Abstract
The development of conventional, siliconbased computers has several limitations, including some related to the Heisenberg uncertainty principle and the von Neumann “bottleneck”. Biomolecular computers based on DNA and proteins are largely free of these disadvantages and, along with quantum computers, are reasonable alternatives to their conventional counterparts in some applications. The idea of a DNA computer proposed by Ehud Shapiro’s group at the Weizmann Institute of Science was developed using one restriction enzyme as hardware and DNA fragments (the transition molecules) as software and input/output signals. This computer represented a twostate twosymbol finite automaton that was subsequently extended by using two restriction enzymes. In this paper, we propose the idea of a multistate biomolecular computer with multiple commercially available restriction enzymes as hardware. Additionally, an algorithmic method for the construction of transition molecules in the DNA computer based on the use of multiple restriction enzymes is presented. We use this method to construct multistate, biomolecular, nondeterministic finite automata with four commercially available restriction enzymes as hardware. We also describe an experimental applicaton of this theoretical model to a biomolecular finite automaton made of four endonucleases.
Keywords:
bioinformatics; DNA; DNA computer; restriction enzymes
Introduction
Biomolecular computers are the answer to problems associated with the development of traditional, siliconbased computers, particularly their miniaturization, as implied by the Heisenberg uncertainty principle, and to limitations in data transfer to and from the main memory by the central processing unit (Amos, 2005Amos M (2005) Theoretical and Experimental DNA Computation. Springer, Berlin, 173 pp.). The first attempt to develop a DNA computer was by Adleman (1994)Adleman L (1994) Molecular computation of solutions to combinatorial problems. Science 226:10211024., who solved some computational problems in a laboratory testtube. Over the next two decades, numerous reports on DNA computing appeared. Some studies have focused on selected, wellknown problems in mathematics and computer science, e.g., the tictactoe algorithm (Stojanovic and Stefanovic, 2003Stojanovic M and Stefanovic D (2003) A deoxyribozymebased molecular automaton. Nat Biotechnol 21:10691074.), the Knight problem (Faulhammer et al., 1999Faulhammer D, Cukras A, Lipton R and Landweber L (1999) Molecular computation: RNA solutions to chess problems. Proc Natl Acad Sci U S A 97:13851389.) or the SAT problem (Lipton, 1995Lipton R (1995) DNA solution of hard computational problems. Science 268:542545.). Other areas of research have attempted to apply DNA computing in medicine, e.g., for cancer therapy (Benenson et al., 2004Benenson Y, Gil B, BenDor U, Adar R and Shapiro E (2004) An autonomous molecular computer for logical control of gene expression. Nature 429:423429.) or ‘miRNA’ level diagnostics (Seelig et al., 2006Seelig G, Soloveichik D, Zhang D and Winfree E (2006) Enzymefree nucleic acid logic circuits. Science 314:15851588.). An interesting trend in DNA computing has been the development of biomolecular solutions for wellknown models in theoretical computer science, such as finite automata, pushdown automata or Turing machines. Although some of this research provided only theoretical solutions without practical laboratory implementation, e.g., biomolecular representations of the Turing machine (Rothemund, 1995Rothemund P (1995) A DNA and restriction enzyme implementation of Turing machines. Discrete Math Theor Comput Sci 27:75120.) or the pushdown automaton (Cavaliere et al., 2005Cavaliere M, Jonoska N, Yogev S, Piran R, Keinan E and Seeman N (2005) Biomolecular implementation of computing devices with unbounded memory. Lect Notes Comput Sci 3384:3549.; Krasinski et al., 2012Krasinski T, Sakowski S and Poplawski T (2012) Autonomous pushdown automaton built on DNA. Informatica 36:263276.), there have been prominent exceptions, including a stochastic automaton (Adar et al., 2004Adar R, Benenson Y, Linshiz G, Rosner A, Tishby N and Shapiro E (2004) Stochastic computing with biomolecular automata. Proc Natl Acad Sci U S A 101:99609965.) and a finite automaton (Benenson et al., 2001Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434., 2003Benenson Y, Adar R, PazElizur T, Livneh Z and Shapiro E (2003) DNA molecule provides a computing machine with both data and fuel. Proc Natl Acad Sci U S A 100:21912196.). These first constructions of DNA computers used one restriction enzyme (RE) as the hardware and DNA fragments as the software and input/output signals. From a biochemical point of view, the DNA computer works by sequentially cutting and joining DNA molecules with the RE FokI and DNA ligase. These DNA computers represent a class of devices known as nondeterministic finite automata that can solve simple computational problems. Benenson et al. (2001)Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434. designed and implemented a model of a twostate twosymbol (Figure 1A) nondeterministic finite state automaton – the simplest model of a computer (Hopcroft et al., 2001Hopcroft J, Motwani R and Ullman J (2001) Introduction to Automata Theory, Languages and Computation. AddisonWesley, Boston, 521 pp.).
