Activatable Tiles: Compact, Robust Programmable Assembly and other Applications

Transcription

1 Activatable Tiles: Compact, obust Pogammable Assembly and othe Applications Umi Majumde, Thomas H. LaBean, John H. eif Depatment of Compute Science, Duke Univesity, Duham, C, USA. {umim, thl, Abstact hile algoithmic DA self-assembly is, in theoy, capable of foming complex pattens, its expeimental demonstation has been limited by significant assembly eos. n this pape we descibe a novel potection/depotection stategy to stictly enfoce the diection of tiling assembly gowth to ensue the obustness of the assembly pocess. Tiles ae initially inactive, meaning that each tile s output pads ae potected and cannot bind with othe tiles. Afte othe tiles bind to the tile s input pads, the tile tansitions to an active state and its output pads ae exposed, allowing futhe gowth. e descibe abstact and kinetic models of activatable tile assembly and show that the eo ate can be deceased significantly with espect to infee s oiginal kinetic tile assembly model without consideable decease in assembly gowth speed. e pove that an activatable tile set is an example of a compact, eo-esilient and self-healing tile-set. e descibe a DA design of activatable tiles and a mechanism of depotection using DA polymeization and stand displacement. e conclude with a bief discussion on some applications of activatable tiles beyond computational tiling, both as a novel concentation system and a catalyst in chemical eactions. 1 ntoduction The potential of self-assembling DA nanostuctues is deived fom the pedictable popeties of DA hybidization as well as fom the assembly s theoetical powe to instantiate any computable patten [3]. infee [1] fomalized this pocess of tiling assembly gowth with his poposed Tile Assembly Model (TAM) which descibes how a complex stuctue can spontaneously fom fom simple components called tiles ; this assembly can also pefom computation. Howeve, the main poblem fo a pactical implementation of TAM based assemblies is that tile additions ae vey eo-pone. Expeiments show that eo ates can be as high as 1% to 8% [4, 5]. The pimay kind of eo encounteed in DA tile assembly expeiments is known as the eo by insufficient attachment [7], which occus when a tile violates the TAM ule stating that a tile may only be added if it binds stongly 1 enough. Thus thee is a mismatch between theoetical models of DA tiles and eality, poviding majo challenges in applying this model to eal expeiments. Thee have been seveal designs of eo-esilient tile sets [6, 8, 7] that pefom poofeading on edundantly encoded infomation [8] to decease assembly eos. hile infee et al. [8] and eif et al. [6, 16] addessed the poblem of deceasing gowth eos 2 in assembly, Chen et al. [7] addessed both gowth and facet nucleation eos 3 by investigating eos by insufficient attachment. Schulman et al. [20] addessed the spontaneous nucleation eo 4 with thei zig-zag tile set. Each of these woks, 1 n the TAM fo tempeatue τ = 2, a tile binds stongly eithe using at least one stong bond o two weak bonds. 2 owth eo happens when a tile with one weak bond (weakly binding tile) attaches at a location whee a tile with two weak bonds could have, and should have, been placed. 3 A facet nucleation eo happens when weakly binding tile attaches to a site whee no tiles should attach at the moment. 4 Spontaneous nucleation eos occu when a lage assembly gows in absence of a seed tile.

2 howeve, addesses only cetain types of eos and poposes a constuction that woks with limited classes of tile sets. Additionally, most of the constuctions esults in a blow up the tile set size by a multiplicative facto, geatly hindeing pactical implementation. This leads to a majo open question in eo-esilient self-assembly: s it possible to design a compact tile set that can addess all thee kinds of eos simultaneously? Ou activatable tile set is an effot towads achieving this ultimate goal. Limitations of Pevious Appoaches towads obust Assembly: Existing eo-esilient tile sets assume diectional gowth. This is a vey stong assumption because expeiments show that eal tiles do not behave in such a fashion. The assumption, howeve, undelies the gowth model in TAM. Thus, a potential solution to minimizing assembly eos is to enfoce this diectionality constaint. Obseve that if we stat with a set of deactivated tiles which activate in a desied ode, we can enfoce a diectional assembly at the same scale as the oiginal one. Such a system can be built with minimal modifications of existing DA nanostuctues [10, 9]. Pevious Appoaches to diect Tiling Assembly Pocedues: The main inspiation fo the idea of activatable tiles has been snaked-poofeading technique of Chen et al. [7], which eplaces each oiginal tile by a k k block of tiles. The assembly pocess fo a block doubles back on itself such that nucleation eo cannot popagate without locally focing anothe insufficient attachment. Can such a gowth ode be enfoced at the oiginal scale of the assembly? Othe motivating wok has been fom Diks et al. [2], who designed a system whee monome DA nanostuctues, when mixed togethe, do not hybidize until an initiato stand is added. Can the idea of tiggeed self-assembly be used in the context of computational DA tiling? The answes to both questions ae yes. The basic scheme in one and two dimensions is shown in Figue 1. The key idea is to stat with a set of potected DA tiles, which we call activatable tiles; these tiles do not assemble until an initiato nanostuctue is intoduced to the solution. The initiato utilizes stand displacement to stip off the potective coating on the input sticky end(s) of the appopiate neighbos [12]. hen the input sticky ends ae completely hybidized, the output sticky ends ae exposed. DA polymease enzyme can pefom this depotection [13], since it can act ove long distances (e.g: acoss tile coe) unlike stand displacement. The newly exposed output sticky ends, in tun, stip the potective laye off the next tile along the gowing face of the assembly. DA polymease enzyme can pefom this depotection, since it can act ove long distances (e.g: acoss tile coe) unlike stand displacement. The newly exposed output sticky ends, in tun, stip the potective laye off the next tile along the gowing face of the assembly. The use of polymease as a long ange effecto is justified because of its successful use in PC, a biochemisty technique often used fo exponentially amplifying DA. PC has been so successful that it has seveal commecial applications including genetic fingepinting, patenity testing, heeditay disease detection, genes cloning, mutagenesis, analysis of ancient DA, genotyping of specific mutations, compaisons of gene expession etc. Many epeated ounds of pime polymeization ae equied in conventional PC. n contast, we ae using only a single ound of pime polymeization (simila to a single ound of PC) to expose the desied sticky ends in ou activatable tiles. Enzyme-fee Activated Tiles: The most elevant pevious wok that has been ecently bought to ou attention is pobably that of Fujibayashi et al. [21, 22]: the Potected Tile Mechanism (PTM) and the Layeed Tile Mechanism (LTM) which utilize DA potecting molecules to fom kinetic baies against spuious assembly. Although this is an enzyme-fee cicuit, in the PTM, the output sticky ends ae not potected and thus they can bind to a gowing assembly befoe the inputs ae depotected and hence cause an eo. n the LTM, the output sticky ends ae potected only by 3 nucleotides each and can be easily displaced causing the above-mentioned eo. Thus only if we can ensue a depotection fom input to output end, eo esilience can be guaanteed. Ou activatable tiles is a small step

