Overall description of Storm

We established three stochastic gene expression models for new mRNA (or new unspliced and spliced mRNA) (Fig 1A) for the inference of the RNA kinetic parameters and thus the RNA velocity in the Storm approach. In Model 1, only transcription and mRNA degradation were considered. In Model 2, we considered transcription, splicing, and spliced mRNA degradation. And in Model 3, we considered the switching of gene expression states, transcription in the active state, and mRNA degradation.

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Fig 1. Schematic overview of Storm.

A. Three models of RNA life cycle considering different biological processes: Model 1 (CSP-Baseline): Reaction dynamics model for new RNA l(t) ignoring the splicing process, where α is the transcription rate and γ_t is the total mRNA degradation rate. Model 2 (CSP-Splicing): Reaction dynamics model of new unspliced and spliced mRNA (u_l(t), s_l(t)) considering the splicing process, where β is the splicing rate, γ_s is the spliced mRNA degradation rate, and α is the same as Model 1. Model 3 (CSP-Switching): Reaction dynamics model of new RNA l(t) considering gene state switching, where α and γ_t are the same as in Model 1, k_on is the rate at which the gene switches from the inactive state to the active state, k_off is the opposite. B. Complete workflow diagram for parameter inference and downstream analysis based on stochastic dynamics of new mRNA considering technical noise. C. Specific parameter inference strategies for one-shot/pulse experiments and steady-state/non-steady-state assumption.

https://doi.org/10.1371/journal.pcbi.1012606.g001

The complete workflow of Storm is demonstrated in Fig 1B. We first analytically solve the new RNA (or new unspliced and spliced mRNA) stochastic dynamics corresponding to the above three models, which are Poisson distribution, independent Poisson distribution and zero-inflated Poisson distribution, respectively. In addition, we model the technical noise as the cell-specific binomial distribution. By integrating the biological noise and the technical noise together, we obtain the distribution for the measured number of new/labeled mRNA molecules (or new unspliced and spliced mRNA molecules), which are cell-specific Poisson distribution, independent cell-specific Poisson distribution and cell-specific zero-inflated Poisson distribution, respectively. Therefore, we call the three models CSP-Baseline, CSP-Splicing, and CSP-Switching for easy memory and use them later to better distinguish these three models. Maximum likelihood estimation (MLE) is used to fit the data and make inferences for the parameters shown in the corresponding models.

To ensure the general applicability of Storm in common nascent RNA labeling schemes, such as one-shot or kinetics/pulse experiments (See Figure 2 of Qiu, et. al [26]), we designed specific estimation strategies for each labeling scheme(Fig 1C). For the one-shot labeling experiments with only new RNA and total RNA data, since there is only one labeling duration and the lack of splicing data, the steady-state assumption under the stochastic dynamics framework is reinvoked to infer parameters (Fig 1C Left). For the one-shot labeling experiments with uu, ul, su and sl RNA, we design a two-stage approach that does not rely on the steady-state assumption. More specifically, we first use scVelo [2] to determine the relative size of the kinetic parameters, and then use CSP-Splicing to determine the absolute size (Fig 1C Left). For kinetics/pulse-labeling experiments with multiple labeling durations, the transient stochastic dynamics framework is used without the steady-state assumption (Fig 1C Right). CSP-Baseline is recommended if the concern is RNA velocity, and CSP-Splicing should be used if the concern also includes splicing dynamics. Although steady-state assumption can also be included, we recommend that non-steady-state approach should be used unless the user has sufficient knowledge of the biological process being studied. Furthermore, the goodness-of-fit index based on deviance commonly used in generalized linear models is utilized to quantify the goodness of fit of our models in kinetics/pulse datasets. The index is then used to select genes that are more consistent with model assumptions for later downstream analysis, such as the enrichment analysis of different gene-specific parameters. Furthermore, we relaxed the previous assumption of constant parameters in genes or cells and assumed that only degradation rates (γ_t in CSP-Baseline and CSP-Switching; γ_s in CSP-Splicing) are constant while the other parameters are cell-specific and depend on the state of gene expression in each cell. This relaxation would be useful for modeling lineage-specific kinetics resulted from hierarchical lineage bifurcation, which is common in cell development. Finally, in order to calculate and visualize the RNA velocity, we reduced the considered stochastic models to derive the deterministic equation for the mean gene expression. The inferred parameters, after filtering with the goodness-of-fit index are then used in RNA velocity analysis and visualization. Notably, to demonstrate Storm’s performance, we conducted systematic comparison with the state-of-the-art method Dynamo [26] for processing metabolic labeling scRNA-seq experiment datasets.

In the continued subsections we will present the details of each step in the Storm workflow, starting from the introduction of our mathematical models.

CSP modeling of counts data with metabolic labeling information

We proposed and analytically solved three aforementioned stochastic gene expression models for the dynamics of new mRNAs (or new unspliced and spliced mRNAs).

For simplicity of modeling, we followed [1, 2] to assume that the genes are independent. In the stochastic gene expression model, the generation of new/labeled mRNA (or new unspliced and spliced mRNA ) is a stochastic process, and we are interested in the evolution of its PMF over time, which is denoted by (1)

In CSP-Baseline and CSP-Splicing, Since the initial value of (or ) is 0, we obtained the following closed-form solution (see “Methods” section). (2) where a(t), b(t) and c(t) are solutions to the corresponding deterministic equation, which means that obeys the Poisson distribution with mean a(t) in CSP-Baseline, and obey independent Poisson distributions with mean b(t) and c(t) in CSP-Splicing. Here α, β are the transcription and splicing rates, and γ_s, γ_t are the spliced and total mRNA degradation rates, respectively. CSP-Switching can be solved similarly analytically under the assumption that the switching rates are much smaller than the transcription and degradation rates (see “Methods” section).

We also specifically modeled technical noise of the measured number of new RNA (or new unspliced and spliced mRNA) molecules in scRNA-seq experiments. Such noises often lead to dropout of RNA measurements during the sequencing process and generally result in variations in sequencing depth across cells. To account for the noise, in Storm we modeled the dropout process of sequencing technology as cell-specific binomial distributions. Adopting the common practice in many preprocessing pipelines through a size factor to normalize the data [1, 2, 10, 13, 26], we assumed that the total numbers of mRNA molecules across all genes in different cells are close. Probabilistically, this assumption implies that where p_j is the probability of mRNA molecules being captured in cell j and n_j is the total number of mRNA molecules across all genes in cell j in scRNA-seq experiments.

Combining the stochastic models for biological and technical noise, we can obtain different formalisms of the distribution for the measured number of new/labeled mRNA molecules l(t) (or new unspliced and spliced mRNA molecules (u_l(t), s_l(t))) in the scRNA-seq experiments (see “Methods” section) for each model. Specifically, in CSP-Baseline, l(t) obeys the cell-specific Poisson (CSP) distribution, that is, (3) where P_n,j(t) is the PMF for the measured number of new mRNA molecules in cell j. In CSP-Splicing, (u_l(t), s_l(t)) obeys the independent cell-specific Poisson (ICSP) distribution, that is, (4) where P_mn,j(t) is the joint PMF for the measured number of new unspliced and spliced mRNA molecules in cell j. The derivation of CSP-Switching is similar (see “Methods” section). We call the above distributions as cell-specific because different cells obey the distributions with different parameters. Finally for labeling efficiency, we did not model it directly but followed Dynamo using GRAND-SLAM [33] to correct the new RNA data in advance.

