How Biochemical Environments Fine-Tune a Redox Process: From

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How Biochemical Environments Fine-Tune a Redox Process: From Theoretical Models to Practical Applications Goedele Roos, Ramon Alain Miranda-Quintana, and Marco Martínez González J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b04736 • Publication Date (Web): 24 Jul 2018 Downloaded from http://pubs.acs.org on July 28, 2018

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The Journal of Physical Chemistry

How Biochemical Environments Fine-tune a Redox Process: From Theoretical Models to Practical Applications

Goedele Roos1,*, Ramón Alain Miranda-Quintana2, Marco Martínez González3 1,*

Corresponding author CNRS UMR 8576, Unité de Glycobiologie Structurale et Fonctionnelle (UGSF) Université de Lille, 1 Sciences et Technologies 50 Avenue de Halley BP 70478 59658 Villeneuve d'Ascq Cedex France [email protected] 2

Department of Chemistry & Chemical Biology; McMaster University; Hamilton, Ontario, Canada

3

a. Laboratory of Computational and Theoretical Chemistry, Faculty of Chemistry, University of Havana, Havana, Cuba; b. Departamento de Química, y Centro de Química Universidade de Coimbra, 3004-535 Coimbra, Portugal

1 ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract In this contribution we give new physical insight on how enzymatic environments influence a redox process. This is particularly important in a biochemical context in which oxidoreductase enzymes and low molecular weight cofactors create a micro-environment, fine-tuning their specific redox potential. We present a new theoretical model, quantitatively backed-up by quantum chemically calculated data obtained for key biological sulphur-based model reactions involved in preserving the cellular redox homeostasis during oxidative stress. We show that environmental effects can be quantitatively predicted from the thermodynamic cycle linking ∆∆G(OX / RED)ref −ligand values to the differential interaction energy ∆∆Gint of the reduced and oxidized species with the environment. Our obtained data can be linked to hydrogen bond patterns found in protein active sites. The thermodynamic model is further understood in the framework of molecular orbital theory. The key insight of this work is that neither the intrinsic properties of a redox couple or of the interacting environment (e.g., ligand) are enough by themselves to uniquely predict reduction potentials. Instead, systemenvironmental interactions need to be considered. This contribution is of general interest as redox processes are pivotal to empower, protect or damage organisms. Our presented thermodynamic model allows a pragmatically evaluation on the expected influence of a particular environment on a redox process, necessary to fully understand how redox processes take place in living organisms.


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Introduction In the search to conceptualize chemical reactions, redox reactions are running behind. Large interest was given to classical organic reactions1 (see

2–7,8–11,12

for reviews), but although

redox processes constitute a class of highly (bio)chemical importance, only recently they fell in the scope of chemical reactivity theory13–16. Redox processes don’t take place in isolation, but in a certain environment, determining the properties of the involved species. This is especially true in a biochemical context, for which a protein or even cellular environment strongly influence redox processes by altering properties (e.g., pKa17,18,18,19) of involved species. The key idea in chemical reactivity theory, which especially holds for conceptual density functional theory, is that reactivity descriptors represent the intrinsic response of the system6,20, independently of other reagents and thus molecular environment. However, it has recently been pointed out that reactivity parameters cannot be defined for isolated species without considering a fragment and/or reservoir interaction16,21–25,26–29. Therefore, a shift in the philosophy of reactivity theory should be made and properties in relation to the environment should be considered16. Redox processes power living organisms, but they are also able to cause irreversible damage30–37. For example, our energy supply under the form of photosynthesis in plants or the synthesis of adenosine triphosphate (ATP) in the electron transport chain is driven by redox reactions. On the other hand, amino acids as cysteine and methionine are very susceptible to oxidative damage altering the activity and structure of sulphur-containing proteins, and organisms depend on redox based repair systems limiting or reversing these harmful effects32,33,38. Interfering with bacterial resistance systems against oxidative stress to target the cellular redox homeostasis can form appealing routes for anti-bacterial drug development32,39– 41

