Open access peer-reviewed chapter
By Anton Wangler, Mark Jonathan Bunse, Gabriele Sadowski and Christoph Held
Submitted: March 22nd 2018Reviewed: July 11th 2018Published: November 5th 2018
DOI: 10.5772/intechopen.80235
Downloaded: 288
The classical approach towards analysing the influence of co-solvents (i.e., cellular molecules that are chemically inert and do not act as reacting agents) on the Michaelis constants of enzyme-catalysed reactions is empirical. More precisely, reaction kinetics is usually mathematically modelled by fitting empirical parameters to experimental concentration vs. time data. In this chapter, a thermodynamic approach is presented that replaces substrate concentrations by thermodynamic activities of the substrates. This approach allows determining activity-based Michaelis constants. The advantage of such activity-based constants K M a over their concentration-based pendants K M obs is twofold: First, K M a is independent of any co-solvent added (while K M obs is not) as long as it does not directly interfere with the reaction mechanism (e.g., inhibitor or activator). Second, known K M a values allow predictions of Michalis constants for different enzymes and reactions under co-solvent influence. This is demonstrated for a pseudo-one-substrate peptide hydrolysis reaction as well as for more complex two-substrate alcohol dehydrogenase reactions.
Understanding the kinetics of enzyme-catalysed reactions is a key aspect not just in the field of biology but also of high relevance for biocatalysis in the industry as enzymes are highly suitable for the production of fine chemicals [1]. The advantage of enzyme catalysis is that high enantioselectivity [2, 3] can often be realised under mild reaction conditions (ambient temperature and pressure).
Key properties for the study of enzyme-catalysed reactions are reaction yield and reaction kinetics. In case of the reaction yield, thermodynamic states an independence of the equilibrium position from the catalyst involved (as long as the catalyst concentration is low) [4, 5, 6, 7]. In contrast, reaction kinetics strongly depends on the catalyst [8, 9]. This means that different enzymes used for the same reaction will cause different kinetic profiles for the considered reaction; this is represented by the experimental (concentration-based) Michaelis constant
In this chapter, an approach is presented to determine
Examples for one-substrate reactions are isomerase reactions where one substrate is converted to another without any change to the chemical composition of the molecule. Examples can be found in glycolysis, one being the reversible conversion of 3-phosphoglycerate (substrate S) to 2-phosphoglycerate (product P) catalysed by phosphoglycerate mutase (enzyme E). The general reaction scheme of a one-substrate reaction is given in Eq. (1).
The kinetics of the reaction according to Eq. (2) is commonly described by the Michaelis-Menten equation including the reaction rate
Eq. (2) is visualised exemplary by plotting of
Qualitative Michaelis-Menten plot of the reaction rate ν plotted over the substrate molality m S according to Eq. (2).
As can be seen from Eq. (2) and Figure 1, the reaction rate follows a hyperbolic curve over increasing substrate concentrations. Further,
Unfortunately, the majority of enzyme-catalysed reactions are not one-substrate reactions; in such cases, the reaction scheme increases in complexity. However, it is often still possible to apply pseudo-one-substrate reaction conditions given that the molality of one substrate is much higher than the molality required to obtain
In Eq. (3), substrates are labelled as S1 and S2; the reaction mechanism (ordered or random) shall not be discussed at this point. In this case, the Michaelis-Menten equation changes to Eq. (4), which contains the Michaelis constants for substrate 1
In the case of a hydrolysis reaction taking place in water as reaction solvent, the molality of substrate 2
To be able to compare reactions from different research groups and further for different enzymes catalysing the same reaction, the Michaelis-Menten equation has to be normalised to the total enzyme concentration
The determination of
Through this linearization, a plot of
As presented in Section 2.1.1, the Michaelis constant is determined based on the molality
To be able to predict co-solvent influences on pseudo-one-substrate reactions, a thermodynamic co-solvent-independent Michaelis constant, further referred to as
In the following, molality-based
To determine
Steps for the prediction of the concentration-based Michaelis constant K M pre under the influence of co-solvents. Predictions are based on the determined activity-based Michaelis constant K M a .
As can be seen, the major aspect for the prediction of the Michaelis constants is the ability to predict the substrate activity coefficients. For this, a physically sound model, namely the electrolyte perturbed-chain statistical associating fluid theory (ePC-SAFT) was used in this work. This model has already been applied successfully to complex mixtures containing low-soluble molecules [24], PEG and salts [25] and electrolytes [20] while also being applied simultaneously to mixtures with up to eight components [4], and thus, it provides a reliable model basis for this work.
