# Inference Method cMFA for multi-omics data integration in microbial community models

Sthyve Junior Tatho Djeanou, Valentina Baldazzi, Simon Labarthe

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Sthyve Junior Tatho Djeanou, Valentina Baldazzi, Simon Labarthe. Inference Method cMFA for multi-omics data integration in microbial community models. 2025. hal-05105798

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Distributed under a Creative Commons Attribution 4.0 International License
cMFA for multi-omics data integration in microbial
community models

Sthyve Tatho¹,²[0009-0001-4227-8970], Valentina Baldazzi³,⁴[0000-0001-9734-9759],
and Simon Labarthe¹,²[0000-0002-5463-7256]

¹ Univ. Bordeaux, INRAE, BIOGECO, F-33610, Cestas, France

² Inria, University of Bordeaux, INRAE, Talence, France

³ Université Côte d'Azur, INRAE, CNRS, ISA, Sophia-Antipolis, France

⁴ Inria, Université Côte d'Azur, INRAE, CNRS, MACBES, Sophia-Antipolis, France
simon.labarthe@inrae.fr, valentina.baldazzi@inrae.fr

**Abstract.** Understanding microbial community functions is challenging due to complex interactions and assembly mechanisms. However, advances in sequencing technologies have enabled the collection of multi-omics data, including population counts and metabolomic or metatranscriptomic profiles. Our main objective is to develop a mathematical model capable of integrating time series of multi-omics data at the community scale. We introduce the community Metabolic Flux Analysis (cMFA) method: a biology-informed inference approach that generalizes classical Metabolic Flux Analysis. This high-dimensional analytical framework aims to estimate metabolic fluxes by integrating multi-omics data. Specifically, we aim to (i) quantify, for each member of the microbial community, their individual contributions to overall community dynamics based on external measurements of metabolite dynamics, and (ii) infer their intracellular distribution of metabolic fluxes. The difficulty here is in accurately inferring latent internal rates from a few observations of community-scale consumption and production rates for extracellular metabolites. We evaluated the cMFA method using synthetic data generated from dynamic models of microbial communities of increasing complexity using dynamic flux balance analysis, based on metabolic models of different Escherichia coli mutants. Synthetic metatranscriptomic data were obtained from internal metabolic fluxes simulated in the dynamic model. To assess the robustness of the method, we benchmarked its performance under varying levels of experimental noise.

**Keywords:** Dynamical system · Applied mathematics · Inference · Biological system.

# 1 Introduction

Microbial communities — whether they reside in the human microbiome, soils, aquatic ecosystems, plants, or other ecosystems — play a central role in numerous biological and environmental processes [10, 2]. However, their study remains challenging due to the complexity of interspecies interactions and metabolic exchanges. Classical experimental and mathematical approaches have difficulty explaining their functioning and
dynamics, in particular for deciphering the specific contribution of each species and their intracellular activity [10, 6].

Thanks to advances in molecular biology, it is now possible to obtain a vast set of omics data (genomics, transcriptomics, metabolomics, etc.)[2, 9, 7, 1]. This abundance of data challenges the relevance of traditional models, which are usually based on macroscopic community-scale reactions but do not account for the internal metabolic processes of microbial cells and remain unable to integrate metatranscriptomic data. This observation calls for the development of new conceptual frameworks better adapted to the complexity of microbial systems.

However, modeling and quantifying interactions within these communities remains a considerable challenge [2]. This is due to the diversity of biochemical pathways, often only partially characterized, and to the many uncertainties inherent to these systems [10, 6]. Moreover, the coherent and relevant integration of multi-omic data into mathematical models poses methodological challenges, both conceptually and computationally. To address these challenges, biologically constrained mathematical models based on metabolic networks and metabolic flux reconstruction appear a promising approach [14, 7]. Indeed, metabolic networks represent a suitable framework for data integration: their genome-scale reconstruction makes it possible to link transcriptomic data (i.e. gene expressions) to metabolic reactions. Furthermore, flux prediction models connect metabolic reactions to metabolite and biomass concentration dynamics[8]. However, if these methods are well established for the metabolism of mono-species, they are still poorly explored at the community level.

In this context, we introduce **cMFA**, for community metabolic flux, a biology-informed metabolic flux inference method integrating multi-omic data, such as microbial abundance, metabolomic data and transcriptomic profiles. This approach overcomes the limitations of purely population-based or strictly stoichiometric models and makes it possible to infer metabolic fluxes and exchanges between microorganisms from community-scale measurements.

