The architecture of redox microdomains: Cascading gradients and peroxiredoxins’ redox-oligomeric coupling integrate redox signaling and antioxidant protection

In the cytosol of human cells under low oxidative loads, hydrogen peroxide is confined to microdomains around its supply sites, due to its fast consumption by peroxiredoxins. So are the sulfenic and disulfide forms of the 2-Cys peroxiredoxins, according to a previous theoretical analysis [Travasso et al., Redox Biology 15 (2017) 297]. Here, an extended reaction-diffusion model that for the first time considers the differential properties of human peroxiredoxins 1 and 2 and the thioredoxin redox cycle predicts important new aspects of the dynamics of redox microdomains. The peroxiredoxin 1 sulfenates and disulfides are more localized than the corresponding peroxiredoxin 2 forms, due to the former peroxiredoxin's faster resolution step. The thioredoxin disulfides are also localized. As the H2O2 supply rate (vsup) approaches and then surpasses the maximal rate of the thioredoxin/thioredoxin reductase system (V), these concentration gradients become shallower, and then vanish. At low vsup the peroxiredoxin concentration determines the H2O2 concentrations and gradient length scale, but as vsup approaches V, the thioredoxin reductase activity gains influence. A differential mobility of peroxiredoxin disulfide dimers vs. reduced decamers enhances the redox polarity of the cytosol: as vsup approaches V, reduced decamers are preferentially retained far from H2O2 sources, attenuating the local H2O2 buildup. Substantial total protein concentration gradients of both peroxiredoxins emerge under these conditions, and the concentration of reduced peroxiredoxin 1 far from the H2O2 sources even increases with vsup. Altogether, the properties of 2-Cys peroxiredoxins and thioredoxin are such that localized H2O2 supply induces a redox and functional polarization between source-proximal regions (redox microdomains) that facilitate peroxiredoxin-mediated signaling and distal regions that maximize antioxidant protection.

Peroxiredoxins are legitimate H 2 O 2 receptors because: (i) owing to their high reactivity (k ≈ 10 7 -10 8 M − 1 s − 1 [20][21][22][23]) and abundance (>50 μM [24]) they capture nearly all the H 2 O 2 supplied to their host cellular compartments at physiological rates; and (ii) they can specifically transmit this signal to a subset of proteins defined, e.g., by topological and electrostatic complementarity.The most abundant peroxiredoxins in the cytosol of human cells are peroxiredoxins 1 and 2 (Prdx1, Prdx2) [24,25].These are 2-Cys peroxiredoxins, whose functional units are homodimers with monomers in an antiparallel conformation, each monomer containing two catalytically active Cys (Fig. 1A).One of these two Cys, the peroxidatic Cys (C P ), is very rapidly oxidized by H 2 O 2 to a sulfenate (Prdx-SO − ).The sulfenate condenses with the other Cys (resolving Cys, C R ) in the opposing monomer, forming a disulfide (Prdx-SS), in what is called the "resolution step".It can also be oxidized to a sulfinate (Prdx-SO 2 − ) by another H 2 O 2 molecule ("hyperoxidation").Whilst the disulfide can be readily reduced to the initial thiolate (Prdx-S − ) by thioredoxin 1 (Trx1-S − ), the sulfinate is slowly reduced to a sulfenate in an ATP-and GSH-dependent reaction catalyzed by sulfiredoxin (Srx).Trx1 that becomes oxidized to Trx1-SS in Prdx reduction is reduced by NADPH, under catalysis by Thioredoxin Reductase (TrxR).
The redox state of Prdx1 and Prdx2 influences their oligomeric state (Fig. 1A).The reduced forms are donut-shaped pentamers of dimers, and these decamers may associate in even higher order structures when hyperoxidized [26][27][28][29].In turn, the disulfide forms dissociate at least

ASK1
Apoptosis signal-regulating kinase  partially into dimers [26][27][28][29].Prdx1-SS decamers can be stabilized through the formation of dimer-to-dimer disulfide bridges via the oxidation of Cys83 under oxidative stress [30], and destabilized by glutathionylation of Cys83 [28].Despite these complexities, studies with a homo-FRET probe suggest that in cells both Prdx1 and Prdx2 substantially dissociate into dimers when oxidized to their disulfide forms [31].Both the large size and the extended hydrodynamic radius (relative to a compact globular shape) of the high molecular weight forms should substantially hinder their mobility in the cytosol relative to that of the dimers.However, the functional significance of these drastic changes in quaternary structure remains unclear.Both Prdx-SO − and Prdx-SS may in principle react with the thiols of target proteins, relaying the oxidizing equivalents.A recent screen [32] found a relatively low overlap between the binders of Prdx1 and Prdx2 in human cells, and distinct preferences for oxidizing their interactors via heterocondensation with Prdx-SO − vs. thiol-disulfide exchange with Prdx-SS.This indicates that signaling through Prdx-mediated redox relays is Prdx-specific and prompts an examination of the underpinnings and functional implications of this specificity.A differential localization of the oxidized forms of these two Prdx may contribute to their distinct specificity.At the low H 2 O 2 supply rates (v sup ) prevailing under basal physiological conditions, the activity of the cytosolic 2-Cys Prdx limit H 2 O 2 's concentration to the sub-nM to low-nM range and half-life to <1 ms [7,33].A H 2 O 2 molecule can diffuse over a range of just ≈0.5 μm within this short half-life.For this reason, under most physiological conditions cytosolic H 2 O 2 is limited to sub-μm-scale domains ("microdomains") around the supply sites.The latter inference from theoretical considerations [6,[34][35][36], was confirmed by experiments using sensitive genetically encoded probes [8,[37][38][39][40], as recently reviewed [41].At these low H 2 O 2 supply rates, the oxidized forms of Prdx1 and Prdx2 are also circumscribed to microdomains [6].This is because, although these peroxiredoxin forms are much longer lived than H 2 O 2 , their diffusion in the cytosol is also much slower due to their large size.
In ref. [42] we examined the spatial distribution of H 2 O 2 and Prdx redox forms using a simple reaction-diffusion model that considered a single 2-Cys peroxiredoxin and neglected the redox cycle of Trx1.The experimental determination of the rate constants for all the main steps in the catalytic cycles of Prdx1 and Prdx2 has been accomplished more recently as result of the work of several groups [20][21][22][23]29,[43][44][45].This information has allowed us to set up the first reaction-diffusion model of the peroxiredoxin-thioredoxin system in the cytosol of human cells that accounts for the differential properties of these Prdx and for the Trx1 redox cycle.Using this model, we examined several questions prompted by the discussion above that were beyond the scope of the previous [42] model: How do the concentration gradients of distinct Prdx species differ?Is oxidized Trx1 also localized?What determines the extent of localization of the various species and the collapse of the gradients?How does the effect of redox state on oligomerization influence the operation of the Prdx -Trx system?
The results reveal steep Prdx-SO − , Prdx-SS and Trx1-SS concentration gradients, substantially steeper for the Prdx1 than for the Prdx2 species.The gradients become shallower and then collapse as the H 2 O 2 supply rate approaches and then surpasses the maximal rate of Prdx-SS reduction by the Trx1/TrxR system.In contrast to the results in Ref. [42], but consistent with a subsequent analysis based on a simpler model [24], this transition is not hysteretic (i.e., trigger-like).Surprisingly, far from H 2 O 2 sources the concentration of reduced Prdx1 even increases with increasing H 2 O 2 supply rate, as a consequence of the coupling between peroxiredoxin's redox and oligomerization states.We discuss the implications of these results for redox signaling and antioxidant protection, and how they help explaining recent experimental observations.

