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Hidden Markov Models
What are the good for?
Morten NielsenCBS
Absolutely nothing!
Objectives
• Introduce Hidden Markov models and understand that they are just weight matrices with gaps
• See the beauty of sequence profiles• Position specific scoring matrices (PSSMs)
• Understand what biological problems are best described using HMM’s– And which are not!
Outline
• What is an HMM– What are they good for?
• How to construct an HMM• How to “score” a sequence to an HMM
– Viterbi decoding• HMM’s that made a difference
– Profile HMMs– TMHMM
• Links to HMM packages
Markov Models
• A model with no memory– What I decide depends only on “state” now, not
on what I have learned in the past
– No dependence on i-1, i-2 …
€
Pi+1 =α i+1,i ⋅Pi
A Markov model?
• No memory• Model generates numbers
– 312453666641
1:1/62:1/63:1/64:1/65:1/66:1/6Fair
1:1/102:1/103:1/104:1/105:1/106:1/2Loaded
0.95
0.10
0.05
0.9
The unfair casino: Loaded dice p(6) = 0.5; switch fair to load:0.05; switch load to fair: 0.1
Why hidden?
• Model generates numbers– 312453666641
• Does not tell which dice was used
• Alignment (decoding) can give the most probable solution/path (Viterby)– FFFFFFLLLLLL
• Or most probable set of states– FFFFFFLLLLLL
1:1/62:1/63:1/64:1/65:1/66:1/6Fair
1:1/102:1/103:1/104:1/105:1/106:1/2Loaded
0.95
0.10
0.05
0.9
The unfair casino: Loaded dice p(6) = 0.5; switch fair to load:0.05; switch load to fair: 0.1
HMM (a simple example)
ACA---ATG
TCAACTATC
ACAC--AGC
AGA---ATC
ACCG--ATC
• Example from A. Krogh• Core region defines the
number of states in the HMM (red)
• Insertion and deletion statistics are derived from the non-core part of the alignment (black)
Core of alignment
.2
.8
.2
ACGT
ACGT
ACGT
ACGT
ACGT
ACGT.8
.8 .8.8
.2.2.2
.2
1
ACGT
.2
.2
.4
1. .4 1. 1.1.
.6.6
.4
HMM construction
ACA---ATG
TCAACTATC
ACAC--AGC
AGA---ATC
ACCG--ATC
• 5 matches. A, 2xC, T, G• 5 transitions in gap region
• C out, G out• A-C, C-T, T out• Out transition 3/5• Stay transition 2/5
ACA---ATG 0.8x1x0.8x1x0.8x0.4x1x1x0.8x1x0.2 = 3.3x10-2
Align sequence to HMM
ACA---ATG 0.8x1x0.8x1x0.8x0.4x1x0.8x1x0.2 = 3.3x10-2
TCAACTATC 0.2x1x0.8x1x0.8x0.6x0.2x0.4x0.4x0.4x0.2x0.6x1x1x0.8x1x0.8 = 0.0075x10-2
ACAC--AGC = 1.2x10-2
Consensus:
ACAC--ATC = 4.7x10-2, ACA---ATC = 13.1x10-2
Exceptional:
TGCT--AGG = 0.0023x10-2
Align sequence to HMM - Null model
• Score depends strongly on length
• Null model is a random model. For length L the score is 0.25L
• Log-odds score for sequence S– Log( P(S)/0.25L)
• Positive score means more likely than Null model
ACA---ATG = 4.9
TCAACTATC = 3.0 ACAC--AGC = 5.3AGA---ATC = 4.9ACCG--ATC = 4.6Consensus:ACAC--ATC = 6.7 ACA---ATC = 6.3Exceptional:TGCT--AGG = -0.97
Note!
Model decoding (Viterby)
• Example: 1245666. What was the series of dice used to generate this output?
1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78
Fair
1:-12:-13:-14:-15:-16:-0.3Loaded
-0.02
-1
-1.3
-0.05Log model
Dynamic programming: computation of scores
T C G C A
T
C
C
A
x
Any given point in matrix can only be reached from three possible positions (you cannot “align backwards”).
=> Best scoring alignment ending in any given point in the matrix can be found by choosing the highest scoring of the three possibilities.
Each new score is found by choosing the maximum of three possibilities. For each square in matrix: keep track of where best score came from.
