Theoretical Analysis of the Two-Electron Transfer Reaction and

Article pubs.acs.org/Langmuir

Theoretical Analysis of the Two-Electron Transfer Reaction and Experimental Studies with Surface-Confined Cytochrome c Peroxidase Using Large-Amplitude Fourier Transformed AC Voltammetry Gareth P. Stevenson,† Chong-Yong Lee,‡ Gareth F. Kennedy,‡ Alison Parkin, § Ruth E. Baker,⊥ Kathryn Gillow,∥ Fraser A. Armstrong, § David J. Gavaghan,*,† and Alan M. Bond*,‡ †

Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom School of Chemistry, Monash University, Clayton, Victoria 3800, Australia § Inorganic Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QR, United Kingdom ⊥ Centre for Mathematical Biology, and ∥Numerical Analysis, Mathematical Institute, 24-29 St. Giles’, Oxford OX1 3LB, United Kingdom ‡

S Supporting Information *

ABSTRACT: A detailed analysis of the cooperative twoelectron transfer of surface-confined cytochrome c peroxidase (CcP) in contact with pH 6.0 phosphate buffer solution has been undertaken. This investigation is prompted by the prospect of achieving a richer understanding of this biologically important system via the employment of kinetically sensitive, but background devoid, higher harmonic components available in the large-amplitude Fourier transform ac voltammetric method. Data obtained from the conventional dc cyclic voltammetric method are also provided for comparison. Theoretical considerations based on both ac and dc approaches are presented for cases where reversible or quasi-reversible cooperative two-electron transfer involves variation in the separation of their reversible potentials, including potential inversion (as described previously for solution phase studies), and reversibility of the electrode processes. Comparison is also made with respect to the case of a simultaneous two-electron transfer process that is unlikely to occur in the physiological situation. Theoretical analysis confirms that the ac higher harmonic components provide greater sensitivity to the various mechanistic nuances that can arise in two-electron surface-confined processes. Experimentally, the ac perturbation with amplitude and frequency of 200 mV and 3.88 Hz, respectively, was employed to detect the electron transfer when CcP is confined to the surface of a graphite electrode. Simulations based on cooperative twoelectron transfer with the employment of reversible potentials of 0.745 ± 0.010 V, heterogeneous electron transfer rate constants of between 3 and 10 s−1 and charge transfer coefficients of 0.5 for both processes fitted experimental data for the fifth to eighth ac harmonics. Imperfections in theory-experiment comparison are consistent with kinetic and thermodynamic dispersion and other nonidealities not included in the theory used to model the voltammetry of surface-confined CcP.

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INTRODUCTION Electrochemical techniques such as cyclic voltammetry have been widely used in investigations of electron transfer reactions in redox active metalloproteins and enzymes.1,2 Yeast cytochrome c peroxidase (CcP) is a well-characterized redox enzyme that is also electrocatalytically active for hydrogen peroxide reduction3−8 and catalytic intermediates have been studied in considerable detail. Hydrogen peroxide reacts with the Fe(III) form of the enzyme to give a form, known as compound I, which contains ferryl heme Fe (Fe(IV)O) and a cation radical located on a tryptophan residue that is located approximately 5 Å from the Fe at closest approach on the proximal side of the heme. Compound I is reduced back to the Fe(III) state by two molecules of cytochrome c (or an © 2012 American Chemical Society

electrode) via an intermediate species known as compound II, which can exist in two forms: [Fe(IV):Trp0] and [Fe(III):Trp+].9,10 A scheme outlining these reactions is shown in Figure 1. Under conditions of low temperatures (0.7 V vs SHE at pH 7). Several studies have been reported using both stationary and rotating disk electrodes.3−8 In the absence of Received: March 28, 2012 Revised: May 16, 2012 Published: May 18, 2012 9864

dx.doi.org/10.1021/la205037e | Langmuir 2012, 28, 9864−9877

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Figure 1. Reaction mechanism (adapted from ref 11) for the cooperative two-electron charge transfer process undertaken by cytochrome c peroxidase.

that for simultaneous two-electron transfer (scheme 2 with two electrons), and single electron transfer (scheme 2 with one electron). Note that the last two models are not genuine options for CcP in its physiological role (a simultaneous twoelectron transfer with cytochrome c is impossible, and a single one-electron transfer will not complete a catalytic cycle), but are included for completeness. The first model will be described in detail and it will then be shown how the mathematics may be simplified to achieve a description of the other two models. The experimental data available on the electrochemistry of CcP, which has so far been restricted to dc methods, will be extended to large-amplitude Fourier transformed ac cyclic voltammetry. This gives access to the higher harmonic components which are kinetically sensitive to the nuances of two-electron transfer processes, and are devoid of capacitive current which plagues the dc method. This allows a far more detailed study to be made of this system than is possible by dc cyclic voltammetry alone. Evans et al. have elegantly surveyed the dc voltammetry of two-electron transfer for the case of homogeneous solutionphase systems.12−15 They also compare the normal ordering of reversible potentials, where the addition or removal of any electron is more difficult than for the electron preceding it, and potential inversion, where the addition or removal of any electron is less difficult than for the electron preceding it.12−15 This survey also includes the cases of simultaneous (“concerted”) two-electron transfer and sequential electron transfer, where the latter involves the existence of an intermediate and the former does not involve an intermediate, although the implication is not that two electrons tunnel together.15 In essence, we now extend Evans’ dc cyclic voltammetric description to the ac version of the two-electron situation prevailing in surface-confined systems, commonly encountered in protein film electrochemistry.7 Due to the large size of enzymes, “nonturnover signals” (expected when substrate is absent) are usually too small to observe; indeed, at best, dc methods yield voltammograms which are dominated by background current, which poses a difficulty in identifying and quantifying any faradaic processes which may be present. We have demonstrated previously the advantages of using large-amplitude FT ac voltammetry for studies involving surface-confined small electron transfer proteins, for which a well-defined faradaic process is observed without requiring any background subtraction procedure.16−20 All of these studies have involved one-electron transfers; thus, we now extend the concepts to the case of the two-electron transfer reaction displayed by CcP. For completeness, we also compare the ac and dc cyclic voltammetric results under surface-confined

hydrogen peroxide, a nonturnover response is observed as a pair of oxidation and reduction peaks close to the region at which H2O2 is reduced.5,6 Both peaks are symmetrical and have a half height width, W1/2, of approximately 60 mV, suggesting that the two one-electron transfers occur in a cooperative manner, i.e., compound II may be metastable. Such an interaction between two separate electron transfer sites may be mechanistically important; it persists when CcP is genetically modified by replacing a distal tryptophan to phenylalanine, which simply raises the two-electron potential to above 0.85 V.6 In order to elucidate further details of the electron transfer processes occurring at an electrode, we have used numerical simulations to analyze the harmonic response expected for a two-electron redox process under conditions of large amplitude Fourier transformed ac cyclic voltammetry and compared the outcome with simulated responses obtained for conventional dc cyclic voltammograms. Our study has been confined to situations where there is no hydrogen peroxide present, thus allowing us to proceed without considering catalytic regeneration of Fe(IV):Trp+ from Fe(III):Trp0. This simplification allows us to avoid consideration of the diffusion of hydrogen peroxide and provides us with an exclusively surface-confined system. Then there are two likely schemes for electrochemical reduction of CcP, represented in Figure 2, where (a) scheme 1

