2.1 The NB distribution can provide a good approximation to the mRNA distribution of the telegraph model even when transcriptional bursting is absent

Consider an idealized scenario where each cell is identical to each other, i.e., for a given gene, there is no cell-to-cell variation in the parameters of the telegraph model (Fig 1a). This implies that all cells are of the same type. Furthermore, assume that all transcripts from each cell can be detected. In this case, the distribution of mRNA counts is given by the steady-state solution of the chemical master equation of the telegraph model:

(1)

where n represents the mRNA count, denotes the Pochhammer symbol, and is the Kummer confluent hypergeometric function [34]. Note that is the gene activation rate, is the gene deactivation rate, ρ is the transcription rate; these rates are non-dimensional because they are normalized by the degradation rate (this convention will be used throughout the paper). Eq (1) is equivalent to the compound Beta-Poisson distribution because the transcript numbers are distributed as follows:

(2)

As mentioned previously, it is well known that if gene expression is bursty then Eq (1) is well approximated by an NB distribution, or its continuous counterpart, the Gamma distribution [36,49]; related derivations for the distribution of protein numbers can be found in Refs. [53,54]. Specifically, transcriptional bursting occurs in the limit taken such that remains constant [49,55]. A simple way to see how this limit leads to an NB distribution makes use of the Beta-Poisson formulation of the telegraph model. From Eq (2), it follows that

(3)

where in the second step we used the standard statistical result: .

It is presently unclear if the converse is true: Does an excellent NB fit to the mRNA count distribution of the telegraph model imply transcriptional bursting?

To answer this question, we proceed as follows. It is straightforward to show using Eq (1) that the mean and variance of mRNA counts for the telegraph model are given by

(4)

where is the averaging operator. In contrast, the mean and variance of an NB distribution are

(5)

By equating and in Eqs (4) and (5) and solving for r and p, we can construct an NB distribution that has the same first and second moments as the mRNA count distribution of the telegraph model:

(6)

This distribution is hereafter referred to as the effective NB distribution denoted as . Of course, this distribution and the telegraph model potentially can differ in their third and higher moments, and hence there is no guarantee that the shapes of the two distributions will generally be similar. We also note that while maximum likelihood estimation or other methods can be used to obtain an effective negative binomial distribution, they often lack closed-form solutions for parameter estimates, which can hinder further quantitative analysis. In contrast, moment matching provides analytical expressions, facilitating deeper insight.

In Fig 2, we compare the distributions given by Eq (1) and Eq (6) for four distinct parameter sets. In each case, the effective NB distribution is practically perfectly aligned with the corresponding telegraph distribution. However, note that only one of these four cases (Fig 2a) corresponds to the classical case of . The three counterexamples (Fig 2b–2d) clearly show that excellent fitting of the telegraph model to an NB distribution does NOT necessarily indicate transcriptional bursting.

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Fig 2. Comparison of the steady-state mRNA distribution of the telegraph model (Eq (1) denoted as ) with the effective NB distribution (Eq (6) denoted as ) for the case of perfectly identical cells (parameters do not vary from cell-to-cell).

The two parameters of the effective NB distribution are chosen so that its first and second moments of mRNA counts exactly agree with those of the telegraph model (the values of the two parameters, and , are stated to 2 decimal places in the figure). In all four parameter cases, the effective NB distribution exceptionally well fits the corresponding telegraph distribution, and yet only in case (a) we have (the classical case of transcriptional bursting). This demonstrates that a good fit of an NB distribution to the telegraph model distribution does not imply the presence of transcriptional bursting.

https://doi.org/10.1371/journal.pcbi.1014014.g002

2.2 An alternative set of conditions under which the NB distribution provides a good approximation to the telegraph model distribution

We start by reparameterizing the gene switching rates as

(7)

where represents the timescale of the transition rates relative to the rate of mRNA degradation and is the fraction of time spent in the active state, which is given by

(8)

Furthermore, for convenience, we denote the telegraph model distribution, Eq (1), by . Given these definitions, we state the following theorem:

Theorem 1: In the limit , the telegraph model distribution and the effective negative binomial distribution (Eq (6)) converge to the same distribution, both with a convergence rate of .

The key idea behind proving this theorem is to show that the Beta and Gamma distributions converge to the same limiting distribution – specifically, a normal distribution – as . This then implies a convergence of the Beta-Poisson (telegraph model) distribution to a Gamma-Poisson (NB) distribution. This convergence is not immediately obvious because the Beta distribution is defined on while the Gamma distribution is defined on . Details of the proof can be found in Methods Sect 4.1.

This proof makes more precise the statement in the caption of SI Fig 4 of Ref [56] which states that the mRNA count distribution of the telegraph model is bimodal (and therefore not similar to an NB distribution) when and ; it is unimodal and resembles an NB distribution when and . In particular, Theorem 1 shows that and are not sufficient by itself to guarantee an excellent fit of the NB distribution to that of the telegraph model. We also note that the conditions identified by the theorem are different from those given by the classical result that requires . In fact, while the classical results cannot explain the excellent fits shown in Fig 2, these can be explained by Theorem 1, because in each of these cases is sufficiently large.

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Fig 3. The mRNA distribution of the telegraph model converges to that of effective NB distribution as the sum of the gene-state switching rates relative to the mRNA degradation rate () increases.

