Gaël Bougaran
email: gael.bougaran@ifremer.fr

Olivier Bernard

Antoine Sciandra

Modeling continuous cultures of microalgae colimited by nitrogen and phosphorus

Keywords: colimitation, phytoplankton, mathematical model, quota

It is well documented that the combination of low nitrogen and phosphorus resources can lead to situations where colimitation of phytoplankton growth arises, yet the underlying mechanisms are not fully understood. Here, we propose a Droop-based model built on the idea that colimitation by nitrogen and phosphorus arises from the uptake of nitrogen. Indeed, since N-porters are active systems, they require energy that could be related to the phosphorus status of the cell. Therefore, we assumed that N uptake is enhanced by the P quota. Our model also accounts for the biological observations that uptake of a nutrient can be down-regulated by its own internal quota, and succeeds in describing the strong contrast for the non-limiting quotas under N-limited and P-limited conditions that was observed on continuous cultures with Selenastrum minutum and with Isochrysis anis galbana. Our analysis suggests that, regarding the colimitation concept, N and P would be better considered as biochemically dependent rather than biochemically independent nutrients.

Nitrogen and phosphorus are major nutrients that are likely to limit growth of phytoplankton in the natural environment [START_REF] Bergstrom | Phytoplankton responses to nitrogen and phosphorus enrichment in unproductive swedish lakes along a gradient of atmospheric nitrogen deposition[END_REF][START_REF] Davey | Nutrient limitation of picophytoplankton photosynthesis and growth in the tropical north atlantic[END_REF][START_REF] Persic | Changes in n and p limitation induced by water level uctuations in nature park kopacki rit (croatia): nutrient enrichment bioassay[END_REF]. Situations of colimitation where both N and P resources are simultaneously low enough to limit growth have also been reported [START_REF] Suttle | Ammonium and phosphate uptake rates, n:p supply ratios, and evidence for n and p limitation in some oligotrophic lakes[END_REF][START_REF] Elser | Phosphorus and nitrogen limitation of phytoplankton growth in the freshwaters of north america: A review and critique of experimental enrichments[END_REF][START_REF] Maberly | Nutrient limitation of phytoplankton and periphyton growth in upland lakes[END_REF][START_REF] Ren | Nutrient limitation on phytoplankton growth in the upper barataria basin, louisiana: Microcosm bioassays[END_REF]Zohary et al., 2005). These two nutrients play a major role in the cell metabolism as they are part of many biochemical processes. Cell nitrogen is mainly used to build proteins, amino acids and nucleic acids while phosphorus is mostly a constituent of nucleic acids and phospholipids [START_REF] Geider | Redeld revisited : variability of c:n:p in marine microalgae and its biochemical basis[END_REF]. From a functional point of view and according to the terminology of Klausmeier et al. (2004a), the assembly machinery is rich in nitrogen and phosphorus, while the acquisition machinery (pigments and porters) is poor in phosphorus but rich in nitrogen. Although nitrogen and phosphorus are important constituents of the biomass, smaller pools of phosphorus in ATP and NADP also play a major role in the energetic processes.

Since the 70s, several studies have focused on how interactions between nutrients could aect the growth of phytoplankton. Two major models were tested to describe multi-nutrient limitation. The multiplicative model assumes that all nutrients for which the corresponding cell quota is not full contribute to growth limitation. This approach assumes that several nutrients can control growth simultaneously. In cases where several nutrient-quotas are suboptimal the multiplicative model can produce very depressed growth predictions. On the other hand, the threshold model (also called the minimum law) states that growth is always controlled by a single nutrient whose quota is the lowest relative to the actual needs of the cell (i.e. the lowest quota relative to the subsistence quota). This model therefore excludes nutrient colimitation for growth. [START_REF] Droop | Some thoughts on nutrient limitation in algae[END_REF][START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF], who worked on the interaction between vitamin B 12 and phosphorus on Monochrysis lutheri, rst demonstrated that growth rate was limited

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by the most limiting nutrient. [START_REF] Rhee | Eects of n:p atomic ratios and nitrate limitation on algal growth, cell composition, and nitrate uptake[END_REF] achieved the same conclusion on Scenedesmus sp. with nitrogen and phosphorus. [START_REF] Terry | Nitrogen and phosphorus requirements of Pavlova lutheri in continuous culture[END_REF], [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] and [START_REF] Leonardos | Elemental and biochemical composition of Rhodomonas reticulata (cryptophyta) in relation to light and nitrate-tophosphate supply ratios[END_REF] also argued in favor of modeling growth rate based on the minimum law.

On the other hand, N-P colimitation has frequently been invoked in the natural environment where it was demonstrated that N and P had to be available together in order to promote growth of the phytoplankton community [START_REF] Maberly | Nutrient limitation of phytoplankton and periphyton growth in upland lakes[END_REF][START_REF] Seppala | Experimental evaluation of nutrient limitation of phytoplankton communities in the gulf of riga[END_REF]Zohary et al., 2005). Some laboratory experiments on monospecic cultures also gave evidence of more complex N-P interactions. For example, [START_REF] Davies | The photosynthetic response of nutrientdepleted dilute cultures of Skeletonema costatum to pulses of ammonium and nitrate; the importance of phosphate[END_REF] demonstrated that carbon xation of N and P starved Skeletonema costatum could be signicantly stimulated only when both NO - 3 and P O 3- 4 were added simultaneously, but was repressed when only NO - 3 was added. [START_REF] Terry | Nitrogen and phosphorus requirements of Pavlova lutheri in continuous culture[END_REF] suggested that, while the minimum law prevailed for most N:P ratios, a synergetic eect on Pavlova lutheri could occur between the two nutrients in a narrow range of N:P ratio.

Because no clear biochemical process of interaction could be identied between nitrogen and phosphorus metabolism, these two nutrients were previously classied as biochemically independent nutrients [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF][START_REF] Saito | Some thoughts on the concept of colimitation: Three denitions and the importance of bioavailability[END_REF].

Recently, however, [START_REF] Agren | The c:n:p stoichiometry of autotrophs -theory and observations[END_REF] hypothesized that the P-content of ribosomes could perform the main link between N-assimilation and P quota. Under this hypothesis, [START_REF] Pahlow | Chain model of phytoplankton p, n and light colimitation[END_REF] proposed a modeling approach of N-P colimitation where they categorized N and P as biochemically dependent, but the mechanisms leading to a physiological state where both nitrogen and phosphorus interfere for growth were not fully elucidated.

Here we propose a Droop-based model based on the idea that colimitation between nitrogen and phosphorus could arise from the uptake of nutrients rather than from the strict growth function. This model is an extension of the model studied by Klausmeier et al. (2004b). The proposed model resulted from a recursive reverse engineering approach, where a mathematical analysis guided asymmetric response between N-limited and P-limited culture are therefore not detailed, and we refer the reader to [START_REF] Bernard | Multimodel analysis of the response of the coccolithophore Emiliania huxleyi to an elevation of pCO 2 under nitrate limitation[END_REF] for an illustration of this approach.

