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Molecular beacon computing model for maximum weight clique problem
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Molecular beacon computing model for maximum weight clique problem Zhixiang Yin ∗, Jianzhong Cui, Chen Zhen School of Science, Anhui University of Science and Technology, Huainan, Anhui, 232001, China Received 1 March 2010; accepted 13 February 2017 Available online xxxxx
Abstract Given an undirected graph with weights on the vertices, the maximum weight clique problem requires finding the clique of the graph which has the maximum weight. The problem is a general form of the maximum clique problem. In this paper, we encode weight of vertex into a unique fixed length oligonucleotide segment and employ sticker model to solve the problem. The proposed method has two distinct characteristics. On one hand, we skip generating initial data pool that contains every possible solution to the problem of interest, the key point of which is constructing the solution instead of searching solution in the vast initial data pool according to logic constraints. On the other hand, oligonucleotide segments are treated like variables which store weights on vertices, no matter what kind of number the weights are, integer or real. Therefore, the proposed method can solve the problem with arbitrary weight values and be applied to solve the other weight-related problem. In addition, molecular beacon is also employed in order to overcome shortcomings of sticker model. Besides, we have analyzed the proposed algorithm’s feasibility. c 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved.
Keywords: DNA computing; Maximum weight clique; Molecular beacon
1. Introduction Molecular beacons are dual-labeled oligonucleotide probes that have a fluorescent dye (reporter) at one end and a fluorescence quencher (usually DABCYL) at the opposite end [10,11]. The probe is designed with a target-specific hybridization domain positioned centrally between short sequences that are self-complementary and are usually unrelated to the target sequence. In the absence of target, the self-complementary domains anneal to form a stem-loop hairpin structure in a unimolecular reaction that serves to bring the fluorescence reporter group into close proximity with the quencher group, and results in quenching of the reporter. In this configuration, the molecular beacon is ‘dark’ (Fig. 1a). Reporter-dye quenchers such as DABCYL and Black Hole Quencher (BHQ) work based on both fluorescence resonance energy transfer (FRET) and the formation of an exciton complex between the fluorophore and ∗ Corresponding author. Fax: +86 554 6682696.
E-mail address: [email protected] (Z.X. Yin). http://dx.doi.org/10.1016/j.matcom.2017.02.003 c 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.
Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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(a) In the absence of target sequence MB is ‘dark’.
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(b) In the presence of target sequence, MB is ‘bright’.
Fig. 1. Schematic diagram of principle of MB.
Fig. 2. A 6-vertex undirected graph with weights on vertices and its complement graph.
the quencher. Thus, DABCYL quenches the reporter dye only in the hairpin form partly due to its relatively short Forster radius. In the presence of target, the central loop domain will hybridize with the complementary target DNA or RNA in a bimolecular reaction, forcing the molecule to unfold; reporter and quencher are now physically separated and the fluorescence of the reporter dye will be restored upon excitation. In this configuration, the molecular beacon is ‘bright’ (see Fig. 1b). The significant advantage of molecular beacon is that they can recognize target sequences with a greater degree of specificity than linear probes. Molecular beacons are readily capable of discriminating between targets that differ by only a single nucleotide. The reason is that the unimolecular hairpin reaction competes with bimolecular probetarget hybridization and serves to reduce the relative stability of undesired imperfect (mismatch) hybridization events. Thermodynamic analysis [1] of molecular beacon reveals that the range of temperatures within which discrimination between perfect matched target sequence and imperfect matched target sequence (even a single nucleotide is unmatched) is wider for molecular beacons than it is for the corresponding linear probes. This is the very reason that molecular beacon is capable of discriminating between targets that differ by only a single nucleotide. The clear advantages of molecular beacons over linear oligonucleotide probes have led to their use in numerous applications ranging from quantitative PCR, to the study of protein–DNA interactions, to the visualization of RNA expression in living cells. Most recently, Yin [15] proposed DNA computing model for 3-SAT problem using molecular beacon, which has initialized a brand-new application field for the molecular beacon. 2. Maximum weight clique problem Assume that G = {V, E, W } is an undirected graph with weights on vertices, where V is the set of vertices and E is the set of edges, W is the weight vector, each component of W , denoted by ωi (i = 1, 2, . . . , n), is the weight of the ith vertex in the graph. ∑ Assume that ωi is a positive real number. For arbitrary clique S of V , the weight of clique is defined as: W (S) = i∈S ωi . If S = Φ, then W (Φ) = 0. Fig. 2 shows a 6-vertex undirected graph with weight vector W = (10.5, 2, 2, 3.2, 3.8, 1)T and its complement graph. The maximum weight clique problem requires finding the clique of the graph which has the maximum weight. Note here, the maximum weight clique is not necessarily the maximum clique in the graph. The problem is a general form of the maximum clique problem. It has been widely applied in pattern recognition and robot technique. The Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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maximum weight clique problem is more difficult than the maximum clique problem, for, every weight of vertex in the graph must be encoded in DNA molecule before computation, i.e., we must encode numerical values into DNA strand. This requirement is widely seen in other weight-related problems, such as, traveling salesman problem and Chinese postman problem, etc. In order to tackle this issue, many weight encoding schemes have been proposed. Narayanan et al. [6] presented a theoretical weight encoding method for traveling salesman problem. Ma et al. [5] proposed a length-based encoding scheme to solve the maximum weight clique problem by operating on plasmids. However, the length of encoded DNA strand is proportional to the weight value. Therefore, their methods are not flexible to represent large weight value. Yamamura et al. [12] presented a concentration control method. Lee [4] proposed a melting temperature control method. Shin et al. [9] proposed a weight encoding method that used fixed length DNA strand for representing integer and real values. Real-valued weight is represented via the relative values of G/C contents against A/T contents. Yin [14] proposed two weight encoding schemes that the length of encoded DNA strand was not longer than 20 or 30 mer. In this paper, we try to represent real valued weights on vertices using fixed length DNA strand while encoding scheme is flexible, general and can be applied to other weight-related problems, more important of all, the method is biologically plausible and easy to implement in real applications. In this paper, we mapped each of vertices and its weights in the given graph into two 15-mer oligonucleotide sequences. These oligonucleotide sequences are then concatenated to form single stranded memory strand. In what follows, we call these sequences vertex sequence, denoted by vi , weight sequence, denoted by ωi . The weight sequence is considered as variables each of which stores the weight value, no matter what kind of the number is, integer or real. The names of weight variables are corresponding weight sequences. For the sake of convenience, the names of these variables are still denoted by ωi . The names of these variables are distinct because the weight sequences are different from one another. The variables are initialized as corresponding weight value in the graph. The access of each of these weight variables is through DNA probes, denoted by Pωi , which are Watson–Crick complements of weight sequences. By introducing the concept of variable and the mechanism of its access, we presented a more general and flexible weight encoding scheme. 