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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling

T0 review · 2 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read CPSDE is a generative pipeline that designs cyclic peptides of all four linkage types directly from a receptor's 3D structure, using an all-atom harmonic SDE with explicit bond modeling and alternating sequence-structure sampling.

desk verdict Novel framework for cyclic peptide design that overclaims direct chemical validity; the raw geometry is never validated and the score approximation is off, but the core idea is worth engaging. read the letter →

arxiv 2505.21452 v1 pith:FJESIL4M submitted 2025-05-27 cs.LG q-bio.BM

Xiangxin Zhou , Mingyu Li , Yi Xiao , Jiahan Li , Dongyu Xue , Zaixiang Zheng , Jianzhu Ma , Quanquan Gu This is my paper
classification cs.LGq-bio.BM
keywords cyclicpeptidedesignharmonicSDEall-atomgenerativemodelatom-bondmodelingsequence-structureco-designroutedsamplingprotein-liganddockingdrug
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces CPSDE, a generative method that designs cyclic peptide ligands by sampling sequence and all-atom 3D structure together, conditioned on a protein target's structure and a user-chosen cyclization chemistry. Existing generative peptide models target linear peptides, and earlier cyclic-peptide efforts cover only one cyclization type, so the paper's aim is to show that one pipeline can cover head-to-tail, head-to-side, side-to-tail, and side-to-side cyclizations. The authors argue that modeling every heavy atom and chemical bond explicitly lets the model reuse abundant linear-peptide and small-molecule complex data instead of requiring scarce cyclic-peptide structures. Their experiments report favorable Rosetta stability and affinity scores for all four types, with two case studies showing stable binding in 100 ns molecular dynamics simulations.

What carries the argument

The load-bearing object is the harmonic SDE built from the ligand chemical graph, with $H = L + \sigma_P^{-2} I$ (graph Laplacian plus a receptor-dependent scalar term). Its forward process $dx_L = -\tfrac12 \beta(t) x_L \, dt + \sqrt{\beta(t)} \Lambda^{1/2} P^\top dw$ has the closed-form perturbation kernel $\mathcal{N}(x_t; x_0 e^{-\frac12 \int_0^t \beta(s)\,ds}, H - H e^{-\int_0^t \beta(s)\,ds})$ and prior $\mathcal{N}(0, H)$, which places bonded atoms close together throughout generation. Around this, the atom73 representation stores each residue as a superposition of possible side chains, and bond edges make the cyclization constraints explicit; routed sampling alternates ATOM SDE denoising with ResRouter residue prediction and rebuilds the full chemical graph at each step.

What would settle it

Take a fixed set of test targets, run CPSDE, and measure the raw (pre-relaxation) cyclization bond lengths and bond angles against crystallographic reference ranges, such as the amide C-N bond near 1.33 Å or disulfide S-S near 2.0-2.5 Å, while counting atomic clashes with the receptor; if a large fraction of raw outputs violates these ranges, the claim that explicit bond modeling enforces cyclization geometry would be refuted.

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Extended reading notes

Core claim

CPSDE casts cyclic peptide design as generating the conditional distribution $P(P\mid T, C)$: the peptide's all-atom structure $P$ given the receptor structure $T$ and the cyclization chemical graph $C$, which specifies the atoms and bonds that close the loop. The structure model ATOM SDE is trained as a harmonic SDE whose noise prior and perturbation kernel are governed by the ligand graph Laplacian $H = L + \sigma_P^{-2} I$, so bonded atoms remain spatially correlated during denoising; the residue predictor ResRouter reads denoised structures and proposes amino acid types. Routed sampling alternates the two models, keeping an atom73 superposition state for free residues and a fixed state for cyclization-constrained atoms, and reassembles the chemical graph at every step. The paper demonstrates that the resulting cyclic peptides match their intended cyclization graphs and reports that best-of-batch designs outperform linear-peptide baselines on Rosetta affinity and diversity, with MD simulations supporting two designed inhibitors.

Load-bearing premise

The load-bearing premise is that a denoising model trained mostly on linear peptides and small molecules will, when conditioned on a cyclization chemical graph during sampling, produce closure bonds with correct lengths and angles and no receptor clashes; the paper's own Appendix J concedes this sometimes fails, reporting inaccurate bond lengths and atomic receptor clashes in generated cyclic peptides.

Editorial extensions

If this is right

  • Cyclization constraints can be supplied as chemical graphs rather than learned from rare cyclic-peptide structures, so the method can in principle be retargeted to new cyclization chemistries without new training data.
  • A single generative pipeline can enumerate all four cyclization types for a target, letting a design campaign compare head-to-tail, head-to-side, side-to-tail, and side-to-side candidates from the same model.
  • Best-of-batch CPSDE designs match or beat the PepFlow linear baseline on Rosetta affinity (mixed set -55.71 vs -47.88 kcal/mol) while keeping diversity around 0.79, suggesting the cyclic outputs are not merely collapsed to one motif.
  • The two MD case studies (100 ns, two repeats each) show the designed cyclic peptides with lower backbone RMSD and more negative MM-PBSA binding free energies than the reference linear peptides, supporting the authors' stability-and-affinity claim.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: since the cyclization chemical graph already encodes arbitrary bonds, the same sampling procedure should extend to polycyclic peptides and non-canonical residues in the loop, although the paper does not generate those cases.
  • Editorial inference: the current protocol requires the user to enumerate cyclization types by hand; a natural extension is a classifier that selects the best cyclization type from pocket geometry, which would turn the method into a one-shot design tool.
  • Editorial inference: the validity rate of raw, unrelaxed outputs is the metric that determines how much of chemical validity the model itself supplies; the Appendix J caveat suggests that the practical pipeline should budget for Rosetta relaxation or enforce bond geometry in post-processing.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper introduces CPSDE, a generative model for target-aware cyclic peptide design. The method combines AtomSDE, a harmonic SDE trained on protein–small-molecule and linear-peptide complexes with an all-atom and bond-level representation, with ResRouter, a residue-type predictor. A "routed sampling" algorithm alternates between denoising the all-atom structure and updating residue types, conditioned on a user-supplied cyclization chemical graph, to generate head-to-tail, head-to-side, side-to-tail, and side-to-side cyclic peptides. The authors evaluate their method with Rosetta stability and affinity scores on 100 receptor pockets, compare with linear-peptide baselines, and provide two case studies with MD simulations. The central claim is that CPSDE is the first generative algorithm capable of directly generating chemically valid cyclic peptides of all four cyclization types informed by a protein target's 3D structure.

Significance. If the central claim holds, the work would be a meaningful advance in computational peptide design, addressing a real gap: most generative peptide models handle linear peptides only, while the four cyclization types and non-canonical bonds are largely unsupported. The harmonic SDE with a graph-Laplacian-based anisotropic covariance is a principled way to inject bond information, and the curated training datasets are a useful resource. The paper also provides honest limitations in Appendix J. However, the significance is currently tempered by two issues: the score approximation in Section 3.2 appears inconsistent with the harmonic kernel, and the paper does not validate the raw generated cyclization geometry, relying instead on Rosetta-relaxed energies. Because the headline contribution is "direct generation of chemically valid cyclic peptides," these points are load-bearing.

major comments (2)
  1. [Section 3.2] The estimated score function displayed after Eq. (4) reads ∇_x log p_t(x) ≈ −(1/√(1−∫_0^s β(u)du))(x − D_θ(x,t)). However, the harmonic perturbation kernel in Eq. (3) and Appendix F has covariance Σ(t) = H − H e^{−∫_0^t β(s)ds} = H(1−e^{−∫_0^t β(s)ds}). For a Gaussian with this covariance, the score is −Σ(t)^{-1}(x − mean) = −H^{-1}(1−e^{−∫β})^{-1}(x − mean). The formula in the paper omits H^{-1} and uses a square root in the denominator; as written it is dimensionally inconsistent with the anisotropic covariance and does not match the derived kernel. The authors should correct this formula or clearly specify the parameterization used in the implementation (e.g., if D_θ predicts noise rather than x_0, the reconstruction loss in Eq. (4) would need to be changed accordingly). Since the reverse SDE in Eq. (2) is the core generative mechanism, this is a load-bearing technical point.
  2. [Appendix J and Section 3.4] The paper's central claim is that CPSDE "directly generates chemically valid cyclic peptides" (Section 1, Figure 2). The training data consist of small-molecule and linear-peptide complexes (Section 4.1), the training objective in Eq. (4) is a per-atom reconstruction loss with no term enforcing the geometry of the newly created cyclization bond, and Algorithm 1 contains no projection step that enforces bond lengths, bond angles, or clash-free ring closure. Appendix J explicitly states that "the generated cyclic peptides may sometimes exhibit invalid conformations, such as inaccurate bond lengths and atomic receptor clashes." All main-table results (Table 1 and Appendix H.1) are reported after Rosetta FastRelax and InterfaceAnalyzer, which can repair poor geometry, so favorable energies do not demonstrate that the raw generated structures satisfy cyclization constraints. To support the headline contribution, the authors should report, on the raw (pre-relaxation) structures, the distributions or pass rates of: (i) the length of the newly formed bond versus the expected value (e.g., 2.0–2.5 Å for disulfide, ~1.3–1.5 Å for amide/ester bonds), (ii) bond angles at the cyclization atoms, (iii) steric clash counts, and (iv) ring-closure distances. Without such evidence, the method is better described as a sequence/coarse-pose generator requiring post-processing, not a direct generator of chemically valid cyclic peptides.
minor comments (7)
  1. [Section 3.1] In the sentence "and and dt is an infinitesimal negative timestep," the second "and" appears to be a typo; it should be "where dt is an infinitesimal negative timestep."
  2. [Eq. (3) and Appendix F] Equation (3) uses the notation \tilde{x}_L in the drift term, while the corresponding SDE in Appendix F is written with x_L. Please unify the notation.
  3. [Section 3.2] The integral notation "∫_0^s β(s)ds" is confusing because the integration variable and the upper limit share the symbol s; please use a different dummy variable (e.g., ∫_0^t β(u)du).
  4. [Table 3] The header "HydrophobicCharged" appears to combine two separate metrics; it should be split into "Hydrophobic Ratio" and "Charged Ratio."
  5. [Algorithm 1] The function name "Supgraph" in line 15 is likely a typo for "Subgraph," matching the text in Section 3.3.
  6. [Appendix G.1] The definition of e_{ij} contains a parenthesis error: it should read e_{ij} ← ϕ_B(∥x_i − x_j∥, b_{ij}) rather than ϕ_B(∥x_i − x_j, b_{ij}\|).
  7. [Section 4.2] The sentence "Among all co-design methods, our method exhibits superior energy performance" is correct only after excluding RFDiffusion, which is marked as not a co-design method; consider making this exclusion explicit to avoid confusion with the later statement that RFDiffusion has the best energy.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generative pipeline is trained on independent complex data and evaluated with external Rosetta scores, and the cyclization graph is an explicit conditioning input rather than a fitted prediction.

full rationale

ATOMSDE is trained to minimize a per-atom reconstruction loss (Eq. 4) on PDBBind and peptide-complex data, and RESROUTER is trained with a cross-entropy residue-type loss (Eq. 5) on peptide data; neither Rosetta's REF2015/InterfaceAnalyzer energies nor the 100 test pockets enter any training objective. The harmonic-SDE prior is adopted from external prior work (Jing et al. 2023; Stark et al. 2023), and the atom73 state from external Protpardelle work (Chu et al. 2024). Self-citations (Chen et al. 2025; Zhou et al. 2024a,b,c,d; Cheng et al. 2024; Ye et al. 2024) appear only in related work, metric definitions, or future-work suggestions and do not carry the central derivation. The cyclization chemical graph C is an explicitly given conditioning input, and Algorithm 1 assembles the output graph from that input; this makes the cyclization 'type' an input label rather than a predicted quantity, but the method's actual output—all-atom coordinates—is not equated by construction with any training target or fitted parameter. Appendix J's admission of occasionally invalid bond lengths and receptor clashes is a real validity gap for the 'chemically valid' phrasing of the contribution, but it is a correctness/soundness concern, not a circularity: the paper does not fit a parameter to the claim and then report that fit as a prediction. No load-bearing self-citation, uniqueness import, or renamed known result was found.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the transfer of a linear and small-molecule-trained model to cyclic topologies, on hand-chosen diffusion hyperparameters, and on computational energy functions as the only evaluation. No new physical entities are introduced. These are the main paid-in assumptions.

free parameters (2)
  • sigma_P (receptor-dependent scalar in harmonic prior) = not specified
    Defines H = L + sigma_P^{-2} I in Section 3.2 and controls the prior coupling of ligand atoms to the receptor. No estimation procedure or value is given; it is a hand-chosen or fit parameter affecting the sampled structures.
  • Noise schedule beta(t) endpoints (beta_min, beta_max) = 0.01, 3.0
    Linear schedule beta(t) = (beta_max - beta_min) t + beta_min chosen in Appendix G.2; affects the diffusion dynamics and sampling trajectories.
assumptions (4)
  • domain assumption SE(3)-equivariant GNN message passing over kNN and chemical-bond graphs captures all interactions needed for correct cyclic peptide placement
    Used throughout Sections 3.2 and G.1 as the backbone architecture; if this representation is insufficient, the predicted structures will be invalid.
  • domain assumption Conditioning the denoiser on the cyclization chemical graph C during reverse sampling is sufficient to enforce a cyclic topology
    The core hypothesis of the routed sampling algorithm in Section 3.4; the paper provides only qualitative post-hoc alignment and admits invalid conformations in Appendix J.
  • domain assumption Rosetta REF2015 total energy and interface energy are valid proxies for peptide stability and binding affinity
    Used in Section 4.1 and Appendix H.1 to select and evaluate all designs; no experimental binding data are presented.
  • domain assumption The atom73 superposition representation from Protpardelle is adequate for side-chain states in free residues
    Borrowed in Section 3.4 and Appendix C from Chu et al. (2024); the paper relies on it without independent validation for cyclic contexts.

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Cite this review

Pith. "Pith review of Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling." pith.science (2026). https://pith.science/paper/FJESIL4M

@misc{pith2026250521452,
  author       = {Pith},
  title        = {Pith review of: Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJESIL4M}},
  note         = {Machine review of arXiv:2505.21452}
}
read the original abstract

Cyclic peptides offer inherent advantages in pharmaceuticals. For example, cyclic peptides are more resistant to enzymatic hydrolysis compared to linear peptides and usually exhibit excellent stability and affinity. Although deep generative models have achieved great success in linear peptide design, several challenges prevent the development of computational methods for designing diverse types of cyclic peptides. These challenges include the scarcity of 3D structural data on target proteins and associated cyclic peptide ligands, the geometric constraints that cyclization imposes, and the involvement of non-canonical amino acids in cyclization. To address the above challenges, we introduce CpSDE, which consists of two key components: AtomSDE, a generative structure prediction model based on harmonic SDE, and ResRouter, a residue type predictor. Utilizing a routed sampling algorithm that alternates between these two models to iteratively update sequences and structures, CpSDE facilitates the generation of cyclic peptides. By employing explicit all-atom and bond modeling, CpSDE overcomes existing data limitations and is proficient in designing a wide variety of cyclic peptides. Our experimental results demonstrate that the cyclic peptides designed by our method exhibit reliable stability and affinity.

Figures

Figures reproduced from arXiv: 2505.21452 by the authors.

Figure 1
Figure 1. Comparative advantages of cyclic peptides over linear peptides. Linear peptides are easily degraded, whereas cyclic peptides are protected against enzyme hydrolysis, allowing them to function more effectively within the human body. Cyclic peptides generally exhibit better stability and affinity. 1 Introduction Therapeutic peptides are a distinct group of pharmaceutical compounds comprising a sequence of precisely ar… view at source ↗
Figure 2
Figure 2. Overview of CPSDE. The generative process is structured as follows: (1) A cyclization type is initially selected, which subsequently determines the associated 2D chemical graph; (2) At time t, given the entire chemical graph defined by both cyclization (highlighted with a yellow shadow) and the predicted residue types, the all-atom structure is initially denoised using ATOMSDE and then re-noised in accordance with t… view at source ↗
Figure 3
Figure 3. Discovery of new SMYD2 cyclic peptide inhibitors via applications of CPSDE. Upper: Visualization of conformational ensembles for the designed peptides sampled by MD. Bottom: RMSD analysis of all heavy atoms within the designed peptides. Design of SMYD2 peptide inhibitors via head-to-tail cy￾clization. SMYD2 is an oncogene that critically regulates tumor-related signaling pathways, making it an attractive target for … view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Discovery of new SET8 cyclic peptide inhibitors via applications of CPSDE. Upper: Visualization of conformational ensembles for the designed peptides sampled by MD. Bottom: RMSD analysis of all heavy atoms within the designed peptides. Herein, CPSDE was employed to des…
Figure 5
Figure 5. Figure 5: shows two cyclic peptide drugs approved in recent 20 years. According to Sharma et al. (2023); Fang et al. (2024); Costa et al. (2023), cyclic peptides can be classified by their cyclization strategies: head-to-tail cyclization (between N￾and C-termini); side-to-side c…
Figure 6
Figure 6. Figure 6: Examples of 3D structures of cyclic peptides and their chemical graphs. The cyclization structures are highlighted in pink. The corresponding cyclization types are: 5I2I E: head-to-tail; 3AV9 X: head-to-tail; 5TXE C: head-to-side; 3QG6 C: side-to-tail; 1U91 C: side-to-…
Figure 7
Figure 7. Figure 7: Statistics on peptide and cyclic peptide lengths in our dataset [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Analysis of cyclic peptide distribution and the proportions of the five cyclization types. with multimodal generative protein language models. Cyclic Peptide Design. Although computational cyclic peptide design is an innovative research area, there are already recent s…
Figure 9
Figure 9. Figure 9: Visualization of reference ligand, linear peptides designed by PepFlow, and cyclic peptides designed by CPSDE. J Limitations and Future Work One limitation is that the generated cyclic peptides may sometimes exhibit invalid conformations, such as inaccurate bond length…
Figure 10
Figure 10. Figure 10: Head-to-tail cyclic peptides designed for target SMYD2. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]
Figure 11
Figure 11. Figure 11: Side-to-side cyclic peptides designed for target SET8. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]
Figure 12
Figure 12. Figure 12: Structure ensembles of SMYD2 from multiple perspectives. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]
Figure 13
Figure 13. Figure 13: Structure ensembles of SET8 from multiple perspectives. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]
Figure 14
Figure 14. Figure 14: Examples of reference peptides, linear peptides designed by PepFlow, and cyclic peptides designed by our method. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png]
Figure 15
Figure 15. Figure 15: Examples of reference peptides, linear peptides designed by PepFlow, and cyclic peptides designed by our method. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_15.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.