Joint Discrete and SE(3) Diffusion for Protein Interface Design

# Joint Discrete and 𝑆⁢𝐸⁢(3) Diffusion for Protein Interface Design
Source: [https://arxiv.org/html/2610.14699](https://arxiv.org/html/2610.14699)
Zhenqiao SongAffiliation:Language Technologies Institute, Carnegie Mellon UniversityCorrespondence to:[zhenqiaosong@cmu\.edu](mailto:[email protected])Lei LiAffiliation:Language Technologies Institute, Carnegie Mellon University

###### Abstract

The design of high\-affinity, target\-specific protein binders is central to the development of modern therapeutics\. However, existing protein\-generation methods remain limited in this setting, as many generated designs fail to simultaneously achieve geometric and physicochemical complementarity at binding interfaces\. To address this challenge, we introduce PIDiff, a physicochemically enhanced multimodal diffusion framework for protein–protein complex design\. PIDiff jointly designs target\-conditioned protein sequences and structures by coupling diffusion in discrete sequence space with diffusion on the continuousS​E​\(3\)SE\(3\)structural manifold\. At its core, PIDiff employs PIFormer, a novel architecture that interleaves geometry\-aware invariant point attention with global sequence self\-attention\. We further curate PIBench, a protein–protein complex dataset with diverse target scaffolds derived from the Protein Data Bank \(PDB\)\. We first pretrain PIDiff on PIBench and subsequently fine\-tune it for two therapeutically relevant applications: minibinder design and antibody design\. PPFrame consistently outperforms baseline methods across sequence design success rate, structure binding energy, and sequence\-structure co\-designability on all tasks\.

###### Keywords:

Machine Learning, ICML

## 1Introduction

Thede novodesign of high\-affinity, target\-specific protein binders is foundational to modern biomedicine, driving transformative advances in therapeutics\([Nelson et al\., 2010](https://arxiv.org/html/2610.14699#bib.bib8)\), diagnostics\([Brennan et al\., 2010](https://arxiv.org/html/2610.14699#bib.bib9)\), and vaccines\([Correia et al\., 2014](https://arxiv.org/html/2610.14699#bib.bib10);[Sesterhenn et al\., 2020](https://arxiv.org/html/2610.14699#bib.bib11)\)\. Central to molecular recognition is the requirement for strict geometric and physicochemical complementarity across the binding interface\([Chothia and Janin, 1975](https://arxiv.org/html/2610.14699#bib.bib12);[Conte et al\., 1999](https://arxiv.org/html/2610.14699#bib.bib13);[Gainza et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib14)\)\. The complementary interface is governed by core energetic drivers: van der Waals forces optimize spatial packing, while hydrophobic and electrostatic interactions enforce physicochemical compatibility\. Although traditional methods have exploited these energetic principles to engage pre\-selected surface patches\([Chevalier et al\., 2017](https://arxiv.org/html/2610.14699#bib.bib15);[Silva et al\., 2019](https://arxiv.org/html/2610.14699#bib.bib16)\), they frequently fail when target proteins lack well\-defined binding pockets, thereby restricting their scope to a narrow range of target topologies\.

Recently, rapid advances in deep generative modeling have significantly expanded binder design capabilities\([Evans et al\., 2021](https://arxiv.org/html/2610.14699#bib.bib18);[Cao et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib17);[Bennett et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib19);[Zambaldi et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib20);[Stark et al\., 2025](https://arxiv.org/html/2610.14699#bib.bib51);[Sappington et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib50)\)\. These approaches have made it possible to design binders across diverse protein targets\. In specialized domains, such as flexible peptides\([Vázquez Torres et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib21)\)and intrinsically disordered proteins\([Wu et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib22)\), deep generative models can yield potent binders without requiring extensive experimental optimization\. Despite these breakthroughs, the overall success rate remains low due to two primary failure modes: the designed structure may fail to bind target interfaces due to the physicochemical or geometric non\-complementarity, and the designed sequence may not fold into the intended structure\. Crucially, most existing frameworks decouple sequence and structure generation or specialize in particular interaction types, leaving challenging, pocketless targets still intractable\.

To address these challenges, we present PIDiff, a physicochemically enhanced multimodal diffusion method to co\-design binder protein sequence and structure for an arbitrary target\. In our framework, proteins are represented as rigid\-body frames, coupling each residue’s spatial backbone coordinates with its local physicochemical properties\. PIDiff employs a unified multimodal diffusion process that operates jointly over discrete amino acid sequence space and the continuousS​E​\(3\)SE\(3\)structural manifold, progressively adding noise to train a joint generative model\. The generative model is powered by PIFormer, an architecture that interleaves invariant point attention\([Jumper et al\., 2021](https://arxiv.org/html/2610.14699#bib.bib1)\)with sequence self\-attention\([Vaswani et al\., 2017](https://arxiv.org/html/2610.14699#bib.bib23)\)to simultaneously capture spatial geometric constraints and amino acid dependencies\. By tightly integrating sequence, structure, and physicochemical properties, PIDiff enforces binding interface complementarity and high sequence\-structure consistency, effectively mitigating key failure modes in binder design to drive higher success rates\.

Concretely, our contributions are listed as follows:

- •We propose PIDiff, a multimodal diffusion method building upon PIFormer to co\-design the binder protein sequences and structures for arbitrary protein targets\.
- •We create PIBench, a general protein\-protein complex dataset across diverse target geometries comprising 602,364 complexes curated from PDB\([Berman et al\., 2000](https://arxiv.org/html/2610.14699#bib.bib49)\)\.
- •Our model pretrained on PIBench outperforms strong baselines, demonstrating a sequence design success rate of 47\.8%, structure binding energy of \-67\.8 and a co\-design consistency of 65\.2%\.
- •We finetune the pretrained PIDiff on two important real\-world applications: target\-protein mini\-binder complex design and antigen\-antibody complex design\. The finetuned models achieve design success rates of40\.7%40\.7\\%and18\.7%18\.7\\%, respectively, demonstrating the framework’s effectiveness in addressing key therapeutic challenges\.

## 2Related Work

Protein\-Protein Complex Design\.Designing target\-specific protein binders with high affinity has been extensively explored using a wide range of approaches\. Conventional experimental approaches rely primarily on directed evolution of natural protein scaffolds\([Vazquez\-Lombardi et al\., 2015](https://arxiv.org/html/2610.14699#bib.bib28)\)or high\-throughput antibody screening from synthetic libraries\([Chao et al\., 2006](https://arxiv.org/html/2610.14699#bib.bib29);[Hackel et al\., 2008](https://arxiv.org/html/2610.14699#bib.bib30)\)\. Although powerful, these methods require considerable experimental effort and do not provide substantial control over the properties of the binder proteins\. Traditional computational methods have sought to model physicochemical properties to accelerate the design process\([Chevalier et al\., 2017](https://arxiv.org/html/2610.14699#bib.bib15);[Fleishman et al\., 2011](https://arxiv.org/html/2610.14699#bib.bib31)\)\. However, their reliance on pre\-defined hotspot residues heavily restricts their scope\. Recent advances in deep learning have significantly improved the prediction and design of protein\-protein interactions\([Evans et al\., 2021](https://arxiv.org/html/2610.14699#bib.bib18);[Humphreys et al\., 2021](https://arxiv.org/html/2610.14699#bib.bib32);[Bryant et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib33);[Pacesa et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib34);[Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\), enabling binder design without prior hotspot specification\([Cao et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib17);[Bennett et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib19);[Chen et al\., 2025](https://arxiv.org/html/2610.14699#bib.bib26)\)\. Nevertheless, the overall success rate remains low due to the difficulty of simultaneously satisfying the strict geometric and physicochemical complementarity essential for stable physical binding interfaces\([Berger et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib35);[Zambaldi et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib20)\)\.

Diffusion Models\.Diffusion models\([Sohl\-Dickstein et al\., 2015](https://arxiv.org/html/2610.14699#bib.bib36)\)have established themselves as a powerful class of generative frameworks\.[Ho et al\. \(2020\)](https://arxiv.org/html/2610.14699#bib.bib2)introduce denoising diffusion probabilistic models, formalizing their connection to score\-based generative modeling\([Song and Ermon, 2019](https://arxiv.org/html/2610.14699#bib.bib37)\)\. Diffusion models have achieved remarkable success in computational protein design\. Early structural approaches focus on backbone design, progressing fromCαC\_\{\\alpha\}trace generation\([Trippe et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib38)\)to full rigid\-bodyS​E​\(3\)SE\(3\)backbone modeling\([Watson et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib24);[Yim et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib5)\)\. In parallel, discrete diffusion formulations are introduced to tackle protein sequence generation\([Alamdari et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib40);[Wang et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib39)\)\. More recent efforts have explored joint sequence\-structure co\-design through multimodal diffusion\([Song et al\., 2025](https://arxiv.org/html/2610.14699#bib.bib42);[Wang et al\., 2025](https://arxiv.org/html/2610.14699#bib.bib41)\)\. However, the application of diffusion models for the sequence\-structure co\-design of protein complexes still remains an open problem\.

