A thermodynamically-constrained mathematical model for the kinetics and regulation of NADPH oxidase 2 complex-mediated electron transfer and superoxide production

Abstract

Reactive oxygen species (ROS) play an important role in cell signaling, growth, and immunity. However, when produced in excess, they are toxic to the cell and lead to premature aging and a myriad of pathologies, including cardiovascular and renal diseases. A major source of ROS in many cells is the family of NADPH oxidase (NOX), comprising of membrane and cytosolic components. NOX2 is among the most widely expressed and well-studied NOX isoform. Although details on the NOX2 structure, its assembly and activation, and ROS production are well elucidated experimentally, there is a lack of a quantitative and integrative understanding of the kinetics of NOX2 complex, and the various factors such as pH, inhibitory drugs, and temperature that regulate the activity of this oxidase. To this end, we have developed here a thermodynamically-constrained mathematical model for the kinetics and regulation of NOX2 complex based on diverse published experimental data on the NOX2 complex function in cell-free and cell-based assay systems. The model incorporates (i) thermodynamics of electron transfer from NADPH to O₂ through different redox centers of the NOX2 complex, (ii) dependence of the NOX2 complex activity upon pH and temperature variations, and (iii) distinct inhibitory effects of different drugs on the NOX2 complex activity. The model provides the first quantitative and integrated understanding of the kinetics and regulation of NOX2 complex, enabling simulation of diverse experimental data. The model also provides several novel insights into the NOX2 complex function, including alkaline pH-dependent inhibition of the NOX2 complex activity by its reaction product NADP⁺. The model provides a mechanistic framework for investigating the critical role of NOX2 complex in ROS production and its regulation of diverse cellular functions in health and disease. Specifically, the model enables examining the effects of specific targeting of various enzymatic sources of pathological ROS which could overcome the limitations of pharmacological efforts aimed at scavenging ROS which has resulted in poor outcomes of antioxidant therapies in clinical studies.

1. Introduction

Reactive oxygen species (ROS; e.g. superoxide: ${\text{O}}_2^{-\cdot }$, hydrogen peroxide: H₂O₂) are known to play a critical role in cell signaling, which regulates gene expression, cell proliferation and division, cell immunity, and cell death [1–4]. They are generated under both physiological and pathophysiological conditions, albeit in excess under pathophysiological conditions, in almost all tissues and organs of multicellular organisms. In addition, when produced in excess, they are known to have injurious effects on biomolecules (e.g. DNA, proteins, and lipids), resulting in organelles, cells, and tissues/organs dysfunction and diseases [5–8].

There are several cellular sources of ROS, but the family of NADPH oxidase (NOX) represents the major non-mitochondrial source [9–12]. There is ample evidence that ROS generated by NOX play a key role in the pathogenesis of many chronic diseases, including atherosclerosis, heart disease, hypertension, diabetic nephropathy, chronic kidney disease, lung fibrosis, cancer, and Alzheimer’s disease, among others [12–18]. Their existence in neutrophils and other phagocytes was first identified by Baehner et al. [19]. It is now well recognized that, in phagocytic cells, the NOX-generated ROS (primarily ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$) play an important role in the cellular innate immune responses [20,21]. The NOX enzyme expressed in phagocytic cells (NOX2) upon stimulation generates a huge amount of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$, so called “respiratory burst”, to kill pathogens [22]. Relevant to the kinetics of this system, which is explored in the present study, the NOX2 enzyme when stimulated in human neutrophils can generate ∼10 nmol ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$/min/10⁶ cells, which requires, remarkably, the transfer of ∼10⁸ electrons/s/cell or a current of ∼16 pA [23–25]. In non-phagocytic cells, the NOX enzymes are of enormous importance with ROS affecting nearly every cellular process yet studied, including transcription factor activation, protein synthesis, cell division and proliferation, nearly all metabolic processes, and apoptosis [1–3,9–15].

Seven NOX isoforms have been identified, each containing at least six trans-membrane helices, two iron-heme groups, a FAD molecule, and a NADPH-binding domain [10,12]. Each of these NOX isoforms differs in composition, modes of activation, and its enzymatic reaction products. NOX2 is the most studied of NOXs, and is found in a variety of cell types of mesodermal origin. It consists of a group of plasma membrane proteins (e.g. gp91^phox, p22^phox) and several cytosolic proteins (e.g. p47^phox, p40^phox, p67^phox, Rac-GTP) (Fig. 1A) [26–32]; note that phox refers to a phagocytic oxidase. The membrane-bound NOX2 components remain dormant in unstimulated cells, but become activated when complexed with the cytosolic components in response to stimulation by various neuroendocrine, paracrine, microorganisms, or inflammatory mediators. The assembled and activated NOX2 complex mediates the transfer of electrons from substrate NADPH to molecular O₂ through the action of an integral membrane-bound protein, namely flavocytochrome b₂₄₅ or b₅₅₈ (FlavoCytB), within the phagocytic oxidase subunits gp91^phox and p22^phox (Fig. 1B) [33–35]. Flavo-CytB contains all the catalytic machinery for oxidation of NADPH to NADP⁺ and reduction of O₂ to ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. However, it is only facilitated when the membrane subunits are complexed with the cytosolic subunits, leading to NOX2 assembly and activation [26–32].

Fig. 1. Proposed kinetic mechanisms for the NOX2 complex-mediated electron flow and superoxide production.

(A) Schematics of unassembled and inactivated NOX2 enzyme showing different cytosolic (p67^phox, p47^phox, p40^phox, and Rac-GTP) and membrane-bound (gp91^phox and p22^phox) subunits. (B) Schematics of an assembled and activated NOX2 complex showing different cytosolic subunits complexed with the membrane-bound subunits, facilitating electron transfer from substrate NADPH to molecular oxygen (O₂) resulting in superoxide $\left({\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }\right)$ production. (C) Schematics shows the five elementary electron transfer reactions and the associated midpoint redox potentials from NADPH through different redox centers of the NOX2 complex, and ultimately to O₂ for ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production. The NOX2 subunit gp91^phox encompass an NADPH binding site and two redox centers, viz. a FAD-containing flavoprotein and a heme-containing heterodimeric cytochrome b₂₄₅ (CytB), together termed as flavocytochrome b₂₄₅ (FlavoCytB). In the first step, two electrons are transferred from NADPH to FAD to reduce FAD to FADH₂. Subsequently, FADH₂ provides two electrons to two heme sites of two CytB_ox in two different steps via the semiradical intermediate FADH^• to produce two CytB_red. Finally, each CytB_red provides an electron to O₂ to form ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ via sequential one electron reduction of O₂ [38] to complete the catalytic cycle. (D) The five-state catalytic scheme corresponding to the electron transfer reactions schematized in Panel C. The five different states (E_n.X, n = 1, …, 5) of the NOX2 complex (E.X) are based on the redox status of FAD and 2CytB. (E) Lumped electron transfer reaction schematics in which the two elementary electron transfer reactions from FADH₂ to 2CytB_ox via FADH^• and from 2CytB_red to 2O₂ are lumped. (F) The three-state catalytic scheme corresponding to the electron transfer reactions schematized in Panel E. The three different states (E_n.X, n = 1, …, 3) of the NOX2 complex (E.X) are based on the redox status of FAD and 2CytB; FADH^•.CytB_ox.CytB_red and FAD.CytB_red. CytB_ox states are neglected from the 5-state catalytic scheme of Panel D.

Several early studies have evaluated the kinetics of NOX2 enzyme and NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production in cell-free and cell-based assay systems, and have reported the K_m of NADPH to vary widely (high K_m of 40–300 μM in cell-free systems vs. as low as K_m of 5–15 μM in whole-cell systems) [36,37]. Nisimoto et al. [38] recently examined the O₂ dependency of NOX2 complex-mediated ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation in intact human neutrophils and cell-free assay systems and found the K_m of O₂ to be 3.1 and 2.3% O₂ (22.1 and 16.4 mmHg), respectively. These values are within the range of K_m values reported for enzymes contributing to cellular housekeeping functions. Also relevant to the current study, it is recognized that the NOX2 enzyme, as most others, are sensitive to pH and temperature, working optimally at neutral pH and physiological temperature. Few early studies have demonstrated high NOX2 enzyme activity in cell-free systems at pH in the vicinity of 7.0, with NADPH as a substrate [39,40]. Subsequently, Morgan et al. reported that the activated NOX2 enzyme-mediated electron current (proportional to ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production) across the plasma membrane is maximal at around pH = 7.2, decreasing at higher or lower pH values [41]. Other related studies have also measured electron current generated by the activated NOX2 enzyme across the plasma membrane at fixed temperature [23–25] and at varying temperatures [42]. Also relevant to the present study, few NOX2 inhibitors have been developed and their modes of actions have been characterized in efforts to assess their potential use as therapeutic agents [43–45].

Although details on the NOX2 enzyme structure, its assembly and activation, electron transfer, and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production have been extensively studied, there is a lack of a quantitative and integrated understanding of the kinetic mechanisms of NOX2 complex-mediated electron transfer and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation, and their regulations by pH, inhibitory drugs, and temperature. To this end, we have developed here a thermo-dynamically-constrained mathematical model to explore the catalytic mechanisms and regulations of NOX2 complex based on a diverse set of published experimental data obtained from cell-free and cell-based assay systems [37–42,45]. The model development and parameterization use strict thermodynamics of elementary electron transfer reactions based on midpoint redox potentials of associated redox centers to constrain some of the kinetic parameters, while fixing some parameters at their expected values and estimating the remaining parameters in a modular and multi-step fashion based on the available data. The proposed model is the first thermodynamically rigorous kinetic model of the complex enzymatic function of NOX2 that describes well all the existing data, and provides critical quantitative understanding of the roles of pH, temperature, and inhibitory mechanisms of various drugs on the NOX2 enzymatic function under physiological and pathophysiological conditions.

2. Materials and methods

Proposed kinetic mechanisms for NOX2 complex-mediated electron transfer, superoxide generation, and regulations by pH and inhibitory drugs:

Upon assembly and activation (Fig. 1B), the NOX2 complex catalyzes the following electron transfer reaction to-wards ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production [46] (see Appendix for further details and thermodynamics):

$$NADPH+2O_2\leftrightarrow NAD{P}^{+}+2O_2^{{-}_{\cdot }}+H^{+}$$ (1)

The five well-described elementary electron transfer reactions resulting ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation are schematized in Fig. 1C, D (see Appendix for further details and thermodynamics). As rationalized in the Appendix, the NOX2 kinetic model based on the catalytic scheme of Fig. 1C, D would be complex containing five unknown parameters for estimation for the base model. In order to reduce the complexity of the model and the number of unknown parameters, we lumped some of the elementary electron transfer reactions, as schematized in Fig. 1E, F, resulting in three unknown parameters for the base model. The corresponding regulation mechanisms by pH and inhibitory drugs are schematized in Fig. 2(A–C).

Fig. 2. Proposed kinetic mechanisms for the regulations of NOX2 complex-mediated electron flow and superoxide production by pH and inhibitory drugs.

