ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES

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1 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA Astrt. Givn st of flxil rnh juntion DNA moluls with stiky-ns (uiling loks), ll hr tils, w onsir th prolm of trmining th propr stoihiomtry suh tht ll stiky-ns oul n up onnt. In gnrl, th stoihiomtry is not uniform, n th gol is to trmin th propr proportion (sptrum) of h typ of moluls prsnt to llow for omplt ssmly. W prtition multists of tils, ll hr pots, into thr lsss: wkly stisfil, stisfil n strongly stisfil oring to possil omponnts tht ssml in omplt omplxs. This lssifition is hrtriz through th sptrum of th pot, whih n omput in PTIME using th stnr Guss-Jorn limintion mtho. 1. Introution Although nturlly ourring DNA molul hs oul hlix strutur, it n onfigur in othr mor omplx struturs,.g.: hirpin, rnh 3- n 4-rm juntion moluls, stik u, trunt othron, t. (s [4, 6, 15, 23, 27]). Ths nwly form moluls hv n propos for omputtionl purposs [8] s wll s sffols for othr struturs [4, 6, 23, 27]. A mol for DNA slf-ssmly introu y Winfr [25] using rigi squr tils hs n stui mor xtnsivly [1, 2, 13, 18, 19, 20]. In this ppr, w onsir nothr mol tht uss flxil tils (h til ompos of singl rnh juntion molul). This mol ws initilly propos in [7] n lort in [12, 9]. Flxil juntion moluls hv n us in xprimnts to otin rgulr grph struturs, suh s th u [4] n trunt othron [23, 27], non-rgulr grph struturs [8], n rntly for prforming omputtion [26]. Th mol using flxil juntion moluls is s on DNA rnh juntion moluls with flxil rms xtning to fr stiky-ns. In this mol, prolm is no in rnh juntion moluls (tils) n solution of th prolm is otin if n only if omplt omplx (omplx without stiky-ns) of pproprit siz n ssml. By imposing rstritions on th numr of typs of tils us in ssmly of omplxs, on n gt DNA omputility lsss tht orrspon to xtnt omplxity lsss. A polynomil rstrition prous prisly th NPTIME quris; no rstrition on th numr of til typs prous th lsss of ll omputl quris [9, 11]. In tst tu, vry molul pprs with high multipliity, on th orr of th Avogro numr, thrfor on n xpt to otin mny kins of omplx struturs, n som, ut not nssrily ll of thm my rprsnt th signt struturs. It n osrv xprimntlly tht portion of th DNA mtril in th pot ns up in inomplt omplxs. Also th pprn of topoisomrs hv n rport [17, 21]. Mthos from rnom grph thory wr us in [12] to invstigt th proility tht rtin siz omplx struturs n otin. In this ppr w onsir th qustion how to ru th th mount of th unsir mtril tht is otin t th n of n xprimnt. W propos mtho tht trmins th orrt stoihiomtry of th moluls in il onitions in orr to ru th inomplt omplxs prsnt t th n of n xprimnt. W show tht th right Dt: Mrh 6, 2009.

2 2 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA proportions of h molul n intifi suh tht t th n of th (thought) xprimnt 1 only fully ssml struturs r otin. A pot is fin to multist of finit numr of typs of tils. For vry pot w fin th sptrum of th pot to th st of vtors of right proportions for th moluls. Th sptrum of pot is onvx sust of Q m (Q is th st of rtionl numrs n m is th numr of istint tils); hn if it is not mpty, it is ithr singlton or infinit. With us of Guss- Jorn limintion lgorithm for mtris, w prov tht th sptrum of givn pot is PTIME omputl polytop. This pross lso intifis th tils tht r uslss, in th sns tht suh tils nnot ppr in omplt omplx. Th sription of th mol proviing th min finitions of omplxs n struturs tht r uilt up y juntion moluls is prsnt in Stion 2. In Stion 3, w lssify th pots in thr tgoris tht istinguish th pots pning on whthr vry juntion (til), or stoky-n n inlu into n ssmly of omplt omplx, or simply whthr th pot givs ris to omplt omplx. Th stoihiomtry is trmin with proportions in whih h typ of molul shoul ppr in mix. Th proportions for givn mix r onsir s oorint ntris of vtor. Th st of suh vtors fins sptrum of givn pot. Th lgri n gomtri proprtis of th sptrum r givn in Stion 4 whr w lso hrtriz th iffrnt lsss of pots oring to th proprtis of th sptrum. 2. Th Mol for Flxil-til Assmly Th min uiling loks for this mol r inspir y th rnh juntion DNA moluls. Ths r synthsiz strlik moluls [22] tht hv flxil rms with stiky-ns (s Figur 1 () to th lft). Eh rm hs two prts: oy n stiky-n xtning from th oy. Th oy prt is oul strn DNA molul, whil th stiky-n prt is singl strn DNA molul. Whn singl strn prts of two rms with omplmntry stiky-ns hyriiz, thy glu th moluls y th stiky-ns, forming mor omplx strutur. ) *t 1 ) * t1 *t 2 *t 2 Figur 1. Aov: Wtson-Crik oning of two DNA juntion moluls. Blow: Juntion grph tht rprsnts oning of th two DNA juntion moluls pit on th lft. For simpliity w ignor som of th thnilitis of th slf-ssml omplxs (.g. w mol squn of stiky-ns with symol) n rprsnt thm s ll grphs. For xmpl, Figur 1 () rprsnts thr- n four-rnh juntion molul nnl long omplmntry 1 Othr thn th ssumption of uniform mixing, th thrmoynmi proprtis of th moluls in th tst tu (pot) r not inlu in th sription of th ssmly pross. Also, rltivly uniform mlting tmprtur for th stiky-ns is ssum.

