Constrained Diffusion for Protein Design with Hard Structural Constraints
Authors: Jacob K Christopher
Austin Seamann
Jingyi Cui
Sagar D Khare
Ferdinando Fioretto
Venue: ICLR 2026 Poster (2026)
Keywords: Constrained Diffusion
Generative Models
Protein Design
Proximal Optimization
Motif Scaffolding
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: This paper introduces a constrained diffusion framework for protein design that enforces functional, stereochemical, and geometric constraints via proximal updates, achieving state-of-the-art results on tasks like motif scaffolding and pocket design.

Abstract: Diffusion models offer a powerful means of capturing the manifold of realistic protein structures, enabling rapid design for protein engineering tasks. However, existing approaches observe critical failure modes when precise constraints are necessary for functional design. To this end, we present a constrained diffusion framework for structure-guided protein design, ensuring strict adherence to functional requirements while maintaining precise stereochemical and geometric feasibility. The approach integrates proximal feasibility updates with ADMM decomposition into the generative process, scaling effectively to the complex constraint sets of this domain. We evaluate on challenging protein design tasks, including motif scaffolding and vacancy-constrained pocket design, while introducing a novel curated benchmark dataset for motif scaffolding in the PDZ domain. Our approach achieves state-of-the-art, providing perfect satisfaction of bonding and geometric constraints with no degradation in structural diversity.

Published as a conference paper at ICLR 2026
CONSTRAINED DIFFUSION FOR PROTEIN DESIGN WITH HARD STRUCTURAL CONSTRAINTS
Jacob K. Christopher University of Virginia csk4sr@virgin edu
Austin Seamann Rutgers University a a1 rut.gerss edu
Jingyi Cui
University of Virginia
cau8rcêvi rginia miiinnaa edd
Sagar Khare
Ferdinando Fioretto"
Rutgers University
University of Virginia
sagar eeggar.khareerutgrss.coo
Fiorettoévirginia.
ABSTRACT
Diffusion models offer a powerful means of capturing the manifold of realistic protein structures, enabling rapid design for protein engineering tasks. However, existing approaches observe critical failure modes when precise constraints are necessary for functional design. To this end, we present constrained diffusion framework for structure-guice -uudedd protein design, ensuring strict adherence to func- tional requirements while maintaining precise stereochemical and geometric feasi- bility. The approach integrates proximal feasibility updates with ADMM decom- position into the generative process, scaling effectively to the complex constraint sets of this domain. We evaluate on challenging protein design tasks, includ- ing motif scaffolding and aacancy-constrainet pocket design, while introducing a novel curated benchmark dataset for motif scaffolding in the PDZ domain. Our approach achieves state-of-the thetter providing perfect satisfaction of bonding and geometric constraints with degradation in structural diversity.
INTRODUCTION
Diffusion models have revolutionized protein engineering with notable successes demonstrated in the design of protein monomers, assemblies, and protein binders against biomolecular targets (Wat- son et al. 2023). In many cases, predefined binding or catalytic motifs are introduced into designed proteins via motif scaffolding but there
guarantees that the generated backbones will accu- rately include the motif (Trippe et al. [2022] Didi er al. (202)). Furthermore, the motifs are typically pre-defined as structural fragments, rather than more physically-
(c.g. hydrogen bonds cho- sen target residues), which narrows the accessible design space (Song et al. (0224). Negative space constraints (e.g. tunnels for substrate access and product egress), while a ubiquitous feature of nat- urally evolved proteins such as enzymes, are not readily incorporated in current generative protein models. These obstacles restrict the scope of design goals accessible to current methods.
These limitations highlight the broader challenge of designing structured objects under strict feasi- bility. Many existing approaches augment diffusion models through constraint guidance (Ho al.) 2020 [H & Salimans, 2022). For example, Gruver et al. (2023) inject soft constraints into discrete sequence prediction for protein design through gradient-based guidance; however, while guidance approaches result in increased feasibility rates, they fail to consistently provide constraint adherent outputs. Others adopt post-processing optimizations which more rigorously target the constraint set. However, these methods rely either simplifications of the highly nonconvex constraint set (e.g., matching an existing ligand template) or result in samples falling outside the data manifold (Bergues et al. 2025) Christopher al. 2024). more effective approach is inject constraint into the generative process, c.g., by projecting intermediate states back to the feasible set (Christopher al., (2024). Yet projecting noisy states early in the sampling process has been show disrupt the diffusion trajectory, potentially biasing samples (Blanke et al. 2025
"Contact author.

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Published as a conference paper at ICLR 2026
(1) predict
2) correct
renoise
prox ih (z))
Figure 1: Illustration of our stochastic proximal sampling for structured-constrainee protein design.
To resolve this tension, this paper proposes to view constrained diffusion through the lens of stochas- tic proximal methods. To enable strict constraint enforcement throughout the diffusion process, while removing the need to project at earlier noisy states, introduce a framework which applies final-state corrections. Proximal steps are applied to a predicted clean posterior, rather than on a noisy intermediate state, and the feasible clean state is then renoised to steer the sampling trajectory along the data manifold, while converging to exact feasibility at the terminal state. A schematic illustration of the proposed scheme is provided in Figure T
Contributions. This paper makes five key contributions: (1) It introduces a stochastic proximal method for constrained diffusion feasibility; (2) based on this view, proposes consensus-based ADMM scheme that separates local stereochemistry from sparse global couplings; (3) it provides a theoretical analysis characterizing the convergence to the constraint set and provides arguments for why final-state projection is preferred over per-step projections; (4) it demonstrates the efficacy of the approach on challenging protein design tasks with nonconvex constraints, including global topol- ogy (e.g., chain closure, ligand binding feasibility) and local stereochemistry (bond lengths, angles, and chirality), achieving perfect constraint satisfaction and providing state-of-the performance; (5) it introduces novel curated benchmark for protein motif scaffolding in PDZ domains, providing the first systematic standard for constrained diffusion methods modular domain engineering.
2 SETTINGS AND BACKGROUND
The goal of de novo protein design is to generate three-dimensional representations which satisfy physical plausibility and functional requirements, such as protein motif scaffolding, binder design, and monomer generation. This task is fundamentally constrained by the physies and chemistry of proteins, where bond geometries must be preserved, chains must remain connected, and higher-level properties such as specific inter-chain interactions or interface complementarity must be realized.
Diffusion-based protein backbone generation. Let Pdata denote the unknown distribution of clean molecular structure representations 2o E IRd (e.g., atomic coordinates). Consider a constrained feasibility set C C RRd, encoding physical and geometric constraints, seeking to sample from the target distribution
Pc & ( 1 (x0 E C),
where 1 1(xu C} is indicator function C. To achieve this, diffusion model learns to recon- struct samples from the data distribution Pdata by coupling forward noising Markov chain with a learned reverse denoising chain (Song al., 2020 Ho et al. 2020
Forward process. Diffusion models construct a Markov chain (=x)T_ starting from Zp ~ Pdata- At each step, Gaussian noise is added according a fixed variance schedule IT This process admits closed-fone marginal distribution, g(x;, 2o) M[ Zo. (1 a.))), where âz I1 - -a). As + T. the distribution converges to an isotropic Gaussian q(TT) ().
Reverse process. A generative model learns to approximate the reverse dynamics, sampling -;-1 given Zt. Since the true reverse kernel q xx) is intractable, a neural network X (,( t) is introduced to parameterize this transition. At inference, the learned reverse transitions
applied iteratively, gradually denoising a random Gaussian vector Dy ~ (O, into a clean structure on- Standard sampling processes will result in outputs distributed approximately pdatas while our
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underlying goal is instead generate outputs distributed in Pc- Generating from pe, requires the sampling procedure to be modified to incorporate constraints, as discussed in the next section.
CHALLENGES WITH CONSTRAINED DIFFUSION
As previously noted, diffusion models natively learn reconstruct samples from unconstrained data distribution Pdata= which is misaligned with the true goal for constrained generation described by Equation x In other domains, this gap is often addressed by formulating constrained sampling as an optimization problem, and recent work has extended this perspective to diffusion models by proposing general framework for inference-time constrained generation (Christopher al., 2024). The approach can be viewed as a sequential optimization problem:
T
min (
Σ (a wo) s.t. De EC 14 t=
where the single stage cost (x,,, 2n) : logp(æ, zo), and the constraint set C captures geo- metric or structural feasibility. Enforcing these constraints typically involves applying projection operator (x;) = arg millgec 1 æt after each reverse diffusion step. This is appealing be- cause it embeds feasibility directly into the generative process; however, enforcing striet projections at every step raises two Nomain--specifi challenges:
1. As observed by Blanke et al. 2025),, projections intermediate states introduces statistical biases. This arises as intermediate samples concentrate near constraint boundaries. This issue has also been reported for soft guidance schemes, where increased weight the guidance terms tends to disrupt the diffusion trajectory and degrading performance (Dhariwal & Nichol 2021
Nichol & Dhariwal] 2021 Ho & Salimans (2022).
