Question

How do you find ${K_m}$ and ${V_{\max }}$ ?

Answer 1

For the calculation of ${K_m}$ and ${V_{\max }}$ we need to first understand the kinetics of an enzyme. As both the asked terms ${K_m}$ and ${V_{\max }}$ are found to be in the Michaelis-Menten equation. Where ${K_m}$ stands for the Michaelis-Menten constant or the concentration of the substrate at which half of the maximum velocity is achieved and ${V_{\max }}$ is the rate of the reaction.

Complete step by step solution:
Michaelis-Menten Kinetic equation is among the best equations or models in explaining the enzyme kinetics. The scientists who developed this model were Leonor Michaelis and Maud Menten.
Initially, it was Victor Henri who suggested that the action of enzyme and substrate is carried by the formation of bonds. However, he fails to explain the details of the reaction. Then Michaelis and Maud Menten investigated the theory of Henri and gave their Michaelis-Menten model.
The final result of Michaelis-Menten equation is ${V_o } = \dfrac{{{V_{\max }}[S]}}{{{K_m} + [S]}}$ where ${K_m}$ stands for the Michaelis-Menten constant or the concentration of the substrate at which half of the maximum velocity is achieved and ${V_{\max }}$is the rate of the reaction and $[S]$ is the concentration of the substrate.
Additional Information: The mathematics of the Michaelis-Menten model is as follows,
Let us say, enzymes (E) and substrate (S) react in equilibrium in such a way that the rate of forward reaction is ${K_1}$ and of backward reaction is ${K_{ - 1}}$ . Now the enzyme-substrate intermediate will form a product with rate as ${K_2}$ .
Then the rate of formation of product is ${V_o } = {K_2}[ES]$
Via equation of steady state it was calculated that $[ES] = \dfrac{{[E][S]}}{{[S] + \dfrac{{{K_{ - 1}} + {K_2}}}{{{K_1}}}}}$
But $\dfrac{{{K_{ - 1}} + {K_2}}}{{{K_1}}}$ is Michaelis-Menten constant and is denoted as ${K_m}$
Thus we can say $[ES] = \dfrac{{[E][S]}}{{[S] + {K_m}}}$ .
Then ${V_o } = {K_2}\dfrac{{[E][S]}}{{[S] + {K_m}}}$
Taking constants ${V_o } = \dfrac{{{V_{\max }}[S]}}{{[S] + {K_m}}}$ or may be written as ${V_o } = \dfrac{{{V_{\max }}[S]}}{{{K_m} + [S]}}$

Note:
The Michaelis-Menten kinetics is not restricted to biochemical reactions only, it found its applications in alveolar clearance dusts, bacteriophage infection, richness of species pools, the irradiance-photosynthesis relationship, clearance of blood alcohol, and many more. This equation also gives the relationship between ion channel conductivity and ligand.