Open Systems - Flux Balance Equations

Our analysis shall be simplified by anticipating ideal mixture of the system. We call such systems homogeneous in space, which means there is no spatial difference in concentrations and we can neglect the spatial extension of the system. To specify the system, we need to decide where we draw the boundary of the system. The boundaries will be open and allows Glucose $Glu$ to be transported into the system and Lactose $LAC$ out of the system. Moreover, $ADP$ and $NAD$ will be consumed by the system, while it produces $ATP$, $NADH$ and $H_2O$. It is important to realise, that while mass inside the system is conserved, every molecule outside the system can appear or disapper with no restriction. We call places where molecules are created out of nowhere a source and places where they disappear a sink. Although there might exist no real sinks and sources, this concept allows us to analyse a part of a larger system by drawing a boundary and not caring about what happens outside.
Another assumption we use is that our hypothetical system operates in a nonequilibrium steady state. When Glucose is supplied initially, the system is whether at equilibrium nor in a nonequilibrium steady state. As time evolves with constant boundary conditions (sources and sinks) the system will reach a configuration, where the uptake and the secretion of the system take place at a constant rate. In fact, all reaction rates which we also call fluxes $v$ occur at a constant rate. This configuration is called a nonequilibrium steady state or flow steady state. By using matrix notation we can represent the system of interest - consisting of $m$ metabolites and $r$ reactions - with the Stoichiometric Matrix, $$\mathbf{N}=\left\lbrace n_{ij} \right\rbrace \quad \text{ for } \quad i=1, \dots , m \quad \text{ and } \quad j= 1, \dots , r$$ where $n_{ij}$ are the stoichiometric coefficients. The change in concentration $\frac{dS_i}{dt}$ of the $i$-th metabolite is therefore $$\frac{dS_i}{dt}=\sum_{j=1}^{r}n_{ij}v_j \hspace{1cm} for \hspace{1cm} i=1, \dots , m.$$ The Stoichiometric Matrix $\mathbf{N}$ for the above system with description of the metabolites and fluxes $v_i$ reads $$\begin{array}{c|cccccccccccc|ccccc|ccc} \text{met.} \backslash \text{reac.} & v_1 & v_2 & v_{-2} & v_3& v_4& v_5& v_6& v_7& v_8& v_9 & v_{10}& v_{11}& v_{12}& v_{13}& v_{14}& v_{15}& v_{16}& v_{17}&v_{18}&v_{19}\\ \hline \text{Glu} & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ \text{G6P} & 1 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{F6P} & 0 & 1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{FDP} & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{DHAP} & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{GAP} & 0 & 0 & 0 & 0 & 1 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{1,3PG} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{3PG} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{2PG} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{PEP} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{PYR} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ \text{LAC} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \text{H+} & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \hline \text{ATP} & -1 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{ADP} & 1 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ \text{NADH} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ \text{NAD} & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \text{H}_2\text{O} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ \end{array}$$ Reactions $v_1$ - $v_{11}$ represent Glycolysis, $v_{12}$ - $v_{16}$ represent cofactor exchange ($ATP$,$ADP$,$NADH$,$NAD$, $\dots$) at the boundary. The assumed nonequilibrium steady state for our system means, that the change in concentration $\frac{dS_i}{dt}$ for each metabolite $i$ of the system is zero $$\frac{dS_i}{dt}=\sum_{j=1}^{r} n_{ij}v_j =\mathbf{N} \mathbf{v}= \mathbf{0} \tag{1}$$ where $\mathbf{v}$ is the so called flux vector. Solving systems of linear equations like $\mathbf{N} \mathbf{v}= \mathbf{0}$ is the major concern of a field in Mathematics called Linear Algebra. The above Flux Balance Equation (1) is underdetermined (more unknowns than equations) and has thereafter no unique solution, but rather a multitude of solutions. Since, the RHS of the system of equations is zero, all the vectors that satisfy this equation (1) form a Vector Space which is called the Nullspace of a Matrix. The Matlab command $null(N)$ gives the vectors in the Nullspace. Adding constraints (e.g. lactose secretion rate) to the problem, leads to a constraint based optimisation problem, that has a unique solution for the fluxvector (see Flux Balance Analysis).