2007

Dissertation

Ph.D.

#### Degree Granting Department

Mathematics and Statistics

#### Major Professor

NataÅ¡a Jonoska, Ph.D.

#### Keywords

DNA, Spectrum, Pot graph, Probabilistic model, Complete complexes, Cyclic molecules

#### Abstract

We analyze a self-assembly model of flexible DNA tiles and develop a theoretical description of possible assembly products. The model is based on flexible branched DNA junction molecules, which are designed in laboratories and could serve for performing computation. They are also building blocks for make of even more complex molecules or structures. The branched junction molecules are flexible with sticky ends on their arms. They are modeled with "tiles", which are star like graphs, and "tile types", which are functions that give information about the number of sticky ends. A complex is a structure that is obtained by gluing several tiles via their sticky ends. A complex without free sticky ends is called "complete complex". Complete complexes are our main interest. In most experiments, besides the desired end product, a lot of unwanted material also appears in the test tube (or pot). The idea is to use the proper proportions of tiles of different types.

The set of vectors that represent these proper proportions is called the "spectrum" of the pot. We classify the types of pots according to the complexes they acan admit, and we can identify the class of each pot from the spectrum and affine spaces. We show that the spectrum is a convex polytope and give an algorithm (and a MAPLE code), which calculates it, and classify the pots in PTIME. In the second part of the dissertation, we approach molecular self-assembly from a graph theoretical point of view. We assign a star-like graph to each tile in a pot, which induces a "pot-graph". A pot-graph is a labeled multigraph corresponding to a given pot type, whose vertices represent tile types. The complexes can be represented by "complex-graphs", and each such graph is mapped homomorphically into a pot-graph. Therefore, the pot-graph can be used to distinguish between pot types according to the structure of the complexes that can be assembled.

We begin the third part of the dissertation with a pot containing uniformly distributed DNA junction molecules capable of forming a cyclic graph structure, in which all possible Watson-Crick connections have already been established, and compute the expectation and the variance of the number of self-assembled cycles of any size. We also tested our theoretical results in wet lab experiments performed at Prof. Nadrian C. Seeman's laboratory at New York University. Our main concern was the probability of obtaining cyclic structures. We present the obtained results, which also helped in defining an important parameter for the theoretical model.

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