Escherichia coli lipid A biosynthesis, drawing parameters from the experimental literature. This model accounts for biosynthesis regulation, which occurs through regulated degradation of the LpxC and WaaA  enzymes. The LpxC degradation signal appears to arise from the lipid A disaccharide concentration, which we deduced from prior results, model results, and new LpxK overexpression results. The model agrees reasonably well with many experimental findings, including the lipid A production rate, the behaviors of mutants with defective LpxA enzymes, correlations between LpxC half-lives and cell generation times, and the effects of LpxK overexpression on LpxC concentrations. Its predictions also differ from some experimental results, which suggest modifications to the current understanding of the lipid A pathway, such as the possibility that LpxD can replace LpxA and that there may be metabolic channeling between LpxH and LpxB. The model shows that WaaA regulation may serve to regulate the lipid A production rate when the 3-deoxy-D-manno-oct-2-ulosonic acid (KDO) concentration is low and/or to control the number of KDO residues that get attached to lipid A. Computation of flux control coefficients showed that LpxC is the rate-limiting enzyme if pathway regulation is ignored, but that LpxK is the rate-limiting enzyme if pathway regulation is present, as it is in real cells. Control also shifts to other enzymes if the pathway substrate concentrations are not in excess. Based on these results, we suggest that LpxK may be a much better drug target than LpxC, which has been pursued most often.Lipid A is a highly conserved component of lipopolysaccharide (LPS), itself a major component of the outer membrane of Gram-negative bacteria. Lipid A is essential to cells and elicits a strong immune response from humans and other animals. We developed a quantitative model of the nine enzyme-catalyzed steps of  Escherichia coli cells, covering about 75% of the cell surface area ,2],3],[3]2],2],,5],,[5],2],,2],.Lipopolysaccharide (LPS) is a glycolipid that forms the major component of the outer leaflet of the outer membrane of most Gram-negative bacteria. It occurs with roughly 1 million copies in ,5]5]. The lipid A, or endotoxin component  . These KDO residues connect to several additional sugar residues, and sometimes also to phosphate, pyrophosphorylethanolamine, or phosphorylcholine residues, which together form the core oligosaccharide . This core then connects to the O-antigen, which is a long polysaccharide that varies widely between different bacterial species and different strains within each species . Of these three components, the lipid A moiety is of particular interest because it is the only component that is essential for cell viability and is highly conserved . These also make its biosynthetic pathway an attractive target for new antibiotics ,9],10],[10]9],9],.LPS comprises three components: lipid A, core oligosaccharide, and O-antigen omponent  includes,2],5],,[5],2],,2],. However, it has received remarkably little quantitative analysis, which is essential for testing the internal consistency of models and for investigating pathway regulation mechanisms. In one modeling study, Kenanov et al.  investigated the elementary flux modes  for the biosynthesis of all E. coli lipids. They found close agreement between the predicted and experimental viability of knock-out mutants. This supported the lipid biosynthesis pathway architecture that they used, which is also the one that is commonly accepted and that which we assume in this work. In other modeling work, we recently simulated the chemical kinetics of the first two steps of lipid A biosynthesis . We found that the second enzyme (LpxC) has sufficient catalytic activity to overcome the first enzyme’s unfavourable equilibrium constant. We are not aware of any other computational models of lipid A biosynthesis.The lipid A biosynthesis pathway architecture, shown in E. coli because E. coli has been the subject of most lipid A research. However, the lipid A biosynthesis pathway is well conserved across Gram-negative bacteria , so our model may be applicable to other Gram-negative bacteria as well. We used published parameters where possible and estimated others as required. Our model can reproduce the observed lipid A production rate and agrees reasonably well with results from several lipid A biosynthesis experiments. These include data that correlated LpxC half lives and cell generation times , and our own experiments on LpxK overexpression. Our model also disagrees with several experimental results. These disagreements highlight potentially interesting biological behaviour such as