1. 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

2. Hidden Markov Models
(O1,H1), (O2,H2),……. (On,Hn) is a sequence of stochastic variables with 2 components - 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 pi =P{H1=i) - often pi 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
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3

3. What is the probability of the data?
The probability of the observed is , which can be hard to calculate. However, these calculations can be considerably accelerated. Let the probability of the observations (O1,..Ok) conditional on Hk=j. Following recursion will be obeyed:
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3

4. What is the most probable ”hidden” configuration?
Let be the sequences of hidden states that maximizes the observed sequence O ie ArgMaxH[ ]. Let probability of the most probability of the most probable path up to k ending in hidden state j.
Again recursions can be found
The actual sequence of hidden states can be found recursively by
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3

5. What is the probability of specific ”hidden” state?
Let be the probability of the observations from j+1 to n given Hk=j. These will also obey recursions:
The probability of the observations and a specific hidden state can found as:
And of a specific hidden state can found as:
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3

6. Fast/Slowly Evolving States Felsenstein & Churchill, 1996
positions 1 n 1
sequences k
slow - rs
HMM:
fast - rf
pr - 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.
Make simpler. Show basic probability expressions.
Likelihood Recursions:
Likelihood Initialisations:

7. Statistical Alignment
Steel and Hein, Holmes and Bruno,2000 C T A
Emit functions:
e(##)= p(N1)f(N1,N2)
e(#-)= p(N1), e(-#)= p(N2)
p(N1) - equilibrium prob. of N
f(N1,N2) - prob. that N1 evolves into N2
An HMM Generating Alignments
# # E
# # E *
* lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) -
# lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) _
# lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) # lb

8. Probability of Data given a pedigree.
Elston-Stewart (1971) -Temporal Peeling Algorithm:
Mother
Father
Condition on parental states
Recombination and mutation are Markovian
Lander-Green (1987) - Genotype Scanning Algorithm:
Mother
Father
Condition on paternal/maternal inheritance
Recombination and mutation are Markovian
Comment: Obvious parallel to Wiuf-Hein99 reformulation of Hudson’s 1983 algorithm

9. Further Examples I Isochore: Gene Finding: Simple Eukaryotic
Churchill,1989,92
Lp(C)=Lp(G)=0.1, Lp(A)=Lp(T)=0.4, Lr(C)=Lr(G)=0.4, Lr(A)=Lr(T)=0.1
poor
rich
HMM:
Likelihood Recursions:
Likelihood Initialisations:
Gene Finding:
Burge and Karlin, 1996
Simple Eukaryotic
Simple Prokaryotic
Make simpler. Show basic probability expressions.

10. Further Examples II Secondary Structure Elements:
Goldman, 1996
Further Examples II
HMM for SSEs: a  L
.909
.0005
.091
.005
.881
.184
.062
.086
.852
.325
.212
.462 a  L
Adding Evolution:
SSE Prediction:
Make simpler. Show basic probability expressions.
Profile HMM Alignment:
Krogh et al.,1994

11. Summary H1 H2 H3 Definition Three Key Algorithms
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
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

12. Recommended Literature
Vineet Bafna and Daniel H. Huson (2000) The Conserved Exon Method for Gene Finding ISMB pp. 3-12
S.Batzoglou et al.(2000) Human and Mouse Gene Structure: Comparative Analysis and Application to Exon Prediction. Genome Research
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
Gravely, BR (2001) Alternative Splicing: increasing diversity in the proteomic world. TIGS
Guigo, R.et al.(2000) An Assesment of Gene Prediction Accuracy in Large DNA Sequences. Genome Research
Kan, Z. Et al. (2001) Gene Structure Prediction and Alternative Splicing Using Genomically Aligned ESTs Genome Research
Ian Korf et al.(2001) Integrating genomic homology into gene structure prediction. Bioinformatics vol17.Suppl.1 pages
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
Stein,L.(2001) Genome Annotation: From Sequence to Biology. Nature Reviews Genetics