(Integrates with Chapter \(13 .\) ) Draw both Lineweaver-Burk plots and Hanes- Woolf plots for an MWC allosteric enzyme system, showing separate curves for the kinetic response in (a) the absence of any effectors, (b) the presence of allosteric activator \(A,\) and \((c)\) the presence of allosteric inhibitor I.

The reaction velocity of enzymes obey Michaelis-Menten kinetics equation, which is given by \( v = \frac{{V_{max}[S]}}{{K_m + [S]}} \), where v is the velocity, \( V_{max} \) is the maximum speed, [S] is the substrate concentration and \( K_m \) is the Michaelis constant. The Lineweaver-Burk plot represents the double reciprocal of this Michaelis-Menten equation, \( \frac{1}{v} = \frac{K_m}{V_{max}}. \frac{1}{[S]} + \frac{1}{V_{max}} \), plotting \( \frac{1}{v} \) (reciprocal velocity) against \( \frac{1}{[S]} \) (reciprocal substrate concentration). The Hanes-Woolf plot is another form of the same equation rearranged to \( \frac{[S]}{v} = K_m . \frac{1}{V_{max}} + \frac{[S]}{V_{max}} \), plotting \( \frac{[S]}{v} \) against [S].

On both types of plots, a straight line is expected under normal conditions (in absence of effectors). When an allosteric activator A is present, the enzymes have higher affinities for the substrates. This means lower \( K_m \) values, which in turn means higher \( \frac{1}{[S]} \) and lower \( \frac{[S]}{v} \) values at the same substrate concentrations, resulting in new lines with steeper slopes on both graphs. When an allosteric inhibitor I is present, the enzymes have lower affinities, indicating higher \( K_m \) values, resulting in new lines with less steep slopes.

Analysis of the slopes and intercepts of the created plots gives information about the kinetic parameters of the enzyme, and how these are affected by regulation from the allosteric effectors. The slope of the line represents \( K_m/V_{max} \), where an increased slope indicates an increased \( K_m \) or reduced \( V_{max} \) or both due to inhibiting effects, while a decreased slope indicates a reduced \( K_m \) or increased \( V_{max} \) or both due to activating effects. The intersect of the line with the y-axis represents \( 1/V_{max} \), where a higher intersect corresponds to a lower \( V_{max} \), and vice versa. This allows for quantitative analysis of the effect of effectors on enzyme activity.