Expected accuracy sequence alignment

Usman Roshan

Optimal pairwise alignment

• Sum of pairs (SP) optimization: find the alignment of two sequences that maximizes the similarity score given an arbitrary cost matrix. We can find the optimal alignment in O(mn) time and space using the Needleman-Wunsch algorithm.

• Recursion: Traceback:

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M(i, j) =

M(i −1, j −1) + s(x i, y j )

M(i, j −1) + g

M(i −1, j) + g

⎧

⎨ ⎪

⎩ ⎪

where M(i,j) is the score of the optimal alignment of x1..i and y1..j, s(xi,yj) is a substitution scoring matrix, and g is the gap penalty

Affine gap penalties

• Affine gap model allows for long insertions in distant proteins by charging a lower penalty for extension gaps. We define g as the gap open penalty (first gap) and e as the gap extension penalty (additional gaps)

• Alignment:– ACACCCT ACACCCC– ACCT T AC CTT– Score = 0 Score = 0.9

• Trivial cost matrix: match=+1, mismatch=0, gapopen=-2, gapextension=-0.1

Affine penalty recursion

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V (i, j) = max{E(i, j),F(i, j), M(i, j)

M(i, j) = V (i −1, j −1) + s(x i,y j )

E(i, j) = max{E(i, j −1) + ext,V (i, j −1) + g}

F(i, j) = max{F(i −1, j) + ext,V (i −1, j) + g}

M(i,j) denotes alignments of x1..i and y1..j ending witha match/mismatch. E(i,j) denotes alignments of x1..i

and y1..j such that yj is paired with a gap. F(i,j) definedsimilarly. Recursion takes O(mn) time where m and n are lengths of x and y respectively.

Expected accuracy alignment

• The dynamic programming formulation allows us to find the optimal alignment defined by a scoring matrix and gap penalties. This may not necessarily be the most “accurate” or biologically informative.

• We now look at a different formulation of alignment that allows us to compute the most accurate one instead of the optimal one.

Posterior probability of xi aligned to yj

• Let A be the set of all alignments of sequences x and y, and define P(a|x,y) to be the probability that alignment a (of x and y) is the true alignment a*.

• We define the posterior probability of the ith residue of x (xi) aligning to the jth residue of y (yj) in the true alignment (a*) of x and y as

Do et. al., Genome Research, 2005

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P(x i ~ y j ∈ a* | x,y) = P(a | x,y)1{x i ~ y j ∈ a}a∈A

∑

Expected accuracy of alignment

• We can define the expected accuracy of an alignment a as

• The maximum expected accuracy alignment can be obtained by the same dynamic programming algorithm

V i j

V i j P x y

V i j

V i j

i j

( , ) max

( , ) ( ~ )

( , )

( , )

=− − +

−−

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

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Do et. al., Genome Research, 2005

Example for expected accuracy

• True alignment• AC_CG• ACCCA• Expected accuracy=(1+1+0+1+1)/4=1

• Estimated alignment• ACC_G• ACCCA• Expected accuracy=(1+1+0.1+0+1) ~ 0.75

Estimating posterior probabilities• If correct posterior probabilities can be computed

then we can compute the correct alignment. Now it remains to estimate these probabilities from the data

• PROBCONS (Do et. al., Genome Research 2006): estimate probabilities from pairwise HMMs using forward and backward recursions (as defined in Durbin et. al. 1998)

• Probalign (Roshan and Livesay, Bioinformatics 2006): estimate probabilities using partition function dynamic programming matrices

Partition function posterior probabilities

• Standard alignment score:

• Probability of alignment (Miyazawa, Prot. Eng. 1995)

• If we knew the alignment partition function then

SaT Mff gappenaltiesij i jija() ln( / )( _ )(,)= +∈∑Pa eSaT() ()/∝PaTe ZTSaT(,) /()()/=

Partition function posterior probabilities

• Alignment partition function (Miyazawa, Prot. Eng. 1995)

• SubsequentlyZ e e eijM SaTaA

S aTaA

sxyTijij

i ii j

i j, ()/ ()/ (, )/,= =⎛⎝⎜⎜ ⎞⎠⎟⎟∈ ∈∑ ∑−−−−

1111

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Z(T) = eS(a ) /T

a∈A

∑

Partition function posterior probabilities

• More generally the forward partition function matrices are calculated as

Z Z Z Z eZZe ZeZZe ZeZZZZijM i jM i jE i jF sxyTijE ijMgT ijEextTijF i jMgT i jFextTij ijMijE ijF

i j, , . , (, )/, , / . /, , / . /, , , ,

( )=+ += += +=++−− −− −−− −− −11 11 111 11 1

Partition function matrices vs. standard affine recursionsZ Z Z Z eZZe ZeZZe ZeZZZZijM i jM i jE i jF sxyTijE ijMgT ijEextTijF i jMgT i jFextTij ijMijE ijF

i j, , . , (, )/, , / . /, , / . /, , , ,

( )=+ += += +=++−− −− −−− −− −11 11 111 11 1

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V (i, j) = max{E(i, j),F(i, j),M(i, j)

M(i, j) = V (i −1, j −1) + s(x i,y j )

E(i, j) = max{E(i, j −1) + ext,V (i, j −1) + g}

F(i, j) = max{F(i −1, j) + ext,V (i −1, j) + g}

Posterior probability calculation

• If we defined Z’ as the “backward” partition function matrices then

Pxy Z ZZ ei j i jM i jM sxyTi j(~) ', , (, )/=−−++11 11

Posterior probabilities using alignment ensembles

• By generating an ensemble A(n,x,y) of n alignments of x and y we can estimate P(xi~yj) by counting the number of times xi is aligned to yj.. Note that this means we are assigning equal weights to all alignments in the ensemble.

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P(x i ~ y j ∈ a* | x,y) = P(a | x,y)1{x i ~ y j ∈ a}a∈A

∑

Generating ensemble of alignments

• We can use stochastic backtracking (Muckstein et. al., Bioinformatics, 2002) to generate a given number of optimal and suboptimal alignments.

• At every step in the traceback we assign a probability to each of the three possible positions.

• This allows us to “sample” alignments from their partition function probability distribution.

• Posteror probabilities turn out to be the same when calculated using forward and backward partition function matrices.

Probalign1. For each pair of sequences (x,y) in the input set

– a. Compute partition function matrices Z(T)– b. Estimate posterior probability matrix P(xi ~ yj) for (x,y)

by

2. Perform the probabilistic consistency transformation and compute a maximal expected accuracy multiple alignment: align sequence profiles along a guide-tree and follow by iterative refinement (Do et. al.).

Pxy Z ZZ ei j i jM i jM sxyTi j(~) ', , (, )/=−−++11 11

V i j

V i j P x y

V i j

V i j

i j

( , ) max

( , ) ( ~ )

( , )

( , )

=− − +

−−

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

1 111

Multiple protein alignment

• Protein sequence alignment: hard problem for multiple distantly related proteins

• Several standard protein alignment benchmarks available: BAliBASE, HOMSTRAD, OXBENCH, PREFAB, and SABMARK

• Benchmark alignments are based on manual and computational structural alignment of proteins with known structure.

Measure of accuracy

• Sum-of-pairs score: number of correctly aligned pairs divided by number of pairs in true alignment.

• Column score: number of correctly aligned columns

• Statistical significance using Friedman rank test

AACAGTAAGT_ _

AACAGTAA_ _GT

Blue: correctRed: incorrectAcc: 2/4=50%

Experimental design

• Methods compared:– Probalign– PROBCONS– MUSCLE– MAFFT

• Probalign temperature parameter trained on RV11 subset of BAliBASE 3.0.

• Default (optimized) parameters for remaining programs

• All experiments performed on CIPRES cluster at SDSC

BAliBASE 3.0

Data Probalign MAFFT Probcons MUSCLE

RV11 69.3 / 45.3 67.1 / 44.6 67.0 / 41.7 59.3 / 35.9

RV12 94.6 / 86.2 93.6 / 83.8 94.1 / 85.5 91.7 / 80.4

RV20 92.6 / 43.9 92.7 / 45.3 91.7 / 40.6 89.2 / 35.1

RV30 85.2 / 56.4 85.6 / 56.9 84.5 / 54.4 80.3 / 38.3

RV40 92.2 / 60.3 92.0 / 59.7 90.3 / 53.2 86.7 / 47.1

RV50 89.3 / 55.2 90.0 / 56.2 89.4 / 57.3 85.7 / 48.7

All 87.6 / 58.9 87.1 / 58.6 86.4 / 55.8 82.5 / 48.5

Method RV11 RV12 RV20 RV30 RV40 RV50 All

MAFFT NS < 0.005 NS NS < 0.005 NS < 0.005

Probcons 0.049 0.0233 NS NS < 0.005 NS < 0.005

MUSCLE < 0.005 < 0.005 0.008 < 0.005 < 0.005 NS < 0.005

Sum-of-pairs and column score accuracies

Friedman rank test P-values

Heterogeneous length data I

Max length /Standard dev.

Probalign MAFFT Probcons MUSCLE

500 / 100 88.4 / 56.6 88.0 / 58.0 86.7 / 51.6 81.5 / 42.5

500 / 200 88.5 / 54.6 87.0 / 51.9 87.2 / 48.9 81.9 / 42.4

1000 / 100 91.4 / 58.1 90.4 / 55.7 89.7 / 51.6 84.3 / 44.1

1000 / 200 90.7 / 55.0 89.3 / 51.4 89.2 / 48.7 83.2 / 42.5

RV40 1000 / 100 (25) 1000 / 200 (20)

92.7 / 59.393.0 / 57.3

91.0 / 54.890.8 / 52.1

89.9 / 48.290.6 / 47.6

BAliBASE datasets with maximum length and minimum devation

BAliBASE datasets with long extensions

Max length /Standard dev.

Probalign MAFFT Probcons

Heterogeneous length data II

Max length /Standard dev.

Probalign MAFFT Probcons

500 / 100 (40) 89.1 / 44.9 87.3 / 49.0 87.4 / 38.6

500 / 200 (21) 88.3 / 43.8 85.0 / 46.4 86.7 / 40.0

500 / 300 (9) 95.3 / 61.0 82.6 / 51.3 87.3 / 46.6

500 / 400 (5) 94.6 / 55.0 72.0 / 38.2 79.8 / 38.0

1000 / 100 (15) 90.2 / 43.3 82.4 / 36.9 85.4 / 27.6

1000 / 200 (12) 89.2 / 38.2 79.7 / 32.4 83.6 / 27.7

1000 / 300 (7) 94.5 / 52.8 78.3 / 42.4 83.9 / 34.6

1000 / 400 (5) 94.6 / 55.0 72.0 / 38.2 79.8 / 38.0

BAliBASE 2.0 reference 6 datasets with max length and minimum deviation