Bacterial strain and growth conditions.

An E. coli K-12 strain AG1 (recA1, endA1, gyrA96, thi-1, hsdR17(r_K^- m_K⁺), supE44, relA1) containing a plasmid (pCA24N) [18] 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 (10g tryptone, 5g yeast extract, 5g NaCl per liter) containing 20 μg/ml of chloramphenicol and when required, protein expression was induced using IPTG (Sigma, UK).

Preparation of cell extracts.

Cell extracts were prepared as described previously [19],[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 (Sigma, UK). The samples were heated for 10 min prior to centrifugation for 5 min. The supernatants were collected for Western blot analysis.

Western blot.

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 (Sigma, UK) 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).

Lipid A biosynthesis pathway.

E. coli lipid A biosynthesis proceeds through nine enzyme-catalyzed steps, which are sometimes referred to as the Raetz pathway [2],[7] (Fig 2). All of these enzymes are constitutively expressed [5]. The pathway has been reviewed several times recently [1],[2],[5] [7],[8], so we only provide a brief summary here, while focusing on the features that are particularly salient to our model.

Lipid A biosynthesis begins with the UDP-N-acetylglucosamine (UDP-GlcNAc) and β-hydroxymyristoyl-ACP substrates. Both substrates are consumed in other metabolic pathways as well [21]: UDP-GlcNAc is a substrate in peptidoglycan synthesis [22],[23] and β-hydroxymyristoyl-ACP is a precursor for phospholipid metabolism [24],[25],[26].

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 in vitro [27], 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 [19],[28],[29],[30]. For this and other reasons, LpxC is likely to be a primary biosynthesis control point [27] (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 [31],[32]. LpxD is similar to LpxA in that they are acyltransferases, and consume the same β-hydroxymyristoyl-ACP substrate [32]. Both LpxD reaction products inhibit the LpxD reaction, acting as either competitive or non-competitive inhibitors against each substrate [32]. 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 fourth and fifth lipid A biosynthesis steps are catalyzed by the peripheral membrane proteins LpxH (EC 3.6.1.54) [33] and LpxB (EC 2.4.1.182) [34]. 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 [34].

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 IV_A [35],[36]. Remarkably, Lipid IV_A has been reported to be an endotoxin agonist in mouse cells and an endotoxin antagonist in human cells [37]. Next, WaaA (previously called KdtA, EC 2.4.99.12/13) sequentially transfers two KDO sugar residues to lipid IV_A to produce KDO₂-lipid IV_A [38],[39],[40]. WaaA has low substrate specificity, with the result that KDO₂-lipid A can act as a competitive inhibitor for this reaction [40], or as another possible WaaA substrate [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 (enzyme, cytidine 5'-monophosphate (CMP), and KDO₂-lipid IV_A) led to detectible concentrations of the KDO-lipid IV_A intermediate [40]. However, the authors only observed trace quantities of lipid IV_A 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.-) [41] and LpxM (EC 2.3.1.-) [42],[43], incorporate lauroyl and myristoyl chains to the KDO₂-lipid IV_A, thus giving the final KDO₂-lipid A product six acyl chains. Cells are still viable without LpxM, or without LpxL and with overexpressed LpxM [44]. 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 [45].

After synthesis, KDO₂-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 [2],[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 [7].

Lipid A biosynthesis regulation.

Lipid A synthesis is regulated, at least in part, through controlled degradation of LpxC [47],[48],[49] and WaaA [50], both performed by FtsH (EC 3.4.24.—). FtsH is an integral membrane AAA-type metalloprotease that degrades a wide variety of proteins. These include heat shock transcription factor RpoH (σ³²), phage λ proteins CII and CIII, and many misfolded proteins [51],[52]. FtsH is an essential protein due to its role in regulating LpxC [47],[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.

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 (denoted FtsH, FtsH*_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 [52],[54], (ii) neither increased nor decreased FtsH degradation of LpxC have a significant effect on the intracellular concentration of RpoH [47] or the activity of WaaA [19], and (iii) LpxC is degraded more slowly at higher temperatures whereas WaaA is degraded more rapidly at higher temperatures [14],[50]. Substrate-specific activation of FtsH for LpxC may occur through YciM acting as an LpxC adapter protein [55].

