Geometry for Structure Modeling

Structure modeling에서 geometry는 그림이 아니라 coordinate, distance, frame, symmetry, constraint를 다루는 언어입니다. Protein, ligand, pocket, complex를 다룰 때는 어떤 값이 좌표계에 의존하고 어떤 값이 보존되어야 하는지 먼저 정해야 합니다.

$$X\in {\mathbb{R}}^{n\times 3},X^{\prime }=RX+t,R\in SO(3),t\in {\mathbb{R}}^3$$

순수 수학으로서의 symmetry와 group action은 Geometry and Symmetry에 둡니다. 여기서는 그 언어를 protein, ligand, pocket, complex의 structure modeling claim에 적용합니다.

Geometry Contract

Structure modeling note는 geometry를 쓰기 전에 아래 contract를 고정해야 합니다.

$${\mathcal{C}}_{\mathrm{geom}}=(\mathcal{U},X,A,F,G,V,M)$$

Part	Meaning	Example
$\mathcal{U}$	biological unit	atom, residue, ligand, pocket, chain, complex
$X$	coordinate tensor	atom coordinates, residue C-alpha coordinates, ligand conformer
$A$	atom or residue identity mapping	atom correspondence, residue index, chain ID
$F$	coordinate frame	global frame, pocket-aligned frame, local residue frame
$G$	allowed symmetry group	rotation, translation, permutation, ligand symmetry
$V$	validity constraints	bond length, chirality, clash, pocket availability
$M$	metric and alignment rule	RMSD, lDDT, clash count, strain, interaction geometry

Without this contract, the same number can support different claims. For example, an RMSD value means different things under ligand-only alignment, pocket alignment, global protein alignment, or symmetry-aware atom matching.

Core Objects

Object	Meaning	Route
Coordinate matrix	atom/residue/point 좌표 $X$	Coordinate frame
Distance	pairwise geometry invariant	Distance geometry
Frame	local coordinate basis or global reference	Coordinate frame
Group action	rotation, translation, permutation이 object에 작용하는 방식	Symmetry group
Invariance	input이 변해도 output scalar가 보존됨	Equivariance
Equivariance	input transform에 맞춰 output도 transform됨	Equivariance

Coordinate Source

Coordinate source changes the meaning of the learning target and the evaluation claim.

Source	Useful for	Main risk
experimental structure	reference geometry, known complex, benchmark target	resolution, missing atoms, alternate conformations
predicted protein structure	scalable target representation	prediction error becomes hidden input noise
docked pose	generated candidate geometry	docking bias or circular evaluation
minimized pose	physically relaxed geometry	force-field assumptions alter the target
generated coordinates	model output	validity and constraint repair must be counted
ligand-defined pocket	local binding-site context	test ligand leakage if unavailable at inference

When coordinates come from different sources, do not treat them as the same label unless the note explains the transformation and filtering policy.

Quantity Type

Quantity	Transform behavior	Example
Scalar invariant	$s(RX+t)=s(X)$	affinity, energy, RMSD after alignment
Vector equivariant	$v(RX+t)=Rv(X)$	force, displacement
Coordinate equivariant	$Y(RX+t)=RY(X)+t$	generated pose, denoised coordinates
Pair distance	unchanged by rigid motion	contact, distance matrix
Local frame	rotates with structure	residue frame, ligand frame

The required model behavior follows the target type:

$$\begin{array}{rlrl}\text{scalar target:}& & f(RX+t)& =f(X)\\ \text{vector target:}& & v(RX+t)& =Rv(X)\\ \text{coordinate target:}& & Y(RX+t)& =RY(X)+t\end{array}$$

Structure Claim Map

Claim	Required geometry check
Pose prediction	coordinate frame, atom mapping, RMSD alignment, clash, strain
Affinity scoring	pose quality와 score meaning을 분리
Structure generation	generated coordinates의 distance, bond, chirality, clash constraint
Equivariant model	scalar output은 invariant, coordinate/vector output은 equivariant
Pocket modeling	pocket definition이 inference time에 사용 가능한 정보만 쓰는지 확인

Metric and Alignment

Geometry metrics are not interchangeable. The alignment rule is part of the metric.

Metric	Good for	Check
ligand RMSD	pose accuracy against a reference ligand	atom mapping, symmetry, alignment rule
pocket RMSD	local receptor movement	residue selection and missing atoms
distance error	invariant geometry comparison	loss of orientation or chirality information
clash count	physical plausibility	atom radii, hydrogens, protonation state
strain or torsion quality	ligand conformer plausibility	force field and tautomer/protonation policy
interaction pattern	contact or pharmacophore consistency	distance thresholds and pocket definition

For a generated set, report the denominator:

$$\text{valid pose rate}=\frac{\#\text{poses passing geometry checks}}{\#\text{generated poses before filtering}}$$

Common Equations

Distance matrix:

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

Invariant scalar:

$$f(RX+t)=f(X)$$

Equivariant coordinate output:

$$g(RX+t)=Rg(X)+t$$

Force-like vector:

$$F_i(RX+t)=R{F}_i(X)$$

Coordinate update:

$$X^{(k+1)}=X^{(k)}+\Delta X_{\theta }(H^{(k)},X^{(k)},E)$$

If $\Delta X_{\theta }$ is coordinate displacement, it must transform equivariantly under rotation. If the model predicts only a scalar score, the score should be invariant to global rigid motion.

Leakage Checks

Check	Why
ligand-defined pocket	test ligand pose가 pocket extraction에 들어가면 deployment보다 쉬워짐
template structure	homolog/template이 split을 넘어가면 structure task가 과대평가됨
coordinate source	predicted, experimental, docked, minimized coordinate를 섞으면 label meaning이 바뀜
atom mapping	generated ligand와 reference ligand의 atom correspondence가 불명확하면 RMSD가 흔들림
alignment rule	global alignment, pocket alignment, ligand-only alignment가 다른 claim을 만듦

Where This Fits

If the note is about	Put it in
group action, metric space, symmetry definition	[[math/geometry-symmetry
equivariant layers or coordinate message passing	[[concepts/geometric-deep-learning/index
protein, ligand, pocket, complex geometry	this page or [[molecular-modeling/structure-based/index
docking pose plausibility	[[concepts/sbdd/pose-quality
model objective and sampler	[[ai/generative-models

Related

• Geometry and Symmetry
• Structure-Based Modeling
• Computational Biology Geometry Route
• Pocket definition contract
• Equivariance