{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T18:22:22Z","timestamp":1774203742433,"version":"3.50.1"},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2018,7,2]],"date-time":"2018-07-02T00:00:00Z","timestamp":1530489600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100013986","name":"UK government","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100013986","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100000265","name":"MRC","doi-asserted-by":"publisher","award":["MR\/M008908\/1"],"award-info":[{"award-number":["MR\/M008908\/1"]}],"id":[{"id":"10.13039\/501100000265","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004440","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["204832\/B\/16\/Z"],"award-info":[{"award-number":["204832\/B\/16\/Z"]}],"id":[{"id":"10.13039\/100004440","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:sec>\n                    <jats:title>Motivation<\/jats:title>\n                    <jats:p>The Gaussian Process Latent Variable Model (GPLVM) is a popular approach for dimensionality reduction of single-cell data and has been used for pseudotime estimation with capture time information. However, current implementations are computationally intensive and will not scale up to modern droplet-based single-cell datasets which routinely profile many tens of thousands of cells.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Results<\/jats:title>\n                    <jats:p>We provide an efficient implementation which allows scaling up this approach to modern single-cell datasets. We also generalize the application of pseudotime inference to cases where there are other sources of variation such as branching dynamics. We apply our method on microarray, nCounter, RNA-seq, qPCR and droplet-based datasets from different organisms. The model converges an order of magnitude faster compared to existing methods whilst achieving similar levels of estimation accuracy. Further, we demonstrate the flexibility of our approach by extending the model to higher-dimensional latent spaces that can be used to simultaneously infer pseudotime and other structure such as branching. Thus, the model has the capability of producing meaningful biological insights about cell ordering as well as cell fate regulation.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Availability and implementation<\/jats:title>\n                    <jats:p>Software available at github.com\/ManchesterBioinference\/GrandPrix.<\/jats:p>\n                  <\/jats:sec>\n                  <jats:sec>\n                    <jats:title>Supplementary information<\/jats:title>\n                    <jats:p>Supplementary data are available at Bioinformatics online.<\/jats:p>\n                  <\/jats:sec>","DOI":"10.1093\/bioinformatics\/bty533","type":"journal-article","created":{"date-parts":[[2018,6,28]],"date-time":"2018-06-28T23:10:31Z","timestamp":1530227431000},"page":"47-54","source":"Crossref","is-referenced-by-count":47,"title":["GrandPrix: scaling up the Bayesian GPLVM for single-cell data"],"prefix":"10.1093","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4618-5018","authenticated-orcid":false,"given":"Sumon","family":"Ahmed","sequence":"first","affiliation":[{"name":"Division of Informatics, Imaging and Data Sciences, Faculty of Biology, Medicine and Health, University of Manchester, Manchester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Magnus","family":"Rattray","sequence":"additional","affiliation":[{"name":"Division of Informatics, Imaging and Data Sciences, Faculty of Biology, Medicine and Health, University of Manchester, Manchester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexis","family":"Boukouvalas","sequence":"additional","affiliation":[{"name":"Division of Informatics, Imaging and Data Sciences, Faculty of Biology, Medicine and Health, University of Manchester, Manchester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2018,7,2]]},"reference":[{"key":"2023013107215178900_bty533-B1","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1016\/j.cpc.2008.11.005","article-title":"Accelerating scientific computations with mixed precision algorithms","volume":"180","author":"Baboulin","year":"2009","journal-title":"Comput. Phys. Commun"},{"key":"2023013107215178900_bty533-B2","first-page":"1533","article-title":"Understanding probabilistic sparse gaussian process approximations","author":"Bauer","year":"2016","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2023013107215178900_bty533-B3","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1038\/ni.3006","article-title":"High-dimensional analysis of the murine myeloid cell