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Chapter 4: Biochemical redox reactions

4.1 Introduction

4.2 Biochemical redox half-reactions, the Faraday conststant and the reduction potential 4.2.1 Defining the reduction potential:

4.2.2 The standard reduction potential is also the midpoint potential of a redox couple

4.3 Determining the value of the midpoint potential 4.4 Solution (ambient) potentials and electrochemical cells 4.5 Example: the potentiometric titration of NAD

4.6 How midpoint potentials are used to depict biochemical electron transfer systems 4.7 Ambient potential in a living cell and oxidative stress

4.7.1 Oxidative stress.

4.8 The pH-dependence of the midpoint potential

4.9 Example: The pH-dependence of the midpoint potential of the NADox/NADred redox

couple

4.10 Thermodynamic reciprocity of interactions between proton binding and reduction potential

4.11 Application: Determining the midpoint potential of E.coli thioredoxin

4.12 Application: A mutation that raises the midpoint potential of the regulatory disulfide the the γ-subunit of the chloroplast ATP synthase from Arabidopsis

4.13 Application: Impact of mutations on the midpoint potential of an [4Fe-4S] cluster in the electron transfer protein:ubiquinone oxidoreductase

4.14 Application: Determining the mitochondrial ambient potential 4.15 Summary

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Chapter 4: Biochemical redox reactions

4.1 Introduction

In Chapter 3 we developed the use of the chemical potential in dealing with

biochemical reactions. This formalism applies to all reactions whether or not they involve hydrolysis of ATP, DNA cleavage or oxidation/reduction changes. However, for the many reactions in chemistry which involve electrons being transferred from one species to another, i.e., reduction and oxidation (hence, redox), there is a specific language and set of parameters that have been developed, namely the concepts of the reduction potential and the “half-reaction” or "half-cell reaction". Chemical and biochemical redox reactions can all, in principle, be carried out by transferring the electrons from the molecule being oxidized to an electrode located in one solution, and then delivering electrons to the molecule being reduced via another electrode located in a separate solution. In many cases, chemical and biochemical reactions can in reality be performed in this manner. The electrical charges need to be able to travel from one electrode to another, and this can be done using a wire, in conjunction with a salt bridge in which ions (e.g., K+ and Cl-) move between the solutions in order to maintain charge neutrality in each solution as electrons are added to one side and removed from the other. A schematic diagram is shown in Figure 4.1. The thermodynamics of such reactions are, of course, the same as we discussed in the last chapter. The parameter of interest remains the transformed reaction Gibbs free energy, , but the terminology used is often that of electrochemistry when dealing with redox reactions. The focus on electron transfer, and the proton transfer reactions which are frequently linked to electron transfer, are particularly helpful in understanding many biochemical reactions.

'

rG

Δ

In addition to biochemical redox reactions, we will also discuss in this chapter the characterization of electron carriers in electron transfer pathways or in other redox processes

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and the characterization of the prosthetic groups within redox-active enzymes. The first group includes c-type cytochromes, quinols, NADH, etc., and the second group includes protein-bound hemes, flavins, Fe/S clusters, disulfides, and many more.

Figure 4.1: Schematic of a redox reaction being carried out in an electrochemical device where electrons from the reductant are delivered to the oxidant through a wire. The maximal electrical work that can be accomplished is equal to the Gibbs reaction free energy of the reaction.

4.2 Biochemical redox half-reactions, the Faraday conststant and the reduction potential

Let’s take a look at the oxidation of NADH by O , a reaction catalyzed by the

mitochondrial respiratory chain. The equation indicates a reaction with oxygen, and it is indicated that we will determine the thermodynamics with respect to dissolved oxygen in the aqueous phase (aq).

2

(4.1)

(4.1) 2

2 NADH + 2 H + O (aq) + R2 NAD + 2 H O 2

+

Chemical reaction vs biochemical reaction notation: The reaction of NADH and O , as it appears in equation is the way one typically would write out a chemical reaction. At constant pH, we need to recall that the proton concentration does not change, so formally in the biochemical reaction, hydrogens and charge need not be conserved. Protons can appear

2 (4.1)

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from or vanish into an infinite proton reservoir. Formally, the proper way to describe this biochemical reaction at constant pH is

(4.2) 2

2 NAD + O (aq) red R 2 NAD + 2 H Oox 2

This can be disorienting, especially if one is not used to it, so it can be excused to balance the reaction so as to keep track of what is going on. However, in using the transformed Gibbs free energy, the protons are not included in the equilibrium expression.

In this reaction, NADH (or NAD ) is the reductant. That is, electrons are taken from NAD and delivered to O , which is the oxidant. The oxidant has the stronger tendency than the reductant to take electrons. We can separate the biochemical reaction in equation

into two half-reactions,

red red 2 (4.2) (4.3) 1 2 2 1 half-reaction 1: O (aq) + 4 e 2 H O half-reaction 2: 2 NAD + 4 eox 2 NAD red

R R

in which we have, again, removed the protons since pH is held constant. The half reactions are written following a convention of placing the oxidant on the left. The net reaction in (4.2) is reaction 1 minus reaction 2 as they are written in (4.3). We can think of these half-reactions as reactions that might take place at the surfaces of the two electrodes in Figure 4.1. In this electrochemical set-up (Figure 4.1) electrons are donated by the oxidation of NAD at one electrode and delivered through a wire to the second electrode, where O is reduced to water. This is a current, and we could get electrical work from the system if we had an electrical device such as a motor inserted into our circuit. The maximal work we could obtain is given by the transformed Gibbs free energy of the reaction, . Remember that this is the maximal work per mole of reaction progress (

red 2

'

rG

Δ

ξ, introduced in the previous chapter) at the particular concentrations present. This is not the amount of work that we could get if we let the reaction run down to equilibrium. The realization that the electrical work is equivalent to

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'

rG

Δ is helpful because is stresses the fact that the chemical driving force for this redox reaction is related to the spontaneous movement of electrons from the reductant (electron donor) to the oxidant (electron acceptor, which oxidizes the reductant). The work capacity of this reaction ( ) is usually expressed in terms of joules, but can also be expressed in terms of electrical work, or volts. One joule is defined as the amount of energy gained when 1 coulomb of charge is moved against a potential of 1 volt, where a coulomb is the amount of charge transported by a current of 1 ampere in 1 second. Recall that the electrical work required to move an amount of charge (Q) from a position where the potential is ψ to a position where the electrical potential is ψ is

' rG Δ 1 2 2 1 ( el w = Ψ − Ψ )Q (4.4)

The amount of absolute charge in 1 mole of electrons is 96,485 coulombs, so the energy of moving 1 mole of electrons, Q = -96,485 coulombs, to a more negative potential, = -1 volt, is (-96,485)(-1) = 96,485 joules. Doing work on the system is positive. This gives us a conversion factor between joules and volts, two different units of energy.

