Low Probability of Initiating nirS Transcription Explains Observed Gas Kinetics and Growth of Bacteria Switching from Aerobic Respiration to Denitrification

Abstract

In response to impending anoxic conditions, denitrifying bacteria sustain respiratory metabolism by producing enzymes for reducing nitrogen oxyanions/-oxides (NO_x) to N₂ (denitrification). Since denitrifying bacteria are non-fermentative, the initial production of denitrification proteome depends on energy from aerobic respiration. Thus, if a cell fails to synthesise a minimum of denitrification proteome before O₂ is completely exhausted, it will be unable to produce it later due to energy-limitation. Such entrapment in anoxia is recently claimed to be a major phenomenon in batch cultures of the model organism Paracoccus denitrificans on the basis of measured e⁻-flow rates to O₂ and NO_x. Here we constructed a dynamic model and explicitly simulated actual kinetics of recruitment of the cells to denitrification to directly and more accurately estimate the recruited fraction (). Transcription of nirS is pivotal for denitrification, for it triggers a cascade of events leading to the synthesis of a full-fledged denitrification proteome. The model is based on the hypothesis that nirS has a low probability (, h⁻¹) of initial transcription, but once initiated, the transcription is greatly enhanced through positive feedback by NO, resulting in the recruitment of the transcribing cell to denitrification. We assume that the recruitment is initiated as [O₂] falls below a critical threshold and terminates (assuming energy-limitation) as [O₂] exhausts. With  = 0.005 h⁻¹, the model robustly simulates observed denitrification kinetics for a range of culture conditions. The resulting (fraction of the cells recruited to denitrification) falls within 0.038–0.161. In contrast, if the recruitment of the entire population is assumed, the simulated denitrification kinetics deviate grossly from those observed. The phenomenon can be understood as a ‘bet-hedging strategy’: switching to denitrification is a gain if anoxic spell lasts long but is a waste of energy if anoxia turns out to be a ‘false alarm’.

Author Summary

In response to oxygen-limiting conditions, denitrifying bacteria produce a set of enzymes to convert / to N₂ via NO and N₂O. The process (denitrification) helps generate energy for survival and growth during anoxia. Denitrification is imperative for the nitrogen cycle and has far-reaching consequences including contribution to global warming and destruction of stratospheric ozone. Recent experiments provide circumstantial evidence for a previously unknown phenomenon in the model denitrifying bacterium Paracoccus denitrificans: as O₂ depletes, only a marginal fraction of its population appears to switch to denitrification. We hypothesise that the low success rate is due to a) low probability for the cells to initiate the transcription of genes (nirS) encoding a key denitrification enzyme (NirS), and b) a limited time-window in which NirS must be produced. Based on this hypothesis, we constructed a dynamic model of denitrification in Pa. denitrificans. The simulation results show that, within the limited time available, a probability of 0.005 h⁻¹ for each cell to initiate nirS transcription (resulting in the recruitment of 3.8–16.1% cells to denitrification) is sufficient to adequately simulate experimental data. The result challenges conventional outlook on the regulation of denitrification in general and that of Pa. denitrificans in particular.

Introduction

A complete denitrification pathway includes the dissimilatory reduction of nitrate () through nitrite (), nitric oxide (NO), and nitrous oxide (N₂O) to di-nitrogen (N₂). Typically, the genes encoding reductases for these nitrogen oxyanions/-oxides (NO_x) are not expressed constitutively but only in response to O₂ depletion, making denitrification a facultative trait [1]. Hence, during anoxic spells, the process enables denitrifying bacteria to sustain respiratory metabolism, replacing O₂ by NO_x as the terminal electron (e⁻) acceptors. Since permanently anoxic environments lack available NO_x, denitrification is confined to sites where O₂ concentration fluctuates, such as biofilms, surface layers of sediments, and drained soil (which turns anoxic in response to flooding).

From modelling denitrifying communities as a homogenous unit to a model of regulation of denitrification in an individual strain

Denitrification is a key process in the global nitrogen cycle and is also a major source of atmospheric N₂O [2]. A plethora of biogeochemical models have been developed for understanding the ecosystem controls of denitrification and N₂O emissions [3]. A common feature of these models is that the denitrifying community of the system (primarily soils and sediments) in question is treated as one homogenous unit with certain characteristic responses to O₂ and concentrations. This simplification is fully legitimate from a pragmatic point of view, but in reality any denitrifying community is composed of a mixture of organisms with widely different denitrification regulatory phenotypes [4]. Modelling has been used to a limited extent to analyse kinetic data for various phenotypes (See [5] and references therein) and for understanding the accumulation of intermediates [6]. To our knowledge, however, no attempts have been made to model the regulation during transition from aerobic to anaerobic respiration in individual strains, despite considerable progress in the understanding of their regulatory networks. It would be well worth the effort, since the regulatory phenomena at the cellular level provide clues as to how denitrification and NO and N₂O emissions therefrom are regulated in intact soils [7]. Explicit modelling of the entire denitrification regulatory network, however, would take us beyond available experimental evidence, with numerous parameters for which there are no empirical values. Considering this limitation, here we have constructed a simplified model to investigate if a stochastic transcriptional initiation of key denitrification genes (nirS) could possibly explain peculiar kinetics of e⁻-flow as Paracoccus denitrificans switch from aerobic to anaerobic respiration [4], [8].

Although denitrification is widespread among bacteria, the α-proteobacterium Pa. denitrificans is the ‘paradigm’ model organism in denitrification research. Recent studies [4], [8], [9] have indicated a previously unknown phenomenon in this species that, in response to O₂ depletion, only a marginal fraction () of its entire population appears to successfully switch to denitrification. In these studies, however, is inferred from rates of consumption and production of gases (O₂, NO_x, and N₂), and a clear hypothesis as to the underlying cause of the low is also lacking. To fill these gaps, we formulated a refined hypothesis addressing the underlying regulatory mechanism of the cell differentiation in response to O₂ depletion. On its basis, we constructed a dynamic model and explicitly simulated the actual kinetics of recruitment of the cells from aerobic respiration to denitrification. The model adequately matches batch cultivation data for a range of experimental conditions [4], [8] and provides a direct and refined estimation of . The exercise is important for understanding the physiology of denitrification in general and of Pa. denitrificans in particular and carries important implications for correctly interpreting various denitrification experiments.

Regulation of denitrification in terms of relevance to fitness

Generally, the transcription of genes encoding denitrification enzymes is inactivated in the presence of O₂. A population undertaking denitrification typically responds to full aeration by completely shutting down denitrification and immediately initiating aerobic respiration [10]. Thus, O₂ controls denitrification at transcriptional as well as metabolic level, and both have a plausible fitness value. The transcriptional control minimises the energy cost of producing denitrification enzymes, and the metabolic control maximises ATP (per mole electrons transferred) because the mole ATP per mole electrons transferred to the terminal e⁻-acceptor is ∼50% higher for aerobic respiration than for denitrification [10].

Denitrification enzymes produced in response to an anoxic spell are likely to linger within the cells under subsequent oxic conditions (although, this has not been studied in detail), ready to be used if O₂ should become limiting later on. However, these enzymes will be diluted by aerobic growth, since the transcription of their genes is effectively inactivated by O₂. Hence, a population growing through many generations under fully oxic conditions will probably be dominated by the cells without intact denitrification proteome. When confronted with O₂ depletion, such a population will have to start from scratch, i.e., transcribe the relevant genes, translate mRNA into peptide chains (protein synthesis by ribosomes) and secure that these chains are correctly folded by the chaperones, transport the enzymes to their correct locations in the cell, and insert necessary co-factors (e.g., Cu, Fe, or Mo). In E. coli grown under optimal conditions, the whole process from the transcriptional activation to a functional enzyme takes ≤20 minutes [11] and costs significant amount of energy (ATP).

