# Methylation analysis with Methyl-IT. Part 2

## An example of methylation analysis with simulated datasets

Part 2: Potential DMPs from the methylation signal

Methylation analysis with Methyl-IT is illustrated on simulated datasets of methylated and unmethylated read counts with relatively high average of  methylation levels: 0.15 and 0.286 for control and treatment groups, respectively. In this part, potential differentially methylated positions are estimated following different approaches.

## 1. Background

Only a signal detection approach can detect with high probability real DMPs. Any statistical test (like e.g. Fisher’s exact test) not based on signal detection requires for further analysis to distinguish DMPs that naturally can occur in the control group from those DMPs induced by a treatment. The analysis here is a continuation of Part 1.

## 2. Potential DMPs from the methylation signal using empirical distribution

As suggested from the empirical density graphics (above), the critical values $H_{\alpha=0.05}$ and $TV_{d_{\alpha=0.05}}$ can be used as cutpoints to select potential DMPs. After setting $dist.name = “ECDF”$ and $tv.cut = 0.926$ in Methyl-IT function getPotentialDIMP, potential DMPs are estimated using the empirical cummulative distribution function (ECDF) and the critical value $TV_{d_{\alpha=0.05}}=0.926$.

DMP.ecdf <- getPotentialDIMP(LR = divs, div.col = 9L, tv.cut = 0.926, tv.col = 7,
alpha = 0.05, dist.name = "ECDF")

## 3. Potential DMPs detected with Fisher’s exact test

In Methyl-IT Fisher’s exact test (FT) is implemented in function FisherTest. In the current case, a pairwise group application of FT to each cytosine site is performed. The differences between the group means of read counts of methylated and unmethylated cytosines at each site are used for testing (pooling.stat=”mean”). Notice that only cytosine sites with critical values $TV_d$> 0.926 are tested (tv.cut = 0.926).

ft = FisherTest(LR = divs, tv.cut = 0.926,
pAdjustMethod = "BH",  pooling.stat = "mean",
pvalCutOff = 0.05, num.cores = 4L,
verbose = FALSE, saveAll = FALSE)

ft.tv <- getPotentialDIMP(LR = ft, div.col = 9L, dist.name = "None",
tv.cut = 0.926, tv.col = 7, alpha = 0.05)

There is not a one-to-one mapping between $TV$ and $HD$. However, at each cytosine site $i$, these information divergences hold the inequality:

$TV(p^{tt}_i,p^{ct}_i)\leq \sqrt{2}H_d(p^{tt}_i,p^{ct}_i)$  [1].

where $H_d(p^{tt}_i,p^{ct}_i) = \sqrt{\frac{H(p^{tt}_i,p^{ct}_i)}w}$ is the Hellinger distance and $H(p^{tt}_i,p^{ct}_i)$ is given by Eq. 1 in part 1.
So, potential DMPs detected with FT can be constrained with the critical value $H^{TT}_{\alpha=0.05}\geq114.5$

## 4. Potential DMPs detected with Weibull 2-parameters model

Potential DMPs can be estimated using the critical values derived from the fitted Weibull 2-parameters models, which are obtained after the non-linear fit of the theoretical model on the genome-wide $HD$ values for each individual sample using Methyl-IT function nonlinearFitDist [2]. As before, only cytosine sites with critical values $TV>0.926$ are considered DMPs. Notice that, it is always possible to use any other values of $HD$ and $TV$ as critical values, but whatever could be the value it will affect the final accuracy of the classification performance of DMPs into two groups, DMPs from control and DNPs from treatment (see below). So, it is important to do an good choices of the critical values.

