1 MARKOV MODELS MARKOV MODELS Presentation by Jeff Rosenberg, Toru Sakamoto, Freeman Chen HIDDEN

2 The Plan Modeling Biological Sequences Markov Chains Hidden Markov Models Issues Examples Techniques and Algorithms Doing it with Mathematica

3 Biological Sequences VVGGLVALRGAHPYIAALYWGHSFCAGSLIAPC URT1_DESRO STGGLFTDITSHPWQAAIFAQNRRSSGERFLCGG TRYP_PIG IVGGYTCAANSIPYQVSLNSGSHFCGGSLINSQWV COGS_UCAPU IVGGVEAVPNSWPHQAALFIDDMYFCGGSLISPEW TRY1_RAT IVGGYTCPEHSVPYQVSLNSGYHFCGGSLINDQWV TRY1_SALSA IVGGYECKAYSQTHQVSLNSGYHFCGGSLVNENWV NRPN_MOUSE ILEGRECIPHSQPWQAALFQGERLICGGVLVGDRW TRY1_BOVIN IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWV FA12_HUMAN

4 Sequences and Models Many biological sequences (DNA/RNA, proteins) have very “subtle” rules for their structure; they clearly form “families” and are “related,” yet simple measures or descriptions of these relationships or rules rarely apply There is a need to create some kind of “model” that can be used to identify relationships among sequences and distinguish members of families from non-members Given the complexity and variability of these biological structures, any practical model must have a probabilistic component: that is, it will be a stochastic model, rather than a mechanistic one. It will be evaluated by the (statistical) accuracy and usefulness of its predictions, rather than the correspondence of its internal features to any corresponding internal mechanism in the structures being modeled.

5 Markov Chains A system with a set of m possible states, S i ; at each of a sequence of discrete points in time t>=0, the system is in exactly one of those states; the state at time t >= 0 is designated by q t ; the movement from q t to q t+1 is probabilistic, and depends only on the states of the system at or prior to t. An initial state distribution π(i) = Prob(q 0 = S i ) Process terminates either at time T or when reaching a designated final state S f

6 Markov Chains of Order N Nth-order Markov chain (N >= 0): transition probabilities out of state q t depend only on the values of q t, q t-1,… q t-(N-1). Typically deal with 1 st -order Markov chain, so only q t itself affects the transition probabilities. In a 1 st -order chain, for each state S j, there is set of m probabilities for selecting the next state to move to: a i,j = Prob(q t+1 = S i | q t = S j ) [1 = 0] If there is some ordering of states such that a i,j = 0 whenever i < j (i.e., no “non-trivial” loops), then this is a “linear” (or “left-to-right”) Markov process “Homogeneous” Markov model: a i,j is independent of t

7 Simple Markov Models Might use a Markov chain to model a sequence where the symbol in position n depends on the symbol(s) in position(s) n-1,…n-N. For example, if a protein is more likely to have Lys after a sequence Arg-Cys, this could be encoded as (a small part of) a 2 nd -order Markov model. If the probabilities of a given symbol are the same for all positions in the sequence, and independent of symbols in other positions, then can use the “degenerate” 0 th -order Markov chain, where the probability of a given symbol is constant, regardless of the preceding symbol (or of the position in the sequence).

8 Hidden Markov Models (HMMs) In a Markov chain (or model), the states are themselves visible; they can be considered the outputs of the system (or deterministically associated with those outputs). However, if each state can emit (generate) any of several possible outputs (symbols) v k, from an output alphabet O of M symbols, on a probabilistic basis, then it is not possible (in general) to determine the sequence of the states themselves; they are “hidden.” Classic example: the “urn” game –A set of N urns (states), each containing various colored balls (output symbols – total of M colors available), behind a curtain –Player 1 selects an urn at random (with Markov assumptions), then picks a ball at random from that urn and announces its color to player 2 –Player 1 then repeats the above process, a total of T times –Player 2 must determine sequence of urns selected based on the sequence of colors announced

9 Additional Parameters for HMMs Now, in addition to the transition probabilities, each state has a prescribed probability distribution to emit or produce a symbol v k from O: b i,k = Prob(v k | S i ) If q t = S i, then the generated output at time t is v k with probability b i,k. So, a HMM is “doubly stochastic” – both the (hidden) state transition process and the (visible) output symbol generation process are probabilistic.