Note that Grün et al. also modeled the scRNA-seq data by integrating biological noise and technical noise [34]. Our work is different from them in the following aspects: (1) Our work models the transient dynamics of new mRNA and solves their distribution for the proposed stochastic models analytically. However, in [34], they instead modeled the total mRNA and derived that the biological noise follows a negative binomial distribution as the steady state of the transcriptional bursting model. (2) Our work accurately models the technical noise as a cell-specific binomial distribution, while they approximated the cell-specific binomial distribution with a Poisson distribution and modeled the capture probability as a random variable subject to the Gamma distribution, which finally leads to a negative binomial distribution (Poisson-Gamma mixture distribution) of the technical noise.

As one-shot labeling experiments are much more convenient than pulse experiments in practice, in the following, we will first demonstrate how Storm can be applied to the one-shot case. We will then extensively show Storm’s power in analyzing the pulse datasets.

Stochastic models combined with steady-state assumptions for one-shot data without splicing information

For one-shot data without splicing information, we designed the corresponding parameter inference method which invokes the steady-state assumption under the stochastic model, focusing specifically on CSP-Baseline (see “Methods” section). Similar steady-state methods of the stochastic model can also be designed for both CSP-Splicing and CSP-Switching as well, although they are not the focus of this paper.

We validated our method in several one-shot datasets (Fig 2 and S1 Fig). We first analyzed a primary human HSPCs datasets from scNT-seq [18]. Both Dynamo and Storm reveal a smooth transition of HSCs into MEP-like and GMP-like cells, which further ramify into Meg/Ery/Bas lineages and Mon/Neu lineages, respectively, which is consistent with the established knowledge of hematopoiesis (Fig 2A). Next, we analyzed the neuronal activity dataset from the scNT-seq study [18] to investigate cellular polarization dynamics after KCl treatment. Dynamo and Storm both revealed a coherent transition that nicely follows the temporal progression from time point 0 to 15, 30, 60 and finally 120 minutes (Fig 2C). We also analyzed a dataset from the sci-fate study [19] in which cell cycle progression and glucocorticoid receptor (GR) activation were explored. Similar to Dynamo, the RNA velocity flow from our method also revealed a sequential transition of cells following the DEX (dexamethasone) treatment times in the first two principal components (PCs) (Fig 2E Left). In the second two PCs, we observed an orthogonal circular progression of the cell cycle (Fig 2E Middle). From the first two UMAP dimensions projected further from the four PCs, we observed a combined dynamics of GR responses and cell cycle progression (Fig 2E Right). We analyzed a dataset from PerturbSci-Kinetics [22], Dynamo and Storm observed similar results (Fig 2F). Additionally we analyzed mouse fibroblast cells dataset from scSLAM-seq [17]. We observed that Dynamo and Storm inferred similar velocities, and they both further discriminated infected from non-infected cells (S1A Fig).

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Fig 2. Stochastic model combined with steady-state assumptions for one-shot data without splicing information.

Storm in this figure refers to the inference strategy of CSP-Baseline model combined with the steady state assumption. A. Streamline projected in the UMAP space plots of primary human HSPCs datasets from scNT-seq [26]. B. Degradation rates γ_t estimated with steady-based method in Storm compared to that of the Dynamo method in the primary human HSPCs datasets. C. Streamline projected in the UMAP space plots of neuronal activity under KCl polarization datasets from scNT-seq [18]. D. Same as B., but for the neuronal activity datasets. E. Streamline plots of the sci-fate dataset [19] reveal two orthogonal processes of GR response and cell-cycle progression. From left to right: streamline plot on the first two PCs, the second two PCs, and the first two UMAP components that are reduced from the four PCs, respectively. The first row is the result of Storm and the second row is the result of Dynamo. F. Streamline projected in the UMAP space plots of the dataset from PerturbSci-Kinetics [22].

https://doi.org/10.1371/journal.pcbi.1012606.g002

Finally, we quantitatively compared the degradation rates γ_t inferred by the two methods (Fig 2B and 2C and S1B and S1D Fig). The inferred results of the two methods are highly correlated, with values of 0.74, 0.76, 0.80, 0.91, and 0.87 respectively on these five datasets. The absolute differences are also low, and the inferred results are mostly distributed around the green line y = x. We think that the possible reason for this is that the steady-state assumption plays a decisive role. Such methods may fail when the steady-state assumption is violated, so it is important to design methods for one-shot experiments that do not rely on the steady state assumption.

Storm analyzes one-shot data with both splicing and labeling information without steady-state assumption

For one-shot data containing both splicing and labeling information, we designed a two-stage parameter inference method that does not depend on the steady-state assumption by first modeling unspliced and spliced RNAs with the dynamic model in scVelo [2] and then unspliced labeled and spliced labeled RNA counts with CSP-Splicing (see “Methods” section). Also distinguishing from scVelo [2] in which velocity genes are picked before running the algorithm, we compute goodness of fit R² to pick well-fitting genes in scVelo [2] for downstream analysis (see “Methods” section).

We validate our method on both simulation and real single cell datasets (Fig 3 and S2 Fig). We first constructed a bifurcated one-shot simulation dataset by following the methods for constructing bifurcated data in SymSim [30] and VeloSim [31] (see “Methods” section). The correct direction of the streamlines started from the right side of the cells on the PCA embedding and then bifurcated into two branches in the middle. We compare the performance of Storm, Dynamo, and a deep learning-based method cellDancer [16] on this simulated dataset, and the results show that only Storm and cellDancer got the correct streamlines (Fig 3A and S2A Fig). In addition we compared the degradation rate values estimated by the different methods with the true values (Fig 3B and 3C). The results show that cellDancer’s estimated values are not in the same order of magnitude as the true values and have a poor correlation (Fig 3B and 3C). This is a difficulty inherent in methods that are missing physical time information. Dynamo and Storm’s estimates are in the same order of magnitude as the true values, but Storm has a lower absolute error and a higher correlation with the true values compared to Dynamo (Fig 3B and 3C). It is also worth noting that the absolute error of the selected genes in Storm is further reduced (Fig 3C), which indicates that the selection strategy is effective. We also compared gene-cell-wise transcription rates. The results show lower absolute errors in estimated transcription rates for Storm and Dynamo (Fig 3D Left and S2C Fig), but much higher for cellDancer (Fig 3D Right). We analyzed the murine intestinal organoid system dataset from scEU-seq [21]. Both Storm, Dynamo and cellDancer observed a bifurcation (Fig 3E and S2B Fig) from intestinal stem cells into the secretory lineage (left) and the enterocyte lineage (right).