## 3Proposed Method: PIDiff

### 3\.1Problem Definition

We model a protein as a joint representation of its amino acid sequence and 3D backbone structure\. Let𝒜\\mathcal\{A\}denote the set of2020standard amino acids\. A protein of lengthLLis represented by a sequence𝒂=\{a1,a2,…,aL\}∈𝒜L\\boldsymbol\{a\}=\\\{a\_\{1\},a\_\{2\},\.\.\.,a\_\{L\}\\\}\\in\\mathcal\{A\}^\{L\}and an associated backbone structure𝒔\\boldsymbol\{s\}\. Following[Jumper et al\. \(2021\)](https://arxiv.org/html/2610.14699#bib.bib1), the backbone geometry is parameterized by a collection of local rigid frames mapping from reference coordinates\[N∗,Cα∗,C∗,O∗\]∈ℝ4×3\[N^\{\*\},C\_\{\\alpha\}^\{\*\},C^\{\*\},O^\{\*\}\]\\in\\mathbb\{R\}^\{4\\times 3\}centered atCα∗=\(0,0,0\)C\_\{\\alpha\}^\{\*\}=\(0,0,0\)\. Specifically, for residuei∈\{1,…,L\}i\\in\\\{1,\\dots,L\\\}, the mainchain heavy atom coordinates\[Ni,Ci,\(Cα\)i\]\[N\_\{i\},C\_\{i\},\(C\_\{\\alpha\}\)\_\{i\}\]are represented by applying a rigid body transformation𝑻i∈S​E​\(3\)\\boldsymbol\{T\}\_\{i\}\\in SE\(3\)to the reference frame:\[Ni,Ci,\(Cα\)i\]=𝑻i⋅\[N∗,C∗,Cα∗\]\[N\_\{i\},C\_\{i\},\(C\_\{\\alpha\}\)\_\{i\}\]=\\boldsymbol\{T\}\_\{i\}\\cdot\[N^\{\*\},C^\{\*\},C\_\{\\alpha\}^\{\*\}\]\. Each frame𝑻i=\(𝒓i,𝒙i\)\\boldsymbol\{T\}\_\{i\}=\(\\boldsymbol\{r\}\_\{i\},\\boldsymbol\{x\}\_\{i\}\)comprises a rotation matrix𝒓i∈S​O​\(3\)\\boldsymbol\{r\}\_\{i\}\\in SO\(3\)and a translation vector𝒙i∈ℝ3\\boldsymbol\{x\}\_\{i\}\\in\\mathbb\{R\}^\{3\}, acting on a spatial coordinate𝒗∈ℝ3\\boldsymbol\{v\}\\in\\mathbb\{R\}^\{3\}via𝑻i⋅𝒗=𝒓i​𝒗\+𝒙i\\boldsymbol\{T\}\_\{i\}\\cdot\\boldsymbol\{v\}=\\boldsymbol\{r\}\_\{i\}\\boldsymbol\{v\}\+\\boldsymbol\{x\}\_\{i\}\. The full backbone structure is thus defined as𝒔=\{𝑻1⋅\[N∗,C∗,Cα∗\],…,𝑻L⋅\[N∗,C∗,Cα∗\]\}\\boldsymbol\{s\}=\\\{\\boldsymbol\{T\}\_\{1\}\\cdot\[N^\{\*\},C^\{\*\},C\_\{\\alpha\}^\{\*\}\],\\dots,\\boldsymbol\{T\}\_\{L\}\\cdot\[N^\{\*\},C^\{\*\},C\_\{\\alpha\}^\{\*\}\]\\\}, with backbone oxygen atoms positioned using the torsion angleψ\\psiaround theCα​–​CC\_\{\\alpha\}\\text\{\-\-\}Cbond\. Furthermore, each residueiiis assigned a physicochemical feature vector𝒇i=\(yi,ci\)\\boldsymbol\{f\}\_\{i\}=\(y\_\{i\},c\_\{i\}\)encoding its hydrophobicityyiy\_\{i\}and chargecic\_\{i\}, yielding a protein feature sequence𝒇=\(𝒇1,𝒇2,…,𝒇L\)\\boldsymbol\{f\}=\(\\boldsymbol\{f\}\_\{1\},\\boldsymbol\{f\}\_\{2\},\\dots,\\boldsymbol\{f\}\_\{L\}\)\.

Given a target protein𝒯=\{𝒂𝒯,𝒔𝒯,𝒇𝒯\}\\mathcal\{T\}=\\\{\\boldsymbol\{a\}^\{\\mathcal\{T\}\},\\boldsymbol\{s\}^\{\\mathcal\{T\}\},\\boldsymbol\{f\}^\{\\mathcal\{T\}\}\\\}of lengthMM, the objective of protein\-protein complex co\-design is to generate a binder proteinℬ=\{𝒂ℬ,𝒔ℬ\}\\mathcal\{B\}=\\\{\\boldsymbol\{a\}^\{\\mathcal\{B\}\},\\boldsymbol\{s\}^\{\\mathcal\{B\}\}\\\}of lengthNNdisplaying complementary physicochemical and geometric properties to target𝒯\\mathcal\{T\}\. This entails jointly sampling the binder sequence𝒂ℬ=\{a1ℬ,…,aNℬ\}\\boldsymbol\{a\}^\{\\mathcal\{B\}\}=\\\{a\_\{1\}^\{\\mathcal\{B\}\},\\dots,a\_\{N\}^\{\\mathcal\{B\}\}\\\}and backbone transformations𝑻ℬ=\{𝑻1ℬ,…,𝑻Nℬ\}\\boldsymbol\{T\}^\{\\mathcal\{B\}\}=\\\{\\boldsymbol\{T\}\_\{1\}^\{\\mathcal\{B\}\},\\dots,\\boldsymbol\{T\}\_\{N\}^\{\\mathcal\{B\}\}\\\}with aligned physicochemical features𝒇ℬ\\boldsymbol\{f\}^\{\\mathcal\{B\}\}\. Formally, we formulate this co\-design task as learning a conditional generative modelpθ​\(ℬ∣𝒯\)p\_\{\\theta\}\(\\mathcal\{B\}\\mid\\mathcal\{T\}\)parameterized byθ\\theta\.

![Refer to caption](https://arxiv.org/html/2610.14699v1/ppframe_recolored.png)Figure 1:Overview of the proposed PIDiff architecture\. At each diffusion time step, the target protein structure is rendered in green, while the binder protein is shown in pink\.
### 3\.2Overall Model Architecture

The overall architecture of PIDiff is illustrated in Figure[1](https://arxiv.org/html/2610.14699#S3.F1)\. PIDiff is a multimodal denoising diffusion probabilistic model \(DDPM\)\([Ho et al\., 2020](https://arxiv.org/html/2610.14699#bib.bib2)\)designed for joint generation of a binder’s primary sequence𝒂ℬ\\boldsymbol\{a\}^\{\\mathcal\{B\}\}and backbone structure𝒔ℬ\\boldsymbol\{s\}^\{\\mathcal\{B\}\}conditioned on a target protein𝒯\\mathcal\{T\}\. The framework models a forward perturbation processqqthat corrupts data stateℬ0∼q⁡\(ℬ0∣𝒯\)\\mathcal\{B\}\_\{0\}\\sim q\(\\mathcal\{B\}\_\{0\}\\mid\\mathcal\{T\}\)into noiseℬ𝕋∼qnoise\\mathcal\{B\}\_\{\\mathbb\{T\}\}\\sim q\_\{\\text\{noise\}\}over𝕋\\mathbb\{T\}steps, alongside a parameterized reverse generative processpθp\_\{\\theta\}:

q\(ℬ1:𝕋∣ℬ0,𝒯\)=∏t=1𝕋q⁡\(ℬt∣ℬt−1,𝒯\)pθ\(ℬ0:𝕋∣𝒯\)=p⁡\(ℬ𝕋\)​∏t=1𝕋pθ​\(ℬt−1∣ℬt,𝒯\)\\small\\begin\{split\}q\(\\mathcal\{B\}\_\{1:\\mathbb\{T\}\}\\mid\\mathcal\{B\}\_\{0\},\\mathcal\{T\}\)&=\\prod\\nolimits\_\{t=1\}^\{\\mathbb\{T\}\}q\(\\mathcal\{B\}\_\{t\}\\mid\\mathcal\{B\}\_\{t\-1\},\\mathcal\{T\}\)\\\\ p\_\{\\theta\}\(\\mathcal\{B\}\_\{0:\\mathbb\{T\}\}\\mid\\mathcal\{T\}\)&=p\(\\mathcal\{B\}\_\{\\mathbb\{T\}\}\)\\prod\\nolimits\_\{t=1\}^\{\\mathbb\{T\}\}p\_\{\\theta\}\(\\mathcal\{B\}\_\{t\-1\}\\mid\\mathcal\{B\}\_\{t\},\\mathcal\{T\}\)\\end\{split\}\(1\)Each latent variableℬt=\(𝒂tℬ,𝒔tℬ\)\\mathcal\{B\}\_\{t\}=\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{t\},\\boldsymbol\{s\}^\{\\mathcal\{B\}\}\_\{t\}\)represents a hybrid state spanning discrete amino acid categories and continuousS​E​\(3\)SE\(3\)transformations\. Model parametersθ\\thetaare optimized by maximizing the standard evidence lower bound on log\-likelihood:

ℒVLB=𝔼q\[logpθ\(ℬ0∣ℬ1,𝒯\)−∑t=2𝕋KL\[q\(ℬt−1∣ℬt,ℬ0,𝒯\)∥pθ\(ℬt−1∣ℬt,𝒯\)\]⏟ℒt\]\\small\\begin\{split\}\\mathcal\{L\}\_\{\\text\{VLB\}\}&=\\mathbb\{E\}\_\{q\}\\Big\[\\log p\_\{\\theta\}\(\\mathcal\{B\}\_\{0\}\\mid\\mathcal\{B\}\_\{1\},\\mathcal\{T\}\)\\\\ &\-\\sum\\nolimits\_\{t=2\}^\{\\mathbb\{T\}\}\\underbrace\{\\mathrm\{KL\}\\left\[q\(\\mathcal\{B\}\_\{t\-1\}\\mid\\mathcal\{B\}\_\{t\},\\mathcal\{B\}\_\{0\},\\mathcal\{T\}\)\\parallel p\_\{\\theta\}\(\\mathcal\{B\}\_\{t\-1\}\\mid\\mathcal\{B\}\_\{t\},\\mathcal\{T\}\)\\right\]\}\_\{\\mathcal\{L\}\_\{t\}\}\\Big\]\\end\{split\}\(2\)By unifying discrete sequence transitions with continuous manifold diffusion, PIDiff enhances sequence\-structure consistency throughout generative sampling\.

