Introduction

Cellular differentiation, a key process in the development of multicellular organisms, is controlled by signaling-modulated gene regulatory networks. Variability in gene expression due to biological noise can act as an additional driving force of cellular differentiation1,2,3,4. Stochastic activation and inactivation of promoters, known as transcriptional bursting5, can induce significant cell-to-cell heterogeneity in gene expression that initiates lineage segregation6,7. For example, an analysis based on single-cell transcriptomic data revealed that retinoic acid-driven differentiation of mouse embryonic stem cells is largely impacted by noise, with a significant increase in gene expression heterogeneity occurring at the exit of pluripotency8. The highly noisy expression of the Dlk1 gene was shown to facilitate the discrimination of two sub-populations of hematopoietic stem cells9. The role of noise in driving cell-to-cell heterogeneity has also been highlighted in the differentiation of neuromesodermal progenitors in zebrafish10, and in the Yan/Pnt network driving Drosophila eye development11. However, the impact of stochasticity has been much less investigated in in vivo developing embryos than in in vitro culture conditions.

During early mouse embryogenesis, the first cell fate decision gives rise to two distinct populations: the Inner Cell Mass (ICM) and the Trophectoderm (TE). The second cell fate decision starts at the 16-cell stage (E3.0), when ICM cells give rise to Epiblast (Epi) and Primitive Endoderm (PrE) cells. While PrE cells will produce extra-embryonic tissues, Epi cells are at the origin of all the cell types that will constitute the future organism. At the protein level, ICM cells are characterized by the co-expression of NANOG and GATA612. This corresponds to the beginning of the blastocyst formation and takes place between the 16C to 32C stage (E3.0-E3.25). ICM cells then asynchronously differentiate between the 32C to 90C stage (E3.25-E3.75) into a random “salt-and-pepper” motif of Epi and PrE cells. Epi cells, some of which already arise between the 32–64C stage (E3.25-E3.5), are characterized by a high level of NANOG and a low level of GATA6, while PrE cells, which arise later at the 64–90C stage  (E3.5-E3.75) are characterized by a low level of NANOG and a high level of GATA613,14,15. Other components also play a crucial role in this decision, such as the FGF/ERK signaling pathway activated by FGF4 secreted by Epi cells. Activation of the FGF/ERK signaling pathway stimulates GATA6 expression and passage to the PrE fate13,16,17,18,19,20. Because ICM cells co-express Nanog and Gata6 while Epi/PrE almost exclusively express one of these genes, a key question relates to the source of initial heterogeneity that drives the onset of specification of ICM cells into Epi or PrE cells. This question is particularly relevant since Epi cells are precursors of embryonic stem cells (ES), that are increasingly used in regenerative medicine21.

The gene regulatory network (GRN) that underlies the ICM to Epi or PrE differentiation is based on a toggle switch between the NANOG and GATA6 transcription factors that also auto-activate (Fig. 1a). Regulation of this core GRN by the ERK pathway, via extra-cellular FGF4 whose production is stimulated by NANOG, plays a key role in the establishment of the salt-and-pepper pattern. Theoretical studies based on dynamical simulations of this GRN modulated by signaling22,23,24,25, or related versions of it20,26, concluded that heterogeneities are required to trigger differentiation. Slight heterogeneities in extracellular FGF4 at the early stage of the differentiation were shown to be necessary and sufficient to reproduce observations performed on wild-type (WT) and mutant embryos23,24. A study analyzing which source of heterogeneity would ensure the most robust differentiation event, in terms of population proportions, led to the prediction that noisy transcription of Nanog was the most suitable candidate to initiate these Fgf4 heterogeneities25.

Fig. 1
figure 1

(a) Simplified Gene Regulatory Network (GRN) driving the differentiation of ICM cells to Epi/PrE cells as proposed in previous modelling studies21,22. (b) Leftmost panel: Schematic view of the evolution of the distribution of gene expression of prototype genes. Rightmost panel: Schematic profiles of the evolution of inter-cellular entropy during the differentiation process. We hypothesize that a gene that initiates the coordination of pluripotency markers alongside Nanog (so-called “Helper genes”), displays a similar inter-cellular entropy profile. (c) Number of cells per cellular stage available for each single cell expression data used in this study.

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Single-cell transcriptomic data indicate that stochastic cell-to-cell expression heterogeneity, followed by signal reinforcement, drives the segregation of ICM cells into Epi and PrE lineages7,27. In agreement with model predictions, NANOG plays a key role in the initiation and coordination of other pluripotency factors to initiate Epi specification. By analyzing single-cell transcriptomic data of WT and Nanog−/−-Gata6−/− double knock-out (DKO) embryos, Allègre et al. (2022) indeed found that ICM cells of DKO embryos exhibit an uncoordinated variability in their transcriptome comparable to that of the undifferentiated ICM cells in WT embryos27. This highlights the essential role played by NANOG in the initiation of Epi specification. Although NANOG is necessary, it is not sufficient to induce differentiation. Indeed, a subset of ICM cells in wild-type (WT) embryos have high levels of Nanog expression at the 16-cell (16 C) and 32-cell (32 C) stages, even though the expression of Fgf4 - the first marker of binary differentiation - is barely detectable. This indicates that other factors are necessary to induce Epi specification. To verify this hypothesis, the authors analyzed the expression of other pluripotency factors (such as Sox2, Klf4 or Oct4), and showed that the interaction levels between these transcription factors were significantly reduced in DKO embryos, supporting an essential role of NANOG in the coordination of these pluripotency factors in Epi specification. Thus, the Epi state seems to be defined by the coordinated expression of Epi/pluripotency markers and NANOG is required to initiate Epi differentiation by enabling this coordinated expression. Allègre et al. (2022) suggested that several genes could act as NANOG-helping factors and that the coincident expression of any of these with NANOG would initiate Epi specification, with none of them being required per se27. The identity of these factors remains to be determined, and it is not known whether several factors are required simultaneously.

