The rate of cellular hydrogen peroxide removal shows dependency on GSH: Mathematical insight into in vivo H2O2 and GPx concentrations

Although its concentration is generally not known, glutathione peroxidase-1 (GPx-1) is a key enzyme in the removal of hydrogen peroxide (H2O2) in biological systems. Extrapolating from kinetic results obtained in vitro using dilute, homogenous buffered solutions, it is generally accepted that the rate of elimination of H2O2 in vivo by GPx is independent of glutathione concentration (GSH). To examine this doctrine, a mathematical analysis of a kinetic model for the removal of H2O2 by GPx was undertaken to determine how the reaction species (H2O2, GSH, and GPx-1) influence the rate of removal of H2O2. Using both the traditional kinetic rate law approximation (classical model) and the generalized kinetic expression, the results show that the rate of removal of H2O2 increases with initial GPxr, as expected, but is a function of both GPxr and GSH when the initial GPxr is less than H2O2. This simulation is supported by the biological observations of Li et al.. Using genetically altered human glioma cells in in vitro cell culture and in an in vivo tumour model, they inferred that the rate of removal of H2O2 was a direct function of GPx activity × GSH (effective GPx activity). The predicted cellular average GPxr and H2O2 for their study are approximately GPxr ≤ 1 μm and H2O2 ≈ 5 μm based on available rate constants and an estimation of GSH. It was also found that results from the accepted kinetic rate law approximation significantly deviated from those obtained from the more generalized model in many cases that may be of physiological importance.

( Figure 1). There are several families of enzymes that remove H 2 O 2 . This network has at least three nodes for peroxide-removal: i. Catalase is the longest known enzyme for removal of H 2 O 2 ; it requires no cofactors in its catalytic mode [6]; ii. the six members of the peroxiredoxin family of enzymes remove H 2 O 2 by reducing it to water and are in general recycled by gathering reducing equivalents from thioredoxin [7,8]; and iii. the glutathione peroxidases rely on glutathione (GSH) for the necessary reducing equivalents.
This study focused only on the effects of GPx and GSH levels on H 2 O 2 removal, assuming the catalase and peroxiredoxin levels were unchanged.

GPx and GSH in removal of H 2 O 2
In 1957 the family of glutathione peroxidases (GPx) was discovered [9]. Currently, at least four members of this family of enzymes are known [10Á12]. They all reduce H 2 O 2 to water (organic hydroperoxides are reduced to water and the corresponding alcohol) with the electrons coming from GSH, a necessary and specific cofactor. The kinetic behaviour of GPx-1 in dilute aqueous solution is best explained by a sequence of simple bimolecular reactions [13Á15]: [GS-GPx]'GSH À! k 3 GPx r 'GSSG'H ' yielding the overall reaction, For bovine GPx-1, the kinetics of this reaction have been well studied and are considered to be a 'pingpong' mechanism with indefinite Michaelis constants, indefinite maximum velocities and no significant product inhibition [10,16Á22]. For this system the effective rate constants are given in Table I. The observations in dilute, buffered solutions lead to the paradigm that in most circumstances, the rate of peroxide removal in vivo is essentially independent of the concentration of GSH [16,18,23]. This assumes low levels of H 2 O 2 (i.e. H 2 O 2 BGPx r BGSH) and, thus, the rate of recycling of GPx r by GSH (equations 2 and 3) is rapid compared to the rate of the reaction of GPx r with H 2 O 2 . Thus, GPx would predominantly exist in its reduced form, which is highly reactive with hydroperoxides (equation 1).  However, recent observations by Li et al. [24] in a cell culture model are not in agreement with the above paradigm. When human cytosolic GPx-1 cDNA was transfected into a set of MnSOD-overexpressing U118 cells (a glioma cell line), they observed that: a. The GSSG content of these cells had a linear direct relation to the product of (GPx activity)) GSH, referred to as effective GPx activity. This is consistent with a higher rate of removal of H 2 O 2 leading to an increase in GSSG; b. Intracellular ROS (oxidation within the cell), as measured by the change in fluorescence of intracellular dichlorofluorescin, had a linear inverse relationship to effective GPx activity. This is consistent with a higher steady-state level of H 2 O 2 ( Figure 2); c. The cell population doubling time had a linear inverse relationship to effective GPx activity, i.e. the greater the effective GPx activity, the faster the cells grew. This observation is coupled to the assumption that a higher effective GPx activity will lower the steady-state level of H 2 O 2 and lead to a more reduced cellular redox environment and increased rate of growth [25]; and d. Most striking is that when the tumourigenicity of this set of cells with varying GPx activity was tested in nude mice, the growth rate of the tumours had a direct, linear relationship to effective GPx activity [24] (Figure 2). This is consistent with the in vitro observations, (aÁc) above, and points to a fundamental role of H 2 O 2 in setting the biological status of cells and tissues [5,25].

