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

1 Introduction

• Existing generative methods largely produce static structures, while dynamics are estimated afterward through prediction or molecular simulation.MD trajectories can differ under identical thermodynamic conditions while representing the same equilibrium distribution, so trajectory-level supervision over-specifies the target.Source: §1 · p. 1
• PhiFold represents equilibrium dynamics with local flexibility, collective geometry, and residue-wise participation, assembled into a valid covariance representation.The decomposition separates distinct dynamical factors and supports explicit control over regional flexibility.Source: §1 · p. 1
• PhiFold leads on local-fluctuation recovery and residue-level dynamical coupling while remaining competitive on collective-motion metrics.Controllable generation shifts regional flexibility without compromising backbone designability.Source: §1 · p. 2
• The framework couples backbone generation with dynamics modeling, using residue-displacement covariance as a time-order-independent summary of equilibrium fluctuations.The covariance captures local and collective information without requiring trajectory-specific variation.Source: §1 · pp. 1–2

2.1 Generative Protein Design

• Backbone generators create novel structures or sequence–structure distributions, but generally do not explicitly model equilibrium dynamics.Some newer approaches use multiple or state-dependent conformations, prescribed motion patterns, or conformational constraints rather than generating equilibrium dynamics for each backbone.Source: §2.1 · p. 2
• PhiFold differs from these approaches by generating backbones with equilibrium dynamics represented through residue-displacement covariance.This represents anisotropic fluctuations, residue couplings, and collective motions rather than discrete states or selected modes.Source: §2.1 · p. 2

2.2 Protein Dynamics Modeling

• MD simulations provide atomistic trajectories but are computationally expensive and sampling-dependent, while harmonic models approximate near-native motions.Gaussian network models capture isotropic fluctuations and residue correlations; anisotropic network models represent directional collective motions.Source: §2.2 · p. 2
• Existing approaches generate ensembles, transition proposals, or trajectories, while DynaProt predicts Gaussian dynamics for a fixed structure.PhiFold instead jointly generates backbones and equilibrium dynamics by directly predicting ensemble covariance.Source: §2.2 · p. 3

3.1 Problem Setup

• After rigid-body alignment, PhiFold uses residue-displacement covariance to represent second-order equilibrium statistics under a local Gaussian approximation.The covariance’s residue-pair blocks describe correlated Cartesian fluctuations; the structured positive-definite approximation regularizes alignment-induced null directions.Source: §3.1 · p. 3

3.2 Equilibrium Covariance Modeling

• The covariance is parameterized by residue-wise marginal fluctuations, a low-dimensional collective-motion representation, and residue-wise collective participation fractions.The collective factor has rank at most K, separating collective-motion geometry from the residue-wise allocation of fluctuation variance.Source: Dynamics Decomposition · p. 3
• The assembly combines low-rank collective covariance with block-diagonal local motion while exactly preserving each prescribed residue-wise marginal covariance.Collective participation fractions also provide an explicit upper bound on normalized cross-residue coupling.Source: Theoretical Properties · p. 4, Covariance Assembly · p. 4
• On ATLAS, the leading K = 20 collective modes capture 95.8% of total variance, and oracle reconstruction has mean relative Frobenius error 0.037.The covariance family is low-rank plus block-diagonal.Source: Theoretical Properties · p. 5

3.3 Coupled Backbone–Dynamics Generation

• Shared backbone-flow representations generate backbone geometry and equilibrium dynamics, rather than estimating dynamics only after generating a static backbone.The dynamics branch uses intermediate flow features and endpoint rotations as local reference frames; predicted marginals, collective geometry, and participation are combined by the constrained assembly.Source: §3.3 · p. 5
• The dynamics branch is trained to recover MD-derived marginal fluctuations, collective-motion subspaces, and residue coupling while retaining differentiable access to the noised backbone state.The pretrained backbone-flow parameters remain fixed, enabling dynamics-guided backbone generation.Source: §3.3 · p. 5

