https://orcid.org/0000-0002-2472-3508
Figures
Abstract
We present a comparative advantage model of carbon–nitrogen exchange in legume–Rhizobium symbiosis that incorporates ATP budget constraints and the energy–mass balance between the host and symbiont. In this framework, the uptake of carbon and nitrogen is limited by the ATP available to each partner, and any imbalance in trade is compensated by adjustments in symbiont biomass. Using empirical estimates of the ATP costs of carbon and nitrogen uptake, together with data on body C:N ratios, the model generates three key predictions, and we prove that they align with empirical results. (i) The condition for the establishment of symbiosis derived from the model is consistent with measured ATP costs in both host and symbiont. (ii) At equilibrium, the model predicts a relatively low carbon supply from the legume and a relatively high nitrogen supply from Rhizobium, in agreement with reported patterns of exchange. (iii) The model further predicts that the proportion of carbon supplied decreases as the host C:N ratio increases, and that the proportion of nitrogen supplied decreases as the symbiont C:N ratio decreases, which are consistent with the empirically observed decline in nodulation during host aging.
Citation: Furukawa T, Iimura T (2026) C–N exchange model of legume–Rhizobium symbiosis incorporating ATP budget constraints and energy–mass balance between the species. PLoS One 21(5): e0349611. https://doi.org/10.1371/journal.pone.0349611
Editor: Nabin Rawal, Nepal Agricultural Research Council, NEPAL
Received: November 13, 2025; Accepted: May 1, 2026; Published: May 22, 2026
Copyright: © 2026 Furukawa, Iimura. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: TF and TI received the Basic Research Fund of Tokyo Metropolitan University (#1D20543 and #1C41104, respectively). The funder’s website is at https://www.tmu.ac.jp. The funder did not play any role in the preparation of this manuscript.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Carbon (C) and nitrogen (N) are two essential elements for all living organisms, and a shortage of either has profound consequences for growth. For plants, soil N is typically limiting, whereas C is readily available through photosynthesis. The agricultural value of legumes as green manure was already recognized in ancient Rome [1]. In the nineteenth century, agronomists discovered that legumes use atmospheric N as a nutrient source, and that root nodules, which contain “bacteria-like” structures [2], are responsible for this fixation. These structures are now known as symbiotic bacteria of the genus Rhizobium and related taxa. Since then, the mechanisms of nodule formation and N fixation have been extensively investigated by plant and microbial biologists, and the legume–Rhizobium symbiosis has also attracted increasing attention from researchers in theoretical fields.
The symbiosis proceeds as follows. When soil N is scarce, legumes release flavonoids that trigger Rhizobium to produce nod factors. In response, root cells initiate nodule formation and encapsulate the bacteria, which differentiate into bacteroids capable of N fixation. Once nodulation is completed, a mutualistic exchange of C and N is established [3]. This process is driven exclusively by the plants [4]. Extensive biochemical research has clarified the energy–mass balance within each species, including the ATP costs of CO₂ fixation and N₂ reduction [5]. However, how the energy–mass balance is maintained between the partners remains poorly understood.
One promising framework for analyzing such plant–microbe interactions is the microbial market model [6], a class of biological market models [7,8] that applies concepts from economics to explain interspecific cooperation. An early example is the work of Schwartz and Hoeksema [9,10], who introduced the classical theory of comparative (or relative) advantage in economics [11] to the study of plant–microbe symbiosis. This theory of economics claims that if each of two countries (trade partners, e.g., England and Portugal) specializes in the production of a good for which it has a relative advantage (e.g., wool cloth for England and wine for Portugal, to borrow the original example), then trade becomes mutually beneficial. Schwartz and Hoeksema [9] used this theory to show how specialization and trade in C and phosphorus can benefit both plants and mycorrhizal fungi. While their model was purely theoretical and alternative approaches have since been proposed [12–14], the comparative advantage microbial market model of Schwartz and Hoeksema [9] remains a widely cited foundation for theoretical work on plant–microbe symbiosis.
