The Synergetic Coupling among the Cellular Antioxidants Glutathione Peroxidase/Peroxiredoxin and Other Antioxidants and its Effect on the Concentration of H2O2

Glutathione peroxidase (GPx), peroxiredoxin (Prx), and catalase are the major antioxidants at the cellular level and protect cell compartments against hydrogen peroxide (H2O2). In addition, they affect cellular processes such as cell signaling by modulating H2O2. In this paper we demonstrate that there is a synergetic coupling between GPxs, Prxs themselves and also with other antioxidants when the GPxs and Prxs are not in their saturated reduced form. This is due to a change in the activity of glutathione peroxidases and peroxiredoxins as a result of a change in the concentrations of other antioxidants. The strength of this synergy depends on the reaction rates and the concentration of these antioxidants. We use a perturbative method to calculate the concentration of H2O2 as function of the production rate of H2O2 and the concentration of various antioxidants. This derivation shows clearly why antioxidants behave in a correlated manner and why any change in the activity of one of them translates to a change in the activity of other antioxidants. Our results show that an increase in the activity of GPxs or Prxs might not be due to a genetic switch but due to an increase in the activity of other antioxidants.

The Glutathione peroxidase family is one of the major antioxidants with eight known members 11 . Different cell types might only have a few of these members and GPxs are not uniformly distributed throughout the cells and may be localized in some specific cell compartments 11 . For instance GPx-1, which is the most abundant one in this family, is found mostly in the cytosol and in the matrix part of the mitochondria 11,13 , GPx-2 and GPx-3 are respectively observed in some types of endothelial cells and in the extracellular matrix 11,14 , GPx-4 mostly protects the biomembrane and the liposomes 11,15,16 and, has mitochondrial, cytosolic and nucleic forms, and finally GPx-5 and GPx-6 are respectively expressed in the epididymis and the olfactory epithelium 11 . GPxs exist in reduced and oxidized forms and the oxidized form needs an intermediate reductase such as glutathione to revert to the reduced form 11 . Therefore the catalytic kinetics of this family is different from catalase.
Peroxiredoxins are among the most abundant proteins in cells with six members 8,17,18 . Prxs are mostly located in the cytosol, however they can be found in other cell compartments such as the peroxisomes and mitochondria. Similar to GPxs, Prxs exist in both reduced and oxidized forms, however, they mostly use thioredoxin (Trx) or other disulfide oxidoreductases to revert back to their reduced forms. Prxs have three distinct types of which, for the typical 2-Cys and atypical 2-Cys, Prx is not conserved; however for the 1-Cys type, Prx is conserved 8 .
Catalase, is the other cellular antioxidant which plays a key role in the detoxification of H 2 O 2 19 . Catalase is predominantly located in the peroxisomes and, as opposed to Prxs and GPxs, does not require any reductase to continue its detoxification process 5,14 .
These antioxidants use their own channels to contribute to the cellular defense system against H 2 O 2 , and it appears that they behave independently. The only common factor among them is the sources of reductase which are predominantly GSH and Trx. In this work, we consider the major antioxidant system of cells and show that, in fact, there is synergy among all the members of GPx and Prx families and also between GPxs and Prxs members and other antioxidants. Furthermore, the activities of GPxs and Prxs are modulated by any other antioxidants in the cells. This phenomenon is a property of the interactions of the antioxidants (which have an oxidized and reduced form) with a reductase which converts their oxidized form to the reduced form. For instance the Prx and GPx families function under these conditions. We apply a perturbative method to the synergy of this antioxidant system and derive an analytical solution for the concentration of H 2 O 2 and the reduced forms of Gpxs and Prxs. These solutions provide an easy way to calculate any effect in the change of antioxidants into the concentrations of H 2 O 2 and antioxidants and also directly show the existence of the synergy among GPxs and Prxs and other antioxidants.