3. Molecular beacon model It is widely recognized that there are two shortcomings in sticker model, compared with surface-based model and self-assembly model. The first problem is that stickers which have annealed to memory strands might fall off during the computation, causing a vital error. In order to overcome this drawback, we adopt PNA molecules as stickers in our method. A peptide nucleic acid (PNA) is a biopolymer and a mimic of nature DNA molecule in which the DNA sugar– phosphate backbone has been replaced by a pseudo-peptide [7]. The backbone is not charged, this confers to this polymers a much stronger binding between PNA/DNA strands than between DNA/DNA strands. It is due to the lack of charge repulsion between PNA and DNA strand. The advantage of PNA molecules over DNA molecules as sticker is their excellent specificity towards target sequences. In addition, it can offer different affinities between PNA/memory strand hybrids and DNA probe/memory strand hybrids. Previously reported DNA algorithm based on sticker model generally utilized different length of sticker and DNA probe in order to create different binding affinities so that separation of the captured memory strands from DNA probes might be performed by thermal method. While, in propose method, separation of captured memory complexes from DNA probes will be quite easy, we can just wash in zero salt environment. More important, it can leave stickers which have annealed to memory strands intact during separation operation. Therefore, the error-rate of the problem is apparently reduced. The second concern is the rate of mis-hybridization, since separation by hybridization is the central mechanism of sticker model and mis-hybridization will lead to false positive or negative solutions. This question was already raised in Roweis’ work [8]. They suggested time-space trade-off method to achieve reliable computation if we used unreliable separation operation. There is another candidate to perform reliable separation operation. It is a hairpin shaped fluorescent DNA probe, named molecular beacon, capable of discriminating between DNA sequences that differ as little as a single base. This is particular useful since separation by hybridization operation may be performed many times in sticker model [2,3,13]. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 3. Illustration of our information representation scheme.
3.1. Information representation scheme In our information representation scheme (Fig. 3), the memory strand, denoted by 5′ − v1 ω1 v2 ω2 · · · vn ωn s − 3′ , is subdivided into 2n + 1 (n is the cardinality of the set of vertices) non-overlapping regions. The length of memory strand is 15(2n + 1)mer. Each of vi (i = 1 · · · n) regions, encodes each vertex in the graph, ωi (i = 1 · · · n) encodes each weight on vertex. The last region s located at the 3′ end of memory strand is a common sequence. The sequence is designed for two purposes. On one hand, if we want to separate memory complexes from other strands we may use the DNA probe (in the attached form) which is Watson–Crick complement of this sequence, denoted by, Ps to capture all the memory complexes, then separate the captured memory complexes from the probe by heating or other methods. On the other hand, the presence of this subsequence means the presence of memory complexes. Therefore, we may use fluorescent probe (in the unattached form), denoted by, Ds and detect this subsequence to report the presence of memory complexes. For maximum weight clique problem, if no sticker is annealed to its matching data bit on a memory strand, that means the bit encoded corresponding vertex in the graph is in the clique, if a sticker is annealed to its matching region on a memory strand then the bit encoded corresponding vertex is out of the clique. The memory complex in Fig. 3 illustrates that vertices v2 , v3 , . . . , vn are in the clique, while vertex v1 is out of the clique. The sticker Svi used in this paper is 10 mer long, consists of two parts: 5 mer Watson–Crick complement of the last 5 mer subsequence of vi , 5 mer Watson–Crick complement of the first 5 mer subsequence of ωi . The polarity of sticker strand is the opposite (3′ –5′ ) to memory strand. Since vertex sequence and weight sequence are encoded adjacent on the memory strand, if a vertex is not in the clique, the weight on the vertex need not be considered anymore. According to information representation scheme, both vertex sequence and weight sequence should be set to ‘cover’ by the sticker. Thus, weight sequence is set to ‘absence’ on the memory strand. The design of sticker also serves another purpose. The annealed sticker on memory strand would not impact on the efficiency of DNA probe, since our probe will undergo conformational change when performing separation operation. 3.2. DNA algorithm for maximum weight clique problem In this subsection, we present our DNA algorithm using operations introduced above for solving maximum weight clique problem. The basic idea for solving maximum weight clique of this paper is firstly finding all cliques, and then finding the clique with the maximum weight. 3.2.1. Encoding Several factors must be taken into careful consideration when designing sequences of memory strand, sticker and probe. For memory strand, sequence design should prevent it from annealing to itself and forming undesired secondary structure, thus leading to regions in memory strand inaccessible. As for DNA separation probe, it should stick specifically to unique region in memory strand as we expect. And last, but most important, the sticker, it is crucial that sticker anneals to specified region in the memory strand not in any other position. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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All sequences used in this paper satisfy constraints: 1. Memory strand sequences contain only As, Ts, and Cs. 2. All Memory strand and probe sequences have no occurrence of 5 or more consecutive identical nucleotides; i.e. no runs of more than 4 As, 4 Ts, 4 Cs or 4 Gs occur in any memory strand or probe sequences. 3. Every probe sequence has at least 4 mismatches with all 15 base alignments of any memory strand sequence (except for its matching region). 4. Every 15 base subsequence of a memory sequence has at least 4 mismatches with all 15 base alignments of itself. 5. No probe sequence has a run of more than seven matches with any eight base alignments of any memory strand sequence (except for its matching value sequence). 6. No memory strand sequence has a run of more than seven matches with any eight base alignments of itself. 7. Every probe sequence has 4, 5, or 6 Gs in its sequence. Constraint 1 is motivated by the assumption that library strands composed only of As, Ts, and Cs will have less secondary structure than those composed of As, Ts, Cs, and Gs. Constraint 2 is motivated by two assumptions: first, those long homo-polymer tracts may have unusual secondary structure and second, that the melting temperatures of probe–memory strand hybrids will be more uniform if none of the probe–memory strand hybrids involve long homo-polymer tracts. Constraints 3 and 5 are intended to ensure that probes bind only weakly where they are not intended to bind. Constraints 4 and 6 are intended to ensure that memory strands have a low affinity for themselves. Constraint 7 is intended to ensure that intended probe–memory strand pairings have uniform melting temperatures. The sequences encoding vertex and its weight in the graph Fig. 2 are listed as follows: v1 : TTCATACACTTACAC v2 : CTATTTATATCCACC v3 : TCCATTTCTCCATAT v4 : ATTTCCAACATACTC v5 : AACTCATACTACTCA v6 : TTCATACACTTACAC s: AACTTCACCCCTATA
ω1 : TTACACCAATCTCTT ω2 : CTCCTACAATTCCTA ω3 : CCCTATTAATCAATC ω4 : ATAACCACAAACTCA ω5 : CTATCCAATAACCTC ω6 : CTCCCAAATAACATT.