3 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 3 stiky-n in on of thir rms, whil Figur 1 () is grph rprsnttion of suh thr- n four-rnh juntion moluls for n ftr nnling. In Figur 1 (), th grphs on th lft rprsnt four-rnh juntion molul with four stiky-ns ll,,, n, n thr rnh juntion molul with stiky-ns â,, n. Th stiky-ns not â n rprsnt omplmntry squns. In th nnling pross, th omplmntry stiky-ns hyriiz, n th rsulting omplx is prout of gluing of ths two str-lik grphs (rprsnt in Figur 1, whr th grphs on th right hv only th stiky-ns, n ). Whn th juntion moluls r flxil, two suh moluls n hyriiz in iffrnt wys. Th flxiility is otin y ing ulg T s in th oy of th juntion n long th squns of th rms, lik in [8, 10] n [17]. With flxiility, w onsir tht ll hyriiztions of rms with omplmntry stiky-ns r possil. In our mol, tst tu with DNA rnh juntion moluls (tils) in it is ll pot. Using pproprit hmil protools, th omplmntry prts of th DNA moluls in th tst tu hyriiz n form mor omplx struturs. W wnt to prform stuy of th ssmly pross n on th possil outoms. For tht, forml mthmtil finition of th omponnts n of th pross is n. For th purpos of ompltion of th ppr w inlu th finitions n kgroun mtril s in [11] Prliminris n Nottion. Th numr of lmnts in finit st A is not y #A. Th miniml lmnt of th st of nturl numrs N is 0 n [n] = {1,2,...,n}. Lt H finit st. A ll multigrph G with lls in H is 4-tupl G = (V,E,η,λ) whr V is finit st of vrtis, E is finit st of gs, η : E P V is funtion from E to th st of two-lmnt susts of V, whih w not P V, n λ : E H is lling funtion. For n g E th two vrtis in th st η() is th st of npoints of th g. As G is multigrph, thr might svrl gs with th sm st of npoints. Not tht this finition os not inlu multigrphs with loops. Whn η is injtiv, G is simply grph. For vrtx v V w fin th nighorhoo of v s N v = { w w = v or E, η() = {v,w} }. W us th nottion: th sugrph of G gnrt y N v (vrtis in N v n ll gs with n points in N v ) is lso not y N v. Th gr of vrtx v V is g(v) = #{ v η() }. A sugrph G of G gnrt y th vrtx sust V is th sugrph tht onsists of vrtis V n ll gs tht hv npoints in V. A sugrph G is gnrt y n g sust E if onsists of ll npoints of th gs in E n gs E. Th lling of th gs is inhrit from G. Givn two ll multigrphs G 1 n G 2 with lls in H 1 n H 2 rsptivly, grph homomorphism from G 1 to G 2, not φ : G 1 G 2, is tripl of funtions φ = (φ v,φ,φ h ) suh tht φ v : V 1 V 2, φ : E 1 E 2 stisfis φ v (η 1 ()) = η 2 (φ ()) n φ h : H 1 H 2 stisfis φ h (λ 1 ()) = λ 2 (φ ()) for h E 1. As is stnr in grph thory, w writ φ for ny of φ v, φ, or φ h whn it is lr from th ontxt whih on of th thr is us. In this ppr, w rquir tht φ h th intity whn H 1 = H 2 = H. A grph homomorphism is onto if oth φ v n φ r onto. Th homomorphism is ll n isomorphism whn oth φ v n φ r ijtions in whih s w ll G 1 n G 2 isomorphi. For st T multist of lmnts of T is pir (T,f) whr f : T N. Th numr f(t) is th multipliity of t T. To s nottion, w writ T for th multist (T,f) tking th isjoint union of th opis of t in T, ut onsir it s st whn pproprit Mol Dsription. W strt with th finition of th stiky-ns. If two stiky-ns hv th sm squns of nulotis, w sy tht thy r of th sm stiky-n typ n w not y H th st of stiky n typs. Eh stiky-n in th tst tu is of rtin stiky n typ, n sin thr r only finitly mny stiky-n typs, H is finit. A stiky-n h is opy of th stiky-n typ h n hs th sm squn on nulotis s th stiky-n typ h. Th Wtson-Crik omplmntrity is mol y funtion θ : H H whih is n involution, i.., θ(θ(h)) = h for ll h H. W ll θ(h) H th omplmntry stiky-n typ to h suh tht

4 4 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA stiky-ns of typs h n θ(h) on. If θ(h) = h thn th stiky ns h n h n glu togthr. For h h H w ssum tht θ(h) h = θ(θ(h)). Thus H n prtition into two sts, H + n H, suh tht if h is n lmnt of H + thn θ(h) is n lmnt of H. W writ (H,θ) for th st of stiky-ns H for whih θ rprsnts th omplmntry funtion n ll this port oning systm or PBS. Furthr, w simplify th nottion y writing ĥ for θ(h) n w fix H. Dfinition 1. Lt (H,θ) port oning systm. A k-til t ovr (H,θ) is grph with lls in H fin y (V t,e t,η t,λ t ) with vrtis V t = { t } J t whr J t = {j 1,...,j k }, gs E t = { 1,..., k }, η t ( i ) = { t,j i } for i = 1,...,k, n lling funtion λ t : E t H stisfying: λ 1 t (h) implis λ 1 t (θ(h)) =. Th lls of lmnts of J t r ll stiky-ns for til t. A til typ of k-til t ovr n PBS (H,θ) is funtion typ(t) = t: H N fin y t(h) = #{ E t λ() = h }. A til is str-lik grph, th vrtx t of k-til t is ll th ntrl vrtx of t n its gr is k. All othr vrtis of t hv gr on n n g inint to on of thm is lso inint to th ntrl vrtx. A k-til n sn s physil rprsnttion of k-rm rnh juntion moluls. A til is k-til for som k. For h h H, ny til of typ t hs xtly t(h) stiky-ns of typ h. In gnrl, til typ n rgr s multist of stiky-n typs (lthough w usully hv t(h) 1 for ll h H, mking t kin to st). Osrv tht h H t(h) = g( t). Th finition of λ implis tht ithr t(h) > 0 or t(θ(h)) > 0 ut not oth. In othr wors, rms of th sm til nnot on. Evry two tils of th sm til typ r isomorphi. For xmpl, Figur 1 () shows xmpls of two tils with grs 4 n 3, rsptivly. Th ntrl vrtx is rprsnt with lk irl n th stiky-ns r init with iffrnt olors n lls. Ths til typs r s follows: t 1 () = 1,t 1 () = 1,t 1 () = 1,t 1 () = 1,t 2 (â) = 1,t 2 ( ) = 1,t 2 (ê) = 1. Dfinition 2. A pot typ ovr PBS (H,θ) is st P of til typs ovr (H,θ). A pot P is multist of tils with til typs in P. For xmpl th pot typ of Figur 1 is P = {t 1,t 2 }. W prsum tht for h pot P thr is n ritrrily lrg numr of tils of givn typ. Lt T finit multist of tils s isjoint union of tils with rtin til typs. Dnot T = { t t T } n with J T = t T J t. Furthrmor, E T = t T E t, n η T n λ T r xtnsions of η t n λ t to omin E T for h t T. Lt G T = ( T J T,E T,η T,λ T ) th isonnt grph whih is th isjoint union of th tils in T. Dfinition 3. Th finit multist of tils T mits grph G T = (T J,E,η,λ) if G T is onnt n thr is n onto grph homomorphism φ : G T G T suh tht φ T is ijtion onto T, φ JT is onto J, n for h j J, φ 1 (j) {j,j } stisfis: φ(j ) = φ(j ) implis λ t () = θ(λ t ( )) whr η t () = { t,j }, n η t ( ) = { t,j } Not tht ll vrtis in J hv grs ithr on or two, s thy r imgs of vrtis of gr on in G T n h n hv t most two primgs. If vrtx j J hs gr two, thn it is inint to two gs, on with ll h n th othr with ll θ(h). Lt J = J f J whr J f onsists of ll vrtis in J of gr on, whih w ll fr n J onsists of ll vrtis in J of gr two, whih w ll on. Informlly, J is th st of oning sits twn two omplmntry stiky-ns. Th stiky-ns of G T vill for furthr oning r lls of th st of fr vrtis J f. It follows tht th rstrition φ t is n isomorphism from til t in G T to th nighorhoo N t in G T [11].

