horn clause computation by self-assembly of dna molecules hiroki uejima masami hagiya satoshi...
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Horn Clause Computation by Self-Assembly of DNA Molecules
Hiroki UejimaMasami Hagiya
Satoshi Kobayashi
Previous Works(SIMD Type Computation) Solution to HPP by Adleman (1994)
For a 7-vertex directed graph Adleman-Lipton paradigm (1995)
Solution candidates are randomly generated. Real solutions are selected from among the genera
ted candidates. Applying a single operation to multiple molecul
es expressing data at once.
Previous Works(Computational Power/Model) The correspondence between forms of DNA m
olecule and computational power based on formal languages.
Various computational models Branching program Turing machine Boolean circuit Random Access Memory Horn clause computation (Kobayashi)
Horn Clause Computation Model by Kobayashi Each molecule corresponds to a
Horn clause. One step of derivation is realized
by one biological operation. SIMD type computation The number of operations is
proportional to the size of problem.
Previous Works(Autonomous Computation) Computation proceeds
autonomously by self-assembly of DNA.
Possible to keep the number of operations constant.
Computation with DNA tiles A simulation of 1-D cellular automata String tiling
Structure of DNA Tile
X
X
X
Y
Y
Z
Z
Z
Y
W
W
W
cf. Winfree’s DNA Tile
Contribution of This Work A Proposal and an analysis of a
new model of DNA computation Based on Horn clause computation Autonomous by self-assembly of DNA
molecules A theoretical research on a
possibility of molecular computation.
Outline of The Algorithm To generate ground Horn clauses by vari
able substitution, using string tiles. The ground clauses are generated randomly
by self-assembly of DNA. This phase proceeds autonomously.
To make a deduction on the ground clauses. This phase also proceeds autonomously.
Horn Clause Usedin This Algorithm A term in a rule is the form f1(…fn(X)…). The arity of a predicate is at most 2. The arity of a function is 1 The variable of the 1st argument of an at
om is X, the 2nd is Y. A fact contains no variables.
Correspondence between DNA and Horn Clause DNA molecule expressing Horn
clause Fact molecule Rule molecule
~Q ~R
P
Q
~Q
P
P ← Q, RP ← QQ
sticky end
The Resolution Principleby Self-Assembly of DNA
~Q
~R
P
Q
~S
~T
P ← Q, R
Q ← S, T
P ← Q, R Q ← S, T
P ← S, T, R
Result Detection To put query molecules in To ligate molecules To detect a circular form
molecule~P P
The query molecule to
detect the fact P
Start!
Self-assembly
Self-assembly
Putting query molecules in
Query molecule
Ligation
Another example of circular molecule
Computational Complexity Time complexity
(The number of operations): constant Space complexity
(The minimum number of molecules to derive a fact): O(2n)
What’s String Tile Proposed by Winfree et al. (2000) String tiling is the collapse of multi-layer
assemblies into simpler superstructures. A string tile has a directed graph inside, t
he edges of the graph corresponds to DNA strands.
The graphs are connected with each other by hybridization of tiles.
Variable Substitutionby Self-Assembly of String Tile
P(f(X), Y) ← Q(X, g(Y))a / Y g(X) / X b / X
P(f(g(b)), a) ← Q(g(b), g(a))
Substitution tile Substitution tileSeed tile
A(f(X),Y) ← B(X, g(Y)), C(X, Y)
g(X) / X b / Xa / Y
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
B(g(b), g(a))
C(g(b), a)
A(f(g(b)), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
A(f(g(b)), a)
B(g(b), g(a))
C(g(b), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
NTM Simulation by Horn Clause Computation
Configuration is expressed by fact. Ss(ft(-1)(ft(-2)(fb(a1))), ft(0)(ft(1)(fb(fb(a2)))))
Transition rule is expressed by rule. Ss’(X, ft(-1)(ft’(0)(Y))) ← Ss(ft(-1)(X), ft(0)(Y)) Ss’(ft’(0)(X), Y) ← Ss(X, ft(0)(Y))
b t(-2) t(-1) t(0) t(1) b b
s
Features of Our Model Autonomous computation keeps
the number of operations constant. Our model is equivalent to non-
deterministic Turing machine. Variable substitution phase are
separated from deduction phase completely.
Advantage of Our Model Close relation to high-level
programming language PROLOG (Horn clause computation)
More suitable for expressing complex algorithms than other models.
Small number of operations(Autonomous computation)
Weak Point of Our Model Error-prone deduction
Term encoding has problem Too long sticky end Biased deduction
Estimation of complexity is not appropriate. Time complexity: Time to reach equilibrium is more
appropriate than the number of operations. Space complexity: More molecules will be required
because multiple proof trees are generated. 3-D conformation of proof tree molecule
Future Works Thermodynamic/kinetic analysis of
autonomous DNA computation Optimization of parameters
according to the analysis Temperature Salt concentration
Analysis of DNA computation as probabilistic algorithm