The impact of the transient uptake flux on bioaccumulation: Linear adsorption and first-order internalisation coupled with spherical semi-infinite mass transport

The uptake of a chemical species (such as an organic molecule or a toxic metal ion) by an organism is modelled considering linear pre-adsorption followed by a first-order internalisation. The active biosurface is supposed to be spherical or semi-spherical and the mass transport in the medium is diffusion-controlled. The analytical solutions for the transient flux and accumulated amounts can be used to discriminate between adsorption and internalisation parameters, which are inseparable in a steady-state flux interpretation. The concentration at the surface of the organism and the (intracellular) uptake flux pass through a maximum before coming to their steady-state values. For any combination of the parameters, the time necessary to reach a diffusive flux, which differs by less than 10% from the eventual steady-state value, can be directly read from a contour plot. For small microorganisms, steady state is usually achieved in a short time and so the usual analysis based on the steady-state flux is a good approximation (except for combinations of large radii, low diffusion coefficients, important adsorption and slow internalisation kinetics). However, interpretation of the cumulative uptake requires (explicit or implicit) consideration of the large transient fluxes arising at short times. By considering an instantaneous steady state approximation, the linear regression of accumulation data outside the transient regime, i.e. at larger measuring times, allows for the discrimination between adsorption and kinetic parameters for small organisms as shown with literature data of lead uptake by Chlorella vulgaris.