4.1.1. Experimental measurement of Vmax, kcat, and K<sup>m</sup>

As discussed in the above sections, the quantification of the initial velocity, the region of the curve wherein velocity does not change with time (up to 10% of the conversion), is of prime importance to estimate the reaction kinetics under the influence of other factors such as product inhibition. This region varies with the type of enzyme, nature of substrate and medium conditions and type of modulator. Hence, for any given system, it cannot be predetermined. The instrument capacity needs, therefore, has to be very precise and effective to detect the signal generated from the build-up of product and to plot it. As the measurement needs to be done at a very constrained range, it may severely compromise on measurements because of the poor instrument capacity.

The linear range for the product of an instrument can be estimated by measuring the response of product at various known concentration and preparing the standard curve. Before the start of experimentation for evaluating enzyme kinetics, it is very essential to optimize the conditions of the reactions so that the enzyme remains stable throughout the reaction progress. If the maximum plateau value of product formed does not reach the same for all levels of tested enzyme, it is likely due to the enzyme instability over time [10, 53].

Conditions required for measuring initial velocity of an enzyme reaction:


For kinase assays, generally, the background can be recorded without the enzyme or the substrate. The condition with highest background level should be used for further estimation of the parameters. In case of EDTA, background is taken with control without EDTA during validation of a kinase assay. Once the assay has been validated, if the background measured with EDTA is the same as that of no enzyme and no substrate control, then EDTA could be used.

Measurement of K<sup>m</sup> and Vmax

Recent studies showed that the mass spectrometric (MS) techniques are applied to study the conversion of substrate into product and their behaviour in the presence of modulators. Matrix-assisted laser desorption ionization-time of flight-mass spectroscopy (MALDI-TOF-MS) is proved as more effective technique when compared to other mass spectrometric techniques, such as electronspray ionization MS, as it showed negligible interference due to the presence of buffers and reagents. MALDI-TOF-MS can quantify the ratio of substrate to product form. Greis et al. studied the phosphorylation catalysed by kinases using this technique for accurate prediction of the kinetics of reaction [93]. This technique has been used in combination with various chromatographic techniques such as capillary isoelectric focusing, frontal affinity chromatography and size exclusion chromatography to analyse the inhibition of the enzyme in direct or indirect manner [93–96]. This method is free of laborious work of

To estimate the enzyme inhibition and relative kinetic parameters, the stepwise process has to follow with the aim of minimum human error, which is discussed in the following sections.

As discussed in the above sections, the quantification of the initial velocity, the region of the curve wherein velocity does not change with time (up to 10% of the conversion), is of prime importance to estimate the reaction kinetics under the influence of other factors such as product inhibition. This region varies with the type of enzyme, nature of substrate and medium conditions and type of modulator. Hence, for any given system, it cannot be predetermined. The instrument capacity needs, therefore, has to be very precise and effective to detect the signal generated from the build-up of product and to plot it. As the measurement needs to be done at a very constrained range, it may severely compromise on measurements

The linear range for the product of an instrument can be estimated by measuring the response of product at various known concentration and preparing the standard curve. Before the start of experimentation for evaluating enzyme kinetics, it is very essential to optimize the conditions of the reactions so that the enzyme remains stable throughout the reaction progress. If the maximum plateau value of product formed does not reach the same for all levels of tested

• Keep the reaction medium condition static (previously optimized or derived from litera-

• Proper analytical technique is developed for the estimation of product formed or substrate

labelling or derivatization and can be done in short period of time.

enzyme, it is likely due to the enzyme instability over time [10, 53].

ture) for the optimum performance of enzyme.

Conditions required for measuring initial velocity of an enzyme reaction:

• Record the signal of the reaction mass at time zero for the background.

• Equilibrate and maintain all the reaction reagents to the optimum temperature.

• Replicate the same reaction for minimum three times to avoid the manual errors.

4.1.1. Experimental measurement of Vmax, kcat, and K<sup>m</sup>

104 Enzyme Inhibitors and Activators

because of the poor instrument capacity.

consumed below 10%.

After primary experiments, wherein the initial velocity conditions have been established and change in velocity with respect to time has been estimated, the substrate concentration should be varied. These data will generate a saturation curve and can be used for the determination of K<sup>m</sup> and Vmax values. The Michaelis Menten kinetic model shows that the K<sup>m</sup> = [S] at Vmax/2. In order for competitive inhibitors in which the substrate competes, the substrate concentration should be maintained around or below the Km. At the substrate concentration higher than the Km, the identification of competitive inhibitors itself becomes more difficult.

For kinase assays, two set of experiments are carried out. In first experiment, the K<sup>m</sup> for ATP is determined at saturating levels of the substrate concentration. Subsequent reactions need to be conducted with optimum ATP concentration (around or below the K<sup>m</sup> value). However, simultaneous determination of K<sup>m</sup> for ATP and specific substrate gives more accurate estimation of kinetic constants with maximum information and any potential co-operativity between substrate and ATP [91]. To achieve steady state, ratio of substrate to enzyme is maintained in between 100 and one million.

