Title: Randomized self-assembly Name: David Doty Affil./Add. Califonia Institute of Technology, Pasadena, CA, USA. Suoted by NSF gants CCF-974, CCF- 6589, and 7694. Keywods: andomized; tile comlexity; linea assembly SumOiWok: 6; Becke, Raaot, Rémila 8; Kao, Schwelle 9; Chandan, Goalkishnan, Reif ; Doty Randomized self-assembly David Doty Califonia Institute of Technology, Pasadena, CA, USA. Suoted by NSF gants CCF-974, CCF-6589, and 7694. Yeas aud Authos of Summaized Oiginal Wok 6; Becke, Raaot, Rémila 8; Kao, Schwelle 9; Chandan, Goalkishnan, Reif ; Doty Keywods andomized; tile comlexity; linea assembly Poblem Definition We use the abstact tile assembly model of Winfee [6], which models the aggegation of monomes called tiles that attach one at a time to a gowing stuctue, stating fom a single seed tile, in which bonds ( glues ) on the tile ae secific (glues only stick to glues of the same tye on othe tiles) and cooeative (so that multile weak glues ae necessay to attach a tile). The geneal idea of andomized self-assembly is to use the inheent andomness of self-assembly to hel the assembly ocess. If multile tyes of tiles ae able to bind to a single binding site, then we assume that thei elative concentations detemine the obability that each succeeds. With caeful design, we can use the same tile set to ceate diffeent stuctues, by changing the concentations to affect what is likely to assemble. Anothe use of andomness is in educing the numbe of diffeent tile tyes equied to assemble a shae. Definitions A shae is a finite, connected subset of Z. A tile tye is a unit squae with fou sides, each side consisting of a glue label (finite sting) and a nonnegative intege stength.
We assume a finite set T of tile tyes, but an infinite numbe of coies of each tile tye, each coy efeed to as a tile. An assembly is a ositioning of tiles on the intege lattice Z ; i.e., a atial function α : Z T. Wite α β to denote that α is a subassembly of β, which means that dom α dom β and α() = β() fo all oints dom α. In this case, say that β is a sueassembly of α. Two adjacent tiles in an assembly inteact if the glue labels on thei abutting sides ae equal and have ositive stength. Each assembly induces a binding gah, a gid gah whose vetices ae tiles, with an edge between two tiles if they inteact. The assembly is τ-stable if evey cut of its binding gah has stength at least τ, whee the weight of an edge is the stength of the glue it eesents (enegy τ is equied to seaate the assembly). The τ-fontie τ α Z \dom α of α (o fontie α when τ is clea fom context) is the set of emty locations adjacent to α at which a single tile could bind stably. A tile system is a tile T = (T, σ, τ), whee T is a finite set of tile tyes, σ : Z T is a seed assembly consisting of a single tile (i.e., dom σ = ), and τ N is the temeatue. An assembly α is oducible if eithe α = σ o if β is a oducible assembly and α can be obtained fom β by the stable binding of a single tile. In this case wite β α (α is oducible fom β by the attachment of one tile), and wite β α if β α (α is oducible fom β by the attachment of zeo o moe tiles). If α is oducible then thee is an assembly sequence α = (α i i k) such that α = σ, α k = α, and fo each i {,..., k }, α i α i+. An assembly is teminal if no tile can be τ-stably attached to it. Wite A[T ] to denote the set of all oducible assemblies of T, and wite A [T ] to denote the set of all oducible, teminal assemblies of T. We also seak of shaes assembled by tile assembly systems, by which we mean dom α if α A [T ], and we conside shaes to be equivalent u to tanslation. We now define the semantics of incooating andomization into self-assembly. Intuitively, thee ae two souces of nondeteminism in the model as defined: ) if α > then thee ae multile binding sites, one of which is nondeteministically selected as the next site to eceive a tile, and ) if multile tile tyes could bind to a single binding site, then one of them is nondeteministically selected. Both concets ae handled by assigning ositive eal-valued concentations to each tile tye; ef. [] gives a full definition that accounts fo both of these. Howeve, in the esults we discuss, only the latte souce of nondeteminism will actually affect the obabilities of vaious teminal assembly being oduced; the binding sites themselves can be icked in an abitay ode without affecting these obabilities. Thus we state hee a simle definition based on this assumtion. A tile concentation assignment on T is a function ρ : T [, ). If ρ(t) is not secified exlicitly fo some t T, then ρ(t) =. If α is a τ-stable assembly such that t,..., t j T ae the tiles caable of binding to the same osition m α, ρ(t then fo i j, t i binds at osition m with obability i ). ρ induces ρ(t )++ρ(t j ) a obability measue on A [T ] in a staightfowad way. Fomally, let α A [T ] be a oducible teminal assembly. Let A(α) be the set of all assembly sequences α = (α i i k) such that α k = α, with α,i denoting the obability of attachment of the tile added to α i to oduce α i (noting that α,i = if the i th tile attached without contention). Then P[α] = k α A(α) i= α i α,i. Wite T (ρ) to denote the andom vaiable eesenting the oducible, teminal assembly oduced by T when using tile concentation assignment ρ.
