{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T18:30:04Z","timestamp":1769106604017,"version":"3.49.0"},"update-to":[{"DOI":"10.1371\/journal.pcbi.1009821","type":"new_version","label":"New version","source":"publisher","updated":{"date-parts":[[2022,2,3]],"date-time":"2022-02-03T00:00:00Z","timestamp":1643846400000}}],"reference-count":41,"publisher":"Public Library of Science (PLoS)","issue":"1","license":[{"start":{"date-parts":[[2022,1,24]],"date-time":"2022-01-24T00:00:00Z","timestamp":1642982400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2019YFA0709501"],"award-info":[{"award-number":["2019YFA0709501"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12071466"],"award-info":[{"award-number":["12071466"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11871465"],"award-info":[{"award-number":["11871465"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>Time series single-cell RNA sequencing (scRNA-seq) data are emerging. However, dynamic inference of an evolving cell population from time series scRNA-seq data is challenging owing to the stochasticity and nonlinearity of the underlying biological processes. This calls for the development of mathematical models and methods capable of reconstructing cellular dynamic transition processes and uncovering the nonlinear cell-cell interactions. In this study, we present GraphFP, a nonlinear Fokker-Planck equation on graph based model and dynamic inference framework, with the aim of reconstructing the cell state-transition complex potential energy landscape from time series single-cell transcriptomic data. The free energy of our model explicitly takes into account of the cell-cell interactions in a nonlinear quadratic term. We then recast the model inference problem in the form of a dynamic optimal transport framework and solve it efficiently with the adjoint method of optimal control. We evaluated GraphFP on the time series scRNA-seq data set of embryonic murine cerebral cortex development. We illustrated that it 1) reconstructs cell state potential energy, which is a measure of cellular differentiation potency, 2) faithfully charts the probability flows between paired cell states over the dynamic processes of cell differentiation, and 3) accurately quantifies the stochastic dynamics of cell type frequencies on probability simplex in continuous time. We also illustrated that GraphFP is robust in terms of cluster labelling with different resolutions, as well as parameter choices. Meanwhile, GraphFP provides a model-based approach to delineate the cell-cell interactions that drive cell differentiation. GraphFP software is available at <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" ext-link-type=\"uri\" xlink:href=\"https:\/\/github.com\/QiJiang-QJ\/GraphFP\" xlink:type=\"simple\">https:\/\/github.com\/QiJiang-QJ\/GraphFP<\/jats:ext-link>.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1009821","type":"journal-article","created":{"date-parts":[[2022,1,24]],"date-time":"2022-01-24T18:42:04Z","timestamp":1643049724000},"page":"e1009821","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":30,"title":["Dynamic inference of cell developmental complex energy landscape from time series single-cell transcriptomic data"],"prefix":"10.1371","volume":"18","author":[{"given":"Qi","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Shuo","family":"Zhang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3511-0512","authenticated-orcid":true,"given":"Lin","family":"Wan","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2022,1,24]]},"reference":[{"issue":"7637","key":"pcbi.1009821.ref001","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1038\/nature21350","article-title":"Scaling single-cell genomics from phenomenology to mechanism","volume":"541","author":"A Tanay","year":"2017","journal-title":"Nature"},{"issue":"5","key":"pcbi.1009821.ref002","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1038\/s41587-019-0071-9","article-title":"A comparison of single-cell trajectory inference methods","volume":"37","author":"W Saelens","year":"2019","journal-title":"Nature 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TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics. In: III HD, Singh A, editors. Proceedings of the 37th International Conference on Machine Learning. vol. 119 of Proceedings of Machine Learning Research. Virtual: PMLR; 2020. p. 9526\u20139536."