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

In response to impending anoxic conditions, denitrifying bacteria sustain respiratory metabolism by producing enzymes for reducing nitrogen oxyanions/-oxides (NOx) 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 NOx. 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’.


Introduction
A complete denitrification pathway includes the dissimilatory reduction of nitrate (NO { 3 ) through nitrite (NO { 2 ), nitric oxide (NO), and nitrous oxide (N 2 O) to di-nitrogen (N 2 ). Typically, the genes encoding reductases for these nitrogen oxyanions/-oxides (NO x ) are not expressed constitutively but only in response to O 2 depletion, making denitrification a facultative trait [1]. Hence, during anoxic spells, the process enables denitrifying bacteria to sustain respiratory metabolism, replacing O 2 by NO x as the terminal electron (e 2 ) acceptors. Since permanently anoxic environments lack available NO x , denitrification is confined to sites where O 2 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 N 2 O [2]. A plethora of biogeochemical models have been developed for understanding the ecosystem controls of denitrification and N 2 O 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 O 2 and NO { 3 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 N 2 O 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 2 -flow as Paracoccus denitrificans switch from aerobic to anaerobic respiration [4,8].
Although denitrification is widespread among bacteria, the aproteobacterium 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 O 2 depletion, only a marginal fraction (F den ) of its entire population appears to successfully switch to denitrification. In these studies, however, F den is inferred from rates of consumption and production of gases (O 2 , NO x , and N 2 ), and a clear hypothesis as to the underlying cause of the low F den is also lacking. To fill these gaps, we formulated a refined hypothesis addressing the underlying regulatory mechanism of the cell differentiation in response to O 2 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 F den . 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 O 2 . A population undertaking denitrification typically responds to full aeration by completely shutting down denitrification and immediately initiating aerobic respiration [10]. Thus, O 2 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 2 -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 O 2 should become limiting later on. However, these enzymes will be diluted by aerobic growth, since the transcription of their genes is effectively inactivated by O 2 . 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 O 2 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 O 2 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 2 -acceptors. When [O 2 ] falls below some critical threshold, the cells will 'sense' this and start synthesising denitrification proteome, utilising energy from aerobic respiration [10]. However, if O 2 is suddenly exhausted or removed, the lack of a terminal e 2 -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 O 2 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 O 2 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 2 -flow rate (i.e., to O 2 +NO x , see Fig. 1), which was estimated on the basis of measured gas kinetics. Had all of the cells switched to denitrification as O 2 exhausted, the total e 2 -flow rate would have carried on increasing, without such a depression. The depression was followed by an exponential increase in the e 2 -flow rate, which was tentatively ascribed to anaerobic growth of a small F den (fraction recruited to denitrification). It was postulated that this fraction escaped entrapment in

