Kinetics of Enzymatic Reactions | Biochemistry

A catalyst — whether chemical or biological — acts by increasing the velocity of reactions. We will now examine the kinetics of enzymatic reactions. The velocity of an enzymatic reaction is generally fol­lowed up by measuring the quantity of substrate transformed per unit time; one can also record the quantity of the product (or one of the products) formed by the reaction.

Samples of the reaction mixture are therefore collected at dif­ferent times and titrations made by chemical, chromatographic, colorimetric or manometric methods depending on each case.

Continuous analysis techniques may be used sometimes, particularly spectrophotometric methods (for ex­ample, one can record the variations of optical density at a given wavelength if the reaction is amenable to it, which is especially the case with NAD or NADP dehydrogenases) or titrimetric methods (for instance, the pH variations occur­ring during the reaction can be balanced by continuous addition of a base or acid, addition which may be automatically controlled by a pH meter).

Enzymatic activity may be expressed in several ways:

1. The enzymatic unit is the quantity of enzyme which catalyses the trans­formation of 1 micromole of substrate in one minute, at 25° in optimal condi­tions of pH and substrate concentration (but in certain cases one millimole, or one milligram of substrate transformed per minute, are also used).

2. The ratio of the number of enzymatic units to the amount (generally to one mg) of protein, gives the specific activity (which, obviously, increases as enzyme purification progresses).

3. The molar activity is the number of moles of substrate transformed (or product formed) per mole of enzyme per minute; for this mode of expression one must have a highly purified enzyme of known molecular weight.

I. Initial Velocity of the Reaction:

If the quantity of substrate transformed is plotted against time, it is observed (Fig. 2-2) that the first portion of the curve is a straight line with constant slope; the curve then bends, i.e. velocity decreases and finally becomes zero when all the substrate which was to be transformed (as a function of the equilibrium of the reaction) has disappeared.

It is therefore advantageous to measure the velocity at the beginning of the reaction, when only a small quantity of the substrate has been transformed and when the amounts of reaction products are sufficiently small for the reverse reaction to be negligible (in practice a large excess of substrate is added); the slope is then maximum and the initial velocity of the reaction (v₀) can be determined.

The reaction velocity (v) at a given time (t) is d[P]/dt (where P is the quantity of product formed in the reaction). It can be determined by measuring the angle of the tangent to the curve at time t. The tangent at the origin gives v₀, the initial velocity of the reaction.

Some factors can influence the velocity of an enzymatic reaction.

II. Effect of Enzyme Concentration on Kinetics:

If several experiments are carried out with increasing quantities of enzymes, it is observed that after a given time (t₁), the quantity of substrate transformed is larger when more enzyme is present, provided one remains in the straight line portion of the curve (i.e. provided initial velocities are measured in each experiment); on the contrary, at time t₂, the proportionality between velocity and enzyme concentration ceases (see fig. 2-3 A).

The variation of initial velocity can also be plotted against enzyme con­centration (after carrying out several experiments with increasing enzyme concentrations): it is observed that within certain limits, initial velocity is proportional to enzyme concentration (see fig 2-3 B); theoretically, for high enzyme concentrations, the velocity curve should bend and velocity should become constant, but in general this phenomenon is not observed because of the limitation imposed by the solubility of protein macromolecules (a 0.01 M solution of a protein of molecular weight 100 000 would require a solution of 1000 g/liter); therefore, in the curve of figure 2-3 B, the plateau is not reached although — the enzyme is a molecule which reacts with the substrate and one would expect that it would influence reaction velocity in the same manner as the substrate (see fig. 2-4).

The curve of figure 2-3 B shows that reaction velocity is nil when enzyme concentration is nil; but it was said earlier that enzymes catalyze reactions which could proceed even in their absence and sometimes with appreciable velocities.

While studying velocity as a function of enzyme concentration, it is usual to carry out a “control” or “blank” test without the enzyme; the values obtained at different times in this blank test are subtracted from the values observed under various enzyme concentrations. The curve obtained with these corrected values represents exclusively the result of the enzymatic action, (and v = 0 for [E] = 0 is therefore justified).

