# Uncompetitive inhibition

An uncompetitive inhibitor interacts with the enzyme-substrate complex, but not with the enzyme alone. For uncompetitive inhibition, the double-reciprocal equation is as follows:

1/V0 = α’/Vmax + Km/Vmax • 1/[S] the reciprocal of V 0 equals alpha prime divided by V max plus K m divided by V max times the reciprocal of the substrate's concentration

where α’ = 1+[I]/K’I alpha prime equals one plus the inhibitor's concentration divided by K I

This equation shows that a double-reciprocal plot of enzyme kinetic data with varying concentrations of an uncompetitive inhibitor should give a straight line with varying y-intercepts, but with the same slopes (Figure 1). In enzyme kinetic assays with an uncompetitive inhibitor, the apparent Km and Vmax will change with increasing inhibitor concentrations [1]. K m and V max will change with increasing inhibitor concentrations.

Figure 1: a) Lineweaver-Burk plot showing the uncompetitive inhibition. b) y-intercepts of each linear regression plotted against the inhibitor concentration.

## Calculating K’I, Km and Vmax K I, K m, and V max

When working with an uncompetitive inhibitor, no parameters can be calculated from the initial Lineweaver-Burk plots. To calculate the different parameters, the "y-intercepts" of these plots must be plotted against the inhibitor concentration:

y-intersectuncompettive = α’/Vmax = 1/Vmax + 1/Vmax • 1/KI • [I] the y-intersect of an uncompetitive inhibition equals alpha prime divided by V max, which also equals the reciprocal of V max plus the reciprocal of V max times the K I times the inhibitor's concentration

This plot should, therefore, result in a straight line with the intercept 1/Vmax and slope 1 / (Vmax • KI ). KI can, therefore, be calculated by dividing the slope of this straight line with the intercept. Vmax can also be calculated from this fit, by taking the reciprocal to the y-intercept. Because the initial Lineweaver-Burk plots had the same slopes, Km / Vmax, Km can now be calculated by multiplying these slopes with Vmax obtained from the plot of y-intercepts against inhibitor concentrations. the reciprocal of V max and slope the reciprocal of V max times the K I. K I can, therefore, be calculated by dividing the slope of this straight line with the intercept. V max can also be calculated from this fit, by taking the reciprocal to the y-intercept. Because the initial Lineweaver-Burk plots had the same slopes, K m divided by V max, so K m can now be calculated by multiplying these slopes with V max obtained from the plot of y-intercepts against inhibitor concentrations.

## Steps of calculating the kinetic parameters when using an uncompetitive inhibitor

• Prepare Lineweaver-Burk plots of the kinetic data and fit the data using linear regression (1 fit per inhibitor concentration).

• Plot the y-intercepts of each of these fits as a function of the inhibitor concentration.

• To calculate Vmax, V max take the reciprocal of the y-intercept of this plot.

• To calculate KI, K I divide the y-intercept of this plot with the slope.

• To calculate Km, go back to the first Lineweaver-Burk plots, take the slope from 1 of these plots (the slope should be the same, or close, for all the plots), and multiply it with Vmax. K m, go back to the first Lineweaver-Burk plots, take the slope from 1 of these plots since the slope should be the same, or close, for all the plots, and multiply it with V max.

## References

1. Lehninger, Albert L.; Nelson, David L.; Cox, Michael M. (2008). Principles of Biochemistry (5th ed.). New York, NY: W.H. Freeman and Company. ISBN 978-0-7167-7108-1.

Theory overview