DNA consists of two long polymers of simple units called nucleotides. Nucleotides are building blocks of DNA and each of them contains three components: sugar, phosphate group and nitrogenous base. There are four different nitrogenous bases which contribute in DNA structure: Thymine(T) and Cytosine(C) which are called pyrimidines; Adenine(A) and Guanine(G) which are called purines. The nucleotides are link together by phosphordiester bonds and form single stranded DNA (ssDNA). Two ssDNA molecules join together to form double stranded DNA (DsDNA) based on complementary rule: “A” always pairs with “T”, and likewise “C” pairs with “G”. In Figure 1, a schematic picture of nucleotide is shown.

DNA computing was initially developed by Leonard Adleman in 1994 [1]. Adleman resolved an instance of Hamiltonian path problem just by handling the DNA molecules. In 1995, Lipton [3] presented a method for solving the satisfiability (SAT) problem. Adleman-Lipton model can be used to solve different NP-complete problems. In Adleman-Lipton model, DNA splints are used for construction of solution space. Adleman [4,5] also presented a molecular algorithm for solving the 3-coloring problem. Chang and Guo [6-8] showed that the DNA operations in Adelman-Lipton model could be used for developing DNA algorithms to resolve the dominating set problem, the vertex cover problem, the maximal clique problem and independent set problem.

The surface-based model was introduced by Liu et al. in 1996 [2]. This model uses DNA molecules attached to a solid surface, instead of DNA molecules floating in a solution. Liu et al. also proposed a surface-based DNA algorithm for solving the satisfiability problem.

In 1996, Roweis et al. [9] introduced the Sticker based DNA computing model and applied it in solving the Minimal Set Cover problem. Perez-Jimenez and Sanchocaparrini [10] used Sticker based DNA computing to resolve knapsack problem, and this model also were applied for breaking the Data Encryption Standard (DES) [11,12]. In our previous work, we also applied sticker model for solving the independent set problem [13].

Other than Adleman-Lipton, surface-based and Sticker based models, other various models are also proposed in DNA computing by researchers. Quyang et al. [14] solved the maximal clique problem using DNA molecules and Restriction endonuclease enzymes. Amos et al. [15,16] described a DNA computation model using restriction endonuclease enzymes instead of successive cycles of separation by DNA hybridization, which can reduce the error-rate of computation. Hagiya et al. [17] proposed a new method of DNA computing that involves a selfacting DNA molecule containing both the input, program, and working memory. In this method, a single-stranded DNA molecule consists of an input segment on the 5’-end, followed by a formula (program) segment, followed by a spacer, and finally with a “head” on the 3’-end that moves and performs the computation. Another method for DNA computation is “computation by self-assembly”. Eric winfree et al. [18-20] introduced a linear and 2-dimentional self-assembly model.

The computing by blocking was introduced by Rozenberg et al. [21] This model uses a novel approach to filter the DNA molecules: Instead of separating the DNA strands to distinct tubes, or destroy and removing the DNA molecules that does not contribute to finding a solution, it blocks (inactivates) them in a way that the blocked strands can be considered as non-existent during the subsequent steps of computation.

In graph theory, a dominating set of a graph G = (V, E), where V is the set of the vertices and E is the set of the edges, is a subset V¹ Í V such that for all u Î V − V¹ there is a v Î V¹ for which (u, v) Î E. The size of a dominating set is the number of vertices it contains. The dominating set problem is to find a minimum size dominating set in G. The dominating set problem has been proved to be a NP—complete problem. For example, the graph in Figure 2 includes 7 vertices and 6 edges.

It is clear that the minimum size dominating set for our graph is {V₄, V₅}, furthermore, the size of the dominating set problem in our graph is 2.

First of all, it is essential to generate the surface-based DNA solution space of our problem. Then, basic biological operations will be used to select legal strands and remove illegal strands from the solution space. It is obvious that a graph with N vertices has 2^N subset of vertices or 2^N possible dominating sets. Furthermore, each possible dominating set can be represented by an N-digit binary number. Also suppose that V¹ is a dominating set of G. If the i^th bit in an N-digit binary number is set to 1, it represents that the i^th vertex is in V¹ but is not in V-V¹. If the i^th bit in an N-digit binary number is set to 0, it represents that the i^th vertex is not in V¹ but is found in V − V¹.

