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Profiles and Sequence Logos Profile Hidden Markov Models Aligning Profiles Multiple Sequence Alignments by Gradual

Sequence Adition Other Ways of Obtaining Multiple Alignments Sequence Pattern Discovery

Profile Hidden Markov Models

Hidden Markow Models:– A hidden Markov model (HMM) is a statisticsl

model,– in which the system being modeled is

assumed to be a Markov process (Memoryless process: its future and past are independent ),

– with hidden states.


Hidden Markow Models:– Has a set of states each of which has limited

number of transitions and emissions,– Each transition between states has an

assisgned probability,– Each model strarts from start state and ends

in end state,


Hidden Markow Models parameters:– A set of finite number of states, Si,– The transition probability from state Si to Sj, aij,– The emission probability density of a symbol ω– in state Si


Hidden Markow Models parameters:– Firstly discuss:

Morkov Models, Markov Assumption


Markow Models and Assumption (cont.):– To understand HMMs:

Talk about weather, Assume there are three types of weather:

– Sunny,

– Rainy,

– Foggy.

Assume weather does not change during the day (if it is sunny it will sunny all the day)


Markow Models and Assumption (cont.): Weather prediction is about the what would be the weather tomorrow,– Based on the observations on the past.


Markow Models and Assumption (cont.): Weather at day n is

– qn depends on the known weathers of the past days (qn-1, qn-2,…)

},,{ foggyrainysunnyqn


Markow Models and Assumption (cont.): We want to find that:

– means given the past weathers what is the probability of any possible weather of today.


Markow Models and Assumption (cont.): For example:

if we knew the weather for last three days was:

the probability that tomorrow would be is:

P(q4 = | q3 = , q2 = , q1 = )


Markow Models and Assumption (cont.):– For example:

this probability could be infered from the statistics of past observations

the problem: the larger n is, the more observations we must collect.

– for example: if n = 6 we need 3(6-1) = 243 past observations.


Markow Models and Assumption (cont.):– Therefore, make a simplifying assumption Markov

assumption: For sequence:

the weather of tomorrow only depends on today (first order Markov model)


Markow Models and Assumption (cont.): Examples:

The probabilities table:



HMM:



Given that day the weather is sunny, what is the probability that tomorrow is sunny and the next day is rainy ?

Markov assumption



If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny?


Markow Models and Assumption (cont.):– Examples:

If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny?

Markov assumption


Hidden Markov Models (HMMs):– What is HMM:

Suppose that you are locked in a room for several days, you try to predict the weather outside, The only piece of evidence you have is whether the

person who comes into the room bringing your daily meal is carrying an umbrella or not.


Hidden Markov Models (HMMs):– What is HMM (cont.):

assume probabilities as seen in the table:



Now the actual weather is hidden from you. You can not directly see what is the weather.



Finding the probability of a certain weather

is based on the observations xi:

},,{ foggyrainysunnyqn



Using Bayes rule:

For n days:



We can omit So:

With Markov assumptions:


Hidden Markov Models (HMMs):– Examples:

Suppose the day you were locked in it was sunny. The next day, the caretaker carried an umbrella into the room.

You would like to know, what the weather was like on this second day.



Calculate 3 probabilities:



Consider the event with highest value. It is most likely to happen.


Hidden Markov Models (HMMs):– Another Examples:

Suppose you do not know how the weather was when your were locked in. The following three days the caretaker always comes without an umbrella. Calculate the likelihood for the weather on these three days to have been



As you do not know how the weather is on the first day, you assume the 3 weather situations are equi-probable on this day and the prior probability for sun on day one is therefore



Assumption:


Hidden Markov Models:– Another Examples:


HMMs to represent a family of sequences– Given a multiple alignment of sequences, we

can use an HMM to model the sequences. – Each column of the alignment may be

represented by a hidden state that produced represented by a hidden state that produced that column.

– Insertions and deletions can be represented by other states


HMMs to represent a family of sequences



http://www.ifm.liu.se/bioinfo/assignments/hmm-profile.png



http://www.ifm.liu.se/bioinfo/assignments


Determining the states of the HMM– The structure is usually fixed and only the

number of “match” states is to be determined


Determining the states of the HMM– An alignment column with no gaps can be

considered as a “match” state considered as a match state.

– An alignment column with a majority of gaps can be considered an “insert” state can be considered an insert state.


Determining the transition probabilities – From a state u the transition to another state v is

represented by t(u.v).– The summation over all states w that are

connected to state u by transition gives one:

– The transition probabilities from a state (except the end state) always add up to 1.


Determining the transition probabilities


Determining the transition probabilities – mu,v: number of transitions from state u to state v

– Transition probabilities t(u,v) can be estimated:


Determining the emission probabilities Emission probabilities in a match or insert state

also adds up to 1.


Determining the emission probabilities eMu: emission probability for a residue from the u

th match state,


Determining the emission probabilities eIu: emission probability for a residue from the u

th insert state,

The probabilities are usually taken from the overal amino acid comparison of a selected data set.


HMMs examples:


HMMs examples:

5 transitions in gap region:• C out, •G out• AC, •CT, •T out• Out transition 3/5• Stay transition 2/5

gap region


Scoring a sequence against a profile HMM– Given a profile HMM, any given path through

the model will emit a sequence with an associated probability,

– The path probability is the product of all transition and emission probabilities along the path.


Scoring a sequence against a profile HMMViterbi algorithm:

– Given a query sequence we can compute the most probable path that will emit that query sequence.



– Another interesting question: What is the probability that a given sequence can be generated by the hidden Markov model

– Solution:Calculated by summing over all possible path giving rise to a given sequence



– Will be applied to profiles HMM similar to:



– At a profile position u:Mu : Match state,

Iu : Insertion state,

Du : Delete state,

t(Mu, Mu+1) : transition probability from state Mu to Mu+1



– During the algorithm:A record must be kept of the highest

probability up to that point in the model and for a given amount of emited sequences.



– During the algorithm:When the sequence up to and

including residue xi has been emited, the highest probability will be written as VDu(xi)



– Dynamic Programming:

COLUMNS: NUMBER OF STATES IN HMM

ROWS:LENGTH OF THE QUERY OR EMITTED SEQUENCE



– Recurrence relations:



– Recurrence relations (log):


Scoring a sequence against a profile HMMViterbi algorithm (EXAMPLE):



Emission Probabilities:

Start Probabilities:



GGCTGATT



G G C T GA T TBeginM1M2I3I3I3I3I3M3End

References

M. Zvelebil, J. O. Baum, “Understanding Bioinformatics”, 2008, Garland Science

Andreas D. Baxevanis, B.F. Francis Ouellette, “Bioinformatics: A practical guide to the analysis of genes and proteins”, 2001, Wiley.

Barbara Resch, “Hidden Markov Models - A Tutorial for the Course Computational Intelligence”, 2010.

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