperature-independent, IPTG-inducible tac-promoter based expression system. These experiments should provide additional information if excessive energy formation and insufficient energy withdrawal for anabolic purposes could be a more general phenomenon during recombinant protein production that can be generalized to other E. coli strains and expression systems. Thus, we analyze here the metabolic response towards production of hFGF-2 in the E. coli K12 strain TG1 using an IPTG-inducible tac-promoter based expression system with special emphasis on the growth inhibitory effect of hFGF-2 production and the energetic status of the cells. Moreover, we pay special attention to the time course data of carbon overflow metabolites such as acetate and pyruvate as well as to time-course data of glycolytic pathway intermediates after induction of hFGF-2 synthesis. In all studies on the metabolic stress response towards induced production of hFGF-2 utilizing different host strain/expression vector combinations the same codon-optimized gene encoding hFGF-2 was used.To resolve these seemingly contradicting findings concerning the energetic status of producing cells, the production of human basic fibroblast growth factor was carried out in the E. coli K12 strain TG1 using a tac-promoter controlled expression system in batch cultures on defined medium with glucose as carbon substrate. A control cultivation was performed using the E. coli K12 strain TG1 transformed with the parental plasmid not containing a gene insert downstream of the tac-promoter.Production of hFGF-2 synthesis was carried out in the E. coli TG1:pJHLΔ neither inhibition of growth and respiratory activity nor accumulation of pyruvate were detectable after IPTG addition  were followed before and after IPTG-induced production of hFGF-2 Fig. . Prior tE. coli BL21(DE3) using a pET-based expression vector reflects similarities to a carbon overfeeding response and is the result of constraints in anabolic pathways and accompanied by accumulation of ATP and ATP degradation products as well as primary metabolites such as pyruvate [E. coli K12 strain TG1.Recently it was shown that the growth inhibitory effect of IPTG-induced human hFGF-2 production in pyruvate . Those sE. coli BL21(DE3) and the pET-based expression system, we also detect growth inhibition, reduced respiratory activity, enhanced acetate formation and accumulation of pyruvate and ATP in response to IPTG-induced production of hFGF-2 employing the E. coli K12 strain TG1 in combination with tac-promoter controlled recombinant gene expression. These findings clearly show that energy excess during recombinant protein production is not unique for the BL21(DE3)/T7 promoter expression vector combination but also observed in E. coli K12 strains consolidating the hypothesis that the “metabolic burden” of recombinant protein production is not energy limitation caused by elevated energy demand for plasmid encoded functions. Thus, the recombinant gene expression associated “metabolic burden” can be related to restrictions in anabolic functions and not to restrictions in catabolic energy generation. The seemingly contradicting findings of a decrease in the intracellular ATP content as well as in the adenylate energy charge in response to temperature-induced production of hFGF-2 [As observed for IPTG induced hFGF-2 production using f hFGF-2  combinedf hFGF-2 , 9 origiE. coli in response to a glucose pulse [E. coli TG1, the strain employed in this study for the production of hFGF-2, after a carbon-upshift  together with a transient burst in respiratory activity [E. coli is adapted to instantaneously increase respiratory ATP generation under conditions of “sudden and unexpected” carbon supply but can also quickly reduce respiratory energy generation if energy utilization cannot keep pace with ATP formation. As these adaptations occur within seconds and minutes they are certainly controlled through allosteric regulation of metabolic enzyme activity most likely through (an) energetic status sensing molecule(s).An immediate but transient increase in the intracellular ATP content in the time scale of seconds was also observed for non-recombinant se pulse . Moreoveactivity . This caactivity . These dCrr from phosphotransferase system) this way contributing to reduced cAMP formation and to long-term changes in the transcription of CRP-cAMP controlled genes [In addition to the analysis of the respiratory activity and energetic cell status, the time-course data of glycolytic pathway intermediates were determined after IPTG induced production of hFGF-2. These analyses revealed a persistent increase in the level of fructose-1,6-bisphosphate (FBP) and pyruvate as well as a decrease in the phosphoenolpyruvate (PEP) level in response to hFGF-2 production Fig. . These fed genes , 16 incled genes .E. coli TG1 to the new steady state level while the FBP level needed approx. one hour to decrease again and adapt to the new steady state conditions [E. coli TG1, cells were able to reduce ATP formation to reach again a constant adenylate energy charge within one hour but maintained a permanent high FBP level indicative of an elevated flux through the glycolytic pathway  were mainly mediated through metabolites which e.g. decrease the influence of transcription control through CRP-cAMP and increase the activity of ArcA this way changing the transcriptome and finally reconstructing the cellular protein and enzyme repertoire [Metabolites can not only modify fluxes through allosteric modification but also through binding to transcription factors. Thus, metabolic flux control through metabolites can be exerted within seconds through allosteric control of enzyme activity but also long-term through modification of transcription factor activity. For example, the long-term changes in carbon and energy metabolism after IPTG-induced production of hFGF-2 in pertoire . These lpertoire .Thus, the metabolic response towards induced recombinant protein