Stochastic Models For Heterogeneous DNA Sequences

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Stochastic Models For Heterogeneous DNA Sequences

• Churchill, G.A.
• Bulletin of Mathematical Biology
• Vol. 51, pp. 79-94,1989.

Presented By Matthew McCall
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DNA Representation

• A typical representation of a DNA sequence is a
  single strand written from the 5 to 3 end.
• Consider two binary representations the
  purine-pyrimidine (AG-TC) and strong-weak
  hydrogen-bonding (GC-AT).
• Together these two yield the same information as
  the single strand representation.

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Stochastic Modeling

• Stochastic Process - A non-deterministic process
  in that the next state of the environment is not
  fully determined by the previous state of the
  environment.
• This is a method for extracting information.
• It doesnt try to mimic the process followed in
  nature, which is far too complex for any simple
  model.

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Markov Chains

• Markov Chains have traditionally been used to
  model sequences.
• They work well when the sequence composition is
  fairly homogeneous but not when the composition
  is heterogeneous.
• Compositional variation between segments of the
  sequence is likely to reflect functional or
  structural differences.

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Proposed Solution

• Sequences are assumed to be composed of
  homogeneous regions, which differ from one
  another.
• As such, each region can be classified into one
  of a finite number of states.
• These states represent the underlying structure
  of the sequence in that region and are assumed to
  develop according to a hidden Markov process.

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The General Case

• A sequence of random variables Yi i1,,n
• Corresponding unobservable states Si
• Denote the sequence of observed outcomes up to
  time t by yty1,,yt
• And similarly the states sts1,,st
• Then the probability of an observation given the
  current state and past observations is
• Pr(ytst,yt-1)
• This is called the observation equation.

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The General Case (cont.)

• The sequence of states, st, cannot be observed
  however they can be inferred from the
  observations.
• The states are assumed to evolve according to a
  set of system equations with the Markov property
• Pr(stst-1) Pr(stst-1)
• The problem addressed here is how to estimate the
  states Si given the observations Yi.
• The result, the smoothed estimate at time t, will
  be denoted Pr(styn).

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Computing Pr(styn)

• Pr(styn) Pr(styt) ? Pr(st1yn)Pr(st1st)/Pr
  (st1yt) dst1
• We can compute each of these terms, so the
  smoothed estimate at time t can be expressed in
  terms of the system equations, Pr(st1st),
  quantities derived from filtering, Pr(styt) and
  Pr(st1yt), and the smoothed estimate at time
  t1, Pr(st1yn).
• So we can employ a recursive algorithm to compute
  the smoothed estimate at each time t.

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Parameter Estimation

• The previous algorithm requires that the
  parameters of the observation and system
  equations be specified.
• These parameters are typically estimated from the
  data.
• The paper suggests an EM algorithm for
  determining the approximate maximum likelihood
  estimate of these parameters.

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Application to DNA Sequences

• Bases of a sequence are viewed as the observed
  outcomes Yi.
• The states Si are assumed to be fixed and
  finite in number.
• Each region then has a specific state.

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The Simplest Case

• Think back to the binary representations
  (strong-weak hydrogen-bonding).
• A sequence of independent binary outcomes
  yt?0,1 which depend on the underlying states
  st?0,1.
• Then the observation equation would be binomial
• Pr(ytstj) (pjyt) (1-pj)(1-yt) , j?0,1
• where p0 Pr(yt1st0) and p1 Pr(yt1st1)

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The Simplest Case (cont.)

• The system equations would be
• Pr(stjst-1i) ?ij
• Define the transition probabilities
• ? Pr(st1st-10)
• ? Pr(st0st-11)
• Both these are assumed to be small, so states
  tend to persist.
• In this case, the size of the different regions
  will have a geometric distribution with means
  equal to the reciprocals of the transition
  probabilities.

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GC Clusters in Yeast mtDNA

• The mitochondrial genome of yeast is 85 kb, which
  appears to have three distinct types of regions
  with differing GC content.
• Intergenic segments consist of large stretches of
  DNA with less than 5 GC content intermingled
  with short stretches with greater than 50 GC
  content.
• The genes themselves have GC content of 18 to 28.

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GC Clusters in Yeast mtDNA

• Respiration deficient mutants, p-, come about
  through a deletion of most of the wild-type DNA
  (p).
• The small amount left is amplified in tandem
  repeats, replicated, and maintained in the
  mitochondrion.
• Some of the p- mutants displace the p DNA in all
  the diploid descendants when mating with
  wild-type strains.
• These are called hypersuppressive p- genomes.

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GC Clusters in Yeast mtDNA

• This paper considers two hypersuppressive and two
  non-suppressive phenotypes.
• Because the segments they consider contain no
  coding regions, they use the binary model.
• The model parameters are set at
• p00.9, p10.5, ??0.01

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Yeast mtDNA Results

• The profiles of the hypersuppressive segments
  share a general structure of GC-rich regions and
  AT-rich regions that differ from the
  non-suppressive segments.
• This suggests that the GC content is in some way
  related to the function of these hypersuppressive
  sequences.

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Conclusion

• In its simplest form, this algorithm is useful
  for studying compositional heterogeneity in DNA.
• The stochastic models proposed are useful in
  extracting information from large and complex
  data sets such as DNA sequence data.
• This algorithm could be used to study
  relationships between DNA primary structure and
  global organization of entire chromosomes.