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DTCGerton Lunter, WTCHGFebruary 10, 2010Hidden Markov models in Computational BiologyOverviewFirst part:Mathematical context: Bayesian NetworksMarkov modelsHidden Markov models

Second part:Worked example: the occasionally crooked casinoTwo applications in computational biology

Third part:Practical 0: a bit more theory on HMMsPractical I-V: theory, implementation, biology. Pick & choose.Part IHMMs in (mathematical) contextProbabilistic modelsMathematical model describing how variables occur together.

Three type of variables are distinguished:Observed variablesLatent (hidden) variablesParameters

Latent variables often are the quantities of interest, and can be inferred from observations using the model. Sometimes they are nuisance variables, and used to correctly describe the relationships in the data.

Example: P(clouds, sprinkler_used, rain, wet_grass)Some notationP(X,Y,Z): probability of (X,Y,Z) occurring (simultaneously)P(X,Y):probability of (X,Y) occurring.P(X,Y|Z): probability of (X,Y) occurring, provided that it is known that Z occurs (conditional on Z, or given Z)

P(X,Y) = Z P(X,Y,Z)P(Z) = X,Y P(X,Y,Z) P(X,Y| Z ) = P(X,Y,Z) / P(Z)X,Y,Z P(X,Y,Z) = 1P(Y | X ) = P(X | Y) P(Y) / P(X) (Bayes rule)IndependenceTwo variables X, Y are independent if P(X,Y) = P(X)P(Y)

Knowing that two variables are independent reduces the model complexity. Suppose X, Y each take N possible values: specification of P(X,Y) requires N2-1 numbers specification of P(X), P(Y) requires 2N-2 numbers.

Two variables X,Y are conditionally independent (given Z) if P(X,Y|Z) = P(X|Z)P(Y|Z).Probabilistic model: exampleP(Clouds, Sprinkler, Rain, WetGrass) =

P(Clouds) P(Sprinker|Clouds) P(Rain|Clouds) P(WetGrass | Sprinkler, Rain)

This specification of the model determines which variables are deemed to be (conditionally) independent. These independence assumptions simplify the model.

Using formulas as above to describe the independence relationship is not very intuitive, particularly for large models. Graphical models (in particular, Bayesian Networks) are a more intuitive way to do the same

Bayesian network: exampleCloudySprinklerRainWet grassP(Clouds) P(Sprinker|Clouds) P(Rain|Clouds) P(WetGrass | Sprinkler, Rain)Rule:

Two nodes of the graph are conditionally independent given the state of their parents

E.g. Sprinker and Rain areindependent given CloudyBayesian network: exampleCloudySprinklerRainWet grassConvention:

Latent variables are openObserved variables are shaded

P(Clouds) P(Sprinker|Clouds) P(Rain|Clouds) P(WetGrass | Sprinkler, Rain)Bayesian network: example

Combat Air Identification algorithm; www.wagner.comBayesian networksIntuitive formalism to develop modelsAlgorithms to learn parameters from training data (maximum likelihood; EM)General and efficient algorithms to infer latent variables from observations (message passing algorithm)

Allows dealing with missing data in a robust and coherent way(make relevant node a latent variable)Simulate data

Markov modelA particular kind of Bayesian network

All variables are observedGood for modeling dependencies within sequences

P(Sn | S1,S2,,Sn-1 ) = P(Sn | Sn-1 ) (Markov property)P(S1, S2, S3, , Sn) = P(S1 ) P(S2|S1 ) P (Sn | Sn-1 )

S1S2S3S4S5S6S7S8Markov modelStates: letters in English wordsTransitions: which letter follows whichS1S2S3S4S5S6S7S8MR SHERLOCK HOLMES WHO WAS USUALLY VERY LATE IN THE MORNINGSSAVE UPON THOSE NOT INFREQUENT OCCASIONS WHEN HE WAS UP ALL .S1=MS2=RS3=S4=SS5=H. P(Sn= y| Sn-1= x ) = (parameters) P(Sn-1Sn = xy ) / P (Sn-1 = x )(frequency of xy) / (frequency of x) (max likelihood)UNOWANGED HE RULID THAND TROPONE AS ORTIUTORVE OD T HASOUT TIVEIS MSHO CE BURKES HEST MASO TELEM TS OME SSTALE MISSTISE S TEWHEROMarkov modelStates: triplets of lettersTransitions: which (overlapping) triplet follows whichS1S2S3S4S5S6S7S8MR SHERLOCK HOLMES WHO WAS USUALLY VERY LATE IN THE MORNINGSSAVE UPON THOSE NOT INFREQUENT OCCASIONS WHEN HE WAS UP ALL .S1=MRS2=RSS3=SHS4=SHES5=HER. P(Sn= xyz| Sn-1= wxy ) = P( wxyz ) / P( wxy )(frequency of wxyz) / (frequency of wxy)THERE THE YOU SOME OF FEELING WILL PREOCCUPATIENCE CREASON LITTLEDMASTIFF HENRY MALIGNATIVE LL HAVE MAY UPON IMPRESENT WARNESTLYMarkov modelStates: word pairsText from: http://www.gutenberg.org/etext/1105