We demonstrate the ability of our model, which is an extension of [START_REF] Bougaran | Modelling continuous cultures of microalgae colimited with nitrogen and phosphorus[END_REF], to produce a correct quantitative and qualitative representation of the experimental data obtained on Selenastrum minutum by [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] and our own data on Isochrysis anis galbana. We further compare the model output to the original model of Klausmeier et al. (2004b). Finally, we discuss the dierent modeling approaches for N-P colimitation and the biological basis for our model.

Material and methods

Isochrysis anis galbana

Isochrysis anis galbana CCAP 927/14 was grown in 2 -L photobioreactor chemostats [START_REF] Bougaran | Transient initial phase in continuous culture of Isochrysis galbana anis Tahiti[END_REF] at dilution rate ranging from 0.1 to 1.1 d -1

under dierent N : P ratios. Cultures were illuminated with continuous light (130 μmol.m -2 .s -1 ), temperature was maintained at 27 • C and pH at 7.2. The Walne medium was modied so as to obtain N-or P-limited medium: for the N-limited medium, NaNO 3 was decreased from 1120 μM to 115 μM (N : P = 0.7); for the P-limited medium, NaH 2 P O 4 was decreased from 159 μM to 8 μM (N : P = 137). Six cultures were grown under N limitation and 4 under P limitation.

All measurements were made on culture samples at steady-state. Chemical analysis of residual N and P concentrations were carried out on a DIONEX ion-chromatography (AS9-HC column). C and N quotas were measured on a

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CHN-elemental analyzer. P quota (denoted q P ) was estimated at steady-state from residual phosphate concentration (1):

q P = p in -p x (1)
where p in is the phosphate concentration in the renewal medium, p is the residual concentration of phosphate in the culture medium and x the carbonbiomass at steady-state.

2.2. Selenastrum minutum [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] grew Selenastrum minutum under phosphate or nitrate limitation in axenic chemostats, with dilution rate ranging from 0.1 to 1.7 d -1 . A constant photon ux density of 100 μmol.m -2 .s -1 was provided at the surface of the culture. Nitrate-limited medium was supplied with 100 μM KN O 3 and 100 μM N a 2 HP O 4 (N : P =1), whereas phosphorus limited medium was enriched with 2000 μM KN O 3 and 10 μM N a 2 HP O 4 (N : P =200). Particulate carbon and nitrogen were measured on an elemental analyzer, whereas particulate phosphorus was analyzed following [START_REF] Afghan | Nutrients[END_REF]. The data of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] were digitalized (using Epson 1640 SU scanner) and the C-based quotas were computed.

Model development

Presentation of the Droop model

The most widespread model to reproduce growth of microalgae under substrate limitation is the Droop model [START_REF] Burmaster | The unsteady continuous culture of phosphate-limited Monochrisis lutheri droop : Experimental and theoretical analysis[END_REF]Droop, 1968b[START_REF] Droop | 25 years of algal growth kinetics. a personnal view[END_REF]. It is widely used since it reproduces the ability of microalgae to uncouple substrate absorption and growth. Its simple structure made it possible to study and mathematically characterize its behavior [START_REF] Bernard | Transient Behavior of Biological Loop Models, with Application to the Droop Model[END_REF][START_REF] Lange | The attractiveness of the Droop equations[END_REF][START_REF] Vatcheva | Experiment selection for the discrimination of semi-quantitative models of dynamical systems[END_REF]. However, when both nitrogen and phosphorus are at suboptimal growth level at the same time, a new model must be designed. In contrast to the Monod model, the Droop model (Droop, 1968b[START_REF] Droop | 25 years of algal growth kinetics. a personnal view[END_REF] states half saturation constant for phosphate uptake (molP.L -1 ) that growth of a phytoplanktonic population characterized by its carbon biomass (x) is related to the cell quota (q) of the limiting nutrient rather than to ambient extracellular concentration. When considering nitrogen in a chemostat the Droop model can be written as follows:

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⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ṅ = Dn in -ρ N (n)x -Dn q N = ρ N (n) -μ(q N )q N ẋ = μ(q N )x -Dx (2)
where q N is the internal carbon-based quota of nitrogen, D is the dilution rate of the chemostat, and n and n in are the concentrations of inorganic nitrogen in the culture medium and in the renewal medium, respectively. The functions ρ N and μ represent inorganic nitrogen uptake and growth rate, respectively. With a phosphorus limitation, the model is written:

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⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ṗ = Dp in -ρ P (p)x -Dp q P = ρ P (p) -μ(q P )q P ẋ = μ(q P )x -Dx (3)
In this model the specic uptake rate ρ and growth rate μ are Michaelis-Menten and Droop functions, respectively:

ρ(s) = ρ m s s+Ks μ(q) = μ(1 -q0 q ) (4)
Where K s is the half saturation constant for uptake, ρ m is the maximum uptake rate, q 0 the minimal subsistence quota of starved cells for the limiting nutrient and μ the hypothetical maximum growth rate obtained for an innite value of the quota for the limiting nutrient.

It can be proved [START_REF] Bernard | Transient Behavior of Biological Loop Models, with Application to the Droop Model[END_REF] that the internal quota for a limiting nutrient stays between two bounds q 0 ≤ q ≤ q exp

(5)

Then, q exp , the maximum quota obtained at maximum growth rate (μ max ) [START_REF] Bernard | Transient Behavior of Biological Loop Models, with Application to the Droop Model[END_REF] can be computed:

q exp = q 0 + ρ m μ (6)
and nally the maximum growth rate is μ max = μ(q exp ).

Remark 1: in the following, we will write :

μ max = μ(1 - q 0 q exp ) (7)
At steady-state, the internal quota is an increasing function of the dilution rate:

q = q 0 μ μ -D (8)
where the denotes the value at steady state. This model has been validated in depth and has been proved to be relevant [START_REF] Droop | 25 years of algal growth kinetics. a personnal view[END_REF][START_REF] Sciandra | The limitations of continuous cultures with low rates of medium renewal per cell[END_REF] to represent conditions of nitrogen

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or phosphorus limitations even in dynamic inuent conditions [START_REF] Bernard | Nonlinear observers for a class of biological systems. application to validation of a phytoplanktonic growth model[END_REF][START_REF] Bernard | Nonlinear qualitative signal processing for biological systems: application to the algal growth in bioreactors[END_REF].