In this study, we extend this line of research by developing a quantitative comparative advantage model of C–N exchange in legume–Rhizobium symbiosis. Specifically, we incorporate hypothetical ATP budget constraints and the energy–mass balance between the species, and we derive unit-free equilibrium ratios that can be compared directly with empirical data. For instance, we obtain supply ratios of C by the legume and N by Rhizobium as functions of body C:N ratios and the C to N exchange ratio, the latter determined by ATP costs of C and N uptake. This framework enables us to generate empirically testable hypotheses and evaluate them against data accumulated over several decades.
The analysis yields three main implications. (i) The condition for the emergence of symbiosis: consistent with the observation that legume–Rhizobium symbiosis occurs only when soil N is limiting [15,16], we show that the relative ATP costs of C and N uptake determine when symbiosis is favored. (ii) The supply ratios of C and N: the model predicts that, at equilibrium, the legume supplies a small fraction of its fixed C, while Rhizobium supplies a large fraction of its fixed N. These predictions are consistent with empirical reports of low C supply by legumes and high N supply by rhizobia. (iii) Dependence on body stoichiometry: the model predicts that the C supply ratio decreases as the legume body C:N ratio increases, and that the N supply ratio decreases as the rhizobial body C:N ratio decreases. These results are consistent with the empirically observed decline in nodulation with host aging, and may also explain why legumes, with their characteristically lower body C:N ratios relative to other plants, are predisposed to form such symbioses.
Two distinctive features of our model should be noted. First, unlike the framework of Schwartz and Hoeksema [9], we account for the shift in rhizobial N acquisition from direct uptake to fixation once symbiosis is established. Second, rather than assuming equilibrium is reached by “price” adjustment of the C to N exchange ratio, we assume adjustment occurs through symbiont biomass, which is more appropriate when resources are demanded in fixed proportions. With ATP budget constraints, equilibrium is attained when energy and mass are simultaneously balanced between host and symbiont. The Edgeworth box, a classical economic tool, is adapted here to illustrate this equilibrium.
2. The model
Let and
be the amounts of N and C taken up by a unit mass of legume per unit time. We consider the legume per unit mass. We assume that Darwinian fitness
of legume is a function of
and
such that
where and
are constants. We have
if
, and
if
: The functional form of Eq 1 represents Liebig’s law of the minimum [17,18]. The constants
and
can be seen as representing “organismal stoichiometry” [19], assuming the relationships to body C:N ratio as
. (Note that efficient uptake of
and
is such that
, namely,
.).
Legume takes up C by fixing CO2 in the air. We assume that unit ATP cost of fixing C is constant and denote it by . In a non-symbiotic state, legume takes up N directly from the soil. We assume that unit ATP cost of direct uptake of N depends on the soil environment and denote it by
, where
is the soil environment parameter such that
if direct uptake is ATP-demanding, and
if not demanding. Thus,
. Assuming that
is the ATP budget of legume per unit time, uptakes
and
of non-symbiotic legume must satisfy
which is the budget constraint of non-symbiotic legume under soil environment s. Note that the ratio represents the cost of N uptake in units of C for legume under soil environment s. We assume that non-symbiotic legume chooses
in the budget constraint that maximizes its fitness Eq 1. See Fig 1, in which budget constraints under
and
are depicted. We call them budget lines (“isolation acquisition isocline” [9]). The absolute values of the slope of budget lines are
and
, respectively. By definition,
. Non-symbiotic legume chooses point AL(0) if
and AL(1) if
, which we call autarky points of legume. The dotted line emanating from the origin
is referred to as optimality line of legume (“optimal consumption vector” [9]). The point
is the specialization point of legume, which we will explain later.
Horizontal axis is uptake of N and vertical axis is uptake of C by a unit mass of legume. The L-shaped lines are contour lines of fitness function and the solid line through the point AL(0) (resp. AL(1)) is budget line under soil environment s = 0 (resp. s = 1) with slope (resp.
). The dotted line emanating from the origin OL is optimality line. The point SL is the specialization point of legume.
