Edinburgh Research Explorer Redox regime shifts in microbially-mediated biogeochemical cycles

. Understanding how the Earth’s biogeochemical cycles respond to environmental change is a prerequisite for the prediction and mitigation of the effects of anthropogenic perturbations. Microbial populations mediate key steps in these cycles, yet are often crudely represented in biogeochemical models. Here, we show that microbial population dynamics can qualitatively affect the response of nutrient-cycles to environmental change. Using simple and generic mathematical models, we ﬁnd 5 that nutrient-limitations on microbial population growth can lead to regime shifts, in which the redox state of a biogeochemical cycle changes dramatically as the availability of a redox-controlling species, such as oxygen or acetate, crosses a threshold (a “tipping point”). These redox regime shifts occur in parameter ranges that are relevant to the sulfur and nitrogen cycles in the present-day natural environment, and may also have relevance to iron cycling in the iron-rich Proterozoic and Archean 10 oceans. We show that redox regime shifts also occur in models with physically realistic modiﬁca-tions, such as additional terms, chemical states, or microbial populations. Our work reveals a possible new mechanism by which regime shifts can occur in nutrient-cycling ecosystems and biogeochemical cycles, and highlights the importance of considering microbial population dynamics in models of biogeochemical cycles. acetate) caused by environmental perturbations can have drastic effects on microbe-mediated biogeochemical cycles. We ﬁrst show that these perturbations can cause regime shifts in redox state 80 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 85 between cyclic system topology and non-linear 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-rich ancient oceans.


Introduction
Metabolic conversions mediated by microbes play a key role in the Earth's biogeochemical cycles (Falkowski et al., 2008;Madigan et al., 2009;Fenchel et al., 1998). For example, microbial nitrogen fixation contributes an estimated 100-200 Tg of nitrogen to the world's oceans annually (Karl et al., 2002), while the microbial decomposition of soil carbon exceeds the anthropogenic contribution of carbon dioxide to the atmosphere by an order of magnitude (Aguilos M et al, 2013). Predicting the response of these cycles to environmental changes, including climate change, is a central current challenge in Earth system science (IPPC, 2013). However, mathematical models for global geochemical cycles often represent microbe-mediated transformations as crude "black boxes" (Allison and Martiny, 2008): for example, microbial decomposition in soil is often represented as a first-order 25 decay process (Todd-Brown et al., 2012;Westrich and Berner, 1984). Indeed, many of the models cited in the most recent IPCC report use linear representations of microbe-mediated processes (IPPC, 2013). 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 (Nikolaki and Tsiamis, 2013). There is thus an urgent need to re-evaluate the 30 role of microbial population dynamics in biogeochemical models (Todd-Brown et al., 2012;Allison et al., 2010).
Here, we use simple mathematical models to show that microbial population dynamics can have important qualitative effects on the response of microbe-mediated biogeochemical cycles to environmental change. Specifically, nutrient limitations on microbial population growth can lead to abrupt 35 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 carniverous pitcher plants (Sirota et al., 2013) to large-scale shifts in terrestrial vegetation cover (Higgins and Scheiter, 2012). These shifts are usually attributed to specific features of the system topology (Scheffer et al., 2009). Our 40 work suggests that for biogeochemical cycles, non-linear 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 1 . 45 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 (Falkowski et al., 2008). Fig. 1 illustrates simplified topologies of the iron, sulfur, carbon and nitrogen cycles (panels a-d) (Falkowski et al., 2008;Fenchel et al., 1998;Galloway et al., 2004;Canfield et al., 2005). To encapsulate the basic topology of these cycles, we begin by considering a simplified two-state model (Fig. 1e), in which 50 an oxidized form of a chemical element (here denoted s o ) is converted via microbial metabolism to a reduced form (s r ), 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 non-trivial regime shifting behaviour. Later in this paper we show that this behaviour is preserved in more realistic models that include features such as spatial heterogeneity, multiple 55 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 (Rickard, 2012), while in the nitrogen cycle, the oxidation of ammonia can be coupled to the reduction of molecular oxygen 60 Fenchel et al. (1998). In this paper, 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 acetetate). Many different chemical species can act as auxiliary electron donors or acceptors; for example acetate or hydro-65 gen 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 behaviour 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 con-70 trol 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 (Shaffer et al., 2009), and on the rate of photosynthesis (López-Urrutia et al., 2006), while the supply of acetate depends on the rate of microbial decomposi-75 tion of organic matter, which can be drastically affected by factors like sewage effluent or phosphorus inflow from agricultural runoff (Conant et al., 2011).
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 microbe-mediated biogeochemical cycles. We first show that these perturbations can cause regime shifts in redox state 80 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 85 between cyclic system topology and non-linear 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-rich ancient oceans.
2 Mathematical models for redox-cycling dynamics.
Our aim is to predict the response of microbe-mediated biogeochemical cycles to changes in the 90 availability of auxiliary electron acceptors and donors, such as oxygen and acetate. We begin with a simple and generic "two-state" representation of a nutrient cycle; later we show that the same phenomena also occur in more complex models. In our two-state model (Fig. 1e), a chemical element is cycled between its oxidized and reduced forms, whose concentrations are denoted by s o and s r respectively. The reduction step s o → s r (blue right-to-left arrow in Fig. 1e) is assumed to be 95 biotic, i.e. mediated by microbial metabolism. This step requires an auxiliary electron donor, such as acetate. The oxidation step s r → s o may occur biotically or abiotically (indicated by the blue and red left-to-right arrows in Fig. 1e), 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 100 the reaction of Fe(III) with sulfide (Canfield, 1989)); further work could extend this model to include such reactions 2 .

