Wet Splicing Systems - CiteSeerX

DIMACS Series in Discrete Mathematics and Theoretical Computer Science

Wet Splicing Systems Elizabeth Laun and Kalluru J. Reddy

This work is dedicated to Tom Head. Abstract. Splicing systems were originally developed as a mathematical or

dry model of the generative capacity of DNA molecules in the presence of appropriate enzymes. The accuracy with which the model predicts the behavior of the corresponding biological or wet splicing system is investigated. A simple example of a wet splicing system is shown to generate in vitro the splicing language predicted by the corresponding dry splicing system.

1. Introduction

In 1987 Tom Head introduced a mathematical model of the generative capacity of linear DNA molecules in the presence of restriction enzymes and a ligase, which he called splicing systems. According to the original model, a splicing system consists of a nite initial set of strings over an alphabet, and a nite set of rules by which the strings can be spliced together to form new strings in addition to the initial strings. The closure of the initial set under the splicing operation generates a splicing language. In his original work Head posed the problem of characterizing the splicing languages. In the same work, Head showed the equivalence of the class of strictly locally testable languages to the class of languages which can be generated by splicing using a restricted, uniform set of rules. Since renamed H-systems, splicing systems have been extensively studied [7, 8, 12, 13, 21, 20, 22, 24, 26]. The generalization of H-systems to include systems having in nite sets of initial strings and rules has proved fruitful with regard to the characterization problem. In particular, G. Paun demonstrated in 1995 that extended H-systems can generate arbitrary recursively enumerable languages [21]. Other interesting and important closure results have been formulated, [7, 22] as have results pertaining to splicing with circular strings in addition to linear strings, as discussed by D. Pixton in [23]. 1991 Mathematics Subject Classi cation. Primary 68Q99, 92B99; Secondary 00A99. Research supported by the NSF under grant number CCR9509831. Research supported by the NSF under grant number CCR9509831. 1

c 0000 (copyright holder)

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ELIZABETH LAUN AND KALLURU J. REDDY

The scienti c community has recently taken great interest in biomolecular models of computation. In particular, Leonard Adleman's seminal 1994 work [3] inspired a surge of research focused on exploring the possibility of using DNA or other biomolecules to solve mathematical problems which are computationally hard [5, 18]. In light of such developments, it seemed worthwhile to demonstrate the viability of Head's model using experimental methods in the laboratory. It was with this aim that an investigation of the feasibility of performing the splicing operation as originally de ned was undertaken. Results of our study indicate that the splicing operation can be easily performed in a single-step laboratory procedure. In this procedure, restriction digestion and religation of linear double-stranded DNA (dsDNA) take place simultaneously in a single buer, producing the new strings which are predicted by the model. While the signi cance of this nding is yet to be realized completely, the importance of experimentally verifying the predictive capability of the H-system model of splicing is clear. We would like to thank T. Head for suggesting this problem. We would also like to thank Zhen Tian for her invaluable technical assistance in the laboratory.

2. De nitions and Examples

Let A be a nite set of alphabet symbols. The free monoid over A is denoted by A*, and the identity element is the empty string, denoted by 1. The free semigroup over A is denoted A+ , as usual. A splicing system S = (A, I, B, C) consists of the nite alphabet A, a nite set I of initial strings in A*, and nite sets B and C of triples (c, x, d) with c, x, and d in A*. Each such triple is called a pattern. For the triple (c, x, d), the string cxd is called a site, and the string x is called a crossing. For the purposes of this discussion it will be helpful to consider the initial set to be linear strings of dsDNA over an alphabet consisting of the nucleotide pairs formed by hydrogen bonding between the bases adenine, guanine, cytosine, and thymine. These nucleotide pairs can be represented by the set f[A/T], [T/A], [C/G], [G/C]g, which we shorten to fa, t, c, gg. Single stranded DNA are denoted by the capitalized alphabet fA, T, C, Gg. It is also helpful to think of a site as the recognition site of a restriction enzyme, and the crossing as the single-stranded overhang which is formed when the enzyme cleaves dsDNA containing the site. Patterns in B are called left-handed, and correspond to recognition sites of restriction enzymes which produce either 5' or blunt end overhangs. Patterns in C are called right-handed, and correspond to the sites of enzymes which produce 3' overhangs. Strings are spliced in L according to the following algorithm: If there are strings ucxdv and pexfq in L, and patterns (c, x, d) and (e, x, f) of the same hand, then the strings ucxfq and pexdv are formed by splicing ucxdv and pexfq together. Again, it is helpful to consider patterns of the same hand which contain the same crossing as corresponding to restriction enzyme sites which produce the same overhangs, i.e. overhangs which allow subsequent religation to occur. The language L = L(S) generated by S consists of the strings in I and all strings in the closure of I under the operation of splicing, i.e. strings which can be generated by splicing in L are adjoined to L. Strings in L(S) which cannot be used for splicing are called the adult strings in L(S).

