## Abstract

Recent single-cell RNA-sequencing 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 time-series single-cell data. Pseudodynamics models population distribution shifts across trajectories to quantify selection pressure, population expansion, and developmental potentials. Applying this model to time-resolved single-cell RNA-sequencing of T-cell 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.

## Additional information

**Publisher’s note:** Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## 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, ZT-I-0007) 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 (2015PG-T1D035), 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 SYS-Stomach 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.

## Author information

### Author notes

### Affiliations

### Authors

### 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.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Fabian J. Theis.

## Integrated supplementary information

### Supplementary Figure 1 Overview of all thymic hematopoietic cells (

*n*= 10,895) observed with single-cell RNA-seq.**(a-f)**Time (**a**-**c**) or louvain group (**d**-**f**) label superimposed on cells in t-SNE based on data processed for louvain clustering (**a**,**d**), in in t-SNE 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 (double-negative T-cells), 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 (double-positive T-cells) are T-cell maturation stages, NCL_ILC: non-conventional lymphoid cells (Immunity, Kernfeld, E. M. et al., 2018) with innate lymphoid cell character, NCL_ydTC: non-conventional lymphoid cells with γδ-T-cell cell character, myeloid: putative myeloid and dendritic cells.**(h)**T-cell 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 single-cell RNA-seq.

Here, we sub-clustered 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 t-SNE based on all thymic hematopoietic cells (

*n*= 10895) observed with single-cell RNA-seq (**a**,**b**) to to show the overall restriction of these markers to the cluster 14_myeloid and across the three sub-clusters (**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 t-SNE (**a**) and by louvain sub-cluster (**c**).**(b, d)**Myeloid cell marker*Lyz2*log expression superimposed on t-SNE (**b**) and by louvain sub-cluster (**d**). See also Supplementary data 1.2 for the further details on this analysis.### Supplementary Figure 3 Myeloid, dendritic cell, leukocyte and T-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 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.**(a-f)**Myeloid and dendritic cell marker genes:*Itgax*(**a**),*Itgam*(**b**),*Adgre1*(**c**),*Lyz2*(**d**),*Cd74*(**e**),*Anpep*(**f**).**(g-j)**Leukocyte and T-cell marker genes:*Ptprc*(**g**),*Cd34*(h),*Notch1*(**i**),*Bcl11b*(**j**).### Supplementary Figure 4 CD3 subunit and T-cell 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.**(a-c)**Cd3 subunits:*Cd3d*(**a**),*Cd3e*(**b**),*Cd3g*(**c**).**(d-i)**T-cell development marker genes:*Flt3*(**d**),*Cd44*(**e**),*Il2ra*(**f**),*Ptcra*(**g**),*Cd8a*(**h**),*Cd8b1*(**i**).### Supplementary Figure 5 αβ T-cell, γδ T-cell 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)**αβ-T-cells marker genes:*Tcra (TCRα)*(**a**),*Tcrb (TCRβ)*(**b**).**(c-k)**γδ-T-cells 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 single-cell RNA-seq.

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: T-cells and non-conventional lymphocytes (*n*= 10872).**(b)**Branch allocation according to diffusion pseudotime.**(c)**Adapted diffusion pseudotime-based branch allocation for pseudodynamics based on a linear cut in the diffusion component 1 versus 2 plane.**(d)**Branching and non-branching 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 single-cell RNA-seq.

**(a-c)**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 αβ-T-cell 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, q-value threshold of 1e-5) either in αβ-T-cell (**b**,*n*= 10079) or the non-conventional 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 (αβ-T-cell lineage) and 2.2.2 (non-conventional 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, q-value threshold of 1e-5) in the αβ-T-cell 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 T-cell branch in cell state bins.

**(a)**Fraction*Cd4/Cd8*positive cells in cell state bins.**(b)**Surface markers expression.**(c)***Bcl2*-family expression.**(d)***Nfat*-family expression.### Supplementary Figure 9 Pseudodynamics model fits to T-cell maturation data with diffusion pseudotime as cell state.