production can be classified into the following consecutive steps: (i) trying to maintain energetic homeostasis by adjusting (respiratory) energy generation through allosteric control and (ii) trying to reorganize the enzyme repertoire through transcriptional control aiming for reduced catabolic carbon processing. In between glycolysis is “self-accelerating” leading to the excessive accumulation of metabolites such as FBP and pyruvate. Under conditions of compromised anabolic capacities, caused for example by excessive recombinant gene transcription, cells might not always be able to reorganize their metabolic enzyme repertoire as required for reduced carbon processing.The situation described above may have some resemblance to the construction of a house: workers are bringing bricks and cement in a certain rate (catabolism) and other workers are constructing the walls (anabolism). If “something” disturbs or slows down the process of wall construction, the situation gets worse if the other workers continue to bring bricks and cement in the same now too fast rate. Solving the problem is possible through different means, for example, by reducing the speed bringing bricks and cement and/or by reducing the disturbing impact.E. coli BL21 (DE3) [Transferring this simplified solution to recombinant protein production can be done, for example, by reducing the carbon feed in controlled fed-batch cultures, as has been shown for 21 (DE3) , 18 as w21 (DE3) . Moreove21 (DE3) –23. InteThe other approach to minimize the metabolic burden is related to the reduction of the “disturbing impact”. Here, most approaches are related to reducing recombinant gene expression, for example by decreasing gene copy numbers, promoter strength or inducer concentration e.g. –26. The E. coli to the non-natural situation of forced recombinant gene expression and/or persistent carbon excess.The cellular response towards recombinant protein production is certainly multifaceted and depends on pre- and post-production conditions as well as on the properties of the recombinant transcript and translated recombinant protein. Recent findings corroborate the hypothesis that the recombinant protein production associated metabolic burden is not the result of limited energy generation but on the opposite an insufficient attempt to reduce catabolic carbon processing because of insufficient carbon withdrawal in anabolic pathways. This response can thus be interpreted as a mal-adaptation of E. coli K12 strain TG1 (DSM6056) was used as host [tac-promoter is described elsewhere [tac-promoter. Cells were grown on a defined medium containing (per liter) 15 g glucose in bioreactor cultures or 10 g glucose in shake flask (pre)-cultures, 8 g (NH4)2SO4, 0.8 g (NH4)2HPO4, 2.7 g KH2PO4, 1 g MgSO4·7H2O, 12 mg Fe(III)citrate, 0.5 mg CoCl2·6H2O, 3 mg MnCl2·4H2O, 0.3 mg CuCl2·2H2O, 0.6 mg H3BO3, 0.5 mg Na2MoO4·2H2O, 1.6 mg Zn(CH3COOH)2·2H2O, 350 mg citric acid·H2O, 1.7 mg EDTA, 4 mg thiamine, and 40 mg ampicillin. Cultivations were carried out in a 2 L bioreactor  at 30 °C, pH 6.6 and a stirring speed of 800 rpm and constant aeration rate of 1.33 vvm. Induction of hFGF-2 synthesis was carried out by the addition of IPTG to a final concentration of 5 mmol L−1 when the biomass concentration reached 3 g L−1. The control strain TG1:pJHLΔ was grown under identical conditions. Cell growth was monitored by measurement of the absorbance at 600 nm (OD600 1 corresponding to approx. 0.5 g/L dry cell mass). Off-gas data  were used to calculate carbon dioxide and oxygen transfer rates as described previously [The  as host . The conlsewhere . The pareviously . For glug, centrifuge 5417, Eppendorf, Hamburg, Germany) and filtration . For measurement of 3-phosphoglycerate (3PG) the pH of the sample was adjusted to pH 3.5–4.0 with KOH. AMP, ADP, and ATP were analyzed by ion-pair HPLC [E. coli determined as 2.15 mL g−1 [Extraction of intracellular metabolites was carried out as described previously , 31 usinair HPLC . All othair HPLC . Since p5 mL g−1 .


Klebsiella pneumoniae (CR-KP) represents an emerging threat to public health. CR-KP infections result in elevated morbidity and mortality. This fact, coupled with their global dissemination and increasingly limited number of therapeutic options, highlights the urgency of novel antimicrobials. Innovative strategies linking genome-wide interrogation with multi-layered metabolic data integration can accelerate the early steps of drug development, particularly target selection. Using the BioCyc ontology, we generated and manually refined a metabolic network for a CR-KP, K. pneumoniae Kp13. Converted into a reaction graph, we conducted topological-based analyses in this network to prioritize pathways exhibiting druggable features and fragile metabolic points likely exploitable to develop novel antimicrobials. Our results point to the aptness of previously recognized pathways, such as lipopolysaccharide and peptidoglycan synthesis, and casts light on the possibility of targeting less explored cellular functions. These functions include the production of lipoate, trehalose, glycine betaine, and flavin, as well as the salvaging of methionine. Energy metabolism pathways emerged as attractive targets in the context of carbapenem exposure, targeted either alone or in conjunction with current therapeutic options. These results prompt further experimental investigation aimed at controlling this highly relevant pathogen.Carbapenem-resistant  Antibiotics are among the most valuable and successful pharmaceutical drug classes. Since the discovery of penicillin in 1928 and the subsequent widespread use of this and other compounds to treat bacterial infections, millions of deaths have been averted globally. As a result, infectious diseases are no longer the leading cause of death in the United States and other industrialized nations as they were at the dawn of the 20th century . NeverthEnterococcus faecium, Staphylococcus aureus, Klebsiella pneumoniae, Acinetobacter