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Fig 3. Storm analyzes one-shot data with both splicing and labeling information without steady-state assumption.

Storm in this figure refers to the inference strategy of CSP-Splicing model combined with scVelo. A. Streamline projected in the PCA space plots of one-shot bifurcation simulation data. Left: Storm; Right: Dynamo. B. Comparison of the estimated degradation rate with the true degradation rate in one-shot bifurcation simulation data. cellDancer uses the average of cell-wise degradation rates. C. Distribution plot of the difference between the estimated degradation rate and the true value, including Storm, Storm (selected) and Dynamo. D. Heat map of absolute error between estimated and true gene-cell-wise transcription rates α of one-shot bifurcation simulation data. Left: Storm; Right: cellDancer. E. Streamline plot in the UMAP space of the murine intestinal organoid system dataset from scEU-seq [21]. F. Streamline projected in the RFP_GFP space plots of cell cycle dataset from scEU-seq [21]. On the left is the result of taking only the data labelled with 15 minutes, and on the right is the data labelled with 30 minutes. G. Comparison of degradation rates (γ_s in Storm and γ_t in Dynamo) in cell cycle datasets with labeling duration of 15 and 30 minutes.

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To demonstrate the precision and robustness of the Storm method in estimating the one-shot dataset, we benchmarked the estimated kinetic parameters of different subsets of the cell cycle pulse-labeling dataset [21], each with a different duration of labeling. On the 15-minute labeling sub-dataset, Storm recovers a transition that matches well with the cell-cycle progression, while the transition recovered by Dynamo is problematic near the M phase (Fig 3F Left). On the 30-minute labeling sub-dataset, both methods recover the cell cycle progression correctly, but the streamlines of our method are considerably smoother compared to those of Dynamo (Fig 3F Right). In addition, we compared the consistency of degradation rates (γ_s in Storm and γ_t in Dynamo) inferred by the two methods between two sub-datasets with different labeling durations (Fig 3G). The results show that they are similar in terms of correlation, but our method is much smaller in terms of mean absolute error. Notably, although Storm shows higher consistency than Dynamo, it is still not satisfactory, perhaps due to the experimental noises from different labeling durations. Therefore, it is crucial to integrate data of different durations of labeling when a kinetic experiment is available. Furthermore, it is equally important to design methods that do not rely on the steady-state assumption and integrate data of different durations for parameter inference.

Statistical analysis of cell cycle dataset based on Storm’s stochastic model

Next, we first performed a goodness-of-fit test of the stochastic model proposed in Storm to a cell cycle dataset from scEU-seq [21] with multiple labeling time points to validate our proposals.

When the fixed labeling duration is t_fixed, a(t_fixed), b(t_fixed) and c(t_fixed) are all fixed constants. We can test whether the number of new mRNA molecules in tscRNA-seq within a fixed labeling duration matches the distribution obtained based on the stochastic models (Eqs (3), (4) and (16)), respectively. A common method of testing whether a dataset obeys a given distribution is the chi-square (χ²) goodness-of-fit test [35]. However, the usual χ² test is not directly applicable because in our case different cells obey different distributions with different parameters. By inspecting the mathematical analysis procedure of the χ² test [36], we constructed a new asymptotic χ² statistics and proposed a modified χ² test for our cell-specific distributions (see “Methods” section).

We used the proposed cell-specific χ² test in the cell cycle dataset from the scEU-seq study [21], in which cells were labeled for 15, 30, 45, 60, 120 or 180 minutes. Because the labeled unspliced mRNA counts u_l(t) were too small to be grouped/binned to create a distribution, hypothesis tests were performed only for CSP and CSZIP distributions and not for ICSP distribution. The results are shown in Table 1. We found that some genes were not well determined (especially for cases with a short duration of labeling) in the sense that these genes had too few new mRNA molecules in the tscRNA-seq experiments, which resulted in very few groupings and perfect fittings. With so few mRNA counts for these genes, we were unable to determine whether they obeyed our proposed distribution or not. Moreover, our results revealed that the CSZIP distribution exhibited a better fit with the data than the CSP distribution when focusing on a fixed time point alone, suggesting that the data are indeed zero-inflated.

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Table 1. The proposed sample-specific hypothesis test results on whether the number of new mRNA molecules in the Cell Cycle dataset obeys the CSP and CSZIP distributions.

UTD means that it is unable to determine because there are too few groupings resulting in zero degrees of freedom, when it is always a perfect fit. The significance level is 0.05.

https://doi.org/10.1371/journal.pcbi.1012606.t001

We next showed the high goodness-of-fit of the CSP and CSZIP distributions on two characteristic genes, namely RPL41 and IL22RA1 with an overall low and high gene expression respectively (Fig 4A). Qualitatively, we found that the expected counts of both the CSP and CSZIP distributions matched well with the observed counts for the gene RPL41. Quantitatively, the results of the cell-specific chi-square test also showed that the CSP or CSZIP distribution was well satisfies in most labeling durations (Fig 4A first row). Similar results were observed for the gene IL22RA1 with significantly higher expression (Fig 4A second row). Therefore, we demonstrated CSP and CSZIP distributions accurately describe these two genes and is thus suitable for modeling the tscRNA-seq datasets.

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Fig 4. Statistical analysis of cell cycle dataset.

A. Observed counts, expected counts of CSP distribution, and expected counts of CSZIP distribution of new mRNA molecules of the two example genes RPL41 and IL22RA1. The first row: Fitting results of the RPL41 gene with a small number of mRNA molecules; The second row: Fitting results of the IL22RA1 gene with a higher number of new mRNA molecules (truncated to 11 for better visualization). P_CSP and P_CSZIP refer to the p-values of the cell-specific chi-square tests with the corresponding distributions. B. Comparison of the total mRNA counts with different labeling durations of the four example genes TSPOAP1, GPRC5A, ADAMTS6 and APEX1. P refers to the p-value of the Chi-square contingency table independence test. C. Results of chi-square independence test for total RNA counts (significance level 0.05). “Same” here means accepting the null hypothesis of the chi-square independence test that total RNA counts with different time durations obey the same distribution. “Different” means the opposite.

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Finally, we found that, for most genes, the number of total mRNA molecules shares the same distribution across different labeling durations. In Fig 4B, we showed the number of total mRNA molecules of four example genes TSPOAP1, GPRC5A, ADAMTS6 and APEX1 is nearly identical across different labeling durations. Quantitatively, we performed a global chi-square independence test on the number of total mRNAs (as distinct from the new mRNAs) with different durations of labeling in all genes and found that, interestingly, there are 72.3% of the genes passed the test at a significance level of 0.05 (Fig 4C). This indicates that a considerable proportion of the number of genes’ total mRNA molecules obeyed the same distribution, consistent with what we observed for the four example genes.

