Theoretical and Experimental DNA Computation (Natural ...

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4.4 Ogihara and Ray’s Boolean Circuit Model 79 The simulation of gates at level k>0 proceeds as follows. We denote by i1 and i2 the indices of the gates that supply the inputs for gi. (1) Input(U) (2) for k =1tod do (3) for each gate gi at level k in parallel do (4) if gi computes ∨ then do (5) if (σ[i1] ∈ U) or (σ[i2] ∈ U) then U ← U ∪ σ[i] (6) else if gi computes ∧ then do (7) if (σ[i1] ∈ U) and (σ[i2] ∈ U) then U ← U ∪ σ[i] (8) end for (9) end for At step (1) the initial tube, U, is created. Then, for each circuit level k>0 (step 2), the gates at level k are simulated in parallel (steps 3 through 8). The simulation of each gate gi at level k is achieved as follows. If gi is an ∨-gate, the string σ[i] ismadepresent 1 in U (i.e., gi evaluates to 1) if either of the strings representing the inputs to gi is present in U (step 5). If gi is an ∧-gate, the string σ[i] ismadepresentinU (i.e., gi evaluates to 1) if both of the strings representing the inputs to gi are present in U (step 7). The simulation then proceeds to the next level. At the termination of the computation, Ogihara and Ray analyze the contents of U to determine the output of the circuit. 4.4.1 Ogihara and Ray’s Implementation In this section we describe the laboratory implementation of the abstract Boolean circuit model just described. The circuit simulated in [113] is depicted in Fig. 4.2. Each gate gi is assigned a sequence of DNA, σ[i], of length L, beginning with a specific restriction site, E. Eachedgei → j is assigned a sequence ei,j that is the concatenation of the complement of the 3’ L/2-mer of σ[i] and the complement of the 5 L/2-mer of σ[j]. In this way, ei,j acts as a “splint” between gi and gj if and only if both σ[i] andσ[j] are present. However, we later highlight a case where this strand design does not hold. The simulation of gates at level 0 (i.e., the construction of the initial tube) proceeds as follows. Begin with a tube of solution containing no DNA. For each input gate xi that evaluates to 1, pour into the tube a population of strands representing σ[i]. If xi evaluates to 0, do not add σ[i] tothetube. We now consider the simulation of gates at level k>0. The simulation of ∨-gates differs from that ∧-gates, so we first consider the case of an ∨-gate, gj, at level k. First, pour into the tube strands representing σ[j]. Then, for each edge i → j pour into the tube strands representing ei,j. Allow ligation to occur. If strands 1 The process by which this is achieved is described in detail in Sect. 4.4.1.
80 4 Complexity Issues 0 1 0 1 x1 x 2 S x 3 g g g x 4 5 6 7 Fig. 4.2. Boolean circuit simulated by Ogihara and Ray Circuit level representing either of the inputs to gj are present, the edge strands will cause strands of length 2L to be formed, consisting of the concatenation of the sequence representing the gi with the sequence representing gj. This process is depicted in Fig. 4.3a. These strands are then separated out by running the solution on a polyacrylamide gel. These strands are then cut with a restriction enzyme recognizing sequence E. This step leaves in solution strands of length L that correspond to gj (i.e., gj evaluates to 1). We now consider the simulation of an ∧-gate, gj, at level k. Again, pour into the tube strands representing σ[j]. Then, for all edges i → j pour into the tube strands representing ei,j. Allow ligation to occur. If strands representing both of the inputs to gj are present, the edge strands will cause strands of length 3L to be formed, consisting of the concatenation of the sequence representing the first input to gj, the sequence representing gj, and the sequence representing the second input to gj (see Fig. 4.3b). Note that this splinting only occurs if the polarity of the edge splints is designed carefully, and we consider this in the next section. The strands of length 3L are again separated out by a gel and cut with the appropriate restriction enzyme. This step leaves in solution strands of length L that correspond to gj (i.e., gj evaluates to 1). If k = d, we omit the gel electrophoresis and restriction stages, and simply run the solution on a gel. If the output gate, s, isan∨-gate, the output of the circuit is 1 if and only if 2L length strands exist in solution. If s is an ∧-gate, the output of the circuit is 1 if and only if 3L length strands exist in solution. Experimental results obtained In [113], Ogihara and Ray report attempts to simulate the circuit previously described using techniques of molecular biology. Their implementation is as 0 1 2
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Natural Computing Series Series Edi
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Martyn Amos Department of Computer
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Preface DNA computation has emerged
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Preface IX acknowledged. Finally, I
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XII Contents 4 Complexity Issues ..
