Application of a mathematical model to clarify the statistical characteristics of a pan-tissue DNA methylation clock

Abstract

DNA methylation clocks estimate biological age based on DNA methylation profiles. This study developed a mathematical model to describe DNA methylation aging and the establishment of a pan-tissue DNA methylation clock. The model simulates the aging dynamics of DNA methylation profiles based on passive demethylation as well as the process of cross-sectional bulk data acquisition. As a result, this study identified two conditions under which the pan-tissue DNA methylation clock can successfully predict biological age: one condition is that the target tissues are sufficiently well represented in the training dataset, and the other condition is that the target sample contains cell subsets that are common among different tissues. When either of these conditions is met, the clock performs well. It is also suggested that the epigenetic age of all samples in the target tissue tends to be either over or underestimated when biological age prediction fails. The model can reveal the statistical characteristics of DNA methylation clocks.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11357-023-00949-5.

Introduction

DNA methylation clocks are used to measure biological age using DNA methylation profiles, and their development is considered to be one of the major achievements in aging research in the past decade [7, 11–13]. The DNA methylation clock is built as a model to predict biological age from the methylation intensity of a selected set of markers by applying ElasticNet regression analysis [42] to multiple methylome samples. The epigenetic age estimated by such a method is strongly correlated with chronological age and serves as a powerful biomarker of aging. The DNA methylation clock is not only expected to serve as a biomarker of aging but also as a way to elucidate the mechanisms underlying aging. A number of experiments have shown that DNA methylation clocks are associated with aging-related physiological phenomena, gene expression patterns, and diseases [34–36].

Another important aim of studies on DNA methylation clocks is the construction of pan-tissue clocks that can be used to infer the biological age of different tissues using a single mathematical equation. Pan-tissue clocks are developed by building predictive models of biological age using datasets of different tissues [11, 28]. The first pan-tissue DNA methylation clocks developed by Horvath were trained using a large amount of methylome data that were derived from various human tissues [11]. This method can predict donor age for a variety of human tissues using a single prediction equation. The findings of these studies suggest that DNA methylation clocks are capable of capturing the universal mechanisms of aging. The next step is to develop a theory that can be used to characterize the specific changes that the DNA methylation clock captures and why Horvath’s pan-tissue DNA methylation clock is established. Elucidation of the theoretical basis and the statistical properties of DNA methylation clocks is likely to facilitate a deeper understanding of the mechanisms underlying aging.

The key to understanding the statistical theory underlying DNA methylation clocks lies in the complex structure of their input data. The cross-sectional data used to train the developed pan-tissue clock is comprised of bulk methylome data with a mixture of cell subsets from different donors of various ages. In the ElasticNet regression analysis, which is widely used for clock construction, the predictive equation is constructed in a data-driven manner. The combination of these factors makes it difficult to interpret the results obtained from DNA methylation clocks. In order to clarify the conditions for the establishment of pan-tissue DNA methylation clocks, it is necessary to evaluate the performance of a developed DNA methylation clock on the simulated datasets generated by the process of acquiring cross-sectional bulk methylome dataset and to clarify their characteristics as a statistical method.

In this study, the author developed a simple mathematical model of DNA methylation aging based on biological knowledge and identified several important statistical properties of Horvath’s pan-tissue DNA methylation clock. The author identified two conditions under which a pan-tissue DNA methylation clock can successfully predict biological age: one condition is that the target tissues are sufficiently represented in the training dataset, and the other condition is that the target sample contains sufficiently common cell subsets. When either of these conditions is met, the epigenetic clock performs well. As another important property of DNA methylation clock, when biological age prediction fails, the epigenetic age of all samples in the target tissue tends to be either overestimated or underestimated. The mathematical model is considered to be useful for examining the statistical properties of DNA methylation clocks.

Result

Concept and outline of the mathematical model

To explore the statistical properties of Horvath’s pan-tissue DNA methylation clock, the author developed a simple mathematical model of cellular DNA methylation aging, focusing on passive DNA demethylation. While a number of markers have been found to be hyper- or hypo-methylated with aging [7, 45, 56], global genomic DNA demethylation is another characteristic pattern of DNA methylation associated with aging [8, 43, 48, 55].

