Determination of initial reaction rates and principal kinetic parameters

As it has been mentioned several times in this chapter, the initial reaction rate of an enzyme-catalysed chemical reaction (V0) is by definition the change of the product concentration as a function of the elapsed time: V0 = d[P]/dt. In the general case of an S→P reaction scheme, an identically valid rate definition is the one that accounts for the decrease in substrate concentration: V0 = −d[S]/dt. When the initial rate V0 is to be measured, the concentration of either the product or the substrate needs to be measured as a function of time.

A chemical reaction is inherently accompanied by changes in electronic structure. Alteration of the electronic structure alters the excitability of the electrons, which in most cases can be detected by various spectroscopic methods. In the simplest case, the optical properties of the product will measurably differ from those of the substrate (and all other components of the solution). In such cases, the formation of the product or the diminution of the substrate can be measured in real time—in other words, a ―progress curve‖ can easily be determined. In an appropriate concentration range (see Chapter 4), the magnitude of the absorbance—or, in the case of fluorescence, the intensity of the emitted light—is linearly proportional to the concentration of the light-absorbing or light-emitting molecule. Based on this, changes in the concentration of the product (or that of the substrate) can be precisely determined in real time.

If neither the product nor the substrate possesses readily measurable spectroscopic properties, there is still an opportunity for a spectroscopic measurement. It can be done if the product can be driven to a second (not necessarily enzymatic) chemical reaction that is instantaneous and results in an optically active (light-absorbing or fluorescent) compound. Moreover, if the second chemical reaction is compatible with the conditions (pH, temperature etc.) of the enzymatic reaction of interest, and its product does not interfere with the enzyme, this more complex approach can also establish a real-time measurement. On the other hand, if the second (coupled) reaction is not compatible with the enzymatic reaction, then—instead of a real-time measurement—a different approach, end time analysis can be applied. In such cases, the enzymatic reaction is usually started in a relatively large volume, and aliquots are withdrawn

192

from the reaction chamber at different time intervals. These aliquots are treated to instantly stop the enzymatic reaction (usually by applying extreme pH or temperature, or strong protein denaturing reagents, e.g. trichloroacetic acid) and then the second, signal-generating chemical reaction is performed.

The principle of real-time initial rate measurements is illustrated in the left panel of Figure 9.3 showing progress curves with different initial substrate concentrations. The reaction is started by adding either the substrate or the enzyme to the solution and, after thorough mixing of the sample, detection is started as soon as possible. A quick start is very important as initial rates are to be measured (cf. V0, a tangent to the curve at the start of the reaction with substrate concentration S4 in Figure 9.3). Naturally, during the enzymatic reaction, the concentration of the substrate will gradually decrease. To this end, the measurement should be performed so that this decrease in substrate concentration should be negligible. This is important because, as illustrated in the left panel of Figure 9.3, decreasing substrate concentration is accompanied by a decreasing reaction rate. Moreover, to make the situation more complicated, the significance of this unwanted effect is itself a function of substrate concentration, as it can be seen on the saturation curve in the right panel of Figure 9.3. At substrate concentrations well below the value of KM, the [S]-V0 function is approximately linear; while at substrate concentrations well above the KM, the V0 is almost independent of [S]. As a consequence, the lower the substrate concentration, the larger the error in the measured rate value at an identical (for example 10 %) percentage of substrate concentration change. For example, at [S] well below KM, a 10 % decrease in [S] will cause an about 10 % decrease in V0; while at [S] well above KM, a 10 % decrease in [S] will cause a much smaller error. As a rule of thumb, the initial rate should be measured so that, during the measurement, the concentration of the substrate does not decrease by more than 10 %. If this requirement is fulfilled, the initial rates will be determined with an error lower than 10 %.

In order to determine the main kinetic parameters, Vmax and KM, the initial reaction rates are measured at various initial substrate concentrations. If substrate concentrations are dispersed in the proper range, V0 values plotted as the function of [S] trace out a typical saturation curve illustrated in the right panel of Figure 9.3.

For accurate determination of the parameters, the values of [S] should cover a wide range.

Naturally, as long as no estimate exists on the value of the KM, iterative trial-and-error sampling of the right range needs to be performed. Once the KM is estimated, the measurements should be set such that the pre-set [S] values cover the ~ 0.2 KM – 5 KM region with at least 8 evenly placed substrate concentrations.

Once a satisfactory number of [S] - V0 data are collected, all information is available to determine the values of the Vmax, kcat and KM parameters.

