I want to interpolate the amount of product formed (as concentration or % of conversion) vs reaction time in a biocatalysis process. The fitting equation should have as (y) the amount of product and as (x) the reaction time. I thought to use as the fitting equation the integrated form of the M&M but I am not able to find the correct mathematical form. Or should I use another equation?
Without any knowledge about the experiment parameters i can only guess which equation might suit your problem. But I'll stick to M&M since thats what you suggested.
Having the product [P] as (y) results in some pretty ugly equations most of the time. I'm assuming no inhibitory effects.
[P] = [S]0 - Km * W { X(t) }
or alternatively (for the % of conversion)
[P]/[S]0 = 1 - Km/[S]0 * W { X(t) }
where [P] is the concentration of product P and [S]0 is the initial concentration of the substrate S.
W is the Lambert-W function (or prodlog). You can try
X(t) = [S]0/Km * exp[ ([S]0 - vmax * t) / Km ]
as an argument for that function. I hope your program can fit this prodlog-function.
Dear Jan,
thank you for your answer.
I tried to use your suggested equation
X(t) = [S]0/Km * exp[ ([S]0 - vmax * t) / Km ]
but there something odd in it.
It can fit the data but I assume that the initial [S] (i.e., at t=0) should be equal to [S]0.
But if you solve the equation for t=0 you obtain:
X(0) = [S]0/Km * exp[ [S]0 / Km ] that is different from [S]0.
Where is the trick?
Gianluca
Oh sorry, I think I was not clear enough in my first reply.
X(t) is not the function you can fit. It is the argument for the Lambert-W function such that the inverse property
X(t) = W { X(t) } * exp( W { X(t) } )
is given.
When we set t = 0 we get
X(t=0) = [S]0/Km * exp( [S]0 / Km )
as you stated. This gives
W { X(t=0) } = [S]0 / Km
by comparison with the above equation. And for our original equation (See the first equation in my first reply):
[P]0 = [S]0 - Km * [S]0 / Km = [S]0 - [S]0 = 0
which is true for t = 0.
Just for your information: This also works for t = ∞ and any t between:
X(t=∞) = [S]0/Km * exp[([S]0 - vmax * ∞) / Km ] = [S]0/Km * 0 = 0
thus
W { X(t=∞) } = 0
thus
[P]∞ = [S]0 - Km * 0 = [S]0
for t = ∞, which is also true (for reactions where product and substrate have the same stoichiometry)!
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