(FEB 2022-11)
31. What is the fold difference between v at [S] =K_m and v at [S] = 1000 K_m where v is the
initial velocity of an enzyme catalyzed reaction, [S] is substrate concentration and K_m is the Michaelis constant?
(1) 1.998                                                    (2) 1000
(3) 2.998                                                    (4) 3.998

The correct answer is (1) 1.998.

Introduction

Michaelis-Menten kinetics describe the relationship between the initial velocity of an enzyme-catalyzed reaction and substrate concentration. A key question is how the initial velocity v0v0 changes when substrate concentration varies greatly relative to the Michaelis constant (KmKm). This article calculates the fold difference in initial velocity between substrate concentrations of KmKm and 1000×Km1000×Km, providing insights into enzyme saturation kinetics and reaction efficiency.

Michaelis-Menten Equation Recap

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

Where:

• v0v0 = initial velocity

• VmaxVmax = maximum velocity

• $[S]$ = substrate concentration

• KmKm = Michaelis constant

Step 1: Calculate $v$ at [S]=Km[S]=Km

vKm=Vmax×KmKm+Km=Vmax×Km2Km=Vmax2vKm=Km+KmVmax×Km=2KmVmax×Km=2Vmax

Initial velocity at substrate concentration KmKm is half maximal velocity.

Step 2: Calculate $v$ at [S]=1000Km[S]=1000Km

v1000Km=Vmax×1000KmKm+1000Km=Vmax×1000Km1001Km=10001001Vmaxv1000Km=Km+1000KmVmax×1000Km=1001KmVmax×1000Km=10011000Vmax

Initial velocity approaches VmaxVmax but slightly less due to substrate not being infinitely saturating.

Step 3: Calculate Fold Difference

Fold difference=v1000KmvKm=10001001VmaxVmax2=10001001×2≈1.998Fold difference=vKmv1000Km=2Vmax10011000Vmax=10011000×2≈1.998

Interpretation

• Increasing substrate from KmKm to 1000Km1000Km nearly doubles initial velocity.

• Saturation kinetics cause diminishing returns beyond substrate concentration near VmaxVmax.

• The fold difference being just under 2 shows that substrate concentration has to increase massively to fully saturate enzymes.

Practical Implications

• Understanding these relationships guides substrate concentration selection in enzyme assays.

• It elucidates enzyme efficiency limits; enzymes reach near-max velocity even at relatively moderate substrate concentrations.

• Useful for drug discovery and metabolic control analysis.

Summary Table

Conclusion

In Michaelis-Menten kinetics, raising substrate concentration from KmKm to 1000Km1000Km approximately doubles the initial velocity from 50% to nearly 100% of VmaxVmax, yielding a fold difference of about 1.998. This exemplifies the dramatic effect of enzyme saturation and is relevant for enzymology study and practical applications.