The very first kinetic measurements date back to times when computers did not exist. At the time, the evaluation of the data presented a considerable challenge. In the right panel Figure 9.3, the [S]

- V0 plot corresponds to a saturation curve, actually a part of a rectangular hyperbola. Most naturally, a simple visual analysis of such a curve would not allow for a reliable determination of the maximum value, Vmax, to which the curve converges. For the very same reason, the substrate concentration belonging to the half-maximal initial rate, i.e. the value of KM, cannot be reliably estimated either.

193

For quantitative analysis of such non-linear relationships, the following procedure was implemented. The mathematical equation describing the model of the relationship was transformed such that the transformed equation had a linear form. This way, when derived values of the experimental data were plotted, it resulted in a straight line instead of a curve. Therefore, the problem was simplified to the task of a linear regression, which could be performed without a computer.

Nowadays, kinetic parameters are calculated using computer programs and non-linear regression.

Nevertheless, in the field of enzyme kinetics, linear plotting of derived values still remained didactically useful, particularly in studies of enzyme inhibitors. Enzyme inhibitors can be classified into several groups based on their mechanism of action. As we will see later, the linearised plots clearly report the type of mechanism of action of the inhibitor.

The most widespread linearised version of the Michaelis-Menten equation (Equation 9.44) was introduced by Hans Lineweaver and Dean Burk. The transformation they proposed leads to Equation 9.52 that was later named after them the Lineweaver-Burk equation. The transformation is done as follows. The reciprocal of both sides of original Equation 9.44 is taken, leading to Equation 9.50:

(9.50)

From this, Equation 9.51 is generated by a simple algebraic transformation that resolves the fraction on the right side of the equation into the sum of two fractions:

(9.51)

Then, on the right side of the equation, the fraction in the second term can be reduced by dividing both the numerator and the denominator by [S]. This results in Equation 9.52:

(9.52)

The plot corresponding to Equation 9.52 is the double reciprocal form of the original saturation curve introduced in Figure 9.3. While, in the original plot, V0 data were plotted as a function of [S], in the double reciprocal Lineweaver-Burk plot, 1/V0 values are plotted as a function of 1/[S].

As illustrated in Figure 9.5, the plot corresponding to Equation 9.52 is a straight line.

194 Figure 9.5. The Lineweaver-Burk double reciprocal plot

According to the general equation of a line in slope-intercept form, Y = aX+b. In our specific case, Y = 1/V0; X = 1/[S]; a = KM/Vmax and b = 1/Vmax. The slope of the line, „a‖ (which is the tangent of the angle between the line and the x axis), provides the value of KM/Vmax; the intercept of the y axis (where the value of X and therefore the value of 1/[S] is 0) provides the value of 1/Vmax; whereas the intercept on the x axis (where the value of Y, therefore that of 1/V0 is 0) provides the value of −1/KM. Based on the slope and any of the two intercepts, both Vmax and KM

can be determined.

This approach is elegantly simple, but this type of data evaluation, without using proper weighing of the individual data, can lead to significant errors. Because of experimental detection limitations, V0 data measured at the lowest [S] are the most likely to be inaccurate. Due to the double reciprocal analysis, these least reliable values will contribute the highest (reciprocal) values, and these will thus have the highest impact on the calculated KM and Vmax parameters.

In the era of computers, data analysis is performed by non-linear regression. In the particular case, an appropriate algorithm based on Equation 9.44 is used. The algorithm iteratively searches for the Vmax-KM parameter pair that, when substituted in the rectangular hyperbolic equation, provides the best (lowest-deviation) fit to the experimental [S] - V0 dataset.

Most fitting programs require initial (estimated) values for the parameters (in this case Vmax and KM) to start the iterative search.

If the total concentration of the enzyme, [E]T, is known during the measurements, then—based on Equation 9.43—not only Vmax but also the turnover number, kcat, can be determined by dividing Vmax by [ET].

It is worth noting that, if the aim is the determination of only the catalytic efficiency (the kcat/KM

quotient), a simple direct measurement can be applied. If the value of V0 is measured at a substrate concentration orders of magnitude lower than the value of the KM and if the total concentration of the enzyme, [E]T, is known, then—based on the same line of thinking that led to Equation 9.48—kcat/KM will equal V0/[E]T[S]. Nevertheless, this type of measurement is only

195

advisable for particular enzyme-substrate pairs. The substrate should convert to a product that provides a highly intense signal, while the enzyme should have high turnover number on that substrate. This is required because, in this kind of measurements, very low levels of product concentration need to be detected.