Extension of Droop's model to handle colimitation

As in Klausmeier et al. (2004a), we propose to modify the Droop model in order to handle colimitation by nitrogen and phosphorus. Klausmeier et al. (2004a) categorized the various growth processes into the acquisition and assembly machineries. The acquisition machinery that combines pigments and membrane porters is N-rich but P-poor; we will however consider that phosphorus may play a major role at the acquisition level since it is involved in the active uptake of both nitrogen (NO - 3 ) and inorganic phosphorus. Conversely, the assembly machinery corresponds to ribosomes and is N and P-rich. Consequently, we now state the general assumptions that uptake of a given nutrient is function of both its external concentration and the internal quota of both nutrients. We also assume that growth rate can be driven by either the nitrogen or the phosphorus quota. The resulting generic model is as follows:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ṅ = Dn in -ρ N (
n, q N , q P )x -Dn ṗ = Dp inρ P (p, q N , q P )x -Dp q N = ρ N (n, q N , q P )μ(q N , q P )q N q P = ρ P (p, q N , q P )μ(q N , q P )q P ẋ = μ(q N , q P )x -Dx (9) This general model structure accounts for various models described in the literature [START_REF] Davidson | An investigation of non-steady-state algal growth. ii. mathematical modelling of co-nutrient-limited algal growth[END_REF][START_REF] Legovic | A model of phytoplankton growth on multiple nutrients based on the michaelis-menten-monod uptake, droop's growth and liebig's law[END_REF]Klausmeier et al., 2004b). In the model of Klausmeier et al. (2004b), the nutrient uptake rates ρ P (p, q N , q P ) and ρ N (n, q N , q P ) are represented by Michaelis-Menten kinetics:

ρ P (p, q P ) = ρ P max (q P ) p p + K SP and ρ N (n, q N ) = ρ N max (q N ) n n + K SN ( 10
)
where the maximum uptake rates ρ P max and ρ N max are down-regulated by

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the corresponding internal quota, using the original inhibition formulation of [START_REF] Gotham | Comparative kinetic studies of phosphatelimited growth and phosphate uptake in phytoplankton in continuous culture[END_REF].

We will base our uptake model on hypotheses diering from the kinetics in equations 10. We will now introduce the chosen expressions for ρ N , ρ P and μ.

Modeling of the growth rate

We based our modeling approach on the assumption that N and P are both involved within a constant ratio in nucleic acids and particularly in RNA. Following the hypothesis that ribosome/RNA intra-cellular concentration is strongly correlated with growth rate (Klausmeier et al., 2004a), we assume that N or P can either limit growth depending on which of the two quotas is the most limiting. This hypothesis is supported by the results of [START_REF] Berdalet | Eects of nitrogen and phosphorus starvation on nucleic acid and protein content of Heterocapsa sp[END_REF] on Heterocapsa sp. who found that both N and P starvation aected the RNA content.

We therefore assume, in line with [START_REF] Droop | Some thoughts on nutrient limitation in algae[END_REF][START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF] and [START_REF] Rhee | Eects of n:p atomic ratios and nitrate limitation on algal growth, cell composition, and nitrate uptake[END_REF], that the most limiting nutrient will drive the growth rate, resulting in a minimum law:

μ = min(μ N (q N ), μ P (q P )) (11)
For sake of simplicity, we chose the following expression:

μ = μ max min 1 -qN0 qN 1 -qN0 qNexp , 1 -qP 0 qP 1 -qP 0 qP exp (12)
where parameters q N exp and q P exp represent the values of nitrogen and phosphorus quota for exponential growth when the growth rate has reached its maximum value μ max .

As a consequence, we will assume in the steady-state analysis that the N-

limited growth rate is μ = μ max 1- q N 0 q N 1- q N 0 q N exp while the P-limited growth rate is μ = μ max 1- q P 0 q P 1- q P 0 q P exp .
Hence, following Droop's theory (see equation 8), the limiting quota q N is an increasing function of the growth rate for N-limited growth, while the limiting

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quota q P is an increasing function of the growth rate for P-limited growth.

When N is in excess, the internal quota of N is called the saturated quota (the same term is used reciprocally for P when P is in excess).

Modeling of the uptake rate

The classical modeling for uptake of an external nutrient is the Michaelis-Menten based modeling of [START_REF] Dugdale | Nutrient limitation in the sea: dynamics, identication and signicance[END_REF].

ρ(p, q P ) = ρ P max (q P ) p p + K p (13)
However, it was demonstrated that the combination of the Droop's equation for the growth rate and the Michaelis-Menten uptake kinetics greatly overestimates the experimental uptake rates for the non-limiting nutrient under extreme N:P ratios [START_REF] Gotham | Comparative kinetic studies of phosphatelimited growth and phosphate uptake in phytoplankton in continuous culture[END_REF]Klausmeier et al., 2004b). Indeed, it is usual to consider that uptake of a nutrient can be down-regulated by its own internal quota [START_REF] Morel | Kinetics of nutrient uptake and growth in phytoplankton[END_REF]. This reasoning leads to a slightly dierent uptake model, as developed by [START_REF] Lehman | The assumptions and rationales of a computer model of phytoplankton population dynamics[END_REF].

We therefore introduced a down-regulation term in the maximum P-uptake rate:

ρ P max (q P ) = ρP max q P Lq P q P Lq P 0 (14

)
where q P L is the hypothetical maximum value for the P quota and ρP max is the maximum value of the uptake rate, when q P = q P 0 . As Flynn (2008) pointed out, there is no particular reason for the critical level q P L to equal the Droopmaximum quota q P exp (and this obviously holds for q NL also). Indeed, the data of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] showed that the maximum achievable quota of a non limiting nutrient could exceed the observed q exp ( 6). For a given dilution rate, [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] dened the luxury consumption capacity as the ratio of the saturated quota to the limiting quota. Luxury uptake was observed to be particularly high for P that could be stored in quantities up to 16 times the actual needs of Selenastrum minutum, while luxury consumption of N did not In conditions of nitrogen limitation and excess P supply (P >> K SP ), we obtain the following from equation ( 9): qP = ρP max q P Lq P q P Lq P 0 -Dq P (15)

and at steady-state, we can compute the saturated quota:

q P = ρP max q P L ρP max + D(q P L -q P 0 ) (16)
It follows that the saturated q P is a decreasing function of the dilution rate. From Droop's theory (see equation 8), the limiting q N is an increasing function of the dilution rate, and consequently the saturated q P is a decreasing function of the limiting q N . This has already been reported by [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF], who observed that the saturated phosphorus quota decreased for N-limited culture (see gure 1).

Remark 2: This conclusion is similar to what can be expected with the model of Klausmeier et al. (2004b), in the situation of excess P supply ( P >> K SP ). Indeed, the kinetics for phosphorus uptake are identical (see equation ( 10)). Since, in both models, at steady state μ = D, equation ( 16) also holds for the model of Klausmeier et al. (2004b), the predicted saturated phosphorus quota decreases with respect to D.

However the situation is dierent for nitrogen. [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] showed that the saturated nitrogen quota of P-limited culture increases with the dilution rate (see gure 1). Our previous reasoning for P-limitation proves that the N uptake rate cannot be regulated solely by the internal nitrogen quota.

Otherwise, it would lead to an equation similar to ( 16), where the saturated q N would decrease with D.

We will now explore another mechanism based on the idea that, as the transport of nitrate is active, it requires energy (i.e. ATP) and cofactors provided

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by the phosphorus pool [START_REF] Crawford | Molecular and physiological aspects of nitrate uptake in plants[END_REF][START_REF] Forde | Nitrate transporters in plants : structure, function and regulation[END_REF]. Thus, the modeling of N uptake must reproduce the fact that the P quota of P-starved cells decreases and, as a consequence, the N uptake also decreases.