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A simple example of a splicing language is the language L=(a)*. This in nite language can be easily seen to be generated by the system S = (A, I, B, C) where A = fag, I is the initial set f 1, aa g, B = f(1, a, 1) g and C = ;. Notice that there are in nitely many splicing systems which can generate (a)*. It is also worth noting that not every regular language is a splicing language. For example, as noted by Gatterdam in [11], the language L= (aa)* cannot be generated by any splicing system having a nite initial set and nite set of rules. The language a*ta*ta* is another such example. We would like to emphasize again that, although the generalizations of the concept of a splicing system by Paun, Pixton, Rozenberg, Salomaa and others permit arbitrary languages as initial sets and arbitrary rule sets, we shall con ne ourselves to the original de nition of an H-system in which both the set of rules and initial strings are nite. Furthermore, we use the original notation as presented by Head in 1987, rather than the notation almost always used in the literature devoted to the mathematical generaliztion of the splicing concept. See [13] for a more complete explanation of the development of the splicing notation.

2.1. The Primary Example. A simple biological example of a splicing system is S = (A, I, B, C) where A is the set fa, g, c, tg introduced above, and I = fgccgcaccggc , caccacgtg g, with ; ; ; and  in fa,c,t,gg* = A*. These initial strings are sequences which appear in the genome of the bacteriophage lambda. The strings are assumed to be dephosphorylated on their 5' ends in order to prevent blunt end ligation. Set B is empty in this example. The substrings gccgcaccggc and caccacgtg appearing in the initial strings are recognition sites for the restriction enzymes Bgl I and Dra III , respectively. The patterns (gccg, cac, cggc) and (cac, cac, gtg) encode speci c actions of these enzymes in the presence of T4 DNA ligase, and together they are the elements in set C. Notice that there are other sites which can be encoded for both of these enzymes. For the purposes of this example, however, only the two patterns above are required. Both restriction sites, when cut, leave 3' CAC single-stranded overhangs which allow religation. There are no other recognition sites for these enzymes in either initial string, so the splicing language L(S) for this example is the set I [ fgccgcacgtg, caccaccggc g. We are motivated to verify experimentally that this language is in fact generated in vitro as predicted by the splicing model. 3. The Experimental Design

The motivation for this experiment was to produce laboratory veri cation of Head's theoretical model of splicing systems. The experiment was designed to test the hypothesis that a particular splicing system will converge to a xed set of strings. The initial set was taken to be two distinct sequences of linear dsDNA, each with dephosphorylated 5' ends, one having a Bgl I restriction site and the other having a Dra III restriction site. The action of iterated cleavage and religation was predicted to result in a dynamical splicing system which would converge to a particular set of adult strings. The experiment was designed so that this progression would be clearly apparent if the molecular (also termed wet ) splicing system behaved as predicted by the mathematical (or dry ) model. There were two major considerations in the experimental design. First, appropriate molecules were chosen such that the predicted outcome could be easily

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ELIZABETH LAUN AND KALLURU J. REDDY

veri ed by standard laboratory techniques. Second, optimal conditions for simultaneous restriction digestion and ligation of dsDNA using the chosen enzymes had to be determined by preliminary experimental investigation. The two substrings of phage lambda DNA which constitute the initial set in the primary example described above were chosen for the following properties: String 1, approximately 1.6 kbp long, was chosen because it contains exactly one Bgl I site, about 1.45 kbp from one end. String 2, which we shall refer to as the 2.1 kbp string, contains exactly one Dra III site, approximately 1.3 kbp from one end. The 2.1 kbp string contains no Bgl I restriction site and the 1.6 kbp string contains no Dra III site. The rst string represents the 1,597 bp fragment of phage lambda DNA between the genome locations 16,196 and 17,793, and the second string represents the 2,137 bp fragment of the phage lambda DNA between locations 40,754 and 42,891. The substrings gccgcaccggc and caccacgtg appear approximately at locations 17,638 and 41,579 in the genome, respectively [25]. Both sites, when cut by the appropriate enzyme, leave 3' overhangs of the sequence CAC which allow subsequent ligation of fragments to occur. Further, if two fragments originating from the two dierent strings are ligated together, neither restriction site is present in the product string, and no subsequent cleavage of these molecules is possible, i.e. these are the adult strings in the splicing language. Two such adult strings can be formed in this system. No blunt end ligation could occur at all, since the initial molecules were dephosphorylated on their 5' ends. In theory therefore, repeated cleavage and ligation steps should reduce the quantity of the original 2.1 and 1.6 kbp molecules, and increase the quantity of the adult molecules, which are approximately 0.98 and 2.7 kbp in length. Such strands would be readily detectable and easily dierentiated by agarose gel electrophoresis and subsequent UV analysis [4]. The initial molecules and the adult product molecules expected after cleavage and religation are shown in Figure 1. It was predicted that an analysis of the dynamical bahavior of this splicing system in its wet form would show a time-related decrease in the initial strings, and a corresponding increase in the adult strings. Intermediate fragments having 3' CAC overhangs were expected as well. The simultaneous restriction digestion and ligation reaction was performed using Bgl I, Dra III and T4 DNA ligase in the ligase buer. The addition of ATP and BSA to enhance the action of T4 DNA ligase and Dra III was necessary. Optimization of the relative quantities of the dierent reagents was explored in a series of preliminary experiments the results of which are omitted here.

4. Laboratory Procedure

The initial two kinds of dsDNA molecules were generated by PCR ampli cation using phage lambda genomic DNA as the template. The primers were twenty base pair exact complements of the ends of the chosen phage lambda DNA substrings of lengths 1.6 and 2.1 kbp, and they were dephosphorylated on their 5' ends to prevent blunt end ligation of the PCR product molecules during the later reaction phase of the experiment. The primers were purchased from Gibco BRL (Gaithersburg, MD). The PCR reactions were run as 100 l aliquots, each containing 0.8 mMdNTPs (0.2 mM each dATP, dCTP, dGTP, dTTP) from Pharmicia, 1.5 mM MgCl2, 1 M each

WET SPLICING SYSTEMS

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INITIAL MOLECULES

GCCGCACCGGC 1442 bp

1.6 kbp

155 bp Bgl I

CACCACGTG 1312 bp

2.1 kbp

825 bp Dra III

ADULT MOLECULES

1442 bp

825 bp

2.7 kbp

1312 bp

155 bp

0.98 kbp

Figure 1. The initial molecules and the adult molecules which are the expected products in the primary example.

forward and reverse primer, 50 ng target lambda DNA from New England Biolabs (Beverly, MA), 0.5 ul Taq polymerase (5 U/ul) from Gibco BRL in 1xPCR buer. The reactions were cycled 40 times at 94C for 1 minute, 55C for 1 minute and 72C for two minutes, ending with 5 minutes at 72. The Peltier Thermal Cycler (MJ Research, Inc., Watertown, MA) was used to perform the PCR. The PCR products were puri ed from 0.7% agarose gel using the QIAEX Gel Extraction Kit by Qiagen. Each QIAquick puri cation column was used to purify two 100 l PCR reactions. The nal puri ed product from each column was eluted in 50 l elution buer (see Appendix). The puri ed PCR products were quantitated

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ELIZABETH LAUN AND KALLURU J. REDDY

