## Figures

## Abstract

Denitrifying bacteria accumulate , NO, and N_{2}O, the amounts depending on transcriptional regulation of core denitrification genes in response to O_{2}-limiting conditions. The genes include *nar*, *nir*, *nor* and *nosZ*, encoding -, -, NO- and N_{2}O reductase, respectively. We previously constructed a dynamic model to simulate growth and respiration in batch cultures of *Paracoccus denitrificans*. The observed denitrification kinetics were adequately simulated by assuming a stochastic initiation of *nir*-transcription in each cell with an extremely low probability (0.5% h^{-1}), leading to product- and substrate-induced transcription of *nir* and *nor*, respectively, via NO. Thus, the model predicted cell diversification: after O_{2} depletion, only a small fraction was able to grow by reducing . Here we have extended the model to simulate batch cultivation with , i.e., , NO, N_{2}O, and N_{2} kinetics, measured in a novel experiment including frequent measurements of . *Pa*. *denitrificans* reduced practically all to before initiating gas production. The production is adequately simulated by assuming stochastic *nar*-transcription, as that for *nirS*, but with a higher probability (0.035 h^{-1}) and initiating at a higher O_{2} concentration. Our model assumes that all cells express *nosZ*, thus predicting that a majority of cells have only N_{2}O-reductase (A), while a minority (B) has -, NO- and N_{2}O-reductase. Population B has a higher cell-specific respiration rate than A because the latter can only use N_{2}O produced by B. Thus, the ratio is low immediately after O_{2} depletion, but increases throughout the anoxic phase because B grows faster than A. As a result, the model predicts initially low but gradually increasing N_{2}O concentration throughout the anoxic phase, as observed. The modelled cell diversification neatly explains the observed denitrification kinetics and transient intermediate accumulations. The result has major implications for understanding the relationship between genotype and phenotype in denitrification research.

## Author Summary

Denitrifiers generally respire O_{2}, but if O_{2} becomes limiting, they may switch to anaerobic respiration (denitrification) by producing -, -, NO- and/or N_{2}O reductase, encoded by *nar*, *nir*, *nor*, and *nosZ* genes, respectively. Denitrification causes transient accumulation of and NO/N_{2}O emissions, depending on the activity of the four reductases. Denitrifiers lacking *nosZ* produce ~100% N_{2}O, whereas organisms with only *nosZ* are net consumers of N_{2}O. Full-fledged denitrifiers are equipped with all four reductases, genetic regulation of which determines accumulation and NO/N_{2}O emissions. *Paracoccus denitrificans* is a full-fledged denitrifying bacterium, and here we present a modelling approach to understand its gene regulation. We found that the observed transient accumulation of and N_{2}O can be neatly explained by assuming cell diversification: all cells expressing *nosZ*, while a minority expressing *nar* and *nir*+*nor*. Thus, the model predicts that in a batch culture of this organism, only a minor sub-population is full-fledged denitrifier. The cell diversification is a plausible outcome of stochastic initiation of *nar-* and *nir* transcription, which then becomes autocatalytic by and NO, respectively. The findings are important for understanding the regulation of denitrification in bacteria: product-induced transcription of denitrification genes is common, and we surmise that diversification in response to anoxia is widespread.

**Citation: **Hassan J, Qu Z, Bergaust LL, Bakken LR (2016) Transient Accumulation of NO_{2}^{-} and N_{2}O during Denitrification Explained by Assuming Cell Diversification by Stochastic Transcription of Denitrification Genes. PLoS Comput Biol 12(1):
e1004621.
https://doi.org/10.1371/journal.pcbi.1004621

**Editor: **Ilya Ioshikhes, Ottawa University, CANADA

**Received: **July 13, 2015; **Accepted: **October 22, 2015; **Published: ** January 5, 2016

**Copyright: ** © 2016 Hassan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

**Data Availability: **All relevant data are within the paper and its Supporting Information files.

**Funding: **The manuscript is part of JH's PhD thesis at the Norwegian University of Life Sciences, which provided the funding together with the Norwegian Research Council. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

**Competing interests: ** The authors have declared that no competing interests exist.

## Introduction

The dissimilative reduction of nitrate () to nitrite (), nitric oxide (NO), nitrous oxide (N_{2}O), and finally to N_{2} (denitrification) is an indispensable process in the nitrogen cycle, returning N to the atmosphere as N_{2}. However, denitrification significantly leaks the gaseous intermediates NO and N_{2}O, both with serious consequences for the environment. N_{2}O catalyses depletion of the stratospheric ozone [1] and causes global warming, contributing ~10% to the anthropogenic climate forcing [2]. Data suggests that since the 1950s, the atmospheric N_{2}O has been increasing, and before being photolysed in the stratosphere, the gas persists for an average ~120 years in the troposphere [3]. ~70% of global N_{2}O emissions are tentatively attributed to microbial nitrification and denitrification in soils [4], where denitrification, generally, is considered a more dominant source [5].

### To mitigate N_{2}O emissions, we need to understand the physiology of denitrifiers

To devise robust strategies for mitigating global N_{2}O emissions, a good understanding of its primary source is imperative, i.e., genetics, physiology, and regulatory biology of denitrifiers. Any knowledge of the environmental controllers of N_{2}O is incomplete without understanding the causal relationships of such controllers at the physiological level [6].

The biogeochemical models developed for understanding the ecosystem controls of denitrification and N_{2}O emissions treat the denitrifying community of soils and sediments as a single homogenous unit with certain characteristic responses to O_{2} and concentrations [6,7]. Natural denitrifying communities, however, are mixtures of organisms with widely different denitrification regulatory phenotypes [8]. The regulatory response of such mixtures is not necessarily equal to the ‘sum of its components’ because there will be interactions, not the least, via the intermediates NO and . Hence, it is probably a mission impossible to predict the regulatory responses of complex communities based on their phenotypic composition. Nevertheless, investigations of the regulation in model organisms like *Pa*. *denitrificans* provide us with essential concepts, enhancing our ability to understand the regulatory responses of mixed communities and to generate meaningful hypotheses. Thus, future biogeochemical models of N_{2}O and NO emissions are expected to have more explicit simulations of the regulatory networks involved, and a first attempt has recently been published [9].