Let and
be the amounts of N and C taken up by a population of Rhizobium per unit time. We consider a size-variable population of Rhizobium. Similarly to legume, we assume that fitness
of Rhizobium is expressed, with organismal stoichiometric constants
and
, by a function
In a non-symbiotic state, Rhizobium takes up N and C both directly from the soil. We assume that unit ATP costs of direct uptake of N and C both depend on the soil environment , and denote them by
and
, with
and
. However, since non-symbiotic Rhizobium lives in a rhizosphere, a very close neighborhood of legume roots, in which soil C:N ratio is relatively kept constant [20], we assume that the uptake cost of N in units of C for Rhizobium is constant (i.e., independent of s), and denote it by
. Let
be the ATP budget of a given population of Rhizobium per unit time. We assume that
is proportionate to the population size, but omit size parameter in the current static analysis for the sake of brevity. The uptakes
and
of non-symbiotic Rhizobium must satisfy
the budget constraint of non-symbiotic Rhizobium under soil environment s. We assume that non-symbiotic Rhizobium chooses in the budget constraint that maximizes its fitness Eq 3. See Fig 2, in which budget constraints under
and
are depicted. We call them budget lines. Due to the assumed constancy of
, budget lines are parallel to each other. Non-symbiotic Rhizobium chooses point AR(0) if
and AR(1) if
, which are the autarky points of Rhizobium. The dotted line emanating from the origin
is optimality line of Rhizobium. The point
represents the maximum amount of N that a given population size of Rhizobium can take by direct uptake when
.
Horizontal axis is uptake of N and vertical axis is uptake of C by a population of Rhizobium. The L-shaped lines are contour lines of fitness function and the solid line through the point AR(0) (resp. AR(1)) is budget line under soil environment s = 0 (resp. s = 1) with slope . The dotted line emanating from the origin OR is optimality line. The point SR(1) represents the maximum amount of N that a given population of Rhizobium can take by direct uptake when
.
(a) Trade in symbiosis
The theory of comparative advantage focuses on the ratios and
, namely, the costs of N uptake in units of C for legume and Rhizobium, and if
then it judges that Rhizobium has a comparative advantage in the uptake of N (and legume has the one in the uptake of C), and predicts that specialization and trade render benefit for the both [9]. It is our hypothesis that
namely, the symbiosis is mutually beneficial when , but not when
. However, there is a point to be considered in legume–Rhizobium symbiosis. As we noted before, Rhizobium changes its mode of N uptake from direct uptake to fixation when entering into symbiosis. Thus, although the budget line emanating from SR(1) in Fig 2 is meaningful as the frontier of N-C uptake for non-symbiotic Rhizobium when
, the point SR(1) cannot be a specialization point of symbiotic Rhizobium. In order for the symbiosis to be really beneficial for Rhizobium, we need one more condition. Assume
and let
(“hat”
) be the unit ATP cost of N fixation by Rhizobium, which we assume is constant. Now, if
namely, if the amount of N obtained by fixation (the right-hand side) is not less than the amount of N obtained by direct uptake when (the left-hand side), then, with
(Eq 5 with
), the mutual benefit of specialization and trade is guaranteed. See Fig 3. The budget line of legume is depicted in Fig 3A and that of Rhizobium in Fig 3B. The point SL is specialization point of legume and SR is that of symbiotic Rhizobium. In Fig 3B, the coordinate of SR(1) is
and the coordinate of SR is
, so Eq 7 is satisfied. Assuming
, let r be an exchange ratio of C to N such that
SL and SR are specialization points of legume and Rhizobium. The lines with slope are trade lines. If
and
, then with exchange ratio r such that
, specialization and trade is beneficial for both legume and Rhizobium.
In our model, this ratio is an exogenously given constant. (See Discussion (b) below regarding the physiological aspect of this ratio). In the figures, the lines with slope −r emanating from SL and SR are trade lines (“trade acquisition isocline” [9]). Under the assumption that legume and Rhizobium maximize their own fitness, they choose the intersection point of the trade line and their own optimality lines. Thus, as can be seen from Fig 3, both legume and Rhizobium can enjoy uptakes that are superior to their autarky points AL(1) and AR(1), and specialization and trade are beneficial for the both.
