The dissipation theory of aging: a quantitative analysis using a cellular aging map

Abstract

We propose a new theory for aging based on dynamical systems and provide a data-driven computational method to quantify the changes at the cellular level. We use ergodic theory to decompose the dynamics of changes during aging and show that aging is fundamentally a dissipative process within biological systems, akin to dynamical systems where dissipation occurs due to non-conservative forces. To quantify the dissipation dynamics, we employ a transformer-based machine learning algorithm to analyze gene expression data, incorporating age as a token to assess how age-related dissipation is reflected in the embedding space. By evaluating the dynamics of gene and age embeddings, we provide a cellular aging map (CAM) and identify patterns indicative of divergence in gene embedding space, nonlinear transitions, and entropy variations during aging for various tissues and cell types. Our results provide a novel perspective on aging as a dissipative process and introduce a computational framework that enables measuring age-related changes with molecular resolution.

Introduction

Aging is a universal biological process, yet dissecting its underlying mechanisms remains an elusive challenge in biology. Numerous theories have sought to explain the aging process, ranging from the programmed aging theory^1,2, phenoptosis³, evolutionary theories^4,5, and damage accumulation^6,7,8. Meanwhile, some scientists have shifted focus from overarching theories to cataloging the observable hallmarks of aging, revealing the complexity and diversity of aging mechanisms across living systems⁹. While these approaches provide important insights, they often limit their investigations to biological observations¹⁰ and a unified framework to understanding aging remains incomplete.

Aging is not merely the passage of time but reflects the dynamic state of the system, encompassing its molecular, cellular, and phenotypic changes over time. In this sense, aging has to be studied as a dynamic process. Biological organisms are essentially a form of dynamical systems composed of numerous interacting components that operate across multiple scales^11,12,13. These components follow complicated rules and feedback loops to maintain homeostasis and functional integrity. Here, we propose to use dynamical systems theories to studycellular and genetics changes during aging - in health and with disease. Specifically, we use ergodic theory and Hopf decomposition theorem to contextualize aging^14,15 and arrive at two distinct components. We present a conservative part which is recurrent, and therefore cycles between the same states in the phase space (the space of all possible states of the system), and the dissipative part which is non-recurrent where genes and cells escape over time and do not return to their initial state¹⁵. We propose that in biological dynamical systems (BDS), what is classically understood as aging represents the predominance of dissipative forces which lead to deviation from recurrent states and increased entropy over time. However, quantifying dissipation in dynamical systems is challenging and requires accurate formalism of the system at hand.

The rise of data-driven approaches has emerged as novel methods to approximate the behavior of complex dynamical systems. Traditional methods for studying biological dynamical systems often rely on simplified models that fail to account for the inherent nonlinearity and high dimensionality of complex processes. Recent advancements in computational techniques, especially in machine learning and data-driven modeling, have made it possible to reconstruct these systems computationally^16,17,18. These data-driven techniques have particular relevance for biological systems, where the complexity of interactions— from molecular to organismal scales—makes traditional mechanistic modeling difficult. We have shown that language models trained on large-scale multimodal data can capture the nuanced dynamics of these systems by embedding complex biological states into interpretable representations¹⁹.

In this paper, we introduce a novel approach for studying aging using data-driven dynamical system modeling. We discuss the mathematical foundations of our theory and describe the use of a transformer-based foundation model to track the changes in the states during aging. We trained this model on a massive corpus of data from single-cell RNA sequencing (scRNA-seq) studies, spanning 171 age groups and multiple tissues (these 171 age groups correspond to discrete chronological bins derived from the dataset’s age labels, covering the full spectrum from embryonic stages to late life). We present a detailed aging map at cellular and molecular resolution, which provides insights into the nature and rate of aging across different tissues and cell types. Specifically, we show dissipation characteristics in the gene embedding space and identify genes with higher amounts of chronological drift and divergence in their embedding. We further analyze the information landscape through an ergodic lens across lifespan and the balance of health and disease by associating and quantifying these changes through entropy production of our model’s predictions. This nuanced and perhaps novel theory and computational framework extends and expands upon our understanding of aging - moving it from a perspective of passive and inevitable decline into an active dynamic process under dissipative and conservative forces.

Results

Modeling and dataset overview

We collected publicly available single-cell RNA sequencing datasets, encompassing over 65 million cells from a wide range of biological sources using the CellxGene platform²⁰. Specifically, the data included 171 distinct age groups, covering the full spectrum from embryonic stages to old ages, and represented 616 different cell types across 215 tissues, providing a comprehensive view of cellular diversity (Fig. 1b). The datasets also span 71 different disease states, allowing us to assess how aging intersects with pathological conditions. These data have been previously standardized and preprocessed to minimize technical batch effects, enhance biological signal consistency, and ensure robustness across different experimental conditions²⁰.

Fig. 1: Overview of theoretical and computational framework.

a Decomposition of aging manifold in time using Hopf decomposition into conservative and dissipative components (b) aggregation of 1528 single-cell RNA-seq dataset encompassing various age groups, tissues, and cell types, (c) modeling of gene expression using a transformer-based model with metadata information and age token, (d) cellular aging map constructed from the output embedding of the model (Created in https://BioRender.com).

We used a multimodal cell foundation model developed previously to map the genotype-phenotype relationship using single-cell transcriptomics data¹⁹. This model integrates gene expression and metadata information such as chronological age labels with gene expression data (Fig. 1c) and creates outputs for each label and gene in the format of an embedding that will be used to construct aging clock at the cellular resolution (Fig. 1d) (see the Methods section for more information on modeling 4).

Cellular Aging Map (CAM) analysis

The model’s output embeddings are high-dimensional vectors that encapsulate rich information about individual tokens and their contextual relationships within the data. These embeddings are uniquely suitable for capturing tissue-specific and nonlinear aging dynamics. To leverage this capability, we constructed a Cellular Aging Map (CAM) by generating embeddings for each cell type and tissue across distinct phases of aging. CAM provides a detailed landscape of how cellular and molecular features evolve over time.

