## 1. Introduction

Diatom–diazotroph associations (DDAs) are symbioses where the diazotrophs (e.g.,

Richelia and

Calothrix sp.) are associated with diatoms (e.g.,

Hemiaulus,

Rhizosolenia, and

Chaetoceros sp.) [

1,

2,

3,

4,

5,

6,

7]. They are widely observed [

5,

8,

9,

10,

11,

12,

13,

14,

15,

16] and predicted [

17,

18,

19] in warm waters of the ocean. The symbiotic diazotrophs form a trichome where generally only one specialized cell, called a heterocyst, fixes dinitrogen (N

_{2}). The remaining cells in the trichome, called vegetative cells, are phototrophic and divide, whereas heterocysts do not. Despite the seemingly ideal combination of cells specialized for carbon (C) and nitrogen (N) acquisition, the trichomes have rarely been observed as free-living organisms in the marine environment [

20,

21]. This indicates that the trichomes receive some essential nutrients, which allow them to grow more efficiently as a part of the symbiosis. Recent studies revealed simplified N pathways in

Richelia [

7] and a significant amount of fixed N transferred to the diatom host from its symbiont [

6]. The exchange of C between the diatom hosts and trichomes has been anticipated, but it has not been clearly demonstrated [

2,

22,

23]. This is in contrast to cyanobacteria–plant symbiosis where the cyanobiont becomes photosynthetically inactive [

23,

24,

25,

26,

27] and C transfer from the host has been directly observed [

23,

24,

28,

29,

30].

In addition to the high rate of N

_{2} fixation, a compilation of observed growth rate [

31] shows a higher mean growth rate for DDAs than other, non-symbiotic, marine cyanobacterial diazotrophs. This enhanced growth is an essential assumption for an ecosystem model to reproduce observed seasonal blooms of DDAs in the oligotrophic ocean [

31]. In general, the marine cyanobacterial diazotrophs grow at approximately 0.3 (d

^{−1}) under nutrient replete diazotrophic cultures [

32,

33,

34,

35,

36], whereas

Richelia in

Rhizosolenia–

Richelia symbiosis can grow as high as 0.87 (d

^{−1}) in diazotrophic conditions [

1,

2]. In addition, in situ studies show that the growth rate of

Crocosphaera (unicellular diazotrophic cyanobacteria) is low (0.001–0.15 (d

^{−1})) in comparison with

Richelia in

Hemiaulus–

Richelia symbiosis, which grew up to 0.59 (d

^{−1}) [

6]. What makes the high rates of N

_{2} fixation and growth possible? Here, we seek to quantify the extent to which the enhanced growth and N

_{2} fixation rates in the trichomes could be caused by the exchange of resources with the host diatom.

To quantitatively examine the host–trichome nutrient exchange, we have developed a coarse-grained model of the

Hemiaulus–

Richelia symbiosis (cell flux model of DDAs: CFM-DDA) adapting relevant parts from previous CFMs [

37,

38,

39,

40,

41], such as an idealized metabolic-flux network constrained by mass, energy, and electron budget. Extensive quantitative characteristics exist for this symbiosis [

6], including cell volume and the number of trichomes per diatom. The availability of these cellular characteristics and their relative consistency make this symbiosis an ideal candidate for modeling. The CFM-DDA model we develop here focuses on C and N metabolisms to quantify growth and N

_{2} fixation (

Figure 1). For most N

_{2}-fixing organisms, oxygen (O

_{2}) metabolism is important, since O

_{2} damages the N

_{2} fixing enzyme, nitrogenase, and may control the rate of N

_{2} fixation [

39,

40,

42,

43,

44]. However, since the trichomes form a heterocyst, a cell with a thick glycolipid layer to minimize O

_{2} influx [

45], we assume that intracellular O

_{2} is managed with normal levels of respiration [

37,

46]. This simplification allows us to focus on the metabolisms of C and N as the basis of the symbiosis. O

_{2} metabolism could straightforwardly be included in our modeling framework if required by future observations.

Here, we resolve only four processes (CFM-DDA,

Figure 1): photosynthesis, biosynthesis, respiration, and N

_{2} fixation. Photosynthesis and biosynthesis are done only by the host diatom and vegetative cells whereas N

_{2} fixation is performed only by the heterocysts. Respiration is done by all cells and adjusts to meet the energetic demand for all the other processes. We scaled the rate of photosynthesis based on the cellular N quota, which was estimated from the typical cell volumes (3493.5 µm

^{3} for a diatom, 18.8 µm

^{3} for a vegetative cell, and 61.0 µm

^{3} for a heterocyst) [

6], empirical volume–C relationship [

47], and an assumed elemental stoichiometry (C:N of 6.6) [

48], as well as the ratio of vegetative–cell:heterocyst of 4:1 and two

Richelia trichomes per diatom based on the available microscopic images [

6,

7,

49,

50] (see Methods). The predicted balance of photosynthesis and metabolic demand for C suggests that a significant amount of C is transferred from the host to the trichome, sustaining its high rate of N

_{2} fixation and enhanced growth.

