Model

We used a simple and interpretable latent variable model for the probability distribution of the counts, with explicit biophysical meaning associated to each latent variable. We assumed cells are asynchronous, and we therefore introduced two latent variables that corresponded to the lineage and time of the cell.

Therefore, in our trajectory model, the gene expression data of each cell x was described as a function of latent variables z = ( l , t )  which specify the lineage l and time t of the cell. By considering a particular parametric class of gene expression dynamics that specify the distribution p ( y | z , θ )  of in vivo counts y, and a sequencing noise model p ( x | y ) , we mapped latent variable z into data space and arrived at an explicit formulation of data distribution that could be trained using the expectation-maximization (EM) algorithm. In other words, we assumed the counts of each cell x had the following density,

where p ( z )  is the distribution of latent variables.

In the following, we specify p ( y | z , θ ) , p ( x | y )  and p ( z ) , which are based on the transcription model, measurement model and sampling measure.

Transcription model.

Since current transcriptomic data can be used to measure the numbers of both nascent and mature transcripts, we considered the production, processing, and degradation of individual RNA molecules in our transcription model, as in previous RNA velocity literature [16].

The distribution of RNA counts in the reaction system (Eq 2) is known [23]. We assumed that the initial distribution of U and S was a product Poisson distributions, which implied that the mean parameter vector λ = ( λ_u , λ_s )  of U and S evolved according to Eq (3).

The functional form of A_l ( t )  reflects our assumptions of gene expression during development. However, explicitly modeling transcription rates as functions of gene expressions is hard and tends to overfit. More fundamentally, we have some prior physical intuition; this function is actually A_l ( λ , u , t ) , where u is some high-dimensional vector of regulator concentrations. These data are not possible to collect using currently available technologies, although this constraint may change in the coming years. In principle, we can write down these equations, and even simulate them (using dyngen [24] or the stochastic simulation algorithm), but we strived to start with an analytically tractable model, particularly to recapitulate and formalize the increasingly popular RNA velocity framework. Therefore, we used a phenomenological model and didn’t consider gene interactions. Rather, the effect of transcriptional regulation on each gene during development was summarized into synchronized state switching and each state had its transcription rate. This formalized transient cell types.

To be able to account for multiple lineages, we assumed cell states S = 0 , 1 , . . . , S formed a directed graph, with each lineage represented as a path of length K on this graph.. The trajectory structure described the graph, and recorded the states s ( l , k ) ∈ S , l = 1 , . . . , L , k = 1 , . . . , K of the L lineages during the K stages. We treated the trajectory structure as known since it can typically be obtained from our previous knowledge of the data. Specifically, for lineage l, we assumed A_l ( t )  is a piecewise constant function: A_l ( t ) = α_s ( l , k ) , where the cellular state index s is determined by the lineage l, and the time interval k to which t belongs, i.e., t ∈ ( τ_k - 1 , τ_k ] . The state index s = s ( l , k )  was determined by the trajectory structure. For example, given the trajectory structure in Fig 1, s ( l = 1 , k = 0 ) = s ( l = 2 , k = 0 ) = 0, s ( l = 1 , k = 1 ) = s ( l = 2 , k = 1 ) = 1, s ( l = 2 , k = 2 ) = 2 and s ( l = 2 , k = 2 ) = 3.

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Fig 1. Chronocell overview.

The input of Chronocell comprises three components: 1) the trajectory structure, which outlines the states each lineage traverses as paths on a directed graph. 2) The sampling assumption, which defines the prior distribution of latent variables, namely lineages and process time, with a default uniform distribution over both; and 3) the scRNA-seq data, consisting of unspliced and spliced count matrices. The Chronocell model consists of a expression model with piecewise-constant transcription rates, and a Bernoulli measurement model. Each state s is associated with a transcription rate α_s for each gene, as well as an exit time τ_k denoting the switching time to the next state, where k is the index for the time segment. The EM algorithm is used for inference, with each iteration alternating between E-steps and M-steps. The results of Chronocell primarily include the estimated parameters and posterior distributions over latent variables for each cell.

https://doi.org/10.1371/journal.pcbi.1012752.g001

Then, the parametric solution of λ was given by

Assuming cells are at steady states at time 0, we haveλ_u ( 0 ) = α_sl , 0 and. Therefore, for each gene, the parameters are θ = ( α , β , γ , τ ) .

Measurement model.

Next, we needed to have a measurement model that connected counts number y in cells to the observed counts x in a single-cell RNA-seq experiment.

We assumed each molecule of mRNA produces Bernoulli ( q )  number of captured RNA molecules, which is also an good approximation of a Poisson model with low mean [25,26]. The capture rate q for each molecule can potentially depend on factors such as cell read depth, gene-specific, and species-specific biases in mRNA capture methods. However, in our model, we assume the capture rate only varies with cell read depth (or cell size), i.e., q = r_i, where r_i represents the read depth (cell size) of cell i. With i as the cell index, j as the gene index, and c as the species index, we have:

where λ_jc is the ODE solution for species c of gene j.

This is equivalent to multiplying the mean of Poisson distributions by a constant r_i. Since usually only the relative value of read depth r_i can be available, we absorbed the mean of r_i into α and infer their product directly:

Inference

Maximum likelihood estimates of parameters by EM algorithm.

We use the expectation–maximization algorithm to estimate model parameters . For simplicity, we discretize the latent variable t with finite regular grid points over the interval: t = t₁ , . . . . , t_M, which basically approximates a continuous measure with a discrete measure. Then, the latent variable z=(l,t) describing lineage and time is discrete, and we write ∑ ⁡ _zi to denote the summation over all L lineages and M time grid points, i.e., . With this, the objective function becomes

where i is the cell index and p ( x_i , z_i | θ )  denotesthe probability of observing x_iwith the latent variable being z_ifor cell i.

