Abstract
Recent singlecell RNAsequencing studies have suggested that cells follow continuous transcriptomic trajectories in an asynchronous fashion during development. However, observations of cell flux along trajectories are confounded with population size effects in snapshot experiments and are therefore hard to interpret. In particular, changes in proliferation and death rates can be mistaken for cell flux. Here we present pseudodynamics, a mathematical framework that reconciles population dynamics with the concepts underlying developmental trajectories inferred from timeseries singlecell data. Pseudodynamics models population distribution shifts across trajectories to quantify selection pressure, population expansion, and developmental potentials. Applying this model to timeresolved singlecell RNAsequencing of Tcell and pancreatic beta cell maturation, we characterize proliferation and apoptosis rates and identify key developmental checkpoints, data inaccessible to existing approaches.
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Code availability
The pseudodynamics model code and the presented examples are available through GitHub (https://github.com/theislab/pseudodynamics) and in the Supplementary Code.
References
 1.
Klein, A. M. et al. Droplet barcoding for singlecell transcriptomics applied to embryonic stem cells. Cell 161, 1187–1201 (2015).
 2.
Taniguchi, K., Kajiyama, T. & Kambara, H. Quantitative analysis of gene expression in a single cell by qPCR. Nat. Methods 6, 503–506 (2009).
 3.
Bandura, D. R. et al. Mass cytometry: technique for real time single cell multitarget immunoassay based on inductively coupled plasma timeofflight mass spectrometry. Anal. Chem. 81, 6813–6822 (2009).
 4.
Haghverdi, L., Büttner, M., Wolf, F. A., Buettner, F. & Theis, F. J. Diffusion pseudotime robustly reconstructs lineage branching. Nat. Methods 13, 845–848 (2016).
 5.
Trapnell, C. et al. The dynamics and regulators of cell fate decisions are revealed by pseudotemporal ordering of single cells. Nat. Biotechnol. 32, 381–386 (2014).
 6.
Qiu, X. et al. Reversed graph embedding resolves complex singlecell trajectories. Nat. Methods 14, 979–982 (2017).
 7.
Wolf, F. A. et al. PAGA: Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. Genome Biol. 20, 59 (2019).
 8.
Saelens, W., Cannoodt, R., Todorov, H. & Saeys, Y. A comparison of singlecell trajectory inference methods: towards more accurate and robust tools. Preprint at bioRxiv https://doi.org/10.1101/276907 (2018).
 9.
Kafri, R. et al. Dynamics extracted from fixed cells reveal feedback linking cell growth to cell cycle. Nature 494, 480–483 (2013).
 10.
Kuritz, K., Stöhr, D., Pollak, N. & Allgöwer, F. On the relationship between cell cycle analysis with ergodic principles and agestructured cell population models. J. Theor. Biol. 414, 91–102 (2017).
 11.
Weinreb, C., Wolock, S., Tusi, B. K., Socolovsky, M. & Klein, A. M. Fundamental limits on dynamic inference from singlecell snapshots. Proc. Natl Acad. Sci. USA 115, E2467–E2476 (2018).
 12.
Schiebinger, G. et al. Optimaltransport analysis of singlecell gene expression identifies developmental trajectories in teprogramming. Cell 176, 928–943.e22 (2019).
 13.
Hashimoto, T., Gifford, D. & Jaakkola, T. Learning populationlevel diffusions with generative RNNs. in International Conference on Machine Learning 48, 2417–2426 (2016).
 14.
Buchholz, V. R. et al. Disparate individual fates compose robust CD8+ T cell immunity. Science 340, 630–635 (2013).
 15.
Cho, H. et al. Modelling acute myeloid leukaemia in a continuum of differentiation states. Lett. Biomath. 5, S69–S98 (2018).
 16.
Segerstolpe, Å. et al. Singlecell transcriptome profiling of human pancreatic islets in health and type 2 diabetes. Cell Metab. 24, 593–607 (2016).
 17.
Haber, A. L. et al. A singlecell survey of the small intestinal epithelium. Nature 551, 333–339 (2017).
 18.
Yui, M. A. & Rothenberg, E. V. Developmental gene networks: a triathlon on the course to T cell identity. Nat. Rev. Immunol. 14, 529–545 (2014).
 19.
Kernfeld, E. M. et al. A singlecell transcriptomic atlas of thymus organogenesis resolves cell types and developmental maturation. Immunity https://doi.org/10.1016/j.immuni.2018.04.015 (2018).
 20.
