A novel gene expression stability metric to unveil homeostasis and regulation

Abstract

We explore the concept of gene expression stability within a homeostatic cell through the gene homeostasis Z-index, a measure that highlights genes under active regulation in response to internal and external stimuli. The Z-index uncovers distinct regulatory activities and patterns driven by a small subset of cells, providing insights that traditional mean-based methods often overlook. By capturing subtle regulatory dynamics, this approach highlights the importance of stability metrics in uncovering detailed gene regulation patterns that underpin cellular adaptation.

Supplementary Information

The online version contains supplementary material available at 10.1186/s13059-025-03810-4.

Background

A homeostatic cell continuously reacts to both internal and external stimuli, a process that often involves the regulation of transcription. While the majority of genes within a homeostatic cell are transcriptionally stable, a small proportion might be in a regulatory or compensatory state, dynamically responding to stimuli. However, identifying these compensatory genes in a homeostatic cell can be challenging.

Gene expression variability is frequently used in gene feature selection. Commonly measured using statistical parameters such as variance or coefficient of variation (CV) [1–7], gene expression variability quantifies the fluctuation of gene expression relative to the mean expression. Genes with higher variability are often deemed more informative due to their pronounced expression fluctuations across diverse cells. However, this perspective ignores the gene’s regulatory dynamics and cannot distinguish between genes that have widespread variability across cells and genes whose variability is attributable to a small proportion of cells that have sharply upregulated expression, thereby skewing the mean expression. In contrast, gene expression stability is much less commonly investigated; it measures the proportion of cells in which a gene’s expression is consistent with the baseline expression status. We posit that genes with lower stability experience differential regulation within specific subsets of cells and are indicative of regulatory heterogeneity.

Several studies explore gene variability and stability across various contexts [2, 8–10]. For example, Kolodziejczyk et al. [10] offer primarily observational insights into how genes from distinct biological mechanisms display differing levels of variability. Lin et al. [9] identify genes with stable expression within homogenous cell groups by employing a two-component mixture model to individually evaluate genes. However, this approach has notable limitations, including its reliance on mixture model estimation accuracy and the absence of cross-gene information, resulting in metrics that lack robust statistical grounding.

Here, we introduce a new measure of gene expression stability, the gene homeostasis Z-index, a robust statistical measure that tests for genes upregulated in a small proportion of cells. We systematically benchmarked the performance of the Z-index against variability metrics that have been shown to be particularly effective in capturing variability across cells (i.e., scran [5], Seurat VST [11], and Seurat MVP) using both simulated and real data. The Z-index unveils genes subject to precise regulation within specific cell subsets, shedding light on their roles in cellular adaptation. For example, we observe organ-specific patterns exemplified by heightened synaptic transmission activities in islets. Furthermore, we uncover regulatory patterns for neuropeptides, such as insulin and somatostatin, which exhibit extreme values within a limited number of cells. These findings underscore the limitations of conventional mean-based approaches and highlight the ability of the gene homeostasis Z-index to surpass these constraints.

Results

K-proportion and gene homeostasis Z-Index identify regulatory genes

As cells of the same type and in the same microenvironment are likely to receive similar stimuli, leading to similar molecular profiles, we can leverage the statistical properties of cell clusters in single-cell genomics data to differentiate between homeostatic and regulatory genes. The expression of homeostatic genes—even those with high variability across cells—should follow a negative binomial distribution. Conversely, regulatory genes should have a pattern of expression that deviates from the negative binomial distribution (i.e., low variability across most cells, with expression in a few cells having an outsized effect on the mean expression).

To better distinguish between homeostatic and regulatory genes, we introduce a gene-specific statistic, “k-proportion,” which represents the percentage of cells with expression levels below an integer value (k) determined by the mean gene expression count (Fig. 1, see Methods for details). Under equivalent mean expressions, regulatory genes display significantly higher k-proportions compared to their homeostatic counterparts. Because the extreme expression levels exhibited by a regulatory gene in a limited subset of cells skew the mean expression upward, the k-proportion—the number of cells with considerably lower expression than the mean—will be higher than if the variation in expression were due to noise.

Fig. 1.

Illustration of k-proportion inflation test and the wave plot. k-proportion is defined as the proportion of counts no greater than k. In A, we show k-proportions as a function of gene mean expected from a negative binomial distribution (curves) and from simulated data (dots), for k = 0, …, 4. For each gene, only one k is tested. The choice of k depends on the gene mean (vertical lines). In the test statistic ${\widehat{\mathrm{\Delta }}}_i={\widehat{g}}_i^{(k)}-F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)$, ${\widehat{g}}_i^{(k)}={\sum }_{j=1}^N{1}_{X_{\mathit{ij}}\le k}/N$ is essentially a nonparametric estimate of the true proportion, robust to the underlying distribution; $F_{\mathit{NB}}\left({\overline{X}}_i,,,\widehat{\varphi }\right)$ is a parametric estimate of the proportion under the assumption of a negative binomial model. The test statistic compares the difference between a nonparametric and a parametric estimate (B, CI = confidence intervals). If the difference is significant, it suggests that negative binomial is not a good fit. Since we test for observed k-proportion being strictly larger, the rejected hypothesis indicates existence of highly expressed outliers, i.e., the gene expression in upregulated in a subset of cells. In a toy example (C, D), we show how regulatory genes expressed in subsets of cells appear in a wave plot, where k-proportion is displayed as a function of gene mean. Regulatory genes surface as anomalies, appearing as distinct outliers above the general trend

The relationship between k-proportion and the mean expression can be illustrated on a “wave plot.” In this visualization, regulatory genes surface as anomalies, appearing as distinct outliers above the general trend, akin to water droplets cast airborne from a wave (Fig. 1). By quantifying the extent of deviation from the homeostatic gene population, we offer a new avenue for characterizing gene regulation directly from single-cell data. We propose a k-proportion inflation test that compares the observed k-proportion with the expected value from a set of negative binomial distributions with a shared dispersion parameter, which is estimated empirically assuming a majority of genes are homeostatic. Due to the asymptotic normality of the proposed test under the null hypothesis, we can obtain a Z-score for each gene, which we term the gene homeostasis Z-index, to serve as a relative measure of gene expression instability; a higher gene homeostasis Z-index indicates more active regulation or compensatory activity (i.e., low stability).

