Structural Biology from Zero · 3: How a Chain Finds Its Shape

The shape of a protein backbone is set largely by two angles per residue, each restricted to particular regions. Even for a chain of one hundred residues, the number of possible conformations remains enormous. Protein folding is governed by the combined effects of many weak interactions, solvent thermodynamics, and a biased energy landscape. This article develops those contributions and explains how they permit folding on biological timescales.

Let me start by making the puzzle quantitative, because the numbers are what make the resolution feel earned.

Levinthal's paradox

Take a modest protein of 100 residues. Give each residue just three accessible backbone conformations — a wild underestimate, but let's be generous to intuition. Then the chain has

$3^{100} \approx 5 \times 10^{47}$

possible shapes. Suppose the chain could try a new conformation every $10^{-13}$ seconds, about as fast as a bond can rearrange. Searching them all would take

$5 \times 10^{47} \times 10^{-13}\ \mathrm{s} = 5 \times 10^{34}\ \mathrm{s},$

which is roughly $10^{17}$ times the current age of the universe. Real proteins fold in microseconds to seconds. Cyrus Levinthal used this discrepancy in 1969 to show that folding cannot be an exhaustive random search. The conformational dynamics must instead be biased toward productive regions of the landscape.

import numpy as np

states, residues, try_time = 3, 100, 1e-13     # states/residue, length, s/attempt
conformations = states ** residues
search_seconds = conformations * try_time
age_universe = 13.8e9 * 365.25 * 24 * 3600      # s

print(f"Conformations: {conformations:.1e}")            # 5.2e47
print(f"Blind search:  {search_seconds:.1e} s")         # 5.2e34 s
print(f"= {search_seconds / age_universe:.0e} x the age of the universe")  # ~1e17
# Proteins fold in ~1e-6 to ~1e0 s. A random search is off by ~40 orders
# of magnitude, so folding must be strongly directed, not blind.

Anfinsen: the answer is in the sequence

Before we can talk about how the chain gets there, we need to know where "there" is. Christian Anfinsen answered that in the experiment that won him the 1972 Nobel Prize. He took ribonuclease, a small enzyme, and unfolded it completely in the test tube by adding a denaturant and breaking its disulfide bonds. Then he removed the denaturant. The enzyme spontaneously refolded — recovering full activity and re-forming exactly the right disulfide bonds — with no help from the cellular machinery that had originally made it.

The conclusion is the thermodynamic hypothesis: for many proteins, the native folded structure is simply the global minimum of free energy accessible to the chain under physiological conditions. All the information needed to specify the fold is already present in the amino acid sequence. Nothing external draws the picture; the sequence, dropped into water, finds its own lowest-energy shape. This is the single most important idea in structural biology, and it is what makes structure prediction — from AlphaFold on down — even conceivable: if the sequence determines the structure, then in principle a good enough model of the physics (or a good enough model learned from data) can read one off the other.

So the target is the free-energy minimum. Now: what shapes that free-energy landscape?

The forces, from weakest reasoning to strongest effect

Folding is governed mainly by non-covalent interactions, with disulfide bonds as an important covalent exception. Their reversibility allows contacts to form and break during folding. The resulting folded state is often only marginally more stable than the unfolded ensemble, which also permits motions needed for function.

Hydrogen bonds

A hydrogen bond forms when a hydrogen covalently attached to an electronegative atom (a donor, usually backbone $\mathrm{N-H}$ or a side-chain $\mathrm{O-H}$ ) is shared with a nearby electronegative atom bearing a lone pair (an acceptor, usually a carbonyl $\mathrm{C{=}O}$ ). The geometry is specific: the donor and acceptor heavy atoms sit about $2.8\text{–}3.2\ \text{Å}$ apart, the hydrogen-to-acceptor distance is roughly $1.8\text{–}2.0\ \text{Å}$ , and the interaction is strongest when the $\mathrm{D-H\cdots A}$ arrangement is close to linear. That directionality is what makes hydrogen bonds structural rather than merely sticky — they only pay off when the geometry is right, so they enforce particular arrangements.

