How to calculate the turnover number?

Question

I understand that Kcat (turnover number) = Vmax/total enzyme concentration. However the formula I have been given is Kcat = specific activity/molecular mass of enzyme.

What is the relationship between these two different formulas - why do they give the same answer?

Definition of Kcat that I am using: specific activity * molecular mass of the enzyme Specific activity = units of enzyme/ mass of protein in mg

Answer 1

Let's do some unit analysis:

$\pu t$ = time

$\pu U$ = units = $\mathrm{mol_{product}/t}$

$\pu s=$ specific activity $\mathrm{= U/g_{protein}}$

Therefore:

$\mathrm{s = mol_{product} / (g_{protien} \times{t})}$

Your sheet says:

$\mathrm{k_{cat}= s \times MW_{protien}}$

And we know that:

$\mathrm{MW_{protein} = g_{protein} / mol_{protein}}$

Substituting definitions of specific activity and MW_protein:

$\mathrm{k_{cat}= mol_{product} \times g_{protein} / (g_{protien} \times t \times mol_{protein})}$

$\mathrm{k_{cat} = mol_{product} / (mol_{protein} \times t)}$

Cancel out the molar terms:

$\mathrm{k_{cat} = t^{-1}}$

This yields the correct unit for kcat ($t^{-1}$). You can think of this as the number of product molecules formed per enzyme molecule, per unit time.