Storm accurately infers kinetic parameters that leads to rich insights of cell cycle via enrichment analysis

In the kinetic experiments, we relied on three stochastic models without the steady-state assumption to infer different set of kinetic parameters using maximum likelihood estimation (see “Methods” section), namely α and γ_t for CSP-Baseline, α, β and γ_s for CSP-Splicing, and α, γ_t and p_off for CSP-Switching. In addition, we defined the goodness-of-fit of each of the three models (see “Methods” section). According to the goodness-of-fit index, we selected genes that were more consistent with the model assumptions for downstream tasks, such as the enrichment analysis and RNA velocity analysis, etc.

Compared with Dynamo [26], the state-of-the-art method for processing tscRNA-seq datasets, our advantages are mainly in the following aspects: (1) Our method does not require steady-state assumptions on the kinetics experiments while Dynamo heavily relies on the steady-state assumptions; (2) Our stochastic model-based approach is more consistent with real biological process, while Dynamo only utilizes the deterministic model of mean value; (3) Our model takes into account all cells in the inference, while the approach based on steady-state assumptions in Dynamo only considers a small number of cells with high expression. In addition, we revealed the difference between the total mRNA degradation rate γ_t and spliced mRNA degradation rate γ_s based on their different physical roles, distinguished them in different models, and finally gave the relationship between these two (see “Methods” section). We noted that in Dynamo, to infer β, γ_t was first inferred when the splicing was ignored, then was inferred using the method based on the steady-state assumption in scVelo [2], and finally was taken as the inference of β upon assuming γ_t = γ_s. However, , while γ_t and γ_s are generally not equal. This point was overlooked in Dynamo, which causes an inaccurate estimate of β. In fact, under the steady-state assumption, β can be directly estimated by using only u_l(t) through the formula u_l(t) = (1 − e^−βt)α/β, similar to the two-step method used in Dynamo to estimate γ_t through since they have similar form. However, we don’t use this method in Storm.

With the above inference methods and insights, we studied a cell cycle dataset from the scEU-seq study [21]. We compared the parameter inference results of the three models (Fig 5A). When splicing was not considered, the inference results based on CSP-Baseline and CSP-Switching were close, with high correlation coefficients, especially in genes with higher goodness of fit (Fig 5A Left). However, whether or not splicing is considered significantly impacts the inference results. The inference results based on CSP-Baseline and CSP-Splicing were quite different, with low correlation coefficients, even in genes with higher goodness of fit (Fig 5A Middle). We speculate that this is due to the assumptions of the two models are incompatible: in CSP-Baseline, γ_t is assumed to be a constant; while in CSP-Splicing, γ_s is assumed to be a constant. But these two assumptions can not be held simultaneously for their different roles in the physical modeling and our analysis results (see “Methods” section). We also compared γ_t and γ_s computed by the CSP-Splicing, and the results showed that γ_s was always greater than γ_t, and the linear correlation between the two was not high (Fig 5A Right). In summary, we showed that kinetic parameters inferred from CSP-Baseline and CSP-Switching but not CSP-Baseline and CSP-Splicing, are consistent.

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Fig 5. Parameter inference and enrichment analysis for the cell cycle dataset.

The inference strategy involved in this figure is for kinetics/pulse data. A. Comparison of parameter inference results of our three stochastic models. From left to right are the comparison of γ_t of CSP-Baseline and CSP-Switching, the comparison of γ_t of CSP-Baseline and CSP-Splicing, the comparison of γ_t and γ_s in CSP-Splicing. The overlapping well-fitted genes were set as the overlap set of genes in the top 40% of the goodness-of-fit for both methods. B. Comparison of inferred parameters between our stochastic models and Dynamo’s method. Left: the comparison of γ_t between CSP-Baseline and Dynamo. Right: the comparison of β between CSP-Splicing and Dynamo. C. Comparison of the goodness-of-fit of the three stochastic models. Left: all highly variable genes. Right: genes in the top 10% of average new mRNA expression in highly variable genes. Here Base refers to the CSP-Baseline model, Splic to the CSP-Splicing model and Switch to the CSP-Switching model. D. Robust analysis. Left: Landscape of CSP-Baseline-based loss functions for the a typical gene WWTR1. Right: Scatter plot of robustness measure and goodness of fit for parameter inference. E. Enrichment analysis results of genes with high gene-wise γ_t, β (top 50%) in well fitted genes (top 40% of goodness of fit). F. Heat map of cell-wise parameters for well-fitted genes. From left to right, cell-wise α based on the CSP-Baseline, cell-wise αp_on based on the CSP-Switching and cell-wise β based on the CSP-Splicing, respectively. Across all three heatmaps, the X-axis is the relative cell cycle position while the order of genes in the y-axis is arranged such that the peak time of each gene increases from the top left to bottom right.

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The inferred total mRNA degradation rates γ_t from Storm and Dynamo are close in well-fitted genes, while CSP-Splicing’s inferred splicing rates β are always larger than those from Dynamo. We compared the inferred results of γ_t in CSP-Baseline with those in Dynamo (Fig 5B Left). Although they were not consistent for some genes, they are quite consistent for the genes with better fitting. We also compared the inference of β in CSP-Splicing with those in Dynamo (Fig 5B Right). The result shows that the inferred β by our approach was usually larger than those in Dynamo, even for the genes with a better fitting. A possible explanation is that the inference of Dynamo ignored the difference between γ_t and γ_s, which made the inferred β smaller. We also compared the goodness-of-fit of the three stochastic models. Overall, they are relatively close (Fig 5C Left). However, when we focused on genes with higher new mRNA levels (top 10%), CSP-Splicing had a better fit (Fig 5C Right). We speculate that this is because genes with higher expression are suitable to be fitted with more complex models.

When the parameter γ_t is small, parameter inference may not be robust enough. However, we found that the genes selected by the goodness-of-fit have robust results. We analyzed the robustness of the parameter inference in the simplest CSP-Baseline model (see “Methods” section). We plotted the landscape of a typical negative log-likelihood loss function based on CSP-Baseline for gene WWTR1 (Fig 5D Left), with the black line corresponding to ∂ℓ(α, γ_t)/∂α = 0 (Eq (83) in the “Methods” section) and blue line corresponding to α = α_cons when holds (Eq (84) in the “Methods” section). The landscape of the loss function shows a fairly flat area around ∂ℓ/∂α = 0, and the two lines almost coincide when γ_t is small, which is consistent with our previous argument. We design a quantitative index to measure the robustness of parameter inference (see “Methods” section) and analyzed the relationship between the robustness measure and the goodness-of-fit (Fig 5D Right). We found that parameter robustness was positively correlated with the goodness of fit and the correlation coefficient was as high as 0.69. Though the reason for this high correlation is not clearly understood in theory, we can utilize this fact to select the genes with high goodness of fit for downstream analysis, which also ensures the results are relatively robust.

We selected the well-fitted genes (top 40% ) and performed enrichment analysis on this fraction according to the magnitude of gene-wise parameters γ_t, β, α and p_off (Fig 5E and S3 Fig). The results of the enrichment analysis showed that these genes were highly correlated with the cell cycle progression.

