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From Surfaces to Volumes: Registered Geometry for Protein Representation Learning

Siyuan Chen, Cai Zhou, Jinrui Zhang, Zhaokang Liang, Taku Komura, Wojciech Matusik, Stephen Bates, Tommi Jaakkola, Wengong Jin, Peter Yichen Chen, Minghao Guo

arXiv:2609.36277v1cs.AI

TL;DR

Protein geometry models capture exposed molecular shape but do not explicitly represent residue-wise internal volume in a shared coordinate system. Protein-TetSphere registers residue-specific volumes to a common tetrahedral reference and Laplacian basis, then integrates those features with surface and chemical information. It improves results across pocket classification, interface prediction, and binder design.

  • Problem

    Surface- and neighborhood-based geometry models do not explicitly represent how enclosed three-dimensional volume is distributed among residues, and irregular residue volumes lack a common parameterization.

    Source: §1 · pp. 1–2

  • Method

    Protein-TetSphere fits residue-specific volumes to a shared fixed-topology tetrahedral reference and encodes their deformations in a common Laplacian basis for integration with surface and chemical features.

    Source: §1 · p. 2

  • Results

    Registered volumetric geometry improves protein recognition, interface prediction, and binder-design performance; pocket balanced accuracy rises from 0.795 ± 0.005 with AtomSurf to 0.826 ± 0.004 with the full model.

    Source: §1 · p. 2, §4.2 · p. 6, §5 · p. 9

  • Takeaways & Limitations

    Across the evaluated settings, registered volumetric geometry complements surface-based representations for protein recognition, interaction prediction, and design.

    Source: §5 · p. 9

    The study focuses on static protein structures; modeling time-varying residue-wise volumes over molecular-dynamics trajectories is left as a future direction.

    Source: §5 · p. 9

Abstract

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Existing protein geometry models typically represent molecular surfaces using local geometric features such as sampled points, normals, and curvature. While effective for capturing exposed molecular shape, these representations do not explicitly model the volumetric organization beneath the surface or provide a consistent coordinate system for residue-wise volumetric structure. We introduce Protein-TetSphere, a registered residue-wise volumetric representation for proteins. Each protein chain is tetrahedralized to obtain local volumetric regions associated with individual residues, which are then registered to a shared fixed-topology tetrahedral reference and represented in a common Laplacian basis. This registration establishes consistent volumetric coordinates across residues, enabling local three-dimensional deformation to be integrated with surface and chemical information in a multimodal protein representation. We evaluate Protein-TetSphere on ligand-binding pocket classification, protein--protein interface prediction, and de novo protein binder design. Across the three tasks, Protein-TetSphere improves ligand-binding pocket balanced accuracy from $0.795$ to $0.826$, Pinder-Pair/Site AUROC from $0.914/0.852$ to $0.932/0.866$, and binder-design success from $14.95\%$ to $19.90\%$ on the BoltzGen Challenge Set and from $27.62\%$ to $32.19\%$ at the ProtDBench backbone level. These results show that registered volumetric geometry provides complementary spatial information beyond molecular surfaces across protein recognition, interaction, and design.

1 INTRODUCTION

Existing geometry representations capture exposed protein shape but do not explicitly describe how volume is organized among residues. Protein-TetSphere registers residue-associated volumes to a shared tetrahedral reference, and the paper evaluates the representation across recognition, interaction, and binder-design tasks.

1 INTRODUCTION

  • Current atom-, neighborhood-, and surface-based representations do not explicitly describe residue-wise enclosed volume, leaving packing, burial, thickness, and spatial extent only indirectly captured.Structurally distinct inhibitors can present similar target-facing interfaces despite different underlying volumetric organization.Source: §1 · p. 1, Figure 1 · p. 2
  • Irregular residue-associated volumes vary in shape and size, so independently constructed local meshes lack the common parameterization needed for consistent residue-wise representation.The introduction hypothesizes that registration to a shared tetrahedral reference can express these heterogeneous geometries as a structured family of deformations.Source: §1 · p. 2
  • Protein-TetSphere fits residue-specific volumes to a shared fixed-topology tetrahedral reference and encodes their deformations in a common Laplacian basis.The resulting volumetric features complement molecular-surface and chemical information.Source: §1 · p. 2
  • The representation is evaluated on ligand-binding pocket classification, protein–protein interface prediction, and de novo binder design, with improvements reported across the tasks.For binder design, it supplies geometric teacher targets to a generative model during training.Source: §1 · p. 2

2 RELATED WORK

Prior protein geometry work represents atomic neighborhoods, molecular surfaces, and volumetric occupancy, while protein pretraining uses sequence and structural signals. Protein-TetSphere adds a shared residue-wise coordinate system for volumetric encoding.