nlms.wb <- nonlinearFitDist(divs, column = 9L, verbose = FALSE, num.cores = 6L)
# Potential DMPs from 'Weibull2P' model
DMPs.wb <- getPotentialDIMP(LR = divs, nlms = nlms.wb, div.col = 9L,
tv.cut = 0.926, tv.col = 7, alpha = 0.05,
dist.name = "Weibull2P")
nlms.wb$T1  ## Estimate Std. Error t value Pr(>|t|)) Adj.R.Square ## shape 0.5413711 0.0003964435 1365.570 0 0.991666592250838 ## scale 19.4097502 0.0155797315 1245.833 0 ## rho R.Cross.val DEV ## shape 0.991666258901194 0.996595712743823 34.7217494754823 ## scale ## AIC BIC COV.shape COV.scale ## shape -221720.747067975 -221694.287733122 1.571674e-07 -1.165129e-06 ## scale -1.165129e-06 2.427280e-04 ## COV.mu n ## shape NA 50000 ## scale NA 50000 ## 5. Potential DMPs detected with Gamma 2-parameters model As in the case of Weibull 2-parameters model, potential DMPs can be estimated using the critical values derived from the fitted Gamma 2-parameters models and only cytosine sites with critical values$TV_d > 0.926$are considered DMPs. nlms.g2p <- nonlinearFitDist(divs, column = 9L, verbose = FALSE, num.cores = 6L, dist.name = "Gamma2P") # Potential DMPs from 'Gamma2P' model DMPs.g2p <- getPotentialDIMP(LR = divs, nlms = nlms.g2p, div.col = 9L, tv.cut = 0.926, tv.col = 7, alpha = 0.05, dist.name = "Gamma2P") nlms.g2p$T1
##         Estimate   Std. Error  t value Pr(>|t|))      Adj.R.Square
## shape  0.3866249 0.0001480347 2611.717         0 0.999998194156282
## scale 76.1580083 0.0642929555 1184.547         0
##                     rho       R.Cross.val                 DEV
## shape 0.999998194084045 0.998331895911125 0.00752417919133131
## scale
##                    AIC               BIC     COV.alpha     COV.scale
## shape -265404.29138371 -265369.012270572  2.191429e-08 -8.581717e-06
## scale                                    -8.581717e-06  4.133584e-03
##       COV.mu    df
## shape     NA 49998
## scale     NA 49998

Summary table:

data.frame(ft = unlist(lapply(ft, length)), ft.hd = unlist(lapply(ft.hd, length)),
ecdf = unlist(lapply(DMPs.hd, length)), Weibull = unlist(lapply(DMPs.wb, length)),
Gamma = unlist(lapply(DMPs.g2p, length)))
##      ft ft.hd ecdf Weibull Gamma
## C1 1253   773   63     756   935
## C2 1221   776   62     755   925
## C3 1280   786   64     768   947
## T1 2504  1554  126     924  1346
## T2 2464  1532  124     942  1379
## T3 2408  1477  121     979  1354

## 6. Density graphic with a new critical value

The graphics for the empirical (in black) and Gamma (in blue) densities distributions of Hellinger divergence of methylation levels for sample T1 are shown below. The 2-parameter gamma model is build by using the parameters estimated in the non-linear fit of $H$ values from sample T1. The critical values estimated from the 2-parameter gamma distribution $H^{\Gamma}_{\alpha=0.05}=124$ is more ‘conservative’ than the critical value based on the empirical distribution $H^{Emp}_{\alpha=0.05}=114.5$. That is, in accordance with the empirical distribution, for a methylation change to be considered a signal its $H$ value must be $H\geq114.5$, while according with the 2-parameter gamma model any cytosine carrying a signal must hold $H\geq124$.

suppressMessages(library(ggplot2))

# Some information for graphic
dt <- data[data$sample == "T1", ] coef <- nlms.g2p$T1$Estimate # Coefficients from the non-linear fit dgamma2p <- function(x) dgamma(x, shape = coef[1], scale = coef[2]) qgamma2p <- function(x) qgamma(x, shape = coef[1], scale = coef[2]) # 95% quantiles q95 <- qgamma2p(0.95) # Gamma model based quantile emp.q95 = quantile(divs$T1$hdiv, 0.95) # Empirical quantile # Density plot with ggplot ggplot(dt, aes(x = HD)) + geom_density(alpha = 0.05, bw = 0.2, position = "identity", na.rm = TRUE, size = 0.4) + xlim(c(0, 150)) + stat_function(fun = dgamma2p, colour = "blue") + xlab(expression(bolditalic("Hellinger divergence (HD)"))) + ylab(expression(bolditalic("Density"))) + ggtitle("Empirical and Gamma densities distributions of Hellinger divergence (T1)") + geom_vline(xintercept = emp.q95, color = "black", linetype = "dashed", size = 0.4) + annotate(geom = "text", x = emp.q95 - 20, y = 0.16, size = 5, label = 'bolditalic(HD[alpha == 0.05]^Emp==114.5)', family = "serif", color = "black", parse = TRUE) + geom_vline(xintercept = q95, color = "blue", linetype = "dashed", size = 0.4) + annotate(geom = "text", x = q95 + 9, y = 0.14, size = 5, label = 'bolditalic(HD[alpha == 0.05]^Gamma==124)', family = "serif", color = "blue", parse = TRUE) + theme( axis.text.x = element_text( face = "bold", size = 12, color="black", margin = margin(1,0,1,0, unit = "pt" )), axis.text.y = element_text( face = "bold", size = 12, color="black", margin = margin( 0,0.1,0,0, unit = "mm")), axis.title.x = element_text(face = "bold", size = 13, color="black", vjust = 0 ), axis.title.y = element_text(face = "bold", size = 13, color="black", vjust = 0 ), legend.title = element_blank(), legend.margin = margin(c(0.3, 0.3, 0.3, 0.3), unit = 'mm'), legend.box.spacing = unit(0.5, "lines"), legend.text = element_text(face = "bold", size = 12, family = "serif")  ## References 1. Steerneman, Ton, K. Behnen, G. Neuhaus, Julius R. Blum, Pramod K. Pathak, Wassily Hoeffding, J. Wolfowitz, et al. 1983. “On the total variation and Hellinger distance between signed measures; an application to product measures.” Proceedings of the American Mathematical Society 88 (4). Springer-Verlag, Berlin-New York: 684–84. doi:10.1090/S0002-9939-1983-0702299-0. 2. Sanchez, Robersy, and Sally A. Mackenzie. 2016. “Information Thermodynamics of Cytosine DNA Methylation.” Edited by Barbara Bardoni. PLOS ONE 11 (3). Public Library of Science: e0150427. doi:10.1371/journal.pone.0150427. # Methylation analysis with Methyl-IT. Part 1 ## An example of methylation analysis with simulated datasets Part 1: Methylation signal Methylation analysis with Methyl-IT is illustrated on simulated datasets of methylated and unmethylated read counts with relatively high average of methylation levels: 0.15 and 0.286 for control and treatment groups, respectively. The main Methyl-IT downstream analysis is presented alongside the application of Fisher’s exact test. The importance of a signal detection step is shown. ## 1. Background Methyl-IT R package offers a methylome analysis approach based on information thermodynamics (IT) and signal detection. Methyl-IT approach confront detection of differentially methylated cytosine as a signal detection problem. This approach was designed to discriminate methylation regulatory signal from background noise induced by molecular stochastic fluctuations. Methyl-IT R package is not limited to the IT approach but also includes Fisher’s exact test (FT), Root-mean-square statistic (RMST) or Hellinger divergence (HDT) tests. Herein, we will show that a signal detection step is also required for FT, RMST, and HDT as well. ## 2. Data generation For the current example on methylation analysis with Methyl-IT we will use simulated data. Read count matrices of methylated and unmethylated cytosine are generated with Methyl-IT function simulateCounts. Function simulateCounts randomly generates prior methylation levels using Beta distribution function. The expected mean of methylation levels that we would like to have can be estimated using the auxiliary function: bmean <- function(alpha, beta) alpha/(alpha + beta) alpha.ct <- 0.09 alpha.tt <- 0.2 c(control.group = bmean(alpha.ct, 0.5), treatment.group = bmean(alpha.tt, 0.5), mean.diff = bmean(alpha.tt, 0.5) - bmean(alpha.ct, 0.5))  ## control.group treatment.group mean.diff ## 0.1525424 0.2857143 0.1331719 This simple function uses the α and β (shape2) parameters from the Beta distribution function to compute the expected value of methylation levels. In the current case, we expect to have a difference of methylation levels about 0.133 between the control and the treatment. ### 2.1. Simulation Methyl-IT function simulateCounts will be used to generate the datasets, which will include three group of samples: reference, control, and treatment. suppressMessages(library(MethylIT)) # The number of cytosine sites to generate sites = 50000 # Set a seed for pseudo-random number generation set.seed(124) control.nam <- c("C1", "C2", "C3") treatment.nam <- c("T1", "T2", "T3") # Reference group ref0 = simulateCounts(num.samples = 4, sites = sites, alpha = alpha.ct, beta = 0.5, size = 50, theta = 4.5, sample.ids = c("R1", "R2", "R3")) # Control group ctrl = simulateCounts(num.samples = 3, sites = sites, alpha = alpha.ct, beta = 0.5, size = 50, theta = 4.5, sample.ids = control.nam) # Treatment group treat = simulateCounts(num.samples = 3, sites = sites, alpha = alpha.tt, beta = 0.5, size = 50, theta = 4.5, sample.ids = treatment.nam) Notice that reference and control groups of samples are not identical but belong to the same population. ### 2.2. Divergences of methylation levels The estimation of the divergences of methylation levels is required to proceed with the application of signal detection basic approach. The information divergence is estimated here using the function estimateDivergence. For each cytosine site, methylation levels are estimated according to the formulas:$p_i={n_i}^{mC_j}/({n_i}^{mC_j}+{n_i}^{C_j})$, where${n_i}^{mC_j}$and${n_i}^{C_j}$are the number of methylated and unmethylated cytosines at site$i$. If a Bayesian correction of counts is selected in function estimateDivergence, then methylated read counts are modeled by a beta-binomial distribution in a Bayesian framework, which accounts for the biological and sampling variations [1,2,3]. In our case we adopted the Bayesian approach suggested in reference [4] (Chapter 3). Two types of information divergences are estimated: TV, total variation (TV, absolute value of methylation levels) and Hellinger divergence (H). TV is computed according to the formula:$TV=|p_{tt}-p_{ct}|$and H:$H(\hat p_{ij},\hat p_{ir}) = w_i[(\sqrt{\hat p_{ij}} – \sqrt{\hat p_{ir}})^2+(\sqrt{1-\hat p_{ij}} – \sqrt{1-\hat p_{ir}})^2]$(1) where$w_i = 2 \frac{m_{ij} m_{ir}}{m_{ij} + m_{ir}}$,$m_{ij} = {n_i}^{mC_j}+{n_i}^{uC_j}+1$,$m_{ir} = {n_i}^{mC_r}+{n_i}^{uC_r}+1$and$j \in {\{c,t}\}$The equation for Hellinger divergence is given in reference [5], but any other information theoretical divergences could be used as well. Divergences are estimated for control and treatment groups in respect to a virtual sample, which is created applying function poolFromGRlist on the reference group. . # Reference sample ref = poolFromGRlist(ref0, stat = "mean", num.cores = 4L, verbose = FALSE) # Methylation level divergences DIVs <- estimateDivergence(ref = ref, indiv = c(ctrl, treat), Bayesian = TRUE, num.cores = 6L, percentile = 1, verbose = FALSE) The mean of methylation levels differences is: unlist(lapply(DIVs, function(x) mean(mcols(x[, 7])[,1]))) ## C1 C2 C3 T1 T2 ## -0.0009820776 -0.0014922009 -0.0022257725 0.1358867135 0.1359160219 ## T3 ## 0.1309217360 ## 3. Methylation signal Likewise for any other signal in nature, the analysis of methylation signal requires for the knowledge of its probability distribution. In the current case, the signal is represented in terms of the Hellinger divergence of methylation levels (H). divs = DIVs[order(names(DIVs))] # To remove hd == 0 to estimate. The methylation signal only is given for divs = lapply(divs, function(div) div[ abs(div$hdiv) > 0 ])
names(divs) <- names(DIVs)

