Malfunctioning of this cooperation may lead to many complex diseases, such as cancer and type 2 diabetes. Timescale differences among these pathways make their integration a daunting task. Metabolic, signaling and gene regulatory networks have three different timescales, such as, ultrafast, fast and slow respectively.in silico/in vivo/in vitro experiments have validated the results (the regulations of glucose transporter 1 (glut1), hexokinase (HK), and hypoxia-inducible factor-1The article deals with this problem by developing a support vector regression (SVR) based three timescale model with the application of genetic algorithm based nonlinear controller. The proposed model can successfully capture the nonlinear transient dynamics and regulations of such integrated biochemical pathway under consideration. Besides, the model is quite capable of predicting the effects of certain drug targets for many types of complex diseases. Here, energy and cell proliferation management of mammalian cancer cells have been explored and analyzed with the help of the proposed novel approach. Previous investigations including  Broadly speaking, biochemical pathways/networks can be categorized as metabolic pathways, gene regulatory networks (GRNs), and signaling pathways. A metabolic pathway is a coherent set of biochemical reactions catalyzed by a number of enzymes. It helps a living organism to transform an initial (source) compound into a final (target) compound and energy. On the other hand, fundamental information processing and control mechanisms in a cell are performed by GRNs. Regulatory genes code for proteins that activate or inhibit the expression of other genes. Thus, a complex web of interactions, called a GRN, in terms of activation and inhibition of genes, is formed. Moreover, signaling pathways contain a series of specific actions in a cell in which a signal is passed from the environment and accordingly the cell responds. These pathways are of diverse nature and are interacting with one another. They also form a hierarchy. For example, an entire organism can be thought of as a huge network of interacting organs, each of which, in turn, is a network of interacting tissues. A tissue is a collection of interacting cells performing similar functions. A cell may be considered as a huge network of interacting components to constitute aforesaid biochemical networks. Moreover, the influence of environment and other factors on enzymes or gene regulation needs to be considered to make a study on how an organism is responsive to environmental changes. Therefore, it is necessary to focus not only on individual processes or pathways but also on their integration. Among several approaches one of the most common is to explore metabolic pathway is Flux Balance Analysis (FBA) Escherichia coli take seconds to minutes to express. It will be longer in the case of mammalian systems. On the other hand, metabolite concentrations can respond in seconds to microseconds Response time of the interconnected complex subsystems may vary from each other. In other words, they can be treated as complex subsystems of different timescales. Biochemical pathways, such as metabolic, signaling and regulatory networks show this kind of phenomena. Timescale differences among metabolic, signaling and gene regulatory networks make their integration a daunting task. Several researchers have tried to model the dynamic behavior of a system by stoichiometric reconstruction of integrated biochemical pathways In this study, we develop a three timescale multiple input and multiple output (MIMO) model based on support vector regression (SVR) and genetic algorithm based controller to simulate the dynamic behavior of integrated signaling, metabolic and gene regulatory networks specifically responsible for mammalian carbon metabolism in normal and cancer cells. Here, the metabolic, signaling and GRN have been mathematically modeled by sets of ODEs with proper three timescale selection, feedback, allosteric effects and perturbation. In this context, we have collected all pathway information under consideration from KEGG database The central carbon metabolic (CCM) network comprises glycolysis, tricarboxylic acid (TCA) cycle, pentose phosphate pathway (PPP) and free fatty acid (FFA) metabolism. Glycolysis consumes glucose to produce pyruvate and energy in the form adenosine triphospate (ATP), although most of the energy is generated from TCA cycle taking pyruvate as input. Here, PPP is involved in macromolecular synthesis including reduced nicotinamide adenine dinucleotide phosphate (NADPH) and ribose 5P. FFA is formed by acetyl CoA of TCA cycle. However, a mammalian tumor cell slows down oxidative phosphorylation by inhibiting TCA cycle activities and produces abnormal amount of lactate. This effect is called “Warburg effect” in vivo and in vitro experiments In accordance with previous investigations including some We have also compared the proposed model with the following theoretical models. In this context, FBA based model Here the proposed model is able to predict that deactivation of glucose-6-phosphate dehydrogenase and ribose 5P isomerase may slow down growth and proliferation of cancer cells. However, glucose-6-phosphate dehydrogenase deactivation may not be able to reduce fermentation and energy supply in malignant cells. On the other hand, though ribose 5P isomerase deactivation may inhibit cell fermentation, it may not get success to stop energy supply in mammalian cancer cells. In this context, deactivation of transketolase and glucose-6-phosphate isomerase may be considered as potential drug targets to resist cell growth, fermentation and proliferation as well as energy supply in human cancer cells. Activation of pyruvate kinase (M2 isoform) may also reduce cancer progression in terms of cell growth, proliferation and fermentation. Besides, it may fail to decrease ATP production in tumor cells. However, deactivation of pyruvate kinase may be a poor choice as a drug target for cancer therapy.In summary, the proposed novel model can tackle three timescale nature of any integrated biochemical pathway comprising metabolic, signaling and gene regulatory networks. Subsequently, it can capture the nonlinear transient dynamics of the integrated network of both normal and perturbed human cells. This has been validated appropriately by some other methods/results 2Here, we describe the proposed methodology for developing the model that will mimic the behavior of an integrated pathway system. The methodology involves six steps. At first, timescales of metabolic, signaling and gene regulatory networks have been selected, and the basic dynamics of the integrated pathway system has been described. Then, we have formulated the state equations for the integrated pathway. In the third step, an appropriate MIMO plant has been developed by tuning its parameters. Subsequently, training dataset has been generated from the appropriately perturbed MIMO plant/system. Thereafter, SVR model is developed to approximate the MIMO plant. Finally, GA controller is applied on the plant as a model predictive controller to capture nonlinear transient cellular dynamics. 2.1In an integrated biochemical pathway system, involving metabolic, signaling and gene regulatory networks, the rate of a metabolic reaction is controlled by an enzyme/protein, which is originated from an expressed gene participating in a gene regulatory network. The expression of this gene is in turn regulated by one or more transcription factors (proteins) which are expressed/activated by signaling molecules in a cascade of signal transduction pathway.c external inputs c dimensional vectors, respectively, for the pathway system.Let Thus the proposed nonlinear state space model for the integrated pathway can be defined ast) derivatives of Escherichia coli take minutes to express. It will be longer in the case of mammalian systems. However, metabolite concentrations can alter in seconds Here We now obtain the fast  randomly . MoreoveIn this context, we have checked whether trial and error based parameter estimation resembles other previous method. For this purpose, we have compared the values of 35 kinetic parameters considered here with those estimated by a method based on hybrid extended Kalman filtering Now, we perturb the normal MIMO plant by knocking out genes  to capture the nonlinear dynamics associated with a particular biological phenomenon  of Thus, the number of inputs to SVR model becomes 2.5From control theoretic point of view, it is difficult to apply online optimization on actual nonlinear MIMO plant with small sampling periods Now, we are going to discuss how a MIMO SVR model can be designed using support vector regression d. For an F of dimension An approximate MIMO SVR model has been developed by combining multiple input single output (MISO) SVR models for each F corresponding to F. On the other hand, d dimensional input space to F. The bias term is represented by F corresponding to Here Based on Vapnik’s Here, the upper limit of tolerable error is a very small positive real quantity Based on the notion of Lagrange multipliers and Karush Kuhn Tucker (KKT) condition Here the term The summation has been carried out over all the support vectors Here, we have experimentally chosen radial basis kernel function with kernel coefficient as 10, regularization parameter as 100 and 2.6In this study, we have applied a GA i.e., the MIMO plant) accepts the controlled input to compute actual output. The GA controller has been developed using two algorithms (•C chromosomes (set of possible solutions) randomly, forming a population. However, it creates chromosomes around i.e., multi-point crossover, to reduce the mutation bias. These steps are carried out for G generations until the fittest chromosome, whose fitness value is greater than a predefined threshold value, has been evolved as an optimal solution •1-norm of the difference between Such a GA based control mechanism has been depicted in gorithms . SupplemFinally, it should be mentioned that the average value of G is 10 for convergence of Algorithm S1, whereas 30 iterations on average are needed for Algorithm S2 to converge.3This section has been divided into three subsections. Firstly, we are going to discuss about MIMO plant validation through analysis of both normal and perturbed behavior of the integrated biochemical pathways under consideration. Secondly, the altered behavior of the integrated biochemical pathways in mammalian cancer cells compared to normal ones will be discussed. Finally, the third subsection deals with the