From Surfaces to Volumes: Registered Geometry for Protein Representation Learning

# Registered Geometry for Protein Representation Learning
Source: [https://arxiv.org/html/2609.36277](https://arxiv.org/html/2609.36277)
## From Surfaces to Volumes: Registered Geometry for Protein Representation Learning

Siyuan Chen Cai Zhou Jinrui Zhang Zhaokang LiangTaku Komura Wojciech Matusik Stephen Bates Tommi JaakkolaWengong Jin Peter Yichen Chen Minghao GuoUniversity of British Columbia Massachusetts Institute of TechnologyCarnegie Mellon University Northeastern UniversityThe University of Hong Kong††thanks:Corresponding author\.

###### Abstract

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 from0\.7950\.795to0\.8260\.826, Pinder\-Pair/Site AUROC from0\.914/0\.8520\.914/0\.852to0\.932/0\.8660\.932/0\.866, and binder\-design success from14\.95%14\.95\\%to19\.90%19\.90\\%on the BoltzGen Challenge Set and from27\.62%27\.62\\%to32\.19%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\.

## 1Introduction

Protein geometry can be represented at multiple levels\. Atomic and residue graphs organize structure through spatial neighborhoods and directional relationships\([Jing et al\., 2021b](https://arxiv.org/html/2609.36277#bib.bib1);[Jing et al\., 2021a](https://arxiv.org/html/2609.36277#bib.bib14)\), while surface\-based methods represent proteins using points or meshes sampled from the molecular surface together with local geometric features such as normals and curvature\([Gainza et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib2);[Sverrisson et al\., 2021](https://arxiv.org/html/2609.36277#bib.bib3);[Wang et al\., 2023](https://arxiv.org/html/2609.36277#bib.bib4)\)\. These representations have been effective for capturing exposed molecular shape and have been widely used for recognizing binding pockets and interaction interfaces\. Atom\-centered geometric models likewise infer interaction sites from local structural neighborhoods\([Tubiana et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib20);[Krapp et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib22)\)\. However, these approaches primarily organize protein geometry through atoms, neighborhoods, or molecular surfaces, and do not explicitly represent how the enclosed three\-dimensional volume is distributed among individual residues\. Consequently, geometric properties such as residue packing, burial, local thickness, and the spatial extent of residue\-associated regions are only indirectly captured\. Two residues may hence exhibit similar exposed surface geometry while occupying substantially different three\-dimensional volumetric contexts\. As illustrated in Figure[1](https://arxiv.org/html/2609.36277#S1.F1), the same trypsin target can accommodate structurally distinct inhibitors such as SFTI\-1 and BbKI through similar target\-facing interfaces despite markedly different underlying volumetric organization\.

Volumetric protein representations based on spatial partitions and voxel grids have demonstrated the utility of internal packing, cavities, and local binding\-site structure\([Rother et al\., 2009](https://arxiv.org/html/2609.36277#bib.bib15);[Day et al\., 2010](https://arxiv.org/html/2609.36277#bib.bib16);[Jiménez et al\., 2017](https://arxiv.org/html/2609.36277#bib.bib17);[Stepniewska\-Dziubinska et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib21)\)\. However, directly extending this idea to residue level introduces a different challenge: residue\-associated volumes are irregular and vary substantially in shape and size, so independently constructed local meshes do not provide a common parameterization\. A useful residue\-wise volumetric representation should therefore preserve local three\-dimensional organization while expressing it in a consistent representation space\. We hypothesize that registering residue\-associated volumes to a shared tetrahedral reference can convert heterogeneous local geometry into a structured family of deformations that can be encoded consistently across a protein\.

We introduce*Protein\-TetSphere*, a volumetric representation for protein learning that can be registered residue\-wise\. Given a protein chain, we first construct a tetrahedral representation of the chain and derive residue\-specific volumetric regions\. Each region is then fitted to a copy of a shared fixed\-topology tetrahedral reference adapted from TetSphere Splatting\([Guo et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib6)\)\. Because all fitted regions share the same connectivity and reference vertex ordering, their deformations can be expressed in a common Laplacian basis\. The resulting coefficients provide a compact and consistent description of local volumetric geometry that can be processed consistently across residues and integrated into protein\-level models\. Protein\-TetSphere complements molecular\-surface and chemical features by providing residue\-wise volumetric information in a shared representation\.

The same representation supports both discriminative and generative protein learning\. We evaluate Protein\-TetSphere on ligand\-binding pocket classification, protein–protein interface prediction, and de novo protein binder design\. For ligand\-binding pocket classification, Protein\-TetSphere describes the local volumetric context and packing of residues forming a candidate pocket \([Section4\.2](https://arxiv.org/html/2609.36277#S4.SS2)\)\. For protein–protein interface prediction, it characterizes residue\-level geometry and packing across interacting chains \([Section4\.3](https://arxiv.org/html/2609.36277#S4.SS3)\)\. Diffusion\-based protein design and inverse\-folding models have established complementary routes for generating backbones and assigning sequences to them\([Watson et al\., 2023](https://arxiv.org/html/2609.36277#bib.bib18);[Dauparas et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib19);[Stark et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib12)\)\. For de novo protein binder design, the registered representation provides geometric teacher targets that guide the internal representations of a generative model during training analogously to[Yu et al\. \(2025\)](https://arxiv.org/html/2609.36277#bib.bib11)\([Section4\.4](https://arxiv.org/html/2609.36277#S4.SS4)\)\. Across these tasks, incorporating the registered volumetric representation improves pocket recognition, interface prediction, and binder\-design performance\. Together, the results demonstrate that explicitly organizing protein volume in a shared residue\-wise coordinate system can provide geometric information complementary to molecular surfaces and offer a general mechanism for incorporating volumetric structure into protein learning\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/teaser_x4.png)Figure 1:Motivation of Protein\-TetSphere\.*Left:*SFTI\-1 and BbKI illustrate how structurally distinct binders can engage the same trypsin target through similar target\-facing interfaces while differing substantially in their residue\-wise volumetric organization\.*Right:*Protein\-TetSphere is evaluated on ligand\-binding pocket classification, protein–protein interface prediction, and de novo protein binder design\.
## 2Related Work

#### Protein geometric representations\.

Protein structures are commonly represented as atom\- or residue\-level graphs, where geometric graph neural networks encode spatial neighborhoods, inter\-residue distances, and directional relationships through scalar and vector features\([Jing et al\., 2021b](https://arxiv.org/html/2609.36277#bib.bib1);[Jing et al\., 2021a](https://arxiv.org/html/2609.36277#bib.bib14)\)\. Surface\-based methods instead learn from points or meshes sampled from the molecular surface\. MaSIF learns surface interaction fingerprints\([Gainza et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib2)\), dMaSIF provides an efficient end\-to\-end surface\-learning pipeline\([Sverrisson et al\., 2021](https://arxiv.org/html/2609.36277#bib.bib3)\), and HMR develops harmonic molecular representations on surface manifolds\([Wang et al\., 2023](https://arxiv.org/html/2609.36277#bib.bib4)\)\. AtomSurf integrates graph and surface encoders, enabling atomic neighborhoods and surface geometry to exchange information throughout the network\([Mallet et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib5)\)\. Other atom\-centered approaches, such as ScanNet and PeSTo, represent local structural neighborhoods or atomic point clouds for predicting binding sites and molecular interfaces\([Tubiana et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib20);[Krapp et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib22)\)\. Together, these methods capture complementary aspects of protein geometry through atomic neighborhoods and molecular surfaces, but they do not explicitly represent the enclosed three\-dimensional volume at the residue level\.

#### Protein volumetric geometry and tetrahedral representations\.

Volumetric descriptions have long been used to analyze protein interiors\. Voronoi partitioning supports measurements of atomic volume, local packing density, and internal cavities\([Rother et al\., 2009](https://arxiv.org/html/2609.36277#bib.bib15)\), while Delaunay tessellation has been used to identify recurring tetrahedral residue\-packing motifs\([Day et al\., 2010](https://arxiv.org/html/2609.36277#bib.bib16)\)\. Grid\-based learning provides another route: DeepSite and Kalasanty represent local protein environments as voxelized three\-dimensional fields for binding\-site prediction\([Jiménez et al\., 2017](https://arxiv.org/html/2609.36277#bib.bib17);[Stepniewska\-Dziubinska et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib21)\)\. These methods demonstrate the value of spatial occupancy and packing information, but they do not provide a shared residue\-wise deformation coordinate system\. In geometric learning, TetSphere Splatting introduces deformable tetrahedral spheres as Lagrangian volumetric primitives\([Guo et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib6)\), and TetCNN studies convolution on tetrahedral meshes\([Farazi et al\., 2023](https://arxiv.org/html/2609.36277#bib.bib7)\)\. We adapt tetrahedral volumetric representations to proteins by deriving residue\-specific regions and registering them to a shared fixed\-topology reference\. This provides a parameterization for consistent residue\-level volumetric encoding in a shared Laplacian basis\.

#### Protein representation pretraining\.

Protein representation pretraining has been explored from both sequence and structure\. Sequence\-based models such as UniRep, ESM, and ProtTrans learn transferable representations from large\-scale unlabeled protein sequences\([Alley et al\., 2019](https://arxiv.org/html/2609.36277#bib.bib24);[Rives et al\., 2021](https://arxiv.org/html/2609.36277#bib.bib25);[Elnaggar et al\., 2022](https://arxiv.org/html/2609.36277#bib.bib26)\)\. Protein language models further show that sequence pretraining can capture structural and functional information\([Rives et al\., 2021](https://arxiv.org/html/2609.36277#bib.bib25)\)\. Structure\-aware approaches provide a complementary direction: GearNet studies structural pretraining\([Zhang et al\., 2023a](https://arxiv.org/html/2609.36277#bib.bib8)\), SiamDiff introduces sequence–structure diffusion objectives\([Zhang et al\., 2023b](https://arxiv.org/html/2609.36277#bib.bib9)\), and EPT explores equivariant multimodal molecular pretraining\([Jiao et al\., 2026](https://arxiv.org/html/2609.36277#bib.bib10)\)\. Large\-scale pretrained sequence representations have also enabled protein structure prediction, as demonstrated by ESMFold\([Lin et al\., 2023](https://arxiv.org/html/2609.36277#bib.bib13)\)\. We similarly pretrain Protein\-TetSphere encoders to learn residue\-wise volumetric representations, which are reused across downstream prediction and de novo binder design\.