Individually a hydrogen bond in water is worth only about $1\text{–}3\ \mathrm{kcal/mol}$ of net stabilization, because the groups could have hydrogen-bonded to water instead. But proteins make many of them, and — crucially — the backbone $\mathrm{N-H}$ and $\mathrm{C{=}O}$ of every residue must be satisfied somehow once buried away from water. The two regular secondary structures from Part 2 are precisely the two tidy solutions to that bookkeeping problem: the α-helix satisfies backbone hydrogen bonds within one strand of chain (each $\mathrm{C{=}O}$ of residue $i$ to the $\mathrm{N-H}$ of residue $i+4$ ), and the β-sheet satisfies them between neighbouring strands. Pauling predicted both in 1951 from the planar peptide bond and the geometry of the hydrogen bond alone, years before either was seen experimentally.

The hydrophobic effect

The hydrophobic effect is a major contribution to the folding of many soluble proteins. It arises from the combined response of water and nonpolar solutes rather than from a simple pairwise attraction between hydrophobic groups.

A nonpolar side chain — a leucine, a phenylalanine — cannot form hydrogen bonds with water. When such a group is exposed, the surrounding water molecules can't simply ignore it; they rearrange into a more ordered, cage-like shell to preserve their own hydrogen-bond network around the intruder. That ordering is an entropic cost: the water loses freedom. When two nonpolar groups instead cluster together, away from water, they present less surface to the solvent, fewer water molecules are forced into cages, and that ordered water is released back into the bulk. The system's entropy goes up.

Burying nonpolar surface changes both solvent entropy and enthalpy, with their relative contributions depending on temperature and solute size. Empirical transfer free energies often scale with buried nonpolar area, using values around $25\ \mathrm{cal\,mol^{-1}\,\text{Å}^{-2}}$ as a useful approximation. This solvent-mediated free-energy gain helps produce the hydrophobic cores of soluble proteins, while hydrogen bonding, packing, electrostatics, and chain entropy influence which compact structures are stable.

Van der Waals packing

Once the core collapses, van der Waals interactions contribute through weak attractions between fluctuating electron clouds, opposed by strong repulsion when atoms overlap. Any single contact contributes only a fraction of a $\mathrm{kcal/mol}$ , but a folded protein has packing density comparable to an organic crystal. Many small contacts together contribute to stability, while tight packing without clashes constrains the accessible folds.

Salt bridges

When a positively charged side chain (lysine, arginine) sits close to a negatively charged one (aspartate, glutamate), the two form a salt bridge — a hydrogen bond reinforced by electrostatic attraction. The Coulomb energy between two charges is

$E = \frac{1}{4\pi\varepsilon_0\,\varepsilon_r}\,\frac{q_1 q_2}{r},$

and the key term is the relative permittivity $\varepsilon_r$ . Bulk water has $\varepsilon_r \approx 80$ and strongly screens charge–charge interactions. Screening can be weaker in a protein interior or binding-site cleft, but the net contribution of a salt bridge also includes desolvation, geometry, nearby groups, and conformational change. A particular salt bridge may therefore be favorable, neutral, or unfavorable. Well-positioned electrostatic contacts can contribute to fold stability and molecular recognition, and they will return when we discuss binding.

Disulfide bonds — the exception

The one genuine covalent player is the disulfide bond: two cysteine thiols oxidized into a $\mathrm{S-S}$ linkage, about $2.05\ \text{Å}$ long and worth roughly $60\ \mathrm{kcal/mol}$ as a bond. But its role in stability is subtler than that big number suggests. A disulfide mainly stabilizes a fold by cross-linking the chain, which lowers the entropy of the unfolded state — a tethered chain has fewer ways to be unfolded, so the folded state is favored by comparison. Disulfides form only in oxidizing environments, which is why they are common in secreted and extracellular proteins (antibodies are held together by them) and rare in the reducing interior of the cell.