5 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 5 Hn, h til is grph mitt y itslf. By onnting k-rm DNA rnh juntion moluls, two onnting rms r onsir s nw g twn th juntions. Similrly, th two-gr vrtis from J in G T r ignor n th two inint gs to h of thos vrtis n sustitut with singl g. This gnrts multigrphs whih w ll omplxs. Dfinition 4. Lt G T grph mitt y multist of tils T. Th omplx C T mitt y T gnrt y G T is multigrph otin from G T with lls in H y () rmovl of vrtis J, n () for h j J, rmovl of th pir 1, 2 with η( 1 ) = { t1,j}, η( 2 ) = { t2,j}, λ( 1 ) H + n ition of n g with η() = { t1, t2 } with ll λ() = λ( 1 ). Th g is si to otin y gluing th gs 1 n 2. Th vrtis J f r ll th stiky-ns of C T, n not y J(C T ). Th typ of th omplx C T is funtion typ(c T ) : H N suh tht typ(c T )(h) = #{ E j J f, η() = { t,j},λ() = h }. Th st of omplxs mitt y pot typ P is th st { C T T, t T,typ(t) P, n C T is mitt y T } Thrfor th st of vrtis in omplx C T is trmin with st T J whr T is th multist of vrtis with nighorhoos isomorphi to tils n th st J r th vrtis tht rprsnt th fr stiky-ns of C T, i.., J = J f. Agin, for h til t, J t = J f, so til is onsir s omplx mitt y itslf, n th typ of th til is th sm s th typ of th omplx. W sy tht C T is mitt y T if thr is grph G T mitt y T tht gnrts C T. Th typ of omplx C T inits th numr for h of th typs of its fr stiky-ns. W ssum tht th ssmly pross ours in n xtrmly ilut solution, so tht whn two omplxs mt, ll of thir omplmntry fr stiky ns join up. Thrfor, w n trt th omplx s if t no tim os it hv omplmntry fr stiky-ns. (This is whr th flxiility of th tils is so ritil.) Thus: Dfinition 5. A omplx C mitt y tils T is stl if, for h h H, ithr typ(c)(h) = 0 or typ(c)(θ(h)) = 0 or oth. Not: In this ppr, ll omplxs r stl unlss othrwis init. Unlik th s of til typs whn h til typ trmins uniqu (up to isomorphism) til, non-isomorphi omplxs n hv th sm typ. Consir th xmpl shown in Figur 2. Thr tils shown in () r (mximlly) onnt to prou two non-isomorphi omplxs of th sm typ mitt y th thr tils. Th stikyns r shown with iffrnt olors, n lls on th gs of th tils init th stiky-n typs. Th grph homomorphism from th thr tils to th two grphs mitt y th tils shown in () is suh tht j 1,j 2 j, k 1,k 2 k, p 1,p 2 p n n 2,n 3 n for th grph to th lft n j 1,j 2 j, k 1,k 2 k, p 2,p 3 p n n 2,n 3 n for th grph to th right. Th two orrsponing omplxs mitt y th tils r pit in (). Thy r otin from th grphs in () y rmovl of th two-gr vrtis j, k, p n n n y onnting th orrsponing two inint gs into singl g. Dfinition 6. A omplx is ll omplt if it hs no fr stiky-ns, i.., if for ll port oning typs h, typ(c)(h) = 0. Th st of ll omplt omplxs mitt y pot typ P is ll th output of th pot typ P n is not with O(P). Hn, omplx C = (T J,E,η,λ) is omplt iff J =, in whih s w writ C = (T,E,η,λ). Th output of th pot typ P is th st of omplxs tht hv rh its finl stg of th ssmly, in th sns tht thy nnot furthr hng thir strutur.

6 6 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA p p 2 1 k 2 m 2 k 1 Θ() Θ() Θ () j 3 p 3 * 1 j 2 Θ() Θ () n 2 *3 *2 j 1 n 3 *1 j k Θ() Θ () p Θ() Θ() m 2 j3 p n Θ() * 2 3 * () 3 () * 1 p 1 j k Θ() Θ() p Θ() Θ () Θ () * 2 m 2 n * 3 j 3 *1 m 2 j 3 Θ() * * 2 3 p 3 () * 1 p 1 j 3 m 2 * 3 Θ () * 2 Figur 2. () Thr tils with thir ntrl vrtx shown s lrgr lk irl, n th fr stiky-ns shmtilly prsnt with iffrnt olors n lls. () Two iffrnt grphs mitt y th thr tils in () otin y onnting th omplmntry ports. () Two (inomplt) omplxs orrsponing to th grph struturs in () mitt y tils in (). Thy r of iffrnt (non-isomorphi) unrlying grphil struturs ut of th sm omplx typ. 3. Pot Typ Clssifition In this stion w prtition th pots into four lsss: unstisfil, wkly stisfil, stisfil n strongly stisfil, oring to possil ssmlis of omplt omplxs n stuy thir proprtis. Dfinition 7. A omplx C = (T J,E,η,λ) is m in omplx C = (T J,E,η,λ ) if T T n thr is grph homomorphism φ from C to C suh tht φ T is th intity. In othr wors, if h is fr in C, thn φ(h) = h, whil if h is on to som h xtning from som til t, thn φ(h) = t. Nturlly, til t is m in omplx C = (T J,E,η,λ) if t T. Now w lssify th pot typs. Dfinition 8. A pot typ P is wkly stisfil if it mits omplt omplx, i., O(P). Othrwis it is unstisfil. A pot typ P is stisfil if, for h h H, thr is omplt omplx C O(P) with C = (T,E,η,λ) suh tht λ() = h for som E. A pot typ P is strongly stisfil if vry omplx tht is mitt y P n m into omplt omplx in O(P).

7 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 7 Th omputtionl output of pot typ P is fin y th st O(P), hn, ruing th numr of inomplt omplxs is of intrst. Thrfor, strong stisfiility is th notion of most immit intrst in our stuy. Dfinition 9. A omplx C = (T J,E,η,λ) is ll k-omplx if #T = k. Lmm 1. A pot typ P is strongly stisfil if n only if for vry til typ t P, thr xists omplt omplx with til t suh tht typ(t) = t. Proof. On implition of th lmm is trivil; if P is strongly stisfil, thn s vry til is omplx, it n m into omplt omplx. W prov th onvrs y inution on th numr of tils in th omplx to m. If omplx onsists of only on til, thn th omplx C to m is itslf th til t. Assum th sttmnt hols for omplxs with k tils; w show tht th lmm hols for ny (k + 1)-omplx. Lt C = (T J,E,η,λ) (k + 1)-omplx with T = {t 0,t 1,...,t k }. If C is omplt thn th lmm hols. Assum it is not, i.., for som h H, typ(c)(h) > 0. Consir th (possily non-stl) omplx C with tils T = {t 1,t 2,...,t k } otin from C y rmovl of til t 0 (Figur 3()). Not tht som of th fr stiky-ns in C might omplmntry to som of thos fr y thing t 0, n in this s C is not stl. Lt J = { thr is n g in C otin y gluing n for som J(t 0 )}. Ths r th stiky-ns fr y thmnt of til t 0. til t 0 Complx C z f () z Complx C f f g () f g * t 0 Complx C* z Complx C 1 f * 1 g Complx C 2 * 1 *2 *1 f g () *t 0 f z * 1 f g () g *2 *t 0 Figur 3. Constrution stps for ming omplx C in omplt omplx in strongly stisfil pot. () Th omplx C with stiky-ns. () Complx C otin from C y rmovl of til t 0 n th til t 0. () Eming of C in omplt omplx C 1 n ming th til t 0 in omplt omplx C 2. () Complt omplx C whih ontins th omplx C s n m omplx. Sin oth C n t 0 r omplxs with lss thn k + 1 tils, y th inutiv hypothsis, thr r omplt omplxs C 1, C 2 O(P) suh tht C n m in C 1 n t 0 n m in C 2 (Figur 3()). Bus oth C 1 n C 2 r omplt, thr r gs in oth of ths omplxs with lls intil to th lls of th gs in J. By Dfinition 4, vry g with ll h in C 1

