3D Structure

3D structure is a modality where examples include coordinates, distances, orientations, surfaces, or spatial relationships. It is central to protein structures, ligand conformers, protein-ligand complexes, point clouds, and geometry-aware vision.

A coordinate representation is:

$$X=(x_1,\dots ,x_N),x_i\in {\mathbb{R}}^3$$

Many tasks should be invariant or equivariant to rigid transforms:

$$x_i^{\prime }=R{x}_i+t$$

where $R$ is a rotation matrix and $t$ is a translation.

Pairwise distances are often used as invariant features:

$$d_{ij}=\Vert x_i-x_j{\Vert }_2$$

Key Ideas

• The absolute coordinate frame is often arbitrary; relative geometry usually carries the physical meaning.
• Coordinates can be represented directly, converted into distances, voxelized, or converted into graphs.
• Some outputs should be invariant scalars; others should be equivariant coordinates or vectors.
• Preprocessing choices such as centering, alignment, cropping, or pocket selection can change the task.
• Chirality and reflection symmetry need special care for molecular settings.

Representation Choices

Representation	Model Sees	Useful For	Main Risk
raw coordinates	atom/residue/point coordinates	coordinate prediction, refinement, equivariant models	arbitrary frame and missing elements
pairwise distances	$d_{ij}=\Vert x_i-x_j{\Vert }_2$	invariant scalar models, contact maps	loses chirality and orientation
local frame features	coordinates relative to anchors	residue or pocket local geometry	frame construction may leak unavailable context
graph from coordinates	nodes plus distance/contact edges	molecular and protein graph models	cutoff or neighbor rule changes input
grid or voxel	discretized density or occupancy	CNN-style spatial models	resolution and alignment artifacts
surface or mesh	geometric boundary	pockets, shape complementarity	surface generation parameters affect claims

Coordinate Contract

For reusable notes, state:

$$\mathcal{X}=(X,A,M,c)$$

where $X\in {\mathbb{R}}^{N\times 3}$ stores coordinates, $A$ maps rows to atoms/residues/points, $M$ is a validity mask, and $c$ stores source context such as experimental, predicted, docked, generated, or simulated.

The transform rule should be explicit:

$$X^{\prime }=X{R}^{\top }+1t^{\top }$$

Scalar outputs such as affinity, energy, class probability, or rank should be invariant:

$$f(X^{\prime })=f(X)$$

Coordinate, force, direction, or vector outputs should be equivariant:

$$g(X^{\prime })=g(X)R^{\top }$$

Structure-Based Risks

• A pocket cropped around a known ligand pose may leak deployment-unavailable information.
• Aligning structures to a template can hide whether the model is truly invariant.
• Missing atoms, unresolved loops, alternate locations, protonation states, and chain selection change the input object.
• Symmetric atoms or repeated residues require explicit mapping before coordinate losses or RMSD.
• Reflection invariance can be wrong for chiral molecules and protein structures.

Practical Checks

• Are coordinates experimental, predicted, simulated, docked, or generated?
• Is the task invariant to rotation, translation, and reflection?
• Are missing atoms, alternate conformations, or low-confidence regions handled?
• Does the model use coordinates, distances, graphs, surfaces, grids, or mixed representations?
• Does evaluation check geometric validity, not only scalar metrics?
• Is the atom/residue/point mapping fixed before computing coordinate losses?
• Is pocket or crop definition available at deployment?

Related

• Modality representation
• PointNet
• NeRF
• Coordinate modeling contract
• Geometry
• Geometric deep learning
• Coordinate frame
• Distance geometry
• Equivariance
• Invariant feature
• Equivariant feature
• Structure
• Protein-ligand complex
• Pose quality