Model construction
The “LOS model” describing LTA4 and oxoETE production in leukocytes includes reactions catalyzed by 5-LO, cPLA2, GPx, HEDH and degradation of HT, oxoETE and LTA4. Since all leukotrienes are synthesized from LTA4, analysis of 5-LO catalyzed LTA4 production was deemed sufficient to evaluate the impact of 5-LO on leukotriene production. Catalytic cycles for each of the enzymes were constructed and rate equations describing the dependence of reaction rate on concentrations of substrates, products and effectors were derived utilizing literature data.
The kinetic model of 5-LO
Known experimental data and hypotheses used for the model development
In this study, we used the following available experimental data and facts on structural and functional properties of 5-LO:
-
1.
AA inhibits 5-LO activity in lipoxygenase reaction at high concentrations (substrate inhibition) [24].
-
2.
LTA4 can be synthesized from exogenous HP [25].
-
3.
5-LO is self-inactivating [25]. Glutathione peroxidase and glutathione protect the enzyme from inactivation and lipid peroxides eliminate the protective effect of glutathione.
-
4.
HT is a reversible inhibitor of 5-LO [21].
-
5.
LTA4 can inactivate 5-LO irreversibly [22].
-
6.
The Fe atom in the catalytic site exists in two possible states Fe2+ and Fe3+. Fe3+ is the catalytically active state. Transition between Fe2+ and Fe3+ states proceeds via oxidation by lipid peroxides, including HP [12]. Reduction from Fe3+ to Fe2+ state can be mediated by redox inhibitors (zileuton).
-
7.
Oxidation of the enzyme (Fe2+ → Fe3+ transition) is influenced by Ca2+ ions [25].
-
8.
Endogenously generated 5-HP is the preferential substrate for the 5-LO mediated LTA-synthase reaction [26].
-
9.
ATP and membrane binding are necessary for 5-LO activation [13].
During model building and simplification, the following assumptions were made:
-
a.
There are two sites for AA binding: the catalytic site and regulatory site. HT is a competitive inhibitor at the catalytic site.
-
b.
Binding of AA to the regulatory site results in formation of dead-end complexes with 5-LO.
-
c.
Binding of AA and HT to the catalytic site are fast reactions in comparison to product formation.
-
d.
Oxidation and reduction of 5-LO (transition between Fe2+ and Fe3+) can result from (i) interaction with lipid peroxides or (ii) “spontaneously” (by means of interaction with oxidative or reducing factors: O2, H2O2, thiol groups etc.).
-
e.
Ca2+ is able to bind to the catalytic site, but it influences only oxidation and reduction reactions, thus kinetic parameters of other reactions of catalytic cycle remain unchanged.
-
f.
Binding of Ca2+ and redox-inhibitors (Z) are fast reactions in comparison to the other reactions of the catalytic cycle.
-
g.
Redox inhibitors are only able to bind to the catalytic site of 5-LO and only when it is not occupied with other factors.
-
h.
The binding of redox inhibitors to the catalytic site blocks the binding of other factors .
-
i.
The regulatory site is able to bind AA only if following conditions are fulfilled: (i) 5-LO is in Fe3+ state and (ii) the enzyme is not bound to a redox-inhibitor.
-
j.
The oxygenase, pseudoperoxydase and LTA4 synthase activities of 5-LO were simplified by not describing electron transfer and oxygen binding. Instead, it was assumed that the oxygen concentration is in saturation, i.e., it is not a parameter of the model.
-
k.
To decrease the number of unknown parameters in our model we have assumed that binding of AA to the regulatory site of 5-LO does not influence the binding of AA, HP and HT to the catalytic site (see Additional file 1: Appendix 1).
-
l.
Interaction with MAP kinases, FLAP, CLP and transport of 5-LO to the nucleus have not been taken into account in the model.