Therefore, we assume that the uptake of nitrogen is enhanced by the phosphorus quota:

ρ N max (q N , q P ) = ρNmax q NLq N q NLq N 0 q Pq P 0 q P Lq P 0 (17

)
We can thus infer that, in transient situations where P becomes limiting, the P-status of the cells could induce a N uptake reduction leading to N-P colimitation. Conversely, when q P is high, the P-driven term allows a maximum potential for N uptake.

Let us compute the saturated quota q N under P limitation. From equation ( 12) in P-limited conditions, the growth rate can be represented by the Droop model:

μ(q P ) = μ(1 - q P 0 q P ) (18)
At steady-state in the chemostat, when N is supplied in excess (i.e. N >> K SN ), we have ρ N max (q N , q P ) = μ(q P )q N : ρNmax q NLq N q NLq N 0 q Pq P 0 q P Lq P 0 = μ(1 -

q P 0 q P )q N (19)
From this equation, the steady-state value of the saturated nitrogen quota can be computed as follows:

q N = ρNmax q P μ(q NL -q N 0 )(q P L -q P 0 ) + ρNmax q P q NL (20)
Thus, the saturated q N is an increasing function of the limiting q P . Since, from equation ( 8), the limiting q P is itself an increasing function of the dilution rate, it follows that the saturated q N is increasing with the dilution rate. It is notable that this conclusion is the opposite to the decreasing behavior of the saturated q P under N limitation.

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Remark 3: If we consider the model of (Klausmeier et al., 2004b) in the situation of excess N supply (N >> K SN ), we get from equation ( 10):

0 = ρ N max q NL -q N q NL -q N 0 -Dq N (21)
and therefore q N = ρNmaxqNL ρNmax+D(qNL-qN0)

. As a consequence, the saturated q N is a decreasing function of the dilution rate. This response is therefore opposite to our model prediction.

Comparison of the model output with the experimental datasets

Qualitative comparison

The chemostat experiments with Isochrysis anis galbana under two extreme N:P ratios gave contrasting patterns for the N and P quotas depending on the limiting nutrient. The steady-state carbon quota of cells was high at low growth rates and decreased to a minimum (0.5 pmol.cell -1 ) at μ=0.8 d -1 . At low growth rates, the steady-state carbon quota appeared to be slightly larger under P limitation (0.9 pmol.cell -1 ) than under N limitation (0.8 pmol.cell -1 ).

The steady-state limiting quota for both N and P increased with growth rate, in agreement with the Droop theory. Steady-state non-limiting P quota was high at low growth rate under N limitation and declined with growth rate. The nonlimiting P quota was always higher than that required to support that growth rate. On the other hand, the steady-state non-limiting N quota increased with growth rate and reached a maximum value for the highest dilution rate tested.

Our results emphasize that luxury consumption capacity was much higher for P than for N. For all the dilution rates tested, the residual concentration of the non-limiting nutrient was high (N > 160 μM and P > 110 μM ), while it was undetectable for the limiting nutrient. [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] observed the same behavior for limiting and nonlimiting quotas on Selenastrum minutum. They specied that ambient phosphorus was present (40-60 μM ) under nitrate limitation, and that nitrate was present (1.25-1.50 mM ) under phosphate limitation.

In both studies, the draw-down for the non-limiting nutrient was incomplete.

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The non-limiting quotas were always at their maxima and were assumed to be saturated. The maximum possible quota for P occurred at the lowest growth rate while the maximum N quota was achieved at μ exp and was equivalent to q N exp .

Model calibration

Model parameters were identied as follows. μ max for Selenastrum minutum was taken from [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF], and for Isochrysis anis galbana it was determined from additional experiments. q N 0 and q P 0 and μN and μP were identied by tting the experimental data to the standard Droop equation ( 7). ρNmax and ρP max were computed from:

ρmax = μμ max μ -μ max q 0 (22)
Finally, q N exp and q P exp where computed from equation ( 6).

The anity constants K SN and K SP for Selenastrum minutum were taken from [START_REF] Nielsen | Growth of plankton algae as a function of n-concentration, measured by means of batch technique[END_REF] and [START_REF] Ahlgren | Growth of oscillatoria agardhii in chemostat culture: 1. nitrogen and phosphorus requirements[END_REF] respectively. They were identied thanks to additional experiments for Isochrysis anis galbana. The values of q P L , q NL , ρN max and ρP max have been adjusted in order to obtain the best t between model predictions and steady-state measurements of q P and q N .

Quantitative comparison

The model output was compared to the experimental datasets for saturated and limiting quotas at steady-state for both Isochrysis anis galbana and Selenastrum minutum [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF]. gure 1 and gure 2 demonstrate the ability of our model to correctly represent the experimental data of the two species both qualitatively and quantitatively (parameter values are given in table 2). The steady-state output of our model correctly reproduces the increasing trend of the limiting q N and q P and the strong asymmetry between the responses of the saturated quotas q N and q P . 6 10 -2 6.5 10 -2 molN. molC -1 q p0

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1.8 10 -3 0.9 10 -3 molP. molC We also tted the experimental data for the two species with the model of Klausmeier et al. (2004b) extended with the uptake inhibition formulation of [START_REF] Gotham | Comparative kinetic studies of phosphatelimited growth and phosphate uptake in phytoplankton in continuous culture[END_REF]. gure 1 shows that, while this model succeeds in tting the N and P limiting quotas and saturated q P correctly, it predicts a decline of the saturated q N with growth rate (as forecasted by our analytical computations and explained in Remark 3), which strongly contrasts with the increasing trend in the experimental data.

Discussion

Asymmetry between N and P uptake

Multi-nutrient limitation has recently become the focus of renewed attention. In the particular case of N and P, there is experimental evidence for an asymmetric response between the N and P non-limiting quotas under P and N limitation respectively. An additional asymmetry is observed for the behavior of q p under P-and N-limited conditions, while the q N trend appears to be irrespective of what nutrient (N or P) is limiting growth. Most modeling approaches for phytoplankton growth have a symmetric formulation and therefore cannot describe the observed asymmetry [START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF]Klausmeier et al., 2004b [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF]. The top gures show q N against μ under nitrogen limitation (a) and phosphorus limitation (b). The bottom gures show q P against μ under nitrogen limitation (c) and phosphorus limitation (d). Solid line : proposed colimitation model with parameter values as in table 2. Dashed line : model of Klausmeier et al. (2004b). and[START_REF] Smith | Optimization-based model of multinutrient uptake kinetics[END_REF]. This was illustrated in the particular case of the model of Klausmeier et al. (2004b) with the experimental data of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF].

