2. Enzyme Kinetics

Parameters of Enzyme Activity

Overview

This section introduces the fundamental parameters that govern enzyme activity, focusing on protein-ligand binding, chemical kinetics, and enzyme kinetics. Understanding these parameters is essential for analyzing how enzymes function and how their activity can be quantified and compared.

Protein-Ligand Binding

Protein-Ligand Binding Interactions

Protein-ligand binding is a reversible process where a protein (P) interacts with a ligand (L) to form a protein-ligand complex (PL). The strength and specificity of this interaction are characterized by association and dissociation constants.

• Association constant (Ka): Measures the affinity of the protein for the ligand.

• Dissociation constant (Kd): Represents the tendency of the complex to dissociate into free protein and ligand. Lower Kd values indicate higher affinity.

Formulas:

• Association constant: $K_a = \frac{[PL]}{[P][L]}$

• Dissociation constant: $K_d = \frac{[P][L]}{[PL]}$

The Value of Kd

Kd values can vary widely, reflecting the diversity of protein-ligand affinities. High-affinity interactions have very low Kd values (e.g., 10-15 M), while lower affinity interactions have higher Kd values (e.g., 10-5 M).

Protein	Ligand	Kd (M)
Avidin (egg white)	Biotin	1 × 10-15
Insulin Receptor – high affinity (human)	Insulin	4 × 10-11
Insulin Receptor – low affinity (human)	Insulin	5 × 10-10
Opioid receptor (rat brain)	Naloxone	3.8 × 10-9
Calmodulin (CaM) (bovine)	Ca2+	1 × 10-8
Nickel-binding protein (NikA, E. coli)	Ni2+	1 × 10-5

Additional info: Kd values are context-dependent and can change with pH, ionic strength, and other environmental factors.

The Langmuir Isotherm: Equilibrium Determination of Kd and n

The Langmuir isotherm describes the fraction of occupied binding sites (θ) as a function of ligand concentration ([L]) and the dissociation constant (Kd).

• Fractional occupancy (θ): $\theta = \frac{[PL]}{[P] + [PL]} = \frac{[L]}{K_d + [L]}$

• For n independent binding sites: $\theta = \frac{n[L]}{K_d + [L]}$

This relationship allows for the determination of Kd and the number of binding sites (n) from equilibrium binding data.

Graphical Determination of Kd and n

Several plots are used to analyze binding data and extract Kd and n:

• θ vs. [L]: Hyperbolic curve, approaches saturation at high [L].

• θ vs. log[L]: Sigmoidal curve, useful for cooperative binding.

• 1/θ vs. 1/[L]: Linear plot (double reciprocal), useful for determining Kd.

• θ/[L] vs. θ: Scatchard plot, linear for single-site binding, nonlinear for multiple or cooperative sites.

Nonlinearity in Scatchard Plots

Scatchard plots are used to analyze binding data. Nonlinearity in these plots can indicate the presence of multiple binding sites or cooperative interactions between sites.

• Linear Scatchard plot: Single class of independent binding sites.

• Nonlinear Scatchard plot: Multiple classes of sites or cooperative binding.

Chemical Kinetics

First Order Reactions

First order reactions involve a single reactant converting to product. The rate depends linearly on the concentration of the reactant.

• General form: A → P, rate constant k1

• Rate law: $v = -\frac{d[A]}{dt} = k_1 [A]$

• Integrated form: $[A]_t = [A]_0 e^{-k_1 t}$

• Plot of ln[A] vs. t yields a straight line with slope -k1.

Reversible First Order Reactions

When the reaction is reversible (A ⇌ P), both forward (k1) and reverse (k-1) rate constants are considered.

• Equilibrium constant: $K_{eq} = \frac{k_1}{k_{-1}}$

• Net rate: $v = k_1 [A] - k_{-1} [P]$

Second Order and Pseudo-First Order Reactions

• Second order: Two reactants (A + B → P), rate law: $v = k_1 [A][B]$

• Pseudo-first order: If [B] is in large excess, the reaction appears first order with respect to [A].

Enzyme Kinetics

Basic Concepts

Enzyme kinetics studies the rates of enzyme-catalyzed reactions and how they change in response to changes in substrate concentration, enzyme concentration, and other factors.

• Enzyme (E) binds substrate (S) to form an enzyme-substrate complex (ES), which then forms product (P).

• General scheme: $E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \overset{k_2}{\rightarrow} E + P$

Setting up the Kinetic Expression

The rate of product formation and the change in concentration of the enzyme-substrate complex can be described by differential equations:

• Rate of product formation: $v = \frac{d[P]}{dt} = k_2 [ES]$

• Change in ES: $\frac{d[ES]}{dt} = k_1 [E][S] - (k_{-1} + k_2)[ES]$

Two Assumptions & a Known

• Rapid Equilibrium Assumption: $\frac{[E][S]}{[ES]} = \frac{k_{-1}}{k_1}$

• Steady-State Assumption: $\frac{d[ES]}{dt} = 0$ (the concentration of ES remains constant over the initial phase of the reaction)

• Known: Total enzyme concentration: $[E]_t = [E] + [ES]$

These assumptions are foundational for deriving the Michaelis-Menten equation, which describes the relationship between reaction velocity and substrate concentration.