description of the effects of six possible drug targets in mammalian cancer cells.3.1Here, we have monitored the concentrations of key molecules during simulation of the proposed three timescale state space model for normal as well as perturbed integrated biochemical pathways related to carbon metabolism. In normal scenario , both glWe have further compared the model with a previous simulation result In comparison with some other mathematical models in analyzing energy management in mammalian cancer cells, we have found that a constraint-based flux balance model in vitro observations from cultured human sertoli cells during insulin deprivation For more validation of the proposed model, we have compared our simulation results with viz., pyruvate dehydrogenase, pyruvate carboxylase, acyl-CoA synthetase, fatty acid synthase, phosphoenolpyruvate carboxykinase 1 and succinyl CoA synthetase, to slow down oxidative phosphorylation through mitochondria leading to “Warburg effect” In the context of present investigation, we have finally simulated the situation of knocking out certain enzymes, Nevertheless, mammalian cancer cells somehow manage both energy, in the form of ATP, and cell proliferation to survive 3.2Here, we have considered the integrated CCM pathway with incorporation of possible mutation responsible for energy management in cancer cells. The necessary pathway and information have been obtained from KEGG database e higher B due to e higher , AKT  and glut1 D and F increases C. Conseqncreases B as wellncreases C. In thincreases A in B7-Hncreases B. This encreases A in Dppancreases B has beencreases A in B7-Hin vivo/in vitro result in vivo/in vitro experiment, performed by Sumi et al. Higher expression levels of phosphofructokinase 1 (PFK1) , phosphoAccording to the simulation results, the enzymes (proteins) catalyzing PPP, such as glucose-6-phosphate dehydrogenase , phospho3.3in vivo and/or in vitro experiments. When the proposed model computationally meets the reference target concentrations of ATP and ribose 5P for GA controller, we have observed that the model has mimicked the altered regulation of cancer cells as previously discussed. At that moment, we have set reference target expression levels (high or low) of certain proteins/enzymes as drug targets using GA controller. Although the model is quite capable of analyzing the effect of any drug target, we have considered only six drug targets among others for our study to restrict the size of present article. Here, we have monitored the effects of deactivating pyruvate kinase (reference expression level 0.02), glucose-6-phosphate dehydrogenase (reference expression level 0.02), transketolase (reference expression level 0.09), ribose 5P isomerase (reference expression level 0.09) and glucose-6-phosphate isomerase (reference expression level 0.03). Finally, the effect of activating pyruvate kinase (reference expression level 0.98) has also been observed and analyzed. •Case 1 (Pyruvate kinase deactivation): According to the simulation results as depicted in i.e., high consumption and less production) (duction) B in mammduction) D cannot •Case 2 (Glucose-6-phosphate dehydrogenase deactivation): Glucose-6-phosphate dehydrogenase is the key enzyme (protein) to utilize G6P through PPP. It leads to producing high amount of NADPH and ribose 5P so that nucleotides and fatty acid can sufficiently be synthesized to maintain intracellular redox homeostasis in vivo/in vitro study has shown how the relative activity (expression level) of glucose-6-phosphate dehydrogenase enzyme controls the NADPH production •Case 3 (Transketolase deactivation): Transketolase has similar importance as glucose 6-phosphate dehydrogenase to maintain cell growth and proliferation leading to metastasis •Case 4 (Ribose 5P isomerase deactivation): We have already discussed that up regulation of ribose 5P isomerase plays an important role in cell growth and proliferation of a cancer patient. A previous study i.e., lactate production as depicted in •Case 5 (Glucose-6-phosphate isomerase deactivation): Glucose-6-phosphate isomerase or phosphoglucose isomerase (PGI) is a glycolytic enzyme that directs G6P flux into the glycolysis branch. As a result, G6P breaks down into fructose 6P. Deactivation of Glucose-6-phosphate isomerase results in slowing down of glucose utilization (lization B as welllization C and ATPlization A productlization D may slolization C and riblization A concent•Case 6 (Pyruvate kinase activation): Activation of pyruvate kinase (e kinase  may slowe kinase C and ribe kinase A confirme kinase B and glue kinase D supporte kinase D also inHere, we are going to discuss the effects of certain rational drug targets on cancer cells in terms of management of energy and cell proliferation from simulation point of view. These results may have significant implications during 4viz., metabolic, signaling and gene regulatory networks, keeping their three timescale nature intact. Here, we have developed the integrated state equations considering appropriate timescales as well as all possible perturbations present in the contemplated integrated biochemical pathway. Besides, depending on the training dataset generated by solving the pathway ODEs, SVR based MIMO model has been developed. The MIMO model can mimic the transient nonlinear dynamic behavior of the integrated biochemical pathway under consideration. Moreover, with the help of the GA controller, the model can predict the effect of drug targets applied to complex diseased cells. In order to investigate the effectiveness of the model, we have used our model to explore how mammalian cancer cells are able to manage their growth, proliferation and energy supply to survive. In this context, “Warburg effect” In this study, we have successfully integrated three types of biochemical pathways, viz., inactive dimer and active tetramer, plays an important role in managing proliferation, growth and energy in mammalian cancer cells. These results have been validated through previous investigations involving in vivo and in vitro experiments According to the results, the proteins or genes HIF-1in vivo/in vitro parameter values are needed, and/or appropriate parameter estimation methods based on the theory of machine learning and control theory can be developed for designing more accurate MIMO plant for an integrated biochemical pathway.Finally, it may be mentioned that the experimental values of different kinetic parameters are still unknown or poorly documented. In this study, we have estimated these values by trial and error based on previous knowledgebase. Some of the parameter values considered here have been seen to be close enough with those estimated by the method of Lillacci et al. AD and RKD initiated control theoretic modeling of metabolic pathways. AD conceptualized the basic idea of three timescale modeling. AD and AB formulated the methodology and implemented it. RKD gave crucial theoretical input. AD and AB wrote the first draft of the manuscript. RKD corrected it. NC read the article and gave fruitful suggestions to edit the manuscript.No funding agency has funded this work.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Using genomic and structural information from SARS-CoV-2, we created a biomass function capturing its amino and nucleic acid requirements and incorporated this into a metabolic model of the human lung cell to predict metabolic perturbations that inhibit virus reproduction. Viruses rely on their host for reproduction. Here, we made use of genomic and structural information to create a biomass function capturing the amino and nucleic acid requirements of SARS-CoV-2. Incorporating this biomass function into a stoichiometric metabolic model of the human lung cell and applying metabolic flux balance analysis, we identified host-based metabolic perturbations inhibiting SARS-CoV-2 reproduction. Our results highlight reactions in the central metabolism, as well as amino acid and nucleotide biosynthesis pathways. By incorporating host cellular maintenance into the model based on available protein expression data from human lung cells, we find that only few of these metabolic perturbations are able to selectively inhibit virus reproduction. Some of the catalysing enzymes of such reactions have demonstrated interactions with existing drugs, which can be used for experimental testing of the presented predictions using gene knockouts and RNA interference techniques. In summary, the developed computational approach offers a platform for rapid, experimentally testable generation of drug predictions against existing and emerging viruses based on their biomass requirements. One of the crucial steps in the virus life cycle is the synthesis of the virus particles within the host cell. This includes the synthesis of viral structural proteins and new genomic material. For these processes, all viruses are fully dependent on their host for the required energy  and builThe observation of virus synthesis dominating the metabolism and physiology of infected cells suggests that it might be possible to manipulate cell metabolism to control the viral infection . Indeed,These experimental findings show that viral biomass synthesis causes significant metabolic flux changes in host cells and that metabolic perturbations can directly alter virus reproduction. Thus, system-level metabolic models could be used to predict what types of metabolic alterations can cause what kinds of impact on virus reproduction. Although modelling of virus reproduction in host cells has mostly taken a kinetic approach, focusing on translation and transcription processes , it has Motivated by the qualitative predictive power of stoichiometric metabolic models and flux analysis, we apply it here to simulate the production of SARS-CoV-2 virus particles as part of the host metabolism and predict metabolic inhibitions against this virus. Given the emerging literature on SARS-CoV-2 infection primarily targeting lung and intestinal cells, we focus here on human lung cells as the host system. We adapt the available human genome-scale metabolic model with a biomass maintenance function based on human lung tissue–derived expression data, and include in this model also a viral biomass reaction, estimated using structural information available from SARS-CoV-2 and related viruses. By optimising flux distributions in this “infected lung cell model” for viral biomass reaction, we were able to predict reactions whose suppression or constraint can theoretically inhibit viral reproduction. We explored the possible impact of these predicted inhibitions