Spatial regularity of auxin pattern in Model B.

<p>(A–L) Wavelength of auxin maxima pattern (<i>L</i>₁) (A, C, E, G, I, and K) and average size of auxin maximum (<i>L</i>₂) (B, D, F, H, J, and L) were determined in Model B1 (A and B), Model B2 (C and D), Model B3 (E and F), Model B4 (G and H), Model B5 (I and J), and Model B6 (K and L). (M–O) Examples of regular patterns with parameter conditions indicated in C and D (M; Model B2 with <i>n</i> = −2.0 and <i>m</i> = −10.0), in I and J (N; Model B5 with <i>n</i> = −4.0 and <i>m</i> = −10.0), and in K and L (O; Model B6 with <i>n</i> = 0.0 and <i>m</i> = 6.0). Numerical simulations were performed in a one-dimensional array of <i>N</i> = 200 (A–L) or 50 (M–O) cells, which are separated each other by apoplast space, by the Euler method with time step Δ<i>t</i> = 0.001 under the periodic boundary condition. Initial values of variables were given by their equilibrium with 1.0% fluctuation. Equations and regulatory functions used are summarized in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.s002" target="_blank">S1 Table</a> with parameter values of <i>K</i> = 2, <i>A</i> = <i>E</i>_<i>p</i> = <i>E</i>_<i>q</i>= <i>G</i>_<i>x</i> = <i>G</i>_<i>p</i> = <i>D</i>_<i>a</i> = <i>V</i> = <i>r</i> = 1.0, <i>p</i> = <i>q</i> = <i>D</i>_<i>x</i> = 10.0, and <i>G</i>_<i>a</i> = 0.2 (A–O).</p

Spatial regularity of auxin pattern in Model O.

<p>(A) The eigenvalues are shown for continuous values of <i>ν</i> in different values of <i>D</i>_<i>a</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e061" target="_blank">Eq 34</a>). (B) The parameter condition that the equilibrium becomes unstable is indicated by the shaded area in the <i>p</i>−<i>D</i>_<i>a</i> plane (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e063" target="_blank">Eq 35</a>). Broken lines indicate <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e064" target="_blank">Eq 36</a> for different values of <i>L</i>_*. (C and D) Wavelength of auxin maxima pattern (<i>L</i>₁) (C) and average size of auxin maximum (<i>L</i>₂) (D) were determined by numerical simulations in <i>p</i>−<i>D</i>_<i>a</i> plane. (E–H) Examples of auxin pattern indicated in C and D with parameter conditions of <i>p</i> = 40.0 and <i>D</i>_<i>a</i> = 100.0 (E), 40.0 (F), 10.0 (G), and 0.2 (H). Numerical simulations were performed in a one-dimensional array of <i>N</i> = 200 cells (C and D) or <i>N</i> = 40 cells (E–H) by the Euler method with time step Δ<i>t</i> = 0.001 under the periodic boundary condition. Initial values of variables were given by their equilibrium with 1.0% fluctuation. Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e005" target="_blank">1</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e007" target="_blank">3</a> and regulatory function <i>φ</i>₀(<i>a</i>_<i>j</i>) = <i>a</i>_<i>j</i>^<i>n</i> are used with parameter values of <i>K</i> = 2, <i>A</i> = <i>E</i>_<i>p</i> = <i>G</i>_<i>p</i> = 1.0, <i>G</i>_<i>a</i> = 0.2, and <i>n</i> = 2.0 (A–H) and <i>p</i> = 1.0 (A).</p

Examples of auxin pattern in two-dimensional space in Models O, A, and B6.

<p>Like the one-dimensional case, spatially regular patterns of auxin maxima can be generated in the two-dimensional space in Model O (A) and Model B6 (C), but cannot in Model A (B). Auxin concentration and PIN1 density are indicated in blue and by the thick magenta lines, respectively. Equations and regulatory functions are used as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.s002" target="_blank">S1 Table</a> with parameter values of <i>K</i> = 6, <i>A</i> = <i>E</i>_<i>p</i> = <i>G</i>_<i>p</i> = 1.0, <i>D</i>_<i>a</i> = <i>p</i> = <i>n</i> = 2.0, <i>G</i>_<i>a</i> = 0.2, and <i>R</i> = 4.0 (A), <i>K</i> = 6, <i>A</i> = <i>E</i>_<i>p</i> = <i>E</i>_<i>q</i> = <i>G</i>_<i>p</i> = <i>D</i>_<i>a</i> = <i>V</i> = 1.0, <i>G</i>_<i>a</i> = 0.2, <i>p</i> = <i>q</i> = 5.0, and <i>R</i> = <i>n</i> = 3.0 (B), and <i>K</i> = 6, <i>A</i> = <i>E</i>_<i>p</i> = <i>E</i>_<i>q</i> = <i>G</i>_<i>p</i> = <i>G</i>_<i>x</i> = <i>D</i>_<i>a</i> = <i>D</i>_<i>x</i> = <i>V</i> = 1.0, <i>G</i>_<i>a</i> = 0.2, <i>p</i> = <i>q</i> = 5.0, and <i>R</i> = <i>m</i> = 3.0 (C). Numerical simulations were performed in two-dimensional sheets of 20 × 20 hexagonal cells by the Euler method with time step Δ<i>t</i> = 0.001 under the periodic boundary condition. Initial values of variables were given by their equilibrium with 1.0% fluctuation.</p

Effects of symplast and apoplast diffusions in Model A.

<p>Wavelength of auxin maxima pattern (<i>L</i>₁) (A and C) and average size of auxin maximum (<i>L</i>₂) (B and D) were determined in the presence of the symplast diffusion (A and B, Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e021" target="_blank">12</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e023" target="_blank">14</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e027" target="_blank">16</a>) or apoplast diffusion (C and D, Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e020" target="_blank">11</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e022" target="_blank">13</a>, <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e023" target="_blank">14</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e028" target="_blank">17</a>), in addition to the simple diffusion between cytoplasm and apoplast (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.g001" target="_blank">Fig 1F</a>). Numerical simulations were performed in a similar manner as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.g004" target="_blank">Fig 4</a> with parameter values of <i>K</i> = 2, <i>A</i> = <i>E</i>_<i>p</i> = <i>E</i>_<i>q</i> = <i>G</i>_<i>p</i> = <i>V</i> = 1.0, <i>q</i> = 10.0, <i>G</i>_<i>a</i> = 0.2, and <i>n</i> = 4.0 (A–D). Equations and regulatory functions used are summarized in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.s002" target="_blank">S1 Table</a>.</p

Spatial regularity control in Model B6.

<p>(A) The eigenvalues are shown for continuous values of <i>ν</i> in different values of <i>D</i>_<i>x</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e078" target="_blank">Eq 43</a>). (B–D) The parameter condition that the equilibrium becomes unstable is shown by shaded area in <i>D</i>_<i>x</i>−<i>D</i>_<i>a</i> (B), <i>V</i>−<i>m</i> (C), and <i>r</i>−<i>m</i> (D) planes (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e079" target="_blank">Eq 44</a>). Broken lines indicate <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.e080" target="_blank">Eq 45</a> for different values of <i>L</i>_*. Regulatory functions as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006065#pcbi.1006065.g008" target="_blank">Fig 8</a> (<i>θ</i>(<i>a</i>_<i>i</i>) = 2<i>a</i>_<i>i</i>^<i>r</i>/<i>a</i>_<i>eq</i>^<i>r</i> + <i>a</i>_<i>i</i>^<i>r</i>), , and ) are used with parameter values of <i>K</i> = 2, <i>A</i> = <i>E</i>_<i>p</i> = <i>E</i>_<i>q</i> = 1.0, and <i>G</i>_<i>a</i> = 0.2 (A–D), <i>p</i> = <i>q</i> = 2.0 (A) or 10.0 (B–D), <i>D</i>_<i>a</i> = 0.1 (A) or 1.0 (C and D), <i>D</i>_<i>x</i> = 5.0 (C and D), <i>G</i>_<i>x</i> = 0.5 (A), 1.0 (B), or 5.0 (C and D), <i>V</i> = 0.1 (A) or 1.0 (B and D), <i>r</i> = 2.0 (A and B) or 4.0 (C), and <i>m</i> = 6.0 (A) or 5.0 (B).</p

Schematic representations of models.

<p>(A) In Model O, auxin is transported between neighboring cells by PIN1, while PIN1 is polarized depending on auxin concentration of neighboring cells. (B) In Model A, auxin is transported between cytoplasm and apoplast by PIN1 and influx carrier, while PIN1 is polarized depending on auxin concentration of neighboring apoplast spaces. (C) In the framework of Model B, an assumed molecule <i>X</i> is incorporated into Model A. Molecule <i>X</i> is expressed in response to cytosolic auxin and diffuses freely between cytoplasm and apoplast. (D) Model B considers various feedback regulations from molecule <i>X</i> to auxin–PIN1 dynamics. (E) In Model B6, PIN1 is polarized depending on <i>X</i> concentration of neighboring apoplast spaces, instead of auxin. (F) In addition to simple diffusion between cytoplasm and apoplast, Models A and B consider direct diffusions between neighboring cells (symplast diffusion) and between neighboring apoplast spaces (apoplast diffusion). (G) Schematic representation of <i>L</i>₁ and <i>L</i>₂: indices for spatial scale of auxin maxima pattern in numerical simulations. <i>a</i>_<i>i</i> (or <i>x</i>_<i>i</i>) and (or ) are auxin (or <i>X</i>) concentrations in cell <i>i</i> and apoplast (<i>i</i>, <i>j</i>), respectively. <i>p</i>_{<i>i</i>,<i>j</i>} is PIN1 density of the membrane toward cell <i>j</i> in cell <i>i</i>.</p

Cost–benefit balance in the symbiosis evolution.

<p>Cost–benefit balance determines the evolution of the symbiotic system by affecting various evolutionary features, such as selection gradient <i>D</i>(<i>x</i>), <i>w</i>(1,0): invasibility of cheaters in a population of mutualists, <i>E</i>(<i>x</i>^*): ESS-stability of partial mutualists, and <i>D</i>(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 <i>D</i>(<i>x</i>)>0 but towards smaller strategies if <i>D</i>(<i>x</i>)<0 (black arrows). Circles indicate singular strategies that are CS (i.e. <i>D</i>(<i>x</i>^*) = 0 and <i>D</i>′(<i>x</i>^*)<0). Filled squares correspond to cheaters (<i>x</i> = 0) or mutualists (<i>x</i> = 1) that are locally ESS-stable (i.e. <i>D</i>(0)<0 or <i>D</i>(1)>0, respectively), and open squares correspond to those that their strategy is not ESS-stable but can coexist with the other strategy.</p

Model for the evolution of the legume–rhizobia symbiosis.

<p>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.</p

Effect of the benefit and cost, assuming a linear benefit function.

<p><b>(A)</b> Theoretically, the benefit function (<i>b_N</i> = 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 <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0093670#pone.0093670.s006" target="_blank">Text S4</a>). 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>(B–D)</b> Pairwise invasibility plots of (B) case (iv), (C) case (v), and (D) case (vi). Rare mutants with strategy <i>y</i> can invade the resident population with <i>x</i> in the gray region (i.e. <i>w</i>(<i>x</i>,<i>y</i>)>0), but cannot in the white region (i.e. <i>w</i>(<i>x</i>,<i>y</i>)<0). <b>(E–H)</b> 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 <i>x</i><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: <i>b</i> = 5.0 (B–H), <i>c</i> = 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, <i>b_N</i> = 0.0, <i>c_N</i> = 0.35, and <i>n</i> = 5.</p

Efficiency of nitrogen fixation.

<p>The efficiency of nitrogen fixation (<i>x_eff</i>) decreases with increasing cost <i>c</i>. <b>(A)</b> This decrease is continuous for a linear cost function (<i>c_N</i> = 0). <b>(B–D)</b> If the benefit function is also linear (<i>b_N</i> = 0), the decrease is discontinuous (arrowheads) at the transition between cases (i) and (ii) <b>(B)</b>, cases (i) and (v) <b>(C)</b>, and cases (iii) and (v) <b>(D)</b>. The parameter regions of cases (i), (ii), (iii), (v) and (vi) are indicated in gray, magenta, blue, purple, and green, respectively. Parameters are: <i>b</i> = 3.5 (A), 1.0 (B), 2.0 (C), and 4.0 (D); <i>b_N</i> = 0.2 (A) and 0.0 (B–D); <i>c_N</i> = 0.0 (A) and 0.5 (B–D); <i>n</i> = 5.</p