Introduction

Crocosphaera is a major unicellular nitrogen-fixer in the ocean1,2,3 and widely used for laboratory studies4,5,6,7,8. The process of nitrogen fixation provides fixed nitrogen to themselves and the environment, supporting their growth and balancing the nitrogen budget in the ocean9,10. On the other hand, Crocosphaera can be a consumer of fixed nitrogen such as ammonium11,12,13,14. Culture studies have shown that Crocosphaera actively consumes available fixed nitrogen in batch (dynamic) cultures11,12 and continuous (steady-state) cultures14. In general, nitrogen fixers seem to prioritize using fixed nitrogen by inhibiting nitrogen fixation15,16. Consuming external fixed nitrogen is advantageous as it bypasses the high energy and electron utilization costs that accompany nitrogen fixation17,18,19. However, despite the availability of fixed forms of nitrogen, nitrogen fixers are observed to fix nitrogen, which would decrease their growth efficiency (here in terms of C)18,19. Empirical evidence shows that both nitrogen fixation as well as the utilization of organic/fixed nitrogen occur concomitantly12,14,20. When two competing strategies are possible, there must be implied trade-offs dictating the balance between the two. A recently developed coarse-grained model of a nitrogen fixer19 shows that also using ammonium will expand their niche compared to only fixing nitrogen. Conversely, the purpose of continuously fixing nitrogen under the presence of fixed nitrogen has not been elucidated. Here we focus on this other side of the question: the effect of fixing nitrogen despite the presence of ammonium.

To numerically examine this question, we have developed a quantitative model for Crocosphaera (Cell Flux Model of Crocosphaera 2: CFM-Croco2), resolving a simple set of molecular pools and minimum representation of elemental fluxes (Fig. 1) (see Methods and Supplementary Methods for details). Although we do not resolve a complex network of metabolisms as in Flux Balance Analysis21,22, we resolve essential metabolisms such as nitrogen fixation, nutrient uptake, respiration, photosynthesis (C fixation) and growth (Fig. 1), following previous models19,23,24,25. The strength of this minimum model is to keep the model efficient and transparent and minimize overlapping metabolic effects. Also, the development process of such minimum models often suggests missing pieces when the model does not reproduce the data. In such a case, we consider what components of metabolism would further improve the model-data fit, based on the current state of knowledge of biochemical pathways. The model results reveal that nitrogen fixation, despite the energy expenditure and allocation of resources, gives Crocosphaera a competitive advantage in both monoculture and in a complex community.

Figure 1
figure 1

Schematics of the model. The green area represents cytoplasmic space, and the cream edge represents the cell membranes. Ovals and rectangular boxes represent inorganic and organic molecules, respectively. Different colors are applied to different elements; yellow, C; pink, N; blue, P. Solid arrows are the elemental fluxes; yellow, C; red, N; blue, P. Black dotted arrows represent positive influences. The black dotted circle represents biosynthesis. CBio, cellular biomass carbon; EPS, extracellular polymeric substances; PRNA, P in RNA; POther, P in other molecular pools; DON, dissolved organic nitrogen; NStore, N storage; NGrowth, N in growth related molecules; NOther, N in other molecules. Fluxes: μ, growth; FPhoto, photosynthesis; FRes, respiration; FEPS, EPS excretion; \({F}_{Fix}^{N}\), nitrogen fixation; FDON, DON excretion; VN, fixed N uptake; VP fixed P uptake. The notations are same as those in Methods and Supplementary Methods.

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Results

Simulating steady state culture

Our model reproduces laboratory data of Crocosphaera grown in Chemostat culture14. The incoming medium has 50 µM ammonium and 20 µM phosphate, and the model simulates these nutrient influxes. These macronutrient concentrations in the media are relatively high. However, the ammonium and phosphate concentrations in the culture are maintained at nanomolar-scale due to cellular uptake, when these nutrients are limiting cellular growth. The model captures the growth dependence of elemental stoichiometry of Crocosphaera, rate of nitrogen fixation, nutrient limitation, and standing stock of N and C (Figs 2 and 3). We further simulated scenarios with doubled nitrogen fixation and with zero nitrogen fixation. The results show increased biomass concentration with nitrogen fixation (Fig. 3E). As the growth rate μ increases, N:C and P:C increase due to investment for growth related molecules (NGrowth and PRNA Fig. 2). The model captures the transition of P limitation to N limitation (Fig. 3A); in the laboratory experiment14 this was observed by a sudden increase in phosphate concentration when the culture became N limited from P limited (Fig. 3B). Here limitation is defined based on which nutrient controls the standing stock of biomass; i.e. if adding N increases biomass, the culture is N limited.

