# From Surfaces to Volumes: Registered Geometry for Protein Representation Learning

arXiv: https://arxiv.org/abs/2609.36277 · HTML: https://mlsanity.com/paper/2609.36277/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 27 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Protein Representation Learning by Geometric Structure Pretraining](https://mlsanity.com/paper/2203.06125/) (2022)
- [Equivariant Graph Neural Networks for 3D Macromolecular Structure](https://mlsanity.com/paper/2106.03843/) (2021)
- [TetSphere Splatting: Representing High-Quality Geometry with Lagrangian Volumetric Meshes](https://mlsanity.com/paper/2405.20283/) (2024)
- [Improving detection of protein-ligand binding sites with 3D segmentation](https://mlsanity.com/paper/1904.06517/) (2019)
- [Pre-Training Protein Encoder via Siamese Sequence-Structure Diffusion Trajectory Prediction](https://mlsanity.com/paper/2301.12068/) (2023)
- [Learning Harmonic Molecular Representations on Riemannian Manifold](https://mlsanity.com/paper/2303.15520/) (2023)
- [TetCNN: Convolutional Neural Networks on Tetrahedral Meshes](https://mlsanity.com/paper/2302.03830/) (2023)
- [AtomSurf : Surface Representation for Learning on Protein Structures](https://mlsanity.com/paper/2309.16519/) (2023)
- [An Equivariant Pretrained Transformer for Unified 3D Molecular Representation Learning](https://mlsanity.com/paper/2402.12714/) (2024)

## Abstract (arXiv)

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.