Algorithms in Bioinformatics

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Algorithms in Bioinformatics

• Lawrence DAntonio
• Ramapo College of New Jersey

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Topics

• Algorithm basics
• Types of algorithms in bioinformatics
• Sequence alignment
• Database Searches

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Algorithm basics

• What is an algorithm?
• Algorithm complexity
• P vs. NP
• NP completeness

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What is an algorithm?

• An algorithm is a step-by-step procedure to solve
  a problem
• The word algorithm comes from the 9th century
  Islamic mathematician al-Khwarizmi

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Algorithm Complexity

• If the algorithm works with n pieces of data and
  the number of steps is proportional to n, then we
  say that the running time is O(n).
• If the number of steps is proportional to log
  n, then the running time is O(log n).

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Example

• Problem find the largest element in a sequence
  of n elements.
• Solution idea Iteratively compare size of
  elements in sequence.

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• Algorithm
• Initialize first element as largest.
• For each remaining element.
• If current element larger than largest, make
  that element largest.

Running time O(n)
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Polynomial Time

• An algorithm is said to run in polynomial time if
  its running time can be written in the form O(nk)
  for some power k.
• The underlying problem is said to be of class P.

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Polynomial Time Examples

• Searching
• Binary Search O(log n)
• Sorting
• Quick Sort O(n log n)

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NP Algorithms

• An algorithm is nondeterministic if it begins
  with guessing a solution to the problem and then
  verifies the guess.
• A problem is of category NP if there is a
  nondeterministic algorithm for that problem which
  runs in polynomial time.

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NP Complete

• A problem is NP-complete if it has an NP
  algorithm, and solutions to this problem can be
  used to solve all other NP problems.
• A problem is NP-hard if it is at least as hard as
  the NP-complete problems

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NP Complete Examples

• Traveling salesman
• Knapsack problem
• Partition problem
• Graph coloring

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P NP ?

• P ? NP
• If P ? NP then NP-complete problems have
  exponential running time.

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Polynomial vs. Exponential
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Algorithms in Bioinformatics

• Algorithms to compare DNA, RNA, or protein
  sequences
• Database searches to find homologous sequences
• Sequence assembly
• Construction of evolutionary trees
• Structure prediction

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Edit operations on sequences

Substitution
Insertion
Deletion
AATAAGC
AAT-AAGC
AATAAGC
ATTAAGC
AATTAAGC
AA-AAGC
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What is sequence alignment?

• Compare two sequences using matches,
  substitutions and indels.
• G A A - - T C A T
• G - T G G - C A -
• 3 matches, 1 substitution, 5 indels

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Complexity of DNA Problems

• 3 billion base pairs in human genome
• Many NP complete problems
• 10600 possible alignments for two 1000 character
  sequences

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Types of sequence alignment

• Determine the alignment of two sequences that
  maximizes similarity (global alignment)
• Determine substrings of two sequences with
  maximum similarity (local alignment)
• Determine the alignment for several sequences
  that maximizes the sum of pairs similarity
  (multiple alignment)

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Significance of Alignment

• Functional similarity
• Structural similarity
• Homology

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Scoring System

• Assign a score for each possible match,
  substitution and indel
• Distance functions Find alignment to minimize
  distance between sequences
• Similarity functions Find alignment to maximize
  similarity between sequences

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Edit Distance

• G A A - - T C A T
• G - T G G - C A -
• Similarity function 1 for match, -1 for
  substitution, -2 for indel
• Score -8

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Dynamic Programming

• Used on optimization problems
• Bottom-up approach
• Recursively builds up solution from subproblem
  optimal solutions

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Dynamic Programming Alignment Algorithm
(Needleman-Wunsch)

• Given sequences a1,a2,,an and b1,b2,,bm to be
  aligned
• Initialize alignment matrix (aligning with
  spaces)
• Entry i,j gives optimal alignment score for
  sequences a1,a2,,ai and b1,b2,,bj (where 1 ? i
  ? n, 1 ? j ? m)

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Computing Alignment Matrix
If a1,a2,,ai and b1,b2,,bj have been
aligned, there are three possible next moves

• Match ai1 with bj1
• Match ai1 with a space
• Match bj1 with a space

Choose the move that maximizes the similarity of
the two sequences
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Global Alignment Matrix
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Optimal Global Alignment
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Alignment Running Time

• Assuming two sequences n characters each
• Running time is O(n2) (each entry of matrix must
  be calculated)

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Variations of Alignment Algorithm

• Gap penalty
• Local alignment
• Multiple alignment

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Gap Penalty

• A gap is a number k of consecutive spaces
• k consecutive spaces are more probable than k
  isolated spaces
• Typical gap penalty function a bk (affine
  gap penalty)
• Here the first space in a gap is penalized ab,
  further spaces are penalized b each.