We assume that the regulatory signal that directs FtsH degradation of LpxC arises from the concentration of lipid A disaccharide. Again, this is based on several findings. First, strains that have decreased LpxA, LpxC, or LpxD function, whether through temperature-sensitive mutants or chemical inhibition, exhibit decreased lipid A content and slowed LpxC degradation [11],[19],[27],[56]. These suggest that the feedback source is downstream of LpxD. Second, chemically inhibiting CMP-KDO production [57] blocks the lipid A biosynthesis pathway at the WaaA point. This was found to cause lipid IV_A accumulation but did not affect LpxC activity [19]. This suggests that the feedback source is upstream of lipid IV_A. Three metabolites fit these two criteria, UDP-2,3-bis(β-hydroxymyristoyl)-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 (Fig 3). This suggested that the feedback source is lipid A disaccharide. Guided by these arguments, we experimentally investigated the effect of over-expressing LpxK; those results further supported the lipid A disaccharide choice, as explained below.

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Fig 3. Lipid A disaccharide accumulation.

Bars indicate intracellular counts of lipid X and lipid A disaccharide at the end of a single cell generation from preliminary model results. The model used the scheme shown in Fig 2 and parameters listed in Table 1, with the exceptions: the FtsH count was set to zero, the LpxC and WaaA counts were set to their steady-state levels in the absence of FtsH degradation (1540 and 978, respectively), and all metabolites were started with zero molecules. Although not shown here, the lipid X count had stabilised at a constant level, while the lipid A disaccharide count was increasing at a constant rate of 924 molecules/s.

https://doi.org/10.1371/journal.pone.0121216.g003

Finally, we assume that the regulatory signal for FtsH degradation of WaaA arises from KDO₂-lipid A that is inside the inner membrane. We chose KDO₂-lipid A rather than its precursors because the two enzymes downstream of WaaA, which are LpxL and LpxM, are non-essential [44], making their substrates unlikely activators. Also, we chose KDO₂-lipid A before it has been transported to the outer membrane, rather than afterwards, because bacterial cells are known to regulate excess lipid A in their outer membrane by shedding it into the environment [58].

Other regulatory signals impinge upon the lipid A biosynthesis pathway as well but are outside of the scope of this work. In particular, modifications to the FabZ and FabI enzymes, used for phospholipid synthesis, have been shown to affect the concentration of LpxC and hence affect the rate of lipid A production [20],[24],[47]. By assuming that FabZ and FabI are maintained at wild-type conditions, and that there are no feedback processes from the lipid A pathway that affect the FabZ or FabI regions of the phospholipid pathway, we were able to legitimately ignore these additional regulatory signals in this work.

Substrate concentrations.

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 [21], [22], [23], [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 [62].

Enzyme abundance.

We used LpxA, LpxD, WaaA, MsbA, and FtsH protein copy numbers from mass spectrometry proteomic data collected on E. coli cytosolic fractions [62]. 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 [62], 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 [13][33][63], along with the assumption that an average E. coli cytoplasm contains approximately 1.9 million protein molecules [64]. Similarly, we calculated LpxL protein counts from protein purification experiments [41] and the estimate that an E. coli membrane contains 580,000 proteins [64].

We estimated the LpxK protein count using information about MsbA. These proteins are co-transcribed [65], implying that their transcripts are synthesized at similar rates. Thus, differences in their expression rates depend on the relative stability of their transcripts and on the translation rates for individual proteins. The MsbA and LpxK transcript half-lives have been reported as 3.2 min and 3.8 min respectively [66], from which we calculated their mean lifetimes as 277 s and 329 s. Given that the average translation rate is 20 amino acids per second [67], and that they comprise 582 [68] and 328 amino acids respectively [36], it should take about 29 s and 16 s for their translations. This means that about 9.6 MsbA proteins and 20 LpxK proteins are translated over the lifetimes of their respective mRNAs. Thus, LpxK is likely to be synthesized about 2.1 times faster than MsbA. Both LpxK and MsbA are membrane proteins, so we assumed that they had similar degradation rates [69]. This meant that the synthesis rate ratio also represented the protein concentration ratio. MsbA has an abundance of about 206 molecules per E. coli cell [62], from which we calculated that the LpxK abundance is about 432 molecules.

