Based on the present-day distribution of photosynthetic bacteria31, we assume a competitive advantage for anoxygenic photosynthetic bacteria in early environments where electron donors such as Fe2+, H2S, or H2 were present. We also assume the contemporaneous existence of environments where cyanobacterial populations could thrive, providing a seedbed for migration. Non-marine waters provide an example of the latter, supported by the branching of non-marine taxa from basal nodes in cyanobacterial phylogenies44,45 and also by the presence of stromatolites in Archean lacustrine successions46, despite the likelihood that many Archean lakes and rivers had low levels of potential electron donors such as Fe2+ and H2S47.
Following Jones et al.40 and Ozaki et al.42, we use Fe (iron) and P (phosphorus) to represent the environment, which is similar to the H2 and P employed in other studies48,49. The logic of this choice is that in Archean oceans, Fe2+ is thought to have been the principal electron donor for anoxygenic photosynthesis50,51, whereas P governed total rates of photosynthesis. (Kasting14 argued that H2 was key to photosynthesis on the early Earth, a view supported by low iron concentrations in some early Archean stromatolites52.). In any event, under the conditions of low P availability thought to have characterized early oceans25,40,49,53,54,55, anoxygenic photosynthesis would have depleted limiting nutrients before alternative electron donors were exhausted. In consequence, rates of photosynthetic oxygen production would be low. As iron availability declined and/or P availability increased, the biosphere would inevitably reach a point where P would remain after Fe2+ had been depleted, expanding the range of environments where cyanobacteria are favored by natural selection42.
Our model keeps track of the abundances of anoxygenic photosynthetic bacteria (APB), x1, cyanobacteria, x2, and three crucial chemicals: iron(II) (Fe2+), y1, phosphate (PO43−), y2, and dioxygen (O2), z. Both types of bacteria require phosphate for reproduction. APB needs iron(II) (or some other suitable reductant) as an electron donor in photosynthesis. The following five equations describe the reproduction and death of APB and of cyanobacteria as well as the dynamics of iron(II), phosphate, and dioxygen:
$${rm{APB}}: {dot{x}}_{1} ={x}_{1}{y}_{1}{y}_{2}-{x}_{1}+{u}_{1} {rm{Cyano}}: {dot{x}}_{2} =c{x}_{2}{y}_{2}-{x}_{2}+{u}_{2} {{rm{Fe}}}^{2+}: {dot{y}}_{1} ={f}_{1}-{y}_{1}-{x}_{1}{y}_{1}{y}_{2}-{y}_{1}z {{rm{PO}}_{4}}^{3-}: {dot{y}}_{2} ={f}_{2}-{y}_{2}-{x}_{1}{y}_{1}{y}_{2}-{x}_{2}{y}_{2} {{rm{O}}}_{2}: dot{z} =a{x}_{2}{y}_{2}-bz-{y}_{1}z$$
(1)
Here, we have omitted to write symbols for those rate constants that, for understanding the GOE, can be set to one without loss of generality (Supplementary Note 1). Each remaining rate constant is a free parameter. Equations (1) thus satisfy redox balance by construction. We are left with a system that has five main parameters: c specifies the rate of reproduction of cyanobacteria; f1 and f2 denote the rates of supply of iron(II) and phosphate, respectively; a denotes biogenic production of oxygen; b denotes geochemical consumption of oxygen. Note that iron(II) and phosphate are also removed by geochemical processes at a rate proportional to their abundance. In addition, iron(II) is used up during anoxygenic photosynthesis, and iron(II) reacts with oxygen and is thereby removed from the system. Phosphate is used up during the growth of APB and cyanobacteria. (We investigate extensions of the model that incorporate bounded bacterial growth rates and organic carbon in Supplementary Note 2 and Supplementary Note 3, respectively.)
We posit iron(II) as the primary electron donor for anoxygenic photosynthesis, and for simplicity of presentation, we refer to y1 and f1 in this context. However, as noted above, y1 and f1 can similarly represent the abundances and influxes of other alternative electron donors, especially dihydrogen (H2)56,57 and hydrogen sulfide (H2S)58. Our model, its analytical solution, and the conclusions that follow hold equally well by considering any of these electron donors or all together.
We also include small migration rates, u1 and u2, which allow for the possibility that APB and cyanobacteria persist in privileged sites from which they can migrate into the main arena of competition. On the Archean Earth, these parameters could have been affected by the flow of water and by surface winds. For the mathematical analysis presented in the main text, we assume that these rates are negligibly small.
The GOE represents the transition from a world dominated by APB (Equilibrium E1) to one that is dominated by cyanobacteria (Equilibrium E2) (Figs. S1, S2). On a slowly changing planet, the abundances of APB and cyanobacteria and of the three chemicals are approximately in steady state. Therefore, we consider the fixed points of Eqs. (1).
