Design of Nucleic Acid Sequences for DNA Computing Based on a Thermodynamic Approach

Title Design of nucleic acid sequences for DNA computing based on a thermodynamic approach

Author(s) Tanaka, Fumiaki; Kameda, Atsushi; Yamamoto, Masahito; Ohuchi, Azuma

Nucleic Acids Research, 33(3), 903-911 Citation https://doi.org/10.1093/nar/gki235

Issue Date 2005

Doc URL http://hdl.handle.net/2115/64504

Type article

File Information gki235.pdf

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Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP Published online February 8, 2005

Nucleic Acids Research, 2005, Vol. 33, No. 3 903–911 doi:10.1093/nar/gki235 Design of nucleic acid sequences for DNA computing based on a thermodynamic approach Fumiaki Tanaka1,*, Atsushi Kameda2, Masahito Yamamoto1,2 and Azuma Ohuchi1,2

1Graduate School of Engineering, Hokkaido University, North 13, West 8, Kita-ku, Sapporo 060-8628, Japan and 2CREST, Japan Science and Technology Corporation, 4-1-8, Honmachi, Kawaguchi, Saitama, 332-0012, Japan

Received October 19, 2004; Revised and Accepted January 20, 2005

ABSTRACT order to satisfy the constraints based on the physicochemical properties of nucleic acids. In particular, it is essential to We have developed an algorithm for designing prevent undesired hybridization. In DNA computing, multiple multiple sequences of nucleic acids that have a sequences need to be designed that do not hybridize non- uniform melting temperature between the sequence specifically with each other (10,11), while in RNA secondary and its complement and that do not hybridize non- structure design, a single sequence needs to be designed that specifically with each other based on the minimum folds into the desired secondary structure (12–15). For free energy (DGmin). Sequences that satisfy these example, 40 DNA sequences of length 15 were designed to constraints can be utilized in computations, various prevent undesired hybridization and used for solving a 20- engineering applications such as microarrays, and variable instance of a three-satisfiability problem (2). In the nano-fabrications. Our algorithm is a random field of microarrays, 69 122 DNA sequences of length 45–47 generate-and-test algorithm: it generates a candidate were designed to hybridize specifically to 24 502 transcripts from Arabidopsis thaliana (4). Furthermore, four DNA sequence randomly and tests whether the sequence sequences of length 26 and four DNA sequences of length satisfies the constraints. The novelty of our algorithm 48 were used to construct a periodic two-dimensional crystal- is that the filtering method uses a greedy search to line lattice (7). These sequences were carefully designed for calculate DGmin. This effectively excludes inappropri- the intended hybridization. Thus, an algorithm/program that ate sequences before DGmin is calculated, thereby generates multiple sequences, which do not hybridize non- reducing computation time drastically when specifically with each other, is useful for various applications compared with an algorithm without the filtering. of nucleic acid. Experimental results in silico showed the superiority We applied a thermodynamic approach to the nucleic acid of the greedy search over the traditional approach design for DNA computing. In particular, we focused on designing a pool P containing n sequences of length l for which based on the hamming distance. In addition, experi- À (i) the duplex melting temperature (TM) is in the range TM to mental results in vitro demonstrated that the experi- þ TM for any pairwise duplex of a sequence in P and its com- mental free energy (DGexp) of 126 sequences plement and (ii) the minimum free energy (DGmin) is greater correlated well with DGmin ( jRj = 0.90) than with the Ã than a threshold (DGmin) in any pairwise duplex of sequences hamming distance ( jRj = 0.80). These results validate in P and any concatenation of two sequences in P plus their the rationality of a thermodynamic approach. We complements except for the pairwise duplex of a sequence in P implemented our algorithm in a graphic user and its complement. Traditional approaches to the sequence interface-based program written in Java. design in DNA computing have approximated the stability between two sequences using the hamming distance (i.e. the number of base pairs) rather than DGmin. However, INTRODUCTION since the hamming distance is only an approximation of the Nucleic acids are now being utilized in computations (1–3), stability, DGmin is preferable for predicting the stability. In various engineering applications such as microarrays (4–6), practice, an RNA secondary structure can be adequately and nano-fabrications (7–9). In these fields of research, called predicted using DGmin rather than the number of base pairs ‘DNA computing’, nucleic acid design or sequence design is a (16). However, the algorithm for calculating the DGmin for crucial problem in engineering using nucleic acids. Nucleic double-stranded DNA requires time complexity O(l3), acid design is deciding the base sequences (e.g. ‘GCTAGCT- where l is the length of the sequence, as is the case with AGGTTTA’, ..., ‘ATCGTACGCTATGTGCA’ in DNA) in secondary structure prediction for single-stranded RNA.

*To whom correspondence should be addressed. Tel: +81 11 716 2111, ext 6498; Fax: +81 11 706 7834; Email: fumiaki@dna-comp.org

The online version of this article has been published under an open access model. Users are entitled to use, reproduce, disseminate, or display the open access version of this article for non-commercial purposes provided that: the original authorship is properly and fully attributed; the Journal and Oxford University Press are attributed as the original place of publication with the correct citation details given; if an article is subsequently reproduced or disseminated not in its entirety but only in part or as a derivative work this must be clearly indicated. For commercial re-use, please contact [email protected] 904 Nucleic Acids Research, 2005, Vol. 33, No. 3

Since all the combinations of three sequences must be evalu- paper, we formulated li such that li = lj = l (0 < i, ated in this design, dozens of sequences cannot be designed j < n À 1), although we can extend our algorithm easily within a reasonable computation time. Another approach that for li „ lj (0 < i „ j < n À 1). We define P as the pool of reduces the computation time is thus needed. Andronescu et al. n sequences, with length l, to be designed. Furthermore, let (13) overcame this drawback in the secondary structure design P = fU0, U1, ..., UnÀ1g and Q = fV0, V1, ..., VnÀ1g such that À þ by hierarchically decomposing the secondary structure into Vi (0 < i < n À 1) is the complement of Ui. TM and TM are smaller substructures and integrating the partial sequences. defined as the lower and upper thresholds of TM for the duplex Ã By using this method, they designed sequences that a tradi- between a sequence and its complement. Moreover, DGmin is tional program cannot. defined as the threshold of DGmin given by the sequence We have developed a random generate-and-test algorithm designer. Thus, our algorithm designs a pool P containing that generates a candidate sequence randomly and tests n sequences of length l for which (i) the duplex TM is in the À þ whether the sequence satisfies the constraints. It stores the range from TM to TM for any duplex of Ui (0 < i < n À 1) and Ã sequence in a sequence pool if and only if it satisfies all Vi, and (ii) DGmin is greater than DGmin for any pairwise duplex the constraints. To reduce the difficulty in computation of sequences in P and any concatenation of two sequences time, we use a greedy search to calculate DGmin. The advant- in P [ Q except for the pairwise duplex of Ui and Vi. age of a greedy search is that it approximates DGmin in less Here, we describe more specifically the combination of 2 time, with time complexity O(l ), than a rigorous algorithm, sequences to be calculated using the DGmin. The sequence 3 which calculates DGmin with time complexity O(l ). The Ui (0 < i < n À 1) is denoted by a string of bases such as 0 1 lÀ1 0 0 DGmin approximated using the greedy search (denoted by ui ui ÁÁÁui (5 to 3 direction), and similarly, Vi (0 < i < 0 1 lÀ1 0 0 DGgre) correlated well with DGmin. The correlation coefficients n À 1) is denoted as vi vi ÁÁÁvi (5 to 3 direction). For 0 0 0 were 0.95, 0.85 and 0.76 at 20mer, 40mer and 60mer lengths, example, if Ui = 5 -AAATTTCCCGGG-3 , then Vi = 5 -CCC- 0 respectively. Furthermore, since DGgre is the upper bound GGGAAATTT-3 . Furthermore, let be the combination for DGmin (i.e. DGmin < DGgre), the sequence such that of sequences X and Y, and XY be the concatenation of se- Ã Ã DGgre < DGmin is sure to satisfy DGmin < DGmin. Therefore, quences X and Y in that order. For example, if Ui, Uj and 0 0 0 using a greedy search excludes in advance most inappropriate Uk (0 < i, j, k < n À 1) are 5 -AAATTT-3 ,5-CCCGGG- 0 0 0 sequences before DGmin is calculated without excluding an 3 and 5 -TCTCTC-3 , respectively, then means the appropriate sequence. With this approach, our algorithm combination of sequences 50-AAATTTCCCGGG-30 and 50- reduces computation time. TCTCTC-30. In our algorithm, the following combinations To evaluate our algorithm, we investigated the effectiveness are considered for the DGmin calculation. of the greedy search filtering. We compared the computation (i) hU U U i (0 < i, j, k < n À 1) time of our algorithm with that of the same algorithm but i j k (ii) hU U V i (0 < i, j, k < n À 1), i „ k without the greedy search filtering. The experimental results i j k (iii) hU V U i (0 < i, j, k < n À 1), i „ j showed that the greedy search filtering reduces the total com- i j k (iv) hU V V i (0 < i, j, k < n À 1), (i „ j) ^ (i „ k). putation time drastically. For example, using the filtering i j k 0 reduced the computation time to 83% for 30 sequences For generality, we use two sequences, S(=s0 s1 ÁÁÁsNÀ1)(5 to 0 0 0 with length 20 and to 87% for 20 sequences with length 15 3 direction) and T(=t0 t1 ÁÁÁtMÀ1)(3 to 5 direction), to À þ Ã when TM ¼ 69:58, TM ¼ 72:58 and DGmin ¼À10:0, and describe the DGmin calculation in detail. The sequences are À þ Ã TM ¼ 60:49, TM ¼ 63:49 and DGmin ¼À7:0, respectively. defined to be antiparallel to each other. Note that any two In addition, we compared the greedy search with the tradi- sequences can be represented by S and T. In this paper, tional approach based on a hamming distance in terms of the S and T represent Ui and the reverse sequence of XjYk filtering performance. We demonstrated that the greedy search (Xj 2fUj, Vjg, Xk 2fUk, Vkg). can filter out inappropriate sequences better than using the The notation si Á tj represents the base pair between the i-th hamming distance. In a laboratory experiment, we investig- base in sequence S and the j-th base in sequence T, hence the ated the correlation coefficient between DGmin and the experi- structure between S and T is a set of base pairs such that each mental free energy (DGexp) and that between the hamming base is paired at most once. In addition, (si Á tj, si0 Á tj0) 0 0 distance and the DGexp using 126 duplexes. The DGexp cor- fðÞ^0 < i < i < NÀ1 ðÞg0 < j < j < MÀ1 is defined as a related better with DGmin ( j R j = 0.90) than with the hamming structure in which base pairs si Á tj and si0 Á tj0 are formed and distance ( j R j = 0.80). the sequences siþ1siþ2 ÁÁÁsi0À1 and tjþ1tjþ2 ÁÁÁtj0À1 do not form To implement our algorithm, we developed a computer any base pairs. The term (x‚si0 Á tj0 ) fx 2fsi‚tjg‚ð0 < i < program called DNA-SDT, a graphic user interface (GUI)- i0 < NÀ1Þ^ðÞg0 < j < j0 < MÀ1 is defined as a structure based application written in Java. This program enables in which base pair si0 Á tj0 is formed and sequence sisiþ1 ÁÁÁ users to design DNA sequences with our algorithm and can si0À1 in the case x ¼ si (tjtjþ1 ÁÁÁtj0À1 in the case x ¼ tj) do not be downloaded freely from the web site (http://ses3.complex. form any base pair. Similarly, ðsi Á tj‚xÞfx 2fsi0 ‚tj0 g‚0ð < i < eng.hokudai.ac.jp/~fumi95/DNA-SDT/index.html). i0 < NÀ1Þ^ðÞg0 < j < j0 < MÀ1 is defined as a structure in which base pair si Á tj is formed and the sequence siþ1siþ2 ÁÁÁsi0 in the case x ¼ si0 (tjþ1tjþ2 ÁÁÁtj0 in the case ALGORITHM x ¼ tj0 ) do not form any base pair. Definition Outline

Let n be the number of sequences to be designed and li Our algorithm is a random generate-and-test algorithm that (0 < i < n À 1) be the length of each sequence. In this generates a sequence randomly and then stores it in the pool if Nucleic Acids Research, 2005, Vol. 33, No. 3 905

and only if the sequence satisfies all the constraints. The is calculated using a greedy search. Both DGmin and DGgre are main feature of our algorithm is that it uses a greedy search calculated based on the nearest-neighbor model, which calcu- for calculating DGmin to filter out inappropriate sequences. lates the total free energy as the summation of the contribu- The advantage of a greedy search is that it approximates tions of various elementary structures (18,19). The elementary DGmin in less time when compared with a well-known structures considered in this paper are stacking base pairs, dynamic programming algorithm (17). Therefore, using bulge loops, internal loops, dangling ends and free ends. the greedy search before the DGmin calculation to exclude The contributions of the stacking base pairs, defined as inappropriate sequences reduces the computation time. Here- si Á tj‚siþ1 Á tjþ1 , to the free energy are calculated using after, the approximated DGmin using a greedy search is denoted 12 parameters reported previously (19). The free energy con- by DGgre. tributions of the loop regions are sequence dependent The algorithm uses three filters: (15,16,20). The free energies of single bulge loops, defined as s Á t ‚s Á t or s Á t ‚s Á t , are calculated (i) T filter: checks whether the T of candidate sequence i j iþ2 jþ1 i j iþ1 jþ2 M M using 64 parameters covering all the possible combinations U ðÞ0 < c < n À 1 and V is in the range from TÀ to Tþ . c c M M of bulged base and flanking base pairs (19). The free energies If not, U is rejected. c of the other loops, bulge loops longer than one and internal (ii) DG filter: checks whether DG is greater than the gre gre loops, are calculated using conventional parameters and equa- threshold, DGÃ , for all the combinations above in P, pro- gre tions (20,21). Bulge loops longer than one are defined as vided that the candidate sequence U ðÞ0 < c < n À 1 or c f s Á t ‚s Á t ^ ðÞgl>3 or s Á t ‚s Á t ^ ðÞl > 3 , V is included in that combination. If the DG of any i j iþl jþ1 i j iþ1 jþl c gre and the internal loops are defined as s Á t ‚s Á t ^ combination is less than or equal to DGÃ , U is rejected. i j iþl jþm gre c ðÞl‚m>2 . The free energies of dangling ends, defined as (iii) DG filter: checks whether DG is greater than the min min ðÞs ‚s Á t , ðÞt ‚s Á t , ðÞs Á t ‚s or ðs Á t ‚ threshold, DGÃ , for all the combinations above in P, 0 1 0 0 0 1 NÀ2 MÀ1 NÀ1 NÀ1 MÀ2 min t Þ, are calculated using 32 parameters covering all the provided that the candidate sequence U ð0 < c < MÀ1 c possible combinations (22). The free ends are defined as n À 1Þ or Vc is included in that combination. If the the sequences s0 ÁÁÁsi and t0 ÁÁÁtj closing with si Á tj such DGmin of any combination is less than or equal to Ã that both si0ðÞpseudoknot-free constraint in the RNA secondary structure prediction. At this point, we do Input: n, l, TÀ , Tþ , DGÃ , DGÃ M M gre min not consider intra-molecular base pairs or the interactions Output: pool P consisting of n sequences with length l. between loop regions. Procedure: (i) Initialize pool P as an empty set. (ii) Iterate the following procedure until P has n sequences. TM filter (a) Generate candidate sequence Uc ðÞ0 < c < n À 1 This filter checks whether a candidate sequence paired to its with length l randomly, then add U to P. À þ c complement has a TM in the range TM to TM. (b) Evaluate U with T filter. If U is rejected, exclude U c M c c from P and return to ii(a). DH TM ¼ þ DS ‚ 1 (c) Evaluate Uc with DGgre filter. If Uc is rejected, exclude R ln ðCT=aÞ Uc from P and return to ii(a). (d) Evaluate Uc with DGmin filter. If Uc passes, leave Uc in where R is the gas constant, CT is the concentration, DH is P; else exclude Uc from P and return to ii(a). the enthalpy and DS is the entropy. Parameter a is set to 1 for self-complementary and to 4 for non-self-complementary. The order of the filters is important. Each candidate Parameters DH and DS are calculated based on the sequence should be evaluated in this order to reduce the com- nearest-neighbor model (18,19). putation time. If pool P has m sequences, the time complexities to evaluate the (m + 1)-th candidate sequence are O(l ), O(m2l2) 2 3 DGgre filter and O(m l ) at the TM, DGgre and DGmin filters, respectively. Ã By evaluating the candidate sequences in ascending order of This filter checks whether DGgre is greater than DGgre for all time complexity, the inappropriate ones can be excluded the combinations as mentioned in Algorithm. sooner. Using a ‘greedy search’ reduces the computation time for calculating DGmin. The greedy search works well because a structure with DGmin tends to include stable helices (i.e. con- Energy model tinuous complementary regions). Therefore, the greedy search The DGmin between S and T is calculated using a dynamic approximates DGmin by iteratively searching for the most programming algorithm (17), while the DGgre between S and T stable helix and fixing the helix. 906 Nucleic Acids Research, 2005, Vol. 33, No. 3

R The greedy search algorithm is as follows: where DGfreeEnd is the free energy of the helix with a R 1. First, calculate the free energies of all helices over the free end and DGcore is that of one without a free end. (d) Search for a region where DGR is minimum over all structure between sequence s0s1 ÁÁÁsNÀ1 and sequence freeEnd helices. This region is freshly denoted as si00 si00þ1 ÁÁÁsk00 t0t1 ÁÁÁtMÀ1. Calculate the free energies of the helices 0 0 0 0 with free ends using the following equations. Here, a (5 ! 3 ) and tj00 tj00þ1 ÁÁÁtl00ðÞ¼j00þk00Ài00 (3 ! 5 ). Further- 0 0 more, the free energies of the regions with and without a helix is denoted by sisiþ1 ÁÁÁsk (5 ! 3 ) and R R 0 0 free end are freshly denoted as DGfreeEnd and DGcore, tjtjþ1 ÁÁÁtlðÞ¼jþkÀi (3 ! 5). R respectively. If DG < DtðÞMÀ1‚tl‚sNÀ1‚sk holds, DGfreeEnd P¼ DGcoreþDs0‚si‚t0‚tj þDtðÞMÀ1‚tl‚sNÀ1‚sk freeEnd kÀ1 fix the base pairs, si00 Á tj00 ‚si00þ1 Á tj00þ1‚ ...‚sk00 Á tl00 , and DGcore ¼ a¼i eS sa‚saþ1‚tjþaÀi‚tjþaÀiþ1 00 00 R DG is the free energy of a helix with free ends; DG update k, l and DGcore to k , l and DGcore þ DGcore, freeEnd core respectively. This means that the base pairs in the region is that without free ends. Ds0‚si‚t0‚tj represents the free energy contribution of the dangling or free end between are energetically favorable. 0 0 0 0 4. Calculate DGgre using sequence 5 -s0s1 ÁÁÁsi-3 and sequence 3 -t0t1 ÁÁÁtj-5 closing with base pair si Á tj. For ðÞî ¼ 0 ðÞj ¼ 0 , DGgre ¼DGcoreþDs0‚si‚t0‚tj þDtðÞþMÀ1‚tl‚sNÀ1‚sk init‚ Ds0‚si‚t0‚tj is zero because there is no dangling or free end. eS s ‚s ‚t ‚t is the free energy of the a aþ1 jþaÀi jþaÀiþ1 where init is the energy penalty for forming double-stranded stacked base pair between sequence 50-s s -30 and a aþ1 DNA. If DG > 0, however, set DG = 0. This is because sequence 30-t t -50. gre gre jþaÀi jþaÀiþ1 the free energies are calculated relative to two non- 2. Search for a minimal value of DG over all helices. freeEnd interacting sequences, for which the free energy is defined Then, fix the base pairs in the helix where DG is mini- freeEnd as zero. Thus, two sequences must remain separate with mum. This region is freshly denoted as s s ÁÁÁs (50 ! 30) i iþ1 k zero free energy rather than form a structure with positive and t t ÁÁÁt (30 ! 50). Furthermore, the free ener- j jþ1 lðÞ¼jþkÀi free energy. gies of the regions with and without free ends are freshly denoted as DGfreeEnd and DGcore, respectively. In the above procedure, the number of iterations for a 3. Iterate the following procedure d times, where d is a para- search, d, is defined as ‘degree’. A structure with degree = 0 meter defined below. has at most one helix. Note that base pairing does not occur (a) IfðÞ^ 0 < iÀ1 ðÞ0 < jÀ1 holds, calculate the free ener- during a greedy search when more than two continuous com- gies of all helices with the loop closing with base pair plementary bases do not exist between two sequences. For si Á tj over the structure between sequence s0s1 ÁÁÁsiÀ1 degree = 1, there are at most three helices. Eventually, for and the sequence t0t1 ÁÁÁtjÀ1. Thus, the free energy degree = d, there are at mostðÞ 1 þ 2 Á d helices. If the length 0 0 of helix, denoted as si0 si0þ1 ÁÁÁsk0 (5 ! 3 ) and of sequences S and T are l and 2 Á l, respectively, the number of 0 0 tj0 tj0þ1 ÁÁÁtl0ðÞ¼j0þk0Ài0 (3 !5 ), is calculated as follows: iterations is at mostðÞl À 1 =3. Thus, the time complexity L L 2 3 DG ¼ DG þ Ds‚s 0 ‚t ‚t 0 of greedy search O degree Á l is Ol at worst. However, freeEnd Pcore 0 i 0 j L k0À1 because the degree increases slowly with the sequence length, DG¼ 0 eS s ‚s ‚t 0 0 ‚t 0 0 þ core a¼i a aþ1 j þaÀi j þaÀi þ1 the time complexity of a greedy search can be in practice eL sk0 ‚si‚tl0 ‚tj ‚ 2 L regarded as Ol . To confirm this, we calculated the degree where DGfreeEnd is the free energy of the helix with a free L for 10 000 random pairs of sequences from 10mer to 100mer in end,DG is that of one without a free end and core steps of 10mer. As shown in Table 1, degree was at most 9 at eL s 0 ‚s ‚t 0 ‚t is the free energy contribution of a k i l j 80mer and 90mer, while it was 33 [=(100 À 1)/3]) at 100mer in bulge or internal loop between sequence s 0 s 0 ÁÁÁs k k þ1 i the worst case. Furthermore, degree was rarely >6(< 1%), and sequence t 0 t 0 ÁÁÁt closing with base pairs l l þ1 j and, for >98% of the 10 000 random pairs, degree was in the sk0 Á tl0 and si Á tj. (b) Search for a region where DGL is minimum over all range 0–4 at any length. Therefore, degree was nearly constant freeEnd regardless of the sequence length on average, indicating that helices. This region is freshly denoted as si0 si0þ1 ÁÁÁsk0 2 0 0 0 0 the time complexity of a greedy search is Ol in practice. (5 ! 3 ) and tj0 tj0þ1 ÁÁÁtl0ðÞ¼j0þk0Ài0 (3 ! 5 ). Further- more, the free energies of the regions with and without a L L free end are freshly denoted asDGfreeEnd and DGcore, DGmin filter L respectively. If DGfreeEnd < Ds0‚si‚t0‚tj holds, fix Ã This filter checks whether the DGmin is greater than DGmin for the base pairs, si0 Á tj0 ‚si0þ1 Á tj0þ1‚...‚sk0 Á tl0 , and update i, j and DG to i0, j0 and DG þ DGL , all the combinations as mentioned in Algorithm. core core core DG between S and T can be decomposed into two terms: respectively. This means that the base pairs in the min region are energetically favorable. DGmin ¼ min fDs0‚si‚t0‚tj þ Vsi‚tj g‚ 2 (c) If ðÞ^k þ 1 < NÀ1 ðÞl þ 1 < MÀ1 holds,calculatethe 0< i < NÀ1 0< j < MÀ1 free energies of all helices with a loop closing with base pair sk Á tl over the structure between sequence skþ1 where Vsi‚tj represents the minimum value of the free en- 0 0 skþ2 ÁÁÁsNÀ1 andsequencetlþ1tlþ2 ÁÁÁtMÀ1.Thus,thefree ergy between sequence sisiþ1 ÁÁÁsNÀ1 (5 ! 3 direction) and 00 00 00 0 0 0 0 energy of helix, denoted as si si þ1 ÁÁÁsk (5 ! 3 )and sequence tjtjþ1ÁÁÁtMÀ1 (3 ! 5 direction) closing with si Á tj. 0 0 tj00 tj00þ1 ÁÁÁtl00ðÞ¼j00þk00Ài00 (3 ! 5 ), is calculated as follows: Recall that Ds‚s ‚t ‚t represents the free energy of the R R 0 i 0 j DG DG Dt ‚t 00 ‚s ‚s 00 0 0 freeEnd ¼P 00core þðÞMÀ1 l NÀ1 k dangling or free ends between sequence s0s1 ÁÁÁsi (5 ! 3 R k À1 0 0 DGcore ¼ a¼i00 eS sa‚saþ1‚tj00þaÀi00 ‚tj00þaÀi00þ1 þ direction) and sequence t0t1 ÁÁÁtj (3 ! 5 direction) closing 00 00 eL sk‚si ‚tl‚tj ‚ with base pair si Á tj. Nucleic Acids Research, 2005, Vol. 33, No. 3 907

Table 1. Distribution of degree for greedy search for 10 000 random pairs of sequences from 10mer to 100mer in steps of 10mer

Degree Sequence length (mer) 10 20 30 40 50 60 70 80 90 100

0 9426 7941 6679 5813 5032 4544 3973 3575 3196 2931 1 560 1838 2783 3238 3638 3723 3953 3856 3861 3863 2 14 204 471 767 1040 1301 1467 1752 1959 2067 3 0 17 57 161 244 336 458 596 698 770 4 0 0 10 16 38 78 106 165 206 242 5 0 0 0 4 7 13 28 43 48 89 6 0 0 0 1 1 5 15 12 24 30 7 0000000047 8 0000000031 9 0000000110

Furthermore, Vsi‚tj is calculated using Figure 1 shows that using our algorithm reduced the computation time drastically. For example, our algorithm needed Vðsi; tjÞ¼minfDðtMÀ1, tj, sNÀ1, siÞ, eSðsi, siþ1, tj, tjþ1Þ 1.3 h to design 30 sequences with length 20 for Ã þVðsiþ1, tjþ1Þ, VBIðsi, tjÞg, 3 DGmin ¼À10:0, while the algorithm without the DGgre filter needed 7.7 h (Figure 1a and b). This means that the DGgre where VBIðsi; tjÞ represents the minimum value of the free filter effectively excludes the sequences that cannot pass energy forming a bulge or internal loops closing with base pair through the DGmin filter. si Á tj. Recall that eS si‚siþ1‚tj‚tjþ1 is the free energy of the A comparison of Figure 1a and b clearly shows the primacy 0 0 Ã Ã stacked base pair between sequence sisiþ1 (5 ! 3 direction) of our algorithm for DGmin ¼À10:0 versus DGmin ¼À13:0. 0 0 and sequence tjtjþ1 (3 !5 direction). The first term represents For instance, our algorithm reduced computation time to Ã the case in which the unpaired end consists of sequences 83% (7.7 ! 1.3 h) for 30 sequences and DGmin ¼À10:0 0 0 0 0 tMÀ1tMÀ2 ÁÁÁtj (5 !3 direction) and sNÀ1sNÀ2 ÁÁÁsi (3 ! 5 while it reduced computation time to 57% (9.1 ! 3.9 h) Ã direction) closing with base pair tj Á si. The second term cor- for 69 sequences and DGmin ¼À13:0. This is because both responds to the case in which the stacked base pair is energet- algorithms can find sequences that satisfy the constraints more Ã Ã ically favorable. In this case, Vsiþ1‚tjþ1 is calculated easily for DGmin ¼À13:0 than for DGmin ¼À10:0. A similar recursively. The third term is calculated using trend is seen in the sequences with length 15 (Figure 1c and d). Therefore, using the DGgre filter enables sequences to be VBI si‚tj ¼ min feL si‚si0 ‚tj‚tj0 þ Vsi0 ‚tj0 g: 4 designed in less time, particularly when the threshold is high. i < i0‚ j < j0 i0Àiþj0Àj>2 Filtering performance of DGgre filter versus Recall that eL si‚si0 ‚tj‚tj0 represents the free energy hamming distance contribution of the loops between sequence s s ÁÁÁs 0 and i iþ1 i The function of the DGgre filter is to filter out the inappropriate sequence tjtjþ1 ÁÁÁtj0 closing with base pairs si Á tj and si0 Á tj0 . sequences from many sequences generated randomly; the The DG is calculated recursively using Equations 2–4 by min promising sequences are then checked using the DGmin filter. dynamic programming. If we compute the VBI term in a 4 This means that the filtering performance of the DGgre filter can straightforward manner, its time complexity is Ol . How- be evaluated using four terms: the number of sequences with ever, this can be reduced to Ol3 using the algorithm of both DGmin and DGgre higher than the threshold (true pos- Lyngsø et al. (23). itive: TP), that with DGmin higher but not with DGgre (false negative: FN), that with DGgre higher but not with DGmin (false positive: FP) and that with neither DGmin nor with DGgre higher RESULTS (true negative: TN) (Figure 2). There is a trade-off between FN and FP, which are the number of sequences incorrectly Effectiveness of DGgre filter excluded by the DGgre filter and that incorrectly passing To evaluate the effectiveness of the DGgre filter, we compared through the DGgre filter, respectively. To evaluate the DGgre the computation time of our algorithm with that of the algo- filter, we compared the filtering performance of the DGgre filter rithm without the DGgre filter. That algorithm checks a ran- with that using the hamming distance (called hamming filter) domly generated sequence by using the TM and DGmin filters with respect to FN such that FP = 0 and to FP such that and then stores the sequence in the pool if and only if the FN = 0 for 10 000 random pairs of sequences with length sequence passes both filters. All the computational experi- 20. Setting FP to 0 means that the DGgre or hamming filter ments described in this section were performed using certainly excludes inappropriate sequences with a DGmin less Ã Windows 2000 on a computer with an Athlon 1.4 GHz than or equal to DGmin. Therefore, the lower the number in FN CPU and 256 MB of memory. The results are shown in such that FP = 0, the better the filter. Similarly, because FN = 0 Figure 1. The computation time grew exponentially because means that the DGgre or hamming filter never excludes appro- Ã the number of three-sequence combinations increased expo- priate sequences with a DGmin greater than DGmin, the filter nentially with the number of sequences. should have fewer sequences in FP such that FN = 0. 908 Nucleic Acids Research, 2005, Vol. 33, No. 3

(a) (b)

(c) (d)

À þ Figure 1. Number of sequences designed versus computation time for two design strategies up to 10 h. In (a) and (b), l = 20, TM ¼ 69:58 and TM ¼ 72:58. In Ã Ã Ã Ã À þ Ã Ã (a), DGgre ¼ DGmin ¼À10:0; and in (b), DGgre ¼ DGmin ¼À13:0. In (c) and (d), l = 15, TM ¼ 60:49 and TM ¼ 63:49. In (c), DGgre ¼ DGmin ¼À7:0; and in Ã Ã (d), DGgre ¼ DGmin ¼À11:0.

Another advantage of the DGgre filter is that it guarantees having the threshold where the number of FN is zero because the DGgre is the upper bound for the DGmin (i.e. DGmin < DGgre). For example, a pair of sequences having a DGgre less than or equal to À5.0 kcal/mol is guaranteed to have a DGmin of at most À5.0 kcal/mol. Thus, the sequences with a DGgre of more than À5.0 kcal/mol include all sequences with a DGmin of more than À5.0 kcal/mol. Therefore, setting the Ã threshold for a DGgre filter (DGgre) to that for the DGmin filter Ã Ã Ã (DGmin) (i.e. DGgre ¼ DGmin) guarantees that the number of FN is zero. This is not the case for hamming filter. Figure 2. Plot of DGgre versus DGmin for 10 000 random pairs of sequences with length 20; DGgreÃ and DGminÃ represent threshold of DGgre and that of DGmin, respectively. Four terms were used for evaluating filtering performance: TP, Comparison between hamming distance and fðDGgreÃ

(a) (b)

Figure 3. Threshold of DGmin versus number of sequences classified as FP and FN. (a) FP such that FN is zero and (b) FN such that FP is zero.

(a) (b)

Figure 4. (a) ÀBP versus experimental free energy. BP represents the number of complementary bases. (b) Predicted minimum versus experimental free energy. In (a) and (b), values and lines are correlation coefficients and regression lines, respectively.

Then, to compare the hamming distance with DGmin,we ours was 0.80. This is because our data were derived from investigated the correlation coefficient with DGexp. To fairly only sequences with a simple structure, i.e. with at most one compare different length sequences, we used the number of loop (single bulge loop). In general, the more loops, the higher complementary bases but not the hamming distance. For the discrepancy between the number of complementary bases example, although sequences 50-GGG-30 and 30-CCC-50 and the experimental stability. Therefore, the correlation be- with three complementary bases and sequences 50-GGGGG- tween the number of complementary bases and the experi- 30 and 30-CCCCC-50 with five complementary bases both have mental stability will be close to theirs with respect to more zero hamming distance, the stability of the latter must be complex sequences, i.e. with more than two loops. These higher than that of the former. Thus, the number of comple- results demonstrate the ability of our program to approximate mentary bases is appropriate for comparing the stability of the experimental stability of double-stranded DNA based on sequences with different lengths. Note that the number of the DGmin calculation. complementary bases is equivalent to the hamming distance for sequences with the same length. The results are shown in Figure 4a and b. In Figure 4a, BP represents the number of DISCUSSION complementary bases. The correlation coefficient, j R j , between ÀBP and DGexp was 0.80, while that between In the previous section, we showed the effectiveness of DGmin and DGexp was 0.90, indicating that the DGmin is better the DGgre filter for filtering out the sequences that cannot than the number of complementary bases (i.e. hamming dis- pass through the DGmin filter. To further address the ques- tance) as a predictor of stability. tion how the DGgre is close to the DGmin, we investigated Bozdech et al. (5) compared the number of complementary theP average prediction error of DGgre, calculated using 10 000 i i bases with the binding energy (the calculation method and i¼1 ðGgreÀGminÞ=10 000. Figure 5 shows that the average nearest-neighbor parameters differed from ours) by using prediction error can be restricted to < 1.0 kcal/mol for 70mer oligonucleotides on microarray. They found that sequences shorter than 50mer, which are frequently used in j R j between the binding energy and relative intensity of DNA computing. hybridization (= intensity of fluorescence) was 0.91, while Figure 6 shows the number of sequences such that DGmin is it was 0.72 between the number of complementary bases equal to DGgre. The number decreased as the sequences be- and the relative intensity of hybridization. This is consistent came longer. For example, DGmin ¼ DGgre in 9250 pairs from with our findings, especially with respect to the correlation 10 000 pairs of sequences at 10mer, while DGmin ¼ DGgre in between the predicted energy (binding energy in theirs, DGmin 157 pairs from 10 000 pairs at 100mer. Therefore, DGgre is a in ours) and the stability derived from the experimental results good predictor of DGmin for short sequences, while it is only an (their j R j was 0.91, while ours was 0.90). With respect to approximation for long sequences. the correlation between the number of complementary bases For comparison with the hamming distance, the correlation and the experimental stability, their j R j was 0.72, while coefficient (jRj) was also calculated. The correlation between 910 Nucleic Acids Research, 2005, Vol. 33, No. 3

designed by the program. Of course, for any given combination of sequences, users can also determine the structure with the DGmin. The program is a GUI-based application written in Java, hence it can be executed on any computer that has the Java Runtime Environment (JRE) installed. Although older ver- sions of JRE supposedly run without problem, the latest version is preferable. We tested the program with JRE 1.4.2 on a Windows 2000 workstation, JRE 1.4.1 on a Windows XP workstation and JRE 1.4.2 on a Turbolinux Workstation 7.0. The program interface consists of two screens: one for FigureP 5. Average prediction error of DGgre, calculated using structure prediction and one for sequence design. The left- 10 000ðDGi ÀDGi Þ=10 000. i¼1 gre min hand side of the window is the screen for structure prediction, which outputs the structure with the DGmin from the two sequences input in the text box. The sequences in the text box can be converted into reverse or complementary sequences. The right-hand side is the screen for sequence design, which outputs the sequences, GC content and TM based on the constraints input in the text boxes. To avoid impossible operations during design, the procedure for sequence design is executed as a separate thread. Therefore, the structure-prediction screen can be operated freely when sequences are being designed. Furthermore, the design procedure can be monitored and interrupted anytime, and the interim results can be output. However, because the Figure 6. Number of sequences such that DGmin ¼ DGgre. button for running the sequence design procedure is locked during design, users cannot design another pool of sequences DGmin and DGgre decreased almost linearly from 0.99 at 10mer in parallel. to 0.63 at 100mer. Because long sequences tend to have more The number of sequences can be set up to 100. If more than helices, the discrepancy increased as the sequences became 100 sequences need to be designed, the user can choose longer. The correlation coefficient between DGmin and the ‘Unlimited’. In this case, the design procedure iterates until hamming distance, which decreased almost linearly from the user interrupts it. The sequence length is restricted at 8mer 0.46 at 10mer to 0.12 at 100mer, was much less than that to 50mer because of the computation time. Although the between DGmin and DGgre. thresholds for DGmin and DGgre can be set freely, using irrel- evant values resulting in zero sequences. We recommend set- Ã Ã ting these thresholds such that DGmin ¼ DGgreðÞ< 0 to guarantee zero false negatives (see Results). IMPLEMENTATION We implemented our algorithm in a program called ‘DNA Sequence Design Tool (DNA-SDT)’. DNA-SDT can be down- SUPPLEMENTARY MATERIAL loaded freely from the web site (http://ses3.complex.eng. Supplementary Material is available at NAR Online. hokudai.ac.jp/~fumi95/DNA-SDT/index.html). It has two functions: sequence design and structure prediction. In sequence design, the program solves the problem of ACKNOWLEDGEMENT designing a pool P containing n sequences of length l for which (i) the duplex T is in the range TÀ to Tþ for any Funding to pay the Open Access publication charges for this M M M article was provided by the Ministry of Education, Science, duplex of Ui ðÞ0 < i < nÀ1 and Vi and (ii) DGmin is greater Ã Sports and Culture, Grant-in-Aid for Scientific Research on than DGmin in any pairwise duplex of sequences in P and any concatenation of two sequences in P [ Q except for the pair- Priority Areas, 2002–2005, 14085201. wise duplex of Ui and Vi. This problem is solved using our Ã algorithm with the threshold for a DGgre filter, DGgre. The À þ Ã Ã REFERENCES parameters, n, l, TM, TM, DGgre and DGmin, are user- 1. Adleman,L.M. (1994) Molecular computation of solutions to defined. DNA-SDT has a table of average TM values at length ave combinatorial problems. Science, 266, 1021–1024. l ðÞ8 < l < 50 [TM ðÞl ] calculated from 10 000 sequences À þ 2. Braich,R.S., Chelyapov,N., Johnson,C., Rothemund,P.W. and generated randomly. Parameters TM and TM are set Adleman,L. (2002) Solution of a 20-variable 3-SAT problem on a ave ave to ½TM ðÞÀl 1:5 and ½TM ðÞþl 1:5, respectively, by default. DNA computer. Science, 296, 499–502. The TM is calculated at 1 mM. 3. Faulhammer,D., Cukras,A.R., Lipton,R.J. and Landweber,L.F. (2000) Molecular computation: RNA solutions to chess problems. Proc. Natl In structure prediction, the program calculates the DGmin Acad. Sci. USA, 97, 1385–1389. between two sequences, then displays the structure with the 4. Rouillard,J.M., Zuker,M. and Gulari,E. (2003) OligoArray 2.0: design of DGmin using a traceback algorithm. Users can thus determine oligonucleotide probes for DNA microarrays using a thermodynamic the structure of any combination of sequences from the pool approach. Nucleic Acids Res., 31, 3057–3062. Nucleic Acids Research, 2005, Vol. 33, No. 3 911

