CZ5226: Advanced Bioinformatics
Lecture 6: HHM Method for generating motifs
Prof. Chen Yu Zong
Tel: 6874-6877
Email: [email protected]
http://xin.cz3.nus.edu.sg
Room 07-24, level 7, SOC1,
National University of Singapore
Problem in biology
• Data and patterns are often not clear cut
• When we want to make a method to recognise a
pattern (e.g. a sequence motif), we have to learn
from the data (e.g. maybe there are other
differences between sequences that have the
pattern and those that do not)
• This leads to Data mining and Machine learning
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A widely used machine learning
approach: Markov models
Contents:
•Markov chain models (1st order, higher order and
inhomogeneous models; parameter estimation;
classification)
• Interpolated Markov models (and back-off models)
• Hidden Markov models (forward, backward and BaumWelch algorithms; model topologies; applications to gene
finding and protein family modeling
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Markov Chain Models
• a Markov chain model is defined by:
– a set of states
• some states emit symbols
• other states (e.g. the begin state) are silent
– a set of transitions with associated
probabilities
• the transitions emanating from a given state define
a distribution over the possible next states
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Markov Chain Models
• Given some sequence x of length L, we can
ask how probable the sequence is given our
model
• For any probabilistic model of sequences, we
can write this probability as
• Key property of a (1st order) Markov chain: the
probability of each Xi depends only on Xi-1
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Markov Chain Models
Pr(cggt) = Pr(c)Pr(g|c)Pr(g|g)Pr(t|g)
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Markov Chain Models
Can also have an end state, allowing the model to
represent:
• Sequences of different lengths
• Preferences for sequences ending with particular symbols
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Markov Chain Models
The transition parameters can be denoted by
where
a xi1 a xi
a xi1 a xi  Pr( xi | xi 1 )
Similarly we can denote the probability of a sequence x as
Where aBxi represents the transition from the begin state
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Example Application
• CpG islands
– CGdinucleotides are rarer in eukaryotic genomes than
expected given the independent probabilities of C, G
– but the regions upstream of genes are richer in CG
dinucleotides than elsewhere – CpG islands
– useful evidence for finding genes
• Could predict CpG islands with Markov chains
– one to represent CpG islands
– one to represent the rest of the genome
Example includes using Maximum likelihood and Bayes’
statistical data and feeding it to a HM model
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Estimating the Model Parameters
• Given some data (e.g. a set of sequences from
CpG islands), how can we determine the
probability parameters of our model?
• One approach: maximum likelihood estimation
– given a set of data D
– set the parameters  to maximize
Pr(D|)
– i.e. make the data D look likely under the model
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Maximum Likelihood Estimation
• Suppose we want to estimate the parameters
Pr(a), Pr(c), Pr(g), Pr(t)
• And we’re given the sequences:
accgcgctta
gcttagtgac
tagccgttac
• Then the maximum likelihood estimates are:
Pr(a) = 6/30 = 0.2
Pr(c) = 9/30 = 0.3
Pr(g) = 7/30 = 0.233
Pr(t) = 8/30 = 0.267
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These data are derived from genome sequences
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Higher Order Markov Chains
• An nth order Markov chain over some alphabet is
equivalent to a first order Markov chain over the
alphabet of n-tuples
• Example: a 2nd order Markov model for DNA can be
treated as a 1st order Markov model over alphabet:
AA, AC, AG, AT, CA, CC, CG, CT, GA, GC, GG, GT,
TA, TC, TG, and TT (i.e. all possible dipeptides)
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A Fifth Order Markov Chain
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Inhomogenous Markov Chains
• In the Markov chain models we have considered
so far, the probabilities do not depend on where
we are in a given sequence
• In an inhomogeneous Markov model, we can
have different distributions at different positions
in the sequence
• Consider modeling codons in protein coding
regions
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Inhomogenous Markov Chains
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A Fifth Order Inhomogeneous
Markov Chain
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Selecting the Order of a
Markov Chain Model
• Higher order models remember more “history”
• Additional history can have predictive value
• Example:
– predict the next word in this sentence fragment
“…finish __” (up, it, first, last, …?)