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Gap Penalty Example

• Use penalty, 1 k
• A - A - C - A
• A C T A T C A
• Score -6
• A A C - - - A
• A C T A T C A
• Score -4

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Local Alignment

• Find conserved regions in otherwise dissimilar
  sequences (e.g., viral and host DNA)
• Smith-Waterman algorithm
• Includes a fourth possibility at each step (dont
  align)

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Local Alignment Example

• Align the following
• G C T C T G C G A A T A
• C G T T G A G A T A C T

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Optimal Local Alignment

• G C T C T G C G A A T A
• C G T T G A G A T A C T
• (G C T C) T G C G A A T A
• (C G T) T G A G - A T A (C T)

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Multiple Alignment

• Find the alignment among a set of sequences that
  maximizes the sum of scores for all pairs of
  sequences
• Dynamic programming run-time for k sequences of
  length n O(k2 2k nk)
• Multiple alignment is NP-complete

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Other Features

• Usually used for protein alignment
• Can be used for global or local alignment

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Multiple Alignment Example
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Multiple vs. Pairwise Alignment

• Optimal multiple alignment does not imply optimal
  pairwise alignment
• AT A -
• A - - T
• - T

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Substitution Matrices

• In homologous sequences certain amino acid
  substitutions are more likely to occur than
  others
• Types of substitution matrices
• PAM
• BLOSUM

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PAM Matrices

• Defines units of evolutionary distance
• 1 PAM unit represents an average of one mutation
  per 100 amino acids
• Start with a set of highly similar sequences and
  compute
• pa probability of occurrence of amino acid a
• Mab probability of a mutating to b

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PAM Matrix Formula

• Entries in a k-PAM matrix

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PAM250 Matrix
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BLOSUM Matrices (Omit)

• Uses log-odds ratio similar to PAM
• Uses short highly conserved sequences
• BLOSUM x matrices created after removing
  sequences that are more than x percent identical
• Better at local alignments

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BLOSUM Matrices

• A motif is a conserved amino acid pattern found
  in a group of proteins with similar biological
  meaning (PROSITE)
• A block is a conserved amino acid pattern in a
  group of proteins (no spaces allowed in the
  pattern) (BLOCKS)

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Motif Example

• Motif obtained from a group of 34 tubulin
  proteins
• MFYW . . FVLIH . FYW . . EGM

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Defining BLOSUM (I)

• BLOSUMn uses blocks that are n identical
  (BLOSUM62 is most common)
• Consider all pairs of amino acids appearing in
  the same column in the blocks

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Defining BLOSUM (II)

• Define n(i,j) to be the frequency that amino
  acids i,j appear in a column pair
• Define e(i,j) to be the frequency that amino
  acids i,j appear in any pair
• Define BLOSUM entry

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PAM vs. BLOSUM

• PAM derived from highly similar sequences
  (evolutionary model)
• BLOSUM derived from protein families sharing a
  common ancestor (conserved domain model)

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Database Searches

• FASTA
• BLAST

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FASTA

• Looks for sequences in a database similar to a
  query sequence
• Heuristic, exclusion method
• Compares query sequence to each database sequence
  (called the text)

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FASTA Algorithm (I)

• Look for small substrings in query and text that
  exactly match (hot spots)
• Find ten best diagonal runs of hot spots

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Hot Spot Example

• E K L A S R K L
• H
• A
• S
• H
• K
• L

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FASTA Algorithm (II)

• Find best local alignment for each run
• Combine these into larger alignment
• Do multiple alignment on query and texts having
  highest score in last step

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BLAST

• Basic Local Alignment Search Tool
• Heuristic, exclusion method
• Computes statistical significance of alignment
  scores

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BLAST Algorithm

• Find all w-length substrings in text that align
  to some w-length substring in query with score
  above a given threshold (called hits)
• Extend these hits as far as possible (segment
  pairs)
• Report the highest scoring segment pairs

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Other Bioinformatics Algorithms

• Palindromes
• Tandem Repeats
• Longest Common Subsequence
• Double Digest (NP complete)
• Shortest Common Superstring (NP complete)

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References

• Clote and Backofen, Computational Molecular
  Biology, Wiley
• Gusfield, Algorithms on Strings, Trees, and
  Sequences, Cambridge University Press
• Mount, Bioinformatics, Cold Spring Harbor Press
• Setubal and Meidanis, Introduction to
  Computational Molecular Biology, PWS
• Waterman, Introduction to Computational Biology,
  CRC Press