## Introduction

Nitrous oxide (N_{2}O) is the third largest contributor to global warming behind carbon dioxide (CO_{2}) and methane. The emission of N_{2}O significantly affects the rate of global warming because it is a greenhouse gas and due to its destructive effect on ozone. Indeed, a molecule of N_{2}O is around 300 times more potent a greenhouse gas than CO_{2} (IPCC 2007, p. 212). Agriculture is responsible for the majority of N_{2}O emissions largely due to the use of nitrate-based fertilizer. Under low oxygen conditions denitrifying soil bacteria, with the ability to respire anaerobically, can reduce nitrate to dinitrogen via nitrite, nitric oxide and N_{2}O in a series of sequential reactions. Depending on the soil type, bacterial populations present and other factors the end product of denitrification may be emission of either N_{2}O or N_{2}. It is therefore important to develop predictive models of denitrification to provide accurate emission estimates for the constituents of the pathway, in particular N_{2}O.

Previous work on modeling denitrification considers the full range of spatial and temporal scales. The review by Heinen (2006) covers over 50 models that address a wide variety of systems including soil, sediment and whole terrestrial ecosystems. When generating the time evolution in these models the time steps can be of the order of 1 month and the denitrification module parameterized by a single rate. While these models integrate a large number of parameters, the systems they seek to describe are nonetheless governed by a plethora of bacterial species. Therefore, the models are approximations to the reaction kinetics because the actual mechanism may only be partially understood, or the bioavailability of essential minerals, for example iron and copper, omitted entirely.

The processes by which microbes emit and consume N_{2}O have been reviewed by Richardson et al. (2009). Specific enzymes in the denitrification pathway have also been the subject of detailed biochemical studies (e.g., Field et al. 2008) together with the effect exerted by genetic regulation (e.g., Bergaust et al. 2012). However, this wealth of information regarding the enzymes, and specifically their kinetic behavior, has yet to be integrated into a robust mathematical model of the chemical reactions. For example, models based on Michaelis–Menten kinetics (e.g., Cornish-Bowden 2012) have been used to supplement experimental work (e.g., Betlach and Tiedje 1981; Xu and Enfors 1996), but in most cases model parameters were chosen arbitrarily and the models used to qualitatively explain their experimental observations. By contrast, other models (e.g., Thomsen et al. 1994) calculate the kinetic parameters so that the model fits their experimental results. However, the models were not used to make more general quantitative predictions regarding the intermediates in the denitrification pathway.

Here we develop a model of denitrification for the intensively studied bacterium, *Paracoccus denitrificans*, for which many of the kinetic parameters in the denitrification pathway are known. Moreover, it has been recently shown that under certain conditions *P. denitrificans* will emit significant amounts of N_{2}O when copper is scarce (Felgate et al. 2012), permitting us to quantitatively model this effect for the first time. In *P. denitrificans*, the following reactions constitute the anaerobic denitrification pathway:

where reactions (1)–(4) are catalyzed by Nitrate reductase (Nar), Nitrite reductase (Nir), Nitric Oxide reductase (Nor), and Nitrous Oxide reductase (Nos), respectively. These reactions consume a total of 10 electrons and 12 protons when two ions are converted to a single molecule of N_{2} gas. Therefore, both electrons and protons must be available if the reductases are to effectively catalyze the reactions. Furthermore, the ability of *P. denitrificans* to successfully reduce to N_{2} hinges on the existence of copper-dependent Nos which is a homodimeric holoenzyme that binds six copper atoms per monomer (Brown et al. 2000). The copper requirement for Nos enzyme activation can be viewed as:

where apo–Nos is the inactive pro-protein prior to copper insertion. It should be noted that there are several forms of the Nos enzyme each with different catalytic activities (Zumft 1997; Table 9; Rasmussen et al. 2002). Here, all references to Nos refer to the most catalytically active form unless stated otherwise.

Following Betlach and Tiedje (1981) and Xu and Enfors (1996) we treat reactions (1)—(4) as a series of enzyme-substrate reactions and apply Michaelis–Menten kinetics. We write the reactions as:

where, for brevity, only the metabolites and enzymes are retained. The Michaelis constant and limiting rate for reaction *i* are K_{M,i} and *V*_{max,i}, with their referenced experimentally derived values shown in Table S1. A mass balance for each metabolite leads to the set of differential equations shown in Table 1. By seeking the steady-state solution, expressions that estimate the reductase concentrations are obtained. Using these values, time courses for and other relevant metabolites can then be calculated. The reductase levels obtained from the low- and high-copper experiments thus provide enzyme levels for further predictive analyses. The equations are shown in Table 1 and the formulation detailed in the next section.

Rate equations | ||

dn_{i} /dt = M_{i-1}−s_{i} M_{i} + D (n_{i,In}−n_{i} ) | i = 1…5 | (5) |

M_{0} = M_{5} = 0, M_{i}(n_{i}) = V_{max,i}n_{i} /(K_{M,i} + n_{i} ) | i = 1…4 | |

Implied experimental enzyme concentrations | ||

e_{i} = (1 + K_{Mi} /n_{i})(M_{i−1} + D (n_{i,In}−n_{i}) − dn_{i} /dt)/(s_{i} k_{cati}) | i = 1…4 | (6) |

Implied steady-state enzyme concentrations | ||

e_{i,ss} = (1 + K_{Mi} /n_{i})(M_{i−1} + D (n_{i,In}−n_{i})) / (s_{i} k_{cat i}) | i = 1…4 | (7) |

Predicted Nos concentration | ||

e_{4,ss}([Cu]) = e_{4,Init}([Cu]) = α[Cu]/(β + [Cu]) | (8) | |

e_{4}(t) = e_{4,Init} exp(−Dt) + e_{4,ss} (1–exp(−Dt)) | (9) | |

Nomenclature | ||

n_{1} = [], n_{2} = [], n_{3} = [NO], n_{4} = [N_{2}O] and n_{5} = [N_{2}] | ||

e_{1} = [Nar], e_{2} = [Nir], e_{3} = [Nor] and e_{4} = [Nos] |

Parameter | Description | Value |
---|---|---|

s_{i} | Substrate stoichiometric constant | 2 for i = 3, else 1 |

K_{M,i} | Michaelis constant for reaction i | Table S1 |

V_{max,i} (= k_{cat,i} e_{i}) | Limiting rate for reaction i | |

k_{cat,i} | Turnover number for reaction i | Table S1 |

D | Dilution rate | 0.05 h^{−1} |

n _{i,In} | Inflow concentration for n_{i} | 20 mmol/L for 0 mmol/L for |

α, β | Calibration parameters | Table S3 |

For a given experiment the model therefore provides a method of predicting the enzyme concentrations and the time courses for the metabolites. The model also facilitates the quantitative prediction of N_{2}O emissions (and the other metabolites) for a prescribed copper concentration.

The model is available for download from http://www.uea.ac.uk/computing/software/modelling-denitrification.