DNA Watson-Crick complementarity in computer science

Transcription

1 DNA Watson-Crick computer science Shnosuke Seki (Supervisor) Dr. Lila Kari Department Computer Science, University Western Ontario Ph.D. thesis defense public lecture August 9, 2010

2 Outle 1 2 Pseudo-primitivity 3 4 An Extended Fe and Wilf s Theorem 5 Extended Lyndon-Schützenberger equation 6

3 Path to the understandg DNA processg 1953 Watson and Crick discovered the DNA double helix. discoveries fundamental processg mechanisms livg organisms like DNA replication Adleman s first experiment molecular computg (DNA computg) 2003 Completion Human Genome Project 8/9/2010 one small step to the understandg DNA encodg

4 Encodg to DNA In molecular computg, DNA strands are ten employed as media for processg, storage, and transmission. DNA strand design Good DNA strands for molecular computg

5 Encodg to DNA In molecular computg, DNA strands are ten employed as media for processg, storage, and transmission. DNA strand design Good DNA strands for molecular computg Knowledge computer science codg theory theory etc.

6 Encodg to DNA In molecular computg, DNA strands are ten employed as media for processg, storage, and transmission. DNA strand design Good DNA strands for molecular computg Knowledge computer science codg theory theory etc. + Particularities DNA 4-letters Watson-Crick

7 Criteria DNA strand design Criteria DNA strand design [Sager & Stefanovic 2006] 1 no strand forms any undesired tra-molecular structure 2 no strand hybridizes with a strand any undesirable manner 3 no strand hybridizes with the WK-complement a strand any undesirable manner

8 Criteria DNA strand design Criteria DNA strand design [Sager & Stefanovic 2006] 1 no strand forms any undesired tra-molecular structure 2 no strand hybridizes with a strand any undesirable manner 3 no strand hybridizes with the WK-complement a strand any undesirable manner

9 DNA double helix & Watson-Crick Watson strand 5 A C T G 3 3 T G A C 5 Crick strand Due to Watson-Crick (WK) : A T, C G, Anti-parallelism, Two WK-complementary DNA sgle strands with opposite orientation can form a DNA double helix.

10 DNA replication DNA replication works the followg way: Watson strand Crick strand

11 DNA replication DNA replication works the followg way: Watson strand Crick strand

12 DNA replication DNA replication works the followg way: Watson strand Crick strand Watson strand Crick strand

13 DNA replication DNA replication works the followg way: Watson strand Crick strand Watson strand Crick strand N.B., the figure above lacks various important features DNA replication such as replication fork, Okazaki fragment, leadg and laggg strands, and DNA ligase. Observation Two WK-complementary strands are equivalent w.r.t. the they encode.

14 A mathematical model the equivalence and its generalization I Defition A mappg θ : Σ Σ is called an antimorphic volution if 1 for any x, y Σ, θ(xy) = θ(y)θ(x) (antimorphism), 2 θ θ is the identity (volutive). Watson-Crick (WK-) volution The antimorphic volution τ defed as τ(a) = T, τ(t) = A, τ(c) = G, τ(g) = C models WK-, and hence, called WK-volution. For stance, τ(actg) = τ(g)τ(act) = = τ(g)τ(t)τ(c)τ(a) = CAGT.

15 A mathematical model the equivalence and its generalization II Formalization the equivalence For any word u, consider u and its WK-complement τ(u) equivalent. Idea for generalization How about regardg u and θ(u) equivalent for any word u and any antimorphic volution θ?

16 Combatorics with affity for DNA computg Ultimate Goal Establish a system with notions and theorems combatorics on which can handle a word and its complement a uniform manner. The ma contribution this thesis We will generalize the classical notions power, repetition, and even primitivity this thesis. Example ACTGCAGTCAGT can be considered repetitive because it can be written as ACTGτ(ACTG) 2.

17 Importance applications Recall the above-mentioned criteria for DNA strand design. Criteria DNA strand design [Sager & Stefanovic 2006] 1 no strand forms any undesired tra-molecular structure 2 no strand hybridizes with a strand any undesirable manner 3 no strand hybridizes with the WK-complement a strand any undesirable manner

18 Importance applications Recall the above-mentioned criteria for DNA strand design. Criteria DNA strand design [Sager & Stefanovic 2006] 1 no strand forms any undesired tra-molecular structure 2 no strand hybridizes with a strand any undesirable manner 3 no strand hybridizes with the WK-complement a strand any undesirable manner Our results this thesis aim at providg uniform treatment 2nd and 3rd criteria.

