Theoretical and Experimental DNA Computation (Natural ...

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3.4 Constructive Models 61
(a)
(b)
Fig. 3.5. (a) Two strings, S and T .(b)Theresultofsplice(S, T )
stituent of the PA-Match operation is a restricted form of the splicing operation
which we denote here by Rsplice, and describe as follows. If S = S1S2
and T = T1T2, then the result of Rsplice(S, T ) is the string S1T2 provided
S2 = T1, but has no value if S2 �= T1.
Leading results of Reif [128], made possible by his PA-Match operation,
concern the simulation of nondeterministic Turing Machines and the simulation
of Parallel Random Access Machines. We can capture the spirit of his
Turing Machine simulation through the Rsplice operation as follows: the initial
tube in the simulation consists of all strings of the form SiSj where Si and
Sj are encodings of configurations of the simulated nondeterministic Turing
Machines, and such that Sj follows from Si after one (of possibly many) machine
cycle. By a configuration we mean here an instantaneous description of
the Turing Machine capturing the contents of the tape, the machine state, and
which tape square is being scanned. If the Rsplice operation is now performed
between all pairs of initial strings, the tube will contain strings SkSl where Sl
follows from Sk after two machine cycles. Similarly, after t repetitions of this
operation the tube will contain strings SmSn where Sn follows from Sm after
2 t machine cycles. Clearly, if the simulated Turing Machine runs in time T ,
then after O(log T ) operations the simulation will produce a tube containing
strings SoSf where So encodes the initial configuration and Sf encodes a final
configuration.
3.4 Constructive Models
In this section we describe a constructive model of DNA computation, based
on the principle of self-assembly. Molecular self-assembly gives rise to a vast
number of complexes, including crystals (such as diamond) and the DNA
double helix itself. It seems as if the growth of such structures are controlled,
at a fundamental level, by natural computational processes. A large body of
work deals with the study of self-organizational principles in natural systems,
and the reader is directed to [40] for a comprehensive review of these.
We concentrate here on the Tile Assembly Model, due to Rothemund and
Winfree [132]. This is a formal model for the self-assembly of complexes such