A Manifold-Based Measure of Transcriptional Entropy for Quantifying Aging in Single Cells

Abstract

Characterizing cellular aging is essential for understanding age-related diseases. While tissue-level studies reveal broad age-associated changes, they often reflect compositional shifts rather than cell-level reprogramming. The cellular damage hypothesis posits that aging involves the accumulation of DNA, chromatin, and other damage across molecular layers, increasing transcriptional entropy. Existing supervised methods for detecting cellular senescence yield cell type–specific senescence scores but rely on labeled data and lack generalizability. Here, we introduce a first-principles framework for quantifying transcriptional entropy in single cells as each cell’s deviation from a transcriptomic manifold, capturing breakdown of transcriptional coordination. This unsupervised approach identifies aging-affected cell types and distinguishes two cellular aging mechanisms: loss of expression precision and activation of stress-response pathways in high entropy cells. Applied to Tabula Muris Senis and SenNet Multiome datasets, transcriptional entropy correlates with chromatin-based mitotic age and highlights regenerative tissue compartments as most affected by aging.

Introduction

Cellular aging is widely believed to arise from the gradual accumulation of molecular damage^1–6. According to the cellular damage hypothesis, damage to the genome propagates disorganization across molecular layers, impairing the coordination of normal cellular functions. DNA damage accumulates even in non-dividing cells such as neurons^7–9 and muscle fibers^10,11, and evidence shows that these lesions, along with persistent chromatin alterations triggered by DNA repair, contribute to widespread dysregulation in gene expression with age^12–15. At the epigenetic level, age-related loss of chromatin accessibility and disruption of repressive domains further amplify transcriptional disarray, with aging cells exhibiting increasing variability in gene regulation across tissues and individuals^13,16–22.

This rise in transcriptomic variability has been the focus of numerous studies seeking to quantify transcriptional entropy as a hallmark of aging or aging-related conditions^{14,21,23–32}. At the bulk sequencing level, several reports have suggested that gene expression becomes more variable with age^{26,27,33–35}, but others have failed to detect a consistent signal^36–38, leaving the extent and significance of transcriptional entropy an open question. Single-cell RNA sequencing has opened the door to more granular assessments of transcriptional noise, allowing recent studies to trace age-related increase in variability from various perspectives. One common strategy is to cluster cells into putatively homogeneous states and compute gene-level or cell-level variances within each cluster^29,39–44. However, this approach depends critically on the resolution of clustering: too coarse and biologically distinct states may be merged, while too fine and technical noise may dominate the variance estimate. On the other hand, multiple approaches estimate cell-level transcriptomic variances or identify aging-related features through carefully curated aging-related gene sets or data collections^{24,28,31,45–51}. In parallel, a growing number of methods quantify biological aging by measuring stochastic epigenetic variation in DNA methylation data using predefined sets of CpG sites^52–61. Predictability of both classes of approaches is typically optimal within cells from similar molecular environments, yet their generalizability across diverse cell types remains limited. As a result, different methods can yield conflicting assessments of age-related variance when applied to the same datasets. This inconsistency is also underscored by recent benchmarking efforts^25,43: existing tools and metrics showed poor agreement across aging datasets and tissues, of which none could consistently recover meaningful age-related variance patterns.

To overcome these limitations, we developed a quantitative measure of transcriptional entropy that can be applied to single-cell RNA sequencing data to detect the loss of transcriptional coordination with aging. We show that transcriptional entropy is consistent with classical definitions of intrinsic transcriptional noise and demonstrates its utility across multiple tissues. In bone marrow, kidney, and liver, we use this measure to isolate both cell types that become disordered with age and gene programs that reflect age-related transcriptional instability. In particular, we test two distinct hypotheses about aging-associated gene programs: whether a gene becomes more disordered with age, and whether its expression is associated with the overall disorder of a cell. Together, these analyses highlight transcriptional entropy as a scalable and interpretable tool for dissecting the molecular basis of aging.

Results

Defining and Quantifying Transcriptional Entropy

We start with a first principle-based definition of transcriptional entropy quantifiable from single-cell transcriptomic data. Our definition bypasses clustering and predefined markers by leveraging the low intrinsic dimension of gene expression. We assume that the transcriptome-wide gene expression vector representing single cells are constrained to lie close to a high dimensional manifold, which captures the coordination of gene regulatory programs. This manifold captures the underlying geometry of cellular states and can be estimated from a low-dimensional embedding of gene expression data. Importantly, individual cells do not lie precisely on this manifold but deviate from it due to both technical and biological noise. Extrinsic biological variation, which captures the differences between cell types and other types of coordinated variation across genes, is captured by distances along the manifold. In contrast, intrinsic biological noise, which refers to idiosyncratic variation that is not coordinated across genes, is captured by deviation of a cell’s true transcriptomic state from the manifold. We refer to the magnitude of this intrinsic biological variation as “transcriptional entropy” (Figure 1A).

Figure 1. Defining and quantifying transcriptional entropy.

A) Conceptual framework for defining transcriptional entropy. A low-dimensional manifold (blue surfaces) is estimated from single-cell RNA data to capture coordinated states across all cells (black dots). Deviations from this manifold reflect both extrinsic variation and intrinsic variation (transcriptional entropy), of which the latter is used to quantify cellular damage and aging. B) For each gene $g$ and cell $c$, a variance-normalized squared residual deviation between inferred expression (${\lambda }_{gc}$) and manifold prediction (${\mu }_{gc}$) is computed. Averaging them across genes yields cell-level entropy. Averaging them across cells yields gene-level entropy. C) In F1 hybrid mice, intrinsic variation in allele-specific counts is estimated through allelic discordance and in total counts is estimated through gene-level entropy. Scatter plot of allelic discordance vs. log-transformed gene-level entropy shows that transcriptional entropy is consistent with classic definitions of intrinsic transcriptional noise. D) Log-transformed cell- (left) and gene-level (right) entropy of ESCC cells exposed to 0 Gy and 12 Gy doses of ionizing radiation.

Our framework distinguishes biological entropy from technical variation by explicitly modeling technical noise and removing its contribution from observed gene expression dispersion^62,63. Then, for each cell, we define its cell-level transcriptional entropy as the squared deviation from its projection on the learned manifold, averaged over genes and adjusted against a gene- and cell-specific expected variance term (Figure 1B). Similarly, gene-level entropy is computed as the average of squared residual deviation from the manifold across cells, standardized against the expected variance for that gene, again after accounting for technical noise (Figure 1B). These cell- and gene-level entropy scores can be compared across individuals from different age strata to identify cell types and genes that become dysregulated with age. See Methods for details.

Transcriptional entropy approximates classic definitions of intrinsic transcriptional noise

In pioneering studies in the early 2000’s, Elowitz et al. articulated the distinction between intrinsic transcriptional “noise” and extrinsic transcriptional variation⁶⁴. In their seminal dual-reporter experiments, two identically regulated fluorescent proteins (e.g., GFP and YFP) were expressed from separate genomic loci in single cells, and the per-cell difference in their expression levels was used to quantify intrinsic transcriptional variation. In this way, they defined intrinsic transcriptional variation as stochastic fluctuations arising from the randomness of transcriptional processes in an otherwise homogeneous intracellular environment. While single-cell RNA sequencing lacks the temporal resolution or controlled reporter design of these experiments, we mimicked the dual-reporter design through allele-specific gene expression analysis in F1 hybrid mice.

