Question

2. Consider the biosynthetic reaction between glutamate and NH₄⁺ to form glutamine at T = 25^o C (R = 8.3145 J / mol∙K),

glutamate + NH₄⁺ --> glutamine

2a. The equilibrium constant for the reaction under standard biochemical conditions is K_eq = 0.00325. Calculate ΔG^o' and please give your answer in kcal/mol (assume RT = 0.6 kcal/mol).

ΔG^o' = __________ kcal/mol

2b. To make the reaction spontaneous, what would the concentration of glutamine have to be maintained at relative to the concentrations of glutamate and NH₄⁺?

Ratio of glutamine concentration to product of glutamate and ammonium concentrations is __________

2c. Show that the biosynthesis of glutamine can be made spontaneous by the addition of ATP, assuming ATP spontaneously hydrolyzes to ADP and P_i with (∆G^o' = -7.3 kcal / mol) at RT = 0.6 kcal/mol. In your final answer, please give your answer in kcal/mol and include all reactants and products.

Write overall reaction: __________________________________________________________________

Overall ΔG^o' = __________ kcal/mol

Answer 1

The equilibrium constant, K for a reaction is related to the standard free energy change, ΔG⁰ as per the expression

ΔG⁰ = -R*T*ln K_eq where R is the gas constant and T is the absolute temperature.

a) Given R*T = 0.6 kcal/mol, we have

ΔG⁰’ = -R*T*ln K_eq

====> ΔG⁰’ = -(0.6 kcal/mol)*ln (0.00325) = -(0.6 kcal/mol)*(-5.72910) = 3.437460 kcal/mol ≈ 3.44 kcal/mol (ans).

b) For the reaction to be spontaneous, we must have ΔG⁰ < 0. Compute the reaction quotient, Q when ΔG⁰ = 0.0 kcal/mol.

We have

ΔG⁰ = ΔG⁰’ + RT*ln Q

===> 0.0 kcal/mol = (3.44 kcal/mol) + (0.6 kcal/mol)*ln Q

===> -(3.44 kcal/mol) = (0.6 kcal/mol)*ln Q

===> -5.73333 = ln Q

===> Q = exp(-5.73333) = 0.00323627 ≈ 0.00324.

We define Q as Q = [Glutamine]/[Glutamate][Ammonium]; therefore, the ratio of the concentration of Glutamine to the product of the concentrations of Glutamate and ammonium is 0.00324 (ans).