Author Summary
In response to oxygen-limiting conditions, denitrifying bacteria produce a set of enzymes to convert NO { 3 /NO { 2 to N 2 via NO and N 2 O. 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 O 2 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. 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 cd 1 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 O 2 (FnrP), NO { 3 /NO { 2 (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 [O 2 ] is sensed by FnrP, which in interplay with NarR induces nar transcription. NarR is activated by NO { 2 (and/or probably by ; thus once a cell starts producing traces of NO { 2 , 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 NO { 3 but various concentrations of NO { 2 only (Table 1). Low probability of initiating nirS transcription. The transcription of nirS is known to be suppressed by O 2 [4,8], but the exact mechanism remains unclear. Circumstantial evidence suggests that it is due to O 2 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 O 2 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), 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 2 -flow rate (i.e., to O 2 +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 2 -flow would have carried on increasing without such a depression. The depression was followed by an exponential increase in the e 2 -flow rate, which was ascribed to anaerobic growth of a small fraction (F den ) of the cells that escaped entrapment in anoxia and carried on growing by denitrification. doi:10.1371/journal.pcbi.1003933.g001 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 (N 2 O reductase encoded by nosZ). The transcription of these genes is regulated by, at least, three FNR-type proteins, which are sensors for O 2 (FnrP), NO { 3 /NO { 2 (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 O 2 inactivates NNR (grey dashed link) [20], and NirS is also unlikely to be functional in the presence of high O 2 concentrations. Hence, for our modelling we hypothesise that the probability of an autocatalytic transcriptional activation of nirS is zero until O 2 falls below a critical concentration O 2 ½ trigger . When O 2 falls below O 2 ½ trigger , 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 NO { 2 -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 O 2 O 2 ½ min À Á is available because the production of the first molecules of NirS will depend on energy from aerobic respiration. doi:10.1371/journal.pcbi.1003933.g002 the ultimate consequence is the elimination of the positive feedback via NO and NNR. When O 2 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 NO { 2 -supplemented medium contains non-biologically formed traces of NO which, once diffused into the cells while O 2 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 O 2 concentration is above a certain threshold. This assumption is backed by empirical data indicating that NO is not produced to detectable levels before O 2 concentration falls below a critical threshold [8,22]. For O 2 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.
O 2 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 O 2 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 10 9 cells mL 21 (roughly the numbers that we are dealing with) is ,10 mm, 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 (N Dz ) and the other without (N D{ ); both equally consume O 2 ½ min (assuming the lack of energy for protein synthesis).
The recruitment of N D{ to N Dz 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][25][26], oxic cultures were sparged with N 2 to remove O 2 and were monitored by measuring optical density (OD 550 ). 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 O 2 , 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.  [4,8]. 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 N 2 ) would leave the medium with only traces of O 2 , 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 (F den ) 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 O 2 after sparging, which was not measured).

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 20uC) 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 KNO 3 or KNO 2 . Prior to inoculation, air in the headspace was replaced with He to remove O 2 and N 2 (He-washing), followed by the injection of no, 1, or 7 headspace-vol.% O 2 . Finally, each vial was inoculated with ,3610 8 aerobically grown cells. . 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 O 2 and NO { 2 (g or aq with a chemical species-name represents gaseous or aqueous, respectively). O 2 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 NO { 2 to N 2 via the intermediates NO and N 2 O (not shown). For monitoring O 2 , CO 2 , N 2 , NO and N 2 O, 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 O 2 and N 2 into the vial (as indicated by the two-way arrows at the top of the figure). 2 ). 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. O 2 diffused from the headspace to the liquid (Fig. 3), where the cells consumed it before switching to denitrification: the stepwise reduction of NO { 2 to N 2 via the intermediates NO and N 2 O (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 O 2 consumption until depletion, followed by transition to denitrification. The denitrification rate increased exponentially till all the NO { 2 present in the medium was recovered as N 2 . The medium contained ample amounts of carbon substrate (34 mM succinate) to support the consumption of all available electron acceptors.
Sampling procedure. To monitor O 2 , CO 2 , NO, N 2 O, and N 2 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 O 2 and N 2 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 (20uC), assuming equilibrium between the headspace and the liquid. However, the O 2 consumption rate was so high that to calculate [O 2 ] 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 O 2 kinetics at the reported sampling times. Transport of O 2 between the headspace and the liquid is modelled using an empirically determined transport coefficient and the solubility of O 2 in water at 20uC. To simulate the metabolic activity (O 2 consumption and N 2 production) and growth, the model divides the cells into two sub-populations: one without and the other with denitrification enzymes (N D{ and N Dz pools, respectively, see Fig. 3). Both equally consume O 2 if present, but N D{ cannot reduce NO { 2 to N 2 . Those N D{ cells that, in response to O 2 depletion, are able to initiate nirS transcription (see Fig. 2) are recruited to the N Dz pool, where N Dz = 0 prior to the recruitment. The recruitment rate (R rec ) is modelled according to a probabilistic function described below (Eqs. 7-8).
The model ignores sampling effect on N 2 (leakage and loss), thus calculating the cumulative N 2 production as if no sampling took place. That is because the experimentally determined N 2 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. O 2 kinetics, II. Population dynamics of N D{ and N Dz , and III. Denitrification kinetics (Fig. 4).