2 RELATED WORK

  • Graph, atom-centered, and surface-based methods capture neighborhoods and exposed molecular geometry but do not explicitly represent enclosed three-dimensional volume at residue level.Examples include surface fingerprints and meshes, graph-and-surface encoders, and local structural-neighborhood approaches.Source: §2 · p. 3
  • Voronoi and Delaunay partitions and voxel grids capture protein packing, cavities, or binding sites, but do not provide shared residue-wise deformation coordinates.Protein-TetSphere adapts tetrahedral representations by registering residue-specific regions to a shared fixed-topology reference and Laplacian basis.Source: §2 · p. 3
  • Sequence and structure pretraining learn transferable protein representations, and Protein-TetSphere pretrains residue-wise volumetric encoders for reuse in prediction and binder design.The related work discusses sequence models, structural pretraining, and equivariant multimodal pretraining as complementary approaches.Source: §2 · p. 3

3 METHOD

Protein-TetSphere fits a tetrahedral volume to each residue, encodes its deformation in a shared spectral basis, and integrates those features with other molecular signals. The pretrained representation is adapted to prediction tasks and used as geometric supervision for binder generation.

3.1 RESIDUE-WISE REGISTERED VOLUMETRIC REPRESENTATION

  • Each residue-specific volume is fitted in two stages: first to its molecular surface, then to its region from whole-chain tetrahedralization, while preserving template connectivity.The fitting progression is illustrated in Figure 2.Source: §3.1 · pp. 3–4, Figure 2 · p. 4

3.2 SPECTRAL ENCODING OF TETSPHERE DEFORMATIONS

  • Shared template topology and vertex ordering allow fitted deformations to use a common Laplacian basis; 64 modes reconstruct fitted volumes within about two percent of a residue’s longest extent.Figure 3 compares fitted TetSpheres with leading-mode reconstructions and visualizes per-vertex error.Source: §3.2 · pp. 4–5

3.3 PROTEIN-TETSPHERE REPRESENTATION PRETRAINING

  • Spectral coefficients are processed as mode tokens and pooled into residue embeddings, then fused with surface and chemical features and processed with molecular-context tokens in a Pairformer-style trunk.Masked multimodal pretraining reconstructs TetSphere deformation coefficients alongside surface, identity, distance, and bond targets.Source: §3.3 · p. 5

3.4 ADAPTATION TO PROTEIN LEARNING TASKS

  • For pocket and interface prediction, pretrained representations augment AtomSurf features or task-specific logits; for binder design, a frozen teacher supplies representation targets during training rather than serving as an inference-time input.The teacher provides single and pair targets to BoltzGen, with learned projectors aligning generator features to the pretrained representation spaces.Source: §3.4 · pp. 5–6

4 EXPERIMENTS

Protein-TetSphere is evaluated on ligand-pocket classification, protein–protein interface prediction, and binder design. The full model improves reported results across these tasks, including binder success on both design benchmarks.

4.2 LIGAND-BINDING POCKET CLASSIFICATION

  • 0.826 ± 0.004 vs AtomSurf’s 0.795 ± 0.005 balanced accuracy: the full model leads on ligand-binding pocket classification.The no-TetSphere control reaches 0.807 ± 0.003, and the full model’s gains over AtomSurf and that control are 0.031 and 0.019.Source: Table 1 · p. 6, §4.2 · p. 6

4.3 PROTEIN–PROTEIN INTERFACE PREDICTION

  • 0.932 ± 0.001/0.866 ± 0.001 vs 0.914 ± 0.002/0.852 ± 0.002 AUROC: the full model improves Pinder-Pair/Pinder-Site over AtomSurf.Pinder-Pair evaluates residue-pair contacts; Pinder-Site evaluates residue-level interfaces, and representative high-confidence contact predictions show 28–30 true contacts per panel under the 5 Å criterion.Source: Table 2 · p. 7, §4.3 · p. 7, Figure 5 · p. 7