Synthesis of denitrification enzymes is rewarding if anoxia lasts long and NO_x remains available, but it is a waste of energy if anoxia is brief. Since the organisms cannot sense how long an impending anoxic spell will last, a ‘bet-hedging strategy’ [12] where one fraction of a population synthesises denitrification enzymes while the other does not may increase overall fitness.

A delayed response to O₂ depletion may lead to entrapment in anoxia

Most, if not all, denitrifying bacteria are non-fermentative and completely rely on respiration to generate energy [13], [14]. This implies that their metabolic machinery will run out of energy whenever deprived of terminal e⁻-acceptors. When [O₂] falls below some critical threshold, the cells will ‘sense’ this and start synthesising denitrification proteome, utilising energy from aerobic respiration [10]. However, if O₂ is suddenly exhausted or removed, the lack of a terminal e⁻-acceptor will create energy limitation, restraining the cells from enzyme synthesis, hence, entrapping them in anoxia. This was clearly demonstrated by Højberg et al. [15], who used silicone immobilised cells to transfer them from a completely oxic to a completely anoxic environment. Such a rapid transition is unlikely to occur in nature; however, the experiment illustrates one of the apparent perils in the regulation of denitrification: the cells that respond too late to O₂ depletion will be entrapped in anoxia, unable to utilise alternative electron acceptors for energy conservation and growth.

Højberg et al.'s [15] observations have largely been ignored in the research on the regulation of denitrification, and it is implicitly assumed that, in response to O₂ depletion, all cells in cultures of denitrifying bacteria will switch to denitrification. Contrary to this, however, Bergaust et al. [4], [8], [16] followed by Nadeem et al. [9] proposed that in batch cultures of Pa. denitrificans, only a small fraction of all cells is able to switch to denitrification. During transition from oxic to anoxic conditions, they observed a severe depression in the total e⁻-flow rate (i.e., to O₂+NO_x, see Fig. 1), which was estimated on the basis of measured gas kinetics. Had all of the cells switched to denitrification as O₂ exhausted, the total e⁻-flow rate would have carried on increasing, without such a depression. The depression was followed by an exponential increase in the e⁻-flow rate, which was tentatively ascribed to anaerobic growth of a small (fraction recruited to denitrification). It was postulated that this fraction escaped entrapment in anoxia by synthesising initial denitrification proteins within the time-window when O₂ was still present, whereas the majority of the cells () failed to do so, thus remained unable to utilise NO_x.

Figure 1. Data generated by batch cultivation of Pa. denitrificans [4] (redrawn).

As the cells transited from oxic to anoxic conditions (Panel A), Bergaust et al. [4] observed a severe depression in the total e⁻-flow rate (i.e., to O₂+NO_x, Panel B), which was taken to indicate that only a fraction of the cells switched to anaerobic respiration (denitrification). Had all of the cells switched, the total e⁻-flow would have carried on increasing without such a depression. The depression was followed by an exponential increase in the e⁻-flow rate, which was ascribed to anaerobic growth of a small fraction () of the cells that escaped entrapment in anoxia and carried on growing by denitrification.

The core hypothesis: A low probability of initiating nirS transcription seems to drive the cell differentiation

Autocatalytic transcription of denitrification genes

In Pa. denitrificans, denitrification is driven by four core enzymes: Nar (membrane-bound nitrate reductase), NirS (cytochrome cd₁ nitrite reductase), cNor (nitric oxide reductase), and NosZ (nitrous oxide reductase, see Fig. 2). The transcriptional regulation of genes encoding these enzymes (nar, nirS, nor and nosZ, respectively) involves, at least, three FNR-type proteins acting as sensors for O₂ (FnrP), / (NarR), and NO (NNR) [10], [17], [18]. NarR and NNR facilitate product-induced transcription of the nar and nirS genes: When anoxia is imminent, the low [O₂] is sensed by FnrP, which in interplay with NarR induces nar transcription. NarR is activated by (and/or probably by ); thus once a cell starts producing traces of , nar expression becomes autocatalytic. The transcription of nirS is induced by NNR, which requires NO for activation; thus once traces of NO are produced, the expression of nirS also becomes autocatalytic. In contrast, the transcription of nor is substrate (NO) induced via NNR, while nosZ is equally but independently induced by NNR and FnrP [19]. Here we are concerned with the dynamics that start with the transcription of nirS, since the experimental treatments that we simulated were not supplemented with but various concentrations of only (Table 1).

Figure 2. The regulatory network of denitrification in Pa. denitrificans.

In Pa. denitrificans, denitrification is driven by four core enzymes: Nar (nitrate reductase encoded by the nar genes), NirS (nitrite reductase encoded by nirS), cNor (NO reductase encoded by nor), and NosZ (N₂O reductase encoded by nosZ). The transcription of these genes is regulated by, at least, three FNR-type proteins, which are sensors for O₂ (FnrP), / (NarR), and NO (NNR). NarR and NNR facilitate product-induced transcription of the nar and nirS genes (see positive-feedback loops), where NNR also counteracts the NO accumulation (negative-feedback loop) [10], [17], [18]. Circumstantial evidence suggests that O₂ inactivates NNR (grey dashed link) [20], and NirS is also unlikely to be functional in the presence of high O₂ concentrations. Hence, for our modelling we hypothesise that the probability of an autocatalytic transcriptional activation of nirS is zero until O₂ falls below a critical concentration . When O₂ falls below , the initial nirS transcription is possibly mediated through a minute pool of intact NNR, crosstalk with other factors, or through non-biological traces of NO found in an -supplemented medium. Regardless of the exact mechanism(s), once nirS transcription is initiated, it will be substantially enhanced by spikes of internal NO emitted from the first molecules of NirS (the positive-feedback loop). The activated positive-feedback will also induce nor and nosZ transcription via NNR (although, the latter can also be induced independently by FnrP [19]), facilitating the synthesis of a full-fledged denitrification proteome. Our model assumes that such recruitment to denitrification will occur with a low probability. We further assume that the recruitment will only be possible as long as a minimum of O₂ is available because the production of the first molecules of NirS will depend on energy from aerobic respiration.

Table 1. The simulated experiment of Bergaust et al [4], [8].

Batch No.	(vol. %)*	(mM)
1	∼0	0.2
2	∼0	1
3	∼0	2
4	1	0.2
5	1	1
6	1	2
7	7	0.2
8	7	1
9	7	2

*Targeted values for initial O₂ in the headspace (where the headspace vol. = 70 mL). The actual initial O₂ measured in the 0, 1, and 7% treatments was 0.012–0.19, 1.2–1.66, 6.6–6.8 vol.%, respectively. The O₂ present in the ∼0% treatments was due to traces of O₂ left behind despite various cycles of evacuation of the headspace air and subsequent flushing of the vials with helium (He-washing).

Low probability of initiating nirS transcription

The transcription of nirS is known to be suppressed by O₂ [4], [8], but the exact mechanism remains unclear. Circumstantial evidence suggests that it is due to O₂ inactivating NNR [20] (dashed link in Fig. 2), but this is not necessary to explain the repression of NirS. There are several mechanisms through which high O₂ concentrations may restrain NirS activity, i.e., through post-transcriptional regulation, direct interaction with the enzyme, or due to competition for electrons. Regardless of the exact mechanism(s), the ultimate consequence is the elimination of the positive feedback via NO and NNR. When O₂ falls below a critical threshold, facilitating NirS activity, this positive feedback would allow the product of a single transcript of nirS to induce a subsequent burst of nirS transcription in response to NO. Such ‘switches’ in gene expression by positive-feedback loops are not uncommon in prokaryotes, and they have been found to result in cell differentiation because the initial transcription is stochastic with a relatively low probability [21].

