Introduction

Oxygenation of the Earth’s atmosphere was traditionally widely viewed to have occurred in two major steps: one in the late Archean and early Proterozoic (between 2.4–2.1 billion years ago (Ga)), known as the Great Oxidation Event (GOE), and another one at the end of the Proterozoic Eon between 0.8–0.6 Ga, referred to Neoproterozoic Oxygenation Event (NOE), coincident with the first major rise of complex, multicellular life1,2,3.

After the GOE, atmospheric oxygen levels are generally accepted to have passed the threshold (~10−6 − 10−5 present atmospheric level (PAL)4) for the formation of mass-independent fractionation of sulfur (MIF-S), as a geochemical proxy for an anoxic atmosphere5,6. And almost certainly reached values > 1% PAL or higher during the geological interval known as the Lomagundi-Jatuli Event that records the most pronounced excursion in the isotopic composition of carbon in marine carbonate (δ13C)1,3,7,8. A wide range of models have been invoked to explain the GOE—from the emergence of cyanobacteria to changes in the cycling of bio-essential elements (e.g., phosphorous)9,10,11. Unlike the GOE, increases in atmospheric oxygen levels during the NOE are poorly constrained. Increases in biological complexity around this time have been commonly invoked as the driver or result of this oxygenation event (e.g., 12). Regardless of the causes of the GOE and NOE, an increase in atmospheric oxygen has been traditionally assumed to have been unidirectional with major steps occurring during the GOE and the NOE1. The rise of land plants has also been proposed to drive a step jump in atmospheric oxygen levels (e.g., 13), consistent with the results from both biogeochemical modeling and proxy records.

Recent evidence from geological records challenges the traditional view of stepwise atmospheric oxygenation. More specifically, results from a suite of geochemical proxies that capture the change in the surface ocean and atmospheric oxygen levels, point toward dynamic atmospheric oxygen14,15,16,17,18,19. For instance, analysis of Paleoproterozoic-age sedimentary rocks suggests fluctuations in the production of MIF-S during and after the GOE—implying swings in atmospheric oxygen levels14. In the Proterozoic Eon, the results from redox-sensitive elements and other geochemical proxies also suggest pulses of surface ocean and atmospheric oxygen levels15,16,20,21,22. Lower proxy-based estimates of atmospheric oxygen are on the low end or below estimates of stable oxygen states driven by 1-D atmospheric models (e.g., 23). Although our picture of Earth’s redox evolution is continually developing, it is becoming increasingly difficult to explain some of the recent insights from the rock record that suggest variable atmospheric oxygen (e.g., 14) and are inconsistent with the traditional view of largely invariant atmospheric oxygen levels during the Proterozoic (e.g., 1).

To provide a new view on the range of stable states of atmospheric oxygen, we conducted a dynamical analysis of global oxygen mass balance. Dynamical system analysis, in its simplest definition, is the study of the state of a system as its main parameter (initial condition, in this case, pO2) is varied. This tool is mainly useful for studying non-linear systems where a small change in a parameter in the system can lead to a substantial change in the state of the system. This approach has not yet been widely used in the Earth Sciences—and more specifically in studying the evolution and dynamic of Earth’s biogeochemical cycles of major elements. However, it is widely used to study non-linear systems across a wide range of fields such as engineering and economics and is ideally suited to explore systems with strong positive and negative feedbacks—like the global oxygen cycle.

We use a simple oxygen mass balance expression as the basis for our analysis:

$$\frac{d({{pO}}_{2})}{{dt}}={J}_{{NPP}}+{J}_{{Hydrogen}}-{J}_{{CH}4,O2}-{J}_{{volc}}-{J}_{{ocean},{sink}}-{J}_{{org},{weath}}-{J}_{{pyr},{weath}}$$
(1)

Here d(pO2)/dt is the change in atmospheric oxygen level in Tmol per year, JNPP, JHydrogen, is the source of oxygen respectively from the burial of organic carbon on land and in the ocean, and the rate of escape of hydrogen atoms to space. The production of oxygen through organic burial is the sum of oxygen production in the ocean and on land. The oceanic oxygen production is calculated by considering the strength of net primary production (NPP (Tmol year−1)), the burial efficiency of organic matter, and a factor that accounts for the effect of anoxia on phosphorous cycling and organic matter burial (see “Methods”). Similar to the previous studies, the terrestrial oxygen production includes the fire feedback which is parameterized as a function of pO2, carbon dioxide level (pCO2), and temperature. The rate of hydrogen escape is a function of atmospheric methane level (pCH4), following previous studies.

The JCH4,O2, Jvolc, Jocean,sink, Jorg,weath, Jpyr,weath, are sinks of oxygen which respectively represent atmospheric methane oxidation, volcanic outgassing (e.g., H2S), oceanic sinks of oxygen (oxidation of sulfide, ferrous iron, and methane), rate of oxidative weathering of organic matter, and pyrite. The rate of atmospheric methane oxidation (JCH4,O2) is similar to previous studies, and it is a function of pO2, pCH4, and an oxidation parameter (Ψ). The rate of oceanic sink of oxygen is parameterized to be a function of NPP and the amount of anoxia in the ocean, which is a function of pO2. Finally, consistent with previous modeling results, the oxidative weathering of pyrite and organic matter was assumed to change, albeit non-linearly, with pO2.

