03-511/711, 15-495/856 Course Notes - Sept 12, 2006

Pairwise alignment continued

Alignment algorithms

The dynamic programs for sequence alignment compute a matrix a[i,j], which gives the scores of the optimal alignments of all prefixes. These algorithms have four components:

• Initialization of the first row and column of a[i,j].
• A recurrence relation for a[i,j], i,j > 1.
• Determination of the score of the optimal alignment from the matrix a[i,j] in o(m-n) time.
• Trace back through the alignment matrix to obtain the optimal alignment in o(m+n) time.

The details of each of these steps are what differentiate global, semi-global and local alignment.

Global alignment with similarity scoring

• p(x,y): similarity of x and y
• p(x,"_"): gap cost
• Score of alignment   =   ∑(p(s'[i], t'[i])), i  =  1..l
  
  
• A simple similarity scoring function that treats all characters equally:
  • p(i, i)   =   M
  • p(i, j)   =   m
  • p(i, "_")   =   g
  
  
• We require that 2g < m < M.   If we allow 2g ≥ m then there will be no substitutions.

Under this simple scoring function, the dynamic programming algorithm for global alignment has the following initialization and recursion steps:

• Initialization
  • a[0,s[i]]  =  a[i-1,0] + g
  • a[t[j],0]  =  a[0,j-1] + g
  
  
  
• Recurrence relation:

  a[i, j]  =  max {	a[i,j-1] + g
  	a[i-1, j-1] + p(i,j)
  	a[i-1,j] + g

  
  

Semiglobal Alignment

Semiglobal alignment is global alignment with no end gap penalties. Some applications include:

• Finding overlaps between fragments for sequence assembly.
• Aligning cDNA's or EST's with genomic DNA to identify gene structure.

The global dynamic programming algorithm can be modified for semi-global alignment as follows:

• Initialization
  • initialize the first row or the first column of a[i,j] to zero, to avoid leading gap penalties.
  
  
• Recurrence relation
  • Same as global.
  
  
• To avoid trailing gap penalties, the score of the optimal semiglobal alignment is MAX_i a[i,n] or MAX_j a[m,j]
  
  
• To avoid trailing gap penalties, start the trace back at the cell in the last row (or column) that with maximum score.    To avoid leading gap penalties, end the trace back any where in the first row (or column) to optimize the score.
  

Local Alignment

• Initialize the first row and column to zero: s[i,0] = t[0,j] = 0 for all i and j
• Recurrence
  
  

  a[i,j]= max  {	a[i-1,j] + g
  	a[i-1,j-1] + p(s[i], t[j])
  	a[i,j-1] + g
  	0

• The score of the optimal alignment is max{ a[i,j]}, where the maximum is taken over all i and all j.
• Trace back starting at a^*[i,j], the cell corresponding to the maximum score. End the trace back when the score reaches zero

Note that :

• There can be more than one optimal alignment
• Suboptimal alignments may be of interest
• M > m > 2g
• Global and semi-global alignments can use distance or similarity functions.
• Local pairwise alignment requires that
  • The scoring function be a similarity function.
  • The similarity matrix, p[i,j], must contain at least one positive value.
  • The expected random alignment score must be negative.