The decarboxylation of lysine catalyzed by lysine decarboxylase has a kcat value of 500 s-1 at 298K, and loss of CO2 is the rate-determining step. What is the free energy of activation for the CO2 loss step? The half-life for the uncatalyzed reaction under the same conditions is 4 billion years (1017 seconds). How much does the enzyme lower the free energy of activation for this reaction? Show your work.

The decrease in free energy is 113.299kJ

Explanation:

K for enzyme catalyzed reaction = 500s^-1

Temperature (T) = 298k

ΔG = ?

ΔG = - 2.303 RT log k

ΔG = (-2.303) (8.314) (298) log 500

ΔG = - 15399.9 J

ΔG catalyzed = - 15. 399kJ

The first order reaction is given as:

t1/2 = 0.693/k

or k = 0.693/t1/2

0.693/10^17

Therefore,

K = 0.693 * 10^-17

Now,

K = 0.693 * 10^-17

T = 298k

ΔG uncatalyzed = ?

ΔG uncatalyzed = - 2.303 RT log k

ΔG uncatalyzed = (-2.303) (8.314) (298) log0.693 * 10^-17

= 97908.1J

ΔG uncatalyzed = 97.9081kJ

Therefore,

The decrease in free energy is:

ΔG uncatalyzed - ΔG catalyzed

97.908 - (-15.399)

= 113.299KJ

The decrease in free energy is 113.299kJ