Fully biotic redox cycles.
If both the oxidative and reductive steps in the redox cycle are mediated by microbes, the dynamics of our two-state model can be represented by the following set of differential equations (in which 105 the dot represents a time rate of change): n or = n or G or (s o , n or ) − dn or (1) n ro = n ro G ro (s r , n ro ) − dn ro (2) The variables in this dynamical system are n ro and n or , the population densities of the oxidizing and 110 reducing microbial populations, respectively, and the concentrations s o and s r of the oxidized and reduced forms of the chemical species being cycled (Table 1 presents a key for this terminology).
Eqs. 1 and 2 describe the microbial population dynamics; the reducing and oxidizing populations have growth rates G or (s o , n or ) and G ro (s r , n ro ) respectively, which depend explicitly on s o and s r , but also depend implicitly on the concentrations of the auxiliary electron donor and acceptor respec-115 tively. Both populations are assumed to be removed from the system at a constant rate d (e.g. due to viral predation and/or washout). Eq. 3 describes changes in the substrate dynamics due to microbial consumption and production; here γ is a yield coefficient, which is assumed for simplicity to be the same for both reactions.
2 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 (Madigan et al., 2009).
In our models, we group together all these "metabolically equivalent" taxa into a single effective population.
The growth rate functions G or and G ro play a crucial role in the model. The microbial growth rate on a limiting nutrient is often described by a Monod function vs/(K + s) where s is the nutrient concentration, v is the maximal growth rate and K is the nutrient concentration at which the growth rate is half-maximal (Ingraham et al., 1983). While other, more complicated growth rate functions have been proposed (Button, 1985) the Monod form encapsulates the key fact that the growth rate is nutrient-dependent at low nutrient concentration but becomes saturated at high nutrient concen-125 tration. For a microbial population performing a redox reaction, the "nutrient" s is likely to be the chemical species being cycled while the concentration of the auxiliary electron acceptor/donor can be implicitly included in the value of the maximal growth rate v.
Importantly, however, in the natural environment, the rate of microbial growth may be limited by other factors such as the availability of carbon or micronutrients, toxin or waste product formation 130 at high densities, or simply competition for space (Hibbing et al., 2010). To account for this in a generic way, we multiply the Monod term by a population density-limitation factor (1 − n/n max ), where the parameter n max sets a maximal population density. This type of logistic population density limitation is a convenient and commonly-used way to encapsulate growth-limitation by factors not explicitly included in the model (Marino et al., 2013;Jones and Lennon, 2010;Berry and Widder, 135 2014). To check the validity of this approach, we also simulated a model in which population growth is instead explicitly limited by availability of an additional nutrient (e.g. carbon). These simulations gave qualitatively similar results to those presented here; see Supplementary Information.
These considerations lead to simple forms for the microbial growth rates in our "two-state" model: in which the parameters are v or and v ro , the maximal growth rates for the reducing and oxidizing microbes respectively, K or and K ro , the concentrations of the chemical species s o or s r at which the growth rate is half-maximal, and n or,max and n ro,max , the maximal densities of the two populations.
Importantly, the concentrations of the auxiliary electron donors and acceptors (e.g. acetate and oxy-145 gen) are implicit in the maximal growth rate parameters v or and v ro : we expect v or to increase with the availability of the auxiliary electron donor, while v ro will increase with the availability of the auxiliary electron acceptor 3 .

