Team:TU Delft-Leiden/Modeling/Techniques/FBA

Flux Balance Analysis Theory

Flux Balance Analysis (FBA) is a method that calculates the fluxes of metabolites through a metabolic network. In order to perform the FBA method, a model of the metabolic network is needed. These models are constructed based on experiments where reactions that occur in a cell are identified. Subsequently, the fluxes through a reaction can be given constraints. However, in practice this is not done very often, as it is difficult to obtain these constraints experimentally. We used two different models, the core model of E. coli, which consists of all the essential reactions of the metabolism and an extended model, the iJO1366 model, which contains 30 times more reactions and 25 times more metabolites [1].

Firstly, all the metabolic reactions of the model are mathematically represented in an m by n matrix, called the stoichiometric matrix (S). For instance, if we analyse a metabolic network that consists of the following two reactions: $$ A + 2 \ C \ \rightarrow \ D $$ $$ 3 \ B \ \rightarrow \ 4 \ A $$ We get the following stoichiometric matrix: $$ \boldsymbol{S} \ = \begin{bmatrix} -1 && 4 \\ 0 && -3 \\ -2 && 0 \\ 1 && 0 \end{bmatrix} $$ So, each row represents one unique compound and each column represents one unique reaction. The values in the stoichiometric matrix are called the stoichiometric coefficients. These coefficients indicate which metabolites are involved in a specific reaction, where the number represents how many molecules of the metabolite are involved in this specific reaction and it is negative when the metabolite is consumed and positive when the metabolite is produced. The stoichiometric coefficient is zero for every metabolite that is not involved in a particular reaction.
Secondly, a vector v is defined which gives the fluxes through all reactions part of the model. Thus, it has a length of n, as there are n reactions.
Thirdly, a vector x is defined which gives the concentrations of all the metabolites of the system, so it has length m.

The basic assumption of FBA is that the system is at steady-state, so that \(\frac{d\boldsymbol{x}}{dt}=\ 0\). Subsequently, the following system of equations is solved: $$ 0 \ = \ \frac{d\boldsymbol{x}}{dt} = \ \boldsymbol{S}\boldsymbol{v} \tag{1}$$ Because there will be more reactions than metabolites in any realistic large-scale metabolic model, there is no unique solution to this system of equations (there are more unknown variables than equations). The set of possible solutions to the system of equations (1) is called the solution space. We can make the solution space smaller by imposing constraints on the system. For instance, we can constrain the maximum or minimum allowable flux of a certain reaction. However, generally the solution space consists of multiple solutions.
To obtain a solution, the FBA method maximizes or minimizes an objective function Z, which is defined by the user. The objective function can be any linear combination of fluxes. The resulting system of equations with constraints and an objective function is optimized by a linear programming algorithm, and a solution is obtained. Still, the solution space can consist of multiple solutions. In this case, the linear programming algorithm will choose one particular solution. The multiple solutions can be explored by using Flux Variability Analysis (FVA), as we will do for our project. [1]

The fact that the solution is at steady state means that the FBA method is not suitable for investigating the changing behaviour of a system over time. However, it is very useful for obtaining insight in often very complex metabolic networks. For our project, we will apply the FBA method to the extracellular electron transport (EET) module [reference theory]. Our goal is to gain insight in the carbon metabolism providing the electrons for our EET module. The results of this analysis can be found in Flux Balance Analysis.