Figure 1.
Schematic representation of model framework.
(A) The evolutionary strategy of an individual rhizobium is its nitrogen fixation activity (0≤x≤1). Rhizobia stochastically inhabit host plants according to pt(x), the probability distribution of their strategy, proliferate in root nodules in a frequency-dependent manner according to Eq. (1), change in their nitrogen fixation ability by mutation, and are released back to the soil following the death of their host plants (Eq. (4)). Proliferation of colonized rhizobia is driven by the benefit (promoting force) and cost (destabilizing force), which depend on the nitrogen fixation activities of the rhizobia. The cost of a single rhizobium is affected by its own strategy xi (i.e. C(xi)) while the benefit is affected by : the average strategy of n colonizers (i.e.
). See text for details. (B and C) Effects of partner choice (B) and mixed nodule (C) on the fitness of rare mutants with nitrogen fixation activity y in the resident population with x. α(x) is the probability that a rhizobium with x is accepted by a host plant, and β is the frequency of nodules inhabited by two rhizobium species. Yellow and blue root nodules are colonized by residents with x and rare mutants with y, respectively.
Figure 2.
Effect of benefit and cost, assuming linear cost function.
(A) Theoretically, the linear cost function (cN = 0) yields three evolutionary outcomes: (i) “No evolution” (gray), (ii) “Maximum evolution” (magenta), and (iii) “Intermediate evolution” (blue). This prediction is consistent with numerical simulations: cases (i), (ii) and (iii) correspond to crosses, squares and circles, respectively. (B–D) Pairwise invasibility plots in (B) case (i), (C) case (ii), and (D) case (iii). Rare mutants with strategy y can invade the resident population with x in the gray region (i.e. w(x,y)>0), but cannot in the white region (i.e. w(x,y)<0). (E–G) Evolutionary dynamics of strategy distribution in (E) case (i), (F) case (ii), and (G) case (iii); darker shades indicate higher frequencies of a strategy. Parameters are: b = 3.0 (B–G), bN = 0.0, c = 0.7 (B and E), 0.1 (C and F), and 0.22 (D and G), cN = 0.0, n = 5.
Figure 3.
Evolution speed of symbiosis establishment.
In case (ii) “Maximum evolution”, an initial population of non-fixing rhizobia (x = 0) evolves to one containing full cooperators (x = 1). The speed of this evolution increases (i.e. cycles required for “Maximum evolution” decrease), as the benefit (b) increases or as the cost (c) reduces. Each data point is the average of five calculations. Parameters are: bN = 0.0, cN = 0.0, n = 5.
Figure 4.
Effect of the benefit and cost, assuming a linear benefit function.
(A) Theoretically, the benefit function (bN = 0) yields six evolutionary outcomes: (i) “No evolution” (gray), (ii) “Maximum evolution” (magenta), (iii) “Intermediate evolution” (blue), (iv) “Co-dependent coexistence” (orange), (v) “Parasitic coexistence by evolutionary branching” (purple), and (vi) “Parasitic coexistence by null mutation” (green) (for derails see Text S4). This prediction is consistent with numerical simulations; crosses, squares, closed circles, open circles, diamonds, and open squares correspond to cases (i), (ii), (iii), (iv), (v) and (vi), respectively. (B–D) Pairwise invasibility plots of (B) case (iv), (C) case (v), and (D) case (vi). Rare mutants with strategy y can invade the resident population with x in the gray region (i.e. w(x,y)>0), but cannot in the white region (i.e. w(x,y)<0). (E–H) Evolutionary dynamics of strategy distribution in (E) case (iv), (F) case (v), (G) case (vi), and (H) case (ii); darker shades indicate higher frequencies of a strategy. Once cheating bacteria with x<0.2 are removed from the coexistence situation (arrowheads), the remaining nitrogen-fixing bacteria can persist stably in case (v), but lose their activities in case (iv). A population of nitrogen-fixing rhizobia can be invaded by cheaters carrying the null mutation (brackets) in case (vi), but not in case (ii). Parameters: b = 5.0 (B–H), c = 0.43 (B), 0.37 (C), 0.3 (D), 0.42 (E), 0.34 (F), 0.28 (G), and 0.26 (H). In all cases, bN = 0.0, cN = 0.35, and n = 5.
Figure 5.
Effects of nonlinear benefit and cost functions.
