Theoretical and Experimental DNA Computation (Natural ...

3.5 Membrane Models 63
tiles within the system is depicted in Fig. 3.6a: we have four rule tiles (labelled
either “1” or “0”) r0,r1,r2,r3, two border tiles (labelled “L” and “R”), and
a seed tile (“S”). Sides depicted with a single line have a binding strength of
1; those with a double line have a binding strength of 2. Thick lines depict a
binding strength of 0.
We impose the following important restriction: a tile may only be added
to the assembly if it is held in place by a combined binding strength of 2. In
addition, a tile with labelled sides (i.e., a rule tile) may only be added if the
labels on its side match those of its proposed neighbor. It is clear that two rule
strands in isolation cannot bind together, as the strength of the bond between
them can only equal 1. Crystallized by the seed tile, a “scaffold” of L and R
tiles emerges to support the assembly, a structure resulting from the binding
strengths associated with their sides. Imagine the situation at the beginning
of this process, where the assembly consists of one S, one L, and one R tile.
The only rule tile that is labelled to match the sides of its neighbors is r2, so
it is added to the complex.
The assembly gradually grows right to left and row by row, with the tiles
in row n>0 representing the binary integer n. The growth of the assembly
is depicted in Fig. 3.6b, with spaces that may be filled by a tile at the next
iteration, depicted by the dashed lines. Note that the growth of the complex is
limited only by the availability of tiles, and that the “northern” and “western”
sides of the assembly are kept exposed as the assembly grows. Notice also how
some row n cannot grow left unless row n − 1 has grown to at least the same
extent (so, for example, a rule tile could not be added to row 2 at the next
iteration, because the binding strength would only equal 1).
It has been shown that, for a binding strength of 2, one-dimensional cellular
automata can be simulated, and that self-assembly is therefore universal [162].
Work on self-assembly has advanced far beyond the simple example given,
and the reader is directed to [163] for a more in-depth description of this. In
particular, branched DNA molecules [142] provide a framework for molecular
implementation of the model. Double-crossover molecules, with the four sides
of the tile represented by “sticky ends”, have been shown to self-assemble into
a two-dimensional lattice [163]. By altering the binding interactions between
different molecules, arbitrary sets of tiles may be constructed.
3.5 Membrane Models
We have already encountered the concept of performing computations by the
manipulation of multi-sets of objects. This style of programming is well established
in computer science, and Petri nets [129] are perhaps its best known
example. Biological systems have provided the inspiration for several multi-set
manipulation models, and in this section we describe a few of them.
As we have seen, abstract machines such as the Turing Machine or RAM
are widely used in studying the theory of sequential computation (i.e, com-