One of the main causes of age-related passive demethylation is the failure to maintain methylation during DNA replication in cell division [22]. Another factor is DNA damage, which also decreases passively DNA methylation [57, 58]. Such passive DNA demethylation due to DNA replication or DNA damage is inevitable in multicellular organisms. Furthermore, in the human pan-tissue clock developed by Horvath [11], the relationship between chronological age and epigenetic age differs before and after the attainment of adulthood (Fig. 1a). Epigenetic age increases logarithmically during early development and at a constant rate after adulthood. This is considered to be due to the fact that the progress of the DNA methylation clock is attributed to cell division [11]. In this model, passive DNA demethylation due to DNA damage or cell division is the main factor driving the monotonous and universal DNA methylation dynamics underlying the DNA methylation clock.

Fig. 1.

Graphical illustration of the mathematical model developed in this study. a Function representing the relationship between chronological age and epigenetic age in Horvath’s clock. Epigenetic age increases logarithmically during early development and at a constant rate after adulthood. b Simulation settings. Four tissue types are considered. Each tissue type is composed of three different subsets of cells, but one of these subsets is common among all tissue types (common subset) such as immune cells. Other subsets include tissue-specific subsets. Each subset has a different methylation profile, damage sensitivity patterns, and molecular pathways. c Dynamics of the methylation profile during aging in a single cell. Passive demethylation occurs randomly across the genome (primary effect). Subsequently, the secondary effects, defined by the molecular pathway, also alter the genome-wide methylation profile. Every year, the single cell experiences primary effects and secondary effects. d The process of acquiring a cross-sectional bulk methylome dataset in an aging study. For each of the cell populations in tissues from individual donors, a single-cell expression profile is simulated during aging. A cross-sectional methylome dataset is then compiled for each sample from the cell populations obtained from donors of randomly selected ages. Bulk methylome data were simulated by adding measurement errors to the average single-cell expression values

To explore the statistical properties of DNA methylation clocks, the author developed the following mathematical model and simulation scheme. This workflow describes the cross-sectional bulk methylome data and the development of DNA methylation clocks by simulating DNA methylation aging after adulthood under various settings. Specifically, this workflow consists of the following three steps: step 1 (initial settings of tissue, cell subset, and molecular network), step 2 (data generation and simulation of aging dynamics), and step 3 (DNA methylation clock development). A full description of the model and simulations in each step are provided in the “Methods” section.

Step 1: initial setting of samples

The sample and tissue settings in the model are shown in Fig. 2b. First, in our model, the author employed four types of tissues (tissue 1, tissue 2, tissue 3, and tissue 4). The cells in each tissue were further classified into three cell subsets: one common cell subset and two tissue-specific cell subsets. An example of a common cell subset is the immune cell, which is commonly found in most human tissues [49]. Tissue-specific cell subsets are cells that are found in specific tissues and are associated with tissue-specific functions. An example of a tissue-specific cell subset is hepatocytes in the liver tissue. The percentage of common cell subsets in the tissue was defined as the model hyperparameter $Common\_Subset\_Proportion$.

Fig. 2.

DNA methylation clock is successfully established when the test tissue is well represented in the training tissue. a Example of DNA methylation clock results on training and test data in the first replicate of the simulations. The correlation (Cor) and error (Err) are shown. The red line represents the y=x line. b Box-and-whisker plot of correlation and error for all 100 replicates. c Relationship between epigenetic age and the damage index estimated based on the sum of passively demethylated methyl groups. A representative scatterplot of the first replicate and a box-and-whisker plot of Pearson’s correlation coefficient for all replicates are shown. d Relationship between epigenetic acceleration and the score of the individual difference of the damage speed. A representative scatterplot of the first replicate and a box-and-whisker plot of Pearson’s correlation coefficient for all replicates are shown. e–g Relationships among epigenetic age, damage accumulation, and donor chronological age at the single-cell level. A representative scatterplot of the first replicate and a box-and-whisker plot of Pearson’s correlation coefficient for all replicates are shown

One hundred methylation markers are set on the genome. The initial DNA methylation level was set to each marker for each cell subset, reflecting that the DNA methylation profile differs among cell subsets and that cell subset-specific DNA methylation exists [20]. In this study, the number of methyl groups takes discrete values between 0 and 500. The proportion of common markers with the same methylation levels and damage susceptibility among cell subsets is denoted by the hyperparameter Common_Marker_Proportion. In addition, as the susceptibility of DNA to damage depends on its position and epigenetic environment in the genome [19], 50% of the methylation markers were designated as being highly sensitive to damage, while the other 50% were designated as having low sensitivity to damage. The common markers have conserved patterns of either high or low damage sensitivity in all cell subsets.

