{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T14:01:22Z","timestamp":1790085682387,"version":"4.0.1"},"reference-count":35,"publisher":"American Chemical Society (ACS)","issue":"18","license":[{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-045"}],"funder":[{"DOI":"10.13039\/501100001502","name":"Department of Atomic Energy, Government of India","doi-asserted-by":"publisher","award":["RTI 4007"],"award-info":[{"award-number":["RTI 4007"]}],"id":[{"id":"10.13039\/501100001502","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/na","name":"Department of Science and Technology, Ministry of Science and Technology","doi-asserted-by":"publisher","award":["CRG\/2023\/001426"],"award-info":[{"award-number":["CRG\/2023\/001426"]}],"id":[{"id":"10.13039\/na","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,9,22]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Characterizing equilibrium conformational ensembles with deep generative models requires understanding whether a model reproduces a target distribution and how it reaches that distribution. Here, we compare two generative routes to molecular conformational sampling, stochastic relaxation and deterministic transport, using denoising diffusion probabilistic models and rectified-flow models across systems of increasing complexity: a multimodal two-dimensional potential, the folded miniprotein Trp-cage, and a high-dimensional dihedral representation of an intrinsically disordered protein. We show that these paradigms differ in end point fidelity and in how distributional error is resolved during sampling. Diffusion models converge through pronounced late-stage stochastic relaxation and robustly recover the configurational breadth across neural architectures. Rectified flow approaches the target distribution through deterministic transport and therefore depends more strongly on architectural expressivity, particularly in heterogeneous, high-dimensional landscapes. Entropy and moment-evolution analyses further show that diffusion more reliably restores the ensemble location and fluctuation structure, whereas rectified flow requires Transformer-level feature mixing to represent transport geometry accurately. These results establish the convergence mechanism as a practical design principle for molecular generative sampling, clarifying when stochastic diffusion provides robustness and when deterministic transport requires higher representational capacity.<\/jats:p>","DOI":"10.1021\/acs.jctc.6c01009","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T11:01:38Z","timestamp":1788778898000},"page":"9343-9359","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Mechanisms\nof Generative Models in Molecular\nConformational Sampling"],"prefix":"10.1021","volume":"22","author":[{"given":"Nagesh","family":"B E","sequence":"first","affiliation":[{"name":"Tata Institute of Fundamental Research , 36\/P, Gopanpally Village, Serilingampally Mandal, Ranga Reddy District , , ,","place":["Hyderabad, Telangana, India, 500046"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1090-5199","authenticated-orcid":true,"given":"Jagannath","family":"Mondal","sequence":"additional","affiliation":[{"name":"Tata Institute of Fundamental Research , 36\/P, Gopanpally Village, Serilingampally Mandal, Ranga Reddy District , , ,","place":["Hyderabad, Telangana, India, 500046"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"issue":"7","key":"2026092209063254900_cit1","doi-asserted-by":"publisher","DOI":"10.1063\/1.5109531","article-title":"Enhanced\nsampling in molecular dynamics","volume":"151","author":"Yang","year":"2019","journal-title":"J. Chem. Phys."},{"issue":"1","key":"2026092209063254900_cit2","doi-asserted-by":"publisher","DOI":"10.33011\/livecoms.4.1.1583","article-title":"Enhanced sampling\nmethods for molecular dynamics simulations","volume":"4","author":"He\u0301nin","year":"2022","journal-title":"Living J. Comput. Mol. Sci."},{"issue":"5","key":"2026092209063254900_cit3","doi-asserted-by":"publisher","first-page":"872","DOI":"10.1016\/j.bbagen.2014.10.019","article-title":"Enhanced\nsampling techniques in molecular\ndynamics simulations of biological systems","volume":"1850","author":"Melo","year":"2015","journal-title":"Biochim. Biophys. Acta, Gen. Subj."},{"issue":"1","key":"2026092209063254900_cit4","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1021\/acs.chemrev.5c00700","article-title":"Enhanced sampling in\nthe age of machine learning: Algorithms\nand applications","volume":"126","author":"Zhu","year":"2026","journal-title":"Chem. Rev."