Evolutionary Dynamics of Nitrogen Fixation in the Legume–Rhizobia Symbiosis

The stabilization of host–symbiont mutualism against the emergence of parasitic individuals is pivotal to the evolution of cooperation. One of the most famous symbioses occurs between legumes and their colonizing rhizobia, in which rhizobia extract nutrients (or benefits) from legume plants while supplying them with nitrogen resources produced by nitrogen fixation (or costs). Natural environments, however, are widely populated by ineffective rhizobia that extract benefits without paying costs and thus proliferate more efficiently than nitrogen-fixing cooperators. How and why this mutualism becomes stabilized and evolutionarily persists has been extensively discussed. To better understand the evolutionary dynamics of this symbiosis system, we construct a simple model based on the continuous snowdrift game with multiple interacting players. We investigate the model using adaptive dynamics and numerical simulations. We find that symbiotic evolution depends on the cost–benefit balance, and that cheaters widely emerge when the cost and benefit are similar in strength. In this scenario, the persistence of the symbiotic system is compatible with the presence of cheaters. This result suggests that the symbiotic relationship is robust to the emergence of cheaters, and may explain the prevalence of cheating rhizobia in nature. In addition, various stabilizing mechanisms, such as partner fidelity feedback, partner choice, and host sanction, can reinforce the symbiotic relationship by affecting the fitness of symbionts in various ways. This result suggests that the symbiotic relationship is cooperatively stabilized by various mechanisms. In addition, mixed nodule populations are thought to encourage cheater emergence, but our model predicts that, in certain situations, cheaters can disappear from such populations. These findings provide a theoretical basis of the evolutionary dynamics of legume–rhizobia symbioses, which is extendable to other single-host, multiple-colonizer systems.


S4.1. Case (i) No evolution
If D(0) < 0 (i.e. c > c 0 = b/n), an initial population with x = 0 cannot be invaded by mutants adopting similar strategies and is therefore stably maintained (Figures 4A and S2,gray). Thus, nitrogen fixation activity does not evolve, and the population remains in the "No evolution" state (i.e. D(0) < 0) in section 3.2.

S4.2. Case (ii) Maximum evolution
Conversely, if D(x) > 0 for 0 ≤ x ≤ 1, maximum nitrogen fixation activity (x = 1) evolves in the absence of null mutation. This condition is summarized as follows: are the determinant and axis, respectively, of the quadratic function nD(x), Because the conditions of Eq. (S4.2) permit both cases (ii) and (vi) (Figures 4A and S2, magenta and green), these cases are divided according to whether the population of the resident population is not invaded by cheaters and is stably maintained.

S4.4. Case (iii) Intermediate evolution
Intermediate between cases (i) and (ii)/(vi), the parameters satisfy both D(0) > 0 and D(x) < 0 for some 0 < x < 1. Thus, singular strategies that satisfy D(x) = 0 exist between 0 and 1, of which the smallest (x * ) is described by This singular strategy x * is always CS because D′(x * ) < 0, indicating that an initial population with x = 0 evolves toward x * . The symbiotic behavior then permits one of two cases depending on the ESS-stability of x * , namely, , the singular strategy x * is both CS and ESS-stable (i.e. D′(x * ) < 0 and E(x * ) < 0), and the population with x * is robust to invasion by mutants adopting similar strategies. In addition, it cannot be invaded by cheaters with x = 0 imposed by the null mutation, because w(x * ,0) < 0 (see Text S4.7). Thus, this condition leads to case (iii) "Intermediate evolution" (i.e. D(0) > 0, D′(x * ) < 0, and E(x * ) < 0) ( Figures 4A and S2, blue).

S4.5. Case (iv) Co-dependent coexistence
Contrary to case (iii), if E(x * ) > 0 (i.e. c < c ESS ) is satisfied, the singular strategy x * is CS but not ESS-stable (i.e. D′(x * ) < 0 and E(x * ) > 0). This situation is known to cause "evolutionary branching", in which the population of x * is invaded by nearby mutants and subsequently splits into two subpopulations. Under most of the numerical conditions investigated in this paper, evolutionary branching leads to stable coexistence of full cooperators (x = 1) and full cheaters (x = 0). These parameter conditions lead to case (iv) or (v) (Figures 4A and S2, orange and purple), depending on the stability of the monomorphic population of cooperators, determined by the sign of D(1). If D(1) < 0 (i.e. c > c 1 ), the population of cooperators cannot maintain their activity against invasion of mutants producing fewer nitrogen resources. This situation leads to case (iv) "Co-dependent coexistence" (i.e. D(0) > 0, D′(x * ) < 0, E(x * ) > 0, and D(1) < 0) ( Figures   4A and S2, orange).

S4.7. Invasibility of cheating rhizobia in case (iii)
The invasibility of cheating bacteria (y = 0) is determined by the sign of In case (iii), we have w(x * ,0) < 0 because the singular strategy x * is ESS-stable (i.e. D(x * ) = 0 and E(x * ) < 0). Thus, in case (iii), a population with x * cannot be invaded by cheaters.

S4.8. Equilibrium proportion of cooperator in cases (iv)-(vi)
In cases (iv)-(vi), emergent cooperators (x = 1) and cheaters (x = 0) coexist. Now suppose that the proportions of cooperators and cheaters alter during a cycle of host plants colonization, proliferation in root nodules, and release to the soil ( Figure 1A). Let the proportions of cooperators and cheaters before the cycle be p and q = 1p, respectively. Then the probability that a host plant is colonized by m cooperators and (n m) cheaters is Given that the average nitrogen fixation in the focal plant is x m n  , the growth rates of cooperators and cheaters are given by Then the proportion of post-cycle cooperators is  p  g 1 (g 0  g 1 ) . If this proportion is equilibrated, we have p = p′ and p + q =1, (S4.10) Solving Eq. (S4.10), we obtain the proportion of cooperators at equilibrium as (S4.11)