It is the summer break in academia and we are either taking our well earned holidays or preparing material for the next session. A publisher has sent us a bunch of text books to look at - they hope we like them and would recommend them to the students for our courses.
Disclaimer: Whilst we are provided with the textbooks for no personal cost, there are no inducements to promote a particular book or publisher. Each book stands (or falls) on it's own merits.
I have an interest in improving the numeracy and basic data literacy amongst our students, so one of the titles appealed to me. It covered basic arithmetic though various applications in the lab. And one of those specific applications is Enzyme Kinetics.
The Michaelis-Menten equation is one of the foundations of kinetic analysis and the results can be represented in various ways. One of the classical method was developed by Lineweaver and Burk in 1934. It is superficially attractive in that the data is transformed from a hyperbolic curve to a straight line but suffers from reciprocal scaling of the errors. As it is a double reciprocal plot (plotting the inverse of one value against the inverse of the other) small errors in measurement at small values in the test tube become huge errors at huge values in the analysis. The plot has many uses for visualising the data though - identification of mechanism of inhibition is one of the classical examples given in introductory undergraduate lectures.
The problem comes when the plot is used for the determination of the constants (Vmax and Km) rather than their visualisation. Because of the reciprocal scaling of the errors, the estimates can be extremely unsafe (how much so we will get to in a moment). This was recognised by Lineweaver and Burk but this is typically left as an unquantified warning to students to be careful about their errors.
Since 1934 there have been other representations of the Michaelis-Menten equation that transform the data to linear forms. The Eadie-Hoftzee and Haynes-Woolf plots both provide linear transformations though interpretation and extraction of the parameters is marginally more complex than a simple reciprocal of the value of the intercepts on the x and y axes.
More recently the advent of computational methods of direct fitting to the equation removes the issue of error scaling completely and there is no real excuse for not using the direct method in any serious experimental study.
I wanted to see just how well each method would be in reconstructing correctly the correct parameters so constructed a small simulation. Taking a fixed Km and Vmax I can calculate the expected values for V at given substrate concentrations. Needless to say with ideal data every method retrieves the expected values perfectly.
In the lab, however, the world is not perfect. I therfore add fixed errors corresponding to measurement errors as one might find on reading a typical UV/Vis spectrometer, and proportional errors such as might be expected from pipetting. I repeat each experiment 10 times and calculate mean/SD for the parameters. The calculations for the Lineweaver-Burk plot are performed with no error weighting, such as might be done in a typical undergrad classroom.
The results are clear. Lineweaver-Burk is giving extremely poor results - so much so that one would never consider it for any analysis.
The simulation can be found here (PDF) (R Source [latex]) if you want to work through it yourself. I recommend the RStudio IDE and you will need appropriate software for Sweave/Latex.
So back to the book. They provide a suitable treatment of the Michaelis-Menten equation but the only experimental treatment for determining the kinetic constants is the Lineweaver-Burk plot, with a cursory half sentence hinting that it may not be suitable. There is no consideration of alternative approaches. It is a pretty damning effort - the book is otherwise quite approachable and could otherwise be recommended, but serious errors where we have to correct the textbooks render it unacceptable. I'm not going to name the book or publisher, but will be contacting the authors directly.
