Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
Authors: Xiangxin Zhou
Mingyu Li
Yi Xiao
Jiahan Li
Dongyu Xue
Zaixiang Zheng
Jianzhu Ma
Quanquan Gu
Venue: ICML 2025 poster (2025)
Keywords: cyclic peptide
peptide design
SDE-based generative model
Primary area: applications

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.

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
Xiangxin Zhou 123 Mingyu Li
Yi Xiao Jiahan Li Dongyu Xue
Zaixiang Zheng'
Jianzhu Ma 46 Quanquan Gu'
Abstract
Linear peptides
Protease
Target
Cyclic peptides offer inherent advantages in phar- maceuticals. 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 pep- tide design, several challenges prevent the devel- opment of computational methods for designing diverse types cyclic peptides. These challenges include the scarcity of 3D structural data on tar- get proteins and associated eyclic 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: AT,MSDE, a generative structure prediction model based on harmonic SDE, and RESROUTER, a residue type predictor. Utilizing a routed sampling algorithm that alternates between these models 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 cyclic peptides. Our experimen- results demonstrate that the cyclic peptides designed by method exhibit reliable stability and affinity.
Higher sta Staloiitg
Decradation
Flexible
Function -
Higher affinity
Cyclic peptides
Protease
Rigid
Target
Redured
Enhanced
degradation
function
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 sequence of precisely arranged amino acids (Driggers al., 2008; Tsomaia, 2015; Zorzi et al., 2017). Peptide drugs tend to exhibit lower toxic- ity, enhanced biological activity, and high target specificity compared to small molecules, superior cellular permeability, lower cost, lower immunogenicity compared to antibody drugs (Driggers et al., 2008). However, conventional linear peptides suffer from a short half-life, limited stability, and susceptibility to hydrolase degradation (Tsomaia, 2015), re- stricting their therapeutic potential and broader use. Unlike linear peptides, shown in Figure 1, cyclic peptides chains of residues that typically form one or two closed loops, often incorporating non non-natural residues. For exam- ple, a single loop can be created by connecting the N- and C-termini with a peptide bond or linking two internal cys- teines through disulfide bond (Camarero & Muir, 1999; Giordanetto Kihlberg, 2014; Kale al., 2018). Such cy- clization enhances their resistance to digestive enzymes and enables them to bind protein surfaces with high affinity in more stable conformations. Traditional methods for discov- ering cyclic peptides involve chemical synthesis and high- throughput screening, both of which are labor-in intteniite and costly. This has led adopting silico approaches such as
Equal contribution BByteDance Seed (Work was done during Xiangxin's 's internship at ByteDance Seed.) "School of Artificial Intelligence, University of Chinese Academy of Sciences 'New Lahoratory of Pattern Recognition (NLPR), State Key Laboratory Multimodal Artificial Intelligence Systems (MAIS), Institute of Automation, Chinese Academy of Sciences (CASIA) "Institute for AI Industry Research, Tsinghua University 'School of Medicine, Shanghai Jiao Tong University "Department of Electronic Engi- neering, Tsinghua University. Correspondence Quanquan Gu <quanquan, gu bytodance.com>>
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
virtual screening (Zotchev al., 2006) and de novo design (Kawamura al., 2017; Peacock & Suga, 2021; Bhardwaj et al., 2022; Garcia Jimenez et al., 2023) to streamline the discovery process.
We introduce CPSDE, the first generative algorithm, our knowledge, capable of directly generating all types of cyclic peptides informed by the 3D structure of protein target, paving the way for developments in peptide- -aased drug discovery.
Cyclic peptides can adopt various structural forms depend- ing on how their amino acid residues are linked together based on different chemical bonds and geometric distances, for example, the distance between two cysteines forming a disulfide bond typically falls from 2.0 to 2.5 Å (Fass, 2012). These structures are classified into four categories (see Fig- ure 2) based on the atoms of residues that form the eyelic structure: (1) Head-to-tail cyclization. The N-terminus (head) of one amino acid forms peptide bond with the C- terminus (tail) of another amino acid, resulting in a closed ring. (2) Side-to-tail cyclization. The side chain of an amino acid is linked to the C-terminus (tail) of the peptide. (3) Head-to-side cyclization. The N-terminus (head) of one amino acid is linked to the side chain of another amino acid. (4) Side-to-side cyclization. The side chain of an amino acid is linked to the side chain of another, forming a cyclic structure that does not involve the head or tail of the peptide backbone. Therefore, incorporating specific chemical and geometric constraints (such as bond lengths, angles, and atom compositions) relating to one or more sets of residues is essential for designing different kinds of cyclic peptides. Recent attempts have tried designing disulfide-linker cyclic peptides based on post-processing strategy (Wang al., 2024a) or head-to-tail cyclic peptides (Rettie et al., 2024) us- ing modified position encoding in protein generative models. However, these methods only consider one specific type of cyclic peptides and do not support other types based on spe- cific constraints. Furthermore, the availability of real-world 3D structural data for protein-ligand complexes involving eyclic peptides is limited, posing challenge to advancing computational cyelic peptide design.
Our approach designs cyclic peptides with robust sta- bility and affinity while maintaining high diversity, un- derscoring its significant potential in drug development and therapeutic innovation.
Through case studies involving molecular dynamics simulations, we demonstrate our method's 'ss practical utility and effectiveness in drug design, reinforcing its applicability and impact in real-world scenarios.
2 Related Work
All-Atom Protein Design, Protein design traditionally in- volves first designing the backbone, followed by sequence design. Diffusion models (Ho et al., 2020; Song et al., 2021) have been applied to protein backbone design (Watson et al., 2023). This process is typically followed by folding models (Dauparas et al., 2022), enabling comprehensive protein design. Recently, there has been shift towards co-design, where both protein sequences and structures are generated jointly (Jin et al., 2022; Luo et al., 2022; Kong et al., 2023; Lisanza et al., 2024; Campbell et al., 2024). However, all-atom' structure modeling is essential for com- prehending protein functionality, such as protein-protein interactions. Consequently, recent research has increas- ingly focused on all-atom protein design to gain a more detailed and accurate understanding. To achieve full-atom antibody design, Kong al. (2024) proposed to use a multi- channel equivariant layer to encode all-atom structures, and Martinkus et al. (2023) introduced backbone and internal generic side chain representation. Chen et al. (2025) adopted a representation that includes amino acid type, backbone structure, and sidechain torsion angles for designing protein complexes. A notable advance in de novo all-atom protein design is Protpardelle (Chu et al., 2024), which suggested modeling a "superposition" over possible side-chain states and introduced the atom73 representation. This inspires us to apply all-atom structures in a similar manner for de novo cyclic peptide design. Unlike Protpardelle, we also incorporate bond modeling, which is crucial for ensuring successful cyclization.
To tackle these challenges, we developed the CPSDE, which generative
comprises two models: a harmonic-SDE--bas structure prediction model named ATOMSDE and residue type predictor named RESROUTER. Unlike leading works (Watson et al., 2023; Yim et al., 2023) that typically use the residue frame representation for protein design, we em- ploy the all-atom and bond representation. Since both linear and cyclic peptides are composed of atoms and bonds, this representation allows us to model interactions at the most fundamental level. It maximizes the use of small molecule and linear peptide data while minimizing reliance on cyclic peptide data. The inclusion of bond modeling effectively ad- dresses the geometric constraints introduced by cyclization. With our designed routed sampling method, we iteratively update both the sequence and structure by alternating be- tween the two models, which enables the generation of all types of cyelie peptides. We highlight our main contribu- tions as follows:
Peptide Design. Peptides, consisting of short chains of amino acid residues, are essential in numerous biologi- cal processes due to their interactions with various target molecules, offering substantial potential in drug discovery, such as targeting undruggable proteins (Hosseinzadeh et al.,
'In some contexts, "all" in "all-atom" is interpreted to encom- pass all types of biomolecules, including small molecules, proteins, and nucleic acids (Krishna et al., 2024). In our work, however, we use "all-atom" with the same meaning as "full-atom", specifically to represent all heavy atoms within proteins.
2

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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
2021). Traditional computational peptide design methods often rely on searching and sampling residues or motifs from chemical databases (Bhardwaj et al., 2016), which and limit the diversity of designed
wet-lab synthesizability. Here we denote the all-atom 3D structure of the eyelic peptide P and the 3D structures of the receptor (i.e., protein target) as T. We define the chemi- cal graph of the cyclization parts as C, which contains atoms as nodes and chemical bonds as edges. Note that since the 3D structures of the cyclization part are unavailable, there is no information about atom positions in C. Our final goal is to model the conditional distribution P(PIT, C), i... gener- ate the cyclic peptides given the 3D receptor structure and the cyclization chemical graph.
can be structures (Cao et al., 2022). In contrast, deep generative models, known for their strong capability in modeling data distributions, have been applied to peptide design and have shown great potential. Several studies have explored de- signing peptide backbones (Boom et al., 2024), designing peptide sequences (Chen et al., 2024), or generating specific peptide structures such as o-helices (Xie et al., 2023; 2024). Lin et al. (2025) proposed target-aware peptide sequence- structure co-design with flow matching on peptide global translation, orientation, backbone torsions, and sequences. Recent advances in full-atom protein design have signifi- cantly improved peptide generation capabilities. For exam- ple, Kong et al. (2024) employed a latent diffusion model on a latent space that encodes full-atom peptide structures. Li et al. (2025) proposed to represent peptides with backbone atoms and side-chain torsion angles, employing flow-based models to generate full-atom peptides given the protein tar- gets. These methods face challenges in adapting to eyelic peptide design because they model protein structures at the residue level, which complicates the incorporation of covalent bonds or non-canonical amino acids essential for cyclization.
We provide basic knowledge on stochastic differential equa- tions (SDE) and SDE-based generative models (Song & Er- mon, 2019; Ho et al., 2020; Song et al., 2021). SDE-based generative models (also known as score-based generative models and diffusion models) learn the data distribution by learning to denoise. The forward SDE injects noise gradu- ally into the data X E Rd and constructs a diffusion process (x, hejo, such that Xy ~ Po and X1 ~ Pr. where po is the data distribution and P1 is the prior distribution:
dx f(x, () dl + g(() dw,
(I)
where f(,()) IRd + Rd is vector-valued function known as drift coefficient, 9(-) IR - R is a scalar function known as diffusion coefficient, and w is the standard Wiener pro- cess (also known as Brownian motion). The induced per- turbation kernel Por (x:(xx))) a Gaussian distribution that can be efficiently sampled. The resultant pi (Xe Xo) typi- cally approximates a Gaussian distribution pi (xx) (i.e., prior distribution), which is independent of xo-
3 Method
In this section, we present CPSDE, a groundbreaking ap- proach for designing cyclic peptides. It features an SDE- based generative structure prediction model, ATOMSDE, and a residue type predictor, RESROUTER, both utilizing all-atom and bond modeling. We begin by defining the overview of
The reverse SDE starts from a sample from the prior dis- tribution, denoises the noisy sample iteratively, and finally produces a generated sample:
cyclic peptide design task and providing 3DE-based generative models in Section 3.1. In Section 3.2, we detail ATOMSDE, based on a harmonic SDE, and in Sec- tion 3.3, we describe RESROUTER, which predicts residue types based on denoised structures. Lastly, in Section 3.4, we explain how to alternate between these models through routed sampling to generate cyclic peptides.
dx = [f(x, t) - v(t)²Vx Px + g(t) dw, (2)
where w a standard Wiener process when time flows back- wards from to O and and di is an infinitesimal negative timestep. Typically, a neural network so so(x,, () is used to ap- proximate the underlying score function Vx log Pe (x().. and it can be trained via the following score matching objective:
3.1 Preliminaries
c = ==E,(A()) (x))) Ex, Soo (x,, t) - Vx, log (x, (xx)
A peptide is specific type of protein, generally composed of fewer than 30 amino acid residues. The type of the i-th residue a; E 21,2.... 20) is determined by its side-chain R group. Thus, the all-atom 3D structure of peptide speci- fies both its sequence and structure, inspiring us to focus on generating 3D coordinates of all atoms for peptide design. Cyclic peptides, particularly their cyclization regions, often include unique inter-residue chemical bonds and occasion- ally non-canonical amino acids. Despite their non-canonical nature, these cyclization regions usually display distinct patterns. Additionally, providing detailed cyclization infor- mation during eyclic peptide design is essential, as it ensures
where X(t) is time-dependent weighting function and t is uniformly sampled over [0, 1].
3.2 ATOMSDE
To accurately model both atomic interactions and bond con- straints, we initially train a docking model named ATOM- SDE using protein-ligand complex data, incorporating both small molecules and peptides as ligands.
In this subsection, for brevity, given a protein- ligand com- plex, we assume there are Nr, atoms whose coordinates are x E N..x3 x3 in the ligand and Np atoms whose coordi- nates are xP E x333 in the protein T. We denote the
3

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Designing Cyclic Peptides via Harmonic SDE with Stom-Bond Modeling
Side-to-tail
Side-to-head
Head-to-tail
Side-to-side
CYCLIC
HyN
COOH
R2
R
R1
R R
R
Chemical graph
RESROUTER ATOMSDE
Cyclization atoms
Denoise
Noise
X
X
X t--t
Cycling
Update AA type
Update AA state
Ser-See
Atom73
Ser
Bockbono
side
chatn
Thr
Gly
Backbone
side chain
siide chain
N residue
73 atoms
Ala
Atom73
Ala Ala
Figure Overview CPSDE. The generative process structured follows: (1) cyclization type 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 yellow shadow) and the predicted residue types, all-atom structure initially denoised using ATOMSDE then re-noised in accordance with the integration step of the reverse-time SDE. The updated structures are then preserved in the Atom73 representation; (3) RESROUTER predicts the residue types not constrained by cyclization. based the chemical graph and the all-atom structures updated using the Atom73 representation Steps (2) and (3) are iteratively executed. The incorporation eyclization chemical graph, with chemical bonds as edges, ensures the generated peptide forms cyclic structure.
denoised structure. Consequently,
chemical graph of the ligand as Gc, where nodes are atoms and edges are chemical bonds. We build generative dock- ing model ATOMSDE learn the conditional distribution (x,
connection information embedded the chemical graph. o o where L
We define H L + cian matrix of chemical graph 9c, D is the degree matrix, A is the adjacent matrix, and оp is receptor-dependent scalar value. The positive definite matrix can be decom- posed as H PAPT where P is an orthogonal matrix (i.c., PPt = I) and = diag( ............ .h. AN,) is diagonal matrix
A is the Lapla-
We choose Variance Preserving (VP) SDE (Ho et al., 2020) instead of Variance Exploding (VE) SDE (Song & Ermon, 2019) for scenario. The reason is that, when is large, VE SDE causes the noisy ligands to drift far from the re- ceptor and introduces different spatial scales between the ligands and receptor, leading to loss of valid interaction with the receptor. Inspired by Jing et al. (2023); Stark al. (2023), introduce harmonic SDE fully leverage the
that contains the eigenvalues. We define the Ge-dependent forward SDE as follows:
1 dxf = -0=---8000 dt + BBIAAAPTT
(3)
where 3(t) is positive time-dependent scalar function that

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atoms, and the others as Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
controls the noise level along the diffusion process. The induced perturbation kernel has an analytic form as (see the derivation in Appendix F):
states of the backbone atoms (i.e, N-Ca-C-O0 and use multilayer perceptron (MLP) to predict the amino acid type. The model ís trained via the following objective:
Pou(x* xx)) = N (xx x - à li s(e)de H He Si P(e)de).
L= 2 N log (a: Do (x 1), Ttt) (5)
where Pe is the model-induced probability for the residue type. ATOMSDE (i.e. Da) is first pretrained and then fixed during the training of RESROUTER.
Given a schedule that satisfies li So B(s)ds = xo, the above perturbation process arrives at the prior distribution Pr (xf) x x at time 1. = 1, which can be efficiently sampled. Intuitively, the anisotropic perturbation process leverages the connection information in the chemi- cal graph Gc. The bonded atoms are initially set close and then gradually perturbed by correlated noises.
3.4 Routed Sampling for Cyclic Peptide Design
With trained ATOMSDE and RESROUTER, we can itera- tively update both the sequence and structure through alter- nate calls to the two models, enabling the generation of all types of cyclic peptides, as shown in Figure 2.
The model is based on an -quiiaaqui neural network (Satorras et al., 2021; Guan et al., 2021), incorporating both k-nearest-neighhoo graph built upon the protein-ligand complex and ligand chemical graph. This design ensures the model is aware of both protein-ligand interactions and atom connections induced by chemical bonds. We leave the details of model architecture design in Appendix G.I.
Given the number of residues within the peptide and the cyclization information, the atoms in a cyclic peptide can be categorized into two classes: The first class is known in terms of their chemical graph (though their 3D structures are not determined), comprising all backbone atoms and the atoms constrained by cyclization. The second class is unknown, consisting of the side chains of canonical amino acid residues not constrained by cyclization. Without loss of generality, consider a specific type of side-to-tail cyclization where the sulfur atom (S) in the side chain of the i-th residue and the carboxylic acid (COOH) group of the C-terminus engage in chemical reactions form C-S thioester bond, In this scenario, the chemical graphs induced by all back- bone atoms (since all canonical amino acid residues share the pattern N-C., -C-) and all atoms of the cysteine (CYS) providing the sulfur for the C-S bond that forms the cyclic structure are known. Conversely, the chemical graph be- tween other atoms, specifically the side chains of residues except for the aforementioned CYS residue, remains un- known. In the remainder of this paper, we refer to the atoms associated with the known chemical graph due cycliza- tion as eyclization free-residue atoms.
We denote the final output of the SE(3)-equivariant neural network as Dx t). For simplicity, we use simple reconstruction loss that is approximately equivariant to the score matching objective as follows:
C = Er, (x(6) (xo (xf)
Da ()) - x [3]. (4)
The estimated score function can then be formulated as
x lexx)( x VI-5 ((s) ds
(x" Do(x",t))
with which we can generate the ligand poses given the lig- and's chemical graph and the receptor's 3D structures by solving the reverse-time SDE in Equation (2).
3.3 RESROUTER
We introduce RESROUTER that predicts the ground-truth residue type given the noisy ligands. The model architec- ture is similar to that of ATOMSDE. Differently, the input and output of the model are modified due to the following considerations.
Inspired by Chu et al. (2024), for the free-residue part, we maintain an atom73 state (i.e., coordinates), a "superposi- tion" for each residue, where all possible amino acid types for this residue share the backbone atoms and Cg, while possessing unique side chain atoms. For the cyclization- constrained part, we maintain a setliiattion-oosstaannn state separately. The general idea of routed sampling is to switch to different collapsed (i.e., specific) atom states for the free- residue part and assemble a new valid chemical graph based on the predicted residue type and cyclization information at each step of solving the reverse-time SDE Specifically, at each step, ATOMSDE denoises the atom coordinates.
With a structure prediction model in hand, we could still not be able to design cyclic peptides, since their sequence unknown. This a classic "chicken-and-ege" problem. This motivates us to alternately denoise the 3D structures and update the residue types. Since the residue type is determined by the side-chain R group, it will provide shortcut for the model to predict the ground-trute residue type if the complete chemical graph of the noisy ligands is input. Thus, we remove the side chain of the canonical amino acid residues except for those that are involved in cyclization.
In practice, for both cyclization-constraie residue atoms, we maintain both denoised states and cur- rent states. This approach is needed because, during the sampling process, the cyclization -ontattion-ontrrinneed atoms and backbone atoms are constantly present, while the side-chain
atoms and free-
The model produces a hidden state (i.e., h) for each atom. For the i-th residue (whose ground-truie truth amino acid type is a.) in a peptide with N residues, we aggregate the hidden
5

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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
atoms of the free residues are sometimes sampled, resulting in them potentially not being adequately updated. Thus, we reuse the previous denoised structure to solve the SDE and align the time (or noise level) of all sampled atoms to the same point. This alignment is necessary because the ATOMSDE expects all input atoms to be at the same noise level, especially when a residue type is sampled after not being sampled for a few steps. Please refer to Appendix E for more details.
serves as an indicator of the stability (or rationality) of a peptide's s 3D binding pose. Hence, we define this energy metric as Stability, We also evaluate the interface binding energy, a crucial metric that indicates the binding affinity of the ligand peptide to its receptor. This assessment is
essential for evaluating the ligand peptide's functionality, particularly when designing peptides for therapeutic appli- cations. We denote this type of energy as Affinity. We also report Diversity, the average of one minus the pair-wise TM-Score (Zhang & Skolnick, 2005) among the designed peptides, reflecting structural dissimilarities. As the refer- ence ligands for the targets in the test set are linear, we do not report metrics that necessitate a reference sequence or structure, such as Amino Acid Recovery (AAR) and Root Mean Square Deviation (RMSD). We selected 100 protein pockets with a large volume for testing. Specifically, these targets have receptors with more than 1,000 surrounding atoms around the reference peptide ligands, serving as re- liable indicators of adequate volume. For each target, all peptide design methods are used to generate a batch of pep- tides, from which we select the most promising (i.c., lowest energy) peptide ligand. We design four types of cyclic types (head-to-tail, head-to-side, side-to-side, and side-to-tail) us- ing our methods, respectively, and we also report the mixed results, where the best promising peptide ligands might ex- hibit different cyclization types. We then report the average and median metrics across all targets. Please refer to Ap- pendix H for more details. This evaluation strategy mirrors practical drug design scenarios where the leading ligand candidates are identified for advancement to the subsequent stages of drug development.
Notably, the two models, ATOMSDE and RESROUTER, do not account for the atom partition introduced by amino acid residues, as they model atoms and bonds at the most fundamental level. The concept of amino acid residues is introduced only during routed sampling to ensure that the free-residue part remains a canonical residue rather than an arbitrary molecule.
Experiments
4.1 Experimental Setup
Dataset. We have curated two datasets of protein-ligand complexes featuring small molecules and peptides as lig- ands, respectively. All complexes with atoms whose el- ements are beyond {C. N, O, F, S, CI, Se, Br} are not included. The small molecule dataset is sourced from PDB- Bind (Wang et al., 2005) and has 14.348 protein-ligant complexes. The peptide dataset is derived from RCSB PDB (Burley et al., 2023), Propedia (Martins et al., 2023) and PepBDB (Wen et al., 2019). It comprises 20,033 protein- ligand complexes, featuring peptide ligands composed of fewer than 30 residues. Samples are clustered by receptor sequence identity of 0.3 to split the dataset into training and validation sets. To train ATOMSDE, we utilize the curated small molecule dataset and subset of the peptide dataset containing ligands with fewer than 200 heavy atoms. To train RESROUTER, we use the whole peptide dataset. Please refer to Appendix D for more details about data.
4.2 Main Results
The results are shown in Table 1. Among all co-design meth- ods, our method exhibits superior energy performance in terms of both stability and affinity and also best diversity. In- terestingly, we find that head-to-tail and head-to-side cyclic peptides show better performance than side-to-tail and side- to-side cyclic peptides. This might be due to the training dataset where C-N bonds (main covalent bonds that form head-to-tail and head-to-side cyelic structures) are more fre- quent than S-S and C-S bonds (main covalent bonds that form side-to-tail and side-to-side cyclic structures). This aligns with the fact that C-N bonds are generally more stable than S-S and C-S bonds in the physical world. Among all methods, RFDiffusion shows the best energy performance but low diversity, as it tends to generate a- a-helices in certain pattern. See Appendix H.5 for ablation studies.
Baselines. As our method is the first cyclic peptide de- sign method based on generative models, we compare our approach with various established methods for linear pep- tide design: RFDiffusion (Watson et al., 2023) generates protein backbones, and sequences are later predicted by ProteinMPNN (Dauparas et al., 2022); ProteinGenerator (Lisanza et al., 2024) improves RFDiffusion by jointly sam- pling backbones and corresponding sequences; PepFlow (Li et al., 2025) is a flow-based full-atom peptide genera- tive model that generates the translation, rotation, and side- chain torsion angles of each residue frame within a peptide; PepGLAD (Kong et al., 2024) is a full-atom peptide design method that utilizes a latent diffusion model.
4.3 Case Studies
Here, we demonstrate how CPSDE can be seamlessly inte- grated into real-world cyclic peptide design pipelines and discuss two scenarios: design of SMYD2 peptide inhibitors via head-to-tail cyclization and SETS inhibitors via side-to- side cyclization.
Evaluation. We use Rosetta (Chaudhury et al., 2010) to compute the total energy of reference ligands, linear pep- tides designed by baseline methods, and cyclic peptides engineered by our approaches. This energy measurement
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
Table Summaay of properties of reference peptides, linear peptides designed by baseline methods, and eyclic peptides designed by CPSDE. (J) / (1) denotes smaller larger number is better.
Stability (4)
Affinity (4)
Method
Co-Design
Peptide Type
Diversity (t)
Avg.
Med.
Avg.
Med.
Reference
N/A
Linear
-672.53
-634.71
-85.03
-78.70
N/A
RFDiffusion
Linear
-633.51
-607.82
-70.30
-61.35
0.55
ProteinGennrator
Linear
-576.39
-55.770
-4.998
-40.39
0.58
PepFlow
Linear
-576.16
-49.331
-47.88
-42.40
0.70
PepGLAD
Linear
-359.44
-310.33
-4.066
-38.56
0.79
CrSDE
Head-to-tail Cyclic
-568.04
-519666
-50.86
-46.62
0.79
CPSDE
Head-to-side Cyclic
-564.81
-508.88
-49.81
-44.16
0.79
CPSDE
Side-to-tail Cyclic
-547.14
-50.200
-4..22
-37.25
0.78
CPSDE
Side-to-side Cyclic
-537.09
479.94
-49.73
-44.71
0.79
CPSDE
Mix
-580.67
-527.80
-55.71
-48.42
0.79
PepFlow
PDB: 4o6f
Our (head-to-tail)
PapFlow
PDB: 4o6r
(head tail)
RMSD (A)
0
20
40
60
80
100
0
i
2
Time (ns)
Density
Figure Discovery of new SMYD2 eyclic 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 cancer therapy (Zheng al., 2022). In particu- lar, SMYD2 attenuates estrogen signaling by weakening ERa-dependent transactivation (Zhang et al., 2013). In- sight into this interaction comes from the SMYD2-ERa
co-crystal structure (PDB: 4O6F), which reveals ERa in U-shaped conformation nestled within SMYD2's deep bind- ing pocket (Jiang al., 2014). The distance between ERo's N-terminal (head) and C-terminal (tail) in this complex is only 5.9 A, finding that has inspired the design of rigid head-to-tail cyclic peptides capitalize this proximity and enhance binding affinity.

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Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
PepFlow
PDB: 1zkk
Our (s-de-to-side)
RepFlow PDB: 12kk Our (side-to-side)
RMSD (A)
0
20
40
60
80
100
0.0
0.5
1.0
Time (ns)
Density
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 design new SMYD2 in- hibitors by generating head-to-tail cyclic peptides. The binding site of ERa (PDB: 4o6f) was selected as the active pocket, and 8 cyclic peptides were generated through head- to-tail cyclization. All peptides demonstrated favorable Rosetta affinity scores (Figure 10), with H2T-6 achieving the most favorable score of -33.9 kcal/mol. To further accu- rately assess its binding stability and affinity, 100 ns molec- ular dynamics (MD) simulations were performed on the H2T-6-SMYD2 complex. More details on the system prepa- ration and simulation protocol of MD are Appendix H.6. The simulations also included the crystallized linear peptide (ground truth) and the HepFlow-generatte linear peptide for comparison. Each simulation was repeated twice to en- sure consistency. As shown in Figure 3, the linear peptide generated by PepFlow exhibited higher flexibility, with an average peptide RMSD of 4.59 Á over the last 50 ns of equilibrated trajectories. In contrast, H2T-6 displayed an average peptide RMSD of 3.05 A, comparable the ground truth linear peptide (1.92 Á). More importantly, binding free energy analysis using MM-PBSA (Wang et al., 2019) revealed that H2T-6 had the highest binding affinity (-24.02 kcal/mol), outperforming both the ground truth peptide (- 19.00 kcal/mol) and the PepFlow-generatee peptide (-7.26 kcal/mol). These results suggest that H2T-6 maintains a stable binding conformation which also exhibits high bind- ing affinity, indicating its potential as candidate SMYD2
inhibitor for further investigation.
Design of SET8 peptide inhibitors via side-to-side ey- clization. SET8 is the only lysine methyltransferase that specifically catalyzes the methylation of histone H4 at the 20th lysine (Qian & Zhou, 2006). SET8-mediated protein modifications are involved in numerous physiological pro- cesses, and its dysregulation is closely linked various human diseases, particularly cancer development and prog- nosis (Yang et al., 2021).
Again, we applied CPSDE to design seed cyclic peptide inhibitors targeting SET8. The 3D structure of SET8 in complex with an H4 peptide (PDB: 1zkk) was used, with the H4 binding site selected as the active pocket (Couture et al., 2005). To explore an alternative cyclization approach, we employed a widely used side-to-side strategy to generate 8 cyclic peptides. Likely, we picked up the top cyclic peptide, $2S-4 (see Figure 11), based on Rosetta affinity score, and performed 100 ns MD simulations with repeats, along with two reference linear peptides. The RMSD and MM- PBSA analyses revealed that S2S-4 not only demonstrated significantly lower peptide RMSD (2.54 Á) compared to the reference linear peptides (ground truth: 4.06 A; PepFlow:
5.23 A), but also exhibited lower binding free energy (S2S-
4: -12.48 kcal/mol; ground truth: -6.39 kcal/mol; PepFlow: -926 kcal/mol). These results indicate that S2S-4 might
serve as potential candidate for later studies on SET8 inhibition.
8

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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
5 Conclusion
In conclusion, CpSDE a generative algorithm capable of producing diverse types cyclic peptides given 3D recep- tor structures, thereby paving the way for advancements peptide-based drug discovery. Our approach enhances drug development by designing stable and high-affinity eyclic peptides. Case studies supported by molecular dynamics simulations validate its practical utility and real-world ap- plicability. Limitations and future work are discussed in Appendix
Acknowledgments
We thank anonymous reviewers for their insightful feedback. We would like extend our gratitude to Yi Zhou and Lihao Wang for their valuable feedback well as to Ellen Wang, Tianze Zheng and Wen Yan for their assistance with the molecular dynamics simulations.
methodology,
Impact Statement
CPSDE contributes to the advancement of computational biology, particularly in the field of cyclic peptide design, by addressing key challenges that have hindered progress this area. By integrating generative structure prediction and se- quence prediction, and high-affinity cyclic peptides, offering valuable applica- tions in drug discovery and therapeutic development. While our primary focus is on the positive applications of CPSDE, such as developing novel peptide-basee based treatments, we ac- knowledge the ethical considerations associated with any powerful generative tool. There is a potential risk of misuse, including the design of harmful bioactive compounds, To mitigate such concerns, emphasize the responsible use of our method and encourage the scientific community to apply it for constructive and beneficial purposes. Our commitment to ethical research practices ensures that CPSDE serves as a force for innovation in pharmaceutical development while upholding societal safety and integrity.
approach enables the design of stable

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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
A Introduction of Cyclic Peptides
Peptides have shown promising capability as therapeutics for protein targets where small molecules struggle to bind, due to their ability to modulate protein-protein interactions (Hosseinzadeh et al., 2021; ; Zorzi et al., 2017). However, traditional linear peptides are usually polar due to exposed acids and amines terminals, limiting their membrane permeability and proteolytic stability, and also restricts their administration options in drug development (Buckton et al., 2021; Merz et al., 2024).
The peptide cyclization is capable of enhancing the conformational stability, increasing binding affinity and specificity for targets (Zorzi et al., 2017). This has fueled the growing interest in eyelie peptide research as efficient therapeutics. Figure shows two cyclic peptide drugs approved in recent 20 years. According Sharma al. (2023); Fang al. (2024); Costa et al. (2023), eyclic peptides
be classified by their eyclization strategies: head-to-tail cyclization (between N- and C-termini); side-to-side cyclization (between two side chains, including peptide stapling, which stabilizes a-helical structures) (Vinogradov et al., 2019); head-to-side and side-to-tail eyclization (between a terminal and side chain); and polycyclization (with multiple cycles). Voclosporin
Lanreotide
Figure 5. Two examples of cyclic peptide drugs. Voclosporin (van Gelder et al., 2022), an analog of ciclosporin, is an immunosuppressant used treat lupus nephritis. Lanreotide (Caplin al., 2014), an analog of somatostatin, is an oncology drug that inhibits growth hormone release and is used to manage carcinoid syndrome.
B Extend Related Work
ligaad Docking, Protein- -igaad docking aims to predict the conformation of protein-ligand complexes, playing a crucial role in drug discovery and design. Compared to traditional score-based or template- docking methods (Eberhardt et al., 2021; Friesner et al., 2004), deep learning- accuracy. A common strategy involves leveraging equivariant or invariant graph neural networks to model both protein and ligand and predict ligand atom positions (Lu et al., 2022: Stärk al., 2022: Pei al., 2024: Qiao et al., 2024). Other approaches, such as Uni-Mol (Zhou al., 2023) and TTAFll All-Atom (Krishna et al., 2024), use positional encoding and attention layers for atom-level representations and predictions. Unlike structure prediction models, Diffdock (Corso et al., 2023) introduces the generative docking framework that views docking as a conditional generation problem, and uses a diffusion model for end-to-end ligand generation. SurfDock (Cao et al., 2024) further incorporates protein surface graphs to refine docking predictions. More recent studies have emphasized flexible docking models, which consider protein conformational variability in docking process. DynamicBind (Lu et al., 2024) and ReDock (Huang et al., 2024) extend the
approaches are faster while achieving comparable
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Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
5i2i_
3av9 X
5txe_C
3qg6_C C
1u91_c
1rgr_ B
3qn7_B
6q1u_D
Figure 6. Examples of 3D structures of cyclic peptides and their chemical graphs. The cyclization structures highlighted in pink. The corresponding cyclization types are: 512IE: head-to- tait: 3AV9.X: head- to-d-tt-tall; STXE_C: head-to-sidee 3QG6.C: side-to-tail 1U91_C: side-to-side; IRGR_B: side-to-side (stapling); 3QN7_B BB and 6QIU_D; polyeyclization.
generative docking paradigm by incorporating diffusion-based modeling of protein structures, enabling joint optimization of both protein and ligand poses. Guan et al. (2025) proposed novel molecular docking framework that simultaneously considers multiple ligands docking protein, enhacing molecular docking accuracy.
Protein Sequence Design. Designing protein sequences that fold into desired structures, known as inverse folding, is fundamental task in protein engineering. Deep learning methods can effectively predict protein sequences given structural priors. These models generally follow two-stage paradigm comprising a structure encoding module and a sequence prediction module. Several models, including GVP (Jing et al., 2021), ESM-IF (Hsu al., 2022), ProteinMPNN (Dauparas al., 2022), and Ingraham et al. (2019), utilize graph neural networks (GNNs) for structural encoding and employ autoregressive decoding for sequence prediction. PiFold (Gao et al., 2023b) adopts similar encoding strategy but directly classifies amino acids for each node. GRADE-IF (Yi et al., 2024) formulates the inverse folding problem as learning the conditional distribution of protein sequence given the backbone structure, and employs discrete diffusion model on protein graph to predict the amino acid type on each node. Besides graph-based approaches, Protein Language Models (PLMs) (Lin et al., 2023; Rives et al., 2021) offer an altermative strategy for inverse folding. Zheng et al. (2023) integrates PLMs as sequence decoders for encoded protein graphs, while Gao et (2023a) and Mao et (2024) incorporate FSM-2 (L.in al., 2023) embeddings refine sequence predictions. ProGen (Madani et al., 2023) shows an autoregressive language model trained massive protein sequence data utilize property tags for controllable sequence generation. Wang et al. (2024b) demonstrate that discrete diffusion serves as a more principled probabilistic framework for large-scale protein language modeling. The resulting diffusion protein language model (DPLM) excels in both sequence generation and representation learning. DPLM-2 (Wang et al., 2025) further extends this discrete diffusioon-bsee paradigm by incorporating tokenized 3D structures thereafter, enabling structure- structure-sequence co-generation as well as any-to-any conditional generation
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
5930
2093
6.0
4033
520
2233
0.30
300
230
2.0
20
25
125
Peptitt Length (resition)
Peptide Lengtth (stom))
Cyclec Peptide Length (resioue)
Cydlc Peptide Length (atom)
Figure 7. Statistics on peptide and cyclic peptide lengths in our dataset.
1000
903
800
Head-to-side
335 Side-to-tall
6.5%
Frequency
600
Linear Peptide 93. 1
Head-to-tall
Cyclic Peptide
Side-to-side
400
15.1%
Polycyclization
221
200
95
49
head-to-tall side-to-tail head-to-side side-to-side policyclization
Figure Analysis cyclic peptide distribution and the proportions five eyclization types.
with multimodal generative protein language models.
Cyclic Peptide Design. Although computational cyclic peptide design is innovative research area, there already recent studies cyclic peptide design that are significantly different from approach (Tang et al., 2025; Rettie et al., 2024). The approach proposed by Tang et al. (2025) is ligand-base based drug design (LBDD) method that models the sequence of cyclic peptides using discrete diffusion, optimized by multiple reward functions. It does not explicitly incorporate the 3D structure of target proteins, whereas our structure-basse based drug design (SBDD) method directly designs ligands based on 3D target structures, Rettie et al. (2024) uses modified RoseTTAFold (Baek al., 2021) and REdiffusion (Watson al., 2023) with cyclic relative positional encoding to generate macrocyclic backbones.
C Protein all-atom representation
Atom14 representation. The atom14 encoding efficiently represents residues using 14 columns to capture atom content, with 14 being the maximum number of heavy atoms the 20 canonical amino acids (e.g., Tryptophan). Empty strings are padded in atoml4 representation when a residue contains fewer than 14 atoms.
Atom37 representation. The atom37 encoding is fixed-dimension representation where each of the 37 heavy atom types in canonical amino acids is assigned unique position in an array. For amino acids missing certain atoms, padding is used to maintain consistent length. The atom37 format includes the following atoms: ['N', 'CA', 'C', 'CB', 'O', 'CG', 'CGI', 'CG2', 'OG', 'OG1', 'SG', 'CD', 'CD1', 'CD2', 'ND1', 'ND2', 'OD1', 'OD2', 'SD', 'CE', 'CEI' 'CE2', 'CE3', 'NE', 'NE1', 'NE2', 'OE1', 'OE2', 'CH2'. 'NHI', 'NH2', 'OH', 'CZ', 'CZ2', 'CZ3', 'NZ', 'OXT'.
Atom73 representation. The atom73 representation, introduced by Chu et al. (2024), indexes the 'N', 'CA', 'C' and 'O' atoms, then independently encodes each amino acid's side chains. Backbone atoms share the same position, while side-chain atoms are assigned to specific residue types. For example, the 'NE2' atom in the 'GLN' residue recorded as "Q-NE2' (where Q is the one-letter code for 'GLN') and occupies a unique column in the atom73 encoding.
D Peptide Dataset
CyclicPepedia. The CyclicPepedia dataset (Liu et al., 2024) contains 9,744 cyclic peptides from multiple sources. It serves as a knowledge database with information on categorization, structural characteristics, pharmacokinetics, physicochemical properties, patented drug applications, and key publications. However, most peptide conformations are generated by RDKit, with only 1,325 eyclic peptides having original 3D structures and 61 containing complex structures. The remaining data only include fingerprints, limiting its application in structure-based drug design.
PPBench2024. PPBench2024 (Lin et al., 2025) is a protein peptide binding dataset sourced from Burley et al. (2023), Martins al. (2023), and Weng et (2020). selects complexes with more than two chains and an interaction distance
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
under 5 Á. Complexes with peptides shorter than 30 amino acids are then filtered. Non-peptide molecules and peptides with unusual bond lengths non-amino acid functional groups are also excluded. This results in total of 15,593 protein-peptide pairs.
PepBench. PepBench is a curated benchmarking dataset designed to train and evaluate protein-peptide binding models in PepGLAD (Kong al., 2024). The training data is sourced from (Berman et al., 2000), while the test set is adopted from Tsaban et al. (2022). To ensure quality, peptides are restricted to lengths between 4 and 25 residues, and receptors to more than 30 residues. Complexes with over 90% sequence similarity are excluded to reduce redundancy. Furthermore, clustering at 40% sequence identity is applied to separate training and test data, removing any training complexes that share clusters with the test set. The final dataset comprises 4,157 training complexes, 114 for validation, and 93 for testing.
PepFlow. The dataset introduced by Li et al. (2025) combines data from Wen et al. (2019) and Wei et al. (2024), resulting in 8,365 protein- peretin-pppideeeee complex structures. Peptides are filtered to include those with lengths ranging from 3 to 25 amino acids and are clustered at 40% sequence identity. From these clusters, 158 structures are selected for the test set, ensuring each cluster has 10 to 50 members. The remaining data is used for training and validation.
Our curation. Our curation has been roughly introduced in Section 4.1. A unique requirement of our method is converting Structured Data Files (SDF files) to PDB files, as the chemical bond information in PDB format is often incomplete. We found that RDKit 2 does not always accurately produce chemical bonds during conversion, as determines bond existence and type based on pairwise atom types and distances. Therefore, we leverage the Chemical Component Dictionary (CCD) ; which describes all residue and small molecule components found in PDB entries, to collect all intra-residue bond information. We use default peptide bonds connect canonical residues with continuous residue indices and the bonds stored in the CONECT information to recover bonds between non-canonical residues and other residues.
E Routed Sampling
The general idea of the proposed routed sampling is that the sequence and structure are alternately updated in the generative process. We present its details in Algorithm In this subsection, we introduce some notations without rigorous definitions but maintain clarity, we provide detailed comments after the lines in the algorithm. The sampling algorithm is inspired by Chu et al. (2024). However, compatible with cyclic structures, especially non-canonical covalent bonds and residues. Several critical functions used in Algorithm will be discussed as follows:
innovatively introduce dynamic chemical graph make the all-atom peptide design
"Extract" and "Cache": As introduced in Section 3.4, due to cyclization, can categorize atoms into two types: constrained atoms and free-residue atoms. We maintain the atom73 states and individual atom states for these types, respectively. Two functions are employed to extract and store atom coordinates based on masks determined by residue types or constrained atom indices.
"Assemble": Each time the residue types are updated, the corresponding side-chain chemical graphs are likewise updated. Consequently, we reassemble each residue's 's chemical graph along with the cyclization chemical graph into a complete peptide chemical graph to serve as input for the models.
"Supgraph"" As introduced in Section 3.3, to avoid residue type information leakage from the side-chain chemical graphs, we remove the side-chain atoms within the free residues from the current peptide chemical graph. "UpdateTTime" and "AlignTime"": As mentioned in Section 3.4, due to the sampling mechanism, updates of side-chain atoms within free residues might not be continuous. In other words, the coordinates of these atoms are occasionally updated, resulting in atoms having different times (or noise levels). Therefore, "U "UpdateTime" is introduced to store the individual time, and "AlignTime" is introduced to align the atom coordinates from different times to the same point. It's important to note that in "AlignTime", no model is involved as the previously eached denoised structures are reused.
F Proof of Prior Distribution Induced by Harmonic SDE
The difference between the widely-used SDE in SDE generative models and our introduced harmonic SDE is that the our perturbation process is anisotropic. Hence, here we provide the derivation of the prior distribution induced by the harmonic SDE in a similar proof by Song et al. (2021).
In Equation (3), we define the Ge-dependent forward SDE as follows:
1 dx -- (()x dt + VB(I)APPTal Pw
hhttps://ww. wttp://wwwwdii..oooo.o org/
hhtpps: h//p:://wwww.//// http://www.b....///ataa/aa/a/aa org//ata/odd
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Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
Algorithm Routed Sampling
Input: number of residues within the ligand peptide N, cyclization Type O, 3D receptor structures T, SDE solver time interval dí, infinitesimal constant
Output: cyclic peptide with its all-atom coordinates Xo, chemical graph Go. amino acid sequence Ao
1: [f] < HarmonicPrior(N. O, T)
>> Initialize time-dependent atom73 state X1 and cyclization state x > Initialize denoised cyclization-relatee atom coordinates x Randomly initialize residue types not constrained by cyclization Derive initial chemical graph Initialize timer that records time for each atom in atom73 state
2: x + Copy(xp)
3: Ai Uniform (20, N)
4: G1 + Assemble(A,,
5: T <-
6: 1+1
7: while t > E do
8:
& Extract( X, U x
D Obtain all-atom x, structure of current noisy peptide > Predict denoised all-atom structure Xo structure
9: X1 - ATOMSDE(X,, G2, t)
x0 <- Cache(X* N0, 0)
xt-dt + Noise (0, Gz 1- dt)
Ge is required by harmonic noise Update X, by saving new structures specific states according A,
X < Cache X,, Xt- As)
13: x x dr + Cache x -d, O)
14: T t Updete'Timer(" T, Ar, dddt)
D> Update the timer for the newly-updated atoms to the latest time Hide side chains of residues not constrained by cyclization 7> Predict sequence based on the denoised structure % Derive new chemical graph given predicted sequence and cyclization Obtain atom-wise time (Atom might have different time) Align atoms with different time t to the same time t t-df according to the reverse- timse harmonic SDE >> Store the time-aligned atom coordinates
15:
g Subgraph Gr, As, O)
At-dt & RESROUTER (Ro (.t)
Ge-de + (asemble(A,-o 0)
18:
t Extract(T, Ar-di)
19:
dt + Extract aa, Ar-dr) U x
20:
xt-d + AliggTime(x--ei &o, te-ac,t-de t)
21:
X, < Cache(Xe X, X1 dt, Atdr)
22:
t t dt
23: end while
X; Extract(X,, Ar) xp
25: Xo ATOMSDE(X,,95, t)
Predict the all-atom structure finally
26: Go + G.
27: Ao + A
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
where (()) is a positive time-dependent scalar function, P is orthogonal matrix (i.e., PPT =I), A diag A1 diagonal matrix that contains the eigenvalues, and H PAPT.
ANL)
We denote the variance of the random variable x as Σ(1), i.e., ( == Cov Cov[x(() for / E [0. 1j. The aforementioned SDE,
characterized by affine drift and diffusion coefficients, allows to employ Eq. (5.51) from Särkkä Solin (2019) derive an ODE that describes the evolution of variance as follows:
dΣ
dt ((t)((A) A E()),
= (() E(t)).
Solving the above ODE, we derive
E(t = + eSe -B(a)da (E(0) H),
Once the boundary condition x is given, we have 0 = 0. Thus, the induced perturbation kernel has an analytic form as:
Puc( = -N::= N(x Jo ((a)ds
He-Jo ((s)do).
Given lim,-1 1o (s)dds = oo, the above perturbation process arrives at the prior distribution -xxx))=((x() H).
G Implementation Details
G.1
Model Architecture
Given noisy sample at time I, two graphs are built for message passing with an SE(3)-eqiivarian neural network, which is parameterized by ΦK, ΦB, b0, ΦH, óg. K de as introduced below. The i-th atom in the complex is attributed with an initial feature h; and the bond ij in the noisy ligand is attributed with an initial feature bij- We first construct a k-nearest neighbor (knn) graph Gк for the complex (i.e., the protein and the noisy ligand at time t), where each ligand atom is connected with the k-nearest atoms in the complex, to capture the protein-ligand interaction:
Ahk + EjeNK(() ox (h,, hy, |xx - xj|1, Eig, t),
where (i) the neighbors of atom in 9K. Eig indicates the edge ij is protein prontin, ligand-ligand or pootein-ligandl edge.
We also leverage the chemical graph Go of the ligand as we have defined previously to make the model aware of the connection information introduced by the chemical bonds:
eij + Ob(lx - xx,, biill), h + EJeNe(eee (h,, hy, Cij. t).
We further aggregate the hidden features of ligand atoms and bonds from these two graphs as follows:
h h, + (n + Ahg), big EveNeOM ΦB h,, hy, euks ekj: t).
Finally, we update the ligand atom positions as follows:
Ax + -EENN xx)xxxx (h,, hj, ||x; - -xyll.t)
AX + EjeNoto x;)xK (h;, hy, ||x; x,l1, X4 - x, + (Ax + Ax+) 1(i E Gc),
where 1(i e Gc} indicates whether atom belongs to the ligand since the protein atom positions are fixed and we only update ligand atom positions.
We denote the final output of the equivaaran neural network as Do(x t), where Do is composed of dx, ΦB, &C, ΦH, ΦE, ,h, te as introduced above.
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Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
G.2 Training Details
We use the same optimizer setting for both ATOMSDE and RESROUTER: AdamW (Loshchilov, 2017) optimizer with constant learning rate 0.0001, betal 0.9, beta2 0.999, and weight decay 0.01. For beta schedule, we use (3(1) = (Amax (min) + min. where Smin = 0.01 and 9aax = 3.0. Note that lim;-1 Je ((s)ds is sufficiently large compared to the variance of our data distribution. To train ATOMSDE, we sample t ~ L(0, 1]. To train RESROUTER, we sample t ~ (4 0.5) due to the fact that the denoised structure output by trained model ATOMSDE at time = 0.5 or more has extremely limited information to determine the residue types. ATOMSDE converges within 48 hours and RESROUTER converges within 18 hours on 8 NVIDIA H100 GPUs.
G.3 Sampling Details
For routed sampling, we divide time interval [0. 1] into 1,000 steps. Inspired by Chu et al. (2024), we skip RESROUTER when / > 0.5 since the structures are too noisy to provide sufficient information for residue type prediction. This approach also accelerates the generative process and reduces the computational cost of inference. When & < 0.5, at cach step, sequence and structures are iteratively updated by RESROUTER and ATOMSDE, respectively.
H Experimental Details
H.1 Relaxation and Energy Estimation
Cyclic peptides offer notable advantages in terms of both system stability and binding affinity. In specific, the stability of a pepetide complex is inversely proportional to its overall free energy, with lower free energy indicating greater stability. To assess this, the Fast lax protocol in PyRosetta (Chaudhury et al., 2010) is employed to relax each complex, after which the total energy is evaluated using the REF2015 scoring function. Binding affinity is measured with the InterfaceAna yzerMover which calculates the binding energy at the interface between the peptide and the target protein within the relaxed complex. An increase in binding energy reflects enhanced peptide binding affinity, suggesting potential funetional improvements.
For each target, lincar peptide methods generate 8 samples with the golden peptide length (i.e., the number of residues within reference linear peptide). Cyclic peptide methods, lacking reference length, enumerate residues from to 20 (or to 23 for side-to-side cyclic peptides), generating samples per length. All reference ligands and samples are relaxed and evaluated using a standard scoring method as described above. For each target and method, we apply the Borda method to select the best ligand, accounting for both stability and affinity, which is then reported in the final results.
H.2 Detailed Experimental Results
We have provided detailed energy measurement results for each target, including the reference ligand, linear peptides designed by baselines, and cyclic peptides designed by our methods, in Tables 6 and 7.
H.3
Inference Speed
We benchmark the average time of generating one peptide for all co-design baselines and our methods on a single NVIDIA (00-XXM4-00GB GPU. The results are reported Table 2. Given that computational drug design does not demand real-time model response, the inference time of our method is deemed acceptable.
Table 2. Generation time of all co-design baselines and our methods.
Method
Peptide Type
Time (s)
ProteinGenerator
Linear
31.80
PepFlow
Linear
12.09
PopGLAD
Linear
4.40
CrSDE
Cyclic
16.88
H.4 Evaluation on Linear Peptide Design
While designing linear peptides is not our primary focus, we compare our method with existing baselines in this task. Unlike the variable-length setting for cyelic peptides, the task of linear peptide design can leverage known reference peptide lengths for target proteins in the test set. For fair comparison across methods, we sample 8 linear peptides per target matching the reference length and relax the complex structure by Rosetta (Chaudhury et al., 2010; Alford et al., 2017). We then apply the Borda method to choose the optimal linear peptides, considering both stability and affinity. We report the average and median for Stability and Affinity of the linear peptides for targets in the test set. We also report the average for the fraction of hydrophobic and charged residues (relevant for specificity) (Ye et al., 2024), DockQ, iRMSD, LRMSD, BSR, and Diversity.
22

---

Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
We DockQ package to compute DockQ, iRMSD, and LRMSD. We follow the definition of binding site ratio (BSR) in
Li et al. (2025). A lower hydrophobicicharged ratio indicates lower risk of non-specific binding (Makowski et al., 2024). The results are reported in Table 3. Notably, the fraction of hydrophobic and charged residues of our designed peptides resembles that of reference. Our method also shows superiority in structural properties.
Table Summary of properties of reference peptides, linear peptides designed by baseline methods and CrSDE. (4)
/ (1) denotes
smaller / larger number is better.
Stability (1)
Affinity (L)
Hydrophohic Ratio (J)
Charged Ratio (l)
DockQ (?)
iRMSD
LRMSD
BSR
Divensity
Method
Co-Design
Pepride Type
Avg.
Mod.
Avg.
Med.
())
(U)
(1)
(1)
Reference
N/A
Linear
-672.53
-634.71
-85.03
-78.70
0.48
0.28
NIA
NiA
N/A
N/A
N/A
REDiffusion
Linear
633.51
-607.82
-70.30
-61.35
0.59
0.27
0.18
5.37
20. 10.10
0.33
0.55
ProteinGeneratoo
Linear
-576.39
-554.70
46.98
44..9
0.53
0.32
0.12
5.56
23.97
0.20
0.58
PepFlow
Linear
16
.498331
.47.88
-2.40
0.17
0.44
2.49
9.42
0.56
0.70
PepGLLAD
Linear
-359.44
-310.33
-55.06
-38.56
0.53
0.25
0.30
2.68
11.99
0.39
0.79
CtSDE
Linear
-567.34
-510.58
-55.48
-49.89
0.45
0.24
0.32
2.36
9.91
0.60
0.77
H.S
Ablation Studies
Effects of RESROUTER. We study the effects of RESROUTER compared with the following two setups: "w/ fix seq" where the residue types are randomly sampled and fixed with only atom coordinates updated during the generative process, "w/ random seq" where the residue types are randomly sampled from uniform distribution instead of predicted by RESROUTER during the generative process. The results are shown in Table 4. It can be observed that both variants perform worse than CPSDE, which demonstrates that RESROUTER can effectively discover critical residue types for protein-ligand interaction. "w/ random seq" performs the worst, possibly because random residue types offer no information gain, and the residue types are updated too frequently. This frequent updating hinders the ATOMSDE from effectively updating the side-chain atoms of the free residues.
Table 4. Ablation study on the effect of RESROUTER.
Stability (1)
Affinity (L)
Method
Avg.
Med.
Avg.
Med.
CPSDE
-568.04
-519.66
-50.86
-46.62
w/ fix seq
-525.73
-439.69
41,56
-38.75
w/ random seq
-521.48
.425.17
-38.64
-39.44
Effects of Harmonic SDE. We study the effects of harmonic SDE. We introduce a variant with isotropic Gaussian as prior and noise distribution (denoted as "w/o Harmonic"). The results are shown in Table 4 and validate the effectiveness of harmonic prior and noise. To explore the underlying reasons, we examined the trajectories of the generative process and found that the harmonic prior provides a good initialization for atom positions, where bonded atoms are located nearby. This feature might be beneficial for routed sampling because some side-chain atom updates can be discontinuous, and such correlated initialization helps mitigate the errors induced by these discontinuous updates.
Table 5. Ablation study on the effect of Harmonic SDE.
Stability (4)
Affinity (L)
Method
Avg.
Med.
Avg.
Med.
CrSDE
-568.04
-519666
-50866
-4.62
w/o Harmonic
-534.38
-439.23
-39.43
-41.08
H.6 System Setup and Protocols of Molecular Dynamics Simulation
To simulate the protein-peptice systems, hydrogen atoms are added, and the dominant protonation state of titratable residues at pH 7 is determined using PropKa in PDB2PQR (Dolinsky et al., 2007). Subsequently, the systems are solvated in 10 A truncated water box, with sodium and chloride ions added to neutralize the system at a concentration of 150 mM to mimic physiological saline. The fTI4SB (Maier et al., 2015) parameter set is applied to proteins and peptides, and the TIP3P model is used for water (Jorgensen et al., 1983; Li et al., 2024).
All simulations were run on RTX 4090 GPUs using the CUDA implemennation of particle-mesh Ewald (PME) molecular
thtpps: :tttps://gii thub. wtm wtwhh...oo//oooraaaaeaaaaaaeaeaa www.ewawwwlereeeeeeeeeeee ner/Dock
23

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
dynamies in Amber22 (Salomon-Ferrer al., 2013). At first, relax each system thoroughly, stages of energy minimization are performed. In the first stage, 2,500 steepest descent and 2,500 conjugate gradient eycles were applied all atoms, with constraints on water molecules and counterions. In the second stage, the same eycles were repeated without constraints. Initial velocities are randomly sampled from Boltzmann distribution. The systems are then heated from 0 K to 310 over 500 ps in the NVT ensemble, using a Langevin thermostat and harmonic restraints of 10.0 keal mol - A A During equilibration 300 K and bar under NPT conditions, harmonic restraints on protein and peptide atoms progressively reduced from 5.0 to 0.1 kcal mol X- in four steps at 0.5 ns intervals, totaling 2.5 ns. All restraints are completely removed during production simulation under 310K and bar, which are maintained using the Langevin thermostat and Berendsen barostat, respectively. A timestep of 4.0 used with hydrogen mass repartitioning (Hopkins al., 2015). Bond lengths are constrained via SHAKE (Ryckaert al., 1977), and non-bonded interactions are cut off 10 Å.
H.7 Visualization of Structure Ensembles Simulated by Molecular Dynamics
We present additional views of the structure ensembles generated by molecular dynamies simulations in Figure 12 and Figure 13.
I Examples of Designed Cyclic Peptides
Here, we present more compelling results of our generated cyclic peptides targeting different receptors in Figures 9, 14 and 15. We find that our generated cyclic peptides consistently exhibit higher or competitive affinities with greater interaction stabilities when binding to the receptor. contrast, linear peptides sampled from PepFlow often result in unstable structures and weaker binding. Furthermore, our designed cyclic peptides not only interact with key receptor regions, similar linear peptides and native peptides, but also establish new, stable, and tight interactions in additional regions. Additionally, generated 3D) eyclic peptides consistently align well with the corresponding 2D chemical graphs, highlighting the effective integration of our two models.
PDB: 7mhz
PepFlow
Our (head-to-tail)
Our (chemical graph)
Affinity: Stability: -187.5
Affinity: -29.8
Affinity: -59.5
Stability: -148.7 -148.77
Stability: -217.9
PDB: 1d8d
PepFlow
Our (head-to-side)
Our (chemical graph)
Affinity:
Affinity: -28.9
Affinity: -44.1 Stability: -1091.3
Stability: -1131.2
Stability: -1049.4
Figure 9, Visualization of reference ligand, lincar peptides designed by PepFlow, and cyclic peptides designed by CPSDE.
Limitations and Future Work
One limitation is that the generated cyclic peptides may sometimes exhibit invalid conformations, such as inaccurate bond lengths and atomic receptor clashes. While Rosetta-basee structure relaxation (Chaudhury et al., 2010; Alford et al., 2017) can refine these structures, it computationally expensive and slow. Additionally, for evaluation, we are currently unable to introduce self-consistency metrics similar to those in protein design (Yim et al., 2024), as there no highly accurate cyclic peptide structure prediction or docking model available.
24

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
Table Stability and affinity of the reference peptide, linear peptides designed by baseline methods, and cyclie peptide designed by method along with the cyclization type.
Reference
RFDillusion
ProteinCieneralor
PepFlow
PepGLAD
Target
Stab.
Ath.
Stab.
Affi
Stab.
Ath.
Stab.
Affi
Stab.
Afti
Stab.
Afti.
Type
Id8d
1131.2
45.1
1092.2
37.2
975.5
43.7
1049.4
28.9
1070.00
-30.1
1091.3
44.1
h2s
1hr8
-417.7
-88.4
-363.9
-78.7
-370.0
-39.3
-4288
-36.8
-00788
-28.5
-313.1
-37.2
x2t
1rg4
-374.0
-140.2
-390.1
-153.6
-289.1
-121.1
-167.5
-69.3
82.6
-67.2
-255.1
-83.5
821
Ivzj
-80.1
-18.77
282.8
-663
356.
-52.8
12.4
-12.22
20.4
-0.6
143.4
-105.3
h2s
Ixoc
-89933
-66.9
-763.8
-40.7
-866.0
-41.2
-886.2
-51.1
-778.8
-30.8
-867.3
-4..7
h2s
Izkk
824.2
47.00
843.0
59.0
-824.0
400.2
-797.3
30.0
439.0
33.5
-93.22
62.9
h21
2anq
556.3
1148.5
-58..7
165.9
-28.55
29.2
428.2
63.5
204.1
-3311
.38.3
91.5
h2s
2mpz
-1136.2
-223.1
-1114.8
-00.77
-1031.2
-91.4
-84.0
-79.3
-1059.0
-244.1
h2s
2vda
223.6
373.1
-25.9
2wqj
-576.8
-123.3
-125.7
2xjz
95.0
-164.0
234.9
-102.2
343.4
-8.11
251.8
-2.00
N/A
NA
375.2
-46.6
h21
Beße
1144.4
-53.3
-1123.9
-0.8
11119.1
39.1
1020.6
20.3
-5515
:27.0
1081.7
-37.4
h21
3och
-569.4
-101.1
-541.2
-78.8
-476.6
-84.8
-446.8
44.00
-231.
-65.0
-436.8
-56.0
h2t
Bewf
1453.3
-36.4
-1425.1
-16.9
-1433.9
-11.7
-1356.4
0.4
-1390.7
-11.9
1484.1
-48.0
h2s
36i
2900.1
105.2
117..1
88.8
-173.3
-34.7
-29.4
30.5
-39.3
100.2
100.3
-55.9
h2t
3h8a
-528.3
-71.1
-388.8
68.6
-446.2
-57.7
-400.1
-8.1
232.1
-38.3
442.8
-45.2
h2s
3j89
919.6
-118.1
99083
-115.3
-876.1
-84.7
-771.6
59.99
N/A
N/A
-744.2
-56.5
h2t
31k4
-263.2
-101.4
-215.7
-80.6
-176.2
-26.4
-256.9
-68.3
N/A
N/A
-223.3
-31.5
h2s
3mhp
1021.0
-93.4
-948.3
-64.0
-851.9
-13.8
-903.4
-3.8
-682.8
-99.5
-849.0
-38.0
h2t
300e
-33800
3p17
104.2
98.1
h2s
3ro2
606.5
-86.2
-376.7
-32.1
-345.0
-31.6
-539.1
-52.5
-117.2
-5.8
-378.1
-2.99
h2s
3ryb
990.0
-4499
-993.
-594
-89411
-34.5
1003.8
-18.2
-857.7
-37.0
-942.6
-47.4
h21
3twt
.65.3
-57.3
-662.1
-23.2
-851 4
-14.4
-936.2
-31.5
-782.6
-31.8
-89.3
-36.6
h2s
3vvs
.91.1
-55.1
632.5
-40.1
634.7
-99.6
-692.
50.3
-63.00
-38.1
-72255
-63.7
h2s
3wy9
618.2
.59.8
522,1
-73.3
610.9
56.6
-23.77
34.0
.13.77
.9.7
-50.8
-24.4
h21
3zha
-404.5
-122.1
-18788
-37.2
-13699
-62.0
-308.1
-60.1
-51.7
-68.2
-261.4
-55.9
h2s
Achg
993600
-112.9
-790.8
-59.4
-809.6
-71.7
-790.1
-66.7
N/A
N/A
-714.0
-64.2
h22
4e7v
.90044
-125.4
N/A
N/A
N/A
NIA
-38.0
-73.0
53.7
.96.0
.473.1
-158.2
h2t
Aedn
800.6
-75.1
-815.0
-75.6
-694.
-24.8
-760.6
-4.33
-521.4
-35.6
-711.4
-53.3
h2t
-1108.6
-118488
4jo6
.25
-81.9
-927.8
-861.7
-40.6
-792.0
-3.66
-131.5
-45.4
$2s
4m1c
-15.44
41.1
-989
-26.8
-16.6
-28.4
-100.1
-33.6
-74.8
-25.5
-102.3
-30.9
h2s
4o6f
439.3
49.99
422.1
66.9
425.3
49.2
.28.8
32.3
.370.9
-21.7
425.6
.4.1
h2t
4po7
.30.22
-2.9
-94.3
.5.0
-244.1
-56.2
-294.1
25.4
266.99
20.9
-61.99
-24.9
h2t
Aque
.91.44
-94.9
-945.3
-67.4
-927.2
-6644
-870.7
-56.3
-198.1
-55.6
-851.1
-54.3
$28
4ugz
-733.4
-96.8
-6528
-30.9
-640.8
-35.5
-557.0
39.6
-555.3
-38.1
-628.2
-48.0
$2s
4wsi
-2
-93.5
-145.6
-58.1
-107
-34.6
-180.7
30.9
-85.3
-45.6
-210.8
-32.0
h2t
4x30
495.0
-28.0
.494.6
30.6
-420.4
-1.3
441.6
468.4
-16.8
-85.4
-83.4
h2
4spd
114.1
-23.3
.6.11
20.3
-78.3
30.0
227.9
6.3
-15.6
115.7
48.9
h2t
-6602
-110.7
-659.0
-662.6
-77.0
-372.5
-57.4
-534.9
-62.6
h2s
Ayjl
1013.7
-30.8
4pp
-19222
-42.2
24.9
-39.5
h2s
5apk
-37.88
-62.1
-240.0
-65.1
-146.3
-6.7
-33.2
-60.7
h2t
Sbmm
469.6
-72.2
337.66
38.4
-375.3
-24.4
-405.0
42.6
-30.6
44.9
414.8
-43.8
h21
Schh
.358.6
.83.3
330.22
87.3
220.6
.663
.3013
92.8
-29.3
65.9
231.5
.6.66
h21
5dhm
-89.5
-155.4
-607.8
-149.4
-451.7
-64.2
-483.1
-39.4
-123.1
-71.9
-424.9
-53.1
h2t
Se2q
977.0
-65.2
866.66
-37.4
839.6
-28.8
932.55
47.5
-819.0
.31.9
.90244
58.5
<2s
Serl
1436.7
-77.3
-1321.0
.4144
-1391.5
-54.9
-1318.7
33.9
.592.4
-37.1
-1313.0
-26.5
h2:
Siyx
744.4
-59.5
-612.4
-7.7
-652.9
-37.8
-659.4
34.77
-564.2
-28.3
-653.3
-38.1
.718.3
-102.9
-64.88
-57.4
-586.4
-33.8
-509.8
27.3
.21188
80.0
-501.0
-36.7
621
5mfg
-1215.8
-42.3
-1171.4
-28.1
-1195.7
-15.5
-1236.5
-2.22
-1049.8
-36.3
-1214.7
-40.0
h2s
5mjy
-1527.3
-79.4
-1538.3
-68.7
-1369.1
-1482.5
-58.8
-1299.2
-43.8
-1425.3
-45.5
h2s
5n4d
.1219.5
Sall
-669.8
-79.2
-677.0
-87.6
-702.1
-91.6
-572.6
-6.00
-354.6
-56.7
-532.3
-44.3
52s
Stxe
-95566
-49.6
-930.2
-18.3
-892.9
-88.2
-883.6
-3.88
-860.2
-38.9
-879.2
-41.1
h2t
5v19
.6224
-128.5
-546.2
-97.0
-$20.6
-72.4
-469.3
-66.0
-58.2
-438.5
-51.6
h2s
5wkf
258.4
-72.2
343.2
-36.4
384.2
-25.3
322.4
-4.88
408.0
-38.8
303.2
-54.4
62t
Swpl
.352.8
.9.6
380,7
-91.3
.28444
.59.4
216.6
.6.4
N/A
N/A
117.33
-5.44
<2s
Syis
416.4
.9.11
.29.11
.34.1
.26..1
-57.8
265.5
.50.1
-149.5
-50.0
-00.1
-6.1
<2x
25

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
Table Stability and affinity of the reference peptide, linear peptides designed by baseline methods, and cyclie peptide designed by our method along with the cyclization type.
Target
Reference
RFDifTusion
ProteinCenerralor
PepElow
PepGLAD
Our
Stab.
Afri
Stab.
Ath.
Stab.
Afri
Stab.
Affi
Stab.
Afñi
Stab.
Athi
Type
5zw6
3361.4
46.9
298..0
-14.3
497.5
29.3
-338.0
41.4
306.9
28.55
-364.7
-1.5
h2s
6bli
-2699.
-104.4
-1231.
-62.2
-1231.4
-46.5
-1147.9
-88.4
N/A
NIA
-1094.2
-39.5
h2s
6cv1
-490.1
119.11
-425.7
-131.3
-284.3
-44.2
-254.6
-38.5
N/A
NIA
-234.9
-36.1
h21
6di8
1850.6
-73.9
-1805.2
-49.3
-1795.9
-4.0
-1783.6
-52.4
-1344
-37.5
-1793.5
-59.9
h21
6dtg
-874,6
-60.3
-792.4
-1.55
-698.6
-27.9
-847.4
-32.4
-783.4
-22.8
-898.9
-52.4
h2s
610k
665.8
80.0
499..4
-5.00
-54..4
46.0
-29.00
33.7
280.6
-5.8
-399.4
38.9
h2s
616d
-71.9
88.3
666.2
47.1
653.4
63.8
674.9
39.8
.99.7
-25.0
-666.4
39.3
h2
6g68
-303.2
-21.66
-128.1
-352.1
-126.1
-83.8
N/A
-15.7
-5799
h2t
6ghr
1342.1
-6699
-1198.8
77.77
-9.8
-1212.9
30.0
-66..1
-1224.1
-50.6
h2t
1323.9
-87.2
1255..5
-37.0
-1222.3
-29.4
-1142.0
26.6
-662.6
-36.7
-1238.7
-5..0
6igk
785.6
104.99
-768.3
84.0
-762.4
-64.6
-698.1
-75.3
447255
-52.7
-680.6
-62.2
h21
6jbk
938.6
-7..7
-933.7
76.5
-88122
42.22
-77.9
50.2
-35.99
-63.7
-847.5
40.55
h2s
6oop
-754,5
-31.1
-672.6
-15.4
-565.0
-31.0
-716.6
-30.7
-552.5
-22.7
-676.5
-34.2
h2s
60m4
-690.0
-67.6
-582.3
49..2
-506.8
-9.4
-662.7
-25.1
-885.6
-22.6
-710.9
-47.6
h2s
6p02
.134.33
2202.1
-1136.5
177.3
1059.6
-150.6
861 6
83.6
-679.6
-81.1
-854.0
94.5
h2s
6peu
.893.2
-78.2
-832.3
-46.2
-736.2
-7.2
-935.1
-61.8
-807.9
-22.6
-860.5
-4..7
s2s
645r
-529.4
-44.9
-585.9
-132.9
-566.2
109.00
-437.5
-98.9
N/A
NIA
-401.3
-60.9
x21
6qsl
-91.22
-4,44
-847.3
-65.4
-871.5
-36.2
-893.4
-38.9
-787.7
-23.0
-903.7
-52.6
h2s
6r16
-330..4
-10..6
-1309.5
-82.5
-1263.7
-4788
-1196.3
-50.6
-440.0
-64.7
-1181.7
-50.5
h2t
6rux
28.1
1069.6
-1151.5
20.3
-1207.8
-37.6
6rxr
.49.00
49.5
.427.0
466.3
33.88
-80.7
499.6
s2t
6sa8
-461.4
-62.0
-307.1
-4..0
-364.0
-2544
-404.2
-35.4
-247.5
-27.8
-424.9
-33.1
h2s
6tsw
1085.1
-44.4
10744.5
-8800
-947.0
400.2
1093.99
-36.1
-544.2
-37.3
-1080.4
-88.9
h2t
Gyln
-393.6
200.3
-358.1
191.0
39.8
-9.77
-153.8 8
-84.7
264.3
-68.4
-142.2
-69.2
h2t
6zw0
-589.9
105.77
-576.7
-15.6
-405.2
-18.5
-558.6
-61.4
-294.6
-38.9
-426.0
-00.5
h2s
7atr
1056.1
66.1
912.3
35.8
1036.8
-3..5
991.2
33..1
.978.0
-33.5
-999.1
49.0
h21
7bek
-561.5
-89.4
-594,4
-9.77
-614.1
-91.1
-486.8
-47.7
-345.7
-49.7
-487.3
-52.1
h2s
7elb
-342.6
-50.3
NIA
N/A
-331.3
-45.8
-276.4
-34.4
-270.9
-37.7
-387.1
-63.7
h2s
78hh
-238.6
-5.33
NIA
N/A
-149.4
-24.6
-186.3
-38.7
-185.7
-35.7
-271.7
-57.3
h2s
7mhz
-187.5
-31.9
N/A
N/A
-74.8
-27.0
-148.7
-29.8
-48.7
-34.2
-217.9
-59.5
h2t
7kkp
36.88
N/A
-2218
-20.4
-27.6
-522.7
-17.2
36.66
h2t
7own
-469.1 I
-5088
NIA
X/A
454.9
-39.1
-464.5
-23.4
-293.0
-27.0
-487.5
-62.8
s2t
7Q66
-375.5
189.6
NIA
X/A
-30.4
-130.5
-17.6
-88.6
241.7
-54.4
-96.6
-80.0
h2s
7ure
41.8
50.4
65.1
61.4
101.5
40.2
120.6
23.2
114.4
.2.22
56.4
40.7
h2s
7vh7
-156.8
.9244
62.6
78.2
.23.8
-3311
47.1
33.2
N/A
NIA
19.1
20.6
h2s
7vwo
-647.0
-93.5
-519.6
-5.2
-435.8
-11.1
-464.2
-44.9
-125.6
-72.4
-523.5
-54.8
h21
7wvx
-314.7
-81.9
-170,0
-54.2
-21133
-45.4
-159.7
-10.5
-177.8
-58.6
-254.4
-64.9
h21
7xxf
-243.6
-73.7
-281.3
-71.4
-259,1
-71.7
-154,1
-54.7
-64.0
-54,4
-115.9
-44.2
h2s
7yal
655.4
-24.22
-632.5
233.3
-24.9
-3.4
-356.3
82.7
859.3
-65.8
-532.1
-241.8
h2s
8dg4
843.5
-91.9
-760.5
.344
654.9
-23.8
621.5
-18.1
-651.9
-41.0
-710.6
40.44
h2i
Property-guided sampling can be incorporated during generation to generate chemically and structurally valid cyclic peptides (Dhariwal & Nichol, 2021; Ho & Salimans. 2022). For example, the process can be conditioned on predefined bond length and angle distributions sample cyclic peptides with specific shapes. Additionally, energy- sampling (Lu et al., 2023; Kulytě et al., 2024) and energy-based preference optimization (Zhou al., 2024a;c;d; Cheng et al., 2024) can guide the generation of low-energy, stable conformations. Techniques and architectures related to AlphaFold could also be leveraged for more accurate atomic interaction modeling (Abramson et al., 2024). For evaluation, we believe it is crucial to validate the cyclic peptides generated by our model in the wet lab, determining their accurate structural conformations and binding modes-an avenue we are actively exploring.
Additionally, we would like to point out that our current method does not explore how to automatically select the best cyclization type for a given receptor, although it can be enumerated. We plan to investigate this aspect in our future work. Other future research includes cyclic ligand peptide design considering flexible protein targets (Zhou et al., 2025) or dual targets (Zhou al., 2024b).
26

---

Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
H2T-1
H2T-2
affinity: -32.9 stability: -324.6
affinity: -32.1 stability: -353.6
H2T-3
H2T-4
affinity: -29.9 stability: -334.4
affinity: -27.3 stability: -351.3
H2T-5
H2T-6
affinity: -26.5 stability: -329.2
affinity: -33.9 stability: -356.9
H2T-7
H2T-8
affinity: -25.7 stability: -353.4
affinity: -26.4
stability: -323.6
Figure 10. Head-to-tail cyclic peptides designed for target SMYD2.
27

---

Designing Cyclic Peptides via Harmonic SDE with Atom- Bond Modeling
S2S-1
S2S-2
affinity: -42.9 stability: -773.8
affinity: -49.0 stability: -541.2
S2S-3
S2S-4
affinity: -43.1 stability: --787.5
affinity: -38.8 stability: -773.6
S2S-5
S2S-6
affinity: -43.1 stability: -297.6
affinity: -43.1 stability: -554.9
S2S-7
S2S-8
affinity: -50.0 stability: -784.7
affinity: -43.2
stability: -542.2
Figure Side-to-side eyclic peptides designed target SET8.
28

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Boot Bond Modeling
PDB: 4o6f
Pepflow
Our (head-to-tail)
Figure 12, Structure ensembles of SMYD2 from multiple perspectives.
29

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
PDB: 1zkk
Pepflow
Our (side-to-iide)
Figure 13. Structure ensembles of SETS from multiple perspectives.
30

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
PDB: 1zkk
PepFlow
Our (head-to-tail)
Our (chemical graph)
Affinity: -47.0 Stability: -824.2
Affinity: -30.0
Affinity: -62.9
Stability: -797.3
Stability: -793.2
PDB: 6rxr
PepFlow
Our (side-to-tail)
Our (chemical graph)
Affinity: -66.1
Affinity: -35.8
Affinity: -192.0
Stability: -449.0
Stability: -466.3
Stability: -499.6
PDB: 4jo6
PepFlow
Our (side-to-side)
Our (chemical graph)
Affinity: -81.9
Affinity: -32.6 Stability: -792.0
Affinity: -45.4
Stability: -925.4
Stability: -759.1
PDB: 5zw6
PepFlow
Our (head-to-side)
Our (chemical graph)
Affinity: -46.9 Stability: -361.4
Affinity: -41.4 Stability: -338.0
Affinity: -51.5
Stability: -364.7
PDB: 7okp
PepFlow
Our (head-to-tail)
Our (chemical graph)
Affinity: -36.8 Stability: -845.2
Affinity: -27.6
Affinity: -36.6
Stability: -793.1
Stability: -857.3
Figure 14, Examples of reference peptides, linear peptides designed by PepFlow, and eyclic peptides designed by method.
31

---

Designing Cyclic Peptides via Harmonic SDE with Atom-Bond Modeling
PDB: 7f6h
PepFlow
Our (head-to-side)
Our (chemical graph)
Affinity: -50.3 Stability: -238.6
Affinity: -38.7
Affinity: -57.3
Stability: -186.3
Stability: -271.7
PDB: 7eib
PepFlow
Our (head-to-side)
Our (chemical graph)
Affinity: -50.3 -344
Affinity: -34.4
Affinity: -63.7
Stabllity: -276.4
Stability: -387.1
PDB: 6trw
PepFlow
Our (head-to-tail)
Our (chemical graph)
Affinity: -44.4 Stability: -1085.1
Affinity: -36.1
Affinity: -48.9
Stability: -1093.9
Stability: -1080.4
PDB: 3e8e
PepFlow
Our (head-to-tail)
Our (chemical graph)
Affinity: -52.3 -ability:-11444
Affinity: -20.3
Affinity: -37.4
Stability:-1024 -1020.6
Stability: -1081.7
PDB: 5yis
PepFlow
Our (side-to-side)
Our (chemical graph)
Affinity: -99.1 Stability: -416.4
Affinity: -50.1
Affinity: -56.1
Stability: -265.5
Stability: -302.1
Figure Examples reference peptides, linear peptides designed by PepFlow, and cyclic peptides designed by our method.
32


Reviews:
[{"review_id": "jjBrEfVxO9", "reviewer": "Reviewer_Rxgy", "summary": "The paper proposes an approach for the design of cyclic peptides using score-based generative models and diffusion. It is termed harmonic SDE, mainly because of conditioning on chemical graph that gives rise to a slightly non-standard forward process. The approach has been evaluated in peptide design against few other approaches, but with a caveat that their outputs are linear rather than cyclic peptides. The main aspects in empirical evaluation deal with stability, affinity, and diversity.\n\n## POST-REBUTTAL ##\nThank you for the hard work. I will increase my score but please do a meaningful revision of the final paper.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1: \"I find the idea of conditioning generative models using chemical graphs interesting, and also the problem of designing cyclic peptides highly relevant for therapeutics purposes. However, the empirical evaluation of both ideas is inadequate.\"**\n\nA1: Our work focuses on designing cyclic peptides instead of linear peptides, although our method is indeed capable of designing the latter as well. Our method is tailored specifically for cyclic peptide design. Designing linear peptides often involves modeling translation, rotation of residue frames, and side-chain torsion angles—techniques that essentially encompass all-atom coordinate modeling, as seen in methods like PepFlow. However, these techniques fall short for cyclic peptide design due to unique geometrical constraints and non-canonical amino acids involved. This challenge motivated us to introduce two modules: AtomSDE and ResRouter. These modules enable explicit modeling of all atoms and bonds, addressing the limitations of previous methods in cyclic peptide design. In the generative process, AtomSDE adjusts atom coordinates based on chemical graphs, while ResRouter refines these graphs by predicting amino acid types. We hope this explanation clarifies the motivation and contributions of our work.\n\nBesides, we have ablated the effect of Harmonic AtomSDE and ResRouter in designing cyclic peptides. Please refer to Appendix H.3. \n\n**Q2: Evaluation of extra boost in linear peptide design.**\n\nA2: While designing linear peptides is not our primary focus, we appreciate your suggestion to compare our method with existing baselines in this task. \n\n|Method|Co-Design|Peptide Type|Stability||Affinity||Diversity|\n|---|-|-|-|-|-|-|-|\n||||Avg.|Med.|Avg.|Med.||\n|Reference||Linear|-672.53|-634.71|-85.03|-78.70||\n|RFDiffusion|N|Linear|-633.51|-607.82|-70.30|-61.35|0.55|\n|ProteinGenerator|Y|Linear|-576.39|-554.70|-46.98|-40.39|0.58|\n|PepFlow|Y|Linear|-576.16|-498.31|-47.88|-42.40|0.70|\n|PepGLAD|Y|Linear|-359.44|-310.33|-45.06|-38.56|0.79|\n|CpSDE|Y|Linear|-567.34|-510.58|-55.48|-49.89|0.77|\n\nOur method excels in designing linear peptides with superior affinity and comparable stability among all co-design methods, while also maintaining considerable diversity. Although our method slightly lags behind RFDiffusion, it's worth noting that RFDiffusion often generates helices and relies (also observed for ProteinGenerator) on a two-stage design pipeline (first designing the backbone, then the sequence). \n\n**Q3: \"Data in crystal form would also allow for evaluating structural properties and reporting against these metrics.\" \"The second aspect is merits of cyclic peptides design which is challenging to evaluate due to the lack of crystal structures of complexes.\"**\n\nA3:  Evaluating energy is more critical than assessing RMSD against crystal structures of known binders, as there can be numerous design alternatives and a low RMSD does not always indicate a superior design model [1]. Therefore, we focus primarily on evaluating stability and affinity from an energy perspective. To further validate our findings, we conduct Molecular Dynamics simulations.\n\nWe have also designed linear peptides as in Q1&A1 and computed the RMSD against known linear peptide binders, shown as follows:\n\n|Method|Co-Design|Peptide Type|RMSD(Avg.)|RMSD(Med.)|\n|---|-|-|-|-|\n|RFDiffusion|N|Linear|2.85|2.27|\n|ProteinGenerator|Y|Linear|3.54|3.12|\n|PepFlow|Y|Linear|1.60|0.98|\n|PepGLAD|Y|Linear|2.17|1.19|\n|CpSDE|Y|Linear|1.43|1.17|\n\nOur method achieves the lowest average RMSD among all methods, along with a comparable median RMSD.\n\n[1] Antigen-specific antibody design via direct energy-based preference optimization, Zhou et al. NeurIPS 2024.\n\n**Q4: \"What fraction of data is actually cyclic peptides and why this dataset would be relevant for generating from that class.\" \"If the data on cyclic peptides is sparse how the model can mine useful signal for completing cyclicazation.\"**\n\nA4: Our method successfully designs cyclic peptides despite the training set containing less than 6% cyclic peptide data. This achievement stems not from relying on this small percentage, but from our approach of explicitly modeling all atoms and bonds, surpassing previous methods in depth. Through this fine-grained model, linear and cyclic peptides are treated equivalently as both are composed of the same fundamental components: atoms and bonds. The cyclization type is predetermined, allowing us to specify the related atom types and bond types accordingly. By incorporating bond modeling, we ensure cyclization occurs in 3D space since bound atoms are naturally drawn closer during the generative process. The ResRouter model equally sidesteps dependency on scarce cyclic peptide examples due to its ability to deduce amino acid types based on the contextual atom arrangement within the peptide and its receptor. Thus, it is this distinctive and comprehensive modeling technique that enables overcoming data limitations in cyclic peptide design.", "claims_and_evidence": "I find the idea of conditioning generative models using chemical graphs interesting, and also the problem of designing cyclic peptides highly relevant for therapeutics purposes. However, the empirical evaluation of both ideas is inadequate. The experiments essentially compare apples-to-oranges in assessing designs that are linear vs cyclic peptides. What would make sense is to have the approach first design only linear peptides and evaluate the extra boost that comes as a result of conditioning on chemical graphs relative to several different architectures/baselines. My understanding is that data in crystal form would also allow for evaluating structural properties and reporting against these metrics vs same approaches without such conditioning. The second aspect is merits of cyclic peptides design which is challenging to evaluate due to the lack of crystal structures of complexes.", "methods_and_evaluation_criteria": "There are two components in the approach: i) ATOMSDE that is a docking model trained using both small molecules and peptides as ligands. It is unclear what fraction of data is actually cyclic peptides and why this dataset would be relevant for generating from that class. ii) RESROUTER that predicts amino-acid from aggregated hidden stats representing backbone atoms. Still the question is if the data on cyclic peptides is sparse how the model can mine useful signal for completing cyclicazation.\n\nPlease see above for more details on evaluation.", "theoretical_claims": "Not applicable", "experimental_designs_or_analyses": "see above", "relation_to_broader_scientific_literature": "Good coverage of related work on generative models for peptides", "essential_references_not_discussed": "Good coverage of related work", "other_strengths_and_weaknesses": "see above", "other_comments_or_suggestions": "see above", "questions_for_authors": "see above", "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": "eLYEQZK5TN", "reviewer": "Reviewer_NNQv", "summary": "This paper describes a generative method to design cyclic peptides given a protein target. The method uses two diffusion models utilized in a coupled fashion. One to generate the structure, the other to predict the sequence.\n\n## Update After Rebuttal\nI thank the authors for addressing my review. I have decided to stay with my rating of 4.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1: \"Comparing the generated ligands to known cyclic peptides.\"**\n\nA1: We conducted a comparison of our method with known cyclic peptides. \n\nVasopressin, a natural cyclic peptide featuring intramolecular disulfide (S-S) bond cyclization, is utilized in the treatment of antidiuretic hormone deficiency, vasodilatory shock, gastrointestinal bleeding, ventricular tachycardia, and ventricular fibrillation [1]. We selected two vasopressin-protein complexes, PDB IDs: 1JK4 and 1YF4, to design cyclic peptides targeting bovine neurophysin II and trypsin, respectively.\n\nThe LEDGF binding site of HIV integrase (HIV-IN) represents a promising target for novel inhibitor development. Prior research has leveraged solution cyclization to discover various head-to-tail cyclic peptides interacting with this site [2]. The PDB IDs are 3A\\*\\* in the following table.\n\nThe results are shown as follows:\n\n|PDB|Ref.||Our||\n|---|-|-|-|-|\n||Stab.|Affi.|Stab.|Affi.|\n|1jk4|-307.23|-43.17|-215.91|-96.67|\n|1yf4|-52.36|-37.60|-47.03|-46.74|\n|3ava|-276.09|-30.49|-246.03|-31.98|\n|3avb|-248.83|-32.31|-233.89|-31.86|\n|3avg|-314.92|-31.15|-276.46|-37.93|\n|3avh|-278.42|-32.41|-255.95|-31.74|\n|3avi|-252.17|-42.72|-222.94|-33.89|\n|3avj|-381.38|-41.21|-334.38|-30.35|\n|3avk|-308.66|-38.18|-281.07|-30.07|\n|3avl|-306.86|-30.78|-220.60|-104.93|\n|3avm|-252.95|-31.78|-227.98|-26.94|\n|3avn|-262.63|-29.71|-228.48|-32.44|\n\nThe results demonstrate that our method successfully designs cyclic peptides with affinity and stability comparable to known cyclic peptides, albeit with slightly lower stability than the reference.\n\n[1] Vasopressin: physiology, assessment and osmosensation, Bichet et al. Journal of Internal Medicine, 2017.\n\n[2] Crystal Structures of Novel Allosteric Peptide Inhibitors of HIV Integrase Identify New Interactions at the LEDGF Binding Site, Peat et al. ChemBioChem, 2011.\n\n**Q2: \"Figure 1 felt unnecessary, to me.\"**\n\nA2: Figure 1 illustrates the therapeutic advantages of cyclic peptides compared to linear peptides. We would like to move this to the appendix.\n\n**Q3: Explain Figure 2, especially the denoising/renoising/ResRouter coupling. \"Some intuition on why the renoising is necessary would be helpful.\"**\n\nA3: The generative process is a reverse SDE, where each Euler step consists of two parts: drift (denoising) and diffusion (renoising). See Equation (2) in the paper where the Wiener process introduces the stochasticity.\n\nIn Figure 2, AtomSDE adjusts the Atom73 coordinates through denoising and renoising, treating the denoised ligand as an intermediate state. ResRouter alters the chemical graph by predicting amino acid types based on the denoised ligand, which provides more useful signals than the noised ligand. Importantly, ResRouter changes only the atom and bond types in the chemical graphs and leaves the atom73 coordinates unchanged. Similar operations are utilized in [3].\n\n[3] An all-atom protein generative model, Chu et al. PNAS, 2024.\n\n**Q4: \"AtomSDE only includes the protein target. How are different pockets specified?\"**\n\nA4: AtomSDE includes both pockets and noisy ligands. Our method aims to design cyclic peptides based on a given binding site. We refer to the protein target as the pockets. This setup aligns with the baselines, such as PepFlow. Specifically, the pockets are defined as residues of the protein target within 10 Angstroms surrounding a known ligand. We will include more details and clarifications in future version of our paper.\n\n**Q5: Explain page 2, line 091.**\n\nA5: Our method models all atoms and bonds, treating linear and cyclic peptides equivalently since both are composed of the fundamental components—atoms and bonds. Assuming AtomSDE is a well-trained docking model, it facilitates the generative process by encouraging two atoms connected by a chemical bond to be positioned closer together appropriately. By predetermining the cyclization type, we specify the related atom and bond types accordingly. Incorporating bond modeling ensures cyclization occurs naturally in 3D space, as bound atoms are drawn closer during the generative process.\n\n\n**Q6: Page 3, line 154: Why are the 3D structures of the cyclization part unavailable?**\n\nA6: The 3D structures of the cyclization segment are unavailable because they belong to what we aim to design.\n\n**Q7: \"In 4.1, it is not clear to me how the training/test splits are generated using the sequence identity.\"**\n\nA7: Samples are grouped by receptor sequence similarity of 0.3 to create separate training and validation sets. In other words, if two samples are more than 30% similar in sequence, they cannot be in the same set. When receptors have multiple chains, the sequences are concatenated together to determine similarity. This method is widely used in previous works.\n\n**Q8: \"Can CpSDE be adapted for use on linear peptides? This would allow for a head-to-head comparison against other methods.**\n\nA8: Please refer to our responses to Reviewer Rxgy's questions 2 and 3.", "claims_and_evidence": "The paper shows comparison with existing methods for linear peptides, and also shows predictions for some example targets. Overall, I think the evidence is adequate, but one evaluation that I would be interested in seeing is a comparison with a known therapeutic cyclic peptide. How well do the ligands generated by CpSDE compare against those?", "methods_and_evaluation_criteria": "Covered above.", "theoretical_claims": "N/A.", "experimental_designs_or_analyses": "The experiment designs were adequate.I did not review them in depth, but referred to them to clarify some points.", "relation_to_broader_scientific_literature": "The key contributions were well-situated within the existing literature in the Related Work section.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "I think the work is innovative and interesting. I would have liked to see more convincing evidence that the benefits have real-world applicability, including, as mentioned above, comparing the generated ligands to known cyclic peptides.", "other_comments_or_suggestions": "Figure 1 felt unnecessary, to me.\n\nOverall, Figure 2 is not very clear, especially the denoising/renoising/ResRouter coupling. Seems to show repetitive denoising without re-noising. In 3.3, it is stated that the ligands are noisy. But in the figure, it shows ResRouter using denoised ligands.\n\nSome intuition on why the renoising is necessary would be helpful.", "questions_for_authors": "* AtomSDE only includes the protein target. How are different pockets specified?\n* Page 2, line 191: The sentence beginning “The inclusion…” could be explained a bit more.\n* Page 3, line 154: Why are the 3D structures of the cyclization part unavailable?\n* In 4.1, it is not clear to me how the training/test splits are generated using the sequence identity.\n* Can CpSDE be adapted for use on linear peptides? This would allow for a head-to-head comparison against other methods.", "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": "aQIeeLLL09", "reviewer": "Reviewer_H4HQ", "summary": "This work tackles the task of cyclic peptide design. Cyclic peptides can have unique advantages in terms of stability and affinity when producing binders compared to other types of peptides or ligands. While there is much work in small molecule as well as protein and peptide generation, there is no prior work on producing cyclic peptides, a gap in the literature, which this paper fills. To this end, the authors propose CpSDE, a method consisting of AtomSDE, a structure prediction model, and ResRouter, a residue type predictor. The two components are called in an alternating manner in a denoising diffusion framework to produce novel cyclic peptides. The approach leverages an explicit all-atom formulation and builds on the atom73 representation with a side chain superposition framework used in previous work. CpSDE also includes explicit bond modeling, and cyclization and target information is given as conditioning. The paper computationally validates the approach through energy-based metrics for stability and validity as well as diversity. Moreover, it runs molecular dynamics simulation for selected cases, showing stable conformations of the generated cyclic peptides, thereby supporting high binding affinity.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1: \"It would be great if the authors would release their curated training dataset as well as models and code for the broader community.\"**\n\nA1: We would like to open source our work to contribute to the community.\n\n**Q2: \"I have some minor wording comments.\"**\n\nA2: Thanks for pointing this out. We will change these words and choose more objective words in the future version of our paper.\n\n**Q3: \"What exactly is the role of $\\sigma_P^{-2}$, when calculating $\\mathbf{H}$?\"**\n\nA3: $\\mathbf{H}=\\mathbf{L}+\\sigma_P^{-2}\\mathbf{I}$. $\\mathbf{L}$ is the Laplacian matrix. An intuitive explanation is that $\\mathbf{L}$ encourages the connected atoms in the graph to be initialized closer. $\\sigma_P$ is the standard derivation of the atom coordinates of the protein pocket.  An intuitive explanation is that $\\sigma_P^{-2}\\mathbf{I}$ encourages the atoms to be initialized more scattered when the pocket itself is large. This reflects a useful prior knowledge, as a pocket typically accommodates a ligand that complements its shape.", "claims_and_evidence": "All claims made in the paper are appropriately supported through convincing experiments.", "methods_and_evaluation_criteria": "All methods and evaluation criteria are appropriate for the problem at hand.", "theoretical_claims": "The paper does not have any complex theorems or proofs, so this question does not apply. The maths around the harmonic SDE and the diffusion framework seems correct.", "experimental_designs_or_analyses": "All experimental designs and analyses seem sound and valid to me. I have no concerns.", "relation_to_broader_scientific_literature": "The authors did an excellent job putting the paper in the context of the broader literature and motivating their approach. The paper has a long list of references and an additional discussion of related work in the appendix.", "essential_references_not_discussed": "I was not able to identify any essential work that was not cited.", "other_strengths_and_weaknesses": "**Strengths:**\n- To the best of my knowledge, this is the first protein/peptide design paper that tackles cyclic peptide generation, thereby filling a gap in the literature. This means that the work can be considered impactful and significant.\n- The chosen methodology relies on existing techniques (harmonic SDE diffusion, graph neural networks, atom73 representation, etc.), which themselves are not novel, but these components are put together in a novel, original and well-motivated way for the task at hand.\n- The quantitative comparisons to existing works show that CpSDE performs on-par with previous works. While one may criticize the work for not achieving state-of-the-art performance across the board on all metrics, all existing works only generate linear peptides (and in the case for RFDiffusion only generate non-diverse simple helices). CpSDE opens up the possibility for cyclic peptides, in contrast to all existing works, which is very innovative.\n- The additional validation based on molecular dynamics simulation that goes beyond simple energy and diversity metrics is very nice and convincing.\n- The paper is very well written and clearly explained, with an excellent introductory section, motivating cyclic peptide design and introducing it in an appropriate manner to the machine learning audience.\n- The quality of the figures and visualizations is excellent.\n- As discussed above, the supplementary material is very comprehensive and leaves no questions open.\n\n**Weaknesses:**\nFrankly, this is a great paper, which I enjoyed reading and reviewing, and I was not able to identify any major flaws or weaknesses that would make me question the work.\n\nConsequently, I applaud the authors to their great work and highly recommend the paper for acceptance.", "other_comments_or_suggestions": "It would be great if the authors would release their curated training dataset as well as models and code for the broader community.\n\nMoreover, I have some minor wording comments:\n- Line 124, \"...a groundbreaking approach...\": I believe it is not up to the authors to decide themselves whether their approach is groundbreaking or not. \"Groundbreaking\" is a very strong word. The community will decide this. I would suggest to change this wording.\n- Line 430, Conclusions, \"...CpSDE is a pioneering...\": Same issue, please tone down the wording and let the community judge whether this is pioneering or not.", "questions_for_authors": "- Line 207: What exactly is the role of $\\sigma_P^{-2}$, when calculating $\\boldsymbol{H}$? The paper only says that this is a \"receptor-dependent scalar value\", but no intuitions are given. It would be great if the authors could explain this better.", "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": "BlEW9t5laT", "reviewer": "Reviewer_NXAJ", "summary": "The paper introduces CPSDE, a new model for designing cyclic peptides using harmonic SDE and explicit atom-bond modeling conditioned on a 3D structure of a protein target. CPSDE comprises two key components: a generative structure prediction model and a residue type predictor. Alternating between these two models, CPSDE iteratively updates sequences and structures. CPSDE can be trained on small molecule and linear peptides, removing the need for abundant cyclic peptide data. It handles cyclization and non-standard amino acids. Experimental results show reliable stability and affinity, and some molecular dynamics simulations further validate the model in real-world design scenarios.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1: \"The main weakness I see for this paper in an ML venue is that it is highly specialized to tackle the problem of cyclic peptide design; arguably this is an important application, but the authors seem to propose a rather complex model to handle the cyclic peptide generation task.\"**\n\nA1: Thank you for recognizing the significance of cyclic peptide design. Our method is complex due to two main challenges: limited data on cyclic peptides; common protein representation based on residue frames that inadequately handle unique geometrical constraints and occasional non-canonical amino acids.\n\nOur approach addresses these issues by employing all atom and bond modeling and two integrated modules: AtomSDE and ResRouter. Both are essential and non-redundant. To our knowledge, this is the first generative model for designing cyclic peptides. We hope this sparks further research and advancements in this important field.\n\n**Q2: Could the authors provide a baseline (be it a non ML approach) to compare the performance? I am aware of two works for cyclic peptide generation that are very recent (as such the authors are not required to include them for comparison)) e.g. [1,2] although the latter cannot model non canonical amino acids.**\n\nThank you for highlighting this. Both [1] and [2] are awesome works, yet they significantly differ from our approach.\n\n- [1] is a ligand-based drug design (LBDD) method that models the sequence of cyclic peptides using discrete diffusion, optimized by multiple reward functions. It doesn't explicitly incorporate the 3D structure of target proteins, whereas our structure-based drug design (SBDD) method directly designs ligands based on 3D target structures.\n\n- [2] uses modified RoseTTAFold and RFdiffusion with cyclic relative positional encoding to generate macrocyclic backbones. We've already cited [2] in our paper. It only supports head-to-tail cyclization due to its residue-level encoding limitations, whereas our work accommodates all four types of cyclic peptides.\n\nBoth works are very recent: [1] was released on November 18, 2024, and [2] on December 23, 2024. As neither has released their code, we are unable to directly compare methods at this time. We will cite these papers and discuss them further in future versions of our paper.\n\nReferences: \n\n[1] PepTune: De Novo Generation of Therapeutic Peptides with Multi-Objective-Guided Discrete Diffusion, Tang et al.\n\n[2] Accurate de novo design of high-affinity protein binding macrocycles using deep learning, Rettie et al.\n\n**Q3: \"It would be helpful to compare a benchmark on a setting which can be handled by other models than CPSDE.\"**\n\nA3: Given that all baseline methods, including ours, can design linear peptides, we compare them under these conditions. Please see our responses to Reviewer Rxgy's Q2 & Q3 for more details. Nonetheless, we continue to emphasize that the principal focus of our work is on cyclic peptide design.\n\n**Q4: \"How does the generation time compare to alternative peptide generative models ? given all-atom and bond modeling, the sampling time might be heavy.\"**\n\nA4: We benchmark the average time of generating one peptide for all co-design baselines and our methods on a single NVIDIA A100-SXM4-80GB GPU. See the results below.\n\n\n| Method | Peptide Type | Time (s) |\n|---|---|---|\n| ProteinGenerator | Linear | 31.80 |\n| PepFlow | Linear | 12.09 |\n| PepGLAD | Linear | 4.40 |\n| CpSDE | Cyclic | 16.88 |\n\nGiven that computational drug design does not demand real-time model response, the inference time of our method is deemed acceptable.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "no theoretical claims", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "Current methods for peptide design, mainly focused on linear peptides as they cannot include non canonical amino acids or are not designed to handle the cyclicity constraint. CPSDE addresses these limitations with atom-bond modeling.", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "Strengths:\n* the problem of cyclic peptide generation conditioned on a target structure is an important problem with few work able to handle both cyclization and non canoncial amino acid. as such the problem tackled is new from an application point of view. the fact that the model can be trained on non cyclic peptide is interesting as well.\n* experimental results show that CPSDE outperforms baseline methods of linear peptides in terms of stability, affinity, and diversity. the authors also performed some MD simulations \n\nWeakness:\n* the main weakness I see for this paper in a ML venue is that it is highly specialized to tackle the problem of cyclic peptide design; arguably this is an important application, but the authors seem to propose a rather complex model to handle the cyclic peptide generation task. \n* moreover, I understand there is not many work existing for cyclic peptide generation but the experimental comparison is done against approaches for linear peptide generation. it is thus complicated to understand how the proposed ML model compares to alternatives. could the authors provide a baseline (be it a non ML approach) to compare the performance ? I am aware of two work for cyclic peptide generation that are very recent (as such the authors are not required to include them for comparison)) e.g. PepTune: De Novo Generation of Therapeutic Peptides with Multi-Objective-Guided Discrete Diffusion Tang et al. and Accurate de novo design of high-affinity protein binding macrocycles using deep learning Rettie et al. although the latter cannot model non canonical amino acids; it would be helpful to compare a benchmark on a setting which can be handled by other models than CPSDE.", "other_comments_or_suggestions": "/", "questions_for_authors": "* How does the generation time compare to alternative peptide generative models ? given all-atom and bond modeling, the sampling time might be heavy.\n\n\n\n\n---- Post rebuttal ----\n\nthank you for your rebuttal; I increased my score; the additional comparison on linear peptide generation is useufl in general to understand how this model compares to other models be it on non cyclic peptides.", "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:

Meta-review: Summary:

The paper introduces CPSDE, a new model for designing cyclic peptides using harmonic SDE and explicit atom-bond modeling conditioned on a 3D structure of a protein target. CPSDE comprises two key components: a generative structure prediction model and a residue type predictor. Alternating between these two models, CPSDE iteratively updates sequences and structures. CPSDE can be trained on small molecule and linear peptides, removing the need for abundant cyclic peptide data. It handles cyclization and non-standard amino acids. Experimental results show reliable stability and affinity, and some molecular dynamics simulations further validate the model in real-world design scenarios.

Strengths:

- The work is innovative and interesting. 
- Idea of conditioning generative models using chemical graphs is interesting.
- The problem of designing cyclic peptides highly relevant for therapeutics purposes. 
- First protein/peptide design paper that tackles cyclic peptide generation.
- The chosen methodology relies on existing techniques but these components are put together in a novel, original and well-motivated way for the task at hand.
- The quantitative comparisons to existing works show that CpSDE performs on-par with previous works. While one may criticize the work for not achieving state-of-the-art performance across the board on all metrics, all existing works only generate linear peptides (and in the case for RFDiffusion only generate non-diverse simple helices). CpSDE opens up the possibility for cyclic peptides, in contrast to all existing works, which is very innovative.
- The additional validation based on molecular dynamics simulation that goes beyond simple energy and diversity metrics is very nice and convincing.
- The paper is very well written and clearly explained, with an excellent introductory section, motivating cyclic peptide design and introducing it in an appropriate manner to the machine learning audience.
- The quality of the figures and visualizations is excellent.
- The supplementary material is very comprehensive and leaves no questions open.
- The problem of cyclic peptide generation conditioned on a target structure is an important problem with few work able to handle both cyclization and non canoncial amino acid. as such the problem tackled is new from an application point of view. 
- The fact that the model can be trained on non cyclic peptide is interesting as well.
- Experimental results show that CPSDE outperforms baseline methods of linear peptides in terms of stability, affinity, and diversity. the authors also performed some MD simulations

Weaknesses:

- The experiments essentially compare apples-to-oranges in assessing designs that are linear vs cyclic peptides. 
- Merits of cyclic peptides design is challenging to evaluate due to the lack of crystal structures of complexes.
- A meaningful revision of the final paper is recommended by reviewer Rxgy.
- Would have liked to see more convincing evidence that the benefits have real-world applicability.
- Would have been good to compare the generated ligands to known cyclic peptides (the authors provided additional evidence in rebuttal).
- The authors seem to propose a rather complex model to handle the cyclic peptide generation task.
- The experimental comparison is done against approaches for linear peptide generation.

Recommendation:

All the reviewers vote for acceptance. I, therefore, recommend to accept the paper and encourage the authors to use the feeback provided to improve the paper for its camera ready version.
Decision: Accept (poster)