Lineweaver–Burk (Double-Reciprocal) Plot

Hans Lineweaver and Dean Burk published this in 1934, and it is important to understand what problem they were solving. Fitting a hyperbola to scattered data requires iterative nonlinear regression, which in 1934 meant days of hand computation. Fitting a straight line requires a ruler. Taking reciprocals of both sides of Michaelis–Menten turns the hyperbola into a line, and suddenly the two constants could be read off a sheet of graph paper in a few minutes. It was a genuine and useful invention.

The geometry is neat. Plot $1/v$ against $1/[S]$ and you get a line whose slope is $K_m/V_{max}$, whose intercept on the vertical axis is $1/V_{max}$, and whose intercept on the horizontal axis is $-1/K_m$. Three readings, two constants, and any two of the three give the third.

Now the honesty, and it is the reason this page exists as a separate one. The double-reciprocal transformation is a bad estimator, and it is bad in a specific and instructive way. Consider what taking a reciprocal does to your experimental error. The measurements at low substrate are the slowest rates, the ones closest to the noise floor of the assay, and therefore the least precise you own. Their reciprocals are the LARGEST numbers on the plot, landing far out on the right-hand side, where a point exerts enormous leverage on the fitted line. Meanwhile your best measurements — the fast rates near saturation, measured with the highest signal-to-noise — have small reciprocals and cluster tightly near the origin, where they barely influence the fit at all. An ordinary unweighted least-squares fit to this plot weights the data almost exactly backwards, and a 5% error on a single slow point can move the fitted $V_{max}$ by tens of percent.

There is a second structural complaint. $V_{max}$ is read at $1/[S] = 0$, which is infinite substrate — you obtain the number you most want by extrapolating off the left-hand edge of your data to a condition you can never create. And $K_m$ is read at a negative substrate concentration, off the other edge, in a region no experiment can occupy. Both are valid algebra and both are strange things to base a measurement on.

The correct modern method is nonlinear regression on the untransformed $v$ against $[S]$ data. Every statistics package does it, most spreadsheets do it, and it weights each point according to its actual precision instead of according to an accident of the transformation. If you are reporting kinetic constants in 2020s work, that is what you should be doing, and reviewers increasingly expect it. Eadie–Hofstee ($v$ against $v/[S]$) and Hanes–Woolf ($[S]/v$ against $[S]$) are both better-behaved linearisations than this one — Hanes–Woolf notably so, because it distributes the error far more evenly — but all three are inferior to fitting the curve directly.

So why is the double-reciprocal plot still drawn everywhere? Because it is legible, and legibility is worth something that accuracy cannot supply. A human eye is extremely good at judging whether points lie on a line and extremely poor at judging whether they lie on the right hyperbola. More importantly, the three classic inhibition mechanisms produce three visibly different pictures on this plot and on no other. A competitive inhibitor pivots the family of lines about a common vertical intercept: $V_{max}$ unchanged, $K_m$ apparently rising. A pure non-competitive inhibitor pivots them about a common horizontal intercept: $K_m$ unchanged, $V_{max}$ falling. An uncompetitive inhibitor, which binds only the enzyme–substrate complex, lowers both by the same factor and gives a family of parallel lines. You can diagnose a mechanism across a room. Use this plot to see the mechanism; do not use it to report the numbers.