Enzyme kinetics.

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 [70],[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 [72]. 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 (e.g. the phosphorylation reaction catalyzed by LpxK is effectively irreversible because ATP is abundant in cells whereas ADP is not).

We used single-substrate Michaelis-Menten kinetics for the LpxC, LpxK, LpxL, LpxM, and MsbA steps. Here, the metabolite flux is (1) where [S] is the substrate concentration, [P] is the product concentration, [E] is the total enzyme concentration, k_cat is the enzyme catalytic rate constant, and K_m is the Michaelis constant. Most of these k_cat and K_m values have been published using data from in vitro experiments (Table 1), although we needed to estimate a few of them. (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 [65]. There are about 432 LpxK molecules per cell and E. coli membranes include about 580,000 individual proteins [64], 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 k_cat is about 2.1 s^-1. We estimated that the LpxK K_m value is 40 μM from a figure presented by Ray and Raetz [35]. Although K_m 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, k_cat, as 0.6 s^-1 from the specific activity of the enzyme in crude lysates [42], much as we did for LpxK. We estimated the LpxM K_m value from data shown in Fig 6 of Clementz et al. [43]. To do so, we simulated Clementz et al.’s experiment using COPASI [15], with the same enzyme and substrate concentrations that they used (0.1 μg/mL and 25 μM for protein and KDO₂-lipid IV_A respectively in a 20 μL reaction mixture), from which we identified the K_m 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 K_m value to 0.021 mM and its k_cat value to 166 s^-1, based on data shown in Fig 6 of [73].

We used single-substrate Michaelis-Menten kinetics with inhibition for the LpxH enzyme. In this case, the metabolite flux is (2) where K_i is the inhibition constant and the other parameters are the same as in Eq 1. Assays conducted on LpxH purified to 60% homogeneity displayed a specific activity of 63.2 μmol/min/mg [33]. 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 [33], to yield its k_cat as about 47 s^-1. The estimation of our K_i value is discussed below.

We used bi-substrate Michaelis-Menten kinetics for the forward LpxA, reverse LpxA, LpxB, and WaaA catalysis of KDO₂-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 (although, this is not true for spatial models [74]). Most bi-substrate enzymatic reactions follow either a sequential or ping-pong mechanism [75]. 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 [76] and LpxD [32], 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 is (3) where [S₁] and [S₂] are the two substrate concentrations, with respective Michaelis-Menten constants K_m1 and K_m2. We needed to estimate these parameters in some cases. (i) The LpxA kinetic parameters were determined previously for the forward reaction [27], [76], but not for the reverse reaction. Thus, we assumed the same K_m values for the reverse reaction as for the forward reaction (we set K_m of ACP to that for β-hydroxymyristoyl-ACP, and K_m of UDP-3-O-[β-hydroxymyristoyl]-NAc to that for UDP-GlcNAc), based upon the likelihood that the enzyme binding affinities are not substantially affected by the acyl group transfer. However, we set the reverse reaction k_cat value to 100 times that of the forward reaction to account for the reaction’s unfavourable equilibrium constant of approximately 0.01 [27]. (ii) We estimated the LpxB k_cat value by starting from the specific activity of LpxB purified to near homogeneity [63], much as we did for LpxH, which resulted in a k_cat value of 129 s^-1. (iii) The WaaA specificity for KDO₂-lipid A is 8.7 fold lower than for lipid IV_A [40]. To account for this, we reduced the k_cat value for the former reaction by 8.7 fold while keeping other reaction constants the same.

Finally, 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 is

(4)

We computed the WaaA k_cat value as 16.7 s^-1 from the specific activity of the purified protein [14]. We computed the inhibition constant from results by Belunis and Raetz [40] 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 (Eq 4), in which case K_m is constant, we derived that at 100 μM of inhibitor and excess CMP-KDO, a 24.1% reduction in the reaction rate implies that K_i is about 0.0317 mM.

LpxC and WaaA synthesis and degradation.

We included translation and degradation reactions for LpxC and WaaA in our model so that we could explore the effects of their regulation via FtsH proteolysis. As in the rest of the model, we accounted for degradative protein turnover within cells, but not protein loss through sequestration into daughter cells or the translation that is required to replace those proteins.