. These examples illustrate the necessity to fully understand how redox processes take place

in a biochemical environment. The broad class of oxidoreductase enzymes and low molecular weight cofactors catalyse the transfer of electrons from donor to acceptor. They all create a particular micro-environment, fine-tuning their specific redox potential42. Numerous studies have appeared describing methods to theoretically obtain redox potentials, both from direct calculations of the free energies as from more conceptual approaches as methods based on the grand canonical ensemble or based on the fractional number of electrons (see Bruschi et al.

43

and references herein for a recent overview). These studies 3 ACS Paragon Plus Environment


often conclude that accurate results – even with very advanced methodologies – can only be obtained if explicit water molecules are included. A key point here is that when the oxidized and/or reduced species interact with the environment, a new supra-system is generated undergoing the reduction or oxidation. The degree to which the environment actively takes part in the redox process is variable, and it will affect the evaluation of the interaction between the (sub)systems and their environment, e.g. using a continuum solvent model allows to include some information regarding the environment interaction, therefore improving the estimates for reduction potentials16. However, none of these previous studies construct a model to predict the effect of a particular environment on the redox potential. Here, we give new physical insight on how the environment influences a redox process by presenting a new theoretical model that quantitatively links the effect of the environment on the redox process, given by

∆∆G(OX / RED)ref −ligand - the difference between the reaction free energy of the isolated reference system and the interacting system - to the differential interaction energy ∆∆Gint of the reduced and oxidized species with the environment. This thermodynamically based model allows a pragmatically evaluation on the expected influence of a particular environment on a redox process, which is in particular important in an enzymatic context. We show that not only the redox pair is affected by the ligand, but that the effect exerted by a given ligand will depend on the redox couple it interacts with. In other words, we cannot rely solely on the intrinsic characteristics of the reduced and oxidized species, or on the intrinsic characteristics of the ligand to predict the redox properties. Only by fully considering the system-environment interactions can we accurately model the variations in the reduction potential E°. We conclude by showing how this thermodynamic model can be further understood in the framework of molecular orbital theory. Our model treats the electron donor and acceptor separately, which is acceptable as many long-range electron transfer reactions in redox proteins occur over donor-acceptor distances of more than 10 Å, giving an essentially zero overlap of donor-acceptor orbitals44,45. To support our presented theoretical model, sulphur-based model reactions are quantitatively studied, as sulphur is the key element of both low molecular weight thiols (e. g., glutathione46 and mycothiol47–49) and of major families of oxidoreductase proteins (e. g., thioredoxins (Trx), peroxiredoxins (Prx) and disulfide binding protein (DsbA, DsbB, DsbC)35,50–52). These


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model reactions represent key reactions involved in preserving the cellular redox homeostasis during oxidative stress35,53 and thus in defending the cell against oxidative stress54–56,30.

Theoretical Background To give physical insight in how the environment influences a redox process, we present a model that predict the influence of the environment by considering the differential interaction of the reduced and oxidized system with the environment. This was already preliminary showed by one of the authors for the oxidation of thiol to sulfenic acid57. Here, we elaborate more on this thermodynamic model and discuss it in terms of molecular orbital (MO) perturbation theory describing the interaction between the molecule (A) and the environment (B).

Scheme 1: Thermodynamic cycle depicting the relationship between complexation energies of any ligand L with the reduced and oxidized species and the ∆G(OX / RED) values.

∆∆G(OX / RED)ref −ligand (pink) equals ∆∆Gint (orange). For a vertical-line representation of this scheme, see SI_Scheme 1.