baumannii, Pseudomonas aeruginosa, and Enterobacter spp.). These pathogens are of particular concern due to their role in infections of human organs  and their ability to evade the action of commonly used antimicrobials (K. pneumoniae (CR-KP) has attracted attention in recent years due to its rapid expansion, high mortality, and morbidity. This pathogen is already considered an epidemic in some countries  in the last decade . The numK. pneumoniae, Blin et al. studied how the variability of metabolic and genomic traits impacts niche specificity and pathogenic specialization of this bacterium, effectively probing the metabolic diversity of Klebsiella  producer resistant to many antibiotics, was used as a reference during this work. Our group has determined its complete genome (https://www.ncbi.nlm.nih.gov/bioproject/) under accession no. PRJNA78291. To enrich functional annotations and increase the quality of the resulting metabolic network, an annotation pipeline based on the standard operating procedures (SOPs) of the Integrative Services for Genomic Analysis (ISGA)   was perfs (ISGA) . The Gens (ISGA) .K. pneumoniae Kp13 was used as a reference, we also determined orthologs among 39 human pathogenic K. pneumoniae strains (including K. pneumoniae Kp13) with full-length genomes (ie. closed assemblies) available in GenBank/NCBI. This was done to focus only on broadly conserved potential targets, avoiding the ranking of proteins with highly druggable features but present only in few bacteria, such that their epidemiological relevance would need to be established. Mauve (Whilst d. Mauve  was usedK. pneumoniae Kp13 metabolic network (Kp13-GEM), we used the PathoLogic module within Pathway Tools v. 18.0 environment    were alsK. pneumoniae or taxonomically related bacteria, as well as the previously cited pathway resources, were also used to inform these decisions, and more details of this iterative process of refining the metabolic model can be found in In addition to filling pathway holes, we also evaluated and refined incomplete metabolic pathways and decided upon the removal or retention of pathway variants , allowing for their accurate representation. Biological and experimental evidence in v in Kp13-GEM is calculated as st is the number of shortest paths between nodes s and t in the network and Qst(v) is the number of shortest paths from nodes s to t using v as an intermediate. High values of BC for a reaction node reflect the participation of this reaction as a mediator of many other metabolic routes, and its blockage could potentially disrupt the metabolic network by altering multiple pathways.The representation of GEMs as a graph allows the identification of nodes that may be subjected to selective pressures in the metabolic context. We used a reaction graph to represent the Kp13 metabolic network . Once a Choke-point analysis was conducted within Pathway Tools. Choke-point reactions are those that either uniquely produce or consume a given product or substrate, respectively . In thisK. pneumoniae bacteria exposed to carbapenems, we reanalyzed a previously published work that used RNA-seq to determine how sublethal concentrations of imipenem treatment affect K. pneumoniae 2 h and 24 h after carbapenem exposure compared to untreated controls , thus representing potential candidates involved in the early drug response that could be synergistically targeted.To capture the transcriptomic landscape of controls . Specifi package , TMM-norK. pneumoniae bacteria and synergizing with carbapenems. For this purpose, Kp13-GEM and the calculated data were integrated into Target-Pathogen, a web platform allowing multi-omics data integration aimed at prioritizing targets (available in http://target.sbg.qb.fcen.uba.ar/patho/)  . First, /patho/) . After rC is “completeness” . Chk is the number of choke-points in the pathway normalized by the total number of reactions present in each pathway. Cy is the maximum betweenness centrality among all reactions in the pathway, normalized by the reaction with maximum centrality in the entire reaction graph representation. H, the off-target score, reflects the results of a BLASTP search of each protein in each pathway against the human proteome database (Gencode v. 17), scaled using 1 - max, such that when a protein has no hit in the human proteome, the value is 1, and if it has e.g. 2 hits, with identities of 40% and 60%, the score is calculated as 0.4 . In this case, H is the average of the off-target score for each protein in the pathway. Finally, Es is the proportion of essential genes in the pathway.Where K. pneumoniae. Equation (2) expresses this strategy:We also sought to incorporate in our prioritization scheme a set of criteria aimed at identifying candidate targets that could be used in a synergic manner for the control of carbapenem-resistant Cp represents the proportion of proteins overexpressed during carbapenem exposure (detailed in the previous Section).where, in addition to the parameters described for Eq. 1, a term denoted by http://target.sbg.qb.fcen.uba.ar/patho/genome/Kp13>. The metabolic model for K. pneumoniae Kp13 (Kp13-GEM) in SBML format is available in All the data generated and integrated in this study, including metabolic annotations and related meta-data, are openly available at the Target-Pathogen web server interface, and can be reached at <We performed a whole-genome-based reconstruction of the Kp13 metabolic network (Kp13-GEM) using Pathway Tools followed by manual curation. The automatic reconstruction of Kp13-GEM with Pathway Tools resulted in a draft metabolic model composed of 386 metabolic pathways, 2,223 reactions , and 1,714 compounds. A total of 1,554 genes/proteins were assigned to the 1,984 enzymatic reactions that compose the Kp13-GEM. We identified 141 pathway holes (reactions without associated genes) in this draft model, which were distributed across 98 metabolic pathways with different degrees of completeness. Refinement of