1. Hidden Markov Model in Biological Sequence Analysis – Part 2
Continued discussion about estimation
Using HMM in sequence analysis:
Splice site recognition
CpG Island
Profile HMM
Alignment
Acknowledgement: In addition to the textbooks, a large portion of the lecture is based on: Computational Methods in Molecular Biology, edited by S. L. Salzberg, D. B.Searls and S. Kasif, pages Elsevier, 1998.

2. HMM E1 E2 A1 A2 p12 p21 b1(A1) p22 p11 Initiation b1(A2) b2(A1) b2(A2)
initial distribution
transition probabilities
emission probabilities

3. How to efficiently calculate emissions:
Hidden Markov Model
Common questions:
How to efficiently calculate emissions:
How to find the most likely hidden state:
How to find the most likely parameters:
Notations:
Let denote the hidden state sequence;
Let denote the observed emission sequence;
Let denote the emission up to time t.

4. Calculating emissions probabilities:
Reminders
Calculating emissions probabilities:
Emissions up to step t chain at state i at step t

5. Reminders
Calculating posterior probability of hidden state:
Emissions after step t chain at state i at step t

6. Posterior state probabilities

7. The Viterbi Algorithm
To find:
So, to maximize the conditional probability, we can simply maximize the joint probability.
Define:

8. The Viterbi Algorithm
Then our goal becomes:
Compare this to dynamic programming pairwise sequence alignment:
At every step, we aim at the maximum
The maximum depends on the last step
At the end, we want the maximum and the trace
So, this problem is also solved by dynamic programming.

9. The Viterbi Algorithm
Time:
States: E *
E *
The calculation is from left border to right border, while in the sequence alignment, it is from upper-left corner to the right and lower borders.
Again, why is this efficient calculation possible?
The short memory of the Markov Chain!

10. The Estimation of parameters
When the topology of an HMM is known, how do we find the initial distribution, the transition probabilities and the emission probabilities, from a set of emitted sequences?
Difficulties:
Usually the number of parameters is not small. Even the simple chain in our example has 5 parameters.
Normally the landscape of the likelihood function is bumpy. Finding the global optimum is hard.

11. Basic idea
There are three pieces: λ, O, and Q. O is observed.
We can iterate between λ and Q.
When λ is fixed, we can do one of two things:
(1) find the best Q given O
or (2) integrate out Q
When Q is fixed, we can estimate λ using frequencies.

12. The Viterbi training
Start with some initial parameters
(2) Find the most likely paths using the Viterbi algorithm.
(3) Re-estimate the parameters.
On next page.
(4) Iterate until the most likely paths don’t change.

13. The Viterbi training
Estimating the parameters.
Transition probabilities:
Emission probabilities:
Initial probabilities: need multiple training sequences.

14. The Baum-Welch is a special case of the EM algorithm.
Baum-Welch algorithm
The Baum-Welch is a special case of the EM algorithm.
The Viterbi training is equivalent to an simplified version of the EM algorithm, replacing the expectation with the most likely classification.
General flow:
(1) Start with some initial parameters
(2) Given these parameters, P(Q|O) is known for every Q. This is done effectively by calculating the forward and the backward variables.
(3) Take expectations to obtain new parameter estimates.
(4) Iterate (2) and (3) until convergence.

15. Baum-Welch algorithm
Reminder:
Estimation of emission probabilities:

16. Baum-Welch algorithm
time t t+1
State i
State j
Emission Ot Ot+1

17. Baum-Welch algorithm

18. Baum-Welch algorithm
Estimation of transition probabilities:
Estimation of initial distribution:

19. Baum-Welch algorithm
Comments:
We don’t actually calculate P(Q|O). Rather, the necessary information pertaining to all the thousands of P(Q|O) is summarized in the forward- and backward- variables.
When the chain runs long, logarithm need to be used for numerical stability.
In applications, the emission is often continuous. A distribution need to be assumed, and the density used for emission probabilities.
The Baum-Welch is a special case of the EM algorithm.
The Viterbi training is equivalent to an simplified version of the EM algorithm, replacing the expectation with the most likely classification.

