Three Dimensional DNA Structures in Computing
Natasa Jonoska, Department of Mathematics
Stephen A. Karl, Department of Biology
Masahico Saito, Department of Mathematics
University of South Florida
Tampa, Florida 33620
Abstract
We show that 3-dimensional graph structures can be used for solving computational problems with DNA
molecules. Vertex building blocks consisting of k-armed (k = 3 or k = 4) branched junction molecules are
used to form graphs. We present procedures for the 3-SAT and 3-vertex-colorability problems. Construction
of one graph structure (in many copies) is sucient to determine the solution to the problem. In our proposed
procedure for 3-SAT, the number of steps required is equal to the number of variables in the formula. For
the 3-vertex-colorability problem, the procedure requires a constant number of steps regardless of the size of
the graph.
Keywords: 3-dimensional structures, DNA-computing, graphs.
1 Introduction
The eld of practical DNA computing opened in 1994 with Adleman's paper (1994), in which a laboratory experiment involving DNA molecules was used to solve a small instance of the Hamiltonian
Path problem. In the follow up paper by Lipton (1995), it was demonstrated how a large class of
NP -complete problems could be solved by encoding the problem in DNA molecules. In particular,
Lipton showed how another famous NP problem, the \satisability" problem (SAT), and subsequently any other NP problem, could be encoded and solved using DNA molecules. Shortly after,
several authors have suggested applications of DNA methodology for computational purposes (for
example Landweber and Baum 1998; Wood in press; Koza et al. 1997; 1998) In many algorithms,
the general approach is to treat DNA molecules as linear strings where much of the information
content is encoded in the order of nucleotides that make up the DNA. Recently reported research
(Seeman at al. 1989; 1991; 1995; 1998) however, demonstrates that it is possible to form higher
order, three dimensional (3D) structures with DNA molecules, and in (Jonoska et al. in press)
it is shown that 3D structures could be used to solve the Hamiltonian Path problem in constant
number of steps regardless of the size of the graph. This showed that the use of 3D DNA structures
can reduce the time and steps needed to identify a solution. Such an approach also is suggested in
(Winfree et al. 1998) where 2-dimensional structures are used.
In this paper we propose solutions to two other famous NP -complete problems, the satisability
(SAT) and the 3-vertex-colorability problems. In both cases, k-armed branched junction molecules
are used to represent the k-degree vertices of the graph. These molecules are much simpler than
previously used vertex building blocks (Jonoska et al. in press). Their construction is somewhat
understood (see Seeman et al. 1989; 1998; Winfree et al. 1998) and consequently, 3D construction
of a graph using these molecules as vertex building blocks might be feasible. For both problems
(SAT and 3-vertex-colorability) we give procedures that use 3D graph structures. In the rst case
the number of steps required for the algorithm is equal to the number of variables in the formula.
This paper has been prepared for the Proceedings of the 4th DNA Based Computing workshop held in
Philadelphia, June 15-19, 1998. A preliminary version of this paper appeared in Paun (1998).
1
In the second case, the procedure requires constant number of laboratory steps, regardless of the
size of the graph. The vertex building blocks also are biochemically characterized (Seeman et al.
1998) and are known to be stable. In this paper we present our procedures under the assumption
that the proposed graph structures are constructed by DNA molecules. At this time, however, we
do not have specic experimental results and, therefore, are unable to discuss the feasibility of these
procedures. The purpose of this paper is to propose potential uses of 3D structures in the eld of
DNA computing and to show that DNA structures similar to those already constructed (Seeman
et al. 1998) could, theoretically, be used in DNA computing.
We start the paper with the description of the SAT problem and we describe our approach for
solving SAT. The third section contains the description of the 3-vertex-colorability problem and
our procedure for its solution. We end our paper with a discussion of potential problems of the 3D
approach in computing.
2 The Satisability (SAT) Problem
2.1 Denition of SAT. Let A = fa1 ; a2; : : :g be a set of boolean variables. A clause is a formula
C = b1 _ b2 _ _ b where _ is the logical \or" and each b is a variable in A or a complement to
a variable in A. The complement of a 2 A is denoted by a. A logical formula is a formula of the
form:
= C 1 ^ C2 ^ ^ C
where each C is a clause.
The satisability problem (SAT) asks whether for a given logical formula, , there is an assignment of f0; 1g to the variables in that would assign to the value of 1.