We extracted allele-specific reads from scRNA-seq data of F1 hybrid mice and treated each allele of a given gene as a replicate of identically regulated units⁶⁵. Some loci exhibit allele-specific biases which are reproducible across cells, which can be adjusted through allele-specific mean subtraction. After this adjustment, the genome-wide pattern of allelic discordance serves as a proxy for intrinsic noise in the classical sense. In parallel, we used our manifold-based approach to compute transcriptional entropy. Note that, while allelic discordance can only be applied to genes with heterozygous loci in the sequenced region, the manifold-based entropy measure can be applied universally. As shown in Figure 1C, allelic discordance, computed from the difference between alleles, correlates strongly with manifold-derived entropy, which is based only on the sum of the alleles (r = 0.46, p < 2.2 × 10⁻¹⁶). This provides empirical support that the latter is indeed measuring intrinsic transcriptional noise. This agreement with classic definitions reinforces the biological relevance of transcriptional entropy and adds conceptual rigor to its application as a molecular metric of aging.

Transcriptional entropy increases with cellular damage in irradiated cells

Before interrogating the heterogeneous and subtle changes expected in natural aging, we began with an extreme laboratory scenario in which transcriptional disruption is expected to be pronounced. Ionizing radiation is a well-characterized inducer of DNA and chromatin damage and provides a controlled perturbation to assess the sensitivity of transcriptomic measures to cellular injury. Using an scRNA-seq dataset on esophageal squamous cell carcinoma (ESCC) cells⁶⁶, we computed both cell-level and gene-level entropy for cells exposed to 0 or 12 Gray doses. As shown in Figure 1D, both entropy scores significantly increase in 12 Gy treated cells from 0 Gy treated cells. This finding supports the view that transcriptional entropy reflects damage-induced breakdown of gene expression coordination: as DNA lesions and chromatin disorganization accumulate, gene expression becomes increasingly stochastic and decoupled from coordinated regulatory programs. The concordance between radiation-induced damage and elevated transcriptional entropy reinforces the view that entropy captures a fundamental aspect of cellular integrity.

Transcriptional entropy increases with age in regenerative cell compartments

We now apply the entropy framework to single-cell RNA sequencing data for the bone marrow of Tabula Muris Senis⁶⁷. Notably, UMAP visualizations indicate that transcriptomic embeddings for most cell types, including multiple hematopoietic lineage cell types, are intermixed across various age strata (Supplementary Figure 1). This absence of age-associated clustering underscores the challenge of identifying aging signatures in heterogeneous and dynamic tissues like bone marrow. We computed the manifold-based cell-level and gene-level transcriptional entropy for all 18 annotated cell types spanning the hematopoietic differentiation hierarchy. Gene- and cell-level entropy measures were estimated separately for each cell type, allowing us to dissect aging dynamics for cells of varying differentiation potential (Supplementary Figure 2, 3). In particular, hematopoietic precursor cells (HPCs) showed a striking, monotonic increase in transcriptional entropy with chronological age at both the cell and gene-levels (ρ = 0.397, p < 2.2 × 10⁻¹⁶, Figure 2A). This trend was robust at the transcriptome level, even though not all individual genes showed age-associated increases in entropy. Intriguingly, the strength of the entropy–age association declined progressively along the hematopoietic differentiation axis (Figure 2B): granulopoietic progenitors retained a positive trend (ρ = 0.34, p < 2.2 × 10⁻¹⁶), while more terminal populations such as monocytes and macrophages exhibited no consistent age-related increase in entropy. This pattern aligns with previous observations that mutational burden accumulates primarily in long-lived hematopoietic stem and multipotent progenitor cells while terminally differentiated granulocytes experienced minimal mutations⁶⁸.

Figure 2. Transcriptional entropy in mouse stem cell compartments.

A) Violin and box plots showing log-transformed cell-level entropy and log-fold change of gene-level entropy with respect to 1-month-old mice by age group in 2144 hematopoietic precursor cells, along with Spearman’s ρ and corresponding p-values to indicate their associations with age. Values in the top and bottom 1% quantiles are considered outliers and are removed from the illustration. Red dashed line denotes zero log-fold change. Pairwise comparisons between age groups were tested using one-sided Wilcoxon rank-sum test (alternative hypothesis: older age group has lower dispersion). Significance levels are annotated above each comparison (NS = not significant, * p < 0.05, ** p < 0.01, *** p < 0.001). B) Violin and box plots showing log-fold change of gene-level entropy with respect to 1-month-old mice by age group across hematopoietic lineages. C) Violin and box plots showing SenMayo score (left) and log-fold change of gene-level entropy relative to the baseline age group (right) for selected stem cell populations. Baseline age groups are indicated in paratheses of each subplot title. D) Violin and box plots showing SenMayo score of hematopoietic precursor cells. E) Scatter plot showing correlation between log-transformed cell-level entropy and SenMayo score of hematopoietic precursor cells. Black solid line denotes the best-fit regression line, with cells above it or below it denoted as low or high entropy. Age proportions for low and high-entropy cells are shown in the stacked bar plots (bottom).

To compare with existing measures of transcriptional variance, we applied published methods that quantify transcriptional variability. Specifically, we calculated Euclidean distances of each cell to the average expressions of its cell type⁴², Euclidean distances to the tissue-wide average restricted to invariant genes⁴², and Scallop scores that quantify the stability of each cell’s cluster assignment with resampling⁴³. None of these approaches revealed any notable age-related shifts, consistent with the interpretation that they conflate extrinsic and intrinsic sources of variation and thus cannot directly capture entropy (Supplementary Figure 4–6).

We extended the analysis to additional tissues profiled in the aging mouse atlas, including lungs, limb muscle, and adipose tissue. Across these tissues, we observed a consistent increase in transcriptional entropy with age in their stem cell compartments (Figure 2C). However, in more differentiated or specialized cell types, there was no consistent relationship between entropy and age (Supplementary Figure 7–12). Specifically, Figure 2C illustrates representative trends in four progenitor populations: hematopoietic stem cells in bone marrow, type II pneumocytes in lung, and mesenchymal stem cells in limb and adipose tissue.

Comparison to SenMayo highlights advantages of a first-principles entropy framework

While literature lacks reliable gene-level markers of senescence, SenMayo serves as a multi-gene composite score that can be computed at the single cell level and that has shown utility in hematopoietic aging⁴⁵. As shown in Figure 2D, SenMayo scores increase with age in hematopoietic precursor cells and correlate strongly with transcriptional entropy, yet this is not universal across progenitor cell types from other tissues. This is expected, as SenMayo was trained primarily on cells from the hematopoietic compartment.

