Graph Polynomials motivated by Gene Assembly

Transcription

1 Colloquium USF Tampa Jan Graph Polynomials motivated by Gene Assembly Hendrik Jan Hoogeboom, Leiden NL with Robert Brijder, Hasselt B

2 transition polynomials assembly polynomial of G w for doc-word w S(G w )(p,t) = s p π(s) t c(s), follow / consistent π / inconsistent never p w = p t+p t+p +pt +p+t p t p t p t p t p t 0 p t 0 p t 0 p 0 t Burns, Dolzhenko, Jonoska, Muche, Saito: Four-regular graphs with rigid vertices associated to DNA recombination (0)

3 contents motivation: models for gene recombination ciliates: -regular graphs + Euler tours graph polynomials concept: interlacing pqpq four worlds, getting more abstract strings rewriting graphs combinatorics matrices linear algebra (set systems) symm diff XOR same operation (pivot), different tools

4 USF Colloquium I Ciliates

5 I ciliates cell structure:. macronucleous. micronucleous 8. cilium wikipedia Franciscosp Unlike most other eukaryotes, ciliates have two different sorts of nuclei: a small, diploid micronucleus (reproduction), and a large, polyploid macronucleus (general cell regulation). The latter is generated from the micronucleus by amplification of the genome and heavy editing.

6 I ciliates MDS IES Ciliates: two types of nucleus gene assembly: splicing and recombination MIC micronucleus MDS macronucleus destined M i IES internal eliminated M M M M M M 9 M M M 8 inverted MAC macronucleus M M M M M M M 7 M 8 M 9 pointers: small overlap

7 I recombine at pointers merge consecutive MDS s align pointers & swap I l I l I r Ir I l I l I r Ir

8 I proposed model no pointers # # # # # Ehrenfeucht, Harju, Petre, Prescott, Rozenberg: Computation in Living Cells Gene Assembly in Ciliates (00)

9 I string pointer model goal: sorting [deleting] pointers inverted... M 9 M M M split u ppu p u ppu invert u pu pu p u pū pu swap u pu qu pu qu p,q u pu qu pu qu in in in }{{} sp in sw result depends on operations?

10 I graph model David M. Prescott. Genome gymnastics: unique modes of dna evolution and processing in ciliates. Nature Reviews Genetics (December 000)

11 I 7 Actin I gene of Sterkiella nova M 9 I p 9 I 9 p 8 M 8 I p I 0 M p 7 I 8 M 7 p M I 7 p M I MIC p I p MAC M I M M I MIC I 0 M I M I M I M I M 7 I M 9 I M I 7 M I 8 M 8 I 9 MAC I 9 I I 8 I 7 M M M 8 M 9 }{{} I I 0, I and I I I -regular graph with Euler circuit

12 I 8 abstraction double occurrence string w defines -regular graph G w + Euler circuit C w or -in -out graph + directed circuit w =

13 I 9 reconnecting at vertex w = segment split 7 0 segment inverted 9 8 &

14 I 0 (a) follows C (b) orientation consistent (c) orientation inconsistent inverting and splitting p q (a) (b) (c) split invert p q interlaced... p... q }{{}... p... q }{{} p... q }{{}... p... q }{{}... segments are swapped Kotzig. Eulerian lines in finite -valent graphs (9)

15 I interlaced symbols rearrangements should not break genome...p...q }{{}...p...q }{{}... interlace graph I(C w ) w = circle graph (bar + loop for orientation)

16 I keeping track of interlaced pairs circular string interlace {} {,} doc string w {} w = w {,} invert swap

17 I keeping track of interlaced pairs -regular graph (circle graph) + Euler cycle interlace graph C I(C) invert local complement C v I(C) v... x...v...y...x... v...y... x...v... x...ȳ...v...y interleaved not interleaved v v y x y x

18 I graph operations G G u looped vertex u N u = N G (u)\{u} local complementation u u N u N u G G {u,v} unlooped edge {u, v} N u = N G (u)\n G (v) N v = N G (v)\n G (u) N b = N G (u) N G (v) edge complementation u v N b u v N b N u N v N u N v invert I(C u) = I(C) v swap I(C {u,v}) = I(C) {u,v} when defined

19 I questions how do these operations interact? dependent on (order) operations chosen? what are the intermediate products?