Nonlinear benefit and cost functions yield the evolutionary outcomes (i) “No evolution” (gray), (ii) “Maximum evolution” (magenta), (iii) “Intermediate evolution” (blue), (iv) “Co-dependent coexistence” (orange), (v) “Parasitic coexistence by evolutionary branching” (purple), and (vi) “Parasitic coexistence by null mutation” (green). Parameter regions of these cases are delineated by the properties of their singular strategy (similar to Text S4); case (i) D(0)<0, case (ii) D(x)>0 for 0<x<1 and w(1,0)<0, case (vi) D(x)>0 for 0<x<1 and w(1,0)>0, case (iii) x* is CS and ESS-stable (i.e. D′(x*)<0 and E(x*)<0), case (iv) x* is CS but not ESS-stable (i.e. D′(x*)<0 and E(x*)>0) and an unstable monomorphic population of cooperators exists (i.e. D(1)<0), and case (v) if x* is CS but not ESS-stable (i.e. D′(x*)<0 and E(x*)>0) and a stable monomorphic population of cooperators exists (i.e. D(1)>0), where x* is the smallest singular strategy (D(x*) = 0 and 0<x*<1). In all cases, n = 5.
Figure 6.
Cost–benefit balance in the symbiosis evolution.
Cost–benefit balance determines the evolution of the symbiotic system by affecting various evolutionary features, such as selection gradient D(x), w(1,0): invasibility of cheaters in a population of mutualists, E(x*): ESS-stability of partial mutualists, and D(1): ESS-stability of mutualists. Thereby, the evolutionary outcomes obtained in our model can be classified into six cases (i)–(vi) according to the cost–benefit balance. For details see text. The selection gradient determines the direction of evolution, such that a monomorphic population evolves towards larger strategies if D(x)>0 but towards smaller strategies if D(x)<0 (black arrows). Circles indicate singular strategies that are CS (i.e. D(x*) = 0 and D′(x*)<0). Filled squares correspond to cheaters (x = 0) or mutualists (x = 1) that are locally ESS-stable (i.e. D(0)<0 or D(1)>0, respectively), and open squares correspond to those that their strategy is not ESS-stable but can coexist with the other strategy.
Figure 7.
Efficiency of nitrogen fixation.
The efficiency of nitrogen fixation (xeff) decreases with increasing cost c. (A) This decrease is continuous for a linear cost function (cN = 0). (B–D) If the benefit function is also linear (bN = 0), the decrease is discontinuous (arrowheads) at the transition between cases (i) and (ii) (B), cases (i) and (v) (C), and cases (iii) and (v) (D). The parameter regions of cases (i), (ii), (iii), (v) and (vi) are indicated in gray, magenta, blue, purple, and green, respectively. Parameters are: b = 3.5 (A), 1.0 (B), 2.0 (C), and 4.0 (D); bN = 0.2 (A) and 0.0 (B–D); cN = 0.0 (A) and 0.5 (B–D); n = 5.
Figure 8.
As the nodule number (n) on a host root increases, the symbiotic relationship evolves less easily. The parameter regions of case (ii) (magenta) and case (iii) (blue) decrease while that of case (i) (gray) increases. However, cheating rhizobia emerge more easily, as shown by the expanding parameter region in which cooperators and cheaters coexist (cases (iv)–(vi); orange, purple, and green). Theoretical and numerical predictions are indicated respectively by gray area and crosses (case (i)), magenta area and closed squares, (case (ii)), blue area and closed circles (case (iii)), orange area and open circles (case (iv)), purple area and diamonds (case (v)), and green area and open squares (case (vi)). Parameters are: b = 5.0, b = 0.0, cN = 0.4.
Figure 9.
Effect of mixed nodule populations.
Our model predicts that mixed nodule populations destabilize the symbiotic relationship (see section 3.8 for details). This prediction is supported by numerical simulations. Theoretical and numerical predictions are indicated respectively by gray area and crosses (case (i)), magenta area and closed squares, (case (ii)), blue area and closed circles (case (iii)), orange area and open circles (case (iv)), purple area and diamonds (case (v)), and green area and open squares (case (vi)). Parameters are: b = 4.0, bN = 0.0, cN = 0.35, n = 5.
Figure 10.
Model for the evolution of the legume–rhizobia symbiosis.
The evolution of the legume–rhizobia symbiosis depends on the cost–benefit balance. As the benefit strengthens relative to the cost, the evolutionary outcome shifts in the following order: (i) “No evolution”, (iii) “Intermediate evolution”, (iv)–(vi) “Coexistence of nitrogen-fixing and cheating rhizobia”, and (ii) “Maximum evolution”. The symbiotic relationship is reinforced by partner fidelity feedback, which strengths the benefit, and by host sanction and partner choice, which diminish the cost. In addition, as the number of nodules on a root increases, symbiotic rhizobia are displaced by selfish cheaters.