Graph Theory and DNA Nanostructures. Laura Beaudin, Jo Ellis-Monaghan*, Natasha Jonoska, David Miller, and Greta Pangborn

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1 Grph Theory nd DNA Nnostructures Lur Beudin, Jo Ellis-Monghn*, Ntsh Jonosk, Dvid Miller, nd Gret Pngborn

2 A grph is set of vertices (dots) with edges (lines) connecting them A grph F A B C 6 (cycle) D C E (tree) K 6 (complete) K 3,4 (biprtite)

3 Wht re self-ssembled DNA nnostructues? A self-ssembled DNA cube nd Octhedron 22 nnometers

4 The moleculr building blocks K-rmed brnched junction molecules D. Luo, The rod from biology to mterils, Mterils Tody, 6 (2003), 38-43

5 Why self-ssembling nnostructures? Biomoleculr computing (Hmilton Cycle/3-St) Nnoelectronics Fine screen filters (lttices) t the nno-size scle Biosensors nd drug delivery mechnisms

6 Biomoleculr computing L. M. Adlemn, Moleculr Computtion of Solutions to Combintoril Problems. Science, 266 (5187) Nov. 11 (1994) Encode question in biologicl structure 2. Apply biologicl process to the structure 3. Be ble to isolte solution to the question from the result of the pplied process

7 The ppliction theory cycle Problems motivted by pplictions in biology New mthemticl theory nd tools Existing mthemticl theory nd tools

8 Communiction is key 1. Explin the biologicl problem to the mthemticin (problem formultion). 2. Develop the necessry nd sufficient formlism to model the problem. 3. Apply/develop mthemticl theory nd tools. 4. Communicte the mthemtics to the biologist in wy tht ctully informs the problem.

9 The fundmentl questions Given trget grph, 1. wht is the minimum number of k-rmed brnched junction molecules tht must be designed to crete the grph? 2. Wht is the minimum number of bond types needed? 3. Wht is the combintoril structure of the molecules in miniml set?

10 Three different lbortory constrints 1. The incidentl construction of grph smller thn G is cceptble 2. The incidentl construction of grph smller thn G is not cceptble but grph with the sme size s G (sme number of edges nd vertices) is cceptble 3. Any grph incidntlly constructed must be lrger thn G. In ll cses, we ssume flexible rmed molecules (bstrct, not embedded, grphs).

11 Definitions ATTCG GGTAACATTCG TAAGCCCATTG TAAGC Sticky end types, b, c, ĉ,, etc. lbel unpired rms sticking off of molecules. Types nd, re complementry sticky ends. A bond-edge is n edge formed by joining two complementry sticky ends. A tile represents brnched junction molecule with specific set of sticky ends. A pot P is set of tiles such tht for ny sticky end type on tile in P, there is sticky end of type on some tile in P A complex is n rrngement of tiles from pot type P with s mny djoined complementry sticky ends s possible with the given tiles A complete complex is complex which hs no undjoined sticky ends

12 Exmple ATTCG GGTAACATTCG TAAGCCCATTG TAAGC Both complete complexes nd incomplete complexes cn be constructed from the this pot P with 4 tiles: P: s ŝ ŝ s t 1 t 2 t 3 t 4 c ĉ c c ĉ t 1 t 2 t 3 t 1 Complete complex ŝ t 2 Incomplete complex

13 Simple constrints 1. A grph G my be constructed s complete complex from pot P if nd only if the number of htted sticky ends of ech type used in the construction of G equls the number of unhtted sticky ends of the sme type tht pper in the construction. 2. The totl number of htted sticky end types must equl the totl number of unhtted sticky end types in complete complex. These constrints drive prity rguments.

14 You try one. How mny tiles to get it, but nothing smller? C 5 C 6 K 4

15 Some things to consider Since every edge in grph G represents the connection of two complementry sticky ends, complete complex will be required to construct G. Since tile cn not represent two vertices of different degree cn represent the sme tile type, t lest the number of different vertex degrees in G re needed. Under the restrictions of scenrio 3, no two djcent vertices cn represent the sme tile type becuse multi-edges nd loops could be formed by swpping sticky ends. or ĉ c ĉ c ĉ c

16 Scenrio 1 exmple The vertex sequence of grph G is the list of vertex degrees in G. For Eulerin grphs the minimum number of tile types is just the number of different digits tht pper in the vertex sequence. This cn be shown by lbeling sticky end types s we follow grphs Euler circuit (lbeling sticky end type for outgoing sticky ends nd for incoming sticky ends). Only 1 bond-edge type is required for Eulerin grphs, nd only s mny tile types s vlencies!

17 Scenrio 2 exmple The minimum number of tile types required to construct cycle such tht no smller grphs cn be constructed out of the tiles is the number of vertices in the cycle C n. n where n is n Even n Odd The bisecting line reflects identicl tile types n 2 The minimum number of bond-edge types in this cse is.

18 Scenrio 3 exmple Complete grphs K n cn only be constructed using n tile types nd n-1 bond-edge types. Since every vertex in complete grph is djcent to every other vertex, no two vertices cn represent the sme tile type under the constrints of scenrio 3. The imge below shows the result of two tiles ( nd b) of the sme type ppering in K n. c ĉ ĉ c b c ĉ ĉ c b or c ĉ c ĉ b A complex other thn K n is formed!

