LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

Molecular Tiling and DNA Self-assembly 73
Fig. 10. Given a set of six distinct tiles whose binding sites are specific to each
pair of tile interaction described by edges in the graph GA (left), notice that G
(right) is not a covering graph for GA since it satisfies conditions (1) − (3) but it
does not satisfy (5) (see text). To see this, consider the mapping p between nodes
of G and GA which is suggested by the labels of the nodes. We want to think of
f in GA as representing a tile Tf with two distinct binding sites, one interacting
with Tc and the other with Td. Nodef1 is linked to two copies of c and node f2
is linked to two copies of d; this means that Tf1 (Tf2), having the same binding
sites as Tf, should bind to Tc1,Tc2 (Td1,Td2). But this is impossible because the
binding would require the existence of two identical sites in Tf1 (Tf2).
Given a set of tiles one would like to characterize the family of closed assemblies,
or equivalently, of covering graphs, if any. An important application is in
the solution of combinatorial problems.
Example 1. [22]. A graph G =(V,E) issaidtobe3-colorable if there is a surjective
function f : V →{a, b, c} such that if v → w ∈ E, thenf(v) �= f(w).
Imagine constructing the graph G with two kinds of molecules, one coding for
the nodes and one for the edges. Node-molecules are branched molecules, where
the number of branches is the degree of the node, and edge-molecules are twobranched
molecules. Each branch of a node-molecule has a sticky end whose
code contains information on the node of the graph that connects to it and on
a color for the node. The n branches of a same node-molecule are assumed to
have the same code. Edge-molecules have two sticky ends and their code contains
information on the origin and target nodes as well as on the colors of such
nodes. The two colors are supposed to be different.
To consider three colors in the physical realization of the graph G, one constructs
a node-molecule for each one of the three colors, together with all possible
combinations of pairs of different colors for edge-molecules.
By combining several identical copies of these molecules and ligating them,
open and possibly closed assemblies will form. Open assemblies are discharged
(this can be done with the help of exonuclease enzymes that digest molecules with
free ends) and closed assemblies, if any, ensure that the graph is 3-colorable. The
only closed assemblies that can be formed in the test tube are covering graphs.