(A) Schematics shows our hypothesized pH regulation mechanism of the electron transfer reactions. We assume rapid equilibrium binding of protons (H⁺) at the states E_n.X (inactive) to form the states E_n.X.H (active and promotes electron transfer reactions) and the states E_n.X.H₂ (inactive), which collectively termed here as (E_n.X)_T, n = 1, 2 and 3. This H⁺ binding modifies the rate constants k_nf and k_nr, by the pH-dependent factors f_nf and f_nr, which enables us to simulate the bimodal effect of pH on the NOX2 complex activity, as observed experimentally. (B) Schematics shows the NOX2 mode of action of the drug GSK2795039 (GSK). The competitive inhibitor GSK binds to the NOX2 state (E₁.X)_T (i.e. competes for the NADPH binding site) in a rapid equilibrium fashion, which modifies the rate constants k_1f and k_3r by the GSK-dependent factors f_1f and f_3r; K_GSK is the binding constant for GSK. (C) Schematics shows the NOX2 mode of action of the drug DPI (diphenyleneiodonium). The uncompetitive inhibitor DPI binds to the NOX2 state (E₂.X)_T (i.e. binds after NADPH is bound) in a rapid equilibrium fashion, which modifies the rate constants k_1r and k_2f by the DPI-dependent factors f_1r and f_2f; K_DPI is the binding constant for DPI.

In Fig. 1, the NOX2 membrane-bound subunits gp91^phox and p22^phox are together represented as E, which encompasses FlavoCytB (flavocy-tochrome b₂₄₅ or b₅₅₈), but is in inactive form (Fig. 1A). Upon binding with the assembled cytosolic subunit complex p47^phox.p40^phox.p67^phox.Rac-GTP (represented as X), the NOX2 enzyme gets assembled and activated to form the enzyme complex E.X (Fig. 1B), which mediates the transfer of electrons from substrate NADPH to molecular O₂ in multiple steps through the action of FlavoCytB within the membrane-bound subunits gp91^phox and p22^phox (Fig. 1C–F). In the first step, two electrons are transferred from NADPH to FAD to reduce it to FADH₂. Subsequently, FADH₂ provides two electrons to the heme sites of two CytB in two sequential steps via the intermediate radical FADH^• to reduce two CytB. In the final step, two electrons are transferred from the heme sites of two reduced CytB to two O₂ molecules in two sequential steps to produce two ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ molecules. The process is referred to as the sequential one-electron O₂ reduction mechanism by FlavoCytB, as proposed recently by Nisimoto et al. [38].

As detailed in the Appendix, the schematics of Fig. 1E, F are simplifications of the schematics of Fig. 1C, D as a result of the lumping of the two elementary electron transfer reactions from FADH₂ to 2CytB_ox via FADH^• and from 2CytB_red to 2O₂, without affecting the overall thermodynamics of the electron transfer reactions. In Fig. 1D, F, the different states of the NOX2 complex E.X are represented by E_n.X, depending on the redox status of FlavoCytB within gp91^phox and p22^phox. Specifically, E₁, E₂, E₃, E₄, and E₅ in Fig. 1D represent FAD.2CytB_ox, FADH₂.2CytB_ox, FADH^•.CytB_ox.CytB_red, FAD.2CytB_red, and FAD.CytB_red.CytB_ox, respectively, while E₁, E₂, and E₃ in Fig. 1F represent FAD.2CytB_ox, FADH₂.2CytB_ox, and FAD.2CytB_red, respectively. Thus, for the simplified three-state model (Fig. 1F), FADH^•.CytB_ox.CytB_red and FAD.CytB_red.CytB_ox states in the complex five-state model (Fig. 1D) are eliminated. The assumption is that the electron transfer rate from FADH^• to CytB_ox is fast compared to that from FADH2 to CytB_ox, so that the contributions of the states FADH^•.CytB_ox.CytB_red and FAD.CytB_red.CytB_ox are accounted for in the forward or reverse rate constants of the relevant electron transfer reactions in Fig. 1F.

As schematized in Fig. 2A, the NOX2 kinetic model includes thermodynamically-balanced, pH-regulated electron transfer reactions from NADPH to O₂ in producing ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$, which actually occurs at multiple redox centers within the NOX2 complex (i.e. the flavin of adenine dinucleotide (FAD) and terminal oxidase cytochrome b₂₄₅ (CytB)). It should be noted here that the resting (inactivated) NOX2 enzyme does not transfer electrons from NADPH to the flavin center at a significant rate, as evidenced by the absence of FAD or CytB reduction [29,33,35]. The pH regulation scheme (Fig. 2A) was developed to reproduce the experimentally-observed bimodal behavior of the NOX2 enzyme activity as a function of pH [39–41]. Specifically, we assume rapid equilibrium binding of protons (H⁺) to E_n.X (inactive states) to form E_n.X.H (active states and promote electron transfer reactions) and E_n.X.H₂ (inactive states), which are collectively termed as the total E_n.X ((E_n.X)_T, n = 1, …, 3). Finally, the NOX2 kinetic model also incorporates differential modes of actions of the drugs GSK2795039 (denoted here as GSK) and diphenyleneiodonium (DPI) on the NOX2 enzyme activity, as schematized in Fig. 2B, C. Specifically, GSK acts as a competitive inhibitor (competes for the NADPH binding site), while DPI acts as an uncompetitive inhibitor (binds after NADPH binding), thereby regulating the NOX2 enzyme activity [45].

Flux expression for NOX2 complex-mediated superoxide production:

At steady state, the rate of change in the enzyme concentration at each state of the assembled and activated NOX2 complex is zero (Fig. 2A–C), and hence the turnover of the enzyme (i.e. reaction flux per unit enzyme concentration; 1/time) can be calculated using any of the rate limiting elementary reaction steps (see Appendix for detailed derivation). This can also be easily computed using our KAPattern tool in MATLAB for generating rate equations for enzymatic reactions [47]. This results in the following expressions for the fractional enzyme states (F₁, F₂, F₃) for the NOX2 complex:

$$F_1=\frac{C_{{\left(E_1.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\frac{f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2}}{Den}$$ (2a)

$$F_2=\frac{C_{{\left(E_2.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\frac{f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}}{Den}$$ (2b)

$$F_3=\frac{C_{{\left(E_3.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\frac{f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\cdot }}}{Den}$$ (2c)

where the common denominator (Den) is defined as:

$$\begin{gathered} Den=f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2} \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2} \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\cdot }} \end{gathered}$$ (2d)

In the above equations, $C_{{\left(E_n.X\right)}_T}$ denotes the total E_n.X concentrations (individual NOX2 complex states) and $C_{{\left(E.X\right)}_T}$ denotes the total E.X concentration (NOX2 complex). The NOX2 reaction flux (concentration/time) can then be derived as the turnover rate of the activated NOX2 complex times the total activated NOX2 complex concentration, which is expressed as:

$$J_{NOX2}=\frac{C_{{\left(E.X\right)}_T}\left(f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}-f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\right)}{\left(\begin{array}{c}{f}_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}+\\ f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2}+\\ f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\end{array}\right)}$$ (3)

where the total NOX2 complex concentration $C_{{\left(E.X\right)}_T}$ is a measure of the FlavoCytB concentration C_FlavoCytB; C_Nr and C_No denote NADPH and NADP⁺ concentrations, respectively; C_O2 and $C_{O{2}^{-\cdot }}$ denote oxygen and superoxide concentrations, respectively; k_nf and k_nr are the rate constants for the nth forward and reverse reactions, respectively; and f_nf and f_nr are the scaling factors for the rate constants k_nf and k_nr due to the binding of NOX2 complex with protons and drugs, as defined in Equation (7) below. The NOX2 reaction flux expression (Equation (3)) is based on sequential one-electron O₂ reduction mechanism by FlavoCytB, similar to that proposal by Nisimoto et al. [38], and hence the flux expression involves C_O2 and $C_{O{2}^{-\cdot }}$, rather than $C_{O2}^2$ and $C_{O{2}^{-\cdot }}^2$ (see Appendix).

Simplified NOX2 flux expression in the absence of reaction products:

Most of the experimental data that are used in the present study for model development involve NOX2 complex-mediated NADPH and O₂ dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. Under such conditions, the NOX2 reaction flux expression can be simplified to:

$$J_{NOX2}=\frac{C_{{\left(C.X\right)}_T}{f}_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}}{\left(\begin{array}{l}{f}_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+\\ f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2}+f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}\end{array}\right)}$$ (4)

which can be rewritten in the form of familiar Michaelis-Menten equation:

$$J_{NOX2}=\frac{{V^{\prime }}_{\max }{C}_{Nr}{C}_{O2}}{\left({K^{\prime }}_{O2}{C}_{Nr}+{K^{\prime }}_{Nr}{C}_{O2}+C_{Nr}{C}_{O2}\right)}$$ (5)

where V′_max is the apparent maximal velocity, and K′_Nr and K′_O2 are the apparent K_m (Michaelis-Menten constants) of NADPH and O₂, respectively. These kinetic parameters are expressed in terms of the reaction rate constants as:

$${V^{\prime }}_{\max }=C_{{\left(C.X\right)}_T}{f}_{2f}{k}_{2f}$$ (6a)

$${K^{\prime }}_{Nr}=\frac{f_{2f}{k}_{2f}}{f_{1f}{k}_{1f}}$$ (6b)

$${K^{\prime }}_{O2}=\frac{\left(f_{2f}{k}_{2f}+f_{2r}{k}_{2r}\right)}{f_{3f}{k}_{3f}}$$ (6c)

NOX2 complex kinetic regulations by pH and inhibitory drugs:

The fractions of total active NOX2 complex in different states facilitating the electron transfer reactions due to regulations by protons and drug molecules can be expressed as:

$$f_H=\left(\frac{C_H}{K_{H1}}\right)/\left(1+\frac{C_H}{K_{H1}}+\frac{C_H^2}{K_{H1}{K}_{H2}}\right)$$ (7a)

$$f_{1f}=f_{3r}=f_H/\left(1+\frac{C_{GSK}}{K_{GSK}}\right)$$ (7b)

$$f_{2f}=f_{1r}=f_H/\left(1+\frac{C_{DPI}}{K_{DPI}}\right)$$ (7c)

$$f_{3f}=f_{2r}=f_H$$ (7d)

where C_GSK, C_DPI and C_H denote the concentrations of GSK, DPI and protons bound to NOX2 complex (different states), respectively; K_GSK, K_DPI, K_H1 and K_H2 represent the respective binding constants for GSK, DPI and protons, as represented in Fig. 2A–C.

Thermodynamic and kinetic constraints for NOX2 model parameters:

The standard midpoint redox potentials $E_{m,j}^0$ 0 for different redox pairs involved in the three lumped elementary electron transfer reactions (Fig. 1F) at pH=7 [29], and the equilibrium constants K_eq,i of the electron transfer reactions derived from these midpoint redox potentials (see Appendix), which are also further corrected for variable pH, are given by:

$$E_{m,Nr/No}^0=-320\hspace{0.17em}mV$$ (8a)

$$E_{m,Fr/Fo}^0=-280\hspace{0.17em}mV$$ (8b)

$$E_{m,Cr/Co}^0=-245\hspace{0.17em}mV$$ (8c)

$$E_{m,O2/O{2}^{{-}_{\cdot }}}^0=-160\hspace{0.17em}mV$$ (8d)

$$K_{eq,1}=K_{eq,E_1\leftrightarrow E_2}=\left(\frac{C_H}{K_H}\right)\exp \left(\frac{2F\left(E_{m,Fr/Fo}^0-E_{m,Nr/No}^0\right)}{RT}\right)$$ (8e)

$$K_{eq,2}=K_{eq,E_2\leftrightarrow E_3}={\left(\frac{K_H}{C_H}\right)}^2\hspace{0.17em}\exp \left(\frac{2F\left(E_{m,Cr/Co}^0-E_{m,Fr/Fo}^0\right)}{RT}\right)$$ (8f)

$$K_{eq,3}=K_{eq,E_3\leftrightarrow E_1}=\exp \left(\frac{2F\left(E_{m,O2/O{2}^{{-}_{\cdot }}}^0-E_{m,Cr/Co}^0\right)}{RT}\right)$$ (8g)

$$K_{eq,NOX2}=\prod _{i=1}^3{K}_{eq,i}=\left(\frac{K_H}{C_H}\right)\exp \left(\frac{2F\left(E_{m,O_2/O_2^{-\cdot }}^0-E_{m,Nr/No}^0\right)}{RT}\right)$$ (8h)

where Nr and No denote NADPH (reduced) and NADP⁺ (oxidized), respectively; Fr and Fo denote FADH₂ (reduced) and FAD (oxidized), respectively; Cr and Co denote the reduced and oxidized forms of FlavoCytB, respectively; K_H = 1 × 10⁻⁷M is the reference proton concentration; F, R and T represent the Faradays constant (96485 J/mol/V), ideal gas constant (8.314 J/mol/K) and temperature (298.15 K), respectively. Ideally, the equilibrium constants in Equation 8(e–h) should also be corrected for temperature variations using the Van’t-Hoff equation [48], which are expected to be small. However, these small corrections could not be accounted for, as the enthalpy changes associated with the individual electron transfer reactions are not known.