### Simulating the cell diversification in response to impending anoxia to analyse its implications for , N_{2}, and N_{2}O kinetics

Dynamic modelling has been used to a limited extent to analyse various denitrification phenotypes; for example, to analyse and reduction and gas-kinetic data for individual strains [10] and mixtures of selected phenotypes [11]; to model the consequence of competition for electrons between denitrification reductases [12,13]; to investigate the control of O_{2} on denitrification enzymes and inhibition of cytochrome *c* oxidase by NO in *Agrobacterium tumefaciens* [14]; and to examine the effect of copper availability on N_{2}O reduction in *Paracoccus denitrificans* [15]. In our previous model [16], we simulated O_{2} and N_{2} kinetics from batch incubations of *Pa*. *denitrificans* [8,17] to test if a postulated cell diversification, driven by stochastic initiation of *nirS*, could explain the N_{2} production kinetics in -supplemented media. The available data also contained -supplemented treatments but and were not monitored, and the experiment provided no information about the N_{2}O kinetics, except that the concentrations were extremely low (below the detection limit of the thermal conductivity detector used). Recently, a neat dataset was generated from batch incubations supplemented with , with frequent measurements of and a more sensitive detection of N_{2}O by an electron capture detector [18]. That encouraged us to extend our previous model and simulate the cell diversification during transition from oxic to anoxic conditions, targeting the regulation of Nar and *c*Nor/NosZ (N_{2}O emissions) in *Pa*. *denitrificans*.

### Regulatory network of denitrification in *Paracoccus denitrificans*

*Pa*. *denitrificans* is a facultative anaerobe capable of reducing all the way to N_{2}:

In response to impending anoxic conditions, the organism sustains respiratory metabolism by producing the membrane-bound cytoplasmic nitrate reductase (Nar), cytochrome *cd*_{1} nitrite reductase (NirS), cytochrome *c* dependent nitric oxide reductase (*c*Nor), and nitrous oxide reductase (NosZ). Transcription of the genes encoding these reductases (*narG*, *nirS*, *norBC*, and *nosZ*, respectively) are regulated by the FNR-type proteins FnrP, NarR, and NNR. FnrP contains a 4Fe-4S cluster for sensing O_{2}, and NNR harbours a NO-sensing haem; NarR, however, is poorly characterised and is most likely a -sensor [19–21]. All these sensors remain inactive during aerobic growth conditions [19].

#### Transcription of denitrification genes in *Pa*. *denitrificans*.

FnrP and NarR facilitate a product-induced transcription of the *nar* genes, and NNR facilitates a product-induced transcription of the *nirS* genes (Fig 1, see P_{1} and P_{2}): Low oxygen concentration ([O_{2}]) activates the self-regulating FnrP, which induces *nar* transcription in coaction with NarR. The self-regulating NarR was previously assumed to be activated by either or [21], but a recent proteomics study indicates that is the activator [19]. Thus once a cell starts producing traces of , *nar* expression becomes autocatalytic. Transcription of *nirS* is induced by NNR, which is apparently inactivated by O_{2} [22,23], but under anoxic/micro-oxic conditions, NNR is activated by NO. Thus, once traces of NO are produced, the expression of *nirS* also becomes autocatalytic [19,20]. In contrast, *nor* transcription is substrate (NO) induced via NNR while *nosZ* is equally induced by NNR or FnrP [24]. High concentrations of NO may constrain *nar* transcription by inactivating FnrP [25] and, like O_{2}, render NosZ dysfunctional by inactivating the Cu_{Z} subunit of the reductase [26], but these observations are ignored in our model because *Pa*. *denitrificans* restricts [NO] to very low levels.

The network is driven by four core enzyme-complexes: Nar (transmembrane nitrate reductase encoded by the *narG* gene), NirS (cytochrome *cd*_{1} nitrite reductase encoded by *nirS*), *c*Nor (NO reductase encoded by *norBC*), and NosZ (N_{2}O reductase encoded by *nosZ*). When anoxia is imminent, the low [O_{2}] is sensed by FnrP, which in some interplay with NarR induces *nar* transcription. NarR is activated by ; thus once a cell starts producing traces of , *nar* expression becomes autocatalytic (see P_{1}). Transcription of *nirS* is induced by NNR, which is activated under anoxic/micro-oxic conditions by NO; thus once traces of NO are produced, the expression of *nirS* also becomes autocatalytic (see P_{2}) [20]. The activated P_{2} will also induce *nor* and *nosZ* transcription via NNR. The transcription of *nosZ*, however, can also be induced equally and independently by FnrP [24]. Micromolar concentrations of NO may inactivate both FnrP [25] and NosZ [26]. These observations, however, are ignored for our modelling because *Pa*. *denitrificans* restricts NO to nanomolar levels.

### Entrapment of cells in anoxia: The underlying hypothesis and modelling

Denitrification proteome, once produced in response to an anoxic spell, is likely to linger within the cells under subsequent oxic conditions, ready to be used if anoxia recurs. But the proteome will be diluted by aerobic growth because the transcription of denitrification genes is inactivated under oxic conditions [20]. Hence, a population growing through many generations under fully oxic conditions is expected to undertake *de novo* synthesis of denitrification enzymes when confronted with anoxia. Batch cultivations of such aerobically raised *Pa*. *denitrificans* provided indirect evidence for a novel claim that, in response to anoxia, only a small fraction of the incubated population is able to produce denitrification proteome [8,17,27,28]. Our dynamic modelling of Bergaust *et al*.*’s* [17] -supplemented incubations corroborated this, suggesting that a probabilistic function (specific probability = 0.005 h^{-1}) resulting in the recruitment of 3.8–16.1% of all cells to denitrification is adequate to explain the measured N_{2} kinetics [16].

Our model was based on the hypothesis that the entrapment of a large fraction in anoxia is due to a low probability of initiating *nirS* transcription, which in response to O_{2} depletion is possibly mediated through a minute pool of intact NNR, crosstalk with other factors (such as FnrP), unspecific reduction of to NO by Nar, and/or through non-biologically formed traces of NO found in a -supplemented medium. Regardless of the exact mechanism(s), once *nirS* transcription is initiated, the positive feedback via NO/NNR (Fig 1, see P_{2}) would allow the product of a single transcript of *nirS* to induce a subsequent burst of *nirS* transcription. The activated positive feedback will also help induce *nor* and *nosZ* transcription via NNR, rapidly transforming a cell into a full-fledged denitrifier. We further hypothesised that recruitment to denitrification will only be possible as long as a minimum of O_{2} is available because, since *Pa*. *denitrificans* is non-fermentative, the synthesis of first molecules of NirS will depend on energy from aerobic respiration.

