bioinformatics workshop, fall 2003 algorithms in bioinformatics lawrence d’antonio ramapo college...

Bioinformatics Workshop, Fall 2003

Algorithms in Bioinformatics

Lawrence D’Antonio

Ramapo College of New Jersey


Topics

• Algorithm basics

• Types of algorithms in bioinformatics

• Sequence alignment

• Database Searches


Algorithm basics

• What is an algorithm?

• Algorithm complexity

• P vs. NP

• NP completeness


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


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).


Example

• Problem: find the largest element in a sequence of n elements.

• Solution idea: Iteratively compare size of elements in sequence.


Algorithm:

1. Initialize first element as largest.

2. For each remaining element.

If current element larger than largest, make that element largest.

Running time: O(n)


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.


Polynomial Time Examples

• Searching

Binary Search: O(log n)

• Sorting

Quick Sort: O(n log n)


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.


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


NP Complete Examples

• Traveling salesman

• Knapsack problem

• Partition problem

• Graph coloring


P = NP ?

• P NP

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


Polynomial vs. Exponential


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


Edit operations on sequences

AATAAGC

ATTAAGC

AAT-AAGC

AATTAAGC

AATAAGC

AA-AAGC

Substitution Insertion Deletion


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


Complexity of DNA Problems

• 3 billion base pairs in human genome

• Many NP complete problems

• 10600 possible alignments for two 1000 character sequences


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)


Significance of Alignment

• Functional similarity

• Structural similarity

• Homology


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


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


Dynamic Programming

• Used on optimization problems

• Bottom-up approach

• Recursively builds up solution from subproblem optimal solutions


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)


Computing Alignment Matrix

• Match ai+1 with bj+1

• Match ai+1 with a space —

• Match bj+1 with a space —

If a1,a2,…,ai and b1,b2,…,bj have been aligned,

there are three possible next moves:

Choose the move that maximizes the similarity of the two sequences


Global Alignment Matrix

— G G A C A

— 0 -2 -4 -6 -8 -10

G -2 1 -1 -3 -5 -7

G -4 -1 2 0 -2 -4

G -6 -3 0 1 -1 -3

C -8 -5 -2 -1 2 0

A -10 -7 -4 -1 0 3

T -12 -9 -6 -3 -2 1


Optimal Global Alignment

G G G C A T

G G A C A —


Alignment Running Time

• Assuming two sequences n characters each

• Running time is O(n2) (each entry of matrix must be calculated)


Variations of Alignment Algorithm

• Gap penalty

• Local alignment

• Multiple alignment


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 + b·k (affine gap penalty)

• Here the first space in a gap is penalized a+b, further spaces are penalized b each.


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


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 (don’t align)


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


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)


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


Other Features

• Usually used for protein alignment

• Can be used for global or local alignment


Multiple Alignment Example

P E A A L Y G R F T - - - I K S D V W

P E S L A Y N K F - - - S I K S D V W

P E A L N Y G R Y - - - S S E S D V W

P E A L N Y G W Y - - - S S E S D V W

P E V I R M Q D D N P F S F Q S D V Y


Multiple vs. Pairwise Alignment

• Optimal multiple alignment does not imply optimal pairwise alignment

AT A -

A - - T

- T


Substitution Matrices

• In homologous sequences certain amino acid substitutions are more likely to occur than others

• Types of substitution matrices* PAM* BLOSUM


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


PAM Matrix Formula

• Entries in a k-PAM matrix

1010 logkab

b

M

p


PAM250 MatrixC S T P A G N D E Q H R K M I L V F Y W

C 12

S 0 2

T -2 1 3

P -3 1 0 6

A -2 1 1 1 2

G -3 1 0 -1 1 5

N -4 1 0 -1 0 0 2

D -5 0 0 -1 0 1 2 4

E -5 0 0 -1 0 0 1 3 4

Q -5 -1 -1 0 0 -1 1 2 2 4

H -3 -1 -1 0 -1 -2 2 1 1 3 6

R -4 0 -1 0 -2 -3 0 -1 -1 1 2 6

K -5 0 0 -1 -1 -2 1 0 0 1 0 3 5

M -5 -2 -1 -2 -1 -3 -2 -3 -2 -1 -2 0 0 6

I -2 -1 0 -2 -1 -3 -2 -2 -2 -2 -2 -2 -2 2 5

L -6 -3 -2 -3 -2 -4 -3 -4 -3 -2 -2 -3 -3 4 2 6

V -2 -1 0 -1 0 -1 -2 -2 -2 -2 -2 -2 -2 2 4 2 4

F -4 -3 -3 -5 -4 -5 -4 -6 -5 -5 -2 -4 -5 0 1 2 -1 9

Y 0 -3 -3 -5 -3 -5 -2 -4 -4 -4 0 -4 -4 -2 -1 -1 -2 7 10

W -8 -2 -5 -6 -6 -7 -4 -7 -7 -5 -3 2 -3 -4 -5 -2 -6 0 0 17


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


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)


Motif Example

• Motif obtained from a group of 34 tubulin proteins

M[FYW] . . F[VLI]H . [FYW] . . EGM


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


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

2

( , )( , ) log

( , )

n i js i j

e i j


PAM vs. BLOSUM

• PAM derived from highly similar sequences (evolutionary model)

• BLOSUM derived from protein families sharing a common ancestor (conserved domain model)


Database Searches

• FASTA

• BLAST


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)


FASTA Algorithm (I)

• Look for small substrings in query and text that exactly match (“hot spots”)

• Find ten best “diagonal runs” of hot spots


Hot Spot Example

E K L A S R K L

H

A *

S *

H

K *

L *


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


BLAST

• Basic Local Alignment Search Tool

• Heuristic, exclusion method

• Computes statistical significance of alignment scores


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


Other Bioinformatics Algorithms

• Palindromes

• Tandem Repeats

• Longest Common Subsequence

• Double Digest (NP complete)

• Shortest Common Superstring (NP complete)


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

bioinformatics workshop, fall 2003 algorithms in bioinformatics lawrence d’antonio ramapo college...

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