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Mammalian E3 is an essential mitochondrial enzyme responsible for catalyzing the terminal reaction in the oxidative catabolism of several metabolites. E3 is a key regulator of metabolic fuel selection as a component of the pyruvate dehydrogenase complex (PDHc). E3 regulates PDHc activity by altering the affinity of pyruvate dehydrogenase kinase, an inhibitor of the enzyme complex, through changes in reduction and acetylation state of lipoamide moieties set by the NAD+/NADH ratio. Thus, an accurate kinetic model of E3 is needed to predict overall mammalian PDHc activity. Here, we have combined numerous literature data sets and new equilibrium spectroscopic experiments with a multitude of independently collected forward and reverse steady-state kinetic assays using pig heart E3. The latter kinetic assays demonstrate a pH-dependent transition of NAD+ activation to inhibition, shown here, to our knowledge, for the first time in a single consistent data set. Experimental data were analyzed to yield a thermodynamically constrained four-redox-state model of E3 that simulates pH-dependent activation/inhibition and active site redox states for various conditions. The developed model was used to determine substrate/product conditions that give maximal E3 rates and show that, due to non-Michaelis-Menten behavior, the maximal flux is different compared with the classically defined kcat.
Dihydrolipoamide dehydrogenase (E3) is one of three catalytic subunits (E1, E2, and E3) found in a number of multienzyme catabolic complexes (1). E3 is responsible for catalyzing the terminal redox reaction, which is the reversible two-electron oxidation of dihydrolipoamide to lipoamide with the concomitant reduction of NAD+ to NADH (2,–4). E3 is a component of the pyruvate dehydrogenase complex (PDHc),2 which oxidizes pyruvate to produce CO2, acetyl-CoA, and NADH (5). The PDHc is at the interface of glycolysis, the tricarboxylic acid cycle, and is reciprocally regulated relative to fatty acid β-oxidation (6, 7).
Its hub-like position in central metabolic pathways has endowed the PDHc with a major regulatory role in mammals with links to metabolic maladaptations (8) related to heart failure (9,–11), diabetes (12, 13), and the Warburg effect in cancer cells (14,–16). An imbalance between glucose/fatty acid substrate selection is a hallmark of the aforementioned diseased states (6, 17). The PDHc is also known to be a critical factor in substrate selection, which is modulated by multiple factors (13, 18).
Mammalian PDHc regulation is governed in large part by its own specific pyruvate dehydrogenase kinase (PDK) and phosphatase (PDP) (19,–21). PDK and PDP bind to the lipoamide-containing L sites (22,–25) located on E2 and the E3-binding protein, a PDHc structural subunit. PDK binding, and thus PDHc activity, is greatly affected by the redox and acetylation state of lipoamide. When reduced, acetylated lipoamide moieties favor PDK binding (23, 24, 26, 27). The redox state of lipoamide moieties in E2 and the E3-binding protein is governed by the forward reductive acetylation activity of E1 and the NADH/lipoamide activity of E3. It has been postulated that E3 activity has primary kinetic control of the lipoamide redox state, because E1 activity is rate-limiting for the overall PDHc reaction (8). Thus, PDK/PDHc regulation via the NAD+/NADH ratio, a general indicator of intracellular redox (28) and bioenergetic status (29), is mediated through the E3 component of the complex.
In vitro E3 component kinetics and redox regulation have been studied from multiple organisms and have demonstrated similar kinetic patterns but with remarkably different regulatory sensitivities to NAD+/NADH ratio and pH (30,–34). Mammalian and bacterial E3 components are the most disparate examples, with E3 from Escherichia coli demonstrating the greatest observed NAD+/NADH catalytic sensitivity (30). Greater NAD+/NADH sensitivity in E. coli E3 (Ec-E3) may be reasoned from a regulatory standpoint because E. coli PDHc lacks the PDK/PDP enzymes, which provide another layer of PDHc regulation in mammals (5, 35). Despite smaller NAD+/NADH kinetic effects in mammalian versus bacterial E3s, PDK activity in the mammalian PDHc is very sensitive to NAD+/NADH (26, 36, 37), mediated through E3.
NAD+/NADH effects are intimately related to pH, as they may be more appropriately modeled as the ratio of NAD+/(NADH + H+), due to the release of a proton during NAD+ reduction (38). Thus, E3 NAD+/NADH regulation is pH-dependent. The reverse E3 reaction consumes NADH, lipoamide, and H+, while producing NAD+ and dihydrolipoamide. Typically, in the reverse direction, initial amounts of added NAD+ to in vitro assays provide an activating effect in acidic pH (30, 39, 40), whereas, ever increasing amounts of NADH cause E3 substrate inhibition (30). These observations span mammalian and bacterial E3s but are most prominent in the archetypical bacterial E3 from E. coli. Although the magnitude of NAD+/(NADH + H+) kinetic regulation differs from E. coli (30), human (33), rat (34), and spinach enzymes (32), we have previously shown that these differences are explained by a single thermodynamically constrained model (41).
Prior E3 analysis (41) demonstrates that NAD+ is mainly an activator in the reverse reaction, with E. coli (30), spinach (32), and human liver (33). Reed (34) also showed, using rat liver E3, that varying amounts of NAD+ lead to kinetic patterns deviating from a simple ping-pong mechanism (34).
Even though the rat liver E3 kinetic study showed no NAD+ activation (34), NAD+ activation was demonstrated with human liver E3 enzyme (33, 41). Interestingly, NAD+ activation was also reported early on by Massey and Veeger (39) using pig heart E3, but little primary data were shown. Muiswinkel-Voetberg and Veeger (40) demonstrated the pH dependence of the initial velocity of pig heart E3 with the reverse reaction and showed activation of the rates by adding 0.1 mm NAD+. However, these data did not include physiologically relevant pH ranges (pH ≥ 7).