We modeled LpxC and WaaA synthesis with zeroth order reaction kinetics, in which the production rate is constant. This approach combines transcription, translation, and any translocation into a single reaction step. We modeled the degradation of these proteins with a first order reaction for degradation that is not catalyzed by FtsH, and also an independent reaction obeying mass action kinetics for degradation that is catalyzed by FtsH (Michaelis-Menten kinetics might be more appropriate, but those parameters cannot be computed from available data). Together, these processes combine to give the net production rate for each of these proteins as (5) where [P] is the concentration of the LpxC or WaaA protein, k_translate is the production rate constant, k_degrade is the rate constant for uncatalyzed degradation, and k_FtsH is the rate constant for FtsH degradation.

To determine the production and degradation parameters for LpxC, we started with results presented in Schäkermann et al [14] 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 k_degrade 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 [47], thus increasing LpxC counts from about 385 molecules to about 1540 molecules. Combining this molecule count with the k_degrade value and the assumption that [FtsH*_LpxC] equaled zero in this mutant, enables Eq 5 to be solved for steady-state conditions to give a k_translate value of about 0.148 molec./s. Next, combining the k_degrade, k_translate, and the wild-type LpxC count of 385 molecules enables Eq 5 to solved for steady-state to give that k_FtsH[FtsH*_LpxC] is 2.89×10^–4 s^-1. We cannot solve for k_FtsH by itself from the available information, but estimate its value below.

We computed the WaaA synthesis and degradation parameters similarly. First, a mutant without FtsH maintained about 72% of its WaaA concentration after 30 minutes [50], from which we computed that k_degrade 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 [50]. This gives the combined degradation rate constant, k_degrade + k_FtsH[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, k_translate, as 0.176 molecules/s.

FtsH activation and inactivation.

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 [62]. 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 to (6) where [FtsH*] represents the concentration of a substrate-specific active form of FtsH, [activator] represents the concentration of the substrate-specific activator (lipid A disaccharide for the LpxC substrate and KDO₂-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.

The FtsH proteolysis rate, for either LpxC or WaaA, depends on three parameters, k_active, k_inactive, and k_FtsH. However, the available experimental data only enabled us to quantify the product k_FtsH[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 (i.e. there are 58 copies of FtsH*_LpxC, 58 copies of FtsH*_WaaA, and 463 copies of inactive FtsH). 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 k_FtsH[FtsH*] products yield k_FtsH of 2.0 mM^-1s^-1 for LpxC and k_FtsH of 6.8 mM^-1s^-1 for WaaA. Secondly, we assumed that both k_inactive 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 k_active from the steady-state version of Eq 6, while substituting in this inactivation rate constant, the assumed FtsH* and FtsH concentrations, and the activator concentrations that arose from simulations in which we fixed the LpxC and WaaA enzyme counts to the values listed in Table 1 (activator counts were 35,600 and 155 molecules, respectively). Results are that k_active 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.

LpxM enzyme count.

Our initial model simulations only produced about 20% of the 1 million lipid A molecules that 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 [62], 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 [43], [45]. 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.

LpxH product inhibition.

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 in vivo [77]. 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 K_i 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 (negative feedback at LpxD prevents accumulation of its product, backing the accumulation up to the LpxC product instead). Thus, we decided to decrease LpxH inhibition by increasing K_i. As mentioned above, mutants with inactive FtsH exhibit 4-fold increased LpxC concentrations [47]; they were also shown to produce 32% more lipid A (Table 1 of [47], comparing their AR3317 at 30°C vs. 42°C, or their AR3289 vs. AR3291). We decided to set the LpxH inhibition constant so that our model would reproduce this result, which turned out to be a K_i 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 K_i 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 [77]. 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.

Mutant with defective LpxA.