system","volume":"15","author":"Becher","year":"2014","journal-title":"Nat. Immunol"},{"key":"2023013107215178900_bty533-B4","doi-asserted-by":"crossref","first-page":"714","DOI":"10.1016\/j.cell.2014.04.005","article-title":"Single-cell trajectory detection uncovers progression and regulatory coordination in human b cell development","volume":"157","author":"Bendall","year":"2014","journal-title":"Cell"},{"key":"2023013107215178900_bty533-B5","doi-asserted-by":"crossref","first-page":"i626","DOI":"10.1093\/bioinformatics\/bts385","article-title":"A novel approach for resolving differences in single-cell gene expression patterns from zygote to blastocyst","volume":"28","author":"Buettner","year":"2012","journal-title":"Bioinformatics"},{"key":"2023013107215178900_bty533-B6","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1038\/nbt.3102","article-title":"Computational analysis of cell-to-cell heterogeneity in single-cell rna-sequencing data reveals hidden subpopulations of cells","volume":"33","author":"Buettner","year":"2015","journal-title":"Nat. Biotechnol"},{"key":"2023013107215178900_bty533-B7","doi-asserted-by":"crossref","first-page":"e1005212.","DOI":"10.1371\/journal.pcbi.1005212","article-title":"Order under uncertainty: robust differential expression analysis using probabilistic models for pseudotime inference","volume":"12","author":"Campbell","year":"2016","journal-title":"PLoS Comput. Biol"},{"key":"2023013107215178900_bty533-B8","first-page":"1425","article-title":"Variational inference for latent variables and uncertain inputs in gaussian processes","volume":"17","author":"Damianou","year":"2016","journal-title":"J. Mach. Learn. Res"},{"key":"2023013107215178900_bty533-B10","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/j.devcel.2010.02.012","article-title":"Resolution of cell fate decisions revealed by single-cell gene expression analysis from zygote to blastocyst","volume":"18","author":"Guo","year":"2010","journal-title":"Dev. Cell"},{"key":"2023013107215178900_bty533-B11","doi-asserted-by":"crossref","first-page":"2989","DOI":"10.1093\/bioinformatics\/btv325","article-title":"Diffusion maps for high-dimensional single-cell analysis of differentiation data","volume":"31","author":"Haghverdi","year":"2015","journal-title":"Bioinformatics"},{"key":"2023013107215178900_bty533-B12","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1038\/nmeth.3971","article-title":"Diffusion pseudotime robustly reconstructs lineage branching","volume":"13","author":"Haghverdi","year":"2016","journal-title":"Nat. Methods"},{"key":"2023013107215178900_bty533-B13","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF01908075","article-title":"Comparing partitions","volume":"2","author":"Hubert","year":"1985","journal-title":"J. Classification"},{"key":"2023013107215178900_bty533-B14","doi-asserted-by":"crossref","first-page":"e117","DOI":"10.1093\/nar\/gkw430","article-title":"TSCAN: pseudo-time reconstruction and evaluation in single-cell rna-seq analysis","volume":"44","author":"Ji","year":"2016","journal-title":"Nucleic Acids Res"},{"key":"2023013107215178900_bty533-B15","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1016\/j.cell.2015.04.044","article-title":"Droplet barcoding for single-cell transcriptomics applied to embryonic stem cells","volume":"161","author":"Klein","year":"2015","journal-title":"Cell"},{"key":"2023013107215178900_bty533-B16","first-page":"1783","article-title":"Probabilistic non-linear principal component analysis with gaussian process latent variable models","volume":"6","author":"Lawrence","year":"2005","journal-title":"J. Mach. Learn. Res"},{"key":"2023013107215178900_bty533-B17","doi-asserted-by":"crossref","DOI":"10.1126\/sciimmunol.aal2192","article-title":"Single-cell rna-seq and computational analysis using temporal mixture modelling resolves th1\/tfh fate bifurcation in malaria","volume":"2","author":"L\u00f6nnberg","year":"2017","journal-title":"Sci. Immunol"},{"key":"2023013107215178900_bty533-B18","first-page":"133","article-title":"Introduction to gaussian processes","volume":"168","author":"MacKay","year":"1998","journal-title":"NATO ASI Series F Comput. Syst. Sci"},{"key":"2023013107215178900_bty533-B19","doi-asserted-by":"crossref","first-page":"E5643","DOI":"10.1073\/pnas.1408993111","article-title":"Bifurcation analysis of single-cell gene expression data reveals epigenetic landscape","volume":"111","author":"Marco","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"2023013107215178900_bty533-B20","first-page":"1","article-title":"GPflow: a Gaussian process library using TensorFlow","volume":"18","author":"Matthews","year":"2017","journal-title":"J. Mach. Learn. Res"},{"key":"2023013107215178900_bty533-B21","doi-asserted-by":"crossref","first-page":"e1003696","DOI":"10.1371\/journal.pcbi.1003696","article-title":"Modeling