The ePC-SAFT equation of state is based on PC-SAFT, developed and proposed by Gross and Sadowski [26] and extended for electrolyte systems by Cameretti et al. [18] ePC-SAFT provides an expression for the residual Helmholtz energy
In Eq. (14), the reference system is seen as a chain of hard spheres, which is represented by the contribution
In Eq. (15), 0i denotes the pure component, which is the reference state at the same temperature T and pressure p as the actual solution of the composition
In this work, a pseudo-one-substrate reaction is presented using the hydrolysis of SPNA catalysed by the enzyme α-CT. The reaction mechanism is given in Scheme 2.
Reaction scheme for the hydrolysis of SPNA catalysed by α-chymotrypsin. Products of the hydrolysis reaction are N-(3-carboxypropanoyl)phenylalanine and p-nitroaniline, respectively.
The kinetic measurements have been discussed already in [7] and are briefly summarised here. Lyophilized powder of α-CT was used as catalyst, and trimethylamine N-oxide (TMAO) and urea were used as co-solvents. Measurements were carried out in Tris-HCl buffer (100 mmol/kgwater tris(hydroxymethyl)aminomethane, pH 8.0). The kinetic measurements of the neat and co-solvent reaction mixtures of the SPNA hydrolysis reactions were performed in a stopped-flow system (HPSF-56 of Hi-Tech Scientific) [27, 28]. In a first step, the substrate stock solution containing SPNA and the respective co-solvent in a 100 mol/kgwater Tris-HCl buffer at pH 8 and the enzyme stock solution containing the respective co-solvent in a 100 mol/kgwater Tris-HCl buffer at pH 8 were prepared and loaded for injection in the measurement cell. After simultaneous injection, the measurement cell was constantly monitored for the extinction at 410 nm wavelength, allowing the determination of the time-dependent change in the 4-NA concentration. The pH values of the stock solutions were measured directly before the start of the reaction to ensure no pH effect on
| Co-solvent | ||
|---|---|---|
| Neat | — | 0.125–1 |
| TMAO | 0.5 | 0.125–1 |
| Urea | 1 | 0.250–1 |
| DMSO | 2.1 | 0.250–1 |
| DMSO | 4.2 | 0.125–1 |
Overview of the measured systems to determine concentration-based Michaelis constants
Enzyme concentration was 8 μmol/kgwater in all kinetic assays.
In a first step, the concentration-based Michaelis constant
Lineweaver-Burk plot for the determination of the concentration-based Michaelis constant of SPNA K M obs at T = 25°C, p = 1 bar and pH = 8 in Tris-HCl buffer [7]. The plot shows experimental data points of the neat measurements (squares) which are obtained from the inverse turnover frequency ν ′ − 1 over the inverse substrate molality of SPNA m SPNA − 1 . K M obs was obtained by linear regression of the experimental data and extrapolation to the abscissa as shown.
As can be seen from Figure 2, a linear relation between the reciprocal molality of the substrate
Note that in a first step, mole-fraction-based activity coefficients were obtained with ePC-SAFT. Eq. (12) was used to convert these into molality-based activity coefficients; these were used throughout this work. In the next step, a plot of the determined
Comparison between experimental concentration-based Michaelis constants K M obs (light grey bars) at T = 25°C, p = 1 bar and pH = 8 in Tris-HCl buffer and the predicted Michaelis constants K M pre (dark grey bars). For the predictions, a constant K M a value of 0.0686 was used and the activity coefficients were predicted with ePC-SAFT based on the parameters from Tables 2 and 3. Reprinted from [7].
| Co-solvent | mco-solvent (mol/kgwater) | ||
|---|---|---|---|
| TMAO | 0.5 | ||
| Urea | 1 | ||
| DMSO | 2.8 | ||
| DMSO | 4.2 |
Comparison between the experimental
As can be seen in Figure 3 and Table 4, an accurate prediction of the co-solvent-induced changes in
As presented in Section 2.1.1 (‘pseudo’) one-substrate reactions occur seldom in enzyme catalysis. Enzyme catalysis often requires co-substrate that is present in a limiting concentration (e.g., NADH, ATP, GTP). A two-substrate reaction can be described by Eq. (16), which cannot be simplified further:
Two-substrate reactions can have a specific binding order attached to them. To account for this, the inhibition constant of S1
Note that Eq. (17) does not show any direct relation between the Michaelis constants and the ordinate, slope or the abscissa of the linearization. In the case of two-substrate reactions, a two-step linearization process is suggested. For this, the molality of one of the substrates, in this case
Exemplary primary plot for a two-substrate reaction obtained from plotting the inverse turnover frequency ν ′ − 1 over the inverse substrate molality of substrate 2 m S 2 − 1 for different pseudo-constant molalities of substrate 1 m S 1 . Molalities m S 1 increase in the order of m S 1 , squares > m S 1 , circles > m S 1 , triangles .
