Heart disease is the leading cause of death in both men and women. Cardiac fibrosis is the uncontrolled accumulation of extracellular matrix proteins, which can exacerbate the progression of heart failure, and there are currently no drugs approved specifically to target matrix accumulation in the heart. Computational signaling network models (SNMs) can be used to facilitate discovery of novel drug targets. However, the vast majority of SNMs are not sex-specific and/or are developed and validated using data skewed towards male in vitro and in vivo samples. Biological sex is an important consideration in cardiovascular health and drug development. In this study, we integrate a cardiac fibroblast SNM with estrogen signaling pathways to create sex-specific SNMs. The sex-specific SNMs demonstrated high validation accuracy compared to in vitro experimental studies in the literature while also elucidating how estrogen signaling can modulate the effect of fibrotic cytokines via multi-pathway interactions. Further, perturbation analysis and drug screening uncovered several drug compounds predicted to generate divergent fibrotic responses in male vs. female conditions, which warrant further study in the pursuit of sex-specific treatment recommendations for cardiac fibrosis. Future model development and validation will require more generation of sex-specific data to further enhance modeling capabilities for clinically relevant sex-specific predictions of cardiac fibrosis and treatment. Cardiac fibrosis, the uncontrolled accumulation of ECM proteins, can exacerbate the progression of HF4. This deposition of ECM proteins is crucial to a patient's initial survival of myocardial infarction (MI) to form the scar tissue that maintains the structural stability of the infarct region5. However, when it continues uncontrolled, it becomes pathologic by reducing ejection fraction and causing a greater chance of arrhythmia due to disruptions in electrical stimulus caused by collagen buildup7.Heart failure (HF) currently afflicts roughly 6.2 million Americans and has a five-year survival rate of only 50%8. However, no current FDA-approved treatments directly target cardiac fibrosis9. Ongoing research focuses on the development of a drug to directly inhibit cardiac fibrosis or even reverse it10. Despite gains in understanding the complex signaling networks of cardiac fibroblasts—the drivers of cardiac fibrosis—finding new anti-fibrotic therapies has remained expensive and tedious, with low efficacy rates in clinical trials11.The current treatment regimen for patients suffering from HF includes drugs such as angiotensin-converting enzyme inhibitors and beta-blockers to reduce their blood pressure and slow their heart rate, which reduces stress on the heart in an attempt to slow the progression of HF15. Currently, these models are limited in scope and, at best, can narrow down a pool of potential drug candidates for further study. As the field of precision medicine grows, it is likely that the accuracy and predictive power of these models will also improve. However, in order to fully leverage the predictive power of these models, it is imperative that the data used to create them is robust across diverse population groups. Unfortunately, experimental studies and clinical trials are historically skewed to male data. For several decades, women of childbearing age were banned from participating in clinical trials16. Even after this ban was reversed in the early 90s, it was not until 2016 that the National Institutes of Health required the use of both male and female animals in preclinical studies17. This has resulted in many FDA-approved drugs having double the adverse drug response rate in women20.The use of computational models to conduct in silico simulations has been used in a variety of ways to accelerate drug discovery24. Most notably, pre-menopausal females are less likely to suffer from MI compared to age-matched men—primarily thought to be due to the cardioprotective role of the ovarian hormone estrogen (E2)22. Additionally, the literature suggests that many adverse drug responses in both males and females occur in a sex-specific manner25. Incorporating sex hormones into computational models could be a starting point to correct for the dearth of sex-specific studies available in the literature and make precision medicine more accurate for all patients. In this study, we aim to make a previously published signaling network model of cardiac fibroblasts sex-specific by incorporating E2 signaling. We then conducted a sex-specific drug screen to analyze divergences in drug response between males and females.Biological sex is an essential factor when considering cardiovascular health27. Specifically, this model was comprised of 10 biochemical and biomechanical inputs, including transforming growth factor beta (TGFβ), angiotensin II (AngII), endothelin 1 (ET 1), mechanical tension, and others; downstream reactions of these inputs culminated in 22 cellular outputs, including alpha-smooth muscle actin (α-SMA), procollagen I and III (proCI and proCIII), several pro-matrix metalloproteinases and tissue inhibitors of metalloproteinases (proMMPs and TIMPs), and proliferation. New nodes and/or reactions were added to the SNM if two independent studies reported activation of inhibition of another node downstream of estrogen signaling. The vast majority of the reactions used to build the model were from cardiac fibroblast specific studies; if not from cardiac fibroblasts, the studies typically reported results from experiments from other cardiac  or fibroblast  cell types. Nearly all of the papers used for model updates used neonate rat cardiac fibroblasts, pooling male and female cells together as their cell type.A previously published signaling network model (SNM) of cardiac fibroblast was expanded to include the ovarian hormone estrogen (E2). The previous model was created via a manual literature search of ~ 300 papers and included 125 nodes  and 174 edges (reactions)45. In addition, 6 already present reactions were altered to include direct estrogen receptor inhibition activity. The added nodes include E2 and its three primary receptors: estrogen receptor alpha (ERα), estrogen receptor beta (ERβ), and g-protein coupled receptor (GPR30)46. Cyclin beta 1 (Cyclinβ1) and cyclin-dependent kinase 1 (CDK1) were added downstream of GPR30 as they were not already included in the fibroblast SNM35. The total network now consists of 132 nodes and 202 edges.After a manual literature search of sex differences in cardiac fibroblasts, we used 18 previous published studies to integrate E2 signaling into a published, large-scale signaling network model of cardiac fibroblasts via 7 new protein nodes and 29 new reaction edges  = 1–f(x), fand = f(x1)f(x2), and for = f(x1) + f(x2)—f(x1)f(x2). Differential equations were constructed using the open-source software Netflux (https://github.com/saucermanlab/Netflux) for MATLAB 47. Cytoscape was used to create all SNM visualizations included in this paper48. All additional validation, perturbation, and drug screen simulations were conducted using MATLAB; finalized codes used in the analysis are available (https://github.com/SysMechBioLab).As previously described, the reactions of the SNM are governed by logic-based ODEs modeled as a system of Hill equations to capture node activity levelsint = 0, ymax = 1, and time constant (τ) = 1, 0.1, or 10 for signaling reactions, transcription reactions, and translation reactions, respectively. Hill coefficient (n) and EC50 values for the new SNM were set at 1.05 and 0.65, respectively, based on a perturbation analysis. Sex-specific models were created by varying the Ymax of the three estrogen receptors  added to the model to mimic the physiological difference of varied expression and activity levels of estrogen receptors in male and female cells29. The Ymax of the three estrogen receptors was set to 0.5 (50% of maximum saturation) and 1 (100% of maximum saturation) in the male and female SNM, respectively.The complete model, including all species (nodes), reactions, and parameter settings, can be found in the Supplementary Table 26. In total, these 47 papers accounted for 118 perturbations of input outputs/intermediates validated by comparing literature experimental results to model predictions according to their change in activity level as 'Upregulation' (Δ Activity ≥ 5%), 'Downregulation (Δ Activity ≤ − 5%), or 'No Change' (− 5% < Δ Activity < 5%).Previous model validation of the cardiac fibroblast SNM conducted with 47 independent papers of direct measurement of model intermediate and output nodes found the SNM to be 81.8% accurate in predicting experimental activity levels of input-outputs  and input-intermediates  found in the literatureTo conduct a sex-specific validation of the updated model, new papers were added to the validation set to account for estrogen and female-specific data. Each of the papers used for validation measured direct output secretion or intermediate signaling response  to a single input stimulus in fibroblasts. In addition, previous perturbations were categorized based on the cell type used in analysis . The newly expanded validation set contained 185 perturbation experiments, which were 39% (n = 73/185) male, 36% (n = 66/185) pooled, 17% (n = 31/185) unreported, and 8% (n = 15/185) female  for an additional 240h.Additional validation was conducted to determine the accuracy of estrogen involvement in signaling pathways by comparing a combination of simultaneous treatments to experimental results. A tension weight of 0.65 was used, and an input stimuli weight of 1 for AngII, ET1, and E2 was used to compare experimental results from Pedram et al. that reported E2 effect on AngII and ET 1 stimulation on α-SMA, fibronectin, proCI, and proCII in pooled neonate cardiac fibroblasts26. The input reaction weight of estrogen was attenuated, and all other input weights were kept at 0.4 for 80 h to simulate basal conditions. A Ymax = 0.1 knockdown of individual nodes for 240 h was used to identify sensitive and influential nodes in the model. Knockdown sensitivity of each node was calculated as the sum of absolute ΔActivityKD levels for the node across all knockdown simulations, and knockdown influence of each node was calculated as the sum of absolute ΔActivityKD levels for all other nodes in the network upon knockdown nodes. Three