5. Bozdech,Z., Zhu,J., Joachimiak,M.P., Cohen,F.E., Pulliam,B. and 14. Andronescu,M., Aguirre-Hernandez,R., Condon,A. and Hoos,H.H. DeRisi,J.L. (2003) Expression profiling of the schizont and trophozoite (2003) RNAsoft: a suite of RNA secondary structure prediction and stages of Plasmodium falciparum with a long-oligonucleotide design software tools. Nucleic Acids Res., 31, 3416–3422. microarray. Genome Biol., 4, R9. 15. Hofacker,I.L. (2003) Vienna RNA secondary structure server. 6. Kane,M.D., Jatkoe,T.A., Stumpf,C.R., Lu,J., Thomas,J.D. and Nucleic Acids Res., 31, 3429–3431. Madore,S.J. (2000) Assessment of the sensitivity and specificity of 16. Mathews,D.H., Sabina,J., Zuker,M. and Turner,D.H. (1999) Expanded oligonucleotide (50mer) microarrays. Nucleic Acids Res., 28, sequence dependence of thermodynamic parameters improves prediction 4552–4557. of RNA secondary structure. J. Mol. Biol., 288, 911–940. 7. Winfree,E., Liu,F., Wenzler,L.A. and Seeman,N.C. (1998) Design and 17. Zuker,M. and Stiegler,P. (1981) Optimal computer folding of large self-assembly of two-dimensional DNA crystals. Nature, 394, RNA sequences using thermodynamics and auxiliary information. 539–544. Nucleic Acids Res., 9, 133–148. 8. Shih,W.M., Quispe,J.D. and Joyce,G.F. (2004) A 1.7-kilobase 18. SantaLucia,J.,Jr (1998) A unified view of polymer, dumbbell, and single-stranded DNA that folds into a nanoscale octahedron. Nature, oligonucleotide DNA nearest-neighbor thermodynamics. Proc. Natl 427, 618–621. Acad. Sci. USA, 95, 1460–1465. 9. Yan,H., Zhang,X., Shen,Z. and Seeman,N.C. (2002) A robust DNA 19. Tanaka,F., Kameda,A., Yamamoto,M. and Ohuchi,A. (2004) mechanical device controlled by hybridization topology. Nature, Thermodynamic parameters based on a nearest-neighbor model 415, 62–65. for DNA sequences with a single-bulge loop. Biochemistry, 43, 10. Arita,M.and Kobayashi,S. (2002)DNAsequence designusing templates. 7143–7150. New Gen. Comput., 20, 263–277. 20. Zuker,M. (2003) Mfold web server for nucleic acid folding and 11. Arita,M., Nishikawa,A., Hagiya,M., Komiya,K., Gouzu,H. and hybridization prediction. Nucleic Acids Res., 31, 3406–3415. Sakamoto,K. (2000) Improving sequence design for DNA computing. 21. Peritz,A.E., Kierzek,R., Sugimoto,N. and Turner,D.H. (1991) In Proceedings of the Genetic and Evolutionary Computation Thermodynamic study of internal loops in oligoribonucleotides: Conference (GECCO-2000), July 8–12, Las Vegas, NV. Morgan symmetric loops are more stable than asymmetric loops. Kaufmann, pp. 875–882. Biochemistry, 30, 6428–6436. 12. Dirks,R.M., Lin,M., Winfree,E. and Pierce,N.A. (2004) Paradigms 22. Bommarito,S., Peyret,N. and SantaLucia,J.,Jr (2000) Thermodynamic for computational nucleic acid design. Nucleic Acids Res., 32, parameters for DNA sequences with dangling ends. Nucleic Acids Res., 1392–1403. 28, 1929–1934. 13. Andronescu,M., Fejes,A.P., Hutter,F., Hoos,H.H. and Condon,A. (2004) 23. Lyngsø,R.B., Zuker,M. and Pedersen,C.N.S. (1999) Fast evaluation of A new algorithm for RNA secondary structure design. J. Mol. Biol., 336, internal loops in RNA secondary structure prediction. Bioinformatics, 607–624. 15, 440–445.