(NOV 2020-11)
53. The Hill equation and its plot describe the following enzyme kinetic behaviours
A. Saturation Kinetics
B. Cooperative Kinetics
C. Log Vi/(V_max — Vi) versus Log [s]
D. Log (V_max — Vi)/Vi versus Log [s]^-1

Which one of the following combination represent correct descriptions?
(1) A and C                                               (2) B and C
(3) B and D                                               (4) A and D

The correct answer is (2) B and C.

Introduction

The Hill equation is pivotal for characterizing cooperative binding in enzymes and receptors, helping to quantify how substrate or ligand binding at one site affects binding at other sites. This cooperative behavior alters the classic Michaelis-Menten kinetics, requiring specialized modeling. The Hill plot, a linear transformation of the Hill equation, enables the determination of the Hill coefficient—a measure of cooperativity. This article explains the key kinetic behaviors described by the Hill equation and the construction and interpretation of the Hill plot.

Hill Equation and Its Significance

• The Hill equation modifies the Michaelis-Menten model by introducing the Hill coefficient, $n$, which indicates the degree of cooperativity.

• When $n>1$, the system shows positive cooperativity; $n<1$ indicates negative cooperativity; $n=1$ reduces to non-cooperative Michaelis-Menten kinetics.

• The equation models the fraction of enzyme or receptor sites occupied by ligand ($\theta$) as a function of substrate concentration $[S]$:

θ=[S]nK+[S]nθ=K+[S]n[S]n

• It reflects mechanisms where binding of one molecule influences binding of others, common in oligomeric enzymes and multi-subunit proteins.

Hill Plot: Linearizing the Equation

• The Hill plot is a method for analyzing cooperative kinetics by plotting:

log⁡(ViVmax−Vi)vslog⁡[S]log(Vmax−ViVi)vslog[S]

• Here, ViVi is the initial velocity, and VmaxVmax is the maximum velocity.

• This log-log plot linearizes the Hill equation, making it easier to estimate the Hill coefficient ($n$) from the slope.

• The plot allows assessment of the degree of cooperativity:

  • Slope $>1$: positive cooperativity

  • Slope $=1$: no cooperativity

  • Slope $<1$: negative cooperativity

Interpretation of Given Options

• A. Saturation kinetics:

  • Generally described by Michaelis-Menten equation, not specifically Hill equation.

• B. Cooperative kinetics:

  • Correct. Hill equation fundamentally models cooperative substrate binding.

• C. log⁡ViVmax−VilogVmax−ViVi vs. $\log [S]$:

  • Correct. This is the canonical Hill plot.

• D. log⁡Vmax−ViVilogViVmax−Vi vs. log⁡[S]−1log[S]−1:

  • Incorrect. This is not a standard form of the Hill plot.

Summary Table

Conclusion

The Hill equation and its plot describe cooperative kinetics (B), and the Hill plot is a graph of log⁡ViVmax−VilogVmax−ViVi versus $\log [S]$ (C). Therefore, option (2) B and C is the correct answer. Understanding these principles is essential for characterizing enzyme kinetics exhibiting cooperativity beyond classical Michaelis-Menten behavior.

This clear explanation empowers students and researchers to apply and interpret Hill kinetics confidently in biochemical studies and examinations.