Organizations: School of Computer Science, Peking University, Beijing, China · School of Electronic and Computer Engineering, Peking University, Shenzhen, China · School of Life Sciences, Tsinghua University, Beijing 100084, China · Department of Automation and BNRist, Tsinghua University, Beijing, China · Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China
Protein design is moving beyond structural correctness toward function-aware design, yet existing generative models typically treat dynamics as a downstream property estimated through simulation or prediction after structure generation. Using MD trajectories as a generative target is also undesirable because stochastic, path-dependent trajectories over-specify the underlying equilibrium ensemble. We introduce PhiFold, a framework for jointly generating protein backbones and their second-order dynamics, represented by residue-displacement covariance. Rather than predicting the quadratically sized full covariance, PhiFold decomposes dynamics into three interpretable components: local flexibility, a low-rank collective-motion representation, and residue-wise collective participation. These components are assembled into a positive-definite covariance matrix with exact marginal consistency, yielding a compact and physically constrained representation of equilibrium dynamics. Across generated proteins, PhiFold improves recovery of local fluctuations and long-range residue coupling while remaining competitive on dominant collective-motion subspaces. It further enables bidirectional control of residue flexibility while preserving backbone designability. By unifying structure generation with an explicit representation of equilibrium dynamics, PhiFold lays a foundation for designing proteins not only by how they look, but also by how they move.
Figures & tables
Figure 1: Motivation and overview of PhiFold. MD trajectories are high-dimensional, path-dependent, and stochastic. PhiFold instead models equilibrium covariance through physically interpretable factors for local flexibility, collective motion, and residue-wise participation.
Figure 2: Physics-structured covariance modeling in PhiFold. Three prediction modules estimate marginal covariance, collective geometry, and participation. These factors are normalized, variance-calibrated, and assembled into a valid joint covariance with exact marginal consistency.
Aspect
Metric
PhiFold
MDGen
ANM
BioEmu
Str2Str
GROMACS
Joint
Post hoc
Post hoc
Post hoc
Post hoc
R1–R2
A. Fine-grained covariance fidelity
Local fluctuation
RMS displacement r↑
0.739
0.516
0.663
0.492
0.480
0.795
RMS displacement NMAE ↓
0.303
0.563
0.838
3.932
4.210
0.241
Marginal eigenvalue r↑
0.682
0.577
0.608
0.536
0.566
0.769
Collective motion
Top-20 subspace overlap ↑
0.428
0.520
0.427
0.356
0.328
0.695
Table 1: Agreement with independent two-replica all-atom GROMACS simulations on the same set of generated proteins. Fine-grained covariance diagnostics compare PhiFold with the representative post hoc estimators MDGen and ANM, while unified ensemble-level metrics compare all methods. All entries report mean performance, and Wasserstein distances are in Å. GROMACS R1–R2 provides an empirical reference for finite-sampling variability and is excluded from method ranking. Best and second-best method results within each evaluation block are shown in bold and underlined , respectively. Additional metrics are reported in the Supplement.
Method
Marginal eig. r↑
Top- 20 subspace overlap ↑
DCCM r↑
Nonlocal DCCM r↑
Long-range DCCM r↑
Valid covariance (%) ↑
(MDGen proxy)
(ANM proxy)
Direct joint
0.316±0.133
0.311±0.073
0.404±0.048
0.163±0.048
0.062±0.065
0.0
Fixed αi=0.5
0.499±0.122
0.414±0.059
0.457±0.116
0.379±0.090
0.345±0.082
100.0
Cholesky
0.484±0.105
0.383±0.051
0.448±0.056
0.273±0.046
0.191±0.060
100.0
Full
0.531±0.131
0.432±0.052
0.513±0.100
0.423±0.086
0.386±0.086
100.0
Table 2: Ablation of the physics-structured covariance parameterization. Values are mean ± standard deviation over 50 generated proteins. Best proxy-recovery results are shown in bold .
Figure 3: Controllable generation of regional flexibility. Solid and dashed curves show signed target- and outside-region Δ RMSF, with positive values indicating the intended direction; shading denotes 95% confidence intervals over 20 matched samples. Open squares show individual GROMACS replicas, and designability changes relative to unconditional generation are reported.
Appendix figures & tables10 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Description
Value
λmarg
Marginal covariance loss weight
1.0
λsub
Subspace alignment loss weight
1.0
λα
PCA participation regularization weight
0.3
λfrob
Joint covariance Frobenius loss weight
0.05
λpair
Local pairwise covariance loss weight
1.0
Knn
Number of local spatial neighbors
8
Appendix
Table 3: Hyperparameter settings.