Kp13-GEM began by manually inspecting the draft GEM using the Pathway Hole Filler (PHF) algorithm. Based on this tool, 41 proteins could be assigned to their respective reaction by filling the holes. During this process, reactions corresponding to 57 metabolic pathways were evaluated, and 26 could be completed . In the last refinement step, using the Autograph methodology and bibliographic evidence, 6 pathways were added, 2 were corrected  and 20 were removed after being identified as false positives during manual curation. The summary of each step of the procedure is found in Klebsiella species. Kp13-GEM presents 386 pathways, 2,175 reactions, 1,997 assigned enzymes, and 1,701 compounds. Compared to E. coli (EcoCyc), the richest manually curated model available to date for any single microbe (K. pneumoniae Kp13 (5.74 Mb) compared to E. coli MG1655 (4.64 Mb), leading to the encoding of more biological functions in the former but also the possibility that false-positive pathways remained in the model even after manual revision. The E. coli model, however, presents a much larger number of transport reactions, reflecting the profound annotation effort put in this model, particularly considering that, in general, automatic genome annotations fail to identify the substrates of most proteins with transport activity , probably as a result of accounting for many specialized metabolic functions displayed by this versatile bacterium, again in line with the high quality of this model’s annotation, which accumulates the equivalent of decades of human effort and literature reconciliation. When compared to models with lower annotation quality in BioCyc, such as Tier 2 and Tier 3, Kp13-GEM presents a higher number of enzymes assigned to reactions, totaling 1,997, while the range for Tier 2 databases was 929-1,111 and of 1,343-1,537 for those in Tier 3 , Tier 2 , and Tier 3 (fully automatic reconstructions).  microbe , Kp13-GEactivity . The numn Tier 3 Table 1.a priori filter to discard proteins for which we could not obtain a representative structural model that harbored at least one druggable pocket. A total of 347 candidate druggable pathways were disclosed using this strategy, representing 89.9% of identified pathways in Kp13-GEM. This massive target space still requires further narrowing, and for this, we took into account essentiality, off-target and metabolic criteria. Once the refinement of Kp13-GEM was finished, we sought to score entire pathways (rather than single proteins) to explore novel molecular targets. We reasoned that when combined therapies are used, targeting more than one protein of the same pathway or yet multiple pathways would lower the chances of resistance development, and once it does, the impact on therapeutic outcome may be decreased . We applA is glycosylated with two 3-deoxy-D-manno-oct-2-ulosonic acid (Kdo) residues, which are added by the WaaA enzyme to produce Kdo2-lipid IVA. We and others have previously identified the potential of lipid A-producing enzymes (expressed from the lpx locus) as attractive molecular targets . Q is a modified nucleoside that is present in certain tRNAs at the wobble position of anticodons in tRNAs for Asn, Asp, Tyr, and His. The de novo biosynthesis of Q is exclusive to eubacteria, and humans have to obtain it from the diet or commensal bacteria , which we also previously identified as an attractive target , is present in both pathways and catalyzes the conversion of octanoate to lipoate. The finding that disruption of lipoylation pathways in Mycoplasma hyopneumoniae leads to bacterial growth inhibition  . Lipoatehibition , coupledhibition , strengtThe S-methyl-5’-thioadenosine (MTA) degradation pathway also appeared top-ranked Table 2.2), the direct precursor of essential cofactors FAD (flavin adenine dinucleotide), and FMN (flavin mononucleotide), plays an important role in reactions involving redox centers, including redox homeostasis, energetic metabolism and protein folding. Six  of the nine enzymes that participate in this metabolic pathway presented a highly druggable score. Of note, this pathway is lacking in mammals, which must obtain riboflavin exogenously. Disruption of the microbial riboflavin biosynthetic pathway, particularly by targeting riboflavin synthase (RibE) and lumazine synthase , has been achieved using chemically produced inhibitors of these enzymes, which were also developed as prospective anti-tuberculosis agents . The reaction that follows, which we identified as a choke-point, is catalyzed by betaine aldehyde dehydrogenase , converting betaine aldehyde into glycine betaine. In Pseudomonas aeruginosa, deletion of betBA resulted in decreased bacterial survival in a mouse lung model, although the mechanism generating the mutant survival defect was not explored , which is involved in the biosynthesis of phosphatidylethanolamine, the major phospholipid of membranes; thymidylate kinase , an enzyme that participates in pyrimidine deoxyribonucleotide phosphorylation; and the gene products of murF (DS = 0.76), murG (DS = 0.8), murD (DS = 0.71) and murE (DS = 0.85), which participate in peptidoglycan biosynthesis, were rediscovered in the current work, and their potential is discussed elsewhere , a key enzyme of the prokaryotic TCA cycle Figure 3exposure , making exposure . InteresK. pneumoniae exposure to carbapenems. Both pathways harbor druggable and essential proteins that are also associated with reactions with elevated betweenness centrality in Kp13-GEM. For example, isocitrate lyase (AceA), a highly druggable (DS = 0.92) and essential protein, which was validated as an antituberculosis target , we prioritized pathways harboring attractive properties from a drug development perspective. Members of these pathways could be developed as candidates in further experimental studies aimed at identifying novel compounds to control K. pneumoniae, in particular during exposure to carbapenems. Other uses of Kp13-GEM include topology-based analyses