Figure 2. Two possible electron transfer mechanisms for cytochrome c peroxidase. (a) Two-electron transfer involving an intermediate as in Figure 1 and (b) two-electron transfer with no intermediate.

represents two single electron transfers via compound II and (b) scheme 2 represents two electrons transferred simultaneously, which is a pathway that cannot be followed when cytochrome c peroxidise reacts by sequential reactions with two molecules of cytochrome c. Comparison of experimental data with the so-called cooperative two-electron transfer (scheme 1) is made against 9865


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conditions. In this context, the pioneering studies by Laviron,21 particularly for reversible electron transfer, need to be noted, as do those by other authors.22

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using data obtained at a very low scan rate. This value for CcP is reported to be 61 mV at 4 °C.11 Since theoretical values at this temperature are 84 and 42 mV for one and two electrons being transferred simultaneously, respectively (see below), it may be assumed that the number of electrons transferred lies between one and two and cannot be correctly deduced from the relationship derived from this model. A further complication is that there may be some broadening due to dispersion (nonhomogeneity among molecules adsorbed on the electrode) or for other reasons. The alternative supposition derived from this analysis of W1/2 at low scan rate is that reduction occurs via a cooperative two-electron transfer, rather than via an electron transfer process where two electrons are transferred simultaneously. A major initial challenge therefore lies in understanding details related to E01 and E02 values (defined in Figure 2) that give rise to W1/2 = 61 mV at a very slow scan rate when an overall two-electron process approaches reversibility. n-Electrons Transferred Simultaneously. Following Laviron21 and others22 we first consider the case of an n electron transfer described by eq 1

EXPERIMENTAL METHODS

Chemicals. Baker’s yeast CcP was isolated and purified according to the method of English et al.23 and stored under liquid nitrogen. Deionized water from a Milli-Q-MilliRho purification system (resistivity 18 M Ω cm) was used to prepare all solutions. In electrochemical studies, 20 mM phosphate buffer adjusted to pH 6.0 by addition of HCl or NaOH (Aldrich) also was present. No further supporting electrolyte was added due to the requirement for a low ionic background to stabilize CcP on the electrode surface. Apparatus and Procedures. Large amplitude Fourier transformed ac cyclic voltammetric experiments were undertaken with instrumentation described elsewhere.24 The applied ac perturbation superimposed onto the dc ramp had an amplitude of 200 mV and a frequency of 3.88 Hz. Unusually low frequency and large amplitude conditions were required because of the slow electrode kinetics (see below). Conventional dc cyclic voltammetric experiments were carried out with an Autolab electrochemical analyzer (Eco-chemie, Utrecht, The Netherlands) equipped with a PGSTAT 20 potentiostat. An all-glass standard three-electrode electrochemical cell was employed with a pyrolytic graphite “edge” (PGE) working, Ag/AgCl (3 M NaCl) reference, and platinum wire auxiliary electrodes. Prior to the voltammetric experiments, the surface was polished with 1 μm alumina suspension on a clean polishing cloth (Buehler). The electrode was then rinsed, sonicated thoroughly, and washed with water. Protein films were obtained by introducing a freshly polished electrode into a 1.0 μM solution of CcP in 20 mM phosphate buffer solution, pH 6.0 at 4 °C. The potential was held at 600 mV for time periods of about 30 s during the course of adsorption of CcP in order to achieve surface coverage of the order of 10 pico-mol cm−2. The electrode carrying the immobilized CcP film was then transferred into a cell containing fresh phosphate buffer solution also at pH 6.0 and 4 ± 1 °C, that had been degassed with high purity nitrogen for at least 10 min. Base-line corrections of dc cyclic voltammograms were undertaken by constructing a cubic spline fit to regions of the voltammogram where faradaic current is negligible and predicting values in the potential region where faradaic current is present. The surface coverage of CcP was calculated from the charge (area under voltammogram) in a dc cyclic voltammogram,22 after subtraction of the background current. Numerical simulations of ac and dc voltammograms (see theory section below) were performed using MATLAB-based software. The protocol for comparing experimental and numerically simulated data was a heuristic one in which unknown parameters of interest were varied until satisfactory agreement was achieved.

E30

A + ne− ⇌ C

(1)

where the coverages of species A and C are represented by ΓA and ΓC, respectively, and E03 is the reversible potential for the process. From the Nernst equation25 (since the process is taken to be reversible) E = E30 +

RT ⎛ ΓA ⎞ ln⎜ ⎟ nF ⎝ ΓC ⎠

(2)

where E is the applied potential, R is the universal gas constant, T is temperature, and F is Faraday’s constant. By convention, voltammetric experiments begin at time t = 0: for times t < 0 the electrode is held at a constant initial potential, Ei, which is sufficiently extreme to ensure that eq 1 does not occur significantly. For t ≥ 0, the time-dependent applied potential E(t) is E(t ) = Ei + vt

(3)

where v is the scan rate. Note that for ac voltammetry we simply add an oscillatory component Eac = ΔE sin(ωt) to the dc ramp, where ΔE is the amplitude and ω is the frequency of the sinusoidal component of the signal. Rearranging eq 2 gives

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ANALYTICAL SOLUTIONS FOR REVERSIBLE ELECTRON TRANSFER IN SURFACE-CONFINED DC VOLTAMMETRY Before proceeding to a consideration of the modeling and numerical simulation of two-electron transfer schemes under conditions of large amplitude ac voltammetry, it is useful to summarize outcomes derived from an analytical solution for the special case of fully reversible electron transfer associated with dc voltammetry. This background material is well-known21,22 but is presented in some detail, as protocols for more complex dc and even ac cases presented below are based on closely related principals. Under conditions of dc voltammetry the apparent number of electrons transferred in a surface-confined process can be estimated on the basis of the peak current width at half of its maximum height, W1/2, under reversible conditions which apply experimentally at zero scan rate or can be approximated by

⎡ nF ⎤ (E − E30)⎥ ΓA = ΓC exp⎢ ⎣ RT ⎦

(4)