In the dashed box, we show that the effective NB distribution (blue solid line) exhibits a lower KL divergence to the telegraph model distribution compared to the Poisson approximation (orange dashed line) as grows. The distributions of the telegraph model (green dots), effective NB (blue solid lines), and Poisson (orange dashed lines) are shown for Points A-E, as indicated in the KL divergence plot. The other parameters are fixed at and . Point A: , Point B: , Point C: , Point D: , Point E: .

https://doi.org/10.1371/journal.pcbi.1014014.g003

In the top left panel of Fig 3, we use the Kullback–Leibler (KL) divergence, defined as to quantify the proximity between the effective negative binomial distribution () and the telegraph model distribution (). For fixed values of ρ and , increasing results in a rapid decrease in the KL divergence, which agrees with Theorem 1. In the same figure, we also show the KL divergence between the Poisson distribution (with its parameter set to the mean of the telegraph model distribution) and the telegraph model distribution. It can be deduced that the fit of the Poisson distribution to the telegraph model distribution also becomes more accurate as increases, although it is always a poorer fit than the effective NB distribution. Of course, in the limit of large , the KL divergence is very small for both effective NB and Poisson distributions and hence, practically speaking, in this limit both will provide an excellent approximation to the telegraph model distribution. These observations are further supported by comparing in Fig 3 the mRNA count distributions predicted by the telegraph model with the effective NB distribution and the Poisson distribution for 5 different values of (these correspond to the points A-E in the plot of the KL divergence versus in the top left corner of Fig 3).

2.3 The NB distribution provides an optimal approximation to the telegraph model distribution for an intermediate range of the sum of the gene switching rates

The distribution comparison in Fig 3 suggests that the effective NB distribution is an optimal fit to the telegraph model distribution in the intermediate range of . This is because: (i) for small , Theorem 1 does not hold. As well, in this case, it is already known that small and imply bimodal distributions of the telegraph model, which clearly cannot be well fitted by the (unimodal) effective NB distribution [56,57]; (ii) for large , the Poisson distribution practically provides an equally good fit as the effective NB distribution, but is preferable by the principle of Occam’s razor because it has one less parameter than the effective NB approximation.

To make this observation more rigorous, we will use a model selection approach. An often used method to select which of two models provides a best fit to the data involves (i) maximizing the likelihood function of each model on the sample data; (ii) evaluating the Bayesian Information Criterion (BIC) for each model which is a function of both the maximum value of the likelihood and the number of parameters; (iii) the optimal model is selected to be the one with the smallest BIC score.

In particular, the BIC is defined as

(9)

where represents the set of count data for , with being the size of the dataset (number of cells). The proposed model is denoted by , its number of parameters by and the distribution that defines the model (its likelihood function) by where θ represents the model parameters. The parameter is the maximum likelihood estimate (MLE) obtained by maximizing the log-likelihood function

In our case, given a parameter set , the count data is simulated by randomly sampling the telegraph model distribution times. The difficulty with this approach is that for each parameter set, there is an infinite number of datasets that can be simulated, and in principle, the algorithm can select a different best model for each dataset. Rather than relying on individual samples, our goal is to associate a unique best model with each set of underlying parameters. The most straightforward way to achieve this is by averaging the BIC over an infinite number of samples, i.e., . However, this approach is computationally intensive, as model parameters must be re-estimated for each sample.

To address this challenge, we introduce a new measure—the approximate expected Bayesian information criterion (aeBIC)—as an efficient estimator of the expected BIC across repeated samples. The aeBIC is defined as

(10)

Here, represents the cross-entropy between two distributions (representing the model and the ground truth), defined as

(11)

where and are the probabilities of observing n molecules in distributions (the ground-truth distribution which for us is the telegraph model distribution) and (the proposed model which can be telegraph, NB or Poisson with parameters θ), respectively. Note that the subscript MLE denotes that the kinetic parameters of the proposed distribution are estimated via maximum likelihood. Specifically, represents the model whose parameters θ minimize Eq (11). If , the cross entropy reduces to the information entropy . Additionally, the cross entropy can be decomposed as [58]

(12)

where is the KL divergence between and . This decomposition implies that the closer the proposed distribution is to the ground-truth distribution , the smaller the KL divergence becomes, and consequently, the cross entropy decreases as well.

In S1 Text, Sect 4.2 we show that the expectation of the BIC over an infinite number of independent samples (each of size ) is related to the aeBIC by:

(13)

This shows that aeBIC serves as an upper bound for the expectation of the BIC. It can also be shown that the equality in Eq (13) is achieved as .

However, in practice, we find that the difference between and is negligible. To demonstrate this, for a fixed set of parameters , we randomly sampled counts from the telegraph model distribution (using its Beta-Poisson formulation in Eq (2)) and repeated this step times. For each trial, we fitted the sampled counts to both Poisson and NB distributions using MLE and computed the BIC for both distributions. The BIC values were then averaged across all datasets to estimate the expectation of the BIC. Additionally, we computed the aeBIC using Eq (10) for both distributions for each parameter set of the telegraph model distribution. The relative error of aeBIC compared to the expectation of BIC was calculated for each set of parameters. This comparison process is illustrated by a cartoon in Fig 4a and the results for numerous parameter sets that cover the range , and are shown in Fig 4b and Fig Aa,b in S1 Text (these parameter sets correspond to a mean number of transcripts that ranges between 0 and 15). Note that, independent of the type of the fitted distribution, the relative error decreases with increasing sample size . Notably, even for small sample sizes, the magnitude of the relative error is very small, and thereby for all intents and purposes, we can equate the aeBIC with the average of the BIC computed over many independent samples. This is very convenient because it is far faster to compute the aeBIC than the BIC since, for the former, the maximum likelihood estimation only needs to be done once using Eq. (11).

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Fig 4. (a) Cartoon illustrating a computational approach to compare the aeBIC (top) with E[BIC] — the expected value of the BIC (bottom).