Biotic-abiotic redox cycles.
If the oxidation step in the redox cycle is instead abiotic, the model has only 3 variables: the pop-150 ulation density of the reducing microbial population n or and the concentrations of the oxidized and 3 By including the auxiliary electron acceptor/donor concentrations as parameters controlling the maximal growth rates, we neglect the possibility that they may be depleted by utilization. This is, however, included in the more realistic versions of the model presented later in the paper.
reduced forms of the chemical species being cycled, s o and s r . In this case, the dynamics of the microbial population n or are still described by Eq. 1, but the chemical dynamics obeẏ Here, the abiotic oxidation rate is described by the function F (s r ). Abiotic oxidation reactions can 155 occur spontaneously (e.g. the abiotic oxidation of hydrogen sulfide (Goldhaber, 2003)), or they can be catalyzed (e.g. some electron transfer processes on mineral surfaces (Schoonen and Strongin, 2005)) or limited by transport processes (Roden, 2004). To account for these factors in a generic way, we assume a Michaelis-Menten form for F (s r ) (Naidja and Huang, 2002): where v a is the maximal abiotic rate constant (which may implicitly depend on a catalyst concentration) and K a is the concentration s r at which the abiotic reaction rate is half-maximal. If K a is large such that K a s r , the reaction rate becomes linear in s r , describing a spontaneous process.

Steady-state solutions.
Analytical predictions for the steady-state population densities and the concentrations of the oxidized 165 and reduced forms of the chemical species being cycled (s o and s r ) can be obtained for both the fully biotic model (Eqs. 1-3) and the biotic-abiotic model (Eqs. 1 and 6). These are given in the Supplementary Information, sections S1-S2.
3 Regime shifts caused by population-density limitation.
Our models allow us to investigate system-level responses to environmental change. We focus on 170 environmental changes that affect the availability of auxiliary electron acceptors or donors, such as temperature-related changes in oxygen solubility (Shaffer et al., 2009), changes in photosynthesis rate, or changes in the abundance or rate of decomposition of organic matter (Conant et al., 2011).
For the fully biotic cycle, the parameters v or and v ro are proxies for the availability of auxiliary electron donors and acceptors respectively. For the biotic-abiotic cycle, the equivalent parameters 175 are v or and v a . 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, s o /s tot , 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 180 auxiliary electron acceptors or electron donors (such as oxygen or acetate) crosses a critical threshold (Fig. 2). These regime shifts happen under circumstances where the total concentration of the chemical element being cycled (s tot = s o + s r ) is high, such that s tot K or , K ro , K a , implying that the microbial population density is limited by factors other than the availability of s o or s r . In contrast, for lower concentrations of the chemical element being cycled, s tot < K or , K ro , K a , the model 185 predicts a more gradual change in system state as the availability of the auxiliary electron acceptor or donor varies.  3.1 Regime shifts also occur in models with spatial heterogeneity and chemical sinks.
The oxidation and reduction steps in natural microbial nutrient cycles are usually spatially separated (Fenchel et al., 1998). Extending our model, we find that our prediction of redox regime shifting behavior is robust to the inclusion of spatial separation between reductive and oxidative zones; in-220 deed, the resulting transport limitation of chemical species s o and s r actually enhances the switching phenomenon (see Supplementary Information section S5).
In the natural environment, coupling between the different redox cycles shown in Fig. 1 may also be important. For example, sulfide reacts with iron to form pyrite, which represents a stable sink for iron and sulfide (Raiswell and Canfield, 2012). We find that our model still produces redox regime 225 shifts when we include extra terms to simulate these sink effects (see Supplementary Information section S6).

Origin of the regime shifts.
The redox regime shifts which we observe in our model arise from the interplay between non-linear population growth, which can be limited by factors other than the chemical species being cycled, 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 250 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 (Alberts et al., 2002). Phosphorylationdephosphorylation 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, decou-255 pling the enzymatic conversion rates from the concentration of substrate (Goldbeter and Koshland, 1981). Although they act on very different length-and time-scales, 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 (Goldbeter and Koshland, 1981), and that the regime 260 shifts observed in our models are equivalent to the ultrasensitive signal responses predicted by this model (see Supplementary Information section 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.
5 Redox regime shifts in a more realistic model.

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Thus far our investigation has focused on a rather simplified model for microbe-mediated biogeochemical cycles. In this simple model, varying v ro and v or 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 270 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 275 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 (Rickard, 2012)); 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 microbes; the model includes explicitly the dynamics of the photosynthesizer population as well as the oxygen Our simulations show that this model indeed undergoes redox regime shifts (Fig. 3b). In particular, the system redox state, as measured by the ratio s o /s tot (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 295 photosynthesizers and hence the oxidiser 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 behaviour in 300 equivalent models where the oxidation step is abiotic (see Supplementary Information, section 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 Supplementary Information section S10).
Since many natural redox cycles involve cycling between more than two redox states (e.g. the 305 nitrogen and sulfur cycles in Fig. 1 6 Conditions for redox regime shifts.

