Mathematical Modeling of Microbial Community Dynamics: A Methodological Review
Abstract
:1. Introduction
2. Background Information
2.1. Modeling Units and Model Classification
2.2. Mathematical Notations
 c = [c_{1}, c_{2},…,c_{I}]: The vector of concentration of J extracellular metabolites (such as substrates and produced metabolites) in environment
 r_{k} = [r_{1},_{k}, r_{2},_{k},…,r_{Jk,k}]: The vector of J_{k} fluxes (or reaction rates) for species k
 S_{k}: (I΄_{k} × J_{k}) Stoichiometric matrix of species k
 x = [x_{1}, x_{1},…,x_{K}]: The vector of relative abundance or biomass concentration of K species
 I = [1,2, …,I]: Indices of I metabolites in environment
 I΄_{k} = [1,2, …,I΄_{k}] Indices of I΄_{k} intracellular metabolites for species k
 J_{k} = [1,2, …,J_{k}] Indices of J_{k} fluxes for species k
 K = [1,2, …,K]: Indices of K species
 Y_{i,k}: The yield of metabolite i for species k
 Y_{x,k}: The biomass yield of species k
 µ_{k}: The growth rate of species k
3. SupraOrganismal Approaches
3.1. Stoichiometric ModelBased Analysis
3.2. Metabolic FunctionBased Dynamic Modeling
4. PopulationBased Models
4.1. Inference of Microbial Interactions
Relation  Examples  

Bidirectional  Mutualism or synergism  ⊕⊕  Biofilm formation to confer antibiotic resistance to the community members [52,53] 
Syntropy (or crossfeeding): Hydrogen transfer between sulfate reducers and methanogens [54]  
Competition  ⊖⊖  Species with similar niches: Paramecium aurelia and Paramecium caudatum [55]  
Antagonism  ⊕⊖  Predation: Ciliates feeding on bacteria [50]  
Parasitism: Bacteria and bacteriophages [50]  
Unidirectional  Commensalism  ⊕⊙  Acetobacter oxydans oxidizing mannitol to produce fructose, which is used by other species such as Saccharomyces carlsbergensis that can metabolize fructose, but cannot mannitol (http://www.eoearth.org/view/article/171918/) 
Mycobacterium vaccae metabolizing cyclohexane to cyclohexanol, which is subsequently used by Pseudomonas species (http://www.eoearth.org/view/article/171918/)  
Amensalism  ⊖⊙  Lactobacilli producing acids that lower the pH of the surrounding environment [50]  
The bread mold Penicillium secreting penicillin that kills bacteria [56]  
Nondirectional  Neutralism  ⊙⊙  Growth of yogurt starter strains of Streptococcus and Lactobacillus in a chemostat [51]: The populations of these strains do not change much regardless of whether cultured separately or together 
4.1.1. Network Inference
4.1.2. Stoichiometric Modeling of Multiple Species
4.2. Nonlinear Regression Models
4.3. ThermodynamicallyBased Models
4.4. TraitBased Modeling
4.5. LotkaVolterra Model
4.6. Evolutionary Game Theory
5. Tools for Simulating Heterogeneity
5.1. Simulation of Spatial Heterogeneity Using PopulationBased Models
5.2. IndividualBased Modeling
5.3. Population Balance Modeling
6. Integrative Modeling Strategies
6.1. Information Feedback
6.2. Indirect Coupling
6.3. Direct Coupling
7. Summary and Recommendations
Approach  Data for Parameter Identification  Inputs for Simulation  Outputs from Simulation  Remarks 

Flux balance analysis (FBA) ([64])  N/A (FBA has no parameters to tune)  x_spe and r_{in}_tot in a certain condition  r_spe in the given condition 

Elementary mode (EM) analysis ([41])  N/A (no parameters to tune)  Information on x_spe and r_{in}_tot in a certain condition  r_spe in the given condition 

Genecentric approach ([42])  e_tot(t,z), x_tot(t,z), and c(t,z) upon (designed) perturbations  e_tot(0,z), x_tot(0,z), c(0,z) ( i.e., initial distributions)  e_tot(t,z), x_tot(t,z), c(t,z), r_tot(t,z) in any new conditions 

Nonlinear regression ([75])  x_spe and c across conditions/times/locations  c at a specific condition/ time/location  x_spe in the given condition/time/location 

Traitbased model ([99])  x_spe(t) and c(t) upon (designed) perturbations  x_spe(0) and c(0) ( i.e., initial conditions)  x_spe(t) and c(t) 

Generalized LotkaVolterra (gLV) model ([103])  x_spe(t) and c(t) (to model the growth rate as a function of c(t))  x_spe(0) and c(0)  x_spe(t) 

Evolutionary game theory ([27])  Understanding or knowledge on the interspecies relationship  x_spe(0) and assumed parameter values  x_spe(t) 

Thermodynamicallybased model ([77])  Information on chemical potentials (or reaction rate values)  x_spe(0) and c(0)  x_spe(∞) and c(∞) ( i.e., values after sufficiently enough time) 

Population balance model (PBM) ([120])  x_spe(t), c(t), and information on population heterogeneity  x_spe(0), c(0), and initial population heterogeneity  x_spe(t), c(t), and population heterogeneity (evolving with time) 

Individualbased model (IbM) ([113])  x_cell(t,z) and c(t,z)  x_cell(0,z) and c(0,z)  x_cell(t,z) and c(t,z) 

Dynamic FBA (dFBA) ([125])  x_spe(t) and c(t)  x_spe(0) and c(0)  x_spe(t), r_spe(t), and c(t) 

Cybernetic model ([129])  x_spe(t) and c(t)  x_spe(0) and c(0)  x_spe(t), r_spe(t), e_spe, and c(t) 

Dynamic multispecies metabolic modeling (DyMMM) ([132])  x_spe(t) and c(t)  x_spe(0) and c(0)  x_spe(t), r_spe(t), and c(t) 

Dynamic OptCom (dOptCom) ([73])  x_spe(t) and c(t)  x_spe(0) and c(0)  x_spe(t), r_spe(t), and c(t) 

Indirect coupling FBA with transport ([122])  x_spe(t,z) and c(t,z)  x_spe(0,z) and c(0,z)  x_spe(t,z), r_spe(t,z) and c(t,z) 

8. Conclusions and Outlook
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Song, H.S.; Cannon, W.R.; Beliaev, A.S.; Konopka, A. Mathematical Modeling of Microbial Community Dynamics: A Methodological Review. Processes 2014, 2, 711752. https://doi.org/10.3390/pr2040711
Song HS, Cannon WR, Beliaev AS, Konopka A. Mathematical Modeling of Microbial Community Dynamics: A Methodological Review. Processes. 2014; 2(4):711752. https://doi.org/10.3390/pr2040711
Chicago/Turabian StyleSong, HyunSeob, William R. Cannon, Alexander S. Beliaev, and Allan Konopka. 2014. "Mathematical Modeling of Microbial Community Dynamics: A Methodological Review" Processes 2, no. 4: 711752. https://doi.org/10.3390/pr2040711