SPINET: Sheaf Protein Inverse Folding Network

Jens Lundsgaard; Colin Mikulski; Zhixuan Yan; Dhananjay Bhaskar

cs.LGSep 28, 2026

SPINET: Sheaf Protein Inverse Folding Network

Authors: Jens Lundsgaard, Colin Mikulski, Zhixuan Yan, Dhananjay Bhaskar

Organizations: Department of Computer Sciences, University of Wisconsin–Madison, Madison, WI, USA · Department of Mathematics, University of Wisconsin–Madison, Madison, WI, USA · Department of Biomedical Engineering, University of Wisconsin–Madison, Madison, WI, USA · Biophysics Graduate Program, University of Wisconsin–Madison, Madison, WI, USA · Data Science Institute, University of Wisconsin–Madison, Madison, WI, USA · Center for Genomic Science Innovation, University of Wisconsin–Madison, Madison, WI, USA · Wisconsin Institute for Translational Neuroengineering, University of Wisconsin–Madison, Madison, WI, USA

Abstract

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.

Figures & tables

Figure 1: SPINET architecture. (a) Backbone coordinates from a molecular dynamics trajectory are converted into residue-level node features and geometric edge features. (b) Sheaf message passing encodes residue interactions within each frame, an RNN aggregates these representations over time, and additional sheaf message-passing blocks refine them before amino acid classification.

Figure 2: SPINET learns a restriction map $\mathcal{F}_{j\leq e_{ij}}$ for each incident node–edge pair. During sheaf message passing, the map transforms the neighbor representation $h_{j}$ into the edge space as $\mathcal{F}_{j\leq e_{ij}}h_{j}$ , where it is combined with the learned edge representation $g_{ij}$ . The resulting message is mapped back to the node space and used to update the residue representation $h_{i}$ ; the updated representation is then concatenated with the previous $h_{i}$ .

Paradigm	Model	Params	top-1 $\uparrow$	top-5 $\uparrow$	top-10 $\uparrow$	Perplexity $\downarrow$
Dynamic	SPINET	1.5M	0.567 $\pm$ 0.060	0.898 $\pm$ 0.038	0.979 $\pm$ 0.017	3.736 $\pm$ 0.676
	TGNN	1.4M	0.319 $\pm$ 0.054	0.712 $\pm$ 0.058	0.898 $\pm$ 0.039	8.872 $\pm$ 1.592
	Traj–MLP (full)	23.1K	0.406 $\pm$ 0.063	0.774 $\pm$ 0.053	0.935 $\pm$ 0.031	6.301 $\pm$ 1.001
	Traj–MLP (dihedrals)	22.5K	0.286 $\pm$ 0.059	0.643 $\pm$ 0.062	0.860 $\pm$ 0.046	9.848 $\pm$ 1.438
	Traj–MLP (coords)	22.7K	0.369 $\pm$ 0.060	0.724 $\pm$ 0.052	0.904 $\pm$ 0.036	7.297 $\pm$ 1.063
Ensemble	DynamicMPNN (re-trained)	4.2M	0.376 $\pm$ 0.063	0.754 $\pm$ 0.054	0.919 $\pm$ 0.034	7.242 $\pm$ 1.228

Table 1: Native-sequence recovery on the 320K mdCATH test split. Top-

k

recovery is the fraction of residues whose native amino acid appears among the

k

highest-ranked predictions; perplexity measures the probability assigned to native residues. Higher recovery and lower perplexity indicate better performance. Parameter counts are shown for each model.

