Question #44974

A biochemist is studying the thermal denaturation of a synthetic polypeptide at $310\text{ K}$. The polypeptide exists in a functional Native state ($\ce{N}$) and a Denatured state ($\ce{D}$), which are in rapid equilibrium. The denatured state further undergoes a secondary equilibrium to form a stable, inactive dimer aggregate ($\ce{A}$). The coupled equilibria are given as:

$\ce{N <=> D} \quad K_1 = 0.5$ $\ce{2D <=> A} \quad K_2 = 50 \text{ M}^{-1}$

Initially, the polypeptide is prepared in a buffer such that the total monomeric concentration is $[P]_0 = 0.18 \text{ M}$ (assuming $100\%$ $\ce{N}$ initially). Once the system reaches thermodynamic equilibrium, what is the percentage of the original polypeptide that remains in the functional Native state ($\ce{N}$)?