Metabolic conversions mediated by microorganisms play a key role in the Earth's biogeochemical cycles . For example, microbial nitrogen fixation contributes an estimated 100–200 Tg of nitrogen to the world's oceans annually , while the microbial decomposition of soil carbon exceeds the anthropogenic contribution of carbon dioxide to the atmosphere by an order of magnitude . Predicting the response of these cycles to environmental changes, including climate change, is a central current challenge in Earth system science . However, mathematical models for global geochemical cycles often represent microbially mediated transformations as crude “black boxes” : for example, microbial decomposition in soil is often represented as a first-order decay process . Indeed, many of the models cited in the most recent IPCC report use linear representations of microbially mediated processes . This simplified picture contrasts strongly with the wealth of data on microbial community diversity and functional complexity which is being generated by recent advances in high-throughput sequencing technology . There is thus an urgent need to re-evaluate the role of microbial population dynamics in biogeochemical models .

Here, we use simple mathematical models to show that microbial population dynamics can have important qualitative effects on the response of microbially mediated biogeochemical cycles to environmental change. Specifically, nutrient limitations on microbial population growth can lead to abrupt changes in redox state in response to a gradual change in an environmental parameter. Sharp transitions, often described as regime shifts, are known to occur in diverse systems in response to diverse stimuli; examples range from aquatic ecosystems in the leaves of carnivorous pitcher plants to large-scale shifts in terrestrial vegetation cover . These shifts are usually attributed to specific features of the system structure (or “topology”; ). Our work suggests that, for biogeochemical cycles, nonlinear effects arising from microbial population dynamics can lead to sharp transitions between broadly oxidized and reduced system states, even for systems with simple topologies. We term this a “redox regime shift”, i.e., a nonlinear transition in the predominant redox state of a biogeochemical cycle in response to a gradual change in an environmental stimulus. In some other studies, the term “regime shift” has been associated with bistability. Here, we use the term simply to describe a sharp response, without any implied bistability.

In a biogeochemical cycle, a chemical element is shuttled between its oxidized and reduced forms in a series of steps that may be biotically or abiotically mediated . Figure  illustrates simplified topologies of the iron, sulfur, carbon, and nitrogen cycles (Fig. 1a–d) . To encapsulate the basic topology of these cycles, we begin by considering a simplified two-state model (Fig. e), in which an oxidized form of a chemical element (here denoted so) is converted via microbial metabolism to a reduced form (sr), which is recycled back to the oxidized form either by a second microbial metabolism or by an abiotic reaction. Although this model is topologically very simple, it reveals an important and nontrivial regime shifting behavior. Later in this paper we show that this behavior is preserved in more realistic models that include features such as spatial heterogeneity, multiple redox states, and explicit coupling to the environment.

A redox reaction in a biogeochemical cycle couples the oxidation/reduction of the element being cycled to the reduction/oxidation of another chemical species. For example, in the sulfur cycle, the microbial reduction of sulfate can be coupled to the oxidation of acetate , while in the nitrogen cycle, the oxidation of ammonia can be coupled to the reduction of molecular oxygen . In this paper, in order to avoid confusion, we refer to the latter chemical species (in these examples acetate or oxygen) as the “auxiliary electron donor/acceptor”. The auxiliary electron donor/acceptor may be supplied from some external source (e.g., oxygen from the atmosphere) or may be generated by another biogeochemical process (e.g., microbial decomposition producing acetate). Many different chemical species can act as auxiliary electron donors or acceptors; for example, acetate or hydrogen can function as the electron donor for reductive reactions, while nitrate or oxygen can function as the electron acceptor for oxidative reactions. The redox-shifting behavior which arises in our models is generic, independent of which chemical species performs the role of auxiliary electron donor/acceptor.

Crucially, if the auxiliary electron acceptor/donor is in short supply then its availability can control the rate of the redox reaction, and hence the flux of the biogeochemical cycle. Moreover, in natural environments, the availability of electron acceptors and donors is strongly dependent on the environmental conditions. For example, in aquatic ecosystems, the supply of oxygen depends on its solubility, which is temperature-dependent , and on the rate of photosynthesis , while the supply of acetate depends on the rate of microbial decomposition of organic matter, which can be drastically affected by factors like sewage effluent or phosphorus inflow from agricultural runoff .