Sequence source	TM-score $\uparrow$	lDDT $\uparrow$	RMSD (Å) $\downarrow$	TM ${}_{\text{tar/dec}}$ $\uparrow$	RMSD ${}_{\text{tar/dec}}$ $\downarrow$	pLDDT ${}_{\text{tar/dec}}$ $\uparrow$	Target favored (%) $\uparrow$
Native	0.824	0.835	2.405	7.244	0.153	1.44	99.0
SPINET	0.760	0.777	2.976	6.912	0.194	1.36	99.5
SPINET (Bo5) †	0.810	0.804	2.360	7.370	0.154	1.42	99.6
TGNN	0.619	0.693	4.576	6.041	0.286	1.16	99.1
Traj–MLP (full)	0.209	0.438	13.231	2.324	0.772	1.03	91.1
MapDiff	0.729	0.765	3.323	6.473	0.219	1.18	97.7

Table 2: Structural recovery and target specificity on the 320K mdCATH test set. TM-score, lDDT, and C α RMSD compare AF3 predictions with representative target conformations. Target/decoy ratios compare recovery under target and unrelated decoy templates; “target favored” is the percentage of domains with a higher TM-score under the target template. Values are medians across the 788 domains evaluable for every method. Higher TM-score, lDDT, and target/decoy TM-score and pLDDT ratios, together with lower RMSD and target/decoy RMSD ratio, indicate better performance. Interquartile ranges and additional metrics appear in Appendix B .

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Subset	Sequence source	TM-like $\uparrow$	lDDT $\uparrow$	RMSD (Å) $\downarrow$	pLDDT $\uparrow$
All states	Native	0.824 (0.615–0.922)	0.835 (0.751–0.887)	2.405 (1.486–4.529)	83.30 (77.08–87.71)
	SPINET	0.760 (0.540–0.871)	0.777 (0.688–0.818)	2.976 (2.054–5.457)	73.17 (69.17–76.82)
	SPINET (Bo5)	0.810 (0.643–0.901)	0.804 (0.735–0.841)	2.360 (1.736–3.892)	73.97 (69.41–77.91)
	TGNN	0.619 (0.362–0.779)	0.693 (0.589–0.744)	4.576 (3.023–8.059)	67.93 (63.78–73.05)
	Traj–MLP (full)	0.209 (0.105–0.478)	0.438 (0.358–0.578)	13.231 (8.425–17.190)	61.44 (55.00–66.85)
	MapDiff	0.729 (0.457–0.861)	0.765 (0.667–0.820)	3.323 (2.170–5.759)	76.81 (71.34–81.48)

Appendix

Table 3: AF3 template-based structural recovery on the 320K mdCATH test set. The all-states analysis reports domain-level median (IQR) over the 788 evaluable domains for every method. Sensitivity analyses retain representative conformations for which the native sequence achieves a TM-score above

0.5

0.7

. Best-of-five (Bo5) selects the SPINET candidate with the highest mean TM-score within each subset. Higher TM-score, lDDT, and pLDDT and lower RMSD indicate better recovery.

Sequence source	Decoy TM-like $\uparrow$	Decoy lDDT $\uparrow$	Decoy RMSD (Å) $\downarrow$	Decoy pLDDT $\uparrow$	$n$ domains
Native	0.104 (0.085–0.125)	0.302 (0.245–0.364)	16.157 (13.565–19.030)	55.04 (42.16–72.48)	796
SPINET	0.106 (0.089–0.122)	0.301 (0.245–0.364)	15.990 (13.547–18.838)	52.22 (41.08–63.39)	796
TGNN	0.102 (0.084–0.118)	0.297 (0.244–0.356)	16.731 (14.031–19.558)	56.61 (44.12–67.05)	796
Traj–MLP (full)	0.091 (0.072–0.109)	0.274 (0.233–0.333)	18.621 (16.117–21.673)	53.90 (43.87–63.31)	796
MapDiff	0.105 (0.084–0.126)	0.303 (0.248–0.362)	16.029 (13.524–19.448)	62.18 (47.76–72.84)	788
PiFold	0.105 (0.086–0.125)	0.306 (0.248–0.365)	16.056 (13.471–19.297)	61.70 (46.78–73.64)	788

Appendix

Table 4: AF3 structural recovery using templates unrelated to the target conformations. Values are domain-level median (IQR); the number of evaluable domains is shown for each method. Low TM-scores and high RMSDs indicate weak recovery of the decoy structures.