The above hypothesis was modelled by segregating the culture into two pools (subpopulations): one for the cells without (N_{D−}) and the other with denitrification enzymes (N_{D+}). Initially, all cells were N_{D−}, growing by consuming O_{2}. As [O_{2}] fell below a certain threshold, N_{D−} recruited to N_{D+} with a constant probability (h^{-1}), assumed to be that of the *nirS* transcriptional activation, and the recruitment halted as O_{2} was completely exhausted, assuming lack of energy (ATP) for enzyme synthesis.

### Underlying assumptions and aims of the present modelling

The present model is an extension of that developed in Hassan *et al*. [16]. Here we have divided the respiring culture into four pools (Fig 2A):

- Z
^{−}: cells without Nar, NirS, and*c*Nor - Z
^{Na}: cells with Nar - Z
^{NaNi}: cells with Nar, NirS, and*c*Nor - Z
^{Ni}: cells with NirS and*c*Nor

All these subpopulations are assumed to scavenge O_{2} (if present) and produce NosZ in response to impending anoxia. The latter because the *nosZ* genes are readily induced by the O_{2}-sensor FnrP [24].

**A.** Cell diversification and growth; **B.** O_{2} kinetics; **C.** Denitrification kinetics. The squares represent state variables, the circles the rate of change of the state variables, the edges (thicker arrows) depict flows into or out of the state variables, the shaded ovals auxiliary variables, and the arrows portray mutual dependencies between the variables. All feedback relationships among the three model sectors could not be shown; however, for illustration the feedback relationships of one sub-population (Z^{−}) are shown (dashed arrows). Within each square (state variable), t_{0} refers to the initial value.

The Z^{−} pool (Fig 2A) contains the inoculum that grows by aerobic respiration. As [O_{2}] falls below a critical threshold [empirically determined, 18], the cells within Z^{−} are assumed to start synthesising Nar with a certain probability and populate the Z^{Na} pool. The aim here is to investigate whether, like for *nirS*, the initiation of *nar* transcription (by some combined activity of FnrP and NarR) can also be explained as a probabilistic phenomenon, quickly differentiating a cell into a full-fledge scavenger through product () induced transcription via NarR (Fig 1, see P_{1}). If so, we were interested to estimate what fraction of the cells is required to adequately simulate the measured data ( production), aiming at scrutinising the general assumption that all cells in batch cultures produce Nar in response to impending anoxia.

Next, when [O_{2}] is further depleted to another critical threshold [18], the Z^{−} and Z^{Na} cells are assumed to initiate *nirS* transcription with a low per hour probability and, thereby, populate the Z^{Ni} and Z^{NaNi} pools, respectively. As explained above for our previous model, NirS + *c*Nor production is assumed to be *a)* coordinated because the transcription of both *nirS* and *nor* is induced by NO via the NO-sensor NNR (Fig 1), and *b)* stochastic because the initial transcription of *nirS* (paving the way for the autocatalytic expression of NirS and substrate-induced *nor* transcription) happens in the absence of NO or at too low [NO] to be sensed by NNR.

Synthesis of denitrification enzymes requires energy, which all the subpopulations can obtain by respiration only. Hence, the initiation of the autocatalytic expression of *nar* and *nirS* (i.e., recruitment to Z^{Na} and Z^{NaNi}/Z^{Ni}, respectively, Fig 2A) depends on the availability of the relevant terminal e^{-}-acceptor(s) above a critical concentration to sustain a minimum of respiration. For Z^{−}, the only relevant e^{-}-acceptors are O_{2} and the traces of N_{2}O produced by Z^{Ni} and Z^{NaNi}. The same applies For Z^{Na}, but in addition, this subpopulation can also obtain energy by reducing , if present. In our previous model [16], we assumed that recruitment to denitrification was sustained by energy from O_{2}-respiration only; not because we simulated -supplemented treatments, and not by N_{2}O because we naively assumed that the pool of this e^{-}-acceptor was insignificant (N_{2}O concentrations were below the detection limit of the system used for those experiments). However, the present model assumes that the recruitment from Z^{−} to Z^{Na} and Z^{−} to Z^{Ni} is sustained by both O_{2}- and N_{2}O-reduction, and the recruitment from Z^{Na} to Z^{NaNi} is sustained by O_{2}-, N_{2}O- and -reduction, when above a critical minimum (). The default value for was set to an arbitrary low value (= 0.44% of the maximum e^{-}-flow rate to O_{2}), and we have investigated the consequences of increasing, decreasing, and setting = 0.

The expressions of *nar* and *nirS* + *nor* (recruitments to Z^{Na} and Z^{NaNi}/Z^{Ni}, respectively, Fig 2A) are modelled as instantaneous discrete-events in each cell, thus ignoring the time-lag from the initiation of gene transcription till the cell is fully equipped with the reductase(s) in question. That is because the lag observed between the emergence of denitrification gene transcripts and the subsequent gas products suggests that the synthesis of denitrification enzymes takes less than half an hour [17,18], which is negligible for our purposes here.

The main purpose of the present modelling is to investigate if a full-fledged model, including all four functional denitrification reductases, could adequately simulate the observed kinetics and stoichiometry of denitrification products [18]. These cultures reduced all available to prior to the onset of gas production and accumulated traces of N_{2}O throughout the anoxic phase, as illustrated in S1 Fig In particular, we were interested to investigate the kinetics, controlled by *nar*- and *nirS* transcription, and to test if the peculiar N_{2}O kinetics (low, but increasing concentrations throughout the anoxic phase) could be explained by our modelled cell diversification.

## Materials and Methods

### An overview of the modelled experiment

#### Batch incubation.

Qu [18] incubated *Pa*. *denitrificans* (DSM-413) at 20°C using 50 mL Sistrom’s [29] medium in 120 mL gas-tight vials. Either succinate or butyrate (5 mM) was used as the main carbon source, enough to secure consumption of all available e^{-}-acceptors. After distribution of the medium, each vial was loaded with a magnetic stirring bar, sterilised through autoclaving, supplemented with 2 mM KNO_{3}, and was tightly sealed. To remove O_{2} and N_{2} from the headspace, the headspace air was evacuated and replaced by helium (He) through several cycles of evacuation and He-filling (He-washing). Some vials were supplemented with oxygen to reach 7 vol.% O_{2} in headspace (treatment designated 7% O_{2}). The remaining vials received no O_{2} (designated 0% O_{2}, although there were traces of O_{2} present despite the He washing). For each treatment (i.e., C source and initial O_{2}), there were three replicates, and each vial was inoculated with 2.2×10^{8} aerobically grown cells.