on the host metabolism itself, as well as the experimental feasibility of implementing the predicted metabolic perturbations using available drug and inhibitor information on metabolic enzymes. Our results indicate that individual and double perturbation of several metabolic reactions from central metabolic pathways can inhibit SARS-CoV-2 reproduction in general and some of these can do so selectively without affecting normal metabolic functions of the host. We highlight these reactions as experimentally testable drug targets for inhibiting SARS-CoV-2 reproduction in human lung cells and provide details of the implemented computational approach for further development.To simulate the production of SARS-CoV-2 virus particles in a human cell, we used an existing, community-developed human genome scale model known as RECON2.2  see the. Within By incorporating the SARS-CoV-2 viral biomass function into the human metabolic model and assuming a minimal media composition , we predict a metabolic flux distribution for optimal virus production in a human cell . We thenResults of the flux variability analysis, where flux range for each reaction is calculated while maintaining an optimal biomass flux to SARS-CoV-2 production.Table S1 To identify reaction perturbations which, when inhibited, can halt or reduce virus production, we systematically simulated a knockout of each flux-carrying reaction. Excluding reactions involved in uptake from media, this analysis highlighted 35 reaction knockouts that can stop virus production and an additional eight reactions that can reduce it below 80% of the original . The forResults of the single and combined reaction perturbations (gene knockout simulations) on SARS-CoV-2 production, as performed on all reactions identified as flux-carrying in the flux variability analysis.Table S2 Considering that it is possible for the effects of single perturbations to be circumvented by re-directing of fluxes, we also explored combined perturbations. We created all possible pairs of flux-carrying reactions according to the flux variability analysis  and simulated the effect of setting their reaction fluxes to zero. This has identified more than 400 reaction pairs, co-inhibition of which results in the reduction of virus optima to 80% or less of the original  and S2. In the case of simulating the minimal media with higher uptake fluxes, we have also identified pairs of completely new enzyme inhibitions, which were not causing any effect on their own (Table S2). Some of these pairs exert their effects by blocking multiple pathways from a given compound and thereby causing disruption in steady state balances in the model. For example, co-inhibition of citrate synthase (CSm) and several other enzymes such as histidase (HISD) totally prevents flux in SARS-CoV-2 biosynthesis reaction by making impossible the mass balance of protons in the cytosol see .In the above discussed analyses, we considered host metabolism as optimised for viral production and evaluated impact of perturbations only on this process. Such metabolic perturbations should also be evaluated for their impact on the normal metabolic functions of uninfected host cells. In previous studies, normal state of metabolism in human cell lines are either represented through a pseudo reaction for cellular maintenance  or throuHere, we expanded from this approach to focus on human lung cells, which are shown to be the primary target of SARS-CoV-2 infections along with intestinal cells The composition and stoichiometries of building block metabolites in the virus biomass and human lung cell–specific maintenance pseudo reactions.Table S3 Results of the flux variability analysis, where flux range for each reaction is calculated while maintaining an optimal flux on the SARS-CoV-2 biomass or human lung cell–specific maintenance pseudo reactions.Table S4 Given the above finding that the same reactions carrying flux for SARS-CoV-2 production also carry flux for host metabolic maintenance, we re-analysed the effects of enzyme perturbations on both virus and the host. We found that many of the previously identified single perturbations limiting virus production also limited significantly the host metabolic maintenance, with only one single perturbation—that involving CYOR—showing more than 5% difference in its impact on virus versus the host (Table S5). The same finding prevailed for double perturbations. The only pairs that displayed 5% or more difference in their effects on virus versus host are those involving CYOR paired with other enzymes (Table S5).Results of the single and combined reaction perturbations (gene knockout simulations) on SARS-CoV-2 biomass and human lung cell–specific maintenance pseudo reactions.Table S5 The limited differential impact of full inhibition of enzymes made us postulate that more refined perturbations could provide a better strategy to just impact SARS-CoV-2 production without affecting the host. In particular, given the differences in optimal metabolic fluxes between virus production and host maintenance states, we argued that there might be flux values for some reaction that are compatible with only one of these states. To explore this possibility, we systematically analysed the flux variability of each reaction given either the optimisation of host maintenance or virus production. This has allowed us to see if any of the reactions would have flux regimes that are only compatible with the optimal host maintenance but not with optimal virus production and then “enforce” such flux regimes on the model. This approach allowed us to identify few single and double reaction perturbations that are fully selective on their effect and solely reduce virus production without causing any impact on the host (see Table S6). They involved a small number of metabolic reactions, namely, aspartate carbamoyltransferase (ASPCTr), carbamoyl-phosphate synthase (CBPS), dihydroorotic acid dehydrogenase (DHORD9), dihydroorotase (DHRTS), orotidine-5-phosphate decarboxylase (OMPDC), orotate phosphoribosyltransferase (ORPT), asparagine synthase (ASNS1), glutamine synthase (GLNS), and threonine deaminase (THRD) and caused up to 27% reduction in SARS-CoV-2 production (Table S6).Results of single and double flux enforcement perturbations on SARS-CoV-2 biomass and human lung cell–specific maintenance pseudo reactions.Table S6 The flux enforcement approach creates further constraints on how the metabolic fluxes in the system can be balanced at steady state. For example, in the optimal flux distribution for SARS-CoV-2 production in the unmodified model, threonine is obtained from the medium through a threonine:leucine shuttle, with both of these amino acids being in relatively similar demand between host and virus requirements. In the case of THRD fluxes enforced to specific ranges, threonine is instead obtained through a threonine:isoleucine shuttle and is now further intertwined with the arginine:isoleucine shuttle, thereby creating a trade-off among these amino acids . CrucialMotivated by this finding, we also simulated the effect of the decreased availability of specific metabolites in the culture medium on the host and virus biomass fluxes. The maximum import rate of the different molecules of the minimal medium we defined was set to values ranging between 0.1 and 10 mmol/gDW/h (where DW stands for grams of dry weight) and host and viral biomass optima compared (Table S7). As the maximum import rate of any molecule of the minimal medium decreases, so does the value of the optimal biomass flux of both host and virus. In some cases, namely, for threonine, glycylphenylalanine, and 3-methyl-2-oxovalerate, this decrease occurs more readily for the virus’ optimal biomass flux than for the host. This could be interpreted that a situation of low availability of the aforementioned molecules in the medium is more detrimental to the virus replication than to the host’s maintenance.The composition of the minimal media used in this study and results of simulations with limitations on different media component fluxes.Table S7 The metabolic approach used here allowed prediction of several reactions and reaction combinations that could limit SARS-CoV-2 production in human cells in general and differentially in human lung cells. The significant ones of these are re-summarised in Here, we have created a stoichiometric biomass function for the COVID-19–causing SARS-CoV-2 virus and incorporated this into a human lung cell genome scale metabolic model. The viral biomass function highlights the key building blocks required to synthesize a SARS-CoV-2 virus and its simulation within the human metabolic model enables predicting optimal flux distributions in the host for sustaining either SARS-CoV-2 reproduction or host maintenance. We used this capability to identify reaction perturbations that can inhibit SARS-CoV-2 reproduction in general or selectively, without inhibiting the host metabolic maintenance. The identified reactions primarily fall onto glycolysis and oxidative phosphorylation pathways and their connections to amino acid biosynthesis pathways. The latter finding is in line with the additional observation we made here that the relative stoichiometries of specific amino acids differ in SARS-CoV-2 biomass versus host cell maintenance estimated using human lung cell protein expression data. Together, these results highlight the possibility of targeting host metabolism for inhibition of SARS-CoV-2 reproduction in human cells in general and in human lung cells specifically.The predictions presented here are based crucially on the structure of the metabolic model as well as the two key assumptions of the flux balance analysis (FBA), namely, the assumptions of metabolic steady state and the optimality of metabolic fluxes towards a specific metabolic function. In the case of the key assumptions of FBA , these aThere is unfortunately not much available literature yet on SARS-CoV-2–infected cells and their metabolism, to allow us detailed comparison between model predictions and experiments. This said, we have identified a pre-print under review, which used a colon epithelial carcinoma cell line as a model system to study impact of SARS-CoV-2 infection on cell physiology . In brieIn terms of model result dependence on the model structure, we used the RECON2.2 model . This cuIn our view, this approach should be more informative than using existing cell-specific models that are not experimentally shown the be SARS-CoV-2 targets, such as the macrophage model, which has been used recently in a pre-print to study SARS-CoV-2 infection . BesidesThe virus life cycle consists of environmental circulation, infection, and subsequent host cell attachment and entry, reproduction within the host cell, and exit for a new round of infection. The presented approach focuses solely on the reproduction within the host cell and the metabolic aspects of that. Although this is a limited focus, reproduction in the host cell is a crucial and essential aspect of the virus life cycle. The importance of this stage is highlighted in several studies, which demonstrate that viruses tend to re-program host metabolism for increased viral production  or encodWe argue that differences in metabolic requirements of a virus versus its host could be a “physiological mismatch” that contributes to this evolutionary dynamic. Before metabolic adaptations happen, however, inhibition of the host metabolism might be a possible strategy to selectively inhibit reproduction of emergent viruses in new hosts. Specific host-based metabolic perturbations have already been shown experimentally to be effective against viruses , whereasIn the development of host-based metabolic strategies to inhibit viruses, metabolic modelling, as presented here, can play a useful role. In particular, our approach can be applied relatively rapidly to any host–virus pair both for existing and emerging viruses, and allow generating experimentally testable hypotheses for virus inhibition. This approach can be applied as long as structural and genomic information can be converted into an estimation of biomass composition for the virus and a suitably detailed metabolic model for the host can be constructed. The former process can be enhanced by further databases of viral structural and genomic information, whereas the latter process would benefit from extending human-focussed efforts such as the human metabolic atlas database  to coverTo identify specific host metabolic reactions that can alter viral production, we make use of a generic human cell genome-scale metabolic model that has been previously developed  and that2O in the scaling) (see the Data Availability section). The remaining elements of the maintenance function were retained as in the RECON2.2 model. The final, lung cell–specific maintenance function is provided in Table S3.As discussed in the main text, when identifying metabolic perturbations that can selectively affect viral production without much affecting the host, the specifics of the used host maintenance representation must be cell specific. To align this representation to a lung cell, we have made use of the gene expression data available from the Human Protein Atlas project . In partNC_045512). For one virus particle, the number of copies of genome (Cg) is assumed to be 1, whereas the number of copies of structural proteins is assumed to be 1,540 for membrane glycoproteins (M), 270 for surface glycoproteins (S), 490 for nucleocapsid phosphoproteins (N), and 5 for each of the other structural proteins of SARS-CoV-2. These protein copy numbers and stoichiometries are estimated from electron microscopy and mass spectrometry studies on other coronaviruses, including SARS-CoV  to represent a rich media . We have used here both this approach and additionally implemented a minimal media containing only essential metabolites, carbon and nitrogen source, and oxygen. The identification of the minimal media was achieved using a linear optimisation based algorithmic approach .The integrated genome-scale metabolic model was simulated using the FBA approach . FBA assapproach , where ahttps://www.ebi.ac.uk/biomodels/MODEL2010280002). All source code used in this work are available on our research group GitHub web pages at: https://github.com/OSS-Lab/FBAhv.All relevant data are presented in the main and supplementary texts. The SBML model of the SARS-CoV-2–infected lung cell is also available at BioModels  under id


The metabolic capabilities of cells determine their biotechnological potential, fitness in ecosystems, pathogenic threat levels, and function in multicellular organisms. Their comprehensive experimental characterization is generally not feasible, particularly for unculturable organisms. In principle, the full range of metabolic capabilities can be computed from an organism's annotated genome using metabolic network reconstruction. However, current computational methods cannot deal with genome-scale metabolic networks. Part of the problem is that these methods aim to enumerate all metabolic pathways, while computation of all  conversions between nutrients and products would suffice. Indeed, the elementary conversion modes  capture the full metabolic capabilities of a network, but the use of ECMs has not been accessible until now. We explain and extend the theory of ECMs, implement their enumeration in ecmtool, and illustrate their applicability. This work contributes to the elucidation of the full metabolic footprint of any cell.  •Elementary conversion modes (ECMs) specify all metabolic capabilities of any organism•Ecmtool computes all ECMs from a reconstructed metabolic network•ECM enumeration enables metabolic characterization of larger networks than ever•Focusing on ECMs between relevant metabolites even enables genome-scale enumeration Understanding the metabolic capabilities of cells is of profound importance. Microbial metabolism shapes global cycles of elements and cleans polluted soils. Human and pathogen metabolism affects our health. Recent advances allow for automatic reconstruction of reaction networks for any organism, which is already used in synthetic biology, (food) microbiology, and agriculture to compute optimal yields from resources to products. However, computational tools are limited to optimal states or subnetworks, leaving many capabilities of organisms hidden. Our program, ecmtool, creates a blueprint of any organism's metabolic functionalities, drastically improving insights obtained from genome sequences. Ecmtool may become essential in exploratory research, especially for studying cells that are not culturable in laboratory conditions. Ideally, elementary conversion mode enumeration will someday be a standard step after metabolic network reconstruction, achieving the metabolic characterization of all known organisms. Clement et al. present a ready-to-use tool that unveils metabolic blueprints of organisms from their reconstructed metabolic networks, through the computation of elementary conversion modes (ECMs). For any cell, the ECMs span all overall conversions from nutrients to new cells and the secretion of products. Therefore, the tool describes all possible effects that an organism may exert on its environment. Metabolism underlies most cellular behaviors. Which chemical compounds a microbe can exploit for growth, which products it can make and at which yields is essential information for understanding the microbe's roles in ecosystems, its responses to varying conditions, and its potentials for biotechnology and bioremediation. In the case of pathogens, metabolic capabilities are informative about the niches in which they can thrive. The functioning of multicellular organisms relies on how the capabilities of different cell types complement each other. A computational method that can enumerate all metabolic capabilities of any cell, from its annotated genome sequence, is therefore of key importance.,In the pre-genomic era, a cell's metabolic capabilities were investigated using experimental physiological information and elemental balancing of nutrients and products. Cellular metabolism was seen as a black box: without having knowledge of the metabolic details, so-called macrochemical equations were calculated which specify the stoichiometry of the conversion of nutrients into biomass (cells) and by-products.All metabolic reactions that can be catalyzed by a cell can be determined from the metabolic-gene annotations of its genome. This allows for the reconstruction of the metabolic network, which can nowadays be done almost purely computationallyBox 1N of a metabolic network. Each column of N captures for one reaction which metabolites are consumed, which are produced, and in what ratios. To facilitate the exposition, we here assume that all reversible reactions are split into a forward and backward reaction, so that all reactions in N are irreversible. Some metabolites are internal to the cell and some metabolites are external; metabolites that occur both inside and outside the cell are considered as two metabolites: one internal and one external. We denote the index set of internal metabolites by Int. The product of the stoichiometric matrix with the vector of reaction rates ECMs are the minimal building blocks of all net conversions by metabolic networks, and were defined by Urbanczik and Wagner.This space is called the conversion cone and should not be confused with the flux cone which comprises all steady-state fluxes. In fact, the conversion cone is the result of multiplying all points in the flux cone with the stoichiometric matrix (see Definition 1.The set of elementary conversion modes (ECMs) is the minimal set of conversionssuch that each steady-state conversion can be written as a positive sum of ECMs, without the production of any external metabolite being canceled in that sum.Some readers might note that this definition of ECMs is similar to the definition of EFMs. This is because both can be defined as elementary vectors  cancel the production of B, since B is produced in the blue conversion and consumed in the green conversion. Therefore, the red conversion is also an ECM: since this conversion does not produce or consume B, a positive combination with the other ECMs does not induce a cancellation. In total, we thus have three conversions, as listed in Now consider the conversion 2 A → BM (red). This conversion can be written as a positive sum of the two previously found ECMs: 2( A→ B ) + ( 2 B → BM ) = ( 2 A → BM ). However, in summing these ECMs, the metabolite In mathematical terms, one could obtain the full set of ECMs by calculating the extreme conversions per orthant, and then taking the union of all these extreme conversions. The requirement that no metabolite production is cancelled, implies that all steady-state conversions can be written as a positive sum of ECMs in which each metabolite is either produced by all ECMs in the sum, or consumed by all ECMs in the sum. In this manner, cancellations no longer occur , defined in 2005 by Urbanczik and Wagner.ECMs can be seen as objects analogous to EFMs: the ECMs form a minimal set that generates all steady-state substrate-to-product conversions, i.e., all