As log function is concave, we can use Jensen’s inequality:

Since is a constant for all z_i, the equality holds and

Our trajectory model makes it possible to write out p ( x_i | z_i , θ )  explicitly:

Therefore, we could use the expectation–maximization algorithm efficiently. Specifically, in the E-step of EM algorithm, we calculated the posterior distribution p_i ( z_i | x_i , θ )  based on p ( x_i | z_i , θ ) ,

In the M-step, based on fixed p_i ( z_i | x_i , θ )  and analytical form of p ( x_i | z_i , θ ) , we optimized θ_j for each gene j separately by maximizing

Both the value and the gradient of F_j can be written out analytically and computed efficiently (Section 2 in S1 Text). Therefore, we could use off-the-shelf quasi-Newton methods for optimizing F_j with respect to θ_j, e.g., ’L-BFGS-B’ method in minimize function provided by Scipy [29,30].

In each step of EM algorithm, we alternated between the expectation and maximization steps. The implementation is based on defining the function that calculates p ( x_ij | z_i , θ ) , so it would be easy to modify p ( x_ij | z_i , θ )  for different models in the future.

A warm start incorporating prior knowledge about data can help algorithm converge to the optimal θ^* quickly. If neither initial parameters nor p ( z | x )  is given, we use random initializations. Multiple runs with different starting points are used to avoid local minima. By default, we tried 100 different random initializations, and run 100 steps for each initialization both for random initialization and warm start.

Read depth estimation.

If we refer to all cell-wise variances as extrinsic noise, then it includes both sequencing noise and a part of biological sotchasticity. This extrinsic noise is shared by covariance between genes and variance of genes. Therefore, we can estimate such extrinsic noise using covariance to leave out the intrinsic noise of genes that are more biologically relevant.

Consider two species a and b. Suppose that in vivo their counts numbers are Y_a and Y_b with Y_a ~ μ_a and Y_b ~ μ_b where μ_a and μ_b are arbitrary distributions. Suppose additionally that during sequencing, the read depth (cell size) r is a random variable on [0,1], and r is independent of Y_a and Y_b. Denote the counts we ultimately get by X_a and X_b, where X_a and X_b are independent given r, Y_a, and Y_b, and  ⁡  [ X_a | r , Y_a ] = r ⁡  [ Y_a ]  and  ⁡  [ X_b | r , Y_b ] = r ⁡  [ Y_b ] .

Therefore,

and

If we assume genes are independent on average, then we can estimate the CV² of read depth () by . We can summarize the extrinsic noise by defining , where the expectation is taken with respect to all possible gene pairs (a,b), and we can estimate ξ by average normalized covariance across genes.

On the other hand, the variance of one gene is

Now apply a second order approximation and suppose  ⁡  [ X | r , Y ] = η₁rY + η₂r²Y and  ⁡  [ Y ] = ϕ₁ ⁡  [ Y ] + ϕ₂ ⁡  [ Y ] ² sequentially.

Then,

and

It’s easy to see that if ϕ₂ = 0, i.e., gene intrinsic variance does not have quadratic terms, the coefficient of quadratic term of variance is also ξ. Therefore, if the quadratic term of the variance of a gene is closed to ξ, we know it is more Poissonian. In other words, it does not differentially expressed and could be used to estimate read depth.

Therefore, our procedure of read depth estimation is: first, estimate read depth CV² and ξ; then, select genes whose quadratic term of the variance is close to ξ; finally, sum them up and normalize by the mean to calculate relative read depth. A similar, but not identical procedure of read depth estimation, was used in [31].

Poisson mixtures model.

For fitting clusters, we used Poisson mixture models. Suppose there are S mixtures with transcription rates α_js, s = 1 , . . . , S for each gene j, and each mixture is at steady state. Thus, the parameters of one gene are and , since only the ratio is identifiable.

Similarly to the Gaussian mixture model, we could use the EM algorithm to infer parameters (including mixture weights) and posteriors.

Analysis

Fisher information.

The Fisher information matrix (FIM) is defined to be

Since

we could calculate FIM numerically with the explicit form of the derivative and the posterior distribution.

Gene selection.

We did not use absolute likelihoods to select dynamical genes, because different genes have different scales and are not directly comparable. Instead, we noticed that the cluster model serves as a natural reference point for comparison, as genes with no dynamics can be fit equally well, if not better, by clusters. Therefore, we decided to use the relative likelihood as a criterion for gene selection.

For trajectory model,

Similarly for clusters model,

We defined the gene-wise likelihood of gene j for trajectory model to be

and for clusters,

The first term is meant to represent the difference in likelihood caused by different flexibility of latent variables of two models.

For gene selection, we compared the gene likelihood of two models and selected genes whose gene likelihood of trajectory model was higher than that of the cluster model.

Simulations

We randomly generated parameters for 200 genes and sampled 2000 cells by default unless otherwise specified. Cells were sampled uniformly over process time and lineages. We then fit the model with the correct trajectory structure and synchronized model if not stated otherwise.

For the simulation parameters, we assumed that the transcription parameters (α,β,γ) followed the log-normal distribution lognormal ( μ , σ ) , where μ is the mean and σ is the standard deviation of the variable’s natural logarithm. We attempted to derive realistic parameters for the distributions from the literature. Hence, we assumed β ~ lognormal ( 2 , 0 . 5 ) , γ ~ lognormal ( 0 . 5 , 0 . 5 )  so that unspliced to spliced ratio was around 0.2 and their distributions resembled those determined from metabolic labeling datasets [32]. The α was assumed to follow lognormal ( 2 , 1 ) , so that the mean of spliced counts was similar to that in scRNA-seq data. We assumed the read depth follows Beta distribution , where μ was the mean and v the variance. For simulations with Gamma noise, we multiplied the mean of Poisson distribution λ by a random variable following Gamma distribution with mean 1 and variance 0.5.

