Enzyme Kinetics: Properties, Classification, and Kinetic Analysis

Enzyme Kinetics

Introduction

Enzyme kinetics is the study of the rates at which enzymatic reactions occur and the factors that affect these rates. Understanding enzyme kinetics is fundamental in biochemistry, as it provides insights into enzyme function, regulation, and mechanisms of catalysis.

General Properties of Enzymes

Definition and Characteristics

• Enzymes are biological catalysts that accelerate chemical reactions without being consumed in the process.

• They increase reaction rates by factors ranging from $10^3$ to $10^{20}$, making reactions proceed at biologically relevant speeds.

• Most enzymes are proteins, but some RNA molecules, called ribozymes, also have catalytic activity.

• Enzyme names typically end with the suffix -ase.

Example: The spontaneous hydrolysis of a dipeptide may take ~7 years, whereas enzyme-catalyzed hydrolysis (by a protease) occurs in ~0.001 seconds.

Advantages of Enzymes

• Function under physiological conditions (aqueous environment, pH ~7, moderate salt concentration).

• Non-enzymatic reactions often require extreme conditions (high temperature, pressure, or non-physiological pH).

• High specificity for substrates, reducing side reactions.

• Enzyme activity is frequently regulated to meet cellular needs.

• Enzymes do not catalyze reactions with a positive free energy change (unfavorable reactions).

Classification of Enzymes

Overview

Enzymes are classified into six major classes based on the type of reaction they catalyze:

1. Oxidoreductases: Catalyze oxidation-reduction (redox) reactions, involving electron transfer. Commonly use electron carriers such as NAD+/NADH. General equation: $A_{(ox)} + B_{(red)} \rightleftharpoons A_{(red)} + B_{(ox)}$ Examples: Dehydrogenase, oxidase, reductase, peroxidase.

2. Transferases: Transfer a functional group from one molecule to another. General equation: $AX + B \rightleftharpoons BX + A$ Examples: Kinase, transaminase, transcarboxylase, transmethylase.

3. Hydrolases: Catalyze the cleavage of bonds by the addition of water. General equation: $AB + H_2O \rightleftharpoons A + B$ Examples: Esterase, phosphatase, peptidase, pyrophosphatase.

4. Lyases: Add or remove groups to form double bonds, or break double bonds by adding groups. Examples: Synthase, decarboxylase, dehydratase, deaminase.

5. Isomerases: Catalyze the interconversion of isomers (structural rearrangements within a molecule). Examples: Racemase, epimerase, cis-trans isomerase.

6. Ligases: Join two molecules together with the input of energy from a nucleotide triphosphate (usually ATP). General equation: $A + B + NTP \rightarrow AB + NDP + P_i$ Examples: Synthetase, carboxylase.

Enzyme Kinetics: Reaction Rates

Uncatalyzed Reactions

• First-order reaction: Involves a single reactant. Rate law: $v = -\frac{d[S]}{dt} = k_{obs}[S]$

• Second-order reaction: Involves two reactants. Rate law: $v = -\frac{d[A]}{dt} = -\frac{d[B]}{dt} = k_{obs}[A][B]$

• Units: First-order ($s^{-1}$), Second-order ($M^{-1}s^{-1}$)

Enzyme-Catalyzed Reactions

• Enzymes bind substrates to form an enzyme-substrate complex (ES), which then converts to product and releases the enzyme.

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

• The binding step is reversible; the catalytic step is often considered irreversible under initial rate conditions.

Transition State and Activation Energy

• Reactions must overcome an energy barrier called the activation energy ($E_a$) to reach the transition state.

• Enzymes lower the activation energy, thereby increasing the reaction rate, but do not change the overall free energy change ($\Delta G$) of the reaction.

Measuring Enzyme Activity and Binding

Ligand Binding

• Protein (P) binds ligand (L) to form a complex (PL): $P + L \rightleftharpoons PL$

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

• Fraction bound: $\frac{[PL]}{[P]_{tot}} = \frac{[L]}{K_d + [L]}$

Enzyme Activity

• Initial velocity ($v_0$) is measured by mixing enzyme and substrate and monitoring product formation over time.