It is convenient to write a \+" instead of an \_" and a \" or a simple concatenation instead
of an \^". In that case the formula becomes = C1 C2 C .
i
k
r
i
r
Consider the example:
= (x + y)(x + y + z)(x + y + z):
()
This formula () has value 1 for the assignments (x; y; z ) 2 f(1; 1; 1); (1; 1; 0); (0; 1; 1); (0; 1; 0)g and
0 for any other assignment.
The SAT problem is one of the standard examples of NP -complete problems (Davis et al.
1995) and all known solutions to these problems require (in the worst case) an exponential increase
in the number of steps with increasing size of the formula. Identifying a solution for even a
moderately simple problem can, therefore, require a prohibitively large number of steps. In fact,
the excitement over DNA as a computational tool rests on the ability to perform in parallel some
of the computational steps. Thus, the number of steps to identify a solution does not increase
exponentially with the size of the graph.
2.2 Solving SAT by constructing 3D graphs with DNA. Lipton, (1995) shows that SAT
is equivalent to the \contact network" problem. The contact network problem is a graph with two
distinguished vertices: the source s and the target t. Each edge in the graph is labeled with x or x
where x is a variable. An edge is considered as connected if the value of the variable assigns value
1 to the edge (if the edge is labeled x then the value x = 0 keeps the edge connected). The SAT
problem for contact network asks whether there is an assignment of the variables that would allow
the source to be connected to the target.
Any logical formula can be transformed in a contact network problem. If = + where and are simpler formulas with contact networks G1 and G2 then the contact network for is
obtained by placing G1 and G2 in parallel (Fig. 1a). If = , then the contact network for is
obtained by placing G1 and G2 in series (Fig. 1b). Similarly, the contact network for the formula
() is presented in Fig. 2.
2
G1
G1
G2
G2
(a)
(b)
Figure 1: Contact network for (a) = + and (b) = When every clause (C ) in the formula has at most 3 variables, the restricted SAT problem,
3-SAT, also is NP -complete (Davis et al. 1994) (the number of variables that appear in the formula
is not necessarily 3). Hence we can concentrate on 3-SAT problems in which the corresponding
contact network problem will contain vertices of degree at most 4.
i
x
y
source
x
x
y
y
z
z
target
Figure 2: Contact network for formula ().
The formula () and its contact network in Fig. 2 will be used as an example in our general
discussion.
Now we show how contact networks can be constructed by DNA to solve 3-SAT problems. First
we list the molecular building blocks for the contact network graph.
Source and target. The endpoints of the contact network (the source and the target) are
presented with a hairpin structured molecule as depicted in Fig. 3. Hydrogen bonds between
the anti-parallel, complementary Watson-Crick (WC) bonds are depicted as dotted segments
between the strands. Polarity of the DNA strands are indicated with arrowheads being placed
at the 3 end. For simplicity, the 3-dimensional helical nature of each molecule is not presented
in any of the gures.
Vertex blocks. Each vertex in the contact network has, at most, degree 4. One edge is a
connection between clauses and the other three are labeled with variables that are included
in a clause as depicted in Fig. 3.
0
Source
Vertex building block
Target
Figure 3: Molecules encoding the source, the target and the vertices for 3-SAT.
If a clause in the formula contains only two variables then 3-armed branched molecules are
3
used for the vertices. In the case when the clause has only one variable then a single double
stranded DNA molecule representing the variable is used.
Edges. Edges of the contact network graph are represented by double stranded DNA molecules
ligated to the vertices. The 3 ends of the DNA strands in the edge molecule end with single
stranded segments of 20 to 30 nucleotides in length that are variable specic. For example,
the single stranded segments of the edge encoding x and encoding x are distinct. The edge
molecules have complementary sequence tails to the corresponding vertex blocks such that
compatible vertex and edge blocks can form WC paired double-stranded DNA segments and
be joined together.
The central part of an edge molecule encodes a site that can be cleaved by a restriction
enzyme. Dierent restriction enzymes are used for dierent variables and complements of the
variables. Hence, if k variables appear in a formula, 2k restriction enzymes are used. For a
variable x, the restriction enzymes that cleave the x-edge and x-edge will be referred to as
(x = 0)-enzyme and (x = 1)-enzyme respectively.
Caps. Caps are used in the proposed solution to \cap o" the cleaved ends of edges. These
caps are not used for building a given graph. The need for this step is explained in the next
section. Caps have the same hairpin shape as sources and targets. If a restriction site used
in an edge consists of the sequence h and k as overhangs, then the corresponding (two) caps
have overhang sequences h and k, respectively.