Figure 4: Empirical analysis for the structured covariance decomposition on ATLAS. (a) Cumulative fraction of empirical covariance variance explained by the leading PCA modes across protein trajectories. (b) Distribution of the fractions of residue-wise marginal variance explained by the largest, intermediate, and smallest eigenvalues. (c) Distribution of protein-level mean and median collective participation. (d) Within-protein dispersion of residue-wise collective participation as a function of protein-level mean participation. Dashed lines indicate dataset-level mean IQR and standard deviation.
Category
Reconstruction
Region
Mean ± Std
p05 / p95
Unconstrained low-rank reference
PCA20 raw
full
0.009 ± 0.007
0.001 / 0.023
PCA20 + exact block tail
full
0.008 ± 0.006
0.001 / 0.019
Structured oracle
Structured + true αi
full
0.037 ± 0.023
0.006 / 0.078
Participation ablation
Structured + αi=1
full
0.046 ± 0.033
0.006 / 0.109
Structured + mean αˉ
full
0.099 ± 0.059
0.013 / 0.201
Appendix
Table 4: Oracle analysis of representation on ATLAS. Relative Frobenius errors are computed against empirical joint covariances. PCA-based reconstructions provide unconstrained low-rank references, whereas structured variants quantify the approximation introduced by the proposed physics-based parameterization and its ablations.
Block
Operation
ϵ
Triggered
Gi
inv. sqrt.
10−6
0/10000
αiΣi
sqrt.
10−8
0/10000
Σi
PSD check
–
0/10000
λ
clipping
10−8
0/1000
Appendix
Table 5: Numerical safeguard audit during inference.
Length
Gi
αiΣi
Σi
Mode
100
9.44e−3
1.33e−1
1.57e−1
4.57e−1
150
5.53e−3
1.22e−1
1.60e−1
4.03e−1
200
4.04e−3
1.42e−1
1.54e−1
3.81e−1
250
3.03e−3
1.45e−1
1.67e−1
3.69e−1
300
2.22e−3
1.37e−1
1.62e−1
3.60e−1
Appendix
Table 6: First percentile of minimum eigenvalues across protein lengths. Columns correspond to Gi , αiΣi , Σi , and mode eigenvalues.
Length
BioEmu
MDGen
Str2Str
PhiFold
(ms / conf.)
(ms / conf.)
(ms / conf.)
end-to-end amortised
ensemble expansion
(ms / conf., K=1000 )
( μ s / conf.)
100
529.286 ± 215.205
61.272 ± 8.027
13998.347 ± 4224.408
1.91 ± 0.05
0.311 ± 0.010
150
846.673 ± 389.510
75.463 ± 22.723
14782.796 ± 4467.001
1.89 ± 0.04
0.404 ± 0.010
200
790.854 ± 33.287
112.116 ± 19.152
16522.789 ± 4985.198
1.90 ± 0.02
0.443 ± 0.004
250
1211.111 ± 36.525
76.746 ± 9.747
18784.649 ± 5625.874
1.90 ± 0.01
0.496 ± 0.012
Appendix
Table 7: Inference cost for generating 1000-frame ensembles. PhiFold reports the end-to-end amortised cost and the incremental ensemble-expansion cost after covariance prediction.
Metric
Value
Designability
80.8%
Mean maximum scTM
0.919
Mean minimum RMSD (Å)
1.785
Diversity (mean pairwise TM-score)
0.377
Appendix
Table 8: Aggregate evaluation over 250 generated structures pooled across all protein lengths.
Metric
Backbone
Ensemble
C α distance validity (%)
100.00±0.00
92.03±1.91
C α –C α distance std. (Å)
0.155±0.003
0.486±0.077
Steric validity (%)
99.85±0.18
98.66±0.96
Angle favored (%)
98.67±0.22
84.92 ± 3.30
Rg/L1/3
3.030±0.061
3.035±0.211
Appendix
Table 9: Structural validity before and after covariance-based ensemble expansion. C α distance validity is computed following FoldFlow with a 0.1 Å tolerance above the ideal distance.
Figure 5: Visualization of generated structural ensemble. 10 conformational samples from each predicted ensemble are aligned and superimposed.