on the network by inducing a reaction graph from the SBML-formatted model, as we have performed. Then, network theory methods can be applied, and topological metrics can be calculated to study relevant aspects of the K. pneumoniae metabolism, compared to those of other bacteria. Our integrative approach, where multiple layers of omics information were overlaid, disclosed a series of candidate pathways to inform target selection in antibiotic drug discovery, with energy metabolism emerging as a dominant biological theme warranting further consideration in the molecular target space of K. pneumoniae.By generating and making extensive use of a refined genome-scale metabolic network for carbapenem-resistant The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/DF, MN, and PR conceived the study design. AP, AT, DF, ES, FS, MN, MP, and PR contributed tools and performed data analysis. AP, DF, MN, MP, and PR drafted the manuscript with input from the other authors. All authors contributed to the article and approved the submitted version.MN was supported by a fellowship from CNPq (process no. 306894/2019-0) and grant by CAPES (process no. 88887.368759/2019-00). CONICET supported fellowships to AP and FS. ES, AT, MP, and DF are members of CONICET. This work was supported by Agencia Nacional de Promoción Científica y Tecnológica  and Universidad de Buenos Aires (20020190200275BA to DF).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.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Neurospora crassa and the flow of material through the three competing pathways of upper glycolysis, the non-oxidative pentose phosphate pathway, and the oxidative pentose phosphate pathway are evaluated as a function of the NADP/NADPH ratio. It is predicted that regulation of phosphofructokinase (PFK) and flow through the pentose phosphate pathway are essential for preventing an extreme level of fructose 1,6-bisphophate accumulation. Such an extreme level of fructose 1,6-bisphophate would otherwise result in a glassy cytoplasm with limited diffusion, dramatically decreasing the entropy and energy production rate and, consequently, biological competitiveness.We report the application of a recently proposed approach for modeling biological systems using a maximum entropy production rate principle in lieu of having in vivo rate constants. The method is applied in four steps: (1) a new ordinary differential equation (ODE) based optimization approach based on Marcelin’s 1910 mass action equation is used to obtain the maximum entropy distribution; (2) the predicted metabolite concentrations are compared to those generally expected from experiments using a loss function from which post-translational regulation of enzymes is inferred; (3) the system is re-optimized with the inferred regulation from which rate constants are determined from the metabolite concentrations and reaction fluxes; and finally (4) a full ODE-based, mass action simulation with rate parameters and allosteric regulation is obtained. From the last step, the power characteristics and resistance of each reaction can be determined. The method is applied to the central metabolism of  While there have been great successes on the atomistic level that have led to the development of multiscale modeling methods that address phenomena from the femtosecond timescale to the microsecond time scale -3, prediMany rate parameters have been measured for a few model organisms, but even for those organisms the rate parameters are determined in vitro and do not reflect the in vivo environment. A general solution to this challenge was proposed in 1985 by E. T. Jaynes : “to preEntropy and maximum entropy are often confusing topics, so it is useful to first define the terms used herein and explicitly state what is meant by maximum entropy. Fundamentally, entropy itself is a measure of probability while maximum entropy is simply a characterization or description of a distribution. An increase in entropy is an increase in probability, but to be clear, one should always explicitly specify the probability distribution being considered.J and K for instance, as a function of the system probability densities, ΔS = − log(Pr(J)/Pr(K)) where Pr(J) and Pr(K) are probability density functions based on, for instance, the multinomial Boltzmann distribution [Entropy production should not be confused with entropy change, although both are associated with changes of state . An entrribution . In bothi, iK is the chemical equilibrium constant and iQ is the reaction quotient. Finding the optimal path for a system is a general problem [Entropy production, on the other hand, is path-dependent. A maximum entropy production path is the thermodynamically optimal or most probable path. The probability density in the case of mass action chemical systems is the multinomial Boltzmann distribution. A change of state due to following a particular path is related to the thermodynamics odds of each respective reaction  problem , but is  problem ,9.  cycle found in nature using stochastic kinetics  and haveNeurospora crassa, a model organism for studying the multi-scale dynamics of circadian rhythms. We find that the pentose phosphate pathway can act in a cyclical manner to produce six NADPH for every glucose consumed instead of the generally expected two NADPH per glucose consumed. Together with recent findings that the oxidative pentose phosphate pathway and upper glycolysis are 180 degrees out of phase in the circadian cycle [In this report, we show that the most thermodynamically probable concentrations can be predicted using a modified version of Marcelin’s 1910 equation describing mass action dynamics using reaction affinities . The firan cycle , suggest2.i ∈ {A, B, C, D} and unsigned stoichiometric coefficients i,αv for each molecular species i in reaction α ∈ {1, −1} with forward reaction α = 1 and the reverse reaction α = −1,in is the count or concentration of species i. The extent of each of the reactions is