There are other available data on the pH-dependent regulation of mammalian E3 (mam-E3) (42,–47); however, none address the NAD+ pH-dependent kinetic regulation. After modeling E3 kinetic data from a number of organisms, we found that lacking mam-E3 data, especially reverse progress curve type data and NAD+/(NADH + H+) effects, weakened mam-E3 model parameterization and predictability in physiological conditions (41).
Therefore, we have collected a large array of pH-dependent progress curves and initial velocity data, in the reverse direction, using different amounts of NAD+ and lipoamide with pig heart E3. We also include independent data sets with the forward reaction that vary pH, NAD+, and dihydrolipoamide. Furthermore, we use spectroscopic methods to determine the redox state of the enzyme through FAD absorbance and fluorescence to gain further insight into E3 regulation.
Efforts to globally fit this data set with our previously defined model (41) revealed the need to modify the model. The result is a new 4-state thermodynamically constrained redox model of pig heart E3, which is able to reproduce a wide array of kinetic and equilibrium data. A unique example includes progress curves demonstrating, to our knowledge for the first time, both reverse E3 acidic NAD+ activation and alkaline NAD+ inhibition in a single consistent data set.
We also demonstrate the physiological kinetic reversibility of E3, which supports E3 kinetic control over the PDHc. Furthermore, in light of the observed non-Michaelis-Menten behavior of E3 (30, 39), we use non-linear optimization to search conditions that produce maximal E3 flux under physiological constraints. Our result demonstrates the difference between the maximum flux and kcat (48).
All chemicals and buffers were purchased from Sigma. Pig heart E3 was purchased from Calzyme (San Luis Obispo, CA) without further purification. All experiments were conducted using Nanopure water. Pig heart E3 was reconstituted in 50 mm potassium phosphate and 0.3 mm EDTA at pH 7. Active site concentration was assessed by monitoring the enzyme-bound FAD spectrum and calculating the concentration based on the molar absorptivity of 11.3 mm−1 cm−1 at 455 nm (47). Stock solutions were aliquoted and stored at −20 °C.
Fluorescence binding assays were conducted in 50 mm potassium phosphate, 0.3 mm EDTA, and adjusted to have an ionic strength of 170 mm using KCl at pH 7. Pig heart E3 (1.2 μm, 2 ml volume) solution was titrated with incremental amounts of a 100 mm NAD+ stock solution at 25 °C. The binding assay was monitored by pig heart E3 FAD fluorescence excited at 455 nm. Pig heart E3 FAD fluorescence emission spectra were collected from 480 to 650 nm using an LS 55 luminescence spectrometer (PerkinElmer Life Sciences). Excitation and emission slit widths were set at 5 and 8 nm, respectively. Fluorescence emission spectra were corrected for dilution.
dl-Lipoamide titrations of pig heart E3 (0.75 μm, 2 ml volume) were conducted in the same general conditions as the NAD+ titration described above; although dl-lipoamide (47 mm) stock was solubilized in a 100% ethanol solution. The final percentage of ethanol in the titration was ≤20%. The dl-lipoamide binding assays were monitored by pig heart E3 protein fluorescence (tryptophan) excited at 280 nm and by pig heart E3 FAD fluorescence excited at 455 nm. E3 protein fluorescence was corrected for dilution and was also corrected for inner filter effects using Equation 1 (49).
Fluorescence binding assays were analyzed using their full spectra according to supplemental Equation S1 and its solution shown in supplemental Equation S2. A detailed description of this method is provided in the supplemental material under Fluorescence Binding Assay Method.
Pig heart E3 kinetic assays were conducted in various initial solutions of NAD+ (0, 0.1, and 0.5 mm), NADH (0.25 and 0.5 mm), dl-lipoamide (0.25, 1, and 3 mm), and pH (4.5, 5.25, 5.65, 6.25, 7, 8, and 8.5). Assays were conducted using 2 nm enzyme, 0.3 mm EDTA, 0.67 mg/ml BSA (50), at 25 °C in 170 mm ionic strength adjusted using KCl. Final ethanol concentrations, due to the dl-lipoamide 100% ethanol solvent, were held constant at 7.5% throughout all assays. Assay pH was buffered using 50 mm citrate (pH 4.5), MES (pH 5.25, 5.65, and 6.25), MOPS (pH 7), and Tris (pH 8 and 8.5). To mitigate unwanted buffer effects, we chose to use the structurally similar MES and MOPS buffers around physiologically relevant pH. We show that MOPS and Tris, although structurally dissimilar, produce no difference in E3 rate at the same pH (supplemental Fig. S10).
Assays were conducted on a Varioskan Flash multimode plate reader (Thermo Fisher Scientific, Waltham, MA) using a 96-well format with 200-μl assay total volumes, which were followed at 340 nm. Calibration curves for NADH and dl-lipoamide were carried out on the plate reader to account for each of their molar absorptivities in the dihydrolipoamide dehydrogenase reaction. Enzyme assays were initiated by the automated injection of 100 μl (half of the total assay volume) of E3 to each well.
Pig heart E3 FAD absorption spectra in Fig. 4, A, D, and E, were obtained using the graph-digitizing software ScanIt 1.06 (amsterCHEM) from Fig. 3 in Ref. 46 and Figs. 1 and and44 in Ref. 47, respectively. Further necessary modifications to these obtained data are described in the supplemental material, under E3 FAD Spectral Analysis). The spectral data in Fig. 4, D and E, were used to estimate the fraction of enzyme that exists in the three major E3 redox states of oxidized, 2e−-reduced charge transfer complex, and 4 e−-reduced state by applying supplemental Equation S3 and its solution supplemental Equation 4. A detailed discussion of how these spectral data were used to extract enzyme fractional states is also provided in the supplemental material, under E3 FAD spectral analysis.
