Alignment and Algorithms

1
Alignment and Algorithms

• Scoring Matrices models of evolution
• Dynamic Program speed
• Pairwise Sequence Alignment, variants
• Significance
• Pattern recognition, Hidden Markov Models (HMM)
• Multiple Sequence Alignment (MSA)
• Protein families
• Structure features

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Scoring Models and Evolution

• Several systems of scoring matrices.
• Statistically based, curated data I.e., based on
  confident alignments of multiple sequences.
• Two best known
• PAM matrices Dayhoff et al., 1967
• BLOSUM matrices Henikoff Henikoff, 1992

3
Scoring matrices, cont.

• Recall using p(a,b) prob. that a (amino acid
  residue) mutated into b.
• PAM (Point Accepted Mutation) uses sequence data
  for proteins which can be aligned with 99
  identical sequence. In such aligned sequences,
  one counts the frequency of each residue, and of
  each pair of residues in a column.

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Scoring matrices, cont.

• Basically, the score s(a,b) from this data is a
  LOD score
• s(a,b) log(p(a,b)/p(a)p(b)).
• Comparison of two models significant
  association vs. independent randomness.
• Pruned scaled to make matrices integer-valued.

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Scoring matrices, cont.

• Interpretation Dayhoff, et al. define a short
  unit of evolutionary time, a PAM unit. Only very
  conservative changes will be seen in the
  composition of evolving proteins in this scale.
• To extend to more distant similarities, used time
  independent Markov assumption.

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Scoring matrices, cont.

• p(a,b) p(a--gtbt1) gives the transition
  probability from a --gt b (or b --gt a we cannot
  keep track of directionality) in one unit of PAM
  time.
• The time independent Markov model assumes
  p(a--gtbt2) in two units is simply the sum over
  all pairs of intermediate one time unit
  transitions
• a --gt c followed by c --gt b

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Scoring matrices, cont.

• If P(1) is the original 20x20 matrix of amino
  acid transition probabilities, then the above
  says simply
• P(n) P(1) x P(1)xx P(1) (n times)
• (matrix multiplication)

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Scoring matrices, cont.

• So, PAM(n) matrix of LOD scores from transitions
  in PAM time n.
• Available n1 to n500 smaller n for more
  closely related sequences.
• Most commonly used 50 to 250.
• Critique accurate for small time, goes off at
  larger times. Compare nucleic acid changes to
  amino acid changes.

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PAM 250 MATRIX
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Scoring matrices, cont.

• BLOSUM Matrices
• From the BLOCKS database.
• BLOSUM BLOCKS Substitution Matrix
• DB of aligned sequence. Conserved regions are
  marked. The BLOSUM(L)
• transition probabilities are figured from
• sequences with L identity of sequence. Score is
  a LOD score again. So, large BLOSUM L lt--gt small
  PAM n.
• Warning corrections for oversampling of closely
  related sequence!

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Scoring matrices, cont.

• If the BLOSUM62 matrix is compared to PAM160
  (it's closest equivalent) then it is found that
  the BLOSUM matrix is less tolerant of
  substitutions to or from hydrophilic amino acids,
  while more tolerant of hydrophobic changes and of
  cysteine and tryptophan mismatches.
• PAM is less sensitive to the effect of amino acid
  return substitutions, over long time periods.

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BLOSUM62 Matrix
13
Dynamic Programming

• Problem find (1) best score and (2) best
  alignment
• Naïve evaluate each alignment and compare
• Too many alignments!
• Need more speed DP

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Dynamic Programming, cont.

• No. of alignments is exponentially large.
• Use problem has a linear (left to right)
  structure to it, from the linearity of proteins,
  nucleic acids.
• Analogous to the problem of distance originally
  arose in CS, edit distance.

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Dynamic Programming, cont.

• The basic algorithms here are
• Needleman-Wunsch (global alignment)
• (2) Smith-Waterman (local alignment)
• BLAST is a heuristic version of SW.
• (3) Many variants.

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Dynamic Programming, cont.

• NW algorithm can be understood pictorially.
• It has three basic steps
• Initialization
• Iteration
• Traceback
• Basic insight you only have to optimize the
  score one step at a time, and can discard looking
  at most alignments.

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Dynamic Programming, cont.

• Missing scoring feature gap penalties.
• Not as well understood as substitutions. Use
  geometric random variable models.
• s(a,-) is independent of a, say s(a,-) -d
  gap penalty
• For longer gaps, two methods
• linear k gaps in a row gives penalty k x (-d).
• affine k gaps in a row gives penalty
• e (k-1) x (-d),
• Where e is the gap initiation penalty.

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Dynamic Programming, cont.