Catalytic cycle of 5-LO
Figure2 shows schematic representation of the enzyme states considered in the model. In the ferric state 5-LO is able to bind any substrates/products/inhibitors at its catalytic and regulatory sites. This state of 5-LO was represented as a square with a triangle underneath the square. The square designates the catalytic site of the enzyme and the triangle represents the regulatory site. The regulatory site can be found in 2 states: free or AA bound. The catalytic site of 5-LO is able to bind AA, HT, HP, PF (non-redox inhibitor), Z (redox inhibitor). As an example Figure2(a) shows the HT and AA bound enzyme state. All the above mentioned compounds compete for the substrate binding part of the catalytic site. In addition to the substrate/product (AA, HP) and competitive inhibitors (HT, PF and Z) the catalytic site of 5-LO is able to bind Ca2+ (as shown in Figure2(a)) which does not compete with AA, HT, HP, PF and Z. Therefore, the catalytic site of the ferric enzyme can be found in a total of 12 states: free of any substrate/competitor and with AA, HT, HP, PF, Z bound all of which can be found with and without Ca2+ bound. On the basis of this analysis we can conclude that catalytic cycle of 5-LO includes 24 potential states (See Additional file2: Appendix Figures A1 and A2). The ferrous state (Fe2+) of 5-LO is not able to bind any substrate/product/inhibitor at the catalytic or regulatory site but it is able to bind Ca2+ at the catalytic site (see Figure2(b)). Thus, total catalytic cycle includes 26 states of 5-LO. Since in derivation of the rate equations describing 5-LO activities we have used new variables representing sums of states of 5-LO, notations for such sums have also been introduced (Figure2(c)). In the equations and text of this paper we have used the simplified notations of the 5-LO states (see Figure1).
Transitions between the states are described in accordance to mass action law and can be either reversible (for example, binding AA to catalytic site) or irreversible (for example, the LTA-synthase reaction). Moreover, these processes can be either relatively fast or slow depending on the values of rate constants obtained from experimental data fitting. This grouping of all processes into two sets (fast and slow processes) allowed us to reduce the initial complexity of the catalytic cycle and derive rate equations describing the operation of 5-LO according to the methods described in[27].
Schematic visualization of the total catalytic cycle is not convenient because of the complexity (26 nodes/states and tenths of transitions between them). To reconstruct a reduced total catalytic cycle and derive rate equations we employed a step-by-step strategy described in Additional file1: Appendix 1. As a result we have developed a reduced catalytic cycle (Figure3) describing oxygenation, dehydration and pseudoperoxidase activities of 5-LO. On the basis of this reduced catalytic cycle, we have derived rate equations (1–7) describing 5-LO mediated AA consumption (V
AAcons
5LO), 5-HP production in oxygenase reaction (V
HPcons
5LO), HT production and HP consumption in pseudoperoxidase reaction (V
HPcons
5LO), all other lipid peroxide (LOOH) consumption in pseudoperoxidase reaction [V
LOOHcons
5LO] and LTA4 production (V
LTAsyn
5LO):
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Where
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(17)
(18)
Since 5-LO undergoes irreversible inactivation, the concentration of active enzyme tends to zero with time and, consequently, all enzyme catalyzed reactions become equal to zero. In vivo, regulatory mechanisms control intracellular 5-LO production de novo[13]. To obtain non-zero steady state concentrations of 5-LO states and, consequently, to derive the rate equations describing the activities of 5-LO, we did not consider self-inactivation of the enzyme and have not taken into account processes responsible for 5-LO production de novo. Under these assumptions the total concentration of active enzyme (F
a
) is equal to total enzyme concentration and F
a
is a parameter of the model. Additionally, identification of model parameters using in vitro experimental data was performed on the basis of a model which takes into account self-inactivation of 5-LO with time (see Additional file1: Appendix 1.5). Under these conditions F
a
represents the sum of active states of 5-LO and changes with time. All rate equations were derived on the basis of the quasi-steady state approach[27].
