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Ashish Goel, ashishg@stanford.edu 1 A simple analysis Suppose complementary DNA strands of length U always stick, and those of length L never stick (eg: U=20, L=10) Suppose you have n randomly chosen DNA strands of length U Probability that there will be some undesired interaction of length at least L · U 2 n 2 4 -L Basis of Adlemans construction and also DNA tiles

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Ashish Goel, ashishg@stanford.edu 2 Tile Systems and Program Size A tile system that uniquely assembles into a shape is analogous to a program for computing the shape Program size: The number of tile types used Goal: Find smallest program size for interesting classes of shapes [Adleman 99][Rothemund and Winfree 00] Can often derive lower bounds on program size Need n distinct tile types to assemble a line of length n (pumping lemma) Need Ω io (log n/log log n) distinct tile types to construct squares of side n (Kolmogorov complexity) Also need a notion of assembly time or running time

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Ashish Goel, ashishg@stanford.edu 3 Tile Systems and Assembly Time d Seed A BC = 2

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Ashish Goel, ashishg@stanford.edu 4 Tile Systems and Assembly Time d Seed A B C A B A B A: 50%, B:30%, C: 20% 0.5 0.3 0.2 0.3 0.5 Assembly time = Average time to go from seed to terminal state

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Ashish Goel, ashishg@stanford.edu 5 Example: Assembling Lines Require n tiles to assemble line of length n Eg. suppose tiles B and F are the same Could pump the tiles BCDE to get infinite line Therefore program size is n For fastest assembly, all tiles must have the same concentration (1/n) Expected assembly time is n 2 Can assemble thicker rectangles (2 £ n, log n £ n, n £ n etc.) faster and with less tile types! BCDE FGHIA BCDEBCDEBCDE

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Ashish Goel, ashishg@stanford.edu 6 Self-assembling Squares Would like to assemble n x n squares as fast as possible and with as few tile types Kolmogorov lower bound of Ω io (log n/log log n) on the program size (i.e. number of different tiles) Assembly time has to be Ω(n) in accretion model Motivation: Simple canonical problem, leads to useful general techniques Also interesting in its own right (e.g. assembling computer memories)

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Ashish Goel, ashishg@stanford.edu 7 Rothemund-Winfree construction Basic Idea: Easy to extend a self-assembled rectangle into a self-assembled square =2 Inefficient to start with a 1 £ n rectangle –Need (n) tile types, (n 2 ) time for lines –Solution: Use counters to make rectangles B C A x z y x z z w w w y

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Ashish Goel, ashishg@stanford.edu 8 Assembly Time Analysis Technique Seed A B C 2 BC A S 12 1 Equivalent Acyclic Subgraph Terminal assembly BC A S 1 2 2 22 1 1 1 Graph G: a vertex for each tile in final assembly, directed edges between adjacent vertices

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Ashish Goel, ashishg@stanford.edu 9 Assembly Time Analysis Technique Let L be the length of the longest path in an equivalent acyclic subgraph of G Often, much smaller than longest path in Markov Chain Let C m be the smallest concentration C i Theorem: The assembly time of the given tile system is O(L/C m ) with high probability Find the right equivalent acyclic subgraph No Markov chains anymore, just combinatorial analysis Analysis technique is tight: there always exists an EAG which gives a tight bound if C i s are identical

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Ashish Goel, ashishg@stanford.edu 10 Proof for identical C i s t i,j : exponential random variable with mean 1/C L = length of longest path in equivalent acyclic graph Number of paths · 3 L For any path P in graph, let t P = i,j 2 P t i,j Assembly-time · sd max P t P E[t P ] · L/C; also sharp concentration (Chernoff bounds) ) Expected assembly time = O(L/C) t i,j = time for tile to attach at position (i,j) after it becomes attachable

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Ashish Goel, ashishg@stanford.edu 11 Squares via counters Self-assemble a log n £ n rectangle by simulating a log n bit counter Start with a seed row that has log n tiles, labeled 0/1 Define an increment operation which adds one to the row Crucial step: assembly must stop when all the tiles in a row are labeled 1 Requires (log n) tile-types to make the seed and (1) other tiles Can be improved to the Kolmogorov optimum of (log n/log log n) Then, use triangulation on one side of the counter to covert the counter into a square

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Ashish Goel, ashishg@stanford.edu 12 Counter tiles

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Ashish Goel, ashishg@stanford.edu 13 Counting example From Cheng, Moisset, Goel; Optimal self-assembly of counters at temperature 2 (basic ideas developed over many papers)

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Ashish Goel, ashishg@stanford.edu 14 Equivalent Acyclic Subgraph for Counters 1111110 ……….. 0000000 1111111 ……….. 0000000 00000000 11111111 01111111 Only time a directed edge is needed from left column 00000000 L(n) = 2 L(n/2) + (log n) = (n) T(n) = L(n)/C m = (n)

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Ashish Goel, ashishg@stanford.edu 15 Constructing squares and Counters Can extend this basic idea to Assemble n £ n squares in time O(n), using O(log n/log log n) tiles (provably optimum) Count optimally in binary using the same assembly time and program size General design and analysis techniques A library of subroutines (counting, base-conversion, triangulating the line) If this is so efficient, why didnt nature learn to count? Possible conjecture: Not evolvable, not robust Open problems: general analysis techniques for assembly time in reversible models?

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