macrochemical equations, while the EFMs form the minimal set that generates all steady-state flux distributions. However, the set of ECMs is much smaller than the set of EFMs: first, because many different EFMs map to the same overall conversion and second, because ECMs are objects in the lower-dimensional space of external metabolite changes rather than in the space of reaction rates. For these reasons, the combinatorial explosion that prohibited the enumeration of all EFMs on a genome-scale network might disappear when enumerating ECMs.Although ECMs were already defined in 2005,Box 2In ecmtool, the user can choose to compute only the stoichiometric relations between a subset of the external metabolites by “hiding” the other external metabolites. The resulting set of ECMs still gives a full summary of these relations and complies with the steady-state assumption on the full metabolic network. The consumption and production of the hidden metabolites still occurs but is not reported. As a result, the reported ECMs are not necessarily mass-balanced, which is emphasized by the question marks in In In the example of 1, resulting in the reaction C → D + T1. Any conversion that uses the reaction of interest produces T1, and its coefficient in the conversion is equal to the reaction rate.Besides hiding metabolites of minor importance, we can keep track of reaction rates of major importance. The tag method, suggested by Urbanczik and Wagner,In Box 3To show how ECM enumeration works in practice, we will here work out some steps of the computation of ECMs for the network given by the following stoichiometry matrix:A, E, and F can only be used as inputs, and which is also shown as the first network of B, i.e., B and therefore already satisfy this constraint. Of the other reactions, B and B. Each pair of a producing and a consuming reaction generates a candidate that satisfies the steady-state constraint, so this gives us •G;•G;•D.The stoichiometry matrix gives a list of generators that generates all conversions before we have imposed the steady-state constraints: All candidates are tested for redundancy by the adjacency test described in which is depicted as the second network in C, D, and G, eventually givingThis process is then repeated for internal metabolites containing all ECMs, namely 2E and F. Hiding these metabolites is done by extending the metabolic network with reactions that create E and F from nothing, and marking the metabolites as internal. We thus getIn E, we see that only B → G, which is added to the new list of generators. When we then impose the steady-state constraint for metabolite F, we get the same candidate B → G, but since this is not a new conversion it is not added to the list of generators. It can thus be seen that hiding metabolites E and F immediately reduces the computational complexity, because now only one reaction from B to G remains while without the hide method there were two such reactions. Moreover, after imposing the remaining steady-state constraints, we find only one ECM: 2A + ?? → G, where the question marks indicate that we do not know whether more metabolites are consumed because this information is hidden.When we now start by imposing the steady-state constraints for metabolite Although it would not give problems in this example, we can in general not hide a metabolite by simply removing it from the network. This is because information about whether the metabolite can be used as an input, as an output, or both would be lost from the computation. With the current method, this information is stored in the directionality of the added reaction.In this work, we unlock the potential of ECMs by making the theory accessible and enumeration possible for any systems biologist. We reformulate and extend the ECM theory of Urbanczik and Wagner,This work contributes to closing the gap between any cell's genotype and phenotype. It offers a computational toolkit for the exhaustive determination of metabolic capabilities, and should be particularly valuable when experimental characterization is impossible because cells cannot be cultured in isolation.The number of ECMs increases much slower with metabolic network size than with the number of EFMs . For exaHelicobacter pylori in a minimal medium  each ECM must be a steady-state conversion, (2) each ECM must be an elementary vector, and (3) each steady-state conversion must be a positive combination of ECMs without metabolites being canceled in the sum.We confirmed that all computed ECMs are steady-state conversions by checking that the net production of internal metabolites equals zero, and that there exists a combination of metabolic reactions that gives rise to the ECM. Then, according to the definition of ECMs given in all steady-state conversions that they can be written as a combination of ECMs? We chose to use the set of EFMs calculated by efmtool. This set spans all possible steady-state flux combinations the metabolic network allows. For each EFM, we then calculated its overall conversion and tried to write this conversion as a combination of ECMs. If we allowed for an error of The third property was more difficult to validate, because how can we prove for Above, we explained and validated that ecmtool finds all ECMs, given an annotated genome. The annotation is necessary for the reconstruction of the metabolic network. Strictly speaking, this minimally requires the annotation of the metabolic genes. Since the annotation of a genome is not always complete, we cannot guarantee that all metabolic capabilities encoded on the genome are found. We can guarantee that all conversions are found of the genome-derived metabolic network.Focusing on the stoichiometric relations between metabolites of major importance by hiding external metabolites of minor importance is a powerful way to scale up the size of metabolic networks that can be dealt with in ecmtool. The ECMs that are obtained now span all possible relations between the non-hidden metabolites but no longer give information about what happens to the hidden metabolites . This aided in exposing the role of oxygen supply in the observed amino acid secretion and carbon polymer synthesis by bacteroids, and in quantitatively reproducing the carbon cost of biological nitrogen fixation.Rhizobia are soil bacteria that can induce formation of nodule structures on plant roots, in which they differentiate into non-dividing bacteroids. Bacteroids fix atmospheric nitrogen into ammonia and make this available to the plant in exchange for carbon in the form of dicarboxylates.Our method enumerates and quantifies, for any organism for which a metabolic reconstruction has been made, all possible stoichiometric relations between substrates, products, and biomass. This method does not rely on any optimality assumption, nor does it require experimentally obtained physiological information. It uncovers the full metabolic capability of an organism, and with that the metabolic footprint that an organism may leave in its environment.,ECM enumeration stands in a long tradition of methods that pursue this goal.Lactobacillus plantarum that was observed experimentally and predicted by FBA.E. coli model to investigate optimized E. coli growth in carbon- and oxygen-limited conditions. Using this approach, they could simplify their analysis by selecting four EFMs that together determined all optimal growth strategies in different glucose- and oxygen-limited conditions. In The enumeration of ECMs facilitates the exploratory study of metabolic networks: investigation of the ECMs could spark new hypotheses and show unexpected connections. It therefore complements optimization approaches like such as flux balance analysis (FBA)Most analyses of metabolic networks require a priori physiological information that is often not available. For example, it is often required to impose constraints on exchange fluxes to choose a reaction rate that needs to be optimized, or at least to know which metabolites can be produced.An overview of all feasible overall reactions might furthermore be useful when studying interacting species, such as crossfeeding species, host-pathogen interactions, or multi-species communities. The possible interactions are determined by what is consumed and produced by the individual species, which is exactly the information offered by the ECMs. Indeed, knowing the capabilities of one and the incapabilities of another might lay bare dependencies on which a stable community is built.E. coli core model and a genome-scale H. pylori model that this method can be used to obtain a much smaller set of conversions that spans all stoichiometric couplings between the user-defined external metabolites. This has not been possible with any other method. Moreover, when we focused only on the relations between glucose, oxygen, and biomass production, the hide method allowed us to scale ECM computation to the genome-scale E. coli model the dual of the dual cone is again the cone: (2)the vectors in the generator representation of a cone form an inequality representation of the dual cone, and vice versa: It remains to be shown how we obtain an inequality representation of Our computation starts from a generator representation of C0 , but by Note that this indirect method heavily relies on the DD method. We found that polcoN. We collect these generators in a matrix i. Imposing such a steady-state constraint means that we take the intersection of the cone Just as the indirect method, the direct method starts with the cone One such iteration thus starts with a set of generators of minimal set of generators. In other words, the cone However, when we combine all generators from the plus-group with the minus-group to create new generators, we will not get a Although performing many linear optimizations is in principle a very slow process, there is an important advantage: the different optimizations can be done completely independently. Therefore, we were able to parallelize this direct method so that it can now be run on large computation clusters.ECM theory focuses on the overall conversions between external metabolites instead of on how these conversions come about internally. This distinction can be exploited to simplify the network even before we start the main computation steps described above. We have implemented several compression steps that together bring large networks back to a workable size. Most of these compression steps were suggested by Urbanczik and Wagner.i we would maximize j, and such that i is infeasible and can be removed from the network. In Section 5.1 of The flux vectors that give