Simulations show Z-index matches or outperforms variability-based methods

Extensive efforts have been made to evaluate variability metrics comprehensively. For example, Zheng et al. [8] critically examined 14 metrics for their ability to capture cell-to-cell variability, identifying scran as particularly effective for this purpose. Building on these findings, we conducted benchmarking analyses using scran alongside alternative metrics such as Seurat VST and Seurat MVP. We excluded CV from the comparison due to its numerical instability in the simulations.

We first simulated data under the null hypothesis, examining various scenarios with different means and dispersion levels. For each scenario, we generated 10 datasets from a negative binomial model, each with 1000 independent genes from 200 cells. We subsequently derived a Z-index for each gene using the k-proportion inflation test.

In analyzing the distribution of method parameters, we found that the distribution of these Z-index values closely approximates a normal distribution with a mean of 0 and a standard deviation of 1, suggesting that the Z-index distribution adheres to a normal distribution asymptotically. Consequently, we find that the type I errors for the k-proportion inflation tests are well-calibrated (Additional file 1: Figure S1). In contrast, the distribution of other variance or dispersion parameters is not obvious. One can only propose arbitrary cut-offs to identify significant discoveries, making type I error challenging to control.

We next assessed the sensitivity and specificity of the k-proportion inflation test under varying conditions. Using a fixed cell number (n = 200), we generated baseline expression data for 5000 genes following a negative binomial distribution with dispersion parameters of 0.5 and a mean expression of 0.25, both empirical estimates based on real data. We introduced 200 inflated genes with outliers with varying magnitude and percentage. We ran nine scenarios varying the expression values for the inflated genes (2, 4, or 8) and the percentage of cells exhibiting upregulation (2%, 5%, or 10%) (Fig. 2). We show the k-proportion statistic captures information beyond the second moment (variance) and is highly correlated with skewness and kurtosis (Additional file 1: Figure S2). Since SCRAN, Seurat MVP, and Seurat VST are not necessarily testing procedures, we used receiver operating characteristic (ROC) curves to assess the overall performance of each method. Specifically, we converted the estimates from values to quantiles and then compared them with true labels of inflated vs. non-inflated genes to obtain sensitivity and specificity measures. A better performing method has an ROC curve closer to the top-left corner, indicating higher sensitivity for a given specificity.

Fig. 2.

Simulation studies comparing Z-index with state-of-the-art variability metrics. Receiver Operating Characteristic (ROC) curves comparing the performance of stability (Z-index) and variance indices (scran, Seurat MVP, and Seurat VST) under varying conditions of outlier values (2, 4, 8) and percentages (2%, 5%, 10%). Each subplot shows the sensitivity versus specificity for each method, highlighting the impact of outlier magnitude and frequency on stability assessment accuracy

We found that when the outlier expression is low, the Z-index typically performs on par with Seurat MVP and SCRAN but surpasses Seurat VST in certain sensitivity ranges. As the expression of outlier increases (i.e., regulation becomes sharper), the performance of the Z-index remains stable, while other methods degrade or shift. The Z-index curve remains consistently higher across all thresholds, reflecting a general advantage. At lower percentages of cells with upregulated genes, the performance differences between Z-index and others are subtle. As the percentage of cells with upregulated genes increases, the differences between the methods are starker—Z-index curves closer to the top-left corner than others—suggesting Z-index has better resilience against increasing biases.

Overall, these results demonstrate that the Z-index exhibits competitive or superior performance across a range of conditions, particularly in scenarios with higher noise levels or larger proportions of cells with upregulated genes. Its robustness and consistency across varying thresholds underscore its utility as a reliable tool for detecting regulatory shifts.

Gene expression stability captures information beyond variability

We next examined the distribution of k-proportions in a dataset of CD34 + cells (Fig. 3) [12]. Through k-means clustering on principal component space, we identified three distinct subgroups within the CD34 + cell population. Based on the subgroup-specific markers, the three subgroups may correspond to megakaryocyte progenitors, antigen-presenting cell progenitors, and early T cell progenitors (Additional file 1: Figure S3-4). We generated wave plots for each subgroup separately and all subgroups combined. As hypothesized, we observed that the empirical distribution of k-proportions in homeostatic genes (most genes) closely aligns with those from negative binomial distributions with the same dispersion level. In relatively homogeneous cell populations, the resulting distribution approximates a Poisson distribution, as evidenced by the dispersion level of 0.163 in subgroup 3. However, as heterogeneity increases, so does the dispersion level. Subgroup 2 has the highest dispersion level (0.526) of the three subgroups, indicating increased variability in gene expression. When considering all subgroups together, the dispersion level is 1.4.

Fig. 3.

