Difference between revisions of "Team:MIT/Modeling"

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. DMMM calculates the concentrations of biomass and extracellular metabolites over time by integrating the set of ODEs \[\frac{dX^i}{dt} = \mu^i X^i\] \[\frac{dC_j}{dt} = \sum_{}^i S_j^iV^iX^i\] where Xi is the biomass concentration of the ith organism, μi is the specific growth rate of the ith organism, Cj is the concentration of jth metabolite in environment, Vi is the flux vector for the ith organism, and Si,j is the subset of stoichiometric matrix of the ith organism corresponding to the jth metabolite. The specific growth rate (μi) of the ith organism, and flux vector Vi are calculated using FBA for each timestep. FBA solves an optimization problem defined by: \[\mu^i = \max c_i^TV^i\] \[\operatorname{Subject to:} S^iV^i=0\] \[lb^i \leq V^i \leq ub^i\] Here, cTi*Vi is the objective function of the ith organism; usually growth maximization is used as the objective function. Si 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 Si*Vi=0 represents the mass balances on each reaction in the organism. lbi and ubi 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 MM.png , where Sj and Sk are the concentration of substrates j and k, respectively, Pk is the concentration of product k, vm and Km are the Michaelis–Menten constants and Ks and Kp are the inhibition constants. The first equation describes purely substrate-limited uptake, the second equation describes uptake inhibition by a preferred substrate and the third equation describes uptake inhibition by a secreted product. These uptake kinetics are organism-specific and usually require a combination of previous knowledge and iterative modelling. 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 an organism, then that organism does not have sufficient nutrients to survive at this time, and death rate must be calculated. Here, the death rate is assumed to be proportional to the penalty calculated by dFBAlab, which is a measure of the infeasibility of the optimization problem. $$\require{mhchem}$$ \ce{SO4^2- + Ba^2+ -> BaSO4 v} from IPython.display import display, Math, Latex display(Math(r'F(k) = \int_{-\infty}^{\infty} f(x) e^{2\pi i k} dx')) display(Math(r'\ce{H2O}'))

\ce{SO4^2- + Ba^2+ -> BaSO4 v}