DNA Structure Notation Operations

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DNA Structure NotationOperations

• Vincenzo Manca
• Dipartimento di Informatica
• Universita di Verona

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10 Years of Molecular Computing

• 1994 Adlemans Experiment
• 1995 Liptons Model
• 1996 Int. Conf. on Math. Linguistics (Marcus)
• 1997 Mangalia (Paun, Head)
• 1998 MFCS Brno (Molecular Computing)
• 1999 (Pauns WMC)
• 2000 DNA6 Leiden
• 2001 DNA7 Tampa (FL) 3-SAT
• 2002 DNA8 Sapporo DNA Duplication
• 2004 DNA10 Milan XPCR Extraction
• 2005 DNA11 Ontario XPCR Recombination

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DNA Computing Motto

• Problem Data and Requirements
• Algorithm Solutions
• Encode data by DNA strands
• Encode algorithms by biotech procedures
• Decode final strands as solutions

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A General schema of combinatorial problem
A set of Requirements for assignments, that
is, sequences 0/1 of some length nThe Space of
possible solutions has E(2,n) elements, but only
some of them satisfy the requirements
Encode assignments by DNA strandsEncode
requirements as biotech protocols that filterthe
strands encoding the true solutions
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Possible Solutions
True Solutions
Solution Extraction In linear time !!!
Space Generation In linear time
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New Trends in DNAC

• DNA Self Assembly (Seeman, Winfree, )
• DNA Automata (Shapiro)
• DNA Algorithms gt new biotech protocols

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A change of perspective
Biotech Protocols
DNA Computing
Computing DNA
Algorithms
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• In the search for implementing algorithms on
  DNA, general algorithmic principles are
  discovered in fundamental biomolecular processes.

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Nucleotides

• H
• - OH
• - H2O

P
P
330 Dalton 1 Dalton 1.64 10-24 1 g. H 6.2
1023 1--- 1 1nm
5
CH2OH
O
1
----
4
----
P
B
3
2
2
5
3
CH2
O
1
1
4
4
H
B
B
O
5
CH2OH
3
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A few grams of DNA the amount of all electronic
information stored in all the world
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Strings

• Strings over an alphabet are sequences of symbols
  of the alphabet
• abbabbba
• On strings a concatenation associative operation
  - - is defined
• (??)? ?(??)
• ? ?? ??
• A language L is a set of strings

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DNA Sequences are Mobile Double Strings

• B A, T, C, G
• B strings over B
• ?i,j
• ?
• s is a ?-strand or s ? or type(s )?
• ? n or mult(?)n

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• Complementation - c (involutive)
• Reverse rev (involutive)
• Mirror mir (involutive)
• mir(?) rev(?c)
• Reverse and Complementation commute
• Hybridization
• ?
• Pairing ?
• ?

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B A, T, C, G BB strings over B
fraction notation Axiom ?
rev(?) rev(?)
? ext Overlap --x-- overlapping
concatenation Z ?-gt up lt- ? down ?-gt ? -gt/
? ?? -gt/ ?
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BilinearityComplementarityAntiparallelism

• The marvelous form

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3
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• Hybridization
• ? mir(?)
• ? ? ? ltgt ? ? ? , ? ? mir(?)
• ? ? ltgt ? ? ? for some ?
• Pairing ? ? gt ? / rev(?)

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• Notation ? / ? ? ?-gt
• ? / mir(?) lt?gt
• ? / ? rev(?) lt- ?
• gt lt?gt ltmir(?)gt
• BB is the set of DNA strings , BB ? B

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A pool P of DNA molecules is a multiset of strands

• i) Set of strands typed by strings
• ii) Set of strings with multiplicities
• P s1?1 , s2?2, .
• P ?1 n1 , ?2 n2, .
• multP(?1) n1 , multP (?2) n2
• s ? P
• ? ? P

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Types of DNA Pools are Languages of BB
Type(T) ? ? BB s ? , s ? T
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Test Tube Operations in DNAC