Tied to each of the fluxes presented in Eq. (1) are previously proposed feedbacks in the coupled carbon, oxygen, and sulfur cycles. For instance, oxygen fluxes from the biosphere are increased when surface oxygen levels are decreased—given that anoxia favors organic carbon burial. In contrast, fires can suppress atmospheric oxygen levels—higher atmospheric oxygen levels favor fires. All of the utilized feedbacks are tied to oxygen fluxes (full parameterization schemes used for the oxygen sinks and sources are presented in the “Method” section). Since the flux of oxygen corresponds to hydrogen escape ( JHydrogen), and atmospheric methane oxidation ( JCH4,O2) depends on the concentration of atmospheric methane, we coupled the oxygen mass balance to a simple mass balance model for atmospheric methane that includes the flux of methane from the ocean as a source of methane and atmospheric methane oxidation and hydrogen escape flux as the sinks of atmospheric methane (pCH4). The details of the parameterization used for the methane mass balance model are also explained in the “Methods” section.

The feedbacks we are exploring have been extensively discussed and there is general consensus that these factors play a critical role in modulating Earth’s atmospheric oxygen levels4,10,24,25,26. There is, however, uncertainty in the strength of each of these feedbacks. To account for these uncertainties, we employed a stochastic approach, exploring the range of previously proposed values for key terms in our oxygen mass balance4,10,23,24,27,28,29. Specifically, in our stochastic approach, instead of using a fixed value for input parameters, the values for the parameters will be sampled randomly within the range suggested in the literature (Table S1), assuming a uniform distribution, and the most probable range of dpO2/dt will be obtained for the full range of pO2 values. We did 1000 simulations for each of the stochastic runs. There was no noticeable difference in the model results with more than 1000 runs, which suggests that our selected number (1000) is sufficiently large to capture the variability in the range of model input parameters. The stochastic approach used in our analysis allows us to be relatively agnostic about the debate over specific parametrizations and explore possible Earth system states given the current consensus on the long-term oxygen cycle. Finally, while our model accounts for the major feedbacks on the global oxygen mass balance, there are other environmental factors (e.g., tectonic activity, continent configuration, etc.) that likely impact atmospheric oxygen levels and have varied throughout Earth’s history and also likely impacted some of the major feedbacks. However, our stochastic approach would likely capture, at least in part, the effect of these factors in altering the feedback strengths.

Results and Discussion

Multiple steady states for pO2

Our results indicate five possible steady states for the atmospheric oxygen levels (dpO2/dt = 0) (Fig. 1). Generally, stable states (states I, III, V; Figs. 1, 2) are defined as the steady-state points (dpO2/dt = 0) where the sign of dpO2/dt changes from positive to negative. In these states, the system tends to remain in equilibrium because the positive-to-negative transition indicates a decrease in pO2 values when starting from values larger than the steady state, eventually reaching stability. Conversely, when starting from values smaller than the steady state, the positive-to-negative transition signifies an increase in pO2 values, pulling the system back toward equilibrium (states I, III, V; Figs. 1, 2). Unstable states are also steady-state points (dpO2/dt = 0) but differ in that the sign of dpO2/dt changes from negative to positive (states II, IV; Figs. 1, 2). In these states, the system tends to move away from equilibrium, as the negative-to-positive transition implies an increase in pO2 values when starting from values larger than the steady state, leading the system away from stability. Conversely, when starting from values smaller than the steady state, the negative-to-positive transition indicates a decrease in pO2 values, pushing the system further from equilibrium.

Fig. 1: The results of stochastic dynamical analysis of global oxygen mass balance.
figure 1

a shows the median of the results of the stochastic analysis for a wide range of environmental and initial conditions. b corresponds to the full range of results for different pO2 steady states (dpO2/dt = 0). The open circles in panel (b) represent points outside the range for each steady-state point. Our results suggest five steady states for the atmospheric oxygen level. The box plot in panel (b) is the result of the stochastic simulation where the middle line in the box corresponds to the median value, the edges of the box correspond to 25th(lower edge) and 75th (upper edge) percentile, and the ends of the whiskers correspond to the minimum and maximum values for each state. The results of the stochastic simulation were filtered to be consistent with the results from geochemical proxies. We only considered results that include at least three steady-state points: a modern-like steady state, a steady state under anoxic condition, consistent with geochemical proxies that indicate an anoxic atmosphere (e.g., mass-independent fractionation of sulfur (MIF-S)) before the GOE, and one steady-state point under the low-oxygen condition that is largely reflective of an environmental conditions that is neither anoxic (lack of MIF-S signal) and nor is modern-like (~1 present atmospheric oxygen level (PAL)).

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Fig. 2: The schematic presentation of the different oxygen steady states.
figure 2

The result from the dynamical analysis of a simple oxygen mass balance suggests four steady states (dpO2/dt = 0) for the atmospheric oxygen level. The first one is a stable state (State I) under anoxic conditions, the second one (State II) is an unstable point, and the third one is the first stable state for oxygen (State III) with an oxygen range of ~10−3 present atmospheric level (PAL). The fourth steady state point is unstable is ~10−1 PAL (State IV). The last stable state is State V with a modern-like atmospheric level. The “Energy” corresponds to the potential energy. The energy was calculated as the surface area under the curve (Fig. 1a) which is the integral of dpO2/dt over the range of pO2 (\(-{\int }_{{{pO}}_{2,i}}^{{{pO}}_{2,f}}\frac{{{dpO}}_{2}}{{dt}}d{p}_{{O}_{2}}\)). The amount of energy depends on the extent of the basin of attraction (a region in which the initial conditions (pO2) lead to the eventual convergence of the system to a specific stable steady state or attractor). State I with the largest basin of attraction has the minimum energy implying a large stability around this state.

