EVOLUTIONARY HMMSBAYESIAN APPROACH TO MULTIPLE ALIGNMENTSiva Theja MaguluriCS 598 SS

Goal

Given a set of sequences and a tree representing their evolutionary relationship, to find a multiple sequence alignment which maximizes the probability of the evolutionary relationships between the sequences.

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Evolutionary Model

Pairwise likelihood for relation between two sequences

Reversibility

Additivity

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Alignment can be inferred from the sequences using DP if Markov condition applies

Joint likelihood of a multiple alignment on a tree

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Alignment Model

Substitution models

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Links Model

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Birth Death process with Immigration ie each residue can either spawn a child or die

Birth rate λ, Death rate µ Immortal link at the left hand side Independent Homogenous Substitution

Probability evolution in Links Model

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Time evolution of the probability of a link surviving and spawning n descendants

Time evolution of the probability of a link dying before time t and spawning n descendants

Probability evolution in Links Model

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Time evolution of the probability of the immortal link spawning n descendants at time t

Probability evolution in Links Model

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Solution of these differential equations is

where

Probability evolution in Links Model

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Conceptually, α is the probability the ancestral residue survives

β is the probability of more insertions given one or more descendants

γ is the probability of insertion given ancestor did not survive

In the limit, immortal link generates residues according to geometric distribution

Links model as a Pair HMM

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Just like a standard HMM, but emits two sequences instead of one

Aligning two sequences with pair HMM, implicitly aligns the sequences

Pair HMM for Links model

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Either the residue lives or dies, spawning geometrically distributed residues in each case

Links model as a Pair HMM

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The path through the Pair HMM is π DP used to infer alignment of two

sequences Viterbi Algorithm for finding optimum

π Forward algorithm to sum over all

alignments or to sample from the posterior,

]Pr[ DA

Multiple HMMs

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Instead of emitting 2 sequences, emit N sequences

2N-1 emit states! Can develop such a model for any tree Viterbi and Forward algorithms use N

dimensional Dynamic programming Matrix Given a tree relating N sequences,

Multiple HMM can be constructed from Pair HMMs so that the likelihood function is

],},{Pr[ TS

Multiple HMMs

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Multiple HMMs

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Composing multiple alignment from branch alignments

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Residues Xi and Yj in a multiple alignment containing sequences X and Y are aligned iff They are in the same column That column contains no gaps for intermediate

sequences No deletion, re-insertion is allowed Ignoring all gap columns, provides and

unambiguous way of composing multiple alignment from branch alignments and vice versa

Eliminating internal nodes

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Internal nodes are Missing data Sum them out of the likelihood function Summing over indel histories will kill the

independence Sum over substitution histories using

post order traversal algorithm of Felsentein

Algorithm

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Progressive alignment – profiles of parents estimated by aligning siblings on a post order traversal – Impatient strategy

Iterative refinement – revisit branches following initial alignment phase – Greedy

Sample from a population of alignments, exploring suboptimal alignments in anticipation of long term improvements

Algorithm

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Moves to explore alignment space These moves need to be ergodic, i.e.

allow for transformation of any alignment into any other alignment

These moves need to satisfy detailed balance i.e. converges to desired stationary distribution

Move 1: Parent Sampling .

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Goal: Align two sibling nodes Y and Z and infer their parent X

Construct the multiple HMM for X,Y and Z

Sample an alignment of Y and Zusing the forward algorithm

This imposes an alignment of XZ and YZ Similar to sibling alignment step of

impatient-progressive alignment

Move 2: Branch Sampling

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Goal: realign two adjacent nodes X and Y Construct the pair HMM for X and Y,

fixing everything else Resample the alignment using the

forward algorithm This is similar to branch alignment step

of greedy-refined algorithm

Move 3: Node Sampling

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Goal: resample the sequence at an internal node X

Construct the multiple HMM and sample X, its parent W and children Y and Z, fixing everything else

Resample the sequence of X, conditioned on relative alignment of W,Y and Z

This is similar to inferring parent sequence lengths in impatient-progressive algorithms

Algorithm

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1. Parent sample up the guide tree and construct a multiple alignment

2. Visit each branch and node once for branch sampling or node sampling respectively

3. Repeat 2 to get more samples

Algorithm

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Replacing ‘sampling by Forward algorithm’ with ‘optimizing by Viterbi algorithm’

Impatient- Progressive is ML version of parent sampling

Greedy-refinement is ML version of Branch and node sampling

Gibbs sampling in ML context

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Periodically save current alignment, then take a greedy approach to record likelihood of refined alignment and get back to the saved alignment

Store this and compare likelihood to other alignments at the end of the run

Ordered over-relaxation

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Sampling is a random walk on Markov chain so follows Brownian motion ie rms drift grows as sqrt(n)

Would be better to avoid previously explored spaces ie ‘boldly go where no alignment has gone before’

Impose a strict weak order on alignments Sample N alignments at each stage and sort

them If the original sample ends up in position k,

choose the (N-k)th sample for the next emission

Implementation and results

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Implementation and results

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A True alignment B impatient progressive C greedy refined D Gibbs Sampling followed by Greedy

refinement E Gibbs sampling with simulated

annealing F Gibbs sampling with over relaxation G without Felsentein wild cards

Discussion

Outlines a very appealing Bayesian framework for multiple alignment

Performs very well, considering the simplicity of the model

Could add profile information and variable sized indels to the model to improve performance

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Questions

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Questions

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What is the assumption that enabled us to use this algorithm, enabling us to avoid the N dimensional matrices of DP ?

What is the importance of immortal link in the Links model ?

References

“Evolutionary HMMs: a Bayesian approach to multiple alignment” - Holmes and Bruno. Bioinformatics 2001

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More results

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More results

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More results

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More results

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Poor performance on 4 is probably because Handel produces a global alignment and doesn’t handle affine gaps

Handle doesn’t incorporate any profile information

Handle cannot use BLOSUM (it’s not additive)