• Notice affine gap does not just depend on the
  position, since you have to know whether you are
  continuing a gap from the left. First example of
  dependency in scoring.
• Key to the iterative step is to realize there are
  only three ways to extend an alignment
• R1 R1 -
• r1 - r1
• We have to update the score by taking the max of
  three possibilities max(s0 s(R1,r1), s0
  s(R1,-), s0 s(-,r1)).
• Then to get the scores of all possible
  subalignments takes just
• about 5 MN computations, instead of an
  exponential number.

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Dynamic Programming, cont.

• To derive the actual alignment requires storage
  one stores a traceback arrow from each position
  in the comparison array.
• This information can be translated into the
  correct alignment.
• Variants
• Smith-Waterman local alignment. Adjustment is to
  0-out and start over again any time a NW score
  goes below zero, and procede. One chooses the
  largest scoreof a segment, and traces back as
  earlier.
• Linear storage saves storage.
• Repeats allows for duplication of segments.

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Dynamic Programming, cont.

• Variants (cont.)
• Non-overlapping Repeats SW, avoiding previous
  best alignments.
• Sampling of highest hits the best alignment
  score maybe an artifact of our scoring system, so
  many alignments may be high scoring.
• Affine gap penalties actually have to have more
  states carried along, a precursor to Hidden
  Markov Models.

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Significance

• E-value in BLAST discussed last time. Equivalent
  to statistical concept of a p-value.
• S score for best alignment is a random variable.
• The p-value for an alignment is the probability
  that such a high score could have been found
  randomly.

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Significance, cont.

• This can be estimated two ways
• Ungapped local alignment Karlin-Altschul
  statistics.
• Generally, KA not available, rely instead on
  simulations take two sequences A and B, and
  randomly shuffle one and recompare. This
  eliminates composition bias in the teest samples,
  but randomizes the rest of the biological
  information in the sequence, namely the order of
  the residues (or nts). SW computes this for
  100s of shuffles.

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HMMs

• These are a special case of a general CS problem
  pattern recognition, or machine learning. Can one
  make the machine a master at recognizing or
  classifying certain things. There are a large
  family of such graphical models, this is one of
  the simplest.
• Basic problem in MB can one recognize different
  functional or structural regions in a protein
  from its a.a. sequence? Its fold? Or less
  specifically, what are the common features of a
  group of proteins.

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HMMs, cont.

• For example, there exist very many G-protein
  coupled receptors, which fall into seven classes.
  Can one classify a new protein into (a) being a
  GPCR, and (b) assigning its class to it from
  sequence data alone?
• Simplest example problem dishonest casino. You
  are given a sequence of observed random variables
  or states of the system (faces of a die), but you
  dont see hidden states (e.g., whether a casino
  is using a fair or biased die). Notice that this
  kind of example has a linear structure to it,
  like our a.a. sequences.

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HMMs, cont.

• The problem is given an observed path of
  (observable states) determine the underlying path
  through the hidden states. I.e., given the record
  of the coin flips, say when the biased coin was
  used and when the fair. Notice that in this model
  we know how many visible states there are and we
  are assuming we know how many hidden states there
  are.
• Of course, we cannot know with cerrtainty when
  the fair/biased coin is used, we must seek a
  probable answer, and the algorithms show how to
  choose a most probable path through the model.
• The harder problem (biological problem) is when
  we do not know what the hidden states are,
  except where we can define them, e.g.,
  structurally. For example, you could run through
  protein a.a. sequence and ask a HMM to decide
  which residues are situated in alpha helices of
  the protein, or when they lie in the cell
  membrane.

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HMMs, cont.

• In the first problem, if we know all the
  parameters, meaning mainly the transition
  problems for the Hidden Markov states, then it is
  easy to find a dynamic programming algorithm for
  finding the best path. This is the Viterbi
  algorithm.
• Sometimes called parsing the hidden markov
  model, because the method originated in speech
  recognition work.
• The harder problem is to figure out what the
  parameters are for the hidden states. This is
  called training the model, and is done by the
  Baum-Welch algorithm. This is a version of the
  so-called EM method, or algorithm (Expectation
  Maximization algorithm).
• BW, or EM, is a heuristic optimization algorithm.
  That is, it starts from a given initial guess for
  the parameters involved and performs a
  calculation which is guaranteed to improve (make
  no worse, actually) the previous guess. One runs
  for a certain time and turns things off by a
  suitable threshold.

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HMMs, cont.

• Finally, a harder problem still is to learn the
  correct architecture of the model. There are
  packages for this in the case of protein
  families/ multiple sequence alignments.
• In the case of structure, for example, one sees a
  classical problem in statistics one can let the
  data tell directly what the patterns should be
  (unsupervised learning) or you could use
  expertise to guide the data. This allows fainter
  structural signals to be heard. Problem is
  this fittingthe data too tightly, making the
  model generated harder to be generally
  applicable? Example of TMHMM and unusual, but
  necessary faint signals.

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Further Notes are available at

• http//www.math.lsa.umich.edu/dburns/547/547syll.
  html
• You can link from there to lectures on sequence
  alignment and hidden Markov models.