The viewer below shows these forces on a real structure: load a small protein and switch on its backbone hydrogen bonds, its salt bridges, and its buried hydrophobic core, each drawn with the actual atoms and distances involved. Turn on salt bridges and you'll see that each one is a pair of nearby, oppositely-charged side chains — the acidic residue (Asp/Glu) drawn in red, the basic one (Lys/Arg/His) in blue, labelled by name and number, with the dashed link measured in ångströms between them. Seeing a helix's rung-by-rung $i \to i{+}4$ hydrogen bonds, or an aspartate–lysine salt bridge measured at a few ångströms, makes these abstractions concrete in a way no equation does. The second tab, the folding funnel, is a live simulation we'll come back to in a moment.

Interactive. Tab 1: the folding forces computed live from deposited PDB coordinates (hydrogen bonds, salt bridges with their charged side chains shown, hydrophobic core, disulfides). Tab 2: a Langevin simulation of many chains descending a tunable energy landscape. Opens full-screen in a new tab.

Adding it up: why folded proteins are barely stable

Folding is a competition, and thermodynamics is the scoreboard. The relevant quantity is the free-energy change on folding,

$\Delta G_{\text{fold}} = \Delta H - T\,\Delta S,$

and both terms are large and fight each other. Forming hydrogen bonds, salt bridges, and van der Waals contacts releases enthalpy ( $\Delta H < 0$ , favorable). But the folding chain also loses an enormous amount of its own configurational entropy — going from a floppy ensemble of $10^{47}$ shapes to essentially one costs a large, unfavorable $-T\Delta S_{\text{chain}}$ . The reason folding wins at all is the hydrophobic effect, whose water-entropy gain shows up on the favorable side and roughly cancels the chain's entropy loss.

What is startling is the size of the final number. Both the favorable and unfavorable contributions are each many tens of $\mathrm{kcal/mol}$ , but they very nearly cancel, and the net stability of a typical folded protein is only

$\Delta G_{\text{fold}} \approx -5\ \text{to}\ -15\ \mathrm{kcal/mol},$

the equivalent of just a few hydrogen bonds. Proteins are marginally stable — a folded protein is a small difference between two large numbers. That sounds like a design flaw, but it is almost certainly a feature: marginal stability keeps proteins flexible enough to move and do chemistry, and lets the cell unfold and degrade them on demand. It is also why a single amino-acid change can tip the balance and destabilize a protein enough to cause disease — recall the one-residue change behind sickle-cell hemoglobin from Part 1.

The small net $\Delta G_{\text{fold}}$ is the result of several larger, non-additive contributions. Folding reduces chain configurational entropy while changing protein–protein, protein–water, and water–water interactions. These terms cannot be assigned reliably by a universal per-residue sum, but experiments show that modest perturbations can shift the net stability by several $\mathrm{kcal/mol}$ . That sensitivity explains why a single mutation can substantially alter folding or stability.

Many small proteins fold in an effectively two-state way — they are either folded or unfolded, with almost no stable half-folded population — which is exactly what you'd expect if the partly-folded states are higher in free energy than either end. You can push the equilibrium around: heat, or adding a denaturant like urea or guanidinium chloride, tilts $\Delta G_{\text{fold}}$ until the protein unfolds, and measuring that tilt is how the numbers above are actually obtained.

From paradox to funnel

We now have the pieces to dissolve Levinthal's paradox. His calculation secretly assumed a flat energy landscape — a golf course where every conformation is equally bad except one tiny hole, so the ball has no choice but to search blindly. But the forces we just catalogued mean the landscape is nothing like flat. Every step that buries a hydrophobic side chain, or completes a backbone hydrogen bond, lowers the free energy a little. Native-like contacts are, on average, more stabilizing than random ones. So the landscape is tilted — shaped like a rugged funnel, wide and high at the top where the unfolded ensemble lives, narrowing to the native state at the bottom.

On a funnel-shaped landscape, local motions are biased on average toward lower-free-energy, native-like regions. Different starting conformations can follow different routes, with intermediates and kinetic traps appearing as local minima. This energy-landscape picture, developed by Bryngelson, Wolynes, Dill, and others through the 1990s, resolves Levinthal's paradox without requiring an exhaustive search through all conformations.