### 3\.3PIDiff Diffusion Process

We formulate the joint forward diffusion process of protein residue sequence𝒂tℬ\\boldsymbol\{a\}\_\{t\}^\{\\mathcal\{B\}\}and backbone structure𝒔tℬ\\boldsymbol\{s\}^\{\\mathcal\{B\}\}\_\{t\}as factorized transitions across individual residues and modalities at each time steptt:

q⁡\(ℬt∣ℬt−1,𝒯\)=∏i=1Nq⁡\(ai,tℬ∣ai,t−1ℬ,𝒯\)⋅∏i=1Nq⁡\(𝒔i,tℬ∣𝒔i,t−1ℬ,𝒯\)\\small\\begin\{split\}q\(\\mathcal\{B\}\_\{t\}\\mid\\mathcal\{B\}\_\{t\-1\},\\mathcal\{T\}\)&=\\prod\\nolimits\_\{i=1\}^\{N\}q\(a^\{\\mathcal\{B\}\}\_\{i,t\}\\mid a^\{\\mathcal\{B\}\}\_\{i,t\-1\},\\mathcal\{T\}\)\\\\ &\\cdot\\prod\\nolimits\_\{i=1\}^\{N\}q\(\\boldsymbol\{s\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{s\}^\{\\mathcal\{B\}\}\_\{i,t\-1\},\\mathcal\{T\}\)\\end\{split\}\(3\)whereai,tℬa\_\{i,t\}^\{\\mathcal\{B\}\}and𝒔i,tℬ=𝑻i,tℬ⋅\[N∗,Cα∗,C∗\]\\boldsymbol\{s\}\_\{i,t\}^\{\\mathcal\{B\}\}=\\boldsymbol\{T\}\_\{i,t\}^\{\\mathcal\{B\}\}\\cdot\[N^\{\*\},C\_\{\\alpha\}^\{\*\},C^\{\*\}\]denote theii\-th residue type and local rigid frame at time steptt, respectively\. This factorization enables efficient noise sampling across different modalities\.

Discrete Sequence Diffusion\.For the sequence components𝒂ℬ\\boldsymbol\{a\}^\{\\mathcal\{B\}\}, state transitions are defined via transition matrices\[𝑸t\]m​n=q⁡\(ai,tℬ=n∣ai,t−1ℬ=m\)\[\\boldsymbol\{Q\}\_\{t\}\]\_\{mn\}=q\(a^\{\\mathcal\{B\}\}\_\{i,t\}=n\\mid a^\{\\mathcal\{B\}\}\_\{i,t\-1\}=m\)\. Expressingai,tℬa\_\{i,t\}^\{\\mathcal\{B\}\}as a one\-hot row vector𝒂i,tℬ\\boldsymbol\{a\}\_\{i,t\}^\{\\mathcal\{B\}\}, the single\-step transition is given by:

q⁡\(𝒂i,tℬ∣𝒂i,t−1ℬ,𝒯\)=Cat⁡\(𝒂i,tℬ,𝒑=𝒂i,t−1ℬ​𝑸t\)\\small q\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\},\\mathcal\{T\}\)=\\operatorname\{Cat\}\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\};\\boldsymbol\{p\}=\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\}\\boldsymbol\{Q\}\_\{t\}\)\(4\)whereCat⁡\(𝒂,𝒑\)\\operatorname\{Cat\}\(\\boldsymbol\{a\};\\boldsymbol\{p\}\)denotes a categorical distribution with probability vector𝒑\\boldsymbol\{p\}\. Starting from𝒂i,0ℬ\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\}, thett\-step marginal and posterior distributions are analytically derived as:

q⁡\(𝒂i,tℬ∣𝒂i,0ℬ,𝒯\)=Cat⁡\(𝒂i,tℬ,𝒑=𝒂i,0ℬ​𝑸¯t\),𝑸¯t=∏j=1t𝑸jq⁡\(𝒂i,t−1ℬ\|𝒂i,tℬ,𝒂i,0ℬ,𝒯\)=Cat​\(𝒂i,t−1ℬ,𝒑=𝒂i,tℬ​𝑸t⊤⊙𝒂i,0ℬ​𝑸¯t−1𝒂i,0ℬ​𝑸¯t​𝒂i,tℬ⊤\)\\small\\begin\{split\}&q\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)=\\operatorname\{Cat\}\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\};\\boldsymbol\{p\}=\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\}\\bar\{\\boldsymbol\{Q\}\}\_\{t\}\),\\hskip 9\.24994pt\\hskip 9\.24994pt\\bar\{\\boldsymbol\{Q\}\}\_\{t\}=\\prod\\nolimits\_\{j=1\}^\{t\}\\boldsymbol\{Q\}\_\{j\}\\\\ &q\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\}\|\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\},\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)=\\text\{Cat\}\\Big\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\};\\boldsymbol\{p\}=\\frac\{\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\}\\boldsymbol\{Q\}\_\{t\}^\{\\top\}\\odot\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\}\\boldsymbol\{\\overline\{Q\}\}\_\{t\-1\}\}\{\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\}\\boldsymbol\{\\overline\{Q\}\}\_\{t\}\{\\boldsymbol\{a\}^\{\\mathcal\{B\}\\top\}\_\{i,t\}\}\}\\Big\)\\\\ \\end\{split\}\(5\)Following[Austin et al\. \(2021\)](https://arxiv.org/html/2610.14699#bib.bib3), we adopt an absorbing\-state transition matrix𝑸t\\boldsymbol\{Q\}\_\{t\}with an absorbing token \[mask\], where each token either remains unchanged or transitions to \[mask\] with probabilityβt\\beta\_\{t\}:

\[𝑸t\]m​n=\{1if​m=n=\[mask\]1−βtif​m=n≠\[mask\]βtif​n=\[mask\],m≠\[mask\]\\small\[\\boldsymbol\{Q\}\_\{t\}\]\_\{mn\}=\\begin\{cases\}1&\\text\{if \}m=n=\\text\{\[mask\]\}\\\\ 1\-\\beta\_\{t\}&\\text\{if \}m=n\\neq\\text\{\[mask\]\}\\\\ \\beta\_\{t\}&\\text\{if \}n=\\text\{\[mask\]\},m\\neq\\text\{\[mask\]\}\\end\{cases\}\(6\)
Continuous Structure Diffusion onS​E​\(3\)SE\(3\)\.To generate backbone structures inS​E​\(3\)NSE\(3\)^\{N\}, we build upon Riemannian score\-based generative modeling\([De Bortoli et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib4)\)\. Following[Yim et al\. \(2023\)](https://arxiv.org/html/2610.14699#bib.bib5), we treat the rotationalS​O​\(3\)SO\(3\)and translationalℝ3\\mathbb\{R\}^\{3\}sub\-manifolds independently, setting the final invariant distribution topSE⁡\(3\)​\(𝑻i,tℬ\)∝𝒰SO⁡\(3\)​\(𝒓i,tℬ\)​𝒩​\(𝒙i,tℬ,𝟎,𝑰3\)p\_\{\\mathrm\{SE\}\(3\)\}\(\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}\)\\propto\\mathcal\{U\}\_\{\\mathrm\{SO\}\(3\)\}\(\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\)\\,\\mathcal\{N\}\(\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\};\\mathbf\{0\},\\boldsymbol\{I\}\_\{3\}\)\. The forward Stochastic Differential Equation \(SDE\) on𝑻i,tℬ=\(𝒓i,tℬ,𝒙i,tℬ\)\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}=\(\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\},\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\)is defined by:

d​𝑻i,tℬ=\[𝟎,−12​𝒙i,tℬ\]​d​t\+\[d​𝑩SO⁡\(3\)\(i,t\),d​𝑩ℝ3\(i,t\)\]\\small d\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}=\\left\[\\mathbf\{0\},\-\\frac\{1\}\{2\}\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\\right\]dt\+\\left\[d\\boldsymbol\{B\}^\{\(i,t\)\}\_\{\\mathrm\{SO\}\(3\)\},d\\boldsymbol\{B\}^\{\(i,t\)\}\_\{\\mathbb\{R\}^\{3\}\}\\right\]\(7\)where𝑩SO⁡\(3\)\\boldsymbol\{B\}\_\{\\mathrm\{SO\}\(3\)\}and𝑩ℝ3\\boldsymbol\{B\}\_\{\\mathbb\{R\}^\{3\}\}denote Brownian motions onS​O​\(3\)SO\(3\)andℝ3\\mathbb\{R\}^\{3\}, respectively\. The score networksθ​\(t,𝑻i,tℬ,𝒯\)s\_\{\\theta\}\(t,\\boldsymbol\{T\}\_\{i,t\}^\{\\mathcal\{B\}\};\\mathcal\{T\}\)is trained via denoising score matching:

ℒstruct=𝔼⁡\[λt​‖∇𝑻i,tℬ​log​pt\|0​\(𝑻i,tℬ∣𝑻i,0ℬ;𝒯\)−sθ​\(t,𝑻i,tℬ,𝒯\)‖2\]\\small\\mathcal\{L\}\_\{\\text\{struct\}\}=\\mathbb\{E\}\\left\[\\lambda\_\{t\}\\left\\\|\\nabla\_\{\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\}\(\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,0\};\\mathcal\{T\}\)\-s\_\{\\theta\}\(t,\\boldsymbol\{T\}\_\{i,t\}^\{\\mathcal\{B\}\};\\mathcal\{T\}\)\\right\\\|^\{2\}\\right\]\(8\)By leveraging independence across coordinates, the conditional score decomposes into rotational and translational components:∇𝑻i,tℬ​log​pt\|0=\[∇𝒓i,tℬ​log​pt\|0,∇𝒙i,tℬ​log​pt\|0\]\\nabla\_\{\\boldsymbol\{T\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\}=\[\\nabla\_\{\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\},\\,\\nabla\_\{\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\}\]\. ForS​O​\(3\)SO\(3\), the perturbation kernelpt\|0p\_\{t\\mid 0\}is governed by the heat kernel onS​O​\(3\)SO\(3\)\([Hsu, 2002](https://arxiv.org/html/2610.14699#bib.bib6)\), enabling analytical score evaluation:

∇𝒓i,tℬ​log​pt\|0​\(𝒓i,tℬ∣𝒓i,0ℬ,𝒯\)=𝒓i,tℬωi,t​log⁡\(\(𝒓i,0ℬ\)⊤​𝒓i,tℬ\)​∂ωf⁡\(ωi,t,t\)f⁡\(ωi,t,t\)f\(ω,t\)=∑l=0∞\(2l\+1\)e−l\(l\+1\)t/2sin⁡\(\(l\+1/2\)​ω\)sin⁡\(ω/2\)\\small\\begin\{split\}&\\nabla\_\{\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\}\(\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)=\\frac\{\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\{\\omega\_\{i,t\}\}\\log\\left\(\(\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,0\}\)^\{\\top\}\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\\right\)\\frac\{\\partial\_\{\\omega\}f\(\\omega\_\{i,t\},t\)\}\{f\(\\omega\_\{i,t\},t\)\}\\\\ &f\(\\omega,t\)=\\sum\_\{l=0\}^\{\\infty\}\(2l\+1\)e^\{\-l\(l\+1\)t/2\}\\frac\{\\sin\(\(l\+1/2\)\\omega\)\}\{\\sin\(\\omega/2\)\}\\end\{split\}\(9\)whereωi,t=ω⁡\(\(𝒓i,0ℬ\)⊤​𝒓i,tℬ\)\\omega\_\{i,t\}=\\omega\(\(\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,0\}\)^\{\\top\}\\boldsymbol\{r\}^\{\\mathcal\{B\}\}\_\{i,t\}\)is the rotation angle in radians\. Forℝ3\\mathbb\{R\}^\{3\}, the translation process follows an Ornstein\-Uhlenbeck process\([Song et al\., 2020](https://arxiv.org/html/2610.14699#bib.bib7)\)with kernelpt\|0\(𝒙i,tℬ∣𝒙i,0ℬ\)=𝒩\(𝒙i,tℬ;e−t/2𝒙i,0ℬ,\(1−e−t\)𝑰3\)p\_\{t\\mid 0\}\(\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,0\}\)=\\mathcal\{N\}\(\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\};e^\{\-t/2\}\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,0\},\(1\-e^\{\-t\}\)\\boldsymbol\{I\}\_\{3\}\), yielding the exact conditional score:

∇𝒙i,tℬlogpt\|0\(𝒙i,tℬ∣𝒙i,0ℬ,𝒯\)=\(1−e−t\)−1\(e−t/2𝒙i,0ℬ−𝒙i,tℬ\)\\small\\nabla\_\{\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\}\\log p\_\{t\\mid 0\}\(\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\\mid\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)=\(1\-e^\{\-t\}\)^\{\-1\}\\left\(e^\{\-t/2\}\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,0\}\-\\boldsymbol\{x\}^\{\\mathcal\{B\}\}\_\{i,t\}\\right\)\(10\)

### 3\.4PIFormer Architecture for the Reverse Generative Process

The reverse generative process parameterizes the conditional transition distributionpθ​\(ℬt−1∣ℬt,𝒯\)p\_\{\\theta\}\(\\mathcal\{B\}\_\{t\-1\}\\mid\\mathcal\{B\}\_\{t\},\\mathcal\{T\}\)to reconstruct uncorrupted sequence\-structure pairs from noisy states\. We parameterize this reverse mapping using PIFormer, a multi\-block neural architecture designed for joint sequence\-structure co\-design\. PIFormer iteratively refines spatial frames and sequence representations acrossKKstacked blocks\. Each block interleaves spatial\-based invariant point attention \(IPA\) layer\([Jumper et al\., 2021](https://arxiv.org/html/2610.14699#bib.bib1)\)with sequence\-based self\-attention layers, effectively modeling 3D geometric interactions alongside global amino acid dependencies\.

Node embeddings𝒉0=\(𝒉10,…,𝒉N^0\)∈ℝN^×dh\\boldsymbol\{h\}^\{0\}=\(\\boldsymbol\{h\}^\{0\}\_\{1\},\\dots,\\boldsymbol\{h\}^\{0\}\_\{\\hat\{N\}\}\)\\in\\mathbb\{R\}^\{\\hat\{N\}\\times d\_\{h\}\}are initialized with residue\-level physicochemical features, while edge embeddings𝒛0∈ℝN^×N^×dz\\boldsymbol\{z\}^\{0\}\\in\\mathbb\{R\}^\{\\hat\{N\}\\times\\hat\{N\}\\times d\_\{z\}\}capture pairwise initial geometry following[Trippe et al\. \(2022\)](https://arxiv.org/html/2610.14699#bib.bib38), whereN^=M\+N\\hat\{N\}=M\+Ndenotes the total residue length of the target\-binder complex\. To enhance sequence modeling capacity, we incorporate pretrained residue representations𝒆∈ℝN^×de\\boldsymbol\{e\}\\in\\mathbb\{R\}^\{\\hat\{N\}\\times d\_\{e\}\}extracted from ESM\-2\([Lin et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib44)\)\. Within thell\-th block \(l∈\{1,…,K\}l\\in\\\{1,\\dots,K\\\}\), representations and rigid frames are updated as follows:

𝒉\(l−0\.75\)\\displaystyle\\boldsymbol\{h\}^\{\(l\-0\.75\)\}=IPALayer⁡\(𝒉\(l−1\),𝒛\(l−1\),𝑻\(l−1\)\),\\displaystyle=\\mathrm\{IPALayer\}\\big\(\\boldsymbol\{h\}^\{\(l\-1\)\},\\boldsymbol\{z\}^\{\(l\-1\)\},\\boldsymbol\{T\}^\{\(l\-1\)\}\\big\),\(11\)𝒉\(l−0\.5\)\\displaystyle\\boldsymbol\{h\}^\{\(l\-0\.5\)\}=LayerNorm⁡\(𝒉\(l−0\.75\)\+𝒉\(l−1\)\)\\displaystyle=\\mathrm\{LayerNorm\}\\big\(\\boldsymbol\{h\}^\{\(l\-0\.75\)\}\+\\boldsymbol\{h\}^\{\(l\-1\)\}\\big\)𝒉\(l−0\.25\)\\displaystyle\\boldsymbol\{h\}^\{\(l\-0\.25\)\}=Transformer⁡\(𝒉\(l−0\.5\)\+𝒆\)\\displaystyle=\\mathrm\{Transformer\}\\big\(\\boldsymbol\{h\}^\{\(l\-0\.5\)\}\+\\boldsymbol\{e\}\\big\)𝒉l\\displaystyle\\boldsymbol\{h\}^\{l\}=LayerNorm⁡\(Linear⁡\(𝒉\(l−0\.25\)\)\+𝒉\(l−0\.5\)\)\\displaystyle=\\mathrm\{LayerNorm\}\\big\(\\mathrm\{Linear\}\(\\boldsymbol\{h\}^\{\(l\-0\.25\)\}\)\+\\boldsymbol\{h\}^\{\(l\-0\.5\)\}\\big\)𝑻~l\\displaystyle\\tilde\{\\boldsymbol\{T\}\}^\{l\}=𝒉l​𝑾f\\displaystyle=\\boldsymbol\{h\}^\{l\}\\boldsymbol\{W\}\_\{f\}𝒓l\\displaystyle\\boldsymbol\{r\}^\{l\}=𝒓l−1\+𝒓l−1⋅𝑻~l\[:3\],𝒙l=𝒙l−1\+𝒓l−1⋅𝑻~l\[3:\]\\displaystyle=\\boldsymbol\{r\}^\{l\-1\}\+\\boldsymbol\{r\}^\{l\-1\}\\cdot\\tilde\{\\boldsymbol\{T\}\}^\{l\}\_\{\[:3\]\},\\boldsymbol\{x\}^\{l\}=\\boldsymbol\{x\}^\{l\-1\}\+\\boldsymbol\{r\}^\{l\-1\}\\cdot\\tilde\{\\boldsymbol\{T\}\}^\{l\}\_\{\[3:\]\}𝑻l\\displaystyle\\boldsymbol\{T\}^\{l\}=\(𝒓l,𝒙l\)\\displaystyle=\(\\boldsymbol\{r\}^\{l\},\\boldsymbol\{x\}^\{l\}\)𝒛l\\displaystyle\\boldsymbol\{z\}^\{l\}=FFN​\(f⁡\(𝒉l,𝒛\(l−1\)\)\)\\displaystyle=\\text\{FFN\}\\big\(f\(\\boldsymbol\{h\}^\{l\},\\boldsymbol\{z\}^\{\(l\-1\)\}\)\\big\)fi​j​\(𝒉l,𝒛\(l−1\)\)\\displaystyle f\_\{ij\}\(\\boldsymbol\{h\}^\{l\},\\boldsymbol\{z\}^\{\(l\-1\)\}\)=Concatenate​\(Linear​\(𝒉il\),Linear​\(𝒉jl\),𝒛i​j\(l−1\)\)\\displaystyle=\\text\{Concatenate\}\\big\(\\text\{Linear\}\(\\boldsymbol\{h\}^\{l\}\_\{i\}\),\\text\{Linear\}\(\\boldsymbol\{h\}^\{l\}\_\{j\}\),\\boldsymbol\{z\}^\{\(l\-1\)\}\_\{ij\}\\big\)where𝑾f∈ℝdh×6\\boldsymbol\{W\}\_\{f\}\\in\\mathbb\{R\}^\{d\_\{h\}\\times 6\}is a trainable projection matrix, and𝑻0=\[𝑻𝒯,𝑻~ℬ\]\\boldsymbol\{T\}^\{0\}=\[\\boldsymbol\{T\}^\{\\mathcal\{T\}\},\\tilde\{\\boldsymbol\{T\}\}^\{\\mathcal\{B\}\}\]concatenates ground\-truth target frames𝑻𝒯\\boldsymbol\{T\}^\{\\mathcal\{T\}\}with perturbed binder frames𝑻~ℬ\\tilde\{\\boldsymbol\{T\}\}^\{\\mathcal\{B\}\}\. FFN denotes the feed\-forward layer\.