19 Primitivity A word w Σ + is called a power u if w u +. Defition A word w Σ + is primitive if w = u k implies k = 1. The (unique) primitive word u Σ + with w u + is called the primitive root w, denoted by ρ(w). Example ACGT is primitive, while ACGTACGT is not; ρ(acgtacgt) = ACGT.

20 θ-primitivity We call a word the set u{u, θ(u)} a θ-power u. Defition We say that a word w Σ + is θ-primitive if for any u Σ +, w {u, θ(u)} k implies k = 1. Example For the WK-volution τ, ACGT is not τ-primitive because GT = τ(ac), and hence, ACGT = ACτ(AC).

21 Defition θ-primitive root Problem Can we defe the θ-primitive root a word w as the unique θ-primitive word t such that w t{t, θ(t)}? The answer is YES. The existence such t is trivial. Its uniqueness will be a corollary extended Fe and Wilf s theorem.

22 A relationship between primitivity and θ-primitivity primitive θ-primitive Proposition For any antimorphic volution θ, a θ-primitive word is primitive. Example ACGT is primitive but not τ-primitive for the WK-volution τ.

23 Defect I A well-known example defect On (x, y)-coordate, the condition y = x decreases the degree freedom by 1 (x is a free variable, and y becomes a bound variable).

24 Defect II For given and a system equations on them, if the system forces some the volved to be powers same word, then the system is said to possess the defect. Example uv = vu implies ρ(u) = ρ(v) (defect on u, v). Example ([Chfrut & Karhumäki 1997]) If a word u{u, v} and a word v{u, v} share a prefix length u + v, then ρ(u) = ρ(v).

25 problem Does a given system word equations force two u, v volved to be {t, θ(t)} for some word t? Theorem ([Czeizler, Kari, & Seki 2010]) ρ(u) = ρ(v) implies that u, v t{t, θ(t)} for some word t, and hence, has a weak defect on u and v.

26 Ma contributions to weak defect We vestigate the weak defect problem contexts 1 (extended) Fe and Wilf s theorem; 2 (extended) Lyndon-Schützenberger equation.

27 Ma contributions to weak defect We vestigate the weak defect problem contexts 1 (extended) Fe and Wilf s theorem; 2 (extended) Lyndon-Schützenberger equation.

28 What is the Fe and Wilf s theorem? u + v gcd( u, v ) u u v v v v v Theorem ([Fe & Wilf 1965]) If a power u and a power v share a prefix length u + v gcd( u, v ), then ρ(u) = ρ(v).

29 An extended Fe and Wilf s theorem α u? u or θ(u) u or θ(u) β v v or θ(v) v or θ(v) v or θ(v) v or θ(v) Problem How long prefix do α u{u, θ(u)} and β v{v, θ(v)} have to share to imply that u, v t{t, θ(t)} for some word t?

30 lcm( u, v ) u u or θ(u) u or θ(u) v v or θ(v) 1 [Czeizler, Kari, & Seki 2010] v or θ(v) If a θ-power u and a θ-power v share a prefix length lcm( u, v ), then u, v t{t, θ(t)} for some word t. Defition θ-primitive root [Czeizler, Kari, & Seki 2010] We can defe the θ-primitive root u, denoted by ρ θ (u), as a unique θ-primitive word t such that u t{t, θ(t)}.

31 2 [Czeizler, Kari, & Seki 2010] Let u, v Σ + with u v. If a θ-power u and a θ-power v share a prefix length 2 u + v gcd( u, v ), then u, v t{t, θ(t)} for some word t.

32 For two positive tegers p, q N (p q), let { lcm(p, q) if q 2 gcd(p, q) b(p, q) = 2p + q gcd(p, q) if q 3 gcd(p, q). extended Fe and Wilf s theorem If a θ-power u and a θ-power v share a prefix length b( u, v ), then u, v t{t, θ(t)} for some word t.

33 Optimality bounds for The bound for FW theorem is strong optimal [Constantescu & Ilie 2005] a sense: for any positive tegers p, q, one can construct u, v such that 1 u = p, v = q, ρ(u) ρ(v), 2 a power u and a power v share a prefix length u + v gcd( u, v ) 1. Problem Is b(p, q) strongly optimal? A partial (trivial?) positive answer b(p, q) is optimal for any p, q with p q = gcd(p, q). Negative answer [Kari & Seki 2010] b(p, q) is not strongly optimal.

