Towards the sequence design preventing pseudoknot formation

1
Towards the sequence design preventing pseudoknot
formation

• 2nd International Workshop on Natural Computing,
  Dec. 10-12, 2007
• Noyori Conference Hall, Nagoya University, Japan

Lila Kari and Shinnosuke Seki
Biocomputing Laboratory Department of Computer
Science, University of Western Ontario, London,
ON, Canada http//www.csd.uwo.ca/lila/,
sseki lila, sseki_at_csd.uwo.ca
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DNA computing brief overview

• Process flow
• Information is encoded into DNA strings over
  adenine (A), cytosine (C), guanine (G), and
  thymine (T)
• Succession of intermolecular reactions among
  encoding DNA strings (bio-operations) in an
  expected manner based on the base-pairing A-T and
  C-G (Watson-Crick complementarity)
• Resulting DNA strings are decoded using such
  techniques as Gel-electrophoresis and PCR
  (polymerase chain reaction).
• Advantages
• Parallelism
• NP-complete problem solvable
• Massive storage capacity
• Energy efficiency

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Watson-Crick complementarity
A
C
C
G
T
A
G
5
3
5
3
T
G
G
A
T
C
C

• C - G, A - T (or A - U in RNAs)
• Two DNA single strands with opposite orientation
  (5 - 3) can bind to each other via
  Watson-Crick complementarity
• Bio-operations of DNA computing strongly depend
  on this biochemical property.

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Adlemans first DNA computing Adleman, 1994
A solution of Hamiltonian Path Problem
(NP-complete problem)

• Hamiltonian Path
• A path which visits each vertex exactly once.
• Hamiltonian Path Problem
• Whether a Hamiltonian path exists in a given
  graph.
• How to encode this problem into a test tube
  called DNA computer?

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Adlemans first DNA computing Adleman, 1994
A solution of Hamiltonian Path Problem
(NP-complete problem)
encode
CTCGAATAGC TCGGATATAC
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Whats the challenges in DNA computing?

• How to design encoding DNA strands?
• Any kind of intramolecular structures are
  undesirable for encoding DNA single strands
• Intramolecular structures deprive strands of
  their ability to interact with another strand
  (bio-operations)
• On the other hand, DNA single strands tend to
  form intramolecular structures for thermodynamic
  stability.

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Intramolecular structures

• E coli transfer-messenger RNA
• hairpin loops
• bulge loops
• internal loops
• multiple loops
• pseudoknots

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Intramolecular structure freeness problem
Formal language theoretic approach

• DNA alphabet
• d-morphic involution
• Antimorphism (cf.
  morphism )
• Involution
• Watson-Crick complementarity
• Antimorphic involution

A
T
C
T
A
G
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Intramolecular structure freeness problem
Formal language theoretic approach
A
C
G
AA
T
G
C

• Hairpin structure
• The hairpin structure is the most well-known, and
  hence intensively investigated intramolecular
  structure (e.g. 2).
• A DNA strand which forms a hairpin can be modeled
  as

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?-bordered words Kari et al. 2007

• ?-border
• v is said to be ?-border of w if
• w is said to be ?-bordered if w has a non-empty
  ?-border otherwise, it is called ?-unbordered.
• A DNA strand which forms a hairpin is modeled as
• Therefore, ?-unbordered words do not form
  hairpins in this sense.
• the set of all ?-borders of w

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Pseudoknots
3
y
?(y)
x
a
y
?
?(x)
?
d
?(y)
s
x
?(x)

• Generic term of cross-dependent structures the
  right figure is the simplest and hence most
  popular type pseudoknot.
• From the viewpoint of formal language, a strand
  which forms a pseudoknot of this type can be
  modeled as
• In this paper, we consider the case where

5
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?-Pseudoknot-bordered words

• ?-pseudoknot-border
• v is said to be ?-pseudoknot-border of w if
• w is said to be ?-pseudoknot-bordered if w has a
  non-empty ?-pseudoknot-border otherwise it is
  called ?-pseudoknot-unbordered.
• In particular, for an antimorphic ?, if xy is a
  ?-pseudoknot-border of w, then
• ?-pseudoknot-unbordered words never form a
  pseudoknot of the type
• the set of all ?-pseudoknot-border
  s of w

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Example of ?-Pseudoknot-bordered words

• Let
• ? Watson-Crick complementarity
• In fact, by letting
• Also by letting

x
?(x)
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?-border and ?-pseudoknot border

• Lemma 2
• Let ? be a d-morphic involution, and
  . Then
• Let be the set of all ?-unbordered
  words.
• Let be the set of all ?-pseudoknot
  unbordered words.
• Proposition 1
• Let ? be a d-morphic involution. Then

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Properties of ?-unbordered words Kari et al.
1997

• Pref(u), Suff(u) the set of all prefixes
  (suffixes) of u.
• Lemma Kari et al. 1997
• Let ? be an antimorphism and . Then
  u is ?-unbordered, i.e, , iff
• Lemma 5
• Let ? be an antimorphism and . Then
• Corollary 2
• Let ? be an antimorphism and . Then

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Properties of ?-pseudoknot unbordered words

• We cannot say generally
• Example 7
• Let
• uu is ?-pseudoknot bordered because for

AATTTTATA
AATTTTATA
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Repetitions of a word and their ?-pseudoknot
unbordered property (cont.)

• Proposition 2
• Let ? be an antimorphism. Then for
  , if
  , then for any ,
• Corollary 3
• For a word ,

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Primitive ?-pseudoknot unborderd words

• A word u is primitive if u wi implies i 1.
• Lemma 7
• Let ? be an antimorphism and .
  If u2 is ?-pseudoknot bordered, then u is
  primitive.
• Corollary 4
• If and it is not primitive,
  then u2 is ?-pseudoknot unbordered, i.e.,
  . This implies
• Example 8
• Let . Neither u
  nor uu are ?-pseudoknot-bordered. This implies
  that

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Primitive ?-pseudoknot unborderd words (cont.)

• Theorem 1
• Let ? be an antimorphism and
  satisfying Then any ?-pseudoknot
  border of u2 is primitive.
• Proposition 3
• Let ? be an antimorphism and .
  If w is a ?-pseudoknot border of u2, then the
  factorization of w into x and y s.t.
  
  is unique.

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Future work

• How to ease the condition
  ?
• The current condition is too strict for practical
  use.
• How to guarantee the ?-pseudoknot unbordered
  property for the concatenation of two different
  ?-pseudoknot unbordered words?
• Concatenation is a basic manipulation used in DNA
  computing.
• How to model more complicated pseudoknots?

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References

• L. Adleman, Molecular computation of solutions to
  combinatorial problems, Science 266, 1021-1024,
  1994
• L. Kari and K. Mahalingam, Involutively bordered
  words, International Journal of Foundations of
  Computer Science, 2007.
• L. Kari and S. Seki, On pseudoknot words and
  their properties, submitted.