0
To form connections in a graph, vertex and edge blocks are combined (along with the source and
target blocks) and their compatible ends are allowed to form double-stranded DNA. Once formed,
the edges are locked together by sealing all open \nicks" in the DNA strands with DNA ligase.
Figure 4 shows a formation of the contact network graph for the formula (). In every such graph
structure, two vertex blocks together with the edges between the vertex blocks represent a clause.
x
y
x
y
x
y
z
z
Figure 4: DNA contact network for formula ().
2.3 Solving the 3-SAT problem:
1. Combine multiple copies of the vertex, edge, source, and target molecules in a single tube.
Allow them to hybridize and then be ligated.
2. Remove partially formed graphs that have open ends. This could be done with an exonuclease
enzyme.
3. For each variable x in the formula repeat the following:
(a) Divide the mix into two tubes.
(b) In the rst tube, add (x = 0)-enzyme and in the second add (x = 1)-enzyme.
(c) Add cap building blocks to both tubes and allow them to hybridize and then be ligated.
This step is depicted in Fig. 5.
(d) Combine the tubes in a single mix.
4
(a)
(b)
(c)
(d)
Figure 5: Capping o cleaved edges.
4. Using PCR primers complementary to the source and the target molecules, separate molecules
that start with source and end with target. If any molecules are recovered, the connection
from source to target is established and 3-SAT problem has a solution.
2.4 Proposition. Let A be the set of molecules that contain a single stranded molecule connecting
both the \source" and the \target" molecules present in the mix after steps 1, 2 and 3 of the
procedure. Then A 6= ; if and only if there is a solution to the 3-SAT problem.
Proof. Let be a logical formula associated with a contact network. Suppose A 6= ;. Then there
is a single stranded molecule connecting the source and target molecule. This means that for every
pair of vertices within a clause, there is an edge connecting them. Such an edge was not cleaved in
Step 3, therefore there is an edge which received the value 1 (the variable x 2 which received 1
or the variable x which received 1). Furthermore, Step 3 ensures that the variable x receives the
value 1 (not cleaved) if and only if x receives 0 (cleaved), and vice versa. Therefore the existence of
a molecule connecting the source and the target implies the assignment of 0 and 1 to the variables
that gives the value 1 to . Thus the 3-SAT problem has a solution.
Conversely, if 3-SAT has a solution, then there are values for the variables that give each of the
clauses value 1. This means that two adjacent vertices within a clause in the corresponding contact
network problem are connected by an edge building block. Note that when an edge is not cleaved,
then it connects the two vertices by a single stranded segment, because the cleaved edges are closed
by cap building blocks. Figure 5 illustrates two vertices that are connected by segments through
the middle and the bottom edges in (Fig. 5 d) after the capping process Step 3 (c) (Fig. 5 a - c).
Therefore for the value 1, the corresponding edge block is not cleaved, and two vertices are
connected by a single strand. Hence the target and the source molecules after Step 2 are connected
by a single stranded DNA string. Thus A 6= ;. 2
In the proof we assumed that only the original graph was constructed. By our construction,
however, the DNA graphs obtained in Step 1 are the original graph as well as its covering spaces.
We will explain this point in Section 4.1.
It is important to note that none of the steps in the procedure depend on the number of vertices
or edges in the graph. The procedure
only on the number of the variables in the formula.
depends
n
In fact, from n variables there are k choices of variables in a clause of k variables and there
are 2 clauses for each choice of k variables. Hence, for a 3-SAT problem with n variables, the
length of the formula can be as long as 34 n(n ? 1)(n ? 2), i.e. a cubic polynomial of n.
In Lipton's algorithm (1995), all possible paths from the source to the target vertices are formed
and then the right answer is extracted. In the above procedure, all possible paths from the source
to the target also are formed. This is done not by obtaining exponentially more distinct molecules,
rather by constructing the graph itself which contains all possibilities. The connections from the
source molecule to the target molecule that remain after Step 3 are solutions to the problem.
k
5
g
1
g
2
g
3
b3 b2 b 1
c
c
c
1
2
a1 a 2 a 3
2-armed
3
e
d
d
d
1
3
3
h2 h 1
y
3
e
2 e
1
4-armed
2
3-armed
h
f f f
1 2 3
Vertex building blocks
x
i3
i
2
i
1
1
x
2
x
y y
3 2 1
3
Edge building block
Figure 6: Building blocks for vertices and edges.
