To calculate Ki from a Lineweaver-Burk plot, you need at least two sets of kinetic data: one without inhibitor and one (or more) with a known concentration of inhibitor. The specific calculation depends on the type of inhibition, because each type changes the plot in a different way. In competitive inhibition, you extract Ki from the change in slope. In uncompetitive inhibition, you extract it from the change in y-intercept. In mixed or non-competitive inhibition, both slope and intercept shift, and you can use either (or both) to solve for Ki.
What Ki Tells You
Ki, the inhibition constant, is the dissociation constant for the inhibitor binding to the enzyme. A smaller Ki means tighter binding and more potent inhibition. It’s reported in molar concentration units (M, mM, µM, etc.) and, unlike IC50, it’s an intrinsic property of the enzyme-inhibitor interaction that doesn’t depend on how much substrate you happened to use in your experiment. That’s what makes Ki the preferred value to report in most biochemical work.
Competitive Inhibition: Ki From the Slope
In competitive inhibition, the inhibitor binds the free enzyme and competes with substrate for the active site. On a Lineweaver-Burk plot, this produces lines that share the same y-intercept (1/Vmax is unchanged) but fan out with different slopes. The more inhibitor you add, the steeper the slope.
The Lineweaver-Burk equation for competitive inhibition is:
1/v = (Km/Vmax)(1 + [I]/Ki) · (1/[S]) + 1/Vmax
The slope of each line equals (Km/Vmax)(1 + [I]/Ki). Since you know the slope of the uninhibited line (Km/Vmax) and the slope of the inhibited line, you can solve for Ki directly:
slope(inhibited) / slope(uninhibited) = 1 + [I]/Ki
Rearranging:
Ki = [I] / (slope(inhibited)/slope(uninhibited) – 1)
You plug in the known inhibitor concentration [I], read the two slopes off your plot, and solve. If you collected data at multiple inhibitor concentrations, you can also make a secondary plot (called a replot) of slope versus [I]. That replot gives a straight line whose x-intercept equals -Ki.
Uncompetitive Inhibition: Ki From the Intercept
An uncompetitive inhibitor binds only the enzyme-substrate complex, not the free enzyme. On a Lineweaver-Burk plot, this produces parallel lines: the slope stays constant, but the y-intercept increases with inhibitor concentration. Both the apparent Vmax and the apparent Km decrease by the same factor.
The y-intercept for the inhibited reaction is:
1/Vmax(app) = (1/Vmax)(1 + [I]/Ki)
So you can solve for Ki the same way you did with the slope in competitive inhibition, but now using the intercepts:
Ki = [I] / (intercept(inhibited)/intercept(uninhibited) – 1)
If your lines aren’t truly parallel, you’re not looking at pure uncompetitive inhibition, and you should consider a mixed inhibition model instead.
Non-competitive Inhibition: Both Slope and Intercept Change
A pure non-competitive inhibitor binds the free enzyme and the enzyme-substrate complex with equal affinity. On a Lineweaver-Burk plot, the lines intersect on the x-axis, meaning Km stays the same but Vmax decreases. The y-intercept for the inhibited reaction follows:
1/Vmax(app) = (1/Vmax)(1 + [I]/Ki)
The calculation is identical in form to the uncompetitive case. Read the y-intercepts, plug in [I], and solve for Ki. The key difference is recognizing which type of inhibition you have: if the lines cross on the x-axis (same Km), it’s non-competitive. If the lines are parallel, it’s uncompetitive. If they cross on the y-axis (same Vmax), it’s competitive.
Mixed Inhibition: Two Constants to Find
Mixed inhibition is the most general case, where the inhibitor binds both the free enzyme and the enzyme-substrate complex but with different affinities. This produces lines that intersect to the left of the y-axis but not on either axis. You now have two inhibition constants to determine: Kic (for binding the free enzyme, extracted from the slope changes) and Kiu (for binding the enzyme-substrate complex, extracted from the intercept changes).
From the slopes, you find Kic using the same approach as competitive inhibition. From the intercepts, you find Kiu using the same approach as uncompetitive inhibition. The intersection point of the lines on the Lineweaver-Burk plot falls at a 1/[S] value equal to -Kic/Kiu, which serves as a quick visual check on your calculations.
Using Secondary Plots for Better Accuracy
Reading slopes and intercepts off a single Lineweaver-Burk plot introduces error, especially since the double-reciprocal transformation compresses data at high substrate concentrations and stretches it at low concentrations. Secondary plots (replots) give more reliable Ki values.
For competitive inhibition, plot the slope of each Lineweaver-Burk line (y-axis) against the corresponding inhibitor concentration (x-axis). You’ll get a straight line. The x-intercept of that line equals -Ki. For uncompetitive or non-competitive inhibition, do the same thing but plot the y-intercept of each Lineweaver-Burk line against [I]. Again, the x-intercept of that secondary line gives you -Ki.
These secondary plots require data at three or more inhibitor concentrations (including zero), which is why well-designed inhibition experiments always test multiple [I] values rather than just one.
A Worked Example
Suppose you’re studying a competitive inhibitor at a concentration of 10 µM. Your uninhibited Lineweaver-Burk line has a slope of 2.0 min/µM, and the inhibited line has a slope of 5.0 min/µM. Using the formula:
Ki = [I] / (slope(inhibited)/slope(uninhibited) – 1)
Ki = 10 µM / (5.0/2.0 – 1)
Ki = 10 µM / 1.5
Ki = 6.7 µM
If you had data at additional inhibitor concentrations, you would repeat this calculation for each, then average the results or use a secondary replot for a more robust estimate.
Ki Versus IC50
You may encounter IC50 values (the inhibitor concentration that cuts activity in half) rather than Ki. IC50 depends on the substrate concentration used in the experiment, so it’s less fundamental than Ki. For competitive inhibition, the two are related by the Cheng-Prusoff equation:
Ki = IC50 / (1 + [S]/Km)
This means IC50 is always larger than Ki when substrate is present. If you only have an IC50 and know Km and [S], you can convert to Ki. But if you already have Lineweaver-Burk data, calculating Ki directly from the plot is more straightforward and avoids the assumptions built into the Cheng-Prusoff conversion.