Poblems The geneal oblem is this: given a shae X Z (a connected, finite set), set the concentations of tile tyes in some tile system T so that T is likely to ceate a teminal assembly with shae X, o close to it. We now state fomal oblems that ae vaiations on this theme. The fist fou oblems use concentation ogamming : vaying the concentations of tile tyes in a single tile system T to get it to assemble diffeent shaes. The last two oblems concen a tile system that only does one thing assemble a line of a desied exected length because in this setting we will equie all concentations to be equal. Howeve, the tile system uses andomized self-assembly to do this with fa fewe tile tyes than ae needed to accomlish the same task in a deteministic tile system. The fist thee oblems concen the self-assembly of squaes, and the oblems ae listed in ode of inceasing difficulty. The fist asks fo a squae with a desied exected width, the second fo a guaantee that the actual width is likely to be close to the exected width, and finally, fo a guaantee that the actual width is likely to be exactly the exected width. Fomally, design a tile system T = (T, σ, τ) such that, fo any n Z +, thee exists a tile concentation assignment ρ : T [, ) such that Poblem. dom T (ρ) is a squae with exected width n. Poblem. with obability at least δ, dom T (ρ) is a squae whose width is between ( ɛ)n and ( + ɛ)n. Poblem. with obability at least δ, dom T (ρ) is a squae of width n. The next oblem genealizes the evious oblems to abitay shaes, while making one elaxation: allowing a scaled-u vesion of a shae to be assembled instead of the exact shae. Fomally, fo c Z + and shae S Z (finite and connected), define S c = { (x, y) Z ( x/c, y/c ) S } to be S scaled by facto c. Poblem 4. Let δ >. Design a tile system T = (T, σ, τ) such that, fo any shae S Z, thee exists a tile concentation assignment ρ : T [, ) and c Z + so that, with obability at least δ, dom T (ρ) is S c. It is easy to see that fo a deteministic tile system to assemble a length n, height line equies n tile tyes. The next oblem concens using andomization to educe the numbe of tile tyes equied, subject to the constaint that all tile tye concentations ae equal. (Without this constaint, a solution to Poblem would tivially be a solution to the next oblem, with an otimal O() tile tyes, but since the solution to Poblem uses diffeent tile tye concentations to achieve its goal, it cannot be used diectly fo this uose). Poblem 5. Let n Z +. Design a tile system T = (T, σ, τ) such that, with tile concentation assignment ρ : T [, ) defined by ρ(t) = fo all t T, dom T (ρ) is a height line of exected length n. As with the case of concentation ogamming, it is desiable fo the line to have length likely to be close to its exected length. Poblem 6. Let n Z + and δ, ɛ >. Design a tile system T = (T, σ, τ) such that, with tile concentation assignment ρ : T [, ) defined by ρ(t) = fo all t T, dom T (ρ) is a height line whose length is between ( ɛ)n and (+ɛ)n with obability at least δ.
