Team:MIT/ModelingDFBA

Dynamic Flux Balance Analysis

C. Hutchinsonii Model Development

E. Coli Model Development

Coculture Simulations and Conclusions

Detailed Description of Modeling Methods

Description of Dynamic Flux Balance Analysis

To model our co-culture, we’re using the Dynamic Multi-species Metabolic Modeling (DMMM) framework, which is just an extension of dynamic flux balance analysis (dFBA) to model a community rather than a monoculture of bacteria (Zhuang et al. 2010). DMMM calculates the concentrations of biomass and extracellular metabolites over time by integrating the set of ODEs \begin{equation} \label{eq:1} \frac{dX_i}{dt} = \mu_i X_i \end{equation} \begin{equation} \label{eq:2} \frac{dS_j}{dt} = \sum_{i=1}^N \nu_j^iX_i, \end{equation} where $X_i$ is the biomass concentration of the ith organism, $\mu_i$ is the specific growth rate of the ith organism, $S_j$ is the concentration of jth metabolite in the extracellular environment, and $\nu^i_j$ is the specific uptake/secretion rate of the jth metabolite due to the ith organism. The specific growth rate $\mu_i$ of the ith organism and the vector of fluxes $V_i$ containing the fluxes of all reactions in the ith organism are calculated using FBA for each timestep. FBA solves an optimization problem defined by: \begin{equation} \label{eq:3} \mu_i = \max c_i^TV_i \end{equation} \[\operatorname{Subject \ to:} A_iV_i=0\] \[lb_i \leq V_i \leq ub_i\] Here, $c^T_iV_i$ is the objective function of the ith organism. Usually growth maximization is used as the objective function, in which case the vector $c^T_i$ contains weights that multiply fluxes corresponding to biomass precursors. $A_i$ is the stoichiometric matrix of the ith organism and contains the stoichiometric coefficients of all the metabolites in each reaction in the whole-genome scale model of the organism. Thus, $A_iV_i=0$ represents the mass balances on each reaction in the organism since intracellular metabolites are assumed to be at steady state. $lb_i$ and $ub_i$ are the flux constraint vectors of ith organism. The flux constraints corresponding to exchange reactions of extracellular metabolites represent the uptake/secretion constraints and must either be unbounded (if the nutrient is not limited or if production is not limited), or calculated using the Michaelis–Menten kinetics method. Typical Michaelis-Menten kinetic expressions are: \begin{equation} \label{eq:4} \nu_{sj} = \dfrac{\nu_mS_j}{K_m+S_j} \end{equation} \begin{equation} \label{eq:5} \nu_{sj} = \dfrac{\nu_mS_j}{K_m+S_j} \dfrac{1}{1+S_k/K_s} \end{equation} \begin{equation} \label{eq:6} \nu_{sj} = \dfrac{\nu_mS_j}{K_m+S_j}\dfrac{1}{1+P_k/K_p} \end{equation} where $S_j$ and $S_k$ are the concentration of substrates $j$ and $k$, respectively, $P_k$ is the concentration of product $k$, $\nu_m$ and $K_m$ are the Michaelis–Menten constants and $K_s$ and $K_p$ are the inhibition constants (Henson and Hanly 2014). Equation $\ref{eq:4}$ describes purely substrate-limited uptake of j, equation $\ref{eq:5}$ describes uptake inhibition by a preferred substrate k, and equation $\ref{eq:6}$ describes uptake inhibition by a secreted product p. These uptake kinetics are organism-specific and usually require a combination of previous knowledge, iterative modelling, and parameter fitting. FBA attempts to calculate the fluxes that maximize the objective function, subject to these two types of constraints (the mass balances and the bounds for the fluxes). If no viable optimal solution is found for biomass maximization, then that organism does not have sufficient nutrients to survive at this time, and death rate must be calculated. We assume that the death rate proportional to the penalty calculated by dFBAlab, which is a measure of the infeasibility of the optimization problem (Gomez et al. 2014).

DFBALab

DFBALab is a MATLAB-based simulator of dFBA that was developed by the Process Systems Engineering Laboratory at MIT (Gomez et al. 2014). DFBALab performs fast and reliable simulation of complex dynamic multispecies cultures by using the LP feasibility problem combined with lexicographic optimization. Instead of using one objective function as in ($\ref{eq:3}$), DFBALab uses a priority list of objectives, with the first as biomass maximization. It is better than other dFBA simulators because it ensures feasibility, when possible, and unique exchange fluxes, and because it is faster. Also, DFBALab can be used with three commercial linear program solvers (CPLEX, Gurobi, and MOSEK) and is compatible with the COBRA toolbox format for genome-scale FBA models. The DFBAlab code can be obtained for free for education and non-profit research purposes at http://yoric.mit.edu/dfbalab. In order to use DFBALab, there are three main MATLAB scripts that the user must create: main.m, RHS.m, and DRHS.m. In main.m, the user must specify the number of microbial species, the genome-scale FBA model for each species, the exchange reactions in each genome-scale model, the list of objectives for each genome-scale model, the initial conditions, the time span of simulation, the plots to generate, and other DFBALab simulation options. The genome-scale FBA models contain the stoichiometric matrices and default reaction bounds, in addition to lists of all the reactions and metabolites. In RHS.m, the user must specify the upper and lower bounds for the exchange fluxes. And in DRHS.m, the user must specify the dynamic mass balances on biomass ($\ref{eq:1}$) and extracellular metabolite ($\ref{eq:2}$) concentrations. Please refer to the DFBALab user manual, which comes with the code, for more details.

References