From the final outputs\(𝒉K,𝑻K\)\(\\boldsymbol\{h\}^\{K\},\\boldsymbol\{T\}^\{K\}\), we compute rotational and translational scores, predict the uncorrupted sequence distribution, and estimate backboneψ\\psitorsion angles:

sθr,i=fr​\(𝒓~iℬ,𝒓iℬ,K\),sθx,i=fx​\(𝒙~iℬ,𝒙iℬ,K\)pθ​\(𝒂^i,0ℬ\)=Softmax⁡\(Linear⁡\(𝒉iK\)\),ψi=fψ​\(𝒉iK\)\\small\\begin\{split\}s^\{r,i\}\_\{\\theta\}&=f\_\{r\}\(\\tilde\{\\boldsymbol\{r\}\}\_\{i\}^\{\\mathcal\{B\}\},\\boldsymbol\{r\}\_\{i\}^\{\\mathcal\{B\},K\}\),\\hskip 9\.24994pts^\{x,i\}\_\{\\theta\}=f\_\{x\}\(\\tilde\{\\boldsymbol\{x\}\}\_\{i\}^\{\\mathcal\{B\}\},\\boldsymbol\{x\}\_\{i\}^\{\\mathcal\{B\},K\}\)\\\\ p\_\{\\theta\}\(\\hat\{\\boldsymbol\{a\}\}\_\{i,0\}^\{\\mathcal\{B\}\}\)&=\\mathrm\{Softmax\}\\big\(\\mathrm\{Linear\}\(\\boldsymbol\{h\}\_\{i\}^\{K\}\)\\big\),\\hskip 9\.24994pt\\psi\_\{i\}=f\_\{\\psi\}\(\\boldsymbol\{h\}\_\{i\}^\{K\}\)\\end\{split\}\(12\)wherefrf\_\{r\}andfxf\_\{x\}compute score function approximations from noisy frames\(𝒓~iℬ,𝒙~iℬ\)\(\\tilde\{\\boldsymbol\{r\}\}\_\{i\}^\{\\mathcal\{B\}\},\\tilde\{\\boldsymbol\{x\}\}\_\{i\}^\{\\mathcal\{B\}\}\)and network\-predicted frames\(𝒓iℬ,K,𝒙iℬ,K\)\(\\boldsymbol\{r\}\_\{i\}^\{\\mathcal\{B\},K\},\\boldsymbol\{x\}\_\{i\}^\{\\mathcal\{B\},K\}\), andfψf\_\{\\psi\}maps node representations toψ\\psitorsion angles\. The predicted distributionpθ​\(𝒂^i,0ℬ\)p\_\{\\theta\}\(\\hat\{\\boldsymbol\{a\}\}\_\{i,0\}^\{\\mathcal\{B\}\}\)parameterizes the clean sequence𝒂i,0ℬ\\boldsymbol\{a\}\_\{i,0\}^\{\\mathcal\{B\}\}, allowing analytical calculation of the reverse categorical posteriorpθ​\(𝒂i,t−1ℬ\|𝒂i,tℬ,𝒂^i,0ℬ,𝒯\)p\_\{\\theta\}\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\}\|\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\},\\hat\{\\boldsymbol\{a\}\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)\.

### 3\.5Training Objective

To achieve effective sequence\-structure co\-design, PIDiff is optimized via a multitask learning objectiveℒt\\mathcal\{L\}\_\{t\}that combines discrete sequence learning, continuousS​E​\(3\)SE\(3\)score\-matching frame losses, and late\-stage auxiliary physical regularization with weightsα\\alpha,γ\\gamma, andω\\omega:

ℒt=α​ℒseq\+γ⁡\(ℒstructr\+ℒstructx\)\+ω​ℒauxℒseq=∑i=1NKL\(q\(𝒂ℬi,t−1∣𝒂ℬi,t,𝒂ℬi,0,𝒯\)∥pθ\(𝒂ℬi,t−1\|𝒂ℬi,t,𝒂^ℬi,0,𝒯\)\)ℒstructr=∑i=1N‖∇𝒓i,tℬ​log​pt\|0​\(𝒓i,tℬ∣𝒓i,0ℬ,𝒯\)−sθr,i‖2,ℒstructx=∑i=1N‖𝒙iℬ−𝒙iℬ,K‖2\\small\\begin\{split\}\\mathcal\{L\}\_\{t\}&=\\alpha\\mathcal\{L\}\_\{\\text\{seq\}\}\+\\gamma\(\\mathcal\{L\}\_\{\\text\{struct\}\}^\{r\}\+\\mathcal\{L\}\_\{\\text\{struct\}\}^\{x\}\)\+\\omega\\mathcal\{L\}\_\{\\text\{aux\}\}\\\\ \\mathcal\{L\}\_\{\\text\{seq\}\}&=\\sum\\nolimits\_\{i=1\}^\{N\}\\mathrm\{KL\}\\left\(q\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\}\\mid\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\},\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)\\;\\parallel\\;p\_\{\\theta\}\(\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\-1\}\|\\boldsymbol\{a\}^\{\\mathcal\{B\}\}\_\{i,t\},\\hat\{\\boldsymbol\{a\}\}^\{\\mathcal\{B\}\}\_\{i,0\},\\mathcal\{T\}\)\\right\)\\\\ \\mathcal\{L\}\_\{\\text\{struct\}\}^\{r\}&=\\sum\\nolimits\_\{i=1\}^\{N\}\\left\\\|\\nabla\_\{\\boldsymbol\{r\}\_\{i,t\}^\{\\mathcal\{B\}\}\}\\log p\_\{t\\mid 0\}\(\\boldsymbol\{r\}\_\{i,t\}^\{\\mathcal\{B\}\}\\mid\\boldsymbol\{r\}\_\{i,0\}^\{\\mathcal\{B\}\},\\mathcal\{T\}\)\-s\_\{\\theta\}^\{r,i\}\\right\\\|^\{2\},\\\\ \\mathcal\{L\}\_\{\\text\{struct\}\}^\{x\}&=\\sum\\nolimits\_\{i=1\}^\{N\}\\left\\\|\\boldsymbol\{x\}\_\{i\}^\{\\mathcal\{B\}\}\-\\boldsymbol\{x\}\_\{i\}^\{\\mathcal\{B\},K\}\\right\\\|^\{2\}\\end\{split\}\(13\)whereℒseq\\mathcal\{L\}\_\{\\text\{seq\}\}maximizes the variational lower bound in Eq\.[2](https://arxiv.org/html/2610.14699#S3.E2), and the frame losses minimize the score\-matching objective in Eq\.[8](https://arxiv.org/html/2610.14699#S3.E8)\. By applying target weighting schedulesλi,tr=1/𝔼⁡\[‖∇log⁡pt\|0​\(𝒓i,t∣𝒓i,0\)‖SO⁡\(3\)2\]\\lambda\_\{i,t\}^\{r\}=1/\\mathbb\{E\}\\big\[\\\|\\nabla\\log p\_\{t\\mid 0\}\(\\boldsymbol\{r\}\_\{i,t\}\\mid\\boldsymbol\{r\}\_\{i,0\}\)\\\|\_\{\\mathrm\{SO\}\(3\)\}^\{2\}\\big\]for rotations andλi,tx=\(1−e−t\)/e−t/2\\lambda\_\{i,t\}^\{x\}=\(1\-e^\{\-t\}\)/e^\{\-t/2\}for translations, the frame objectives simplify to the unweighted formulations above\.