Then churls their thoughts (although their eyes were kind) To thy fair appearance lies To side this title is impanelled A quest of thoughts all tenants to the sober west As those gold candles fixed in heaven's air Let them say more that like of hearsay well I will drink Potions of eisel 'gainst my strong infection No bitterness that I was false of heart Though absence seemed my flame to qualify As easy might I not free

When thou thy sins enclose! That tongue that tells the story of thy love Ay fill it full with feasting on your sight Book both my wilfulness and errors down And on just proof surmise accumulate Bring me within the level of your eyes And in mine own when I of you beauteous and lovely youth When that churl death my bones with dust shall cover And shalt by fortune once more re-survey These poor rude lines of life thou art forced to break a twofold truth Hers by thy deeds Hidden Markov modelHMM = probabilistic observation of Markov chainAnother special kind of Bayesian network

Si form a Markov chain as before, but states are unobservedInstead, yi (dependent on Si) are observedGenerative viewpoint: state Si emits symbol yiyi do not form a Markov chain (= do not satisfy Markov property)They exhibit more complex (and long-range) dependencies

S1S2S3S4S5S6S7S8y1y2y3y4y5y6y7y8Hidden Markov modelNotation above emphasizes relation to Bayesian networksDifferent graph notation, emphasizing transition probabilities P(Si|Si-1). E.g. in the case Si {A,B,C,D}:

Notes: Emission probabilities P( yi | Si ) not explicitly representedAdvance from i to i+1 also implicitNot all arrows need to be present (prob = 0)

S1S2S3S4S5S6S7S8y1y2y3y4y5y6y7y8ABDCPair Hidden Markov modelS11S21S31S41S51z1S12S22S23S24S25z2S31S32S33S34S35z3y1y2y3y4y5Pair Hidden Markov modelS11S21S31S41S51z1S12S22S23S24S25z2S31S32S33S34S35z3y1y2y3y4y5Normalization:paths psp(1) sp(N)y1yA z1zB P(sp(1),,sp(N),y1yA,z1zB) = 1 N = N(p) = length of pathStates may emit a symbol in sequence y, or in z, or both, or neither (silent state).If a symbol is emitted, the associated coordinate subscript increases by one. E.g. diagonal transitions are associated to simultaneous emissions in both sequences.A realization of the pair HMM consists of a state sequence, with each symbol emitted by exactly one state, and the associated path through the 2D table.

(A slightly more general viewpoint decouples the states and the path; then the hidden variables are the sequence of states S, and a path through the table. In this viewpoint the transitions, not states, emit symbols. The technical term in finite state machine theory is Mealy machine; the standard viewpoint is also known as Moore machine)Inference in HMMsSo HMMs can describe complex (temporal, spatial) relationships in data. But how can we use the model?

A number of (efficient) inference algorithms exist for HMMs:Viterbi algorithm: most likely state sequence, given observablesForward algorithm: likelihood of model given observablesBackward algorithm: together with Forward, allows computation of posterior probabilitiesBaum-Welch algorithm: parameter estimation given observables Summary of part IProbabilistic modelsObserved variablesLatent variables: of interest for inference, or nuisance variablesParameters: obtained from training data, or prior knowledgeBayesian networksindependence structure of model represented as a graphMarkov modelslinear Bayesian network; all nodes observedHidden Markov modelsobserved layer, and hidden (latent) layer of nodesefficient inference algorithm (Viterbi algorithm)Pair Hidden Markov modeltwo observed sequences with interdependencies, determined by an unobserved Markov sequencePart IIExamples of HMMsDetailed example:The Occasionally Crooked CasinoDirk Husmeiers slideshttp://www.bioss.sari.ac.uk/staff/dirk/talks/tutorial_hmm.pdf

