Molecular Computation Approach to Compete Dijkstra’s Algorithm

Molecular Computation Approach to Compete Dijkstra’sAlgorithmZ. Ibrahim*, Y. Tsuboi *, O. Ono * and M. Khalid✝* Department of Electrical and Electronic EngineeringIkuta Campus, Meiji UniversityKanagawa, JapanE-mail: (zuwairie, tsuboi, ono)@isc.meiji.ac.jp✝Center of Artificial Intelligence and Robotics (CAIRO)Universiti Teknologi Malaysia Kuala Lumpur, Malaysia E-mail: marzuki@utmkl.utm.myAbstractDNA molecules are composed of single or double DNA fragments or often called oligonucleotides or strands.Nucleotides form the basis of DNA. A single-stranded fragment has a phospho-sugar backbone and four kinds of bases denoted by the symbols A, T,G, and C for the bases adenine, thymine,guanine, and cytosine respectively.These four nucleic acids, which can occur in any order combined in Watson-Crick complementary pairs to form a double strand helix of DNA. Due to the hybridization reaction, A is complementary with T and C is complementary with G.As an example, any sequence oligonucleotides, such as 5’-ACCTG-3’has a complementary sequence, 3’-TGGAC-5’.Digits 5’ and 3’denotes orientation of a DNA oligonucleotides. This complementary makes DNA a unique data structure for computation and can be exploited in many ways.Deoxyribonucleic Acid or DNA computing is an emerging field that bridging the gap between chemistry,molecular biology, computer science, and mathematics. In this paper,the potential of DNA computing to compete with Dijkstra’s algorithm is investigated in detail that considers the computation of single source shortest path of a network or graph as a point of reference. The DNA-based computation is designed in such a way that every path is encoded by oligonucleotides and the path’s length is directly proportional to the length of oligonucleotides. Using these properties, gel electrophoresis is performed in order to separate the respective DNA molecules according to their length.Thus, the shortest strands representing the shortest path could be extracted.We will describe in detail in this paper the overall biochemical laboratory procedures involved during the computation.Dijsktra’s algorithm [2] computes the shortest paths from a source vertex to every other vertex in a graph, the so-called single source shortest path (SSSP)problem. In this work,Dijkstra’s algorithm is emphasized because of its undisputable widely applied for various kinds of theoretical and practical applications.Dijkstra’s algorithm is frequently used because of its time bound O (m +n log n ),implemented by using a binary heap [3],remains the best for computing single source shortest path problem with non-negative weighted graphs [4]. For example, Dijkstra’s algorithm has been integrated with Artificial Intelligence (AI)knowledge-based approach and case-based reasoning for the shortest path computation of a road network [5],optimization for recognition of articulated objects [6],during the segmenting computation [7], and for path finding with the lowest sum of weights between source and sink in point-to-point router for industrial design rules [8].1IntroductionIn bioinformatics,mathematical approach is regarded as a common tool for solving biological problems. Contradict to the bioinformatics, DNA computing is a new research of interest where biological tools are employed as a new medium for computation.As of 1994, Leonard M.Adleman [1] launched a novel approach to solve the Hamiltonian Path Problem (HPP) with seven vertices by DNA molecules.While in conventional silicon-based computer,information is stored as binary numbers in silicon-based memory, he encoded the information of the vertices by a randomly DNA sequences.During the computation,gigantic memory capacity and massively parallelism inherent in DNA computing is exploited to make a brute force search on a big problem space in constant or polynomial-time. The output of the computation, also in the form of DNA molecules can be read and printed by electrophoretical fluorescent method.Although Dijkstra’s algorithm has extensively been used to compute the single source shortest path of a graph,tremendous amount of effort to improve the Dijkstra’s algorithm is reported to date for several reasons.One of the reasons is that, the time required finding the optimum pathbecomes remarkably long when the search scope is broad, thus the Dijkstra’s method is not suitable for real-time problems. Hence, Noto and Soto [9] extend the conventional Dijkstra’s algorithm for obtaining, in a specified time, a path that is as close as possible to the path obtained by Dijkstra’s algorithm. The performance of Dijkstra’s algorithm also depends on how it is implemented. For example, by avoiding the conventional data structure of microcomputer, a 2-level bucket data structure is used instead [10]. In that research, the relationship between the implementation performance and the number of bucket level used is studied.As a result, the best number of levels suitable to be used in practice is defined.Nevertheless, these widespread extensions or improvement works are within the non-molecular or mathematical scope and normally performed on sequential silicon based computer as computing platform. Due to this fact, the