bi-modality improves characterization of cell cycle on gene expression in single cells","volume":"10","author":"McDavid","year":"2014","journal-title":"PLoS Comput. Biol"},{"key":"2023013107215178900_bty533-B22","doi-asserted-by":"crossref","first-page":"979.","DOI":"10.1038\/nmeth.4402","article-title":"Reversed graph embedding resolves complex single-cell trajectories","volume":"14","author":"Qiu","year":"2017","journal-title":"Nat. Methods"},{"key":"2023013107215178900_bty533-B23","first-page":"1939","article-title":"A unifying view of sparse approximate gaussian process regression","volume":"6","author":"Qui\u00f1onero-Candela","year":"2005","journal-title":"J. Mach. Learn. Res"},{"key":"2023013107215178900_bty533-B24","volume-title":"Gaussian Processes for Machine Learning","author":"Rasmussen","year":"2006"},{"key":"2023013107215178900_bty533-B25","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.1093\/bioinformatics\/btw372","article-title":"Pseudotime estimation: deconfounding single cell time series","volume":"32","author":"Reid","year":"2016","journal-title":"Bioinformatics"},{"key":"2023013107215178900_bty533-B26","first-page":"10.1101\/276907","article-title":"A comparison of single-cell trajectory inference methods: towards more accurate and robust tools","author":"Saelens","year":"2018","journal-title":"bioRxiv"},{"key":"2023013107215178900_bty533-B27","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1038\/nature13437","article-title":"Single-cell RNA-seq reveals dynamic paracrine control of cellular variation","volume":"510","author":"Shalek","year":"2014","journal-title":"Nature"},{"key":"2023013107215178900_bty533-B28","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.stem.2015.07.013","article-title":"Single-cell RNA-seq with waterfall reveals molecular cascades underlying adult neurogenesis","volume":"17","author":"Shin","year":"2015","journal-title":"Cell Stem Cell"},{"key":"2023013107215178900_bty533-B29","first-page":"1257","article-title":"Sparse gaussian processes using pseudo-inputs","author":"Snelson","year":"2006","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2023013107215178900_bty533-B30","first-page":"567","article-title":"Variational learning of inducing variables in sparse gaussian processes","author":"Titsias","year":"2009","journal-title":"International Conference on Artificial Intelligence and Statistics"},{"key":"2023013107215178900_bty533-B31","first-page":"844","article-title":"Bayesian gaussian process latent variable model","author":"Titsias","year":"2010","journal-title":"International Conference on Artificial Intelligence and Statistics"},{"key":"2023013107215178900_bty533-B32","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1038\/nbt.2859","article-title":"The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells","volume":"32","author":"Trapnell","year":"2014","journal-title":"Nat. Biotechnol"},{"key":"2023013107215178900_bty533-B33","doi-asserted-by":"crossref","first-page":"3530","DOI":"10.1105\/tpc.112.102046","article-title":"Arabidopsis defense against botrytis cinerea: chronology and regulation deciphered by high-resolution temporal transcriptomic analysis","volume":"24","author":"Windram","year":"2012","journal-title":"Plant Cell"},{"key":"2023013107215178900_bty533-B34","doi-asserted-by":"crossref","first-page":"14049","DOI":"10.1038\/ncomms14049","article-title":"Massively parallel digital transcriptional profiling of single cells","volume":"8","author":"Zheng","year":"2017","journal-title":"Nat. Commun"},{"key":"2023013107215178900_bty533-B35","article-title":"Topslam: waddington landscape recovery for single cell experiments","author":"Zwiessele","year":"2016","journal-title":"bioRxiv"}],"container-title":["Bioinformatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/35\/1\/47\/48962689\/bioinformatics_35_1_47.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article-pdf\/35\/1\/47\/48962689\/bioinformatics_35_1_47.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T05:02:06Z","timestamp":1675141326000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/bioinformatics\/article\/35\/1\/47\/5047752"}},"subtitle":[],"editor":[{"given":"Oliver","family":"Stegle","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2018,7,2]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1,1]]}},"URL":"https:\/\/doi.org\/10.1093\/bioinformatics\/bty533","relation":{"has-preprint":[{"id-type":"doi","id":"10.1101\/227843","asserted-by":"object"}]},"ISSN":["1367-4803","1367-4811"],"issn-type":[{"value":"1367-4803","type":"print"},{"value":"1367-4811","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2019,1,1]]},"published":{"date-parts":[[2018,7,2]]}}}