For simulations under the desynchronized model, gene-wise τ_k was sampled from a uniform distribution on , where T_k corresponds to the global τ_k in the synchronized model, and Δτ is the smallest interval length ensuring that τ_k retains its order.

To vary the sampling distributions, we generated a Gaussian distribution with a random mean and standard deviation 0.05. We then sampled both from the Gaussian and the uniform distribution, and blended them together in different proportions. The percentages of time sampled from a Gaussian ranged from 0 to 1, increasing by 0.1 increments.

To characterize the time errors, we calculate root mean squared error (RMSE) for the posterior mean of process time in comparison to the true time. To calculate the errors of parameters, we divide the absolute error of α by the square root of true values and β , γ by true values, so that errors of different genes are more comparable. We refer to these as normalized errors throughout the text.

Real datasets preprocessing

To estimate the squared coefficient of variation of the read depth , we calculated the covariance matrix of all genes with nonzero means, which was divided by the mean squared and averaged across gene pairs to calculate the mean normalized covariance as an estimate of ξ. We then selected Poissonian genes whose variances are close to baseline variance with reasonably large mean (var < 1 . 2 ( μ + ξμ² ) , μ > 0 . 01). However, as some genes can be co-regulated and, therefore, correlated, we calculated the mean normalized covariance of Poissonian genes and repeated selecting new Poissonian genes until the mean normalized covariance no longer changed. This typically occurred after two iterations. Finally, we normalized the sum of counts of the selected Poissonian genes by their mean to obtain the relative read depth estimates, which were then used as fixed parameters during the fitting process.

Based on simulations of factors of informativeness of the data, including numbers of cells and genes, the average mean of α parameters as well as β and γ ratio, we determined the procedure of filtering genes for fitting: we applied count mean thresholds (0.02 for unspliced and 0.1 for spliced), filtered out genes with a small unspliced to spliced ratio (), and finally, selected genes with a variance larger than 1.2 times the baseline variance, i.e., var > 1 . 2 ( μ + ξμ² ) , where μ is the mean, var is the variance, and ξ is the read depth CV². For cell cycle data, we also constrained genes to the Gene Ontology term “cell_cycle" (GO:0007049). We occasionally adjusted the variance threshold to 1.5 in order to end up with 50–200 genes. Then trajectory and Poisson mixture models were fit on those genes.

Use of other methods

We have also used dyngen to generate simulation data [24]. We use a bifurcation backbone for generation of 10000 cells and 200 genes following its vignette.

For Monocle 3 and diffusion pseudotime, we always provide the correct root cells whose simulation times are 0 for simulations. For real dataset, we set to the cells from the cell type that is expected to be the progenitor. In Monocle 3, when the cells form disconnected clusters, only cells on the partition that includes the root cell has finite pseudotime. We only consider those cells and discard other cells with infinite pseudotime for comparison with simulation time. For slingshot, we manually set the true start cluster based on the simulation time for simulation and the cell type in real datasets. For veloVI, we use the mean latent time across genes for comparison with simulation time.

A trajectory model generalizing cellular states

We begin by defining the trajectory as a dynamical process underlying all cells, with potentially different lineages/branches within this process. Cells are then assumed to be sampled from various, unobserved, time points along this process. Thus, the latent variable z = ( t , l )  is introduced to account for such heterogeneity due to process time (t) and lineage (l), and z follows a sampling distribution determined by the specific biological system and experimental conditions. Consequently, the probability distribution of the data we obtain is a mixture of cells over time and lineage,

(1)

where x is the data and θ is the set of parameters that define the trajectory. This is the common framework of trajectory inference, and developing a trajectory model requires defining the gene dynamics along the lineage and the process time P ( x | z , θ ) , as well as the sampling distribution of cells over the process P ( z ) .

To define the dynamical process, we first state our transcription model. We consider only transcription, splicing and degradation reactions in cells, and assume only transcription rates are time-dependent (Fig 1 Gene expression):

(2)

where A_l ( t )  is the transcription rate function for lineage l at time t, and β and γ are the splicing, and degradation rates. The chemical master equation describing the evolving distribution of the above biochemical reaction network has an analytical solution [23]. If we assume the initial distribution to be Poisson, the solution remains Poisson with the means (λ_u and λ_s) of U and S evolving according to the following ordinary differential equations (ODEs) of RNA velocity:

The modeling of transcription rate A_l ( t ) is motivated by the common abstraction of cellular differentiation as cell state transitions: each lineage is abstracted as a series of switches in cellular states over time. The series of switches are specified by the given trajectory structure, which includes a directed graph of cell states where each lineage corresponds to a path (Fig 1 Input Structure). We introduce one transcription rate per gene for each cellular state (α). Switching is assumed to be instantaneous and occurs at an unknown but fixed time (τ) with the first switch leaving initial state 0 to occur at τ ₀. Without loss of generality, we consider the entire process to start at time 0 (τ ₀=0) and have a time length of 1 (e.g. τ ₂=1 in Fig 1). Consequently, the transcription rate function A _l  ( t )  of lineage l is simplified as piecewise constant functions of the process time over  [ 0 , 1 ]  (Fig 1 Model). This piecewise constant function is defined in the limiting regime where transcriptional state switching (such as expression of master regulatory factors and changes of chromatin state) precedes gene expression and has a much faster time scale. Thus, the piecewise constant function serves as a reasonable approximation when the time scale of transcription rate changes is comparable to or larger than the mRNA half-life. It also directly reduces to discrete cell clusters in the fast dynamic limit, i.e., when dynamical timescale ( and) is much smaller than sampling intervals, for example, the total time length divided by cell number, n, under a uniform sampling distribution. This connection to cluster models enables us to interpolate between discrete cell states and continuous dynamics.