metabolic channeling and additional sources of pathway regulation.Here, we build on the prior experimental and modeling work to present a quantitative model of the central steps of lipid A biosynthesis. This model is specific to . This level of detail was determined to be adequate because preliminary simulations using spatial stochastic simulations (with Smoldyn ) and non-spatial stochastic simulations (with StochKit ) yielded essentially identical results. The COPASI files are available as supplementary information and will be submitted to the BioModels database.Simulations were performed with non-spatial deterministic methods using the COPASI software E. coli K-12 strain AG1 , supE44, relA1) containing a plasmid (pCA24N)  bearing E. coli LpxK-GFP gene fusion (to the C-terminus) was obtained from the National BioResource Project (NIG) Japan. Cells were grown at 30°C in LB media  containing 20 μg/ml of chloramphenicol and when required, protein expression was induced using IPTG .An ,20]20]. Briefly, an overnight culture was inoculated into fresh LB containing different concentrations of IPTG at an OD600 of 0.05 and grown to mid log phase (OD600 = 0.5). The respective cultures were normalized to the same OD600 of 0.5. 3 ml of normalized culture was centrifuged at 13,000 rpm for 1 min and the cell pellets re-suspended in 100 μl of 2x Laemmli sample buffer . The samples were heated for 10 min prior to centrifugation for 5 min. The supernatants were collected for Western blot analysis.Cell extracts were prepared as described previously 20 μl of each sample were loaded onto a 10% SDS-polyacrylamide gel. Following electrophoresis, proteins were transferred to a PVDF membrane using the Bio-Rad Trans-Blot Turbo system. An LpxC antiserum generated in rabbit (a generous gift from Prof. Franz Narberhaus) and a secondary anti-rabbit peroxidase-linked antibody  were used for immunodetection at dilutions of 1:20000 and 1:10000 respectively. Blots were developed using the ECL chemiluminiscent reagents (Bio-Rad) and the signals detected using the ChemiDoc MP system (Bio-Rad).E. coli lipid A biosynthesis proceeds through nine enzyme-catalyzed steps, which are sometimes referred to as the Raetz pathway ,7]7]  and β-hydroxymyristoyl-ACP substrates. Both substrates are consumed in other metabolic pathways as well in vitro, suggesting that the reaction products are not committed to proceed on through the lipid A pathway but may instead revert back into the pathway substrates. The product is then deacetylated by LpxC (EC 3.5.1.108) in an essentially irreversible reaction, making this the first committed pathway step ,28],,. For this and other reasons, LpxC is likely to be a primary biosynthesis control point  (and is a prime drug target), as discussed below. The third pathway enzyme, LpxD (EC 2.3.1.191), incorporates a second hydroxymyristate moiety onto the lipid A precursor ,32]32]. LpxD is similar to LpxA in that they are acyltransferases, and consume the same β-hydroxymyristoyl-ACP substrate . Both LpxD reaction products inhibit the LpxD reaction, acting as either competitive or non-competitive inhibitors against each substrate . We simplified this by only including non-competitive inhibition by UDP-2,3-bis(β-hydroxymyristoyl)-D-glucosamine. Ignoring inhibition by ACP had minimal effect on our results because we fixed its concentration.The first three steps of the lipid A pathway occur in the cytoplasm. First, LpxA (EC 2.3.1.129) acylates UDP-GlcNAc with β-hydroxymyristoyl-ACP. This reaction has an unfavourable equilibrium constant of 0.01  and LpxB (EC 2.4.1.182) . LpxH cleaves most of the UDP moiety to leave just a single phosphate on the remaining lipid portion, which is called lipid X. Feedback regulation has not been proposed before for LpxH but proved necessary for our model to achieve steady-state behaviour, as described below. Then, LpxB combines lipid X with the preceding lipid metabolite, UDP-2,3-bis(β-hydroxymyristoyl)-D-glucosamine, to form lipid A disaccharide .The fourth and fifth lipid A biosynthesis steps are catalyzed by the peripheral membrane proteins LpxH (EC 3.6.1.54) A,36]36]. Remarkably, Lipid IVA has been reported to be an endotoxin agonist in mouse cells and an endotoxin antagonist in human cells . Next, WaaA  sequentially transfers two KDO sugar residues to lipid IVA to produce KDO2-lipid IVA,39],,40]; in the latter case, the reaction produces “alternate lipid A”, which we define as having more than 2 KDO sugar residues. In addition, the WaaA reaction has been shown to be reversible, based on the finding that in vitro combinations of the reaction products , and KDO2-lipid IVA) led to detectible concentrations of the KDO-lipid IVA intermediate . However, the authors only observed trace quantities of lipid IVA even after prolonged incubations, thus, the forward reaction is likely to be strongly thermodynamically favourable, and in which case, the in vivo back-reaction rate is probably negligible. For this reason, our model treats WaaA catalysis as being irreversible. Finally, the “late acyltransferases,” LpxL (EC 2.3.1.