perturbation conditions were conducted: (1) male , (2) female post-menopausal , and (3) female pre-menopausal . Heat maps of the total knockdown effect on the SNM nodes were created, and the top ten influential and sensitive nodes were determined for each condition.A network perturbation analysis was conducted in MATLAB, as previously described, to identify influential signaling nodes under different estrogen conditions12. In that prior study, 121 drugs were identified from DrugBank to connect with nodes in the model, totaling 36 unique drug-target interactions spanning agonists and antagonists, competitive and non-competitive50. In our current study, we employed the same framework to perform a sex-specific drug screen using the three experimental conditions used for the perturbation analysis . A static application of drug was administered (w = 0.85) with profibrotic stimuli  for each of the three experimental conditions. To compare potential changes to cardiac fibroblast activity due to drug treatment, a Matrix Content Score was determined for each condition as MCS = Actvitymatrix + Activityprotease + Activityinhib where Activitymatrix = (proCI + proCIII + fibronectin + periostin + osetopontin + LOXL1)/6; Activityprotease = /7; and Activityinhib = (TIMP1 + TIMP2 + PAI1)/3.In previous work, Zeigler et al. developed an approach to simulate the effects of known drugs on their cardiac fibroblast network model26. Additionally, 7 new papers were added to the validation set to account for estrogen and female-specific signaling57. Once disaggregated, the validation set contained 185 perturbation experiments, which were 39% (n = 73/185) male, 36% (n = 66/185) pooled, and 17% (n = 31/185) unreported, and 8% (n = 15/185) female. For the scope of this paper, the 31 perturbations validated with experiments in the literature that did not report cell sex were not included in the rest of the analysis.To assess if model accuracy was influenced by sex, a sex-specific analysis was conducted using the sex-specific SNMs for model simulations compared with sex-specific experimental data reported in the literature Fig. . Previou26. When compared with 54 independent studies, 77% (119/154 simulations) of model predictions matched experimental results in the literature. Model predictions of estrogen treatment specifically were 59% (24/41 simulations) accurate across experimental conditions. Model predictions were the most accurate for the male SNM  and least accurate for the female SNM . The pooled SNM was 77% accurate (51/66 simulations). It is important to note that the female SNM model validation was only conducted with 15 simulations from 3 papers. This is because very few papers report female-specific data for cardiac fibroblasts and is likely the cause of the lower accuracy of the female SNM compared to the male and polled SNMs.Model predictions were grouped into three categories: increase, decrease, and no change. High accuracy of the model was relatively maintained after the integration of E2 in the SNM 51. Additionally, the inclusion of E2 into the model did not impact the accuracy of model predictions for other major cellular inputs, including AngII and TGFB. The lower accuracy of the model in predicting interleukin 1 (IL1), interleukin 6 (IL6), and neutrophil elastase (NE) effect on cellular outputs is consistent with the previously published SNM's limitations26.Notably, the model was accurate in predicting the divergent effect of estrogen treatment on male and female proCI and proCIII production, with E2 treatment increasing collagen production in males and decreasing it in females49. The model predictions were consistent with in vitro results of the study, accurately predicting an increase in fibronectin, collagen 1, collagen III, and α-SMA production when stimulated with AngII or ET 1. Likewise, when these fibrotic agonists are paired with E2, the model predicts a return to control level values of the measured outputs. E2 stimulus alone also produced relatively little changes from control level predictions similar to experimental results. The only instance in which the model did not qualitatively match experimental outputs was that it failed to predict a substantial increase in α-SMA production for cells stimulated with ET-1, and therefore, the effects due to combined treatment with ET-1 and E2 were minimal.The experiments used in the validation set were all an analysis of a singular biochemical stimulus on cardiac fibroblasts. To determine if the model could accurately predict interactions between E2 and other fibrosis-driving input conditions, we compared simulation predictions against experimental data from Pedram et al. that investigated the effect of estrogen in conjunction with the fibrotic agonists' angiotensin II (AngII) and endothelian-1 (ET−1) on pooled male + female neonate cardiac fibroblasts Fig. 49. The m58.A perturbation analysis was conducted to computationally uncover the downstream effect of attenuating estrogen levels in the sex-specific SNMs Fig. . In atteThe top ten influential and sensitive nodes were determined for each condition Fig. B–C. TensIL6 and fibronectin were sensitive across each estrogen perturbation. CBP, proMMP12, periostin, mTORC2, CREB, epac, p70S6K, and thrombospondin4 were in the top ten sensitive nodes only for the female pre-menopause condition. ProMMP14, proliferation, TNC, proMMP2, NFKB, AP1, and proMMP9 were sensitive in the female post-menopause and male condition. Only p38 and PP1 were sensitive in just the female-post or male estrogen perturbations, respectively.12. The effect of the drugs on fibrotic factors was analyzed when stimulated with various input stimuli with the same experimental conditions used in the perturbation analysis. The resulting changes in output expression levels indicate that drug effects tend to follow similar patterns for the post-menopausal female and male conditions. In contrast, the pre-menopausal female condition exhibits a more unique pattern. In particular, the pre-menopausal female model predicts more profibrotic drug effects and fewer anti-fibrotic drug effects. This divergence is most dramatic in the case of NFKB + TNFa combinatory antagonism, which predicts increases in matrix production and matrix content in the pre-menopausal female model but decreases in matrix production and matrix content in the post-menopausal female and male models. Individual treatments of TNFa antagonism or IL6 antagonism are also predicted to increase matrix content in pre-menopausal condition without affecting the other conditions. In total, six different drug strategies were predicted to be anti-fibrotic for post-menopausal female and male conditions with neutral or profibrotic effects in the pre-menopausal case.A sex-specific drug screen Fig.  was condDeveloping a therapeutic strategy to target cardiac fibrosis has remained elusive due in part to the complex signaling mechanisms of cardiac fibroblasts, which can be influenced by biochemical factors, mechanical forces, and biological sex. In this study, we aimed to develop sex-specific network models of cardiac fibroblast signaling by integrating estrogen signaling into a previously published SNM of cardiac fibroblast biochemical and biomechanical pathways to understand sex-specific divergences in fibrotic signaling networks. Additionally, we conducted a sex-specific drug screen to demonstrate the model's potential for facilitating the discovery of sex-specific treatment recommendations.27. The accuracy of both the female SNM and the estrogen pathways was 15–20% lower than the total SNM accuracy. This is likely because there are limited studies available in the literature that can be used to build and validate the model. Female data are consistently underreported in the literature, and even when female samples are included in analysis, almost no studies report the effect of fibrotic agonists on female cardiac fibroblasts outside the context of estrogen59. Therefore, it limited our analysis of the female SNM accuracy in predicting the effect of the other model inputs, such as AngII and TGFB, on outputs and intermediates. The NIH requires the use of both male and female cells and animals in funded research. However, the vast majority of studies that adhere to these guidelines do not disaggregate their results by sex60. In the future, we recommend sex-specific reporting to help fill in the gaps in our understanding of sex-specific fibrotic signaling.Integration of estrogen into the SNM did not negatively alter its previously high predictive power in simulating experiments from the literature. The male SNM was the most accurate, which is unsurprising since the previously published versions of the model were built and validated using reported experiments that were overwhelmingly conducted with male cells64. Model predictions identified numerous other proteins that were sensitive to the different estrogen perturbations. Still, there is currently a lack of sufficient experimental data to determine if these findings are valid in vitro or in vivo.The results of the perturbation analysis uncovered potential regulatory nodes that are different between males and females pre- and post-menopause, which therefore warrant further study. Not only did these results highlight differences across distinct pathways, but several notable differences were even found across different isoforms in the same family. For example, the matrix metalloproteinase MMP12 exhibited heightened sensitivity in the pre-menopausal female condition, while MMP2, 9, and 14 sensitivities were heightened in the post-menopausal female and male conditions. MMPs are highly involved in fibrotic turnover with diverse functions involved in matrix proteolysis, cell adhesion, feedback signaling, and tissue mechanics, and there is emerging evidence of differences between male and female cells dependent on MMP type65. More recently, studies have exhibited the potential use of Thalidomide to prevent excessive cardiac remodeling in heart failure, but to our knowledge, no studies have investigated the role E2 may play in negating its potential benefits67.The sex-specific drug screen clearly exhibited instances of varied drug response due to higher estrogen levels in the female pre-menopause SNM. For example, NFKB + TNFa combinatory antagonism predicted increases in matrix production and matrix content in the pre-menopausal female model but the