2 1 (Ψ − Ψ )

(4.5) 1 volt = 96,485 joules

The conversion factor is called the Faraday constant, F.

(4.6) = 96,485 coulombs/mol

F

In the system pictured in Figure 4.1, electrical current will move from left (NAD ) to right (O ), which means that the electric potential of the electrode on the left is more negative than that on the right. Since electrical work is nonPV work, this means that it is equivalent to reversible work ( ). The maximal electrical work per mole of reaction progress (the extent of reaction parameter, ξ) must be equal to the transformed Gibbs free energy of the reaction under the defined conditions, as it would proceed if both reactants were

red 2

el rev

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present in the same solution. As the reaction is written in equations and , we can see that (4.2) (4.3) (4.7) ' ( right left) 4(96, 485) r Q Ψ − Ψ = − ΔΨ = Δ G

The negative sign comes from the charge (Q) being negative, and the 4 is the absolute value of the stoichiometry number of the electrons as the reaction has been written in (4.2) and (4.3), four electrons per mole of O2. We will use the notation “νe ”to indicate the absolute

value of the stoichiometry number, so in this case, νe =4. The spontaneous direction of reaction is from left to right, and the value of is negative. The direction of the current flow (NAD

(4.2) '

rG

Δ

red to O2) is also from left to right as we have drawn our device in Figure

4.1, towards the more positive electrode. The potential to do work is given by the voltage difference between the two electrodes and this work potential must be equivalent to .

From equation , since . Clearly, if we know

the value of the transformed Gibbs free energy of reaction, we can readily calculate the potential difference between the two electrodes in the setup in Figure 4.1.

' rG Δ (4.7) ' 0 it follows that ( ) 0 rG right left Δ < Ψ − Ψ >

The reason for going through all of this is to emphasize the reality that redox reactions can and often are examined using an electrode as either an electron source (reductant) or an electron acceptor (oxidant). Let us now convert the expression for the reaction free energy to units of volts. If we generalize equation (4.7) we see that

' ' e r r e F G or G F ν ν − ΔΨ = Δ Δ ΔΨ = − (4.8)

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Figure 4.2: Transformed Gibbs reaction free energy converted to an electric potential difference for a redox reaction for a 1-electron and 2-electron reaction. This is a plot of equation (4.8).

Each of the four electrons drops down the potential ΔΨ, so the total reaction free energy is equal to the votage drop, converted to units of joules, multiplied by the number of moles of electrons, 4 in this case. Figure 4.2 shows a plot of the relationship in (4.8). For a 1-electron reaction, the slope of the line shows that 1 kJ is equivalent to about 10 mV. For a 2-electron reaction, the slope if half.

By dividing the expression for ' by

rG

Δ νe F we get the following.

' ' 2 ' ' 2 ' ' 2 [ ] ln [ ][ ( )] [ ] ln where 4 [ ][ ( )] [ ] ln 4 [ ][ ( )] o ox r r red o ox r r e e e e red o ox red NAD G G RT NAD O aq NAD G G RT F F F NAD O aq NAD RT E E F NAD O aq ν ν ν ν Δ = Δ − Δ Δ = − = − = ) (4.9)

where E' is the electric potential difference between the two electrodes ( '

E = ΔΨ and E'o is

the electric potential difference under standard state conditions (1 M of each reactant,

298.15K, pH 7, specified ionic strength). We can calculate the value of E'o from the values of the transformed Gibbs free energies of formation for reaction (4.2).

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' ' ' 2 2 ' ' ' ' ' 2 2 2 2(1059.11) 2( 155.6) 2(1120.09) (16.4) 449.56 / ( 449.56) 1.164 volts 4(96485) o o o ox red o r f NAD f H O f NAD f o r o r o o r e G G G G G G G kJ mol G E F ν Δ = Δ + Δ − Δ − Δ Δ = + − − − Δ = − Δ − = − − = 'o O (4.10)

If, for example, the concentrations of reduced and oxidized NAD are the same and the concentration of O2(aq) is 250 μM (2.5 x 104 M), then the potential between the electrodes

would be ' ' 4 2 ' ' [ ] (8.31)(298) 1 ln 1.165 ln( ) [ ][ ( )] 4(96485) 2.5 10 1.165 0.053 1.112 volts o ox e red NAD RT E E F NAD O aq x E E ν − = − = − = − = (4.11)

Note that a negative value of converts to a positive value of , and both indicate a spontaneous reaction direction from left to right as the reaction is written (electrons flowing towards the more positive side). In this example, there is a strong driving force for the reaction as written in

'

rG

Δ '

E

(4.2) to proceed from left to right. The numbers confirm what is obvious, which is that NADH is a strong reductant for oxygen.

4.2.1 Defining the reduction potential:

The half-reactions defined in (4.3) each contain the oxidized and reduced form of a reactant, such as NADox and NADred. These constitute a “redox couple”. Every redox

reaction, such as (4.2), involves two redox couples. Depending on the conditions of the reaction, the spontaneous direction of the redox reaction will be from the reduced form of one of the redox couples to the oxidized form of the second redox couple. The convention is to

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compare the thermodynamics of "redox couples" on the basis of their reduction potentials, which we will now define. Let's generalize by splitting the following reaction

(4.12) ox red red ox

A + B R A + B into two half-reactions.

(4.13) 1 ox red 1 ox red A + A B + B e e e e ν ν − − R R

The convention in dealing with biochemical half-reactions is to always write them with the electrons on the left, i.e., the reaction direction from left to right is a reduction. The transformed Gibbs free energy of reaction for (4.12) is given by

' ' ln[ ][ [ ][ o red ox r r ox red A B G G RT A B Δ = Δ − ] ] (4.14)

which we can also write as

' ' ln[ ][ [ ][ o red ox e ox ] ] red A B RT E E F A B ν = − (4.15)

Equation (4.15) is called the Nernst equaton. We will split equation (4.14) into two parts, corresponding to the half-reactions in (4.13).

' ' ' ln[ ] ln[ ] [ ] [ ] o o red red r r A r B ox ox A G G RT G RT A B B ⎡ ⎤ ⎡ ⎤ Δ = Δ⎥ ⎢− Δ − ⎣ ⎦ ⎣ ⎦ (4.16)

The two expressions on the right in (4.16) can be related to the half-reactions in (4.13). We can now define a transformed reaction Gibbs free energy for each half-reaction.

' ' ' ' [ ] ln [ ] [ ] ln [ ] o o red r A r A ox red r B r B ox A G G RT A B G G RT B ⎡ ⎤ Δ = Δ ⎣ ⎦ ⎡ ⎤ Δ = Δ ⎣ ⎦ (4.17)

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The expressions in (4.17) can also be obtained by starting with the half-reactions in (4.13) and using the procedures described in the Chapter 3, considering the electron to be formally one of the reactants, and assigning the electron a chemical potential of zero.

The standard state transformed Gibbs reaction free energy of the half-reactions can be obtained from the corresponding Gibbs free energies of formation.