## 3Method

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/pipeline.png)Figure 2:TetSphere fitting and multimodal pretraining\.*Top:*per\-residue TetSpheres are fitted \(steps00–30003000\) and represented by Laplacian coefficientsCrC\_\{r\}\.*Bottom:*surface, chemical, and TetSphere features are fused with nucleotide, ligand, and bond inputs, processed by a Pairformer\-style single–pair trunk, and pretrained with geometry, identity, and pairwise structure objectives\.### 3\.1Residue\-Wise Registered Volumetric Representation

Given a protein chain and its molecular surface, we construct one fitted tetrahedral mesh for each residue\. Let𝒯0=\(V0,𝒦\)\\mathcal\{T\}\_\{0\}=\(V\_\{0\},\\mathcal\{K\}\)denote a fixed\-topology tetrahedral\-mesh template sphere\([Guo et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib6)\)\. For residuerrwith atom set𝒜r\\mathcal\{A\}\_\{r\}, we define its geometric center as

cr=1\|𝒜r\|​∑a∈𝒜rxa\.c\_\{r\}=\\frac\{1\}\{\|\\mathcal\{A\}\_\{r\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\_\{r\}\}x\_\{a\}\.\(1\)A copy of the template is placed atcrc\_\{r\}and subsequently deformed to match the residue\-local molecular geometry; Figure[2](https://arxiv.org/html/2609.36277#S3.F2)\(top\) shows this fitting progression over the course of the optimization\.

The fitting proceeds in two stages\. We first construct a residue\-local molecular surface using Molecular Surface \(MSMS\)\([Sanner et al\., 1996](https://arxiv.org/html/2609.36277#bib.bib27)\), and fit the template boundary to this surface using bidirectional point\-to\-point nearest\-neighbor correspondences\. The resulting mesh is then refined against the residue region extracted from a tetrahedralization of the complete protein chain, using point\-to\-triangle closest\-point correspondences\. In both stages, the template vertices are iteratively updated while preserving the tetrahedral connectivity𝒦\\mathcal\{K\}, producing a residue\-specific fitted tetmesh

𝒯r=\(Vr,𝒦\)\.\\mathcal\{T\}\_\{r\}=\(V\_\{r\},\\mathcal\{K\}\)\.\(2\)Further implementation details are provided in Appendix[A](https://arxiv.org/html/2609.36277#A1)\.

### 3\.2Spectral Encoding of TetSphere Deformations

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/laplacian_mode_ablation.png)Figure 3:Truncated\-basis reconstruction\.Fitted TetSpheres and leading\-MMLaplacian reconstructions for Tyr35 of 3KH5 and Leu380 of 4ZE2\. Colors indicate per\-vertex error; values report mean error relative to residue extent\.Because all fitted TetSpheres share the same template topology and vertex ordering, their deformation fields can be represented in a common spectral basis\. We construct an undirected graph from𝒦\\mathcal\{K\}by connecting every pair of vertices that co\-occur in a tetrahedron\. LetAAandDDdenote the resulting adjacency and degree matrices, respectively\. We define the graph Laplacian as

and compute its eigendecomposition

We retain the firstMMnon\-constant eigenvectorsUM∈ℝN×MU\_\{M\}\\in\\mathbb\{R\}^\{N\\times M\}to obtain a compact basis for the registered deformation fields; the leading modes are shown in Figure[2](https://arxiv.org/html/2609.36277#S3.F2)\(top right\)\. 64 modes already reconstruct the fitted volumes to within about two percent of the residue’s longest extent \(Figure[3](https://arxiv.org/html/2609.36277#S3.F3)\)\.

LetΔ​Vr∈ℝN×3\\Delta V\_\{r\}\\in\\mathbb\{R\}^\{N\\times 3\}denote the displacement of the fitted TetSphere from its rest\-pose template\. We project this displacement onto the shared Laplacian basis\. To express the resulting deformation vectors in a consistent residue\-centered orientation, we construct a right\-handed local backbone frameRrR\_\{r\}from the N, Cα, and C atoms and define

Cr=UM⊤​Δ​Vr​Rr∈ℝM×3\.C\_\{r\}=U\_\{M\}^\{\\top\}\\Delta V\_\{r\}R\_\{r\}\\in\\mathbb\{R\}^\{M\\times 3\}\.\(5\)Each row ofCrC\_\{r\}gives the three\-dimensional coefficient of a shared deformation mode in the residue\-local backbone frame\. The matrix compactly describes residue\-specific volumetric geometry\. Eigenvector sign conventions and reconstruction details are provided in Appendix[A](https://arxiv.org/html/2609.36277#A1)\.

### 3\.3Protein\-TetSphere Representation Pretraining

We first encode the residue\-level spectral coefficientsCrC\_\{r\}into a TetSphere embedding\. Rather than flatteningCrC\_\{r\}, we treat each spectral mode as a token, as illustrated in the lower panel of Figure[2](https://arxiv.org/html/2609.36277#S3.F2)\. For residuerrand modemm, we construct

hr,m=ϕC\(Cr\(m,:\)\)\+em\+ϕλ\(log\(1\+λm\)\),h\_\{r,m\}=\\phi\_\{C\}\\\!\\left\(C\_\{r\}\(m,:\)\\right\)\+e\_\{m\}\+\\phi\_\{\\lambda\}\\\!\\left\(\\log\(1\+\\lambda\_\{m\}\)\\right\),\(6\)whereeme\_\{m\}is a learned mode embedding andλm\\lambda\_\{m\}is the corresponding Laplacian eigenvalue\. A lightweight Transformer processes the mode tokens\{hr,m\}m=1M\\\{h\_\{r,m\}\\\}\_\{m=1\}^\{M\}, which are aggregated by learned\-query pooling into a residue\-level TetSphere embeddingtrt\_\{r\}\. The TetSphere embedding is then fused with residue\-local surface and chemical representations to form the initial protein residue features\.

To incorporate molecular context, we process these residue features together with nucleotide and ligand tokens using a shared Pairformer\-style single–pair trunk\. Pair features encode structural relations including inter\-token distance, chain and polymer relations, and relative protein geometry\. The trunk jointly updates single representationssis\_\{i\}and pair representationszi​jz\_\{ij\}, allowing local surface, chemical, and volumetric information to interact with the surrounding biomolecular assembly\.

We pretrain the model with masked multimodal reconstruction\. During training, selected inputs from the surface, chemical, TetSphere, molecular\-identity, and pairwise streams are withheld, and the model is trained to reconstruct the corresponding local and relational targets from the remaining modalities and assembly context\. In particular, the TetSphere branch reconstructs the spectral deformation coefficients, while the other objectives supervise complementary surface, molecular\-identity, distance, and bond information\. Further pretraining details are provided in Appendix[B](https://arxiv.org/html/2609.36277#A2)\.

### 3\.4Adaptation to Protein Learning Tasks

We evaluate a multimodal protein representation that integrates TetSphere volumetric geometry with surface and chemical information on ligand\-pocket classification, protein–protein interface prediction, and de novo protein binder design\.

For ligand\-binding pocket and interface prediction, the protein structure is available at inference time\. For ligand\-binding pocket classification, we concatenate the frozen pretrained single representationssis\_\{i\}with the corresponding AtomSurf residue features before the original graph\-input block, while keeping the remaining task formulation unchanged\. For PPI prediction, we retain the native AtomSurf site and pair heads and add task\-specific residual adapters that predict site\- and pair\-level logit corrections fromsis\_\{i\}, using a symmetric combination of endpoint representations for residue pairs\.

De novo protein binder design has a different information regime because the geometry of the structure being designed is not available as an input at inference time\. We therefore use REPresentation Alignment \(REPA\)\([Yu et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib11)\)rather than directly providing Protein\-TetSphere features to the model\. REPA introduces an auxiliary training objective that encourages the internal representations of a generative model to match features produced by a frozen pretrained teacher, transferring the teacher’s representation without requiring it at inference time\.

We apply this strategy to BoltzGen\([Stark et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib12)\), using the pretrained multimodal model to provide geometric representation targets\. During training, the pretraining model encodes the known protein structures and provides single and pair targets\(siT,zi​jT\)\(s\_\{i\}^\{T\},z\_\{ij\}^\{T\}\)for the corresponding BoltzGen states\(siG,zi​jG\)\(s\_\{i\}^\{G\},z\_\{ij\}^\{G\}\)\. Learned projectors map the generator features into the pretrained representation spaces, and the alignment losses are added to the native BoltzGen objective:

ℒ=ℒBoltzGen\+λs​ℒalignsingle\+λz​ℒalignpair\.\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{BoltzGen\}\}\+\\lambda\_\{s\}\\mathcal\{L\}\_\{\\mathrm\{align\}\}^\{\\mathrm\{single\}\}\+\\lambda\_\{z\}\\mathcal\{L\}\_\{\\mathrm\{align\}\}^\{\\mathrm\{pair\}\}\.\(7\)The pretraining model is used only during training and is not required during design\. Task\-specific implementation details are provided in Appendices[C](https://arxiv.org/html/2609.36277#A3)–[E](https://arxiv.org/html/2609.36277#A5)\.

