Why the substrate inhibition model sometimes leads to "ambiguous" fits

The enzyme substrate inhibition model built into Prism is a standard model, not one we derived. For many data sets, it works fine. For some, Prism reports that the fit is ambiguous. 

Here is the equation as entered into Prism

It is easier to understand if rearranged.

Vmax controls the height of the peak (but doesn't equal the Y value at the peak).

When X values are small, the last term approaches zero so the early parts of the curve (small X values) are determined by the value of Km.

Conversely, when X values are large, the first term approaches zero, so the later parts of the curve (higher X values) are determined by the value of Ki.

The middle part of the curve is determined by both Ki and Km.  

There are two reasons why the fit would result in an "ambiguous" result. 

• There are not data points at a wide enough range of X values. There need to be X values less than Km and greater than Ki, ideally lots of X values. If you only have data in the middle of the curve, there simply is no information to separately determine Km and Ki. 
• The shape of the curve, while it looks like a substrate inhibition curve, simply isn't quite right. In other words, the data really don't comply with the substrate inhibition model.