Fill in scores one row at a time, starting in upper left corner of matrix, ending in lower right corner.
score(x,y) = max
score(x,y-1) - gap-penalty
score(x-1,y-1) + substitution-score(x,y)
score(x-1,y) - gap-penalty
Model decoding (Viterby)
• Example: 1245666. What was the series of dice used to generate this output?
1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78
Fair
1:-12:-13:-14:-15:-16:-0.3Loaded
-0.02
-1
-1.3
-0.05Log model
1 2 4 5 6 6 6
F -0.78
L Null -3.08
Model decoding (Viterby)
1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78
Fair
1:-12:-13:-14:-15:-16:-0.3
Loaded
-0.02
-1
-1.3
-0.05Log model
1 2 4 5 6 6 6
F -0.78 -1.58
L Null -3.08 -3.88
€
log(PL (4)) = −1− 0.05 − 3.08 = −4.13 or
log(PL (4)) = −1−1.3 −1.58 = −3.88
Model decoding (Viterby)
1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78
Fair
1:-12:-13:-14:-15:-16:-0.3Loaded
-0.02
-1
-1.3
-0.05Log model
1 2 4 5 6 6 6
F -0.78 -1.58
L Null -3.08 -3.88
Identify what series of dice was used to generate this output?
Model decoding (Viterby)
1:-0.782:-0.783:-0.784:-0.785:-0.786:-0-78
Fair
1:-12:-13:-14:-15:-16:-0.3Loaded
-0.02
-1
-1.3
-0.05Log model
1 2 4 5 6 6 6
F -0.78 -1.58 -2.38 -3.18 -3.98 -4.78 -5.58
L Null -3.08 -3.88 -4.68 -4.78 -5.13 -5.48
Series of dice is FFFFLLL
HMM’s and weight matrices
• In the case of un-gapped alignments HMM’s become simple weight matrices
.2
.8
.2
ACGT
ACGT
ACGT
ACGT
ACGT
ACGT.8
.8 .8.8
.2.2.2
.2
1
ACGT
.2
.2
.4
1. .4 1. 1.1.
.6.6
.4
HMM construction
X
.8
.2
ACGT
ACGT
ACGT
ACGT
ACGT
ACGT.8
.8 .8.8
.2.2.2
.2
11. 1. 1. 1.1.
HMM construction
ACA---ATG sco = 0.8x1x0.8x1x0.8x1x1x1x0.8x1x0.2 = 3.3x10-2 or
Log-sco = log(0.8)+log(0.8)+log(0.8)+log(1)+log(0.8)+log(0.2)
HMM’s and weight matrices
• In the case of un-gapped alignments HMM’s become simple weight matrices
• To achieve high performance, the emission frequencies are estimated using the techniques of – Sequence weighting– Pseudo counts
HMMs. What are they good for?
• Weight matrices do not deal with insertions and deletions
• In alignments, this is done in an ad-hoc manner by optimization of the two gap penalties for first gap and gap extension
• HMM is a natural frame work where insertions/deletions are dealt with explicitly
Profile HMM’s
• Alignments based on conventional scoring matrices (BLOSUM62) scores all positions in a sequence in an equal manner
• Some positions are highly conserved, some are highly variable (more than what is described in the BLOSUM matrix)
• Profile HMM’s are ideal suited to describe such position specific variations
What goes wrong when Blast fails?
• Conventional sequence alignment uses a (Blosum) scoring matrix to identify amino acids matches in the two protein sequences
Alignment scoring matrices
• Blosum62 score matrix. Fg=1. Ng=0?
L A G D S D
F
I
G
D
S
L
Alignment scoring matrices• Blosum62 score matrix. Fg=1. Ng=0?
• Score =2+6+6+4-1=17
L A G D S D
F 0 -2 -3 -3 -2 -3
I 2 -1 -4 -3 -2 -3
G -4 0 6 -1 0 -1
D -4 -2 -1 6 0 6
S -2 1 0 0 4 0
L 4 -1 -4 -4 -2 -4
LAGDSI-GDS
What goes wrong when Blast fails?
• Conventional sequence alignment uses a (Blosum) scoring matrix to identify amino acids matches in the two protein sequences• This scoring matrix is identical at all positions in the protein sequence!