The assumption of constant coefficients is often violated because of the time-dependent kinetics and multiple lineages [11]. Many works relaxed the constant coefficient assumption and inferred cell-specific parameters to overcome this issue [10, 13, 16, 26]. In our proposal, we take a post-processing step to get the cell-specific parameters after inferring all parameters through previous procedures. We relaxed the constant coefficient assumption and proposed a method to infer cell-specific parameters except the constant degradation rate γ_t or γ_s, i.e., we inferred cell-specific α in CSP-Baseline, cell-specific α × p_on in CSP-Switching, and the cell-specific α and β in CSP-Splicing (see “Methods” section). This partial constant coefficient assumption had support from the study in [21], which showed that the degradation rate of most genes was independent of time. Finally, We plotted heat maps of the cell-wise α (based on CSP-Baseline), α × p_on (based on CSP-Switching) and β (based on CSP-Splicing) for the well-fitted genes (Fig 5F). The results show that cells in the same cell cycle phase usually have closer kinetic parameters.

Storm improves the robustness and accuracy of time-resolved RNA velocity analysis

Our three stochastic models described the evolution of the PMF (or joint PMF) of the number of new mRNA (or new unspliced and spliced mRNA) molecules over time for different settings. To estimate RNA velocity of single cells, only the evolution of the mean value over time will be considered, which requires us to reduce the stochastic models to the corresponding deterministic models (see “Methods” section).

Based on the deterministic model derived for the mean corresponding to the three stochastic models, we inferred the relevant parameters for computing different types of RNA velocity for different models. In Models 1 and 3, we computed the total RNA velocity because the splicing process was ignored. In CSP-Splicing, we calculated both total RNA velocity and spliced RNA velocity (see “Methods” section). Note that because the new RNA velocity mostly reflects the metabolic labeling process of RNA and does not reveal RNA biogenesis, it is thus not used. In addition, a derived relationship between γ_t and γ_s suggests that the total RNA velocity can be computed based on either or . In practice, we used the former approach by default.

We compared the streamlines of the total RNA velocity of our three models with that of Dynamo on the cell cycle scEU-seq dataset (Fig 6A). Almost all streamlines from our models correctly reflect the cell cycle progression, except that part of them from CSP-Splicing had a minor flaw in the M phase and CSP-Switching in the S phase. In addition, we found both CSP-Splicing and Dynamo’s spliced RNA velocity (S4A Fig) did not get entirely correct streamline results. The streamlines of CSP-Splicing were problematic in the M-G1 phase, while the streamlines of Dynamo were problematic in the S phase. We speculate that this is probably due to the fact that new unspliced mRNAs have rather low expression levels, frustrated with many dropouts and very sparse data, resulting in unreliable inferences of the parameter β and inaccurate RNA velocities.

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Fig 6. RNA velocity analysis of the cell cycle dataset.

The inference strategy involved in this figure is for kinetics/pulse data. A. Comparison of total RNA velocity streamline visualizations between three stochastic methods and Dynamo in cell cycle dataset. B. Comparison of average correctness of total velocity in gene expression space and RFP_GFP space. The p-values are given by the one-sided Wilcoxon test. Here Base refers to the CSP-Baseline model, Splic to the CSP-Splicing model and Switch to the CSP-Switching model. C. Similar to B, comparison of velocity consistency. D. The duration time (unit: hour) of each cell cycle phase of the human RPE1-FUCCI system based on Storm’s CSP-Baseline and Dynamo. E. Total RNA velocity streamlines calculated using Storm’s CSP-Baseline with gene-wise parameters (instead of using gene-cell-wise parameters except for the degradation rate). F. The smoothed expression of DCBLD2 in different cells. G. Comparison of total RNA velocity in DCBLD2 between CSP-Baseline and Dynamo. H. Phase portraits of new-total RNA planes of DCBLD2 of CSP-Baseline and Dynamo. Quivers correspond to the total (x-component) or new (y-component) RNA velocity calculated by the different methods.

https://doi.org/10.1371/journal.pcbi.1012606.g006

We also quantitatively benchmarked the average correctness and consistency of the velocities in different methods in the original gene expression space and low-dimensional space (here the RFP_GFP space is used which corresponds to the Geminin-GFP and Cdt1-RFP-corrected signals of RPE1-FUCCI cells)(Fig 6B and 6C and S4B and S4C Fig). The definition of correctness and consistency of velocity is given in the “Methods” section. In the gene expression space, the average correctness and consistency of the total RNA velocity of CSP-Baseline, CSP-Splicing, and CSP-Switching are significantly better than that of Dynamo (Fig 6B and 6C Left), while the spliced RNA velocity of CSP-Splicing has slightly lower consistency than that of Dynamo (S4B and S4C Fig Left). In the RFP_GFP space, the average correctness of total RNA velocity of all methods are significantly higher compared to that in the gene expression space, and simpler methods tend to improve more. The average correctness of CSP-Baseline is highest at this time (Fig 6B Right). However, the average correctness of the CSP-Splicing’s spliced RNA velocity still perform slightly worse than Dynamo’s (S4B Fig Right). In contrast, the total RNA velocity consistency of CSP-Baseline and CSP-Splicing is significantly better than that of Dynamo (Fig 6C) and the spliced RNA velocity consistency of CSP-Splicing is also significantly better than that of Dynamo (S4C Fig Right). Overall, the CSP-Baseline-based total RNA velocity has the highest average correctness and consistency, and significantly outperforms Dynamo, while the CSP-Splicing-based spliced RNA velocity was close to Dynamo quantitatively.

To demonstrate the significance of inferring time-resolved velocities with physical units, we calculated the duration time of each cell cycle phase of the human RPE1-FUCCI system based on the total RNA velocities (see “Methods” section, Fig 6D). Indeed, the human RPE1-FUCCI system has a cell-cycle time of about 21 hours (about 6 hours for G1-S phase, 8 hours for S phase, 4 hours for G2-M phase, 1 hour for M phase and 2 hours for M-G1 phase) [37].

To demonstrate the value of using gene-cell-wise parameters (except degradation rates), we visualized the streamlines of total RNA velocity based on gene-cell-wise parameters and those based only on gene-wise parameters (Fig 6E and S4D Fig). We observed that the streamlines of CSP-Baseline and CSP-Switching in the S to G2-M phase are incorrectly reversed (Fig 6E and S4D Fig Right), and the streamlines of CSP-Splicing are also less smooth and accurate than those when gene-cell-wise parameters are used (S4D Fig Left).

We now illustrate the advantages of our method in the estimation of kinetic parameters and the calculation of RNA velocity with two example genes: DCBLD2 and HIPK2. In gene DCBLD2, the cells at M and M-G1 have the highest overall expression and the correct RNA velocity should be negative (Fig 6F). However, Dynamo returned the positive velocity, which is problematic (Fig 6G Right). In contrast, CSP-Baseline, CSP-Switching and CSP-Splicing all returned negative velocities (Fig 6G Left and S4E Fig). We speculated one possible explanation is that the expression of the gene DCBLD2 has not yet reached a steady state. Consistent results were also observed from phase portraits of new-total RNA planes of DCBLD2 (Fig 6H and S4F Fig). For gene HIPK2, similarly, cells in phase M and M-G1 have the highest expression overall and the correct velocity should be negative (S4I Fig), but Dynamo and CSP-Baseline both returned positive velocities while CSP-Switching got the correct results (S4G and S4H Fig). We speculated one possible explanation for this is that the expression switch plays an important role in HIPK2.