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Our results from the stochastic simulation were filtered to be consistent with the geochemical proxies. Specifically, we only considered the results from the stochastic simulation that include a steady state point under high oxygen modern-like conditions, a steady state point under anoxic conditions (pO2 < 10−5 present atmospheric level (PAL)) which is consistent with the presence of anoxic atmosphere during Earth’s early stage (e.g., mass-independent fractionation of sulfur), and a steady-state point between the two end members of high and anoxic conditions that would generally correspond to the geologic times where there is no evidence for anoxic nor high oxygen conditions.

The results from the stochastic analysis of the global oxygen mass balance suggest that atmospheric oxygen can be in balance under essentially anoxic conditions (state I; ~8 × 10−9 (25th percentile) to 2 × 10−7 (75th percentile) with median ~3 × 10−8 and minimum and maximum values of ~1 × 10−13 and 4 × 10−7 PAL), very low oxygen conditions (state II; 1 × 10−5 to 1 × 10−4 with median ~3 × 10−5 and minimum and maximum values of 3 × 10−7 and 2 × 10−4 PAL), low oxygen condition (state III: ~1 × 10−3 to 1 × 10−2 with median ~3 × 10−3 and minimum and maximum values of ~8 × 10−5 and 3 × 10−2 PAL), relatively high oxygen condition (state IV: ~1 × 10−1 to 4 × 10−1 with median ~2 × 10−1 and minimum and maximum values of ~10−2 and 7 × 10−1 PAL), and high, near modern oxygen conditions (state V: 9 × 10−1 to 1.1 × 10° with median ~10° and minimum and maximum values of 7 × 10−1 and 1.3 × 10° PAL) (Fig. 1b). To explore the extent of stability in each stable state, we calculated the energy corresponding to each steady state point. The energy was calculated as the surface area under the curve (Fig. 1a) which is the integral of dpO2/dt over the range of pO2 (−\({\int }_{{{pO}}_{2,i}}^{{{pO}}_{2,f}}\frac{{{dpO}}_{2}}{{dt}}d{p}_{{O}_{2}}\)) (Fig. 2). The amount of energy depends on the extent of the so-called basin of attraction (a region in which the initial conditions (pO2) lead to the eventual convergence of the system to a specific stable steady state or attractor). For instance, state I with the largest basin of attraction has the minimum energy implying large stability around this state (Fig. 2).

According to our results, the only stable states occur under states I, III, and V. The first stable state, state I, emerges under anoxic conditions, followed by an unstable state (state II) with an atmospheric oxygen value around ~10−5 PAL. State III is another stable state which is followed by an unstable state (state IV). Under the unstable states, a small perturbation in the global oxygen mass balance would lead to a shift toward adjacent stable states (states I, III, and state V). The expression of the state shifts would likely be high levels of variance in atmospheric oxygen levels around a relatively stable baseline condition. The results also show that modern-like high oxygen conditions of state V can only be achieved if the atmospheric oxygen level passes the pO2 threshold for the unstable state IV (~10−1 PAL) (Figs. 1, 2). This means that there is a rather large “basin of attraction” for State III which suggests that under a wide range of parameter values (Table S1) that can represent a wide range of environmental conditions and a wide range of initial conditions (pO2 values), the system moves dynamically toward State III. The basin of attraction for State V starts from the maximum pO2 at State IV, which means that reaching State V requires an increase in the pO2 level above this value.

To further explore the influence of each feedback on the global oxygen mass balance, we conducted a sensitivity analysis. Specifically, to test the sensitivity of our results for the steady-state pO2 levels to the strength of each feedback, we fixed the value of each oxygen feedback to its maximum value used in the stochastic analysis and recorded the change in the median of dpO2/dt vs pO2 (Figure S2). The magnitude of change in the value for a steady-state point for pO2 in response to a change in the rate of oxygen production or consumption indicates the relative importance of that feedback in driving the steady-state point for pO2. For instance, according to Fig. 2S, a change in the rate of hydrogen escape would result in a major shift in the value of the first steady-state point for pO2 with no substantial change on other steady-state points for pO2. This result suggests an important role of hydrogen escape in driving the first pO2 steady-state point. According to the results from the sensitivity analysis, the rate of hydrogen escape and the atmospheric oxidation of methane have the largest influence on the emergence of the first and second steady-state for pO2 under anoxic conditions (Figure S2), consistent with the previous work30. Specifically, a change in the rate of hydrogen escape and atmospheric methane oxidation can result in a shift or disappearance of the first and second steady-state points for atmospheric oxygen levels (Figure S2). The major role of the rate of hydrogen escape on the first stable state for the atmospheric oxygen level (state I) indicates the key role of hydrogen escape in stabilizing the atmospheric oxygen level on the early Earth. According to our results, an increase in the rate of atmospheric methane oxidation influences the emergence of the second steady-state point (state II) around 10−5 PAL, consistent with the modeling results from previous studies (e.g., 30).