The model schematic is shown in Fig. 1A, where the overall forward E3 reaction is shown in Reaction 1.
In Reaction 1, A, B, P, Q, and H+ represent dihydrolipoamide, NAD+, lipoamide, NADH, and a proton, respectively. This model assumes that pig heart E3 is composed of four major redox states, including fully oxidized (S1), hydride-reduced disulfide (S2), hydride-reduced FAD (S3), and a hydride-reduced disulfide and FAD state (S4). We also assume that substrates, products, and protons bind randomly and are in rapid equilibrium with each redox state. Binding polynomials are used to determine the fraction of bound and free forms of the enzyme (Equation 2).
Dihydrolipoamide/lipoamide (A/P) and NADH/NAD+ (Q/B) pairs are considered to bind to two separate sites on E3 as is reasoned through existing structural data (Fig. 1B). The (A/P) and (Q/B) sites are denoted as α and β, respectively. In our experiments, a dl-lipoamide mixture was used, as is the case with datasets that were obtained from the literature. A discussion of this in regard to the kinetic model can be found in the supplemental material under Lipoamide Stereochemistry.
As demonstrated previously (41), we have allowed equilibrium dissociation constants to vary depending on the redox status of the enzyme to more accurately account for redox effects on the enzyme-ligand interaction. Different enzyme redox states are indicated in the binding polynomials by use of their numbered states (in Equation 2, x = states 1, 2, 3, or 4). Binding polynomials (Equation 2) for protons (Dh), representing the active site thiolate and base, are accounted for and assumed to be most significant in the instance of the hydride-reduced disulfide state (S2).
The kinetic equations are shown in matrix form in the supplemental Equation S5 and were solved analytically by computing the pseudo inverse of the state-transition matrix using MATLAB (2014b) Symbolic Math Toolbox. The solution to supplemental Equation S5 provides the distribution equations for each enzyme state, which are used to create a flux expression for the net production of NADH (Q) (Equation 4).
Thermodynamic loop constraints were derived using the two inner loops (Fig. 1A), described in detailed elsewhere (41). The resulting constraints are shown in Equation 5 and were used to eliminate two rate constants from the flux expression, k5 and k9.
The overall equilibrium constant for the dihydrolipoamide dehydrogenase reaction (Equation 6) is required for the above constraint and was normalized to pH 7 by multiplying the right hand side of Equation 6 by 107. Previously (41), the apparent equilibrium constant at pH 7 was estimated to be 0.0766, using Equation 7, from the midpoint potentials of the lipoamide/dihydrolipoamide (−0.287 V) and NAD+/NADH (−0.320 V) couples determined at 25 °C (2, 47).
In Equation 7, n is the number of electrons transferred; F is Faraday's constant (96,485 C mol−1); ΔEm is the midpoint potential difference; R is the gas constant (8.315 J K−1 mol−1), and T is temperature is in Kelvin. As was discussed previously (41), there are indeterminate errors associated with midpoint potentials (52) that are amplified through the exponential relationship between ΔEm and Keq (Equation 7). Also in relation to this fact, we have found additional pH-dependent equilibrium data for E3 in the Goldberg et al. database (53), which reports an apparent equilibrium constant of 0.27 at pH 7. We allowed variation in Keq to encompass experimental variability, where the boundary for the apparent Keq at pH 7 was set between 0.03 and 0.4. Details of model fitting are described in the supplemental material under Fitting.
NAD+/NADH and Lipo/DHL ratios can be optimized to produce a physiological maximal flux by using a fixed total amount of both lipoamide and NAD of 10 and 3 mm, respectively. These values were obtained from literature estimations (1, 54), and with specified NAD+/NADH and Lipo/DHL ratios were used to solve for the corresponding oxidized and reduced lipoamide and NAD concentrations needed to satisfy the conditions. For instance, in the case of NAD, the total NAD concentration (NADT) and ratio are given in Equation 8. Substitution and rearrangement yields Equation 9 and concentrations for both NAD+ and NADH.
Using the above constraints, the NAD+/NADH and Lipo/DHL ratios were optimized using the best-fit rate and equilibrium dissociation constants (Table 1) for both mam-E3 and Ec-E3 (Figs. 1111–14); the Ec-E3 kinetic model and parameters have been determined in a prior publication using a 3-state redox model (41).
Substrate/product ratios were optimized using the same global and local minimizers as the data fitting described in the supplemental material. Initial ratio values were randomly chosen between the following boundaries. The NAD+/NADH ratio was given the upper bounds of 102 (Ec-E3) and 25 (mam-E3) and lower bounds of 10−2 (Ec-E3) and 25−1 (mam-E3). The Lipo/DHL ratio was given the upper and lower bound of 1012 and 10−12, respectively, for both Ec-E3 and mam-E3.
NAD+/NADH boundaries were chosen to encompass a range that more than spans observed in vivo estimates (55,–57) to allow a wide range that may apply to normal, abnormal, or diseased states (58). Lipo/DHL ratios are less documented and thus are given less stringent bounds. Enzyme fractional states (Figs. 7 and and111111–14) were calculated with defined NAD+/NADH, Lipo/DHL, and pH values using distribution equations (59).
Substrate/product optimization was carried out by either allowing pH to vary (Figs. 11 and and12;12; Table 2) or fixing pH at different values (Figs. 13 and and14).14). Optimizations that allowed pH to vary (Figs. 11 and and12)12) are shown in comparison with kcat calculations described below and were given very wide boundaries for NAD+/NADH and Lipo/DHL (upper bound, 1012; lower bound, 10−12) to probe flux optima beyond physiological constraints.
