Upscaling diffusion through first-order volumetric sinks: a homogenization of bacterial nutrient uptake

In mathematical models that include nutrient delivery to bacteria, it is prohibitively expensive to include a pointwise nutrient uptake within small bacterial regions over bioreactor length-scales, and so such models often impose an effective uptake instead. In this paper, we systematically investigate how the effective uptake should scale with bacterial size and other microscale properties under first-order uptake kinetics. We homogenize the unsteady problem of nutrient diffusing through a locally periodic array of spherical bacteria, within which it is absorbed. We introduce a general model that could also be applied to other single-cell microorganisms, such as cyanobacteria, microalgae, protozoa, and yeast and we consider generalizations to arbitrary bacterial shapes, including some analytic results for ellipsoidal bacteria. We explore in detail the three distinguished limits of the system on the timescale of diffusion over the macroscale. When the bacterial size is of the same order as the distance between them, the effective uptake has two limiting behaviours, scaling with the bacterial volume for weak uptake and with the bacterial surface area for strong uptake. We derive the function that smoothly transitions between these two behaviours as the system parameters vary. Additionally, we explore the distinguished limit in which bacteria are much smaller than the distance between them and have a very strong uptake. In this limit, we find that the effective uptake is bounded above as the uptake rate grows without bound; we are able to quantify this and characterise the transition to the other limits we consider