The model proposed in the present paper is based on the idea, also developed by others [START_REF] Agren | The c:n:p stoichiometry of autotrophs -theory and observations[END_REF][START_REF] Davidson | An investigation of non-steady-state algal growth. ii. mathematical modelling of co-nutrient-limited algal growth[END_REF][START_REF] Pahlow | Chain model of phytoplankton p, n and light colimitation[END_REF], that a non-limiting nutrient can control the limitation exerted by another Oschlies ( 2009), but contrary to the classical assumption that N and P lead to independent limitation, we conclude that these two nutrients would be better considered as biochemically-dependent nutrients, as dened by [START_REF] Arrigo | Marine microorganisms and global nutrient cycles[END_REF] and [START_REF] Saito | Some thoughts on the concept of colimitation: Three denitions and the importance of bioavailability[END_REF].

Model prediction at physiological limit boundaries

Phytoplankton cell nutrient quota at a given growth rate in the chemostat can vary between two boundaries. Indeed, if the growth is limited by the nutrient being considered, the internal quota is at its minimum level (Droop, 1968a). When this nutrient is in excess and growth is limited by another nutrient, cells accumulate the non-limiting nutrient up to a maximal quota. Hence, we can dene the physiological limit boundaries [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] as the maxi-

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mal and minimal attainable quota values for a given growth rate. Accordingly, the luxury consumption capacity, for a given growth rate can be dened from these physiological limit boundaries as the ratio of the maximum quota to the minimum quota. Under the extreme N:P input-ratio tested experimentally with both Selenastrum minutum and Isochrysis anis galbana, we assumed that the quota of the non-limiting nutrient was saturated, while the quota of the limiting nutrient was restricted to its minimum value for the given growth conditions. Therefore we can assume that the physiological limit boundaries were reached.

Under very low N:P input ratio, the model ts the data, with the saturated q P decreasing as the N-limited growth rate increases, according to equation ( 16). On the contrary, the experimental data of the saturated q N under P-limitation turned out to be an increasing function of the growth rate for both Selenastrum minutum and Isochrysis anis galbana. The saturated-q N behavior matches the hypothesis of [START_REF] Droop | 25 years of algal growth kinetics. a personnal view[END_REF] that a non-limiting quota follows the Droop equation with a higher apparent q N 0 . As our analysis pointed out, the model of Klausmeier et al. (2004b), which only integrates the inhibiting eect of q N over N uptake, predicts a decreasing trend for the saturated-q N with the growth rate, as is observed for the saturated-q P under N-limiting conditions (see equation 16). Therefore, the limiting eect of q P on N uptake is necessary to t the physiological boundaries correctly (equation 20). The asymmetric functions for N (structural) and P (functional) and particularly the energetic role of P at the acquisition level were therefore critical in our modeling approach. Thus, although the acquisition machinery is P-poor, P appears to play a major role in the uptake of NO - 3 .

The phenomenon of luxury consumption of inorganic nutrients by phytoplankton has been widely documented [START_REF] Ketchum | The absorption of phosphate and nitrate by illuminated cultures of Nitzschia closterium[END_REF][START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF]. gure 3a shows that the capacity of N-luxury consumption decreases with growth rate, as does P (gure 3b). The capacity of N-luxury consumption has been reported to be lower than that of P [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF][START_REF] Terry | Nitrate and phosphate uptake interactions in a marine prymnesiophyte[END_REF] and our model succeeds in describing this dierence.

Model prediction inside the physiological limit boundaries

In cases where N:P input ratio is closer to the critical ratio for each species, one might suspect dierent behavior for the non-limiting quota, as it may not reach saturation. Indeed, the asymmetric structure of our model results in an asymmetric behavior for the non-limiting q N and q P , as well as in uptake-control contrasts depending on (1) whether N or P is limiting and (2) the growth rate.

q P varies according to the description of Klausmeier et al. (2004b): under P- limited growth rate or near the critical ratio, uptake-control is exerted solely by the low P-resource (i.e. P/(P + K p ) < 1) and q P builds-up according to the Droop model (gure 4b for N:P=10-100).

Conversely, under N-limited growth rate, the model predicts that q P may either increase or decrease, depending on the dilution rate and the N:P-input ratio in the chemostat. As long as q P -saturation is not achieved, uptake control

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is solely exerted by the low P resource. q P increases with growth rate because P resource increases (i.e.P /(P + K p ) < 1). Higher N-limited growth rates can lead to situations where the P-uptake capacity then becomes limiting and q P decreases and converges towards q P exp (see gure 4b for N:P= 2).

Inside the physiological limit boundaries, N-uptake control under P-limited growth rate is even more complex because it is also governed by q P : at low P- limited growth rate, N uptake is strongly forced by both the limited N-resource (i.e.N/(N + K n ) < 1) and the low ATP-pool that limits the activity of N porters (gure 4a). Since both the available N and P resources increase with the P-limited growth rate, q P increases and favors uptake of the less limiting N, which results in an increase of q N with growth rate. The mechanistic basis of our model therefore contrasts with the hypothesis of Klausmeier et al. (2004a) that P-limitation favors resource acquisition, at least for nitrate.

There are several other datasets in the literature that describe the response of q P and q N to growth rate under varying N:P ratio, but few are as comprehensive as the work of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF], or investigated such low N:P ratios. Indeed, as far as we know, only the studies of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] and [START_REF] Rhee | Phosphate uptake under nitrate limitation by Scenedesmus sp. and its elcolgical implications[END_REF] provided evidence that the physiological limit boundaries were reached. [START_REF] Healey | Interacting eects of light and nutrient limitation on the growth rate of synechococcus linearis (cyanophyceae)[END_REF] investigated a rather large range of N:P input ratio (2.5-100) in his study on the interacting eects of light and nutrient limitation on Synechococcus linearis. The data showed an increasing trend for both N-and Plimiting quotas with growth rate. The experimental non-limiting q N was con- rmed to be an increasing function of the growth rate, with a rather higher apparent q N 0 , as predicted by our model and observed by [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF], but not by others [START_REF] Laws | Nutrient-and light-limited growth of thalassiosira uviatilis in continuous culture, with implications for phytoplankton growth in the ocean[END_REF]. Nevertheless, given the scatter in the data, the non-limiting q P of Synechococcus linearis exhibited no particular trend in the range of the growth rate tested.

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m a n u s c r i p t [START_REF] Goldman | Steady-state growth and chemical composition of the marine chlorophyte Dunaliella tertiolecta in nitrogen-limited continuous cultures[END_REF] demonstrated that N-limited continuous cultures of Dunaliella tertiolecta exhibit increasing (limiting) q N and (non-limiting) q P with growth rate under N:P ratios ranging from 5 to 15. However, the authors clearly specied that virtually all the inuent P O 3- 4 was stripped from the medium at all the values of μ up to 1.1 d -1 . This makes it possible to infer that the physiological limit boundary was not attained and that q p was not saturated, even under the lowest N:P ratio tested. Calculations based on the dataset of [START_REF] Healey | Interacting eects of light and nutrient limitation on the growth rate of synechococcus linearis (cyanophyceae)[END_REF] seem to indicate that ambient P O 3- 4 was remaining under N limitation. Nevertheless, since [START_REF] Healey | Interacting eects of light and nutrient limitation on the growth rate of synechococcus linearis (cyanophyceae)[END_REF] did not give any direct indications for ambient P O 3- 4 it can hardly be ascertained whether q p was saturated or not.