Based on our results from the sensitivity test, the emergence of the third and fourth steady-state points under low-oxygen conditions (states III & IV) is controlled by the oceanic oxygen production ( JNPP) and the sink of oxygen in the ocean and on land. An increased rate of oceanic oxygen production ( JNPP) would shift the imbalance value of oxygen (dpO2/dt) to positive values, which would influence the appearance of steady-state points under low-oxygen conditions. At the same time, according to our results, while under anoxic conditions (<10−5 PAL) the oxidative weathering of pyrite and organic matter is essentially ineffective in controlling the global oxygen mass balance (Figures S2, S3), an increased rate of oxidative weathering under low-oxygen condition would dramatically change the oxygen imbalance value which would impact the emergence of steady-state points under low-oxygen conditions. Of the two steady-state points under low-oxygen conditions (states III & IV), the first one is the result of an increase in oxygen production in the wake of the evolution of oxygenic photosynthesis while the rate of oxidative weathering is low. An increase in the atmospheric oxygen level would result in an increase in the rate of oxidative weathering which would result in a negative imbalance for the oxygen. A continued increase in the production of oxygen in the ocean would have resulted in a decrease in anoxia in the ocean which would have acted as positive feedback in the global oxygen mass balance and have resulted in the emergence of the state IV. Such a state is, however, unstable, which means that a change in the oceanic oxygen mass balance (e.g., a change in the oxygen production through net primary production or marine oxygen consumption) would shift the system to the adjacent stable state (state III or state V), depending on the directionality of the change. Our results also suggest a minor role of enhanced phosphorous recycling under anoxic conditions in driving the emergence of steady-state points under low-oxygen conditions, caused by low-sulfate conditions under low-oxygen conditions.

Our new results on the potential for multiple stable states can be placed in the context of widely held views of atmospheric oxygen evolution. For instance, some foundational modeling work has suggested the presence of a bi-stability in the atmospheric oxygen level, with one stable state less than 10−5 PAL, another stable state above 5 × 10−3 PAL, and an unstable state in between these two stable states30. Our results are consistent with this work—where there are two stable states (states I and III) before and after around 10−5 PAL with an unstable state around 10−5 PAL (state II; Fig. 1). Our results for the range of atmospheric oxygen at unstable state II (1 × 10−5 to 1 × 10−4 with median ~3 × 10−5 and minimum and maximum values of 3 × 10−7 and 2 × 10−4 PAL) are largely consistent with the previous modeling results. The results from the photochemical modeling suggest instability of atmospheric oxygen between 5 × 10−8 and 5 × 10−4 PAL that are largely consistent with our results for the range of atmospheric oxygen level at unstable state II23.

Our results for the range of stable states under low-oxygen conditions is also consistent with the some previous results. For instance, the results from a one-dimensional time-dependent photochemical model suggest a stable range of atmospheric oxygen above ~5 × 10−4 to 5 × 10−3 PAL23. This range is consistent with the range of atmospheric oxygen resulting from our stochastic simulation for state III (state III: ~1 × 10−3 to 1 × 10−2 with median ~3 × 10−3). The results from another modeling study also indicated the presence of three stable states, one under anoxic conditions, one under low-oxygen conditions, and one under high-oxygen conditions27, which is roughly consistent with our results here. However, these studies did not focus on the possibility and extent of the fluctuation in the atmospheric oxygen level under low-oxygen conditions, which is increasingly supported by recent lines of evidence (e.g., 14,31). Our results suggest the Earth system can reach a steady state at high oxygen conditions (state IV; ~10−1 PAL), as suggested by previous work27, but that such a state is largely unstable (Fig. 2). Our results are consistent with the range of steady-state oxygen levels suggested by the previous studies. However, our analysis can provide further insight into the nature of such steady-state points, the range of variations in atmospheric conditions within these Earth system states, and the transitions between the states.

We can directly compare these stable states to estimates of atmospheric oxygen levels from the rock record. Prior to the rise of cyanobacteria, the escape of hydrogen atoms to spaces was the only source of oxygen in the atmosphere32. This oxygen production pathway would have been significant under very low oxygen conditions, especially if there were high atmospheric methane levels32. The increased pO2 level caused by significant hydrogen escape flux would result in a decrease in the atmospheric methane level (pCH4) which would, in turn, result in a reduction in the hydrogen escape flux and lead to the first stable state of atmospheric oxygen (State I; Fig. 1). The wide range of values for state I also suggests the possibility of large fluctuations and dynamic conditions in the atmospheric oxygen level under anoxic conditions.

The evolution of cyanobacteria would have added another source of oxygen to the global oxygen mass balance, resulting in a shift to a higher atmospheric oxygen level. Increased oxygen production is likely to have transitioned the Earth system to the second steady state point of atmospheric oxygen (State II; Fig. 1). Such a state, however, unlike the first state, is unstable, and the system would only occupy such state transiently (e.g., a tipping point) and a small change in the rate of oxygen production or consumption would result in a transition into the adjacent stable states (state I and state III).

According to our results, oscillations in MIF-S production, suggested by the recent results from geochemical proxies14,33, can be explained by a fluctuation in the atmospheric oxygen level in both states I and II. Specifically, the maximum value in state I is ~2 × 10−7 PAL which is higher than the minimum value suggested for the threshold of atmospheric oxygen for caseation in the production of MIF-S (e.g., 23) which implies that it is possible that a fluctuation in the atmospheric oxygen level under state I result in an oscillation in the production of MIF-S signal. Suggested higher values for the threshold of atmospheric oxygen level for MIF-S formation (~10−6 PAL) are also consistent with the range of atmospheric oxygen level at state II which suggests that the apparent fluctuation in MIF-S signal can alternatively be attributed to an oscillation in the atmospheric oxygen level at unstable state II. Regardless, according to this framework, the GOE can then be defined as a departure from unstable State II and the transition to the second stable state for atmospheric oxygen level (state III: Fig. 1).