However, since the number of the graph structures is divided by 2 with each repetition of Step 3
(a), we need at least 2n copies (n is the number of variables) of the graph. This will increase the
volume or weight of DNA needed concomitant with the size of the graph.
3 The 3-vertex-colorability problem
3.1 Denition of 3-vertex-colorability. Let G be a graph with vertices V and edges E . The
graph G is 3-vertex-colorable if there is a surjective (onto) function f : V ! fa; b; cg such that if
two vertices v; w 2 V are adjacent (connected by an edge) then f (v) 6= f (w). The n?colorbility is
dened similarly.
This is an N P -complete problem which has previously been addressed in the DNA computing
literature (see Adleman 1996; Amos et al. 1998 for example). As with other algorithms, the full
power of the three dimensional feature of the problem has not been explored. Here we show how
a 3D graph structure of DNA could be used to solve the problem. In particular, we show that the
whole graph can be constructed by DNA if and only if a solution to the problem exists.
3.2 Construction of graphs by DNA.
Building blocks for the vertices. As in the previous application, if a vertex has degree k
then a k-armed branched molecule is used for its building block. Examples of 2-, 3-, and
4-armed branched molecules are presented in Fig. 6. The 30 ends of the k-armed branched
molecules end with single stranded extensions. These extensions are 30 to 45 nucleotides long
and consist of three parts each 10 to 15 nucleotides long. The rst and the third parts (for
example x1 and x3 where x 2 fa; : : : ; ig in Fig. 6) are specic encodings for the edge that is
represented by the given arm of the molecule. The middle part of the encoding is the same
for all arms of the vertex molecule and represents the color of the vertex. For each vertex
three blocks are needed (each corresponding to one of the three possible colors of the vertex).
Edges. Each edge is a regular double-stranded molecule. The 30 ends of the molecule end with
single stranded segments that are complementary to the corresponding \arm" of the vertex
that is incident to the edge. Hence, the rst and the third part of the encoding at the 30 end
are x1 and x3 (see Fig. 6) which are WC-complementary to x1 and x3 (x 2 fa; : : : ; ig) of the
corresponding \arm" of the incident vertex molecule. The middle part of the encoding x2
is complementary to the color of the incident vertex, but here, the color sequence x2 at one
30 end is dierent from the color sequence y2 at the other 30 end.
For each edge we have exactly six double stranded molecules, each representing a pair of
distinct colors at the endpoints of the edge molecule.
6
To form the graph, all edge molecules and all vertex building blocks are combined and their
compatible ends are allowed to form double-stranded DNA. Once formed, the edges are locked
together by sealing all open \nicks" in the DNA strands with DNA ligase. Three-dimensional DNA
structures that do not contain open ends are referred to as graph structures (Fig. 7).
3.3 Proposition. For a given graph G, a graph structure can be formed by vertex building blocks
and edge molecules if and only if G is 3-vertex-colorable or 2-vertex-colorable.
Proof: The proof follows directly from the encodings of the single stranded 30 ends of the edge
molecules and the vertex building blocks. If a graph structure is formed, then the endpoints of
each edge molecule are \colored" distinctly. Since the colors at the 30 ends of the arms at a vertex
building block are the same, only one color is associated with each vertex. The converse is equally
straight forward. If G is 3-vertex-colorable, a graph structure can be constructed from the building
blocks as follows: if a vertex v has color a, choose the vertex building block for v to have the color
encodings at the arms a. If an edge e is incident to vertices v with color a and w with color b,
choose an edge molecule for e with v-end colored a and w-end colored b. These molecules can form
a graph structure with no open ends. Again, we assumed that multiple copies of the graphs are
constructed. We will discuss the problem of covering graphs in Section 4.1. 2
An example of a graph with a possible 3-vertex-coloring and its corresponding DNA graph
structure are given in Fig. 7. The three colors of the vertices are represented by circles, triangles
and squares surrounding the vertices. In the gure of the DNA graph structure, the colors are
represented by dierent shades.
4
00
11
0
1
00
1
0
011
1
0
1
3
11
00
00
11
1
0
5
00
11
1
0
00
11
0
1
0
1
1
0
2
6
1
1
0
0
1
11
00
0
1
0
1
1
00
1
00
11
1
0
0
1
0
1
11
00
00
11
0
1
1
0
0
1
0
1
0
1
11
00
0
1
00
11
0
1
Figure 7: A coloring of a graph and its corresponding DNA graph structure.