Model and parameter estimates
The considered reaction network is represented in Fig. 1B, and the corresponding parameters are shown in Table 1.The main considerations underlying the choice of geometry, reactions and parameter values are explained below.
Geometry.We consider a 10 µm-thick (except where otherwise noted) cell layer with a symmetry axis at the center, delimited by a membrane on each side (Fig. 1C).This geometry allows the spatial distribution of the various chemical species to be modeled considering a single spatial dimension.Regions adjacent to the H 2 O 2 sources (i.e., to the membranes) will be denoted as (source-)"proximal", whereas the regions farthest from the sources (i.e., at the center of the considered spatial domain) will be denoted as "distal" (Fig. 1C).
H 2 O 2 supply is assumed to occur equally at both membranes, at a constant rate (v sup ).H 2 O 2 likely crosses membranes passively in some cell types and organelles [51], through channels such as peroxiporins in others [52][53][54][55][56], and may also be produced at the surface of some internal membranes.The model is agnostic about these alternative modes of H 2 O 2 supply.Analysis of theoretical and experimental evidence in the literature [7,42,[57][58][59][60][61][62][63] (Supplementary Information Section 1, SI1) suggests that basal v sup in human cells are no higher than the low μM s − 1 .
Higher values are likely during oxidative stress, although achieving 100s of uM s − 1 may be unlikely.
H 2 O 2 -clearing pathways other than reaction with Prdx1 and Prdx2 were aggregated in a single first-order process with rate constant k Alt = 100 s − 1 .This value is in the range of estimates that include the activities of peroxiredoxin 6, glutathione peroxidases, and catalase for a variety of human cell lines [24].
Catalytic cycle of Prdx1 and Prdx2.The core reaction network includes the catalytic cycles of both the major 2-Cys peroxiredoxins in the cytosol of human cells, according to an independent-sites model.A modest intra-dimer cooperativity of Prdx2 [46] and likely also of Prdx1, can be neglected without major impact on the conclusions of this work.The choice of values for the rate constants of each step is justified below.
The resolution rate constant for Prdx1-SO − is taken as the mean between quite similar values in recent literature: 12.9 s − 1 [22], 9 s − 1 [23], 11-12 s − 1 [44].For Prdx2-SO − , this was determined in Ref. [46], which reports that the formation of the first disulfide in a dimer approximately halves the resolution rate constant of the sulfenate at the second site.Because this work mainly addresses the events at low oxidative loads, we adopted the rate constant that pertains to dimers still devoid of disulfides.The adopted value is also in the range of other recent determinations: 0.25 s − 1 [23], 0.64 s − 1 [22], 0.2-0.3s − 1 [44].
Thioredoxin disulfide reduction follows Henri-Michaelis-Menten kinetics.We assumed that the enzyme is saturated with NADPH, given its low K M (NADPH) = 6 μM [71].The chosen V Max is in the range of estimated activities for many human cell lines [24].However, in aligning the results with the v sup ranges discussed above one must keep in mind that some cells have V Max values one order of magnitude lower [24].
Diffusion.For H 2 O 2 we used the diffusion constant determined for a hydrogel with the approximate viscosity of the cytosol [50].We estimated the diffusion constants for the peroxiredoxin decameric (i.e., with the peroxidatic cysteines in thiolate, sulfenic and sulfinic forms) and dimeric (disulfide) forms from the experimentally determined diffusion constant for Immunoglobulin G in the cytosol [72] by applying the expression , with MW Prdx replaced by the molecular weight of a Prdx decamer or of a dimer, respectively.The previous expression follows from the Stokes-Einstein relationship by assuming spherical proteins.The estimated diffusion constant for Trx1 is a mean of the values obtained by applying the expression above to the diffusion constant obtained for myoglobin diffusion in rat myocardium cells (MW Myo = 17.7 kDa, D Myo = 42.4 μm 2 s -1 [73]) and that obtained for 10 kDa Dextran in amphibian oocytes (D Dex = 25 μm 2 s -1 [74]).These estimates are admittedly rough.However, because under the conditions of interest for this work the maximal concentrations and characteristic lengths of the concentration gradients scale approximately as D − 1/2 , the uncertainty in the diffusion constants has a limited effect on the results.
The reference values adopted for the total concentrations of Prdx1, Prdx2 and Trx1 (Table 1) are within the range and proportions of estimated values for HEK293 and other human cell lines [24].