Since the total coverage of the electrode, Γ0, is constant, then ΓA + ΓC = Γ0. This means ΓA =

ΓC =

1+

Γ0 − nF exp⎡⎣ RT (E

− E30)⎤⎦

Γ0 nF ⎡ 1 + exp⎣ RT (E − E30)⎤⎦

and

(5)

Finally, we can differentiate either ΓA or ΓC in eq 5 with respect to time to get the faradaic current, If, associated with surfaceconfined electron transfer of a reversible process 9866


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dΓ dΓA = −nFa C dt dt nF ⎡ exp⎣ RT (E − E 0)⎤⎦ Γ0n2F 2va

and

If = nFa =

RT

(1 +

nF exp⎡⎣ RT (E

− E 0)⎤⎦

ΓC = 2

)

(6)

Γ0n2F 2va 4RT

⎛ dΓ dΓ ⎞ If = Fa⎜ A − C ⎟ ⎝ dt dt ⎠

and that W1/2

=

(8)

(19)

⎡ F ⎤ dx −Fv (E − E10)⎥ , so that x x = exp⎢ = ⎣ RT ⎦ dt RT

(20)

dy ⎡ F ⎤ −Fv (E − E20)⎥ , so that y y = exp⎢ = ⎣ RT ⎦ dt RT

(21)

Since we are taking E = Estart − vt

0.084 n

(9)

ΓA =

at 4 °C. Note that this relationship in eqs 8 and 9 can only be applied in the situation of full reversibility of the electron transfer process. It is also clear why W1/2 for the simultaneous twoelectron transfer is half that of a single electron transfer, i.e., 84 mV (n = 1) and 42 mV (n = 2) at 4 °C. Two Electrons Transferred Cooperatively. For a cooperative two-electron transfer, the analysis is somewhat more complicated. The reactions can be written as E10

A + e− ⇌ B E 20

dΓC Γ Fv y(1 + 2x) = 0 dt RT (1 + y + xy)2

(12)

E = E20 +

RT ⎛ ΓB ⎞ ln⎜ ⎟ F ⎝ ΓC ⎠

(13)

If =

(14)

⎡ −F ⎤ ΓC = ΓB exp⎢ (E − E20)⎥ ⎣ RT ⎦

(15)

Using the fact that ΓA + ΓB + ΓC = Γ0, where Γ0 is constant, along with eqs 14 and 15 gives

Γ F 3av 2 y(1 − xy)(1 − y + 8x + xy) dI f = 02 2 dt RT (1 + y + xy)3

F Γ0 exp⎡⎣ RT (E − E10)⎤⎦ −F F 1 + exp⎡⎣ RT (E − E10)⎤⎦ + exp⎡⎣ RT (E − E20)⎤⎦

d2If

(16)

dt

as well as Γ0 ΓB = F −F 0 ⎤ ⎡ 1 + exp⎣ RT (E − E1 )⎦ + exp⎡⎣ RT (E − E20)⎤⎦

Γ0F 2av y(1 + 4x + xy) RT (1 + y + xy)2

(24)

In order to establish a formula for W1/2 in the cooperative two-electron case that can be compared to that which applies when two electrons transfer simultaneously, it is necessary to ensure that only a single peak is found. This outcome will certainly be achieved when E01 ≤ E02, but when E01 > E02 two peaks could be present. When only a single peak is found in the current then we know that a maximum will occur at (E01 + E02)/ 2. However, as the difference between E01 and E02 increases, the single peak will be split into two resolved peaks yielding a local minimum at (E01 + E02)/2. In order to establish the difference in E0 values needed for this to occur the second derivative of If can be calculated. The point at which this switches between being negative and positive will be located where peak splitting occurs. The first and second derivatives of If are

where ΓA, ΓB, and ΓC are the surface coverages of A, B, and C, respectively. These two equations can be rearranged to give ⎡ F ⎤ ΓA = ΓB exp⎢ (E − E10)⎥ ⎣ RT ⎦

(23)

After some rearrangement the following expression for If is obtained

Now the Nernst equation must hold for two situations RT ⎛ ΓA ⎞ ln⎜ ⎟ F ⎝ ΓB ⎠

(22)

−Γ0Fv xy(2 + y) dΓA and = dt RT (1 + y + xy)2

(11)

E = E10 +

Γ0xy Γ0 and ΓC = 1 + y + xy 1 + y + xy

the derivatives can be written as

(10)

B + e− ⇌ C

ΓA =

(18)

we need the derivatives of ΓA and ΓC. In order to simplify this process we use a change of variables

(7)

RT ≃ 3.53 nF

F −F 1 + exp⎡⎣ RT (E − E10)⎤⎦ + exp⎡⎣ RT (E − E20)⎤⎦

In order to evaluate the Faradaic current If

where a is the electrode area. Note that, initially, the current is negative in this situation since we are dealing with a reduction process. Equation 6 allows us to conclude, as have others,21,22 that the maximum possible faradaic current is Ifmax =

−F Γ0 exp⎡⎣ RT (E − E20)⎤⎦

2

=

−Γ0F 4av 3 ⎧ x 3y 2 (y + 16) − x 2y(4y 2 + 13y + 64) ⎨ (1 + y + xy)4 R3T 3 ⎩ ⎪

⎪

+ (17)

(25)

x(y 3 + 8y 2 − 13y + 16) + y 2 − 4y + 1 ⎫ ⎬ (1 + y + xy)4 ⎭ (26)

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We now need to evaluate the second derivative at (E01 + E02)/ 2 and find the value of (E01 − E02) at which its sign changes. Since the denominator of the second derivative is always positive we only need to consider the numerator. A further simplification arises from the definitions of x and y. The point we are interested in is E = (E01 + E02)/2. This means that y = 1/x (which does indeed yield an extremum of If since its first derivative will trivially be zero in this case) and leads us to consider the function f(x), which is equivalent to considering the second derivative of If f (x) = 16x 3 + 12x 2 − 1

when they are governed by the Butler−Volmer formalism. The benefit of having the analytical solution described above is that we can check our model in the limit of reversibility. The mechanism considered theoretically now consists of two-electron transfer steps, from a redox active species confined to the electrode surface. This scenario for cooperating species provides three different oxidation states A, B, and C. The mechanism can be represented by the following equations: 1 k red

A + e− HoooI B 1 kox

(27)

2 k red

The function f(x) is zero at x = 1/4, with coincident zeros at x = −1/2. However, since x is defined to be an exponential, we must disregard the roots at x = −1/2. The root at x = 1/4 does represent a point where the function switches between being positive and being negative, and is therefore the point at which the single peak splits into two, or equivalently where two peaks merge into a single peak. Our definition of x and the fact that we are considering E = (E01 + E02)/2, means that

