Theoretical and Experimental DNA Computation (Natural ...

5.5 Evaluation of Adleman’s Implementation 115
mean that the experiment will not successfully scale up. We consider these
issues in later sections.
5.5 Evaluation of Adleman’s Implementation
We describe later how the various multi-set operations described in the previous
section may be realized thorough standard DNA manipulation techniques.
However, it is convenient at this point to emphasize two impediments to effective
computation by this means. The first hampers the problem size that
might be effectively handled, and the second casts doubt on the potential for
biochemical success of the precise implementations that have been proposed.
Naturally, the strings making up the multi-sets are encoded in strands
of DNA in all the proposed implementations. Consider for a moment what
volume of DNA would be required for a typical NP-complete problem. The
algorithms mentioned earlier require just a polynomial number of DNA manipulation
steps. For the NP-complete problems there is an immediate implication
that an exponential number of parallel operations would be required
within the computation. This in turn implies that the tube of DNA must contain
a number of strands which is exponential in the problem size. Despite the
molecular dimensions of the strands, for only moderate problem sizes (say, n
∼ 20 for the Hamiltonian Path problem) the required volume of DNA would
make the experiments impractical. As Hartmanis points out in [76], if Adleman’s
experiment were scaled up to 200 vertices the weight of DNA required
would exceed that of the earth. Mac Dónaill also presents an analysis of the
scalability of DNA computations in [53], as do Linial and Linial [97], Lo et al.
[101], and Bunow [38].
We note that [19] has described DNA algorithms which reduce the problem
just outlined; however, the “exponential curse” is inherent in the NP-complete
problems. There is the hope, as yet unrealized (despite the claims of [24])
that for problems in the complexity class P (i.e. those which can be solved
in sequential polynomial time) there may be DNA computations which only
employ polynomial sized volumes of DNA.
We now consider the potential for biochemical success that was mentioned
earlier. It is a common feature of all the early proposed implementations that
the biological operations to be used are assumed to be error free. An operation
central to and frequently employed in most models is the extraction of DNA
strands containing a certain sequence (known as removal by DNA hybridization).
The most important problem with this method is that it is not 100%
specific, 1 and may at times inadvertently remove strands that do not contain
the specified sequence. Adleman did not encounter problems with extraction
because in his case only a few operations were required. However, for a large
problem instance, the number of extractions required may run into hundreds,
1 The actual specificity depends on the concentration of the reactants.