authors found that there is no, at this time, any research carried out to compute the single source shortest path of a graph by using another medium of computation such as DNA computing, even though from the first day DNA computing is launched,researchers made use of it to solve a lot of combinatorial problems. Hence, in this paper, an attempt to improve or rather to compete the Dijkstra’s algorithm by introducing DNA computing-based approach is proposed.This study is motivated from our previous research works carried out to compute an instance of1-center problem [11]. We have improved the encoding style presented in [12] to deduce a new one. By using a previously developed method for computing the1-center problem, every single path is encoded by its respective oligonucleotide. The paths are encoded in such a way that the length is directly proportional to the length of oligonucleotides. All the DNA strands are then mixed in a test tube and hybridization, concatenation, and ligation are allowed to occur to make sure that a vast number of combination paths are produced in the test tube. PCR is important in this study to amplify sequentially only the strands with a certain start and end vertex.Since the weight of every path is relatively varied to the length of DNA molecules, it is able to extract the shortest path by means of gel electrophoresis.It is then followed by graduated PCR in order to determine the intermediate vertices exist in between the start and end vertex, if any. The intermediate vertices could be recognized by looking at the gradual bands,which is showed by gel electrophoresis. It is expected that the technique developed on the present work enabled us to compute the single source shortest path of a graph as Dijkstra’s algorithm.2The Dijkstra’s algorithmDijkstra’s algorithm solves the single source shortest path problem on a weighted graph, provided all edge weights are nonnegative [13]. The input to the Dijkstra’s algorithm is a weighted, undirected graph G=(V, E,Z) as depicted in Figure 1.V is a set of elements called vertices,E is the setof edges, and Z : E o㧾 assigns a real weight to eachedge. Set㧾 consists of all paths between every pair of node. Weight can be negative or nonnegative. We assume that Z : E o㧾 assigns only nonnegative weight. All nodes in G are reachable from s where V is a fixed source node. A path is a finite sequence of non-repeatednodes P = (sX0,X1, ……, X k ), such that, for 0 d i d k, there exists a connection from X i to X i+1, i.e. e i= (X i, X i+1) E. The weight Z(P) of a path P = (X0,X1, ……, X k ) is defined as:¦ 1kiieP ZZ(1)Figure 1:Input graphLet is the set of verticesS VX for which the distance from tos X has been computed and Q is the set of vertices VX for which the distance from s to X has not yet been computed. Let G(,s X) is the current estimate of the weight from vertex s to vertex X. On the other hand,S)(X is a predecessor of VX. In other words, the vertex through which the shortest distance from tos X was established.Initially,0),(ssG;szX;G(,s X) = f;Q = V;while (0zQ){find Qu with the smallest ),(usG;uSS ;uQQ ;for everyX adj []uif G(,s X)!),(usG + ),(XZ u;thenG(,s X) = ),(usG + ),(XZ u;S)(X=;u}Dijkstra’s algorithm maintains a set of vertices whose final shortest path weights from the source have already been calculated, along with a complementary set of verticesTable 1: Output graph represented by a matrix whose shortest path weights have not yet been determined.The algorithm repeatedly selects the vertex with theminimum current shortest path estimate among those whose shortest path weights have not yet been determined. It updates the weight estimates to all vertices adjacent to the currently selected vertex.This updating is commonly referred to as “relaxation”of the edges between these vertices. The vertex is then added to the set of vertices whose final shortest path weights have been calculated.It continues to do this until all vertices’final shortest path weights have been calculated and the set of vertices whose shortest path weights have not been determined is empty. The summary of the Dijkstra’s algorithm can be expressed as follows [9]:StartNodeIntermediate NodeEndNodeMinimumWeightDistanceX1X5X227X1-X310X1X3X422X1-X5143Computing with DNA3.1Generation of N0SolutionStep 1: Let V be the total number of nodes in the graph and E the total number of paths in the graph. The random sequence strands correspond to all nodes(and its complements) and distances (and its complements) are created. Let O_i (i = 1,.....,V)and i_O(i = 1,....., V) be the fixed length random sequences correspond to all nodes in the graph and its complements respectively. In the same way, let O_d (d = 1,.....,E) and d_O (d = 1,.....,E)be the variable length random sequences correspond to all the distances and its complements respectively.The length l of O_d and d_O will be directly proportional to the weight distances. For instance, let say, for a constant proportional increase of 2,all the random sequences are placed in Table 2 and Table3. For the sake of convenience, in Table 2 and Table 3, in order to avoid clumsy presentation, we will use symbolic representation of the distance sequences. Recall that these sequences would be occurred in any