using a UV-visible spectrophotometer. The concentration of the 1.6 kbp DNA was approximately 36 ng/l in puri ed form, and the concentration of the 2.1 kbp DNA was approximately 34 ng/l. A volumetric ratio (1.6 kbp puri ed PCR DNA/2.1 kbp puri ed PCR DNA ) of 0.7196 was calculated to yield approximately equal molarity of the two types of initial molecules. In preliminary experiments, however, a volumetric ratio of 0.625 of the two PCR products yielded good results. A preliminary run of the simultaneous restriction digestion and ligation experiment was performed, and the results of this reaction are shown in Figure 2. The reaction contained 15 l puri ed 1.6 kbp DNA, 24 l puri ed 2.1 kbp DNA, 4 l Bgl I restriction enzyme (8U/l), 10 l Dra III restriction enzyme (3U/l), 10 l T4 DNA ligase, 0.5 l Bovine Serum Albumin (10g/l), 20 l 10mM rATP, 12 l 10x T4 Ligase Buer and 24.5 l sterile water. The enzymes were obtained from Strategene, and the BSA came from NEBiolabs. This reaction was incubated at 22C overnight. The reaction product was then puri ed according to the puri cation protocol in the Appendix. Approximately half (11 l) of the puri ed reaction product was then run on a 1.0% agarose gel stained with ethidium bromide. See Figure 2. A time sequence study of the primary simultaneous restriction digestion and ligation reaction, called Reaction T, was then performed. The results of the experiment appear in Figure 3. Reaction T was a 600 l solution containing 105 l of puri ed 1.6 kbp DNA, 168 l of puri ed 2.1 kbp DNA, 28 l Bgl I restriction enzyme (8U/l), 70 l Dra III restriction enzyme (3U/l), 45 l T4 DNA ligase (all enzymes from Strategene), 2.5 l Bovine Serum Albumin (10g/l)from NEBiolabs, 80 l 10mM rATP, 60 l 10x T4 DNA ligase buer and 41.5 l sterile water. The reaction was initially incubated at room temperature (22C). Aliquots of 145 l were removed from the reaction mixture after 5, 15, and 60 minutes, respectively. The rst three samples were immediately stored at ?70C. The remaining 145 l sample was incubated approximately 20 additional hours at 16C. All the samples were then heated to 65 C for 15 minutes and puri ed according to the puri cation protocol in the Appendix. Slightly more than half (about 14 l) of each puri ed sample was then run on a 1.0% agarose gel stained with ethidium bromide. The gel photograph of the time sequence reaction appears in Figure 3. All photographs were taken using Polaroid 55 Professional lm and a Fotodyne Polaroid MP-4 Land Camera apparatus.

5. Results and Discussion

The restriction enzymes chosen apparently work with good eciency at the concentrations used and in the T4 DNA ligase buer. The Bgl I and Dra III began to cleave the initial strands of DNA in T4 DNA ligase buer within 5 minutes after Reaction T was begun, as seen in Lane 3 of the gel in Figure 3. Digestion of the 2.1 and 1.6 kbp fragments apparently continued throughout the reaction, although the majority of both initial fragments were cleaved within an hour at room temperature, as apparent from Lane 5. The resulting DNA pieces with overhangs appeared within 5 minutes, as seen in Lane 3. The ligase worked well at 22C, as expected, and the fragments with overhangs began to religate immediately. The 5 minute aliquot in Lane 3 of the gel appearing in Figure 3 showed some apparent 2.7 and 0.98 kbp adult products of religation. Within 60 minutes the two adult DNA strands of lengths 2.7 and 0.98 kbp were clearly visible (Lane 5). The complete

WET SPLICING SYSTEMS

Preliminary Reaction: Simultaneous restriction digestion and ligation of 1.6 kbp and 2.1 kbp dsDNA fragments. Lanes 1 and 4 are the standard molecular weight markers (phage lambda DNA cut with Hind III and BstE II, respectively). Lane 2 contains the initial dsDNA strings of lengths 2.1 and 1.6 kbp. Lane 3 contains the products after overnight incubation at room temperature. Note that the expected adult strings of lengths 2.7 and .98 kbp appear. The faint bands appearing in Lane 3 are unreligated fragments from restriction digestion of the initial strings.

Figure 2.

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ELIZABETH LAUN AND KALLURU J. REDDY

Reaction T: Time sequence study of simultaneous restriction digestion and ligation of 1.6 kbp and 2.1 kbp dsDNA fragments. Lanes 1 and 7 are standard molecular weight markers (phage lambda DNA cut with Hind III and BstE II, respectively). Lane 2 contains the initial dsDNA strings of lengths 2.1 and 1.6 kbp. Lanes 3, 4 and 5 contain the products after incubation at room temperature for 5, 15 and 60 minutes, respectively. Lane 6 contains the product strings after overnight incubation at 16 C. Note that the bands containing the expected adult strings of lengths 2.7 and .98 kbp appear increasingly bright as time passes, while the bands of both the original dsDNA and the unreligated fragments grow fainter with time.

Figure 3.