Finally we generated a simulated non-steady-state pulse data (S5 Fig). Both the CSP-Baseline and Dynamo produced the correct streamlines (S5A Fig). However, the error between the degradation rate estimated by CSP-Baseline and the true value was lower compared to Dynamo, and the error was further reduced for the well-fitted genes selected by goodness-of-fit (S5B and S5C Fig).

Derivation of three stochastic dynamical models

Here we developed three stochastic models for the dynamical description of new mRNAs: Model 1 (CSP-Baseline): a stochastic dynamical model of new mRNA involving only metabolic-labeling transcription and degradation; Model 2 (CSP-Splicing): a stochastic dynamical model of new unspliced and spliced mRNA involving metabolic-labeling transcription, splicing and spliced mRNA degradation; and Model 3 (CSP-Switching): a stochastic dynamical model of new mRNA involving gene state switching, metabolic-labeling transcription and degradation.

Chi-square goodness-of-fit test for cell-specific distributions at a fixed time

We would construct an asymptotic χ² statistic for the data with common distribution type but sample-specific parameters. This goodness-of-fit test is to assess whether the null hypothesis that the considered data, at a fixed labeling duration, obeys the proposed distribution can be accepted.

We first divided the value range of the considered data into c classes. According to the range that the samples fall in, we got n independent categorically distributed random samples X_i ∈ {1, 2, …, c} for i = 1, 2, …, n with sample dependent parameter p_i, respectively. An equivalent representation for the categorical variable X_i is to denote X_i = (X_ij)_{j = 1, …, c} ∈ {e₁, …, e_c}, where e_j = (δ_jk)_{k = 1, …, c} is the indicator vector for j = 1, …, c. Correspondingly, the parameter p_i = (p_i1, …, p_ic)^T is a c-dimensional vector with non-negative elements and sums to one, which is defined as (17) This implies that Var(X_ij) = p_ij(1 − p_ij) and . Therefore, the covariance matrix of random vector X_i is (18) For sample i, we defined the truncated random vector and truncated vector , which is the first c − 1 components of X_i and p_i, respectively. The covariance matrix of is the submatrix consisting of the upper-left (c − 1) × (c − 1) block of Σ_i, denoted by , which can be written as (19) where is the diagonal matrix formed by the components of .

Define , and , and let (20) Below we would show that χ² is an asymptotic chi-square statistic with degrees of freedom c − 1. First note that (21) then the covariance (22) Let . When n goes to infinity, Y_n converges in distribution to the normal distribution N(0, I_c−1) according to the central limit theorem for the independent sum of random variables. Thus, converges in distribution to a chi-square distribution with degrees of freedom c − 1.

In summary, we proposed a new asymptotic χ² statistic for sample-specific distributions. For a fixed labeling duration t_fixed, a(t_fixed), b(t_fixed) and c(t_fixed) are all constants, the proposed χ² statistics can be used to test whether the new mRNA sequencing data are consistent with the CSP, ICSP and CSZIP distributions based on Models CSP-Baseline, CSP-Splicing and CSP-Switching, respectively. In addition, since there are one, two and two parameters to be inferred in CSP, ICSP and CSZIP distributions, respectively, the same number of degrees of freedom should be subtracted. Following [28], we ensured that the expected count np_j ≥ 0.25 in each group when determining the group value ranges. Finally, we take p-value as 0.05 in the computation.

Parameter inference in one-shot experiments with steady state assumption

In the one-shot experiments where we only observe new RNA l_j(t) and total RNA r_j(t) data for one labeling duration t, we had to invoke the steady-state assumption for the total RNA.

When the dynamics of total RNA in CSP-Baseline is at steady state, i.e., (23) where is the invariant PMF of the true expression of total RNA. At this point is a Poisson distribution with α/γ_t as the mean and from [3] we know that at this point the PMF of the true expression of old RNA is a Poisson distribution with as the mean. From Eq (9) we know that when technical noise is considered, the observed old RNA counts obey a similar CSP distribution (24)

At this point, we obtained the distributions of the new RNA and old RNA observations so that parameter inference can be performed using the MLE. Notice that the distribution of new RNA counts is not independent of total RNA counts, whereas the distribution of new RNA and old RNA counts is independent. We want to maximize the log-likelihood function (25) where is the probability of X = n for a Poisson-distributed random variable X with mean λ. When ∂ℓ/∂α = 0 and ∂ℓ/∂γ_t = 0, the likelihood function is maximized and it can be solved analytically (26) where 〈⋅〉 means the population average defined by (27) Since here it is for the one-shot data set, K = 1. Note that Eq (26) is similar to the formula in Dynamo [26] for estimating the parameters for one-shot data. The difference is that this formula averages the raw counts, while the method in Dynamo averages the smoothed data.

Parameter inference in one-shot experiments without steady state assumption

In one-shot dataset containing both labeling and splicing information, i.e. unspliced unlabeled RNA u_u,j, unspliced labeled RNA u_l,j, spliced unlabeled RNA s_u,j and spliced labeled RNA s_l,j information is observed, we can make parametric inference without relying on the steady-state assumption.

The method is divided into two steps; in the first step, we sum unspliced unlabeled RNA and unspliced labeled RNA to obtain unspliced RNA, and unspliced labeled RNA and spliced labeled RNA to obtain spliced RNA. Then, we use the dynamic model without relying on steady state assumption in scVelo proposed by Bergen et al. [2] to infer the observation time of cells t_obs,j, switching time t_s, degradation rate γ_s,scv, and splicing rate β_scv. Despite the problem of scale invariance, the absolute magnitude of these values is not physically meaningful, but can still provide useful information for inferring the absolute magnitude of the parameters, e.g., the value of β_scv/γ_s,scv is meaningful, and in addition the cells with t_obs,j less than t_s are in the on state.

In the second step, we integrate the results from the first step and the labeling information to determine the absolute size of the parameter. We define the notation to be the set of cells in the on state. In CSP-Splicing, the ul_j and sl_j of cells in the on state obey ICSP distribution (28) We want to maximize the log-likelihood function (29) which is equivalent to (30) This problem is not well-defined, but it is after adding the result (31) from the first step. By solving the system of nonlinear equations consisting of equations (30) and (31), we can obtain the absolute magnitudes of α, β, and γ_s.

Since modeling assumptions in scVelo are often violated, we selected only well-fitting genes for use in the second stage and downstream analyses. We use R² as the goodness-of-fit, which is defined as (32) Due to the dropout effect, cells with expression close to 0 are not used in the actual calculation of R². More specifically the rule is (u_j, s_j) < (max(u_j)/5, max(s_j)/5).