The folding funnel tab of the interactive above lets you watch this directly: many chains are released from different points on a rugged landscape and converge on the same basin, and as you flatten the landscape toward Levinthal's golf course that convergence falls apart — almost no chain finds the native state.

Why folding is cooperative — and fast

Two features of real folding fall out of the funnel picture and deserve to be named, because they come up constantly.

The first is cooperativity. Why do small proteins fold in an all-or-nothing, two-state way, rather than gradually assembling one piece at a time? Because the stabilizing interactions are mutually reinforcing. A half-formed hydrophobic core is worse than useless: it exposes greasy surface to water (paying the entropic cost) without yet reaping the packing and hydrogen-bonding rewards that only a complete core provides. Partly folded states therefore sit higher in free energy than either the fully unfolded or fully folded ensembles, so the population piles up at the two ends and folding looks like a sharp, switch-like transition. Cooperativity is the thermodynamic signature of a funnel with a single deep basin.

The second is speed, and the funnel explains that too — but it also predicts which proteins fold fast. Because folding is a biased descent rather than a search, folding rates are governed less by protein size than by the topology of the native contacts. The contact order — how far apart along the sequence the residues that touch in the folded structure tend to be — turns out to predict folding rate remarkably well: proteins whose stabilizing contacts are mostly local (helices, which bond $i$ to $i{+}4$ ) fold in microseconds, while those relying on long-range contacts (large β-sheets, where distant strands must find each other) fold orders of magnitude slower. Along the way, many proteins pass through a molten globule — a compact intermediate that has already formed much of its secondary structure and buried its core, but whose side chains are still loosely packed and mobile. The molten globule is one of the bumps on the funnel wall: real, detectable, and on-pathway.

When folding goes wrong

Because stability is marginal and the funnel is rugged, folding is not foolproof, and its failures are medically enormous. A chain can fall into the wrong basin and misfold; misfolded or partly folded chains often expose hydrophobic surface that should have been buried, and those sticky surfaces make them aggregate. The most notorious aggregates are amyloid fibrils — highly ordered, stacked β-sheet assemblies (a "cross-β" structure) that are implicated in Alzheimer's, Parkinson's, and type II diabetes, and that in the case of prions can even template their own misfolding infectiously.

The cell fights this constantly. A class of proteins called molecular chaperones (Hsp70, the GroEL/GroES chamber, and others) don't fold their clients directly so much as give crowded, aggregation-prone chains a protected space and repeated chances to find the funnel, spending ATP to do it. There is a second way the cell tilts the odds: many proteins begin folding co-translationally, while the chain is still being extruded from the ribosome. Because the ribosome releases the sequence strictly N-terminus first, the early part of the chain can start folding before the later part even exists — a vectorial, staged process that can steer a chain down its funnel more reliably than dumping the whole sequence into solution at once, the way Anfinsen's experiment did. That folding needs this kind of active management inside the cell is a reminder that Anfinsen's clean test-tube result — sequence alone determines structure — is the thermodynamic ideal, while the kinetics, in the crowded and vectorial reality of a cell, sometimes need help.

Where this leaves us

A protein sequence, dropped into water, folds because the hydrophobic effect drives its greasy residues into a tightly packed core, releasing enough water entropy to pay for the chain's own loss of freedom; because backbone hydrogen bonds, satisfied economically as helices and sheets, organize the collapsed chain; and because van der Waals packing and a scattering of salt bridges and disulfides fine-tune and lock the result. The whole balance is delicate — a few $\mathrm{kcal/mol}$ — and the search succeeds not by luck but because evolution has shaped the energy landscape into a funnel that drains to a single native state. Levinthal's impossible search never had to happen.

We now know what a protein is made of and why it folds. What we have been describing loosely as "the fold" or "the native structure" actually has a precise, layered vocabulary — helices and sheets, the way they pack into domains, the way domains assemble into machines. Next time we make that hierarchy exact: the four levels of protein structure, from the local hydrogen-bond patterns that give helices and sheets their geometry, up to the symmetric multi-subunit assemblies that do most of the real work in a cell.