One of the most fascinating areas of study in chemical kinetics is enzyme This chapter presents the basic mathematical treatment of enzyme kinetics and. How to read enzyme kinetics graphs (and how they’re made). Km and Vmax. Competitive and noncompetitive inhibitors. ABSTRACT. Procedures to define kinetic mechanisms from catalytic activity measurements that obey the. Michaelis-Menten equation are.

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After the thermodynamic interpretation, we will now review the fundamental relationships of enzyme kinetics. The product will be denoted as P as previously. When the topic of enzyme kinetics first emerged, almost nothing was known about the physical nature of enzymes and the possible mechanisms of rate enhancement. Let us start with a thought experiment considering the dependence of the rate of a non-catalysed chemical reaction as a function of reactant concentration.

In other words, the rate of the reaction is linearly proportional to the concentration of the reactant S. In typical cases, when an tiletype catalyses the same reaction, the rate is enhanced by orders of magnitude. However, very importantly, the plot of the [S]-V function would be principally different in this latter case. At a constant enzyme concentration, [E], and in the range of low [S] values, increasing [S] would result in an almost linear increase of the rate, V.

However, as [S] is increased even further, V would not increase to the same extent and it would ultimately approximate a maximal value limit Figure 9. The first kinetic model that successfully explained this phenomenon was introduced by Leonor Michaelis and Maud Menten. Their presumption, which nowadays might seem trivial, was revolutionary in their time. They assumed that the enzyme directly interacts with the substrate in a stoichiometric manner, the interaction results in a well-defined intermediate complex, and the interaction leads filetypr thermodynamic equilibrium.

This scheme is kinehics by Equation 9. As a tribute to this first successful model, ES has been named the Michaelis complex. The above simplest scheme is based on the following assumptions. The interaction between the substrate and tiletype enzymes ifletype generates the ES complex leads to a quasi- equilibrium; and the reaction is instantaneous, i.

Accordingly, for this first reaction step, the simple model introduces only an equilibrium constant, K Sand kinetis is not concerned with the two rate constants that determine K S.

### Enzyme Kinetics

K S —which, in the case of non-covalent E-S interaction, is a dissociation constant—is defined by Equation 9. According to this model, the catalytic rate constant, k cat that corresponds to the rate of the decomposition of ES towards the product, is much lower than the non-defined rate constants corresponding to K S.

Accordingly, the rate of ES decomposition towards the product is so low that at least in the time frame of the measurement it does not affect the quasi-equilibrium concentrations of [E], [S] and [ES]. Let us see how the initial rate of the reaction depends on substrate concentration if the starting assumptions apply. The rate equation corresponding to the scheme introduced in Equation 9.

This equation refers to a first-order reaction in which the rate of the reaction is proportional to the concentration of only a single entity, in this case the ES complex:. The scheme does not consider filetypd opposite reaction, i. This is because the model focuses on the very beginning of the reaction when the concentration of the product is negligible. It is therefore of utmost importance that the rates defined in this model are always initial rates that correspond to the theoretical zero time point of the reaction.

The concentration of ES is not pre-set by the experimenter, but it can be determined experimentally kinwtics the right model is established. In the following steps, we will transform Equation 9.

To do so, we need to express ES concentration as a function of the pre-set enzyme and substrate concentrations. Let us start with Equation 9. Then let us consider a self-explanatory relationship shown in Equation 9. Finally, both sides are divided by the multiplying factor of [ES], which results in Equation 9.

By these algebraic transformations, ES concentration has been expressed as a function of the experimentally pre-set enzyme and substrate concentrations and that of the equilibrium dissociation constant.

If the starting conditions are set such that the total substrate concentration exceeds the total enzyme concentration by orders of magnitude, the amount of substrate getting into the ES complex will be negligible compared to the total amount of substrate.

Consequently, the free substrate concentration at the beginning of the reaction will practically equal the total substrate concentration.

This way, both the total enzyme concentration, [E] Tand the free substrate concentration, [S], will be experimentally-set known parameters. In the next step, based on Equation 9. Note that the highest achievable initial reaction rate, denoted as V maxwill be achieved when all enzyme molecules are incorporated into the ES complex.

In this case, the substrate saturates the enzyme molecules. According to this, the k cat [E] T product in Equation 9. By taking this into account, we can formulate Equation 9. This equation is a so-called rectangular hyperbola function that has the following general description: P 1 is also the horizontal asymptote of the hyperbola, the maximal value of Y that the graph of the function approaches as X tends to infinity.