Gene expression stability analysis on Zheng et al. CD34 + cells. A UMAP show three subgroups in the CD34 + cells. B Library size vs gene coverage for each individual cell. C Wave plots of observed k-proportion vs gene mean for subgroups 1, 2, 3, and all subgroups combined. Genes with top significant Z-index are labeled by their names. The k-proportion vs gene mean from a Poisson distribution and from the fitted negative binomial are shown. D Heatmap visualization of selected regulatory genes for cells from different subgroups. EComparison of Z-index with variability metrics across CD34 + subgroup 2 cells: SCRAN variance, Seurat MVP, Seurat VST, and CV (lognormal CV².)

In all the scenarios, we consistently observed “droplets,” which signify genes undergoing active regulation (Fig. 3C). Figure 3D displays selected genes for each cell population that have a significant Z-index after adjusting for multiple comparisons using false discovery rate. In subgroup 1, H3F3B, GSTO1, TSC22D1, CLIC1, LYL1, and FAM110A are under regulation in a subset of cells, indicating cellular oxidant detoxification activities. In subgroup 2, PRSS1 and PRSS3 are upregulated in a small portion of cells, revealing digestive activities. In subgroup 3, genes associated with cell-killing activities, NKG7 and GNLY, are upregulated. Such patterns are not observed in normalized data (Additional file 1: Figure S5).

When considering genes significantly upregulated in the combined analysis, we observed active regulation of HLA and RPL families occurring exclusively in subgroup 2, indicating cytoplasmic translation and processing of exogenous peptide antigen activities. Additionally, MAP3K7CL is upregulated in subgroup 1, suggesting this cell population might transmit signals downstream in response to extracellular stimuli through the MAPK signaling pathway. Overall, we show Z-index can be applied to both homogeneous and heterogenous cell groups to identify sharply upregulated genes.

We further compared Z-index estimates with four different variability metrics (scran, Seurat MVP, Seurat VST, and CV/lognormal CV²) across CD34 + subgroup 2 cells (Fig. 3E). There is a positive relationship between Z-index and variability for most genes. As Z-index increases and stability decreases, variability tends to increase; however, the strength and shape of this correlation differ across methods. SCRAN variance shows a wide range of variability values across Z-index values. Seurat MVP exhibits a compressed range of variability compared to SCRAN, with a moderate correlation between variability and Z-index. Seurat MVP is conservative in its variability estimates, emphasizing stability more than dynamic fluctuations. Seurat VST increases linearly as Z-index increases, with a more systematic gradient compared to SCRAN. Outliers with high variability are still prominent, but moderate variability patterns are less scattered. CV displays the weakest correlation between variability and Z-index. Variability values are heavily suppressed, particularly for genes with higher Z-index. Overall, cases with extreme Z-index values are consistently identified by SCRAN variance and Seurat VST, making them reliable for detecting outliers. However, these methods face power issues for capturing variability in the rest of the gene set, particularly for moderately variable genes or those not classified as highly unstable. The presence of genes with high variability but not high stability underscores the need for complementary methods to fully capture the complexity of gene expression dynamics. This approach is critical for identifying both outliers and genes involved in subtle, but biologically significant, processes.

As SCRAN variance, Seurat MVP, and Seurat VST all perform different types of transformation and normalization, we further investigated the differences between stability and variability quantified from UMI counts directly, using 1011 β cells from an islet donor. After correcting for multiple comparisons, we identified 372 genes out of 18,513 coding genes that have a significant homeostasis Z-index and with a k-proportion inflated over 10%. In genes like SYT16, RBP4, PCDH7, and IAPP, “droplets” show leading inflated k-proportions on a wave plot and exhibit moderate variance and CV² (Fig. 4). Interestingly, genes with lower stability can have higher variance under the same mean expression level, while genes with high variability can remain relatively stable across the entire cell population. Among the genes with top variance but relative stability are LRRTM4, G6PC2, NRG3, CNTN4, and CNTN5 (Additional file 1: Figure S6). We also investigated the relationship between Z-index, variance, and CV. Interestingly, we find no monotonic relationship across all genes, indicating that these metrics depict different properties. However, for genes with a fixed mean expression, those with a high Z-index exhibit greater variance. Upon a detailed examination of the expression patterns and distributions of the top inflated genes across cells (Fig. 4C, D), we observed significant variations in both the proportions and expression levels of upregulated genes. This observation underscores the flexibility of the Z-index, as it can capture a broad spectrum of expression patterns originating from localized upregulation, without any prerequisite assumption of the underlying distribution.

Fig. 4.

Gene expression stability analysis on pancreatic β cells from one islet donor. Top 15 genes from the inflation test are denoted in red in all plots. A K-proportion, variance, and CV² as a function of gene mean (log 10 scale). B Scatterplots of Z-index vs. variance and Z-index vs. CV². C Heatmap of UMI counts for top 15 inflated genes across all cells. D Z-index values derived from separate tests on spliced UMI count and unspliced UMI counts. E Violin plot shows the heterogeneity of count distribution across the top 6 inflated genes. F Wave plot shows k-proportions derived from long non-coding RNAs (lincRNA), mitochondrial genes’ RNAs, and ribosomal genes' RNAs, which all can be fitted with the same negative binomial distribution as the coding genes

Velocity analysis has gained popularity for studying gene expression dynamics by distinguishing between nascent (i.e., unspliced) and mature (i.e., spliced) RNA molecules [13]. This inspired us to assess the Z-index for spliced and unspliced counts (Fig. 4D). Intriguingly, the k-proportions for these counts each adhere to a distinct group of negative binomial distributions, both maintaining the same dispersion level (Additional file 1: Figure S7). Furthermore, we find that upregulation of gene expression can be achieved through different avenues: some genes amplify their spliced counts, others their unspliced counts, and yet others both. For instance, among the top 15 genes, HSPA1A and IER3 only show inflated spliced counts in specific cell subsets. Conversely, UNC5C, UNC5D, and MARCH1 primarily demonstrate an increase in unspliced counts. Genes like GCG, IAPP, SYT16, RBP4, and PDK4 exhibit elevations across the board. These findings emphasize the nuanced nature of gene expression upregulation, suggesting influences beyond mere transcriptional boosts, possibly involving independent mechanisms like accelerated splicing or reduced degradation.