We further examined the relationship between transcriptional entropy and the SenMayo score by performing direct cell-by-cell comparison for the hematopoietic stem cells, where both measures performed well (Figure 2E). As expected, the two metrics were significantly correlated, consistent with their shared association with chronological age (r = 0.22, p < 2.2 × 10⁻¹⁶). Importantly, cells with low SenMayo score but high transcriptional entropy predominantly came from old mice. Within each SenMayo score strata, we observed a significant association between age and transcriptional entropy: among cells with similar SenMayo scores, those with higher entropy primarily originated from 24- and 30-month-old mice, the oldest cohorts in the dataset. We also see this effect for type II pneumocytes in lungs, the one other tissue where the two measures are highly correlated (Figure 2C, Supplementary 13–15). This enrichment of high entropy cells from older mice, after stratifying by SenMayo score, shows that transcriptional entropy more effectively identifies highly aged cells, possibly including those that fall outside the decision boundary of a marker set learned specifically on bone marrow cells. These findings underscore the potential of entropy to capture transcriptomic disarray in aging cells without reliance on designated senescence markers.

Kidney aging: transcriptional entropy reveals segment-specific decline in proximal tubule cells

Next, we applied the transcriptional entropy framework to the aging rat kidney, where we used 10x Multiome to simultaneously profile transcriptome and chromatin accessibility within single cells. Kidney samples were collected from rats at their 16, 30, 56, and 82 weeks of age, and our analysis focused on the 12075 proximal tubule (PT) cells identified by clustering and manual annotation using canonical markers. PT cells from all age groups were uniformly intermixed in the transcriptomic embedding, regardless of the dimensionality reduction method employed, underscoring the subtlety of aging-related transcriptomic changes and the challenge of isolating aging cell states based solely on global expression profiles (Figure 3A).

Figure 3. Transcriptional entropy of PT cells in aging rat kidney.

A) Experimental design and UMAP visualization of 41,394 kidney cells with scMultiome profiles from 16- to 82-week-old rats. Cell types and ages of 20,873 PT cells are highlighted in the UMAP. B) Violin and box plots showing log-transformed cell-level entropy and log-fold change of gene-level entropy with respect to 16-week-old rats by age group for S1 to S3 PT cells. C) In parallel with transcriptional entropy, EpiTrace mitotic age score is computed by quantifying accessibility of aging-associated differentially methylated loci. Scatter plot shows the correlation between log-transformed cell-level entropy and EpiTrace mitotic age score of PT cells, with marginal density plots by age group to the side. Red dashed line denotes the best-fit regression line. D) Line plots showing log-transformed gene-level entropy across age groups for 29 PT segment identity markers. E) Scatter plots showing correlations between normalized expressions of two inflammation and stress-related markers, CCL2 and CD44, and log-transformed cell-level entropy adjusted for total counts of each cell. Black dashed line denotes the best-fit regression line. F) Significantly enriched pathways for genes with gain in transcriptional entropy (top) and activation in high-entropy cells (bottom). Dot size denotes −log10(p-value) and x-axis denotes normalized enrichment score (NES).

Despite the lack of overt clustering by age, there is a significant increase in transcriptional entropy with age in PT cells, both at the cell- and gene- levels (Supplementary Figure 16). This upward trend is particularly significant between 16 to 30 weeks, and between 30 to 56 weeks. The proximal tubule can be anatomically and functionally divided into three segments: segment 1 (S1), segment 2 (S2), and segment 3 (S3), each characterized by distinct marker genes and physiological roles. PT cells can be assigned to different segments based on segment-specific marker genes. Upon stratifying cells by segment (Figure 3B), we found that the age-associated increase in transcriptional entropy was pronounced in segments 2 and 3, but not in 1. By contrast, published methods estimating transcriptional variability did not capture this aging trend (Supplementary Figure 17–19). Our findings suggest that S1 PT cells are relatively protected from age-related stressors, whereas S2 and S3 cells are more susceptible to age-induced transcriptomic dysregulation.

These findings align with existing literature indicating differential vulnerability among proximal tubule segments. S2 and S3 segments are more prone to ischemic injury and nephrotoxic damage, reflecting their higher metabolic activity and exposure to reabsorption stressors^69–73. In contrast, S1 segments exhibit greater resilience due to their distinct protective mechanisms and cellular environments: located more proximal to the cortical labyrinth, S1 cells receive comparably more oxygen supply than S2 and S3 cells to counter their high metabolic demand, thereby reducing ischemic injury risk. Furthermore, S3 segments have been shown to possess regenerative capacity in response to recurring injuries^74–76. Thus, our finding that transcriptional entropy is higher in the more injury prone, regenerative tissue compartment aligns with our observations of elevated entropy in stem cell populations in Tabula Muris Senis data.

We hypothesized that increased transcriptional entropy in the aging rat kidney should be associated with molecular drift at the level of chromatin organization. To test this, we leveraged the matched chromatin accessibility data generated alongside the cellular transcriptomes and applied EpiTrace, a recently published method for estimating cellular mitotic age from chromatin accessibility profiles⁶¹. EpiTrace computes a global aging score based on the accessibility of loci previously identified as differentially methylated with age in bulk tissue methylation studies, offering a chromatin-derived readout of replicative history. The original EpiTrace study provided strong evidence that this score could reliably stratify cells by mitotic age across diverse contexts.

We compared EpiTrace mitotic age estimates with transcriptional entropy scores at single-cell resolution (Figure 3C, Supplementary Figure 20). As shown in Figure 3C, the two measures are significantly correlated across biological age groups (r = 0.40, p < 2.2 × 10⁻¹⁶), despite being derived from entirely distinct molecular modalities and conceptual frameworks. Even among PT cells derived from organisms of the same chronological age, particularly at 16 and 30 weeks, transcriptional entropy and EpiTrace scores remain significantly associated, suggesting that both capture an underlying dimension of cellular aging during the development and proliferation of PT cells (Supplementary Figure 21). This cross-modal concordance reinforces the interpretation of transcriptional entropy as a mechanistically grounded marker of cellular aging.

Age-linked gene programs: increase in transcriptional entropy and cellular response

To further interpret the mechanistic underpinnings of transcriptional entropy, we turned our attention to identifying entropy-associated genes and pathways. In this analysis, we asked two questions for each gene: (1) does the gene exhibit an increase in transcriptional entropy with age — indicating its expression becomes less predictable across cells, and (2) is the gene’s expression correlated with cell entropy level — indicating that its activation is linked to cellular response to transcriptional disarray. These two modes of association capture different biological phenomena and should not be conflated. We hypothesize that genes involved in maintaining stable cellular identities, such as PT segment markers, should lose regulatory precision over time and therefore exhibit increased gene-level entropy with age. In contrast, genes involved in inflammation and senescence pathways are often induced in response to damage or stress, and thus for these genes, we do not necessarily expect an increase in their gene-level entropy. Instead, we postulate that their expression levels should positively correlate with cell-level entropy scores, as cells with high transcriptional entropy are more likely to exhibit hallmarks of senescence. Thus, this analysis allows us to disentangle two mechanisms of gene-age association: one reflecting the loss of expression precision and the other reflecting programmatic activation in damaged or disordered cellular states. The distinction between these two modes provides a nuanced framework for interpreting how different biological pathways contribute to aging at the molecular level.