Possible large fluctuation under low-oxygen conditions

The Proterozoic, post-GOE world would have been trapped under low oxygen conditions with the possibility of large swings in atmospheric oxygen level. The absence of evidence for MIF-S for Proterozoic rock record (early Paleoproterozoic excepted) and the absence of evidence for modern-like oxygen conditions, suggests that atmospheric oxygen during the Proterozoic would have been between 10−5 PAL to modern atmospheric oxygen level (Fig. 3). Within this wide possible range for atmospheric oxygen levels, our results show only two steady states with only one stable state (State III: ~10−3 – 10−2 PAL) (Fig. 3B). Our stochastic simulation suggests that at stable State III atmospheric oxygen level can vary approximately more than two orders of magnitude (~ minimum and maximum of ~8 × 10−5 and 3 × 10−2 PAL), which suggests that atmospheric oxygen levels can take any values within this range without falling out of the stability zone (Fig. 1). This result opens the possibility for orders of magnitude swings in the atmospheric oxygen level under low-oxygen conditions. According to our results, the evidence for higher oxygen values beyond the ~10−2 PAL, as the upper threshold for oxygen level at State III, can only be explained by reaching State IV. This state, however, is unstable, meaning that relatively high oxygen conditions at State IV would have been ephemeral and a small change in the rate of oxygen production or consumption in the ocean and or on land would force the system to move dynamically to adjacent stable states (state III or state V; Figs. 1, 2). Taken together, our finding suggests that there could have been large fluctuations in atmospheric oxygen levels during the Proterozoic, likely within the range of about 10−4 to 10−2 PAL (Fig. 3B). This conclusion is, arguably, in accordance with some geological evidence (e.g., sulfur isotope systematics14 and Cr isotope values34, among other redox proxies1).

Fig. 3: The geological history of atmospheric oxygen level.
figure 3

a corresponds to the records of mass-independent fractionation of sulfur (MIF-S)5,33, as a proxy for an anoxic atmosphere. And b shows the change in atmospheric oxygen through time. The blue boxes with a line denote the trajectory for atmospheric oxygen level based on the results in this study and the gray boxes correspond to the traditional trajectory of atmospheric oxygen level1,52. The range of values in blue boxes is the same as the range presented in Fig. 1b where the lower and upper edge of each blue box correspond to the 25th and 75th percentile of the data that resulted from the stochastic simulation. Our results suggest a possibility of a large fluctuation in the atmospheric oxygen level for most of the Earth’s history. The rise of atmospheric oxygen in the late Proterozoic and early Phanerozoic, coincident with the rise and radiation of the complex life marks the end of the fluctuations in the atmospheric oxygen level and transition into stable State V with modern-like oxygen level. The high atmospheric oxygen during Lomagundi, after GOE, can be explained by either atmospheric oxygen in state V (~10° PAL) or reaching the maximum value for the atmospheric oxygen level in stable state III (~10−2 PAL).

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According to our results, high-oxygen conditions during the Lomagundi-Jatuli event (LJE) can be attributed to two stable states of III and V. Specifically, considering that there is no strong evidence to date for large oscillation of atmospheric oxygen during the LJE that can be linked to an unstable state of atmospheric oxygen (e.g., state IV), the suggested high oxygen level during LJE can be explained by either an atmospheric oxygen level in state III or V. The maximum value of atmospheric oxygen in state III is around 10−2 PAL which suggests that an increase in the rate of oxygen production during LJE can be explained by a rise in atmospheric oxygen level to the maximum value of state III. Alternatively, a high atmospheric oxygen level during this time can be explained by reaching a modern-like atmospheric oxygen level (state V). Regardless, our results suggest large shifts in atmospheric oxygen levels across the LJE transition.

Late Proterozoic and transition to the modern-like pO2 level

The late Proterozoic and early Phanerozoic times marked the end of the large fluctuation in the atmospheric oxygen level and the transition into the last stable state of atmospheric oxygen. Specifically, our results suggest that an increase in pO2 levels would have resulted in moving from the stable State III to the stable State V with the modern-like pO2 value. This increase in oxygen levels could have been caused by either a decrease in oxygen sinks in the ocean or land or an increase in the oceanic oxygen production caused by enhanced marine primary production35,36, which is consistent with the results from geochemical proxies that are suggestive of a transition into roughly modern-like atmospheric oxygen level around this time37.

Strong stabilizing feedback of plant evolution on pO2

The evolution of plants in the Phanerozoic likely helped stabilize atmospheric oxygen around modern value. To explore the impact of plant evolution on the atmospheric oxygen level, we conducted two stochastic simulations with and without plants. We explored the effect of the rise of plants by removing the flux of carbon burial from land plants from the total oxygen production flux (JNPP; Eq. 2 in the “Methods” section). According to our results, the median atmospheric oxygen level without plants would be around 0.4 atmospheric oxygen level (PAL). This result shows that the rise of plants resulted in an increase in the oxygen production flux and helped with stabilizing the atmospheric oxygen level near modern value (Fig. 4). Similar to the results from previous studies (e.g., 13), our results also suggest that, by exerting negative feedback such as fire, plants may also help prevent large swings in atmospheric oxygen levels.

Fig. 4: The impact of plant evolution on the steady-state atmospheric oxygen level.
figure 4

According to our results, the rise of plants on land would have helped with stabilizing the atmospheric oxygen level near modern value.