## 4Experiments

### 4\.1Tasks and Evaluation Protocol

We evaluate the multimodal Protein\-TetSphere representation on three tasks: ligand\-binding pocket classification, protein–protein interface prediction, and de novo protein binder design\. For ligand\-binding pocket classification, we follow the MaSIF\-ligand benchmark introduced by MaSIF\([Gainza et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib2)\), use AtomSurf\([Mallet et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib5)\)as the surface\-aware baseline, and report balanced accuracy\. For protein–protein interface prediction, we follow the AtomSurf protocol on the clustered PINDER split\([Kovtun et al\., 2024](https://arxiv.org/html/2609.36277#bib.bib23)\)and evaluate both residue\-pair contact prediction \(Pinder\-Pair\) and residue\-level interface prediction \(Pinder\-Site\) using AUROC\. For de novo protein binder design, we transfer the pretrained representation to BoltzGen\([Stark et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib12)\)through representation alignment and evaluate on the BoltzGen Challenge Set and ProtDBench\([Liu et al\., 2026](https://arxiv.org/html/2609.36277#bib.bib28)\)\.

We train a separate multimodal pretraining model for each task\. For ligand\-binding pocket classification and protein–protein interface prediction, the model is trained on the corresponding benchmark split, with validation used for checkpoint selection\. For de novo protein binder design, pretraining uses the released BoltzGen corpus\. Detailed task\-specific protocols are provided in Appendices[B](https://arxiv.org/html/2609.36277#A2)–[E](https://arxiv.org/html/2609.36277#A5)\.

### 4\.2Ligand\-binding pocket Classification

Ligand\-binding pocket classification asks which of seven cofactors occupies a given protein binding pocket\. As shown in Table[1](https://arxiv.org/html/2609.36277#S4.T1.fig1), AtomSurf achieves a balanced accuracy of0\.795±0\.0050\.795\\pm 0\.005, while the full model improves this to0\.826±0\.0040\.826\\pm 0\.004\. To distinguish the contribution of registered volumetric geometry from that of an additional trainable representation pathway, we include\+ Ours \(w/o TetSphere\)\. This control retains the trainable Tet token and the same downstream fusion pathway, but receives no structure\-specific TetSphere input, reaching0\.807±0\.0030\.807\\pm 0\.003\. The full model therefore improves over AtomSurf by0\.0310\.031and over this control by0\.0190\.019, supporting that the registered volumetric information contributes beyond the additional latent capacity alone\.

Table 1:Test balanced accuracy for ligand\-binding pocket classification, reported as the mean and standard deviation over five random seeds\.![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_pocket_complementarity_fad.png)\(a\)FAD– native ligand
0 of 53 atoms inside the shells
![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_pocket_complementarity_adp.png)\(b\)ADP– too small
2 of 27 atoms inside the shells
![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_pocket_complementarity_hem.png)\(c\)HEM– intersects
10 of 43 atoms inside the shells

Figure 4:Shape comparison in the 6BD9 FAD pocket\.FAD, ADP, and HEM are compared within the same pocket; numbers report ligand heavy atoms inside the fitted TetSphere shells\.Figure[4](https://arxiv.org/html/2609.36277#S4.F4)illustrates how the registered volumetric geometry captured by TetSphere can provide complementary cues for ligand\-binding pocket recognition\. In this FAD\-binding pocket, the native FAD remains compatible with the fitted residue\-wise volumes, whereas ADP under\-occupies the pocket and HEM intersects the fitted shells\. Unlike a surface description alone, TetSphere explicitly represents the three\-dimensional organization of the pocket interior through registered residue\-wise volumes, making differences in volumetric occupancy directly accessible to the model\. Consistent with this geometric distinction, the full model correctly identifies the pocket as FAD\-binding\.

### 4\.3Protein–Protein Interface Prediction

We evaluate protein–protein interface prediction on clustered PINDER\([Kovtun et al\., 2024](https://arxiv.org/html/2609.36277#bib.bib23)\), using Pinder\-Pair for residue\-pair contacts and Pinder\-Site for residue\-level interfaces\. As shown in Table[2](https://arxiv.org/html/2609.36277#S4.T2), AtomSurf achieves AUROC of0\.914±0\.0020\.914\\pm 0\.002on Pinder\-Pair and0\.852±0\.0020\.852\\pm 0\.002on Pinder\-Site\.\+ Ours \(w/o TetSphere\)uses the same residual\-adapter architecture and training protocol but no structure\-specific TetSphere input, reaching0\.919±0\.0020\.919\\pm 0\.002and0\.855±0\.0010\.855\\pm 0\.001\. The full model improves to0\.932±0\.0010\.932\\pm 0\.001and0\.866±0\.0010\.866\\pm 0\.001, gains of0\.0180\.018and0\.0140\.014over AtomSurf and0\.0130\.013and0\.0110\.011over the no\-TetSphere control\. These results support the contribution of registered volumetric geometry beyond additional latent capacity\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_4mis.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_5xor.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_1pdk.png)\(a\) 4MIS\(b\) 5XOR\(c\) 1PDKFigure 5:Representative Pinder\-Pair interface predictions from our model\. Chain A is shown in red and chain B in blue, with TetSphere volumes highlighting residues involved in predicted contacts\. Each panel shows 28–30 predicted contacts, all of which are true contacts under the55Å heavy\-atom criterion, illustrating the correctness of our high\-confidence contact predictions in these examples\.Unlike residue\-site prediction, Pinder\-Pair requires identifying specific residue–residue contacts across the two chains, rather than simply locating interface residues\. Each TetSphere captures residue\-local volumetric deformation in a shared parameterization, and the pretrained representation integrates these signals with neighboring residues\. Together, they provide an interface view complementary to the surface and chemical features used by AtomSurf\. Figure[5](https://arxiv.org/html/2609.36277#S4.F5)shows representative high\-confidence Pinder\-Pair predictions, where the predicted contacts form coherent and spatially complementary regions across the interacting chains\.

Table 2:Protein–protein interface prediction on clustered PINDER in the holo setting\. Results are test AUROC \(mean±\\pms\.d\.\) over five seeds\.
### 4\.4De Novo Protein Binder Design

In de novo binder design, the binder geometry is not available before generation, so TetSphere cannot be supplied directly as an input\. We therefore align BoltzGen hidden states to the single and pair representations of the multimodal pretraining model during training, while retaining the native BoltzGen generation objective\. The pretraining model is used only during training\.

We evaluate on two ten\-target benchmarks, ProtDBench and the BoltzGen Challenge Set\. For ProtDBench, we follow the original evaluation setting, generating 32 backbones for each of 150 \(target, binder\-length\) conditions and eight ProteinMPNN sequences per backbone, yielding 4,800 backbones and 38,400 sequence candidates per method\. We report both backbone\-level and sequence\-level pass rates\. For the BoltzGen Challenge Set, each method generates 2,000 candidates using five seeds per target and 40 designs per seed\. All generation budgets and evaluation criteria are held fixed across methods\. Full protocols are provided in Appendix[E](https://arxiv.org/html/2609.36277#A5)\.

Table 3:Binder\-design success across four training arms\.Rows prefixed with\+\+are cumulative, with a shared denominator within each block\. Denominators are 38,400 ProtDBench sequences, 4,800 ProtDBench backbones, and 2,000 Challenge Set candidates\. Best and second\-best values are shaded dark and light green\. The±\\pmvalues are bootstrap standard deviations over designs with targets fixed\. See Appendix[E](https://arxiv.org/html/2609.36277#A5)for details\.The decomposition localizes where the endpoint gap emerges; it is not an additive causal attribution across criteria\.

As shown in Table[3](https://arxiv.org/html/2609.36277#S4.T3), Ours improves the Challenge Set pass rate from14\.95%14\.95\\%for BoltzGen to19\.90%19\.90\\%, the highest among the four arms\. On ProtDBench, the backbone\-level pass rate increases from27\.62%27\.62\\%to32\.19%32\.19\\%, again the highest result\. At the sequence level, Ours reaches14\.67%14\.67\\%, improving over BoltzGen \(13\.99%13\.99\\%\) and remaining comparable to\+Cont\.\(14\.80%14\.80\\%\)\.

We include two controls to separate volumetric supervision from continued training and representation alignment\.\+Cont\.continues training BoltzGen for the same budget without alignment, whereas\+ Ours \(w/o TetSphere\)uses the same alignment objective and schedule without structure\-specific TetSphere input\. The no\-TetSphere control reaches11\.73%11\.73\\%sequence\-level and26\.46%26\.46\\%backbone\-level success on ProtDBench and16\.90%16\.90\\%on the Challenge Set\. In comparison, the full model reaches14\.67%14\.67\\%,32\.19%32\.19\\%, and19\.90%19\.90\\%, respectively, showing that registered volumetric geometry provides additional gains beyond continued training and representation alignment alone\.

The gains are not confined to a small subset of targets\. As shown in Figure[6](https://arxiv.org/html/2609.36277#S4.F6), Ours outperforms BoltzGen on 7 of 10 ProtDBench targets and 6 of 10 Challenge Set targets, and improves over\+Cont\.on 9 and 7 targets, respectively\. Relative to the no\-TetSphere control, Ours performs better on 6 of 10 ProtDBench targets and 7 of 10 Challenge Set targets\. Thus, the aggregate improvement reflects broadly distributed gains across both benchmark panels rather than a small number of outlier targets\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/binder_benchmarks_bars_x4.png)Figure 6:Target\-level binder\-generation success rates, with the same four arms as Table[3](https://arxiv.org/html/2609.36277#S4.T3): the released BoltzGen checkpoint, the continued\-training control, the no\-TetSphere control, and Ours\. The left panel reports ProtDBench at the backbone level, where a backbone counts as a success if any of its eight redesigned sequences is accepted; the right panel reports final hard\-filter success on the BoltzGen Challenge Set, where each candidate carries a single sequence\. These correspond to the backbone\-level success row and to the final cumulative row of the Challenge Set block in Table[3](https://arxiv.org/html/2609.36277#S4.T3)\. Both panels share a common vertical scale so rates can be read across benchmarks, and both retain all ten targets rather than collapsing the comparison to one aggregate; response is strongly target\-dependent in both\.The filter decomposition in Table[3](https://arxiv.org/html/2609.36277#S4.T3)further shows where the gains emerge\. On ProtDBench, Ours improves both interface pTM and interface PAE, increasing the fraction satisfying the first three criteria jointly from18\.20%18\.20\\%to23\.16%23\.16\\%\. The bound–unbound RMSD gate reduces this advantage at the sequence level, but the gain remains clear at the backbone level, where success increases from27\.62%27\.62\\%to32\.19%32\.19\\%\. This suggests that the improved interface geometry translates into a larger fraction of backbones with at least one successful sequence\.