GSSG
In the above study of Li et al. [24], over-expression of MnSOD and genetic modifications with respect to GPx-1 resulted in higher fluxes of H 2 O 2 and various levels of GPx-1 in the cells. Because of the linear relationships with respect to [GPx][GSH] seen in Figure 2, these modifications appear not to have caused any significant changes in catalase or peroxiredoxin. Thus, the work of Li et al. serves as a reference for our modelling efforts to understand the GPx1-GSH-H 2 O 2 system.

Objective
The objective of this work is to examine the rate of removal of H 2 O 2 with respect to the kinetic rate behaviour of GPx-1 and GSH. Justification of the kinetic model is possible by using the in vivo observations of Li et al. [24] to: (1) determine when the rate-results from the kinetic models are consistent with the observed effective GPx activity dependency; and (2) estimate the probable range of average cellular GPx and H 2 O 2 in the cell lines investigated. To do this, we employed both the generalized and the classical approaches to express the kinetic rate behaviour involved in the GPx1-GSH-H 2 O 2 -system (equations 1Á3) and extract concentration dependency from the overall system time constant, t (also termed turnover time or biological 'average life' [26]). Finally, the variation of the classical model results from those of the general model was examined within this framework.

Generalized mathematical description of the removal of H 2 O 2 by GPx
Often in determining the rate of removal of hydrogen peroxide, the concentration of GSH is assumed to be constant [27]. Invoking this approximation and assuming spatial independence, the transient behaviour of species described by equations (1Á3) are a set of non-linear ordinary differential equations (ODEs) that describe the rates of change in the concentration of each species, equations (5Á10). Here C i represents the concentration of species i.  . Effective GPx activity is 'GPx-activity' (or GPx) as measured by standard activity assay [44] multiplied by the concentration of GSH. The units are somewhat arbitrary (AU); using typical expressions of the activity of GPx (mU/mg protein) and for GSH levels (nmol/mg protein) units for effective GPx activity would be mU?nmol (mg protein) (2 . Figure adapted from [55]. From a mathematical viewpoint, the experimental observations of Li et al. [24] can now be compared to the concentration dependency of the rate of removal of H 2 O 2 for initial masses of H 2 O 2 , GPx and GSH introduced to the system (termed impulse response). These masses are described as equivalent initial concentrations. Since effective GPx activity proposed by Li et al. is the GPx activity coupled with GSH, we represent this as the product of initial GPx r and GSH concentrations, [GPx r ] 0 )[GSH] 0 . This approximation is used to represent effective GPx activity for the purpose of investigating our kinetic rate models.

Classical approximation of the rate of removal of H 2 O 2 by GPx
Because of the inherent non-linearity of the generalized expressions for the rate of removal of H 2 O 2 , a traditional kinetic rate law approximation (the classical model) is typically used. The classical model, in fact, is derived from the generalized rate expressions. Using a steady-state approximation, assuming that the enzyme concentration is lower than the substrate concentration, the rate of change of all substrateenzyme intermediates are negligible, the relationship between the initial rate, n 0 , total enzyme concentration, e, and initial substrate concentrations, S i , for an enzymatic reaction with two substrates is approximated as [28]: where F i 's are functions of reaction rate constants, k i 's. This approximation can be obtained from the general model (equations 5Á10) by invoking several approximations for the kinetic rate model for the GPx1-GSH-H 2 O 2 system. Starting with equations (5Á10), by assuming constant concentrations of intermediates (equations 7 and 9, set to zero) and manipulating equation (6), one can obtain the classical rate expression for removal of H 2 O 2 , [16,29]: where, And This classical expression results in a rate that is constant and depends only on the initial concentrations.
In this study, both the generalized and classical models are used to evaluate the rate of H 2 O 2 removal. A comparison of relevant similarities and differences are provided.