For each cell subset, the model generates a matrix of molecular pathways that specify how demethylation at one site affects the methylation levels at other sites. This matrix is generated by a graph-generation algorithm as a weighted-directed graph with scale-free topology since biological networks are generally known to have a scale-free topology [41, 50–52]. The similarity of the graphs obtained for each subset is parameterized by Network_Similarity_Score. This score takes values between 0 and 1, with larger values representing more similar molecular pathways.

Step 2: data generation and simulation of aging dynamics

The 100 samples of each of the tissues, i.e., tissue 1 to tissue 4, were simulated (total 400 samples). From each tissue sample, 40 cells were simulated, and the labels of cell subsets were randomly assigned to each single cell according to the composition of the cell subsets of each tissue. Each cell undergoes passive demethylation and secondary effects every year.

In the primary effect of passive demethylation, methyl groups at the markers are lost randomly with a probability that is proportional to the product of the assigned damage sensitivity of each marker and the scores of the individual differences in damage accumulation speed. Individual differences in the speed of DNA demethylation reflect the genetic effects associated with each donor in each sample [30]. Let $\mathbf{m}$ and $\mathbf{d}$ be the vectors of the number of current methyl groups and the number of methyl groups lost in the marker regions, respectively. Then, the updated methylation value ${\mathbf{m}}^{\prime }$ can be estimated as follows:

$$\begin{array}{c}\hfill {\mathbf{m}}^{\prime }=\mathbf{m}-\mathbf{d}\end{array}$$ 1

The next step is to calculate the secondary effects of DNA demethylation based on the causal molecular pathways. The updated methylation value ${\mathbf{m}}^{\prime \prime }$ can be estimated as follows:

$$\begin{array}{c}\hfill {\mathbf{m}}^{\prime \prime }={\mathbf{m}}^{\prime }-\text{Round}(\alpha \ast \mathbf{Ad}+\mathbf{e})\end{array}$$ 2

where $\mathbf{A}$ is the weighted adjacency matrix of molecular pathways of the causal network assigned to each cell. $\mathbf{e}$ is a random variable. $\alpha$ is a hyperparameter that controls the size of the secondary effect of DNA demethylation (default ¥$\alpha$ = 10). $\text{Round}$ is a function that rounds to integer values.

Since epigenetic dysregulation progresses with aging [53, 54], the weighted adjacency matrices $\mathbf{A}$. are updated. Thus, in each year, each element of the matrix is multiplied by 0.8 with a probability of 0.1.

This sequence of dynamics will be simulated in a cell population composed of all donors from 20 to 80 years of age. The cross-sectional bulk data are obtained as follows (Fig. 1d). Given that the bulk measurement values of a sample can be obtained as the mean of the single-cell measurements in the cell population plus the measurement error [21, 46].

One age is randomly selected for each sample. The methylation values are quantified as the percentage of methyl groups. The author calculated methylation intensity as the percentage of methyl groups after simulating methylation and demethylation using discrete values per cell because the number of methyl groups on the genome is a discrete value. The bulk methylation intensity can then be calculated as the mean of the single-cell data at this selected age plus the measurement error, which is generated from a normal distribution. As a result, the cross-sectional bulk methylome data from donors of various ages have been simulated.

Step 3: DNA methylation clock development

The DNA methylation clock experiment was then applied to the simulated bulk methylome dataset from multiple tissues. This was performed using the following two treatments: “mixture split” and “separate split.” The “mixture split” treatment randomly splits the 400 total samples into 300 training samples and 100 test samples. In this treatment, the training data and the test data come from the same tissues. In the “separate split,” the clock was trained on 300 samples from the tissue 1 to tissue 3 types, i.e., 100 samples from each tissue type. The 100 samples from tissue 4 were used as test data. A “separate split” allows us to evaluate the clock performance for tissues that were not used in the training dataset. The DNA methylation clock model constructed using the training dataset was applied to the test dataset, and its performance was evaluated.

In the test tissues of the training data, the DNA methylation clock works stably even under conditions of high sample heterogeneity

The author explored the statistical properties of the pan-tissue DNA methylation clock by conducting experiments with different simulation settings. First, the author examined the performance of the DNA methylation clock in the tissues included in the training data of the “mixture split” experiment. The author used 100 datasets as replicates with randomly assigned Common_Subset_Proportion, Common_Marker_Proportion and Network_Similarity_Score hyperparameters under different random number seeds. In these simulations, Common_Subset_Proportion was the same in all tissues. The proportion of tissue-specific cell subsets in each tissue was set by normalizing two pairs of uniform random numbers to the sum of 1 - Common_Subset_Proportion. As in the study of Horvath [11], DNA methylation clock performance for each replicate was quantified by correlation and error. Correlation is taken to Pearson’s correlation coefficient between chronological and epigenetic age. Error is defined as the absolute median deviation between chronological age and epigenetic age. Figure 2a shows an example of the DNA methylation clock in the training data and the test data in the first replicate. Figure 2b shows a box-and-whisker plot of the correlation and error for all 100 replicates. The findings show that the DNA methylation clock is stable and performs well, regardless of the combination of the parameters of sample heterogeneity.

