Randomized Self-Assembly for Exact Shapes

Transcription

1 Randomized Self-Assembly fo Exact Shapes David Doty Depatment of Compute Science Iowa State Univesity Ames, IA 500, USA Abstact Woking in Winfee s abstact tile assembly model, we show that a constant-size tile assembly system can be pogammed though elative tile concentations to build an n n squae with high pobability, fo any sufficiently lage n. This answes an open question of Kao and Schwelle Randomized Self-Assembly fo Appoximate Shapes, ICALP 008), who showed how to build an appoximately n n squae using tile concentation pogamming, and asked whethe the appoximation could be made exact with high pobability.. INTRODUCTION log n log log n Self-assembly is a tem used to descibe systems in which a small numbe of simple components, each following local ules vening thei inteaction with each othe, automatically assemble to fom a taget stuctue. Winfee [] intoduced the abstact Tile Assembly Model atam) based on a constuctive vesion of Wang tiling [9], [0] as a simplified mathematical model of Seeman s wok [5] in utilizing DNA to physically implement self-assembly at the molecula level. In the atam, the fundamental components ae un-otatable, tanslatable squae tile types whose sides ae labeled with glue labels and stengths. Two tiles placed next to each othe inteact if the glue labels on thei abutting sides match, and a tile binds to an assembly if the total stength on all of its inteacting sides exceeds the ambient tempeatue, equal to in this pape. See Section fo a fomal definition. Winfee [] demonstated the computational univesality of the atam by showing how to simulate an abitay cellula automaton with a tile assembly system. Building on these connections to computability, Rothemund and Winfee [] investigated the minimum numbe of tile types needed to uniquely assemble an n n squae. Utilizing the theoy of Kolmoov complexity, they show ) that fo any alithmically andom n, Ω tile types ae equied to uniquely assemble an n n squae, and Adleman, Cheng, Goel, and Huang [] exhibit a constuction showing that this lowe bound is asymptotically tight. Real-life implementations of the atam involve at the pesent time) ceating tile types out of DNA doublecossove molecules [], copies of which can be ceated at an exponential ate using the polymease chain eaction PCR) [4]. PCR technology has advanced to the point whee it is automated by machines, meaning that copies of tiles ae easy to supply, wheeas the numbe of distinct tile types is a pecious esouce, costing much moe lab time to ceate. Theefoe, effot has been put towads developing methods of pogamming tile sets though methods othe than had-coding the desied behavio into the tile types. Such methods include tempeatue pogamming [6], [8], which involves changing the ambient tempeatue though the assembly pocess in ode to alte which bonds ae possible to beak o ceate, and staged assembly [5], which involves pepaing diffeent assemblies in diffeent test tubes, which ae then mixed afte eaching a teminal state. Each of these models allows a single tile set to be eused fo assembling diffeent stuctues by pogamming it with diffeent envionmental conditions affecting the behavio of the tiles. The model used in this pape is known as tile concentation pogamming. If the tile assembly system is nondeteministic if intemediate assemblies exist in which moe than one tile type is capable of binding to the same position and if the solution is wellmixed, then the elative concentations of these tile types detemine the pobability that each tile type will be the one to bind. Tile concentations affect the expected time befoe an assembly is completed such a model is consideed in [] and [], fo instance), but we ignoe such unning time consideations in the pesent pape. We instead focus on using the biased andomness of tile concentations to guide a pobabilistic shape-building alithm, subject a cetain kind of geometic space bound ; namely, that the alithm must be executed within the confines of the shape being assembled. This estiction follows fom the monotone natue of the atam: once a tile attaches to an assembly, it neve

2 detaches. Chandan, Gopalkishnan, and Reif [4] show that a one-dimensional line of expected length n can be assembled using Θ log n) tile types, subject to the estiction that all tile concentations ae equal. Futhemoe, they show that this bound is tight fo all n. Becke, Rapapot, and Rémila [] show that thee is a single tile assembly system T such that, fo all n Z +, setting the tile concentations appopiately causes T to assemble an n x n squae, such that n has expected value n. Howeve, n will have a lage deviation fom n with non-negligible pobability. Kao and Schwelle [7] impove this esult by constucting, fo each δ, ɛ > 0, a tile assembly system T such that setting the tile concentations appopiately causes T to assemble an n x n squae, whee ɛ)n n + ɛ)n with pobability at least δ, fo sufficiently lage n Z +. Kao and Schwelle asked whethe a constant-sized tile assembly system could be constucted that, though tile concentation pogamming, would assemble a squae of dimensions exactly n n, with high pobability. We answe this question affimatively, showing that, fo each δ > 0, thee is a tile assembly system T such that, fo sufficiently lage n Z +, thee is an assignment of tile concentations to T such that T assembles an n n squae with pobability at least δ. Kao and Schwelle also asked whethe abitay finite connected shapes, possibly scaled by facto c N depending on the shape) by eplacing each point in the shape with a c c block of points, could