Finite automata with two states. (A) All possible eight transition rules for a twostate, twosymbol nondeterministic finite automaton. Computer programming involved the selection of some of these transition rules for the initial and final (accepted) states. (B) Graph representing an example of a twostate, twosymbol finite automaton A_{1} that accepts words with at least one symbol “b”.
Conventionally, finite automata (finite state machines) are used as controllers for electromechanical devices such as automatic doors and supermarket entrances (Sipser, 2006Sipser M (2006) Introduction to the Theory of Computation. Thomson Course Technology, Boston, 431 pp.), as well as for many household devices such as dishwashers, electronic thermostats, digital watches and calculators. They can also be used as probabilistic tools to predict financial market prices and to recognize patterns in data analysis. Finite automata consist of a control unit equipped with a reading head and an input tape that, in a finite state, can read input words built of symbols from a finite set (called alphabet). The software of a finite automaton consists of transition rules that determine the sequence of states during computation. In each step, the automaton reads one symbol to the right of the input word and then changes its state according to the current transition rule. The input word is accepted if the automaton is in one of the final states after reading the whole word. Finite automata are generally represented in the form of graphs that allow one to display the relationship between objects (Sipser, 2006Sipser M (2006) Introduction to the Theory of Computation. Thomson Course Technology, Boston, 431 pp.). Figure 1B shows an example of a twostate twosymbol finite automaton A_{1} in the form of a graph. The state diagram has two states labeled s_{0} and s_{1}. The initial (starting) state is s_{0} – indicated by an arrow pointing to it from nowhere. The accepted state is s_{1} and is denoted with a thick circle. The arrows referred to as transitions show the relationship between states. When an automaton receives an input string (input word) such as aabab, it first processes this string and then produces an output to accept or reject. For example, the input word aabab can be processed by automaton A_{1} as follows: 1) Start action in state s_{0}. 2) Read first symbol a from input word and move from state s_{0} to s_{0}. 3) Read second symbol a from input word and move from state s_{0} to s_{0}. 4) Read symbol b and move from state s_{0} to s_{1}. 5) Read symbol a from input word and move from state s_{1} to s_{1}. 6) Read symbol b from input word and move from state s_{1} to s_{1}, and finally, 7) Accept input string because automaton A_{1} has read the whole input string aabab and is in accepted state s_{1}.
The biomolecular finite state machine proposed by Benenson et al. (2001)Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434. implemented the above scheme of computation using molecules and DNA processing proteins. The laboratory implementation of this DNAbased computer included one restriction enzyme (FokI), DNA oligonucleotides as transition molecules, input signals and T4 DNA ligase. The restriction enzyme FokI recognized the GGATG sequence (all DNA sequences are presented in the 5′ → 3′ direction, unless stated otherwise) and made an asymmetrical cut in doublestranded DNA. The automaton’s two symbols (a and b) and terminator t that signals the end of the word were coded by doublestranded DNA molecules of six base pairs in length (Figure 2A). Each of the input molecules had a FokI recognition site and represented an input word consisting of the symbols a and b. They also contained flanking sequences to bind the enzyme and to detect the final state of computation. Single stranded overhangs produced by FokI in the input molecule represented not only a symbol, but also a state of the machine (Figure 2B) (Benenson et al., 2001Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434.).
An example of an input molecule representing the word aba. The symbol t in the input word allows the detection of the final product of computation. (A) and (B) Before and after the first cut with endonuclease FokI, respectively. bp – base pairs.
