Transient biouptake flux and accumulation by microorganisms: The case of two types of sites with Langmuir adsorption

The uptake of a chemical species by an aquatic microorganism is modelled considering two kinds of sites where Langmuirian adsorption is followed by first order internalisation kinetics. Simpler models, such as only one internalisation route (while most of the adsorption takes place on non-internalising sites) or a linear isotherm for adsorption on one or both sites, become limiting cases of this double-Langmuirian model. The model considers the sites located on the spherical (or semi-spherical) surface of the organism, and takes diffusion from the medium into explicit account. The numerical solution for the internalisation flux shows a maximum. We provide an estimate for the time needed to reach a certain proximity to steady state. The transient solution confirms that the analytical expressions for the steady-state flux are usually valid and that the accumulated amounts reflect the impact of the short-time uptake. The Instantaneous Steady-State Approximation (ISSA), where an intercept of the linear regression of accumulated amount as a function of time is interpreted as an adsorbed amount, can be critically assessed with the transient numerical code for two cases: (i) when the total burden of metal on the cell is the input data and (ii) when an extraction procedure provides further information on the adsorbed and internalised amount.