-)  and LpxM (EC 2.3.1.-) ,43]43], incorporate lauroyl and myristoyl chains to the KDO2-lipid IVA, thus giving the final KDO2-lipid A product six acyl chains. Cells are still viable without LpxM, or without LpxL and with overexpressed LpxM . In cold-adapted E. coli, the LpxL function is replaced by LpxP (EC 2.3.1.-), which incorporates a palmitoleate instead of the laurate, presumably as a way of adjusting membrane fluidity .The remaining four steps of lipid A biosynthesis are catalyzed by integral membrane enzymes. LpxK (EC 2.7.1.130) is a kinase that phosphorylates lipid A disaccharide to produce lipid IV2-lipid A is joined to core oligosaccharide and then flipped from the inner leaflet of the inner membrane to the outer leaflet of the inner membrane by MsbA (EC 3.6.3.39), an ABC transporter ,46]46]. Next, several enzymes add the O-antigen to form LPS, and then transport the LPS on to the outer leaflet of the outer membrane .After synthesis, KDO,48],,52]. FtsH is an essential protein due to its role in regulating LpxC ,53]53]. Lipid A biosynthesis regulation is less well established than is the synthetic pathway, so we describe these aspects of our model in more detail.Lipid A synthesis is regulated, at least in part, through controlled degradation of LpxC LpxC, and FtsH*WaaA, respectively). This assumption of substrate-specific FtsH activation is supported by several findings: (i) FtsH degradation of RpoH and λ CII is substrate-specific, accomplished through separate adapter proteins ,54]54], (ii) neither increased nor decreased FtsH degradation of LpxC have a significant effect on the intracellular concentration of RpoH  or the activity of WaaA , and (iii) LpxC is degraded more slowly at higher temperatures whereas WaaA is degraded more rapidly at higher temperatures ,50]50]. Substrate-specific activation of FtsH for LpxC may occur through YciM acting as an LpxC adapter protein .We assume that FtsH can reversibly convert between an inactive state, an active state for degrading LpxC, and a different active state for degrading WaaA -D-glucosamine, lipid X, and lipid A disaccharide. Of these, UDP-2,3-bis(β-hydroxymyristoyl)-D-glucosamine was already regulated by product inhibition, which makes its concentration a poor indicator of pathway flux and hence a poor candidate. Preliminary simulations that represented FtsH knock-out mutants, and hence did not include FtsH regulation, exhibited lipid A disaccharide accumulation . Unless specified otherwise, we assumed a 30 minute (1800 s) doubling time, which is the experimental value in rich media , 61]61]. For the most part, we did not account for protein synthesis, protein degradation, or cell volume growth during a cell generation. These approximations are legitimate because metabolic enzyme concentrations for constitutive enzymes likely remain constant over the course of the cell cycle. We modeled the interactions of individual substrates and enzymes for a single , 22], , , 25]25], so their relative concentrations are controlled by factors outside our simulation. We kept their levels constant at 2 million molecules (5 mM) throughout most simulations, which led to substrate saturation conditions. Additionally, we fixed the ACP level to 1024 molecules, based on proteomic results .We investigated lipid A metabolism with either excess or limiting concentrations of the UDP-GlcNAc, β-hydroxymyristoyl-ACP, and CMP-KDO substrates. The stoichiometric ratios of these substrates to lipid A are 2:1, 4:1, and 2:1, respectively, due to consumption of multiple copies and/or lipid dimerization by LpxB. The first two of these substrates are also precursors for other biosynthetic pathways E. coli cytosolic fractions . Several of these are integral membrane proteins, so their experimental protein counts are likely to be lower limits for their true counts in a cell. We used the same proteomic data source for the copy number of LpxM , another integral membrane protein, but then increased the count 20-fold as described below. We calculated LpxC, LpxH, and LpxB protein counts from protein purification data 33]63][63]33][6[63]33]71]. However, it was legitimate here because our model includes regulatory feedbacks that extend over most of the pathway length. These are alternative ways to enable models