Localized redox relays as a privileged mode of cytoplasmic hydrogen peroxide signaling

Abstract

Hydrogen peroxide (H₂O₂) is a key signaling agent. Its best characterized signaling actions in mammalian cells involve the early oxidation of thiols in cytoplasmic phosphatases, kinases and transcription factors. However, these redox targets are orders of magnitude less H₂O₂-reactive and abundant than cytoplasmic peroxiredoxins. How can they be oxidized in a signaling time frame? Here we investigate this question using computational reaction-diffusion models of H₂O₂ signaling. The results show that at H₂O₂ supply rates commensurate with mitogenic signaling a H₂O₂ concentration gradient with a length scale of a few tenths of μm is established. Even near the supply sites H₂O₂ concentrations are far too low to oxidize typical targets in an early mitogenic signaling time frame. Furthermore, any inhibition of the peroxiredoxin or increase in H₂O₂ supply able to drastically increase the local H₂O₂ concentration would collapse the concentration gradient and/or cause an extensive oxidation of the peroxiredoxins I and II, inconsistent with experimental observations. In turn, the local concentrations of peroxiredoxin sulfenate and disulfide forms exceed those of H₂O₂ by several orders of magnitude. Redox targets reacting with these forms at rate constants much lower than that for, say, thioredoxin could be oxidized within seconds. Moreover, the spatial distribution of the concentrations of these peroxiredoxin forms allows them to reach targets within 1 μm from the H₂O₂ sites while maintaining signaling localized. The recruitment of peroxiredoxins to specific sites such as caveolae can dramatically increase the local concentrations of the sulfenic and disulfide forms, thus further helping these species to outcompete H₂O₂ for the oxidation of redox targets. Altogether, these results suggest that H₂O₂ signaling is mediated by localized redox relays whereby peroxiredoxins are oxidized to sulfenate and disulfide forms at H₂O₂ supply sites and these forms in turn oxidize the redox targets near these sites.

Graphical abstract

Highlights

• •2-Cys peroxiredoxins impose strong cytoplasmic and transmembrane H₂O₂ gradients.
• •Maximal cytoplasmic H₂O₂ concentrations are insufficient to oxidize Tyr phosphatases.
• •Prx inhibition modestly increases maximal H₂O₂ concentration, blunts gradients.
• •Sulfenate and disulfide Prx strongly exceed H₂O₂ concentrations, are also localized.
• •Prx recruitment causes modest local H₂O₂ depletion, helps Prx outcompete H₂O₂.

1. Introduction

Hydrogen peroxide (H₂O₂) is a key intermediate in many signaling pathways in mammalian cells [1]. Its best-studied signaling effects are mediated by the oxidation of thiolates in transcription factors [2], kinases [3], [4] and phosphatases [5], [6], [7], [8]. However, whereas the redox active thiolates in these targets react with H₂O₂ with rate constants in the range of 9−164 M⁻¹ s⁻¹ [5], [6], [9], [1], the cell cytoplasm contains very abundant peroxiredoxins I and II (PrxI, PrxII) [10], [11], which react with H₂O₂ with rate constants in the range of 10⁷–10⁸ M⁻¹ s⁻¹ [12], [13]. And even in the absence of these and other catalysts, such as glutathione peroxidases and catalases, the $\approx 50\mathrm{\mu }\mathrm{M}$ glutathione thiolate in cells would outcompete those redox targets for H₂O₂ [14]. Explaining how those poorly reactive targets can be acted upon by H₂O₂ despite the strong competition by other agents is one enduring open question in redox biology [15], [16].

Three main types of explanations were proposed, which are not mutually exclusive [17]. First, there may be strong intracellular H₂O₂ concentration gradients. Sufficiently high H₂O₂ concentrations to oxidize the less reactive targets might be attained near very localized sites of H₂O₂ production or entry into the cytoplasm despite the mean cytoplasmic concentration being very low [15]. Indeed, estimates of H₂O₂ mean diffusion length [15] as well as recent mathematical models [18], [19] predict the occurrence of strong concentration gradients if H₂O₂ sources are localized. And observations with ratiometric H₂O₂ probes targeted to various cellular membranes support the existence of microdomains of elevated H₂O₂ concentration in the cytoplasm of live cells treated with growth factors [20]. However, despite the evidence for strong cytoplasmic H₂O₂ concentration gradients, the following question is still unclear. Q1: Are H₂O₂ concentrations attained near the supply sites sufficient to oxidize the above mentioned targets?

Second, peroxiredoxins may be transiently inactivated during signaling, allowing H₂O₂ to accumulate. The first such proposal was the floodgate hypothesis [21], which posits that the oxidation of the less reactive H₂O₂ targets is facilitated by the oxidation of eukaryotic 2-Cys peroxiredoxins to redox-inactive sulfinic and sulfonic forms (hyperoxidation). This hypothesis is supported by observations that overexpressing 2-Cys peroxiredoxins in mammalian cells blocks peroxide activation of nuclear factor kappa-light-chain-enhancer of activated B cells (NF-κB) [22], [23], and that treatment of cells with tumor necrosis factor causes substantial hyperoxidation of PrxII [24]. Furthermore, it could explain why eukaryotic 2-Cys peroxiredoxins are much more susceptible to hyperoxidation than their bacterial homologues [21]. This susceptibility is due to structural features that are absent in the latter peroxiredoxins but widely conserved among the former ones [21], suggesting that it is selectively favored in eukaryotes. However, hyperoxidation was undetectable during mitogenic signaling [25], [26], [27], and unnecessary for ASK1 activation in H₂O₂-induced apoptosis [28]. More recently, it was hypothesized that localized hyperoxidation of the 2-Cys peroxiredoxins sharpens H₂O₂ concentration gradients, contributing to increased H₂O₂ concentrations near sites of H₂O₂ supply [29]. The localization of hyperoxidation could explain the difficulty in detecting it in the experiments above, but this localized floodgate hypothesis remains untested. Further support for the role of localized inhibition of the peroxidase activity of 2-Cys peroxiredoxins in redox signaling was provided by Woo et al. [27]. These authors showed that upon stimulation of a wide variety of cell lines by growth factor or immune receptors the peroxidase activity of PrxI associated to cell membranes is inhibited by phosphorylation. However, it is unclear if this localized inhibition can substantially affect local H₂O₂ concentrations in the presence of abundant PrxII, which also associates to the platelet-derived growth factor (PDGF) receptor upon PDGF stimulation [25], and when active cytoplasmic 2-Cys peroxiredoxins can readily diffuse to the inhibition sites. Although PrxI phosphorylation promotes PDGF-dependent tyrosine phosphorylation of cellular proteins [27], this does not necessarily imply a direct inhibition of protein tyrosine phosphatases (PTP) by H₂O₂. For instance, PrxI inhibition should lead to local accumulation of sulfenic or disulfide PrxII, which in turn could oxidize a PTP, thereby inhibiting it. Overall, the following question needs to be addressed. Q2: Can a localized inactivation of the 2-Cys peroxiredoxins cause a sufficient elevation of local H₂O₂ concentrations to directly oxidize the less reactive targets in a signaling time frame?

Third, the action of H₂O₂ may be mediated by redox relays whereby peroxiredoxins and/or peroxidases act as initial H₂O₂ sensors and then oxidize the end targets [15], [30], [31]. Such a relay was first described for the peroxiredoxin Orp1 and the transcription factor Yap1 in the yeast Saccaromyces cerevisiae [32], and other similar relays in yeasts were described meanwhile [33], [34], [35]. More recently, both PrxI [28] and PrxII [2], [36] were shown to engage in redox relays in the cytoplasm of mammalian cells. This is an attractive hypothesis that could also explain the specificity of redox signaling. However, the following question needs to be clarified. Q3: Can in general the oxidized forms of cytoplasmic 2-Cys peroxiredoxins accumulate sufficiently to outcompete H₂O₂ in oxidizing the less reactive targets?

Conversely to a role in relaying oxidizing equivalents to redox targets, peroxiredoxins can protect moderately reactive thiolates near sites of H₂O₂ supply against oxidation. Kang et al. [37] have shown that PrxII is recruited to the vascular endothelial growth factor (VEGF) receptor 2 (VEGFR2) upon VEGF stimulation of endothelial cells, protecting this receptor against oxidation of its Cys1199 and Cys1206 residues. More recently the same group [38] showed that in proliferating cells PrxI associates with the centrosome protecting it from H₂O₂ during interphase. It is then inhibited by phosphorylation during early mitosis, thereby facilitating oxidative inactivation of centrosome-bound phosphatases. Q4: But, under what conditions can a local accumulation of the 2-Cys peroxiredoxins protect reactive thiols in other proteins from oxidation and what extent of protection is achievable?

