CZ5226: Advanced BioinformaticsCZ5226: Advanced Bioinformatics

Lecture 6: HHM Method for generating motifsLecture 6: HHM Method for generating motifs

Prof. Chen Yu ZongProf. Chen Yu Zong

Tel: 6874-6877Tel: 6874-6877Email: Email: csccyz@nus.edu.sgcsccyz@nus.edu.sghttp://xin.cz3.nus.edu.sghttp://xin.cz3.nus.edu.sg

Room 07-24, level 7, SOC1, Room 07-24, level 7, SOC1, National University of SingaporeNational University of Singapore

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Problem in biologyProblem 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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Contents:

•Markov chain models (1st order, higher order andinhomogeneous models; parameter estimation; classification)

• Interpolated Markov models (and back-off models)

• Hidden Markov models (forward, backward and Baum-Welch algorithms; model topologies; applications to genefinding and protein family modeling

A widely used machine learning approach: Markov models

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Markov Chain ModelsMarkov 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 ModelsMarkov 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 ModelsMarkov Chain Models

Pr(cggt) = Pr(c)Pr(g|c)Pr(g|g)Pr(t|g)

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Markov Chain ModelsMarkov 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 ModelsMarkov Chain Models

ii xx aa1

)|Pr( 11 iixx xxaa

ii

The transition parameters can be denoted by

where

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 ApplicationExample 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 ParametersEstimating 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 EstimationMaximum Likelihood Estimation

• Suppose we want to estimate the parameters Pr(a), Pr(c), Pr(g), Pr(t)

• And we’re given the sequences: accgcgctta gcttagtgactagccgttac

• Then the maximum likelihood estimates are:

Pr(a) = 6/30 = 0.2 Pr(g) = 7/30 = 0.233Pr(c) = 9/30 = 0.3 Pr(t) = 8/30 = 0.267

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These data are derived from genome sequences

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2020

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Higher Order Markov ChainsHigher 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 ChainA Fifth Order Markov Chain

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Inhomogenous Markov ChainsInhomogenous 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 ChainsInhomogenous Markov Chains

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A Fifth Order InhomogeneousA Fifth Order InhomogeneousMarkov ChainMarkov Chain

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Selecting the Order of aSelecting the Order of aMarkov Chain ModelMarkov 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)Hidden Markov models (HMMs)

Given say a T in our input sequence, which state emitted it?

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Hidden Markov models (HMMs)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 searchingHMM-based homology searching

Transition probabilities and Emission probabilities

Gapped HMMs also have insertion and deletion states

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Profile HMMProfile HMM: m=match state, I-insert state, d=delete state; go from: 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”. left to right. I and m states output amino acids; d states are ‘silent”.

d1 d2 d3 d4

I0 I2 I3 I4I1

m0 m1 m2 m3 m4 m5

Start End

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HMM-based homology searchingHMM-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 HMMThe Parameters of an HMM

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HMM for Eukaryotic Gene FindingHMM for Eukaryotic Gene Finding

Figure from A. Krogh, An Introduction to Hidden Markov Models for Biological Sequences

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A Simple HMMA Simple HMM

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Three Important QuestionsThree 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?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?How Likely is a Given Sequence?

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How Likely is a Given Sequence?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:How Likely is a Given Sequence:The Forward AlgorithmThe 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:How Likely is a Given Sequence:

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The forward algorithmThe forward algorithm• Initialisation:

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

probability that we’re in start state and have observed 0 characters from the sequence

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Forward algorithm exampleForward algorithm example

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Three Important QuestionsThree 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:Finding the Most Probable Path:The Viterbi AlgorithmThe 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:Finding the Most Probable Path:The Viterbi AlgorithmThe 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:Finding the Most Probable Path:The Viterbi AlgorithmThe Viterbi Algorithm

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Three Important QuestionsThree 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 TaskThe 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 ParametersLearning 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:Learning Parameters:The Baum-Welch AlgorithmThe 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 stepThe Expectation step

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The Expectation stepThe Expectation step

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The Expectation stepThe Expectation step

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The Expectation stepThe Expectation step

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The Expectation stepThe 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 stepThe Expectation step

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The Backward AlgorithmThe Backward Algorithm

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The Expectation stepThe Expectation step

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The Expectation stepThe Expectation step

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The Expectation stepThe Expectation step

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The Maximization stepThe Maximization step

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The Maximization stepThe Maximization step

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The Baum-Welch AlgorithmThe Baum-Welch AlgorithmInitialize 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