(4.18) ' ' ' ' ' ' ' ' ' ' ( ) ( ) red ox red ox red ox red ox o o o o o r A A A f A f A o o o o o r B B B f B f B G G G G μ μ μ μ Δ = − = Δ − Δ Δ = − = Δ − Δ G G 'o B G

For the full reaction (4.12) the standard state Gibbs free energy of reaction can be written as

(4.19) 'o 'o

rG rGA r

Δ = Δ − Δ

Divide (4.17) through by νe Fto convert to units of volts to obtain the following.

' ' ' ' [ ] ln [ ] [ ] ln [ ] o o red A A e o red B B e o A RT E E F A B RT E E F B ν ν ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ x x 'o B (4.20) In (4.20), 'o and A

E E are defined as the standard reduction potentials of the redox couples Aox/Ared and Box/Bred, respectively.

' ' ' ' o o r A A e o o r B B e G E F G E F ν ν Δ = − Δ = − (4.21)

We also note that since 'o 'o ,

rG rGA r Δ = Δ − Δ 'o B G 'o B

(4.19), then for the full reaction (4.12) 'o 'o

A

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The minus sign in front of 'o B

E in (4.22) results from the convention of writing the half– reactions with the oxidized form on the left, as in (4.13). The full reaction is equal to reaction A minus reaction B in(4.13). The quantitative relationship between ΔrGA'o and EA'oin (4.21)is exactly the same as shown in Figure 4.2. To get a better feeling for equation (4.20), we will convert to a log10 instead of natural log, and assume T = 298.15K, to get

' ' 59log[ ] (mV units for ' and ' ) [ ] o red o A A e ox A A A E E A ν = − E E (4.23)

Assuming a standard reduction potential of +100 mV, the data in Figure 4.3 for a 1-electron and 2-electron reaction. For a 1-electron reaction the slope is -59mV per log unit, or per order of magnitude change in the ratio of [

[ ] red ox A A ]

. The slope is half this value for a 2-electron reaction, about -30 mV/log unit.

Figure 4.3: Plot of equation (4.23) assuming the temperature is 298K and 'o 100 A

E = mV .

For a 1-electron reaction, the slope is 59 mV/log unit, and for a 2-electron reaction, the slope is about 30 mV/log unit. This is the change in the reduction potential for every 10-fold change in the ratio [

[ ] red ox A A ]

. The larger this ratio, the better the reducing power, or the more negative the value of the solution potential.

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4.2.2 The standard reduction potential is also the midpoint potential of a redox couple

In equation (4.20), when 50% of A has been reduced, then [Ared] = [Aox] and the logarithmic

term is equal to zero. At this point,

(4.24) ' ' when 50% of A is reduced. o A A E =E

For this reason, the standard reduction potential is also referred to as the midpoint potential

of the redox couple, and is designated as , the potential at which half of the redox couple is reduced and half oxidized. If we had an electrode maintained at a potential of

, submerged in a solution of “A” , at equilibrium half of “A” would be reduced. Often the pH is indicated, and the superscript prime indicates constant pH. If no pH is designated, it should be assumed the

' , m pH E ' , m pH E ' m

E refers to pH 7. It is important to recognize that for biochemical redox reactions, it is conventional to define the standard state as pH 7, whereas for chemical reactions, the usual definition of the standard state concentration (activity) of 1 M is used.

4.3 Determining the value of the midpoint potential

Values of many standard reduction potentials (or midpoint potentials) are tabulated , and some are shown in Table 4.1(1-3). Most of the redox couples shown in Table 4.1 are

involved in enzyme catalyzed reactions in E. coli (4)as well as in many other organisms.

Note that these all apply to standard conditions at pH 7 ([H+] = 10-7 M).

Redox couple e ν Standard reduction potential( ' or ,7 o m 'o E E ),mV O2/H2O 4 815 3 2 NO / NO− − 2 420 2 4 NO / NH− + 6 360 O2/H2O2 2 295 DMSO/DMS1 2 160 TMAO/TMA 2 130 ubiquinone/dihydro-ubiquinol 2 110 fumarate/succinate 2 30 menaquinone/dihydro-menaquinol 2 -80 glucose/gluconate 2 -140 oxaloacetate/malate 2 -165 pyruvate/L-lactate 2 -185

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dihydroxyacetone phosphate/

glycerol-3-phosphate 2 -190

acetaldehyde/ethanol 2 -195

NADox/NADred 2 -320

H+/H 2 2 -420 CO2/formate 2 -430 acetate/acetaldehyde 2 -580 acetate/pyruvate 2 -700 1dimethylsulfide (DMS)

2trimethylamine N-oxide (TMAO); trimethylamine (TMA)

If not, they can be determined either from existing data or experimentally. Three approaches are given below.

Method 1: One way is to calculate values from the Gibbs free energies of formation of the reduced and oxidized forms of the redox couple. Many of these are tabulated. For example, 'o E 1 ' ' ' / 3 3 ' / ' / NAD + 2 e NAD (1120.09 10 ) (1059.11 10 ) 2(96485) 0.316 volts or -316 mV red ox ox red ox red ox red ox red o o f NAD f NAD o NAD NAD e o NAD NAD o NAD NAD G G E F x x E E ν − ⎡Δ − Δ ⎤ ⎣ ⎦ = − − = − = − R (4.25)

Note that we must convert kilojoules to joules by multiplying the transformed Gibbs free energies of formation by 1000. The same exercise can be done for the standard reduction potential of the O2/H2O redox couple, yielding

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2 2 2 2 2 2 2 2 1 2 2 ' ' ' / 3 3 ' / ' / O + 4 e 2 H O 2 2( 155.6 10 ) (16.4 10 ) 4(96485) 0.848 volts or 848 mV o o f H O f O o O H O e o O H O o O H O G G E F x x E E ν − ⎡ Δ − Δ ⎤ ⎣ ⎦ = − − − = − = R (4.26)

Method 2: The equilibrium constants of many biochemical redox reactions are also

tablulated, many determined experimentally. If one can determine the equilibrium constant for a reaction involving two redox couples, and if one knows the midpoint potential of one of the redox couples, then the second is easily calculated. For the generalized reaction in (4.12), the equilibrium constant can be expressed in terms of the standard reduction potentials or midpoint potentials. ' ' ' ' / / ' ( ) ' o o e r o o e AoxAred BoxBred FE G RT RT F E E RT K e e K e ν ν Δ − − = = = (4.27)

Method 3: A third way is to experimentally determine the potential developed between the redox couple of interest and a reference redox couple. The convention is to report standard reduction potentials versus the standard hydrogen electrode (SHE). The standard hydrogen electrode is a platinum electrode that is in contact with hydrogen gas at a pressure of 1 bar and an aqueous solution of 1 M protons. Either hydrogen gas can be oxidized to yield protons or protons can be reduced to form hydrogen gas at this electrode. The convenience of this esoteric choice of the standard hydrogen electrode is that the reduction potential of the H+/H2

redox couple, + . This is because 2 o H /H E = 0 2 0 o H

μ = for hydrogen gas, the most stable form of the element under standard conditions and, by definition, the standard state chemical potential

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of a solution of 1 M protons . Hence, the measured potential for any redox couple in the standard state (pH 7, 1 M concentrations) in relation to the standard hydrogen electrode is simply its standard reduction potential.