#### and gas measurement.

Gases (CO_{2}, O_{2}, NO, N_{2}O, and N_{2}) were monitored by frequent sampling of the headspace, using an improved version of the robotised incubation system [30]. In short, the system draws gas samples from the headspace (peristaltic pumping) via the septum pierced by a needle, filling three loops used for injecting samples into the two GC columns and the chemiluminescence NO analyser. The sample drawn is replaced by He (reversing the peristaltic pump), thus securing ~1 atm pressure. The primary improvements of the new system are a more sensitive detection of N_{2}O (by an electron capture detector), lower sampling volumes (~1 mL), and lower leaks of O_{2} and N_{2} through the sampling system (4 nmol O_{2} and 12 nmol N_{2} per sampling, which is ~20% of that for the old system).

To extract samples for measuring without tampering the original vials, identical (parallel) vials were prepared for each treatment. Using sterile syringes, samples of 0.1 mL were regularly drawn from the liquid-phase of the parallel vials and immediately analysed for .

Results for one of the treatments are shown in S1 Fig, illustrating the complete reduction of to prior to the onset of significant N-gas production. In previous experiments [17], N_{2}O concentrations were below the detection limit of the system, but thanks to the new system, the N_{2}O kinetics were monitored with a reasonable precision.

### The model

The model is constructed in Vensim DSS 6.2 Double Precision (Ventana Systems, inc. http://vensim.com/) using techniques from the field of system dynamics [31].

#### Cell diversification and growth.

The respiring population is divided into four subpopulations, according to their reductases (Fig 2A): 1) Z^{−}: cells without Nar, NirS, and *c*Nor; 2) Z^{Na}: cells with Nar; 3) Z^{NaNi}: cells with Nar, NirS, and *c*Nor; and 4) Z^{Ni}: cells with NirS and *c*Nor. All the subpopulations are assumed to equally respire O_{2,} if present, and express *nosZ* in response to oxygen depletion [24]. Z^{−} contains the inoculum (= 2.2×10^{8} cells) that grows by aerobic respiration. As O_{2} is depleted, the Z^{−} cells populate the other pools by producing Nar and/or NirS + *c*Nor.

The recruitment from Z^{−} to Z^{Na} (R_{Na}, Fig 2A) takes place first:
(1)

(cells h^{-1})

where *r*_{Na}*(O*_{2},*N*_{2}*O)* is a conditional specific probability (h^{-1}) for any Z^{−} cell to initiate *nar* transcription (quickly transforming it into a scavenger through autocatalytic gene expression, see Fig 1, P_{1}):
(2)

(h^{-1})

where r_{Na} (h^{-1}) is a constant specific probability for a cell to initiate *nar* transcription once O_{2} concentration in the aqueous-phase ([O_{2}]_{aq}, mol L^{-1}) falls below a critical concentration ([O_{2}]_{na}), empirically determined as the [O_{2}]_{aq} (= 4.75×10^{−5} mol L^{-1}) at the outset of accumulation in the medium [18]. The second condition for a cell to produce first molecules of Nar is a minimum of e^{-}-flow to an e^{-}-acceptor (, mol e^{-} cell^{-1} h^{-1}), assumed to generate minimum ATP required for protein synthesis. and (mol e^{-} cell^{-1} h^{-1}) are the cell-specific velocities of e^{-}-flow to O_{2} and N_{2}O, respectively. The latter is weighed down by 0.5 because mole ATP per mole e^{-} transferred to /NO_{x} is lower for denitrification than for aerobic respiration [17,20]. For a Z^{−} cell, and are not considered here, since such a cell is assumed to have no NirS and *c*Nor.

The fraction of the cells that successfully produces Nar (F_{Na}) is calculated based on the integral of the recruitment (Eq 1):
(3)

(dimensionless)

where *t*_{Na} is the time-window available for the recruitment. In theory, *t*_{Na} is the time-period when (Eq 2). Since the e^{-}-flow to N_{2}O started after all had been reduced to (S1 Fig), the recruitment based on would be inconsequential for the simulated (and measured) kinetics. Therefore, to calculate the functional F_{Na} actually responsible for producing , we ignored the N_{2}O-sustained recruitment, thus considering *t*_{Na} to be the time when .

Next, the cells within Z^{Na} and Z^{−} are recruited to Z^{NaNi} and Z^{Ni} (R_{NaNi} and R_{Ni}, respectively, Fig 2A), as they are assumed to stochastically initiate *nirS* transcription, paving the way for NO/NNR mediated autocatalytic expression of *nirS* + *nor* (Fig 1). In principle, the rates of both these recruitments are modelled as that of the recruitment from Z^{−} to Z^{Na} (Eqs 1 and 2): *a)* Both trigger as O_{2} falls below another critical concentration ([O_{2}]_{ni}), low enough to activate NNR to induce *nirS* transcription; [O_{2}]_{ni} (= 1.16×10^{−5} mol L^{-1}) is empirically determined as the O_{2} concentration at the outset of NO accumulation [18]. *b)* Both continue as long as a minimum of e^{-}-flow to the relevant terminal e^{-}-acceptor is possible, sustaining the respiratory metabolism to generate ATP for protein synthesis:
(4)

(h^{-1})

where r_{Ni} is a constant specific probability (h^{-1}) for the initiation of *nirS* transcription. and are multiplied with 0.5 for the same reasons as described for Eq 2.

The recruitment from Z^{−} to Z^{Ni} (R_{Ni}, Fig 2A) is modelled as a product of Z^{−} and a conditional specific probability, *r*_{Ni}*(O*_{2},*N*_{2}*O)*, which is different from Eq 5 only in that is omitted, since Z^{−} do not possess Nar:
(6)

(h^{-1})

The fraction that successfully produced NirS + *c*Nor (F_{Ni}) is calculated based on the integral of R_{NaNi} and R_{Ni}:
(8)

(dimensionless)

where *t*_{NaNi} is the duration of the recruitment from Z^{Na} to Z^{NaNi}, i.e., when (Eqs 4 and 5), F_{Na} is the fraction recruited to the pool of Nar positive cells (Z^{Na}, Eq 3), and *t*_{Ni} is the duration of the recruitment from Z^{−} to Z^{Ni}, i.e., when (Eqs 6 and 7).

