Contributions to computational phylogenetics and algorithmic self-assembly

This dissertation addresses some of the algorithmic and combinatorial problems at the interface between biology and computation. In particular, it focuses on problems in both computational phylogenetics, an area of study in which computation is used to better understand evolutionary relationships, and algorithmic self-assembly, an area of study in which biological processes are used to perform computation. The first set of results investigate inferring phylogenetic trees from multi-state character data. We give a novel characterization of when a set of three-state characters has a perfect phylogeny and make progress on a long-standing conjecture regarding the compatibility of multi-state characters. The next set of results investigate inferring phylogenetic supertrees from collections of smaller input trees when the input trees do not fully agree on the relative positions of the taxa. Two approaches to dealing with such conflicting input trees are considered. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. The second is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. We give fixed-parameter tractable algorithms for both approaches. We then turn to the algorithmic self-assembly of fractal structures from DNA tiles and investigate approximating the Sierpinski triangle and the Sierpinski carpet with strict self-assembly. We prove tight bounds on approximating the Sierpinski triangle and exhibit a class of fractals that are generalizations of the Sierpinski carpet that can approximately self-assemble. We conclude by discussing some ideas for further research

Hierarchical Growth is Necessary and (Sometimes) Sufficient to Self-Assemble Discrete Self-Similar Fractals

In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based model which only allows for a single tile to attach to a growing assembly at each step, there are no tile assembly systems capable of self-assembling the discrete self-similar fractals known as the "H" and "U" fractals. We then show that in a related model which allows for hierarchical self-assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile assembly systems which self-assembles the "U" fractal and conjecture that the same holds for the "H" fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the models and greater understanding of the power of hierarchical assembly

Self-Assembling Rulers for Approximating Generalized Sierpinski Carpets

Discrete self-similar fractals have been studied as test cases for self-assembly, both in the laboratory and in mathematical models, since Winfree exhibited a tileset in the abstract Tile Assembly Model (aTAM) with which the Sierpinski triangle self-assembles. More recently, Kautz and Lathrop showed that the Sierpinski carpet, along with infinitely many of its generalizations, also self-assembles in the aTAM. In all of these cases, the self-assembly is in the weak sense in which the structure that self-assembles is the entire first quadrant, but those tiles corresponding to the desired set S are recognizably labeled. In contrast, strict self-assembly requires that tiles are placed at every location corresponding to a point in S, and only at those locations. It is currently unknown whether any self-similar fractal strictly self-assembles in the aTAM. Thus motivating the development of techniques to approximate fractal structures with strict self-assembly. Ideally, an approximation of a set S would be an in-place approximation, i.e., a set X ⊃ S with the same fractal dimension as S that strictly self-assembles in such a way that those tiles corresponding to the desired set S are recognizably labeled. Prior to the results presented here, only the Sierpinski triangle was known to have an in-place approximation due to the construction of Lutz and Shutters. The main result of this paper is that every generalized Sierpinski carpet has an in-place approximation in the aTAM. We exhibit a construction that, given a set of parameters specifying a generalized Sierpinski carpet, encodes the necessary algorithm into a tileset. The key to our construction is the use of rulers to control the self-assembling fractal structure. These rulers have the same functional purpose as optimal counters, but are able to control the self-assembly of a fractal structure without distorting it. To verify the fractal dimension of the resulting structure, we prove a result of independent interest on the dimension of sets embedded into discrete fractals. We also give a conjecture on the approximability of self-similar fractals