The software (transition rules) was coded by DNA transition molecules (Figure 3B), containing the FokI recognition sequence, spacers and sticky ends of a length characteristic for FokI. Each of the transition molecules consisted of four parts that were DNA sequences made of nucleotides identified as p_{1}, p_{2}, p_{3} and p_{4} (Figure 3A). Part p_{1} of a transition molecule was singlestranded DNA while parts p_{2}, p_{3} and p_{4} were doublestranded DNA (Figure 3A,B). Each part of the transition molecule was encoded by a different sequence of nucleotides and had a characteristic length: k for p_{1}, l for p_{2}, m for p_{3} and n for p_{4}. The first part, p_{1} (a sticky end) of a transition molecule, was complementary to the singlestranded part of an input molecule and represented a pair < state, symbol > in a biomolecular finite automaton. The second part, p_{2} (a spacer part), allowed control of the depth of cutting into the input molecule. The third part, p_{3} (a restriction site), contained the sequence of nucleotides specific for a particular endonuclease and enabled this restriction enzyme to act. The last part, p_{4} (an additional part), aided ligation of the restriction enzyme to the whole DNA molecule because long DNA molecules are cut better by restriction enzymes. Parts p_{1}, p_{2} and p_{4} did not contain the restriction enzyme cleavage site present in part p_{3}.
Transition molecules. (A) The parts of transition molecules. N indicates nitrogenous bases: A (adenine), T (thymine), G (guanine) and C (cytosine). (B) All possible transition rules and transition molecules in the twostate, twosymbol biomolecular automaton presented by Benenson et al. (2001)Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434.. (C) Construction of the detection molecules for states s_{0} and s_{1}.
In the biomolecular computer described by Benenson et al. (2001)Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434. there were additional elements (detection molecules) that, in laboratory experiments, recognized the final state of computation. These molecules consisted of sticky ends (AGCG and ACAG, representing the chosen final states s_{0} and s_{1}, respectively) and an additional doublestranded fragment of DNA (the total length in each case being 161 bp and 251bp, respectively) (Figure 3C).
The finite automaton described above was produced in the laboratory by incubating FokI, transition molecules, detection molecules and input molecules in a single tube. The computation process was initiated by cutting an input molecule with FokI. In each cycle of the computation process, a transition molecule combined with the sticky ends of an input molecule followed by the sealing of two phosphodiester bonds by DNA ligase. FokI could then cut within the next symbol and produce a sticky end representing a new < state, symbol > pair (Figure 4). This biomolecular computer was limited to two states and used only one restriction enzyme (FokI). Since the initial description, other modifications have been incorporated into DNAbased computers (Unold et al., 2004Unold O, Troc M, Dobosz T and Trusiewicz A (2004) Extended molecular computing model. WSEAS Trans Biol Biomed 1:1519.; Soreni et al., 2005Soreni M, Yogev S, Kossoy E, Shoham Y and Keinan E (2005) Parallel biomolecular computation on surfaces with advanced finite automata. J Am Chem Soc 127:39353943.; Chen et al., 2007Chen P, Jing L Jian Z, Lin H and Zhizhou Z (2007) Differential dependence on DNA ligase of type II restriction enzymes: a practical way toward ligasefree DNA automaton. Biochem Biophys Res Commun 353:733737.) to improve their potential in biomedical sciences (Benenson et al., 2004Benenson Y, Gil B, BenDor U, Adar R and Shapiro E (2004) An autonomous molecular computer for logical control of gene expression. Nature 429:423429.; Seelig et al., 2006Seelig G, Soloveichik D, Zhang D and Winfree E (2006) Enzymefree nucleic acid logic circuits. Science 314:15851588.), including the use of two restriction enzymes (Krasinski and Sakowski, 2008Krasinski T and Sakowski S (2008) Extended Shapiro finite state automaton. Found Comput Decision Sci 33:241255.; Krasinski et al., 2013Krasinski T, Sakowski S, Waldmajer J and Poplawski T (2013) Arithmetical analysis of biomolecular finite automaton. Fund Inform 128:463474.).
Based on these reports, we hypothesized that the number of states in a DNAbased computer could be extended by increasing the number of restriction enzymes. To assess this hypothesis, a set of appropriate transition molecules would need to be constructed. In this study, we developed all transition rules for 162 transition molecules (Supplementary material Tables S1S8) in a biomolecular ninestate, twosymbol nondeterministic finite automaton M (Figure 5A) with four restriction endonucleases. We describe the results for the laboratory implementation of a biomolecular automaton involving four endonucleases (BaeI, BbvI, AcuI and MboII). While preparing this model, we noted that the construction of transition molecules was relatively difficult and required an appropriate method to rapidly encode the particular transition molecules. We also present an algorithm for the construction of transition molecules in biomolecular automata with multiple restriction enzymes. This algorithm was used to construct a multistate biomolecular nondeterministic finite automaton with multiple commercially available restriction enzymes as hardware.