To assess the predictive capabilities of our model, we first tested its ability to estimate the chronological age of samples. The age predictor is obtained by masking the age token for each sample and letting the model predict molecular age from the learned vocabulary distribution, analogous to masked language modeling. The model outputs a probability distribution over age labels, and the token with the highest probability taken as the predicted molecular age. Using randomly selected 100,000 samples, stratified to ensure coverage across all tissues, cell types, and age bins, we evaluated the model’s predictions by calculating the z-scored age gap, defined as the difference between the predicted and true ages normalized by the standard deviation. In most cases, the model accurately predicted the cell age, yielding a z-scored age gap close to zero. However, in certain instances, the model predicted significantly higher or lower ages for the samples (Fig. 2a), indicating accelerated or decelerated aging trajectories in those cells. These deviations suggest that the model captures underlying variations in gene expression that reflect processes moving alongside chronological aging and whose effects might be seen as accelerating of decelerating cellular aging. In particular, the distribution of age gaps varied between tissues and cell types, which motivated the development of CAM.

Fig. 2: Construction of aging clock using age prediction from the model.

a A 100,000 subsample of all healthy tissues (left), healthy samples from respiratory airway tissue (middle), and healthy samples from breast tissue (right), z-scored gap represents the difference between the model’s prediction and chronological age labels. Panels report the explained variance (EV) of the predictor with respect to chronological age, computed as EV = 1 − SSE/SST using the model outputs $\hat{y}$ against labels y (i.e., relative to a predict-the-mean null). Negative R² represents predictions that are poorly linearly calibrated for a given subset, even when a monotonic association exists. Individual cells follow a spectrum of correlation of predicted molecular age with chronological age from those with (b) high correlation coefficients that follow chronological age labels, (c) to cell types with low correlation coefficients that deviate from the chronological age labels. We therefore report Spearman’s ρ (with two-sided p-value) as the primary measure of monotonic association; the overlaid line is an ordinary least-squares fit shown only for visualization. Subsets with negative EV (e.g., Breast; Retinal rod cell; Fibroblast of Mammary Gland) still show significant ρ, indicating nonlinear/heterogeneous aging trajectories rather than absence of age-related signal.

For instance, predictions for the respiratory airway tissue demonstrated high correlation, with a Pearson coefficient of 0.93, and most samples from this tissue were predicted with a z-scored age gap near zero (Fig. 2a middle panel). This high degree of concordance suggests a relatively uniform rate of aging and possibly stable molecular aging dynamics within the respiratory airway. It might also reflect the rapid turnover of cells in the respiratory tract, an organ system with its own embedded pluripotent stem cells, and one where the time scale of cell turnover is so much faster than the scale of diffusive aging²¹. In contrast, predictions for breast tissue exhibited higher deviations from the labeled chronological age with low explained variance reflecting heterogeneous rate of aging with certain cellular populations exhibiting accelerated or decelerated aging trajectories compared to the expected chronological benchmarks. These observations align with previous findings of tissue-specific aging dynamics^{22,23,24,25,26}, and suggest the role of epigenetic phenomena and hormonal cycling, and add more resolution to identify the aging heterogeneity within a tissue.

To further explore aging heterogeneity within tissues, we analyzed the model’s age predictions for each cell type individually. This detailed CAM investigation revealed a spectrum of behaviors across different cell populations. Certain cell types, such as Naive T cells, regulatory T cells, and macrophages, exhibited high correlation between predicted and chronological age (Fig. 2b). This could indicate that these cell types experience relatively uniform aging trajectories, with minimal heterogeneity in gene expression patterns over time. On the contrary, other cell types, including retinal rod cells, central memory CD4-positive alpha-beta T cells, and fibroblast of mammary gland cells displayed lower correlation coefficients, explained variance, indicative of high heterogeneity in their aging profiles (Fig. 2c). This variability predicted age and actual age suggests that age is a contextual construct and that our model can differentiate cells governed by time alone from those with dominant additional forces, e.g. discriminating frontline immune cells whose molecular control is systematic and progressive from context-driven cells that are more influenced by environment, exposures, or inflammation, and therefore adding noise to age prediction. Naive T cells are generated by the thymus at a rate that declines with age, and naive B cells similarly show age-related subtype shifts with immune aging. These cells should therefore track with and even be a marker of chronological age. Negative R² values shows that aging trajectories are heterogeneous and cannot be well-approximated by a single global linear mapping. This reflects the inherent dynamics of tissue- and cell-type-specific aging which leads to broader dispersion in molecular age and divergence from chronological age. Naive T cells are generated by the thymus at a rate that declines with age, and naive B cells similarly show age-related subtype shifts with immune aging. These cells should therefore track with and even be a marker of chronological age.

Next, we analyzed the embeddings of tissues and cell types to explore their association with aging, calculating the similarity score between the embeddings of tissue or cell type tokens and the age token. This similarity score reflects the extent to which a tissue or cell type’s molecular profile is influenced by the aging process, as represented by the age token in our model. Higher similarity scores indicate a stronger association with aging, suggesting that the corresponding tissues or cell types are more susceptible to age-related changes. Conversely, lower scores suggest resistance to aging or a slower aging rate based on our dataset.

To further elucidate the tissue- and cell-type-specific patterns of aging, we compared their embeddings against the age token and ranked each tissue or cell type by its similarity score (Fig. 3). Boxplots in Fig. 3a show the top age-associated tissues (left) and the least age-associated tissues (right). Among the highest-scoring tissues were the yolk sac, breast, spleen, liver, bone marrow, and prostate, all exhibiting elevated similarity to the age token. This suggests that these tissues undergo more pronounced age-related molecular changes, as they are high-turnover anatomo-physiologic elements in synchrony with systemic aging. By contrast, tissues such as the adrenal gland, eye, blood, and esophagus clustered at the lower end of the similarity spectrum, indicating comparatively reduced susceptibility to—or delayed onset of—molecular aging signals, where there is low turnover and unique aging trajectories

Fig. 3: Similarity analysis reveals tissues that are more susceptible to or are independent of the aging process.

a Top age-related tissues that have higher dependency to age token (left) and tissues that are not dependent to age token (right), (b) Top age-related cell types with age token (left), and least age-related cell types, c Dynamics of changes in aging using z-scored age gap during different stages of lifespan for age groups every 5 years (left), all samples age labels without grouping (right).

