1 DNA Structure Notation Operations Vincenzo Manca Dipartimento di Informatica Universita’ di Verona
2 10 Years of Molecular Computing 1994 Adleman’s Experiment * 1995 Lipton’s Model * 1996 Int. Conf. on Math. Linguistics (Marcus) 1997 Mangalia (Paun, Head) 1998 MFCS Brno (Molecular Computing) 1999 (Paun’s 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
3 DNA Computing Motto Problem: Data and Requirements Algorithm: Solutions Encode data by DNA strands Encode algorithms by biotech procedures Decode final strands as solutions
4 A General schema of combinatorial problem A set of Requirements for “assignments”, that is, sequences 0/1 of some length n The Space of possible solutions has E(2,n) elements, but only some of them satisfy the requirements Encode assignments by DNA strands Encode requirements as biotech protocols that filter the strands encoding the true solutions
5 Space Generation In linear time Solution Extraction In linear time !!!
6 New Trends in DNAC o DNA Self Assembly (Seeman, Winfree, …) o DNA Automata (Shapiro) o DNA Algorithms ==> new biotech protocols
7 Biotech Protocols Algorithms DNA Computing Computing DNA A change of perspective
8 In the search for implementing algorithms on DNA, general algorithmic principles are discovered in fundamental biomolecular processes. In the search for implementing algorithms on DNA, general algorithmic principles are discovered in fundamental biomolecular processes.
9 1’ 2’ 3’ 4’ 5’ O P B CH 2 1’ 2’ 3’ 4’ 5’ O P CH 2 OH H 1’ 2’ 3’ 4’ 5’ O P B CH 2 OH -H - OH = - H 2 O B Nucleotides ~330 Dalton 1 Dalton = g. H = ’--- 1’ = ~ 1nm A few grams of DNA = the amount of all electronic information stored in all the world ----
10Strings 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
11 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
12 Complementation - c (involutive ) Reverse rev (involutive) Mirrormir ( involutive ) mir( )= rev( c ) Reverse and Complementation commute Hybridization|| ] [ ] [ Pairing
13 B = {A, T, C, G} BB* = strings over B : fraction notation Axiom : = rev( ) rev( ) ext Overlap -- x -- overlapping concatenation Z -> up up <- down -> ->/ = ->/
14 Bilinearity Complementarity Antiparallelism The marvelous form 5’ 3’
15 Hybridization : || mir( ) ] [ , mir( ) ] [ , mir( ) ] [ ] [ for some Pairing : ] [ ==> / rev( )
16 Notation / = = -> / mir( ) = / mir( ) = / = rev( ) = <- / = rev( ) = <- ===> = ===> = BB* is the set of DNA strings, BB* B*
17
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, ….} mult P ( 1) = n1, mult P ( 2) = n2 s P P
19 Types of DNA Pools are Languages of BB* Type(T) = { BB* | s : , s T }
20 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
21 S TRAND H YBRIDIZATION
22
23
24 Polymerase Extension
25 DNA Ligase ’’ ’’ ’’ ’’ Ligase Joins 5' phosphate to 3' hydroxyl ’’ ’’
26 Ligase Catenation
27
28 More Complex Operations Amplification (PCR) Sequencing Restriction (R. Enzymes) Clonation (Plasmide Transinfection)
29 PCR: Polymerase Chain Reaction
30 Exponential Linear h( h( longshort PCR with 3’ sticky end
31 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 : which appears within the first two steps.
32 T of type L Operation T’ of type L’
33 Mathematically Test Tube Operations Type (T) = L means that 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
34
35 DNA Test Tube Machine Register Machines where: - Registers are Test Tubes (multisets of strands instead of numbers) - DNA Test Tubes operations - DNA Test Tubes operations (instead of arithmetic operations)
36 Adleman’s 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)
37 Adleman - Lipton’s Extract Model In 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
38 Adleman’s Graph
39 i c j c Node i = i i Arc ij = mir( i j) Ai BiBj Bj ’ Ai ’ ii ii Adleman’s Encoding | i| = | i| = 10 i, j = 1, …, 7
40 Adleman’s Algorithm Generation of hamiltonian paths from v1 to v7 Generation of hamiltonian paths from v1 to v7 Generate paths of G (hybridization/ligation) Perform PCR of primers Perform PCR of primers 0, mir( 6) Separate paths of length 140 (7 x 20) for J := 1 to 7 do Select strands where occurs for J := 1 to 7 do Select strands where j j occurs output remaining strands
41 MIX and Split Method Generation of space solutions of N variables 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
42 Lipton’s Algorithm 3-Sat(N, M) o Generate N-space solutions in T o For J = 1 To M T1 := Extract [T, L(1,J)] T := T - T1 T2 := Extrtact[T, L(2,J)] T := T - T2 T3 := Extract[T, L(3,J)] T := Merge(T1, T2) T := Merge(T, T3) o Detect T oif T , then take a clone and sequence it (Solution) oelse “Unsolvable Problem”
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
A Formulation of the DNA Extraction Problem P Given an input pool P of heterogeneous DNA strands with the same length and with the same prefix and suffix, and given a string P [ ] P P [ ] Provide an output pool P [ ] such that all and only the types of -superstrands of P are represented in P [ ].
§ In other words, extraction of -superstrands of P means To provide a pool P [ ] such that for any two strings : P P [ ] i.e. the strings represented in P [ ] are all and only the -superstrings belonging to P.
46 Cross Pairing PCR ShortlyXPCR
47 XPCR provides an efficient method for affix concatenation of double strands (Head’s null context splicing rule) N.B. Genome Sequencing is related to Affix Concatenation Closure
Melting + Hybridization Polymerase Extension h( )
Melting + Hybridization Polymerase Extension h( )
50
Linear Amplification h( ) Linear Amplification Exponential Amplification h( )
52
53 XPCR was tested in many different situations XPCR was tested in many different situations in pools generated by recombination of 22 strands of lengths between in pools generated by recombination of 22 strands of lengths between
RhoA XPCR Lane 2: RhoA of 582 bp Lane 3: of 253 bp Lane 4: XPCR of = 606 bp Starts at position -229 of RhoA
55 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 < n do R1 := R1 + separate(P1, m) P2 := PCR(P2, , mir( )) For each m < n do R2 := R2 + separate(P2, m) Q := mix(R1, R2) Q := PCR(Q, , mir( )) Q := separate(Q, n +| | + | |) Output Q
56 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). Experimental Check
57 Experimental Check Our extraction is correct and complete in the sense that: Our extraction is correct and complete in the sense that: 1. XPCR-Extraction selected only -superstrands -superstrands 2. XPCR-Extraction selected all kinds of -superstrands ( 1, 2, 3 …- superstrands). -superstrands ( 1, 2, 3 …- superstrands).
58 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)
59 Applications o XPCR in generation of space solutions o XPCR in in vitro mutagenesis o XPCR in gene extraction
60
61 XPCR −Mutagenesis(P, , ) = 1. let P : { } 2. input Q : { } 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 XPCR Mutagenesis
62 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)
63 Ongoing Research XPCR Clonation Dry DNA Computing