Enzymatic Kinetics

As for any other chemical reactions, enzymatic reactions depend on several factors: of course, they depend on the concentration of substrate and enzyme, and also on physical conditions (e.g., temperature), presence of inhibitors, etc. The equation that describes the kinetics of the reaction can hence be complex, depending on the number of factors that are included.

The most simple and famous equation is the MichaelisMenten equation (after the names of Leonor Michaelis and Maud Menten who first proposed the quantitative theory for enzyme kinetics in 1913). The actual formulation was derived by Briggs and Haldane. In this theory, only the two most important factors are included: the concentration of substrate (S) and the concentration of enzyme (E). Moreover, it is based on the following hypotheses:

• the binding reaction that produces the enzyme-substrate complex is reversible and at steady state; and

• the reaction that gives the final product is irreversible.

Hence, it is possible to represent the mechanism of the enzymatic reaction by

dS dt

dt dP

— = k3-ES dt where k1 and k2 are, respectively, the constant rates of direct and inverse reaction of enzyme-substrate complex (ES), and k3 the constant rate ofproduction ofproduct (P).

Due to the steady-state assumptions, the concentration of the enzyme-substrate complex can be assumed to be constant:

Hence,

Due to the fact that catalysts are not consumed in the reaction, the total amount of enzyme is constant:

and substituting this result in the previous equation,

Finally, the kinetics of the product and the substrate can be expressed as dP _ dS S _ S

where the maximum rate of the reaction (^max) depends on the total amount of enzyme available (E0) and the rate with which the enzyme transforms the substrate. This is the well-known Michaelis-Menten equation; it describes a hyperbolic saturation curve with two parameters: the ma imum rate, which is reached when all the enzyme is bound with substrate (i.e., in the case of a large e cess of substrate), and the constant kS, called half-saturation constant (because when substrate concentration is equal to this value, the reaction rate is half of the ma imum).

Given this formulation, the specificity factor (r) can be easily derived and assuming that both substrates (say A and B) are scarce (i.e., A << KA),

MA,max KB

The Michaelis-Menten equation is the basis for all enzymatic kinetics, but several other factors influence the processes and can be included. Temperature is one of the most important environmental factors that influence all reactions' kinetics: indeed, the higher the temperature the lower is the energy needed for activation. This is usually described by the Arrhenius law, f (T) — dT~T"{, where Tref is the reference temperature (usually 20 °C) and 0 a parameter typical for each reaction. This is undoubtedly true also for enzymatic processes, but only when temperature is lower than 45-50 °C. At this temperature, due to the increased mobility of atoms, proteins start to modify the rigid structure: the active site can hence lose one of the most important factors that allow us to bind the substrate. The catalytic capability of enzyme as well as the rate of the reaction are then reduced. This aspect has a relatively low ecological importance, due to the high temperature needed for the modification of proteins' structure: in any case, this aspect cannot be simply neglected because of the increasing anthropic pressure on the environment that can locally increase the temperature.

Enzymatic activity can also be reduced by the action of two different types of inhibitors: the competitive and the noncompetitive.

In the first case, the kinetics of the enzyme-inhibitor complex need to be added to the classical MichaelisMenten approach:

where I is the inhibitor, and EI the enzyme-inhibitor complex.

Assuming that also the latter reaction is at steady state with equilibrium constant equal to K, the kinetics of product (and substrate) are as follows:

The presence of the inhibitor does not reduce the maximum rate (still depending on the total amount of enzyme available E0), but it increases the half-saturation constant: this means that under the same conditions the reaction proceeds slower when inhibitor is present. The reduction of the rate depends directly on inhibitor's concentration and inversely on the equilibrium of the enzyme-inhibitor complex.

The slowing down of the reaction can be insignificant in case of a large excess of substrate: the inhibitor has low probability to find free active sites and then the reaction is fostered at the usual rate.

Noncompetitive inhibition has an opposite behavior. In this case, the kinetics can be described by the following system:

EI EIS

Assuming that all reversible reactions are at the steady state, that they are all independent, and naming kS and ki, respectively, the equilibrium constant of the creation of the complexes enzyme-substrate and enzyme-inhibitor, the kinetics of product (and substrate) are dP^ dS _ kyEo S ^ Mmax S

dt - dt - (1 + (I/k)) ? S + ks - (1 + (I/k)) ? S + ks

In this case the maximum rate is reduced by the inhibitor, while the half-saturation constant is the same; hence, the reaction is definitely slowed down: indeed, even if the increase of substrate speeds up the reaction, the rate will always be lower than in normal conditions.