Interestingly, the model predicts striking modications for q p behavior near the limit physiological boundary even for a small increase of the N:P-input ratio in the range 1 to 10. The relatively high values for the minimal N:P-input ratio tested by [START_REF] Goldman | Steady-state growth and chemical composition of the marine chlorophyte Dunaliella tertiolecta in nitrogen-limited continuous cultures[END_REF] (N:P = 5) and [START_REF] Healey | Interacting eects of light and nutrient limitation on the growth rate of synechococcus linearis (cyanophyceae)[END_REF] (N:P = 2.5) could explain why none of the authors observed the typical decrease in the saturated q P at the physiological boundary, as predicted by the model. Indeed, it is clear from gure 4b that under input ratio values of 2 and 5, q p increases for low growth rates and decreases for higher growth rates. In his experiment on P uptake under nitrate limitation by Scenedesmus.sp in a chemostat, [START_REF] Rhee | Phosphate uptake under nitrate limitation by Scenedesmus sp. and its elcolgical implications[END_REF] investigated N:P ratios as low as 1:1. Unfortunately, due to the scatter in the data for q P and because the quota is given as cell quota rather than carbon-based quota, it is dicult to ascertain whether these data support the model's decreasing trend for saturated q P or not.

Our model highlights an important feature for q P : conversely to q N , the q P behavior is very sensitive to the N:P input ratio. On one hand, a slight increase of extremely low N:P input ratio can result in striking modications in the q P behavior. On the other hand, under N-limited growth but close to the critical ratio, q P remains rather constant with growth rate (see gure 4 for N:P= 5). growth rate under dierent N:P input ratio ranging from 1 to 100. Numbers represent the values of the N:P input ratio. Dotted-lines represent situations where uptake is substratelimited (S/(S + K S ) < 0.9). The solid lines represent situations where the uptake-capacity is limiting.

This result has signicant implications for experiments aiming to examine N-P colimitation, which should be designed so as to guarantee that the physiological limit boundaries are actually attained.

Figure 5 shows simulation results for the variations of q N : q P against growth rate under dierent input N:P ratios ranging from 1 to 100. At low growth rates q N : q P remains equivalent to the input ratio as long as the input ratio falls within the physiological limit boundaries and neither of the two nutrients is limiting. Increase in growth rate results in alteration of the q N : q P ratio driven by the physiological limit boundaries to the quota, as we described above for extreme N:P input ratios.

Comparison with other N-P colimitation models

The mechanisms of N-P colimitation remain ambiguous, because N and P are involved together in many biochemical processes. Dierent modeling approaches were thus proposed for N-P colimitation over the last decade. Some q :q (mol N . (mol P) ) N P Figure 5: Steady-state predictions for the q N : q P ratio against growth rate under dierent N:P input ratio ranging from 1 to 100. Numbers represent the values of the N:P input ratio. of these converged towards the hypothesis that interaction between nutrients takes place at the acquisition level, which characterizes the interface between the environment and the intracellular compartment [START_REF] Flynn | The importance the form of the quota curve and control of non-limiting nutrient transport in phytoplankton models[END_REF]Klausmeier et al., 2004b[START_REF] Klausmeier | A model of exible uptake of two essential resources[END_REF][START_REF] Smith | Optimization-based model of multinutrient uptake kinetics[END_REF]. On the other hand, both [START_REF] Agren | The c:n:p stoichiometry of autotrophs -theory and observations[END_REF] and [START_REF] Pahlow | Chain model of phytoplankton p, n and light colimitation[END_REF] developed models based on the idea that colimitation occurs in the assembly machinery.

Several models of colimitation have been based on the Droop theory and since [START_REF] Droop | Some thoughts on nutrient limitation in algae[END_REF][START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF] and [START_REF] Rhee | Eects of n:p atomic ratios and nitrate limitation on algal growth, cell composition, and nitrate uptake[END_REF] demonstrated that the multiplicative law generally underestimates growth rate, the growth function was often represented as a minimum law rather than the multiplicative one [START_REF] Davidson | An investigation of non-steady-state algal growth. ii. mathematical modelling of co-nutrient-limited algal growth[END_REF]Klausmeier et al., 2004b;[START_REF] Smith | Optimization-based model of multinutrient uptake kinetics[END_REF]. In addition, an internal quota down-regulation of the uptake system was sometimes incorporated into the model to prevent overestimation of the quota for the non-limiting nutrient [START_REF] Davidson | An investigation of non-steady-state algal growth. ii. mathematical modelling of co-nutrient-limited algal growth[END_REF][START_REF] Gotham | Comparative kinetic studies of phosphatelimited growth and phosphate uptake in phytoplankton in continuous culture[END_REF]Klausmeier et al., 2004b). [START_REF] Smith | Optimization-based model of multinutrient uptake kinetics[END_REF] [START_REF] Pahlow | Linking chlorophyll-nutrient dynamics to the redeld n:c ratio with a model of optimal phytoplankton growth[END_REF] where phytoplankton allocate internal N between surface uptake sites and internal enzymes in the same proportion for all nutrients. This model was in good agreement with the tted data of [START_REF] Droop | The nutrient status of algal cells in continuous culture[END_REF] and [START_REF] Rhee | Phosphate uptake under nitrate limitation by Scenedesmus sp. and its elcolgical implications[END_REF], except at the extreme N:P ratios where it did not perfectly agree with the data of Rhee.

Interestingly, the model of Klausmeier et al. (2004b) reproduced the behavior of q P under N and P limiting conditions, with general trends resembling those of our model. Nevertheless, as our analysis pointed out, the same behavior was assumed for q N , although non limiting q N was experimentally demonstrated to be an increasing function of the growth rate, even under very high N:P ratio [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF][START_REF] Healey | Eect of phosphorus deciency on 2 algae growing in chemostats[END_REF][START_REF] Terry | Nitrate and phosphate uptake interactions in a marine prymnesiophyte[END_REF].

Another important feature for models of N-P colimitation is their ability to describe the asymmetry between non-limiting N and P quotas. Hence, the modeling approaches can be categorized into two broad types depending on whether they include a symmetric or an asymmetric formulation for the dierent nutrients. The classical multiplicative formulation of multi-nutrients models induces a symmetric behavior for the dierent nutrients and is, therefore, inappropriate for describing N-P colimitation. Conversely, as discussed by [START_REF] Pahlow | Chain model of phytoplankton p, n and light colimitation[END_REF], the threshold model, which links growth rate to the quota of a suboptimal resource, could principally achieve asymmetry but this is not mechanistically based and would restrict the range of q N : q P ratios observed experimentally. [START_REF] Flynn | The importance the form of the quota curve and control of non-limiting nutrient transport in phytoplankton models[END_REF] approached the asymmetry of the non limiting q P and q N observed by Elri and Turpin (1985) for Selenastrum minutum with a twostate model, where the uptake functions were of the classical Michaelis-Menten type for the limiting nutrients, and were multiplied by a function depending on q N and q P in order to reproduce the kinetic of the non limiting quota.

