Question

When glucose, \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{11}\), is metabolized to \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) in living systems, the first step is the conversion of glucose to glucose-6-phosphate. $$ \begin{array}{l} \text { glucose }+\mathrm{H}_{3} \mathrm{PO}_{4} \longrightarrow \text { glucose-6-phosphate }+\mathrm{H}_{2} \mathrm{O} \\ \Delta G^{\circ}=13.4 \mathrm{~kJ} \end{array} $$ This nonspontaneous reaction can be made to proceed by coupling it with the hydrolysis of ATP. $$ \mathrm{ATP}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{ADP}+\mathrm{H}_{3} \mathrm{PO}_{4} \quad \Delta G^{\circ}=-31.0 \mathrm{~kJ} $$ (a) Write an equation for the coupled reaction and calculate \(\Delta G^{\circ} .\) (b) Estimate the number of moles of ATP required to convert glucose to glucose-6-phosphate.

Short answer

Answer: One mole of ATP is needed to convert one mole of glucose to glucose-6-phosphate.

Step-by-step solution

(a) Writing the coupled reaction equation and calculating the new \(\Delta G^{\circ}\) value

To write the equation for the coupled reaction, we simply sum the reactions given in the exercise together. $$ (\text{glucose}+\mathrm{H}_{3} \mathrm{PO}_{4}\longrightarrow\text{glucose-6-phosphate}+\mathrm{H}_{2} \mathrm{O}) + (\mathrm{ATP}+\mathrm{H}_{2} \mathrm{O}\longrightarrow\mathrm{ADP}+\mathrm{H}_{3} \mathrm{PO}_{4}) $$ Now, we can see that \(\mathrm{H}_{3} \mathrm{PO}_{4}\) and \(\mathrm{H}_{2} \mathrm{O}\) will cancel out: $$ \text{glucose}+\mathrm{ATP}\longrightarrow\text{glucose-6-phosphate}+\mathrm{ADP} $$ Now, to calculate the new \(\Delta G^{\circ}\) value for the coupled reaction, we just add the \(\Delta G^{\circ}\) values of the given reactions: $$ \Delta G^{\circ}_{\text{coupled}} = \Delta G^{\circ}_1 + \Delta G^{\circ}_2 = 13.4 \mathrm{~kJ} + (-31.0 \mathrm{~kJ}) = -17.6 \mathrm{~kJ} $$ So, the coupled reaction has a \(\Delta G^{\circ}\) value of -17.6 kJ, which means it's spontaneous.

(b) Estimating the number of moles of ATP needed to convert glucose to glucose-6-phosphate

As we have written the coupled reaction equation, we can see that the reaction between glucose and ATP is a one-to-one reaction, meaning that one mole of glucose reacts with one mole of ATP. Therefore, to convert one mole of glucose to glucose-6-phosphate, one mole of ATP is required.