Sector I: O 2 kinetics
Structural-basis for the O 2 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 (  (Table 1) and modelled as a function of transport (Tr O2 ) between the headspace and the liquid [28]: Units: mol vial 21 h 21 where k t (L vial 21 h 21 ) is the empirically determined coefficient for the transport of O 2 between the headspace and the liquid (See Table 2  In addition, changes in O 2HS 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 O 2HS is removed and replaced by He (dilution), but the sampling also results in a marginal leakage of O 2 through the tubing and membranes of the injection system. Eq. 2 shows how the model calculates the net change in O 2HS DO 2 S ð Þ À Á as a result of each sampling:    where Gr D{ (cells vial 21 h 21 ) is the (aerobic) growth rate, and R rec (cells vial 21 h 21 , Eq. 7) is the rate of recruitment of N D{ to the N Dz pool.
Gr D{ is modelled as: The rate of recruitment. The rate of recruitment (R rec , see h 21 where r den (h 21 ) is a constant representing the specificprobability of the recruitment, O 2 ½ trigger is the O 2 concentration above which the transcription of nirS is effectively suppressed by O 2 , and O 2 ½ min is the O 2 concentration assumed to provide minimum energy for the initial transcription to result in functional NirS. Once the first molecules of NirS are produced while ½ trigger , 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).  As for O 2 ½ min , 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 O 2 ½ min = 1610 29 mol L 21 , which would sustain an aerobic respiration rate equivalent to 0.4% of the empirically determined v max O2 ð Þ (assuming our estimated K m O2 ð Þ = 2.5610 27 mol L 21 , Table 2). As modelled, the time-window for the recruitment to denitrification depends on the time taken to deplete O 2 ½ LP from O 2 ½ trigger to O 2 ½ min (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 r den and F den will not be affected by including it in the model. Therefore, the recruitment (Eq. 7) is modelled as an instantaneous event.
Calculation of F den : The fraction of the cells recruited to denitrification. F den is calculated based on the integral of the recruitment (Eq. 7): Dimensionless where r den (h 21 , see Eqs. 7-8 and Fig. 5) is the specificprobability for the recruitment of a cell to denitrification, t t is the time when [O 2 ] in the liquid falls below O 2 ½ trigger (the concentration below which r den triggers), and t m is the time when [O 2 ] in the liquid falls below O 2 ½ min (the concentration below which r den is assumed to be zero). Hence, effectively, F den expresses the probability for any cell to switch to denitrification within the time-frame t m {t t .
The pool of the cells carrying denitrification proteome. The pool of the cells carrying denitrification proteome (N Dz , see Fig. 4B) is initialised with zero cells, and its population dynamics are modelled as:  The N Dz cells are assumed to have the same ability as N D{ to grow by aerobic respiration; their aerobic growth rate is formulated as:

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 (N Dz ) control the consumption rate of . The denitrification intermediates NO and N 2 O are not explicitly modelled, as they accumulated to miniscule concentrations only [4,8].   Table 2 for a summary of the parametric values and their sources and Table 3 for the initial values assigned to the state variables.

Parameterisation
Most of the parameter values used in the model are well established in the literature (See Table 2  Sector II: Population dynamics of the cells without (N D{ ) and with (N Dz )denitrification proteome The initial number of cells N D{ t 0 ð Þ 3610 8 cells vial 21 [4,8] The initial number of denitrifying cells N Dz t 0 ð Þ 0 cells vial 21 Assumption Sector III: Denitrification Kinetics Table 5 molN vial 21 [4,8] Initial N 2 in the headspace N 2 t 0 ð Þ 0 molN vial 21 [4,8] doi:10.1371/journal.pcbi.1003933.t003 estimated O 2 ½ trigger between 0.1-12 mM, but recent batch incubation data from Pa. denitrificans have provided a more precise estimate in the range 8.8-10.7 mM (average = 9.75 mM) [22]. The model, however, is not sensitive to O 2 ½ trigger within the latter range because of a high velocity of O 2 depletion. O 2 ½ min ( = 1610 29 mol L 21 ) 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 O 2 ½ min by one order of magnitude (See Results/Discussion).

