Hanes–Woolf plot

In biochemistry, a Hanes–Woolf plot, Hanes plot, or plot of ${\displaystyle a/v}$ against ${\displaystyle a}$ is a graphical representation of enzyme kinetics in which the ratio of the initial substrate concentration ${\displaystyle a}$ to the reaction velocity ${\displaystyle v}$ is plotted against ${\displaystyle a}$. It is based on the rearrangement of the Michaelis–Menten equation shown below:

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

Hanes plot of a/v against a for Michaelis–Menten kinetics

where ${\displaystyle K_{\mathrm {m} }}$ is the Michaelis constant and ${\displaystyle V}$ is the limiting rate.^[1]

J. B. S. Haldane stated, reiterating what he and K. G. Stern had written in their book,^[2] that this rearrangement was due to Barnet Woolf.^[3] However, it was just one of three transformations introduced by Woolf. It was first published by C. S. Hanes, though he did not use it as a plot.^[4] Hanes noted that the use of linear regression to determine kinetic parameters from this type of linear transformation generates the best fit between observed and calculated values of ${\displaystyle 1/v}$, rather than ${\displaystyle v}$.^[4]: 1415

Starting from the Michaelis–Menten equation:

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

we can take reciprocals of both sides of the equation to obtain the equation underlying the Lineweaver–Burk plot:

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

which can be multiplied on both sides by ${\displaystyle {a}}$ to give

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

Thus in the absence of experimental error data a plot of ${\displaystyle {a/v}}$ against ${\displaystyle {a}}$ yields a straight line of slope ${\displaystyle 1/V}$, an intercept on the ordinate of ${\displaystyle {K_{\mathrm {m} }/V}}$and an intercept on the abscissa of ${\displaystyle -K_{\mathrm {m} }}$.

Like other techniques that linearize the Michaelis–Menten equation, the Hanes–Woolf plot was used historically for rapid determination of the kinetic parameters ${\displaystyle K_{\mathrm {m} }}$, ${\displaystyle V}$ and ${\displaystyle K_{\mathrm {m} }/V}$, but it has been largely superseded by nonlinear regression methods that are significantly more accurate and no longer computationally inaccessible. It remains useful, however, as a means to present data graphically.

See also

• Michaelis–Menten kinetics
• Lineweaver–Burk plot
• Eadie–Hofstee plot
• Direct linear plot