be assembled fom a constant tile set though concentation pogamming. Ou constuction answeing the fist question computes the binay expansion of n with high pobability in a self-assembled ectangle of height Olog n) and width On / ). By assembling this stuctue within the seed block of the constuction of [7], ou constuction can easily be combined with that of [7] to answe this question affimatively as well, by eplacing the numbe n with a pogam that outputs a list of points in the shape, and using this as the seed block of the constuction of [7]. We omit a detailed constuction in this extended abstact. This pape is oganized as follows. Section povides backgound definitions and notation fo the atam and tile concentation pogamming. Section povides the constuction and poof of coectness. Section 4 concludes the pape, discusses pactical limitations of the constuction and potential impovements, and suggests non-squae stuctues that can be assembled with the same technique.. THE TILE ASSEMBLY MODEL AND TILE CONCENTRATION PROGRAMMING We give a bief sketch of the Tile Assembly Model that is adequate fo eading this pape. Moe details and discussion may be found in [8], [], [], []. Ou notation is that of [8], which povides a self-contained intoduction to the Tile Assembly Model fo the eade unfamilia with the model. All logaithms in this pape ae base. We wok in the -dimensional discete space Z. Define the set U {0, ),, 0), 0, ),, 0)} to be the set of all unit vectos, i.e., vectos of length in Z. We wite [X] fo the set of all -element subsets of a set X. All gaphs in this pape ae undiected gaphs, i.e., odeed pais G V, E), whee V is the set of vetices and E [V ] is the set of edges. Intuitively, a tile type t is a unit squae that can be tanslated, but not otated, having a well-defined side u fo each u U. Each side u of t has a glue with label label t u) a sting ove some fixed alphabet Σ and stength st t u) a nonnegative intege specified by its type t. Two tiles t and t that ae placed at the points a and a + u espectively, bind with stength st t u) if and only if label t u), st t u)) label t u), st t u)). In ou figues, we follow Winfee s convention of epesenting stength-0 bonds with dashed lines, stength- bonds with single lines, and stength- bonds with double lines. Given a set T of tile types, an assembly is a patial function α : Z T, with points x Z at which α x) is undefined intepeted to be empty space, so that dom α is the set of points with tiles. α is finite if dom α is finite. Fo assemblies α and α, we say that α is a subassembly of α, and wite α α, if dom α dom α and α x) α x) fo all x dom α. α is a single-tile extension of α if α α and dom α dom α is a singleton set. In this case, we wite α α+ m t), whee { m} dom α dom α and t α m) A gid gaph is a gaph G V, E) in which V Z and evey edge { a, b} E has the popety that a b U. The binding gaph of an assembly α is the gid gaph G α V, E), whee V dom α, and { m, n} E if and only if ) m n U, ) label α m) n m) label α n) m n), and ) st α m) n m) > 0. An assembly is τ-stable, whee τ N, if it cannot be boken up into smalle assemblies without beaking bonds of total stength at least τ; i.e., if evey cut of G α has weight at least τ, whee the weight of an edge is the stength of the sides of tiles

3 that it connects. In contast to the model of Wang tiling, the nonnegativity of the stength function implies that glue mismatches between adjacent tiles do not pevent a tile fom binding to an assembly, so long as sufficient binding stength is eceived fom the sides of the tile at which the glues match. Self-assembly begins with a seed assembly σ typically assumed to be finite and τ-stable) and poceeds asynchonously and nondeteministically, with tiles adsobing one at a time to the existing assembly in any manne that peseves stability at all times, fomally modeled as follows. A tile assembly system TAS) is an odeed tiple T T, σ, τ), whee T is a finite set of tile types, σ : Z T is the seed assembly, satisfying dom σ <, and τ N is the tempeatue, equal to in this pape. A finite) assembly sequence in a TAS T T, σ, ) is a finite sequence α α i i k) of assemblies in which α σ, each α i+ is a single-tile extension of α i, and each α i is τ-stable. The esult of an assembly sequence is es α) α k. We wite A[T ] to denote the set of all esults of assembly sequences of T, known as the poducible assemblies of T. An assembly α is teminal, and we wite α A [T ], if no tile can be stably added to it. Let T T, σ, τ) be a TAS. A tile concentation assignment on T is a function ρ : T [0, ). 4 If ρt) is not specified explicitly fo some t T, then ρt). If α : Z T is a τ-stable assembly such that t,..., t j T ae the tiles capable of binding to Thee ae multiple senses in which a tile system can be nondeteministic. The tivial sense is that the location of attachment, if thee is moe than one candidate, is selected nondeteministically. Such systems may still be deteministic in a stonge sense that they will lead to a unique final assembly. We employ a stonge vesion of nondeteminism in which the tile capable of binding to a single position of an assembly is not fixed; the andomized alithm we implement elies on this choice being made accoding to the tile concentations. A tile set can be pogammed with diffeent inputs though selection of an appopiate seed assembly. In this pape, we wish to model the situation in which, once wok has been done once to ceate a single tile set, the tile set can be pogammed entiely though adjustment of tile concentations. Hence, ou esult is stated in tems of the existence of a tile assembly system, with a fixed seed assembly in fact, a single seed tile), that can be used to constuct squaes of any size, solely by adjusting the tile concentations. [8] gives a teatment of the model that allows fo infinite assembly sequences, and indeed ou constuction may esult in an infinite assembly sequence, though with pobability 0. We simplify the pesentation by consideing only finite assembly sequences. 