8 8 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA (or C 2 ) is otin y intifying (gluing) two on-gr vrtis, h ing jnt to ntrl vrtx of til, ll s -vrtx. For n g f J, lt f 1 n f 2 gs in C 1 n C 2 rsptivly, suh tht f 1 is otin y gluing f 1 with som othr g whr f 1 n f 2 is otin y gluing two gs f 2 n f 2 f 2 J(t 0) with λ(f 2 ) = θ(λ(f 1 )). Thn λ(f 1) = λ(f 2 ). Furthr, for i = 1,2, st η(f i ) = { i,j i }, η(f i ) = { i,j i } suh tht λ(f i) = λ(f i ) = θ(λ(f i )). By onstrution, 2 is th ntrl vrtx of t 0, t0 (ntrl vrtis inint to on suh pir of gs r init in Figur 3()). W form nw omplt omplx C y rmoving pirs of gs f 1 in C 1 n f 2 in C 2 n ing gs f1 n f 2 with η(f 1 ) = { 1, t0 } n η(f2 ) = { 1, 2} with lls λ(f1 ) = λ(f 2 ) = λ(f 1) = λ(f 2 ) (s Figur 3()). By onstrution, th grph C is m in C sin ll ons twn t 0 n C hv n rstlish, n lso C is omplt, us thr r no fr stiky-ns from C 1 n C 2 tht hv not n on. Lmm 1 implis tht ll strongly stisfil pot typs r lso stisfil n tht ll stisfil pot typs r wkly stisfil. But th onvrs is not nssry tru. Figur 4() shows pot typ tht is stisfil, ut not strongly stisfil, sin tils of til typs t 3 n t 4 n nvr m into omplt omplx. Th pot typ of Figur 4() is n xmpl of pot typ tht is wkly stisfil, ut not stisfil sin th stiky-n typ n nvr prt of ny finit omplt omplx. () t 1 t t 3 t 4 2 () t 1 t 2 t 3 t 4 Figur 4. () Stisfil pot typ tht is not strongly stisfil (til typs t 3 n t 4 nnot prt of omplt omplx) () wkly stisfil pot typ tht is not stisfil (th stiky-n nnot prt of omplt omplx). Not tht th numr of stiky-n typs os not pn on th numr of til typs. Th pot typs in ll thr xmpls in Fig 5 r strongly stisfil; in th first xmpl th numr of til typs n stiky-n typs r qul; in th son on th numr of til typs r lss thn fr stiky-n typs; n in th thir xmpl th numr of til typs is grtr thn stiky-n typs. Nottion. For th rst of th ppr w rsrv m = #P, i.., m rprsnts th numr of til typs in pot typ P n P = {t 1,t 2...,t m }. Th numr of stiky-ns is 2n = #H with H = {h 1,...,h n,ĥ1,...,ĥn}. As w ssum tht h omplx is stl omplx, to h omplx typ C w ssoit vtor z C = (z 1,z 2,...,z n ) from Z n suh tht z i : H + Z, whr z i = typ(c)(h i ) typ(c)(ĥi).

9 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 9 () t 1 t 2 t 3 () t 1 t 2 () t 1 t 2 t 3 Figur 5. Strongly stisfil pot typs: () #P = #H + () #P < #H + () #P > #H +. Thrfor, ithr typ(c)(h i ) > 0 or typ(c)(ĥi) > 0, ut not oth. If typ(c)(h i ) > 0, thn z i > 0, n if typ(c)(ĥi) > 0, thn z i < 0. Thus in this sns z C givs informtion out th rmining fr stiky n typs on th omplx C. Sin til is omplx, for vry t P n vry h H, if t(h) > 0 thn t(ĥ) = 0. In this s, for vry til t, w ssoit vtor z t = (z t (h 1 ),z t (h 2 ),...,z t (h n )) from Z n suh tht z t (h i ) = t(h i ) if t(h i ) > 0 t(h i ) if t(ĥi) > 0 0 othrwis. 4. Sptrum of Pot 4.1. Dfinitions. Rports on DNA ssmlis (.g., [15, 21, 23, 27]) init tht whn on runs n xprimnt, th sign omplxs r only prt of th struturs tht show up in th pot; thr my lot of inomplt omplxs. Suh inomplt struturs my inrs th rror rt n th ost of th xprimnt. If th stoihiomtry in th tst tu is not wll lirt, i.., if th rtio of th moluls us in th mix os not orrspon to th potntil us in th finl ssmly, thn unr ny xprimntl onitions, inomplt omplxs shoul xpt. In this stion w propos mtho whih thortilly (ignoring ll ynmi onsirtions suh s thos in [14]) llows possiility tht only omplt omplxs r ssml (ssuming tht ssmly ours in il onitions in wll mix ilut pot). W ll th st of vtors of th rtios of th moluls llowing suh omplt ssmly th sptrum of th pot. Furthr, w giv n lgorithm for omputing th sptrum using stnr linr lgr mthos. W show tht th losur of sptrum of pot in Euliin sp is simplx, whos vrtis orrspon to onnt omplt omplxs. Hr, through invstigting susts of ffin sps, w otin hrtriztion of th sptrum of givn lss of pot typ. Dfinition 10. Givn pot typ P, th sptrum of P is th st S = S(P) of ll vtors r = (r t Q : t P) suh tht: (1) for h t, r t 0, n t P r t = 1, (2) for h h, t P r t t(h) = t P r t t(ĥ), i.., t P r t (t(h) t(ĥ)) = 0 In sptrum, for h h H thr r s mny stiky-ns of typ ĥ s thr r of typ h. Whn til t is onsir s omplx, w n us vtor z t (h) = t(h) t(ĥ), n thn th