It is noteworthy that k-proportions derived from long non-coding RNAs (lincRNA), mitochondrial genes’ RNAs, and ribosomal genes’ RNAs all can be fitted with the same set of negative binomial distributions as the coding genes in the nucleus (Fig. 4F). The shared distribution may hint at underlying biological or biochemical processes that govern RNA production or stability across diverse RNA types.

Gene homeostasis Z-index helps identify regulatory gene modules

K-proportion is a non-parametric statistic that is highly sensitive to outliers. As shown in Fig. 4, both IAPP and GCG are identified as outliers relative to the baseline negative binomial distribution. However, their gene expression patterns differ. Specifically, IAPP and GCG both exhibit high expression levels in a limited subset of cells, but these subsets are distinct. This suggests that the Z-index is effective in identifying anomalies, but further multi-gene analyses are needed downstream to better understand the underlying patterns.

In this dataset, we examined whether the upregulation of genes occurred within the same group of cells by analyzing their correlations (Fig. 5). The results reveal distinct gene modules with strong associations, each linked to specific biological functions. One module comprising neuregulins, ROBO, and ROR1 is involved in beta-cell communication, structural organization, and survival. This module supports islet integrity and adaptive responses to cellular stress. Another module, encompassing genes such as INS, IAPP, HSP90, HLAs, and granins, integrates core beta-cell functions, including insulin secretion, amyloid management, stress protection, and immune interactions, reflecting the central role of beta cells in glucose metabolism and immune recognition. A separate cluster, dominated by JUN and several heat-shock proteins (HSPs), functions as a stress-response module. This module plays a critical role in protecting beta cells from damage by regulating inflammation, mitigating oxidative stress, and promoting protein stability under metabolic overload. Additionally, REGs and IL32 collectively form another highly correlated gene module that balances regeneration and inflammation. REGs contribute to beta-cell repair, while IL32 highlights its dual role in amplifying immune responses and potentially causing cell damage. These findings indicate that individual beta-cell subsets may adopt specialized roles influenced by their unique microenvironments. Collectively, these gene modules define critical aspects of islet beta-cell biology, ranging from structural integrity and metabolic regulation to stress adaptation and immune modulation.

Fig. 5.

Characterization of selected regulatory genes in β cells from one donor. A Correlation of UMI counts of selected genes across all cells. Genes are grouped into modules by their correlations (orange squares). B Heatmap visualization of gene modules identified from A. C Example genes that exhibit extreme values. This deviation from a homogeneous distribution underscores a potential flaw in the current analytical methodologies that heavily rely on mean values

Notably, INS expression is upregulated, albeit within a specific subset of cells, in tandem with the upregulation of HSP90s and granins. Instead of following a negative binomial distribution, the INS expression pattern exhibits numerous extreme values. To illustrate, 50% of the total INS expression is from 13.7% of cells, while another 25% of expression is attributable to a mere 5% of cells (Fig. 5C). This skewed contribution to the mean expression underscores a potential flaw in the current analytical methodologies that heavily rely on mean values. Our observation suggests there may be a baseline expression of INS across beta cells—indicating a latent potential for insulin production. The rapid synthesis in response to external stimuli, such as glucose levels or nervous system signals, seems to be a specialized function of cell subsets. The regulatory patterns observed for other important endocrine factors, IAPP, SST, and GCG, mirror this principle, where localized extreme values significantly contribute to their overall high expression in β cells.

Gene homeostasis Z-index as a portable self-normalizing metric for differential stability analysis

Variability metrics such as SCRAN Variance, Seurat MVP, Seurat VST, and CV do not have well-characterized asymptotic distributions. This limitation arises because these metrics often depend on data-specific transformations and empirical estimates that vary across samples and datasets. Without a known asymptotic distribution, statistical testing for significance is challenging because it requires assumptions about the underlying variability distribution, which may not hold. The lack of rigorous probabilistic foundations forces the use of arbitrary cut-offs (e.g., top 5% most variable genes) to classify genes as high- or low-variability. As a consequence, the reliance on hard cut-offs can lead to poor control of Type I errors, i.e., falsely identifying genes as highly variable due to noise or extreme cases. This compromises the reproducibility and statistical rigor of variability analyses, particularly in scenarios with high noise or heterogeneous data. In contrast, the stability Z-index leverages a standardized normal distribution, enabling robust hypothesis testing and controlled Type I error rates. In this section, we showcase how to utilize the Z-index to perform differential stability analysis across multiple-sample single-cell studies.

Here we provide a specific example focusing on β cells from a dataset derived from pancreatic islets of 29 individuals, including type 2 diabetic (T2D) and non-diabetic (ND) cases, where data integration and cell-type identification have already been completed. For β cells in each individual sample, we applied the proposed k-proportion inflation test to compute the Z-index for each gene. We calculated the Z-index using the expression data of all genes from β cells of that individual as the baseline, effectively normalizing gene expression relative to the individual's overall gene expression profile. This normalization accounts for biological or technical factors influencing overall expression levels or variability within a sample. Genes with Z-scores significantly deviating from the baseline are hypothesized to be under active regulation. The Z-index thus serves as a quantitative measure of active regulation for each gene within a specific cell type of an individual sample. Because Z-index values asymptotically follow a standard normal distribution, differential stability analysis between T2D and ND samples can be performed using a t-test.