Length
Condition
Target bbRMSD
Outside bbRMSD
Localization ratio
Δ Ordered
ANM RMSIP top-20
Design- ability
Mean max scTM
(Å)
(Å)
(pp)
(%)
100†
Uncond.
–
–
–
–
–
100
0.957
Flex
5.850±6.509
2.089±2.619
2.80
+0.13
0.827±0.144
100
0.956
Rigid
1.954±1.654
0.766±1.123
2.55
+0.63
0.914±0.079
100
0.948
150
Uncond.
–
–
–
–
–
100
0.957
Flex
4.271±2.864
2.340±1.247
1.83
−0.75
0.837±0.097
95
0.945
Appendix
Table 10: Backbone-level effects and designability under regional guidance. Backbone displacement, localization, Δ Ordered, and ANM RMSIP are defined relative to matched unconditional samples and are therefore not applicable to the unconditional rows. Strict designability is the percentage passing the strict designability criterion, and scTM is the mean maximum self-consistency TM-score. † Length-100 regional RMSDs use min_outside_frac=0.15 and a 20-residue outside-region Kabsch anchor, so their regional decomposition should be interpreted cautiously.
Motivation: Generative models for protein backbone design have to simultaneously ensure geometric validity, sampling efficiency, and scalability to long sequences. However, most existing approaches rely on iterative refinement, quadratic attention mechanisms, or post-hoc geometry correction, leading to a persistent trade-off between computational efficiency and structural fidelity. Results: We present Physics-Informed Mamba (PI-Mamba), a generative model that enforces exact local covalent geometry by construction while enabling linear-time inference. PI-Mamba integrates a differentiable constraint-enforcement operator into a flow-matching framework and couples it with a Mamba-based state-space architecture. To improve optimisation stability and backbone realism, we introduce a spectral initialization derived from the Rouse polymer model and an auxiliary cis-proline awareness head. Across benchmark tasks, PI-Mamba achieves 0.0% local geometry violations and high designability (scTM = 0.91±0.03, n = 100), while scaling to proteins exceeding 2,000 residues on a single A5000 GPU (24 GB).
Tianyu Wu, Lin Zhu
Center for Biophysics and Quantitative Biology, University of Illinois Urbana-Champaign, 600 S Mathews Ave., 61801, IL, USA · School of Information Science, University of Illinois Urbana-Champaign, 501 E Daniel St., 61820, IL, USA
De novo functional protein design aims to generate protein sequences that realize specified biochemical functions without relying on evolutionary templates, enabling broad applications in biotechnology and medicine. Existing approaches adopt either direct function-to-sequence mapping or decoupled structure-sequence generation strategies but often fail to achieve functionality and foldability simultaneously. To address this, we propose CodeFP, a Co-generative protein language model for de novo Functional Protein design that simultaneously decodes sequence and structure tokens, thereby enabling superior simultaneous realization of functionality and foldability. CodeFP utilizes functional local structures to enrich functional semantic encodings, overcoming the suboptimal translation of flat encodings into structure tokens, while introducing auxiliary functional supervision to alleviate training ambiguity stemming from the one-to-many structure-to-token mapping. Extensive experiments show that CodeFP consistently achieves average improvements of 6.1% in functional consistency and 3.2% in foldability over the strongest baseline.
Xinrui Chen, Yizhen Luo, Siqi Fan +1
Institute for AI Industry Research (AIR), Tsinghua University · Department of Computer Science and Technology, Tsinghua University · PharMolix Inc.
De novo protein generation has transformative potential in therapeutic design, enzyme engineering, and synthetic biology. While diffusion-based and flow matching approaches have achieved progress, they typically operate at single resolution and lack mechanisms for incorporating functional constraints. We introduce ProHiFlo, a hierarchical flow matching framework with three innovations: (1) coarse-to-fine generation that models backbone geometry before refining to all-atom coordinates, reducing computational cost while maintaining accuracy; (2) functional guidance leveraging pretrained predictors to steer generation toward desired properties without retraining; (3) adaptive SE(3)-equivariant architecture for efficient multi-scale processing. Experiments on unconditional generation, motif scaffolding, and functional design demonstrate state-ofthe-art performance while requiring 4 fewer sampling steps. On enzyme active site scaffolding, ProHiFlo achieves 58.9% success rate compared to 41.2% for RFDiffusion.
Chuanzhen Wang, Meade Cleti, Pete Jano
3Tongji University · 1Arizona State University · University of Wisconsin-Madison