Nondeterministic seedless oritatami systems and hardness ... · cotranscriptional folding (oritatami means folding in Japanese). (Left) 3D Image of a tile generated by RNA origami

Nondeterministic seedless oritatami systems and hardness. . .

Nondeterministic seedless oritatami systems andhardness of testing their equivalence

Yo-Sub Han1 Hwee Kim1∗ Makoto Ota2 Shinnosuke Seki2

1Yonsei University

2University of Electro-Communications

The 22nd International Conference onDNA Computing and Molecular Programming

Presented by Hwee Kim (Yonsei University, UEC) DNA 2016 1 / 17

Nondeterministic seedless oritatami systems and hardness. . . Introduction

RNA Origami to Oritatami System (OS)

RNA Origami (Geary et al. (2014))




Cotranscriptional folding—folding occurs during transcription




Oritatami System is a mathematical model of computation bycotranscriptional folding (oritatami means folding in Japanese).

(Left) 3D Image of a tile generated by RNA origami (Right) Conformation thatrepresents the tile




RNA Origami Oritatami SystemNucleotides Beads

Primary structure Sequence of beadsconnected by a line

h-bonds between nucleotides InteractionsCotranscriptional folding rate Delay

Resulting secondary structure Conformation

beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure








beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure








beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure








beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure








beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure








beadinteraction

a

b

b

b b

bd

c e(a, d), (c, e)

interaction rules

conformation

abbcbdbbe

⇒

primary structure



Dynamics of OS (Geary et al. (2015))

We assume that the seed σ of the system is given as the initialconformation. Then, each bead of the primary structure w stabilizes(at a point of the triangular lattice) as follows:

1 The bead look ahead up to next d beads, where d is the delay ofthe system.

2 The first bead stabilizes as to maximize the number of interactionsthat the lookahead of d beads forms (Note that only the first beadstabilizes, not the whole lookahead of d beads.).

3 The ruleset H denotes which pair of beads forms an interactionwith each other.

The arity α denotes the maximum number of interactions that a beadcan form.




Example (Delay-3 Glider)

The seed is given as the red line. The primary structure isdbeabcdbeabc · · · . The ruleset is {(a,e), (c,d)}.

a

b

c

dbe






a

b

c

d b e






a

b

c d

b

e






a

b

c d






a

b

c d b

e aa






a

b

c d

b

ea






a

b

c d

b

e

a






a

b

c d

b






a

b

c d

b

eab






a

b

c d

b

e






a

b

c d

b

e a

b

c d

b

e a

b

c



Dynamics of OS

The OS is deterministic if every bead is stabilized uniquely. Otherwise,the system is nondeterministic.

Example (Nondeterminism)

The seed is given as the red line. The primary structure is def . Theruleset is {(d ,a), (d ,b), (f ,a), (f ,b), (f , c)}.

a

b

c



Dynamics of OS




a

b

c

a

b

c

d

ef d

e

f



Dynamics of OS




a

b

c

a

b

c

d

ef d

e

f



Dynamics of OS




a

b

c

a

b

c

d

ef d

e

f



Dynamics of OS




a

b

c

a

b

c

d

ef d

e

f



Previous Works

OS is Turing complete (Geary et al. (2015)).We can construct a binary counter (Geary et al. (2016)).Under the restriction on parameters (delay and arity), we canretrieve a ruleset from the given conformation in polynomial time.Otherwise, the problem is NP-complete (Ota and Seki (2016)).


Nondeterministic seedless oritatami systems and hardness. . . Problem Verification

What We Have Done

For a given DNF (Disjunctive Normal Form) formula, we propose anOS that checks whether the given formula is a tautology or not.

A DNF formula ϕ is written as∨

1≤i≤n Ci for some clausesC1, . . . ,Cn that is a logical AND (∧) of some of the Booleanvariables v1, . . . , vm and their negations.A Boolean formula is a tautology if the formula is evaluated toTRUE on all possible assignments.

Example

A DNF formula (a ∧ b) ∨ (a ∧ c) is not tautology, since the formulareturns FALSE when b = FALSE. A DNF formula(a ∧ b) ∨ (a ∧ ¬b) ∨ (¬a ∧ b) ∨ (¬a ∧ ¬b) is a tautology.

The DNF tautology problem is coNP-complete.


Nondeterministic seedless oritatami systems and hardness. . . Designing OS

Designing OS—Big Picture

ClausesC1 C2 · · · Cn

We regard each column as a clause.





Variables

v1

v2

...

vm

We regard each row as a variable.





Variables

v1

v2

...

vm

We choose random values for variables and propagate values right.





Variables

v1

v2

...

vm

We evaluate the value of the clause and propagate the result down.





Variables

v1

v2

...

vm

Check if one of these is TRUE

We check if one of the clauses returns TRUE.





Variables

v1

v2

...

vm

Check if one of these is TRUE

A circuit module with 2-bits

inputs and 2-bits outputs

The essential module




e-Buffer

Formatterf-BufferFormatter

TT

F

e-Buffer


F F

punsat

Formatter

T T T

Formatter

F F F

e-Buffer

f-Buffer

e-Buffer

f-Buffer

Assignor

psat

Starter

Verifier

Turner

Turner

C1 C2 C3

v2 ⇒

Assignor

v1 ⇒ e-Buffer

f-Buffer

e-Buffer

f-Buffer

T

F

T

Evaluator(n-type)

Evaluator(n-type)

Evaluator(p-type)

Evaluator(n-type)

Evaluator(e-type)

Evaluator(p-type)

U U*

origin

An overview of the NOS that evaluates a given DNF formula(¬v1 ∧ ¬v2) ∨ (v1 ∧ ¬v2) ∨ (v2).