• Denature (Melting)
• Renature (Hybridization, Annealing)
• Mix
• Split
• fish (by Affinity)
• Remove
• length
• Separate (Gel Electrophoresis)
• Ligate (Ligase)
• Extend (Polymerase)
• Synthetize (Oligos)
• Infix

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STRAND HYBRIDIZATION
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Polymerase Extension
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DNA Ligase
?
?
?
?
?
?
?
?
?
?
Ligase Joins 5' phosphate to 3' hydroxyl
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Ligase Catenation
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More Complex Operations

• Amplification (PCR)
• Sequencing
• Restriction (R. Enzymes)
• Clonation (Plasmide Transinfection)

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PCR Polymerase Chain Reaction
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PCR with 3 sticky end
h(a)
b
h(b)
a
long
short
Linear
Exponential
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PCR Lemma
Given a pool P of type ??? and two primers ?,
? that hybridize with ? and ? respectively ( ?
? ? ).If the extensions e1 and e2 of the two
primers with the relative single strands overlap,
then an exponential amplification of ??? strands
happens which has the blunt form lte1 Z
exte2gtwhich appears within the first two steps.
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Operation
T of type L
T of type L
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MathematicallyTest Tube Operations

• Type (T) L
• means that
• Types of strands of T constitute the language L
• Given some test tubes as arguments with some
  types
• provide as results
• Test tubes with other types

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DNA Test Tube Machine

• Register Machines where
• - Registers are Test Tubes
• (multisets of strands instead of numbers)
• - DNA Test Tubes operations
• (instead of arithmetic operations)

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Adlemans Problem
Given a Graph (of seven nodes) Find (if there
are) The paths from two given nodes
(0,6) Passing once for every node (hamiltonian
paths)
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Adleman - Liptons Extract ModelIn Combinatorial
Problems

• The Generation of all possible solutions
• in linear time
• The Extraction of true solutions
• in linear time
• Extraction is performed in a number of sub-steps
  and each of them selects all the strands that
  include a sub-strand of a given type

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Adlemans Graph
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Adlemans Encoding
Ai Bi
Bj
Bj Ai
Node i ?i ?i
?i
?i
Arc ij mir(?i ?j)
?ic ?jc
i , j 1, , 7
?i ?i 10
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Adlemans Algorithm

• Generation of hamiltonian paths from v1 to v7
• Generate paths of G (hybridization/ligation)
• Perform PCR of primers ?0, mir(?6)
• Separate paths of length 140 (7 x 20)
• for J 1 to 7 do Select strands where ?j?j
  occurs
• output remaining strands

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MIX and Split Method

• Generation of space solutions of N variables
• Merge X1 and ?X1 in a tube T
• Split T into A and B
• For J 2 To N
• Extend strands of A with XJ
• Extend strands of B with ?XJ
• Merge A and B into T
• Split T into A and B
• Merge A and B

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Liptons Algorithm 3-Sat(N, M)

• Generate N-space solutions in T
• For J 1 To M
• T1 Extract T, L(1,J)
• T T - T1
• T2 ExtrtactT , L(2,J)
• T T - T2
• T3 ExtractT , L(3,J)
• T Merge(T1, T2)
• T Merge(T, T3)
• Detect T
• if T? ?, then take a clone and sequence it
  (Solution)
• else Unsolvable Problem

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DNA Extraction

• Strands of type ? are called ?-strands
• (or instances of ?)
• A ?-strand with ? including ? as substring is
  called a ?-superstrand (? is a ?-superstring)
• Problem
• Extract all the ?-superstrands of a pool P

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A Formulation of the DNA Extraction Problem
Given an input pool P of heterogeneous DNA
strands with the same length and with the same
prefix and suffix, and given a string ? Provide
an output pool P? such that all and only the
types of ?-superstrands of P are represented in
P? .
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In other words, extraction of ?-superstrands of P
means To provide a pool P? such that for any
two strings ? ? ??? ? P ltgt ??? ? P? i.e.
the strings represented in P? are all and only
the ?-superstrings belonging to P.
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Cross Pairing PCR

