# Statistical method inference cMFA for multi-omics data integration in microbial community models

Sthyve Junior Tatho Djeanou, Simon Labarthe, Valentina Baldazzi

► To cite this version:

Sthyve Junior Tatho Djeanou, Simon Labarthe, Valentina Baldazzi. Statistical method inference cMFA for multi-omics data integration in microbial community models. JOBIM 2025 - Journées Ouvertes en Biologie Informatique et mathématiques, Jul 2025, Bordeaux, France. hal-05163368

HAL Id: hal-05163368

https://inria.hal.science/hal-05163368v1

Submitted on 15 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
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## INTRODUCTION

### Biology-informed inference from multi-omics data

<figure><img src="image_3.png" /><figcaption>Population count data</figcaption></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Metabolomics data</figcaption></figure>

<caption>Metatranscriptomic data</caption>

<figure><img src="image_7.png" /><figcaption>Metabolic network</figcaption></figure>

<figure><img src="image_8.png" /></figure>

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Biology
Inform
Inference
Problem

<figure><img src="image_10.png" /><figcaption>Interpretation</figcaption></figure>

<caption>modelling</caption>

<figure><img src="image_11.png" /><figcaption>Microbial Community</figcaption></figure>

Understanding natural microbial communities presents significant challenges due to the complex interactions and metabolite exchanges between microorganisms. To gain deeper insights into these dynamics, we are focusing on:

* Developing mathematical models based on metabolic networks.

* Inferring metabolic fluxes using biology-informed high-dimensional analysis methods to integrate multi-omics data.

**Scientific Question:**

How can mathematical models help us better understand the functioning of microbial communities ?

* Internal flux reconstruction based on some external reaction rates.

* Estimating the contribution of each microorganism to the community.

## COMMUNITY METABOLIC FLUX ANALYSIS (CMFA)

### METABOLIC NETWORK

A metabolic network refers to the interconnected set of biochemical reactions and pathways that occur within a cell or organism.

<figure><img src="image_12.png" /><figcaption>Sequence reconstruction</figcaption></figure>

<figure><img src="image_13.png" /></figure>

S : stoichiometric matrix

$$S\nu = 0$$

### COMMUNITY MFA (cMFA)

<figure><img src="image_14.png" /></figure>

### MATHEMATICAL FORMULATION

Find, for $t \ge 0$, for a set $\mathcal{M}$ of $N_m$ metabolites and for a set $\mathcal{B}_m$ of $N_b$ micro-organisms the flux vector

$$\hat{v}(t) = (v^{(\hat{b}_1)}(t), \dots, v^{(\hat{b}_{N_b})}(t)) \text{ of dimension } N_r$$

such that

$$\hat{v}(t) := \arg \min_{v_t} \sum_{m \in \mathcal{M}} \left\| R_{m,t} - \sum_{b \in \mathcal{B}_m} \chi_t^{(b)} v_{m,t} \right\|^2 + \lambda \mathcal{R}(T\nu)$$

$$\text{subject to: } \begin{cases} S^{(b)} \cdot v^{(b)} \\ C_{\min}^{(b)} \le v^{(b)} \end{cases} = 0 \quad \text{for } b \in \mathcal{B}_m$$

Regression problem is:

* High-dimensional ($N_r \sim 2500 \times N_b$)

* Highly-underdetermined ($N_r \gg N_m$)

* Biology-informed ($S^{(b)}$)

* Constrained by metatranscriptomics $\mathcal{R}(T\nu)$: where $T$ is the reaction weight reflecting the level of gene expression.

* Regularized by a lasso ($\mathcal{R}(\nu) = \|\nu\|_1$) penalty to promote sparsity

### CO-CULTURE DEFINITION

<figure><img src="image_15.png" /><figcaption>Different biological interaction of E. coli</figcaption></figure>

### BENCHMARK METHODOLOGY

<figure><img src="image_16.png" /></figure>

### GENERATING DATA: DFBA

Dynamic Flux Balance Analysis (DFBA) is a mathematical modeling approach to predict the metabolites fluxes in metabolic networks.

* **Model origin**

BIGG Models
Database iJO1366.

$$\begin{aligned} \partial_t b_i &= B \hat{v}_i (M, b) b_i \\ \partial_t M_j &= \sum_{i \in C} \hat{v}_{i,j} (M, b) b_i \end{aligned}$$

* **model adjustment**

knocking out specific
genes and reactions

$S.T$

$$\hat{v}_i = \arg \max B v_i \begin{cases} S_i \hat{v}_i = 0 \\ C_M \le \hat{v}_i^s \le 0 \\ 0 \le \hat{v}_i^p \le C_{max} \\ -C_{max} \le \hat{v}_i^{in} \le C_{max} \end{cases}$$

### IMPLEMENTATION

1. NumPy, Pandas for data handling and structure.

2. PyGAM for spline smoothing.

3. Matplotlib for visual representations.

4. COBRApy for constraint-based modeling.

5. OSQP for quadratic programming problems.

## RESULTS

### INFERRING INDIVIDUAL FLUXES: DFBA VS CMFA

<figure><img src="image_17.png" /><figcaption>Internal fluxes cMFA vs dFBA</figcaption></figure>

<figure><img src="image_18.png" /></figure>

### BENCHMARK ON METHOD ROBUSTNESS

<figure><img src="image_19.png" /></figure>

## CONCLUSION & PERSPECTIVES

**Conclusion** : cMFA is robust to noise, metabolic import rates, and exchange rate inference with both high- and low-quality data. It scales well to large communities, but internal flux inference is less reliable with lower-quality data.

**Perspectives** : Large Community, missing metabolites, benchmark on real data.

## REFERENCES

1. Brunner et al., 2023, PLoS Comput. Biol. 19, e1011661

2. Junior, M.L. et al. (2020). PLoS Comput. Biol., 16, e1007099.

3. Gerken Starepravo et al., 2022, Digit. Chem. Eng. 2, 100011

4. Lecomte et al., 2023, Metab. Eng. doi.org/10.1016/j.ymben.2024.02.014

5. Mahadevan et al., 2002, Biophys. J. 83, 1331–1340

<figure><img src="image_20.png" /></figure>