The pH-dependency of the rate constants are appropriately determined based on thermodynamics and whether H⁺ is a reactant or a product of the associated electron transfer reaction. In reaction 1 of Fig. 1F, H⁺ is a reactant, and in reaction 2, 2H⁺ are products. The model accounts for these H⁺ effects through modifications of the forward or reverse rate constants, which are constrained by equilibrium constants, which are functions of pH (Equation 8(e–h)). As shown below, k_1r and k_2r are dependent on pH, in addition to k_1f. The ratio k_1f/k_1r=K_eq,1 is proportional to (C_H/K_H), while the ratio k_2f/k_2r=K_eq,2 is proportional to (K_H/C_H)². In addition, all the rate constants are further modified by pH-dependent function f_H, defined in Equation (7a), as determined by the kinetic schematics of Fig. 2A. The pH-dependency of the rate constant k_1f as (C_H/K_H)^0.5 rather than (C_H/K_H) is motivated by the pH-dependent kinetic data (NOX2 K_m of NADPH as a function of pH [39]); k_1r is set appropriately to satisfy the thermodynamic constraint below. The rate constants are constrained by the following kinetic and thermodynamic constraints:

$$k_{1f}=\frac{k_{2f}}{K_{Nr}}{\left(\frac{C_H}{K_H}\right)}^{0.5}$$ (9a)

$$k_{1r}=\frac{k_{1f}}{K_{eq,E1\leftrightarrow E2}}$$ (9b)

$$k_{2r}=\frac{k_{2f}}{K_{eq,E2\leftrightarrow E3}}$$ (9c)

$$k_{3f}=\frac{\left(k_{2f}+k_{2r}\right)}{K_{O2}}$$ (9d)

$$k_{3r}=\frac{k_{3f}}{K_{eq,E3\leftrightarrow E1}}$$ (9e)

where K_Nr and K_O2 are the intrinsic K_m of NADPH and O₂ for NOX2 at pH=7, respectively. The apparent V_max and apparent K_m of NADPH and O₂ for NOX2 at variable pH and drug concentrations, defined in Equation (6), can then be expressed as:

$${V^{\prime }}_{\max }=C_{{\left(E.X\right)}_T}{k}_{2f}{f}_H/\left(1+\frac{C_{DPI}}{K_{DPI}}\right)$$ (10a)

$${K^{\prime }}_{Nr}=K_{Nr}{\left(\frac{K_H}{C_H}\right)}^{0.5}\left(1+\frac{C_{GSK}}{K_{GSK}}\right)/\left(1+\frac{C_{DPI}}{K_{DPI}}\right)$$ (10b)

$${K^{\prime }}_{O2}=K_{O2}\left(k_{2f}/\left(1+\frac{C_{DPI}}{K_{DPI}}\right)+k_{2r}\right)/\left(k_{2f}+k_{2r}\right)$$ (10c)

It is clear from Equation (10) that the drug DPI decreases both the apparent V_max and apparent K_m of NADPH and O₂ for NOX2, while the drug GSK only increases the apparent K_m of NADPH for NOX2. In the absence of drugs, the apparent V_max and apparent K_m of NADPH and O₂ for NOX2 at variable pH can be simplified as:

$${V^{\prime }}_{\max }=C_{{\left(E.X\right)}_T}{k}_{2f}{f}_H$$ (11a)

$${K^{\prime }}_{Nr}=K_{Nr}{\left(\frac{K_H}{C_H}\right)}^{0.5}$$ (11b)

$${K^{\prime }}_{O2}=K_{O2}$$ (11c)

Temperature dependency of NOX2 complex kinetics:

The effect of temperature on the kinetics of NOX2 complex was modeled by multiplying the elementary electron transfer reaction rate constants by the following temperature correction coefficient based on the Arrhenius principle [48]:

$$T_{fact}=Q_{10}^{\raisebox{1ex}{$\left(T_2-T_1\right)$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}$$ (12)

where T₁ = 25 °C at which the reaction rate constants are defined, T₂ is the temperature (°C) at which the reaction rate constants are calculated, Q₁₀ = 2.5 is the temperature correction factor for the increase of reaction rate for every 10 °C temperature change. This approach of accounting for the temperature effect was motivated by the results of a study by Morgan et al. [42] on the temperature dependency of the NOX2 enzyme activity in stimulated human eosinophils while changing the temperature rapidly during the peak of the respiratory burst. We corrected the kinetic parameter k_2f for the temperature change effect as k_2f = T_fact × 10, and this way all the rate constants have temperature effects as per Equation (9).

Numerical estimation of NOX2 model parameters:

The developed NOX2 kinetic model contains several unknown parameters (Table 1). We used the values of the standard midpoint redox potentials involved in the elementary electron transfer reactions and the corresponding equilibrium constants (Equation (8)) to constraint the reverse reaction rate constants k_1r, k_2r, and k_3r (Equation (9)). In addition, we fixed some kinetic parameters at their stipulated values (i.e. k_2f = 10 sec⁻¹ which is a measure of the maximal reaction velocity V_max and cannot be distinguished from C_FlavoCytB; K_H = 1 × 10⁻⁷ as the reference proton concentration). To estimate the remaining unknown kinetic parameters, we used a modular multi-step optimization technique (e.g. sequential use of the available experimental data [37–42,45] related to electrons transfer, pH effect, drugs action, and temperature effect) of the combined least-square objective function defined below to fit the model solution to the experimental data:

$$\underset{\varphi }{\min }E\left(\varphi \right),E\left(\varphi \right)=\sum _{k=1}^{N_{\exp }}\frac{1}{N_{data,k}}\left({\sum _{j=1}^{N_{data,k}}\left(\frac{J_{j,k}^{data}-J_{j,k}^{\mathrm{mod}el}\left(\varphi \right)}{\max \left(J_{j,k}^{data}\right)}\right)}^2\right)$$ (13)

where N_exp is the number of experiments and N_data,k is the number of data points in the kth experiment; $J_{j,k}^{data}$ are the experimental data and $J_{j,k}^{\mathrm{mo}\text{del}}\left(\varphi \right)$ are the corresponding model solutions that depend on the values of the model parameters ϕ. The accuracy and robustness of the model fitting to the experimental data were assessed based on the value of the mean square residual error E (ϕ) and the sensitivities of E (ϕ) or J (ϕ) to perturbations in the optimal parameter estimates. All the model parameters and their values (fixed or estimated) are summarized in Table 1.

Table 1.

Estimated parameter values for NOX2 kinetic model (NOX2 complex-mediated electron transfer and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production).

Parameter	Definition/Comment	Value	Units	Reference
C(E.X)T	Total active NOX2 complex concentration (known from specific experimental conditions)	Vary depending on the experiment/assay	μM	[37–40,45]
KNr	Km of NADPH for NOX2 complex at pH = 7 (estimated based on NADPH-dependent data on ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production)	Vary depending on the experiment/assay 40–300 (Cell-free assays) 5–15 (Cell-based assays)	μM	[37,39,40]
KO2	Km of O2 for NOX2 complex at pH = 7 (estimated based on O2 dependent data on ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$	46 (Cell-free assays) 60 (Cell-based assays)	μM	[38]
CH	Proton (H+) concentration (vary depending on experiment/assay)	10−pH	M
KH	Reference 10−pH at which Keq’s or $E_m^0$'s of electron transfer redox reactions were obtained (fixed)	10–7.0	M	[37–40,45]
KH1	First H+ binding constant (estimated based on pH dependent data on ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production)	10–7.52	M	[40]
KH2	Second H+ binding constant (estimated based on pH dependent data on ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production)	10–6.69	M	[40]
k1f	Forward rate constants from state (E1.X) to state (E2.X); dependent on CH/KH (calculated)	(k2f /KNr )(CH /KH )0.5	1/(μM.s)	Eq. 9a
k1r	Reverse rate constants from state (E2.X) to state E1.X; proportional to (CH/KH)−0.5 (calculated)	$k_{1f}/K_{eq,E_1\leftrightarrow E_2}$	1/(μM.s)	Eq. 9b
k2f	Forward rate constant from state (E2.X) to state E3.X (fixed)	10.0	1/s	[37–40,45]
k2r	Reverse rate constant from state E3.X to state (E2.X); proportional to (CH/KH)2 (calculated)	$k_{2f}/K_{eq,E_2\leftrightarrow E_3}$	1/s	Eq. 9c
k3f	Forward rate constant from state E3.X to state E1.X; dependent on CH/KH via k2r (calculated)	(k2f+k2r)/KO2	1/(μM.s)	Eq. 9d
k3r	Reverse rate constant from state E1.X to state E3.X; dependent on CH/KH via k2r (calculated)	$k_{3f}/K_{eq,E_3\leftrightarrow E_1}$	1/(μM.s)	Eq. 9e
KDPI	DPI binding constant for NOX2 complex (estimated)	0.78	μM	[45]
KGSK	GSK binding constant for NOX2 complex (estimated)	0.55	μM	[45]

3. Results

In this section, we present the parameterization and simulation results of the integrated kinetic model of NOX2 complex-mediated electron transfer and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production, and their regulations by pH, temperature, and inhibitory drugs. As described below, we estimated the values of the unknown model parameters (Table 1) by fitting the model solution to a diverse set of published experimental data (Figs. 3 and 4) [37–42,45]. The parameterized and validated/corroborated model was then used to predict emergent properties of the NOX2 enzyme system (i.e. NADPH and O₂-dependent bimodal effects of pH on NOX2 enzyme activity, fractional enzyme states, and associated kinetic parameters as well as alkaline pH-dependent NADP⁺ inhibition of NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production and the effect of temperature on NOX2 enzyme activity) (Figs. 5–8).

Fig. 3. Model fittings to available experimental data on NADPH and O₂ dependent superoxide generations via the assembled and activated NOX2 complex, and their regulations by pH.

(A–C) Model simulations (lines) are compared to experimental data (symbols) on the effects of varying concentrations of NADPH on relative NOX2 fluxes (JNOX2; the rates of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production via the NOX2 complex normalized with respect to the maximal rate), studied either in permeablized polymorphonuclear neutrophils (PMNs) (cell-based system) [37] (A) or in isolated plasma membrane (PM) preparations from PMNs (cell-free system) [37,40] (B,C) under different NOX2 stimulation conditions. In Bauldry et al.’s experiments [37], 1 ml cuvette buffer contained multiples of 1 × 10⁶ cells or its equivalent PM protein. Since 1 × 10⁶ cells is equivalent to approximately 1.5 pmol of FlavoCytB, its concentration in the cuvette was assumed to be multiples of 1.5 pmol/ml = 1.5 nM, which was used as the total activated NOX2 complex concentration for the model simulations. Similar assumptions were made for the analysis of Suzuki et al.’s experimental data [40]. (D) Model simulations (lines) are compared to experimental data (symbols) on the effects of varying concentrations of O₂ on relative NOX2 fluxes (JNOX2; the rates of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production via the NOX2 complex normalized with respect to the maximal rate), studied either in intact PMNs (cell-based system) or in isolated PM preparations from PMNs (cell-free system) under different NOX2 stimulation conditions [38]. In Nisimoto et al.’s experiments [38], since 0.8 ml cuvette buffer contained about 4.2 × 10⁵ cells or its equivalent PM protein and NOX2 was activated differently, FlavoCytB concentration in the cuvette was assumed to be multiples of (0.42/ 0.8) × 1.5 pmol/ml ≈ 0.8 nM, which was used as the total activated NOX2 complex concentration for the model simulations. Percentage O₂ in the cuvette was first converted to partial pressure of O₂ (P_O2), and then to dissolved O₂ using [O₂] = α_O2 × P_O2, where α_O2 = 1.65 μM/mmHg and P_O2 = (150/21) × (%O₂) mmHg. (E) Model simulation (line) is compared to experimental data (symbols) [40] demonstrating the bimodal behavior of the assembled and activated NOX2 complex activity, measured as the rate of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production (normalized with respect to the maximal rate) in isolated PM preparations from PMNs upon pH changes ranging from acidic to neutral to alkaline values (pH = 5–10), with high or saturating substrate NADPH and O₂ concentrations (250 μM each) and FlavoCytB = 1.5 nM. (F) Model simulation (line) on the apparent K_m of NADPH for the NOX2 complex is compared to experimental data (symbols) [39] for pH values ranging from 5 to 8. The model simulations are made with fixed NOX2 complex concentration (assembled and activated equivalent to 1.5 nM FlavoCytB), with unlimited O₂ concentration (250 μM) in the cuvette. In all plots (A–F), JNOX2 is appropriately normalized to have unit NOX2 complex maximal activity for the purpose of model comparisons to the experimental data. The estimated apparent K_m values and other experimental details are also indicated in the respective figure insets.

Fig. 4. Model fittings to available experimental data on NADPH dependent superoxide generations via the assembled and activated NOX2 complex, and their regulations by the inhibitory drugs DPI and GSK.

(A) Model simulations (lines) are compared to experimental data (symbols) on the kinetic profiles of the drug DPI as an uncompetitive inhibitor of the NOX2 complex [45]. In Hirano et al.’s experiments [45], the effects of various concentrations of the drug DPI on the NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production (JNOX2; normalized with respect to the maximal rate) were studied in the isolated cell membrane preparations (cell-free system; recombinant NOX2) at high O₂ and pH = 7.4. (B) Model simulations (lines) are compared to experimental data (symbols) on the kinetic profiles of the drug GSK as a competitive inhibitor of the NOX2 complex [45], with the experimental details the same as that for Panel A. (C, D) Model predictions of the inhibitory profiles of the drugs DPI and GSK (relative NOX2 flux as % of control vs. log10 of drug concentrations) for different concentrations of NADPH in the experimental solution. (E, F) Model simulations of the time courses of NADPH depletions and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ productions (simulated as decreasing concentrations of NADPH and increasing concentrations of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$) in response to different increasing concentrations of the uncompetitive inhibitor drug DPI and competitive inhibitor drug GSK. The model parameters are the same as that for panels (A, C) and (B, D). (G–J) Model simulations of NOX2 apparent K_m of NADPH and V_max with respect to NADPH as functions of the drug DPI (G, H) and GSK (I, J) concentrations at high O₂ and pH = 7.4.

Fig. 5. Model predictions demonstrating multi-dimensional profiling of the NADPH, O₂ and pH dependent superoxide generations and the corresponding NOX2 complex states.

(A, B) Model simulations depict bimodal behavior of the NOX2 complex activity, simulated as the rate of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production (NOX2 flux; nM/s) upon pH changes ranging from acidic to neutral to alkaline values (pH = 5–9) with either the substrate NADPH or O₂ varying ranging from limited to saturated levels (5–250 μM) in the absence of the products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. When NADPH is varied, O₂ is maintained at saturated level (250 μM) and vice-versa. (C, D) Model simulations depict transitions of the assembled and activated NOX2 complex states, expressed as fractions of the total NOX2 complex (F₁, F₂, F₃), as functions of pH changes ranging from acidic to neutral to alkaline values (pH = 5–9) with the substrate NADPH ranging from limited to saturated levels (5–250 μM) at high saturated level of O₂ (250 μM) in the absence of the products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. (E, F) Model simulations depict transitions of the assembled and activated NOX2 complex states (F₁, F₂, F₃) as functions of pH changes ranging from acidic to neutral to alkaline values (pH = 5–9) with the substrate O₂ ranging from limited to saturated levels (5–250 μM) at high saturated level of NADPH (250 μM) in the absence of the products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. The model simulations are made with fixed NOX2 complex concentrations (assembled and activated equivalent to 1.5 nM FlavoCytB) with parameters corresponding to the cell-free system, as reported in Table 1.

Fig. 8. Model predictions demonstrating the effects of temperature on the kinetic parameters of the NOX2 complex governing electron transfer and superoxide production.

(A, B) Model simulations depict variations of the apparent K_m of NADPH for the NOX2 complex as functions of varying O₂ concentration (5–250 μM) and pH (5–9) for a range of temperature (T = 18–43 °C) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. (C, D) Model simulations depict variations of the apparent K_m of O₂ for the NOX2 complex as functions of varying NADPH concentration (5–250 μM) and pH (5–9) for a range of temperature (T = 18–43 °C) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. (E, F) Model simulations depict the effects of varying temperature (T = 18–43 °C) on the apparent V_max with respect to O₂ or NADPH for the NOX2 complex as functions of varying NADPH or O₂ concentrations (5–250 μM) at pH = 7.4 (E) and varying pH (5–9) at high saturating levels of NADPH and O₂ (250 μM) (F) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. These apparent kinetic parameters are calculated with fixed NOX2 complex concentration (assembled and activated equivalent to 1.5 nM FlavoCytB) with the intrinsic kinetic parameters values, as reported in Table 1, corresponding to the cell-free system.

Kinetics of NOX2 complex-mediated electron transfer, superoxide production, and regulation by pH:

Fig. 3 shows that the NOX2 kinetic model (Equation (4) or (5)) fits well the published experimental data [37–40] on NADPH, O₂ and pH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation via the assembled and activated NOX2 complex. These fits provided the values of the rate constants for the elementary electron transfer reactions in Fig. 2A, subject to thermodynamic and kinetic constraints with some parameters fixed appropriately (Table 1). Bauldry et al. [37] developed a cell-based assay system to activate NOX2 in situ in permeabilized neutrophils at pH = 7.4 to determine the substrate NADPH affinity for NOX2, which was more physiologically appropriate than that obtained using data from isolated membrane preparations (usually several fold greater). As shown in Fig. 3A, the NOX2 kinetic model was able to fit the above data very well for three different stimulation conditions, resulting in higher V_max values (a measure of enzyme activity) with higher NOX2 stimulations (not apparent in the figure due to normalization of the rates) and appropriate K_m values (3.9–7.3 μM at pH 7.4; a measure of the affinity of the enzyme for the substrate). Furthermore, Fig. 3B, C shows that the same model fits well published experimental data on NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production via the activated NOX2 complex observed in isolated membrane preparations [37,40] at different pH with approximately 10–50 fold higher K_m values for NADPH. Interestingly, under isolated membrane preparation conditions, the model was able to capture the increased velocities of NOX2 with higher stimulations, but with the same apparent K_m value (Fig. 3C). In addition, we observed differences in the K_m values for NADPH from one experiment (Fig. 3B) to the other (Fig. 3C) in isolated membrane preparations, suggesting differences in the conditions used for experiments (e.g. pH). Similarly, as shown in Fig. 3D, the developed NOX2 kinetic model also fitted very well the activated NOX2 enzyme kinetics in permeablized neutrophils and isolated membrane preparations with varying O₂ concentrations when performed under saturated NADPH at pH = 7.4 [38]. The model fits revealed small variations in the apparent K_m values of O₂ for NOX2 enzyme prepared under different experimental conditions (see the figure caption for further descriptions of the experiments).

The published kinetic data from Suzuki et al. [40] in isolated membrane preparations was used to characterize the bimodal behavior of the NOX2 enzyme activity under varying pH conditions and identify the pH regulation mechanism and dependence of the rate constants on pH (Fig. 3E, F). The NOX2 kinetic model with postulated pH regulation mechanism (Fig. 2A; Equation (4) or (5)) was able to capture the bimodal behavior of the NOX2 enzyme activity with pH variations, with the optimal activity near pH 7 (Fig. 3E), as observed by Suzuki et al. [40]. The authors used stimulated granule rich fragments (GRF) from human neutrophils and found that the GRF activity was dependent on pH and the activation status (resting vs. stimulated) of the neutrophils. The same model was also able to fit well the effects of pH variations on NOX2 K_m of NADPH (Fig. 3F), indicating that the affinity of NADPH for NOX2 changes significantly with pH, as observed experimentally by Babior et al. in isolated membrane preparations [39].

Regulation of NOX2 enzyme kinetics by inhibitory drugs:

The NOX2 kinetic model (Equation (4) or (5)) was then used to simulate the recently published experimental data [45] regarding the action of an uncompetitive inhibitor, DPI, as well as a competitive inhibitor, GSK, by properly accounting for their bindings to the NOX2 enzyme states (Fig. 2B, C). The model simulations are shown in Fig. 4. Hirano et al. [45] investigated the modes of inhibitory actions of the drugs DPI and GSK in cell-free membrane-isolated activated NOX2 complex by determining the concentration-dependent effects on NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production. With proper DPI and GSK binding to the NOX2 enzyme states (i.e. GSK binding to (E₁.X)_T and DPI binding to (E₂.X)_T; Fig. 2B, C), the NOX2 kinetic model was able to describe well these kinetic data (Fig. 4A, B). In addition, these model fits provided appropriate values for the binding constants of these two drugs and the NOX2 enzyme states characterizing the drug inhibitions.

Using the estimated values of the model parameters, we simulated the resultant effects of these drugs on the NOX2 enzyme kinetics, as shown in Fig. 4C, D. Here, the model predictions plotted with logarithmic changes in the drug concentrations against the fraction of NOX2 enzyme activity clearly show a sigmoidal inhibitory effects of both drugs. However, the inhibition with GSK was markedly more dependent on the availability of NADPH (Fig. 4D) than that with DPI for higher NADPH, specifically at concentration greater than 10 μM (Fig. 4D). Furthermore, the experimental results of Hirano et al. [45] showed that DPI decreased both the K_m of NADPH and V_max with respect to NADPH for NOX2, while GSK only increased the K_m value without affecting the V_max value. The model was able to simulate such behavior as shown in Fig. 4G, H for DPI and Fig. 4I, J for GSK (also see Equation (10)). Finally, Fig. 4C, F shows the model predictions of the concentration-response curves of the drugs DPI and GSK on NADPH depletion in comparison with ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production in a time-dependent manner (for 0–60 min) in a typical isolated membrane preparation experiment. These temporal profiles clearly show that the increased concentration of these drugs slowed down the consumption rate of NADPH utilized to produce ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$, which agrees well with the raw fluorescent measurements of Hirano et al. [45].

Model predictions of key emergent properties of the NOX2 enzyme system:

Enzyme activities are sensitive to pH changes and operate optimally at certain pH. Several studies in the literature demonstrated the bimodal effects of pH on the NOX2 enzyme activity [39–41]. One such study is that of Suzuki et al. [40], discussed above (Fig. 3E), and in a similar study, the NOX2 complex-generated electron current across the plasma membrane was reported to be maximal at around pH = 7.2, and to decrease at higher and lower pH values [41]. We have incorporated the effects of altered pH into the NOX2 kinetic model via its effects on the thermodynamics of electron transfer (redox) reactions as well as via its effects on the NOX2 enzyme states (E_n.X)_T (Fig. 2A). Using the model parameter values estimated from the experimental data in Fig. 3 for the cell-free assay system, the NOX2 kinetic model was used to simulate the enzyme functionality over a wide range of pH values (5–9) as well as NADPH or O₂ concentrations (5–250 μM), as shown in Fig. 5. As illustrated in Fig. 5A, B, the model simulations are consistent with the bimodal behavior of the NOX2 enzyme activity with respect to pH alterations over a wide range of NADPH and O₂ variations with the optimal activity near pH = 7.0 at saturated NADPH and O₂ levels, as has been experimentally observed [40,41]. Unexpectedly, however, the optimal activity was shifted to lower pH values at lower concentrations of NADPH or O₂, indicating modifications in the optimal NOX2 enzyme activity in acidic environments (Fig. 5A, B).

We further used the NOX2 kinetic model to determine the effects of substrate variations and pH upon the transitions of intermediate enzyme states for the activated NOX2 complex. As shown in Fig. 5C, D, the model predicted that the NOX2 complex prefers to stay in the enzyme state (E₂.X)_T at a lower pH (acidic), whereas at a higher pH (alkaline) it prefers to be in the enzyme state (E₁.X)_T, transitioning from (E₁.X)_T to (E₂.X)_T state at a level close to the neutral pH of 7 depending on NADPH concentrations. We observed that under these conditions of varying pH and NADPH, and at a constant saturated levels of O₂, the fractional levels of (E₃.X)_T remain minimal (shown in green). In contrast, when we used a constant saturating concentration of NADPH (250 μM) and varied pH and O₂ concentrations, as shown in Fig. 5E, F, the fractional levels of the state (E₃.X)_T were drastically increased with O₂ under alkaline conditions. These predictions show that the NOX2 enzyme activity highly depends on the availability of its substrates together with the environment under which it operates which has not been previously quantitatively appreciated.

Alterations in the apparent K_m's of substrates NADPH (Fig. 6A, B) and O₂ (Fig. 6C, D) were simulated to study their relationships to these substrates and pH, and their effects on NOX2 enzyme kinetics. This was done over a range of pH (5–9) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. These simulations revealed that the apparent K_m of NADPH remained low (high NADPH affinity) and largely independent of O₂ under acidic conditions. However, this apparent K_m was highly dependent on O₂ when pH increased to basic conditions leading to increased apparent K_m and higher NADPH required for the NOX2 complex binding (Fig. 6A, B). In contrast, the simulations for the apparent K_m of O₂ indicate that it remains high under acidic conditions and slightly drops as NADPH level decreases from high to low values. Interestingly, as pH increased to basic conditions, the model predicted an overall reduction in the apparent K_m of O₂ for NOX2 with its dependency on NADPH (Fig. 6C, D). The simulations of the dependency of apparent V_max for NOX2 at different pH with varying O₂ or NADPH suggest optimal enzyme activity near pH 7.0 (simulations similar to Fig. 5A). Furthermore, the maximum velocities decreased as pH was altered in either direction from the neutral value indicating that pH exerts bimodal effects on both the NOX2 reaction flux and maximal reaction velocity (V_max).

Fig. 6. Model predictions demonstrating multidimensional profiling of the kinetic parameters of the NOX2 complex governing electron transfer and superoxide production.

(A–B) Model simulations depict variations of the apparent K_m of NADPH for the NOX2 complex as functions of varying pH (5–9) and O₂ concentration (5–250 μM) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. (C–D) Model simulations depict variations of the apparent K_m of O₂ for the NOX2 complex as functions of varying pH (5–9) and NADPH concentration (5–250 μM) in the absence of the reaction products NADP+ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$. These apparent kinetic parameters are calculated with fixed NOX2 complex concentration (assembled and activated equivalent to 1.5 nM FlavoCytB) with the intrinsic kinetic parameters values, as reported in Table 1, corresponding to the cell-free system.

The full NOX2 kinetic model (Equation (3)) was also used to investigate the effects of product accumulation and the associated product inhibition of NOX2 reaction flux (i.e. NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production) at different pH levels. As shown in Fig. 7A, C, the model simulations show that the NOX2 reaction flux was minimally affected by the concentration of the product NADP⁺ at pH of 6.3, but was strongly affected by an increase to an alkaline pH of 7.7, with an intermediate effect at neutral pH of 7.0. The corresponding effects on the individual NOX2 complex states (NOX2 enzyme fractions) are illustrated in Fig. 7B, D showing increased sensitivities to NADPH for the NOX2 complex state transitions from (E₁.X)_T state to (E₂.X)_T state at pH = 6.3 vs. that at pH = 7.7 with increased NADP⁺ concentrations. On the contrary, the generated ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ as a product did not show any effect on the NOX2 reaction flux in terms of the product inhibition at any pH (results not shown). This model observation could be due to the fact that the respective reaction of ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production (reaction from (E₃.X)_T to (E₁.X)_T) is always thermodynamically favored in the forward direction (i.e. high $K_{eq,E_3\leftrightarrow E_1}$).

Fig. 7. Model predictions demonstrating the effects of NADP⁺ on NADPH-dependent superoxide generation and the corresponding NOX2 complex states at high O₂ and different pH.

(A, B) Model simulations depict pH-dependent product NADP⁺ inhibition of the substrate NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production via the assembled and activated NOX2 complex at high saturating level of O₂ (NADPH = 0:5:250 μM; NADP⁺ = 0:10:500 μM; pH = 6.3, 7.0, 7.7; and O₂ = 250 μM). (C, D) Model simulations depict transitions of the assembled and activated NOX2 complex states, expressed as fractions of the total NOX2 complex (F₁, F₂, F₃), as functions of the substrate NADPH and product NADP⁺ variations ranging from limited to excess levels (NADPH = 0:5:250 μM; NADP⁺ = 0:10:500 μM) at high saturating level of O₂ (250 μM) for three different levels of pH (6.3, 7.0, 7.7). The model simulations were generated using a fixed NOX2 complex concentration (assembled and activated equivalent to 1.5 nM FlavoCytB) and with values of model parameters corresponding to the cell-free system, as reported in Table 1.

Regulation of NOX2 complex kinetics by temperature:

Temperature affects the NOX2 enzyme activity by slightly modulating the availability of the intermediate enzyme states through the alterations of the elementary electron transfer reactions equilibrium constants (Equation (8)). In addition, motivated by the data of a study by Morgan et al. [42], we further incorporated the effect of temperature on the NOX2 enzyme activity by multiplying the electron transfer reaction rate constants with the temperature correction coefficient (Equation (12)). The model predictions for the effects of temperature ranging 18–43 °C on the NOX2 kinetic parameters (apparent K_m of NADPH and O₂ and apparent V_max) governing the NOX2 enzyme kinetics are shown in Fig. 8. These predictions suggest that the apparent K_m of NADPH and O₂ for NOX2 increase with increasing concentrations of O₂ (Fig. 8A) and NADPH (Fig. 8C), respectively, but are independent of temperature. Similarly, the apparent K_m of NADPH and O₂ are significantly increased and decreased with increasing pH, respectively, but are independent of temperature (Fig. 8B, D). Unlike the apparent K_m of NADPH and O₂, the apparent V_max for NOX2 appears to be highly dependent on temperature with velocities increasing with increasing temperature. However, they are significantly dependent on the availability of substrates, with highly decreased enzyme velocities at high temperatures and low substrate concentrations, which increased as the substrate concentrations increased at a constant pH value of 7.4 (Fig. 8E). In contrast, when pH was altered from acidic to alkaline values, the enzyme maximum velocities were drastically decreased with increased temperature with the observation of optimal rates at pH 7 (Fig. 8F).

4. Discussion

The major ROS (${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$, H₂O₂)-producing systems within cells are cytochrome P450 oxidases, uncoupled nitric oxide synthase (NOS), mitochondrial electron transport chain (ETC), xanthine oxidase (XO), and NOX [5–7,49–51]. Production of ROS by ETC is the byproduct of oxidative phosphorylation, and is largely known to be deleterious to the cell when produced in excess [5–7]. In contrast, NOXs are the only known enzyme family in the plasma membrane with the sole function of generating ROS upon assembly and activation, and have a major role in both physiological and pathophysiological conditions [1–3,9–15]. Although several kinetic models explaining the plausible mechanisms for production and detoxification of mitochondrial ROS are well elucidated [52–55], to the best of our knowledge, no such kinetic model exists for NOXs (e.g. NOX2; the highly orchestrated, widely expressed, and most studied NOXs) that can explain the detailed kinetics of ROS production from this major non-mitochondrial source. Therefore, in this study, we developed and validated a thermodynamically-constrained mathematical model for the NOX2 complex-mediated electron transfer, ROS production, and regulations by pH, temperature, and inhibitory drugs.

An important aspect of our NOX2 kinetic model is its ability to fit well published kinetic data from multiple studies under a range of experimental conditions with appropriate model parameter values. In this regard, we note here that a model cannot fit data from various sources under different experimental conditions with suitable model parameter values if it does not incorporate appropriate kinetic mechanisms, no matter how many parameters it contains. Interestingly, the effect of pH on the NOX2 enzyme activity has a unique bimodal behavior, while the effects of the two distinct drugs GSK and DPI have very different behaviors. As shown, the model simulations are consistent with the experimentally-observed bimodal behavior of the NOX2 enzyme activity upon pH variations, and suggested alkaline pH-dependent product inhibition of the NOX2 reaction flux due to NADP⁺ accumulation at high pH. The model enables predictions that are important for quantitative understanding of the function and behavior of this key enzyme under physiological and pathophysiological conditions. The model also enables testing of the modes of actions and kinetics of different inhibitory drugs targeting the NOX2 enzyme system. Such analyses enable examination of the effects of targeting specific enzymatic sources of pathological ROS, which could overcome the limitations of pharmacological efforts aimed at scavenging of ROS, including the poor outcomes of antioxidant therapies in clinical studies [56,57].

Model captures NOX2 complex-mediated electron flow and superoxide production, which is regulated by pH:

The model exhibits remarkable ability to simulate the NADPH and O₂-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation via the assembled and activated NOX2 complex as characterized from diverse experimental data in different assay systems at varying levels of pH (Fig. 3) [37–42]. The model predicted higher V_max values with higher NOX2 stimulations, but with slightly different K_m values for NADPH for three different stimulation conditions in a study by Bauldry et al. [37] in a cell-based assay system in situ permeabilized neutrophils at pH = 7.4 (Fig. 3A). Indeed, the estimated average K_m value of ∼6 μM at pH = 7.4 for NADPH from this in situ experimental system appears to be more appropriate than those found in isolated membrane preparations (K_m value usually 10–15 fold greater; Fig. 3B), as suggested by Bauldry and colleagues [37]. Similarly, Suzuki et al. [40] used activated GRF to measure K_m of > 0.25 mM NADPH and a V_max of > 2.0 nmol NADP⁺/10⁷ PMN-eq/minute. We were able to capture these observations with a similar apparent K_m value for NADPH (0.32 mM) (Fig. 3C) indicating differences in the NOX2 enzyme kinetics in isolated membrane conditions despite the same mechanism of action. These differences in the K_m values may be attributed to different affinities of the enzyme for the substrate NADPH due to differences in the molecular interactive forces between the enzyme and the substrate in these different experimental systems (cell-free vs cell-based assay systems).

The ability of NOX2 kinetic model to capture the NOX2 enzyme kinetics under different experimental conditions was tested in other ways, specifically by varying pH and reaction substrates. For example, unlike the experiments discussed above which were performed at a saturating O₂ level, Nisimoto et al. [38] used stimulated plasma membrane-enriched fractions and cytosolic fractions from human neutrophils equilibrated with N₂/O₂ gas mixtures to explore the regulation of NOX2 enzyme activity during alterations of O₂ availability. The experimentally observed K_m of O₂ was different for different experimental systems with the K_m of PMN intact cells averaging 3.1% (22 mmHg) and that of cell-free system averaging 2.3% (16 mmHg). As illustrated in the results above, our model was able to very well capture the NOX2 enzyme kinetics under these conditions with varying concentrations of O₂ at high NADPH and at pH 7.4, and provided appropriate estimates for the apparent K_m of O₂ (59 and 43 μM, respectively).

The NOX2 complex-mediated ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production occurs through transfer of electrons from NADPH to O₂ across the plasma membrane, where these electrons alkalinize the external solution while the protons acidify it. It has been found using human eosinophils that NOX2 can translocate electrons, generating a trans-membrane current of ∼30–40 pA at 37 °C at pH ≈ 7.2, and this current is reduced at both lower and higher pH [42]. Similarly, our model predicted bimodal behavior of NOX2 enzyme activity upon pH variations. This is of considerable relevance since there are inconsistencies in studies describing the effects of pH due to differences in the techniques used to quantify these effects and the use of cell-free vs. cell-based assay systems [58,59]. It has been reported that the NOX2 enzyme activity in cell-free assay systems is inhibited at both high and low pH, and functions optimally near a neutral pH of 7.0 [39,40]. This aspect (bimodal pH effect) was modeled by hypothesizing that protons bind to the (E_n.X)_T state of the NOX2 complex such that (E_n.X.H)_T is the active state and (E_n.X)_T and (E_n.X.H₂)_T are inactive states (Fig. 2A). It was then determined if the model prediction with this pH regulation mechanism is consistent with the published kinetic data of Suzuki et al. [40] obtained from studies of isolated membranes activated by PMA with high NADPH and O₂ (Fig. 3E). In addition to capturing the results of those studies, our model of the protonation mechanism for the NOX2 complex was able to capture the experimental data from Babior et al. [39] who evaluated the variation of the apparent K_m of NADPH with respect to varying pH (Fig. 3F). The computed model results of these different conditions have nicely validated our proposed kinetic mechanism of the NOX2 complex-mediated ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ generation.

The NOX2 kinetic model as parameterized using the values corresponding to the cell-free NOX2 complex systems (shown in Table 1) was used to further understand the kinetic behavior of NOX2 under a wide range of experimental conditions. The results of these model simulations not only fit the observed bimodal behavior of the NOX2 enzyme activity with pH variations, but also predicted that the pH at which the optimal NOX2 complex activity occurs depends on the concentrations of its participating substrates (Fig. 5A, B). The model results were also consistent with the simulated fractional enzyme states which revealed that these intermediate enzyme states undergo transformation based on the availability of substrates NADPH and O₂ with the majority of NOX2 residing in a particular state depending on the condition (Fig. 5C–F). Finally, the NOX2 enzyme behavior was also characterized with respect to alterations in the apparent K_m and V_max as functions of varying pH (pH = 5–9), O₂ (0–250 μM), and NADPH (5–250 μM) in the absence of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ (Fig. 6). Together, these simulations clearly demonstrated that pH plays an important role in the NOX2 enzyme kinetics and its affinity for substrates. We conclude that the optimal activity occurs at pH 7 under saturating substrate conditions, and can shifts to the acidic pH at low substrate concentrations.

Despite others having studied the NOX2 enzyme kinetics under different experimental conditions, none of these studies have elucidated the effects of the reaction products NADP⁺ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ on the enzyme activity. We therefore used the present model to evaluate the role of these reaction products under different substrate and medium conditions, in particular pH. As shown, the model predicted that the pH-dependent product (NADP⁺) inhibition of the NADPH-dependent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production is significantly higher at alkaline pH and saturated O₂. The model predicts that optimal product inhibition occurs at a pH of 7.7, rather than 7.0 (Fig. 7A, C). The observed changes in the intermediate enzyme fractions with product therefore indicate that NADP+ alters the enzyme transition states to alter the enzyme activity under different simulation conditions (Fig. 7B, D).

Regulation of NOX2 enzyme kinetics by inhibitory drugs:

One of the important features of the NOX2 kinetic model is the incorporation of mechanisms of several inhibitors, specifically the drug molecules GSK and DPI (Fig. 4). Based on the experimental data of Hirano et al. [45] and the hypothesized inhibitory mechanisms for these drugs, the molecule GSK was shown to bind with the enzyme state (E₁.X)_T, whereas DPI was shown to bind to the enzyme state (E₂.X)_T in the NOX2 complex model to accurately describe the kinetic data. The model simulations showed decreased K_m of NADPH and V_max with respect to NADPH using DPI as an inhibitor, while GSK increased the K_m value without affecting the V_max value for NOX2 (Fig. 4G–J; Equation (10)). These results are consistent with a mechanism of uncompetitive inhibition with DPI and competitive inhibition with GSK. Interestingly, the model also predicts that the percentage inhibition of the NOX2 reaction flux via GSK is dependent upon NADPH while DPI is not (Fig. 4C, D). An earlier study by Doussiere et al. [60] suggests the heme of CytB is a potential target for the drug DPI for its inhibition of the NOX2 enzyme activity. However, this study did not include kinetic data of the type shown in Fig. 4A (i.e. NOX2 enzyme activity as a function of O₂ level for different doses of DPI) which is needed to assess the possibility that the mechanism of action of DPI is by acting on the (E₃.X)_T state (FAD-2CytB_red) of the NOX2 complex. Thus, our NOX2 kinetic model could not test this alternative action of DPI on the NOX2 enzyme kinetics.

The NOX inhibitors have established the roles of the NOX enzyme in both normal and pathological states [43–45]. Unfortunately, human antioxidant supplementation trials have not been found to be therapeutically very effective [56,57], and other approaches that can better target specific sites or pathways of ROS production (e.g. NOX) are being sought [43–45,61–63]. As emphasized in those studies, an ideal NOX inhibitor would be active and functional in both cell-free and cell-based systems, does not have an intrinsic antioxidant activity, and is NOX isoform selective with minimal toxicity. The significance of the proposed NOX2 kinetic model lies in its capacity to mechanistically and quantitatively test the modes of actions and kinetics of different pharmacological drug inhibitors that could be used to target the NOX enzymes for therapeutic purposes.

Regulation of NOX2 enzyme kinetics by temperature:

From the aforementioned discussions, it is apparent that, mostly steady-state measurements at fixed temperatures examining the activated NOX2 complex-mediated ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production (or the equivalent electron current generation across the plasma membrane) have been conducted. Only few studies have examined the effects of varying temperature on the activated NOX2 complex kinetics [42,64–68]. The NOX2 enzyme is shown to be optimally active at sub-physiological temperatures, also showing a bimodal temperature dependence with loss of function at higher temperatures and significant inhibition at lower temperatures. The temperature at which the optimal activity occurs depends on the experimental system used in the study. The results of the most recent study by Morgan et al. [42] suggested that the steady-state NOX2 enzyme activity is strongly dependent on temperature between 20 and 30 °C, with the Arrhenius activation energy E_a of 25.1 kcal mol⁻¹ (Q₁₀ = 4.2). In contrast, the NOX2 enzyme activity was found weaker in response to a rapid temperature increase to 37 °C with E_a only 14 kcal mol⁻¹ (Q₁₀ = 2.2). The developed NOX2 kinetic model was used to simulate the effects of varying temperature (T = 18–43 °C) on the NOX2 apparent K_m of NADPH and O₂ and apparent V_max with respect to NADPH and O₂ with varying levels of NADPH and O₂ (0–250 μM) and pH (5–9) with a fixed physiologically-relevant Q₁₀ value of 2.5 (Fig. 8). These simulations found that the apparent K_m of NADPH and O₂ was independent of temperature at all pH, but the apparent V_max increased with increasing temperature. The model cannot simulate the loss of NOX2 function scenario at high temperatures (above 37 °C), since it does not account for the denaturation process of the NOX2 enzyme at high temperatures.

Significance of the present study and future directions:

The main objective of the present study was to develop and parameterize a thermodynamically-constrained mechanistic kinetic model of the NOX2 enzyme system, as no such model exists in the literature. The NOX2 enzyme plays an important role in the cellular oxidative stress, one of the major factors in several human diseases. A well-characterized kinetic model of the NOX2 enzyme system will allow us to better understand the critical role of ROS in cellular oxidative stress, together with the other contributors to ROS production and detoxification. Therefore, this NOX2 kinetic model will be an important component (module) of an integrated computational model of the mitochondrial and cellular ROS handling systems (production and scavenging) that will be essential for furthering our understanding the critical roles of ROS (and NOX2) in mitochondria and cell physiology and pathophysiology.

The parameterized and validated/corroborated NOX2 kinetic model provides several important insights into the functioning (i.e. emergent properties) of the NOX2 enzyme system upon assembly and activation (Figs. 5–8), which can be experimentally tested. These include (i) the bimodal behavior of the NOX2 enzyme activity upon pH variations ranging from alkaline values to acidic values, (ii) NADPH-dependent shifting of the peak NOX2 enzyme activity upon pH variations, (iii) alkaline pH-dependent NADP⁺ product inhibition of the NOX2 enzyme activity, and (iv) temperature effects on the NOX2 enzyme activity in cell-free and cell-based assay systems. Experimental verification of these model predictions is important, and hence could be undertaken in future studies, which will provide further understanding of the NOX2 enzyme system.

The NOX2 enzyme activity is regulated in part by its multistep assembly and activation process, involving a group of plasma membrane proteins (i.e. gp91^phox and p22^phox) and several cytosolic proteins (i.e. p47^phox, p40^phox, p67^phox and Rac-GTP) [26–32] (Fig. 1A, B). However, in the current study, our primary objective was the development and validation of a mechanistic and quantitative understanding of the NOX2 complex-mediated electron transfer and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production, and the regulatory effects of pH, inhibitory drugs, and temperature under the assumption that the NOX2 enzyme is already assembled and activated. Incorporating the kinetics of NOX2 assembly and activation process into the proposed NOX2 electron flow and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production model will enhance the mechanistic and quantitative understanding of the NOX2 enzyme system, but is beyond the scope of the present study, and will be considered in future studies.

Summary:

The NOX2 kinetic model described in this manuscript is the first thermodynamically-constrained mechanistic mathematical model that incorporates diverse kinetic details on the NOX2 complex function. The model accounts for the effects of pH, inhibitory drugs, and temperature in addition to thermodynamics of electron transfer and subsequent ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production. The key model hypotheses and simulation outcomes illustrated in this manuscript include (i) competitive and uncompetitive inhibitory effects of the drugs GSK and DPI on ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production enabling detailed pharmacological profiling of these drugs on the NOX2 enzyme activity; (ii) altered NOX2 enzyme activity upon pH and temperature variations; and (iii) product inhibition of the NOX2 enzyme activity via NADP⁺ accumulation at higher pH levels without having such inhibitory effect with ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ accumulation. Among many potential applications, the model provides a mechanistic and quantitative framework to investigate the critical role of the NOX2 enzyme system in ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production regulating diverse cellular mechanisms in health and disease, such as NOX2 and mitochondrial crosstalk/interaction in ROS induced ROS release in oxidative stress and diseases.

APPENDIX

This Appendix provides the mathematical derivations of the NOX2 reaction flux expression and the regulations by pH and inhibitory drugs in the presence of the reaction products. We rationalize the assumptions and mathematical derivations through the following steps.

Overall NOX2 Electron Transfer Reaction and Thermodynamics:

The overall two-electron transfer reaction for NADPH oxidation and O₂ reduction towards NADP+ and superoxide $\left({\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }\right)$ productions, mediated by the assembled and activated NOX2 enzyme complex, is given by:

$$Nr+2O_2\underset{k_r}{\overset{k_f}{\rightleftharpoons }}No+2O_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }+H^{+}$$ (A1)

where Nr and No denote NADPH (reduced) and NADP⁺ (oxidized), respectively; k_f and k_r (both are in the units of 1/M².s) denote, respectively, the forward and reverse rate constants of the overall reaction. The equilibrium constant (K_eq, unitless) for the overall reaction is defined by:

$$K_{eq}=\frac{k_f}{k_r}=\left(\frac{K_H}{C_H}\right)\exp \left(-\frac{2F\left(E_{m,Nr/No}^0-E_{m,O_2/O_2^{-\cdot }}^0\right)}{RT}\right)$$ (A2)

where C_H = 10^−pH M is the proton concentration and K_H = 10⁻⁷ M is the proton binding constant; $E_{m,Nr/No}^0$ and $E_{m,O_2/O_2^{-\cdot }}^0$ denote, respectively, the standard midpoint redox potentials for half-reactions involving NADPH oxidation and O2 reduction at pH=7; F, R and T denote the Faradays constant (96485 J/mol/V), ideal gas constant (8.314 J/mol/K), and temperature (298.15 K), respectively.

Elementary NOX2 Electron Transfer Reactions and Thermodynamics:

The two-electron transfer from NADPH to 2O₂ occurs via different redox centers of the NOX2 enzyme complex, as schematized in Fig. 1C. The elementary electron transfer reactions based on the kinetic scheme of Fig. 1D can be written as:

$$\text{Reaction}\hspace{0.17em}\text{1:}\hspace{0.17em}Nr+Fo.\hspace{0.17em}\hspace{0.17em}2Co+H^{+}\underset{k_{1r}}{\overset{k_{1f}}{\rightleftharpoons }}No+Fr.\hspace{0.17em}2Co$$ (A3)

$$\text{Reaction}\hspace{0.17em}\text{2a:}\hspace{0.17em}Fr.\hspace{0.17em}2Co\underset{k_{2ar}}{\overset{k_{2af}}{\rightleftharpoons }}Fs.\hspace{0.17em}Co.\hspace{0.17em}Cr+H^{+}$$ (A4)

$$\text{Reaction}\hspace{0.17em}2\text{b:}\hspace{0.17em}Fs.\hspace{0.17em}Co.\hspace{0.17em}Cr\underset{k_{2br}}{\overset{k_{2bf}}{\rightleftharpoons }}Fo.\hspace{0.17em}2Cr+H^{+}$$ (A5)

$$\text{Reaction}\hspace{0.17em}\text{3a:}\hspace{0.17em}Fo.\hspace{0.17em}2Cr+O_2\underset{k_{3ar}}{\overset{k_{3af}}{\rightleftharpoons }}Fo.\hspace{0.17em}Cr.\hspace{0.17em}Co+O_2^{-\hspace{0.17em}\cdot }$$ (A6)

$$\text{Reaction}\hspace{0.17em}3\text{b:}\hspace{0.17em}Fo.\hspace{0.17em}Co.\hspace{0.17em}Cr+O_2\underset{k_{3br}}{\overset{k_{3bf}}{\rightleftharpoons }}Fo.\hspace{0.17em}2Co+O_2^{-\hspace{0.17em}\cdot }$$ (A7)

where Fr, Fs and Fo denote FADH2 (reduced), FADH· (semiradical) and FAD (oxidized), respectively; Cr and Co denote the reduced and oxidized forms of CytB, respectively; F.2C denotes the FlavoCytB (flavocytochrome b); k_nf and k_nr denote, respectively, the forward and reverse rate constants for the individual elementary electron transfer reactions; k_1f and k_1r are in the units of 1/M.s, k_2af, k_2bf, k_2ar and k_2br are in the units of 1/s, and k_3af, k_3bf, k_3ar and k_3br are in the units of 1/M.s. The equilibrium constants (unitless) for the five elementary electron transfer reactions are defined by:

$$K_{eq,1}=\frac{k_{1f}}{k_{1r}}=\left(\frac{C_H}{K_H}\right)\exp \left(-\frac{2F\left(E_{m,Nr/No}^0-E_{m,Fr/Fo}^0\right)}{RT}\right)$$ (A8)

$$K_{eq,2a}=\frac{k_{2af}}{k_{2ar}}=\left(\frac{K_H}{C_H}\right)\exp \left(-\frac{F\left(E_{m,Fr/Fs}^0-E_{m,Cr/Co}^0\right)}{RT}\right)$$ (A9)

$$K_{eq,2b}=\frac{k_{2bf}}{k_{2br}}=\left(\frac{K_H}{C_H}\right)\exp \left(-\frac{F\left(E_{m,Fs/Fo}^0-E_{m,Cr/Co}^0\right)}{RT}\right)$$ (A10)

$$K_{eq,3a}=\frac{k_{3af}}{k_{3ar}}=\exp \left(-\frac{F\left(E_{m,Cr/Co}^0-E_{m,O_2/O_2^{-\cdot }}^0\right)}{RT}\right)$$ (A11)

$$K_{eq,3b}=\frac{k_{3bf}}{k_{3br}}=\exp \left(-\frac{F\left(E_{m,Cr/Co}^0-E_{m,O_2/O_2^{-\cdot }}^0\right)}{RT}\right)$$ (A12)

where $E_{m,red/ox}^0$’s denote the standard midpoint redox potentials for half-reactions involving different reduced-oxidized pairs at pH=7. Note that $E_{m,Fr/Fo}^0=\left(E_{m,Fr/Fs}^0+E_{m,Fs/Fo}^0\right)/2$. It can be easily shown from Equation (A2) and (A8-A12) that:

$$K_{eq}=\frac{k_f}{k_r}=\frac{k_{1f}.\hspace{0.17em}{k}_{2af}.\hspace{0.17em}{k}_{2bf}.\hspace{0.17em}{k}_{3af}.\hspace{0.17em}{k}_{3bf}}{k_{1r}.\hspace{0.17em}{k}_{2ar}.\hspace{0.17em}{k}_{2br}.\hspace{0.17em}{k}_{3ar}.\hspace{0.17em}{k}_{3br}}=K_{eq,1}.\hspace{0.17em}{K}_{eq,2a}.\hspace{0.17em}{K}_{eq,2b}.\hspace{0.17em}{K}_{eq,3a}.\hspace{0.17em}{K}_{eq,3b}$$ (A13)

Lumped NOX2 Electron Transfer Reactions and Thermodynamics:

Clearly, the kinetic model of the NOX2 enzyme complex based on the five elementary electron transfer reactions (Equation (A3–A7)) would contain 10 kinetic parameters (k_1f, k_1r, k_2af, k_2ar, k_2bf, k_2br, k_3af, k_3ar, k_3bf, k_3br), subject to 5 thermodynamic constraints (Equation (A8–A12)), resulting in 5 unknown kinetic parameters for estimation. In order to reduce the complexity of the model and the number of unknown kinetic parameters for estimation, we can lump reactions 2a and 2b and reactions 3a and 3b, resulting in the following two lumped two-electron transfer reactions:

$$\text{Reaction}\hspace{0.17em}\text{2:}\hspace{0.17em}Fr.\hspace{0.17em}\hspace{0.17em}2Co\underset{k_{2r}}{\overset{k_{2f}}{\rightleftharpoons }}Fo.\hspace{0.17em}\hspace{0.17em}2Cr+2H^{+}$$ (A14)

$$\text{Reaction}\hspace{0.17em}3:\hspace{0.17em}Fo.\hspace{0.17em}\hspace{0.17em}2Cr+2O_2\underset{k_{3r}}{\overset{k_{3f}}{\rightleftharpoons }}Fo.\hspace{0.17em}\hspace{0.17em}2Co+2O_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$$ (A15)

The forward and reverse rate constants for these two lumped two-electron transfer reactions (k_2f and k_2r are in the units of 1/s and k_3f and k_3r are in the units of 1/M².s) are such that the corresponding equilibrium constants (unitless) are defined by:

$$K_{eq,2}=\frac{k_{2f}}{k_{2r}}=\frac{k_{2af}.\hspace{0.17em}\hspace{0.17em}{k}_{2bf}}{k_{2ar}.\hspace{0.17em}\hspace{0.17em}{k}_{2br}}={\left(\frac{K_H}{C_H}\right)}^2\hspace{0.17em}\exp \left(-\frac{2F\left(E_{m,Fr/Fo}^0-E_{m,Cr/Co}^0\right)}{RT}\right)$$ (A16)

$$K_{eq,3}=\frac{k_{3f}}{k_{3r}}=\frac{k_{3af}.\hspace{0.17em}\hspace{0.17em}{k}_{3bf}}{k_{3ar}.\hspace{0.17em}\hspace{0.17em}{k}_{3br}}=\exp \left(-\frac{2F\left(E_{m,Cr/Co}^0-E_{m,O_2/O_2^{-\cdot }}^0\right)}{RT}\right)$$ (A17)

The lumped electron transfer reactions 1–3 (Equation (A3, A14, A15)) constitute the kinetic scheme of Fig. 1E, or equivalently Fig. 1F, which contain 6 kinetic parameters (k_1f, k_1r, k_2f, k_2r, k_3f, k_3r), subject to 3 thermodynamic constraints (Equation (A8, A16, A17)), resulting in 3 unknown kinetic parameters for estimation. Thus, the electron transfer reactions of Equation (A3, A14, A15) (Fig. 1E or 1F) was to derive the kinetic model of the NOX2 enzyme complex.

Derivations of NOX2 Reaction Flux Expression:

Correlating Fig. 1E and F, it is easy to see that Fo. 2Co, Fr. 2Co, and Fo. 2Cr denote E₁.X, E₂.X and E₃.X, respectively. The following ordinary differential equations (ODEs), describing the rate of change of enzyme concentration in each of the three enzyme states, are derived from the reactions of Equation (A3, A14 and A15) based on the kinetic scheme of Fig. 1F:

$$\frac{d{C}_{E_1.X}}{dt}=-k_{1f}{C}_{Nr}{C}_{E_1.X}-k_{3r}{C}_{O{2}^{-\hspace{0.17em}\hspace{0.17em}\cdot }}^2{C}_{E_1.X}+k_{1r}{C}_{No}{C}_{E_2.X}+k_{3f}{C}_{O2}^2{C}_{E_{3.X}}$$ (A18)

$$\frac{d{C}_{E_2.X}}{dt}=k_{1f}{C}_{Nr}{C}_{E_1.X}-k_{1r}{C}_{No}{C}_{E_2.X}-k_{2f}{C}_{E_2.X}+k_{2r}{C}_{E_3.X}$$ (A19)

$$\frac{d{C}_{E_3.X}}{dt}=k_{3r}{C}_{O{2}^{-\hspace{0.17em}\hspace{0.17em}\cdot }}^2{C}_{E_1.X}+k_{2f}{C}_{E_2.X}-k_{2r}{C}_{E_3.X}-k_{3f}{C}_{O2}^2{C}_{E_3.X}$$ (A20)

The total enzyme concentration C_E.X based on the principle of mass conservation is given by:

$$C_{E.X}=C_{E_1.X}+C_{E_2.X}+C_{E_3.X}$$ (A21)

where $C_{E_1.X}$, $C_{E_2.X}$, and $C_{E_3.X}$ denote the concentrations of E₁. X or Fo. 2Co, E₂. X or Fr. 2Co, and E₃. X or Fo. 2Cr, respectively, which are the three different enzyme states representing different redox status of the NOX2 enzyme complex; C_Nr and C_No are the concentrations of Nr and No, respectively; C_O2 and $C_{O{2}^{-·}}$ are the concentrations of O₂ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$, respectively; k_nf and k_nr (n = 1, 2, 3) denote the forward and reverse rate constants for the three elementary rate limiting electron transfer reactions (Equation (A3, A14, A15)).

At steady state, the rate of change of relative portion of the NOX2 enzyme complex at each state is zero. Thus, the flux of the overall NOX2 reaction (Equation (A1)) can be calculated from any of the three elementary rate limiting electron transfer reactions. Now, Equation (A18–A21) at steady state can be rewritten in the following normalized form as:

$$-\left(k_{1f}{C}_{Nr}+k_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\right)\frac{C_{E_1.X}}{C_{E.X}}+k_{1r}{C}_{No}\frac{C_{E_2.X}}{C_{E.X}}+k_{3f}{C}_{O2}^2\frac{C_{E_2.X}}{C_{E.X}}=0$$ (A22)

$$k_{1f}{C}_{Nr}\frac{C_{E_1.X}}{C_{E.X}}-\left(k_{1r}{C}_{No}+k_{2f}\right)\frac{C_{E_2.X}}{C_{E.X}}+k_{2r}\frac{C_{E_3.X}}{C_{E.X}}=0$$ (A23)

$$k_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\frac{C_{E_1.X}}{C_{E.X}}+k_{2f}\frac{C_{E_2.X}}{C_{E.X}}-\left(k_{2r}+k_{3f}{C}_{O2}^2\right)\frac{C_{E_3.X}}{C_{E.X}}=0$$ (A24)

$$\frac{C_{E_1.X}}{C_{E.X}}+\frac{C_{E_2.X}}{C_{E.X}}+\frac{C_{E_3.X}}{C_{E.X}}=1$$ (A25)

In matrix notation, Equation (A22–A25) can be rewritten as:

$$\left[\begin{array}{ccc}-k_{1f}{C}_{Nr}-k_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2& K_{1r}{C}_{No}& k_{3f}{C}_{O2}^2\\ k_{1f}{C}_{Nr}& -k_{1r}{C}_{No}-k_{2f}& k_{2r}\\ k_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2& k_{2f}& -k_{2r}-k_{3f}{C}_{O2}^2\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}\frac{C_{E_1.X}}{C_{E.X}}\\ \frac{C_{E_2.X}}{C_{E.X}}\\ \frac{C_{E_3.X}}{C_{E.X}}\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\end{array}\right]$$ (A26)

Based on the solution of the above system of linear equations (Equation (A26)), the relative portion of the NOX2 enzyme complex concentration in each state is:

$$F_1=\frac{C_{E_1.X}}{C_{E.X}}=\left(k_{1r}{k}_{2r}{C}_{No}+k_{2f}{k}_{3f}{C}_{O2}^2+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}^2\right)/Den$$ (A27)

$$F_2=\frac{C_{E_2.X}}{C_{E.X}}=\left(k_{1f}{k}_{2r}{C}_{Nr}+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}^2\right)/Den$$ (A28)

$$F_3=\frac{C_{E_3.X}}{C_{E.X}}=\left(k_{1f}{k}_{2f}{C}_{Nr}+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\right)/Den$$ (A29)

where the common denominator (Den) in Equation (A27–A29) is given by:

$$Den=k_{1f}{k}_{2f}{C}_{Nr}+k_{1f}{k}_{2r}{C}_{Nr}+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}^2+k_{1r}{k}_{2r}{C}_{No}+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}^2+k_{2f}{k}_{3f}{C}_{O2}^2+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2$$ (A30)

The elementary rate limiting electron transfer reaction fluxes can be written as the flux of forward reaction – flux of reverse reaction, based on reactions in Equation (A3, A14 and A15):

$$J_1=k_{1f}{C}_{Nr}{C}_{E_1.X}-k_{1r}{C}_{No}{C}_{E_2.X}$$ (A31)

$$J_2=k_{2f}{C}_{E_2.X}-k_{2r}{C}_{E_3.X}$$ (A32)

$$J_3=k_{3f}{C}_{O2}^2{C}_{E_3.X}-k_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2{C}_{E_1.X}$$ (A33)

The overall NOX2 reaction flux can then be obtained from any of the rate limiting reaction fluxes as (J₁ = J₂ = J₃ = J_NOX2):

$$J_{NOX2}=\frac{C_{E.X}\left(k_{1f}{k}_{2f}{k}_{3f}{C}_{Nr}{C}_{O2}^2-k_{1r}{k}_{2r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\right)}{\left(\begin{array}{c}{k}_{1f}{k}_{2f}{C}_{Nr}+k_{1f}{k}_{2r}{C}_{Nr}+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}^2+k_{1r}{k}_{2r}{C}_{No}+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}^2\\ +k_{2f}{k}_{3f}{C}_{O2}^2+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\end{array}\right)}$$ (A34)

where C_E.X = C_FlavoCytB represents the concentration of the total active NOX2 enzyme complex.

Simplifications for Sequential O₂ Reduction by CytB:

During O₂ reduction and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ production, two electrons are transferred from two reduced CytB to two molecules of O₂ in two sequential steps without cooperativity. However, our derivation above is based on the assumption that two electrons are transferred from two reduced CytB to two molecules of O₂ in a single step with full cooperativity. To account for this discrepancy, we assume the following O₂ and ${\text{O}}_2^{-\hspace{0.17em}\hspace{0.17em}\cdot }$ dependencies of the forward and reverse rate constants k_3f and k_3r:

$$k_{3f}={k^{\prime }}_{3f}/C_{O2}\hspace{1em}\text{and}\hspace{1em}{k}_{3r}={k^{\prime }}_{3r}/C_{O{2}^{-\hspace{0.17em}\cdot }}$$ (A35)

Substituting Equation (A35) in Equation (A34) and dropping out the prime notation, the NOX2 reaction flux can then be expressed as:

$$J_{NOX2}=\frac{C_{E.X}\left(k_{1f}{k}_{2f}{k}_{3f}{C}_{Nr}{C}_{O2}-k_{1r}{k}_{2r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}^2\right)}{\left(\begin{array}{c}{k}_{1f}{k}_{2f}{C}_{Nr}+k_{1f}{k}_{2r}{C}_{Nr}+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}+k_{1r}{k}_{2r}{C}_{No}+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}\\ +k_{2f}{k}_{3f}{C}_{O2}+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\end{array}\right)}$$ (A36)

The corresponding NOX2 enzyme complex states (Equation A27–A29) can be expressed as:

$$F_1=\frac{C_{E_1.X}}{C_{E.X}}=\left(k_{1r}{k}_{2r}{C}_{No}+k_{2f}{k}_{3f}{C}_{O2}+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}\right)/Den$$ (A37)

$$F_2=\frac{C_{E_2.X}}{C_{E.X}}=\left(k_{1f}{k}_{2r}{C}_{Nr}+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}\right)/Den$$ (A38)

$$F_3=\frac{C_{E_3.X}}{C_{E.X}}=\left(k_{1f}{k}_{2f}{C}_{Nr}+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\right)/Den$$ (A39)

where the common denominator (Den) in Equation (A37–A39) is given by:

$$Den=k_{1f}{k}_{2f}{C}_{Nr}+k_{1f}{k}_{2r}{C}_{Nr}+k_{1f}{k}_{3f}{C}_{Nr}{C}_{O2}+k_{1r}{k}_{2r}{C}_{No}+k_{1r}{k}_{3f}{C}_{No}{C}_{O2}+k_{2f}{k}_{3f}{C}_{O2}+k_{2r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{2f}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+k_{1r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}$$ (A40)

Regulations by pH and Inhibitory Drugs:

Considering the postulated proton and drug bindings to the three different NOX2 enzyme complex states E₁.X, E₂.X and E₃.X, the forward and reverse reaction rate constants k_nf and k_nr can be scaled appropriately to account for the effects of the proton and drug binding factors f_nf and f_nr, as schematized in Fig. 2A, B and 2C, and as detailed in the main texts:

$$k_{nf}={k^{\prime }}_{nf}{f}_{nf}\hspace{1em}\text{and}\hspace{1em}{k}_{nr}={k^{\prime }}_{nr}{f}_{nr}$$ (A41)

Substituting Equation (A41) into Equation (A36) and dropping out the prime notation, the NOX2 reaction flux can then be expressed as:

$$J_{NOX2}=\frac{C_{{\left(E.X\right)}_T}\left(f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}-f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\right)}{\left(\begin{array}{c}{f}_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}+\\ f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2}+\\ f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\end{array}\right)}$$ (A42)

The corresponding NOX2 enzyme complex states (Equation (A37–A39)) can be expressed as:

$$F_1=\frac{C_{{\left(E_1.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\left(f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}\right)/Den$$ (A43)

$$F_2=\frac{C_{{\left(E_2.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\left(f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+f_{1f}{k}_{1f}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2}\right)/Den$$ (A44)

$$F_3=\frac{C_{{\left(E_3.X\right)}_T}}{C_{{\left(E.X\right)}_T}}=\left(f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }}\right)/Den$$ (A45)

where the common denominator (Den) in Equation (A43–A45) is given by:

$$\begin{gathered} Den=f_{1f}{k}_{1f}{f}_{2f}{k}_{2f}{C}_{Nr}+f_{1f}{k}_{1f}{f}_{2r}{k}_{2r}{C}_{Nr}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{Nr}{C}_{O2} \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+f_{1r}{k}_{1r}{f}_{2r}{k}_{2r}{C}_{No}+f_{1r}{k}_{1r}{f}_{3f}{k}_{3f}{C}_{No}{C}_{O2}+f_{2f}{k}_{2f}{f}_{3f}{k}_{3f}{C}_{O2} \\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+f_{2r}{k}_{2r}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+f_{2f}{k}_{2f}{f}_{3r}{k}_{3r}{C}_{O{2}^{-\hspace{0.17em}\cdot }}+f_{1r}{k}_{1r}{f}_{3r}{k}_{3r}{C}_{No}{C}_{O{2}^{-\hspace{0.17em}\cdot }} \end{gathered}$$ (A46)