Chemical kinetics

Enzymes are proteins with molecular weights of the order of 10 ⁴ to 10 ⁶ and catalyze most of the reactions that occur in living beings. The action of enzymes is very specific, catalyzing only a certain class of reactions. Furthermore, very small concentrations of enzyme are often sufficient to produce enormous catalytic activity on large amounts of substrate. The molecule on which the enzyme acts is called a substrate. The substrate binds to the active site of the enzyme, forming the enzyme-substrate complex. In this complex the substrate is transformed into the product, at which point it is released from the enzyme. There are many mechanisms to explain enzyme catalysis, here we will consider the simplest one, the Michaelis Menten mechanism, which is: \begin{equation} E+S\rightleftharpoons ES \rightarrow E+P \end{equation} Where E represents the enzyme , S the substrate, ES the enzyme-substrate complex and P the final product. The speed of the reaction will be given by the expression, $r=\frac{d[P]}{dt}=k_2[ES]$

Applying the steady state approximation to the enzyme-substrate complex, ES: \begin{equation} \frac{d[ES]}{dt}=0=k_1[E]_0[S]-k_1[ES][S]- k_{-1}[ES]-k_2[ES] \end{equation} Solving for [ES], \begin{equation} [ES]_{ee}=\frac{k_1[E]_0[S]}{k_1 [S]+k{-1}+k_2} \end{equation} Substituting into the kinetic equation \begin{equation} r=\frac{k_2k_1[E]_0[S]}{k_1[S]+k_{- 1}+k_2} \end{equation} Dividing by $k_1$ \begin{equation} r=\frac{k_2[E]_0[S]}{[S]+\frac{k_{-1}+k_2} {k_1}}=\frac{k_2[E]_0[S]}{[S]+K_m} \end{equation} Where $K_m$ is the Michaelis constant: $K_m=\frac{k_{-1} +k_2}{k_1}$.

The Michaelis Menten equation can be simplified depending on the relationship between $K_m$ and $[S]$

• If $K_m>>[S] \;\;\; \Rightarrow \;\;\; r=\left(k_2/K_m\right)[E]_0[S]$. First order reaction, since $[E]_0$ is constant for every reaction.
• If $K_m<<[S] \;\;\; \Rightarrow r=k_2[E]_0$. In this situation we find zero order kinetics. At high substrate concentrations the enzyme becomes saturated and the rate no longer depends on the substrate concentration.