1. Alignment of biological sequences
Bioinformatics
Alignment of biological sequences
UL, 2017, Juris Viksna

2. Topics Short review about sequence comparison:
biological motivation to compare sequences
sequence similarity criteria
DP basic algorithm for distance computation between sequences
Global and local sequence comparisons
similarity matrices and gap penalties
modified algorithms that use gap penalties
local sequence comparison
Similarity matrices
how to obtain them
relations between similarity matrices and sequence evolution
suitability for matrices for specific sequences

3. Comparison of biological sequences
Two sequence comparisons (pairwise alignment):
the formulation of the problem
DP algorithm (match = 1, mismatch = 1, gap = 2)
gloabal and local comparisons
affine gap penalties
similarity matrices
Multiple alignment
the formulation of the problem (SOP)
Star alignment
relation with phylogenetic trees, progressive alignment
Sequence classification: profiles and moitifs
profile matrices
HMM (Hidden Markov Models)

4. Why we need to compare sequences?
Genome is already sequenced (assume...)
There are methods that predict DNA coding regions (genes)
What are biological functions of these genes??
We can find out what protein (sequence) gene encodes
But we still do not know what this protein does...
However we can search for known proteins with similar sequences and such that functions of these proteins are known
We want to find out something about proteins in humans
The best approach is “experimental”, but tricky with humans...
But we can try to use similar protein (e.g. in mice) and start our experiments with them

5. Basic assumptions
Will consider proteins/RNA/DNA just as sequences in correspondingly 20 and 4 letter alphabets
Aims of comparison:
to find out how similar the sequences are (some similarity measure)
to find “common motif” of sequences (alignment)
Regarding algorithmic complexity two distinctive cases:
comparison of two sequences (relatively easy)
simultaneous comparison of n sequences (complexity grows exponentially with n)
In this lecture we will consider the problem of comparison of two sequences

6. Nucleotides and DNA [Watson, Crick 1953]
For us DNA is a sequence in 4 letter alphabet
[Adapted from Y.Guo]

7. Proteins
For our purposes we will treat proteins as sequences in 20 symbol
alphabet
[Adapted from R.Shamir]

8. From DNA to proteins Each codon consists of 3 nucleotides Mutations:
Substitution: (changes a single aa)
Insertion / Deletion: “frame shift”
(change all subsequent aa)
NB! Insertion / Deletion might be a multiple of 3...
“Silent mutation” – DNA changed, but not aa
“Nonsense mutation” - creates “stop” codon

9. Genetic code
Genetic code
Completely worked out in 1962

10. Evolution of sequences
Mutations are a natural process of DNA evolution
DNA replication errors:
substitutions
insertions
deletions
Similarity between sequences:
indicates their common ancestral origin
indicates similarity of biological functions
Well, this is of course simplification: the change of protein
function will determine whether the organism will have
offsprings and the changed gene will survive
Protein sequence similarity is closely associated with
similarity of DNA coding regions
} indels

11. Sequence evolution Each codon consists of 3 nucleotides Mutations:
Substitution: (changes a single aa)
Insertion / Deletion: “frame shift”
(change all subsequent aa)
NB! Insertion / Deletion might be a multiple of 3...
“Silent mutation” – DNA changed, but not aa
“Nonsense mutation” - creates “stop” codon

12. Sequence evolution
ggcatt
agcatt
agcata
agcatg
agccta
aggatt
gacatt

13. Sequence homology Homologs - evolved from the common ancestor
Orthologs - the same function in different organisms
Paralogs - similar function in the same organism

14. Orthologs vs paralogs
[Adapted from R.Shamir]

15. How to compare sequences?
Given two proteins:
>sp|P69905|HBA_HUMAN Hemoglobin
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPT
TKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDM
PNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLP
AEFTPAVHASLDKFLASVSTVLTSKYR
>tr|Q61287|Q61287_MOUSE Hemoglobin
MVLSGEDKSNIKAAWGKIGGHGAEYVAEALERMFASFP
TTKTYFPHFDVSHGSAQVKGHGKKVADALASAAGHLDD
LPGALSALSDLHAHKLRVDPVNFKLLSHCLLVTLASHH
PADFTPAVHASLDKFLASVSTVLTSKYR
How to assess their similarity?

16. Sequence alignment - BLAST

17. Sequence alignment - BLAST

18. Sequence alignment – the results we expect

19. Sequence alignment - SSEARCH

20. Sequence alignment - SSEARCH

21. Sequence alignment - scores
sequence similarity/identity (%) This is well-defined for aligned sequence parts
“Score” (usually very method-specific in absolute value)
p-value – probability that alignment with given score or higher is found by chance Normally the given values are only approximations
Expect(E)-value (a parameter that describes the number of hits one can "expect" to see by chance when searching a database of a particular size) The lower the better. But short similar sequences might have comparatively high values (E-value decreases exponentially with Score)
Z-score – number of standard deviations from mean value

22. Z-score

23. Z-score

24. How to align two sequences - BLAST
Find two exact similarity regions (usually 4 aa each)
Try to join and extend these match until score falls below threshold
Anyway, how we should do this “correctly”?

25. The “Manhattan Tourist” problem
Visit as many sights as possible starting from top-left corner and moving just down or right

26. Longest common subsequence
Given two sequences A and B find a longest possible sequence C
that is subsequence of both A and B (such C does not need to be unique)
Example:
A = GGATATCGGGCGAT
B = ATTCCCCCGCCCTA
C = ATTCGCA or ATTAGCT
How can we find it?

27. LCS – dynamic programming solution
A = a1 a2an B = b1 b2bm
c(i,k) - length of LCS of a1 a2ai and b1 b2bk
0, if i = 0 or k = 0
c(i–1,k–1) +1, if i, k > 0 and ai = bk
max{c(i, k–1), c(i–1, k)}, if i, k > 0 and ai  bk
c(i, k) =

28. LCS – example
A = GADTAMAWGRAMMA
B = GAGAWKIAMM

29. LCS - example  G A D T M W R  G A W K I M

30. LCS - example  G A D T M W R  G A W K I M 1

31. LCS - example  G A D T M W R  G A W K I M 1 2

32. LCS - example  G A D T M W R  G A W K I M 1 2 3

33. LCS - example  G A D T M W R  G A W K I M 1 2 3 4

34. LCS - example  G A D T M W R  G A W K I M 1 2 3 4

35. LCS - example  G A D T M W R  G A W K I M 1 2 3 4

36. LCS - example  G A D T M W R  G A W K I M 1 2 3 4

37. LCS - example  G A D T M W R  G A W K I M 1 2 3 4 5

38. LCS - example  G A D T M W R  G A W K I M 1 2 3 4 5 6

39. LCS - example  G A D T M W R  G A W K I M 1 2 3 4 5 6 7

40. LCS - example  G A D T M W R  G A W K I M 1 2 3 4 5 6 7

41. LCS - example LCS: GAWAAMM Alignment: GA-DTAMAW—GRAMMA G A D T M W R  G A W K I M 1 2 3 4 5 6 7
LCS: GAWAAMM Alignment: GA-DTAMAW—GRAMMA
GAG----AWKI—AMM-

42. Edit distance  Levenshtein 1966
Minimal number of operations that transforms one sequence into another
insert, delete, substitute (1 simbols)
Edit distance is 0 (sequences are identical) or positive
For example “AIMS” & “AMOS”: (distance=2 for all three solutions)
AIMS AMOS 
AIM-S
A-MOS
AIMS
AMOS
[Adapted from D.Gilbert]

43. Edit distance
Given two sequences A and B find a the smallest possible number of Insertion, Deletion and Substitution operations that chnages A to B
Example:
A = GGATATCGGGCGAT
B = ATTCCCCCGCCCTA
[G][G]AT[A]T[C][C][C][C]CG[G-C][G-C]C[G-C]C[A]T[A]
ED = 12?
How can we find it?

44. Edit distance A = a1 a2an B = b1 b2bm
e(i,k) – lenght of ED for sequences a1 a2ai and b1 b2bk
i, if k = 0
k, if i = 0
e(i–1,k–1), if i, k > 0 and ai = bk
min{e(i–1,k–1),e(i,k–1),e(i–1,k)}+1,if i,k > 0 and ai  bk
e(i, k) =

45. { ED - modifications e(i,0) = i e(0,j) = j e(i-1,j)+ t e(i,j-1)+ t
If you interested in result «up to a sign» it does not matter whether min or max is used.
min is more natural for ED, max for LCS.
max is also the usual choice for sequence comparison.
tij – probability that aa ai changes to aa bj
e(i,j)= min {
e(i-1,j)+ t
e(i,j-1)+ t
e(i-1,j-1) + t(ai,bj)
e(i,0) = i
e(0,j) = j
For ED:
t = 1
t(ai,bj) = 0 if ai=bj
t(ai,bj) = 1 if ai  bj
For «inverse» LCS:
t = 0
t(ai,bj) = 1 if ai=bj
t(ai,bj) = 0 if ai  bj

46. Substitution (similarity) matrices
A R N D C Q E G H I L K M F P S T W Y V A R N D C Q E G H I L K M F P S T W Y V
Similarity Matrix
Most popular:
PAM
Blossum
Gonnet
The one shown is
BLOSSOM 62
(almost :)
"Traditional assumption": substitution score > 0 for substitutions
that are more frequent as random ones, and < 0 for less frequent
than random ones.

47. Sequence similarity as the longest path problem
We can treat matrix
as graph with weighted edges.
The problem then translates to finding path with the largest/smallest weight in Directed Acyclic Graph.

48. Complexity of similarity computation
Size of matrix: nm
Computing of value for each cell: const
Total time: (nm)
Total memory: (nm)
Notice that if we want just score only two rows are needed.
In this case the required memory: (nm)
However, if we also need the alignment (and we usually do)?

49. Edit distance in linear space?

50. Interpretation of comparison results
Alignment grid (edit graph).
Every alignment is a path from (0,0) to (n,m).

51. Interpretation of comparison results

52. Needleman-Wunsch algorithm

53. Global and local alignments

54. Global and local alignments
Using LCS the best local alignment will have the same score as the best global alignment (however the alignment might be «better»)
Using ED best local alignments are likely to be for sequences with length 1 
Local comparison/alignment «starts to work» if scoring is somewhere between the two above – there are extra points for each match and penalty points for each mismatch

55. Computing local aligments
Just allow a «free ride» from each node to the top-left vertex

56. Computing local aligments

57. Computing local aligments
In this case global alignment has better score, but misses «conserved domain»

58. Global and local comparisons
GLOBAL best alignment of entirety of both sequences
For optimum global alignment, we want best score in the final row or final column
Are these sequences generally the same?
Needleman Wunsch
find alignment in which total score is highest, perhaps at expense of areas of great local similarity
LOCAL best alignment of segments, without regard to rest of sequence
For optimum local alignment, we want best score anywhere in matrix
Do these two sequences contain high scoring subsequences
Smith Waterman
find alignment in which the highest scoring subsequences are identified, at the expense of the overall score
[Adapted from R Altman]

59. Alignment with gap penalties

60. Alignment with gap penalties

61. Finding gapped alignments
Add yet another «gap» edge between
each vetex and each of its «gap predecessors»
However there are (nm(m+n)) of them 

62. Finding gapped alignments

63. Finding gapped alignments

64. Finding gapped alignments

65. Finding local gapped alignements
[Adapted from M.Craven]

66. Gap penalties vin general case
The computation requires time though O(n3)...
[Adapted from M.Craven]

67. Substitutiom matrices
Margaret Dayhoff ( )
First woman in the field of Bioinformatics

68. Substitutiom matrices
Dayhoff, M.O., Schwartz, R. and Orcutt, B.C. (1978). "A model of Evolutionary Change in Proteins". Atlas of protein sequence and structure (volume 5, supplement 3 ed.). Nat. Biomed.
Res. Found. pp. 345–358.

69. Frequencies (probabilities) of amino acids
         1978        1991
L       0.085       0.091
A       0.087       0.077
G       0.089       0.074
S       0.070       0.069
V       0.065       0.066
E       0.050       0.062
T       0.058       0.059
K       0.081       0.059
I       0.037       0.053
D       0.047       0.052
R       0.041       0.051
P       0.051       0.051
N       0.040       0.043
Q       0.038       0.041
F       0.040       0.040
Y       0.030       0.032
M       0.015       0.024
H       0.034       0.023
C       0.033       0.020
W       0.010       0.014
Frequencies (probabilities) of amino acids

70. Substitution matrices as mutation probabilities
Therefore:
score  1 (or log score  0)
the better score, the better alignment

71. Substitution matrices as mutation probabilities
Currently we assume that there are no gaps in alignments
[Adapted from M.Craven]

72. Substitution matrices as mutation probabilities
[Adapted from M.Craven]

73. Substitution matrices as mutation probabilities
This gives extra ”scoring points” for each
matched symbol.
[Adapted from M.Craven]

74. PAM matrices
[Adapted from M.Craven]

75. PAM matrices
[Adapted from M.Craven]

76. PAM matricas
[Adapted from M.Craven]

77. PAM matrices A PAM (Percent Accepted Mutation) is one
accepted point mutation on the path between
two sequences, per 100 residues.
Most frequently used PAM250
Obtained from PAM1 by matrix multiplication...

78. PAM matrices - problemsQh Qm
T years
ancestor
 “The common ancestori", is actually unknown

79. PAM matrices - problems
ancestor ~  A C Q
shT PAMs
Evolution distances will be different for different pairs...

80. PAM-250

81. BLOSOM matrices
BLOSUM62 is the BLAST default
[Adapted from M.Craven]

82. How to align two sequences - BLAST
Find two exact similarity regions (usually 4 aa each)
Try to join and extend these match until score falls below threshold

83. BLOSOM matrices

84. Relationship between PAM & BLOSUM

85. Substitution matrices
The best matrix depends from evolutionary distance
In general the similarity score should be proportional to logarithm of probability of having common ancestor
The exact «right procedure» for computation of matrices is not trivial

86. Protein evolution rates
Mutation frequencies are fairly stable, but still could differ for different groups of proteins:
fibrinopeptides > hemoglobin > cytochrome > Hystone
For longer proteins mutations rates might be different in different sequence regions.

87. Protein evolution rates

88. Substitution matrices matrices - problems
If we observe a substitution a  b between two sequences this actually could mean:
a  b
a  x  b
a  x  y  b
As a result the computed probabilities will not be
“exactly right"...

89. Nucleotide substitution matrices
Probably describe better to mutation processes, but not the mutations that could survive during evolution.
These tend to be much simpler, since they can not reflect the role of specific nucleotide position.

90. What about aligment scores?
p-value – probability that alignment with given score or higher is found by chance
E-value – average number of alignments with given score and higher
Assuming all match probabilities to be equal to p, p-value could be computed using the fact that probability to have k matches from n is equal to:
𝑛 𝑘 𝑝 𝑘 (1−𝑝) 𝑛−𝑘 .
Such probabilities correspond to binomial distribution.
E-value can be derived from p-value and database size.

91. Probability distributions
For larger n binomial distribution can be aslo well approximated by normal distribution.
Both of them are easy to use to compute p-values.
However...

92. Computing of p-values However situation are made much more complex by:
Use of local, not global, alignments
Use of similarity matrices with different probabilities for matches/mismatches
Use of gap penalties
The computations of exact distributions becomes non-realistic, still
a good approximations exist that deals with all these additional requirements.
Still, these rely on assumption that probabilities of protein sequences are determined just by probabilities of amino acids.
This may result to p-values being considerably lover than «real probabilities» to reach specific alignment score.

93. P-Values P(s > S) = .01 Likewise for P=.001 and so on.
P-value of .01 occurs at score threshold S (392 below) where score s from random comparison is greater than this threshold 1% of the time
Likewise for P=.001 and so on.
[Adapted from M.Gerstein]

94. ROC (Receiver Operating Characterisctic) curves
We will consider proteins to be homologs, if their similarity exceed some threshold t
true positives (tp) - s(a,b)  t and a, b are homologs
false positives (fp) - s(a,b)  t and a, b are not homologs
true negatives (tn) - s(a,b) < t and a, b are not homologs
false negatives (fn) - s(a,b) < t and a, b are homologs
Sensitivity = tp/n = tp/(tp+fn)
Specificity = tn/n = tn(tn+fp)

95. ROC curves

96. ROC curves
100%
Coverage (roughly, fraction of sequences that one confidently “says something” about)
Thresh=10
Thresh=20
[sensitivity=tp/n=tp/(tp+fn)]
Thresh=30
Different score thresholds
Error rate (fraction of the “statements” that are false positives)
Two “methods” (red is more effective)
[Specificity = tn/n =tn/(tn+fp)]
error rate = 1-specificity = fp/n
[Adapted from M.Gerstein]

97. Homology (>10%) clusters of CATH 2

98. Multiple sequence alignments
Very similar DP recurions work, however in time (nN), where N is number of sequences
Not tractable for usual requirements on N
In practice many heuristic methods are used that «work well» but do not guarantee the optimal result

99. Time required for sequence comparisons
Smith-Waterman algorithm (1981)
Local comparisons
Linear gap penalties
Use of substitution matrices
Is (n2) time practically acceptable?
Protein length - around 300 aa
Comparison of 2 proteins: op. ( sec at 1 GHz)
Protein database entries
Database search: op, 100 sec
Comparison of two databases op, 25 years :(

100. Heuristic methods - FASTA

101. Heuristic methods - FASTA
Hash table of short words in the query sequence
Go through DB and look for matches in the query hash (linear in size of DB)
K-tuple determines word size (k-tup 1 is single aa)
Lipman & Pearson 1985
VLICT = _
VLICTAVLMVLICTAAAVLICTMSDFFD
[Adapted from M.Gerstein]

102. The FASTA Algorithm 4 steps:
use lookup table to find all identities at least ktup long, find regions of identities
rescan 10 regions (diagonals) with highest density of identities using PAM250
join regions if possible without decreasing score below threshold
rescore ala Smith-Waterman 32 residues around initial region (Note: doesn’t save alignment)

103. FASTA parameters ktup = 2 for proteins, 6 for DNA
init1 Score after rescanning with PAM250 (or other)
initn Score after joining regions
opt Score after Smith-Waterman

104. FASTA algorithm
[Fig.1 of Pearson and Lipman 1988]

105. FASTA algorithm
[Adapted from D Brutlag]

106. Heuristic methods - BLAST
Altschul, S., Gish, W., Miller, W., Myers, E. W. & Lipman, D. J. (1990). Basic local alignment search tool. J. Mol. Biol. 215,
Indexes query and DB
Starts with all overlapping words from query
Calculates “neighborhood” of each word using PAM matrix and probability threshold matrix and probability threshold
Looks up all words and neighbors from query in database index
Extends High Scoring Pairs (HSPs) left and right to maximal length
Finds Maximal Segment Pairs (MSPs) between query and database
Blast 1 does not permit gaps in alignments
[Adapted from M.Gerstein]

107. Heuristic methods - BLAST

108. BLAST algorithm
Keyword search of all words of length w from the in the query of length n in database of length m with score above threshold
w = 11 for nucleotide queries, 3 for proteins
Do local alignment extension for each found keyword
Extend result until longest match above threshold is achieved
Running time O(nm)
[Adapted from S.Daudenarde]

109. BLAST algorithm keyword Neighborhood words neighborhood
Query: KRHRKVLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLKIFLENVIRD
GVK 18
GAK 16
GIK 16
GGK 14
GLK 13
GNK 12
GRK 11
GEK 11
GDK 11
Neighborhood
words
neighborhood
score threshold
(T = 13)
extension
Query: 22 VLRDNIQGITKPAIRRLARRGGVKRISGLIYEETRGVLK 60
+++DN +G + IR L G+K I+ L+ E+ RG++K
Sbjct: 226 IIKDNGRGFSGKQIRNLNYGIGLKVIADLV-EKHRGIIK 263
High-scoring Pair (HSP)
[Adapted from S.Daudenarde]

110. Original BLAST Dictionary Alignment Output All words of length w
Ungapped extensions until score falls below some threshold
Output
All local alignments with score > statistical threshold
[Adapted from S.Daudenarde]

111. Original BLAST: Example
w = 4
Exact keyword match of GGTC
Extend diagonals with mismatches until score is under 50%
Output result:
GTAAGGTCC
GTTAGGTCC
A C G A A G T A A G G T C C A G T
C T G A T C C T G G A T T G C G A
From lectures by Serafim Batzoglou (Stanford)

112. Gapped BLAST

113. Gapped BLAST

114. Gapped BLAST: Example GTAAGGTCC-AGT GTTAGGTCCTAGT
Original BLAST exact keyword search, THEN:
Extend with gaps in a zone around ends of exact match until score < threshold then merge nearby alignments
Output result:
GTAAGGTCC-AGT
GTTAGGTCCTAGT
A C G A A G T A A G G T C C A G T
C T G A T C C T G G A T T G C G A
From lectures by Serafim Batzoglou (Stanford)

115. Gapped BLAST : Example GTAAGGTCC-AGT GTTAGGTCCTAGT
Original BLAST exact keyword search, THEN:
Extend with gaps in a zone around ends of exact match until score < threshold then merge nearby alignments
Output result:
GTAAGGTCC-AGT
GTTAGGTCCTAGT
A C G A A G T A A G G T C C A G T
C T G A T C C T G G A T T G C G A
From lectures by Serafim Batzoglou (Stanford)

116. BLAST - Programms blastp
compares an amino acid query sequence against a
protein sequence database
blastn
compares a nucleotide query sequence against a
nucleotide sequence database
blastx
compares a nucleotide query sequence translated
in all reading frames against a protein sequence database
tblastn
compares a protein query sequence against a
nucleotide sequence database dynamically translated in all reading
frames
tblastx
compares the six-frame translations of a nucleotide
query sequence against the six-frame translations of a nucleotide
sequence database. Please note that tblastx is extremely slow and
cpu-intensive

117. PSI-BLAST – position-specific matrices and transitive sequence comparison

118. PSSM – position-specific scoring matrix

119. Iterated PSI-BLAST

120. Iterated PSI-BLAST

121. General Protein Search Principles
Choose between local or global search algorithms
Use most sensitive search algorithm available
Original BLAST for no gaps
Smith-Waterman for most sensitivity
FASTA with k-tuple 1 is a good compromise
Gapped BLAST for well delimited regions
PSI-BLAST for families
Initially BLOSUM62 and default gap penalties
If no significant results, use BLOSUM30 and lower gap penalties
FASTA cutoff of .01
Blast cutoff of .0001
Examine results between exp and 10 for biological significance
Ensure expected score is negative
Beware of hits on long sequences or hits with unusual aa composition
Reevaluate results of borderline significance using limited query region
Segment long queries ³ 300 amino acids
Segment around known motifs
(some text adapted from D Brutlag)

122. Links to databases and search tools
UniProt/Swiss-Prot – “main” protein sequence database:
Text queries, BLAST search, etc
(EBI site)
(PIR site)
Text queries, SSearch, ClustalW
Search tools:
(pair alignments)
FASTA, BLAST, SSearch etc
(multiple alignments)
ClustalW, Tea-Coffee etc
FASTA, SSearch – searches and software downloads

123. Links to databases and search tools
Protein structure database (PDB):
Ensembl genome browser: