Exam 2 Lecture 6

Enzyme Kinetics and Michaelis-Menten Model

Overview of Reaction Rates

• Reaction Rate Changes with Substrate Concentration.

  • At high substrate concentrations, the reaction rate increases.

  • This condition is characterized as saturating conditions.

• In saturating conditions:

  • All enzyme is bound as enzyme-substrate complex (ES).

  • Reaction rate does not depend on substrate concentration anymore; it becomes zero order with respect to substrate concentration.

Concepts and Definitions

• Zero Order Reaction:

  • The reaction rate is constant and does not depend on the substrate concentration.

  • All enzyme active sites are occupied.

  • As substrate is converted into product, new substrate binds immediately.

Course Announcements

• Discuss Chapter 7 in class today.

• Upcoming quiz in discussion, either today or tomorrow.

• Discussion worksheet consists of two parts:

  • Part 1: Michaelis-Menten kinetics (to be covered today).

  • Part 2: Reversible inhibition (to be covered Wednesday).

• Deadline for the worksheet is extended to Friday.

• A short problem set is also due on Friday night.

Michaelis-Menten Kinetics

• Recent learning focuses on the Michaelis-Menten model for enzyme kinetics:

  • Involves the enzyme binding a substrate, converting it to product, and releasing the product.

  • Important to focus on initial rates and product concentration being low to ignore product concentration in kinetics.

Kinetic Scheme

• The simplified kinetic scheme:

  • Enzyme (E) + Substrate (S) ⇌ Enzyme-Substrate complex (ES) → Product (P) + Free Enzyme (E)

Reaction Rate Expression

• The reaction rate ($V_0$) is dependent on:

  • Substrate concentration

  • Two constants:

  • Vmax: Defines the asymptote of the reaction rate as substrate concentration increases.

  • Km (Michaelis constant): Defines the curvature in the hyperbola of the graph of reaction rate vs. substrate concentration.

• Primary focus is to derive an expression relating the reaction rate to total enzyme concentration ([E]ₒ) and total substrate concentration ([S]).

Deriving First Order Rate Equation

• For product formation ($V_0$), we express it in terms of the enzyme-substrate complex concentration ([ES]):

  • $V<em>0 = k</em>2[ES]$

Steady State Assumption

• This assumption states that the concentration of the enzyme-substrate complex remains constant throughout the reaction.

  • Mathematically, this means:

  • $rac{d[ES]}{dt} = 0$

  • Rate formation of ES equals rate of disappearance of ES:

  • Formation: $k_1[E][S]$

  • Disappearance: $(k<em>{-1} + k</em>2)[ES]$

Analyzing Concentrations Over Time

• Concentrations of substrate ([S]), product ([P]), total enzyme ([E]ₒ), and enzyme-substrate complex ([ES]) change over time:

  • At $t = 0$: High substrate concentration, $[P] = 0$.

  • As the reaction progresses:

  • Substrate concentration decreases, product concentration increases.

  • Total enzyme concentration remains constant throughout the reaction as a characteristic of catalysts.

  • [ES] plateaued over approximately 95% of the reaction time, demonstrating the steady-state condition.

Calculating Michaelis Constant (Km)

• When $k<em>2 ext{ (chemistry step)}$ is much smaller than $k</em>{-1} ext{ (dissociation step)}$,

  • $K<em>m = rac{k</em>{-1} + k<em>2}{k</em>1} ext{ simplifies to } K<em>m ext{ approximately } k</em>{-1}$.

• Lower Km values indicate a stronger enzyme-substrate binding affinity.

• Example values:

  • Chymotrypsin: Km = 5 mM

  • Arginine tRNA synthetase substrates measured for relative affinities.

Total Enzyme Concentration

• Total enzyme concentration is the sum of free enzyme and enzyme-substrate complexes:

  • $[E]ₒ = [E] + [ES]$.

Deriving Michaelis-Menten Equation

• Substituting into $V<em>0 = k</em>2[ES]$ and rearranging leads to the Michaelis-Menten equation:

  • $V<em>0 = rac{V</em>{max}[S]}{[S] + K_m}$

Turnover Number (Kcat)

• Kcat indicates the number of substrate molecules converted to product per unit time when the enzyme is saturated.

  • Given by: $K<em>{cat} = rac{V</em>{max}}{[E]ₒ}$.

• Typical values of Kcat include:

  • Carbonic anhydrase: 600,000 products/min

  • Lysozyme: lower rates compared to carbonic anhydrase.

Catalytic Efficiency

• Catalytic efficiency is represented by the ratio of $rac{K<em>{cat}}{K</em>m}$.

• High numbers indicate effective enzyme kinetics.

• Conditions leading to large ratios:

  • Small Km values (high affinity);

  • High Kcat values (fast reactions).

Concept of Catalytic Perfection

• Limited upper bound of $rac{K<em>{cat}}{K</em>m}$ at $10^8$ to $10^9$.

• Enzymes achieving this rate are considered to have catalytic perfection as reaction rates match diffusion limits.

Enzyme Statistics and Analysis

• Discussion of reaction velocity profiles of different enzymes:

  • Comparing velocities at low substrate concentrations versus high concentrations.

• Examples illustrating how to estimate reaction velocities, Kcat, and Km using experimental data.

• Discussion on fraction of enzyme active sites bound to substrate as a direct measure of efficiency under saturation conditions.

Lineweaver-Burk Plot

• Reciprocal transformation of the Michaelis-Menten equation provides a linear form for easier analysis.

  • $rac{1}{V<em>0} = rac{K</em>m}{V<em>{max}} rac{1}{[S]} + rac{1}{V</em>{max}}$

Summary of Key Points

• Understanding of Km and Kcat is critical for analyzing enzyme behavior and interactions.

• Efficient enzyme function depends on relative values of these constants, which indicate affinity and speed.

Key Practice Problems and Takeaways

• Solve problems related to enzyme kinetics and experiment designs.

• Ability to calculate and interpret Km and kcat values through provided data and reactions.

Future Topics

• Upcoming discussions on inhibitors and their effects on enzyme kinetics, building from these principles currently established.