attain steady-state, and in fact are typically more effective than reversible reactions . Also, most of the lipid A reactions are likely to be nearly irreversible, due to either favourable energetics or much more abundant substrates than products .We modeled all pathway reactions using single-substrate or bi-substrate Michaelis-Menten mechanisms. We ignored reaction reversibility in most cases. This can lead to misleading results in metabolic models because it ignores feedback effects that arise from product inhibition and hence can prevent models from attaining a steady-state kcat is the enzyme catalytic rate constant, and Km is the Michaelis constant. Most of these kcat and Km values have been published using data from in vitro experiments (i) The specific activity of LpxK in crude E. coli membrane extract was estimated to be 22 nmol/min/mg in a plasmid-containing strain but 7-fold lower in wild type . There are about 432 LpxK molecules per cell and E. coli membranes include about 580,000 individual proteins , so the LpxK purity in crude membrane is about 0.074%. Thus, the pure wild-type specific activity is about 4 μmol/min/mg, from which kcat is about 2.1 s-1. We estimated that the LpxK Km value is 40 μM from a figure presented by Ray and Raetz . Although Km estimations from crude samples are prone to inaccuracies when the substrate can be catalyzed by other enzymes in the lysates, there is no evidence of such competition for the LpxK substrate. (ii) We estimated the LpxM catalytic rate constant, kcat, as 0.6 s-1 from the specific activity of the enzyme in crude lysates , much as we did for LpxK. We estimated the LpxM Km value from data shown in et al. . To do so, we simulated Clementz et al.’s experiment using COPASI , with the same enzyme and substrate concentrations that they used (0.1 μg/mL and 25 μM for protein and KDO2-lipid IVA respectively in a 20 μL reaction mixture), from which we identified the Km value that corresponded to their figure results at a time of 30 minutes. (iii) We treated the MsbA catalyzed translocation of KDO2-lipid A across the inner membrane as another Michaelis-Menten process, setting its Km value to 0.021 mM and its kcat value to 166 s-1, based on data shown in .We used single-substrate Michaelis-Menten kinetics for the LpxC, LpxK, LpxL, LpxM, and MsbA steps. Here, the metabolite flux iseriments , althougKi is the inhibition constant and the other parameters are the same as in . This implies that the pure enzyme specific activity is about 105 μmol/min/mg, which is combined with the LpxH molecular weight of 26.8 kDa , to yield its kcat as about 47 s-1. The estimation of our Ki value is discussed below.We used single-substrate Michaelis-Menten kinetics with inhibition for the LpxH enzyme. In this case, the metabolite flux is2-lipid A steps. Treating the forward and reverse LpxA reactions as independent irreversible reactions is legitimate in non-spatial models, such as ours, because doing so does not introduce any new approximations . Most bi-substrate enzymatic reactions follow either a sequential or ping-pong mechanism . In the sequential mechanism, the enzyme forms a ternary complex with both substrates before catalyzing the reaction. In the ping-pong mechanism, the enzyme binds one substrate, forms one product, and then binds the second substrate and forms the second product. The only bi-substrate reactions in the lipid A pathway that have been investigated in sufficient detail to determine mechanisms are the steps catalyzed by LpxA  and LpxD , both of which were found to follow the sequential mechanism. Lacking further experimental evidence, we assumed sequential mechanisms for the other bi-substrate reactions in the lipid A pathway as well. The sequential mechanism metabolite flux is1] and [S2] are the two substrate concentrations, with respective Michaelis-Menten constants Km1 and Km2. We needed to estimate these parameters in some cases. (i) The LpxA kinetic parameters were determined previously for the forward reaction , 76]76], but not for the reverse reaction. Thus, we assumed the same Km values for the reverse reaction as for the forward reaction , based upon the likelihood that the enzyme binding affinities are not substantially affected by the acyl group transfer. However, we set the reverse reaction kcat value to 100 times that of the forward reaction to account for the reaction’s unfavourable equilibrium constant of approximately 0.01 . (ii) We estimated the LpxB kcat value by starting from the specific activity of LpxB purified to near homogeneity , much as we did for LpxH, which resulted in a kcat value of 129 s-1. (iii) The WaaA specificity for KDO2-lipid A is 8.7 fold lower