The chemical reactions that take place in the cells are catalyzed by enzymes. These important catalysts are specific to a particular reaction. Overall, they are extremely versatile in the sense that some thousands of enzymes now known carry out all those different reactions indispensable for the cell, such as hydrolysis, polymerization, oxidation reductions, etc. Furthermore, these proteins do not behave as passive surfaces on which the reactions take place, but are rather complex molecular machines that operate through a series of very different mechanisms.
The study of enzymatic kinetics began in 1902 when Adrian Brown reported his research on the rate of hydrolysis of sucrose catalyzed by the enzyme of the yeast invertase (now known as β-fructofuranosidase):
Sucrose + H2O → glucose + fructose
He showed that when the sucrose concentration is much higher than that of the enzyme, the rate of reaction becomes independent of the sucrose concentration; that is, the velocity is of zero-order with respect to sucrose.
Brown, therefore, thought that the overall reaction was composed of two elementary reactions, in which the substrate first formed a complex with the enzyme which subsequently decomposed into products and free enzyme:
According to this model, when the substrate concentration becomes high enough to convert the enzyme into the ES form completely, the second step of the reaction becomes that which limits the speed, and the overall reaction is insensitive to further increases in the concentration of the substrate. Here, ES indicate the enzyme, the substrate, the enzyme-substrate complex, and the products, respectively. The general expression of the speed of this reaction is:
v = d [P] / dt = k2 [ES]
The overall production speed of [ES] is given by the difference between the speeds of the elementary reactions that lead to its formation and that determine its disappearance:
d [ES] / dt = k1 [E] [S] - k-1 [ES] - k2 [ES]
This equation cannot be explicitly integrated without making some simplifying assumptions. The two possibilities are 1.
It is taking balance. In 1913, Leonor Michaelis and Maude Menten assumed that:
K-1 >> K2, that is, that the first stage of the reaction could reach equilibrium.
Ks = k-1 / k1 = [E] [S] / [ES]; Ks represents the dissociation constant of the first stage of the enzymatic reaction. After this assumption, the equation d [ES] / dt = k1 [E] [S] - k-1 [ES] - k2 [ES] can be integrated.
Assumption of the steady-state. With the exception of the initial part of the reaction (transient phase), which occurs in a very short time after mixing the enzyme with its substrate, [ES] remains constant until the substrate begins to run out. Consequently, the synthesis speed of ES must be equal to the demolition speed for almost the entire time of the reaction; That is, [ES] remains in a stationary state. It can therefore be assumed that [ES] remains constant and therefore: d [ES] / dt = 0. The values of [ES] and [E] are generally not easily quantifiable directly, but the amount of total enzyme [E] T is instead always known:
[E] T = [E] + [ES]
At this point, the velocity equation of our enzymatic reaction can be derived by combining the equation d [ES] / dt = k1 [E] [S] - k-1 [ES] - k2 [ES] with the assumption of the steady-state, the equation d [ES] / dt = 0 with [E] T = [E] + [ES], we have: k1 ([E] T - [ES]) [S] = (k-1 + k2) [ES]
After appropriate rearrangements it will become:
[ES] (K-1 + k2 + k1 [S]) = k1 [E] T [S]
By dividing both terms by k1 and solving by [ES],
[ES] = [E] T [S] / KM + [S]
Where KM, known as Michaelis constant, is defined by:
KM = k-1 + k2 / k1
The initial reaction rate, based on [E] T and [S], will be:
v0 = (d [P] / dt) t = 0 = k2 [ES] = k2 [E] T [S] / KM + [S]
The use of the initial velocity instead of the velocity as a whole minimizes some complications, such as the effects of the possible reversibility of the reactions, the inhibition of the enzyme by the product, and the possible progressive inactivation of the enzyme itself.
The maximum speed of a reaction, on the other hand, is reached only when the enzyme is saturated by a high concentration of the substrate, that is, it is completely in the form [ES]:
Vmax = K2 [E] T
So by combining v0 = (d [P] / dt) t = 0 = k2 [ES] = k2 [E] T [S] / KM + [S] to the equation Vmax = k2 [E] T we obtain:
v0 = Vmax [S] / KM + [S]
The latter expression, the Michaelis-Menten equation, is the basis of enzymatic kinetics and describes a rectangular hyperbola. Hence, we can conclude by saying that the speed of an enzymatic reaction at any instant is determined by the ratio between two constants Vmax and KM and the concentration of the substrate at that instant.
Michaelis' constant, KM, operationally can be defined in a very simple way. At the substrate concentration where [S] = KM, from the equation v0 = Vmax [S] / KM + [S] it is obtained that v0 = vmax / 2 and therefore, KM corresponds to the substrate concentration at which the reaction rate is half of the maximum. If an enzyme consequently has a KM with a small value, it will reach maximum catalytic efficiency at a rather small substrate concentration. The KM value, however, varies considerably with the type of enzyme and with the nature of the substrate. Temperature and pH can also influence the constant value. Michaelis' constant can be expressed as:
KM = k-1 / k1 + K2 / k1 = Ks + k2 / k1
Since Ks is the dissociation constant of the Michaelis complex, if Ks decreases, the affinity of the enzyme for the substrate increases. KM is also a measure of the affinity of the enzyme for its substrate, if the ratio k2 / k1 is very small compared to Ks, i.e., k2> k-1.