🧲 Big Picture: Why This Chapter Matters

The defining feature of enzyme catalysis is that the enzyme binds the substrate in a specific complex.

But binding in solution is not just about direct attraction between enzyme and substrate — you must also consider:

• 💧 Water displacement
• 🔄 Entropy changes
• ⚖️ The fact that observed binding energies are small differences between large energetic terms

This chapter first explains the forces, then explains how they translate into binding energies.

A. Interactions Between Nonbonded Atoms

All intermolecular forces are fundamentally electrostatic, but we classify them into types.

1️⃣ Electrostatic Interactions

These occur between:

• Charged ions
• Permanent dipoles
• Induced dipoles

📏 Distance Dependence

The strength depends on distance ( r ) and dielectric constant ( D ):

So:

• Ion–ion interactions are long-range
• Dipolar interactions are short-range

⚡ Role of the Dielectric Constant (D)

• Water: D ≈ 80
• Protein interior: D ≈ 2–4

A dielectric reduces effective charge via polarization (illustrated in Figures 11.1 and 11.2).

Inside a dielectric: q_ = q/D

This means:

• Charges interact much more weakly in water
• Much more strongly inside proteins

🖥 Electrostatics in Proteins

Calculating electrostatics is difficult because:

• Proteins are heterogeneous
• There’s a protein–water interface

But modern methods (Warwicker–Watson algorithm, DelPhi) divide the system into tiny cubes (~1 ų) and solve classical electrostatics numerically.

From Table 11.1, calculated electrostatic interaction energies match experiments within ~10%.

Important note:

• Local D can exceed 80 due to water alignment and focusing effects.

2️⃣ Nonpolar (van der Waals / Dispersion) Forces

These are weak but universal.

The total potential energy between atoms:

U = rac{A}{r^{12}} - rac{B}{r^6}

• ( r^{-12} ): strong short-range repulsion (electron cloud overlap)
• ( r^{-6} ): attractive London dispersion

📊 Figure 11.3 shows the classic potential well.

🧪 Polarizability Matters

Dispersion forces arise from temporary induced dipoles.

• Oxygen: low polarizability
• Sulfur: high polarizability

(Table 11.2 shows polarizabilities and B values.)

📏 van der Waals Radii (Table 11.3)

Atoms have characteristic radii:

• H: 1.44 Å
• O: ~1.7 Å
• C: 1.9 Å
• S: 2.0 Å

Contact distance = sum of radii.

Wells are shallow → distances vary ±0.1 Å easily.

📈 Additivity

Though weak individually:

• A methylene group contributes ~8.4 kJ/mol
• In lysozyme, a sugar contributes ~–58 kJ/mol via van der Waals contacts

So dispersion forces are weak but add up significantly.

3️⃣ Hydrogen Bonds 💧

Structure:

• Donor X–H
• Acceptor (usually O or N)

Typical geometry:

• N···O: 2.85–3.00 Å
• O···H: ~1.8–2.0 Å

Strength:

• 12–38 kJ/mol (3–9 kcal/mol)
• Amide–amide ~21 kJ/mol

Strong enough to matter. Weak enough to break rapidly.

⏱ Dissociation Rates

Using transition state theory:

• 3 kcal/mol → 4 × 10¹⁰ s⁻¹
• 6 kcal/mol → 3 × 10⁸ s⁻¹
• 9 kcal/mol → 2 × 10⁶ s⁻¹

Hydrogen bonds are dynamic.

🔬 Low-Barrier Hydrogen Bonds?

As donor and acceptor get closer:

• Double well → single well
• Possibly stronger bonds

Still debated whether important in catalysis.

🧠 Oxyanion Hole Example

Backbone NH groups stabilize developing negative charge in serine protease transition states.

Hydrogen bonds can stabilize transition states, not just ground-state binding.

4️⃣ Force Fields (Equation 11.3)

Used in simulations:

E_ = ext{bond stretch} + ext{angle bend} + ext{dihedral} + ext{van der Waals} + ext{electrostatics}

Includes:

• Harmonic bond stretching
• Angular bending
• Dihedral rotation
• r⁻¹² – r⁻⁶ terms
• Coulomb electrostatics

Hydrogen bonds are included in electrostatics.

