Biochemistry: A Short Course 7

Enzyme Kinetics

Introduction to Reaction Rates

Enzyme kinetics is the study of the rates at which biochemical reactions occur and the factors that affect these rates. The velocity of a reaction depends on the concentration of reactants and a proportionality constant known as the rate constant.

• First-order reactions: Reaction rate is directly proportional to the concentration of a single reactant. The rate constant has units of s-1.

• Second-order (bimolecular) reactions: Involve two reactants. The rate constant has units of M-1s-1.

Example: Many enzymatic reactions in biochemistry are bimolecular, involving an enzyme and a substrate.

The Michaelis-Menten Model

Describing the Kinetics of Many Enzymes

The Michaelis-Menten model is fundamental for understanding how enzymes catalyze reactions. It describes how the rate of product formation depends on substrate concentration and enzyme properties.

• Enzyme-substrate complex (ES): An enzyme (E) binds to a substrate (S) to form ES, which can either dissociate back to E and S or proceed to form product (P).

• Michaelis-Menten equation:

$V_0 = \frac{V_{\text{max}} [S]}{[S] + K_M}$

• V0: Initial reaction velocity

• Vmax: Maximum velocity when the enzyme is saturated with substrate

• KM: Michaelis constant, a measure of the substrate concentration at which the reaction velocity is half of Vmax

Turnover number (kcat): The number of substrate molecules converted to product per unit time by a single enzyme molecule when fully saturated with substrate. Typical values range from 1 to 104 s-1.

Enzyme efficiency: The ratio $k_{cat}/K_M$ indicates how efficiently an enzyme converts substrate to product.

Derivation of the Michaelis-Menten Equation

The model assumes the formation of an enzyme-substrate complex as a necessary intermediate:

$E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES \underset{k_{-2}}{\overset{k_2}{\rightleftharpoons}} E + P$

• k1: Rate constant for ES formation

• k-1: Rate constant for ES dissociation to E and S

• k2: Rate constant for ES conversion to E and P

Under the steady-state assumption, the concentration of ES remains constant over time:

• Rate of ES formation: $k_1 [E][S]$

• Rate of ES breakdown: $(k_{-1} + k_2)[ES]$

Setting these rates equal and solving for [ES]:

$[ES] = \frac{[E]_T [S]}{[S] + K_M}$

where $[E]_T$ is the total enzyme concentration and $K_M = \frac{k_{-1} + k_2}{k_1}$.

Substituting [ES] into the expression for initial velocity gives the Michaelis-Menten equation:

$V_0 = V_{\text{max}} \frac{[S]}{[S] + K_M}$

Allosteric Enzymes

Catalysts and Information Sensors

Allosteric enzymes are a special class of enzymes whose activity can be regulated by molecules binding at sites other than the active site. These enzymes do not follow Michaelis-Menten kinetics and often have multiple active sites.

• Cooperativity: Allosteric enzymes display cooperative binding, resulting in a sigmoidal (S-shaped) curve when plotting reaction velocity against substrate concentration.

• Regulation: Allosteric enzymes can be activated or inhibited by regulatory molecules, allowing fine control of metabolic pathways.

Example: Aspartate transcarbamoylase (ATCase) is an allosteric enzyme involved in nucleotide biosynthesis.

Single-Molecule Enzyme Studies

Studying Enzymes One Molecule at a Time

Advances in technology allow the study of individual enzyme molecules, revealing a distribution of behaviors rather than just average properties. This approach provides insights into enzyme mechanisms and heterogeneity that are not accessible through traditional ensemble measurements.

• Single-molecule methods: Techniques such as fluorescence microscopy can track the activity of individual enzyme molecules.

• Significance: These studies help uncover rare events and mechanistic details important for understanding enzyme function.

Clinical Relevance: Enzyme Deficiency and Disease

Cystathioninuria and Enzyme Activity

Cystathioninuria is a rare metabolic disorder caused by a deficiency in the enzyme cystathionase, which catalyzes the conversion of cystathionine to cysteine and α-ketobutyrate. This reaction requires the coenzyme pyridoxal phosphate, derived from vitamin B6 (pyridoxine).

• Detection: High levels of cystathionine in blood and urine indicate cystathioninuria.

• Clinical significance: Originally thought to be associated with intellectual disabilities, recent research suggests cystathioninuria is generally benign.

• Coenzyme role: Some cases are due to decreased affinity for pyridoxal phosphate, treatable with vitamin B6 supplementation.

Example: Loss of enzyme activity due to coenzyme deficiency is also seen in other metabolic disorders, such as methylmalonic acidemia.

Key Terms and Concepts

• Michaelis-Menten equation: $V_0 = \frac{V_{\text{max}} [S]}{[S] + K_M}$

• Michaelis constant (KM): $K_M = \frac{k_{-1} + k_2}{k_1}$

• Turnover number (kcat): Number of substrate molecules converted per enzyme per second

• Allosteric enzyme: Enzyme regulated by molecules binding at sites other than the active site, often showing cooperativity

• Steady-state assumption: The concentration of the enzyme-substrate complex remains constant during the reaction