The simple form of the transcription rate function lead to a tractable model and facilitates inference and analysis. In fact, it affords explicit solutions for the distribution and for its derivatives with respect to parameters. Ideally, gene regulatory networks involved in cell differentiation would be modeled, but with current (transcriptomic) data types it is difficult to include gene interactions and to model transcription rates as (protein-mediated) functions of other genes with accuracy. Thus, we assume that the dynamics of different genes are independent and that all correlations are absorbed into the shared latent process time. Additionally, for simplicity, we can assume all genes are fully synchronized, as in the synchronized model, where τ is the same for all genes. However, the desynchronized model, which allows for different τ values for each gene, is also available. In summary, our trajectory model is suitable for capturing the coordinated global gene expression changes instead of the detailed gene dynamics.

After deriving an explicit distribution of in vivo counts, we turn to the measurement model. We assume simple binomial sampling of each molecule, and that the average binomial sampling probability, i.e., read depth, varies between cells but remains the same for all molecules in one cell. Then, in vitro counts remain Poisson but with means adjusted by read depth. Instead of using normalized data, we estimated read depth using the total UMI counts of near-Poissonian genes that are not used for the inference, and incorporated it into the count distribution. Therefore, we arrive at an analytical form of the conditional probabilistic distributions P ( x | z , θ )  of counts from a dynamic process by specifying the trajectory structure, gene expression model, transcription rate functions, and scRNA-seq measurement model.

The remaining part of specifying the sampling distribution P ( z )  is crucial, because it breaks the scale invariance of parameters and ensures the identifiability of the model: multiplying the transcription, splicing and degradation rates by the same constant leads to the same marginal likelihood if the sampling distribution can be changed. Ideally, the specification of the sampling distribution depends on our knowledge of the studied biological system and the experimental design. For example, for stem cells that constantly divide and differentiate, we may assume a uniform sampling distribution over  ( 0 , 1 ]  with a point mass on time 0, where the point mass represents the fraction of time that cells spend in the initial proliferative state. On the other hand, for time series data, we may assume the sampling distribution of cells are centered around their captured time points with some variances. However, in practice, since there is no obvious principled way to determine such distributions, we just assume a uniform sampling distribution over process times  ( t > 0 )  by default, but the weight on time point 0 and different lineages can be identifiable and be updated in the inference.

A trajectory is thus defined with the above dynamical process and sampling distributions. With the parameterized form of the probabilistic distribution of counts, we can employ the Expectation-Maximization (EM) algorithm to estimate model parameters and posterior distributions of process times and cell lineages by maximizing Evidence Lower Bound (ELBO) with either warm start or random initialization (Section Inference). The synchronized model was used by default, where all genes share the same τ. Since fitting the desynchronized model from scratch is more challenging, it is recommended to begin the fitting process using the results from the synchronized model. For desynchronized models, we also introduce a penalty term proportional to the squared difference between gene-wise τ _k and the global τ _k to encourage synchronization, and the coefficient for this penalty term is set as a parameter, with a default value of 0. We tested the EM algorithm and inference on simulations generated from our trajectory model (Section Simulations). Both the synchronized and desynchronized models are identifiable, and the parameters can be recovered accurately under reasonable conditions (S2 and S3 Figs). We will elaborate on these conditions in Section Identifying failure scenarios reveals the fragility of inference.

By having an explicitly parameterized distribution of raw counts, we can easily interpret results and systematically assess the model under a more principled framework. Since parameters all have biophysical meanings, we can not only directly interpret them but also validate their accuracy by comparing them to orthogonal experiments that measure the same parameters. Furthermore, we can directly select DE genes by fold changes in transcription rates across states, after filtering genes by goodness of fit (Section Gene selection). The performance can be quantified through parameter errors, aiding in the identification of both confident and uncertain scenarios.

Not only are the parameters and results more interpretable, but we can also compare different models systematically. Regarding false positives from clustered data, we can evaluate when a trajectory model is no longer appropriate by comparing it to a cluster model using standard model selection methods like AIC (S4 Fig). Importantly, the posterior distributions and multiple minima of AIC scores can hint at the lack of continuity (S4 Fig), serving as a retrospective metric when model selection methods are compromised by unaccounted noise. We also demonstrate model selection on a disconnected trajectory, which includes both a single cluster and a bifurcation (S5 Fig). Our representation of the trajectory structure as paths on a directed graph naturally includes this scenario. We compare the results of the true model with those of a cluster model and a connected structure. AIC and BIC can correctly identify the true structure when compared to the two incorrect models (S5 Fig).

Demonstrating performance of Chronocell on ground-truth simulations

Since the true trajectory structure may differ from both our prior knowledge and the initial structure used, we first demonstrate the inference pipeline under a slightly incorrect trajectory. The inferred parameters provide insights into the true structure, and we then apply model selection to identify the correct model. Additionally, we include non-variable genes to assess the performance of the differential expression (DE) procedure.