},{"issue":"1","key":"pcbi.1009821.ref017","doi-asserted-by":"crossref","first-page":"3222","DOI":"10.1038\/s41467-021-23518-w","article-title":"Generative modeling of single-cell time series with PRESCIENT enables prediction of cell trajectories with interventions","volume":"12","author":"GHT Yeo","year":"2021","journal-title":"Nature Communications"},{"issue":"7","key":"pcbi.1009821.ref018","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/s41576-021-00341-z","article-title":"Statistical mechanics meets single-cell biology","volume":"22","author":"AE Teschendorff","year":"2021","journal-title":"Nature Reviews Genetics"},{"issue":"3","key":"pcbi.1009821.ref019","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s00205-011-0471-6","article-title":"Fokker-Planck Equations for a Free Energy Functional or Markov Process on a Graph","volume":"203","author":"SN Chow","year":"2012","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"10","key":"pcbi.1009821.ref020","doi-asserted-by":"crossref","first-page":"4929","DOI":"10.3934\/dcds.2018215","article-title":"Entropy dissipation of Fokker-Planck equations on graphs","volume":"38","author":"SN Chow","year":"2018","journal-title":"Discrete & Continuous Dynamical Systems\u2014A"},{"key":"pcbi.1009821.ref021","unstructured":"Li W. A study of stochastic differential equations and Fokker-Planck equations with applications. Georgia Institute of Technology; 2016. Available from: http:\/\/hdl.handle.net\/1853\/54999."},{"issue":"2","key":"pcbi.1009821.ref022","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1038\/s41576-020-00292-x","article-title":"Deciphering cell\u2013cell interactions and communication from gene expression","volume":"22","author":"E Armingol","year":"2021","journal-title":"Nature Reviews Genetics"},{"issue":"1","key":"pcbi.1009821.ref023","doi-asserted-by":"crossref","first-page":"1088","DOI":"10.1038\/s41467-021-21246-9","article-title":"Inference and analysis of cell-cell communication using CellChat","volume":"12","author":"S Jin","year":"2021","journal-title":"Nature Communications"},{"key":"pcbi.1009821.ref024","volume-title":"Applied Optimal Control: Optimization, Estimation and Control","author":"AE Bryson","year":"1975"},{"key":"pcbi.1009821.ref025","first-page":"6571","volume-title":"Advances in Neural Information Processing Systems","author":"RTQ Chen","year":"2018"},{"issue":"13","key":"pcbi.1009821.ref026","doi-asserted-by":"crossref","first-page":"3970","DOI":"10.1016\/j.celrep.2017.12.017","article-title":"Developmental Emergence of Adult Neural Stem Cells as Revealed by Single-Cell Transcriptional Profiling","volume":"21","author":"SA Yuzwa","year":"2017","journal-title":"Cell Reports"},{"issue":"10","key":"pcbi.1009821.ref027","doi-asserted-by":"crossref","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","article-title":"Fast unfolding of communities in large networks","volume":"2008","author":"VD Blondel","year":"2008","journal-title":"Journal of Statistical Mechanics: Theory and Experiment"},{"issue":"7","key":"pcbi.1009821.ref028","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1016\/j.cell.2019.05.031","article-title":"Comprehensive Integration of Single-Cell Data","volume":"177","author":"T Stuart","year":"2019","journal-title":"Cell"},{"issue":"6","key":"pcbi.1009821.ref029","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1038\/s42256-021-00333-y","article-title":"Simultaneous deep generative modelling and clustering of single-cell genomic data","volume":"3","author":"Q Liu","year":"2021","journal-title":"Nature Machine Intelligence"},{"issue":"1","key":"pcbi.1009821.ref030","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1038\/msb.2010.71","article-title":"Dynamic interaction networks in a hierarchically organized tissue","volume":"6","author":"DC Kirouac","year":"2010","journal-title":"Molecular Systems Biology"},{"issue":"7","key":"pcbi.1009821.ref031","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1038\/s41592-021-01171-x","article-title":"The triumphs and limitations of computational methods for scRNA-seq","volume":"18","author":"PV Kharchenko","year":"2021","journal-title":"Nature 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