},{"issue":"3","key":"2026092209063254900_cit5","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/0304-4149(82)90051-5","article-title":"Reverse-time\ndiffusion equation models","volume":"12","author":"Anderson","year":"1982","journal-title":"Stochastic Processes\nAppl."},{"key":"2026092209063254900_cit6","unstructured":"Sohl-Dickstein, E. A. W. J.; Maheswaranathan, N.; Ganguli, S. In  Deep Unsupervised\nLearning Using Nonequilibrium Thermodynamics, Proceedings\nof the 32nd International Conference on Machine Learning; JMLR.org, 2015; pp 2256\u20132265."},{"key":"2026092209063254900_cit7","volume-title":"Generative Modeling by Estimating\nGradients of the Data Distribution","author":"Song","year":"2019"},{"key":"2026092209063254900_cit8","unstructured":"Ho, J.; Jain, A.; Abbeel, P. In  Denoising Diffusion Probabilistic\nModels, Proceedings of the 34th International Conference\non Neural Information Processing Systems; Curran Associates\nInc.: Red Hook, NY, USA, 2020."},{"key":"2026092209063254900_cit9","unstructured":"Dhariwal, P.; Nichol, A. In  Diffusion Models Beat\nGans on Image Synthesis, Proceedings of the 35th International\nConference on Neural Information Processing Systems; Curran Associates Inc.: Red Hook,\nNY, USA, 2021."},{"key":"2026092209063254900_cit10","doi-asserted-by":"crossref","unstructured":"Lim, S.; Yoon, E.; Byun, T.; Kang, T.; Kim, S.; Lee, K.; Choi, S. In  Score-based\nGenerative Modeling through Stochastic Evolution Equations in Hilbert\nSpaces, Proceedings of the 37th International Conference\non Neural Information Processing Systems; Curran Associates\nInc.: Red Hook, NY, USA, 2023.","DOI":"10.52202\/075280-1645"},{"key":"2026092209063254900_cit11","unstructured":"Mohamed, S.; Rezende, D. J. In  Variational Inference\nwith Normalizing Flows, Proceedings of the 32nd International\nConference on Machine Learning; PMLR, 2015; pp 1530\u20131538."},{"key":"2026092209063254900_cit12","unstructured":"Bettencourt, J.; Ricky, D. D.; Chen, T. Q.; Rubanova, Y. In  Neural Ordinary Differential\nEquations, Advances in Neural Information Processing Systems\n31; NeurIPS, 2019."},{"key":"2026092209063254900_cit13","unstructured":"Liu, Q.; Gong, C.; Liu, X.\n          Flow straight and fast: Learning to\ngenerate and transfer data with rectified flow. 2022, arXiv:2209.03003v1. arXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2209.03003."},{"key":"2026092209063254900_cit14","unstructured":"Ben-Hamu, H.; Nickel, M.; Le, M.; Lipman, Y.; Chen, R. T. Q.\n          Flow matching for generative\nmodeling. 2023, arXiv:2210.02747v2. arXiv.org\ne-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2210.02747."},{"key":"2026092209063254900_cit15","unstructured":"Burnaev, E.; Rout, L.; Korotin, A.\n          Generative modeling with optimal\ntransport maps. 2022, arXiv:2110.02999v2.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2110.02999."},{"key":"2026092209063254900_cit16","unstructured":"Herberg, E.\n          \n          Lecture notes:\nNeural network architectures. 2023, arXiv:2304.05133v2.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2304.05133."},{"key":"2026092209063254900_cit17","unstructured":"Prieur, C.; Lazar, M.; Robu, B.\n          A unified representation of neural\nnetworks architectures. 2025, arXiv:2512.17593v3.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2512.17593."},{"key":"2026092209063254900_cit18","first-page":"477","article-title":"Neural Networks:\nFrom the Perceptron to Deep Nets","volume-title":"Spin\nGlass Theory and Far Beyond","author":"Lucibello","year":"2013"},{"key":"2026092209063254900_cit19","unstructured":"Le, Q. V.; Sutskever, I.; Vinyals, O. In  Sequence to Sequence\nLearning with Neural Networks, Advances in Neural Information\nProcessing Systems 27; NeurIPS, 2014."},{"key":"2026092209063254900_cit20","unstructured":"Vaswani, A.; Shazeer, N.; Parmar, N.; et al. In  Attention\nis All you Need, Advances in Neural Information Processing\nSystems 30; NeurIPS, 2017."},{"issue":"6457","key":"2026092209063254900_cit21","doi-asserted-by":"publisher","DOI":"10.1126\/science.aaw1147","article-title":"Boltzmann generators:\nSampling equilibrium states of many-body systems with deep learning","volume":"365","author":"Noe\u0301","year":"2019","journal-title":"Science"},{"key":"2026092209063254900_cit22","unstructured":"Xie, Y.; Winkler, L.; Sun, L.; et al.\n          Enhanced diffusion\nsampling: Efficient rare event sampling and free energy calculation\nwith diffusion models. 2026, arXiv:2602.16634v2.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2602.16634."