Designing OS—Assigning Random Values

Q) How can we encode a randomly assigned value?A) We exploit nondeterminism to differentiate the height of the lastbead.

12f2

11f2

10f2

1p1/n1/e1

1a

3fb

1a

FALSE

1a

6a

7a

8a

9a

4a

10eb

11eb

12eb

12f2

11f2

10f2

1p1/n1/e1

1a

3fb

3a

4a

1a

2a

5a

4a

1a

2a

6a

7a

8a

9a

4a

10eb

11eb

12eb

TRUE

4a

4a

3a

2a

5a

2a

The Assignor module randomly assigns the value for each variable.



Designing OS—Evaluating Clauses

We use 4 glider-like conformations for evaluators.

1 6 7 12

FALSE1 10 11 12

TRUE

3 4 9 10 9 107 8

1 1

2 2

2

2

3 3

3

4 4

4

5

5

5 5

6

6

6

7

7

8 8

8

9

9 10

11

11 11

(a) (b)

(c) (d)

Not yet decided(*)Unsatisfied(U)

Not yet decided(*)Unsatisfied(U)

FALSE FALSE FALSE

TRUETRUE TRUE




The module should fold differently according to the relationshipbetween Ci and vj , the previous evaluation and the value of vj .Relationship Prev. evaluation vj Output Conformationvj is in Ci ∗ TRUE ∗ (d)

FALSE U (a)U TRUE U (c)

FALSE U (a)¬vj is in Ci ∗ TRUE U (c)

FALSE ∗ (b)U TRUE U (c)

FALSE U (a)neither vj nor ¬vj ∗ TRUE ∗ (d)is in Ci FALSE ∗ (b)

U TRUE U (c)FALSE U (a)




Example

When vj = TRUE and vj is in Ci

3fb/7fb 9β10β 4β 3βinput U

3fb/7fb 9β10β 8β 7βinput ∗

1µ

10eb

11eb

12eb

10eb

11eb

12eb1µ




Example




1µ

2µ

3µ 4µ10eb

11eb

12eb

10eb

11eb

12eb2µ 3µ 4µ 1µ

2µ

3µ 4µ

2µ 3µ 4µ




Example




1µ

2µ

3µ 4µ10eb

11eb

12eb

10eb

11eb

12eb1µ 2µ 3µ

5µ

6µ

4µ

5µ6µ




Example




1µ

2µ

3µ 4µ10eb

11eb

12eb

10eb

11eb

12eb1µ 2µ 3µ

5µ

6µ

4µ

5µ6µ

7µ

8µ

9µ 7µ 8µ 9µ




Example




1µ

2µ

3µ 4µ10eb

11eb

12eb

10eb

11eb

12eb1µ 2µ 3µ

5µ

6µ

4µ

5µ6µ

7µ

8µ

9µ 7µ 8µ 9µ10µ

11µ

12µ

10µ

11µ

12µ



Designing OS—Verifying the Result


punsat

Formatter f-Buffer

psat

Verifier

f-Buffer

U U*




The verifier “probes” the result ∗.

10f2 9f2 8f2 7f2

7v 8v 9v 10v

15v

16v

17v

15v16v17v

10f2 9f2 4f2 3f2

7v 8v 9v 10v

15v

16v17v

15v16v17v

14v 14v

input Uinput ∗




For the equivalence problem (We started from here!), we can add onerule for the last probe to be equivalent if and only if the formula is atautology.

10f2 9f2 4f2 3f2

7v 8v 9v 10v

15v

16v17v

15v16v17v

14v

input U for C1

10f2 9f2 4f2 3f2

7v 8v 9v 10v

15v2

16v217v2

15v216v217v2

14v

This case does not happen

when the formula is tautology!

input U for C1


Nondeterministic seedless oritatami systems and hardness. . . Conclusions

Conclusions and Future Works

Oritatami System (OS) is a mathematical model of computation bycotranscriptional folding.We designed an OS that solves the DNF tautology problem.

We exploited nondeterminism to encode 2 different valuesrandomly.We designed a circuit module handling 2-bits inputs and 2-bitsoutputs.We proved that the OS equivalence problem is coNP-complete.

Future worksCan we exploit nondeterminism to encode more than 2 cases atonce?Can we propagate (and process) n-ary information rather than abit?. . . and many theoretical and practical questions—this is TerraIncognita!



Thank You!



References I

Geary, C., Meunier, P., Schabanel, N., and Seki, S. (2015). Efficientuniversal computation by greedy molecular folding. CoRR,abs/1508.00510.

Geary, C., Rothemund, P. W. K., and Andersen, E. S. (2014). Asingle-stranded architecture for cotranscriptional folding of RNAnanostructures. Science, 345:799–804.

Geary, C. W., Meunier, P., Schabanel, N., and Seki, S. (2016).Programming biomolecules that fold greedily during transcription. InProceedings of the 41st International Symposium on MathematicalFoundations of Computer Science, pages 43:1–43:14.

Ota, M. and Seki, S. (2016). Rule set design problems for oritatamisystem. Theoretical Computer Science (accepted).


Documents

Nondeterministic seedless oritatami systems and hardness ... · cotranscriptional folding (oritatami means folding in Japanese). (Left) 3D Image of a tile generated by RNA origami