MaximumLikelihood Estimation of Substitution Heterogeneity through Clustering

1
,
Maximum-likelihood estimation of substitution
heterogeneity through clustering Zhang Zhang and
Jeffrey P. Townsend Department of Ecology and
Evolutionary Biology, Yale University, New Haven,
Connecticut, United States of America
Summary Detecting substitution heterogeneity is
of great importance for locating divergent and
conservative regions along DNA and protein
sequences. Heterogeneous regions correspond to
nonuniform features and imply different
evolutionary process arising from different
functional constraints or selections. Here we
propose a maximum-likelihood method to detect
substitution heterogeneity in sequences. The
method uses divide and conquer to cluster
sequences into heterogeneous regions of different
substitutions. To determine whether a cluster is
deviated significantly from the sequences, the
method adopts several criteria, such as
Likelihood Ratio Test (LRT), Akaike Information
Criterion (AIC) and its variation (AICc), and
Bayesian Information Criterion (BIC). The method
does not need a priori knowledge for clustering
or the number of clusters, and particularly, it
is more accurate testified by the application to
several real data and can be applied to
comparative evolutionary studies. In addition, we
recommend that criteria with a consideration of
sampling size (AICc and BIC) be used in all
circumstances.

• Introduction
• Substitutions along sequences are concentrated in
  some regions and/or relatively sparse in others.
• Why should we care
• Substitution heterogeneity provides
• clues to sequence function
• implications of sequence structure
• evidence of interesting evolutionary phenomena
• What is the question
• Substitutions are not uniform.
• Positive selection is often acting on small
  regions along sequence.
• Heterogeneity of substitution within genes is
  not well accounted for by existing methods that
  average selective pressure across all sites under
  the assumption that all sites evolve at the same
  rate, or estimate selective pressure at each site
  under a given statistical distribution.
• Lack of a statistical method to detect
  clustering within discrete linear sequences when
  there is no priori specification of cluster size
  or the number of clusters.
• How we can do it
• Here we propose a method based on maximum
  likelihood estimation (MLE) to detect regional
  clusters with different substitution
  heterogeneity.

• Model selection
• To examine whether a clustered model best fits a
  sequence, our method adopts several different
  criteria for model selection.
• Likelihood Ratio Test (LRT) is a most popular
  strategy for model selection. The maximized
  log-likelihoods of the null (lnL0) and the
  clustering (lnLc) models should be asymptotically
  distributed as a ?2 with two degrees of freedom.
  That is, the p-value lt the significance level
  (usually 0.05), indicates that the clustering
  model fits the data significantly better than the
  null model and vice versa.
• Akaike Information Criterion (AIC) represents
  the Kullback-Leibler distance between a true
  model and an examined model and quantifies the
  information lost by approximating the true model,
  where L is the maximized likelihood value and k
  is the number of parameters. We define k0 and kc
  as the number of parameters under the null
  hypothesis and the clustering model, respectively
  and thus k00 and kc2.
• AICc, a modification of AIC, allows for
  sampling size (n) as well as k and L, especially
  for clustering short sequences into heterogeneous
  regions which probably involves more biases.
• Bayesian Information Criterion (BIC), similar
  to AICc, is a function of its maximized
  log-likelihood, number of parameters and sampling
  size.
• Algorithm implementation
• The new method uses divide and conquer approach
  to detect all possible clusters among sequences.
  After locating the first cluster, the method
  partitions sequences into three sub-sequences and
  then repeats the same analysis for these three
  sub-sequences, until all segments of the sequence
  have failed to demonstrate clustering (see Figure
  2).

• Results
• We used our method to detect clusters along the
  Drosophila Adh gene within five species of
  Drosophila melanogaster species subgroup (D.
  melanogaster, D. sechellia, D. simulans, D.
  yakuba and D. erecta) and identified
  heterogeneous clusters with different
  substitution (see Table 1 and Figure 3).

• Materials and methods
• Alignment
• Suppose that two or more aligned sequences have N
  sites and for each site, 0 represents identical
  and 1 represents variant. Therefore, the aligned
  sequences can be denoted as
• Clustering model
• The null hypothesis assumes no heterogeneous
  cluster among sequences. Consequently, the
  likelihood of the null model (without clusters)
  is calculated as
• where n is the number of variant sites.
• Under the clustering model, the entire
  sequence is partitioned into three regions and
  the central region is considered as the
  heterogeneous cluster (see Figure 1).
• Figure 1 Illustration of a cluster among sequence
  X, suppose that cs and ce are the start position
  and end position of cluster, respectively, and
  ns, nc and ne are the number of variant sites in
  the beginning, central and ending regions,
  respectively, where n ns nc ne.
• The likelihood of the clustering model is
  formulated as

Acknowledgments We thank Zheng Wang, Francesc
López, Aleksandra Adomas, Gina Wilpiszeski and
Andrea Hodgins-Davis for valuable discussions.
This work is supported by a grant from the
National Institute of General Medical Sciences at
the U.S. National Institutes of Health (GM068087).
Further information Please contact
Zhang.Zhang_at_yale.edu and Jeffrey.Townsend_at_yale.edu
. The proposed method has been implemented in the
program MLCluster that is freely available at
www.yale.edu/townsend/software.html. A PDF
version of the poster can be obtained at
www.yale.edu/townsend/Poster/MLCluster.pdf.
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