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time sequence indicates a progressive decrease in the quantities of the initial 2.1 and 1.6 kbp strings, and a progressive increase in the quantities of the adult strings, as expected. Repetitions of this experiment have produced results which indicate that the general dynamical behavior of such a system is quite predictable. The quantities of the initial molecules decreases over time, until virtually all initial molecules disappear. The intermediate fragments with overhangs rst increase in apparent quantity as the initial molecules are cleaved by the restriction enzymes during the rst minutes of the reaction, and then the quantities of these intermediate fragments decrease as they are used up during the religation process, forming adult molecules which cannot be recleaved. The adult molecules show a steady increase in quantity throughout the reaction, and are apparently not involved in further interactions with other molecules or enzymes. Somewhat surprising was the speed with which the restriction digestion occurred at room temperature. After only 5 minutes, cleavage of the initial molecules began to occur. The high concentrations of these enzymes relative to the amount of DNA in the reaction may account for this unexpectedly favorable result. Variations of the experiment, however, have yielded slight variations in the speed with which ligation appears to occur. Factors such as ligase concentration in the reaction, and temperature seem to be important. Dierent results might be observed with other restriction enzymes, particularly those which cleave optimally in dierent salt solutions. In all cases, the ligation reaction required the addition of ATP to the reaction mixture, as evident from preliminary work not presented here. This splicing system behaved as predicted by the dry mathematical model proposed by Head. The intial set consisting of the two dsDNA strings was dynamically transformed by the process of cleavage and religation into the predicted set of adult strings of dsDNA. The adult strands showed no evidence of further cleavage, and the initial strings gradually disappeared from the reaction mixture over time. The intermediate fragments were visible as quickly as ve minutes after the reaction was initiated. Their concentrations initially appeared to steadily increase until they reached a maximum concentration in the reaction, after which time the quantities of the intermediate fragments in the reaction mixture appeared to monotonically diminish. The apparent quantities of the adult strings steadily increased throughout the duration of the reaction. The general dynamical behavior of the system suggested by these results was veri ed by several repetitions of the experiment, the details of which we do not include here.

6. Conclusions and Suggestions for Further Research

The results of these experiments verify that Head's mathematical model accurately predicts the splicing language generated by the wet splicing system for the example we investigated. The splicing language predicted by the dry splicing model was generated in vitro. Although no unexpected strings were generated, we would like to nd conditions under which complete religation of the intermediate fagments readily occurs. Further studies using other wet splicing systems are necessary. In particular, we expect that nding compatible buer solutions may be a non-trivial matter,

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ELIZABETH LAUN AND KALLURU J. REDDY

depending upon the restriction enzymes chosen. We propose to attempt this type of experiment using two restriction enzymes having dierent restriction sites producing identical palindromic overhangs, in which case we would expect to see four intermediate fragments which have sticky ends, 10 intermediate dsDNA with no sticky ends, and four adult molecules which could not be recleaved by either restriction enzyme. An example of such a pair of restriction enzymes is BamH I and Bgl II. Experiments utilizing chemically synthesized dsDNA should also be performed, since the possibility of using the splicing operation as a computational tool may necessarily depend on the ability to use arbitrary intial sets of strings. We also propose to verify by experiment that the splicing model can predict the behavior of a splicing system which is designed to produce an in nite set of adult strings. Again, such a step seems reasonable and necessary for an ongoing investigation of the splicing system as a computational tool. In conclusion, we have initial results which seem promising. Further research must be done in the areas suggested above, as well as in areas which will broaden the scope of the investigation of the splicing system model.

7. Appendix

Elution Buer : 10mM Tris-HCl, 0.1mM EDTA Standard TE Buer : 10mM Tris-HCl, 1.0mM EDTA

Puri cation Protocol:

Qiagen Buer QX1 was added to the reaction product in a 5:1 (vol:vol) ratio. This mixture was loaded into a QIAquick column and spun in a standard table-top centrifuge for 1 minute at 13,000 rpm. The eluent was poured o, and .75 ml of Qiagen Bufer PE was added. The sample was allowed to stand 2-3 minutes, then the column was spun 1 minute at 13,000rpm, the eluent discarded, and the column was respun for another minute. The elution buer was added (25 l), and the column was allowed to sit 5 minutes before the DNA was eluted by spinning the column at 14,000rpm for 5 minutes.

References

[1] IEEE International Conference on Evolutionary Computation, Indiana University, Purdue University, Indianapolis, Illinois, April13{16, 1997, Special Session on DNA-based Computation. [2] Paci c Symposium on Biocomputing, 1997. [3] Leonard M. Adleman, Molecular computation of solutions to combinatorial problems, Science 266 (1994), 1021{1024. [4] A. Ausubel, R. Brent, R.E. Kingston, D.D. Moore, J.G. Seidman, J.A. Smith, and K. Struhl, Current Protocals in Molecular Biology, Greene Publishing Associates and WileyInterscience, 1994. [5] Dan Boneh, Christopher Dunworth, and Richard J. Lipton, Breaking DES using a molecular computer, Tech. Report CS-TR-489-95, Princeton University, May 1995. [6] Erzsebet Csuhaj-Varju, Rudolf Freund, Lila Kari, and Gheorghe Paun, DNA computation based on splicing: universality results, In Hunter and Klein [14]. [7] Karel Culik II and Tero Harju, The regularity of splicing systems and DNA, 16th International Colloquium on Automata, Languages and Programming (Berlin), Lecture Notes in Computer Science, vol. 372, Springer Verlag, 1989, pp. 222{233.