4.1 Experiments Setup

• PhiFold trains on 1,390 ATLAS chains split by protein, then evaluates 50 designable generated proteins across five lengths against independent two-replica GROMACS simulations.The pretrained FoldFlow backbone trunk is frozen while only the dynamics heads are optimized.Source: §4.1 · p. 5, §4.2 · p. 5

4.2 Main Results

• PhiFold leads eight of twelve reported metrics and provides the strongest cross-scale fidelity, with strong local-fluctuation recovery and competitive collective-motion measures.It leads all three local-fluctuation metrics and ranks first on five of six unified ensemble metrics.Source: Table 1 · p. 6, §4.2 · p. 6

4.3 Ablation Study

• Direct covariance regression produces no positive-definite outputs; Cholesky remains inferior to the full model, and fixed α=0.5 particularly reduces residue-coupling recovery.The ablation compares variants on 50 generated proteins using proxy recovery for collective subspaces and residue coupling.Source: §4.3 · p. 6, Table 2 · p. 7

4.4 Controllable Generation of Regional Flexibility

• Flexibility guidance raises target RMSF by 0.104–0.221 Å and rigidity guidance lowers it by 0.025–0.101 Å, while outside-region changes generally stay within 0.033 Å.Aggregated designability changes remain within five percentage points and mean scTM changes within 0.017.Source: Figure 3 · p. 7, §4.4 · p. 7

• PhiFold jointly generates backbones and residue-displacement covariances, separating local flexibility, collective motion, and residue-wise participation while ensuring valid covariances with exact marginal consistency.Source: §5 · p. 7
• The framework is particularly suited to near-equilibrium fluctuations, domain-scale collective motion, and long-range dynamical coupling.Source: §5 · p. 7
• Amino-acid identity, sidechain interactions, energetic frustration, and environment can reshape state populations and large-amplitude transitions, so PhiFold is not a replacement for sequence-resolved atomistic modeling.Joint sequence–backbone–dynamics generation and multimodal, temporally resolved ensembles are identified as future directions.Source: §5 · p. 7

• The objectives use MD-derived marginal covariances and leading K collective eigenvectors, with separate supervision for marginals and collective covariance.Residues failing Cholesky decomposition are excluded from the marginal loss.Source: A Training Objectives · p. 10
• An auxiliary basis-invariant loss supervises the raw collective representation, while PCA-derived participation targets supervise residue-wise participation.The passage specifies implementation clamps for Cholesky diagonals and per-residue loss values.Source: A Training Objectives · p. 10
• Pairwise loss supervises local off-diagonal covariance blocks using each residue’s eight nearest spatial neighbors, while assembly preserves marginal blocks without an extra consistency loss.Source: A Training Objectives · p. 10

• Local-flexibility evaluation compares predicted and simulated RMSF patterns within each protein and assesses correlation of marginal-block eigenvalues to capture directional variance.The per-target RMSF correlation is averaged across targets.Source: B Evaluation Metrics · pp. 10–11
• Residue coupling is assessed by correlations between predicted and simulated coupling matrices over nonlocal residue pairs, with long-range coupling restricted to pairs more than 8 Å apart initially.Source: B Evaluation Metrics · p. 11
• Collective-motion agreement uses leading eigenspace overlap, with top-20 overlap computed per protein and then averaged across evaluated proteins.Source: B Evaluation Metrics · p. 11
• Ensemble agreement is assessed with per-residue Gaussian W2 distances and empirical W2 in top-two-PC space, using MD-PCA or Joint-PCA projections and common-size subsamples optimally matched by the Hungarian algorithm.Weak contacts are native pairs whose contact probability is below 0.9.Source: B Evaluation Metrics · p. 11