The kinetic model of phospholipase A2
Calcium-dependent phospholipase A2 (cPLA2) catalyzes the production of AA from phospholipids (PL) in the cell membrane. Elevations in the cellular calcium concentration significantly stimulate cPLA2 activity. On the basis of several models of the enzyme developed previously and available experimental data[28, 29] we have derived the rate equation for cPLA2 to be (V
AA
PLA 2, see Additional file1: Appendix 2):
(19)
where
(20)
The kinetic model of glutathione peroxidase
GPx enzyme reduces HP to HT. This reaction requires glutathione as a cofactor[7, 8]. The stoichiometry of the reaction catalyzed by the enzyme is as follows:
(21)
GPx catalyzes this reaction in accordance to the Ping-Pong mechanism and the derivation of the rate equation for GPx is given in Additional file1: Appendix 3:
(22)
where
(23)
(24)
The kinetic model of 5-hydroxyeicosanoid dehydrogenase
HEDH catalyzes the conversion of HT into oxoETE. The mechanism of HEDH is considered as Bi-Bi Ping-Pong, with NADP as the second substrate[9]. Derivation of the rate equation for HEDH is given in Additional file1: Appendix 4:
(25)
where
(26)
(27)
The LTA4 and oxoETE synthesis model (“LOS model”)
To build the “LOS model” we utilized the rate equations describing the activities of 5-LO, phospholipase A2, glutathione peroxidase and 5-hydroxyeicosanoid dehydrogenase as given above. The kinetic scheme of the “LOS model” is shown in Figure1. In accordance with the scheme, AA binding to 5-LO is converted to HP via the lipoxygenase reaction. As a result a complex of 5-LO and HP is formed. HP can be either released from the complex (V
HPsyn
5LO) or used to form LTA4 via the LTA4-synthase reaction V
LTAsyn
5LO. Additionally, LTA4 can be produced from free HP in the absence of AA (see sequence of reversible reaction V
HPsyn
5LO and irreversible reaction V
LTAsyn
5LO). To present all these process correctly the intermediate state of the enzyme (Ca)F
HP
(AA) (complex 5-LO with HP) was added to the kinetic scheme. The concentration of the state (Ca)F
HP
(AA) is not a variable of the model, i.e. there are no differential equations describing the time dynamics of (Ca)F
HP
(AA). However, in accordance with the quasi-steady state approach chosen to describe 5-LO operation in the “LOS model” (and, consequently, applied to derive rate equations of various 5-LO activities) concentration of state (Ca)F
HP
(AA) is expressed in terms of variables of the “LOS model” (see Additional file1: Appendix 1).
To avoid unlimited accumulation of metabolites resulting from constant influx of AA we have introduced processes of degradation of HT, oxoETE and LTA4 (V
HTd
, V
LTAd
and V
oxoETEd
) in the model. The reaction rates of these processes are described in accordance with mass action law (Additional file1: Appendix 5). Additionally, concentrations of PL, lipid peroxide LOOH and its reduced product LOH, reduced (GSH) and oxidized (GSSG) glutathione, and reduced and oxidized forms of NADPH are considered as parameters of the model, i.e., do not change with time. The values for the intracellular concentrations of GSH, GSSG, NADPH and NADP were taken from the following sources[30–34]. The concentration of LOOH has either been chosen on the basis of known experimental conditions or has been varied to describe various oxidative states of the cells.
Based on all the above assumptions the system of differential equations describing the “LOS model” is presented below:
(28)
(29)
(30)
(31)
(32)
Description of the parameters of the “LOS model” and experimental data used for their identification
According to assumption L of the section “Known experimental data and hypotheses used for the model development”, some parameters were equated with each other (see 1.4). Thus, for 5-LO 17 independent parameters remained, among them 11 equilibrium constants and 6 rate constants. Additionally, 5 parameters for GPx, and 8 parameters for HEDH needed to be identified. Several of the values of the parameters have been directly taken from other literature sources- e.g. the Michaelis constant for glutathione (K
m
GSH) for glutathione peroxidase reaction[7], the rate constant of LTA4 and HT degradation[23]. The values for other parameters were chosen on the basis of the best coincidence between modeling results and corresponding experimental data. To select the values of the parameters we used the algorithm of fitting based on the Hook-Jeeves method[35] implemented in the DBSolve Optimum package[36]. As a criterion of fitness, the following function was used:
(33)
Here, n is the total number of experimental points, is the experimentally measured value of the variable or reaction rate, v
i
is the value of the variable or reaction rate calculated based on the model at a point corresponding to the experimental ones.
Given the complexity of the model, simultaneous identification of parameter estimates would be challenging. Therefore, parameter identification was performed individually for each enzyme by fitting to literature data sets pertinent to the specific enzyme. For example, the parameters of 5-LO were identified via fitting of the 5-LO model against more than 10 experimentally measured curves (76 experimental points)[22, 24–26, 37, 38], 4 unknown parameters of GPx were identified on the basis of 12 experimentally measured points[39] and the parameters of HEDH have been fitted against 47 experimental points[9].