Finite automata with nine states. (A) All possible 162 transition rules for a ninestate, twosymbol nondeterministic finite automaton. On each arrow the symbols a and b should be placed. These 162 transition rules are coded by DNA molecules – see all 162 transition molecules in Tables S1S8. (B) Graph showing an example of a fourstate, twosymbol finite automaton M_{1}. State s_{2} corresponds simultaneously to the initial and final states. This fourstate automaton requires the autonomous action of four restriction enzymes and alternative splicing by the restriction enzymes BaeI, BbvI, AcuI and MboI. (C) An example of a ninestate automaton (s_{2} – initial state, s_{1} – final state).
Materials and Methods
Synthetic DNA
Synthetic DNA sense (α) and antisense (β) oligonucleotides (200 nmol, lyophilized) were produced by Genomed (Warsaw, Poland). The oligonucleotides were used to obtain doublestranded DNA molecules of appropriate length with sticky ends for software input and output. An example of the construction of a DNA molecule representing the input word abba using sense (α) and antisense (β) oligonucleotides is described in the section ‘Construction of DNA computer elements’ below.
The oligonucleotide sequences for construction of the input molecules (input word abba) were abba(α): 5′AATTCTAACGCGACTAATCAGCATCAGCCGACTATATTAGTTGTCATCGC3′, and abba(β): 5′GGCCGCGATGACAACTAATATAGTCGGCTGATGCTGATTAGTCGCGTTAG3′. The oligonucleotide sequences for the detection molecule were: detect(α): 5′AATTCGTTTATTAGTTGTCATCGC3′ and detect(β): 5′GGCCGCGATGACAACTAATAAACG3′. The oligonucleotide sequences for the software (transition molecules) were: T122(α): 5′AATTACTACTGTA CCCTAGTTATTAGTTGTCATCGC3′,
T122(β): 5′GGCCGCGATGACAACTAATAACTAGGGTACAGTAGT3′, T162(α):
5′AATTGAAGACGCTGATCCACGCCCTACTACTGTACCCTGGGGACCCCCCG3′, T162(β): 5′GGC CCGGGGGGTCCCCAGGGTACAGTAGTAGGGCGTGGATCAGCGTCTTC3′, T107(α): 5′AATTCTGAAG AGCTCGTTAGCTCTCTTC3′, T107(β): 5′GGCCGA AGAGAGCTAACGAGCTCTTCAG3′, T24(α): 5′AA TTGCAGCAGCTCTCATACTTTAGATTGCCTTCAG3′, and T24(β): 5′GGCCCTGAAGGCAATCTAAAGTATGAGAGCTGCTGC3′.
The transition molecules, input molecule and detection molecule were prepared by annealing pairs of oligonucleotides: abba(α) and abba(β), detect(α) and detect(β), T122(α) and T122(β), T162(α) and T162(β), T107(α) and T107(β), and T24(α) and T24(β). Annealed pairs of oligonucleotides had additional sticky ends (AATT and GGCC) that enabled the insertion of DNA molecules in LITMUS 38i plasmids (see ‘Construction of DNA computer elements’).
Enzymes
The restriction enzymes AcuI, BaeI, BbvI, MboII, BtgzI and T4 DNA ligase were obtained from New England Biolabs (Ipswich, MA, USA). T4 polynucleotide kinase (PNK) was from Fermentas Thermo Scientific (Grand Island, NY, USA).
Chemicals and plasmid vectors
LITMUS 38i plasmids were obtained from Fermentas Thermo Scientific. Plasmid miniprep kits and gel extraction kits were from Axygen (Union City, CA, USA). The Perfect 100 bp DNA ladder was from EurX (Gdansk, Poland). This ladder contained 13 bands with fragments sizes of 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1500, 2000 and 2500 bp. For easy reference, the 500 bp and 1000 bp bands are brighter than the other bands in the ladder. All other chemicals and bacterial media were from SigmaAldrich (St. Louis, MO, USA).
Construction of DNA computer elements
The DNA library was constructed using LITMUS 38i plasmids as the collection of DNA molecules to represent the computer elements that had been stored and propagated in Escherichia. coli. Briefly, singlestranded oligonucleotides labelled according to the represented components of the automaton (the input word, detection molecule and transition molecules) were phosphorylated and annealed (by heating and slowly lowering the temperature) to form doublestranded DNA fragments. The oligonucleotide mixture was mixed with a larger fragment of LITMUS 38i plasmid digested with EcoRI and EagI. After overnight ligation with 40 U of T4 ligase, the ligase reaction mixture was used to transform E. coli strain DH5α (F– Φ80 lacZΔM15 Δ(lacZYAargF) U169 recA1 endA1 hsdR17 (rK, mK+) phoAsupE44 λthi1 gyrA96 relA1) by the heat shock method. After DNA analysis of the colonies, the best clone was chosen and used for large scale DNA preparation with a plasmid prep kit (Axygen), according to the manufacturer’s instructions. Prior to the experiment, the appropriate automaton DNA components were obtained by PCR followed by RE digestion. We used REs to form appropriate “coding” DNA ends (AcuI, BaeI, BtgZI and MboII) and Taq polymerase to form the second, “noncoding” (in terms of automaton) DNA ends that contained A overhangs at the 3end. Each DNA molecule thus had one coding end that was complementary to some DNA transition molecules and one noncoding end that was incompatible with any other DNA molecules in the assay tube. This procedure eliminated the possibility of accidental, random joining of DNA automaton molecules. All molecules were purified by gel extraction prior to the experiments.
We illustrate the above scheme with a concrete example, including the method used to prepare the DNA molecule representing the input word abba:

Step 1: Sense (α) and antisense (β) oligonucleotides representing input word abba were placed in the assay tube.
Sense oligonucleotide:

abba (α) : 5′AATTCTAACGCGACTAATCAGCATCAGCCGACTATATTAGTTGTCATCGC3′
Antisense oligonucleotide:

abba (β) : 3′GATTGCGCTGATTAGTCGTAGTCGGCTGATATAATCAACAGTAGCGCCGG5′


Step 2: Pairs of oligonucleotides [abba(α) and abba(β)] were annealed to obtain doublestranded DNA fragments with additional sticky ends (AATT and GGCC) that enabled the insertion of DNA molecules into LITMUS 38i plasmids.

abba (α) : 5′AATTCTAACGCGACTAATCAGCATCAGCCGACTATATTAGTTGTCATCGC3′

abba (β) : 3′GATTGCGCTGATTAGTCGTAGTCGGCTGATATAATCAACAGTAGCGCCGG5′


Step 3: Doublestranded DNA fragments were cloned into LITMUS 38i plasmids (digested with EcoRI and EagI).

Step 4: The LITMUS 38i plasmids were subsequently propagated in E. coli.

Step 5: PCR was used to obtain many copies of intermediate DNA molecules (with A overhangs at the 3end).

Step 6: The restriction enzyme (BtgzI) was used to form an appropriate “coding” (in relation to the automaton) DNA end.
Finally, we obtained the input word abba:
200 bpAATTCTAACGCGACTAATCAGCATCAGCCG
A200 bpTTAAGATTGCGCTGATTAGTCGTAGTCGGCTGAT
PCR reaction
DNA molecules for the computer were obtained by PCR using a Perpetual OptiTaq PCR master mix (Eurx) in conjunction with the primers shown in Table 1. The PCR mixture (25 μL) consisted of 1.25 U of Perpetual OptiTaq DNA polymerase, 1x reaction buffer (1.5 mM MgCl_{2}), 0.2 mM of each dNTP and 0.5 μM of upstream and downstream primer. The PCR conditions were as follows: initial denaturation step at 95 °C for 3 min, 30 cycles of 95 °C for 30 s, 60 °C (annealing temperature) for 30 s and 72 °C for 30 s, and a final extension step at 72°C for 5 min. PCR was done in a model PTC100 thermal cycler (MJ Research Inc., Waltham, MA, USA). The PCR products were subsequently digested with an appropriate RE and the samples then run on 2% agarose gels and stained with ethidium bromide (0.5 μg/ml).
Transition molecules were prepared with primer_2 and primer_3 and had a final length (after digestion with RE and gel purification) of ~110 bp. The detection molecule was prepared with primer_2 and primer_4 and had a final length of 404 bp. The word molecule was prepared with primer_5 and primer_6 and had a final length of 230 bp.
Computation reactions
Autonomous and programmable cleavage of DNA molecules by the four endonucleases was observed in one test tube. This reaction was run for 2 h in CutSmart buffer (New England Biolabs) supplemented with Sadenosylmethionine at 37 °C. The reaction tube contained a set of DNA fragments representing the input molecules, transition molecules and detection molecules, 1 U of each enzymes and 40 U of T4 DNA ligase. The reaction product was purified with phenol, chloroform and izoamyl alcohol (25:24:1, v/v), precipitated with ethanol and separated by electrophoresis on a 2% agarose gel. The control sample was similar to the test samples except for the absence of REs and ligase. The reactions started with ligation of the transition molecule with an input word. After the cyclic reactions of digestion followed by ligation, a final DNA fragment (the rest of the input molecule) joined to the detection molecule yielded a 614 bp DNA fragment that was detected by agarose gel electrophoresis.
Results and Discussion
An algorithmic method for the construction of transition molecules
The issue of how to effectively construct transition molecules in biomolecular finite automata is complex and becomes more difficult when several restriction enzymes are used. To address this problem, the paper’s first author developed an algorithm to construct transition molecules in biomolecular automata with multiple restriction enzymes, as described below.
The main idea of this general method relies on dividing the set of states Q of finite automaton M into disjoint subsets of states Q_{i} ⊂ Q (Figure 6) and assigning only one restriction enzyme e_{i} ∈ E (where E={e_{1},...,e_{r}} is the set of restriction enzymes) to each Q_{i} in the following way. Any transition rule with the target state s in Q_{i} is achieved by the enzyme e_{i}. The source state may be arbitrary state s in Q (Figure 7A).
Schematic illustration of the method. The method relies on dividing the set of Q states of a finite automaton M into disjoint subsets of states and assigning only one restriction enzyme to a particular subset.
