Post on 02-Jan-2016
description
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!