**(a-b)**Cross validation results (leave-one-time-point-out) of pseudodynamics fits on T-cell maturation with diffusion pseudotime as cell state.**(a)**Overall prediction error on withheld data by regularization parameter value omega.**(b)**Regularized log-likelihood value of prediction at held-out time point (prediction error) by regularization parameter value omega and time point.**(c-i)**Observed density, model fit to full data (simulation) and imputed density (simulation_cv) on T-cell 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 (leave-one-time-point-out 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 T-cell maturation data with diffusion pseudotime cell state.

**(a-c)**Maximum likelihood estimator spline fit of birth-death (**a**) and drift parameter (**b**) and diffusion parameter (**c**) by regularization parameter (rho).### Supplementary Figure 11 Monocle2 embedding of thymic hematopoietic cells observed with single-cell RNA-seq.

**(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 beta-selection:*Rorc*,*Bcl2l1*(*Bcl-xL*) and*Bcl2*.**(d-k)**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 single-cell RNA-seq and pseudodynamics model fits to this Monocle2 cell state.

**(a,b)**Distribution of cells by sample across cell state (monocle2 pseudotime) on T-cell 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 Drop-seq 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 T-cell lineage cells.**(d-f)**Pseudodynamics parameter fits to monocle2 cell state: (**d**) Birth-death, (**e**) drift and (**f**) diffusion parameter estimates for three different regularisation parameters (rho).### Supplementary Figure 13 Rag1 and Rag2 knockout cells fall into the wild-type developmental manifold of T-cell maturation.

**(a-c)**Diffusion map projections of the combined wild-type and knock-out data sets show that the knockout cells lie within the wild-type manifold in each projection. KO: knockout, WT: wild-type.**(d)**The overall structure of the manifold is conserved between a diffusion map computed just on wild-type and one computed on wild-type and mutant samples: Scatter plot of diffusion pseudotime coordinates of wild-type cells in wild-type only (WT) and in wild-type with mutants (WT+RagKO) data sets.### Supplementary Figure 14 Developmental arrest of T-cell maturation in Rag1 and Rag2 knockout mice at β-selection.

**(a,b)***Rag1*and*Rag2*knockout mice have T-cell populations which are delayed in developmental progress along T-cell maturation measured with diffusion pseudotime compared to age-matched wild-type mice. The p-value is the result of the one-sided Kolmogorov-Smirnov (KS) test which was used to test whether the knock-out empirical cumulative density function of the knock-out lies below that of the wild type cells. (**a**) Rag2 knockout (Rag2KO) (*n*= 423 cells) and wild-type (*n*= 971 cells) samples at E14.5. (**b**)*Rag1*knockout (Rag1KO) (*n*= 1565 cells) and wild-type (*n*= 2484 cells) samples at E16.5.**(c,d)**Mean marker gene expression by sample of age matched wild-type (WT) and*Rag1*/*Rag2*knockout (Rag1KO, Rag2KO) embryos at E14.5 (**c**) and E16.5 (**d**).**(e)**Beta-selection point estimators by regularization parameter (reg).**(f)**Scatter plot of diffusion pseudotime coordinates obtained based on wild-type and on merged wild-type 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.

**(a-f)**Kernel density estimator of observed density (shaded) and best model fit (lines) of β-cell population density across cell state by time point for models with state- (s) or state- and time-dependent (st) birth-death rates.**(g)**Observed number of β-cells 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

### Reporting Summary

### Supplementary Video 1

Pseudodynamics fit to in vitro differentiation of embryonic stem cells

### Supplementary Video 2

Pseudodynamics fit to T-cell maturation

### Supplementary Video 3

Cell flux captured with pseudodynamics during T-cell maturation

### Supplementary Data 1

Single-cell RNA-seqm analysis workflows

### Supplementary Data 2

Gene-wise 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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