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[8] K. L. Denningho and R. W. Gatterdam, On the undecidability of splicing systems, Internat. J. Comput. Math. 27 (1989), 133{145. [9] Rudolf Freund, Erzsebet Csuhaj-Varju, and Franz Wachtler, Test tube systems with cutting/recombination operations, In Paci c Symposium on Biocomputing [2]. [10] Rudolf Freund, Lila Kari, and Gheorghe Paun, DNA computation based on splicing: The existence of universal computers, Journal of the ACM, To appear. Also Technical Report 185-2/FR-2/95, TU Wien, 1995. [11] R. W. Gatterdam, Splicing systems and regularity, Internat. J. Comput. Math. 31 (1989), 63{67. [12] Tom Head, Splicing systems and DNA, Lindenmayer Systems: Impacts on Theoretical Computer Science, Computer Graphics and Developmental Biology (Grzegorz Rozenberg and Arto Salomaa, eds.), Springer Verlag, Berlin, Heidelberg, New York, 1992, pp. 371{383. [13] Tom Head, Gheorghe Paun, and Dennis Pixton, Generative mechanisms suggested by DNA recombination, Handbook of Formal Languages (Grzegorz Rozenberg and Arto Salomaa, eds.), vol. 2, Springer Verlag, Berlin, Heidelberg, New York, October 1996. [14] Lawrence Hunter and Teri Klein (eds.), Biocomputing: Proceedings of the 1996 paci c symposium, Maui, HI, World Scienti c Publishing Co., Singapore, January 1996. [15] IEEE Computer Society Technical Committee on Pattern Recognition and Machine Intelligence (PAMI), Proceedings of the First International Symposium on Intelligence in Neural and Biological Systems, Herndon, VA, IEEE Computer Society Press, May 1995. [16] Natasa Jonoska and Steven A. Karl, Ligation Experiments in Computing with DNA, In IEEE International Conference on Evolutionary Computation [1], Special Session on DNA-based Computation. [17] Lila Kari, DNA computing: Arrival of biological mathematics, Math. Intelligencer 19 (1997), no. 2, 9{22. [18] Richard J. Lipton, DNA solution of hard computational problems, Science 268 (1995), 542{ 545. [19] A. Mateescu, Gheorghe Paun, Grzegorz Rozenberg, and Arto Salomaa, Simple splicing systems, Discrete Applied Mathematics (1996). [20] Gheorghe Paun, The splicing as an operation on formal languages, In Proceedings of the First International Symposium on Intelligence in Neural and Biological Systems [15], pp. 176{180. [21] , Regular extended H systems are computationally universal, Journal of Automata, Languages, Combinatorics 1 (1996), no. 1, 27{36. [22] Dennis Pixton, Splicing in abstract families of languages, in preparation. , Linear and circular splicing systems, In Proceedings of the First International Sym[23] posium on Intelligence in Neural and Biological Systems [15], pp. 181{188. [24] , Regularity of splicing languages, Discrete Appl. Math. 69 (1996), no. 1{2, 99{122. [25] F.W. Stahl R.W. Hendrix, J.W. Roberts and R.A. Weisberg, Lambda II, Cold Spring Harbor Laboratory. [26] Rani Siromoney, K.G. Subramanian, and V. Rajkumar Dare, Circular DNA and splicing systems, Proceedings of Parallel Image Analysis, 2nd International Conference ICPIA '92, Ube, Japan, 21-23 Dec 1992. (Ube, Japan) (A. Nakamura, M. Nivat, A. Saoudi, P.S.P. Wang, and K. Inoue, eds.), Lecture Notes in Computer Science, no. 654, Springer Verlag, Berlin, Heidelberg, New York, 1992, pp. 260{273. Department of Mathematics, Binghamton University, Binghamton, NY 13905

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Department of Biological Sciences, Binghamton University, Binghamton, NY 13905

E-mail address :

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