The aeBIC utilizes a single score to select the best model distribution (telegraph, NB or Poisson) given that the ground-truth mRNA distribution is that of the telegraph model. The BIC method assigns a score to each different sample of simulated data from the telegraph model and then all these scores are averaged leading to E[BIC]. (b) The relative error (RE) of aeBIC compared to E[BIC] for two distributions (Poisson and NB) as a function of sample size for 10 parameter sets (see Table A in S1 Text for the values of and ; ρ is fixed to 15). Error bars show the standard error of the mean. (c) Phase diagram showing the regions of parameter space where the telegraph, NB and Poisson distributions are selected as optimal by the aeBIC, given that the ground-truth mRNA distribution is that of the telegraph model. Here is the sample size, is the sum of gene-state switching rates normalised by the degradation rate of mRNA, and is the fraction time spent in the active state. The fraction of the total parameter space occupied by the region where the NB distribution is optimally selected is shown on the plots. Note that the transcription rate is fixed to which implies that the maximum mean number of transcripts in the phase plots is 15.

https://doi.org/10.1371/journal.pcbi.1014014.g004

Next, we use Eqs (10)–(11) to compute the aeBIC and identify the best-fitted model (the one with the smallest aeBIC) among Poisson, NB and telegraph model distributions across the parameter space (while keeping the transcription rate ρ constant), given that the ground-truth data is generated by the telegraph model. Note that this model selection can be performed for different sample sizes because the aeBIC is a function of . Note also that the mean mRNA is proportional to since the mean mRNA of the telegraph model is (Eq (4)) and ρ is fixed in our analysis. The results are shown in Fig 4c — the model associated with the smallest aeBIC score corresponds to the model most frequently selected by the BIC calculated over many independent samples of the same size, thus providing a further test of the validity of the aeBIC approach (Table B in S1 Text). In particular, aeBIC preferentially selects the telegraph model distribution for small , while the Poisson distribution is preferred for large . This agrees with the distribution comparison in Fig 3 which suggested that the effective NB distribution is an optimal fit to the telegraph model distribution in an intermediate range of . The NB distribution is more likely to be selected as decreases, which is consistent with the fact that the telegraph model distribution converges to an NB distribution when the gene spends most of its time in the inactive state ( is small). Interestingly, the shape and area of the crescent-shaped region where the NB distribution is an optimal model remains largely unchanged as the sample size increases, but its position shifts horizontally. In particular, the fraction of space where the NB distribution is the optimal model increases monotonically from to as the sample size increases over four orders of magnitude, from to cells (Fig G in S1 Text).

Note that an alternative (and faster) way to obtain the phase space plots in Fig 4c is to calculate the aeBIC with the value of the cross-entropy computed using moment matching between the proposed and the ground-truth distribution (instead of MLE). Specifically, the aeBIC using moment-matching is defined as

(14)

where is determined by evaluating Eq (11) with parameters θ determined by moment-matching between the proposed distribution () and the ground-truth distribution (). In Fig B in S1 Text top left corner, we demonstrate that these two methods lead to very similar minimum values for the cross entropy and hence there is no appreciable difference on the model selection phase plots in Fig 4c. On a MacBook Air with an Apple M2 chip and 16 GB memory, generating a single phase space plot with this method ( pairs of ) takes about 68 seconds (CPU time) and uses only 3.6 MB of memory, demonstrating its computational efficiency.

2.4 Cell-to-cell variability in transcript capture probability significantly affects the region of parameter space where the NB distribution is optimally selected

Thus far, we have assumed that scRNA-seq data can be directly explained by the telegraph model. In practice, however, not all transcripts are captured. During the reverse transcription stage, each transcript is captured with probability . UMIs are attached to molecules before PCR, enabling PCR duplicates to be collapsed during downstream processing and thereby correcting amplification bias. Finally, during sequencing, the sequencing depth determines the detection probability () of each UMI-labeled transcript. The observed UMI counts for a gene therefore represent the number of transcripts successfully captured, labeled, and detected. Collectively, the overall probability linking the true transcript count to the observed count is denoted as (including the effect of and ). Therefore, it is necessary to account for technical noise in the data. A simple way to do this is to assume that the number of zeros in the data is artificially high and to generate simulated scRNA-seq data using a zero-inflated telegraph model (a distribution that is a sum of a Dirac-delta function and the telegraph model distribution). While this approach or variants of it are widespread [14,18,19,59] and attempts have been made to explain zero-inflation using mechanistic models [60], it is not ideal because (i) the downsampling of transcripts due to imperfect capture results in an increase of not only zeros but also 1’s, 2’s, etc; (ii) there is evidence that scRNA-seq data with UMIs barely suffer from zero-inflation [6,61]. A more principled and increasingly used model to explain the effect of imperfect capture is the binomial capture model [10,38,62]. This model assumes that each transcript is captured and observed with a probability (Fig 5a). For current standard droplet-based sequencing technologies, the capture probability typically ranges between 0.05 and 0.3, depending on the sequencing technique used [11,63–65]. Extrinsic noise, such as variability in transcription rates, is a major contributor to cell-to-cell heterogeneity in gene expression [66,67]. However, in most cases, variability in capture probability and transcription rate cannot be disentangled mathematically (see S1 Text, Sect 4.3), unless additional experimental controls, such as spike-ins, are employed [8]. Therefore, in the following, we focus on variability in transcript capture probability.

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Fig 5. (a) Schematic illustrating the binomial capture model for scRNA-seq.

Transcripts in each cell are captured with some probability . This causes a downsampling of the distribution of mRNA counts. (b) Phase diagram showing the regions of parameter space where the telegraph, NB and Poisson distributions are selected as the optimal ones by the aeBIC, given that the ground-truth mRNA distribution is that of the telegraph model. The phase diagrams are shown for three different values of (values stated next to the plots). Here is the sample size, is the sum of gene-state switching rates normalised by the degradation rate of mRNA, and is the fraction time spent in the active state. The fraction of the total parameter space occupied by the region where the NB distribution is optimally selected is shown on the plots. Note that the transcription rate is fixed to which implies that the maximum mean number of transcripts in the phase plots is and for the phase plots in rows 1, 2 and 3, respectively. These phase plots do not appreciably change if aeBIC is determined using moment-matching instead of MLE (Fig B Text).

https://doi.org/10.1371/journal.pcbi.1014014.g005

For simplicity, first we consider the case where the capture probability does not vary from cell to cell and from gene to gene, i.e., the distribution of the capture probability is a Dirac-delta function, . We investigated the impact of on model selection outcomes (using aeBIC) and the results are shown in Fig 5b. Note that in this case in Eq (10) is the telegraph model distribution Eq (1) with ρ rescaled to [37,38]. Comparing Fig 4c with Fig 5b, we see that for a fixed sample size, the fraction of parameter space where the NB distribution is preferentially selected is almost constant for capture probabilities in the range . The accuracy of scRNA-seq techniques is constantly improving, with commonly used platforms such as 10x having a capture probability that exceeds [65]. Hence, we can conclude that if the capture probability is roughly the same from one cell to another, then the universality of the NB distribution in scRNA-seq datasets has little to do with the low capture efficiency.