than for lipid IVA. To account for this, we reduced the kcat value for the former reaction by 8.7 fold while keeping other reaction constants the same.We used bi-substrate Michaelis-Menten kinetics for the forward LpxA, reverse LpxA, LpxB, and WaaA catalysis of KDOFinally, we used bi-substrate Michaelis-Menten kinetics with inhibition for the LpxD and WaaA steps. Assuming the sequential mechanism again, which was shown to be the correct mechanism for LpxD, the metabolite flux iskcat value as 16.7 s-1 from the specific activity of the purified protein . We computed the inhibition constant from results by Belunis and Raetz  which showed that 100 μM of lipid A inhibited the WaaA reaction by 24.1% in vitro. Their experimental conditions involved purified enzyme and substrates, thus excluding the possibility of FtsH playing a role in the inhibition. Whilst assuming a non-competitive inhibition . Together, these processes combine to give the net production rate for each of these proteins aset al which showed that LpxC in wild-type E. coli can have a half-life of 120 minutes under nutrient and temperature conditions that lead to rapid growth. We assumed that the long LpxC half-life arose because FtsH was essentially inactive under these conditions. This implies that the uncatalyzed LpxC degradation has a half-life of about 120 minutes and kdegrade is about 9.62×10–5 s-1. Separately, it has been shown that cells with inhibited FtsH activity, using a temperature sensitive mutant, exhibit 4-fold elevated LpxC concentrations , thus increasing LpxC counts from about 385 molecules to about 1540 molecules. Combining this molecule count with the kdegrade value and the assumption that [FtsH*LpxC] equaled zero in this mutant, enables ktranslate value of about 0.148 molec./s. Next, combining the kdegrade, ktranslate, and the wild-type LpxC count of 385 molecules enables kFtsH[FtsH*LpxC] is 2.89×10–4 s-1. We cannot solve for kFtsH by itself from the available information, but estimate its value below.To determine the production and degradation parameters for LpxC, we started with results presented in Schäkermann , from which we computed that kdegrade is about 1.8×10–4 s-1. Next, the half-life of WaaA in wild-type cells under optimal growth conditions is about 10 minutes . This gives the combined degradation rate constant, kdegrade + kFtsH[FtsH*WaaA], as approximately 1.15×10–3 s-1, implying that the catalyzed degradation rate constant is about 9.8×10–4 s-1. Combining the total degradation rate with the wild-type WaaA abundance of 153 molecules per cell gives the protein translation rate, ktranslate, as 0.176 molecules/s.We computed the WaaA synthesis and degradation parameters similarly. First, a mutant without FtsH maintained about 72% of its WaaA concentration after 30 minutes . We did not account for FtsH sequestration through activation for other degradation targets, such as RpoH or misfolded proteins. We modeled FtsH activation and inactivation with mass action kinetics, meaning that we treated net FtsH activation towards a specific substrate according to2-lipid A for the WaaA substrate), and [FtsH] represents the concentration of inactive FtsH. The first term on the right hand side represents the activation rate and the second represents the inactivation rate.Our model treats FtsH activation in a substrate-specific manner, but with the constraint that the total FtsH count per cell is conserved at 579 molecules kactive, kinactive, and kFtsH. However, the available experimental data only enabled us to quantify the product kFtsH[FtsH*], for each substrate, with values given above. Thus, the system is underdetermined, with multiple possible combinations of parameters values that are each equally good at agreeing with the available data. We addressed this by making a few assumptions. First, we assumed that during growth on rich media, 10% of the total FtsH is activated for degradation of LpxC, 10% is activated for degradation of WaaA, and 80% is inactive . This is intuitively sensible because it assumes a reasonably large reservoir of inactive FtsH to allow for strong regulatory control and other proteolysis tasks. Combining this assumption with the prior values for the kFtsH[FtsH*] products yield kFtsH of 2.0 mM-1s-1 for LpxC and kFtsH of 6.8 mM-1s-1 for WaaA. Secondly, we assumed that both kinactive values equal 0.1 s-1. This gives the active states a 10 s lifetime, which is fast enough to enable rapid control. We then solved for kactive from the steady-state version of kactive is 0.14 mM-1s-1 for LpxC and 32.3 mM-1s-1 for WaaA. Note that the assumptions made here do not affect the model’s steady-state condition at all, making them necessary