Steer, Select, and Stratify: Training-Free Sampling for Protein Motif Scaffolding

Motif scaffolding embeds functional sites with specified geometry into protein scaffolds for therapeutics and biocatalysis. RFdiffusion is widely used for this task. However, two challenges remain: obtaining successful designs on difficult tasks and covering diverse successful structures within a finite candidate budget. We further diagnose the first challenge into two bottlenecks: generated-motif errors are associated with design failure, while accurate motif geometry alone does not ensure structural recovery after sequence design and refolding. For the second challenge, independent sampling can repeatedly produce similar successful structures, limiting finite-batch coverage. We therefore propose ${\mathrm{S}}^3$-Scaffold (Steer, Select, and Stratify), a training-free sampling framework with three corresponding components: gated gradient guidance, rollout reweighting, and Exploration via Marginal-Preserving Stratification (EMPS). Gated gradient guidance restores truncated geometric gradients without changing the denoiser's forward computation and uses clean predictions to guide scaffold updates gated by denoising time and motif error. Rollout reweighting combines scaffold consistency and refolded-motif errors from sequence design and refolding into a parent score for allocating independent continuations. EMPS jointly stratifies scaffold layouts and a geometry-dependent process coordinate of translational noise without additional model evaluations. Before rollout reweighting, we prove that EMPS preserves matched-reference marginals and expected success count while providing expected coverage of fixed successful categories no worse than independent or single-axis sampling. For the full framework, we establish finite-batch lower bounds on expected coverage under anchor-retention and successful-continuation conditions. Across all 30 MotifBench tasks at the standard budget of 100 backbones per task, ${\mathrm{S}}^3$-Scaffold raises RFdiffusion's published score from 21.95 to 30.27, solved tasks from 17 to 22, and successful clusters from 205 to 225. Component ablations and mechanism diagnostics support the complementary roles of geometric control, downstream feedback, and coordinated exploration.