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
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DNA and splicing (circular) Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. di Milano - Bicocca ITALY Dipartimento di Informatica e Applicazioni,
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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}