Mathematical modeling has proved useful in assessing previous hypotheses and suggest new ones about the operation of thiol redox systems [39], [16], [40], [18], [19], [41]. Here we draw on the present knowledge of the reactivity and cellular concentrations of 2-Cys peroxiredoxins, sulfiredoxin (Srx) and thioredoxin 1 (Trx1) [5], [6], [12], [13], [42], [9], [1] and apply an integrative computational approach to address the four questions raised above.

Our analyses show the following. At low to moderate localized H₂O₂ supply rates commensurate to mitogenic signaling, a sharp H₂O₂ concentration gradient with a length scale of $~0.3\mathrm{\mu m}$ is established. However, local H₂O₂ concentrations remain insufficient to oxidize PTPs in a mitogenic signaling time frame (Q1). Further, only drastic inhibitions of the peroxiredoxins' peroxidase activity can have a substantial effect on the local H₂O₂ concentration near supply sites (Q2), and this comes at the cost of dampening the cytoplasmic H₂O₂ concentration gradient. On the other hand, the concentrations of sulfenic and disulfide forms of Prx exceed those of H₂O₂ by several orders of magnitude, and also show strong gradients. Redox targets able to react with these forms at rate constants $\sim 200{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ could be oxidized in minutes near the H₂O₂ supply sites (Q3). The recruitment of peroxiredoxins to specific sites such as caveolae has a modest effect ($\sim 20\%$ decrease) on local H₂O₂ concentrations (Q4). However, it can have a dramatic effect on the local concentrations of oxidized forms of the peroxiredoxins, further helping these species outcompete H₂O₂ for oxidation of redox targets. Altogether, these results suggest that sulfenate and/or disulfide forms of Prx mediate the oxidation of redox targets within $\sim 1\mathrm{\mu }\mathrm{m}$ of H₂O₂ supply sites under conditions consistent with mitogenic signaling.

At higher H₂O₂ supply rates, the system can show a hyperoxidation catastrophe, in keeping with a recent analysis [43]. At a critical H₂O₂ supply rate Prx abruptly becomes almost fully hyperoxidized and the H₂O₂ gradient collapses, allowing H₂O₂ to penetrate deeply into the cell and oxidize some of the most reactive targets. Recovery from such a state can start only below a lower critical H₂O₂ supply rate (hysteresis) and proceeds over a period of hours. The global floodgate hypothesis may thus find its place as part of a stress response in this context.

2. Models and methods

The reference mathematical model implemented in this work focuses on the early stages of H₂O₂ signaling, and describes the processes and geometries depicted in Fig. 1. It embodies the following main simplifications. First, it neglects intracellular H₂O₂ sources. It is expectable that in early signaling H₂O₂ produced externally by NADPH oxidases is supplied to the cytoplasm mainly by permeation across the membrane [20]. Second, cytoplasmic H₂O₂ sinks other than 2-Cys peroxiredoxins are neglected. It is estimated [16], [18] that in most cells such sinks consume a small fraction of the H₂O₂, and their activity is insufficient to sustain substantial H₂O₂ gradients. Furthermore, recent quantitative proteomic studies of multiple human cell lines and tissues [11] consistently show that glutathione peroxidases are one to two orders of magnitude less abundant than PrxI and PrxII. The neglect of alternative H₂O₂ sinks is a conservative assumption with respect to the H₂O₂ attained in the cytoplasm. Third, it neglects thioredoxin oxidation, as the redox capacity of most cells is sufficient to keep thioredoxin reduced in absence of strong oxidative stress [16], [44], [40], [18]. Fourth, it accounts for a single 2-Cys peroxiredoxin whose concentration represents the sum of PrxI and PrxII, the two 2-Cys peroxiredoxins that are abundant in the cytoplasm of most mammalian cells [11]. The reference model considers that the single peroxiredoxin has the kinetic properties of PrxII. This is both because this peroxiredoxin has been more extensively characterized than PrxI, and because this is a conservative assumption with respect to most questions investigated in this work. The value we adopt for the rate constant for H₂O₂ reduction by thiolate peroxiredoxin ($k_1=1.0\times {10}^2\mathrm{\mu }{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ [13]) is less conservative than that determined in ref. [45] ($13\mathrm{\mu }{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$). However, even the former value has been determined at a sub-physiological temperature (${25}^{\mathrm{o}}\mathrm{C}$), and is likely still conservative. In subsequent sections we will show that the main results are robust with respect to these assumptions.

Fig. 1.

Processes and geometry simulated: A: H₂O₂ permeates the membrane with permeability constant $k_p$ and then oxidizes PS⁻ to PSO⁻ and PSO⁻ to PSO⁻₂, with rate constants k₁ and k₂ respectively. PSO⁻₂ is reduced back to PSO⁻ with rate constant k₃ and PSO⁻ condenses with Prx's resolving Cys to yield PSS, with rate constant k₄. Thioredoxin reduces PSS to PS⁻, with rate constant k₅. B: We consider a section of a cell described by a rectangular prism with base area A, half-length 5 μm and a symmetry plane parallel to the top and bottom membranes. H₂O₂ is assumed to permeate both these membranes uniformly. This geometry models a cell in a cell layer and permits reducing the spatial problem to one dimension. Namely, along the normal to the plane of the permeant membranes. C: Scheme for the 3D simulation box of H₂O₂ concentration near a receptor. The region of the caveolae where the various peroxiredoxin forms can bind has a radius of 0.1 μm. The binding of 2% of the peroxiredoxin in the considered cylinder to this small region implies that the local concentration is 250-fold higher than in the bulk, considering a volume element with the depth of 10 nm. The H₂O₂ is able to enter through the top surface of the cylinder. In all other surfaces we implement a zero flux boundary condition for H₂O₂. With respect to the peroxiredoxin, all boundaries of this simulation box are set to zero-flux. The various forms of peroxiredoxins bound to the membrane are laterally immobile, and unbind with a spatially uniform rate constant. For parameter values please see Table 1.

The concentrations of the various chemical species in the cytoplasm are a function of both time and space, and therefore their evolution can be obtained through the following set of reaction diffusion equations:

$$\begin{array}{ccc}{\displaystyle \frac{\partial c_{\mathrm{H}}}{\partial t}}& =& D_{\mathrm{H}}{\nabla }^2{c}_{\mathrm{H}}-k_1{c}_{{\mathrm{PS}}^{-}}{c}_{\mathrm{H}}-k_2{c}_{{\mathrm{PSO}}^{-}}{c}_{\mathrm{H}},\\ {\displaystyle \frac{\partial c_{\mathrm{P}{\mathrm{S}}^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{\mathrm{P}{\mathrm{S}}^{-}}-k_1{c}_{{{\mathrm{PS}}^{-}}^{}}{c}_{\mathrm{H}}+k_5{c}_{\mathrm{P}\mathrm{S}\mathrm{S}}{c}_{\mathrm{T}\mathrm{r}\mathrm{x}},\\ {\displaystyle \frac{\partial c_{\mathrm{P}\mathrm{S}{\mathrm{O}}^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}^{-}}+k_1{c}_{{\mathrm{PS}}^{-}}{c}_{\mathrm{H}}-k_2{c}_{{\mathrm{PSO}}^{-}}{c}_{\mathrm{H}}& +k_3{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}_2^{-}}-k_4{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}^{-}},\\ {\displaystyle \frac{\partial c_{\mathrm{P}\mathrm{S}{\mathrm{O}}_2^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}_2^{-}}+k_2{c}_{{\mathrm{PSO}}^{-}}{c}_{\mathrm{H}}-k_3{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}_2^{-}},\\ {\displaystyle \frac{\partial c_{\mathrm{P}\mathrm{S}\mathrm{S}}}{\partial t}}& =& D{\nabla }^2{c}_{\mathrm{P}\mathrm{S}\mathrm{S}}+k_4{c}_{\mathrm{P}\mathrm{S}{\mathrm{O}}^{-}}-k_5{c}_{\mathrm{P}\mathrm{S}\mathrm{S}}{c}_{\mathrm{T}\mathrm{r}\mathrm{x}},\end{array}$$ (1)

where $c_{\mathrm{H}}(\mathbf{r},t)$, $c_{{\mathrm{PS}}^{-}}(\mathbf{r},t)$, $c_{{\mathrm{PSO}}^{-}}(\mathbf{r},t)$, $c_{{\mathrm{PSO}}_2^{-}}(\mathbf{r},t)$, $c_{\mathrm{PSS}}(\mathbf{r},t)$, and $c_{\mathrm{Trx}}(\mathbf{r},t)$ are the concentrations of H₂O₂, of the peroxiredoxin forms, and of reduced thioredoxin, respectively. The reference values of the diffusion and rate constants are as in Table 1. The diffusion constant for all peroxiredoxin forms $(\text{M}\text{W}=230\text{k}\text{D}\text{a})$ was estimated from the experimentally determined diffusion constant for Immunoglobulin G (IgG, $\text{MW}=153\text{kDa}$) in the cytoplasm [46] by applying the expression

$$D_{\mathrm{Prx}}=D_{\mathrm{IgG}}\sqrt[3]{{\mathit{MW}}_{\mathrm{IgG}}/{\mathit{MW}}_{\mathrm{Prx}}},$$

as per the Stokes-Einstein relationship. The estimated value is of the same order of magnitude and just slightly higher than those recently determined for untranslating polysomes [47]. The diffusion constant for H₂O₂ [48] was determined for diffusion in buffer. The same authors determined a 5-fold lower diffusion constants for diffusion in hydrogels with a viscosity similar to that of the cell cytoplasm. Where pertinent we will discuss the consequences of the lower value of this diffusion constant. In Eqs. (1) we model the diffusion of the peroxiredoxin forms and of the H₂O₂ according to Fick's laws [49].