These models are calibrated on experimental data.

B. Binding Energies of Proteins and Ligands

Binding = small net difference between:

• Enzyme–ligand interactions
• Water–enzyme interactions
• Water–ligand interactions
• Entropy changes

This is why binding energy is hard to compute.

1️⃣ The Hydrophobic Effect 🌊

Not really a “bond” — it’s an entropy effect.

Water forms hydrogen-bond networks. Hydrophobic solutes force water into ordered cages → entropy decreases.

When hydrophobes cluster:

• Water is released
• Entropy increases
• Binding becomes favorable

Two Contributions:

1. Entropy gain of water
2. Additional dispersion forces (higher polarizability in hydrocarbons)

Surface Tension Model (Figure 11.4)

Using surface tension analysis:

Two cases:

Case A: Transfer to bulk hydrocarbon

ΔG = –Aγ_HC/W

Case B: Transfer into preformed protein cavity

ΔG = –2Aγ_HC/W

So:

✨ Hydrophobic binding is proportional to buried surface area. ✨ Binding is stronger in preformed cavities.

Experimental Result

Transfer energy ≈ 80–100 J/mol per Ų buried.

Hydrophobic effect drives:

• Protein folding
• Helix formation
• Molecular recognition

Hansch Equation (Hydrophobicity Constant π)

Defined as:

pi = log(P/P_0)

Where:

• P = partition coefficient (octanol/water)
• π measures hydrophobicity of substituent

Incremental free energy:

Delta G = 2.303RTpi

Example: Each –CH₂– adds:

• π = 0.5
• ≈ 2.84 kJ/mol

Additive for non-conjugating groups.

Aqueous Solvation Energies (Table 11.5)

Wolfenden measured transfer to:

• Cyclohexane
• Vapor

Key patterns:

Very hydrophobic:

• Leu, Ile, Val

Very hydrophilic:

• Arg, Asp, Glu

Arg:

• –62 kJ/mol (cyclohexane → water)
• –83 kJ/mol (vapor → water)

These data are crucial for understanding:

• Mutation effects
• Denatured state energetics

2️⃣ Hydrogen Bond Inventory Model

Consider reaction (Equation 11.10):

Count hydrogen bonds before and after binding.

If:

• Same number and strength on both sides → ΔH ≈ 0 (isoenthalpic)

Thus:

💡 Hydrogen bonds in solution often don’t change enthalpy much.

Why Binding Still Occurs?

Because of:

• Release of ordered water
• Entropy gain

Experimental Evidence

Removing a hydrogen bond donor/acceptor weakens binding by:

• Only 2–6 kJ/mol

Even though individual bond strength is ~20 kJ/mol.

Reason: Water compensates energetically.

Salt Bridges

Example: NH₃⁺ + COO⁻

Inventory shows:

• Two hydrogen bonds on each side
• Electrostatics in water are weak

Thus salt bridges are often nearly isoenthalpic in water.

🔥 Core Takeaways

1. All intermolecular forces are electrostatic in origin.
2. Water competes strongly with enzyme–ligand interactions.
3. Hydrogen bonds in solution are often isoenthalpic.
4. The hydrophobic effect is primarily entropy-driven.
5. Binding energy = small difference between large opposing terms.
6. Buried surface area is a major determinant of hydrophobic binding.
7. Dispersion forces are weak but additive.
8. Electrostatics are highly sensitive to dielectric environment.
9. Force fields approximate these interactions in simulations.
10. Entropy is often as important as enthalpy in binding.

🧠 Conceptual Summary

Binding in biology is not about forming “strong new bonds.”

It is about:

• ⚖️ Exchanging interactions with water
• 💧 Releasing ordered solvent
• 📉 Minimizing entropic penalties
• 🧲 Exploiting complementary surfaces
• 📦 Burying hydrophobic area

The magic of enzymes lies in how they manipulate these delicate energetic balances.