Of course, for some scRNA-seq methods the capture probability may be very low and then in that case the technical noise will be so large that there will be a significant impact on model selection. For example, in the case of and sample size in Fig 5b, the Poisson distribution emerges as the optimal model across most of the parameter space, with the NB distribution being selected only in a limited region. The telegraph model, however, is never identified as the optimal choice under these conditions. This is expected because the bimodal character of the telegraph model distribution at small becomes less obvious at lower capture probabilities — downsampling causes the two peaks of the distribution to become closer together or even to merge and hence the best-fit distributions are exclusively unimodal (Poisson or NB distributions).

The wide distribution of the total counts (from all genes) per cell in typical scRNA-seq datasets (see for, e.g., Fig 1A of [17]) strongly suggests that the capture probability varies considerably across cells (Fig 6a). To address this, we simulate scRNA-seq data by sampling from the underlying mRNA distribution given by Eq (2) with ρ rescaled to , and that varies between cells according to three different distributions (Dirac-delta function and two different Beta distributions), all of which have mean but with different coefficients of variation (CV = 0 for the Dirac-delta function, and and for the two different Beta distributions). See Fig 6b for a plot of the three distributions. Note that the Beta distribution was chosen because it samples numbers in the range , a necessity given is a probability. The aeBIC was used to select between the standard telegraph model distribution (no assumption of variability of rates or capture probability between cells), NB and Poisson models with the ground-truth (measured) mRNA distribution given by that of the telegraph model with distributed according to a Beta distribution (S1 Text, Sect 4.4). Note that the aeBIC remains an accurate predictor of the expectation of the BIC computed from MLE and hence suitable for model selection even when the transcript numbers are heavily influenced by technical noise (Fig Ac in S1 Text). The phase plots generated by aeBIC-based model selection are shown in Fig 6c. We find that for fixed sample size , an increase in the CV of the Beta distribution, leads to very small changes in the fraction of parameter space where the telegraph model distribution is selected, but the fraction of parameter space where the NB or Poisson distributions are selected changes significantly. In particular, an increase in the cell-to-cell variability in the capture probability tends to favor the selection of the NB distribution over the Poisson distribution. This is because variability in the capture probability increases the Fano factor (variance divided by the mean) of mRNA distributions (S1 Text, Sect 4.5) — Fano factors larger than 1 can be captured by an NB distribution but the Poisson distribution has a fixed Fano factor of 1. For example, for a sample size of cells, as the CV of the Beta distribution of is increased from 0 to (keeping the mean constant at 0.3), the fraction of space where the telegraph model distribution is optimal decreases slightly from to , where the NB distribution is optimal increases from to , and where the Poisson distribution is optimal decreases from to .

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Fig 6. (a) Schematic illustrating the binomial capture model for scRNA-seq with a probability of mRNA capture, , that varies between cells according to some distribution.

(b) We consider three different distributions all with mean but with varying coefficient of variation (CV): (i) Dirac() with CV = 0; (ii) Beta() with CV = 0.11; (iii) Beta() with CV = 0.21. (c) Phase diagram showing the regions of parameter space where the telegraph, NB and Poisson distributions are selected as the optimal ones by the aeBIC, given that the ground-truth mRNA distribution is that of the telegraph model with effective transcription rate where is sampled from the 3 distributions mentioned above. Here is the sample size, is the sum of gene-state switching rates normalised by the degradation rate of mRNA, and is the fraction time spent in the active state. The fraction of the total parameter space occupied by the region where the NB distribution is optimally selected is shown on the plots. Note that the transcription rate is fixed to which implies that the maximum mean number of transcripts in the phase plots is .

https://doi.org/10.1371/journal.pcbi.1014014.g006

2.5 In NB-optimal parameter space, burst parameter estimation accuracy is typically low, but ranking genes by burst frequency remains reliable

We also seek to understand if any biologically relevant interpretation can be imparted to the two parameters of the NB distribution fitted to the scRNA-seq data. Consider the ideal scenario in which we can perfectly correct for technical noise. In regions of parameter space where the NB distribution is optimally selected, can we obtain accurate estimates of the burst size and burst frequency from the two parameters of the best fitted NB distribution?

The aeBIC Eq (10) used for model selection depends on the choice of the ground-truth model describing the distribution of the data () and the model(s) () to be fitted to this data. In our case, is the telegraph model with effective transcription rate , and distributed according to a Beta distribution, . Previously, we assumed that the models are given by the standard telegraph, NB and Poisson distributions. These models do not have any information on the distribution of and hence this is the case where model selection using aeBIC is done without correcting for technical noise. In contrast, a perfect correction for technical noise is possible if instead the models are corrected for technical noise, i.e., the mRNA distributions of the telegraph model, NB and Poisson models are integrated over the distribution (this of course assumes perfect knowledge of the parameters a and b characterizing the technical noise). For a derivation of these distributions, see Sect 4.4. The differences between the conventional models and those with sampled from a distribution are illustrated in Fig 7a.

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Fig 7. (a) Illustration of the differences between the standard and technical-noise-corrected models; (b) Phase diagrams produced by aeBIC model selection based on standard or corrected models.

