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##### DNA and splicing (circular)

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**DNA and splicing**(circular) circular Paola Bonizzoni, Clelia De Felice, Giancarlo Mauri, Rosalba Zizza Dipartimento di Informatica Sistemistica e Comunicazioni, Univ. di Milano - Bicocca ITALY Dipartimento di Informatica e Applicazioni, Univ. di Salerno, ITALY Circular splicing, definitions State of the art Our contributions Works in progress**We apologize...**<<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) theoretical results**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” Unconventional models of computation SPLICING**LINEAR**SPLICING CIRCULAR**CIRCULAR SPLICING**restriction enzyme 2 restriction enzyme 1 ligase enzymes**Circular languages: definitions and examples**w, w A*, w~ w w=xy, w = yx • Conjugacy relation on A* abaa 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* ~ Cir(L) = {~w | w L} (circularization of L) L L C (A linearization of C, i.e. Cir(L)=C ) {w A*| ~w C}=Lin(C) C (Full linearization of C)**Definition:**FA~ ={ C A~| L A*, Cir(L) = C, L FA, FA Chomsky hierarchy} Theorem [Head, Paun, Pixton] C Reg ~ Lin (C) Reg**Circular splicing systems**(A= finite alphabet, I A~ initial language) Paun’s definition SCPA = (A, I, R) R A* | A* $ A* | A* rules r = u1| u2 $ u3 | u4 R ~hu1u2, ~ku3u4 A~ u2 hu1 u4ku3 ~ u2 hu1 u4ku3 Definition A circular splicing language C(SCPA) (i.e. a circular language generated by a splicing system SCPA ) is the smallest circular language containing I and closed under the application of the rules in R**Other definitions of splicing systems**(A= finite alphabet, I A~ initial language) Head’s definition SCH = (A, I, T) T A* A* A* triples A~ ~hpxq, ~kuxv (p, x, q ), (u,x,v) T vkux ~hpx vkux q qhpx Pixton’s definition SCPI = (A, I, R) R A* A* A* rules A~ ~h, ~h (, ;), (, ; ) R ~ h h h h **Problem:**Characterize C(Reg, Fin) FA~ C(Fin, Fin) class of circular languages C= C(SCPA) generated by SCPA with I and R both finite sets. Theorem [ Paun96] F{Reg~, CF~, RE~} R +add. hyp. (symmetry, reflexivity, self-splicing) C(F, Fin) F Theorem [Pixton95-96] F{Reg~, CF~, RE~} R Fin+add. hyp. (symmetry, reflexivity) C(Reg~, Fin)Reg~, C(F, Reg) F**Circular finite splicing languages and Chomsky hierarchy**CS~ CF~ C(Fin, Fin) Reg~ ~((aa)*b) ~(an bn) ~(aa)* I= ~ab ~1, R={a | b $ b | a} I= ~aa ~1, R={aa | 1 $ 1 | aa}**Our contributions**Reg~ C(Fin, Fin) C(Fin, Fin) Fingerprint closed star languages Reg~ X*, X regular group code cyclic languages Cir (X*) X finite weak cyclic, other examples ~ (a*ba*)***Our contributions (continued)**Comparing the three definitions of splicing systems C(SCH ) C(SCPA ) C(SCPI ) ~ (a*ba*)*, ~ ((aa)*b) = ... ?**Definition**Star 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* X=b ab*a X= a*ba* • (a*ba*)* = X***c**q0 q0 y x z x’ y’ z’ Fingerprint closed languages Definition For any cycle c, L contains the Fingerprints of c Fingerprint of a cycle cnc L power of the cycle, where the internal cycles are crossed a finite number of times i n y , j n x c=(x(y(zz’)jy’)i x’)nc**Theorem**Fingerprint closed star languages C(Fin,Fin) Sketch Take SCPA = (A, I, R) with I=Cir({successful path containing fingerprint of cycles}) R={1 | 1 $ 1 | ƒ | ƒ fingerprint of cycle c, for any cycle c} Star languages fingerprint closed (for example X=b ab*a) • X*, X regular group code (for example X=Ad ) • X finite, Cir(X*) Star languages not fingerprint closed (a*ba*)*but not generated!!!**Not Star Languages in C(Fin, Fin)**new! Cyclic Languages Definition Cyclic(z) ={(~(z* p)) | p Pref (Lin( ~z))} Example 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(abc)= ~(abc)*a~(abc)*ab ~(abc)*b ~(abc)*bc ~(abc)*c ~(abc)*ca**Theorem**For any z, |z|>2, z unbordered word, then Cyclic(z) C(Fin,Fin) i.e. z uA* A*u 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 }**Other circular regular splicing languages**• ~(abc)*a~(abc)*ab ~(abc)*b ~(abc)*bc ~(abc)*c ~(abc)*ca ~(abc)*ac Cyclic(abc) weak cyclic languages • Cyclic (abca) .... bordered word...**Works 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 .**Conclusions**DNA6 auditorium Thanks!