0 o H μ + = 2 ' ' / / ' ' / ox red ox red o o o measured A A H H vsSHE o o measured A A vsSHE E E E E E + = − = (4.28)

It is useful to keep in mind that the sign of the standard reduction potential refers to whether the redox couple will be more reducing (negative value of ) or more oxidizing (positive value of ) than the proton/hydrogen couple in the standard hydrogen

electrode. Also, a negative means that current will flow from the electrode measuring the redox couple of interest to the standard hydrogen electrode, and a positive means current will flow from the standard hydrogen electrode to the reactants in the setup in Figure 4.1. Note that in Table 4.1, the biochemical definition of the standard potential for the H

' / ox red o A A E ' / ox red o A A E ' / ox red o A A E ' / ox red o A A E +/H 2

couple is -420 mV vs SHE. This is because the biochemical definition of the standard state is at pH 7, or [H+] = 10-7 M. At 298K, going from 1 M to 10-7 M is a change of 7 log units, or -7(∼60 mV/log unit) = -420 mV (see Figure 4.3).

The standard hydrogen electrode is convenient from a computational viewpoint since the midpoint potential of the H+/H2 redox couple is zero. However, from a practical

viewpoint, the standard hydrogen electrode is not convenient at all. Instead, it is common to use either a saturated calomel reference electrode or a silver chloride reference electrode. These are readily purchased and are packaged with a salt bridge and porous glass frit, ready to be inserted into the electrochemical solution.

The calomel electrode uses the redox couple of mercury metal (liquid) and Hg2Cl2.

1 2 2

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The name derives from the fact the Hg2Cl2 is also called “calomel”. The reduction potential

depends on the concentration of chloride, and these reference electrodes are most often used with a saturating solution of KCl. At room temperature, . Hence, if a calomel reference electrode is used, one can simply add +241 mV to the potential obtained to the value versus the standard hydrogen electrode.

241 versus SHE

calomel

E = + mV

Another choice as reference electrode is the silver chloride electode. This uses the redox couple of silver metal and silver chloride.

1 0

AgCl + e − R Ag (s) + Cl−

As with the calomel electrode, the silver chloride electrode reduction potential depends on the concentration (activity) of chloride, and is routinely used with saturated KCl solution. The solution potential of the silver chloride electrode at room temperature is +205 mV vs SHE. For any biochemical reaction, the data obtained are always converted to values versus the standard hydrogen electrode by adding 205 mV to the value obtained with the Ag/AgCl reference electrode. To experimentally determine the midpoint potential of a redox-active biochemical substance, it is necessary to use an electrochemical cell and to manipulate the solution potential, as described in the following section.

4.4 Solution (ambient) potentials and electrochemical cells

Let’s consider a simple electrochemical cell containing a biochemical redox couple of interest. In this example we have two electrodes and the device is conceptually identical to that shown in Figure 4.1. One electrode is in direct contact with the solution containing the material being studied. The second electrode is the reference electrode which is in contact with the electrochemical solution through a salt bridge. The most commonly used reference electrodes are the saturated calomel electrode and the silver chloride electode, discussed in the previous section. The voltage measured between the two electrodes (Figure 4.4) will be

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dependent on the reduction potential of the redox couple in solution and the reduction potential of the reference electrode.

Figure 4.4: Schematic of a simple electrochemical cell. This version has two electrodes. The solution must be made anaerobic because O2, being a strong oxidant, will interfere

with the system. Argon gas is frequently used to flush the system. The working electode is often platinum gauze, increasing the surface area that can react with redox-active solution components. The reference electrode is usually a saturated calomel electrode or a silver chloride electrode. Mediators are required to convey electrons between most biochemical reagents and the working electrode.

What happens if we have more than one redox couple present in the same solution at equilibrium? At equilibrium, the reduction potentials of all the redox couples must be the same, and this reduction potential will be monitored by the electrode that is in

electrochemical contact with the solution. This is called the solution potential or ambient potential and is designated as Eh. If the solution potentials are not the same for the redox

couples, this indicates that the solution is not in equilibrium.

Mediators help attain equilibrium: It is almost aways necessary to include mediators in the electrochemical solution since most biochemical compounds will not readily react at the surface of the electrode. The mediators are selected based on their ability to undergo redox chemistry at the electrode surface and also by their ability to equilibrate with the biochemical redox couples in solution. The mediators are themselves redox couples, existing in reduced and oxidized forms, and they are each characterized by a midpoint potential, 'o

m

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solution potential is far from the 'o m

E value of a particular mediator, the concentration of either

the value of [ [ ] red ox A A ] m

for the mediator will be either very small ( 'o h

E >>E ) or very large

( 'o

h m

E <<E ). In the first instance, this means the [Ared] is very small and in the second case,

[Aox] is very small. Under these conditions, the rate by which the mediators can transfer

electrons and help reach equilibrium will be very slow. For this reason, a number of mediators with a range of 'o

m

E values is often present in the electrochemical solution in addition to the biochemical redox couple(s) being studied. A list of several mediators is shown in Table 4.2. Mediator/Reductant/Oxidant 'o m E potassium ferricyanide +430 p-benzoquinone +280 2,6-dichlorophenol indophenol +217 mV 2,5-dimethyl benzoquinone +180 phenozine methosulfate +80 mV ascorbate +30 duroquinone +5 methylene blue +11 mV menadione 0 pyocyanine 34 mV 2,5-dihydroxy-p-benzoquinone -60 anthroquinone -100 indigo carmine -125 mV anthroquinone 1,5-disulfonate -170 9,10-anthraquinone 2,6-disufonic acid -185 mV anthroquinone 2-sulfonate -225

benzyl viologen -350

dithionite -3861

1The midpoint value of dithionite is very dependent

on pH and also concentration. See (5)

Potentiometric titrations: One can perform a potentiometric titration by changing the solution potential while, simultaneously, monitoring the red

ox A

A ratio of the redox couple of

interest using some chemical or spectroscopic methods. The electrochemical cells are constructed to facilitate removing samples at different Eh values or to determine the

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must be able to change the solution potential systematically to do this. Most commonly, an electrochemical cell such as that schematically shown in Figure 4.4 is used, along with a calomel or silver chloride reference electrode. There are several ways to manipulate the solution potential. Regardless of which method is used, one is changing the ratio red

ox

A

A for all of the redox couples in solution and, thus, changing the solution potential.

1. One can add reductant (e.g., a buffered solution of dithionite) or oxidant (e.g., a buffered solution of ferricyanide) to change the solution potential.