Consider the half reaction for the reduction:



OX--nL (ε) + me-→ RED --nL (ε)

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(1)

with L the ligand, n the number of ligands present and m the number of exchanged electrons (n and m can be equal or different), RED the reduced species and OX the oxidized species, a thermodynamic cycle (Scheme 1)17 can be formulated, showing that the ligand effect on the free energy of oxidation can be linked to the free energy changes upon ligand coordination around RED or OX. The ∆G associated to reaction (1) can be calculated as

∆Gligand = ∆G(OX − − L / RED − − L) = G(RED − − L) − G(OX − − L)

(2)

and

∆∆G(OX / RED)ref −ligand = ∆Gref − ∆Gligand

(3)

with ref pointing to the reference system having no explicit ligands present, thus

∆Gref = ∆G(OX / RED) = G(RED) − G(OX )

(4)

As shown in the cycle of Scheme 1, ∆∆G(OX / RED)ref −ligand can also be calculated as ∆∆Gint , which is the difference between the interaction energies of the RED and OX systems with ligand L:

∆∆G(OX / RED) ref −ligand = ∆G(OX + L) − ∆G(RED + L) = ∆∆Gint

(5)

with

∆G( A + L) = G( A − − L) − G( A) − G(L)

(6)

It is interesting to notice that the term corresponding to G( L) will cancel in Eq. (5), therefore the analysis can be carried out in an exact but more simplified way by using the following expression as Eq. (6):

∆G( A + L) = G( A − − L) − G( A)

∆∆Gint can be approximately calculated as follows: 6 ACS Paragon Plus Environment

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∆∆Gint ≈ ∆E(OX + L) − ∆E(RED + L) + ∆∆Gint,solv

(7)

with ∆E , the gas phase complexation energies and ∆∆Gint,solv the difference in solvation free energy between the oxidized and reduced species in complex with the ligand.

∆∆Gint,solv = ∆G(OX − − L) solv −∆G(RED − − L) solv

(8)

In this approximation, the deformation energy upon going from gas phase to aqueous solution and the entropy change T ∆Sint is neglected. This is a fair approximation57–60 as the deformation energies are expected to be small16,58,59 and the entropy term ∆S int is expected to be similar between the reference reaction and the reaction in the presence of ligands59 and thus this term vanishes when ∆∆Gint is calculated.

Summarizing, we obtain

∆∆G(OX / RED)ref −ligand = ∆G(OX / RED)ref − ∆G(OX / RED)ligand = G( RED) − G(OX ) − G( RED − − L) + G(OX − − L) (9)

= ∆∆Gint = ∆G(OX + L) − ∆G(RED + L) ≈ ∆E(OX + L) − ∆E(RED + L) + ∆G(OX

− −

L) solv −∆G(RED − − L) solv

which converts to Eq. (10) when invoking solution standard states (in the examples discussed in the result section, all species, i. e. OX, RED, L and their complexes are solvated in aqueous solution).

∆∆G 0 (OX / RED)ref −ligand = ∆G 0 (OX / RED)ref − ∆G 0 (OX / RED)ligand = G 0 (RED) − G 0 (OX ) − G 0 ( RED − − L) + G 0 (OX

− −

L) (10)

0 = ∆∆Gint

= ∆G 0 (OX + L) − ∆G 0 (RED + L) ≈ ∆E(OX + L) − ∆E(RED + L) + ∆G 0 (OX − − L) solv −∆G 0 ( RED − − L) solv



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If the interaction is better with the oxidized species, then ∆∆G(OX / RED) ref −ligand < 0 and the ligand favors the oxidation, vice versa, if the interaction is better with the reduced species, then

∆∆G(OX / RED) ref −ligand > 0 and the reduction is favored compared to the reference

reaction. In the next section, we discuss biochemically relevant examples to support and to illustrate Scheme 1 and Eq. (9).

Computational Details Examples to support our proposed thermodynamic model are calculated, partially based on data from Olah et al.57 and Roos et al.60. Additional data are calculated using Gaussian0961 or Gaussian0362 for free energies of solvation. Equation (10) is the heart of our model, in which for this study G 0 is the free energy in aqueous solution and E the energy calculated in gas phase. In the calculations, the gas phase standard state of 1 atm and the solution phase standard state of 1 mol l-1 H+ is used, indicated with the superscript ‘ 0 ’.

