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I am learning about why it is important to measure Km and Vmax for each experimental setup because measuring the "apparent" Km and Vmax includes enzyme inhibitions of which one might not be aware.

However, what if there are entire reactions going on that are consuming the product or substrate that are not known? Are these accurately aggregated into the Km and Vmax measurements?

Here are some of the links that I read through:

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Have you heard of something known as "Occam's Razor" ?

It says when you have multiple possible explanations/hypotheses then select the one which is simplest (i.e least number of assumptions)

Same with mathematical models. Chemical kinetics models usually assume first order unless there is some evidence against it. Similarly, for enzyme kinetics, as long as you don't think there is something anomalous going on, use Michaelis-Menten model.

What if there are entire reactions going on that are consuming the product or substrate that are not known

Many physiological enzyme catalysed reaction are not Michaelis-Menten. If you happen to have an insight about it or if the simulations dont match the experimental data, you can revise the model.

However, Michaelis-Menten model for for competitive inhibition exists, as you yourself indicate in the link.

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As Arthur Kornberg said: "Don't waste clean thinking on dirty enzymes."

Discovering an assay for a biological event in a cell-free extract opens the way to its molecular resolution and reconstitution (Commandment I). Trying to devise a mechanism with a crude extract, even with ingenious experiments, is generally a waste of effort. An extract is too dirty; purification is the only way to go.

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The answer is no really, but some variants might allow you to study inhibition.

Michaelis Menten kinetics are experiments which try to characterize the catalysis characteristics of a reaction but to do so the numbers are obtained at concentrations of substrate that you dont find in a cell.

Vmax is defined as the maximum rate at which the reaction is run in question. To measure this the substrate concentration is also maximal - the binding sites of enzymes must be fully occupied during the entire experiment. So by definition there is no competitive reaction possible here.

Km is a binding constant obtained where the reaction rate is found to be half Vmax. This definition is free of any competing reactions. You can use Michaelis Menten kinetics to study competitive inhibition though and by comparing a reaction rate and comparing it to the reaction rate (called Vo) you can derive a Ki which gives you how much your inhibitor has affected substrate binding.

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