In the present study, we used computational approaches to identify a set of genes that may participate in Epi specification when coincidentally expressed with Nanog. Because of the primary role played by cell-to-cell heterogeneity in ICM differentiation, we first estimated the temporal evolution of the inter-cellular entropy of expression of key genes involved in this process28, using several single-cell datasets. The inter-cellular entropy of a gene driving the differentiation process is expected to first increase and then decrease, reflecting the progressive passage through a pluripotent, progenitor and finally differentiated state. Computation of inter-cellular entropy is generally carried out using the Shannon entropy. This requires a discretization of data, for which several methods have been suggested, including those based on the number of cells in the experiments3, or the Bayes-Block discretization procedure29. These studies revealed an initial increase followed by a decrease in the mean inter-cellular entropy across all genes. Similar conclusions have been reached upon estimation of the changes in digital entropy30. Thus, we first checked whether the temporal profile of the entropy of expression of the driver gene Nanog, exhibits the same evolution. Then, using the inter-cellular entropy evolution to identify genes whose variability in expression plays a key role in the differentiation of ICM cells, we searched for other genes displaying a similar profile (see rightmost panel of Fig. 1b for a schematic representation).

Although the Shannon entropy is simple to interpret – large values of this entropy reflect a high degree of randomness in the underlying data–, discretization and binning procedures can bias the results when data are limited. Because gene expression datasets for early mammalian development typically involve a small number of cells, we used inter-cellular differential entropy, which is a continuous analog to the Shannon entropy. Our computation of differential entropy relies on the fitting of the empirical mRNA distributions to gamma distributions28,31, which are directly related to a minimal model of transcriptional bursting and involve only two parameters32,33. This circumvents the challenging issues of binning required for the Shannon entropy. As illustrated in the leftmost panel of Fig. 1b, the changes in inter-cellular entropy associated with differentiation are expected to be associated but not limited to a transient spreading of the gamma distribution. To strengthen the computations based on relatively low numbers of cells available in each gene expression dataset for the ICM to Epi/PrE differentiation process in mice (Fig. 1c), we computed and analyzed the inter-cellular differential entropy using five available sets of expression data, consisting of single-cell RT-qPCR and single-cell RNA-seq datasets14,27,34,35,36. This allowed us to identify several genes, which we call candidate Nanog-helper genes, displaying peak-shaped temporal profiles of inter-cellular differential entropy (henceforth referred to as inter-cellular entropy) similar to that of Nanog.

In the second part of this study, we performed network inference to unravel the relationships between Nanog and the selected genes using CARDAMOM37,38. We found that the candidate Nanog-helper genes identified through the examination of the temporal profiles of differential entropy all have a mutually positive influence with Nanog. This network motif is known to enforce the decision of fate switching39 and to promote the occurrence of transient coordinated states of high expression40. As these two properties characterize the passage towards the Epi state, we concluded that the genes identified, Pecam1, Sox2 and Hnf4a, could act as Nanog-helper genes in the specification of ICM cells.

Results

Preliminary analyses

To capture cell-to-cell variability, we rely on single-cell observations8,41,42, i.e. scRT-qPCR and scRNA-seq datasets14,27,34,35,36 for which the number of cells at each cellular stage can be found in Fig. 1c. We focused on the 21 genes that are common to all datasets, listed in Supp. Table 1. This small set of genes includes the main known components of the GRN governing the ICM to Epi/PrE differentiation. The datasets have been pre-processed to allow computation of differential entropy, as described in the Methods section. As detailed in the Transformation to mRNA counts sub-section, we did not normalize the threshold cycle values (Ct) of the scRT-qPCR data using reference genes. Indeed, as shown in Fig. 2a, the variability on Ct values is similar for reference genes and for genes of interest. Moreover, the level of expression of reference genes vary with the cellular stage. Thus, normalizing the scRT-qPCR data would bias the computation of entropy of the genes of interest by including a component imputable to the variability of the reference genes. For scRNA-seq datasets, we used the raw counts. Because we do not analyze gene expression profiles individually but rather focus on the time evolution of inter-cellular entropy, the absence of normalization does not qualitatively affect the results shown below.