4 Note in paticula that we do not equie ρ to be a pobability measue on T. ρ induces a pobability measue as descibed late on subsets of tiles in T that contend nondeteministically to bind, but thee may be moe than one such subset, and the elative concentation of a tile fom one subset to that of anothe is ielevant, since they do not compete. the same position m of α, 5 then fo i j, t i ρt binds at position m with pobability i) ρt )+...+ρt j).6 ρ induces a pobability measue on A [T ] in the obvious way. 7 Fo p [0, ] and X Z, we say X stictly self-assembles in T ρ) with pobability at least p if P α A [T ][dom α X] p. That is, T self-assembles into a shape equal to X with pobability at least p. Note that diffeent two assemblies may have the same shape though they might assign diffeent tile types to the same position.. CONSTRUCTION OF THE TILE SET This section is devoted to poving the following theoem, which is the main esult of this pape. Fo all δ > 0 and n Z + log δ 8, define δ log 0.94, c δ + log log δ 8 log 0.77, and k n log n +, and define bn, δ) max { δ, δ} kn+c + cδ + k n. Theoem.. Fo all δ > 0, thee is a tile assembly system T δ T, σ, ) such that, fo all integes n bn, δ), thee is a tile concentation assignment ρ n : T [0, ) such that a tanslation of the set { x, y) x, y {,..., n} } stictly self-assembles in T δ ρ n ) with pobability at least δ. Note that fo any fixed δ > 0, bn, δ) On / ) the constant in the O) depending on δ), whence n bn, δ) fo all sufficiently lage n... Intuitive Idea of the Constuction Kao and Schwelle intoduced a basic pimitive in [7] efining a lowe-pecision technique descibed in []), called a sampling line. The sampling line allows tile concentations to encode a natual numbe whose binay epesentation can be pobably appoximately epoduced. Kao and Schwelle utilize the sampling line 5 Moe pecisely, if α+ m t i ) is τ-stable fo some m dom α and all i {,..., j}, but α + m t) is not τ-stable fo any t T {t,..., t j }. 6 This quantity is the conditional pobability that t i attaches to position m, given that one of t,..., t j will bind to position m at the cuent stage of self-assembly. We do not use pobabilities to model the choice of which position m eceives a tile; any fai assembly sequence see [8] fo a definition) will suffice. 7 Fomally, let α A [T ] be a poducible teminal assembly. Let Aα) be the set of all assembly sequences α α i i k) such that es α) α, with p α,i denoting the pobability of attachment of the tile added to α i to poduce α i noting that p α,i if the i th tile attached without contention). Then P[α] k α α Aα) i i p α,i. Although this definition assigns each fontie location to be equally pobable to eceive a tile that is the souce of the tem ), in the constuctions of this pape, any fai assembly α i sequence will wok, even one that biases the choice of fontie location attachment away fom unifom.

4 to encode n N by an appoximation n N such that ɛ)n n +ɛ)n with pobability at least δ. The idea of ou constuction is as follows. We will appoximate only numbes m small enough that the sampling line appoximation has sufficient space to be an exact computation of m with high pobability. The constuction of Kao and Schwelle can be thought of as estimating n by, in a sense, pobabilistically counting to n using independent Benoulli tials with appopiately fixed success pobability; i.e., the pobabilities ae used to estimate an appoximate unay encoding of n, which is conveted to binay by a counte. Repesenting n in unay, of couse, takes space n, and ecoveing it pobabilistically fom tiles subject to andomization equies using much moe than space n to ovecome the eo intoduced by andomization. Kao and Schwelle use an ingenious technique to spead this estimation out into the cente of the n n squae being built, affoding On ) space to appoximate n closely. Howeve, that constuction lacks the space to compute n exactly, which equies much moe than n Benoulli tials applying the standad Chenoff bound to the Kao- Schwelle sampling line achieves an uppe bound of On 5 ) tials to achieve a sufficiently small estimation eo. Hence, attempting to use a sampling line diectly to compute n would esult in a line containing many moe tiles than the n tiles that compose an n n squae, and no amount of twisting the line will cause it to fit inside the boundaies of the squae. We split n s binay expansion bn) b b... b log n + {0, } into thee subsequences b b 4 b 7..., b b 5 b 8..., and b b 6 b 9..., each of length about log n, and intepet these binay stings as natual numbes m, m, m n / to be estimated. The poblem of estimating n is educed to that of estimating these thee numbes. At the same time, we intoduce a new sampling line technique that can exactly estimate a numbe m with high pobability using only Om ) tials. 