opposite response in the post-menopausal female and male models. The NFKB + TNFa combinatory antagonism was modeling Thalidomide, a drug that in the 1950s caused severe congenital disabilities in thousands of children after expecting mothers took it to alleviate morning sickness68. Although our drug screen does capture this likelihood of higher adverse effects in pre-menopausal women, it is possible that the integration of estrogen signaling alone was not enough to account for the sex-specific differences in fibrotic response to different therapeutics between the male and female post-menopausal models.The differences between the male and female post-menopause SNMs were much more subtle compared to the female pre-menopause SNM. Although there were slight differences in the intensity of the effect of the drugs in up or downregulating fibrotic factors in a sex-specific manner, the results did not provide enough evidence to elucidate mechanisms that account for the adverse drug responses that are more likely to occur in females compared to men. A study that investigated the effect of age on adverse drug response in both men and women found that adverse events were the highest for women aged 30–59 but that adverse events being higher in women than men persisted throughout life69. These attenuations were not accounted for and would likely play a role in the downstream signaling response. Future studies could address this by testing with more levels of estrogen and integrating feedback loops for estrogen and its receptors.Many of the limitations of the sex-specific models designed in this study are due to the lack of sex-specific data, particularly female data, available to build and validate the model. An additional limitation of the current models is their lack of quantitative data to infer input parameter values. Without these inputs, the models can only make semiquantitative predictions that can be qualitatively compared to experimental literature but have little clinical significance. Because of this constraint, the model was only validated with estrogen levels of high , medium , and low . However, like other hormones, estrogen levels can fluctuate from person to person and throughout life. Additionally, females undergoing menopause are often considered perimenopausal for several years (median of 4) as hormone levels gradually change70. This could impact the accuracy of the pooled SNM as it is mainly validated with neonatal experiments. The male and female SNMs models were validated with adult cardiac fibroblasts and should be minimally affected by differences between adult and neonatal cardiac fibroblasts.Additionally, much of the data used to build the SNM were from neonatal cardiac fibroblasts, not adult cardiac fibroblasts. While neonatal cardiac fibroblasts have their experimental advantages, such as being cheaper and quicker to acquire, recent research also indicates there could be significant differences in cellular regulation, especially related to regenerative capacity, which is related to cardia remodeling72. Future studies should address this by integrating genomic or transcriptomic data into the model to account for additional differences due to biological sex in addition to gonadal hormones73.A final limitation of the sex-specific SNMs was the use of only estrogen signaling to model sex differences due to the limited literature available that investigates hormonal responses in cardiac fibroblasts. Not only are there additional sex hormones  that could be incorporated in the future, but there are also known biological sex dimorphisms that are independent of gonadal hormones. Research shows that genetic differences between males and females can also contribute to cardiovascular disease presentation and severityWe integrated estrogen signaling into a cardiac fibroblast SNM in order to make sex-specific predictions of cardiac fibrosis. These models were validated to be 77% accurate in predicting experimental outcomes from the literature, making them a valuable tool to further study the sex dimorphisms in cardiac fibroblasts. Estrogen's cardioprotective effects were evident in the drug screen of the pre-menopausal SNM. Differences between the male and female pre-menopausal conditions perturbation analysis and drug screen indicate several potential regulatory mechanisms that warrant further study. Future model development and validation will require more generation of sex-specific data to further enhance modeling capabilities for clinically relevant sex-specific predictions of cardiac fibrosis and treatment.Supplementary Tables.


E. coli core model. Additionally, we compare the robustness and the complexity of 30 different models of metabolism, connecting structural and biological properties. Our findings show that antibiotic resistance is associated with high structural robustness, while the complexity can distinguish between eukaryotic and prokaryotic organisms.Metabolic networks are probably among the most challenging and important biological networks. Their study provides insight into how biological pathways work and how robust a specific organism is against an environment or therapy. Here, we propose a directed hypergraph with edge-dependent vertex weight as a novel framework to represent metabolic networks. This hypergraph-based representation captures higher-order interactions among metabolites and reactions, as well as the directionalities of reactions and stoichiometric weights, preserving all essential information. Within this framework, we propose the communicability and the search information as metrics to quantify the robustness and complexity of directed hypergraphs. We explore the implications of network directionality on these measures and illustrate a practical example by applying them to a small-scale  A metabolic network ,2,3,4,5 Given the network structure of metabolism, many researchers have attempted to characterize and understand it through network theory. It has been shown that graphs whose nodes are metabolites and are connected by chemical reactions have a scale-free distribution  and haveThis article aims to lay the foundation for a theory of metabolic networks based on hypergraphs. We describe the method by which each metabolic network can be represented as a hypergraph and introduce two applicable measures, namely, communicability and search information.The work is organized as follows. In In this section, we give a formal definition of metabolic hypergraphs and introduce the notation that is used to characterize them.V such that different nodes interact with each other if and only if they belong to the same hyperedge. Thus, unlike traditional graphs, where edges connect pairs of nodes, hyperedges represent interactions involving multiple nodes. If the dimension A hypergraph To interpret metabolic networks as hypergraphs, we first need to define a special type of hypergraph introduced by Chitra et al. . A hyperIn this paper, we deal with directed hypergraphs, which are an extension of directed graphs. In a directed hypergraph, each hyperedge is associated with a direction similar to the direction of an arrow connecting two vertices in a directed graph. In this context, a hyperedge v. Similarly, v is contained. We also use Additionally, we define N vertices and M hyperedges, the incidence matrix is the matrix Given a directed hypergraph H = E. coli and propose a directed graph with reactions as nodes that take into account the directionality of the reactions, highlighting the difference with the undirected counterparts.In this article, we focus on metabolic networks. A metabolic network  is a setS can be rewritten using the EDVW matrix However, reactions are intrinsically higher-order interactions since they can occur only when all reactants are present. In There are different techniques for studying metabolic networks. Popular methods employ kinetic metabolic models ,40 and sAG=S^TS^ ,38, wherposed in  already posed in . In thisThe main advantage of the metabolic hypergraph framework we propose is that it captures all the physical properties that a metabolic network displays: the directness of reactions, the higher-order interactions, and the chemical properties like mass conservation, due to the inclusion of weights. This framework represents a link between network theory and biology. Another commonly employed method for analyzing large-scale metabolic network models is constraint-based metabolic modeling, such as flux balance analysis. Flux balance analysis (FBA) is used to obtain steady-state reaction rates that are consistent with a metabolic network and linear constraints on the reaction rates, without necessitating any knowledge about the kinetic parameters . FBA is We remark that the previous graph representation of metabolic networks can be seen as a pairwise projection of a metabolic hypergraph. For example, the RAG is an undirected projection of the hypergraph, as in , and the.json format is used. The data contain information on metabolites, reactions, and genes. Metabolites are identified by a Bigg ID, consisting of an abbreviation defining their type, for example, “h” for hydrogen and “ATP” for adenosine triphosphate, and a subscript indicating the compartment to which they belong. Regarding the reactions, in addition to their IDs, the metabolites belonging to them are given, with their respective stoichiometric coefficients. We work in the convention in which a metabolite with a positive stoichiometric coefficient is a product; otherwise, it is a reactant. In the BiGG dataset, the direction of the reactions is also determined using the parameters “lower_bound” and “upper_bound”. These parameters are associated with each reaction and correspond to the maximal flux of metabolites that can flow through. Values of lower_bound Mycobacterium tuberculosis H37Rv have some reactions with lower_bound Synechococcus elongatus PCC 7942, the bounds are obtained experimentally. We decide to proceed with this convention, but using relaxed reaction boundaries to include these reactions is also a valid option.In our experiments, metabolic hypergraphs are generated from the stoichiometry matrix of the models stored in the BiGG database . We analAll the metabolites present in the BiGG models were kept; we did not discard dead-end metabolites.Lastly, we highlight that some hyperedges may have an empty tail or head. These hyperedges correspond to reactions