Lineweaver-Burk plots
In this example, we'll make a combination graph commonly used to characterize enzyme activity: a curve of initial velocity vs. substrate concentration, sometimes referred to as a Michaelis-Menten plot, with an inset Lineweaver-Burk plot.
Lineweaver-Burk analysis is one method of linearizing substrate-velocity data so as to determine the kinetic constants Km and Vmax. One creates a secondary, reciprocal plot: 1/velocity vs. 1/[substrate].
Now that programs such as Prism easily do nonlinear regression, the best way to determine Km and Vmax is to fit a hyperbola directly to the substrate-velocity data. Yet the Lineweaver-Burk plot continues to be a useful visual tool, particularly because of its characteristic shifts in the presence of various types of inhibitors. So we'll create a Lineweaver-Burk plot with data points derived from double-reciprocal transformation, but we'll superimpose a line based upon nonlinear regression analysis, so that it reflects the best possible estimates of Kd and Bmax.
Enter the Enzyme Activity Data
When you launch Prism, the Welcome to GraphPad Prism dialog appears. Choose to Create an XY data table.
In the lower half of the Welcome dialog, tell Prism how to format the first data table. For the X values, select Numbers. For the Y columns, choose the format that fits your data; to keep data entry in this example short, we'll use Enter and plot a single Y value for each point.
When you click OK, Prism displays the new table.
For each point on the substrate-velocity curve, enter the concentration of substrate into the X column and the value for initial velocity into the Y column. The units are up to you — use whatever is meaningful. Just remember that Prism will graph the data and compute the binding parameters in whatever units you've used on your table. Place labels over the X and Y columns as a reminder:
Here are the data for our example:
Suppose that this data is uncorrected for a constant background value of 3.4, which you determined from your assay measurement in the absence of substrate (row 1). You can use Prism to make the background correction (subtract 3.4 from each Y value). Click Analyze, then choose Remove baseline and column math from the Transform, Normalize... menu.
In the Parameters: Remove Baseline and Column Math dialog, tell Prism where the baseline is (in this example, the mean of the first 1 rows; note that you also have the option to put "baseline" values in the bottom row(s) and that, when appropriate, you may exclude those rows from the analysis). Choose the option Difference: Value - Baseline under Calculation and the option to Create a new graph of the results.
The corrected data are produced on a Results sheet as follows:
In this example, we did a baseline correction by subtracting the Y value in the first row from all the other Y values. Prism gives you other options — you can subtract a constant value specified in a parameters dialog rather than on the data table, or you can subtract a different number on each row.
Since we requested it as part of the "Remove baseline" analysis, Prism makes a new graph of the corrected data. To view the new graph, choose the last (most recently created) graph listed in the "Graphs" section of the Navigator panel.
The graphs in this step-by-step example include some other formatting changes not covered here—to learn more, consult the Prism User's Guide.
Fit a Curve to the Substrate-Velocity Data to Compute Km and Vmax
With the baseline-corrected plot (the last graph listed in the "Graphs" section of the Navigator panel) in view, click the Analyze button. Choose the XY analyses category, and select Nonlinear regression (curve fit). Click OK.
In the Parameters: Nonlinear regression dialog box, choose Enzyme kinetics - Velocity as a function of substrate. From the options, select Michaelis-Menten. Click OK to exit the parameters set-up and complete the curve fit. The curve should appear on the graph.
A fitted curve is treated by Prism as a separate data set (different from the underlying points). In this instance, we started the curve-fit analysis from the graph, so Prism knows where we want the curve to go. At other times, that may not be the case. When you do a curve fit and get a corresponding Results sheet, but can find no curve on your graph, click Change... Data Sets on Graph. The resulting dialog shows which data sets are included on the graph. Click Add to open the Add Data Sets to Graph dialog, from which you can select the appropriate curve and add it to the graph.
Click the Results section, where Prism displays the best-fit values for the kinetic constants.
If your results don't agree with those shown here, check to be sure that you did the curve fit starting from the baseline-corrected Michaelis-Menten graph.
