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

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):

$V_{\mathit{AAcons}}^{5LO}=k_{\mathit{lo}}\frac{AA}{K_{\mathit{AA}}{\Delta }_Z}\frac{F_a}{{\Delta }_{\mathit{tot}}}$

(1)

$V_{\mathit{LTAsyn}}^{5LO}=k_{\mathit{LTAsyn}}{\Delta }_{\mathit{HP}}\frac{Fa}{{\Delta }_{\mathit{tot}}\delta }$

(2)

$V_{\mathit{HPsyn}}^{5LO}=k_3\frac{F_a}{{{}^{\Delta }}_{\mathit{tot}}}\left(\frac{HP}{{\Delta }_Z{K}_{d3}}{\delta }_{HP-{\Delta }_{\mathit{HP}}}\right)$

(3)

$V_{\mathit{HPcons}}^{5LO}=\frac{F_a}{{{}^{\Delta }}_{\mathit{tot}}}{\rho }_1\left(HP{\Delta }_{\mathit{redox}}-\frac{HT}{K_{\mathit{ox}}}\right)$

(4)

$V_{\mathit{HPsyn}}^{5LO}=V_{\mathit{HPcons}}^{5LO}$

(5)

$V_{\mathit{LOOHcons}}^{5LO}=\frac{F_a}{{{}^{\Delta }}_{\mathit{tot}}}{\rho }_1\left(LOOH{\Delta }_{\mathit{redox}}-\frac{LOH}{K_{\mathit{ox}}}\right)$

(6)

$V_{\mathit{LOHsyn}}^{5LO}=V_{LOOHcons\text{'}}^{5LO}$

(7)

Where

${\Delta }_{\mathit{tot}}=\delta L+{\Delta }_{\mathit{HP}}+{\Delta }_{\mathit{redox}}$

(8)

$\Delta HP=\frac{k_{\mathit{lo}}\frac{AA}{k_{\mathit{LTAsyn}}}+\frac{k_3}{k_{d3}}HP\delta HP}{{\Delta }_Z\left(\frac{{}^{k}LTAsyn}{\delta HP}+k_3\right)}$

(9)

${\delta }_L=1+\frac{AA}{K_{\mathit{AA}}^i{△}_Z}+\delta HP\left(\frac{AA}{K_{\mathit{AA}}}+\frac{HT}{K_{\mathit{HT}}}+\frac{PF}{K_{\mathit{PF}}}\right)\frac{1}{{\Delta }_Z}$

(10)

$\delta HP=1+\frac{AA}{{k^i}_{\mathit{AA}}}$

(11)

${\Delta }_{\mathit{redox}}=\frac{\frac{{\rho }_2}{K_r}+\frac{{\rho }_1}{K_{\mathit{ox}}}*\left(LOH-HT\right)+{\rho }_7*Z}{\left({\rho }_2+{\rho }_1*\left(LOOH+HP\right)\right)}\frac{{△}_Z^{\mathit{Ca}}}{{△}_2^{\mathit{Ca}}}$

(12)

${\rho }_1=k_{\mathit{ox}}+k_{ox2}\frac{Ca}{K_2^{\mathit{Ca}}}$

(13)

${\rho }_2=k_r+k_{r2}\frac{Ca}{K_2^{\mathit{Ca}}}$

(14)

${\rho }_7=k_{\mathit{ing}}\left(1+\frac{Ca}{K_3^{\mathit{Ca}}}\right)$

(15)

${\Delta }_Z=1+\frac{Z}{Kdz}$

(16)

${△}_z^{\mathit{Ca}}=1+\frac{Z}{K_{\mathit{dz}}}+\frac{Ca}{K_3^{\mathit{Ca}}}+\frac{Z}{K_{\mathit{dz}}}*\frac{Ca}{K_3^{\mathit{Ca}}}=\left(1+\frac{Z}{K_{\mathit{dz}}}\right)*\left(1+\frac{Ca}{K_3^{\mathit{Ca}}}\right)$

(17)

${△}_2^{\mathit{Ca}}=1+\frac{Ca}{K_2^{\mathit{Ca}}}$

(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

The kinetic model of glutathione peroxidase

The kinetic model of 5-hydroxyeicosanoid dehydrogenase

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.

Description of the parameters of the “LOS model” and experimental data used for their identification

Here, n is the total number of experimental points,${}_i\overline{v}$ 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].