For both, the ground-truth model for observed data is the telegraph model with distributed according to the distribution which has mean and CV=. The transcription rate ρ is fixed to 15; the maximum mean number of transcripts is . The labels “Tele”, “NB” and “Pois” denote the regions selected using the aeBIC procedure with corrected models. The dashed lines demarcate the same regions but using the aeBIC procedure with standard models. The “Pois” area is divided into a white part (where both aeBIC procedures select the Poisson distribution) and a grey part (where the aeBIC with standard models selects the NB distribution while the aeBIC with corrected models selects the Poisson distribution). The heatmap shows the magnitude of the relative errors in the estimated burst frequency () and burst size () in the NB-optimal region (using the aeBIC with corrected models). The errors are computed using Eq (17) — note that this approach assumes full knowledge of the distribution of probability capture, an ideal case. In the plots, denotes sample size, is the sum of gene-state switching rates normalised by the degradation rate of mRNA, and is the fraction time spent in the active state.

https://doi.org/10.1371/journal.pcbi.1014014.g007

In Fig 7b we contrast model selection using conventional and technical-noise-corrected models for the case where is distributed according to a distribution which has a mean of and CV. We find that the regions of parameter space where the conventional and corrected telegraph models are selected are very similar. However, the region where the corrected NB distribution is selected (rainbow-colored region) is smaller than the region where the conventional NB distribution is selected — in fact, there is a region (shown in grey) where the conventional NB distribution and the corrected Poisson distribution are selected, i.e., in this region the apparent NB character of the mRNA count distribution is purely due to technical noise. The discrepancies between the two approaches are negligible for small sample sizes () but significant for large samples .

Next, in the parameter space region where the corrected NB distribution is selected as the optimal model, we estimate the two parameters of the corrected NB distribution by the method of moments. We equate the first two moments of the latter distribution with those of the model simulating the data, i.e., the corrected telegraph model. This mimics the fitting of the corrected NB distribution to the data using the method of maximum likelihood in the limit of large sample sizes. In Sect 4.5 we show that this procedure leads to where and are exactly as given by the effective NB distribution in Eq (6). Clearly, because and do not depend solely on the true burst frequency and burst size of the telegraph model [68], the latter two parameters cannot be directly estimated by moment matching.

This issue can be circumvented by interpreting the best-fitting corrected NB distribution as equal to the steady-state NB distribution solution of the bursty gene expression circuit:

(15)

where s is a geometrically distributed random burst size with mean and the burst frequency is [69]. This model has experimental support [26] and its steady-state distribution is exactly the same as that of the telegraph model in the limit of transcriptional bursting, i.e., when the inactivation rate is much larger than the activation rate.

Hence the described inference procedure leads to estimated burst frequency and size that are given by

(16)

Since we know that the true burst frequency of the telegraph model is and the true burst size is , it follows that the relative errors are given by

(17)

In Fig 7b we show a heatmap of the relative errors in the NB-optimal region (using the aeBIC with technical-noise-corrected models). The magnitude of these errors vary significantly and are smallest when is small, i.e., when and . Hence, this implies that even though the NB distribution may be optimally selected, nevertheless this does not guarantee the accuracy of the estimated burst parameters. Indeed, the lack of accuracy is not very surprising given that the NB distribution of the reaction scheme in 15 is only a good approximation to the telegraph model distribution when [49] and that we have previously established that it is possible to have excellent NB fits even when this inequality does not hold (Fig 2). Note that the burst size can generally be estimated more reliably than the burst frequency — this is also apparent from Eq (17) which implies and .

Next, we sought to understand whether reliable information of some type can be extracted from the burst parameters, even though their absolute values are generally inaccurate. In particular, we seek to understand whether ranking of genes by their burst parameter values can be accurate since this is based on relative rather than absolute information. To answer this question, we used the following protocol:

1. Randomly sample two pairs of parameter sets and , each associated with a different gene, from the parameter space region where the NB distribution (corrected for technical noise) is selected as the optimal model for distributed according to the Dirac , and distributions. For a sample size of , these regions are shown in purple in Fig C in S1 Text. Note that the transcription rate ρ is fixed to 15 in all cases.
2. Calculate the true burst frequency and size for each gene. For the first gene, we denote the true burst frequency and burst size as and , respectively; for the second gene, and , respectively. These relationships follow from Eq (7).
3. For each gene, use the method of moments to fit an NB distribution integrated over the distribution of to the distribution of the data, i.e., telegraph model distribution integrated over the distribution of . This leads to the estimated burst frequency-size pairs for each gene: and which are given by Eq (16).
4. Compute the ground-truth ratios of burst sizes for the pair of genes as . Note that by placing the smaller burst size in the numerator, we guarantee . The ground-truth ratio for the burst frequencies can be computed similarly.
5. Compute the estimated ratio of burst sizes for the pair of genes , preserving the original pair order used for the ground-truth ratio of burst sizes. That is, if then ; if then . The estimated ratio for the burst frequencies can be computed similarly.

In Fig 8a we show a plot of versus . Each point in this scatter plot corresponds to a randomly selected pair of genes. Points are classified by their color: (i) blue if indicates a reversal of pair order which implies that burst parameter inference fails to preserve the relative magnitude of the burst parameters; (ii) orange if and indicates that the order of the estimated burst parameters is the same as that of the ground truth parameters but that the estimation leads to an amplification of the difference between the burst parameters of the two genes; (iii) green if and indicates that the order of the estimated burst parameters is the same as that of the ground truth parameters but that the estimation leads to a reduction of the difference between the parameters of the two genes.

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Fig 8. (a) Scatter plot of the estimated ratio of burst parameters of a pair of genes and of the ground-truth ratio of the burst parameters of the same pair of genes.