34 A new bound for For p, q with p > q 2 gcd(p, q), let gcd(p, q) b (p, q) = b(p, q). 2 an improved [Kari & Seki 2010] Let u, v with u > v 2 gcd( u, v ). If a θ-power u and a θ-power v share a prefix length b ( u, v ), then u, v t{t, θ(t)} for some word t.

35 Optimality the new bound A partial positive answer For any p, q with p > q = 2 gcd(p, q), b (p, q) is optimal. iff condition for b (p, q) to be optimal for (p, q) Let u, v with u > v 3 gcd( u, v ), ρ θ (u) ρ θ (v), and gcd( u, v ) = 1. A θ-power u and a θ-power v share a prefix length b ( u, v ) 1 iff either 1 u = (ab(ba) i b) m ab and v = ab(ba) i b, or 2 u = [a(ba) i (ab) i+1 a] m a(ba) i ab and v = a(ba) i (ab) i+1 a for some m 1, i 0, and a, b Σ s.t. a = θ(a) and b = θ(b). Consequently, b (p, q) is NOT strongly optimal (e.g., it is not optimal for (9, 5)).

36 Optimality the new bound A partial positive answer For any p, q with p > q = 2 gcd(p, q), b (p, q) is optimal. iff condition for b (p, q) to be optimal for (p, q) Let u, v with u > v 3 gcd( u, v ), ρ θ (u) ρ θ (v), and gcd( u, v ) = 1. A θ-power u and a θ-power v share a prefix length b ( u, v ) 1 iff either 1 u = (ab(ba) i b) m ab and v = ab(ba) i b, or 2 u = [a(ba) i (ab) i+1 a] m a(ba) i ab and v = a(ba) i (ab) i+1 a for some m 1, i 0, and a, b Σ s.t. a = θ(a) and b = θ(b). Consequently, b (p, q) is NOT strongly optimal (e.g., it is not optimal for (9, 5)).

37 Ma contributions to weak defect We vestigate the weak defect problem contexts 1 (extended) Fe and Wilf s theorem; 2 (extended) Lyndon-Schützenberger equation.

38 Lyndon-Schützenberger equation Lyndon-Schützenberger equation [Lyndon & Schützenberger 1962] For u, v, w Σ and non-negative tegers l, n, m 0, an equation the form u l = v n w m is called the Lyndon-Schützenberger equation. Solution [Lyndon & Schützenberger 1962, Lothaire 1983, Harju & Nowotka 2004] For l, n, m 2, the equation u l = v n w m implies u, v, w t + for some word t.

39 An extended Lyndon Schützenberger equation Let us extend the LS-equation as: u 1 u l = v 1 v n w 1 w m, where u 1,..., u l {u, θ(u)}, v 1,..., v n {v, θ(v)}, and w 1,..., w m {w, θ(w)}. Problem Fd conditions on l, n, m under which the exls equation (l, n, m) possesses the weak defect, i.e., implies u, v, w {t, θ(t)} + for some word t.

40 exls equations without the weak defect In [Czeizler, Czeizler, Kari, & Seki 2009], we provided examples to verify the followg. Proposition ([Czeizler, Czeizler, Kari, & Seki 2009]) If one l, n, m is 2, then exls equations (l, n, m) does NOT possess the weak defect. As a result, all l, n, m must be at least 3 for exls equations (l, n, m) to possess the weak defect.

41 exls equations with weak defect I Proposition ([Czeizler, Czeizler, Kari, & Seki 2009]) s ( 6, 3, 3) possess the weak defect. u 1 u 2 u 3 u i u l v 1 v 2 v 3 v n w 1 w m Symmetry enables us to assume that v 1 v n w 1 w m. Sce l 6, this means v 1 v n 1 2 u 1 u l u 1 u 2 u 3. Thus, u 1 u l and v 1 v n share a prefix length at least max(3 u, 3 v ) 2 max( u, v ) + m( u, v ).

42 exls equations with weak defect II Pro The common prefix between u 1 u l and v 1 v n is long enough to apply to obta u, v {t, θ(t)} for some θ-primitive word t. Hence, w 1 w m {t, θ(t)}. Usg the lcm-variant, we can conclude w {t, θ(t)}. The same pro technique works for exls equations (5, 5, 5) or (5, 4, 7). Proposition ([Czeizler, Czeizler, Kari, & Seki 2009]) s (5, 5, 5) and (5, 4, 7) possess the weak defect.