3.4 Procedure:
1. Combine multiple copies of all vertex building blocks with all edge molecules in a single tube
and allow the complementary ends to hybridize and be ligated.
2. Remove partially formed 3D DNA structures with open ends that have not been matched.
This could be done by using an exonuclease enzyme.
3. Remove by gel electrophoresis the graphs that are larger than the original graph formed in
the above steps. This step will be explained in Section 4.1.
If there are graph structures remaining in the tube, then we conclude that the graph is
3-colorable. Note that all 2-colorable graphs also are 3-colorable (we thank Alexander Hartmenik for pointing this out to us).
7
The number of laboratory steps in this procedure again does not depend on the number of
vertices (or edges) in the graph. Once the building blocks are formed, the procedure needs only
four steps to perform.
It was suggested by Junghuei Chen that electron microscopy combined with gel electrophoresis
might be used to detect the graph structures formed by DNA. Note that to the right in Fig. 7, the
graph structure depicted forms a knot as single stranded DNA. Such knot structures have been
used to study recombination enzymes (Wasserman and Cozzarelli 1986), and similar techniques for
detecting the knot structures might be useful here. There are graphs, however, that do not form a
single stranded knot, but rather form links (multicomponent circles), and partially formed graphs
may contain smaller knots and links as well.
4 Discussion
4.1 Covering graphs. In Jonoska et al. (in press) it was shown that whenever a graph is
constructed by vertex and edge DNA molecules as building blocks, a covering space of the graph
is possible (regarding the graph as a 1-complex, see below). Here, we also are faced with that
possibility. Unlike the situation with the Hamiltonian path problem when formation of such graphs
could lead to false conclusions, this is not true for the solution of SAT problem.
The denition of a covering spaces can be found in standard text books in Topology (see
Munkres 1975 for example). Briey, a covering p : X~ ! X satises the condition that every point
of X has a small neighborhood U in X such that p?1 (U ) consists of the same (homeomorphic)
copies of U . Since the building blocks connect locally and possibly build bigger pieces globally, we
may get covering spaces of the given graph (regarded as 1-complex).
Figure 8: A 3-colorable covering graph of a non-colorable graph.
In SAT the construction of the contact network graph is oriented from the source to the target.
If a covering graph is formed within the initial step of the computation, then this structure will
be connected and in Step 3 (a), the whole structure will appear in a single tube. This means,
either connections with variable x are removed from the whole structure or connections with x
are removed from the whole structure. Hence, the connection from the source to the target will
be established correctly even if the connection is part of a covering graph structure. A Lemma in
covering space theory (Lemma 4.1 in p337, Munkres 1975) states that any path in a space lifts to
a path in a covering space. Therefore, if there is a solution to SAT, then the solution is realized in
a covering graph as well, and conversely, if there is a path in a covering graph from a source to
a target, then it gives a solution to SAT.
8
For the 3-colorability problem, however, covering graphs can lead to false positives since it is
possible that an original graph is not 3-colorable but a covering graph is. Since all vertices are
connected to all other vertices, the example in the bottom of Fig. 8 is obviously not a 3-colorable
graph. The top of the gure is a covering graph, which is 3-colorable. Thus such possibilities need
to be excluded from our solution. This is addressed in 3.4 Step 3 of the process we described. In
this step, graphs of larger sizes (e.g., covering graphs) are eliminated by gel electrophoresis.
a
v’’
v
v
v’
a
b
e
e
d
c
b
c
d
(a)
(b)
Figure 9: Perturbation of a vertex.
4.2 Implementation and drawbacks.
Although we believe the theoretical procedures described are empirically possible, little specic experimentation is available to conrm this proposition. Three-dimensional DNA structures
are relatively well known and their construction and stability has been investigated (Gussow and
Clackson 1989). There are, however, very few application of general laboratory procedures to the
manipulation of complex, 3-dimensional DNA structures. Enzymatic processes are known to be
error prone and the optimization of experimental protocols most likely will take considerable effort. Below, we have outlined several of the potential experimental diculties of the proposed
procedures. We are grateful to anonymous referees for providing us with the inspiration for this
section.
Building the graph structures. The idea of using dierent building blocks to depict the entire
graph structure was inspired by the work of N. Seeman and his research group (Seeman et
al. 1998) where they describe the construction of dierent 3D DNA structures. The building
blocks proposed here are one of the simplest forms of the molecules they described (see also
Winfree et al. 1998). Furthermore, a plastic DNA model for a portion of a vertex building
block is built in Jonoska et al. (in press), indicating that the similar structures proposed in
this paper may be feasible.
