Many signaling and regulatory molecules within cells exist in very few copies per cell. Any process affecting even limited numbers of these molecules therefore has the potential to affect the dynamics of the biochemical networks of which they are a part. This sensitivity to small copy-number changes is what allows stochasticity in gene expression to introduce a degree of randomness in what cells do. While this randomness can be suppressed, it does not appear to be so in many biological systems, at least not to the maximum degree possible. This suggests that this randomness is not necessarily detrimental to cell populations, as it can produce qualitatively new behaviours in genetic networks which may be utilized by cells.<br /><br />In this thesis, two other mechanisms are investigated which, through their interaction with low copy-number molecules, are able to produce qualitatively different dynamics in genetic networks: the stochastic partitioning of molecules in cell division, and the direct interaction of two low copy-number molecules. For this, a novel simulator of chemical kinetics is first presented, designed to simulate the dynamics of genetic circuits inside growing populations of cells. It is then used to study a genetic switch where one repressive link is formed by direct interaction between RNA molecules. This arrangement was found to decouple the stability of the two noisy attractors of the network and the speeds of the state transitions. In other words, it allows the network to have two equally-stable noisy attractors, but differing state transition speeds.<br /><br />Next, the cell-to-cell diversity in RNA numbers (as quantified by the normalized variance) of a single gene over time in a growing model cell population was studied as a function of the division synchrony. In the model, synchronous cell divisions introduce transient increases in the cell-to-cell diversity in RNA numbers of the population, a prediction which was verified using single-molecule measurements of RNA numbers. Finally, the effects of the stochastic partitioning of regulatory molecules in cell division on the dynamics of two genetic circuits, a switch and a clock, were studied. Of these two circuits, the switch has the most dramatic changes in its dynamics, brought on by the inevitable negative correlation in molecule numbers that sister cells inherit. This negative correlation can allow a cell population to partition the phenotypes of the individual cells with less variance than a binomial distribution.<br /><br />These results advance our understanding of the different behaviours that can be produced in genetic circuits due to these two mechanisms. Since they produce unique behaviours, these mechanisms, and combinations thereof, are expected to be used for specialized purposes in natural genetic circuits. Further, since the downstream effects of these mechanisms may be more predictable than, e.g., modifying promoter sequences, they may also be useful in the design and implementation of future synthetic genetic circuits with specific behaviours.<br />