Fig 4A shows a state in which legume and Rhizobium specialize and trade. This is an Edgeworth’s box diagram [21], in which Fig 3A and Fig 3B are merged in such a way that the origin OR of Rhizobium is at the top-right corner and the specialization points SL and SR of legume and Rhizobium are superimposed at the top-left corner, S. The height of the box then equals the amount of C fixed by a unit mass of legume. The width equals the amount of N fixed by a given population size of Rhizobium. Note that every point in the diagram represents an allocation of fixed N and C to legume and Rhizobium, and that any point in the interior of the shaded rectangle in Fig 4A is a mutually beneficial allocation of N and C (compared to staying at their autarky points). In Fig 4A, where demands for N and C by legume and Rhizobium are marked by the open circles, there is an excess demand for N (and an excess supply for C), because total N demanded exceeds the width of the box (and total C demanded is less than the height). Thus, some adjustment is needed. We assume that supply–demand imbalance is adjusted by a body mass adjustment, specifically by the adjustment of the population size of Rhizobium such that it is increased (resp. decreased) whenever the excess supply (resp. excess demand) of C exists (see Discussion (c) below for the dynamical model of this process). Fig 4B depicts the equilibrium state of such a mass adjustment process. In our current static setting, it is characterized by the intersection of three rays (two optimality lines and the trade line) at some point E. There, we have (1) legume and Rhizobium both specialize in their products, and, at point E, (2) both are maximizing their fitness on the trade line, and (3) supply–demand imbalance is cleared. Thus, point E represents a feasible and mutually optimal allocation of N and C. The fitness of both legume and Rhizobium are improved at E than at their autarky points AL(1) and AR(1). We here refer to E as the equilibrium of symbiosis.
The line with slope is the trade line. Point E in (B) represents the equilibrium of trade, where both legume and Rhizobium maximize their fitness on the trade line, and total demand for N equals total supply of N (similarly for C). Note that, at E, both legume and Rhizobium are strictly better off compared to at their autarky points AL(1) and AR(1). (Here,
is assumed).
(b) Equilibrium values (formulae)
Let h be the height of the box of Fig 4B, which is the amount of C fixed by a unit mass of legume; i.e., . Note that this is equal to the value of
at point S and constant in the model. Let
and
be the slopes of the optimality line of legume and that of Rhizobium; i.e.,
and
. Under our assumption, these are equal to the body C:N ratios of legume and Rhizobium, respectively. Using these variables and the exchange ratio r in the range of Eq 8, the equilibrium uptake of the N and C by legume,
, and that of Rhizobium,
, are:
The equilibrium supplies of C by legume, , and N by Rhizobium,
, are:
The equilibrium width of the box, namely, the width of the box of Fig 4B, is:
See S1 Appendix for the derivations of these formulae.
3. Results
(a) The condition for symbiosis
We now assess the inequalities in Eqs 6–7. Recall that and
. To evaluate each component, we count the number of ATP and ATP equivalents that are required to take up (and convert to if necessary) a specific form of C or N: glyceraldehyde 3-phosphate (G3P) (C3H7O6P) for legume’s C, glucose (C6H12O6) for non-symbiotic Rhizobium’s C, and ammonium (NH4+) for all N. As to the ATP equivalents of reductants, we assume 3 for NADPH, 2.5 for NADH, 1.5 for reduced form of ferredoxin (Fdred), by default.
First, during photosynthesis, legume fixes aerial CO2 to G3P in Calvin–Benson cycle, in which 9 ATP and 6 NADPH are used for each molecule of G3P [22]. For there are three C in a G3P, ATP cost of C for legume is mol ATP/mol C, namely,
. Non-symbiotic legume’s major N sources are nitrate (NO3-) and NH4+ [23]. In aerobic soil, where Rhizobium inhabits [24], the dominant form of N is NO3- [25]. Since the use of NO3- is more ATP-demanding than that of NH4+, we let
and
be the ATP cost of uptake of N via NO3- and via NH4+, respectively. Now, NO3- is taken up actively by nitrate transporters. Root respiration studies revealed that 1–3 mol ATP are consumed for each molecule of NO3- [26]. It is then converted to nitrite (NO2-) and then to NH4+ using 1 NAD(P)H and 6 Fdred [22], where 11.5–12 mol ATP/mol N are used. The use of NO3- also invokes synthesis of transporter [27] and passing of NO2- across plastid membrane that requires energy. Although there is no precise measurement of ATP requirements for them, we may add extra 1–2 mol ATP/mol N, in particular when N is extremely scarce. We thus let
. Meanwhile, NH4+ is taken up passively when its external concentration is high, and actively by using ammonium transporters when the concentration is low [28]. In the former case, the ATP requirement is negligible, but in the latter it requires 1 mol ATP/mol N to use proton pump, so we estimate that 0.5 mol ATP/mol N is needed, i.e.,
. Therefore, with
, we have
and
.
