Introduction

Numerous studies on single cells have shown that individual molecules (genes, proteins or metabolites), within an iso-genic and -phenotypic cell population, can display highly variable expression levels. For example, immunofluorescence flow cytometry showed that Sca-1 expressions in multipotent murine hematopoietic cells follow a Gaussian-like distribution1 and the monitoring of green fluorescent proteins in Escherichia coli displayed fluctuations in their expression levels over time2.

Such heterogeneous or noisy characteristics have shown to play pivotal roles for the survival of species to diverse environmental conditions or for cell fate decisions3,4,5. Notably, it was demonstrated that regulating stochastic noise in the levels of comK in Bacillus subtilis was necessary to control cell fate decision under nutrient-deficient conditions6. For Caenorhabditis elegans, the intestinal cell fate process from early embryonic lineage was shown to be regulated by the variability in end-1 expression, providing the basis for incomplete penetrance7. These studies have identified crucial single molecules that regulate heterogeneity or variability of single cells within a population. However, little work has been performed to investigate global responses, comprising the entire spectrum of molecular species, within single cells. In particular, the extent of transcriptome-wide expressions noise in the early mammalian development has yet to be determined.

In this paper, to understand global gene expression structure and noise patterns of single cells during mammalian developmental stages, we investigated transcriptome-wide RNA-Seq expressions of several cells during human8 and mouse9 embryogenesis. A total of 7 human and 10 murine cell origins, from oocytes to blastocysts, were analysed using high-dimensional statistical techniques, such as correlation metrics10,11,12,13,14,15,16, Shannon entropy17,18,19 and noise analyses20.

Results

Single cell transcriptome expressions scatter increases along development stages

To observe gene expression variability between 2 single cells at each developmental stage, we plotted pair-wise distributions of single cell transcriptomes (Figure 1). For human, we noticed global expressions scatter is tightly constrained up to 2-cell stage, after which the scatter widened, especially for lowly expressed genes. For mouse, the scatter widened noticeably from middle of 2-cell stage. These data suggest that transcriptome-wide expression distributions become more variable along the developmental stages.

Figure 1
figure 1

Gene expression structures for developmental stages.

Gene expression distributions, estimated from kernel density estimation (kde2d R function) of all genes expressions (RPKM) between all possible pairs of single cells in human and mouse, from oocytes to blastocysts. m is the total number of single cells and n is the total number of genes.

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To better understand the variability and the effects of technical and biological noises, we performed transcriptome-wide correlation (similarity) analysis10,11,12,13,14 by comparing the expressions of two cells from the same cell origin. Although large expressions scatters are observed, especially for late developmental stages (Figure 1), the global averaged Pearson correlation coefficients between single cells of the same stage, as expected20, are generally high (Figure 2A, dotted lines). However, the correlation coefficients between cells of distinct origins, are significantly lower (Figure 2A, solid lines and Supplementary Fig. S1 online). We further probed for non-linear relationships between the transcriptomes of cells of the same stage or different stages and computed Spearman ranking metrics (non-linear monotonic relationships), distance correlation15 (strict statistical dependence) and maximum information coefficient16 (linear and non-linear associations) (see Materials and Methods). Remarkably, all metrics revealed similar trends compared with Pearson correlations (Supplementary Fig. S2 online). These results indicate that the global transcriptional program of developmental cells clearly deviates along the stages in time, with faster rate of deviation occurring for mouse when compared with human (Figure 2A, solid lines).

Figure 2
figure 2

Phase transition in single cell transcriptomes.

(A) Pearson correlation, R, between transcriptomes of cells of the same development stage (dotted lines) and or between transcriptomes of zygote and other stages (solid lines) for human and mouse. (B) Shannon entropy (H) of single cell transcriptomes (average for m cells, error bars indicate 1 s.d.).

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Next, we assessed the diversity of single cell transcriptomes using Shannon entropy, which measures the disorder of a high-dimensional system, where higher values indicate increasing disconnection between variables and zero value indicates order17,18,19 (Materials and Methods). For both human and mouse, Shannon entropies remained low in early stages but gradually increased from 2-cell (human) or 4-cell (mouse) stage, to reach high values for morula and blastocyst (Figure 2B). This result, therefore, shows the disconnection or diversity of transcriptome-wide expressions increases during mammalian development.