Equations
The concentrations of the chemical species in the cytosol depend on time and distance from the membrane as mathematically described by the following system of reaction-diffusion partial differential equations, whose parameters are described in Table 1: All the concentrations have zero-flux boundary conditions except for that of H 2 O 2 at the membrane side of the domain (x = 0).Here, the flux of H 2 O 2 entering the domain per unit time is: where L is the distance from the membranes to the cell center.

Methods
The equations were solved using the method of lines [75], in which a PDE is discretized in all but one dimension.Here we discretized the spatial dimension.This reduced the system to only one continuous M. Griffith et al.
dimension (time), allowing solutions to be computed via numerical integration techniques for ordinary differential equations (ODEs).We use a second-order central finite difference for the second order spatial derivative as well as a second-order central difference for the first order spatial derivative in the boundary conditions in order to preserve second-order accuracy in spatial derivatives across the whole system.A uniform mesh in space with size Δx = 0.1 μm is used.
To solve the resultant system of ODEs, we used the LSODA solver within the scipy package in Python [76].This solver uses a combination of the Adams and BDF methods with automatic stiffness detection and switching between both methods.LSODA is a wrapper to the Fortran solver from ODEPACK [77].It switches automatically between the non-stiff Adams method and the stiff BDF method with adaptive time stepping.The method was originally detailed in Ref. [78].
This integration method provided excellent total mass conservation in the system, with only a fractional loss (0.01 %) in the simulations with fastest concentration variations.The method of lines approach also allows the solver to adaptively change the time step as the simulation progresses.This means that small time steps are used initially when the system is subject to large changes, to preserve good mass conservation in the system, while larger time steps are used when the system approaches the steady state.We use this to obtain the results that follow in the next section.

Dynamics of the concentrations after the onset of H 2 O 2 supply
We first examine the time course of the spatial distribution of the various species following the onset of a constant v sup = 10 μM s − 1 H 2 O 2 supply rate, starting with the Prdx1, Prdx2 and Trx1 pools fully reduced.For this v sup and the reference parameters in Table 1, there is a modest oxidation of these pools (Fig. 2A,B,C).The dynamics of the concentrations of H 2 O 2 and of the oxidized forms of Prdx1, Prdx2 and Trx1 unfolds over a wide range of time scales, as follows.The H 2 O 2 concentration gradient establishes and reaches a quasi-steady-state within 1 ms, as a consequence of the fast diffusion of this species and of the high concentrations and reactivity of Prdx1 and Prdx2 (Fig. 2L).The concentrations of Prdx1-SO − and Prdx1-SS approach their quasi-steadystate in the 0.1 s time scale (Fig. 2D,F).In turn, the concentrations of the corresponding Prdx2 forms do so with some delay, in the seconds time scale, their gradients fully settling only by ≈10 s (Fig. 2E,G).The slower dynamics of these Prdx2 species is mainly due to this peroxiredoxin's lower resolution rate constant (k C2 ) relative to Prdx1's.The concentration of Trx1-SS settles in approximately the same time scale as Prdx1-SS in the proximal regions (Fig. 2H), but only by ≈10 s at the distal one (Fig. 2H inset).Finally, the concentrations of Prdx1-SO 2 − and Prdx2-SO 2 − take nearly 1 h to settle to a steady state, due to the slow Srxcatalyzed reduction of these species.This process is too slow to sustain significant Prdx-SO 2 − concentration gradients (Fig. 2J and K).Nevertheless, the concentrations of Prdx1-SO 2 − and Prdx2-SO 2 − remain nM, because at this low v sup sulfinylation of the sulfenic Prdx species cannot significantly compete with the condensation reaction.

At low oxidative loads Prdx1 disulfides and sulfenates are more localized than Prdx2's
The steady state spatial distributions of the various species deserve particular attention due to their relevance for redox signaling.For v sup < 50 μM s − 1 , H 2 O 2 , the peroxiredoxins' sulfenic forms, as well as the peroxiredoxins' and Trx1's disulfide forms, are all localized near the H 2 O 2 sources (Fig. 3A,B,D,E).Concretely, the concentrations of H 2 O 2 , Prdx1-SO − , Prdx2-SO − , Prdx1-SS, Prdx2-SS and Trx1-SS decrease to half their values adjacent to the membrane 0.35 μm, 0.57 μm, 1.2 μm, 0.65 μm, 1.4 μm and 1.1 μm away, respectively (Fig. 3B).(Due to the uncertainties about the diffusion constants of the proteins and other factors highlighted in SI2, the absolute values presented in this section are merely indicative.The qualitative relationships are robust, though.)The localization of these Prdx oxidized forms agrees with earlier predictions from a simpler model that considered only a single peroxiredoxin [6].The present model reveals, additionally, a stronger localization of Prdx1-SO − relative to Prdx2-SO − (Fig. 3D), which is due to the ≈20-fold faster resolution step of the former species.Also as a result of Prdx2's slower resolution step, the concentration of Prdx2-SO − in the proximal regions is 2.7-fold that of Prdx1-SO − , despite the total concentration of Prdx2 being just 27 % of Prdx1's, and the Prdx2-SO − /Prdx1-SO − concentrations ratio increases to ≈400-fold 5 μm away (Fig. 3D, inset).
Prdx1-SS is also more localized than Prdx2-SS.This primarily reflects the stronger localization of the former's precursor species, and is just slightly enhanced by the faster reduction of Prdx1-SS by Trx1 relative to Prdx2-SS.Despite this faster reduction, the proximal concentration of Prdx2-SS is just 45 % of Prdx1-SS's, though distally Prdx2-SS is 65.-fold more concentrated than Prdx1-SS (Fig. 3D, inset).In turn, the localization of Trx1-SS (Fig. 3E) primarily reflects the distribution of Prdx1-SS and fast reduction via TrxR.In contrast to the other oxidized peroxiredoxin forms, Prdx1-SO 2 − and Prdx2-SO 2 − are distributed nearly uniformly throughout the aqueous space (Fig. 3B).Unlike the other reduced forms of Prdx1, Prdx2 and Trx1, Prdx1 2 -S − is strongly localized to the H 2 O 2 sources (Fig. 3C inset).
The amplitudes (Fig. 4A) and length scales (Fig. 4B) of the concentration gradients remain almost unchanged over v sup values up to one order of magnitude lower than the system's maximal rate of Prdx-SS reduction (Fig. 4C), which in this case is approximately V Max .But as v sup further approaches this capacity, the gradients of the oxidized species become progressively shallower, and those of the reduced species become steeper (Fig. 4A,C,D).At v sup values very near V Max , the concentrations of Prdx1/2-SO − are highest between the proximal and distal regions (Fig. 4D).At even higher v sup , the gradients of the Prdx and Trx1 species vanish (Fig. 4A,E).The considered k Alt = 100 s − 1 alternative sink makes the H 2 O 2 concentration decrease by just 33 % from the proximal to the distal region under these conditions (Fig. 4A,E).