Very recently, [START_REF] Pahlow | Chain model of phytoplankton p, n and light colimitation[END_REF] developed a chain model that includes P-control of N-assimilation in terms of the anity-based uptake function, and where q N ultimately determines growth rate. These authors based their mod-

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eling approach on the idea that the P content of ribosomes could act as the mechanistic link between q P and N assimilation, as proposed by Agren (2004).

This model succeeds in describing the asymmetric behavior of q P and q N for a range of experimental datasets for dierent phytoplankton species.

Here we propose another mechanism: that active N uptake can be regulated by the cellular P tied to the ATP-pool. Although the P fraction is low in the acquisition machinery, there are strong indications in the literature that nutrient uptake can compete with C-xation for ATP. [START_REF] Falkowski | Nitrate uptake in marine phytoplankton: Energy sources and the interaction with carbon xation[END_REF] showed that a pulse of either NO - 3 or NH + 4 resulted in a transitory decrease in carbon xation per unit chlorophyll a for natural phytoplankton. Additional experiments with an uncoupler of the two light reactions led him to infer that cyclic photophosphorylation is a primary energy source for NO - 3 uptake. [START_REF] Davies | The photosynthetic response of nutrientdepleted dilute cultures of Skeletonema costatum to pulses of ammonium and nitrate; the importance of phosphate[END_REF] also performed pulse-experiments on simultaneously N-and P-starved Skeletonema costatum. The authors reached the same conclusion as [START_REF] Falkowski | Nitrate uptake in marine phytoplankton: Energy sources and the interaction with carbon xation[END_REF] that a pulse of either NH + 4 or NO - 3 resulted in a decrease in the steady-state rate of carbon xation. On the other hand, the joint addition of N and P resulted in a large stimulation of carbon xation. [START_REF] Davies | The photosynthetic response of nutrientdepleted dilute cultures of Skeletonema costatum to pulses of ammonium and nitrate; the importance of phosphate[END_REF] assumed that the uptake of nitrogen was itself phosphate limited and that, under conditions of extreme and continuing phosphorus limitation, the repression of carbon xation would be expected to be long-lasting. A similar decrease of carbon xation rates for dierent inorganic nutrients has also been reported by several other investigators [START_REF] Glover | Physiological responses of sargasso sea picoplankton to nanomolar nitrate perturbations[END_REF][START_REF] Smalley | Feeding in the mixotrophic dinoagellate ceratium furca is inuenced by intracellular nutrient concentrations[END_REF][START_REF] Beardall | A comparison of methods for detection of phosphate limitation in microalgae[END_REF]. Thus, pulse-experiments and particularly that of [START_REF] Davies | The photosynthetic response of nutrientdepleted dilute cultures of Skeletonema costatum to pulses of ammonium and nitrate; the importance of phosphate[END_REF] support our hypothesis that, under P limitation, competition for ATP occurs between N uptake and dark reactions.

One may suspect a more direct relationship between N and P uptake. Indeed, [START_REF] Terry | Nitrate and phosphate uptake interactions in a marine prymnesiophyte[END_REF] demonstrated an asymmetric interaction between N and P uptake on Pavlova lutheri, where P uptake could inhibit N uptake, whereas N uptake had no eect on P uptake. The severity of P O 3- 4 -uptake inhibition of

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NO - 3 -uptake increased with the preconditioning N:P ratio ( i.e. the lower the q P , the higher the N-uptake inhibition). Terry (1982) postulated that the in- hibition of NO - 3 -uptake by P O 3- 4 -uptake may result from the allocation of a limited energy resource rather than from a direct action of P O 3- 4 on the NO - 3 uptake system. As reected by our uptake modeling, under P-limited steadystate where q P is low, P uptake could directly compete with N uptake for the limited available energy.

Additionally, [START_REF] Perry | Phosphate utilization by an oceanic diatom in phosphoruslimited chemostat culture and in oligotrophic waters of central north-pacic[END_REF] showed that the ATP:C ratio in Thalassiosira pseudonana was consistently lower in P-limited than in N-limited chemostat cultures, while the ATP:P ratio remained unaected. [START_REF] Hunter | Atp and chlorophyll a as estimators of phytoplankton carbon biomass[END_REF] and [START_REF] Laws | Variability in ratios of phytoplankton carbon and rna to atp and chlorophyll a in batch and continuous cultures[END_REF] also recorded with Thalassiosira weissogii that, under P limitation, the ATP: C ratio was about 30 to over 50 percent lower than under N limitation , especially for low growth rate. It is worth noting that, in the two latter studies, ATP analyses were corrected for alkaline phosphatase activity [START_REF] Karl | Eects of alkaline phosphatase activity on nucleotide measurements in aquatic microbial communities[END_REF] which may induce a drop in ATP:C ratio under P limitation. [START_REF] Hunter | Atp and chlorophyll a as estimators of phytoplankton carbon biomass[END_REF] concluded that "the relationship between ATP:C ratios and growth rate is highly dependent on the factor limiting growth, at least at moderate or low growth rates". These results conrm q P as an indi- cator of the ATP pool level in the cell.

Our model species that under N limitation, the high P quota enhances N uptake, which we interpret as a high availability of ATP for cellular functions.

However, investigation of the ATP pool by itself is not sucient to reect the metabolic rate within the cell : competition for ATP allocation between the different cell functions depends on the balance between (1) the level of metabolic under N-limiting conditions and that (2) it is rather stable with growth rate under P limiting conditions (except perhaps for very low growth rates), (3) being generally lower under P limitation than under N limitation, especially at low growth rates. In the study of [START_REF] Hunter | Atp and chlorophyll a as estimators of phytoplankton carbon biomass[END_REF] with Thalassiosira weissogii, A AT P under P limitation is only half that under N limitation for the lowest growth rates tested. The same trends are observed in [START_REF] Laws | Variability in ratios of phytoplankton carbon and rna to atp and chlorophyll a in batch and continuous cultures[END_REF], although the scatter in the data makes the conclusions more delicate. Hence, one can infer that under N limitation, (1) competition for energy between the cellular functions is lower than under P limitation and that (2) competition is an increasing function of growth rate, as ATP availability is reduced due to the energy demand growing faster than the ATP pool. Moreover, [START_REF] Dai | Relationship of energy charge and toxin content of Microcystis aeruginosa in nitrogen-limited or phosphorous-limited cultures[END_REF] recently demonstrated that the adenylate energy charge (the metabolic energy stocked in the adenine nucleotide pool) of Microcystis aeruginosa is an increasing function of the P-limited growth rate. From these various biological observations, one can expect that the restricted ATP-pool under P-limited growth should enhance the competition for ATP of (1) N uptake with carbon xation and (2) between N and P-uptake systems. We therefore conclude that A c c e p t e d m a n u s c r i p t

the behavior of q P with growth rate qualitatively reects the competition for energy between the cellular functions and these ndings justify the q P regula- tion in our N uptake modeling approach (17).