# Data frame with the Hellinger divergences from both groups of samples samples
l = c(); for (k in 1:length(divs)) l = c(l, length(divs[[k]]))
data <- data.frame(H = c(abs(divs$C1$hdiv), abs(divs$C2$hdiv), abs(divs$C3$hdiv),
abs(divs$T1$hdiv), abs(divs$T2$hdiv), abs(divs$T3$hdiv)),
sample = c(rep("C1", l[1]), rep("C2", l[2]), rep("C3", l[3]),
rep("T1", l[4]), rep("T2", l[5]), rep("T3", l[6]))
)

Empirical critical values for the probability distribution of H and TV can be obtained using quantile function from the R package stats.

critical.val <- do.call(rbind, lapply(divs, function(x) {
hd.95 = quantile(x$hdiv, 0.95) tv.95 = quantile(x$TV, 0.95)
return(c(tv = tv.95, hd = hd.95))
}))

critical.val
##       tv.95%    hd.95%
## C1 0.7893927  81.47256
## C2 0.7870469  80.95873
## C3 0.7950869  81.27145
## T1 0.9261629 113.73798
## T2 0.9240506 114.45228
## T3 0.9212163 111.54258

### 3.1. Density estimation

The kernel density estimation yields the empirical density shown in the graphics:

suppressMessages(library(ggplot2))

# Some information for graphic
crit.val.ct <- max(critical.val[c("C1", "C2", "C3"), 2]) # 81.5
crit.val.tt <- min(critical.val[c("T1", "T2", "T3"), 2]) # 111.5426

# Density plot with ggplot
ggplot(data, aes(x = H, colour = sample, fill = sample)) +
geom_density(alpha = 0.05, bw = 0.2, position = "identity", na.rm = TRUE,
size = 0.4) + xlim(c(0, 125)) +
xlab(expression(bolditalic("Hellinger divergence (H)"))) +
ylab(expression(bolditalic("Density"))) +
ggtitle("Density distribution for control and treatment") +
geom_vline(xintercept = crit.val.ct, color = "red", linetype = "dashed", size = 0.4) +
annotate(geom = "text", x = crit.val.ct-2, y = 0.3, size = 5,
label = 'bolditalic(H[alpha == 0.05]^CT==81.5)',
family = "serif", color = "red", parse = TRUE) +
geom_vline(xintercept = crit.val.tt, color = "blue", linetype = "dashed", size = 0.4) +
annotate(geom = "text", x = crit.val.tt -2, y = 0.2, size = 5,
label = 'bolditalic(H[alpha == 0.05]^TT==114.5)',
family = "serif", color = "blue", parse = TRUE) +
theme(
axis.text.x  = element_text( face = "bold", size = 12, color="black",
margin = margin(1,0,1,0, unit = "pt" )),
axis.text.y  = element_text( face = "bold", size = 12, color="black",
margin = margin( 0,0.1,0,0, unit = "mm")),
axis.title.x = element_text(face = "bold", size = 13,
color="black", vjust = 0 ),
axis.title.y = element_text(face = "bold", size = 13,
color="black", vjust = 0 ),
legend.title = element_blank(),
legend.margin = margin(c(0.3, 0.3, 0.3, 0.3), unit = 'mm'),
legend.box.spacing = unit(0.5, "lines"),
legend.text = element_text(face = "bold", size = 12, family = "serif")
)
The above graphic shows that with high probability the methylation signal induced by the treatment has H values $H^{TT}_{\alpha=0.05}\geq114.5$. According to the critical value estimated for the differences of methylation levels, the methylation signal holds $TV^{TT}_{\alpha=0.05}\geq0.926$. Notice that most of the methylation changes are not signal but noise (found to the left of the critical values). This situation is typical for all the natural and technologically generated signals. Assuming that the background methylation variation is consistent with a Poisson process and that methylation changes conform to the second law of thermodynamics, the Hellinger divergence of methylation levels follows a Weibull distribution probability or some member of the generalized gamma distribution family [6].

## References

1. Hebestreit, Katja, Martin Dugas, and Hans-Ulrich Klein. 2013. “Detection of significantly differentially methylated regions in targeted bisulfite sequencing data.” Bioinformatics (Oxford, England) 29 (13): 1647–53. doi:10.1093/bioinformatics/btt263.
2. Hebestreit, Katja, Martin Dugas, and Hans-Ulrich Klein. 2013. “Detection of significantly differentially methylated regions in targeted bisulfite sequencing data.” Bioinformatics (Oxford, England) 29 (13): 1647–53. doi:10.1093/bioinformatics/btt263.
3. Dolzhenko, Egor, and Andrew D Smith. 2014. “Using beta-binomial regression for high-precision differential methylation analysis in multifactor whole-genome bisulfite sequencing experiments.” BMC Bioinformatics 15 (1). BioMed Central: 215. doi:10.1186/1471-2105-15-215.
4. Baldi, Pierre, and Soren Brunak. 2001. Bioinformatics: the machine learning approach. Second. Cambridge: MIT Press.
5. Basu, A., A. Mandal, and L. Pardo. 2010. “Hypothesis testing for two discrete populations based on the Hellinger distance.” Statistics & Probability Letters 80 (3-4). Elsevier B.V.: 206–14. doi:10.1016/j.spl.2009.10.008.
6. Sanchez R, Mackenzie SA. Information Thermodynamics of Cytosine DNA Methylation. PLoS One, 2016, 11:e0150427.