# The chemostat: from dynamical systems to possibilistic networks

Valentina Soto Minutti

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Valentina Soto Minutti. The chemostat: from dynamical systems to possibilistic networks. Modeling and Simulation. 2022. hal-04327695

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# The chemostat : from dynamical systems to
possibilistic networks.

Student : Valentina SOTO MINUTTI

Université Toulouse III - Paul Sabatier

Mention Biodiversité, Écologie et Évolution

M2 Modélisation des Systèmes Écologiques

2022-2023

## Tutors

Jérôme HARMAND<fnref n="1" /> - Élie LE-QUÉMÉNER<fnref n="1" /> - Alain RAPAPORT<fnref n="2" /> - Cédric GAUCHEREL<fnref n="3" />

<footnote n="1">¹LBE - Laboratoire de Biotechnologie de l'Environnement - Narbonne, France</footnote>

<footnote n="2">²UMR MISTEA - Mathématique, Informatique et Statistique pour l'Environnement et
l'Agronomie - Montpellier, France</footnote>

<footnote n="3">³UMR AMAP - botAnique et Modélisation de l'Architecture des Plantes et des végétations -
Montpellier, France</footnote>
To my parents, Marcia and Fernando, who have unconditionally been there for me.

To Marco, who has always believed in me and will be graduating with me.

I'm grateful for the kindness and opportunities given by those who supported my career journey.
## Acknowledgments

I would like to thank Jérôme for the chats and explanations on the board, Élie for his patience, valuable daily feedback and code debugging, Alain for the math classes, talks and chess games and Cédric for teaching me the basis of the possibilistic method. I am equally thankful to the LBE for hosting me, in particular to all the PhD students, interns and researchers I met from different cultures and disciplines who helped me gain a better understanding of microbiology and chemical engineering.

I am especially thankful for the diversity of lessons learned during this internship which prepared me for the job of a researcher.

## Contributions

The formalisms of the approach and an idea of the expected results for the simple models were already chosen before the internship. The student built the algorithms. Among the specific tools to do this in the MSE minor, was the knowledge of dynamical systems, programming, time series analysis, Bayesian statistics, informatic tools for science (LateX, Python and R). Adaptive dynamics were fundamental for the biological interpretation of the models.
Abstract

Understanding microbial ecosystem dynamics is a challenge in biological sciences. Traditional approaches face uncertainties due to sensitivity to initial conditions and parameters, among others. This study explores translating complex dynamical systems to qualitative models in order to assist in their synthesis. We use the chemostat model in continuous culture in order to investigate the fidelity of discretization when translating differential equations into a set of qualitative states and transitions. Our hypothesis is that this transformation can (1) yield insights beyond traditional methods for two-dimensional models and (2) be extended to more complex models, which are analytically hard to study. The project focuses on predicting (1) the possible states of a system and (2) all transition rules guiding state changes. This methodology is developed for simple models, revealing tendencies not found in quantitative methods, and is also successfully applied in complex models. However, the biological interpretation of the transition rules remains a work in progress since the development of this methodology still requires confrontation with empirical data.

Résumé

La compréhension des dynamiques au sein des écosystèmes microbiens est un défi en biologie. Les approches traditionnelles sont sensibles aux fluctuations des conditions initiales et des paramètres ? Cette étude a pour objectif the traduire des systèmes dynamiques complèxes en modèles qualitatifs, afin de contribuer synthétiser leur comportement. Notre hypothèse est que cette transformation peut (1) amener des nouvelles perspectives pour les modèles à deux dimensions et (2) être étendue à des modèles plus complèxes, difficiles à étudier analytiquement. Le projet se focalise dans la prédiction (1) les états possibles du système et (2) la totalité des transitions entre ces états. Cette méthodologie est développée à partir de modèles simples, en révélant des tendances non observées dans d'autres approches, et est aussi appliquée avec succès dans des modèles plus complèxes. Pourtant, l'interprétation biologique des règles est un travail en cours qui devra faire objet de calibration sur des données expérimentales.
# Contents

## 1 Introduction 7

1.1 Laboratoire de Biotechnologie de l'Environnement (LBE) 7

1.2 On the study and modelling of microbial growth 8

1.3 Starting point 8

1.3.1 About the latest quantitative methods : limitations 10

1.4 Benefits of an integrative approach 10

1.5 Objectives, questions and hypothesis 12

## 2 Materials and Methods 12

2.1 Introduction to the three chosen models 13

2.1.1 Simple model: a single species growing in a substrate 13

2.1.2 Intermediate model: Two species and in competition for a substrate 13

2.1.3 Third model: a nitrifying chemostat 14

2.2 Reminder of the chemostat growth functions 15

2.3 Concrete example of qualitative description with a simple model 16

2.4 Translating Ordinary Differential Equations into qualitative networks 18

2.4.1 Determining qualitative variables 18

2.4.2 Generating sets of parameters 19
<table><tr><td>2.4.3</td><td>Generating initial conditions</td><td>19</td></tr><tr><td>2.4.4</td><td>Simulations</td><td>20</td></tr><tr><td>2.4.5</td><td>Coupling of the signs</td><td>20</td></tr><tr><td>2.4.6</td><td>Binary transition matrix</td><td>22</td></tr><tr><td>2.4.7</td><td>Boolean Networks: Translating transitions into generic rules</td><td>23</td></tr><tr><td>2.4.8</td><td>Verification step with Jupyter EDEN interface</td><td>23</td></tr></table>

## 3 Results 24

<table><tr><td>3.1</td><td>One species and one substrate</td><td>24</td></tr><tr><td>3.2</td><td>Two species and one substrate</td><td>26</td></tr><tr><td>3.3</td><td>Two species and three substrates</td><td>26</td></tr><tr><td>3.4</td><td>Verification with Jupyter EDEN server</td><td>27</td></tr></table>

## 4 Discussion 28

<table><tr><td>4.1</td><td>Important remarks</td><td>28</td></tr><tr><td>4.2</td><td>Did the discretization choice allow for a faithful translation?</td><td>30</td></tr><tr><td>4.3</td><td>Practicality of rule-based approaches in biology</td><td>30</td></tr><tr><td>4.4</td><td>Perspectives: Limits and necessary improvements</td><td>30</td></tr><tr><td>4.5</td><td>Conclusion</td><td>32</td></tr></table>

## A Appendix 37
B Chemostat equations 37

B.1 Simple model: one species and one substrate 37

B.1.1 Rules 37

B.2 Haldane trajectory 38

C Supplementary materials about the models 39

C.1 Model with two species and one substrate 39

C.1.1 Transition matrix: Model with two species and one substrate 39

C.1.2 Rules 42

C.2 Nitrifying chemostat 45

C.2.1 Experimental device 45

C.2.2 State combinations for the model with two species and three substrates 46
# 1 Introduction

Microbial communities significantly influence global biogeochemical cycles, plant-nutrient relationships and interactions among diverse taxa [19]. In disciplines destined to serve human societies, the natural dynamics of microorganisms play a central role in human biology, agri-food systems and environmental biotechnologies. However, the **modelling of microbial ecosystems** is an ongoing challenge for the scientific community, in both **fundamental and applied research** [38].