Haghverdi, L., Buettner, F. & Theis, F. J. Diffusion maps for highdimensional singlecell analysis of differentiation data. Bioinformatics 31, 2989–2998 (2015).
 21.
Vosshenrich, C. A. J. et al. A thymic pathway of mouse natural killer cell development characterized by expression of GATA3 and CD127. Nat. Immunol. 7, 1217–1224 (2006).
 22.
Ribeiro, V. S. G. et al. Cutting edge: Thymic NK cells develop independently from T cell precursors. J. Immunol. 185, 4993–4997 (2010).
 23.
Tang, Y. et al. Emergence of NKcell progenitors and functionally competent NKcell lineage subsets in the early mouse embryo. Blood 120, 63–75 (2012).
 24.
Cook, A. M. Proliferation and lineage potential in fetal thymic epithelial progenitor cells. Edinburgh Research Archive, 1–195 (2010).
 25.
Germain, R. N. Tcell development and the CD4CD8 lineage decision. Nat. Rev. Immunol. 2, 309–322 (2002).
 26.
Qiu, W.L. et al. Deciphering Pancreatic Islet β Cell and α Cell Maturation Pathways and Characteristic Features at the SingleCell Level. Cell Metab. 25, 1194–1205.e4 (2017).
 27.
Bader, E. et al. Identification of proliferative and mature βcells in the islets of Langerhans. Nature 535, 430–434 (2016).
 28.
Herbach, N., Bergmayr, M., Göke, B., Wolf, E. & Wanke, R. Postnatal development of numbers and mean sizes of pancreatic islets and betacells in healthy mice and GIPRdn transgenic diabetic mice. PLoS One 6, e22814 (2011).
 29.
Hija, A. et al. G0G1 Transition and the restriction point in pancreatic βcells in vivo. Diabetes 63, 578 (2014).
 30.
Scaglia, L., Cahill, C. J., Finegood, D. T. & BonnerWeir, S. Apoptosis participates in the remodeling of the endocrine pancreas in the neonatal rat. Endocrinology 138, 1736–1741 (1997).
 31.
Waddington, C. H. Organisers and Genes (Cambridge Univ. Press, 1940).
 32.
McKenna, A. et al. Wholeorganism lineage tracing by combinatorial and cumulative genome editing. Science 353, aaf7907 (2016).
 33.
Spanjaard, B. et al. Simultaneous lineage tracing and celltype identification using CRISPR–Cas9induced genetic scars. Nat. Biotechnol. 36, 469 (2018).
 34.
La Manno, G. et al. RNA velocity of single cells. Nature 560, 494–498 (2018).
 35.
Cohen, S. D., Hindmarsh, A. C. & Dubois, P. F. CVODE, a stiff/nonstiff ODE solver in C. Computers in Physics 10/2, 138–143 (1996).
 36.
Fröhlich, F., Theis, F. J., Rädler, J. O. & Hasenauer, J. Parameter estimation for dynamical systems with discrete events and logical operations. Bioinformatics 33, 1049–1056 (2017).
 37.
Stapor, P. et al. PESTO: Parameter EStimation TOolbox. Bioinformatics 34, 705–707 (2018).
 38.
Raue, A. et al. Lessons learned from quantitative dynamical modeling in systems biology. PLoS One 8, e74335 (2013).
 39.
Stubbington, M. J. T. et al. T cell fate and clonality inference from singlecell transcriptomes. Nat. Methods 13, 329–332 (2016).
 40.
Satija, R., Farrell, J. A., Gennert, D., Schier, A. F. & Regev, A. Spatial reconstruction of singlecell gene expression data. Nat. Biotechnol. 33, 495–502 (2015).
 41.
Macosko, E. Z. et al. Highly parallel genomewide expression profiling of individual cells using nanoliter droplets. Cell 161, 1202–1214 (2015).
 42.
Waltman, L. & van Eck, N. J. A smart local moving algorithm for largescale modularitybased community detection. Eur. Phys. J. B 86, 471 (2013).
 43.
Wolf, F. A., Angerer, P. & Theis, F. J. SCANPY: largescale singlecell gene expression data analysis. Genome Biol. 19, 15 (2018).
