# Species Coexistence in Nitrifying Chemostats: A Model of Microbial Interactions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Nitrifying Chemostat Conditions

^{−1}) as the nitrogen source, complemented with a mineral solution.

#### 2.2. Microbial Community Measurements

#### 2.3. Model Development

_{in}denotes the concentration of the nutrient in the input flow, with dilution rate D. The function µ(S(t)) is the growth rate of the population and the yield factor is Y. Thereafter, to simplify the notations, the time-dependence of variables is often omitted.

_{A}and X

_{B}represent the concentrations of AOB and NOB respectively, while ${\mathsf{\mu}}_{A}\left({S}_{1}\right)$ and ${\mathsf{\mu}}_{B}\left({S}_{2}\right)$ are the growth rates of X

_{A}and X

_{B}, respectively. As suggested in [31], they are supposed to be Monod functions and are thus written as ${\mathsf{\mu}}_{A}\left({S}_{1}\right)={\mathsf{\mu}}_{max1}\frac{{S}_{1}}{{S}_{1}+{K}_{s1}}$ and ${\mathsf{\mu}}_{B}\left({S}_{2}\right)={\mathsf{\mu}}_{max2}\frac{{S}_{2}}{{S}_{2}+{K}_{s2}}$.

_{ij}represents the influence of the species j on the growth rate of species i. This influence can be positive, negative or nil according to the sign and the value of a

_{ij}.

_{ij}) corresponds to that of the well-known Lotka–Volterra model which is a widely used model in general ecology. Our model differs from the classic Lotka–Volterra model in that it is coupled to a mass-balance model (the classic chemostat model) which includes resource dynamics. Because linear systems are easier to identify, using a linear “species by species” relationship allows a simpler way of predicting species interactions. However, in doing so, the influence of resource density is neglected. An example of the use of such an interaction coefficient structure to describe microbial interactions within a cheese microbial community is given in [34]. It should also be noted that, for a given species j, the interaction terms are assumed to be constant. We are aware that this assumption may be questionable from a purely biological point of view. However, as explained in the next section, only the sign of interactions will finally be characterized. Said differently, we rather characterize the distributions in which these parameters live, instead of their precise values.

#### 2.4. Identification of the Parameters of Microbial Interaction

_{1}or S

_{2}, were studied for each of the two nitrifying functions, thus considering only four species in total.

_{A}= n

_{B}= 2, there still remains a large number of parameters to be identified (interaction parameters between the four considered species, that is to say 16 parameters, plus the degrees of freedom of the kinetics, representing eight additional parameters, thus a total of 24 parameters). To minimize this number, we adopt, in the present study, a kind of “neutral” hypothesis in considering that within a function, all the species exhibit identical kinetics and that the only difference in growth was due to their interaction. This assumption can be justified taking into account the following arguments. Without interactions, considering the same growth rates for either AOBs or NOBs, all species would exhibit exactly the same dynamics: their coexistence would be “guaranteed” (of course assuming that initial biomass concentrations are nonzero). However, it is not what is observed in practice (cf. the data we use from [31]: we will come back on these data later on but it will be clearly shown that the dynamics of AOB1 and AOB2, and of NOB1 and NOB2, are significantly different). Now, with different growth rates and again no interactions, the competitive exclusion principle applies and coexistence is simply not possible. Again, it is actually not what is observed in practice. In other terms, the kinetics associated with each species—the rate at which species i consumes its main limiting resource— should be considered to be modulated by interactions.

_{ij}such that the outputs of model #2 match the simulated outputs of model #1 (S

_{1}, S

_{2}, S

_{3}and X

_{T}= X

_{A}+ X

_{B}= AOB

_{1}+ AOB

_{2}+ NOB

_{1}+ NOB

_{2}). This optimization problem can be reformulated as follows: find the parameters of ${{\mathsf{\mu}}_{i}|}_{i=1..4}$ under the constraints (4) and the interaction parameters a

_{ij}in the model #2 such that:

- First, we randomly ran 2000 optimizations (using more did not improve the results (i.e., did not enable us to further decrease the final confidence intervals computed for each identified parameter) with different initial conditions for the parameters to be estimated (all unknown parameters in the right-hand terms of the above equations). In other terms, we identified all the parameters with different initial conditions 2000 times;
- In a second step, we have proceeded to a statistical analysis of the results, keeping only the best sets of optimized parameters that are the sets for which the SSR was, at most, 5% greater than the best solution over the 2000 optimizations. Instead of keeping only the best one, we proceeded in this way for two reasons: (i) The problem to be solved is complex, notably because the model is nonlinear and the number of unknowns is high. Thus, we did not necessarily obtain the same parameter values for each of the 2000 optimizations. In other words, the problem we consider here is said to be “non-identifiable”. It means that, given the structure of the model considered, a unique set of parameters cannot be found from the available data. In other words, there exist “hidden relationships” between parameters—possibly nonlinear—which prevent the optimization algorithm from converging towards a unique global minimum and, consequently, from delivering a unique set of optimized parameters (cf. for instance [35] or [36] for more information about this concept); (ii) in the results, either the parameters are centered around 0 with a large standard deviation (indicating either no interaction between the corresponding species or an undetermined interaction) or the signs of the identified interactions are always the same whatever the sets of optimized parameters considered and, thus, the proposed statistical analysis of the results to obtain qualitative insights about microbial interactions in the ecosystem is considered without considering precise parameter values. In the sequel, this procedure will be referred to as a “Monte-Carlo-like” approach;
- Finally, following the procedure described above, 520 sets of optimized parameters over the 2000 runs were kept and analyzed. The results are presented and discussed in the following sections.

#### 2.5. Remarks

- An important question that may arise is the following: Why are outputs of the model #1 used in the optimization problem (in the evaluation of the left-hand terms of (5) and (6)) rather than experimental data available in [31]? The essential reason is that experimental data are corrupted by noise. It is well known in system identification that, by definition, noise is not informative. Since the problem posed here is very sensitive to data (recall, in particular, that the problem is non-identifiable), it is better to use noise-free data [35,36]. Here, completely noise-free data generated with the simulation of model #1 (itself optimized with respect to real data in Dumont’s work, cf. [31]) were used instead of using real data.
- As noted before, the sum of the four species considered within model #2 represented on average about 55% of the total biomass in the experiments published in [31]. For practical purposes, before proceeding to parameter identification, the relative abundances of these four species were re-normalized in such a way that their sum equaled the total biomass as measured in the experiments. In other words, considering that model #2 (which describes the dynamics of four species) is equivalent to saying that the two biological reactions involved are now only attributable to these four species and that the remaining detected species (41 − 4 = 37 in the data considered over the 525 days of experiments) are simply ignored.
- Finally, another question that may be posed is related to the parameter identification results. One may ask whether a solution to the optimization problem should be to fix all interaction parameters to zero or not. In fact, this solution cannot give a better fit than any other because there is no reason to have X
_{A}= X_{1}+ X_{2}and X_{B}= X_{3}+ X_{4}. In other terms, individual concentrations of the four AOBs and NOB species are taken explicitly into account in the criterion used to fit model #2 parameters while only the total biomass concentration was considered in the optimization criterion to optimize model #1 parameters. Since the criterions are different, model parameters (i.e., interaction terms) are tuned so that the differences between the two sides of Equations (5) and (6) are minimized.

## 3. Results

#### 3.1. Coexistence of 2 + 2 Species on 1 + 1 Growth-Limiting Resources by Microbial Interaction

#### 3.2. Web of Microbial Interactions

**a**= +0.47. These two ammonium-oxidizing phylotypes exercise interactions of both facilitation and competition with both of the nitrite-oxidizing phylotypes. However, a stronger, mostly negative interaction (

_{22}**a**= −6.36,

_{24}**a**= −1.23 and

_{34}**a**= +4.81) of all bacteria with NOB2 is to be noted which seems to be compensated for by a strong intra-specific relationship of NOB2 with itself (

_{14}**a**= +3.26/Bimodal distribution). Concerning interactions between the two nitrite-oxidizing phylotypes, NOB1 imposes a high negative interaction on NOB2 (

_{44}**a**= −1.23) while NOB2 forces a moderately negative relationship on NOB1 (

_{34}**a**= −0.84). A moderate intra-specific competition appears for NOB1 (

_{43}**a**= −0.18/Bimodal distribution) whereas a high intra-specific facilitation appears for NOB2 (

_{33}**a**= +3.26). Concerning the type of interactions which maintain NOB in relation to AOB, a supplementary “balanced structure” appears. Indeed, if we take into account only the signs of interaction, there are as many positive as negative interactions. Whereas NOB1 presents fairly positive interactions with AOBs (

_{44}**a**= +0.44 and

_{31}**a**= +0.11/Bimodal distributions), NOB2 seems to interact rather negatively with AOBs (

_{32}**a**= −0.67/Bimodal distribution and

_{41}**a**= −0.35).

_{42}## 4. Discussion

_{ij}) with a classic model of a nitrifying chemostat. To the best of our knowledge, it is one of the first models of microbial ecosystems which tries (i) to integrate the dynamics of several interacting bacterial in the chemostat and (ii) tunes its parameters to match data together with the modeling of functions (here, nitritation and nitratation). To date, the prime focus of predator–prey models has been applied to models describing food webs in which non-trophic interactions, such as competition, facilitation and biotic disturbance, have been largely ignored [15]. Because food web models focus—by their very nature—exclusively on trophic interactions, they assume implicitly that predation is the most important process regulating community structure and dynamics. Moreover, while models of complex food webs incorporate only competition among species, they generally ignore any form of resource competition among the species (cf. [38,39]). This can be explained by the fact that, in food webs, nutrients flow via trophic links and, for this reason, trophic interactions have a fundamental character due to the principle of mass conservation.

_{ij}. However, an example of a configuration in which such interactions could develop is as follows. Consider—for some unknown reason—that AOB1 and NOB1, on the one hand, and AO2 and NOB2, on the other hand, develop specific interactions (for instance in forming ”bi- specific” flocs). Because of their proximity, AOB1 and NOB1 (resp. AOB2 and NOB2) could develop specific biotic interactions while each pair would now be in competition with the other. If such an explanation remains highly speculative, it is not inconceivable that such interactions are developing in a complex natural ecosystem.

^{+}between AOBs. Unfortunately, some interactions, such as NOB2/NOB2, as well as the low abundance of NOBs compared to AOBs, often underlined in the literature, cannot be explained.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

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**Figure 1.**Distributions of the optimized interaction parameters for model #2 after Monte-Carlo-like optimization: (

**a**) a

_{11}to a

_{14}; (

**b**) a

_{21}to a

_{24}; (

**c**) a

_{31}to a

_{34}; (

**d**) a

_{41}to a

_{44}and for the two ammonia oxidizing bacteria (AOB) and nitrite-oxidizing bacteria (NOB) of model #2 (

**e**) μ

_{max}

_{1}, μ

_{max}

_{2}, K

_{s}

_{1}and K

_{s}

_{2}.

**Figure 2.**Simulations of main outputs of models #1 and #2 in appropriate concentration units (obtained with the mean values of the optimized parameters given in Table 1 for model#1 and Table 4 for model #2, the mean values over the 520 values were used as well for parameters with bimodal distributions). X

_{A}, X

_{B}, S

_{1}(a negative part for the y-axis was considered for clarity), S

_{2}and S

_{3}concentrations from the top to the bottom, predictions of model #1 in red crosses, predictions of model #2 in green where X

_{A}= X

_{1}+ X

_{2}and X

_{B}= X

_{3}+ X

_{4}.

**Figure 4.**Interaction web (based on the statistical analysis of the signs of interactions). +, − and 0 correspond to positive, negative and neutral interactions, respectively. The size of bacteria “oval greys” and the thickness of lines indicate the trends of the bacterial abundances and the levels of interaction (based on the mean values reported in Table 3), respectively.

**Table 1.**Parameters of model #1 (adapted from [31]).

μ_{max}_{1} (1/day) | K_{s}_{1} (mg/L) | Y_{A} | μ_{max}_{2} (1/day) | K_{s}_{2} (mg/L) | Y_{B} |
---|---|---|---|---|---|

0.81 | 0.17 | 0.26 | 0.26 | 0.16 | 0.01 |

μ_{max}_{1} (1/day) | K_{s}_{1} (mg/L) | μ_{max}_{2} (1/day) | K_{s}_{2} (mg/L) |
---|---|---|---|

0.828 | 0.147 | 0.18 | 0.026 |

**Table 3.**Mean values of interaction parameters of model #2 (“B” stands for “bimodal distribution”, the sign of the interaction being specified by B/+ or B/−).

a_{11} | a_{12} | a_{13} | a_{14} | a_{21} | a_{22} | a_{23} | a_{24} |

−0.087 | ~0 | −0.697 | 4.815 | ~0 | 0.476 | 0.995 | −6.362 |

a_{31} | a_{32} | a_{33} | a_{34} | a_{41} | a_{42} | a_{43} | a_{44} |

B/+ | B/+ | B/− | −1.225 | B/− | −0.351 | −0.837 | B/+ |

**Table 4.**A particular set of optimized parameters taken randomly from the final optimized parameter sets.

μ_{max}_{1} (1/day) | K_{s}_{1} (mg/L) | μ_{max}_{2} (1/day) | K_{s}_{2} (mg/L) |
---|---|---|---|

0.81 | 0.17 | 0.26 | 0.016 |

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**MDPI and ACS Style**

Dumont, M.; Godon, J.-J.; Harmand, J.
Species Coexistence in Nitrifying Chemostats: A Model of Microbial Interactions. *Processes* **2016**, *4*, 51.
https://doi.org/10.3390/pr4040051

**AMA Style**

Dumont M, Godon J-J, Harmand J.
Species Coexistence in Nitrifying Chemostats: A Model of Microbial Interactions. *Processes*. 2016; 4(4):51.
https://doi.org/10.3390/pr4040051

**Chicago/Turabian Style**

Dumont, Maxime, Jean-Jacques Godon, and Jérôme Harmand.
2016. "Species Coexistence in Nitrifying Chemostats: A Model of Microbial Interactions" *Processes* 4, no. 4: 51.
https://doi.org/10.3390/pr4040051