Each of the straight lines in Figure 4 has its own slope (
Eqs. (18) and (19) again show a linear correlation between
Exemplary secondary plot for Orprim (left) and Slprim (right) over the reciprocal pseudo-constant molalities of m S 1 − 1 derived from the primary plot given in Figure 4.
The relations shown in Eqs. (20)–(23) are finally used to determine
The determination of activity-based Michaelis constants
From Eq. (24), a primary plot is created as described in Section 3.1.1 in which
Predictions for the co-solvent influence on
Steps for the prediction of the concentration-based Michaelis constants K MS 1 pre and K MS 2 pre under the influence of co-solvents. Predictions are based on the determined activity-based Michaelis constants K MS 1 a and K MS 2 a .
In this work, the reduction of acetophenone by two different enzymes, ADH 270 and ADH 200, was investigated as model reaction for a two-substrate reaction. The reaction scheme is given in Scheme 4. Kinetic data for the ADH 270 were taken from [23].
Reaction scheme for the reduction of acetophenone to 1-phenylethanol with the co-substrate nicotinamide adenine dinucleotide in its protonated form (NADH+H+) and its deprotonated form (NAD+) catalysed by two different genetically modified alcohol dehydrogenases recombinant from E. coli (evo-1.1.270; evo-1.1.200).
2-[4-(2-hydroxyethyl)piperazin-1-yl]ethanesulfonic acid (HEPES) and polyethylene glycol 6000 (PEG 6000) were purchased from VWR. Acetophenone (ACP) and NADH were purchased from Sigma Aldrich. Sodium hydroxide was purchased from Bernd Kraft GmbH. The genetically modified enzyme alcohol dehydrogenase 200 (evo-1.1.20) expressed recombinant in E. coli was purchased from Evoxx. All chemicals were used without further purification, and all samples were prepared using Millipore water from the Milli-Q provided by Merck Millipore as stated in the chemical provenance (Table 5). Kinetic results using the genetically modified enzymes alcohol dehydrogenase 270 (evo-1.1.270) were taken from [23].
| Compound | Purity | CAS | Supplier |
|---|---|---|---|
| 2-[4-(2-hydroxyethyl)piperazin-1-yl]ethanesulfonic acid (HEPES) | >99% | 7365-459 | VWR |
| Polyethylene glycol 6000 | — | 25322-68-3 | VWR |
| Acetophenone (ACP) | >99% | 98-86-2 | S |
| NADH | >97% | 606-68-8 | S |
| Sodium hydroxide | >98% | 1310-73-2 | BK |
| Alcohol dehydrogenase 200 (evo-1.1.200) | 30% | evo-1.1.200 | E |
Chemical provenance table for the components measured in this work.
S = Sigma Aldrich Chemie GmbH; VWR = VWR International GmbH; BK = Bernd Kraft GmbH; E = Evoxx technologies GmbH.
Reactions were carried out in an HEPES buffer (0.1 mol/kgwater) at pH 7. The pH of the buffer and each sample was measured using a pH electrode from Mettler Toledo (uncertainty ±0.01) and adjusted with sodium hydroxide when required. For measurements of the co-solvent influence, 17 wt.% of PEG 6000 was added to the buffer. In the first step, the substrate solutions of ACP were prepared in equal number to the different NADH concentrations measured. Buffer was added to 5 ml Eppendorf cups, and ACP was added gravimetrically afterwards using the XS analytical balance provided by Mettler Toledo (uncertainty ±0.01 mg). Eppendorf cups were filled to the maximum capacity in order to decrease losses of ACP to the vapour phase. The ACP stock solutions were preheated in an Eppendorf ThermoMixer C at 25°C. ACP concentrations of the neat reaction were 20, 30 and 40 mmol/kgwater and 60, 80 and 100 mmol/kgwater for the PEG 6000 measurements, respectively. NADH was added gravimetrically to the ACP solution after preheating. Each sample was prepared directly before measurements due to reported long-term instability of NADH in solution [32]. NADH concentrations were chosen to be 0.15, 0.2, 0.25, 0.3, 0.35 and 0.4 mmol/kgwater. The enzyme stock solution was prepared by gravimetrically adding 1 wt.% of enzyme to 2 ml of buffer with direct storage on ice for the period of all measurements to ensure enzyme stability and activity. To initiate the kinetic measurements, 20 mg of the enzyme solution was transferred into a quartz cuvette SUPRASIL TYP 114-QS from Helma Analytics which was preheated to 25°C while being placed in an Eppendorf Biospectrometer. After addition of 1 g of the substrate solution containing ACP and NADH, the measurement of the extinction over time at 340 nm wavelength was initiated.