10 10 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA son prt of th finition n rwrittn (2 ) r t z t (h) = 0. t P Thr r finit numr of tils in givn pot typ, so th proportion of h til is rtionl numr. For th prtil purposs w onsir S(P) Q m, Q ing th st of rtionl numrs. From th finition of th sptrum, it n osrv tht th sptrum n rprsnt m s n intrstion of th hyprpln H 1 = {x R m : x i = 1} with th krnl of th linr trnsformtion (2) in Dfinition 10. Thrfor th sptrum is sust of th unit u S(P) [0,1] m. Rmrk. If pot hs mixtur of tils whos proportions orrspon to vtor in th sptrum, thn th finition for sptrum sys tht, in il onitions, us thr is onto-on orrsponn twn stiky-ns n orrsponing omplmnts, ll stiky-ns oul on, n thrfor mixtur with only omplt omplxs oul otin. If th us proportion of th moluls is not in th sptrum, thr r no onitions unr whih t th n of th xprimnt only omplt omplxs will prsnt in th tst tu. Convrsly, givn omplt omplx C, thn thr is vtor in th sptrum whos ntris r prisly th rtios of th numr of tils of givn typ vs th totl numr of tils in C. If thr r til typs tht o not ppr in C thn th orrsponing ntris in th sptrum vtor r 0. Lt Ĉ st of omplt omplxs in O(P) with i numr ( of tils of ) typ t i (som i my 0), totl numr of tils us y omplxs in Ĉ n lt s = 1,..., m, w sy tht s orrspons to Ĉ. Lmm 2. For pot typ P, O(P) if n only if S(P). ( ) Proof. Lt C O(P) omplt omplx n s = 1,..., m vtor tht orrspons to {C}. Sin C is omplt omplx, for vry h H thr r s mny stiky-ns of typ h m mong th tils in C s of typ ĥ. Consquntly m i t i (h) = i t i (ĥ), or y iviing oth sis of th qulity y th totl numr of tils, it follows tht m i t i(h) = m i ti(ĥ). By th finition of th sptrum it follows tht th vtor s is n lmnt of th sptrum. Convrsly, if S(P), thr xists nonzro vtor r = (r i : t i P) of rtionl numrs in S(P). Eh oorint of r n writtn s r i = q i for q i 0 n i > 0 oth intgrs. Lt i = lm( 1, 2,... m ), so tht for h i, p i i = for n intgr p i. Thus r = (r 1,r 2,...,r m ) = (p 1 q 1,p 2 q 2,...,p m q m ). A st of stl omplxs tht ontins totl of p i q i tils of typ t i, i [m], is st of omplt omplxs. Lmm 2 in ft, sys tht pot typ is wkly stisfil if n only if its sptrum is non-mpty. Exmpl 1. Th sptrum of th pot typ givn in Figurs 4 (), () h ontining four til typs, is th solution of th following systms of qutions for r 1,r 2,r 3,r 4 0. () r 1 + r 2 + r 3 + r 4 = 1 () r 1 + r 2 + r 3 + r 4 = 1 r 1 r 2 + 2r 3 = 0 r 1 r 2 + r 3 + r 4 = 0 r 1 r 2 + 2r 4 = 0 r 1 r 2 = 0 r 1 r 2 = 0 r 3 r 4 = 0. Both systms hv th sm solution, i.., oth pot typs hv th sm sptrum S = { ( 1 2, 1 2,0,0) }, ut th first pot typ is stisfil, whil th son is only wkly stisfil. Ths two xmpls

11 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 11 show tht sptr lon nnot us to istinguish twn wkly stisfil pot typ n strongly stisfil pot typ. In th ov xmpl, th sol point of th sptrum mnts tht ln stoihiomtry is otin if on uss qul numr of moluls of th first two typs t 1,t 2, ut no moluls from th othr two typs. Points (vtors) in th sptrum of givn pot typ provi proportions for ln stoihiomtry, ut thy o not giv ny informtion out th typ of omplt omplxs t th n of th ssmly. W n only ssum tht in vry ilut solution, th smllst omplxs in O(P) r most likly to ppr Gomtri Rprsnttion of th Sptrum. In this stion w us onpts from Linr Progrmming n w rfr th rr to [5, 16] for goo introution to th sujt. It is known tht th onvx hull of st S is th smllst onvx st ontining S [3]. Also, vry polytop is th onvx hull of its xtrm points n th xtrm points r th vrtis of th polytop [3]. Turning to linr progrmming, givn systm of inqulitis i,1 x i,n x n i, i = 1,...,n (whih w n rvit s Ax ), si fsil solution is point suh tht A hols with mximl numr of qulitis in th systm. It is known tht vtor is n xtrm point of th onvx polyhron S = {r : Ar =, r 0} (A is n m mtrix, rnk(a) = n < m, R n, n r R m ) if n only if is si fsil solution to Ar [3, 16]. Dnot y H m th intrstion of Q m + = {(r 1,r 2,r 3,...,r m ) : r i Q, r i 0 for i [m]} with th hyprpln r 1 + r r m = 1. Dfinition 10 implis tht th sptrum of pot with m til typs is sust of th st H m n th hyprplns t Pr t z t (h) = 0, for h h. Th losur of th intrstion of ths hyprplns n R m + is polytop, i.., it is onvx hull of its vrtis. Thrfor th following proprty hols. Proposition 1. Th sptrum S(P) of pot typ P with P = m n orrsponing st of stiky-ns H with H + = n is n intrstion of n hyprplns n th st H m. Morovr th losur of sptrum in Euliin sp is onvx hull whos vrtis r rtionl points. Proposition 2. Th sptrum S(P) of givn pot P is ithr mpty, singlton or n infinit st. Proof. Th sptrum of n unstisfil pot is mpty (Lmm 2). Sin th sptrum is ns in onvx hull n inlus th vrtis of tht onvx hull, if it ontins two points thn it ontins t lst two vrtis, n hn vry rtionl point twn thos vrtis, so th sptrum is infinit. Proposition 3. Lt P = {t 1,t 2,...,t m } pot typ n S(P) its sptrum. For vry xtrm ( k1 point s =, k 2,..., k ) m of S(P), thr xists omplt omplx C = (T,E,η,λ) O(P) with 1 2 m k i i tils of typ t i, for i [m] whr = lm( 1, 2,..., m ) n g(k j, j ) = 1, for j [m]. Proof. First not tht, y th finition of sptrum n s S(P), w hv m k i i = 1. It follows tht th stiky-ns of k i i tils of typ t i n ll on suh tht th rsult is st of omplt