Table 1
Processes considered and reference values of the corresponding parameters.

Values
Refs.M. Griffith et al.

Saturation of the capacity of the Trx1/TrxR system to reduce Prdx disulfides determines the breakdown of localization
Fig. 5 shows the influence of the various enzyme activities and concentrations on the breakdown of the H 2 O 2 concentration gradient.For the reference conditions, the main determinant of the maximal v sup that cells can sustain while keeping a strong H 2 O 2 concentration gradient (v sup,crit ) is the maximal rate of Prdx-SS reduction (Fig. 5A).
Here, this rate approximately matches the TrxR activity (V Max ), but Trx1 depletion from the cytosol may further decrease v sup,crit (Fig. 5B).In turn, the total concentrations of Prdx control the amplitude (Fig. 5E and  F) and length scale of the H 2 O 2 gradients, but have almost no influence on v sup,crit .The comparatively modest activity of alternative H 2 O 2 sinks has a marginal influence on the H 2 O 2 gradients at v sup < v sup,crit but it determines the amplitude of the more modest H 2 O 2 gradients at higher v sup (Fig. 5D).Finally, the Srx activity virtually does not influence the H 2 O 2 gradients (Fig. 5C).
Effects of these enzyme activities and protein concentrations on the gradients of the oxidized Prdx and Trx1 species follow similar trends (Fig. S1).Of note, the Trx1-SS concentration gradient attains its maximal amplitude at intermediate v sup values below v sup, crit (Fig. S1, last row).

Cytosol's redox polarity increases as H 2 O 2 supply rates approach the critical value
The steady state response of the various species to the H 2 O 2 supply rate (Fig. 6) is another important consideration for redox signaling.Near the H 2 O 2 sources the Prdx1, Prdx2 and Trx1 pools remain largely reduced over v sup values up to the low μM s − 1 range (Fig. 6A,B,C,black lines).The Prdx2-S − pool starts being significantly depleted in these locations for v sup > 5 μM s − 1 , as the slow resolution step begins to be limiting and Prdx2 accumulates mostly as Prdx2-SO − (Fig. 6E, black line).In turn, the Prdx1-S − and Trx1-S − pools only begin to be significantly depleted at higher v sup (>15-20 μM s − 1 ) (Fig. 6A,C, black lines), by virtue of the higher resolution rate constant of Prdx1-S − and of the high TrxR activity catalyzing Trx1-SS reduction, respectively.As v sup increases, both peroxiredoxins accumulate predominantly in sulfenic form (Fig. 6D and E, black lines).However, as v sup further approaches v sup,crit the disulfide forms gain prominence as a consequence of the local depletion of the Trx1-S − pool (Fig. 6F,G,H black lines).At v sup in the range of ½-to 2-fold v sup,crit , a large fraction Prdx1 locally accumulates in the still poorly characterized reduced but presumably peroxidatically inactive [45] "Prdx1 2 -S − " form (Fig. 6I).
Away from the H 2 O 2 sources, substantial depletion of the Prdx-S − and Trx1-S − pools occurs only at higher v sup , approaching the considered 180 μM s − 1 TrxR activity, but is much more abrupt (Fig. 6A,B,C, dashed red lines).The Prdx1-S − concentration even increases over this v sup range, for reasons that will be addressed in the next section.The oxidized forms of both Prdx and of Trx1 only attain μM local concentrations at v sup values very close to v sup,crit (Fig. 6D-H, dashed red lines).Therefore, as v sup increases up to v sup,crit the redox polarity of the cytosol, as expressed by spatial differences in the availability of reducing At H 2 O 2 supply rates beyond v sup,crit , μM H 2 O 2 concentrations invade the cytosol (Fig. 6L), and the cytosolic concentration gradients of all the Prdx and Trx species vanish.Concurrently, a large and increasing fraction of both peroxiredoxins accumulates in sulfinic form, throughout the cytosol, decreasing the oxidative load on Trx1 (Fig. 6C, H, J, K).
Considering that Prdx1 2 -S − is decameric (Fig. S2) or that it is instantaneously converted to (active) Prdx1-S − (Fig. S3) does not significantly influence the results above.
In erythrocytes [79] and in some arterial cells [17], Prdx2 is the dominant cytosolic 2-Cys Prdx.In order to examine the consequences of distinct relative abundances of Prdx1 and Prdx2, we repeated the simulations underlying Fig. 6 after swapping the total concentrations of these two proteins (Fig. S4).The spatio-temporal dynamics in this case does not qualitatively differ from that in cells where Prdx1 is the dominant cytosolic 2-Cys Prdx, but it does differ in the following important quantitative aspects.The Trx1-S − pool does not get as strongly depleted as v sup reaches v sup,crit (Fig. S4C), because a larger fraction of the Prdx pool accumulates in sulfinic form and the Prdx disulfides do not accumulate as much (compare Figs. S4F,G,J,K to the corresponding panels in Fig. 6, black lines).The distal Prdx1/2-S − pools are depleted more gradually, and Prdx2-SO − and Prdx2-SS accumulate to μM concentrations as v sup increases (compare Figs. S4A,B,E,G to the corresponding panels in Fig. 6, dashed red lines).These outcomes are mainly a consequence of the longer lifetime of Prdx2-SO − relative to Prdx1-SO − , allowing the former to penetrate farther from the H 2 O 2 sources.

