Binding Energy and Enzyme Rate I was preparing for the Canadian Chemistry Olympiad and I ran into the following question on one of the past exams:


  If enzyme A binds to the substrate 25 times stronger than enzyme B, what is the ratio of the catalytic rate between enzyme A and enzyme B if the energy of the two transition states is identical? What is the difference in activation energy between the two reactions?


I feel as if I can make a qualitative prediction, but cannot derive a quantitative prediction. I suspect, since enzyme A bind more strongly to the substrate than enzyme B, it would have a greater activation energy, and thus result in a slower reaction.

  

I suspect that it is necessary to use the equation
$$ \begin{align}
k = \frac{kT}{h} e^{-\Delta G^\ddagger/RT}
\end{align} $$

But the relationship between the two activation energies does not seem simple

$$\begin{cases}
\Delta G^\ddagger_A = \Delta G_{E + S \rightarrow T} - 25\Delta G_\text{bind} \\
\Delta G^\ddagger_B = \Delta G_{E + S \rightarrow T} - \Delta G_\text{bind}
\end{cases}$$

Plugging this into the the original equation, and solving for $k_B/k_A$, I get the following:

$$ \frac{k_B}{k_A} = \frac{e^{\Delta G_\text{bind}/RT}}{e^{25\Delta G_\text{bind}/RT}} = e^{-24\Delta G_\text{bind}/RT}$$

Since $\Delta G_\text{bind} &lt; 0$, $k_B/k_A &gt; 1$, supporting my qualitative prediction. Yet this does not give me a useful ratio, since it is dependent on the binding energies of enzymes A and B. This makes me suspect I am missing/overlooking something, any insight would be appreciated.
Your intuition that the enzyme binding more strongly to substrate will be slower is correct, this is called ground-state stabilization.  I think where you might be making an error is how you handle the factor of 25.  The factor of 25 probably refers to the ratio of equilibrium constants between E and ES for the two enzymes, not the ratio of binding energies.  So you need to start with delta G = - RT ln Keq, and compare for 2 different Keq's.  Then use your equation to estimate the difference in rate constants.