2. Additionally, intermediate feasibility requires projecting highly noisy states onto complex, non- convex constraints. which can result in solutions trapped in local minima (Pardalos Vavasis 1991 Boyd & Vandenberghe/ (2004), disrupting the diffusion performance while simultancously limiting the efficacy of feasibility updates.
Imposing constraints precisely on noisy samples may be reasonable under convexity assumptions, where the theoretical provisions apply, but within protein generation settings characterized by non- convex constraints, it is necessary to develop inference-tine strategies which do not fundamentally rely on æ, E C. In the next section, we present our proposed method, designed intentionally with this principle in mind.
4 REVERSE DIFFUSION AS PROXIMAL OPTIMIZATION
To effectively sample constraint compliant outputs æo ~ Pdata (20)) (ax E C) (Equation x we design inference-time method that converges to C, where the terminal distribution mo minimizes KL(TO Pc). Under this framing, a single reverse step of a diffusion sampler is viewed as an opti- mization problem in which the denoiser provides a data-driven "anchor", and feasibility is enforced by penalizing the distance to the constraint set.
Since Pdata is only available through denoiser, the reverse process is realized incrementally. Each state 1 of the reverse diffusion is composed of three stages: (1) predict the clean structure á from the current noisy state 2t, (2) correct this prediction by proximal operator prox, n. that enforces feasibility, and (3) renoise the corrected clean structure with the forward kernel, denoted FWD(:,)), to obtain the next noisy sample. The procedure is outlined in Figure and summarized as:
predict
FWD
Øt-1-
This modus operandi has strong theoretical properties discussed in Section 6 First, description of each step is detailed.
1. Clean state prediction. In protein design applications, it is common to employ 2p co-prdiiction parameterization. This design is intentional, as enables adaptation from pretrained folding mod- els like RoseTTFold, reusing their architectures and learned weights for initialization (Baek al.) 2021). At reverse time t E T, T,..., the model takes a noisy latent x; and predicts a clean structure,
X (I2,, 1) 4/
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providing approximation of the final state. As t - 0, predictions improve in accuracy noise signal reduces. The availability of the predictor x is convenient as its output can be leveraged in next step to restore feasibility.
2. Feasibility step (proximal projection). Next, feasibility requirements are applied on the pre- dicted clean state. A feasible estimate is produced through proximal map:
a/= = prox n2.9 (4%) == arg min
||
abll2 + g(ar)
21t.
where > O is step size determined by the degree of trust in the denoiser's prediction at step t and g 3xdd + RR. is feasibility potential.
A natural first choice would be to take g as the indicator function of the feasible set. However, this would require exact projections onto a potentially nonconvex set, which can be ill-defined when ath lies far from C. To avoid instability, the hard indicator is replaced by its Moreau envelope (Boyd & Vandenberghe, 2004), yielding the smooth penalty
Xe distc(x)² 2
g(x)
with de
where distc is distance metric from the feasible set (e.g., in SE(3), it could be infyec || y The parameter de > plays the role of an inverse smoothing radius: as de + oo the penalty enforces exact feasibility, and for finite x it softly biases toward C.
Forward renoising. Having obtained a feasible estimate, the next step reintroduces noise by sampling from the forward marginal at t conditioned on the corrected clean sample ztj
zt = FWD(#), E) = 1 + Ot-1E
(4)
with e (O, I). This guarantees that, conditioned on #.. the marginal of 2;-1 matches the forward diffusion at time step t - 1. Note that as - 0 and ag - 1, the Markov chain terminates clean œi 2.. If g acts as in indicator function, exact feasibility is recovered, while otherwise Zo becomes arbitrarily close to C as X increases.
Selecting the schedule. It instructive to connect the proximal subproblem (Equation 2 prob- abilistic reasoning. By modeling the network's 's clean error at step t as Gaussian with variance neI:
p( x.) exp
|m0 ab(1²)
21je
then interpreting the penalty y as a soft prior of the form xx exp(-g(20)), Equation (2) computes the per-step MAP estimate zi of the clean state. The subsequent renoising step. Equation (4. reinstantiates the correct stochasticity for the reverse chain while anchoring it to constraint set C.
Because of shrinks over time, it is natural schedule A grow, so that feasibility becomes domi- nant only when the model's s L is accurate. Similarly, if 1): of_1 the trust weight can be directly connected the diffusion variance, and the clean proximal problem remains on the same scale.
This predict-prox pet-t-roo---niiss step is the stochastic analogue of proximal gradient step. As elaborated in Section 6 the predict-roo--enoiss cycle both respects the diffusion dynamics and guarantees convergence to feasible terminal states.
5 DECOUPLING GLOBAL TOPOLOGY FROM LOCAL GEOMETRY VIA ADMM
The constraint set C captures a strong coupling between local stereochemical variables and global variables governing topology and long-range residue interactions. Because residues that are far apart in sequence may lie adjacent in the folded structure, enforcing global constraints thus necessitates coordinated updates that can significantly impact nearby stereochemisty. For instance, we observe 3-strand can cause the associated residues to shift
that applying non-covalent bond constraints substantially, degrading the fidelity of this local geometry (Budyak et al., 2024). These interdepen- dencies make the proximal step computationally complex. However, the presence of these separable local and global constraints confers structure to the problem and thus presents an opportunity to exploit it, enabling the use of decomposition approaches.
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Consider that feasible point be equivalently represented constraints Clocal capture properties that are applicable all backbone design tasks (c.g.. adherence to stereochemical bond lengths and angles between consecutive atoms and residues). The global constraints Cglobal are problem-specine functionals: for example, in our first experiment, this con- straint set defines bond lengths and angles between specific nnighhhbgooing residues, characterizing non-covalent bonds which are necessary for protein-ligand pocket design.
Clocal n Cglobal- The local
Following this intuition, the feasibility potential is decomposed as g(x) glocal (2)) Igllball (2)), where
x 2
X diste (x)²
Nlocal (x) cissst crea
Oglobal
and flocal. Iglobal R" + IR U {+oo}. These functions are proximable; squared distance penalties adopted, which treated indicator functions when X + Hence, proximal update is reframed as:
(1())) = arg min 1
|| 2ill 2 + Glocal (x) + (global(x)). 21):
:
=:((a))
Crucially, we define the local block F to include the both repairs stereochemistry and stays close to d/ while the global block G focuses on long-range feasibility. We solve this by consensus ADMM on
term so that the local step
mün Fly) + G(z) y,z
s.t. y/ = 2,
(5)
with scaled dual variable u and penalty p > O. This leads to the proximal splitting form of ADMM (Douglas-Rachford (Parikh et al.
2014), with the update
y ++] proxp', pt fzy u u²),
(6a)
.+1 = prox,* pt clx + u M u ++ = u + yet zk k+1
(6b)
(6c)
where A is an iteration counter and yd z" at E Raxd, Here y and are two copies of the backbone, associated with F and respectively, and the dual variable u accumulates their mismatch. At convergence the iterates satisfy y recovering the minimizer of F + G; in practice it is only necessary to take a single sweep per diffusion step, but warm-starting across steps ensures the two copies remain close.
These updates can, thus, interpreted as applying ADMM to the consensus problem, where the dual variable carries forward residuals between local and global feasibility corrections. In practice, this is implemented by minimizing the associated augmented Lagrangian. For clarity, we present only the proximal form here but detail the explicit augmentented Lagrangian in Appendix G
THEORETICAL ANALYSIS
Now, we show that the samples generated by the proposed stochastic proximal method come with feasibility guarantees. In the following we assume that the constraint set C is prox-regular and defer all proofs in Appendix H
We start by providing a bound on the feasibility guarantees attained by the generated final sample.
Theorem 6.1. Consider a feasibility potential g(x) = 2 dist (x=) defined as in Equation Then, the proximal minimizer To satisfies:
diste (wo) <
diste (Zo))
22Am
feasibility
The inequality shows that the proximal step contracts the violation by (2Amm))-1/2 guiding the reverse process towards the constraint Then, t + 0 and Xene co, the corrected iterate converges arbitrarily close to the constraint set.
The following result provides rationale for how schedule A
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1. Standard Diffusion: RFDiffusion conditioned on relevant motifs or structures, as it is often used Published as a conference paper at ICLR 2026
Theorem 6.2. Let K be finite number such that E Eldisto A St with a non decreasing ot yields:
K for all t, then choosing
K
K
E c (#))
S
and hence
I [dist (#o)
2c,
2c1
As consequence, tightening c₁ towards the end guarantees decreasing expectation the viola- tions, leading to arbitrary small terminal violations. Note also that taking 9i as the identity function the constraint set C' gives exact feasibility.
Beyond these quantitative feasibility bounds, the per-step proximal subproblem needs to be well- posed. In particular, the question becomes whether (possibly local) minimizer of the proximal mapping exists under the modeling assumptions.
Theorem 6.3. Consider the proximal subproblem
prox (a) = 2m
| 2bll + 2 diste(x)?
with d O and C C 3x3 nonempty and closed. Then:
Existence. prox continuous and coercive, hence attains a global minimizer for every Thy- In particular, arg min prox * Ø.
2. Local uniqueness. If d lies within the prox-regularity neighborhood of C, then the projection [I((a)) single-valued. Moreover, if prox, a is strongly convex in a neighborhood of Ilc (#)), then the proximal minimizer is unique within that neighborhood.
The analysis shows that fcasibility improves monotonically as A tightens relative to Tee while the proximal subproblem remains well-posed due the quadratic distance term. In practice, this pro- vides principled guidelines for selecting schedules which balance denoiser trust against constraint enforcement. The next section demonstrates that the theoretical foundations translate tangible performance improvements in protein backbone design.
EXPERIMENTS
We conduct our evaluation on two key tasks for protein motif scaffolding in the PDZ domain and vacancy-constraineed pocket design for unconditional generations, described in more details in Sec- tions 7.1 and
Baselines. The comparison includes state-of-the-arr structure- design methods incorporating constraint conditioning. We use RFDiffusion (Watson et al.) 2023) as the underlying backbone across all baselines and method, although note that any method evaluated could be extended to other backbone structure diffusion models (Yim et al.) 2023) Cutting et al.) 2025). RFDiffusion is selected, it's considered a state-of-the-art approach for the tasks considered in this work:
in practice.
2. Recentering of Mass Guidance: RFDiffusion with conditioning to bias the generation towards particular Cartesian coordinate where bond interactions cavity-defining residues enforce constraint satisfying formations (Braun et al. 2024). Conditioning is fine-tuned to find the lowest average constraint violation per sample prior to beginning experiments (see Appendix B).
3. Constraint-Guine Diffusion (CGD): RFDiffusion with constraint- based guidance Sequential Monte Carlo sampling (Lee et al., (025). This SOTA method uses importance sampling with guided rate matrix to define weights and periodic resampling based on constraint violations.
Metrics. The following metrics are adopted to evaluate the performance of our method:
1. Constraint Satisfaction:
The percentage of samples satisfying all domain-specife constraints. For Section 7.. we verify presence of the described hydrogen-bond pairing the backbone through DSSP (Kabsch & Sander 1983). while Section 7.2 computes constraints via Cartesian coordinate bounds.
2. Structure Realism: Percentage of samples containing secondary structures within constrained regions while maintaining general backbone realism (c.g., 3-sheets remain less than ten residues, inter-residue distances and angles preserved).

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BN youee why
soly cesee
Structure X
Giobal Bistonce Cocatr alst
(a) Ours
(b) RFDiffusion (Standard)
Figure 2: Visualization of randomly selected samples generated by (a) our proximal method and by
(b) Standard RFDiffusion on our introduced PDZ domain benchmark.
3. Usable Percentage:
Percentage of samples passing above conditions for Structure Realism and Constraint Violation. Indicates frequency of generating a structure that satisfies physical plausibility and functional requirements.
Radius of Gyration: Average radius of gyration across generated backbones, measuring the overall spatial compactness of the structure. Lower values typically correspond more compact, globular folds, while higher values indicate extended or unfolded conformations.
5. Diversity: Percentage of samples which are both useable and satisfy minimum root mean squared error between all other samples (2 A). Higher diversity indicates better coverage of pos- sible structure.
Additionally, we note that further details on all experimental setups are provided in Appendix
7.1 PDZ DOMAIN: NON-COVALENT BONDS
Antibodies widely used in molecular biology to target specific proteins, but their large size and extracellular restriction limit their utility. particularly in intracellular contexts. In addition, their binding interfaces, often mediated by flexible loop regions, pose challenges for computational de- sign. As an alternative, biology employs small modular protein domains (e.g., WW, SH2, SH3, PH, and PDZ), which provide more designable binding modes. PDZ domains, for instance. recognize unstructured C-terminal motifs of partner proteins, typically through 8-shect-like hydrogen bond contacts. These interactions are generally weak and promiscuous. serving primarily protein lo- calization. To enhance affinity and specificity, Huuag et al.) (2009) engineered concatenated PDZ fusions with other small domains, demonstrating improved performance. Protein diffusion mod- els extend this concept by enabling conditioning both PDZ domains and their peptide ligands, thereby facilitating the de novo design of concatenated architectures.
Dataset construction. To benchmark constrained diffusion for PDZ engineering, we collected all resolved PDZ/PDZ binding motif (PBM) complexes from the RCSB Protein Data Bank (Berman et al. 2000). Seventy-two structures were initially retrieved and manually curated remove entries with unresolved regions or peptides too short for recognition, yielding fifty-two usable complexes. To expand the existing PDZ domain in reliable way make additional contacts with the target PBM, the N- and C-termini had to be rearranged. Each structure was processed to reposition ter- mini closer to the bound peptide by introducing cut in a loop adjacent to the ligand, trimming the original termini, and applying vanilla RFDiffusion to in-paint the resulting gap. Candidate back- bones were filtered to exclude chain breaks and non-3-sheet pairings. Sequences for the redesigned regions were generated using ProteinMPNN, followed by structure prediction with AlphaFold2. Pre- dicted models were retained only they satisfied stringent criteria: (i) self coff-coossttency RMSD the RFDiffusion backbone <2.5 A, (ii) mean PLDDT >90, and (iii) peptide RMSD <2.0 A. After filter- ing, 31 high-confidence PDZ designs remained for benchmarking. While all 31 structures are valid targets for this approach, six of them (PDB IDs: 2AWW, 2G2L, 4JOG, 6UBH, 6Y90, 8CN3) con- tain short peptide ligands or peptides with geometrically restrictive residues such as prolines. These structures remain in the benchmark set, but they are distinguished in the results having poor-posed ligands. More details are provided in Appendix C novel contribution of this work.
This benchmark for constrained diffusion is also
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RFDiffusion Recenter
Ours
Standard
CGD
Constraint Satisfaction (%)
0.0
0.0
0.0
100.0
Structure Realism (%) [t]
(32.0)
(18.7)
(38.2)
21.0
Usable Percentage (%) [t]
21.0
Radius of Gyration (A) [L]
(13.6)
(13.2)
(16.2)
12.4
Diversity (%) [t]
N/A
N/A
N/A
18.8
Table 1: Comparison to structure-based design baselines, Standard (Watson et al. 2023). J. Recenter
(Braun al. 2024), and CGD (Lee al., (0255), and ours, for the PDZ domain. Results reported across 31 31,000 samples for each baseline, highlighting best and second best results. Parentheses indicate when statistics are computed over unusable structures.
Task description. Given a target PDZ domain and its peptide ligand, the objective to design an additional domain concatenated to the PDZ that contributes new stabilizing contacts with the pep- tide. This requires satisfying global inter-chain constraints. Specifically, because the peptide forms 8-sheet-like contacts with the PDZ domain, aim to create complementary set of 3-sheet-like like in- teractions on the opposite face of the peptide using the engineered concatenated domain. Constraints enforeed to ensure valid bond lengths and angles, while also ensuring local stereochemisty is pre- served. Success is evaluated by the metrics aforementioned, where global feasibility is defined by ideal bond lengths (2.9 = 0.2 Å), C=O... N angles (155 10%). and C..N.... (120 = 10%).
Results. Table provides results on the PDZ benchmark comparing our constrained diffusion ap- proach (visualized in Figure (a)) the baseline methods. Notably, across nearly hundred thousand samples generated for the three baselines, not one sample perfectly satisfied the bonding distance and angle constraints. The baselines frequently generate incorrect secondary structures, as illustrated in Figure (b), making it implausible that generations will effectively bind with the pep- tide ligand. While recentering and CGD perform well in terms of the local geometric requirements captured by the structure realism measurement, they are unable to cope with the global requirement of non-covalent bonding between the PDZ. backbone continuation and the peptide ligand, as reflect by the constraint satisfaction rates. While ceustraint-gudded diffusion satisfies the bond distance constraints for some generations, it is never able to generate residues which appropriately meet the angle requirements. Importantly, the baselines yield usable generations for any of the 31 struc- contrast, our method achieves state-of-the- results, generating usable
tures in the benchmark. structures in 21.0% % of total generations (and up 83.0% for well-posed ligands), markedly outper- forming the existing baselines. In addition to observing perfect constraint satisfaction, we outper- form all methods substantially in radius of gyration and diversity metrics. These results highlight the unique ability of our approach to handle both local stereochemical properties while enforcing global functional constraints, providing a vastly more viable approach to protein engineering under specific property requirements and design constraints.
7.2 MOLECULE ENCAPSULATION VACANCY CONSTRAINTS
Several recent approaches have improved the usefulness of diffusion-based protein generation with the integration all-atom models and catalytic site scaffolding (Krishna et al. 2024 Ahern et al.) 2025 Braun et al.) 2024). A key remaining limitation is precise spatial control over where new structure is placed. Adjusting the diffusion origin can help, but explicit user control over inclu-
(a) Standard
(b) Recentering
(c) Constraint-Guided
(d) Ours
Figure 3: Visualization of randomly selected samples for the molecule encapsulation experiment; green parts of the structure fall within feasible regions, while the red parts violate the constraints.
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RFDiffusion
Our
Standard
Constrint Satisfaction (8) [t]
27.4
100.0
Structure Realism (8) ft)
(100.0)
(100.0)
93.8
97.8
Usable Percentage (%) [1]
0.0
20.5
24.2
97.8
Radios Gyration (A) [1]
(15.2)
(14.3)
23.9
26.6
14.8
Diversity (1)
NIA
NIA
20.5
24.2
97.8
Table 2: Comparison to structure-based design baselines: Standard (Watson et al. 2023). Recenter (Braun et al., (024), and CGD <Lee et al., 2025). and ours, for molecule encapsulation. Results reported across 4000 samples, highlighting best and second best results. Parentheses indicate when statistics are computed over unusable structures.
sion/exclusion volumes would better enable tasks such as shaping binding grooves, or membrane-embeddde features. For example, Braun et al. (2024) approximated pocket formation by inserting placeholder a-helix occupy volume during generation and delet- ing afterward. Generalizing this idea to geometric volume constraints, hard inclusion masks and forbidden regions, could provide finer control and higher success rates in targeted protein design.
pockets, peptide-
Task description. Given a fixed spatial environment defined by a rectangular box with an internal conical exclusion zone, the goal is to design protein backbones which fall exclusively in this noncon- vex region while preserving the local geometries and secondary structures. Feasible structures are characterized by all atoms falling within the defined box (20 À 40 Ax 40 Å), while simultaneously avoiding the exclusion zone introduced by the displacement (visualized in Figure
Results. Figure 3 provides visualization of representative samples from each baseline. Standard diffusion and recentering of mass guidance perform similarly in this domain, yielding realistic struc- tures but failing to satisfy the functional requirements any out of 1000 samples generated for each. We observe slightly different failure modes: standard diffusion generally violates the box constraint, while recentering more often violates the vacancy constraint. The recentering is effective at keep- ing the structures inside the box, but it cannot capture the exclusion zone. Table 2 reports much stronger performance for constraint guideed diffusion, especially when augmented with recentering, which generates feasible samples 27.4% of the time, resulting in 24.2% usable samples. However, is worth noting that the qualitative performance suffers, as the radius of gyration is much higher than other baselines. This is often indicative of structures which contain unfolded conformations, ultimately undermining structural stability and realism. In comparison, our method reports perfect constraint satisfaction, producing 4x as many usable samples the nearest baseline, with im- pressive 97.8% success rate. Furthermore, it maintains radii of gyration comparable to standard diffusion, indicating the generated samples combine structural plausibility, compactness, and fold coverage.
RELATED WORK
While existing de novo protein structure design models produce plausible generations, as we have shown in this paper, sampled protein backhones frequently violate inter-atomic bond lengths, angles, or chain closure requirements, often necessitating the generation of tens of thousands of candidates
obtain handful of viable designs (c.g.. Watson al. backbone generators such as RFDiffusion (Watson et al. 2023 Ahern et al.) (025) provide major advances in functional conditioning, outputs still require post hoc filtering to ensure stereochemical correctness, and current pipelines continue to rely heavily on rejection sampling. Recent models such as Genie and OriginFlow further illustrate the growing interest in diffusion and flow-based approaches for backbone generation (Lin et al.) 2024 Yan et al. 2025); however, these methods exhibit similar limitations, with generated back backbones often violating global constraints and requiring substantial downstream filtering.
2023 Sappington et al. (024).. Although
Training-time methods have been proposed to address these issues by embedding structural con- straints into generative models (Eguchi et al. 2022 Lutz al. 2023). However, because protein design requires task-specific constraints, model trained on one constraint set does not generalize, making broad applicability impractical without retraining. Furthermore, training-time approaches typically provide only distributional guarantees, biasing samples on average rather than ensuring per-sample feasibility. Models such as ReQFlow (Yue et al. 2025) and FoldFlow-2 (Huguet al.

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2024 provide valuable tools, but likewise do not directly enforce hard geometric constraints, instead
incorporating them as soft biases.
Inference-time approaches have emerged as strong alternative, enforcing per-sample compliance and removing the need for model retraining, Diffusion guidance methods were first introduced for soft constraint imposition but are fundamentally limited, offering only probabilistie bias rather than guaranteed adherence to the constraint set (Ho & Salimans] (022). While these techniques have im- proved performance for protein backbone generation, with models such as Chroma (Ingraham al.) 2023) leveraging conditioning on specific substructures, as we have shown in our experiments, they are often ineffective in providing consistent constraint satisfaction. Overcoming these limitations requires generative models which can effectively integrate these constraints into the design process, as presented in this work.
CONCLUSION
Motivated by the significant challenge of integrating functional design constraints into protein en- gineering tasks, this paper present constrained diffusion framework for structure-guided design. By applying proximal feasibility updates with ADMM decomposition, the approach couples local stereochemical property enforcement with global utility requirements. To assess the quality of ex- isting solutions as compared to the methodology presented in this paper, the work introduces a novel curated benchmark for protein motif scaffolding in PDZZ domains, providing the first standard for constrained diffusion methods in modular domain engincering. Evaluation reports state-of-the- results across motif scaffolding and vacancy-constraingg pocket design, illustrating the ability of this approach to generate high quality proteins which adhere to precise domain-centri constraints.
ETHICS STATEMENT
This work develops methods for constrained generative modeling in protein design. Our method im- proves feasibility in backbone design tasks, which holds significant potential to accelerate existing protein engincering pipelines. To mitigate risks of potential misuse, this paper restricts evaluation to safe, publicly available structural benchmarks, curated from the Protein Data Bank. All implemen- tations follow standard open science practices, ensuring safe and transparent research
REPRODUCIBILITY STATEMENT
The code and benchmark are released in the supplementary material along with instructions to guide the reproduction of the results presented in this paper. Methodological details are described exten- sively in the paper and accompanying appendix. Section 7 Appendix B Appendix [F describe the specific constraints, hyperparameters, models, and hardware used for evaluation pipelines.
ACKNOWLEDGMENTS
This research is partially supported by NSF awards 2533631, 2401285, 2334936, and 22268 16, by DARPA under Contract No. #HR001 1252E005, by the National Institutes of Health, National In- stitute of General Medical Sciences, under Award Number T32 GM135141, and Sargassum BioRe- finery (SaBRe) Center, project Schmidt Sciences' Virtual Institute of Feedstocks of the Future (VIFF), with support from the Foundation for Food & Agriculture Research. The authors acknowl- edge the Research Computing at the University Virginia. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF, DARPA, or the National Institutes of Health.
REFERENCES
Woody Ahern Jason Yim. Doug Tischer, Saman Salike, Seth M Woodbury, Donghyo Kim, Indrek Kalvet. Yakov Kipnis, Brian Coventry, Han Raut Altae Tran, et al. Atom level enzyme active site scaffolding using rfdiffusion2. bioRxiv, pp. 2025-04, 2025.
10

---

Published as a conference paper at ICLR 2026
Minkyung Back, Frank DiMaio, Ivan Anishchenko, Justas Dauparas, Sergey Ovchinnikov, Gyu Rie Lec, Jue Wang. Qian Cong. Lisa N Kinch, Dustin Schaeffer, et al. Accurate prediction of protein structures and interactions using three-track neural network. Science, 373(6557):871- 876, 2021.
Noémie Bergues, Arthur Carré, Paul Join-Lambert, Brice Hoffmann, Arnaud Blondel, and Hamza Tajmouati. Template-guis guided 3d molecular pose generation via flow matching and differentiable optimization. arXiv preprint arXiv:2506.06300. 2025.
Helen M Berman, John Westbrook, Zukang Feng, Gary Gilliland, Talapady N Bhat, Helge Weissig, Πlya Shindyalov, and Philip Bourne. The protein data bank. Nucleic acids research, 28(1): 235-242, 2000.
Matthieu Blanke, Yongquan Qu, Sara Shamekh, and Pierre Gentine. Strietly constrained generative modeling via split augmented langevin sampling. arXiv preprint arXiv:2505. 2025.
Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004.
Markus Braun, Adrian Tripp, Morakot Chakatok, Sigrid Kaltenbrunner, Celina Fischer, David Stoll, Aleksandar Bijelic, Wael Elaily, Massimo G Totaro, Melanie Moser, al. Computational enzyme design by catalytic motif scaffolding. bioRxiv, pp. 2024-08, 2024.
Ivan L Budyak, Lihua Huang, and Rina K Dukor. Higher order structure characterization of two in- terdomain disulfide bond variants a bispecific monoclonal antibody. Journal of Pharmuceutical Sciences, 113(8):251122655) 2024.
Sidhartha Chaudhury, Sergey Lyskov, and Jeffrey J Gray. Pyrosetta: script-based interface for im-
plementing molecular modeling algorithms using rosetta. Bioinformatics, 26(5):689-691, 2010.
Jacob K Christopher, Stephen Back, and Nando Fioretto. Constrained synthesis with projected diffusion models. Advances in Neural Information Processing Systems, 37:89307-89333, 2024.
Daniel Cutting, Frédéric Dreyer, David Errington, Constantin Schneider, and Charlotte M Deane. De novo antibody design with se (3) diffusion. Journal of Computational Biology, 32(4):351-36) 2025.
Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780-8799 2021.
Kieran Didi, Francisco Vargas, Simon V Mathis, Vincent Dutordoir, Emile Mathieu, Urszula J Ko- morowska, and Pietro Lio. A framework for conditional diffusion modelling with applications in motif scaffolding for protein design. arXiv preprint arXiv:2312.099336 2023.
Raphael R Eguchi, Christian A Choe, and Po-Ssu Huang. Ig. vae: Generative modeling of protein structure by direct 3d coordinate generation. PLoS computational biology, (86))e1011271, 2022.
Nate Gruver, Samuel Stanton, Nathan Frey, Tim GJ Rudner, Isidro Hotzel, Julien Lafrance-Vannn Arvind Rajpal, Kyunghyun Cho, and Andrew Wilson. Protein design with guided discrete diffusion. Advances in neural information processing systems, 36:
2023.
Jonathan Ho and Tim Salimans.
Classifier-free diffusion guidance.
arXiv preprint
arXiv:2207. 12598, 2022.
Jonathan Ho. Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
Jin Huang, Koki Makabe, Matthew Biancalana, Akiko Koide, and Shohei Koide. Structural basis for exquisite specificity of affinity clamps, synthetic binding proteins generated through directed domain-interface evolution. Journal of molecular biology, 392(5); 2009.
Guillaume Huguet, James Vuckovic, Kilian Fatras, Eric Thibodeau-LLuufe Pablo Lemos, Riashat Islam, Chenghao Liu, Jarrid Rector-Brood Brooks, Tara Akhound- -aaound-Sadeghh Michael Bronstein, et al. Sequence-augee augmented se (3)-flow matching for conditional protein generation. Advances in neural information processing systems, 37:33007-30036, 2024.

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Published as a conference paper at ICLR 2026
John B Ingraham. Max Baranov, Zak Costello, Karl W Barber, Wujie Wang, Ahmed Ismail, Vincent Frappier, Dana M Lord, Christopher Se--Thww-Hing Erik R Van Vlack, et al. Illuminating protein space with programmable generative model. Nature, 623(7989):1070-10778 2023.
Wolfgang Kabsch and Christian Sander. Dictionary of protein secondary structure: pattern recog- nition of hydrogen-bonded and geometrical features. Biopolymers: Original Research on Biomolecules, 22 1983.
Rohith Krishna, Jue Wang, Woody Ahern, Pascal Sturmfels, Preetham Venkatesh, Indrek Kalvet, Gyu Rie Lee, Felix S Morey-Burrows, Ivan Anishchenko, Ian R Humphreys, et al. Generalized biomolecular modeling and design with rosettafold all-atom. Science, xead6523), 2024.
Cheuk Kit Lee, Paul Jeha, Jes Frellsen, Pietro Lio, Michael Samuel Albergo, and Francisco Var- gas. Debiasing guidance for discrete diffusion with sequential monte carlo. arXiv preprint arXiv:2502.06079 2025.
Yeqing Lin, Minji Lee, Zhao Zhang, and Mohammed AlQuraishi. Out of many, one: Design- ing and scaffolding proteins at the scale the structural universe with genie 2. arXiv preprint arXiv:2405. 15489, 2024.
Isaac D Lutz, Shunzhi Wang. Christoffer Norn, Alexis Courbet, Andrew J Borst, Yan Ting Zhao, Annie Dosey, Longxing Cao, Jinwei Xu, Elizabeth M Leaf, et al. Top-down design of protein architectures with reinforcement learning. Science, 38066642) 2023.
Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. In International conference on machine learning, pp. 8162-8171. PMLR, 2021.
Panos M Pardalos and Stephen A Vavasis. Quadratic programming with one negative eigenvalue is np-hard. Journal of Global optimization, 1(1):15-22, 1991.
Neal Parikh, Stephen Boyd, et al. Proximal algorithms. Foundations and trends@ in Oplimization, 1(3):127-23,, 2014.
Isaac Sappington, Martin Toul, David Lee, Stephanie A Robinson, Inna Goreshnik, Clara Mc- Curdy, Tung Ching Chan, Nic Buchholz, Buwei Huang, Dionne Vafeados, et al. Improved protein binder design using beta-pairing targeted rfdiffusion. bioRxiv, pp. 2024-10, 2024.
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint MrXiv:2011.134555 2020.
Zhenqiao Song, Yunlong Zhao, Wenxian Shi, Wengong Jin, Yang Yang, and Lei Li. Generative en- zyme design guided by functionally important sites and moll-moleeclee substrates. arXiv preprint arXiv:2405.08205. 2024.
Brian L Trippe, Jason Yim, Doug Tischer, David Baker, Tamara Broderick, Regina Barzilay, and Tommi Jaakkola. Diffusion probabilistic modeling of protein backbones in 3d for the motif- scaffolding problem. arXiv preprint arXiv:2206. 04100.111111111 19, 2022.
Joseph L Watson, David Juergens, Nathaniel R Bennett, Brian L Trippe, Jason Yim, Helen E Eise- nach, Woody Ahern, Andrew J Borst, Robert J Ragotte, Lukas F Milles, et al. De novo design of protein structure and function with rfdiffusion. Nature, 620(7976):1085 1000--0000 2023.
Junyu Yan, Zibo Cui, Wenqing Yan, Yuhang Chen, Mengchen Pu, Shuai Li, and Sheng Ye. Robust and reliable de novo protein design: A flow-matching- protein generative model achieves remarkably high success rates. bioRxiv, pp. 2025 (04, 2025.
Jason Yim, Brian L Trippe. Valentin De Bortoli, Emile Mathieu, Arnaud Doucet, Regina Barzilay, and Tommi Jaakkola. Se (3) diffusion model with application to protein backbone generation. arXiv preprint 200.v:2302.02227 2023.
Angxiao Yue, Zichong Wang, and Hongteng Xu. Regflow: rectified quaternion flow for efficient and high-quality protein backbone generation. anXiv preprint arXiv:2502. 14637, 2025.
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A LIMITATIONS AND FUTURE WORK
In scientific domains such as protein engineering, exact constraint satisfaction is essential for both physical realism (local constraints) and functional utility (global constraints). However, it is valu- able to acknowledge several important trade-offs associated with the adoption of constrained meth- ods. First, enforcing feasibility inherently introduces increased computational complexity. Sampling from pe is a fundamentally harder task than sampling from Pdata especially under the assumption of hard constraint satisfaction, as reflected reported runtimes (Appendix [F. While runtime is worth considering, especially as the method scales, it is important to note that when sampling from pc, approach offers the most efficient runtime, providing much better performance than the current re- jection sampling paradigm. Although not applicable in experimental settings, if sampling from Pdata is sufficient, is not computationally tractable to adopt constrained diffusion methods.
As second caveat concerns theoretical assumptions. Our analysis assumes prox-regularity of the constraint set to provide formal guarantees. This condition is standard in proximal theory, but it may not apply for highly nonconvex constraints encountered in protein design tasks. In such cases, vergence guarantees do not explicitly apply, and solutions are instead justified empirically. However, the development of formal guarantees for nonconvex constraint sets remains an open and largely un- explored direction, as existing optimization theory cannot provide guarantees for general nonconvex sets.
Finally, while our experimental evaluation illustrates tite--of-theaar performance on two highly nontrivial and practically significant protein engineering problems, necessarily leaves many other design challenges open for future exploration. The problems we consider already capture central difficulties in enforcing exact feasibility under realistic structural and functional constraints, making them representative of some of the hardest settings encountered in practice. Extensions such as plac- ing global constraints on higher-level properties (c.g., polarity of specific regions in the structure) or imposing specific formation constraints secondary structures (c.g., controlling the radius of 3-barrel) are complimentary next steps. We view these as exciting opportunities for future work, with the present study establishing critical foundation for handling such broader classes of design challenges.
B EXPERIMENTAL DETAILS
schodule
PDZ Domain
00
linic (< tol, 3.0 A)
45
lincar in f, from dy
7e-
SO13;
15-42
0/02 1.6
Molecule Encupsalatinn
f, from on to dy
S013:
Table 3: Experiment hyperparameters.
All hyperparameters are reported in Table 3 Both tasks use RFDiffusion defaults: T 50 steps, linear 3 schedule from 10 10-2 to 0.07, SO(3) schedule 1.5-2.5, coordinate scaling 0.25, and Gaus- sian noise E [0.02, 1.5). Proximal multipliers e₁ tighten from er (finite) to c across the trajectory. Other hyperparameters X and r) can be derived by Theorem 6.2
B.1 PDZ DOMAIN: NON-COVALENT BONDS
The evaluation is conducted across our newly introduced dataset, including 31 distinct PDZ struc- tures. For each structure, generate 1000 samples with each of the baselines, yielding total of 31,000 designed proteins. From these generations, we assess the overall performance of each method, leveraging PyRosetta conduct the final evaluation of the generations (Chaudhury et al. 2010
Recenter of Mass Guidance. Prior to running the evaluation, we search the local space surround- ing the peptide ligand to empirically optimize the selected center of mass. We generate ten samples for each selected point in space, providing representative data for the performance on different con- ditioning. Selection is determined based on the lowest average constraint violation (capturing angles and bond lengths between the peptide ligand and the sampled generations). After completing this search, this center of mass is used for all runs the particular motif.
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Constraint-Guider Diffusion. Adopting the Sequential Monte Carlo conditioning proposed by Lee et al. (2025), we set the sample weighting via the constraint violation of the clean predicted state Ti-- For the evaluation, we fix the number of particles P 200, this seems to balance po- tential performance and overall runtime. We adopt multinomial resampling function and introduce an inverse temperature parameter, B, which controls the sharpness of the resampling distribution. Candidate weights are computed as
Wi O exp (-8 (c: min 8,)
where c denotes the score of constraint violation i. Earlier in the diffusion process, B = 30 remains high, keeping the resampling fairly stochastic, but we lower this 3 as - 0 to improve selection for the final state and increase the likelihood of non-covalent bonds forming.
Constrained Diffusion (Ours). As described in Section our projection is implemented through an ADMM derived decomposition. We assign y be consecutive residues which fall between the start and end of the bonding points on the peptide ligand. For instance, if the peptide has six covalent bonding sites, y composed of five residues. In this case, O-N and N-O bonds are placed on the first, third, and fifth residues. Then, is composed of the linker preceding (typically six residues), and the chain continuation following y (typically 60-100 residues). Additionally, we note that our approach leverages constraint guidance for this experiment, as our approach is complementary to these guidance methods.
B.2 MOLECULE ENCAPSULATION VACANCY CONSTRAINTS
For this setting, we evaluate across 1000 samples for each of the baselines. We use a fixed box of size 20 A x 40 x 40 Å, centering the apex of the cone Å above the bottom face of the box, with a half-angle of 25% As we identify that many baseline samples violate only the width constraints of the box (40 Ax 40 Å), which is not integral to the application, we do not report these violations in our evaluation provided in Figure 2
Recenter of Mass Guidance. The center of mass is positioned the center of the box, encour- aging the sample to (1) remain inside the box and (2) maximize contact with the cone shaped vacancy. Similar to the previous setting, the guidance is tuned small set of sample prior to generating outputs for evaluation; however, for this experiment the center mass fixed during tuning, only searching over the guidance strength parameter.
Constraint-Guide Diffusion. We adopt an identical setup for the constraint guidance as de- scribed for the PDZ domain, but leveraging identical constraint violation measure:
g(x) = inf ||
y1
yeC
which captures the sum of the distance of each atom from the feasible region. We similarly adopt a particle count of p 200, and follow an identical schedule to the previous experiment for the inverse temperature.
Constrained Diffusion (Ours). In this setting, all atomic positions are subject to the global con- straints, making y x, removing the need for explicit variable splitting in this setting. While we note that the vacancy constraint is indeed nonconvex, the projection operator can be represented in closed form, as it is composed of simple convex shapes; this alleviates concerns surrounding the tractability of the projection operator, which is the primary motivation for decoupling global and local requirements. To preserve local geometric properties and secondary structures, we extend this operator by enforcing rigid-body consistency secondary structure segments, while adjoining linker regions are adjusted to absorb residual displacements. This construction ensures that the pro- jection both satisfies global constraints and maintains local stereochemical integrity. Additionally, we note that our approach leverages constraint guidance and recentering of mass for this experiment, our approach is complementary these guidance methods.
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Additional Results. To supplement the results reported in the paper, we include additional anal- ysis of the secondary structure preservation rates for samples generated by each baseline and by method. This metric quantifies the percentage of residues assigned to canonical secondary structure state in the relaxed models. Higher values indicate improved preservation of fold-level features.
RFDiffusion
Ours
Standard
Recenter
CGD
Secondary Structure (%)
80.8
82.5
83.5
85.0
Table Secondary structure preservation across design methods.
In addition to the primary metrics reported in the main text, we assess the structural plausibility of generated designs using a secondary structure preservation metric. This measure assessed using PyRosetta to determine the percentage of residues which fall in helix, sheet, or turn conformations in the relaxed structures (Chaudhury et al. 2010). As shown Table our method achieves the high- est secondary structure frequency, suggesting that the model maintains global fold characteristics more reliably than the baselines.
C DATASET CONSTRUCTION
Cutoff: 90
350
Cutoff: 0
150
300
Count
200
50
PLDOT
Content
Meen: 1.40
Mean: 1.25
Outoff: 2.5
Cutoff:
Court
2.00
2.3
Self-Consistency RMSD
Peptide RMSD
Figure 4: Dataset filtering details expanded.
Details for the filtering are included Figure For every candidate structure per target PDZ-PBM loop closure, AlphaFold models were constructed. To determine the confidence of each model, predicted Local Distance Difference Test (PLDDTT) was collect. The cutoff for PLDDT for a viable loop closure was set to >90 (top left). The former N- and C- termini form a 3-sheet pairing, to conserve this, a filter >60% the residues being 3-sheet was used (top right). ProteinMPNN was used to generate potential sequences that conform to generated backbones for the in-painted region. After having AlphaFold predict the structure given the new sequence, RMSD was measured between the generated backbone and the predicted structure. A cutoff of <2.5A was used (bottom left). To ensure the peptide was not interfered with after predicting the new structure with AlphaFold, stringent cutoff of <1 RMSD was used (bottom right)
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D COMPARISON TO PROJECTED DIFFUSION MODELS
As discussed in Section prior work on constrained generative sampling has proposed Projected Diffusion Models @Christopher et al. 2024). These models augment the sampling process by im- posing projection onto the feasible set after each diffusion step. While theoretical guarantees have been derived for convex constraint sets, and empirical observations have illustrated that these prop- erties hold for certain nonconvex sets, the paper has argued critical limitations for kep-then-projed approaches when applied to challenging settings as explored in this work. This section presents empirical justification of these aforementioned claims.
Projected Diffusion Model
Ours
Local Global
Global
Constraint Satisfaction (%)
0.0 (Failed to Converge)
100.0
100.0
Structure Realism (%)
N/A
0.0
97.8
Usable Percentage (%)
0.0
97.8
Radius of Gyration (A)
N/A
(13.1)
14.8
Diversity (%)
N/A
N/A
97.8
Table 5: Comparison to Projected Diffusion Models on the Molecule Encapsulation setting.
The results in Table 5 reaffirm the limitations of integrating intermediate projection steps when working with high-dimensional data subject complex nonconvex constraints. While this setting it was possible to exactly satisfy the global constraints, the resulting structures are unusable, with high concentrations of atoms placed on the edges of the box and cone boundaries. When also enforcing the local constraints explicitly, the projection operator never converged on any samples or timesteps.
E HYPERPARAMETER SENSITIVITY ANALYSIS
To assess the robustness of our method to hyperparameter choices, we performed two ablation stud- ies: (1) varying constraint weight schedule X, and (2) varying the number of ADMM iterations per diffusion step. All experiments were conducted on the PDZ domain design target 6ubh, a well-posed example from our benchmark set suited constrained generation.
À
10
À
100
1000
= inf
Constraint Satisfaction (%)
100.0
100.0
100.0
100.0
Structure Realism (%)
80.0
100.0
100.0
90.0
Usable Percentage (%)
80.0
100.0
100.0
90.0
Radius of Gyration (Å)
10.58
10.53
10.56
10.34
Diversity (%)
80.0
100.0
100.0
90.0
Table 6: Ablation over constraint-weight schedules À on PDZ domain sample 6ubh.
As illustrated by T increasing the initial weight of \T yields improvements in structural real- ism and usable sample percentage. This trend aligns with the interpretation that stricter constraint enforcement at early timesteps guides the diffusion trajectory toward the feasible manifold, improv- ing geometrie quality. However, increasing this too strictly, results in over constraining early states, analogous to placing an overly large weight on the diffusion-guidance terms as discussed in Section
The next analysis centers on determining the optimal number of ADMM iterations diffuson step. As described in Section 5 a single ADMM sweep per diffusion step is sufficient in practice. In fact, the ablation suggests that multiple iterations degrade quality. The failure case that emerges over additional ADMM iterations actually always tied to the formation of a 8-sheet across the binding sites; additional iterations do not result in local stercochemical properties breaking, but the analysis suggests that they do lower the liklihood of secondary structure conformations. This is because higher wieght is placed the ADMM optimization than the diffusion updates, leading
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ADMM
ADMM =
ADMM
Constraint Satisfaction (%)
100.0
100.0
100.0
Structure Realism (%)
80.0
70.0
40.7
Usable Percentage (%)
80.0
70.0
40.7
Radius of Gyration (A)
10.58
10.72
11.24
Diversity (%)
80.0
70.0
40.7
Table 7: Ablation on the number of ADMM iterations per diffusion step.
feasible generations but not ones which fall closely the data manifold. Furthermore, note that additional ADMM steps also become impractical because adding these iterations scales the runtime nearly linearly.
F RUNTIME
Benchmarks were run on single slice of an NVIDIA A100 Multi-Instance GPU (MIG) to effi- ciently parallelize sampling across available hardware. We posit that overall runtimes could be improved by scaling the available resources, but this is beyond the scope of this study; for instance, both constraint-guided diffusion and our approach could be significantly parallelized to increase efficiency, they operate up to P 200 particles. Note that reducing P will accelerate per- formance, providing a tunable trade-off between speed and performance. Hence, we expect that if resource constraints could be ignored, these methods could be accelerated by multiple orders of magnitude.
Standard
Recenter
CGD
Ours
PDZ Domain
Time Per Usable Sample (min)
935.2
Molecule Encquoaaation
Time Per Usable Sample (min)
142.4
98..
Table Reported runtimes on both settings. Analysis considers the average time per usable sample.
Our method provides the best runtimes per usable sample. In protein design experiments, this is more important metric than raw runtime, as only usable samples are relevant when considering over- all performance. Hence, under the assumption that functional requirements are necessary, which is specifically the applications this work targets, the proposed method is indeed the most computa- tionally efficient!
G ADMM
This section expands the discussion of the ADMM formulation introduced in Section Begin by recalling the consensus formulation our proximal update:
proxp+o (y. == arg min F(y) + G(z) 3.z
s.t. =
where F includes the data anchor term and local fcasibility requirements, and encodes global fcasibility.
+ Ae dist Creat (2)2
F(y)
|1
2/612
27)
A
G(z) = 2 distcrlobal (z)2
Recall that the consensus initialization y 00 = 21 E R3xn, which aligns the start of the iterates with the denoiser's prediction.
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Classical ADMM results (see Parikh al. 2014) guarantee convergence to minimizer of F + G under convexity assumptions. In our case, convexity does not strictly hold, but in practice this algorithm is often effectively applied to nonconvex settings, achieving feasible solutions.
The formulation yields the augmented Lagrangian, introducing the dual variable u and a penalty parameter p
Cy z, u) Fly) + G(z) u 2) +
z|1?
Then, expanding the ADMM updates introduced in Equation yields:
++1 arg min F(y) + _ y x u 11
y
*+1 arg min G(z) + 5 Ely + 211
u ++1 u k+1 =k11
These updates are equivalent the proximal splitting form presented in Section 5 with prox, realized by the y-update and proxg by the z-update. The dual variable u accumulates the resid- ual mismatch between the two copies, driving them toward consensus. convergence, y 2, recovering the minimizer of F + G.
H COMPLETE PROOFS
Proof of Theorem 6.1
Claim: distc(
diste c (#t)
Vhene
Lemma H.1. Let d(z)
1
A distc(=)?. By optimality of ay o(ā)) S (IIc(())).
2m
Lemma holds as the minimizer's objective value can't 't exceed the value at any feasible point, in particular at the projection of a-j
Then, expanding from the lemma:
1at6 #b + 2 dist < 2m
(1()(())) - *bol + 2 diste (#)c)))))
Since b dist (#f)) this term can then be omitted.
|
+ + A dist c
<
1 [Tc 1/6
27p
2n..
Dropping the non-negative first term gives:
distc S
1 ([Tc((6)) 2b
2m
or equivalently
diste( S vhem
1 diste (26b)
Proof of Theorem 6.2
Claim: B[ddistc(zo)]] < K
20,
Begin by assuming E (istt((f) < K, and the schedule is defined such that X Ct
By Theorem 6.1 it is known that
diste (x() < V2 distc((())
and thus, the squared expectation
TP. [dist ( 2 2xem E(diste(
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Then, by our assumption,
E diste (xb)²
distc(a)))
$ 2Amme
K 27
Substituting our schedule X
n yields:
E distt (idistc(#6)))) - 2cr
By applying Corollary 6.2 time
dissee (zx) )
201
Thus, if the schedule of is decreasing, the expected violation converges as - 0.
Proof of Theorem 6.3 Claim: Existence of a local solution for the proximal mapping.
Existence: ng is continuous and coercive, hence attains a global minimizer for every t/g. In particular, arg min prex * Ø.
First, assume Me, d O, C 3 nonempty, closed. For ease of notation:
(x(x)) = 2np x atl + d
dist)(
Since the distance from the constraint is strictly non-negative.
(xx)
2m. || 1
1
2m ||=|| + 2m
1
1 1 (xc 20) + 2m
|
Since the term ||>|||
oc, + is t..
also lower semicontinuous since (i) the quadratic anchor term 2n, x - lf1l is continuous, and (ii) the squared distance to the closed set is lower semicontinuous and finite everywhere.
Since t is coercive and lower semicontinuous, it obtains global minimum.
arg min 4 + Ø
Local uniqueness. If ih lies within the prox-regularity neighborhood of C. then the projection Ic(āg) is single-valued. Moreover, if prox,.g is strongly convex in neighborhood of II (1)). then the proximal minimizer is unique within that neighborhood.
Now, assume C is prox-regular all points in neighborhood X of xx [e((()) Prox-regularity then implies:
Single valued projection. The projection Ie(()) is single valued and Lipschitz the neighborhood 2/ C X.
Smooth distance function. Thus, distc(-)² is C on U with gradient:
V ( ddist(((())) = x [T())))
x Eu.
Let be small enough that the above holds and at E U. Consider d restricted U. gradient
2 = _ x 2b)) + (e lc(())),
x EEU,
and its (generalized) Hessian can be bounded below using the Jacobian of IIc. particular, since Ic is nonexpansive in UI,
(x) v)
|e + Xe((T (TTc()))))
%e ||
n
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for all and almost every æ E U. Thus 4, is locally p-strongly convex on U with μ 1/me 0.
Local strong convexity implics a unique critical point in 24, hence a unique local minimizer of d in U. Since 2, is coercive with at least one global minimizer, and Φ is in the prox-regularity neighborhood so the minimizer lies in U, the proxima minimizer unique in that neighbor- hood.
20


Reviews:
[{"review_id": "3Mk5MAGsvl", "reviewer": "Reviewer_muax", "summary": "This paper introduces a framework for incorporating hard structural constraints into diffusion-based protein design. The authors identify that existing guidance methods often fail to satisfy precise, non-convex constraints required for functional design. They propose viewing the reverse diffusion process through the lens of proximal optimization. Their empirical results on PDZ scaffolding and molecule encapsulation show state-of-the-art constraint satisfaction (near 100% vs. about 0% for the baselines).", "questions": "- How sensitive is the proposed approach to the schedule $\\lambda$, the number of iterations, etc.?\n- How does the proposed method perform compared to the method in Christopher et al. (2024)?\n- Did you try a baseline projecting $x_t$ directly? Did it fail as expected?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The overall idea of the proposed method is creative and explores a new direction for diffusion models. I see the main strengths of the paper as follows:\n- strong results: the 100% satisfaction on PDZ vs 0% for strong baselines appears meaningful\n- method: framing the reverse step as proximal optimization is a neat idea, and applying the correction to the estimated $\\hat{x}_0$ instead of the noisy $x_t$ is well-motivated\n- benchmark: the PDZ benchmark designed for the evaluation seems like a meaningful contribution in its own right", "weaknesses": "I believe there are a few minor weaknesses:\n- baselines: while RFDiffusion is a good baseline, the authors should compare to the method in Christopher et al. (2024) [1] given the similarity between the proposed methods.\n- the paper does not present any sensitivity studies for hyperparameters. How sensitive is the proposed approach to the schedule $\\lambda$, the number of ADMM iterations, etc.?\n\n[1]  Christopher et al., Constrained Synthesis with Projected Diffusion Models, 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2cjdTQIqZP", "reviewer": "Reviewer_ausW", "summary": "This paper presents a constrained diffusion framework for precision engineering of detailed molecular features in proteins. Authors validate their approach in silico on a contributed tasks and novel curated benchmark datasets.", "questions": "See weaknesses for potential discussion points.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- This is definitely an interesting and strong contribution focusing on a slightly neglected aspect that is however of very high practical & theoretical relevance in precision engineering of protein architectures & binding sites.\n- The proposed solution is benchmarked against convincing baselines and authors report significant improvements.\n- Code & benchmark data is released.", "weaknesses": "- I feel a benchmark is scarce, it’s a relevant problem but I would welcome an effort to introduce more interesting cases. There are many problems requiring precision engineering that could show that the method is robust to different cases. E.g. one widely recognised problem used to study precision protein engineering for long time is parametric protein design with coiled coil bundles.\n- While the technical / implementation aspects of the method are properly described I think the formulation and description of the proposed benchmark dataset is less understandable (I suppose even for struct/bio readearship, not mentioning purely ML audience). I suggest to rewrite this section so it reads more clear (also minor thing - the PBM is never defined in the paper).\n- Finally, I believe being able to come up with backbones that satisfy the complex geometric criteria is an important contribution I have one follow up q to the authors - are these backbones designable? I miss the experiment that would prove (in silico would naturally be enough given the venue) that indeed we can design compatible sequences that will fold into structures that are predicted by the authors e.g. with the ProteinMPNN / AF2 pipeline that authors already set up.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "X5E8Bavmkt", "reviewer": "Reviewer_TyWr", "summary": "This paper presents a constrained diffusion framework for protein design, aiming to enforce hard structural and functional constraints.1 The core method is a \"predict-prox-renoise\" stochastic proximal method, which applies feasibility corrections to the predicted clean state $\\hat{x}_{0}^{t}$ rather than the noisy state $x_t$.1 The authors also propose an ADMM decomposition to decouple local and global constraints.1 The method is evaluated on two tasks: motif scaffolding (using a newly introduced PDZ benchmark) and vacancy-constrained pocket design, claiming 100% constraint satisfaction rates where baselines achieve 0%.", "questions": "see weakness", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Novel and well-motivated \"predict-prox-renoise\" method for hard constraints.   \n\n(Superficially) perfect 100% constraint satisfaction on complex tasks.   \n\nContributes a new, curated benchmark dataset for PDZ motif scaffolding.", "weaknesses": "Fails to cite or compare against any true SOTA competitors in motif scaffolding (e.g., Genie 2 , OriginFlow ), invalidating its SOTA claim.   \n\nThe \"Ours\" method in Exp 2 is a combination of the new method and the baselines , making results impossible to attribute.   \n\nThe ADMM method  is not general , and the theory  does not apply to the problem.   \n\n 0% success for all baselines is not credible.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "onnwgXu9Yk", "reviewer": "Reviewer_dnLp", "summary": "This author proposed a method to solve a gap in protein backbone generation: enforcing hard functional and stereochemical constraints during diffusion-based design. The authors rproposed emthod use reverse diffusion with a predict–prox–renoise procedure: (i) predict a clean structure, (ii) apply a proximal feasibility update (with a Moreau–envelope penalty that tightens over time), and (iii) re-noise to stay on the data manifold. They further decouple local stereochemistry from global topology via a consensus ADMM split, enabling separate proximal updates that are warm-started across steps. Theoretical results bound constraint violations and motivate schedules for the penalty and trust parameters. Empirically, the method is evaluated on (1) a new PDZ motif-scaffolding benchmark and (2) vacancy-constrained pocket design with box/void geometry. Results claim perfect constraint satisfaction and strong usable rates vs. RFdiffusion baselines and constraint-guided SMC, while maintaining structural diversity and reasonable compactness.", "questions": "1. Beyond PDZ scaffolding and geometric vacancy control, which biologically grounded constraints (e.g., catalytic triad geometry, specific H-bond networks) most motivate hard-constraint enforcement, and how will the framework extend to them?\n2. Can you quantify how often unconstrained RFdiffusion fails due to global vs. local constraint violations in real pipelines, to better motivate where hard constraints matter most?\n3. How does enforcing exact feasibility impact downstream functional success (e.g., binding affinity predictions) compared with soft guidance—any surrogate metrics supporting the motivation? \n4. The Moreau-envelope penalty transitions toward hard constraints as λ→∞. How sensitive are outcomes to the λt and ηt schedules, and do you provide heuristics or automatic tuning beyond the variance-linked choice?\n5. In the ADMM split, how do you select which residues belong to the “local” vs. “global” blocks in more complex topologies, and what is the computational cost per sweep as structure size grows?\n6. On the PDZ benchmark, your method attains 21% usable overall (up to 83% for favorable cases). What characteristics define “well-posed” ligands, and how general is that 83% across targets?\n7. In the vacancy-constraint experiment, could you provide additional quality metrics (e.g., secondary-structure preservation per segment, clash scores) to complement constraint satisfaction and radius of gyration?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The predict–prox–renoise design is simple, modular, and theoretically grounded; the scheduling guidance (λ↑ as σ↓; η tied to diffusion variance) is clear and actionable. ADMM splitting of local vs. global constraints is well-motivated for proteins (stereochemistry vs. long-range couplings) and yields a practical proximal scheme with consensus guarantees under mild conditions. Theorems quantify contraction of constraint violations and give scheduling rules that drive terminal feasibility, aligning with the method’s design. On the PDZ benchmark, baselines produce no usable designs whereas the proposed method yields 21% usable (up to 83% for well-posed ligands) with perfect constraint satisfaction and better diversity/compactness; similarly strong wins appear in the vacancy-constraint task.", "weaknesses": "Feasibility guarantees rely on prox-regularity; highly nonconvex sets common in protein design may violate this, limiting formal guarantees. Proximal/ADMM steps add overhead and may require careful tuning of λ/η/ρ schedules; guidance on sensitivity and robustness across architectures and tasks could be expanded. Results are strong on PDZ and a geometric vacancy constraint, but broader functional constraints (e.g., catalytic geometry, interface polarity, multi-chain assemblies) and end-to-end sequence design with wet-lab validation remain future work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)