On the Challenge Set, Ours shows its largest advantage at the binder\-only RMSD stage: cumulative survival reaches23\.50%23\.50\\%compared with17\.45%17\.45\\%for BoltzGen, while the median binder\-only RMSD decreases from1\.4921\.492to1\.3341\.334Å\. This advantage is retained through the subsequent composition filters and leads to the final19\.90%19\.90\\%pass rate\. Representative successful designs are shown in Figure[12](https://arxiv.org/html/2609.36277#A5.F12)in Appendix[E](https://arxiv.org/html/2609.36277#A5)\.

The higher success rate is not accompanied by reduced structural diversity\. Ours achieves higher diversity\-adjusted cluster pass rates than BoltzGen and\+Cont\.on both benchmarks and is highest among all four arms at TM0\.80\.8\. Although the no\-TetSphere control is more diverse at TM0\.60\.6, its lower ProtDBench success shows that cluster count alone does not reflect useful design quality\.

Taken together, these results show that volumetric alignment improves backbone\-level success and key geometric quality criteria while preserving structural diversity, with particularly clear gains in interface quality on ProtDBench and binder\-local geometry on the Challenge Set\.

## 5Conclusion and Limitations

We introduce Protein\-TetSphere, a registered residue\-wise volumetric representation of protein geometry\. Whole\-chain tetrahedralization provides residue\-specific volumetric regions, which are fitted to a shared fixed\-topology reference and encoded in a common Laplacian basis\. Ligand\-binding pocket and PPI prediction evaluate these representations through matched AtomSurf comparisons, while ProtDBench and the BoltzGen Challenge Set evaluate their use as geometric supervision for binder generation\. Across these settings, registered volumetric geometry consistently complements surface\-based representations and improves protein recognition, interaction prediction, and design\.

The current study focuses on static protein structures\. A natural extension is to model Protein\-TetSphere representations over molecular\-dynamics trajectories, capturing time\-varying residue\-wise volumetric geometry and conformational change\. Another direction is to develop multiscale registered representations for biomolecular assemblies, combining residue\-level volumes with domain\-, complex\-, and partner\-level geometry\. These extensions could broaden volumetric protein representations toward dynamic modeling, interface design, and functional protein generation\.

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## Appendix AConstruction and Implementation Details

This appendix specifies the deterministic implementation of the Protein\-TetSphere construction in Section[3](https://arxiv.org/html/2609.36277#S3)\.

### A\.1Surface Construction and Coordinate Convention

For each protein chain, we construct the molecular surface from its present heavy atoms; no ligand, nucleic acid, solvent, neighboring chain, atom completion, or OXT synthesis is applied\. The MSMS wrapper uses its standard PDB\-to\-xyzrn conversion and invokes MSMS with a1\.51\.5Angstrom probe\. For a chain\-level MSMS vertex setSS, the center and normalized coordinate are

c=\|S\|∑x∈S−1⁡x,x¯=q⁡\(x−c\),q=0\.04\.c=\|S\|^\{\-1\}\\sum\_\{x\\in S\}x,\\qquad\\bar\{x\}=q\(x\-c\),\\qquad q=0\.04\.\(8\)The chain surface is centered by subtractingccbefore it is written; target loaders apply the factorqq\. World\-space coordinates are recovered asx=x¯/q\+cx=\\bar\{x\}/q\+c\.

### A\.2MSMS and fTetWild Targets

For each protein, the surface builder uses its atomic structure to assign a local atom set to each residue\. For a given residue, this set includes the atoms of the current residue, the previous residue’sCCandOOatoms, and the next residue’sNNatoms\. MSMS is then run on this set to obtain the local residue surface\. The overlapping atoms only close the local peptide\-bond surface and do not create additional residue rows\.

The centered chain\-level OBJ is then tetrahedralized with fTetWild\.

LetYYbe the TetWild volume vertices and letArA\_\{r\}be the coordinates of the chain atoms belonging to residuerr, after subtractingcc\. Each volume vertex receives the label

ℓ⁡\(y\)=arg⁡minr⁡mina∈Ar⁡‖y−a‖2\.\\ell\(y\)=\\mathop\{\\arg\\min\}\_\{r\}\\;\\min\_\{a\\in A\_\{r\}\}\\\|y\-a\\\|\_\{2\}\.\(9\)To make the decomposition cover the interface neighborhood, a radius\-1\.01\.0ball query onYYpropagates every label to nearby volume vertices\. For each residuerr, the splitter keeps every tetrahedron with at least one vertex in the expanded label set, compacts its vertices, and extracts its boundary triangles\. The resulting residue targets can overlap, especially near peptide and chain interfaces, but share the same centered coordinates\.

### A\.3Reference Template and Fitting

Our reference template contains 532 vertices and 1,727 tetrahedra\. For each aligned residue, the current mesh constructor uses a fixed rest scales=0\.6s=0\.6and initializes one disconnected copy asVr,0=s​V0\+x¯rV\_\{r,0\}=sV\_\{0\}\+\\bar\{x\}\_\{r\}, without supplying an additional template rotation\. Tetrahedron indices are offset by 532 for each successive copy, yieldingR×532R\\times 532vertices andR×1,727R\\times 1\{,\}727tetrahedra\.

The local\-surface and chain\-level fitting procedures use the same fixed topology\. In the local MSMS fit, each residue copy is aligned to its local MSMS target using bidirectional point\-to\-point nearest\-neighbor displacements\. For a predicted boundary vertexviv\_\{i\}and its residue targetSrS\_\{r\}, the forward displacement is proportional toNNSr⁡\(vi\)−vi\\operatorname\{NN\}\_\{S\_\{r\}\}\(v\_\{i\}\)\-v\_\{i\}\. Each target point additionally contributes a backward displacement from its nearest predicted boundary vertex within the same residue group\. The forward and backward fields are averaged within each direction and combined with equal weightswf=wb=0\.5w\_\{f\}=w\_\{b\}=0\.5\.

In the chain\-level fTetWild fit, each predicted vertex is matched directly to the residue\-labeled triangular surface extracted from the complete\-chain fTetWild mesh\. A batched bounding\-volume hierarchy \(BVH\) returns the closest point on the corresponding target triangles\. This fit therefore uses a one\-sided point\-to\-triangle displacement from each predicted vertex to its closest target surface point; it does not construct a backward target\-to\-prediction correspondence\.

The fitting displacementdicpd\_\{\\mathrm\{icp\}\}is combined with the smooth\-barrier displacementdregd\_\{\\mathrm\{reg\}\}computed from the rest pose:

g=−\(dicp\+4×10−5​dreg\),V←V−0\.1​AdamUniform⁡\(g\),g=\-\\bigl\(d\_\{\\mathrm\{icp\}\}\+4\\times 10^\{\-5\}d\_\{\\mathrm\{reg\}\}\\bigr\),\\qquad V\\leftarrow V\-0\.1\\,\\operatorname\{AdamUniform\}\(g\),\(10\)with smooth\-energy coefficient3×10−4/R3\\times 10^\{\-4\}/R\. The two fitting procedures use budgets of 1,000 and 3,000 vertex updates, respectively\. The chain\-level fTetWild fit continues from the in\-memory local MSMS result and resets the AdamUniform state\. Both fitting procedures use convergence\-based early termination within these maximum budgets\.

After each update, a Gauss–Seidel projection moves vertices to repair inverted or collapsed tetrahedra using at most 10 projection sweeps\. The configured relative floor is0\.10\.1times the absolute signed volume of each rest tetrahedron\. Before serialization, a stricter cleanup performs up to eight rounds of 50 additional projection sweeps\.

### A\.4Spectral Basis and Local Frames

The graph used for the spectral descriptor has one binary edge for every pair of template vertices that co\-occurs in a tetrahedron\. We setAi​j=1A\_\{ij\}=1for these edges,Di​i=∑jAi​jD\_\{ii\}=\\sum\_\{j\}A\_\{ij\}, andL=D−AL=D\-A\. The template graph is required to have exactly one zero eigenvalue\. We retain the first 64 non\-constant eigenvectors\. For determinism, the sign of each retained eigenvectorumu\_\{m\}is chosen so that its value atpm=arg⁡maxj​\|um​\(j\)\|p\_\{m\}=\\arg\\max\_\{j\}\|u\_\{m\}\(j\)\|is positive\.

For residue backbone coordinatesnrn\_\{r\},ara\_\{r\}\(Cα\\alpha\), andcrc\_\{r\}\(C\), we construct the right\-handed frame

xr\\displaystyle x\_\{r\}=cr−ar∥cr−ar∥2,\\displaystyle=\\frac\{c\_\{r\}\-a\_\{r\}\}\{\\lVert c\_\{r\}\-a\_\{r\}\\rVert\_\{2\}\},\(11\)y~r\\displaystyle\\widetilde\{y\}\_\{r\}=\(nr−ar\)−\(\(nr−ar\)⊤​xr\)​xr,\\displaystyle=\(n\_\{r\}\-a\_\{r\}\)\-\(\(n\_\{r\}\-a\_\{r\}\)^\{\\top\}x\_\{r\}\)x\_\{r\},\(12\)yr\\displaystyle y\_\{r\}=y~r∥y~r∥2,zr=xr×yr,Rr=\[xr​yr​zr\]\.\\displaystyle=\\frac\{\\widetilde\{y\}\_\{r\}\}\{\\lVert\\widetilde\{y\}\_\{r\}\\rVert\_\{2\}\},\\qquad z\_\{r\}=x\_\{r\}\\times y\_\{r\},\\qquad R\_\{r\}=\[x\_\{r\}\\;y\_\{r\}\\;z\_\{r\}\]\.\(13\)For each fit,

Δ​Vr=Vr−s​V0q,Cr=U64⊤​Δ​Vr​Rr\.\\Delta V\_\{r\}=\\frac\{V\_\{r\}\-sV\_\{0\}\}\{q\},\\qquad C\_\{r\}=U\_\{64\}^\{\\top\}\\Delta V\_\{r\}R\_\{r\}\.\(14\)The omitted constant Laplacian mode corresponds to global translation, which is therefore removed by the projection\. Mode/channel means and standard deviations are estimated from valid training residues only and reused unchanged for validation and test examples\. The inverse low\-pass reconstruction is

V^r=s​V0\+q​U64​Cr​Rr⊤\.\\widehat\{V\}\_\{r\}=sV\_\{0\}\+qU\_\{64\}C\_\{r\}R\_\{r\}^\{\\top\}\.\(15\)
The multimodal pretraining architecture is described in Appendix[B](https://arxiv.org/html/2609.36277#A2)\.