19 Proof techniques Get upper bounds by finding set of tiles tht suffice to build the grph. Lower bounds/unwnted grphs re hrd. A combintion of number theory nd liner lgebr, on equtions determined by equivlence of htted nd unhtted sticky ends of given type in complete complex. n n 1 n E.g. tiles t1 = { } t2 = ˆ, suffice for K n for n even, in Scenrio 2. To show tht no smller grph on m vertices results from x tiles of type 1 nd y tiles of type 2, we show this hs unique solution: x+ y = m n n x = m xn ( 1) + y 1 = y y = mn n n However, x nd y must be integers, so this is contrdiction. ( )

20 Tble A: Minimum Tile Types T1 ( G) Scenrio 1 Generl grph G Scenrio 2 K n,m with n m Scenrio 3 Trees K n,m K-regulr grphs C n T 1 (C n ) = 1. K n T 1 (K n ) = 1 if n is even, nd T 1 (K n ) = 2 if n is odd. Trees K n,n Trees T2 = minimum number of tile types required if complexes of smller size thn the trget grph re llowed. The number of different vertex degrees T 1 (G) the number of different even vertex degrees + 2*(the number of different odd vertex degrees). The number of different vertex degrees T 1 (G) the number of different vertex degrees + 1. T 1 (K n,m ) = 1 if n=m nd even, nd T 1 (K n,m ) = 2 otherwise. T 1 (G) = 1 if n is even, nd T 1 (G) = 2 if n is odd. ( ) G = minimum number of tile types required if complexes of the sme size s the trget grph, but not smller, re llowed. T 2 (T) = the number of different lesser size subtree sequences. C n T n (C n ) =. K n T 2 (K n ) = 2 if n is even, nd T 2 (K n ) = 3 if n is odd. T 2 (K ) = 2 if gcd(m,n)=1, nd T n,m 2 (K n,m ) = 3if gcd(m,n)>1. 2 T 2 (K n,n ) 3. ( ) T = minimum number of tile types required if complexes of the sme size s (or 3 G smller thn) the trget grph re not llowed. T 3 (T) = the number of induced subtree isomorphisms. C n T 3 (C n )= n K n T 3 (K n ) = n. K n,m T 3 (K n,m ) = min(n,m)+1.

21 Tble B: Minimum Bond-Edge Types Scenrio 1 Generl grph G Scenrio 2 Scenrio 3 Trees C n B n 2 2 (C n )=. K n B 2 (K n )= 1 if n is even, nd B 2 (K n )= 2 if n is odd. K n,m B1 ( G) = minimum number of bond-edge types required if complexes of smller size thn the trget grph re llowed. B1 ( G) ( ) B2 B3 = 1 for ll grphs. G = minimum number of bond edge types required if complexes of the sme size s the trget grph, but not smller, re llowed. B 2 (T) = the number of different sizes of lesser size subtrees. B 2 (K n,m )= 1 if gcd(m,n)=1, nd B 2 (K n,m )= 2 if gcd(m,n)>1. ( ) G = Minimum number of bond edge types required if complexes of the sme size s (or smller thn) the trget grph re not llowed. C n B n 2 3 (C n ) =. K n B 3 (K n ) = n 1. Thus fr, the sme pots hve chieved both minimum tile types nd minimum bond-edge types, but we don t know if this is lwys possible.

22 Pending Vrious lttices, both 2 nd 3 dimensionl (s incomplete complexes?) Tubes (C m x P n ) (ditto) C m x C n Vrious Pltonic nd Archimeden solids

23 And whole other kettle of fish Sme set up nd questions, but now ssume rigid rmed molecules i.e. fixed rottion (or loction) of the sticky end types bout tile vertex. Edge-length constrints becuse the helixes hve to twist, if we cll twist unit, ech edge is of integer length. Rigid edges.

24 A different ssembly method zipping together single strnds of DNA (not llowed) N. Jonosk, N. Sito, 02

25 A chrcteriztion A theorem of C. Thomssen specifies precisely when grph my be constructed from single strnd of DNA, nd theorems of Hongbing nd Zhu to chrcterize grphs tht require t lest m strnds of DNA in their construction. Theorem: A grph G my be constructed from single strnd of DNA if nd only if G is connected, hs no vertex of degree 1, nd hs spnning tree T such tht every connected component of G E(T) hs n even number of edges or vertex v with degree greter thn 3.

26 You never know. Oriented Wlk Double Covering nd Bidirectionl Double Trcing Fn Hongbing, Xuding Zhu, 1998 The uthors of this pper cme cross the problem of bidirectionl double trcing by considering the so clled grbge collecting problem, where grbge collecting truck needs to trverse ech side of every street exctly once, mking s few U-turns (retrctions) s possible.

27 Bibliogrphicl References F. Hongbing, X Zhu, Oriented Wlk Double Covering nd Bidirectionl Double Trcing, J. Grph Theory 29 (1998) N. Jonosk, G. L. McColm, A. Stninsk. The Spectrum of Pot with DNA molecules, University of South Florid Deprtment of Mthemtics. N. Jonosk, G. L. McColm, A. Stninsk. The Grph of Pot with DNA molecules, University of South Florid Deprtment of Mthemtics D. Luo, The rod from biology to mterils, Mterils Tody, 6 (2003), C. Thomssen, Bidirectionl Retrction-Free Double Trcings nd Upper Embeddbility of Grphs, J. Combin. Theory, Ser. B 50 (1990) Acknowledgement: The project described ws supported in by the Vermont Genetics Network through NIH Grnt Number 1 P20 RR16462 from the INBRE progrm of the Ntionl Center for Reserch Resources, nd by Ntionl Security Agency Stndrd Grnt.