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(1
i
j
≤| α |
) is denoted by
α [
i
,
j
]
and is called a substring of
α
. In particular, it
is called a prefix if i
=
1, and a suffix if j
= | α |
. A string with prefix
α
and suffix
β
is
also denoted by
α ... β
.The complementation function, denoted by the superscript
by A c
T c
C c
G c
Γ by
=
,
=
,
=
,
=
c ,isdefinedon
Γ
T
A
G
C . It extends naturally to
c
c
c
c . Another operation on
Γ is the reverse
= λ
( αβ )
= α
the conditions
λ
and
β
operation rev such that, for every X
Γ
, rev
(
X
)=
X , rev
( α
X
)=
Xrev
( α )
,and
rev
.
Reversing and complementation operations are involutive and commute ,thatis:
( λ )= λ
rev
(
rev
( α )) = α
c
c
( α
)
= α
and
c
c
rev
( α
)=(
rev
( α ))
.
The mirroring of
α
is defined by:
c
mir
( α )=
rev
( α
)
as ¯
and is an involutive operation. We abbreviate mir
.
In DNA molecules, an intrinsic concatenation verse is given which goes from the
Phosphate to the Oxydryl (that respectively correspond to Carbon 5 and 3 positions
in the sugar DNA backbone). This verse is symbolically denoted by an arrow at one
side of non-null strings
( α )
α
α
,asin:
α
that can be seen as an indication of the extremity where the Oxydryl terminal is
located (while Phosphate is at the other extremity). However, when we omit the
arrow, the usual reading from left to right is assumed, which corresponds to the
5
3 verse.
An exact pairing (symmetric) relation
Γ such that:
||
is defined over
α || β
if
β =
mir
( α )
.
respectively. Let
h hybr be a non-null length that we call hybridization threshold (it represents a bi-
ological parameter we leave unspecified). If
Let
γ
be the longest string such that
α
and
β
include
γ
and ¯
γ
| γ | >
h hybr , then we say that
α
and
β
pair by hybridizing on
γ
(or shortly, that they hybridize ) and write:
α ] γ [ β .
In this case a double DNA string composed of
α
and
β
is defined, that will be
denoted by:
α
rev
( β ) .
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