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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
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We apologize...
ltltAn important aspect of this years meeting can
be summed up us SHOW ME THE EXPERIMENTAL
RESULT! gtgt
(T. Amenyo, Informal Report on 3rd Annual
DIMACS Workshop on DNA Computing,
1997)
theoretical results
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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
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LINEAR
SPLICING
CIRCULAR
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CIRCULAR SPLICING
restriction enzyme 2
restriction enzyme 1
ligase enzymes
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Circular languages definitions and examples
w, w? ?A, w w ? ? wxy, 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)
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Definition
FA C ? A ? L ? A, Cir(L) C, L ? FA, FA
? Chomsky hierarchy
Theorem Head, Paun, Pixton
C ? Reg ? Lin (C) ? Reg
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Circular splicing systems
(A finite alphabet, I? A initial language)
Pauns 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
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Other definitions of splicing systems
(A finite alphabet, I? A initial language)
Heads definition
SCH (A, I, T)
T? A ? A ? A triples
? A
(p, x, q ), ( u,x,v) ? T
hpxq ,
kuxv
vkux
hpx vkux q
q hpx
SCPI (A, I, R)
Pixtons definition
R? A ? A ? A rules
? A
(?, ?? ? ), (??, ? ?? ) ? R
?h ,
?? h?
h ? h? ??
h? ??
h ?
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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? Finadd. hyp. (symmetry, reflexivity)
C(F, Reg) ? F
C(Reg, Fin)?Reg,
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Circular finite splicing languages and Chomsky
hierarchy
CS
CF
C(Fin, Fin)
Reg
((aa)b)
(an bn)
(aa)
I ab ? 1, Ra b b a
I aa ? 1, Raa 1 1 aa
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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
(aba)
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Our contributions (continued)
Comparing the three definitions of splicing
systems
C(SCH ) ? C(SCPA ) ? C(SCPI )
?
(aba), ((aa)b)
... ?
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Star languages
Definition
L ? A is star language if L is regular,
closed under conjugacy relation and LX, with
X regular
Proposition
SCPA(A,I,R), I ? Cir(X) ? C(SCPA) ? Cir (X)
Consistence easily follows!!!
Examples

• (b(aba)) X

Xb ? aba
X aba

• (aba) X

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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
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Theorem
Fingerprint closed star languages ?? C(Fin,Fin)
Sketch
Take SCPA (A, I, R) with
ICir(successful path containing fingerprint of
cycles)
R1 1 1 ƒ ƒ fingerprint of cycle c,
for any cycle c
Star languages fingerprint closed
(for example Xb ? aba)

• X, X regular group code

(for example XAd )

• X finite, Cir(X)

Star languages not fingerprint closed
(aba) but not generated!!!
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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
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Theorem
For any z, zgt2, z unbordered word, then
Cyclic(z) ? C(Fin,Fin)
i.e. z ? uA? Au
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)))
Rz 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
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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...

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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 .

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Conclusions
DNA6 auditorium
Thanks!