Chapter 18, Problem 18.111QP

Adenosine triphosphate, ATP, is used as a free-energy source by biological cells. (See the essay on page 624.) ATP hydrolyzes in the presence of enzymes to give ADP:

$\begin{array}{ll}\text{ATP}(aq)+{\text{H}}_2\text{O}(l)\stackrel{}{\to }\hfill & \text{ADP}(aq)+{\text{H}}_2{\text{PO}}_4{}^{-}(aq);\hfill \\ \hfill & \Delta G°=-30.5\text{kJ/mol at 25}°\text{C}\hfill \end{array}$

Consider a hypothetical biochemical reaction of molecule A to give molecule B:

$\text{A}(aq)\stackrel{}{\to }\text{B}(aq);\Delta G°=+15.0\text{kJ/mol at 25}°\text{C}$

Calculate the ratio [B]/[A] at 25°C at equilibrium.

Now consider this reaction “coupled” to the reaction for the hydrolysis of ATP:

$\begin{array}{ll}\text{A}(aq)+\text{ATP}(aq)+{\text{H}}_2\text{O}(l)\stackrel{}{\to }\text{B}(aq)+\hfill & \text{ADP}(aq)+\hfill \\ \hfill & {\text{H}}_2{\text{PO}}_4{}^{-}(aq)\hfill \end{array}$

If a cell maintains a high ratio of ATP to ADP and H₂PO₄⁻ by continuously making ATP, the conversion of A to B can be made highly spontaneous. A characteristic value of this ratio is

$\frac{[\text{ATP}]}{[\text{ADP}][{\text{H}}_2{\text{PO}}_4{}^{-}]}=500$

Calculate the ratio [B][A] in this case and compare it with the uncoupled reaction. Compared with the uncoupled reaction, how much larger is this ratio when coupled to the hydrolysis of ATP?