Our model assumes such stochastic recruitment to denitrification, triggered by an initial nirS transcription occurring with a low probability. This initial transcription is possibly mediated by a minute pool of intact NNR and/or through crosstalk with other factors, such as FnrP. A -supplemented medium contains non-biologically formed traces of NO which, once diffused into the cells while O₂ is low, will activate background levels of NNR and, thereby, may also increase the probability of triggering nirS transcription.

For this modelling exercise, we do not need a full clarification of the mechanisms involved but only to assume that the probability of an autocatalytic transcriptional activation of nirS would be practically zero as long as O₂ concentration is above a certain threshold. This assumption is backed by empirical data indicating that NO is not produced to detectable levels before O₂ concentration falls below a critical threshold [8], [22]. For O₂ concentrations below this threshold, the model assumes a low (but unknown) probability for each cell to initiate the autocatalytic transcription of nirS, paving the way for the rest of the denitrification proteome.

O₂ is required for the initial production of NirS

We further assume that the recruitment to denitrification will only be possible as long as a minimum of O₂ is available because the synthesis of first molecules of NirS will depend on energy from aerobic respiration.

Can NO produced within one cell help activate the autocatalytic transcription of nirS in the neighbouring cells?

It is perhaps less obvious that the autocatalytic transcriptional activation of nirS takes place only within the NO-producing cell because NO diffuses easily across membranes [23]. However, the average distance between the cells in a culture with 10⁹ cells mL⁻¹ (roughly the numbers that we are dealing with) is ∼10 µm, which is ∼10 times the diameter of a cell. This implies that an NO molecule produced by a cell has a much higher probability to react with and activate the NNR inside the same cell than to do so in another one.

Modelling the cell differentiation

To represent the batch cultivation conducted by Bergaust et al. [4], [8], the model explicitly simulates growth of two sub-populations, one with denitrification enzymes () and the other without (); both equally consume O₂, but cannot reduce NO_x to N₂. Once oxygen concentration in the liquid falls below a critical level [22], the cells within are assumed to initiate nirS transcription (and thereby ensure recruitment to ) with a rate described by a probabilistic function: (cells h⁻¹), where is assumed to be an dependent probability (h⁻¹) for any cell within to initiate nirS transcription (leading to a full denitrification capacity). When falls below , triggers and holds a constant value as long as is above a critical minimum . For , is zero (assuming the inactivation of NNR by O₂); is also zero for (assuming the lack of energy for protein synthesis).

The recruitment of to is simulated as an instantaneous event; thus, the model does not take into account the time-lag between the initiation of nirS transcription and the time when the transcribing cell has become a fully functional denitrifier. This simplification is based on the evidence that this lag is rather short. Experiments with E. coli [11] under optimal conditions suggest lags of ∼20 minutes between the onset of transcription and the emergence of a functional enzyme. In Pa. denitrificans [8], [22], the lag observed between the emergence of denitrification gene transcripts and the subsequent gas products suggests that the time required for synthesising the enzymes is within the same range.

Employing the model to understand ‘diauxic lags’ between the aerobic and anaerobic growth-phases

In a series of experiments with denitrifying bacteria (Pseudomonas denitrificans, Pseudomonas fluorescens, Alcaligenes eutrophus and Paracoccus pantotrophus) [24]–[26], oxic cultures were sparged with N₂ to remove O₂ and were monitored by measuring optical density (OD₅₅₀). All the strains except Ps. fluorescens went through a conspicuous ‘diauxic lag: a period of little or no growth’ [26]; the OD remained practically constant during the lag period, lasting 4–30 hours, which was eventually followed by anaerobic growth.

To understand the diauxic lag, Liu et al. [24] used the common assumption that all cells would eventually switch to denitrification. They constructed a simulation model based on the assumption that all the cells contained a minimum of denitrification proteome (even after many generations under oxic conditions). This minimum would allow them to produce more denitrification enzymes when deprived of O₂, albeit very slowly due to energy limitation. The time taken to effectively produce adequate amounts of denitrification enzymes ( = the diauxic lag) was taken to be a function of the initial amounts of these enzymes per cell. Although their model may possibly explain short time-lags, it appears unrealistic for lag phases as long as 10–30 hours [25] because to produce such long lags, conceivably, the initial enzyme concentration would be less than one enzyme molecule per cell, which is mathematically possible but biologically meaningless.

The model presented in this paper provides an alternative explanation for the apparent diauxic lags: a sudden shift from fully oxic to near anoxic conditions (by sparging with N₂) would leave the medium with only traces of O₂, which would be quickly depleted due to aerobic respiration. As a consequence, the available time for initiating the synthesis of denitrification proteome would be marginal, allowing only a tiny fraction () of the cells to switch to denitrification. This marginal fraction would grow exponentially from the very onset of anoxic conditions, but it would remain practically undetectable as measured (OD) for a long time, creating the apparent 4–30 h lag. The length of the lag depends on the fraction of the cells switching to denitrification. To demonstrate this alternative explanation, we adjusted our model to the reported conditions and simulated the experiment of Liu et al [24]. The model produced qualitatively similar ‘diauxic lags’ in the simulated cell density (OD), although the time length of the lag could be anything (depending on assumptions regarding the residual O₂ after sparging, which was not measured).

Materials and Methods

An overview of the modelled experiment: Batch incubations in gas-tight vials

Bergaust et al. [4], [8] studied aerobic and anaerobic respiration rates in Paracoccus denitrificans (DSM413). The cells were incubated (at 20°C) as stirred batches in 120 mL gastight vials, containing 50 mL Sistrom's medium [27] (Fig. 3). The medium was supplemented with various concentrations of KNO₃ or KNO₂. Prior to inoculation, air in the headspace was replaced with He to remove O₂ and N₂ (He-washing), followed by the injection of no, 1, or 7 headspace-vol.% O₂. Finally, each vial was inoculated with ∼3×10⁸ aerobically grown cells.

Figure 3. An overview of the modelled system: batch incubation in a gas-tight vial.

The experiment: The stirred Sistrom's medium [27] was inoculated with aerobically grown Pa. denitrificans cells, which were provided with different concentrations of O₂ and (g or aq with a chemical species-name represents gaseous or aqueous, respectively). O₂ is consumed by respiration, driving its transport from the headspace to the liquid. Once the aerobic respiration becomes limited, the cells may switch to denitrification (recruitment), reducing to N₂ via the intermediates NO and N₂O (not shown). For monitoring O₂, CO₂, N₂, NO and N₂O, a robotised incubation system [28] was used, which automatically takes samples from the headspace by piercing the rubber septum. Each sampling removes a fraction (3–3.4%) of all gases in the headspace, but it also involves a marginal leakage of O₂ and N₂ into the vial (as indicated by the two-way arrows at the top of the figure). The model: The model operates with two sub-populations: one without and the other with denitrification enzymes ( and , respectively). Both consume O₂ if present, but cannot reduce NO_x. The cells may be recruited to the pool as falls below a critical threshold. The rate of recruitment () is modelled as a probabilistic function: (cells h⁻¹), where represents an O₂ dependent specific-probability (h⁻¹) for any cell to initiate nirS transcription (leading to the synthesis of a full-fledged denitrification proteome).

Treatments selected for simulation

Only -supplemented treatments (Table 1) were selected for this modelling exercise for two reasons. First, was not monitored; hence, results of the -supplemented treatments could not provide exact estimates of anaerobic respiration rates (due to an unknown transient accumulation of ). Second, by excluding the treatments requiring Nar, we could single out and focus on the regulation of the other key enzyme NirS.