Anderson et al. [27] 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 k_cat 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 [47]. 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 (Table 1), which is a reduction of 98%. Our results were unaffected by whether we decreased the LpxA reverse reaction k_cat value or not, or even if the reverse reaction was removed altogether. This limitation is clarified by noting that the reaction velocity of the LpxA step is simply k_cat[LpxA] in these conditions, from Eq 1, which works out to 57,000 product molecules per generation; these need to dimerize to form lipid A, meaning that the LpxA step limits lipid A production to only 29,000 molecules per generation, in close agreement with our model result. Thus, Anderson et al.’s [27] observation that LpxA in the SM101 strain exhibits a 150-fold reduced k_cat value but reduces the lipid A production rate by only 30%, is not compatible with our model. This is because their results cannot arise from our assumed LpxA protein count or k_cat parameters. The disagreement implies that these parameters are incorrect by more than an order of magnitude, which seems unlikely, or potentially, there is an alternate biosynthetic pathway; for example, perhaps LpxD can catalyze what is normally the LpxA step.

Inhibition of LpxC.

CHIR-090 is a powerful antibiotic that controls E. coli and Pseudomonas aeruginosa growth with efficacy that is similar to the popular drug ciprofloxacin [30]. Its minimum inhibitory concentration (MIC) on E. coli, meaning the lowest drug concentration required to inhibit visible growth, is between 0.20 and 0.25 μg/ml [9], [20]. Barb et al. showed that CHIR-090 acts by inhibiting LpxC through the two-step mechanism [9] (7) where LpxC-I represents a complex between LpxC and the CHIR-090 inhibitor, and I* represents an enzyme/inhibitor isomer [9]. They reported the reaction kinetics parameters as k₄/k₃ = 4 nM, k₅ = 1.9 min^-1, and k₆ = 0.18 min^-1 [9]. 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 (the reversibility of inhibitor binding implies that, at steady-state, LpxC is sequestered into and released from complexes at the same rate). On the other hand, the inhibitor induces a very strong transient effect. Fig 4A shows the amount of lipid A produced by the model over the course of 30 minutes, in which all metabolite concentrations started at steady-state and then CHIR-090 was added at time 0. We assumed that the CHIR-090 MIC corresponds to the inhibitor concentration that reduces lipid A production by 50%, based on results that show that this is when cells become non-viable [11]. From the data shown in Fig 4A, our model cells became non-viable when the free intracellular CHIR-090 concentration exceeds 0.31 molecules/cell, which is 0.76 nM. Clearly, molecule counts are discrete, implying that these are time-averaged quantities. This MIC corresponds closely with the in vitro inhibition constant of CHIR-090, K_i*, which is 0.5 nM [9]. 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.

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Fig 4. Comparison of model with experiment.

(A) Effect of CHIR-090 antibiotic on lipid A production. The model (Fig 2 plus Eq 7) was started with all metabolites at their steady-state concentrations without CHIR-090. Then, antibiotic was added and the total amount of lipid A produced over the following 30 minutes was quantified, shown with the solid black line. The free antibiotic concentration, quantified as the average number of uncomplexed CHIR-090 molecules/cell, was kept constant. The black dashed line shows our estimate of the MIC for the intracellular antibiotic concentration and the red dashed line shows the antibiotic inhibition constant [9]. (B) Correlation between LpxC half-life and cell generation time. The experimental data (red circles) are from Schäkermann et al. [14], who varied generation times using different growth conditions and then quantified LpxC half-lives. The model data (black diamonds) were collected by varying the LpxC half-lives (and LpxC kcat) and then quantifying the generation times, defined here as the time required to produce 1 million lipid A molecules. Lines are least-difference best-fits to the data using the function y = c₁/x + c₂, primarily to guide the eye. (C) Effect of overexpressing LpxC on the LpxC half-life (black curve, left axis) and on the lipid A production rate, measured relative to the wild-type production rate (blue curve, right axis). The dashed line shows the wild-type condition using the LpxC translation rate from Table 1.

https://doi.org/10.1371/journal.pone.0121216.g004

Correlation between LpxC half-life and cell generation time.

Schäkermann et al. [14] 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. [14], we used Eq 6 to compute the concentration of FtsH*_LpxC that would produce it (here, k_translate was 0, and k_degrade and k_FtsH were the values from Table 1). Next, we changed our standard model in two ways: we adjusted the LpxC k_cat value to account for the given growth temperature according to data shown in Fig 2 of Jackman et al. [10], and we adjusted the rate of FtsH activation for LpxC degradation, k_active, until the model exhibited the desired FtsH*_LpxC concentration. The rationale for the latter change is that cells might regulate LpxC half-lives by altering the FtsH activation rate. Finally, we computed the cell generation time from the steady-state lipid A production rate in this adjusted model, under the assumption that a generation time is determined by how long a cell requires to produce 1 million lipid A molecules. Fig 4B shows that the model results agree reasonably well with the experimental results, which supports our model. We suspect that our model underestimates cell generation times with long LpxC half-lives because it does not account for other cell processes, which may limit cell division rates at these fast growth rates.

