Problem description

The threestep model describes the variation of the metabolite concentrations with time. (See Figure 8, solid arrows represent mass flow, dashed arrows represent kinetic regulation, arrow ends represent activation, blunt ends inhibition, S and P are the pathway substrate and product and are held at constant concentrations shown in Table 2, M$_1$ and M$_2$ are intermediate metabolites of the pathway, E$_1$, E$_2$ and E$_3$ are the enzymes, G$_1$, G$_2$ and G$_3$ are the mRNA species for the enzymes.) The threestep model is a nonlinear biochemical dynamic model formed by 8 ODEs, which is shown in Figure 9.

Figure 8: The threestep metabolic pathway model

Figure 9: ODE system of threestep model

$$\begin{eqnarray*} \frac{dG_1}{dt} & = & \frac{V_{1}}{1 +\left(\frac{P}{Ki_{1}}\r... ...right)\cdot (M_2 - P)}{1 + \frac{M_2}{Km_5} +\frac{P}{Km_6}} \\ \end{eqnarray*}$$

As shown in the Figures 8 and 9, M$_1$, M$_2$, E$_1$, E$_2$, E$_3$, G$_1$, G$_2$ and G$_3$ reprensent the concentrations of the species involved in the pathway model and S and P keep fixed initial values for each experiment (see Table 2). Thus the optimization problem is to fit the 36 kinetic parameters (see Figure 9 and Table 5), which are divided into two different classes: Hill coefficients, allowed to vary within the rage ( $10^{-1}, 10^{+1}$) and all the others allowed to vary within the range ( $10^{-12}, 10^{+6}$).

The global optimization problem is stated as the minimization of an unweighted distance measure between experimental and predicted values of the $8$ state variables:

$$\index{ThreeStep model!Objective function} J = \sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}\{[Y_{pred}(i)-Y_{exp}(i)]_j\}^2$$ (8)

where $n$ is the number of data for each experiment, $m$ is the number of experiments, $Y_{exp}$ represents the known experimental data, and $Y_{pred}$ is predicted data using the model with a given set of the $36$ parameters. Here $w_{ij}$ is set to $1$ which means unweighted.

The so-called experiment data (i.e., pseudoexperiment data) are generated by simulation from a set of chosen parameters (see Nominal values in Table 5). Thus, there are 16 different pseudoexperiments (simulations) in which the initial concentrations of S and P are listed in Table 2. The advantage of pseudoexperiment or simulation is devoid of measurement noise.

Table 2: Pseudoexperiment: combinations of S and P

PS	$0.1$	$0.46416$	$2.1544$	$10$
$0.05$	1	2	3	4
$0.13572$	5	6	7	8
$0.36840$	9	10	11	12
$1.0$	13	14	15	16

For more information, please read the original paper $^{\mbox{\scriptsize\citep{THREESTEP2003}}}$.