Determining enzyme kinetics for systems biology with nuclear magnetic resonance spectroscopy

Abstract

Enzyme kinetics for systems biology should ideally yield information about the enzyme's activity under in vivo conditions, including such reaction features as substrate cooperativity, reversibility and allostery, and be applicable to enzymatic reactions with multiple substrates. A large body of enzyme-kinetic data in the literature is based on the uni-substrate Michaelis-Menten equation, which makes unnatural assumptions about enzymatic reactions (e.g., irreversibility), and its application in systems biology models is therefore limited. To overcome this limitation, we have utilised NMR time-course data in a combined theoretical and experimental approach to parameterize the generic reversible Hill equation, which is capable of describing enzymatic reactions in terms of all the properties mentioned above and has fewer parameters than detailed mechanistic kinetic equations; these parameters are moreover defined operationally. Traditionally, enzyme kinetic data have been obtained from initial-rate studies, often using assays coupled to NAD(P)H-producing or NAD(P)H-consuming reactions. However, these assays are very labour-intensive, especially for detailed characterisation of multi-substrate reactions. We here present a cost-effective and relatively rapid method for obtaining enzyme-kinetic parameters from metabolite time-course data generated using NMR spectroscopy. The method requires fewer runs than traditional initial-rate studies and yields more information per experiment, as whole time-courses are analyzed and used for parameter fitting. Additionally, this approach allows real-time simultaneous quantification of all metabolites present in the assay system (including products and allosteric modifiers), which demonstrates the superiority of NMR over traditional spectrophotometric coupled enzyme assays. The methodology presented is applied to the elucidation of kinetic parameters for two coupled glycolytic enzymes from Escherichia coli (phosphoglucose isomerase and phosphofructokinase). 31P-NMR time-course data were collected by incubating cell extracts with substrates, products and modifiers at different initial concentrations. NMR kinetic data were subsequently processed using a custom software module written in the Python programming language, and globally fitted to appropriately modified Hill equations.

Figure 1

A work flow diagram.

Figure 2

³¹P NMR (a) The effect of EDTA on the line shapes of ATP using ³¹P NMR. Spectra were collected with a 90° pulse angle and repetition time of 1 s (0.5 s acquisition time, 0.5 s relaxation delay). ATP concentration was 5 mM; (b) MgCl₂ titration of FBP, ATP and ADP and the effect on ³¹P NMR spectral offset and line shape. Spectra were collected with a 60° pulse angle and repetition time of 1.3 s (0.8 s acquisition time, 0.5 s relaxation delay). FBP, ATP and ADP concentrations were 10 mM, and the indicated concentration of MgCl₂ (in mM) was added. All other parameters are described in Section 3.3. Raw NMR FID data are included as supplementary material.

Figure 3

Spline fits of ³¹P-NMR Phosphofructokinase data. Data were acquired using a 90° pulse angle to collect 100 transients per FID using a repetition time of 1 s (0.5 s acquisition time, 0.5 s relaxation delay). Top row: Progress curves representing NMR peak integrals (G6P , ATP , FBP , ADP , PEP ) are fitted with splines (G6P , F6P , ATP , FBP , ADP , PEP ). Inhibitor assays containing PEP are shown in the last two blocks. Note that with the exception of the second-last assay, F6P concentrations are inferred from equilibrium with G6P via PGI. Bottom row: Respective rates derived from spline-fitted NMR data. Dual colour lines indicate an average of two respective rates. For comparison, the rate calculated by the irreversible Hill equation (---, Table 1: PFK) at the specific substrate, product and effector concentrations is shown. The Hill equation parameters were the same throughout and obtained from a global fit of all the time courses shown. Rates are normalised to total protein. Raw NMR FID data, as well as NMR peak integrals and spline data, are included as supplementary material.

Figure 4

Phosphoglucose Isomerase. (a) Example of a ³¹P-NMR time course of a PGI reaction using a cell extract incubated with an initial concentration of 8.5 mM F6P and no G6P, collected at a 60° pulse angle over 47 min (0.8 s acquisition, 0.5 s relaxation). Additional NMR parameters are described in Section 3.3. In this reaction, F6P was converted in reverse to G6P as the reaction approached equilibrium. The time course is not shown to full equilibration; final concentrations were 5 and 2.8 mM for G6P and F6P respectively. TEP is an internal standard; (b) Reversible Michaelis-Menten equation (see Table 1) fitted to PGI progress curves derived from NMR data: equilibrium values are represented by the red contour line (), arrows indicate both the metabolite concentrations and the direction of reaction as each time course progresses towards equilibrium (→→→). The rate was normalised to total protein concentration. Substrate and product concentration axes are in logarithmic scale. R²= 0.99.