12 12 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA omplxs. Dnot this st Ĉ. Th numr of tils us y omplxs in st Ĉ is m #T = k m i k i = =. i i W n to show tht Ĉ onsists of singl omplt omplx. Assum tht Ĉ n prtition to two non-mpty susts Ĉ1 n Ĉ2 with p i n q i th numrs of tils of typ t i us y omplxs in Ĉ1 n Ĉ2, rsptivly. Lt u n v th totl numr of tils ppring in( omplxs in Ĉ1 n Ĉ2, rsptivly. Th sptrum points orrsponing to p1 Ĉ 1 n Ĉ2 r s 1 = u, p 2 u,..., p ) ( m q1 n s 2 = u v, q 2 v,..., q ) m. Sin Ĉ is isjoint union of Ĉ1 v n Ĉ2, thn #T = u + v, n sin oth Ĉ1 n Ĉ2 r nonmpty, u < n v <. First, w osrv tht s 1 s 2. Othrwis, suppos tht s 1 = s 2, so tht p i u = q i for h i, w v hv k i = p i + q i i u + v = p i + v u p i = p i u + v u, so k i = p i i u = q ( i v, n s = p1 u,..., p ) m = s 1 = s 2. For h i, u lt r i = g(p i,u), n so p i = r i k i, n u = r i i. Thn i u for vry i, so lm( 1, 2,..., m ) u, i.., u, whih ontrits u <. Hn, s 1 s 2. Nxt w osrv tht s n writtn s onvx omintion of s 1 n s 2 with u u + v s 1 + v ( u + v s p1 + q 1 2 = u + v,..., p m + q ) ( m k1 =,..., k ) m = s. u + v 1 m This ontrits th ft tht s is n xtrm point. So, Ĉ nnot prtition into two non-mpty susts, thrfor Ĉ must hv singl omplx. ( k1 Rmrk 1. To vry xtrml point s =, k 2,..., k ) m, thr might mor thn on omplx 1 2 m with k i i tils of typ t i, pning on th numr of possil onntions twn th tils. Similrly, thr might mor thn on omplt omplx ssoit with givn point from th sptrum of givn pot typ. Dfinition 11. Lt P pot typ with its sptrum S = S(P). Th omplt omplxs orsponing to xtrm points in S(P) r ll xtrml omplxs for P. Dfinition 12. A omplx C with st of tils T(C) is ll miniml omplt omplx if thr is no omplt omplx C with st of tils T(C ) stisfying T(C ) T(C). From Proposotion 3, it follows tht vry onnt xtrm omplx is lso miniml omplt omplx. But th onvrs is not nssry tru. Th following xmpl shows tht not ll miniml omplxs r xtrm omplxs s sn in Figur 6. Exmpl 2. Consir th pot typ in Figur 5 (). Its sptrum is th st of solutions to th following systm of qutions. r 1 + r 2 + r 3 = 1 3r 1 2r 2 r 3 = 0. Hn, th sptrum is S(P) = {(u,4u 1,2 5u) : 1 4 u 2 }. Th xtrm points r ( 5 1 s 1 = 4 4),0, 3 ( 2 n s 2 = 5, 3 ) 5,0, whih r solutions to th following systms. [ ] [ ] [ r1 1 = r 2 0 for r 1 0, r 2 0,r 3 0. ], [ ][ ] [ r1 1 = r 3 0 ], [ ][ ] [ r2 1 = r 3 0 ]

13 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES 13 Thr miniml omplt omplxs for this pot typ r shown in Figur 6. t 2 () t 1 t 3 t 3 t 3 t t 1 1 t 2 t 2 () t 1 t 2 t 3 Figur 6. ) Th xtrm omplt omplxs for th pot typ givn in Figur 5 () ) A omplt omplx tht is miniml, ut not xtrm omplt omplx. ( 1 Complxs C 1 n C 2 r xtrml orrsponing to xtrml points 4,0, 3 ( 2 n 4) 5, 3 ) 5,0, ( 1 rsptivly. Th sptrum point orrsponing to C 3 is 3, 1 3, 1, n C 3 is not xtrml, ut 3) miniml. Dfinition 13. Lt C omplt omplx with tils T n lt s = (r 1,...,r m ) sptrum point orrsponing to C. A til-numr vtor for C is th vtor = ( 1,..., m ) whr i = r i #T. Proposition 4. Lt P = {t 1,t 2,...,t m } pot typ. Th til-numr vtor of vry omplt omplx in O(P) is linr omintion of th til-numr vtors of th xtrml omplt omplxs. Proof. Lt S(P) th sptrum of P with l xtrm points s i, i [l], n C 1, C 2,..., C l xtrm omplt omplxs orrsponing to th xtrm points. Lt C with st of tils T omplt omplx uilt from tils typs in P, lt s = (r 1,r 2,...,r m ) th vtor in S(P) orrsponing to C n lt = ( 1, 2,..., m ) = #T s th til-numr vtor for C. For i [l], lt s i = (r1 i,ri 2,...,ri m ), whr ri j 0 for h j, xtrm points for S(P). Lt m i (i [l]) th numr of tils in C i. Thn for h i = i j whr (i 1,...,i m ) = i is th til-numr vtor for C i. Thus, s i = 1 i i. By Proposition 1, s n writtn s onvx omintion of th xtrm points in S(P), sy s = = #T s = l l µ i s i n #T µi i i. j=1 l µ i = 1. Thn Thrfor th til-numr vtor of C n writtn s linr omintion of th til-numr vtors of th xtrml omplt omplxs. Th prvious proposition stts tht th til-numr vtor of vry omplt omplx is linr omintion of th til-numr vtors of th xtrm omplxs. Although til-numr vtors hv intgr ntris y finition, this linr omintion is not nssrily n intgr omintion. For givn pot typ P, th st of omplt omplxs, O(P) n hrtriz through th xtrm omplt omplxs. Corollry 1. If th sptrum onsists of only on point, S(P) = {(r t1,r t2,...,r tm )}, with r ti > 0 for ll i [m], thn P is strongly stisfil.

14 14 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA Proof. If r ti > 0 for ll i [m], thn th omplt omplx orrsponing to th xtrm point from th sptrum ontins tils of h typ. If th sptrum onsists of only on point, thn tht point must n xtrm point. By Proposition 3 thr is miniml omplt omplx orrsponing to th xtrm point ontins tils of typs t i for whih r ti > 0. So, y Lmm 1, P is strongly stisfil. Corollry 1 sys tht if th sptrum S(P), of givn pot typ P, onsists of only on point with positiv oorints, thn vry omplt omplx in O(P) ontins tils of h typ, n hn P is strongly stisfil. Lt onsir othr xmpls for th sptr of strongly stisfil pot typs. In two-imnsionl sp (orrsponing to pot typ with xtly two til typs) th sptrum is prt of th lin sgmnt (r 1 + r 2 = 1, 0 r 1 1, 0 r 2 1) onnting th points (0,1) n (1,0). So non-mpty sptrum is ithr point of tht lin sgmnt, or it is th ntir lin sgmnt. Th sptrum is th ntir lin sgmnt if n only if omplmntry stiky-ns r llow on th sm til. Bus (1,0) n (0,1) r in th sptrum whn til of th first typ forms omplt omplx, n til of th son typ forms omplt omplx. Howvr, th finition for tils ssurs tht tils hv no omplmntry stiky-ns, thrfor th sptrum in two-imnsionl s lwys onsists of only on point. In thr-imnsionl sp, (orrsponing to pot typ with xtly thr til typs) th 3 sptrum is within th intrstion of th plns r i z ti (h) = 0, th pln r 1 +r 2 +r 3 = 1 n th susp {(r 1,r 2,r 3 ) r i 0 for i {1,2,3}}. Thrfor, th sptrum of pot with thr tils is sust of th tringl {(r 1,r 2,r 3 ) r 1 + r 2 + r 3 = 1} {(r 1,r 2,r 3 ) r i 0 for i {1,2,3}} (whih is th tringl tht onnts points (1, 0, 0), (0, 1, 0) n (0, 0, 1)). Similrly s in two imnsions th sptrum is th ntir st if n only if omplmntry stiky-ns on singl til r llow, i.., whn til of th first typ forms omplt omplx, til of th son typ forms omplt omplx, n til of th thir typ forms omplt omplx. As this sitution is not llow so, th sptrum in thr imnsionl s onsists of only on point or lin sgmnt. W gnrliz ths osrvtions. Proposition 5. Th sptrum, S(P), of th pot typ P = {t 1,t 2,...,t m } is onvx hull with vrtis {(1,0,0,...,0), (0,1,0,...,0),...,(0,0,...,0,1)} if n only if h til with typ in P is omplt omplx. Proof. Assum th sptrum S(P), of th pot typ P = {t 1,t 2,...,t m } is onvx hull with vrtis {(1,0,0,...,0),(0,1,...,0),...,(0,0,...,1)}. Proposition 4 shows tht for h i [m] th omplx C i orrsponing to th sptrum point of ll 0s with on 1 in th ith position onsists of tils of typ t i n is onnt omplt omplx. Hn, for ths sptrum points, for h m h, z ti (h) = z tj (h) = 0. Thus, h til forms omplx in O(P). j=1 Convrsly, ssum tht h til from typ in P is omplt omplx. Thn, for h h H z ti (h) = 0, n thrfor th son onition in th finition of th sptrum is trivilly stisfi. So, th sptrum of P is: S(P) = {(r 1,r 2,...,r m ) : r i 0, for i [m] n m r i = 1}, whih is polytop with vrtis {(1,0,0,...,0), (0,1,...,0),...,(0,0,...,1)}. As stoihiomtry is our intrst, th points in th sptrum r vtors with rtionl ntris, ut w usully pit its losur in Euliin sp whih is ompt sust of R m.

