| This thesis is concerned with two separate subjects; (i) self-assembly, and (ii) model reduction in biology; it is divided into these parts accordingly. The underlying question addressed in the first part is can we direct naturally occurring self-assembly such that a desired structure is the only one that forms? This problem is broken up into two parts: (i) the enumeration of all structures that can be self-assembled, and (ii) the derivation of a mechanism under which any one of those structures becomes the only one that forms. In the second part, approximation techniques are applied to models of biological systems, yielding reduced models that are quantitatively accurate. The reduced models offer the following advantages over the originals: (i) they can be analyzed analytically, (ii) they contain many fewer effective parameters, and (iii) they are directly testable, falsifiable, and predictive.;Self-assembly. The overarching question of self-assembly is, can we harness the natural assembly process so that 'human-made objects' are made spontaneously? As a model system, the self-assembly of nano- and micro-meter sized colloidal particles is considered. Graph theory and geometry is used to derive a method capable of enumerating a provably complete set of self-assemblable colloidal structures, which corresponds to all sphere packings. A method for directing the self-assembly of nearly any one of those structures is then derived.;Model reduction in biology. Biological systems have many components. Because of this, it has become common to model such networks using large systems of coupled ordinary differential equations. However, there is as yet no simple way of determining how solutions to large systems of equations depend on their parameters or components. Here, it is shown that large biological signaling networks can be reduced to systems involving a few equations and effective parameters using the method of dominant balance, which was developed and refined within fluid mechanics. The effective parameters lump the system's many components together, yielding a simplified system that contains within it information on all of the many components. The reduced system quantitatively agrees with the original, and demonstrates which features of the signaling network are essential. |