Second, non-symbiotic Rhizobium can utilize NO3- and NH4+. We assume the same costs of NO3- and NH4+ uptakes as those of legume’s, namely, and
. Non-symbiotic Rhizobium takes up various forms of C including glucose (C6H12O6) [29]. To take up glucose, Rhizobium uses ATP-binding cassette (ABC) transporters [30], which consume 2 ATP per glucose. For there are six C in a glucose, ATP cost of C for Rhizobium is then 0.33 mol ATP/mol C. We assume it as a baseline and let
. With
, this leads to
. Also, our assumption
implies, with
, that 8.94
.
To sum up, if we use the median for , we have
i.e., the inequalities in Eq 6 are satisfied. It remains to verify the inequality in Eq 7, or, what amounts to the same thing, the inequality
As to the value of , namely, ATP cost of N-fixation, symbiotic Rhizobium fixes one molecule of aerial N2 to two molecules of ammonia (NH3) using nitrogenase. In this process, 16 ATP and 8 Fdred are used [22]; here, it is thought that 2 ATP are required per Fdred [31]. NH3 to NH4+ is zero cost. Thus, we assume that 32 mol ATP/ mol N2 is required, namely,
. Comparing this figure to the possible range of
,
, we may conclude that Eq 13 is being satisfied. This result suggests that the condition for symbiosis as predicted by the comparative advantage argument is supported by the ATP cost data of host and symbiont.
(b) Equilibrium values
Our model predicts various equilibrium values (Eqs 9–11). Here, we assess the values by checking two unit-free ratios: the supply ratio of C by legume and that of N by Rhizobium
, assuming that
and
are equal to the C:N ratios of legume and Rhizobium, respectively. Now, the C:N ratio of hairy vetch clover is 11:1, legume hay 17:1, and mature alfalfa 25:1 [32]; sweet clover 25.9:1 and blue lupin 20:1 [33]. We therefore assume that
is between 11 and 26. The C:N ratio of soil microbes is 8:1 on average [32]. The C:N ratio of Rhizobium meliloti, however, is between 11:1 and 12:1 [34]. The
value is thus likely between 11 and 12. It is our hypothesis that legume and Rhizobium trade under the exchange ratio r in the range of Eq 8. Thus, using our estimates
and
, we assume r in the range
. It is interesting to note that the theoretical cost of biological N fixation in units of C, namely
, falls within this range. Table 1 shows possible ranges of
and
for
1 (the lower bound),
, and
(the upper bound), assuming
and
. As can be seen from the expressions, supply ratio of C (resp. N) increases (resp. decreases) as the exchange ratio r increases.
Legume supplies in vegetative state about 9.9 to 21.8% of the C that it fixes to Rhizobium [35], whereas Rhizobium supplies over 93% of its fixed N to legume [36]. The predictions of the model are that legume supplies 5–14% of its fixed C and Rhizobium supplies 85–88% of its fixed N (see Table 1). Although the predicted ratio of C supply is slightly lower than the reported one, it seems that the tendency of a small supply ratio of C and a large supply ratio of N is reproduced here.
(c) Effects of the C:N ratios of host and symbiont
By partially differentiating the supply ratios by
and
by
, we have:
Thus, the higher the value of is, the lower the supply ratio of C is; the lower the value of
is, the lower the supply ratio of N is.
