Thermodynamic Activity-Based Michaelis Constants

2.1.1. Concentration-based approach

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 ν, the maximum reaction rate νmax, the Michaelis constant KMobsand the substrate molality mSin mol/kg_water.

Eq. (2) is visualised exemplary by plotting of νover mSin Figure 1.

As can be seen from Eq. (2) and Figure 1, the reaction rate follows a hyperbolic curve over increasing substrate concentrations. Further, KMobsdefines the shape of the curve as it is the concentration of substrate at which the reaction velocity becomes half of its maximal value 0.5νmax. Based on this, the importance of KMobsbecomes obvious. If the value of KMobsis low compared to mSrequired to reach ν=νmax, it can be deduced that the cellular concentration of the substrates will also be close to KMobsas a significant increase in mS(e.g., fivefold) will never increase νmore than by a factor of 2 [9]. Thus, the knowledge of KMobsis of high importance for biology and for technical applications of enzyme-catalysed reactions.

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 νmax. These conditions are obtained if substrate simultaneously presents the reaction solvent, which is the case for hydrolysis reactions. A general scheme for a two-substrate reaction is given in Eq. (3).

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 KMS1obsand substrate 2 KMS2obsas well as the inhibition constant KiS1obs, which defines the reaction mechanism [8, 9].

In the case of a hydrolysis reaction taking place in water as reaction solvent, the molality of substrate 2 mS2(water) is usually two to three orders of magnitude higher than the molality of substrate 1, which gets cleaved by the enzyme. Rearranging Eq. (4) leads back to the Michaelis-Menten equation under this assumption shown in Eqs. (5)–(7).

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 mEaccording to Eq. (8).

The determination of kcatand KMobsis usually performed by measuring reaction rates for different substrate concentrations as shown in Figure 1. While this approach is common, it also poses a lot of difficulties and causes high uncertainties. The determination of kcatrequires that the solubility of the substrate has to be higher than the molality mSthat is required for reaching kcat. Lowering the enzyme concentration and thus the required molality mSoften causes diffusion limitations that might lead to highly uncertain kinetic constants. Another possible issue is contrary, as a reaction might require high concentrations of an expensive substrate to determine kcat. To overcome these possible limitations, the Lineweaver-Burk equation is commonly applied for the determination of the kinetic constants [19].

Through this linearization, a plot of ν′−1over mS−1yields the kinetic constants: The slope (Sl) and the ordinate (Or) of the obtained linear fit can be used to determine KMobsand kcatas given in Eqs. (10) and (11).

2.1.2. Activity-based approach

As presented in Section 2.1.1, the Michaelis constant is determined based on the molality mS. If co-solvents are added to the neat reaction mixtures, the experimental procedure has to be performed also for the changed conditions. From the perspective of process design, this poses a huge cost-intensive and time-consuming approach towards finding suitable co-solvents for the desired application.

To be able to predict co-solvent influences on pseudo-one-substrate reactions, a thermodynamic co-solvent-independent Michaelis constant, further referred to as KMa, has to be determined. KMais a constant value, which does not depend on co-solvent given that the co-solvent does not disturb the reaction mechanism (e.g., co-solvent acts as inhibitor or activator) and that the co-solvent has no denaturing effect on the enzyme. KMacan be determined under neat (co-solvent-free) conditions by replacing the molality mSin Eq. (9) with thermodynamic activities of the substrate aS. The latter are accessible by multiplying the concentration of a substrate by the respective concentration-based activity coefficient (molality-based γSm, mole-fraction-based γSxor molarity-based γSc) as shown in Eq. (12) [20, 21, 22]:

In the following, molality-based γS^m will be used to analyse the data. Replacing molalities in Eq. (9) with activities leads to an activity-based Lineweaver-Burk equation:

To determine KMa, the experimental ν′−1values, which were determined for different mS−1values, are further plotted over the reciprocal substrate activity aS−1. It is noteworthy that this procedure does not change the value of kcat[7, 23]. Under the assumption that the addition of a co-solvent only changes non-covalent interactions between the substrate and the other components in the reaction mixture, KMais assumed to be a constant value. That is, any observed change in KMobsis directly reflected in γS^m. With this knowledge, a prediction of KMobsunder co-solvent influence becomes possible. For this, two hypothetical molalities of the substrate mShypare chosen randomly. Afterwards, activity coefficients of the substrate in the co-solvent system are predicted and the respective activities are calculated. Further, a random value of kcat(e.g., value of the neat reaction) is chosen. Note that kcatis a factor that cancels out during the linearization to determine KMpre(see Eq. (13)). In the next step, the predicted activities together with KMaand kcatare used to predict ν′−1according to Eq. (13). Predicted ν′−1values are afterwards plotted over the chosen reciprocal molalities mShyp−1. In the final step, the predicted concentration-based Michaelis constant KMpreis determined according to Eq. (9). The process to determine KMpreis illustrated in Scheme 1.

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 arescalculated from different contributions as shown in Eq. (14):

In Eq. (14), the reference system is seen as a chain of hard spheres, which is represented by the contribution ahc. The perturbations to this hard-chain reference system are accounted for in ePC-SAFT by the molecular dispersive interactions, characterised by the Van der Waals energy incorporated in adispand by the associative hydrogen bonding forces represented in aassoc. As an addition for electrolyte systems, the Coulomb interactions based on the Debye-Hückel equation are expressed by aion. Based on ares, fugacity coefficients φcan be calculated which allow determining the activity coefficients γS^x using Eq. (15).

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 x→. This means that activity coefficients can be estimated independent of the amount of components, temperature and pressure of the solution regarded. Eq. (15) is finally used with Eq. (12) to obtain the molality-based γS^m.

3.1.1. Concentration-based approach

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 KiS1obsbased on the Haldane relation was accounted for in this work; if KiS1obsis lower than KM,S1obs, an ordered mechanism is present in which S1 has to bind first [8, 31]. The Lineweaver-Burk linearization of Eq. (16) leads to Eq. (17):

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 mS2, is chosen to be at least 50 times higher than mS1. Under this assumption, a so-called primary plot can be created. For this, different levels of mS2that are regarded to be constant over the short reaction time are chosen for varying mS1; then, a family of straight lines are obtained as shown exemplarily in Figure 4.

Each of the straight lines in Figure 4 has its own slope (Slprim) and ordinate (Orprim); both, Slprimand Orprimare a function of mS2as shown in Eqs. (18) and (19).

Eqs. (18) and (19) again show a linear correlation between Slprimand mS2−1as well as between Orprimand mS2−1, respectively. This allows for another linearization step represented in two secondary plots in Figure 5.

SlSlsec, OrSlsec, SlOrsecand OrOrsecobtained from the secondary plots are defined according to Eqs. (20)–(23):

The relations shown in Eqs. (20)–(23) are finally used to determine KMS1obs, KMS2obs, kcatand KiS1obs.

3.1.2. Activity-based approach

The determination of activity-based Michaelis constants KM,S1aand KM,S2afor two-substrate reactions is analogous to pseudo-one-substrate reactions. As for the pseudo-one-substrate reaction, molalities in Eq. (17) are replaced with activities as shown in Eq. (24):

From Eq. (24), a primary plot is created as described in Section 3.1.1 in which ν′−1is plotted over aS1−1. Afterwards, the two secondary plots are created by plotting the Orprim and Slprim of the primary plot over aS1−1to finally obtain the activity-based kinetic constants KiS1a, KMS1aand KMS2aas described for the concentration-based approach in Section 3.1.1.

Predictions for the co-solvent influence on KMS1obsand KMS2obsare performed in analogy to pseudo-one-substrate reactions: Two molalities of S1 mS1hypfor two molalities of S2 mS2hyphave to be chosen; then, the required activity coefficients have to be predicted in order to create a predicted primary plot; the secondary plots are then constructed by plotting Orprim and Slprim over the chosen reciprocal molalities mS2hyp−1. In a final step, the predicted Michaelis constants KMS1preand KMS2preare obtained from the secondary plots. The prediction process is illustrated in Scheme 3.