• Steady-state is reached when the concentration of ES remains constant.

• Initial rates are measured at various substrate concentrations to analyze enzyme kinetics.

Michaelis-Menten Kinetics

Michaelis-Menten Equation

The Michaelis-Menten equation describes the relationship between the initial velocity ($v_0$) and substrate concentration ($[S]$):

• $v_0 = \frac{V_{max}[S]}{K_m + [S]}$

• $V_{max}$: Maximum velocity when the enzyme is saturated with substrate.

• $K_m$: Michaelis constant; substrate concentration at which $v_0 = \frac{V_{max}}{2}$.

• $k_{cat}$: Turnover number; number of substrate molecules converted to product per enzyme per second at saturation.

• Catalytic efficiency: $\frac{k_{cat}}{K_m}$

Lineweaver-Burk Plot

The Michaelis-Menten equation can be linearized for graphical analysis:

• Lineweaver-Burk (double reciprocal) plot: $\frac{1}{v_0} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}$

• Y-intercept: $\frac{1}{V_{max}}$; X-intercept: $-\frac{1}{K_m}$

Comparison: Michaelis-Menten vs. Ligand Binding

Michaelis-Menten (Enzyme)	Ligand Binding (Protein)
$E + S \rightleftharpoons ES$	$P + L \rightleftharpoons PL$
$K_m = \frac{[E][S]}{[ES]}$	$K_d = \frac{[P][L]}{[PL]}$
Fraction bound: $\frac{[ES]}{[E]_{tot}} = \frac{[S]}{K_m + [S]}$	Fraction bound: $\frac{[PL]}{[P]_{tot}} = \frac{[L]}{K_d + [L]}$

Enzyme Kinetic Parameters

• Michaelis constant ($K_m$): Apparent substrate affinity; lower $K_m$ indicates higher affinity.

• Turnover number ($k_{cat}$): Maximum number of substrate molecules converted per enzyme per second.

• Maximal velocity ($V_{max}$): Rate when enzyme is saturated with substrate.

• Catalytic efficiency ($k_{cat}/K_m$): Measures enzyme performance at low substrate concentrations; higher values indicate greater efficiency.

Allosteric Regulation

Allosteric Activators and Inhibitors

• Allosteric enzymes are regulated by molecules that bind at sites other than the active site, causing conformational changes that affect activity.

• Allosteric activators increase enzyme activity, often lowering the apparent $K_m$ (higher affinity).

• Allosteric inhibitors decrease enzyme activity, often raising the apparent $K_m$ (lower affinity).

• Allosteric enzymes often display sigmoidal (S-shaped) velocity vs. substrate concentration curves, rather than the hyperbolic curves of Michaelis-Menten enzymes.

Summary Table: Enzyme Classes

Class	Reaction Type	Example Enzymes
Oxidoreductase	Redox (electron transfer)	Dehydrogenase, oxidase
Transferase	Group transfer	Kinase, transaminase
Hydrolase	Hydrolysis (bond cleavage with water)	Esterase, phosphatase
Lyase	Addition/removal to double bonds	Decarboxylase, synthase
Isomerase	Isomerization	Racemase, epimerase
Ligase	Bond formation with ATP hydrolysis	Synthetase, carboxylase

Key Equations

• Michaelis-Menten: $v_0 = \frac{V_{max}[S]}{K_m + [S]}$

• Lineweaver-Burk: $\frac{1}{v_0} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}$

• Fraction bound (enzyme): $\frac{[ES]}{[E]_{tot}} = \frac{[S]}{K_m + [S]}$

• Fraction bound (protein-ligand): $\frac{[PL]}{[P]_{tot}} = \frac{[L]}{K_d + [L]}$

• Catalytic efficiency: $\frac{k_{cat}}{K_m}$

Applications and Examples

• Enzyme kinetics is essential for drug development, understanding metabolic pathways, and diagnosing enzyme deficiencies.

• "Kinetically perfect" enzymes, such as some transaminases, operate at the diffusion limit, meaning every substrate encounter leads to product formation.