Hidden Markov Models in Bioinformaticshein/hmm.pdf · Hidden Markov Models in Bioinformatics 14.11...
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Hidden Markov Models in Bioinformatics 14.11 60 min
Definition
Three Key Algorithms
• Summing over Unknown States
• Most Probable Unknown States
• Marginalizing Unknown States
Key Bioinformatic Applications• Pedigree Analysis• Isochores in Genomes (CG-rich regions)• Profile HMM Alignment• Fast/Slowly Evolving States• Secondary Structure Elements in Proteins• Gene Finding• Statistical Alignment
Hidden Markov Models
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10H1
H2
H3
(O1,H1), (O2,H2),……. (On,Hn) is a sequence of stochastic variables with 2components - one that is observed (Oi) and one that is hidden (Hi).
The marginal discribution of the Hi’s are described by a Homogenous Markov Chain:
•Let pi,j = P(Hk=i,Hk+1=j)
•Let πi =P{H1=i) - often πi is the equilibrium distribution of the Markov Chain.
•Conditional on Hk (all k), the Ok are independent.
•The distribution of Ok only depends on the value of Hi and is called the emit function
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e(i, j) = P{Ok = i Hk = j)
What is the probability of the data?
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10
H1H2H3
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PO5 = iH5 = 2 = P(O5 = i H5 = 2) PO4
H4 = j
H4 = j∑ p j,i
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P(r O ) = P(r
H ∑r O
r H )P(
r H )The probability of the observed is , which can
be hard to calculate. However, these calculations can be considerablyaccelerated. Let the probability of the observations (O1,..Ok)conditional on Hk=j. Following recursion will be obeyed:
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POk = iHk = j
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i. POk = iHk = j = P(Ok = i Hk = j) POk−1
Hk−1 = j
Hk−1 = j∑ p j,i
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ii. PO1 = iH1 = j = P(O1 = i H1 = j)π j (initial condition)
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iii. P(O) = POn= iHn= j
Hn= j∑
What is the most probable ”hidden” configuration?
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H61 =max j{H6
1 * p j,1 *e(O6,1)}O1 O2 O3 O4 O5 O6 O7 O8 O9 O10
H1H2H3
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P{OH}Let be the sequences of hidden states that maximizes the observedsequence O ie ArgMaxH[ ]. Let probability of the mostprobability of the most probable path up to k ending in hidden state j.
Again recursions can be found€
Hkj
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H*
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i. Hkj = maxi{Hk−1
i pi, j}e(Ok, j)
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ii. H1j = π je(O1,1)
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iii. Hk−1* = {i Hk−1
i pi, je(Ok, j) = HkHk
*}
The actual sequence of hidden states can be found recursively by
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Hk*
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H5* = {i H5
i * pi,1 *e(O6,1) = H61}
What is the probability of specific ”hidden” state?
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Qkj = P(Ok Hk+1 = i)p j,iQk+1
i
Hk+1= i∑
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10
H1H2H3
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P{Hk = j) = PkjQk
j /P(O)
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P{H5 = 2) = P52Q5
2 /P(O)
Let be the probability of the observations from j+1 to n given Hk=j.These will also obey recursions:
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Qkj
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P{O,Hk = j) = PkjQk
jThe probability of the observations and a specific hidden state canfound as:
And of a specific hidden state can found as:
Fast/Slowly Evolving StatesFelsenstein & Churchill, 1996
n1positions
seque
nces
k
1
slow - rsfast - rfHMM:
• πr - equilibrium distribution of hidden states (rates) at first position
•pi,j - transition probabilities between hidden states
•L(j,r) - likelihood for j’th column given rate r.
•L(j,r) - likelihood for first j columns given j’th column has rate r.