Acknowledgements
We would like to express our gratitude towards Ping Xu and Kashfia Neherin for mouse husbandry and tissue preparation. F.J.T. acknowledges financial support by the Graduate School QBM, the German Research Foundation (DFG) within the Collaborative Research Centre 1243, Subproject A17, by the Helmholtz Association (Incubator grant sparse2big, ZTI0007) and by the Chan Zuckerberg Initiative DAF (advised fund of Silicon Valley Community Foundation, 182835). R.M. acknowledges financial support by the Leona M. and Harry B. Helmsley Charitable Trust (2015PGT1D035), a Charles H. Hood Foundation Child Health Research Award, the Glass Charitable Foundation and the National Institutes of Health (1DP3DK111898, R01 AI132963, UC4 DK104218). J.H. acknowledges financial support by the German Research Foundation (DFG) (HA 7376/1–1) and the German Federal Ministry of Education and Research (BMBF) within the SYSStomach project (01ZX1310B). H.L. acknowledges financial support by the Helmholtz Society and German Center for Diabetes Research (DZD e.V.). D.S.F. acknowledges financial support by a German research foundation (DFG) fellowship through the Graduate School of Quantitative Biosciences Munich (QBM) (GSC 1006) and by the Joachim Herz Stiftung.
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Contributions
F.J.T. conceived the study; F.J.T. and J.H. conceived the model and supervised analyses; R.M., H.L. and D.S.F. conceived the experiments; A.K.F. implemented the models and performed the parameter estimation in all examples; D.S.F. and E.M.K. performed the initial computational analysis of the thymus data; R.M.J.G. performed the experiments for the thymus study; D.S.F. performed the initial computational analysis of the pancreas data; A.B.P. and M.B. performed the experiments for the pancreas study; and D.S.F., A.K.F, J.H. and F.J.T. wrote the manuscript with assistance from all authors.
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Integrated supplementary information
Supplementary Figure 1 Overview of all thymic hematopoietic cells (n = 10,895) observed with singlecell RNAseq.
(af) Time (ac) or louvain group (df) label superimposed on cells in tSNE based on data processed for louvain clustering (a,d), in in tSNE based on data processed for diffusion pseudotime analysis used throughout the paper (b,e) and in diffusion map based on data before filtering of putative myeloid or dendritic cells (c,f). See also Supplementary data 1.2 for the further details on this analysis. (g) Cell type labels for louvain groups from marker genes. stem: hematopoietic stem cells, DN2a/b (doublenegative Tcells), phase 2 (Nat. Rev. Immunol. 14, Yui, M. A. & Rothenberg, E. V., 2014), phase 3 (Nat. Rev. Immunol. 14, Yui, M. A. & Rothenberg, E. V., 2014) and DP (doublepositive Tcells) are Tcell maturation stages, NCL_ILC: nonconventional lymphoid cells (Immunity, Kernfeld, E. M. et al., 2018) with innate lymphoid cell character, NCL_ydTC: nonconventional lymphoid cells with γδTcell cell character, myeloid: putative myeloid and dendritic cells. (h) Tcell development phase across merged louvain groups from (g). (i) Abstracted graph from paga (bioRxiv, Wolf F. A. et al., 2017) at a connectivity threshold of 0.02 with cluster identity superimposed. See also Supplementary data 1.2 for the further details on this analysis.
Supplementary Figure 2 Myeloid and dendritic cell subclusters of the thymic hematopoietic cells observed with singlecell RNAseq.
Here, we subclustered cluster 14_myeloid from Supplementary Fig. 2a into three clusters, again with louvain clustering. We show the expression of a dendritic and a myeloid cell marker across the tSNE based on all thymic hematopoietic cells (n = 10895) observed with singlecell RNAseq (a, b) to to show the overall restriction of these markers to the cluster 14_myeloid and across the three subclusters (c, d) to show the heterogeneity in this cluster. The violin plots in (c, d) are based on kernel density estimates and on n = 130 cells in group 0, n = 118 in group 1 and n = 102 in group 2. (a, c) Dendritic cell marker Cd74 log expression superimposed on tSNE (a) and by louvain subcluster (c). (b, d) Myeloid cell marker Lyz2 log expression superimposed on tSNE (b) and by louvain subcluster (d). See also Supplementary data 1.2 for the further details on this analysis.
Supplementary Figure 3 Myeloid, dendritic cell, leukocyte and Tcell marker gene expression by Louvain group.