given by ζ1 and ζ−1, such that when ζ1 = 0, the system is in the state where neither of the An and Bn reactants have turned into products. When ζ1 = 1, the system is in the state where a stoichiometric amount of the An and Bn reactants have turned into products such that there are now Cn + vC,1 and Dn + vD,1 products and An − vA,1 and Bn − vB,1 reactants. The net extent of each reversible reaction is ζ1 − ζ−1 ≡ ζnet1, and ζ−1 − ζ1 ≡ ζnet−1,. Consequently, k1 and k−1 are the rate constants of the forward and the reverse reaction, respectively. Using the signed stoichiometric coefficients γi,1 such that γi,1 = −|vi,1| for reactants and γi,1 =|vi,1| for products, the time-dependence of any molecular species i is,Consider a reversible chemical reaction with molecular species Although kinetics and thermodynamics are alternate formulations of the law of mass action, eA1(in)/RT, which is the exponential of the thermodynamic driving force A1(in) on reaction 1, and a purely kinetic component, eA1(in)/RT is the odds ratio of the forward reaction to the reverse reaction. The odds are reciprocally related such that eA1/RT = eA−1/RT−. The odds of reaction 1 then change on a time scale determined by A1(in) > 0) will be associated with slower changes in the odds of the forward reaction than the odds of the opposing reaction since the respective relaxation times of the odds are inversely related through the relation,Z reactions, the time-dependence of chemical species i is given by,Generalizing to a large system consisting of αc is a constant,A convenient thermodynamic optimization procedure can be obtained using the Marcelin formulation of α such that cα = c removes any kinetic bottlenecks in phase space of the system such that the relative dynamics are governed only by the thermodynamics,That is, the Marcelin Equation sets each of the relaxation rates of the forward and reverse forces to the same rate. Assuming the same relaxation rate for each respective force for all reactions α is given by,A simulation using S of M metabolites × Z reactions and a vector of the net reaction fluxes Z, the usual steady state condition applies,Accordingly, at the thermodynamically-optimal steady state, the net flux through a reaction is proportional to the net thermodynamic odds. The thermodynamically-optimal steady state is one of many possible kinetic steady states, but its dynamics are such that the state is both kinetically and thermodynamically stable. That is, given a stoichiometric matrix αK is the equilibrium constant and αQ is the reaction quotient for reaction α. Given Z by Z diagonal matrices of forward and reverse equilibrium constants, K+ and K−, and diagonal matrices of the respective forward and reverse reaction quotients, Q+ and Q−, the thermodynamically-optimal steady state is the product of the stoichiometric matrix and the net thermodynamic odds of each reaction,The thermodynamic odds of a reaction in c in α, as will be shown below.That is, the system is thermodynamically stable as well as kinetically stable. Furthermore, since the parameter i, in and reducing the difference between predicted and observed values by optimizing a loss function for each reaction α. As a loss function, we have chosen the log ratio of the observed values to the predicted values of the M(α) reaction products of reaction α,It is possible that the thermodynamically-optimal steady state may not have metabolite levels at physiologically realistic values. For example, inference of a steady state model of central metabolism will predict concentrations of fructose 1,6-bisphosphate in excess of 1 M (see below). If the cell were to produce these large concentrations, the cytoplasm would become glassy and practically no diffusion would occur. Hence the entropy production rate would actually approach zero. The issue is that the model assumes that diffusion will remain sufficient regardless of solute concentrations. This is easily remedied without including diffusion in the model by comparing predicted concentrations or counts of metabolite αL = 0, and when the predicted values are greater or less than the observed values αL > 0 or αL < 0, respectively. Unfortunately, experimental measurements of metabolite concentrations are hard to obtain; even when available, metabolomics data sets are sparse. Part of the reason is that the chemical properties of small molecules vary widely, so no single experimental design can discriminate and measure each molecular species. Sparsity is also due to the fact that biologically relevant metabolite concentrations may span several orders of magnitude, which is often greater than typical instrument dynamic ranges. Instead, one can simply use rule-of-thumb estimates for metabolite concentrations in place of experimentally observed values. Based on mass spectrometry estimates of absolute concentrations of metabolites [αL > 0 may indicate that the system needs to be modulated or regulated such that the metabolite concentrations stay within a physiological range. The desired regulation can be implemented using any appropriate function, such as a Hill equation, a logistic function or a hyperbolic function. Parameters for the regulation functions are then estimated and the simulations and analysis are repeated, adjusting parameters each time, until a physiological level of metabolite concentrations is achieved. This resulting steady state is thermodynamically optimal conditioned on the required regulation.When the observed and predicted values agree, abolites , a reasoαλ of reaction α, the rate constants are inferred for each reaction as follows. The net flux αλ is the activity of the enzyme catalyzing reaction α as a function of the regulation. For example, solving for the rate constants of reaction α = 1 from λ1 = 1 (no regulation) in From this steady state with reasonable metabolite concentrations and activity when we make inferences based on incomplete information, we should draw them from that probability distribution that