Aerobic respiration followed by denitrification

O₂ diffused from the headspace to the liquid (Fig. 3), where the cells consumed it before switching to denitrification: the stepwise reduction of to N₂ via the intermediates NO and N₂O (not shown). Headspace concentrations of gases were monitored by frequent sampling (every 3 hours). A typical result is shown in Fig. 1A, illustrating the increasing rate of O₂ consumption until depletion, followed by transition to denitrification. The denitrification rate increased exponentially till all the present in the medium was recovered as N₂. The medium contained ample amounts of carbon substrate (34 mM succinate) to support the consumption of all available electron acceptors.

Sampling procedure

To monitor O₂, CO₂, NO, N₂O, and N₂ in the headspace for respiring cultures, Bergaust et al. [4], [8] used a robotised incubation system, which automatically takes samples from the headspace by piercing the rubber septum (Fig. 3). The auto-sampler is connected to a gas chromatograph (GC) and an NO analyser (For details, see [28]). The system uses peristaltic pumping, which removes a fraction (3–3.4%) of all the gases in the headspace and then reverses the pumping to inject an equal amount of He into the headspace, thus maintaining ∼1 atmosphere pressure inside the vial. Sampling also involves a marginal leakage of O₂ and N₂ into the headspace (∼22 and ∼60 nmol per sampling, respectively) through tubing and membranes of the injection system.

Calculation of gases in the liquid

Concentrations of gases in the liquid were calculated using solubility of each gas at the given temperature (20°C), assuming equilibrium between the headspace and the liquid. However, the O₂ consumption rate was so high that to calculate [O₂] in the liquid, its transport rate (from the headspace to the liquid) had to be taken into account.

An overview of the model

The model effectively represents the physical phenomena mentioned above, so as to ensure that the simulation results match the measured data for the right reasons. Net effect of sampling (dilution and leakage) is included in the simulation of O₂ kinetics at the reported sampling times. Transport of O₂ between the headspace and the liquid is modelled using an empirically determined transport coefficient and the solubility of O₂ in water at 20°C. To simulate the metabolic activity (O₂ consumption and N₂ production) and growth, the model divides the cells into two sub-populations: one without and the other with denitrification enzymes ( and pools, respectively, see Fig. 3). Both equally consume O₂ if present, but cannot reduce to N₂. Those cells that, in response to O₂ depletion, are able to initiate nirS transcription (see Fig. 2) are recruited to the pool, where  = 0 prior to the recruitment. The recruitment rate () is modelled according to a probabilistic function described below (Eqs. 7–8).

The model ignores sampling effect on N₂ (leakage and loss), thus calculating the cumulative N₂ production as if no sampling took place. That is because the experimentally determined N₂ accumulation (which is to be compared with the model predictions) was already corrected for the net sampling effect.

The model is developed in Vensim DSS 6.2 Double Precision (Ventana Systems, Inc. http://vensim.com/) using techniques from the field of system dynamics [29]. The model is divided into three sectors: I. O₂ kinetics, II. Population dynamics of and , and III. Denitrification kinetics (Fig. 4).

Figure 4. A stock and flow diagram of the model's structure.

The squares represent the state variables, the circles the rate of change in the state variables, the shaded ovals the auxiliary variables, the arrows dependencies between the variables, and the edges represent flows into or out of the state variables. A. The panel represents the structure that governs the O₂ kinetics. Briefly, it shows that O₂ in the vial's headspace () is transported () to the liquid-phase (), where it is consumed () by both and populations with an identical cell-specific velocity of O₂ consumption (). represents net marginal changes in due to sampling. B. The panel represents the structural basis for population dynamics of the cells without () and with () denitrification enzymes. Briefly, it shows that both the populations are able to grow by aerobic respiration ( and , respectively). The growth rate of , however, is primarily based on denitrification (). Initially,  = 0 and is populated through recruitment () of the cells from , where the recruitment is a function of and an [O₂] dependent specific-probability of the recruitment for any cell. C. The panel represents the structural basis for the /N₂ kinetics. Briefly, it illustrates that control the consumption rate of (), recovered as N₂, in proportion to a cell-specific velocity of consumption ().

Sector I: O₂ kinetics

Structural-basis for the O₂ kinetics is mapped in Fig. 4A: the squares represent the state variables, the circles the rate of change in the state variables, the shaded ovals the auxiliary variables, the arrows mutual dependencies between the variables, and the edges represent flows into or out of the state variables. Briefly, Fig. 4A (left to right) shows that O₂ in the vial's headspace () is transported () to the liquid-phase (), where it is consumed () by both the and populations (lacking and carrying denitrification enzymes, respectively) in proportion to an identical cell-specific velocity of O₂ consumption (). represents net marginal changes in due to sampling. Below we present equations and a detailed explanation of the structural components shown for this sector.

O₂ in the headspace

(, mol vial⁻¹) is initialised by measured initial concentrations (Table 1) and modelled as a function of transport () between the headspace and the liquid [28]:

(1)

Units: mol vial⁻¹ h⁻¹

where (L vial⁻¹ h⁻¹) is the empirically determined coefficient for the transport of O₂ between the headspace and the liquid (See Table 2 for parametric values and their sources), (mol L⁻¹ atm⁻¹) is the solubility of O₂ in water at 20°C, (atm) is the partial pressure of O₂ in the headspace, and (mol L⁻¹) is the O₂ concentration in the liquid-phase .

Table 2. Model parameters.

	Description	Value	Units	Reference
Sector I: O2 Kinetics
	Dilution: the fraction of O2 replaced by He during sampling	0.035	Unitless	[28]
	Solubility of O2 in water (20°C)	0.00139	mol L−1 atm−1	[37]
	The O2 transport coefficient between headspace and liquid	1.62	L vial−1 h−1	[28]
	O2 leakage into the vial during each sampling	2.04×10−8	mol vial−1	[28]
	The time taken to complete each sampling	0.017	h	[28]
	The half saturation constant for O2 consumption	2.5×10−7	mol L−1	Model-based estimation
	The maximum cell-specific velocity of O2 consumption	1.33×10−15	mol cell−1 h−1	[4], [8]
Sector II: Population dynamics of the cells without ( ) and with ( ) denitrification proteome
	[O2] in the liquid below which the recruitment to halts	1×10−9	mol L−1	Assumption
	[O2] below which the recruitment to triggers	9.75×10−6	mol L−1	[22]
	The specific-probability of recruitment of a cell to	0.0052	h−1	Model-based estimation
	The growth yield per molN	5.79×1013	cells molN−1	[4], [8]
	The growth yield per mol O2	15×1013	cells mol−1	[4], [8]
Sector III: Denitrification Kinetics
	The half saturation constant for reduction	4×10−6	molN L−1	[33], [34]
	The maximum cell-specific velocity of reduction	1.83×10−15	molN cell−1 h−1	[4], [8]
General
	Universal gas constant	0.083	L atm K−1 mol−1	–
	Temperature	293.1	K	[4], [8]
	Headspace volume	0.07	L vial−1	[4], [8]
	Liquid-phase volume	0.05	L vial−1	[4], [8]

In addition, changes in due to sampling are included at the reported sampling times. The robotised incubation system [28] used in the experiment monitors gas concentrations by sampling the headspace, where each sampling alters the concentrations in a predictable manner: a fraction of is removed and replaced by He (dilution), but the sampling also results in a marginal leakage of O₂ through the tubing and membranes of the injection system. Eq. 2 shows how the model calculates the net change in as a result of each sampling:

(2)

mol vial⁻¹ h⁻¹

where (mol vial⁻¹) is the O₂ leakage into the headspace, (dilution) is the fraction of replaced by He, and (h) is the time taken to complete each sampling. is negative if is greater than 0.58 µmol vial⁻¹ and marginally positive if it is less than that.