The cluster labels are explained in Supplementary Fig. 1. The violin plots are based on kernel density estimates of log gene expression and are based on the following number of cells per louvain group: 0_stem: n = 1314, 1_DN2a_1: n = 906, 2_DN2a_2: n = 588, 3_Phase2_1: n = 568, 4_Phase2_2: n = 625, 5_Phase2_3: n = 687, 6_Phase2_4: n = 1050, 7_Phase3_1: n = 1234, 8_Phase3_2: n = 784, 9_DP_1: n = 463, 10_DP_2: n = 759, 11_DP_3: n = 881, 12_NCL_ydTC: n = 402, 14_myeloid: n = 350,13_NCL_ILC: n = 284. (af) Myeloid and dendritic cell marker genes: Itgax (a), Itgam (b), Adgre1 (c), Lyz2 (d), Cd74 (e), Anpep (f). (gj) Leukocyte and Tcell marker genes: Ptprc (g), Cd34 (h), Notch1 (i), Bcl11b (j).
Supplementary Figure 4 CD3 subunit and Tcell development marker gene expression by Louvain group.
The cluster labels are explained in Supplementary Fig. 1. The violin plots are based on kernel density estimates and are based on the following number of cells per louvain group: 0_stem: n = 1314, 1_DN2a_1: n = 906, 2_DN2a_2: n = 588, 3_Phase2_1: n = 568, 4_Phase2_2: n = 625, 5_Phase2_3: n = 687, 6_Phase2_4: n = 1050, 7_Phase3_1: n = 1234, 8_Phase3_2: n = 784, 9_DP_1: n = 463, 10_DP_2: n = 759, 11_DP_3: n = 881, 12_NCL_ydTC: n = 402, 14_myeloid: n = 350,13_NCL_ILC: n = 284. (ac) Cd3 subunits: Cd3d (a), Cd3e(b), Cd3g (c). (di) Tcell development marker genes: Flt3 (d), Cd44 (e), Il2ra (f), Ptcra (g), Cd8a (h), Cd8b1 (i).
Supplementary Figure 5 αβ Tcell, γδ Tcell and natural killer cell marker gene expression by Louvain group.
The cluster labels are explained in Supplementary Fig. 1. The violin plots are based on kernel density estimates and are based on the following number of cells per louvain group: 0_stem: n = 1314, 1_DN2a_1: n = 906, 2_DN2a_2: n = 588, 3_Phase2_1: n = 568, 4_Phase2_2: n = 625, 5_Phase2_3: n = 687, 6_Phase2_4: n = 1050, 7_Phase3_1: n = 1234, 8_Phase3_2: n = 784, 9_DP_1: n = 463, 10_DP_2: n = 759, 11_DP_3: n = 881, 12_NCL_ydTC: n = 402, 14_myeloid: n = 350,13_NCL_ILC: n = 284. (a,b) αβTcells marker genes: Tcra (TCRα) (a), Tcrb (TCRβ) (b). (ck) γδTcells and natural killer cell marker genes: Tcrg (TCRγ) (c), Tcrd (TCRδ) (d), Ifng (e), Il17a (f), Il2rb (g), Ncr1 (h), Klrd1 (i), Klrb1b (j), Klrb1f (k).
Supplementary Figure 6 Cell type partitioning in the diffusion map of the thymic hematopoietic cells observed with singlecell RNAseq.
DC: diffusion component. (a) Diffusion map on all thymic hematopoietic cells (n = 10895). Colour: branch allocation according to diffusion pseudotime. (b,c,d) Diffusion map on gated thymic hematopoietic cells: Tcells and nonconventional lymphocytes (n = 10872). (b) Branch allocation according to diffusion pseudotime. (c) Adapted diffusion pseudotimebased branch allocation for pseudodynamics based on a linear cut in the diffusion component 1 versus 2 plane. (d) Branching and nonbranching region allocation for pseudodynamics: Cells are in the branching region if they have a diffusion pseudotime coordinate smaller than 0.25 and are not early thymic progenitors.
Supplementary Figure 7 Surface marker and transcription factor expression across cell state in thymic hematopoietic cells observed with singlecell RNAseq.