Best-fit kinetic parameters (Table 1) using mam-E3 data (Fig. 6), for the 4-state redox model, and best-fit kinetic parameters for Ec-E3 described in Moxley et al. (41) were used to calculate the pH-dependent kcat in both the forward (Figs. 11A and and1212A; top) and reverse (Figs. 11B and and1212B) directions. The forward and reverse kcat values were derived by taking the limit of the flux expression as the substrates, in either direction, approach infinity (48). The corresponding enzyme state fractions that result from infinite substrate concentrations were calculated by taking the limit of each enzyme state distribution equation as substrates approach infinity (Figs. 11, A and B, and and12,12, A and B, bottom).
As described by Cornish-Bowden (48), the kcat is not a true flux maximum but a limit in mathematical terms. These derivations and calculations were used to compare with the true optimizations of flux in NAD+/NADH, Lipo/DHL, and pH (Figs. 11, C and D, and and12,12, C and D), albeit with a finite NAD and lipoamide pool of 3 and 10 mm, respectively (1, 54). These flux optimizations also differed from those shown in Figs. 13 and and1414 by allowing a much wider range of NAD+/NADH and Lipo/DHL to further probe E3 flux optima (upper bound, 1012; lower bound, 10−12, for both ratios). In addition, pH was an adjustable parameter bound between 4 and 9.
Fig. 2 summarizes an equilibrium titration of oxidized E3 (S1 in Fig. 1A) with NAD+. Utilizing the entire spectral dataset, accurate estimates of the fraction of NAD+ bound to the enzyme are computed by assuming linear superposition of NAD+ bound and free enzyme spectra (60). In this method, we assume that the first spectrum represents the unbound species, and the final spectrum, in the titration, represents the bound species. We solve for the term α (supplemental Equation S2), which provides the fraction of bound protein, and a fit to each spectrum (Fig. 2B). From here, α as a function of NAD+ (Fig. 2C) can be globally fitted with all other data sets (see below) and provides a very sensitive experiment reporting on NAD+ binding affinity to the E3-oxidized state (or S1 in Fig. 1A). Details of this method are explained in the supplemental material, under Fluorescence Binding Assay.
Reed (34) reported that both NAD+ and lipoamide binding to the oxidized form of E3 (Fig. 2A) would be consistent with the Lineweaver-Burk plot patterns they observed. Because we could not find any reports of lipoamide binding experiments in the literature, we conducted equilibrium titration experiments with dl-lipoamide and oxidized E3 (Fig. 3). In this experiment, E3 protein fluorescence was excited at 280 nm and was monitored from 294 to 540 nm (Fig. 3A). We observed apparent fluorescence quenching between 294 and 400 nm (Fig. 3A, black to blue), apparent fluorescence enhancement between 400 and 500 nm (Fig. 3A, black to green), and static fluorescence beyond 500 nm (Fig. 3A, black to yellow).
Apparent fluorescence quenching between 294 and 400 nm can be partially attributed to primary and secondary inner filter effects (49), resulting from the absorbance of dl-lipoamide from 280 to 440 nm (Fig. 3B, main solid lines). dl-Lipoamide molar absorptivity (m−1 cm−1) in this region was determined (Fig. 3B, inset), where the predicted absorbances are shown (Fig. 3B, main dashed lines). The dl-lipoamide molar absorptivity was then used to correct for inner filter effects using Equation 1 resulting in the corrected data (Fig. 3A, black to blue dashed lines). Inner filter effect corrected spectra (Fig. 3A, black to blue dashed lines) were then analyzed by applying the linear least squares method described in the supplemental material, where αi was determined as a function of dl-lipoamide (Fig. 3C).
Fluorescence enhancement between 400 and 500 nm is believed to be attributed to dl-lipoamide fluorescence (Fig. 3D). This conclusion was reached by exciting (280 nm) a solution of dl-lipoamide alone as a function of increasing dl-lipoamide, which showed increases in fluorescence between 400 and 500 nm. Fluorescence beyond 400 nm was attributed to E3 FAD fluorescence (Fig. 3E) by conducting a separate titration exciting E3 FAD fluorescence at 455 nm, whereas while titrating with dl-lipoamide showed little effect on the spectra. This result is consistent with the lipoamide-binding site being distal to the isoalloxazine ring of the FAD cofactor. This result coincides with the structural data (Fig. 1B) and a two-site mechanism (one site each for NAD+/NADH and Lipo/DHL) that we used to model the kinetic data (see “Experimental Procedures” for model description).
Additional information on ligand affinity and the redox equilibrium properties of pig heart E3 was obtained by analyzing existing FAD spectra of a NAD+ equilibrium titration on the E3 charge transfer complex (Fig. 4A) (46) and a dihydrolipoamide titration of oxidized E3 (Fig. 4D) (47). The NAD+ equilibrium titration (Fig. 4A) was conducted on the E3 charge transfer complex at pH 5.8 (apparent Keq = 0.0048) (46) and reports on the binding affinity of NAD+ to the enzyme state S2 (Fig. 1A). A singular value decomposition of the spectral data illustrates that the spectra are mostly composed of two species (Fig. 4B), where we assume that these species represent NAD+-free and -bound charge transfer complex. These spectra were then analyzed, as the fluorescence titrations described above, to obtain a binding curve as a function of NAD+ (Fig. 4C).
The redox status of pig heart E3 in the presence of increasing concentrations of dihydrolipoamide was monitored through FAD absorbance spectra (Fig. 4D) (47). Using absorbance spectra of known E3 major redox states (Fig. 4E), the fractions of E3 redox states as a function of dihydrolipoamide (Fig. 4F) can be determined from the titration spectra. (See supplemental material under E3 FAD spectral analysis.)