## Appendix BMultimodal Masked\-Pretraining Architecture

The multimodal pretraining model learns contextual residue representations within a biomolecular assembly\. Protein residues integrate local surface geometry, amino\-acid\-derived chemistry, and registered TetSphere deformations, while nucleotide and ligand tokens provide partner context when present\. Each modality is first encoded according to its own input contract, after which the resulting tokens interact through shared single and pair states\. Throughout this section,*multimodal pretraining model*refers to this representation\-learning model\.

LetPP,QQ, andAAbe the numbers of protein\-residue, DNA/RNA nucleotide, and ligand\-atom tokens, respectively, and letN=P\+Q\+AN=P\+Q\+A\. The encoder returns contextualized states at both token and token\-pair resolution,

Zprotein∈ℝP×256,Zall∈ℝN×256,z∈ℝN×N×128\.Z\_\{\\mathrm\{protein\}\}\\in\\mathbb\{R\}^\{P\\times 256\},\\qquad Z\_\{\\mathrm\{all\}\}\\in\\mathbb\{R\}^\{N\\times 256\},\\qquad z\\in\\mathbb\{R\}^\{N\\times N\\times 128\}\.\(16\)Protein residues are the primary representation targets\. Partner tokens supply assembly context and receive type\-specific identity supervision, but are not assigned synthetic protein\-surface, TetSphere, or chemistry inputs\.

### B\.1Type\-Specific Tokenization

Table[4](https://arxiv.org/html/2609.36277#A2.T4)summarizes the modality contracts, masks, and reconstruction targets\. Each protein residue is represented by three parallel views\. The Surface branch encodes 16 residue\-local representative points and four normalized surface descriptors with three invariant message\-passing blocks\. Its nearest\-neighbor graph is constructed independently within each residue and, because a residue contributes at most 16 points, is complete: each point attends to all other points of the same residue\. Information therefore cannot pass between residue surfaces before masking and fusion\. Attention pooling combines the point states with the descriptor projection to obtain a 256\-dimensional Surface token\.

The chemistry branch projects hydropathy and charge to the shared width and contextualizes them with two pair\-biased encoder layers\. Because these attributes are derived from amino\-acid identity, they are masked jointly with the corresponding amino\-acid target to prevent direct identity leakage\. The TetSphere branch treats the 64 normalized three\-channel Laplacian coefficients as an ordered mode sequence\. Each coefficient is projected to a 128\-dimensional mode token and augmented with learned mode\-index and eigenvalue embeddings\. Two four\-head Transformer layers and learned\-query pooling then produce a 256\-dimensional TetSphere token\. Distinct learned states represent deliberately masked inputs and genuinely unavailable inputs\. Masking on this branch acts at mode resolution rather than on whole residues: a residue is first selected with probability0\.150\.15, and within each selected residue a further15%15\\%of its valid modes are drawn at random\. The drawn coefficients are zeroed and flagged, while the remaining modes of the same residue stay visible, so the decoder must exploit correlations across the spectrum rather than infer a residue shape from nothing\. The reconstruction loss is independent of this draw and is evaluated on every valid mode of every valid residue\.

A learned residue query attends to the Surface, chemistry, and TetSphere tokens through two residue\-local fusion blocks; the updated query becomes the initial protein state\. DNA/RNA nucleotides and ligand atoms use separate type\-specific encoders\. Their identity, molecule\-type, CCD, and validity features form 80\-dimensional inputs that are independently projected by two\-layer MLPs to the shared 256\-dimensional token width\.

Table 4:Canonical inputs, encoders, masking policies, and reconstruction targets of the multimodal pretraining model\. Task\-specific overrides are described below\.
### B\.2Joint Single–Pair Contextualization

Protein, nucleotide, and ligand states are restored to the native crop order to forms\(0\)∈ℝN×256s^\{\(0\)\}\\in\\mathbb\{R\}^\{N\\times 256\}\. For each ordered token pair, a 42\-dimensional structural featuregi​jg\_\{ij\}records distance radial\-basis values, signed polymer\-relative positions, chain and entity relations, center validity, the direction fromiitojjin tokenii’s valid protein frame, relative protein\-frame rotation, and the associated validity flags\. Unavailable coordinates or frames contribute zero\-valued geometry together with explicit validity indicators\. Ordered molecule\-type and ligand\-bond embeddings provide complementary categorical context\.

Pair initialization combines this fixed structural anchor with a content\-derived interaction,

zi​jmm=ϕpair​\(\[si\+sj,\|si−sj\|,si⊙sj\]\),ϕpair:768→256→128,z\_\{ij\}^\{\\mathrm\{mm\}\}=\\phi\_\{\\mathrm\{pair\}\}\\\!\\left\(\[s\_\{i\}\+s\_\{j\},\\,\|s\_\{i\}\-s\_\{j\}\|,\\,s\_\{i\}\\odot s\_\{j\}\]\\right\),\\qquad\\phi\_\{\\mathrm\{pair\}\}:768\\rightarrow 256\\rightarrow 128,\(17\)zi​j\(0\)=LayerNorm⁡\(ϕstruct​\(gi​j\)\+etype⁡\(i\),type⁡\(j\)\+ebond⁡\(i,j\)\+zi​jmm\)\.z\_\{ij\}^\{\(0\)\}=\\operatorname\{LayerNorm\}\\\!\\left\(\\phi\_\{\\mathrm\{struct\}\}\(g\_\{ij\}\)\+e\_\{\\mathrm\{type\}\(i\),\\mathrm\{type\}\(j\)\}\+e\_\{\\mathrm\{bond\}\(i,j\)\}\+z\_\{ij\}^\{\\mathrm\{mm\}\}\\right\)\.\(18\)Four Pairformer blocks then update the single and pair states jointly\. Each block applies a pair transition, triangle\-context aggregation, and a gated residual from the fixed structural anchor, followed by pair\-biased attention and a transition on the single states and a single\-to\-pair refresh\. Thus, protein–protein and protein–partner interactions are represented by the same geometry\-aware trunk rather than separate interface modules\. Final normalization producesZallZ\_\{\\mathrm\{all\}\}, and selecting its protein rows givesZproteinZ\_\{\\mathrm\{protein\}\}for protein\-specific downstream use\.

### B\.3Masked Reconstruction Objective

Masking forces each representation to recover withheld information from the remaining modalities and assembly context\. Masks are sampled deterministically for each training draw and applied before the corresponding local or identity encoder\. Amino\-acid identity and chemistry share a 20% residue mask; the Surface input uses an independent 15% residue mask; the TetSphere input uses a 15% residue mask followed by a 15% mask over the valid modes of each selected residue; and nucleotide, ligand\-atom, and ligand\-component identities use 20% masks\. When a ligand component is selected, its CCD input is hidden on all constituent atoms before their contextualized states are pooled for component prediction\.

The canonical objective contains seven weighted terms:

ℒpretrain=\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{pretrain\}\}=\{\}1\.00​ℒTet\+0\.20​ℒSurface\+0\.15​ℒdistance\+0\.25​ℒAA\\displaystyle 1\.00\\,\\mathcal\{L\}\_\{\\mathrm\{Tet\}\}\+0\.20\\,\\mathcal\{L\}\_\{\\mathrm\{Surface\}\}\+0\.15\\,\\mathcal\{L\}\_\{\\mathrm\{distance\}\}\+0\.25\\,\\mathcal\{L\}\_\{\\mathrm\{AA\}\}\+0\.20​ℒNA\+0\.25​ℒligand​\-​id\+0\.10​ℒbond\.\\displaystyle\+0\.20\\,\\mathcal\{L\}\_\{\\mathrm\{NA\}\}\+0\.25\\,\\mathcal\{L\}\_\{\\mathrm\{ligand\\text\{\-\}id\}\}\+0\.10\\,\\mathcal\{L\}\_\{\\mathrm\{bond\}\}\.\(19\)ℒTet\\mathcal\{L\}\_\{\\mathrm\{Tet\}\}andℒSurface\\mathcal\{L\}\_\{\\mathrm\{Surface\}\}reconstruct Laplacian coefficients and surface descriptors with Smooth\-L1 loss\.ℒSurface\\mathcal\{L\}\_\{\\mathrm\{Surface\}\}is restricted to masked residues, whereasℒTet\\mathcal\{L\}\_\{\\mathrm\{Tet\}\}supervises the complete valid coefficient field rather than only the drawn modes; the error on the masked subset alone is tracked as a diagnostic\.ℒdistance\\mathcal\{L\}\_\{\\mathrm\{distance\}\}regresseslog⁡\(1\+di​j\)\\log\(1\+d\_\{ij\}\)on sampled valid mixed\-token pairs\.ℒAA\\mathcal\{L\}\_\{\\mathrm\{AA\}\}is 20\-class cross entropy;ℒNA\\mathcal\{L\}\_\{\\mathrm\{NA\}\}combines base and nucleotide\-CCD cross entropies in a0\.7:0\.30\.7\{:\}0\.3ratio; andℒligand​\-​id\\mathcal\{L\}\_\{\\mathrm\{ligand\\text\{\-\}id\}\}combines element, atom\-name, and component\-CCD cross entropies in a0\.5:0\.25:0\.250\.5\{:\}0\.25\{:\}0\.25ratio\.ℒbond\\mathcal\{L\}\_\{\\mathrm\{bond\}\}is five\-class cross entropy over no\-bond, single, double, triple, and aromatic labels\. Each term is normalized by its own globally reduced valid\-target count\. An absent partner modality therefore contributes neither numerator nor count and does not alter the stated loss weights\.