Here, we show that changes in the supply of auxiliary electron acceptors or donors (such as oxygen or acetate) caused by environmental perturbations can have drastic effects on microbially mediated biogeochemical cycles. We first show that these perturbations can cause regime shifts in redox state for simple, spatially homogenous models. We then demonstrate that the same phenomena can also occur in more realistic models which include features such as explicit supply of auxiliary electron acceptors or donors via microbial metabolism, intermediate redox states, and spatial heterogeneity (such that the nutrient supply is limited by transport processes). These regime shifts do not depend sensitively on the detailed structure of our equations or model, but instead result from the interplay between cyclic system topology and nonlinear microbial population growth requiring multiple nutrients. These redox regime shifts are predicted to occur in parameter ranges relevant to the natural sulfur and nitrogen cycles, and may also be relevant to iron cycling in the iron-containing ancient oceans.

Our aim is to predict the response of microbially mediated biogeochemical cycles to changes in the availability of auxiliary electron acceptors and donors, such as oxygen and acetate. We begin with a simple and generic “two-state” representation of a biogeochemical cycle; later we show that the same phenomena also occur in more complex models. In our two-state model (Fig. e), a chemical element is cycled between its oxidized and reduced forms, whose concentrations are denoted by so and sr, respectively. The reduction step so→sr (blue right-to-left arrow in Fig. e) is assumed to be biotic, i.e., mediated by microbial metabolism. This step requires an auxiliary electron donor, such as acetate. The oxidation step sr→so may occur biotically or abiotically (indicated by the blue and red left-to-right arrows in Fig. e), and requires an auxiliary electron acceptor, such as oxygen. We have not included the possibility of an abiotic reduction reaction in our model because these are typically minor reactions at ambient temperatures in the natural environment (with the notable exception of the reaction of Fe(III) with sulfide; e.g., ); further work could extend this model to include such reactions. It is important to note that, in reality, a given biogeochemical function may be performed by many coexisting microbial species (taxa); for example many different genetically distinct taxa can use acetate to reduce sulfate . In our models, we group together all these “metabolically equivalent” taxa into a single effective population.

Our models allow us to investigate system-level responses to environmental change. We focus on environmental changes that affect the availability of auxiliary electron acceptors or donors, such as temperature-related changes in oxygen solubility , changes in photosynthesis rate, or changes in the abundance or rate of decomposition of organic matter . For the fully biotic cycle, the parameters vor and vro are proxies for the availability of auxiliary electron donors and acceptors, respectively. For the biotic–abiotic cycle, the equivalent parameters are vor and va. We quantify the response of the ecosystem to changes in auxiliary electron donor or acceptor abundance via the steady-state fraction of the oxidized chemical species, so/stot, which acts as a proxy for the global redox state of the system.

Our main result is that, for both the fully biotic and the biotic–abiotic models, our model can undergo regime shifts: sharp changes in the predominant redox state of the system as the availability of auxiliary electron acceptors or electron donors (such as oxygen or acetate) crosses a critical threshold (Fig. ). These regime shifts happen under circumstances where the total concentration of the chemical element being cycled (stot=so+sr) is high, such that stot≫Kor,Kro,Ka, implying that the microbial population density is limited by factors other than the availability of so or sr. In contrast, for lower concentrations of the chemical element being cycled, stot<Kor,Kro,Ka, the model predicts a more gradual change in system state as the availability of the auxiliary electron acceptor or donor varies.

Figure a and c show results for the fully biotic cycle model; in Fig. a, we vary the maximal reducer growth rate vor, mimicking a change in auxiliary electron acceptor abundance, while in Fig. c we vary the maximal oxidizer growth rate vro, mimicking a change in auxiliary electron donor abundance. As expected, these perturbations lead to profound changes in the system's global redox state (as measured by the fraction of the oxidized chemical species so), from oxidized to reduced (Fig. a) or vice versa (Fig. c). Crucially, the sharpness of this transition increases as we increase the total abundance of the chemical species being cycled, stot. When stot is large enough to saturate the growth rates of the relevant microbial populations (stot≫Kor,Kro), we obtain a “switch-like” response, which we term a redox regime shift. For the fully biotic cycle, the model prediction is symmetric with respect to changes in vor and vro (electron donor and acceptor; compare Fig. a and c). Consequently, it is the ratio of vor / vro (mimicking a change in the ratio of auxiliary electron donor/acceptor abundance) that drives the behavior of the model.

Figure b and d show equivalent results for the biotic–abiotic cycle model. In this case also, the model predicts regime shifts in response to both increasing auxiliary electron donor or acceptor availability (Fig. b and d, respectively), for large concentrations of the chemical species being cycled (stot≫Kor,Kro,Ka). However, in contrast to the situation for the fully biotic cycle, here the responses to changes in auxiliary electron acceptor and donor are qualitatively different in shape (compare Fig. b and d). This is because the biotic and abiotic reaction rates (the two terms in Eq. 6) have different functional dependences on s.

