Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104

On the Computational Power of DNA Annealing and Ligation

In [20] it was shown that the DNA primitives of Separate, Merge, and Amplify were not sufficiently powerful to invert functions defined by circuits in linear time. Dan Boneh et al [4] show that the addition of a ligation primitive, Append, provides the missing power. The question becomes, "How powerful is ligation? Are Separate, Merge, and Amplify necessary at all?" This paper proposes to informally explore the power of annealing and ligation for DNA computation. We conclude, in fact, that annealing and ligation alone are theoretically capable of universal computation

Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

Modelling the evolution of biological complexity with a two-dimensional lattice self-assembly process

Self-assembling systems are prevalent across numerous scales of nature, lying at the heart of diverse physical and biological phenomena. Individual protein subunits self-assembling into complexes is often a vital first step of biological processes. Errors during protein assembly, due to mutations or misfolds, can have devastating effects and are responsible for an assortment of protein diseases, known as proteopathies. With proteins exhibiting endless layers of complexity, building any all-encompassing model is unrealistic. Coarse-grained models, despite not faithfully capturing every detail of the original system, have massive potential to assist understanding complex phenomenon. A principal actor in self-assembly is the binding interactions between subunits, and so geometric constraints, polarity, kinetic forces, etc. can often be marginalised. This work explores how self-assembly and its outcomes are inextricably tied to the involved interactions through the use of a two-dimensional lattice polyomino model. %Armed with this tractable model, we can probe how dynamics acting on evolution are reflected in interaction properties. First, this thesis addresses how the interaction characteristics of self-assembly building blocks determine what structures they form. Specifically, if the same structures are consistently produced and remain finite in size. Assembly graphs store subunit interaction information and are used in classifying these two properties, the determinism and boundedness respectively. Arbitrary sets of building blocks are classified without the costly overhead of repeated stochastic assembling, improving both the analysis speed and accuracy. Furthermore, assembly graphs naturally integrate combinatorial and graph techniques, enabling a wider range of future polyomino studies. The second part narrows in on implications of nondeterministic assembly on interaction strength evolution. Generalising subunit binding sites with mutable binary strings introduces such interaction strengths into the polyomino model. Deterministic assemblies obey analytic expectations. Conversely, interactions in nondeterministic assemblies rapidly diverge from equilibrium to minimise assembly inconsistency. Optimal interaction strengths during assembly are also reflected in evolution. Transitions between certain polyominoes are strongly forbidden when interaction strengths are misaligned. The third aspect focuses on genetic duplication, an evolutionary event observed in organisms across all taxa. Through polyomino evolutions, a duplication-heteromerisation pathway emerges as an efficient process. This pathway exploits the advantages of both self-interactions and pairwise-interactions, and accelerates evolution by avoiding complexity bottlenecks. Several simulation predictions are successfully validated against a large data set of protein complexes. These results focus on coarse-grained models rather than quantified biological insight. Despite this, they reinforce existing observations of protein complexes, as well as posing several new mechanisms for the evolution of biological complexity

A Cross-disciplinary Framework for the Description of Contextually Mediated Change

We present a mathematical framework (referred to as Context-driven Actualization of Potential, or CAP) for describing how entities change over time under the influence of a context. The approach facilitates comparison of change of state of entities studied in different disciplines. Processes are seen to differ according to the degree of nondeterminism, and the degree to which they are sensitive to, internalize, and depend upon a particular context. Our analysis suggests that the dynamical evolution of a quantum entity described by the Schrodinger equation is not fundamentally different from change provoked by a measurement often referred to as collapse, but a limiting case, with only one way to collapse. The biological transition to coded replication is seen as a means of preserving structure in the fact of context-driven change, and sextual replication as a means of increasing potentiality thus enhancing diversity through interaction with context. The framework sheds light on concepts like selection and fitness, reveals how exceptional Darwinian evolution is as a means of 'change of state', and clarifies in what sense culture, and the creative process underlying it, are Darwinian.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:q-bio/051100