Parameter inference in kinetics experiments

In the kinetics experiments, we observed data l_j(t_k) (or (u_l,j(t_k), s_l,j(t_k))) for new mRNA (or new unspliced and spliced mRNAs) with different labeling durations. We assumed that there are K labeling durations t_k for k = 1, 2, …, K, and the number of cells with labeling duration t_k is n_k. We utilized the MLE to infer the unknown parameters in different models without relying on steady-state assumptions.

In CSP-Baseline, we need to maximize the log-likelihood function (33) It is equivalent to minimizing the following loss function (34) The optimum of the loss is achieved when the gradient equals 0. Utilizing the concrete expression of a(t) in CSP-Baseline, we got . Then ∂L(α, γ_t)/∂α = 0 has a closed form solution (35) Another component of the Euler-Lagrange equation ∂L/∂γ_t = 0 has no closed form solution, so we need to solve γ_t by numerical iterations. We took the initial value of γ_t as the solution from Dynamo [26] under the steady-state assumption. Denote it as γ_t,0, and correspondingly, we take the initial value of α as α₀ = α(γ_t,0).

In CSP-Splicing, we need to maximize the log-likelihood function (36) which is equivalent to minimizing the loss function (37) Utilizing (12), we got ∂b(t)/∂α = (1 − e^−βt)/β and when β ≠ γ_s. The case for β = γ_s is similar. So ∂L(α, β_t, γ_s)/∂α = 0 has a closed form solution (38) However ∂L/∂β = 0 and ∂L/∂γ_s = 0 have no closed form solution, and we need to solve these equations by iterations. The choice of initial values is similar to the CSP-Baseline case. We took the initial value of β and γ_s as the solution from Dynamo [26] under the steady-state assumption, which we denoted as β₀, γ_s,0. And then the initial value of α is taken as α₀ = α(β₀, γ_s,0).

In CSP-Switching, we need to maximize the log-likelihood function (39) where ZIP(λ, p_off)|_n ≔ Prob(X = n) is the probability of X = n for a ZIP-distributed random variable X with parameters λ and p_off. It is equivalent to minimizing the loss function (40) Similar as before, we chose the initial value of γ_t, denoted as γ_t,0, based on the steady state assumption, and chose the moment estimator (41) and (42) as the initial values of p_off and α.

According to the biological meaning of the parameters, we added the constraints 0 < α < 10α₀, 0 < β < 10β₀, 0 < γ_t < 10γ_t,0, 0 < γ_s < 10γ_s,0 and 0 < p_off < 1, and we called the SLSQP optimizer in SciPy to solve the above optimization problem.

Goodness-of-fit test for the distribution evolution in time

In ordinary least squares (OLS) linear regression, people often use (43) to define the goodness of fit, where y_i is the sample observation, is the model prediction, and is the sample mean. For the generalized linear model (GLM), the R² can be defined using the deviance D and null deviance D₀ [29], (44) where , ℓ₀ and ℓ_s denotes the log-likelihood function of the model with parameter , the null model (that is, fitted with only the intercept), and the saturated model (that is, fitted with one parameter per sample), respectively. can be seen as a generalization of R², which is equal to R² when the model is a least squares linear regression [29]. Finally, to overcome the disadvantage of adding more parameters without reducing (similar to R²), we used adjusted (denoted as ) as the goodness of fit of different models, which is defined as (45) where d_D and are the degrees of freedom of D and D₀, respectively. To more intuitively explain the usefulness of , we named it goodness of fit of model in the mian text instead of using adjusted deviance R².

In CSP-Baseline, ℓ_s has the closed form (46) To calculate ℓ₀, we need to maximize the log-likelihood function (47) where a₀ is the intercept. The problem has a closed form solution a₀ = 〈l_j(t_k)〉/〈p_j(t_k)〉. In addition, d_D = N − 2 and , where N is the number of cells.

In CSP-Splicing, ℓ_s has the closed form (48) To calculate ℓ₀, we need to maximize the log-likelihood function (49) where b₀ and c₀ are intercepts and have closed form solutions b₀ = 〈u_l,j(t_k)〉/〈p_j(t_k)〉 and c₀ = 〈s_l,j(t_k)〉/〈p_j(t_k)〉, respectively. In addition, d_D = 2N − 3 and .

In CSP-Switching, to calculate ℓ_s, we need to maximize the log-likelihood function (50) When p_off is equal to zero, Eq (50) is maximized, and the closed form solution of ℓ_s is (51) To calculate ℓ₀, we need to maximize the log-likelihood function (52) Similar to solving Eq (40), p_off,0 and a₀ were initialized using moment estimators with additional constraints 0 < p_off < 1 and 0 < a < 10a₀. We then called the SLSQP optimizer in SciPy to solve the problem. In addition, d_D = N − 2 and . Before projecting high-dimensional RNA velocities to low dimensions for visualization, we first pick genes with higher goodness-of-fit. The default setting is that the top 40% of genes are picked, and this percentage can be specified by the user.

Post-processing for cell-specific parameters

In our cell-specific modeling of gene expression, we only assumed that γ_t (in CSP-Baseline and CSP-Switching) and γ_s (in CSP-Splicing) are constants over cells and are inferred based on the corresponding stochastic models, while the other parameters are cell-specific and continuously dependent on gene expressions. This relaxed assumption implies that only the degradation rate is common to all cells, and only cells with similar gene expressions have similar other parameters (due to continuous dependence). To realize this assumption, we first constructed the k-nearest neighbor (kNN) graph of cells by a data preprocessing. The cell-specific parameter inference was performed by applying the inference to the kNN graph for each cell with local constant parameter assumption and already inferred degradation rates. In other deterministic model-based methods to infer RNA velocity (either unspliced/spliced-based or new/total-based) [1, 2, 26], they also construct similar kNN graph and perform kNN smoothing on the data based on this graph. This post-processing step of ours can be seen as the generalization of the usual kNN smoothing to our stochastic setting, as we model discrete counts. The inference details for our three models were shown as below.

In CSP-Baseline, we have (53) where denotes the set of top k (default is 30) cells that have the most similar gene expressions as the jth cell with labeling duration t_k (including itself) and . Assuming that γ_t has been inferred, we can obtain a local estimator (54) by using the MLE. Define . Then the cell-specific transcription rate α_j(t_k) has a closed form solution (55)

In CSP-Splicing, we have (56) Similarly, assuming γ_s has been inferred, and defining the local estimators (57) we have (58) which is a nonlinear system. We have (59) To solve β_j(t_k), we set its initial value as previously inferred β by global constant assumption. We then call the foot function in SciPy to solve the nonlinear equation (59) to get β_j(t_k). The α_j(t_k) has a closed form solution (60) In summary, in CSP-Splicing, we can infer the cell-specific transcription rate α_j(t_k) and splicing rate β_j(t_k).