This stage is a representation pretrainer, not a coordinate or sequence generator\. It uses no MSA, atom diffusion, explicit interface cross\-attention, autoregressive decoder, or full\-coordinate reconstruction objective\. The amino\-acid classifier is a masked identity head rather than a sequence decoder, and the chemistry prediction head is retained for diagnostics but is excluded from the canonical objective\. Absolute coordinates enter only through the shared\-frame pair geometry and pair\-distance target; TetSphere remains a translation\-free deformation descriptor expressed in residue\-local frames\.

### B\.4Task\-Specific Input Contracts

The shared single–pair trunk is adapted to the three task settings through task\-specific input contracts\. For any pair used for bond reconstruction, the input bond type is replaced by a learned mask embedding before pair initialization, so the bond target cannot be copied from the input\.

#### Ligand\-binding pocket classification\.

For MaSIF\-ligand pretraining, ligand tokens are retained only as masked context: element, atom\-name, and CCD identity fields are replaced by mask values, while ligand identity, component, and bond targets are disabled\. The ligand coordinate array is retained only to define the pocket instance\. Downstream prediction uses protein geometry alone; ligand coordinates determine fixed pocket membership and supervision rather than model features\.

#### Protein–protein interface prediction\.

For PPI, the pair representation keeps its 42\-channel width, but 30 geometry channels are zeroed for cross\-chain pairs before any learnable pair module consumes them\. These are the distance RBF channels\[0:16\]\[0\{:\}16\], local direction\[28:31\]\[28\{:\}31\], relative rotation\[31:40\]\[31\{:\}40\], and direction/rotation validity flags\[40:42\]\[40\{:\}42\]\. The 12 topology channels\[16:28\]\[16\{:\}28\]remain available\. The same mask is applied to the protein pair path used by the chemistry encoder and to the full mixed\-token pair path; cross\-chain attention remains enabled by the token\-valid pair mask\.

#### De Novo Protein Binder Design\.

Binder design retains the native BoltzGen inputs and conditioning pathway, including the provided target structure and benchmark\-specific design conditions\. No ligand\-specific identity mask or PPI cross\-chain geometry mask is applied\. During fine\-tuning, the multimodal pretraining model provides fixed single and pair representation targets for alignment; it is not required at generation time\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_pocket_adaptation.png)Figure 7:Ligand\-pocket adaptation pipeline\.Adaptation of the pretrained residue representation to ligand\-binding pocket classification\. The native AtomSurf molecular surface and residue graph enter unchanged; in parallel, the frozen pretrained encoder produces residue\-level single representationssis\_\{i\}\. The two are concatenated per residue and consumed by the existing AtomSurf graph\-input block, so the fused features then follow the original surface–graph encoder and ligand\-classification head without further modification\. The same downstream path is used for the full and\+ Ours \(w/o TetSphere\)variants; the two differ only in the structure\-specific input supplied to the frozen pretrained encoder\. The right\-hand column shows four of the seven cofactor classes\.

## Appendix CLigand\-binding pocket classification

Given a binding site on a protein, the task is to identify which of seven cofactors occupies it: ADP, COA, FAD, HEM, NAD, NAP, or SAM\. We use the MaSIF\-ligand benchmark\([Gainza et al\., 2020](https://arxiv.org/html/2609.36277#bib.bib2)\), comprising 1,634 training, 202 validation, and 418 test binding pockets\. We retain the standard sequence\-based split and report balanced accuracy over the seven ligand classes\.

For ligand\-binding pocket classification, we train the multimodal pretraining model using only the training split and select the checkpoint on the validation split\. To prevent ligand\-identity leakage during pretraining, the ligand element, atom\-name, and CCD\-identity fields are masked, while ligand\-identity and component reconstruction targets and the ligand\-bond target are disabled\. The only ligand\-specific quantity retained is the coordinate array associated with the pocket instance\. For downstream classification, the model uses protein\-only geometry: ligand coordinates are used only to define the fixed pocket membership and its class label, consistent with the AtomSurf task formulation\. Additional masking details are given in Appendix[B](https://arxiv.org/html/2609.36277#A2)\.

For downstream prediction, we follow the AtomSurf framework\([Mallet et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib5)\)\. We implement the baseline using the official public AtomSurf codebase\. Because the exact task architecture used to obtain the results reported in the original paper is not included in the public release, our absolute baseline values may differ slightly from the published numbers\. We therefore base all comparisons on the matched implementation used consistently across our ablation variants\. AtomSurf jointly encodes the molecular surface and residue graph\. We keep the pretrained encoder frozen and concatenate its residue\-level single representations with the native AtomSurf residue\-graph input features stored ingraph\.x, before the AtomSurf encoder\.

More precisely, for residueii, letxigraph∈ℝ31x\_\{i\}^\{\\mathrm\{graph\}\}\\in\\mathbb\{R\}^\{31\}denote the native AtomSurf residue\-graph input feature and letsi∈ℝ256s\_\{i\}\\in\\mathbb\{R\}^\{256\}denote the frozen pretrained single representation\. We form

x~i=\[xigraph;si\]∈ℝ287,\\widetilde\{x\}\_\{i\}=\[x\_\{i\}^\{\\mathrm\{graph\}\}\\,;\\,s\_\{i\}\]\\in\\mathbb\{R\}^\{287\},and usex~i\\widetilde\{x\}\_\{i\}in place of the original residue\-graph input to AtomSurf’s first input block, which projects it to the native128128\-dimensional graph width\. The subsequent surface–graph encoder blocks and the ligand\-classification head are unchanged\. The same fusion path is used for the full and w/o TetSphere variants; only the structure\-specific input supplied to the pretrained encoder differs\.

The AtomSurf baseline and our model use the same downstream split, optimization protocol, checkpoint\-selection criterion, and evaluation cohort\. Neither model uses language\-model features\. For\+ Ours \(w/o TetSphere\), we retain the same downstream fusion architecture and replace only the pretrained representation source with the corresponding No\-Tet variant\. This control isolates the contribution of the registered volumetric geometry from that of the additional representation pathway and model capacity\.

For optimization, the pretraining model is initialized from scratch and trained for 20 epochs with batch size 1 and a maximum of 448 residues per example\. We use AdamW with learning rate3×10−43\\times 10^\{\-4\},β1=0\.9\\beta\_\{1\}=0\.9,β2=0\.999\\beta\_\{2\}=0\.999, weight decay10−410^\{\-4\}, bf16 autocast, and gradient clipping at 1\.0\. The pretraining checkpoint is selected using the validation split only\. The selected model is then used for full\-residue inference to export256256\-dimensional residue\-level single representations, which are kept fixed during downstream training\.

All downstream AtomSurf models are initialized from scratch and trained for 200 epochs with a global batch size of 8 using Adam with learning rate10−310^\{\-3\},β1=0\.9\\beta\_\{1\}=0\.9,β2=0\.99\\beta\_\{2\}=0\.99, zero weight decay, and gradient clipping at 1\.0\. We use a PolynomialLR schedule with 10 warmup epochs and a minimum learning rate of10−810^\{\-8\}\. The ligand\-classification objective is standard seven\-class cross entropy\. All variants use the same distributed\-training configuration\. A checkpoint is saved after every epoch, and the final model is selected by validation balanced accuracy\. The test split is used only for the final evaluation and does not participate in feature normalization, pretraining checkpoint selection, or downstream model selection\.

Figure[8](https://arxiv.org/html/2609.36277#A3.F8)shows representative test pockets correctly classified by the full model but misclassified by the AtomSurf baseline\. The examples span multiple ligand classes and are selected for visual clarity\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_3t4n_adp.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_3pzc_coa.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_3oc4_fad.png)\(a\) 3T4N – ADPours✓\\checkmarkAtomSurf: SAM×\\times\(b\) 3PZC – COAours✓\\checkmarkAtomSurf: HEM×\\times\(c\) 3OC4 – FADours✓\\checkmarkAtomSurf: SAM×\\times![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_3dy5_hem.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_1kqn_nad.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ligand_classification_example_3av6_sam.png)\(d\) 3DY5 – HEMours✓\\checkmarkAtomSurf: FAD×\\times\(e\) 1KQN – NADours✓\\checkmarkAtomSurf: NAP×\\times\(f\) 3AV6 – SAMours✓\\checkmarkAtomSurf: ADP×\\timesFigure 8:Test pockets our representation classifies correctly and native AtomSurf does not\.TetSphere surfaces are colored by distance to the native ligand \(red close, blue far\); labels give the true class and the class AtomSurf predicts instead\.
## Appendix DProtein–Protein Interface Prediction

Given two protein chains in their bound conformations, the task is to predict their interaction interface\. Following AtomSurf\([Mallet et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib5)\), we consider two settings\. Pinder\-Pair predicts whether a residue pair across the two chains is in contact, whereas Pinder\-Site predicts whether an individual residue belongs to the interface\. Both tasks are evaluated using AUROC over continuous prediction scores\.

Our experiments follow the clustered PINDER split, with 42,220 training clusters and 1,958/1,955 validation/test cluster representatives\([Kovtun et al\., 2024](https://arxiv.org/html/2609.36277#bib.bib23)\), and evaluate in the holo setting\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_adaptation.png)Figure 9:PPI residual adaptation pipeline\.Residual adaptation of the pretrained residue representation for protein–protein interface prediction\. AtomSurf consumes the molecular surface and residue graph of the bound complex and produces the baseline Pinder\-Site and Pinder\-Pair logitsℓi\\ell\_\{i\}andℓL​R\\ell\_\{LR\}\. In parallel, the frozen pretraining model yields residue\-level single representations, from which a site adapter predicts a per\-residue correctionΔi\\Delta\_\{i\}and a pair adapter predicts a residue\-pair correctionΔL​R\\Delta\_\{LR\}from a symmetric combination of the two projected endpoints\. Each correction is added to the corresponding AtomSurf logit, so the AtomSurf branch is left unchanged\. The complex shown is barnase–barstar \(PDB 1BRS\); on the molecular surface, residues within55Å of the partner chain are highlighted\.The public AtomSurf release does not include the exact task\-specific interface labels used for the originally reported PPI results\. We therefore define a residue pair as positive when any heavy\-atom distance between the two residues is below55Å\. This definition is applied identically to AtomSurf and our model\. Because it differs from the private labeling used for the originally reported AtomSurf results, our absolute scores should be interpreted within the matched comparison reported here rather than compared directly with the published AtomSurf numbers\.

