A hybrid cybernetic model can be given in a general form as follows:

dx = SxZrMc + V(хш — x)

dV

dt

where x is the vector of nx concentrations of extracellular components in the reactor (such as substrates, products and biomass), Sx is the (nxxnr) stoichiometric matrix, and Z is the (nrxnz) EM matrix, rM is the vector of nz fluxes through EMs, Fin and Fout are volumetric feed rates at the inlet and outlet, V is the culture volume, xIN is the vector of nx concentrations of extracellular components in the feed. Eq. (1) can also represent batch operation by setting Fin = FnUT = 0 (i. e., V is constant), and fed-batch systems by setting FnUT = 0. In chemostat operations, Fin = FnUT = F, and F/V is often given as dilution rate D. With Z normalized with respect to a reference substrate, rM implies uptake fluxes through EMs. Fluxes through EMs are given as below:

where the subscript j denotes the index of EM, vM, j is the cybernetic variable controlling enzyme activity, eM, j and eMj are the enzyme level and its maximum value, respectively, and rMnj is the kinetic term. Enzyme level eM, j is obtained from the following dynamic equation, i. e.,

where the first and second terms of the right-hand side denote constitutive and inducible rates of enzyme synthesis, and the last two terms represent the decrease of enzyme levels by degradation and dilution, respectively. In the second term of the right-hand side, uM, j is the cybernetic variable regulating the induction of enzyme synthesis, b is the fraction of internal resources (such as DNA, RNA, protein, lipid and other components) involved in the enzyme synthesis process, and rME j is the kinetic part of inducible enzyme synthesis rate. In the third and fourth terms, pMj and ji are the degradation and specific growth rates, respectively.

The cybernetic control variables, uMj and vM, j are computed from the following the "Matching Law" and the "Proportional Law"(Kompala et al., 1986; Young & Ramkrishna, 2007), respectively:

where the return-on-investment Pj denotes the carbon uptake flux through the jth EM.

The structure of HCMs is illustrated using Fig. 2.1. In this tutorial example, we get three EMs from the network. The uptake flux is split into three individual fluxes thorough EMs, which are catalyzed by enzymes Ej, E2 and E3, respectively. HCMs view that the uptake fluxes are optimally distributed (by the cybernetic variables u and v) among three EMs for maximizing a metabolic objective function (such as the carbon uptake flux or growth rate). The uptake and excretion rates are represented by nonnegative combinations of individual fluxes through EMs.