Transcriptome-wide noise increases during development stages

To further understand the effects of increasing entropy and diversity in single cell transcriptomes during embryogenesis, we quantified single cells' expressions scatter by computing transcriptome-wide average noise (renamed as total noise), , i.e. summing the squared coefficient of variation21, defined as the variance (σ2) of expression divided by the square mean expression (μ2), for all genes (i) between all possible pairs of single cells (Materials and Methods). We observed that is low during initial embryonic cell differentiation, but increases at later stages with significant increase from 2- to 4-cell stage onwards (Figure 3A). We also compared total noise for embryonic stem, normal somatic and cancer cells and found similar values as obtained for later stage development cells (Supplementary Fig. S3 online). These data indicate that total noise stabilises at ~0.7 and may not increase further.

Figure 3
figure 3

Transcriptome-wide noise patterns.

(A) Total noise () of single cells for each development stage (average for m cells, error bars indicate 1 s.d.). (B) Noise (η2) vs. mean (μ) expression patterns for each development stage. (C) Plots of α and β against cell stage. The insert illustrates how α and β are determined.

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So far, we have analysed the entire transcriptome without setting any expression cut-off. It is known that lowly expressed genes in single cells are dominated by stochastic and/or technical noises, which reduce their between cells correlation values, while highly expressed genes show more deterministic expressions22. Thus, we delineated total noise to investigate average noise for every group of 500 genes, between pairs of single cells, as a function of mean expressions (μ) starting from the lowest expressions.

As expected, we noticed noise is relatively high for the lowly expressed portion of the transcriptome (η2(μ) ~ 2 for μ < 0.1 for all cell types) (Figure 3B). This portion of transcriptome is usually discarded due to low signal-to-noise ratios. Considering noise patterns above this threshold (μ ≥ 0.1), we observed all patterns followed the relationship, η2(μ) = α/μ + β, where α and β are proportionality constant and asymptotic value, respectively (Supplementary Fig. S4 online). That is, noise scales with the inverse of mean values reaching asymptotic values at higher expressions for all cell types and species, including other embryonic stem cells, somatic cells and human cancer cells23 (Supplementary Fig. S3 online). Notably, this relationship was also previously observed for other high-throughput datasets21,24.

Fitting the noise pattern of each cell type and species, we observed α and β increase during the human development process; oocyte, zygote and 2-cells stages show similar patterns, 4-cell stage shows transitional values, while 8-cell, morula and blastocyst stages reach higher values. Similar trends were also found for mouse, nevertheless, the transition between low and high values occurred earlier at middle of 2-cell stage (Figure 3C). These results show that transcriptome-wide noise increased along the developing stages and stabilises from 8-cell stage. Note that other grouping sizes, e.g. 50, 100, 1000 genes (Supplementary Fig. S5 online), did not affect the overall increase in transcriptome-wide noise along the developing stages.

Deciphering the origins of noise patterns using a stochastic transcriptional model

To understand the transcriptional mechanisms governing increasing noise patterns along the development process, we utilised a single cell transcriptional model based on Gillespie stochastic algorithm. We simulated transcriptome data constituting 20,000 gene ‘units’, where each gene dynamics is governed by 3 ordinary differential equations with 5 kinetic parameters25,26; transcription rate (s), degradation rate (δ), promoter activation (kon) and deactivation (koff) rate constants, where both continuous gene promoter activation (koff = 0) and quantal (bursty) dynamics (koff > 0) can be simulated (Figure 4A). The transcriptional amplification process27, i.e. number of transcripts produced per activation event, is controlled by φ.

Figure 4
figure 4

Simulations of transcriptome-wide expressions.