EVVFIGDSLVQLMHQC
X X X
X X X
AGDS.GGGDS
When Blast works!
1PLC
._
1PLB._
When Blast fails!
1PLC
._
1PMY._
Sequence profiles
• In reality not all positions in a protein are equally likely to mutate
• Some amino acids (active cites) are highly conserved, and the score for mismatch must be very high
• Other amino acids can mutate almost for free, and the score for mismatch should be lower than the BLOSUM score
• Sequence profiles can capture these differences
ADDGSLAFVPSEF--SISPGEKIVFKNNAGFPHNIVFDEDSIPSGVDASKISMSEEDLLN TVNGAI--PGPLIAERLKEGQNVRVTNTLDEDTSIHWHGLLVPFGMDGVPGVSFPG---I-TSMAPAFGVQEFYRTVKQGDEVTVTIT-----NIDQIED-VSHGFVVVNHGVSME---IIE--KMKYLTPEVFYTIKAGETVYWVNGEVMPHNVAFKKGIV--GEDAFRGEMMTKD----TSVAPSFSQPSF-LTVKEGDEVTVIVTNLDE------IDDLTHGFTMGNHGVAME---VASAETMVFEPDFLVLEIGPGDRVRFVPTHK-SHNAATIDGMVPEGVEGFKSRINDE----TKAVVLTFNTSVEICLVMQGTSIV----AAESHPLHLHGFNFPSNFNLVDPMERNTAGVPTVNGQ--FPGPRLAGVAREGDQVLVKVVNHVAENITIHWHGVQLGTGWADGPAYVTQCPI
Profile HMM’s
Conserved
Core: Position with < 2 gaps
Deletion
Insertion
Non-conserved
Must have a G Any thing can match
HMM vs. alignment
• Detailed description of core– Conserved/variable positions
• Price for insertions/deletions varies at different locations in sequence
• These features cannot be captured in conventional alignments
Profile-profile scoring matrix
1K
7C
.A
1WAB._
Profile HMM’s
All M/D pairs must be visited once
L1- Y2A3V4R5- I6
P1D2P3P4I4P5D6P7
Example. Sequence profiles
• Alignment of protein sequences 1PLC._ and 1GYC.A• E-value > 1000• Profile alignment
– Align 1PLC._ against Swiss-prot– Make position specific weight matrix from
alignment– Use this matrix to align 1PLC._ against 1GYC.A
• E-value < 10-22. Rmsd=3.3
Example continued
Score = 97.1 bits (241), Expect = 9e-22 Identities = 13/107 (12%), Positives = 27/107 (25%), Gaps = 17/107 (15%) Query: 3 ADDGSLAFVPSEFSISPGEKI------VFKNNAGFPHNIVFDEDSIPSGVDASKIS 56 F + G++ N+ + +G + +Sbjct: 26 ------VFPSPLITGKKGDRFQLNVVDTLTNHTMLKSTSIHWHGFFQAGTNWADGP 79 Query: 57 MSEEDLLNAKGETFEVAL---SNKGEYSFYCSP--HQGAGMVGKVTV 98 A G +F G + ++ G+ G VSbjct: 80 AFVNQCPIASGHSFLYDFHVPDQAGTFWYHSHLSTQYCDGLRGPFVV 126
Rmsd=3.3 ÅModel redStructure blue
HMMs. What are they good for II
• Trans membrane helix proteins
HMMs. What are they good for II
• Transmembrane helix proteins
TMHMM. A. Krogh, 2001
Gene Finding
HMM packages
• HMMER (http://hmmer.wustl.edu/)– S.R. Eddy, WashU St. Louis. Freely available.
• SAM (http://www.cse.ucsc.edu/research/compbio/sam.html)– R. Hughey, K. Karplus, A. Krogh, D. Haussler and others, UC Santa
Cruz. Freely available to academia, nominal license fee for commercial users.
• META-MEME (http://metameme.sdsc.edu/)– William Noble Grundy, UC San Diego. Freely available. Combines
features of PSSM search and profile HMM search.
• NET-ID, HMMpro (http://www.netid.com/html/hmmpro.html)– Freely available to academia, nominal license fee for commercial
users.– Allows HMM architecture construction.
• EasyGibbs (http://www.cbs.dtu.dk/biotools/EasyGibbs/)– Webserver for Gibbs sampling of proteins sequences
