There are three factors involved in the activity, the substrate acting as a substrate, the substrate acting as an inhibitor and another inhibitor for which you want to measure the Ki.
One way to do it is to measure the dissociation constant of the inhibitor-enzyme complex by some biophysical method. In most cases, the dissociation constant Kd is the same as the Ki.
The other way is by steady-state kinetics, which is more complicated.
For the kinetics approach, the first thing you need to know is the kinetic rate equation for the enzyme reaction, including terms in the denominator for both substrate inhibition and inhibition by the other compound. For information about these equations, try Segel's or Leskovac's books on steady-state kinetics.
https://www.amazon.com/Enzyme-Kinetics-Behavior-Equilibrium-Steady-State/dp/0471303097
https://link.springer.com/book/10.1007/b100340
There are two ways to use this information.
1. Derive the Cheng-Prusoff relationship (IC50/Ki) algebraically from the kinetic rate equation, then measure the IC50 and calculate the Ki. In order to do this calculation, you need to have first measured the kinetic constants for the substrates and the Vmax.
2. Perform a global steady state kinetic analysis by varying substrate and inhibitor concentrations and fit all the kinetic constants including the Ki.
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