O₂ in the liquid-phase

(, mol vial⁻¹, see Fig. 4A) is initialised by assuming equilibrium with at the time of inoculation . is modelled as a function of its transport into the liquid (, Eq. 1) and consumption rate (, mol vial⁻¹ h⁻¹), where the latter is modelled as a function of total cell numbers and the cell-specific velocity of O₂ consumption:

(3)

mol vial⁻¹ h⁻¹

where and (cells vial⁻¹, see Sector II for details) are the cells without and with denitrification enzymes, respectively, and (mol cell⁻¹ h⁻¹) is the cell-specific velocity of O₂ consumption. Thus, we assume that the and cells have the same potential to consume O₂.

is modelled as a Michaelis-Menten function of O₂ concentration:

(4)

mol cell⁻¹ h⁻¹

where (mol cell⁻¹ h⁻¹) is the maximum cell-specific velocity of O₂ consumption (determined under the actual experimental conditions), (mol L⁻¹) is the O₂ concentration in the liquid-phase, and (mol L⁻¹) is the half saturation constant for O₂ reduction.

Sector II: Population dynamics of the cells without () and with () denitrification proteome

Fig. 4B represents the structure governing the population dynamics of and . Briefly, the figure shows that both the populations are able to grow by aerobic respiration ( and , respectively). Initially,  = 0 and is populated through recruitment () of the cells from the pool, where the recruitment is a product of and an [O₂] dependent specific-probability (h⁻¹) of the recruitment (, see Eqs. 7–8). The growth rate of is primarily based on denitrification (), but the cells that are recruited before O₂ is completely exhausted also grow by consuming the remaining traces of O₂. Below we present equations and a detailed explanation of the structural components shown for this sector.

The pool of the cells lacking denitrification proteome

The pool of the cells lacking denitrification proteome () is initialised with 3×10⁸ cells vial⁻¹. The population dynamics of are modelled as:

(5)

cells vial⁻¹ h⁻¹

where (cells vial⁻¹ h⁻¹) is the (aerobic) growth rate, and (cells vial⁻¹ h⁻¹, Eq. 7) is the rate of recruitment of to the pool.

is modelled as:

(6)

cells vial⁻¹ h⁻¹

where (mol cell⁻¹ h⁻¹, Eq. 4) is the cell-specific velocity of O₂ consumption, and (cells mol⁻¹) is the cell yield per mole of O₂ (determined under the actual experimental conditions).

The rate of recruitment

The rate of recruitment (, see Fig. 4B) of the cells from to is modelled as:

(7)

cells vial⁻¹ h⁻¹

where (h⁻¹) represents the conditional specific-probability for any cell to be recruited to denitrification, modelled as a function of O₂ concentration in the liquid-phase (, see Fig. 5):

(8)

Figure 5. Modelling of (h⁻¹) as a function of .

A. The panel shows the O₂ concentration in the liquid-phase falling as a result of aerobic respiration. B. The panel shows the probability for a cell to switch to denitrification (, h⁻¹) modelled as a function of . (Panels A & B) is the concentration below which is assumed to trigger (due to withdrawal of the transcriptional control of O₂ on denitrification [22]), whereas is assumed to be the concentration below which terminates (due to lack of energy for enzyme synthesis). The double-headed arrow (at the bottom of Panel A) illustrates the limited time-window () available for the cells to switch to denitrification.

h⁻¹

where (h⁻¹) is a constant representing the specific-probability of the recruitment, is the O₂ concentration above which the transcription of nirS is effectively suppressed by O₂, and is the O₂ concentration assumed to provide minimum energy for the initial transcription to result in functional NirS. Once the first molecules of NirS are produced while , the transcription of nirS will be greatly enhanced through positive feedback by NO, paving the way for a full-scale production of denitrification proteome [10] (See Introduction and Fig. 2 for details).

( = 9.75×10⁻⁶ mol L⁻¹) is the empirically determined at the outset of NO accumulation: Bergaust et al. [8] estimated between 0.1–12 µM, but recent Pa. denitrificans batch incubation data have provided a more precise estimate between 8.8–10.7 µM (average = 9.75 µM) [22].

As for , we lack empirical basis for determining the parameter value, but sensitivity of the model to this parameter was tested (See Results/Discussion). Our simulations were run with  = 1×10⁻⁹ mol L⁻¹, which would sustain an aerobic respiration rate equivalent to 0.4% of the empirically determined (assuming our estimated  = 2.5×10⁻⁷ mol L⁻¹, Table 2).

As modelled, the time-window for the recruitment to denitrification depends on the time taken to deplete from to (Fig. 5); for obvious reasons, the length of this time-window depends on the cell density.

The lag observed between the emergence of denitrification gene transcripts and the subsequent gas products is as short as 20 minutes [8], [22], which is insignificant in the sense that the estimations of and will not be affected by including it in the model. Therefore, the recruitment (Eq. 7) is modelled as an instantaneous event.

Calculation of : The fraction of the cells recruited to denitrification

is calculated based on the integral of the recruitment (Eq. 7):

(9)

Dimensionless

where (h⁻¹, see Eqs. 7–8 and Fig. 5) is the specific-probability for the recruitment of a cell to denitrification, is the time when [O₂] in the liquid falls below (the concentration below which triggers), and is the time when [O₂] in the liquid falls below (the concentration below which is assumed to be zero). Hence, effectively, expresses the probability for any cell to switch to denitrification within the time-frame .

The pool of the cells carrying denitrification proteome

The pool of the cells carrying denitrification proteome (, see Fig. 4B) is initialised with zero cells, and its population dynamics are modelled as:

(10)

cells vial⁻¹ h⁻¹

where (cells vial⁻¹ h⁻¹, Eq. 7) is the recruitment rate, (cells vial⁻¹ h⁻¹) the denitrification-based growth and (cells vial⁻¹ h⁻¹) the aerobic growth rate.

is modelled as:

(11)

cells vial⁻¹ h⁻¹

where (molN cell⁻¹ h⁻¹, see Eq. 15) is the cell-specific velocity of reduction, and (cells molN⁻¹) is the growth yield per molN of as the e⁻-acceptor (determined under the actual experimental conditions).

The cells are assumed to have the same ability as to grow by aerobic respiration; their aerobic growth rate is formulated as:

(12)

cells vial⁻¹ h⁻¹

where (mol cell⁻¹ h⁻¹, see Eq. 4) is the cell-specific velocity of O₂ consumption, and (cells mol⁻¹) is the growth yield per mole of O₂ as the e⁻-acceptor.

Sector III: Denitrification kinetics

The structure controlling the denitrification kinetics is mapped in Fig. 4C. Briefly, the figure shows that the cells with denitrification proteome () control the consumption rate of (), recovered as , in proportion to a cell-specific velocity of consumption (). The denitrification intermediates NO and N₂O are not explicitly modelled, as they accumulated to miniscule concentrations only [4], [8].

and

The pool (molN vial⁻¹) is initialised by measured initial concentrations (Table 1), and the pool is initialised with zero molN vial⁻¹. and N₂ kinetics are modelled as:

(13)

molN vial⁻¹ h⁻¹

where is the consumption rate of :

(14)

molN vial⁻¹ h⁻¹

where (cells vial⁻¹) represents the denitrifying cells, and (molN cell⁻¹ h⁻¹) is the cell-specific velocity of reduction, which is modelled as a function of using the Michaelis-Menten equation:

(15)

molN cell⁻¹ h⁻¹

where (molN cell⁻¹ h⁻¹) is the maximum cell-specific velocity of consumption (determined under the actual experimental conditions), (molN L⁻¹) is the concentration in the liquid-phase, and (molN L⁻¹) is the half saturation constant for reduction.

See Table 2 for a summary of the parametric values and their sources and Table 3 for the initial values assigned to the state variables.

Table 3. Initial values for the state variables.