Figure 5

Spline fits of Phosphoglucose Isomerase data. Top row: Time courses of the PGI reaction were acquired by incubating a cell extract with various starting concentrations of substrate G6P () and product F6P () and monitoring reaction progress using ³¹P NMR with a 90° pulse angle and 1 s repetition time (1.0 s acquisition, 0.0 s relaxation) with 80 transients per FID. Other parameters are as described in Section 3.3. Progress curves derived from NMR peak integrals were fitted with splines (G6P , F6P ). Bottom row: The respective averaged rates of the fitted splines are plotted (dual colours indicate average of two respective rates) with the rate of the fitted kinetic equation included (---, Table 1: PGI). Rates were normalised to total protein content. Raw NMR FID data, as well as NMR peak integrals and spline data, are included as supplementary material.

Figure 6

Example of a Phosphofructokinase ³¹P-NMR time course. As high Mg²⁺ concentrations can lead to line-broadening and an obscured spectrum, data were collected with no additional Mg²⁺ (beyond the trace amounts left from the growth medium) for better resolution. A pulse angle of 60° and a repetition time of 1.3 s (0.8 s acquisition time, 0.5 s relaxation delay) was used. 10 mM triethyl phosphate (TEP) is included as an internal standard. All other parameters are described in Section 3.3. (a) Full NMR spectrum. Initial concentrations were 14 mM G6P, 3 mM F6P, 13 mM ATP. The first few FIDs collected before the lock signal had stabilised, have been excluded; (b) Expansion of the sugar-phosphate region (4.0 to 1.5 ppm); (c) Expansion of the nucleoside phosphate region (−5 to −10 ppm).

Figure 7

Phosphofructokinase: irreversible bi-substrate Hill equation globally-fitted to aggregated ³¹P-NMR progress curves (see Table 1 for equation and fitted parameters). Rate is normalised to total protein. Substrate concentration axes are in logarithmic scale. Arrows indicate both the metabolite concentrations and the direction of the reaction for individual time courses (→→→). PEP inhibitor assay data have been excluded from the plot as only two variables can be visualised simultaneously. R²= 0.96.

Figure 8

Phosphofructokinase enzyme-coupled kinetic assay. ATP saturation curves at different F6P concentrations were generated using the coupled enzyme assay system as described in Section 3.5. Points represent initial rate data and are fitted with a standard irreversible Michaelis-Menten kinetic equation. Error bars represent experimental replicates (n = 3) and are S.E.M. F6P: 10 mM (), 5mM (), 2.5mM (), 1.25mM (), 0.625 mM ().

Figure 9

A simulation of a two-enzyme NMR time course involving PGI and PFK, beginning with initial measured concentrations. (a) Model schematic. Kinetic parameters are as described in the text. To accurately approximate experimental conditions, the model consisted of three reactions in addition to the two glycolytic enzymes: both ATP and ADP were in rapid-equilibrium reactions with MgATP and MgADP (1 mM free Mg²⁺ , K_eqvalues were 10₄ and 10³ respectively [57,58]), and MgADP was consumed by an elementary first-order hydrolysis reaction producing AMP + Pi + Mg²⁺ (k = 2 × 10⁻⁴); see text for details; (b) Simulated time course concentrations (G6P , F6P , ATP , FBP , ADP ) compared with experimental time course data (G6P , F6P ., ATP , FBP , ADP ). The time course started with only F6P and ATP present as substrates. Note: for parameter fitting, F6P was assumed to be in equilibrium with G6P via the PGI reaction, and as such no quantified F6P data are included in this figure, except for an initial concentration. AMP and orthophosphate are not shown.

Figure 10

(a) Phosphoglucose Isomerase and (b) Phosphofructokinase parameter fitting performed after deleting a number of closer-to-equilibrium data points. To observe the effect of losing data from the latter part of the time courses on the fitting process, parameter fitting was performed as before using the Levenberg-Marquardt algorithm after a series of truncations had been made to the spline-fitted datasets, starting at the end of each time course, i.e., closer to equilibrium (original data in Figure 5 and Figure 3; fitted kinetic equations as in Table 1). Error was calculated as before (Table 1, footnote b) but with two variations: the first method involved rescaling the degrees of freedom to reflect the degree of truncation of the data (--), the second method retained the original degrees of freedom (-). This approach to error estimation was adopted to be able to distinguish between two sources of error: that due to losing data generally, and that due to specifically losing closer-to-equilibrium data. Twelve and twenty data points were deleted sequentially from the PGI and PFK data, respectively (representing up to 40 % and 24 % of the length of the longest respective time courses). Fitted parameters (-).