Distance Geometry

Distance geometry represents a structure through pairwise distances rather than absolute coordinates. It is useful when the coordinate frame is arbitrary but relative spatial relationships matter.

For coordinates $X=(x_1,\dots ,x_N)$, the distance matrix is:

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

Under a rigid transform $x_i^{\prime }=R{x}_i+t$:

$$D_{ij}^{\prime }=\Vert x_i^{\prime }-x_j^{\prime }{\Vert }_2=D_{ij}$$

So $D$ is invariant to rotation and translation.

Why It Matters

• Protein contact maps, residue distances, molecular conformers, and pocket-ligand contacts can be expressed through distances.
• Distance features are stable when absolute orientation is irrelevant.
• Distance matrices can support graph construction, edge features, and geometric constraints.

Limits

Distance-only representations can lose information:

• Orientation can be missing.
• Chirality can be ambiguous if reflections are not constrained.
• Multiple coordinate sets can share similar distances under noise or missing entries.
• Long-range distance matrices scale as $O(N^2)$.

Validity Conditions

Not every matrix is a Euclidean distance matrix. A valid distance matrix should be symmetric, nonnegative, zero on the diagonal, and geometrically realizable:

$$D_{ij}=D_{ji},D_{ii}=0,D_{ij}\ge 0.$$

After double-centering, the Gram matrix should be positive semidefinite for exact Euclidean distances:

$$B=-\frac{1}{2}J{D}^{(2)}J⪰0.$$

Predicted distances can violate these constraints, so a model may need projection, coordinate reconstruction, or geometry-aware losses.

From Distances to Coordinates

Classical multidimensional scaling starts from squared distances and builds a centered Gram matrix:

$$B=-\frac{1}{2}J{D}^{(2)}J,J=I-\frac{1}{N}11^{\top }$$

If $B$ is positive semidefinite, coordinates can be recovered from its top eigenvectors:

$$B=U\Lambda U^{\top },X=U_k{\Lambda }_k^{1/2}$$

This shows the link between distances, inner products, and coordinates.

Modeling Uses

Use	Representation	Evaluation Risk
contact prediction	thresholded distances	threshold choice can dominate score
conformer generation	pairwise atom distances	chirality and local chemistry may be invalid
protein structure	residue distance map	missing residues and chain policy matter
docking	protein-ligand distance constraints	known pose or ligand-defined pocket can leak
graph construction	radius or kNN edges	cutoff changes topology and model input

Practical Checks

• Are all distances observed, or only local/contact distances?
• Does the task need orientation, handedness, or local frames?
• Are distances used as labels, input features, graph edges, or constraints?
• Does the model output a valid geometry, not only accurate pairwise distances?
• Are predicted distances symmetric and geometrically realizable?
• Are chirality, orientation, and atom/residue mapping needed for the task?

Related

• Geometry
• Contact map
• Graph construction
• Coordinate frame
• Invariant feature
• Coordinate modeling contract
• 3D structure