Modeling of 5-LO kinetics

The “LOS model” and parameter values identified during model building enabled us to reproduce various experimental data on the kinetics of 5-LO. Due to the large number of fits only selected representative examples are presented in the main text. The values of the parameters obtained are given in Table1. Figure4 demonstrates a model generated curve fitted to the experimentally measured dependencies of HP production rate on AA[24]. Based on this it was concluded that the model of 5-LO satisfactorily described the observed non-monotonic behavior. Figure5 demonstrates the simulated time series of total concentration of HP and HT fitted to literature data[37]. Additionally, our model satisfactorily fitted experimental data on LTA4 production from endogenous and exogenous HP (see Figure6)[26] and the Ca²⁺ dependence of 5-HP production by 5-LO on Ca²⁺ concentration[38]. In the latter example the model satisfactorily reproduced both the EC50 value (2–3 μM) and the non-monotonic shape of the experimentally measured dependence (see Figure7). Other results of fitting are summarized in Additional file1: Appendix 6.

Parameter	Value	Units
V max PLA 2	450	μM/min
K m PLA 2 _ APC	20	μM
K Ca PLA 2	0.1	μM
K AA	10.7	μM
K 2 Ca	7.11	mM
K 2 Ca	14.4	μM
K ox	100	μM
K r	5.8 10-7	-
K AA i	552	μM
K HT	0.54	μM
K d3	1.3 10-4	-
k lo	4.6 103	1/min
k 3	0.34 103	(min· μM)-1
k ox	2.7 10-4	(min· μM)-1
k ox2	67.2	(min· μM)-1
k r	2.54 10-4	1/min
k r2	4.4 10-5	1/min
k LTAsyn	5.4 10-4	1/min
k ing	16450	1/min
K dz	36.6	μM
K PF	0.126	uM
K GSSG gpx	0.072	μM
k cat gpx	0.489	1/min
K NADP hedh	2.9	μM
K NADPH hedh	2.69	μM
K oxoETE hedh	1.67	μM
K HT hedh	0.332	μM
k 1 hedh	88.34	1/min
k 2 hedh	1724	1/min
k - 1 hedh	31.5	1/min
k - 2 hedh	8.08	1/min
k HTd	0.001	1/min
k LTAd	0.07	1/min
k oxoETEd	0.005	1/min

The next step of model evaluation examined its ability to reproduce experimentally measured data which had not been used for parameter identification. As an example we selected a dataset describing the influence of glutathione on 5-LO. In accordance with the experimental data[40], the effect of glutathione on 5-LO products in a cell-free extract strongly depends on the concentration of AA. Under conditions of low AA concentrations, the reaction rate of 5-LO is inversely proportional to the glutathione concentration. When AA concentration is high, the rate of 5-LO catalyzed reactions do not decrease even at high concentrations of GSH. Figure8 demonstrates that our “LOS model” satisfactorily reproduces this threshold influence of GSH on the total concentration of “5-LO metabolites” (sum of HP, HT and LTA4) at various concentrations of AA. Our model explains this phenomenon (threshold like response to GSH increase at various AA concentrations) as driven by the lack of lipid peroxides in this cell-free extract system under conditions of low AA concentrations. In this case the main agent responsible for 5-LO oxidation is HP (see reaction r6 in Figure3). HP acts as an electron acceptor in the pseudo-peroxidase reaction converting 5-LO from inactive (Fe²⁺) to active (Fe³⁺) state. Under conditions of low AA concentrations, 5-LO is unable to produce a sufficient quantity of HP molecules to drive the transition of the enzyme to active form. Additionally data from experiments with PMN homogenates[40] demonstrated that the addition of GSH can, however, suppress the activation of 5-LO, as long as the AA concentration remains below a critical limit.

Modeling of the response to redox and non-redox inhibitors of 5-LO

We applied the “LOS model” to study the influences of redox and non-redox inhibitors on the operation of 5-LO. Inhibitor PF has been profiled in stimulated human whole blood (HWB) against several relevant human targets including 5-LO, 12-LOX, 15-LO and COX enzymes[18]. The compound completely inhibited the synthesis of the 5-LO products (HT, oxoETE, LTB4 and LTE4) with estimated IC50s between 100 and 190 nM. These data demonstrate that the non-redox inhibitor PF is 6–10 times more potent than zileuton.

The mechanism of 5-LO inhibition by the redox and non-redox inhibitors has been modeled under LTA4 steady-state concentration (Additional file1: Appendix 1). Parameter values describing the kinetic properties of redox and non-redox 5-LO inhibitors were chosen in such a way to provide satisfactory coincidence between the IC50 and IC80 measured experimentally[18] and those calculated by the “LOS model”.

Figure9 demonstrates that the derived values of parameters (Table1) for zileuton on 5-LO allowed the model to satisfactorily reproduce experimentally measured production of the sum of HP and HT[3]. Moreover, the “LOS model” qualitatively reproduced experimental kinetic data describing the application of redox and non-redox inhibitors in vitro. Experiments in a crude cell lysates 5-LO system containing 10 μM of peroxide (13(S)-HpODE) examined the addition of 10 μM of zileuton or PF (reference). The “LOS model” satisfactory reproduced peroxide consumption following zileuton, and the absence of effect following PF application (Figure10) due to its non-redox mechanism[18].

Thus, comparison of the model simulations to two sets of experimental data which were not used in parameter identification enabled us to conclude that our model adequately described the influence of inhibitors on the system behavior.

Difference between redox and non-redox inhibitors

where LTA 4 _st ^inh , oxoETE _st ^inh ,are steady-state concentrations of LTA4 and oxoETE at a specified inhibitor concentration LTA 4 _st ^inh = 0, oxoETE _st ^inh = 0 are steady-state concentrations of LTA4 and oxoETE in the absence of inhibitor.

We have simulated how the LTA4 and oxoETE dose response depends on the LOOH level and found that both were influenced significantly (Table2). Indeed, assuming LOOH level equal to 0 (no oxidative stress) we have simulated how steady state concentrations of LTA4 and oxoETE depend on zileuton and PF compound concentrations (Figures1112, solid lines). Figure11 demonstrates that model fits satisfactorily ex vivo experimental data[18] on the dependence of LTA4 and oxoETE on PF concentration (Tables3,2). The solid lines on Figures1112 also show that the potency of zileuton to inhibit oxoETE production is the same as for LTA4 production, but it is ten times lower than the potency of PF compound to inhibit both rates (see Tables3,2).

Peroxide concentration (μM)	IC50 for inhibition by Zileuton (nM)		IC50 for inhibition by PF-4191834(nM)
	For LTA4	For 5oxoETE	For LTA4	For 5oxoETE
0	980	962	122	125
5*	860	1145	125	129
10**	860	1410	143	146
100	1450	12670	290	300

*- corresponds to healthy subjects.

** - corresponds to asthmatic subjects.

Assuming a level of LOOH in healthy controls of 5 µM (see Figures11–12, (dashed lines)) we have found that curves describing level of inhibition of LTA4 and oxoETE show a similar tendency but in case of inhibition with zileuton a new feature was observed. The increase in LOOH level from 0 to 5 µM was accompanied by a decrease in zileuton potency for inhibition of oxoETE synthesis (see Figure12, dashed line) in comparison with its potency to inhibit LTA4 synthesis (IC50_LTA4<IC50_oxoETE). This effect is more pronounced when setting the [LOOH] equal to 10 µM (approximating that of asthmatic patients, Figures11–12, dotted lines). To further illustrate this effect we simulated the impact of a supra-physiological LOOH concentration of 100 µM on the inhibitory properties of compounds (dash-dot lines in Figures11b and12b).

Discussion of redox-inhibitor properties at high oxidative stress

Zileuton donates an electron to the active (Fe³⁺) state of 5-LO to form the inactive ferrous (Fe²⁺) state and a zileuton radical (see reaction r₇ in Figure3). The zileuton radical is eliminated without subsequent reduction to zileuton[42]. To activate 5-LO it is necessary to oxidize the inactive ferrous (2+) state, i.e. an electron donated to an acceptor such as HP or LOOH. Therefore, under conditions of low LOOH only HP is able to oxidize 5-LO. If zileuton is applied, 5-LO is converted to the inactive ferrous state, the rate of HP synthesis decreases and, as a consequence, the enzyme is completely inhibited because of lack of its oxidation. In the absence of other sources of lipid peroxides and HP, 5-LO is completely inhibited by zileuton.

Under conditions of high LOOH, the transformation of ferrous 5-LO state into ferric state can proceed in two possible ways: either via HP or LOOH reduction (see reaction r ₆ in Figure3). When zileuton is applied, the concentration of ferrous state of 5-LO increases. However, high level of LOOH may compensate for the lack of HP in the pseudo-peroxidase reaction that leads to maintenance of level of the active 5-LO state and, as a consequence, the lipoxygenase reaction occurs. Under these conditions the rate of pseudo-peroxidase reaction increases and both HP and LOOH are reduced in this reaction converting 5-LO into active ferric state. As a consequence of increased HP consumption in the pseudo-peroxidase reaction, its concentration decreases, leading to a decrease in LTA4 production. At the same time increases in HT concentration produced in the pseudo-peroxidase reaction leads to further increase in oxoETE production.