{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T12:48:18Z","timestamp":1791809298792,"version":"4.3.4"},"reference-count":37,"publisher":"American Chemical Society (ACS)","issue":"19","license":[{"start":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T00:00:00Z","timestamp":1790035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100021171","name":"Basic and Applied Basic Research Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2021A1515110418"],"award-info":[{"award-number":["2021A1515110418"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,10,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Bayesian flow networks (BFNs) jointly model atomic coordinates and atom types for structure-based drug design, but their generative dynamics remain poorly understood. We show that the two channels resolve on different terms: atom types pass through a sharp commitment transition at tc1, whereas coordinate precision improves smoothly with no critical point of its own, reaching sub-\u00c5ngstrom resolution only at a stated threshold tc2. Extreme-value theory predicts the type channel\u2019s posterior half-maximum in closed form, and the predicted 1\/\u03b21 scaling holds across seven \u03b21 values (R2 = 0.996) and over 13 points spanning three BFN architectures and two domains, though measured times run 11%\u201320% below the predicted absolute values. Exploiting that structure, phase-aware iterative refinement aligns three rounds to these boundaries and trains no auxiliary regressor, reward function, or property predictor. On 100 CrossDocked2020 test pockets at K = 200, it reaches a mean Vina Dock of \u22128.72 kcal\/mol. Gradient-guided methods that need such training stay ahead by a small margin: over the 100-pocket intersection, the paired gap to MolJO (\u22128.98) is 0.25 kcal\/mol (95% CI [\u22120.13, +0.60], paired Wilcoxon p = 0.022); equivalence testing at a preregistered \u00b10.5 kcal\/mol margin does not establish equivalence within it. CByG (\u22129.16) leads by 0.44 kcal\/mol on unpaired means. Phase-aware iteration improves the Round-1 to Round-3 mean by 1.34 kcal\/mol (paired Wilcoxon p = 4.80 \u00d7 10\u201318, Cohen\u2019s dz = \u22121.59) and shifts the median by \u22121.44 kcal\/mol, a shift Best-of-K resampling cannot produce. Paired on the pocket, an audited rerun passes every PoseBusters check 4.7 percentage points more often than MolJO.<\/jats:p>","DOI":"10.1021\/acs.jcim.6c01538","type":"journal-article","created":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T15:04:30Z","timestamp":1790089470000},"page":"12346-12359","source":"Crossref","is-referenced-by-count":0,"title":["Hierarchical Phase\nTransitions in Bayesian Flow Networks\nEnable Training-Free Molecular Design"],"prefix":"10.1021","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-6742-6320","authenticated-orcid":true,"given":"Liangji","family":"Zhou","sequence":"first","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuanqiu","family":"Chen","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tongtong","family":"Chen","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yijuan","family":"Huang","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qihao","family":"Pan","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Si","family":"Li","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chenyu","family":"Gou","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Meiqiong","family":"Fan","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu","family":"Gao","sequence":"additional","affiliation":[{"name":"the Sixth Affiliated Hospital of Sun Yat-sen University , , No. 26 of Yuancun Second Cross Road , ,","place":["Guangzhou, Guangdong, China, 510655"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,9,22]]},"reference":[{"key":"2026101208074673800_cit1","unstructured":"Ho, J.; Jain, A.; Abbeel, P.