While we discussed various mathematical frameworks so far, a single approach alone cannot comprehensively describe the dynamic nature of microbial communities. For example, constraint-based approaches such as cFBA can simulate interspecies metabolic interactions, but they cannot predict interactions under dynamic environmental conditions. Conversely, gLV models are able to simulate community dynamics well but do not provide direct mechanistic interpretations on the microbial interactions that vary in space and time. In this regard, synergistic integration of different mathematical tools appears to be a promising strategy. Integrative use of more than two approaches can be attempted at three different levels: (i) information feedback; (ii) indirect coupling; and (iii) direct coupling. Information feedback is the weakest form of integration, through which one can take advantage of information generated from different approaches without coupling in order to facilitate model development. Indirect coupling implies that the dynamic simulation of a model uses the outputs of another model that are previously generated through independent simulations; on the other hand, in direct coupling, different modeling frameworks are merged into an expanded single platform. This section provides examples of the current practices of integrative modeling approaches.

#### 6.2. Indirect Coupling

Model integration is referred to indirect coupling if the outputs obtained from the “independent” simulation of one framework are fed into another model during its simulation. A good example is the integration of a genome-scale network with a reactive transport model by Sheibe

et al. [

122] for the study of

in situ uranium bioremediation by

Geobacter species (

i.e.,

Geobacter sulfurreducens). In this system considering a single organism, they repeatedly ran FBA to generate a look-up table that provides the nutrient uptake and growth rates in various possible environmental conditions. Then, the resulting look-up table was referenced at every time step and every grid cell throughout the dynamic simulation of a reactive-transport model. The prediction of condition-specific biomass yield was predicted using FBA by constraining the uptake rates of the three nutrients (acetate, Fe (III), and NH

_{4}) that limit the species growth. In order to generate a look-up table, they chose 10 different concentration levels of these key nutrients, and for each of 1000 combinations, they performed FBA. Consequently, they could successfully predict acetate concentration and U (VI) reduction rates in a field trial of

in situ uranium bioremediation. Obviously, this indirect coupling results in reduced computation time in comparison to direct coupling that runs both FBA and reactive-transport simulation for their interaction at every time step. As environmental conditions were discretized over coarse meshes, interpolation within the look-up table is required to get the rates between pre-specified conditions. While performing this interpolation process at every time step/grid will be time-consuming, particularly in case of field-scale simulations, one can minimize the look-up table by containing the fluxes of key metabolites only, instead of the full flux vector.

Figure 10 illustrates the concept of indirect coupling between FBA and a reactive-transport model. In principle, the same approach can be applied to microbial communities, although the look-up table generation using cFBA and the interpolation would require substantially higher computational power.

**Figure 10.**
Indirect coupling between a reactive-transport model with FBA for the dynamic simulation of a single organism growth. One should first generate a look-up table through the repeated running of FBA with a large number of different sets of nutrient uptake rates (which is 1000 in total in this example) generated by discretizing the range of each uptake (**top**). During the dynamic simulation of the reactive-transport model, the growth rate and metabolite production rates are updated at each time/location from the look-up table (**bottom**).

**Figure 10.**
Indirect coupling between a reactive-transport model with FBA for the dynamic simulation of a single organism growth. One should first generate a look-up table through the repeated running of FBA with a large number of different sets of nutrient uptake rates (which is 1000 in total in this example) generated by discretizing the range of each uptake (**top**). During the dynamic simulation of the reactive-transport model, the growth rate and metabolite production rates are updated at each time/location from the look-up table (**bottom**).

The integration of FBA with dynamic modeling can be considered in a much simpler context—well-mixed conditions—using the framework called dynamic FBA (dFBA) [

123]. In a study of bioprocesses that produce bioethanol from glucose and xylose, Hanly and Henson modeled simple consortia composed of two organisms that are (i) non-interacting [

124] and (ii) indirectly interacting [

125]. The non-interacting consortium was composed of wild-type

S. cerevisiae that consumes glucose only and mutated

E. coli capable of consuming xylose alone. The indirectly interacting model included (wild-type)

S. cerevisiae and

Pichia stipites (also known as

Scheffersomyces stipitis) that consume both glucose and xylose. In the latter, there were two types of competition–interspecies and intraspecies. While interspecies competition for glucose could be described simply by Michaelis-Menton kinetics, the authors had to incorporate glucose inhibition to describe intraspecies competition between the consumptions of glucose and xylose in

P. stipitis (

Figure 11). dFBA has also been used in other applications, e.g., for exploring bacterial diversity and their metabolic interactions [

126]. In balanced growth conditions where uptake rates change in time but the biomass (and other metabolites) yield from a substrate is constant, the dFBA can be implemented as a form of indirect coupling by referring to the pre-calculation of an FBA solution, without having to solve the LP problem at every time step.

**Figure 11.**
Uptake kinetics for the dynamic FBA (dFBA) simulation of the dynamics of non-interacting (**left**) and indirectly interacting (**right**) consortia. Glc and Xyl denote the concentrations of glucose and xylose, respectively. For the sake of simplicity, product (i.e., ethanol) inhibition term is neglected.

**Figure 11.**
Uptake kinetics for the dynamic FBA (dFBA) simulation of the dynamics of non-interacting (**left**) and indirectly interacting (**right**) consortia. Glc and Xyl denote the concentrations of glucose and xylose, respectively. For the sake of simplicity, product (i.e., ethanol) inhibition term is neglected.

Modeling microbial consortia using indirect coupling can also be done by integrating EM analysis with a dynamic framework such as cybernetic models. The framework called hybrid cybernetic model (HCM) [

127,

128] identifies a relevant subset of EMs as metabolic options for accommodating the metabolic shift in individual species. Geng

et al. [

129] applied HCM to model a situation where the culture medium contains four sugars (

i.e., glucose, xylose, mannose, and galactose) which are competitively consumed by three yeast strains (

i.e.,

S. cerevisiae,

P. stipitis, and

Kluyveromyces marxianus). Consumption patterns of these sugars vary depending on the organism.

S. cerevisiae consumes glucose, mannose, and galactose, but not xylose. Among three hexoses fermentable by

S. cerevisiae, galactose is the least preferred substrate, while glucose and mannose are preferably consumed. On the other hand,

P. stipitis and

K. marxianus ferment all four sugars, the consumption of which starts with the pair of glucose and mannose, followed by the pair of xylose and galactose. Without having to incorporate empirical inhibition terms, the competitive consumption of four sugars by each organism was modeled in a simpler form based on the cybernetic control variables (

Figure 12).

The integration of dynamic population-based models (e.g., gLV model) with cFBA would be another possible form of indirect coupling, yet we could not find an appropriate example in the literature. As an input, cFBA requires information of species abundance, the dynamic change of which can be provided from the independent simulation of a gLV model. Thus, this integration enables the prediction of the change of flux distributions at each time step within individual species and the community. These predictions are beyond the level achievable using gLV model or cFBA alone.

**Figure 12.**
Uptake kinetics for the hybrid cybernetic model (HCM) simulation of the consortia composed of three yeast strains (Saccharomyces cerevisiae, Pichia stipitis, and Kluyveromyces marxianus) growing on four different carbon sources. Glc, Man, Gal, and Xyl denote the concentrations of glucose, mannose, galactose, and xylose, respectively. The symbols e and v represent enzyme level and its activity. For the sake of simplicity, self-substrate inhibition term is neglected.

**Figure 12.**
Uptake kinetics for the hybrid cybernetic model (HCM) simulation of the consortia composed of three yeast strains (Saccharomyces cerevisiae, Pichia stipitis, and Kluyveromyces marxianus) growing on four different carbon sources. Glc, Man, Gal, and Xyl denote the concentrations of glucose, mannose, galactose, and xylose, respectively. The symbols e and v represent enzyme level and its activity. For the sake of simplicity, self-substrate inhibition term is neglected.

#### 6.3. Direct Coupling

In complex systems containing multiple interacting species and many constraining environmental variables, simulation of microbial community dynamics using look-up tables to indirectly couple FBAs and dynamic models will become unattractive. Fang

et al. [

130], therefore, applied a “direct” coupling of a genome-scale metabolic network with a reactive-transport model. A reactive-transport model dynamically interacts with FBA at each time step to obtain reaction rates required for solving differential equations (

Figure 13).

**Figure 13.**
Implementation of direct coupling between FBA and a reactive-transport model.

**Figure 13.**
Implementation of direct coupling between FBA and a reactive-transport model.

In contrast to dFBA-based approaches which assume quasi-steady state on intracellular metabolites, King

et al. [

131] coupled a reactive transport model with a dynamic model of intracellular kinetics based on a simplified network. Resat

et al. [

113] also considered intracellular dynamics (using an even simpler network) to describe the cellular dynamics using an IbM framework (similar to BacSim [

117]). An IbM was directly coupled with a three dimensional reactive-transport model.

Direct coupling examples addressed above were focused on a single organism, while Resat

et al. [

113] also considered the simulation of two species. The framework developed by Zhuang

et al. [

132], on the other hand, demonstrated the extended application of dFBA to ecological settings. Instead of directly obtaining substrate uptake rate as determined by kinetic equations, they use kinetic equations as upper bounds of uptake rates [

133]. To simulate the dynamic change of

Rhodoferax and

Geobacter species, which are acetate oxidizing Fe (III)-reducers competing in anoxic subsurface environments, they identified uptake kinetics for metabolites that affect the growth of species, including acetate, ammonia, and Fe (III) for two species and imposed them as upper bounds in respective FBA implementations. They termed this method as the Dynamic Multispecies Metabolic Modeling (DyMMM) framework. The same method has also been applied to the community of

Geobacter and sulfate-reducing bacteria (7SRBs) [

134].

Figure 14 shows the procedures of implementing DyMMM.

**Figure 14.**
Implementation of the Dynamic Multispecies Metabolic Modeling (DyMMM) framework proposed by Zhuang

et al. [

132] for the dynamic simulation of microbial consortia.

**Figure 14.**
Implementation of the Dynamic Multispecies Metabolic Modeling (DyMMM) framework proposed by Zhuang

et al. [

132] for the dynamic simulation of microbial consortia.

Zomorrodi

et al. [

73] proposed a general framework for the dynamic simulation of microbial community by extending OptCom. In contrast to DyMMM that considers a community-level objective alone, d-OptCom solves a bi-level optimization problem for which both species- and community-level objectives should be specified. d-OptCom also considers uptake kinetics as upper bounds of fluxes, similarly to DyMMM. If species-level objectives are eliminated from the bi-level optimization formulation, the structure between d-OptCom and DyMMM becomes similar while the former solves nonlinear dynamic programming based on an implicit Euler discretization.