CONTINUOUS AND DISCRETE MODELS OF COOPERATION IN COMPLEX BACTERIAL COLONIES

Abstract

In this paper, we study the effect of discreteness on various models for patterning in bacterial colonies (finite-size effect) and present two types of models to describe the growth of the bacterial colonies. The first model presented is the Communicating Walkers model (CWm), a hybrid model composed of both continuous fields and discrete entities — walkers, which are coarse-graining of the bacteria; coarse-graining may amplify the discreteness inherent to the biological system. Models of the second type are systems of reaction diffusion equations, where the branching of the pattern is due to non-constant diffusion coefficient of the bacterial field. The diffusion coefficient represents the effect of self-generated lubrication fluid on the bacterial movement. The representation of bacteria by a density field neglects their discreteness altogether. We implement the discreteness of the bacteria by introducing a cutoff in the growth term at low bacterial densities. We demonstrate that the cutoff does not improve the models in any way. The cutoff affects the dynamics by decreasing the effective surface tension of the front, making it more sensitive to anisotropy and decreasing the fractal dimension of the evolving patterns. We compare the continuous and semi-discrete models by introducing food chemotaxis and repulsive chemotactic signaling into the models. We find that the growth dynamics of the CWm and the growth dynamics of the Non-Linear Diffusion model (one of the continuous models) are affected in the same manner. From such similarities and from the insensitivity of the CWm to implicit anisotropy, we conclude that even the increased discreteness, introduced by the coarse-graining of the walkers, is small enough to be neglected. There are advantages and disadvantages to the two types of models. Employing both of them in parallel enables us to conclude that the discreteness of the bacteria does not significantly affect the growth dynamics (no finite-size effect).