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Introduction “Where a calculator
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Introduction 3 Even though their un
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1 DNA: The Molecule of Life “All
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1.3 DNA as the Carrier of Genetic I
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1.3 DNA as the Carrier of Genetic I
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Synthesis 1.4 Operations on DNA 11
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Gel electrophoresis 1.4 Operations
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5’ 3’ A T A G A G T T 3’ TCA
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1.4 Operations on DNA 17 Others may
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Vector Strand to be inserted (targe
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1.5 Summary 1.6 Bibliographical Not
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24 2 Theoretical Computer Science:
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26 2 Theoretical Computer Science:
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Page 61 and 62: 46 3 Models of Molecular Computatio
Page 87 and 88: 72 4 Complexity Issues is that, alt
Page 89 and 90: 74 4 Complexity Issues The weak par
Page 91 and 92: 76 4 Complexity Issues 4.3 A Strong
Page 93: 78 4 Complexity Issues Ω = {∧,
Page 97 and 98: 82 4 Complexity Issues connecting a
Page 99 and 100: 84 4 Complexity Issues Level simula
Page 101 and 102: 86 4 Complexity Issues At level D(S
Page 103 and 104: 88 4 Complexity Issues xANDy)) and
Page 105 and 106: 90 4 Complexity Issues Input matrix
Page 107 and 108: 92 4 Complexity Issues memory acces
Page 109 and 110: 94 4 Complexity Issues denoted by P
Page 111 and 112: 96 4 Complexity Issues The final st
Page 113 and 114: 98 4 Complexity Issues 1)), ADDR[])
Page 115 and 116: 100 4 Complexity Issues Size(STI)=O
Page 117 and 118: 102 4 Complexity Issues best attain
Page 119 and 120: 104 4 Complexity Issues 10. LDC 0 1
Page 121 and 122: 106 4 Complexity Issues ACC[] := bi
Page 124 and 125: 5 Physical Implementations “No am
Page 126 and 127: 5.3 Initial Set Construction Within
Page 128 and 129: Vertex 1 Vertex 2 Vertex 3 ¡ ¡ ¡
Page 130 and 131: 5.5 Evaluation of Adleman’s Imple
Page 132 and 133: 5.6 Implementation of the Parallel
Page 134 and 135: 5.8 Experimental Investigations 119
Page 138 and 139: Create initial strand library Confi
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Experiment 25. New full-length chai
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5.8 Experimental Investigations 131
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5.8 Experimental Investigations 133
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5.9 Other Laboratory Implementation
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5.11 Bibliographical Notes 145 whic
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148 6 Cellular Computing of truly i
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150 6 Cellular Computing 6.2 Succes
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152 6 Cellular Computing are “glu
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154 6 Cellular Computing x y z x (a
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156 6 Cellular Computing 6.7 Biblio
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158 References 14. Martyn Amos, Ste
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160 References 53. Dónall A. Mac D
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162 References 92. D. Knuth. The Ar
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164 References 135. Kensaku Sakamot
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Index ALGOL, 102 AND, 23-24, 42, 87
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iological, 74, 114, 150 Feynman, R.
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Petrinet,63 phage, 18-20 phosphate,
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Natural Computing Series W.M. Spear
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