A prologue to protein structure prediction

Proteins are a major class of biological macromolecules. They carry out many of the catalytic, structural, transport, signaling, and regulatory functions required by cells. Much of this functional diversity begins with the enormous number of possible amino-acid sequences. A sequence constrains how a polypeptide folds, moves, and interacts with other molecules, although its environment and binding partners can also affect the structures it adopts.

Reflecting their diverse functions, proteins also have diverse shapes and sizes. Some proteins, such as hemoglobin, are globular, whereas others, such as collagen, are fibrous. A protein’s structure is closely connected to what it can do [Mattaini, 2020].

Protein structure prediction asks a deceptively simple question:

Given a protein’s amino-acid sequence, what three-dimensional structure is it likely to have?

From a computer-science point of view, we want to map a 1D sequence to a 3D geometric object.

Background on amino acids

So what is an amino acid? An amino acid is a small organic molecule that can serve as a building block of proteins. The standard amino acids share an alpha carbon bonded to an amino nitrogen, a carboxyl carbon, a hydrogen atom, and a variable side chain $\ce{R}$.

The conventional neutral scaffold is often written $\ce{NH2-CH(R)-COOH}$, although proline’s side chain closes back onto the amino nitrogen. Protonation states depend on the chemical environment; near physiological pH, free amino acids are commonly represented as zwitterions with positively charged amino ($\ce{NH3^+}$) and negatively charged carboxylate groups ($\ce{COO^-}$) [Reddy, 2026].

The side chain $\ce{R}$ is what mostly distinguishes one amino acid from another. Different side chains have different sizes, charges, polarities, and chemical behaviors.

For our purposes, the important idea is simple:

A protein sequence is not a sequence of interchangeable symbols. Each symbol corresponds to a chemically different residue.

You can inspect the amino acids in the interactive figure below.

The peptide bond

Amino acids are connected into chains by peptide bonds. A peptide bond is the covalent bond between the carbonyl carbon of one amino acid and the amino nitrogen of the next.

For formal atom accounting, peptide-bond formation can be pictured as removing $\ce{OH}$ from the carboxyl group of one amino acid and $\ce{H}$ from the amino group of another. The remaining atoms form a $\ce{C(=O)-N}$ linkage and $\ce{H2O}$. This bookkeeping picture explains which atoms remain in the chain; it is not a description of the ribosome’s reaction mechanism.

You can compare the free components and the connected product below.

Suppose we have the chain

Once amino acids have been incorporated into a chain, the units are called amino-acid residues or residues for short. In this example, methionine is one residue in the chain $\ce{MGSAKE}$.

N-terminus and C-terminus

Once many residues have been linked together, the result is a polypeptide chain.

A linear polypeptide chain has direction:

• The N-terminus is the end containing the terminal amino group.
• The C-terminus is the end containing the terminal carboxyl group.
• Protein sequences are conventionally written from the N-terminus to the C-terminus.

For our example,

• $\ce{M}$ is the N-terminal residue.
• $\ce{E}$ is the C-terminal residue.
• $\ce{M}$ is residue 1, $\ce{G}$ is residue 2, and so forth.

This N-terminus to C-terminus ordering is important because AlphaFold2 receives an ordered sequence, not an unordered collection of residues.

The four levels of protein structure

Before discussing each level, let us first look at a real protein structureShow sidenoteThe structure contains local patterns such as $\alpha$-helices and $\beta$-strands, together with the global arrangement that packs them into one fold.. The figure below shows crambin (PDB 1CRN), a compact 46-residue plant proteinShow sidenoteThe Protein Data Bank, or PDB, stores experimentally determined three-dimensional structures of proteins, nucleic acids, and other biological macromolecules..

The terms primary, secondary, tertiary, and quaternary describe different levels of protein organization.

The amino-acid sequence is the primary structure. Recurring backbone patterns, such as $\alpha$-helices and $\beta$-strands, form secondary structure. The complete arrangement of one chain is its tertiary structure. When multiple polypeptide chains assemble, their arrangement is the quaternary structure of the complex [Branden et al., 1999].

Primary structure

The primary structure is the ordered 1D amino-acid sequence. Reusing our example,

This is the primary structure of the example chainShow sidenoteYou can track the current structural level in the figure below. .

Order matters. $\ce{MGSAKE}$ and $\ce{MEASKG}$ are different sequences and generally correspond to different polypeptides.

From a machine-learning point of view, the target sequence is the starting input. It provides each residue’s amino-acid identity and order. Once the residues are indexed, we can compute sequence separation, such as $j-i$, but the corresponding separation in 3D is still unknown.

Secondary structure: local backbone patterns

Secondary structure refers to recurring local conformations of the protein backbone, commonly characterized by their backbone geometry and hydrogen-bonding patternsShow sidenote.

The two most common regular patterns are:

• $\alpha$-helix: Notice the dotted lines in the top right of the diagram. In a standard $\alpha$-helix, the carbonyl oxygen of residue $i$ hydrogen-bonds with the backbone $\ce{N-H}$ group of residue $i+4$. Repeating this pattern coils a nearby stretch of the chain into a helix.
• $\beta$-sheet: It looks like folded paper or a zig-zagging plane. It is formed by multiple segments of the protein chain (called $\beta$-strands) lining up next to each other. Those strands can be far apart in the sequence even though they are neighbors in 3D.

