Eadie–Hofstee diagram

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 often omitted, because although Haldane and Stern credited Woolf with the underlying equation, it was just one of the three linear transformations of the Michaelis–Menten equation that they initially introduced. However, Haldane indicated latter that Woolf had indeed found the three linear forms: "''In 1932, Dr. Kurt Stern published a German translation of my book "Enzymes", with numerous additions to the English text. On pp. 119-120, I described some graphical methods, stating that they were due to my friend Dr. Barnett Woolf.'' ..''Woolf pointed out that linear graphs are obtained when $v$ is plotted against $v x^$, $v^$ against $x^$, or $v^x$ against $x$, the first plot being most convenient unless inhibition is being studied''."

Derivation of the equation for the plot

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

:$v =$

in which $V$ is the rate at substrate saturation (when $a$ approaches infinity, or ''limiting rate'', and $K_\mathrm$ is the value of $a$ at half-saturation, i.e. for $v = 0.5V$, known as the ''Michaelis constant''. Eadie and Hofstee 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:

:$= +$ · $$

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

:$v = V - K_\mathrm$ · $$

which shows that a plot of $v$ against $v/a$ is a straight line with intercept $V$ on the ordinate, and slope $-K_\mathrm$ (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.

Attribution to Augustinsson

The plot is occasionally attributed to Augustinsson and referred to the Woolf–Augustinsson–Hofstee plot or simply the Augustinsson plot. However, although Haldane, Woolf or Eadie are not explicitly cited when Augustinsson introduces the $v$ versus $v/a$ equation, both the work of Haldane and of Eadie are cited at other places of his work and are listed in his bibliography.

Effect of experimental error

Experimental error is usually assumed to affect the rate $v$ and not the substrate concentration $a$, so $v$ is the dependent variable. As a result, both ordinate $v$ and abscissa $v/a$ 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 have compared the suitability of the various 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 $K_\mathrm$ and $V$, 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 $v$ vales, from 0 to $V$ 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.

See also

* Michaelis–Menten equation
* Lineweaver–Burk plot
* Hanes–Woolf plot

Footnotes and references

{{DEFAULTSORT:Eadie-Hofstee diagram
Diagrams
Enzyme kinetics

Biotechnology
Molecular biology