Interplay within microbial communities impacts ecosystems on several scales, and elucidation of the consequent effects is a difficult task in ecology. In particular, the integration of genome-scale data within quantitative models of microbial ecosystems remains elusive. This study advocates the use of constraint-based modeling to build predictive models from recent high-resolution -omics datasets. Following recent studies that have demonstrated the accuracy of constraint-based models (CBMs) for simulating single-strain metabolic networks, we sought to study microbial ecosystems as a combination of single-strain metabolic networks that exchange nutrients. This study presents two multi-objective extensions of CBMs for modeling communities: multi-objective flux balance analysis (MO-FBA) and multi-objective flux variability analysis (MO-FVA). Both methods were applied to a hot spring mat model ecosystem. As a result, multiple trade-offs between nutrients and growth rates, as well as thermodynamically favorable relative abundances at community level, were emphasized. We expect this approach to be used for integrating genomic information in microbial ecosystems. Following models will provide insights about behaviors (including diversity) that take place at the ecosystem scale.

All relevant data are within the paper and its Supporting Information files.

Microbial organisms comprise approximately 50% of the Earth’s biomass [

In the last decade, great advances have been made in the development of high-throughput techniques that enable the study of the metagenomics, meta-transcriptomics, and meta-metabolomics of natural communities. Such techniques provide ‘omics-scale information for organisms, from which it is possible to identify specific molecules (

We propose to overcome this challenge by using recent systems biology approaches for the prediction of quantitative behaviors of single organisms based on genome-scale data [

Genome-scale descriptions, in this context, are provided by metabolic networks. A metabolic network summarizes the set of biochemical reactions encoded by the genome of a given organism. Two reactions are linked within a metabolic network if the substrate of one reaction is the product of the other. Such genome-scale descriptions of organisms are currently applied in systems biology for the purpose of investigating physiology [

Quantitative analyses utilize such metabolic networks as inputs for constraint-based models (CBMs) in order to infer physiological features based on a genome-scale description [

Several attempts have been made to model the metabolic network of microbial communities. Rodrígez

To overcome such a weakness, more elaborated modeling approaches have been proposed. Zomorrodi and collaborators [

In this study, we propose a complementary model, to investigate the general case of microbial ecosystems. Based on Pareto optimality [

Specifically, following previous works, we implemented a multi-objective flux balance analysis method [

For the sake of MO-FBA and MO-FVA illustration, this study models a microbial ecosystem comprising three distinct phenotypes: a primary producer,

The genomic data for a particular microorganism describes a set of genes, allowing the identification of enzymes and related reactions. Reactions produce metabolites that are used as substrates in subsequent reactions; such interplay constitutes a _{1}, …, _{r}) stands for the flux vector, _{j} is the flux of reaction _{j} for all

(A) _{ij} of the matrix corresponds to the stoichiometric coefficient of metabolite _{i} in reaction _{j}, with reactants as negative and products positive. Exchange reactions and exchange metabolites are placed in the right and inferior section of the matrix, respectively. Therefore, submatrix

Under steady-state conditions, the continuous supply of metabolites from the media is facilitated by exchange reactions at a constant rate (dark gray eclipses and dashed lines in _{i} = _{i} ≤ _{i} ≤ _{i}, resulting in a model described as a set of constraints. Such models are termed CBMs. CBMs usually comprise more reactions than metabolites; therefore, these models are undetermined in that when a solution

Flux balance analysis (FBA) is one of the most widely used approaches for the identification of points of interest in the flux space [^{⊺}

The set of all optimal flux distributions, _{j}:

Both FBA and FVA are today state-of-the-art tools to explore CBMs [

FBA and FVA utilize constraints derived from mass conservation laws; however, it is possible to exploit thermodynamic laws to derive constraints in order to obtain further insights into the behavior of a metabolic system [_{i} (expressed in J.mol^{−1}), which quantifies the potential to perform chemical work. Chemical potentials depend on metabolite concentration according to _{i} is the molar concentration, ^{0} is the standard chemical potential (dependent on temperature, pressure, and ionic strength); these are usually tabulated [_{r}
_{j} ≤ 0 for a spontaneous reaction. In the following, we note the Gibbs energy of reaction as a difference of potentials, _{j} ≐ Δ_{r}
_{j}.

Under NESS conditions, the entropy balance implies that ^{⊺}_{ς} = ^{⊺}_{ξ}, where _{ς} represents the internal portion of fluxes, _{ξ} boundary fluxes, and _{j} and _{i}, respectively. The term ^{⊺}_{ξ} represents the

The integration of such equations into general CBMs is not straightforward, as in most of applications, concentrations _{i} are not known; therefore, these must be introduced as variables. As a result of non-linear expressions, CBM formulations using these constraints are generally more complex to solve [

In general, optimization problems are aim at determining ^{⊺}

The first algorithm to solve a LP, which was proposed in 1947 by Dantizg [

In order to model a microbial community, each strain is considered a single compartment [^{σ}, which is formed by the stoichiometric matrices of each single organism. Accordingly, for a community of ^{l},

For the sake of illustration, an ecosystem may be considered to comprise three microbial strains. (A) According to the metabolic model, each microorganism is considered a separate compartment, depicted here in green, orange, and purple. Metabolic networks are linked via an additional compartment, termed the “pool” (blue)), which sums up all external metabolites exchanged between organisms and the environment. (B) depicts the Stoichiometric Matrix ^{σ}, where each compartment is colored accordingly, with their corresponding _{1}, _{2} and _{3} define the “objective space”.