We used a two-lineage trajectory and uniformly sampled 10,000 cells over lineages and process times with biological plausible parameters extracted from literature [32] (Section Simulations). 100 out of 200 genes are variable (Fig 2a). We fit under a slightly wrong assumption of the trajectory structures and assumed all genes are variable (Fig 2b). For random initialization, we initialized randomly 100 times and picked the one with highest ELBO score (Fig 2c). We also fit with warm start by initiating the fitting process from correct cell clusters grouped by true time and lineages. Both types of initialization were able to converge to the ELBO with true parameters, and random initialization yielded a slightly higher ELBO compared to warm start, with a negligible difference (Fig 2d). For the following analysis, we used the fitting results of random initialization. The posterior distributions correctly recapitulate the time and lineages of cells, with a root mean squared error (RMSE) around 0.05 for the posterior mean of process time in comparison to the true time (Fig 2e). This means that the error in time of a cell is around 5% of the total time length of the trajectory in average. However, the error is not completely uniform: as cells in the second interval are closer to the steady state, they have more spread posteriors and larger errors. Since the posteriors of each cell are accurate, the posteriors averaged over cells also resemble the empirical distribution of the true time (Fig 2f).

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Fig 2. Demonstration of inference on simulation.

a The ground truth trajectory structure. Cells jump to the next state (2) from starting state (1) at τ₀ = 0, and then bifurcate into two lineages with different ending states (3 and 4) at τ₁. The process ends at τ₂ = 1. Out of 200 total genes, 100 genes are non variable with the same distributions along time. b The falsely assumed structure that does not know the first two states are supposed to be merged into one. All genes are assumed to vary along time. c The ELBO scores of 100 random initializations (blue dots) compared to those of warm start (red line). The x axis is the Pearson’s correlation between the mean process time of each random initialization and the true time. d ELBO scores over fitting iterations of both warm start (red line) and the best random initialization (blue line), with the ELBO calculated with true parameters (gray line) as reference. e Heatmaps of inferred posterior distributions. x axis is time grids, and y axis is cells aligned by their true times with true transcription states on the left. The intensity of color indicates the weights of posterior distributions of cells on the grids. Heatmap of cells from τ₀ to τ₁ use a gray color palette. Heatmap of cells from τ₁ to τ₂ use purple color palette. Heatmap of cells from τ₂ to τ₃ of first lineage use blue, and those of second lineage use red. RMSE stands for root mean square errors. f The averaged posterior distributions across cells (dark blue) and true empirical distribution (gray) of process time. g Inferred parameters values compared to true values. For non variable genes, only are identifiable and compared. Error is mean normalized error as described in the text, and the mean is computed across genes. h The confusion matrix for gene selection. i α values of selected genes over states. j Two models and the distribution of the chosen one by (train) ELBO, AIC, BIC and test ELBO, calculated on 20 samples each with a different set of parameter.

https://doi.org/10.1371/journal.pcbi.1012752.g002

The parameters are also recovered accurately (Fig 2g). For variable genes, all parameters are identifiable, while for non-variable genes, only α and the ratio of β to γ are identifiable. We noticed a trend that the absolute errors of α scale with the square root of true values, while for β and γ, they scale with the true values (S6a Fig). This trend also appeared in other simulations (S2b and S2c Fig). Therefore, to calculate the errors of parameters, we divide the absolute error of α by the square root of true values and β , γ by true values, so that errors of different genes are more comparable. We refer to them as normalized errors in the text. The parameter α tends to be estimated with higher accuracy compared to β and γ (Fig 2g). This aligns with intuition because, although parameters are identifiable in our trajectory model, the Evidence Lower Bound (ELBO) is nevertheless insensitive to proportional changes in β and γ, which have minimal impact on the phase portrait in the unspliced and spliced space of each gene. To confirm this, we used the true model to calculate the Fisher information matrix and the smallest eigenvalues of each gene (S6b Fig). The corresponding eigenvectors describe the flattest direction of the ELBO. To validate that the flattest direction primarily lies in the β and γ parameters, we add the corresponding eigenvectors to the true parameters after normalizing it by the square root of eigenvalues, and calculate the new ELBO with the modified parameters. Indeed the resultant changes in ELBO are indeed small and β and γ values were mainly varied (S6c Fig), which confirms that the β and γ are harder to estimate accurately (Fig 2g).

To select genes whose dynamics are well-fit by the model, we evaluate their goodness of fit by comparing the gene-wise likelihood with that of clustering models (Section Gene selection). We adopt this relative likelihood criterion because absolute likelihoods of different genes are not directly comparable and clusters model serves as a natural reference point for comparison to filter genes without continuous dynamics. All 91 genes selected belong to the variable class (Fig 2h). The similar transcription rates of states 1 and 2 of selected genes would also suggest us that those two states could be merged into a single one, if we didn’t know the true structure (Fig 2i). Therefore, we applied model selection methods to compare the ground truth model and the assumed model. We generated another 20 parameter sets, fit under both models, and computed in-sample ELBO scores (ELBO), AIC, BIC, and out-of-sample ELBO scores (test ELBO). The ELBO is always better in model B due to overfitting. All of the last three metrics (AIC, BIC and test ELBO) favor the true model most of the time (90%) (Fig 2j). However, we want to emphasize that model selection worked because simulations were strictly generated under our trajectory model. In reality, if true transcription rates are far away from piece-wise constant functions, the resulting deterministic noise can introduce bias in AIC/BIC and cross-validation, leading them to prefer more complex models [33].

