SPONGE

(Simple Phytoplankton Optimal Nutrient Gathering Equations)

by S. Lan Smith

Ecosystem Change Research Program

Frontier Research Center for Global Change

The SPONGE is a new model to describe the uptake kinetics of multiple nutrients by phytoplankton (algae). This page briefly introduces the model and discusses some implications and possible applications.

The SPONGE is based on optimal allocation of intra-cellular resources (specifically nitrogen) for efficient uptake of the growth-limiting nutrient. The growth-limiting nutrient is defined as the nutrient with the lowest "cell quota" (intra-cellular concentration) relative to its respective minimum value, according to the classic "quota model" (Droop, 1968; Caperon, 1968).

To skip the Introduction and go directly to the description of the SPONGE, click here.

Background

The Michaelis-Menten equation is the most widely applied model for uptake kinetics (and one of the most widely applied equations in biological modeling in general). It can be derived explicitly for the case of a single enzyme processing a single substrate. However, nutrient uptake is not such a simple, single-enzyme process. Nevertheless, this equation has proven a good approximation for uptake of various nutrients (and other substrates, e.g., organic matter by bacteria).

Motivation for a new model for multi-nutrient uptake kinetics

　　　　　　　 To examine the coupled biogeochemical cycles of various nutrients and carbon, ecosystem models must account for the uptake of multiple nutrients. Straightforward application of separate Michaelis-Menten equations for each nutrient would perhaps seem like a reasonable way to proceed. Using this approach, uptake of any nutrient would be independent of all others, and would not depend on whether or not that nutrient limits growth.

　　　　　　　 However, experiments have shown that uptake of phosphorus is greater when it is growth-limting than when it is not (Rhee, 1974). Furthermore, when ratios of ambient nutrient concentrations differ greatly from the ideal ratio for phytoplankton (when one or more nutrients is supplied in great excess), straightforward application of separate Michaelis-Menten equations greatly over-estimates uptake of non-limiting nutrients (Droop, 1974; Gotham and Rhee, 1981a, b).

　　　　　　　 To better describe uptake of multiple nutrients over wide ranges of nutrient ratios, models have been developed which reduce (inhibit) uptake rate of any nutrient as its cell quota increases (e.g., Gotham and Rhee, 1981a,b). However, such models are considerably more complex, with more parameters (adjustable constants that must be fit to data).

　　　　　　　 Inspired by the success of the “optimization-based” (or “optimality-based”) model of Pahlow (2005) for describing various complex processes in phytoplankton growth, I decided to investigate whether such an approach might yield better results (or at least a simpler model) for multi-nutrient uptake kinetics.

Optimal Uptake of a Single Nutrient

Pahlow presented a new optimality-based equation for uptake of a single nutrient, which was an extension of the affinity-based uptake model of Aknes and Egge (1999):

Aksnes and Egge (1999) showed that if the affinity, A, and maximum uptake rate, Vmax, are constant, their equation is equivalent to the Michaelis-Menten equation.

Pahlow (2005) extended this by considering separately the surface uptake sites and internal enzymes (or other hardware) required for uptake:

This leads to a slightly different equation:

The optimality comes in here:

How does this relate to the Michaelis-Menten equation?

　　　　　　　 So one can choose parameters such that Pahlow’s optimal uptake equation will give the same uptake rate as the Michealis-Menten equation for any fixed nutrient concentration (with corresponding fixed optimal allocation, fA). The difference between the models will then be increase as nutrient concentration deviates from that fixed value. However, that is not the only way that one could match the two equations.

To examine the minimum difference between the two equations, I conducted a simple fitting exercise. I fit Pahlow’s uptake model to a Michaelis-Menten equation, over different ranges of nutrient concentration:

As expected, over narrow ranges of concentration, the equations can give very similar results. Over wider ranges however, Pahlow’s model will yield greater uptake of nutrient. Of course the real test of the models would be to fit each to measured uptake rates. Here I have only illustrated the minimum difference between the two uptake models.

Multi-Nutrient Optimal Uptake

Considering separate sets of uptake hardware for each nutrient, one could imagine many possible strategies for optimizing uptake. I actually tried many, but I’ll only discuss two.

First, I considered what would be the best solution (truly optimal). To maximize growth rate, the phytoplankton should maximize uptake of whatever nutrient limits their growth rate, while still getting enough of all other nutrients. If resources could be diverted from uptake of one nutrient to another, the truly optimal allocation would be to use the minimum necessary resources for uptake of each non-limiting nutrient (just enough to ensure that it does not become growth-limiting) and to use all remaining resources for uptake of the limiting nutrient. This would mean that phytoplankton would always tend to maintain their optimal nutrient ratios (the ratios defining the co-limitation point for growth). However, even at steady state, the ratios of nutrients in biomass (which must equal the ratios of uptake rates at steady state) do vary widely and depend on the ratio of ambient nutrient concentrations (Droop, 1974: Rhee, 1974). This optimal ratio (or “critical ratio”) can vary with growth rate (Terry et al., 1985), but such variation could not account for the much wider changes in the composition of biomass as a function of ambient nutrient ratios (Droop, 1974: Rhee, 1974).　 Therefore, this strategy must be rejected. Alas, evolution is not perfect, and many biological systems are less than perfectly adapted.