Figure 2
figure 2

Model data comparison of N:C and P:C of Crocosphaera culture for multiple scenarios. (A) N:C. (B) P:C. Points are the data from the experiment14. The data of N:C are based on (Total N – DON – NH4+)/POC. The data of P:C are based on POP/POC. Bacterial contamination was negligible14. Blue curves, default run of the model manually fitted to the data. Green curves, model run with doubled nitrogen fixation. Red curves, model run with zero nitrogen fixation. (C,D) Allocation of nitrogen and phosphorus to different functionalities. Gray shadings indicate P limitation in the default nitrogen fixation; white areas are N limitation. Note: DON, dissolved organic N; POC, particulate organic C; POP, particulate organic P.

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Figure 3
figure 3

Model data comparison for different nitrogen fixing capacities. (A) Nutrient limitation (B) PO43− concentrations. (C) Nitrogen fixation rates (normalized by POC). (D) Total nitrogen concentration. (E) Biomass carbon concentration. The upper left legend applies to all the figure panels. Points represent the data from the experiment14. The nitrogen fixation data are based on μ and the concentration difference between total N and ammonium in the incoming media. Blue curves, default run of the model manually fitted to the data. Green curves, model run with doubled nitrogen fixation. Red curves, model run with zero nitrogen fixation. Gray shadings indicate P limitation with default nitrogen fixation; white areas are N limitation.

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The model predicts that the ammonium is fully consumed, even when P is limited, due to the luxury uptake of N for N storage. This resonates with the experimental data, where the ammonium was observed at nanomolar concentrations. The model indicates that Crocosphaera do not accumulate quantitatively significant P in the cell based on the excess P, but they do store extra N (Fig. 2C,D), keeping the free intracellular N concentration small. The lack of P storage is inferred, as inclusion of P storage in the model was unnecessary to fit the data. In particular, the data do not show a significant change in the trend with a change from P limitation to N limitation, and PRNA was sufficient to express the data. It has been shown that Crocosphaera produce cyanophycin4,6 and this result corroborates such storing capacity.

With increasing growth rate, NGrowth (growth related protein) increases (Fig. 2C). This leads to increases in nitrogen fixation with increased growth rate (Fig. 3C). Despite such changes, the total nitrogen in the culture decreases, since cells are flushed away at a higher rate (Fig. 3D). Total C decreases more strongly with the growth rate (Fig. 3E) due to increasing N:C of the cells.

A doubled rate of nitrogen fixation increases the nitrogen storage resulting in higher N:C under P limitation (Fig. 2A). Under N limitation, N:C is unchanged. Instead, the cellular density is increased, leading to higher total N and C (Fig. 3D,E). Also, notably, it shifts the range of limitation (Fig. 3A); e.g. N limitation is above ~ 0.15 (d−1) for the default run, but above ~0.23 (d−1) for the doubled nitrogen fixation. This indicates that the range of N limitation is narrowed down due to nitrogen fixation. Biomass C increases at growth rates where this shift occurs, but it stays unchanged where it is originally P limited. When nitrogen fixation does not occur, cells can still grow using the available ammonium. However, the stored N decreases under P limitation leading to lower N:C (Fig. 2A). Under N limitation, without nitrogen fixation, the total population decreases leading to lower biomass C in the culture (Fig. 3E).