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Our analysis provides a clear set of criteria that need to be satisfied for redox regime shifts to occur.
These are: 1. 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 5 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. 6 Multiplicative Monod kinetics is the most widely used method of modelling microbial growth limitation by multiple substrates (Moore et al., 2002;Jin and Bethke, 2005). However we note that Liebig's law of the minimum provides an alternative (Saito Mak A.;Goepfert, 2008), which would not affect our qualitative results.
concentration of another nutrient (see Supplementary Information Section 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.
2. The total concentration of the element being cycled must be high enough to saturate the growth rates of the microbial reducers and oxidisers (or the abiotic oxidation reaction): s tot K or , K ro , K a . This ensures that the growth of the redox-cycling populations will become saturated 320 with respect to s, causing a switch-like response to changes in auxiliary electron acceptor or donor availability (as in Fig. 2).
3. The growth rates of the redox-cycling populations must be unsaturated with respect to the concentrations of the auxiliary electron acceptor and/or donor, so that the system responds to changes in auxiliary electron acceptor or donor availability.

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7 Are redox regime shifts likely in the natural environment?
We now assess whether these conditions, required for redox regime shifts, are likely to be prevalent in the natural environment.

Condition 1: A factor exists that ultimately limits population density.
In the natural environment, there are many possible limiting factors for microbial population density.

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Microbial growth requires sources not only of energy but also of carbon, nitrogen, phosphorus, sulfur and other, trace, biomass components (Madigan et al., 2009;Ingraham et al., 1983). For redoxcycling microbial populations, the redox reaction provides an energy source, but cannot satisfy all the requirements for formation of biomass. It is thus almost inevitable that growth is ultimately limited by the availability of biomass components rather than the redox species. Indeed, carbon 335 limitation is common in microbial soil/sediment communities (Demoling et al., 2007), while in ocean communities nitrogen or phosphorus is often growth-limiting (Mills et al., 2008).

Condition 2: High concentration of the chemical element being cycled.
Our second condition states that s tot K or , K ro (i.e. the oxidizer/reducer growth rate must be saturated with respect to s r or s o ). To assess whether this condition is fulfilled in the natural environment, populations, s tot K or , K ro . For example, marine sulfate reducers are generally not limited by sul-fate, because sulfate is highly abundant (indeed it is the second most abundant anion in the oceans (Goldhaber, 2003)). In contrast, our data survey suggests that redox regime shifts are unlikely to be associated with carbon cycles, because the typical half-saturation constant for methanogenesis is large relative to typical environmental concentrations of acetate.

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For the iron cycle, our survey suggests that redox regime shifts are unlikely in modern-day environments, but may have occurred in the past. While modern oceanic concentrations of dissolved Fe 2+ ions are low, the ancient oceans may have contained high concentrations of Fe 2+ (≈ 1mM), suggesting that redox regime shifts could have occurred in the iron-rich Archean or Proterozoic iron cycles (Canfield, 1998). Condition 3 states that, for biotic redox reactions, the concentration of the auxiliary electron donor or acceptor must be low enough that changes in their availability affect the growth rate of the microbial reducers/oxidizers (i.e. the oxidative and reductive microbial metabolic reactions must be unsaturated with respect to the auxiliary electron acceptor/donor).

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Biotic reduction processes often take place in the presence of strong competition for auxiliary electron donor, for example, sulfate-reducing microbes typically compete with methanogens for acetate (Muyzer and Stams, 2008). The concentration of acetate in freshwater sediments is typically about 1µM (Roden and Wetzel, 2003) but can be as high as 100µM (Burdige, 2002). This compares to approximate half-saturation constants for growth with respect to acetate of 70µM for sulfate re-365 duction and 12µM for methanogenesis (Roden and Wetzel, 2003;Ingvorsen et al., 1984), suggesting that indeed these reactions are very likely to be unsaturated with respect to acetate.
For oxidative processes, oxygen is the most widely used auxiliary electron acceptor. The supply of oxygen is expected to be rate-limiting for growth in oxygen-poor environments (which are becoming more common in the coastal oceans) (Diaz and Rosenberg, 2008). The half-saturation constant with 370 respect to oxygen for bacterial sulfide oxidation is 1 − 20µM (Klok et al., 2012;González-Sánchez and Revah, 2007), and while the concentration of oxygen in oxygen-saturated (i.e. fully aerated) water is 0.3mM (Kamyshny et al., 2011), significant competition for oxygen means that the concentration is much lower in many environments (Shaffer et al., 2009). It is interesting to note that oxygen concentrations were also low in the Proterozoic and Archean oceans (Canfield, 1998).