We performed differential stability analysis within 35,551 cells, by comparing T2D (n = 10) with ND (n = 19) cases. Notably, genes in the T2D-affected β cells showed a tendency for decreased stability, as indicated by smaller Z-index values compared to ND cases. Many of these genes are linked to neuronal activities, including PDE4B, NCAM2, SEMA3A, DPP10, CHODL, UNC5C, PTH2R, and EPB41L4A, constituting a list of 116 genes with Z-index differences greater than 5. To identify key master regulators of these 116 genes, we employed ChEA3 [14], a transcript factor enrichment analysis tool that integrates multiple omics data. The top 10 transcription factors identified were NPAS3, ZNF385B, DBX2, ZNF385D, MEIS2, RORB, PURG, ZNF804A, TBX18, and PRRX1. Notably, NPAS3, DBX2, and MEIS2 are known to regulate genes involved in neural development [15–17]. Importantly, chromosomal abnormalities affecting the coding potential of these genes have been associated with conditions such as schizophrenia, cognitive disability, and obesity. Notably, few of these TFs are documented in the JASPER database. They can be overlooked if using a motif-based approach, even with paired ATAC-seq data.

Across 13,103 genes, the results after applying a 5% FDR threshold revealed 1,302 significant differentially stable (DS) genes, 207 significant DE genes, and 34 genes that were significant in both analyses (Fig. 6A). The larger number of significant DS genes highlights the sensitivity of DS analysis in detecting subtle differences in gene regulation across subsets of cells. By contrast, DE analysis primarily captures differences in average gene expression levels. The distinction between these methods underscores their complementary roles, as they probe fundamentally different aspects of gene behavior.

Fig. 6.

Gene expression stability analysis on pancreatic beta cells from T2D and ND donors. A Scatterplot comparing differential expression analysis using gene mean vs. differential stability analysis using Z-index (log fold-change vs. differences in Z-index). Differential stability hits are shown in red, differential expression hits are shown in green, and double hits are shown in purple. B GO Enrichment plots for top biological process categories for DS and DE genes. C Waterfall plots for G6PC2 and PDE4B. The x-axis represents individual donors, while the y-axis reflects the expression of the gene measured in UMIs. The plot uses color intensity to denote the percentage of cells within each donor expressing the gene at a given UMI count

To broaden the scope of functional insights, we reduced the significance threshold to a p-value of 5%, thereby increasing the number of genes included in the enrichment analysis (3763 vs 2610 genes). This revealed marked differences in the biological pathways associated with DS and DE genes (Fig. 6B). Genes identified by DE analysis are primarily enriched in neuronal functions, including axon development, cell growth, and synapse organization. On the other hand, DS genes are enriched in pathways related to glucose metabolism, such as glycoprotein biosynthetic processes, glycosylation, and the regulation of small GTPase-mediated signal transduction. These findings emphasize that DS and DE analyses capture distinct aspects of gene function and regulation, providing complementary insights into β-cell biology.

Among the DS genes, those upregulated in ND include GRAMD2B, HADH, CNTN4, MLXIPL, HS6ST3, CASR, STX1A, RASGRF1, and others. These genes show higher Z-index values and greater stability in ND, suggesting impaired regulation in T2D under pathological conditions. These genes are expressed in fewer cells and at lower levels in T2D samples compared to ND samples, consistent with disrupted functionality. Many of these genes have notable associations with glucose metabolism. For example, MLXIPL, also known as ChREBP (Carbohydrate-Responsive Element-Binding Protein), is a transcription factor that regulates the expression of genes involved in glycolysis and lipogenesis in response to glucose levels. HADH, an enzyme essential for the β-oxidation pathway of fatty acid metabolism, plays a critical role in glucose metabolism. Similarly, STX1A encodes a protein involved in the fusion of insulin-containing granules with the plasma membrane in pancreatic β-cells, a key step in insulin secretion. The impaired regulation of these genes in T2D suggests they may play critical roles in maintaining normal glucose homeostasis, which is disrupted under disease conditions.

In contrast, DS genes upregulated in T2D include PDE4B, SAMD5, AGMO, MSR1, ZPLD1, ALOX12B, SEMA3A, and NCAM2. These genes exhibit lower stability in T2D, indicating perturbed regulation within subsets of cells under pathological stress. The lower stability of these genes reflects increased regulatory heterogeneity, which could be a compensatory mechanism or a consequence of cellular stress in T2D.

Overall, the findings from DS and DE analyses illustrate their complementary nature. While DE analysis identifies genes with significant changes in average expression, DS analysis uncovers genes with regulatory disruptions, even when average expression levels remain relatively stable.

To enhance the analysis of gene expression stability, we introduce a waterfall plot that effectively visualizes differences in stability across individual samples. As shown in the example of Fig. 6C, this plot highlights the expression levels of G6PC2 across individual donors, categorized into ND and T2D groups. The cascading structure of the waterfall plot provides an intuitive representation of gene stability across donors and conditions. T2D samples are depicted by shorter cascades, indicating fewer cells reach high expression. In contrast, ND samples are characterized by longer tails and outlier values, which reflect sharp regulatory shifts and greater variability in the percentage of cells expressing G6PC2 at specific UMI levels. Biologically, G6PC2 is a critical gene that modulates the sensitivity of β-cells to glucose by influencing their response to fasting glucose concentrations. Its expression affects the threshold for insulin secretion, making it a key player in glucose homeostasis. G6PC2 is ranked at the 6th DS gene that shows lower stability in T2D vs. ND. However, it was not detected in DE after FDR control. Similarly, we observed PDE4B with increased regulation in T2D vs. ND, ranked at 1 st in DS genes. However, we did not detect it in DE. PDE4B is an enzyme that degrades cyclic adenosine monophosphate (cAMP), a second messenger that mediates various cellular responses, including the regulation of glucose metabolism in pancreatic β-cells. Excessive PDE4B activity could reduce cAMP levels too much, impairing insulin secretion and contributing to glucose intolerance or insulin resistance—key features of metabolic disorders such as T2D.

