Pattern Formation by Bacteria

Abstract

A uniform distribution of histidine auxotrophic Salmonella Typhimurium on an agar gel exhibits a spatial pattern of growth in responsto a diffusing front of histidine. We formulate a mathematical model of this system which describes the histidine concentration (H(r,t)) at radius r and time t in a petri dish, the concentration of the growth medium’s buffer (G(r,t)) and the size of the bacterial population (B(r,t)). Histidine diffuses and is taken up by growing cells, the buffer also diffuses but it is neutralized by acids produced as byproducts of cell growth, and the bacterial population is fixed, as in a stiff top layer of agar.

The uptake of histidine and the depletion of buffer are modelled by Michaelis-Menten kinetics. The cell growth is modelled as having a hysteretic dependence on the histidine and buffer concentrations: Specifically, cell growth competence depends on combinations of G and H. Cell growth continues until the combination is reduced to a threshold value T_off at which cell growth stops. Growth does not begin again until a threshold value T_on (> T_off) is reached.

Typical experiments begin with a uniform distribution of buffer (G(r,0) = G^o) and bacteria (B(r,0) = B^o), and a drop of histidine solution placed in the center of a petri dish (H(r,0) = H^o H(R’ - r)*, where R’ is the drop’s radius). With such initial conditions, the mathematical model is shown to have a central disc of high growth which is surrounded by a ring of even higher growth. This central region is in turn enclosed by concentric rings of heavy growth which are separated by bands of low cell growth. Similar growth patterns are observed in experiments of this kind which have been performed in conjunction with J. Roth and M. Schmid.

This work supported in part by NSF Grant No. MCS-7605985

These results were presented at Mathematical Biology Meeting, Universität Heidelberg, July, 1979.