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Biomolecular Computers with Multiple Restriction Enzymes

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Biomolecular Computers with Multiple Restriction Enzymes Genetics and Molecular Biology, 40, 4, 860-870 (2017) Copyright © 2017, Sociedade Brasileira de Genética. Printed in Brazil DOI: http://dx.doi.org/10.1590/1678-4685-GMB-2016-0132 Research Article Biomolecular computers with multiple restriction enzymes Sebastian Sakowski1, Tadeusz Krasinski1, Jacek Waldmajer2, Joanna Sarnik3, Janusz Blasiak3 and Tomasz Poplawski3 1Faculty of Mathematics and Computer Science, University of Lodz, Lodz, Poland. 2Logic, Language and Information Group, Department of Philosophy, University of Opole, Opole, Poland. 3Department of Molecular Genetics, University of Lodz, Lodz, Poland. Abstract The development of conventional, silicon-based 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 frag- ments (the transition molecules) as software and input/output signals. This computer represented a two-state two-symbol finite automaton that was subsequently extended by using two restriction enzymes. In this paper, we pro- pose 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. Received: May 17, 2016; Accepted: May 16, 2017. Introduction of biomolecular solutions for well-known models in theo- Biomolecular computers are the answer to problems retical computer science, such as finite automata, associated with the development of traditional, silicon- pushdown automata or Turing machines. Although some of based computers, particularly their miniaturization, as im- this research provided only theoretical solutions without plied by the Heisenberg uncertainty principle, and to limita- practical laboratory implementation, e.g., biomolecular tions in data transfer to and from the main memory by the representations of the Turing machine (Rothemund, 1995) central processing unit (Amos, 2005). The first attempt to or the pushdown automaton (Cavaliere et al., 2005; develop a DNA computer was by Adleman (1994), who Krasinski et al., 2012), there have been prominent excep- solved some computational problems in a laboratory test- tions, including a stochastic automaton (Adar et al., 2004) tube. Over the next two decades, numerous reports on DNA and a finite automaton (Benenson et al., 2001, 2003). These computing appeared. Some studies have focused on se- first constructions of DNA computers used one restriction lected, well-known problems in mathematics and computer enzyme (RE) as the hardware and DNA fragments as the science, e.g., the tic-tac-toe algorithm (Stojanovic and Ste- software and input/output signals. From a biochemical fanovic, 2003), the Knight problem (Faulhammer et al., 1999) or the SAT problem (Lipton, 1995). Other areas of point of view, the DNA computer works by sequentially research have attempted to apply DNA computing in medi- cutting and joining DNA molecules with the RE FokI and cine, e.g., for cancer therapy (Benenson et al., 2004) or DNA ligase. These DNA computers represent a class of de- ‘miRNA’ level diagnostics (Seelig et al., 2006). An inter- vices known as nondeterministic finite automata that can esting trend in DNA computing has been the development solve simple computational problems. Benenson et al. (2001) designed and implemented a model of a two-state two-symbol (Figure 1A) nondeterministic finite state au- Send correspondence to Tomasz Poplawski. Department of Molec- ular Genetics, University of Lodz, Pomorska 141/143, 90-236 Lodz, tomaton – the simplest model of a computer (Hopcroft et Poland. E-mail: [email protected]. al., 2001). Sakowski et al. 861 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 double-stranded DNA. The automa- Figure 1 - Finite automata with two states. (A) All possible eight transi- ton’s two symbols (a and b) and terminator t that signals the tion rules for a two-state, two-symbol nondeterministic finite automaton. end of the word were coded by double-stranded DNA mol- Computer programming involved the selection of some of these transition ecules of six base pairs in length (Figure 2A). Each of the rules for the initial and final (accepted) states. (B) Graph representing an input molecules had a FokI recognition site and represented example of a two-state, two-symbol finite automaton A1 that accepts words with at least one symbol “b”. an input word consisting of the symbols a and b. They also contained flanking sequences to bind the enzyme and to de- tect the final state of computation. Single stranded over- Conventionally, finite automata (finite state ma- hangs produced by FokI in the input molecule represented chines) are used as controllers for electromechanical de- not only a symbol, but also a state of the machine (Figure vices such as automatic doors and supermarket entrances 2B) (Benenson et al., 2001). (Sipser, 2006), as well as for many household devices such The software (transition rules) was coded by DNA as dishwashers, electronic thermostats, digital watches and transition molecules (Figure 3B), containing the FokI rec- calculators. They can also be used as probabilistic tools to ognition sequence, spacers and sticky ends of a length char- predict financial market prices and to recognize patterns in acteristic for FokI. Each of the transition molecules data analysis. Finite automata consist of a control unit consisted of four parts that were DNA sequences made of equipped with a reading head and an input tape that, in a fi- nucleotides identified as p1, p2, p3 and p4 (Figure 3A). Part nite state, can read input words built of symbols from a p1 of a transition molecule was single-stranded DNA while finite set (called alphabet). The software of a finite automa- parts p2, p3 and p4 were double-stranded DNA (Figure ton consists of transition rules that determine the sequence 3A,B). Each part of the transition molecule was encoded by of states during computation. In each step, the automaton a different sequence of nucleotides and had a characteristic reads one symbol to the right of the input word and then length: k for p1, l for p2, m for p3 and n for p4. The first part, changes its state according to the current transition rule. p1 (a sticky end) of a transition molecule, was complemen- The input word is accepted if the automaton is in one of the tary to the single-stranded part of an input molecule and final states after reading the whole word. Finite automata represented a pair < state, symbol > in a biomolecular finite are generally represented in the form of graphs that allow automaton. The second part, p2 (a spacer part), allowed one to display the relationship between objects (Sipser, control of the depth of cutting into the input molecule. The 2006). Figure 1B shows an example of a two-state two- third part, p3 (a restriction site), contained the sequence of symbol finite automaton A1 in the form of a graph. The state nucleotides specific for a particular endonuclease and en- diagram has two states labeled s0 and s1. The initial (start- abled this restriction enzyme to act. The last part, p4 (an ad- ing) state is s0 – indicated by an arrow pointing to it from ditional part), aided ligation of the restriction enzyme to the nowhere. The accepted state is s1 and is denoted with a thick whole DNA molecule because long DNA molecules are cut circle. The arrows referred to as transitions show the rela- better by restriction enzymes. Parts p1, p2 and p4 did not tionship between states. When an automaton receives an in- contain the restriction enzyme cleavage site present in part put string (input word) such as aabab, it first processes this p3. string and then produces an output to accept or reject. For In the biomolecular computer described by Benenson example, the input word aabab can be processed by autom- et al. (2001) there were additional elements (detection mol- aton A1 as follows: 1) Start action in state s0. 2) Read first ecules) that, in laboratory experiments, recognized the final symbol a from input word and move from state s0 to s0.3) Read second symbol a from input word and move from state s0 to s0. 4) Read symbol b and move from state s0 to s1. 5) Read symbol a from input word and move from state s1 to s1. 6) Read symbol b from input word and move from state s1 to s1, and finally, 7) Accept input string because au- tomaton A1 has read the whole input string aabab and is in accepted state s1. The biomolecular finite state machine proposed by Benenson et al. (2001) implemented the above scheme of computation using molecules and DNA processing pro- Figure 2 - An example of an input molecule representing the word aba. teins. The laboratory implementation of this DNA-based The symbol t in the input word allows the detection of the final product of computer included one restriction enzyme (FokI), DNA computation. (A) and (B) Before and after the first cut with endonuclease 862 DNA computer Figure 3 - Transition molecules.
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