Fig. 2
figure 2

(a) \(\:{C}_{t}\) values from the Allègre and Guo scRT-qPCR analyses for Nanog, Pou5f1 and two reference genes used in each study (Rpl30 and Rps17). The variations in the level of expression of the reference genes are of the same order as those of the genes of interest in the two datasets. (b) PCA graphs of the two main components for all datasets, considering the 8C, 16C, 32C, 64C and 90C stages for the 21 genes common to all datasets. Except for Goolam and Posfai’s dataset, a bifurcation event where an initially homogeneous population divides into two sub-populations as the developmental stage increases is visible, despite the reduced number of data and the minimal pre-processing procedure (see Methods). The rightmost panel shows, for Allègre’s dataset, that the two sub-populations are composed of two almost orthogonal sets of vectors, one defined by Epi lineage markers (Nanog, Fgf4, Pecam1, Sox2 and Bmp4), the other by PrE lineage markers (Fgfr2, Sox17, Pdgfra and Gata4). (c) Spearman correlation coefficients for Allègre’s and Guo’s datasets, using the pre-processing procedure described in the Methods section. The increase in the numbers of correlations and anticorrelations aligns with observations reported in the original studies (see text).

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Given the non-standard way of treating the expression data and the reduced number of genes considered, we first checked the ability of the datasets after transformation to capture specification of ICM cells into two distinct populations of Epi and PrE cells using Principal Component Analysis (PCA). A bifurcation event where an initially homogeneous population divides into two sub-populations as the developmental stage increases is visible, except for Goolam and Posfai’s dataset, which includes cells from the trophectoderm. For the others, the emergence of two populations is more visible as the number of cells increases. As shown in Supp. Figure 1 and Supp. Figure 2, there are hints of these two populations arising in Goolam and Posfai’s dataset as well, although not as clearly. However, in all cases, the two emerging populations correspond to cells expressing Fgf4 (supposedly Epi cells) and to cells that do not express this gene (supposedly PrE cells) as shown in Supp. Figure 1. Analysis of the two sub-populations in Allègre’s dataset reveals that they are composed of two almost orthogonal sets of vectors, one defined by Epi lineage markers (notably Nanog, Fgf4, Pecam1, Sox2 and Bmp4), and the other by PrE lineage markers (notably Fgfr2, Sox17, Pdgfra and Gata4) as shown in the rightmost panel of Fig. 2b. These vectors remain similar for Guo and Wang’s dataset (Supp. Figure 2). As expected, in Goolam’s and Posfai’s datasets, the vectors are less clustered, but we note that Nanog and Gata6 point towards opposite directions, supporting the appearance of two sub-populations. We thus conclude that the selection of genes, together with the proposed data transformation, allows capturing cell specification although with a varying degree of robustness.

To further confirm that the transformation of data performed here does not bias the intrinsic interactions between genes, we computed Spearman correlation coefficients as was done in Allègre’s and Guo’s study (Fig. 2c). As expected, we observed an increase in the number of correlations and anticorrelations as differentiation progresses. Similarly to the observations summarized in Supp. Tables 1&2 in Allègre et al. (2022)27, we found a positive correlation between Nanog and Pou5f1 at the 16C stage, a positive correlation between Nanog and Fgf4, Klf2, Klf4, Sox2, Pou5f1 and Pecam1 at the 32C stage, and a positive correlation between Sox2 and Klf2, Klf4 and Pecam1 at the 32C stage. We also recovered the observed anti-correlation between Sox2 and Fgfr2 at the 32C stage. The analysis of Guo’s dataset revealed anti-correlations between Fgf4 and Fgfr2, and between Fgf4 and Sox17 at the 32C stage. At the 64C stage, computation of the Spearman correlations indicate an anti-correlation between Fgf4 and Pdgfra, Gata4 and Gata6, and between Nanog and Gata4 and Gata6. These observations all align with the analysis performed by Guo et al. (2010) (See Fig. 5a therein)14. We thus conclude that our data pre-processing for both the scRT-qPCR and scRNA-seq datasets and the genes used in this study captures the differentiation process of interest. However, PCA and correlation analyses do not allow a clear characterization of possible Nanog-helper genes.

Nanog shows a robust temporal profile of inter-cellular entropy

For hematopoietic differentiation, it has been shown that the genes playing a key role in the associated GRN display an increase in Shannon entropy at the exit of the pluripotency stage, followed by a decrease when cells acquire a differentiated state30,43. To build on these results while circumventing the complications inherent to the computation of Shannon entropy when data are limited (see Introduction), we here use differential entropy that relies on fitting the mRNA distributions to gamma distributions28,31,32,33 as detailed in the Methods section. Both scRT-qPCR and scRNA-seq data for Epi/PrE specification were well fitted by gamma distributions, as illustrated by four typical examples of distributions and associated cumulative distribution functions (CDF) in Fig. 3a. Kolmogorov-Smirnov (KS) statistics and p-values for all datasets at each available cell stage are shown in Fig. 3b. In most cases, KS statistic values are relatively small, reflecting a good adequacy between empirical and theoretical cumulative distribution functions. These small KS values are associated with large p-values (> 0.05), which correspond to a small ‘non-significant’ deviation between the two cumulative distribution functions as can be seen in Fig. 3a. Despite the high accuracy of the fits compared to the empirical distributions, some fits require further inspection. We show examples of such fits, with larger KS statistics and smaller p-values, in Supp. Fig. 3. As can be seen for Sox2 at the 16C stage in Wang’s dataset, or Sox2 at the 64C stage in Posfai’s dataset, the fits are still adequate and capture the behavior of the empirical mRNA distribution. The CDF is, however, more sensitive to outliers because of the presence of many zero expression values. Note that the sparse number of cells implies that a few outliers largely decrease the p-value of an otherwise accurate fit. Nevertheless, comparison between data-based and fitted gamma distribution functions (Fig. 3a and Supp. Fig. 3) shows that fitting gamma distributions to both single-cell RT-qPCR datasets after our pre-processing steps is accurate. The good quality of the fits confirms that this transformation gives rise to a quantity proportional to mRNA counts (see Methods).

