Enzymatic Activity Measurements

3.2.3.1 Michaelis-Menten Kinetics

The catalytic efficiency of free and adsorbed enzymes can be quantified by applying the Michaelis-Menten kinetics. [203] This model proposes a two-step reaction which contains a pre-equilibrium between free enzymes and enzymes occupied with substrate as first step: [203]

𝐸𝐸+𝑆𝑆 𝐸𝐸𝑆𝑆^𝑘𝑘�⎯� 𝐸𝐸^cat +𝛽𝛽

𝑘−1�⎯�

𝑘+1�⎯� (3.12)

where E signifies the enzyme and S the substrate, respectively. The complex between enzyme and substrate is described by ES and P is the product released from the enzyme active site after conversion.

The rate constant k+1 and k-1 is the rate constant of the formation and cleavage of the ES complex, respectively. The chemical processes following the ES complex formation are described by the overall rate constant kcat. By using this reaction mechanism and by assuming that a steady-state concentration of the ES complex is established, the Michaelis-Menten equation is derived. The latter reads [203]

𝜐𝜐= 𝜐𝜐_max[𝑆𝑆]

𝐾𝐾m+ [𝑆𝑆] (3.13)

where Km is the Michaelis constant, υ is the initial rate of the catalysed reaction, and υmax is the maximum initial rate at infinite substrate concentration [S].

Figure 3.11 a shows the initial rate υ of the hydrolysis of o-nitrophenyl-β-D-glucopyranoside (o-NPG) as function of o-NPG concentration which is catalysed by free β-D-glucosidase. This reaction is of pseudo-first order and is known to follow the Michaelis-Menten kinetics. The reaction rate increases with increasing [S] and asymptotically approaches its maximum rate υmax. This is caused by saturation of the enzyme active sites with the substrate and, thus, increasing of [S] does not increase the reaction rate above this threshold. The maximum rateis related to kcat via υmax= kcat [E] with [E] as enzyme concentration since kcat is referred to as the turnover number for the enzyme, i.e., the number of catalytic turn over events per enzyme molecule that occur per unit of time. [203] Consequently, kcat

defines the maximum rate of the reaction with infinite availability of the substrate to the enzyme´s active site. Moreover, kcat is very sensitive to changes of the enzyme structure and changes of the solution conditions. Thus, this parameter serves as direct measure for any perturbation of the tertiary

Figure 3.11: a) Plot of the initial rate υ of the hydrolysis of o-nitrophenyl-β-D-glucopyranoside (o-NPG) as function of the o-NPG concentration. The hydrolysis was catalysed by native β-D-glucosidase in 10 mM MOPS buffer pH 7.2 at 293 K. b) Lineweaver-Burk plot of the hydrolysis of o-NPG catalysed by free β-D-glucosidase in 10 mM MOPS buffer pH 7.2 at 293 K.

structure and the microenvironment which may be induced by the adsorption of enzymes to the microgel network.

The Michaelis constant equals the substrate concentration that provides a reaction rate that is half of the maximum rate υmax. Additionally, for reactions consisting of a fast established equilibrium between enzyme, substrate and enzyme-substrate complex, i.e., kcat << k-1, the Km is identical to the dissociation constant KS = k-1/k1. [203] Under these specific conditions, Km is a measure for the substrate binding affinity to the enzyme active site and the stability of the ES complex. For enzyme-substrate systems with values of Km larger or smaller than KS the Michaelis constant still serves as relative measure of the stability of the ES complex.

In this thesis, the kinetic constants Km and kcat are determined from non-linear curve fitting of untransformed experimental data to the Michaelis-Menten equation (3.13). However, linearised plots of the kinetic data are helpful when discussing the results and comparing the kinetic parameters of free and adsorbed enzyme. The most commonly used method for linearising the enzyme kinetic data is the method of Lineweaver and Burk. Therefore, the reciprocal of the Michaelis-Menten equation (3.13) is calculated and rearranged to give the basic equation of the double-reciprocal Lineweaver-Burk plot:

[203]

1 𝜐𝜐 = 𝐾𝐾_m

𝜐𝜐max

1 [𝑆𝑆] +

1

𝜐𝜐max (3.14)

Thus, the plot of 1/υ versus 1/[S] results in a straight line with Km/ υmax as slope and 1/ υmax as intercept (Figure 3.11 b). This form of data presentation easily visualises any differences in the kinetic parameters as changes of the slope and intercept are extracted from the plot at first sight.

3.2.3.2 Temperature- and pH-Dependence of the Catalytic Activity

The catalytic efficiency of enzymes shows a non-monotonic temperature profile with a distinct temperature maximum specific for each enzyme. Below the temperature maximum, most enzyme

reactions obey the Arrhenius equation which relates kcat to the activation energy of the transition state of the enzyme-catalysed reaction Ea: [203-204]

𝑘𝑘_cat=𝐴𝐴 exp�−𝐸𝐸_a

𝑅𝑅𝑅𝑅� (3.15)

where A is a pre-exponential constant and represents the probability of the reaction to take place and R is the gas constant. At higher temperature, heat-induced denaturation of the protein sets in and compromises the enhancement of the catalytic efficiency predicted by eq. (3.15). Thus, the increase of kcat and the decrease of the fraction of active enzymes with increasing temperature results in a temperature maximum of the catalytic efficiency. The position of this maximum defines the temperature stability of the enzyme. In many cases, it is located between 40 and 50 °C, that is, slightly above the body temperature. [204] However, the temperature maximum is determined by the time-dependent heat inactivation of the protein. Consequently, it is shifted to lower temperatures for longer incubation times of the enzyme at the corresponding temperature. [204]

In order to gain information about the temperature-dependent catalytic activity of enzymes and the influence of immobilisation of enzymes on the latter, the investigation should be restricted to the temperature range over which protein heat denaturation is not significant. Moreover, it is essential to use kcat for this analysis, since the binding affinity, i.e., Km, is also dependent on temperature. [205]

Thus, the analysis of the initial rate υ for one substrate concentration will not suffice to analyse the temperature-dependence of the enzyme activity. Instead, kcat has to be determined for each temperature because it is not affected by the temperature-dependence of Km. The temperature-dependent analysis of enzymes in the free and bound state is described in section 4.2.2.

In addition to the temperature, the enzyme activity exhibits a significant dependence on the pH value of the solution. [203-204] The shape of the pH-dependent activity curve is determined by two effects:

i) the role of ionic groups in the stabilisation of the tertiary structure of the protein and ii) the involvement of acid-base amino acid groups in substrate binding and in the catalytic steps of the reaction. Generally, the protein conformation is stable over a relatively broad pH range and denaturation occurs at extremely low and high pH values. [204] However, in most enzymes acid-base groups are part of the active site and the protonation state of these groups is essential for the binding and conversion of the substrate. Thus, the enzymatic activity is maximised over a narrow range of pH in which the protonation state of the catalytic residues is optimum. For example, β-D-glucosidase [206] shows a sharp pH optimum at pH ~7, whereas lysozyme [207] exhibits a broader pH optimum around pH ~5. Due to this strong sensitivity of the catalytic activity towards the pH value of the environment, enzymes may be exploited as local pH meters. Thus, enzymes adsorbed to the microgel network in their native structure may be used to detect local pH changes of charged microgels which are probably caused by their electrostatic potential. Results regarding this analysis can be found in section 4.4.4.