Generating knotted polymer and protein structures by machine learning.

Knotted molecules occur naturally and can be designed to obtain unique biological and materials properties. While knotted molecular conformations can be produced using conventional molecular simulation methods, here we develop a diffusion-based machine-learning framework that directly generates polymer and protein conformations with specified knot types. Although diffusion models have achieved remarkable success in image generation and other domains, steering the diffusion process toward a desired molecular topology presents a distinct challenge. To address this, we integrate a knot-type classifier into the diffusion model, which guides the generative process toward the specified topology. Among several architectures tested, a Transformer-based classifier performs best, achieving over 99% accuracy in recognizing knot types for polymers with variable chain lengths. With this classifier-guided framework, the diffusion model generates polymer conformations that not only exhibit the desired knot types but also reproduce key structural statistics of the training ensembles, including the radius of gyration and knot size distributions. We further integrate this approach with an existing protein backbone diffusion model to generate knotted protein structures. Sequence-structure consistency tests support the structural plausibility of the generated designs. These results provide a potential route for designing knotted polymers and proteins.