1. Expected accuracy sequence alignment
Usman Roshan

2. 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:
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

3. 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
ACCT T AC CTT
Score = Score = 0.9
Trivial cost matrix: match=+1, mismatch=0, gapopen=-2, gapextension=-0.1

4. Affine penalty recursion
M(i,j) denotes alignments of x1..i and y1..j ending with
a match/mismatch. E(i,j) denotes alignments of x1..i
and y1..j such that yj is paired with a gap. F(i,j) defined
similarly. Recursion takes O(mn) time where m and n
are lengths of x and y respectively.

5. 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.

6. 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

7. 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
Do et. al., Genome Research, 2005

8. Example for expected accuracy
True alignment
AC_CG
ACCCA
Expected accuracy=( )/4=1
Estimated alignment
ACC_G
Expected accuracy=( ) ~ 0.75

9. 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

10. Partition function posterior probabilities
Standard alignment score:
Probability of alignment (Miyazawa, Prot. Eng. 1995)
If we knew the alignment partition function then

11. Partition function posterior probabilities
Alignment partition function (Miyazawa, Prot. Eng. 1995)
Subsequently

12. Partition function posterior probabilities
More generally the forward partition function matrices are calculated as

13. Partition function matrices vs. standard affine recursions

14. Posterior probability calculation
If we defined Z’ as the “backward” partition function matrices then

15. 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.

16. 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.

17. Probalign 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
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.).

18. 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.

19. 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
Blue: correct
Red: incorrect
Acc: 2/4=50%
AACAGT
AA_ _GT
AACAGT
AAGT_ _

20. 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

21. BAliBASE 3.0 Sum-of-pairs and column score accuracies 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
Friedman rank test P-values
Method
RV11
RV12
RV20
RV30
RV40
RV50
All
MAFFT NS
< 0.005
Probcons
0.049
0.0233
MUSCLE
0.008

22. Heterogeneous length data I
BAliBASE datasets with maximum length and minimum devation
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
BAliBASE datasets with long extensions
Max length /
Standard dev.
Probalign
MAFFT
Probcons
RV / 100 (25)
1000 / 200 (20)
92.7 / 59.3
93.0 / 57.3
91.0 / 54.8
90.8 / 52.1
89.9 / 48.2
90.6 / 47.6

23. Heterogeneous length data II
BAliBASE 2.0 reference 6 datasets with max length and minimum deviation
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)