2. One can use a potentiostat, which is a device that uses a third electrode to add or remove electrons from solution using an external source of electrons, and in this way alter the solution potential.

3. One can use a dominant redox couple which will equilibrate with the system to be studied, and whose total concentration is substantially greater than that of other redox-active

components in the solution. One adds a known amount of [Ared] and a known amount of

[Aox]. In this way, during the equilibration, the [

[ ] red ox A A ]

ratio for the dominant redox couple remains essentially fixed (since it is present at much higher concentration than any other redox couple), and determines the solution potential. The solution potential can be readily calculated by using equation (4.20) if the values of 'o and

A

E νe are known for the dominant

redox couple. The concentrations of all the other redox couples will equilibrate to be consistent with the solution potential.

If one has, for example, two redox couples present at equilibrium, the solution potential, Eh, must be the same as the reduction potentials of each redox couple.

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' ' ' ' ln[ ] ln[ ] [ ] [ ] o h A B o red red h A B e ox e o E E E or A B RT RT E E E F A F B ν ν = = x ⎡ ⎤ ⎡ ⎤ =⎥ ⎢= − ⎣ ⎦ ⎣ ⎦ (4.29)

Note that in equation (4.29) the electron stoichiometry numbers νe are those that apply for each redox couple separately. From equation (4.23) we can see that for a 1-electron reaction, a change of the solution potential by about 60 mV will change the ratio of [

[ ] red ox A A ] by 10-fold, increasing the ratio for -60 mV, and decreasing it for a change of +60 mV.

4.5 Example: the potentiometric titration of NAD

Now let’s look at an example of how the equations we have derived can be used to determine the value of a midpoint potential as well as the number of electrons transferred in a half-reaction. Figure 4.5 illustrates simulated data of a potentiometric titration of NAD, which shows the fraction of NAD that is reduced as a function of the solution potential, Eh.

We expect the data to fit to the following equation. ' ' ln[ [ ] o red h NAD m e o NAD RT E E E F NAD ν = = − ] x (4.30) In practice it is common to switch to from the natural logarithm to log10.

' 2.303 log[ [ ] o red h m e o NAD RT E E F NAD ν = − ] x (4.31) At 298K, 2.303 (2.303)(8.31)(298) 0.059 volts or 59 mV 96485 RT F = = (4.32)

Therefore, with this value inserted, assuming 298K we get (using mV units) ' 59log[ [ ] o red h m e o NAD E E NAD ν = − ] x (4.33)

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By determining the fraction of NAD that is reduced as a function of Eh, we can determine the

values of both and 'o

e E

ν NAD experimentally.

The data are plotted in two ways in Figures 4.5 and 4.6. In Figure 4.5, the percentage of the total NAD that is reduced is plotted as a function of Eh. The reduction of NAD can be

determined by monitoring its optical absorbance, making this a “spectro-electrochemical titration”. The value of presenting data in this way is that one can readily see that the over the range of Eh values the NAD has gone from fully oxidized to fully reduced.

Figure 4.5: Potentiometric titration of NAD showing the fraction of NAD that is reduced (NADred/NADtotal) as a function of the solution potential (Eh). The midpoint is

about -320 mV (vs SHE).

Since this is an equilibrium measurement, it should make no difference in which direction one does the titration, reducing or oxidizing. In practice, it is important to demonstrate reversibility to be sure that equilibrium has been attained at each point. The potential at which 50% of the NAD has been reduced, readily seen by inspecting the plot in panel A, is equal to the midpoint potential of NAD under the conditions being examined. At pH 7, this is about –320 mV.

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In Figure 4.6, log[ [ ] red ox NAD NAD ]

is plotted versus the solution potential, Eh. We expect from

equation (4.33) to get a straight line where the intercept will be 'o m

E and the slope will be 59

e

ν −

. The data do fit a straight line with a slope of –30 mV, indicating, as we expect, the

e

ν = 2 and this is a 2-electron redox reaction. The solution potential where

[ ] log 0 [ ] red ox NAD

NAD = is the midpoint potential

Figure 4.6: The same data as in Figure 4.5 plotted as the logarithm of the ratio red

ox

NAD

NAD . The data fit a straight line with a slope of -30 mV/log unit, consistent with 2

e

ν = .

4.6 How midpoint potentials are used to depict biochemical electron transfer systems

Any redox couple that has a more negative standard reduction potential will be a stronger reductant than any redox couple whose standard reduction potential is more positive (see Table 4.1). This applies, of course, to standard state conditions (pH 7, 1 M

concentrations). So, for example, NADred is the reductant for O2. The standard potential for

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' 2 2 ' ' / / ' ' 848 ( 316) 1164 mV o ox red o o O H O NAD NAD o o E E E E E = − = − − = (4.34)

Notice that we do not multiply ' in /

ox red

o

NAD NAD

E (4.34) by a factor of two because of the difference in the stoichiomety numbers for the electrons (νe =2for the NAD couple and 4 for the O2 half-reaction. This is because the reduction potentials essentially are already

normalized per electron.

Figure 4.7 is an example of the use of reduction potentials in the biochemical literature. This shows the "Z scheme" describing the energetics of the light-driven reactions in plant photosynthesis. The various redox components that make up the photosynthetic electron transfer chain are all located according to their standard reduction potentials. More negative values are interpreted as "higher energy", meaning that they are better reductants. The Z scheme shows the role of the two photosynthetic reaction centers, photosystem I (PSI and photosystem II (PSII). The absoption of a photon of light results in creating an excited state of chlorophyll P680, which becomes a very strong reductant. The reduction potential is decreased about 1.5 volts, and the electron is transferred through a chain of redox active groups whose reduction potentials get progressively more positive. The ChlP680+/ChlP680 redox couple has a more positive reduction potential than O2/H2O, and oxidizes water to O2,

with intermediates being a Mn cluster and a tyrosine. After a second light reaction in in photosystem I, followed by another linear chain of redox reactions, the end product is NADPred. We will not go into any further details, but just point out that this kind or scale is

frequently used to represent electron transport chains in biochemistry. The tendency is that electrons are transferred from redox couples with more negative reduction potentials to those with more positive potentials. Since these are standard state potentials, one must beware that

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the true reduction potentials will be altered by concentrations of the reduced and oxidized forms. However, in many instances, the redox centers are fixed within proteins or protein complexes, so there is no change in concentration possible as electrons are transferred within a complex with fixed geometry. However, in the functioning electron transport chain, the ratios of each redox component, red

ox A

A , will be determined by the steady state concentrations.

If Aredis oxidized very rapidly by the next redox component along the chain then

red ox

A << A and the reduction potential will be considerably more positive (i.e., better oxidant)

than indicated by the midpoint potential. Although there are a number of exceptions (6),

generally, electron transfer is in the direction towards components with the more positive midpoint potential. The thermodynamics of electron transfer reactions are no different than other biochemical reactions and must always progress towards the minimum Gibbs free energy. We will consider the kinetics of electron transfer reactions in a separate chapter.