The units for Vmax are the units that you used when entering Y values on the original data table. The units for Km are the units you used when entering the X values.
Display Kinetic Constants on the Graph
If you wish, you can paste cells from this results table to your graph, so that the kinetic constants will be displayed there and will also be linked to the results sheet. That way, if you change the saturation binding data later, Vmax and Km will be automatically updated both on the results sheet and on the graph. If you don't like the way the pasted data are displayed, you can change that. Suppose that you wish to paste the data, but make your own labels and customize the display. Select a cell on the results sheet containing the value you want to paste. Choose Edit... Copy. Switch to the graph, and choose Edit.. Paste Table. Double-click on the embedded table to bring up formatting options.
Here is a finished example:
At this point, you may wish to change some of your original activity data (data table) and observe how the values of Vmax and Km change automatically on the results sheet and on the graph.
Prepare the Lineweaver-Burk Transforms
From the Results sheet containing the baseline-corrected substrate-velocity data, click Analyze, then select Transform, Normalize... and Transform.
In the next dialog, Parameters: Transform, choose to transform both X and Y, replacing them with 1/X and 1/Y, respectively. Here are the settings for the transforms:
The transformed data are shown on a new results sheet (the original data are unchanged in the "Data tables" section of your project). Prism will carry over the original column titles, which will now be incorrect, so you can edit them if you wish:
Notice that the top row of the results table is blank. Prism leaves individual cells blank when it has been asked to perform an illegal math operation, in this case to evaluate the quantity 1/0, in that cell
Complete the Lineweaver-Burk Plot
The default Lineweaver-Burk plot appears as the most recently generated graph (now the last graph listed in the Navigator panel). You will notice that there is no line superimposed over the points. We recommend that you add a line to your graph that accurately reflects the Kd and Bmax values that we found using nonlinear regression. This will be a straight line running from the X intercept to the point predicted by the nonlinear regression analysis for the lowest non-zero substrate concentration (i.e., the highest value of 1/[s]). Referring to the results sheet for your nonlinear regression analysis and the original substrate-velocity data table, note that the coordinates for the X-axis intercept are
X = -1/Km = -1/22.27 = -0.0449
Y = 0
and the coordinates for the upper-right end of the line are
X = 1/[Smin] = 0.2
Y = (1/Vmax)(1+Km/[Smin]) = (1/52.21)(1+22.27/5) = 0.1045
Switch to the table section of your project, choose New Table, and create a new XY data table. Fill in the table using the coordinates above:
Note the name of this data table. You may wish to rename it to something more meaningful, such as "L-B line data". Now switch back to the Lineweaver-Burk plot (the graph we created in the previous section). Double-click on the graph to bring up the Format Graph dialog, then click on the Data Sets on Graph tab and click Add... then select the correct data set from the list in the next dialog box.
Click OK to complete the process.
This results in the Y and X intercepts being plotted as points. Format this data set to make the points invisible but to connect them with a line: Double-click on the graph to bring up the Format Graph dialog. Adjust the options as seen below to remove symbols and add a connecting straight line.
Here is the graph, with axis titles edited:
Create a Composite Graph (Layout)
Switch to the Layout section of your project, and click New Layout. Choose the Landscape orientation and the Inset arrangement:
When you click OK, you'll see a layout with two placeholders. Double-click on each in turn, assigning your saturation binding plot to the larger placeholder, and your Lineweaver-Burk plot to the smaller placeholder. Now you can then individually select, then move and resize, each graph so that the two fit together nicely. Alternatively, you can drag and drop your graphs to the appropriate locations.
Note that as you resize the Lineweaver-Burk plot, Prism does a certain amount of reproportioning of the graph-adjusting font sizes and line thicknesses — as appropriate. But you will probably want to make some modifications of your own to further improve its readability.
To make any modifications, switch to the appropriate graph sheets — don't try to make them on the layout. After you make each change, switch back to the layout and check the result there.
Keywords: michaelis menten lineweaver burk plot