3 Potection Potection nput nput nput Signal fo Depotection by DA polymeization OE DMESOAL DEPOTECTO AD Potection 2 nput S2 Potection 2 nput S2 nput Signal fo nput nput 2 S4 Depotection by Stand S4 Displacement nput nput S4 nput Signal fo S4 S2 S4 Depotection by DA S2 S4 nput polymeization nput Potection 2 nput nput TO DMESOAL DEPOTECTO AD Figue 1: Ou basic scheme of depotection fo one and two dimensional assembly. Hee we have used the standad TAM notation fo tiles and input and output pads. The oval padding ove the input and output pads denotes potetcion fom hybidization. towads this goal. Ou esults and the Oganization of the Pape: Section 1 intoduced the notion of depotection and discussed the need fo activatable tiles in computational assemblies. n Section 2, we povide abstact and kinetic models fo activatable tiles that build on infee s oiginal TAMs, with the pimay diffeence being that each tile now has an associated finite state machine. e analyze potential souces of eo in activatable TAM and compae both eo ate and gowth speed with that of the oiginal TAM. n Section 3 we obseve that since tiling assembly gowth happens at the oiginal scale of the assembly with low eo ates, activatable tiles can povide compact eo-esilience. n Section 3 we also pove that activatable tiles can povide compact self-healing by epaiing a hole of cetain size with high pobability befoe backwad assembly gowth can stat, assuming suitable values of kinetic paametes. n Section 4, we descibe the DA design of an example one dimensional activatable tile and its depotection using stand displacement and DA polymeization. n Section 5 we extend this design to the two dimensional case. n Section 6 we obseve that the applications of activatable tiles ae not limited to computational assemblies and discuss a novel concentation/sensing system based on activatable tiles. e also biefly descibe how activatable tiles can be used fo catalyzing chemical eactions. n Section 7 we conclude the pape with some open questions and futue wok. 2 The Activatable Tile Assembly Models An abstact model is a theoetical abstaction fom eality that is often easie to wok with conceptually as well as mathematically. Thus developing an abstact activatable tile assembly model will help us descibe the mechanism of tiling assembly gowth with activatable tiles as well as analyze potential souces of eo in the pocess. Since infee has aleady established the famewok fo tiling assembly models with his TAMs, we build ou abstact Activatable Tile Assembly Model (aatam) and the kinetic Activatable Tile Assembly Model (katam) discussed in this section on infee s abstact and kinetic TAMs espectively[1]. f the eade is aleady familia with the infee models we ecommend him/he to skip section 2.1, othewise ead Section 2.1 whee we pesent an oveview of infee s models. 2.1 Oveview of infee s Tile Assembly Model The oiginal abstact Tile Assembly Model (atam): The key featues of this model ae [Figue 2]: (i) The unit of assembly is a squae which is also called a tile, (ii) Each side of a tile is labeled by a bond-type, each of which associated with a stength function. Ou bond stengths ae null, weak and stong, with an associated stength of 0, 1 and 2 espectively. (iii) A tile set is a finite set of tile types. A tile type can be used an abitay numbe of times duing the self-assembly

4 Figue 2: (a)abstact Tile (b)the Siepinski Tiangle Tiling (c) atam gowth (d) A ktam gowth. pocess. (iv) Assembly begins with the seed tile. (v) A tile may be added to the assembly if it can be placed to match one o moe sides with a total bond stength geate than o equal to 2 (also called a τ = 2 system). (vi) nput sides of a tile t ae the sides of t with which the tiles initially bind in the assembly, while popagation sides (output sides) ae the sides which popagate infomation by binding to neighboing tiles. (vii) Tile additions in the atam ae non-deteministic, but, in many cases, the assembly occus in a locally deteministic assembly sequence [14]. Fo instance, the Siepinski Tiangle tile set has a locally deteministic assembly sequence. The tile set and the atam gowth of Siepinski Tiangle fomation by self-assembly is shown in Figues 2b and 2c espectively. The ule tiles coespond to the computational tiles, while the bounday tiles povide input in the two dimensions and the seed is esponsible fo nucleating the assembly. The atam povides an elegant famewok fo descibing how to pogam tiles to fom computational assemblies. The esulting assembly, howeve, is eo-fee only because pefect gowth is assumed. To model vaious types of eos, infee poposed the kinetic Tile Assembly Model (ktam) [15] which is based on the physical chemisty of DA tiles in solution. e discuss the ktam in the following paagaph. The oiginal kinetic Tile Assembly Model: ktam essentially models the self-assembly pocess as a continuous-time Makov chain whee tiles can be added to a location at a ate f (popotional to thei concentation) but tiles fall off at a ate,b detemined by the total stength b of bonds holding them to the neighbos. The model assumes that the concentations of each tile type ae held constant thoughout the whole self-assembly pocess and moeove, that the concentations of all tiles ae equal [15]. Thus if mc is the logaithm of any tile concentation in solution, then f = pe mc, p gives the time-scale constant. Futhe, if se be the unit bond fee enegy, then,b = pe bse. infee demonstated [15] that the optimal opeating envionment is when f =,2 fo a τ = 2 system. Hence,1 f and tiles adding to the assembly with a stength of 1 fall off vey quickly. A ktam gowth of Siepinski Tiangle Assembly is shown in Figue 2d. infee [15] also suggested that if the gowth of the tiling assembly is faste than the time equied to locally establish equilibium at the gowth sites, tiles (iespective of whethe they ae matched o not) will become embedded and fozen in the inteio of the aggegate with an out-of-equilibium distibution. This is known as the kinetic tapping model and povides a way to elate the net gowth ate and the eo ate ɛ. 2.2 The abstact Activatable Tile Assembly Model (aatam) n the simplest vesion of activatable tiles, the idea is to stat with a set of potected ule tiles 5 so that the tiles do not assemble until the pe-assembled initiato assembly consisting of a seed tile 6 and multiple bounday tiles 7 is intoduced in the mixtue. n the moe complex vesion, the initiato is the seed tile alone and the bounday tiles have a potection-depotection scheme simila to that of the ule 5 ule tiles ae esponsible fo computation in algoithmic self-assembly. 6 Seed tile nucleates the assembly. 7 Bounday tiles povide two dimensional input fo computation by self-assembly.

