This model represents a glycolysis inspired pathway (the upper part of the glycolysis) with different physiological constraints on enzyme synthesis as described in Bartl et al.. A specific enzyme, here denoted by $u$, usually catalyses a metabolic reaction, expressed in terms of the stoichiometric matrix and the metabolites, here denoted by $x$: The dynamical model can be written as a system of differential equations:

$$\begin{align*} \dot{x}_1 &= - \dfrac{k_1x_1}{x_1+k_M}u_1 \\ \dot{x}_2 &= \dfrac{k_1x_1}{x_1+k_M}u_1 - \dfrac{k_2x_2}{x_2+k_M}u_2 \\ \dot{x}_3 &= \dfrac{k_2x_2}{x_2+k_M}u_2 - \dfrac{k_3x_3}{x_3+k_M}u_3 \\ \dot{x}_4 &= \dfrac{k_2x_2}{x_2+k_M}u_2 + \dfrac{k_3x_3}{x_3+k_M}u_3- \dfrac{k_4x_4}{x_4+k_M}u_4 \\ \dot{x}_5 &= \dfrac{k_4x_4}{x_4+k_M}u_4 \end{align*}$$

with $x_1(0)=S_1$, $x_2(0) = S_2$, $x_3(0) = S_3$, $x_4(0) = S_4$, $x_5(0)=S_5.$ The model is considered to be fully observable, i.e., $y_1=x_1$, $y_2=x_2$, $y_3=x_3$, $y_4=x_4$, $y_5=x_5$, and $u_1,u_2,u_3,u_4$ are independent variables.

The pharmacokinetics model is a two compartment model that embodies the ligands of the macrophage mannose receptor, and it is represented mathematically as a system of differential equations of the form:

$$\begin{align*} \dot{x}_1 &= \alpha_1(x_2-x_1) - \dfrac{k_av_mx_1}{k_ck_a+k_cx_3+k_ax_1},\\ \dot{x}_2 &= \alpha_2(x_1-x_2),\\ \dot{x}_3 &= \beta_1(x_4-x_3) - \dfrac{k_av_mx_3}{k_ck_a+k_cx_3+k_ax_1}, \\ \dot{x}_4 &= \beta_2(x_3-x_4), \end{align*}$$

where $x_1(0) = c_0$, $x_2(0)=0$, $x_3(0) = \gamma x_4(0)=0$, $x_1$ represents the enzyme concentration in plasma, $x_2$ its concentration in compartment, $x_3$ is the plasma concentration of the mannosylated polymer that acts as a competitor of glucose oxidase for the mannose receptor of macrophages, and $x_4$ is the concentration of the same competitor in the part of the extravascular fluid of the organs accessible to this macromolecule.

This is a full polynomial form of the model, given in Enzyme kinetics:

$$\begin{align*} \dot{x}_1 &= -bx_1 + ax_5\\ \dot{x}_2 &= \alpha x_1 - \beta x_2\\ \dot{x}_3 &= \gamma x_2 - \delta x_3\\ \dot{x}_4 &= \sigma x_4 x_6 ( \gamma x_2 - \delta x_3) \\ \dot{x}_5 &= -\sigma x_4 x_5^2 x_6 ( \gamma x_2 - \delta x_3)\\ \dot{x}_6 &= -x_6^2( \gamma x_2 - \delta x_3) \end{align*}$$

with $x_1(0) = 0.3617$, $x_2(0) = 0.9137$, $x_3(0)=1.3934$, $x_4(0) = x_3(0)^{\sigma}$, $x_5(0)=\dfrac{1}{A+x_3(0)^{\sigma}}$, $x_6(0)=\dfrac{1}{x_3(0)}.$

The model describes the oscillations in enzyme kinetics. The state variable $x_1$ represents an enzyme concentration whose rate of synthesis is regulated by feedback control via a metabolite $x_3$, and $x_2$ regulates the synthesis of $x_3$. It is characterised by a rational kinetics consisting of a Hill-like term, and it is given by:

$$\begin{align*} \dot{x}_1 &= -bx_1 + \dfrac{a}{A+X_3^\sigma},\\ \dot{x}_2 &= \alpha x_1 - \beta x_2,\\ \dot{x}_3 &= \gamma x_2 - \delta x_3 \end{align*}$$

with $x_1(0) = 0.3617$, $x_2(0) = 0.9137$, $x_3(0)=1.3934.$

The following two-dimensional single-input single-output system represents a chemical reactor model

$$\begin{align*} \dot{x}_1 &= u(Ce -x_1) - rx_1 \\ \dot{x}_2 &= rx_1 - ux_2 \\ y &= x_1 - x_2, \end{align*}$$

where coefficients are in $\mathbb{R}$, $x_1$ and $x_2$ denote the reactant and product concentrations, respectively. The input $u$ corresponds to the input flow of reactant, $r$ and $C_e$ denote kinetic and reactor parameters.