Key Results 4 The solutions to Poblems 4 use temeatue seed G S tile systems. The solutions to Poblems 5 and 6 concentation concentation use a temeatue tile system (thee is no need fo cooeative binding in one dimension). exected length l = / Figue shows a simle tile system with thee tile tyes that can gow a line of any desied seed G G G G G G S exected length to the ight of the seed tile; this is the basis fo the solutions to Poblems,,, Fig.. A andomized temeatue τ = tile and 4. The length of the line has a geometic distibution, with exected value contolled by the ete nondeteministically to bind to the ight of system that can gow a line of any desied exected length l by setting =. Two tiles com- l atio of the concentations of G and S. Figue the line (using stength glues, indicated by double black lines), one of which stos the gowth, shows the solution to Poblem, due to Becke, while the othe continues, giving the length of the Remilá, and Raaot []. It is essentially the tile line (not counting the seed) a geometic distibution with exected value l. system fom Figue (tile tyes A and B ae analogous to G and S in Figue ) augmented with a constant numbe of exta tiles that can assemble the squae to be as high as the line is long. Fig.. A tile system that gows a squae of any desied exected width. Figue taken fom [4]; stength glues ae indicated by two lines between the tiles. The seed is labeled S, and C A and C B esectively eesent the concentations of A and B. is used the same way as in Figue, and c eesents total concentation of all othe tile tyes, since [4] assumed that concentations of all tile tyes must sum to. Kao and Schwelle [4] showed a solution to Poblem, and Doty [] imoved thei constuction to show a solution to Poblem. Hee, we descibe only the latte constuction, since the two shae simila ideas, and the latte constuction solves both oblems. Figue shows an imovement to the tile system of Figue, which will be the stating oint fo the solution. It also can gow a line of any desied exected length. Howeve, by using multile indeendent stages of gowth, each stage having a geometic distibution, the esulting assembly is moe likely to have a length that is close to its exected length. Moe tile tyes ae needed fo moe stages, but only a constant numbe of stages ae equied. In aticula, if the exected length is chosen to be midway between any two consecutive owes of two, i.e., midway in the inteval [ a, a ) fo abitay a N, with = stages, the obability is at most.5 that the actual seed seed G G concentation G G S concentation G S G concentation G G S concentation S exected length / G G G G concentation G G S concentation G Fig.. A tile system that gows a line of a given length with geate ecision than in Figue. stages each have exected length /, making the exected total length /, but moe tightly concentated about that exected length than in the case of one stage. length is outside the inteval [ a, a ). So although the length is not contolled with exact ecision, the numbe of bits needed to eesent the length is contolled with exact ecision (with high obability), using a constant numbe of tile tyes. Figue 4 shows a tile system T with the following oety: fo any bit sting s (equivalently, any natual numbe m if we assume the most significant bit of s is ), thee is a tile concentation assignment that causes T to gow an assembly of height O(log m), width O(m ), such that the tile tyes in the ue-ight cone of the G G S
assembly encode s. The bottom ow is the tile system fom Figue, with identical stength glues on the noth of the tiles (othe than the final sto tile on the ight). 5 concentation = m l l concentation = ml l l k l k samling tiles Concentations of G i 's and S i 's ensue S 5 almost cetainly is laced within this inteval.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, seed G G G S G G G G G S G G G G G G S G 4 S 4 S 5 gowth signal to sto at next owe of m in binay most significant k bits least significant l bits ignoed samling ow Fig. 4. Comuting the binay sting (equivalently, the natual numbe m = ) fom tile concentations. Fo bevity, glue stengths and labels ae not shown. Each column incements the imay counte, eesented by the bits on the left of each tile, and each gay tile incements the samling counte, eesented by the bits on the ight of each tile. The numbe of bits at the end is l + k, whee c is a constant coded into the tile set, and k deends on m, and l = k + c. The most significant k bits of the samling counte encode m. In this examle, k = and c =. Figue 5 