Identifying failure scenarios reveals the fragility of inference

Given the accuracy on perfect data, we sought to characterize the impact of different factors on inference accuracy and identify potential failure scenarios that could lead to unreliable results. We first study the requirements on some obvious factors like the number of cells, the number of genes, and the means of counts. Relatively small numbers of cells and genes appear to be sufficient (S7 and S8 Figs). As the number of cells increases, parameter errors decrease, while time errors remain the same. On the other hand, time errors decrease with an increasing number of genes while parameter errors remain the same. Additionally, counts means must be sufficiently high to enable accurate inference, which necessitates adequate sequencing depth (S9 Fig). We notice that when the trajectory structure becomes more complex, the requirement for count means also increases: Chronocell needs higher means for similar accuracy (S10 Fig). Furthermore, when count means are low, increasing the number of cells can actually compromise the accuracy, underscoring the critical importance of obtaining sufficient counts.

As we use piecewise-constant functions to approximate transcription rates, we also test how this simplification impacts results when transcription rates are, in fact, not piecewise-constant. We generate simulations using piecewise-exponential functions for transcription rates, which ranges from almost linear to almost piecewise as the rate constants increase. We fit the model with Chronocell under the piecewise-constant assumption, and the inference accuracy remains satisfactory when the rate constants are comparable to or larger than the mRNA half-life (S11 Fig).

As unaccounted noise is prevalent in scRNA-seq datasets, we then characterize the effects of noise on inference. The first type of noise is cell-wise read depth (or cell size) that influences all genes similarly. The inference is sensitive to such noise: when read depth variance is not correctly accounted for in the fitting, inference results may be highly inaccurate when its squared coefficient of variation (CV²) exceeds 0.1 (S12a Fig). As the CV² typically observed in real datasets exceeds 0.1 (S17 Fig), an accurate estimate of cellwise read depth is critical. Considering this fact, instead of simply using total counts as normalization factors, we use normalized covariance between genes to decompose extrinsic noise (influencing all genes) and intrinsic noise (gene specific). We estimate the read depth CV² across cells from the normalized covariance between genes. Subsequently based on the read depth CV², we subtract the extrinsic variance caused by the read depth from total variance and identify Poissonian genes whose remaining variances (intrinsic variances) are close to their means (variance  <  1.2 mean). We then estimate cell-wise read depth using the sum of those Poissonian genes. Different sets of genes are used for estimating the CV² of read depth and fitting trajectories. In reality, these read depth estimates correlate well with total counts number for most datasets, with one interesting exception (S17 Fig).

On the other hand, gene specific noise seems to have less impact on inference. We added gene-wise gamma noise in simulation which generates negative binomial distributions in steady states and approximates bursting noise. Parameter errors gradually increase as CV² of Gamma noise increases but remain reasonably small even with a CV² of 1 (S12b Fig). However, while it may not completely undermine the fitting process, it can lead to failures in model selection. When repeating the model selection procedures in Figs S4 and 2 after introducing Gamma noise, the AIC/BIC and cross-validation metrics favor the wrong models that are more complex than the true one (S12c and S12d Fig). This highlights the importance of providing reasonable trajectory structure to the fitting based on prior knowledge, and exploring alternative methods for quality control against false positives caused by clusters.

Another probable factor in real datasets that can lead to suboptimal results is insufficient dynamics. Intuitively, for a dynamical model to be appropriately fit, the data must capture a significant amount of transient dynamics to recapitulate the evolving processes over time, without which the data start to resemble discrete clusters. Such cases can occur in at least two possible scenarios: fast timescales and concentrated sampling distributions, both of which result in clusters in the extreme. Therefore, false positives caused by clusters are naturally included as a component of identifying unreliable results.

The first situation, fast timescales, arises when mRNA half-lives are significantly shorter than the timescale of biological processes, thus cells are mostly near steady state and provide little information about the intermediate dynamics. By setting the length of the time interval to unity and increasing β and γ, which is equivalent to increasing processes timescale with unchanged mRNA half-lives, we found that ideally, the mean value of γ should not fall out of the range of 1 to 10, and the mean ratio of γ to β over genes should not be too small (S13 Fig). Hence, if the average half-life of spliced mRNA is approximately 30 minutes [32], snapshot data sampled from processes involving steps exceeding 10 hours are no longer suitable.

Furthermore, given an appropriate timescale, the sampling distribution still has to cover the region where the transient dynamics occur. Imagine, for example, that all cells are from one unknown time point; there is no way for a trajectory to be inferred. Instead, a cluster should be used to fit the data. Thus, it is crucial to determine the minimum level of uniformity required in the sampling distribution and verify whether this requirement is met. By gradually changing sampling distributions from a uniform distribution to a Gaussian with a random mean, we generated datasets with sampling distributions that exhibit decreasing levels of uniformity, which was quantified using entropy (S14a Fig). As the sampling distribution deviates from uniformity, the errors of process time and parameters quickly increase as expected (S14b Fig). In fact, even when the sampling distribution is not far away from a uniform one with entropy 4, the errors are big enough that the results are completely wrong (S14b Fig). Further, even using the true sampling distribution as a prior for a warm start with correct initialized time cannot mitigate the lack of dynamics: the errors in parameter estimations remained substantial (S14c Fig). Therefore, sufficiently transient dynamics is an inherent requirement for trajectory inference even when perfect prior information is provided.

In summary, a suitable dataset for fitting process time needs to contain enough dynamic information as well as limited noise. This stringent requirement for a successful trajectory inference highlights the need for suitable datasets and careful model assessment. Straightforwardly, we could make a consistency check by verifying if the remaining noise, parameter values, and uniformity of the average posterior distribution fall into the appropriate range determined in simulations. However, when a result is incorrect, it may not necessarily exhibit large unexplained noise, high splicing/degradation rates, or a concentrated cellular distribution over process time, as it could inadvertently fit undesired patterns and output plausible results. Thus, inconsistency is a sufficient but not necessary indicator of unreliable results.