10 Bayesian Aspects of HMM Usage Given an HMM M, we can relatively easily calculate the probability of occurrence of an arbitrary output sequence, s: P(s | M) However, we often want to determine the underlying set of states, transition probabilities, etc. (the model) that is “most likely” to have produced the output sequence s that we have observed: P(M | s) Bayes’ Formula for sequence “recognition”: P(M | s) = P(s | M) P(M) / P(s) Very hard to find this “absolute” probability: depends on specific a priori probabilities we are unlikely to know) Instead, make it a “discrimination” problem: define a “null model” N, find P(M | s) / P(N | s)

11 Issues in Using HMMs Model architecture/topology Training –Selecting an appropriate training set –Finding an “optimal” HMM that fits that set –Must avoid overfitting Scoring (for HMM construction, sequence recognition) –How likely that our sequence was generated by our HMM? –Versus some “null model” – this converts a very difficult recognition problem into a tractable discrimination problem –Score is the “relative likelihood” for our HMM Efficiency of evaluation –Pruning the search: Dynamic programming –Using log-odds scores

12 A Simple HMM for Some DNA Sequences State (S i ) Transition Probabilities (a i,j ) Emission Probabilities (b i,k )

13 HMMs and Multiple Alignments Can convert a multiple alignment into an HMM: –Create a node for each column in which most sequences have an aligned residue –Columns with many missing letters go to Insert states –Emission probabilities are computed from the relative frequencies in the alignment column (for Match states), usually with aid of a regularizer (to avoid zero-probability cases) –Emission probabilities for Insert states are taken from background frequencies Can also create a multiple alignment from a linear HMM: –Find Viterbi (most likely) path in the HMM for each sequence –Each match state on that path creates a column in the alignment –Ignore or show in lower case letters from insert states –Setting transition probabilities is equivalent to setting gap penalties in sequence alignment - more an art than a science.

14 Protein Sequences Alignment ALYW GHSFCAGSL AIFAKHRRSPGERFLCGGIL AIYRRHRG-GSVTYVCGGSL AIFAQNRRSSGERFLCGGIL ALFQGE------RLICGGVL ALFIDD------MYFCGGSL AIYHYS------SFQCGGVL SLNS GSHFCGGSL Three states 1.Match states 2.Insertion states 3.Deletion states

15 M1M1 M2M2 M3M3 M4M4 I4I4 I3I3 I2I2 I1I1 I0I0 D1D1 D3D3 D2D2 D4D4 Topology of Profile HMM M1M1 M2M2 M3M3 M4M4 I4I4 I3I3 I2I2 I1I1 I0I0 D1D1 D3D3 D2D2 D4D4 Match states Insertion states Deletion states M i -> M i+1 M i -> I i M i -> D i+1 I i -> M i+1 I i -> I i I i -> D i+1 D i -> M i+1 D i -> I i D i -> D i+1

16 Regularizers For avoiding overfitting to training set Substitution matrices –Identify more likely amino acid substitutions, reflecting biochemical similarities/differences –Fixed for all positions in a sequence; one value for a given pair of amino acides Pseudocounts –For protein sequences, typically based on observed (relative) frequencies of various amino acids –“Universal” frequencies or position/type dependent values Dirichlet mixtures –Probabilistic combinations of Dirichlet densities Densities over probability distributions: i.e., the probability density of various distributions of symbols (in a given sequence position) Used to generate data-dependent pseudocounts

17 Algorithms for HMM Tasks (1) 3 Major Problems: –Determine HMM parameters (given some HMM topology and a training set) –Calculate (relative) likelihood of a given output sequence through a given HMM –Find the optimal (most likely) path through a given HMM for a specific output sequence, and its (relative) likelihood Forward/Backward Algorithm –Used for determining the parameters of an HMM from training set data –Calculates probability of going forward to a given state (from initial state), and of generating final model state (member of training set) from that state –Iteratively adjusts the model parameters

18 Algorithms for HMM Tasks (2) Baum-Welch (Expectation-maximization, EM)Algorithm –Often used to determine the HMM parameters –Can also determine most likely path for a (set of) output sequence(s) –Add up probabilities over all possible paths –Then re-update parameters and iterate –Cannot guarantee global optimum; very expensive Forward Algorithm –Calculates probability of a particular output sequence given the HMM –Straightforward summation of product of (partial path) probabilities Viterbi Algorithm –Classical dynamic programming algorithm –Choose “best” path (at each point), based on log-odds scores –Save results of “subsubproblems” and re-use them as part of higher-level evaluations –More efficient than Baum-Welch

19 HMMs for Protein/Gene Sequence Analysis Using any of various means, identify a set of related sequences with conserved regions Make 1 st -order Markov assumptions: transitions independent of sequence history and sequence content (other than at the substitution site itself) Construct a HMM based on the set of sequences Use this HMM to search for additional members of this family, possibly performing alignments –Search by comparing fit to HMM against fit to some null model For phylogenetic trees, also concerned with the length of the paths involved and with shared intermediate states (sequences)

20 Very Simple Viterbi min = DNA Sequence Alignment – with value of +1 for match, -1 for mismatch, -2 for “gap”; remember “best” at each step

21 Very Simple Viterbi: Traceback

22 Onward to Mathematica!