rise to the ECMs should satisfy the steady-state and the irreversibility constraints. If we can prove that a reaction can never be active in a solution that meets both of these constraints, this reaction can be safely removed. In principle, the feasibility of a reaction can be tested by running a linear optimization: for reaction We can delete redundant reactions; a reaction is called redundant if it can be written as a conical combination of other reactions. Because the function of these reactions can always be replaced by the combination of the other reactions, it does not add functionality to the network and can therefore be removed. Redundant reactions in systems with fewer than about 10,000 reactions can be removed using a program called redund from lrslib,A, and say that there are several other reactions producing or consuming A. We prove in Section 5.3 of A is canceled. After doing this for all reactions connected to A, A. This implies that no reaction flux is possible through A cannot be compensated by another reaction. Therefore, we can delete both A from the network without affecting the ECM results.Each reversible reaction can be used to cancel itself and one metabolite it connects to. Say that a reversible reaction, Sometimes an internal metabolite can only be produced and not consumed, or vice versa. In this case, the reaction flux through the reactions connected to this metabolite has to be zero in any steady-state solution. Therefore, we can delete the metabolite and all connecting reactions without affecting the set of ECMs.A, but that there are several reactions that consume A. Then, again without affecting the set of ECMs, we can add A is exactly canceled. The reaction A, and can therefore not be active in a steady-state solution. We can thus cancel both A.Say that we have a reaction A. In Section 5.5 of A. After doing this, A, so that we can delete both A from the network. Since we compensated for the action of A cycle is a set of reactions that can be used in a certain ratio such that nothing is produced or consumed. Say that the reactions https://github.com/SystemsBioinformatics/ecmtool, and can additionally be installed through the Python package manager pip. The direct and indirect computation methods are both available within the program. A manual is available as Section 11 of We implemented our algorithms in a publicly available Python program called ecmtool. It is freely available on GitHub at


Hormones of the brain-pituitary-peripheral axis regulate metabolism, gonadal maturation, and growth in vertebrates. In fish, reproduction requires a significant energy investment to metabolically support the production of hundreds of eggs and billions of sperms in females and males, respectively. This study used an LC-MS-based metabolomics approach to investigate seasonally-related changes in metabolic profile and energy allocation patterns in female goldfish liver. We measured basal metabolic profile in female goldfish at three phases of the reproductive cycle, including 1) Maximum growth period in postovulatory regressed phase, 2) mid recrudescence in fish with developing follicles, and 3) late recrudescence when the ovary contains mature ovulatory follicles. We also investigated changes in the liver metabolism following acute treatments with GnRH and GnIH, known to be involved in controlling reproduction and growth in goldfish. Chemometrics combined with pathway-driven bioinformatics revealed significant changes in the basal and GnRH/GnIH-induced hepatic metabolic profile, indicating that metabolic energy allocation is regulated to support gonadal development and growth at different reproductive cycles. Overall, the findings support the hypothesis that hormonal control of reproduction involves accompanying metabolic changes to energetically support gonadotropic and somatotropic activities in goldfish and other oviparous vertebrates. Ovariant manner . There it manner . Similart manner . Thus, vCarassius auratus). Fish were imported by a local supplier  from a fish farm in Pennsylvania , where they were reared under the natural daylight and temperature cycles. Each experimental group was assigned 20 fish to ensure that a minimum of six female fish could be used in this study. We used fish at three stages of the annual reproductive cycle, including regressed/maximal growth phase (July-August), mid recrudescence (December- January) and late recrudesce (March-April). At arrival, fish were acclimatized for 4–7 days in a flow through system in 25 L tanks kept under the daylight and temperature corresponding to the environmental conditions. Fish were fed a commercial fish diet once a day to satiation . A buffered MS-222 solution  was used to anesthetize the fish before intraperitoneal injection and euthanization following the protocols approved by the University of Calgary animal care committee (protocol #AC19-0161) under the guidelines of the Canadian Council of Animal Care. Ovarian stages were assessed by visual inspection after euthanization before tissue collection. Photographs of ovaries used at different gonadal stages are shown in In all studies, we used sexually mature  4–6 inch (average weigh 34.6 g) goldfish (2) was purchased from Bachem . Goldfish GnIH  was synthesized by the University of Calgary Peptide Services . Before injection, hormones were dissolved in phosphate-buffered saline (PBS). GnRH-III, GnIH, and PBS (control) were administered intraperitoneally twice, at 9 am (T-0 h) and 9 pm   based on a 7-fold cross-validation permutation testing method was used to assess the significance of the PLS-DA models. Models with a p-value <0.05 were considered significant.ectively . PLS-DA p-values are also shown in We used Variable Importance in Projection (VIP) > 1 as a variable selection method and threshold to identify significant metabolites in the PLS-DA models. Metabolites with VIP score >1 were selected for each model to generate heatmaps and pathway analysis . The clustered heatmaps display the relative concentration of metabolites and an overview of the metabolic profile for various treatment groups to visualize changes in metabolite concentrations. Hierarchical clustering was conducted on the rows/variables of the data matrix and the dendrograms cluster metabolites that undergo similar changes in abundance. Finally, differences in the levels of metabolites in various groups were tested by one-way ANOVA, followed by Tukey’s post-hoc analyses that identify differences between groups . To furtp-value adjusted for multiple comparisons using False Discovery Rate (FDR) < 0.05 indicates a significant difference between groups.Danio rerio) KEGG pathway library as a reference.The concentration levels of the VIP > 1 metabolites identified for every PLS-DA model were used to conduct pathway analysis and determination of metabolic pathways involved in different treatment groups. In the present study, we used the MetPA metabolomics pathway analysis offered by the Metaboanalyst 4.0 platform as an analytical tool . Pathwaym/z and retention times standard library led to the identification of 71 metabolites in the aqueous phase of liver extracts. Examples of peak detection and measurements are shown in n = 10), mid recrudescence (n = 10), and late recrudescence (n = 10). We also used a quality control group   and by CV-ANOVA (p = 2.75 × 10−5) , and network topology analysis that estimates the relative importance of metabolites based on their relative position in a given pathway (pathway impact). The results of the pairwise pathway analysis shown in de novo purine synthesis during mid recrudescence, compared to regressed and late recrudescence  and malonate (first step product of fatty acids synthesis) that indicate a significant increase of these metabolites in mid compared to late recrudescence  . The PCA, n = 5) . The sco, n = 5) . Statist × 10−5) . To furt0−5) (−5 . In the 0−5) (−5 . Given t0−5)  (−5 . The sta0−5) (−5 . Further0−5) (−5 . Additio0−5) (−5 . The int0−5) (−5  and mult0−5)  (−5 . The ana0−5)  for each reproductive phase to investigate differences in the metabolic profiles of the three groups tested  and mid recrudescence  compared to the control  suggest a stimulatory action of GnIH on these metabolic pathways  and docosahexaenoic acid (DHA; The pathway analysis demonstrates that GnRH injections alter carbohydrate and energy metabolism in late recrudescence . Single cid DHA; .Results from this study characterize the liver metabolic profile of female goldfish at three stages of the reproductive cycle. In the regressed gonadal phase, carbohydrate and amino acid metabolism were enhanced, possibly supporting anabolic processes related to growth. On the other hand, the liver metabolism of mid-recrudescent fish is characterized by elevated lipid synthesis during the peak of vitellogenesis and lipid mobilization from the liver to support maximal gonadal maturation. The metabolic profile of goldfish during late recrudescence revealed a high concentration of antioxidant molecules such as uric acid and glutathione and reduced ketone bodies. These molecules may play a vital role in the final stages of the gonadal cycle. In the present study, we also demonstrated that GnRH and GnIH are involved in the multifactorial regulation of metabolism in supporting growth and reproduction in female goldfish. Results indicate that acute treatment with GnRH and GnIH induce metabolic changes depending on the season and mode of administration. GnRH treatments had similar effects during the regressed phase and mid recrudescence. Double injection with GnRH enhanced pathways related to growth and reproduction in regressed and mid recrudescence, respectively. On the other hand, GnRH treatments during late recrudescence stimulated pathways supporting vitellogenesis and gonadal maturation. GnIH showed an additive effect of single and double injections and changed the metabolic profile resembling the previous reproductive stage. During the regressed phase, GnIH treatments stimulated metabolic pathways related to the final stages of reproduction. In contrast, GnIH promoted a shift toward growth processes during mid recrudescence. Finally, during late recrudescence, GnIH stimulated metabolic pathways characteristic of the mid recrudescence metabolic profile. Overall, our results provide new information on the metabolic changes that accompany the growth and reproductive processes during the annual reproductive cycle of female goldfish and provide an insight into the role of GnRH and GnIH in the multifactorial control of growth and reproduction in goldfish and other oviparous species.