(A) Top: single cell transcriptional model. Bottom: promoter activity (left and middle panels) and gene expressions (right panels) for continuous and quantal simulations, without and with transcriptional amplification. (B) Simulations (panel 1, black curves) using continuous activation, no amplification and no extrinsic noise. Addition of extrinsic noise only (panel 2) and with low transcriptional amplification (panel 3) allowed better fit of simulations to early development stages patterns (e.g. oocyte, brown curve). Higher amplification with extrinsic noise (panel 4). Higher amplification and quantal activation with extrinsic noise (panel 5). Panel 5 shows a good fit with late stage developmental cells. (C) Simulated noise vs. mean expression patterns for all developmental stages in human (top) and mouse (bottom). Note that kon = 0.5 was used for mouse simulations and kon = 0.3 for human.

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Using the model, we simulated gene expressions for various conditions by controlling the 5 kinetic parameters. Firstly, simulations were performed by choosing the transcription (s) and degradation (δ) rate parameters for each gene from statistical distributions found in experimental data28,29 (Supplementary Fig. S6 online). The three other parameters were kept at default constant values (φ = 1, koff = 0 and kon = 0.5). In this theoretical setting, generated noise is entirely intrinsic (and Poisson) and decreased with mean expressions with α constant and β = 0 for the whole range of gene expressions (Figure 4B, panel 1). It is conceivable that none of the development stages follow this idealised condition, which does not contain noise due to extrinsic or other non-Poisson factors, such as technical noise30,31.

To consider expression-independent noise in our simulations, such as technical and environmental variability, we included additive and multiplicative random white noise32,33 to our simulated expressions. As a result, we observed an increase of β, while α remained unchanged (Figure 4B, panel 2). This is because the white noises do not interfere with the transcriptional regulation.

Although none of the development stages fitted this pattern, we observed the expression-independent noise values at higher expression levels of early stages could be achieved by setting a certain threshold of random white noise, such as β = 0.03 for human and 0.02 for mouse (Supplementary Fig. S4 online). We, therefore, set these values as the level of extrinsic noise for all the subsequent simulations, assuming it remains unchanged across each cell type for a particular species.

Next, we increased the number of transcripts produced per activation event, which resulted in amplification of the scatter in expressions (Supplementary Fig. S7 online), with corresponding increase in noise structure (Figure 4B, panel 3). Since φ controls transcriptional amplification, it is conceivable that increasing φ will proportionally increase α. However, β was not affected.

For human, the simulated noise structures matched experimental patterns of oocyte, zygote and 2-cell stages for low φ and 4-cell stages for high φ (Figure 4C). For later stages, although higher values of φ improved the simulation results, as noted earlier, β could not be increased to fit the experimental patterns (Figure 4B, panel 4). For mouse, the trend of increasing over developmental stages is also observed (Figure 3B), however, the values are generally higher for early stages (Figure 4C and Supplementary Fig. S8 online). Like human data, the mouse simulations also did not match for the values of β (Supplementary Fig. S8 online).

To improve the simulation results, that is to specifically increase β values, we next explored the parameters governing bursty transcriptional dynamics: kon and koff. Previous experiments in mouse ES cells34 suggest 0.1 ≤ kon ≤ 2.5 h−1 (median, 0.5 h−1) and 3 ≤ koff ≤ 200 h−1 (median, 14 h−1). To obtain β values that fit the human experimental patterns of later development stages, we initially set kon = 0.5 and increased koff to 5. However, the value for β obtained from simulations was still too low and further increase of koff did not produce major increase in β. We, therefore, reduced the value of kon such that kon = 0.3 to obtain a good fit (Figure 4B, panel 5). For mouse, kon = 0.5 and koff > 0.5 were required to fit the experimental β values. Again, further increases of koff did not significantly increase β (Supplementary Fig. S8 online).

Overall, from these simulations, we concur that distinct transcriptional mechanisms govern transcriptome-wide expressions during embryonic development. Notably, we showed that α is governed solely by transcriptional amplification whereas β is controlled by both extrinsic noise and quantal transcriptional activation.

Discussion

Recent studies on single cells have shown that individual molecules (genes, proteins or metabolites), within a homogenous cell population, can display highly variable expression values1,2,6,7,35,36. This variability has been linked to biological noise or the stochastic nature of molecular network regulations37,38,39. However, there is a general lack in the investigation of global regulatory mechanisms at an omics-wide scale for single cell behaviour. Studying global properties has been instrumental in interpreting collective mechanisms of living organisms, for example, the innate immune response to invading pathogens13 or the attractor states of cell differentiation process40. Here, to understand the global noise patterns of single developmental cells, we investigated RNA-seq transcriptome-wide expressions of oocytes to blastocysts in human and mouse.