Figure 4.7: The Z scheme diagram, illustrating the use of reduction potentials of a series of redox couples involved in electron transfer chains. Each redox active participant is placed at a height in the diagram corresponding to its reduction potential with H2O/O2 at about +0.8 volts. Light

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electron-deficient P680 Chlorophyll is a stronger oxidant than O2, and water is oxidized to form O2 as the

electrons from water re-reduce chlorophyll P680. Govindjee website

Figure 4.8 shows another example, illustrating the photosynthetic scheme for green sulfur bacteria (Chlorobiaceae). In this case, light activates a bacteriochlorophyll (P840) which

becomes a strong reductant, reducing the primary acceptor (A0), which is a modified

bacteriochlorophyll. Electrons then flow to a quinine-like molecule, A1, and then, via several

Fe/S centers to a ferredoxin. There are two options at this point. There is a cyclic electron transfer pathway in which the electrons are passed to a menaquinone within the membrane and, eventually, reduces the oxidized P840. This pathway includes the bc1 complex which

couples the electron transfer reaction to the generation of a transmembrane proton electrochemical gradient, Δp in this diagram. This will be discussed in the next chapter. Alternatively, the ferredoxin can reduce NADPox to NADPred. In this non-cyclic pathway,

electrons from the oxidation of H2Sare used to reduce P840, yielding elemental sulfur, which

can be further oxidized by these organisms to sulfate.

Figure 4.8: Schematic of the photosynthetic electron transfer pathways of the green sulphur bacteria. The redox components are placed according to their midpoint potentials. Light (hν) excites the P840 active-site chlorophyll, which

becomes a strong reductant, initiating electron transfer which can be either cyclic or non-cyclic. (Figure is Fig. 5.5 in (7))

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4.7 Ambient potential in a living cell and oxidative stress

In addition to the many metabolic redox reactions, there are many other redox-dependent processes in both eukaryotic and prokaryotic cells, including protein folding, transcriptional regulation, enzyme regulation and signal transduction. Whereas in the

laboratory one is usually striving to reach equilibrium to make a measurement, in living cells, as we have already discussed the reactant concentrations are maintained in a steady state that is distinctly not at equilibrium. The concentrations of the reduced and oxidized forms of redox couples within cells is determined by the rates by which they are produced and utilized in a myriad of biochemical reactions. We saw, for example, in Section 3.13, the results of one mathematical model of glycolysis showing that many, but not all of the reactions were close to equilibrium conditions. In general, in some sets of biochemical redox reactions will be near equilibrium because the rates of the reactions are fast compared to the reactions coupling the reactions to others taking place within the cell.

One natural set of barriers to rapid redox equilibration within cells are the boundaries between intracellular compartments and organelles(8, 9). Hence, the ambient potential of the

cytoplasm of a mammalian cell is distinct from that of the nucleus or that of the endoplasmic reticulum or the mitochondrion. However, even within these organelles, not all the redox reactions are necessarily maintained at or even near a single solution potential. In the cytoplasm, for example, there are sets of reactions that are equilibrated with the NADred/NADox redox couple, and another set of reactions equilibrated with the

NADPred/NADPox couple. These are not necessarily in equilibrium with each other because of

the kinetics of the reactions linking these reaction networks(9).

Many cellular processes that are redox-regulated depend on the redox status of disulfide bonds between cysteines is key enzymes or transcription factors, for example. The formation of disulfide bonds may also be simply part of the protein folding process required for forming

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a stable, native protein. One has an equilibrium between the reduced and oxidized cysteine pair within a protein

(4.35) 1

2 protein(CySSCy) + 2 e− Rprotein(CySH)

We can also refer to this as the cysteine/cystine or sulfhydryl/disulfide redox couple. As this reaction has been written, the reduced cyteines are assumed to be protonated, but this depends entirely on the pH. Protons have not been included to balance the reaction to emphasize that this reaction occurs at constant pH.

Figure 4.9: The structure of glutathione. Oxidation forms a disulfide-linked dimmer.

In eukaryotic cells, there are two major systems which determine the redox status of

Prot(CySH)2 /ProtCySSCy in proteins: 1) glutathione, a tripeptide with one cysteine (Figure

4.9), which can exist in either a reduced (GSH) or oxidized form (GSSG); and 2)

thioredoxins (Trx) or proteins within the thioredoxin family(10). Thioredoxins are small

proteins which contain a pair of cysteines in an exposed loop (see Figure 4.10) which can also be either reduced or oxidized, Trx(SH)2/TrxSS. There are a number of different

thioredoxins with specific roles, as well as proteins with thioredoxin folds or domains that are redox-active. In mammalian systems, free cysteine circulates in the plasma and the redox status of this cyteine pool is the major determinant of the equilibrated redox couples that are extracellular.

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Figure 4.10: Structure of the reduced form of yeast thioredoxin 1 from yeast (Saccharomyces cerevisiae). (Figure is from (11). )

The reported “redox status” of a cell or cellular compartment is usually determined experimentally by the redox status of one of the key redox couples listed above(9, 10). Of

course, the effective solution potential will be different for various compartments within a eukaryotic cell, such as the cytoplasm, mitochondrion, endoplasmic reticulum, etc.

Depending on the process, the ambient potential of interest may be one reported by

glutathione, but one of the other key dominant redox couples (such as thioredoxin or NAD) may be more significant. Figure 4.11 shows some representative ambient potentials for different eukaryotic cellular compartments. Generally, the mitochondrion and cell cytoplasm are considered to be “reducing environments”, and this is supported by the quantitative measurement of the steady state redox potentials indicated in Figure 4.11.

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Figure 4.11: Estimates of the ambient potentials of cellular compartments in a eukaryotic cell, including the circulating blood plasma. Different dominant redox couples are indicated. These values will be dependent on the physiological state of the cell (Figure is from (9)).

The glutathione-linked solution potential within the mitochondrion is about -300 mV, which is significantly more reducing than the GSH/GSSG potential in the cytoplasm (-260 mV for proliferating cells). These values will depend on the physiological state. The

endoplasmic reticulum, which is where protein disulfides are made, is much more oxidizing, with a GSH/GSSG potential of about -150 mV. In general, protein disulfides are rarely found within the cytoplasm but are much more common in secreted proteins. However, the

formation of disulfides as part of protein folding, is not a spontaneous reaction with O2 in

most cases, but is catalyzed by specific enzymes(10, 12, 13), both in eukaryotic cells (in the

endoplasmic reticulum) and in prokaryotes (in the periplasm of Gram negative bacteria).

