# Modelling the dynamics of Salmonella infection in the gut at the bacterial and host levels

Coralie Muller, Arie Wortsman, Pablo Ugalde-Salas, Clémence Frioux, Simon Labarthe

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## To cite this version:

Coralie Muller, Arie Wortsman, Pablo Ugalde-Salas, Clémence Frioux, Simon Labarthe. Modelling the dynamics of Salmonella infection in the gut at the bacterial and host levels. JOBIM 2022 - Journées Ouvertes en Biologie, Informatique et Mathématiques, Jul 2022, Rennes, France. hal-03839374

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Submitted on 4 Nov 2022

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Distributed under a Creative Commons Attribution 4.0 International License
# Modelling the dynamics of Salmonella infection in the gut at the bacterial and host levels

Coralie MULLER<fnref n="1" />, Arie WORTSMAN<fnref n="1" />, Pablo Andres UGALDES SALAS<fnref n="1" />, Clémence FRIOUX<fnref n="1" />, and Simon LABARTHE<fnref n="2" />,<fnref n="1" />

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## Context : a multidisciplinary project

► We want to model the dynamics of a host-microbiota-pathogen system through the dFBA method. **Dynamic FBA** (dFBA) integrates the fluxes calculated in FBA into a system of **ordinary differential equations (ODEs)**, enabling us to model the temporal dimension of metabolic simulations. In order to speed up the computation of the FBA models needed for a thorough numerical exploration, the metabolic models will be approximated by a **metamodel method** [1].

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## Colonocyte metabolic network curation

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► We extracted and cleaned reactions occurring in colonocytes (epithelial cell) from a metabolic network of the whole human body [2].

► A long manual curation process was performed in order to clean up and make the model usable. It consists of **five main steps**:

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## Harmonization

► We model the host-microbiota-pathogen system by gathering three metabolic models: *Faecalibacterium prausnitzii* [3] as a representative butyrate producing **commensal bacterium**, *Salmonella Enterica* serovar Thiphimurium [4] for the **enteric pathogen**, these models previously used in CEMRACS[1] project, and a metabolic model of **colonocyte** as human [2] **epithelial cell**.

► In order to make the simulation with the 3 models, we have **harmonized** the networks between them on **two points**:

Different IDs for the same metabolite and different name for compartments between networks

Fix fluxes for essential reactions in the right compartments for the exchanges

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**Healthy gut:**

- **Production** of butyrate by commensal bacteria

- **Consumption** of **but** and **O2** in TCA cycle

- **Good** production of **biomass**

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## Validation colonocyte model

► We want to simulate two main cases to **validate** our model

**Colonized gut:**

- **NO** production of **but** by commensal bacteria

- **Accumulation** of **O2** and transfert to [luLI]

- **Decreased** biomass production

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**Keywords:**

*Gut microbiota - Metabolic network - Biological system*

*System of ODE - Metamodelling - Numerical metabolic modelling*

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## Simulation: FBA

► For the simulation, we consider that the system works by maximizing biomass production. Also, we assume that the system is in a stationary state. This way, can model the system by solving the linear **optimization** problem:

$$ \min_{\nu \in \mathbb{R}^{N_r}} \nu_{\text{biomass}} $$

$$ A \cdot \nu = 0 $$

$$ c_{\min} \leq \nu \leq c_{\max} $$

► We assume all three of the system's compartment's follow the FBA assumptions. We also define the function that maps constraints to the optimal flux according to FBA [5] by:

$$ \mathcal{F}: \mathbb{R}^{N_{up}} \rightarrow \mathbb{R}^{N_r} $$

$$ c^{up} \rightarrow \nu^* $$

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## Simulation : dFBA

► Our model assumes that FBA holds for each instant [5]. Then, for simulating it for a period of time, it is necessary to solve FBA on each compartment. After this timestep, an ODE system models the fluxes between different compartments, until the next step...

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► How does the equations of the **ODE system look like**?. For the concentration of a component, we have the following form :

$$ \partial_t s = \frac{Q_{in}s_{in} - Q_{out}s}{V} + \text{biological and chemical reactions} + \text{transport to other compartment} $$

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## Mathematical model of the system

► Luminal compartment: $\partial_t S_{th} = (\mathcal{F}_{S_{th},1}(m_l, S_{th}, F_{prau}) - \rho n_l - D_{S_{th}})S_{th}$

$$ \partial_t F_{prau} = (\mathcal{F}_{F_{prau},1}(m_l, S_{th}, F_{prau}) - \rho n_l - \alpha \frac{O_{2l}}{K_{O2} + O_{2l}} - D_{F_{prau}})F_{prau} $$

$$ \partial_t n_l = \gamma_n(n_e - n_l) - d_n n_l - D_{n_l} $$

$$ \partial m_l = D(m_i n - m_l) + \mathcal{F}_{S_{th},m_l}(m_l, S_{th}, F_{prau}) S_{th} + \mathcal{F}_{F_{prau},m_l}(m_l, S_{th}, F_{prau}) F_{prau} + \beta m_l O_{2l} + \text{diag}(\gamma) T_r(m_e, m_l) $$

$$ \begin{gathered} \partial_t n_e = C_{but,n} n_e \left( n_e - L_n \frac{but_e}{K_{but} + but_e} \right) (L_n - n_e) - d_n n_e + \gamma_n(n_m - n_e) + VF(S_{th}) \\ \partial_t NO_e = \gamma_{NO}(NO_e - NO_e) + VF(S_{th}) \end{gathered} $$

$$ \partial_t O_{2e} = \mathcal{F}_{ent,O_2}(NO_e, )_{2e}, but_e) - d_{O_2} O_{2e} + L_{O_2} + \gamma(O_{2l} - O_{2e}) $$

$$ \partial_t but_e = \mathcal{F}_{ent, but}(NO_e, )_{2e}, but_e) + \gamma_{but}(but_m - but_e) $$

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## Next steps

► Each time iteration of the model implies an optimization step in a high dimensional space, which means that the simulation has a very high computational cost.

Are there ways to solve this?

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► Is there a big trade-off between computation cost and precision?

► There are still reactions in the colonocyte network that do not behave as expected. Mainly, the consumption of butyrate in the network has some problems we've been looking at.

Solving this is a work in progress.

**References:**

[1] Clémence Frioux et al. Accelerating metabolic models evaluation with statistical metamodels: application to Salmonella infection

models. working paper or preprint, April 2022.

[2] Ines Thiele et al. Personalized whole-body models integrate metabolism, physiology, and the gut microbiome. Molecular Systems

Biology, 16(5):1-24, 2020.

[3] El-Semman et al. "Genome-scale metabolic reconstructions of Bifidobacterium adolescentis L2-32 and Faecalibacterium prausnitzii

A2-165 and their interaction." BMC systems biology 8.1 (2014): 1-11.

[4] Raghunathan et al. Constraint-based analysis of metabolic capacity of Salmonella typhimurium during

host-pathogen interaction. BMC Syst Biol 3, 38 (2009). https://doi.org/10.1186/1752-0509-3-38

[5] Ove Øyås and Jörg Stelling. Genome-scale metabolic networks in time and space. Current Opinion in Systems Biology, 8:51-8, 2018.

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coralie.muller@inria.fr
arie.wortsman-zurich@inria.fr

<footnote n="1">¹Inria University of Bordeaux, 200 avenue de la Vieille Tour, 33405, Talence, France</footnote>

<footnote n="2">² INRAe UMR BIOGECO, 69 route d'Arcachon, 33612, Cestas, France</footnote>