Sequence source	Target/decoy ratio				Favored (%)				$n$ domains
Sequence source	TM $\uparrow$	lDDT $\uparrow$	RMSD $\downarrow$	pLDDT $\uparrow$	TM	lDDT	RMSD	pLDDT	$n$ domains
Native	7.244 (5.586–8.900)	2.654 (2.035–3.475)	0.153 (0.085–0.313)	1.44 (1.05–1.94)	99.0	98.5	98.5	87.1	788
SPINET	6.912 (5.275–8.271)	2.439 (1.875–3.206)	0.194 (0.119–0.370)	1.36 (1.13–1.78)	99.5	98.6	98.9	95.7	788
SPINET (Bo5)	7.370 (5.706–8.865)	2.595 (1.964–3.326)	0.154 (0.097–0.282)	1.42 (1.14–1.83)	99.6	98.9	99.6	94.5	788
TGNN	6.041 (3.836–7.391)	2.201 (1.670–2.871)	0.286 (0.175–0.536)	1.16 (1.04–1.49)	99.1	97.0	98.2	92.4	788
Traj–MLP (full)	2.324 (1.329–4.969)	1.471 (1.154–2.274)	0.772 (0.455–0.921)	1.03 (1.00–1.28)	91.1	87.2	86.4	78.7	788

Appendix

Table 5: Target-decoy structural specificity on the 320K mdCATH test set. Values are domain-level median (IQR) over the 788 domains evaluable for every method. Ratios compare recovery under target and decoy templates; “favored” gives the percentage of domains with better recovery under the target template. Higher TM-like, lDDT, and pLDDT ratios and lower RMSD ratios indicate greater target specificity.

Temperature	top-1 $\uparrow$	top-5 $\uparrow$	top-10 $\uparrow$	Perplexity $\downarrow$
450K	0.700 $\pm$ 0.082	0.958 $\pm$ 0.037	0.993 $\pm$ 0.011	2.356 $\pm$ 0.643
413K	$0.644\pm 0.074$	$0.934\pm 0.039$	$0.988\pm 0.014$	$2.841\pm 0.663$
379K	$0.598\pm 0.068$	$0.910\pm 0.039$	$0.982\pm 0.016$	$3.330\pm 0.672$
348K	$0.572\pm 0.062$	$0.896\pm 0.040$	$0.979\pm 0.017$	$3.613\pm 0.663$
320K	$0.564\pm 0.063$	$0.890\pm 0.042$	$0.976\pm 0.019$	$3.712\pm 0.671$

Appendix

Table 6: Sequence recovery and perplexity across mdCATH simulation temperatures. Higher top-

k

recovery and lower perplexity indicate better sequence prediction.

Paradigm	Model	Params	top-1 $\uparrow$	top-5 $\uparrow$	top-10 $\uparrow$	Perplexity $\downarrow$
Dynamic	SPINET	1.5M	0.518 $\pm$ 0.051	0.878 $\pm$ 0.033	0.972 $\pm$ 0.015	4.261 $\pm$ 0.667
	TGNN	1.4M	0.231 $\pm$ 0.069	0.591 $\pm$ 0.080	0.820 $\pm$ 0.053	18.788 $\pm$ 6.644
	Traj–MLP (full)	23.1K	0.310 $\pm$ 0.053	0.693 $\pm$ 0.049	0.887 $\pm$ 0.037	8.575 $\pm$ 1.126
	Traj–MLP (dihedrals)	22.5K	0.268 $\pm$ 0.050	0.621 $\pm$ 0.052	0.844 $\pm$ 0.039	10.206 $\pm$ 1.305
	Traj–MLP (coords)	22.7K	0.276 $\pm$ 0.044	0.672 $\pm$ 0.053	0.868 $\pm$ 0.038	9.751 $\pm$ 1.186
Ensemble	DynamicMPNN (re-trained)	4.2M	$0.338\pm 0.058$	$0.726\pm 0.059$	$0.904\pm 0.033$	$8.272\pm 1.499$

Appendix

Table 7: Sequence recovery and perplexity on the ATLAS test split for SPINET , its ablations, and static and ensemble baselines. Higher top-

k

recovery and lower perplexity indicate better performance.

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