15 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES Figur 7. Th losur of th sptrum of th pot typ givn in Figur 5 () is lin sgmnt; th losur of th sptrum of th pot typ givn in Figur 5 () is th tringl oun y th ott lins long with its intrior Algri Rprsnttion of th Sptrum. Th sptrum is th intrstion of n hyprplns (for h h, t Pr t z t (h) = 0) n H m. Hn it is th solution of n homognous n on non-homognous qutions with m vrils ovr Q +. (3) r 1 + r r m = 1 z t1 (h 1 )r 1 + z t2 (h 1 )r z tm (h 1 )r m = 0 z t1 (h 2 )r 1 + z t2 (h 2 )r z tm (h 2 )r m = z t1 (h n )r 1 + z t2 (h n )r z tm (h n )r m = 0. A stnr n n ffiint wy to solv this systm is y th Guss-Jorn limintion pross, whih trnsforms th ugmnt mtrix of systm (3) into th row-hlon form. Th omputtionl omplxity of solving systm (3) with Guss-Jorn limintion is O(m 2 n) [5]. From Exmpl 1 it n sn tht points (vtors) of th sptrum for stisfil n wkly stisfil pots (ut not nssrily strongly stisfil pots) my hv zro oorints. If th sptrum of strongly stisfil pot typ is singlton, thn ll of its oorints r positiv (Proposition 6). Dfinition 14. Lt A st of m imnsionl vtors. Th support of A is th st supp(a) = {i [m] : thr xists vtor u = (u 1,u 2,...,u m ) A suh tht u i 0}. In othr wors, if i / supp(a), thn th i th oorint of vry point in A is 0. Proposition 6. Suppos S(P) is th sptrum of givn pot typ P with P = m. Thn: (): supp(s(p)) = [m] if n only if P is strongly stisfil. (): supp(s(p)) [m] if n only if P is wkly stisfil ut not strongly stisfil. Proof. (): supp(s(p)) = [m] if n only if vry til n prt of omplt omplx, whih y Lmm 1 mns tht th pot typ is strongly stisfil. (): supp(s(p)) [m] if n only if thr is oorint tht is zro in vry vtor of th sptrum, i.., t lst on til typ nnot m into omplt omplx, so th pot is not strongly stisfil, ut it is wkly stisfil sin it hs nonmpty sptrum.

16 16 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA Dfinition 15. Lt P(H,θ) = {t 1,...,t m } pot typ. Th m imnsionl vtors l h = (l 1 (h),l 2 (h),...,l m (h)) suh tht { 1 if ti (h) 1 or t l i (h) = i (ĥ) 1 0 othrwis r ll stiky-n vtors. Proposition 7. Clssifition of pot typs into wk stisfiility, stisfiility n strong stisfiility is in PTIME. Proof. Lt P = {t 1,t 2,...,t m } pot typ with m til typs n n stiky-n typs. Th sptrum S(P) is otin y solving systm (3) of n + 1 qutions with m vrils. If thr is solution with ll positiv oorints, y Proposition 6, th pot is strongly stisfil. If thr is solution to th systm (3) n supp(s(p)) [m], thn th sptrum is nonmpty n y Lmm 2 n Proposition 6, th pot is wkly stisfil ut not strongly stisfil. Thrfor wk stisfiility n strong stisfiility r in PTIME. Now suppos tht th pot is wkly stisfil ut not strongly stisfil, i.., supp(s(p)) [m]. Consir th stiky-n vtors l h = (l i (h) : t i P) for P. If thr xists stiky-n h H suh tht l i (h) = 0 for ll i supp(s(p)), thn h oul not m into ny omplt omplx, hn th pot is wkly stisfil, ut not stisfil (othrwis th pot is stisfil). To hk whthr th pot is stisfil, on ns to hk whthr th stiky-n vtors r orthogonl to th sptrum. This is otin through th ot prout twn th stiky-n vtors n gnri vtor v of th sptrum, i.., v is in th intrior of th sptrum (trt s mnifol). Th pot is stisfil if n only if l h v > 0 for h h H. Hn, th omputtionl omplxity for lssifying pot typs oring to thir typ of stisfiility is O(m 2 n) + O(mn) = O(m 2 n), i.., it is in PTIME. Corollry 2. A pot typ P(H,θ) is stisfil if n only if for vry h H, l i (h) = 1 for som i supp(s(p)). Exmpl 3. Th pot typs givn in Figur 4 hv th sm sptrum onsisting of th singl vtor s = ( 1 2, 1 2,0,0) Thrfor supp(p) = {1,2}, lthough on of thm (Figur 4()) is stisfil, whil th othr (Figur 4() ) is only wkly stisfil. Proposition 7 hlps to lssify thm. For th pot typ givn in Figur 4 ) th stiky-n vtors r: l = (1,1,1,0), l = (1,1,0,1). Th ot prouts twn th sptrum n th stiky-n vtors r: s l = 1, n s l = 1. Thrfor, this pot typ is stisfil. For th pot typ givn in Figur 4 ) th stiky-n vtors r: l = (1,1,1,1), l = (1,1,0,0), n l = (0,0,1,1). Th ot prouts twn th sptrum n ths vtors r: s l = 1, s l = 1, ut s l = 0. Thrfor, this pot typ is not stisfil. Proposition 7 provis stright-forwr wy to omput th stoihiomtry vtors for givn pot typ n to lssify th pot typ into on of th lsss: non-stisfil, wkly stisfil, stisfil n strongly stisfil. A Mpl progrm tht omputs th sptrum of givn pot typ, th support, n lssifis th pot typ in on of th stisfiility lsss is givn in [24]. Th input of th progrm r th numr of til typs m, th numr of stiky-n typs n, sptrum mtrix = [ p,q ] of siz (n + 1) (m + 1), whr p,q = z tq (h p 1 ) for p = 2,...,n + 1, q [m], 1,i = 1 for i [m + 1], i,m+1 = 0 for i = 2,...,n + 1, n stiky-n vtor mtrix L = [l p,q ] of siz n m whr l p,q = 1 if t q (h p ) > 0 or t q (ĥp) > 0, othrwis l p,q = 0. Th Mpl progrm outputs th support of th pot, th sptrum of th pot n lssifis th pot typ.

