Probabilistic Analysis of Pattern Formation in Monotonic Self-Assembly

Inspired by biological systems, self-assembly aims to construct complex structures. It functions through piece-wise, local interactions among component parts and has the potential to produce novel materials and devices at the nanoscale. Algorithmic self-assembly models the product of self-assembly as the output of some computational process, and attempts to control the process of assembly algorithmically. Though providing fundamental insights, these computational models have yet to fully account for the randomness that is inherent in experimental realizations, which tend to be based on trial and error methods. In order to develop a method of analysis that addresses experimental parameters, such as error and yield, this work focuses on the capability of assembly systems to produce a pre-determined set of target patterns, either accurately or perhaps only approximately. Self-assembly systems that assemble patterns that are similar to the targets in a significant percentage are “strong” assemblers. In addition, assemblers should predominantly produce target patterns, with a small percentage of errors or junk. These definitions approximate notions of yield and purity in chemistry and manufacturing. By combining these definitions, a criterion for efficient assembly is developed that can be used to compare the ability of different assembly systems to produce a given target set. Efficiency is a composite measure of the accuracy and purity of an assembler. Typical examples in algorithmic assembly are assessed in the context of these metrics. In addition to validating the method, they also provide some insight that might be used to guide experimentation. Finally, some general results are established that, for efficient assembly, imply that every target pattern is guaranteed to be assembled with a minimum common positive probability, regardless of its size, and that a trichotomy exists to characterize the global behavior of typical efficient, monotonic self-assembly systems in the literature.

[1]  Jack H. Lutz,et al.  The Tile Assembly Model is Intrinsically Universal , 2011, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[2]  Max H. Garzon,et al.  Models of massive parallelism: analysis of cellular automata and neural networks , 1995 .

[3]  Grigory Tikhomirov,et al.  DNA-based programming of quantum dot valency, self-assembly and luminescence. , 2011, Nature nanotechnology.

[4]  D. Stefanovic,et al.  Training a molecular automaton to play a game. , 2010, Nature nanotechnology.

[5]  David Doty,et al.  Randomized Self-Assembly for Exact Shapes , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[6]  P. Rothemund.  Folding DNA to create nanoscale shapes and patterns , 2006, Nature.

[7]  Sung Yong Park,et al.  DNA-programmable nanoparticle crystallization , 2008, Nature.

[8]  Pierre-Etienne Meunier,et al.  Non-cooperative Algorithms in Self-assembly , 2014, UCNC.

[9]  Matthew J. Patitz,et al.  Intrinsic universality in tile self-assembly requires cooperation , 2013, SODA.

[10]  S. Glotzer,et al.  Self-Assembly of Patchy Particles. , 2004, Nano letters.

[11]  N. Seeman,et al.  Design and self-assembly of two-dimensional DNA crystals , 1998, Nature.

[12]  Leslie G. Valiant,et al.  A theory of the learnable , 1984, STOC '84.

[13]  Erik D. Demaine,et al.  Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM , 2013, STACS.

[14]  Scott M. Summers.  Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming , 2011, Algorithmica.

[15]  Matthew J. Patitz,et al.  Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue , 2011, DNA.

[16]  Erik Winfree,et al.  The program-size complexity of self-assembled squares (extended abstract) , 2000, STOC '00.

[17]  Erik Winfree,et al.  Proofreading Tile Sets: Error Correction for Algorithmic Self-Assembly , 2003, DNA.

[18]  Erik Winfree,et al.  Self-assembly of carbon nanotubes into two-dimensional geometries using DNA origami templates. , 2010, Nature nanotechnology.

[19]  D. Ingber,et al.  Self-assembly of three-dimensional prestressed tensegrity structures from DNA , 2010 .

[20]  Jack H. Lutz,et al.  Strict self-assembly of discrete Sierpinski triangles , 2007, Theor. Comput. Sci..

[21]  D. Y. Zhang,et al.  Engineering Entropy-Driven Reactions and Networks Catalyzed by DNA , 2007, Science.

[22]  Matthew J. Patitz,et al.  Asynchronous Signal Passing for Tile Self-assembly: Fuel Efficient Computation and Efficient Assembly of Shapes , 2013, UCNC.

[23]  Eytan Domany,et al.  Equivalence of Cellular Automata to Ising Models and Directed Percolation , 1984 .

[24]  Shawn M. Douglas,et al.  Self-assembly of DNA into nanoscale three-dimensional shapes , 2009, Nature.

[25]  Robert T. Schweller,et al.  Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D , 2009, SODA '11.

[26]  John H. Reif,et al.  Tile Complexity of Linear Assemblies , 2012, SIAM J. Comput..

[27]  Stephen Wolfram,et al.  Theory and Applications of Cellular Automata , 1986 .

[28]  Ashish Goel,et al.  Running time and program size for self-assembled squares , 2001, STOC '01.

[29]  Matthew J. Patitz,et al.  Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[30]  Jehoshua Bruck,et al.  Neural network computation with DNA strand displacement cascades , 2011, Nature.

[31]  Hanqiong Hu,et al.  Directed self-assembly of block copolymers: a tutorial review of strategies for enabling nanotechnology with soft matter. , 2014, Soft matter.

[32]  Mildred B. Wohlford.  Two Hands are Better than One , 1951 .

[33]  Ming-Yang Kao,et al.  Reducing tile complexity for self-assembly through temperature programming , 2006, SODA '06.

[34]  Matthew J. Patitz,et al.  Limitations of Self-assembly at Temperature One , 2009, DNA.

[35]  Matthew J. Patitz.  An Introduction to Tile-Based Self-assembly , 2012, UCNC.

[36]  Lila Kari,et al.  The power of nondeterminism in self-assembly , 2011, SODA '11.

[37]  Hao Yan,et al.  Organizing DNA origami tiles into larger structures using preformed scaffold frames. , 2011, Nano letters.

[38]  A. Turberfield,et al.  A DNA-fuelled molecular machine made of DNA , 2022 .

[39]  M. Sahani,et al.  Algorithmic Self-Assembly of DNA , 2006 .

[40]  Russell Deaton,et al.  DNA-linked nanoparticle building blocks for programmable matter. , 2011, Angewandte Chemie.