Simulating a simple dynamic ecological model

By using a set of parameters obtained from the steady state simulation, we run a simple ecosystem model (Figs 4, 5, S1 for up to day 500 and Fig. S2 for up to day 1000). Here we simulate additional non-nitrogen-fixing phytoplankton and zooplankton to represent a minimum ecosystem as used in resource competition theory26,27,28,29. We have prescribed higher nutrient uptake for non-nitrogen-fixing phytoplankton to give them an advantage to compensate for the lack of nitrogen fixation as conventionally modeled27,28,29,30,31. Biomass C became stable at approximately day 40 (Fig. 4A,B), where Crocosphaera is limited by P and non-nitrogen-fixing phytoplankton are limited by N, representing a common situation in the open ocean (Fig. 4C,D) where nitrogen fixers and non-nitrogen fixers co-exist27,28,29. The growth rate at day 500 was 0.217 and 0.382 for Crocosphaera and non-nitrogen-fixing phytoplankton respectively (Fig. S1), which are within the observation range32,33,34,35. The ratio of these growth rates is 1.76, a value similar to those previously parameterized in global ecosystem models31,36,37.

Figure 4
figure 4

Simulated co-existence of Crocosphaera and non-nitrogen-fixing phytoplankton. (A,B) Concentrations of Biomass C of Crocosphaera and non-nitrogen-fixing phytoplankton, respectively. (C,D) Nutrient limitation of Crocosphaera and non-nitrogen-fixing phytoplankton, respectively. The legend in (A) applies to all the figure panels. Blue curves are the default run with nitrogen fixation. Red dotted curves are the run without nitrogen fixation.

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Figure 5
figure 5

Ammonium uptake, fraction of nitrogen fixation and ammonium concentrations. (A,B) N uptake by Crocosphaera and non-nitrogen-fixing phytoplankton respectively. (C) Fraction of nitrogen fixation of all the N sources for Crocosphaera. (D) Ammonium concentration. The legend in (A) applies to all the figure panels. Blue curves are the default run with nitrogen fixation. Red dotted curves are the run without nitrogen fixation.

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To test the effect of nitrogen fixation, we turned off nitrogen fixation of Crocosphaera. The model predicts a larger population of Crocosphaera when nitrogen fixation is occurring (Fig. 4A). Interestingly, the model also shows decreasing non-nitrogen-fixing phytoplankton when we allow active nitrogen fixation by Crocosphaera. With nitrogen fixation, we predict higher total uptake of N by Crocosphaera as a community (Fig. 5A), due to increased population/biomass (Fig. 4A). As a result, the N available for non-nitrogen-fixing phytoplankton decreases, lowering their population (Fig. 4B) and the amount of total N they take up (Fig. 5B).

The model predicts a relatively low fraction of nitrogen fixation (~30%) (Fig. 5C), despite low concentrations of available ammonium (Fig. 5D). These concentrations resemble those under N limitation14. Under laboratory conditions, Crocosphaera is known to grow diazotrophically when there is no added nitrogen. Thus, the result of this low fraction of nitrogen fixation is likely due to continuous addition of nitrogen to the system. Such situations may be common in the marine environment due to continuous remineralization via the microbial loop38.

Discussion

Competitive view of Crocosphaera

Nitrogen fixers are often described as a provider for the environment since they provide fixed nitrogen to other organisms39,40. It is true that nitrogen fixation is essential in balancing lost fixed nitrogen9 and in a relatively long time scale, nitrogen fixation supports the community by providing bioavailable nitrogen41. Also, it is true that nitrogen fixers can grow by themselves by only using dinitrogen6,14,42 and excretion of N containing molecules is observed (25% ~ 50% of fixed nitrogen)39,43. These facts often leave an impression that nitrogen fixers actively stimulate the growth of other phytoplankton by providing fixed nitrogen (Fig. 6A). However, our study shows that nitrogen fixers can also be nitrogen limited (Fig. 3) and, within a short time scale and distance, compete with other non-nitrogen-fixing phytoplankton for fixed nitrogen (Fig. 6B). The nitrogen limitation of Crocosphaera is supported by nitrogen depletion in culture14 and our prediction of increased cellular biomass with nitrogen fixation (Fig. 3D,E).

Figure 6
figure 6

An emerged competitive view of Crocosphaera based on this study. (A) A general long-time view and (B) proposed short-time competitive view of how nitrogen fixation by Crocosphaera influences N fluxes and plankton population within a short time scale. Cro., Crocosphaera; Phy., non-nitrogen-fixing phytoplankton; N, fixed nitrogen; dotted arrows, fluxes; thick arrows, influence of nitrogen fixation. The differences are in red in (B).

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In nitrogen limiting environments, Crocosphaera increases their population with nitrogen fixation, until the population becomes limited by another nutrient (here phosphorus). This increases the community uptake of nitrogen by Crocosphaera, limiting the nitrogen sources in the environment, and ultimately decreasing the population of other non-nitrogen-fixing phytoplankton (Fig. 6B). Since this effect might be overcome with excretion, we tested excreting 50% of fixed nitrogen to the environment. This makes the growth of Crocosphaera N limited and the growth rate is decreased, which was accompanied by decreased NGrowth and PRNA. However, the model shows that the population of non-nitrogen-fixing phytoplankton is still lower than in the case without nitrogen fixation. Also, we have tested a maximum uptake rate of non-nitrogen-fixing phytoplankton 10 times higher than that of Crocosphaera, which resulted in a growth rate for non-nitrogen-fixing phytoplankton of ~6.5 times higher than Crocosphaera. However, if we allow Crocosphaera’s nitrogen fixation, the population of non-nitrogen-fixing phytoplankton was still lower than the case with zero nitrogen fixation.

The predicted fraction of nitrogen fixation of 30% is rather low given that Crocosphaera can grow diazotrophically. Since this value is based on a single chemostat experiment with a fixed resource concentration of ammonium, chemostat cultures of Crocosphaera with various resource concentrations of fixed nitrogen might be useful to test our prediction. However, recent observations show little link between the presence of Crocosphaera and primary productivity44, supporting the presented competitive view. We note that there are multiple sources of fixed nitrogen even in the oligotrophic gyres where Crocosphaera is observed: atmospheric deposition45,46, active remineralization by members of the microbial loop38 and occasional upwelling47. Such N sources may decrease the fraction of nitrogen fixation, making Crocosphaera competitive. Also, this result may reflect that Crocosphaera take up fixed nitrogen during the day, but they fix nitrogen at night; leading to a certain balance between the two and the model reflects the average over the diel period. In addition, we note that typical values of nitrogen fixation and N content per cell from compiled data of Trichodesmium48 suggest that just fixing nitrogen at a typically observed rate can only support their growth of ~1 year−1 (see Supplementary Methods), significantly lower than the observed growth rate (Maximum growth rate of ~ 0.14 d−1 [ref.48]). This may indicate that other marine nitrogen fixers may actively use external fixed nitrogen and compete with non-nitrogen fixers.

In ecological models, this type of competition has not been considered, with the assumption that nitrogen fixers grow 100% diazotrophically anywhere in the modeled ocean31,37,49, likely based on forced diazotrophic growth in laboratory studies and low concentrations of fixed nitrogen42,50,51,52. However, our study shows that this may not always be true. Including the competitive aspect of nitrogen fixers may lead to a different model output; e.g. increased abundance of nitrogen fixing organisms and a decrease of non-nitrogen-fixing phytoplankton.

Where this type of competition happens and where not is still a question. This can be related to the concentration of fixed nitrogen and the time scale of its resource input. It can also be related to the resource ratio of fixed nitrogen to other nutrients (e.g. phosphorus and iron)27,28. To study that, chemostat cultures of Crocosphaera with various resource concentrations may be useful to further constrain the effects of N on nitrogen fixation. Additionally, CO2 might influence the growth53. Thus, to isolate the effect of nutrients on nitrogen fixation, CO2 concentration should be maintained stable or the concentration of CO2 must be measured frequently to clarify its daily fluctuation. Furthermore, growing Crocosphaera and non-nitrogen-fixing organisms e.g. Synechococcus spp. in chemostat culture and observing the flux of N under various resource nutrient concentrations can be useful. In this case, both the flux of N2 and dissolved N must be traced separately. Also, in the field, using labeled N (e.g. 15NH4+) and observing its fate would clarify the competition. To do that, a possible strategy might be isolating Crocosphaera with flow cytometry and measuring 15N with mass spectrometry or NanoSIMS.