We previously demonstrated (41) that a 3-state redox model consisting of oxidized, 2e−, and 4e− reduced states can simulate NAD+ activation effects as they vary among different E3 homologs. E. coli and spinach E3 data sets each showed that NAD+ acts only as an activator in acidic pH, which is then neutralized at increasing pH values (see Figs. 3F and and44C in Ref. 41), although the model best-fit for spinach E3 simulated a small amount of NAD+ inhibition at pH values above 7.5 (see Fig. 4C in Ref. 41).
Although a reasonable amount of data exist for E. coli (30) and spinach (32) E3 at more neutral and basic pH values, using various NAD+ concentrations, data sets in these conditions using mammalian sources of E3 are lacking (41). To gain more insight, we used the best-fit parameters for a human liver E3 kinetic data set, discussed previously (41), to simulate higher NAD+-containing conditions as a function of pH (supplemental Fig. S1). These simulations predict NAD+ inhibition at near neutral to basic pH values (supplemental Fig. S1B).
In light of these observations and lack of available mammalian E3 data, we collected pH-dependent kinetic data with pig heart E3 using a wider pH range that encompasses acidic, neutral, and basic values (Fig. 5). In doing so, we also observed activation by NAD+ at acidic pH (Fig. 5, A and D) and inhibition as the pH approaches more basic values (Fig. 5, B and D). This effect is more clearly seen in a difference plot (Fig. 5F), where positive and negative values indicate activation and inhibition, respectively. Altogether, these data demonstrate a pH-dependent switch between E3 NAD+ activation and inhibition near a pH of 6 (Fig. 5F).
The previously described 3-state redox model (41), parameterized using human liver E3 data (33), is able to capture the pH dependence of both the forward and reverse E3-catalyzed reaction and predicts product inhibition of NAD+ at near neutral to more basic pH values (supplemental Fig. S1). But the model fails to reproduce our data from the pig heart enzyme to the 3-state redox model (Fig. 5C, dashed line, and see 3-state redox model fitting).
To circumvent the lack of fit to our data (Fig. 5, circles) to the 3-state redox model (Fig. 5, dashed lines), we first implemented a 6-state model that accounted for a documented (44) change in oligomeric state as a function of pH with pig heart E3. This model is described in supplemental Fig. S3. We discovered that this model can fit the core pH-dependent NAD+ activation/inhibition phenomenon with pH dependence of the forward reaction (supplemental Fig. S4), but it is significantly more complicated, adding six rate constants to the core 3-state model.
The simpler 4-state mechanism of Fig. 1A is better supported by the literature (47, 61, 62) and active site structure (Fig. 1B). This model, which accounts for the oxidation/reduction of the FAD cofactor (see FAD cofactor in Fig. 1B), is able to better fit the data (Fig. 5, solid lines). We also accurately fitted this data set using a 4-state redox model where the equilibrium dissociation constants (Kd) are not allowed to vary with the redox state of the enzyme (supplemental Fig. S2, solid lines), which suggests that the dissociation constants do not need to change as a function of E3 redox state to explain NAD+-positive and -negative regulation.
A 4-state redox-dependent model simulates the pH-dependent NAD+ activation/inhibition phenomenon of the reverse E3 reaction and forward pH-dependent data (Fig. 5, solid lines). To better characterize the NAD+ activation/inhibition phenomenon, we gathered more reverse E3 kinetic data, while also incorporating the fitting of the equilibrium spectroscopic experiments (Fig. 6).
In addition, to more accurately simulate forward E3 kinetics, we obtained forward pig heart E3 initial velocity data (Fig. 6O) from Ref. 63, which varied NAD+ in different concentrations of dihydrolipoamide. Dihydrolipoamide equilibrium titration data (47), described above, were also incorporated into this large data set by numerically integrating individual titration points to long times, which provides model consistency with FAD redox states as a function of dihydrolipoamide (Fig. 6M). Equilibrium titrations, described in Figs. 2 and and4,4, were incorporated into the fitting using fractional occupancies (Equation 3) to simulate the binding of NAD+ to enzyme states S1 (Fig. 6Q)and S2 (Fig. 6R) and dl-lipoamide binding to enzyme state S1 (Fig. 6P).
Fig. 6 shows the best fit of the 4-state redox-dependent Kd model. We also fit this large data set using a 4-state redox-independent Kd model (supplemental Fig. S5), so that equilibrium dissociation constants for substrates and products do not depend on the redox state of the enzyme. The 4-state redox-independent model primarily has trouble fitting the forward initial rates as a function of pH with all other data sets (supplemental Fig. S5 N).
The average difference between model fits and mean data shown in Fig. 6 is 12.6%. By computing the difference between model simulations and the error/uncertainty range of the data, the average deviation between model and data is 3.1%. Of 455 simulation points, 90 lie outside the range of uncertainty in the data. However, although the model fits are good, they are imperfect and do show biased error with respect to data in Fig. 6, G–L. The fits to these data could be improved by adding additional states to the model. The potential impact of these imperfections in the model fits is explored below.
Because our experiments use a dl-lipoamide mixture, a simple modification to the 4-state redox model to account for this mixture was used to investigate whether this detail in the model would improve the fit to the data (supplemental Figs. S11 and S12). We were unable to improve the fit accounting for this fact in the model, however. A discussion of this model and results canbe found in the supplemental material under “Lipoamide Stereochemistry.”