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L(j,f) = (L(j-1,f)pf , f + L(j-1,s)ps, f )L( j , f )
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L(j,s) = (L(j-1,f)pf ,s + L(j-1,s)ps,s)L( j,s)Likelihood Recursions:
Likelihood Initialisations:
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L(1,s) = π sL(1,s)
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L(1,f) = π f L(1, f )
- # # E # # - E** λβ λ/µ (1− λβ)e-µ λ/µ (1− λβ)(1− e-µ) (1− λ/µ) (1− λβ)
-# λβ λ/µ (1− λβ)e-µ λ/µ (1− λβ)(1− e-µ) (1− λ/µ) (1− λβ)
_# λβ λ/µ (1− λβ)e-µ λ/µ (1− λβ)(1− e-µ) (1− λ/µ) (1− λβ)
#- λβ
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1− λβe−µ
1−e−µ
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λβe−µ
1− e−µ
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(µ − λ)β1− e−µ
An HMM Generating Alignments
Statistical AlignmentSteel and Hein,2000 + Holmes and Bruno,2000
C
T
C A C
Emit functions:e(##)= π(N1)f(N1,N2)e(#-)= π(N1), e(-#)= π(N2)
π(N1) - equilibrium prob. of Nf(N1,N2) - prob. that N1evolves into N2
Probability of Data given a pedigree.Elston-Stewart (1971) -Temporal Peeling Algorithm:
Lander-Green (1987) - Genotype Scanning Algorithm:
Mother Father
Condition on parental states
Recombination and mutation are Markovian
Mother Father
Condition on paternal/maternal inheritance
Recombination and mutation are Markovian
Comment: Obvious parallel to Wiuf-Hein99 reformulation of Hudson’s 1983 algorithm
Further Examples I
Simple Prokaryotic Simple EukaryoticGene Finding:Burge and Karlin, 1996
Isochore:Churchill,1989,92
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L(j,p) = (Lj−1,p pp,p + L(j-1,s)ps, f )Pp (S[ j]), L(j,r) = (L(j-1,r)pp,r + L(j-1,r)pr,r)Pr (S[ j])Likelihood Recursions:
Likelihood Initialisations:
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L(1,p) = π pPp (S[1]), L(1,r) = π rPr (S[1])
poorrichHMM: Lp(C)=Lp(G)=0.1, Lp(A)=Lp(T)=0.4,
Lr(C)=Lr(G)=0.4, Lr(A)=Lr(T)=0.1
Further Examples IISecondary Structure Elements:Goldman, 1996
Profile HMMAlignment:Krogh et al.,1994
.462.212.325
.852.086.062L
.184.881.005β
.091.0005.909α
Lβα
αα ββL L L
HMM for SSEs:
Adding Evolution: SSE Prediction:
Summary O1 O2 O3 O4 O5 O6 O7 O8 O9 O10
H1H2H3
DefinitionThree Key Algorithms• Summing over Unknown States• Most Probable Unknown States• Marginalizing Unknown StatesKey Bioinformatic Applications• Pedigree Analysis• Isochores in Genomes (CG-rich regions)• Profile HMM Alignment• Fast/Slowly Evolving States• Secondary Structure Elements in Proteins• Gene Finding• Statistical Alignment
Recommended LiteratureVineet Bafna and Daniel H. Huson (2000) The Conserved Exon Method for Gene Finding ISMB 2000 pp. 3-12
S.Batzoglou et al.(2000) Human and Mouse Gene Structure: Comparative Analysis and Application to Exon Prediction. Genome Research.10.950-58.
Blayo, Rouze & Sagot (2002) ”Orphan Gene Finding - An exon assembly approach” J.Comp.Biol.
Delcher, AL et al.(1998) Alignment of Whole Genomes Nuc.Ac.Res. 27.11.2369-76.
Gravely, BR (2001) Alternative Splicing: increasing diversity in the proteomic world. TIGS 17.2.100-
Guigo, R.et al.(2000) An Assesment of Gene Prediction Accuracy in Large DNA Sequences. Genome Research 10.1631-42
Kan, Z. Et al. (2001) Gene Structure Prediction and Alternative Splicing Using Genomically Aligned ESTs Genome Research 11.889-900.
Ian Korf et al.(2001) Integrating genomic homology into gene structure prediction. Bioinformatics vol17.Suppl.1 pages 140-148
Tejs Scharling (2001) Gene-identification using sequence comparison. Aarhus University
JS Pedersen (2001) Progress Report: Comparative Gene Finding. Aarhus University
Reese,MG et al.(2000) Genome Annotation Assessment in Drosophila melanogaster Genome Research 10.483-501.
Stein,L.(2001) Genome Annotation: From Sequence to Biology. Nature Reviews Genetics 2.493-