A parallel analysis of cell-type embeddings (Fig. 3b) revealed a similar dichotomy. Cell populations, including basal cells, decidual cells, lumina epithelial, hepatocytes, and various immune-related cell types (e.g., Kupffer cells, cortical epithelial cells, and follicular B cells) ranked among the most age-associated. Their elevated similarity scores signal a strong coupling between these cell types’ transcriptomic profiles and aging. Conversely, a distinct set of cell types—such as certain interneuron subpopulations and stem/progenitor cells—displayed comparatively lower scores, pointing to more resilient or less dynamically changing expression patterns with advancing age in cells with longer half-lives.

To investigate how these age-associated signatures translate into overall aging trajectories, we computed the age gap for each sample, defined as the difference between the model-predicted molecular age and chronological age. Figure 3c (left) depicts the distribution of these age gap z-scores in 5-year increments across the adult lifespan. Notably, the violin plots demonstrate broad heterogeneity at most ages, potentially due to individual-level variability in molecular signatures. This could also be due to some tissues and cell types within the same chronological age bracket exhibiting different molecular ages. Overall, the dynamics of z-score show underlying biological or environmental factors that modulate the aging process.

The line plot in Fig. 3c (right) captures the dynamics of these z-score age gaps across increasing chronological age in a higher resolution. Although individual trajectories vary, an overall trend emerges in which the magnitude of the age gap tends to increase in early decades, suggesting that the rate of molecular aging accelerates over initial stages of life. However, in later stages (specifically after 4th decade), the average aging gap becomes negative, which indicates deceleration in the aging process. These results demonstrate the nonlinear nature of the aging process, which is consistent with previous studies²⁷. This motivates track aging in various decades separately to extract contributing molecular shifts during each stage.

Temporal drift in age embedding space

In the Methods section we demonstrated that embeddings capture rich contextual information as to the changes occurring during aging. Unlike gene expression, which is represented as a scalar value of gene activity, embeddings encode dynamic relationships and contextual interactions within the genetic landscape. These embeddings are high-dimensional vectors that incorporate both gene activity, genetic interactions, and also the temporal context in which these changes occur, which makes them particularly well-suited for studying the complexity of aging dynamics.

One method to capture the temporal dynamics of the aging process is with similarity analysis of embedding vectors. We can calculate cosine similarity among age embeddings to understand the relationship between different age stages, and between age embeddings and gene embeddings to inform the role of genes for each stage. By calculating the cosine similarity between gene embeddings and age embeddings across different stages of the lifespan, we observed distinct variations in the roles that genes play at each stage (Fig. 4a−c). Specifically, the temporal similarity trajectories of individual genes suggest that their relevance to the aging process is tissue and stage-specific.

Fig. 4: Embedding-based analysis reveals temporal and tissue-specific gene dynamics across the aging process.

a Cosine similarity between gene embeddings and age embeddings across the lifespan highlights tissue-dependent and cell type-specific aging trajectories. Selected genes are shown for blood and lung tissues, as well as macrophages and regulatory T cells, illustrating stage-specific gene relevance from adolescence to late life. b Temporal similarity profiles of representative genes in healthy individuals exhibit diverse aging trajectories. c Comparison of gene similarity trajectories between healthy and diseased aging for selected genes shows structured, multi-phase relevance during healthy aging and disruption during disease conditions.

For example, for blood tissue, genes like SPATA42 and PRF1 contribute more to aging during adolescence, but in later stages, for example, between 70 and 80 years old, TRAJ32 and NDST2 have a higher genetic impact on blood aging. A similar pattern was observed for other tissues like lung, and also specific cell types, such as macrophages and regulatory T cells (Fig. 4a).

Genes have distinct patterns in their contribution to aging. Genes like IFITM3, which is involved in antiviral defense, show relatively stable similarity across the lifespan, suggesting constitutive or aging-independent activity. (Fig. 4b). Other genes exhibit sharply peaked interactions at distinct time points and still others maintain broader relevance across multiple age stages. MKRN1 and SAFB2 displayed transient similarity peaks in early to mid-life, suggesting potential regulatory roles during periods of developmental or reproductive maturation (Fig. 4b). In contrast, genes such as MCTP1 showed increasing similarity with later age embeddings, implying potential involvement in aging-associated maintenance or stress-response pathways.

We also compared the gene embedding variations during healthy and diseased aging (Fig. 4b). Specifically, SESN1, HEBP2, and ATR, exhibited complex similarity trajectories with multiple local maxima in healthy cases. However, these trajectories were altered in the presence of age-related disease. For example, ATR showed diminished association during adulthood in diseased samples, but in old age showed higher similarity, potentially reflecting impaired DNA repair mechanisms first and then overcompensation. These shifts suggest that aging-associated diseases disrupt the coordinated temporal regulation of key protective genes, which can guide how pathology may override or derail otherwise adaptive aging programs.

Aging Is a dissipative process

The CAM analysis demonstrates that chronological time is an inadequate aging metric that cannot sufficiently capture the biological aging process, as while these chronology and biology are concordant in some cells and tissues, in others they diverge significantly and in nonlinear manner. Dynamical systems theory is ideal to investigate this seeming paradox and the complexity of aging. Dissipation in particular, in this context, can provide a framework for quantifying the loss of system stability and resilience over time, offering a more holistic perspective on the progression of biological aging. In the next section, we integrate this theoretical approach with our empirical findings to elucidate how tissues and cells transition through different stages of aging and how this process may differ across biological systems. This approach moves beyond static chronological measures and toward a dynamic and system-level understanding of aging.

Gene embeddings in relation to the age token offer insight into the evolution of the dynamical system manifold in aging. As outlined in the Methods section, we categorized genes into two classes - dissipative and conservative - following from the classical Hopf decomposition. While the former exhibit high temporal drift in the embedding space, reflecting their sensitivity to aging and dynamically shifting contextual relationships, the latter conservative genes demonstrate minimal drift, maintaining stable dynamics over time, suggesting consistent roles in cellular processes throughout their aging process.

Our analysis revealed distinct examples as to how genes segregate into these two categories. Genes such as MKRN1, SESN1, and ATR were identified as conservative, as their embeddings remained stationary in the embedding space across all stages of life. Conversely, genes such as ALDH3B1, NR2C2, and HERPUD1 displayed higher drift from their initial embeddings, categorizing them as dissipative (Fig. 5a). Herein lay an important learning. Interestingly, the classification of genes as conservative or dissipative appears independent of their biological roles. Both types of genes are involved in a wide range of cellular processes, including DNA repair, stress response, and oxidative metabolism. This observation suggests that conservation and dissipation are universal characteristics intrinsic to the aging process and occur across diverse molecular pathways. This approach allows us to move beyond static measures of gene expression and explore the broader molecular interplay that defines biological aging.

Fig. 5: Metrics of dissipation during aging.

Extracting conservative and dissipative genes based on temporal drift in gene embedding space during aging in (a) healthy and (b) diseased conditions. c Distribution of gene embedding drift to calculate the dissipation threshold. d Statistical significance of conservative and dissipative genes after permutation test, (e) phase portrait of recurrence vs divergence.

Our analysis revealed alterations in the regulatory dynamics of gene embeddings for cells in diseased as opposed to healthy conditions. For example, ATR characterized as a conservative gene with stable embeddings in healthy cells, exhibited a marked increase in drift within diseased samples (Fig. 5b). This shift suggests that ATR, which plays a critical role in DNA damage response and cell cycle regulation, may be forced or directed to adopt a different expression pattern under pathological conditions. A similar pattern was observed for SAFB2, a regulatory gene involved in chromatin organization and stress response, which also displayed increased drift with disease. These findings further extend the implications of our model and imply a broader trend where some perhaps vulnerable genes with stable behavior in healthy tissues become destabilized in the presence of disease. In contrast other genes might be able to resist such forces and even attain compensatory functionality. TDP2 classified as a dissipative gene due to its high variability in healthy samples attained conservative embeddings in diseased conditions (Fig. 5b). This unexpected shift in stability could suggest a potential disease-responsive mechanism that provides regulatory constraint of variability of genes like TDP2, which might preserve essential cell function during stress or damage.

We label genes as dissipative or conservative based on their maximum embedding drift and verify this drift is statistically significant with a permutation test. Every gene’s embedding drift is computed as the maximal temporal drift $\mathop{\max }\nolimits_{t\in {\mathcal{T}}}{D}_{g}(t)$ (refer to the Methods section ?? for more details). For all 5917 genes we consider, we build a null distribution of the temporal drift metric after shuffling the age labels, computing the metric and repeating this process 1000 times. We compared the observed embedding drift to their corresponding null distributions, and computed Z-scores and FDR-adjusted empirical p-values. We computed both analytical p-values using the normal approximation and empirical p-values based on the permutation distribution. While visualizations of the analytical p-values were helpful for interpretation, all reported p-values and statistical decisions were based on the empirical values, which make no distributional assumptions. The test further validates our selection of dissipative genes based on the statistical significance of their embedding drift (Fig. 5c). The majority of dissipative genes showed drift scores lying in the extreme positive tail of their null distribution (mean Z = 5.37; 65% of genes with Z > 3). Benjamini-Hochberg FDR correction confirmed significance with a mean adjusted p-value = 0.005. We also see that very few of the conservative genes have a statistically significant temporal drift (Fig. 5d). This permutation test allows us to reject false positives or falsely dissipative genes which have a very sparse embedding space structure of equidistant embeddings and always present a large temporal drift regardless of the order. Lastly, a phase portrait of gene trajectories and their divergence versus recurrence shows our dissipative genes clustering for these dynamical state metrics (Fig. 5e).

Aging is closely linked to changes in molecular order and structure, with profound implications for the organization of biological systems. From an information theory perspective, it has been proposed that aging is driven by the progressive loss of youthful epigenetic information which essentially leads to increased disorder and reduced fidelity in biological processes²⁸. However, despite this compelling hypothesis no concrete metrics have been established to quantify this information loss systematically.

To address this gap we use entropy as a metric to quantify how aging reshapes our understanding of the information landscape of gene expression. Entropy, as a measure of uncertainty or disorder in a system, provides a valuable framework for understanding the dynamics of information loss during aging. Specifically, we calculate Shannon entropy based on the model’s prediction of each token in the dataset after masking. This process generates a probability distribution for every token over the entire vocabulary, representing the model’s uncertainty about the masked token. By applying Shannon entropy to this distribution, we quantify the degree of uncertainty or disorder associated with each prediction. Crucially, we condition these predictions on the age token to capture how the model’s uncertainty changes across the aging spectrum (refer to the Method section ?? for more details). This approach allows us to track the evolution of the information landscape as a function of age. In this context, higher entropy indicates greater uncertainty and suggests a loss of molecular specificity, aligning with the dissipative nature of aging and progressive information degradation.

We performed a detailed analysis across tissue and cell types and present kidney and endothelial cells as polar forms of entropy dynamics. Healthy kidney cells show a clear trend: entropy increases progressively with age, with the rate of increase accelerating later in life (Fig. 6c). Furthermore, the distribution of entropy across different age groups shows a marked rise in entropy variation, which could indicate heightened heterogeneity, potentially signaling either greater uncertainty or increased plasticity in molecular organization within aging kidney tissue (Fig. 6e).

Fig. 6: Entropy analysis during aging in various tissues and cell types.

Average Shannon Entropy value for (a) endothelial cell for health and diseased conditions, (b) lung tissue for health and disease, (c) kidney tissue in health condition, (d) changes in entropy distribution during aging for healthy blood tissue, and (e) healthy kidney tissue (Post-fertilization refers to the embryonic stages that occur after zygote formation and initial development.

In contrast, endothelial cells exhibited a different pattern with no consistent trend of entropy increase across the aging process (Fig. 6a) but a variability in entropy production at different ages. At specific life stages—such as the mid-thirties, early fifties, and early sixties—entropy sharply but only transiently increased. These “entropy spikes” were followed by subsequent reductions in entropy, suggesting episodic disruptions in molecular organization that are later compensated for, stabilized or even repaired. This dynamic behavior in endothelial cells highlights a more complex, stage-specific response to aging, distinct from the progressive changes observed in other tissues like the kidney and follows a pattern that has been previously observed in lipid clocks^29,30. Disease-transformed endothelial cells which consistently exhibited higher entropy compared to their healthy counterparts, demonstrating a higher loss of molecular specificity and an increase in structural or functional disorders (Fig. 6a). This heightened entropy suggests that disease conditions exacerbate the underlying molecular chaos within endothelial cells.

Discussion

While classic approaches have provided important insights, a more comprehensive understanding of aging might well be gained by integrating perspectives from diverse disciplines. Early innovators like Raymond Pearl and Ludwig von Bertalanffy adopted such cross-disciplinary views. In “Biology of Death” (1922)^31,32, Pearl proposed the “rate of living” theory, linking energy expenditure and metabolic rates to organismal lifespan. Bertalanffy also proposed a system theory mathematical framework to describe organism’s growth over time³³ aligning with Strehler’s general theory of aging³⁴. These early theories pioneered the use of more system-level characterization of biological systems in studying aging, paving the way for more integrated biological modeling.

Building on this legacy, we re-conceptualize aging as a dynamic, systems-level process, providing a comprehensive exploration of cellular and molecular changes and novel theoretical framework that leverages emerging advanced modeling techniques. We model biological organisms as dynamical systems governed by both recurrent (conservative) and non-recurrent (dissipative) forces, drawing on principles from ergodic theory. In this formulation aging emerges from the predominance of dissipative dynamics, leading to deviations from homeostatic, cyclic behaviors and a progressive rise in entropy over time. Because an explicit formulation of aging dynamics is not possible, we introduce a novel computational method to learn the dynamics from data implicitly at the cellular level.

Our cellular aging map generates contextualized embeddings for each gene, tissue, and cell type across the lifespan. These embeddings capture nonlinear and tissue-specific aging trajectories, enabling the identification of differential aging rates across biological contexts. Model predictions revealed differential rates of aging across tissues and cell types and showed populations with accelerated, decelerated, or resistant aging dynamics. Using embeddings drift analysis, we distinguished between conservative and dissipative genes during both health and disease. We applied Shannon entropy analysis to quantify dissipation and as an indication of molecular disorder and aging-associated information loss in various tissues. Collectively, these findings support aging as a temporally and spatially heterogeneous process and offer tools to quantify and interpret aging at the cellular and molecular levels.

Crucially, this entropy-based framework enables us to move beyond simplistic narratives of aging as steady decline, random breakdown, or uniform loss. Indeed, entropy increases consistently in kidney cells, and in endothelial cells, particularly under disease pressure—but also, to a lesser extent, during healthy aging. This progressive rise in entropy suggests that aging and disease interact synergistically, compounding molecular disorganization and uncertainty. The combination of disease-related stress and age-associated deterioration likely drives the continuous entropy elevation seen in diseased endothelial systems, contributing to their reduced resilience and functional decline.

In contrast, healthy endothelial cells exhibit non-linear entropy dynamics, marked by episodic spikes rather than continuous rise (Fig. 6a). This suggests an adapative capacity to buffer and recover from local instabilities. Such elasticity has been appreciated in endothelial cells which in their native state or broadly uncommitted and capable of profound elasticity that only erodes as disease dominates³⁵. Herein lies a more dynamic view of aging, not only as a passive descent into disorder continuous decay propelled by disease, but as a balance of adaptive or compensatory recovery in the face of micro-disruptions and cumulative applied stresses.

Our study has several limitations. A primary constraint is reliance on learning-derived embeddings which while powerful in capturing complex relationships among genes, tissues, and aging stages, may limit biological interpretability and obscure direct mechanistic insights. Although complementary analyses such as entropy and temporal drift assessments were employed to enhance understanding, the inherent abstraction of these embedding spaces still limits complete causal inference. Additionally, the sensitivity of our gene discrimination and dataset biases, such as potential under representation of certain tissues, cell types, or age ranges, could restrict generalizability. Entropy analyses revealed interesting differences across tissues and cell types, though their interpretation may be confounded by the distinction between inherent data uncertainty and model uncertainty. Moreover, our focus primarily on transcriptional dynamics could leave gaps in fully capturing rapid or transient aging processes and other layers of the aging landscape, such as epigenetic, proteomic, and metabolomic changes. Addressing these limitations with further experimental validation and more comprehensive multi-omics integration will help refine our approach and broaden the applicability of the cellular aging map.

In summary, by expanding theoretical and computational framework for aging and refining its applications with additional datasets, our work lays the foundation for an integrative approach to quantify the progressive loss of stability and resilience during aging. We propose aging as a dissipative, entropy-increasing process governed by the shifting balance of conservative and dissipative forces across time, tissue, and context. This framework not only deepens our conceptual understanding of aging but also provides tools to identify molecular targets, support therapeutic development, and advance personalized interventions aimed at mitigating age-related decline.

Methods

Background

The study of dynamical systems has traditionally focused on identifying a set of differential equations that describe the time evolution of the system. While this approach works well for simple systems, it becomes increasingly complex as the number of degrees of freedom grows, necessitating new tools and methods of analysis. In recent years, and with the advent of neural networks, there has been a paradigm shift in modeling dynamical systems. Data-driven modeling is increasingly being used to model and predict the behavior of dynamical systems, without explicitly solving the governing equations, especially for complex systems where the governing differential equations cannot be derived easily^36,37. In this section, we provide methods used to implement a data-driven approach in modeling aging dynamics.

In the modeling of aging, when the system has a large number of degrees of freedom, the complexity of dynamics is extremely high. In such cases, the presence of multiple positive Lyapunov exponents—often indicative of chaos, shifts the focus from predicting individual trajectories to studying the evolution of the dynamical system as a whole^{15,32,38,39,40,41,42,43,44}. The emphasis moves to understanding the manifold on which the system evolves over time, to capture the overall behavior rather than the specifics of single-particle dynamics. In the context of dynamical systems, a manifold is a mathematical space that locally resembles Euclidean space and serves as a geometric framework to capture the global structure of the dynamics.

These manifolds, commonly referred to as Lagrangian Coherent Structures (LCS), can be generalized to time-dependent manifolds that evolve smoothly over time. To formalize this, consider a phase space denoted by ${\mathcal{P}}$, which describes the state space of the system, and a time interval ${\mathcal{T}}=[{t}_{0},{t}_{N}]$, representing the duration of interest. The dynamics of the system are governed by a time-dependent flow map:

$$F(t;{t}_{0}):{\mathcal{P}}\to {\mathcal{P}},$$

(1)

where F(t; t₀)(x₀) represents the position of the trajectory starting at ${{\bf{x}}}_{0}\in {\mathcal{P}}$ at the initial time t₀, after evolving under the flow for a time t − t₀. For simplicity, we may assume t₀ = 0, allowing the notation to reduce to:

$$F(t):{\mathcal{P}}\to {\mathcal{P}}.$$

(2)

An invariant manifold${\mathcal{M}}\subset {\mathcal{P}}$ satisfies:

$$F(t;{t}_{0})({\mathcal{M}})={\mathcal{M}},\quad \forall t\in {\mathcal{T}},$$

(3)

where every point on ${\mathcal{M}}$ remains on the manifold under the flow of the system. This property makes invariant manifolds fundamental in understanding the geometric structure of dynamical systems, as they often represent boundaries between qualitatively different behaviors or act as organizing centers for trajectories.

In a more general setting, we are often interested in the evolution of the manifold itself over extended time periods. To capture this dynamic, we consider a time-dependent family of manifold ${\mathcal{M}}(t)\subset {\mathcal{P}}$ that changes smoothly with time. We introduce another flow map $H(t):{\mathcal{M}}\left({t}_{0}\right)\to {\mathcal{M}}(t)$ that captures the intrinsic changes of the manifold:

$${\mathcal{M}}(t)=H\left(t,{\mathcal{M}}\left({t}_{0}\right)\right)$$

(4)

where H(t) maps the manifold at the initial time t₀, denoted by ${\mathcal{M}}({t}_{0})$, to its configuration at a later time t, represented by ${\mathcal{M}}(t)$. The evolution of the manifold can then be expressed as:

$${\mathcal{M}}(t)=H(t,{\mathcal{M}}({t}_{0})).$$

(5)

This formulation allows us to distinguish between the intrinsic changes of the manifold and the evolution of the points within the manifold itself. The two flow maps, F(t), which describes the evolution of individual points in ${\mathcal{P}}$, and H(t), which captures the intrinsic deformation of ${\mathcal{M}}$, can be combined to provide a comprehensive description of the dynamics. Specifically, the combined map can be expressed as F(t)∘H(t), representing the total dynamical evolution of the manifold. However, in this work, our primary focus is on the map H(t), which operates on large time scales and governs the long-term intrinsic evolution of the manifold. Understanding H(t) is key to characterizing the structural dynamics of the system over extended periods.

Modeling the manifold of gene expression

In the case of the dynamics of a gene expressions across a lifetime we are dealing with a dynamical system with many degrees of freedom that creates a changing manifold with time. To model this dynamic we rely on samples taken from different time points from a large population that captures the diversity of this manifold too.

In this dynamical system, we have the gene expressions and phenotypes associated with them as the total degrees of freedom that change over time. Let G = (g₁, ⋯ , g_n) be the set of genes and Ph = (ph₁, ⋯ , ph_m) be the set of phenotypes, then all degrees of freedom are X = G ∪ Ph. Where X represents the union of the gene expressions and phenotypes. The evolution of this system over time gives rise to a dynamic manifold ${\mathcal{M}}(t)\subset {\mathcal{P}}$, where ${\mathcal{P}}$ is the phase space spanned by X. The structure and evolution of ${\mathcal{M}}(t)$ reflect both intrinsic changes in gene expression and their phenotypic consequences.

To capture and model this complex interplay, the flow map H(t) introduced earlier becomes essential. It represents the evolution of the manifold, incorporating not only temporal dynamics but also variability within the population. By analyzing H(t), we aim to uncover the long-term dynamics and patterns in gene expression and their associated phenotypic outcomes. To model H(t, x₁, ⋯ , x_n), in which x_i ∈ X and $t\in {\mathcal{T}}$, we use a Masked Language Model (MLM) approach. More specifically, we are using BERT (Bidirectional Encoder Representations from Transformers)⁴⁵ to reconstruct the degrees of freedom as inputs for each time step by randomly masking 85% of them. Using the MLM the whole dynamics can not be described but it is still able to find the time relationships between the degrees of freedom which is the task of looking at the time evolution of the phase space manifolds.

Model architecture

The MLM method used here is a powerful tool that can learn contextual relationships among input features in a self-supervised manner. This approach is particularly suited to our problem because gene expressions and phenotypes are inherently interdependent. For example, changes in one gene’s expression can cascade into changes in other genes or phenotypes. Moreover,in real-world biological data, we often encounter incomplete datasets with missing gene expressions or phenotypic measurements at certain time points. This mimics the challenges of reconstructing the full manifold ${\mathcal{M}}(t)$ from sparse or incomplete observations. Also, this method is scalable for high-dimensional data involved here. We will go over the data and modeling details in the next few sections.

More specifically, a transformer-based MLM architecture was employed to capture complex interactions between genes and their temporal dynamics. The model used a multi-head self-attention mechanism to learn intricate dependencies between genes. The model was designed to predict masked tokens representing gene expression values, conditioned on the presence of an age token, as well as other biological covariates like tissue type and disease state. This architecture enabled the model to achieve context-aware learning of gene-gene, gene-age, and gene-phenotype relationships, thereby allowing a more holistic understanding of the underlying dynamics of aging. Positional embedding was used to represent each gene uniquely by encoding the positional information relative to other genes within the dataset. This representation allowed the model to capture spatial relationships and dependencies between genes. By incorporating positional embedding, the model could learn not just the gene-specific expression values but also contextualize these values in relation to other genes, thereby improving its ability to capture the underlying structure and interactions within the gene regulatory network. Chronological age was provided as a learnable token to the transformer model, enabling the embedding space to encode tissue- and cell-type-specific relationships with age.

Data collection

Our study analyzed over 65 million cells using publicly available single-cell RNA sequencing datasets, representing 616 cell types from 215 tissues across 171 age groups, from embryonic to old age. It also included data on 71 disease states, providing insights into the interplay between aging and pathology. Measured with 20 different assays, the datasets were standardized and preprocessed through normalization, quality control, and batch correction to ensure biological consistency and minimize technical variability²⁰.

Input embeddings

The input embeddings were crafted to integrate multidimensional biological data, combining gene expression profiles with age tokens and other biological metadata to provide a comprehensive representation of the cellular state. Specifically, each embedding was a concatenation of several components. First, the gene expression profiles served as the foundational input, encoding the expression levels of genes in a given cell or tissue. Each profile was normalized and standardized to ensure comparability across samples and minimize technical variability. Next, a discrete representation of the sample’s age (labeled as chronological age in the dataset) was incorporated to capture the temporal progression in gene regulation.

By explicitly embedding age as a feature, the model was conditioned to recognize age-dependent changes in gene expression, enabling it to map temporal trajectories of biological processes. Additional contextual information, such as tissue type and disease state, was included to account for environmental and physiological variations. These metadata were embedded as learnable tokens, allowing the model to disentangle age-related effects from other factors influencing gene expression. Each gene’s position within the dataset was encoded using a positional embedding strategy. This approach captured the spatial and regulatory relationships between genes, enabling the model to account for dependencies and interactions within the gene regulatory network. The combined embeddings were fed into the transformer model, where they underwent further transformation through the multi-head self-attention mechanism.

Training scheme and evaluation

The MLM was trained using a masked token prediction task, where portions of gene expression vectors were masked, and the model was tasked with reconstructing them based on contextual information. Model performance was evaluated using reconstruction accuracy and the coherence of generated embeddings across age groups. Embeddings for age tokens, phenotype tokens, and gene tokens were extracted to assess their drift and divergence. In the following section, we will discuss the type of analysis done on these output embeddings.

Hopf decomposition, conservative and dissipative genes

While reconstructing genes alone provides insight into the MLM’s predictive ability, our primary interest lies in understanding how the relationships between genes and phenotypes evolve over time. Due to the high dimensionality of the original space, direct visualization and analysis of the dynamical manifold are not feasible. Instead, we focus on the embedding space produced by the MLM, which provides a compact and tractable representation of the manifold.

The key question is whether the embedding space faithfully captures the properties of the dynamical manifold. To address this, we use the BERT embedding function $E(t,{x}_{1},\cdots \,,{x}_{n}):{\mathcal{T}}\times X\to {{\mathbb{R}}}^{d}$, where d is the dimension of the embedding space. Replacing the original flow map H with the embedding function E, prior research⁴⁶ shows that BERT embeddings are robust and possess the Lipschitz continuity property:

$$\left\Vert E\left({x}_{1},{t}_{1}\right)-E\left({x}_{2},{t}_{2}\right)\right\Vert \le L\left(\left\Vert {x}_{1}-{x}_{2}\right\Vert +\left\vert {t}_{1}-{t}_{2}\right\vert \right)$$

(6)

where L is the Lipschitz constant. This property implies that small changes in the input state x or time t correspond to small changes in the embedding space.

This robustness is critical because it ensures that the embedding space accurately reflects the dynamics of the original manifold, allowing us to analyze temporal patterns, relationships between genes and phenotypes, and population-wide variations. The Lipschitz continuity guarantees that the embeddings provide a faithful representation of the system’s evolution, forming the foundation for the subsequent analysis. Now we can further decompose the embedding space to gain deeper insights into the dynamics of the system. To achieve this, we employ the Hopf Decomposition from ergodic theory, which provides a powerful framework for understanding the behavior of dynamical systems.

In the study of dynamical systems, the Hopf Decomposition offers a way to partition the phase space X with respect to an invertible and non-singular mapping T: X → X. This decomposition divides the phase space into a disjoint union: X = C ∪ D in which C represents the conservative part and D represents the dissipative part. These components capture distinct aspects of the system’s dynamics:

• Conservative Component (C): In this region of the phase space, the dynamics preserve the underlying structure without permanent deformation. In essence, the measure of subsets remains invariant under the mapping T. Formally, given a measure μ of the non-singular dynamical system with the mapping T: X → X, the conservative property is defined as:

  $$\mu \left({T}^{-1}(\sigma )\right)=\mu (\sigma ),$$

  (7)

  where σ ⊆ X and T⁻¹ represents the pre-image under the mapping T. This property is also referred to as a measure-preserving dynamical system.

• Dissipative Component (D): In this region, the dynamics exhibit dissipation, with certain subsets of the phase space gradually “wandering” away over time. Specifically, the dissipative part contains a wandering set, defined as a subset σ ⊆ D that eventually has no intersection with its earlier states under repeated applications of T. This is expressed mathematically as:

  $$\mu \left({T}^{n}(\sigma )\cap \sigma \right)=0,$$

  (8)

  for n≥1, where Tⁿ represents the n-fold application of the mapping T.

In our modeling of the dynamical system we can write the dynamical embeddings $E(t)\in {{\mathbb{R}}}^{d}$ as conservative and dissipative parts:

$$E(t)={E}_{C}(t)+{E}_{D}(t)$$

(9)

In this context, E_C(t) and E_D(t) represent the conservative and dissipative components, respectively. It is worth noting that this is just one possible approach to representing the embedding space. However, it is preferred because it highlights the genes as states, making them easier for us to study.

The conservative component should not change much with time x_C(t) ≈ x_C(t + δt). This can be justified because we shown that the modeling is Lipschitz continuous:

$$\left\Vert {E}_{C}(t)-{E}_{C}(t+\delta t)\right\Vert \le L\left(\left\Vert {x}_{C}(t)-{x}_{C}(t+\delta t)\right\Vert +\delta t\right)$$

(10)

where L is a constant. Since x_C(t) changes little, the E_C(t) remains relatively stable. On the other hand, for the dissipative component x_D(t) which changes significantly over time, the embedding variance:

$$\left\Vert {E}_{D}(t)-{E}_{D}(t+\delta t)\right\Vert \ge {L}^{{\prime} }\left\Vert {x}_{D}(t)-{x}_{D}(t+\delta t)\right\Vert$$

(11)

where $L^{\prime}$ reflects the sensitivity of embeddings to dissipative components. The significant changes in x_D(t) leads to higher variance in E_D(t).

In the context of our dynamical system, the transformation under study is the change in gene embeddings over age. By analyzing these embeddings, we can classify genes into two categories: conservative genes, which exhibit stability in their embeddings over time, and dissipative genes, which show significant changes.

Gene trajectories

For every cell i that expresses gene g our model produces a gene embedding ${E}_{g,t}^{i}$ for it’s age group t. We define our gene trajectories s_g(t) which represent the temporal drift of gene embeddings as the average cosine similarity d_cos computed between gene embeddings at time t and t₀ across multiple samples or cells C to reduce variance.

$${s}_{g}(t)=\frac{1}{{C}^{2}}\mathop{\sum }\limits_{i=1}^{C}\mathop{\sum }\limits_{j=1}^{C}{d}_{\cos }({E}_{g,t}^{i},\,{E}_{g,{t}_{0}}^{j})$$

(12)

Conservative Genes (G _c)

conservative genes are those whose embeddings remain relatively stable across the aging process. This stability indicates that the underlying biological processes associated with these genes are less affected by age-related changes. Formally, we define the set of conservative genes as:

$${G}_{C}=\left\{g\in G| \mathop{\max }\limits_{t\in {\mathcal{T}}}{D}_{g}(t)\le \delta \right\},$$

(13)

where D_g(t) represents the embedding drift of gene g at time t, defined as:

$${D}_{g}(t)=\left\Vert {s}_{g}(t)-{s}_{g}({t}_{0})\right\Vert,$$

(14)

with s_g(t) denoting the embedding trajectory of gene g at age t and s_g(t₀) as the baseline embedding trajectory (at the early lifespan age groups t₀). The threshold δ > 0 is a predefined value, determined through statistical analysis, such as the 90th percentile of the $\mathop{\max}\nolimits_{t\in {\mathcal{T}}}{D}_{g}(t)$ distribution across all genes. These genes show minimal divergence from their baseline embedding, reflecting biological stability and potentially fundamental roles in maintaining homeostasis.

Dissipative Genes (G _D)

Dissipative genes, in contrast, are those whose embeddings exhibit significant drift over time. They capture processes that are highly dynamic and responsive to age-related changes. The set of dissipative genes is defined as:

$${G}_{D}=G\setminus {G}_{C}$$

(15)

where G is the complete set of genes. This classification ensures that any gene not meeting the stability criterion for G_C is categorized as dissipative.

To quantify the variability of each gene’s embeddings over the aging process, we compute the variance of embeddings across all sampled time points:

$${\sigma }_{g}^{2}=\frac{1}{N}\mathop{\sum }\limits_{i=1}^{N}{\left\Vert {E}_{g,{t}_{i}}-{\bar{E}}_{g}\right\Vert }^{2},$$

(16)

where N is the total number of time points sampled, and ${\bar{E}}_{g}$ is the mean embedding of gene g, calculated as:

$${\bar{E}}_{g}=\frac{1}{N}\mathop{\sum }\limits_{i=1}^{N}{E}_{g,{t}_{i}}.$$

(17)

This variance metric provides an additional layer of insight into the dynamical behavior of genes, allowing us to rank genes based on their temporal stability or variability. Conservative genes will exhibit low variance, while dissipative genes will show high variance, highlighting their active role in dynamic processes. This decomposition into conservative and dissipative genes has some implications for understanding aging-related processes that we will discuss in the results section.

We further quantify the Hopf decomposition of our gene trajectories with additional phase space behavior metrics: Recurrence and Divergence.

Recurrence metric

For the recurrence of a gene trajectory s_g(t), we compute the recurrence ratio as the proportion of time-points that return within ϵ of a previous state:

$${R}_{i,j}^{(g)}=\Theta \left(\epsilon -\left\Vert {s}_{g}(i)-{s}_{g}(j)\right\Vert \right)$$

(18)

$$R{R}^{(g)}=\frac{1}{{N}^{2}}\sum _{i,j}{R}_{i,j}^{(g)}$$

(19)

Where ${R}_{i,j}^{(g)}$ is an element of the binary recurrence matrix which defines when a trajectory returns to a ϵ-neighborhood for time points i and j. Θ is the Heaviside step function. We choose a large enough ϵ in order to observe heterogeneity of recurrence within dissipative genes which typically all show a recurrence rate of 0 for a small ϵ.

Divergence metric

We choose the standard deviation of the gene trajectory s_g(t), which gives us a sense of the spread of the trajectory or its dispersion:

$${\sigma }_{({s}_{g})}=\sqrt{\frac{1}{{\mathcal{T}}}\mathop{\sum }\limits_{t=0}^{{\mathcal{T}}}{\left\Vert {s}_{t}-\bar{s}\right\Vert }^{2}}$$

(20)

These metrics are visualized in Fig. 5e with a 2D Kolmogorov-Smirnov test showing the dissipative genes cluster and the gene split is statistically significant with respect to these dynamical state metrics.

Entropy metrics offer a complementary perspective on the dynamics of aging by quantifying the uncertainty and complexity in the system. In our model, we analyze entropy changes to understand how the predictability of certain tokens, such as tissue or cell type, evolves over time. These metrics provide insights into the system’s variability and stability during the aging process.

To + calculate entropy, we focus on the conditional entropy of each masked token, where the model predicts the masked token’s value based on the surrounding context. For instance, a masked token might correspond to a specific tissue or cell type. Using the model’s predictions, we extract the probability distribution over all possible values (output of the softmax function) and compute the entropy using Shannon entropy. Mathematically, the entropy for a token x is given by:

$$H(x)=-\mathop{\sum }\limits_{i=1}^{k}P(x={x}_{i}| {\rm{context}})\log P(x={x}_{i}| {\rm{context}}),$$

(21)

where P(x = x_i∣context) is the predicted probability of token x taking the value x_i, given the surrounding context,k is size of the dictionary including all tokens, H(t) is entropy and measures the uncertainty associated with the prediction of token x.

By calculating the entropy for each token at different time points, we can track changes in uncertainty and identify patterns associated with aging. Specifically, tokens with low entropy indicate high confidence in predictions, suggesting stability and consistency in the corresponding biological processes. On the contrary, tokens with high entropy reflect greater uncertainty, potentially pointing to dynamic or less predictable biological processes.