for running simulations but irrelevant to the results that we present below.The FtsH proteolysis rate, for either LpxC or WaaA, depends on three parameters, Our initial model, defined using literature parameter values where available and our best estimates elsewhere, exhibited a lipid A production rate that was much too low. Also, several internal metabolite concentrations accumulated to very high levels. We addressed these problems with two model adjustments.E. coli cells actually produce per generation. This did not change even if we removed FtsH degradation of LpxC and WaaA and all negative feedbacks. We found that this slow production rate arose from LpxM acting as a bottleneck in the pathway, as seen by its substrate increasing linearly over time, rather than stabilizing at a steady-state level. This may indicate that cells have more than the 186 LpxM proteins that proteomic research on the cell cytoplasm indicated , which would not be surprising because LpxM is an integral membrane protein. Alternatively, it may be that other enzymes acylate LpxM’s substrate in parallel to LpxM; in particular, LpxL and LpxP can catalyze essentially the same lipid A synthesis reaction , . Assuming the former explanation, we increased the LpxM molecule count 20 fold, from 186 to 3720. This removed substrate accumulation upstream of LpxM and caused the model to produce lipid A at about 1 million molecules per generation.Our initial model simulations only produced about 20% of the 1 million lipid A molecules that in vivo. We found that the accumulation arose because LpxH rapidly diverted UDP-2,3-bis(β-hydroxymyristoyl)-D-glucosamine towards lipid X, while LpxB only consumed lipid X in a 1:1 ratio with UDP-2,3-bis(β-hydroxymyristoyl)-D-glucosamine. For this reason, we assumed that LpxH is regulated through product inhibition, as described above. The experimental lipid X count was reproduced when we set Ki to 0.2 μM; however, this value is unusually small and it created a bottleneck in the pathway, which we observed as accumulation of the LpxC product and decreased lipid A production . Thus, we decided to decrease LpxH inhibition by increasing Ki. As mentioned above, mutants with inactive FtsH exhibit 4-fold increased LpxC concentrations ; they were also shown to produce 32% more lipid A . We decided to set the LpxH inhibition constant so that our model would reproduce this result, which turned out to be a Ki value of 0.015 mM. This value was large enough that it did not cause the LpxC product to accumulate in wild-type cells. However, the LpxC product still accumulated in cells without FtsH due to the higher metabolite flux through the LpxC step and the lack of product inhibition at this step. This Ki value caused the steady-state lipid X concentration to be about 22,000 molecules, in either wild-type or FtsH mutant cells, which is much larger than the 2000 that were observed experimentally . We found that this difference cannot be eliminated simply by adjusting enzyme kinetic parameters without creating large metabolite accumulations, which suggests that this region of the biosynthesis pathway includes dynamics that are not in our model. For example, the difference could arise from metabolic channeling between the LpxH and LpxB enzymes.Our initial simulations also resulted in high lipid X concentrations, which rose over the course of several cell generations and stabilized at about 400,000 copies. In contrast, 2000 copies were observed  showed that cells that have defective LpxA copies, which have at least 150-fold lower LpxA specific activities, have 5- to 10-fold increased LpxC concentrations and an LPS content that is reduced by approximately 30% (their strain SM101 at 30°C). We modeled this perturbation by decreasing the LpxA kcat value by 150 fold, to 0.048 s-1. Our simulation resulted in a 3.9 fold increase in LpxC levels, in reasonably close agreement with experiment. This is essentially the maximum LpxC increase that our model can produce under any condition, arising from our assumption that cells with completely inactive FtsH exhibit 4-fold higher LpxC concentrations . However, our model only produced 24,000 lipid A molecules per generation, rather than the 1.1 million molecules that it produces with our standard parameters  on E. coli, meaning the lowest drug concentration required to inhibit visible growth, is between 0.20 and 0.25 μg/ml , 20]20]. Barb et al. showed that CHIR-090 acts by inhibiting LpxC through the two-step mechanism . They reported the reaction kinetics parameters as k4/k3 = 4 nM, k5 = 1.9 min-1, and k6 = 0.18 min-1. They did not report separate parameters for k3 and k4, but their results only becomes logical