We used Oligo software (data now shown) to confirm their uniform thermodynamic behavior (nearest-neighbor model) and DNA/RNA secondary structure prediction software Mfold, freely available at http://www.bioinfo.rpi.edu/ applications/mfold/old/dna/form1.cgi[23] to conform that no severe secondary structure of memory strand occurs (data now shown). 3.2.2. DNA algorithm The difference between our method and Head’s method is we do not use restriction enzyme to cleave the double stranded DNA molecules instead we use sticker to cover memory strand, therefore, no enzyme is needed in our method. Since the complement graph contains all edges missing in the original graph, the pairs of vertices that is adjacent in the complement graph cannot be in the same clique. Therefore, for each edge vi v j ∈ G c , we divide the tube T containing memory complexes into two new tube Ti and T j , add sicker Si in Ti and S j in T j . According to the information representation scheme, memory strand with bit vi on means all cliques that do not contain vertex vi , with bit v j on means cliques that do not contain vertex v j . That is to say, all cliques not containing vi are in tube T j , all cliques not containing v j are in tube Ti . Combine tube Ti and T j into tube T . Repeat these operations until every edge of complement graph is considered. Then tube T must contain all the maximal cliques. Perform next Sort operation to classify all the memory complexes representing all the cliques (utilize DNA probes Pωi to determine the presence or absence of weight sequences ωi on the memory strand). Finally, with the initial value of weight variables we will find the maximum weight clique. The following is the DNA algorithm: 1. Synthesize memory strands in the form of 5′ − v1 ω1 v2 ω2 · · · vn ωn s − 3′ , and pour these strands into tube T . Synthesize molecular beacon probes, denoted by, Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pvn , Pωn , Ps , Ds . Note here, probe Ps , Ds both correspond to the bit region s (common sequence) except that Ps is attached one while Ds is unattached one. The loop sequence of molecular beacon is the Watson–Crick complement of corresponding bit sequence on memory strand. The stem sequence is the same for all probes. Fluorescein is used as fluorophore and DABCYL as quencher. Attach Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pvn , Pωn , Ps to the surface. A detailed protocol for synthesizing molecular beacon Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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is available on the worldwide web at http://www.molecular-beacon.org. Synthesize PNA molecules, denoted by, Sv1 , Sv2 , . . . , Svn . 2. For every edge vi v j ∈ G c Divide (T, Ti , T j ) Set (Ti , vi ) and Set (T j , v j ) T = ∪(Ti , T j ). EndFor 3. Sort(T ) 4. Detect
3.3. Application of molecular beacon in the proposed algorithm In the proposed algorithm, molecular beacon probe is employed in Sort and Detect operation. In former case, it functions as reliable separation probe (in the attached form). In later case, it serves as report probe (in the unattached form). In this subsection, we show in detail how to solve an instance of maximum weight clique problem (Fig. 2). Step 1: Synthesize identical memory stands with moderate amount and pour these strands into test tube T . Synthesize molecular beacon probes, Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pv6 , Pω6 , Ps , Ds . Note here, probe Ps , Ds both correspond to the bit region s (common sequence) except that Ps is attached one while Ds is unattached one. Synthesize PNA molecules, denoted by, Sv1 , Sv2 , . . . , Sv6 . Following is the encodings used in this paper: Memory strand: 5′ -TTCATACACTTACAC-TTACACCAATCTCTT-CTATTTATATCCACC-CTCCTACAATTCCTATCCATTTCTCCATAT-CCCTATTAATCAATC-ATTTCCAACATACTC-ATAACCACAAACTCAAACTCATACTACTCA-CTATCCAATAACCTC-TTCATACACTTACAC-CTCCCAAATAACATTAACTTCACCCCTATA-3′ PNA stickers: Sv1 : 5′ -TGTAAGTGTA-3′ Sv2 : 5′ -AGGAGGGTGG-3′ Sv3 : 5′ -TAGGGATATG-3′ Sv4 : 5′ -GTTATGAGTA-3′ Sv5 : 5′ -GATAGTGAGT-3′ Sv6 : 5′ -GGGAGGTGTA-3′ . Molecular beacon probes: Pv1 : 5′ -CGCTC GTGTAAGTGTATGAA GAGCG-3′ Pω1 : 5′ -CGCTC AAGAGATTGGTGTAA GAGCG-3′ Pv2 : 5′ -CGCTC GGTGGATATAAATAG GAGCG-3′ Pω2 : 5′ -CGCTC TAGGAATTGTAGGAG GAGCG-3′ Pv3 : 5′ -CGCTC ATATGGAGAAATGGA GAGCG-3′ Pω3 : 5′ -CGCTC GATTGATTAATAGGG GAGCG-3′ Pv4 : 5′ -CGCTC GAGTATGTTGGAAAT GAGCG-3′ Pω4 : 5′ -CGCTC TGAGTTTGTGGTTAT GAGCG-3′ Pv5 : 5′ -CGCTC TGAGTAGTATGAGTT GAGCG-3′ Pω5 : 5′ -CGCTC GAGGTTATTGGATAG GAGCG-3′ Pv6 : 5′ -CGCTC GTGTAAGTGTATGAA GAGCG-3′ Pω6 : 5′ -CGCTC AATGTTATTTGGGAG GAGCG-3′ Ps : 5′ -CGCTC TATAGGGGTGAAGTT GAGCG-3′ Ds : 5′ -CGCTC TATAGGGGTGAAGTT GAGCG-3′ . The underlined sequences above are 15 mer loop sequences bounded by 5 bps stem sequence of molecular beacon probes. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 4. Illustration of Sort operation.
Step 2: For the edge v1 v3 in the complement graph, divide tube T into T1 and T3 , add sticker Sv1 into T1 , Sv3 into T3 , and then anneal. These operations serve to set bit regions v1 in tube T1 , v3 in tube T3 on. Then use molecular beacon probe Pv1 to separate those memory complexes with v1 bit region off in T1 , use molecular beacon probe Pv3 to separate those memory complexes with v3 bit region off in T3 . These operations serve to see that every bit region is correctly set on. Because those memory complexes with bit region off must be captured by the probes, and the captured memory complexes are discarded. Use molecular beacon probe Ps to separate all memory complexes in T1 and T3 from added stickers. Wash in zero salt concentration, collect the captured memory complexes and pour them again into tube T . These operations serve to exclude extra, unhybridized stickers in the tubes, although surplus stickers do not influent the next cycle of computation. At this stage, tube T contains memory complexes that encode correct cliques. If there are not edges any more in the graph, the cliques are {v2 , v3 , v4 , v5 , v6 } or {v1 , v2 , v4 , v5 , v6 }. Similar operations are performed until every edge in the complement graph is processed. At this stage, tube T contains memory complexes that encode all cliques: {v1 , v5 }, {v2 , v5 }, {v3 , v5 }, {v4 , v5 }, {v5 , v6 }, {v1 , v2 , v5 }, {v1 , v4 , v5 }, {v2 , v3 , v5 }, {v3 , v4 , v5 }, {v3 , v5 , v6 }, {v4 , v5 , v6 } and {v3 , v4 , v5 , v6 }. Step 3: Use probes Pω1 , Pω2 , Pω3 , Pω4 , Pω5 , and Pω6 to sort memory complexes. When this step terminates, 26 tubes are produced. Tube Tk (0 ≤ k ≤ 63) contains memory complexes with the known weight of clique. Next we only need to detect the presence of the memory complexes in each of them. Fig. 4 illustrates in detail the operations. Step 4: Add probes Ds in Tk (0 ≤ k ≤ 63) to detect the presence or absence of common sequence s, fluorescent image, the detected fluorescent signal means there is target sequence s in the tube, i.e., there exist memory complexes in Tk . Return maximum weight clique and terminate the algorithm. Fig. 5 illustrates in detail the operations. 4. Discussion As far as complexity concerned, for a n vertices and m edges maximum weight clique problem, we need to encode 2n + 1 oligonucleotide sequences (n is the cardinality of the set of vertex in the graph). Therefore, 2n sequences are used for encoding vertices and their weight, and one common sequence is used for separation and detection. The memory strand is the concatenation of the sequences (note here, once these sequences are chosen, the corresponding sequences of stickers and DNA probes are thus assured according to Watson–Crick complementary Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 5. Illustration of Detect operation.
rule), thus encoding complexity is o(n). Since edges to be considered are n(n − 1)/2 − m at most, the time complexity is o(n 2 ). In order to determine maximum weight clique of the graph, 2n tubes are needed in the Sort operation. Several aspects should be taken into careful consideration to see that the final result can be obtained, though some of these concerns have been addressed before. The top priority is that stickers may fall off from memory strand during the course of the computation. In the proposed method, we have chosen PNA molecule as sticker. The motivation of using PNA molecule as sticker comes from the fact that decreasing salt concentration will cause PNA sticker/memory strand hybrids to bind more strongly while DNA probe/memory strand hybrids to bind weakly. Therefore, in the proposed method, separation captured memory complex from DNA probe increases the binding affinity of sticker to memory strand rather than decreasing, compared with previously reported DNA algorithm based on sticker model. Thus, the possibility that sticker might fall off from memory strand might be decreased. But, this requirement sacrifices the kinetics of probe. In short, reasonable trade-off must be made between binding affinity of PNA stickers’ to memory strand and kinetics of probe. The second concern is the rate of mis-hybridization, since separation by hybridization is the central mechanism of sticker model. We have employed novel molecular beacon probe to perform reliable separation. The third concern is that strand lost due to memory strand might stick to the wall of tubes during computation, especially, combine operation. At present, this is a common question in test tube based DNA computing. We reasonably assume that the lost DNA strand is a minute fraction of the total which would be unimportant to whole computation. Alternatively, we may increase the amount of memory strand and sticker to compensate the strand lost during the computation. The fourth concern is the design of memory and stickers. A well designed encoding strategy for stickers may improve sticker specificity, thus improve the overall performance of our method. Similarly, good encodings for memory strand may prevent it from annealing to itself and forming unexpected secondary structure, causing data bits inaccessible. Encoding problem has been addressed extensively in DNA computing. We have not addressed the problem here. Acknowledgments The authors would like to thank every author appeared in the references. This project is supported by the National Natural Science Foundation of China (Grant No. 61672001). References [1] G. Bonnet, S. Tyagi, A. Libchaber, F.R. Kramer, Thermodynamic basis of the enhanced specificity of structured DNA probes, Proc. Natl. Acad. Sci. USA l96 (1999) 6171–6176. [2] X.H. Fang, X.J. Liu, S. Schuster, W.H. Tan, Designing a novel molecular beacon for surface-immobilized DNA hybridization studies, J. Am. Chem. Soc. 121 (1999) 2921–2922. [3] T. Fumiaki, K. Atsushi, M. Yamamoto, Design of nucleic acid sequences for DNA computing based on a thermodynamic approach, Nucleic Acids Res. 33 (2005) 903–911. [4] J.Y. Lee, S.Y. Shin, T.H. Park, Solving traveling salesman problems with DNA molecules encoding numerical values, BioSystems 78 (2004) 39–47. [5] R.N. Ma, Q. Zhang, L. Gao, J. Xu, Using DNA to solve the maximum weight clique of graphs, Acta Electron. Sin. 32 (1) (2004) 13–16. [6] A. Narayanan, S. Zorbalas, DNA algorithms for computing shortest paths, in: Proceedings of the Genetic Programming, Morgan Kaufmann, 1998, pp. 718–723. [7] E.P. Nielsen, E.H. Michael, R.H.B. Egholm, B.H. Ole, Sequence-selective recognition of DNA by strand displacement with a thyminesubstituted polyamide, Science 254 (1991) 497–1500. [8] S. Roweis, E. Winfree, R. Burgoyne, N. Chelyapov, M. Goodman, P. Rothemund, L. Adleman, A sticker based architecture for DNA computation, in: Proceedings of the Second Annual Meeting on DNA Based Computers, in: DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, Providence, PI, 1996.
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[9] S.Y. Shin, B.T. Zhang, S.S. Jun, Solving traveling salesman problems using molecular programming, in: Proceedings of the Congress on Evolutionary Computation, IEEE Press, 1999, pp. 994–1000. [10] S. Tyagi, D.P. Bratu, F.R. Kramer, Multicolor molecular beacons for allele discrimination, Nature Biotechnol. 6 (1998) 49–53. [11] S. Tyagi, F.R. Kramer, Molecular Beacons: probes that fluoresce upon hybridizatio, Nature Biotechnol. 4 (1996) 303–308. [12] M. Yamamura, Y. Hiroto, T. Matoba, Solutions of shortest path problems by concentration control, Lect. Notes Comput. Sci. 2340 (2002) 231–240. [13] G. Yao, W. Tan, Molecular beacon-based array for sensitive DNA analysis, Anal. Biochem. 331 (2004) 216–223. [14] Z.X. Yin, DNA Computing in Graph and Combination Optimization, Science Press, 2004, pp. 57–72 (in Chinese). [15] Z.X. Yin, F.Y. Zhang, J. Xu, DNA computing based on molecular beacon, J. Biomath. 18 (2003) 497–502.
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Original articles
Molecular beacon computing model for maximum weight clique problem Zhixiang Yin ∗, Jianzhong Cui, Chen Zhen School of Science, Anhui University of Science and Technology, Huainan, Anhui, 232001, China Received 1 March 2010; accepted 13 February 2017 Available online xxxxx
Abstract Given an undirected graph with weights on the vertices, the maximum weight clique problem requires finding the clique of the graph which has the maximum weight. The problem is a general form of the maximum clique problem. In this paper, we encode weight of vertex into a unique fixed length oligonucleotide segment and employ sticker model to solve the problem. The proposed method has two distinct characteristics. On one hand, we skip generating initial data pool that contains every possible solution to the problem of interest, the key point of which is constructing the solution instead of searching solution in the vast initial data pool according to logic constraints. On the other hand, oligonucleotide segments are treated like variables which store weights on vertices, no matter what kind of number the weights are, integer or real. Therefore, the proposed method can solve the problem with arbitrary weight values and be applied to solve the other weight-related problem. In addition, molecular beacon is also employed in order to overcome shortcomings of sticker model. Besides, we have analyzed the proposed algorithm’s feasibility. c 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights ⃝ reserved.
Keywords: DNA computing; Maximum weight clique; Molecular beacon
1. Introduction Molecular beacons are dual-labeled oligonucleotide probes that have a fluorescent dye (reporter) at one end and a fluorescence quencher (usually DABCYL) at the opposite end [10,11]. The probe is designed with a target-specific hybridization domain positioned centrally between short sequences that are self-complementary and are usually unrelated to the target sequence. In the absence of target, the self-complementary domains anneal to form a stem-loop hairpin structure in a unimolecular reaction that serves to bring the fluorescence reporter group into close proximity with the quencher group, and results in quenching of the reporter. In this configuration, the molecular beacon is ‘dark’ (Fig. 1a). Reporter-dye quenchers such as DABCYL and Black Hole Quencher (BHQ) work based on both fluorescence resonance energy transfer (FRET) and the formation of an exciton complex between the fluorophore and ∗ Corresponding author. Fax: +86 554 6682696.
E-mail address: [email protected] (Z.X. Yin). http://dx.doi.org/10.1016/j.matcom.2017.02.003 c 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights 0378-4754/⃝ reserved.
Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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(a) In the absence of target sequence MB is ‘dark’.
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(b) In the presence of target sequence, MB is ‘bright’.
Fig. 1. Schematic diagram of principle of MB.
Fig. 2. A 6-vertex undirected graph with weights on vertices and its complement graph.
the quencher. Thus, DABCYL quenches the reporter dye only in the hairpin form partly due to its relatively short Forster radius. In the presence of target, the central loop domain will hybridize with the complementary target DNA or RNA in a bimolecular reaction, forcing the molecule to unfold; reporter and quencher are now physically separated and the fluorescence of the reporter dye will be restored upon excitation. In this configuration, the molecular beacon is ‘bright’ (see Fig. 1b). The significant advantage of molecular beacon is that they can recognize target sequences with a greater degree of specificity than linear probes. Molecular beacons are readily capable of discriminating between targets that differ by only a single nucleotide. The reason is that the unimolecular hairpin reaction competes with bimolecular probetarget hybridization and serves to reduce the relative stability of undesired imperfect (mismatch) hybridization events. Thermodynamic analysis [1] of molecular beacon reveals that the range of temperatures within which discrimination between perfect matched target sequence and imperfect matched target sequence (even a single nucleotide is unmatched) is wider for molecular beacons than it is for the corresponding linear probes. This is the very reason that molecular beacon is capable of discriminating between targets that differ by only a single nucleotide. The clear advantages of molecular beacons over linear oligonucleotide probes have led to their use in numerous applications ranging from quantitative PCR, to the study of protein–DNA interactions, to the visualization of RNA expression in living cells. Most recently, Yin [15] proposed DNA computing model for 3-SAT problem using molecular beacon, which has initialized a brand-new application field for the molecular beacon. 2. Maximum weight clique problem Assume that G = {V, E, W } is an undirected graph with weights on vertices, where V is the set of vertices and E is the set of edges, W is the weight vector, each component of W , denoted by ωi (i = 1, 2, . . . , n), is the weight of the ith vertex in the graph. ∑ Assume that ωi is a positive real number. For arbitrary clique S of V , the weight of clique is defined as: W (S) = i∈S ωi . If S = Φ, then W (Φ) = 0. Fig. 2 shows a 6-vertex undirected graph with weight vector W = (10.5, 2, 2, 3.2, 3.8, 1)T and its complement graph. The maximum weight clique problem requires finding the clique of the graph which has the maximum weight. Note here, the maximum weight clique is not necessarily the maximum clique in the graph. The problem is a general form of the maximum clique problem. It has been widely applied in pattern recognition and robot technique. The Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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maximum weight clique problem is more difficult than the maximum clique problem, for, every weight of vertex in the graph must be encoded in DNA molecule before computation, i.e., we must encode numerical values into DNA strand. This requirement is widely seen in other weight-related problems, such as, traveling salesman problem and Chinese postman problem, etc. In order to tackle this issue, many weight encoding schemes have been proposed. Narayanan et al. [6] presented a theoretical weight encoding method for traveling salesman problem. Ma et al. [5] proposed a length-based encoding scheme to solve the maximum weight clique problem by operating on plasmids. However, the length of encoded DNA strand is proportional to the weight value. Therefore, their methods are not flexible to represent large weight value. Yamamura et al. [12] presented a concentration control method. Lee [4] proposed a melting temperature control method. Shin et al. [9] proposed a weight encoding method that used fixed length DNA strand for representing integer and real values. Real-valued weight is represented via the relative values of G/C contents against A/T contents. Yin [14] proposed two weight encoding schemes that the length of encoded DNA strand was not longer than 20 or 30 mer. In this paper, we try to represent real valued weights on vertices using fixed length DNA strand while encoding scheme is flexible, general and can be applied to other weight-related problems, more important of all, the method is biologically plausible and easy to implement in real applications. In this paper, we mapped each of vertices and its weights in the given graph into two 15-mer oligonucleotide sequences. These oligonucleotide sequences are then concatenated to form single stranded memory strand. In what follows, we call these sequences vertex sequence, denoted by vi , weight sequence, denoted by ωi . The weight sequence is considered as variables each of which stores the weight value, no matter what kind of the number is, integer or real. The names of weight variables are corresponding weight sequences. For the sake of convenience, the names of these variables are still denoted by ωi . The names of these variables are distinct because the weight sequences are different from one another. The variables are initialized as corresponding weight value in the graph. The access of each of these weight variables is through DNA probes, denoted by Pωi , which are Watson–Crick complements of weight sequences. By introducing the concept of variable and the mechanism of its access, we presented a more general and flexible weight encoding scheme. 3. Molecular beacon model It is widely recognized that there are two shortcomings in sticker model, compared with surface-based model and self-assembly model. The first problem is that stickers which have annealed to memory strands might fall off during the computation, causing a vital error. In order to overcome this drawback, we adopt PNA molecules as stickers in our method. A peptide nucleic acid (PNA) is a biopolymer and a mimic of nature DNA molecule in which the DNA sugar– phosphate backbone has been replaced by a pseudo-peptide [7]. The backbone is not charged, this confers to this polymers a much stronger binding between PNA/DNA strands than between DNA/DNA strands. It is due to the lack of charge repulsion between PNA and DNA strand. The advantage of PNA molecules over DNA molecules as sticker is their excellent specificity towards target sequences. In addition, it can offer different affinities between PNA/memory strand hybrids and DNA probe/memory strand hybrids. Previously reported DNA algorithm based on sticker model generally utilized different length of sticker and DNA probe in order to create different binding affinities so that separation of the captured memory strands from DNA probes might be performed by thermal method. While, in propose method, separation of captured memory complexes from DNA probes will be quite easy, we can just wash in zero salt environment. More important, it can leave stickers which have annealed to memory strands intact during separation operation. Therefore, the error-rate of the problem is apparently reduced. The second concern is the rate of mis-hybridization, since separation by hybridization is the central mechanism of sticker model and mis-hybridization will lead to false positive or negative solutions. This question was already raised in Roweis’ work [8]. They suggested time-space trade-off method to achieve reliable computation if we used unreliable separation operation. There is another candidate to perform reliable separation operation. It is a hairpin shaped fluorescent DNA probe, named molecular beacon, capable of discriminating between DNA sequences that differ as little as a single base. This is particular useful since separation by hybridization operation may be performed many times in sticker model [2,3,13]. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 3. Illustration of our information representation scheme.
3.1. Information representation scheme In our information representation scheme (Fig. 3), the memory strand, denoted by 5′ − v1 ω1 v2 ω2 · · · vn ωn s − 3′ , is subdivided into 2n + 1 (n is the cardinality of the set of vertices) non-overlapping regions. The length of memory strand is 15(2n + 1)mer. Each of vi (i = 1 · · · n) regions, encodes each vertex in the graph, ωi (i = 1 · · · n) encodes each weight on vertex. The last region s located at the 3′ end of memory strand is a common sequence. The sequence is designed for two purposes. On one hand, if we want to separate memory complexes from other strands we may use the DNA probe (in the attached form) which is Watson–Crick complement of this sequence, denoted by, Ps to capture all the memory complexes, then separate the captured memory complexes from the probe by heating or other methods. On the other hand, the presence of this subsequence means the presence of memory complexes. Therefore, we may use fluorescent probe (in the unattached form), denoted by, Ds and detect this subsequence to report the presence of memory complexes. For maximum weight clique problem, if no sticker is annealed to its matching data bit on a memory strand, that means the bit encoded corresponding vertex in the graph is in the clique, if a sticker is annealed to its matching region on a memory strand then the bit encoded corresponding vertex is out of the clique. The memory complex in Fig. 3 illustrates that vertices v2 , v3 , . . . , vn are in the clique, while vertex v1 is out of the clique. The sticker Svi used in this paper is 10 mer long, consists of two parts: 5 mer Watson–Crick complement of the last 5 mer subsequence of vi , 5 mer Watson–Crick complement of the first 5 mer subsequence of ωi . The polarity of sticker strand is the opposite (3′ –5′ ) to memory strand. Since vertex sequence and weight sequence are encoded adjacent on the memory strand, if a vertex is not in the clique, the weight on the vertex need not be considered anymore. According to information representation scheme, both vertex sequence and weight sequence should be set to ‘cover’ by the sticker. Thus, weight sequence is set to ‘absence’ on the memory strand. The design of sticker also serves another purpose. The annealed sticker on memory strand would not impact on the efficiency of DNA probe, since our probe will undergo conformational change when performing separation operation. 3.2. DNA algorithm for maximum weight clique problem In this subsection, we present our DNA algorithm using operations introduced above for solving maximum weight clique problem. The basic idea for solving maximum weight clique of this paper is firstly finding all cliques, and then finding the clique with the maximum weight. 3.2.1. Encoding Several factors must be taken into careful consideration when designing sequences of memory strand, sticker and probe. For memory strand, sequence design should prevent it from annealing to itself and forming undesired secondary structure, thus leading to regions in memory strand inaccessible. As for DNA separation probe, it should stick specifically to unique region in memory strand as we expect. And last, but most important, the sticker, it is crucial that sticker anneals to specified region in the memory strand not in any other position. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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All sequences used in this paper satisfy constraints: 1. Memory strand sequences contain only As, Ts, and Cs. 2. All Memory strand and probe sequences have no occurrence of 5 or more consecutive identical nucleotides; i.e. no runs of more than 4 As, 4 Ts, 4 Cs or 4 Gs occur in any memory strand or probe sequences. 3. Every probe sequence has at least 4 mismatches with all 15 base alignments of any memory strand sequence (except for its matching region). 4. Every 15 base subsequence of a memory sequence has at least 4 mismatches with all 15 base alignments of itself. 5. No probe sequence has a run of more than seven matches with any eight base alignments of any memory strand sequence (except for its matching value sequence). 6. No memory strand sequence has a run of more than seven matches with any eight base alignments of itself. 7. Every probe sequence has 4, 5, or 6 Gs in its sequence. Constraint 1 is motivated by the assumption that library strands composed only of As, Ts, and Cs will have less secondary structure than those composed of As, Ts, Cs, and Gs. Constraint 2 is motivated by two assumptions: first, those long homo-polymer tracts may have unusual secondary structure and second, that the melting temperatures of probe–memory strand hybrids will be more uniform if none of the probe–memory strand hybrids involve long homo-polymer tracts. Constraints 3 and 5 are intended to ensure that probes bind only weakly where they are not intended to bind. Constraints 4 and 6 are intended to ensure that memory strands have a low affinity for themselves. Constraint 7 is intended to ensure that intended probe–memory strand pairings have uniform melting temperatures. The sequences encoding vertex and its weight in the graph Fig. 2 are listed as follows: v1 : TTCATACACTTACAC v2 : CTATTTATATCCACC v3 : TCCATTTCTCCATAT v4 : ATTTCCAACATACTC v5 : AACTCATACTACTCA v6 : TTCATACACTTACAC s: AACTTCACCCCTATA
ω1 : TTACACCAATCTCTT ω2 : CTCCTACAATTCCTA ω3 : CCCTATTAATCAATC ω4 : ATAACCACAAACTCA ω5 : CTATCCAATAACCTC ω6 : CTCCCAAATAACATT.
We used Oligo software (data now shown) to confirm their uniform thermodynamic behavior (nearest-neighbor model) and DNA/RNA secondary structure prediction software Mfold, freely available at http://www.bioinfo.rpi.edu/ applications/mfold/old/dna/form1.cgi[23] to conform that no severe secondary structure of memory strand occurs (data now shown). 3.2.2. DNA algorithm The difference between our method and Head’s method is we do not use restriction enzyme to cleave the double stranded DNA molecules instead we use sticker to cover memory strand, therefore, no enzyme is needed in our method. Since the complement graph contains all edges missing in the original graph, the pairs of vertices that is adjacent in the complement graph cannot be in the same clique. Therefore, for each edge vi v j ∈ G c , we divide the tube T containing memory complexes into two new tube Ti and T j , add sicker Si in Ti and S j in T j . According to the information representation scheme, memory strand with bit vi on means all cliques that do not contain vertex vi , with bit v j on means cliques that do not contain vertex v j . That is to say, all cliques not containing vi are in tube T j , all cliques not containing v j are in tube Ti . Combine tube Ti and T j into tube T . Repeat these operations until every edge of complement graph is considered. Then tube T must contain all the maximal cliques. Perform next Sort operation to classify all the memory complexes representing all the cliques (utilize DNA probes Pωi to determine the presence or absence of weight sequences ωi on the memory strand). Finally, with the initial value of weight variables we will find the maximum weight clique. The following is the DNA algorithm: 1. Synthesize memory strands in the form of 5′ − v1 ω1 v2 ω2 · · · vn ωn s − 3′ , and pour these strands into tube T . Synthesize molecular beacon probes, denoted by, Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pvn , Pωn , Ps , Ds . Note here, probe Ps , Ds both correspond to the bit region s (common sequence) except that Ps is attached one while Ds is unattached one. The loop sequence of molecular beacon is the Watson–Crick complement of corresponding bit sequence on memory strand. The stem sequence is the same for all probes. Fluorescein is used as fluorophore and DABCYL as quencher. Attach Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pvn , Pωn , Ps to the surface. A detailed protocol for synthesizing molecular beacon Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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is available on the worldwide web at http://www.molecular-beacon.org. Synthesize PNA molecules, denoted by, Sv1 , Sv2 , . . . , Svn . 2. For every edge vi v j ∈ G c Divide (T, Ti , T j ) Set (Ti , vi ) and Set (T j , v j ) T = ∪(Ti , T j ). EndFor 3. Sort(T ) 4. Detect
3.3. Application of molecular beacon in the proposed algorithm In the proposed algorithm, molecular beacon probe is employed in Sort and Detect operation. In former case, it functions as reliable separation probe (in the attached form). In later case, it serves as report probe (in the unattached form). In this subsection, we show in detail how to solve an instance of maximum weight clique problem (Fig. 2). Step 1: Synthesize identical memory stands with moderate amount and pour these strands into test tube T . Synthesize molecular beacon probes, Pv1 , Pω1 , Pv2 , Pω2 , . . . , Pv6 , Pω6 , Ps , Ds . Note here, probe Ps , Ds both correspond to the bit region s (common sequence) except that Ps is attached one while Ds is unattached one. Synthesize PNA molecules, denoted by, Sv1 , Sv2 , . . . , Sv6 . Following is the encodings used in this paper: Memory strand: 5′ -TTCATACACTTACAC-TTACACCAATCTCTT-CTATTTATATCCACC-CTCCTACAATTCCTATCCATTTCTCCATAT-CCCTATTAATCAATC-ATTTCCAACATACTC-ATAACCACAAACTCAAACTCATACTACTCA-CTATCCAATAACCTC-TTCATACACTTACAC-CTCCCAAATAACATTAACTTCACCCCTATA-3′ PNA stickers: Sv1 : 5′ -TGTAAGTGTA-3′ Sv2 : 5′ -AGGAGGGTGG-3′ Sv3 : 5′ -TAGGGATATG-3′ Sv4 : 5′ -GTTATGAGTA-3′ Sv5 : 5′ -GATAGTGAGT-3′ Sv6 : 5′ -GGGAGGTGTA-3′ . Molecular beacon probes: Pv1 : 5′ -CGCTC GTGTAAGTGTATGAA GAGCG-3′ Pω1 : 5′ -CGCTC AAGAGATTGGTGTAA GAGCG-3′ Pv2 : 5′ -CGCTC GGTGGATATAAATAG GAGCG-3′ Pω2 : 5′ -CGCTC TAGGAATTGTAGGAG GAGCG-3′ Pv3 : 5′ -CGCTC ATATGGAGAAATGGA GAGCG-3′ Pω3 : 5′ -CGCTC GATTGATTAATAGGG GAGCG-3′ Pv4 : 5′ -CGCTC GAGTATGTTGGAAAT GAGCG-3′ Pω4 : 5′ -CGCTC TGAGTTTGTGGTTAT GAGCG-3′ Pv5 : 5′ -CGCTC TGAGTAGTATGAGTT GAGCG-3′ Pω5 : 5′ -CGCTC GAGGTTATTGGATAG GAGCG-3′ Pv6 : 5′ -CGCTC GTGTAAGTGTATGAA GAGCG-3′ Pω6 : 5′ -CGCTC AATGTTATTTGGGAG GAGCG-3′ Ps : 5′ -CGCTC TATAGGGGTGAAGTT GAGCG-3′ Ds : 5′ -CGCTC TATAGGGGTGAAGTT GAGCG-3′ . The underlined sequences above are 15 mer loop sequences bounded by 5 bps stem sequence of molecular beacon probes. Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 4. Illustration of Sort operation.
Step 2: For the edge v1 v3 in the complement graph, divide tube T into T1 and T3 , add sticker Sv1 into T1 , Sv3 into T3 , and then anneal. These operations serve to set bit regions v1 in tube T1 , v3 in tube T3 on. Then use molecular beacon probe Pv1 to separate those memory complexes with v1 bit region off in T1 , use molecular beacon probe Pv3 to separate those memory complexes with v3 bit region off in T3 . These operations serve to see that every bit region is correctly set on. Because those memory complexes with bit region off must be captured by the probes, and the captured memory complexes are discarded. Use molecular beacon probe Ps to separate all memory complexes in T1 and T3 from added stickers. Wash in zero salt concentration, collect the captured memory complexes and pour them again into tube T . These operations serve to exclude extra, unhybridized stickers in the tubes, although surplus stickers do not influent the next cycle of computation. At this stage, tube T contains memory complexes that encode correct cliques. If there are not edges any more in the graph, the cliques are {v2 , v3 , v4 , v5 , v6 } or {v1 , v2 , v4 , v5 , v6 }. Similar operations are performed until every edge in the complement graph is processed. At this stage, tube T contains memory complexes that encode all cliques: {v1 , v5 }, {v2 , v5 }, {v3 , v5 }, {v4 , v5 }, {v5 , v6 }, {v1 , v2 , v5 }, {v1 , v4 , v5 }, {v2 , v3 , v5 }, {v3 , v4 , v5 }, {v3 , v5 , v6 }, {v4 , v5 , v6 } and {v3 , v4 , v5 , v6 }. Step 3: Use probes Pω1 , Pω2 , Pω3 , Pω4 , Pω5 , and Pω6 to sort memory complexes. When this step terminates, 26 tubes are produced. Tube Tk (0 ≤ k ≤ 63) contains memory complexes with the known weight of clique. Next we only need to detect the presence of the memory complexes in each of them. Fig. 4 illustrates in detail the operations. Step 4: Add probes Ds in Tk (0 ≤ k ≤ 63) to detect the presence or absence of common sequence s, fluorescent image, the detected fluorescent signal means there is target sequence s in the tube, i.e., there exist memory complexes in Tk . Return maximum weight clique and terminate the algorithm. Fig. 5 illustrates in detail the operations. 4. Discussion As far as complexity concerned, for a n vertices and m edges maximum weight clique problem, we need to encode 2n + 1 oligonucleotide sequences (n is the cardinality of the set of vertex in the graph). Therefore, 2n sequences are used for encoding vertices and their weight, and one common sequence is used for separation and detection. The memory strand is the concatenation of the sequences (note here, once these sequences are chosen, the corresponding sequences of stickers and DNA probes are thus assured according to Watson–Crick complementary Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.
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Fig. 5. Illustration of Detect operation.
rule), thus encoding complexity is o(n). Since edges to be considered are n(n − 1)/2 − m at most, the time complexity is o(n 2 ). In order to determine maximum weight clique of the graph, 2n tubes are needed in the Sort operation. Several aspects should be taken into careful consideration to see that the final result can be obtained, though some of these concerns have been addressed before. The top priority is that stickers may fall off from memory strand during the course of the computation. In the proposed method, we have chosen PNA molecule as sticker. The motivation of using PNA molecule as sticker comes from the fact that decreasing salt concentration will cause PNA sticker/memory strand hybrids to bind more strongly while DNA probe/memory strand hybrids to bind weakly. Therefore, in the proposed method, separation captured memory complex from DNA probe increases the binding affinity of sticker to memory strand rather than decreasing, compared with previously reported DNA algorithm based on sticker model. Thus, the possibility that sticker might fall off from memory strand might be decreased. But, this requirement sacrifices the kinetics of probe. In short, reasonable trade-off must be made between binding affinity of PNA stickers’ to memory strand and kinetics of probe. The second concern is the rate of mis-hybridization, since separation by hybridization is the central mechanism of sticker model. We have employed novel molecular beacon probe to perform reliable separation. The third concern is that strand lost due to memory strand might stick to the wall of tubes during computation, especially, combine operation. At present, this is a common question in test tube based DNA computing. We reasonably assume that the lost DNA strand is a minute fraction of the total which would be unimportant to whole computation. Alternatively, we may increase the amount of memory strand and sticker to compensate the strand lost during the computation. The fourth concern is the design of memory and stickers. A well designed encoding strategy for stickers may improve sticker specificity, thus improve the overall performance of our method. Similarly, good encodings for memory strand may prevent it from annealing to itself and forming unexpected secondary structure, causing data bits inaccessible. Encoding problem has been addressed extensively in DNA computing. We have not addressed the problem here. Acknowledgments The authors would like to thank every author appeared in the references. This project is supported by the National Natural Science Foundation of China (Grant No. 61672001). References [1] G. Bonnet, S. Tyagi, A. Libchaber, F.R. Kramer, Thermodynamic basis of the enhanced specificity of structured DNA probes, Proc. Natl. Acad. Sci. USA l96 (1999) 6171–6176. [2] X.H. Fang, X.J. Liu, S. Schuster, W.H. Tan, Designing a novel molecular beacon for surface-immobilized DNA hybridization studies, J. Am. Chem. Soc. 121 (1999) 2921–2922. [3] T. Fumiaki, K. Atsushi, M. Yamamoto, Design of nucleic acid sequences for DNA computing based on a thermodynamic approach, Nucleic Acids Res. 33 (2005) 903–911. [4] J.Y. Lee, S.Y. Shin, T.H. Park, Solving traveling salesman problems with DNA molecules encoding numerical values, BioSystems 78 (2004) 39–47. [5] R.N. Ma, Q. Zhang, L. Gao, J. Xu, Using DNA to solve the maximum weight clique of graphs, Acta Electron. Sin. 32 (1) (2004) 13–16. [6] A. Narayanan, S. Zorbalas, DNA algorithms for computing shortest paths, in: Proceedings of the Genetic Programming, Morgan Kaufmann, 1998, pp. 718–723. [7] E.P. Nielsen, E.H. Michael, R.H.B. Egholm, B.H. Ole, Sequence-selective recognition of DNA by strand displacement with a thyminesubstituted polyamide, Science 254 (1991) 497–1500. [8] S. Roweis, E. Winfree, R. Burgoyne, N. Chelyapov, M. Goodman, P. Rothemund, L. Adleman, A sticker based architecture for DNA computation, in: Proceedings of the Second Annual Meeting on DNA Based Computers, in: DIMACS: Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, Providence, PI, 1996.
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Please cite this article in press as: Z.X. Yin, et al., Molecular beacon computing model for maximum weight clique problem, Mathematics and Computers in Simulation (2017), http://dx.doi.org/10.1016/j.matcom.2017.02.003.