The lower mobility of prdx decamers relative to dimers helps protect distal regions from H 2 O 2
The source-distal concentration of Prdx1-S − actually increases with the H 2 O 2 supply rate up to v sup,crit (Fig. 7A, top panel black dashed line).Near v sup,crit , the distal Prdx1-S − concentration is ~ 16 % higher than in the absence of a H 2 O 2 supply.Moreover, the overall pools of both Prdx1 and Prdx2 accumulate distally at v sup just below v sup,crit (Fig. 7B and C).These surprising phenomena are due to the interplay between the redox polarity of the environment and the increased mobility of the Prdx-SS dimers relative to the Prdx-S − decamers.More concretely (Fig. 7F), as v sup increases and a Trx1 redox gradient develops, Prdx-SS dimers originate mainly near the H 2 O 2 sources whereas Prdx-S − decamers are regenerated mainly far from the sources, where Trx1-S − remains abundant.Because the large toroidal decamers diffuse more slowly than the dimers, they are more likely to accumulate near their production sites.As a consequence, the overall Prdx pool also accumulates distally.In agreement with this explanation, these phenomena do not occur when the peroxiredoxin disulfide forms are assumed to have the same diffusion constant as the other peroxiredoxin forms (Fig. 7A,D,E).The longer lifetime of Prdx2-SO − relative to Prdx1-SO − attenuates these As a consequence of the phenomena described in the previous paragraph, at a v sup that approaches v sup,crit , the distal H 2 O 2 concentration is up to 31 % lower than it would otherwise be (Fig. 7A, inset in bottom panel), and the mean H 2 O 2 concentration over the cytosol is also slightly lower (Fig. S5).
Considering that Prdx1 2 -S − is decameric (Fig. S6) or that it is instantaneously converted to (active) Prdx1-S − (Fig. S7) does not significantly influence the results above.In cells such as human hepatocytes that have substantially lower TrxR activity the phenomena described in this section occur at substantially lower v sup , and are even more pronounced (Fig. S8).

Redox microdomains have a layered architecture around localized H 2 O 2 sources
We drew on the recent experimental determination of the differential kinetic parameters for the redox cycles of human Prdx1 and Prdx2 (Fig. 1, Table 1) to explore the compositional architecture and dynamics of redox microdomains.The considered concentrations/activities of Prdx, Trx1, TrxR, sulfiredoxin and alternative H 2 O 2 sinks are typical of human cell lines, and their quantitative relationships are such as to yield these cells' stereotypical response to H 2 O 2 as discussed in Ref. [24].For computational tractability, we considered a geometry where H 2 O 2 is uniformly supplied to the cytosol from two parallel membranes.However, the qualitative results are expected to apply as well to other geometries where H 2 O 2 is supplied at discrete sources, such as from endosomes [37], mitochondria [40,80], the endoplasmic reticulum [37,81,82] or peroxisomes [83].
The results show that for H 2 O 2 supply rates corresponding to basal conditions to moderate stress (SI1) steep concentration gradients of H 2 O 2 and of the oxidized forms of Prdx1, Prdx2 and Trx1 readily establish once H 2 O 2 is locally supplied to the cytosol, and reach a steady state in a few seconds (Fig. 2).Unlike the gradients of the other oxidized species, those of the Prdx sulfinic species virtually vanish at steady state (Fig. 2J and K), due to these forms' slow turnover.
As a consequence of Prdx1-SO − 's faster resolution step, the steady state concentrations of Prdx1-SO − and Prdx1-SS decline by 50 % of their maxima at approximately half the distances from H 2 O 2 sources relative to their Prdx2 counterparts (Fig. 3B).The Prdx disulfides decay by 50 % just slightly farther from the H 2 O 2 sources than the respective precursor sulfenates, despite the former diffusing faster (as dimers) than the latter.This happens due to the fast reduction of the disulfide forms by Trx1, resulting in half-lives of just 10-70 ms.In turn, Trx1-SS decays to half of its maximum concentration at about the same distance from the H 2 O 2 supply sites as Prdx2-SO − , for the reference TrxR activity.Importantly, all these concentrations decrease approximately exponentially with sources, the relevant oxidized Prdx species may already be too rarefied (nM concentrations, Fig. 3D inset) to oxidize redox targets in the fewminutes time frame observed in experiments (e.g., [11,[84][85][86]).[Note that the fastest known oxidation of other proteins by Prdx-SS, that of Trx1-S − , has rate constants in the 10 6 M − 1 s − 1 range (Table 1), and no comparably fast oxidations by Prdx-SO − are known.]Therefore, these species can't directly actuate targets anchored at distant sites.For diffusible targets, most of the Prdx-mediated signal transduction must occur proximal to H 2 O 2 sources, and the resulting target modifications must be stable enough to reach possibly distant actuation points such as gene promoters.Secondly, because the concentrations of oxidized species decrease nearly exponentially with the distance from the H 2 O 2 sources, the seemingly small differences in the gradients' length scales in Fig. 4B translate into orders-of-magnitude differences in gradient amplitude and concentration a few μm away (Fig. 3D inset, Fig. 4A).Thus, for the reference conditions and a 1 μM s − 1 H 2 O 2 supply rate, the Prdx2-SO − /Prdx1-SO − concentration ratio grows from 2.7 to ≈400 when comparing proximal regions to distal ones 5 μm away (Fig. 3D); and whilst at the proximal regions the concentration of Prdx1-SS is 2.2-fold higher than Prdx2-SS's, it is 65-fold lower at the distal regions.Therefore, for similarly reactive targets, Prdx1-SO − and Prdx1-SS should act closer to the H 2 O 2 sources than the corresponding Prdx2 forms.This remains true over a wide range of H 2 O 2 supply rates, almost up to those causing the collapse of the gradients (Fig. 4).
Nevertheless, the above-characterized localization may not fully avoid interference between the potentially thousands of discrete sites releasing H 2 O 2 to the cytosol (in mitochondria, peroxisomes, endocytic vesicles, cell membrane, etc.).Whether mediation of redox relays by site-targeted protein scaffolds, such as recently described for the Prdx2-STAT3 relay [87], can avoid such interference is currently under investigation.
It must be noted that the absolute values of the gradient length scales and amplitudes above are merely indicative, for the reasons discussed in SI2.However, the conclusions about the relative localization of Prdx1 versus Prdx2 forms are robust.