For each retained system, the positive Pinder\-Pair examples are the unique residue pairs satisfying this contact criterion\. We keep all positive pairs and sample an equal number of non\-contact pairs from the complement of the residue\-pair Cartesian product\. For Pinder\-Site, a residue is labeled as an interface residue if it appears in at least one positive cross\-chain residue pair\. We collect the unique interface residues on each chain and independently sample an equal number of non\-interface residues from the remaining residues on that chain\. The sampling seed is derived deterministically from the split and system identifier\. Training samples are resampled by epoch, whereas validation and test samples use a fixed sampling state\.

The task\-specific pretraining model is trained on the PINDER training split, with validation used for checkpoint selection\. Because holo structures directly expose the relative geometry between the two chains, we mask cross\-chain geometric signals during pretraining to prevent direct leakage of the contact labels\. Specifically, cross\-chain distance, local\-direction, relative\-rotation, and corresponding validity channels are zeroed before the learnable pair modules\. The remaining topology channels encode non\-geometric relations such as polymer\-relative position and chain/entity relationships, while cross\-chain attention remains enabled\. Additional details are provided in Appendix[B](https://arxiv.org/html/2609.36277#A2)\.

For downstream prediction, the pretraining model provides one residue\-level single representation

si∈ℝ256s\_\{i\}\\in\\mathbb\{R\}^\{256\}for each protein residue\. Because PPI contains both residue\-level and residue\-pair\-level prediction tasks, we use task\-specific residual adapters rather than directly fusing the pretrained single representation with the AtomSurf residue feature, as in ligand\-binding pocket classification\. This allows the same pretrained single representations to support residue\-wise Pinder\-Site prediction and a symmetric residue\-pair construction for Pinder\-Pair prediction\. Pretrained pair representations are not passed directly to either downstream head\.

The original AtomSurf Pinder\-Site and Pinder\-Pair pipelines provide the baseline logits, while the pretrained single representations are used to predict additive task\-specific corrections\.

For Pinder\-Site, each residue representation is projected independently:

uisite=fsite​\(si\),fsite:256→32,u\_\{i\}^\{\\mathrm\{site\}\}=f\_\{\\mathrm\{site\}\}\(s\_\{i\}\),\\qquad f\_\{\\mathrm\{site\}\}:256\\rightarrow 32,wherefsitef\_\{\\mathrm\{site\}\}consists of LayerNorm, a linear projection, SiLU, and a second LayerNorm\. A residual head maps the resulting 32\-dimensional feature through a32→64→132\\rightarrow 64\\rightarrow 1multilayer perceptron to produce

Δisite\.\\Delta\_\{i\}^\{\\mathrm\{site\}\}\.The final site logit is

ℓisite=ℓiAtomSurf\+Δisite\.\\ell\_\{i\}^\{\\mathrm\{site\}\}=\\ell\_\{i\}^\{\\mathrm\{AtomSurf\}\}\+\\Delta\_\{i\}^\{\\mathrm\{site\}\}\.
For Pinder\-Pair, the two residue representations are projected using a separate task\-specific adapter:

uLpair=fpair​\(sL\),uRpair=fpair​\(sR\),fpair:256→32\.u\_\{L\}^\{\\mathrm\{pair\}\}=f\_\{\\mathrm\{pair\}\}\(s\_\{L\}\),\\qquad u\_\{R\}^\{\\mathrm\{pair\}\}=f\_\{\\mathrm\{pair\}\}\(s\_\{R\}\),\\qquad f\_\{\\mathrm\{pair\}\}:256\\rightarrow 32\.We then construct the symmetric interaction feature

qL​R=\[uLpair\+uRpair;\|uLpair−uRpair\|;uLpair⊙uRpair\]∈ℝ96\.q\_\{LR\}=\\left\[u\_\{L\}^\{\\mathrm\{pair\}\}\+u\_\{R\}^\{\\mathrm\{pair\}\}\\,;\\,\\left\|u\_\{L\}^\{\\mathrm\{pair\}\}\-u\_\{R\}^\{\\mathrm\{pair\}\}\\right\|\\,;\\,u\_\{L\}^\{\\mathrm\{pair\}\}\\odot u\_\{R\}^\{\\mathrm\{pair\}\}\\right\]\\in\\mathbb\{R\}^\{96\}\.This construction is invariant to exchanging the two residues\. A residual head mapsqL​Rq\_\{LR\}through a96→64→196\\rightarrow 64\\rightarrow 1multilayer perceptron to produce

ΔL​Rpair,\\Delta\_\{LR\}^\{\\mathrm\{pair\}\},and the final pair logit is

ℓL​Rpair=ℓL​RAtomSurf\+ΔL​Rpair\.\\ell\_\{LR\}^\{\\mathrm\{pair\}\}=\\ell\_\{LR\}^\{\\mathrm\{AtomSurf\}\}\+\\Delta\_\{LR\}^\{\\mathrm\{pair\}\}\.
The final linear layers of both residual heads are initialized to zero, so the residual branches initially contribute zero correction and the model starts from the original AtomSurf predictions\. The full model and\+ Ours \(w/o TetSphere\)use the same downstream residual architecture; they differ only in whether structure\-specific TetSphere information is provided to the pretraining model\. This control tests whether the improvement comes from the registered volumetric geometry itself rather than from adding an additional pretrained representation pathway\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_4hsr.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_7vpr.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/ppi_interface_1ije.png)\(a\) 4HSR\(b\) 7VPR\(c\) 1IJEFigure 10:Further predicted interface regions, drawn exactly as in Figure[5](https://arxiv.org/html/2609.36277#S4.F5)\. These three complexes have longer chains than the three shown in the main text \(216216–535535,280280–281281and9090–438438residues, against9797–215215\), so more unrelated structure falls in frame behind each patch; the ribbon context is drawn faint and segments that would pass in front of the envelopes are removed\. Each panel draws3030predicted contacts and all3030are true contacts\.For PPI, we train the multimodal pretraining model from random initialization on the clustered PINDER training split for 60 epochs\. We use a batch size of 2 and AdamW with learning rate10−410^\{\-4\}and weight decay10−210^\{\-2\}, with bf16 precision\. Each training example contains at most 512 residues across the two chains\. Complexes within this limit are used in full; for larger complexes, we apply interface\-centered cropping\. Residues within1515Å of the partner chain define candidate interface anchors, from which one anchor on each chain is sampled and contiguous sequence windows are extracted with a nominal single\-chain neighborhood size of 256 residues\. The crop is resampled across epochs\. The PPI\-specific masking and reconstruction objectives, including cross\-chain geometry masking and TetSphere partial\-mode denoising, follow Appendix[B](https://arxiv.org/html/2609.36277#A2)\. The resulting model is used to export256256\-dimensional residue\-level single representations for downstream training\.

For downstream optimization, all matched PPI experiments use the same fixed data manifest, sampling protocol, and validation\-based model\-selection rule\. All downstream variants are trained jointly on Pinder\-Site and Pinder\-Pair for up to 100 epochs with batch size 4 across five random seeds\. We use Adam with learning rate10−310^\{\-3\},β1=0\.9\\beta\_\{1\}=0\.9,β2=0\.99\\beta\_\{2\}=0\.99, zero weight decay, and gradient clipping at 1\.0\. The learning rate is controlled by ReduceLROnPlateau with factor0\.50\.5, patience 5, and minimum learning rate10−410^\{\-4\}, using validation Pinder\-Site AUROC as the scheduler signal\.

The training objective is the equally weighted sum of the Pinder\-Site and Pinder\-Pair binary cross\-entropy losses\. Both terms use BCEWithLogits with a positive\-class weight computed from the globally aggregated positive and negative element counts,

w\+=N−N\+N\+,w\_\{\+\}=\\frac\{N\-N\_\{\+\}\}\{N\_\{\+\}\},and are normalized by the corresponding global valid\-element counts\. Model selection is based exclusively on validation Pinder\-Pair AUROC\.

## Appendix EDe Novo Protein Binder Design

De novo protein binder design differs from the two prediction tasks above because the geometry of the binder being designed is not available before generation\. BoltzGen\([Stark et al\., 2025](https://arxiv.org/html/2609.36277#bib.bib12)\)is conditioned on the provided target structure together with benchmark\-defined design conditions, including target chains, binding\-site constraints when applicable, and binder\-length specifications\. Consequently, the binder TetSphere representation, and hence the representation of the completed target–binder complex, cannot be supplied directly at inference\. We therefore transfer the pretrained geometric representation to BoltzGen through representation alignment during training\.

For this task, we train the multimodal pretraining model described in Appendix[B](https://arxiv.org/html/2609.36277#A2)on the released BoltzGen PDB data\. Starting from a fixed manifest of 219,627 records, we apply the native BoltzGen record\-level filters:SizeFilter\(1–300 chains\),DateFilter\(release date before June 1, 2023\), andResolutionFilter\(00–99Å\), with predefined validation records exempted from the training filters\. After filtering, 193,326 records remain in the train/validation split\. We further exclude records without protein chains or without ready protein chains, yielding 189,300 usable assemblies: 188,902 for training and 398 for validation\.