4.4 DE NOVO PROTEIN BINDER DESIGN

  • Because binder geometry is unavailable before generation, the model aligns BoltzGen hidden states to pretrained single and pair representations during training while retaining BoltzGen’s generation objective.The pretraining model is used only during training.Source: §4.4 · p. 7
  • 19.90% vs 14.95% on the Challenge Set and 32.19% vs 27.62% at ProtDBench backbone level: Ours leads BoltzGen on both.These are the highest results among the four training arms.Source: Table 3 · p. 8, §4.4 · p. 8
  • The no-TetSphere control reaches 11.73% sequence-level and 26.46% backbone-level ProtDBench success and 16.90% Challenge Set success; Ours reaches 14.67%, 32.19%, and 19.90%.Controls separate volumetric supervision from continued training and representation alignment.Source: §4.4 · p. 8
  • On two ten-target design benchmarks, gains span targets: Ours beats BoltzGen on 7/10 ProtDBench and 6/10 Challenge Set targets, and +Cont. on 9/10 and 7/10.The target-level comparisons use the same four arms and retain all ten targets in both panels.Source: §4.4 · p. 8, Figure 6 · p. 9
  • On ProtDBench, the first three criteria jointly rise from 18.20% to 23.16%, but the bound–unbound RMSD gate narrows the sequence-level advantage.Backbone-level success nevertheless rises from 27.62% to 32.19%; on the Challenge Set, binder-only RMSD-stage survival is 23.50% vs 17.45% and median RMSD is 1.334 vs 1.492 Å.Source: §4.4 · p. 9
  • Across both benchmarks, Ours improves key geometric quality criteria while preserving structural diversity; it has the highest diversity-adjusted cluster pass rates at TM 0.8.The no-TetSphere control is more diverse at TM 0.6, but has lower ProtDBench success.Source: §4.4 · p. 9

5 CONCLUSION AND LIMITATIONS

Protein-TetSphere registers residue-wise volumes to a shared tetrahedral reference and uses their Laplacian representation alongside surface and chemical information. The study reports improvements across protein recognition, interaction prediction, and binder design, while focusing on static structures.

5 CONCLUSION AND LIMITATIONS

  • Registered residue-wise volumes are fitted to a shared fixed-topology reference and encoded in a common Laplacian basis.The representation is evaluated alongside surface and chemical information.Source: §5 · p. 9
  • Protein-TetSphere is evaluated through matched AtomSurf comparisons for pocket classification and PPI prediction, and as geometric supervision for binder generation.The design evaluations use ProtDBench and the BoltzGen Challenge Set.Source: §5 · p. 9
  • The study focuses on static protein structures; proposed extensions include molecular-dynamics trajectories and multiscale representations of biomolecular assemblies.These directions could broaden volumetric representations toward dynamic modeling, interface design, and functional protein generation.Source: §5 · p. 9

A CONSTRUCTION AND IMPLEMENTATION DETAILS

The appendix details a deterministic construction pipeline for residue-wise volumetric targets, their fitting to a shared tetrahedral template, and their spectral representation.

A.1 SURFACE CONSTRUCTION AND COORDINATE CONVENTION

  • The chain surface is built from present heavy atoms using a 1.5 Angstrom probe, centered, and converted between normalized and world coordinates.The construction excludes ligands, nucleic acids, solvent, neighboring chains, atom completion, and OXT synthesis.Source: §A.1 · p. 13

A.2 MSMS AND FTETWILD TARGETS

  • Residue targets combine current-residue atoms with neighboring peptide atoms, then select tetrahedra near propagated residue labels from the chain-level fTetWild volume.The radius-1.0 label expansion can produce overlapping residue targets in shared centered coordinates.Source: §A.2 · p. 13

A.3 REFERENCE TEMPLATE AND FITTING

  • Each residue receives a disconnected copy of the 532-vertex, 1,727-tetrahedron reference, initialized at rest scale 0.6.Local MSMS fitting uses bidirectional point-to-point displacements; chain-level fitting uses one-sided closest-point-to-triangle displacements.Source: §A.3 · p. 13
  • The two fitting procedures use maximum budgets of 1,000 and 3,000 vertex updates, with convergence-based early termination and smooth-energy coefficient 3 × 10−4/R.The chain-level fit continues from the in-memory local fit and resets the AdamUniform state.Source: §A.3 · p. 14
  • After updates, Gauss–Seidel projection repairs inverted or collapsed tetrahedra, followed by up to eight cleanup rounds before serialization.The update projection uses at most 10 sweeps and a relative floor of 0.1 times each rest tetrahedron's absolute signed volume.Source: §A.3 · p. 14

A.4 SPECTRAL BASIS AND LOCAL FRAMES

  • The template graph's Laplacian supplies the first 64 non-constant eigenvectors, while omitting the constant mode removes global translation.Mode/channel normalization statistics come from valid training residues and are reused unchanged for validation and test examples.Source: §A.4 · p. 14

B MULTIMODAL MASKED-PRETRAINING ARCHITECTURE

The multimodal pretraining model combines protein-residue surface, chemical, and registered TetSphere views with nucleotide and ligand partner context. Type-specific encoders form tokens that are jointly contextualized through geometry-aware single and pair states.

