My belief the the competitive inhibition equation and the classic inhibition terms are toxic ideas which have stifled research is based on the expanded derivation of the inhibition term which demonstrates it is an inverted binding curve. For the competitive inhibition equation this inversion explains the required linear change in substrate affinity associated with this equation as the inversions of the MM equation produces a straight line which has been used to determine Km and Vmax for years.
I posted this recently as an answer to another question but I think its relevant enough to be of general interest to all researchers so here it is.
Stop worrying about competitive or noncompetitive inhibition both equations are flawed based on the failure of their derivations to distinguish between the effect the inhibitor has, and the binding equilibrium.
This is particularly a problem with the competitive inhibition equation where a failure to complete the derivation obscures the inhibition terms relationship with the inhibitor binding curve.
To complete the derivation
1+[I]/Ki must be multiplied by Ki,
to produce (Ki +[I])/Ki.
Next zero in the form of [I] - [I] must be added to the bottom to give you
(Ki +[I])/(Ki + [I] - [I]).
Next the whole term must be divided by Ki +[I].
This ends up producing a term that shows the binding curve for the inhibitor associating with the enzyme is
1/(1- ([I] / Ki + [I])
One portion in the denominator
([I] / Ki + [I])
Is the same binding curve used to describe all binding interactions, here describing the fraction of the enzyme population bound be inhibitor
When you subtract it from 1, the other part of the denominator, you are defining the portion of the enzyme population not bound by the inhibitor.
(1- ([I] / Ki + [I])
and when you have it as an inverted term multiplied by the Km you are essentially claiming the effect the competitive inhibition has on Km is not related to inhibitor binding but results from the portion of the enzyme population free of inhibitor.
This is a case of a faulty equation being generated through improper derivation and the failure to segregate the effect the inhibitor has on the enzyme from the equilibrium defining the formation of the enzyme inhibitor complex.
In noncompetitive inhibition this is not as much of a problem as dividing Vmax by 1+[I]/K means you are multiplying Vmax by (1- ([I] / Ki + [I]) giving you
Vmax- Vmax ([I] / Ki + [I])
This means when an enzyme is inhibited by the inhibitor it is shut off but once again this results from a failure in the derivation to acknowledge the effect the inhibitor has on the enzyme so this inhibition term is strictly based on the inhibitor and enzymes binding equilibrium. To fix this the second Vmax simply has to be changed to a delta term which allows the binding to change the rate according to what is experimentally observed rather than forcing the rate to zero.
Vmax- (Delta)Vmax ([I] / Ki + [I])
Now the silly idea that competitive inhibition is the only way to effect substrate affinity can be fixed as something the blocks the active site really has no ability to change how well the substrate binds to it. Changes in substrate affinity most likely result from altered active site geometry so you are talking about a shift between two substrate binding affinities. Therefore, the easiest way to do that is to define the change in substrate affinity the same way we did with the noncompetitive equation above
Km- (Delta)Km ([I] / Ki + [I])
And then use both terms in the MM equation to define any inhibition or activation you may come across. Or you can continue to search the literature for equations that define everything using separate equations which use inhibition terms that don't distinguish between binding equilibrium and effect.
As for your claim of data fitting the competitive inhibition equation I doubt that's what you have as on reanalysis I don't believe there has ever been evidence for an inhibitor forcing km linearly to infinity especially using some sort of jedi mind trick like influencing the interaction between your substrate and enzyme by dividing it by the fraction of uninhibited enzyme.
But if you still think you have data that fits the competitive inhibition equation, I would suggest global data fitting of the equation to your data and fitting all the classic inhibition equations and of course the one I just mentioned then compare the fits with the AIC and see what you get.
If you want you can use my peerJ article where I made some excel templates for global fitting to all of these equations and the ones that were derived when people started noticing how terrible they were like the partial competitive, partial noncompetitive, partial mixed noncompetitive etc.
Hey if you want the full derivation of the competitive inhibition equation and the equation that should replace it for describing modifiers of substrate affinity check out my video here
https://www.youtube.com/watch?v=hgWy0PIvksw
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