Table 1 Computational methods for scoring genome wide DNA methylation heterogeneity

Model-based (MeH)

Abundance based

\({\left(\sum_{i=1}^{R}{a}_{i}^{2}\right)}^{-1}\in \left\{1,\dots ,{2}^{w}\right\}\)

\(a:\) methylation patterns

Counting distinct methylation patterns

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Pairwise-similarity based

\({\left(\sum_{i=1}^{R}\sum_{j=1}^{R}{{d}_{ij}p}_{ij}^{2}\right)}^{-1/2}>0\)

\(p:\) methylation patterns

Considering pairwise similarity between patterns

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Phylogenetic-tree based

\({\left(\sum_{i=1}^{B}{L}_{i}{a}_{i}^{2}\right)}^{-1}>0\).5

\(a:\) methylation patterns

Considering the total similarity among all patterns

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Other methods

Methylation-concurrence [18, 22]

\(\frac{\sum_{c=1}^{C}{\omega }_{c}}{\sum_{c=1}^{C}{\omega }_{c}+ \sum_{m=1}^{M}{\omega }_{m}+ \sum_{u=1}^{U}{\omega }_{u}}\in [\mathrm{0,1})\)

\(\omega :\) reads covering CG sites

Measuring the methylation concurrence between patterns

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Proportion of Discordant Reads (PDR) [18, 20]

\(\frac{{\sum }_{r\in {R}_{c}}I(\exists i,j \epsilon r \mathrm{s}.\mathrm{t}. {x}_{j,r}\ne {x}_{i,r})}{|{R}_{c}|}\in [\mathrm{0,1}]\)

\(r:\) reads covering CG sites

Counting distinct methylation patterns among reads

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Methylation entropy [17, 18, 23, 24]

\(\frac{1}{w}\sum_{k}-{a}_{k}{log}_{2}{a}_{k}\in [\mathrm{0,1}]\)

\(a:\) methylation patterns

Measuring the chaos among the reads of different methylation patterns

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Epipolymorphism [18, 19]

\(1-\sum_{k}{a}_{k}^{2}\in [\mathrm{0,1})\)

\(a:\) methylation patterns

Estimating the probability of observing two different patterns at random

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Fraction of Discordant Read Pairs (FDRP) [18]

\(\frac{{\sum }_{{r}_{s}\in {R}_{c}}{\sum }_{{r}_{t}\in {R}_{c},t>s}I(\exists i\in \left\{{r}_{s}\cap {r}_{t}\right\} \mathrm{s}.\mathrm{t}. {x}_{i,{r}_{s}}\ne {x}_{i,{r}_{t}})}{(\genfrac{}{}{0pt}{}{|{R}_{c}|}{2})}\in [\mathrm{0,1}]\)

\(r:\) reads covering CG sites

Calculating pairwise disagreement of between any two reads

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Quantitative FDRP (qFDRP)[18]

\(\frac{{\sum }_{{r}_{s}\in {R}_{c}}{\sum }_{{r}_{t}\in {R}_{c},t>s}\frac{{\sum }_{i\in {\{r}_{s}\cap {r}_{t}\}}I({x}_{i,{r}_{s}}\ne {x}_{i,{r}_{t}})}{|\{{r}_{s}\cap {r}_{t}\}|}}{(\genfrac{}{}{0pt}{}{|{R}_{c}|}{2})}\in [\mathrm{0,1}]\)

\(r:\) reads covering CG sites

Quantifying the similarity of paired-methyl reads by Hamming distance

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Methylation Haplotype Load (MHL) [18, 21]

\(\frac{{\sum }_{l=0}^{L}(l+1)\frac{{\sum }_{r\in {R}_{c}}{\sum }_{i=1}^{\left|r\right|-l}I({x}_{i,r}=1\wedge \dots \wedge {x}_{i+l.r}=1)}{{\sum }_{r\in {R}_{c}}\left|r\right|-l}}{{\sum }_{l=0}^{L}l+1}\in [\mathrm{0,1}]\)

\(r:\) reads covering CG sites

Estimating the fraction of strings that are fully methylated for all possible lengths

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