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<
ϕ
γψ
>
which produces the DNA string
depicted in Fig. 2.36, where
γ
is the
ϕ
,
ψ
are the strings
,
¯
and
¯
substring of
ϕ
γ
of
ψ
,and
ϕ
ψ
without the substrings
γ
and
¯
γ
, respectively.
Fig. 2.35
Two DNA single strings that overlap
Fig. 2.36
The overlap concatenation of two strings
<
ϕ
>
and
<
ψ
>
from the (overlap)
hybridization displayed in Fig. 2.35
Lemma 2.1 (PCR
Lemma).
Let
P
be
a
DNA
pool
such
that
Type
(
P
)=
¯
{
α
rev
(
β
)
,
γ
,
δ
}
and let:
¯
γ
][
δ
,
γ
][
β
,
¯
δ
][
α
.
In the following cases:
¯
i)
γ
][
ext
(
δ
,
α
)
,
¯
ii)
δ
][
ext
(
γ
,
β
)
,
¯
iii) ext
(
γ
,
β
)
and ext
(
δ
,
α
)
overlap,
PCR
exponentially amplifies blunt strings, where primers hybridize at extremal
regions of their single strands, having the following forms, respectively:
(
P
,
n
)
¯
¯
ext
(
δ
,
ext
(
γ
,
ext
(
δ
,
α
)))
(2.2)
¯
ext
(
γ
,
ext
(
δ
,
ext
(
γ
,
β
)))
(2.3)
¯
<
ext
(
γ
,
β
)
> <
ext
(
δ
,
α
)
>.
(2.4)
Proof.
By the hypotheses, in the cases i) and ii) the extensions (2.2) and (2.3) have
to be proper (giving results different from the primers). It is easy to check that they
are blunt double strings including the primers in the extremal parts of their strands,
therefore they are seeds of exponential amplifications. In the last case iii), the sit-
uation of Fig. 2.35 occurs, then the overlap concatenation (2.4) (realized by the