The proportionality between reaction velocity and enzyme concentration has important practical applications because it enables the estimation of the relative concentrations of a given enzyme in cell homogenates without the necessity of purifying the enzyme.

One has, on the one hand, v = Const. X [E], and on the other hand v = [P]/t, therefore:

If the same time is used to study the reaction, t is a constant and the product t X Const., i.e., the product of two constants will give another constant (which may be denoted by C). In these conditions enzyme concentration is therefore directly proportional to the quantity of product formed ([E] = [P]/C).

Let us consider for example, a homogenate which can yield x mg of P in a given time; if another homogenate can yield 2x mg of P in the same time, we conclude that it contains twice the quantity of enzyme catalysing the formation of P.

Such measurements of relative enzyme concentrations are carried out sys­tematically (on blood for example) in clinical laboratories, because certain variations enable the diagnosis of a pathological condition.

In research laboratories, these determinations often represent the initial tests carried out while studying the regulation of the biosynthesis of an enzyme. It must be noted however, that the increase of active enzyme concentration can be due to either an increased synthesis of enzyme or an activation of the pre-existing enzyme molecules.

III. Effect of Substrate Concentration on Kinetics:

If enzyme concentration is maintained constant while substrate concentra­tion [S] is varied, a rapid increase of velocity is observed first; but if [S] continues to increase, the curve bends and for higher values of [S], velocity ceases to increase and tends asymptotically to a maximum value (V_max), as can be seen in figure 2-4.

1. Enzyme-Substrate Complex:

For a better understanding of the changes taking place when substrate and enzyme concentrations are varied, one must study in more details the reaction catalyzed by an enzyme.

When this reaction is written (admitting of course, that there can be more than one substrate and more than one product), we get the false impression that the enzyme will modify the velocity of this reaction by its mere presence in the medium.

But in reality it par­ticipates in the reaction by forming transitorily a specific enzyme-substrate complex. The existence of such complexes was suggested early this century, but it is only about thirty years later that their reality could be proved, first in the case of the peroxidase of horseradish (whose porphyrin prosthetic group has a characteristic absorption spectrum which is modified during the forma­tion of the complex with the substrate), then for other enzymes, by various methods, using physicochemical techniques like the study of visible or ultraviolet absorption spectra, fluorescence, nuclear magnetic resonance etc. or by equilibrium dialysis.

The modes of interactions between the 2 partners of the complex are now determined very accurately by X-ray diffraction analysis of crystals of enzyme-substrate complexes.

The basic reaction describing the enzymatic catalysis is written:

This representation is valid for the initial period of the reaction when con­centration of P is sufficiently low for the reverse reaction to be neglected. The quantity of P formed will depend directly on the E —S complex concentration so that one is led to study the variations of E — S concentration as a function of the increase of [S]. Posing the problem in this manner, it is clear that in the beginning, if [S] increases, [E-S] will also increase and reaction velocity will therefore increase.

But if [S] further increases, it is obvious that [E – S] cannot continue to increase beyond a certain maximum value which depends on the quantity of enzyme available. [E] becomes the limiting factor of the formation of the E —S complex, and an increase of [S] ceases to affect v: the plateau is reached (see fig. 2-4).

2. Michaelis-Menten Equation:

These two authors carried out a satisfactory quantitative study of the varia­tions of the velocity of an enzymatic reaction as a function of substrate concentra­tion.

This study is based on the representation:

E + S ⇋ E – S →E + P

and on the fact that equilibrium between E, S and E — S is a rapid process compared to the reaction E — S → E + P (which therefore controls the velocity of the enzymatic reaction).

For the study of the velocity of a reaction, the Menten-Michaelis equation offers the advantage that it is not necessary to know either [E] or [E — S], concentrations which are often difficult to deter­mine (and which were even more so in 1913 at the time of these two authors).

