Chemical Kinetics and Enzyme Catalysis: Study Notes

Chemical Kinetics

General Rate Equations

Chemical kinetics studies the rates of chemical reactions and how they depend on various factors, such as concentration and temperature.

• Generalized Reaction: aA + bB → cC + dD

• Rate Equation: The rate of consumption of A is given by: $V_0 = -\frac{d[A]}{dt} = k[A]^x[B]^y$ where k is the rate constant, and x, y are the reaction orders with respect to A and B.

Measuring Initial Rate

The initial rate is determined by the slope of the concentration vs. time curve at the earliest time points.

• Initial Rate (V0): Slope of [A] vs. time at t = 0.

• Why Initial Rate? It avoids complications from product accumulation and reverse reactions.

Order of Reaction

The order of a reaction describes how the rate depends on reactant concentrations.

• First Order: Rate depends linearly on one reactant ($V_0 = k[A]$).

• Second Order: Rate depends on the product of two reactant concentrations ($V_0 = k[A][B]$ or $V_0 = k[A]^2$).

• Zero Order: Rate is independent of reactant concentration ($V_0 = k$).

Graphs of [A] vs. Time

• First Order: Exponential decay of [A] over time.

• Second Order: Faster decay; curve is steeper than first order.

• Zero Order: Linear decrease of [A] over time.

Graphs of V0 vs. [A]

• First Order: Linear relationship.

• Second Order: Parabolic relationship.

• Zero Order: Horizontal line (rate is constant).

Enzyme Kinetics: Michaelis-Menten Equation

Introduction

The Michaelis-Menten equation describes the rate of enzymatic reactions with a single substrate. It is fundamental in biochemistry for understanding enzyme behavior.

• Equation: $V_0 = \frac{k_{cat}[E]_t[S]}{K_m + [S]} = \frac{V_{max}[S]}{K_m + [S]}$ where [E]t is total enzyme concentration, [S] is substrate concentration, Km is the Michaelis constant, and Vmax is the maximal rate.

Derivation and Mechanism

The Michaelis-Menten mechanism involves substrate binding, chemical transformation, and product release.

• Steps:

  1. Substrate binding: $E + S \xrightleftharpoons[k_{-1}]{k_1} ES$

  2. Chemical step: $ES \xrightarrow{k_2} E + P$

• Steady-State Assumption: The concentration of ES remains constant during the initial phase of the reaction.

• Michaelis Constant: $K_m = \frac{k_{-1} + k_2}{k_1}$

Key Parameters

• Vmax: Maximum rate achieved when the enzyme is saturated with substrate. $V_{max} = k_{cat}[E]_t$

• kcat (Turnover Number): Number of substrate molecules converted to product per enzyme per second. $k_{cat}$ is a first-order rate constant.

• Km: Substrate concentration at which the reaction rate is half of Vmax. Unit: M.

Special Cases

• When $[S] \ll K_m$: $V_0 \approx \frac{k_{cat}[E]_t}{K_m}[S]$ (first order with respect to [S])

• When $[S] \gg K_m$: $V_0 \approx k_{cat}[E]_t = V_{max}$ (zero order with respect to [S])

• When $[S] = K_m$: $V_0 = \frac{V_{max}}{2}$

Lineweaver-Burk Plot

A double-reciprocal plot linearizes the Michaelis-Menten equation:

• Equation: $\frac{1}{V_0} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}$

• Slope: $\frac{K_m}{V_{max}}$

• Y-intercept: $\frac{1}{V_{max}}$

• X-intercept: $-\frac{1}{K_m}$

Enzyme Specificity Constant

• Definition: $\frac{k_{cat}}{K_m}$ measures enzyme efficiency.

• Upper Limit: About $10^8$ to $10^9$ M-1s-1, limited by diffusion.

Reversible Enzyme Inhibitors

Reversible inhibitors bind non-covalently to enzymes and decrease the reaction rate by affecting Vmax, Km, or both.

• Types:

  • Competitive: Inhibitor binds to the active site, competing with substrate.

  • Uncompetitive: Inhibitor binds only to the ES complex.

  • Mixed: Inhibitor binds to both E and ES.

• Effect: Competitive inhibition increases Km, uncompetitive decreases both Vmax and Km, mixed affects both variably.

Summary Table: Michaelis-Menten Parameters

Parameter	Definition	Units	Significance
Vmax	Maximum velocity	M/s	Rate at enzyme saturation
Km	Michaelis constant	M	[S] at 1/2 Vmax
kcat	Turnover number	s-1	Max number of reactions per enzyme per second
kcat/Km	Specificity constant	M-1s-1	Enzyme efficiency

Additional info:

• Steady-state assumption is crucial for deriving the Michaelis-Menten equation.

• In cells, [S] is often close to Km, making enzyme rates sensitive to substrate concentration changes.

• Enzyme-catalyzed reactions can act as biological switches due to this sensitivity.