\n          Denoising Diffusion Probabilistic\nModels.  Advances in Neural Information Processing\nSystemsCurran Associates Inc., Red Hook, NY, USA2020, 6840\u20136851."},{"key":"2026101208074673800_cit2","unstructured":"Song, Y.; Sohl-Dickstein, J.; Kingma, D. P.; Kumar, A.; Ermon, S.; Poole, B.\n          Score-Based Generative Modeling through\nStochastic Differential Equations.  International\nConference on Learning Representations. 2021; arXiv:2011.13456."},{"key":"2026101208074673800_cit3","unstructured":"Hoogeboom, E.; Satorras, V. G.; Vignac, C.; Welling, M.\n          Equivariant Diffusion\nfor Molecule Generation in 3D.  Proceedings\nof the 39th International Conference on Machine Learning (ICML). 2022; pp 8867\u20138887, arXiv:2203.17003."},{"key":"2026101208074673800_cit4","unstructured":"Guan, J.; Qian, W. W.; Peng, X.; Su, Y.; Peng, J.; Ma, J.\n          3D Equivariant Diffusion for Target-Aware Molecule\nGeneration and Affinity Prediction.  International\nConference on Learning Representations (ICLR). 2023; arXiv:2303.03543."},{"key":"2026101208074673800_cit5","unstructured":"Guan, J.; Zhou, X.; Yang, Y.; Bao, Y.; Peng, J.; Ma, J.; Liu, Q.; Wang, L.; Gu, Q.\n          DecompDiff: Diffusion Models\nwith Decomposed Priors for Structure-Based Drug Design.  Proceedings of the 40th International Conference on Machine\nLearning (ICML). 2023; pp 11827\u201311846, arXiv:2403.07902."},{"key":"2026101208074673800_cit6","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1038\/s43588-024-00737-x","article-title":"Structure-based drug design with equivariant diffusion\nmodels","volume":"4","author":"Schneuing","year":"2024","journal-title":"Nat. Comput. Sci."},{"key":"2026101208074673800_cit7","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1016\/j.chembiol.2003.09.002","article-title":"The Process\nof Structure-Based Drug Design","volume":"10","author":"Anderson","year":"2003","journal-title":"Chem. Biol."},{"key":"2026101208074673800_cit8","unstructured":"Peng, X.; Luo, S.; Guan, J.; Xie, Q.; Peng, J.; Ma, J.\n          Pocket2Mol: Efficient Molecular Sampling Based on\n3D Protein Pockets.  Proceedings of the 39th\nInternational Conference on Machine\nLearning (ICML). 2022; pp 17644\u201317655, arXiv:2205.07249."},{"key":"2026101208074673800_cit9","doi-asserted-by":"publisher","first-page":"2701","DOI":"10.1039\/D1SC05976A","article-title":"Generating 3D molecules conditional\non receptor binding sites with deep generative models","volume":"13","author":"Ragoza","year":"2022","journal-title":"Chem. Sci."},{"key":"2026101208074673800_cit10","unstructured":"Corso, G.; Sta\u0308rk, H.; Jing, B.; Barzilay, R.; Jaakkola, T.\n          DiffDock: Diffusion\nSteps,\nTwists, and Turns for Molecular Docking.  International\nConference on Learning Representations. 2023; arXiv:2210.01776."},{"key":"2026101208074673800_cit11","doi-asserted-by":"crossref","unstructured":"Jing, B.; Corso, G.; Chang, J.; Barzilay, R.; Jaakkola, T.\n          Torsional Diffusion for Molecular\nConformer Generation. Advances in\nNeural Information Processing Systems2022, 24240\u201324253.","DOI":"10.52202\/068431-1760"},{"key":"2026101208074673800_cit12","article-title":"Bayesian Flow Networks","author":"Graves"},{"key":"2026101208074673800_cit13","unstructured":"Song, Y.; Gong, J.; Qu, Y.; Zhou, H.; Zheng, M.; Liu, J.; Ma, W.-Y.\n          Unified Generative Modeling of 3D Molecules via Bayesian Flow Networks.  International Conference on Learning Representations. 2024; Oral. arXiv:2403.15441."},{"key":"2026101208074673800_cit14","unstructured":"Qu, Y.; Qiu, K.; Song, Y.; Gong, J.; Han, J.; Zheng, M.; Zhou, H.; Ma, W.