B MULTIMODAL MASKED-PRETRAINING ARCHITECTURE

  • Protein residues combine surface geometry, amino-acid-derived chemistry, and registered TetSphere deformations, while nucleotide and ligand tokens provide partner context when present.Protein residues are the primary representation targets; partner tokens receive type-specific identity supervision without synthetic protein-surface, TetSphere, or chemistry inputs.Source: B MULTIMODAL MASKED-PRETRAINING… · p. 14

B.1 TYPE-SPECIFIC TOKENIZATION

  • Each protein residue receives parallel Surface, chemistry, and TetSphere tokens, which a learned residue query fuses into its initial protein state.The Surface branch uses 16 residue-local points and four descriptors; the TetSphere branch encodes 64 normalized three-channel Laplacian coefficients as mode tokens.Source: §B.1 · p. 15

B.2 JOINT SINGLE–PAIR CONTEXTUALIZATION

  • Geometry-aware ordered token-pair features initialize pair states, and four Pairformer blocks jointly update single and pair states for protein and partner interactions.The shared trunk supplies protein-specific downstream states by selecting the protein rows after final normalization.Source: §B.2 · p. 16

B.3 MASKED RECONSTRUCTION OBJECTIVE

The objective masks modality inputs so representations recover withheld information from remaining modalities and assembly context. Its weighted losses reconstruct geometry and predict identities and relationships; this stage pretrains representations rather than generating sequences or coordinates.

B.3 MASKED RECONSTRUCTION OBJECTIVE

  • Training masks amino-acid identity and chemistry together at 20% of residues, Surface independently at 15%, and TetSphere at selected residues and valid modes.Nucleotide, ligand-atom, and ligand-component identities also use 20% masks.Source: §B.3 · p. 16
  • The canonical objective combines weighted TetSphere, Surface, distance, amino-acid, nucleotide, ligand-identity, and bond losses, normalized by each term's valid-target count.TetSphere reconstructs the complete valid coefficient field; Surface reconstruction is restricted to masked residues.Source: §B.3 · p. 16
  • This stage is a representation pretrainer, not a coordinate or sequence generator, and uses no MSA, atom diffusion, or full-coordinate reconstruction objective.TetSphere is a translation-free deformation descriptor in residue-local frames.Source: §B.3 · p. 16

B.4 TASK-SPECIFIC INPUT CONTRACTS

The task-specific input contracts control which information is available to each task. They mask potential target leakage while preserving each task’s intended inputs.

B.4 TASK-SPECIFIC INPUT CONTRACTS

  • Before pair initialization, bond-reconstruction inputs replace the bond type with a learned mask embedding so the bond target cannot be copied.Source: §B.4 · p. 16
  • For pocket classification, ligand identity fields are masked and ligand identity, component, and bond targets are disabled; ligand coordinates define pocket membership only.Source: §B.4 · p. 17
  • For PPI, 30 cross-chain geometry channels are zeroed while 12 topology channels and cross-chain attention remain available.Source: §B.4 · p. 17
  • Binder design retains BoltzGen’s native inputs and conditioning; fixed pretrained single and pair targets support training alignment but are not needed at generation.Source: §B.4 · p. 17

C LIGAND-BINDING POCKET CLASSIFICATION

This section specifies a seven-class cofactor-pocket task and adapts AtomSurf by adding frozen pretrained residue representations. Ligand information is restricted to defining the pocket and its label, not model features.

C LIGAND-BINDING POCKET CLASSIFICATION

  • The task classifies pockets among seven cofactors using 1,634 training, 202 validation, and 418 test pockets, with balanced accuracy as the metric.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 17
  • Ligand identity fields and reconstruction targets are masked during pretraining; downstream prediction uses protein geometry, with ligand coordinates defining fixed pocket membership and its class label.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 17
  • The matched AtomSurf baseline uses the public codebase because the original task architecture is unavailable, so absolute baseline values may differ from published results.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 17
  • The frozen pretrained residue representation is concatenated with AtomSurf residue-graph features, while its surface–graph encoder and ligand-classification head remain unchanged.The same fusion path is used for the full and w/o TetSphere variants; only the structure-specific input to the pretrained encoder differs.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 17, Figure 7 · p. 18
  • The matched variants share downstream data splits, optimization, checkpoint selection, and evaluation cohort; the no-TetSphere control isolates the registered-volume input from the added representation pathway and capacity.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 17
  • Figure 8 presents test pockets classified correctly by the full model but misclassified by AtomSurf, with examples spanning multiple ligand classes.The examples are selected for visual clarity.Source: C LIGAND-BINDING POCKET CLASSIFICATION · p. 18

D PROTEIN–PROTEIN INTERFACE PREDICTION

The PPI evaluation predicts both residue-level interfaces and cross-chain residue contacts on clustered PINDER. The downstream design adds task-specific residual corrections to AtomSurf predictions using pretrained residue representations.

D PROTEIN–PROTEIN INTERFACE PREDICTION

  • Pinder-Pair predicts cross-chain residue contacts and Pinder-Site predicts interface residues; both are evaluated by AUROC on the clustered PINDER split in the holo setting.Source: D PROTEIN–PROTEIN INTERFACE PREDICTION · p. 18
  • The public AtomSurf release lacks the original task-specific interface labels, so labels use a 5 Å heavy-atom contact rule and scores are interpreted within the matched comparison.The same label definition is applied to AtomSurf and the proposed model; results should not be compared directly with the published AtomSurf numbers.Source: D PROTEIN–PROTEIN INTERFACE PREDICTION · p. 18
  • Because holo structures reveal inter-chain geometry, cross-chain geometric signals are masked during pretraining to prevent direct leakage of contact labels, while topology channels and cross-chain attention remain.Source: D PROTEIN–PROTEIN INTERFACE PREDICTION · p. 20
  • Pretrained single representations feed separate site and pair adapters that predict additive corrections to AtomSurf logits; pretrained pair representations are not passed directly to either head.The pair adapter uses a symmetric residue-pair construction, while the site adapter operates on individual residues.Source: D PROTEIN–PROTEIN INTERFACE PREDICTION · p. 20
  • The full model and no-TetSphere control share the residual architecture and differ only in whether the pretrained model receives TetSphere information.This control tests the registered volumetric input against an additional pretrained representation pathway.Source: D PROTEIN–PROTEIN INTERFACE PREDICTION · p. 20
  • Figure 10 shows 30 predicted contacts in each of three longer-chain complexes, and all 30 contacts in each panel are true contacts.The longer chains place more unrelated structure behind each interface patch than the main-text examples.Source: Figure 10 · p. 21

E DE NOVO PROTEIN BINDER DESIGN

Binder design cannot use the designed binder’s geometry as an input, so the approach transfers pretrained representations to BoltzGen during training. Evaluation uses separate benchmark protocols and fixed generation budgets.

E DE NOVO PROTEIN BINDER DESIGN

  • Because binder geometry is unavailable before generation, pretrained geometric representations are transferred to BoltzGen through alignment during training rather than supplied as inference inputs.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 21
  • A pretrained model provides fixed single and pair representation targets, while projectors align BoltzGen trunk states and the native generation objective remains in use.Alignment errors are masked over valid tokens and ordered pairs; positions without targets remain part of the native objective.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 22
  • The pretrained model is used only to provide training targets, leaving BoltzGen’s original generation inputs and inference procedure unchanged.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 22
  • Four training conditions compare released BoltzGen, continued training without alignment, alignment without TetSphere, and the full TetSphere representation.These controls separate continued training, representation alignment, and the registered volumetric representation.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 22
  • The ten-target ProtDBench and BoltzGen Challenge Set are evaluated separately, with benchmark conditions and screening criteria fixed across methods; success rates measure computational screening, not experimental binding.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 22
  • Challenge Set hard-filter survival is measured against the fixed 2,000-candidate request budget, and Table 3 reports cumulative survival through sequential criteria.The criteria include backbone RMSD and specified amino-acid fraction limits.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 23
  • At TM 0.6, all-candidate clustering was 29.56% for Ours on ProtDBench and 38.25% on Challenge Set, versus 17.92% and 23.95% for BoltzGen.ProtDBench diversity rates are intended for within-benchmark training-arm comparisons, not comparisons across benchmarks.Source: E DE NOVO PROTEIN BINDER DESIGN · p. 24

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