The 4-state redox-dependent Kd model, using best-fit parameters (Table 1), was used to calculate a physiological mam-E3 4-state redox steady-state distribution and flux surface (Figs. 7 and and8).8). These calculations used a physiologically relevant total NAD concentration of 3 mm (54) and lipoamide concentration of 10 mm (1). NAD+/NADH ratios were also chosen based on a range that spans conditions that may represent normal, abnormal, or diseased redox states (58), and were constrained by total NAD and lipoamide concentrations using Equations 8 and 9.
The calculated mam-E3 4-state redox steady-state distribution (Fig. 7) has a similar profile to a previously calculated distribution (41), parameterized by a much smaller data set using human liver E3 data (33, 41). The model is validated by experimental observations using stopped-flow technology (46, 61) that show the redox state-3 (S3), or the FAD reduced state, is present in very low fractions relative to the remaining enzyme redox states.
We calculated the mam-E3 physiological flux surface by both treating the flux as a function of NAD+/NADH and pH, with fixed Lipo/DHL (Fig. 8A), and as a function of NAD+/NADH and Lipo/DHL at pH 7.2 (Fig. 8C). The flux surface was calculated to gain insight into conditions that either produce forward (positive) or reverse (negative) E3 flux. These calculations can then predict the point at which the readily reversible mam-E3 switches direction and magnitude in flux.
Cross-sections from the multidimensional mam-E3 flux surface at different fixed pH (Fig. 8B) and Lipo/DHL (Fig. 8D) show NAD+/NADH ratios that cross a zero flux (equilibrium) threshold, demonstrating inflection points of E3 forward and reverse flux. For example, the calculated mam-E3 flux is near zero at NAD+/NADH ratios of 2.5, 1.5, and 1, at fixed pH values of 7, 7.2, and 7.5, respectively, using a fixed Lipo/DHL ratio of 1 (Fig. 8, B and D). Therefore, mam-E3 has the potential to catalyze its reaction in both directions with similar magnitude at or near physiological pH, NAD+/NADH, and reasonable steady-state NAD and lipoamide pools. The steady-state value of the Lipo/DHL ratio is, of course, also an important dictator of E3 flux direction and magnitude (Fig. 8D).
To explore how the error and uncertainty in model fits to data affect these model predictions, we randomly perturbed adjustable parameter values to obtain 1000 new parameter sets within a 10% boundary of the best-fit values from Table 1 (see supplemental material for parameter perturbation). The maximum and minimum model output from the parameter sets shows a wider span of the model simulation, but the qualitative behavior of the model is unchanged. (Compare supplemental Fig. S8 with Fig. 6.) The effects of this parameter perturbation on the predictions shown in Fig. 8 are illustrated in supplemental Fig. S9, revealing that these predictions are largely insensitive to the uncertainty in the model parameterization.
In comparison of the calculated mam-E3 flux surface (Fig. 8), we also computed the Ec-E3 flux surface (Fig. 9) using best-fit parameter values with the previously described (41) 3-state redox model. These calculations illustrate a stark difference between mammalian and E. coli E3 enzymes in their net flux as a function of NAD+/NADH, Lipo/DHL, and pH. Mam-E3 easily toggles between forward and reverse net flux (Fig. 8A), whereas Ec-E3 catalytically favors the forward reaction (Fig. 9A), requiring higher NAD+/NADH and Lipo/DHL ratios for net reverse catalysis at pH 7.2 (Fig. 9D). Despite this observation, we find that after increasing the Lipo/DHL ratio to 5 and/or the pH to 7.5, Ec-E3 is clearly able to switch catalytic direction (Fig. 10) within reasonable physiological conditions. Additional flux surface plots for both mam-E3 (supplemental Fig. S6) and Ec-E3 (supplemental Fig. S7) are given and further illustrate the flux as a multidimensional function of NAD+/NADH, Lipo/DHL, and pH.
Maximal flux is typically determined by calculating the kcat for a specified condition (48, 59, 64). But the kcat is not the true maximum (48). It is the limit of the flux as enzyme substrates approach infinity. To illustrate this point, we calculated the kcat for mam-E3 as a function of pH using the 4-state redox model (using best-fit parameters in Table 1) in both the forward and reverse direction (Fig. 11, A and B, top); some of these results are included in Table 2. This calculation reveals that the reverse kcat produces rates less than the observed turnover (Fig. 6, G–L) at a given pH. The enzyme state fractions corresponding to each of these limits were also calculated (Fig. 11, A and B, bottom). In comparison, this calculation was also carried out with Ec-E3 (Fig. 12, A and B) using the 3-state redox model (41), producing a similar result that we have mentioned elsewhere (41).
Despite kcat producing a submaximal flux in the reverse direction for both mammalian (Fig. 11B) and E. coli E3 (Fig. 12B), the forward kcat (Figs. 11A and and1212A) does not suffer from this issue. These calculations then illustrate the point that without substrate inhibition or possibly mechanistic randomness (64), this limit will produce the maximum amount of substrate-bound state (i.e. ES) leading to maximal flux. However, mechanisms that have a non-hyperbolic dependence of the rate on the substrate concentration (in this case in the reverse direction) need to be analyzed by other optimization methods rather than the traditional limit of substrate concentration.
To address this issue, mammalian and E. coli E3 reverse flux was maximized (Fig. 13, A and B), using the best-fit rate constants and equilibrium constants (Table 1) and by varying NAD+/NADH and Lipo/DHL (Ec-E3 best-fit rate constants were determined previously (41)). Importantly, NAD+/NADH and Lipo/DHL ratios were constrained as described under “Experimental Procedures” according to total physiological NAD and lipoamide concentrations. Thus, the maximal mam-E3 and Ec-E3 flux was determined by varying the NAD+/NADH and Lipo/DHL at different fixed pH values (Fig. 13) according to physiological constraints.
