modelling genomes
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Modelling genomes. Gil McVean Department of Statistics, Oxford. Why would we want to model a genome?. To identify genes Protein-coding RNA Small RNAs To identify regulatory elements Transcription factor binding sites Enhancers To classify genome content Repeat DNA Unique sequence - PowerPoint PPT PresentationTRANSCRIPT
Modelling genomes
Gil McVeanDepartment of Statistics, Oxford
Why would we want to model a genome?
• To identify genes– Protein-coding– RNA– Small RNAs
• To identify regulatory elements– Transcription factor binding sites– Enhancers
• To classify genome content– Repeat DNA– Unique sequence
• To understand the processes that shape genomes– Mutation– Recombination– Duplication– Rearrangement– Natural selection
Non-coding DNA Start codon Codon Non-coding DNATERMs t
A rather simple model for a protein-coding gene
T: 30%C: 20%A: 30%G: 20%
ATG: 100% AAA: 1/61%AAC: 1/61%AAG: 1/61%…TTT: 1/61%
T: 30%C: 20%A: 30%G: 20%
TAA: 30%TAG: 40%TGA: 30%
ST
AT
ES
EM
ISS
ION
S
Define model
Explore properties
Estimate parametersfrom data
Test goodness-of-fit
Refine model
A ‘genome’ model is like any other statistical model
Hidden Markov Models in bioinformatics
• The model of a gene just described can be thought of as a hidden Markov model (HMM)
– The underlying states evolve in a Markov fashion, but we observe features (the DNA sequence) emitted by those states
• You will remember that there are lots of nice computational properties of hidden Markov models that we can use for inference
– Finding a most likely sequence of states– Calculating posterior probabilities of a given state at a given position
• There are also various algorithms we can use to estimate parameters of HMMs (e.g. ML estimation by EM)
• How would you use the model of a gene to find new genes?– How well do you think it would do?
Making useful HMMs in bioinformatics
• To be useful, HMMs for genes have to incorporate many features– Regulatory sequences– Intron-splicing features– Correlations and biases in amino acid and base composition
• A REALLY important feature to capture is their evolution– Important parts of genes and genomes evolve slower due to constraint
Searching for homology
• If we compare human and chimpanzee sequences they are approximately 98.8% identical at the DNA level. It is ‘easy’ to identify which parts of the genome in humans correspond to which parts in chimps
• If we compare human with, say mouse, we can see some parts that are similar, and other parts where there is only vague or even no obvious similarity.
• When measuring evolution, we need to identify regions that are homologous
– Homology means similarity by descent
• Traditionally, the problem of identifying homology has been intrinsically linked to the problem of alignment
Alignment of PFEMP1 proteins from P. falciparum
The simplest problem: aligning two sequences
• Suppose we have just two protein sequences that we want to align
• In evolution, three types of event can happen– Mutation to new amino acids– Insertion of new amino acids– Deletion of amino acids
• We want to work out which amino acids in the two sequences are homologous – i.e. related to each other through shared ancestry
WAKISWEEKS
W—AKIS
WEEK-S What do the ‘-’s really mean?
How can we construct an alignment algorithm?
• What we want to do is to look at every possible alignment and choose the one that is ‘best’
• What we have to do is to find an efficient algorithm that can search every possible alignment and that has an objective measure as to what ‘best’ means
• A natural approach is to make a model of alignments, parameterise it and find the alignment that maximises the likelihood
• Although the problem sounds hard we can solve it using a hidden Markov model structure
Xi
Yj
How does is work?
• Suppose residues Xi and Yj are aligned to each other
• Three things could happen next– The next two residues in each sequence could also align (A) – A gap could be introduced in sequence X
(B)– A gap could be introduced in sequence Y
(C)
• We can parameterise the probabilities of each event
XiXi+1
YjYj+1
Xi-
YjYj+1
XiXi+1
Yj-(A) (B) (C)
The full algorithm
• We need to consider similar transitions for the cases when residue Xi is aligned to a gap after residue Yj, and when Yj is aligned to a gap after Xi
• We need to specify various probabilities– The probability of inserting a gap– The probability of extending a gap– The probability of finishing the alignment– The probability of observing an aligned pair of residues (20x20)– The probability of observing a residue aligned to a gap (20)
• Once specified we can use the Viterbi and Forward/Backward algorithms to identify ML alignments, sample from the posterior or calculate posterior probabilities
Xi-a…Xi
Yj …-
Xi …-
Yj-a…Yj
H
D
H
Xi+1
)()()()1( 1 iHDHDHHHH Xeqifqifif
Transition probabilities= qij
Emission probabilities= ek(Xi+1 )
In alignment the state space is two-dimensional (residue i aligned to residue j)
The forward algorithm
H
D
H
D
Xi+1Xi
H
D
Xi-1
jkjjikk qivXeiv )(max)()1( 1
The Viterbi algorithm
A traceback matrix is used to keep track of the best partial alignments
An example
• Suppose the gap opening and extension parameters are 0.2 and 0.5 respectively. There is a 80% chance of observing a match, a 20/19% chance of observing any given mismatch and a 5% chance of observing each unaligned amino acid (We can ignore termination for the moment)
• The BEST alignments are given below, each of which has log likelihood of -16.84, or 31% of the total likelihood (lnlk = -15.67).
• In many real situations, the best alignment represents only a fraction of the total likelihood
W—AKIS
WEEK-S
WA-KIS
WEEK-S
Posterior decoding
• Using the forward-backward algorithm we can calculate the posterior probability that any residue is aligned to any other, or that a given residue is in a gap state
X1 X2 X3 X4 X5Y1
Y2
Y3
Y4
Y5
X1 X2 X3 X4 X5Y1
Y2
Y3
Y4
Y5
Conditional on X2-Y3
Extending the method
• Originally, alignment algorithms (Needleman and Wunsch, 1970; Smith and Waterman, 1981; Gotoh 1982) were not explicitly defined as hidden Markov models
– Finite-state automata (FSA)
• There have been many extensions to the original idea– Local alignment– Repeat alignment– Protein family identification– Gene finding– Multiple alignment
• The alignment algorithm is very much a workhorse of bioinformatics, as an alignment is needed or almost all subsequent analyses (e.g. phylogenetic tree reconstruction, population genetic inference)
– However, relying on a single alignment is not always a great idea
Doing away with alignment
• For most problems, the alignment is not of primary interest
• The natural thing to do is to integrate over alignments (as in the FB algorithm) to estimate parameters of interest
• The key problem is that there is no computationally efficient algorithm for statistical multiple alignment. All widely-used methods use heuristic approaches
Gene conversion and var gene diversity in P. falciparum
• Multiple alignment methods typically assume the sequences are related to each through an evolutionary tree
• For the case of multi-gene families, this may not be the case, because gene conversion between copies can lead to mosaic structures
• If we wish to learn about the processes of conversion, a natural approach is to model the mosaicism
– In the case of var genes, the sequences are so diverged that we also need to consider the problem of alignment
Mosaic alignment
• We could model the n+1th sequence as a mosaic of the previous n
• We can calculate the likelihood of observing a given sequence by summing over all possible mosaic structures and their alignment
• We can also identify the most likely mosaic structure and calculate the expected number of recombination events
– Repeating the procedure for all sequences provides a way of assessing the importance of mosaicism within the family
Extensive mosaicism within the var family