How to measure K<sup>m</sup>


The reaction product is measured at various times for eight different levels of substrate. The product generated (Y-axis) is plotted against the reaction time (X-axis) with each curve of different concentration of substrate. The slope of the line is the initial velocity (v) of the reaction curve.

These resulting initial velocities (Y-axis) are plotted against the concentration of substrate (Xaxis) and fitted with a non-linear regression analysis in a rectangular hyperbola (Figure 2). The K<sup>m</sup> is one-half the maximum velocity determined under saturating substrate concentrations.

The linear plots are generated to determine the kinetic constants besides fitting the data by non-linear regression such as Lineweaver–Burk plot wherein the reaction rates (1/vo) are plotted against reciprocal substrate concentrations (1/[S]) with y-intercept equivalent to 1/ Vmax; and the slope of Km/Vmax. Various linearization plots such as Eisenthal–Cornish–Bowden plot, Dixon plot and Hanes plot can also be used to estimate kinetic and inhibition constant (see Section of Enzyme Kinetics).

Optimization experiments

Published literature information should be used in selecting these factors. The following parameters should be optimized prior to kinetic study so that the enzyme remains stable throughout the experiments.


The assay conditions should be validated to avoid the loss of enzyme activity due to the wrong selection of buffer, pH or temperature [97].

#### 4.2. Assay conditions for bisubstrate reaction

The plotting of Lineweaver Burk plot is more important to distinguish between a sequential reaction mechanism (plotted lines intersect each other) and a double-displacement or 'pingpong' kinetic mechanism (plotted lines remain parallel). In some cases, these graphs gave preliminary evidences in different types of sequential kinetic constants. In the process of predicting kinetic and inhibition constant for the bisubstrate reaction, it is compulsory to understand the mechanism of the reaction to predict the appropriate rate equations. The data were generated by plotting the reciprocals of the initial rate of product formation versus [A] at the differing [B] and vice-versa. Most importantly, the values of kinetic parameters such as K<sup>m</sup> and Vmax change for substrate A with the change in substrate B concentration [12, 41, 63]. In case of random sequential reaction, the double reciprocal plot of rate expressions gives the apparent value of slopes and intercept which include the term for concentration B.

$$\frac{1}{\frac{1}{\sigma\_0}} = \frac{1}{[A]} \left( \frac{K\_{\text{ia}} K\_{\text{b}}}{V\_{\text{max}}[B]} + \frac{K\_{\text{b}}}{V\_{\text{max}}} \right) + \frac{K\_{\text{a}}}{V\_{\text{max}}[B]} + \frac{1}{V\_{\text{max}}} \quad [B] = \text{constant} \tag{73}$$

The double reciprocal plot gives the confirmation of the mechanism followed by the reaction. But, as the slope <sup>K</sup>iaK<sup>b</sup> <sup>V</sup>max½B� <sup>þ</sup> <sup>K</sup><sup>b</sup> <sup>V</sup>max and intercept <sup>K</sup><sup>a</sup> <sup>V</sup>max½B� <sup>þ</sup> <sup>1</sup> <sup>V</sup>max terms involve the concentration term, it signifies the dependency on the concentration B. Also, it is noted that the slope and intercept are the linear equations and can be plotted against 1/[B]. When the intercept is plotted against 1/[B], it gives the values of 1/Vmax and Ka/Vmax and this plot is called as intercept of intercept (Figure 15). At different concentration of substrates various different forms of enzyme exist in the system. To obtain the intercept of intercept of the secondary plot, concentration of both A and B are maintained at higher concentrations which give rise to higher concentration of EAB Form.

To calculate the Ka/Vmax of desired enzymatic reaction, the EA form of enzyme should be predominant. Hence, the slope of the intercept is calculated by maintaining the concentration of A at high levels while concentration of B is kept low. The plot of slope versus 1/[B] will give the values of K<sup>b</sup> and Kia, which are measured graphically (Figure 16).

Figure 15. Secondary plot of intercept versus reciprocal of concentration of B.

plot, Dixon plot and Hanes plot can also be used to estimate kinetic and inhibition constant

Published literature information should be used in selecting these factors. The following parameters should be optimized prior to kinetic study so that the enzyme remains stable

The assay conditions should be validated to avoid the loss of enzyme activity due to the wrong

The plotting of Lineweaver Burk plot is more important to distinguish between a sequential reaction mechanism (plotted lines intersect each other) and a double-displacement or 'pingpong' kinetic mechanism (plotted lines remain parallel). In some cases, these graphs gave preliminary evidences in different types of sequential kinetic constants. In the process of predicting kinetic and inhibition constant for the bisubstrate reaction, it is compulsory to understand the mechanism of the reaction to predict the appropriate rate equations. The data were generated by plotting the reciprocals of the initial rate of product formation versus [A] at the differing [B] and vice-versa. Most importantly, the values of kinetic parameters such as K<sup>m</sup> and Vmax change for substrate A with the change in substrate B concentration [12, 41, 63]. In case of random sequential reaction, the double reciprocal plot of rate expressions gives the

apparent value of slopes and intercept which include the term for concentration B.