Results/Discussion
The specific-probability (r den , h 21 ) 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 r den = 1 h 21 . In consequence, the simulated N 2 accumulation (molN vial 21 ) showed gross overestimation as compared to the measured for all the treatments (as illustrated for some randomly selected ones in Fig. 6).
To find a more adequate value, r den was calibrated to produce the best possible match between the simulated and measured N 2 through optimisation. (The optimisation was carried out in Vensim DSS 6.2 Double Precision, http://vensim.com/). Table 4 presents the optimal r den for each treatment; no consistent effect of initial [O 2 ] and [NO { 2 ] 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 N 2 (See Figs. 7, 8, and 9). This indicates that the simulations with r den = 0.0052 should provide a reasonable approximation of F den (the fraction recruited to denitrification) during the actual experiment.
Sensitivity analysis. O 2 ½ min (the O 2 concentration below which the recruitment is arrested) was arbitrarily chosen to be 1610 29 mol L 21 . In order to evaluate the sensitivity of the model to this parameter, we tested the model performance by increasing and decreasing O 2 ½ min by one order of magnitude. For each parameter value, we estimated r den for the individual vials by optimisation (as outlined in the foregoing paragraph). A good fit was obtained for both the O 2 ½ min values, but the optimisation resulted in slightly different r den values. Increasing O 2 ½ min by a Figure 6. Comparison of the measured [4,8] and simulated data assuming r den = 1 h 21 . Assuming a single homogeneous population, as we forced our model to achieve 100% recruitment to denitrification by setting the specific-probability of recruitment (r den ) to 1 h 21 , the simulated N 2 accumulation (molN vial 21 ) 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.% O 2 in the headspace and initial NO { 2 is shown above each panel. doi:10.1371/journal.pcbi.1003933.g006 factor of 10 (to 1610 28 mol L 21 ) resulted in 18-38% higher r den estimates (average = 28% 6stdev 10). Decreasing O 2 ½ min by a factor of 0.1 (to 1610 210 mol L 21 ) resulted in 5-17% lower r den estimates (average = 11% 6stdev 6).