Table 1.

Model parameters.

Process	Rate	Parameter	Value	Ref.
${\mathrm{PS}}^{-}+{\mathrm{H}}_2{\mathrm{O}}_2⟶{\mathrm{PSO}}^{-}+{\mathrm{H}}_2\mathrm{O}$	$k_1{c}_{\mathrm{H}}{c}_{{\mathrm{PS}}^{-}}$	$k_1$	$1.0\times {10}^2\mathrm{\mu }{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$	[13]
${\mathrm{PSO}}^{-}+{\mathrm{H}}_2{\mathrm{O}}_2⟶{\mathrm{PSO}}_2^{-}+{\mathrm{H}}_2\mathrm{O}$	$k_2{c}_{\mathrm{H}}{c}_{{\mathrm{PSO}}^{-}}$	$k_2$	$0.012\mathrm{\mu }{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$	[42]
${\mathrm{PSO}}_2^{-}⟶{\mathrm{PSO}}^{-}$	$k_3{c}_{{\mathrm{PSO}}_2^{-}}$	$k_3$	$1.0\times {10}^{-4}$ s−1	[40]
${\mathrm{PSO}}^{-}⟶\mathrm{PSS}$	$k_4{c}_{{\mathrm{PSO}}^{-}}$	$k_4$	1.7 s−1	[42]
$\mathrm{PSS}+\mathrm{Trx}1{\mathrm{S}}^{-}⟶{\mathrm{PS}}^{-}+\mathrm{Trx}1\mathrm{SS}$	$k_5{c}_{\mathrm{PSS}}{c}_{\mathrm{Trx}}$	$k_5$	$0.21\mathrm{\mu }{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$	[13]
Permeation of H2O2	$k_{\mathit{p}}A(c_{\mathrm{H}}^0-c_{\mathrm{H}}(0,t))$	$k_p$	$7.8\times {10}^{-5}\mathrm{m}{\mathrm{s}}^{-1}$	See text
Diffusion constant of H2O2		$D_H$	$1.8\times {10}^{-9}{\mathrm{m}}^2{\mathrm{s}}^{-1}$	[48]
Diffusion constant of peroxiredoxin		$D$	$5.2\times {10}^{-13}$ m2 s−1	See text
Concentration of thioredoxin		$c_{\mathrm{Trx}}$	$10\mathrm{\mu }\mathrm{M}$	[11]

Per-cell H₂O₂ removal rate constants determined by Wagner et al. [50] allow to estimate the permeability constants in the range ${10}^{-7}-{10}^{-5}\mathrm{m}{\mathrm{s}}^{-1}$. In order to be very conservative about intracellular H₂O₂ concentrations, which in the models under appreciation are approximately proportional to the permeability constant, we adopt the higher value in Table 1.

The thioredoxin concentration is set at the constant value $c_{\mathrm{Trx}}(\mathbf{r},t)=10\mathrm{\mu }\mathrm{M}$, which is typical for mammalian cells [11]. We will also examine the effects of lower thioredoxin concentrations.

We solve these equations in the rectangular prism domain shown in Fig. 1B. The boundary conditions differ between the two ends of the domain: the membrane side at $x=0\mathrm{\mu }\mathrm{m}$, and the cell center side, at $x=5\mathrm{\mu }\mathrm{m}$. All concentration fields are solely function of the distance to the membrane. They have zero-flux boundary conditions except for the concentration of H₂O₂ at the membrane side of the domain. The amount of H₂O₂ that enters the domain through the membrane per unit time is $k_{p}A(c_{\mathrm{H}}^0-c_{\mathrm{H}}(0,t))$, where $c_{\mathrm{H}}^0$ is the extracellular H₂O₂ concentration, and $c_{\mathrm{H}}(0,t)$ is the intracellular H₂O₂ concentration adjacent to the membrane at time t. This condition is imposed as a boundary condition in the spatial derivative of $c_{\mathrm{H}}$ at the membrane:

$${\frac{\partial c_{\mathrm{H}}}{\partial x}|}_{x=0}=-\frac{k_p}{D_{\mathrm{H}}}(c_{\mathrm{H}}^0-c_{\mathrm{H}}(0,t)).$$ (2)

We use these boundary conditions to solve the Eqs. (1) in the considered domain, until we obtain the stationary profiles for the different concentration fields. The equations are integrated using a finite difference scheme with $\mathrm{\Delta }x=0.01\mathrm{\mu m}$, on the order of magnitude of the diameter of a peroxiredoxin dimer ($\sim 8.0$ nm, according to PdB file 4DSQ.pdb).

In order to analyze the potential protective effect of peroxiredoxin recruitment against oxidation of specific sites we consider two additional models. In the first one, we describe the binding of all peroxiredoxin species to the membrane by adding to Eqs. (1) the concentrations of the bound species. We consider that all peroxiredoxin forms have the same binding constant, $k^{+}$, to the membrane and the same unbinding constant, $k^{-}$. The ratio between the values of these constants is set such that 2% of the peroxiredoxin is bound to the membrane in the absence of H₂O₂ gradients. In this setting the stationary state does not depend on the numerical values of $k^{+}$ or $k^{-}$, only on their ratio. The mathematical model thus becomes:

$$\begin{array}{lll}{\displaystyle \frac{\partial c_{\mathrm{H}}}{\partial t}}& =& D_{\mathrm{H}}{\nabla }^2{c}_{\mathrm{H}}-k_1(c_{{\mathrm{PS}}^{-}}+{\delta }_0{c}_{{\mathrm{PS}}^{-}}^s)c_{\mathrm{H}}-k_2(c_{{\mathrm{PSO}}^{-}}+{\delta }_0{c}_{{\mathrm{PSO}}^{-}}^s)c_{\mathrm{H}}\\ {\displaystyle \frac{\partial c_{{\mathrm{PS}}^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{{\mathrm{PS}}^{-}}-k_1{c}_{{\mathrm{PS}}^{-}}{c}_{\mathrm{H}}+k_5{c}_{\mathrm{PSS}}{c}_{\mathrm{Trx}}+{\delta }_0(k^{-}{c}_{{\mathrm{PS}}^{-}}^s-k^{+}{c}_{{\mathrm{PS}}^{-}})\\ {\displaystyle \frac{\partial c_{{\mathrm{PSO}}^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{{\mathrm{PSO}}^{-}}+k_1{c}_{{\mathrm{PS}}^{-}}{c}_{\mathrm{H}}-k_2{c}_{{\mathrm{PSO}}^{-}}{c}_{\mathrm{H}}+k_3{c}_{{\mathrm{PSO}}_2^{-}}-k_4{c}_{{\mathrm{PSO}}^{-}}+{\delta }_0(k^{-}{c}_{{\mathrm{PSO}}^{-}}^s-k^{+}{c}_{{\mathrm{PSO}}^{-}})\hfill \\ {\displaystyle \frac{\partial c_{{\mathrm{PSO}}_2^{-}}}{\partial t}}& =& D{\nabla }^2{c}_{{\mathrm{PSO}}_2^{-}}+k_2{c}_{{\mathrm{PSO}}^{-}}{c}_{\mathrm{H}}-k_3{c}_{{\mathrm{PSO}}_2^{-}}+{\delta }_0(k^{-}{c}_{{\mathrm{PSO}}_2^{-}}^s-k^{+}{c}_{{\mathrm{PSO}}_2^{-}})\\ {\displaystyle \frac{\partial c_{\mathrm{PSS}}}{\partial t}}& =& D{\nabla }^2{c}_{\mathrm{PSS}}+k_4{c}_{{\mathrm{PSO}}^{-}}-k_5{c}_{\mathrm{PSS}}{c}_{\mathrm{Trx}}+{\delta }_0(k^{-}{c}_{\mathrm{PSS}}^s-k^{+}{c}_{\mathrm{PSS}})\end{array}$$ (3)