(ac) Surface marker expression across cell state. (a) Natural cubic spline fits to log expression data with cell state covariate (df=10). Shown are markers that are traditionally used to distinguish αβTcell stages. Vertical lines indicate the right boundaries of cell stages in cell state. (b,c) Heatmap of all surface marker genes that are differentially expressed in cell state (log counts with limma (Nucleic Acid Res. 43, Ritchie, M. E. et al., 2015), natural cubic spline model with df = 4, qvalue threshold of 1e5) either in αβTcell (b, n = 10079) or the nonconventional lymphocyte lineage (c, n = 1070). Genes are ordered separately for both lineages by the cell state at which their expression is maximal. Lists of all genes in these two heatmaps are supplied in Supplementary data 2.2.1 (αβTcell lineage) and 2.2.2 (nonconventional lymphocyte lineage). The gene annotation used is supplied in Supplementary data 2.3. The underlying differential expression analysis results are supplied Supplementary data 2.4. (d,e) Transcription factor expression across cell state. (d) Natural cubic spline fits to log expression data with cell state covariate (df = 10). (e) Heatmap of all transcription factor genes that are differentially expressed in cell state (log counts with limma (Nucleic Acid Res. 43, Ritchie, M. E. et al., 2015), natural cubic spline model with df = 4, qvalue threshold of 1e5) in the αβTcell lineage. Genes are ordered by peak time. A list of all genes in this heatmap is supplied in Supplementary data 2.1. The gene annotation used is supplied in Supplementary data 2.3. The underlying differential expression analysis results are supplied Supplementary data 2.4.
Supplementary Figure 8 Mean gene expression of marker gene groups in αβ Tcell branch in cell state bins.
(a) Fraction Cd4/Cd8 positive cells in cell state bins. (b) Surface markers expression. (c) Bcl2family expression. (d) Nfatfamily expression.
Supplementary Figure 9 Pseudodynamics model fits to Tcell maturation data with diffusion pseudotime as cell state.
(ab) Cross validation results (leaveonetimepointout) of pseudodynamics fits on Tcell maturation with diffusion pseudotime as cell state. (a) Overall prediction error on withheld data by regularization parameter value omega. (b) Regularized loglikelihood value of prediction at heldout time point (prediction error) by regularization parameter value omega and time point. (ci) Observed density, model fit to full data (simulation) and imputed density (simulation_cv) on Tcell lineage at a given time point. The cell state shown is the cell state used in the main text linearly scaled into the interval [0,0.9] and extended to 1. Accordingly, there are no observations in (0.9,1]. The imputed density is the model fit of a model trained on all remaining time points with a regularization parameter of 10 (leaveonetimepointout cross validation). (c) E13.5, (d) E14.5, (e) E15.5, (f) E16.5, (g) E17.5, (h) E18.5, (i) P0. (j) Population size estimates. Observed (points) and simulated (line) total size of thymic hematopoietic compartment in a thymic lobe with 95% confidence interval on simulated data (shaded) and observed data with one standard deviation around the mean as error bars. The population size observations are based on 5 replicates for t = 12.5 to t = 17.5 and on two replicates for t = 18.5 and t = 19.5. Replicates are independent measurements based on separate thymus samples for t = 12.5 to 18.5 and are independent measurements based on the two lobes of a single thymus for t = 19.5.
Supplementary Figure 10 Pseudodynamics parameter fits across regularization hyperparameters for Tcell maturation data with diffusion pseudotime cell state.
(ac) Maximum likelihood estimator spline fit of birthdeath (a) and drift parameter (b) and diffusion parameter (c) by regularization parameter (rho).
Supplementary Figure 11 Monocle2 embedding of thymic hematopoietic cells observed with singlecell RNAseq.
(a,b) Monocle2 pseudotime (a) and time (b) superimposed on monocle2 embedding based on n = 10705 cells. (c) Gene expression as function of pseudotime with spline interpolation (Nat. Methods, Qiu X. et al., 2017) of genes related to betaselection: Rorc, Bcl2l1 (BclxL) and Bcl2. (dk) Cell counts in hexagonal cell state bins in monocle2 embedding by clusters of thymocytes (defined in Supplementary Fig. 2b) with n = 1314 cells in 0_stem (d), 1494 cells in 1_DN2a (e), n = 2930 cells in 2_Phase2 (f), n = 2018 cells in 3_Phase3 (g), n = 2103 cells in 4_DP (h), n = 117 cells in 5_NCL_ILC (i), n = 402 cells in 6_NCL_ydTC (j), n = 327 cells in 7_myeloid (k).
Supplementary Figure 12 Monocle2 pseudotime assignment (cell state) of thymic hematopoietic cells observed with singlecell RNAseq and pseudodynamics model fits to this Monocle2 cell state.