When a gene is markedly upregulated, it can often be detected by both DE and DS analyses. However, genes significant in DS may not always be identified through DE due to differences in statistical power. For instance, in this dataset, HADH is significant in DS but does not pass the FDR threshold in DE analysis (Fig. 6D). HADH plays a critical role as a key enzyme in the mitochondrial β-oxidation pathway, breaking down fatty acids into acetyl-CoA. This pathway is particularly vital during fasting or periods of low glucose availability, where HADH activity facilitates energy production by enhancing fatty acid oxidation. By reducing the body's reliance on glucose, HADH helps maintain stable blood glucose levels. Dysfunction or deficiency in HADH can impair fatty acid oxidation, forcing a greater dependence on glucose as the primary energy source, which may contribute to hypoglycemia.

Discussion

In the field of single-cell research, the focus has primarily centered around studying the characteristics of cell types with uniform and consistent behaviors. Consequently, genes that exhibit more varied and inconsistent behaviors have often been overlooked [18–20]. However, these very genes, tightly regulated in specific cell subsets, hold immense importance as they enable cells to respond and adapt to diverse internal and external stimuli within their microenvironment. Recognizing the fundamental role of these genes in maintaining cellular homeostasis, we introduced gene expression stability. Gene expression stability focuses on the consistent baseline expression status of a gene across different cells in a sample, providing insights into which genes undergo sharp regulation within specific subsets. The gene homeostasis Z-index serves as a quantitative measure of gene expression instability. High scores indicate active regulation or compensatory activity, shedding light on which genes are actively regulated in response to stimuli. Using the Z-index, we observed significant regulatory undertakings in generic defense mechanisms, such as cellular stress response and inflammation. Crucially, we observe the regulatory patterns for important neuropeptides, including INS, IAPP, and SST, are highly localized, showing extreme values in a limited number of cells.

A key conceptual advancement of this study is the proposal that instability, as measured by the Z-index, can be regarded as a molecular phenotype. While variability analyses have traditionally been used for gene feature selection, the ability to quantify stability provides direct insight into regulatory dynamics within heterogeneous populations. This reframing of stability as a phenotype underscores its potential as a marker for identifying genes and pathways actively responding to environmental or pathological stimuli. Stability reflects a gene’s ability to maintain its baseline expression across diverse cellular contexts, and deviations in stability could reveal novel insights into disease mechanisms and regulatory processes.

Insulin secretion is a tightly regulated process that responds rapidly to changes in glucose concentration. In the context of T2D, one central issue is impaired insulin secretion and function. This makes T2D an ideal disease to demonstrate the effects of the Z-index and its relevance to the disease state. Our study points to the importance of β cells and their altered stability in T2D. When applied to T2D data, DS analysis highlighted genes not captured by DE analysis, underscoring the distinct properties assessed by these methods. For example, HADH was identified as a significant DS gene but not as a DE gene. As a key enzyme in the mitochondrial β-oxidation pathway, HADH plays a crucial role in maintaining energy homeostasis under fasting conditions. Its altered stability in T2D reflects regulatory disruptions that may not manifest as changes in mean expression but have significant implications for cellular metabolism. Similarly, GRAMD2B and STX1A were significant in DS analysis, with implications for glucose metabolism and insulin secretion, respectively, but were not detected through DE. Conversely, some genes, such as PDE4B, were significant in both DS and DE analyses but ranked differently, illustrating how DS provides a complementary view focused on regulatory consistency rather than absolute expression levels. These findings reinforce the idea that DS uncovers a “new molecular phenotype,” capturing regulatory dynamics that are not apparent through mean-based analyses alone.

Looking forward, the application of DS analysis extends beyond T2D to other diseases characterized by heterogeneous cell populations and dysregulated gene expression. Inflammatory diseases and neurodegenerative disorders, for instance, often exhibit subtle regulatory changes within specific cell subsets that are missed by DE analysis. By identifying genes with altered stability, the Z-index could serve as a powerful tool for uncovering novel disease-associated regulatory mechanisms. Additionally, integrating DS analysis with single-cell multi-omics data could provide a deeper understanding of the interplay between transcriptional regulation, epigenetic modifications, and metabolic states across various disease contexts. This expanded application of the Z-index has the potential to reveal new therapeutic targets and biomarkers, advancing our understanding of complex diseases.

Conclusions

Our study introduces gene expression stability as a novel concept in single-cell research, emphasizing its role in uncovering genes that exhibit nuanced regulatory behaviors often overlooked by traditional methods. The gene homeostasis Z-index serves as a quantitative measure of instability, highlighting genes undergoing significant regulation within specific cellular subsets. By reframing stability as a molecular phenotype, we illuminate its potential to identify genes and pathways crucial for disease mechanisms and therapeutic targeting.