shows a high-level oveview of the entie tile system that assembles an n n squae, solving Poblem. Using simila ideas to Figue 4, one can encode thee diffeent numbes m, m, m N into the tile concentations. We choose these numbes to be such that each m i = O(n / ), and each of thei binay exansions, intewoven into a single bit sting, is the binay exansion of n. Then each tile at the ue ight of Figue 4 encodes not one but thee bits of n, o equivalently each encodes an octal digit of n. These bits ae then used to assemble a counte that counts fom n down to as it gows noth, and a constant set of tiles (simila to Figue exand this counte to gow about as fa east as the counte gows noth, ceating an n n squae that suounds the assembly of Figue 4. Since Since the width of the stuctue to the ight is O(n / ), sace is left ove hee fo sufficiently lage n, and it is filled in by the fille tiles emanating fom the east wall of the squae. quadule counte estimating m, m, and m Figue 4 m m m shift off c least significant bits fille tiles fill in squae base-8 counte counts down fom n to, altenating left-moving decement by two ows and ightmoving coy ows to detect undeflow and sto at convet n to octal isolate most significant half of bits by sliding makes in fom each side until they meet 5 fille tiles 7 7 7 7 7 5 5 5 7 7 7 7 5 5 5 7 5 5 5 otate n uwads fille tiles fille tiles Fig. 5. High-level oveview of the entie constuction solving Poblem, not at all to scale. Fo bevity, glue stengths and labels ae not shown. The double counte numbe estimato of Figue 4 is embedded with two additional countes to ceate a quadule counte estimating m, m, and m, shown as a box labeled as Figue 4 in the above figue. In this examle, m = 4, m =, and m = 5, eesented vetically in binay in the most significant 4 tiles at the end of the quadule counte. Concatenating the bits of the tiles esults in the sting, the binay eesentation of 859, which equals n k 4 fo n = 87, so this examle builds an 87 x 87 squae. Once the counte ends, c tiles (c = in this examle) ae shifted off the bottom, and the to half of the tiles ae isolated (k = 4 in this examle). Each emaining tile eesents thee bits of n, which ae conveted into octal digits, otated to face uwads, and then used to initialize a base-8 counte that builds the east wall of the squae. Fille tiles cove the emaining aea of the squae. m i = O(n / ), and the tiles of Figue 4 ceate a stuctue of height O(log m i ) and width O(m i ) = O(n / ), the squae is sufficiently lage to contain the tiles of Figue 4. Finally, the tiles of Figue 4 ae used in a diffeent way to solve Poblem 4, shown in Figue 6. Given a finite shae S, Soloveichik and Winfee [5] use an inticate constuction of a seed block that unacks, fom a set of tile tyes that deend
on S, a single-tae Tuing machine ogam {, } that oututs a binay sting bin(s) eesenting a list of the coodinates of S. The width of the seed block is then c, chosen to be lage enough to do the unacking, and also lage enough to accommodate the simulation of by a tile set that simulates single-tae Tuing machines. Once this seed block is in lace, a tile set then assembles the scaled shae by caying bin(s) though each block. The ode in which blocks ae assembled is detemined by a sanning tee of S, so that any blocks with an ancesto elationshi have a deendency, in that the ancesto must be (mostly) assembled befoe the descendant, wheeas blocks without an ancesto elationshi can otentially assemble in aallel. S fille tiles fille tiles S fille tiles double counte estimato S fille tiles fille tiles binay eesentation of list of oints in S comutation of above image fom [5] Fig. 6. On the left is the seed block used to elace the seed block of [5], fom which the constuction of [5] can assemble a scaled vesion of the shae S (encoded by a binay sting eesenting the list of coodinates, also labeled S in the figue). S is outut by the single-tae Tuing machine ogam. is estimated fom tile concentations as in Figue 4, then fou coies of it ae oagated to each side of the block, whee it is executed in fou otated, but othewise identical, comutation egions. When comleted, fou coies of the binay eesentation of S bode the seed block, which is sufficient fo the constuction of [5] to assemble a scaled vesion of S using a sanning tee of S as shown on the ight. 6 We elace the seed block tiles of [5], which deend on S, with a single tile system that oduces the ogam fom tile concentations, and use the emainde of the tile set of [5] unchanged. This is illustated in Figue 