},{"issue":"6761","key":"2026092209063254900_cit23","doi-asserted-by":"publisher","DOI":"10.1126\/science.adv9817","article-title":"Scalable emulation of\nprotein equilibrium ensembles with generative deep learning","volume":"389","author":"Lewis","year":"2025","journal-title":"Science"},{"key":"2026092209063254900_cit24","unstructured":"Tan, C. B.; Bose, A. J.; Lin, C.; et al.\n          Scalable\nequilibrium sampling with sequential boltzmann generators. 2025, arXiv:2502.18462v3. arXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2502.18462."},{"issue":"11","key":"2026092209063254900_cit25","doi-asserted-by":"publisher","DOI":"10.1063\/5.0249683","article-title":"A comparison\nof probabilistic generative\nframeworks for molecular simulations","volume":"162","author":"John","year":"2025","journal-title":"J. Chem.\nPhys."},{"issue":"11","key":"2026092209063254900_cit26","doi-asserted-by":"publisher","DOI":"10.1063\/5.0279756","article-title":"How good is generative\ndiffusion model for enhanced\nsampling of protein conformations across scales and in all-atom resolution?","volume":"163","author":"Bera","year":"2025","journal-title":"J. Chem. Phys."},{"key":"2026092209063254900_cit27","doi-asserted-by":"publisher","first-page":"8735","DOI":"10.1039\/D5SC00108K","article-title":"Accurate prediction of the kinetic sequence of physicochemical\nstates using generative artificial intelligence","volume":"16","author":"Bera","year":"2025","journal-title":"Chem. Sci."},{"issue":"21","key":"2026092209063254900_cit28","doi-asserted-by":"publisher","first-page":"E4758","DOI":"10.1073\/pnas.1800690115","article-title":"Developing a molecular dynamics force\nfield for both folded and disordered protein states","volume":"115","author":"Robustelli","year":"2018","journal-title":"Proc. Natl. Acad. Sci. U.S.A."},{"key":"2026092209063254900_cit29","doi-asserted-by":"crossref","unstructured":"Li, J.; Li, B.; Guo, Z.; et al. In  Scaling Molecular\nDynamics\nwith ab initio Accuracy to 149 Nanoseconds Per Day, Proceedings\nof the International Conference for High Performance Computing, Networking,\nStorage, and Analysis; IEEE Press, 2024.","DOI":"10.1109\/SC41406.2024.00036"},{"issue":"18","key":"2026092209063254900_cit30","doi-asserted-by":"publisher","first-page":"4032","DOI":"10.1021\/acs.jpcb.3c00982","article-title":"Conformational plasticity\nin \u03b1-synuclein and how crowded environment\nmodulates it","volume":"127","author":"Menon","year":"2023","journal-title":"J. Phys. Chem. B"},{"key":"2026092209063254900_cit31","unstructured":"Ba, J.; Kingma, D. P.\n          Adam: A method\nfor stochastic optimization. 2014, arXiv:1412.6980v9.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.1412.6980."},{"key":"2026092209063254900_cit32","unstructured":"Hutter, F.; Loshchilov, I.\n          Decoupled weight\ndecay regularization. 2017, arXiv:1711.05101v3.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.1711.05101."},{"key":"2026092209063254900_cit33","volume-title":"PyTorch: An Imperative Style,\nHigh-performance Deep Learning Library","author":"Paszke","year":"2019"},{"key":"2026092209063254900_cit34","unstructured":"Shlens, J.\n          \n          Notes on kullback-leibler\ndivergence and likelihood. 2014, arXiv:1404.2000v1.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.1404.2000."},{"key":"2026092209063254900_cit35","unstructured":"Schwing, A. G.; Guo, P.\n          Variational rectified\nflow matching. 2025, arXiv:2502.09616v1.\narXiv.org e-Print archive. https:\/\/doi.org\/10.48550\/arXiv.2502.09616."}],"container-title":["Journal of Chemical Theory\nand Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.acs.org\/doi\/pdf\/10.1021\/acs.jctc.6c01009","content-type":"application\/pdf","content-version":"vor","intended-application":"unspecified"},{"URL":"https:\/\/pubs.acs.org\/jctcce\/article-pdf\/22\/18\/9343\/68401504\/acs.jctc.6c01009.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.acs.org\/jctcce\/article-pdf\/22\/18\/9343\/68401504\/acs.jctc.6c01009.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,22]],"date-time":"2026-09-22T13:13:37Z","timestamp":1790082817000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.acs.org\/jctcce\/article\/22\/18\/9343\/5422249\/Convergence-Mechanisms-of-Generative-Models-in"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,7]]},"references-count":35,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2026,9,7]]},"published-print":{"date-parts":[[2026,9,22]]}},"URL":"https:\/\/doi.org\/10.1021\/acs.jctc.6c01009","relation":{},"ISSN":["1549-9618","1549-9626"],"issn-type":[{"value":"1549-9618","type":"print"},{"value":"1549-9626","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,7]]}}}