Firstly, by studying the distribution of gene expressions between single cells, we observed the expression scatter increased from 2-cell to 4-cell stage onwards in both human and mouse (Figure 1). Next, we examined the Pearson correlation and Shannon entropy for each developmental stage. Again, we observed that expressions become more variable from the 2-cell stage (Figure 2). Subsequently, the global noise character of single cells was investigated by quantifying the squared coefficient of expression variations over mean expression values. Here, we observed clear transition of noise patterns occurring between 2-cell and 8-cell stage (Figure 3).

To understand the noise patterns, we developed a stochastic transcriptional model and estimated the parameter values to match each developmental cell pattern (Figure 4). From the model, we concur that the early developmental stages are mainly dominated by low transcriptional activity. For these stages, the number of transcripts produced per activation event, φ, is low. The lower overall transcription in oocytes and early zygote is consistent with i) transcriptional silencing and ii) stochastic degradation of maternal RNA that has been observed from oocytes to 4-cell stage in humans41,42. Transcriptional silencing is likely due to chromatin condensation state that prevents transcriptional machinery from reaching gene promoters41,43,44.

To track the gene expression profiles of common maternal45,46 and zygotic genes45, we plotted their relative expressions (Supplementary Fig. S9 online) and found 2 and 3 major temporal clusters, respectively, for 137 maternal and 116 zygotic highly expressed genes (Figure 5 and Table S1 online). Notably, the maternal genes (e.g. Cdh3, Dppa5, Mos, Npm2, Zp1, Zp2) showed dominant decay profiles of RNA expressions, indicating lack of transcription process (Figure 5A). However, the zygotic genes and genes expressed in embryonic stem cells47,48 (e.g. Klf4, Lin28a, Myc, Nanog, Pou5f1, Sox2) showed transcriptional process significantly increasing after the 4-cell stage (Figure 5B). The high transcriptions can be due to instructive signaling pathways, or multiple rounds of transcription reinitiation by RNA polymerase26,49,50. The observation of high transcriptome-wide noise for the middle stage developmental cells indicates the generation of heterogeneity in gene expressions between individual cells. Such heterogeneity has been shown to be necessary for cell fate diversifications1,6.

Figure 5
figure 5

Gene expression profiles of maternal and zygotic genes during development.

Average gene expression profiles of (A) maternal (2 clusters) and (B) zygotic (3 clusters) genes during human embryo development. Clusters were obtained using k-means clustering. From the initial lists of genes obtained from Xue et al.45 (Table S1), we retained maternal genes with expression values reaching peak before 4-cell stages and zygotic genes with expression values reaching peak from 8-cell stage onwards. Right panels show the expression values in all development stages of individual maternal genes (Cdh3, Dppa5, Mos, Npm2, Zp1 and Zp2; obtained from Kocabas et al.46) expressed in oocytes and zygotic genes involved in later embryonic stages and stem cells (Myc, Klf4, Pou5f1 (Oct4), Sox2, Lin28a and Nanog; obtained from Takahashi et al.47 and Yu et al.48).

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For the later stage developmental cells, on top of high transcriptional process, the cells possess quantal activation of most transcription factors, or are subject to more extrinsic variability such as phenotypic diversity among individual cells. These factors increase the general expression scatter and noise levels. However, investigating expression-independent random noise in our single cell transcriptional model simulations suggest that the levels of extrinsic and/or technical noise in our RNA-Seq data for all cells are relatively low ( ~ 0.25). That is, the relatively high levels of noise for later stages stem from quantal activation rather than technical biases, or in certain cases, such as blastocyst cells, may result from phenotypic variability, as blastocysts consist of different cellular subtypes. Conversely, since phenotypic variability among more homogenous 8-cell stage is similar to blastocyst (Figure 1), we believe that quantal promoter activation is crucial for the increase of noise scatter along development stages. Notably, such quantal promoter activation has been noted to occur for single cell organisms such as E. coli3 and has been shown to be important for the cell fate decision of B. subtilis6.

