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 Equation: Mixed-model inhibition

## Introduction

The mixed model is a general equation that includes competitive, uncompetitive and noncompetitive inhibition as special cases. The model has one more parameter than the others, and this parameter tells you about the mechanism of inhibition.

## Step by step

Create an XY data table. Enter substrate concentration into the X column, and enzyme activity into the Y columns. Each data set (Y column) represents data collected in the presence of a different concentration of inhibitor, starting at zero. Enter these concentrations into the column titles. Be sure to enter concentrations, not logarithms of concentration.

After entering data, click Analyze, choose nonlinear regression, choose the panel of enzyme kinetics equations, and choose Mixed model enzyme inhibition.

## Model

VmaxApp=Vmax/(1+I/(Alpha*Ki))

KmApp=Km*(1+I/Ki)/(1+I/(Alpha*Ki))

Y=VmaxApp*X/(KmApp + X)

The parameter I is the concentration of inhibitor, a value you enter into each column title. This is constrained to equal a data set constant.

The parameters Alpha, Vmax, Km and Ki are shared, so Prism fits one best-fit value for the entire set of data.

## Interpreting the parameters

Vmax is the maximum enzyme velocity absence of inhibitor, expressed in the same units as Y.

Km is the Michaelis-Menten constant, expressed in the same units as X. It describes the interaction of substrate and enzyme in the absence of inhibitor.

Ki is the inhibition constant, expressed in the same units as I, which you entered into the column titles.

Alpha determines mechanism. Its value determines the degree to which the binding of inhibitor changes the affinity of the enzyme for substrate. Its value is always greater than zero.

When Alpha=1, the inhibitor has equal affinitity for the enzyme and the enzyme-subtrate conmples. This is identical to noncompetitive inhibition.

When Alpha>1, the inhibitor preferentially binds to the free enzyme.

When Alpha is very large, binding is almost entirely to the free enzyme, and the mixed-model approaches competitive inhibition.

When Alpha<1, the inhibitor preferentially binds to the enzyme-substrate complex.

When Alpha is very small (but greater than zero), the inhibitor binds almost entirely to the enzyme-substrate complex and the mixed model becomes approaches an uncompetitive model.

## Terminology

Copeland suggests avoiding the term mixed model, because it is easy to get confused and think that refers to inhibition by a mixture or two or more drugs. Instead, he refers to the model on this page as noncompetitive. With this way of thinking, the case where alpha = 1.0 is simply a special case of noncompetitive inhibition.

## Reference

Equation 3.2 in: RA Copeland, Evaluation of Enzyme Inhibitors in Drug Discovery, Second edition, Wiley 2013. ISBN: 978-1-118-48813-3.