20 I move to adjacency Z -matrices {}?? PPT

21 I 7 principal pivot transform A = ( X V\X X P Q V\X R S ) A X = ( X V\X ) X P P Q V\X RP S RP Q partial inverse ( ) x A = y ( x y ) iff A X ( x y ) = ( x y ) PPT matches edge & local complement Geelen. A Generalization of Tutte s Characterization of Totally Unimodular Matrices. JCTB (997) P = ( p ) p, P = ( p q ) p 0 q 0

22 I 8 ( x (partial inverse) A y ) = ( x y ) iff A X ( x y ) = ( x y ) Thm. (A X) Y = A (X Y) symmetric difference when defined any sequence involving all pointers: A {p,p } {p n } = A V = A Cor. does not depend on order of operations (!)

23 I 9 four worlds doc strings circle graphs binary matrices set systems {,{},{},{,}, {,},{,,} } invert local compl ppt {} xor {} { {},,{,},{}, {,,},{,} } swap, edge compl {,} {,} { {,,},{,},{,}, {},{}, }

24 USF Colloquium II Graph Polynomials

25 II graph invariants colours 80 acyclic orientations

26 II graph invariants u v u v chromatic polynomial χ G (t) = t(t )(t ) (t 7 t +7t 0t +9t 8t +77t ) u v uv χ G (t) = χ G+uv (t)+χ G/uv (t) χ G (t) = χ G e (t) χ G/e (t) deletion & contraction # acyclic orientations 80 = ( ) V G χ G ( )

27 II graph polynomials definitions recursive / closed form combinatorial / algebraic evaluations combinatorial interpretation polynomials Tutte (one to rule them all...) Martin (on -in -out graphs) assembly (Ciliates) transition pol interlace (DNA reconstruction) and their relations!

28 II Tutte: deletion & contraction loops and parallel edges diamond graph: T D = x +x +xy+x+y +y x +x +x+y x (x+y) x +x+y x +xy x xy+y x+y y xy x y

29 II Tutte polynomial T G (x,y) = A E G = (V,E), G[A] = (V,A) k(a) connected components in G[A] (x ) k(a) k(e) (y ) k(a)+ A V rank nullity circuit rank deletion & contraction T G (x,y) = no edges xt G/e (x,y) bridge e yt G e (x,y) loop e T G e (x,y)+t G/e (x,y) other T G (x,y) = x i y j with i bridges and j loops extended to matroids

30 II Tutte everywhere recipe theorem: deletion-contraction implies Tutte evaluation (Tutte-Grothendieck invariant) χ G (t) = ( ) V c(g) t c(g) T G ( t,0) T D = x +x +xy+x+y +y χ D (t) = t(t )(t ) (check Wolfram alpha) evaluations: T G (,0) acyclic orientations = ( ) V χ G ( ) T G (,) forests T G (,) spanning forests T G (,) spanning subgraphs T G (0,) strongly connected orientations

31 II 7 Martin (-in -out) G counting components:, 7,,.

32 II 8 transition system T( G) (graph state) half-edge connections at vertices Martin polynomial Martin polynomial of -in -out digraph G m( G;y) = (y ) k(t) c( G) T T( G) G c( G) components k(t) circuits for transition system T k :, 7,,. (y ) 0 +7(y ) +(y ) +(y ) Thm. () recursive form () Tutte connection () evaluations Pierre Martin, Enumérations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck, PhD thesis, 977

33 II 9 Martin () recursive formulation v (y ) 0 +7(y ) + (y ) +(y ) graph reductions: glueing edges Def. m( G;y) = for n = 0 m( G;y) = ym( G ;y) cut vertex m(g;y) = m( G v;y)+m( G v; y) without loops y +y G y y y y y y y

34 II 0 Martin () Tutte connection plane graph G, with medial graph G m Thm. m( G m ;y) = T(G;y,y) proof: deletion-contraction glueing edges

