Problem description

The HIV model describes the mechanism of irreversible inhibition of HIV proteinase. The enzyme is only active in a dimer form and the product is a competitive inhibitor for the substrate and the inhibitor is irreversible It is described in Figure 15, where E represent the enzyme, S and P represent substrate and product respectively.

Figure 15: The HIV proteinase model

[Schema of HIV proteinase model]

[Equations of HIV proteinase model]

$$\begin{eqnarray*} I + E & {\scriptstyle{k_{on}} \atop \scriptstyle\longleftright... ...p \scriptstyle\longleftrightarrow} \atop \scriptstyle{k_p} & EP \end{eqnarray*}$$

In this model, the mechanism is described with an ODE system made up of $9$ differential equations (see Figure 16). The problem is to estimate a number of rate constants of the mechanism of irreversible inhibition of HIV proteinase. Data on the inhibition of the dissociative dimeric proteinase from HIV are used. Note that at this time the data are real experimental data, so that there exist measurement error.

Figure 16: ODE system of HIV model

$$\begin{eqnarray*} \frac{dM}{dt} & = & -2\cdot{}k_{md}\cdot{}M\cdot{}M + 2\cdot{}... ...t{}EI - k_{de}\cdot{}EI \\ \frac{dEJ}{dt} & = & k_{de}\cdot{}EI \end{eqnarray*}$$

As shown in Table 6, the real experimental data are produced from five time courses at four different inhibitor concentrations measured fluorimetrically. The HIV proteinase (assay concentraion $0.004 \mu\mathbf{M}$) is added to a solution of an irreversible inhibitor and a fluorogenic substrate ( $25 \mu\mathbf{M}$). Five assays are conducted, at four different concentrations of the inhibitor ( $0, 0.0015, 0.003$ and $0.004 \mu\mathbf{M}$ in replicate). The fluorescence changes are monitored for 1 hour in each experiment.

Table 6: HIV experiment

Dataset

	Inhibitor
	($\mu\mathbf{M}$)

Parameter

	Initial
	value

Optimum

	Element of
	variables vector

bounds

-

-

$k_{md}(\mu\mathbf{M}^{-1}\cdot{}s^{-1})$

$0.1$

-

-

-

$-$

-

-

-

$k_{dm}(s^{-1})$

$0.001$

-

-

-

$-$

-

-

-

$k_{on}(\mu\mathbf{M}^{-1}\cdot{}s^{-1})$

$100$

-

-

-

$-$

-

-

-

$k_s(s^{-1})$

-

$46.349292$

x[0]

$10^{-6}$

$-$

$10^{+6}$

-

-

$k_{cat}(s^{-1})$

-

$5.491365$

x[1]

$10^{-6}$

$-$

$10^{+6}$

-

-

$k_p(s^{-1})$

-

$269.804443$

x[2]

$10^{-6}$

$-$

$10^{+6}$

-

-

$k_i(s^{-1})$

-

$0.000177$

x[3]

$10^{-6}$

$-$

$10^{+6}$

-

-

$k_{de}(s^{-1})$

-

$0.000582$

x[4]

$10^{-6}$

$-$

$10{+6}$

-

-

$\epsilon{}P$

$0.024$

-

-

-

$-$

-

A

$0$

$24.637840$

x[5]

$10$

$-$

$40$

B

$0.0015$

$23.456802$

$-$

x[6]

C

$0.003$

$27.159763$

$-$

x[7]

D

$0.004$

$16.190568$

$-$

x[8]

E

$0.004$

$24.672660$

$-$

x[9]

A

$0$

$0.005387$

x[10]

$0.002$

$-$

$0.006$

B

$0.0015$

$0.005183$

$-$

x[11]

C

$0.003$

$0.006000$

$-$

x[12]

D

$0.004$

$0.004119$

$-$

x[13]

E

$0.004$

$0.003051$

$-$

x[14]

A

$0$

$-0.004763$

x[15]

$-0.1$

$-$

$0.1$

B

$0.0015$

$-0.004950$

$-$

x[16]

C

$0.003$

$-0.017078$

$-$

x[17]

D

$0.004$

$-0.007473$

$-$

x[18]

E

$0.004$

$0.002483$

$-$

x[19]

$-$

Also as shown in Table 6, the problem is set to fit the rate constants $k_s, _{cat}, k_p, k_i$ and $k_{de}$ to the experimental data. According to the original papers $^{\mbox{\scriptsize\citep{DYNAFIT1996,GEPASI1998}}}$, it is assumed that there is a certain degreen of uncertainty in the value of the initial concentrations of substrate and enzyme, and that the offset (baseline) of the fluorimeter is not exactly zero. Therefore, there are a total of $20$ adjustable parameters (given in Table 6). The allowed ranges are also listed in the table.

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

$$\index{HIV model!Objective function} J = \sum_{i=1}^{n}\sum_{j=1}^{m}w_{ij}\{[Y_{pred}-Y_{exp}]_{ij}\}^2$$ (9)

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 $20$ parameters ( $Y_{pred} = \epsilon{}P$). Here $w_{ij}$ is set to $1$ which means unweighted.

For more information, please read the original paper $^{\mbox{\scriptsize\citep{DYNAFIT1996,GEPASI1998}}}$.