Methods

$\mathrm{k}$-proportion and the $\mathrm{k}$-proportion inflation test

We denote the number of cells as$N$, the number of genes as$M$, and let the observed gene expression matrix (UMI counts) be denoted as$X$. Each element indexed by $X_{\mathit{ij}}$ represents the UMI count of the $i$-th gene in the $j$-th cell. We define $k$-proportion as the percentage of cells with gene expression less than a defined integer value$k$,$g_i^{(k)}=P(X_{\mathit{ij}}\le k)$. Our goal is to test whether the underlying true $k$-proportion is significantly higher than the expectation under a negative binomial model, i.e., $H_0:$ ${\mathrm{\Delta }}_i=g_i^{(k)}$-$F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)=$ 0, where $F_{\mathit{NB}}$ is the cumulative distribution function under a negative binomial distribution, ${\lambda }_i$ is the mean and $\varphi$ is the dispersion level, i.e.,$F_{\mathit{NB}}\left(k\right)=P_{\mathit{NB}}(X_{\mathit{ij}}\le k)$. Under the alternative hypothesis, $k$-proportion is inflated due to active regulation in a small subset, i.e., $H_A:$ $g_i^{(k)}$-$F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)>0$. Since$g_i^{(k)}$, mean ${\lambda }_i$ and dispersion level $\varphi$ are unknown, we replace these parameters by their estimates, respectively. Observed $k$-proportion for $i$-th gene for a given$k$, is computed as${\widehat{g}}_i^{(k)}={\sum }_{j=1}^N{1}_{X_{\mathit{ij}}\le k}/N$. The observed mean expression is estimated as ${{\widehat{\lambda }}_i=\overline{X}}_i={\sum }_{j=1}^N{X}_{\mathit{ij}}/N$. Our test statistic compares the difference between observed $k$-proportion and the fitted value under a negative binomial model:${\widehat{\mathrm{\Delta }}}_i={\widehat{g}}_i^{(k)}-F_{\mathit{NB}}\left({k;\lambda }_i,=,{\overline{X}}_i,,,\varphi ,=,\widehat{\varphi }\right)$. The estimate of $\varphi$ and choice of $k$ will be discussed as follows.

Estimation of dispersion level

Within the same cell type from a single cell experiment, we observed that the empirical distribution of $k$-proportions in most genes (homeostatic) closely aligns with $k$-proportions that are generated from negative binomial distributions with the same dispersion level across genes. Since the dispersion level captures the variability introduced by technical factors and biological factors, the observation confirms that these factors affect cells measured in the same single cell experiment the same way. We estimated the dispersion level by minimizing the square error loss between fitted $k$-proportions and observed values across all genes: $\widehat{\varphi }=\text{arg}{\mathit{min}}_{\varphi }{\sum }_{i=1}^M{({\widehat{g}}_i^{(k)}-F_{\mathit{NB}}\left({k;\overline{X}}_i,,,\varphi \right))}^2$. We assume homeostatic genes outnumber regulatory genes by a magnitude; thus, no filtering on genes was performed.

Choice of $\mathrm{k}$

Since we use $k$-proportion as a measure reflecting the portion of cells with considerably lower expression, it is easy to see that for a given mean expression, there can be multiple choice of $k$. Intuitively, we can compute any proportion of values smaller than the gene mean. However, the choice of $k$ must be carefully balanced. If $k$ is too small, it lacks sensitivity to discern meaningful differences. Conversely, if $k$ is too large, it fails to adequately represent low-expression behavior.

This work builds upon our earlier research on zero-proportion inflation. For a comprehensive derivation and discussion, please refer to our prior publication. Here, we summarize key findings from that work. For homogeneous cell populations, we demonstrated that the Poisson distribution serves as a robust model for most genes, supported by rigorous model comparisons. Under the null hypothesis, we assume that the observed zero proportion equals the expected rate of zeros derived from a Poisson distribution. For heterogeneous populations, we extend this framework with a finite Poisson mixture model as the alternative hypothesis.

$$H_0:p=e^{-{\lambda }_g}\&H_A:p=\sum _k{\pi }_k{e}^{-{\lambda }_{g_k}}$$

Practically, we do not explicitly estimate ${\pi }_k$; instead, we simply test if the observed is larger than the expected $p$ with the estimated gene mean ${\lambda }_g$. It might seem counterintuitive that this test statistic does not fully leverage the specification of the alternative hypothesis; mixture parameters ${\pi }_k$; are not estimated. However, the function of zero-proportion against the mean satisfies Jensen’s inequality trivially, $p$ under the alternative hypothesis is always greater than that under $H_0$. As a result, we do not need to estimate ${\pi }_k$ to test the alternative hypothesis. We show that the proportion of zeros is always larger than expected under various alternative hypotheses.

Now, regarding the k-proportion test, we need to consider several factors: (1) We test only once for each gene to avoid dependent hypotheses and complications in controlling false discovery. (2) The power of eligible k varies; for instance, testing for zero proportion has limited power for highly expressed genes unlikely to exhibit zeros. Hence, k is a function of the mean. (3) We aim for a simple, dataset-independent procedure, opting for Poisson over Negative Binomial to avoid reliance on dispersion parameter estimation. (4) We maintain Jensen’s inequality, ensuring that for various $k$- proportions (e.g., 1-prop, 2-prop), $p$ under the alternative hypothesis is always greater than under $H_0$​. The regions of $k$- proportions satisfying Jensen’s inequality decrease monotonically as a function of the mean, as shown in Additional file 1: Figure S8.

In summary, our procedure for selecting $k$ for each gene will trivially depend on the mean. If the mean is$X$, $k$ is selected such that $k+\sqrt{k}<X<k+1+\sqrt{k+1}$.

Gene homeostasis Z-index

After determining $k$ and $\widehat{\varphi }$, our test statistic is the difference between observed and expected $k$-proportion: ${\widehat{\mathrm{\Delta }}}_i={\widehat{g}}_i^{(k)}-F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)$. The detection of regulatory genes is now equivalent to a one-tailed test against the null, ${\mathrm{\Delta }}_i=0$. Under the alternative, ${\mathrm{\Delta }}_i>0$. A Z-index (Z-score) can be computed for each gene as $z_i={\widehat{\mathrm{\Delta }}}_i/\sqrt{\text{Var}({\widehat{\mathrm{\Delta }}}_i)}$. We can calculate the variance of ${\mathrm{\Delta }}_i$ using the delta method, which leads to the following equation:

$${\text{Var}(\mathrm{\Delta }}_i)=\left(1,+,o,\left(1\right)\right)\frac{1}{N}[g_i^{\left(k\right)}\left(1,-,g_i^{\left(k\right)}\right)+{\left(\frac{\partial F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)}{\partial {\lambda }_i}\right)}^2{\text{Var}(\text{X}}_{\mathit{ij}})-2\frac{\partial F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)}{\partial {\lambda }_i}\text{Cov}(1_{X_{\mathit{ij}}\le k},{\text{X}}_{\mathit{ij}})]$$

.

The first term accounts for the variability of $g_i^{\left(k\right)}$, the second term accounts for the variability of $F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)$ and the last term is the covariance between $g_i^{\left(k\right)}$ and $F_{\mathit{NB}}\left({k;\lambda }_i,,,\varphi \right)$. We compute $\text{Var}({\widehat{\mathrm{\Delta }}}_i)$ by plugging in estimates under the null, i.e., $\text{Var}\left({\widehat{\mathrm{\Delta }}}_i\right)=\frac{1}{N}[F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)\left(1,-,F_{\mathit{NB}},\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)\right)+{\left(\frac{\partial F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)}{\partial {\lambda }_i}\right)}^2{\text{Var}(\text{X}}_{\mathit{ij}})-2\frac{\partial F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)}{\partial {\lambda }_i}\text{Cov}(1_{X_{\mathit{ij}}\le k},{\text{X}}_{\mathit{ij}})]$(see Appendix for details). A $p$-value can be derived from the Z-score using the standard normal distribution. We used Benjamini Hochberg method to adjust for multiple comparisons across genes.

Interpretation

In the test statistic ${\widehat{\mathrm{\Delta }}}_i={\widehat{g}}_i^{(k)}-F_{\mathit{NB}}\left(k,;,{\overline{X}}_i,,,\widehat{\varphi }\right)$, ${\widehat{g}}_i^{(k)}={\sum }_{j=1}^N{1}_{X_{\mathit{ij}}\le k}/N$ is essentially a nonparametric estimate of the true proportion, which is robust to the underlying distribution; $F_{\mathit{NB}}\left({\overline{X}}_i,,,\widehat{\varphi }\right)$ is a parametric estimate of the proportion, under the assumption of a negative binomial model. Our test statistic compares the difference between a nonparametric and a parametric estimate to see whether there is significant discrepancy between the two. If the difference is significant, it suggests that negative binomial is not a good fit. Since we test for observed $k$-proportion being strictly larger, the rejected hypothesis indicates existence of highly expressed outliers.

Datasets and pre-processing

Throughout the analysis, we used publicly available single-cell sequencing data from the 10X protocols. All the data sets were analyzed after their own filtering processes. No extra filtering was performed.

In Zheng et al. dataset [12], only cells that are labeled as CD34 + were selected for analysis. There are 277 cells with 31,482 genes. We performed UMAP on log-transformed data. We used k-means to separate cells into 3 subtypes using their UMAP coordinates. We applied the k-proportion inflation test on UMI counts for each subtype and with all subtypes combined.

For the T2D dataset, there are 160,954 cells with 32,746 genes, from 29 donors. In the original dataset, cells are annotated into 15 cell types/subpopulations. For each gene in each cell, we have UMI counts for all transcripts, spliced and unspliced. We extended the k-proportion inflation analysis to beta cells from the subtype labeled as β₂ cells, which contained 35,551 cells.

Downstream analysis

Cell annotation analysis was performed using the findmarker() function in Seurat. The GO enrichment analysis was performed using ClusterProfiler. Transcription factor enrichment analysis was performed using ChEA3. The visualization was performed using ComplexHeatmap [21].

Data and code availability

Data from Zheng et al. [12] were downloaded from the 10X website: http://support.10xgenomics.com/single-cell/datasets. The curated CD34 + subset with subtype labels and the β-cell subset from the T2D study analyzed here—including processed data sufficient to reproduce this manuscript’s results (count matrices and metadata for the selected cells)—is available at Zenodo (https://doi.org/10.5281/zenodo.16854251) [22]. The full dataset generated for the T2D project, including additional subsets of the β-cell subset and non-β cells, will be released with a separate, forthcoming publication under accession GSE234313 [23]; controlled raw sequencing reads will be available via NIH dbGaP (accession to be provided in that article). The RegulationIndex R package implementing the methods described here is available under the BSD 3-Clause license (https://github.com/ChenMengjie/RegulationIndex). The analysis code used to reproduce the figures is archived at the same Zenodo record: https://doi.org/10.5281/zenodo.16854251 [22].

Authors’ contributions

M.C. conceived this work, developed the methods, performed the analyses, and wrote the paper.

Funding

The work was supported by National Institutes of Health grant R01 GM126553, R01 HG011883, and HG012927, and additional grant no. NSF 2016307 and Sloan Research Fellowship to M.C.

Declarations

Ethics approval and consent to participate

Ethics approval is not applicable to this study.

Competing interests

The author declares no competing interests.