Theoretical and Experimental DNA Computation (Natural ...

More documents

Recommendations

Info

3.5 Membrane Models 67 objects, and are of the form before → after, meaning “evolve every instance of before into an instance of after.” Note that, as we are considering multi-sets, this rule may be applied to multiple objects. Evolution rules are represented by a pair (u, v) of strings over the alphabet V . v may be either v ′ or v ′ δ,where δ is a special symbol not in V . v ′ is a string over {ahere,aout,ainj } where j is a membrane identifier. ahere means “a copy of a remains in this region”, aout means “send a copy of a through the membrane and out of this region”, and ainj means “send a copy of a through the membrane of the region labelled j” (i.e., place a copy of a in membrane j, noting that this is only possible if the current region featuring this rule contains j). When the special symbol δ is encountered, the membrane defining the current region (assuming it is not the skin membrane) is “dissolved”, and the contents of the current region are placed in the “parent” region (with reference to Fig. 3.8, if membrane 5 were to be dissolved, then region 3 would contain the accessible regions 4, 6 and 7, whereas only regions 4 and 5 are accessible with membrane 5 intact). In order to simplify the notation, we omit the subscript “here”, as it is largely redundant for our purposes. Thus the rule a → ab means “retain a copy of a here and create a copy of b here”, whereas a → bδ means “transform every instance of a into b and then dissolve the membrane.” Note that objects on the left hand side of evolution rules are “consumed”, or removed, during the process of evaluation. We may also impose priorities upon rules. This is denoted by >, andmay be read as follows, using the example (ff → f) > (f → δ): “transform ff to f as often as possible (halving the number of occurrences of f in the process) until no instances of ff remain, and then transform the one remaining f to δ, dissolving the membrane.” We assume the existence of a “clock” that synchronizes the operation of the system. At each time step, the configuration of the system is transformed by the application of rules in each region in a nondeterministic, maximally parallel fashion. This means that objects are assigned to rules nondeterministically in parallel, until no further assignment is possible (hence maximal). This series of transitions from one configuration to another forms the computation. A computation halts if no rules may be applied in any region (i.e., nothing can happen). The result of a halting computation is the number of objects sent out through the skin membrane to the outside environment. We now give a small worked example, taken from [118] and depicted in Fig. 3.9. This P system calculates n 2 for any given n ≥ 0. The alphabet V = {a, b, d, e, f}. Region 3 contains one copy each of objects a and f, andthree evolution rules. No objects are present in regions 1 and 2, so no rules can be applied until we reach region 3. We iterate the rules a → ab and f → ff n times in parallel, where n ≥ 0 is the number we wish to square. This gives n copies of b and 2 n copies of f. Wethenusea → bδ instead of a → ab, replacing the single a with b and dissolving the membrane. This leaves n + 1 copies of b and 2 n+1 copies of f in region 2; the rules from region 3 are “destroyed” and
68 3 Models of Molecular Computation 1 2 3 (f f a f a a a b b δ f f f b d d de f) > (f δ) e e out Fig. 3.9. Example P system the rules from region 2 are now used. The priority relation dictates that we must use the rule ff → f as often as possible, so in one time step we halve the number of copies of f and, in parallel, transform b n+1 to d n+1 . In the next step, using d → de, n+1 copies of e are produced, and the number of copies of f is once again halved. This step is iterated n times (enforced by the priority relation), producing n + 1 copies of e at each step, and then f → δ dissolves membrane 2. All objects are deposited in the skin membrane, which contains a single rule for e. In one step, all copies of e are sent out of the system using e → eout, the number of copies of e equalling the value of n 2 .Atraceofthe execution of this system for n = 4 is given in Fig. 3.10. Since the initial development of P systems, several variants have appeared in the literature. One of these, described in [116], abandons the use of membrane labels and the ability to send objects to precise membranes, and instead assigns an electrical “charge” to objects and membranes. The motivation behind this change to the basic P system is that sending objects to specificallylabelled membranes is “non-biochemical” and artificial. In the modified P system, objects pass through membranes according to their charge; positively charged objects enter an adjacent negatively charged region (if there are several candidate regions, the transition is selected nondeterministically) and a negatively charged object enters a positively charged region. Neutral objects can only be sent out of the current region; they cannot pass through “inner” membranes. In order to achieve universality, the new variant of the P system is augmented with an additional operation; the action of making a membrane thicker (the membrane dissolving operation is retained). This compensates for the loss of membrane labels by providing an alternative method for controlling the communication of objects through membranes.
Page 2 and 3:
Natural Computing Series Series Edi
Page 4:
Martyn Amos Department of Computer
Page 8 and 9:
Preface DNA computation has emerged
Page 10:
Preface IX acknowledged. Finally, I
Page 13 and 14:
XII Contents 4 Complexity Issues ..
Page 16 and 17:
Introduction “Where a calculator
Page 18 and 19:
Introduction 3 Even though their un
Page 20 and 21:
1 DNA: The Molecule of Life “All
Page 22 and 23:
1.3 DNA as the Carrier of Genetic I
Page 24 and 25:
1.3 DNA as the Carrier of Genetic I
Page 26 and 27:
Synthesis 1.4 Operations on DNA 11
Page 28 and 29:
Gel electrophoresis 1.4 Operations
Page 30 and 31:
5’ 3’ A T A G A G T T 3’ TCA
Page 32 and 33: 1.4 Operations on DNA 17 Others may
Page 34 and 35: Vector Strand to be inserted (targe
Page 36: 1.5 Summary 1.6 Bibliographical Not
Page 39 and 40: 24 2 Theoretical Computer Science:
Page 61 and 62: 46 3 Models of Molecular Computatio
Page 81: 66 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 and 94: 78 4 Complexity Issues Ω = {∧,
Page 95 and 96: 80 4 Complexity Issues 0 1 0 1 x1 x
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 136 and 137:
Page 138 and 139:
Create initial strand library Confi
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Experiment 25. New full-length chai
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
5.9 Other Laboratory Implementation
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160:
5.11 Bibliographical Notes 145 whic
Page 163 and 164:
148 6 Cellular Computing of truly i
Page 165 and 166:
150 6 Cellular Computing 6.2 Succes
Page 167 and 168:
152 6 Cellular Computing are “glu
Page 169 and 170:
154 6 Cellular Computing x y z x (a
Page 171 and 172:
156 6 Cellular Computing 6.7 Biblio
Page 173 and 174:
158 References 14. Martyn Amos, Ste
Page 175 and 176:
160 References 53. Dónall A. Mac D
Page 177 and 178:
162 References 92. D. Knuth. The Ar
Page 179 and 180:
164 References 135. Kensaku Sakamot
Page 182 and 183:
Index ALGOL, 102 AND, 23-24, 42, 87
Page 184 and 185:
iological, 74, 114, 150 Feynman, R.
Page 186 and 187:
Petrinet,63 phage, 18-20 phosphate,
Page 188:
Natural Computing Series W.M. Spear
show all

Theoretical and Experimental DNA Computation (Natural ...

Create successful ePaper yourself

Delete template?

Save as template?