5 Figue 3: (a-i)oiginal Abstact ule Tile, (a-ii)potected Abstact Tile, (b)diffeent states associated with the activatable ule tile, (c)state Tansition Diagam fo Activatable ule Tile, (d-i)oiginal Abstact Bounday Tile along x axis, (d-ii)potected Abstact Bounday Tile along x-axis, (e)diffeent states associated with the Activatable Bounday Tile along x-axis, (f)state Tansition Diagam fo Activatable Bounday Tile along x- axis, (g-i)oiginal Abstact Bounday Tile along y axis, (g-ii)potected Abstact Bounday Tile along y-axis, (h)diffeent states associated with the Activatable Bounday Tile along y-axis, (i)state Tansition Diagam fo Activatable Bounday Tile along y-axis. ote that the oval padding on the sides of cetain tiles in the Figue is used to denote potection of tile pads fom hybidization. Sides of tiles without labels indicate absence of sticky ends. tiles. The aatam is simila to the oiginal abstact TAM (atam) due to infee [1] except that each tile type t has an associated finite state machine (FSM) M t and hence, each tile has a state. efe to Figue 3 fo the est of this discussion. The new abstact ule tile is shown in Figue 3(a-ii). Unlike the oiginal tile [Figue 3(a-i)], it has all its sides potected. The states in the FSM M t aise fom the pesence o absence of potection on the fou sides of the tile type t (as shown in Figue 3(b)). The state tansition diagam is shown in Figue 3(c). The idea is to stat with a completely potected ule tile and at the end have a tile simila to one descibed in infee s atam once its input ends ae popely bound to appopiate neighbos in a gowing assembly. A tile of type t is inet in state t until it is activated on its input pad n1 t o n2 t by the tiling assembly suface (coesponding states S2 t and t ). The bound input pad in tun activates the emaining input which can bind to an appopiate adjacent tile on the gowing face of the cystal (S4 t ). n case thee is no such neighbo available, the potection (P 1 t o P 2 t ), which is pat of the tile until the outputs ae depotected, coves the inputs again and the tile leaves the assembly ( t ). ith at most one of the input pads bound, (ecall that outputs ae not available fo binding until both input pads ae matched) thee can be at most one weak bond between the tile and the assembly. A tile in aatam abides by the tempeatue τ = 2 ule just as in atam and hence this tile dissociates. hen both inputs ae matched, the long ange effecto (LE) depotects the output pads (S4 t ). 8 8 n the moe complex vesion of activatable tile set, the seed is the initiato tile in the assembly expeiments and consequently does not have any potection on any of its sides. Bounday tiles still need to be potected [Figue 3(d-ii) and 3(g-ii)]. The coesponding states ae shown in Figues 3(e) and 3(h). Depotection is simple fo bounday tiles. Since the bonds on the east side of the x-axis bounday tiles (B x) and that on the south side of the y-axis bounday tiles (B y) ae stong bonds, matches on those pads can tigge the depotection of the noth and west pads fo B x as well as B y [Figues 3(f) and 3(i)].

6 Figue 4: Continuous time Makov Chain associated with the katam (Specifically the Kinetic Tapping Model). 2.3 The kinetic Activatable Tile Assembly Model (katam) The katam is based on infee s oiginal model ktam, but due to the the stochastic natue of the potection on all sides of the tile, additional eos need to be modeled. Theefoe we need moe fee paametes than just f and,b fo tiling assembly gowth. Figue 5 shows the diffeent states possible in the finite state machine fo the katam and Figue 4 shows the state tansition diagam. n addition to the assumptions of ktam, the main assumptions of katam ae: (i)the input potection is only evesible while the output pads ae still potected, (ii) potection is ievesible, meaning once a tile is completely depotected, it cannot etun to the stage whee evey side of the tile has a potective cove. Monomes in solution ae thus eithe entiely potected o entiely depotected. e stat with an empty gowth site(). Completely potected tiles can be added to it at a ate f, popotional to thei concentation (ecall mc is the logaithm of the concentation). This event coesponds to state S6, S8 and S2, depending on whethe the tile has 0, 1 o 2 input matches at its gowth site. n katam, tiles binding at the gowth site come in anothe flavo too: They may be completely depotected (i.e. as the tiles in the oiginal ktam). The eason fo modeling these depotected tiles is that even a tile with both inputs coectly matched can be knocked off the gowth site afte output depotection. These tiles, howeve, ae added to the gowth site, at a diffeent ate f that will be shown late to be much smalle than f. This is the tansition pobability to states 0, S7 and S5 fom. Late in this section we will discuss how we can deive f fom the fee paametes. Futhe, tiles in states S2, S6 and S8 fall off at a ate,0 since they ae completely potected and still not bound to the tiling assembly. The ate of dissociation,b fom states S7, 0 o S5 depends only on the extent of input matches (just as in oiginal ktam) and hence ae,0,,1 and,2 espectively. ith one input match, the tile in S8 (S2) tansitions to S9 () at the ate of dp (depotection) and etuns to S8 (S2) at the ate of p (potection). This tile state coesponds with monomes with one depotected input and one potected input; if the second input is also matched, then it is depotected at the ate of dp (S4). f, howeve, thee is a mismatch fo the second input, eithe the potective cove 9 falls back on the inputs at the ate p (S9 S8) o the tiles come off the gowth site at the ate,1. ote that dp and p ae fee paametes whose value depends on the expeimental situation. hen both inputs ae matched, the output pads(s5) ae depotected at the ate dp out. Just as with dp and p, dp out is a fee paamete that depends on the expeimental situation. Tiles can, howeve, fall off fom the gowth site, while in any state at a ate that depends on the extent of binding. Tiles with moe than two bindings (thee o fou) can fall off the gowth site, too, but at a consideably lowe ate of,3 o,4. Thus we do not show these tansitions in Figue 3(ight). 9 The design of these tiles ae such that the potection is pat of the nanostuctue until the outputs ae depotected.

