Optimization-based model of multinutrient uptake kinetics

We present a new, optimization-based model for uptake kinetics of multiple nutrients, which has the same number of parameters (two for each nutrient) as the Michaelis–Menten model. We fit this model and an existing inhibition-based model to data from chemostat experiments at various flow rates (under extreme limitation by both nitrogen [N] and phosphorus [P]) and compared these models and the Michaelis–Menten model to an independent data set for the same species in a chemostat at various N : P input ratios (at constant flow rate). Our model fit the data well, with a slightly higher square error than the much more complex inhibition model. We also successfully applied our model to a data set for a different species under various degrees of vitamin B12and Plimitation. Our model agrees with measured cell quotas of nonlimiting nutrients when supply ratios differ greatly from the optimal ratio for phytoplankton, whereas the Michaelis–Menten model greatly overestimates the uptake of nonlimiting nutrients at these extreme nutrient supply ratios. The key to our model’s success is the optimization of uptake for the limiting nutrient, which results in distinct behavior for limiting versus nonlimiting nutrients, without additional parameters; phytoplankton allocate their internal resources (nitrogen) to optimize uptake of the limiting nutrient, but not in response to changes in ambient nutrient ratios. For ambient nutrient ratios that are very different from the optimal ratio of phytoplankton, straightforward application of separate Michaelis–Menten equations for multiple nutrients greatly overestimates the uptake rates of nonlimiting nutrients compared with data from chemostat experiments (Droop 1974; Rhee 1974). Uptake of the same nutrient is faster when it is limiting than when it is nonlimiting (Rhee 1974; Gotham and Rhee 1981a,b). Droop (1974) developed a parameterization for uptake of nonlimiting nutrients to match his observations, and Gotham and Rhee (1981a,b) developed an inhibition-based model in which the maximum uptake rate of a nutrient is a decreasing function of its cell quota (internal concentration). Both of these approaches yield more accurate uptake rates for nonlimiting nutrients, but both add parameters, which must be determined separately for various nutrients and even for the same nutrient with different ratios of ambient nutrient concentrations. We present a new optimization-based model for uptake kinetics of multiple nutrients. Our uptake model is an extension of the single-nutrient optimal-uptake equation of Pahlow (2005), which is itself an extension of the affinitybased uptake model of Aksnes and Egge (1991). For this study we embedded both our uptake model and the inhibition model of Gotham and Rhee (1981a,b) into a model of phytoplankton growth on multiple nutrients (Legovic and Cruzado 1997; Klausmeier et al. 2004) and applied the resulting models to simulate chemostat experiments. We fit both models to data from chemostat experiments at extreme nitrogen : phosphorus (N : P) input ratios at various flow rates. We also compared both models to an independent data set for the same species in a chemostat over a range of less extreme N : P ratios at fixed flow rate. We compared our model and that of Droop (1974) to a data set for a different species under various degrees of limitation by P and vitamin B12, at various flow rates (Droop 1974). Thus, we quantitatively compared the models’ agreement with data under various conditions of limitation by N, P, and vitamin B12.