Slides 1-15

Recommended reading:Slides 16-23: the Forward and Backward algorithm, and posteriorsApplications in computational biologyDirk Husmeiers slides:http://www.bioss.sari.ac.uk/staff/dirk/talks/tutorial_hmm_bioinf.pdf

Slides 1-8: pairwise alignmentSlides 12-16: Profile HMMsPart IIIPracticalsPractical 0: HMMsWhat is the interpretation of the probability computed by the Forward (FW) algorithm?The Viterbi algorithm also computes a probability. How does that relate to the one computed by the FW algorithm?How do the probabilities computed by FW and Backward algorithms compare?Explain what a posterior is, either in the context of alignment using an HMM, or of profile HMMs.Why is the logarithm trick useful for the Viterbi algorithm? Does the same trick work for the FW algorithm?

Practical I: Profile HMMs in context

Practical I: Profile HMMs in contextLookup protein sequence of PRDM9 in the UCSC genome browserSearch Intropro for the protein sequence. Look at the ProSite profile and sequence logo. Work out the syntax of the profile (HMMer syntax), and relate the logo and profile.Which residues are highly conserved? What structural role do these play? Which are not very much conserved? Can you infer that these are less important biologically?Read PMID: 19997497 (PubMed). What is the meaning of the changed number of zinc finger motifs across species? Relate the conserved and changeable positions in the zinc fingers to the INTERPRO motif. Do these match the predicted pattern?Read PMID: 19008249 and PMID:20044541. Explain the relationship between the recombination motif and the zinc fingers. What do you think is the cellular function of PRDM9? Relate the fact that recombination hotspots in Chimpanzee do not coincide with those in human with PRDM9. What do you predict about recombination hotspots in other mammalian species? Why do you think PRDM9 evolves so fast?

Background information on motif finding:www.bx.psu.edu/courses/bx-fall04/phmm.ppthttp://compbio.soe.ucsc.edu/SAM_T08/T08-query.htmlPractical II: HMMs and population genetics

Practical II: HMMs and population geneticsRead PMID: 17319744, and PMID: 19581452What is the difference between phylogeny and genealogy? What is incomplete lineage sorting?The model operates on multiple sequences. Is it a linear HMM, a pair HMM, or something else?What do the states represent? How could the model be improved?Which patterns in the data is the model looking for? Would it be possible to analyze these patterns without a probabilistic model? (Estimate how frequently (per nucleotide) mutations occur between the species considered. What is the average distance between recombinations?)How does the method scale to more species?

Practical III: HMMs and alignment

Practical III: HMMs and alignmentPMID: 18073381What are the causes of inaccuracies in alignments?Would a more accurate model of sequence evolution improve alignments? Would this be a large improvement?What is the practical limit (in terms of evolutionary distance, in mutations/site) on pairwise alignment? Would multiple alignment allow more divergent species to be aligned?How does the complexity scale for multiple alignment using HMMs, in a nave implementation? What could you do to improve this?What is posterior decoding and how does it work? In what way does it improve alignments, compared to Viterbi? Why is this?

Practical IV: HMMs and conservation: phastCons

Practical IV: HMMs and conservation: phastConsRead PMID: 16024819What is the difference between a phyloHMM and a standard HMM?How does the model identify conserved regions? How is the model helped by the use of multiple species? How is the model parameterized?The paper uses the model to estimate the fraction of the human genome that is conserved. How can this estimate be criticized?Look at a few protein-coding genes, and their conservation across mammalian species, using the UCSC genome browser. Is it always true that (protein-coding) exons are well conserved? Can you see regions of conservation outside of protein-coding exons? Do these observations suggest that the model is inaccurate?Read PMID: 19858363. Summarize the differences of approaches of the new methods and the old phyloHMM.Practical V: Automatic code generation for HMMs

Practical V: Automatic code generation for HMMshttp://www.well.ox.ac.uk/~gerton/Gulbenkian/HMMs and alignments.doc. Skip sections 1-3.

Implementing the various algorithms for HMMs can be hard work, particularly when a reasonable efficiency is required. Library implementations are however neither fast nor flexible enough. This practical demonstrates a code generator that takes the pain out of working with HMMs.

This practical takes you through an existing alignment HMM, and modifies it to identify conserved regions ( la phastCons)

Requirements: a Linux system, with Java and GCC installed.

Experience with C and/or C++ is helpful for this tutorial.