$$\begin{array}{lll}{\displaystyle \frac{{\mathit{dc}}_{{\mathrm{PS}}^{-}}^s}{\mathit{dt}}}& =& -k_1{c}_{{\mathrm{PS}}^{-}}^s{c}_{\mathrm{H}}(0,t)+k_5{c}_{\mathrm{PSS}}^s{c}_{\mathrm{Trx}}-k^{-}{c}_{{\mathrm{PS}}^{-}}^s+k^{+}{c}_{\mathrm{PS}{}^{-}}(0,t)\\ {\displaystyle \frac{{\mathit{dc}}_{{\mathrm{PSO}}^{-}}^s}{\mathit{dt}}}& =& k_1{c}_{{\mathrm{PS}}^{-}}^s{c}_{\mathrm{H}}(0,t)-k_2{c}_{{\mathrm{PSO}}^{-}}^s{c}_{\mathrm{H}}(0,t)+k_3{c}_{{\mathrm{PSO}}_2^{-}}^s-k_4{c}_{{\mathrm{PSO}}^{-}}^s-k^{-}{c}_{{\mathrm{PSO}}^{-}}^s+k^{+}{c}_{{\mathrm{PSO}}^{-}}(0,t)\\ {\displaystyle \frac{{\mathit{dc}}_{{\mathrm{PSO}}_2^{-}}^s}{\mathit{dt}}}& =& k_2{c}_{{\mathrm{PSO}}^{-}}^s{c}_{\mathrm{H}}(0,t)-k_3{c}_{{\mathrm{PSO}}_2^{-}}^s-k^{-}{c}_{{\mathrm{PSO}}_2^{-}}^s+k^{+}{c}_{{\mathrm{PSO}}_2^{-}}(0,t)\\ {\displaystyle \frac{{\mathit{dc}}_{\mathrm{PSS}}^s}{\mathit{dt}}}& =& k_4{c}_{{\mathrm{PSO}}^{-}}^s-k_5{c}_{\mathrm{PSS}}^s{c}_{\mathrm{Trx}}-k^{-}{c}_{\mathrm{PSS}}^s+k^{+}{c}_{\mathrm{PSS}}(0,t)\end{array}$$

where $c_{{\mathrm{PS}}^{-}}^s(t)$, $c_{{\mathrm{PSO}}^{-}}^s(t)$, $c_{{\mathrm{PSO}}_2^{-}}^s(t)$ and $c_{\mathrm{PSS}}^s(t)$ are the volume concentrations of the species bound to the membrane. We assume that all the bound peroxiredoxin species have the same reactivity towards H₂O₂ and thioredoxin as the unbound counterparts. The binding of the peroxiredoxin species depends only on the concentration of the unbound species adjacent to the membrane ($c_{{\mathrm{PS}}^{-}}(0,t)$, $c_{{\mathrm{PSO}}^{-}}(0,t)$, $c_{{\mathrm{PSO}}_2^{-}}(0,t)$ and $c_{\mathrm{PSS}}(0,t)$). Accordingly, in Eqs. (3) the coefficient δ₀ is equal to 1 next to the membrane and equal to 0 everywhere else.

In the second model for peroxiredoxin localization the same reaction-diffusion equations are solved within a cylindrical domain with azimuthal symmetry (Fig. 1C). In this setting, we consider that the binding of the peroxiredoxin species to the membrane can only occur in a circular region representative of a caveola. In this model, the bound species are not able to diffuse within the cell membrane. This system was simulated in the (r,z) plane with a second order finite difference scheme. The increments $\mathrm{\Delta }x$ used were identical to those in the 1D counterparts.

3. Results

3.1. Peroxiredoxins prevent deep penetration of H₂O₂ into the cell cytoplasm

We start by modeling the penetration of H₂O₂ in the cell for a given fixed concentration of extracellular H₂O₂, $c_{\mathrm{H}}^0$. We first consider that all the peroxiredoxin is initially in reduced form. Under these conditions, for 2-Cys peroxiredoxin concentrations typical of most cells [11], the H₂O₂ concentration at steady state drops steeply near the membrane (Fig. 2A). Furthermore, the steady-state concentration of H₂O₂ adjacent to the membrane can be almost two orders of magnitude lower than the extracellular H₂O₂ concentration (Fig. 2B).

Fig. 2.

H₂O₂penetration in the cell: The H₂O₂ can penetrate the cell depending on the amount of cytoplasmic PS⁻ and on the extracellular H₂O₂ concentration. A: Concentration of H₂O₂ as a function of the distance from the membrane in the case of c⁰_PS−=50 μM and c⁰_H=0.4 μM. In this situation the total amount of H₂O₂ inside the cell is small and it is mainly located in the neighborhood of the membrane. B: Order of magnitude of the ratio between the H₂O₂ concentration adjacent to the membrane and the extracellular H₂O₂ concentration as a function of the initial cytoplasmic PS⁻ concentration and of the extracellular H₂O₂ concentration. Two regimes can be clearly observed: for high extracellular H₂O₂ concentrations the H₂O₂ oxidizes all the PS⁻ and invades the cell, while for low extracellular H₂O₂ concentrations the cytoplasmic PS⁻ is able to reduce the incoming H₂O₂. C: Except for low cytoplasmic PS⁻ concentrations, in the regime of low extracellular H₂O₂ concentrations the depth at which the cytoplasmic H₂O₂ concentration decays to 37% of its value near the membrane is in the range 0.5–0.7 µm, almost independently of $c_{\text{H}}$ and $c_{{\text{PS}}^{-}}$. Black dots indicate the collapse of the cytoplasmic gradient. D: Order of magnitude of the ratio between the H₂O₂ concentration adjacent to the membrane and the extracellular H₂O₂ concentration when initially all the peroxiredoxin is hyperoxidized. Each dot represents a separate simulation for the pertinent values of c⁰_H and initial peroxiredoxin concentration. The green shaded area marks the transition region between both regimes in panel B, highlighting the occurrence of hysteresis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

This decay of the H₂O₂ concentration near the membrane is approximately exponential, similar to previous results for a simpler model [18], [19]. The typical width of the membrane neighborhood where H₂O₂ is concentrated has a weak dependence on both $c_{\mathrm{H}}^0$ and the concentration of peroxiredoxin in thiolate form (Fig. 2C), as the following approximation also shows. For low $c_{\mathrm{H}}^0$, only a small fraction of the cell's ${\mathrm{PS}}^{-}$ is oxidized, and therefore the steady state concentration of ${\mathrm{PS}}^{-}$ in the cell is much higher than that of the other redox forms (Fig. 3A). Therefore, the first equation in (1) can be approximated as ${\partial }_t{c}_{\mathrm{H}}=D_{\mathrm{H}}{\partial }_x^2{c}_{\mathrm{H}}-k_1{c}_{\mathrm{H}}{c}_{{\mathrm{PS}}^{-}}$, and $c_{{\mathrm{PS}}^{-}}$ is approximately equal to its initial value. The steady state for $c_{\mathrm{H}}$ is then approximately given by

$$c_{\mathrm{H}}(x)\sim c_{\mathrm{H}}(0)\exp \left(-x/\lambda \right),$$ (4)

where

$$c_{\mathrm{H}}(0)\sim \frac{c_{\mathrm{H}}^0}{1+{\displaystyle \frac{\sqrt{D_{\mathrm{H}}{k}_1{c}_{{\mathrm{PS}}^{-}}}}{k_p}}},$$ (5)

and

$$\lambda =\sqrt{\frac{D_{\mathrm{H}}}{k_1{c}_{{\mathrm{PS}}^{-}}}}$$ (6)

is the length scale of the decay of the H₂O₂ concentration. Considering a $c_{{\mathrm{PS}}^{-}}\approx 50\mathrm{\mu }\mathrm{M}$ and the parameter values in Table 1, one obtains $\lambda \approx 0.60\mathrm{\mu }\mathrm{m}$. Considering the 5-fold lower value of D_H obtained [48] for diffusion in a hydrogel with a viscosity approaching that of the cytoplasm one obtains $\lambda \sim 0.27\mathrm{\mu }\mathrm{m}$, which should be more representative of the situation in vivo. The order of magnitude of $\lambda$ will be the same over the physiological range of peroxiredoxin concentrations, due to the shallow dependence of $\lambda$ on $c_{{\mathrm{PS}}^{-}}^{-1/2}$.