8 Since m, m, m n /, estimating m, m, and m will equie On / ) tials, 8 As opposed to the Om 5 ) tials that would be equied by the Kao-Schwelle sampling line. It is possible to use Kao and Schwelle s oiginal sampling line to estimate seven numbes log n + the length of the binay expansion of n), and the six numbes m - m 6 log n + encoded by length- substings of n s binay expansion, 6 each small enough that m 5 i on) and to use these numbes to econstuct n and fom that, build an n n squae. A staightfowad and tedious analysis of the constants involved eveals that such a technique can be used to constuct n n squaes fo n 0 8. We achieve much moe feasible bounds on n 0 7 fo δ 0.0) using the techniques intoduced in this pape, and indeed, bette bounds than those equied by Kao and Schwelle to appoximate n, whose constuction achieves, fo instance, a 0.0, 0.0)-appoximation only fo n 0, accoding to thei analysis. which fits within the width of an n n squae fo sufficiently lage n. Intuitively, the eason that estimating m, m, and m ceates an impovement ove estimating n diectly is that the space needed fo the unay encodings of numbes whose binay length is one-thid that of n s does not scale linealy with that length; the unay encoding of these numbes scales with n /, not n/, whence a quadatic incease in the space needed fo pobabilistic ecovey emains sufficiently small On / )) that thee such decodings easily fit into space n... Pobabilistic Decoding of a Natual Numbe using a Sampling Line In this section, we descibe how to exactly compute a positive intege m pobabilistically fom tile concentations that ae appopiately pogammed to epesent m. In ou final constuction, the sampling line will estimate not one but thee integes m, m, and m, as descibed in Section., by embedding additional bits into the tiles. Howeve, fo the sake of claity, in this section, we descibe how to estimate a single positive intege m, and then descibe in Section.. how to modify the constuction and set the pobabilities to allow thee numbes to be estimated simultaneously on a single sampling line. seed seed G G concentation p G G stop S concentation p expected length /p Figue. The potion of the basic Kao-Schwelle sampling line that contols its length. Two tiles compete nondeteministically to bind to the ight of the line, one of which stops the gowth, while the othe continues, giving the length of the line a geometic distibution. The basic length-contolling potion of the Kao- Schwelle sampling line is shown in Figue. 9 A hoizontal ow of tiles foms to the ight of the seed. Two tiles, G ) and S stop ) nondeteministically connect to the ight end of the line; G continues the gowth, while S stops the gowth. If S has concentation 9 Ou desciption of the Kao-Schwelle sampling line is incomplete, as discussed in the next paagaph. G G G stop S 4

5 p [0, ] and G has concentation p, then the length L of the line is a geometic andom vaiable with expected value /p. By setting p appopiately, E[L] can be contolled, but not pecisely, since a geometic andom vaiable may have a deviation fom the expected value that is too lage fo ou puposes. Kao and Schwelle allow a thid tile type to bind within the sampling line, which does the actual sampling fo computing a natual numbe, but ou constuction splits this sampling into a sepaate set of tiles that foms above the line. The sampling potion is discussed in Section... Fo the pesent time, we estict ou discussion to contolling the length of the line.... Moe Pecisely Contolling the Sampling Line Length: Ou al is to contol L, the length of the sampling line, such that, by setting tile concentations appopiately, we may ensue that L lies between a and a with high pobability, fo an a Z + of ou choosing which will be influenced by the numbe n we ae estimating). That is, we may ensue that the numbe of bits equied to epesent L is computed pecisely, even if the exact value of L vaies widely within the inteval [ a, a ). We then attach a counte a goup of tiles that measues the length of the line by counting in binay to the noth of the line that measues L until the final stopping tile. The stop signal is not intended to stop the counte immediately, but athe to signal that the counte should continue until it eaches the next powe of i.e., the next time a new most significant bit is equied and then stop. Hence, we may choose an abitay powe of and set tile concentations to ensue that the counte counts to that value and then stops. To incease the pecision with which we contol L, we use not one but many stages of and stop tiles, G, S, G, S,..., G, S. The constuction is shown in Figue. G i and S i each compete to bind to the ight of S i and G i. S i signals a tansition to the next stage i+, with S stopping the gowth of the line afte stages. Theefoe, the sequence of tiles to the ight of the seed is a sting descibed by the egula expession G S G S... G S. Each S i has concentation p, and the emaining G i tiles each have concentation p. The length L of the line is a negative binomial andom vaiable 0 with paametes, p see [9]) with expected value /p by lineaity of expectation; i.e., its length is the numbe of Benoulli tials equied befoe exactly successes, povided each Benoulli tial has success pobability p. Let N, R N and p [0, ]. A binomial andom vaiable BN, p) the numbe of successes afte N Benoulli tials, each having success pobability p) is elated to a negative binomial andom vaiable N R, p) the numbe of tials befoe exactly R successes) by the elationships P[N R, p) < N] P[BN, p) > R],.) P[N R, p) > N] P[BN, p) < R]..) Thus, Chenoff bounds that povide tail bounds fo binomial distibutions can be applied to negative binomial distibutions via.) and.). To cause L to fall in the inteval [ a, a ), we must set its expected length L by setting p /L) to be such that the th success occus when the line has length in the inteval [ a, a ). Note that pn is the expected numbe of successes in the fist N tiles of the line; i.e., it is the expected numbe of successes in exactly N Benoulli tials. We