MARKOV MODELS

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MARKOV MODELS
HIDDEN

• Presentation
• by
• Jeff Rosenberg, Toru Sakamoto, Freeman Chen

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The Plan

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

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

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Markov Chains

• A system with a set of m possible states, Si at
  each of a sequence of discrete points in time
  tgt0, the system is in exactly one of those
  states the state at time t gt 0 is designated by
  qt the movement from qt to qt1 is
  probabilistic, and depends only on the states of
  the system at or prior to t.
• An initial state distribution p(i) Prob(q0
  Si)
• Process terminates either at time T or when
  reaching a designated final state Sf

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Markov Chains of Order N

• Nth-order Markov chain (N gt 0) transition
  probabilities out of state qt depend only on the
  values of qt, qt-1, qt-(N-1).
• Typically deal with 1st-order Markov chain, so
  only qt itself affects the transition
  probabilities.
• In a 1st-order chain, for each state Sj, there is
  set of m probabilities for selecting the next
  state to move to ai,j Prob(qt1 Si qt
  Sj) 1 lt i lt m, t gt 0
• If there is some ordering of states such that
  ai,j 0 whenever i lt j (i.e., no non-trivial
  loops), then this is a linear (or
  left-to-right) Markov process
• Homogeneous Markov model ai,j is independent
  of t

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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 2nd-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 0th-order Markov chain,
  where the probability of a given symbol is
  constant, regardless of the preceding symbol (or
  of the position in the sequence).

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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) vk, 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

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Additional Parameters for HMMs

• Now, in addition to the transition probabilities,
  each state has a prescribed probability
  distribution to emit or produce a symbol vk from
  O bi,k Prob(vk Si)
• If qt Si, then the generated output at time t
  is vk with probability bi,k.
• So, a HMM is doubly stochastic both the
  (hidden) state transition process and the
  (visible) output symbol generation process are
  probabilistic.

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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)

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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

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A Simple HMM for Some DNA Sequences
State(Si)
Emission Probabilities (bi,k)
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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.

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Protein Sequences Alignment
ALYW-------GHSFCAGSL AIFAKHRRSPGERFLCGGIL AIYRRHRG
-GSVTYVCGGSL AIFAQNRRSSGERFLCGGIL ALFQGE------RLIC
GGVL ALFIDD------MYFCGGSL AIYHYS------SFQCGGVL SLN
S-------GSHFCGGSL

• Three states
• Match states
• Insertion states
• Deletion states

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Topology of Profile HMM
Match states
Deletion states
Insertion states
Mi -gt Mi1 Mi -gt Ii Mi -gt Di1
Di -gt Mi1 Di -gt Ii Di -gt Di1
Ii -gt Mi1 Ii -gt Ii Ii -gt Di1
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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

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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

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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

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HMMs for Protein/Gene Sequence Analysis

• Using any of various means, identify a set of
  related sequences with conserved regions
• Make 1st-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)

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Very Simple Viterbi
DNA Sequence Alignment with value of 1 for
match, -1 for mismatch, -2 for gap remember
best at each step
min -1099
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Very Simple Viterbi Traceback
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Onward to Mathematica!