Chemistry:Secondary plot (kinetics)

In enzyme kinetics, a secondary plot uses the intercept or slope from several Lineweaver–Burk plots to find additional kinetic constants. For example, when a set of v by [S] curves from an enzyme with a ping–pong mechanism (varying substrate A, fixed substrate B) are plotted in a Lineweaver–Burk plot, a set of parallel lines will be produced.

The following Michaelis–Menten equation relates the initial reaction rate v₀ to the substrate concentrations [A] and [B]:

[math]\displaystyle{ \begin{align} \frac{1}{v_0} &= \frac{ K_M^A}{v_\max {[}A{]}}+\frac{ K_M^B}{v_\max {[}B{]}}+\frac{1}{v_\max} \end{align} }[/math]

The y-intercept of this equation is equal to the following:

[math]\displaystyle{ \begin{align} \mbox{y-intercept} = \frac{ K_M^B}{v_\max {[}B{]}}+\frac{1}{v_\max} \end{align} }[/math]

The y-intercept is determined at several different fixed concentrations of substrate B (and varying substrate A). The y-intercept values are then plotted versus 1/[B] to determine the Michaelis constant for substrate B, [math]\displaystyle{ K_M^B }[/math], as shown in the Figure to the right. The slope is equal to [math]\displaystyle{ K_M^B }[/math] divided by [math]\displaystyle{ v_\max }[/math] and the intercept is equal to 1 over [math]\displaystyle{ v_\max }[/math].

Secondary Plot of enzyme system Horseradish Peroxidase and o-Phenylenediamine (with hydrogen peroxide as the second substrate)

Secondary plot in inhibition studies

A secondary plot may also be used to find a specific inhibition constant, K_I.

For a competitive enzyme inhibitor, the apparent Michaelis constant is equal to the following:

[math]\displaystyle{ \begin{align} \mbox{apparent } K_m=K_m\times \left(1+\frac{[I]}{K_I}\right) \end{align} }[/math]

The slope of the Lineweaver-Burk plot is therefore equal to:

[math]\displaystyle{ \begin{align} \mbox{slope} =\frac{K_m}{v_\max}\times \left(1+\frac{[I]}{K_I}\right) \end{align} }[/math]

If one creates a secondary plot consisting of the slope values from several Lineweaver-Burk plots of varying inhibitor concentration [I], the competitive inhbition constant may be found. The slope of the secondary plot divided by the intercept is equal to 1/K_I. This method allows one to find the K_I constant, even when the Michaelis constant and v_max values are not known.