LieFlower: Controlling Protein Dynamics via Lie-Guided Discrete Flows

Many diseases, including cancer and neurodegeneration, arise from mutation-driven changes in protein dynamics that shift conformational ensembles toward dysfunctional regimes rather than from a single aberrant structure. Existing generative protein design methods largely condition on static structures or sequence-derived features, leaving no mechanism to design sequences that actively modulate non-equilibrium motion. We introduce LieFlower, a dynamics-conditioned generative framework that integrates discrete flow matching with Lie group representations of molecular dynamics to control protein behavior at the sequence level. We encode protein trajectories using truncated log-signatures, elements of a free Lie algebra that capture causal ordering and non-equilibrium flux, and we quantify conformational shifts by projecting log-signature differences along a global reference direction defined through a holonomy-aware Baker-Campbell-Hausdorff composition. This construction yields scalar targets that measure directional dynamical change, which we predict directly from sequence using a surrogate model and use to guide discrete generative flows without requiring simulation at inference time. We evaluate LieFlower by generating both targeted mutations and peptide binders that induce directional shifts in protein dynamics, including peptide-mediated modulation of mutant p53, demonstrating a general framework for dynamics-aware sequence design beyond static structure-based models.