We introduce the mathematical model and evaluate its predictions through a systematic benchmark based on simulated data. In particular, we highlight how this method allows to clarify the metabolic role of each microbial actor in the community, and we assess the robustness of its predictions in different ecological scenarios and for an increasing level of noise.

# 2 Material and Methods

## 2.1 Description of the cMFA approach

The cMFA model extends the classical *Metabolic Flux Analysis* (MFA) approach, as formalized in [5], which allows for modeling the steady state of a metabolic network by estimating internal fluxes from exchange flux measurements. This model takes as input the consumption rates ($R_s$), production rates ($R_p$), and the organism's growth rate ($R_b$), as illustrated in Fig. 1.

We generalized this formalism to microbial communities, as presented in Equation (1), which enables to simultaneously account for multiple organisms and metabolites, as illustrated in Fig. 1.
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** a) Sketch of the unicellular MFA method. Based on consumption ($R_s$) and production ($R_p$) rates, the intracellular flux distribution among the cell metabolic network is reconstructed (blue arrows). b) Sketch of the cMFA method. From community scale consumption ($R_s$) and production ($R_p$) rates and individual growth rates, individual consumption and production are reconstructed (blue arrows), together with intracellular metabolism (brown and green arrows).</figcaption></figure>

*Mathematical model.* For a set of metabolites $\mathcal{M}$ of cardinality $N_m$ and a set of organisms $\mathcal{B}_m$ of size $N_b$ described by their respective metabolic networks, the problem consists in estimating, for each time $t \ge 0$, the whole set of metabolic fluxes $\hat{v}(t)$ gathering intracellular and exchanges fluxes of every individuals, by minimizing the difference between community-scale measured ($R_t$) and predicted rates. Namely, for a set of $N_b$ metabolic networks taken as inputs, we set the following optimization problem:

$$ \hat{v}(t) := \arg \min_{v_t} \sum_{m \in \mathcal{M}} \|R_{m,t} - \sum_{b \in \mathcal{B}_m} \chi_t^{(b)} v_{m,t}^{(b)}\|_2^2 + \lambda \mathcal{R}(v) \quad (1) \\ \text{subject to:} \quad \begin{cases} S^{(b)} \cdot v^{(b)} = 0 \\ C_{min}^{(b)} \le v^{(b)} \le C_{max}^{(b)}, \end{cases} \quad \text{for } b = 1, \dots, N_b $$

In this formulation, $R_{m,t}$ and $v_{m,t}^{(b)}$ represent respectively the observed community-scale consumption rate and the per capita export flux for micro-organism $b$ of metabolite $m$ at time $t$, while $\chi_t^{(b)}$ denotes the biomass density of the organism $b$ at time $t$ which is assumed observed in the data. The matrix $S^{(b)}$ is the reaction matrix, i.e. the matrix which specifies for each metabolite of the system its stoichiometry in the reactions it is involved in. The term $\mathcal{R}$ is a regularization term to be specified (see below), $C_{min}^{(b)}$ and $C_{max}^{(b)}$ represents prior estimates of lower and upper bounds constraining the metabolic fluxes, respectively. These bounds allow specification of whether a reaction is blocked, imported, exported, or reversible. For intracellular and production reactions, this bounds are defined a priori in the metabolic network of the corresponding microorganism and represent biological and chemical constraints on reaction rates. For substrate metabolite $m$ consumed by multiple organisms, the minimum bound substrate import is adjusted proportionally to each organism's biomass to reflect fair competition for the shared resource:
$$C_{min}^{(b)} = \max \left( \nu_{intr}, -\frac{m}{\Delta_t \cdot \sum_{b \in B_m} \chi_b} \right) \quad (2)$$

This constrained penalized inference problem is inherently under-determined, as the number of reactions to be estimated ($N_r$) far exceeds the number of available metabolite measurements ($N_m \ll N_r$). To overcome this indeterminacy, additional biological knowledge must be incorporated.

*Prior biological information.* Biology-informed problem is achieved through different ways.

First, structural constraints imposed by the stoichiometry of the metabolic network (via the equation $S^{(b)}\nu^{(b)} = 0$), define the steady-state condition of microbial metabolism, ensuring that there is no net accumulation of metabolites in the system over time. Second, a principle of metabolic parsimony is applied through the regularization term by choosing a lasso-type term $\mathcal{R}(\nu) = \|\nu\|_1$ which promotes sparsity in the metabolic flux distributions. Finally, using metatranscriptomic data, we knock out reactions that are not activated in the data. This allows for the regulation of metabolic fluxes in the model in response to gene expression levels. Namely, we set

$$C_{min}^{(b)} := \bar{C}_{min}^{(b)} T_t^{(b)} \le \nu_t^{(b)} \le C_{max}^{(b)} := \bar{C}_{max}^{(b)} T_t^{(b)}$$

where $\bar{C}_{min}^{(b)}$ and $\bar{C}_{max}^{(b)}$ are nominal constraints for internal reactions of micro-organism (b), and $T_t^{(b)}$ is a boolean vector indicating if the expression of the corresponding reaction, i.e. of the reaction-associated genes taking into account gene-protein-reaction rules, is above a certain threshold.

## 2.2 cMFA numerical implementation

*Preprocessing: data smoothing.* In microbial ecology, fine estimates of time rates are difficult, due to experimental specificities. First, as measurements are destructive, time-series are not obtained from paired time samples from the same microbial community: rather, they are made of replicates of punctual time sampling of different communities, with the same ecological specificities. Second, the rate of the time sampling is often coarse. To overcome this difficulties, we apply a spline smoothing to the dataset. Namely, we search for a time-varying function $S(t) = \sum_{i=0}^{n-1} B_{i,k,\tau}(t) \cdot c_i$, where $B_{i,k,\tau}$ are B-spline basis functions of degree $k$ for a set of knots $\tau$, and $c_i$ are the coefficients that determine the contribution of each basis function in approximating the spline.

The coefficients $c_i$ are determined by minimizing the following functional:

$$J(c) = \frac{1}{2} \sum_{i=1}^{N_r} \|S(t) - X_{i,t}\|_2^2 + \frac{\lambda_s}{2} \int_a^b \left( \frac{d^2 S(t)}{dt^2} \right)^2 dt$$

where $N_r$ and $X_{i,t}$ are respectively the number of replicates and the measured metabolites and microbial densities. The term $\lambda_s$ controls the smoothness of the curve.
The spline degree is set to 3, so that the smoothing process is left with two hyperparameters to be selected: $\tau$ and $\lambda_s$. This data smoothing is a generalized additive problem which is solved with the dedicated pyGAM library version 0.9.1 [12], which provides the maximum likelihood estimate and confidence cone for splines, together with selected hyperparameters. The inferred spline is directly used to estimate the concentrations of each state variable over time. The smoothness of the spline allows the computation of exact derivatives using SciPy [15] with the `scipy.derivative()` method, hence giving an estimate of growth, consumption and production rates represented in the model (Fig. 2). Finally, the respective upper and lower bounds of the confidence cones are used for metabolite and microbial densities in eq. (2) to constrain the substrate import fluxes, allowing a more flexible modeling of nutrient uptake.

**Vectorized form.** The optimization problem is vectorized, leading to the following formulation:

$$ 
\begin{array}{l}
\forall t \ge 0, \quad \hat{v}(t) := \arg\min_{\nu_t} \|R_t - B_t\nu_t\|_2^2 + \lambda \mathcal{R}(\nu_t) \\
\text{subject to:} \quad \left\{ \begin{array}{l} S \cdot \nu_t = 0 \\ \nu_{\min} \le \nu_t \le \nu_{\max} \end{array} \right. \\
\text{with} \\
\nu_{\min} = C_{\min} \mathcal{T}_t, \quad \nu_{\max} = C_{\max} \mathcal{T}_t, \quad B_t\nu_t = \sum_{b \in \mathcal{B}_m} \chi_t \nu_t
\end{array} 
\quad (3) 
$$

Where for all time $t$ and for all microorganisms $B_t$ represents the sparse matrix of biomass levels, $R_t$ the vector of measured rates, and $\nu_t$ the unknown vector of fluxes.

**Renormalization.** Due to the presence of significant disparities in the magnitudes of the variables, left and right normalization matrices where defined with

$$ \sigma_{1t}^{-1} = \frac{1}{R_t}, \quad \sigma_{2t} = \frac{R_t}{B_t} $$

so that after renormalization, the dimensionless problem becomes

$$ 
\begin{array}{l}
\hat{v}_t = \arg\min_{\tilde{v}_t} \|\tilde{R}_t - \tilde{B}_t\tilde{v}_t\|_2^2 + \lambda \mathcal{R}(\sigma_{2t}\tilde{v}_t) \\
\text{subject to:} \quad \left\{ \begin{array}{l} \tilde{S}\tilde{v}_t = 0 \\ \tilde{v}_{\min} \le \tilde{v}_t \le \tilde{v}_{\max} \end{array} \right.
\end{array} 
\quad (4) 
$$

with,

$$ \tilde{R}_t = \sigma_{1t}^{-1} R_t, \quad \tilde{B}_t = \sigma_{1t}^{-1} B_t \sigma_{2t}, \quad \tilde{v}_t = \sigma_{2t}^{-1} \nu_t $$

and

$$ \tilde{S} = S\sigma_{2t}, \quad \tilde{v}_{\min} = \sigma_{2t}^{-1}\nu_{\min}, \quad \tilde{v}_{\max} = \sigma_{2t}^{-1}\nu_{\max} $$

This renormalization harmonizes the scales of fluxes, particularly for external exchange reactions, thus facilitating numerical solving and improving solver convergence.

The resulting problem is a high dimensional sparse quadratic programming model, which is solved with the dedicated Python package OSQP [13] version 0.6.7.
## 2.3 Benchmark construction

*Benchmarking increasingly complex biological interactions and noise levels.* A synthetic dataset is constructed, representing different ecological interactions of increasing complexity. We simulate three microbial communities, each composed of two microorganisms (see Fig. 2 left panel): 1) Competition: both microorganisms share a limiting resource but produce different metabolites; 2) Co-production: both share a limiting resource, consume individual resources and co-produce the same metabolites; 3) Cross-feeding: one organism benefits from the metabolic by-product of the other. Additionally, we assess the method’s robustness to noise by applying increasing perturbation to the synthetic data (see below).

*Synthetic bacterial and metabolite densities.* To model *Escherichia coli* metabolism, we took the BIGG model iJO1366 (http://bigg.ucsd.edu/models/iJO1366) which is the model of *Escherichia coli* str. K-12 substr. MG1655. We manually modified it by knocking out the genes and reactions that were identified in [11] as improving some amino acids production (valine and alanine) from glucose, galactose and xylose substrates in order to produce the different ecological scenarios (see Fig. 2).

Community growth is simulated using the dynamic flux balance analysis (dFBA) method [8], which predicts temporal changes in metabolic fluxes. Simulations are performed using the dfba_sampling package [4] version 0.2.1 that can be found at https://gitlab.inria.fr/slimmest/python-dfba-sampling, which builds upon the Python library COBRApy [3] version 0.29.1. Initial conditions of the dynamic system are chosen to reflect biologically relevant values, expressed in mmol. The study is conducted over a time interval ranging from $t_{initial} = 0$ to $t_{final} = 9$ time units.

*Data noising.* To obtain realistic and replicated experimental data, time points are extracted at $N = 10$ equidistant time points and a controlled noise $\epsilon(t)$ is added to the dFBA outputs at each time point with three replicates. This noise follows a multiplicative normal distribution so that the variation is proportional to the data value and thresholded.

$$D_{final}(t) = \max(0, D_{dFBA}(t) \times (1 + \epsilon(t))), \text{ with, } \epsilon(t) \sim \mathcal{N}(0, \sigma^2)$$

where $D_{dFBA}$ are the dFBA outputs and standard deviation $\sigma$ characterizes the intensity of the noise.

To assess cMFA robustness to different levels of noise, we define $\sigma = q\sigma_b$, where $\sigma_b = 1$ is the maximal noise standard deviation. The basal noise level $q = 6\%$ is used for the different ecological scenarios (see Fig. 3) and $q \in \{[1\%, 5\%, 10\%, 20\%, 30\%, 40\%, 50\%, 60\%, 70\%, 80\%, 90\%, 100\%]\}$ are used for the data noise benchmark (see Fig. 4). For each noise level, we replicated the whole benchmark pipeline (i.e. data noising, pre-processing with spline-smoothing and cMFA inference) $n = 7$ times, allowing to display inference variability.

Final synthetic ground truth data can be observed in Fig. 2, right panel, where we can observe for each ecological situation the dFBA solution (plain lines), the noisy data set (dots, with 3 replicates at 10 evenly spaced time steps), and the spline reconstruction (dotted lines) with its confidence cone.
<figure><img src="image_4.png" /><figcaption>**Fig. 2. Synthetic data for cMFA benchmark.** Left panel: the different cases of increasingly complex ecological interactions are sketched. We represent the two bacteria and their main substrate or products. GLU: glucose. ALA: alanine. VAL: valine. GAL: galactose. XYL: xylose. ACE: acetate. Right panel: for each ecological situation, the synthetic data are represented: dFBA solution (plain lines), noisy data set which is cMFA input (dots, with 3 replicates at 10 evenly spaced time steps), and the spline reconstruction (dotted lines) with its confidence cone, as reconstructed by cMFA pre-processing step.</figcaption></figure>
*Synthetic metatranscriptomics Data:* Synthetic metatranscriptomic data are obtained from the internal fluxes computed during the dFBA simulations so that we have complete knowledge of whether a reaction is active or inactive, as we know the exact values of the internal fluxes. We assume that if the absolute value of the flux $\mathcal{F}_r^b$ of reaction r for bacteria b is less than $10^{-15}$, then the flux is considered null (inactive). In other words, we set

$$ \mathcal{T}_t^{(b)}(r) = \mathbb{1}_{\mathcal{A}_t^{(b)}}(r) \text{ where } \mathcal{A}_t^{(b)} = \{r \in \mathbf{R} \mid |\mathcal{F}_r^b(t)| > 10^{-15}\} $$

# 3 Results

## 3.1 Reconstruction of Exchange Fluxes

The first expected result consists in evaluating the quality of the reconstruction of exchange reactions such as the consumption and production rates of external metabolites and the growth rates of each microorganism. Initially, the available information is at the community scale. E.g., in the *Competition* case, we only know the overall metabolic activity of the community, that is, the total amount of glucose consumed by both microorganisms simultaneously. However, since one produces valine and the other alanine, and that the abundance of each micro-organism is known, one can expect to achieve a metabolic deconvolution to reconstruct individual glucose consumption. In Fig. 3 top-left panel, we can observe that the cMFA model achieves a very good reconstruction of exchange fluxes ($R^2$: 0.97). We can observe at time 7 a discrepancy in biomass reconstruction. We stress that comparison here are done between dFBA outputs, taken as gold standard, and cMFA, which accumulates uncertainties from data noise, data preprocessing with spline-smoothing and cMFA inference itself. Conversely, we note a very good agreement at the other time steps, indicating that data noise is completely removed by cMFA: the metabolic model informing the inference allows to find a trade-off between noise added to metabolic substrates and products, which is close to the dFBA solution. In other word, cMFA denoises the data by finding metabolically-coherent outputs. Overall consumption in time of each bacteria can be computed (see Fig. 3 center panel): bacteria 1 consumes 56.8% of the glucose, compared to 43.2% for the latter.

A similar analysis can be conducted for the *Co-production* case. In this case, the two microorganisms individually consume two different metabolites (galactose and xylose) while competing for glucose. Additionally, both are strong acetate producers. This situation presents a double indeterminacy: they partially consume the same resource and produce the same metabolite. We then expect that their different growth rates will help to deconvolve the data. The cMFA model successfully reconstructs the exchange fluxes with a coefficient of determination of 0.97 for all the time (see Fig. 3 middle left panel), revealing that 41.2% of the glucose and 46.3% of the acetate are consumed and produced, respectively, by the xylose-consuming microorganism, versus 58.8% of glucose consumed and 53.7% of acetate for the other (see middle center panel Fig. 3).

Finally, we consider the *Cross-feeding* case, which is biologically widespread but computationally complex, since the extracellular level of the metabolite grounding cross-feeding, here alanine, constantly evolves in time due to constant production and consumption by the cross-feeders. Despite this complexity, our model achieves a score
of 0.97 on exchange reactions, demonstrating its robustness. So, of the total alanine
metabolism, 32% is attributed to production by bacteria 1, which consumes glucose
and produces alanine, while 68% corresponds to its consumption by bacteria 2 (see
Fig. 3 middle panel). This confirms the initial observation that alanine is consumed in
greater amounts than it is produced (see Fig. 2 right panel), knowing that we initially
had alanine present in the medium. In this scenario, cross-feeding is active all along the
simulation which explains the absence of a plateau phase — a behavior that the model
successfully captures.

## 3.2 Reconstruction of Internal Fluxes

While the reconstruction of exchange fluxes is directly constrained by observed data, internal fluxes depend solely on biology-informed terms in cMFA, such as the bounds imposed on reactions or the mass-conservation constraint. In the absence of additional data, the inferred solution may favor certain metabolic pathways over others, without necessarily reflecting the true underlying biology. This highlights the importance of integrating meta-transcriptomic data to guide the model toward a biologically plausible solution.

The results shown in Fig. 3 right panel illustrate a comparison between simulated internal fluxes and those inferred by our method : for each community, all internal reactions (over 5000 in total, approximately 2500 per microorganism) are displayed for all time steps. The obtained score is highly satisfactory (respectively 0.99, 0.93, and 0.94), demonstrating that the model is able to reconstruct the internal metabolism of each microorganism within each community.

## 3.3 Benchmark on data noise

Since the data used for cMFA are obtained from three replicates of dFBA output per-
turbation by multiplicative noise, we assess the robustness of the model with respect
to the level of noise. Noise is injected at various intensities, ranging from low (1%)
to high (100%) proportion of a maximal noise (see Sect. 2.3), and $n = 7$ repetitions
of the benchmark are conducted. The $R^2$ score is taken as summary statistic of cMFA
accuracy, with differentiated $R^2$ for exchange and intracellular fluxes.

As expected, the $R^2$ scores for exchange reactions are generally higher than those
for internal reactions for all ecological cases (Fig. 4). This is because exchange reac-
tions are directly constrained by observed data, whereas internal reaction fluxes are
constrained by biological knowledge (i.e. metabolic network), and covariable such as
metatranscriptomic data. However, this biological information is incomplete, since the
binary constraint used to describe metatranscriptomic in cMFA is boolean and fails to
quantitatively represent gene expression, assigning the same activity level to all active
reactions. If cMFA predictions are very accurate with low noise ($R^2$ close to 1), we can
observe a degradation of $R^2$ scores with noise levels, which remains limited for simple
ecological scenario (*Competition*, $R^2$ around 0.8 even for high noise level) and more
pronounced for more complex situations (*Co-production* and *Cross-feeding*, $R^2$ around
0.4 for high noise level).
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Comparison between measured and inferred fluxes between the dFBA ground truth and the cMFA estimate. We represent in the different rows the 3 different ecological scenarios of the benchmark. **Left panel:** exchange fluxes. The reconstruction of the exchange flux is displayed at each time step for the different micro-organisms (dots or star lines) and the different metabolites and biomass (colored lines). Unmarked lines are common to both microorganisms, i.e., community-scale measurements and reconstructions. **Center panel:** individual contribution. The total relative individual contribution of each microorganism to the consumption or the production to a shared metabolite is displayed. Namely, we represent for a shared metabolite *m* and a bacteria *b* the value $\int_0^T \nu_{m,t}^{(b)} dt / (\sum_{b_i} \int_0^T \nu_{m,t}^{(b_i)} dt)$. **Right panel:** intracellular fluxes reconstruction. The dFBA fluxes of all model reactions (x-axis) are displayed against the corresponding cMFA reconstruction (y-axis). The bisector $y = x$ is displayed (red line), together with $R^2$ and RMSE, for bacteria 1 (dots) and 2 (stars).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Impact of increasing levels of noise on model performance across different communities. For each noise level, the mean *R*² score is displayed along with a 95% confidence interval computed from *n* = 7 replicates of the benchmark.</figcaption></figure>

# 4 Conclusion

We introduced the cMFA method, a biology-informed inference method for metabolic flux reconstruction at the community scale. The method, which is formulated as a quadratic programming model, is solved using state-of-the-art solvers and benchmarked upon synthetic data produced with dFBA for three different scenarios of increasing ecological complexity and for increasing level of noise. The method showed a good accuracy and robustness in these different situations. The cMFA method allows for multi-omic time series integration at the community scale.

**Acknowledgments.** This study was funded by the French National Research Agency (ANR) under the project MISTIC (ANR-22-PEAE-0011), as part of the PEAE 2022 program.

**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

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