20. Common questions in HMM-based sequence analysis
HMM is very versatile in terms of coding sequence properties by model structures.
Given a sequence profile (HMM with parameters known), find the most likely hidden state path from a sequence.
A harder question:
Given a bunch of (aligned) training sequence segments,
- find the most plausible HMM structure
-- train the parameters of the HMM.

21. Splice site recognition E: exon I: intron 5: 5’splice site
Nature Biotech. 22(10): 1315

22. CpG Island
In the four character DNA book, the appearance of “CG” is rare, due to some issues related to the probability of mutations.
However, there are regions where “CG” is more frequent. And such regions are mostly associated with genes. (Remember, only 5% of the DNA sequence are genes. )
……GTATTTCGAAGAAT……

23. CpG Island A+ C+ G+ T+ G- A- C- T-
Within the “+” and “-” groups, the transition probabilities are close to shown in the table. A small probability is left for between “+” and “-” transitions.

24. CpG Island
Transition probabilities are summarized from known CpG islands.
To identify CpG island from new sequences:
Using the most likely path: run the Viterbi algorithm. When the chain passes through any of the four “+” states, call it an island
Using the posterior decoding: run the forward- and backward- algorithm. Obtain the posterior probabilities of each letter being in a CpG island.

25. Profile HMM --- a simple case
Motif: functional short sequence that tends to be conserved.

26. Profile HMM --- a simple case
For (1) accounting for sequence length, (2) computational stability, Log odds ratios are preferred.

27. Profile HMM --- a simple case

28. Profile HMM --- the general model
Deletion
Insertion

29. Profile HMM --- the general model

30. Profile HMM --- the general model
Avoid over-fitting with limited number of observed sequences.
Add pseudocounts proportional to the observed frequencies of the amino acids.
And add pseudocounts to transitions.

31. Profile HMM --- searching with profile HMM
Given a profile HMM, how to find if a short sequence is such a motif?
Use the Viterbi algorithm to find the best fit and the log odds ratio:

32. Multiple sequence alignment
(1) From a known family of highly related sequences, find the most conserved segments which have functional and/or structural implications.
Pandit et al. In Silico Biology 4, 0047

33. Multiple sequence alignment
(2) From a known family of sequences that are related only by very small motifs (that contribute to the function), find the real motif from vast amounts of junk.
This is a harder problem that draws statisticians.
Science 262,

34. Heuristic methods for global alignment
Scoring: Sum of Pairs (SP score)
Assume columns of the alignment is independent from each other
For each column, the score is the sum of the scores between all possible pairs
For the overall alignment, the score is the sum of the column scores

35. Heuristic methods for global alignment
Multi-dimensional dynamic programming:
2N-1 choices at each step. Impractical with more than a few sequences.

36. Heuristic methods for global alignment
Progressive alignment: when exhaustive search is hard, go stepwise.

37. Heuristic methods for global alignment
MSA: reducing multi-dimensional dynamic programming search space based on fast methods like progressive alignment.

38. Heuristic methods for global alignment
CLUSTALW: profile alignment rather than purely pairwise
Key improvement: Once an aligned group of sequences is established, position-specific information from the group is used to align a new sequence to it.

39. Heuristic methods for global alignment
Alignment by profile HMM training.
We have studied the scoring system by known HMM. The difficulty here is to train the HMM from unaligned sequences.
Baum-Welch and Viterbi can be trapped by local optima. Stochastic methods like simulated annealing can help.

40. Heuristic methods for global alignment
Alignment by profile HMM training.
For global alignment, the number of matching states is ~mean(sequence length).
The motif discovery we will talk about also utilizes the HMM in essence.
The difference is in motif models, only a very small number of (5~20) matching states is expected. The vast amount of data is generated from a relatively simple background chain.

41. Modeling genes with HMM
Genes account for only 5% of the human genome.
Within each gene, there are exons (that code for protein sequence) and introns (that are skipped during transcription).
Characteristics of the exons, introns and the borders between them allow people to identify putative genes from long DNA sequences.
HMM is a model that can incorporate such characteristics.