Horn Clause Computation by Self-Assembly of DNA Molecules

1
Horn Clause Computation by Self-Assembly of DNA
Molecules

• Hiroki Uejima
• Masami Hagiya
• Satoshi Kobayashi

2
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
  generated candidates.
• Applying a single operation to multiple molecules
  expressing data at once.

3
Previous Works(Computational Power/Model)

• The correspondence between forms of DNA molecule
  and computational power based on formal
  languages.
• Various computational models
• Branching program
• Turing machine
• Boolean circuit
• Random Access Memory
• Horn clause computation (Kobayashi)

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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.

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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

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Structure of DNA Tile
Z
X
Z
X
Y
W
Z
X
Y
W
Y
W
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cf. Winfrees DNA Tile
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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.

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Outline of The Algorithm

• To generate ground Horn clauses by variable
  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.

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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 atom is X,
  the 2nd is Y.
• A fact contains no variables.

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Correspondence between DNA and Horn Clause

• DNA molecule expressing Horn clause
• Fact molecule
• Rule molecule

sticky end
P
P
Q
R
Q
Q
Q
P ? Q
P ? Q, R
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The Resolution Principleby Self-Assembly of DNA
P
P ? Q, R Q ? S, T
P ? Q, R
P ? S, T, R
Q
R
Q ? S, T
Q
S
T
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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
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Start!
15
Self-assembly
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Self-assembly
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Putting query molecules in
Query molecule
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Ligation
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Another example of circular molecule
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Computational Complexity

• Time complexity
• (The number of operations) constant
• Space complexity
• (The minimum number of molecules to derive a
  fact) O(2n)

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Whats 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, the
  edges of the graph corresponds to DNA strands.
• The graphs are connected with each other by
  hybridization of tiles.

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Variable Substitutionby Self-Assembly of String
Tile
P(f(X), Y) ? Q(X, g(Y))
g(X) / X
b / X
a / Y
Substitution tile
Seed tile
Substitution tile
P(f(g(b)), a) ? Q(g(b), g(a))
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A(f(X),Y) ? B(X, g(Y)), C(X, Y)
g(X) / X
b / X
a / Y
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A(f(g(b)), a) ? B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a) ? B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a) ? B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a) ? B(g(b), g(a)), C(g(b), a)
A(f(g(b)), a)
C(g(b), a)
B(g(b), g(a))
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A(f(g(b)), a) ? B(g(b), g(a)), C(g(b), a)
A(f(g(b)), a)
C(g(b), a)
B(g(b), g(a))
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NTM Simulation by Horn Clause Computation
t(-2)
t(-1)
t(0)
b
t(1)
b
b
s

• 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))

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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.

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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)

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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

33
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