In addition, the author examined the biological interpretation of the epigenetic age on this mathematical model. First, the author observed a correlation between epigenetic age and the accumulated damage, which was quantified as the sum of passively dropped methylations in all cells in a sample (Fig. 2c). The findings of the model suggest that the DNA methylation clock captures the cumulative loss of methylation groups in the sample due to cell division or DNA damage. Next, the author considered the epigenetic acceleration associated with this model. Epigenetic acceleration is defined as the difference between epigenetic age and chronological age and represents how much the epigenetic age is accelerated in each sample compared to the chronological age, and is associated with the risk of various diseases [27] and is influenced by genetic differences among individuals [30]. There was a moderate correlation between the epigenetic acceleration and the score of individual differences in the speed of DNA damage (Fig. 2d). The results suggest that epigenetic age can capture tissue damage quantified by the cumulative DNA demethylation caused by cell division and DNA damage.

The dynamics of DNA methylation aging in the simulation were then examined at the single-cell level. Ten cells from each sample were randomly selected to examine the association between donor chronological age and cellular damage. The cumulative genome-wide sum of methyl groups passively lost in each cell was used as a measure of accumulated cellular damage. This cellular damage index correlated well with donor chronological age (Fig. 2e; representative sample and 100-replicate boxplot). It was also suggested that, as the chronological age of the donor increased, both epigenetic age and cellular damage index increased monotonically; however, the variance also increased with age. This result is consistent with that reported for the increase in cell-to-cell variability during aging [31, 32, 37].

In addition, the author calculated the epigenetic age of each individual cell. The trained DNA methylation clock was applied to the single-cell DNA methylation data. The findings suggested that single-cell epigenetic age is a good predictor of the donor’s chronological age in cases where there is no age-related change in the composition of cellular subsets (Fig. 2g; representative sample and 100-replicate boxplot). Also, the variation in single-cell epigenetic age was greater for older donors. These observations are consistent with recently published results from single-cell DNA methylation clocks based on single-cell bisulfite sequencing data [44]. In addition, single-cell epigenetic age was suggested to capture damage accumulation in individual cells (Fig. 2f; representative sample and 100-replicate boxplot). In the case of the application of the model trained by bulk data to single-cell data, a little bit smaller correlation was observed between epigenetic age and chronological age. It is considered that the pan-tissue DNA methylation clock averages out stochastic variation by using bulk data, allowing robust aging prediction based on accumulated genomic stress.

The author conducted the simulations with different sizes of the secondary effect of passive DNA demethylation ($\alpha$ = 1, 10, 25). Here, this simulation considered the situation with the highest heterogeneity among tissues, where Common_Subset_Proportion is 0 for all tissues, Common_Marker_Proportion is 0, and Network_Similarity_Score is 0. The leftmost columns in Fig. 3 show an example histogram of the correlation coefficients between the bulk-level expression and chronological age for all markers. When small secondary effects were considered ($\alpha =1$), the majority of markers showed a negative correlation with age due to the primary effect of passive DNA demethylation. Using the default setting ($\alpha =10$), both markers with positively and negatively correlated with chronological age exist, but the distribution was slightly shifted in the negative direction. And a large number of markers with correlation coefficients distributed near zero, which has shown that the simulated bulk data set also contains many markers with little association with age. The histograms of the large secondary effect ($\alpha =25$) are almost symmetrical. Performance results show that the DNA methylation clock performs consistently well in the “mixture split” treatment, although performance decreases slightly when the secondary effects are large.

Fig. 3.

Relationship between DNA methylation clock performance and the magnitude of the primary and secondary effects of DNA demethylation (mixture split). Three situations were considered: $\alpha =1$ (weak secondary effect), $\alpha =10$ (medium secondary effect, default setting), and $\alpha =25$ (strong secondary effect). The left column is an example histogram of the correlation coefficients between the bulk-level expression and chronological age for all markers in the first replicates. The second and third columns show boxplots of the predicted performance (correlation and error, respectively) of the DNA methylation clock for all 100 replicates. The fourth column shows a box-and-whisker plot of the correlation coefficient between epigenetic age and accumulated damage for all 100 replicates in the test data

On the other hand, in the result of the same simulated dataset under the “separate split” treatment, the size of the secondary effect affects the results (Supplementary Fig. S1). In the $\alpha$ = 1 simulation, epigenetic age correlated well with chronological age and cumulative damage, but large errors were observed. When $\alpha$ = 10 or 25, the correlation coefficients were lower and the errors were much higher than those obtained in the “mixture split” treatment. The findings suggested that information on random and passive demethylation alone is not sufficient for a pan-tissue clock to be established and that some inter-tissue or inter-subset conservation would be required.

The relationship between the number of simulated cells and performance is shown in Supplementary Fig. S2. The author randomly set the number of simulated cells to integers ranging from 2 to 100 and ran 400 iterations. The higher the number of simulated cells, the better the performance. The relationship is non-linear, with the performance improvement slowing down around 20 cells. The actual bulk sample contains a large number of cells, and simulating all of them would be impractical in terms of computation time. These results suggest that the default setting of 40 cells as the number of cells in this study is sufficient.

While there is a trade-off between data set size and computational time, this study investigated the statistical properties of the DNA methylation clock using the small sample size and number of markers. Larger data sets can be generated by changing the parameter settings. As an example, simulations with a larger dataset where the 500 samples of each of the tissues (total 2000 samples) and the number of markers is 4000 were simulated. The results were comparable to the results in the default settings (Supplementary Fig. S3). This model can be used not only for the DNA methylation clock but also for other aging methylome analyses; it is possible to generate datasets with different sizes depending on the objectives of the study.

Common cell subset is important in the application to tissues that were not included in the training data

We investigated which of the parameters related to sample heterogeneity (Common_Subset_Proportion, Common_Marker_Proportion, and Network_Similarity_Score) are essential for establishing a pan-tissue DNA methylation clock that works under the “separate split” treatment conditions.

First, to evaluate the importance of the common cell subset, the author simulated the performance of the DNA methylation clock for different Common_Subset_Proportion parameters. In this experiment, Common_Subset_Proportion parameters were randomly assigned the same value for all tissues. The most stringent situation consisted of Network_Similarity_Score having a score of 0 and Common_Marker_Proportion having a score of 0 was set up. Figure 4a shows the relationship between Common_Subset_Proportion and DNA methylation clock performance in the test tissue samples. The results suggest that the presence of a certain percentage of common cell subsets is a sufficient requirement for establishing a DNA methylation clock, even when the conservation of both the DNA methylation profile and the molecular pathway among subsets is short. To better clarify how the common cell subset should be distributed to optimize performance in the training and test tissue samples, the author performed another 100 simulations in which all tissues were randomly assigned different Common_Subset_Proportion parameters. Figure 4b shows the relationship between the performance and the absolute deviation in the mean Common_Subset_Proportion parameters in the training and test tissues. This score was not correlated with performance indices. Figure 4c shows the relationships between performance and Common_Subset_Proportion in the test tissue. Common_Subset_Proportion in the test tissue is associated with performance indices, suggesting that the DNA methylation clock performs better when the test tissue contains a common cell subset sufficiently.

Fig. 4.

Relationship between Common_Subset_Proportion and DNA methylation clock performance (scatterplots with Spearman’s correlation coefficient. a Scatterplots of Common_Subset_Proportion and the performance index. Common_Subset_Proportion is the same among all tissues. Each dot represents a replicate. b Relationship between the absolute deviation in the average Common_Subset_Proportion obtained for the training and test tissues, and the performance index. Common_Subset_Proportion differs among all tissues in this experiment. c Relationship between Common_Subset_Proportion in the test tissues and the performance index. Common_Subset_Proportion differs among all tissues in this experiment

We examined how the performance of the DNA methylation clock is affected by Network_Similarity_Score and Common_Marker_Proportion when Common_Subset_Proportion is 0 in all tissues. Supplementary Fig. S4(a) shows the performance index for each of the combinations of parameters where Network_Similarity_Score and Common_Marker_Proportion take values 0, 0.2, 0.4, 0.6, 0.8, and1.0; the average value of five replicates is shown for each pattern. The figure shows that the clocks perform well only when Network_Similarity_Score is high ($>=0.8$), suggesting that the existence of a common cell subset is necessary for the establishment of a pan-tissue clock.

We examined the effect of Network_Similarity_Score on the performance of the DNA methylation clock (Supplementary Fig. S4(b)). When Common_Subset_Proportion is 0.25 in all tissues and Common_Marker_Proportion is 0.5, the DNA methylation clock was established when Network_Similarity_Score was high. Also, the author examined the effect of Common_Marker_Proportion on the performance of the DNA methylation clock when Common_Subset_Proportion is 0.25 and Network_Similarity_Score is 0.5 (Supplementary Fig. S4(c)). Although there was a correlation between Common_Marker_Proportion and performance, the error was high, even when Common_Marker_Proportion was high.

The obtained results suggest that, for optimal performance, the DNA methylation clock should contain common cell subset fractions sufficiently. In addition, a high degree of molecular pathway conservation among cell subsets improves performance.

When biological age prediction fails, the epigenetic age of all samples in the target tissue tends to be either overestimated or underestimated

The findings showed that having a cell subset that is common to all samples is important for the DNA methylation clock to work as a pan-tissue clock. Therefore, the author examined how epigenetic age is affected by bias in a situation where the biological age prediction by the DNA methylation clock fails. Here, the author simulated a situation in which the proportion of common cell subsets is zero and the conservation of pathways and DNA methylation profiles among subsets are also low (Network_Similarity_Score=0.2, Common_Marker_Proportion=0.2, Common_Subset_Proportion=0). The histograms showing the correlation between chronological age and epigenetic age for 100 replicates showed both high positive and high negative correlations (Fig. 5a). The scatterplot of the error and the median value of epigenetic acceleration are shown in Fig. 5b. It was found that the predicted epigenetic age tends to be markedly overestimated or underestimated in the test tissue when the errors are large (see Fig. 5c for example cases). If the DNA methylation clock fails to predict age, then all of the samples of that tissue are either estimated to be too young or too old.

Fig. 5.

The situation when the establishment of the DNA methylation clock fails. a Histogram of correlation coefficients between epigenetic age and chronological age for 100 replicates. b Scatterplot of the error and the median of epigenetic acceleration (Med acc) for 100 replicates. c Examples of which epigenetic age is estimated as being either too young or too old in all samples of the test tissue

This result well explains the recent reports of epigenetic age prediction for mouse muscle stem cell populations by Hernando-Herraez et al. [37]. In their previous study, they applied a DNA methylation clock to a mouse muscle stem cell population purified by cytometry and reported that the epigenetic age of stem cells from old donors was higher than that of young donors, but the estimated epigenetic age was much younger than the chronological age. The cytometry-purified muscle stem cell population contains no immune cell subset, and the composition of the cell subsets in their samples is very different from the tissues that they used for training. Furthermore, the state of DNA methylation and molecular pathways of the stem cells also differ from those of the non-stem cells that were abundant in the normal tissues used for training. Their observations thus likely reflect the statistical nature of the DNA methylation clock rather than the biological nature of the stem cells.

Age-related drift in common cell subsets contributes to the establishment of a DNA methylation clock

Aging causes changes in the composition of the cell subsets in a tissue [33]. Therefore, the author simulated a situation in which the common cell subset increases by modifying the simulation algorithms related to the cellular sampling process with age-weighted bootstrap resampling in the calculation of bulk measurements. The author performed simulations for 20 cells of all subsets. W_j was used as the weight of the j-th cell subset, and 20+ W_j(Age - 20) cells were resampled by bootstrapping and the average was taken.

We compared the results of 50 simulations in each scenario with or without cell subset drift. Under the drift scenario, the author set $W\_1=2,W\_2=0$, and $W\_3=0$. Under the no drift scenario, the author set $W\_1=W\_2=W\_3=0$. In both scenarios, Network_Similarity_Score and Common_Marker_Proportion were both set to 0.5. The drift of the common subset proportion improved the correlation between epigenetic age and chronological age (Welch’s t-test P value: $2.96\times {10}^{-4}$) and accumulated damage (Welch’s t-test P value: $8.03\times {10}^{-3}$), although the contribution of drift to the improvement in the performance index error was not significant (Welch’s t-test P value: $3.98\times {10}^{-1}$) (Fig. 6a). These results are consistent with a previous study which reported that age-related changes in the proportion of immune cells contribute to the establishment of the DNA methylation clock [38].

Fig. 6.

Age-related drifts in cell subset fractions contribute to improving DNA methylation clock performance. Box-and-whisker plot of performance index for 50 simulations of each scenario with and without drift. The rightmost panel shows the correlation between epigenetic age and the damage quantified as the sum of passively dropped methylations in all cells in a sample

Discussion

This study identified two conditions under which pan-tissue DNA methylation clocks can effectively predict biological age: one condition is that the target tissues need to be sufficiently represented in the training dataset, and the other condition is that the target sample should contain cell subsets that are common among samples. When either of these conditions is met, the epigenetic clock performs well. However, the findings showed that when biological age prediction is unsuccessful, the epigenetic age of all of the samples in the target tissue tends to be either overestimated or underestimated. This finding is important for the application and biological interpretation of DNA methylation clocks. For example, it is dangerous to conclude that a sample is aging either faster or slower based solely on the epigenetic age of tissues that are not used to train the clock. Epigenetic aging studies of such samples need to consider the possibility that the obtained predictions are incorrect. For such samples, it would be worthwhile to develop a new dedicated clock or to employ multiple aging biomarkers. It is suggested that a simple mathematical model of DNA methylation dynamics during aging is useful for examining the statistical properties of DNA methylation clocks.

An important point of this study is that the mathematical model developed in this study is both a qualitative and conceptual model. Although numerous hyperparameters were employed in this model, their true values are unknown. In addition, hundreds of thousands of genome-wide DNA methylation markers are known to exist, and simulating all of them is computationally expensive. And while parameters such as the number of effective markers on the genome and available sample sizes can differ drastically between humans, mice, or other species, DNA methylation clocks seem to be quite stable across these different conditions. The primary aim of our research is to create a minimum model that elucidates the mathematical structure of the epigenetic clock, independent of the species or the specific subject under investigation. The results of our model are consistent with the observations of previous studies and provide new insights into the DNA methylation clock.

The model in this study is not limited to pan-tissue clocks but simulates bulk methylome data composed of samples of different origins and is applicable to age prediction, in general, using DNA methylome data. Recently, a pan-mammalian DNA methylation clock has also been developed [59]. The findings of this study can also be used to develop and interpret pan-mammalian clocks that construct clocks from samples of different species’ origin.

In recent years, aging clocks have been developed using a variety of omics data, including data from the transcriptome, proteome, metabolome, and the DNA methylome [40]. Since these aging clocks also employ cross-sectional bulk data, they are considered to share the methodological nature DNA methylation-based DNA methylation clocks. Consequently, developing a simple mathematical model of DNA methylation aging and DNA methylation clocks is considered to be useful for examining the statistical properties of not only DNA methylation clocks but also other omics clocks.

Methods

Detailed simulation scheme

The simulation scheme employed in this study consists of the following three steps: step 1 (initial settings), step 2 (data generation and simulation of aging dynamics), and step 3 (DNA methylation clock development). The details of the simulation in each step are described below.

Step 1: initial settings

We employed four types of tissue in this model. The cells of each tissue could be further classified into three cell subsets: one cell subset common among tissue types and two tissue-specific cell subsets. Here, the percentage of the common cell subsets in the tissues was defined as the simulation parameter $Common\_Subset\_Proportion$.

For each cell subset, the initial conditions are as follows: A total of 100 markers exist in the genome, and each marker can have a maximum of Max_CpG methyl groups. First, as the initial state of the common cell subset, each marker is randomly assigned a number of methyl groups ranging from 0.5 ¥times Max_CpG - CpG_Window to 0.5 ¥times Max_CpG + CpG_Window; for this simulation, Max_CpG=500 and CpG_Window=100 were used. For all markers, DAMAGE_PROB is a vector representing sensitivity to demethylation, which is randomly assigned to be either highly sensitive to DNA damage (50 %) or less sensitive (50 %). For high-sensitivity markers, DAMAGE_PROB=$5.0\times {10}^{-3}$, for low-sensitivity markers, DAMAGE_PROB=$1.0\times {10}^{-3}$. The setting used for the common cell subset was randomly assigned. Next, the markers of Common_Marker_proportion were randomly selected to be cell subset-specific markers. The initial settings for markers in the tissue-specific subsets were determined by randomly sorting the marker methylation values and DAMAGE_PROB employed for the common cell subset.

The causal networks for cell subsets can be given under arbitrary network similarity as follows: The causal network between methylation markers can be defined as the adjacency matrix of a weighted directed graph. The causal networks with arbitrary network similarity are generated by adding together the adjacency matrices of the common and cell subset-specific networks. The random weighted directed graph was generated as described below using the sample_pa function in the “igraph” package in R. Since biological networks typically have a scale-free topology, the power attribute is set to 1 to make the network scale-free. In addition, in the sample_pa function, the “m” attribute is a parameter that controls the number of edges in the network and can be set arbitrarily. The author defined Network_Similarity_Score as a parameter that expresses the network conservation between cell subsets and is expressed as a value between 0 and 1. A larger Network_Similarity_Score indicates that the causal network is more similar among cell subsets. The m attributes of common and cell subset-specific networks are defined as $\frac{10\ast Network\_Similarity\_Score}{Network\_Similarity\_Score+1}$ and $\frac{10}{Network\_Similarity\_Score+1}$, respectively. To give edges with direction, each element of the symmetric adjacency matrix was replaced by 0 with a probability of 0.5. Edge weights were assigned values sampled from a standard normal distribution. This procedure yields a weighted causal network among markers with a scale-free topology for each cell subset.

Step 2: data generation and simulation of aging dynamics

The 100 samples for each tissue type were simulated for tissue 1 to tissue 4. From each tissue sample, 40 cells were simulated where the author randomly assigned a cell subset according to the composition of the cell subsets of each tissue. The initial methylation value of each cell in a sample was taken as the initial value of the cell subset plus noise. The noise is sampled from a standard normal distribution and rounded to an integer value. When setting the initial conditions and after computing the primary and secondary effects of demethylation, replace the number of methyl groups by 0 if it is less than 0 and by MAX_CpG if it is greater than MAX_CpG.

The primary effect of passive demethylation can then be calculated as follows: First, for the i-th sample, the j-th subset, and the k-th marker, the probability of losing a methyl group is DAM_SPEED[i] * DAM_PROB[jk], where DAM_SPEED[i] is the parameter of individual differences in the speed of DNA demethylation in each sample. The parameters are sampled from a normal distribution with a mean of 0 and a standard deviation of 0.05. Here, the author set an upper limit (1.25) and a lower limit (0.75) to limit DAM_SPEED within a certain range. For this simulation, for each individual cell, the author simulated the number of methyl groups that are lost as follows: Let $\mathbf{m}$ and $\mathbf{d}$ be the vectors of the number of current methyl groups and those of loss in the marker regions, respectively. Then, the calculation and the update of the methylation value from $\mathbf{m}$ to ${\mathbf{m}}^{\prime }$ can be conducted as follows:

$$\begin{array}{c}\hfill {\mathbf{m}}^{\prime }=\mathbf{m}-\mathbf{d}\end{array}$$ 3

The next step is the calculation of the secondary effects of DNA demethylation based on causal molecular pathways and the update from ${\mathbf{m}}^{\prime }$ to ${\mathbf{m}}^{\prime \prime }$ as follows:

$$\begin{array}{c}\hfill {\mathbf{m}}^{\prime \prime }={\mathbf{m}}^{\prime }-\text{Round}(\alpha \ast \mathbf{Ad}+\mathbf{e})\end{array}$$ 4

where $\mathbf{A}$ is the weighted adjacency matrix of the causal network of a cell subset, $\mathbf{e}$ is a random variable, $\alpha$ is a hyperparameter that controls the size of the secondary effect of DNA demethylation, and $\text{Round}$ is a function that rounds to integer values.

Finally, the author updated the weighted adjacency matrices of the causal networks after the end of each year to account for the decay of signaling pathways with aging. Suppose that each of the cellular causal network weights is multiplied by BR_ST with probability BR_FREQ. Here, the author set BR_FREQ=0.1 and BR_ST=0.8. This sequence of dynamics will be simulated in a cell population of all donors from 20 to 80 years of age.

The cross-sectional bulk data are obtained as follows: First, the cell population of one age is randomly selected for each sample. Bulk methylation intensity is then calculated as the mean of the single-cell data plus the measurement error. The measurement error is generated from a normal random number with a mean of 0 and a standard deviation of 5. The methylation values are quantified as the percentage of methyl groups, so the author divided them by MAX_CpG. As a result, cross-sectional bulk methylome data from donors of various ages have been simulated.

Step 3: pan-tissue DNA methylation clock construction using ElasticNet regression

The training of the DNA methylation clocks was performed by ElasticNet regression on the training dataset using the glmnet package in R [47], following the Horvath clock methodology. First, the lambda parameter in the ElasticNet regression model was determined by cross-validation by applying the cv.glmnet function with family=gaussian and alpha=0.5. Next, the glmnet function with family=gaussian and alpha=0.5 is applied under this optimal lambda to construct a predictive model of age. The constructed model was applied to the test dataset, and its performance was evaluated.

Supplementary Information

Below is the link to the electronic supplementary material.

Code availability

This is a simulation study and does not use real data. The code used in this study is available at https://github.com/DaigoOkada/epiclo_simulation.

Declarations

Competing interests

The author declares no competing interests.

Ethics approval

Not applicable.