– now predict it given more history
• “Fast guys finish __”
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Hidden Markov models (HMMs)
Given say a T in our input sequence, which state emitted it? 27
Hidden Markov models (HMMs)
Hidden State
• We will distinguish between the observed parts of a
problem and the hidden parts
• In the Markov models we have considered previously, it
is clear which state accounts for each part of the observed
sequence
• In the model above (preceding slide), there are multiple
states that could account for each part of the observed
sequence
– this is the hidden part of the problem
– states are decoupled from sequence symbols
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HMM-based homology searching
Transition probabilities and Emission probabilities
Gapped HMMs also have insertion and deletion
states
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Profile HMM: m=match state, I-insert state, d=delete state; go from
left to right. I and m states output amino acids; d states are ‘silent”.
d1
d2
d3
d4
I0
I1
I2
I3
I4
m0
m1
m2
m3
m4
Start
m5
End
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HMM-based homology searching
• Most widely used HMM-based profile searching tools
currently are SAM-T99 (Karplus et al., 1998) and HMMER2
(Eddy, 1998)
• formal probabilistic basis and consistent theory behind gap
and insertion scores
• HMMs good for profile searches, bad for alignment (due to
parametrisation of the models)
• HMMs are slow
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Homology-derived Secondary Structure of Proteins
Sander & Schneider, 1991
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The Parameters of an HMM
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HMM for Eukaryotic Gene Finding
Figure from A. Krogh, An Introduction to Hidden Markov Models for Biological Sequences
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A Simple HMM
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Three Important Questions
• How likely is a given sequence?
the Forward algorithm
• What is the most probable “path” for
generating a given sequence?
the Viterbi algorithm
• How can we learn the HMM parameters given
a set of sequences?
the Forward-Backward
(Baum-Welch) algorithm
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How Likely is a Given Sequence?
• The probability that the path is taken and the
sequence is generated:
• (assuming begin/end are the only silent states
on path)
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How Likely is a Given Sequence?
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How Likely is a Given Sequence?
The probability over all paths is:
but the number of paths can be exponential in the
length of the sequence...
• the Forward algorithm enables us to compute
this efficiently
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How Likely is a Given Sequence:
The Forward Algorithm
• Define fk(i) to be the probability of being in
state k
• Having observed the first i characters of x
we want to compute fN(L), the probability of
being in the end state having observed all
of x
• We can define this recursively
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How Likely is a Given Sequence:
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The forward algorithm
probability that we’re in start state
• Initialisation:
and have observed 0 characters from
the sequence
f0(0) = 1 (start),
fk(0) = 0 (other silent states k)
• Recursion:
fl(i) = el(i)k fk(i-1)akl
(emitting states),
fl(i) = k fk(i)akl (silent states)
• Termination:
Pr(x) = Pr(x1…xL) = f N(L) = k fk(L)akN
probability that we are in the end state and have observed the entire sequence
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Forward algorithm example
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Three Important Questions
• How likely is a given sequence?
• What is the most probable “path” for
generating a given sequence?
• How can we learn the HMM parameters
given a set of sequences?
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Finding the Most Probable Path:
The Viterbi Algorithm
• Define vk(i) to be the probability of the most probable
path accounting for the first i characters of x and ending
in state k
• We want to compute vN(L), the probability of the most
probable path accounting for all of the sequence and
ending in the end state
• Can be defined recursively
• Can use DP to find vN(L) efficiently
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Finding the Most Probable Path:
The Viterbi Algorithm
Initialisation:
v0(0) = 1 (start), vk(0) = 0 (non-silent states)
Recursion for emitting states (i =1…L):
Recursion for silent states:
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Finding the Most Probable Path:
The Viterbi Algorithm
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Three Important Questions
• How likely is a given sequence?
(clustering)
• What is the most probable “path” for
generating a given sequence? (alignment)
• How can we learn the HMM parameters
given a set of sequences?
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The Learning Task
• Given:
– a model
– a set of sequences (the training set)
• Do:
– find the most likely parameters to explain the training
sequences
• The goal is find a model that generalizes well to
sequences we haven’t seen before
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Learning Parameters
• If we know the state path for each training
sequence, learning the model parameters is simple
– no hidden state during training
– count how often each parameter is used
– normalize/smooth to get probabilities
– process just like it was for Markov chain models
• If we don’t know the path for each training
sequence, how can we determine the counts?
– key insight: estimate the counts by considering every
path weighted by its probability
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Learning Parameters:
The Baum-Welch Algorithm
• An EM (expectation maximization) approach, a
forward-backward algorithm
• Algorithm sketch:
– initialize parameters of model
– iterate until convergence
• Calculate the expected number of times each
transition or emission is used
• Adjust the parameters to maximize the likelihood
of these expected values
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The Expectation step
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The Expectation step
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The Expectation step
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The Expectation step
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The Expectation step
• First, we need to know the probability of the i th
symbol being produced by state q, given sequence x:
Pr( i = k | x)
•Given this we can compute our expected counts for
state transitions, character emissions
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The Expectation step
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The Backward Algorithm
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The Expectation step
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The Expectation step
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The Expectation step
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The Maximization step
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The Maximization step
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The Baum-Welch Algorithm
Initialize parameters of model
• Iterate until convergence
– calculate the expected number of times each transition or
emission is used
– adjust the parameters to maximize the likelihood of these
expected values
• This algorithm will converge to a local maximum (in the
likelihood of the data given the model)
• Usually in a fairly small number of iterations
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