​A student was asked to plot a graph representing enzyme kinetic data for initial velocity, v₀, and substrate concentration, [S] using any of the equations given below. The student used an equation for which neither X-axis nor Y-axis had independent variables. Which one of the following equations might the student have used?​

Correct option is C

The basic Michaelis-Menten equation:

​$$v_0 = \frac{V_{\text{max}} [S]}{K_m + [S]}$$

where:

V₀ = Initial velocity ( Dependent on [S])

[S] : Substrate concentration (Independent Variable)

V_max and K_m: Constants

In most plots:

Either [S] or $$\frac{1}{[S]}$$ is the independent variable (x- axis)

and V₀ and $$\frac{1}{[V_0]}$$ is plotted on y- axis

We are looking for an equation that does not plot a true independent variable on either axis—in other words, both axes depend on experimental data or a mixture of variables.

Option Analysis:

1. Option 1: $$\frac{1}{v_0} = \left( \frac{K_m / V_{\text{max}}}{[S]} \right) + \left( \frac{1}{V_{\text{max}}} \right)$$

   ​This is the Lineweaver-Burk equation: plots $$\frac{1}{V_0}$$​ vs. $1/[S]$, both are derived from experimental variables, but [S] is independent.

   Option 2: $$\frac{[S]}{v_0} = \frac{[S]}{V_{\text{max}}} + \frac{K_m}{V_{\text{max}}}$$

   ​This is the Eadie-Hofstee equation: plots [S]/v0[S]/v_0$$\frac{[S]}{V_0}$$​ vs. $[S]$, so [S] is still on an axis, i.e., independent.

   Option 3: $$\frac{v_0}{[S]} = \left( \frac{V_{\text{max}}}{K_m} \right) - \frac{v_0}{K_m}$$

   ​This equation involves both v0v_0$$V_0$$​ and $[S]$ on both sides — neither axis is strictly independent.
   → This is a rearranged form, similar to the Woolf-Augustinsson-Hofstee plot, which plots $$V_0$$​ vs. $$\frac{V_0}{[S]}$$

   ​Hence, both axes are combinations of dependent variables.

   Option 4: $$v_0 = \frac{V_{\text{max}} [S]}{K_m + [S]}$$
   

Original Michaelis-Menten form. Clearly, [S] is the independent variable.

Correct Answer: Option 3

It uses a plot where neither axis (neither v0v_0$$V_0$$​ nor $$\frac{V_0}{[S]}$$​ is an independent variable, as both are derived from experimental data.

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