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Markov models and applications Sushmita Roy BMI/CS 576 www.biostat.wisc.edu/bmi576/ sroy@biostat.wisc.edu Oct 15 th, 2013

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Key concepts Markov chains – Computing the probability of a sequence – Estimating parameters of a Markov model Hidden Markov models – States – Emission and transition probabilities – Parameter estimation – Forward and backward algorithm – Viterbi algorithm

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Why Markov models? So far we have assumed in our models that there is no dependency between consecutive base pair locations This assumption is rarely true in practice Markov models allow us to model the dependencies inherent in sequential data such as DNA or protein sequence

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Applications of Markov models Genome annotation – Given a genome sequence find functional untis of the genome Genes, CpG islands, promoters.. Sequence classification – A Hidden Markov model (Profile HMMs) can represent a family of proteins Sequence alignment

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Markov models Provide a generative model for sequence data A Markov chain is the simplest type of Markov models Described by – A collection of states, each state representing a symbol observed in the sequence – At each state a symbol is emitted – At each state there is a some probability of transitioning to another state

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Markov chain model notation X t denote the state at time t a ij =P(X t =j|X t-1 =i) denotes the transition probability from i to j

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A Markov chain model for DNA sequence A TC G begin state transition.34.16.38.12 transition probabilities

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can also have an end state; allows the model to represent – a distribution over sequences of different lengths – preferences for ending sequences with certain symbols Markov chain models begin end A TC G

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Computing the probability of a sequence from a first Markov model Let X be a sequence of random variables X 1 … X L representing a biological sequence from the chain rule of probability

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Computing the probability of a sequence from a first Markov model Key property of a (1 st order) Markov chain: the probability of each X t depends only on the value of X t-1 This can be written as the initial probabilities or a transition probability from the begin state

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Example of computing the probability from a Markov chain begin end A tc g What is P(CGGT)?

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Learning Markov model parameters Model parameters – Transition probabilities Estimated via maximum likelihood

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Estimating model parameters Assume we have some training data that we know was generated from a first order Markov model Need to estimate transition probabilities – P(X t |X t-1 )

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Estimating model parameters P(X) P(X t |X t-1 ) Number of times x follows G

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Estimating parameters example Assume the following training data ACCGCGCTTA GCTTAGTGAC TAGCCGTTAC Fill in the zeroth and first order transition probability values P(A)6/30 P(T)8/30 P(G)7/30 P(C)9/30 0/5 2/53/5 1/72/70/74/7 A T G C AT G C Do we really want to set this to 0?

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Laplace estimates of parameters instead of estimating parameters strictly from the data, we could start with some prior belief for each for example, we could use Laplace estimates where represents the number of occurrences of character i using Laplace estimates with the sequences GCCGCGCTTG GCTTGGTGGC TGGCCGTTGC pseudocount

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Estimation for 1 st order probabilities Using Laplace estimates with the sequences for first order probabilities GCCGCGCTTG GCTTGGTGGC TGGCCGTTGC

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Using Markov chains to classify sequences as CpG islands or not CpG islands are isolated regions of the genome corresponding to high concentrations of CG dinucleotides – Range from 100-5000 bps For example regions upstream of genes or gene promoters Given a short sequence of DNA, how can we classify it as a CpG island?

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Build a Markov chain model for CpG islands Learn two Markov models – One from sequences that look like CpG (positive) – One from sequences that dont look like CpG (negative) – Parameters estimated from ~60,000 nucleotides +ACGT A.18.27.43.12 C.17.37.27.19 G.16.34.38.12 T.08.36.38.18 -ACGT A.30.21.28.21 C.32.30.08.30 G.25.24.30.21 T.18.24.29 CpGNot CpG G is much more likely to follow A in the CpG positive model

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Using the Markov chains to classify a new sequence Let y denote a new sequence To use the Markov models to classify y we compute the log odds ratio The larger the value of S(y) the more likely is y a CpG island – Note: must also normalize by the length

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Distribution of scores for CpG and non CpG examples CpG

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Applying the CpG Markov chain models Is CGCGA a CpG island? – Un-normalized Score= – =3.3684 Is CCTGG a CpG island? – Un-normalized Score=0.3746

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Extensions to Markov chains Hidden Markov models (Next lectures) Higher-order Markov models Inhomogeneous Markov models

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Order of a Markov model Describes how much history we want to keep First order Markov models are most common Second order n th order

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Higher order Markov models An n th order Markov model over alphabet A is equivalent to a first order Markov model A n A n corresponds to the alphabet of n-tuples For example – A 2 nd order Markov model over A,T,G,C – And a 1 st order Markov model over AA,AT,AG,AC,TA,TG,TC,TT,GA,GT,GC,GG,CA,CT,CC,CG – Are equivalent However higher the order the more parameters we need to estimate

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A first order Markov chain for a second-order Markov chain with {A,B} alphabet AA BA BB AB

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Inhomogeneous Markov chains So far the Markov chain has the same transition probability for the entire sequence Sometimes it is useful to switch between Markov chains depending on where we are in the sequence For example, recall the genetic code that specifies what triplets of bases code for an amino acid – There are different levels of redundancy for each the first, second or third positions – This could be modeled by three Markov chains

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Inhomogeneous Markov chain Let 1, 2 and 3 denote the three Markov chains – So our transition probabilities look like – a ij 1, a ij 2, a ij 3 Let x 1 be in codon position 3 Probability of x 2, x 3.. would be Exercise: write down the terms for x 5 and x 6

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Summary Markov models allow us to model dependencies in sequential data Markov models are described by – states, each state corresponding to a letter in our observed alphabet – Transition probabilities between states Parameter estimation can be done by counting the number of transitions between consecutive states (first order) – Laplace correction is often applied Often used for classifying sequences as CpG islands or not Extensions of Markov models – Order – Inhomogeneous Markov models

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