B + e− HoooI C 2 kox

4RT ln(2) F

1 k red

A + n1e− HoooI B 1 kox

Ifmax

B + n2e− HoooI C 2 kox

kjox

RT ⎛ x1 ⎞ ln⎜ ⎟ F ⎝ x2 ⎠

(29)

⎞ ⎛ αjF [E(t ) − Ej0 − R uItot]⎟ k red j = kj0exp⎜ − ⎠ ⎝ RT

(36)

⎛ ⎞ F koxj = kj0 exp⎜(1 − αj) [E(t ) − Ej0 − R uItot]⎟ ⎝ ⎠ RT

(37)

where αj is the charge transfer coefficient for process j. The redox active molecules attached to an electrode have a total surface coverage, Γ. If we let θ1 be the proportion that is in state A and let θ2 be the proportion in state C, then by the law of mass conservation, the proportion of the surface covered by species B must equal (1−θ1−θ2). We can now describe reactions in eqs 32 and 33 in a similar mathematical form to that given elsewhere.16,17 By assuming a Langmuir isotherm in which interactions between adsorbed molecules are negligible, and that all adsorption sites are equivalent, we can write

(30)

where x1 > x2 > 0, and x1 and x2 both satisfy the following equation: x 4 − x 3 − 9x 2 − x + 1 = 0

(35)

kjred

where and are the potential-dependent heterogeneous charge-transfer rate constants for oxidation and reduction (where j = 1 relates to eq 32 and j = 2 relates to eq 33). If Butler−Volmer kinetics apply25 for each electrochemical reaction (j = 1 or 2) then

(28)

In order to calculate W1/2 for this case, we need to find E such that If = Imax f /2. For E01 = E02 = E0 and following analogous methods used for the single electron transfer case, we can establish that W1/2 =

(34)

2 k red

Consequently, if E01 − E02 ≤ 4RT In(2)/(F) ≈ 66 mV for T = 277 K, then only a single peak will be found, and under these conditions it is sensible to deduce a value for W1/2. For our cooperative two-electron transfer, and if E01 = E02 = E0, the maximum of If will be at E = E0 which means that x = y = 1. Using eq 24 with x = y = 1 2Γ F 2av = 0 3RT

(33)

which take place on the electrode surface. In the general case allowed for in simulations, eqs 32 and 33 are replaced by

⎡ F ⎤ 1 exp⎢ (E 0 − E10)⎥ = ⎣ 2RT 2 ⎦ 4 so that E10 − E20 =

(32)

(31)

Therefore x1 ≈ 3.57 and x2 ≈ 0.28, where we have ignored the other two negative roots since x must always be positive. This means that W1/2 ≈ 61 mV at 4 °C, the value reported11 for CcP at low scan rate. Although the derivation and mechanism differ, the theoretical study by Plichon and Laviron26 on thin layer voltammetry gives rise to the same normalized outcomes deduced for the surface-confined conditions we have described in the absence of chemical reactions coupled to electron transfer.

dθ1 1 1 = kox (1 − θ1 − θ2) − k red θ1 dt

(38)

dθ2 2 2 = k red (1 − θ1 − θ2) − kox θ2 dt

(39)

Initially, all species are in the form of A; therefore, θ1 = 1 and θ2 = 0. In the general case, used to simulated the voltammogram, the number of electrons transferred in either of the reactions given in eqs 32 or 33 may have either sign so that both oxidation and reduction reactions are covered by the treatment that follows. The total current can be expressed as the sum of the faradaic current and charging current. Thus

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MATHEMATICAL MODEL FOR COOPERATIVE KINETICS There are many situations where the species being reduced or oxidized will not display the full reversibility assumed above. The presence of kinetic control requires us to model the reaction by considering the rates of oxidation and reduction

Itot(t ) = If (t ) + Ic(t )

(40)

where 9868


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⎛ dθ dθ ⎞ If = Fa Γ⎜ 1 − 2 ⎟ ⎝ dt dt ⎠

capacitance, the theoretical impact of these parameters will be analogous to that shown elsewhere with respect to solution phase processes; for example see.31 dc Cyclic Voltammetry. Case 1: Sequential Electron Transfer. Figure 3 contains numerically simulated dc cyclic

(41)

and

Ic = Cdl

dE dt

(42)

where Cdl is the double layer capacitance and Γ is as described before. Note the negative sign for the second term of the faradaic current. This is required since the molecules represented by θ2 are undergoing oxidation which, by convention, provides a positive current. Nondimensional Variables. As described in other papers,27−30 for the purpose of efficiency of the simulation, we recast the problem in terms of nondimensional variables. This gives us the following nondimensional model: 0 dθ1 = κ10[(1 − θ1 − θ2)e(1 − α1)(ε(τ) − ε1 − ρu ιtot) dτ 0

− θ1e−α1(ε(τ) − ε1 − ρu ιtot)]

(43) Figure 3. Simulated dc cyclic voltammograms for two sequential, reversible one-electron transfer processes, with a separation in E01 and E02 of 0 to 500 mV (E01 ≥ E02 and the processes are symmetrically located about 0 V). Parameters used in the simulations: v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, a = 0.1 cm2, Γ = 10.0 pmol cm−2, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω. In order to simulate two reversible processes, k01 = k02 = 10 000 s−1 and α1 = α2 = 0.50.

0 dθ2 = κ20[(1 − θ1 − θ2)e−α2(ε(τ) − ε2 − ρu ιtot) dτ 0

− θ2e(1 − α2)(ε(τ) − ε2 − ρu ιtot)]

(44)

where at τ = 0, θ1 = 1, and θ2 = 0. Also ιtot = ιc + ιf

ιtot

⎛ dθ dθ ⎞ dε = γdl + ⎜ 1 − 2⎟ ⎝ dτ dτ dτ ⎠

(45)

voltammograms for two sequential, reversible one-electron charge transfer process that may be associated with a surfacebound redox protein when the separation of E01 and E02 lies between 0 and 500 mV, and the effect of Ru and Cdl is not taken into account. The reversible case was simulated using a very large value of k0. As expected,21,22 when the separation of the E0 values is decreased, the processes start to merge, and when less than ∼75 mV, only one peak is observed by visual inspection. For the sequential case with well-resolved reversible electron transfer processes, each process corresponds to the theory for a single electron transfer. The simulated voltammograms shown correspond well with the analytical solutions. Thus W1/2 for each will be 84 mV at 4 °C (eq 9) with n = 1 as applies in Figure 3. For simultaneous transfer of two electrons reversibly, W1/2 is 42 mV, as described by the analytical solution for this case (eq 9) with n = 2. For the cooperative, reversible twoelectron transfer case with E01 = E02, W1/2 is predicted to be approximately 61 mV. These characteristic differences in the three types of electron transfer, available for very large k0 values or at very slow scan rate, can be exploited to establish which model best describes an experimental data set. The outcomes for the surface-confined case considered in this paper and that for thin layer voltammetry26 are identical, when both are normalized. Figure S1 shows the effect of introducing kinetics (α = 0.5 in this and all other figures) in a scenario where E01 > E02 with E01 = 0.25 V and E01 = −0.25 V. Four different regimes are considered: both processes are reversible (red line), both processes are quasi-reversible (green line), the first process is reversible and the second is quasi-reversible (black line), and the first process is quasi-reversible and the second is reversible (yellow line). Other parameters used in the simulation are given in the figure caption. The behavior shown in Figure S1 is easy to understand due to the fact that the two processes are well separated in terms of their E0 values and, therefore, can be