7 Out2t Out2t Out2t Out2t Out2t Out1t O t n1t n1t n1to Out1t t n1t n1to Out1t t n1t n1to Out1t Out1t t n1t n1to t n1t n1to 0 Out4t Out3t t n1t n1t CPT with two De-potected n4t Empty owth Site(S) Coect Potected Tile(CPT) in S CPT with one De-potected nput in S Completely De-potected nput in S S2 Coect Tile(CT) S4 POSSBLE STATES TH ZEO PUT MATCH S5 POSSBLE STATES TH OE PUT MATCH Out4t Out3t t n3t n1to n4t Potected Tile with zeo input matches in S S6 Out4t Out3t t n3t n1to n4t Completely De-potected Tile with zeo input matches S7 POSSBLE STATES TH TO PUT MATCHES Out4t Out3t t n1t n1to n4t Out4t Out3t t n3t n1to Potected Tile with one input match(pomt) in S S8 Out4t Out3t t n1t n1to n4t POMT with one De-potected nput in S S9 Out4t Out3t t n3t n1to Figue 5: The diffeent states associated with the tansition diagam fo the katam. Out4t Out3t t n3t n1t Completely Depotected Tile with one input match Figue 6: Kinetic Paametes fo an example depotection system. 2.4 Kinetic Paametes fo an Example Depotection System The chain of equations in Figue 6 shows how to deive the vaious paametes fo a cetain implementation of depotection using a DA polymease enzyme (modeled as a ievesible pocess) and stand displacement technique (modeled as a evesible pocess). 10 Conside the gowth site S in an assembly A and suppose tile T 1 can bind at S. T 1 P ot, the potected vesion of T 1, aives at S at a ate f (coesponding ate constant k f ). Since all its pads ae potected, it can leave the gowth site at a ate,0 (coesponding ate constant k,0 ). ithout loss of geneality, suppose the south end input (e.g. input 1) binds, and tigges a signal making the east end input ( input 2) available fo binding and, if this hybidization is successful, the outputs become uncoveed. nput 1 has an exposed toehold, facilitating displacement of the potection stand P 1. nput 1 of T 1 P ot hybidizes with A via stand displacement of P 1, to fom AT 1 n1bnd (fowad and backwad ate constants ae k sd1 and k sd1 espectively). P 1 is now fee to displace P 2, the potection of input 2, using a toehold egion foming AT 1 n2t oeexp (fowad and backwad ate constants ae k sd2 and k sd2 espectively). Once input 2 is exposed by stand displacement, it hybidizes with A, foming AT 1 n2bnd (fowad and backwad ate constants ae k sd3 and k sd3 espectively); the complement P of the pime P, which was held in a haipin loop on the output potection stand P 3, is now made available. The pime binds to P 3, foming AT 1 nbnd P at the ate f (t can dehybidize at the ate,1 ); DA polymease enzyme E next binds at the 3 end of P foming AT 1 nbnd P E (fowad and backwad ate constants ae k poly and k poly espectively) and extends it to the output ends with a ate constant k ext. 11 n the next step when P ext P 3 dissociates, the outputs of tile T 1 in the assembly AT 1 OutExp ae exposed. Completely depotected tiles (T 1 UnP ot ) can fall off S at a ate b,2 dictated by thei concentation (ate constant k b,2 ; these tiles ae the sole souce of eos in assembly. They can cause an eo by attaching to a gowth site in anothe assembly A with a single match (A T 1 UnP ot ). e deive the ates of the diffeent eactions. k sd1, k sd2 and k sd3 ae the fowad ate constants of 10 Fo the subsequent discussion, note that the backwad ate is denoted by the negation of fowad ate. Fo instance, if k poly be the fowad ate of association of the DA polymease enzyme to the pime, k poly denotes the ate of dissociation of the polymease enzyme. 11 The polymease extension of the pime hybidized to the potection stand, is modeled as an ievesible atomic pocess fo simplicity. f the exonuclease activity duing polymeization occus at a easonably low ate, then the assumption is quite justified.

8 stand displacement while k sd1, k sd2 and k sd3 ae the coesponding backwad ate constants. n k geneal, sdi k sdi = e /kt fo i = 1, 2, 3, whee is the fee enegy change in the duplex fomation fo the toehold egion and is calculated in a simila manne as se. Futhe, Thompson et al. [18] estimated fom empiical data the aveage time taken pe base-pai migation to be of the ode of 100 µ seconds. Thus k sdi and k sdi, fo i = 1, 2, 3 can be estimated fom the coesponding toehold length. Once at least one of the inputs ae hybidized, the tile can dissociate with a ate constant k,i, whee i = 1, 2 depending on how many inputs ae cuently bound. ecall that, at this stage, we assume that T 1 etuns to its oiginal potected state (T 1 P ot ) when it falls off the assembly. This coesponds to ate,i = k f e ise. The pime hybidizes at a ate f = k f [P ] = k f e mc. e assume that the polymease enzyme initially binds with a ate constant of k poly and dissociates with k poly, but the subsequent polymeization afte binding is ievesible. The polymeization once begun occus with a ate constant of k ext. Since mathematical teatment of pe-steady state kinetics is quite difficult when exonuclease activity is included, eukayotic DA polymeization is often studied in only steady state. n geneal, let k cat and k exo be the catalytic ate of DA polymeization and exonuclease eactions, k 1 and k 1 epesent the association and dissociation ates, espectively of nucleotide binding and n be the numbe of consecutive nucleotide incopoation allowed, then using steady state kinetic analysis [17], the concentation of the tiles whose both outputs ae exposed due to the polymeization of the potection stand ([AT 1 OutExp ]) can be evaluated given the concentation of 1 the tiles with pimes bound to P 3, ([AT 1 nbnd P ]). Specifically, k ext = 1+(1+ Km )Σn 1 i=0 ( kexo ) k n i ( Km cat )n i 1 ( is the nucleotide concentation and K m = k 1+k cat k 1 ). The deivation of k ext is as follows: if D i and D i epesent the concentation of the polymeized pime, i bases long and the complex polymeized i- me with the next nucleotide to be incopoated bound in its position but not yet catalyzed espectively (both ae complexes with DA polymease), then dd i dt = k cat D i 1 +k 1D i +k exod i+1 k exo D i k 1 D i and dd i dt = k 1 D i (k 1 +k cat )D ddn i fo i = 0,..., n 1 and dt = k cat D n 1 k exod n. The last equation ensues that the polymeization stops at the end of n bases. Futhe D 1 = 0. Hence solving fo these equations in steady state whee each dd i dt and dd i dt is zeo, we can obtain the equation fo k ext The Kinetic Tapping Model n the context of the abstact tile system, the kinetic tapping model monitos a paticula gowth site. As tiles attach to the neighboing gowth sites, the tile cuently in the monitoed gowth site feezes thee pemanently at the effective gowth ate (even if it has one o moe mismatches among its fou binding sides). The kinetic tap model can be used to find the pobability that the coect tile is in the gowth site when the site feezes. n addition to the states descibed in Figue 5, the model has the sink states Fozen Coect(FC) and Fozen ncoect(f) 12 [Figue 3(ight)]. n this model, the pobability of an eoless step in the assembly is the pobability of a tile tansitioning to FC at t. e compae the gowth speed and the eo ate with that of the oiginal infee model. Since thee ae many fee paametes in the kinetic model, such as f,,b, p etc we decease the dimensionality of the paamete space by clumping some of the paametes togethe e.g. p, dp and dp out. This is done by computing the ate at which tiles become completely depotected afte eaching a gowth site, thus neglecting the intemediate states in Figue 3(ight). This coesponds to the ate at which a tile eaches state S5 if it is in. e call this ate eff and assume that eff is a function of se such that eff = k f e ( 2+ɛ 1) se whee ɛ 1 is a constant and 0 < ɛ 1 < 1. ote that eff is simila to f in the oiginal ktam. Based on the continuous time Makov Chain (CTMC) in Figue 3(ight), we can evaluate eff as eff = dp dp dp out f ( dp +,0 ) ( p+ dp +,1 ) ( dp out +,2 +. p) 12 A gowth site can only be fozen if the output pads of the tile sitting in that gowth site ae available fo binding. Hence the tansitions to FC and F ae only fom S5, S7 and 0 and not fom S6 o S9.