In line with the first question, we curated a set of 29 PT segment identity markers and observed significant age-dependent increases in their gene-level entropy, particularly across 16, 30, and 56 weeks (Supplementary Table 1). This observation indicates that PT cells progressively lose segment fidelity during initial stages of aging (Figure 3D). Addressing the second question, we identified two inflammation and stress-related markers whose normalized expression levels showed significantly positive correlations with cell-level entropy scores, after adjusting for total count differences across cells and multiple testing: CCL2, a chemokine secreted by stressed or injured PT cells to mediate inflammation (r = 0.71, p = 8.3 × 10⁻⁵)⁷⁷, and CD44, a fibrosis-related marker that characterizes maladaptive, high-entropy states of PT cells in chronic kidney diseases (r = 0.23, p = 2.4 × 10⁻⁵)⁷⁸, highlighting the coordinated activation of inflammation and stress-response programs in high-entropy cellular states (Figure 3E). Together, these findings suggest that transcriptional entropy, whether assessed at the gene level or cell level, captures key aging-related signatures of PT cells: the breakdown of segment distinctions and the activation of pro-inflammatory pathways.

Beyond these marker-level observations, we next systematically examined pathway-level associations with transcriptional entropy in PT cells (Figure 3F). We ranked all genes by the magnitude of their age-dependent changes in entropy using Spearman’s ρ and identified 5,022 genes with significantly positive changes. Pathway enrichment analyses revealed strong representation of apoptosis and senescence-related pathways such as p27 and p53 signaling, the caspase pathway, and the senescence-associated secretory phenotype (SASP); mitochondrial pathways such as calcium ion homeostasis, biogenesis, and the TCA cycle; and DNA repair processes, including nucleotide excision repair and homologous recombination (Supplementary Table 2). The involvement of these pathways suggests that the loss of expression precision not only intersects directly with canonical aging hallmarks but also captures key molecular alterations such as mitochondrial dysregulation and genome instability. At the cell level, using correlation values for 994 genes whose expression levels were significantly associated with transcriptional entropy (Supplementary Figure 22), we found enrichment in pathways related to immune activation and extracellular remodeling, including antigen processing and presentation, cytokine activity, leukocyte chemotaxis, and extracellular matrix organization (Supplementary Table 3). These results indicate that disordered cellular states are characterized by coordinated activation of inflammatory and fibrotic responses, consistent with the emergence of maladaptive programs during aging in PT cells.

Liver aging: transcriptional entropy captures broad dysregulation of gene expression beyond cell mitotic age

We next applied the transcriptional entropy framework to study liver aging. To jointly profile transcriptional and chromatin-level aging, we performed 10x single-cell Multiome analysis on liver tissue collected from young (4 months) and old (22 months) mice (Figure 4A). After quality control, we obtained 183,005 high-quality liver cells across both age groups. Our analysis focused on the 66,816 high-quality hepatocytes identified through clustering and marker-based annotation.

Figure 4. Transcriptional entropy of hepatocytes in aging mouse liver.

A) Experimental design and violin and box plots showing log-transformed cell-level and gene level entropy of 66,816 hepatocytes with scMultiome profiles from young (4 month) and old (22 month) mice. B) Scatter plot showing correlation between log-transformed cell-level entropy and EpiTrace mitotic age score of hepatocytes, with marginal density plots by age group to the side. Black dashed line denotes the best-fit regression line. C) Schematic of hepatic zonation including periportal (zone 1), midlobular (zone 2), and pericentral zones (zone 3) and line plots showing log-transformed gene-level entropy of 21 zonation identity markers in their respective zones. Genes with age-related entropy increase are labeled. Black dots indicate mean entropy and I-shaped bars indicate 95% confidence intervals. D) Histograms showing the log-transformed mean cell-level entropy gain with age across zonation bins, with bins corresponding to each hepatic zone highlighted. E) Significantly enriched pathways for genes with gain in transcriptional entropy (top) and activation in high-entropy cells (bottom). F) Summary of significantly enriched pathways for genes with zone-specific gain in transcriptional entropy (top) and activation in zone-specific high-entropy cells (bottom) across zone 1–3.

Similar to other tissues, transcriptomic embeddings of hepatocytes via standard workflows showed no distinct clustering by age (Supplementary Figure 23), as aging-related transcriptomic changes are subtle and distributed. Despite this, transcriptional entropy, quantified at both the gene and cell levels, was significantly elevated in hepatocytes from old mice relative to young controls (p < 2.2 × 10⁻¹⁶, Figure 4A). As for the kidney, we further computed mitotic age by applying EpiTrace to the cell-matched chromatin accessibility data (Supplementary Figure 24). EpiTrace scores were significantly positively correlated with transcriptional entropy (Figure 4B), echoing the alignment observed in the aging kidney. By contrast, only 1 of the 3 published methods were able to capture the elevated transcriptional variance (Supplementary Figure 25–27).

Despite its strong statistical significance (r = 0.31, p < 2.2 × 10⁻¹⁶), the correlation between transcriptional entropy and EpiTrace’s estimate of mitotic age is moderate, reflecting the difference in what they capture. Transcriptional entropy increases uniformly across hepatocytes from old mice. In contrast, EpiTrace scores in old mice exhibit a bimodal distribution, revealing a subpopulation of hepatocytes with elevated mitotic age while another remains comparable to that of young mice. As hepatocytes are largely post-mitotic, this bimodal pattern likely reflects partial regenerative activity in aging livers: the subpopulation with noticeably higher EpiTrace scores may correspond to cells that have undergone replication during regeneration, whereas the remainder retain mitotic histories similar to young hepatocytes. Thus, the two measures capture distinct aspects of the aging process: EpiTrace reflects cumulative mitotic history, while transcriptional entropy captures a broader dysregulation of gene expression independent of cell division.

Liver aging: transcriptional entropy reveals zone-specific vulnerability

The liver is organized into lobules with hepatocytes arranged along defined spatial zones that perform distinct metabolic and detoxification functions, starting at the periportal region (zone 1), extending through the midlobular region (zone 2), and ending at the pericentral region (zone 3). We explored how aging affects hepatocyte spatial identity by analyzing a curated set of 21 gene markers for zone identity (Supplementary Table 4). Gene-level transcriptional entropy estimates revealed zone-specific aging effects: pericentral and midlobular markers showed the strongest increases in entropy with age, while most periportal markers remained relatively stable (Figure 4C). A notable exception among periportal markers was SDHD, a key gene in mitochondrial oxidative metabolism, which exhibited substantial increase in entropy in old cells. These results suggest that hepatocyte zonal specialization becomes increasingly unstable with age, particularly in zones 2 and 3.

To further assess the change in hepatocyte zonal integrity with age, we assigned each cell a zonation score defined as the ratio of total pericentral marker expression to the combined expression of pericentral and periportal markers. As expected, zone 1 (periportal) and zone 3 (pericentral) markers showed high expressions near 0 and 1, respectively (Supplementary Figure 28). Zone 2 (midlobular) markers, which are not used in the zonation score calculation, are low near 0 and 1 and high in the middle, serving as an independent validation that this score effectively captures the zonation gradient (Supplementary Figure 29). We then ranked all cells by their zonation score and divided them into 10 equal sized bins. Then, we averaged the cell-level entropy for all cells in each zonation bin, separately for young and old mice. Across both age groups, we observed that the mean cell-level entropy and the proportion of high-entropy cells (top 5%) varied systematically across the zonation axis and tracked closely with one another (Supplementary Figure 30). Within each zonation bin, old cells consistently showed significantly higher entropy than their young counterparts, and this difference was not attributable to changes in cell proportions, which remained close to 50:50 (Supplementary Figure 31). Interestingly, the magnitude of this age-related entropy increase exhibits significant spatial variations: moderate in the periportal zone, elevating to a peak in the midlobular zone, declining towards pericentral, then peaking again at the terminal of the pericentral zone (zone 3) (Figure 4D). These patterns highlight the spatial differences in age-associated transcriptional dyscoordination, which will be examined in greater detail in the subsequent zone-specific analysis.

Liver aging: hepatocytes with elevated entropy engage in senescence, inflammatory, and stress-response programs

To examine the association between transcriptional entropy and the aging-related processes of senescence and inflammation, we curated 138 genes enriched in senescence and pro-inflammation processes and assessed the correlation of their expression with cell-level entropy, following the same approach applied to aging kidney PT cells (Supplementary Table 4). In total, 29 (21.0%) of the markers showed significant positive correlations, which is significantly higher than the background frequency of 8.8%, and the frequency among the zonation markers genes (4.7%). Note that this analysis is different from the gene-level entropy analysis on the zonation markers shown in Figure 4C: while that analysis assessed the increase in dysregulation at the gene level, this analysis identified genes whose expressions correlate with cell-level entropy. As previously discussed when analyzing the aging rat kidney, these two approaches interrogate distinct aspects of aging-related functional decline.

The high numbers of senescence- and pro-inflammation genes with significantly positive associations with cell-level entropy supports a close connection between increased transcriptional entropy and activation of stress and senescence programs in hepatocytes (Supplementary Figure 32). Example genes include AXL, a cell differentiation- and carcinogenesis-associated gene whose expression is heightened in parallel to chronic liver disease progression⁷⁹, CCND2, a cell cycle regulator that was previously found to increase its expression in senescent hepatocytes⁸⁰, IL1B, which promotes hepatic inflammation through inflammasome activation in mice⁸¹, and MMP9, which is involved in extracellular matrix remodeling and contribute to hepatic fibrosis and apoptosis during injury⁸². Overall, genes whose expression levels increased in high-entropy cells (Supplementary Figure 33) were enriched for inflammatory and immune signaling pathways (e.g., cytokine–cytokine receptor interaction, chemokine signaling, interferon response), cell death programs (e.g., apoptotic signaling, necroptosis), autophagy-related processes (e.g., macroautophagy, selective autophagy), and RNA stress responses (e.g., cytoplasmic stress granule formation) (Figure 4E, Supplementary Table 5). These findings support the conclusion that hepatocytes with elevated transcriptional entropy engage stress-response and inflammatory programs characteristic of dysfunctional or damaged cellular states.

In contrast, genes with the greatest increase in transcriptional entropy (a.k.a. increase in dysregulation) with age were enriched in RNA and ribosome-related processes (e.g., rRNA processing and modification, ribosome biogenesis) and metabolism (e.g., glutathione metabolism, metabolism of xenobiotics by cytochrome P450) (Figure 4E, Supplementary Table 6). The increased entropy of RNA and ribosome pathways suggests reduced precision of proteostasis with age. The increased entropy in antioxidant and detoxification pathways points to a breakdown in the coordination of hepatocellular defense mechanisms, potentially rendering aged hepatocytes more susceptible to oxidative stress and xenobiotic toxicity.

Liver aging: zone-specific gene entropy and functional alterations with age

Using the zonation scores computed for each cell, we next examined how aging-related changes differ across spatial zones of the liver lobule. We first computed zone-specific gene-level entropy to identify genes exhibiting the greatest loss of transcriptional coordination with age. Notably, genes with the largest entropy increases were not necessarily those highly expressed ones, indicating that age-related transcriptional variability is not only driven by expression level (Supplementary Figure 34). We also extracted differentially expressed genes between high and low entropy cells within each zone to characterize zone-specific functional reprogramming accompanying transcriptional dysregulation. Below, we highlight genes identified through both approaches.

Starting with the periportal zone, aging hepatocytes showed greater entropy across genes involved in rRNA maturation and modification, miRNA metabolism, and mitochondrial electron transport, reflecting instability in the transcriptional and translational machinery that coordinates RNA processing and mitochondrial function. Notably, ASS1, involved in protein catabolism and hepatic lipid metabolism^83–85, showed the greatest age-related entropy increase in periportal hepatocytes (log FC = 0.47), suggesting heightened transcriptional variability in core anabolic programs. In parallel, high-entropy periportal hepatocytes exhibited activation of pathways in energy production and oxidative metabolism, including steroid and cholesterol biosynthesis, fatty acid oxidation, amino acid (e.g., L-serine, tryptophan) metabolism, and carbohydrate homeostasis. Enrichment of protein synthesis and trafficking processes, such as translation initiation, protein targeting, COPII-coated vesicle budding, and retrograde vesicle-mediated transport, were also observed in periportal hepatocytes likely as an adaptive compensation towards rising metabolic demand (Figure 4F, Supplementary Table 7, 8).

Pericentral hepatocytes also displayed a widespread metabolic stress profile. At the gene level, increase in transcriptional entropy was primarily associated with endoplasmic reticulum (ER) signatures, including organization, vesicle-mediated transport, protein localization and exit, stress response, and unfolded protein response (UPR). As the cellular center of protein folding, modification, and secretion, ER is expected to withstand a particularly heavy metabolic load in the pericentral zone that specializes in xenobiotic metabolism and detoxification. Upon ER stress in the form of accumulation of unfolded and misfolded proteins, UPR is triggered to restore its homeostasis and promote cell survival⁸⁶. Moreover, we observed elevated activity of pathways regulating integrated stress response signaling, apoptotic processes, and programmed necrotic cell death, suggesting activation of broader cellular damage response programs. Examples of genes with age-related entropy increase include GLUL (log FC = 0.26), a gene involved in glutamine synthesis and ammonia detoxification^87,88, and IQGAP2 (log FC = 0.86), a gene linked to cellular apoptosis and suppression of hepatocellular carcinoma⁸⁹. In parallel, activation of stress-response programs was evident in high-entropy pericentral hepatocytes. For instance, peroxisome-associated programs, including protein import, membrane transport, and organization, were prominently enriched. As a key oxidative organelle, peroxisome mediates metabolism of fatty acids and detoxification of reactive oxygen species (ROS) to reduce oxidative stress and maintain redox homeostasis^90,91. Autophagic processes such as autophagosome assembly, lysosomal and nuclear microautophagy, and late nucleophagy were also upregulated, reflecting elevated cellular recycling in response to damaged proteins and organelles. Concurrent enrichment of amino acid starvation response and target of rapamycin (TOR) signaling pathways further indicates adaptive nutrient sensing and energy conservation under the high metabolic burden of aging pericentral hepatocytes. Together, these findings indicate an uneven functional engagement of detoxification and metabolic programs resulting from an imbalanced cellular stress specific to the pericentral cells (Figure 4F, Supplementary Table 7, 8).

Finally, our data showed that the midlobular zone — the principal regenerative reservoir of hepatocytes — was most strikingly altered. We found that genes with increased transcriptional entropy were associated with tissue regeneration, cell cycle transition, telomere maintenance, and multiple RNA regulatory mechanisms, such as RNA methylation, translation, and mRNA splicing. For example, we detected significant age-related entropy increase in AXIN2 (log FC = 4.74), a WNT-responsive gene that contributes to liver homeostasis and repair^92,93. As WNT signaling is normally enriched in pericentral hepatocytes⁹⁴, its increase in entropy in midlobular zone is consistent with our finding of age-related loss of zone identity. Increased transcriptional entropy with age was also observed in TERT (log FC = 1.35), a critical gene for regenerative capacity through maintenance of metabolic balance and telomere integrity^95–97. Correspondingly, pathway enrichment analysis revealed that high-entropy midlobular hepatocytes were enriched in pathways related to genome maintenance and transcriptional regulation (e.g., DNA repair, histone acetylation, transcription, and RNA processing), proteostasis and post-translational control (e.g., protein ubiquitination, stabilization, modification, and phosphorylation), and stress response and cellular homeostasis (e.g., ROS response, acetyl-CoA metabolism, and transition metal ion homeostasis). These pathways focus on maintaining genomic integrity, protein quality, and metabolic balance, which are core processes underpinning the regenerative capacity of midlobular hepatocytes. Together, these results demonstrate that aged zone 2 hepatocytes exhibit erosion of reliable regenerative plasticity that normally defines their compartment, offering a mechanistic basis for the well-documented decline in liver regeneration with age (Figure 4F, Supplementary Table 7, 8).

In summary, trends observed across the three spatial zones of hepatocytes reflect their underlying functional specialization. Affected pathways in zone 1 and zone 3 hepatocytes align with longstanding evidence that they experience distinct but significant stressors with age. Periportal cells engage in oxidative metabolism and protein synthesis and operate under oxygen-rich conditions that favor metabolic rigidity and transcriptional stability, whereas pericentral cells are exposed to hypoxia and high concentrations of xenobiotic metabolites, rendering them prone to injury and stress responses. In parallel, the entropy peak in the midlobular zone echoes our findings in the bone marrow, where long-lived precursor populations such as hematopoietic stem cells show the most prominent age-associated increases in entropy. In the liver, midlobular (zone 2) hepatocytes have been shown to contribute disproportionately to tissue renewal throughout homeostasis, and our results raised the possibility that this regenerative compartment is particularly vulnerable to transcriptional disorganization with age^96,98.

Discussion

Together, these findings establish transcriptional entropy as a robust estimator of intrinsic gene expression noise, extending its classic definition⁶⁴ to modern single-cell transcriptomic data. The measure is rigorously validated across multiple biological systems, exhibiting strong cross-modal concordance with chromatin-based estimates of mitotic age and revealing loss of cellular identity and activation of inflammatory responses during aging through marker and pathway-level analysis. Unlike existing approaches, transcriptional entropy does not rely on curated gene markers or supervised training, instead offering a first principles-based method for quantifying cellular aging that can be applied to unseen cell types and tissue contexts. As single-cell atlases continue to grow, transcriptional entropy may serve as a scalable and universal tool for mapping aging dynamics, identifying vulnerable cell populations, and informing interventions aimed at preserving cellular integrity over time.

An important implication of this work is that transcriptional entropy offers a principled way to prioritize cell populations most vulnerable to aging-related dysfunction. Across tissues, we observed that entropy highlights compartments known to exhibit early molecular disruption or functional decline with age, even when they are not easily identifiable by conventional marker-based approaches. In the context of age-related diseases, these results suggest that entropy can serve as a molecular screen to identify susceptible cell types, guiding both mechanistic studies and therapeutic targeting.

A consistent theme across tissues is the heightened entropy observed in regenerative compartments — such as hematopoietic stem cells, segment 3 proximal tubule cells, zone 2 hepatocytes, and mesenchymal stem cells — relative to more terminally differentiated cells. These compartments are defined by their long lifespan and proliferative potential, rendering them uniquely exposed to the cumulative effects of molecular damage. Our observation that these same cells exhibit the largest increases in transcriptional entropy suggests that the very systems responsible for tissue maintenance may be particularly susceptible to stochastic dysregulation. While this may reflect their intrinsic vulnerability, it may also point to selective pressure for tighter transcriptional control in differentiated cells. Disentangling these effects, and understanding whether increased entropy causally impairs regenerative potential, remains an important avenue for future research.

In stem cell populations, increased entropy likely reflects age-related degradation of regulatory systems that normally maintain transcriptional fidelity. However, the same accumulated damage can also fuel clonal evolution, which intensifies with age⁹⁹. Such clonal dynamics give rise to genetically distinct lineages that can propagate into mature blood cells, a phenomenon clinically recognized as clonal hematopoiesis of indeterminate potential (CHIP)¹⁰⁰. Transcriptional entropy does not capture such heterogeneity: If clonal mutations induce coherent, reproducible expression changes across descendant cells, they would be absorbed into the learned manifold and do not contribute to the entropy score. The entropy measure is designed to estimate the magnitude of completely idiosyncratic and uncoordinated stochasticity. Thus, it captures a distinct dimension of aging, one that reflects intrinsic molecular disorder rather than structured heterogeneity driven by clonal expansion (Supplementary Figure 35).

At the gene level, we provide a conceptual framework for classifying aging-associated transcriptional changes along two axes: genes whose expression becomes more disordered with age (i.e., increasing gene-level entropy), and genes whose expression levels are associated with the entropy of the cell (i.e., activated in high-entropy cells). These categories capture distinct biological processes. The former reflects a breakdown in regulatory precision, and we find it enriched among genes defining cell identity such as zonation markers in the liver. The latter likely represents genes activated in response to cellular damage or dysregulation, including those involved in inflammation, stress response, and cell death. These axes are not redundant: they identify genes with distinct roles in aging biology and provide complementary entry points for understanding how aging reshapes our cells. Moving forward, integrating transcriptional entropy with spatial transcriptomics, lineage tracing, and perturbation experiments may offer powerful new ways to parse the causes and consequences of aging at cellular resolution.

Methods

Data preprocessing

The computation of transcriptional entropy takes as input a gene by cell raw UMI count matrix generated from single-cell RNA sequencing. By the nature of scRNA-seq data, many genes have nonzero counts in only few or zero cells. These extremely sparsely observed genes are not expected to be informative and are thus excluded from our analysis. As data quality varies across datasets, exact details for quality control are provided in the later data sections. Before computing transcriptional entropy, we apply a library size normalization to each cell, where a size factor is computed, defined as the cell’s total UMI count divided by the mean total UMI count across all cells, and the count for each gene is divided by this size factor.

Transcriptional entropy

Let $Y_{gc}$ denote the observed, library-size normalized UMI count for gene $g$ in cell $c$. Previous works have shown that UMI-based scRNA-seq data have technical noise that can be approximated by a Poisson^101,102 distribution with a library size scaling factor, and thus, in line with previous expression models^63,103–106, we assume that $Y_{gc}$, conditioned on the true expression ${\lambda }_{gc}$ of gene $g$ in cell $c$, has distribution:

$$Y_{gc}\mid {\lambda }_{\mathrm{g}\mathrm{c}}\sim \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left(s_c{\lambda }_{gc}\right)$$

where $s_c$ is the cell-specific size factor. We don’t observe the true expression $\left({\lambda }_{gc}\right)$, but as described in the main text, variations in ${\lambda }_{gc}$ across all cells can be decomposed into two components: extrinsic variation, which is captured by a manifold fitted to the data, and intrinsic variation, which is the remaining unpredictable component. To model this, we use a Gamma prior on ${\lambda }_{gc}$:

$${\lambda }_{gc}\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left({\mu }_{gc},v_{gc}\right),$$

where ${\mu }_{gc}$ denotes the prior mean and $v_{gc}$ denotes the prior variance. We assume that ${\mu }_{gc}$ is the “predictable” component of gene expression, while the deviation in ${\lambda }_{gc}$ from ${\mu }_{gc}$ is the unpredictable component. We denote the estimate of ${\mu }_{gc}$ from the data by ${\hat{\mu }}_{gc}$, i.e., ${\hat{\mu }}_{gc}$ is what we have been referring to as the “manifold” fit. The estimate ${\hat{\mu }}_{gc}$ can be achieved through various existing dimensional reduction methods^103,107,108. Here, we adopt the method from SAVER, which uses the glmnet function in $R$ with cross-validation to choose the degree of shrinkage. Then, given ${\hat{\mu }}_{gc}$, the true expression of gene $g$ in cell $c$ can be estimated via its Bayesian posterior mean, which essentially evaluates to a weighted average of the prior mean ${\hat{\mu }}_{gc}$ and the observed data $Y_{gc}$⁶³. Given both $\{{\hat{\lambda }}_{gc}\}$ and $\{{\hat{\mu }}_{gc}\}$, transcriptional entropy quantifies the deviations $\{{\hat{\lambda }}_{gc}-{\hat{\mu }}_{gc}\}$, which are summarized at the gene or cell level. Since the expected deviations vary according to ${\mu }_{gc}$, we need to standardize the observed deviations based on some variance model. Thus, we introduce a variance-scaled squared deviation, ${\delta }_{gc}$:

$${\delta }_{gc}=\frac{{\left({\hat{\lambda }}_{gc}-{\hat{\mu }}_{gc}\right)}^2}{{\hat{v}}_{gc}}$$

Here, ${\hat{v}}_{gc}$ serves as a normalization factor that adjusts for baseline variability expected of each gene in each cell. The estimation of $v_{gc}$ is based on one of three candidate models — constant coefficient of variation (CV), constant Fano factor (FF), and constant variance, as described below:

$$\mathit{constant}\mathit{CV}:v_{gc}={\mu }_{gc}^2/{\varphi }_g^{cv}$$

$$constantFF:v_{gc}={\mu }_{gc}/{\varphi }_g^F$$

$$constantvariance:v_{gc}={\varphi }_g^v$$

Marginal likelihood of the observed counts, given ${\hat{\mu }}_{gc}$, is calculated for each model, and the one that attains the highest likelihood is chosen. Then, ${\hat{v}}_{gc}$ is computed based on this model and the maximum likelihood estimate of the appropriate dispersion parameter, ${\hat{\varphi }}_g$.

The manifold-based prediction ${\hat{\mu }}_{gc}$ and the baseline variance ${\hat{v}}_{gc}$ are both computed through models fitted on the full cell population to provide a common reference for evaluating intrinsic variation across diverse cellular contexts. Examples of how these parameters are computed are introduced in later sections.

Finally, we summarize the variance-scaled squared deviation ${\delta }_{gc}$ into cell- and gene-level transcriptional entropy measures. At the cell level, entropy is calculated by averaging ${\delta }_{gc}$ across genes while fixing $c$:

$${\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}_c^{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}=\frac{1}{G}\sum _{g=1}^G{\delta }_{gc}$$

This reflects the degree to which cell $c$’s transcriptome as a whole diverges from its manifold prediction. At the gene level, entropy is calculated by averaging residuals across cells within a pre-defined stratum $S$ (e.g., a given cell type in a given age group) to quantify the intrinsic variability of each gene specific to the specific subpopulation $S$:

$${\mathrm{E}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}_{g,S}^{\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}}=\frac{1}{\left|S\right|}\sum _{c\in S}{\delta }_{gc}$$

To ensure reliable estimation, gene-level entropy is only calculated for strata $S$ containing at least 10 cells in our analyses. Overall, cell-level entropy allows direct assessment of transcriptional disorder for a given cell, while gene-level entropy enables identification of genes and relevant pathways that exhibit increase in dyscoordination within specific cell strata.

Allele-specific entropy analysis

We downloaded Smart-seq2 scRNA-seq libraries for 188 primary mouse fibroblasts for two alleles (CAST/EiJ × C57BL/6J) and preprocessed to remove genes with low total counts⁶⁵. We constructed a unified matrix by summing the counts of two alleles across all cells and used it to compute transcriptional entropy for each gene and compare it against allelic discordance calculated by following the standard formula.

Ionizing radiation damage entropy analysis

Single-cell RNA-seq count matrices were downloaded for 133 ESCC cells treated with different doses of ionizing radiation (41 with 0 Gy, 92 with 12 Gy)⁶⁶. All cells and genes passed the pre-processing. Gene- and cell-level transcriptional entropy were separately computed for 0 Gy and 12 Gy cells, revealing significant differences in transcriptional dysregulation.

Tabula Muris Senis entropy analysis

Raw droplet-based gene count matrices were downloaded from Tabula Muris Senis, a single-cell transcriptomic atlas of aging mouse tissues⁶⁷. Selected tissues for our analysis include bone marrow, lung, limb muscle, and adipose tissue, with cells sampled at 1, 3, 18, 21, 24, and 30 months of age. Annotations for cell type and age groups were directly taken from the accompanying metadata. In the original study, extensive quality control has already been conducted across all tissues for the raw count matrices: genes not expressed in at least 3 cells and then cells without at least 250 detected genes were removed. Additionally, cells with fewer than 2500 total UMI counts were excluded in droplet-based libraries. Therefore, we only performed one additional quality control step to make sure that entropy calculation applies only to expressed genes: across all cells within each tissue, genes with zero total counts were removed.

To compute transcriptional entropy, we estimated the manifold-based prediction ${\mu }_{gc}$ and the baseline variance $v_{gc}$ based on models fit on combined cell types within each tissue. The latent true expression ${\lambda }_{gc}$ was inferred separately for each cell type × age group stratum (e.g., 18-month-old hematopoietic precursor cells). Cell-level entropy was calculated by averaging the residuals across all retained genes in each cell, while gene-level entropy was calculated once for each stratum in each gene. Note that some of the strata were missing, since not all cell types were present across all ages and strata containing fewer than 10 cells were also excluded.

Three clustering-based methods that we used to compare against transcriptional entropy are from decibel (version 1.0) and Scallop R package (version 1.3.0)⁴³. Briefly, we used decibel to compute Euclidean distance from each cell to its cell type expression average and Euclidean distance from each cell to its tissue-wide expression average using a set of invariant genes. Invariant genes are selected by their coefficients of variation, of which the details can be found in the package documentation. Scallop is used to compute a membership score, which is the frequency of each cell getting assigned to its most frequently assigned cluster label. This score quantifies transcriptional stability and should be examined in the opposite way to other measures.

SenMayo is a set of 125 gene signatures proposed as robust cross-tissue and cross-species markers of cellular senescence and SASP activity⁴⁵. In our analysis, SenMayo scores were calculated by extracting the inferred true expressions of these genes and computing enrichment scores using the escape R package (version 2.4.0)¹⁰⁹. These scores were then compared against cell-level entropy across stem cell populations in each tissue.

Aging rat kidney entropy analysis

We extracted 10X scMultiome data for 4 kidney samples from 16-, 30-, 56-, and 82-week-old rats, respectively. Standard preprocessing through Seurat was performed to annotate 24 cell types across all 41394 cells¹¹⁰. For quality control, we removed cells with fewer than 250 detected genes or fewer than 2500 total UMI counts, as well as genes expressed in fewer than 5 cells. All analyses were conducted over the subset of 20873 PT cells from segment 1, 2, and 3 post quality control. Transcriptional entropy and clustering-based variance estimates for comparison were computed in the same procedure as described in the previous section.

EpiTrace is a method that estimates mitotic age of single cells by measuring chromatin accessibility in a set of reference “clock-like” genomic loci, as these loci are characterized by decreased heterogeneity in accessibility during cellular aging⁶¹. We used the paired ATAC data and selected the “iterative” option to compute mitotic age score. This option iteratively searches for additional loci showing best correlations between their accessibility and estimated mitotic age, starting from the initial reference set of clock-like differentially methylated loci. Since the original method used reference loci from human, we performed liftover from human genome (hg18) to rat genome (rn7) using the easylift R package (version 1.6.0)¹¹¹ before applying the method.

To search for pathways enriched in genes with gains in transcriptional entropy and with activation in high-entropy cells, we performed gene set enrichment analysis with the fgsea R package (version 1.34.2)¹¹². Pathway databases used include KEGG¹¹³, Gene Ontology¹¹⁴, and MSigDB Canonical Pathways¹¹⁵. For pathways related to genes with gains in transcriptional entropy, we ranked all genes by their Spearman’s ρ values across age groups. For pathways related to genes with activation in high-entropy cells, we ranked all genes by the correlation values between their log-normalized expressions and cell-level entropy across all cells. In both cases, enrichment p-values were adjusted for multiple testing using the Benjamini-Hochberg procedure. Pathways with an FDR-adjusted p-value ≤ 0.05 were considered significant.

Aging mouse liver entropy analysis

We collected 10X scMultiome data for 6 samples in young mouse liver and 10 samples in old mouse liver with respectively equal distribution of males and females. Standard preprocessing through Seurat was performed for cell type annotation, where all cells annotated as hepatocytes were retained for downstream analysis. For quality control, we first identified and removed all doublets using the DoubletFinder R package (version 2.0.4)¹¹⁶. Then, we applied the same procedure as before by removing cells with fewer than 250 detected genes or fewer than 2500 total UMI counts, as well as genes expressed in fewer than 5 cells, leaving us 66816 hepatocytes with adequate transcriptome coverage.

Transcriptional entropy, clustering-based variance estimates for comparison, and pathways related to genes with gains in transcriptional entropy and activation in high-entropy cells were all computed following the same procedure as for PT cells.

Zonation scores were calculated for all cells as the ratio of total pericentral marker expression to the combined expression of pericentral and periportal markers. Cells with strict 0 or 1 zonation scores are removed, and the rest are divided into 10 equal-sized bins, where we considered 0–20% bins to be periportal, 40–60% bins to be midlobular, and 80–100% bins to be pericentral. To find zone-specific genes with gain in transcriptional entropy, we computed gene-level entropy for all genes expressed by young and old cells within each zone separately and retained all genes with positive log-fold changes between the age groups. To find zone-specific genes with activation in high-entropy cells, we extracted differentially expressed genes between low (bottom 20%) and high (top 20%) entropy cells within each zone. Identified genes were then searched for their enriched pathways using the enrichR R package (version 3.2)¹¹⁷. KEGG¹¹³ and Gene Ontology¹¹⁴ were used as databases. Similarly, enrichment p-values were adjusted for multiple testing using the Benjamini-Hochberg procedure. Pathways with an FDR-adjusted p-value ≤ 0.05 were considered significant.

Supplementary Material

Supplementary Figure 1-35 These figures are kept together in the INClock_paper_supplementary.docx.

Supplementary Table 1 List of PT segment identity markers.

All 29 PT segment identity markers are recorded.

Supplementary Table 2 Genes with significant expression changes across ages and their enriched pathways in PT cells.

Sheet 1 includes all genes ranked by their Spearman’s ρ values across age groups of PT cells. Sheet 2 includes significantly enriched pathways identified with these genes.

Supplementary Table 3 Genes significantly correlated with cell-level entropy and their enriched pathways in PT cells.

Sheet 1 includes all genes significantly correlated with cell-level entropy of PT cells. Sheet 2 includes significantly enriched pathways identified with these genes.

Supplementary Table 4 Hepatocyte zonation markers and senescence and proinflammation markers.

Sheet 1 records all 21 hepatocyte zonation markers (9 periportal, 4 midlobular, 8 pericentral). Sheet 2 records all curated 138 senescence and pro-inflammation markers.

Supplementary Table 5 Genes significantly correlated with cell-level entropy and their enriched pathways in hepatocytes.

Sheet 1 includes all genes significantly correlated with cell-level entropy of hepatocytes. Sheet 2 includes significantly enriched pathways identified with these genes.

Supplementary Table 6 Genes with significant expression changes across ages and their enriched pathways in hepatocytes.

Sheet 1 includes all genes ranked by their log-fold changes in gene-level entropy across old vs. young hepatocytes. Sheet 2 includes significantly enriched pathways identified with these genes.

Supplementary Table 7 Genes with significant zone-specific positive expression changes across ages and their enriched pathways in hepatocytes.

Sheet 1 includes all genes with positive log-fold changes in gene-level entropy across old vs. young hepatocytes in each zone. Sheet 2 includes significantly enriched pathways identified with these genes by zone.

Supplementary Table 8 Genes significantly correlated with zone-specific cell-level entropy and their enriched pathways in hepatocytes.

Sheet 1 includes all genes significantly positively correlated with cell-level entropy of hepatocytes in each zone. Sheet 2 includes significantly enriched pathways identified with these genes by zone.