Fig. 3.

Spatial distribution of peroxiredoxin species: A: Concentration profiles of H₂O₂, PS⁻, PSO⁻, PSS and PSO⁻₂ as a function of the distance from the membrane in the case of $c_{\text{PS}}-(0)=50\mu M$ and c⁰_H=0.1 μM. For this low H₂O₂ influx rate, only a small fraction of PS⁻ is oxidized to PSO⁻ and PSS. The inset highlights the different orders of magnitude of the concentration of the various species and the strong gradients of $c_{{\text{PSO}}^{-}}$ and $c_{{\text{PSS}}^{}}$. The color code is as indicated in panel B. Note that in the main plot the concentration of H₂O₂ is multiplied by 5000 for visualization in the same scale. B: Same as A but for c⁰_H=0.4 μM. In this case c_PS^-- is slightly depressed near the membrane, and $c_{{\text{PSO}}^{-}}$ and $c_{{\text{PSS}}^{}}$ are elevated. Due to the very low reduction rate constant for PSO⁻₂, the concentration of this form becomes almost constant over the cytoplasm. C: Same as A but for c⁰_H=1.0 μM. For these high extracellular concentrations, the H₂O₂ invades the cell and oxidizes most of the PS⁻ to PSO⁻₂. D: Ratio between the average concentration of the various forms of peroxiredoxin and the initial concentration of PS⁻. Each point is colored according to the relative fractions of peroxiredoxin in each form. Pure red and blue indicate that 100% of the peroxiredoxin is in the PS⁻ or PSO⁻₂ forms, respectively. Pure green indicates that 100% of the peroxiredoxin is in the PSO⁻+PSS forms. Comparing this figure to Fig. 2B, we observe that when the outside H₂O₂ concentration is very high, all the peroxiredoxin is hyperoxidized and the H₂O₂ is able to penetrate the whole cell. For low extracellular H₂O₂ concentrations, most of the peroxiredoxin is in its reduced form. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.2. Cytoplasmic H₂O₂ concentration gradients coexist with concentration gradients of sulfenic and disulfide peroxiredoxin forms

Importantly, coexisting with the cytoplasmic H₂O₂ gradients there are also strong concentration gradients of ${\mathrm{PSO}}^{-}$ and PSS (Fig. 3A, B). However, this is not the case of ${\mathrm{PSO}}_2^{-}$, whose concentration remains uniformly low throughout cytoplasm. The reasons for these differences among the spatial distributions of the various peroxiredoxin forms are as follows. H₂O₂ rapidly oxidizes ${\mathrm{PS}}^{-}$ to ${\mathrm{PSO}}^{-}$ near the membrane, and ${\mathrm{PSO}}^{-}$ is rapidly converted to PSS before is has time to diffuse far away. Reduction of PSS to ${\mathrm{PS}}^{-}$ is also too fast to permit this species to diffuse far away from the membrane. In turn, both the formation of ${\mathrm{PSO}}_2^{-}$ and its reduction to ${\mathrm{PSO}}^{-}$ are much slower, allowing ${\mathrm{PSO}}_2^{-}$ to diffuse throughout the cytoplasm.

In a narrow range of $c_{\mathrm{H}}^0$ approaching the upper border of the blue region in Fig. 2B, the concentration of ${\mathrm{PS}}^{-}$ close to the membrane drops strongly (Fig. 3B) and the approximation of constant $c_{{\mathrm{PS}}^{-}}^{}$ breaks down. Due to the lower ${\mathrm{PS}}^{-}$ concentration, both the H₂O₂ concentration near the membrane (Fig. 2B) and the penetration length (Fig. 2C) increase in this range of $c_{\mathrm{H}}^0$. However, the increase in the H₂O₂ concentration near the membrane in this range of $c_{\mathrm{H}}^0$ is relatively modest.

3.3. Cytoplasmic H₂O₂ gradients collapse at high H₂O₂ supplies

As characterized more extensively elsewhere for a model neglecting concentration gradients [43], in cells where a high peroxiredoxin reduction capacity coexists with limited alternative H₂O₂ sinks the system shows bi-stability and hysteresis. This dynamic characteristic manifests itself as a run-away hyperoxidation of nearly all the peroxiredoxin and a drastic increase in cytoplasmic H₂O₂, once the extracellular H₂O₂ concentration exceeds a critical value ($c_{\mathrm{H}}^{0*}$). Once this new steady state is established, recovery starts only after $c_{\mathrm{H}}^0$ decreases below another lower critical value ($c_{\mathrm{H}}^{0**}<c_{\mathrm{H}}^{0*}$) and takes several hours. The reaction-diffusion model considered in the present work shows this behavior as well. When the extracellular H₂O₂ concentration reaches a critical value $(c_{\mathrm{H}}^{0*})$, the peroxiredoxin accumulates in hyperoxidized form (Fig. 3D). As a consequence, the cytoplasmic concentration gradients of H₂O₂, ${\mathrm{PS}}^{-}$, ${\mathrm{PSO}}^{-}$ and PSS collapse (Fig. 3C), and H₂O₂ concentrations become much higher throughout all the cytoplasm (Fig. 2B). The transition between these regimes is quite abrupt, and $c_{\mathrm{H}}^{0*}$ increases approximately as the square root of the initial concentration of peroxiredoxin in the cell (Fig. 2B).

Remarkably, if one starts with a system where the peroxiredoxin is initially hyperoxidized, the critical values of $c_{\mathrm{H}}^0$ $(c_{\mathrm{H}}^{0**})$ below which the peroxiredoxins are again able to sustain a H₂O₂ gradient are considerably lower than $c_{\mathrm{H}}^{0*}$ (Fig. 2D). The system thus presents hysteresis.

Note that the actual values of $c_{\mathrm{H}}^{0*}$ and $c_{\mathrm{H}}^{0**}$ depend strongly on the permeability of the membrane and on the activity of alternative H₂O₂ sinks. Here we are neglecting these sinks and considering a high permeability constant (see Section 2). Therefore, we expect that in most cells these critical concentrations are higher than depicted in Fig. 2, Fig. 3.

3.4. The concentration of H₂O₂ near the membrane is insufficient to oxidize PTPs

Known rate constants for direct oxidation of PTPs by H₂O₂ are $k_{\mathit{ox}}\le 164{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$, which raises the question of whether H₂O₂ concentrations sufficient to directly oxidize PTPs are attainable in the cytoplasm. In order to frame this question, we first examine how the maximal cytoplasmic H₂O₂ concentrations relate to the extracellular concentrations, peroxiredoxin concentrations and cytoplasmic gradients to highlight a fundamental trade-off between signal localization and the maximal cytoplasmic H₂O₂ concentration. The latter concentration occurs adjacent to the membrane. Under conditions where only a small fraction of the peroxiredoxin is oxidized it is approximated by Eq. (5). The product $k_{\mathit{sink}}=k_1{c}_{{\mathrm{PS}}^{-}}^{}$ in this equation represents the pseudo-first-order rate constant for H₂O₂ consumption in the cytoplasm, and therefore the following conclusions apply irrespective of the main cytoplasmic process contributing to $k_{\mathit{sink}}$. Considering a generic eukaryotic cell with $10\mathrm{\mu }\mathrm{m}$ width, for signaling to be localized the length scale of the cytoplasmic H₂O₂ should be $<1\mathrm{\mu }\mathrm{m}$. As per Eq. (6) this implies $k_{\mathit{sink}}>310{\mathrm{s}}^{-1}$, conservatively considering $D_{\mathrm{H}}=3.1\times {10}^{-10}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ as for H₂O₂ diffusion in a hydrogel [48]. Because even for the most permeable cells $k_p<{10}^{-5}\mathrm{m}{\mathrm{s}}^{-1}$, Eq. (5) implies $c_{\mathrm{H}}(0)/c_{\mathrm{H}}^0<1/30$. Therefore, in most cells the maximal cytoplasmic H₂O₂ concentrations are one to two orders of magnitude lower than the extracellular ones under conditions that warrant localized signaling. Furthermore, because $\sqrt{D_{\mathrm{H}}{k}_1{c}_{{\mathrm{PS}}^{-}}^{}}/k_p⪢1$, for fixed $D_{\mathrm{H}}$ and $k_{\mathit{p}}$ Eqs. (4), (5) imply that $c_{\mathrm{H}}(0)\propto \lambda$. Hence, any decrease in $k_{\mathit{sink}}$ will proportionately increase the length scale of the gradient. Moreover, the approximate proportionality of $c_{\mathrm{H}}(0)$ to $k_{\mathit{sink}}^{-1/2}$ implies that only very drastic inhibitions of cytoplasmic H₂O₂ clearance can have a substantial impact on the H₂O₂ concentrations near supply sites. For instance, in order increase $c_{\mathrm{H}}(0)$ 2-fold Prx would have to be at least 75% inhibited.