Simulations predicted a stabilising effect of reduced sucrose biosynthesis capacity and increased capacity of starch biosynthesis on carbon assimilation under transient heat stress. Model predictions were experimentally validated by quantifying plant growth under such stress conditions. In conclusion, this suggests that Fourier polynomials represent a predictive mathematical approach to study dynamic plant-environment interactions.Quantification of system dynamics is a central aim of mathematical modelling in biology. Defining experimentally supported functional relationships between molecular entities by mathematical terms enables the application of computational routines to simulate and analyse the underlying molecular system. In many fields of natural sciences and engineering, trigonometric functions are applied to describe oscillatory processes. As biochemical oscillations occur in many aspects of biochemistry and biophysics, Fourier analysis of metabolic functions promises to quantify, describe and analyse metabolism and its reaction towards environmental fluctuations. Here, Fourier polynomials were developed from experimental time-series data and combined with block diagram simulation of plant metabolism to study heat shock response of photosynthetic CO Fourier polynomials can be used to model the stress response of carbon balancing in plants. This work suggests that Fourier polynomials could be used to predict plant responses in varied environmental conditions. In a metabolic context, ODE models are frequently applied to simulate enzyme kinetic reactions and, by this, to explain dynamics of observed metabolite concentrations. Kinetic models, based on ODEs, have frequently been applied in a broad field of biological research, e.g., in the context of metabolic engineering of microbial systems and strain design2, disease research3, and plant metabolism6. In contrast to the high diversity of application fields, the principle of ODE kinetic modeling remains conserved: based on genome sequence information or biochemical evidence from literature a biochemical reaction network is established, substrate and product concentrations are quantified, and enzyme kinetics are applied to calculate reaction rates within a metabolic system. Enzyme kinetic parameters, e.g., velocity under substrate saturation (Vmax) or substrate affinity (KM) of Michaelis–Menten equations, are experimentally determined and used for computationally assisted parameter estimation. A well-defined and experimentally validated kinetic ODE model enables computational simulation and prediction of complex system behavior. A clear limitation of such an approach, however, is the requirement of kinetic parameters which are (frequently) difficult and/or expensive to quantify. Initiatives like KiMoSys, a public repository of published experimental data, summarize and concentrate data on metabolites, protein abundance, and fluxes providing a solid database for model construction and initial development7. Yet, under highly dynamic conditions, e.g., in a fluctuating environment, it still remains a challenge to resolve system dynamics on an enzyme kinetic level. This is due to the high dynamics of metabolites, transcripts, protein levels, and enzyme activities9. Although being laborious, the development and optimization of ODE kinetic models provide an important and informative mathematical method to study biochemical system behavior. Simultaneously, however, diverse problems might occur with solving and applying such models due to uncertainties about parameters, model structure, kinetic rate laws or parameter sensitivities11. Depending on the research question focused on by a study, an explicit knowledge about enzymatic activities and their dynamics might not be essential to derive a mathematical description of metabolite dynamics. For example, metabolic fluxes might be estimated by tracing labeled atoms or molecules in a metabolic pathway system12. While only very limited information about single enzyme activities or kinetics can be derived from flux estimations, they still provide comprehensive insights into metabolic states and pathway activities, also on a large scale13. Beyond, algorithms and user interfaces have been developed which enable the combination of flux data with relative metabolite levels14.Capturing dynamics in biological systems by mathematical terms is the general aim of biomathematical modeling. Differential equations represent an adequate strategy to describe dynamics over space and time. Ordinary differential equations (ODEs) and partial differential equations (PDEs) have been successfully applied to reveal biological system dynamics and to develop predictive models of growth rates, transcription, translation, or metabolic processes15. Similar to flux analysis, dynamics of metabolite concentrations in time-series experiments were used in this approach to derive a time-continuous mathematical function to identify regulatory cascades in metabolic pathways. This approach made use of spline interpolations which were composed of cubic polynomials which were fitted to adjacent pairs of data points in a time-series data set. While such an approach is suitable for accurate data fitting, underlying mathematical functions are frequently not related to biological function and, thus, are less predictive than enzyme kinetic models. In the present study, we developed a mathematical model based on Fourier polynomials to simulate and analyze dynamics of photosynthesis and carbohydrate metabolism under transient heat exposure by function superposition. Model simulations indicated a significant impact of sucrose and starch biosynthesis on the stabilization of carbon assimilation and growth under elevated temperatures.For estimating metabolic functions, i.e., the sum of synthesizing and degrading/consuming reactions of metabolite pools, under dynamic environmental conditions we have previously suggested a method for implicit estimation of metabolic functions−1 gDW−1). Carbon balance Eq. 1) comprised the summation of NPS rates and negative starch rates, balance Eq. 2) additionally comprised summation of negative sugar rates  which was left from photosynthetically assimilated CO2 after starch synthesis, e.g., for sugar biosynthesis or biomass production. Further, BE2 revealed residual net carbon flux after additional sugar biosynthesis.A block diagram model of the central carbohydrate metabolism of plants was developed to integrate experimental data on net photosynthesis (NPS), starch, and sugar metabolism Fig. . The netinput , and 1,2 Eqs. ).1\documak and bk represent the Fourier coefficients for NPS, starch, and sugar equations. ω is the fundamental frequency of the signal . This Fourier polynomial-based balance equation model was applied to simulate dynamics of carbohydrate metabolism in plants of Arabidopsis thaliana, accession Columbia-0, under transient heat exposure. In addition, NPS and carbohydrate dynamics were recorded and simulated in a starch-deficient mutant pgm1 and a mutant with a deficiency in sucrose biosynthesis capacity, spsa1. Coefficients of Fourier polynomials are provided in the supplements . In contrast to significant genotype-effects in net CO2 assimilation under temperature fluctuation, transpiration rates were similar in Col-0, spsa1 and pgm1 under each tested condition and no significant genotype effect was detected , electron transport rates (ETR), photochemical (qP), and non-photochemical quenching parameters (qN) were determined by pulse-amplitude modulation before and after transient exposure to 40 °C , photochemical (qP), and non-photochemical quenching (qN) parameters recorded within rapid light curves . This drop did not change significantly among the different temperatures. In the recovery phase after the heat shock (8 h), plants increased their starch content depending on the temperature they were subjected to. Plants of both genotypes treated with 32 °C increased their starch concentration by ~40% between 6 and 8 h , but no further change was observed after the recovery phase at 8 h . In the recovery phase, the sucrose concentration in the heat-treated pgm1 plants returned to a level comparable to control conditions  or spsa1 under control conditions , while exhibiting very low variance . In addition, in Col-0 glucose levels were increasing above the level of control plants already during heat exposure to 40 °C at 6 h (P < 0.001). Fructose dynamics in Col-0 were similar to the glucose dynamics in Col-0 . In pgm1, hexose levels decreased significantly after temperature treatment , with the