define ɛ and ɛ so that L + ɛ) a ɛ ) a and the two eo pobabilities deived below ae appoximately equal; ɛ and ɛ suffice. The event that L < a is equivalent to the event that a Benoulli tials ae conducted with expected numbe of successes p a ) with at least successes. By.) and the Chenoff bound [9, Theoem 4.4, pat ], P [ L < a ] P [ > + ɛ)p a ] ) p a e ɛ +ɛ) +ɛ ) a e ɛ /L +ɛ) +ɛ ) e ɛ +ɛ) +ɛ ) /+ɛ) e ɛ +ɛ) +ɛ < a /+ɛ) a ) 0 The tem negative is misleading; a negative binomial andom vaiable is bette descibed infomally) as the invese of a binomial andom vaiable, if one thinks of a binomial andom vaiable as being like a function that maps a numbe of Benoulli tials to a numbe of successes. A negative binomial andom vaiable maps a numbe of successes to the numbe of tials necessay to achieve that numbe of successes. The event that L a is equivalent to the event that a Benoulli tials ae conducted with expected numbe of successes p a ) with fewe than successes. To bound the pobability that L is too lage, we use.) and the Chenoff bound fo deviations below the 5

6 seed G S G S... G stop S concentation p concentation p concentation p concentation p concentation p concentation p expected length /p seed G G G G S G G S G G G G G... G G G stop S Figue. The potion of the sampling line of ou constuction that contols its length. stages each have expected length /p, making the expected total length /p, but moe tightly concentated about that expected length than in the case of one stage. mean [9, Theoem 4.5, pat ], P [L a ] P [ ɛ )p a ] ) p a e ɛ ɛ ) ɛ ) a e ɛ /L ɛ ) ɛ ) e ɛ ɛ ) ɛ By the union bound, ) /+ɛ) e ɛ ɛ ) ɛ < a /+ɛ) a ) P[L [ a, a )] < 0.94.) Theefoe, by setting sufficiently lage, we can exponentially decease the pobability that L falls outside the ange [ a, a ), independently of a. Fo example, letting leads to P [ L [ a, a ) ] < Since is a constant depending only on δ, it can be encoded into the tile types as shown in Figue.... Computing a Numbe Exactly using a Sampling Line: As stated peviously, ou al is that, with a sampling line of length Om ), we can exactly compute a numbe m. The idea is shown in Figue, and is inspied by the sampling line of Kao and Schwelle [7] but can estimate a numbe moe pecisely using a given length, as well as having a length that is itself contolled moe pecisely by the technique of Section... The length-contolling potion of the sampling line of length L will contol a counte placed above the sampling line, which counts to the next powe of geate than L, a. This counte will eventually end up with a total bits befoe stopping. Let k be the maximum numbe of bits needed to epesent m k will be about log n in ou application), and let l a k. We fom a ow above the ow descibed in Section.., which does the sampling. To implement the Benoulli tials that estimate m, one of two tiles A the gay tile in Figue ) o B the white tile in Figue ) nondeteministically binds to evey position of this ow. Set the concentation of A to be ml + l and the concentation of B to be ml + l a. We embed a a second counte the sampling counte within the pimay counte. Wheneve A appeas, the sampling counte incements, and when B appeas it does not change. Let M be the andom vaiable epesenting the final value of the sampling counte. Then M is a binomial andom vaiable with E[M] m l + l. We will choose k and l so that the most significant k bits of the sampling counte will almost cetainly epesent m. Intuitively, the least significant l bits of M absob the eo. This will occu if m l M < m + ) l. Note that m < k. Let ε m. Then the Chenoff bound [9, Theoems 4.4/4.5, pat ] and the union bound tell us that P [ M m + ) l o M < m l] P [M + ε)e[m] o M < ε)e[m]] e E[M]ε / + e E[M]ε / < e ml m) / + e ml m) / e l m + e l m < e l k / + e l k / Let c N be a constant. By setting l k + c, the pobability of eo deceases exponentially in c: P [ M m + ) l o M < m l] < e c / + e c / < 0.77 c..4) Fo instance, letting c 6 bounds the left-hand side of.4) below The numbe of samples is a k+c O k ) ). Since m < k, integes m such that m n can be 6

7 concentation m l l concentation ml l l k l k sampling tiles Concentations of G i 's and S i 's ensue S 5 almost cetainly is placed within this inteval.,0,0,0,0,0,0,0,0,0,0,0,0,0,0,, 0,,0,0,0,0,0,0,0,0 0,0 0,0 0,0 0, 0, 0, 0, 0,,,,,,,,0,0 0,0,0,0,0,0 0,0 0, 0, 0,,,,, 0, 0, 0, 0,0,0,0,0,0 0, 0, 0, 0,,,,0,0 0,0,0,0 0,0 0,,, 0, 0,0,0,0 0,0 0,,, 0, 0,,,0 0,0 0,,, 0,0 0,0,0, 0, 0,,0,0 0,0,0 0,0, 0,,0 0,, 0,,0 0,0, 0,,0 0,0,0 0,, 0,,0 0,,0 0,0, 0,0,0 0,,0 0,, 0,0, 0, seed G G G S G G G G G S G G G G G G S G 4 S 4 G 5 G 5 G 5 G 5 G 5 S 5 gowth signal to stop at next powe of m in binay most significant k bits least significant l bits ignoed sampling ow Figue. Computing the natual numbe m fom tile concentations using a sampling line. Fo bevity, glue stengths and labels ae not shown. Each column incements the pimay counte, epesented by the bits on the left of each tile, and each gay tile incements the sampling counte, epesented 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 depends on m, and l k + c. The most significant k bits of the sampling counte encode m. In this example, k and c. pobably exactly computed using much fewe than n Benoulli tials, and can theefoe be computed by a sampling line without exceeding the boundaies of an n n squae... Computing n Exactly We have shown how to compute a numbe m exactly using a sampling line of length Om ) and width Olog m). To compute n, the dimensions of the squae, we must compute m, m, and m, which ae the numbes epesented by the bits of the binay expansion of n at positions conguent