In biochemistry, an Eadie–Hofstee diagram (more usually called an Eadie–Hofstee plot) is a graphical representation of the Michaelis–Menten equation in enzyme kinetics. It has been known by various different names, including Eadie plot, Hofstee plot and Augustinsson plot. Attribution to Woolf is misleading, because although Haldane and Stern[1] credited Woolf with the underlying equation, it was just one of the three linear transformations of the Michaelis–Menten equation that he derived.

## Derivation of the equation for the plot

The simplest equation for the rate ${\displaystyle v}$ of an enzyme-catalysed reaction as a function of the substrate concentration ${\displaystyle a}$ is the Michaelis-Menten equation, which can be written as follows:

${\displaystyle v={{Va} \over {K_{\mathrm {m} }+a}}}$

in which ${\displaystyle V}$ is the rate at substrate saturation (when ${\displaystyle a}$ approaches infinity, or limiting rate, and ${\displaystyle K_{\mathrm {m} }}$ is the value of ${\displaystyle a}$ at half-saturation, i.e. for ${\displaystyle v=0.5V}$, known as the Michaelis constant. Eadie[2] and Hofstee[3] independently transformed this into straight-line relationships, as follows: Taking reciprocals of both sides of the equation gives the equation underlying the Lineweaver–Burk plot:

${\displaystyle {1 \over v}={1 \over V}+{K_{\mathrm {m} } \over V}}$ · ${\displaystyle {1 \over a}}$

This can be rearranged to express a different straight-line relationship:

${\displaystyle v=V-K_{\mathrm {m} }}$ · ${\displaystyle {v \over a}}$

which shows that a plot of ${\displaystyle v}$ against ${\displaystyle v/a}$ is a straight line with intercept ${\displaystyle V}$ on the ordinate, and slope ${\displaystyle -K_{\mathrm {m} }}$ (Hofstee plot). In the Eadie plot the axes are reversed, but the principle is the same. These plots are kinetic versions of the Scatchard plot used in ligand-binding experiments.[4]

## Augustinsson plot

The plot is occasionally attributed to Augustinsson,[5] but the paper in question is not listed by Web of Science or at the journal web site, and appears to be unobtainable. It is impossible to know, therefore, whether this was an independent derivation or a citation of Eadie's paper (given that both authors worked on cholinesterases).

## Effect of experimental error

Experimental error is usually assumed to affect the rate ${\displaystyle v}$ and not the substrate concentration ${\displaystyle a}$, so ${\displaystyle v}$ is the dependent variable.[6] As a result, both ordinate ${\displaystyle v}$ and abscissa ${\displaystyle v/a}$ are subject to experimental error, and so the deviations that occur due to error are not parallel with the ordinate axis but towards or away from the origin. As long as the plot is used for illustrating an analysis rather than for estimating the parameters, that matters very little. Regardless of these considerations various authors[7][8][9] have compared the suitability of the various plots plots for displaying and analysing data.

## Use for estimating parameters

Like other straight-line forms of the Michaelis–Menten equation, the Eadie–Hofstee plot was used historically for rapid evaluation of the parameters ${\displaystyle K_{\mathrm {m} }}$ and ${\displaystyle V}$, but has been largely superseded by nonlinear regression methods that are significantly more accurate and no longer computationally inaccessible.

## Making faults in experimental design visible

As the ordinate scale spans the entire range of theoretically possible ${\displaystyle v}$ vales, from 0 to ${\displaystyle V}$ one can see at a glance at an Eadie–Hofstee plot how well the experimental design fills the theoretical design space, and the plot makes it impossible to hide poor design. By contrast, the other well known straight-line plots make it easy to choose scales that imply that the design is better than it is.

## Footnotes and references

1. ^ Haldane, J B S; Stern, K G (1932). Allgemeine Chemie der Enzyme. Dresden and Leipzig: Steinkopff. pp. 119–120.
2. ^ Eadie, GS (1942). "The Inhibition of Cholinesterase by Physostigmine and Prostigmine". J. Biol. Chem. 146: 85–93.
3. ^ Hofstee, BHJ (1959). "Non-Inverted Versus Inverted Plots in Enzyme Kinetics". Nature. 184 (4695): 1296–1298. doi:10.1038/1841296b0.
4. ^ Srinivasan, Bharath (2020-10-08). "Explicit Treatment of Non Michaelis-Menten and Atypical Kinetics in Early Drug Discovery". dx.doi.org. Retrieved 2021-04-19.
5. ^ Augustinsson, K. B. (1948). "(title unknown)". Acta Physiol. Scand. 15 Supplement: 52.
6. ^ This is likely to be true, at least approximately, though it is probably optimistic to think that ${\displaystyle a}$ is known exactly.
7. ^ Dowd, JE; Riggs, DS (1965). "A Comparison of Estimates of Michaelis–Menten Kinetic Constants from Various Linear Transformations". J. Biol. Chem. 240: 863–869. PMID 14275146. }}
8. ^ Atkins, GL; Nimmo, IA (1975). "A comparison of seven methods for fitting the Michaelis–Menten equation". Biochem. J. 149 (3): 775–777. doi:10.1042/bj1490775. PMC 1165686.
9. ^ Cornish-Bowden, A. Fundamentals of Enzyme Kinetics (4th ed.). Weinheim, Germany: Wiley-Blackwell. pp. 51–53. ISBN 978-3-527-33074-4.