PhiFold: Towards Dynamic Protein Design with Physics-Structured Covariance Modeling

Yutian Liu; Mujie Lin; LanqianZhang; Meng Fan; Chang Liu; ZhiweiNie; Siwei Ma

cs.AISep 26, 2026

PhiFold: Towards Dynamic Protein Design with Physics-Structured Covariance Modeling

Authors: Yutian Liu, Mujie Lin, LanqianZhang, Meng Fan, Chang Liu, ZhiweiNie, Siwei Ma

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

Abstract

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.

A. Fine-grained covariance fidelity
Aspect	Metric	PhiFold	MDGen	ANM	BioEmu	Str2Str	GROMACS
		Joint	Post hoc	Post hoc	Post hoc	Post hoc	R1–R2
Local fluctuation	RMS displacement $r$ $\uparrow$	$\mathbf{0.739}$	$0.516$	$\underline{0.663}$	$0.492$	$0.480$	$0.795$
	RMS displacement NMAE $\downarrow$	$\mathbf{0.303}$	$\underline{0.563}$	$0.838$	$3.932$	$4.210$	$0.241$
	Marginal eigenvalue $r$ $\uparrow$	$\mathbf{0.682}$	$0.577$	$\underline{0.608}$	$0.536$	$0.566$	$0.769$
Collective motion	Top-20 subspace overlap $\uparrow$	$\underline{0.428}$	$\mathbf{0.520}$	$0.427$	$0.356$	$0.328$	$0.695$

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. $r$ $\uparrow$	Top- $20$ subspace overlap $\uparrow$	DCCM $r$ $\uparrow$	Nonlocal DCCM $r$ $\uparrow$	Long-range DCCM $r$ $\uparrow$	Valid covariance (%) $\uparrow$
	(MDGen proxy)		(ANM proxy)
Direct joint	$0.316\pm 0.133$	$0.311\pm 0.073$	$0.404\pm 0.048$	$0.163\pm 0.048$	$0.062\pm 0.065$	$\mathit{0.0}$
Fixed $\alpha_{i}=0.5$	$0.499\pm 0.122$	$0.414\pm 0.059$	$0.457\pm 0.116$	$0.379\pm 0.090$	$0.345\pm 0.082$	$\mathbf{100.0}$
Cholesky	$0.484\pm 0.105$	$0.383\pm 0.051$	$0.448\pm 0.056$	$0.273\pm 0.046$	$0.191\pm 0.060$	$\mathbf{100.0}$
Full	$\mathbf{0.531\pm 0.131}$	$\mathbf{0.432\pm 0.052}$	$\mathbf{0.513\pm 0.100}$	$\mathbf{0.423\pm 0.086}$	$\mathbf{0.386\pm 0.086}$	$\mathbf{100.0}$

Table 2: Ablation of the physics-structured covariance parameterization. Values are mean

\pm

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

\Delta

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
$\lambda_{\mathrm{marg}}$	Marginal covariance loss weight	$1.0$
$\lambda_{\mathrm{sub}}$	Subspace alignment loss weight	$1.0$
$\lambda_{\alpha}$	PCA participation regularization weight	$0.3$
$\lambda_{\mathrm{frob}}$	Joint covariance Frobenius loss weight	$0.05$
$\lambda_{\mathrm{pair}}$	Local pairwise covariance loss weight	$1.0$
$K_{\mathrm{nn}}$	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 $\pm$ Std	p05 / p95
Unconstrained low-rank reference	PCA20 raw	full	0.009 $\pm$ 0.007	0.001 / 0.023
Unconstrained low-rank reference	PCA20 + exact block tail	full	0.008 $\pm$ 0.006	0.001 / 0.019
Structured oracle	Structured + true $\alpha_{i}$	full	0.037 $\pm$ 0.023	0.006 / 0.078
Participation ablation	Structured + $\alpha_{i}=1$	full	0.046 $\pm$ 0.033	0.006 / 0.109
Participation ablation	Structured + mean $\bar{\alpha}$	full	0.099 $\pm$ 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	$\epsilon$	Triggered
$G_{i}$	inv. sqrt.	$10^{-6}$	0/10000
$\alpha_{i}\Sigma_{i}$	sqrt.	$10^{-8}$	0/10000
$\Sigma_{i}$	PSD check	–	0/10000
$\lambda$	clipping	$10^{-8}$	0/1000

Appendix

Table 5: Numerical safeguard audit during inference.

Length	$G_{i}$	$\alpha_{i}\Sigma_{i}$	$\Sigma_{i}$	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

G_{i}

\alpha_{i}\Sigma_{i}

\Sigma_{i}

, and mode eigenvalues.

Length	BioEmu	MDGen	Str2Str	PhiFold
	(ms / conf.)	(ms / conf.)	(ms / conf.)	end-to-end amortised	ensemble expansion
				(ms / conf., $K{=}1000$ )	( $\mu$ s / conf.)
100	529.286 $\pm$ 215.205	61.272 $\pm$ 8.027	13998.347 $\pm$ 4224.408	1.91 $\pm$ 0.05	0.311 $\pm$ 0.010
150	846.673 $\pm$ 389.510	75.463 $\pm$ 22.723	14782.796 $\pm$ 4467.001	1.89 $\pm$ 0.04	0.404 $\pm$ 0.010
200	790.854 $\pm$ 33.287	112.116 $\pm$ 19.152	16522.789 $\pm$ 4985.198	1.90 $\pm$ 0.02	0.443 $\pm$ 0.004
250	1211.111 $\pm$ 36.525	76.746 $\pm$ 9.747	18784.649 $\pm$ 5625.874	1.90 $\pm$ 0.01	0.496 $\pm$ 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 $\alpha$ distance validity (%)	$100.00\pm 0.00$	$92.03\pm 1.91$
C $\alpha$ –C $\alpha$ distance std. (Å)	$0.155\pm 0.003$	$0.486\pm 0.077$
Steric validity (%)	$99.85\pm 0.18$	$98.66\pm 0.96$
Angle favored (%)	$98.67\pm 0.22$	84.92 $\pm$ 3.30
$R_{g}/L^{1/3}$	$3.030\pm 0.061$	$3.035\pm 0.211$

Appendix

Table 9: Structural validity before and after covariance-based ensemble expansion. C

\alpha

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	$\Delta$ Ordered	ANM RMSIP top-20	Design- ability	Mean max scTM
			(Å)	(Å)		(pp)		(%)
$100^{\dagger}$	Uncond.	–	–	–	–	–	100	0.957
	Flex	$5.850\pm 6.509$	$2.089\pm 2.619$	2.80	$+0.13$	$0.827\pm 0.144$	100	0.956
	Rigid	$1.954\pm 1.654$	$0.766\pm 1.123$	2.55	$+0.63$	$0.914\pm 0.079$	100	0.948
150	Uncond.	–	–	–	–	–	100	0.957
	Flex	$4.271\pm 2.864$	$2.340\pm 1.247$	1.83	$-0.75$	$0.837\pm 0.097$	95	0.945

Appendix

Table 10: Backbone-level effects and designability under regional guidance. Backbone displacement, localization,

\Delta

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

\texttt{min\_outside\_frac}=0.15

and a 20-residue outside-region Kabsch anchor, so their regional decomposition should be interpreted cautiously.

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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 =

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