In CSP-Switching, we have (61) When computing RNA velocity, we only need to know α_j(t_k)(1 − p_off,j(t_k)) as a whole, and not their respective values (see next subsection). To simplify the computation, we used the moment estimation instead of MLE, and got (62) Similarly, assuming γ_t has been inferred, α_j(t_k)(1 − p_off,j(t_k)) has a closed form solution (63)

Reduction from stochastic to deterministic models for RNA velocity

We used discrete counts data in the proposed parameter inference and goodness-of-fit calculation via stochastic models. However, when we need to compute and visualize the RNA velocity, we should take the reduction from stochastic to deterministic models to get the mean velocity. Below we would show the reduction process and reveal the connection between the stochastic and their corresponding deterministic models.

In CSP-Baseline, let us denote the mean value of by , which is defined as . From Eq (5) we can obtain the deterministic equation after suitable algebraic manipulations (64) Similarly, the mean value of total RNA satisfies the equation (65) Since the initial value of is zero, we got (66)

In CSP-Splicing, the marginal PMFs of and are (67) respectively. The mean values of and have the form and . From the CME (10), we can obtain (68) and (69) Similarly, we can derive the equations for the mean values of total unspliced and spliced mRNA : (70) Since the initial value of is (0, 0), we got (71) and (72)

Similar to CSP-Baseline, in CSP-Switching, and satisfy the equations (73) Since the initial value of is zero, we got (74)

Computation of RNA velocity

To ease the notation, we denoted the new mRNA after data preprocessing by , defined as which is different from the true expression , the discrete counts data l(t), and the notation in the post-processing subsection. We would also use the notation , and with similar definition.

In CSP-Baseline, only the total RNA velocity can be obtained due to the lack of the splicing stage. From Eq (65), we have (75) where is the number of total mRNA molecules of the jth cell labeled with length t_k after data preprocessing.

In CSP-Splicing, we add the two equations in Eq (70) to obtain (76) and thus get the equation for total RNA velocity (77) In addition, in CSP-Splicing, we can also calculate the spliced RNA velocity by the following equation (78)

Similar to CSP-Baseline, the total RNA velocity in CSP-Switching can be obtained by the equation (79)

Relationship between γ_t and γ_s and its implications

The difference between Eqs (65) and (76) implies the difference between the total mRNA degradation rate γ_t and spliced mRNA degradation rate γ_s. After suitable manipulations, we had the relation between γ_t and γ_s as below (80) Therefore, we naturally got a method to infer γ_t when γ_s is known. Specifically, we first performed a zero-intercept linear regression (81) to get the slope k. Then we computed γ_t by γ_t = γ_s/k. Therefore, we can also infer γ_t and compute the total RNA velocity by Eq (75) in CSP-Splicing.

We would also like to point out that CSP-Baseline and CSP-Splicing are incompatible upon assuming that γ_t and γ_s are both constants. These two assumptions usually do not hold simultaneously. Otherwise, from Eq (80) we knew that is a constant, which is equivalent to that is a constant, i.e., is a constant. But this is only true when β and γ_t are equal.

Robust analysis of the parameter inference in the CSP-Baseline

When γ_tt is small, holds, then (82) which implies that from the mean perspective the nonlinear fitting of α and γ_t degenerated into a linear fitting of α at this point. For a more precise analysis, let , we have ∂ℓ(α, γ_t)/∂α = 0 is equivalent to (83) But when holds, ∂a(t)/∂α ≈ t, then we have (84) is a constant, which we denoted by α_cons. In addition, to quantitatively measure the robustness of inference on γ_t, since the optimal parameter is always located where the gradient is zero, we defined the l₁-norm of the derivative of the loss function with respect to γ_t restricted to ∂ℓ/∂α = 0 (i.e. black line), (85) as a measure of robustness. Since the half-life of the total mRNA molecules is usually not less than half an hour, we took γ_t,max = 1.5.

Definition of correctness and consistency of velocity

The correctness of cell velocities is defined as follows: Consider the cell i with position x_i and velocity v_i. Define its one-step extrapolated position as x_i + v_i. We say that v_i is correct (correctness index = 1) if the cell j closest to the extrapolated position x_i + v_i ranks after i in the temporal ordering. Otherwise the correctness does not hold and we set the correctness index to be 0. Thus the average correctness refers to the percentage of correct velocities. Because the boundaries of the cell cycle being estimated are not clear and sharp, we did not use the RNA velocity benchmark metric cross-boundary direction (CBDir) proposed by Qiao et al. [12] and widely used for comparison of RNA velocity methods.

The consistency means the extent to which the velocity of one cell is consistent with the velocities of its neighboring cells, and we use the average correlation coefficient proposed in scVelo [2] to measure this consistency.

Calculation of the duration of each cell cycle phase

After the total RNA velocities are obtained, we can evaluate the time of each phase of a cell cycle based on them. Specifically, we first pick k cells (i = 1, 2, …, k) whose relative positions are closest to 0 as a cell group, calculate their average expression and velocity as the initial expression x⁰ and velocity v⁰, and extrapolate the state of the cell group with a short time step dt, that is, x¹ = x⁰ + v⁰dt. We then search for another k cells (i = 1, 2, …, k) which are closest to the extrapolated state x¹, set the majority of the phase of these k cells to the phase of x¹, and set their average velocity as v¹ for the second cell group. Next, the extrapolation and local k-cells group identification step can be repeated until a given threshold of the relative position is exceeded. In the actual calculation, we set k = 300, dt = 0.01, and the threshold of the relative position to be 88% quantile of all relative positions. The above approach for processing the cell groups instead of cells themselves is to reduce the data noise by local averaging.

Generation of simulation data

In this paper we generate two simulation datasets, one of which is bifurcated one-shot dataset following VeloSim’s [31] flow, and the other is a non-steady-state kinetics dataset following scVelo’s [2] model.

The bifurcated one-shot dataset is generated as described below. First we set the maximum observation time to T and the labeling time t_l. The observation times for half of the cells were generated randomly with a uniform distribution [t_l, T/2], and the other half of the cells were divided into two equal parts, which were generated randomly with a uniform distribution [T/2, T]. We then followed SymSim [30] and VeloSim [31] to generate cell extrinsic variability factor(EVF)s and gene effect vector and used theme to generate the cell-gene-wise transcription rate α. Splicing rate β and degradation rate γ_s are gene-wise and were generated from uniform distributions of [0, 0.5] and [0, 5], respectively. After all kinetic parameters were generated, we used the Gillespie algorithm to generate raw uu, ul, su and sl RNA counts data. We did not consider technical noise when generating the simulated data and therefore set the size factor of all cells to 1.

The non-steady-state kinetics dataset is based on the model in scVelo [2], but ignores the splicing process. First we set the maximum observation time T and the K labeling times t₁, t₂ ⋯ t_K. The observation times for cells are randomly generated with a uniform distribution [t_K, T], and the labeling times of the cells were randomly selected with equal probability from t₁ to t_K. Transcription rate α and degradation rate γ_t, both of which are gene-wise, were randomly generated with a uniform distribution [0.5, 1] and [0, 0.5], respectively. To generate non-steady-state data, we set the switching time t_s = 0.5ρ/γ_t, where ρ is a random number generated with a uniform distribution [0, 1]. Finally similarly we use Gillespie algorithm to generate raw new and total RNA counts data and do not consider technical noise.