Considering the $c_{\mathrm{H}}(0)/c_{\mathrm{H}}^0$ ratio estimated above, a $c_{\mathrm{H}}^0=0.5\mathrm{\mu }\mathrm{M}$ would take $\sim 5.6$ days to oxidize 63% of a very reactive PTPs residing at the cytoplasmic face of the membrane. Therefore, the results above indicate that in the regime where steep cytoplasmic gradients are maintained, H₂O₂ concentrations are far too low to directly oxidize PTPs in a mitogenic signaling time frame. Indeed, numerical results considering more generic conditions (Fig. 4A) fully substantiate this view: in this regime the oxidation of the PTPs could take weeks to years even at the points of highest H₂O₂ concentration. At higher $c_{\mathrm{H}}^0$ values above $c_{\mathrm{H}}^{0*}$ the H₂O₂ invades the cytoplasm, as per the previous section, and the time for oxidation decreases. [These results remain qualitatively the same if the higher condensation rate constant estimated for peroxiredoxin I [43] is considered (Fig. S1A).] However, although peroxisome-sequestered catalase and remaining glutathione thiolate cannot sustain a cytoplasmic H₂O₂ gradient where NADPH depletion prevents the action of peroxiredoxins and peroxidases, they can still substantially delay the oxidation of PTPs by H₂O₂ [14].

Fig. 4.

H₂O₂capacity for signaling: A: Minimum time scale for direct oxidation of PTPs by H₂O₂. The inverse of the pseudo-first-order rate constant for the direct oxidation of the active form of a PTP by H₂O₂ with k_ox=164 M^-1s^-1 near the membrane is color coded as indicated. Values give the time needed to oxidize 63% of the target. Note the extremely slow oxidation under conditions that permit strong H₂O₂ gradients. Especially in the high hyperoxidation regime, times for oxidation are underestimated due to the neglect of alternative H₂O₂ sinks. B: Ratio between average intracellular concentrations of PSO⁻ and H₂O₂. Note the very high ratios (>1000) under conditions that permit strong H₂O₂ gradients. Results for the ratios between average intracellular concentrations of PSS and H₂O₂ are qualitatively similar. (The reader is referred to the web version of this article for a color version of this figure.).

Although the results above pertain to a scenario where H₂O₂ permeates uniformly across all the membrane, the maximal H₂O₂ concentrations remain insufficient to quickly oxidize PTPs even in the extreme scenario where all the permeation is localized in a specific channel with a 50 nm diameter (Fig. S1B,C). Therefore, PTPs cannot be directly oxidized by H₂O₂ under conditions that avoid extensive peroxiredoxin hyperoxidation and keep the cytoplasm protected against excessive H₂O₂.

The hysteretic behavior described above is abolished if cells have (a) lower capacity to reduce PSS and/or (b) more-active alternative H₂O₂ sinks. In case (a) high extracytoplasmic H₂O₂ concentrations cause the cytoplasmic 2-Cys peroxiredoxins to accumulate in disulfide form instead (Fig. S2A), which is also accompanied by the collapse of the H₂O₂ gradients (Fig. S2B). In case (b) peroxiredoxin oxidation increases more gradually with the H₂O₂ supply rate, and the alternative H₂O₂ sinks may eventually be able to sustain a cytoplasmic H₂O₂ concentration gradient. However, cytoplasmic H₂O₂ concentrations will be even lower and oxidation of PTP by H₂O₂ even slower in this case.

3.5. Peroxiredoxins show good signaling properties for moderate H₂O₂ pulses

Interestingly, under all conditions where strong intracellular concentration gradients of H₂O₂ are sustained the concentrations of ${\mathrm{PSO}}^{-}$ and PSS exceed those of H₂O₂ by several orders of magnitude (Fig. 4B). As a consequence, these oxidized peroxiredoxin forms can oxidize PTPs and other redox targets much faster than H₂O₂ if their reactivity with these targets is comparable to that of H₂O₂.

All the results presented so far pertain to situations where the extracellular H₂O₂ concentration is sustained long enough for the system to approach a steady state. However, H₂O₂ production is very dynamic in the seconds and minutes subsequent to growth factor binding to receptors [25], [27], [20]. Therefore, in order to further assess the signaling potentialities of ${\mathrm{PSO}}^{-}$ and PSS vs. H₂O₂ in this signaling context we simulated pulses of extracellular H₂O₂ concentration. The pulse shown in Fig. 5A attains a maximum extracellular H₂O₂ concentration of $\approx 0.75\mathrm{\mu }\mathrm{M}$. The brief pulse causes little ($<0.05\%$) hyperoxidation, but substantial peroxiredoxin oxidation (Fig. 5C,F). The results highlight two desirable properties of these peroxiredoxin forms as signaling intermediates. First, and as observed for the steady-state results, they attain much higher cytoplasmic concentrations than H₂O₂ (compare Fig. 5B,D). Second, they propagate the signal deeper into the cell than H₂O₂ (compare Fig. 5E,G). Namely, whereas the concentration of H₂O₂ decays near-exponentially with depth, high ${\mathrm{PSO}}^{-}$ and PSS concentrations are attained even $0.5\mathrm{\mu }\mathrm{m}$ from the membrane (Fig. 5D,G), allowing the oxidation of potential targets even at that depth.

Fig. 5.

Response to extracellular H₂O₂pulse. A: Time course of the pulse. B, C, D: Time course of cytoplasmic concentrations of H₂O₂, PS⁻, and PSO⁻, respectively, at depths 0 (violet) to $5\mathrm{\mu }\mathrm{m}$ (red) from the membrane ($0.5\mathrm{\mu }\mathrm{m}$ steps). E, F, G: Spatial profiles of cytoplasmic concentrations of H₂O₂, PS⁻, and PSO⁻, respectively, at times 0 (violet) to 180 s (red) from the membrane (5 s steps). The violet lines in panels E-G reflect the initial concentration change over depth. The time courses and spatial profiles for PSS are very similar to those for PSO⁻. Note the different scales for extracellular H₂O₂, cytoplasmic H₂O₂ and peroxiredoxin concentrations. The dashed gray line marks the time at which the cytoplasmic H₂O₂ concentration reaches its maximum. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

These good signaling properties deteriorate for stronger pulses. A pulse attaining a maximum extracellular H₂O₂ concentration of $5.9\mathrm{\mu }\mathrm{M}$ already causes $\approx 35\%$ hyperoxidation (Fig. S3E), and substantial H₂O₂ penetration in the cytoplasm (Fig. S3F). The concentrations of ${\mathrm{PSO}}^{-}$ and PSS start to decrease earlier than for the weaker pulse, and still in the ascending phase of the pulse, due to gradual hyperoxidation (Fig. S3D). The concentration gradients of these species are temporarily flattened out. Furthermore, the recovery of the peroxiredoxin redox status is strongly delayed, due to the slow reduction of ${\mathrm{PSO}}_2^{-}$ (Fig. S3C,D,E).

A pulse attaining a maximum extracellular H₂O₂ concentration of $12\mathrm{\mu }\mathrm{M}$ triggers the hyperoxidation “catastrophe” described above (Fig. S4). Virtually all peroxiredoxin is hyperoxidized within 1 min (Fig. S4E), all the cytoplasmic gradients collapse (Fig. S4F-H). Although maximal ${\mathrm{PSO}}^{-}$ and PSS concentrations similar to the previous pulse are attained at $\approx 10\mathrm{s}$ after the onset of the pulse, the concentrations of these species fall precipitously thereafter and nearly vanish after $\approx 1\min$ (Fig. S4D). Recovery of the peroxiredoxin redox state after the pulse occurs in a time scale of hours.

3.6. Peroxiredoxin recruitment causes modest local H₂O₂ depletion

A small fraction (up to a few percent) of PrxI and PrxII is able to associate to the cellular membrane [51], [25], [27]. Moreover, PrxII is recruited to VEGFR2 upon VEGF stimulation of endothelial cells, protecting this receptor against oxidative inactivation by H₂O₂ [37]. The most straightforward explanation for this protection is that the co-localization of PrxII locally depletes H₂O₂. In order to assess the extent of local H₂O₂ depletion that is achievable in this way, we model two different scenarios. In the first scenario we consider that 2% of all the peroxiredoxin is reversibly and uniformly bound to the membrane, and that all redox states of peroxiredoxin show similar binding and unbinding rate constants. This system is described by Eqs. (3). Similarly to the system analyzed in the previous sections, the H₂O₂ concentration either drops exponentially near the membrane, or invades the cell, depending on the value of $c_{\mathrm{H}}^0$ and on the total concentration of peroxiredoxin. The binding of this small fraction of peroxiredoxin to the membrane does not alter the critical values $c_{\mathrm{H}}^{0*}$ (compare Figs. 6A and 3B). However, it decreases the concentration of H₂O₂ adjacent to the membrane under the conditions where the peroxiredoxin is able to sustain a strong cytoplasmic H₂O₂ concentration gradient (i.e., where $c_{\mathrm{H}}^0<c_{\mathrm{H}}^{0*}$). This decrease is in the range of 10–25%, and is the stronger the higher the initial $c_{{\mathrm{PS}}^{-}}^{}$ and the lower $c_{\mathrm{H}}^0$ are (Fig. 6A).

Fig. 6.

Effects of peroxiredoxin recruitment: (1) Local H₂O₂depletion. A: Decrease in H₂O₂ concentration near the membrane if 2% of the peroxiredoxin is uniformly distributed at the membrane. B: Decrease in H₂O₂ concentration near caveolae if 2% of the peroxiredoxin is concentrated at one caveola. For the explored range of conditions local H₂O₂ concentrations can decrease by up to 25%. (The reader is referred to the web version of this article for a color version of this figure.).

In the second scenario 2% of all peroxiredoxin is bound specifically to one caveola. We considered a cylindrical 3D simulation box as depicted in Fig. 1C. We ran the simulation in the regime of interest, i.e., for high initial concentration of ${\mathrm{PS}}^{-}$ ($c_{{\mathrm{PS}}^{-}}(\mathbf{r},0)=80\mathrm{\mu }\mathrm{M}$) and relatively low $c_{\mathrm{H}}^0$ ($0.40\mathrm{\mu }\mathrm{M}$). The 1D results for the previous scenario indicate that these parameter values yield a moderate protection of the receptor by the bound peroxiredoxins, with little ${\mathrm{PSO}}_2^{-}$ accumulation. In the current 3D scenario, the concentration of ${\mathrm{PSO}}_2^{-}$ is also negligible, and a steady state is achieved by 3 s of simulation time. In this simulation, the steady state depends strongly on the magnitude of the binding and unbinding constants. We chose values of $k^{+}=1.0\times {10}^4{\mathrm{s}}^{-1}$ and $k^{-}=39.2{\mathrm{s}}^{-1}$, which guarantee the binding of 2% of the peroxiredoxin to the membrane.

Under these conditions H₂O₂ is appreciably depleted near the caveola (Figs. 6B and 7A). However, even this very strong concentration of peroxiredoxin at a caveola decreases the local H₂O₂ concentration adjacent to the membrane by no more than $\approx 25\%$, similar to the 1D simulation. This happens despite the local concentration of ${\mathrm{PS}}^{-}$ at the caveolae ($\approx 10\mathrm{mM}$, Fig. 7B) strongly exceeding the concentration at the membrane in the previous scenario, which should apparently lead to a stronger protection of the receptor. However, as the ${\mathrm{PS}}^{-}$ at the caveloa is oxidized by incoming H₂O₂ the quick binding of new ${\mathrm{PS}}^{-}$ molecules depletes them in the cytoplasm around the caveola (Fig. 7C), owing to their relatively slow diffusion. In turn, this allows more H₂O₂ to diffuse from neighboring regions in the cytoplasm, thereby attenuating the effectiveness of protection. Remarkably, in the plane of the membrane not only the bound ${\mathrm{PS}}^{-}$ but also the bound PSS and ${\mathrm{PSO}}^{-}$ are strongly concentrated at the caveola region; that is, within $0.10\text{μm}$ from the receptor (Fig. 7B). Furthermore, the concentrations of unbound PSS and ${\mathrm{PSO}}^{-}$ are also elevated in the cytoplasm near the caveola (Fig. 7D). Therefore, the recruitment of peroxiredoxins to caveolae can strongly favor the peroxiredoxin-mediated oxidation of redox targets over their direct oxidation by H₂O₂.

Fig. 7.

Effects of peroxiredoxin recruitment (2): Distribution of H₂O₂and oxidized peroxiredoxin for the case where 2% of the peroxiredoxin is bound at one caveola. A: Distribution of H₂O₂ close to the caveola. B: Distribution of the various peroxiredoxin forms at the membrane. C: Distribution of non-bound PS⁻ close to the caveola. We observe that in the caveola vicinity there is a minimum in the PS⁻ concentration. D: Distribution of non-bound PSO⁻ close to the caveola. The distribution of PSS is qualitatively similar. The concentrations of these peroxiredoxin forms is maximal next to the caveola. (The reader is referred to the web version of this article for a color version of this figure.).

4. Discussion

The results above address several key questions about H₂O₂ signaling and the participation of cytoplasmic 2-Cys peroxiredoxins in this process. Consistent with previous estimates [15], [40], models [18], [19] and experimental observations [20], we found that typical concentrations of 2-Cys peroxiredoxins are sufficient to sustain strong H₂O₂ concentration gradients in the cytoplasm of mammalian cells. As a result, in the absence of oxidative stresses able to extensively oxidize these peroxiredoxins most of the H₂O₂ in the cell cytoplasm is contained within a few tenths of μm of the supply sites (Fig. 2A,C). The existence of such steep H₂O₂ concentration gradients could explain how effective antioxidant protection can be achieved despite the high local H₂O₂ production rates involved in redox signaling [15]. However, the H₂O₂ concentrations attained even at the apex of these gradients are by and large insufficient to substantially oxidize PTPs in a mitogenic signaling time frame. Conservatively, we considered very high values for the H₂O₂ permeability constant and for the PTP oxidation rate constant, assumed diffusion constants for peroxiredoxin consistent with the mobility of its decameric form in a crowded cellular environment, and neglected alternative cytoplasmic H₂O₂ sinks. But even under these conservative assumptions oxidation of the PTPs would occur in a time scale of weeks (Fig. 4A) or longer. In an extreme scenario where all the H₂O₂ would permeate trough a single channel with 50 nm diameter the local H₂O₂ concentrations would oxidize PTPs in a time scale of several hours (Fig. S1B,C), still too slow for mitogenic signaling. This outcome ensues from the fundamental trade-off demonstrated in Section 3.4. Namely, given physiologically plausible H₂O₂ membrane permeabilities and diffusion constants, any agent consuming H₂O₂ fast enough to localize it within $\approx 1\mathrm{\mu }\mathrm{m}$ of a permeant membrane also generate a $>30$-fold trans-membrane gradient.

At moderately high H₂O₂ supply rates, peroxiredoxins can become substantially oxidized near the membrane, thus allowing for somewhat higher local H₂O₂ concentrations (Fig. 3B) at the cost of shallower cytoplasmic concentration gradients (Fig. 2C). However, the effect is modest (Fig. 2B,C) and restricted to a narrow range of H₂O₂ supply rates. These results also imply that the local inhibition of peroxiredoxins' peroxidatic activity – be it by hyperoxidation [21], phosphorylation [27] or non-covalent [40] – can only significantly increase local H₂O₂ concentrations if it substantially decreases the total cytoplasmic peroxidatic activity. Otherwise, the local inhibitory effect is efficiently counteracted by the diffusion of active peroxiredoxin from the bulk. However, whereas PrxI is inhibited by phosphorylation but not very susceptible to hyperoxidation, PrxII is susceptible to hyperoxidation but not inhibited by phosphorylation [27]. Because the concentrations of these two peroxiredoxins in the cytoplasm of mammalian cells are broadly similar [11], neither of these inhibitory mechanisms in isolation can dramatically increase local H₂O₂ concentrations. Indeed, it follows from Eq. (5) that a 50% decrease in total peroxidase activity will increase the H₂O₂ concentration near the membrane by at most $1-\sqrt{0.5}=29\%$. And as per Eq. (4) this will come at the cost of extending the length scale of H₂O₂ gradient by 29% as well.

At H₂O₂ supply rates slightly higher than those discussed in the previous paragraph, the peroxiredoxins become fully oxidized and H₂O₂ can then invade all the cytoplasm. But even in this situation where the interior of the cytoplasm would no longer be protected the oxidation of PTPs would be slow (Fig. 4A). A simple calculation shows that in order to oxidize 50% of the most reactive PTPs characterized to date ($k_{\mathit{ox}}=164{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ [6]) in 10 min a H₂O₂ concentration of $\log (2)/(164{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}\times 600\mathrm{s})=7\mathrm{\mu M}$ would be required. The oxidation of other PTPs in the same time frame would require H₂O₂ concentrations in the order of $100\mathrm{\mu }\mathrm{M}$. However, sustained $7\mathrm{\mu }\mathrm{M}$ extracellular H₂O₂ is sufficient to make Jurkat T cells apoptosise [52]. It is thus very unlikely that such high concentrations are attained anywhere in the cytoplasm.