Importantly, the behavior of our model does not depend strongly on its other parameters. In particular, the total microbial population density is not important for the results of Fig. , as we show analytically in Sect. S3. For the fully biotic cycle, the steady-state solution of the model depends only on the ratio of the maximal population density parameters (nro, max/nor, max) and not on the absolute values of the maximal population density nro, max and nor, max. Moreover the ratio nro, max/nor, max affects only the threshold point at which the regime shift happens, not the qualitative switching behavior (see Sect. S3 for more details). Thus we expect to see redox regime shifts across environments with very different microbial population densities, even for systems with very large microbial populations, and for those where the sizes of the oxidizing and reducing populations are different, as long as the microbial population density is ultimately limited by a factor other than the concentration of the chemical element being cycled. It is important to note, however, that the timescale over which the system responds to environmental change does depend on the population density; for large populations, the system responds more slowly. For the biotic–abiotic cycle, the mathematical results are slightly more complicated but the conclusions are broadly similar (see Sect. S3).

Interestingly, the system-scale regime shifts that we observe in our biogeochemical models can be mapped directly onto a well-known molecular-scale phenomenon in intracellular biochemical signaling networks. In biological cells, responses to environmental signals are often mediated by phosphorylation–dephosphorylation cycles, in which a target enzyme is activated by addition of a phosphate group, and deactivated by removal of the phosphate group; the kinase and phosphatase enzymes mediating these reactions act in opposition to each other . Phosphorylation–dephosphorylation cycles can exhibit “zero-order ultrasensitivity”, in which they respond extremely sensitively to changes in the level of signal, because the enzymes have become saturated, decoupling the enzymatic conversion rates from the concentration of substrate . Although they act on very different length and timescales, biogeochemical cycles are topologically similar to phosphorylation–dephosphorylation cycles. In fact, one can show mathematically that our models, in the steady state, map exactly onto the classic Goldbeter–Koshland model for phosphorylation–dephosphorylation cycles , and that the regime shifts observed in our models are equivalent to the ultrasensitive signal responses predicted by this model (see Sect. S7). This raises the interesting possibility of mapping molecular-level dynamic phenomena onto biogeochemical models more generally – a direction that may prove fruitful in future work.

Thus far our investigation has focused on a rather simplified model for microbially mediated biogeochemical cycles. In this simple model, varying vro and vor was assumed to be analogous to varying the availability of auxiliary electron acceptors (such as oxygen or nitrate) and electron donors (such as acetate or hydrogen), respectively. In reality, however, auxiliary electron acceptors or donors may be supplied, or utilized by, other biotic or abiotic processes, and thus we expect their concentrations to vary with the system dynamics. We now introduce a more ecologically realistic model in which the concentrations of the auxiliary electron acceptor/donor are explicitly represented, and allowed to vary. For this model, we find the same redox regime-shifting behavior as in the simple model described previously.

Specifically, we focus on an example in which acetate is the auxiliary electron donor and oxygen is the auxiliary electron acceptor. We suppose that acetate is produced by microbial decomposition of organic matter (long-chain organics such as lignin or cellulose; ); we represent explicitly in the model not only the concentration of acetate but also the population density of the decomposer population. Likewise, we suppose that oxygen is generated by photosynthetic microorganisms; the model includes explicitly the dynamics of the photosynthesizer population as well as the oxygen concentration. External environmental inputs control the population dynamics of the decomposers and the photosynthesizers; these inputs are the organic matter concentration and the light intensity, respectively. Our model is shown schematically in Fig. 3a; we assume that oxidative and reductive processes occur in different spatial zones, represented by boxes and coupled by chemical transport of so and sr. For simplicity, we consider transport only of the chemical species being cycled (so and sr); allowing transport of oxygen/acetate would cause spatial shifting of the redox zones, which, although interesting, would be better investigated in a model with more detailed spatial resolution. In the model, the growth rate of the oxidizing microbial population (nro) is assumed to depend on the concentrations of both sr and the auxiliary electron acceptor (i.e., oxygen), via a multiplicative Monod term, with explicit population density limitation, and the equivalent scenario holds for the reducer population. Multiplicative Monod kinetics is the most widely used method of modeling microbial growth limitation by multiple substrates . However we note that Liebig's law of the minimum provides an alternative , which would not affect our qualitative results. Our model also includes a linear loss term for the auxiliary electron acceptors or donors, which represents competitive consumption by other populations. Full details and dynamical equations for this model are presented in Sect. S8.