Overall, our investigations on the transcriptome-wide expressions of the early mammalian developmental stages reveal increasing variability and noise patterns across the mammalian development process. These data suggest different stages of the cell differentiation process can be better understood by investigating the transcriptome-wide noise patterns. In conclusion, our systemic approach provides novel insights into the transcriptome-wide expression and noise patterns for development cells and the underlying nature of the transcriptional mechanisms.

Methods

Single cell datasets

Single cell RNA-Seq datasets were downloaded from Gene Expression Omnibus (GEO) database from previously published data for 7 human8 (GSE36552) and 10 mouse9 (GSE45719) developmental stages. All datasets were obtained through Illumina sequencing systems. Each dataset contains the RPKM values (Reads Per Kilobase Mapped) for n ~ 20,000 genomic features, which is proportional to the number of transcripts of coding and non-coding genes (and splicing variants).

Correlation analyses

To quantify transcriptome variability we utilised correlation metrics, which are widely employed to compare global relationships between high-throughput datasets10,11,12,13,14.

The Pearson correlation between two transcriptomes, X and Y, containing n gene expressions, is obtained by , where xi and yi are the ith observation in the vectors X and Y respectively, μX and μY, the average values of each transcriptome and σX and σY, the corresponding standard deviations. The Spearman correlation coefficient between transcriptomes X and Y is defined by , where rx,i and ry,i are the ranks of the ith observations xi and yi, in vectors X and Y respectively. Both correlations were computed using the cor function of R stats package (http://www.r-project.org/).

Pearson and Spearman correlations respectively measure linear and non-linear monotonic relationships between two vectors, where R = 1 (respectively ρ = 1) if the two vectors are identical and R = 0 (respectively ρ = 0) if there is no linear or monotonic relationships between the vectors. However, both metrics do not detect other non-linear relationships, therefore null correlation values do no imply statistical independence. To obtain a more stringent measure of statistical dependence between transcriptomes X and Y, we used Distance Correlation15 (dCor), where dCor(X, Y) = 0 if and only if the two vectors are statistically independent. Maximum Information Coefficient16 (MIC) can also be used to detect other types of non-linear associations between the transcriptomes, by calculating mutual information of the vectors using an automated non-parametric approach for binning. The computation of Distance Correlation values was performed using the dcor function of the R energy package and the computation of Maximum Information Coefficient with the mine function of the R minerva package, with default parameters.

Entropy analysis

To assess the diversity of single cell transcriptomes, we used Shannon entropy17,18,19. Shannon entropy measures the disorder of a high-dimensional system, where higher values indicate increasing disorder. Entropy of each single cell transcriptome, X, is defined as , where p(xi) represents the probability of gene expression value x = xi. Entropy values were obtained through binning approach and the number of bins, b = 26, was determined from the data using Doane's rule51, such as , where gX is the skewness of the expression distribution of each sample and . The computation of entropy values was performed using the maximum likelihood implementation (entropy.empirical) of the R entropy package.

Transcriptome-wide average noise

To quantify between single cells' expressions scatter, we computed Transcriptome-wide average noise for each cell type, defined as , where n is the number of genes and is the pairwise noise of the ith gene (variability between any two cells), defined as , where m is the number of cells and is the expression noise of the ith gene, defined by the variance divided by the squared mean expression18 in the pair of cells (j,k), such as , where μijk = (xij + xik)/2 is the average value of the ith gene in the pair of single cells (j,k) and , is the corresponding variance.

Transcriptome-wide noise patterns

To elucidate transcriptome-wide noise patterns, we sorted the transcriptome into groups of w = 500 genes from low to high expression values for each pair of cells (j,k). We formed G = n/w groups and obtained the average gene expression of each group for each pair of cells, and the average gene expression noise of all genes contained in the group, , for each pair of cells. We finally obtained characteristic whole transcriptome noise patterns and by averaging the patterns of all pairs of cells.