43 with l = 4, 5 The followg stronger statement requires results combatorics on. Theorem ([Czeizler, Czeizler, Kari, & Seki 2009]) s ( 5, 3, 3) possess the weak defect. More useful results combatorics on have been obtaed [Kari, Masson, & Seki 2009] to address some cases left open [Czeizler, Czeizler, Kari, & Seki 2009]. Theorem [Kari, Masson, & Seki 2009] s ( 4, 3, 3) possess the weak defect.

44 Summary on the exls equation l n m weak defect how to prove YES YES [Czeizler, Czeizler, Kari, & Seki YES combatorial arguments YES [Czeizler, Czeizler, Kari, & Seki YES combatorial arguments [Kari, Masson, & Seki 2009] OPEN? 2 NO examples 2 NO [Czeizler, Czeizler, Kari, & Seki 2 NO

45 Open problems I Open problems about the 1 Fd an optimal bound for (p, q) for which b (p, q) is not optimal. 2 Extend the problem settg further as: how long prefix do a θ-power u 1, that u 2,..., that u n have to share to imply that u 1, u 2,..., u n t{t, θ(t)} for some word t?

46 Open problems II Open problems about the exls equation 1 Solve exls equations u 1 u 2 u 3 = v 1 v n w 1 w m with n, m 3. 2 Fd a condition under which exls equations with l = 2 are solved positively. 3 Extend the exls equation further as u 1 u l = v 11 v 1n1 v 21 v 2n2 v k1 v knk, where k 2, l, n 1,..., n k 1, u 1,..., u l {u, θ(t)}, and v i1,..., v i {v i, θ(v i )}.

47 Open problems III Other open problems 1 ExFW theorem and positive answers to exls equations are part weak defect theorem. Investigate and establish the weak defect theorem.

48 Information my publication I The followg is a list my publications related to the topics troduced this talk. The list all my publications is available on my website: 1 E. Czeizler, E. Czeizler, L. Kari, and. An extension the Lyndon Schützenberger result to pseudoperiodic. In Proc. DLT 2009, LN 5583, Sprger (2009) E. Czeizler, L. Kari, and. On a special class primitive. Theoretical Computer Science 411(3) (2010)

49 Information my publication II 3 L. Kari, B. Masson, and. Properties pseudo-primitive and their applications. Submitted. 4 L. Kari and. An improved bound for an extension Fe and Wilf theorem. Fundamenta Informaticae 101(3) (2010)

50 I [Chfrut & Karhumäki 1997] C. Chfrut and J. Karhumäki. Combatorics. In G. Rozenberg and A. Salomaa (eds.), Handbook Formal Languages, vol.1, pp Sprger-Verlag, [Constantescu & Ilie 2005] S. Constantescu and L. Ilie. Generalized Fe and Wilf s theorem for arbitrary number periods. Theoretical Computer Science 339(1) (2005)

51 II [Czeizler, Czeizler, Kari, & Seki 2009] E. Czeizler, E. Czeizler, L. Kari, and. An extension the Lyndon Schützenberger result to pseudoperiodic. In Proc. DLT 2009, LN 5583, Sprger (2009) [Czeizler, Kari, & Seki 2010] E. Czeizler, L. Kari, and S. Seki. On a special class primitive. Theoretical Computer Science 411(3) (2010) [Fe & Wilf 1965] N. J. Fe and H. S. Wilf. Uniqueness theorem for periodic functions. Proc. American Mathematical Society 16 (1965)

52 III [Harju & Nowotka 2004] T. Harju and D. Nowotka. The equation x i = y j z k a free semigroup. Semigroup Forum 68 (2004) [Kari, Masson, & Seki 2009] L. Kari, B. Masson, and S. Seki. Properties pseudo-primitive and their applications. Submitted. [Kari & Seki 2010] L. Kari and. An improved bound for an extension Fe and Wilf theorem. Fundamenta Informaticae 101(3) (2010)

53 IV [Lothaire 1983] M. Lothaire. Combatorics on Words. Encyclopedia Mathematics and its Applications, 17, Addison-Wesley Publishg Co., [Lyndon & Schützenberger 1962] R. C. Lyndon and M. P. Schützenberger. The equation a M = b N c P a free group. Michigan Mathematical Journal, 9: , [Sager & Stefanovic 2006] J. Sager and D. Stefanovic. Designg nucleotide sequences for computation: A survey constrats. In Proc. DNA 11, LN 3892, Sprger, 2006.

54 Thank you for your kd attention.