if this complexation reaction comes to equilibrium reasonably quickly. Thus, we assume here that k4 = 0.1 s-1, from which we compute that k3 = 25000 mM-1s-1. We added this mechanism to our model to see if it would exhibit the same inhibitory effect. Firstly, we noticed that fixing the free inhibitor concentration to a constant value does not affect the steady-state biosynthesis pathway at all. This seems reasonable because the LpxC count is not fixed in our model, but arises from the LpxC translation and degradation rates, which the inhibitor does not affect . On the other hand, the inhibitor induces a very strong transient effect. . From the data shown in in vitro inhibition constant of CHIR-090, Ki*, which is 0.5 nM . Taking the in vivo MIC as 0.25 μg/ml, and using the CHIR-090 molecular weight of 437.4 g/mol, shows that the MIC is 570 nM of extracellular CHIR-090. Combining this with the intracellular MIC suggests that the intracellular CHIR-090 concentration is about 1000 fold lower than the extracellular concentration.CHIR-090 is a powerful antibiotic that controls et al.  showed that faster growing cells, such as those grown in rich media and/or at higher temperatures, stabilize LpxC more rapidly than slowly growing cells. We simulated these experiments as a way of validating our model. We only considered Schäkermann et al.’s results for rich medium because all of our model parameters were estimated from experiments performed with rich medium. For each LpxC half-life value from Table 2 of Schäkermann et al. , we used LpxC that would produce it . They also found that overexpressing LpxC with 0.01% arabinose resulted in a protein half-life of about 11 minutes .Führer . The 0.1% arabinose induction experiment is harder to match because this resulted in 1.7 fold more LPS, whereas LpxC overexpression could not produce more than 1.36 fold more lipid A in our model due to LpxD acting as a bottleneck. A possible explanation for this difference is that Führer et al. found that LpxC overexpression led to longer cell generation times , in addition to the effects mentioned above. Thus, it may be that cells produce lipid A only 36% faster than normal, but they take twice as long to divide, leading to 1.7 fold more LPS in cells. Together, these results show that our model agrees qualitatively with Führer et al.’s LpxC overexpression experiments, and it could agree quantitatively as well, but this cannot be assessed currently with the available experimental data.We simulated arabinose induction by increasing our model’s LpxC translation rate constant and observing its effect on the lipid A production rate and LpxC half-life . As in t observed that E. coli grown on minimal medium had a 150 minute generation time. Replicating this in our model was straightforward. As above, we used the assumption that a cell generation is the time that the model takes to produce 1 million lipid A molecules. Our model reproduces the 150 minute generation time when any one substrate limits the lipid A production rate, with 16,000 UDP-GlcNAc molecules, or 43 β-hydroxymyristoyl-ACP molecules, or 510 CMP-KDO molecules. It also reproduces the 150 minute generation time when all three substrates are partially limiting, with 80,000 UDP-GlcNAc, 210 β-hydroxymyristoyl-ACP, and 2400 CMP-KDO molecules (about 5 times the prior numbers). Thus, substrate limitation does increase generation times in the model, as expected.It is well known that organisms grow more slowly when nutrients are limited. For example, Taniguchi et al.  in that the low substrate concentrations in the model led to decreased lipid A disaccharide counts, which caused slower LpxC degradation, whereas Schäkermann et al. showed that substrate limitation leads to faster LpxC degradation. They found that growth in minimal medium leads to an LpxC half-life of about 10 minutes . This appears to occur because substrate limiting conditions lead to increased (p)ppGpp alarmone concentrations, which increase FtsH expression . Our model could agree with these data as well, but only if the total FtsH count was increased roughly 25-fold, to about 14,000 molecules . This modified model exhibited a 10 minute LpxC half-life and a 160 minute generation time, in close agreement with results presented in Schäkermann et al. The cost of achieving this agreement was that the high FtsH concentration led to fast WaaA degradation, which made WaaA catalysis of CMP-KDO the rate-limiting step as observed by the fact that lipid IVA accumulated rapidly. The large FtsH expression increase and the lipid IVA accumulation suggest that our model is incorrect for substrate limiting conditions. In particular, achieving the combination