lowest values being reached at 40 °C. After recovery (8 h), hexose levels did not change significantly from the 6 h time point in the 32 °C plants. After exposure to 36 and 40 °C, however, hexose levels increased significantly from 6 to 8 h (P < 0.001).Col-0 and l-0 Fig. . In spsa1), and sugar metabolism (BE2), are affected by transient heat exposure, derivatives were built with respect to time compensation of reduced CO2 assimilation rates under 36 and 40 °C , changes in starch and sugar dynamics became obvious in integrals of BEnts Fig. . In this1 and BE2 can predict whole-plant performance under heat, the surface of leaf rosettes were quantified before and after heat exposure , detected within a rapid light curve protocol, showed a significant increase during heat exposure suggesting a differential photosystem and/or thylakoid organization compared to Col-0 and pgm1. Further, compared to Col-0, spsa1 might have had a reduced rate of photorespiration and/or mitochondrial respiration during heat exposure. While it remains speculation from our study, a higher starch accumulation rate in spsa1 might result in a lowered respiration rate under heat because carbon equivalents may be fixed more efficiently. The observation of a destabilized NPS rate in starchless pgm1 plants would support the stabilizing role of starch biosynthesis under transient heat exposure. Comparison of transpiration revealed similar rates across all genotypes, which suggests that observed NPS effects are unlikely due to differential stomata closure and/or secondary effects like leaf cooling22. However, previous reports under ambient conditions have shown that SPS knockout mutants have rather enhanced than lowered dark respiration rates which do not directly support the hypothesis of NPS stabilization by starch biosynthesis23. Another explanation might be a secondary effect of the spsa1 mutation on rubisco and/or rubisco activase which, to our knowledge, has not been shown in current literature but which needs to be proven in future studies.In temperate regions, plants are frequently exposed to a changing temperature regime, and these changes might occur both over short- and long-time scales. For example, the temperature typically changes between day and night, and beyond, the temperature might also change transiently within the diurnal light and dark period. While temperature acclimation of plants typically can be observed after days of exposure to non-lethal cold or heatpgm1 showed a differential pattern at 22 and 32 °C but became similar in its fructose profile to Col-0 and spsa1 under 36 and 40 °C. In mature Arabidopsis leaves, fructose levels are significantly affected by invertases that catalyze the hydrolysis of sucrose and release free hexoses24, and by fructokinase catalyzing ATP-dependent phosphorylation which yields fructose-6-phosphate25. As fructose and glucose profiles differed in the present study, this cannot (solely) be explained by invertase reactions which release equimolar concentrations of both hexoses. However, differential regulation of hexokinase and fructokinase could explain the different hexose profiles. Fructokinase yields the direct substrate for glycolysis, TCA cycle, and mitochondrial respiration. In a previous study that analyzed transcript levels in Arabidopsis thaliana under combined drought and heat stress found increased transcripts for both hexokinase and fructokinase26. Although the experimental design differed significantly from this study, together with other findings this suggests a central role of hexose phosphorylation in heat stress response and acclimation17. As leaf respiration rates typically increase under elevated temperature27, observed consistent fructose dynamics might be due to a relatively high rate of glycolytic consumption under transient heat exposure.While sucrose and glucose metabolism showed a dynamic and differential accumulation profile between 32, 36, and 40 °C experiments, dynamics of fructose concentrations were most conserved across all temperature treatments and, remarkably, also across genotypes see Fig. . An init2 assimilation rates with starch and sugar turnover allows for balancing of the central carbohydrate metabolism. In this context, Fourier polynomials support the functional and time-continuous estimation of dynamics of metabolism. Integration and differentiation of Fourier polynomials is straightforward, and, at the same time, provides a comprehensive mathematical framework that is applied in diverse fields of natural sciences and engineering30. As described in the block diagram model , spsa1 and pgm1  were grown on a 1:1 mixture of GS90 soil and vermiculite in a climate chamber under short-day conditions . The spsa1 line was confirmed via PCR to be homozygous and activity was found to be decreased to 30-50% of the wildtype Col-0  and grown for 2 further weeks under short-day conditions with the same settings as in the climate chamber. After 6 weeks, on the day of sampling, the temperature in the growth cabinet was kept at 22 °C during the first 2 h in the light . Then, in three independent experiments, the temperature was increased to (i) 32 °C, (ii) 36 °C, or (iii) 40 °C for a total of 4 h . In the control experiment, the temperature was set constantly to 22 °C. Between 6 and 8 h, i.e., until the end of the light period, the temperature was set to 22 °C in all experiments. A graphical representation of the experimental setup is provided in Supplementary Fig. Plants of ® Junior-PAM . Plants were dark incubated at 22 °C for 15 min prior to measurements. After dark incubation, Fv/Fm was determined by applying a saturating light pulse (photosynthetic photon flux density (PPFD) = 4000 µmol m−2 s−1). A rapid light curve protocol was applied to quantify ETR, qP and qN under increasing PPFD36. Sequentially, every 20 s, actinic irradiance was increased from 0 up to 2250 µmol photons m−2 s−1 .Maximum quantum yield of photosystem II (Fv/Fm) and electron transport rates (ETR) were quantified by pulse-amplitude modulation (PAM) using a WALZ® growth cabinet using a WALZ® GFS-3000FL system equipped with measurement head 3010-S . Temperature, light, and humidity control of the measurement head were set to follow ambient conditions, i.e., to follow surrounding growth cabinet conditions. A summary of recorded temperature, light, and humidity curves is provided in the supplement , incubated with the prepared sample for 30 min at 40 °C and absorbance was determined photometrically at 620 nm. Glucose amount was determined photometrically by a coupled hexokinase/glucose 6-phosphate dehydrogenase assay resulting in NADPH + H+ at 340 nm. For fructose quantification, phosphoglucoisomerase was added to the reaction mixture after glucose determination.Soluble sugars sucrose, glucose, and fructose were determined from dried ethanol extracts after dissolving in water. After incubation with 30% KOH at 95 °C, sucrose was quantified using an anthrone assay. Anthrone was dissolved in 14.6 M H37. In brief, freeze-dried leaf tissue was suspended in extraction buffer containing 50 mM HEPES–KOH (pH 7.5), 10 mM MgCl2, 1 mM EDTA, 2.5 mM DTT, 10% (v/v) glycerol and 0.1% (v/v) Triton-X-100. Following incubation on ice, extracts were incubated for 30 min at 25 °C with a reaction buffer containing 50 mM HEPES–KOH (pH 7.5), 15 mM MgCl2, 2.5 mM DTT, 35 mM UDP-glucose, 35 mM F6P, and 140 mM G6P. Reactions were stopped by adding 30% KOH and heating to 95 °C. Sucrose was determined photometrically after incubation with anthrone in H2SO4.The activity of sucrose phosphate synthase (SPS) was determined using the anthrone assaywww.r-project.org)38. Fourier series fitting was done within MATLAB® (www.themathworks.com), and block diagram models were created in Simulink® (www.themathworks.com). Plant total leaf surface was quantified using the Fiji software)39 with the SIOX plugin (https://imagej.net/plugins/siox). The sample size was chosen according to  measurement and growth capacities . Replicates represent biological replicates which were treated independently from each other to test and validate reproducibility. Plants were grown in pots with randomized order within the climate chamber to minimize or exclude any position or sampling effect. Samples were randomly chosen for molecular analysis. Investigators were blinded to group allocation during data collection.Statistical analysis was performed in R and R Studio (Further information on research design is available in the Supplementary InformationDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Data 5Supplementary Data 6Reporting Summary