It turns out that the large uncertainty of results can be a better indicator. As both noise and limited dynamics tend to diminish or obscure the difference of scores like ELBO between correct and incorrect outcomes, they introduce multiple comparable maxima and make the fitting results unstable. The resultant large uncertainty can be measured by two different approaches. First, as each random initialization outputs a different ELBO/AIC score and cell ordering, we can inspect the distribution of scores with respect to a summary parameter of cell orderings, which describes the global landscape of the score function. Ideal scenarios usually give one distinct maximum of ELBO (or minimum of AIC) at the correct ordering of cells (correlation around one), while failure scenarios usually lead to multiple comparable maxima of ELBO (or minima of AIC) at different cell orderings beside the correct one (S15b Fig). We use average precision (AP) of the 100 random initializations to quantify the distinctiveness of the correct outcomes and summarize the uncertainty. One random initialization is considered correct if its resultant mean process time correlates well with that of the best one, e.g., a Pearson’s correlation of 0.8 (Section Uncertainty assessment). Ideal cases lead to AP close to one and low AP indicates instability but not vice versa, which means low AP is a sufficient indicator for instability (S15b Fig). Second, the uncertainty can also be revealed by standard bootstrap analysis. We generated 100 sets of resampled data and computed the correlation in process time between the original and each resampled set (Section Uncertainty assessment). In failure scenarios the results of original data and resampled data often differ and correlations are scattered with large variance. On the other hand, in ideal scenarios, process time estimates of resampled results agree with the original one and correlations are centered around one (S15c Fig).

Uncovering distinct underlying cellular distributions in process time

We applied Chronocell to a variety of datasets with different anticipated sampling distributions over time. For those datasets, we estimated read depth as described in Section Read depth estimation (S17 Fig), and then filtered genes for fitting based on their means, variances and unspliced to spliced ratios (Section Real datasets preprocessing). The trajectory structures are determined based on prior knowledge. Random initialization was always performed and its uncertainty was assessed by both AP and bootstrapping. Warm start was applied as well if cell type annotations were available, which were used to initialize the fitting process. The corresponding clusters (Poisson mixtures) model was fit with the same read depth and used for comparison. The genes were selected based on goodness of fit, in the same manner as in the simulation (Section Gene selection). Then, DE genes are selected based on the fold changes in transcription rates across states.

We first tested our method on the T cells of PBMC (Peripheral Blood Mononuclear Cells) dataset from 10x Genomics, which is typically expected to exhibit a few distinct clusters (S18a Fig). Indeed, though the AIC of trajectory model is lower than clusters model likely due to unaccounted noise, the scores of 100 random initializations display multiple minima: multiple different cell orderings result in similar AIC values (S18b Fig). The low average precision (0.28) of random initializations suggests clusters model would be more suitable for PBMC. Serving as a negative control, it confirms our ability to reject unreliable results on real datasets, even when standard model selection methods are invalidated by incorrect modeling of noise.

The second dataset contains a snapshot collection of glutamatergic neuronal lineage cells in a developing human forebrain (Fig 3a) [16], which is presumed to capture cells along a continuous trajectory. However, the unstable result of random initializations indicates its unreliability, as results with reversed directions yield comparable AIC scores (Fig 3b), resembling simulations that lack dynamics information (S15b Fig). This observation is further supported by examining the cellular posterior distributions obtained through warm start with cell type annotations, where the average posterior distribution reveals that cells are concentrated around starting time τ₀, i.e., the initial state, with a low entropy (Fig 3c). Therefore, both the inconsistency indicated by the concentrated posterior and instability indicated by comparable peaks of AIC suggest this is not a suitable dataset for Chronocell and likely lacks enough dynamics.

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Fig 3. Inference results for Forebrain data.

a Schematics of Forebrain data and PCA plot of cells colored by cell type annotations. b The AIC scores of the trajectory and cluster models. The x axis is the mean process time correlations of 100 random initializations (blue dots). The AIC scores of random initializations are compared to those of warm start (red line) as well as 3 Poisson mixture model (yellow line). AP stands for average precision. c Posterior distributions of process time of the trajectory model with warm start. Cells were ordered in y axis by their inferred mean process time and the left bar displays their cell types using the colors in a. The below histogram shows average posterior distribution averaged over cells. The entropy of the average posterior distribution was calculated using its weights on the 100 discretized time grids.

https://doi.org/10.1371/journal.pcbi.1012752.g003

The third dataset contains erythroid lineage cells during mouse gastrulation collected from multiple time points (Fig 4a) [34]. Biological time availability offers a valuable means to evaluate results by comparing the posterior distributions of cells to their corresponding physical times, which is particularly useful since real snapshot datasets lack a definitive ground truth. The AIC scores of random initializations show a clear minimum at a correct direction that align with cell type annotations (S19a Fig), and the average precision of random initializations is 0.79 which is notably higher than those of PBMC and Forebrain datasets (Figs S18b and 3b). The process times of bootstrap samples are reasonably stable as well (S19a Fig). The fit dynamics were able to explain most of the variance, leaving the CV² of unexplained noise of most genes under 1 (S19c Fig). Therefore, the Erythroid dataset appears to be a suitable dataset for our trajectory model. The best result from random initialization is used for analysis (Fig 4b), and cellular posterior distributions confirm that cells have a broad distribution over process time (Fig 4c). Furthermore, the posterior distributions of cells do not differ significantly until E7.5, after which they roughly progress along the process time in sync with real-time progression (Fig 4d). This observation aligns with the understanding that erythroid differentiation in a mouse embryo is believed to begin around embryonic day 7.5 (E7.5) [35]. Although the alignment with physical time is not perfect, it still suggests that our trajectory model can successfully capture the correct trend of process time, even when assuming a uniform sampling distribution for all cells.

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Fig 4. Inference results for Erythroid data.

a Schematics of Erythroid data and PCA plot of cells colored by cell type annotations. b The fitted trajectory structure and inferred mean process time from random initialization indicated in blue on the same PCA plot as in a. c Posterior distributions of process time. Cells were ordered in y axis by their inferred mean process time and the left bar displays their cell types using the colors in a. The below histogram shows average posterior distribution over cells. The entropy of the average posterior distribution was calculated using its weights on the 100 discretized time grids. d Averaged posterior distribution across cells from different experimental time points. n is the number of cells. e α values of 24 selected genes over states. f Phase plots of top 5 DE genes of 24 selected genes. The x axis is the raw unspliced counts and y axis is the raw spliced counts. The blue curve is the fit mean of product Poisson distributions of unspliced and spliced counts over process time, and its darkness corresponds to the value of process time.

https://doi.org/10.1371/journal.pcbi.1012752.g004

After applying our gene selection procedure based on relative likelihood and discarding genes with extreme values, we ended up with 24 (49%) genes (Fig 4e). Subsequently for demonstration, we chose the top 5 DE genes (Cpox, Smim1, Abcg2, Rbpms, Prtg) with the largest fold change of transcription rates, and plotted their phase portraits (Fig 4f). Interestingly, four of them (Cpox [36], Smim1 [37], Abcg2 [38], Rbpms [39]) were reported to be directly relevant to erythroid development, which illustrates that selecting DE genes based on inferred transcription rates is a straightforward and effective approach.

Based on the results from datasets ranging from clusters to trajectories, it becomes evident that real datasets can display a spectrum of continuity in cellular process time distribution. Therefore, it is critical to assess the quality of inference and verify the requirements for reliable results are indeed met.

Degradation rates estimates agree with metabolic labeling data

In addition to validating the process time, we also sought to validate the inferred parameters, which motivated us to use a metabolic labeling dataset from scEU-seq, comprising human retinal pigment epithelial (RPE1) cells undergoing cell cycles (Fig 5a) [40]. Metabolic labeling of new mRNA allows for the estimation of degradation rates from cells with varying labeling times, enabling a comparison with our parameter estimations.

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Fig 5. Inference results for cell cycle data.

a Schematics of Cell cycle data and scatter plot of the Geminin-GFP and Cdt1-RFP of RPE1 cells colored by cell type annotations. b The fit trajectory structure and inferred mean process time from random initialization indicated in blue on the same scatter plot as in a. c Posterior distributions of process time. Cells were ordered in y axis by their inferred mean process time and the left bar displays their cell types using the colors in a. The below histogram shows average posterior distribution averaged over cells. The entropy of the average posterior distribution was calculated using its weights on the 100 discretized time grids. d α values of 84 selected genes over states. e Comparison of γ estimates for 84 selected genes with estimates derived from metabolic RNA labeling data. CCC stands for concordance correlation coefficient. n is the number of genes for which estimates are available in each respective paper.

https://doi.org/10.1371/journal.pcbi.1012752.g005

In the trajectory structure, we specified that the initial and final states were identical. Given our assumption that the initial state is at a steady state, capturing the cyclic nature of the cell cycle poses an additional requirement that cells must also approach a steady state in the last interval as well, which happened to be reasonably satisfied by our results. In line with the expectation that cell cycle is a highly dynamic process, the random initialization provides a relatively stable result: the AIC exhibits a single minimum that aligns with the direction of cell cycle progression (S20a Fig), and bootstrap samples mostly align with the result of the original one (S20b Fig). Starting from the result of random initialization, a desynchronized model was fit to enhance the accuracy of parameter estimations (S20c Fig). The resulting fit maintains alignment with the cell cycle progression (Fig 5b), and the CV² of the unexplained noise mostly remains below 1 for most of the genes (S20c Fig). The RPE1 dataset was metabolically labeled for different lengths of time but posterior distributions of RPE1 cells with different labeling times do not show significant differences, as a negative control in contrast to the erythroid data (S20e Fig).

The posterior distributions successfully capture the cyclic nature of the data. The cell types ordered by mean process time exhibit a cyclic pattern (Fig 5c); the phase plots of marker genes confirm the cycling dynamics of fit means of unspliced and spliced counts (S20f Fig); the starting and ending values of fit means of most of genes match with each other (S20g Fig). As both axes are cyclic, it is possible to connect the opposite edges and transform the two-dimensional cell-by-time grids posterior distributions into a torus on the surface of which the posterior distributions look like a circle (Fig 5c). Based on cell type annotation, total RNA counts number (S20h Fig), and marker genes dynamics (S20e Fig), we can roughly assign the three intervals to G2/M, G1/S and S/G2 phase, and mitosis happens shortly after τ₀.

After selecting genes by relative likelihood and discarding genes with extreme values, we ended up with 84 (46%) genes (Fig 5d). We compared the degradation rates of 84 selected genes to those derived by scEU-seq [40] and TimeLapse-seq [41]. For scEU-seq, as we neglected the changes in degradation rate along cell cycle, we used the averaged degradation rates in S10c Fig of Battich et al. [40]. We observed moderate correlations between our degradation rate estimates and the respective estimates from Battich et al. and Schofield et al. (Fig 5e), and these correlations exhibited a magnitude similar to the correlation between the Battich et al. and Schofield et al. estimates (S20i Fig), which suggests that the parameters inferred by Chronocell indeed possess a meaningful biophysical interpretation.