The important distinction is that an $\alpha$-helix is built from one continuous stretch of sequence, whereas neighboring $\beta$-strands may come from distant parts of the chain.

Secondary-structure labels describe recurring backbone geometry and hydrogen-bonding patterns. A residue can therefore have a local role, such as belonging to a helix, and a global role determined by how that helix is positioned relative to the rest of the protein.

Tertiary structure: the complete fold of one chain

Tertiary structure is the overall three-dimensional arrangement of a single polypeptide chain. It describes how helices, sheets, loops, and side chains are positioned relative to one anotherShow sidenote.

Describing tertiary structure requires long-range relationships. For example:

Residue 15  ─...─...─...─...─ Residue 120

These residues are far apart in sequence, but folding may bring them close together in space:

sequence distance: large
spatial distance:  small

This is one of the central difficulties of protein-structure prediction. Local sequence neighborhoods are not enough. The model must infer which distant residues become spatial neighbors and how they are oriented relative to one another.

That observation will later motivate one of AlphaFold2’s central design choices: representing relationships between pairs of residues, not only features attached to individual residues.

Quaternary structure: multiple chains together

Many functional proteins contain more than one polypeptide chain. Each chain has its own primary, secondary, and tertiary structure. The arrangement of multiple chains is called the quaternary structureShow sidenote.

For example:

Chain A + Chain B + Chain C → protein complex

In structure files, chains are commonly labelled A, B, C, and so on. A chain is also often called a subunit when it is part of a multi-chain complex.

The original AlphaFold2 system described in 2021 primarily targeted single-chain prediction [Jumper et al., 2021]. Predicting a multi-chain complex is a related problem, but it introduces additional questions: which chains interact, where their interfaces lie, and how the chains are arranged.

From a chain to three-dimensional geometry

The four structural levels tell us what scale we are describing. To understand the output of a structure-prediction model, we also need a more mechanical description of how the atoms are arranged.

A structure model records atomic coordinates, but those coordinates are one representation of the molecule rather than a preferred pose in space. We will approach that representation in two steps: first through the internal rotations of the chain, then through three-dimensional coordinates and local frames.

Each residue contributes three backbone atoms: the peptide nitrogen $\ce{N}$, the alpha carbon $\ce{C}\alpha$, and the carbonyl carbon $\ce{C'}$ [Branden et al., 1999]. For residue $i$, we will write them as $\ce{N}_i$, $\ce{C}\alpha_i$, and $\ce{C'}_i$. The side chain branches from $\ce{C}\alpha_i$, while the peptide bond connects $\ce{C'}_i$ to $\ce{N}_{i+1}$.

A backbone with rotatable joints

Bond lengths and bond angles fluctuate, but they stay close to strongly preferred values. If we temporarily hold them fixed, most of the backbone’s remaining flexibility comes from rotations around bonds. These rotations are described by torsion angles, also called dihedral angles.

A dihedral angle is determined by four consecutively bonded atoms. The first three atoms define one plane, the last three define another, and the signed angle between those planes measures the twist around the bond shared by the middle two atoms. For an internal residue $i$, the three backbone torsions are:

• $\phi_i$ (phi) uses $\ce{C'}_{i-1}-\ce{N}_i-\ce{C}\alpha_i-\ce{C'}_i$ and measures rotation around $\ce{N}_i-\ce{C}\alpha_i$.
• $\psi_i$ (psi) uses $\ce{N}_i-\ce{C}\alpha_i-\ce{C'}_i-\ce{N}_{i+1}$ and measures rotation around $\ce{C}\alpha_i-\ce{C'}_i$.
• $\omega_i$ (omega) uses $\ce{C}\alpha_i-\ce{C'}_i-\ce{N}_{i+1}-\ce{C}\alpha_{i+1}$ and measures rotation around the peptide bond $\ce{C'}_i-\ce{N}_{i+1}$.

With this indexing, $\phi_1$ is undefined because there is no preceding residue, while $\psi_N$ and $\omega_N$ are undefined because there is no following residue [IUPAC-IUB Commission on Biochemical Nomenclature, 1970].

The angles $\phi$ and $\psi$ provide most of the backbone’s conformational freedom, so we will focus on themShow sidenoteThe peptide bond has partial double-bond character, which makes it nearly planar and strongly restricts $\omega$. Most peptide bonds are near $180^\circ$ in the trans configuration. The cis configuration, near $0^\circ$, is uncommon but occurs more often before proline than before other residues. . Side chains introduce additional torsions, conventionally named $\chi_1,\ldots,\chi_4$ when present [Branden et al., 1999].

Rotating $\phi$ or $\psi$ changes the direction of the backbone, but these angles cannot vary freely. Some choices bring non-bonded atoms too close together, creating a steric clashShow sidenoteAtoms have a finite effective size. When two non-bonded atoms are forced much closer than their preferred separation, repulsion rises steeply; structural models call such an implausibly close contact a steric clash..