Our graph has 7 vertices and 128 possible dominating sets. Furthermore, the solution space has 128 distinct molecules, which have designed and synthesized as

shown in Figure 3. The length of these DNA molecules is 55 nucleotides, consisting of a unique 20 nucleotides sequence at each end (binding sites for PCR primers), and a 15 nucleotides hybridization sequence in the middle. As discussed before, excellent specificity and discrimination based on single-base mismatch is obtained using 15 mer sequences, for this reason, we considered 15 nucleotides for hybridization sequence. The graph of our problem have 7 vertices, thus, the 7 central nucleotides in the 15 mer hybridization sequence were synthesized as a combinatorial set with two possibilities at each position: (G + C) (G + C) (G + C) (G + C) (A + T) (A + T) (A + T), representing 2⁷ = 128 distinct molecules in solution space. The adjacent 4 nucleotides at the two ends of the hybridization sequence have unique and constant sequences in order to limit the size of the solution space to 128 molecules.

The following algorithm is proposed for solving the dominating set problem:

1) Prepare solution space by designing and synthesis of appropriate DNA strands which are immobilized on a surface.

2) For i = 1 to n, where n is the number of vertices in the graph G.

a) mark (i, 1);

b) For each vertex V_j which have adjacency to V_i;

c) mark (j, 1);

d) destroy-unmarked;

e) unmark.

3) Cleave remaining immobilized DNA strings from surface.

4) Amplify by PCR.

5) Input DNA molecules to tube T₀.

6) For i = 0 to n – 1 For j = i down to 0 Separate (T_j, i + 1) → (T_(j+1)’, T_j)

Combine (T_j+1, T_j+1, T_(j+1)’)

7) Read T₁; else if it was empty then:

Read T₂; else if it was empty then:

Read T₃; else if it was empty then:

.

.

.

Read T_n–1; else if it was empty then:

Read T_n.

According to the steps in the algorithm, the dominating set problem can be resolved by surface-based DNA computation in polynomial time.

Step 2 of the algorithm is executed n times (n is 7 in our graph). Step 2a, marks the DNA strands which contain the vertex V_i and the DNA strands which do not contain the vertex V_i remain unmark. From the definition of dominating set, the unmarked DNA strands represent sets V − V¹, which do not contain the vertex V_i. If there is no vertex adjacent to V_i, then step 2d will destroy the unmarked DNA strands. Otherwise, step 2c will be executed z times, where z is the number of vertices adjacent (directly connected by an edge) to V_i. Each time step 2c is executed, it marks DNA strands which contain V_j (subsets which contain vertices that have adjacency to V_i). Furthermore, the remaining unmarked DNA strands consist of all of the strands which do not contain V_i and V_j, or we can say it contains vertices which do not have adjacency to V_i. Thus, the unmarked DNA strands are illegal strands and should be destroyed. For all vertices, similar processing is also performed, therefore, at the end of step 2, illegal strands will be destroyed and only legal strands (representing dominating sets) will be remain in solution space.

During steps 3 to 5, legal DNA strands which remain on surface at the end of step 2, are cleaved from surface, amplified by using PCR, and finally, all of them are transferred to tube T₀.

In step 6 of the algorithm, we applied 2 operations: separation and combination. Here, we briefly discuss about these operations:

Separate (T_a, i) ® (T_b, T_c), this operation creates two new tubes T_b and T_cT_b contains the DNA molecules having the i^th bit value 1 (T_b = +(T_a, i)) and T_c contains the DNA molecules having the i^th bit value 0 (T_c = –(T_a, i)).

Combine (T_a, T_b, T_c): the DNA molecules from the tubes T_b and T_c are combined to form a new tube T_a, simply the contents of T_b and T_c are poured to tube T_a. (T_a= T_b ∪ T_c).

By the execution of step 6, the DNA strands which represent the Æ subset are placed in tube T₀, the DNA strands represent the subsets which contain only one vertex are placed in tube T₁, the DNA strands represent the subsets which contain two vertices are placed in tube T₂, the DNA strands represent the subsets which contain 3 vertices are placed in tube T₃, and so on.

In step 7, all of the tubes (from T₁ to T_n) are evaluated for presence of DNA strands , and the first tube which is not empty and contains DNA strands represent minimum

size dominating set. In our example, tube T₁ is empty and devoid of any DNA strands. The first tube which contains DNA molecules is tube T₂ which represent the subset {V₄, V₅}. Hence, the minimum size dominating set in our graph is 2.