	Symbol	Value	Units	Reference
Sector 1: O2 Kinetics
Initial O2 in the headspace		See Table 5	mol vial−1	[4], [8]
Initial O2 in the liquid-phase		Equilibrium with	mol vial−1	Assumption
Sector II: Population dynamics of the cells without ( ) and with ( )denitrification proteome
The initial number of cells		3×108	cells vial−1	[4], [8]
The initial number of denitrifying cells		0	cells vial−1	Assumption
Sector III: Denitrification Kinetics
Initial in the liquid-phase		See Table 5	molN vial−1	[4], [8]
Initial N2 in the headspace		0	molN vial−1	[4], [8]

Parameterisation

Most of the parameter values used in the model are well established in the literature (See Table 2). However, somewhat uncertain parameters include , , , and the assumed parameter :

Pa. denitrificans has three alternative terminal oxidases [30] with ranging from nM to µM [31], [32], so we decided to estimate by fitting our model to the data. Unfortunately, Bergaust et al.'s [4], [8] ∼0% O₂ treatments data, for which is relevant, has technical problems (needle clogging and/or high O₂ leakage during sampling). Therefore, we estimated ( = 2.5×10⁻⁷ mol L⁻¹) by aptly simulating our model against another ∼0% O₂ data-set produced by batch cultivations of Pa. denitrificans under similar experimental conditions [22].

is given in the literature as 4–5 µM [33], [34]. The model, however, does not show any considerable sensitivity to this parameter even within a range as wide as 0.1–10 µM because the simulated experiments were operating with much higher [].

( = 9.75×10⁻⁶ mol L⁻¹) is empirically determined as the at the outset of NO accumulation: Bergaust et al. [8] estimated between 0.1–12 µM, but recent batch incubation data from Pa. denitrificans have provided a more precise estimate in the range 8.8–10.7 µM (average = 9.75 µM) [22]. The model, however, is not sensitive to within the latter range because of a high velocity of O₂ depletion.

( = 1×10⁻⁹ mol L⁻¹) is assigned an arbitrary low value, since we lack any empirical estimation/data to support it. To compensate for the uncertainty, we conducted a sensitivity analysis exploring the consequences of increasing or decreasing by one order of magnitude (See Results/Discussion).

Results/Discussion

The specific-probability (, h⁻¹) of recruitment of a cell to denitrification

To test the assumption of a single homogeneous population, we forced our model to achieve 100% recruitment to denitrification by setting  = 1 h⁻¹. In consequence, the simulated N₂ accumulation (molN vial⁻¹) showed gross overestimation as compared to the measured for all the treatments (as illustrated for some randomly selected ones in Fig. 6).

Figure 6. Comparison of the measured [4], [8] and simulated data assuming  = 1 h⁻¹.

Assuming a single homogeneous population, as we forced our model to achieve 100% recruitment to denitrification by setting the specific-probability of recruitment () to 1 h⁻¹, the simulated N₂ accumulation (molN vial⁻¹) showed considerable overestimation as compared to that measured. To illustrate this, the simulated and measured data are compared here for some randomly chosen treatments. Initial vol.% O₂ in the headspace and initial is shown above each panel.

To find a more adequate value, was calibrated to produce the best possible match between the simulated and measured N₂ through optimisation. (The optimisation was carried out in Vensim DSS 6.2 Double Precision, http://vensim.com/). Table 4 presents the optimal for each treatment; no consistent effect of initial [O₂] and [] was found on the optimal results. The average for all the treatments = 0.0052, which appears to give reasonable fit between the simulated and measured N₂ (See Figs. 7, 8, and 9). This indicates that the simulations with  = 0.0052 should provide a reasonable approximation of (the fraction recruited to denitrification) during the actual experiment.

Table 4. Specific-probability of recruitment of a cell to denitrification () estimated for each batch culture by optimisation (best match between the simulated and measured N₂ kinetics).

Batch No.	Treatment*: (vol.%) (mM)	Optimal (h−1)
1	∼0, 0.2	0.0066
2	∼0, 1	0.0059
3	∼0, 2	0.0029
4	1, 0.2	0.0033
5	1, 1	0.0062
6	1, 2	0.0020
7	7, 0.2	0.0018
8	7, 1	0.0117
9	7, 2	0.0066
		Avg. = 0.0052

*Treatment refers to the initial concentration of O₂ in the headspace (measured as headspace vol.%) and the initial concentration of in the medium (mM).

Figure 7. Simulations of the treatments with ∼0 vol.% using  = 0.0052 h⁻¹.

The figure compares the measured and simulated O₂ depletion (mol vial⁻¹) and N₂ accumulation (molN vial⁻¹) for the ∼0 vol.% O₂ treatments of Bergaust et al. [4], [8], i.e., the vials with near-zero O₂ in the headspace () at the time of inoculation. Separate plots are shown for each initial concentration of (0.2, 1, and 2 mM). The measured initial O₂ was somewhat erratic due to episodes of needle clogging and/or high O₂ leakage during sampling, so the initial used in the simulations is chosen somewhat ad lib so that the simulated O₂ depletion coincides with that measured. The discrepancy compared to the measured O₂ seems to be significant for 2 mM treatment. That is most likely due to the inhibitory effect of nitrite on aerobic respiration, which is not taken into account; all simulations are run with an identical . Near exhaustion, the simulated O₂ increases slightly at each sampling time; that is due to the leakage of O₂ via the injection system exceeding dilution of the headspace (with He) during each sampling.

Figure 8. Simulations of the treatments with 1 vol.% using  = 0.0052 h⁻¹.

The figure compares the measured and simulated O₂ depletion (mol vial⁻¹) and N₂ accumulation (molN vial⁻¹) for the treatments with 1 vol.% O₂ in the headspace () at the time of inoculation; separate plots are shown for each initial concentration of (0.2, 1, and 2 mM). At each sampling time, the simulated O₂ is visibly reduced; that is because sampling implies 3.4% dilution of the headspace (with He). This contrasts with the simulations of the treatments with low O₂ (Fig. 7), where the leakage of O₂ into the system is more dominant.

Figure 9. Simulations of the treatments with 7 vol.% using  = 0.0052 h⁻¹.

The figure compares the measured and simulated O₂ depletion (mol vial⁻¹) and N₂ production (molN vial⁻¹) for the treatments with 7 vol.% O₂ in the headspace () at the time of inoculation; separate plots are shown for each initial concentration of nitrite (0.2, 1, and 2 mM). At each sampling time, the simulated O₂ is visibly reduced because of sampling, which results in 3.4% dilution of the headspace (with He).

Sensitivity analysis

(the O₂ concentration below which the recruitment is arrested) was arbitrarily chosen to be 1×10⁻⁹ mol L⁻¹. In order to evaluate the sensitivity of the model to this parameter, we tested the model performance by increasing and decreasing by one order of magnitude. For each parameter value, we estimated for the individual vials by optimisation (as outlined in the foregoing paragraph). A good fit was obtained for both the values, but the optimisation resulted in slightly different values. Increasing by a factor of 10 (to 1×10⁻⁸ mol L⁻¹) resulted in 18–38% higher estimates (average = 28% ±stdev 10). Decreasing by a factor of 0.1 (to 1×10⁻¹⁰ mol L⁻¹) resulted in 5–17% lower estimates (average = 11% ±stdev 6).