Mammalian E3 showed a reverse flux maximum at a pH near 6 (Fig. 13A), although the maximum for E. coli is near 7.5 (Fig. 13D). For pH below 6, NAD+/NADH ratios varied from 1 to 7 for mam-E3 but remain fixed at 0.04 for higher pH (Fig. 13B). However, Ec-E3 prefers more oxidized NAD+/NADH ratios for maximum reverse activity across all pH ranges, where NAD+/NADH values between 12 and 60 are most optimal around neutral pH (Fig. 13E). The NAD+/NADH ratios below pH 6 result in higher S2 and lower S4 fractions to optimize the reverse mam-E3 flux (Fig. 13C). In the case of Ec-E3, the reverse flux is maximized by more evenly distributing the enzyme redox states (Fig. 13F). The mam-E3 and Ec-E3 forward flux was also maximized at different fixed pH values (Fig. 14), demonstrating differences between mam-E3 and Ec-E3 fractional states, when the forward flux has been maximized.
Reverse/forward mammalian and E. coli E3 fluxes were also maximized by varying NAD+/NADH and Lipo/DHL, using wider boundaries for substrate/product ratios (Figs. 11 and and12;12; Table 2), without fixing pH. This approach illustrates that wider substrate/product ratios do allow the forward E3 flux to increase significantly for mam-E3 (compare Figs. 11C (top) with with1414A) but not for Ec-E3 (compare Fig. 12C (top) to Fig. 14D). However, the reverse E3 flux does not benefit from going outside physiological substrate/product ratios (compare Figs. 11D with with1313A) for mam-E3 or Ec-E3 (compare Figs. 12D (top) with with1313D (top)).
Early reports (39, 40) of pig heart E3 reverse reaction kinetics demonstrated a unique NAD+ product activation phenomenon, which is especially prominent at low pH and higher NADH concentrations. However, these earlier reports (39, 40) lacked assay conditions that varied NAD+/NADH or lipoamide and did not probe physiological pH values.
NAD+ product activation has also been observed in E. coli (30), spinach (32), and human liver (33) E3, in which we previously derived a 3-state redox kinetic model that accurately simulated these data sets (41). By analyzing mammalian data sets from rat (34) and human (33) liver E3, it was determined that the available data were not sufficient to accurately characterize mam-E3 NAD+/NADH pH-dependent regulation (41).
To address this void, pig heart E3 reverse kinetics were assayed in 162 initial conditions (Fig. 6, G–L) varying NAD+/NADH, dl-lipoamide, and pH ranging from 4.5 to 8.5, in physiological ionic strength (170 mm) using KCl as a background salt. We also conducted NAD+ and dl-lipoamide equilibrium titrations on oxidized pig heart E3 to more directly identify equilibrium dissociation constants that would be consistent with our kinetic model (Figs. 2 and and3).3). Other more sophisticated literature-derived (46, 47) active site (FAD) spectroscopic experiments were analyzed by linear algebraic methods (Fig. 4). These data sets (Fig. 4) were also incorporated into our large experimental ensemble (Fig. 6) to maintain consistency with observations that directly probe E3 active site redox states. Furthermore, forward reaction pH-dependent and dihydrolipoamide/NAD+-dependent data sets, from human liver (33) and pig heart (63) E3 (Fig. 6, N and O), were added to accurately model mam-E3 kinetic reversibility.
Our newly collected pig heart E3 reverse kinetic assays have demonstrated a NAD+ inhibitory effect in the physiological pH range (pH 7–7.5), rather than the activation effect traditionally reported at low pH (39, 40). Generally, we find that at a pH near 6, NAD+ switches between an activator and inhibitor (Fig. 5). We have not been able to find this effect demonstrated with any previous E3 kinetic data sets. NAD+ inhibition effects are more physiologically relevant in the grand scheme of PDHc regulation, because it is this type of regulation that manifests near neutral pH. Moreover, product inhibition is ubiquitous among enzyme kinetic characterization studies and is typically expected (59). Although we were unable to find reverse E3 kinetic data demonstrating a NAD+ inhibitory effect, we discovered that our previous 3-state model (41) predicts NAD+ inhibition (supplemental Fig. S1B) at physiological pH by extrapolating [NAD+] >100 μm using a previously determined (41) best-fit parameter set derived from fitting human liver E3 kinetic data (33, 41). Despite similar 3-state model predictions based on human liver E3 parameters to our pig heart E3 data, we were unable to accurately fit our data using the 3-state model (Fig. 5).
To account for the lack of fit to the data with the 3-state model, we developed a 4-state redox model (Fig. 1A) that accounts for a chemical step that was previously simplified in the 3-state redox model (41). The 4-state redox model accounts for the internal electron transfer between the active site disulfide and the FAD cofactor (Fig. 1, A and B). This added step decomposes the 2e− reduced state in the original 3-state redox model (41), creating an additional state. This mechanism is able to simulate both reverse reaction NAD+ activation/inhibition and reaction forward pH dependence (Fig. 5, solid lines).
All model simulations were subjected to thermodynamic constraints (2, 51) and to a global fitting (Fig. 6) approach rather than the more simple (and inconsistent) approach of individual data set fitting, which is ubiquitous in the field of enzyme kinetics (66).
Alternative parameter sets that fit the data (Fig. 6) equally as well from random independent starts are difficult to find. This result is consistent with parameters that are more constrained than our previous result in fitting a small human liver E3 data set (33, 41) that produced numerous best-fit examples with an extremely wide range of parameter values (41). Thus the breadth of the data set reported here facilitates a level of precision in model identification that was not possible based on previously available kinetic data (41).