Dynamic and patchy ocean environment and meaning of storage and higher population

We point out that the ocean is highly patchy with numerous occasional upwelling regions54,55 and nutrient-depleted zones56, making the distribution of nutrients and plankton vary significantly throughout the ocean. Such patchiness may lead to chaotic distribution of nitrogen fixers and variable rates of nitrogen fixation. Within a smaller scale than generally modeled (e.g. 1° × 1°), there is a spectrum of nitrogen resources; in one place, there is active influx of fixed nitrogen while in other places there are zero or negative fluxes. In reality, such variable nitrogen fluxes may make it complex to determine whether Crocosphaera compete or help other organisms, which might be one cause of elusive nitrogen fixation rates57. Increasing the resolution of modeled grids to resolve eddies58 might be useful. Also, simply allowing fixed nitrogen uptake for modeled nitrogen fixers may cause some changes. Once these are combined, the ecological model may reproduce such observed patchiness in nitrogen fixation.

Given such a dynamic environment, our predicted increased cell densities and storage may be an advantage in sustaining species. When the environment shifts from P limitation to N limitation, with N storage, Crocosphaera may continue growing at a high rate until the storage is depleted. Higher population/larger niche may lead to a higher chance of survival as a group upon facing different zooplankton. Also, molecular studies suggest P scavenging by Crocosphaera59,60. Given these implications, it is surprising that Crocosphaera does not seem to accumulate excess phosphorus, the behavior often seen in other phytoplankton61,62,63,64. A recent modeling study showed that nitrogen fixers are limited by P in the Atlantic Ocean65. However, our model is calibrated to a strain from the Pacific Ocean (Crocosphaera watsonii PS0609A)14 where the nutrient is considered replete65. Lack of P storage might be a result of continued phosphorus repletion; a strain from the Atlantic Ocean might have P storing capacity. Alternatively, since the cellular space is limited, Crocosphaera might have chosen to use space for other purposes that are more important for their survival, such as photosystems, nitrogenase, or other nutrient storage.

Conclusions

Based on a chemostat culture experiment, we have developed a model of Crocosphaera that combines uptake of available fixed nitrogen and nitrogen fixation. We have tested hypothetical conditions where the rate of nitrogen fixation is increased or is zero. The model indicates that increasing nitrogen fixation increases N storage, or their population, depending on nutrient limitation. We then simulate a simple ecological situation where Crocosphaera and non-nitrogen-fixing phytoplankton co-exist. The model suggests that Crocosphaera compete for N sources with non-nitrogen-fixing phytoplankton; increasing rate of nitrogen fixation can decrease the population of phytoplankton within a small timescale and distance. Our model results can be further tested by extensive laboratory measurements as well as field observations. Given the effect on N fluxes and phytoplankton population dynamics, reflecting the competitive aspects of Crocosphaera may be essential in predicting their roles, and those of other nitrogen fixers, in the changing environment.

Methods

Steady state model

In this section, we describe the model with equations, describing fundamental equations used in the steady state model and dynamic model (Additional details are in Supplementary Methods). Nomenclature with units, values used for adjustable and fixed parameters, initial values in the dynamic model, and used data from the chemostat experiment14 are provided in Tables S1–S7. The model includes three N molecules (NGrowth, NOther, NStore) and two P molecules (PRNA, POther). NGrowth includes growth related molecules rich in N, such as nitrogenase and proteins for photosynthesis and biosynthesis. Also, PRNA (P contained in RNA) positively influences biomass production as well, since a large part of RNA is involved in protein synthesis66. In addition, we included N storage (NStore) for the cell to accumulate excess N. Other molecules in N and P are included in NOther and POther respectively, representing basic need of N and P for maintaining cell viability23,67.