All parameter sets are sampled from the region of parameter space where the technical-noise-corrected NB distribution is optimally selected using the aeBIC approach (purple regions in Fig C in S1 Text). Points marked as blue are those pairs of genes for which the estimation led to an incorrect ordering of genes by the size of the burst parameter. The order was correctly inferred for gene pairs corresponding to orange and green points. Perfect ratio estimation is shown by the solid red line; gene pairs corresponding to orange (green) points overestimate (underestimate) the distance between the burst parameters of the gene pairs. (b) Distributions of the relative errors in burst frequency and burst size for those pairs of genes for which the order was correctly inferred (orange and green points) in (a). Note that the inference and model selection approach here assumes full knowledge of the distribution of probability capture, an ideal case.

https://doi.org/10.1371/journal.pcbi.1014014.g008

The results show that in many instances the ranking of a pair of genes by the size of their burst frequency is correctly estimated — these are the orange and green points in Fig 8a which imply gene ranking accuracy by burst frequency in of cases. However, the ranking of genes by burst size is significantly less accurate: it is correct in merely of all cases, though it is better than expected by chance. In Fig 8b we show the relative errors of the burst parameters for the pairs of genes that were correctly ranked (orange and green points in (a)). This verifies that generally relative errors and ranking accuracy are not related to each other since the ranking can be accurate and yet the relative errors can be large.

The main limitations of our approach are that we assume: (i) perfect knowledge of the distribution of capture probability; (ii) the information in the first two moments of the transcript count distribution is enough to accurately infer the burst parameters; (iii) we exclusively perform parameter inference using the technical-noise corrected NB model. To overcome these limitations, in Sect 4.6, we describe an approach which simulates the workflow of an actual scRNA-seq experiment: (i) synthetic count data is generated for thousands of genes in cells, which includes noise from both biological and technical sources; technical noise is modeled by a Beta distributed transcript capture probability; (ii) a maximum likelihood-based approach is used to infer the burst frequency and size, and to select between the technical-noise corrected telegraph, NB and Poisson models derived in Sect 4.4. Note that this approach relies on using the measured total counts from all genes in a cell as a proxy for the (unknown) capture probability for that cell; the approach also does not simply use the first two moments of the observed distribution of counts, but the full distribution information and hence overcomes some of the known limitations of moment-matching approaches [70]; (iii) for those genes for which the technical-noise corrected NB model is selected as optimal by the BIC, the burst frequency and size are estimated separately from the technical-noise corrected telegraph and NB models. These are compared to the ground-truth values to calculate the relative errors. In addition, the percentage of pairs of genes that are correctly ranked according to the magnitude of their burst frequency and size, are calculated for each model. The whole methodology is illustrated in Fig Da in S1 Text. We considered three variations of this procedure, according to the generation of synthetic data in step (i):(a) cells, and the distribution of is fixed to ; (b) cells, and the distribution of is fixed to ; (c) cells, and the distribution of is fixed to .

The results for (a) are shown in Fig D in S1 Text and for (b) and (c) are shown in Fig E in S1 Text. We find that for each fixed value of , the relative errors of the burst frequency and size increase with the fraction of time that the gene spends in the active state . This agrees with the results shown in Fig 7. Interestingly, even though the genes analyzed are those for which the technical-noise corrected NB model is selected as the optimal model by BIC, the parameter estimates are more accurate using the technical-noise corrected telegraph model (compare blue and green bars in Fig Db, Fig Ea and Fig Ec in S1 Text). In addition, a high percentage of pairs of genes are correctly ranked according to the magnitude of their burst frequency, both by the technical noise-corrected NB and telegraph models; however, ranking by the burst size was much less reliable (see the piecharts in Fig Dc, Fig Eb and Fig Ed in S1 Text). The rankings also agree with the results shown in Fig 8. Hence, overall, the results obtained from the MLE-based inference and BIC verify the accuracy of those previously obtained using moment-matching and aeBIC.

2.6 Analysis of experimental data reveals a substantial fraction of genes best fitted by NB but not transcriptionally bursty

Finally, we sought to determine how many genes that are best fitted by the negative binomial (NB) distribution are truly transcriptionally bursty, and whether this fraction is significant. Prior review papers, such as [71], provide ranges of switching rates ( and ) that partially address this question. To answer it more comprehensively, we analyzed scRNA-seq data from mouse fibroblasts [72], preprocessing the dataset to ensure reliability (see S1 Text, Sect 4.7). After preprocessing, 670 cells and 21,684 genes were retained.

We fitted the three -modified models in Fig 7a to this dataset using MLE. The distribution of was estimated via kernel density estimation, and the computation of count distributions over followed the procedure described in Sect 4.8, consistent with the approach outlined in Supplementary Information Section VI of Ref [73]. After parameter inference, we applied BIC to select the best-fitting model for each gene and found that ∼80% of genes were best fitted by the NB model (Fig 9a).

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Fig 9. Analysis of mouse fibroblast scRNA-seq data reveals that many genes best fitted by the NB distribution are not transcriptionally bursty.

(a) Model selection using BIC on 21,684 genes (670 cells) after preprocessing shows that ∼80% of genes are best fitted by the NB model. (b,c) For these NB-fitted genes, Bayesian inference of was performed using the -modified telegraph model, with reliability assessed by the confidence interval criterion (CI range/median <0.4). The fraction of NB-fitted genes classified as transcriptionally bursty depends on the threshold chosen for : 0.1 in (b) and 0.25 in (c). In both cases, a substantial fraction of genes are best fitted by the NB model yet are not transcriptionally bursty.

https://doi.org/10.1371/journal.pcbi.1014014.g009

For these NB-fitted genes, we refined the inference using Bayesian estimation with the Turing.jl package in Julia, fitting each gene’s count distribution with the -modified telegraph model. We then quantified the confidence interval (CI) for . If the CI range (2.5%–97.5%) divided by the median was less than 0.4 (i.e., ± 20% around the median), we considered the estimate of reliable. This yielded 14,722 genes, of which the inferred parameters have been summarized and deposited on GitHub (see Data availability statement).

We next computed the fraction of genes classified as transcriptionally bursty under different thresholds for (0.1 in Fig 9b and 0.25 in Fig 9c). In both cases, a considerable fraction of genes were best fitted by the NB model yet not transcriptionally bursty. This result highlights the importance of our theoretical framework for correctly interpreting NB fits in scRNA-seq data.

4.1 Proof of Theorem 1

It is well known that the probability distribution of the Beta-Poisson process

is identical to the steady-state distribution of the telegraph model given by Eq (1) [35]. Another standard result is that the probability distribution of the Gamma-Poisson process

is identical to that of a negative binomial distribution . In what follows we shall fix these two parameters to be

(18)

Note also that the definition of the Gamma distribution, , that we use is .

Proposition 7 in Ref. [96] states that for two mixed Poisson distributions, denoted as and , the convergence holds if and only if where → denotes convergence in distribution. Consequently, to prove Theorem 1, it suffices to demonstrate that the distribution of the random variable x converges to that of the random variable as .

To proceed, we first determine the density function of the transformed random variable . This is computed as

which precisely corresponds to the density function of .

To show , we need to establish the convergence

(19)

as . Rather than directly proving Eq (19), we instead demonstrate that both distributions converge to a common limiting distribution, denoted as Dist, i.e.,

(20)

It turns out that Dist is a normal distribution.

Next, we show that the Gamma distribution in Eq (20) converges to a normal distribution as . To achieve this, we introduce the moment generating function (MGF) for the random variable ,

Consider the linear transformation

where

and . The MGF of the transformed random variable is

Using the MGF of the Gamma distribution, this simplifies to

As , , the expression further reduces to

which is the MGF of the standard normal distribution . Therefore, we conclude that

with convergence rate .

Finally, we show that Beta ( σ_on , σ_off ) = Beta ( f_onN_σ , ( 1 − f_on ) N_σ ) → 𝒩 ( μ , σ )  as . Note that we use the definition of the Beta distribution, , given by

(21)

where B ( α , β ) = Γ ( α ) Γ ( β ) / Γ ( α + β ) . The density function of this Beta distribution at the point is given by

(22)

Using Stirling’s formula, as , the term in Eq (22) reduces to

(23)

Next, we rewrite the term in Eq (22) as

Noting that and σ / ( 1 − μ ) → 0 as , we expand around using a Taylor series expansion, yielding

(24)

where

and

Using Eqs (22)–(24), we obtain

which corresponds to the probability density function of the normal distribution , with a convergence rate of . This concludes the proof of Theorem 1.

4.2 Mathematical relationship between aeBIC and BIC

Since the data in set (the set of count data for ) are randomly sampled from the ground-truth distribution , the expectation of the BIC (Eq (9)) is given by

(25)

where the inequality follows from the fact that the expectation of the maximum is greater than or equal to the maximum of the expectation. A proof of this inequality is as follows.

For any fixed θ, it is true that

So taking expectations on both sides yields

Because this is true for any θ, it follows that

Furthermore, we note that

(26)

according to Eq (11). Note that the first step in Eq (26) is a reformulation of the log-likelihood function: the term is an indicator function equal to 1 if and only if , and hence . From Eqs (10), (25), and (26), it follows that

(27)

indicating that aeBIC serves as an upper bound for the expectation of the BIC. For a fixed distribution with fixed parameters, as , which further suggests

By noting that

one can conclude that

as .

4.3 Breakdown of variability involved in transcription

Variability in scRNA-seq data arises from three major sources: intrinsic noise, extrinsic noise, and technical noise. The first two correspond to biological variability, whereas the latter is introduced by sequencing procedures.

Intrinsic noise mainly reflects stochastic promoter and chromatin activity, which is well captured by the gene-state switching dynamics of the telegraph model, as supported experimentally in [67]. The probability generating function (PGF) of the steady-state mRNA distribution under this model is

where is given in Eq (1).

Extrinsic noise is primarily attributed to cell-to-cell variation in transcription rates [66,67], manifested as heterogeneity in ρ.

Technical noise arises during sequencing, most notably from variability in transcript capture efficiency. Assuming each transcript is independently captured with probability [10,62], the observed counts in each cell follow the PGF

(28)

This result follows from the general property that binomial downsampling of a distribution with PGF produces a new PGF (see [49], SI p. 9 for the special case ; for a more general discussion see [97]).

Importantly, Eq (28) shows that the cell-specific transcription rate ρ and capture probability always appear as a product, , whose distribution can in principle be calculated from knowledge of the joint distribution . This implies that, at the level of transcript count distributions, variability in transcription rates and variability in capture probability are mathematically indistinguishable. For this reason, in the following we focus our discussion on variability in transcript capture probability.

4.4 Derivation of the observed count distributions under the assumption of Beta-distributed capture rates

4.4.1 Telegraph model.

If the capture rate follows a Beta distribution, , defined by

the PGF of the observed counts is given by integrating Eq (28) over the distribution of , which yields a generalized hypergeometric function

(29)

Finally, we obtain the closed-form probability of observing n counts

(30)

Note that this method of accounting for technical noise, i.e., via compound distributions, is essentially the same as that used to account for static extrinsic noise [98,99].

4.4.2 Negative binomial model.

The PGF of negative binomial distribution, , is given by

If is the same in all cells, the PGF of the observed counts becomes

If we have a Beta distribution of then we integrate the above expression over the Beta distribution, yielding

(31)

Finally, we obtain the probability distribution of the observed counts

4.4.3 Poisson model.

The PGF of the Poisson distribution, , is given by

If is the same in all cells, the PGF of the observed counts becomes

If we have a Beta distribution of then we integrate the above expression over the Beta distribution, yielding

Finally, we obtain the probability distribution of the observed counts

(32)

4.5 Inference of burst parameters assuming ideal conditions

We consider ideal conditions: (i) infinite sample size so that the measured moments of mRNA numbers do not suffer from finite sample size effects and thus the method of moments can be used for reliable inference; (ii) knowledge of the distribution describing cell-to-cell variability of the capture probability; this means that we can completely correct for technical noise.