In general, increases with age [37]. For example, young alfalfa hay has a C:N ratio of 13:1, while that of mature alfalfa is 25:1 [32]. The C:N ratio of R. meliloti decreases with age [34], i.e.,
decreases with age. The two inequalities in Eq 14 then predict that the supply ratios of C and N both decrease with age (the change in the total amount of fixed N is indefinite because
and
; see Table 2). The nodulation of legumes decreases with age [35], which may be explained by the predictions of decreasing trade with age.
It should also be noted that the C:N ratios of legumes are lower than those for other plants (for example, non-legume plants: barley straw 85:1, oat straw 70:1, corn stalks 60:1 [38]; legume plants: sweet clover 25.9:1 and blue lupin 20:1 [33]). The low values of legume suggest that legumes are more suitable for symbiosis with N-fixing partners than other plants.
4. Discussions
(a) On the model
We have shown a comparative advantage microbial market model of C–N exchange in legume–Rhizobium symbiosis extending Schwartz and Hoeksema’s [9] model. A distinctive feature of our model is that it endogenously determines the ratio of fixed N to fixed C (i.e., the aspect-ratio of the Edgeworth box) and their supply ratios, given the body C:N ratios and some fixed C to N exchange ratio. ATP costs of uptake were used to determine the possible range of C to N exchange ratio. We have derived some implications of the model and assessed them against data accumulated over the last several decades. The following results were obtained: (i) The condition for the emergence of symbiosis predicted by the comparative advantage argument is supported by empirical ATP cost data of host and symbiont. (ii) The model predicts that supply ratio of C by legume is small and that of N by Rhizobium is large, which is consistent with reported figures. (iii) The model also predicts that the supply ratio of C decreases as the C:N ratio of legume increases; the supply ratio of N decreases as the C:N ratio of Rhizobium decreases, which explains the decrease in nodulation with aging. In our view, the result (ii) merits special attention because the characteristically imbalanced supply ratios are being explained by a few robust, stoichiometric quantities such as body C:N ratios and C to N exchange ratio. Of course, however, these predictions are still theoretical in nature, and further experimental validation is required. See also S2 Appendix for the supplementary discussion about the model.
(b) On the physiology of exchange ratio (r) and ATP costs
We used ATP requirements as a common measure of biological constraint. The required numbers of ATP and ATP equivalents were counted using G3P, glucose, and NH4+ as the basis of metabolism The feasible range of the exchange ratio r, in particular, was determined by these ATP costs. Physiologically, no unified theory for this ratio r has yet been established [39], but it is generally considered to be governed by environmental carbon availability—particularly atmospheric concentrations [40]—as well as the abundance of other elements, such as phosphorus [41]. In this study, we treat this ratio as an exogenously given constant; otherwise, the equilibrium may not be well defined.
However, in addition to the processes described here, the period from the initiation to the breakdown of nodule formation includes other costly processes on the plant side, such as nodule development and maintenance. Based on the methodology we have developed, we aim to construct a more detailed model of the entire nodule life cycle that incorporates these processes. We also want to note that similar approaches may be possible for the study of other types of symbiosis, such as the relationship between plant and fungi.
(c) Body mass adjustment process
In our static setting, we have derived the equilibrium of symbiosis E by the intersection of three rays. Here, we show a body mass adjustment process that converges to this equilibrium. Let be the ATP budget of Rhizobium of population size
, and assume
(the budget increases with the population size). The total supply of N by Rhizobium of population size
is then
, and the total supply of C by legume is constantly
. The demand for C is the C-coordinate of the intersection of the trade line and the respective optimality line, so it is
for legume and
for Rhizobium. Consider a process described by a first order ordinary differential equation
i.e., the population size increases (resp. decreases) with the excess supply (resp. excess demand) of C. Since , we have
, and
for every
, so this process is globally asymptotically stable. Also, we have
if and only if
is such that
(Eq 11), as expected.
Supporting information
S1 Appendix. Derivation of the equilibrium values (Eqs 9–11).
https://doi.org/10.1371/journal.pone.0349611.s001
(PDF)
S2 Appendix. Supplementary discussion about the model.
https://doi.org/10.1371/journal.pone.0349611.s002
(PDF)
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