Transition rule and molecule. (A) A transition rule from the source state to the target state. (B) and (C) Construction of a type 1 and type 2 transition molecule, respectively.
This approach generates two types of transition molecules:
Type 1  a transition from any state in subset Q_{i} to any state in the same subset Q_{i} is implemented by a transition molecule that satisfies the following conditions: part p_{3} (restriction site) of a transition molecule is characteristic of endonuclease e_{i} and part p_{1} (sticky end) has length k_{i} characteristic for the same endonuclease e_{i} (Figure 7B).
Type 2  a transition from any state in subset Q_{j} to any state of subset Q_{i}, i ≠ j, is implemented by a transition molecule that satisfies the following conditions: part p_{3} (restriction site) of a transition molecule is characteristic of endonuclease e_{i} and part p_{1} (sticky end) has length k_{j} characteristic for endonuclease e_{j} (Figure 7C).
Part p_{2} (spacer part) of the transition molecule allows control of the depth of cutting into the input molecule and its length l depends on the state of the biomolecular automaton in which we want to transit after reading the next symbol of the input word. The calculation of l is a simple arithmetical task that involves the length of the codes in the input molecules and the distances from the restriction site in the given endonuclease. Part p_{4} is of fixed length n for all transition molecules and its length depends on biochemical reactions.
This method has an additional property that relies on the possibility of expanding the number of states in a given model of finite automata (for instance, from a sixstate to a ninestate automaton). The addition of a new restriction enzyme e_{r+1}, while leaving the actual transitions unchanged, allows to add new states (which form a new set Q_{r+1}), and to construct new transition molecules (from states of Q_{r+1} and to states of Q_{r+1}) according to two types of transition molecules: Type 1 and Type 2.
A multistate finite automaton with multiple restriction enzymes
The algorithmic method described above allows the construction of transition molecules for a given model of biomolecular automata by using multiple endonucleases. As the main application of this method, we decided to construct an optimal version for codes of symbols that were six base pairs in length. The model of a sixstate nondeterministic finite automaton (Krasinski and Sakowski, 2008Krasinski T and Sakowski S (2008) Extended Shapiro finite state automaton. Found Comput Decision Sci 33:241255.) with two endonucleases (BbvI, AcuI) (Figure 8B,C) was extended to a ninestate nondeterministic finite automaton M (162 transition molecules are presented in Tables S1S8) by including the REs BaeI and MboII (Figure 8A,D). These REs produce four sticky ends of lengths k_{1}=5, k_{2}=4, k_{3}=2 and k_{4}=1. The two symbols (a and b) are encoded by doublestranded DNA molecules of six base pairs in length (Figure 9). By using the procedure described by Krasinski et al. (2013)Krasinski T, Sakowski S, Waldmajer J and Poplawski T (2013) Arithmetical analysis of biomolecular finite automaton. Fund Inform 128:463474., we could calculate the maximal number of states p with the formula: p = n – k + 1 (where n is the length of symbol codes and k is the length of the sticky ends). For example, if we use only one RE with k_{1}=5, a maximum of two states can be achieved. To create more states (up to nine) four REs that produce four different sticky ends are required.
The action of restriction enzymes (A) BaeI, (B) BbvI, (C) AcuI and (D) MboII. Y indicates a pyrimidine whereas R indicates a purine.
Schematic diagram of the laboratory implementation of automaton M_{1} using four endonucleases (BaeI, MboII, AcuI and BbvI) on the word abba. The transition molecules allow alternating and autonomous cleavage of DNA molecules that represent the input molecule. A detector is required for recognition of the final product of computation.