Parameters: Initial concentrations and reaction rate constants
In developing the model, we first need a range of concentrations that bracket expected physiological values. Using the data of Li et al. [24], we estimate the range of GSH in the five cell lines ( Figure 2) to be 0.12Á0.44 mM. Thus, we used the initial concentrations of 0.1Á0.6 mM for GSH (Table II) Most GPx is determined to be in its reduced form (!99%) from both in vivo studies [18] and mathematical simulations [27]. Therefore, we assumed all GPx in our model to be initially in the reduced form, GPx r . Estimated cellular concentrations of GPx vary from 0.2 mM in red blood cells [18] to values of 2.5 mM and 6.7 mM derived from mathematical models [27,30]. Rat liver cytosolic GPx-1 has been estimated to be 5.8 mM from Se of 0.46 ppm [31]; total GPx (monomer) in mitochondria and in the luminal space of endoplasmic reticulum is estimated to be 10 mM and 0.32 mM, respectively [32]. These values may be an over-estimate as we now know Table II. Initial concentrations used for the GPx model.
Rate constants for equations (1Á3) have been determined in dilute buffer solutions [16,18,23]. These rate constants vary depending on conditions such as the buffer-salt and pH of the solution. Rate constants used (Table I) represent estimates of the effective intracellular rate constants for the three principal steps of the GPx catalytic cycle [30].

Time constant for the removal of H 2 O 2
In order to search for ranges of possible physiological GPx r and H 2 O 2 for cell lines under conditions used by Li et al. [24], time-dependent numerical solutions given by our model of the GPx1-GSH-H 2 O 2 system are correlated to the observations of Li et al. As shown in Figure 2, the data of Li et al. present a linear relation between the effective GPx activity and the relative cellular H 2 O 2 . This biological observation can be compared to the concentration dependency of the rate of removal of H 2 O 2 . The dependency is generally reflected in an analytical solution for the overall system time constant, t (turnover time), provided that the model is linear. The overall rate by which the system evolves is dominated by this approximated time constant in the system. Thus, the functional dependency of t will allow us to understand the kinetic behaviour of the GPx1-GSH-H 2 O 2 system.
However, because of the non-linearity of the rate equations associated with the removal of H 2 O 2 (due to the coupling of time-dependent concentrations of species in the terms on the right-hand side of each expression (equations 5Á10), a closed-form solution does not exist. For non-linear systems, t can be approximated.

Relating overall system time constant to effective GPx activity
To meet our objectives, we have determined the dependency of effective GPx activity on t for the chosen range of initial GSH, GPx r and H 2 O 2 concentrations. Specifically, this is when t is inversely proportional to effective GPx activity, consistent with the observations of Li et al. [24], Then, comparing these values to acceptable physiological conditions for the genetically-modified cells used by Li et al. [24], we will pose possible ranges of average cellular GPx and H 2 O 2 . The initial conditions for variables held constant are shown in Table II given by the classical approach is independent of time, t can be directly calculated by integrating equation (12).

Numerical methods
All equation-sets were solved with initial concentrations and rate constants, listed in Tables I and II. Species rate expressions, shown in equation (5Á10), are therefore numerically integrated by using the IMSL (International Mathematical and Statistical Library) DIVPAG (double-precision initial value problem solver using either Adam-Moulton's or Gear's method) coded using Fortran [40Á42].