6. Choose c to be sufficiently lage that can be simulated within the taezoidal egion of the c c block of Figue 6, and also sufficiently lage that the constuction of Figue 4 has sufficient oom to estimate the binay sting fom tile concentations in the cente egion (the double counte estimato ) of Figue 6. Once this is done, the constuction of [5] can take ove and assemble the entie scaled shae S c. The otion of the constuction of [5] that achieves this is a constant-size tile set, so combined with the esented constuction emains constant. This solves Poblem 4. Finally, Poblems 5 and 6 have solutions due to Chandan, Goalkishnan, and Reif [], which we now exlain intuitively (the actual analysis is a bit tickie but is close to the following intuitive agument). Figue 7 shows an examle of a solution to Poblem 5 fo Fig. 7. Examle of solution to Poblem 5 fo the case of exected length 9. the case of exected length n = 9. Each T ib tile tye has an east glue, g i, that matches two tile tyes T (i )A and R (i )A. Thee ae O(log n) stages (5 stages in this case). Each stage has obability to eithe decement the stage o eset back to the highest stage. The numbe n is ogammed into the system by choosing each stage to have eithe o tiles. Given that we ae in stage i, to make it fom stage i to stage without esetting means that i consecutive unbiased coin flis must come u heads, which we exect to take i flis befoe haening. Thus we exect stage i to aea i times; this means that stage i s exected contibution to the total length is eithe i o i, deending on whethe it
has o tiles. The eason this woks to encode abitay natual numbes n is that evey natual numbe can be exessed as n = log n i= b i i, whee b i {, }. Since thee ae a constant numbe of tile tyes e stage, this imlies that the numbe of tile tyes equied is O(log n). This solves Poblem 5. To solve Poblem 6, it suffices to concatenate k indeendent assemblies of the kind shown in Figue 7, whee k is a constant that, if chosen sufficiently lage based on δ (the desied eo obability), solves Poblem 6 since it inceases the numbe of tile tyes equied. In addition to oving that this woks, Chandan, Goalkishnan, and Reif [] also show a moe comlex constuction with even shae bounds on the obability that the length diffes vey much fom its exected value. 7 Oen oblems The constuction esolving Poblem shows that fo evey δ, ɛ >, a tile set exists such that, fo evey n N, aoiately ogamming the tile concentations esults in the self-assembly of a stuctue of size O(n ɛ ) O(log n) whose ightmost tiles eesent the value n with obability at least δ. (In the tile system descibed, ɛ = /, and it could be made abitaily close to by estimating moe than numbes at once.) Is this otimal? Fomally, say that a tile assembly system T = (T, σ, ) is δ-concentation ogammable (fo δ > ) if thee is a (total) comutable function : A [T ] N (the eesentation function) such that, fo each n N, thee is a tile concentation assignment ρ : T [, ) such that P[(T (ρ)) = n] δ. In othe wods, T, ogammed with concentations ρ, almost cetainly self-assembles a stuctue that eesents n, accoding to the eesentation function, and such a ρ can be found to ceate a high-obability eesentation of any natual numbe. Question. Is the following statement tue? Fo each δ >, thee is a tile assembly system T and a eesentation function : A [T ] N such that T is δ-concentation ogammable and, fo each ɛ > and all but finitely many n N, P[ dom T (ρ) < n ɛ ] δ. If so, what is the smallest bound that can be witten in lace of n ɛ? Recommended Reading. Floent Becke, Ivan Raaot, and Eic Rémila. Self-assembling classes of shaes with a minimum numbe of tiles, and in otimal time. In FSTTCS 6: Foundations of Softwae Technology and Theoetical Comute Science, ages 45 56, 6.. Haish Chandan, Nikhil Goalkishnan, and John H. Reif. Tile comlexity of linea assemblies. SIAM Jounal on Comuting, 4(4):5 7,. Peliminay vesion aeaed in ICALP 9.. David Doty. Randomized self-assembly fo exact shaes. SIAM Jounal on Comuting, 9(8):5 55,. Peliminay vesion aeaed in FOCS 9. 4. Ming-Yang Kao and Robet T. Schwelle. Randomized self-assembly fo aoximate shaes. In ICALP 8: Intenational Colloqium on Automata, Languages, and Pogamming, volume 55 of Lectue Notes in Comute Science, ages 7 84. Singe, 8. 5. David Soloveichik and Eik Winfee. Comlexity of self-assembled shaes. SIAM Jounal on Comuting, 6(6):544 569, 7. Peliminay vesion aeaed in DNA 4. 6. Eik Winfee. Algoithmic Self-Assembly of DNA. PhD thesis, Califonia Institute of Technology, June 998.