of rapid LpxC degradation and relatively low lipid A disaccharide concentrations, without increasing the FtsH concentration 25-fold, requires additional LpxC degradation regulation.However, these results disagree with those of Schäkermann et al. As mentioned above, a preliminary model that did not include FtsH feedback regulation exhibited lipid A disaccharide accumulation , which l, 81]81], we defined flux control coefficients asi represents one of the 10 pathway enzymes, J represents the biosynthesis rate of lipid A at the outer membrane, and Ei represents the count of enzyme i. We quantified flux control coefficients by starting the model at steady-state, varying an enzyme count by 5%, and observing the effect on the lipid A production rate. When we set the three pathway substrates to saturating concentrations and used the “open-loop” case, in which we removed negative feedback through FtsH but kept the wild-type enzyme counts from ,55]55]. However, the “closed-loop” case, in which we replaced the feedbacks through FtsH and thus returned to a better model for the wild-type pathway, showed no sensitivity to LpxC concentration perturbations . Finally, limiting KDO-CMP shifted control to WaaA in the open-loop case, and to MsbA in the closed-loop case. These results show that control of pathway flux is typically localized to relatively few enzymes, but ones which depend on the substrate concentrations.Yet different enzymes controlled pathway flux when we reduced substrate concentrations. For each substrate, we set its concentration to the value that led to about 0.5 million lipid A molecules produced per generation and then computed flux control coefficients for each enzyme. Limiting UDP-GlcNAc shifted control to LpxA. This contrasts the view that LpxA does not affect pathway flux, simply because it is reversible and has an unfavorable equilibrium constant Next, we investigated the sensitivity of lipid A biosynthesis on enzyme counts for the case of large perturbations, assuming saturating substrate concentrations. For each enzyme, we determined what fraction of the wild-type count  would le leads to the obvious question of why it is regulated in addition to LpxC. From our sensitivity analysis, one answer may be that WaaA regulation is used to control pathway flux when the CMP-KDO concentration is limiting  Our initial model produced lipid A at 20% of the observed rate and exhibited LpxM substrate accumulation, which suggest that LpxM is actually present at a much higher concentration than was measured with proteomic methods . (ii) High modeled levels of lipid X suggest that the LpxH kinetics are regulated by product inhibition, and also that LpxH and LpxB may form a complex that performs metabolic channeling. (iii) The model exhibited 4-fold increased LpxC when LpxA was made defective, instead of the observed 5–10 increase ; this may arise from experimental differences, or due to unmodeled aspects of LpxC regulation, such as transcription regulation. (iv) The model exhibited a 98% reduction in the lipid A synthesis rate when LpxA was defective, instead of the 30% that was observed experimentally , which suggests the presence of alternate metabolic routes around LpxA, such as by LpxD. (v) The model underestimated cell generation times when LpxC had a very long half-live , which suggest that the growth rates of fast growing cells are not limited by the lipid A production rate. (vi) The model could exhibit the experimentally observed LpxC half-life and cell generation time when substrates were limited , but only when the total FtsH concentration was increased 25-fold; this suggests that there are additional mechanisms that regulate LpxC degradation. None of these speculations are proven by the model. Instead, they are possible solutions to situations in which the experiments that we drew on to create the model do not agree with experiments that were used to test the model. They provide hypotheses for further experimental investigation.Quantitative disagreements between our model and experiments are particularly useful because they show which model assumptions are likely to be incorrect. These errors then enable new insights about the true system behaviour. In particular: , which might make cells unable to counteract its inhibition. Additionally, when we accounted for feedback regulation, we found that the lipid A production rate is particularly sensitive to the LpxK concentration Click here for additional data file.S2 FileParameters are same as in S1 with the inclusion of (CPS)Click here for additional data file.


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


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