For simplicity, we used μ = μ(g) and η2 = η2(g) in the main and following texts. As a result, the reported curve for each stage is the average pattern of all single cells pairs. We fitted noise as function of mean expressions, η2 = f(μ),

using nonlinear least squares piecewise curve fitting with the nls R function, to obtain the values of α and β for each pair of cells in all development stages.

Transcriptome simulations

Each transcriptome model consists of a set of 20,000 genes and each gene's expression is obtained from the stochastic simulation52 of a telegraph process25, such as the system of rate equations that governs gene expression xi of the ith gene is

where the promoter activation of each gene is defined by a 2-state model, active (Ai) and inactive (Ii) states and the transition rates between states are defined by two parameters, kon,i and koff,i. si is the transcription rate when the promoter is active, φi is the transcription amplification factor and δi is the degradation rate constant. The distribution of RNA degradation rate constants, δi, was obtained from transcriptome-wide RNA half-lives data in differentiating mouse ES cells28 and human B cells29 and fitted to a lognormal distribution with mean (log scale) and standard deviation parameters, μ = −2.24 and σ = 0.61 for mouse and μ = −1.58 and σ = 0.73 for human (Supplementary Fig. S6 online).

RNA transcription rates, si, were estimated from the same data28,29, such as si = xiδi (kon,i + koff,i)/(φikon,i), where xi is the gene expression value (read counts) of the ith gene, taken from a Zipf's law distribution with exponent r = 0.8 to fit the expression range in our data (Supplementary Fig. S6 online, inserts). As a result, we fitted the estimated values using a lognormal distribution for xiδi, with parameters μ = 0.17 and σ = 2.67 for mouse and μ = 0.73 and σ = 2.53 for human (Supplementary Fig. S6 online). Since the model simulates integer read counts expressions, to obtain corresponding RPKM values, we multiplied our simulated values by a normalization constant, Γ, defined as the average ratio between RPKM values of all genes and corresponding number of reads (Γ = 0.03 for the human dataset and 0.12 for mouse).

To generate transcriptome-wide expressions, we assigned different values for degradation rate constant, δi and transcription rate, si, for each gene and set the values of φi, kon,i and koff,i, identical for all genes. To test variable values of φi, kon,i and koff,i for each gene, we generated transcriptome-wide expressions using a Poisson distribution for φi with parameter λ = φi. Our result showed no noticeable difference in the patterns between fixed value or Poisson distributed φi.

Similarly we simulated transcriptomes with promoter activation kinetics that vary between genes. We estimated the distributions of kon,i and koff,i from previously observed experimental distributions of promoter ‘on’ and ‘off’ time intervals34 (τon,i and τoff,i). From the data, we observed the distributions of τon,i and τoff,i could be approximated by an exponential distribution with parameter λ = 10 and a lognormal distribution with parameters μ = 0.69 and σ = 1 respectively. Since kon,i = 1/τoff,i and koff,i = 1/τon,i53, we found median kon,i = 0.5 h−1 (0.1 ~ 2.5 h−1 range) and median koff,i = 14 h−1 (3 ~ 200 h−1). We then compared the simulations using fixed kon,i = 0.5 and koff,i = 14 or variable kon,i and koff,i and found no significant change in the patterns.

To account for non-intrinsic variations, we introduced different levels of additive and multiplicative white (Gaussian) noise to the simulated data32,33. Additive noise is achieved by adding a different random value to each gene in each cell and multiplicative noise, by multiplying all gene expressions in the same cell by a random number such as, , where is the expression of ith gene in the jth cell including non-intrinsic noise, εij represents additive noise for the and ith gene in the jth cell and ωj is the multiplicative noise that affects the jth cell. εij is chosen from a normal distribution (εij ∈ N(0,ε logxij)) with mean and standard deviation parameters, μ = 0 and σ = ε logxij, where ε represents the level of additive noise. ωj is log-normal distributed, such as ωi ∈ N(0,logω) with parameters μ = 0 and σ = ω respectively and ω is the level of multiplicative noise. We used ε = 0.08 and ω = 0.1 to simulate non-intrinsic noise of human dataset (equivalent to β ~ 0.03) and ε = 0.06 and ω = 0.1 for mouse (β ~ 0.02).