• Shortly
• XPCR

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XPCR provides an efficient method for affix
concatenation of double strands (Heads null
context splicing rule)
?
?
?
?
?
?
?
N.B. Genome Sequencing is related to Affix
Concatenation Closure
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?
?
?
?
h(?)
Melting Hybridization
?
Polymerase Extension
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?
?
?
?
h(?)
Melting Hybridization
?
Polymerase Extension
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?
?
?
?
?
h(?)
?
?
?
h(?)
h(?)
Exponential Amplification
Linear Amplification
Linear Amplification
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• 
  
  
  
  
  XPCR was tested in many different situations
  
• in pools generated by recombination of 22
  strands of lengths between 10 - 20

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RhoA XPCR
Lane 2 RhoA ???? of 582 bp Lane 3 ??? of 253
bp Lane 4 XPCR ????? of 582253 -229 606 bp ?
Starts at position -229 of RhoA
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XPCR DNA Extraction

• XPCR-Extract(P, ?)
• L length(P) , R1 ? , R2 ?
• For each n ? L do
• Q separate(P, n)
• P infix(Q, ?, ?)
• (P1, P2) split(P)
• P1 PCR(P1, ?, ?)
• For each m lt n do R1 R1 separate(P1, m)
• P2 PCR(P2, ?, mir(?))
• For each m lt n do R2 R2 separate(P2, m)
• Q mix(R1, R2)
• Q PCR(Q, ?, mir(?))
• Q separate(Q, n ? ?)
• Output Q

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Experimental Check

• Consider a pool P of ??-strands that are
• either ?-superstrands or ?-superstrands, and
• where all ?-superstrands are either
• ?1-superstrands, ?2-superstrands, or
• ?3-superstrands (? ? ?, ?1 ? ?2 ? ?3 15 bp).

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Experimental Check

• Our extraction is correct and complete in the
  sense that
• 1. XPCR-Extraction selected only
• ?-superstrands
• 2. XPCR-Extraction selected all kinds of
• ?-superstrands (?1, ?2 , ?3 - superstrands).

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Gamma Extraction
Lane 2 ?? strands of 120 bp (? 15 bp) Lane 3
?? of 45 bp Lane 4 XPCR ?? and ?? 150 bp Lane
5 PCR(?, ? a.s.) (? at -45) Lane 6 PCR(?, ?
a.s.) Lane 7 PCR(?1, ? a.s.) (?1 at -125) Lane
8 PCR(?2, ? a.s.) (?2 at -75)
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Applications

• XPCR in generation of space solutions
• XPCR in in vitro mutagenesis
• XPCR in gene extraction

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XPCR Mutagenesis
XPCR -Mutagenesis(P, ? , ? ) 1. let P
lt???gt 2. input Q lt?-20,-1 ? ?1, 20gt
3. (P1, P2) split(P) 4. P1 PCR(P1, 1,
20, mir(?-18,-1)) 5. P2 PCR(P2, 1, 20,
mir(?-20,-1)) 6. P1 separate(P1, ? ) 7. P2
separate(P2, ? ) 8. P1 mix(P1,Q) 9. P1
PCR(P1, ?1, 18, mir(?1, 20)) 10. P1
separate(P1, ? ? 20) 11. P mix(P1,
P2) 12. P PCR(P, ?1, 20,mir(?-20,-1)) 13.
P separate(P, ? ? ?) 14. output P
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XPCR Mutagenesis
Figure 10 Electrophoresis results Lane 1
molecular size marker ladder (100bp) Lane 2
amplification of strand ? (230bp) Lane 3
amplification of strand ? (229bp) Lane 4
amplification of strand ?-18, -1 ? ?1,20
(188bp) Lane 5 cross pairing amplification of ?
and ?-18, -1 ? ?1,20 (400bp) Lane 6 cross
pairing amplification of ? and ? ? ?1,20
(609bp) Lane 7 RhoAwt (582bp), lane 8 positive
control by PCR(? , ?-20, -1) (354 bp)
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Ongoing Research

• XPCR Clonation
• Dry DNA Computing