Complexity of graph self-assembly in accretive systems and self-destructible systems

Self-assembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano-scale structures. Recent years have seen convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretical studies of self-assembly utilize mathematical models with two limitations: (1) only attraction, while no repulsion, is studied; (2) only assembled structures of two dimensional square grids are studied. In this paper, we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs. This allows for the study of a more general class of self-assembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the self-destructible graph assembly model, and identify a fundamental problem in them: the sequential construction of a given graph. We refer to it as the Accretive Graph Assembly Problem (AGAP) and the Self-Destructible Graph Assembly Problem (DGAP), in the respective models. Our main results are: (i) AGAP is NP-complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5; (ii) counting the number of sequential assembly orderings that result in a target graph (#AGAP) is #P-complete; and (iii) DGAP is PSPACE-complete even if the maximum degree of the graph is restricted to 6 (this is the first PSPACE-complete result in self-assembly). We also extend the accretive graph assembly model to a stochastic model, and prove that determining the probability of a given assembly in this model is #P-complete.

[1]  Matthew Cook,et al.  Self-Assembled Circuit Patterns , 2003, DNA.

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

[3]  Erik Winfree,et al.  One Dimensional Boundaries for DNA Tile Self-Assembly , 2003, DNA.

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

[5]  G. Whitesides,et al.  Self-Assembly of Mesoscale Objects into Ordered Two-Dimensional Arrays , 1997, Science.

[6]  Erik Winfree,et al.  Complexity of Compact Proofreading for Self-assembled Patterns , 2005, DNA.

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

[8]  Chengde Mao,et al.  Self-assembly of hexagonal DNA two-dimensional (2D) arrays. , 2005, Journal of the American Chemical Society.

[9]  Ashish Goel,et al.  Error Free Self-assembly Using Error Prone Tiles , 2004, DNA.

[10]  R. Robinson.  Undecidability and nonperiodicity for tilings of the plane , 1971 .

[11]  Erik Winfree,et al.  Self-healing Tile Sets , 2006, Nanotechnology: Science and Computation.

[12]  Sudheer Sahu,et al.  A Self-assembly Model of Time-Dependent Glue Strength , 2005, DNA.

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

[14]  Erik Winfree,et al.  Self-replication and Evolution of DNA Crystals , 2005, ECAL.

[15]  N. Jonoska,et al.  Three dimensional DNA structures in computing. , 1999, Bio Systems.

[16]  Hao Wang.  Proving theorems by pattern recognition — II , 1961 .

[17]  E. Winfree,et al.  Algorithmic Self-Assembly of DNA Sierpinski Triangles , 2004, PLoS biology.

[18]  Eric Klavins.  Toward the Control of Self-Assembling Systems , 2003, Control Problems in Robotics.

[19]  Ming-Yang Kao,et al.  Complexities for generalized models of self-assembly , 2004, SODA '04.

[20]  J. Reif,et al.  Construction, analysis, ligation, and self-assembly of DNA triple crossover complexes , 2000 .

[21]  A. Alan Middleton.  Computational Complexity of Determining the Barriers to Interface Motion in Random Systems , 1999 .

[22]  A. Turberfield,et al.  Engineering a 2D protein-DNA crystal. , 2005, Angewandte Chemie.

[23]  Natasa Jonoska,et al.  Computation by Self-assembly of DNA Graphs , 2004, Genetic Programming and Evolvable Machines.

[24]  J. Reif,et al.  Directed nucleation assembly of DNA tile complexes for barcode-patterned lattices , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Eric Klavins,et al.  Graph grammars for self assembling robotic systems , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[26]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[27]  Richard J. Lipton,et al.  DNA Based Computers , 1996 .

[28]  Antonio Bicchi,et al.  Control Problems in Robotics , 2003, Springer Tracts in Advanced Robotics.

[29]  Ashish Goel,et al.  Combinatorial optimization problems in self-assembly , 2002, STOC '02.

[30]  David Reguera,et al.  Viral self-assembly as a thermodynamic process. , 2002, Physical review letters.

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

[32]  Sudheer Sahu,et al.  Compact Error-Resilient Computational DNA Tiling Assemblies , 2004, DNA.

[33]  Erik Winfree,et al.  Programmable Control of Nucleation for Algorithmic Self-assembly , 2004, DNA.

[34]  Erik Winfree,et al.  Complexity of Self-assembled Shapes , 2004, DNA.

[35]  Ashish Goel,et al.  Invadable self-assembly: combining robustness with efficiency , 2004, SODA '04.

[36]  P W Rothemund,et al.  Using lateral capillary forces to compute by self-assembly , 2000, Proc. Natl. Acad. Sci. USA.

[37]  Natasa Jonoska,et al.  Self-assembling DNA Graphs , 2002, DNA.

[38]  Natasa Jonoska,et al.  A Computational Model for Self-assembling Flexible Tiles , 2005, UC.

[39]  Yuriy Brun,et al.  DNA triangles and self-assembled hexagonal tilings. , 2004, Journal of the American Chemical Society.

[40]  Qi Cheng,et al.  Linear Self-Assemblies: Equilibria, Entropy and Convergence Rates , 2003 .