has the maximum entropy permitted by the information we do have [Finally, for the readers that are interested in such details, the inference of rate parameters in this manner meets the criteria that Jaynes laid out for MAXENT:  do have . Here, w do have .For complex systems such as biological systems, the power is the change in free energy with respect to time,α,netζ with time such that the rate αA and the extent of the reaction αζ,For a reaction, the rate dependence is due to the net change in the extent of a reaction αR and conductance αC of reaction α can be calculated at steady state as,αG is the free energy change across the reaction, which is equal in value to the reaction affinity αA for a system at steady state with a large number of particles. These latter equations should not be taken to imply that there is a linear relationship between flux and a change in free energy (or flux and resistance). A reaction at steady state has high resistance if the change in free energy is large for the steady state flux relative to other reactions in the pathway. An example of this will be discussed in the This relationship is useful for comparing different chemical processes, as will be seen below. Likewise, the resistance The equations described above for both the optimization and simulation were implemented in the language C in a software program called Boltzmann, which is available as open source code under a Berkeley Software Distribution (BSD) style license . The ODEϵ, ionic strength I, and pH of the cell cytoplasm, assumed to be ϵ = 0.78, I = 0.25, and pH = 7.0 [Chemical potentials were obtained from component contribution methods ,30 and apH = 7.0 .Neurospora crassa metabolic model originally from Dreyfuss et al. [The compartmentalized s et al.  was updas et al. . The mod2 are set to non-equilibrium values of 2 mM and 0.1 mM, respectively. Likewise, the cofactors CoA, ATP, ADP, orthophosphate, NAD, NADH, NADP and NADPH are also fixed boundary species; these concentrations were taken from a mass spectrometry analysis of absolute metabolite concentrations from Bennett et al. [In order to maintain a non-equilibrium state, boundary concentrations for the initial reactant glucose 6-phosphate and final product COt et al.  The redot et al. .Analysis of simulation data is included in supplementary information/computational notebooks using Python.3.Neurospora crassa [A metabolic model of a crassa  was usedA map of the flux through the system is shown in i in an aqueous solution with dielectric constant ϵ, ionic strength I, and constant pH. The values of ϵ, I, pH are chosen to match those in the cell environment, which in this case are assumed to be ϵ = 0.15, I = 78 and pH = 7.0. The non-equilibrium boundary conditions for the initial reactant glucose and final product CO2, and with those for the cofactors CoA, ATP, ADP, orthophosphate, NAD, NADH, NADP and NADPH prevent the intermediates of glycolysis and the TCA cycle from reaching equilibrium.The maximum entropy production optimization results in metabolite levels tending towards their most probable (Boltzmann) distribution such that,−22M for mitochondrial oxaloacetate at the low end to 106M for fructose 1,6-bisphosphate and 109M for mitochondrial acetyl CoA at the high end. While a concentration of 10−22M might be reasonable in a cell , concentrations above 10−3M to 10−2M are usually not physiologically realistic [−3M, the cytoplasm could become so glassy that diffusion of even small molecules would decrease below the level at which metabolism could operate.The resulting maximum entropy production rate concentrations are shown in ealistic . For insHowever, the issue is not whether the maximum entropy production rate formulation is appropriate for biological systems, but rather that any model is always a reduced representation of a real system . In a co−3M. The results are shown in This discrepancy between predicted metabolite concentrations and physiological expectations can be taken advantage of, however, to infer points of post-translational enzyme regulation. To do this, a loss function is formulated , similarNeurospora crassa, PFK in Neurospora crassa is inhibited by high concentrations of ATP [The loss function does not help to identify how these reactions/enzymes might be regulated. In experimental studies of s of ATP . Early ss of ATP ,38. We c−3M) and reanalysis of the loss function for each reaction finds that only the succinyl-CoA synthetase reaction resulting in the production of succinate has a loss-function value, L = 3.8 above the expected value of 0.0. Given that succinate is highly soluble, however, this mild increase of the loss function is reasonable. The predicted concentration of succinate is 3.4 mM. For comparison, the concentration of succinate in exponentially growing E. coli under similar conditions was estimated to be 0.57 mM [After regulation of PFK and PDHm are applied, the maximum entropy optimization produces the steady state concentrations shown in  0.57 mM .In addition to changes in the metabolite concentrations, application of regulation to PFK and PDHm results in reduced flux through both glycolysis and the TCA cycle, as expected. In this case, the flux is reduced by just over 2-fold (6.53/2.86).Yarrowia lipolytica [With reasonable steady-state metabolite concentrations in hand, rate constants can be inferred from the simulation using polytica , which hpolytica ,40-42.From the predicted flux values and reaction-free energies, the power characteristics and resistance of each reaction can be estimated. Shown in E. coli in which NADP/NADPH = 2.1 · 10−6/1.2 · 10−4 = 0.0175 [E. coli study such that NADP/NADPH = 2.6 · 10−3/8.3 · 10−5 = 31, resulting in a driving force of ~−31.2 KJ/mol on these same dehydrogenase reactions.Rather than simply not extracting this available energy from glucose at a high rate due to already high levels of ATP, it is reasonable to expect that the energy from glucose is utilized elsewhere. Proteomics studies