LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

220 Nataˇsa Jonoska, Shiping Liao, and Nadrian C. Seeman
machine [24]. Several successful experiments performing computation have been
reported, most notably the initial successful experiment by Adleman [1] and the
recent one from the same group solving an instance of SAT with 20 variables
[3]. Successful experiments that have confirmed computation such as the binary
addition (simulation of XOR) using triple cross-over molecules (tiles) have been
reported in [16]. In [6] a 9-bit instance of the “knight problem” has been solved
using RNA and in [8] a small instance of the maximal clique problem has been
solved using plasmids. Theoretically it has been shown that by self-assembly of
three dimensional graph structure many hard computational problems can be
solved in one (constant) biostep operation [10,11].
Nano Devices. Based on the B-Z transition of DNA, a nano-mechanical device
was introduced in [17]. Soon after, “DNA fuel” strands were used to produce
devices whose activity is controlled by DNA strands [26,28]. The PX-JX2 device
introduced in [26] has two distinct structural states, differing by a half-rotation;
each state is obtained by addition of a pair of DNA strands that hybridizes with
the device such that the molecule is in either the JX2 state or in the PX state.
We consider it a natural step at this point to use the PX-JX2 device and
DNA self-assembly to develop a programmable finite state machine. A simulation
of a finite state automaton that uses duplex DNA molecules and a restriction
endonuclease to recognize a sequence of DNA was reported in [2]. Unfortunately
this model is such that the DNA representing the input string is “eaten up”
during the process of computation and no output (besides “accept-reject”) is
produced.
In this paper we exploit the idea of using DNA tiles (TX molecules) that
correspond to Wang tiles to simulate a finite state machine with output, i.e., a
transducer. Theoretically we show that by composition of transducers we can
obtain all recursive functions and hence all computable functions (see Sections
3, and 4). This result is not surprising considering the fact that iterations of
a generalized sequential machine can simulate type-0 grammars [14,15]. However,
the explicit connection between recursive functions and Wang tiles has not
been described before. Also, although the dynamics of transducers have been
considered [22], the composition of transducers as means to obtain class of recursive
functions represents a new way to tie together recursive functions, tiles
and transducers. Moreover, in this paper, the purpose of Wang tiles is not to
tile the plane, as they usually appear in literature, but merely to facilitate our
description of the use of DNA TX molecules to simulate transducers and their
composition. In particular, we assume the existence of “boundary colors” at some
of the tiles which do not allow tiling extension beyond the appearance of such a
color. The input tiles contain input “colors” on one side and a boundary color
on the opposite side, such that the computation is performed in the direction of
the input color. This is facilitated with composition colors on the sides (Section
4).
We go beyond two dimensions in this paper. We show how PX-JX2 DNA
devices (whose rotation uses three dimensional space) can be incorporated into
the boundary such that the input of the transducer can be programmed and