4.5.1 Moments of the measured mRNA counts in scRNA-seq data.

We assume that the distribution of the measured counts from a gene of interest is given by the telegraph model with capture probability sampled from a distribution, i.e., Eq (30).

Given a PGF of observed counts , the first two orders of moments can be derived as

(33)

Substituting Eq (29)into Eq (33), we obtain expressions for the mean and the variance of the measured count distribution

(34)

and

(35)

It follows that the Fano factor of mRNA counts (variance divided by the mean) is given by

(36)

where and are the mean and the coefficient of variation (standard deviation divided by the mean) of the distribution of the capture probability, .

4.5.2 Moments of the best fit model.

We assume that we are in a region of parameter space where aeBIC selects the NB distribution with Beta-distributed capture probability as the optimal model. Substituting Eq (31) into Eq (33), we find expressions for the mean and variance of this model

(37)

and

(38)

4.5.3 Fitting the model to the data and parameter inference.

Since we are assuming that the parameters of the Beta distribution, i.e., a and b, are known, the only parameters to be inferred are r and p. These two unknown parameters can be determined exactly by matching the mean and variance of the optimal model Eqs (37)–(38) with the mean and variance of the data Eqs (34)–(35). This leads to two simultaneous equations for r and p which when solved lead to expressions for the inferred two parameters of the NB distribution

Note that these expressions do not have any dependence on the parameters of the technical noise (a and b), which indeed show that the inference method has perfectly corrected for technical noise. In fact, r and p determined in this way are none other than those parameterizing the effective NB distribution Eq (6) that best fits the conventional telegraph model.

4.6 Parameter inference without exact knowledge of the capture rate distribution

Here, we describe an inference approach that simulates the workflow of an actual scRNA-seq experiment including the subsequent analysis of the data. Note that in the inference and model selection parts of the procedure, we do not assume that the distribution of the capture probability is known — rather we use the distribution of the measured total counts per cell as a proxy.

4.6.2 Parameter inference and model selection.

• Compute the total observed count for each cell j as
• Estimate the normalization factor using the known mean capture rate (e.g., obtained via spike-in controls) [37]:
• Use kernel density estimation (KDE) to estimate the distribution of the normalization factors.
• For each gene i, estimate model parameters by minimizing the negative log likelihood across cells:(39)

where the likelihood is computed by marginalizing over β:

(40)

where for each candidate model is given by

1. – Telegraph: , with ;
2. – Negative Binomial: , with ;
3. – Poisson: , with .

• Evaluate the integral in Eq (40) using Gauss quadrature for computational efficiency. Minimize the negative log likelihood using the Nelder-Mead algorithm.
• Compute the Bayesian Information Criterion (BIC) for model selection based on the log-likelihood in Eq (39).
• If the NB model is selected, estimate the burst size and the burst frequency from the parameter estimates of the telegraph and negative binomial models. For the telegraph model, the burst size is and the burst frequency is . For the NB model, the estimates are given by for the burst frequency and for the burst size.
• Calculate the relative errors in the burst parameter estimates given the actual values used for simulating the genomic data. Randomly select sets of two pairs of burst size and burst frequency estimates. In each set, rank genes by the magnitude of the burst frequency and separately by the burst size. Calculate the percentage of rankings that are flipped compared to the true ranking.

4.7 Preprocessing of scRNA-seq dataset

We obtained the raw UMI count matrix for mouse fibroblasts from the file ss3_n682_fibs_umiCounts.rds, available at https://github.com/sandberg-lab/lncRNAs_bursting/tree/main/data. To ensure data quality, we applied standard quality control (QC) procedures consisting of two filtering steps. (i) Cell filtering: cells were removed if fewer than 30% of genes had nonzero counts, thereby excluding cells with insufficient transcript coverage. (ii) Gene filtering: genes were removed if they were expressed (nonzero counts) in fewer than 1% of all cells, thereby excluding genes with highly sparse expression. The resulting dataset was then used for downstream statistical modeling and analysis.

4.8 MLE-based parameter inference

Given observed mRNA counts , the likelihood of the dataset under kinetic parameters φ is

Parameter inference is performed by minimizing the negative log-likelihood,

(41)

We optimize using the Nelder–Mead algorithm implemented in the Optim.jl package in Julia. This derivative-free approach avoids costly gradient evaluations of while maintaining robustness to initialization. Unless otherwise noted, we adopt a gradient tolerance of g_tol = and a maximum of iterations = 1000, following the default settings of the optimize command.

To account for variability in capture probability across cells, we define the normalized capture probability

(42)

where is the total transcript count of cell i. The distribution of β, denoted , is not known a priori and is approximated by KDE from the observed values , yielding an estimate .

Efficient computation of requires integrating over the latent variability in β. Instead of computing this integral directly, we approximate it using Gauss–Legendre quadrature:

(43)

where

Here and , and are the quadrature nodes and weights generated using gausslegendre. The probability is evaluated using the three -modified models shown in Fig 7, with replaced by β. Under this formulation, the inferred parameters – ρ for the telegraph model, b for the NB model, and λ for the Poisson model – are effectively scaled by the mean capture probability , which is typically unknown.

Algorithm 1 summarizes the numerical procedure for MLE under variable capture probability.

Algorithm 1. MLE-based parameter inference accounting for variability.

Input: Number of cells , mRNA counts , and total mRNA counts for each cell i

Output: Inferred kinetic parameters

1: Compute normalized capture probabilities

2: Estimate the density via KDE using the samples

3: Generate Gauss–Legendre quadrature nodes and weights using gausslegendre

4: Initialize parameter values ϕ

5: While convergence criterion not met do

6:  Evaluate for each cell i and quadrature node

7:  Compute using Eq (43)

8:  Evaluate the loss via Eq (41)

9:  Update ϕ using the Nelder–Mead algorithm

10: End While

11: Return