## 1.1 Laboratoire de Biotechnologie de l'Environnement (LBE)

In the context of applied research, bioprocesses employ **already-existing** biological processes in an artificial setting. The laboratory in which the internship took place (LBE), intends to develop the concept of **environmental biorefinery**, which aims to transform biomass into renewable non-fossil fuels, as a byproduct of microbial metabolism, such as bacteria and microalgae, which have the capacity to produce biofuels, often considered as one of the most valuable alternatives to fossil fuels[39] [4]. Their research also focuses on **the treatment of residues**, such as solid and liquid effluents derived from industrial activities and household waste. Residues **can be valorized** if subjected to microbial anaerobic digestion for producing biogas and organic crop fertilizers [30], while also promoting the reusing of residual matter [36]. Luckily, the increasing diversity of anthropogenic pollutants found in water, threatening aquatic and terrestrial ecosystems, **can** also be addressed, since **biodegradation can** be performed by certain strains of yeasts, bacteria, fungi and algae [18].
## 1.2 On the study and modelling of microbial growth

Generally, study and development of these processes is attempted through numerically powerful tools, such as dynamic simulations, which allow predicting performances based on (ODE) that represent the evolution of microbial biomass and metabolism as a response to a substrate input.

Traditionally, **metabolism** was studied by observing the **growth of microorganisms** in a closed medium, also known as **batch culture** [22]. However, batch culture was not suited for studying certain phenomena, especially since the nutrients run out after consumption [25]. This is why, **continuous culture** in a chemostat (Fig. 1), where microorganisms grow in a medium with a constant inflow and outflow of nutrients and substrate, was the chosen setting for our modelling approach.

> The main objective of this project is to introduce qualitative modelling in the context of environmental bioprocesses, and to evaluate its relevance for representing microbial ecosystems and their behavior.

## 1.3 Starting point

Complex models have been the focus of several scientists' work throughout history, as can be the study of **substrate uptake kinetics**, and **predator-prey interactions**.

The studying of these systems has contributed to the understanding of time-consuming phenomena and **fundamental questions** about physiology and evolutionary mechanisms such as species co-evolution or ecological successions [12], even reaching the molecular level [10]. This contributed greatly to bridging the gap between experimental microbiology and *in-situ* studies performed by ecologists [14]. Generally, these subjects are already
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Schematic illustration of a continuous culture. The sterile medium is pumped into the chemostat vessel ($Q_{in}$) along with sterile air. The stirring mechanism allows for equal access of all microorganisms to the nutrients, which is why the medium is assumed to be homogeneous. Simultaneously, the outflow of media and cells is collected ($Q_{out}$). The clear blue area represents the gaseous phase, whereas the clear yellow area represents the liquid phase where microorganisms grow. Adapted from [23].</figcaption></figure>

approached with **quantitative models** which are based on **systems of dynamical equations**.

Why consider another model for the chemostat, given the existing variety of modelling approaches? The reason lies in the essential significance of the chemostat's **emergent properties** for enhancing these methods. Currently, these properties remain insufficiently understood [27].
1.3.1 About the latest quantitative methods : limitations

In microbiology, the ‘omics’ era has accelerated the development of theoretic bio-
chemical models [12] [5], which are primarily based on differential equations. Indeed,
an understanding of microscopic mechanical properties became recently possible through
molecular dynamics simulation [28], but even the most popular models are found by users
to be excessively complex and time-consuming [11] and model parameterization is still
one of the biggest challenges in system biology [26]. Experimental measurements are also
made in spaced time intervals. Thus, time delays between experimental measurements
force a discrete nature on data. The intrinsic constraint linked to the structure of data
could explain why classical approaches lack support to match the increasing complexity
of ecosystem modelling.

**1.4 Benefits of an integrative approach**

When it comes to understanding biological phenomena, predictions are more robust
when a plurality of methods is used [15], which justifies the interest in finding an inte-
grative approach.

This method is meant to yield a **visual intuitive representation** of a system of
asynchronous events condensing information on inter-species interaction and on abi-
otic factors. Ultimately, this could allow for an **efficient inference** of the system's
possible **states** and **transitions**. The aim is to bridge the gap between Ordinary Dif-
ferential Equations (ODE)'s **well-established methodology** with this approach's
**expressive power** even as the number of interactions increases and the system
becomes more intricate.
**Qualitative descriptions** are an intuitive way of representing a system's main attractors [32]. Building this abstract representation is a relatively easy task if the model is **not complex**, or has fewer effects than more realistic models, and is well-studied in literature.

**Symbolic methods** have proven helpful when trying to evaluate the existence of irreversible states in a system's behavioural analysis [2] and are the main basis of this approach. The system is viewed as a **network of nodes**, corresponding to the realization of **events**, interconnected through observable or non-observable **transitions** [13]. Since they were born as a response to computational complexity, their intuitive aspect is pursued by building a **generic graphic description of a parameterised system**.

Among symbolic methods, **Boolean networks** [17, 33] are used as a simple, yet expressive formalism [1] to model state transition systems [31]. They are useful for representing **intertwined feedback loops** in an ecosystem, since handling them otherwise can be counter-intuitive and sometimes impossible [34]. These networks generate a **highly abstract** representation of large systems while performing an exhaustive coverage of the space of states [32], allowing for the intuitive identification of the system's main **attractors** [37]. Among Boolean networks, **Petri-Nets** [29] are currently used for describing the topology of **non-sequential** discrete event systems since one of their major attributes is their ability to detect errors in a short number of observations [13].

Inspired by these formalisms, 'Ecological Discrete Event Networks' was developed with the purpose of assisting scientists as a tool to **discriminate between different scenarios** in order to choose the most suitable model.
## 1.5 Objectives, questions and hypothesis

Since the greatest difficulty in switching between quantitative and qualitative models is finding a **faithful** qualitative translation of the system, the central question is:

Is it possible to find a good discretization that allows a faithful translation of a system of differential equations into a *possibilistic* language?

When attempting to find an accurate generic expression of complex dynamical systems, other than the ones traditionally used, **theoretical data** was used. After a **translation of quantitative data into qualitative data**, the **hypothesis** is that it is possible to:

(1) Obtain a generic qualitative representation of a simple chemostat model (two dimensions) and explore trying to find additional information absent in ordinary differential equation (ODE) approaches.

(2) Utilize this approach to observe trends for models that are more complex and have more than two dimensions, making them harder to study using traditional methods.

The main objective in this project is to implement algorithms capable of predicting, from the **initial conditions** of the system and a set of parameters

- (1) the set of **possible states** of the system

- (2) the **transition rules** between these states.