As shown in ^{1} to ^{k} are used to construct a diagonal block matrix. Each block is linked to a ^{l}, for _{q} to _{n} for metabolites _{q} to _{n} between the Pool and the external environment, is additionally set (bottom right in ^{σ}_{i} and _{i}, these constraints describe a solution flux space, as depicted in

Each compartment above corresponds to an organism with a specific objective function _{k}. Accordingly, the following multi-objective optimization problem, for analyzing flux balance conditions (MO-FBA), may be defined:
_{1}, …, _{k})^{⊺} are the objective functions of the

Interpretation of MO-FBA can be done in terms of growth rates and resources used to produce such growth. Indeed, if one of the members of the ecosystem decreases its growth rate, more resources are available for other members. According to their particular physiologies, they can use these new available resources to increase their own biomass. A guideline containing three ideal cases for two guilds is provided in

Given a particular point _{j} with _{j} may be determined by solving the following LPs:
_{j}, _{1}, …, _{k}].

Biological systems are hypothesized to favor thermodynamic states where entropy production is maximal [_{j} for each reaction, with _{j}, _{j}] of the flux _{j} near the Pareto optima

In 1906, Vilfredo Pareto in his

Let

A point ^{w} is a ^{w}). Therefore, a (weak) Pareto optimum is the image of a (weak) efficient solution. Note that all efficient solutions are also weakly efficient solutions but no vice-versa. The collection of Pareto optimal points is termed

Approaches for solving MOPs have been reviewed, for example, by [

The most well known approach is the “weighted sum approach”, wherein the weighted sum of the objective functions is optimized, _{k}
_{k}(_{k} ≥ 0 and at least one _{k} > 0. If

Another commonly used approach is the “_{j}(_{i}(_{i},

Not all approaches rely on scalarization: for MOLPs, a set of algorithms describing the shape of the image of efficient points,

The various approaches to studying microbial communities have been recently reviewed by Biggs

In order to illustrate the application of the present approach, we modeled the microbial ecosystem of hot spring microbial mats [

The model comprises three guilds of microorganisms of the SYN, FAP, and SRB phenotypes. Organics acids produced by SYN may be utilized by FAP and SRB. FAP is capable of fixing carbon by anoxygenic photosynthesis. Under anoxygenic fermentation conditions, FAP is additionally capable of producing hydrogen, which, in turn, may be used by SRB.

Using the available compartment model of this system, as described in [^{−1}) as maximal growth rate [

Glycolate is produced by the use of O_{2} instead of CO_{2} by the Rubisco enzyme; the flux ratio between the use of O_{2} and CO_{2} varies between 0.03 and 0.07. This restriction was included linearly in the model by fixing a ratio of 0.03 between SYN reactions

Excess photosynthate producing during the day is stored as polyglucose (PG) by SYN. PG is fermented at night, producing several organic acids that accumulate in the media and are integrated as biomass mostly under light conditions [

For each of the exchanged metabolites, standard Gibbs energies for biological conditions were obtained from [_{2}: 6CO_{2}+6H_{2}O _{6}H_{12}O_{6}. The assumption that this reaction approaches equilibrium at standard biological conditions (_{hv} = 68.6 kJ.mol^{−1} (^{3} and 10^{−3} M, and therefore chemical potential equals ^{3}) ≈ 20 (kJ.mol^{−1}) for T = 75°Celsius. For water and _{H2O} = dg_{hv} = 0.

For each guild, a metabolic model was built in MATLAB and an ecosystem stoichiometric matrix ^{σ} was constructed, as described above. MO-FBA was carried out using BENSOLVE [

From methods discussed in Biggs

All scripts are available in

SYN, SRB, and FAP growth rates are represented in a 3-dimensional space, in each axis, respectively, in

(A) shows a 3D Pareto front, in yellow, describing the maximal growth rates of SYN, FAP, and SRB (in terms of units per hour, h^{−1}), when considered as a system. It is evident that a decrease in the growth rate of one organism results in an increase in that of the other two, but not necessarily in equal proportions (see ^{−1}) and 1.11 (h^{−1}), respectively. In (B), (C), (D), and (E), the Pareto front was projected onto the triangular surface formed by P1, P2, and P3. (B), (C), and (D) shows the respective growth rates for SYN, FAP, and SRB, respectively. (E) shows the sum of the three growth rates, which represent the total biomass of the ecosystem.

The results show that when each guild grows at its maximal rate, no biomass is produced by the other guilds. The sum of the growth rates is always minimal in vertices (blue areas in

Multi-objective FVA was performed in the P4 and P5 regions to explore NH_{3} import and export fluxes between guilds (_{3} that is not used to build biomass is excreted. This point is emphasized in

(A) shows NH_{3} maximal and minimal fluxes for SYN, FAP, SRB, and pool compartments (green, yellow, purple, and blue respectively) for extreme points P4 and P5. The export of NH_{3} by SYN is correlated with a drop in their growth rate; similarly, increases in NH_{3} intake are correlated with increases in the growth rates of FAP and SRB. (B) Three sections selected for the illustration of MO-FVA; (C) Mean values of the minimal and maximal fluxes over selected sections of NH_{3}, CO_{2}, acetate, and glycolate (columns) for each section (rows).

Nitrogen uptake by FAP and SRB occurs solely from ammonia that is available in the pool compartment; therefore, these strains compete for its intake. When SRB is not growing (superior panel in _{3} is taken up mainly by FAP (both minima and maxima are negative, implying an intake from the pool). Small amounts that are not taken up by FAP may be either taken up by SRB (maximal rate value is null and minimal rate negative, which depicts a possible import) or excreted to the external environment (pool maximal rate value is positive and minimal rate value is null, which depicts a possible export to the media). When SRB is growing (inferior panel of

In order to analyze the relationships between the growth rate of each strain and nitrogen- or carbon-related fluxes, we performed a MO-FVA as described in Computational Procedures, focusing on exchange reactions. For the purpose of illustration, we highlighted three sections from 225 calculated, as shown in

For NH_{3} exchange reactions, high growth rates of SYN are related to lower levels of ammonia export, which represents a limiting factor for FAP and SRB growth rates. This results in the two strains competing for its use (

SYN consumes approximately the same amount of CO_{2} under all relative abundance conditions (see second column in

The present results additionally emphasize that FAP and SRB produce relatively small amounts of CO_{2} at low growth rates. However, when the growth rate of FAP increases, the maximal excretion of CO_{2} reduces, whereas its minimal excretion increases; these data indicate the theoretical efficiency of carbon management, as experimentally reported by [

As discussed previously, the direct integration of thermodynamic constraints into MO-FBA and MO-FVA formulations is complex. Instead, we used the thermodynamic optimization problem stated in as a post-treatment analysis. Considering fluxes as computed by MO-FVA in 5 151 points of Pareto front (as a result of which growth rates are also determined), we estimated the corresponding maximal

(A) Description of the chemical motive force (kJ.gr^{−1}.DW^{−1}.h^{−1}) for each point of the Pareto front; red regions indicate thermodynamically favored growth rates, while the points where the solver does not reach the optimal criteria are shown in white. The obtained surface appears smooth, without sudden changes in neighboring values. (B) Description of the overall community biomass distribution based on the growth rate of each strain, with a particular emphasis on regions supported by experimental measurements showing a SYN: FAP ratio of between 1.5 and 3.5.

Results show that higher

Given that all surface showed positive values, all regions are feasible from a thermodynamic viewpoint. Under the hypothesis that a biological system prefers configurations in which entropy production is maximal, it is expected that an ecosystem would favor growth rates with higher

We compared growth rates and flux predictions of MO-FBA and MO-FVA with those obtained by a comparable approach (OptCom [_{3}, acetate, glycogen, and CO_{2}, are described in _{3}, acetate, glycogen, and CO_{2} fall into the range predicted by MO-FVA. Without constraining SYN biomass (point O1), OptCom does not reach the maximal biomass optimum. However, when SYN biomass is increasingly constrained (points O2 to O11), the total biomass increases. This suggests the existence of local optima in the OptCom general formulation for this model.

The composition of a community that function in a constant environment can be also assessed using the approaches proposed in [

As reported in previous studies, in particular [

Unlike previous works that consider multiple objectives, our approach does not rely either on assumptions about ecosystem behaviors, such as maximization of the total ecosystem biomass, ([

For illustration purposes, we applied MO-FBA to the daytime part of the diurnal cycle of the microbial hot spring mat system [

MO-FVA results show that nitrogen flux is correlated to growth rates, and that the three guilds compete for their usage. In contrast, CO_{2} consumption and glycolyte and acetate production by SYN do not seem to be correlated with its growth rate, indicating that these processes are not carbon-limited. Reduced carbon, represented by acetate, appears as being the main carbon flux in the system for FAP and SRB, and becomes a limiting nutrient for FAP at high growth rates. This result is consistent with those of [

By coupling MO-FVA results with chemical potentials, we were able to analyze thermodynamic constraints and study favored conditions of the Pareto front by comparing their respective maxima

Nevertheless, further refinement of the thermodynamic calculations is warranted. In particular, the calculation of

Despite the above limitations, we consider the present form of the modeling approach as fruitful guidance to gain qualitative as well as quantitative data for the metabolic interplay between various species in an ecosystem. This method paves the way for improved contextualization of other -omics datasets in microbial ecology by providing a mechanistic description of species co-occurrence

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A Stoichiometric Matrix of each guild used, along with thermodynamic data considered.

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The convention used is the same for

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MB is supported by CNRS and Region Pays de la Loire funding (GRIOTE project). This study was supported by ANR (IMPEKAB, ANR-15-CE02-001-03). We would like to thank Editage (