-Y.\n          MolCRAFT: Structure-Based\nDrug Design in Continuous Parameter Space.  Proceedings of the 41st International Conference on Machine Learning\n(ICML). 2024; pp 41749\u201341768, arXiv:2404.12141."},{"key":"2026101208074673800_cit15","unstructured":"Qiu, K.; Song, Y.; Yu, J.; Ma, H.; Cao, Z.; Zhang, Z.; Wu, Y.; Zheng, M.; Zhou, H.; Ma, W.-Y.\n          Empower Structure-Based Molecule Optimization\nwith Gradient Guided Bayesian Flow Networks.  International Conference on Machine Learning (ICML). 2025; arXiv:2411.13280."},{"key":"2026101208074673800_cit16","doi-asserted-by":"crossref","unstructured":"Choi, S.; Kim, H.; Park, C.; Lee, D.; Lee, S.; Kim, Y.; Park, H.; Kwon, S.; Jo, Y.; Park, S.\n          Controllable 3D Molecular Generation for\nStructure-based Drug Design Through Bayesian Flow Networks and Gradient\nIntegration. 2025; arXiv:2508.21468 [q-bio.BM].","DOI":"10.52202\/085713-4187"},{"key":"2026101208074673800_cit17","doi-asserted-by":"crossref","unstructured":"Zhong, Q.; Yu, H.; Lin, Y.; Shen, W.; Zeng, L.; Hu, J.\n          SculptDrug: A Spatial Condition-Aware Bayesian Flow\nModel for Structure-based\nDrug Design.  Proceedings of the AAAI Conference\non Artificial Intelligence. 2026; pp 1623\u20131631, arXiv:2511.12489 [cs.LG].","DOI":"10.1609\/aaai.v40i2.37139"},{"key":"2026101208074673800_cit18","unstructured":"Qiu, K.; Song, Y.; Fan, Z.; Liu, P.; Zhang, Z.; Zheng, M.; Zhou, H.; Ma, W.-Y.\n          Piloting Structure-Based\nDrug Design via Modality-Specific Optimal Schedule.  International Conference on Machine Learning (ICML). 2025; arXiv:2505.07286."},{"key":"2026101208074673800_cit19","doi-asserted-by":"publisher","first-page":"9957","DOI":"10.1038\/s41467-024-54281-3","article-title":"Dynamical\nregimes of diffusion models","volume":"15","author":"Biroli","year":"2024","journal-title":"Nat. Commun."},{"key":"2026101208074673800_cit20","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2408799121","article-title":"A phase transition\nin diffusion models\nreveals the hierarchical nature of data","volume":"122","author":"Sclocchi","year":"2025","journal-title":"Proc.\nNatl. Acad. Sci. U. S. 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Model."},{"key":"2026101208074673800_cit29","author":"Harris","year":"2023","journal-title":"Benchmarking\nGenerated Poses: How\nRational is Structure-based Drug Design with Generative Models?"},{"key":"2026101208074673800_cit30","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1103\/RevModPhys.54.235","article-title":"The Potts\nmodel","volume":"54","author":"Wu","year":"1982","journal-title":"Rev. Mod. Phys."},{"key":"2026101208074673800_cit31","unstructured":"Snell, C.; Lee, J.; Xu, K.; Kumar, A.\n          Scaling LLM Test-Time Compute Optimally\ncan be More\nEffective than Scaling Model Parameters.  International\nConference on Learning Representations (ICLR). 2025; Oral presentation. arXiv:2408.03314."},{"key":"2026101208074673800_cit32","doi-asserted-by":"crossref","DOI":"10.1021\/acs.jcim.5c02316","article-title":"Test-Time Training Scaling\nLaws for Chemical Exploration in Drug Design","author":"Thomas","year":"2025"},{"key":"2026101208074673800_cit33","article-title":"Inference-Time Alignment in Diffusion Models with\nReward-Guided Generation: Tutorial and Review","author":"Uehara","year":"2025"},{"key":"2026101208074673800_cit34","doi-asserted-by":"publisher","first-page":"4200","DOI":"10.1021\/acs.jcim.0c00411","article-title":"Three-Dimensional Convolutional Neural\nNetworks and a Cross-Docked Data Set for Structure-Based Drug Design","volume":"60","author":"Francoeur","year":"2020","journal-title":"J. Chem. Inf. 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