Beyond data collection, model derivation, and parameterization, we also probed model predictions in flux and enzyme states as a function of NAD+/NADH, Lipo/DHL, and pH to gain insight into E3 kinetics (Figs. 7 and and8).8). Calculation of E3 redox states as a function of NAD+/NADH, Lipo/DHL, and pH (Fig. 7) reveals an interesting prediction that is consistent with stopped-flow experiments (46). The calculated mam-E3 state-3 (S3), irrespective of most conditions, is present at very low fractions (Fig. 7). This model prediction is consistent with stopped-flow spectroscopic studies, which were unable to observe this state due to its very transient nature using pig heart E3 (46).
To better understand E3 kinetic reversibility, we calculated the mam-E3 steady-state flux surface (Fig. 8) as a function of NAD+/NADH and pH, at Lipo/DHL = 1, which demonstrates a readily reversible catalytic surface. Of particular interest here are the conditions at which this enzyme switches direction in flux because of the kinetic control E3 has on lipoamide (in PDHc L2 domain) redox status (8); the lipoamide redox state is important for PDK binding (22) and thus PDHc phosphorylation status. Therefore, PDHc NAD+/NADH sensing is mediated through E3 and is closely linked to PDHc phosphoryl regulation.
Our 4-state model predicts, using constant NAD and lipoamide pool sizes (see “Experimental Procedures”), at physiological pH that NAD+/NADH values between 1 and 2.5 would begin to change E3 flux direction with a Lipo/DHL ratio set to one (Fig. 8B). In comparison with mam-E3, we also calculated the Ec-E3 flux surface (Fig. 9) and noticed a significant difference in the kinetic reversibility of these homologs in the simulated conditions. These simulations demonstrate the kinetic difficulty of Ec-E3 reverse (Fig. 9) catalysis compared with mam-E3 (Fig. 8). In this respect, Lipo/DHL ratios are very important in dictating flux direction and magnitude, whereas higher Lipo/DHL ratios for Ec-E3 relative to mam-E3 (Figs. 8D, ,9,9, and and1010 and supplemental Fig. S7) are more essential for reverse Ec-E3 catalysis. After increasing Lipo/DHL and pH, we found that Ec-E3 also demonstrates a reversible catalytic flux (Fig. 10), simulated using E. coli intracellular pH and NAD+/NADH ratios (55, 67). Sole E3 NAD+/NADH ratio regulation of the E. coli PDHc is expected due to the lack of PDK and PDP enzymes (30, 35).
In order for mam-E3 to kinetically control the lipoamide redox state, and thus PDK binding affinity (22), it needs to effectively catalyze both forward and reverse directions in physiological conditions. It follows then that the demonstration of a readily reversible mam-E3 physiological catalytic surface (Fig. 8) supports the overall mechanism of mam-E3 kinetic regulation of L2 domain redox status, important for PDK binding affinity and subsequent phosphoryl regulation (26).
Although rate laws have been developed for a large number of classical models of enzyme mechanisms (64), our analysis of E3 kinetics required the introduction of a novel mechanism to capture pH-dependent NAD+ product activation (30, 33, 39, 40) and inhibition (shown here in Fig. 5), as well as pH and NAD+-dependent progress curve lag phases (30, 33, 39, 68), substrate inhibition (30, 39), and pH-dependent oligomeric state regulation (44), as discussed elsewhere. In this model, as for many complex enzymatic mechanisms, the conventional method (kcat) to represent the optimal rate (48) fails to reveal the true catalytic maximum.
For example, in the case of simple substrate inhibition, it is traditional to ignore inhibitory terms to produce a kcat value. This value, however, does not reflect the actual mechanism, and thus it is a misrepresentation of the actual limit of enzymatic turnover. In the case of E3, calculating the theoretical reverse maximum flux with pH-dependent substrate inhibition (in NADH), pH-dependent product activation/inhibition (in NAD+), randomness in substrate/product binding, and more than one pathway to product requires solving a multivariate optimization problem to find maximal enzymatic flux.
Our results show that the forward E3 maximal flux with mam-E3 (Fig. 11C, top) and Ec-E3 (Fig. 12C, top) is similar to kcat (Figs. 11A and and12A,12A, top), whereas the reverse E3 maximal flux (Figs. 11D and and12D,12D, top), conditions, and enzyme states (Fig. 11D and and12D,12D, bottom) are much different from the reverse kcat (Figs. 11B and and12B,12B, top; Table 2). Our optimized conditions are constrained by reasonable physiological NAD and lipoamide pools (see “Experimental Procedures”) and NAD+/NADH and Lipo/DHL ratios, where kcat does not consider these constraints and produces results that may be submaximal and non-physiological in the extent of extrapolated infinite substrate (48).
We believe that the general numerical approach of optimizing enzyme substrate/product ratios, using a rigorously parameterized enzyme kinetic model, will be useful for understanding enzyme flux optima especially in the context of a mechanism that produces non-Michaelis-Menten behavior (hyperbolic deviation (64)). Furthermore, more physiologically feasible enzyme states and optimal conditions are revealed with the proper constraints. This approach should be of general interest to fields that rely on enzyme optimization such as enzymology and enzyme engineering, as well as more broadly scoped fields including systems biology and bioengineering.
M. A. M. designed and conducted the experiments. M. A. M. and J. N. B. analyzed the experiments and wrote computer code for analysis. M. A. M. wrote the paper. J. N. B., D. A. B., and M. A. M. revised the paper. All authors approved the final manuscript.
M.A.M. acknowledges Dr. Kalyan Vinnakota for help with the Varioskan Flash multimode plate reader.
*This work was supported by National Institutes of Health Grants R01-DK095210, R01-HL072011, T32-HL094273, and K99-HL121160. The authors declare that they have no conflicts of interest with the contents of this article. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
This article contains supplemental Equations S1–S5, Figs. S1–S12, and additional references.
2The abbreviations used are: