e above equation, ߜሺݔே, ݕேሻ is the current (the last) local

t, which needs to be optimised while ∑

ߜሺݔ௜, ݕ௜ሻ

ேିଵ

௜ୀଵ

is all the

ts before the last pair of residues from two sequences. It is

that ∑

ߜሺݔ௜, ݕ௜ሻ

ேିଵ

௜ୀଵ

has been optimised. For any local alignment,

e distance equation can be re-written as below, where ݔ௠ and ݕ௠

the residues for the current local alignment while ∑

ߜሺݔ௜, ݕ௜ሻ

௠ିଵ

௜ୀଵ

r the alignment for two sequences from the first pair of aligned

ll the (m-1)^th pair of aligned residues,

݀ሺݔ^௠, ݕ^௠ሻൌ෍ߜሺݔ௜, ݕ௜ሻ൅ߜሺݔ௠, ݕ௠ሻ

௠ିଵ

௜ୀଵ

(7.5)

ose a representative (parental) sequence without an insertion or a

s expressed as

ݔൌሺݔଵ, ݔଶ, ⋯, ݔேሻ

(7.6)

a sequence can become an evolutionary sequence with the

s, or the deletions or the insertions. Suppose there are two

s for an alignment. They are ݔଵݔଶ⋯ݔெ and ݕଵݕଶ⋯ݕே. There

e incremental alignments in the forward propagation stage.

a local alignment between two sequences x and y has reached a

hat two partially or previously aligned sequences are

௠ ^{ሺ݉൏ܯሻ and ݕ}ଵ^ݕଶ^⋯ݕ௡ ^{ሺ݊൏ܰሻ. There are therefore}

g residues from two sequences x and y as the alignment

es. Two sequences with the remaining residues are denoted by

ାଶ^⋯ݔெ ^{and ݕ}௡ାଵ^ݕ௡ାଶ^⋯ݕே^{. The first remaining residue of}

x is ݔ௠ାଵ and the first remaining residue of sequence y is ݕ௡ାଵ.

the likely candidates to be picked up for the next alignment.

, a deletion or an insertion may happen anywhere in two

s according to the evolutionary theory. This means that there are

nment choices. The first is to align ݔ௠ାଵ with ݕ௡ାଵ. The second

n ݔ௠ାଵ with a gap. The third is to align ݕ௡ାଵ with a gap. The