Modelling One-Dimensional Non-Uniform Growth with Applications to Cylindrical Yeast Colonies

Abstract

Biologists have shown that yeast can be restricted to grow vertically upwards from an agar plate to form cylindrical colonies. It is known that cell proliferation within the cylindrical yeast colonies is nutrient driven. However, the cell behaviour within the colony is not fully understood. Yeast colonies are not well mixed cultures and the cells throughout the colony will not have equal access to nutrient. This results in non-uniform domain growth within the colonies. Furthermore, the height of the cylindrical yeast colonies was found to grow linearly in time. We present a discrete cellular automaton and a continuous partial differential equation model to predict the cellular behaviour and cell growth within the cylindrical yeast colonies. We provide a general method for determining the average trajectories of initial cells in a non-uniformly growing domain using cellular automata and obtain closed form solutions for some particular cases of interest. Furthermore, we provide a numerical approximation to the pathlines of individual cells using a reaction-diffusion-advection PDE model that couples domain length, nutrient concentration and cell density on a non-uniformly growing domain. We compare our numerical approximation to the experimental results of Vulin et al. (2014) to predict the cell behaviour within the cylindrical yeast colonies. It was found that only a fixed number of cells at the base of the colony can proliferate. The cylindrical yeast colonies grow linearly as a result of this.