A − − L denotes the molecule in complex with the ligand; A denotes the bare molecule and

A + L indicates the transfer of A to A − − L . All A, L and A − − L complexes are fully optimized, without freezing of ligand positions, for details Olah et al.

57

and Roos et al.

60

. A, L and

A − − L complexes are solvated in aqueous solution using the implicit IEF-PCM solvation model.

G 0 values are calculated using the method presented in Ho et al. 63. This methodology was successfully applied before to give reduction potentials in good agreement with experiment64– 66

. G 0 values are obtained as the sum of MP2/6-311++G(d,p) gas phase free energies G 0 gas

0 and the IEF-PCM21 free energies of solvation ∆Gsolv (thus: G 0 = G 0 gas + ∆G 0 solv ). Here,

proper conversion between the gas phase and solution standard states are made by applying the RT ln(

RT ) conversion term (with R universal gas constant of 0.082057338 L atm K-1 P

mol-1 and T room temperature of 298.15 K), thus: G 0 gas (1M) = G 0 gas (1atm) +1.9 . 8 ACS Paragon Plus Environment

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For example, G 0 (OX ) is calculated as G 0 (OX ) = G 0 gas (OX ,1atm) + 1.9 + ∆G 0 solv (OX ) ; giving for Eq. (10) :

∆∆G 0 (OX / RED) ref −ligand 0 0 0 0 = Ggas ( RED,1atm) + 1.9 + Gsolv ( RED)  − Ggas (OX ,1atm) + Gsolv (OX )  0 0 0 − Ggas ( RED − − L,1atm) + 1.9 + Gsolv (RED − − L)  + Ggas (OX

−−

0 L,1atm) + 1.9 + Gsolv (OX

−−

L) 

0 are calculated at the HF/6-31+G(d) level63,67, using UAHF radii, consistent with the ∆Gsolv

parameterization in IEF-PCM68. The gas phase interaction energies ∆E are calculated at the MP2/6-311++G(d,p) (for the CH3SO- --nL (ε) + 2H+ (aq) + 2e- → CH3S---nL (ε) + 2H2O reduction), or at the B3LYP/6.

-

311++G(d,p) level (for the CH3S --nL + e- → CH3S --nL

and CH3SSCH3--nL + e- →

.

CH3SSCH3 ---nL).

4. Results and Discussion

Validation of the thermodynamic model

First, the oxidation of thiolate CH3S- to sulfenic

acid CH3SO- is considered. Sulfenic acid formation, also called sulfenylation is representative for a wide variety of physiological processes. It serves as intermediate in the redox regulation of signaling pathways31,69–71, and in de novo disulfide bond formation53 and is formed under oxidative stress72,73. The essential anti-oxidant proteins peroxiredoxins (Prxs) effectively scavenge reactive oxygen species (ROS) during which ROS are reduced through a cysteine-based reaction mechanism52,74. During this process, the active site cysteine of Prx is oxidized to sulfenic acid. In our preceding study on the thiol sulfenylation thermodynamics of human 2-Cys peroxiredoxin thioredoxin peroxidase B (Tpx-B)52,75,76 the effect on the thiol/sulfenic acid oxidation potential of conserved active site residues was found to be consistent with the hydrogen bonds formed with the active site cysteine in QM/MM optimized Tpx-B structures57. Here, we quantitatively validate this observation from Eq. (9).