involved in the transportation of metabolites from the outside of the cell to the inside or vice versa. For example, EX_h2o_e  cell has a similar complexity to a yeast cell (Saccharomyces cerevisiae). That is expected. Eukaryote cells have similar metabolic pathways. The additional complexity in human metabolism is due to multi-cellularity, which is not accounted for in this study.The complexity of metabolic networks is anticipated to be quite similar across organisms since they share many common reactions and metabolic pathways. Nevertheless, some differences are expected in the metabolism of aerobic and anaerobic organisms, as well as between eukaryotes and prokaryotes. Aerobic and anaerobic organisms should have a different metabolism because of the different ways they produce energy, while eukaryotes and prokaryotes have significantly different cell structures. With this in mind, we measure the average search information of the 30 different metabolic hypergraphs and report the results in hrocytes  that mayMetabolic networks are very large and complex systems. For this reason, it is important to build a framework able to unite biology and network theory. Many successful studies have represented metabolic networks as graphs with metabolites as nodes, reactions as nodes, or both. Taking a step further, with the employment of hypergraphs, we are able to capture what all of these previous graph representations were missing, the higher-order interactions of reactions. In this paper, we show how metabolic networks are naturally mapped into hypergraphs. In particular, the stoichiometry matrix can be viewed as a weighted incidence matrix of a directed hypergraph with edge-dependent vertex weight. No information is lost representing metabolic networks as hypergraphs: the higher-order interactions between metabolites, the directionalities of reactions, and the stoichiometric weights are all included.Within this novel framework, we propose two measurements to characterize a hypergraph’s robustness and complexity. We apply them to directed hypergraphs with EDVW, but the generalization to undirected and unweighted hypergraphs is straightforward. This approach allows for analysis at the local scale, with communicability and access and hide information, and at the global scale, with natural connectivity as a measure of robustness and average search information as a measure of complexity. We comment on the complications introduced by directionality and how they can be reflected in the measures. To illustrate the practical application of our framework and metrics, we present an example using the e_coli_core model. This small-scale metabolism demonstrates how our metrics operate locally, and it offers valuable insights into the behavior of metabolic hypergraphs. At the global scale, we compare 30 different BiGG models in robustness and complexity, leading to some interesting results. We show that the metabolisms of organisms that have evolved resistance to antibiotics are associated with hypergraphs that display high robustness. Furthermore, we observe that eukaryotic and prokaryotic organisms have different complexity values.In our analysis of complexity, we excluded the source and sink reactions because they create problems when computing the search information (they are unreachable hyperedges). Another possibility could be to add a boundary node, representing the environment around the cell, that links the sinks with the sources. In this way, the search information is no longer ill defined and the external reactions could be included in the analysis. However, the introduction of such externally may have undesired effects on the measures, like introducing new and biologically unmotivated shortest paths. It is worth mentioning that an additional boundary node could be crucial when incorporating hypergraph dynamics in the model. A possibility for future works could be modifying the definition of the average search information and the probability of taking a step in the hypergraph. Here, we consider a walk biased by the stoichiometric weights, but more options could be explored. One possibility is to define the probabilities based on the communicability measure or on the rates computed via flux balance analysis ,43. IndeWe believe that this framework represents a promising approach to bridging network theory and biology. We hope that it may serve as a starting point, potentially reaching experts in the field who could further refine and utilize these findings to obtain more biological insights.


Background: Whole-genome models (GEMs) have become a versatile tool for systems biology, biotechnology, and medicine. GEMs created by automatic and semi-automatic approaches contain a lot of redundant reactions. At the same time, the nonlinearity of the model makes it difficult to evaluate the significance of the reaction for cell growth or metabolite production. Methods: We propose a new way to apply the global sensitivity analysis (GSA) to GEMs in a straightforward parallelizable fashion. Results: We have shown that Partial Rank Correlation Coefficient (PRCC) captures key steps in the metabolic network despite the network distance from the product synthesis reaction. Conclusions: FBA-PRCC is a fast, interpretable, and reliable metric to identify the sign and magnitude of the reaction contribution to various cellular functions. Genome-scale metabolic models (GEMs) that combine functional annotation of the genome with available metabolic knowledge are an valuable tool for modern computational and systems biology ,2. GEMs FBA is a computational approach used to analyze and predict the metabolic behavior of an organism using GEM . FBA is Kinetic modeling involves creating mathematical models that describe the behavior of complex systems, such as chemical reactions, biological processes, or ecological systems. These models often contain a large number of parameters, each representing a specific aspect of the system, such as reaction rates, enzyme concentrations, or external inputs. However, not all of these parameters are equally important for the behavior of the system. Some parameters may have little influence on the system trajectory, and their values may be difficult or impossible to estimate from experimental data. These unimportant parameters are called unobservable parameters. Identifying unobservable parameters is important because it can help simplify the model and reduce the number of unknowns that need to be estimated. One approach to identifying unobservable parameters is global sensitivity analysis (GSA), which estimates how variations in each parameter value affect the behavior of the system as a whole . GSA canOne approach to identifying unobservable parameters in FBA is Flux Variability Analysis (FVA). FVA estimates the range of possible flux values for each reaction in the network, while keeping the objective near its optimal value. This allows researchers to identify reactions that are essential for the network’s behavior, which have narrow flux distribution, as well as reactions that can be varied without significantly affecting the network’s output. However, FVA does not provide information about flux interactions and how fluxes influence the objective value when it is far from optimum, which can be important for understanding the behavior of complex networks. To analyze flux interactions, Kelk and colleagues have developed a method called CoPE-FBA , which uRecently, a global sensitivity analysis (GSA) of constraint-based models was published in the literature . This tySobol variance-based sensitivity analysis is a powerful tool for quantifying the contribution of individual parameters and interactions between parameters to the variability of the model output. It is based on the decomposition of the variance of the model output into contributions from individual parameters, as well as combinations of parameters. This allows the authors to identify which parameters have the greatest impact on the output, and to quantify the degree to which the interactions between parameters affect the model behavior. This method of GSA is computationally expensive and requires the development of a complex computational infrastructure. This may limit its applicability in some contexts, particularly for models that are computationally intensive or have a large number of parameters. Moreover, it may require specialized expertise in order to implement and analyze the results of the method.Despite its limitations, Sobol variance-based sensitivity analysis remains a powerful tool for GSA and can provide valuable insights into the behavior of complex models. It is important for researchers to carefully consider the trade-offs between computational cost and analytical power when selecting a method for GSA, and to choose a method that is well-suited to the specific needs of their study.In response to the limitations of Sobol variance-based sensitivity analysis, a new approach has been proposed for estimating objective function sensitivity to flux boundary values using Partial Rank Correlation Coefficient (PRCC) calculations . The PRCOne advantage of the PRCC approach is that it does not require extensive coding and can be implemented using standard flux balance analysis (FBA) tools, such as the Cobrapy toolbox . The calAnother benefit of the PRCC approach is that the sensitivity coefficient is signed, allowing researchers to distinguish between parameters that positively or negatively influence the objective function. This provides additional insight into the behavior of the model and can help guide the selection of interventions or modifications to the system.To calculate the PRCC sensitivity coefficient, a set of random points in the parameter set is sampled using the Sobol low discrepancy sequence, as described in previous work ,26,27,28Marino and co-authors  provide In summary, the PRCC approach provides a computationally efficient and flexible alternative to Sobol variance-based sensitivity analysis for conducting GSA in constraint-based models. Its ease of implementation and ability to provide signed sensitivity coefficients make it a valuable tool for studying the behavior of complex systems.w is the objective coefficient vector and The metabolic network with is (FBA) , some opThere is a special type of reaction in the constraint-based modelling called ‘boundary reactions’, which usually describe the exchange of metabolites between the internal ‘cell’ and the external ‘environment’.Define parameter space: for non-boundary irreversible reactions only one parameter Generate a set of quasi-random low-discrepancy points in the parameter space. Update parameters (reaction bounds) and find the optimal objective value for each point in the parameter space.Calculate Partial Rank Correlation Coefficient (PRCC) for each parameter and objective value. The statistical significance of the PRCC value is estimated as described by Marino et al. . The sufOur approach consists of three steps:E. coli str. K-12 substr. W3110 WGMM was taken from BiGG database [The toy model  was creaFor network distance calculations, all metabolites participating in more than 20 reactions in any compartment, except amino acids, succinate, PEP, and fructose-6-phospate, were removed. Network distance was calculated as the number of reaction steps between the node of interest and objective reaction. Calculations were performed with R package ‘igraph’ version 1.3.5 .The Sobol quasi-random low-discrepancy sequence was generated with the python Quasi-Monte Carlo submodule of the SciPy v.1.7.3 . Model sPRCC calculations were performed with R package ‘sensitivity’ version 1.28.0 . TDCC vaAll simulations were performed on the OIST HPC cluster with 8CPU and 64GB per job. Sobol points generation, application to the reaction boundaries and optimization of objectives were performed in chunks of 8192 per job. Calculations of the PRCC sensitivity coefficients were performed on 262,144 Sobol points in chunks of 10 features per job. Convergence of the calculation was controlled by TDCC between consecutive datasets different in 8192 Sobol points. The TDCC value between 262,144 and 253,952 was 0.909. The average execution time was 30 min per job for the Sobol point calculations and 7 h per job for the PRCC calculation.Techniques such as Flux Balance Analysis (FBA) utilize the stoichiometry matrix of the reaction system to estimate steady-state fluxes, which are compatible with the viable state of the system. In FBA, an objective is optimized over the steady-state flux vectors, usually by maximizing the flux through certain reactions. The behavior of constraint-based models is controlled by parameters, such as reaction flux boundaries. Our approach estimates the sensitivity of the model’s objective function to these boundary values.To construct the parameter space, we consider that reversible reactions have two boundaries, whereas irreversible reactions have only one, with the lower bound usually set to zero. Boundary reactions, which describe the transport of matter through the model boundary, require special treatment. To evaluate the sensitivity of the objective function to the presence of various nutrients, all boundary reactions are considered reversible, contributing two parameters to the parameter space.As an example, we consider the toy model described in the Kelk paper  Figure , consistE. coli str. K-12 substr. W3110 (BiGG iEC1372_W3110). Using lower bounds for reversible and boundary reactions and upper bounds for all reactions in the model, we obtained a total of 3730 parameters. The objective coefficient was set to one for the lysine exchange reaction EX_lys__L_e_u. We simulated over 262 K points to obtain reliable values for the PRCC coefficients. Out of 3730 parameters, 55 were significant at the 1% threshold, with 19 lower and 36 upper bounds  and aspartate kinase (lysC) increased lysine titers by 78.1% and 123.6%, respectively [p-value < 1 × 10−16), and for the reversible ASPK reaction, the PRCC coefficients for its upper and lower bounds are 0.0037  and −0.004 , respectively. Although FBA-PRCC identifies the reactions important for lysine production and their contribution, the order is different from the experimental data. For instance, DAPDC has a higher PRCC value but a lower increase in lysine production. However, it is important to note that lysine biosynthesis is highly regulated in the cell, as mentioned in [ectively . In our ioned in , and acc+ production or by shifting NAD/NADH balance.The majority of the 15 parameters that are negatively correlated with lysine production correspond to redox balance by decreasing HUnlike the recently published Sobol variance-based sensitivity analysis for conducting GSA in constraint-based models , the FBAIn this work, we have presented a new, fast, and parallelizable framework for estimating the sensitivity coefficients of reaction boundaries in constraint-based models. We demonstrated the performance of our framework using a 27-reaction toy model and the whole-genome metabolic reconstruction of E. coli metabolism. This is the first time that sensitivity coefficients have been calculated for all boundary values in the whole-genome model. Previous analyses, such as those published in , have beOne area where our approach could be particularly useful is in identifying new antibacterial drug targets. We can do this by identifying reaction boundaries that negatively correlate with biomass production and then finding inhibitors for these reactions. Additionally, we can model combinational therapy by analyzing the FBA-PRCC of perturbed models where certain reactions are inhibited or blocked, similar to our analysis in our previous work, Lebedeva et al. .In the future, we plan to expand the application of our approach by exploring its potential in engineering chimeric bacterial cells or microbial communities. To achieve this, we plan to combine our FBA-PRCC method with our metagenomic analysis platform, ASAR . By doinOverall, we believe that our approach has the potential to be a powerful tool for both fundamental research and practical applications in biotechnology and medicine.


High uric acid is associated with gout, hypertension, metabolic syndrome, cardiovascular disease, and kidney disease. URAT1 (SLC22A12), originally discovered in mice as Rst, is generally considered a very selective uric acid transporter compared to other closely-related kidney uric acid transporters such as OAT1  and OAT3 (SLC22A8). While the role of URAT1 in regulating human uric acid is well-established, in recent studies the gene has been linked to redox regulation in flies as well as progression of renal cell carcinoma. We have now identified over twenty metabolites in the Urat1 knockout that are generally distinct from metabolites accumulating in the Oat1 and Oat3 knockout mice, with distinct molecular properties as revealed by chemoinformatics and machine learning analysis. These metabolites are involved in seemingly disparate aspects of cellular metabolism, including pyrimidine, fatty acid, and amino acid metabolism. However, through integrative systems metabolic analysis of the transcriptomic and metabolomic data using a human metabolic reconstruction to build metabolic genome-scale models (GEMs), the cellular response to loss of Urat1/Rst revealed compensatory processes related to reactive oxygen species handling and maintaining redox state balances via Vitamin C metabolism and cofactor charging reactions. These observations are consistent with the increasingly appreciated role of the antioxidant properties of uric acid. Collectively, the results highlight the role of Urat1/Rst as a transporter strongly tied to maintaining redox homeostasis, with implications for metabolic side effects from drugs that block its function. Disorders of uric acid are widespread and are associated with gout, kidney stones, metabolic syndrome, hypertension, kidney disease, and cardiovascular disease . Under nHowever, while deletion of Urat1/Rst in knockout (KO) mice resulted in the expected effect on renal urate handling, this effect was modest . BecauseHere, we have performed a more targeted analysis of the Urat1 knockout, which has revealed the presence of 8 new metabolites. Interestingly, this set of metabolites is different from metabolites that are elevated in the Oat1 and Oat3 knockout mice; chemoinformatics and machine learning analyses revealed that the metabolites also differ in their general structures and molecular properties. To understand the overall metabolic impact of these alterations, we first performed standard metabolite-based pathway analysis. Analysis of the resulting models revealed seemingly disparate changes in metabolism from loss of Urat1. At the single pathway level, loss of Urat1 was implicated in metabolism of fatty acids, amino acids, pyrimidines, and sugars. Nevertheless, deeper analysis using multi-omics constraint-based modeling with Constraint-based Reconstruction Analysis (COBRA) of genome-scale models (GEMs) revealed that the parts of the different sub-systems that were affected were related to maintenance of the redox state. Moreover, increases in catalase and peroxidation reactions were noted with Urat1/Rst loss. In light of the widely held view of uric acid as an antioxidant, it is, to our knowledge for the first time, possible to appreciate the implication of the redox state changes revealed by multiscale metabolic reconstructions. These results set the stage for new studies on how endogenous biochemical pathways modulated by Urat1, including fatty acid and sugar metabolic pathways involved in metabolic syndrome, regulate redox state at the cell, tissue and whole organism level. Furthermore, since inhibitors of Urat1 are increasingly used to treat human hyperuricemia, the clarification of the role of Urat1 in individual in vivo metabolic pathways as well as overall redox state should be helpful understanding potential drug-metabolite interactions and unexpected toxicity from URAT1 inhibiting drugs.ad libitum access to food and water. Serum was obtained from the whole blood. Mice were adults at least 2 months of age. These animals have been described in previous publications [All experimental protocols involving the use of animals were approved by the UC San Diego Institutional Animal Care and Use Committee (IACUC). All animals were handled in accordance with the Institutional Guidelines on the Use of Live Animals for Research. Adult WT and Urat1 (Rst) knockout males  were houications .Microarray profiling and analyses were performed as previously described . BrieflyIndividual, unpooled samples were measured by the Metabolon analytical system . The samples were subjected to ultrahigh performance liquid chromatography-tandem mass spectroscopy (UPLC-MS/MS) utilizing an ACQUITY ultra-performance liquid chromatography (UPLC)  and a Q-Exactive high resolution/accurate mass spectrometer interfaced with a heated electrospray ionization (HESI-II) source and Orbitrap mass analyzer operated at 35,000 mass resolution . To calculate the p-values, two-way analysis of variance (ANOVA) testing was used .p < 0.05 in the Urat1/Rst knockout plasma  and for narrowing down to a set of relevant molecular properties have been previously described in considerable detail ,22. Six t plasma . This wan Orange .p < 0.05 for the principle analysis . Metabolomic data constraints test with flux variability analysis (FVA) [▸If KO/WT > 1 and the metabolite could only be secreted, then the metabolite exchange was constrained with a non-zero lower bound (10% of maximum secretion)▸If KO/WT > 1 and the metabolite could only be taken up, then the metabolite exchange was constrained with a lower bound that was greater than the minimum▸If KO/WT < 1 and the metabolite could only be secreted, then the metabolite exchange was constrained with an upper bound that was less than the maximum▸If KO/WT < 1 and the metabolite could only be taken up, then the metabolite exchange was constrained with upper bound that was less than zero (90% of maximum uptake)Multiple genome-scale metabolic reconstructions were used to construct a scaffold for data mapping and simulation including Recon3D and multiple microbial reconstructions. Recon3D is the most recent iteration of the human metabolic network reconstruction and has more than twice the number of genes as Recon1 and more than three times the number of biochemical reactions/transporters than Recon1 ,25. The ribed in  and implribed in . The mouribed in . A minimis (FVA)  to classp < 0.001 for the Kolmogorov-Smirnov test. The gpSampler function in the CobraToolbox was used to characterize the steady state solution space with n model reactions as described above). Reaction co-sets were calculated from these points with a 0.95 Pearson correlation coefficient cutoff. Given the fact that reaction order can affect the composition of a minority of reactions in GIMME, we employed a strategy similar to the Model Building Algorithm  KO and WT models. Additionally unconstrained uptake/secretion was permitted for oxygen, sodium, potassium, iron, magnesium, bicarbonate, protons, and water. Prior studies analyzing murine kidney transporter KO models comparing model content with different objective functions including biomass, urea, and ATP production resulted in models with similar composition and size. ATP production was used, given that the transcriptomic data analysis was performed on the organ level (as opposed to single-cell) transcriptomics. Following construction of two GIMME models (WT and KO), the feasible solution space of each model was sampled. Differentially active reactions between WT and KO were than computed from the normalized sampled feasible flux states , requirithm (MBA ), in whiModel specific connectivity analyses were performed by modifying the stoichiometric matrix, S, as defined above. First S was converted to a binary matrix, (Sb·SbT) . Multipl(Sb·SbT) . Model mper se.The Urat1  KO mouse has previously been described . It has ric acid ,34. In tWe have now analyzed the plasma metabolome of the Urat1 KO using a targeted approach measuring over 500 identifiable molecules. This provides a portrait of the systemic physiological changes due to the loss of that portion of urate reabsorption dependent upon Urat1 function. To create a more comprehensive portrait of local as well as systemic changes from the loss of Urat1, we first analyzed the metabolite alterations in KO mouse serum. We then sought to determine the molecular properties of the metabolites accumulating in the Urat1 (Rst) knockout mice and used machine learning methods to determine whether particular sets of molecular properties could help distinguish metabolites altered in the Urat1KO compared with those altered in the Oat1KO and Oat3KO. Finally, to determine functional metabolic consequences of Urat1 KO, we constructed context-specific GEMs using transcriptomic and metabolomic data, comparing the metabolic flux states in WT versus KO mice. Such reconstructions of metabolic networks based in multi-omics data can define, in the context of in vivo gene deletion , alterations in metabolic capacities at multiple scales that are not evident from conventional pathway analysis. Indeed, it can, as is the case here, reveal how seemingly unrelated pathways cooperate, or function in a complementary manner, in the context of the whole tissue or organism.p < 0.05 (and 22 metabolites for p < 0.1)  in the Urat1KO plasma for p < 0.1) . These iThe “molecular properties” of Urat1KO metabolites were analyzed using ICM Chemist Pro (Molsoft) and then compared to molecular properties of Oat1KO and Oat3KO metabolites . The metInstead of individual molecular properties, when we examined a set of molecular properties, the differences between the metabolites altered in the Urat1KO and the two Oat knockouts became even more evident. The chosen set of 8 molecular properties  was narrowed down from a much larger list of molecular properties by analyzing information gain using the Rank widget in the Orange data mining and machine learning software as well as various distributions of single molecular properties. Because the dataset was unbalanced (with only 20 Urat1KO metabolites compared to approximately 4 times as many in the Oat1KO and Oat3KO), for machine learning we compared all 20 Urat1KO metabolites to 20 randomly sampled from the Oat1KO and 20 randomly sampled from the Oat3KO). FreeViz visualization  of the dIn order to simultaneously analyze the metabolomic and transcriptomic alterations in a biologically integrated context and to discern how these disparate alterations in metabolic pathways fit together at the level of the whole tissue and organism, as well as to overcome the fact that a relatively small number of metabolites were altered in the Urat1KO, we used a well-validated systems approach for analysis of metabolism . Genome-The plasma metabolomics data from the Urat1KO can be viewed as a systemic analysis of altered metabolism in the Urat1KO. Nevertheless, the transporter mediating reabsorption of uric acid and possibly other organic anions is almost exclusively in the apical membrane of the proximal tubule of the kidney; thus, loss of the transporter is likely to affect intracellular levels of metabolites and thereby result in altered gene expression in kidney tissue. The metabolic reconstruction provides a portrait of altered local metabolism in these normally Urat1-expressing cells/tissue after deletion of the gene. This view, then, would complement the conventional analyses of altered individual pathways obtained by analyzing the plasma metabolites changed in the Urat1KO. Thus, in order to assess the transcriptomic and metabolomic changes in the Urat1KO on cellular flux states, we used constraint-based analysis approach with GEMs using Recon3D . The paip < 0.1 to determine whether we still observed a dominant effect on the redox state. The pattern persisted with redox cofactors dominating the connectivity analysis  dominated the list of reactions that were present in KO but not WT . Focusinanalysis . Althouganalysis , it is ianalysis . Comparianalysis . CrucialHigh uric acid levels are associated with gout, urate kidney stones, progression of renal disease, cardiovascular disease, hypertension, and metabolic syndrome. URAT1 has generally been held to be a highly selective urate transporter ,10,11,45URAT1 (SLC22A12), originally Rst in mice, is a member of the SLC22 transporter family . The SLCWhile original studies of the Urat1/Rst KO mouse in 2008 also supported a role in urate transport, the alterations in urate levels were not as great as expected from certain human studies, and it was suggested at the time that Urat1 might have roles beyond the transport of urate . Indeed,p < 0.1 and 8 at the cutoff of p < 0.05) such endogenous metabolites in the Urat1KO mice and, interestingly, there is only very modest overlap with the Oat1 metabolomics data and very minimal overlap with the Oat3 metabolomics data. In previously published work, Oat1 and Oat3 KO mice had much larger numbers of metabolites altered in comparison to WT animals; reflecting the broader spectrum of metabolic involvement, including multi-organ interactions [We have identified approximately 20 (at the less stringent cutoff of ractions ,51. ThesHowever, the affected pathways based on standard pathway analysis, were not connected in an obvious way to each other or to uric acid metabolism. That led us to perform an integrated transcriptomic and metabolomic data analysis using GEMs in order to reconcile the disparate metabolic alterations into a broader portrait of changes in metabolism at the level of the kidney and the whole animal. There are have been a growing number of examples for the use of GEMs to integrate and analyze complex physiological responses to drug treatments and gene knockouts ,54,55,56Our analyses provide much needed physiological context to the widely-held view of uric acid as a modulator of redox state ,46. HoweThere are other implications as well. The link to sugar and fatty acid pathways related to ROS detoxification in turn may also provide a potential mechanism for the dysregulation seen in certain types of metabolic syndrome, a common and major feature of which is altered uric acid levels. In this regard, it is noteworthy that another important human urate transporter, SLC2A9, may, as seems to be the case with URAT1, play a role in carbohydrate metabolism.In summary, while URAT1 does not broadly regulate metabolites in vivo as is the case for the closely related multispecific transporters OAT1 and OAT3, neither does it appear to regulated only one metabolite, as is generally emphasized in the literature. The ability to interact with multiple substrates is apparent from the growing list of drugs and metabolites interacting with the transporter in vitro ,57,58,59The results set the stage for a whole new range of future studies on the physiological roles of URAT1-including the potential metabolic side effects of drugs blocking URAT1 function as well as the connection between URAT1 function and pathways involved in metabolic syndrome and pathophysiological states in which the cellular redox state plays a critical role.


However,the widespread adoption of this tool is hindered by limited softwareavailability and computational efficiency. Currently, the most widelyaccepted 13C-flux tools, such as INCA and 13CFLUX2, aredeveloped in a closed-source environment. While several open-sourcepackages or software are available, they are either computationallyinefficient or only suitable for flux estimation at isotopic steadystate. To address the need for a time-efficient computational toolfor the more complicated flux analysis at an isotopically nonstationarystate, especially for understanding the single-carbon substrate metabolism,we present FreeFlux. FreeFlux is an open-source Python package thatperforms labeling pattern simulation and flux analysis at both isotopicsteady state and transient state, enabling a more comprehensive analysisof cellular metabolism. FreeFlux provides a set of interfaces to manipulatethe objects abstracted from a labeling experiment and computationalprocess, making it easy to integrate into other programs or pipelines.The flux estimation by FreeFlux is fast and reliable, and its validityhas been confirmed by comparison with results from other computationaltools using both synthetic and experimental data. FreeFlux is freelyavailable at https://github.com/Chaowu88/freeflux with a detailed online tutorial and documentation provided at https://freeflux.readthedocs.io/en/latest/index.html. It is a powerfultool used in synthetic biology, bioengineering, and medical scienceto understand metabolic phenotypes and the response of a natural oraltered biological system.1 The approachinvolves minimizing the computed and experimentally measured labelingpattern  of metabolitespropagated via a metabolic network from isotopically labeled tracersubstrates to identify the best estimate of intracellular metabolitefluxes under a quasi-steady state assumption that metabolic fluxesand isotopomer distributions are stationary.3 However,this approach is not suitable for autotrophic systems that consumesingle-carbon (C1) substrates such as CO2, CO, or CH4 because the labeling pattern of metabolites eventually becomesfixed irrelevant of fluxes. To solve this, isotopically nonstationary(INST) MFA was proposed, which involves measuring a series of transientMDVs at various time points to capture flux information.5 However, 13C INST MFA requires costly experimental andanalytical procedures and presents significant computational challengesfor nonspecialists, which impedes its widespread use. Several computationaltools have been developed to facilitate data analysis, such as 13CFLUX2,6 INCA,7 Metran,8 OpenFLUX2,9 OpenMebius,10 WUflux,11 and FiatFlux,12 but they are either not open-source or builton the MATLAB platform, which can be cost-prohibitive. Additionally,most of the tools are primarily used for flux estimation at isotopicsteady state. Recently, Python-based tools such as FluxPyt,13 mfapy,14 and influx_s15 have been developed, although increasing thecomputational efficiency remains a challenge for transient MFA.16 to enable high-throughput metabolicphenotyping and can be used for validation. We assessed the performanceof FreeFlux by comparing its flux estimation results at steady stateand transient state with those obtained using other 13CMFA tools in Escherichia coli and Synechocystis models, respectively, using both syntheticand experimental data. To assist users, we provide an online tutorialand comprehensive documentation. By making this package freely available,we aim to enhance the accessibility of 13C fluxomics methodologyto a broad user base, ultimately benefiting the community of metabolicengineering and synthetic biology at large.As we enter the era of big data, the rate at which we are acquiringvast amounts of metabolic information is increasing rapidly. However,the development of new tools to process and analyze these data isnot keeping up with this pace. Therefore, each new approach to fluxomicmodeling and analysis requires corresponding advancements in tooldevelopment. We present here FreeFlux, a Python package designed toenable fast and reliable fluxomic phenotyping for both isotopicallysteady state and transient state. The package follows an object-orienteddesign, providing a collection of classes and methods for handlingMDVs, metabolites, reactions, metabolic models, and conducting labelingpattern simulations and flux estimation at both isotopically steadystate and nonstationary state. FreeFlux is highly flexible and canbe easily integrated into other pipelines or frameworks. For instance,it can generate synthetic training data sets that relate measurableMDVs and flux distributions for machine learning or deep learningmodels13C MFA using FreeFlux begins with building a modelwhich consists of biochemical reactions with atom mapping indicatingthe correspondence of carbons in the reactants and products of eachreaction .18 The feasible range of fluxes obtained by FVA can be usedto constrain the sampling space for the initial guess in flux estimation.To simulate the MDV of metabolites, the Simulator requires a definedflux distribution and substrate labeling strategy of the metabolicnetwork. In the case of dynamic MDVs in the 13C nonstationarystate, additional input of metabolite pool sizes is necessary to simulateMDVs as a function of time. The core function of 13C MFA,flux estimation, is performed by the Fitter/InstFitter object of themodel. In addition to the labeling information of the substrates,experimental measurements such as the MDVs at steady state or thetime-course MDVs at transient state, as well as exchange metabolicfluxes such as the substrate uptake rate and product secretion rate,are required. If prior knowledge is available, bounds can be set forthe estimates to assist in the convergence of fitting. Various statisticalanalyses can be conducted on the fitting results, including the χ2 test and normality test of the residuals. The symmetric confidenceintervals (CIs) can be calculated using a local estimation method.19 The contribution matrix at convergence can becomputed and the sensitivity analysis can be performed to evaluatethe importance of measurements. Additionally, FreeFlux offers a MonteCarlo method for estimating the CIs, which provides a more accurateestimation of the uncertainties.reaction 1.By calE. coli model20 in isotopically steady state, and a Synechocystis sp. PCC6803 model5 in a transient state. The network reactions of both modelsare shown in Tables S1 and S6, respectively.To generate the synthetic data, we used the optGpSampler21 of the COBRApy toolbox22 to sample a reference flux distribution, and simulated the MDVsof target metabolite fragments using FreeFlux with a specified labelingstrategy (we assume a constant pool size of metabolites for transientMDVs). The simulated MDVs were added with 1 mol % uncertainties. Thesynthetic data of E. coli and Synechocystis are listed in Tables S2, S3, S7, and S8, respectively.To assess the performance of our computationaltool, we first usedsynthetic data to conduct flux estimation using two well-establishedmodels: an E. coli and Synechocystis models using syntheticdata show good agreement with the reference, as indicated by the smallmean absolute error (MAE) of 3.0 for the E. coli model and 3.6 for the Synechocystis model, and coefficient of determination (R2) of 0.997 and 0.990, respectively. In the E. coli model, the three reactions with the largestestimated CIs are atpm: ATP → ATP.ex, ppc: phosphoenolpyruvate+ CO2 → oxaloacetate and ppck: oxaloacetate + ATP→ phosphoenolpyruvate + CO2. Stoichiometrically,ppc and ppck form a futile cycle of carbon that consumes only ATPand are coordinated with the atpm reaction to maintain the energybalance. This suggests that the estimated fluxes of these reactionsare not individually constrained for a unique solution, explainingthe uncertainty in their flux estimates. In the Synechocystis model, the CIs estimated  and transient state (Synechocystis) and compared with the results obtained from Metran8 and INCA.7 We obtained the experimentaldata for the E. coli model from Crownet al.,23 in which a mixture of [1-13C] glucose and [U-13C] glucose (4:1) was usedas tracers. We corrected the experimentally measured MDVs from proteinogenicamino acids for the natural abundance of isotopomers with FreeFlux,and the corrected values are listed in Table S4. We also listed the measured fluxes in Table S5. For the Synechocystis model,we used the time-course MDVs experimentally measured by Gopalakrishnanet al.24 with natural isotope abundancecorrected. The measurements used for flux estimation are providedin Tables S9 and S10. To achieve betterfitting, we considered the dilution effect25 of intracellular CO2 and unlabeled amino acids for the E. coli model, and metabolically inactive intermediatesfor the Synechocystis model duringflux estimation, consistent with previously reported data.23 The comparison of calculated and experimental MDVs at convergenceof both models is provided in Figure S1 and S2, respectively.The performance of FreeFlux was further evaluated usingexperimental data at isotopic steady state , we calculateda contribution matrix, which quantifies the relative impact of eachmeasurement to the uncertainty of each flux estimate.2613C INST MFA. For instance, the MDVsof the DHAP_123 metabolite fragment collected at the initial threetimepoints  account for over 10% of the variancesin flux estimation for both Calvin–Benson–Bassham (CBB)cycle reactions, prk and rbc1, catalyzed by phosphoribulokinase andribulose-1,5-bisphosphate carboxylase, as well as for the intersectingreactions, tpi and gapdh, catalyzed by triose-phosphate isomeraseand glyceraldehyde-3-phosphate dehydrogenase, respectively, whichconnect the CBB cycle and glycolysis. Similarly, the measurement at20 s of sedoheptulose-7-phosphate significantly impacts the CBB cyclereactions catalyzed by transaldolase , sedoheptulose-1,7-bisphosphatealdolase (sba), and sedoheptulose-bisphosphatase (sbp). This evidencehighlights the need for accurate determination of labeling patternsof intermediates within the initial time window in rapidly metabolizingupstream pathways. In comparison, the flux uncertainties of tricarboxylicacid (TCA) cycle reactions are primarily dictated by measurement errorsof TCA metabolites, including citrate, succinate, and fumarate. Interestingly,measurements of these metabolites conducted after 90 s notably contributeto the uncertainty of pathway flux estimates, consistent with theTCA cycle’s minor metabolic activity in cyanobacteria.27 Moreover, photorespiration-relevant reactions(gt and glyk) that convert glyoxylate to 3-phosphoglycerate, heavilyrely on the measurements of multiple metabolites including 3-phosphoglycerate,phosphoenolpyruvate and glycerate. These redundant measurements arebeneficial as they reduce reliance on a single mandatory measurementof a certain metabolite for an accurate estimation of flux in the corresponding pathway. Overall,guided by the computed contribution matrix, meticulous measurementsof metabolites that have a substantial impact on the pathway fluxof interest become crucial for reliable analysis.tal data 3D. G-ValE. coli and Synechocystis models with synthetic and experimentaldata, respectively. Synechocystis modelin under 30 min and estimated 95% CIs in approximately 2 h (100 runs)and 8.5 h (500 runs), respectively. These time efficiencies are notableconsidering the typically time-intensive nature of flux estimationin the 13C INST state and contrast favorably against otheravailable optimization-based tools for 13C MFA  method is used to establish the functional relationship betweenMDV and flux distribution and to identify the minimal set of precursorsthat contribute to the labeling pattern of a target EMU, which significantlyreduces the variables and computation complexity.28 The EMU network decomposition is implemented using an adjacencymatrix29 and is performed in parallel withmultiple starting EMUs, which is wrapped in the “decompose_network”method. The resulting EMU networks are combined by size and storedas Pandas DataFrame objects, which can be exported as required.FreeFlux used a modelof a metabolicnetwork consisting of biochemical reactions to perform MFA. The Modelclass is used to create an instance of the network, and reactionscan be added individually or read from a file using the “read_from_file”method. The basic information about the reactions, such as reactantstoichiometry, reaction reversibility, and tracer atoms, should beprovided. The metabolic flux of a reversible reaction is split intoforward and backward flux, which are combined with the fluxes of irreversiblereactions to form “total fluxes”. By specifying a distributionof total fluxes and a labeling strategy , the forward problem of 13C] glucose, 20% [U-13C]glucose, and 100% [U-13C] glutamine. The MDV simulationfor the isotopic steady state is performed by the “simulate”method of a Simulator object created from the Model and an InstSimulatorobject for the 13C INST state. This yields a single isotopomerdistribution and a trajectory of isotopomer distributions with time,respectively.Simulation of labeling patterns is essentialin optimizing the selection of isotope tracers and designing experimentsfor maximum information gain. The simulator requires inputs such asa defined flux distribution, metabolite concentrations, and labelingstrategy. For each labeled substrate, the fraction and isotopic purityof every added isotopomer should be specified. FreeFlux supports strategiesfor multiple labeling substrates, for example, by the addition ofa mixture of 80% [1-Prior to flux estimationusing labelingdata, FreeFlux provides constraint-based analyses, that is, FBA andFVA through the Optimizer module. FBA seeks an optimal intracellularflux distribution maximizing/minimizing the objective, such as biomassformation. On the other hand, FVA is used to identify the feasiblerange of fluxes while maintaining an optimal or suboptimal of theobjective under the same constraints of uptake and production ratesas FBA. The results of FVA are stored in an FVAResults object andare used to determine the sampling region of the initial guess offlux estimation.13C MFA, flux estimation is performed by aFitter object. In addition to feasible flux ranges and labeling strategy,the Fitter requires inputs of measured MDV of metabolite fragmentsat steady state and experimental exchange fluxes such as substrateuptake rate, product secretion rate, and growth rate. Extra boundsfor fluxes can also be set. By calling the “solve” method,FreeFlux attempts to find the best estimates of fluxes by minimizingthe weighted residual sum of squares between simulated and measuredMDVs and exchange fluxes through a nonlinear and nonconvex optimizationprocess.19 Flux estimationfor 13C INST state is accomplished by the InstFitter, whichinstead requires a series of MDVs measured at various time points.During the optimization process, the InstFitter solves systems ofordinary differential equations (ODEs) to simulate the transient MDVswith respect to time at each iteration (Supporting Information Methods). A noncasual first-order-hold equivalentmethod is used for numerical approximation of the ODE solutions whichsignificantly improved the computation efficiency24 (Supporting Information Methods).FreeFlux provides two freely available solvers for the optimizationproblem: “ralg” and “slsqp”, which applythe r-algorithm with adaptive space dilation30 and the sequential least squares programming,31 respectively. To obtain the best flux estimation, usersare encouraged to repeat calling the “solve” methodwith a randomized initial guess.As the primary function of FreeFlux and thecore task of 2 testand normal probability plot, can be used to determine the goodness-of-fitand normality of the weighted residuals. FreeFlux calculates the estimatedCIs by a local method using the Hessian matrix at convergence. TheHessian matrix is further used to determine the contribution of measurementvariances to the uncertainties of estimates and the sensitivitiesof estimates to the perturbation of measurements.19 The method “solve_with_confidence_intervals”is also provided by FreeFlux to evaluate the uncertainties using MonteCarlo simulation, in which measurements are added with Gaussian noises,the computation is repeated for a specified number of times, and theCIs are obtained from quantiles of the resulting set of estimates.The fitting routine is completedwhen convergence criteria aresatisfied, such as tolerance for termination or the maximum numberof iterations. Outputs are stored in a FitResults/InstFitResults object,which provides methods for comparing the simulated and measured MDVsand fluxes. Statistical metrics, such as the χ13C enrichment due to dilutionfrom the inoculum, inactive intracellular metabolite pools, or mediacomposition. To obtain a more accurate flux estimation, a dilutionparameter, also known as the G-value, is incorporated to compensatefor the effect.25 FreeFlux can calculate G-values for both measured metabolites, such as amino acidsand intermediates, as well as unmeasured, such as CO2,by introducing several pseudo reactions to the network. Detailed informationon how to use this feature is available in the online documentationof FreeFlux.As labeling experiments progress, metabolites can experience asignificant reduction in 13C MFA. FreeFlux provides the MDV class and a range of methods tocreate and manipulate an MDV instance. The “conv” method,which can also be achieved using the overloaded “*”operator, performs a convolution operation between two MDVs, yieldingthe merged distribution of isotopomers of substrates of a condensationreaction. Additionally, FreeFlux includes the “correct_for_inoculum”and “correct_for_natural_abundance” methods, which correcta raw measurement of MDV for the unlabeled fraction introduced byinoculum and the natural isotope abundance of specific element atomsand their numbers, respectively.26 The“get_natural_MDV” function calculates the natural MDVof an unlabeled EMU with n carbon atoms, while the “get_substrate_MDV”function estimates the MDV of an EMU from a labeled substrate. Notably,besides its application in 13C fluxomics, the MDV classcan be used for data processing in qualitative labeling experiments.For more information on the functions and usage of FreeFlux, pleaserefer to our online tutorial.Thefractional abundance of isotopomers of a metabolite fragment,also known as MDV, is a crucial computation unit for