## 3. Conclusions

DDAs are major N

_{2} fixers in the ocean whose rate of N

_{2} fixation is quantitatively significant [

6], but the connection between their metabolic rates and symbiotic association are unknown. With a simple mechanistic model of cellular metabolisms of

Hemiaulus–

Richelia symbiosis, we predict that the observed high rates of N

_{2} fixation and growth of the trichomes [

6,

31] are supported by the C transfer from the host diatom, which is qualitatively consistent with the observations of plant–cyanobacteria symbiosis [

23,

24,

28,

29,

30]. Our model also explicitly accounts for the C cost for N

_{2} fixation, which is a central factor in the competitive fitness of diazotrophs relative to other plankton. The growth rate handicap by DDAs is commonly expressed as a constant factor in ecological and biogeochemical models [

17,

18,

19], whereas our model shows that it is dependent on the molecular exchanges. Similarly, the model enables various cell sizes and number of the trichome per diatom, as well as the ratio of vegetative cells to the heterocysts, allowing the material exchanges and their metabolic advantages to be computed from mechanistic considerations. Such model flexibility allows the expression of diverse DDAs and can be used to study how such diversity helps DDAs to acquire their roles as significant sources of bioavailable N.

## 4. Methods

The CFM-DDA (

Figure 1) is based on the following core equation, a steady-state solution for the time dependences of each C and N pool (see

Supplementary Methods and Figure S1 for the derivation):

where

${F}_{Pho}^{D}$ and

${F}_{Pho}^{V}$ (pmol C d

^{−1} cell

^{−1}) are the daily rate of per-DDA photosynthesis by diatoms (

Dia) and vegetative cells (

Veg), respectively,

μ is the growth rate (d

^{−1}),

${Q}_{C}^{V}$,

${Q}_{C}^{H}$, and

${Q}_{C}^{D}$ are the cellular C quotas of

Veg,

Het (heterocysts), and

Dia per DDA,

${Q}_{N}^{V}$,

${Q}_{N}^{H}$, and

${Q}_{N}^{D}$ are their respective N quotas per DDA,

E is the ratio of respiration to biosynthesis, and

${Y}_{C:N}^{N2fix}$ is a conversion term from N

_{2} fixation to its C cost (mol C mol N

^{−1}).

This equation represents the balance between the C supply (left-hand side) and C consumption (right-hand side). The terms on the left are the photosynthesis from each organism providing C to the system. The first term on the right-hand side represents the consumption of C based on biosynthesis (making new cells) and respiration for supporting it. The second term represents the C cost based on N_{2} fixation.

We scale the rates and quotas based on the cell volume (

V) (µm

^{3}). To convert

V to C quotas, we have used a power relationship based on the compilation of various phytoplankton species [

47]. This study suggests different relationships for non-diatom phytoplankton and diatoms: (pg C cell

^{−1}) = 0.216 ×

V^{0.939} and (pg C cell

^{−1}) = 0.288 ×

V^{0.811}, respectively. Thus, we used the former equation for

${Q}_{C}^{V}$ and

${Q}_{C}^{H}$ and the latter for

${Q}_{C}^{D}$. We convert these C quotas to N quotas (

${Q}_{N}^{V}$,

${Q}_{N}^{H}$, and

${Q}_{N}^{D}$) based on the Redfield ratio of 6.6 C:1 N, following previous studies [

6,

48].

E is obtained based on the energy balance between biosynthesis and respiration with an energy transfer efficiency of 0.6 [

85].

${Y}_{C:N}^{N2fix}$ is based on the sum of the C costs for providing electron and energy to N

_{2} fixation [

37,

38] with the same energy transfer efficiency as that for biosynthesis (0.6) [

85] (see details in

Supplementary Methods). As we have now defined all the values for the left-hand side, we can obtain (

${F}_{Pho}^{D}+{F}_{Pho}^{V}$) as a solution of Equation (1). To partition

${F}_{Pho}^{D}$ and

${F}_{Pho}^{V}$, we assume that the rates of photosynthesis are proportional to the cellular N quotas, since the size of the N quota indicates the enzyme availability for photosynthesis. We have used the averaged cell volume for each cell from observations of

Hemiaulus and

Richelia [

6] ignoring two extraordinary large symbioses and a ratio of diatom:trichome of 1:2 to represent commonly observed relations between

Hemiaulus and

Richelia based on microscopic images [

6,

7,

49,

50] except for

Figure 5, where the cell volume and number of the trichomes are varied: 1490–4680 (µm

^{3}) (a range from the observation [

6] after ignoring the two large outliers) and 1–5 (between minimum and significantly higher value than generally observed), respectively. We used a ratio of

Veg:

Het of 4:1 to represent typically observed

Richelia trichomes in

Hemiaulus based on microscopic images [

6,

7,

49,

50].