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A substantial improvement in the efficiency and reliability of PCR was obtained
when the process was realized by means of heat-stable DNA polymerase, such
as Taq polymerase (an enzyme isolated from the bacterium Thermus aquaticus).
Namely, in this way the three phases realizing the denaturation, primer hybridiza-
tion, and polymerase extension can be realized by alternately heating and cooling
the PCR sample to a defined series of temperature steps (depending on the length
and nucleotide composition of the hybridization regions of the involved strands).
Polymerase Chain Reaction, which is a milestone in DNA recombinant technol-
ogy [1], shows many interesting combinatorial properties. In fact, it is in some cases
a complicated process, and, when it is used in non-standard ways, it yields very
complex behaviors. Very often, anomalies are ascribed to experimental noise, but,
if we frame PCR within a rigorous symbolic notation, then non-trivial combinato-
rial aspects appear and, under suitable hypotheses, a formula can be derived which
describes the general form of sequences that are exponentially amplified. This ap-
proach is not of merely mathematical interest. On the contrary, it can suggest new
methods that enjoy biological relevance for in vitro DNA manipulation. In fact,
starting from DNA computing problems [46], we investigated specific methods for
DNA extraction and recombination [28, 30, 26]. In these attempts, where theoretical
issues were supported by the experiments, we realized that a special kind of PCR,
called
Cross Pairing PCR
or
XPCR
for short, can be the basis for new algorithms
that solve a wide class of DNA extraction and recombination problems. Within a
basic DNA symbolic notation, we will show a crucial computational property of
Polymerase Chain Reaction, we call
PCR Lemma
, which suggested to us the idea of
Cross Pairing PCR.
Ta b l e 2 . 4
PCR Algorithm
PCR
(
P
,
n
)=
let
Type
(
P
)=
{<
γ
...
δ
>,
γ
,
¯
δ
},
n
integer;
input
P
;
for
i
=
1
,
n
do
begin
P
:
=
denature
(
P
)
;
P
:
=
hybridize
(
P
)
;
P
:
=
ext end
(
P
)
;
end
;
output
P
.
the DNA pool obtained by applying to the pool
P
the procedure of Table 2.4 (the usual PCR process). The parameter
n
denotes the
number of the fundamental PCR steps (also called PCR cycles, which we avoid
We denote by
PCR
(
P
,
n
)
