Oritatami, a model of cotranscriptional foldingusers.ics.aalto.fi/sseki/presen/20150304_turku_presentation.pdfA U C G A U G C G A U (For reference) folding with minumum free energy

Oritatami, a model of cotranscriptional folding

Shinnosuke Seki

TUCS seminarMarch 4th, 2015

Is folding hard?

Most structures in the nature, however complex and intricate, areobtained from a linear genetic code by folding, e.g.,

RNA sequences→ large chain of amino-acids→ proteins.

Predicting the most likely folding of an input sequence is known tobe NP-hard.

Is the nature stubbornly solving such hard problems?

Cotranscriptional foldingRNA origami

Design of ssRNA origami that self-assembles RNA tiles

[Geary, Rothemund, Andersen 2014]

Cotranscriptional folding

A T C G C A T C G

Template DNA sequence

T7 RNA polymerase

T7 RNA polymerase

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(For reference) folding with minumum free energy


A T C G C A T C G


T7 RNA polymerase

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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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A T C G C A T C G


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Oritatami system

Free energy

RNA primary structure is modeled as a sequence over Σ.

RNA secondary structure (conformation) is modeled as a pair ofp a non-self-crossing directed path on the triangular grid

that is labeled by a primary structureR A subset of {(i , j) | The i-th and j-th vertices on p with j ≥ i + 2

that are adjecent to each other on the grid}.

Ex.) Two conformations a primary structure GCAAGCUCUACG may take.

G

C

A A

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C U

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U A

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GG

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A A G

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C U A

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Hydrogen bonds

Oritatami system

Free energy

RNA primary structure is modeled as a sequence over Σ.

RNA secondary structure (conformation) is modeled as a pair ofp a non-self-crossing directed path on the triangular grid

that is labeled by a primary structureR A subset of {(i , j) | The i-th and j-th vertices on p with j ≥ i + 2

that are adjecent to each other on the grid}.

Ex.) Two conformations a primary structure GCAAGCUCUACG may take.

G

C

A A

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C U

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U A

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GG

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C U A

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Hydrogen bonds

Oritatami systemFree energy

Principle of thermodynamics

Secondary structures with smaller free energy are more stable.Hence, among all possible secondary structures, a primarystructure folds into the one(s) with smallest free energy.

Ex.) The right conformation has more hydrogen bonds, and hence, more

stable, that is, has smaller free energy.

G

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Hydrogen bonds

Oritatami systemDefinition

An oritatami system is a 6-tuple Ξ = (Σ,R, α,w , σ, δt), whereR ⊆ Σ× Σ aRb means that a bead of type a ∈ Σ can form

a hydrogen bond with a bead of type b ∈ Σ.α ∈ N Beads can form at most α bonds.w ∈ Σ∗ ∪ Σω a primary structureσ an initial conformation called seed.δt ∈ N delay time

If w is periodic, we say that the oritatami system is cyclic.

Oritatami systemFolding process and nondeterminism

Delay time δt = 2

δt = 3

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Primary structure w = GCUCUACG


There are various ways to elongate the current conformation (blue)by transcribing the next 2 letters GC of the primary structure.

Delay time δt = 2

δt = 3

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Delay time δt = 2

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Delay time δt = 2

δt = 3

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Delay time δt = 2

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Delay time δt = 2

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As such, it folds like this.

Delay time δt = 2

δt = 3

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When the delay time becomes 3, the determination of how to foldrefers to the next 3 letters GCU instead.

Delay time δt = 2 δt = 3

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Hence, it folds nondeterministically in the two ways.


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Hence, it folds nondeterministically in the two ways.


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Cyclic oritatami systemBinary counter

reverse0 seed 1

reverse01

reverse 1 0

reversetowards next increment

adder subunit reverse subunit

This primary structure is a repetition of adder subunit and reversesubunit. Repetitive primary structures can be transcribed easilyfrom a cyclic DNA sequence.

Cyclic oritatami systemBinary counter (screenshot)

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Cyclic oritatami system is Turing complete

TheoremThe class of cyclic oritatami system with delay time 3 is Turingcomplete.

Proof.We design an oritatami system to emulate a cyclic tag system.

