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A smoothed EM-algorithm for DNA methylation profiles from sequencing-based methods in cell lines or for a single cell type

  • Lajmi Lakhal-Chaieb , Celia M.T. Greenwood , Mohamed Ouhourane , Kaiqiong Zhao , Belkacem Abdous and Karim Oualkacha EMAIL logo
Published/Copyright: October 23, 2017
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Statistical Applications in Genetics and Molecular Biology
From the journal Statistical Applications in Genetics and Molecular Biology Volume 16 Issue 5-6

Abstract

We consider the assessment of DNA methylation profiles for sequencing-derived data from a single cell type or from cell lines. We derive a kernel smoothed EM-algorithm, capable of analyzing an entire chromosome at once, and to simultaneously correct for experimental errors arising from either the pre-treatment steps or from the sequencing stage and to take into account spatial correlations between DNA methylation profiles at neighbouring CpG sites. The outcomes of our algorithm are then used to (i) call the true methylation status at each CpG site, (ii) provide accurate smoothed estimates of DNA methylation levels, and (iii) detect differentially methylated regions. Simulations show that the proposed methodology outperforms existing analysis methods that either ignore the correlation between DNA methylation profiles at neighbouring CpG sites or do not correct for errors. The use of the proposed inference procedure is illustrated through the analysis of a publicly available data set from a cell line of induced pluripotent H9 human embryonic stem cells and also a data set where methylation measures were obtained for a small genomic region in three different immune cell types separated from whole blood.

Keywords: DNA methylation; EM-algorithm; Kernel smoothing; MethylC-Seq; RRBS

Acknowledgement

The postdoctoral salary of Mohamed Ouhourane was gratefully supported by the Natural Sciences and Engineering Research Council of Canada through individual discovery research grants to Belkacem Abdous and Karim Oualkacha and by the Fonds de recherche du Québec – Santé through individual grant # 31110 to Karim Oualkacha. Thanks also to the Ludmer Centre for Neuroinformatics and Mental Health.

Appendix

The complete data is Δc={(Xi,Yi,Li,Si),i=1,⋯,N} and the corresponding log–likelihood function is

lc=∑i=1N(1−si){yilog(p0)+(xi−yi)log(1−p0)}+∑i=1Nsi{yilog(p1)+(xi−yi)log(1−p1)}+∑i=1N{silog[1−π(li)]+(1−si)log[π(li)]}.

We begin by estimating π(l) as a piecewise constant function. Without loss of generality, let [0, L) be the set of the entire genomic positions and consider the partition 0=L0<L1<⋯<LC=L. Let π(l)=∑c=1CγcI(Lc−1≤l<Lc). The parameters of the model become p0, p1, γ1,⋯,γC. We propose to estimate these parameters using an EM-algorithm. The E-step and M-step of this iterative algorithm are :

  • E-step: Compute

    θ^i1=E[Si|Xi=xi,Yi=yi,p^0,p^1,π^(l)]=[1−π^(li)]g(yi|xi,p^1)π^(li)g(yi|xi,p^0)+[1−π^(li)]g(yi|xi,p^1)θ^i0=1−θ^i1,

  • M-step: Compute

    (7)p^0=∑i=1Nθ^i0yi∑i=1Nθ^i0xip^1=∑i=1Nθ^i1yi∑i=1Nθ^i1xiγ^c=∑i=1Nθ^i0I(Lc−1≤li<Lc)∑i=1NI(Lc−1≤li<Lc)π^(l)=∑c=1C∑i=1Nθ^i0I(Lc−1≤li<Lc)∑i=1NI(Lc−1≤li<Lc)I(Lc−1≤l<Lc)

Note that (7) can be written as

π^(l)=∑i=1Nθ^i0wi∑i=1Nwi,

with wi=∑c=1CI(Lc−1≤l,li<Lc) is the number of CpG sites that are in the same interval than l. This is a Nadaraya-Watson estimator with the kernel function K(l,li)=∑c=1CI(Lc−1≤l,li<Lc), which is equal to one if l and li are in the same interval and 0 otherwise. A generalisation of this estimator to an arbitrary kernel function is given by

wi=∑i=1NK(l−lih),

where K is a kernel function and h a bandwidth that controls the smoothness of the estimators.

Furthermore, one can also generalise (7) using local polynomial techniques, which yield π^(l)=ea^0/(1+ea^0), where a^0 is the first element of (a^0,a^1,⋯,a^m) and this set is obtained by minimizing

∑i=1NK(li−lh){θ^i0−ea0+a1(li−l)+⋯+am(li−l)m1+ea0+a1(li−l)+⋯+am(li−l)m}2.

Here, m is the order of the polynomial we use to locally approximate log{π(l)/[1−π(l)]}.

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Supplemental Material:

The online version of this article offers supplementary material (https://doi.org/10.1515/sagmb-2016-0062).

Published Online: 2017-10-23
Published in Print: 2017-11-27

©2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Approximate maximum likelihood estimation for population genetic inference
  4. A smoothed EM-algorithm for DNA methylation profiles from sequencing-based methods in cell lines or for a single cell type
  5. Modifying SAMseq to account for asymmetry in the distribution of effect sizes when identifying differentially expressed genes
  6. A statistical method for analysing cospeciation in tritrophic ecology using electrical circuit theory
  7. Multivariate association between single-nucleotide polymorphisms in Alzgene linkage regions and structural changes in the brain: discovery, refinement and validation
  8. Bayesian estimation of differential transcript usage from RNA-seq data
  9. A Bayesian hierarchical model for identifying significant polygenic effects while controlling for confounding and repeated measures
https://doi.org/10.1515/sagmb-2016-0062
Keywords for this article
DNA methylation; EM-algorithm; Kernel smoothing; MethylC-Seq; RRBS

Articles in the same Issue

  1. Frontmatter
  2. Research Articles
  3. Approximate maximum likelihood estimation for population genetic inference
  4. A smoothed EM-algorithm for DNA methylation profiles from sequencing-based methods in cell lines or for a single cell type
  5. Modifying SAMseq to account for asymmetry in the distribution of effect sizes when identifying differentially expressed genes
  6. A statistical method for analysing cospeciation in tritrophic ecology using electrical circuit theory
  7. Multivariate association between single-nucleotide polymorphisms in Alzgene linkage regions and structural changes in the brain: discovery, refinement and validation
  8. Bayesian estimation of differential transcript usage from RNA-seq data
  9. A Bayesian hierarchical model for identifying significant polygenic effects while controlling for confounding and repeated measures
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