Self-replication and evolution of DNA crystals

Is it possible to create a simple physical system that is capable of replicating itself? Can such a system evolve interesting behaviors, thus allowing it to adapt to a wide range of environments? This paper presents a design for such a replicator constructed exclusively from synthetic DNA. The basis for the replicator is crystal growth: information is stored in the spatial arrangement of monomers and copied from layer to layer by templating. Replication is achieved by fragmentation of crystals, which produces new crystals that carry the same information. Crystal replication avoids intrinsic problems associated with template-directed mechanisms for replication of one-dimensional polymers. A key innovation of our work is that by using programmable DNA tiles as the crystal monomers, we can design crystal growth processes that apply interesting selective pressures to the evolving sequences. While evolution requires that copying occur with high accuracy, we show how to adapt error-correction techniques from algorithmic self-assembly to lower the replication error rate as much as is required

Capabilities and limits of compact error resilience methods for algorithmic self-assembly in two and three dimensions

Abstract. Winfree’s pioneering work led the foundations in the area of errorreduction in algorithmic self-assembly [26], but the construction resulted in increase of the size of assembly. Reif et. al. contributed further in this area with compact error-resilient schemes [15] that maintained the original size of the assemblies, but required certain restrictions on the Boolean functions to be used in the algorithmic self-assembly. It is a critical challenge to improve these compact error resilient schemes to incorporate arbitrary Boolean functions, and to determine how far these prior results can be extended under different degrees of restrictions on the Boolean functions. In this work we present a considerably more complete theory of compact error-resilient schemes for algorithmic self-assembly in two and three dimensions. First we consider two-dimensional algorithmic self-assembly. We present an error correction scheme for reduction of errors from ɛ to ɛ 2 for arbitrary Boolean functions in two dimensional algorithmic self-assembly. Then we characterize the class of Boolean functions for which the error reduction can be done from ɛ to ɛ 3, and present an error correction scheme that achieves this reduction. Then we prove ultimate limits on certain classes of compact error resilient schemes: in particular we show that they can not provide reduction of errors from ɛ to ɛ 4 is for any Boolean functions. Further, we develop the first provable compact error resilience schemes for three dimensional tiling self-assemblies. We also extend the work of Winfree on selfhealing in two-dimensional self-assembly [25] to obtain a self-healing tile-set for three-dimensional self-assembly.

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

Abstract. 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 see convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretic 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 one fundamental problem in them: the sequential construction of a given graph, referred to as Accretive Graph Assembly Problem (AGAP) and Self-Destructible Graph Assembly Problem (DGAP), respectively. Our main results are: (i) AGAP is ¤¦ ¥-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 §¨ ¥-complete; and (iii) DGAP is ¥�©�¥����� �-complete even if the maximum degree of the graph is restricted to 6 (this is the first ¥�©�¥����¨ �-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 §� ¥-complete.

A self assembly model of time-dependent glue strength

Abstract Self-assembly is a ubiquitous process in which small objects selforganize into larger and complex structures. In 2000, Rothemund and Winfree proposed a Tile Assembly Model as a mathematical model for theoretical studies of self-assembly. We propose a refined self-assembly model in which the glue strength between two juxtaposed tiles is a function of the time they have been in neighboring positions. We then present an implementation of our model using strand displacement reactions on DNA tiles. Under our model, we can demonstrate and study catalysis and self-replication in the tile assembly. We then study the tile complexity for assembling shapes in our model and show that a thin rectangle of size k × N can be assembled using O((log(N)) / log log(N)) types of tiles, demonstrating the glue model has additional capabilities over the prior tiling assembly model. We also describe a method to implement with DNA tiles our model of time-dependant glue strength