Figure 1. Averaging over cellular inputs by diffusion.

Microbes use extracellular secretions for external digestion and other functions. The extracellular concentration of secretions at any location depends on the secretions of each cell in the population and the rate at which the secretions diffuse across space. When diffusion rates are low, then the concentration near a particular cell depends mainly on the secretion rate of that cell. When diffusion rates are high, then the concentration near a particular cell depends on the average secretion rate over all the cells in the common spatial neighborhood. (A) Concentration of an extracellular secretion in a low-diffusion environment. Each light-colored circle shows the location of a cell. The color index expresses the effective number of cells contributing to each spatial location. In this case, each cell is mainly affected by its own secretion rate. (B–E) Increasing diffusion rate causes a rise in the effective number of cells contributing to the extracellular concentration at each location. With high diffusion, the concentration depends on the average secretion rate over many different cells. If cells reduce their secretions in response to higher nearby extracellular concentration, then the total concentration may not change as the number of contributing cells per location increases. Instead, the same total may be achieved by a greater number of smaller inputs, in which case stochastic fluctuations in each input have less consequence for fluctuations in extracellular concentration. As the consequence declines for fluctuations of individual cells, one may expect weaker regulatory control per cell, and thus greater stochasticity per cell.