Theoretical and Experimental DNA Computation (Natural ...

144 5 Physical Implementations
1 = 0 plus carry 1 in binary). The DNA sequences used represent not only
binary strings but also allow for carries and the extension of DNA strands to
represent answers. Guarnieri et al. use sequences that encode a digit in a given
position and its significance, or position from the right. For example, the first
digit in the first position is represented by two DNA strands, each consisting of
a short sequence representing a “position transfer operator”, a short sequence
representing the digit’s value, and a short sequence representing a “position
operator.”
DNA representations of all possible two bit binary integers are constructed,
which can then be added in pairs. Adding a pair involves adding appropriate
complementary strands, which then link up and provide the basis for
strand extension to make new, longer strands. This is termed a “horizontal
chain reaction”, where input sequences serve as templates for constructing
an extended result strand. The final strand serves as a record of successive
operations, which is then read out to yield the answer digits in the correct
order.
The results obtained confirmed the correct addition of 0 + 0, 0 + 1, 1 +
0, and 1 + 1, each calculation taking between 1 and 2 days of bench work.
Although limited in scope, this experiment was (at the time) one of the few
experimental implementations to support theoretical results.
The tendency of DNA molecules to self-anneal was exploited by Sakamoto
et al. in [135] for the purposes of solving a small instance of SAT. The authors
encode the given formula in “literal strings” which are conjunctions of the
literals selected from each SAT clause (one literal per clause). A formula is
satisfiable if there exists a literal string that does not contain any variable
together with its negation. If each variable is encoded as a DNA subsequence
that is the Watson-Crick complement of its negation then any strands containing
a variable and its negation self-anneal to form “hairpin” structures.
These can be distinguished from non-hairpin structure-forming strands, and
removed. The benefit of this approach is that it does not require physical
manipulation of the DNA, only temperature cycling. The drawback is that it
requires 3 m literal strings for m clauses, thus invoking once again the scalability
argument.
Algorithmic self-assembly (as described in Chap. 3) has been demonstrated
in the laboratory by Mao et al. [103]. This builds on work done on
the self-assembly of periodic two-dimensional arrays (or “sheets”) of DNA
tiles connected by “sticky” pads [161, 163]. The authors of [103] report a
one-dimensional algorithmic self-assembly of DNA triple-crossover molecules
(tiles) to execute four steps of a logical XOR operation on a string of binary
bits.
Triple-crossover molecules contain four strands that self-assemble through
Watson-Crick complementarity to produce three double helices in roughly a
planar formation. Each double helix is connected to adjacent double helices
at points where their strands cross over between them. The ends of the core
helix are closed by hairpin loops, but the other helices may end in sticky ends