The activity of the peroxiredoxin disulfide reduction system determines the H 2 O 2 supply rates at which the gradients collapse
The amplitudes and length scales of the concentration gradients described above are almost invariant over aggregate H 2 O 2 supply rates (v sup ) up to one order of magnitude lower than the maximal rate at which the Trx/TrxR system can reduce Prdx-SS (Fig. 4).Under these conditions, the H 2 O 2 supply rate at each discrete source determines the local concentrations of H 2 O 2 , Prdx and Trx species, as long as the sources are separated enough to avoid mutual interference.At higher v sup values, the Prdx-S − pools become gradually more depleted.As a consequence, the H 2 O 2 's half-life increases, allowing these molecules to diffuse farther and thus flatten the gradient.The gradients of the resulting oxidized species can be no steeper than those of the respective precursors, and therefore become shallower as well.When the aggregated v sup (i.e., the sum of the supply rates at all sources) surpasses the cytosol's Prdx-SS reduction capacity, the peroxiredoxins become completely oxidized.Because the activity of the alternative sinks is too low to sustain a significant H 2 O 2 gradient, all the gradients then collapse and H 2 O 2 invades the cytosol at μM concentrations.Of all the enzyme activities and protein concentrations considered in the model, only the TrxR activity, and to a lesser extent the total Trx1 concentration, control the value of v sup at which the gradients collapse (Fig. 5).In turn, the Prdx concentration is the main variable controlling the amplitude and length scale of the H 2 O 2 gradient, the most abundant Prdx having the greatest influence.
Although the TrxR activity hardly influences the H 2 O 2 concentrations at low v sup , it gains a substantial influence as v sup approaches TrxR's apparent maximal rate (Fig. 5A).This happens for the following reason.At these v sup the TrxR becomes progressively less able to keep up with the flux of Trx1-SS generated by the reduction of Prdx1/2-SS.As a consequence, the Trx1-S − pool becomes depleted, and Prdx1/2-SS reduction becomes the rate-limiting step in the Prdx catalytic cycle.This causes these Prdx to accumulate as Prdx1/2-SS, depleting the Prdx1/2-S − pools.Under these conditions, the abundance of the latter pools becomes strongly dependent on TrxR activity, and therefore so does the H 2 O 2 concentration.These theoretical results are in full These results also help explaining why, in contrast to other redox relays [87,88], the Prdx1-ASK1 redox relay does not need the help of a scaffold protein to operate [89].In this redox relay, Prdx1-SS oxidizes ASK1's C250 which then forms disulfide-bonded multimers that become active as a mitogen-activated protein kinase kinase kinase, eventually triggering apoptosis [89][90][91][92].However, Trx1-S − binds ASK1, hindering access to C250, thereby preventing its oxidation [93,94], and also reducing the disulfide-bonded multimers, inactivating them [91].Therefore, the Prdx1-SS -ASK1 relay can only operate and effectively oxidize ASK1 when the Trx1-S − pool is strongly depleted.Under these conditions, the gradients collapse and ≈75 μM Prdx1-SS becomes available throughout the cytosol (Fig. 6F).At this concentration, Prdx1-SS can oxidize ASK1 in seconds, given the 3 × 10 4 M − 1 s − 1 rate constant [89] for this interaction.
The computational predictions for H 2 O 2 supply rates in excess of v sup,crit are also in keeping with experimental observations of the cellular responses to H 2 O 2 stress.As v sup further increases, a large and increasing fraction of both peroxiredoxins accumulates in sulfinic form throughout the cytosol, decreasing the oxidative load on Trx1 (Fig. 6C, H, I, J).A similar phenomenon has been experimentally documented for the peroxiredoxin Tpx1 in the fission yeast Schizosaccharomyces pombe [95].The relief of the oxidative load on thioredoxin caused by Tpx1 hyperoxidation proved essential to allow the former protein to assist the repair of oxidative damage and ensure cell survival following exposure to high H 2 O 2 concentrations [95].Moreover, because Prdx1/2-SO 2 − can function as holdases that prevent the aggregation of unfolding client proteins, their accumulation to high concentrations may favor cell survival [96,97].However, it is unclear if such high v sup are ever attained in human physiology.The breakdown of localization with increasing v sup is associated with a smooth threshold-like response.This supports the predictions [24] for cells with a similar protein composition, based on a coarse-grained homogeneous-system kinetic model that did not explicitly consider the distinct 2-Cys peroxiredoxins in the cytosol of human cells.In turn, the model in Ref. [6] predicted that the breakdown of spatial localization is associated to a v sup range where two stable steady states coexist, leading to a hysteretic (i.e., trigger-like) response.This is likely a consequence of the approximations underlying the latter model, though it may occur in some cell types (further discussion in SI3).

Localized H 2 O 2 supply and fast kinetics relative to diffusion polarize the cytosolic redox environment
Perhaps the most striking feature from the analysis of the steady state response to localized H 2 O 2 supply (Fig. 6) is the development, with increasing v sup , of a stark redox polarization between H 2 O 2 -sourceproximal and distal regions.In proximal regions, the local Prdx1/2-S − and Trx1-S − pools are gradually depleted as v sup increases beyond ≈5 μM s − 1 , and Prdx and Trx1 disulfides, as well as Prdx sulfenates, accumulate to high-μM concentrations.These local conditions are suitable for the operation of Prdx-and/or Trx1-mediated redox relays.In contrast, in distal regions the Prdx and Trx1 pools remain highly reduced up to v sup values approaching the cytosolic capacity to reduce Prdx1/2-SS, and only then abruptly collapse.These local reducing conditions minimize oxidative damage to local cellular components, but are unsuitable for the operation of the above-mentioned redox relays due to These conclusions hold as well if Prdx1 2 -S − is decameric or instantaneously converts to Prdx1-S − (Figs.S3 and 4).

Peroxiredoxins' redox-oligomeric coupling enhances the cytosol's redox polarization
The just-discussed redox polarization is a consequence of H 2 O 2 and oxidized Prdx and Trx1 forms in this systemsave for the Prdx sulfinic formshaving short lifetimes that do not allow them to diffuse far from the H 2 O 2 sources.A coupling between Prdx's redox and oligomeric states is not necessary for such a redox polarization, but further enhances it, as explained below.
Although the extent to which Prdx1/2-SS dissociates into dimers in vivo remains uncertain, even a partial dissociation has the following two surprising consequences: (i) the concentration of Prdx1-S − at distal sites even increases with increasing v sup , and (ii) the Prdx pool partially migrates away from H 2 O 2 sources towards distal regions (Fig. 7).The emergence of a sustained intracellular concentration gradient of a total protein pool has been observed [98] and theoretically explained [99] in the context of kinase-phosphatase systems.For this phenomenon to occur, the alternative forms of the protein must have distinct diffusion coefficients, and at least one of the reactions interconverting them must be spatially localized [99].
In the present context, these phenomena significantly decrease H 2 O 2 concentrations at distal sites (Fig. 7A, bottom panel) and slightly decrease the total H 2 O 2 concentration in the cytosol (Fig. S5), with the greatest effect for stresses that bring the Prdx-S − pool to the verge of collapse.In turn, their effect on the source-proximal H 2 O 2 concentration under the same circumstances is minimal (Fig. 7A, gray line in the inset of the bottom panel).Therefore, under oxidative stress a higher mobility of the Prdx-SS relative to other Prdx redox states allocates the dwindling Prdx-S − pool to where it can have the greatest protective effect.It is uncertain if the high v sup at which these phenomena are predicted for the tumor cell line simulated in this work are relevant for human physiology.However, these phenomena are predicted to occur at substantially lower v sup in differentiated cells, which tend to have much lower TrxR activities (Fig. S8).Other factors that may modulate them and/or affect the attending estimates are discussed in SI4.

Concluding remarks
In summary, this work highlights that redox microdomains are not just sites of H 2 O 2 accumulation.They are also enriched in oxidized Prdx and Trx species that are relevant for signal transduction.The relative abundances of these species change, and their concentrations exponentially decay, with increasing distance from the H 2 O 2 sources.As a consequence, a strong redox polarization of the cytosol between H 2 O 2source-proximal and distal regions emerges.The same likely happens in the mitochondrial matrix, and in the lumens of other organelles that contain abundant peroxiredoxins and peroxidases.Despite the small dimensions in some directions, the eccentric and convoluted shapes of some of these organelles prompt the generation of substantial concentration gradients if the reactive species are supplied at discrete and widely separated sites.
A coupling between redox and oligomeric states of the 2-Cys Prdxs, even if partial, further enhances that redox polarization by promoting the accumulation of the active reduced forms distally from H 2 O 2 sources under oxidative stress.
In the cytosol, the sulfenic and disulfide Prdx1 forms are more localized than the corresponding Prdx2 species.However, the oxidation of regulatory targets in redox relays that operate in the absence of oxidative stress by any of these peroxiredoxins must occur primarily in the vicinity of H 2 O 2 sources.This is unlike redox relays that operate under stress conditions where the gradients collapse, such as that of Prdx1-ASK1.
The main factor determining what H 2 O 2 supply rates cells tolerate until redox gradients collapse is the capacity of the peroxiredoxin disulfide reduction system, which in most human cell lines is determined by the TrxR activity.At H 2 O 2 supply rates approaching this capacity, the concentrations of H 2 O 2 and of oxidized Prdx and Trx1 species become very sensitive to the TrxR activity, in contrast to what is observed at H 2 O 2 supply rates up to the low μM s − 1 .
Altogether, these results highlight the importance of protein diffusion limitations and modulation for redox signaling and antioxidant protection.

Fig. 1 .
Fig. 1. (A) Redox cycle of 2-Cys peroxiredoxins and its coupling to oligomeric states.Monomers associated in the same dimer are indicated by different shades of the same color.The dashed circle points to a diagram of the two active sites' redox state in a dimer.Dimers whose active sites are converted to the disulfide form, tend to dissociate from the decamers.In turn, decamers in the sulfinic form may associate into even higher order structures, which are not considered in the present model (prepared with the help of BioRender).(B) Modeled reaction scheme.The kinetic parameters for each process are indicated near the respective arrows.In the case of bimolecular reactions, the product of the rate constant by the co-reactant concentration are shown.Parameter meanings and their reference values are indicated in Table 1 (C) Modeled geometry.H 2 O 2 is uniformly supplied over each membrane (thick black lines).The dashed line marks an axis of symmetry.The simulated domain corresponds to the region between the left-hand side membrane and the axis of symmetry.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 2 .
Fig. 2. Time course of the concentrations in the proximal (black) and distal (dashed red) regions after the onset of a 10 μM s ¡1 H 2 O 2 supply rate.All the other parameters as in Table 1.Prdx1, Prdx2 and Trx1 are fully reduced at t = 0.The inset in panel H shows the time course of the concentration of Trx1-SS at the center of the cell.Note the logarithmic time scale.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 3 .
Fig. 3. Spatial concentration profiles at steady state for a constant low (1 μM s ¡1 ) H 2 O 2 influx.The membrane is located at l= 0 and the center of the cell at l= 5 μm.(A) H 2 O 2 concentration.The inset shows the concentration in logarithmic scale.Note the exponential decay with the distance from the membrane.(B) Concentrations of the oxidized Prdx and Trx species scaled by their values adjacent to the membrane.The gray dashed horizontal line marks a 50 % decreased concentration relative to that at the proximal region.(C) Concentrations of the reduced Prdx and Trx species.The inset shows the distribution of the Prdx1 2 -S − in logarithmic concentration coordinates.(D) Concentrations of the sulfenic and disulfide Prdx species.Color/dashing convention is the same as in (B).The inset shows the concentrations in logarithmic scale.Note the nearly exponential decrease with the distance from the membrane, the cross-over between the Prdx1-SS and Prdx2-SS concentrations, and the extremely low concentrations of the Prdx1 species towards the distal region.(E) Trx1-SS concentration.The inset shows the concentration in logarithmic scale.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 4 .
Fig. 4. Dependence of the concentration gradients of H 2 O 2 , Prdx and Trx1 species on v sup.(A) Amplitude of the gradients, evaluated as the ratio between the concentration at the source-proximal regions and that at the distal region.The vertical gray dashed line marks V Max (TrxR).(B) Distance from the membrane at which the concentrations decrease by 50 %.(B, C, D) Spatial concentration profiles at the v sup values indicated by the arrows under panel B. The color/dashing code is the same as in Fig. 3.Note the different concentration scales in the various panels.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 5 .
Fig. 5. Effect of the protein concentrations and enzyme activities on the breakdown of the H 2 O 2 concentration gradient.The plots show the H 2 O 2 gradient amplitudes as a function of the H 2 O 2 supply rate for 0.5-(thin lines), 1-(medium lines) and 2-fold (thick lines) the reference value of each parameter.In the case of Trx1 T , the thinnest of the four lines shows the effect of decreasing this parameter to 0.25-fold its reference value.Note the logarithmic scales.

Fig. 6 .
Fig. 6.Steady-state concentrations nearest (black) and farthest (dashed red) the H 2 O 2 sources, as a function of the H 2 O 2 supply rate.Note the logarithmic v sup scale.The inset in panel L shows the concentrations in logarithmic scale.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 7 .
Fig. 7. H 2 O 2 -induced Prdx-S ¡ accumulation at distal sites.(A) Influence of the H 2 O 2 supply rate on the Prdx1-S − (top panel) and H 2 O 2 (bottom) concentrations at the most distal (dashed lines) and proximal (solid lines in top panel) sites from the H 2 O 2 sources, considering Prdx1-SS diffusing as dimers (black lines) or as decamers (cyan lines).The inset in the bottom panel shows the percent difference in H 2 O 2 concentrations at the proximal (gray) and distal (dark red) sites for Prdx1-SS diffusing as dimers vs. diffusing as decamers.The vertical dashed lines mark the v sup value causing the greatest percent H 2 O 2 concentration decrease at the distal site.Distribution of Prdx1 (B, D) and Prdx2 (C, E) species considering Prdx-SS diffusing as dimers (B, C) or decamers (D, E), with v sup = 171 μM s − 1 , which yields near maximal distal Prdx1-S − accumulation.The dashed black lines mark the average total Prdx concentration.Note the overall accumulation of Prdx1 and Prdx2 towards distal sites in (B, C). (F) Prdx1-S − (green) and Prdx1-SS (red) production rates and percentage of Trx1 as Trx1-S − (blue, axis at the right side) as function of the distance from the H 2 O 2 sources for v sup = 171 μM s − 1 with Prdx1-SS diffusing as dimers.Note the excess Prdx1-SS production proximal from the sources and excess Prdx1-S − production distal from the sources.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) Work financed by the European Regional Development Fund, through COMPETE2020-Operational Program for Competitiveness and Internationalization, and Portuguese funds via FCT-Fundação para a Ciência e a Tecnologia, under projects UIDB/04539/2020, UIDP/ 04539/2020, LA/P/0058/2020, UIDB/00324/2020, UIDP/00313/ 2020, UIDB/04564/2020 and UIDP/04564/2020.Matthew Griffith has been supported by the University of Bath and a NERC GW4+ Doctoral Training Partnership studentship from the UK Natural Environment Research Council (grant no.NE/L002434/1).