The model is manually parameterized to reproduce the chemostat culture conditions under which known concentrations of ammonium and phosphate were continuously added, when Crocosphaera watsonii PS0609A was grown14. The model is based on the minimum set of parameters, each of which has exclusive influence on model results, allowing us to narrow down the parameter values (Fig. S3; shown with a sensitivity study68). Chemostat cultures allow more realistic growth conditions, as nutrients can be gradually added at a steady rate, rather than in batch culture, where nutrient concentrations start high and are incrementally depleted during growth. Also, since the cellular growth rate can be controlled by adjusting the dilution rate, it allows us to model growth-rate dependency of cellular parameters. The study also offered a wide range of parameters (various elemental compositions and nitrogen fixation) for various specific growth rates and reported active nitrogen fixation despite the continuous addition of ammonium.

Once we parameterized the model, we manually changed the rate of nitrogen fixation to evaluate its effect on their biomass concentrations. In the environment, the interplay between competing strategies is likely influenced by competition with other microorganisms. These inter-organism interactions are challenging to reproduce under laboratory conditions, but undoubtedly influence growth rates due to competition for common resources.

The model resolves C, N and P fluxes and consists of coarse-grained macromolecules in N and P (Fig. 1). To simulate a chemostat culture where Crocosphaera grows under continuous addition of ammonium and phosphorus, we use fundamental balances of cellular quotas, cell densities, and dissolved nutrients. We recognize a distinct diurnal cycle of Crocosphaera4,5,6. However, to consider a steady state and to keep the model simple, we focus on the daily average of metabolisms and cellular quotas. Additionally, this decreases the number of free parameters.

$$\frac{1}{{C}_{Bio}}\frac{{dC}_{Bio}}{dt}={F}_{Photo}-\mu -{F}_{EPS}-{F}_{Res}$$
(1)
$$\frac{{dQ}_{N}}{dt}={V}_{N}+{F}_{Fix}^{N}-{\mu Q}_{N}-{F}_{DON}$$
(2)
$$\frac{{dQ}_{P}}{dt}={V}_{P}-{\mu Q}_{P}$$
(3)
$$\frac{dX}{dt}=\mu X-DX$$
(4)
$$\frac{d[EPS]}{dt}={F}_{EPS}X-D[EPS]$$
(5)
$$\frac{d[N]}{dt}=D({[N]}_{in}-[N])-{XV}_{N}$$
(6)
$$\frac{d[DON]}{dt}={F}_{DON}X-D[DON]$$
(7)
$$\frac{d[P]}{dt}=D({[P]}_{in}-[P])-{XV}_{P}$$
(8)

[Equation 1]~[Eq. 3] represent balances of cellular biomass C, CBio, and cellular quotas of N and P quotas, QN and QP, respectively. CBio is a balance of photosynthesis FPhoto, growth µ, EPS (extracellular polymeric substances) excretion FRes and respiratory loss FEPS [Eq. 1]. Here, cellular excretion was not considered since it was not able to be constrained by the data. QN is balanced by nitrogen uptake VN, nitrogen fixation \({F}_{Fix}^{N}\), growth, and nitrogen excretion \({F}_{DON}\) [Eq. 2]. QP is simply a balance of uptake VP and growth. [Eq. 4] shows that if cell density X is balanced by growth and dilution D. [Eq. 5]~[Eq. 8] are the balances of EPS and dissolved nutrients. EPS and DON ([EPS] and [DON] represents their concentrations respectively) are the balances of excretion (FEPS and FDON respectively)and dilution ([Eq. 5] and [Eq. 7]). It is possible that the uptake of DON may occur and we define \({F}_{DON}\) as net excretion that represents a balance of uptake and excretion of DON. Ammonium and phosphate are balances of nutrient flow and uptake ([Eq. 6] and [Eq. 8]), where [j] and [j]in represent the concentration of j (here N (ammonium) or P (phosphate)) in the chemostat culture and incoming medium. We solve these equations assuming a steady state (d/dt = 0). A detailed solution of the model is in the Supplementary Material.

