RINGER: Conformer Ensemble Generation of Macrocyclic Peptides with Sequence-Conditioned Internal Coordinate Diffusion

Thank you very much for your thoughtful feedback on our work. Your critiques have helped strengthen the manuscript substantially. As outlined below, we have addressed the weaknesses you raised through additional experiments and discussions. We have also responded to your questions to provide clarification.

Responses to Weaknesses:

1. We would like to clarify the novelty of RINGER with respect to FoldingDiff to highlight key differences:

• a) FoldingDiff performs unconditional sampling of protein backbones with a focus on designability. No side-chain information is included in the model, nor is FoldingDiff able to model side chains. This unconditional sampling cannot be easily controlled or steered based on fixed residues. Furthermore, evaluation of unconditional sampling is limited because generated samples cannot be compared directly with a ground truth.

• b) In contrast, RINGER performs conditional sampling based on complete structure including side-chain information. Conditional geometry sampling has largely been used in the context of protein structure prediction, where a single static structure is generated rather than sampling a diverse ensemble. RINGER integrates side-chain information to generate complete, all-atom ensembles with high fidelity.

• c) Architecture – RINGER leverages a transformer architecture like FoldingDiff. However, FoldingDiff uses a standard relative positional encoding for 1D sequences, and operates at the residue level. RINGER enables modeling at the atomic level, and hence can be applied to additional macrocycle chemotypes such as replacement of ring atoms. Furthermore, we introduce a cyclic positional encoding that ensures a naturally shift-invariant representation for homodetic peptides (which have no obvious ‘start’ position given their symmetry). We appreciate the idea of an ablation study and will try to finalize such a study by the November 22 deadline.

• d) Problem Domain – Fundamentally, macrocycles represent a unique biophysical modality that have challenging dynamics. RINGER is the first macrocycle-geometry diffusion model to the best of our knowledge.

2. You raise a fair point on the challenges of macrocycle generation compared to other molecules. We agree it is worthwhile to explore whether methods designed for other molecules could be effective here. We have expanded the introduction to better motivate the distinct challenges macrocycles present, including their vast conformational space and complex cyclic constraints. We now also benchmark against two recent state-of-the-art methods, Torsional Diffusion and DMCG, designed for small molecules. We believe this presents a more comprehensive benchmarking of RINGER, providing empirical evidence that specialized methods are needed for accurate macrocycle generation. This helps justify RINGER's novel contributions tailored to this class of molecules.

3. i) The focus of our work is on complete, all-atom conformer generation. We performed studies on unconditional generation to validate that our underlying approach is able to capture the underlying, complex backbone distributions for macrocycles, rather than to claim that we perform SOTA backbone generation; hence we do not believe further benchmarking for unconditional generation is warranted. ii) Similarly, the purpose of reporting detailed backbone metrics is to better understand the ability of these methods to capture accurate backbone geometries; this complements the all-atom metrics and allows us to dissect whether backbone- or side-chain conformations are driving performance. Backbone conformation is a critical determinant of macrocycle discovery and design (see refs 1-3 below by Rezai et al, Bhardwaj et al, and Hosseinzadeh et al). Finally, outside of our studies in the appendix, there currently exist no methods for backbone-specific conformer generation.

[1] Rezai et al. Conformational flexibility, internal hydrogen bonding, and passive membrane permeability: successful in silico prediction of the relative permeabilities of cyclic peptides. J Am Chem Soc, 2006

[2] Bhardwaj et al. Accurate de novo design of membrane-traversing macrocycles. Cell, 2022

[3] Hosseinzadeh et al. Comprehensive computational design of ordered peptide macrocycles. Science, 2017

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Thank you for your thoughtful summary and positive comments on our work. We are pleased you found the architectural innovations tailored for macrocycle generation to be novel and effective. We also appreciate you highlighting the significance of this task for drug discovery and the quality of the manuscript. Your comments have helped strengthen our manuscript, and we are grateful for the suggestions to include additional ablation studies and benchmarking against the latest diffusion models. As outlined below, we have incorporated these valuable recommendations to better showcase the contributions of our work.

Response to Weaknesses:

1. We appreciate the idea of an ablation study and will try to finalize such a study by the November 22 deadline.

2. We agree that benchmarking against the latest diffusion models is valuable. We have incorporated comparisons to Torsional Diffusion and DMCG in the revised manuscript. These results show that RINGER outperforms these methods, providing further evidence for its strengths in diffusing over internal coordinates.

Response to Questions:

1. The redundant torsional angles arise from the cyclic nature of the backbone, where traversing around the ring leads to three dihedral angles between the same four atoms. For an N-membered ring, there are N atoms with 3N (Cartesian) degrees of freedom (x, y, and z for each atom). Converting this to internal coordinates, we remove 3 global translational and 3 global rotational degrees of freedom to end up with only 3N-6 independent degrees of freedom. There are a number of coordinates one can choose to span this 3N-6 dimensional space: A natural choice would be to choose N-1 ring bond distances, N-2 ring bond angles, and N-3 ring bond torsions. We say that the additional 2 bond angles and 3 torsions are "redundant". In fact, there is no unique way of choosing the N-2 bond angles and N-3 torsions, which is why we select the entire 2N-dimensional space (bond angles and torsions, but excluding fixed bond distances) as a natural, yet redundant, representation for the ring.

2. To select the 0.01 rTFD threshold, we found that the rTFD(pre-opt., post-opt.) distribution was bimodal with the split between the two modes being approximately 0.01 (with most of the density <0.01). This indicates that a small fraction of optimizations were able to find Cartesian coordinates that closely satisfy the bond distance constraints at the cost of distorting the torsions to an unacceptable degree.

3. Non-rotatable side chains like phenyl rings have limited degrees of freedom, so they are well-modeled by pre-defined ideal geometries (not necessarily generated from RDKit). Explicitly modeling them in the diffusion process would increase complexity without much benefit.

4. We limited the training data to the 30 lowest-energy conformers for simplicity of training and found good performance relative to adding additional conformers. Our results demonstrate that using 30 conformers allows us to generalize to entire ensembles of hundreds to thousands of conformers; and hence provides a parsimonious choice of training data.

5. Apologies if this was unclear – RINGER already achieves excellent recall prior to xTB optimization whereas other methods require xTB optimization to improve recall. This demonstrates that the underlying diffusion approach is capable of directly generating diverse ensembles (with recall being a measure of this diversity).

We hope these address any concerns you may have!

Thank you very much for your feedback. We appreciate your recognition of the novelty of applying diffusion models to macrocycles. As outlined below, we have addressed your concerns through text clarifications, additional details, and examples.

Responses to Weaknesses:

1. We have extensively updated the methods section in the Supplementary file to add more detail around our method, including a diagram and flowchart in the appendix to more clearly define the key steps and architecture. We will also provide an open-source implementation of RINGER upon accceptance.

2. Thank you for the feedback on task ambiguity. Our paper tackles the key problem of conformer generation, which generates the three dimensional configuration of a molecule given its molecular identity. Hence, we generate its complete, three dimensional structure from its 2D connectivity. To reduce ambiguity, we have updated the abstract and text to clearly state that RINGER takes 2D graphs as inputs and outputs 3D Cartesian coordinates. We believe this also is reflected in the updated system diagram and is now unambiguous.

3. We've updated the text throughout to incorporate plain language that states how prior heuristics- and physics-based methods are insufficient to model macrocycle conformer generation.

Thank you again for your helpful comments to improve our manuscript. We believe the updates and additions to the manuscript, along with the additional results from additional experiments suggested by other reviewers, have significantly strengthened the paper. We hope you will consider raising your score.

Thank you very much for your assessment. We appreciate your recognition of the importance of this problem. As outlined below, we have provided additional details and clarifications to address your concerns.

Responses to Weaknesses:

1. Studying molecular systems with internal coordinates has been well studied, but the majority of literature for diffusion over internal coordinates (angles and torsions) have been recent. Aside from the main works we cite (e.g., Torsional Diffusion, FoldingDiff, etc.), are there specific references you believe we are missing? Although these studies have used diffusion over internal coordinates, we believe our approach and its application are distinct and provide a clear contribution to the ML community, in particular: a) FoldingDiff focuses on unconditional generation, and does not address the same task for generating ensembles of a given molecule, nor is it able to explicitly model all-atom geometries and side chains. Additionally, it employs a residue-centric representation instead of our atom-centric one. b) We have added Torsional Diffusion as a baseline, over which we demonstrate considerable improvement. This highlights the significance of addressing coupled ring constraints through new architecture design.

2. We agree that the most interesting part is the study of highly-coupled internal coordinates present within ring systems, as nearly all prior work follows the rigid rotor hypothesis, where one can treat torsions as freely-rotable and modeled independently (Torsional Diffusion, etc.). We apologize if there is any confusion on this part - each atom is parameterized by a corresponding angle and torsion along with atomic features; we are able to define a consistent reference frame through the macrocycle backbone by following the standard N-to-C directionality for peptide chains. To supplement Section 3.3, we have added a new figure in the appendix to explicitly demonstrate which angles and torsions are extracted, and we have updated the wording in our main text to better explain how all backbone angles and torsions are extracted.

3. Reassembly of the Cartesian structure is straightforward given the fixed bond distances along with the generated angles and torsions: After post-processed optimization, which only sets the ring atom coordinates, all additional atomic coordinates are constructed sequentially starting from the backbone atom positions using the NeRF algorithm with fixed bond distances and the RINGER-predicted side-chain angles and torsions.

4. The idea to study multi-ring systems (e.g., two-ring systems) is an interesting direction that we would love to pursue. However, we believe it was critical to first focus on single-ring systems, thus addressing this missing gap in the field of macrocycles. Moreover, single-ring macrocycles cover the majority of cyclic peptides derived from platforms based on mRNA display. In addition to this, there are currently no extensive datasets that contain conformer ensembles for multi-ring macrocyclic peptides. Multi-ring systems introduce interesting topologies, and we agree that this introduces challenges for defining new encoding schemes. Finally, we point out that two-ring and fused-ring peptide systems are used as a strategy to staple peptides into locked conformations - hence this problem is more akin to the protein structure prediction problem of identifying several low-energy conformers rather than a diverse conformer ensemble. We believe that this is an interesting direction that we hope to pursue in follow up studies. We also point out that our atom-centric sequence design is flexible enough to allow RINGER to model non-homodetic macrocycles, e.g., rings containing other heteroatoms and functional groups.

We believe the additional details and clarifications significantly strengthen the manuscript. We hope you will consider raising your rating based on our responses and our updated manuscript that includes more extensive benchmarking and improved implementation details. We sincerely appreciate your thoughtful feedback.

Thanks for the response which addressed most of my concerns. I do value the practical side of this work to solve the ring structures in conformer generation.

I still have one confusion about how you parameterize the angle and dihedral, you mentioned for each atom you have the angle and dihedral feature and you also mentioned you defined a frame along the backbone for the cycle. I am still very much confused, e.g. an angle, you need two intersecting vectors or three atoms and for dihedrals, you need two intersecting planes or four atoms, how do you define it atom-wise? I would highly recommend you describe in further detail and in the ideal case provide an illustration geometrically. (Figure 5 does look good but it only shows the learning part, but not how it makes sense geometrically).

I will raise my score to 6 after addressing this issue.