The fraction recruited to denitrification (F den )
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 F den . Their estimation was effectively F den~N Dz tex ð Þ N tex ð Þ , where t ex is the time when O 2 is exhausted, N Dz (cells vial 21 ) is the number of actively denitrifying cells estimated by the measured rate of denitrification (molN h 21 ) divided by the cell-specific denitrification (molN cell 21 h 21 ), and N is the total number of cells estimated on the basis of O 2 consumption. Although this equation indisputably estimates the fraction of the cells that was actively denitrifying at the time t ex , it is a biased estimate of the 'true' F den because the number of cells does not remain constant through the recruitment phase: N D{ (the cells without denitrification enzymes) and N Dz will both grow until O 2 is depleted, but N Dz will grow faster because their growth is supported by both O 2 and NO x . As a result, the estimation of F den by this equation might be too high. Table 4. Specific-probability of recruitment of a cell to denitrification (r den ) estimated for each batch culture by optimisation (best match between the simulated and measured N 2 kinetics).  Besides, the experimental estimation is prone to error because of infrequent sampling, since the sampling time does not necessarily coincide with t ex .
In contrast, our model directly and more precisely calculates F den (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 F den and the time-span of the recruitment (t m {t t ) along with the F den estimations of Bergaust et al [8,16].
In the ,0% O 2 treatments, F den is supported by the sampling leaks of O 2 . Due to low cell density in the ,0% O 2 treatments (initial O 2 = 1.5-2 mmol), the O 2 leakage into the vial during sampling (every 3 hours) caused oxygen concentrations to exceed O 2 ½ min for 0.1-2.4 hours. This resulted in various spikes of recruitment after the initial O 2 was depleted. The recruitment through these spikes amounted to, on average, ,19% of F den in the ,0% O 2 treatments. F den ,,100%. The model's estimations of F den (Table 5) corroborate the suggestion of Bergaust et al. [8,16] and Nadeem et al. [9] that in batch cultures of Pa. denitrificans F den remains far below 100%. According to Bergaust et al. [8,16], F den was 2-21% (average = 10%), whereas the model estimated it between 3.8-16.1% (average = 8.2%).
F den is inversely related to cell density. Bergaust et al. [16] argued that as the velocity of O 2 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 O 2 depletion. Simulation results (Table 5) support this: high initial O 2 concentrations resulted in high cell densities at the time of O 2 depletion, shortening the time-span for the recruitment to denitrification, hence resulting in the low F den .
Underlying cause of the low F den . F den remains low because of a) the limited time-window available to the cells for the recruitment and b) the low r den (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 N 2 as the cultures had reached different cell densities (OD 550 = 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 O 2 kinetics sector (Fig. 4A). Firstly, the net sampling loss of O 2HS (DO 2 S ð Þ ) 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 O 2HS down to very low levels ( = 1610 29 mol vial 21 ). Since we lack information about the exact concentration of O 2 immediately after the sparging, the present exercise is only qualitative.
Liu et al. [24] inoculated the culture to have an initial OD 550 = 0.07, which would correspond to ,6.5610 9 cells vial 21 [4,8]. We used this number to initialise the N D{ pool (shown in Fig. 4B The 'diauxic lag' is plausibly the initial growth phase of a minute F den (fraction recruited to denitrification). As the experiment of Liu et al. [24] was simulated with the model's estimated r den = 0.0052 h 21 (specific-probability of recruitment), F den 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 (N D{ zN Dz ) 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 O 2 concentrations after sparging (resulting in a higher F den ). Fig. 10B isolates the OD of N Dz 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 N D{ to the N Dz pool, followed by the exponential growth-phase of N Dz .
This exercise serves to illustrate that the 'diauxic lags' observed [24][25][26] may simply be a result of low recruitment to denitrification in response to sudden removal of O 2 . 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 O 2 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 O 2 depletion.
Model-based hypothesis: Initial O 2 determines the timespan to denitrify all NO { 2 to N 2 in a batch Two sensitivity analyses were run to investigate the system's response to initial O 2 in the headspace, O 2HS t 0 ð Þ: one corresponding 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 (N D{ ) and with (N Dz ) 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 N Dz and show them on a logarithmic scale so that the exponential growth of N Dz , right from the onset of anoxic conditions, becomes visible. The graph initially shows a quick recruitment of the cells from the N D{ to the N Dz pool, followed by the exponential growth-phase. doi:10.1371/journal.pcbi.1003933.g010 ð Þ ), i.e., the denitrifying cells (N Dz ) but without aerobic and NO x -based growth, and E. Cumulated N 2 . The cumulated N 2 reached stable plateaus at nearly the same time for all the runs (Panel E), despite the fact that the time taken to deplete O 2 below O 2 ½ trigger decreased with increasing O 2 ½ LP t 0 ð Þ (Panel A). Thus, once denitrification was initiated, the rates increased with increasing initial O 2 ½ LP due to an increasing population of oxygen-grown cells (Panels B-D). The fraction of the cells recruited to denitrification (F den ) declined with increasing initial O 2 concentration (not shown), but this was not sufficient to compensate for the increasing number of oxygen-raised cells. doi:10.1371/journal.pcbi.1003933.g012 O 2 ½ LP t 0 ð Þ (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 O 2 ½ LP t 0 ð Þ due to an increasing population of oxygen-grown cells (Fig. 12B-D). F den (Eq. 9) declined with increasing O 2 ½ LP t 0 ð Þ (F den = 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 oxygenraised cells.
If the model is run without any initial O 2 , there would be no recruitment and, hence, no denitrification. Verification of this in batch cultures is difficult because traces of O 2 remain after Hewashing 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 O 2 depletion, only a small fraction (F den ) of the entire population is able to switch to denitrification [4,8,9]. The evidence is based on indirect analyses of e 2 -flow rates to O 2 and NO x during the transition of the cells from aerobic to anaerobic respiration. To provide a direct and refined estimation of F den , 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 F den 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 (F den ) of 0.0052 (h 21 ) for a cell to switch to denitrification is sufficient to robustly simulate the measured denitrification gas kinetics. The model estimated the resultant F den 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 F den .

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)