9 Based on the continuous time Makov Chain (CTMC) in Figue 3(ight), we can evaluate eff as eff = dp dp dp out f ( dp +,0 ) ( p+ dp +,1 ) ( dp out +,2 + p). This fomulation is deived as follows: the ate of moving fom to is dp f ( dp +,0 ). Similaly, the ate at which tiles tansition fom to S4, given the ate of tansfe fom to is dp dp f ( dp +,0 ) ( p+ dp +,1 ). Similaly one can obtain the ate of tansitioning fom S4 to S5 given the effective ate of tansition fom to S4. One pimay assumption in katam is that,1 > f > eff >,2. Futhe, we assume,1 = e se,,2 = e 2se, eff = e ( 2+ɛ 1) se, f = e ( 2+ɛ 1+ɛ 2 ) se fo 0 < ɛ 1, ɛ 2 < 1. Fo simplicity of the model, we can ensue that ɛ 2 ɛ 1 by adjusting the kinetic paametes in the depotection system (e.g. toehold length in the stand displacement events, nucleotide concentation and template length fo polymeization etc). Hence eff,2. Anothe impotant assumption we make is that DA polymeization has been modeled as ievesible and, hence, at equilibium evey tile is completely depotected. Based on these assumptions we conclude the following claim: Claim 2.1 ith espect to the oiginal ktam, the eo ate in katam can be significantly deceased ɛ without a consideable decease in the speed of the gowth of the assembly since ɛ old = e ɛ 1 se and > e ɛ 2 se. old Poof Since we epesent the state of a gowth site in Figue 4, thee is a tansition fom state S5 to state when the tile afte output depotection leaves the gowth site. The state of the tile howeve, has changed as indicated in Figue 6 (fom T 1 P ot to T 1 UnP ot ). Thus, the tansition fom S4 to S5 is ievesible. This implies that at t (steady state), all potected monomes will pass though state S5. Hence e mc is an uppe bound on the faction of tiles in state S5 at t. 13 Based on the CTMC in Figue 4, the pobability of a tile leaving state S5 is whee, the effective eff + f,2 +,2 gowth ate, is given by = eff + f,2. Hence, the expected potion of completely depotected tiles that leaves S5 is,2 +,2 e mc, by lineaity of expectation. ecall that the ate of tile addition is solely dependent on the concentation of the tiles. Hence the ate at which a completely depotected tile binds to a gowth site is f = k,2 f +,2 e mc. Obseve that we have f =,2 f <,2 f eff since f > 0. Fo simplicity, we will use f,2 eff as an estimate of f since the bound can be made tight using the assumption ɛ 2 ɛ Thus f = e( 2+ɛ 2) se. e now analyze the pobability of an eoless assembly step based on the kinetic tapping model in Figue 3(ight). Let p i (t) be the pobability that i is the state t seconds afte the gowth site has appeaed, assuming the site has not yet been fozen. Thus if we stat with an unit concentation of tile in, it accumulates diffeentially in F C and F. Based on the CTMC in Figue 3(ight), we compute the pobability of an eoless step fo the example assembly of Siepinski Tiangle Patten geneation (Figue 2) as p F C ( ) = eff + f +,2 1 eff + f + 2 f +,2 + 4 f +,1 +,0 1+2( f f + eff, since (i) thee )( +,2 +,1 ) is only one coect tile fo any gowth site (this tile ends up in FC), two tiles with one binding site match and the emaining fou tile types have no matching binding sites (these tiles end up in F) and (ii),1 > f > eff >,2. Hence eo ate is ɛ = 1 p F C ( ) 2( f f + eff )( +,2 +,1 ) 13 This is a loose uppe bound because at t, most gowth sites ae fozen and the tiles in those gowth sites cannot leave and contibute to the expected numbe of tiles leaving S5. The concentation of completely depotected tiles, howeve, is maximum at steady state. 14 ɛ 2 ɛ 1 implies eff,2. Thus out of eff tiles eaching state S5 in unit time, only,2 leave. Only these tiles which leave the gowth site afte complete depotection can come back to any gowth site with a ate constant of k f. Hence if eff,2 then f should be much less than eff and we can safely neglect the contibution fom f in the denominato +,2 while computing the value of f.

10 fo small ɛ. Simplifying, ɛ = 2 e (ɛ 1 ɛ 2 )se 1+e (ɛ 1 ɛ 2 )se e (1 ɛ 1)se (1+e (ɛ 1 ɛ 2)se ) 1+e (1 ɛ 1 )se +e (1 ɛ 2 )se e se 2e (1 ɛ 2) se, neglecting e se (e ɛ 1 se + e ɛ 2 se 1). ecall that eo in infee s model is ɛ old = 2e mc se = 2e (1 ɛ 1 ɛ 2 ) se [15]. Thus ɛ ɛ old = e ɛ 1 se. The gowth speed in katam, = k f e ( 2+ɛ 1) se (1 + e (ɛ 1 ɛ 2 ) se e ɛ 1 se ) and gowth speed in ktam, old = f,2 = k f e ( 2+ɛ 1+ɛ 2 ) se (1 e (ɛ 1+ɛ 2 ) se ). Thus we have old = e ɛ 2 se (1+e (ɛ 1 ɛ 2 )se e ɛ 1 se ) (1 e (ɛ 1 +ɛ 2 )se ) > e ɛ 2 se. Since ɛ 2 ɛ 1, we conclude that the assembly eo ate can be significantly deceased with slight decease in the gowth speed. ote that thee is a lot of slack in ou analysis. Futhe, since thee ae multiple fee paametes in addition to mc and se in katam, the exact coelation between eo ate and gowth speed is still an open question. 3 Compact Poofeading with Activatable Tiles Activatable tiles povide eo-esilience to a gowing assembly by enfocing diectional gowth. deally the output ends ae neve available until the coesponding input ends ae completely hybidized, thus peventing both eos by insufficient attachment as well as nucleation eos. Thee is a small pobability, howeve, of eos by insufficient attachment caused by tiles that leave a gowth site afte output depotection. Futhemoe, the computation still occus at the oiginal scale, unlike Chen s snaked poofeading technique [7] which inceases the lattice size by a multiplicative facto of k 2. Hence, activatable tiles indeed povide compact eo-esilience. Since the seed is the only completely unpotected tile when the assembly begins and the concentation of completely unpotected ule o bounday tiles existing in solution at any given time is vey low, activatable tiles can also pevent spontaneous nucleation and enfoce contolled gowth. 15 e can fomally pove that activatable tiles ae indeed an instance of compact poofeading. Soloveichik et al. [14] gave a concise definition of compact poofeading in and we adapt it to ou ATAM: Definition iven a small constant 0 < q < 1, a sequence of deteministic tile systems {T 1, T 2, T 3,...} is a compact poofeading scheme fo patten P if (i) T poduces the full infinite patten P unde the aatam, (ii) T has poly(log ) tile types (poly(n) denotes n O(1) ) and (iii) T poduces the coect initial potion of the patten P with pobability at least q in time O(poly(log)) in the katam fo some value of the fee paametes in the model. Theoem 3.1 An activatable Tile System A is a compact poofeading scheme. Poof Let the tile system in atam be T and the activatable tile system be A. A is same as T except that each tile type has an associated finite state automata. Since in aatam activatable tiles can bind to a gowth site only if they can bind stongly enough (just as in atam), A can poduce the whole system coectly unde aatam so the fist condition is satisfied. Moeove, A = T, the only diffeence being that we stat the assembly with potected vesion of T. Since this wok is concened with only deteministic tile systems, the agument of Soloveichik et al. [14] applies and we need only constant numbe of tile types so long the tile set has a locally deteministic assembly sequence. The agument fo the thid condition is simila to that of Chen et al. [7]. n this model, eos ae only caused by insufficient attachments; these eos ae caused by tiles dissociating fom gowth sites afte thei output potection has been stipped off. n an insufficient attachment event, fist an unpotected monome (with a single binding site match) attaches at the ate of f. Howeve, befoe this tile is knocked off at the ate of,1, a second tile (potected/unpotected) can attach to the fist tile at the ate f + eff. Thus, based on the coesponding CTMC [Figue 7] we can say that the ate of an insufficient attachment is insuf = f ( f + eff ) = e ( 3+ɛ 1+ɛ 2 ) se 1+e (ɛ 1 ɛ 2 )se,1 + f + eff 1+e (1 ɛ 1 )se +e (1 ɛ 2 )se, since an insufficient attachment happens as soon as the gowth site tansitions to eithe state 2 o Contolled gowth is defined to be the gowth occuing fo paamete values in a cetain pat of the kinetic paamete space, such that (i) gowth does occu, (ii) eos ae ae and (iii) gowth not seeded by the seed tile is ae [15].

