Nutrient uptake rate as a function of cell size and surface transporter density: A Michaelis-like approximation to the model of Pasciak and Gavis

Pasciak and Gavis were first to propose a model of nutrient uptake that includes both physical transport by diffusion and active biological transport across the cell membrane. While the Pasciak–Gavis model is not complicated mathematically (it can be expressed in closed form as a quadratic equation), its parameters are not so easily interpretable biologically as are the parameters of the Michaelis–Menten uptake model; this lack of transparency is probably the main reason the Pasciak–Gavis model has not been adopted by ecologically oriented modelers. Here I derive a Michaelis-like approximation to the Pasciak–Gavis model, and show how the parameters of the latter map to those of the Michaelis-like model. The derived approximation differs from a pure Michaelis–Menten model in a subtle but potentially critical way: in a pure Michaelis–Menten model, the half-saturation constant for nutrient uptake is independent of the density of transporter (or “porter”) proteins on the cell surface, while in the Pasciak–Gavis model and its Michaelis-like approximation, the half-saturation constant does depend on the density of porter proteins. The Pasciak–Gavis model predicts a unique relationship between cell size, nutrient concentration in the medium, the half-saturation constant of porter-limited nutrient uptake, and the resulting rate of uptake; the Michaelis-like approximation preserves the most important feature of that relationship, the size at which porter limitation gives way to diffusion limitation. Finally I discuss the implications for community structure that are implied by the Pasciak–Gavis model and its Michaelis-like approximation.