Fig. 3
figure 3

(a) Representative examples of histograms and fits to gamma distributions of the mRNA counts. As shown in the lower panels, the theoretical cumulative distribution functions are close to the empirical ones, indicating the good quality of the fit. (b) Kolmogorov-Smirnov (KS) coefficients and p-values of the fits to gamma distributions for the 21 genes considered, for all datasets at each available cell stage. All p-values above 0.05 are shown in the same yellow.

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The inter-cellular differential entropies computed with the gamma distributions of the main components of the GRN driving ICM differentiation (Fig. 1a) are shown for each dataset in Fig. 4a. Accuracies of the predictions are quantified using bootstrapping (see Methods). The inter-cellular entropy of Nanog shows an increase with a maximum reached at the 32C stage, followed by a decrease onwards, for all datasets involved. This robustness in the temporal profile of Nanog cell-to-cell heterogeneity is in line with the main role played by this transcription factor in the emergence of the population of Epi cells. It also confirms that driver genes display a surge in inter-cellular entropy also during in vivo development in mice, as was shown for in vitro erythrocytic differentiation3. For Gata6, inter-cellular entropy decreases from the 16C stage onwards, in agreement with the proposed mechanism of PrE specification that is triggered by the appearance of Epi cells22. As for Fgfr2 and Fgf4, the analysis does not reveal a consistent pattern of entropy evolution.

Fig. 4
figure 4

(a) Evolution of the inter-cellular differential entropy of expression of the main genes driving the specification of ICM cells into the Epi or PrE fate. Only Nanog displays a peaked shape profile in entropy, in agreement with its driving role in this process. Each dataset is represented by a different color, and error bars correspond to confidence intervals obtained by bootstrapping. The shaded region between the 16C and 64C stage in Nanog’s entropy profile represents the region of interest where the entropy profiles are compared. (b) Comparison of the evolution of inter-cellular differential entropy between Nanog and each of the 20 common genes considered, from the 16C to the 64C stage. A square indicates that the entropy profile in a given dataset is similar for the two genes. (c) Evolution of the inter-cellular differential entropy of Hnf4a, Pecam1 and Sox2 between the 16C to 64C stage. Only these three genes show the peak shaped entropy in at least 4 datasets, when considering the mean inter-cellular entropy computed over 1000 bootstraps.

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Identification of candidate Nanog-helper genes based on the entropy profiles

We then computed the inter-cellular entropies of the genes common to all datasets at each cellular stage. Results are shown in the form of heatmaps for every dataset in Supp. Fig. 3. By nature, trends in differential entropy can only be compared qualitatively44. To identify genes assisting Nanog in Epi specification, we focused on the 16C, 32C and 64C stages, since ICM to Epi specification occurs during these stages and then qualitatively compared the entropy profiles of the 20 genes at these stages to that of Nanog27. The genes with the same qualitative evolution – that is, those whose entropy increases from the 16C to 32C and then decreases from the 32C to 64C, just as Nanog’s entropy (as shown in the rightmost panel of Fig. 4a) – were selected as possible candidate genes helping Nanog during Epi specification. Due to the limited number of cells in each dataset, we also imposed as a selection criterion for a candidate helper gene that the evolution of its inter-cellular entropy is consistent in at least 4 datasets, although this probably excludes valid candidates.

Surprisingly, very few genes display the same inter-cellular entropy profile as Nanog in this timeframe, and results were dataset dependent, as shown in Fig. 4b. Three genes emerged from this analysis: Hnf4a, Pecam1 and Sox2. Their entropy profiles between the 16C to 64C stage are shown in Fig. 4c. Pecam1 and Sox2 are known to be related to the Epi fate, the former as a marker and the latter as an active player. Indeed, Sox2 codes for the transcription factor SOX2, which is important for the emergence of the ICM lineage45. More generally, it mediates FGF4 signaling and FGFR2 expression27 while regulating NANOG46. It is also a central player in the formation and maintenance of pluripotency in the ICM47 and in stem cell populations48. At the 32C stage, the levels of expression of Pecam1 and Sox2 are correlated with those of Nanog.

By contrast, Hnf4a is not known as an Epi factor. It codes for the zinc finger transcription factor HNF4A and has never been related directly to Nanog. It is a marker of the endoderm layer in the implanting blastocyst49 known to be necessary for the survival of the embryonic ectoderm50 and the formation of the visceral yolk sac51. Furthermore, HNF4A is regulated by GATA652. Interestingly, Hnf4a was up regulated in an experiment involving the presence of a FGFR inhibitor, alongside most markers of the Epi cell fate, although it is not known as a Epi marker14.

In conclusion, based on the analysis of inter-cellular entropy of expression during Epi specification, we found out that three of the common genes robustly display the same temporal profile of variability as Nanog. Based on previous observations highlighting the requirement for coincident expression between Nanog and other unidentified helper genes for Epi specification27, we concluded that Pecam1, Sox2 and Hfn4a are suitable candidates that may act as Nanog-helper genes.