The procedures presented assume that desired building blocks are already available. This is
one of the dicult and complex part of the procedures. However, the 3-armed and 4-armed
branched molecules already have been constructed in the laboratory and it has been reported
that they are fairly stable (Seeman et al. 1989; 1998). It is certainly necessary to study the
complexity and the stability of the k-armed branched molecules for k greater than 4. We
are not aware of any experiments producing distinct k-armed branched molecules in large
numbers, and a potentially large number of such distinct molecules are needed for carrying
out both of the procedures described here.
Nonetheless, using k-armed (k > 4) branched molecules is certainly not necessary in either of
the procedures described here: (1) the 3-SAT requires only 4-armed branched molecules and
every SAT problem is equivalent to a 3-SAT problem by a simple renaming of the variables
(see for example Davis et al. 1994) and (2) there is a surjective homomorphism from a graph
with vertices of degree at most three to any other graph. Statement (2) follows form a simple
9
graph construction of substituting every vertex of degree 4 or higher with vertices of degree at
most three. For example in Fig. 9a a vertex v with degree 5 is depicted. In Fig. 9b, the same
vertex is split into three vertices v; v0 ; v00 each of degree 3. Two new edges connecting v with
v 0 and v 0 with v 00 are also introduced. The old 5 edges incident to v are split between v and
the new vertices v0 and v00 . In checking the 3-vertex-colorability using the new graph (having
vertices with degree at most 3) we need to encode the edges connecting the new vertices (v
with v0 and v0 with v00 ) all with the same color.
In (Seeman et al. 1989; 1991; 1995; 1998) k-armed junctions are used to construct a tetrahedron and a cube by DNA. The construction was very dicult and appropriate reaction
conditions were hard to determine. This indicates that construction of graph structures by
DNA requires extensive experimentation to determine reliable reaction conditions. Even after such conditions are obtained, it might be dicult to apply regular molecular biological
techniques on graph structures. Only a small fraction of DNA building blocks may form such
graph structures, limiting the size of the problem severely. We also do not have experimental
estimates on how many copies are needed to carry out the procedures.
Ligation. If ligation of nicks used in procedures succeeds with probability p (p < 1) at each
site, then a structure of n nicks only has probability only pn of being completely sealed
(Ausubel et al. 1993).
Use of exonuclease. This procedure, used in Step 2 of both problems, most likely is not
100% ecient and some open structures (i.e., incomplete or incorrect graphs) may not be
removed from the reaction mix resulting in false positives. Alternatively, exonuclease may
digest closed structures potentially resulting in false negatives. Furthermore, the dynamics
of exonuclease action on complex branched DNA molecules has not been characterized and
is poorly understood.
Using PCR to amplify DNA paths. Polymerase chain reaction amplication of intentionallycomplex 3D DNA structures has not been well characterized and may prove dicult. There
is, however, some empirical evidence indicating that this is not a likely limiting step (at least
for smaller graphs). PCR amplication of complex DNA structures is not unprecedented.
After isolation from bacterial, small, circular plasmid DNA molecules are known to exist in
several conformational states; linear, circular, concatemeric circles, and super coiled. Direct
amplication of these somewhat complex molecules is routinely performed and known to be
ecient (Gussow and Clackson 1989).
Gel electrophoresis. Step 4 of the 3-vertex-colorability algorithm requires that a large number
of graph structures be successfully constructed. Furthermore, the number of complete graph
structures is likely considerably smaller than the mass of all incomplete DNA structures used.
This requires the separation of \potential" solutions from incomplete ones. We have chosen gel
electrophoresis because of its ability to sort DNA by size. Commonly, electrophoresis is used
to separate linear segments of DNA based on their nucleotide length. Gel electrophoresis is,
however, also capable of separating DNA based on its 3-dimensional conformation somewhat
independent of length (Wasserman and Cozzarelli 1986). Unforturnately, this is, again, a
poorly characterized technique. Nonetheless, we believe that it may be a useful addition to
algorithms of DNA computing.
Regardless of the above, we are encouraged by the progress of others in the construction of 3D
DNA structures. The purpose of this contribution has been to illustrate how these structures can
be used in DNA computing and to stimulate debate and empirical application of graph theory and
3D information processing to molecular computation.
10
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