To enforce physical contact fidelity during final structure generation, we activate two auxiliary terms,ℒaux=ℒstructatom\+ℒstructpair\\mathcal\{L\}\_\{\\text\{aux\}\}=\\mathcal\{L\}\_\{\\text\{struct\}\}^\{\\text\{atom\}\}\+\\mathcal\{L\}\_\{\\text\{struct\}\}^\{\\text\{pair\}\}, during late denoising time steps \(t<0\.25t<0\.25\):

ℒstructatom\\displaystyle\\mathcal\{L\}\_\{\\text\{struct\}\}^\{\\text\{atom\}\}=14​N​∑i=1N∑m∈𝔸‖𝒙i,mℬ−𝒙i,mℬ,K‖2,𝔸=\{N,Cα,C,O\}\\displaystyle=\\frac\{1\}\{4N\}\\sum\_\{i=1\}^\{N\}\\sum\_\{m\\in\\mathbb\{A\}\}\\left\\\|\\boldsymbol\{x\}\_\{i,m\}^\{\\mathcal\{B\}\}\-\\boldsymbol\{x\}^\{\\mathcal\{B\},K\}\_\{i,m\}\\right\\\|^\{2\},\\hskip 9\.24994pt\\mathbb\{A\}=\\\{\\mathrm\{N\},\\mathrm\{C\}\_\{\\alpha\},\\mathrm\{C\},\\mathrm\{O\}\\\}\(14\)ℒstructpair\\displaystyle\\mathcal\{L\}\_\{\\text\{struct\}\}^\{\\text\{pair\}\}=1S​∑1≤i≠j≤N∑m,n∈𝔸𝕀⁡\(dm​ni​j<τ\)​‖dm​ni​j−d^m​ni​j‖2\\displaystyle=\\frac\{1\}\{S\}\\sum\_\{1\\leq i\\neq j\\leq N\}\\sum\_\{m,n\\in\\mathbb\{A\}\}\\mathbb\{I\}\(d^\{ij\}\_\{mn\}<\\tau\)\\left\\\|d^\{ij\}\_\{mn\}\-\\hat\{d\}^\{ij\}\_\{mn\}\\right\\\|^\{2\}S\\displaystyle S=∑1≤i≠j≤N∑m,n∈𝔸𝕀⁡\(dm​ni​j<τ\)\\displaystyle=\\sum\_\{1\\leq i\\neq j\\leq N\}\\sum\_\{m,n\\in\\mathbb\{A\}\}\\mathbb\{I\}\(d^\{ij\}\_\{mn\}<\\tau\)whered∗d^\{\*\}denotes the pairwise atomic distances, andτ\\tauspecifies the interaction distance cutoff\.

### 3\.6Self\-Conditioned Inference

During inference, conditioned on the target protein’s sequence𝒂𝒯\\boldsymbol\{a\}^\{\\mathcal\{T\}\}, structure𝒔𝒯\\boldsymbol\{s\}^\{\\mathcal\{T\}\}, and physicochemical features𝒇𝒯\\boldsymbol\{f\}^\{\\mathcal\{T\}\}, PIDiff conditionally co\-samples the binder sequence𝒂ℬ\\boldsymbol\{a\}^\{\\mathcal\{B\}\}and backbone structure𝒔ℬ\\boldsymbol\{s\}^\{\\mathcal\{B\}\}to yield high\-affinity complexes\. Sequence generation initializes all binder positions in an absorbing state \[mask\], fixing residue types and updating physicochemical features𝒇ℬ\\boldsymbol\{f\}^\{\\mathcal\{B\}\}upon state transition\. Structure generation utilizes Euler\-Maruyama geodesic random walks onS​E​\(3\)SE\(3\)over𝕋\\mathbb\{T\}steps\([De Bortoli et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib4)\)\. Translations receive Gaussian updates from𝒩⁡\(𝟎,𝑰3\)\\mathcal\{N\}\(\\mathbf\{0\},\\boldsymbol\{I\}\_\{3\}\), while rotations update along the Lie algebraS​O​\(3\)SO\(3\)tangent space as𝒓i,t′=𝒓i,t​∑j=13δj​𝒆j\\boldsymbol\{r\}^\{\\prime\}\_\{i,t\}=\\boldsymbol\{r\}\_\{i,t\}\\sum\_\{j=1\}^\{3\}\\delta\_\{j\}\\boldsymbol\{e\}\_\{j\}withδj∼i\.i\.d\.𝒩⁡\(0,1\)\\delta\_\{j\}\\stackrel\{\{\\scriptstyle\\text\{i\.i\.d\.\}\}\}\{\{\\sim\}\}\\mathcal\{N\}\(0,1\)and𝒆1,𝒆2,𝒆3\\boldsymbol\{e\}\_\{1\},\\boldsymbol\{e\}\_\{2\},\\boldsymbol\{e\}\_\{3\}are orthonormal basis vectors\. To maximize structural fidelity, we implement a self\-conditioned inference scheme at each time steptt: PIFormer first performs an intermediate forward pass to updateCαC\_\{\\alpha\}positions, then recalculates spatial pair embeddings to refine sequence likelihoods and rigid frames in a second pass\.

## 4Experiments

Table 1:Model performance on general protein\-protein complex design task\. Our PIDiff achieves the best sequence design success rate, structure binding energy and sequence\-structure co\-designability\.### 4\.1PIBench Construction

To establish a foundation model for protein\-protein complex design that can be seamlessly adapted to real\-world applications, we curate a comprehensive dataset called PIBench\. Following the data processing pipeline of AlphaFold3\([Abramson et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib43)\), we extract 301,182 chain\-pair interfaces from the PDB\. Complete processing workflows are detailed in Appendix[A\.1](https://arxiv.org/html/2610.14699#A1.SS1)\. By treating both chains within each interface symmetrically as target\-binder pairs, we construct a benchmark comprising 602,364 complexes across diverse scaffolds\. To prevent data leakage, sequences are clustered at 50% identity, yielding 28,884 distinct clusters\. We reserve 10 clusters each for validation and testing, allocating the remaining clusters for training\. Detailed data statistics are given in Appendix Table[4](https://arxiv.org/html/2610.14699#A1.T4)\.

### 4\.2Experimental Setup

Implementation Details\.The PIFormer comprises five blocks, each containing one IPA layer and six self\-attention layers\. Parameters for the self\-attention layers are initialized from the pretrained 150M ESM\-2 model\. Node and edge embedding dimensions are set to 640 and 320, respectively, bringing the total parameter count of PIDiff to 240M\. The diffusion process is configured for 1,000 steps\. In inference, for each target protein, we generate 10 binder candidates and rank them by their complex pLDDT scores, selecting the top\-kkcandidates\. More implementation details are provided in Appendix[B](https://arxiv.org/html/2610.14699#A2)\.

Baseline Models\.We compare the proposed PIDiff against the following representative protein complex co\-design methods: \(1\)RFD\+ProteinMPNN \(IF\)first applies RFDiffusion\([Watson et al\., 2023](https://arxiv.org/html/2610.14699#bib.bib24)\)to design a binder protein structure based on the given target and then uses ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)to generate the corresponding sequence\. \(2\)APM\([Chen et al\., 2025](https://arxiv.org/html/2610.14699#bib.bib26)\)is an all\-atom generative model for protein complex design\. \(3\)Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)is a fully atomistic binder generation method showing strong performance on protein\-binding protein design\.

Evaluation Metrics\.To comprehensively evaluate PIDiff, we assess generated complexes across three dimensions: sequence quality, structure reliability, and sequence\-structure consistency\.Sequence Evaluation:We evaluate designed sequences using five metrics: \(1\)ipTMvia AlphaFold3 \(AF3\) to measure interface confidence; \(2\)pLDDTvia AF3 to assess complex structure stability; \(3\)Success Rate \(SR\)defined as the fraction of designed candidates that simultaneously achieve anipTM\>0\.8\\mathrm\{ipTM\}\>0\.8and apLDDT\>80\\mathrm\{pLDDT\}\>80; \(4\)Diversity, measured as the amino acid variation among candidates for the same target; and \(5\)Novelty, calculated as the sequence dissimilarity between candidates and reference sequences\.Structure Evaluation:We assess designed structures via: \(6\)Binding Energy \(Δ​G\\Delta G\), computed via Rosetta\([Chaudhury et al\., 2010](https://arxiv.org/html/2610.14699#bib.bib45)\)to estimate interface energetic stability, where lower values indicate stronger binding propensity, and \(7\)Designability, defined as the fraction of designs achieving a self\-consistency RMSD \(scRMSD\\mathrm\{scRMSD\}\)<2​Å<2\\,\\text\{\\AA\}\([Trippe et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib38)\)\.Sequence\-Structure Consistency:We measure joint consistency via \(8\)Co\-designability \(Consistency\), defined as the proportion of generated binder sequences whose AF3\-folded structure matches the corresponding designed structures within2​Å2\\,\\text\{\\AA\}RMSD\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\.

Table 2:Model performance on target\-protein mini\-binder complex design task\. Consistency denotes sequence\-structure co\-designability\. Our PIDiff achieves the best average sequence design success rate, structure binding energy and sequence\-structure co\-designability\.
### 4\.3General Protein\-Protein Complex Design

This task evaluates the foundational capability of PIDiff to design protein\-protein complexes across diverse target scaffolds\. We train and evaluate PIDiff on our curated PIBench\. Top\-1 candidate performance for all methods is summarized in Table[1](https://arxiv.org/html/2610.14699#S4.T1)\.PIDiff demonstrates the best performance in generating high\-affinity, structurally consistent binders across a wide spectrum of targets\.Specifically, PIDiff achieves top performance across all metrics spanning sequence quality, structural reliability, and sequence\-structure co\-designability, reaching a sequence design success rate of 47\.83%, an interface binding energy of−67\.8\-67\.8, and a co\-designability score of 65\.22%\. Although sequence diversity and novelty scores are slightly lower than those of Proteína\-Complexa, PIDiff achieves higher structural diversity \(Appendix Table[7](https://arxiv.org/html/2610.14699#A3.T7)\)\. This indicates that PIDiff successfully generates binders with flexible loop configurations and distinct subdomain topologies, whereas binders from Proteína\-Complexa have almost invariant secondary structure cores\.

### 4\.4Target Protein\-Mini Binder Complex Design

Datasets\.We curate experimentally validated target\-protein mini\-binder complexes across ten diverse target classes from[Bennett et al\. \(2023\)](https://arxiv.org/html/2610.14699#bib.bib19)\. For targets with more than 50 complexes, we partition the data using a random 8:1:1 train/validation/test split\. For targets with fewer complexes, all instances are reserved for testing to evaluate model generalization in a zero\-shot setting\. Comprehensive dataset statistics are provided in Appendix Table[5](https://arxiv.org/html/2610.14699#A1.T5)\.

Main Results\.The results for target\-protein mini\-binder complex design are summarized in Table[2](https://arxiv.org/html/2610.14699#S4.T2), with target\-specific breakdowns provided in Appendix Tables[8](https://arxiv.org/html/2610.14699#A3.T8)–[17](https://arxiv.org/html/2610.14699#A3.T17)\.Overall, PIDiff demonstrates superior capacity for generating novel, high\-affinity binders across diverse target classes\.Specifically, PIDiff achieves higher sequence design success rates on 7 out of 10 targets, superior structure binding energy across all 10 targets, and improved co\-designability on 7 out of 10 targets\. Notably, PIDiff also yields higher novelty scores, reversing the trend observed in our general complex design task\. We attribute this shift to the distinctions between the fine\-tuning dataset and PIBench, enabling PIDiff to navigate a broader candidate space for novel binder discovery\.

Table 3:Model performance on antigen\-antibody complex design task\. Our PIDiff achieves the best sequence design success rate, structure binding energy and sequence\-structure co\-designability\.
### 4\.5Antigen\-Antibody Complex Design

Dataset\.Following[Jin et al\. \(2022\)](https://arxiv.org/html/2610.14699#bib.bib46), we curate4,2614\{,\}261complete antigen\-antibody complexes from the Structural Antibody Database \([Dunbar et al\. \(2014\)](https://arxiv.org/html/2610.14699#bib.bib47), SAbDab\) by removing cases lacking a light chain or antigen\. Train/validation/test splits \(8:1:1 ratio\) are constructed by clustering individual CDR sequences at 40% identity using MMseqs2\([Steinegger and Söding, 2017](https://arxiv.org/html/2610.14699#bib.bib48)\), yielding 1,034 \(CDR\-H1\), 1,418 \(CDR\-H2\), and 2,254 \(CDR\-H3\) clusters\. Full split statistics are provided in Appendix Table[6](https://arxiv.org/html/2610.14699#A1.T6)\.

Main Results\.Average performance across distinct CDR cluster splits for antigen\-antibody complex design is summarized in Table[3](https://arxiv.org/html/2610.14699#S4.T3), with CDR\-specific breakdowns detailed in Appendix Tables[18](https://arxiv.org/html/2610.14699#A3.T18)–[20](https://arxiv.org/html/2610.14699#A3.T20)\.Overall, PIDiff outperforms all baselines in generating high\-affinity and novel antibodies\.Specifically, PIDiff achieves top performance across sequence design success rate, CDR H2/H3 novelty, interface binding energy \(Δ​G\\Delta G\), and sequence\-structure co\-designability\. Crucially, both the designability and co\-designability of PIDiff exceed95%95\\%, demonstrating exceptional structure fidelity and tight sequence\-structure consistency\.

\(a\) Model Scaling

\(b\) Physicochemical Properties

\(c\) Self\-Consistency

\(d\) Structure Learning

\(e\) Binder Finetuning

![Refer to caption](https://arxiv.org/html/2610.14699v1/1stx_v2.png)

\(f\) General Protein Complex

![Refer to caption](https://arxiv.org/html/2610.14699v1/insulinr_v2.png)

\(g\) Binder Design

![Refer to caption](https://arxiv.org/html/2610.14699v1/7ucg.png)

\(h\) Antibody Design

Figure 2:Ablation study: \(a\) \- \(e\), and case study with target protein \(f\) PDB 1STX\.E, \(g\) InsulinR, \(h\) HIV\-1 \(7UCG\)\. Target proteins are shown in blue, the folded structures for the designed binder sequences in green, and the designed binder structures in gray, with light chain in pink for antibody\.

## 5Analysis

### 5\.1Does Model Scale Help?

PIDiff scales effectively with parameter capacity\.We benchmark our 240M PIDiff against a smaller model \(76M\) configured with 4 blocks and a hidden dimension of 480\. As shown in Figure[2](https://arxiv.org/html/2610.14699#S4.F2)\(a\), scaling capacity improves sequence success rate, structural designability, and co\-designability on PIBench\. These results highlight the scalability of our architecture, indicating that performance scales predictably with model capacity\.

### 5\.2Effect of Physicochemical Complementarity Constraints

As stable protein\-protein interactions depend on complementary geometries and physicochemical properties, we conduct an ablation study by training a variant of PIDiff without physicochemical features on PIBench\. As illustrated in Figure[2](https://arxiv.org/html/2610.14699#S4.F2)\(b\), omitting these constraints leads to a significant performance degradation across both sequence and structural metrics\.These findings confirm that explicitly incorporating physicochemical features is critical for stable interface design\.

### 5\.3Influence of Self\-Consistency

To evaluate the role of self\-consistency during inference, we conduct an ablation by disabling this mechanism on the general protein\-protein complex design task\. As shown in Figure[2](https://arxiv.org/html/2610.14699#S4.F2)\(c\), removing self\-consistency yields a measurable drop in both structural designability and co\-designability\.This validates self\-consistency as an effective inference\-time strategy for generating reliable structures and reinforcing sequence\-structure co\-design alignment\.

### 5\.4Impact of Joint Co\-Design and Auxiliary Losses

To isolate the contributions of the joint sequence\-structure co\-design mechanism and auxiliary supervision, we evaluate a structure\-only ablation \(PIDiff w/o seq\) alongside a model variant trained without auxiliary loss terms \(PIDiff w/o auxiliary loss\)\. As illustrated in Figure[2](https://arxiv.org/html/2610.14699#S4.F2)\(d\), both components enhance structural designability, with sequence\-structure co\-design exerting a significantly stronger effect\. We attribute these gains to the stricter physical regularization provided by both mechanisms,confirming that they are vital for designing reliable structures\.

### 5\.5Impact of Supervised Fine\-Tuning

To evaluate how domain\-specific adaptation affects downstream performance, we compare our pretrained PIDiff on the mini\-binder design task\. As shown in Figure[2](https://arxiv.org/html/2610.14699#S4.F2)\(e\), targeted fine\-tuning improves interface binding energy \(Δ​G\\Delta G\) across 6 out of 10 targets, slightly promoting the mean energy score from−48\.98\-48\.98to−49\.85\-49\.85\. Full results are provided in Appendix Table[21](https://arxiv.org/html/2610.14699#A3.T21)\.These results demonstrate that while our pretrained model already possesses robust zero\-shot design capabilities, supervised fine\-tuning further enhances binding affinity for specialized applications\.

### 5\.6Designing Diverse and High\-Affinity Protein Complexes

Figures[2](https://arxiv.org/html/2610.14699#S4.F2)\(f\)–\(h\) showcase qualitative case studies across general complex, mini\-binder, and antibody design tasks\. Spanning distinct target topologies, the generated candidates consistently satisfy stringent quality thresholds: ipTM\>0\.8\>0\.8, pLDDT\>85\>85, interface binding energy<−35<\-35, and co\-design RMSD<2​Å<2\\,\\text\{\\AA\}\.This highlights PIDiff’s ability to co\-design highly consistent binder sequences and structures against diverse target scaffolds\.

## 6Conclusion

In this work, we introduce PIDiff, a physicochemically enhanced multimodal diffusion method for joint binder protein sequence\-structure co\-design\. Powered by PIFormer, an architecture combining IPA layer with sequence\-level self\-attention, PIDiff effectively couples 3D geometry with amino acid dependencies\. Across general protein complex generation as well as specialized mini\-binder and antibody design tasks, PIDiff consistently surpasses state\-of\-the\-art baselines across multiple metrics\. Overall, PIDiff demonstrates strong potential as a robust and scalable tool for computational binder discovery\. A potential limitation is the absence of wet\-lab validation, which is an essential step planned for our future research to further validate the model’s practical utility\.

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## Appendix

## Appendix ASupplementary Dataset Information

### A\.1PIBench Curation Details

We curate a large\-scale general protein\-protein complex dataset by identifying chain\-pair interfaces, following the data processing pipeline introduced in AlphaFold3\([Abramson et al\., 2024](https://arxiv.org/html/2610.14699#bib.bib43)\)\. The curation process begins by applying a series of quality filters to identify valid PDB entries:

- •Retain structures with a reported resolution of 9Å or better\.
- •Remove polymer chains containing unknown residues\.
- •Exclude protein chains where consecutiveCαC\_\{\\alpha\}atoms are separated by more than 10Å\.
- •Filter out polymer chains with fewer than four resolved residues\.
- •Eliminate clashing chains, defined as chains with more than 30% of their atoms located within 1\.7Å of an atom in another chain\. When two chains clashed: - –The chain with the higher percentage of clashing atoms is removed\. - –If the clashing percentage is equal, the chain with fewer total atoms is removed\. - –If both chains have the same number of atoms, the chain with the larger chain ID is removed\.

This filtering process result in 68,891 valid PDB entries\. From these entries, chain\-pair interfaces are identified, defined as pairs of chains with a minimum heavy atom separation of less than 5Å\. Furthermore, to ensure efficient processing, all complexes are filtered to exclude sequences longer than 1,024 residues, finally resulting in a total of 301,182 chain\-pair interfaces\.

### A\.2PIBench Data Statistics

Table 4:Data statistics for PIBench\.We provide the data statistics for the curated PIBench in Table[4](https://arxiv.org/html/2610.14699#A1.T4)\.

### A\.3Data Statistics for Target\-Protein Mini\-Binder Complex Dataset

Table 5:Detailed data statistics for curated target\-protein mini\-binder complex dataset\.We collect experimentally confirmed positive target\-protein mini\-binder complexes against ten targets with diverse structural properties from[Bennett et al\. \(2023\)](https://arxiv.org/html/2610.14699#bib.bib19)\. The detailed data statistics for the ten target proteins are reported in Table[5](https://arxiv.org/html/2610.14699#A1.T5)\.

### A\.4Data Statistics for Antigen\-Antibody Complex Dataset

Table 6:Detailed data statistics for antigen\-antibody complex dataset\.We detail the data statistics of antigen\-antibody complex dataset for each CDR clustering in Table[6](https://arxiv.org/html/2610.14699#A1.T6)\.

## Appendix BSupplementary Implementation Details

Diffusion Schedules\.Continuous spatial noise scheduling is parameterized across both Euclidean and rotational manifolds\. For translation vectors inℝ3\\mathbb\{R\}^\{3\}, the variance schedule scales fromσmin=0\.1\\sigma\_\{\\min\}=0\.1toσmax=20\.0\\sigma\_\{\\max\}=20\.0\. We use a coordinate scaling factor of 0\.1\. For rotation matrices inS​O​\(3\)SO\(3\), the schedule scales fromσmin=0\.1\\sigma\_\{\\min\}=0\.1toσmax=1\.5\\sigma\_\{\\max\}=1\.5\.

Architecture & Hyperparameters\.The Invariant Point Attention \(IPA\) modules are configured with88attention heads, utilizing88query/key points and1212value points per head\. Loss hyper\-parameters are set toα=1\.0\\alpha=1\.0,γ=1\.0\\gamma=1\.0, andω=0\.25\\omega=0\.25, and the thresholdτ\\taufor the pairwise distance constraint is set to0\.250\.25\.

Training Setup\.We train PIDiff on a single NVIDIA B200 GPU for up to1\.21\.2million steps\. Optimization is performed with a learning rate of11e−4\-4and a batch capacity of1,5361\{,\}536tokens per step\. Validation is executed per5,0005,000gradient steps\.

## Appendix CSupplementary Experimental Results

### C\.1Structure Diversity on the General Protein\-Protein Complex Design Task

Table 7:Structure diversity on the general protein\-protein complex design task, which is quantified as1−TM​\-​score1\-\\mathrm\{TM\\text\{\-\}score\}averaged across generated candidate structures\.Structural design diversity is quantified as1−TM​\-​score1\-\\mathrm\{TM\\text\{\-\}score\}averaged across generated candidate structures\. We interpret these diversity scores across three regimes: values above0\.50\.5represent high overall structural diversity; scores between0\.30\.3and0\.50\.5correspond to structures within the same superfamily that feature distinct loop configurations or subdomain topologies; and scores below0\.30\.3denote closely related structures sharing nearly identical secondary structure cores\. As reported in Table[7](https://arxiv.org/html/2610.14699#A3.T7), PIDiff achieves the highest structural diversity score, demonstrating a capacity to sample diverse loop structures and topological variations rather than collapsing onto rigid backbone templates\.

### C\.2Detailed Target\-Protein Mini\-Binder Complex Design Results

The detailed target\-protein mini\-binder complex design results for different targets are provided in Table[8](https://arxiv.org/html/2610.14699#A3.T8)\-[17](https://arxiv.org/html/2610.14699#A3.T17)\.

Table 8:Model performance on target\-protein mini\-binder complex design for target FGFR2\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves the best performance across sequence evaluation, structural evaluation and sequence\-structure consistency metrics\. Additionally, PIDiff also obtains the highest novelty score\. Altogether, the results validate PIDiff is capable of designing high\-affinity and novel protein complexes\.Table 9:Model performance on target\-protein mini\-binder complex design for target InsulinR\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves the highest sequence design success rate, structure binding energy and sequence\-structure co\-designability, validating its ability in simultaneously designing high\-quality and consistent protein\-protein complex sequences and structures\.Table 10:Model performance on target\-protein mini\-binder complex design for target PDGFR\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves better structure design performance, but lags behind Proteína\-Complexa on both sequence design quality and sequence\-structure design consistency\.Table 11:Model performance on target\-protein mini\-binder complex design for target TGFb\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves better performance on both sequence and structure design quality\. However, the sequence\-structure co\-designability of PIDiff is a bit lower than that of Proteína\-Complexa\.Table 12:Model performance on target\-protein mini\-binder complex design for target VirB8\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff obtains better sequence and structure design quality as well as sequence\-structure consistency\.Table 13:Model performance on target\-protein mini\-binder complex design for target EGFR\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves better structure binding energy and sequence\-structure co\-designability\. However, the sequence design success rate of PIDiff is zero, which might be due to the lack of deep pockets of EGFR targets and the zero\-shot setting of this class\.Table 14:Model performance on target\-protein mini\-binder complex design for target H3\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff obtains higher sequence design success rate and lower structure binding energy, confirming that PIDiff has the capability to design high\-affinity binders even in zero\-shot setting\.Table 15:Model performance on target\-protein mini\-binder complex design for target IL7Ra\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves higher sequence design success rate, lower structure binding energy and higher sequence\-structure co\-designability\. Note that the designability and co\-designability of PIDiff on this target are both 100%, validating the strong binder design capability of PIDiff in the zero\-shot setting\.Table 16:Model performance on target\-protein mini\-binder complex design for target Tie2\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. PIDiff achieves higher structure design quality and sequence\-structure consistency\.Table 17:Model performance on target\-protein mini\-binder complex design for target TrkA\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\. Similarly, PIDiff achieves better sequence design success rate, structure binding energy and sequence\-structure consistency, confirming its ability in designing good binders in the zero\-shot setting\.
### C\.3Detailed Antigen\-Antibody Complex Design Results for Each CDR Clustering

We provide the detailed results of antigen\-antibody complex design for each CDR clustering in Table[18](https://arxiv.org/html/2610.14699#A3.T18)\-[20](https://arxiv.org/html/2610.14699#A3.T20)\.

Table 18:Model performance on antigen\-antibody complex design task for CDR\-H1 clustering\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\.Table 19:Model performance on antigen\-antibody complex design task for CDR\-H2 clustering\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\.Table 20:Model performance on antigen\-antibody complex design task for CDR\-H3 clustering\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\. IF denotes the inverse folding model ProteinMPNN\([Dauparas et al\., 2022](https://arxiv.org/html/2610.14699#bib.bib25)\)and Complexa denotes Proteína\-Complexa\([Didi et al\., 2026](https://arxiv.org/html/2610.14699#bib.bib27)\)\.
### C\.4Detailed Binder Design Performance on the Pretrained Model

Performance metrics for the pretrained PIDiff on target\-protein mini\-binder complex design are presented in Table[21](https://arxiv.org/html/2610.14699#A3.T21)\. Notably, even without domain\-specific adaptation, PIDiff exhibits robust zero\-shot capabilities across diverse targets, most prominently in sequence\-structure consistency, where it achieves100%100\\%co\-designability across all evaluated targets\. Subsequent supervised fine\-tuning on a curated mini\-binder dataset further elevates overall design quality, markedly boosting the mean sequence design success rate from8\.72%8\.72\\%to40\.77%40\.77\\%\. These results confirm that while pretraining imparts strong zero\-shot structural alignment, targeted fine\-tuning substantially enhances sequence\-level binding optimization\.

Table 21:Pretrained model performance on target\-protein mini\-binder complex design task\. \(↑\\uparrow\): the higher the better\. \(↓\\downarrow\): the lower the better\.