Each of the populations will grow depending on the rates of e^{-}-flow to the various e^{-}-acceptors they are able to use:
(9)
(10)
(11)
(12)

(cells h^{-1})

where (cells mol^{-1} e^{-} to X = O_{2} or /NO_{x}) is the growth yield determined under the actual experimental conditions, and (mol e^{-} cell^{-1} h^{-1}) is the cell-specific velocity of e^{-}-flow to X (O_{2} or /NO_{x}), which depends on the concentration of the e^{-}-acceptor (see Eqs 17, 20 and 28). For and , a restricted velocity () is used so that when electrons flow to O_{2}, , and simultaneously, the total *v*e^{−} per cell does not exceed the maximum electrons that the TCA cycle () can deliver per hour (see Eqs 21 and 22).

#### O_{2} kinetics.

O_{2} is initially present in the headspace (, mol, initialised according to the experiment, see Table 1) but is transported to the liquid-phase () due to its consumption therein (Fig 2B). The transport rate () is modelled according to Molstad *et al*. [30]:
(13)

(mol h^{-1})

where k_{t} (L h^{-1}) is the empirically determined coefficient for the transport of gas between the headspace and the liquid, (mol L^{-1} atm^{-1}) is the solubility of O_{2} in water at 20°C, (= [O_{2}]_{g} × R × T, atm) is the partial pressure of O_{2} in the headspace, and [O_{2}]_{aq} (mol L^{-1}) is the O_{2} concentration in the liquid .

In addition, the model simulates the changes in due to sampling. The robotised incubation system used 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 O_{2} through the tubing and membranes in the injection system. The net change in (ΔO_{2(S)}) as a result of each sampling is calculated as:
(14)

(mol h^{-1})

where (mol vial^{-1}) is the O_{2}-leakage into the headspace, D (dilution) is the fraction of each headspace gas removed and replaced by equal amount of He, and t_{s} (h) is the time taken to complete each sampling. ΔO_{2(S)} is negative if is high and marginally positive at very low oxygen concentrations.

O_{2} in the liquid-phase (, mol, Fig 2B) is initialised by assuming equilibrium with at the time of inoculation (). The dynamics of are modelled as a function of transport between the headspace and the liquid (, Eq 13) and its reduction rate (, mol h^{-1}):
(15)
(16)

(mol h^{-1})

where Z^{−}, Z^{Na}, Z^{NaNi}, and Z^{Ni} (cells) are all the sub-populations present (described above); thus, we assume that all cells have the same potential to consume O_{2}. (mol cell^{-1} h^{-1}) is the cell-specific velocity of O_{2} consumption, obtained by the velocity of e^{-}-flow to O_{2} , where is modelled as a Michaelis-Menten function of oxygen concentration:
(17)

(mol e^{-} cell^{-1} h^{-1})

where (mol e^{-} cell^{-1} h^{-1}) is the maximum velocity of e^{-}-flow to O_{2} per cell (determined under the actual experimental conditions), [O_{2}]_{aq} (mol L^{-1}) is the O_{2} concentration in the liquid-phase, and (mol L^{-1}) is the half-saturation constant for O_{2} reduction.

#### Denitrification kinetics.

The ** and pools (mol, Fig 2C) are initialised according to the experiment (Table 1; = 0). The kinetics of these nitrogen oxyanions () are modelled as:
(18)
(19)**

(mol h^{-1})

where (mol h^{-1}) is the reduction rate, Z^{Na} + Z^{NaNi} (cells) is the total number of cells with Nar, Z^{NaNi} + Z^{Ni} (cells) is the total NirS active population, and (mol cell^{-1} h^{-1}) is the cell-specific velocity of consumption, obtained by the velocity of e^{-}-flow to . The latter is modelled as a Michaelis-Menten function of concentration:
(20)

(mol e^{-} cell^{-1} h^{-1})

where (mol e^{-} cell^{-1} h^{-1}) is the maximum velocity of e^{-}-flow to per cell (determined under the actual experimental conditions), (mol L^{-1}) is the concentration in the aqueous-phase, and (mol L^{-1}) is the half-saturation constant for reduction.

The velocity of and consumption had to be restricted () to ensure that when electrons flow to O_{2}, and simultaneously, the total *v*e^{−} per cell does not exceed an estimated maximum delivery of electrons from the TCA cycle (). In competition for electrons, O_{2} is prioritised [20], followed by and , respectively [18]:
(21)
(22)

(mol e^{-} cell^{-1} h^{-1})

where is the realised e^{-}-flow to , limited either by available or the availability of electrons (due to competition with O_{2}); is the realised e^{-}-flow to . Such competition for electrons was not implemented for and because at the onset of NO-, N_{2}O- and N_{2} production, the total velocity of e^{-}-flow to all available e^{-}-acceptors (as predicted by the enzyme kinetics alone) never exceeded .

Gas consumption and production takes place in the aqueous phase, but the gases are transported between aqua and the headspace depending on their concentrations in the two phases. Each gas in aqua, X_{aq} (molN, Fig 2C), is modelled as a function of production, consumption (not applicable to N_{2}), and the net transport, where N_{2}O_{aq} and are initialised with zero, and NO_{aq} is initialised with a negligible 1×10^{−25} mol to avoid division by zero (in Eq 28).
(23)
(24)
(25)

(molN h^{-1})

where (molN h^{-1}) is the relevant /NO_{x} reduction rate, and represents the gas transport rate between aqua and the headspace (Eq 29; N.B. < 0 for the net transport from aqua to the headspace).