11 Figue 7: Continuous Time Makov Chain fo eo due to insufficient attachment in katam Figue 8: (Left) (Fom top to bottom) Abstact 1D Tile, Unpotected Abstact Tile, Potected Abstact Tile, (ight) 2D oiginal and potected abstact tiles Ou goal with espect to a paticula gowth site is to buy the coect tile k levels deep befoe an insufficient attachment event occus. 16 n othe wods, if we have a k k squae whose left bottom cone location is occupied by this tile, then the k k squae completes befoe an insufficient attachment event occus. This puts the tile unde consideation into a k-fozen state. The pocess of tile attaching o detaching in a 2D assembly can be modeled as a andom walk. 17 ote that the fowad gowth (tile association at the output ends of the cuent tile) happens at the ate of eff + f while the backwad gowth (dissociation of the cuent tile) has a ate of,2. Thus, the aveage ate of gowth (the mean of fowad and backwad ates) is 1 2 ( eff + f +,2) and the expected time taken fo this k k squae to gow is O( k4 ) since in a 2D andom walk, we have to take k4 steps in expectation in ode to cove k 2 locations. Thus, fo any small ɛ insuf, we can find a constant c insuf such that, with pobability 1 ɛ insuf, no insufficient attachment happens at this specific location but a coect tile becomes k-fozen within time O( k4 k4 ). n othe wods, gowth site is filled coectly and buied k levels deep in O( k4 < c insuf insuf. Hence, fo a given k, such that with high pobability a given ) time. Fo constant kinetic paametes and k, this time is also constant. Hence we can use the same agument as Adleman et al. [19] that the squae is completed in expected O() time. Compact Self-healing with Activatable Tiles: The impact of activatable tiles goes beyond the compact eo-esilience which is a pimay concen fo fault toleant self-assembly. n case of goss extenal damage, e.g. a hole is ceated in a gowing tiling assembly, activatable tiles can epai the 16 The time taken fo single tile attachment is O( 1 1 eff ) which is less than insuf. 17 The stochastic pocess of tile attachment and detachment in self -assembly has often been modeled as a andom walk [7]. Futhe this is simila to the lattice gas model whee modeling inteactions as andom walks is quite well established.

12 Figue 9: (a) 1D Abstact Tile System fo testing sequential assembly. n the potected system, unlike the unpotected countepat whee none of the tiles have any potection, 1, 2 and O have potection layes, (b) The desied assembly with initiato tile and S2 espectively, (c) Possible eoneous assemblies if the assembly is not foced to go fom the initiato tile to the output tile. damage with minimal eo by enfocing diectional gowth. Since the oiginal, self-assembled lattice was fomed by algoithmic accetion in the fowad diection, only fowad egowth is capable of ebuilding the coect stuctue. The potected monomes in the solution ensue a fowad diectional accetion. Thee is a small pobability, howeve, of backwad gowth fom the unpotected monomes that wee once pat of the oiginal tiling assembly and dissociated afte outputs ae depotected. The likelihood is compaatively small since the fowad eaction ate depends on concentation of the monomes and the potected tiles ae much moe abundant than thei unpotected countepats. Theoem 3.2 A damaged hole of size S (whee S is small compaed to the size of the desied patten) is epaied befoe backwad gowth can occu in the katam with high pobability in time O(S 2 ) fo appopiately set values of the fee paametes in the model. Poof Sketch Obseve that the maximum ate of eo due to backwad gowth is bounded by f while the fowad ate of gowth is eff + f. e will now show how to estimate the value of se equied to epai a hole of size S, whee size is defined to be the numbe of tiles. Obseve that > f. Using the same technique as in Theoem 3.1, the hole can be epaied in O( S2 ) by a 2D andom walk on the set of S tile positions on the 2D plane. ext, we need to guaantee no backwad gowth happens duing this inteval. e can ague that fo any small ɛ heal (0 < ɛ heal < 1), we can find a constant c heal such that with pobability 1 ɛ heal, S2. Fo a given S, we can compute se so that thee is no < c heal f backwad gowth when a hole of size at most S gets epaied in O(S 2 ) time assuming constant kinetic paametes. 4 DA Design of One Dimensional Activatable Tiles The DA design of one dimensional (1D) activatable tiles is vey helpful in undestanding the moe complex DA design of two dimensional (2D) activatable tiles. t is also motivated by the need fo a potection stategy fo tiles that self-assemble into a 1D lattice, such as the bounday of the computational tiling. Hence we fist descibe the DA design of 1D activatable tile. To ease undestanding, we make use of thee levels of abstaction to descibe the DA tile design: at the highest level of abstaction, we descibe the depotection stategy using a finite state machine; in the next level, we explain the same mechanism with an example; in the lowest level, the design desciption involves actual DA sequences. The 1D abstact tile is a squae with a single input and single output. Evey tile has a tile coe (which is common to all tiles) and a unique pai of sticky ends encoding the input and the output espectively [Figue 8(Left)]. An unpotected tile is the same as the oiginal tile while its

13 Figue 10: A Finite State Automata fo the depotection of the output sticky end of a 1D tile only when its input sticky end is coectly matched. Figue 11: A high-level eaction pathway fo depotection of a potected tile. potected countepat has a laye coveing its sticky ends. The toehold, an exposed pat of the input sticky end, facilitates stand displacement and consequently depotection [Figue 8(Left)]. Conside a simple example system compising of five tiles. Thee ae two stat tiles, and S2, two intemediate tiles, 1 and 2 and one output tile O [Figue 9]. deally in the pesence of, the desied assembly ode is 1 O, while in the pesence of S2, the desied assembly ode is S2 2 O [Figue 9]. n the absence of potection, howeve, such diectional gowth cannot be guaanteed [Figue 9]. The goal is to pevent eoneous gowth by stating with potected tiles and depotecting them only afte they have been attached to the gowing assembly. The following subsections discuss how such sequential assembly is ensued. 18 High Level Desciption of the Design: The key idea is fist pesented in the fom of a finite state machine [Figue 10(Left)]. Since the input/output sticky ends ae potected by a potection stand, an inactive set of tiles co-exist in the solution and do not self-assemble into one dimensional lattices (). hen an the initiato stand (which may be pat of a lage nanostuctue) is intoduced in the mixtue, the fome displaces the potection stand at the input end of the tile with matching sticky ends at any gowth site, esulting in a patially active tile with a coectly bound input sticky end and potected output sticky end (). n pesence of a suitable pime, the potection stand (now fee at the input end but still hybidized to the output sticky end) can act as the template fo DA polymeization (S4-S6). hen the polymeization completes, the potection stand is stipped off the output sticky end leaving it to initiate anothe depotection. Thus the assembly poceeds fom input to output end at all times ( active state (S7)). 18 Eos in 1D and 2D ae vey diffeent. Single tile mismatches, a majo souce of eo in 2D tile assemblies, do not happen in 1D systems. nstead, eos ae intoduced when assembly does not occu in the desied fashion due to ambiguous binding sites as shown in Figue 9.

14 Stage 0 Tile 1(Potected) M A P B E S2 A B H Stage 1 A B H A B M P E A B H S2 A B H Tile 2(Unpotected) Toehold hybidization Stage 2 M S2 A A P B B H A B H E Hybidization of sticky ends by displacement of the potection stand Stage 3 M S2 P Pime binding to now available template (potection stand) P A B E P A B H A H B Stage 4 Polymease Pime polymeization and gadual de-potection of output sticky end due to the stipping of the template stand M A P B E M A P A B H S2 A B H Stage 5 M M A S2 A B H Exposed output sticky end A Complete polymeization of the pime and dehybidization of P potection stand fom the output sticky end B E P A B H Figue 12: eaction Pathway fo depotection of Tile 1 by Tile 2 at the expeimental level. e futhe explain the depotection stategy using the example system fom above; Figue 11 shows the coesponding eaction pathway. nitially, all the tiles (1, 2 and O) in the solution ae potected (Stage 0) and they do not inteact until the stat tile is intoduced. next displaces the potection laye at the input end of the intemediate tile 1 (Stage 1). Once the potection laye at the input end of 1 is feed, the pime in the solution can hybidize with it and DA polymease enzyme can extend it all the way up to the output end, thus exposing the output sticky end (Stage 2 and 3). Tile O is depotected and hybidized to 1 in the next cycle. Expeimental Design: The simplest DA sequence designs ae shown in Figue 13. The tile coe can be simply a double-standed DA with the sticky ends being the single standed ovehang extending out of the double standed potion. The sticky ends ae potected by the potection stand M. Fo adjacent tiles, the potection stand needs to be aanged in a diffeent manne so as to satisfy both constaints on the diection fo sticky end matching as well as template fo polymeization [Figue 13], esulting in two kinds of tile types. e exclude the textual desciption of the detailed design due

15 Pime(P) # of bases depends on the width of the tile coe Template fo extension by Polymease ((M)[Potection Stand] Template T=B +P ++A +M++E ( to ) Quenche(DABCYL) Fluoophoe(FAM) Complement of the Pime in Solution(P ) Toehold S2A S2B H2 A B H1 Sticky End 2(S2) Sticky End 1() 6 3 E Pime(P) # of bases depends on the width of the tile coe and length of sticky ends Template fo extension by Polymease ((M)[Potection Stand] Template T=E+A +P ++B +M+ ( to ) Quenche(DABCYL) Fluoophoe(FAM) Complement of the Pime in Solution(P ) Toehold S2A S2B H2 A B H1 Sticky End 2(S2) Sticky End 1() 3 E 6 Figue 13: ((Top)Potection Stategy fo Tile 1, (Bottom)Potection Stategy fo its neighbo Tile 2. numbe of bases fo each section is shown in pink. The Figue 14: (a)ule Abstact DX Tiles fo XO computation, (b)desied Tiling Assembly stating with the input stand (c)eoneous owth when diectionality constaint is not enfoced. to space constaints. Some of the key featues ae (i) The 3 base potion (E) at the 3 end of the potection stand pevents polymeization of the toehold H1, (ii) The potion of the potection stand which hybidizes to the pime P is held tightly in a haipin loop of six bases between two subpotions of the input sticky end, (iii) The fluoophoe and the quenche fo detection puposes, ae positioned such that the flouophoe is quenched only when coect tiles hybidize. ow suppose, Tile 2 is aleady depotected and pat of an assembly. How is Tile 1 depotected by Tile 2? Figue 12 descibes a eaction pathway. One can veify whethe the tile system is assembling as desied by obseving the patten in the fluoescence data. ative gel electophoesis can be used to find whethe the dominant assembly in pesence of the initiato tile is the one desied. 5 DA Design of Two Dimensional Activatable Tiles Ou DA design fo 2D activatable tiles is a diect extension of ou 1D activatable tiles. Befoe giving a sequence design fo activatable tiles, just as we did in the 1D desciption, we fist descibe the potection/depotection stategy using an abstact vesion of DX tiles [11]. The 2D abstact DX

16 Figue 15: High Level eaction pathway fo depotection in two dimensions tile is a squae with two inputs and two outputs; evey tile has a tile coe (common to all tiles) and unique sticky ends encoding the input and the output. [Figue 8(ight)]. An unpotected tile is the same as the oiginal tile wheeas its potected countepat has a laye coveing its sticky ends. One of the tile inputs (input 1) has a few bases exposed at the end, facilitating stand displacement and, consequently, depotection [Figue 8(ight)]. This exposed potion is efeed to as a toehold. The othe input (input 2), howeve, is completely hybidized to a potection stand which is sepaate fom the output potection stand. The idea is that the input 2 sticky end cannot hybidize until the input 1 is coectly hybidized. The toehold of input 2 is exposed only when its potection stand is displaced by the potection stand fom input 1. hen input 2 hybidizes completely, it fees the potection stand coveing the outputs fom the input end. DA polymease enzyme then exposes the output sticky ends. Example computational system in 2D: XO computation Fo this computation, the output is 1 only when exactly one of the inputs is 1. Figue 14a shows DX ule tiles implementing the computation and Figue 14b shows an example gowth. n absence of potection, howeve, such diectional gowth of tiling assembly cannot be guaanteed. An instance of an eoneous system is shown in Figue 14c. The goal is to enfoce a sequential assembly to avoid ambiguities shown in Figue 14c by using the novel potection-depotection scheme. High Level Desciption of the Design: e descibe a high level vesion of depotection using XO computation as an example system (Figue 15). Assuming tiles T 2 and T 3 ae pat of a gowing tiling assembly, ideally anothe T 3 should bind, at thei output ends. This esults in the output sticky end 0b of bound tile T 3 (pat of a tiling assembly) displacing the potection stand ove input sticky end 0b of the potected monome T 3. The potection stand fom the input 0b of tile T 3 next displaces the toehold potection of the 0a input fo tile T 3. n the following step the output sticky end 0a of bound tile T 2 displaces the potection on 0a input of potected tile T 3. Once the potection laye at this input end is feed, the pime in the solution can hybidize with it and DA polymease enzyme

