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

arXiv: https://arxiv.org/abs/2609.32309 · HTML: https://mlsanity.com/paper/2609.32309/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics](https://mlsanity.com/paper/2302.01170/) (2023)
- [Protein Conformation Generation via Force-Guided SE(3) Diffusion Models](https://mlsanity.com/paper/2403.14088/) (2024)
- [Str2Str: A Score-based Framework for Zero-shot Protein Conformation Sampling](https://mlsanity.com/paper/2306.03117/) (2023)
- [Learning residue level protein dynamics with multiscale Gaussians](https://mlsanity.com/paper/2509.01038/) (2025)

## Abstract (arXiv)

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.