Modelling uniaxial non-uniform yeast colony growth: Comparing an agent-based model and continuum approximations

Biological experiments have shown that yeast can be restricted to grow in a uniaxial direction, vertically upwards from an agar plate to form a colony. The growth occurs as a consequence of cell proliferation driven by a nutrient supply at the base of the colony, and the height of the colony has been observed to increase linearly with time. Within the colony the nutrient concentration is non-constant and yeast cells throughout the colony will therefore not have equal access to nutrient, resulting in non-uniform growth. In this work, an agent based model is developed to predict the microscopic spatial distribution of labelled cells within the colony when the probability of cell proliferation can vary in space and time. We also describe a method for determining the average trajectories or pathlines of labelled cells within a colony growing in a uniaxial direction, enabling us to connect the microscopic and macroscopic behaviours of the system. We present results for six cases, which involve different assumptions for the presence or absence of a quiescent region (where no cell proliferation occurs), the size of the proliferative region, and the spatial variation of proliferation rates within the proliferative region. These six cases are designed to provide qualitative insight into likely growth scenarios whilst remaining amenable to analysis. We compare our macroscopic results to experimental observations of uniaxial colony growth for two cases where only a fixed number of cells at the base of the colony can proliferate. The model predicts that the height of the colony will increase linearly with time in both these cases, which is consistent with experimental observations. However, our model shows how different functional forms for the spatial dependence of the proliferation rate can be distinguished by tracking the pathlines of cells at different positions in the colony. More generally, our methodology can be applied to other biological systems exhibiting uniaxial growth, providing a framework for classifying or determining regions of uniform and non-uniform growth.