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Taken together, our stochastic results from dynamical analysis of global oxygen mass balance suggest three stable states for atmospheric oxygen level: one under anoxic conditions ~10−8 PAL, one around 3 × 10−3 PAL, and one near modern value (~10° PAL). According to our results, the atmospheric oxygen level could have fluctuated greatly for most of the Earth’s history until the transition into the last stable state in the late Proterozoic and early Phanerozoic, where the oxygen level finally reached its modern value. The transition into state V with the modern-like oxygen level is coincident with the major rise and radiation of complex life, reinforcing the need to explore the role of instability in the global oxygen cycle played in triggered biological innovations in early animals38.

Methods

Global oxygen mass balance

To explore the range of stable states for the atmospheric oxygen level, we used a simple expression for the global oxygen mass balance:

$$\frac{d({{pO}}_{2})}{{dt}}={J}_{{NPP}}+{J}_{{Hydrogen}}-{J}_{{CH}4,O2}-{J}_{{volc}}-{J}_{{ocean},{sink}}-{J}_{{org},{weath}}-{J}_{{pyr},{weath}}$$
(2)

Here d(pO2)/dt is the change in atmospheric oxygen level in the unit of Tmol per year, JNPP, JHydrogen,are the sources of oxygen respectively from burial of organic carbon on land and in the ocean, and the rate of escape of hydrogen atoms to space. The JCH4,O2, Jvolc, Jocean,sink, Jorg,weath, Jpyr,weath, are sinks of oxygen which respectively represent methane oxidation in the atmosphere, volcanic outgassing, the oceanic sink of oxygen (oxidation with sulfide, iron, and methane), rate of oxidative weathering of organic matter and pyrite. Below we detail the parameterization scheme used for each parameter in Eq. (1). It should also be noted that to account for the uncertainty associated with the choice of values of the model input parameters, we took a stochastic approach wherein the values for each modeling input parameter were selected randomly assuming a uniform distribution and the most probable range of results were obtained (Fig. 1b).

JNPP

The JNPP as a total carbon burial on land and in the ocean can be expressed as:

$${J}_{{NPP}}={J}_{{NPP},{ocean}}+{J}_{{NPP},{land}}$$
(3)

The flux of carbon burial in the ocean, JNPP,ocean, can be parameterized as:

$${J}_{{NPP},{ocean}}=\varepsilon * {NPP} * {area} * {R}_{c/p}$$
(4)

Where ε is the efficiency of the carbon pump, NPP is the net primary production, and the last term is the area of the ocean. The value of NPP is multiplied by a productivity factor to account for a lower productivity at lower oxygen levels39,40:

$${NPP}={{NPP}}_{{modern}} * {f}_{{prod}}$$
(5)

We used high burial efficiency under anoxic conditions, consistent with the modeling and experimental results as well as the modern observations that indicate an increase in the burial efficiency of carbon under anoxic conditions41,42. The fraction of oxic condition is calculated as one minus the fraction of anoxia (1- fanoxic). We considered full anoxia ( fanoxic = 1) under very low oxygen conditions (pO2 = 10−13 PAL), consistent with results from geochemical proxies, and modern-like anoxia for modern-like pO2 (fanoxic ~ 0.0001 to 0.001). The fraction of anoxia for pO2 less than 10−5 PAL was less than 0.05, consistent with the results from geochemical proxies and the biogeochemical modeling that suggest weak ocean oxygenation under the atmospheric oxygen threshold for production of mass-independent fractionation of sulfur (e.g., 43,44). The fraction of anoxia in the ocean for other initial conditions (pO2) was varied between these two end members. To account for the uncertainty involved in choosing the modeling parameters, the value for each parameter was selected in a stochastic manner with the assumption of uniform distribution for the range of each parameter.

We also assume lower rates of oxygen production at lower atmospheric oxygen levels, following previous studies (e.g., 27). Considering a simple mass balance framework for atmospheric oxygen which considers oxygen production through organic matter burial and oxygen consumption through the reaction of oxygen with reduced species as well as the oxidative weathering of pyrite and organic (JO2,production = JO2,sink). Given that the rate of oxygen consumption is a function of atmospheric oxygen level and at lower atmospheric oxygen level, the rate of oxygen consumption would be lower too (e.g., oxidative weathering of organic matter which is further discussed below), to maintain a low-atmospheric condition, the rate of oxygen production would also need to be lower at lower atmospheric oxygen level. This relationship is like shifts in nutrient availability under low-oxygen conditions (e.g., 27).

The term Rc/p is the ratio of organic carbon to organic phosphorous normalized to modern (oxic) conditions. The term Rc/p accounts for the effect of change in redox condition and expansion of anoxia on phosphorous recycling and organic matter burial. Specifically, under anoxic conditions, the release of phosphorous would be enhanced, which would result in an increase in marine primary production and organic matter burial11,45. The link between anoxia and enhanced phosphorous availability however depends on the availability of sulfide which controls the availability of dissolved iron (e.g., 46). Given the interaction between phosphorous and dissolved iron, an increase in sulfide availability would result in a decrease in the iron sink of phosphorous(e.g., 46). Thus, an expansion in the anoxia can foster a higher rate of organic matter burial only if the rate of sulfide production through sulfate reduction is high which requires high sulfate levels. We followed the previous studies11,45 to parameterize the term Rc/p and included a factor that accounts for sulfate availability as well:

$${R}_{c/p}=\left(\frac{\frac{({\frac{C}{P}})_{{anoxic}} * {\left(\frac{C}{P}\right)}_{{oxic}}}{[{O}_{2}]/{[{O}_{2}]}_{{modern}} * {\left(\frac{C}{P}\right)}_{{anoxic}}+(1-{[O_{2}]}/{[{O}_{2}]}_{{modern}}) * {\left(\frac{C}{P}\right)}_{{oxic}}}}{{\left(\frac{C}{P}\right)}_{{oxic}}}\right) * {R}_{{SO}4}$$
(6)