Mutually positive interactions between Nanog and candidate Nanog-helper genes

To confirm this finding and shed light on the possible mechanism underlying this influence on the specification process, we investigated their impact in the underlying GRN using a network inference method applied to the same datasets. Previous analyses have shown that genes playing a key role in differentiation often display a surge in variability during specification (see Introduction). However, it has not yet been tested whether, conversely, key genes can be identified based on their peak-shaped inter-cellular entropy profile. To investigate this question and find out how the genes identified by entropy estimations interact within the minimal GRN shown in Fig. 1a, we used CARDAMOM, a network inference algorithm able to infer GRNs from time-stamped single-cell expression data37,38. The method crucially exploits a particular mechanistic model of transcriptional bursting and the resulting gamma distributions. When selecting the main genes of the minimal model shown in Fig. 1a to build the network, CARDAMOM recovered all the interactions proposed in our previous studies (Fig. 5a). It should be noted that the inferred network predicts interactions from the levels of expression of the genes and does not include ERK signaling. As such, inferred interactions on Fgf4 correspond to influences on Fgf4 expression. The activation of Nanog by Fgf4 could be seen as a shortcut for other underlying activations that are not considered in the subset of genes considered to infer the GRN (i.e. Fgf4 → Nanog is inferred instead of Fgf4 → other genes → Nanog). In the same way, cross-inhibition between Gata6 and Fgf4, activation of Nanog by Fgf4 and inhibition of Gata6 by Fgfr2 cannot be compared with the GRN that considers signaling, and most probably correspond to indirect interactions mediated by other genes not considered in the inference process.

Fig. 5
figure 5

Inferred Gene Regulatory Networks using CARDAMOM from a combination of the 5 single-cell time-stamped datasets. Green arrows represent activation and red arrows represent inhibitions. The width of the arrows represents interaction strength, with a larger width being associated to a larger interaction strength. Gene abbreviations are described in Supp. Table 2. The ‘Stm’ node stands for stimulus node and represents a perturbation in the environment of the cells, inducing them to evolve towards a new steady state37,38. (a) Inferred GRN when using only the principal genes proposed in previous modelling studies Nanog - Gata6 - Fgf4 and Fgfr222,23. Interactions inferred by CARDAMOM are in agreement with original assumptions of the model. (b) Inferred GRN when using the principal genes and the candidate Nanog-helper gene Sox2. Sox2 appears in a reciprocal activation loop with Nanog. The same behavior is found for Bmp4, Fgf4, Hnf4a, Id2, Klf2, Pdgfa, Pdgfra and Pecam1, as discussed in the main text. (c) Time decomposition of the networks inferred by CARDAMOM when considering all the genes common to all data sets and for which the entropy profiles were estimated. For clarity, only interactions with Nanog are shown.

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When the proposed helper gene Sox2 was added to the main genes of the minimal model mentioned above, a reciprocal activation with Nanog was revealed (Fig. 5b). Because this small motif enforces the decision of fate switching39 and promotes the occurrence of transient coordinated states of high expression40, it confirms that Sox2 coincident expression with Nanog favors the passage to the Epi state. Closer inspection reveals the two activations are simultaneous, i.e. they are detected at the same cell stage, notably the 32C stage, which corresponds to the time of appearance of Epi cells27. The same results were obtained when Bmp4, Hnf4a, Klf2 or Pecam1 were individually added to the main GRN, as shown in Supp. Table 2. Other genes, such as Fgf4, Id2, Pdgfa and Pdgfra also show these double activations, although they do not necessarily appear simultaneously at the same stage, nor at the 32C stage specifically as shown in Supp. Table 2.

CARDAMOM was then run considering the whole set of common genes whose entropy profile was determined in the previous section. The interactions inferred between Nanog and all the other genes considered are shown separately for the different cell stages in Fig. 5c. Sox2, Pecam1 and Hnf4a robustly display a reciprocal activation with Nanog at the 32C stage. Importantly, as shown in Fig. 5c, the double activation patterns between Nanog and the helper-genes precede the appearance of the toggle switch between Nanog and Gata6, which agrees with their roles in driving Epi specification. Another Epi factor, Pou5f1 was found to activate and be activated by Nanog at the 32C stage. This gene codes for the OCT4 transcription factor, which together with SOX2, cooperates with NANOG on enhancers to maintain cell pluripotency in mouse embryonic stem cells53. Its role as a Nanog-helper gene is thus highly plausible, suggesting that it may not have been detected properly by the method based on the entropy profile.

Other genes identified when considering the smaller network, namely Bmp4, Fgf4, Klf2, Id2, Pdgfa, and Pdgfra, which displayed double activation profiles, either do not exhibit these double activation profiles, or these are not simultaneous, or these are not observed at the 32C stage, when considering the whole set of common genes. The robustness of their identification is thus dependent on the number of genes involved for the inference. Only Hnf4a, Pecam1 and Sox2 display simultaneous double activation profiles at the 32C stage regardless of how many genes are used for the inference, revealing the robust behavior of the genes identified via the estimation of the inter-cellular entropy.

Discussion

In the present study, we have used single-cell transcriptomic data from scRT-qPCR and scRNA-seq experiments, to analyze the ICM to Epi/PrE differentiation in preimplantation mouse embryos. The aim was to identify genes favoring the transition to the Epi state when coincidentally expressed alongside Nanog. Focusing on the genes common to the five datasets considered, PCA on all datasets captured the bifurcation event corresponding to the differentiation process under investigation. Spearman correlations on the transformed scRT-qPCR datasets agreed with the previously reported correlations and anticorrelations between gene expression levels. These preliminary analyses have thus confirmed that the data considered are adequate, despite the limited number of genes and cells involved. Moreover, they have established that the pre-processing methods used, which, for scRT-qPCR datasets do not include normalization of Ct values with respect to reference genes, capture the essential relations between genes during ICM to Epi/PrE differentiation. However, these analyses also revealed significant differences between datasets and did not allow the identification of candidate Nanog-helper genes via standard analysis only. To this end, we resorted to the computation of inter-cellular differential entropy. Estimation of the inter-cellular differential entropy revealed a robust peak-shaped temporal profile for Nanog, in agreement with the known role of this gene as a driver of Epi specification. Indeed, the precocious Epi cells then induce the specification of neighboring cells in PrE through Fgf4 signaling. This mechanism results in the salt-and-pepper arrangement of Epi and PrE cells in the blastocyst22. In agreement with this scenario, Gata6, the transcription factor associated with the PrE fate, does not display a peak-shaped profile in inter-cellular entropy. Three genes among those analyzed displayed an inter-cellular entropy profile similar to that of Nanog: Pecam1, Sox2 and Hfn4a. Based on the findings of Allègre et al. (2022)27 suggesting that the coincident expression of Nanog and still to be identified factors would initiate Epi specification, we identified Pecam1, Sox2 and Hfn4a as candidate Nanog-helper genes. Network inference using CARDAMOM has shown that all three are connected to Nanog by a reciprocal activation loop, with the two branches being simultaneously active at the 32C stage, preceding the toggle switch between Nanog and Gata6. Because such mutual reinforcement in expression is known to induce coordinated expression and to support fate switching decisions39,40, Pecam1, Sox2 and Hfn4a most probably correspond to factors triggering Epi specification when coincidently expressed with Nanog.

Indeed, Sriram et al.39 have shown that a reciprocal activation loop – also called mutual activation – permits the enforcement of cell fate choice in Candida albicans. Bifurcation analysis of the underlying GRN shows that bistability in the level of the master regulator Wor1 occurs on a wide range of kinetic parameters. Changing the strength of the mutual regulations directly influences the lower threshold values at which the concentration of the master regulator Wor1 starts exhibiting bistability. While mutual inhibition increases this threshold, mutual activation decreases it. Mutual activation also extends the domain of bistability by increasing the value of the upper threshold at which bistability disappears. The interplay between the two feedback loops gives plasticity to this system and makes it robustly reproducible. It is straightforward to apply this to the ICM to Epi/PrE differentiation; Nanog being the master regulator in this case27, which forms the negative feedback loop with Gata6 and the mutual activation loops with the candidate helper-genes Hnf4a, Pecam1 and Sox2. Furthermore, we have identified that the reciprocal activation loops with the candidate Nanog-helper genes precede the toggle switch with Gata6. According to Schuh et al.40 who performed extensive stochastic simulations of abstract GRNs characterized by different interactions, reciprocal activation loops favor co-bursting and the appearance of transient states of coordinated high expression of the genes involved in the reciprocal activation loops. Thus, the reciprocal activation loops play a double role in 1) the initiation of the differentiation, by creating transient coordinated states of high expression surpassing the threshold necessary for the initiation of the differentiation process, while 2) enforcing and locking the cellular decision once the threshold has been passed.

To quantify cell-to-cell heterogeneity, we have estimated differential entropy that can be directly computed from the parameters of the gamma distributions fitted to mRNA expression data. Entropy is more commonly computed using the Shannon entropy. Two types of entropy can be computed: intra-cellular or inter-cellular entropy. Intra-cellular entropy relates to the variability of the transcriptome of a single-cell and is connected to the stemness and pluripotency level of a cell14,28,42,54. Intra-cellular entropy shows mixed behaviors, with some studies reporting an increase54,55 and others a decrease56 during early development. To explain these apparently contradictory results, it has been argued that a decrease in intra-cellular entropy is observed during late stages of differentiation when cells are subject to an increase in regulatory constraints and thus become more specialized. In the earlier stages however, an increase in cell fate pluripotency, associated with an increase in intra-cellular entropy, could be necessary to ensure that all cell fates are attainable28. In contrast, inter-cellular entropy, which captures the cell-to-cell variability of expression of a given gene in a cell population, has always been reported to exhibit a peak-shaped behavior for genes driving cell specification. As found here for Nanog, it increases from the pluripotent to the progenitor stage and then decreases from the progenitor to the differentiated stage.

Entropy sorting, another mathematical framework based on Shannon entropy, has been used to distinguish genes indicative of cell identity57. Like for digital entropy30, scRNA-seq data are discretized into two groups for each gene: a group representing the active state, where the gene is expressed, and a group representing the inactive state, where the gene is not expressed. The algorithm is particularly successful in clustering cells based on their identity, allowing it to define cell types with great precision at a given time point. However, this method does not consider dynamics.

Analysis of the most delta-entropic genes, i.e. genes for which the entropy difference between two consecutive stages is the largest, could complement the present approach. Indeed, for hematopoietic differentiation, Dussiau et al. (2022) have shown that the most delta-entropic genes are those involved in lineage specification, whereas the genes with the highest variation in mean expression are related to mechanisms of cell survival43. Such analysis needs entropy values to be comparable in absolute values, which is not possible for differential entropy, and thus could not be done in this study44. Another interesting point would be to compare the temporal profiles of differential entropy of Nanog and candidate Nanog-helper genes between WT embryos and embryos knocked-out for specific genes.

It has been suggested that Epi factors exhibit redundancy27,58. In the same line, we hypothesize that the number of Nanog-helper genes is most probably larger than three. As identified by CARDAMOM from the reciprocal activation loop, Pou5f1 could be one of them although it was not firmly detected on the basis of its entropy profile. Moreover, as our innovative analysis was carried out with the 21 genes common to the five datasets considered, some significant genes are probably missing. From a biological point of view, gene transcription is known to be inactive most of the time59,60. The probability of having simultaneous bursts of transcription of two genes is thus expected to be low, which would call for the existence of several Nanog-helper genes. In the same line, experimental validation of the candidate Nanog-helper genes identified here would be delicate. When knocked-down individually, the absence of a Nanog-helper gene is only expected to provoke some delay in the average specification time. The validation of this effect would thus require a fine tuning of the observation window around the 32C stage, when Epi cells start to appear, performed on many embryos.

It would be instructive to include candidate Nanog-helper genes in existing models of Epi and PrE specification20,23,25,61. Because all models require some source of heterogeneity to initiate specification, simulations of these extended GRN would allow the identification of the conditions under which a coordinated noise pattern in the expression of Nanog and another gene would be sufficient to trigger Epi specification. This scenario is also expected to affect the robustness of differentiation in terms of populations and differentiation timings. Altogether, combination of experimental approaches, data analyses and modeling are required to address the critical role of gene expression heterogeneity in driving cell differentiation in early development.

Methods

Transformation to mRNA counts

For scRT-qPCR datasets, we first transform the \(\:{C}_{t}\) data into quantities that are proportional to mRNA counts. Based on Richard et al. 20163, we define the number \(\:{m}_{k,l}\) of gene l mRNA molecules in cell k by:

$$\:{m}_{k,l}={n}_{wells}\times\:D\times\:{2}^{{C}_{t,threshold}-A-{C}_{{t}_{k,l}}}$$
(1)

where \(\:{n}_{wells}\) is the number of wells in the experiment, \(\:D\) is the sampling coefficient, \(\:{C}_{t,threshold}\) is the detection threshold, \(\:A\:\) is a constant number of pre-amplifications, and \(\:{C}_{{t}_{k,l}}\) is the normalized \(\:{C}_{t}\) value using spikes and references genes:

$$\:{C}_{{t}_{k,l}}=\widehat{{C}_{{t}_{k,l}}}-\left(\overline{{C}_{{t}_{k,l}}}-\overline{{C}_{{t}_{0}}\:}\right)$$
(2)

In the above equation \(\:\widehat{{C}_{{t}_{k,l}}}\) is the raw \(\:{C}_{t}\) value for gene j, cell i, \(\:\overline{{C}_{{t}_{k,l}}}\) is the i-th cell mean spike value and \(\:\overline{{C}_{{t}_{0}}}\) is the global mean spike value. As shown in Fig. 2a, using reference genes would bias the computation of entropy. Because we do not have access to the spike values, we use the raw values \(\:\widehat{{C}_{{t}_{k,l}}}\) to compute the “pseudo-mRNA counts”. Furthermore, we do not have access to the sampling coefficient, therefore the final transformation for scRT-qPCR datasets is:

$$\:{m}_{k,l}={n}_{wells}\times\:{2}^{{C}_{t,threshold}-{A}-\widehat{{C}_{{t}_{k,l}}}}\:\:$$
(3)

The values used for each scRT-qPCR dataset are described below in the Datasets and Pre-processing subsection.

For scRNA-seq datasets, we simply use the raw counts.

Datasets and pre-processing

Five datasets of single cell gene expression levels in developing embryos before implantation were used. A brief description of these datasets and the pre-processing procedures are given below. The number of cells available in the analysis is shown in Fig. 1c.

  • Allègre et al. (2022) scRT-qPCR dataset27. The dataset combines single-cell expression levels of 48 genes in 18 individual cells isolated at the 16-cell (16C) stage and the single-cell expression levels of 98 genes from 98 individual cells at the 32-, 64-, and 90- (32C, 64C, 90C) cell stage. Some expression values are thus missing at the 16C stage. The expression cut-off \(\:{C}_{t,threshold}\) is 35, \(\:{n}_{wells}\) is either 48 or 96 depending on the cell stage considered, and \(\:{A}_{pre-amplification\:}\:\)is 18. The pseudo-mRNA counts were computed using Eq. (3) as described above. This transformed dataset was used for PCA, correlation, entropy estimation and CARDAMOM analysis.

