The model describes the first multi-gene loop identified in the Arabidopsis circadian clock that comprises a negative feedback loop, in which two partially redundant genes, Late Elongated Hypocotyl (LHY) and Circadian Clock Associated 1 (CCA1), repress the expression of their activator, Timing of CAB Expression 1 (TOC1). A minimal mathematical representation of the system requires 7 coupled differential equations and 29 parameters. The differential equations involve Michaelis-Menten kinetics that describe enzyme-mediated protein degradation, and Hill functions that describe some transcriptional activation terms. The model is given by:

$$\begin{align*} \dot{x}_1 &= n_1 \dfrac{x_6}{g_1+x_6} - m_1 \dfrac{x_1}{k_1 + x_1} + q_1x_7u(t),\\ \dot{x}_2 &= p_1x_1 - r_1x_2 + r_2x_3 - m_2 \dfrac{x_2}{k_2 + x_2},\\ \dot{x}_3 &= r_1x_2 - r_2x_3 - m_3 \dfrac{x_3}{k_3 + x_3},\\ \dot{x}_4 &= n_2\dfrac{g_2^2}{x_2^2+x_3^2} - m_4 \dfrac{x_4}{k_4 + x_4},\\ \dot{x}_5 &= p_2x_4 - r_3x_5 + r_4x_6 - m_5 \dfrac{x_5}{k_5 + x_5},\\ \dot{x}_6 &= r_3x_5 - r_4x_6 - m_6 \dfrac{x_6}{k_6 + x_6},\\ \dot{x}_7 &= p_3 - m_7 \dfrac{x_7}{k_7 + x_7} - (p_3 + q_2x_7)u(t) \end{align*}$$

with $x_i(0)=0, i=1,...,7$.

The system, that could describe a biochemical reaction network, is represented by twenty differential equations, twenty two parameters, and all the states are assumed to be measured:

$$\begin{align*} \dot{x}_1&=-\dfrac{v_{max}x_1}{(k_m+x_1)}-p_1x_1 + u(t),\\ \dot{x}_{2}&=p_{1}x_{1}-p_{2}x_{2},\\ \dot{x}_{3}&=p_{2}x_{2}-p_{3}x_{3},\\ \dot{x}_{4}&=p_{3}x_{3}-p_{4}x_{4},\\ \dot{x}_{5}&=p_{4}x_{4}-p_{5}x_{5},\\ \dot{x}_{6}&=p_{5}x_{5}-p_{6}x_{6},\\ \dot{x}_{7}&=p_{6}x_{6}-p_{7}x_{7},\\ \dot{x}_{8}&=p_{7}x_{7}-p_{8}x_{8},\\ \dot{x}_{9}&=p_{8}x_{8}-p_{9}x_{9},\\ \dot{x}_{10}&=p_{9}x_{9}-p_{10}x_{10},\\ \dot{x}_{11}&=p_{10}x_{10}-p_{11}x_{11},\\ \dot{x}_{12}&=p_{11}x_{11}-p_{12}x_{12},\\ \dot{x}_{13}&=p_{12}x_{12}-p_{13}x_{13},\\ \dot{x}_{14}&=p_{13}x_{13}-p_{14}x_{14},\\ \dot{x}_{15}&=p_{14}x_{14}-p_{15}x_{15},\\ \dot{x}_{16}&=p_{15}x_{15}-p_{16}x_{16},\\ \dot{x}_{17}&=p_{16}x_{16}-p_{17}x_{17},\\ \dot{x}_{18}&=p_{17}x_{17}-p_{18}x_{18},\\ \dot{x}_{19}&=p_{18}x_{18}-p_{19}x_{19},\\ \dot{x}_{20}&=p_{19}x_{19}-p_{20}x_{20},\\ \end{align*}$$

where $x_i, i = 1, …, 20$ are the states. The unknown parameters are $p_i, i = 1, …, 20$, plus the two Michaelis-Menten parameters $ν_{max}$ and $k_m$. All the states are assumed to be measured.

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