A2.

Assume the following enzymatic reaction scheme of reversible competitive inhibition:
$$
\mathrm{E}+\mathrm{I} \stackrel{k_{i}}{\rightleftharpoons} \mathrm{EI}
$$

Assuming elementary reactions and that reactions are fast in both directions so that the reaction is at equilibrium at all times (Michaelis-Menten), prove that the rate of product production is given by the following formula:
$$
r_{p}=\frac{k_{2} C_{m} C_{s}}{C_{1}+k_{m}\left(1+\frac{C_{i}}{k_{1}}\right)}
$$

Where,
$$
k_{m}=\frac{k_{-1}}{k_{1}}
$$
and $k_{1}$ and $k_{-1}$ are synthesis and dissociation constants respectively of the intermediate complex ES. $k_{1}$ is the enzyme complex dissociation constant. $k_{2}$ synthesis constant of the product $\mathrm{P}\left(C_{e 0}\right.$ is the initial enzyme concentration and $C_{n}, C_{i}$ are the substrate and inhibitor concentrations respectively, in steady state).

Question

A2.

Assume the following enzymatic reaction scheme of reversible competitive inhibition:
$$
\mathrm{E}+\mathrm{I} \stackrel{k_{i}}{\rightleftharpoons} \mathrm{EI}
$$

Assuming elementary reactions and that reactions are fast in both directions so that the reaction is at equilibrium at all times (Michaelis-Menten), prove that the rate of product production is given by the following formula:
$$
r_{p}=\frac{k_{2} C_{m} C_{s}}{C_{1}+k_{m}\left(1+\frac{C_{i}}{k_{1}}\right)}
$$

Where,
$$
k_{m}=\frac{k_{-1}}{k_{1}}
$$
and $k_{1}$ and $k_{-1}$ are synthesis and dissociation constants respectively of the intermediate complex ES. $k_{1}$ is the enzyme complex dissociation constant. $k_{2}$ synthesis constant of the product $\mathrm{P}\left(C_{e 0}\right.$ is the initial enzyme concentration and $C_{n}, C_{i}$ are the substrate and inhibitor concentrations respectively, in steady state).

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Define Michaelis-Menten kinetics for the system**: Michaelis-Menten kinetics describes the rate of enzymatic reactions where a substrate S is converted to a product P by an enzyme E. The formation of the enzyme-substrate complex ES is a key step. ‖ ‖ 2 ⋮ **Write the equilibrium expressions**: For the enzyme-substrate complex and the enzyme-inhibitor complex, the equilibrium expressions are given by $K_m = \frac{k_{-1}}{k_1}$ for ES and $K_i = \frac{k_{-3}}{k_3}$ for EI. ‖ ‖ 3 ⋮ **Apply the steady-state assumption**: The steady-state assumption implies that the concentration of the enzyme-substrate complex ES is constant over time, which leads to the Michaelis-Menten equation. ‖ ‖ 4 ⋮ **Derive the modified Michaelis-Menten equation**: Considering competitive inhibition, the presence of the inhibitor affects the availability of the enzyme to bind to the substrate. The modified equation accounts for the inhibitor by adding a term that includes the inhibitor concentration $C_i$ and its dissociation constant $K_i$. ‖ ‖ 5 ⋮ **Express the rate of product formation**: The rate of product formation $r_p$ is proportional to the concentration of the ES complex and the rate constant $k_2$. Using the modified Michaelis-Menten equation, express $r_p$ in terms of $k_2$, $C_{E0}$, $C_S$, $C_I$, $K_m$, and $K_i$. ‖ ‖ 6 ⋮ **Solve for $r_p$**: By substituting the expressions for $K_m$ and $K_i$ into the modified Michaelis-Menten equation, solve for $r_p$ to obtain the given formula. ‖ ∻Answer∻ ⚹ The rate of product formation in the presence of a competitive inhibitor is given by the formula $r_p = \frac{k_2 C_{E0} C_S}{K_m(1 + \frac{C_I}{K_i}) + C_S}$, where $K_m = \frac{k_{-1}}{k_1}$ and $K_i = \frac{k_{-3}}{k_3}$. ⚹ ∻Key Concept∻ ⚹ The key concept is understanding the Michaelis-Menten kinetics and how it is modified in the presence of a competitive inhibitor. ⚹ ∻Explanation∻ ⚹ The explanation involves applying the steady-state assumption to derive the rate of product formation, which is affected by the presence of an inhibitor that competes with the substrate for the enzyme binding site.◊What is the significance of the Michaelis-Menten kinetics in enzyme reactions involving reversible competitive inhibition?⍭ Generate me a similar question◊⚹

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