Consider an enzymatic mechanism in which product formation is reversible $($i$.$e$.$ enzyme can catalyze conversion of product to substrate$)$:

a$.$ Derive an expression for the net rate of product formation.

b$.$ Compare this rate to the rate observed if product formation is irreversible.

c$.$ Evaluate the rate $($with reversible product formation$)$ in the limit of low product concentration.

d$.$ Evaluate the rate $($with reversible product formation$)$ in the limit of a low rate constant for product conversion to enzyme$-$substrate complex.

$($Hint:

$E+$SharrESharrE$+P$

Use $k_1$ and $k_{-1}$ for the first reversible reaction and $k_2$ and $k_{-2}$ for the second reversible reaction.

Use the quasi$-$steady$-$state assumption for $[$ES$],$ solve for $[$ES$],$ then obtain an expression for $\frac{d[P]}{dt}=k_2[ES]-k_{-2}[E][P]$

$($b$-$d$)$ should give you the same result as the irreversible product formation $($or the standard MM kinetics$))$