# 2 Materials and Methods

Firstly, the three **ODE models** that were qualitatively translated will be introduced, followed by a brief reminder of the chosen **growth functions** and a **concrete example** to present the approach. Lastly, a **detailed explanation of the steps** used to perform the translation will be shown.
## 2.1 Introduction to the three chosen models

### 2.1.1 Simple model: a single species growing in a substrate

This model studies the dynamics of $s$ (substrate) and $b$ (biomass) over time, where a microbial species metabolizes a single substrate. The continuous culture system's dynamics were supposed to follow the **Chemostat equations** (see Appendix A).

Once this equations are adapted, the system becomes :

$$ egin{cases} \frac{ds}{dt} = D(S_{in} - s) - \mu(s)b \\ \frac{db}{dt} = \mu(s)b - Db \end{cases} \quad (1) $$

where $D$ is the dilution rate ($h^{-1}$), $S_{in}$ is substrate influx ($g/L/h$), and $\mu(s)$ is the specific growth function of the species ($1/h$).

The summarizing power of phase portraits can be high for simple models (i.e. two-dimensions), making it easier to verify it. However, in more complex models, the **higher number of variables and parameters, increases their complexity** and makes their mathematical analysis harder to perform.

### 2.1.2 Intermediate model: Two species and in competition for a substrate

The previously created algorithm was **modified** in order to create a slightly similar model, except for the fact that instead of one species and one substrate, this model
assumes that **two species** share the substrate a common food source in a bioreactor :

$$
\left\{
\begin{aligned}
\frac{ds}{dt} &= D(S_{in} - s) - (\mu_A(s)b_A + \mu_B(s)b_B) \\
\frac{db_A}{dt} &= \mu_A(s)b_A - Db_A \\
\frac{db_B}{dt} &= \mu_B(s)b_B - Db_B
\end{aligned}
\right.
\qquad (2)
$$

where s, b_A and b_B stand for the concentration of (g/L) of substrate and biomass of species A and B respectively, and μ_A and μ_B indicate the specific growth function of each species (b_A and b_B respectively) in (1/h).

In continuous culture, leaching (i.e. the loss of biomass) can happen if the dilution
rate is superior to their maximum growth rate. In order to explore the faithfulness of this
approach when studying this model, different cases were tested to compare leaching and
equilibrium in qualitative networks (D > μmax of species A and B).

2.1.3 Third model: a nitrifying chemostat

A more complex model with commensalism between two microbial species was also studied using results from a previous publication [5]. Two distinct functional groups of bacteria are represented according to their functional type, i.e. *Ammonia Oxidizing Bacteria* (*AOB*) and *Nitrite Oxidizing Bacteria* (*NOB*) (see Appendix Fig. 14). The ODE system for this model is:
$$ 
\left\{
\begin{aligned}
\frac{db_A}{dt} &= (\mu_A(s_1) - D)b_A \\
\frac{db_B}{dt} &= (\mu_B(s_2) - D)b_B \\
\frac{ds_1}{dt} &= (S_{in} - s_1)D - \frac{1}{YA}\mu_A(s_1)b_A \\
\frac{ds_2}{dt} &= -s_2D + \frac{1}{YA}\mu_A(s_1)b_A - \frac{1}{YB}\mu_B(s_2)b_B \\
\frac{ds_3}{dt} &= -s_3D + \frac{1}{YB}\mu_B(s_2)b_B
\end{aligned}
\right.
$$

where YA and YB represent the yield constant of each species, which is the amount of biomass produced per every unit of substrate consumed for each functional group, and s1, s2, s3 denote the concentrations of substrate introduced in the reactor, substrate produced by the first functional group and substrate produced by the second functional group, respectively.

## 2.2 Reminder of the chemostat growth functions

Each of the three models was tested with three distinct specific growth functions (Haldane, Monod and Contois) in order to observe the contrasts.

Among the many growth functions used to understand microbial growth, Jacques Monod's growth function (1950) (Fig. 2) is by far the most used in literature [21]. Alternatively, microbial growth rate occasionally drops as a response to excess substrate concentration [3], known as the **inhibitory effect**. In such cases, then the growth function discovered by John B.S. Haldane (1930) is the most adequate. In order to take into account microbial biomass for substrate production, Contois's growth function was also tested. It is possible to visualize an example of the growth of a single species according
to each expression below (Fig. 2).

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Curves of Monod, Contois, and Haldane Growth functions respectively. It is possible to see the effect of the inhibitory constant on Haldane's growth function.</figcaption></figure>

## 2.3 Concrete example of qualitative description with a simple model

In the case of a simple chemostat, **one species** grows while consuming a single **substrate**, given a set of initial conditions and parameters. It is possible to **qualitatively describe** the system using the derivative of the variables. Depending on whether the variables are increasing or decreasing, the sign of the derivative is used to characterize the state of the system:

* 'increasing' (+)

* 'stabilizing or decreasing' (-)

Based on this example, it is possible to qualitatively summarize the transitions of the
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Example of one simulation of two trajectories of the chemostat with initial conditions $b_0 = 10, s_0 = 5$. Time series of $b$ (microorganism concentration) and $s$ (substrate concentration) evolving over time using the Haldane growth function. It is possible to see that after optimal concentration, microorganism growth is inhibited, as is assumed by the Haldane growth function (parameters: $D = 0.40; \mu_{max} = 5; Sin = 100; KI = 4; Ks = 50; s_0 = 5; b_0 = 10$).</figcaption></figure>

system with a **transition rule**, which illustrates the transition between one state of the system and another :

$$ds^+, db^+ \gg db^- \quad (3)$$

Here, $s$ (substrate) and $b$ (biomass) can be qualified as increasing and decreasing. The symbol ‘»’ separates **conditions** and **resulting** states, in that order. A condition and a resulting state are each viewed as **single qualitative states**, even when constituted by several decreasing and increasing variables. To avoid redundancy, if a variable does not change after a transition, it is omitted in the resulting state. Following this symbolic rule, it is possible to interpret that

If substrate and biomass are increasing, then biomass can decrease.

where **If** precedes the state, or condition, that **can** possibly induce the transition towards the resulting state. By using this method, the quantitative trajectories were
translated to a qualitative representation, summarized by a rule.

Since this is a particular trajectory of the system of dynamical equations given a set of initial conditions, the interest of this project was to extract the main trends when a **wide range of initial conditions is simulated**. Translating this **simple, well-known** model was useful to build a **network** of states, in which it is possible to circulate by using rules. **Once the translation between quantitative and qualitative models was deemed faithful, the model was made increasingly complex** by adding more differential equations.

## 2.4 Translating Ordinary Differential Equations into qualitative networks

### 2.4.1 Determining qualitative variables

The number of qualitative variables is different for each model:

- **Model: One species, one substrate**

- Qualitative variables: $ds, db$

- State of the variable: $ds^+, ds^-, db^+, db^-$

- Combination of initial conditions: $s_0, b_0$

- **Model: Two species, one substrate**

- Qualitative variables: $ds, db_A, db_B$

- State of the variable: $ds^+, ds^-, db_A^+, db_A^-, db_B^+, db_B^-$

- Combination of initial conditions: $s_0, b_{A0}, b_{B0}$

- **Model: Two species, three substrates**
- Qualitative variables: $ds_1, ds_2, ds_3, db_A, db_B$

- State of the variable: $ds_1^+, ds_1^-, ds_2^+, ds_2^-, ds_3^+, ds_3^-, db_A^+, db_B^-$

- Combination of initial conditions: $s_{10} - s_{20} - s_{30} - b_{A0} - b_{B0}$

where $ds, ds_1, ds_2, ds_3, ds_4, ds_5$ stand for substrate variation, and $db_A, db_B$ stand for biomass variation of two different species. The sense of variation, or state of the variable, is given by the sign.

## 2.4.2 Generating sets of parameters

In order to choose the values of the parameters, a **pseudo-random sampling from a Uniform distribution** ranging from 0 to $S_{in}$ was performed since there was no prior information about the distribution. After a large number of repetitions (n = 100), a mean is calculated, obtaining an averaged value for each of the general parameters $D$ (Dilution rate), $S_{in}$ (Substrate influx), but also for the specific growth function parameters $\mu_{max}$ (Maximum specific growth rate), $K_i$ (Inhibition constant), and $K_s$ (Saturation constant).

This method, also known as Monte-Carlo method, was chosen is that it is a robust algorithm to optimize the 'estimation' of each parameter [16].

## 2.4.3 Generating initial conditions

For each model and each growth function, initial conditions for substrate and biomass were also drawn from a Uniform distribution, ranging from the minimal initial concentration (0.01) and a chosen maximal inoculation value, inferior to the total volume of the reactor. For the first model, a **combination** of two values was drawn while **three** for the second model and **five** for the last model. This was repeated a number of times
(n = 10 000) in order to build a list of initial conditions.

### 2.4.4 Simulations

Each model was simulated for the list of initial conditions using an already implemented differential equation system solver in R (package *deSolve*).

> For simplicity and convenience regarding calculation time, it was more practical to study the **sign of each variable's derivative between two roots**. The objective was to describe the system's transitions by **successions of monotonies** of the state variables' derivatives.

An already implemented function was used to **detect asymptotic behaviour** and avoid false changes in the sign of the derivative for each set of initial conditions. If the slope between the two chosen time steps is considered inferior to a certain threshold (1e-10), it is considered null.

At each switch in **monotonic behavior** *behavior tendencies*, the sign of the slope (1 if positive, 0 if negative) of all state variables were coupled (Fig. 4).

A specific threshold to induce a change in a state of the system could also have been chosen, as done in the majority of qualitative approaches could have been chosen, yet this is another source of difficulty [9].

### 2.4.5 Coupling of the signs

> Since the derivatives of an ODE system vary in parallel, the **sign** of the derivatives were coupled **for each simulation**.
Since the trajectories were transformed from continuous to binary, depending on the sign of the slope of the state variables (+ or 1 if positive, - or 0 if negative), a different number of combinations was obtained for each of the three models.

For the simple model, 4 possible combinations ('0-0', '0-1', '1-0', '1-1') were obtained, and for the intermediate model, 8 possible combinations ('000', '100', '010', '110', '001', '101', '011', '111'). For the complex model (i.e. three substrates and two different species) there were 32 possible states (see Appendix).

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Schematic illustration of the coupling of monotonic behaviour of the trajectories for a given example. It is possible to see the name of the variable, in this case s and b), and the state that she is going through (positive or negative).</figcaption></figure>

In the example above, the **successions of monotonies** of the trajectories of the two derivatives can be described as follows :

$$ds \xrightarrow{1,1} ds^+, ds^+ \qquad (4)$$

$$db \xrightarrow{1,0} db^+, ds^- \qquad (5)$$
When the trajectories are **coupled**, the system can be described by a **qualitative rule**, expressing its **transition** from one state to another :

$$ \underline{ds^+, db^+} \rightarrow \underline{ds^+, db^-} \qquad (6) $$

$$ \text{State 1} \qquad \text{State 2} \qquad (7) $$

### 2.4.6 Binary transition matrix

The previous example was **specific** to a set of initial conditions of the simple model.

In order to **summarize the observed transitions for the entire set of initial conditions**, a **binary transition matrix**, representing the **presence** and **absence** of a transition by '1' and '0' respectively. The vertical states represent the **condition** needed to produce an **outcome** (Fig. 5).

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Schematic representation of a transition matrix confronting conditions (coupled states of each state variable) and possible outcomes (coupled states of each state variable). When a transition is encountered among the simulated trajectories, '1' is added to the corresponding row i and column j. This process is irreversible and cumulative and is repeated with as many initial conditions as possible.</figcaption></figure>

As an additional tool to obtain information about the **occurrence** of an observed transition, meaning if it is **rare** or not, the transitions were accumulated and normalized in relation to the total number of transitions, as a percentage. However, this constitutes
ongoing work that was not fully resumed.

### 2.4.7 Boolean Networks: Translating transitions into generic rules

It was then possible to use the binary transition matrix to build an **interaction graph** (package igraph).

For a reminder, according to this approach, a **node** is a **set of states** (ex.: 'ds⁺, db⁺' in which 'biomass and substrate increase'). The transition between two nodes is represented by **edges**, or **rules**.

### 2.4.8 Verification step with Jupyter EDEN interface

So far, the model simulations and the transition rules have been obtained using *RStudiodio*. A previously implemented tool was used in order to obtain further information. Interaction networks were thus **also** generated **in parallel** using *Python* in a *Jupyter* interface since the code itself can be quite tedious to handle [35]. This platform aims to model ecosystems qualitatively with any given set of rules. The chosen EDEN approach can be characterized by (1) its **non-deterministic** nature since trajectories are not declared and the outcome will depend on the rules [8] but also by (2) its **asynchronous** nature, which can be beneficial since simultaneous changes from one node to the next can be multiple [34]. Whenever a node has no possible following node, it is said to be a deadlock or a state with no possible outcome.
After uploading the **rules obtained in R with a specific notation** to this platform, the possibilistic networks could also be built. Since this platform is able to return a transition matrix, an algorithm was created to import it back into R and compare it to the original matrix, just to verify that the approaches are analogous.

This was done by altering the *Jupyter* matrix to obtain the same format and then performing a matrix subtraction. This part will mostly serve for the **continuation** of the project after the internship is over. The resulting networks are visible in the Appendix.

# 3 Results

Interaction networks were built for the three models. As a reminder, the objectives were to (1) **Simulate** a theoretic chemostat under different growth assumptions, (2) **Translate** this system into a network of discrete events, (3) **Test** it on well-known models easy to analyse and (4) **Explore** more intricate models.

## 3.1 One species and one substrate

(1) **Graphic Network**: No differences were found between Monod and Contois (Fig. 6) in the number of transitions ($n=4$) and two supplementary transitions were found for Haldane ($n=6$).

(2) **Rules**: Two supplementary rules were found for Haldane, in comparison with the two other growth functions (Fig. 6).
This model can be qualitatively studied by a **phase portrait** (Figs. 6a, 6b, 6c).
Since qualitative states and transitions can be deduced from it, it is possible to
represent it by using a **boolean network** (Figs. 6d, 6e, 6f).

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Comparison between the qualitative well-known method of phase portraits (upper row) and qualitative networks (middle). Comparison between quantitative and qualitative approaches of the simple model. (2a) Phase portrait qualitatively summarizing the possible transitions of a simple chemostat with one species growing with a substrate. It is possible to observe the four states of the system (‘−’, ‘+−’, ‘++’, ‘−+’). (2b) The first and second characters stand for *ds* and *db*, respectively. The bottom row shows the frequency of transitions.</figcaption></figure>

Each of the transitions is recapitulated as lists of rules (see Appendix).

The list of trajectories was identical for Monod and Contois (Fig. 7). Four of those trajectories were also observed for Haldane ('01', '11-01', '01-00-10', '10-11-01'), yet a
specific transition was observed solely in this case ('01-11') (see Appendix, Fig.10).

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Trajectories of substrate and biomass as a function of time representing the 8 possible successions of states found for the simple model. Note that some states are rare and some states lead systematically to other states.</figcaption></figure>

## 3.2 Two species and one substrate

(1) **Graphic network and rules:** When a supplementary species is added, the interactions differ between the three growth functions (Fig. 8). Also, the rules are increasingly specific when one species is added. Configurations according to the value of *D* with respect to the maximum growth rate of each species.

## 3.3 Two species and three substrates

(1) **Graphic network and rules:** When a supplementary species is added, the interactions differ between the three growth functions, as shown in the transition matrix. Also, the rules are increasingly specific when one species is added.
<figure><img src="image_9.png" /><figcaption>**Fig. 8.** Possibilistic networks representing all the possible state transitions of the system for the model, where two species share a substrate. Each network corresponds to one of the three tested growth functions.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Possibilistic network of the states of the system. The characters correspond to $db_A$, $db_B$, $ds_1$, $ds_2$ and $ds_3$ respectively.</figcaption></figure>

## 3.4 Verification with Jupyter EDEN server

The transition rules were extracted from the transition matrix using R. These rules were then uploaded to the Jupyter platform, where the algorithm attributes a rule to each transition (see Appendix).
# 4 Discussion

## 4.1 Important remarks

(1) It is interesting to see that for the model with one species and one substrate, Monod and Contois growth functions seem to **behave similarly** until the model becomes complex. Haldane's growth function shares 4 of their trajectories but has a trajectory that is largely present (11–01). This means that in half of the transitions, biomass still increases even if the substrate is decreasing. This could be a lag phase in which bacterial growth is still metabolizing the substrate right before inhibition. However, it could also be a mistake due to the chosen way to describe a system with the sign of the derivatives.

(2) The most striking feature of this study is that a possibilistic network is that it is possible to know **how many times a transition is encountered**, which makes possible networks more informative for models with more than two dimensions. This is why it is necessary to complete it with a **cumulative frequency** for each transition. Some of these frequencies are sometimes very close to 0. This is likely to be a numerical calculation problem. However, it might be one of those rare shifts observed in intricate biological systems.

(3) Furthermore, the **frequencies can yield information analogous to the size of the basins of attraction**. Said otherwise, the size of the area between the axis and the nullclines of a phase portrait could be translated to the number of times a trajectory ends up in a particular state (for example leaching). In this regard, it could be interesting to repeat the sampling several times in order to create thresholds, so that it becomes possible to **differentiate between a rare transition and a numeric artefact**, linked to oscillations when asymptotic behaviour is encountered. This means that if after a large
number of samples, the transition is encountered in a small number of cases, it is most
likely not an observable transition.

(4) The fourth most striking feature is that if the understanding of the **frequencies** improves, it might become viable to **revert back to the configuration of the phase portrait** and thus **have more accurate estimations of the parameters** of a model based on experimental data. This is useful since more often than not, parameters are drawn from discrete measurements for the most part, but also fitted theoretically [20], which can cause uncertainty when trying to build predictions. This could also **reduce** the **impact of overfitting and ill-conditioning**, a small error when choosing the set of parameters can lead to magnified errors in the conclusions [7].

(5) Haldane's growth functions have fewer possible successions of transitions than Contois and Monod in the simple model. If confronted with experimental data, this could be the most interesting result since when a certain variation in a time series (i.e. fluctuations of substrate or biomass), is observed it would be possible to state that these can not be found in 'Monod' and 'Contois', not to say that the microbial species is likely to follow a **transition that can only be seen if there is an inhibitory effect**.

(5) The approach is also **partially verified for the three-dimensional model**, since leaching and equilibrium are still well represented in the qualitative networks. However, the model of the nitrifying chemostat is not only too complex to interpret when it comes to its rules but is also demanding in calculation time. Thus, the number of initial conditions had to be smaller in comparison to the other tested models. This is why this model **served** to test the approach on five-dimensional models but not to withdraw conclusions.
## 4.2 Did the discretization choice allow for a faithful translation?

As expected, the qualitative network reflected the transitions seen in the phase portrait. Nevertheless, a **finite list of trajectories** was found due to the translation of the system. As expected, the number of transitions between states increases with the complexity of each model. It is however remarkable that given the number of simulations and the degree of nestedness, **it is still possible to simplify the whole system to a finite number of rules**.

## 4.3 Practicality of rule-based approaches in biology

Even if the resulting EDEN networks seem hardly intuitive to read at first sight, the recapitulation of the list of rules as well as its verbal explanation could be very useful for biologists and other experts. Indeed, **rule-based modelling** languages have proved useful for people who are not necessarily familiar with the mathematical formalisms of a model [24]. Most experimental biologists are still **not familiarized** with software and modelling tools [6]. In addition, because it is based on **graphical** formalisms, it can yield a source of simplification for biochemical networks, which can more often than not imply unbearable combinatorial complexity ([6]). Indeed, some interpretations can be made for each of the rules (see Appendix), yet this becomes almost impossible for the nitrifying chemostat model.

## 4.4 Perspectives: Limits and necessary improvements

The possibilistic approach provides a qualitative simple approach. However, further work needs to be done to **confirm the veracity** of the transition rules. The simulation of
parameters using Monte-Carlo methods is robust, but **calibration** seems to be a crucial
stage when working with parameter-sensitive models.

Also, when retrieving the slopes, if the slope's value was within a threshold value,
non-null values were rounded to 0. This can be a **source of bias** since the entire method
is based on the transitions of states for each variable within all trajectories. Furthermore,
Monod's growth function is mainly empirical and not at all mechanistic, which could
substantially alter the results [21]. Most importantly, the way in which the space of
states is explored was through simulated groups of initial conditions, depending on the
number of state variables. These conditions are sensitive to the combinations made, but
also to the size of the time step. Altogether, these assumptions could alter the result,
given the sensitivity to parameter fluctuations of ODE approaches. The major issue is
the strong dependence on the discretization step size. Indeed, fake transitions are found
when the step is too large. Even if this approach allows for complexity modelled with
insufficient information it has non-negligible limitations since (1) the space of states can
easily increase in complexity [2] (2) transitions between states are also highly dependent
on the chosen time bounds since when too large, they might become obsolete [2].

In additional measurements which were not presented in this report, the possibilistic
approach is still however sensitive to parameter fluctuation, since a small change in a
single parameter (< 0.5) can cause bigger or smaller changes depending on the nature of
the parameter. The most drastic changes occur for small variations in Ks and Ki for the
two least complex models, even if the rest of the parameters remain fixed. It could be
possible however to make a global stability analysis, in order to build a larger network
which illustrates the ramifications when the system is not parameterized, leading to a more
generic approach. The predicted rules will differ, but it is still interesting to have generic
rules in a parameterized system. However, this made the testing on theoretical data more difficult, because when generated randomly and analyzed, the qualitative network could be associated with a certain growth function, but not with certainty, since the same growth function would yield different results if parameters were changed.

## 4.5 Conclusion

Overall, the goal was to find an alternative method to navigate the states of the system, in this case, a theoretical chemostat. This method was intended to serve as a simplifying tool. This algorithm is now able to perform an **automatic** translation of a system of dynamical equations (quantitative) to a Boolean network (qualitative). In the first case, variations are assumed to be synchronous. In the latter case, variations are supposed to follow **transition rules**. These transitions can be described in order to have a better grasp of rare and systematic variations. However, this method is applicable but not yet verifiable, since it has not been confronted with empirical data properly. Also, further testing of the effect of parameter variation needs to be done, since it is **not generic**, due to its dependence on parameter values.

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# A Appendix

## B Chemostat equations

$$ egin{cases} \frac{ds}{dt} = D(S_{in} - s) - \sum \mu_i(s)b_i \\ \frac{db}{dt} = \mu_i(s)b_i - Db_i \end{cases} \quad (8) $$

where $b_i$ is the biomass of species $i$, $\mu_i$ its specific growth rate, $D$ the dilution rate and $S_{in}$ is the substrate inflow. The constants represent a concentration (mass/volume) except for the dilution rate (mass/volume/time).

### B.1 Simple model: one species and one substrate

#### B.1.1 Rules

**Monod and Contois growth function**

$$ \mathbf{R1} : ds^{-}, db^{+} \rightarrow db^{-} $$

If biomass is increasing and the substrate is decreasing due to consumption, biomass **can** eventually decrease too.

$$ \mathbf{R2} : ds^{-}, db^{-} \rightarrow ds^{+} $$

If the substrate is decreasing and biomass is decreasing, the substrate **can** increase due to lack of consumption.

$$ \mathbf{R3} : ds^{+}, db^{-} \rightarrow db^{+} $$

If the substrate is increasing and biomass is stable, biomass **can** start growing again.

$$ \mathbf{R4} : ds^{+}, db^{+} \rightarrow ds^{-} $$

If substrate and biomass are both increasing, substrate **can** decrease due to consumption.
## Haldane growth function

$$R1: ds^{-}, db^{+} \rightarrow db^{-}$$

If biomass decreases and substrate increases, substrate **can** decrease because of consumption (asynchronous).

$$R2: ds^{-}, db^{-} \rightarrow ds^{+}$$

If biomass is increasing and the substrate is decreasing because of consumption, biomass **can** eventually decrease too.

$$R3: ds^{+}, db^{-} \rightarrow db^{+}$$

If substrate is decreasing and biomass is decreasing, substrate **can** increase because of lack of consumption.

$$R4: ds^{+}, db^{+} \rightarrow ds^{-}$$

If substrate is increasing and biomass is stable, biomass **can** start growing again.

$$R5: ds^{+}, db^{+} \rightarrow ds^{-}$$

If substrate is decreasing and biomass increases, substrate **can** still continue to increase (asynchronous).

$$R6: ds^{+}, db^{+} \rightarrow ds^{-}$$

If substrate and biomass are both increasing, substrate **can** decrease because of consumption.

## B.2 Haldane trajectory

<figure><img src="image_11.png" /><figcaption>Fig. 10. Trajectory specific to Haldane Growth function.</figcaption></figure>
# C Supplementary materials about the models

## C.1 Model with two species and one substrate

<figure><img src="image_12.png" /><figcaption>**Fig. 13.** As a reminder, when the dilution rate is inferior to the growth rate of microorganisms, these can grow properly. Otherwise, these will be leached by the current. This is represented in the state of equilibrium (a) and leaching (c) reflected in the possibilistic networks.</figcaption></figure>

### C.1.1 Transition matrix: Model with two species and one substrate
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Possibilistic networks extracted from the Jupyter EDEN platform, already implemented before the internship, representing the possible state transitions of the system for the model with a single species growing in a substrate. Each node corresponds to a state of the system, connected by rules. Only the increasing variables, ON, are shown. If the state is nan none of the variables are increasing.</figcaption></figure>

<table><caption>Table 1 – Frequency of the transitions (in percentage of the total number of transitions) with Monod's growth function for the model with two species and a substrate. The diagonal was erased to plot the networks.</caption><thead><tr><th></th><th>000</th><th>100</th><th>010</th><th>110</th><th>001</th><th>101</th><th>011</th><th>111</th></tr></thead><tbody><tr><th>000</th><td>0.00</td><td>16.61</td><td>0</td><td>0</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>100</th><td>0.00</td><td>8.50</td><td>0</td><td>0</td><td>0.00</td><td>10.58</td><td>0.00</td><td>0.00</td></tr><tr><th>010</th><td>0.00</td><td>0.00</td><td>0</td><td>0</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>110</th><td>0.00</td><td>0.00</td><td>0</td><td>0</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>001</th><td>15.21</td><td>0.00</td><td>0</td><td>0</td><td>8.71</td><td>1.06</td><td>0.00</td><td>0.00</td></tr><tr><th>101</th><td>0.00</td><td>0.00</td><td>0</td><td>0</td><td>2.32</td><td>8.13</td><td>0.00</td><td>1.62</td></tr><tr><th>011</th><td>0.00</td><td>0.00</td><td>0</td><td>0</td><td>21.78</td><td>0.00</td><td>0.52</td><td>0.00</td></tr><tr><th>111</th><td>0.00</td><td>0.00</td><td>0</td><td>0</td><td>0.00</td><td>0.00</td><td>4.96</td><td>0.00</td></tr></tbody></table>