Figure 3 presents a comparison of the three different kinetics. The kinetics discussed involves only one substrate, that is, describes the production of one (or more) product starting from only one reactant (e.g., the dissociation of carbonic acid in carbon dioxide and water in lungs fostered by carbonic anhydrase). Many enzymatic processes require more than one substrate in order to achieve the product (e.g., the hydration of carbon dioxide always catalyzed by carbonic anhydrase). The kinetics of these reactions are more complex. The simpler case is when different substrates do not interact; hence, any substrate limits the reaction with a saturation curve. The final production rate is equal to the product of the maximum production rate (always depending only on the total amount of enzyme) and any rate-limiting effects, that is, dP rr Sj dt- ^ n S-msj [1]

— No inhibition —Competitive inhibition —Noncompetitive inhibition

— No inhibition —Competitive inhibition —Noncompetitive inhibition

Figure 3 Comparison between the enzymatic kinetics without inhibition (black line), in presence of competitive inhibitor (red line) and in the presence of noncompetitive inhibitor (blue line). The maximum rate is evidenced by dashed line and the half-saturation constant by an arrow: it is clear that the presence of noncompetitive inhibitor halves the maximum rate, while competitive inhibitor doubles the half-saturation constant.

Figure 3 Comparison between the enzymatic kinetics without inhibition (black line), in presence of competitive inhibitor (red line) and in the presence of noncompetitive inhibitor (blue line). The maximum rate is evidenced by dashed line and the half-saturation constant by an arrow: it is clear that the presence of noncompetitive inhibitor halves the maximum rate, while competitive inhibitor doubles the half-saturation constant.

Another simple model is derived by the Liebig's law of the minimum, stating that the crop growth in controlled by the scarcest resource. In the case of enzymatic processes, this means that only the substrate with the lowest concentration (compared to the specific half-saturation constant) will limit the whole reaction, that is, dP

Equation [1] gives a stronger limitation, due to the multiplicative form of a series of numbers lower than 1. When all substrates except one are in excess, this difference can be neglected, reflecting the fact that only one substrate really limits the reaction rate.

If the substrates interact with each other and/or intermediate complexes are generated during the reaction, the kinetics are more complex, and depend on the number of substrates and intermediate complex involved in the reaction.

In some reactions, the amount of enzyme is directly linked to the amount of product: this usually happens when the enzyme is produced by the product itself. For instance, microorganisms produce the enzymes that are necessary for their growth. In this case the total enzyme E0 cannot be considered constant as in the MichaelisMenten approach: it is assumed that the enzyme is directly proportional to the product; hence, the maximum rate is expressed as being linearly dependent on the product itself.

In the case of microbial growth, it is necessary to add another hypothesis to the classical Michaelis-Menten formulation: indeed, the product, that is, microbial biomass, is destined to decay. While in the classical formulation the fate of the product was totally irrelevant, in this case, due to the link between product and enzyme, it is fundamental. Given these new assumptions the standard equations for substrate and product can be used, considering organic matter (S) as substrate and microbial biomass (X), as the product. Two new parameters need to be introduced: the yield factor (Y) and a mortality rate for microbes (kd). The yield factor is the biomass synthesized per unit of substrate consumed, and considers that the stoichiometry of the biomass production is usually different from the ideal ratio (1:1) used in the MichaelisMenten equation. The mortality rate takes into account the consumption of the product. Finally, the system equation is dS _ 1 S

where ^max is the maximum specific growth rate and is usually considered dependent from other factors (e.g., temperature).

Time

Figure 4 Example of Monod dynamics: in black the substrate, in red the biomass of bacteria. At the beginning biomass grows exponentially, consuming substrate. When substrate concentration decreases drastically, growth slows down and then stops before that substrate finishes. Once the substrate is completely consumed, biomass starts to decay exponentially.

Time

Figure 4 Example of Monod dynamics: in black the substrate, in red the biomass of bacteria. At the beginning biomass grows exponentially, consuming substrate. When substrate concentration decreases drastically, growth slows down and then stops before that substrate finishes. Once the substrate is completely consumed, biomass starts to decay exponentially.

This system is the well-known Monod kinetics for microbial growth, whose trends are shown in Figure 4.

Monod derived these equations studying the growth of a culture of heterotrophic bacteria Escherichia coli on a substrate of lactose. With little or no changes, the same system can also be used to describe growth of other microorganisms. For instance, phytoplankton growth can be expressed by a similar kinetics:

In this case, the maximum growth rate (^max) depends on temperature (T) and light (L). Furthermore, phytoplank-ton growth is a process that requires more than one substrate (nutrients): the most important ones are carbon, nitrogen, and phosphorus. While carbon is usually largely present in the environment (directly as dissolved CO2, or as carbonic acid - that is dissociated in CO2 and H2O by the enzyme anhydrase), nitrogen and phosphorus can both act as limiting factors and hence they need to be included in the equation as described above for multisubstrate processes.

See also: Adsorption; Biodegradation; Decomposition and Mineralization.

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