4.7.1 Oxidative stress.

Oxidative stress (14)describes pathological situations usually resulting from the

production of reactive oxygen species (ROS), which includes hydrogen peroxide (H2O2),

superoxide (O2-), peroxynitrites (OONO-), organic hydroperoxides (ROOH) and hydroxyl

radicals (HO•). These are “pro-oxidants” and can promote the oxidation of cellular components, resulting in disease states. Reactive oxygen species can be generated by elements of the respiratory chain in mitochondria. Reaction of reactive oxygen species with the glutathione pool will result in reducing the concentration of reduced glutathione and may

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result in lowering the total concentration of glutathione. A cascade of redox consequences results, leading to various pathological conditions, depending on the context.

4.8 The pH-dependence of the midpoint potential

The majority of redox or electron transfer reactions in biochemistry are accompanied by proton transfer reactions, such as the reduction of a disulfides in (4.35). For example, if there is a proton binding site on reactant A, the reduction of Aox to Ared will increase the negative

charge on the molecule and, if there is a proton binding site available, the positively charged proton might bind. In principle, each reactant in (4.12) could be comprised of multiple protonated species, as discussed in the previous chapter. Hence, reactant Aox will consist of a

mixture of Aox , Aox(H+)1, Aox(H+)2, Aox(H+)3 etc. up to some maximum number of bound

protons, depending on the number of available sites. Upon reduction, Ared will, similarly, be

comprised of a distribution of protonated species. This is important because the reduction potential of each different protonated species may be unique. It is likely to be easier to reduce a more protonated species since it carries more postive charge, i.e., the protonated species will have a more positive reduction potential. However, by using the transformed thermodynamic functions, assuming a constant pH, we need not be concerned about the specifics of the distributions of the protonated species in order to define the basic thermodynamics. This assumes, as we did in defining the transformed Gibbs free energy function, that the

protonation reactions are rapid and the protonated species are always equilibrated. However, it is clear that upon reduction, the distribution of the protonated species may change and, more important for us at this time, the average number of bound protons may also change.

In Chapter 3 we derived the expression in equation (3.47) describing the

pH-dependence of the transformed Gibbs reaction free energy, ', and the same relationship is

rG

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also valid for the pH-dependence of the standard state transformed Gibbs reaction free energy, 'o. rG Δ ' ' , , ( ) 2.303 ( ) o r r H T P G RT N pH ξ ⎡∂ Δ ⎤ = Δ ⎥ ⎣ ⎦ (4.36)

In this expression, recall that is the change in the number of bound protons (per mole of reaction progress) for the reaction under the specified conditions. Let’s go back to reaction

rNH

Δ

(4.12), but assume that there are coupled protonation reactions. Substituting from equation (4.22), we get ' ' ' ' , , , , ( ) ( ) 2.303 ( ) ( ) o o r A r B r H T P T P G G RT N pH pH ξ ξ ⎡∂ Δ ⎤ ⎡∂ Δ ⎤ − = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ Δ (4.37)

or for each half reaction

' ' ' ' , , , , ( ( )) 2.303 ( ) ( ) ( ( )) 2.303 ( ) ( ) o o m r H e T P m r H e T P E A RT N A pH F E B RT N B pH F ξ ξ ν ν ⎡ ⎤ = − Δ ⎢ ⎥ ∂ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = − Δ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ (4.38)

where and are the changes in number of bound protons (per mole) for each half reaction, and

( ) rNH A Δ ΔrNH( )B (4.39) ( ) ( ) rNH rNH A rNH Δ = Δ − Δ B

For each half-reaction, we can substitute numerical values at 298K in equation (4.38) to get the following.

'

' , ,

( ( )) change in protons bound

59 59

( ) number of electrons transferred

o m r H e T P E A N pH ξ ν ⎡Δ = − = − • ⎢ ⎥ ∂ ⎢ ⎥ ⎣ ⎦ (4.40)

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' ' 2 1 , , 59 ( 2 ) o o r H m pH m pH e N 1 E E pH ν pH Δ = − − (4.41)

This says that the pH-dependence of the midpoint potential can be used to determine the change in the number of bound protons for a redox half-reaction, provided that we know the number of electrons (νe ) involved in the reaction.

4.9 Example: The pH-dependence of the midpoint potentials of the NADox/NADred and

oxaloacetate/malate redox couples.

We will consider as an example the redox equation

(4.42) L-malate + NAD oxaloacetate + NAD

half-reaction 1: NAD + 2 e NAD

half-reaction 2: oxaloacetate + 2 e malate

ox red ox red − − R R R

This reaction is catalyzed by malate dehydrogenase and is part of the TCA cycle. We saw in

Section 4.5 that both the and 'o

e Em

ν values for a NAD half-reaction could be experimentally determined by potentiometric titration, as in Figure 4.5. We can now determine the value of

for the reduction of NAD by plotting the value of the midpoint potential

rNH Δ ' ( o m ) E NAD as a

function of pH by using equation (4.40). The result is a straight line, shown in Figure 4.12, with a slope of -30 mV per pH unit. The fact that this is a straight line means that is not changing over the pH range being examined (pH 5 to 9). The fact that the slope of the line in

Figure 4.12 is not zero means that there is a change of protonation of NAD upon reduction, as we already know. The slope of –30 mM/pH unit tells us that

rNH Δ r H e N ν Δ is 1/2, so over the full range of pH (since

1

rNH

Δ =

2

e

ν = ). The midpoint potential of NAD gets more negative as the pH is increased. NADred is a better reductant at higher pH.

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(4.43)

-1 + + -1 +

ox red

NAD +2e +H RNAD or NAD +2e +H RNADH

Also shown in Figure 4.12 is the pH-dependence of the midpoint potential of the oxaloacetate (OAA)/malate redox couple. From Table 4.1, we see that '

,7 165

o m

E = − mV for this redox couple, which is also indicated in Figure 4.12. The pH-dependence of the

midpoint potential is a straight line with a slope of -59 mV/log unit. Since this is a 2-electron reaction (νe =2), we conclude from equation (4.40) that ΔrNH =2 throughout the pH

range.

At any pH, the difference between the two lines is

' ' ' , , , ' ' , ( / ) ( / o o o m pH m pH ox red m pH o o r m pH e

E E NAD NAD E OAA malate

G E F ν Δ = − Δ Δ = − ) (4.44)

Under standard state conditions (1 M concentrations) the reaction would go in the opposite direction than indicated in (4.42). Under metabolic steady state conditions, however, the direction of the reaction is as written, as required by the TCA cycle to produce NADred.

Figure 4.12: The pH-dependence of the NADred/NADox and the OAA/L-malate

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4.10 Thermodynamic reciprocity of interactions between proton binding and reduction potential

In the example of NAD, described above, the oxidized form remained deprotonated and the reduced form remained fully protonated throughout the pH range. Now let’s look at a situation where we have a redox couple, Aox/Ared, in which both the oxidized and the

reduced forms can bind 1 proton, but that the proton affinity, or pK, is shifted upon reduction. We will assign the pK of the oxidized form, pKox a value of pH 6. Upon

reduction from Aox to Ared, the proton affinity is greater so it will become protonated at a

higher pH value (lower [H+]). Furthermore, we will specify that we are able to measure the extent to which A is reduced, but cannot distinguish whether it is protonated or

unprotonated. The experiment will be to change the solution potential, and measure the apparent midpoint potential, , at a series of different pH values, as in Figure 4.12. The question is how does the apparent midpoint potential vary with pH from pH 4 to 10, a span that encompasses both the pK of the reduced and oxidized forms of the redox couple. This is a useful problem to examine because we will introduce some of the procedures that will be used throughout the text when approaching problems dealing with thermodynamics in a number of different contexts.