on Neurospora show that the enzymes of upper glycolysis and the oxidative pentose phosphate pathway oscillate with the circadian cycle but are 180° out of phase with each other . Consequ= 0.0175 . This raAs expected, the flow of material increases through the pentose phosphate pathway as the NADP/NADPH ratio increases, as shown in 4.The maximum entropy principle can be used to predict concentrations, obtain optimal rate constants, and calculate the characteristics of biological circuits including energy, power, flux and resistance for individual reactions and entire pathways. In the model of central metabolism used here, enzyme kinetics are represented as the summary reaction of the catalytic process. Dobovišek et al. have shown that the maximum entropy production principle can be used to evaluate enzyme kinetics, as well ,46. In tThe calculated rate constants are optimal for the growth conditions used in the simulation. Similarly, organisms will have rate constants that are optimal for the conditions to which they have adapted. Ideally, to compare predicted metabolite concentrations to those from a wet lab experiment, the organism should be cultured over many generations to ensure adaptation to the laboratory conditions.One could calculate experimentally based in vivo rate constants for many reactions, as well, following the same procedure outlined above but using experimentally measured concentrations of metabolites free (unbound) in the cytoplasm . But thiPark et al. have recently argued that one can calculate free energies of reaction from the respective whole cell concentrations because most cellular metabolites are in free form . The argUsing measured whole-cell concentrations of metabolites as the upper bound for metabolites free in solution can also be employed to infer regulation rather than using a rule-of-thumb , as was done here. The experimentally determined concentrations would give more precise estimates of regulation, although the rule-of-thumb values worked well in this study for demonstration purposes.It is assumed that any predicted concentrations far away from the data are due to the incompleteness of the model and not the maximum entropy production principle. In the cases studied here, it is clear that the predicted high levels of some metabolites would reduce the ability of molecules to diffuse in the cell. In cases such as this, regulation may be inferred by comparison to experimental data or expectations. But it can also be expected that branching reactions for metabolites that are not included in a model may impact the predictions. Branching per se at the point where the metabolite is an intermediate does not necessarily impact metabolite concentrations because the metabolite level is a function of the inward and outward flux producing and consuming the metabolite. The total inward and outward flux may not change when there is a branching reaction if the boundary conditions have not changed; in this case, the output flux splits but the total outward flux may be conserved.i) may be used in many reactions, the impact on flux can be multiplicative.Likewise, changing the boundary conditions would change the overall flux through all pathways. The ability to completely characterize each individual reaction with regard to flux, energy dissipation, power and resistance can also lead to a more complete characterization of biological circuits, as is done for electrical circuits, and even allow for the use of sophisticated control theory analyses of the operations of a cell. Such developments could be extremely useful for synthetic biology, in that the addition to a cell of an engineered circuit often results in decreased growth and other unintended consequences. Being able to fully characterize the impact of the new circuit before implementing it in the laboratory could dramatically change the design process and success rate.Yarrowia lipolytica, Wasylenko et al. used 13C-metabolic flux analysis (MFA) in a control strain and an engineered strain to understand where NADPH was primarily being produced, which turned out to be the oxidative pentose phosphate pathway [For example, in an effort to increase the production of fatty acids in  pathway . MFA is Neurospora crassa.Finally, there is a need for faster optimization methods to reach the steady state. While optimization using αc is reaction-independent allows one to find the maximum entropy production distribution at a non-equilibrium steady state. However, one concern is that this would imply that all reactions in the eventual simulation would occur on the same timescale. If all the processes are represented at the same level, for instance that of elementary reactions, then the inferred rates will likely be sufficiently representative of the true processes. However, if one mixes elementary reactions and summary reactions (that represent larger processes) in the model, then it could be the case that there is not a realistic separation of time scales. Summary reactions representing larger processes, for instance, could be rescaled by αnc instead of αc, where αc represents the average transit time through a reaction at the faster scale and n is the number of elementary reactions represented by the summary reaction. Multi-scale simulations can be used in which fast and slow processes are run independently based on the assumption that the fast processes reach steady state quickly on the time scale of the slow processes. If the processes to be represented are scaled appropriately, for instance by αnc, the processes do not need to be run independently but can be run together using computational singular perturbation methods [Scaling up to a genome-scale model will likely bring new challenges. The time scale of metabolism spans the range from milliseconds for individual reactions to on the order of an hour for secondary metabolites to reach steady state. The assumption in  methods  or other methods . FurtherNotebookS1NotebookS2NotebookS3TableS1TableS2