Results
In the model section we obtained the expression for the concentration of reduced form of GPxs and Prxs. Interestingly the concentration of reduced forms of GPxs and Prxs depend on the concentration of other antioxidants and by increasing the concentration of one of the other antioxidants the concentrations of the reduced forms of GPxs and Prxs increase. GPxs and Prxs activity depends on concentration of the reduced form of GPxs and Prxs, hence by increasing the concentration of other antioxidants, GPxs and Prxs activity increases. This increase in the GPxs and Prxs activity means that there is a synergetic coupling among GPxs, Prxs and other antioxidants. The derived formula provides a unique way for predicting the concentration of H 2 O 2 and the activity of GPxs and Prxs and how a change in any of the antioxidants affects the overall rearrangement of antioxidant activity.
In this section we use numerical methods to exactly solve the kinetic equations which govern the detoxification of H 2 O 2 in order to investigate the existence of coupling between antioxidants and how this could modulate GPxs and Prxs activities. Since discussing all of the antioxidants together only obfuscates the problem we focus on special cases in which the activity of one antioxidant varies with a change of concentration in another, but hold the concentrations of all other antioxidants constant. Two of the interesting cases we discuss in this paper are the coupling between GPx1, Prx3 and Prx2. These antioxidants are three major antioxidants in the mitochondria and their activity plays a key role in some signaling processes, and also in the detoxification of H 2 O 2 and protection of mDNA.
To understand if there is any correlation between these antioxidants we investigate the change in the concentration of the reduced form of Prx3, as a function of GPx1. To be crystal clear, in our paper, the activity of each enzyme is associated with the concentration of the reduced form of that enzyme. In Fig. 1a we plot the concentration of the reduced form of Prx3 as a functions of Prx3 and GPx1. Interestingly, we observe that by changing the concentration of GPx1, Prx3 activity changes. Hence changing Gpx1 concentration not only changes the concentration of H 2 O 2 but also changes Prx3 activity. This can result in a higher decrease in the concentration levels of H 2 O 2 by Prx3. Therefore there is a synergetic coupling between GPx1 and Prx3 which amplify the Prx3 activity. In Fig. 1b we plot Prx2 activity as a function of the concentration of Prx2 and GPx1. Again, we observe the same phenomenon as for Prx3, namely that by increasing the concentration of GPx1, Prx2 activity increases and we conclude that there is a synergetic coupling between Prx2 and GPx1.
To find out in which regions the effect of this synergetic coupling to GPx1 is more significant than the effects of Prx3 or Prx2 by themselves, we evaluate the rate of change in Prx3 and Prx2 activity as functions of the concentration of GPx1 and Prx3 and Prx2 respectively while keeping the concentration of Prx2 and Prx3 respectively constant. Mathematically for Prx3/GPx1 coupling, we calculate Scientific RepoRts | 5:13620 | DOi: 10.1038/srep13620 while keeping the concentration of Prx2 constant. If the rate of change of Prx3 activity is dependent more on the concentration of GPx1, then the vectors will be more tilted towards the GPx1 axis. However, if the rate of change in Prx3 activity is more dependent on Prx3 then we would expect to observe vectors more tilted towards the Prx3 axis. In Fig. 2a we plot the rate of change in Prx3 as a function of the concentration of GPx1 and Prx3. We observe that at low concentrations of Prx3, the vectors have a big vertical component and by increasing the concentration of Prx3 they started tilting towards the vertical axis. The reason for this behavior is the competition between the increases in the activity of Prx3 by synergy verses the increase in the activity of Prx3 because of the increase in the concentration of Prx3. At low concentration of Prx3, Prx3 is dominantly in the oxidized form. Hence adding Prx3 to the system directly increases the concentration of reduced form of Prx3. However, adding GPx1 only converts a small fraction of Prx3 from oxidized form to the reduced form via synergy. In opposite, for high concentration of Prx3, the amount of oxidized form of Prx3 is already high and adding Prx3 makes a small change in the total concentration of reduced form of Prx3. In this case adding GPx1 could convert the same amount of oxidized Prx3 to the reduced form and compete with Prx3. That is why we observe that at high concentration of Prx3 the vectors are more tilted towards the Gpx1 axis and the effect of synergy becomes more predominant. In Fig. 2b we plot the rate of change in the Prx2 activity as a function of the concentrations of GPx1 and Prx2 for a given concentration of Prx3. Similar to the case of GPx1/Prx3, we observe that the synergetic coupling become important at high concentration of Prx2 and as the concentration of GPx1 increases this becomes smaller. Interestingly, in both Prx3 and Prx2 we observe that the rate of change of Prx3 or Prx2 activity with respect to the concentration of Prx3 or Prx2 shows a minimum. This minimum corresponds to the transition between an increase in Prx3 or Prx2 activity as a result of increase in the concentration of Prx3 or Prx2, respectively, or synergy. In  To clarify the discussion in the previous paragraph we plot the transition from the synergy dominate region to the normal region (discussed in the above paragraph) as a function of Trx in Fig. 3. This plot shows that by increasing the concentration of Trx the transition point of both Prx3 and Prx2 is shifted towards lower Prx3 and Prx2 values, respectively. However, for Prx2 this change is very slow at low concentration of Trx. This decrease in the transition concentration of Prx3 and Prx2 is due to the increase in the reduced form of Prx3 and Prx2 which cause the transition point is shifted to lower values of Prx3 or Prx2.
To understand these synergetic behaviors, in Fig. 4 we plot the concentration of H 2 O 2 as a function of the concentration of GPx1 and Prx3 or Prx2 corresponding to the presented results in Figs 1 and 2. By increasing the concentration of GPx1 or Prx3 or, Prx2 the concentration of H 2 O 2 decreases, however, the rate of this decrease depends on the types of antioxidants. For instance, GPx1 has a higher capacity to detoxify H 2 O 2 as compared to Prx3 and Prx2. It should be noted that the decrease in the concentration of H 2 O 2 corresponds to the increase in the activity of Prx3 and Prx2. This explains the reason behind the observed synergetic coupling between GPx1 and Prx3/Prx2. In fact by increasing the concentration of GPx1 the concentration of H 2 O 2 decreases. This decrease in the concentration of H 2 O 2 converts the oxidized form of Prx3 and Prx2 to the reduced forms and increases the Prx3, Prx2 activity.