to mod, mod, and 0 mod, espectively. To compute all thee of these numbes, we embed two exta sampling countes into the double counte, in addition to the sampling counte descibed in Section., to ceate a quaduple counte. This equies 8 sampling tiles instead of, in ode to epesent each of the possible outcomes of conducting thee simultaneous Benoulli tials, each tial used fo estimating one of m, m, o m. Given i {,, }, let b i {0, } denote the outcome of the i th of thee simultaneous Benoulli tials, and let p i b i ) denote the pobability we would like to associate with that outcome. As noted in Section.., the values of the c i s ae given by p i ) mil + l, a and p i 0) p i ). Since each of the thee simultaneous Benoulli tials is independent, we can calculate the appopiate concentation of the tile epesenting the thee outcomes by multiplying the thee outcome pobabilities togethe. Then the equied concentation of the tile epesenting outcomes b, b, b is given by p b ) p b ) p b ). Once the values m, m, and m ae computed, we must emove the c least significant bottom) bits fom the bottom of the pimay counte. Since c is a constant depending only on δ, it can be encoded into the tile types. We must then emove the bottom half of the emaining bits. At this point, the concatenation of the bits on the tiles epesent the binay expansion of n. Rathe than expand them out to use thee times as many tiles, we simply tanslate each of them to an octal digit, giving the octal epesentation of n, with one octal digit pe tile eplacing the thee bits pe tile. Finally, this epesentation of n is otated 90 degees counte-clockwise, used as the initial value fo a decementing, upwadsgowing, base-8 counte, and used to fill in an n n squae using the standad constuction []. Rotating n to face up stats the counte k+ tiles fom the bottom of the constuction so fa. Futhemoe, testing whethe the counte has counted below 0 equies counting once beyond 0, using moe ows than the stating value of the counte. Theefoe, to ensue that exactly an n n squae is fomed, the value n k 4, athe than n exactly, is pogammed into the tile concentations to seve as the stat value of the upwads-gowing counte. An outline of this constuction is shown in Figue 4. It is outine to veify that the choice of the paametes c, k, and descibed just befoe Theoem. ae sufficient to obtain the bound bn, δ) of Theoem.. A simulated implementation of this tile assembly system using the ISU TAS Tile Assembly Simulato [0] is available at lnsa/softwae. html. The tile set uses appoximately c+4 tile types, whee and c ae calculated fom δ as above. Isolating the most significant half of the bits can be done using a tile set simila to the alithm one might use to pogam a single-tape Tuing machine to compute the function 0 n 0 n. 7

8 fille tiles fill in squae quaduple counte estimating m, m, and m m m m shift off c least significant bits base-8 counte counts down fom n to 0, altenating left-moving decement by two ows and ightmoving copy ows to detect undeflow and stop at 0 convet n to octal isolate most significant half of bits by sliding makes in fom each side until they meet Since the width of the stuctue to the ight is On / ), space 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. Figue fille tiles 5 5 otate n upwads fille tiles fille tiles Figue 4. High-level oveview of the entie constuction, not at all to scale. Fo bevity, glue stengths and labels ae not shown. The double counte numbe estimato of Figue is embedded with two additional countes to ceate a quaduple counte estimating m, m, and m, shown as a box labeled as Figue in the above figue. In this example, m 4, m, and m 5, epesented vetically in binay in the most significant 4 tiles at the end of the quaduple counte. Concatenating the bits of the tiles esults in the sting 00000, the binay epesentation of 859, which equals n k 4 fo n 87, so this example builds an 87 x 87 squae. Actually, 87 is too small to wok with ou constuction, so the counte will exceed length 87, but we choose small numbes to illustate the idea moe clealy.) Once the counte ends, c tiles c in this example) ae shifted off the bottom, and the top half of the tiles ae isolated k 4 in this example). Each emaining tile epesents thee bits of n, which ae conveted into octal digits, otated to face upwads, 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. 8

9 4. CONCLUSION We have descibed how a single tile set in Winfee s abstact tile assembly model, appopiately pogammed by setting tile concentations, exactly assembles an n n squae with high pobability, fo any sufficiently lage n. The focus of the pesent pape is on conceptual claity. We have theefoe descibed the simplest i.e., easiest to undestand, but not necessaily smallest) vesion of the tile assembly system that achieves the desied asymptotic esult that an n n squae assembles with high pobability fo sufficiently lage n. We now obseve that this theoetical esult could be impoved in pactice by complicating the tile set. Ou implementation of the tile set uses appoximately c + 4 tile types, whee, fo example, and c 7 ae sufficient to achieve eo pobability δ 0.0. The tiles ae so numeous because of the need to simultaneously epesent 4 bits in a tile, in addition to infomation such as the significance of the bit MSB, LSB, o inteio bit), and doing computation such as addition, which equies tiles that can handle the 8 possible input bit + cay signals. Putting togethe a few such modules of tile sets esults in thousands of tiles befoe too long. The numbe of tile types could be educed by splitting the estimation of m, m, and m into thee