Overexpression of LpxC.

Führer et al. [48],[49] cloned the lpxC gene into an inducible expression vector and then induced with 0.01% or 0.1% arabinose, which overexpressed the LpxC enzyme. They did not quantify the extent of overexpression, but comparable cells increased protein expression by 100 to 200 fold with 0.01% arabinose induction [78]. Führer et al. found that LpxC overexpression increased LPS amounts in cells, by about 1.27 fold and 1.7 fold in cells induced with 0.01% and 0.1% arabinose respectively (our estimates from Fig 4 of [49]). They also found that overexpressing LpxC with 0.01% arabinose resulted in a protein half-life of about 11 minutes [48].

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 (Fig 4C). As in the experiment, overexpressing LpxC led to increased lipid A production and shorter LpxC half-lives. However, these effects stopped changing once LpxC was overexpressed about 4-fold, in contrast to the 100 to 200 fold overexpression that the experiments may have produced. At higher than 4-fold overexpression, the LpxC product accumulated in our model because LpxD became a bottleneck; this prevented further changes to the LpxC lifetime and lipid A production rate. Nevertheless, if one assumes a more modest experimental overexpression, then our results agree well with the 0.01% arabinose induction experiment. In particular, at 2.8 fold overexpression, our model shows a 1.27 fold increase of lipid A production and a 14 minute LpxC lifetime, which is reasonably close to the 11 minutes that Führer et al. observed [48]. 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 [48], 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.

Substrate limitation.

It is well known that organisms grow more slowly when nutrients are limited. For example, Taniguchi et al. [79] 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.

However, these results disagree with those of Schäkermann et al. [14] 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 [14]. This appears to occur because substrate limiting conditions lead to increased (p)ppGpp alarmone concentrations, which increase FtsH expression [80]. 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 (we also used 250,000 UDP-GlcNAc, 650 β-hydroxymyristoyl-ACP, and 7600 CMP-KDO molecules, which are about 15 times higher than the single-substrate limitation values presented above). 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 IV_A accumulated rapidly. The large FtsH expression increase and the lipid IV_A 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.

Overexpression of LpxK stabilizes LpxC.

As mentioned above, a preliminary model that did not include FtsH feedback regulation exhibited lipid A disaccharide accumulation (Fig 3), which led us to propose that this metabolite is the feedback source for LpxC degradation. If this is the case, then it follows that LpxK overexpression should reduce lipid A disaccharide concentration, which would down-regulate LpxC degradation and lead to higher LpxC concentrations. We tested this hypothesis experimentally. Fig 5A shows that this is indeed the case. LpxC concentrations increased substantially with LpxK overexpression, even under modest IPTG induction. We tested the same perturbation in our model, finding exactly the same results (Fig 5B). These results are consistent with the assignment of lipid A disaccharide as the feedback source. In contrast, the opposite correlation between LpxC concentration and LpxK overexpression would be expected if the feedback source were downstream of LpxK, so our experiments provide strong evidence against that possibility. Similarly, only a very weak correlation would be expected if the feedback source were further upstream of LpxK, meaning at or before the lipid X metabolite, due to the near irreversibility of the LpxB enzyme, so our experiments provide strong evidence against those possibilities as well. Thus, our experiments strongly indicate that lipid A disaccharide is the feedback source for activating FtsH for LpxC degradation.

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Fig 5. Overexpression of LpxK increases LpxC concentration.

(A) Lower row shows LpxK bands on an SDS-PAGE gel, arising from overexpression induced with the amount of IPTG shown at the top of each column. The upper row shows the resultant LpxC bands on a Western blot for the same induction levels. (B) Model prediction of LpxC overexpression arising from LpxK overexpression. The model was that shown in Fig 2 but with different LpxK enzyme counts, at steady-state.

https://doi.org/10.1371/journal.pone.0121216.g005

Lipid A synthesis sensitivity on enzyme concentration.