The reduction of NO to N_{2}O (Rr_{NO}) and N_{2}O to N_{2} () is modelled likewise as a function of the number of relevant cells and the velocity of e^{-}-flow to NO and N_{2}O (mol e^{-} cell^{-1} h^{-1}), respectively:
(26)
(27)

(molN h^{-1})

where *v*_{NO} and are obtained by the velocity of e^{-}-flow to NO and N_{2}O, respectively . is modelled as a Michaelis-Menten function of [N_{2}O]_{aq}, similarly as that of O_{2}, , and (Eqs 17 and 20), but is modelled assuming a cooperative binding of two NO molecules with *c*Nor to form N_{2}O [32]:
(28)

(mol cell^{-1} h^{-1})

where (mol e^{-} cell^{-1} h^{-1}) is the empirically determined maximum velocity of e^{-}-flow to NO per cell, [NO]_{aq} (mol L^{-1}) is the NO concentration in the liquid-phase, and K_{1NO} & K_{2NO} (mol L^{-1}) are the equilibrium dissociation constants for the *c*Nor/NO- and *c*Nor/(NO)_{2} complex, respectively.

The transport of NO, N_{2}O, and N_{2}between the liquid and the headspace (Eqs 23–25) is modelled as:
(29)

(molN h^{-1})

where k_{t} is the empirically determined coefficient for the transport of each gas between the headspace and the liquid, k_{H(N)} (molN L^{-1} atm^{-1}) is the solubility of NO, N_{2}O, or N_{2} in water at 20°C, P_{N} (= [N]_{g}×R×T, atm) is the partial pressure of each gas in the headspace, and [N]_{aq} (mol L^{-1}) represents the concentration of each gas in the liquid-phase.

The amount of NO and N_{2}O in the headspace (, molN, Fig 2C) is a function of transport (Eq 29) and the disturbance by gas sampling. The latter is simulated as discrete events at time-points given as input to the model (equivalent to the sampling times in the experiment):
(30)

(molN h^{-1})

where ΔNO_{x(S)} is the net change in the amount of (molN), D (dilution) is the fraction of each gas removed and replaced by equal amount of He, and t_{s} (h) is the time taken to complete each sampling. For N_{2}, the model ignores the sampling loss because the N_{2} production data to be compared with the model output are already corrected for the sampling disturbance [30]. Thus, the model estimates somewhat higher N_{2} concentrations than that experienced by the organisms, which is acceptable, since the concentration of N_{2} is unlikely to have consequences for the metabolism.

### Parameterisation

Most of the parameter values used in the model are well established in the literature (see Table 2); however, uncertain parameters include , , , and .

(Eq 17). *Pa*. *denitrificans* has three haem-copper terminal oxidoreductases [39] with ranging from nM to µM [40,41], so we decided to estimate the parameter value by optimising for the low [O_{2}] treatments data. Vensim was used for the optimisation, where = 2.25×10^{−7} neatly simulated the O_{2} depletion for both the succinate- and butyrate-supplemented treatments.

**.** *In vitro* studies of NosZ from *Pa*. *denitrificans* estimate the values for = 5 μM at 22°C and pH 7.1 [42] and 6.7 μM at 25°C and pH 7.1 [43]. When our model was simulated with in this range, given our empirically estimated [24], the simulated N_{2}O reached concentrations much higher than that measured (see Results/Discussion). A more adequate parameter value (= 0.6 μM) was found by optimising in Vensim. The value is within the range determined for soil bacterial communities [44].

(Eq 17) could be estimated using the empirically determined cell yield per mole of electrons to O_{2} (, cells per mol e^{-}) and the maximum specific growth rate (μ, h^{-1}): . We are confident about the yields for the two C-substrates used, but the empirically determined μ for the butyrate treatments is suspiciously low (= 0.067 h^{-1}), providing = 2.45×10^{−15} mol e^{-} cell^{-1} h^{-1}. Simulations with this value grossly underestimated the rate of O_{2} depletion compared to measured, which forced us to estimate the parameter value by optimisation: = 4.42×10^{−15} and 4.22×10^{−15} mol e^{-} cell^{-1} h^{-1} for the succinate- and butyrate treatments, respectively. These values give μ = 0.22 and 0.12 h^{-1}, respectively: for the succinate treatments, the value is very close to that empirically determined (= 0.2 h^{-1}); for the butyrate treatments, the value seems more realistic than 0.067 h^{-1}.

(Eqs 2, 5 and 7) is the per cell velocity of e^{-}-flow to O_{2} () assumed to generate minimum ATP required for synthesising the initial molecules of denitrification enzymes. Since we lack any empirical or other estimations for this parameter, it is arbitrarily assumed to be the when [O_{2}]_{aq} reaches 1 nM. At this concentration, is determined by the Michaelis-Menten equation , using and given above. The values obtained for the succinate- and butyrate-supplemented treatments = 1.96×10^{−17} and 1.87×10^{−17} mol e^{-} cell^{-1} h^{-1}, respectively, which for both the cases is 0.44% of . To investigate the impact of on the model behaviour (r_{Na} and r_{Ni}, Eqs 1, 2, 4, 5, 6 and 7), sensitivity analyses were performed by simulating the model with corresponding to [O_{2}]_{aq} = 5×10^{−9}, 5×10^{−10}, and 0 mol L^{- 1} (see Results/Discussion).

## Results/Discussion

### Low probabilistic initiation of *nar* transcription, resulting in the fraction of the population with Nar < 100%

To test the assumption of a single homogeneous population with all cells producing Nar in response to O_{2} depletion, we simulated the model with the specific probability for a Z^{−} cell to initiate *nar* transcription (r_{Na}) = 4 h^{-1}. This resulted in 98% of the cells possessing Nar within an hour (see Eqs 1–3). Evidence suggests that less than half an hour is required to synthesise denitrification enzymes [17,18], but an hour’s time is assumed here to allow margin for error. The results show that, for all the treatments, the simulated production (mol vial^{-1}) grossly overestimates that measured (Fig 3).

To test the assumption of a single homogeneous population with almost all cells expressing *nar* in response to O_{2} depletion, we forced our model to achieve 98% Nar-positive cells (Z^{Na}) within an hour by setting the specific-probability of initiating *nar* transcription (r_{Na}) = 4 h^{-1}. This resulted in grossly overestimated rates of accumulation for all treatments (grey curves). In contrast, we simulated the model with r_{Na} = 0.035 h^{-1} obtained through optimisation, resulting in a reasonable agreement with measurements for all treatments (except for an apparent time frameshift for the Butyrate, 7% O_{2} treatment).