þ

and intercept <sup>K</sup><sup>a</sup>

The double reciprocal plot gives the confirmation of the mechanism followed by the reaction.

term, it signifies the dependency on the concentration B. Also, it is noted that the slope and intercept are the linear equations and can be plotted against 1/[B]. When the intercept is plotted against 1/[B], it gives the values of 1/Vmax and Ka/Vmax and this plot is called as intercept of intercept (Figure 15). At different concentration of substrates various different forms of

Ka Vmax½B� þ 1 Vmax

<sup>V</sup>max½B� <sup>þ</sup> <sup>1</sup> Vmax  ½B� ¼ constant (73)

terms involve the concentration

KiaK<sup>b</sup> Vmax½B�

<sup>V</sup>max½B� <sup>þ</sup> <sup>K</sup><sup>b</sup> Vmax  þ Kb Vmax

(see Section of Enzyme Kinetics).

Optimization experiments

106 Enzyme Inhibitors and Activators

throughout the experiments.

• Bovine serum albumin

• pH

• Salts, for example NaCl, KCl

• Divalent cations, for example Ca2+, Mg2+, Mn2+

• Detergents, such as Triton, CHAPS and DMSO

selection of buffer, pH or temperature [97].

1 v0 ¼ 1 ½A�

But, as the slope <sup>K</sup>iaK<sup>b</sup>

4.2. Assay conditions for bisubstrate reaction

• Reducing agents, such as β-ME, DTT and glutathione

• Buffer source, for example HEPES versus acetate buffer

Figure 16. Secondary plot of slope versus reciprocal of concentration of B.

Similarly, same kind of analysis can be performed to find out the Vmax and Kb/Vmax by keeping high levels of B and A. The plot of intercept of intercept, i.e. intercept versus 1/[A], will give the values of 1/Vmax and Kb/Vmax. The slope plotted against the 1/[B] was studied for deriving the values of K<sup>a</sup> and Kib by analysing at high levels of B and low concentrations of A (building up the concentration of EB). These secondary plots elaborate various aspects of enzyme kinetics some of which are discussed in the kinetics section.

#### 4.3. Software used in kinetic data analysis

The data-fitting process can be accomplished by using a software program that provides nonlinear regression-fitting capability. Various programs such as Kaleidagraph are developed wherein the user put the experimental data along with probable fitting function for the prediction of rate constants. Some other software such as dynafit, mathematica, sigmaplot and prism are applied for the estimation of rate expression and effect of inhibition or activator on the enzyme [98]. Alternatively, some of the programs like Enzfitter have a predefined library of equations which can also be used for the prediction kinetics of enzyme. While using the software for data prediction, one should be able to differentiate in errors added due to the lack of fit and pure error as they contribute as a source of error [99]. Although both sources of error normally contribute to the sum of squares of deviations from a model, they can be separated. The inconsistencies between replicate observations are unaffected by the choice of model and thus allow calculations of how much of the total sum of squares is due to the pure error, and from this one can calculate the contribution of lack of fit [57, 80, 100].

The basic assumption during the development of related software is that the rate of reaction is zero in the absence of substrate. The buffer solution used for the reaction should be selected so that the pKa is not greater than 1 unit of pH that the desired one. Generally, the operational pH should be less than pKa. This will maintain the desired configuration required for the optimal activity of the enzyme. Also, the used buffer system should not react with the enzyme used for the catalysis or the buffer interfering with the analysis method [10, 52, 88]. Primary assumptions that any developed software have done are as follows:


7. The overall enzyme reaction rates and mode of inhibition are depended upon the intermolecular forces between enzyme subunits, substrate or inhibitor.

While deriving the kinetics and mechanistic parameters in appropriate conditions at various points in iterative manner, various well established programs make some primary assumption during the development of the rate expression. For example, the popular program sigmaplot can fit Michaelis-Menten data very easily, but if used in its default state it incorporates assumptions that: (1) the errors in the observed rates are subjected to a normal (Gaussian) distribution and substrate concentration are exactly known; and (2) all of the rates have the same standard deviation and are independent of each other as magnitude of error in one rate measurement do not affect the measurement of any other rate.

Some of the assumptions like no deviation in standard deviation cause the problem that need to be eliminated from the model. Very less number of software databases allows doing so. Hence, the selection of the software to process the data has to be done very meticulously. It is always preferable to derive the rate expression and related constant manually with desired set of assumptions for error free fitting of data which may be crossed checked with the computerized data fitting. The computer-based data fitting serves as boon to researchers when more complex and complicated rate expression are observed (e.g. bisubstrate reaction with inhibition and termolecular reactions). In the gist, software-based studies of kinetics become essential part of the system [2].