<table><caption>Table 2 – Frequency of the transitions (in percentage of the total number of transitions) with Haldane's growth function for the model with two species and a substrate. The diagonal was erased to plot the networks.</caption><thead><tr><th></th><th>000</th><th>100</th><th>010</th><th>110</th><th>001</th><th>101</th><th>011</th><th>111</th></tr></thead><tbody><tr><th>000</th><td>0.00</td><td>16.21</td><td>0.00</td><td>0.00</td><td>3.05</td><td>0</td><td>0.00</td><td>0.00</td></tr><tr><th>100</th><td>0.00</td><td>1.96</td><td>0.00</td><td>14.54</td><td>0.00</td><td>0</td><td>0.00</td><td>0.00</td></tr><tr><th>010</th><td>15.88</td><td>0.00</td><td>8.37</td><td>0.66</td><td>0.00</td><td>0</td><td>0.00</td><td>0.00</td></tr><tr><th>110</th><td>0.00</td><td>0.00</td><td>7.99</td><td>7.21</td><td>0.00</td><td>0</td><td>0.00</td><td>0.04</td></tr><tr><th>001</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0</td><td>7.71</td><td>0.00</td></tr><tr><th>101</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0</td><td>0.00</td><td>0.00</td></tr><tr><th>011</th><td>0.00</td><td>0.00</td><td>16.28</td><td>0.00</td><td>0.00</td><td>0</td><td>0.02</td><td>0.00</td></tr><tr><th>111</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0</td><td>0.09</td><td>0.00</td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** Possibilistic networks extracted from the *Jupyter EDEN* platform, already implemented before the internship, representing the possible state transitions of the system for the model with two species and a substrate. Each node corresponds to a state of the system, connected by rules. If the state is 'nan', there are no increasing variables.</figcaption></figure>
<table><thead><tr><th></th><th>000</th><th>100</th><th>010</th><th>110</th><th>001</th><th>101</th><th>011</th><th>111</th></tr></thead><tbody><tr><th>000</th><td>0.00</td><td>25.20</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>100</th><td>0.00</td><td>20.58</td><td>0.00</td><td>5.23</td><td>0.00</td><td>1.54</td><td>0.00</td><td>0.00</td></tr><tr><th>010</th><td>7.23</td><td>0.00</td><td>0.18</td><td>1.29</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><th>110</th><td>0.00</td><td>0.00</td><td>0.11</td><td>6.51</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.02</td></tr><tr><th>001</th><td>11.07</td><td>0.00</td><td>0.00</td><td>0.00</td><td>2.20</td><td>1.92</td><td>0.00</td><td>0.00</td></tr><tr><th>101</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>1.21</td><td>2.45</td><td>0.00</td><td>0.07</td></tr><tr><th>011</th><td>0.00</td><td>0.00</td><td>5.68</td><td>0.00</td><td>7.27</td><td>0.00</td><td>0.06</td><td>0.00</td></tr><tr><th>111</th><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.19</td><td>0.00</td></tr></tbody></table>

Table 3 – Frequency of the transitions (in percentage of the total number of transitions) with Contois's growth function for the model with two species and a substrate. The diagonal was erased to plot the networks.

### C.1.2 Rules

#### Monod growth function

$$R1: ds^{-}, db_{A}^{-}, db_{B}^{+} \rightarrow db_{B}^{-}$$

If B's threshold is insufficient, the species can eventually decrease in concentration.

$$R2: ds^{-}, db_{A}^{-}, db_{B}^{-} \rightarrow ds^{+}$$

If total biomass decreases, substrate can increase.

$$R3: ds^{+}, db_{A}^{-}, db_{B}^{+} \rightarrow ds^{-}$$

If B consumes the substrate, the substrate can decrease.

$$R4: ds^{-}, db_{A}^{+}, db_{B}^{+} \rightarrow db_{A}^{-}$$

If substrate concentration decreases, it can be inferior to A's threshold, but not to B's.

$$R5: ds^{+}, db_{A}^{-}, db_{B}^{-} \rightarrow db_{B}^{+}$$

If substrate concentration decreases, it can be superior to B's minimal threshold for growth.

$$R6: ds^{-}, db_{A}^{-}, db_{B}^{+} \rightarrow ds^{+}$$

If substrate and species A concentrations decrease, species B can grow (since $K_{iB} < K_{iA}$).

$$R7: ds^{+}, db_{A}^{+}, db_{B}^{+} \rightarrow ds^{-}$$

If total biomass increases, substrate can decrease.

$$R8: ds^{+}, db_{A}^{-}, db_{B}^{+} \rightarrow db_{A}^{+}$$

If substrate increases, it can reach A's break-even concentration.
Contois growth function

$$
\textbf{R1 : } ds^{-}, db_{A}^{+}, db_{B}^{-} \rightarrow db_{A}^{-}
$$

If A is growing, B is excluded and substrate is decreasing, species A **can** stop growing ($s < Ks_A$).

$$
\textbf{R2 : } ds^{-}, db_{A}^{-}, db_{B}^{+} \rightarrow db_{B}^{-}
$$

If B is growing, A is stagnant and the substrate is decreasing, species B **can** stop growing
($s < Ks_B < Ks_A$).

$$
\textbf{R3 : } ds^{-}, db_{A}^{-}, db_{B}^{-} \rightarrow ds^{+}
$$

If substrate is decreasing but also total biomass, and consumption, substrate **can** increase.

$$
\textbf{R4 : } ds^{+}, db_{A}^{+}, db_{B}^{-} \rightarrow ds^{-}
$$

If substrate is increasing A is growing and B is decreasing, substrate **can** decrease
($\mu_{\max A} > \mu_{\max B}$).

$$
\textbf{R5 : } ds^{-}, db_{A}^{+}, db_{B}^{+} \rightarrow db_{B}^{-}
$$

If substrate is decreasing and total biomass is growing, species B **can** be excluded
($\mu_{\max A} > \mu_{\max B}$).

$$
\textbf{R6 : } ds^{+}, db_{A}^{-}, db_{B}^{-} \rightarrow db_{A}^{+}
$$

If substrate is increasing and total biomass is decreasing, species A **can** exclude B
($\mu_{\max A} > \mu_{\max B}$).

$$
\textbf{R7 : } ds^{-}, db_{A}^{+}, db_{B}^{-} \rightarrow ds^{+}
$$

If substrate is decreasing, species A is growing and B is excluded, substrate **can** accumulate
again.

$$
\textbf{R8 : } ds^{+}, db_{A}^{-}, db_{B}^{+} \rightarrow ds^{-}
$$

If substrate increases, species B is growing and A is stagnant, substrate **can** decrease ($b_{B0} > b_{A0}$).

$$
\textbf{R9 : } ds^{-}, db_{A}^{+}, db_{B}^{+} \rightarrow db_{A}^{-}
$$

If substrate decreases and total biomass increases, species A **can** decrease ($Ks_B < Ks_A$ and $s < Ks_A$).

$$
\textbf{R10 : } ds^{+}, db_{A}^{-}, db_{B}^{-} \rightarrow db_{B}^{+}
$$

If substrate increases and total biomass is stagnant, B **can** start growing (inoculation time and $b_{B0} > b_{A0}$).

$$
\textbf{R11 : } ds^{-}, db_{A}^{-}, db_{B}^{+} \rightarrow ds^{+}
$$

If substrate decreases, species A decreases and B increases, substrate **can** accumulate again ($Ks_A > Ks_B$).

$$
\textbf{R12 : } ds^{+}, db_{A}^{+}, db_{B}^{+} \rightarrow ds^{-}
$$

If substrate and total biomass increase along with consumption, substrate concentration **can**
decrease.
$$
\mathbf{R13} : \mathbf{ds}^{+}, \mathbf{db}_{A}^{+}, \mathbf{db}_{B}^{-} \rightarrow \mathbf{db}_{B}^{+}
$$

If substrate increases, species A is growing and B is decreasing, species B **can** increase (co-existence).

$$
\textbf{R14 : } ds^{+}, db_{A}^{-}, db_{B}^{+} \rightarrow db_{A}^{+}
$$

If substrate increases, species B is growing and A is stagnant, species A **can** start growing ($Ks_B < Ks_A < s$).