Pure equilibria
In the absence of APB and cyanobacteria, the abiotic equilibrium abundances of iron(II) and of phosphate are given by f1 and f2, respectively, and there is no oxygen in the system. If f1f2 > 1, then APB can emerge. Subsequently, the system settles to Equilibrium E1, where only APB are present and there is still no oxygen. E1 is stable against invasion of cyanobacteria if
$${f}_{1}-{f}_{2},> ,frac{(c+1)(c-1)}{c}.$$
(2)
This condition can be fulfilled if the influx of iron, f1, is large enough, or if the influx of phosphate, f2, is small enough. The term on the right-hand side of the inequality is an increasing function of the reproductive rate, c, of cyanobacteria.
If cf2 > 1, then the system admits another equilibrium, E2, where only cyanobacteria are present and oxygen is abundant. Equilibrium E2 is stable against invasion of APB if
$$a(c{f}_{2}-1),> ,(b+c)({f}_{1}-c).$$
(3)
The left-hand side of the inequality is positive. If the right-hand side is negative (that is, if f1 < c), then the condition certainly holds. If the right-hand side is positive, then the condition can be fulfilled if the influx of phosphate, f2, is large enough, or if the production of oxygen, a, is large enough. In other words, the dominance of cyanobacteria after the GOE can be guaranteed by a sufficiently large supply of phosphate or sufficiently large production of oxygen. It may or may not be possible for the proportional removal rate of oxygen, b, to become small enough for the condition to be fulfilled.
Mixed equilibrium
If Conditions (2) and (3) are either both satisfied or both not satisfied, then the system also admits an interior equilibrium, (hat{E}). If Conditions (2) and (3) are both satisfied, then Equilibrium (hat{E}) is unstable; if those conditions are both not satisfied, then Equilibrium (hat{E}) is a stable mixed equilibrium where both types of bacteria coexist. Equilibrium (hat{E}) is characterized by the stable coexistence of APB and cyanobacteria if
$$b,> ,c(a-1).$$
(4)
Condition (4) is understood as follows. If b is sufficiently large, then there is not enough atmospheric oxygen for rusting to render E2 stable against invasion of APB before E1 loses stability; the result is stable coexistence. But if b is sufficiently small, then rusting causes E2 to become stable before E1 becomes unstable. The critical value of b therefore depends on the input of atmospheric oxygen for Equilibrium E2; it is an increasing function of the reproductive rate of cyanobacteria and of their rate of production of oxygen.
If a < 1, then bistability is not possible. In this case, for Equilibrium E2, dioxygen is depleted by rusting before there is any significant loss of iron(II). As a result, E2 cannot gain stability before E1 loses stability, regardless of the values of b or c.
Figure 2 shows, for different values of b, the behavior of the system as a function of f1 and f2.
High values of f1 and low values of f2 promote stability of E1 and instability of E2. Low values of f1 and high values of f2 promote instability of E1 and stability of E2. a If the proportional consumption rate of oxygen, b, is large, then intermediate values of f1 and f2 lead to both E1 and E2 being unstable, with Equilibrium (hat{E}) corresponding to stable coexistence. b For an intermediate value of b, either E1 is stable with E2 unstable, or E1 is unstable with E2 stable. c If b is small, then intermediate values of f1 and f2 lead to both E1 and E2 being stable.
Transition from Equilibrium E
1 to Equilibrium E
2
The transition between Equilibria E1 and E2 can be achieved by reducing the supply of iron(II), f1, since such a reductant is required for anoxygenic photosynthesis. When this happens, we lose the stability of E1 and gain the stability of E2.
The transition is gradual if b > c(a − 1). Figure 3 shows gradual oxygenation due to decreasing f1. In this case, the transition occurs via the mixed equilibrium, (hat{E}), where both types of bacteria coexist (Fig. 4). A subsequent increase in f1 can cause APB to regain dominance (Fig. S3a).
Equilibrium E1 (APB dominate) loses stability and Equilibrium E2 (cyanobacteria dominate) gains stability when f1 drops below ({f}_{1}^{* }) and (f_1^{prime}), respectively. We set f2 = 80, c = 10, a = 10, b = 100, and u1 = u2 = 10−3. a We simulate Eqs. (8) from Supplementary Note 1 with α1 = α2 = β1 = β2 = 1, and we set f1 = 100 − 40(t/105). t* denotes the time at which Equilibrium E1 loses stability. b There is stable coexistence of both types of bacteria for (f_1^{prime} ,<,{f}_{1},<,{f}_{1}^{* }).
For values of f1 = 17 (a), 12 (b), 11 (c), 10 (d), 9 (e), and 4 (f), the stable equilibrium (green dot) moves continuously from a world that is dominated by APB to one that is dominated by cyanobacteria. Parameter values are f2 = 10, c = 1, a = 10, b = 12, u1 = u2 = 1, and α1 = α2 = 1. The GOE is gradual.
Alternatively, if b < c(a − 1), then the transition is sudden (i.e., discontinuous). Figure 5 shows rapid oxygenation due to decreasing f1. In this case, E2 is already stable before E1 loses stability (Fig. 6). This results in bistability and hysteresis: Once the world is dominated by cyanobacteria, moderate fluctuations in the supply rate of iron would no longer change the status quo (Fig. S3b).
<div class="c-article-section__figure js-c-reading-companion-figures-item" data-test="figure" data-container-section="figure" id="figure-5" data-title="The GOE can be triggered by a decline in the influx of iron(II) and is sudden if b
Equilibrium E2 (cyanobacteria dominate) gains stability and Equilibrium E1 (APB dominate) loses stability when f1 drops below (f_1^{prime}) and ({f}_{1}^{*}), respectively. We set f2 = 80, c = 10, a = 10, b = 80, and u1 = u2 = 10−3. a We simulate Eqs. (8) from Supplementary Note 1 with α1 = α2 = β1 = β2 = 1, and we set f1 = 100 − 40(t/105). t* denotes the time at which Equilibrium E1 loses stability. b Bifurcation plots reveal bistability for ({f}_{1}^{* },<,{f}_{1},<,{f}_{1}^{prime}).
For f1 = 73 (a), there is a single stable equilibrium (green dot) describing a world dominated by APB. For values of f1 = 65 (b), 62 (c), 51 (d), and 47 (e), there is a second stable equilibrium (green dot) describing the dominance of cyanobacteria, and in addition, there is an unstable equilibrium (red dot). The unstable equilibrium moves as the value of f1 changes. For f1 = 40 (f), the only stable equilibrium is the one where cyanobacteria dominate. Parameter values are f2 = 10, c = 1, a = 10, b = 1, u1 = u2 = 1, and α1 = α2 = 1. The GOE is triggered by a saddle-node bifurcation and is sudden.
The effects of increasing f2 are nearly identical to those of decreasing f1. Increasing the supply of phosphate results in loss of stability of E1 and gain of stability of E2. This is because as f2 rises, APB proliferate, inducing a concomitant depletion of iron reserves. The transition is gradual if b > c(a − 1) (Fig. S4) or sudden if b < c(a − 1) (Fig. S5). The critical values of f1 and f2 are robust to changes in u2 (Figs. S6a, S6b).
Another possibility is that the GOE resulted from an increase in the parameter c, which denotes the reproductive rate of cyanobacteria, as affected by biological mutations. We cannot exclude the possibility that cyanobacterial performance and, therefore, primary production increased as a function of genetic innovations; however, the observation that even today oxygenic photosynthesis by cyanobacteria is limited when alternative electron donors are present places limits on such speculation. The parameter c could also be affected by geophysical or geochemical properties unrelated to oxygen consumption and independent of iron(II) or phosphate flux, such as temperature, pH, salinity, or availability of trace nutrients or other resources. The transition can be gradual (Fig. S7) or sudden (Fig. S8), depending on whether b > c(a − 1) or b < c(a − 1) when c is such that Equilibrium E1 becomes unstable. Similar to the critical values of f1 and f2, the critical value of c for triggering a GOE is robust to changes in u2 (Fig. S6c).
Yet another possibility is that the GOE was triggered by an increase in parameter a, which measures the production rate of oxygen (Fig. S9), or by a reduction in parameter b, which denotes the proportional consumption rate of oxygen (Fig. S10). For this transition to occur, however, it is essential that u2 is sufficiently large. Moreover, the critical values of a and b are strongly dependent on the magnitude of u2. If a is not large enough, then it is not possible for a reduction in b to trigger a GOE, regardless of how small b becomes (Fig. S11).
A GOE resulting from an increase in a or a decrease in b is necessarily sudden. This is because as a rises or b declines, Equilibrium E2, which is characterized by abundance of oxygen, may eventually gain stability, while Equilibrium E1 remains stable. If a becomes sufficiently large or b becomes sufficiently small, then E1 may cease to exist, and a saddle-node bifurcation results in rapid oxygenation.
Effects of migration rate
The migration rates, u1 and u2, have negligible effects on the abundance of APB for Equilibrium E1 and on the abundances of cyanobacteria and oxygen for Equilibrium E2. The principal effects of the migration rates are to determine the abundance of APB for Equilibrium E2 and the abundances of cyanobacteria and oxygen for Equilibrium E1. As such, u1 and u2 control the magnitude of the decline in APB across the GOE and the magnitude of the rise in cyanobacteria and oxygen across the GOE (Figs. S12, S13).
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