Using the method described here, we divided the set of nine states Q into four disjoint subsets of states: Q_{1}={s_{0},s_{1},s_{2}}, Q_{2}={s_{3},s_{4},s_{5}}, Q_{3}={s_{6},s_{7}} and Q_{4}={s_{8}}. To each subset we assigned only one restriction enzyme: BbvI to subset Q_{1}, AcuI to subset Q_{2}, BaeI to subset Q_{3} and MboII to subset Q_{4}. We distinguished two types of transition molecules: Type 1 – those with sticky ends of a length characteristic for the endonuclease that were assigned to a particular subset and Type 2 – those with sticky ends of a length not characteristic for the endonuclease that were assigned to a particular subset. All possible transition molecules for the biomolecular nondeterministic ninestate twosymbol finite automaton are shown in Supplementary material Tables S1S8.
Experimental assessment of the automaton with multiple restriction enzymes
We tested the action of automaton M_{1} in Figure 5B by running it on the accepted input word abba. These experiments focused on the key automaton element that is essential to the action of automata, namely, the autonomous and alternating action of four REs (Figure 9). If a sticky end CGTT is obtained in terminator t of the input word then the detection molecule will ligate to the input molecule. Since the detection molecule had no restriction sequence characteristic for any of the REs, DNA molecules 614 bp long were obtained (the previous steps produced much shorter fragments, as seen in Figures 9 and 10). Detection of the 614 bp fragment in gel electrophoresis indicated the acceptance of the input word by the automaton. The positive result of our experiment (Figure 10) proved that a multistate biomolecular automaton may act with four endonucleases. Based on this experiment, we conclude that it is possible to construct more complex finite automata using several restriction enzymes.
Experimental testing of automaton M_{1} running on the accepted word abba. The final product of computation was 614 bp long (see Figure 9). Abbreviations: 1 – the result of computation using four endonucleases and DNA ligase. 2 – the result of computation without the endonucleases and ligase (control experiments). M – 100 bp DNA ladder. Final product (614 bp) – DNA molecule that represented termination of the computational process in the final state s_{2}. Detector (404 bp) – DNA molecule that recognized the final state of computation. Intermediate products (~360 bp) – intermediate DNA molecule formed during the biochemical reaction. Input word abba (230 bp) – DNA molecule that represented the word abba. Transitions (~120 bp) – DNA molecule that represented transition molecules.
The general scheme for preparing the automaton components differed from that of Benenson et al. (2001Benenson Y, PazElizur T, Adar R, Keinan E, Livneh Z and Shapiro E (2001) Programmable and autonomous computing machine made of biomolecules. Nature 414:430434., 2003)Benenson Y, Adar R, PazElizur T, Livneh Z and Shapiro E (2003) DNA molecule provides a computing machine with both data and fuel. Proc Natl Acad Sci U S A 100:21912196.. Based on our approach, we propose to build a “DNA library”, a collection of DNA molecules representing computer elements that is stored and propagated in a population of E. coli through molecular cloning. Once prepared, the DNA molecules can be used at a later stage. Figure 11 summarizes the procedures for obtaining the various computer components.
A general scheme for preparing all the automaton components (input, transition, and detection molecules).
Conclusions and perspectives
The main problem with the DNA computer constructed by Ehud Shapiro’s group at the Weizmann Institute of Science was its complexity. Scaling up their DNA computer was limited by the number of states. For this reason, we focused our efforts on trying to build a more complicated DNA computer – a multistate finite automaton. Endonucleases such as FokI (with four sticky ends) allow the construction of a DNA computer with at most threestates. This is a sufficient size for analysis of the five genes of smallcell lung cancer (Benenson et al., 2004Benenson Y, Gil B, BenDor U, Adar R and Shapiro E (2004) An autonomous molecular computer for logical control of gene expression. Nature 429:423429.), although cancers are often caused by many more genes (frequently > 5). In this case, our biomolecular computer with multiple restriction enzymes could be useful for studying cancers caused by multiple genes.
The results described here show that it is possible to construct a biomolecular computer with multiple endonucleases and that this computer can act autonomously in a wet lab. Our model can be used to calculate certain algorithms, such as for vending machines that require a ninestate option for their solution automaton; this complexity cannot be dealt with using the twostate automaton described by Ehud Shapiro’s group. To a large extent, the complexity of computation with biomolecular finite automata is limited by the complexity of finite state machines that can typically only calculate simple algorithms (in polynomial time).
To prove the feasibility of our theoretical model in the wet lab we have presented the results of the laboratory implementation of a finite automaton with multiple endonucleases (BbvI, AcuI, BaeI and MboII). These experiments focused on the key element essential to the action of automata, namely, the autonomous and alternating action of multiple (four) endonucleases in one test tube. One of the endonucleases (BaeI) cuts doublestranded DNA molecules in both directions (to the left and right). Our experiments provide a new way of using endonucleases that cut DNA molecules in both directions, thereby allowing the implementation of more powerful computational devices, e.g., pushdown automata.
The algorithmic method described here for the construction of transition molecules in biomolecular automata with multiple restriction enzymes is an ad hoc approach to assembling multiple restriction enzymes for the construction of biomolecular computers. This method allows the rapid construction of the main element (transition molecules) of a biomolecular finite automaton and can be used in the future to construct other computational models, e.g., pushdown automata or Turing machines made of biomolecules. An additional interesting property of this model is the possibility of increasing the number of states in the previously prepared model by adding restriction enzymes and appropriate encoding of the transition molecules. As an example of this approach, all transition molecules for a ninestate finite state automaton were encoded using commercially available restriction enzymes.
The model described here provides a basis for constructing other computational models that can be used to solve a variety of problems, such as the biomolecular Turing machines with the use of the endonuclease BaeI.
Acknowledgments
This project is supported by the National Science Centre of Poland (NCN, grant no. DEC2011/01/B/NZ2/03022).
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Supplementary material
The following online material is available for this article:
Table S1  Transition molecules for the subset of states Q_{1}={s_{0},s_{1},s_{2}}  Type 1
Table S2  Transition molecules for the subset of states Q_{1}={s_{0},s_{1},s_{2}}  Type 2
Table S3  Transition molecules for the subset of states Q_{2}={s_{3},s_{4},s_{5}}  Type 1
Table S4  Transition molecules for the subset of states Q_{2}={s_{3},s_{4},s_{5}} Type 2
Table S5  Transition molecules for the subset of states Q_{3}={s_{6},s_{7}}  Type 1
Table S6  Transition molecules for the subset of states Q_{3}={s_{6},s_{7}} Type 2
Table S7  Transition molecules for the subset of states Q_{4}={s_{8}}  Type 1
Table S8  Transition molecules for the subset of states Q_{4}={s_{8}}  Type 2
Comments to Supplementary Tables S1–S8
In all tables (Table S1 to S8), we present only important relevant parts (parts p_{1} and p_{3} – Figure 3A) of transition molecules. In practice, transition molecules were completed using fragments of the LITMUS 38i plasmid attached to the lefthand side (part p_{4} – Figure 3A) and by a sequence of nucleotide substitutions within block N (part p_{2} – Figure 3A). Each of the transition molecules contained A overhangs at the 3end that remained after Taq polymerase action (PCR) used to construct each molecule. An example of a completed transition molecule from Table S1 (No. 1) is presented below.

Transition molecule T1 from Table S1 (No. 1, T1:):
5′GCAGCNN 3′
3′CGTCGNNCAGC5′

A completed transition molecule T1 – with parts p_{2} and p_{4}:
5′ 100 bp AATTGCAGCCT 3′
3′A 100 bp TTAACGTCGGACAGC5′

Associate Editor: Guilherme Correa de Oliveira
Publication Dates

Publication in this collection
23 Oct 2017 
Date of issue
OctDec 2017
History

Received
17 May 2016 
Accepted
16 May 2017