(a,b) Distribution of cells by sample across cell state (monocle2 pseudotime) on Tcell lineage. Colour: time point in days after fertilisation. (a) Kernel density estimate of union of all samples per time point (n = 442 at t = 12.5, n = 1795 at t = 13.5, n = 1052 at t = 14.5, n = 1013 at t = 15.5, n = 2616 at t = 16.5, n = 1966 at t = 17.5, n = 882 at t = 18.5, n = 939 at t = 19.5). (b) Box plot of each sample per time point (n = {152, 58, 232} at t = 12.5; n = {628, 476, 691} at t = 13.5; n = {508, 544} at t = 14.5; n = {437, 576} at t = 15.5; n = {870, 918, 828} at t = 16.5; n = {975, 991} at t = 17.5; n = {455, 427} at t = 18.5; n = {422, 517} at t = 19.5). Here, pseudotime coordinates are computed based on all replicates. Replicates are independent Dropseq samples which are based on separate thymus samples, the two replicates at P0 are based on the two lobes of a single thymus. The center of each boxplots is the sample median, the whiskers extend from the upper (lower) hinge to the largest (smallest) data point no further than 1.5 times the interquantile range from the upper (lower) hinge. (c) Scatter plot of diffusion pseudotime cell state versus moncle2 cell state (pseudotime) on Tcell lineage cells. (df) Pseudodynamics parameter fits to monocle2 cell state: (d) Birthdeath, (e) drift and (f) diffusion parameter estimates for three different regularisation parameters (rho).
Supplementary Figure 13 Rag1 and Rag2 knockout cells fall into the wildtype developmental manifold of Tcell maturation.
(ac) Diffusion map projections of the combined wildtype and knockout data sets show that the knockout cells lie within the wildtype manifold in each projection. KO: knockout, WT: wildtype. (d) The overall structure of the manifold is conserved between a diffusion map computed just on wildtype and one computed on wildtype and mutant samples: Scatter plot of diffusion pseudotime coordinates of wildtype cells in wildtype only (WT) and in wildtype with mutants (WT+RagKO) data sets.
Supplementary Figure 14 Developmental arrest of Tcell maturation in Rag1 and Rag2 knockout mice at βselection.
(a,b) Rag1 and Rag2 knockout mice have Tcell populations which are delayed in developmental progress along Tcell maturation measured with diffusion pseudotime compared to agematched wildtype mice. The pvalue is the result of the onesided KolmogorovSmirnov (KS) test which was used to test whether the knockout empirical cumulative density function of the knockout lies below that of the wild type cells. (a) Rag2 knockout (Rag2KO) (n = 423 cells) and wildtype (n = 971 cells) samples at E14.5. (b) Rag1 knockout (Rag1KO) (n = 1565 cells) and wildtype (n = 2484 cells) samples at E16.5. (c,d) Mean marker gene expression by sample of age matched wildtype (WT) and Rag1/Rag2 knockout (Rag1KO, Rag2KO) embryos at E14.5 (c) and E16.5 (d). (e) Betaselection point estimators by regularization parameter (reg). (f) Scatter plot of diffusion pseudotime coordinates obtained based on wildtype and on merged wildtype and mutant mouse samples. The dependency can be approximated with a smooth function (blue line: natural cubic spline fit of degree five). Yellow line: identity function.
Supplementary Figure 15 Pseudodynamics model fit to the continuous pancreatic beta cell maturation data set.
(af) Kernel density estimator of observed density (shaded) and best model fit (lines) of βcell population density across cell state by time point for models with state (s) or state and timedependent (st) birthdeath rates. (g) Observed number of βcells in black with one standard deviation around the mean as error bars and population size fits by model with the following number of replicates per time point: n = 3 at t = 0, n = 3 at t = 5.5, n = 3 at t = 10.5, n = 8 at t = 11.5, n = 3 at t = 15.5, n = 5 at t = 46.5. Replicates were separately measured in one unique animal per replicate.
Supplementary information
Supplementary Figures and Text
Supplementary Figures 1–15 and Supplementary Notes 1–3
Supplementary Video 1
Pseudodynamics fit to in vitro differentiation of embryonic stem cells
Supplementary Video 2
Pseudodynamics fit to Tcell maturation
Supplementary Video 3
Cell flux captured with pseudodynamics during Tcell maturation
Supplementary Data 1
Singlecell RNAseqm analysis workflows
Supplementary Data 2
Genewise results tables
Supplementary Data 3
Pseudodynamics input
Supplementary Data 4
Pancreatic beta cell data
Supplementary Software
Pseudodynamics code used in this study.
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Fischer, D.S., Fiedler, A.K., Kernfeld, E.M. et al. Inferring population dynamics from singlecell RNAsequencing time series data. Nat Biotechnol 37, 461–468 (2019). https://doi.org/10.1038/s4158701900880
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