3.2.1. Chemicals

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].

3.2.2. Kinetic assays

Reactions were carried out in an HEPES buffer (0.1 mol/kg_water) 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/kg_water and 60, 80 and 100 mmol/kg_water 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/kg_water. 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.

3.3.1. ADH 270

In a first step, the primary plot for the ACP reduction catalysed by ADH 270 was determined under neat conditions. For this, ν′−1is plotted over mNADH−1for pseudo-constant mACPlevels of 20, 30 and 40 mmol/kg_water in Figure 6.

A linear correlation of ν′−1over mNADH−1can be observed from Figure 6. As described in Section 3.1.1, this correlation is used for the creation of the secondary plots shown in Figure 7.

Figure 7 shows the required linear correlation between Orprim and Slprim over mACP−1required for the determination of the Michaelis constants KM,NADHobsand KM,ACPobsaccording to Eqs. (20)–(23). Applying the relations given in Eqs. (20)–(23), the Michaelis constant of NADH KM,NADHobs=0.37±0.09mmol/kgwaterand of ACP KM,ACPobs=18.56±3.23mmol/kgwaterwere determined [23]. Afterwards, activity coefficients for NADH and ACP were predicted for the respective molalities; these ePC-SAFT predictions are based on the pure-component and binary interaction parameters listed in Tables 6 and 7. Based on the activities of NADH aNADHand ACP aACP, the respective primary and secondary plots were created.

In analogy to the concentration-based approach, activity-based Michaelis constants were determined (Section 3.1.2) to be KM,NADHa=5.649·10−8and KM,ACPa=0.640. As can be seen, activity-based Michaelis constants can be completely different from their concentration-based pendants; they even do not have any unit due to the definition of the activity as shown in Eq. (12). In the next step, a prediction of the co-solvent influence of 17 wt.% of PEG 6000 on KM,ACPobsand KM,NADHobswas performed as described in Section 3.2.1. These predictions were compared to experimental results given in Figure 8 and Table 8.

As can be seen from Figure 8 and Table 8, predictions of the co-solvent influence of 17 wt.% of PEG on KM,NADHobsand KM,ACPobsof the ACP reduction catalysed by ADH 270 are in very good agreement with experimental data. Upon addition of 17 wt.% PEG 6000, KM,NADHobsdecreased by a factor of two, while KM,ACPobsincreased by a factor of 2.5. Both trends were predicted accurately with ePC-SAFT. This shows that the influence of PEG 6000 on the KMobsvalues is caused by non-covalent molecular interactions between the co-solvent and the substrates instead of co-solvent-enzyme interactions.

3.3.2. ADH 200 and comparison to ADH 270

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 KM,ACPobsand KM,NADHobs. However, it becomes obvious from Table 9 that ADH 200 shows a completely different kinetic profile under neat conditions.

Table 9 shows that KM,NADHobs(KM,ACPobs) using ADH 200 are 9 times (2 times) lower than KM,NADHobsKM,ACPobsusing ADH 270 for identical conditions. Nevertheless, this is an expected behaviour. It can be further observed from Table 9 that also the activity-based Michaelis constants KM,NADHaand KM,ACPaare different for ADH 200 and ADH 270 for the ACP reduction at same conditions. The prediction results of the influence of 17 wt.% PEG 6000 on KMobsof the reaction catalysed by ADH 200 are given in Figure 9.

Figure 9 shows that ePC-SAFT is able to predict the change of KM,NADHobsand KM,ACPobsunder the influence of 17 wt.% of PEG 6000 for ADH 200 in good agreement with experimental data. The same ePC-SAFT parameters were used as for the prediction of the same reaction catalysed by ADH 270. This is a further validation of our approach as it shows that predictions are possible independent of the enzyme catalysing the reaction. For both enzymes, ADH 200 and ADH 270, the co-solvent influences on the substrate activities were the key for predicting the change in KMobs.