Question #45002

An apoenzyme $E$ binds a fat-soluble vitamin $V$ to form a holoenzyme. At $340\text{ K}$, the apoenzyme undergoes reversible thermal denaturation:

$\ce{E_{native} <=> E_{denatured}}$

The thermodynamic parameters for this denaturation are $\Delta H^\circ = 259.0\text{ kJ/mol}$ and $\Delta S^\circ = 800\text{ J/(K\cdot mol)}$.

The intrinsic binding constant of $V$ to the specific hydrophobic pocket of $\ce{E_{native}}$ is $K_B = 9.1 \times 10^7\text{ M}^{-1}$ at $340\text{ K}$. The denatured enzyme, lacking the specific 'lock and key' pocket, only binds $V$ via non-specific hydrophobic interactions with $K_{NS} = 1.0 \times 10^5\text{ M}^{-1}$ at $340\text{ K}$.

Calculate the standard Gibbs free energy of effective binding ($\Delta G^\circ_{eff}$) for the vitamin to the enzyme mixture at $340\text{ K}$.

(Given: $2.303 RT = 6.5\text{ kJ/mol}$ at $340\text{ K}$; $\log_{10}(9.1) = 0.96$; $\log_{10}(9.01) = 0.95$. The effective binding constant is defined as $K_{eff} = \frac{[\ce{EV_{native}}] + [\ce{EV_{denatured}}]}{([\ce{E_{native}}] + [\ce{E_{denatured}}])[\ce{V}]}$)