The same structure can have different coordinates

At the scientific level, protein structure prediction asks for a plausible three-dimensional structure of a polypeptide chain with amino-acid sequence

where $\mathcal{A}$ is the alphabet of standard residue types. For example, the six-residue sequence $\ce{MGSAKE}$ has $N=6$.

What would it mean for a model to return a structure? For each residue $i$, let $\mathcal{H}(a_i)$ be the set of heavy atoms present in residue type $a_i$. The main geometric output is

In plain language, we predict a three-dimensional coordinate for every non-hydrogen atom represented in the polypeptide chain. A simplified coordinate table looks like this:

Atom              x        y        z
Residue 1 N       ...
Residue 1 C-alpha ...
Residue 1 C       ...
Residue 1 O       ...
Residue 1 C-beta  ...
Other side-chain atoms ...
Residue 2 N       ...
...

These numbers are measured in a global coordinate system: a chosen origin together with three perpendicular axes. We could rotate a structure on the screen or move it to the other side of the scene without changing any bond, angle, or spatial relationship within the molecule.

Suppose every atomic coordinate $x$ is transformed in the same way:

where $Q\in\operatorname{SO}(3)$ is a rotation and $s\in\mathbb{R}^3$ is a translationShow sidenoteA matrix in $\operatorname{SO}(3)$ has orthonormal columns and determinant equal to $1$. It preserves lengths and angles without reflecting space.. The individual coordinate triples change, but the protein’s internal geometry does not. For example, the distance between atoms $x_a$ and $x_b$ remains:

Bond lengths, bond angles, and torsion angles are likewise unchanged. The two coordinate sets therefore describe the same structure in different global poses.

Why a reflection is not another pose

Why did we require $\det(Q)=1$? A matrix with orthonormal columns and determinant $-1$ reverses orientation: it describes a reflection, possibly followed by a rotation. A reflection still preserves distances, so distance alone cannot tell a structure from its mirror image. It does, however, reverse handednessShow sidenoteThe labels L and D describe an amino acid’s configuration by a formal relationship to L- or D-glyceraldehyde; they do not mean that the molecule literally points left or right. By convention, an unmarked amino-acid symbol in a protein sequence denotes the L configuration when the residue is chiral. Glycine is achiral. [IUPAC-IUB Joint Commission on Biochemical Nomenclature, 1984] .

An object is chiral if it cannot be placed on top of its mirror image using only rotations and translations [International Union of Pure et al.]. Your left and right hands illustrate the exact geometric distinction: they have the same pattern of distances, but no rotation of one hand makes it coincide with the other. They have opposite handedness.

The same distinction appears in amino acids. Except for glycine, the $\ce{C}\alpha$ atom in a standard amino acid is bonded to four different groups, giving the residue a particular three-dimensional handedness. Reflecting all of its coordinates reverses that arrangement. The result is the mirror-image configuration, not the original residue viewed from another direction.

A reflection preserves all pairwise distances. Distances alone therefore cannot distinguish a protein from its mirror image; a complete geometric description must retain some information about direction or handedness.

Let each residue carry its own coordinate system

Global coordinates contain the geometry we need, but they also contain an arbitrary choice we do not care about: a global origin and its axes. We would like to describe where an atom lies relative to a residue, in a way that does not change when the whole molecule is moved.

The three ordered, non-collinear backbone atoms $\ce{N}_i$, $\ce{C}\alpha_i$, and $\ce{C'}_i$ provide enough information to attach a local coordinate frame to residue $i$. We can place its origin at $\ce{C}\alpha_i$, point one axis toward $\ce{C'}_i$, and use $\ce{N}_i$ to determine the backbone plane. There are two possible directions perpendicular to that plane, so we choose one handedness convention and use it for every residue. The main AlphaFold2 article will develop the exact construction; here, we only need the idea that every residue receives its own origin and axes.

Let the frame of residue $i$ be represented by a rotation $R_i$ and translation $t_i$. Here, $t_i$ is the global position of the frame’s origin, and the columns of $R_i$ are unit vectors pointing along its three local axes. The frame maps a point with local coordinates $u$ into global coordinates:

Because $R_i$ is a rotation, $R_i^{-1}=R_i^\top$. We can therefore express a global point $x$ in residue $i$‘s frame by reversing the transformation:

This equation answers a concrete geometric question: where is atom $x$ when residue $i$ is used as the origin and its backbone determines the axes?

Now move the whole protein by a rotation $Q$ and translation $s$. The atom and residue frame move together:

The coordinates seen from residue $i$ do not change:

The global coordinates changed, but the atom’s position relative to the residue did not. A local frame removes the arbitrary global pose while preserving directional information that distances alone would discard.

Torsion angles and local frames now have separate roles. Torsion angles describe how bonded parts of the chain turn around particular bonds. A residue-local frame describes the position and orientation of other atoms or residues from that residue’s point of view.

In the Torsion angles view, compare the $\alpha$-helix and $\beta$-strand presets and inspect which four atoms define each angle. Then switch to Local frame and toggle the global pose. The molecule moves, but the coordinates measured in the selected residue’s frame remain the same.

References