' ,

o m pH

E

To begin, let's identify the number of different molecular species we have in our solution. These are

Aox : oxidized, not protonated

AoxH+: oxidized, protonated

Ared : reduced, not protonated

Aox H+ : reduced, protonated

These four species are related by several equations. The protonation reactions are written in the direction of deprotonation, and the equilibrium constant is a proton dissociation

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equilibrium constant, used to define the pK. Note that the pH at which half of the species (e.g., Aox) is protonated, (e.g., [Aox] = [A Hox

+]), then [H+] = K

ox, or pH = pKox. Also, since

we are dealing with chemical species and not biochemical components, we have dropped the “primes” over the thermodynamic parameters.

+ +

ox ox red red

'

1) A A + A H + A + A H

2) protonation of the oxidized species: A [A ][ ]

; ( ) log

[ ]

3) protonation of the reduced species: A

total ox ox o ox ox r ox ox ox ox red re A H H H K G H pK K A H A H + + + + + + = + = Δ = − R R ' 1 1 [A ][ ] ; ( ) log [ ]

4) reduction of the oxidized species: A

[ ] [ ] 59log ; ( ) 2.303 log [ ] [ ] 5) reduc d o red

red r red red red

red ox red o red o red h m r m ox ox H H K G H pK K A H e A A A E E G e RT FE A A + + + + − − + = Δ = − + = − Δ = − = − R 1 o 2 2

tion of the protonated oxidized species: A

[ ] [ ] 59log ; ( ) 2.303 log [ ] [ ] ox red o red o red h m r m ox ox H e A H A H A H E E G e RT FE A H A H + − + + + − + + + = − Δ = − = − R 2 o (4.45)

In equations 4 and 5, above, it is assumed that the temperature is 298K, in order to get the value of 59 mV. In these equations, Eh is the solultion potential, experimentally determined.

It is usual to use units of mV in place of volts, but remember to use volts when changing units to joules.

The protonation and redox reactions above can be put into a simple thermodynamic cycle, shown in Figure 4.13. We can see from Figure 4.10 that there are two different pathways to go from the oxidized species Aox to the protonated, reduced species, AredH+.

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Since, the transformed Gibbs reaction free energy is a state function, the free energy change must be identical no matter which way we go. Aox can be either protonated first and then

reduced, or reduced first and then protonated. The the free energy change will be the same. Hence, we can conclude that

(4.46) 2 1 ( ) ( ) - ( ) ( rGox H rG e rG e rGred H + − − −Δ + Δ = Δ + Δ +)

The negative sign in front of the reaction free energy terms for the protonation reactions come from the fact that we defined these in the direction of deprotonation in (4.45). From (4.46) it follows that 2 1 ln 2.303 log 59( ) ( ) red red ox ox o o ox red m m K K RT RT F K F pK pK E E = − = − K (4.47)

Figure 4.13: Thermodynamic cycle showing two equivalent pathways of going from Aox to AredH+(indicated by the red arrows).

Equation (4.47) tells us that if the difference in the midpoint potentials between the

protonated and deprotonated forms is specified, this also defines the difference between the pK values of the reduced and oxidized forms. This is more readily seen in a free energy diagram, Figure 4.14, which shows the drop in the standard state molar free energy as

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reactants are converted to products. We will encounter these diagrams at many points in the text, particularly when we discuss the thermodynamics of ligand binding.

Figure 4.14: Free energy diagram showing the relative values of the standard state molar Gibbs free energy values of the system in different chemical states. Each level is labelled by the species present. For example, the top line (Aox + H+ + e- ) stands for the

sum of the standard state chemical potentials for each of the species, (

ox o o A o H e μ +μ + +μ −),

etc. The bottom line is the level of the standard state chemical potential of AredH+. The

two sets of vertical lines on the left show the situation in which reduction increases the proton affinity of molecule A. By necessity, protonation of molecule A must also increase the affinity for the electron, indicated by the larger magnitude of the drop in free energy associated with reduction of AoxH+ compared to Aox. The two sets of

vertical lines on the right depict the situation where reduction of molecule A has no effect on the proton affinity, and vice versa. The coupling free energy quantifies the mutual influence between the protonation and electron transfer reactions.

One important concept that is easily understood in terms of a free energy diagram is the idea of thermodynamic coupling or cooperativity. In the current problem, we have stipulated that reduction of the molecule A results in increasing the affinity for proton binding. Let’s rearrange equation (4.46) (4.48) 2( ) 1( ) ( ) ( ) o o o o rG e rG e rGred H rGox H − − + Δ − Δ = Δ − Δ +

The difference in the reaction free energy of reducing the protonated and deprotonated forms is exactly matched by the difference of the reaction free energy of protonating the

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reduced and oxidized forms. These differences are called the coupling free energy, . o rGcoupling Δ (4.49) 2( ) 1( ) ( ) ( o o o o o rGcoupling rG e rG e rGred H rGox − − + Δ = Δ − Δ = Δ − Δ H+)

If the reaction free energy of binding a proton is favored by, say -20 kJ/mol by reduction, then the binding of a proton will, by necessity, make the reduction more favorable by the same -20kJ/mol. If reduction has no influence on the protonation, then protonation will have no influence on the reduction potential, i.e., o 0

rGcoupling

Δ = and there is no

cooperativity. This is called reciprocity, and is a form of cooperativity. This concept is encountered very frequently in biochemical reactions and in ligand binding. The coupling free energy is shown on the free energy diagram in Figure 4.14 by showing the case where it is assumed that there is no coupling ( o 0

rGcoupling

Δ = ) on the right side. The standard state free energy of the system is lower due to the addition of the favorable (negative) coupling free energy, which stabilizes A Hred +relative to Aox, shown by the lower standard state free

energy in the diagram.

The equations in (4.45) represent a specific model that we can use to simulate data or, if we were really doing an experiment, to fit to data. Although we have equations defining

1 o m E and 2 o m

E , these cannot be directly measured. Instead, we measure the "apparent" or transformed midpoint potential, . We are now back to the “prime” in because we are keeping the pH constant during the reaction and grouping species in pseudo-isomer groups that differ only by the state of protonation. This is simply the transformed

thermodynamic parameter, as we discussed in the previous chapter. The value of is what we can actually measure by a potentiometric titration since we have no way to know whether the molecule is protonated or not. It is useful to see how is related to the

non-' , o m app E Em app'o, ' , o m app E ' , o m app E

References

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