17 ON STOICHIOMETRY FOR THE ASSEMBLY OF FLEXIBLE TILE DNA COMPLEXES Conluing Rmrks Dtrmining th stoihiomtry for omplx tst tus with lrg numr of iffrnt typs of moluls is iffiult. It pns on th mutully nnling proprtis of th moluls s wll s th thrmoynmil proprtis of th mix whih my iffr for iffrnt xprimntl onitions. Furthr, s lrgr omplxs form in th tst tu, thir thrmoynmil proprtis hng n furthr ssmly my pn of th ntropi onitions in th tst tu. In this ppr w fous on DNA moluls with flxil rnh juntions tht xtn with stiky-ns n th nnling rltions twn ths stiky-ns. Th onsir stoihiomtry is otin s st of vtors with ntris orrsponing to th rtios of th molulr prsn in th tst tu. Th finition of til os not llow omplmntry stiky-ns to prsnt on sm til. All rsults, howvr, hol rgrlss of this onstrint, n our hoi to inlu this proprty in th til finition ws mostly gui y th simpliity n th lgn of th rgumnts. Th sptrum strutur n th lssifition of th pot typs rmin unhng without this onstrint. Th rsults on stoihiomtry vtors, i.., th sptrum, o not onsir ny thrmoynmil proprtis n w ssum tht ssmly pross hppns in il onitions, mning, it is possil for ll vill stiky-ns to on with thir omplmnts. Any slf-ssmly in n tul xprimnt will hv to tk into ount th thrmoynmil proprtis whih shoul proly trmin mpirilly. W liv th finings in this ppr provi strting point for suh n xprimntl nlysis. Th mtho prsnt hr pns on th oning rltions of th stiky-ns n is gnrl nough suh tht this pproh, w liv, n sily ppli to othr oning rltions s wll. Aknowlgmnt. Authors thnk Stphn W. Sun n Dvi A. Rson for proviing vlul suggstions. Th work is support in prt y NSF grnts CCF # n CCF # Rfrns [1] L.M. Almn, Q. Chng, A. Gol, M-D. Hung, D. Kmp, P. Moisst Espns, P.W.K. Rothmun. Comintoril optimiztion prolms in slf-ssmly, STOC 02 Proings, Montrl Qu, Cn, [2] L.M. Almn, J. Kri, L. Kri, D. Rishus. On th iility of slf-ssmly of infinit rions, Proings of FOCS 2002, IEEE Symposium on Fountions of Computr Sin, Wshington (2002) [3] A. Bronst. An Introution to Convx Polytops, Springr-Vrlg, (1983). [4] J.H. Chn, N.C. Smn. Synthsis from DNA of molul with th onntivity of u, Ntur 350 (1991). [5] L. Coopr, D. Stinrg. Mthos n Applitions of Linr Progrmming, W. B. Sunrs Compny, (1974). [6] R. P. Goomn, I. A. T. Shp, C. F. Trin, C. M. Ern, R. M. Brry, C. F. Shmit, A. J. Turrfil. Rpi Chirl Assmly of Rigi DNA Builing Bloks for Molulr Nnofrition, Sin 310 (2005). [7] N. Jonosk, S. Krl, M. Sito. Thr imnsionl DNA struturs in omputing, BioSystms 52 (1999) [8] N. Jonosk, P. S-Aryn, N.C. Smn. Computtion y slf-ssmly of DNA grphs, Gnti Progrmming n Evolvl Mhins 4 (2003) [9] N. Jonosk, G.L. MColm. A omputtionl mol for slf-ssmling flxil titls, C. S. Clu t l. (s) Springr LNCS 3699 (2005). [10] N. Jonosk, G.L. MColm. Slf-ssmly y DNA Juntion Moluls: Th Thortil Mol, Fountions of Nnosin. J.Rif (i) (2004). [11] N. Jonosk, G.L. MColm, Complxity Clsss for Slf-Assmling Flxil Tils to ppr in Thortil Computr Sin, (2009)

18 18 N. JONOSKA, G. L. MCCOLM, AND A. STANINSKA [12] N. Jonosk, G. MColm, A. Stninsk. Expttion n vrin of slf-ssml grph struturs, A. Cron, N. Pir (s) Springr LNCS 3892 (2006) [13] M-Y. Ko, V. Rmhnrn. DNA slf-ssmly for onstruting 3D oxs. Algorithms n Computtions, ISAC 2001 Prings, Springr LNCS 2223 (2001) [14] S. A. Kurtz, S. R. Mhny, J. S. Royr, J. Simon. Ativ trnsport in iologil omputing. L. Lnwr n E. Bum (s) DIMACS Vol 44 (1997) [15] C. Mo, W. Sun, N.C. Smn. Dsign two-imnsionl DNA holliy juntion rrys visuliz y tomi for mirosopy, Journl of Amrin Chmil Soity 121(23) (1999) [16] M. J. Pnik. Linr Progrmming: Mthmtis, Thory n Algorithms, Kluwr Ami Pulishrs, (1996). [17] J. Qi, X. Li, X. Yng, N.C. Smn. Ligtion of tringls uilt from ulg 3-rm DNA rnh juntions, Journl of Amrin Chmil Soity 120 (1996) [18] J. H. Rif, S. Shu, P. Yin. A slf-ssmly mol of tim-pnnt glu strngth, A. Cron, N. Pir (s) Springr LNCS 3892 (2006). [19] P.W.K. Rothmun, P. Ppkis, E. Winfr. Algorithmi slf-ssmly of DNA Sirpinski tringls, PLoD Biology 2 (12) 424 (2004). [20] P.W.K. Rothmun, E. Winfr. Th progrm-siz omplxity of slf-ssml squrs, Proings of 33r ACM mting STOC 2001, Portln, Orgon, My (2001) [21] P. S-Aryn, N. Jonosk, N.C. Smn. Slf-ssmly of grphs rprsnt y DNA hlix xis topology, Journl of Amrin Chmil Soity 126(21) (2004) [22] N. C. Smn, J. H. Chn n N. R. Kllnh. Gl Eltrophorti Anlysis of DNA Brnh Juntions, Eltrophorsis 10 (1989) [23] W.M. Shih, J.D. Quisp, G.F. Joy., A 1.7-kilos singl strn DNA fols into nnosl othron, Ntur 427 (2004) [24] A. Stninsk, A Thortil Mol for Slf-ssmly of Flxil Tils Ph.D. Thsis, Univrsity of South Flori, Tmp, [25] E. Winfr, Algorithmi Slf-Assmly of DNA, PhD thsis, Clth, Jun [26] G. Wu, N. Jonosk, N.C. Smn Constrution of DNA nno-ojt irtly monstrts omputtion sumitt. [27] Y. Zhng, N.C. Smn. Th onstrution of DNA trunt othron, Journl of Amrin Chmil Soity 116(5) (1994) Dprtmnt of Mthmtis Univrsity of South Flori, Tmp, FL E-mil rss: {jonosk,molm}@mth.usf.u E-mil rss: n.stninsk@hlmholtz-munhn.