Fractional Calculus in Epigenetics: Modelling DNA Methylation Dynamics Using Mittag–Leffler Function

DNA methylation is an epigenetic modification where a methyl group is added to a DNA molecule, typically at the cytosine base within a CpG dinucleotide. This process can influence gene expression without changing the underlying DNA sequence. Essentially, methylation can act like a switch that regulates which genes are active in a cell. DNA methylation (DNAm) models often describe the dynamic changes of methylation levels at specific DNA sites, considering methylation and demethylation processes. A common approach involves representing the methylation state as a continuous variable, and modelling its change over time or in response to various factors using differential equations. These equations can incorporate parameters such as the methylation and demethylation rates, factors like DNA replication, the influence of regulatory proteins, and other related parameters. Understanding DNAm dynamics in relation to age is crucial for elucidating ageing processes and developing biomarkers. This work introduces a theoretical framework for modelling DNAm dynamics using a fractional calculus approach, extending standard models based on the integer-order differential equations. The proposed fractional-calculus representation of the methylation process, defined by the fractional-order differential equation and its solution based on the Mittag–Leffler function, provides improved results compared to the standard model that uses a first-order differential equation, which contains an exponential function in its solution, in terms of the comparison criteria (sum of absolute errors, sum of squared errors, mean absolute percentage error, R-squared, and adjusted R-squared). Moreover, the Mittag–Leffler model provides a more general representation of DNAm dynamics, making the standard exponential model only one specific case.