(JUNE 2008)
30. For an enzyme catalyzed reactions exhibiting Michalis Menten equation what would be increase in substrate concentration to increase the rate of reaction from 10 % of Vmax to 90 % of Vmax
(1) 81 fold                                                  (2) 8 fold
(3) 4 fold                                                    (4) 2 fold

The correct answer is (1) 81 fold.

Introduction

Michaelis-Menten kinetics describe the relationship between enzyme reaction velocity and substrate concentration. Understanding how substrate concentration changes affect reaction velocity is crucial in enzymology, drug design, and metabolic regulation. This article calculates the fold increase in substrate concentration required to increase the enzymatic rate from 10% to 90% of VmaxVmax and explains the significance of this change using the Michaelis-Menten model.

Michaelis-Menten Equation Recap

The equation relating initial velocity V0V0 to substrate concentration $[S]$ is:

V0=Vmax[S]Km+[S]V0=Km+[S]Vmax[S]

Where

• V0V0 = initial velocity

• VmaxVmax = maximum velocity

• KmKm = Michaelis constant

• $[S]$ = substrate concentration

Calculations

Step 1: Substrate concentration at 10% VmaxVmax

Set V0=0.1VmaxV0=0.1Vmax:

0.1Vmax=Vmax[S1]Km+[S1]0.1Vmax=Km+[S1]Vmax[S1]

Simplify by dividing both sides by VmaxVmax:

0.1=[S1]Km+[S1]0.1=Km+[S1][S1]

Cross multiply:

0.1(Km+[S1])=[S1]0.1(Km+[S1])=[S1]0.1Km+0.1[S1]=[S1]0.1Km+0.1[S1]=[S1]

Subtract 0.1[S1]0.1[S1] from both sides:

0.1Km=0.9[S1]0.1Km=0.9[S1]

Solve for [S1][S1]:

[S1]=0.1Km0.9=Km9[S1]=0.90.1Km=9Km

Step 2: Substrate concentration at 90% VmaxVmax

Set V0=0.9VmaxV0=0.9Vmax, similarly:

0.9=[S2]Km+[S2]0.9=Km+[S2][S2]0.9(Km+[S2])=[S2]0.9(Km+[S2])=[S2]0.9Km+0.9[S2]=[S2]0.9Km+0.9[S2]=[S2]0.9Km=0.1[S2]0.9Km=0.1[S2][S2]=0.9Km0.1=9Km[S2]=0.10.9Km=9Km

Step 3: Fold increase in substrate concentration

Fold increase=[S2][S1]=9KmKm9=81Fold increase=[S1][S2]=9Km9Km=81

Interpretation

• To increase the reaction velocity from 10% to 90% of VmaxVmax, substrate concentration must be increased 81-fold.

• This demonstrates the non-linear relationship between substrate concentration and reaction rate typical of Michaelis-Menten kinetics.

Practical Significance

• Explains why enzymes respond sensitively to substrate changes near low concentrations.

• Guides experimental design to achieve desired enzymatic rates.

• Informs pharmacokinetic models for drug metabolism and enzyme-targeted therapies.

Summary Table

Conclusion

Using the Michaelis-Menten equation, increasing substrate concentration from Km99Km to 9Km9Km increases enzyme reaction rate from 10% to 90% of VmaxVmax, corresponding to an 81-fold increase in substrate concentration. This highlights the enzyme’s saturation kinetics and its implications for both biological systems and experimental enzyme studies.