Here (C/P)anoxic, and (C/P)oxic, are the C to P ratios under anoxic, and oxic conditions suggested by the previous studies ((C/P)anoxic = 1100; (C/P)anoxic = 250)11,45. The [O2] and [O2]modern are, respectively, the seawater concentration of oxygen at different pO2 levels and modern conditions. We used the previous earth system modeling that relate the seawater oxygen concentration to various pO2 levels47.

The term RSO4 accounts for the effect of sulfate level on phosphorous availability, which is parameterized as a Monod-type formulation, similar to the parameterization scheme suggested for the rate of sulfate reduction:

$${R}_{{SO}4}=\frac{[{{SO}}_{4}]}{[{{SO}}_{4}]+{k}_{m,{SO}4}}$$
(7)

Where [SO4] and km,SO4 denote the seawater sulfate concentration and the half-saturation constant for sulfate reduction. We assumed sub-mM levels for seawater sulfate under low oxygen conditions (pO2 < 10−2 PAL) and mM range and super mM range of seawater sulfate for higher pO2 levels, consistent with the range of seawater sulfate suggested by the previous studies under low pO2 and high pO2 levels.

We utilized a parameterization from previous studies to calculate the flux of carbon burial on land13,48:

$${J}_{{NPP},{land}}=2 * E * {V}_{{NPP}}\left({O}_{2}\right) * {V}_{{NPP}}\left(T\right) * {V}_{{NPP}}\left(C{O}_{2}\right) * \frac{{k}_{{fire}}}{{k}_{{fire}}-1+{ignit}}$$
(8)

In this formulation, the strength of carbon burial on land is a function of pO2 (VNPP(O2)), temperature (VNPP(T)), and carbon dioxide partial pressure (VNPP(CO2)). The parameter “E” corresponds to evolutionary forcing, whose value was from the previous works and factor 2 is for normalization13,48. The last term in the equation corresponds to the fire feedback on organic burial on land. Following the previous works, the terms in Eq. (5) were parameterized as follows:

$${V}_{{NPP}}\left({O}_{2}\right)=1.5-0.5 * {{pO}}_{2}$$
(9)
$${V}_{{NPP}}\left(T\right)=1-{\left(\frac{T-25}{25}\right)}^{2}$$
(10)
$${V}_{{NPP}}\left({{CO}}_{2}\right)=\frac{{{pCO}}_{2}-{{pCO}}_{2,\min }}{{{pCO}}_{2,1}+(\, {{pCO}}_{2}-{{pCO}}_{2,\min })}$$
(11)

Where T is the temperature in the unit of degrees Celsius, pCO2 is CO2 atmospheric level in ppm, pCO2,min denotes the minimum atmospheric CO2 concentration in which plants can grow, and pCO2,1 is a constant13,29,48,49.

In the fire feedback term, kfire is a constant and term “ignit” is a linear function of pO2:

$${ignit}=\max (586.2 * {{pO}}_{2}-122.102,0)$$
(12)

J Hydrogen and J CH4,O2

Following previous works, we considered the escape flux of hydrogen atoms to space to be a function of methane partial pressure (pCH4)4,30,32:

$${J}_{H{{ydrogen}}}=\alpha .\mu .p{{CH}}_{4}$$
(13)

Where α is a constant and µ denotes the number of moles in 1 atmosphere. It should be noted that the number of moles in 1 atm depends on the relative abundance of gases. For instance, a CO2-rich atmosphere contains less moles compared to an H2-rich atmosphere. To account for this caveat the value of α was varied in our stochastic simulation.

The atmospheric concentration of methane, pCH4, would be calculated by solving the following methane mass balance equation:

$$\frac{d({\, {pCH}}_{4})}{{dt}}={J}_{{CH}4}-{J}_{{CH}4,O2}-{J}_{{Hydrogen}}$$
(14)

Here JCH4, is the net flux of biogenic methane from the oceans, JCH4,O2 is the rate of atmospheric methane oxidation, and JHydrogen, as explained above, is the hydrogen escape flux.

The net flux of biogenic methane from the ocean is considered to be a function of the net primary production which regulates the amount of organic matter available for methanogenesis, the extent of anoxia in the ocean, and a coefficient, which reflect the amount of methane being oxidized in the ocean which itself is the function of oxygen availability. At higher atmospheric oxygen, the value of this coefficient increases, reflecting an increase in the oceanic oxidant reservoir (e.g., sulfate and oxygen concentrations) which would impact the rate of methane oxidation in the ocean.

$${J}_{{CH}4}={\alpha }_{1}(1-\varepsilon ) * {NPP} * ({\, f}_{{anoxic}} * {area})$$
(15)

Parameters α1, ε, NPP, and fanoxic, respectively, denote the coefficient for methane production, burial efficiency of organic matter, net primary production, and the areal fraction of ocean anoxia.

