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DNA and splicing (circular)
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  1. 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

  2. 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

  3. 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

  4. LINEAR SPLICING CIRCULAR

  5. CIRCULAR SPLICING restriction enzyme 2 restriction enzyme 1 ligase enzymes

  6. 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)

  7. Definition: FA~ ={ C A~|  L  A*, Cir(L) = C, L  FA, FA  Chomsky hierarchy} Theorem [Head, Paun, Pixton] C  Reg ~ Lin (C)  Reg

  8. 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

  9. 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 

  10. 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

  11. 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}

  12. 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*)*

  13. Our contributions (continued) Comparing the three definitions of splicing systems C(SCH )  C(SCPA )  C(SCPI )  ~ (a*ba*)*, ~ ((aa)*b) = ... ?

  14. 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*

  15. 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

  16. 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!!!

  17. 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

  18. 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 }

  19. 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...

  20. 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 .

  21. Conclusions DNA6 auditorium Thanks!