The sulfenic and/or disulfide forms of 2-Cys peroxiredoxins oxidize redox targets in vitro and in vivo forming redox relays [34], [35], [28], [2], [36]. Remarkably, the computational analyses presented here highlight three important advantages of peroxiredoxin-mediated oxidation of redox targets (i.e., redox relays) over direct oxidation of the targets by H₂O₂. First, the concentrations of ${\mathrm{PSO}}^{-}$ and PSS exceed those of H₂O₂ by several orders of magnitude under conditions where H₂O₂ is contained near its supply sites (Fig. 4B). Therefore, even if these peroxiredoxin forms are just as reactive with the redox targets as H₂O₂ they can oxidize them much faster. However, the case of thioredoxin ($k_5=2.1\times {10}^5{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$ [13]) illustrates that PSS can oxidize some targets at very high rate constants. Second, there are also important concentration gradients of these peroxiredoxin forms, thus retaining the capacity for localized signaling. Third, unlike the H₂O₂ gradients, those of ${\mathrm{PSO}}^{-}$ and PSS plateau at the membrane (Figs. 3A,B and 5G). This further favors the oxidation of the targets and allows the redox signal to propagate deeper into the cytoplasm. The latter may be important in some signaling contexts where the targets are located in intracellular membranes [53], [54]. The penetration of these peroxiredoxin forms in the cytoplasm is less problematic from the point of view of antioxidant protection than that of H₂O₂ because their interactions are expected to be specific. These considerations suggest a “localized redox relay” model of redox signaling (Fig. 8). This model is consistent with the observations by Klomsiri et al. [55] of localized formation of reversibly oxidized protein forms near H₂O₂ sources in LPA-stimulated PC3 and SKOV3 cells.

Fig. 8.

H₂O₂signaling through localized redox relays. At H₂O₂ (blue) supply rates commensurate with mitogenic signaling typical 2-Cys peroxiredoxins in the cytoplasm of mammalian cells sustain steep gradients but keep H₂O₂ concentrations very low throughout. This outcome is determined by the high cytoplasmic concentrations and high reactivity of these proteins. But these factors together with the limited rates at which the sulfenic and disulfide forms of these proteins (red) are reduced also determine that the concentrations of these forms strongly exceed those of H₂O₂. The oxidized peroxiredoxin species also show steep concentration gradients over the cytoplasm, here represented by level curves in red lines. Therefore, the oxidation of redox targets (green) mediated by them is also restricted to a vicinity of the H₂O₂ supply sites. But unlike the H₂O₂ concentration gradients (level curves in blue), those of the peroxiredoxin species plateau near the H₂O₂ supply sites, allowing the redox signal to penetrate slightly deeper into the cytoplasm. The competitive advantage of the peroxiredoxin sulfenic or disulfide forms over H₂O₂ in oxidizing targets at specific sites (e.g., membranes, caveolae or endosomes) can be strongly amplified by localizing even a small fraction of the cytoplasmic peroxiredoxins to those sites. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The localized oxidation of the Prx could regulate targets in at least the following four alternative ways. First, ${\mathrm{PSO}}^{-}$ and PSS can directly oxidize some targets [33], [34], [28], [35], [2]. Second, because these Prx forms can be glutathionylated (PSSG) [56], [57], PSSG might in turn glutathionylate redox targets either directly or mediated by glutaredoxin, as suggested in ref. [14]. S-glutathionylation is a well-established regulatory post-translational modification for many proteins, and this mode of operation would mirror similar proposals with glutathione peroxidases as peroxide sensors and glutaredoxins as key players [58], [59], [60], [61]. However, the presently available information about the kinetics of PSSG deglutathionylation does not permit to determine if this species is also localized. Third, the transfer of oxidizing equivalents to targets might also be mediated by Trx [62]. Fourth, a conformational change induced by Prx oxidation might either enable or hinder a non-covalent interaction with a target [63]. Kinetic experiments in vitro will be instrumental for testing if ${\mathrm{PSO}}^{-}$, PSS and/or PSSG can outcompete H₂O₂ for actuation of redox targets.

The results above highlight that despite small and very diffusible, H₂O₂ cannot diffuse far away from production sites due to the very high activity of cytoplasmic peroxiredoxins, and the redox signal is actually carried to distal sites by proteins.

Although under conditions of mitogenic signaling hyperoxidation has just a minor effect on local H₂O₂ concentrations and weakens the cytoplasmic concentration gradient, at higher H₂O₂ the opening of the hyperoxidation “floodgate” could be part of a stress response. The analysis in ref. [43] predicts that in cells with abundant peroxiredoxin reduction capacity and limited alternative H₂O₂ sinks the peroxiredoxin/thioredoxin/sulfiredoxin system shows bi-stability and hysteresis. The simulations in Fig. 2, Fig. 3 show that the predicted bi-stability and hysteresis persist when concentration gradients are taken into account. Furthermore, they show that the onset of the high-hyperoxidation state is accompanied by the collapse of the cytoplasmic H₂O₂ and peroxiredoxin concentration gradients, and by very low ${\mathrm{PSO}}^{-}$ and PSS concentrations despite the high H₂O₂ concentrations. Under these particular conditions, signal transmission through redox relays is interrupted and H₂O₂ may become able to oxidize some intermediate-reactivity targets deep into the cell. This predicted behavior is reminiscent of experimental observations for Schizosaccharomyces pombe, where exposure to high H₂O₂ concentrations caused strong hyperoxidation of the Tpx1 peroxiredoxin, interruption of redox-relay signaling to the Pap1 transcription factor and accumulation sufficient cytoplasmic H₂O₂ to trigger the stress response mediated by mitogen-activated protein kinase Sty1 [33].

A small fraction of the cytoplasmic 2-Cys peroxiredoxins can be recruited to the cellular membrane [51], [25], [37] or to the centrosome [38], where they can protect sensitive proteins against oxidation by H₂O₂ [37], [38]. Our simulations of the effects of localizing 2% of the cytoplasmic peroxiredoxin (Fig. 6, Fig. 7) show a modest ($\sim 25\%$) decrease in local H₂O₂ concentrations despite the very high (up to 10 mM) local concentration of peroxiredoxin considered. Moreover, the effect is quantitatively similar irrespective of the peroxiredoxin binding uniformly to the membrane or just at caveolae. This is because the fast binding of ${\mathrm{PS}}^{-}$ to the membrane in replacement of oxidized molecules depletes ${\mathrm{PS}}^{-}$ from the neighboring cytoplasm, facilitating the inflow of H₂O₂ into this region.

On the other hand, the recruitment of peroxiredoxin to specific sites can originate very high local concentrations of ${\mathrm{PSO}}^{-}$ and PSS. This effect can be further amplified if the H₂O₂ supply is also localized at the same sites, for instance through permeation by aquaporins [64], [65]. Oxidation by ${\mathrm{PSO}}^{-}$ and PSS of targets also recruited to the same sites could thus be very fast. Likewise, a local inactivation of PrxI as described in ref. [27] although having a modest effect on local H₂O₂ concentrations could effectively block signal transmission through PrxI-mediated redox relays. Therefore, the possibility that the recruitment of peroxiredoxins to specific sites plays a more important role in facilitating and regulating the operation of localized redox relays than on modulating local H₂O₂ concentrations deserves further consideration. The observed protective effect of PrxII against VEGFR2 oxidation by H₂O₂ can as well be explained in this framework: the selective binding of oxidized PrxII, but not ${\mathrm{PrxIIS}}^{-}$, to VEGFR2 might protect the sensitive thiols in the latter protein by blocking access to H₂O₂ and/or by inducing a conformational change that hindered its reactivity.

Although the geometry addressed in this work applies most properly to situations where H₂O₂ permeates the apical and basolateral membranes of a cell in a cell layer, the qualitative results are valid for other geometries. Namely, for H₂O₂ release from endosomal vesicles or other intracellular membranes. Likewise, the key features determining the occurrence and functional advantages of localized redox relays over direct oxidation of targets by H₂O₂ are common to most eukaryotic cells known to date.

Author contributions

AS, FSA, RDMT designed analyses; AB, FSA, PA, RDMT performed analyses; AS, RDMT wrote the paper.

Competing financial interests

The authors declare no competing financial interests.

Appendix A. Supplementary data

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