The biological results presented above motivated the asymmetric formulation for the two uptake functions. Our model assumes that no interactions occur between N status and P uptake and that, therefore, P uptake is solely governed by the internal P quota, as observed by [START_REF] Terry | Nitrate and phosphate uptake interactions in a marine prymnesiophyte[END_REF]. Indeed, the model tted suciently well with the experimental data that it was concluded that such an additional interaction was not necessary to the model. Additionally, under phosphorus limitation, it was suggested that the constrained energy pool was preferentially allocated to P uptake rather than N uptake or carbon xation.

There are various indications of competition for energy between the two functional machineries and between the active N-and P-uptake systems. These observations, together with the demonstration of [START_REF] Rhee | Eects of n:p atomic ratios and nitrate limitation on algal growth, cell composition, and nitrate uptake[END_REF] that the minimum law correctly predict growth rate under dierent N:P ratios, led us to (1) consider the N-P interaction at the acquisition level rather than at the assembly level and (2) choose the minimum law for growth rate in our modeling of N-P colimitation.

Our analysis suggests that important traits of phytoplankton N-P colimitation can be captured under the assumption that competition for energy is the main link between N and P. Nevertheless, the complexity and the diversity of N-P interactions raise the question of multi-level interactions and of the pre-eminent process for N-P colimitation. Further investigations are needed to elucidate these particular points. Experiments investigating energy charge, RNA content and short-term uptake measurements on chemostat cultures under varying and extreme N:P ratios should oer new insights. As our analysis pointed out, in agreement with [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF], the achievement of quota saturation (related to the range of N:P ratio explored) is a crucial condi-A c c e p t e d m a n u s c r i p t tion for a better understanding of phytoplankton N-P colimitation. Under these extreme conditions, it would be interesting to conrm the model prediction that phytoplankton stoichiometry deviates from the input N:P ratio, particularly at very low dilution rates.

Conclusion

We proposed a new model of N-P colimitation. This Droop-based model hypothesizes that the interaction between N and P should be considered at the acquisition level rather than the assembly level. Indeed, we conclude that the Droop function between growth rate and quota remains unaected by the N-P interaction, although N quota is largely determined by the cellular P status.

The P control on N uptake through energy availability needed for NO - 3 porters combined with a negative retroaction of quota level on nutrient uptake allowed the model to reproduce the experimental data of [START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF] and of experimental trials in the present study, made under extreme N:P input ratios.

Furthermore, our uptake-based approach emphasizes that assimilation did not have to be modeled separately from uptake to reproduce the asymmetry between N and P. We therefore suggest that, regarding the colimitation concept, N and P would better be considered as dependent nutrients rather than biochemically independent ones.

As several authors already pointed out [START_REF] Flynn | The importance the form of the quota curve and control of non-limiting nutrient transport in phytoplankton models[END_REF][START_REF] Klausmeier | A model of exible uptake of two essential resources[END_REF][START_REF] Leonardos | Elemental and biochemical composition of Rhodomonas reticulata (cryptophyta) in relation to light and nitrate-tophosphate supply ratios[END_REF] there is a critical need for new experimental data considering N and P under dierent and extreme input ratios and growth rates.

  model structure so that the model nally satises some qualitative properties. However, for sake of clarity, only the nal version of the model is presented and justied, and we demonstrate that its mathematical behavior structurally (i.e. independently of the choice of parameter values) matches the data trend. The mathematical developments that guided the modications of the Droop model in order to end up with a colimitation model that presents an

Figure 1 :

 1 Figure1: Comparison of the two models output with the experimental results for Selenastrum minutum at steady-state[START_REF] Elri | Steady-state luxury consumption and the concept of optimum nutrient ratios : a study with phosphate and nitrate limited Selenastrum minutum (chlorophyta)[END_REF]. The top gures show q N against μ under nitrogen limitation (a) and phosphorus limitation (b). The bottom gures show q P against μ under nitrogen limitation (c) and phosphorus limitation (d). Solid line : proposed colimitation model with parameter values as in table 2. Dashed line : model ofKlausmeier et al. (2004b).

Figure 2 :

 2 Figure 2: Comparison of the model output to the experimental results for Isochrysis anis galbana at steady-state. The top two gures show q N against μ under nitrogen limitation (a) and phosphorus limitation (b). The bottom gures show q P against μ under nitrogen limitation (c) and phosphorus limitation (d). Parameter values used for the model are given in table 2

Figure 3 :

 3 Figure 3: Model simulation of N (a ) and P (b ) luxury consumption capacities computed at steady-state against growth rate. The luxury consumption capacities were calculated for two extreme N:P input ratios (i.e. 1 and 100).

Figure 4 :

 4 Figure4: Model predictions for the steady-state N quota (a ) and P quota (b ) against growth rate under dierent N:P input ratio ranging from 1 to 100. Numbers represent the values of the N:P input ratio. Dotted-lines represent situations where uptake is substratelimited (S/(S + K S ) < 0.9). The solid lines represent situations where the uptake-capacity is limiting.

  used an alternate nutrient uptake model, built on the anity-based uptake model(Aksnes and Egge,

Figure 6 :

 6 Figure 6: Computed steady-state A AT P for Thalassiosira weissogii under P limiting (+) and N limiting conditions (o). Data from Hunter and Laws (1981) (a) and from Laws et al. (1983) (b).

Table 1 :

 1 Parameters used for model of N-P colimitation.

	Parameter Meaning (units)
	μ max	maximum growth rate (day -1 )
	n in p in	nitrate concentration in the renewal medium (molN.L -1 ) phosphate concentration in the renewal medium (molP.L -1 )
	q P exp q N 0 q P 0 q NL q P L ρ N ρ P ρN max ρP max K sN	phosphorus quota at μ max (molP.molC -1 ) nitrogen subsistence quota (molN.molC -1 ) phosphorus subsistence quota (molP.molC -1 ) hypothetical maximum nitrogen quota (molN.molC -1 ) hypothetical maximum phosphorus quota (molP.molC -1 ) nitrate uptake rate (molN.molC -1 .day -1 ) phosphate uptake rate (molP.molC -1 .day -1 ) nitrate maximum uptake rate (molN.molC -1 .day -1 ) phosphate maximum uptake rate (molP.molC -1 .day -1 ) half saturation constant for nitrate uptake (molN.L -1 )
	K sP

x carbon-biomass (mol.L -1 ) μ hypothetical growth rate when quota is innite (day -1 ) D dilution rate (day -1 ) n ambient nitrate concentration (molN.L -1 ) p ambient phosphate concentration (molP.L -1 ) q N nitrogen quota (molN.molC -1 ) q P phosphorus quota (molP.molC -1 ) q N exp nitrogen quota at μ max (molN.molC -1 )

Table 2 :

 2 Parameter values used for model of N-P colimitation in Selenastrum minutum and

	Isochrysis anis galbana.
	Parameter	Value	Unit
	μ max	S. minutum I. a. galbana 1.68 1.5	day -1
	q n0

Acknowledgments: This paper presents research results supported by the ANR-06-BIOE-014 Shamash project. The authors are grateful to Julie Marchetti for the experimental dataset with Isochrysis anis galbana.