(46)

While all simulations were undertaken using these nondimensional variables, outcomes are presented using parameters similar to those used in the experiments in order to simplify comparison of theory and experiment.

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THEORETICAL RESULTS Numerical simulations that mimic the voltammetry expected of a surface-bound protein, such as CcP, having two mutually dependent electron transfer steps are considered. In this scenario species A is considered to be the one that is reduced via a one-electron charge transfer step to produce an intermediate, species B, which is followed by a further oneelectron transfer step to generate species C. Consequently, the problem is presented in terms of two sets of E0, k0, and α values. Given that the second electron transfer can only happen after the first has taken place, we can divide this problem of cooperative two-electron transfer into two cases: (a) the first process has a less negative E0 value than the second, so that sequential electron transfer occurs, and (b) the first process has an E0 value which is more negative than the second. The latter will be referred to as the potential inversion case, as suggested by Evans in his analysis of solution phase voltammetry.12 In this case, even though the E0 value of the second process is reached, no electrons can transfer until the E0 value of the first process is reached. Finally, we will present a comparison of voltammetry for a system based on the simultaneous two-electron transfer case with one set of E0, k0, and α values, noting that this situation is modeled by setting k02 = 0 in the section above. For ease of presentation, noting that our theoretical simulations do not include uncompensated resistance or double layer 9869


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considered independently in this non-potential-inverted situation. As the E0 values converge, the two peaks will begin to merge, but unlike the case in Figure 3, the precise wave-shapes are now determined by the relative magnitudes of the k0 values. Case 2: Potential Inverted (Cooperative) Electron Transfer. In the potential inversion case, with reduction being considered, E01 ≤ E02. In this scenario, as the applied potential progresses from the initial value toward E02, no current will be detected due to the fact that species B is not yet present and therefore cannot be reduced in this potential region. However, as the applied potential approaches E01, species A is reduced to species B, finally allowing B to be reduced to C. As shown in Figure 4, this means that only a single reduction (and Figure 5. Comparison of dc voltammograms for three different mechanisms. The black line represents a one-electron transfer process with k02 = 0 s−1 and E0 = 0 V; the red line represents two electrons transferring simultaneously, with k02 = 0 s−1 and E0 = 0 V; and the blue line represents two electrons transferring cooperatively, with E01 = E02 = 0 V. Other parameters used in the simulations are v = 0.02 V s−1, Estart = 0.25 V, Ereverse = −0.25 V, a = 0.1 cm2, Γ = 10.0 pmol cm−2, α1 = α2 = 0.5, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω. Unless otherwise stated k0 values are taken to be 10 000 s−1.

transfer, with n1 = 1, n2 = 1, and E01 = E02 = 0 V. Other parameters are given in the figure caption. Clearly, all three mechanisms give different dc voltammetric responses when the processes involved are reversible. This supports the outcome of the scenarios given above via analysis of the W1/2 values. FT ac Cyclic Voltammetry. Case 1: Sequential Electron Transfer. As in the dc case, we begin with a consideration of the situation where the two-electron transfers are sequential, i.e., E01 > E02. Figure S3 contains numerically simulated, fourth to seventh harmonic components of ac voltammograms where the separation between E01 and E02 lies between 0 and 50 mV and both processes are reversible. Other parameters are stated in the figure caption. These higher order harmonics are of greatest interest in experimental studies because of the absence of background current, so only these cases are shown although similar features in terms of faradaic current apply to the first three harmonics. Only a small range of E01 separations is displayed since, in the dc voltammetry section above, we established that larger separations simply led to two resolved single electron transfers. Analogous to the dc results is the prediction that, as the separation between E01 and E02 is reduced to zero, the peak heights increase. Figures 6 and 7 show the effect of kinetics on FT-ac voltammetry. Figure 6 shows the dc component and the first three harmonics when the two-electron transfer processes are well separated and one of the processes is quasi-reversible, whereas Figure 7 shows the first six harmonics under a different set of parameter values. In both cases, behavior analogous to that seen in the dc case (Figure S1) is exhibited. In a dc experiment we would rely on interpreting one single voltammogram per experiment, whereas with the ac version we can analyze all of the harmonics as is shown in Figure 7; the higher the harmonic the greater the sensitivity to differences in the kinetics. Case 2: Potential Inverted (Cooperative) Electron Transfer. We now turn our attention to the situation where E01 ≤ E02, the potential inverted (cooperative) electron transfer case. Figure S4 presents the sixth and seventh harmonics of an FT-ac voltammetric simulation for the reversible case. The results are

Figure 4. Simulated dc cyclic voltammograms for two potential inverted, reversible, one-electron transfer processes, with a separation in E01 and E02 between 0 and 500 mV (where E02 ≥ E01 and they are symmetrically located about 0 V). Parameters used in the simulations: v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, a = 0.1 cm2, Γ = 10.0 pmol cm−2, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω. In order to simulate two reversible processes, k01 = k02 = 10 000 s−1, and α1 = α2 = 0.50.

oxidation) peak is found in contrast to the two that may be detected in the sequential case. The more inverted the processes, the narrower and taller the peak, and in fact if the inversion is extreme, the simulated voltammogram approaches that predicted for simultaneous two-electron transfer. Figure S2 presents simulations for the inverted case with E01 = −0.25 V, E02 = 0.25 V, and varying the electrode kinetics. The same four scenarios are considered as for Figure S1, and as expected, only a single reduction and a single oxidation peak are evident, in contrast to the two seen in Figure S1. Examination of Figure S2 (black and yellow lines) reveals that the forward and backward scans exhibit different peak heights arising from the unequal k0 values. Scanning in the forward direction results in the reversible process not being allowed to proceed until the irreversible process has occurred, producing a narrow, tall peak; scanning in the reverse direction will result in the irreversible process waiting for the reversible process to occur therefore giving a more spread-out current response with a lower peak height. Comparison of Three Possible Mechanisms of Electron Transfer. Figure 5 shows reversible dc voltammograms for the three different mechanisms of electron transfer with n1 and n2 notation now introduced from use of eqs 34 and 35: single electron transfer, with n1 = 1, n2 = 0, k20 = 0 s−1, and E01 = 0 V; simultaneous two-electron transfer, with n1 = 2, n2 = 0, k02 = 0 s−1, and E01 = 0 V; and cooperative two-electron 9870


