# Microbial Interactions as Drivers of a Nitrification Process in a Chemostat

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Definition

- n: the number of OTU considered.
- ${n}_{i},\phantom{\rule{0.166667em}{0ex}}i\in \{1,2\}$: the number of OTU in functional group ${G}_{i}$. In the example ${G}_{1}$ corresponds to ammonia oxidizing bacteria (AOB) and ${G}_{2}$ corresponds to nitrite oxidizing bacteria (NOB).
- Let m be an interger then $\left[m\right]:=\{1,\dots ,n\}$.
- ${x}_{i}$: is the concentration of OTU i measured in $[g/l]$. $i\in \left[n\right]$.
- x: vector ${({x}_{1},\dots ,{x}_{n})}^{\top}$.
- ${s}_{1}$: concentration of substrate 1 in $[g/l]$. In the example ${s}_{1}$ represents ammonium.
- ${s}_{2}$: concentration of substrate 2 in $[g/l]$. In the example ${s}_{2}$ represents nitrite.
- ${s}_{3}$: concentration of substrate 3 in $[g/l]$. In the example ${s}_{3}$ represents nitrate.
- ${s}_{in}$: entry concentration of substrate 1 in $[g/l]$. May depend on time ${s}_{in}={s}_{in}\left(t\right)$.
- s: vector ${({s}_{1},{s}_{2},{s}_{3})}^{\top}$. Referred to as metabolites.
- ${\mathcal{I}}_{i}(t,x)$: Interaction function of OTU $i\in \{1,\dots ,n\}$.
- ${\mu}_{i}(s,x)$: growth function of OTU $i\in \{1,\dots ,n\}$.
- $\mu =({\mu}_{1}(s,x),\dots ,{\mu}_{n}(s,x))$ vector containing the growth function of every OTU.
- D: dilution rate of the continuous reactor in $[1/day]$. May depend on time $D=D\left(t\right)$.
- ${y}_{i}$: yield of grams of OTU i formed per gram of substrate consumed.
- ${y}_{{s}_{i}/{x}_{j}}$: yield of grams of substrate ${s}_{i}$ consumed/produced per gram of OTU j formed. If negative it represents consumption, if positive it represents production.
- Y: matrix containing all yields such that ${Y}_{ij}={y}_{{s}_{i}/{x}_{j}}$.
- For integers ${m}_{1}$ and ${m}_{2}$ and $a\in \mathbb{R}$, ${a}_{{m}_{1}\times {m}_{2}}$ represents a matrix of ${m}_{1}$ rows and ${m}_{2}$ columns with a in every entry.
- Let m be an integer then ${I}_{m}$ is the identity matrix of size m.
- Let M be a matrix, then ${M}_{i\u2022}$ represents the i-th row of matrix M.
- Let S be a finite set with $m\in \mathbb{N}$ elements. Then $\left|S\right|:=m$.
- Given a vector $v=({v}_{1},\dots ,{v}_{n})\in {\mathbb{R}}^{n}$, the function $diag\left(v\right)$ stands for:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill diag:{\mathbb{R}}^{n}\to & {\mathbb{M}}_{n\times n}\left(\mathbb{R}\right)\\ \hfill \phantom{\rule{1.em}{0ex}}& \\ \hfill v\to & \left(\begin{array}{cccc}{v}_{1}& 0& \dots & 0\\ 0& {v}_{2}& \ddots & \vdots \\ \vdots & \ddots & \ddots & 0\\ 0& \dots & 0& {v}_{n}\end{array}\right)\end{array}\end{array}$$

#### 2.1. Stoichiometric Equations

**Hypothesis**

**1**

**(H1).**

#### 2.2. Mass Balance Equations

#### 2.3. Kinetic Equations

## 3. Mathematical Analysis

#### 3.1. Properties of the System

**Lemma**

**1.**

**Lemma**

**2.**

#### 3.2. Equilibrium Points

**Definition**

**1.**

**Definition**

**2.**

**Hypothesis**

**2**

**(H2).**

#### 3.2.1. Both Functional Groups Are Present

**Hypothesis**

**3**

**(H3).**

#### 3.2.2. Washout of ${G}_{2}$

**Hypothesis**

**4**

**(H4).**

#### 3.2.3. Washout of ${G}_{2}$

**Hypothesis**

**5**

**(H5).**

#### 3.3. Stability: Operating and Ecological Diagrams

Algorithm 1: Algorithm for Evaluating the Possible Equilibrium Points of System (3). |

#### 3.3.1. Case Study 1: 1 AOB and 1 NOB

#### 3.3.2. Case Study 2: 2 AOB and 2 NOB

#### Remarks

## 4. Generalized Approach for Modelling Interactions

**Hypothesis**

**6**

**(H6).**

- $\mathcal{I}\left({(0,\dots ,0)}^{\top}\right)={(1,\dots ,1)}^{\top}$
- There is an open set $\mathsf{\Omega}\subset {\mathbb{R}}^{n}$ such that $\mathcal{I}\in {C}^{1}\left(\mathsf{\Omega}\right)$.

#### 4.1. Unravelling the Interaction Function

- First, because the interest is testing the idea that interactions could be driving the system. Therefore adding a penalization in the objective function for each control to remain near 1 can be seen as an attempt to explain data without any interaction. In other words, if the control terms are found to drift from 1, it means that interactions are necessary to explain the system dynamics.
- Second, to force a regularized control. Otherwise note that u is linear in (25), therefore if the integral cost does not have a non-linear expression of u the optimal control will be of a bang-bang type with possibly singular arcs [25]. Since the objective is to find a differentiable expression of $I\left(x\right)$ the addition of the regularization term is deemed necessary.

#### 4.2. Proof of Concept

## 5. Application

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AOB | Ammonia oxidizing bacteria |

NOB | Nitrite oxidizing bacteria |

OTU | Operational Taxonomic Unit |

## Appendix A. Proofs of Properties of the System

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

## Appendix B. Deduction of Equilibrium Points

#### Appendix B.1. Both Functional Groups Are Present

#### Appendix B.2. Washout of G_{2}

#### Appendix B.3. Jacobian of the System

#### Appendix B.4. Stability Analysis with no Interactions

**Coexistence:**

**Washout of ${x}_{2}$**

**Washout**

**Jacobian of the system:**

## Appendix C. Tracking Problem Reformulation and Details

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**Figure 1.**Interaction matrices. Note how the presence of x

_{1}affects very negatively x

_{2}in (

**b**), with respect to other interactions. The terms in the diagonal entries of the matrix represent intraspecies interactions, while the terms off the diagonal represent the interspecies interactions. (

**a**) Interaction matrix of model (3) with no interactions. (

**b**) A non-zero interaction matrix of model (3).

**Figure 2.**(

**a**) Operating diagram of model (3) with no interactions (interaction matrix represented by Figure 1a). (

**b**) Operating diagram of model (3) with interactions represented by Figure 1b. Note how (

**b**) has a much larger zone where partial nitrification takes place. This is due to the negative interaction of x

_{1}on x

_{2}.

**Figure 3.**Interaction matrices for each case for a consortia of 4 bacterial species where x

_{1}and x

_{2}are AOB and x

_{3}and x

_{4}are NOB. Parameters a

_{11}and a

_{12}were modified in (

**b**,

**c**), respectively.

**Figure 4.**Ecological diagrams (ED). The different zones represent the combination of surviving species in the steady state. PN takes place when neither x

_{3}nor x

_{4}are present. CN takes place if x

_{3}or x

_{4}are present. (

**a**) ED from interaction matrix on Figure 3a. (

**b**) ED from interaction matrix on Figure 3b. In the legend (1) and (2) represent the two different stable equilibria in each zone. (

**c**) ED from interaction matrix on Figure 3c. Note that in (

**b**) two stable equilibria exist for each zone.

**Figure 5.**Synthetic data generated by model (3), with parameters from case study 1. Note the effects of the increased input

_{sin}generated in day 150.

**Figure 6.**Asterisks represent the synthetic data, while the continuous lines represent the method’s output. The method is able to reconstruct the metabolites pattern, from the biomasses concentrations.

**Figure 7.**Asterisks represent the synthetic data, while the continuous lines represent the method’s output. The method reconstructs a continuous trajectory from the synthetic data.

**Figure 10.**Simulation of system (25) when u = 1, with functions as in (27). Data points are represented by a star. The continuous line represents the simulation.

**Figure 11.**Results on applying the tracking method to a nitrification experiment when regrouping OTU in their functional groups. Data points are represented by asterisks. The continuous line represents the tracking procedure results.

**Figure 12.**The tracking procedure applied to the observed biomass (asterisks) regrouped in two functional groups.

**Figure 14.**Results on applying the tracking method to a nitrification experiment when all OTU are tracked independently. Data points are represented by a star. The continuous line represents the tracking procedure results.

Yields per Biomass Formed | $\mathit{j}\in {\mathit{G}}_{1}$ | $\mathit{j}\in {\mathit{G}}_{2}$ |
---|---|---|

${y}_{{s}_{1}/{x}_{j}}$ | $-{\displaystyle \frac{1}{{y}_{j}}}$ | 0 |

${y}_{{s}_{2}/{x}_{j}}$ | $\frac{1}{{y}_{j}}$ | $-{\displaystyle \frac{1}{{y}_{j}}}$ |

${y}_{{s}_{3}/{x}_{j}}$ | 0 | $\frac{1}{{y}_{j}}$ |

Kinetic Parameters | ${\overline{\mathit{\mu}}}_{\mathit{i}}$ [1/day] | ${\mathit{K}}_{\mathit{i}}$ [g/L] | $\frac{1}{{\mathit{y}}_{\mathit{i}}}$ [gr/gr] |
---|---|---|---|

${x}_{1}\in {G}_{1}$ | 0.77 | 0.7 | 3.98 |

${x}_{2}\in {G}_{2}$ | 1.07 | 0.3 | 16.12 |

**Table 3.**Kinetic parameters of model (3) from Dumont et al. [16].

Case Study 2 Kinetic Parameters | ${\mathit{\mu}}_{\mathit{i}}$ [1/day] | ${\mathit{K}}_{\mathit{i}}$ [mg/L] | $\frac{1}{{\mathit{y}}_{\mathit{i}}}$ [gr/gr] |
---|---|---|---|

${x}_{1}\in {G}_{1}$ | 0.828 | 0.147 | 3.85 |

${x}_{2}\in {G}_{1}$ | 0.828 | 0.147 | 3.85 |

${x}_{3}\in {G}_{2}$ | 0.18 | 0.026 | 100 |

${x}_{4}\in {G}_{2}$ | 0.18 | 0.026 | 100 |

Kinetic Parameters | ${\mathit{\mu}}_{\mathit{i}}$ [1/day] | ${\mathit{K}}_{\mathit{i}}$ [g/L] | $\frac{1}{{\mathit{y}}_{\mathit{i}}}$ [gr/gr] |
---|---|---|---|

${x}_{1}\in {G}_{1}$ | 1.97 | $7\times {10}^{-1}$ | 4.49 |

${x}_{2}\in {G}_{2}$ | 1.87 | $5.4\times {10}^{-1}$ | 45.51 |

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**MDPI and ACS Style**

Ugalde-Salas, P.; Ramírez C., H.; Harmand, J.; Desmond-Le Quéméner, E.
Microbial Interactions as Drivers of a Nitrification Process in a Chemostat. *Bioengineering* **2021**, *8*, 31.
https://doi.org/10.3390/bioengineering8030031

**AMA Style**

Ugalde-Salas P, Ramírez C. H, Harmand J, Desmond-Le Quéméner E.
Microbial Interactions as Drivers of a Nitrification Process in a Chemostat. *Bioengineering*. 2021; 8(3):31.
https://doi.org/10.3390/bioengineering8030031

**Chicago/Turabian Style**

Ugalde-Salas, Pablo, Héctor Ramírez C., Jérôme Harmand, and Elie Desmond-Le Quéméner.
2021. "Microbial Interactions as Drivers of a Nitrification Process in a Chemostat" *Bioengineering* 8, no. 3: 31.
https://doi.org/10.3390/bioengineering8030031