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Taken together, this analysis suggests that the redox regime shifts predicted by our model are likely to be relevant in the present-day natural environment, with respect to the sulfur and nitrogen cycles, and may also have played a role in iron cycling in the iron-rich Proterozoic and Archean oceans.

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How likely are the changes in auxiliary electron acceptor/donor concentrations that could trigger redox regime shifts in biogeochemical cycles? Focusing on oxygen as the most significant natural auxiliary electron acceptor, oxygen concentrations in oceans or inland water bodies can be affected by temperature changes (for example, a 4.8 • C global temperature increase has been predicted to cause a 68% reduction in the mean oceanic oxygen concentration (Shaffer et al., 2009)) and by 385 perturbations which affect the balance between photosynthesis and oxygenic respiration, such as eutrophication (which can lead to drastic increases of biomass, generating "oxygen minimum zones" (Diaz and Rosenberg, 2008)). Furthermore, over Phanerozoic time pO 2 has varied between 15-37%, which represents a variation large enough to generate redox regime shifts (Berner, 1999).
The availability of auxiliary electron donors (such as acetate, lactate or hydrogen) is expected to 390 be altered by changes in the rate of organic matter degradation, which has been predicted to increase with temperature (Conant et al., 2011), and is also sensitive to changes in the abundance of organic matter due to sewage or phosphorus influx (Todd-Brown et al., 2012). Changes in electron donor availability could also arise due to competition effects, such as reductive degradation of pollutants (Beaudet et al., 1998), or perturbations in other biogeochemical cycles. This raises the interesting 395 possibility that a redox regime shift in one biogeochemical cycle could trigger shifts in others, due to changes in the level of competition for auxiliary electron donors.

Discussion
Microbial populations are key mediators of the Earth's biogeochemical cycles (Falkowski et al., 2008). Our work shows that microbial population dynamics can have important consequences for 400 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 limting 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 405 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 410 redox regime shifts may be common in the natural environment.
Regime shifts are a well-known phenomenon in many ecosystems (Scheffer et al., 2009) including microbial ecosystems (Bürgmann et al., 2011). They are known to occur in biogeochemical cycles (Blodau and Knorr, 2006) and have played an important role in the Earth's history; a notable example being the rapid transition to an oxic atmosphere around 2.3 Ga (Lenton and Watson, 2011). Our work 415 suggests a new mechanism by which regime shifts may occur in microbe-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 420 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 asso-425 ciated 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 (Benton and Twitchett, 2003). More generally, our work reveals that microbial population dynamics can lead to qualitative changes in the behaviour of biogeochemical cycles, with significant system-level consequences. Better understanding of microbial   (Falkowski et al., 2008;Fenchel et al., 1998;Galloway et al., 2004;Canfield et al., 2005), together with the model investigated in the first part of this work (e). In all panels, oxidation reactions proceed to the right, and reduction reactions proceed to the left. Biologically catalysed (metabolic) reactions are shown in blue, and abiotic reactions are shown in red. We note that abiotic reduction reactions are not shown, as these are minor reactions in the natural environment (but can be included in our model, see Supplementary Information section S1). We also note that many intermediate chemical states are not shown (particularly for the nitrogen and sulfur cycles) but inclusion of extra states does not affect our conclusions; see Supplementary Information. In panel e, sr and so represent the reduced and oxidized forms of the chemical element being cycled.  Figure 2. Redox regime shifts in model nutrient cycles. The global redox state, as measured by the oxidized fraction so/stot, predicted by the steady-state solution of the model equations for the fully biotic cycle (a and c, Eqs. 1-3) or the biotic-abiotic cycle (b and d, Eqs. 1, 2 and 6) is plotted as a function of parameters that form proxies for the degree of reductive or oxidative driving. These parameters are: for reductive driving, the maximal growth rate of the reductive population, vor (a and b, keeping vor fixed at 2h −1 or va = 0.2µMh −1 ), and, for oxidative driving, either the maximal growth rate of the oxidative population vro (c, keeping vro fixed at 2h −1 ) or the maximal abiotic oxidation rate va (d, also with vro = 2h −1 ). The results show a shift between oxidized and reduced ecosystem states as a threshold in reductive or oxidative driving is crossed; the sharpness of this transition increases with the concentration of the chemical species being cycled, stot (shown in the colorbar).