Sequence Similarity

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Sequence Similarity
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Why sequence similarity

• structural similarity
• gt25 sequence identity ? similar structure
• evolutionary relationship
• all proteins come from lt 2000 (super)families
• related functional role
• similar structure ? similar function
• functional modules are often preserved

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Muscle cells and contraction
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Actin and myosin during muscle movement
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Actin structure
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Actin sequence

• Actin is ancient and abundant
• Most abundant protein in cells
• 1-2 actin genes in bacteria, yeasts, amoebas
• Humans 6 actin genes
• ?-actin in muscles ?-actin, ?-actin in
  non-muscle cells
• 4 amino acids different between each version
• MUSCLE ACTIN Amino Acid Sequence
• 1 EEEQTALVCD NGSGLVKAGF AGDDAPRAVF PSIVRPRHQG
  VMVGMGQKDS YVGDEAQSKR
• 61 GILTLKYPIE HGIITNWDDM EKIWHHTFYN ELRVAPEEHP
  VLLTEAPLNP KANREKMTQI
• 121 MFETFNVPAM YVAIQAVLSL YASGRTTGIV LDSGDGVSHN
  VPIYEGYALP HAIMRLDLAG
• 181 RDLTDYLMKI LTERGYSFVT TAEREIVRDI KEKLCYVALD
  FEQEMATAAS SSSLEKSYEL
• 241 PDGQVITIGN ERFRGPETMF QPSFIGMESS GVHETTYNSI
  MKCDIDIRKD LYANNVLSGG
• 301 TTMYPGIADR MQKEITALAP STMKIKIIAP PERKYSVWIG
  GSILASLSTF QQMWITKQEY
• 361 DESGPSIVHR KCF

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A related protein in bacteria
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Relation between sequence and structure
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A multiple alignment of actins
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Gene expression
CCTGAGCCAACTATTGATGAA
CCUGAGCCAACUAUUGAUGAA
PEPTIDE
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Biomolecules as Strings

• Macromolecules are the chemical building blocks
  of cells
• Proteins
• 20 amino acids
• Nucleic acids
• 4 nucleotides A, C, G, ,T
• Polysaccharides

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The information is in the sequence

• Sequence ? Structure ? Function
• Sequence similarity
• ? Structural and/or Functional similarity
• Nucleic acids and proteins are related by
  molecular evolution
• Orthologs two proteins in animals X and Y that
  evolved from one protein in immediate ancestor
  animal Z
• Paralogs two proteins that evolved from one
  protein through duplication in some ancestor
• Homologs orthologs or paralogs that exhibit
  sequence similarity

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Protein Phylogenies

• Proteins evolve by both duplication and species
  divergence

duplication
orthologs
paralogs
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Evolution
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Evolution at the DNA level
Mutation
Deletion
ACGGTGCAGTTACCA
SEQUENCE EDITS
AC - - - - CAGTCACCA
REARRANGEMENTS
Inversion
Translocation
Duplication
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Evolutionary Rates



next generation
OK



OK



OK



Changes in non-functional sites are OK, so will
be propagated
X



X



Still OK?



Most changes in functional sites are deleterious
and will be rejected
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Sequence conservation implies function
Proteins between humans and rodents are on
average 85 identical
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Sequence Alignment
AGGCTATCACCTGACCTCCAGGCCGATGCCC TAGCTATCACGACCGCGG
TCGATTTGCCCGAC
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC-
-GACCGC--GGTCGATTTGCCCGAC
Definition Given two strings x x1x2...xM, y
y1y2yN, an alignment is an assignment of
gaps to positions 0,, M in x, and 0,, N in y,
so as to line up each letter in one sequence
with either a letter, or a gap in the other
sequence
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What is a good alignment?

• Alignment
• The best way to match the letters of one
  sequence with those of the other
• How do we define best?
• Alignment
• A hypothesis that the two sequences come from a
  common ancestor through sequence edits
• Parsimonious explanation
• Find the minimum number of edits that transform
  one sequence into the other

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Scoring Function

• Sequence edits
• AGGCCTC
• Mutations
  
• AGGACTC
• Insertions
• AGGGCCTC
• Deletions
• AGGCTC
• Scoring Function
• Match m
• Mismatch s
• Gap d
• Score F ( matches) ? m ( mismatches) ? s
  (gaps) ? d

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How do we compute the best alignment?
AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA
Too many possible alignments O( 2MN)
AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC
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Alignment is additive

• Observation
• The score of aligning x1xM
• y1yN
• is additive
• Say that x1xi xi1xM
• aligns to y1yj yj1yN
• The two scores add up
• F(x1M, y1N) F(x1i, y1j)
  F(xi1M, yj1N)
• Key property optimal solution to the entire
  problem is composed of optimal solutions to
  subproblems Dynamic Programming

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Dynamic Programming

• Construct a DP matrix F MxN
• Suppose we wish to align
• x1xM
• y1yN
• Let
• F(i, j) optimal score of aligning
• x1xi
• y1yj

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Dynamic Programming (contd)

• Notice three possible cases
• xi aligns to yj
• x1xi-1 xi
• y1yj-1 yj
• 2. xi aligns to a gap
• x1xi-1 xi
• y1yj -
• yj aligns to a gap
• x1xi -
• y1yj-1 yj

m, if xi yj F(i, j) F(i-1, j-1)
-s, if not
F(i, j) F(i-1, j) d
F(i, j) F(i, j-1) d
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Dynamic Programming (contd)

• How do we know which case is correct?
• Inductive assumption
• F(i, j 1), F(i 1, j), F(i 1, j 1) are
  optimal
• Then,
• F(i 1, j 1) s(xi, yj)
• F(i, j) max F(i 1, j) d
• F(i, j 1) d
• Where s(xi, yj) m, if xi yj -s, if not

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Example

• x AGTA m 1
• y ATA s -1
• d -1

F(i,j) i 0 1 2 3 4
A G T A
0 -1 -2 -3 -4
A -1 1 0 -1 -2
T -2 0 0 1 0
A -3 -1 -1 0 2
Optimal Alignment F(4, 3) 2 AGTA A - TA
j 0
1
2
3
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The Needleman-Wunsch Algorithm

• Initialization.
• F(0, 0) 0
• F(0, j) - j ? d
• F(i, 0) - i ? d
• Main Iteration. Filling-in partial alignments
• For each i 1M
• For each j 1N
• F(i-1,j-1) s(xi, yj) case 1
• F(i, j) max F(i-1, j) d case
  2
• F(i, j-1) d case 3
• DIAG, if case 1
• Ptr(i,j) LEFT, if case 2
• UP, if case 3
• Termination. F(M, N) is the optimal score, and
• from Ptr(M, N) can trace back optimal alignment

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Performance

• Time
• O(NM)
• Space
• O(NM)
• Possible to reduce space to O(NM) using
  Hirschbergs divide conquer algorithm

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Substitutions of Amino Acids

• Mutation rates between amino acids have dramatic
  differences!

How can we quantify the differences in rates by
which one amino acid replaces another across
related proteins?
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Substitution Matrices

• BLOSUM matrices
• Start from BLOCKS database (curated, gap-free
  alignments)
• Cluster sequences according to gt X identity
• Calculate Aab of aligned a-b in distinct
  clusters, correcting by 1/mn, where m, n are the
  two cluster sizes
• Estimate
• P(a) (?b Aab)/(?cd Acd) P(a, b) Aab/(?cd
  Acd)

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Gaps are not inserted uniformly
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A state model for alignment
M (1,1)
Alignments correspond 1-to-1 with sequences of
states M, I, J
I (1, 0)
J (0, 1)
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC-
-GACCGC-GGTCGATTTGCCCGACC IMMJMMMMMMMJJMMMMMMJMMMM
MMMIIMMMMMIII
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Lets score the transitions
s(xi, yj)
M (1,1)
s(xi, yj)
s(xi, yj)
Alignments correspond 1-to-1 with sequences of
states M, I, J
-d
-d
I (1, 0)
J (0, 1)
-e
-e
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC-
-GACCGC-GGTCGATTTGCCCGACC IMMJMMMMMMMJJMMMMMMJMMMM
MMMIIMMMMMIII
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A probabilistic model for alignment

• Assign probabilities to every transition (arrow),
  and emission (pair of letters or gaps)
• Probabilities of mutation reflect amino acid
  similarities
• Different probabilities for opening and
  extending gap

M (1,1)
I (1, 0)
J (0, 1)
-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC--- TAG-CTATCAC-
-GACCGC-GGTCGATTTGCCCGACC IMMJMMMMMMMJJMMMMMMJMMMM
MMMIIMMMMMIII
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A Pair HMM for alignments
log(1 2?)
M P(xi, yj)
log Prob(xi, yj)
log(1 ?)
log(1 ?)
log ?
log ?
log ?
log ?
I P(xi)
J P(yj)
Highest scoring path corresponds to the most
likely alignment!
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How do we find the highest scoring path?

• Compute the following matrices (DP)
• M(i, j) most likely alignment of x1xi with
  y1yj ending in state M
• I(i, j) most likely alignment of x1xi with
  y1yj ending in state I
• J(i, j) most likely alignment of x1xi with
  y1yj ending in state J
• M(i, j) log( Prob(xi, yj) )
• max M(i-1, j-1) log(1-2?),
• I(i-1, j) log(1-?),
• J(i, j-1) log(1-?)
• I(i, j) max M(i-1, j) log ?,
• I(i-1, j) log ?

log(1 2?)
M P(xi, yj)
log Prob(xi, yj)
log(1 ?)
log(1 ?)
log ?
log ?
I P(xi)
J P(yj)
log ?
log ?
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The Viterbi algorithm for alignment

• For each i 1, , M
• For each j 1, , N
• M(i, j) log( Prob(xi, yj) )
• max M(i-1, j-1) log(1-2?),
• I(i-1, j) log(1-?),
• J(i, j-1) log(1-?)
• I(i, j) max M(i-1, j) log ?,
• I(i-1, j) log ?
• J(i, j) max M(i-1, j) log ?,
• I(i-1, j) log ?
• When matrices are filled, we can trace back from
  (M, N) the likeliest alignment

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One way to view the state paths State M

y1
yn
x1

xm
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State I

y1
yn
x1

xm
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State J

y1
yn
x1

xm
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Putting it all together
States I(i, j) are connected with states J and M
(i-1, j) States J(i, j) are connected with
states I and M (i-1, j) States M(i, j) are
connected with states J and I (i-1, j-1)

y1
yn
x1

xm
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Putting it all together
States I(i, j) are connected with states J and M
(i-1, j) States J(i, j) are connected with
states I and M (i-1, j) States M(i, j) are
connected with states J and I (i-1, j-1) Optimal
solution is the best scoring path from top-left
to bottom-right corner This gives the likeliest
alignment according to our HMM

y1
yn
x1

xm
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Yet another way to represent this model
Ix
Ix
BEGIN
END
Iy
Iy
Mx1
Mxm
Sequence X
We are aligning, or threading, sequence Y through
sequence X Every time yj lands in state xi, we
get substitution score s(xi, yj) Every time yj
is gapped, or some xi is skipped, we pay gap
penalty