order of nucleotides and its length will be directly proportional to the distances. At the end of this step,oligonucleotides dO_and i_O are synthesized.Step0:Mark the starting pointStep1:Calculate the movement from the starting point to each node connected to the starting point and markthe node to which that the value is the smallest. Step2:Calculate the movement cost for movement between the starting point and each nodeconnected to the marked node and mark the nodefor which that value is the smallest.Step3:Repeat Step 2 until the destination is marked. Consider an input undirected graph in Figure 1. The output graph where vertex X1 is chosen as a fixed source vertex isshown in Figure 2. Another way is to present the graph as a matrix. This can be shown in Table 1 where the start node, intermediate nodes, end nodes and minimum weight distances are listed to represent the output graph.Note that the objective of this paper is to compete the Dijkstra’s algorithm by introducing a molecular approach based on DNA computing where solving the single source shortest path problem is important as a point of reference.While the mathematical formulation of Dijkstra’s algorithm is frequently used and the computation is performed onmicrocomputer in sequential fashion, it is found that the matrix-based representation of the output graph as in Table 1 can be well adapted to perform such computation by DNA, mostly, in parallel fashion.Step 2: Let i be the start node of a path,d the path’s distance and j the end node of that path.Oligonucleotides representing every path between two nodesin the graph aresynthesized as follows:synthesize oligonucleotidesO_i-dψ j_start as ALL O_i + ALL O_d + HL O_jO_j-dψ i_start as ALL O_j + ALL O_d + HL O_iO_i-dψ j as HR O_i + ALL O_d + HL O_jO_j-dψ i as ALL O_j + ALL O_d + HL O_iwhere ‘ALL’represents the whole DNA strand,‘HL’ theleft half part, ‘HR’ the right half part and ‘+’ is a join. Foreach path, connecting two identical nodes, each of thenodes could be a starting node or end node interchangeably.Similarly, the starting node could be the beginning of thepath or rather a connective node in a path. Thus,fouridentical oligonucleotides will be created for each edge. Figure 2:The output of computation.In this case, vertexX1 is selected as a fixed source vertexAs such, if O_i = TATT then i _O = ATAA; if O_j = G CG G thenj _O = CG CC and if O_d = ACG ACG TTthen d _O = TCGTGCAA. Then the path O_i-d ψ j_start is TATTACGACGTTGC (ALL O_i + ALL O_d + HL O_j ), the path O_j-d ψ i_start is GCGGACGACGTTTA (ALL O_j + ALL O_d + HL O_i ), the path O_i-d ψj is TTACGACGTTGC (HR O_i + ALL O_d + HL O_j ) and lastly the path O_j-d ψ i is GGACGACGTTTA (HR O_j + ALL O_d + HL O_i ).At the end of this step, 28 identical edge strands will be created by a combination of the nodes sequences and distance sequences. Because of this combination, the edges strands O_i-d ψ j ,O_i-d ψ j_start ,O_j-d ψ i , and O_j-d ψ i_start listed in Table 4 are influenced by the additional nodes bases and hence, unable to handle the weight of each edges. The additional bases would cause errors during the computation because the additional bases composed in the edge sequences reflect the weight value of edges.To compensate the additional bases, the strands corresponds to distance sequences are shorten according to the additional bases and the node sequences in the edge strands are left unchanged.All the modified oligonucleotides O_i-d ψ j (modified ),O_i-d ψ j_start (modified ),O_j-d ψi (modified ), and O_j-d ψ i_start (modified ) are listed in Table 5. This modification however, decreases the generality of the approach because the weight of edges less than 3 is not allowed.Thus, the lower bound of the proposed approach exists.Figure 7: Gel electrophoresis output where lane M is DNA size marker. Lane 1and 2are used for the testedDNA moleculesTable 2:Fixed length node random sequences Nodes DNA Code,O_i (5’o 3’)Complement,iO _(3’o 5’)v 1TATT ATAA v 2GCGG CGCC v 3GTCG CAGCv 4AGAA TCTT v 5TCCGAGGCStep 3: All modified oligonucleotides O_i-d ψ j (modified ),O_i-d ψ j_start (modified ),O_j-d ψi (modified ), and O_j-d ψi_start (modified ) followed by all complement oligonucleotides (i O _and d O _) are inserted into a test tube and DNA ligase reaction is performed in which random routes through the graph are formed. This operation is depicted in Fig.8.In this way, the initial solution,N 0,which contains a large set of random routes of the graph are generated.After that, in order tomake sure that no sticky end strands exist in the solution,polymerase enzyme is used for the extension where the sticky end strands are altered to become blunt end strands.Table 3: Variable length distance random sequences 3.2DNA-based computational designAt this moment, an initial solution N 0is already obtained and it is time to filter out the right routes among the vast random routes of the graph.away the unwanted strands but rather copying the right strands exponentially by incredibly sensitive PCR process. Next, the proposed DNA-based computational is designed according to the following steps. The overall step performed during computation is shown in Figure9.i.Select a particular node as an ending node and find theshortest path from the starting node,X1 to the endnode.ii.Store the strands representing shortest path into a solution.iii.Find the intermediate nodes between the start node,X1 and the particular end node, if any.Step 4: Produce another four solutions from the initial solution,N0.N 0Annealing and LigationO_i-d j(modified)O_i-d j_start(modified)O_ j-di(modified)O_ j-d i_start(modified)O_i O_dFigure 8: Performing annealing and ligation processes Step four is performed by merely divides the solution N 0equally into four solutions. In this work, it is assumed that all the strands are spread to all areas in the solution. For step five, in order to select the routes that begin with the start node,X 1, and a particular end node, for each test tube obtained in the previous step,polymerase chain reaction (PCR) is performed. Primers employed during the PCR process are shown in Figure 10. As a result, four solutions,N 12,N 13,N 14, and N 15 are produced.For better understanding, lets take a look closely at the molecules of solution N 14only. After the PCR operations are accomplished, there should be a numerous number of strands representing the beginning node,X 1 and the end node,X 4. Some of them are shown in Figure 11.Next, gel electrophoresis technique will separates these strands in term of length and the shortest length is chosen in order to obtain the shortest path. As a result, the strands X 1o X 3o X 4 will be chosen. The same procedure will be executed to the solution N 12,N 13,and N 15 results in solutions consisting the shortest strands X 1o X 5o X 2,X 1o X 3, and X 1o X 5 respectively. The strand X 1o X 5o X 2,X 1o X 3,X 1o X 3o X 4, and X 1o X 5composing the solution N 12,N 13,N 14, and N 15respectively are shown in Figure 12.Figure 9: Overall step involved during computationN 1N 14N 15N 13N 12PCR with primer 5'-TATT-3' and 3'-AG-5'.Perform gel electrophoresisPCR with primer 5'-TATT-3' and 3'-TC-5'.Perform gel electrophoresisPCR with primer 5'-TATT-3' and 3'-CG-5'.Perform gel electrophoresisPCR with primer 5'-TATT-3' and 3'-CA-5'.Perform gelelectrophoresisFinally, the intermediate nodes between the start node,X 1,and the end node, if any, are searched. For this,we rely on graduated PCR,which is employed to the product of the previous operation.G raduated PCR allows one to “print”the result of the computation. For the sake of explanation,the strand X 1o X 3o X 4 is taken again for instance.Basically, the solution N 14 consists of shortest strands which encode the beginning node,X 1, and the end node, X 4.However, there is no information regarding the intermediate node.Figure 10: N 0is divided into four test tubes namely N 12,N 13,N 14, and N 15. PCR process is executed to each solution with primers representing the start node and 3’half complement of end nodes. Next, for each test tube, the gel electrophoresis is applied for separating the strands by length in order to search for the shorteststrandsG raduated PCR is performed by running three different PCR operations with the use of O 1 as the left primer and2O ,3O ,5O , and 3’ half of 4O as the right primerindividually.It is expected that for the solution N 14containing the strand X 1o X 3o X 4, 54 base pair (bp),graduated PCR will produce band of x , 28,x , and 54 in successive lanes of gel electrophoresis as depicted in Figure 13. Meaning that, there is an intermediate node, X 3 in between the start node X 1and the end node X 4. The symbol x denotes the absence of a band corresponding to the omission of node strand X 2and X 5along that strand. It is possible that after the graduated PCR, there is no intermediate node, referring to the strand X 1o X 5. In the case of larger network, it is also not impossible that more than one intermediate node occurred.In that case, the order of the intermediate network can be read and provided by the location of bands of gel electrophoresis.4ConclusionsThis paper has proposed a method that applies for the first time ever,DNA computing in determining the shortest path from a source vertex to every other vertex in a graph asDijkstra’s algorithm. The output is expected to be similar as the Dijkstra’s algorithm. The authors carried out the first attempt to compete the Dijkstra’s algorithm by using DNA as a medium for computation. The contents of this paper also imply a possibility to verify the proposed procedures by experimental work.Although this paper just considered a small and less complicated graph as an instance, the comprehensive discussions demonstrated the capability of the proposed DNA-based computation to compete the Dijkstra’s algorithm.While, DNA computing is still an infant research area compared to that matured conventional silicon-based computation, it seems to be hard, at this moment, to replace the Dijkstra’s algorithm by DNA computing approach in practice. Although the practical benefits of DNA computing are still questionable and the vast majority of work to date has been theoretical, there may have been many allusions to potential uses of this emerging computational paradigm in many fields.We believe that an attempt to compete the Dijkstra’s algorithm is indispensable as the first step for future applications.References[1]L. 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