In a first step, the primary plot for the ACP reduction catalysed by ADH 270 was determined under neat conditions. For this,
Primary plot based on Eq. (17) for the ACP reduction catalysed by ADH 270 under neat conditions at T = 25°C, p = 1 bar and pH = 7 in HEPES buffer [23]. The reciprocal turnover frequency normalised to the total enzyme concentration ν ′ − 1 is plotted over the reciprocal initial molality of NADH m NADH − 1 for ACP molalities of 20 (triangles), 30 (squares) and 40 mmol/kgwater (circles). Lines represent the respective fit lines required for further data analysis.
A linear correlation of
Secondary plots based on Eqs. (18) and (19) for the ACP reduction catalysed by ADH 270 under neat conditions at T = 25°C, p = 1 bar and pH = 7 in HEPES buffer [23]. Left: Ordinates Orprim of the fit lines resulting from the primary plot are plotted over the reciprocal initial molality of ACP m ACP − 1 . Right: Slopes Slprim of the fit lines resulting from the primary plot are plotted over the reciprocal initial molality of ACP m ACP − 1 . Lines represent the respective fit lines required for further data analysis.
Figure 7 shows the required linear correlation between Orprim and Slprim over
| Component | mi (−) | σi (Å) | ||||
|---|---|---|---|---|---|---|
| Water [29] | 1.204 | [A] | 353.95 | 1:1 | 2425.7 | 0.0451 |
| ACP [4] | 3.40 | 3.65 | 322.00 | 1:1 | 0 | 0.0451 |
| NADH [33] | 27.27 | 2.21 | 260.72 | 8:8 | 358.2 | 0.0001 |
| PEG [25] | MPEG·0.05 | 2.90 | 204.60 | 4:4 | 1799.8 | 0.020 |
| Na+ [34] | 1 | 2.82 | 230 | — | — | — |
| OH− [34] | 1 | 2.02 | 650.00 | — | — | — |
ePC-SAFT pure-component parameters.
[A]
| Binary pair | kij (−) |
|---|---|
| Water-ACP [4] | 0.0330 |
| Water-NADH [33] | −0.0585 |
| Water-PEG [25] | [A] |
| Water-Na+ [34] | [B] |
| Water-OH− [34] | −0.25 |
| Na+-OH− [34] | 0.649 |
ePC-SAFT binary interaction parameters kij.
[A]
[B]
In analogy to the concentration-based approach, activity-based Michaelis constants were determined (Section 3.1.2) to be
Comparison between the experimentally measured Michaelis constants K M obs of ACP and NADH under neat reaction conditions (white bars) and under the influence of 17 wt.% PEG 6000 (striped bars) for the reduction of ACP catalysed by ADH 270 at T = 25°C, p = 1 bar and pH = 7. The grey bars present the prediction of the respective K M pre based on the determined activity-based K M a from the experimental neat data [23]. Required activity coefficients were calculated with ePC-SAFT based on the parameters from Tables 6 and 7.
| Neat | — | — | ||
| 17 wt.% PEG 6000 |
Overview of the Michaelis constants under neat reaction conditions and the comparison between predicted
| Enzyme | ||||
|---|---|---|---|---|
| ADH 270 [23] | ||||
| ADH 200 [this work] |
Comparison between the Michaelis constants of NADH and ACP for the reduction of ACP for neat reaction conditions at T=25 °C, p=1 bar and pH=7 in HEPES buffer. Two different enzymes were used as catalyst, ADH 270 and ADH 200. Activity coefficients required for the prediction of
As can be seen from Figure 8 and Table 8, predictions of the co-solvent influence of 17 wt.% of PEG on
To further validate this approach, the ACP reduction was also investigated with ADH 200 as catalyst. This step is important to support the hypothesis that co-solvent-substrate interactions determine the co-solvent influence on
Table 9 shows that
Comparison between the experimentally measured Michaelis constants K M obs for ACP and NADH from this work under neat reaction conditions (white bars) and under the influence of 17 wt.% PEG 6000 (striped bars) for the reduction of ACP catalysed by ADH 200 at T = 25°C, p = 1 bar and pH = 7. The grey bars present the predicted values for K M pre based on the determined activity-based K M a from the experimental neat data. Required activity coefficients were predicted with ePC-SAFT based on the parameters from Tables 6 and 7.
Figure 9 shows that ePC-SAFT is able to predict the change of
In this work, it was found that experimental Michaelis constants
AW and CH gratefully acknowledge the financial support of DAAD (project number 57340264) funded by the Federal Ministry of Education and Research (BMBF). Further, this work was supported by the Cluster of Excellence RESOLV (EXC 1069) funded by the Deutsche Forschungsgemeinschaft (DFG).
The authors declare no conflict of interest. Note that reference [23] is still under second review.
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