The P 2 parameter is K S. Note that this equation is in a perfect accordance with the experimental observations regarding the [S]-V 0 relationship illustrated in Figure 9. Accordingly, in the [S] range where the substrate concentration is orders of magnitude lower than the value of K sthe initial reaction rate will be linearly proportional to substrate concentration, exactly as the experiments show.

In other words, in this substrate concentration range the reaction is a pseudo -first order reaction in respect of the substrate. Accordingly, if we consider only this [S] range, we get equation 9. In the substrate concentration region where [S] exceeds K S by orders of magnitude, the substrate saturates the enzyme, i.

Accordingly, in this substrate concentration range, the initial reaction rate is practically independent of the substrate concentration, i.

This is why the [S]-V 0 plot illustrated in Figure 9. When the substrate concentration equals the value of K Sthe initial reaction rate is the half of the V max value. In a simple descriptive way, the deduced Equation 9.

Yet, it leads to serious theoretical contradictions. The more efficient the enzyme, the less rational the initial assumptions of the above model. If the enzyme is highly efficient, the rate of ES fileyype into product should kineyics very high. If so, that process should interfere with the presumed quasi-equilibrium between the enzyme, the substrate and the ES complex.

Therefore, the assumption of equilibrium for the first step of the reaction renders the model ill-suited for describing the action of genuinely efficient enzymes. The second—and equally significant—problem is that this first model also contradicts the thermodynamic bases of catalysis.

K S is a dissociation constant and, as such, it defines the affinity, i.

In the first model, K S describes how strongly the enzyme binds the substrate or, in other words, how stable the ES complex is. The lower the K Sthe more stable the complex. Moreover, based on the model, the lower the K Sthe more effective the enzyme. This is because a low K S means that the enzyme reaches half-maximal reaction rate at low substrate concentration.

But there is a discrepancy here. By increasing the stability of the interaction between the enzyme and the substrate, the reaction rate should decrease because the enzyme would stabilise the substrate in the ground state. Naturally, the enzyme must bind the substrate, but it should not bind it too tightly. Instead, the enzyme should bind tightly the transition state, thereby decreasing the activation free enthalpy of the reaction. Due to these contradictions, the first kinetic model had to be developed further.

The improved model accounts for the two additional kinetic rate constants that were left out from the first model. One of these, denoted as k 1corresponds to the formation of the ES complex from free enzyme and substrate. The other, denoted as k -1corresponds to the reverse reaction, the dissociation of the ES complex towards enzyme and substrate. The improved scheme is illustrated in Equation 9. According to the improved model, when the solutions of the enzyme and the substrate are mixed, instead of a dynamic equilibrium, a steady-state will develop very quickly, almost instantaneously.

Depending on the exact initial conditions, the steady-state may last long, i. The steady-state requires equal rates for ES generation and ES decomposition. This is illustrated in Figure 9. The reaction is triggered by adding the substrate. Proportions of the figure are not realistic, as otherwise various parts of the figure could not be shown on the same page.

In reality, the steady-state can be reached in milliseconds and it can last for minutes.

## Basics of enzyme kinetics graphs

Decomposition of the ES complex can happen on two different routes: The development of the steady-state is illustrated in Figure 9. Although the steady-state modification of the model was suggested by George Edward Briggs and John Burdon Sanderson Haldane, the improved model is still being referred to as the kinetic model of Michaelis and Menten and the derived equation as the Michaelis-Menten equation.

In this equation, the ES complex forms at a rate k 1 [E][S]; it decays back towards the substrate at a rate k- 1 [ES] and decomposes towards the product at a rate k 2 [ES].

In the steady-state, the rate of ES formation and the sum of the two types of ES decomposition rates are equal in magnitude and, thus, the concentration of the ES complex does not significantly change. In the next section, Equation 9. The improved model has the same initial requirement: Klnetics rate of the reaction can be easily formulated, as shown in Equation 9. By rearranging Equation 9. Just as it was done in the first model, the free enzyme concentration is expressed as the difference of the total enzyme kintics and the concentration of the ES complex Equation 9.

Then the [ES] term-containing parts are organised to the same side, resulting in Equation 9.

Then, on the right side of the equation, both the numerator and the denominator are divided by the k 1 rate constant, which results in Equation 9. In the denominator filegype Equation 9. This quotient has been defined as the Michaelis constant, with the abbreviation K M. Replacing the quotient for K M yields Equation 9. Note that K M is defined as the ratio of the two-direction decay rate of the ES complex and the one-direction formation rate of the complex.

In other words, it quantifies the instability of the ES complex. As already mentioned, in the case of Equation 9. Therefore, the k 2 [E] T product in Equation 9.