distinct geometical egions, so that each tile is equied to emembe less infomation. This would complicate the tile set, as it would equie moe shifting ticks to ensue sufficient oom fo all countes, and would equie binging the bits back togethe again at the end, but it would likely educe the numbe of tile types. A lage value of n is equied to achieve a pobability of success at least δ) fo easonably small δ; n > is equied to estimate n with 99% chance of coectness. This shotcoming can be compensated in a numbe of ways. In a simila spiit to the linea speedup theoem, moe than thee simultaneous Benoulli tials may be conducted with each sampling tile. Fo example, conducting 6 Benoulli tials with each sampling tile would estimate two bits of m,..., m with pe sampling tile, athe than one bit, halving the equied length of the sampling line. This would esult in a pohibitively lage tile set, howeve; as the numbe of tile types inceases exponentially with the numbe of simultaneous Benoulli tials pe tile type. A conceptually simple and pactically moe feasible impovement is to use 0/-valued tile concentations to simulate tile type pogamming i.e., designing tile types specially to build a paticula size squae, as in []) fo small values of n, by including tile types that deteministically constuct an n n squae fo each small n, setting concentations of those tiles to be and setting concentations of all othe tiles to be 0. Though this solution lacks the feel of tile concentation pogamming, it is likely that eal-life implementations of tile concentation pogamming will need to use such had-coding ticks fo smalle stuctues that lack the space to cay out the amount of sampling equied to econstuct pecise inputs solely fom tile concentations. An altenate impovement to the tile set would be to combine the pesent technique with the Kao-Schwelle technique of building a sampling line inside of a squae, to moe efficiently use the n space available to cay out the estimation. Howeve, squae-building is not necessaily the only application of this technique, as discussed next. The pimay novel contibution of this pape is a tile set that, though appopiate tile concentation pogamming, foms a thin stuctue of length On / ) and height Olog n), whose ightmost tiles encode the value of n. The numbe n could be used to assemble useful stuctues othe than squaes. Fo the task of building a squae, this constuction wastes the n space available above the thin ectangle, but fo computing othe stuctues, it may be advantageous that the ectangle is kept thin. Fo instance, biochemists outinely use filtes e.g., Millipoe Ultafiltation Membanes) and poous esins [] to sepaate poteins based on size, in ode to isolate one paticula potein fo study. The ability to pecisely contol the size of the filte holes o esin beads would allow fo moe tageted filteing of poteins than is possible at the pesent time. DNA is likely too eactive with amino acids to be used as the substate fo such a stuctue, so an implementation of the tile assembly model not based on DNA would be equied fo such a technique. Similaly, polyacylamide gel electophoesis [6], anothe technique fo disciminating biological molecules on the basis of size, equies molecula mass size makes, which ae contol molecules of known molecula mass, in ode to compae against the molecule of inteest on the gel. At the pesent time, By patitioning n s binay epesentation into t athe than thee subsequences, fo t N a constant, the numbe of tials needed to estimate n is On /t ). Howeve, the constant factos in the O) incease, making the technique even less feasible fo small values of n. But if some application equies an asymptotically vey shot line, the line can be made length On ε ) fo any ε > 0 using this technique. 9

10 some natually-occuing molecules of known mass ae used, but thei masses ae not contollable, and the ability to quickly and easily assemble molecules of pecisely a desied taget mass would be useful in expeiments equiing mass makes that diffe fom the standads. Again, DNA is a special case in which this idea is unnecessay, since pecise standads have been developed fo DNA gels e.g., Novagen DNA Makes). But the tile assembly model may one day be implemented using substances that ae appopiate fo a potein gel. Acknowledgments. I thank Jack Lutz fo helpful advice duing the pepaation of this pape, and Julie Hoy fo suggesting potential biochemical applications. I also thank Pavan Adui and Sikanta Tithapua fo allowing me to audit thei classes on andomness and computation, whose subject matte poved vey useful in devising the techniques of this pape. This eseach was suppoted in pat by National Science Foundation Gants , , and CCF: REFERENCES [] L. Adleman, Q. Cheng, A. Goel, and M.-D. Huang, Running time and pogam size fo self-assembled squaes, in STOC 0: Poceedings of the thity-thid annual ACM Symposium on Theoy of Computing. New Yok, NY, USA: ACM, 00, pp [] F. Becke, I. Rapapot, and E. Rémila, Self-assembling classes of shapes with a minimum numbe of tiles, and in optimal time, in Foundations of Softwae Technology and Theoetical Compute Science FSTTCS), 006, pp [] D. M. Bollag, Gel-filtation chomatogaphy, Methods in Molecula Biology, vol. 6, pp. 9, Novembe 994. [Online]. Available: Abstact/doi/0.85/ : [4] H. Chandan, N. Gopalkishnan, and J. H. Reif, The tile complexity of linea assemblies, in 6th Intenational Colloquium on Automata, Languages and Pogamming, vol. 5555, 009. [5] E. D. Demaine, M. L. Demaine, S. P. Fekete, M. Ishaque, E. Rafalin, R. T. Schwelle, and D. L. Souvaine, Staged self-assembly: nanomanufactue of abitay shapes with O) glues, Natual Computing, vol. 7, no., pp , 008. [6] M.