Error decomposition

• The representation separates approximation into low-rank collective covariance and a residue-local residual, preserving residue-wise marginal covariance exactly.Consequently, local fluctuation magnitude and anisotropy incur no approximation error; approximation remains in the correlated component.Source: C Error Decomposition and Empirical… · p. 12

Variance approximation

• Residue-wise collective variance is allocated through a scalar participation coefficient, an interpretable and controllable factorization that can deviate from empirical covariance.This variance-approximation source is distinct from error caused by discarding collective dimensions.Source: C Error Decomposition and Empirical… · p. 12

Empirical validation

• 0.037 mean relative Frobenius error is achieved using empirical marginals, top-20 collective factors, and residue-specific participation, versus 0.009 for unconstrained PCA20.Replacing residue-specific participation with a protein-wise mean raises the error to 0.099.Source: C Error Decomposition and Empirical… · p. 12, Figure 4 · p. 13

Collective approximation

• The median cumulative variance explained by the leading five and twenty modes is 0.882 and 0.958, respectively, indicating a concentrated low-dimensional collective subspace.The discarded covariance spectrum determines the irreducible collective-truncation error, rather than the parameterization.Source: C Error Decomposition and Empirical… · p. 12

Empirical validation

• The largest eigenvalue explains 75.9% of residue-wise marginal variance on average, while the smallest explains 7.0%, supporting full three-dimensional marginal covariance.Figure 4 summarizes eigenvalue contributions to marginal variance and the distribution of collective participation.Source: C Error Decomposition and Empirical… · p. 12, Figure 4 · p. 13

D Numerical Stability of the Covariance Assembly

• Eigenvalue floors safeguard inverse-square-root, square-root, and spectral operations during covariance assembly when intermediate blocks are ill-conditioned.The safeguards apply to the row Gram matrix, allocated collective covariance, and mode-level eigenvalues.Source: D Numerical Stability of the… · p. 13
• No numerical floor was activated across 50 generated proteins, 10,000 residue blocks, and 1,000 collective modes; intermediate spectra stayed above the applied thresholds.At lengths 100 and 300, the first percentile of row-Gram minimum eigenvalues was 9.4×10^-3 and 2.2×10^-3, respectively, over three orders of magnitude above the 10^-6 floor.Source: D Numerical Stability of the… · pp. 13–14

E.1 Inference Efficiency

• PhiFold predicts the backbone and equilibrium covariance once, then expands conformers through covariance-based sampling instead of repeated neural sampling.The end-to-end amortised cost includes one-time prediction spread over 1,000 conformers, while expansion cost measures additional conformers after covariance construction.Source: §E.1 · p. 14

E.2 Designability and Diversity

• 80.8% of 250 evaluated structures satisfy the designability criterion, and mean pairwise TM-score is 0.377, indicating diversity close to FoldFlow.The backbone network is inherited from FoldFlow and remains frozen during training.Source: §E.2 · pp. 14–15

E.3 Structural validity of generated ensembles

• Covariance-expanded conformations preserve similar structural characteristics while increasing conformational variability, without unphysical unfolding or structural collapse.The evaluation covers Cα distance validity, clashes, stereochemistry, and global compactness.Source: §E.3 · p. 15, Table 9 · p. 15, Figure 5 · p. 15

F Backbone-Level Effects and Designability under Regional Guidance

• Regional guidance produces greater RMSD inside than outside the selected region, while high ANM subspace overlap indicates largely preserved global collective-motion organization.Flexibility guidance generally reduces ordered secondary structure and rigidity guidance increases it; length-100 regional decomposition is interpreted cautiously due to its small outside-region alignment anchor.Source: F Backbone-Level Effects and… · p. 16, Table 10 · p. 16
• Guided and unconditional samples have broadly comparable designability; protein length has a larger effect than guidance condition, with designability decreasing for longer proteins.Regional guidance introduces only modest additional changes relative to the unconditional baseline.Source: F Backbone-Level Effects and… · p. 16, Table 10 · p. 16