The fraction recruited to denitrification ()

A refined estimation with the presented model

Bergaust et al. [8], [16] and Nadeem et al. [9] used data from batch cultivations of Pa. denitrificans, as illustrated in Fig. 1, to assess . Their estimation was effectively , where is the time when O₂ is exhausted, (cells vial⁻¹) is the number of actively denitrifying cells estimated by the measured rate of denitrification (molN h⁻¹) divided by the cell-specific denitrification (molN cell⁻¹ h⁻¹), and N is the total number of cells estimated on the basis of O₂ consumption. Although this equation indisputably estimates the fraction of the cells that was actively denitrifying at the time , it is a biased estimate of the ‘true’ because the number of cells does not remain constant through the recruitment phase: (the cells without denitrification enzymes) and will both grow until O₂ is depleted, but will grow faster because their growth is supported by both O₂ and NO_x. As a result, the estimation of by this equation might be too high. Besides, the experimental estimation is prone to error because of infrequent sampling, since the sampling time does not necessarily coincide with .

In contrast, our model directly and more precisely calculates (Eq. 9) by a) explicitly simulating the actual kinetics of the recruitment of the cells to denitrification (in contrast to estimating total and denitrifying cell numbers from gas kinetics) and b) avoiding aerobic and anaerobic growth of the cells. Table 5 shows the model's estimations of and the time-span of the recruitment () along with the estimations of Bergaust et al [8], [16].

Table 5. The model's and Bergaust et al.'s [16] estimations of the fraction recruited to denitrification ().

Batch No.	(vol.%) (mM)	(µmol)*	Model-based Estimations		Estimations of [16]
			**
1	0, 0.2	2	25.8	0.141	0.19
2	0, 1	1.5	29.2	0.161	0.21
3	0, 2	1.7	27.2	0.156	0.19
4	1, 0.2	50.1	10.1	0.052	0.03
5	1, 1	37.8	11.1	0.056	0.07
6	1, 2	38.4	11.3	0.057	0.04
7	7, 0.2	199	7.4	0.038	0.02
8	7, 1	200	7.4	0.038	0.07
9	7, 2	200	7.4	0.038	0.08
				Avg. = 0.082	Avg. = 0.1

*Refers to the initial values of O₂ in the headspace () used in the simulations. The values show some inconsistency for the treatments corresponding to the same vol.% because of traces of O₂ left behind after He-washing.

** is the time when [O₂] in the liquid falls below ( = 9.75 µM [22], the concentration below which recruitment of the cells to denitrification is assumed to trigger), and is the time when [O₂] in the liquid falls below ( = 1 nM, a practically zero concentration below which the recruitment is assumed to terminate). Due to low cell density in the ∼0% O₂ treatments, the O₂ leakage into the vial during sampling (every 3 hours) caused oxygen concentration to exceed for 0.1–2.4 hours. This resulted in various recruitment spikes after the initial O₂ was depleted. If such recruitment is omitted,  = 0.126, 0.142, and 0.133 for the treatments 1, 2, and 3, respectively.

In the ∼0% O₂ treatments, is supported by the sampling leaks of O₂

Due to low cell density in the ∼0% O₂ treatments (initial O₂ = 1.5–2 µmol), the O₂ leakage into the vial during sampling (every 3 hours) caused oxygen concentrations to exceed for 0.1–2.4 hours. This resulted in various spikes of recruitment after the initial O₂ was depleted. The recruitment through these spikes amounted to, on average, ∼19% of in the ∼0% O₂ treatments.

<<100%

The model's estimations of (Table 5) corroborate the suggestion of Bergaust et al. [8], [16] and Nadeem et al. [9] that in batch cultures of Pa. denitrificans remains far below 100%. According to Bergaust et al. [8], [16], was 2–21% (average = 10%), whereas the model estimated it between 3.8–16.1% (average = 8.2%).

is inversely related to cell density

Bergaust et al. [16] argued that as the velocity of O₂ depletion is proportional to cell density, the time-frame available for the cells to produce (necessary initial) denitrification proteome would be inversely related to the cell density at the time of O₂ depletion. Simulation results (Table 5) support this: high initial O₂ concentrations resulted in high cell densities at the time of O₂ depletion, shortening the time-span for the recruitment to denitrification, hence resulting in the low .

Underlying cause of the low

remains low because of a) the limited time-window available to the cells for the recruitment and b) the low (specific-probability of the recruitment), presumably due to a low probability of initiating nirS transcription (subsequently reinforced through positive feedback by NO).

Simulation of the ‘diauxic lag’

To investigate whether the recruitment of a small fraction of the cells to denitrification could explain the ‘diauxic lag’ observed by Liu et al. [24], we used our model to simulate the conditions they reported for their experiment. In short, Liu et al. [24] incubated Ps. denitrificans (ATCC 13867) in oxic batch cultures, which were sparged with N₂ as the cultures had reached different cell densities (OD₅₅₀ = 0.05–0.17). The sparging resulted in apparent diauxic lags, i.e., periods with little or no detectable growth. The length of such lags increased with the cell density present at the time of sparging.

Structural amendments and parameterisation of the model

To tentatively simulate their experiment, two changes were made in the O₂ kinetics sector (Fig. 4A). Firstly, the net sampling loss of () was omitted, since it was specifically set up for the robotised incubation system [28] used by Bergaust et al [4], [8]. Secondly, a sparging event was introduced, which immediately takes down to very low levels ( = 1×10⁻⁹ mol vial⁻¹). Since we lack information about the exact concentration of O₂ immediately after the sparging, the present exercise is only qualitative.

Liu et al. [24] inoculated the culture to have an initial OD₅₅₀ = 0.07, which would correspond to ∼6.5×10⁹ cells vial⁻¹ [4], [8]. We used this number to initialise the pool (shown in Fig. 4B). They used ( = 157 µmolN vial⁻¹) instead of , so we replaced the pool (Fig. 4C) by the pool, initialised it accordingly, and adjusted Eqs. 11 and 15: In Eq. 11, was replaced with the cell yield per molN of as the e⁻-acceptor ( = 9.65×10¹³ cells molN⁻¹ [4], [8]). In Eq. 15, was replaced with the maximum cell-specific velocity of consumption ( = 2×10⁻¹⁵ molN cell⁻¹ h⁻¹), calculated using the maximum specific NO_x-based growth rate ( = 0.322 h⁻¹) reported for their experiment. Finally, in Eq. 4, was calibrated ( = 2.28×10⁻¹⁵ mol cell⁻¹ h⁻¹) with the reported maximum specific aerobic growth rate ( = 0.342 h⁻¹).

The ‘diauxic lag’ is plausibly the initial growth phase of a minute (fraction recruited to denitrification)

As the experiment of Liu et al. [24] was simulated with the model's estimated  = 0.0052 h⁻¹ (specific-probability of recruitment), turned out to be 1.1% for the treatment sparged at h = 1.1 and 0.2% for the one sparged at h = 2.55. Simulations of the total cell density () for these cases(Fig. 10A) showed long apparent lags comparable to 10–30 h lag phases observed in their later experiments [25]. However, lags in the range that Liu et al. [24] observed ( = 3 and 6 h for sparging at h = 1.1 and 2.55, respectively) could be achieved by our model by assuming higher residual O₂ concentrations after sparging (resulting in a higher ). Fig. 10B isolates the OD of for the simulated treatments and shows them on a logarithmic scale so that their exponential growth, right from the onset of anoxic conditions, becomes apparent. The figure initially shows a quick recruitment of the cells from the to the pool, followed by the exponential growth-phase of .

Figure 10. Simulation of the ‘diauxic lags’ observed by Liu et al [24].

A. The panel shows cumulated OD (optical density) of the cells without () and with () denitrification enzymes for the simulated experiment of Liu et al. [24], where one treatment was sparged at time = 2.55 h and the other at 1.1 h. The simulations show, qualitatively, similar ‘lags’ in the two ODs as observed by the experimenters. These apparent lags are due to exponential growth of a minute fraction of the cells that successfully switched to denitrification. The growth of this fraction remains practically undetectable (the “lag” phase) until it reaches a level comparable to the large population trapped in anoxia. B. This panel isolates the ODs of and show them on a logarithmic scale so that the exponential growth of , right from the onset of anoxic conditions, becomes visible. The graph initially shows a quick recruitment of the cells from the to the pool, followed by the exponential growth-phase.

