Biochemistry: Chapter 3 — Enzymes

3.1 What are Enzymes? Catalytic Principles & Transition-State Stabilization #

Enzymes are biological catalysts—predominantly proteins and specialized catalytic RNAs (ribozymes)—that accelerate chemical reactions by factors of 10⁶ to 10¹⁷ under mild physiological conditions (37°C, neutral pH, aqueous solution) without being consumed or permanently altered in the process.

To understand how enzymes operate, one must distinguish thermodynamics from kinetics:

• Thermodynamic Feasibility (Δ G°’): Enzymes do NOT alter the overall standard free energy change (Δ G°’) of a reaction, nor do they alter the chemical equilibrium (K_eq). The starting free energy of substrates and final free energy of products remain identical.
• Kinetic Acceleration (Δ G^ddagger): Enzymes accelerate reactions exclusively by lowering the free energy of activation (Δ G^ddagger) required to reach the unstable, high-energy transition state (X^ddagger).

The free energy profile of an enzyme-catalyzed reaction versus an uncatalyzed reaction is displayed in Figure 3.1, illustrating the depression of the activation energy barrier through transition-state stabilization.

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3.1.1 The Induced-Fit Model #

Emil Fischer’s historical “Lock-and-Key” model envisioned the enzyme’s active site as a rigid template complementary to the substrate. However, Linus Pauling and J. B. S. Haldane proved that an enzyme perfectly complementary to its substrate would act as a thermodynamic trap, stabilizing the substrate and increasing the activation barrier!

Daniel Koshland resolved this by proposing the Induced-Fit Model:

1. The active site is dynamic and flexible, complementary not to the substrate, but to the transition state (X^ddagger).

2. Substrate binding induces a cooperative conformational restructuring of the enzyme, aligning catalytic residues and straining substrate bonds into the transition-state geometry.

The dynamic conformational shift of an enzyme active site upon substrate binding is shown in Figure 3.2.

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3.1.2 Fundamental Catalytic Mechanisms #

Enzymes employ four major chemical strategies:

1. Approximation and Orientation: Substrates are brought together in the active site with precise stereochemical alignment, eliminating translational and rotational entropy.

2. General Acid-Base Catalysis: Proton transfers are mediated by ionizable amino acid side chains (His, Asp, Glu, Lys) rather than relying on scarce H_3O⁺ or OH^– at neutral pH.

3. Covalent Catalysis: A transient covalent bond is formed between an active-site nucleophile (e.g., reactive Ser195 in chymotrypsin) and the substrate.

4. Metal Ion Catalysis: Bound divalent metal cations (Zn²⁺, Mg²⁺, Mn²⁺) polarize bonds, stabilize developing negative charges, or coordinate water molecules.

3.2 Enzyme Kinetics: The Michaelis–Menten Model #

When initial reaction velocity (v₀) is measured against substrate concentration [S], enzymes exhibit a rectangular hyperbolic saturation curve. At low [S], velocity is directly proportional to [S] (first-order kinetics); as active sites become saturated, velocity approaches an asymptotic maximum (V_max, zero-order kinetics).

This hyperbolic kinetic saturation profile is illustrated in Figure 3.3, defining the graphical determination of V_max and K_m.

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3.2.1 Derivation of the Rate Equation #

Consider the classic single-substrate scheme:

E + S underset{k_-1}{overset{k₁}{⇌}} ES overset{k₂}{longrightarrow} E + P

Under the Briggs–Haldane Steady-State Assumption, the concentration of the intermediate [ES] complex remains constant over the measurement period:

k₁ [E][S] = (k_-1 + k₂) [ES]

Defining the Michaelis Constant (K_m):

K_m = frac{k_-1 + k₂}{k₁}

Substituting [E]_t = [E] + [ES] and defining V_max = k₂ [E]_t, the initial velocity is given by the Michaelis–Menten Equation:

v₀ = frac{V_max [S]}{K_m + [S]}

#### Significance of Kinetic Constants:

• K_m (Michaelis Constant): The substrate concentration at which reaction velocity is exactly half-maximal (v₀ = V_max / 2). It reflects the affinity of enzyme for substrate; a low K_m indicates high affinity.
• k_cat (Turnover Number): The number of substrate molecules converted to product per active site per unit time when saturated (k_cat = V_max / [E]_t, units: s^-1).
• k_cat / K_m (Catalytic Efficiency): The apparent second-order rate constant under conditions of low substrate concentration ([S] ≪ K_m). The theoretical upper limit is bounded by aqueous diffusion (10⁸ to 10⁹ M^-1·s^-1); enzymes operating at this threshold (catalase, acetylcholinesterase, carbonic anhydrase) have reached kinetic perfection.

3.3 Linearization & Inhibition Modes: Lineweaver–Burk Analysis #

Taking the double reciprocal of the Michaelis–Menten equation yields the Lineweaver–Burk Equation:

(1 / v₀) = frac{K_m}{V_max} · (1 / [S]) + frac{1}{V_max}

Plotting 1/v₀ versus 1/[S] yields a straight line where the y-intercept is 1/V_max, the x-intercept is -1/K_m, and the slope is K_m / V_max.

The fundamental geometry of the Lineweaver-Burk double reciprocal plot is displayed in Figure 3.4.

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3.3.1 Modes of Reversible Inhibition #

Reversible inhibitors bind non-covalently and fall into three primary classes:

#### 1. Competitive Inhibition

The inhibitor resembles the substrate and competes directly for the active site, forming an inactive EI complex.

• V_max is unchanged: At saturating substrate concentrations ([S] → infty), substrate outcompetes the inhibitor.
• Apparent K_m increases: (K_m‘ = α K_m). More substrate is needed to achieve half-maximal velocity.
• Lineweaver–Burk profile: Lines intersect at the same point on the y-axis (1/V_max).

Figure 3.5 shows the Lineweaver-Burk plot of competitive inhibition, demonstrating identical y-intercepts with shifting x-intercepts.

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#### 2. Uncompetitive Inhibition

The inhibitor binds only to the enzyme-substrate complex (ES), forming a dead-end ESI complex.

• V_max decreases: (V_max‘ = V_max / α’). Active ES complexes are depleted.
• Apparent K_m decreases: (K_m‘ = K_m / α’). Binding of inhibitor pulls the E + S ⇌ ES equilibrium to the right.
• Lineweaver–Burk profile: Parallel lines with identical slopes.

#### 3. Non-Competitive (Mixed) Inhibition

The inhibitor binds to an allosteric site on both free enzyme (E) and ES complex with equal affinity.

• V_max decreases: Catalytic machinery is disabled.
• K_m remains unchanged: Substrate binding affinity is unaffected.
• Lineweaver–Burk profile: Lines intersect on the negative x-axis (-1/K_m).

3.4 Catalytic Mechanisms: The Chymotrypsin Catalytic Triad #

Chymotrypsin is a pancreatic serine protease that cleaves peptide bonds on the carboxyl-terminal side of bulky aromatic residues (Phe, Tyr, Trp).

Chymotrypsin utilizes a Catalytic Triad consisting of Serine 195, Histidine 57, and Aspartate 102.

The complete multi-step chemical mechanism of peptide bond cleavage by chymotrypsin is depicted in Figure 3.6, detailing the acyl-enzyme intermediate and oxyanion hole stabilization.

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3.4.1 Step-by-Step Catalytic Cascade #

1. Substrate Binding: The aromatic side chain docks into chymotrypsin’s hydrophobic S1 pocket, positioning the scissile peptide bond next to Ser195.

2. Nucleophilic Attack: Asp102 polarizes His57, which acts as a general base to abstract a proton from Ser195. This generates a potent alkoxide ion on Ser195 that attacks the substrate’s carbonyl carbon.

3. First Tetrahedral Intermediate & Oxyanion Hole: The carbonyl oxygen acquires a negative charge, stabilized by hydrogen bonds from backbone amide nitrogens of Gly193 and Ser195 (the oxyanion hole).

4. Acyl-Enzyme Intermediate: His57 acts as a general acid, protonating the leaving amino terminus. The first peptide fragment leaves, leaving the N-terminal portion covalently linked to Ser195 as an acyl-enzyme intermediate.

5. Deacylation by Water: Water enters; His57 abstracts a proton from water, generating a hydroxide ion that attacks the acyl-enzyme, forming the second tetrahedral intermediate.

6. Product Release: His57 protonates Ser195, releasing the second peptide fragment and regenerating the catalytic triad.

3.5 Allosteric Regulation & Covalent Modification #

Metabolic flux is coordinated by allosteric enzymes positioned at committed regulatory steps. Allosteric enzymes display sigmoidal (S-shaped) velocity curves resulting from cooperative subunit communication.

Figure 3.7 illustrates allosteric transitions between the low-affinity T-state and the high-affinity R-state, driven by allosteric activators and feedback inhibitors.

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3.5.1 Aspartate Transcarbamoylase (ATCase) #

ATCase catalyzes the committed step of pyrimidine biosynthesis in E. coli:

• Feedback Inhibition: CTP (the pyrimidine end-product) binds allosteric regulatory subunits, stabilizing the inactive T-state and shifting the curve rightward.
• Allosteric Activation: ATP (a purine) binds the same regulatory sites, stabilizing the active R-state and shifting the curve leftward, balancing purine and pyrimidine pools.

3.6 Curiosity Spark & Clinical Insights #

3.7 Key Takeaways & High-Yield Summary #

• Transition-state stabilization: Enzymes accelerate reactions by depressing Δ G^ddagger; they do not alter Δ G°’ or equilibrium constants.
• Michaelis–Menten constants: K_m is the substrate concentration at v₀ = V_max/2; k_cat/K_m measures catalytic efficiency and is diffusion-limited (10⁸–10⁹ M^-1s^-1).
• Inhibition kinetics: Competitive increases apparent K_m (V_max unchanged); uncompetitive decreases both K_m and V_max; non-competitive decreases V_max (K_m unchanged).
• Catalytic triad: Ser195 (covalent nucleophile), His57 (general acid-base), and Asp102 (electrostatic orienter) work cooperatively in chymotrypsin.

3.8 Review & Self-Assessment Questions #

1. An enzyme has a K_m of 2.0 mM and a V_max of 100 µmol·min^-1·mg^-1. Calculate the reaction velocity when [S] = 0.5 mM.

• Answer: Using v₀ = frac{V_max[S]}{K_m + [S]} = (100 × 0.5 / 2.0 + 0.5) = (50 / 2.5) = mathbf{20 µmol·min^-1·mg^-1}.

2. Why does mutating Asp102 to Alanine in chymotrypsin reduce catalytic efficiency by over 10⁴-fold even though Asp102 never contacts the substrate?

• Answer: Asp102 forms a low-barrier hydrogen bond with His57, polarizing its imidazole ring and stabilizing the positive charge that develops when His57 abstracts a proton from Ser195. Mutating Asp102 disables His57’s ability to act as a general base, preventing the generation of the nucleophilic Ser195 alkoxide ion.