We investigated the sensitivity of lipid A synthesis on enzyme counts in several ways. First, we investigated the effect of small enzyme concentration variations. Using the methods of metabolic control analysis [70], [81], we defined flux control coefficients as (8) where i represents one of the 10 pathway enzymes, J represents the biosynthesis rate of lipid A at the outer membrane, and E_i 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 Table 1, the pathway flux was solely controlled by LpxC (Fig 6A black bars). This is consistent with the view that LpxC is the rate-limiting step of lipid A synthesis [48],[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 (Fig 6A red bars). Instead, the feedback caused the LpxC enzyme count to return to its steady-state level, which meant that the perturbation did not affect the lipid A production rate. LpxK was the sole enzyme that controlled pathway flux in the closed-loop case.

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Fig 6. Sensitivity of lipid A production rate on enzyme abundance.

(A) Black bars show enzyme control coefficients for the open-loop case, in which wild-type enzyme counts were assumed but FtsH regulation was disabled. Red bars show enzyme control coefficients for the closed-loop case, in which FtsH regulation was enabled. (B) Enzyme abundance reductions that led the model to produce 0.5 million lipid A molecules per cell generation for the open-loop (black bars) and closed-loop (red bars) cases.

https://doi.org/10.1371/journal.pone.0121216.g006

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 [27]. Limiting β-hydroxymyristoyl-ACP shifted control to LpxD, LpxH, and LpxB (60%, 20%, and 20%, respectively). 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.

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 (Table 1) would lead to 0.5 million lipid A molecules produced in 30 minutes. In the open-loop case, the production rate was most sensitive to the LpxC and LpxK enzyme counts (Fig 6B black bars). In the closed-loop case, as before, LpxC perturbations were ineffectual because its concentration was regulated through feedback. As a result, the pathway was most sensitive to LpxK (Fig 6B red bars).

WaaA Regulation.

Katz and Ron’s finding that the concentration of WaaA is regulated through degradation by FtsH [50] 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 (Fig 7A). In this situation, the lipid A synthesis rate is insensitive to small changes in the LpxC concentration, making that less useful for regulation, but is controlled by the WaaA concentration instead. Also, reducing lipid A production when CMP-KDO is limiting would conserve KDO for other uses. For example, KDO can be catalyzed by KDO aldolase to produce D-arabinose and pyruvate [82]. Although it has been suggested that CMP-KDO synthesis is the rate-limiting step to lipid A synthesis [83], the strong effects of LpxC and the other upstream enzymes, discussed above, indicate that this is not the normal case. Additionally, our model shows that CMP-KDO limitation leads to rapid lipid IV_A accumulation, with no apparent correction mechanism. Together, these results point to the likelihood that WaaA regulation helps control the lipid A production rate when the CMP-KDO concentration is low, but probably not so low as to be rate-limiting.

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Fig 7. WaaA regulation.

(A) The flux control coefficient of WaaA as a function of the CMP-KDO substrate concentration. This is for the open-loop case, in which wild-type enzyme counts were assumed but FtsH regulation was disabled. (B) The ratio of alternate lipid A to normal lipid A (KDO₂-lipid A) as a function of the number of WaaA proteins. All parameters are the same as in Table 1, except that the WaaA proteolysis rate constant was changed in order to alter the WaaA degradation rate and hence the WaaA steady-state copy number. The maximum enzyme count shown arose from no FtsH mediated proteolysis. The dashed line indicates the wild-type count, from Table 1.

https://doi.org/10.1371/journal.pone.0121216.g007

A second possible explanation for WaaA’s regulation is that it might decrease reactions with undesirable substrates [50], brought about by the fact that WaaA has a low substrate specificity. As mentioned previously above, WaaA can glycosylate a wide range of lipid acceptors, including KDO₂-lipid A in particular [39], [40], which we included in our model architecture as production of alternate lipid A. We ran simulations with different rate constants for WaaA degradation by FtsH. We found that slower WaaA degradation led to higher steady-state enzyme counts, which then led to higher relative amounts of alternate lipid A (Fig 7B). Thus, our model agrees with suggestions that WaaA regulation might help regulate the lipid A molecule composition [50]. This mechanism functions even if CMP-KDO is in excess.