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Figure 6. Simulated FT-ac voltammograms with variable k0 values (dc component and first three harmonics shown) where E01 = 0.25 V and E02 = −0.25 V (sequential). Blue dashed lines indicate both k0 values are 10 000 s−1; black lines indicate k01 = 10 000 s−1 and k02 = 0.1 s−1; red lines indicate k01 = 0.1 s−1 and k02 = 10 000 s−1; and green dashed lines indicate both k0 are 0.1 s. Other parameters are v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, ΔE = 100 mV, f = 9 Hz, a = 0.1 cm2, Γ = 10.0 pmol cm−2, α1 = α2 = 0.5, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω.

harmonics and can therefore easily be distinguished. In contrast, with the first harmonic, the wave-shapes for simultaneous and cooperative two-electron transfer are comparable. However, in moving to the higher harmonics, the differences in the voltammograms become obvious. The decrease in current in the higher harmonics is larger for the case of cooperative two-electron transfer relative to the simultaneous two-electron transfer process. Differences in the ac harmonics for the cooperative and simultaneous two-electron transfer mechanisms allows a distinction between them to be readily made.

again analogous to those seen in the dc voltammograms shown in Figure 4, in the sense that as the amount by which the two E0 values are inverted increases, the taller and narrower the peaks become. Figure 8 shows simulated results for the inverted case where E01 = −0.25 V and E02 = 0.25 V, and k0 values are varied. Again, the results parallel those shown for the dc case in Figure S2, but as well as the dc component of the data, we now have a much richer data set with which to compare experimental data. Comparison of Three Possible Mechanisms of Electron Transfer. Figure 9 provides simulations for the ac first to sixth harmonics for three different mechanisms of electron transfer: single electron transfer, with n1 = 1, n2 = 0, k02 = 0 s−1, and E01 = 0 V; simultaneous two-electron transfer, with n1 = 2, n2 = 0, k02 = 0 s−1, and E01 = 0 V; and cooperative two-electron transfer, with n1 = 1, n2 = 1, and E01 = E02 = 0 V. Other parameters are given in the figure caption. Clearly, the single electron transfer case is different from the other two mechanisms in all

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COMPARISON OF EXPERIMENTAL AND NUMERICALLY SIMULATED RESULTS FOR CCP In solution phase systems, comparisons of simulated and experimental data often lie within experimental error, providing the correct mechanism has been employed in the simulation.32,33 In contrast, for surface-confined processes, usually the 9871


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The midpoint potential of the voltammogram in Figure 10, derived from the coverage of the oxidation and reduction peak potentials gives a value of ∼735 mV (vs the standard hydrogen electrode), and is in good agreement with the literature value.35 At this slow scan rate, the process should approximate that predicted for a reversible electron transfer. The calculated peak width (for both oxidation and reduction) at half the peak current (i.e., W1/2) is 60 ± 2 mV, again in good agreement with the literature,11 which shows that single-electron transfer is not occurring, since theoretically W1/2 ≈ 84 mV at 4 °C. Also, simultaneous two-electron transfer is unlikely since, theoretically, W1/2 ≈ 42 mV at 4 °C. However, when the experimentally obtained value of W1/2 = 60 ± 2 mV is compared to that for the theoretically calculated cooperative two-electron transfer of approximately 60 mV, with E01 = E02, excellent agreement is observed. dc data obtained as a function of scan rate, the so-called trumpet plot, are shown in Figure 11. The use of faster scan rate dc conditions should allow departure from reversibility to be assessed and kinetic analysis undertaken. This plot displays the peak potential associated with the oxidation current (top half of the plot) and the peak reduction current (bottom half of the plot) versus scan rate. Three data sets are shown: the experimental data, data from a numerical simulation based on two cooperative electrons transferred with equal E0 values, and data from a numerical simulation based on simultaneous twoelectron transfer. As the scan rate is increased, the separation between the oxidation and reduction peaks increases. The fact that the separation increases with scan rates as low as 0.1 V s−1 indicates that the electrode kinetics are relatively slow. It is also clear that the experimental data are more closely reproduced by the cooperative two-electron transfer numerical simulation, with E01 = E02 = 0.745 V (hence equal to the midpoint potential) and k01 = k02 = 10 s−1 and other parameters as given in the figure caption, than by a simultaneous two-electron model with a single E0 = 0.745 V and a single k0 = 10 s−1. However, the trumpet plot (as well as W1/2) represents experimental analysis of data obtained at only two potentials and is therefore of limited value. Given that many combinations of E01, E02, k01 and k02 are available, it is also necessary to confirm, or otherwise, that E01 = E02 and k01 = k01 is a unique solution. This requires a good fit for all data obtained at all potentials. Thus, a good fit to the trumpet plot for the n = 2 model may also be obtained with a k0 value larger than the 10 s−1 used in the plot shown in Figure 11; however, the shape of the voltammogram would be far removed from that of the experimental data. Also noteworthy is the observation that the difference in the oxidation and reduction peak potentials does not converge to zero at the lowest scan rates used, as predicted theoretically. Thus, anomalies are found, but the poor faradaic-to-background current limits the ability to provide the details relevant to the nuances. FT ac Cyclic Voltammetry. The use of FT ac voltammetry in analyzing CcP is experimentally challenging, because of the low signal-to-noise ratio encountered when the rates of electron transfer are slow, as appears to apply in this case. In order to access the higher harmonics in a system with slow kinetics, the applied signal in the ac component requires use of a relatively low frequency (f = 3.88 Hz) and a relatively large amplitude (ΔE = 200 mV). The slow kinetics of CcP is reflected by comparison of current magnitudes found in previous studies on surface-confined proteins17,19,20 which exhibit relatively faster electron transfer rates, allowing an applied voltage with a lowest

Figure 7. Simulated FT-ac voltammograms with variable k0 values (harmonics three to six shown) where E01 = 0.005 V and E02 = −0.005 V (sequential). Red lines indicate k01 = 10 000 s−1 and k02 = 0.1 s−1; blue lines indicate k01 = 0.1 s−1 and k02 = 10 000 s−1; and black lines indicate both k0 are 0.1 s. Other parameters are v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, ΔE = 100 mV, f = 9 Hz, a = 0.1 cm2, Γ = 10.0 pmol cm−2, α1 = α2 = 0.5, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω.