To find a reasonable parameter value, we optimised r_{Na} for the 0% O_{2} treatments, so that the simulated production matches that measured. The results (Table 3) suggest that a low probabilistic initiation of *nar* transcription (average r_{Na} = 0.035 h^{-1}) is adequate to simulate the measured kinetics (Fig 3). In the Butyrate, 7% O_{2} treatment (Fig 3B), the simulated starts earlier, but the rate of accumulation is similar to that measured.

Once O_{2} falls below a certain threshold, the production of Nar is assumed to trigger with r_{Na} = 0.035 h^{-1} and last until a minimum of respiration is sustained by the e^{-}-flow to O_{2} and N_{2}O ( and ), assumed to fulfil the ATP needs for Nar production (Eqs 1 and 2). But the production of Nar sustained by was inconsequential for simulating the measured production, since was already exhausted when N_{2}O started accumulating (i.e., when > 0). For this reason, the fraction that produced Nar (F_{Na}, Eq 3 and Table 4) is calculated as functional (= 0.23–0.43) and theoretical (= 0.56–0.81), where the first is the fraction actually responsible for production (sustained by ), but the latter also incorporates the fraction that produced Nar after the exhaustion of (sustained by as well as ). The rationale behind calculating the theoretical F_{Na} is the empirical data indicating that Nar transcription is not turned off in response to depletion [18]. Although our model cannot test the theoretical F_{Na}, but the functional F_{Na} suggests that, contrary to the common assumption, the measured kinetics can be neatly explained by only 23–43.3% of the population producing Nar in response to O_{2} depletion.

### Very low probabilistic initiation of *nirS* transcription

When we optimised the specific probability of *nirS* transcriptional activation (r_{Ni}, see Eqs 4, 5, 6 and 7) to fit the measured data, the average r_{Ni} = 0.004 h^{-1} (Table 3) adequately simulated the measured depletion and N_{2} accumulation (Fig 4). The recruitment to denitrification lasted for 19.5–47.3 h, i.e., the time when [O_{2}] was below a critical concentration and the velocity of e^{-}-flow to O_{2} and the relevant /NO_{x} remained above a critical minimum (Eqs 4, 5, 6 and 7). The resulting fraction recruited to denitrification (F_{Ni}, see Eq 8 and Table 4) was 0.08–0.18, the bulk of which depended on the e^{-}-flow to and N_{2}O (instead of aerobic respiration).

Each panel compares the measured depletion (sub-panel) and N_{2} accumulation (main panel; n = 3–4) with simulations. The simulations are carried out with an optimised specific-probability of *nirS* transcriptional initiation (average r_{Ni} = 0.004 h^{-1}, Eqs 4, 5, 6 and 7), allowing 7.7–22.1% of the population to produce NirS + *c*Nor (Eq 8) during the available time-window (= 19.5–47.3 h).

To test whether the measured data could be explained without the recruitment sustained by and N_{2}O respiration, we also simulated the model with the recruitment as a function of O_{2} alone and re-optimised r_{Ni}, which on average increased to 0.012 h^{-1} (providing F_{Ni} = 0.083–0.35). This was expected since O_{2} is exhausted rather quickly, shrinking the time-window available for the recruitment. Comparatively, these simulations were less satisfactory: using the average r_{Ni} = 0.012 h^{-1} generally resulted in larger deviations than for the default simulations (S2 Fig), and the optimal r_{Ni} for individual treatments varied grossly (50% higher values for the ~0% O_{2} treatments than for the 7% O_{2} treatments). This contrasts the default simulations, where the optimal r_{Ni} values for individual treatments were quite similar.

### Sensitivity of r_{Na} and r_{Ni} to

Recruitment to denitrification (both *nar* and *nirS* transcription) is assumed to continue only as long as the combined e^{-}-flow to O_{2,} and N_{2}O is greater than (Eqs 1, 2, 4, 5, 6 and 7). To test the model’s sensitivity to this parameter, we estimated r_{Na} and r_{Ni} by optimisation for different values of , relative to the default value = 1.95×10^{−17} mol e^{-} cell^{-1} h^{-1}. For all cases, the model was able to adequately simulate the measured N_{2} kinetics by moderate adjustments of r_{Na} and r_{Ni}. Table 5 shows the average optimal values of r_{Na} and r_{Ni}, obtained by fitting the simulated N_{2} kinetics to the data, for different values of . S3 Fig shows adequate simulations of the measured N_{2} kinetics assuming = 0, with optimised r_{Na} = 0.033 h^{-1} and r_{Ni} = 0.0033 h^{-1}. Thus, although assuming > 0 appears logical, it is not necessary to explain the measured data.

### N_{2}O kinetics

To simulate the N_{2}O kinetics, we used = 5.5×10^{−15} mol e^{-} cell^{-1} h^{-1}, empirically determined under similar experimental conditions as simulated here [24], and adopted the literature values for [= 5 and 7 μM 42,43, respectively]. But with = 5 μM, the model predicted N_{2}O accumulation ~10–20 times higher than measured for the ~0% and ~2–3 times higher for the 7% O_{2} treatments (Fig 5). This forced us to simulate the model with the parameter value estimated by optimisation, providing the average = 0.6 μM.

Each main panel (A–D) compares the measured N_{2}O (single vial results) with the default simulation using the parameter values given in Table 2, i.e., = 0.6 μM (estimated through optimisation) and = 5.5×10^{−15} mol e^{-} cell^{-1} h^{-1} [24]. In contrast, each inserted panel shows the simulated N_{2}O assuming 1) N_{2}O consumption only by the cells producing N_{2}O (Z^{NaNi} + Z^{Ni}), and 2) the literature value for = 5 μM [42]. The results show that the default simulation best explains the measured N_{2}O kinetics, assuming its production by a small fraction (Z^{NaNi} + Z^{Ni}) and consumption by the entire population (Z^{−} + Z^{Na}+ Z^{NaNi} + Z^{Ni}).