  • Guo et al. (2010) scRT-qPCR dataset14. The dataset contains the single-cell expression values of 48 genes from 387 individual cells isolated at four developmental stages (8C, 16C, 32C and 64C), from mouse embryos encompassing two differentiation events (formation of the TE and ICM, and formation of the Epi and PrE). The expression cut-off \(\:{C}_{t,threshold}\) is 28, \(\:{n}_{wells}\) is 48, and \(\:{A}_{pre-amplification\:}\)is 18. The pseudo-mRNA counts were computed using Eq. (3) as described above. Since our purpose is to study the ICM to Epi/PrE differentiation event, we further used the supplementary file S5 in Guo et al. 2010 to remove presumed outside cells (and therefore TE progenitors) based on the \(\:{C}_{t}\)value of Id2 since Id2 expression is a marker of TE cells14. Most outer cells have \(\:{C}_{t,Id2}\) levels above 24 and were thus removed. This leaves us with 195 individual cells. This transformed dataset was used for PCA, correlation and entropy estimation. The dataset was slightly transformed to be used in CARDAMOM in order to be quantitatively comparable to the other datasets. The quantitative differences are probably due to the fact that spikes and sampling coefficients used in Guo’s study are not available and would be necessary for the datasets to be quantitatively comparable. Thus, for CARDAMOM, \(\:{n}_{wells}\) is 1, and \(\:{A}_{pre-amplification\:}\)is 0.

  • Goolam et al. (2016) scRNA-seq dataset34. The dataset contains the single-cell expression values of 41 480 genes from 124 individual cells isolated at 5 developmental stages (2C, 4C, 8C, 16C and 32C) encompassing the two first differentiation events (formation of the TE and ICM, and formation of the Epi and PrE). Data were obtained using Smart-seq2. We used TPMs for PCA and correlation analysis, while raw counts were used for entropy estimation and CARDAMOM.

  • Posfai et al. (2017) scRNA-seq dataset35. The dataset contains the single-cell expression values of 16 379 genes from 106 individual cells isolated at 3 developmental stages (16C, 32C and 64C). The study focuses on the formation of the TE and ICM, but data also includes the formation of Epi and PrE. Data were obtained using Smart-seq2. We used TPMs for PCA and correlation analysis, and raw mRNA counts for entropy estimation and CARDAMOM.

  • Wang et al. (2021) scRNA-seq dataset36. The dataset contains the single-cell expression values of 37 405 genes from 124 individual cells isolated at 7 developmental stages (1C, 2C, 4C, 4LC, 8C, 16C, 32C and 64C) encompassing two differentiation events (formation of the TE and ICM, and formation of the Epi and PrE). We used TPMs for PCA and correlation analysis, and raw mRNA counts for entropy estimation and CARDAMOM.

We restricted the analysis to genes that are common to all datasets. The full list of these 21 genes is given in Supp. Table 1. We also restricted ourselves to the 8C to the 90C stage in all datasets. For the scRNA-seq datasets, presumed outside cells were not removed from the analysis due to a lack of quantitative criterion to do so in the original studies.

Correlations - PCAs - fitting of mRNA counts to gamma distribution

Spearman correlations were computed using the cor() function of the dplyr package on R version 4.3.062. To generate the PCA plots we used the prcomp() function of the stats package on R. We fitted the mRNA count distribution to gamma distributions using the fitdist() function of the fitdistr package, developed by Delignette-Muller et al. 2015, for R version 4.3.0, using the method of moments estimation. The KS-statistics and their p-values were computed using the ks.test() function available on R.

Entropy Estimation

To estimate the differential entropy of various single-cell RT-qPCR and scRNA-seq gene expression datasets we used the property that gene expression is typically subject to transcriptional bursting5, leading to mRNA molecules of a gene following a gamma distribution28,31,32,33. The number of mRNA molecules have hence been fitted to a gamma distribution. The entropy of the gamma distribution, determined by its parameters \(\:\alpha\:\) and \(\:\beta\:\), is given by

$$\:{H}_{diff,\gamma}={\alpha}-ln\left(\beta\right)+ln\left(\varGamma\left(\alpha\right)\right)+\left(1-\alpha\right)\Psi\left(\alpha\right)$$
(4)

where α is the shape parameter, β the rate parameter of the gamma distribution, Γ(x) is the gamma function, and Ψ(x) is the digamma function. We use the method of moments to obtain the parameters of the distribution. We computed the mean \(\:\mu\:\) and standard deviation \(\:\sigma\:\:\) of the empirical data. These two moments are related to the parameters of the gamma distribution by

$$\:\alpha\:=\frac{{\mu\:}^{2}}{{\sigma\:}^{2}}\:\text{a}\text{n}\text{d}\:\beta\:=\frac{\mu\:}{{\sigma\:}^{2}}$$
(5)

To obtain a measure of uncertainty on the estimation of entropy, we performed bootstrapping. For each dataset, at every cellular stage, and for every gene, we resampled the same number of data (corresponding to the initial number of cells at that stage, in that dataset), computing their mean and standard deviation, relating them to the parameters of the gamma distribution, then computing the inter-cellular entropy. This was repeated 1000 times for every gene, at every cellular stage, in each dataset, to obtain the distributions of inter-cellular entropy. The resulting mean entropy was plotted for these 1000 re-samples, and the error was taken to be the standard variation of the 1000 entropy estimates.

CARDAMOM

The CARDAMOM software was cloned from https://github.com/eliasventre/cardamom as of 10/10/2024. It was run using python 3.9.13.

CARDAMOM inference was performed by merging the 5 single-cell datasets used in the present study. To this end, we transformed Guo et al.’s (2010) dataset4, as explained in the Datasets and Pre-processing subsection, to make it quantitatively similar to other datasets when compared on PCAs and UMAPs.