As mentioned above, the velocity of the enzymatic reaction is proportional to the E —S complex concentration, therefore v = k₃ [E—S]. As for the maximum velocity (V _max), it is observed for a substrate concentration such that the entire enzyme is bound to the substrate; the maximum value of [E — S] is equal to the total enzyme concentration [E_T]: V_max — k₃ [E_T].

We can write:

The equilibrium constant or Michaelis constant for the dissociation of the complex E — S is:

where [E] is the free enzyme concentration, equal to [E_T] — [E — S], Substitut­ing [E] by this value, we can write:

This may be written in the form:

It may be noted that, having determined experimentally the variations of v as a function of [S], one can use the Michaelis-Menten equation to obtain the value of V_max and calculate K_m.

The knowledge of enzyme concentration is not necessary; K_m can therefore be determined on incompletely purified prepara­tions. If for a given substrate, an identical K_m is found in two preparations, it may be assumed that they contain the same enzyme.

Taking v = V_max/2, we have:

The Michaelis constant (or equilibrium constant of the dissociation of the E -S complex) is therefore equal to the substrate concentration for which velocity is half the maximum velocity (see fig. 2-4). In general, the K_m values range between 10^-2 M and 10^-8 M. It must be noted that K_m is a measure of the affinity of the enzyme for its substrate.

The stronger the E — S interaction, the larger the quantity of enzyme combined with its substrate in the form E — S, and the smaller the quantity of free enzyme; therefore [E] will be small and [E — S] large; consequently, K_m will be small.

The affinity of an enzyme for a substrate is equal to 1/K_m. If K_m is large (a high substrate concentration is therefore required to obtain a velocity equal to V_max/2), affinity is low; if K_m is small, affinity is high (a small substrate concentration is sufficient to have a velocity equal to V_max/2).

Examining the Michaelis-Menten equation written in the form one may note that when [SJ is very large compared to K_m, K_m can be neglected, and v tends to V_max. On the contrary, when [S] is very small compared to K_m, v is equal to V_max , which shows that v is proportional to [S].

This does confirm the experimental results represented by the curve of figure 2-4.

Enzymatic catalysis comprises two steps:

1) The formation of E—S which implies the recognition of S by the active site of the enzyme. This first step is characterized by the K_m which, as mentioned earlier, is equal to

2) The chemical operation of the decomposition of E – S into E + P.

This second step is characterized by V_max.

v can be plotted as a function of [S] on the basis of experimental results; a branch of an equilateral hyperbola is obtained (figure 2-4). V_max (at least an approximate value) is obtained for large values of [S].

To determine K_m the point V_max/2 is plotted on the vertical axis and from the corresponding point of the curve a perpendicular is drawn to the horizontal axis. But the value of V_max being only approximate, the determinations of V_max/2 and thence K_m are rather inaccurate.

Furthermore, a large number of experimental points are needed to trace such a curve, particularly at the low substrate concentrations where precision is not very high. But there is another graphical representation which requires only a small number of experimental points obtained at high substrate concentrations where precision in the determination of initial velocity is much greater.

3. Lineweaver-Burk Representation:

The Michaelis-Menten equation may be written:

The latter form is an equation of the type y = ax + b (where 1/v and 1/[S] are the variables y and x, while K_m/V_max and 1/ V_max are the constants a and b). Then, by plotting the values of 1/v (the reciprocals of velocities determined experimentally) as a function of 1/[S], we obtain a straight line which — as seen in figure 2-5 — intersects the vertical axis (1/v) at the point 1/V_max (since 1/[S] = 0, we have 1/v = 1/V_max).

This straight line intersects the horizontal axis (1/[S]) at the point – 1/K_m; if 1/v = 0, we have:

This graphical representation is more convenient for determining the Michaelis constant (K_m) and the maximum velocity (V_max) of the reaction for a given enzyme concentration. Moreover, as will be seen later, it permits the distinction between a competitive inhibitor and a non-competi­tive inhibitor.