35 II Martin () evaluations m( G;y) = T T( G) (y )k(t) c( G) G -in -out digraph and n = V( G) G a( G) anti circuits m( G;y) = y +y m( G; ) = Thm. m( G; ) = ( ) n ( ) a( G) m( G;0) = 0, when n > 0 m( G; ) number of Eulerian systems m( G;) = n m( G;) = k m( G; ) for odd k third connection

36 USF Colloquium III Rearrangement Polynomials

37 III rearrangement vs. Martin Assembly polynomial Interlace polynonial connected to Martin polynomial interlace graph local and edge complement

38 III transition polynomials assembly polynomial of G w for doc-word w S(G w )(p,t) = s p π(s) t c(s), follow / consistent π / inconsistent never p w = p t+p t+p +pt +p+t p t p t p t p t p t 0 p t 0 p t 0 p 0 t Burns, Dolzhenko, Jonoska, Muche, Saito: Four-regular graphs with rigid vertices associated to DNA recombination (0)

39 III weighted polynomials transition polynomials W = (a, b, c) transition T defines partition V,V,V eg wrt fixed cycle weight W(T) = a V b V c V M(G,W;y) = T T( G) polynomial a b c Martin 0 (-way) assembly 0 p Penrose 0 - W(T)y k(t) c( G) F. Jaeger: On transition polynomials of -regular graphs (990)

40 III de Bruijn graphs How to apply de Bruijn graphs to genome assembly, Compeau, Pevzner & Tesler

41 III interlace polynomial de Bruijn Graphs for DNA Sequencing originally recursive definition simple graph G (with loops) interlace polynomial (single-variable, vertex-nullity) q(g;y) = (y ) n(a(g)[x]) X V(G) as Tutte, but: vertices vs. edges, algebraic vs. combinatorial Arratia, Bollobás, Sorkin: The interlace polynomial: a new graph polynomial (000) Aigner, van der Holst: Interlace polynomials (00) Bouchet: TutteMartin polynomials and orienting vectors of isotropic systems (99)

42 III recursive formulation local & edge complement Def. q(g;y) = if n = 0 q(g;y) = yq(g\v;y) q(g;y) = q(g\v;y)+q((g v)\v;y) v isolated (unlooped) v looped q(g;y) = q(g\v;y)+q((g e)\v;y) e = {v,w} unlooped edge q = y +y+ \b y +y a d c \a y d c \c {c,d}\c d y a b d c b\b a d c y+ {a,c}\a \a a\a y d c d c y+ \c c\c d y d \d d\d

43 III 7 Cohn-Lempel-Traldi e e e e D D D -regular graph G with Eulerian system C k circuit partition of E(G), partition vertices: D follows C D orientation consistent D orientation inconsistent Thm. Then k c(g) = n((i(c)+d )\D )

44 III 8 for circle graphs q(i(c);y) = X V(G) (y ) n(a(i(c))[x]) k c(g) = n((i(c)+d )\D ) = m( G;y) = (y ) k(t) c( G) T T( G) Thm. m( G;y) = q(i(c);y) w =

45 III 9 basic properties Thm. q(g;y) = q(g v;y) q(g;y) = q(g e;y) q(g;y) = q(g X;y) v looped e unlooped edge G[X] nonsingular Thm. m( G; ) = ( ) n ( ) a( G) q(g; ) = ( ) n ( ) n(a(g)+i) m( G;0) = 0, when n > 0 q(g;0) = 0 if n > 0, no loops m( G; ) #Eulerian systems q(g; ) #induced subgraphs with odd number of perfect matchings m( G;) = n q(g;) = n m( G;) = k m( G; ) odd k q(g;) = k q(g; ) odd k

46 III 0 what did we do? defined and studied these polynomials for -matroids (some things become less complicated that way) nullity corresponds to minimal size two directions deletion and contraction need another minor for third direction thank you ciliates!

47 III conclusion when studying new polynomials look back at old ones connections to recursive formulations, and special evaluations THANKS