We used the previous parameterization to simulate the rate of atmospheric methane oxidation30. Specifically, the rate of methane oxidation is assumed to be a function of the partial pressure of both atmospheric oxygen and methane multiplied by an oxidation coefficient.

$${J}_{{CH}4,O2}=\Psi .\left(\mu p{O}_{2}\right).{\left(\mu p{{CH}}_{4}\right)}^{0.7}$$
(16)

Here µ denotes the number of moles in 1 atmosphere. The oxidation parameter, Ψ, is parameterized as log Ψ = p1 x 4 + p 2 x 3 + p3 x 2 + p4 x 1 + p5, where x = log(µ pO2) and coefficients p1 = 0.00286, p2 = −0.1608, p3 = 3.216, p4 = −26.53, p5 = 76.04 are resulted from fitting a curve to the previously published data (Figure S1). The details of the goodness of the fit, along with the range of values and their confidence intervals are shown in Figure S1.

While our parameterization scheme for the rate of atmospheric methane oxidation is similar to previous studies, some assumptions in this parameterization can potentially cause uncertainty in the rate of methane oxidation. For instance, this parameterization makes assumptions about the age of the Sun as well as about the surface temperature, both of which effects can potentially influence the photochemistry of CH4 and O2. The assumption about the age of the sun can impact the Lyman alpha UV flux, an important factor for kinetics of CH4 photolysis CH4 and the Earth’s surface temperature can impact the water vapor content of the troposphere with impacts on the abundance of OH and the rate of methane oxidation. To account for the uncertainty associated with these assumptions, we varied the kinetics of atmospheric methane oxidation in our stochastic simulation.

Finally, considering that the methane mass balance equation depends on pO2, we first solved the methane mass balance equation and obtained the fixed points (steady states) of the methane mass balance equation, and then used those fixed points to solve the oxygen mass balance equation.

J ocean,sink

The oceanic sink of oxygen was a function of the strength of marine net primary production and fraction of anoxia, as well as the seawater oxygen concentration (e.g., 27,45,50).

$${J}_{{ocean},{sink}}=k1 * (1-\varepsilon ) * {NPP} * ({\, f}_{{anoxic}} * {area}) * \left(1-\frac{[{O}_{2}]}{[{O}_{2}]+{k}_{m,O2}}\right)$$
(17)

Where [O2] and km,O2 denote the seawater oxygen concentration and the half-saturation constant for aerobic respiration. As explained above, we used the results from previous earth system modeling to relate the pO2 to seawater oxygen concentrations. The constant “k1” is a factor that accounts for variability in the pool of reducing compounds (e.g., sulfide, iron, methane) in the ocean. The constant “k1” reflects the relative comparison between the rate of oxygen consumption by reducing species. For instance, a value of 0.1 means that the rate of oxygen consumption by reducing species is about an order of magnitude lower than the maximum rate of anaerobic respiration. Under anoxic and low-oxygen conditions of Archean and Proterozoic, this value would have likely been low. For instance, most of the sulfide resulting from sulfate reduction would have precipitated as pyrite, consistent with the results from sulfur isotope records that suggest a near-one value for a fraction of pyrite (fpyr). Our results for the Jocean,sink for the modern-like condition are consistent with the rate of oxygen uptake in the modern oxygen minimum zone. Specifically, using our formulation for the rate of oxygen consumption (Eq. 16 in the “methods” section), and considering the modern organic matter burial efficiency (~0.005), the modern net primary production (~100 gram/m2/year), and the modern areal fraction of anoxia (~0.005), the order of magnitude rate estimate for the rate of oxygen consumption would be around 0.02 tera mol per year. And using the measured rate of sulfate reduction in the modern oxygen minimum zones which is on the order of 0.1 mmol/m2/day, and the areal fraction of anoxia in the ocean and considering that half of the reduced sulfur would become oxidized with oxygen, the rate of oxygen consumption through reaction with sulfide would become around 0.03 tera per mol per year which is comparable with the value resulted from our estimate. Finally, to account for the uncertainty, this value along with other values used in this equation and elsewhere was varied within a wide range in our stochastic simulation (Table S1).

J org,weath and J org,pyr

Consistent with previous modeling results, the oxidative weathering of pyrite and organic matter was assumed to change with pO228,29,48,51. The previous modeling results show that at a certain pO2 level, the oxidative weathering of organic matter and pyrite would be complete and equal to their modern values. This result suggests that above a certain pO2 level, the weathering flux would no longer be controlled by pO2. With this in mind, we assumed that the weathering flux of organic matter and pyrite changes with pO2 and followed a Michaelis-Menten equation:

$${F}_{{org},{weath}}={F}_{{weath}1}.\frac{{{pO}}_{2}}{{k}_{{weath}1}+{{pO}}_{2}}\frac{{NPP}}{{{NPP}}_{{modern}}}$$
(18)
$${F}_{{pyr},{weath}}={F}_{{weath}1}.\frac{{{pO}}_{2}}{{k}_{{weath}1}+{{pO}}_{2}}$$
(19)

Here Fweath1 denotes the modern values for pyrite and organic matter weathering, and kweath1 is the atmospheric oxygen level above which the weathering of pyrite and organic matter is complete (Table S1). To account for the effect of the amount of organic matter produced in the ocean, the oxidative rate of organic matter would be multiplied by the ratio of NPP to its modern value (NPPmodern). Finally, to account for the variability in choosing the parameters, we took a stochastic approach where modeling parameters were randomly sampled within a specified range supported by literature (Table S1), and the most probable range of stable states of pO2 was calculated.