agreement is less satisfactory, because of imperfections in the model.17,20,34 The surface-confined voltammetry of CcP is no exception, so that ranges for kinetic and thermodynamic parameters are reported rather than single values, for reasons that are stated below. It should be noted that the differences apply equally to the dc method but due to the high sensitivity of the ac harmonics, the nuances of these effects attributed to nonideality can be more clearly established. dc Cyclic Voltammetry. Figure 10 shows the dc cyclic voltammogram obtained with a slow scan rate of v = 20 mV s−1 for CcP confined on a PGE electrode in contact with 20 mM phosphate buffer solution at 4 °C. This voltammogram is typical for dc cyclic voltammetry of a surface-confined protein in the sense that the capacitive current dominates the relatively small oxidation and reduction faradaic curves. To assist in achieving clarity in the analysis of the peak signal, the capacitive current is subtracted using the method described in the Experimental Methods to give the modified data set also shown in Figure 10. In the first study of the nonturnover signals,11 the oxidative and reductive peaks were the same size, and the apparent size difference in Figure 10 is believed to be an artifact due to baseline subtraction, a common complication in dc voltammetry of noncatalytic signals and a reason why FT ac is so useful, because harmonics can be accessed which are background free. 9872


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Figure 8. Simulated ac voltammograms (dc component and first three harmonics shown) with variable k0 values, where E01 = −0.25 V and E02 = 0.25 V (inverted). Blue lines indicate both k0 are 10 000 s−1; black lines indicate k01 = 10 000 s−1 and k02 = 0.1 s−1; red lines indicate k01 = 0.1 s−1 and k02 = 10 000 s−1; and green dashed lines indicate both k0 are 0.1 s. Other parameters are v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, ΔE = 100 mV, f = 9 Hz, a = 0.1 cm2, Γ = 10.0 pmol cm−2, α1 = α2 = 0.5, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω.

frequency of 9.0 Hz and smaller amplitudes in the range of 80− 100 mV to be employed. Figure 12 shows the fifth to eighth harmonic voltammograms derived from the reduction of CcP confined on a graphite PGE electrode, obtained using similar experimental conditions as for the dc cyclic voltammetric case. The first four harmonics are not shown here since capacitive current is present. Initially, numerical simulations are attempted for the case of cooperative two-electron transfer, with n1 = 1, n2 = 1, k01 = k02 = 8 s−1, and E01 = E02= 0.745 V; and from simultaneous two-electron transfer, with n1 = 2, k01 = k02 = 8 s−1, and E01 = E02= 0.745 V. As noted elsewhere α was fixed at 0.5 in these and all other simulations; other parameters are stated in the figure caption. It is now very clear that the cooperative two-electron transfer model fits the data far better than the simultaneous two-electron transfer one. However, the agreement between simulation and experimental data is not perfect, and the level of agreement achieved is harmonic dependent.

It is noteworthy that previous determinations of the apparent electron transfer rates for the case of a cooperative two-electron transfer of CcP employed models that differ from the present theoretical considerations. One report used the cyclic voltammetric method and was based on a consideration of the model of n-electrons transferred simultaneously (Figure 2), which resulted in the value of n being 1.44.35 The assumption of a single E0 value yielded a k0 value of ∼5.7 s−1. Another study employed a steady state rotating disk electrode configuration and reported an optimum k0 value to be around 7 s−1.36 Interestingly, that model considered the case of two E0 and k0 values, with the analysis of the catalytic signals suggesting that the first reduction process was very fast, while the second reduction process had a k0 value of 7.6 s−1. From a theoretical standpoint it is obviously possible to have a difference in the two E0 and the two k0 values of cooperative electron transfer, as suggested in the literature.36 However, our data imply that this is not the situation prevailing with CcP. The 9873


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Figure 11. Comparison of experimental and simulated “trumpet plots’’ to show the divergence of the oxidative and reductive peak potentials with increasing scan rate. (■) represent the reduction and oxidation parts of the experimental data; (◊) represent a simulation based on two cooperative electrons transferring, with E01 = E02 = 0.745 V; (○) represent data from a simulation of simultaneous two-electron transfer, with k02 = 0 s−1 and E0 = 0.745 V. Other parameters used in the simulations are v = 0.005 to 1.20 V s−1, Estart = 0.5 V, Ereverse = 1.0 V, a = 0.1 cm2, Γ = 3.5 pmol cm−2, all α = 0.5, T = 277 K, Cdl = 0 F cm−2, Ru = 200 Ω, with k0 = 10 s−1 for values not already defined.

8 s−1 but with the same E0 values as the experimental data. It is immediately evident that there is no combination of parameters that precisely fits all of the harmonics. For example, with k01 = k02 = 8 s−1 the fifth harmonic closely matches the experimental data, whereas for the seventh harmonic, the peak current of the simulated voltammogram with k01 = k02 = 3 s−1 only fits moderately well. Simulated voltammograms with unequal k0 values (k01 = 8 s−1, k02 = 3 s−1) introduce significant asymmetry; experimental data have a high level of symmetry implying the k0 values are very similar. The ac data are only consistent with a cooperative two-electron transfer model, with both E0 values being within 10 mV of each other and the k0 values also being essentially equal in order that both the current magnitudes and wave shape symmetry match experimental data. Furthermore, k0 values for both processes need to lie in the range of 3 to 10 s−1 to provide satisfactory agreement with experimental data. The reasons for not being able to fit the higher harmonics of the ac data precisely with a single set of parameters are multifold: first, rather than each molecule having a single k0 and E0 value, it is believed that surface-confined CcP enzymes exhibit a range of values, dispersed, possibly in a Gaussian fashion, about a mean value.38 The impact of kinetic dispersion is expected to be harmonic dependent, as found in this study. Kinetic and thermodynamic dispersion is known to be profound in the case of enzymes adsorbed onto a highly heterogeneous PGE electrode as employed here. For example, adsorbed enzymes adopt random orientations leading to variable distances between the electroactive site and the electrode, which in turn gives rise to distant dependent kinetic dispersion (this is analyzed in ref 39). Second, a Langmuir isotherm and noninteraction of centers is assumed. Other isotherms, such as Frumkin, may be employed, but no compelling reason to introduce this additional complexity is available, given the other artifacts present. Third, the impact of the use of Butler−Volmer theory may be significant, and possible improvements may be achieved by the use of Marcus−