Haldane growth function

$$
\mathrm{R1} : \mathrm{ds}^{-}, \mathrm{db}_{A}^{+}, \mathrm{db}_{B}^{-} \rightarrow \mathrm{db}_{A}^{-}
$$

If substrate decreases, species A **can** decrease (KsB).

$$
\textbf{R2 : } ds^{-}, db_{A}^{-}, db_{B}^{-} \rightarrow ds^{+}
$$

If biomass decreases, consumption decreases and substrate concentration can increase.

$$
\textbf{R3 : } ds^{+}, db_{A}^{+}, db_{B}^{-} \rightarrow ds^{-}
$$

If species A grows, substrate concentration **can** decrease, because of high consumption (KsA > KsB).

$$
\textbf{R4 : } ds^{-}, db_{A}^{+}, db_{B}^{+} \rightarrow db_{B}^{-}
$$

If species A grows, it **can** competitively exclude B.

$$
\textbf{R5 : } ds^{+}, db_{A}^{-}, db_{B}^{-} \rightarrow db_{A}^{+}
$$

If species A's break-even concentration is reached and B is already excluded, the species **can** grow.

$$
\textbf{R6 : } ds^{-}, db_{A}^{+}, db_{B}^{-} \rightarrow ds^{+}
$$

If the break-even concentration is met for species A and B is already excluded, the substrate **can**
increase.

$$
\mathbf{R7} : \mathbf{ds}^{-}, \mathbf{db}_{A}^{-}, \mathbf{db}_{B}^{-} \rightarrow \mathbf{db}_{B}^{+}
$$

If species A has already been excluded, species B **can** grow ($Ks_B < Ks_A$).

$$
\textbf{R8 : } ds^{-}, db_{A}^{-}, db_{B}^{+} \rightarrow db_{A}^{+}
$$

If substrate decreases below A's Kᵢ but is still superior to Ks, A **can** grow (Ks < s < Kᵢ).

$$
\textbf{R9 : } ds^{+}, db_{A}^{+}, db_{B}^{+} \rightarrow ds^{-}
$$

If total biomass concentration increases, consumption increases and substrate **can** decrease.

$$
\textbf{R10 : } ds^{+}, db_{A}^{+}, db_{B}^{-} \rightarrow db_{B}^{+}
$$

If substrate concentration continues to increase despite species A's growth, species B **can** grow again.
## C.2 Nitrifying chemostat

### C.2.1 Experimental device

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Schematic representation of a nitrifying chemostat with two functional species ($b_A$ and $b_B$). Nitrification is a two-step aerobic microbial process involving ammonium oxidation to nitrate, facilitated by aerobic ammonium-oxidizing bacteria (AOB) followed by nitrite oxidation to nitrate, carried out by aerobic nitrite-oxidizing bacteria (NOB) (for the full article, see [5]).</figcaption></figure>
### C.2.2 State combinations for the model with two species and three substrates

As a reminder, the five state variables are the three substrates ($s_1, s_2, s_3$) and the two species biomass ($x_A$ and $x_B$), respectively. Positive derivatives are represented by 1 and negative derivatives are represented by 0.

```
00000 00001 00010 00011 00100 00101 00110 00111 01000 01001
01010 01011 01100 01101 01110 01111 10000 10001 10010 10011
10100 10101 10110 10111 11000 11001 11010 11011 11100 11101
11110 11111
```
$$
R1: ds_1^-, ds_2^-, ds_3^-, dXA^-, dXB^+ \Rightarrow dXB^-
$$

$$
R2: ds_1^-, ds_2^-, ds_3^+, dXA^+, dXB^- \Rightarrow ds_3^-
$$

$$
R3: ds_1^-, ds_2^+, ds_3^-, dXA^+, dXB^- \Rightarrow ds_2^-
$$

$$
R4: ds_1^-, ds_2^+, ds_3^-, dXA^+, dXB^+ \Rightarrow ds_2^-, dXB^-
$$

$$
R5: ds_1^-, ds_2^+, ds_3^+, dXA^-, dXB^- \Rightarrow ds_2^-, ds_3^-, dXA^+
$$

$$
R6: ds_1^+, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow ds_1^-
$$

$$
R7: ds_1^+, ds_2^+, ds_3^+, dXA^-, dXB^- \Rightarrow ds_1^-, ds_2^-, ds_3^-, dXA^+
$$

$$
R8: ds_1^-, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow dXB^+
$$

$$
R9: ds_1^-, ds_2^+, ds_3^+, dXA^-, dXB^+ \Rightarrow ds_2^-, ds_3^-, dXA^+
$$

$$
R10: ds_1^+, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow ds_1^-, dXB^+
$$

$$
R11: ds_1^+, ds_2^-, ds_3^-, dXA^+, dXB^+ \Rightarrow ds_1^-
$$

$$
R12: ds_1^+, ds_2^+, ds_3^+, dXA^-, dXB^+ \Rightarrow ds_1^-, ds_2^-, ds_3^-, dXA^+
$$

$$
R13: ds_1^-, ds_2^-, ds_3^+, dXA^+, dXB^- \Rightarrow dXA^-
$$

$$
R14: ds_1^-, ds_2^-, ds_3^+, dXA^+, dXB^+ \Rightarrow dXA^-
$$

$$
R15: ds_1^-, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow ds_3^+
$$

$$
R16: ds_1^-, ds_2^-, ds_3^+, dXA^+, dXB^+ \Rightarrow dXB^-
$$

$$
R17: ds_1^-, ds_2^+, ds_3^+, dXA^-, dXB^+ \Rightarrow ds_2^-, dXA^+, dXB^-
$$

$$
R18: ds_1^-, ds_2^+, ds_3^+, dXA^+, dXB^- \Rightarrow ds_2^-
$$

$$
R19: ds_1^+, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow ds_1^-, ds_3^+
$$

$$
R20: ds_1^+, ds_2^-, ds_3^+, dXA^+, dXB^- \Rightarrow ds_1^-
$$

$$
R21: ds_1^+, ds_2^+, ds_3^+, dXA^-, dXB^+ \Rightarrow ds_1^-, ds_2^-, dXA^+, dXB^-
$$

$$
R22: ds_1^-, ds_2^-, ds_3^-, dXA^+, dXB^- \Rightarrow ds_3^+, dXB^+
$$

$$
R23: ds_1^-, ds_2^-, ds_3^-, dXA^+, dXB^+ \Rightarrow ds_3^+
$$

$$
R24: ds_1^-, ds_2^-, ds_3^+, dXA^+, dXB^- \Rightarrow dXB^+
$$