BioDynaSpec: Harmonic-Guided Spatio-Spectral Autoregressive Diffusion for Protein Dynamics Generation

Generating long-horizon molecular dynamics (MD) is difficult due to error accumulation in time-domain autoregressive models, which causes drift, and fixed step-size constraints on temporal resolution. We propose **BioDynaSpec**, which reformulates protein dynamics as spatio-spectral generation: **Independent Windowed Fourier Decomposition (IWFD)** decomposes trajectories into window-wise spectral representations, and a generator combines low-to-high frequency autoregression with diffusion denoising to reconstruct continuous motion. This formulation is motivated by **a local near-equilibrium view of protein dynamics**: after per-window alignment, fluctuations around an anchor conformation are better characterized in spectral space, where local mode structure is more explicit than in frame-wise coordinates. To improve cross-residue and cross-frequency consistency, we introduce **Inter-Residue Frequency Coupling (IRFC)**, a learnable Gaussian distance bias in attention that injects a resonance-inspired structural prior. On ATLAS, BioDynaSpec improves 250-frame trajectory generation with $R_{250}=1.509$ Å, where $R_s$ denotes the mean per-frame C$\alpha$-RMSE over the first $s$ frames after alignment, reducing error by 60.4\% versus MDGEN and 57.2\% versus ProAR, while achieving the best PCA-2D displacement-profile correlation and stepwise distribution matching. For equilibrium conformational sampling, it achieves Root Mean $W_2=1.31$, MD PCA $W_2=0.90$, and Joint PCA $W_2=1.19$, improving over the next best method by 50.03\%, 35.25\%, and 47.58\%, respectively. It also improves near-equilibrium local-dynamics and covariance consistency, achieving PCA-PSD-LogCorr $=0.817$ and CFRE $=0.989$, corresponding to a 21.9\% gain and a 36.7\% reduction over the next best method, respectively. The source code is available at https://github.com/Linmj-Judy/BioDynaSpec.git.

Proteins do not stay still: they bend, vibrate, and shift shape over time, and these motions are often essential for how they work in the body. Simulating these motions accurately is important, but existing AI methods often drift off course when asked to predict many future steps in a row. Our method, called BioDynaSpec, tackles this by looking at protein motion in a different way. Instead of predicting each future structure one frame at a time, it breaks motion into short time windows and represents each window in terms of its underlying movement patterns, similar to separating a sound into different frequencies. This makes slow, large-scale motions and fast, local vibrations easier to model together. We also add a physics-inspired mechanism that helps the model respect how nearby parts of a protein influence one another. On a large benchmark dataset, BioDynaSpec predicts long protein motion sequences more accurately than previous methods and produces more realistic collections of protein shapes. These results suggest that combining machine learning with physically meaningful motion representations can make protein dynamics generation more reliable.