This exercise serves to illustrate that the ‘diauxic lags’ observed [24]–[26] may simply be a result of low recruitment to denitrification in response to sudden removal of O₂. This is possibly a more plausible explanation than suggested by the authors and further elaborated by Hamilton et al. [35], claiming that there is a true lag caused by extremely slow production of denitrification enzymes due to energy limitation. Our explanation of the apparent diauxic lag is corroborated by a chemostat culturing experiment conducted by Bauman et al [36]: A steady state carbon (acetate) limited continuous culture with Pa. denitrificans was made anoxic and monitored for denitrification gene transcription, N-gas production, and acetate concentrations. A transient (8–10 h) peak of acetate accumulation after O₂ depletion suggested an apparent diauxic lag in the metabolic activity, but denitrification started immediately and increased gradually throughout the entire ‘lag’ period. They further observed that the number of denitrification gene transcripts peaked sharply during the first 1–2 hours. These observations are in good agreement with our model.

The aforestated observation of Liu et al. [24] that the length of the apparent lags increased with the aeration period (or the cell density at the time of sparging) is also in agreement with our model demonstrating that the time available for the cells to switch to denitrification is inversely related to the cell density at the time of O₂ depletion.

Model-based hypothesis: Initial O₂ determines the timespan to denitrify all to N₂ in a batch

Two sensitivity analyses were run to investigate the system's response to initial O₂ in the headspace, : one corresponding to a range of initial [O₂] in the liquid-phase below (see Eqs. 7–8) and the other for a range much higher than . All other model parameters and initial values remained as listed in Tables 2 and 3, respectively. The exercise helps illustrate the relative importance of aerobic growth versus the recruitment () in determining the time taken to deplete the pool.

Sensitivity analysis (1)

Sensitivity analysis (1) was run with three within a very low range, starting from a concentration marginally below :

1.  = 2.02×10⁻⁵ mol vial⁻¹ ,

2.  = 1.01×10⁻⁵ mol vial⁻¹ ,

3.  = 5.04×10⁻⁶ mol vial⁻¹

This is rather a simple case demonstrating that increasing within this low range (Fig. 11A) will result in increasing rates of denitrification (Fig. 11D) by increasing the number of aerobically grown cells (, Fig. 11B) and, thus, the rate of recruitment (, Fig. 11C).

Figure 11. Sensitivity analysis (1): Varying initial O₂ in the headspace within a low range.

The figure shows the impact of varying within a low range on: A. O₂ concentration in the liquid-phase , B. The number of aerobically growing cells (), which do not possess denitrification enzymes, C. The rate of recruitment of to denitrification (), and D. N₂ accumulation. Marked in Panel A, is the below which triggers, and is the below which terminates. In Panel C, the spikes of recruitment (following the initial recruitment) are due to spikes of O₂ by sampling, causing to transiently exceed . The model predicts that reducing within a low range (Panel A) will lower the number of aerobically grown cells (Panel B) and, thereby, the recruitment rate (Panel C), thus increasing the time taken to deplete (slower N₂ accumulation, Panel D).

Sensitivity analysis (2)

Sensitivity analysis (2) was run with three initial O₂ concentrations much higher than :

1.  = 2×10⁻⁴ mol vial⁻¹ ,

2.  = 1.19×10⁻⁴ mol vial⁻¹ ,

3.  = 3.84×10⁻⁵ mol vial⁻¹

In this case, the cumulated N₂ reached stable plateaus at nearly the same time for all the runs (Fig. 12E), despite that the time taken to deplete O₂ below decreased with increasing (Fig. 12A), reducing the time available to the cells for switching to denitrification (See Fig. 5). Thus, once denitrification was initiated, the rates increased with increasing due to an increasing population of oxygen-grown cells (Fig. 12B–D). (Eq. 9) declined with increasing ( = 0.058, 0.041 and 0.028 for runs 3, 2 and 1, respectively), but this was not sufficient to compensate for the increasing number of oxygen-raised cells.

Figure 12. Sensitivity analysis (2): Varying initial O₂ in the headspace (()) within a high range.

The figure shows the impact of varying within a range much higher than (the [O₂] below which recruitment of the cells to denitrification is assumed to trigger) on: A. O₂ concentration in the liquid-phase , B. The number of aerobically growing cells (), which do not possess denitrification enzymes, C. The rate of recruitment of to denitrification (), D. The number of cells as a result of the recruitment alone (), i.e., the denitrifying cells () but without aerobic and NO_x-based growth, and E. Cumulated N₂. The cumulated N₂ reached stable plateaus at nearly the same time for all the runs (Panel E), despite the fact that the time taken to deplete O₂ below decreased with increasing (Panel A). Thus, once denitrification was initiated, the rates increased with increasing initial due to an increasing population of oxygen-grown cells (Panels B–D). The fraction of the cells recruited to denitrification () declined with increasing initial O₂ concentration (not shown), but this was not sufficient to compensate for the increasing number of oxygen-raised cells.

If the model is run without any initial O₂, there would be no recruitment and, hence, no denitrification. Verification of this in batch cultures is difficult because traces of O₂ remain after He-washing of the batches. However, we (Bergaust et al., unpublished data) have been able to demonstrate that the aerobically grown Pa. denitrificans cells are indeed entrapped in anoxia if transferred to anoxic conditions as instantaneously as in the experiments conducted by Højberg et al. [15].

Conclusion

The prevailing wisdom in denitrification research is that, under impending anoxic conditions, all cells in a batch culture of denitrifying bacteria will switch to denitrification. However, recent experiments with batch cultures of Pa. denitrificans have provided evidence that, in response to O₂ depletion, only a small fraction () of the entire population is able to switch to denitrification [4], [8], [9]. The evidence is based on indirect analyses of e⁻-flow rates to O₂ and NO_x during the transition of the cells from aerobic to anaerobic respiration. To provide a direct and refined estimation of , we constructed a dynamic model and directly simulated kinetics of recruitment of the cells to denitrification. We first formulated a hypothesis as to the underlying regulatory mechanism of cell differentiation under approaching anoxia. Briefly, it is that the low is due to a low probability of initiating transcription of the nirS genes, but once initiated, the transcription is greatly enhanced through autocatalytic positive feedback by NO, resulting in the recruitment of the transcribing cell to denitrification. Then, as we implemented this hypothesis in the model, the simulation results showed that the specific-probability () of 0.0052 (h⁻¹) for a cell to switch to denitrification is sufficient to robustly simulate the measured denitrification gas kinetics. The model estimated the resultant between 3.8–16.1% only (average = 8.2%). The phenomenon may be considered as a ‘bet-hedging’ regulation ‘strategy’ [12]: the fraction switching to denitrification benefits if the anoxic spell is long and NO_x remains available, whereas the non-switching fraction benefits, by saving energy required for the protein synthesis, if the anoxic spell is short. The strategy has important implications for the interpretation of numerous experiments on Pa. denitrificans and other denitrifying organisms, as this study has illustrated by presenting a more plausible explanation of the apparent diauxic lags [24] on the basis of the low .

Supporting Information

Dataset S1

contains a Vensim simulation model (Hassan_et_al_2014.mdl) used in this study along with two files (7%_Oxygen_2mM_Nitrite.vdf and Measured_Data) containing simulated and measured data, respectively.

(ZIP)

Data Availability

The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and its Supporting Information files.

Funding Statement

This work is funded by Norwegian University of Life Sciences, financed by the Ministry of Education and Research, Norway. No institution has, in any way, influenced the outcome of this work. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.