DNA Segmentation

1
DNA Segmentation

• Presented by Ming-Te Cheng
• IFT 6299 - Algorithmique de lADN
• Autumn 2004
• November 15, 2004

2
Overview

• Introduction
• Segmentation Models
• Segmentation Methods
• Discussion

3
Introduction

• Statistical analysis of DNA sequences are
  motivated by 3 areas of exploration
• DNA sequence data offer an extremely fine view
  where traditional methods of variation analysis
  can be extended
• DNA sequence data allow fine-tuning and
  organization of genetic process
• Comparison of sequences between species demands
  methods of determining similarities in evolution
  or function

4
Introduction

• Large chunks of the genome are sequenced, where
  the functionality of many sequences are unknown
• Scientists rely on homology (similarity) to
  analyze unknown sequences with previously well
  studies small sequences
• Need methods of describing and assessing
  sequences that provide useful characterizations
• Ex Segmentation models

5
Segmentation Models

• Assumption Sequences can be partitioned into a
  number of segments
• Each segment has a certain degree of internal
  homogeneity (or similarity)
• Ex Isochores
• Large segments (gt 300 kb) of DNA belonging to a
  number of classes defined by different GC
  levels and by fairly homogeneous base
  compositions

6
Segmentation Methods

• Common techniques
• Moving Window
• Maximum Likelihood Estimation
• Hidden Markov Models
• Recursive Segmentation

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Segmentation Methods Moving Window

• Most commonly used algorithm in biology community
• Straightforward implementation
• Calculate density of a sequence feature of
  interest within a window
• Move window along sequence
• Recalculate density again

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Segmentation MethodsMoving Window

• Drawbacks
• Arbitrary choice of window size and moving
  distance
• If window size is too large, local fluctuations
  that contain significant biological information
  may be averaged out
• If moving distance is too long, one domain can be
  split between two windows and its distinctive
  feature may be lost

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Segmentation MethodsMaximum-Likelihood Estimation

• Algorithm that computes the maximum likelihood
  estimate for the number of changed segments

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Segmentation Methods Maximum-Likelihood
Estimation

• Let X1,,Xn represent a sequence of independent
  random letters from an alphabet
• Let every Xi be one of two known distributions,
  specified by the probabilities and
• Changed segment is a segment a,b of indices
  where for all
• Unchanged segment is a segment a,b of indices
  where for all

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Segmentation Methods Maximum-Likelihood
Estimation

• Let xi be the observed values of Xi for sequence
• Let C be a non-intersecting set of hypothetical
  changed segments
• Let z (z1,,zn) be the indicator vector for C

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Segmentation Methods Maximum-Likelihood
Estimation

• Likelihood function is can be written as
• Log-likelihood can be written as
• First term represents log-likelihood of null
  hypothesis that there are no changed segments
• Second term represents log-likelihood ratio of
  the alternative hypothesis

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Segmentation Methods Maximum-Likelihood
Estimation
14
Segmentation MethodsHidden Markov Models

• Example of Markov Model

15
Segmentation MethodsHidden Markov Models

• Example of Hidden Markov Model

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Segmentation MethodsHidden Markov Models

• Assumes that different segments can be classified
  into a finite set of state, where the nucleotide
  data in each state follows a probability
  distribution

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Segmentation MethodsHidden Markov Models

• Let the finite number of r states underlying the
  observations be denoted by Si
• Let the states follows a Markov process with
  transition matrix
• System of equations for the hidden chain can be
  written as

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Segmentation MethodsHidden Markov Models

• Likewise, system of equations for the
  observations can be written as
• where yi (yi,1,,yi,m) represent vector of m
  possible observed outcomes, and where each
  observation is associated with one of the states

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Segmentation MethodsHidden Markov Models

• With the system equations for hidden chain and
  observations, the smoothing equations can be
  derived
• and be used to plot the homogeneous regions in
  the sequence

20
Segmentation MethodsHidden Markov Models
21
Segmentation MethodsRecursive Segmentation

• Assumes that sequences exhibit hierarchical
  patterns (possibility of subdomains)
• It is possible to apply a filter to convert the
  original four-base DNA sequence into k-symbol
  sequence
• Ex S(strong) C,G and W(weak) A,T

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Segmentation MethodsRecursive Segmentation

• Divide-and-conquer approach is applied
• For k-symbol sequence of length N, calculate each
  position i (0 lt i lt N) the entropy H of the whole
  sequence, entropy Hl of the subsequence on the
  left side of the partition point, and entropy Hr
  of the subsequence on the right side.

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Segmentation MethodsRecursive Segmentation

• Entropy equations as defined by (Shannon 1948)
• where Nj, Nj,l, and Nj,r are the counts of
  symbol j in the whole, left, and right sequence,
  respectively

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Segmentation MethodsRecursive Segmentation

• Maximized Jensen-Shannon divergence was chosen to
  measure the heterogeneity of the sequence
• If divergence is large enough, the sequence is
  heterogeneous and should be segmented
• Equation is recursively applied for both the left
  and the right subsequence, as long as the
  calculated divergence value stays above the given
  threshold (similar to constructing a binary tree)

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Segmentation MethodsRecursive Segmentation

• Alternate approach to determining stopping
  criterion involves finding a model at the border
  between underfitting models (those that do not
  fit the data well) and overfitting models (those
  that fit the data too well by using too many
  parameters)
• Bayesian Information Criterion (BIC) was used to
  balance goodness-of-fit of the model to data

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Segmentation MethodsRecursive Segmentation

• Alternate approach to determining stopping
  criterion involves finding a model at the border
  between underfitting models (those that do not
  fit the data well) and overfitting models (those
  that fit the data too well by using too many
  parameters)
• Bayesian Information Criterion (BIC) was used to
  balance the goodness-of-fit of the model to
  data
• L is the likelihood of the model, K the number
  of free parameters, and N the sample size

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Segmentation MethodsRecursive Segmentation

• Two models can be compared
• Modelling the sequence as one single random
  sequence
• Modelling it as two random subsequences with
  different base compositions
• In order for recursive segmentation to continue,
  the following must apply
• where k is the number of different symbols in
  the sequence

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Segmentation MethodsRecursive Segmentation

• Alternate recursive segmentation algorithm
  condition can be used to define the segmentation
  strength s, i.e.
• Recursive segmentation process can be continued
  as long as s gt s0, where s0 is predefined by the
  user

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Segmentation MethodsRecursive Segmentation
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Segmentation MethodsRecursive Segmentation
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Segmentation MethodsRecursive Segmentation
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Discussion

• DNA sequences can be assumed to have segments
  where each has a degree of homogeneity
• A number of statistical methods can be used to
  identify and analyse these segments
• Isochores
• CpG islands
• Replication origin and terminus
• Complex patterns in telomeres
• Coding-noncoding borders
• Other statistical methods for analysing DNA
  segmentation do exist, each with varying degrees
  of success
• Bayesian approach
• Walking Markov
• Change-point methods

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References

• Braun J.V., Müller H.-G. Statistical methods
  for DNA sequence segmentation, Statistical
  Science, 13142-162, 1998.
• Duda R.O., Hart P.E., Stork D.G. (2001) Pattern
  Classification, New York John Wiley Sons, Inc.
• Churchill, G.A. Stochastic models for
  heterogeneous DNA sequences, Bulletin of
  Mathematical Biology, 5179-94, 1989.
• Csürös M. Algorithms for finding
  maximal-scoring segment sets, Proc. WABI, 2004.
• Li W., et al. Applications of recursive
  segmentation to the analysis of DNA sequences,
  Computational Chemistry, 26491-510, 2002.