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Redox potentials are traditionally formulated for reduction processes and thus we consider the reverse of the sulfenylation reaction, which is the reduction of sulfenic acid to thiolate: CH3SO- --L (ε) + 2H+ (aq) + 2e- → CH3S---L (ε) + 2H2O

(11)

Neutral polar and non-polar ligands L are taken to resemble residues from the active site of Prx57 (e. g. CH4 can serve as a model for valine, while CH3OH mimics threonine). 0 for this example is approximately calculated as follows (Table 1), cfr. Eq.(7): ∆∆Gint

0 0 (12) ∆∆Gint = ∆G°(RSO − + L) − ∆G°(RS − + L) ≈ ∆E(RSO − + L) − ∆E(RS − + L) + ∆∆Gint,solv

Table 1 supports the thermodynamic cycle of Scheme 1 by numerically showing that ligands which interact more strongly with the CH3SO- system than with the CH3S- system are characterized by a more negative ∆∆G°(RSO − / RS − )ref −ligand value and thus favor sulfenylation. Ligands interacting more strongly with the CH3S- system than with the CH3SOsystem lead to positive ∆∆G°(RSO − / RS − )ref −ligand values and therefore disfavor sulfenylation. These findings can now be linked to what was found in the QM/MM study57. For example, in the Tyr43Ala mutation, the S- form of the active site cysteine (Cys51) becomes stabilized compared to the wild type as more hydrogen bonds are made to the S- form in the Tyr43Ala mutation than in the WT type, while in the SO- form the same number (4) of hydrogen bonds are made to the protein in both WT and Thr48Ala. Therefore, Tyr43 favors sulfenylation. Also Thr48 favors sulfenylation by stabilizing Cys51SO- over Cys51S- . Only one hydrogen bond to S of Cys51S- is formed, while both S and O of Cys51SO- interact to Thr48. Val50 does not have a significant effect on sulfenylation in agreement with the absence of interactions between Val50 and Cys51.


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Table 1: ∆∆G°(RSO − / RS − )ref −ligand and ∆∆G 0 int for the Sulfenylation Reaction CH3S---nL (ε) + H2O2 → CH3SO- --nL (ε) + H2O

∆∆G°

nL

−

−

(RSO / RS )ref −ligand

∆E

∆E

CH3SO- --nL

CH3S---

0 ∆∆Gint,solv

∆∆G 0 int

nL

[1] CH4

4.3

-3.3

-1.9

3.1

1.7

C2H4

4.3

-6.1

-4.1

3.7

1.7

benzene

5.8

-9.2

-6.8

4.9

2.5

H2O

-3.3

-21.0

-14.8

-0.5

-6.7

CH3COOH*

7.6

-28.0

-33.1

-0.1

5.0

CH3OH

-2.1

-22.7

-15.6

0.45

-6.7

CH3CONH2 -2.7

-31.2

-22.2

2.1

-6.9

NH3

-11.0

-7.5

-0.45

-4.0

-1.2

All values are given in kcal/mol. [1] Data taken from ref.

57

The electron capture reactions by the thiyl radical (Eq. (13)) and disulfide (Eq. (14)) are discussed as second example (Table 2). Vital redox processes in for example redox signalling pathways30 or in ROS removal and free radical scavenger reactions56 involve continuous cycles of thiol/disulfide switches in which thiyl radicals and disulfide radical anions can be intermediate products30,35,53,56,70. Disulfide and thiyl radicals are also intermediates in electron capture and electron transfer dissociation mass spectroscopy (ECD and ETD) experiments77, further illustrating their pivotal role. To fully understand thiyl and disulfide radical formation, the effects of the biochemical surrounding environment needs to be illuminated. (Table 2). For the following reactions, .

-

CH3S --nL + e- → CH3S --nL

(13) .

CH3SSCH3--nL + e- → CH3SSCH3 ---nL

(14)

∆∆G°ref −ligand and ∆∆G 0 int values are calculated (Table 2).



The ligands L are chosen to represent an enzymatical environment and are the non-polar (CH4 and C2H4), polar neutral (H2O, CH3OH, NH3, CH3COOH and CH3CONH2) or positively charged (NH4+) ligands and monovalent cations (Na+, K+ and Li+ ). The ∆∆G 0 int values (Table 2) explain that i) Na+ and Li+ favour the electron capture by the thiyl radical while disfavouring the electron uptake by the disulfide bond and ii) K+ has no effect on the electron capture by the thiyl radical, while disfavouring the electron uptake by the disulfide. ∆∆G 0 int is positive (or zero for K+) in the case of the electron capture reaction by the thiyl radical (reaction (13)) and negative for the electron capture by disulfides (reaction (14)). This means that for the thiyl radical, Na+ and Li+ interact more strongly with the reduced system compared to the oxidized system, while K+ interacts equally well with both the oxidized and reduced form. The opposite is true in the case of the disulfides. Here, Na+, K+ and Li+ interact less strongly with the reduced than with the oxidized species. For electron capture by disulfides in the presence of charged ligands, the electron is partially captured by the positively charged ligand. This not only holds for the cations, but also for NH4+. The assistance of the positive sites in the electron captures is accompanied with the lengthening of the S--L distance upon reduction60, and as such a weakening of the interaction. The disulfide and thiyl electron capture examples show clearly that a same ligand type can have a different effect on the reduction potential E° depending on the redox couple60. This conclusion can be generalized as from the thermodynamic cycle (Scheme 1), the ligand effect arises not from the properties of the isolated ligand, but from the interaction between reagent and ligand. This interaction depends on the properties of both the ligand and of the reagent and thus properties of an isolated ligand cannot uniquely determine E°.


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.

-

Table 2: ∆∆G°ref −ligand and ∆∆Gint values for the CH3S --nL + e- → CH3S --nL and .

CH3SSCH3--nL + e- → CH3SSCH3 ---nL reductions. .

-

CH3S --nL + e- → CH3S --nL

nL

∆∆G°ref −ligand ∆E

∆E -

0 ∆∆Gint,solv

∆∆Gint

CH3S --nL

CH3S --nL

[1] 1 CH4

-2.2

-1.9

0.0

-4.0

-2.1

2 CH4

-5.5

-3.8

-0.1

-8.4

-4.7

1 H2 O

2.4

-16.5

-3.5

-10.6

2.4

2 H2 O

2.2

-31.6

-6.9

-20.4

4.3

-16.7

-3.5

-10.2

3.0

0.7

-8.4

-2.1

-6.9

-0.6

7.1

-153.8

-27.7

-121.3

4.7

2.3

-133.7

-18.9

-112.1

2.7

-0.9

-114.2

-11.7

-102.4

0.02

1 CH3OH

4.1

1 NH3 1 Li+ 1 Na+ 1 K+

.-

-

CH3SSCH3--nL + e → CH3SSCH3 --nL

nL

∆∆G°ref −ligand ∆E

∆E .

CH3SSCH3 ---L [1]

0 ∆∆Gint,solv

∆∆Gint

CH3SSCH3-L

1 CH4

-1.5

-1.0

0.0

-2.8

-1.8

1 H2 O

2.6

-10.1

-3.2

-4.7

2.2

5.4

-10.9

-2.6

-1.9

6.4

1.1

-5.5

-2.2

-2.0

1.3

-22.6

-120.3

-18.1

-110.5

-8.3

-22.2

-130.9

-33.9

-114.9

-17.9

-17.0

-120.2

-21.5

-110.2

-11.4

-102.0

-15.9

-101.9

-15.8

1 CH3OH 1 NH3 1 NH4+ 1 Li+ 1 Na+ 1 K+

-20.7 All values are given in kcal/mol. [1] Data taken from ref. 60



The pKa dependence of the disulfide/thiol reduction potential is well documented17,18,19 and can now be understood in the light of the thermodynamic model given by Scheme 1 and Eq. (9). When the cysteine pKa decreases, the associated disulfide/thiol reduction potential increases. This was shown for Trx active site mutants and DsbA for which a linear correlation between the cysteine pKa and the disulfide/thiol reduction potential was found78. Empirical relations between the cysteine pKa and the disulfide/thiol reduction potential have been proposed for Trx-type oxidoreductases18. Cysteine pKa’s tend to decrease when hydrogen bonds are formed to the cysteine sulfur (see ref.17 for a review and extra references). Hydrogen bonds indicate a substantial interaction with the reduced species for which the thermodynamic model shows a favored reduction (high reduction potential). From the above examples it is clear that the ligand effect can be interpreted as the differential stabilization of the reduced and oxidized species and Scheme 1 allows for a pragmatically evaluation on the expected influence of a particular environment on reduction/oxidation potentials. Not only a qualitative relation between ∆∆G 0 (OX / RED)ref −ligand and ∆∆G 0 int exists, but ∆∆G 0 (OX / RED)ref −ligand can be directly estimated from ∆∆G 0 int , as Figure 1 shows a fair correlation between both values. The correlation improves when only one reaction is considered, consistent with the approximations made to calculate ∆∆G 0 int (cfr. Eq. (7)), which become more exact within a particular reaction.


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Figure 1: Correlation Between ∆∆G 0 (OX / RED)ref −ligand and ∆∆G 0 int

a) all three reactions; b) reduction of CH3SO- to CH3S-; c) electron capture by CH3S; d) electron capture by CH3SSCH3.

Molecular orbital model The thermodynamic model shows that the environmental effects can be interpreted as the differential interaction of the reduced and oxidized system with the environment and thus their differential stabilization. This can be understood in terms of molecular orbital (MO) perturbation theory describing the interaction between the molecule (A) and the environment (B). As the HOMO-LUMO interaction between ligand and environment is expected to be dominant, we need to construct the shifted interaction orbitals, which are the perturbed HOMO ε H and LUMO ε L orbitals (Scheme 2). If the interaction with the environment is good (weak), the perturbation ∆ε will be large (small).



Scheme 2: The HOMO-LUMO interaction between molecule A and environment B yields the perturbed HOMO ε H and LUMO ε L interaction orbitals, which are, compared to the unperturbed orbitals χ A and χ B , the shifted interaction orbitals due to the perturbation ∆ε.

Depending on how large the perturbation for the oxidized species ∆ε ox versus the one for the reduced ∆ε red species is (and thus, depending on how strong the interaction of the oxidized and reduced species with the environment is), the environment will favour, disfavour or have no effect in accordance with the thermodynamic model (Scheme 1). To fully understand Figure 2, one needs to imagine that a redox process can be conceptually split into an electron transfer and a reorganization step. For example, during a reduction process, discussed in Figure 2 (an analogous diagram can be drawn for the oxidation process), the oxidized partner A will take up one electron in its LUMO orbital from the reduced interaction partner C performing the reduction (this is the electron transfer step). This LUMO orbital is then relaxed to the HOMO orbital of its reduced counterpart A’ (this is the reorganization step). As A and A’ are in interaction with the environment, the LUMO and HOMO interaction orbitals are shifted by ∆ε L,ox and ∆ε H,red (respectively) compared to the isolated reference case. When comparing ∆ε L,ox with ∆ε H,red , three cases can be distinguished: a ∆ε H,red < ∆ε L,ox (blue in Figure 2): the environment interacts more strongly with the oxidized species than with the reduced counterpart, disfavouring the reduction, but supporting the oxidation. This means that the environmentally induced perturbation of the LUMO orbital ∆ε L,ox of the oxidized counterpart will be larger than the perturbation of the HOMO orbital ∆ε H,red of the reduced counterpart. Compared to the 16 ACS Paragon Plus Environment

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unperturbed reference case, the driving force for reduction will be smaller, while the relaxation energy will be larger. Since here, ∆ε H,red < ∆ε L,ox , the net effect will be a smaller energy gain upon reduction and thus a disfavoured reduction compared to the isolated reference state. b ∆ε H,red > ∆ε L,ox (red in Figure 2): the environment interacts more strongly with the reduced species than with the oxidized counterpart, disfavouring the oxidation, but supporting the reduction. This means that the environmentally induced perturbation of the LUMO orbital ∆ε L,ox of the oxidized counterpart will be smaller than the perturbation of the HOMO orbital of the reduced counterpart ∆ε H,red . Compared to the unperturbed reference case, the driving force for reduction will be smaller, while the relaxation energy will be larger. Since here, ∆ε H,red > ∆ε L,ox , the net effect will be a larger energy gain upon reduction and thus a favoured reduction compared to the isolated reference state. c ∆ε L,ox = ∆ε H,red (black in Figure 2): the environment interacts equally strong with oxidized and reduced species and has no effect on the oxidation. ∆ε L,ox = ∆ε H,red and thus, compared to the unperturbed reference case, the environment has no net effect as the decrease in driving force will be the same as the increase of the relaxation energy.



Figure 2: Reduction of an Oxidized Species A by Partner C Performing the Reduction, in the Presence of a Perturbing Environment B.

The reference case of the unperturbed, isolated molecule in vacuum is presented in grey, blue corresponds to case a, red to case b and black to case c discussed in the HOMO LUMO main text. The perturbed HOMO ε red and LUMO ε ox,P interaction orbitals arise ,P

from the perturbations ∆ε H,red and ∆ε L,ox .


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Conclusion The take home message of this study is that neither the intrinsic properties of a redox couple nor these of the interacting environment (e.g., ligand) are enough by themselves to uniquely predict reduction potentials. Instead, we need to consider the system-environment interactions to accurately model the variations in the reduction potential. The environment effects can be quantitatively predicted from the thermodynamic cycle linking ∆∆G(OX / RED) ref −ligand values to the differential interaction energy ∆∆Gint of the reduced and oxidized system with the environment. Our obtained data can be linked to calculated hydrogen bond patterns found in protein active sites, and as such giving new physical insight in experimental data, like the well documented pKa dependence of the disulfide/thiol reduction potential. Therefore, our thermodynamic model allows a pragmatically evaluation on the expected influence of a particular environment on a redox process, necessary to fully understand how redox processes take place in living organisms. The thermodynamic model can be understood from molecular orbital theory. What is left is to include the effect of the interaction partner in the model and to find a model to estimate the correction values of isolated systems (in the framework of the MO model, to find the correction ∆ε on the orbital energies) from first principles. A particularly attractive approach to do this would be to incorporate these perturbations in more realistic E vs. N curves.



Acknowledgements The authors thank Paul Ayers and Andreas Savin for helpful discussions. GR is a CNRS researcher. The authors declare no competing financial interest.


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TOC image 49x27mm (300 x 300 DPI)

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Figure 1: Correlation Between ∆∆G°(OX/RED)ref-ligand and ∆∆G°int a) all three reactions; b) reduction of CH3SO- to CH3S-; c) electron capture by CH3S·; d) electron capture by CH3SSCH3. 96x57mm (600 x 600 DPI)


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Reduction of an Oxidized Species A by Partner C Performing the Reduction, in the Presence of a Perturbing Environment B.

90x72mm (600 x 600 DPI)



Thermodynamic cycle depicting the relationship between complexation energies of any ligand L with the reduced and oxidized species and the ∆G(OX/RED) values. ∆∆G(OX/RED)ref- ligand (pink) equals ∆∆Gint (orange). For a vertical-line representation of this scheme, see SI_Scheme 1. 61x46mm (300 x 300 DPI)


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The HOMO-LUMO interaction between molecule A and environment B yields the perturbed HOMO εH and LUMO εL interaction orbitals, which are, compared to the unperturbed orbitals χA and χB, the shifted interaction orbitals due to the perturbation ∆ε. 56x55mm (600 x 600 DPI)


How Biochemical Environments Fine-Tune a Redox Process: From

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