Cyclic oritatami system is Turing completeModified cyclic tag system

A cyclic tag system (cts) is a Turing-complete binary-stringrewriting system which consists of an initial word u ∈ {0, 1}∗ and alist of productions v1, v2, . . . , vn ∈ {0, 1}∗ considered sequentiallyin this order, cycling back to v1 after vn being considered.Its rewriting proceeds as follows:

1. Examine the leftmost letter of the current word.

2. If it is 1, then append the current production at the end of theword.

3. Delete the examined (leftmost) letter.

Skipping cts

It skips the next production after appending the current one in thecase of the leftmost letter being 1.

Cyclic oritatami system is Turing completeTools

Glider

•

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Gliders for oritatami systems with delay time 3.(Left) A glider proceeds rightward according to the rule(2,−3), (5, 0), (8, 3) ∈ R. It can make a turn, due to somehardcoding in R or collision against environments.(Right) Forward-swept wing (fsw) glider.

Cyclic oritatami system is Turing completeTools

Geometrical encoding

Our cts emulator encodes letters (0/1) of the current wordgeometrically by bumps (0) and dents (1).

0 1

Cyclic oritatami system is Turing completePrimary structure

Assume that the skipping cts to be emulated has n productionsv1, v2, . . . , vn ∈ {0, 1∗.

The primary structure of our cts emulator consists of the subunitsof the following two kinds:

Production It encodes one of the productions and plays the roleof appending it to the current word.

Reversal-read-copy (r2c) As the name suggests, it plays threeroles: reversal, reading a letter, and copying letters.

The primary structure is of the form:

α1x1α2x2 · · ·αnxn,

where x1, . . . , xn are r2c and αi encodes the i-th production vi .

Cyclic oritatami system is Turing completeReading the prefix of the form 0∗1

0

1

1

Reversal0

Reading 0

Reading 1

n = 3 production subunits are colored.

Cyclic oritatami system is Turing completeAppending the encoded current production at the end

••

•

Glid

eror

swit

chba

ckof

3hbe

ads

p4hp5h

p6h •p10h

One copy unit

•

p(w−18−4(`−|vi |))h

•p(w−4)h

Cyclic oritatami system is Turing completeRelated open problems

Arity-1

Our cts emulator requires multiple arity. Is the class of oritatamisystems with arity 1 still Turing complete?

Minimum alphabet size

What is the smallest alphabet Σ with which oritatami systems canbe Turing complete?

Minimum period

What is the shortest period with which cyclic oritatami systemscan be Turing complete?

Future projects on oritatami systems

There is a broad unexplored frontier of oritatami systems andcotranscriptional folding.

I Design of oritatami systems to self-assemble structures andmechanisms useful in nano-engineering.

I Algorithms to convert folding designs into oritatami systemsand their computational complexity

I Development of oritatami simulator (Pierre-Etienne Meuniermade the first one) and oritatami CAD

I Optimization of oritatami systems design

I Intrinsic universality

I Stochastic oritatami systems

I In-vitro implementation of oritatami systems in laboratories

I Oritatami GWAP (game with a purpose).

Oritatamists

Cody Geary(CalTech)

Pierre-Etienne Meu-nier(Aalto Univ.)

Nicolas Schabanel(LIAFA, Univ. ParisDiderot)

University ofElectro-Communications

I National University specializedin computer and physicalscience, engineering, andtechnology.

I Notable alumni includesI Seinosuke Toda (Toda’s

theorem)I Sumio Iijima (inventor of

carbon nanotubes)I Ken Kutaragi (father of

PlayStation)

I Just 15 mins train ride from thecentral Tokyo.

University ofElectro-Communications

Hiro Ito Satoshi Kobayashi

JSPS Fellowship to Japan

The Academy of Finland has funded researcher mobility with theJapan Society for the Promotion of Science (JSPS) since 1988.

I All scientific disciplines

I Duration 12-24 monthsI It covers

I travel costs between Finland and JapanI monthly grant of 362,000 JPY (about 2700 EUR)I settling-in allowance of 200,000 JPY

Thank you very much for your attention!

This talk is supported by the Academy ofFinland, Postdoctoral Researcher Grant No.13266670/T30606.

References I

C. Geary, P-E. Meunier, N. Schabanel, S. Seki.Efficient universal computation by molecular co-transcriptionalfolding.submitted.

C. Geary, P. W. K. Rothemund, E. S. Andersen.A single-stranded architecture for cotranscriptional folding ofRNA nanostructures.Science 345: 799-804, 2014.

Documents

Oritatami, a model of cotranscriptional foldingusers.ics.aalto.fi/sseki/presen/20150304_turku_presentation.pdfA U C G A U G C G A U (For reference) folding with minumum free energy