🧠 Protein Folding – Thermodynamics, Cooperativity & Stability

1️⃣ Leventhal’s Paradox – Why Proteins Don’t Fold Randomly

If a 100-residue protein had only 2 conformations per residue, it would have:

2^{100} approx 10^{30}

possible conformations.

If it sampled each conformation in 10⁻¹³ s, it would take 10⁹ years to find the native state.

But real proteins fold in seconds to minutes.

✅ Conclusion:

Proteins do not search randomly. Folding is guided.

2️⃣ Hydrogen Bonds Guide Folding

You asked:

H-bonding of secondary structure guides folding?

Yes — but with nuance.

In early folding:

• Local hydrogen bonds form quickly
• α-helices and β-sheets appear early
• These act as nucleation points

These pre-formed secondary structures reduce conformational space.

⚠ Important: Hydrogen bonds do not drive folding alone — they guide structure formation within the larger thermodynamic landscape.

Hydrophobic collapse and entropy of water are equally critical.

3️⃣ Force Fields & Folding Simulations

Modern molecular dynamics simulations include:

• Bond stretching
• Angle bending
• Torsion angles
• Electrostatics
• Van der Waals interactions

These are encoded in force field equations.

Small proteins (~40 aa) can now fold in silico in milliseconds.

4️⃣ Chemical Denaturation – Guanidinium Chloride

You asked:

Can guanidinium chloride replace hydrogen bonds?

Yes — partially.

Guanidinium chloride:

• Forms hydrogen bonds
• Solubilizes hydrophobic residues
• Stabilizes unfolded state

Water also forms hydrogen bonds — but water alone does not denature because:

• It does not solubilize hydrophobic cores as effectively.
• Guanidinium stabilizes exposed peptide backbone better.

5️⃣ Two-State Folding Model

Simple case:

N ightleftharpoons U

Only two states:

• Fully folded
• Fully unfolded

The sharp transition indicates:

🔁 Cooperative process Either folded or unfolded — no stable intermediates.

❓ Your Question:

2-state process means 2 independent units that have different denaturation?

No.

Two-state = one cooperative unit.

If a protein has multiple domains:

• Each domain may unfold separately
• Then you get multiple transitions
• That is not a two-state system

6️⃣ Thermal Denaturation

You asked:

Protein unfolding by heating related to interactions vibrating?

Yes.

Heating increases molecular motion:

• Ionic interactions weaken
• Dipole interactions weaken
• Hydrogen bonds break
• Hydrophobic packing loosens

When thermal energy exceeds stabilizing interactions → unfolding occurs.

7️⃣ Gibbs Free Energy & Stability

Delta G = Delta H - TDelta S

If:

• ΔG < 0 → folded favored
• ΔG > 0 → unfolded favored

You asked:

Free energy positive → unfolded?

Yes — for folding reaction.

If ΔG_folding > 0 → folding is unfavorable → protein unfolds.

Magnitude of Stability

Typical folding ΔG: -20 ext{ to } -60 ext{ kJ/mol}

This equals only a few hydrogen bonds!

⚠ Important: Proteins are marginally stable.

8️⃣ Do Hydrogen Bonds “Give” Gibbs Free Energy?

Not directly.

Hydrogen bonds contribute to:

• ΔH (enthalpy)

But ΔG depends on both:

• ΔH (bond formation)
• ΔS (entropy change)

You cannot look at hydrogen bonds alone.

9️⃣ Entropy & Enthalpy Contributions

Enthalpy (ΔH)

Favorable:

• Hydrogen bonds
• Electrostatic interactions
• Hydrophobic packing

Unfavorable:

• Breaking interactions with water

Entropy (ΔS)

Now the important correction:

You asked:

Unfolded → ΔG decreases because entropy bigger than enthalpy? But aromatic rings exposed reduce entropy?

This is where many students get confused.

Key distinction:

There are two entropies involved:

1️⃣ Chain entropy

• Unfolded chain = very flexible
• High entropy
• Folding reduces chain entropy (unfavorable)

2️⃣ Water entropy

• Hydrophobic residues force water to become ordered
• This decreases water entropy
• When folding buries hydrophobic residues:
  • Ordered water is released
  • Water entropy increases
  • Favorable

So what dominates?

For folding:

• ΔH is favorable
• Chain entropy is unfavorable
• Water entropy is strongly favorable

Net result: Delta G < 0

Important Correction to Your Thought

You said:

Aromatic ring exposed → entropy decreases → unfolded unfavorable?

Correct locally for water.

But in unfolded state:

• Chain entropy is very large and positive
• Many water interactions exist

The full balance determines stability.

It is not dependent on one amino acid. It is a global thermodynamic balance.

🔥 10️⃣ Heat Capacity & Calorimetry

In Differential Scanning Calorimetry (DSC):

We measure heat capacity vs temperature.

When protein unfolds:

• Energy goes into breaking bonds
• Temperature does not increase as expected
• Heat capacity spikes

You asked:

Some of the temperature unfolds the protein?

Correct — but more precisely:

Energy input goes into:

• Breaking non-covalent bonds
• Not increasing temperature

The area under the peak = ΔH_unfolding

1️⃣1️⃣ Heat Capacity & Protein Size

You asked:

Smaller protein → lower heat capacity? Why?

Because:

• Larger proteins expose more surface upon unfolding
• More solvent interaction change
• Larger ΔCp

Heat capacity scales roughly with:

• Nonpolar surface area
• Protein size

Smaller protein → smaller hydrophobic core → smaller heat capacity change

1️⃣2️⃣ Temperature Dependence of Stability

ΔH and ΔS are temperature dependent.

At melting temperature (Tm):

Delta G = 0

So: T_m = rac{Delta H}{Delta S}

❄ Cold Denaturation

Important concept from your file:

Proteins can also unfold at low temperature.

Because:

• Hydrophobic effect weakens at low temperature
• Water entropy effect changes

Thus:

• ΔG becomes positive again
• Cold denaturation occurs

This surprises many students.

1️⃣3️⃣ Why ΔG is Small but Folding is Strongly Favored

Even if ΔG = -10 kcal/mol:

K = e^{-ΔG/RT}

At 298K:

→ ~10⁷ folded per unfolded

Small energy difference = huge equilibrium shift.

📊 Summary of Energy Contributions

Net result: Small negative ΔG → strong folded population.

🧬 Final Conceptual Picture

Protein folding is:

• Guided by early secondary structure
• Driven by hydrophobic effect
• Stabilized by hydrogen bonds
• Cooperative
• Marginally stable
• Temperature dependent
• Both heat and cold sensitive