Notes on Non-Linear Regression

The Excel file that we are making for non-linear regression works by applying your data to the Michaelis-Menten function. The function is shown in the upper-left corner of the spreadsheet. There are two parameters for this function: Km and Vmax. The initial values for these parameters are found in a box with red borders. You may set these to values of interest to you and the spreadsheet will calculate the fit of your data to those parameters. Your data go in the yellow-filled cells, and the green numbers to the right of that are the calculated values for the non-linear regression fitting. An r2 (coefficient of correlation) value is given to indicate the goodness of fit of your data to the function with those initial parameters. The closer this r2 comes to 1.0 the better is your fit. A 95% fit (0.95) is desirable but even a 30% fit can be meaningful.

You could insert your data and manually try some values for Km and Vmax to see how well the fit is, and by trying different numbers you could arrive at something close. Sometimes this is desirable...and, if your computer has not got a full installation of Excel with Solver, you have to use that pathway exclusively.

Look at the curve shown and decide which of the two parameters seem the most-off the mark. The Vmax is a rate faster than anything you measured, so if you have rates above Vmax, obviously the Vmax value needs to be increased! The Km value should be near a place that would give half of Vmax on the function. Change the one necessary parameter until you get a good r2 fit.

In general a competitive inhibitor increases the Km, a non-competitive inhibitor decreases the Vmax, and a co-participant increases the Vmax. Starting with the control (catechol dilution series) Km and Vmax as starting points, try all three of these ideas with your analogs to see what fits best.

However, Excel has an add-in function called the Solver that can try Km and Vmax values for you and arrive at a best fit which is achieved when SSresid is minimized. This add-in function must be installed with Excel for the Solver to show up in the Tools menu. If your installation does not apear to have this choice in the Tools menu, then you need to go back to the original Excel CD and install the Solver from the Data Analysis Value Pack or download it from its corporate website.

The Solver will start with your initial values and try others over a range. It keeps track of the values for Km and Vmax that give the minimum SSresid (residual error). When it has found those values, it can display them on the graph in the spreadsheet. If your data are "nice" the solver will give you good fits to reasonable values.

What if my data aren't so nice?

The Solver is a great computing tool to zero in on a certain target, but it is, alas, a dumb computer. With certain data sets it will ignore excellent fits striving for better, ultimately obtaining the least residual error with nonsense values for Km and Vmax. Obviously, if you scale up those two parameters to ridiculously high values, all the data fit close together near the origin...giving an outstanding fit to a nonsense function!

So if your data include some outliers, using Solver can be problematic. What I recommend is this. Use the Solver to arrive at a nice fit for your control dilution series for Catechol alone. This will hopefully give you a large r2 value and therefore solid Vmax and Km parameters to go with those data. Then load in your problematic data set...the set that has a dilution series of catechol poisoned by the structural analog. Rather than using the Solver, put your Km and Vmax values from the unpoisoned Catechol fit. The R2 value next to your poisoned rates will indicate the goodness of fit to your "control" parameters. If the fit is good then this inhibitor is not inhibiting at all; it is innocuous! If the fit is poor, then perhaps you need to see what needs fixing.

Another approach is to let Solver do all the graphs for all of the analogs...then look for a Vmax that might explain them all. Set that Vmax, then in the Solver dialog box, force Solver to change ONLY the Km. This will arrive at a best fit Km for the fixed Vmax. If the r2 value is good enough, then you can compare the Kms obtained to see if any of your isomers are indeed competitive inhibitors.

As always, when there is doubt or problem...visit your instructor for assistance! This will be most helpful to you if you arrange for that well in advance of any due-date for output!

 



This page © Ross E. Koning 1994.



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