-Y. Kao and R. T. Schwelle, Reducing tile complexity fo self-assembly though tempeatue pogamming, in Poceedings of the 7th Annual ACM-SIAM Symposium on Discete Alithms SODA 006), Miami, Floida, Jan. 006, pp , 007. [7], Randomized self-assembly fo appoximate shapes, in Intenational Colloqium on Automata, Languages, and Pogamming ICALP), se. Lectue Notes in Compute Science, L. Aceto, I. Damgåd, L. A. Goldbeg, M. M. Halldsson, A. Inglfsdtti, and I. Walukiewicz, Eds., vol. 55. Spinge, 008, pp [Online]. Available: conf/icalp/icalp008-.html#kaos08 [8] J. I. Lathop, J. H. Lutz, and S. M. Summes, Stict selfassembly of discete Siepinski tiangles, Theoetical Compute Science, vol. 40, pp , 009. [9] M. Mitzenmache and E. Upfal, Pobability and Computing. Cambidge Univesity Pess, 005. [0] M. J. Patitz, Simulation of self-assembly in the abstact tile assembly model with ISU TAS, in 6th Annual Confeence on Foundations of Nanoscience: Self-Assembled Achitectues and Devices Snowbid, Utah, USA, Apil )., 009. [] P. W. K. Rothemund, Theoy and expeiments in alithmic self-assembly, Ph.D. dissetation, Univesity of Southen Califonia, Decembe 00. [] P. W. K. Rothemund and E. Winfee, The pogamsize complexity of self-assembled squaes extended abstact), in STOC 00: Poceedings of the thity-second annual ACM Symposium on Theoy of Computing. New Yok, NY, USA: ACM, 000, pp [] P. W. Rothemund, N. Papadakis, and E. Winfee, Alithmic self-assembly of DNA Siepinski tiangles, PLoS Biology, vol., no., pp , 004. [4] J. Sambook and D. Russell, Molecula Cloning: A Laboatoy Manual. Cold Sping Habo Laboatoy Pess, 00. [5] N. C. Seeman, Nucleic-acid junctions and lattices, Jounal of Theoetical Biology, vol. 99, pp. 7 47, 98. [6] A. Shapio, E. Viñuela, and J. M. J., Molecula weight estimation of polypeptide chains by electophoesis in SDS-polyacylamide gels, Biochem Biophys Res Commun., vol. 8, pp , Novembe 967. [7] D. Soloveichik and E. Winfee, Complexity of selfassembled shapes, SIAM Jounal on Computing, vol. 6, no. 6, pp , 007. [8] S. M. Summes, Reducing tile complexity fo the self-assembly of scaled shapes though tempeatue pogamming, Computing Reseach Repositoy, Tech. Rep , 009. [Online]. Available: http: //axiv.og/abs/ [9] H. Wang, Poving theoems by patten ecognition II, The Bell System Technical Jounal, vol. XL, no., pp. 4, 96. [0], Dominoes and the AEA case of the decision poblem, in Poceedings of the Symposium on Mathematical Theoy of Automata New Yok, 96). Polytechnic Pess of Polytechnic Inst. of Booklyn, Booklyn, N.Y., 96, pp. 55. [] E. Winfee, Alithmic self-assembly of DNA, Ph.D. dissetation, Califonia Institute of Technology, June

11 . Choice of Paametes APPENDIX We now deive the settings of vaious paametes equied to achieve a desied success pobability and deive lowe bounds on n necessay to allow the space equied by the constuction. To ensue pobability of failue at most δ, we pick, the numbe of stages of stopping tiles that must attach befoe the pimay counte is sent the stop signal, so that 0.94 δ 4 as in.): log δ 8. log 0.94 Fo example, choosing achieves pobability of eo δ/4 in ensuing the counte stops between the numbes a and a ) at most To ensue that each of m, m, and m ae computed exactly, we set c, the numbe of exta bits used in the pimay counte beyond k, such that e c / + e c / δ 4, as in.4), o moe simply, such that 0.77 c δ 4 ; i.e., set c + log log δ 8 log 0.77 Fo example, choosing c 7 achieves pobability of eo δ/4 in ensuing that m is computed coectly) at most in fact, at most ). By the union bound, the length of the sampling line and the values of m, m, and m ae computed with sufficient pecision to compute the exact value of n with pobability at least δ. The example values of and c given above achieve δ 0.0. The choices of and c imply a lowe bound on the value of n necessay to allow sufficient space to cay out the constuction. Clealy the counte must each at least value, since thee ae diffeent stopping stages. The moe influential facto will be the value c, which doubles the space necessay to un the counte each time it is incemented by. n equies log n + bits to epesent, but ou estimation will be a sting of length the next highest multiple of above log n +. Theefoe, each of m, m, and m equies log n + k bits to epesent. Recall that the pimay counte will have height k + c and count to k+c so long as k+c ). Then, c columns ae equied to shift off the constant c bits fom the least significant bits of the counte, and k columns ae equied to shift off the least significant half of the bits of the counte to. isolate the k most significant bits. k columns ae needed to tanslate the goups of thee bits into octal and to otate this sting to face upwads fo the squae-building counte. Hence, the total length equied along the bottom of the squae to compute n is max{, k+c } + c + k. Expanding out the definitions of, k, and c deived above gives the lowe bound bn, δ) on n descibed in Theoem.. Fo sufficiently lage n and small enough δ, is much smalle than k+c, so the latte tem dominates. Fo example, to achieve pobability of eo δ 0.0 equies n > 8,000,000. Accoding to peliminay expeimental tests, in pactice, a smalle value of c is equied than the theoetical bounds we have deived. Fo example, if the desied eo pobability is δ 0.0, setting c 7 satisfies the analysis given above, but in expeimental simulation, c appeas to suffice fo pobability of eo at most 0.0, and educes the space equiements by a facto of 7 6. In this case, n 9000 can be computed by a constuction that will stay within the 9000 x 9000 squae.