(SEPT 2022-1)
34. A student was asked to plot a graph representing enzyme kinetic data for initial velocity, v_O , 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 equation might the student have used?
(1) 1/ v_O = (K_m/V_max) 1/[S]+ 1/V_max)
(2) [S]/v_O = [S]/V_maxx + (K_m /V_max)
(3) v_O / [S] = (V_max / K_m) – v_O / Km
(4) v_O = V max [S] / K_m + [S]

 

The correct answer is (3) v0[S]=VmaxKm−v0Km[S]v0=KmVmax−Kmv0.

---

Introduction

Enzyme kinetics often involves transforming variables to linearize the Michaelis-Menten equation for easier extraction of kinetic constants. In some cases, the chosen equations produce plots where neither the x-axis nor y-axis is fully independent, meaning both variables share dependent components. Identifying such cases helps avoid misinterpretations and selects suitable plotting strategies. This article explains which enzyme kinetic equation produces a plot without independent variables on either axis, emphasizing its formation and utility.

---

The Equations and Their Variables

The four given equations for enzyme kinetics involve initial velocity v0v0, substrate concentration $[S]$, and constants VmaxVmax and KmKm:

1. Lineweaver-Burk equation:

1v0=KmVmax1[S]+1Vmaxv01=VmaxKm[S]1+Vmax1

• X-axis: 1[S][S]1 (independent)

• Y-axis: 1v0v01 (dependent)

2. Hanes-Woolf equation:

[S]v0=[S]Vmax+KmVmaxv0[S]=Vmax[S]+VmaxKm

• X-axis: $[S]$ (independent)

• Y-axis: [S]v0v0[S] (dependent)

3. Eadie-Hofstee type rearrangement:

v0[S]=VmaxKm−v0Km[S]v0=KmVmax−Kmv0

• X-axis: v0v0 (dependent variable)

• Y-axis: v0[S][S]v0 (dependent variable)

Both axes depend on v0v0, so neither axis contains an independent variable.

4. Michaelis-Menten original equation:

v0=Vmax[S]Km+[S]v0=Km+[S]Vmax[S]

• X-axis: $[S]$ (independent)

• Y-axis: v0v0 (dependent)

---

Why Does Equation (3) Fit the Criteria?

• Both axes in equation (3) depend on v0v0.

• This dependence means the student using equation (3) plots a graph where neither axis is independently controlled.

• Such plots are also called ratio plots and are useful in certain kinetic analyses but have unique interpretative traits.

---

Significance of Equation (3) in Enzyme Kinetics

• Rearranged form of the Michaelis-Menten equation.

• Useful in Eadie-Hofstee plots, where velocity is plotted against velocity normalized by substrate concentration.

• Provides a linear plot from which KmKm and VmaxVmax can be derived.

---

Summary Table of Plot Types

---

Conclusion

Among the provided equations, only equation (3)—v0[S]=VmaxKm−v0Km[S]v0=KmVmax−Kmv0—yields a plot where neither the X nor Y axis represents an independent variable, hence fitting the student’s scenario. This plot type facilitates enzyme kinetics analysis but requires careful interpretation due to both variables’ interdependence.