Second, I considered a simpler strategy, in which the amount of internal resource (nitrogen) available for uptake of each nutrient is assumed fixed. Then the only problem is how to allocate this resource between surface sites and internal enzymes. In this simple strategy, the allocation is assumed to be determined only by the ambient concentration of the growth-limiting nutrient.

Here’s an illustration of what this means:

　　　　　　　 Of course this has been all just fun and games so far, with no comparison to data. Is this SPONGE model any good? Here’s a test, with the data of Rhee (1974) from experiments in chemostats at extreme N:P ratios.

　　　　　　　 The model with SPONGE kinetics fits the data much better than the model using Michaelis-Menten kinetics. Under P limitation, it also agrees better with the data than the more complex model of Gotham and Rhee (1981a,b). Under N-limitation, however, Gotham and Rhee’s model fits the data better (as it should, with two more parameters per nutrient).

　　　　　　　 Here’s another test, with experiments by Droop (1974) for a different algae under various degrees of limitation by Vitamin B12 and phosphorus. Droop’s model used Michaelis-Menten kinetics for uptake of the limiting nutrient, and added one more parameter to set uptake of each non-limiting nutrient as a function of the uptake rate of the limiting nutrient. (Thus it has one more parameter per nutrient compared to the SPONGE.)

But how does this SPONGE model work? Here’s a breakdown for the N-limited experiments of Rhee (1974) at input N:P ratio =1:1.

Conclusions

　　　　　　　 Although there is no direct proof for the assumed optimization of uptake hardware, based on my results I propose the hypothesis that phytoplankton do optimize their uptake hardware for changes in concentration of limiting nutrient, but not for changes in ratios of nutrient concentrations. Differences between our model’s predictions and those of other models suggest several means of testing this hypothesis with experiments.

Quantitative comparisons show that our SPONGE model agrees well with data from both N- and P- limited chemostat experiments at extreme nutrient ratios and at ratios more typical of natural environments. It also agrees well the data of Droop (1974) for a different species under various degrees of both vitamin B12- and P-limitation. Although it agrees with data for these nutrients, there is reason to believe that uptake of trace metals and other micronutrients may differ, based on genetic differences among species and on the relative abundances of nutrients in phytoplankton versus seawater (Quigg et al., 2003). For multi-nutrient applications, at least with macronutrients and possibly others, I propose this model as a more versatile alternative to Michaelis-Menten uptake kinetics.

Implications

　　　　　　　 For uptake of a single nutrient, the difference between Michaelis-Menten kinetics and Optimal Uptake kinetics increases with the variability (or range) of nutrient concentration. For multiple nutrients, the difference between Michaelis-Menten kinetics and the SPONGE will also increase with the variability in the ratios of nutrient concentrations (or supply).

　　　　　　　 The simplicity of Optimal Uptake kinetics, for single- or multi-nutrient applications means that it is easy to substitute these equations into existing ecosystem models. In large scale models, Optimal uptake kinetics should give different patterns of biomass (or chlorophyll) and primary production, in both space and time, than Michaelis-Menten kinetics. The SPONGE model should especially be tested in applications with extreme nutrient ratios (e.g., coastal zones with heavy nutrient loading). It could also yield different results in areas where nitrogen fixation and denitrification alter nutrient ratios.

References:

Aksnes, D. L. and J. K. Egge. 1991. A theoretical model for nutrient uptake in phytoplankton. Mar. Ecol. Prog. Ser. 70: 65–72

Caperon, J. 1968. Population growth response of iscochrysis galbana to nitrate variation at limiting concentrations. Ecology 49: 866–872

Droop, M. R. 1968. Vitamin B12 and marine ecology. 4. the kinetics of uptake, growth and inhibition of Monochrysis lutheri. J. Mar. Biol. Ass. U. K. 48: 689–733

Droop, M. R. 1974. The nutrient status of algal cells in continuous culture. J. Mar. Biol. Ass. U.K. 54: 825–855

Gotham, I. J. and G.-Y. Rhee. 1981a. Comparative kinetic studies of nitrate-limited growth and nitrate uptake in phytoplankton in continuous culture. J. Phycol. 17: 309–314

Gotham, I. J. and G.-Y. Rhee. 1981b. Comparative kinetic studies of phosphate-limited growth and phosphate uptake in phytoplankton in continuous culture. J. Phycol. 17: 257–265

Pahlow, M. 2005. Linking chlorophyll-nutrient dynamics to the redfield N:C ratio with a model of optimal phytoplankton growth. Mar. Ecol. Prog. Ser. 287: 33–43.

Quigg, A., Z. V. Finkel, A. J. Irwin, Y. Rosenthal, T.-Y. Ho, J. R. Reinfelder, O. Schofield, F. M. M. Morel, and P. G. Falkowski. 2003. The evolutionary inheritance of elemental stoichiometry in marine phytoplankton. Nature 425: 291−294.

Rhee, G.-Y. 1974. Phosphate uptake under nitrate limitation by scenedesmus sp. and its ecological implications. J. Phycol. 10: 470–475

Terry, K. L., E. A. Laws, and D. J. Burns. 1985. Growth rate variation in the N:P requirement ratio of phytoplankton. J. Phycol. 21: 323–329