The complexity seen in biological and soft systems often precludes a first principles approach. In order to gain a good understanding of such complex systems, simplification is needed. As systems become larger, the interplay between the underlying mechanisms and details leads to complex system-wide behaviour. Very often this behaviour will be common to a large group of otherwise unrelated systems. Simple model systems can represent a wide variety of such systems, even though the underlying chemistry is different. Such toy models are immensely instructive in the understanding of more complex systems.
In this thesis we describe and analyse toy models for two different soft and living systems. In the first part we consider an experimental model system that is representative of a broad class of materials consisting of ordered membranes. We study the self-assembly of a mixture of SDS and β-cyclodextrin (β-CD) into concentric hollow microtubes using small-angle x-ray scattering. After a concentration dependent waiting time we observe the appearance of hollow cylinders that grow inward. The distribution of waiting times follows the non-linear scaling with SDS/ β-CD concentration that is predicted by classical nucleation theory for a two-dimensional critical nucleus. Moreover, when the experimental time is rescaled according to classical nucleation theory, the entire trajectory of inward growth collapses onto a single curve, indicating that a nucleation process determines the entire kinetics of inward growth. The mechanism of inward growth can therefore be explained by the successive nucleation of new, discrete cylinders inside previous existing ones, constricted in their size by the size of the original tube.
In the second part we build upon a toy model based on equilibrium binding of ligands to a template, as a model for transcription regulation. Transcription initiation is a complex process involving multiple steps, which, in its most simplified form, can be described in three steps: (1) binding of RNAP to the promoter, (2) (irreversible) isomerization to an open complex, followed by (3) escape of the open complex to form an RNAP complex active in transcription. When the rearrangement of RNAP and transcription factors is fast compared to the formation of an open complex, we can assume that the rate at which the open complex is formed – the first kinetically significant step in the transcription process – is proportional to the occupation probability of the promoter by RNAP. Thermodynamic theory, based on the toy model of ligand adsorption to a template, has been developed to calculate this probability.
Such thermodynamic models are traditionally derived in the limit of genes in isolation, within a canonical ensemble. However, individual regulatory proteins are typically charged with the simultaneous regulation of a battery of different genes. As a result, when one of these proteins is limiting, competitive effects have a significant impact on the transcriptional response of the regulated genes. We present a general framework for the analysis of any generic regulatory architecture that accounts for the competitive effects of the regulatory environment by isolating these effects into an effective concentration parameter.