Our simulations show that this model indeed undergoes redox regime shifts (Fig. 3b). In particular, the system redox state, as measured by the ratio so/stot (shown by color in Fig. 3b), changes sharply in response to changes in either organic matter availability (which stimulates the decomposer population and hence the reducer population), or to changes in light intensity (which stimulates the photosynthesizers and hence the oxidizer population). As organic matter availability increases at fixed light intensity (vertical dashed line in Fig. 3b), the redox state of the system changes sharply from oxidized to reduced (red to purple). Likewise, as the light intensity increases for fixed organic matter concentration (horizontal dashed line in Fig. 3b), the redox state also undergoes a regime shift, in this case from reduced (purple) to oxidized (red). We observe similar regime-shifting behavior in equivalent models where the oxidation step is abiotic (see Sect. S8). We have also shown that the qualitative behavior of the model is not dependent on the strength of the loss term representing competition for auxiliary electron acceptors/donors (see Sect. S10).

Since many natural redox cycles involve intermediate steps between the most oxidized and most reduced states (e.g., the nitrogen and sulfur cycles in Fig. 1), we have also simulated a version of the model which includes a redox state intermediate between so and sr. This model also shows regime shifts between predominantly oxidized and predominantly reduced system states (see Sect. S9).

Our analysis provides a clear set of criteria that need to be satisfied for redox regime shifts to occur. These are as follows:

The density of the redox-cycling microbial populations must ultimately be limited by a factor other than the concentration of the chemical element being cycled. This factor could be the concentration of another nutrient (see Sect. S4), or space limitation. It is important to note, however, that the population density need not be small; large populations are also predicted to show regime shifts, albeit with longer response times.

Thus far we have established our model, demonstrated that the predicted redox regime shift is robust as complications are introduced, and defined the conditions required for redox regime shifts to occur. We now assess whether these conditions are likely to be prevalent in the natural environment.

Microbial populations are key mediators of the Earth's biogeochemical cycles . Our work shows that microbial population dynamics can have important consequences for the response of biogeochemical cycles to environmental changes. Under circumstances where the microbial population density is limited by factors other than the concentration of the chemical being cycled (e.g., by the concentration of another limiting nutrient), our models predict that redox-cycling systems can undergo regime shifts in their predominant redox state in response to small changes in the availability of auxiliary electron acceptors or donors (such as oxygen and acetate), which drive the oxidative and reductive redox-cycling reactions, respectively. These regime shifts arise from the interplay between the nonlinearity of microbial population dynamics, multiple nutrient limitation, and the cyclic system topology. Diverse environmental perturbations are expected to affect the availability of auxiliary electron acceptors and donors, including temperature-mediated changes in oxygen solubility and changes in organic matter abundance due to eutrophication, suggesting that these redox regime shifts may be common in the natural environment.

Regime shifts are a well-known phenomenon in many ecosystems , including microbial ecosystems . They are known to occur in biogeochemical cycles and have played an important role in the Earth's history – a notable example being the transition to an oxic atmosphere around 2.3 Ga . Our work suggests a new mechanism by which regime shifts may occur in microbially mediated biogeochemical cycles. This mechanism is identified here in a very simple and generic model but also shown to exist in more realistic models. Further work could extend our models to include detailed spatial or temporal dynamics and/or additional environmental variables such as temperature or pH.

Our analysis also predicts clear criteria for the conditions under which redox regime shifts should be expected. By analyzing parameter values for a range of natural environments, we show that these criteria are likely to be satisfied for the natural sulfur and nitrogen cycles. This phenomenon may also be relevant for iron cycling in the Archean or Proterozoic oceans, due to their much lower oxygen concentrations and potentially much higher concentrations of iron than present-day oceans. Indeed, redox regime shifts may even help to explain changes in the Earth's biogeochemical cycles associated with mass extinction events, such as the rise in ocean sulfide levels during the end-Permian extinction event (251 Ma), which is believed to have poisoned the oceans and killed as much as 90 % of all macroscopic species on Earth . More generally, our work reveals that microbial population dynamics can lead to qualitative changes in the behavior of biogeochemical cycles, with significant system-level consequences. Better understanding of microbial population dynamics is vital for accurate prediction of the effects of anthropogenic changes on the Earth's systems, both on small and large scales.