The measured N_{2}O shows a conspicuous increase throughout the entire active denitrification period, and this phenomenon is neatly captured by the model. The reason for this model prediction is that the number of N_{2}O producing cells (Z^{NaNi} + Z^{Ni}, Fig 2A) is low to begin with compared to the number of N_{2}O consuming cells (Z^{−} + Z^{Na}+ Z^{NaNi} + Z^{Ni}), but the fraction of N_{2}O producers will increase during the anoxic phase for two reasons: one is the recruitment to Z^{NaNi} & Z^{Ni}, another is the fact that the model predicts approximately three times faster cell-specific growth rate for Z^{NaNi} & Z^{Ni} than for Z^{−} & Z^{Na} ( is identical for all groups, while and are both zero for Z^{−} & Z^{Na} but for Z^{NaNi} & Z^{Ni}, it holds that ≈ > _{.} To illustrate this phenomenon, we ran the model, assuming that the Z^{−} & Z^{Na} cells had no N_{2}O reductase, resulting in *a)* constant N_{2}O concentration throughout the entire anoxic phase and *b)* much higher N_{2}O concentrations than measured (Fig 5). The overestimation is a trivial result, easily avoidable by increasing or decreasing moderately. However, the prediction of a constant N_{2}O concentration is clearly in conflict with the experimental data, and no parameterisation could force the model to reproduce this phenomenon other than the differential expression of denitrification genes.

Hence, although there is room for further refinements, our default assumption regarding differential expression of NirS and NosZ explains the observed N_{2}O kinetics: 1) abrupt initial accumulation to very low levels due to recruitment of relatively small numbers to the N_{2}O producing pools (Z^{NaNi} & Z^{Ni}), and 2) increasing N_{2}O concentration due to recruitment and faster cell-specific growth of Z^{NaNi} & Z^{Ni} than that of the cells only consuming N_{2}O (Z^{−} + Z^{Na}).

This modelling exercise sheds some light on the possible role of regulatory biology of denitrification in controlling N_{2}O emissions from soils. If all cells in soils had the same regulatory phenotype as *Pa*. *denitrificans*, their emission of N_{2}O would probably be miniscule, and soils could easily become strong net sinks for N_{2}O because the majority of cells would be ‘truncated denitrifiers’ with only N_{2}O reductase expressed. It remains to be tested, however, if the regulatory phenotype of *Pa*. *denitrificans* is a rare or a common phenomenon among full-fledged denitrifiers. We foresee that further exploration of denitrification phenotypes will unravel a plethora of response patterns.

### Conclusion

Using dynamic modelling, we have demonstrated that the denitrification kinetics in *Pa*. *denitrificans* can be adequately explained by assuming low probabilistic transcriptional activation of the *nar* and *nirS* genes and a subsequent autocatalytic expression of the enzymes. Such autocatalytic gene expressions are common in prokaryotes, rendering a population heterogeneous because of the stochastic initiation of gene transcription, with a low probability [45]. For N_{2}O kinetics, our hypothesis was that *a)* the gas is produced by a fraction of the incubated population that is able to initiate *nirS* transcription with a certain probability, leading to a coordinated expression of *nirS* + *nor* via NO [16], and *b)* N_{2}O is consumed by the entire population because, in response to anoxia, *nosZ* is readily induced by FnrP [24]. Our model corroborated this hypothesis by reasonably simulating the N_{2}O kinetics with the specific-probability of *nirS* transcriptional activation = 0.004 h^{-1}, resulting in 7.7–22.1% of the population producing NirS + *c*Nor (hence N_{2}O), but all cells producing NosZ (hence equally consuming N_{2}O).

## Supporting Information

### S1 Dynamic Model. The folder contains the dynamic model used in this study ‘Hassan_et_al_2015_Pa._denitrificans.mdl’.

The model requires Vensim (Double Precision), which is available at http://vensim.com/free-download/. The zip folder also contains files with the empirical data; these files are automatically loaded into the model when it is run.

https://doi.org/10.1371/journal.pcbi.1004621.s001

(ZIP)

### S1 Fig. *Pa*. *denitrificans* gas and kinetics.

Typical gas kinetics (O_{2}, NO, N_{2}O, N_{2}) and accumulation in *Pa*. *denitrificans* during the transition from aerobic respiration to denitrification; batch cultures, n = 3; 20°C; Sistrom’s medium; 2 mM KNO_{3} and 7 vol% initial O_{2} in the headspace. All the available (100 μmol vial^{-1}) was recovered as before the onset of N-gas production. In previous experiments [17], N_{2}O concentrations were below the detection limit of the system, but thanks to a new system with electron capture detector, the N_{2}O kinetics were monitored with reasonable precision. Adapted from [18].

https://doi.org/10.1371/journal.pcbi.1004621.s002

(TIF)

### S2 Fig. Comparison of measured and simulated data assuming stochastic initiation of *nirS* transcription with aerobic respiration being the only energy source for producing NirS + *c*Nor.

In each panel, the measured depletion (sub-panel) and N_{2} accumulation (main panel; n = 3–4) are compared with simulations. The simulations here are to be compared with the default simulations (Fig 4), which were run assuming that the coordinated NirS + *c*Nor production (via *nirS* transcriptional activation) is sustained by the energy generated by O_{2} as well as and/or N_{2}O reduction. The default simulations provided an average specific-probability of *nirS* transcriptional activation (r_{Ni}) = 0.004 h^{-1} (Eqs 4, 5, 6 and 7) by optimisation, allowing 7.7–22.1% of the population to produce NirS + *c*Nor (Eq 8) in 19.5–47.3 h. To match the measured data here, the average r_{Ni} had to be raised to 0.012 h^{-1}, since the time available for the enzyme synthesis shrank (= 3.5–16 h) due to a rapid exhaustion of O_{2}. Comparatively, the assumption that the ATP from and/or N_{2}O reduction should help cells produce denitrification enzymes seems more plausible and provide better agreement with the measured data.

https://doi.org/10.1371/journal.pcbi.1004621.s003

(TIF)

### S3 Fig. Measured vs. simulated N_{2} kinetics assuming = 0.

The default simulations are carried out assuming that for a cell to produce first molecules of Nar and NirS, a minimum of e^{-}-flow to an available e^{-}-acceptor (, mol e^{-} cell^{-1} h^{-1}) is necessary to generate a minimum of ATP required for protein synthesis (Eqs 1, 2, 4, 5, 6 and 7). Although assuming > 0 seems logical, the measured N_{2} kinetics are adequately simulated here with = 0. This shows that the assumption is not necessary to explain the measured data.

https://doi.org/10.1371/journal.pcbi.1004621.s004

(TIF)

## Author Contributions

Conceived and designed the experiments: LLB LRB. Performed the experiments: ZQ LLB. Analyzed the data: JH LRB. Contributed reagents/materials/analysis tools: JH. Wrote the paper: JH LRB.

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