Discussion
Observation of the synergetic coupling among the major antioxidants glutathione peroxidase family and peroxiredoxin family is an interesting and fundamental phenomenon which impacts our understanding of the way these antioxidants regulate H 2 O 2 . It shows that antioxidants may not be working as individual entities to detoxify H 2 O 2 but that they act in harmony and in a coordinated fashion to protect cells against oxidants. Moreover, this coordinated manner of regulating H 2 O 2 levels permits more intricate control of H 2 O 2 concentration levels, which is critical for one of the roles of H 2 O 2 , as a cell signaling species.  The source of the observed synergy is the coupling of these antioxidants to the same substrate, namely H 2 O 2 . When the production of this substrate is high, the conversion of the oxidized forms of GPxs or Prxs to their reduced forms cannot match the rate of conversion of their reduced forms to oxidized forms. Hence, a portion of the GPxs or Prxs remain in their oxidized form. An increase in the concentration of each antioxidant would decrease the concentration of H 2 O 2 and as a result increase the concentration of the reduced form of Gpxs and Prxs which would lead to an increase in their activity. This increase in the activity of GPxs and Prxs through an increase in the concentration of other antioxidants, translates into the synergetic behavior in these systems. In other words, at low concentrations of H 2 O 2 all Prxs and GPxs are completely in their reduced forms, hence there is no synergy among them. However at high concentrations of H 2 O 2 , which could correspond to a high production rate of H 2 O 2 or low concentration of antioxidants, Prxs and GPxs are not in their fully oxidized form and any variation in the concentration of GPxs could increase the activity of Prxs. We should comment that the synergy between these antioxidants could be small because of the reaction rates of Gpxi and Prxi. For example our calculation shows that the synergy is small between Prx5 and Gpx1 for all the concentration range of these antioxidants.
The observed synergy is not limited to GPxs and Prxs and other agents, which are antioxidants could reduce H 2 O 2 at the cellular level, and also have a synergetic coupling with the GPxs and Prxs. To include these terms the reaction rates of these antioxidants with H 2 O 2 need to be added into the current formalism.
In our calculations we neglected the spatial distribution of all involved species. Although this approximation provides a good description for phenomena that do not depend on the spatial distribution of H 2 O 2 such as the average concentration of H 2 O 2 , in some cases a spatio-temporal model may be necessary. For example if one is interested in investigating the cell signaling which affects a receptor on the cell membrane or the distribution of H 2 O 2 around the peroxisome, then one needs to consider the spatial distribution of H 2 O 2 . However, because of the fast diffusion of H 2 O 2 we would not expect inhomogeneity, except for parts of the cells which are close to the sources of H 2 O 2 or membranes. We are currently developing a model which takes into account the spatial distribution of H 2 O 2 and other antioxidants.
In conclusion, we have developed a theoretical framework to investigate the detoxification of H 2 O 2 by antioxidants to see if there is any coupling among the antioxidants. By using a perturbative method we have analytically derived the concentration of H 2 O 2 as a function of the concentration of other antioxidants and of the production rate of H 2 O 2 . Using the derived concentration of H 2 O 2 we have shown that the concentrations of the oxidized forms of GPxs and Prxs depend on the concentrations of other antioxidants. This shows that at high concentrations of H 2 O 2 in which GPxs and Prxs are not in their saturated reduced forms any change in the concentrations of any antioxidants would modulate the reduced form of GPxs and Prxs. This proves that there is synergetic behavior between antioxidants and GPxs and Prxs. By using a numerical method we have solved the detoxifying equations for GPx1, Prx3 and Prx2 to show that this synergetic behavior does, in fact, exist among these antioxidants. We also demonstrated that there are regions in which this synergetic behavior plays a dominant role. Our results reveal a new way of understanding and thinking about the behavior of the antioxidant system in cells and may help in developing drugs to either protect normal cells or eliminate the abnormal and malignant cells.

Methods
In our model we consider a single cell with GPxs, Prxs, and catalase as the dominant antioxidants and we assume that there is a production of H 2 O 2 inside the cell with a given rate. We find a quasi-steady state in which the concentrations of all involved species remain constant over the relevant time scale. Because a cell has dimensions on the micrometer scale and the diffusion constant of H 2 O 2 is on the order of 1000 µm s 2 the detoxification of each of the antioxidants is not limited by H 2 O 2 accessibility. Therefore we assume that each antioxidant detoxifies H 2 O 2 based on its detoxifying potential although the antioxidant may be confined to a special compartment such as the peroxisome. Hence, we ignore spatial dependency in our model, and for antioxidants (which are confined to a compartment) we include the volume ratio of the compartment to the whole cell in order to capture some of the spatial effects. Besides, since the concentration of GSH is much higher than the concentration of GPxs, the change in the concentration of GSH is negligible and we assume that it remains constant on the relevant time scale. The Prx (typical 2-Cys and atypical 2-Cys): Prx (1-Cys): We use the chemical reactions in equations (1) to (10) to write the kinetics equations for detoxifying H 2 O 2 assuming that H 2 O 2 is produced at the rate P h o 2 2 inside the cell. where V i is the volume of compartment i which contains antioxidant i, V is the total volume of cell, C gpx