DNA and splicing

(circular)

Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. di Milano - Bicocca ITALY

Dipartimento di Informatica e Applicazioni, Univ. di Salerno, ITALY

Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza

Circular splicing, definitions

State of the art

Our contributions

Works in progress

<<An important aspect of this year’s meeting can be summed

up us: SHOW ME THE EXPERIMENTAL RESULT! >> (T. Amenyo, Informal Report on 3rd Annual

DIMACS Workshop on DNA Computing, 1997)

We apologize...

theoretical results

Before Adleman experiment (1994)...Before Adleman experiment (1994)...

Tom Head 1987 (Bull. of Math. Biology)

“ Formal Language Theory and DNA:an analysis of the generative capacity of

specific recombinant behaviors”

SPLICINGUnconventional

models of computation

SPLICINSPLICINGG

LINEARLINEAR

CIRCULACIRCULARR

CIRCULAR SPLICING

restriction enzyme 1

restriction enzyme 2

ligase enzymes

Circular languages: Circular languages: definitions and definitions and examplesexamples

• Conjugacy relation on A* w, w A*, w ~ w w=xy, w = yx

Example abaa, baaa, aaab,aaba are conjugate

• A~ = A* ~ = set of all circular words ~w = [w]~ , w A*

• Circular language C A ~ set of equivalence classes

A* A* ~

L Cir(L) = {~w | w L} (circularization of L)

CL

C{w A*| ~w C}= Lin(C)(Full linearization of C)

(A linearization of C, i.e. Cir(L)=C )

FA~ ={ C A~ | L A*, Cir(L) = C, L FA, FA Chomsky hierarchy}

Definition:

Theorem [Head, Paun, Pixton]

C C Reg Reg ~ Lin (C) Lin (C) Reg Reg

Paun’s definition

Circular splicing systemsCircular splicing systems(A= finite alphabet, I A~ initial

language)

SCPA = (A, I, R) R A* | A* $ A* | A* rules

~hu1u2 ,~ku3u4 A~

r = u1 | u2 $ u3 | u4 R

u2 hu1 u4ku3 ~ u2 hu1 u4ku3

DefinitiDefinitionon

I and closed under the application of the rules in R

A circular splicing language C(SCPA) (i.e. a circular language generatedby a splicing system SCPA ) is the smallest circular language containing

Other definitions of splicing Other definitions of splicing systemssystems

Head’s definition SCH = (A, I, T) T A* A* A* triples

A~

(p, x, q ), ( u,x,v) T

vkux ~ hpx vkux q

~hpxq ,~kuxv

q hpx

(A= finite alphabet, I A~ initial language)

SCPI = (A, I, R)

A~

(, ; ), (, ; ) R

~ h h

~h ,~ h

h

Pixton’s definition R A* A* A* rules

h

Problem:

Theorem [ Paun96]

Characterize

FA~ C(Fin, Fin)

C(Reg, Fin)

class of circular languages C= C(SCPA) generated by SCPA with I and R both finite sets.

F{Reg~, CF~, RE~}

R +add. hyp. (symmetry, reflexivity, self-splicing)

Theorem [Pixton95-96] R Fin+add. hyp. (symmetry,

reflexivity)

C(F, Fin) F

F{Reg~, CF~, RE~}

C(F, Reg) FC(Reg~, Fin)Reg~,

Circular finite splicing languages Circular finite splicing languages and Chomsky hierarchyand Chomsky hierarchy

CS~

CF~

Reg~

~((aa)*b)

~(aa)*~(an bn)

I= ~aa ~1, R={aa | 1 $ 1 | aa} I= ~ab ~1, R={a | b $ b | a}

Our Our contributionscontributions

Reg~

Fingerprint closedstar languages

X*, X regulargroup code

Cir (X*)X finite

cyclic languages

weak cyclic,other examples

~ (a*ba*)*

Reg~ C(Fin, Fin)

Our contributions Our contributions (continued)(continued)

Comparing the three definitions of splicing

systems

C(SCH ) C(SCPA ) C(SCPI )

~ (a*ba*)*, ~ ((aa)*b)

= ... ?

Star languagesStar languages

L A* is star language if L is regular, closed under

conjugacy relation and L=X*, with X regular

Proposition:SCPA=(A,I,R), I Cir(X*) C(SCPA) Cir (X*)

“Consistence” easily follows!!!

Examples

• (b*(ab*a)*)* = X*

• (a*ba*)* = X*

X=b ab*a

X= a*ba*

Definition

Fingerprint closed Fingerprint closed languageslanguagesDefinitionDefinition

For any cycle c, L contains the Fingerprints of c

Fingerprint of a cycleFingerprint of a cycle cnc L

power of the cycle, where the internal cycles are crossed a finite number of times

c=(x(y(zz’)jy’)i x’)nc i n y , j n x

c

q0

x’

x

y’

y z

z’

q0

Fingerprint closed star languages C(Fin,Fin)

Theorem

I=Cir({successful path containing fingerprint of cycles})R={1 | 1 $ 1 | ƒ | ƒ fingerprint of cycle c, for any cycle c}

Star languages not fingerprint closed

(a*ba*)* but not generated!!!

Star languages fingerprint closed• X*, X regular group code

• X finite, Cir(X*)

Sketch

Take SCPA = (A, I, R) with

(for example X=b ab*a)

(for example X=Ad )

Not Star Languages in C(Fin, Not Star Languages in C(Fin, Fin)Fin)new!

Definition

Cyclic(z) ={(~(z* p)) | p Pref (Lin( ~z))}

Example

Cyclic(abc)= ~(abc)*a ~(abc)*ab

~(abc)*b ~(abc)*bc

~(abc)*c ~(abc)*ca

z = abc A*

Lin ( ~ z) =Lin ( ~ abc) ={abc, bca,cab}

Pref(Lin ( ~ z)) =Pref(Lin ( ~ abc)) =Pref({abc, bca,cba}) = {a, ab, b, bc, c, ca}

Cyclic Languages

Theorem

Cyclic(z) C(Fin,Fin)

The proof is quite technical ...

Example (continued)

Cyclic (abc) is generated by SCPA = (A,I,R) where I,R are defined as follows

I={~ ((abc)i p | 0 i 3, p Pref(Lin(~

(abc))) }R={z ab | z $ z | ca z, z ab | z $ z b | c z, z ca | z $ z $ bc z,

z a | z $ z | b z, z b | z $ z $ c z , z c | z $ z | a z }

For any z, |z|>2, z unbordered word, then

i.e. z uA* A*u

Other circular regular splicing Other circular regular splicing languageslanguages

• ~(abc)*a ~(abc)*ab ~(abc)*b ~(abc)*bc ~(abc)*c ~(abc)*ca

Cyclic(abc)~(abc)*ac

weak cyclic languagesweak cyclic languages

• Cyclic (abca) .... bordered word...

Works in progressWorks in progress

• Characterize Reg~ C(Fin, Fin)

• Characterize FA~ C(Fin, Fin)

• C(SCPI) = Star languages

• Additional hypothesis

r= u1 | u2 $ u3 | u4 in R

• Reflexive: r’ = u1 | u2 $ u1 | u2

• Symmetric: r” = u3 | u4 $ u1 | u2

• Self-splicing: From ~ xu1u2yu3u4 ,

with r,r” as above, generates ~u4 xu1 , ~u2yu3 .

DNA6auditorium

Thanks!