Dissertation/Thesis Abstract

On the mathematics of self-assembly
by Reishus, Dustin, Ph.D., University of Southern California, 2009, 175; 3389617
Abstract (Summary)

Self-assembly is the ubiquitous process by which simple objects come together under simple rules to form more complex objects. Self-assembly occurs in nature to produce structures of extraordinary complexity. In the future, it may be possible to harness the power of self-assembly to manufacture useful devices in enormous quantities at little cost. In order to do so, it would be valuable to have a deep understanding of self-assembly, at both theoretical and practical levels.

I first describe experimental work with DNA self-assembly. DNA is an ideal substance to use in experimental self-assembly: It has well-understood structure; it has readily-available tools to synthesize, manipulate, and visualize it; and it has "programmable" interactions with other molecules of DNA. I describe two self-assembling DNA complexes that can further self-assemble into regular lattices.

Mathematical models of self-assembly have been created to aid in the analysis of the power and limits of self-assembly. I explore decidability questions in a mathematical model of self-assembly known as the tile assembly model. I prove the undecidability of distinguishing self-assembling systems in which infinite structures can be assembled from systems in which only finite structures can be assembled.

Many self-assembly processes are rooted in chemistry. The event-systems model generalizes the classical theory of chemical thermodynamics and places the kinetic theory of chemical reactions on a firm mathematical foundation. I prove that many of the expectations acquired through empirical study are warranted.

Finally, I use the event-systems model to explore questions in pure mathematics. The atomic hypothesis in chemistry (the theory that every substance is composed of a unique set of atoms) is analogous to the fundamental theorem of arithmetic in mathematics (the theory that every natural number is the product of a unique set of primes). I exploit this analogy by creating event-systems in which the basic components are natural numbers that can ‚ÄĚreact" through multiplication. Important thermodynamic properties such as temperature and pressure have purely mathematical implications in these systems. In particular, the pressure at equilibrium is the Riemann zeta function's value of the temperature of the system.

Indexing (document details)
Advisor: Adleman, Leonard
Commitee: Finkel, Steven, Huang, Ming-Deh, Rothemund, Paul
School: University of Southern California
Department: Computer Science
School Location: United States -- California
Source: DAI-B 71/01, Dissertation Abstracts International
Subjects: Applied Mathematics, Computer science
Keywords: Event-systems, Riemann zeta function, Self-assembly, Tile assembly model
Publication Number: 3389617
ISBN: 978-1-109-56354-2
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