Figure 9. Comparison of simulated FT-ac voltammograms for different mechanisms (harmonics one to six). The black line represents one electron transferring with k02 = 0 s−1 and E0 = 0 V; the red line represents two electrons transferring simultaneously, with k02 = 0 s−1 and E0 = 0 V; and the blue line represents two electrons transferring cooperatively, with E01 = E02 = 0 V. Other parameters used in the simulations are v = 0.02 V s−1, Estart = 0.5 V, Ereverse = −0.5 V, ΔE = 100 mV, f = 9 Hz, a = 0.1 cm2, Γ = 10.0 pmol cm−2, α1 = α2 = 0.5, T = 277 K, Cdl = 0 F cm−2, and Ru = 0 Ω. Unless otherwise stated k0 values are taken to be 10 000 s−1.

higher harmonic components are known to be very sensitive to the subtleties of the electrode process,37 and that applies in the present case. Figure 13 considers the simulated cases for k01 = k02 = 3 s−1; k01 = 8 s−1, k02 = 6 s−1; k01 = 8 s−1, k02 = 3 s−1 and k01 = k02 =

Figure 10. dc cyclic voltammogram of CcP confined on an edge-plane pyrolytic graphite electrode in contact with 20 mM phosphate buffer solution at pH 6.0. Raw data (blue line) and the background-corrected response (black line). The temperature was 4 °C, and the scan rate was 20 mV s−1. 9874


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Figure 13. Comparison of FT-ac voltammograms for CcP confined to an edge-plane pyrolytic graphite electrode in contact with 20 mM phosphate buffer solution at pH 6.0, and simulated responses for cooperative electron transfer. Black lines represent experimental data; red lines represent simulated data with k01 = 3 s−1, k02 = 3 s−1, and E01 = E02 = 0.745 V; blue lines represent simulated data with k01 = 6 s−1, k02 = 8 s−1, and E01 = E02 = 0.745 V; green lines represent simulated data with k01 = 3 s−1, k02 = 8 s−1, and E01 = E02 = 0.745 V; pink lines represent simulated data with k01 = 8 s−1, k02 = 8 s−1, and E01 = E02 = 0.745 V; yellow lines represent simulated data with k01 = 8 s−1, k02 = 8 s−1, E01 = 0.740 V, and E02 = 0.740 V (all showing the fifth to the eighth harmonics). Experimental parameters were T = 277 K, v = 0.00931 V s−1, ΔE = 200 mV and f = 3.88 Hz. Simulation parameters as per experimental ones along with a = 0.06 cm2, Γ = 3.5 pmol cm−2, all α = 0.5, Cdl = 100 μF cm−2, and Ru = 200 Ω.

Figure 12. Comparison of FT-ac voltammograms for CcP confined to an edge-plane pyrolytic graphite electrode in contact with 20 mM phosphate buffer solution at pH 6.0, and simulated responses. Black lines represent experimental data; red lines represent simulated data obtained from cooperative two-electron transfer with k01 = k02 = 8 s−1 and E01 = E02 = 0.745 V; blue lines represent simulated data from simultaneous two-electron transfer, with k01 = 8, k02 = 0 s−1 and E01 = 0.745 V (all showing the fifth to the eighth harmonics). Experimental parameters were T = 277 K, v = 0.00931 V s−1, ΔE = 200 mV, and f = 3.88 Hz. Simulation parameters as per experimental ones along with a = 0.06 cm2, Γ = 3.5 pmol cm−2, all α = 0.5, Cdl = 100 μF cm−2, and Ru = 200 Ω.

Hush theory, but again the additional complexity is not justified at this stage. Finally, there is an intermediate in Figure 1 that is implied to exist in two forms, that is

and kinetic dispersion. The large background current correction needed in this method makes full wave shape analysis problematic, but again a complete model with theory based on a single set of parameters is not achieved.

keq

Fe(IV): Trp0 ⇌ Fe(III): Trp+

(47)

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It is possible to introduce an equilibrium constant, keq as in eq 47, and associated rate constants to specify details of their distribution. Acid−base reactions also may accompany the redox chemistry. In the present treatment, it is assumed that all such steps are reversible and hence accommodated into the E01 and E02 terms. Commentary by Evans in his review15 of the solution phase scenario are directly relevant to the surfaceconfined case. The dc cyclic voltammetric data also clearly show features of nonideality that parallel those seen by the ac method with a full analysis of wave shape and peak current magnitude as a function of the harmonic. Thus, it is noted that although the dc W1/2 value is close to the value predicted theoretically at slow scan rate, the peak-to-peak separation does not converge to zero at slow scan rate, again consistent with thermodynamic

CONCLUSIONS As far as we are aware, this is the first time that the largeamplitude ac voltammetric approach has been used for the study of a cooperative two-electron transfer process in a surface-confined system. However, detailed studies are available by Evans and others for the solution soluble case under dc conditions.12−15 Many of the findings in the solution phase case have parallels in surface-confined systems. The results provide a significant extension in terms of complexity to previous studies on surface-confined systems such as azurin (one electron system)17,19 and two-center ferredoxins (two independent electron transfers).20 Access to more detailed information on the reduction of CcP is important, since this biologically important metalloprotein has physiological catalytic properties. 9875


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(9) Coulson, A. F. W.; Erman, J. E.; Yonetani, T. Studies on Cytochrome c Peroxidase: XVII. Stoichiometry and mechanism of the reaction of compound ES with donors. J. Biol. Chem. 1971, 246, 917− 924. (10) Liu, R.-Q.; Miller, M. A.; Han, G. W.; Hahm, S.; Geren, L.; Hibdon, S.; Kraut, J.; Durham, B.; Millett, F. Role of Methionine 230 in Intramolecular Electron Transfer between the Oxyferryl Heme and Tryptophan 191 in Cytochrome c Peroxidase Compound II. Biochem. 1994, 33, 8678−8685. (11) Mondal, M. S.; Fuller, H. A.; Armstrong, F. A. Direct Measurement of the Reduction Potential of Catalytically Active Cytochrome c Peroxidase Compound I: Voltammetric Detection of a Reversible, Cooperative Two-Electron Transfer Reaction. J. Am. Chem. Soc. 1996, 118, 263−264. (12) Evans, D. H.; Hu, K. Inverted potentials in two-electron processes in organic electrochemistry. J. Chem. Soc., Faraday Trans. 1996, 92, 3983−3990. (13) Evans, D. H. The kinetic burden of potential inversion in twoelectron electrochemical reactions. Acta Chem. Scand. 1998, 52, 194− 197. (14) Kraiya, C.; Evans, D. H. Investigation of potential inversion in the reduction of 9,10-dinitroanthracene and 3,6-dinitrodurene. J. Electroanal. