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SPINET: Sheaf Protein Inverse Folding Network

Jens Lundsgaard, Colin Mikulski, Zhixuan Yan, Dhananjay Bhaskar

arXiv:2609.34153v1cs.LGq-bio.BM

TL;DR

Most inverse-folding models use a single structure or discrete conformations, although protein function involves motion between states. SPINET predicts sequences from molecular-dynamics trajectories using spatial sheaf message passing and recurrent temporal aggregation. It outperforms evaluated static and ensemble baselines in sequence recovery, with structural recovery on mdCATH favoring target conformations over decoys.

  • Problem

    Most inverse-folding models use a single static structure, while discrete ensembles do not capture evolving local interactions and intermediate states along a molecular-dynamics trajectory.

    Source: §1 · p. 1

  • Method

    SPINET predicts sequences from molecular-dynamics trajectories by combining sheaf message passing within frames with recurrent aggregation across frames.

    Source: §1 · pp. 1–2

  • Results

    56.7% top-1 recovery on mdCATH, versus 44.5% for the strongest static baseline and 40.7% for the strongest ensemble baseline; target-template recovery exceeded decoy-template recovery for 99.5% of evaluable domains.

    Source: §6 · p. 8

  • Takeaways & Limitations

    The dynamics-aware formulation takes a step toward designing proteins that undergo prescribed conformational changes.

    Source: §1 · p. 2

    Structural evaluation supplied target conformations as AF3 templates and therefore does not establish experimental folding or motion; structural recovery was assessed on mdCATH only.

    Source: §6 · p. 8

Abstract

from arXiv · show

Proteins change shape as they function, yet most inverse folding models predict amino acid sequences from a single, fixed backbone. A central challenge in protein engineering is to design proteins that undergo specific motions, which requires accounting for how their structures change over time. This motivates inverse protein folding conditioned on protein motion. We introduce SPINET, which predicts sequences from molecular dynamics trajectories. It uses cellular sheaves to represent residue interactions within each frame and recurrent units to integrate information across frames, then predicts all amino acids in a single pass. We evaluate SPINET on mdCATH and ATLAS, where it outperforms all evaluated static and ensemble baselines in sequence recovery. On mdCATH, it achieves 56.7% top-1 recovery, compared with 44.5% for the strongest static baseline and 40.7% for the strongest ensemble baseline. We also evaluate whether the predicted sequences are compatible with conformations sampled along the target trajectory. On mdCATH, they achieve a median TM-score of 0.760, and structural recovery favors target conformations over unrelated decoys for 99.5% of test domains.

1 Introduction

Most inverse-folding models use a single static structure or discrete conformations, leaving molecular-dynamics trajectories unmodeled. SPINET predicts sequences from trajectories by combining learned spatial sheaf message passing with recurrent temporal aggregation, and reports improved sequence recovery plus analyses of structural compatibility and learned representations.

1 Introduction

  • Protein motion motivates seeking sequences compatible with prescribed conformational changes, since proteins move between states during catalysis, transport, signaling, and allostery.The introduction frames this as a further goal in protein design.Source: §1 · pp. 1–2
  • Most inverse-folding models condition on one static structure, while discrete-state approaches still omit evolving local interactions and intermediate states along a molecular-dynamics trajectory.The paper states that, to its knowledge, no inverse-folding model had directly taken such a trajectory as input.Source: §1 · p. 1
  • SPINET predicts sequences directly from molecular-dynamics trajectories by combining sheaf message passing within frames, recurrent integration across frames, and further sheaf refinement before residue classification.The approach targets motion-conditioned inverse folding rather than predictions from a static structure or discrete ensemble.Source: §1 · pp. 1–2
  • 56.7% top-1 sequence recovery on mdCATH, with SPINET outperforming static and ensemble baselines; it also achieves state-of-the-art sequence recovery on ATLAS.On mdCATH, perplexity is 3.74; on ATLAS, SPINET reaches 51.8% top-1 recovery and perplexity 4.261, versus 23.1% and 18.788 without learnable sheaf restriction maps.Source: §1 · p. 2
  • The evaluation combines sequence-recovery and perplexity comparisons on mdCATH and ATLAS with structural recovery on mdCATH using representative trajectory conformations and unrelated decoy templates.The reported structural analysis finds predictions specifically compatible with the target conformational ensemble rather than arbitrary templates.Source: §1 · p. 2
  • Preliminary analysis suggests differences in learned sheaf-Laplacian spectral gaps correspond to differences in protein flexibility.The paper presents learned restriction maps and their associated sheaf Laplacians as analyzable representations.Source: §1 · p. 2

2 Related Work

Prior inverse-folding approaches vary in how they encode geometry or generate sequences, but generally condition on static backbones or discrete conformations. SPINET instead integrates information across molecular-dynamics trajectory frames to recover all residue identities in one pass.

2 Related Work

  • Graph-based inverse-folding models differ in their geometric representations and aggregation, but condition predictions on one static structure.Examples include GVP-GNN, PiFold, and ScFold.Source: §2 · p. 2
  • MapDiff iteratively denoises an amino-acid sequence, while retaining a fixed-backbone structural input.Its masked-residue refinement changes sequence generation, not the structural input setting.Source: §2 · p. 2
  • DynamicMPNN generates sequences autoregressively from multiple conformations encoded in a shared representation, treating them as discrete states rather than a trajectory.It therefore incorporates multiple target states without using the evolving intermediate interactions of molecular dynamics.Source: §2 · p. 2
  • IF-MD uses molecular-dynamics samples to optimize sequences for desired kinetic properties, rather than to recover a sequence directly from an input trajectory.The paper therefore does not treat it as a directly comparable inverse-folding baseline.Source: §2 · p. 2
  • SPINET conditions sequence recovery on molecular-dynamics trajectories and predicts amino-acid identities at all positions in a single pass, without autoregressive decoding.Its trajectory input distinguishes it from approaches that use one fixed backbone or a set of discrete conformations.Source: §2 · p. 2, Figure 1 · p. 3

3 Background

The background introduces cellular sheaves as graph-based structures with node and edge vector spaces linked by restriction maps. Their sheaf Laplacian generalizes the standard graph Laplacian under the stated identity-map condition.

3 Background

  • A cellular sheaf assigns a finite-dimensional vector space to each graph node and edge, plus a linear restriction map for each node-edge incidence.In SPINET, all these spaces have dimension d, so each restriction map is a matrix in R^d×d.Source: §3 · p. 3
  • For a graph with n nodes and stalk dimension d, the sheaf Laplacian is an nd × nd block matrix.The supplied passages identify diagonal and off-diagonal blocks but do not provide their full expressions.Source: §3 · p. 3
  • When d = 1 and every restriction map is the identity, the sheaf Laplacian reduces to the standard graph Laplacian D − A.Source: §3 · p. 3

4 Methods

SPINET applies sheaf message passing to each trajectory frame, aggregates residue representations over time with an RNN, and performs further sheaf message passing before classifying residues.

4.1 Node and Edge Features

  • Node features encode backbone geometry in each residue’s local frame, while edge features encode relative positions, frame similarities, and bond-versus-proximity status.Source: §4.1 · p. 4

4.2 Learnable Restriction Maps

  • Learned restriction maps use node and directed-edge features to give each node–edge incidence its own transform.The maps are unconstrained, and their outputs support sheaf message passing.Source: §4.2 · p. 4

4.3 Spatiotemporal Embedding

  • SPINET performs sheaf message passing within each frame, processes frame-wise residue representations in temporal order with an RNN, then applies further sheaf message-passing blocks.The post-temporal sheaf is learned from aggregated representations, allowing neighboring residues to exchange trajectory-wide information.Source: §4.3 · pp. 4–5

4.4 Residue Classification and Loss

  • After the final message-passing block, a classification head predicts amino-acid classes and training minimizes cross-entropy against native residues.Source: §4.4 · p. 5

5 Results

The results evaluate SPINET’s sequence recovery and structural compatibility on mdCATH, with additional sequence recovery on ATLAS. They also examine target-versus-decoy specificity, window-conditioned candidates, and a preliminary spectral-gap–flexibility relationship.

5 Results

  • On mdCATH, SPINET reached 56.7% top-1 recovery, versus 44.5% for MapDiff and 40.7% for fine-tuned DynamicMPNN.The primary evaluation uses 5,398 domains and a 70/15/15 split by CATH topology.Source: §5 · pp. 5–6, Table 1 · p. 6
  • Best-of-five window-conditioned selection raised median TM-score from 0.760 to 0.810, but is an oracle upper bound rather than a direct single-sequence baseline comparison.The selected candidate had the highest mean TM-score across five representative conformations.Source: §5 · p. 7
  • SPINET achieved a median TM-score of 0.760 and RMSD of 2.98 Å, exceeding static baselines in global structural recovery.Scores are medians across 788 domains evaluable for every method; five representative conformations were selected independently of predicted sequences and held fixed across methods.Source: §5 · pp. 6–7, Table 2 · p. 7
  • Target-template TM-score exceeded decoy-template TM-score for 99.5% of domains, with a median TM-score ratio of 6.91.Larger TM-score ratios and smaller RMSD ratios indicate greater preference for the target.Source: Table 2 · p. 7, §5 · p. 7
  • On ATLAS, SPINET reached 51.8% top-1 recovery and perplexity 4.26, versus 41.3% and 6.47 for fine-tuned DynamicMPNN.DynamicMPNN was the strongest evaluated baseline by top-1 recovery on that split.Source: §5 · p. 7

5.1 Interpretability of learned restriction maps

  • Examples with the largest learned sheaf-Laplacian spectral gaps generally show greater RMSF than those with the smallest gaps, but dataset-wide validation remains necessary.Figure 3 displays three proteins at each end of the spectral-gap range and labels their RMSF across the trajectory.Source: §5.1 · p. 8, Figure 3 · p. 8

6 Conclusion

SPINET uses molecular-dynamics trajectories for inverse folding, with results supporting sequence recovery compatible with sampled target conformations. Its structural evaluation and use for new design paths remain bounded by unresolved experimental and trajectory-robustness questions.

6 Conclusion

  • 56.7% top-1 recovery on the 320 K mdCATH test set exceeded the strongest static baseline at 44.5% and the strongest ensemble baseline at 40.7%.Predicted sequences had a median TM-score of 0.760; target-template recovery exceeded decoy-template recovery for 99.5% of evaluable domains.Source: §6 · p. 8
  • Structural evaluation used AF3 templates, so it does not establish that predicted sequences experimentally fold into target conformations or undergo the prescribed motion.Structural recovery was assessed on mdCATH only; ATLAS evaluation covered sequence recovery.Source: §6 · p. 8
  • Performance under substantial trajectory truncation or frame shuffling was not tested; structural evaluation on ATLAS and tests with shorter, sparse, or reordered trajectories remain future work.The paper also identifies experimental assessment of folding, dynamics, and function for sequences designed along modified or generated paths as future work.Source: §6 · pp. 8–9
  • Applying SPINET to a new design target requires a plausible motion path before the intended sequence is known.Paths could be modified from an existing protein's motion or proposed by sampling models, backbone generators, simulations, or manual prescription; evaluating designed sequences experimentally remains future work.Source: §6 · p. 9

A.1 SPINET Hyperparameters and Training

SPINET training uses fixed hyperparameters, while baseline training follows model-specific procedures. DynamicMPNN's MD-derived conformer pairs may be less distinct than the states for which that model was designed.

A.1 SPINET Hyperparameters and Training

  • SPINET training used d=8, c=16, b=6, ε=7.5 Angstroms, k=32, learning rate 0.001, 20 epochs, and time subsequences of length 128 with step 32.The batch size was fewer than 3,000 nodes; TGNN and Traj–MLP ablations used the same hyperparameters wherever applicable.Source: §A.1 · p. 11

A.2 Baseline Model Training

  • Static baselines used relaxed PDB structures and the same splits as dynamic models, with model-specific epoch limits, batch sizes, and learning rates.PiFold stopped after 10 epochs without validation-perplexity improvement; MapDiff was trained and evaluated only on mdCATH because of high computational expense.Source: §A.2 · p. 11
  • DynamicMPNN used conformer pairs selected from MD trajectories; published two-state weights were also fine-tuned with the same conformer-selection procedure.The training procedure sampled frames at intervals of 4 for mdCATH and 25 for ATLAS, then selected pairs from a pool of 10 frames with high pairwise TM-scores.Source: §A.2 · p. 11
  • MD-derived conformer pairs may be less distinct than the conformational states DynamicMPNN was designed to use.The authors note that more trajectory frames can be nearly identical when an MD trajectory spans a limited conformational range.Source: §A.2 · p. 11

B Structural Recovery and Target-Decoy Specificity on mdCATH

Structural recovery on mdCATH favors SPINET under target templates, while decoy-template recovery is weak. Results vary with the conformations evaluated and include a post-hoc best-of-five selection.

B Structural Recovery and Target-Decoy Specificity on mdCATH

C Sequence Recovery by Amino Acid Class

The amino-acid-class analysis reports per-residue average F1 scores on the mdCATH test split. SPINET performs particularly well for glycine and proline, while GVP is relatively strong for arginine and threonine.

C Sequence Recovery by Amino Acid Class

D Sequence Recovery Across mdCATH Simulation Temperatures

Across mdCATH simulation temperatures, SPINET’s top-1 recovery rises and perplexity falls at higher temperatures; the comparison used an earlier model variant.

D Sequence Recovery Across mdCATH Simulation Temperatures

E Sequence Recovery on ATLAS

On the ATLAS test split, SPINET outperforms evaluated baselines in sequence recovery, while its top-1 recovery is lower than on mdCATH.

E Sequence Recovery on ATLAS

  • After 18 of 1,938 trajectories were excluded for data-loading issues, 1,920 proteins were randomly split into training (1,152), validation (384), and test (384) sets.The authors chose random splitting because ATLAS was assembled to represent diverse protein structures, unlike mdCATH’s topology-grouped split.Source: E Sequence Recovery on ATLAS · p. 14
  • TGNN top-1 recovery falls from 31.9% on mdCATH to 23.1% on ATLAS, a larger decrease than SPINET’s.The authors suggest that the sheaf-based model remains effective on the smaller ATLAS dataset.Source: E Sequence Recovery on ATLAS · p. 14
  • 51.8% top-1 recovery and 4.261 perplexity on ATLAS; SPINET outperforms evaluated baselines, but trails mdCATH’s 56.7% top-1 recovery.Table 7 reports results for SPINET, its ablations, and static and ensemble baselines.Source: Table 7 · p. 14, E Sequence Recovery on ATLAS · p. 14
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