17 Figue 16: (Details of a potected DA tile fo two dimensional tiling (umbe of bases in each stand is shown in pink.). can extend it to the output ends, thus exposing the output sticky ends one by one (output 2 followed by output 1). Expeimental Design: A sequence design of such an activatable tile is not difficult. Figue 16(ight) shows the details of the potection stategy fo an expeimental validation. Hee, one can use a DX o a TX tile [10, 11] as the tile coe, since they ae vey compact; a compact tile will impove the likelihood of the stand displacement and polymeization events that ultimately depotect the tile and make it available fo hybidization. Figue 16(ight) also gives the potection stategy fo the input and output sticky ends. The toehold of input 1 hybidizes with the coect sticky end (pat of a lage nanostuctue) and eventually displaces the input 1 potecto. The Y potion of the input 1 potecto stand then hybidizes with the input 2 toehold potecto. Eventually, the input 2 toehold Hn2 is exposed by stand displacement, thus facilitating anothe stand displacement by the output sticky end (pat of a lage nanostuctue) that binds with input 2 sticky end. Once the potection stand 2 is feed fom the input side, it hybidizes with the pime P. ext, DA polymease enzyme pesent in the solution extends the pime and eventually pulls the potection stand(template) fist out of output 2, SOut2, and next out of output 1, SOut1. 6 Othe Applications of Activatable Tiles Beyond thei applications to computational tiling, activatable tiles can be used fo building sensing and concentation systems [Figue 17]. Fo instance, a type of modified activatable tile that has a docking site (e.g. a DA o A aptame binding site) specific to this taget molecule can be designed. nitially, the tiles ae in the inactive state; neithe they ae bound to a taget molecule no they ae assembled togethe. hen a taget molecule binds to the tile s docking site, the tile tansitions fom an inactive to an active state. Tiles in the active state can assemble. As the activated tiles assemble, the taget molecules ae concentated making an excellent concentation system. Fo added functionality, one can attach metallic nanopaticles (P) to the tiles o taget molecules. ith the P, the assembly of activatable tiles detects the pesence of a taget molecule in solution (based on the coloimetic output) and behaves as a nano-scale senso. Activatable tiles can also be used fo eaction catalyzation. Suppose that fo some small k, the goal is to place in close poximity, k distinct types of small, taget molecules to initiate o catalyze a chemical eaction. k distinct modified activatable tiles can be designed with a docking location that povides a binding site fo one of the distinct taget molecules. The tiles undego a state tansition fom inactive to active only when they ae caying thei taget molecules. Once activated, these k distinct tiles assemble into a small tiling lattice, putting the taget molecules in close poximity, and allowing them to eact. n addition, some of the eaction poducts can be used to make the tiles disassemble and etun to the inactive state, allowing the tiles to be eused. Obseve that the location of the binding

18 Concentation System (based on the assembly of the Activated Tiles) Sensing System based on Coloimetic due to the aggegation of nanopaticles Tile activated by Taget Molecule aget lecule cule ite S2 Metalic anopaticle Figue 17: A concentation and sensing system with activatable tiles. Potected Sticky End Potected Sticky End Potected Sticky End S2 S4 S4 Taget Molecule Potected M1 s Taget Molecule Taget Molecule Taget Molecule Sticky Docking Potected M2 s Docking Site Ms Docking Site M4 s Docking Site End Site D1 Sticky End D2 D3 D4 Potected Tile 1 Potected Tile 2 Potected Tile 3 Potected Tile 4 Potected Sticky End Poduct S2 Potected Tile 1 M1 Potected Tile 2 S4 Potected Tile 3 S4 M2 M3 M4 Poduct eaction catalyzed by Activatble Tiles Potected Tile 4 M1 M2 M3 M4 S2 M1 M2 Tile 1 Activated by M1 Tile 2 Activated by M2 M3 S4 Tile 3 Activated by M3 M4 S4 Tile 4 Activated by M4 M2 M3 M1 M4 Figue 18: eaction catalyzation with activatable tiles. site has a majo ole to play in this catalyzation pocess. The binding site on the same face of each tile type is so designed that afte assembly, the molecules, bound to the tiles will be close to each othe. They ae neve bound inside the lattice and theefoe, the eaction can neve become slowe. Figue 18 shows such a eaction catalyzation fo k = 4. eaction catalyzation is quite an established subfield of chemisty whee eithe the catalyst eithe holds the eactant molecules in close poximity to each othe and thus inceases the eaction ate (heteogeneous catalyst) o it can eact with the eactants to fom the poducts and is eventually eleased fom the poducts (homogeneous catalyst). An example of a heteogeneous catalyst is finely divided ion in the Habe pocess of manufactuing ammonia while that of a homogeneous catalyst is chloine fee adicals in the beakdown of ozone. Obseve that in ou case, activatable tiles behave both as homogeneous and heteogeneous catalysts since not only they paticipate in the eaction, but also the eaction takes place on thei sufaces. Although this is quite a novel idea, the concept of DA diected chemisty has been exploed quite extensively in the ecent yeas (See [23]). 7 Conclusion The eduction of eos in computational tiling assemblies eliminates a majo oadblock in the development of applications of pattened DA lattices, allowing, fo example, the constuction of complex nano-electonic cicuits. n this pape, we have descibed a novel tile design which uses stand displacement and DA polymeization to impove the obustness in computational assembly without inceasing the scale of the assembly. One of the key featues of ou design is that although depotection enfoces sequentiality, the massive paallel advantage of self-assembly continues to be thee. e developed abstact and kinetic models fo activatable tiles that allow us to compae eo ates and gowth speed with that of infee s oiginal kinetic model. e showed that activatable tiles can povide obust assembly gowth in the same scale as the oiginal assembly and can even epai small amounts of damage assuming suitable values of the model s kinetic paametes. These esults show that an activatable tile set is indeed a compact eo-esilient and self-healing tile set. e futhe descibed a DA design fo activatable tiles based on these models. Additionally we obseved that