Horn Clause Computation by Self-Assembly of DNA Molecules Hiroki Uejima Masami 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 generated candidates. Applying a single operation to multiple molecules expressing data at once.

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)

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 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.

Horn Clause Used in This Algorithm A term in a rule is the form f 1 ( … f n (X) … ). The arity of a predicate is at most 2. The arity of a function is 1 The variable of the 1 st argument of an atom is X, the 2 nd 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, R P ← Q Q sticky end

The Resolution Principle by 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

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

Variable Substitution by Self-Assembly of String Tile P(f(X), Y) ← Q(X, g(Y)) a / Y g(X) / Xb / X P(f(g(b)), a) ← Q(g(b), g(a)) Substitution tile Seed tile

A(f(X),Y) ← B(X, g(Y)), C(X, Y) g(X) / X b / X a / Y

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. S s (f t(-1) (f t(-2) (f b (a 1 ))), f t(0) (f t(1) (f b (f b (a 2 ))))) Transition rule is expressed by rule. S s ’ (X, f t(-1) (f t ’ (0) (Y))) ← S s (f t(-1) (X), f t(0) (Y)) S s ’ (f t ’ (0) (X), Y) ← S s (X, f t(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