The multimodal pretraining model is trained from random initialization for 160 epochs with batch size 4 using AdamW with learning rate3×10−43\\times 10^\{\-4\}, weight decay10−410^\{\-4\}, and gradient clipping at 1\.0\. Training uses bf16 precision\. The learning rate uses a 32,000\-step warmup, corresponding to5%5\\%of the configured schedule, followed by cosine decay to one tenth of the peak learning rate\. The checkpoint is selected using the total validation loss and is kept fixed during downstream BoltzGen training\. Given a training complex with known structure, the resulting model produces residue\-level single and pair representations\(siT,zi​jT\)\(s\_\{i\}^\{T\},z\_\{ij\}^\{T\}\), which serve as representation targets for the corresponding internal BoltzGen states\.

We initialize the generator from the released BoltzGen checkpoint and align the final single and ordered\-pair representations of the BoltzGen trunk to the pretrained representation\. BoltzGen produces

siG∈ℝ384,zi​jG∈ℝ128,s\_\{i\}^\{G\}\\in\\mathbb\{R\}^\{384\},\\qquad z\_\{ij\}^\{G\}\\in\\mathbb\{R\}^\{128\},while the pretraining model provides

siT∈ℝ256,zi​jT∈ℝ128\.s\_\{i\}^\{T\}\\in\\mathbb\{R\}^\{256\},\\qquad z\_\{ij\}^\{T\}\\in\\mathbb\{R\}^\{128\}\.The single projector consists of LayerNorm\(384\)\(384\), a linear layer from 384 to 384, GELU, and a linear layer from 384 to 256\. The pair projector consists of LayerNorm\(128\)\(128\)followed by a linear layer from 128 to 128\.

For a projected BoltzGen representationuuand its corresponding pretraining targetvv, both vectors areℓ2\\ell\_\{2\}\-normalized over the feature dimension, and the alignment error is

ℓalign​\(u,v\)=1−u¯𝖳​v¯\+0\.05​SmoothL1⁡\(u¯,v¯\)\.\\ell\_\{\\mathrm\{align\}\}\(u,v\)=1\-\\bar\{u\}^\{\\mathsf\{T\}\}\\bar\{v\}\+0\.05\\,\\operatorname\{SmoothL1\}\(\\bar\{u\},\\bar\{v\}\)\.The single and pair alignment losses are masked means over valid native tokens and ordered token pairs for which a corresponding pretraining target is available\. Positions without a valid target are excluded from the alignment term but remain part of the native BoltzGen objective\. The complete training objective is

ℒ=4\.0​ℒdiffusion\+0\.05​ℒdistogram\+0\.05​ℒalignsingle\+0\.05​ℒalignpair\.\\mathcal\{L\}=4\.0\\,\\mathcal\{L\}\_\{\\mathrm\{diffusion\}\}\+0\.05\\,\\mathcal\{L\}\_\{\\mathrm\{distogram\}\}\+0\.05\\,\\mathcal\{L\}\_\{\\mathrm\{align\}\}^\{\\mathrm\{single\}\}\+0\.05\\,\\mathcal\{L\}\_\{\\mathrm\{align\}\}^\{\\mathrm\{pair\}\}\.
For downstream optimization, we fine\-tune the entire BoltzGen trunk together with the representation projectors\. We use AdamW with a peak learning rate of2×10−52\\times 10^\{\-5\}for the newly initialized projectors and2×10−62\\times 10^\{\-6\}for the BoltzGen trunk\. Both parameter groups use a 1,024\-step linear warmup followed by cosine decay to one tenth of their respective peak learning rates over 20 epochs\. Training is performed on eight H100 GPUs with 8,192 samples per epoch, batch size 1 per rank, diffusion multiplicity 8, a 512\-token crop, and bf16 mixed precision\.

The pretraining model is used only to provide representation targets during training and is not required at inference, leaving the original BoltzGen generation inputs and native inference procedure unchanged\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/binder_alignment.png)Figure 11:Representation alignment for de novo protein binder design\.During training, the fixed multimodal pretraining model provides single and pair representation targets for known target–binder complexes\. Learned projectors map the corresponding BoltzGen trunk representations into these target spaces, where the alignment loss is applied\. The pretraining model and alignment projectors are used only during training; generation follows the native BoltzGen inference pipeline\.We evaluate four training conditions\.BoltzGenis the released baseline\.\+Cont\.continues training from the released checkpoint using the same data and training schedule but without representation alignment\.\+ Ours \(w/o TetSphere\)uses the same representation\-alignment procedure as the full model, but the pretraining model receives no structure\-specific TetSphere input\.Oursuses the full pretrained representation with TetSphere geometry\. Together, these controls separate the effects of continued training, representation alignment, and the registered volumetric representation\.

We evaluate on the ten\-target ProtDBench and BoltzGen Challenge Set following their respective standard evaluation protocols\. The two benchmarks are evaluated separately and are not pooled into a single pass rate\. Within each benchmark, target specifications, binder\-length conditions, generation budgets, random\-seed policy, and screening criteria are held fixed across methods\. Reported success rates therefore measure computational screening outcomes under the corresponding benchmark protocol rather than experimental binding\.

#### ProtDBench budgets and criteria\.

Each arm generates 32 backbones in every\(target,binder\-length\)\(\\text\{target\},\\text\{binder\-length\}\)cell\. The ten targets contribute 150 cells in total \(BHRF1 and SC2RBD nine each; IL7RA, PDL1, TrkA, and TNFa fifteen each; IR and H1 seventeen each; VEGFA and IL17A nineteen each\), giving 4,800 backbones per arm\. ProteinMPNN \(v\_48\_020, original weights, sampling temperature10−410^\{\-4\}, cysteine omitted\) then designs eight sequences per backbone, yielding 38,400 sequence candidates per arm\.

A candidate passes the AF2\-IG Easy conjunction when normalized pLDDT\>0\.8\>0\.8, interface pTM\>0\.5\>0\.5, unscaled interface PAE<10\.85<10\.85, and bound–unbound RMSD<3\.5<3\.5Å\. The unbound prediction is run only for candidates that satisfy the first three criteria, so the bound–unbound RMSD criterion is evaluated conditionally\. A backbone is counted as successful when at least one of its eight designed sequences passes\.

#### Challenge Set budgets and criteria\.

Each arm requests 2,000 candidates: ten targets, five generation seeds per target, and 40 designs per seed\. Each candidate consists of one backbone and the single sequence produced by the BoltzGen inverse\-folding stage, so backbone\- and sequence\-level success coincide\. The final hard\-filter rate is defined with respect to this fixed request budget: its numerator is the sum of the per\-target hard\-filter counts, and its denominator is the 2,000 requested candidates\. We track the number of unique designed sequences separately as an audit statistic and do not use it to renormalize the reported rate\. All 2,000 requested candidates in each reported arm are sequence\-unique\.

The released BoltzGen hard filter is applied sequentially\. A candidate must contain no unknown residue; complex, design\-subset, and binder\-only backbone RMSD must each be≤2\.5\\leq 2\.5Å; and alanine, glycine, glutamate, leucine, and valine fractions must not exceed0\.300\.30,0\.200\.20,0\.200\.20,0\.300\.30, and0\.200\.20, respectively\. Table[3](https://arxiv.org/html/2609.36277#S4.T3)reports this cascade as cumulative survival: each row gives the percentage of the 2,000 requested candidates still passing after that criterion together with all criteria above it\. The rows therefore share a single denominator, decrease monotonically, and are comparable across arms\. We report the cascade this way rather than as per\-stage retention conditional on the preceding stages, because a conditional rate is computed over a different surviving population for each arm and cannot distinguish a genuinely weaker stage from one that simply received more marginal candidates from upstream\.

![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/binder_design_1jqdprot.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/binder_design_3qkgprot.png)![Refer to caption](https://arxiv.org/html/2609.36277v1/figures/binder_design_7aahprot.png)\(a\) 1JQD\(b\) 3QKG\(c\) 7AAHFigure 12:Binders designed by our model on three BoltzGen Challenge Set targets\. The designed binder is drawn as ball\-and\-stick inside its own TetSphere volumes \(blue\), and the target residues it contacts are shown as TetSphere volumes \(red\); the remainder of the target is a secondary\-structure cartoon\.
#### Uncertainty and omitted rows\.

The±\\pmvalues in Table[3](https://arxiv.org/html/2609.36277#S4.T3)are standard deviations from 2,000 nonparametric bootstrap replicates, with resampling performed within targets and with ProtDBench backbones kept together with their eight designed sequences\. The target panel is held fixed\. Diversity statistics are reported without bootstrap uncertainty because resampling duplicates distort cluster counts\.

#### Structural diversity\.

Generated binder chains are extracted from each complex and clustered separately within each target using Foldseekeasy\-clusterin TM\-align mode with a coverage threshold of0\.80\.8and the stated TM\-score threshold\. Cluster counts are then summed across targets\.

For the diversity\-adjusted cluster pass rate reported in Table[3](https://arxiv.org/html/2609.36277#S4.T3), we count Foldseek clusters among successful designs and normalize by the total generation budget of the corresponding benchmark: 4,800 backbones for ProtDBench and 2,000 candidates for the Challenge Set\. ProtDBench contains multiple binder\-length conditions within each target, which naturally increases its cluster count; diversity rates are therefore intended for comparisons across training arms within a benchmark, rather than across the two benchmarks\.

For completeness, clustering all generated candidates rather than only successful designs gives, at TM0\.60\.6,17\.92%17\.92\\%,22\.04%22\.04\\%,36\.19%36\.19\\%, and29\.56%29\.56\\%on ProtDBench and23\.95%23\.95\\%,26\.30%26\.30\\%,58\.15%58\.15\\%, and38\.25%38\.25\\%on the Challenge Set for BoltzGen,\+Cont\.,\+Ours \(w/o TetSphere\), and Ours, respectively\. At TM0\.80\.8, the corresponding values are38\.81%38\.81\\%,47\.92%47\.92\\%,63\.90%63\.90\\%, and66\.17%66\.17\\%on ProtDBench and42\.15%42\.15\\%,44\.05%44\.05\\%,75\.10%75\.10\\%, and71\.15%71\.15\\%on the Challenge Set\.