## On the Computational Power of DNA Annealing and Ligation (1995)

Venue: | DNA Based Computers, volume 27 of DIMACS |

Citations: | 76 - 18 self |

### BibTeX

@INPROCEEDINGS{Winfree95onthe,

author = {Erik Winfree},

title = {On the Computational Power of DNA Annealing and Ligation},

booktitle = {DNA Based Computers, volume 27 of DIMACS},

year = {1995},

pages = {pages},

publisher = {American Mathematical Society}

}

### Years of Citing Articles

### OpenURL

### Abstract

In [Winfree] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [Boneh] show that the addition of a ligation primitive, Append, provides the missing power. The question becomes, "How powerful is ligation? Are Separate, Merge, and Amplify necessary at all?" This paper proposes to informally explore the power of annealing and ligation for DNA computation. We conclude, in fact, that annealing and ligation alone are theoretically capable of universal computation. 1 Introduction When Len Adleman introduced the paradigm of using DNA to solve combinatorial problems [Adleman], his computational scheme involved two distinct phases. To solve the directed Hamiltonian path problem, he first mixed together in a test tube a carefully designed set of DNA oligonucleotide "building blocks", which anneal to each other and are ligated to create long strands of DNA representing paths t...

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Citation Context ...t energy state (or in our case, the lowest free-energy state at a given temperature) uniquely represents the answer to our computation. This is closely related to the approach taken by J. J. Hopfield =-=[Hopfield]-=- in his seminal work on neural networks. In our case the lowest energy configuration is one where every rule molecule has all four sticky ends bound. Given the presence of the initial molecule, this c... |

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Citation Context ...de, in fact, that annealing and ligation alone are theoretically capable of universal computation. 1 Introduction When Len Adleman introduced the paradigm of using DNA to solve combinatorial problems =-=[Adleman], his-=- computational scheme involved two distinct phases. To solve the directed Hamiltonian path problem, he first mixed together in a test tube a carefully designed set of DNA oligonucleotide "buildin... |

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Citation Context ...], Chapter 11). 3 This is, clearly, highly speculative, but we hope not ungrounded. 4 Blocked cellular automata are a 1D version of what Toffoli and Margolus call partitioning cellular automata in 2D =-=[Toffoli]-=-. 5 If the table contains multiple entries for a given pair of read symbols, then the BCA is said to be nondeterministic. Figure 2. Operation of a blocked cellular automaton. x y g(x,y) f(x,y) t t+1 t... |

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Citation Context ...e weakness of single-stranded DNA and other factors, it is not clear how this approach will scale. Other authors have proposed DNA implementations of Turing Machines directly (e.g. [Beaver], [Smith], =-=[Rothemund]-=-). The approaches vary from using PCR to relying on restriction enzymes. These approaches show promise, although the reliability and efficiency of the steps is unclear. Furthermore, single-tape, singl... |

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Citation Context ...esis and analysis of DNA molecules, and thus also much energy. Our proposal is possibly the most nearly implementable example of the principle that computation is free, but input and ouput are costly =-=[Bennett]. Why use -=-the DAE structure for rule molecules? Clearly the particular choice of molecule is not of intrinsic importance to the idea of this construction. The logical essence is to have an "H"-shaped ... |

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Citation Context ...; however due to the weakness of single-stranded DNA and other factors, it is not clear how this approach will scale. Other authors have proposed DNA implementations of Turing Machines directly (e.g. =-=[Beaver]-=-, [Smith], [Rothemund]). The approaches vary from using PCR to relying on restriction enzymes. These approaches show promise, although the reliability and efficiency of the steps is unclear. Furthermo... |

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Citation Context ...ct, BCA are computationally universal. A BCA with k+3s symbols can simulate in linear time the operation of a Turing Machine with k tape symbols and s head states -- the proof is analogous to that in =-=[Lindgren]-=-. Thus we can conclude that a BCA can be used to answer any question which can be phrased in terms of a computer program. Small BCA have been designed which sort lists of integers, compute primes, and... |

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Citation Context ...h.edu Abstract In [Winfree] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al =-=[Boneh] show that-=- the addition of a ligation primitive, Append, provides the missing power. The question becomes, "How powerful is ligation? Are Separate, Merge, and Amplify necessary at all?" This paper pro... |

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Citation Context ...ncentration of salts. For example, a rule of thumb is that the difference in melting temperature between a perfectly matched structure and an imperfectly matched structure is 1 degree per 1% mismatch =-=[Wetmur]-=-. (F) This is the simplest DNA branched junction. The assembly of these structures consists of course of sequential steps; only the end product is shown. This 3-armed junction is probably floppy. Howe... |

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Citation Context ...of DNA Annealing and Ligation May 25, 1995 Erik Winfree y Computation and Neural Systems California Institute of Technology Pasadena, California 91125, USA email: winfree@hope.caltech.edu Abstract In =-=[Winfree]-=- it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [Boneh] show that the addit... |

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Citation Context ...ology. Two dimensions offers several advantages, however, such easier design of efficient computations. Perhaps more importantly, in higher dimensions it becomes easier to design error-tolerant rules =-=[Gacs]-=-; intuitively, point defects in 2D can be filled-in from adjacent correctly-computed cells, while in 1D a point defect severs communication between the left and right side. Open question: Can the DNA ... |

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Citation Context ...r the more awkward branched junction constructions we were originally considering. 18 Suggested by Len Adleman, private communication. possibility, along the lines investigated by Robinson and Seeman =-=[Robinson]-=-, would be to conjugate nanowire onto individual rule molecules, such that when the rule molecules fit together, an electrical circuit is formed. This proposal differs from Robinson and Seeman's sugge... |

13 |
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Citation Context ...ccur, in which the crossover point drifts right or left. (H) This is the most complicated structure we will consider. We will put it to good use later. It has been found to be fairly rigid and planar =-=[Fu1]. Note the-=- sticky ends. Other related double-crossover junctions are possible, depending upon the number of half-turns present in the helical regions. Ned Seeman calls this molecule "DAE" for double-c... |

9 |
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Citation Context ...ll continue to grow. However, it is also possible to design the cellular automaton such that all cells go into a special state to halt computation at the same time (the Firing Squad Problem, see e.g. =-=[Yunes]-=-), thereby allowing us to design linear pieces of DNA which fit into the gaps at the final level of the lattice, so that it cannot grow further. This may make extraction of the final tape configuratio... |

5 |
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Citation Context ...raph. After this ligation phase, there ensue n steps of affinity purification, whereby exactly the strands representing Hamiltonian paths are separated into a test tube ("the answer"). Richa=-=rd Lipton [Lipton]-=- subsequently refined the formalism for DNA-based computation. He did away with Adleman's first phase, ligation, and replaced it by starting all computations with a fixed set of DNA strands representi... |

4 |
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Citation Context ...but the outermost strands in a lattice. It may be better to reverse the order of the ligase and resolvase steps. 12 Although a resolvase has been shown to cut crossovers in double-crossover molecules =-=[Fu2]-=-, it is unknown whether the enzyme will be functional on the inner strands in the lattice. However, the enzyme may be able to, at diminished speed, work from the edges in. 4.3 Analysis and Estimates. ... |

3 |
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Citation Context ...day junction. However, it is important for the construction of the lattice that the two linear pieces in the "H" be planar; Holliday junctions have been shown to prefer a (flexible) 60 ffi s=-=kew angle [Eis]-=-. The chemically linked strands imagined above have not yet been characterized. The reason we propose the large double crossover crossovers less accessible to resolvase. 15 We optimistically require o... |