Results and discussion
Mathematical ranges of concentrations demonstrating effective GPx activity dependency In Figure 3    Based on our generalized mathematical model, there exist sets of initial GPx r and GSH concentrations within all ranges studied where t is generally inversely proportional to [GPx r ] 0 [GSH] 0 for the removal of H 2 O 2 , agreeing with the findings of Li et al. [24] shown in Figure 2 and the relationship expressed in Equation (16). This linear relationship between t and [GPx r ] 0 [GSH] 0 is clearly visible for the following cases:   [24]. Reported levels of GSH and activities of GPx of other cells are compared with those of the U118 cells.
Typical levels of GSH in cells range from 1Á10 mm [25]. From the data of Li et al. [24] on the level of GSH in U118 cells and a cellular volume of 2.4 pL (F.Q. Schafer, unpublished), we estimated the range of GSH in the five cell lines of Figure 2 to be 0.12Á 0.44 mm. This is 10-times smaller than concentrations typically observed in proliferating cells.
The measured activity of GPx in the set of cells studied ranged from 15Á65 mU/mg protein (using the assay and unit definition of [43]).
[GPx] is considered to be at lower levels in tumour cells and cancer [6,44Á48]. These values are comparable to the range of values published for other cancer cell lines, e.g. PC-3 cells, 18 mU/mg protein [49]; MCF-7, 38 mU/mg protein; MDA-MB231, 98 mU/mg protein; and MCF-10A, 218 mU/mg protein [50]. These comparisons point to the low levels of GSH in U-118 cells as being a contributor to Li et al.'s [24] observation that peroxide levels and tumour growth are a function of (GPx activity))[GSH].
The time constant results provided by the general model indicate that if the possible intracellular concentration of H 2 O 2 is in the range of 5Á50 mM, then the physiological concentration of GPx is likely to be between 0.1Á10 mM. However, as mentioned above, the upper limit for intracellular [H 2 O 2 ] in normal cells is proposed to be Â700 nm [37,38]. However, the genetically-modified glioma cells used by Li et al. [24] over-expressed MnSOD by as much as 5-fold. This increase in MnSOD will likely increase the steady-state concentration of H 2 O 2 [1] It should be noted that actual concentrations may vary from those proposed by our model. This is because the modelling results are a consequence of the selected reaction rate constants and initial concentrations used in Equations (1Á3).
Finally, it is important to recognize that, in our modelling of the removal of H 2 O 2 by the GPx-GSH-H 2 O 2 system, spatially dependent concentrations were not considered and cellular averages were used. However, gradients in the intracellular concentrations clearly exist [6,37,51] and can result in local dominance of the rate of removal of H 2 O 2 that can alter our predicted cellular average concentrations.
Deviations of the classical model from the general model results  ] profiles for both the general (solid lines) and classical (dotted lines) models are presented on a semi-log plot (Figure 4(a)). The [H 2 O 2 ] from the classical model is calculated by integrating the rate expression shown in Equation (12). The time taken for 63% decay (which is Â t) in both models agrees relatively well for the three cases where [GPx r ] 0 is 0.1, 0.5 and 1 mM (as also shown in Figure 3(d)). For example, in the case where [GPx r ] 0 is 1 mM, although t's given for both models are close, the times predicted for 10% decay by the two models are more than an order of magnitude different. The rates of removal of H 2 O 2 at 1 ms given by the two models, as shown in Figure 4 (2) and (3), are much slower compared to the H 2 O 2eliminating step. The reaction rate constant for Equation (2) is three orders of magnitude smaller than the rate constant for equation (1); the rate constant for Equation (3) is very near that of equation (1). Thus, Equation (2) would be a rate-limiting reaction in the recycling of GPx r . In cases with lower [GPx r ] and [GSH], the slow recycling effect becomes more significant at earlier times during the process.
Nevertheless, these discrepancies are based on the set of initial concentrations used, as illustrated in Figure 4 Since the classical rate expression is derived by invoking the steady-state approximation on GPx o and GS-GPx, the rate given by the classical model should be in agreement with this steady-state rate given by the general model, as seen in Figure 4(b).
Finally, modelling the removal of H 2 O 2 by the GPx-GSH-H 2 O 2 system is a multi-scale problem and is spatially dependent. The time scale for removal of H 2 O 2 is on the order of milliseconds [27,52] whereas cell growth is on the order of days. Therefore, small differences in modelling solutions could significantly impact long-term predicted behaviour. For this reason, the classical approach to expressing the rate of enzymatic reactions should be used with caution, especially when addressing more complex systems.

Conclusions
With the use of kinetic modelling, we have investigated the removal of H 2 O 2 by GPx. Our goal was to examine the concentration dependency of intracellular H 2 O 2 removal to understand the anomalies in the findings of Li et al. [24]. They observed that biochemical parameters related to the removal of H 2 O 2 in genetically-modified U118-9 cells were a function of effective GPx-activity; most striking was their observation that the rate of tumour growth in an animal model was directly related to effective GPx activity. Using mathematical modelling, with sets of reaction rate constants and initial species concentrations taken from the literature, we found that: . . but, while offering useful simplicity, under certain conditions, the classical approach can result in substantial differences from the more general form over long time periods.
In the future, to further examine this system, the current lumped parameter mathematical model should be refined to include spatial dependency and H 2 O 2 generation. Issues of transport properties, such as species diffusivities and membrane permeability, and reaction rate constants, perhaps due to the crowded environment [53,54], need to be investigated. A direct coupling of cell growth to H 2 O 2 residence time is required to connect mathematical simulation to biological observations.
Mathematical modelling made it possible to quantitatively study the time constants (turnover time) associated with the removal of H 2 O 2 by GPx, providing insight into a biological observation that could not be approached experimentally. Finally, modelling demonstrates that the paradigm established from the kinetic-observations in dilute aqueous buffer do not always hold in the complex milieu of the cell.