CBio does not appear in the steady state solution, since under the steady state, dCBio/dt = 0. We use fluxes and quotas normalized by CBio to avoid repeated appearance of CBio following previous studies23,24,67,69,70. Total QN consists of growth related proteins NGrowth, constant components, Nother, which includes N for maintaining cells to be viable71, and N storage Nstore:

$${Q}_{N}={N}_{Growth}+{N}_{Other}+{N}_{Store}$$
(9)

NGrowth includes proteins for nitrogen fixation (nitrogenase), photosynthesis (photosystems), andother biosynthetic processes such as the synthesis of proteins, nucleic acids, carbohydrates and lipids. These proteins are shown to be significant in magnitude in proteomic studies5,72,73. We did not explicitly represent the molecular allocation to nutrient acquisition since it has been shown that the protein allocation to membrane transport is relatively small (less than 10%)74 and it has been predicted to be even smaller in a molecular allocation model75. However, we note that other models show the potential significance of molecules for nutrient-acquisition70,76,77 and thus more molecular and proteomic evidence would be needed. It has been known that cellular N contains a growth-rate-dependent part and a constant (maintenance/essential) part23,78,79,80,81, and the model reflects these; NGrowth for the former and NOther for the latter. Also, storage molecules are recognized in various phytoplankton including Crocosphaera6,82,83,84,85, and we include this concept in the model as well. In the model, NGrowth linearly influences the rate of growth, photosynthesis and nitrogen fixation (see Supplementary Methods). QP consists of P in RNA PRNA and P in other relatively constant molecules POther, including phospholipids in cellular membranes.

$${Q}_{P}={P}_{RNA}+{P}_{Other}$$
(10)

Here POther includes a relatively constant P pool such as DNA and P in lipid membranes. RNA is observed to have a strong growth-rate dependency63,66,86, and to reflect that, we separate it from other P pools. Since a large part of RNA is involved in protein synthesis, in the model, it positively influences the rate of protein synthesis (see Supplementary Methods).

Dynamic model

To test the effect of nitrogen fixation on biomass concentration in a more realistic environment, we simulated an ecological situation where Crocosphaera exists with other non-nitrogen-fixing phytoplankton. The model consists of fundamental balances of phytoplankton densities Xi, zooplankton densities Xzoo, cellular N and P quota of phytoplankton \({Q}_{N}^{i}\) and \({Q}_{P}^{i}\) respectively, and the concentration of inorganic nutrients in the culture [j]:

$$\frac{d{X}_{i}}{dt}={X}_{i}{\mu }_{i}-{X}_{Zoo}{G}_{i}$$
(11)
$$\frac{d{X}_{Zoo}}{dt}={X}_{Zoo}({G}_{Cro}+{G}_{Phy})-{m}_{2}{X}_{Zoo}^{2}$$
(12)
$$\frac{d{Q}_{N}^{i}}{dt}={V}_{N}^{i}-{\mu }_{i}{Q}_{N}^{i}+{F}_{Fix}^{N}$$
(13)
$$\frac{d{Q}_{P}^{i}}{dt}={V}_{P}^{i}-{\mu }_{i}{Q}_{P}^{i}$$
(14)
$$\frac{d[j]}{dt}={V}_{j}^{Cro}{X}_{Cro}+{V}_{j}^{Phy}{X}_{Phy}+{S}_{j}$$
(15)

where i is phytoplankton type (either Crocosphaera (Cro) or non-nitrogen-fixing phytoplankton (Phy)) and j is inorganic nutrient (either N, ammonium or P, phosphate), Gi represents grazing of phytoplankton i and V and S represents uptake and source terms. The term, m2 is a square mortality rate of zooplankton as used in a recent marine ecological model37. The equations are solved using the finite-difference method. µi is solved based on pseudo-steady state assumption where cellular components represent the steady state solution. \({F}_{Fix}^{N}\) is zero for non-nitrogen-fixing phytoplankton. Gi and \({V}_{j}^{i}\) are based on KTW (kill-the-winner) theory87 and Monod kinetics88, respectively (details for µi, Gi and \({V}_{j}^{i}\) are in Supplementary Methods). The KTW method considered commonly observed active prey-switching behavior of zooplankton89,90,91, which are known to stabilize ecosystems92,93. Monod kinetics is a widely used equation for nutrient uptake, which well represents the general saturating relationship between nutrient uptake and concentration94,95,96.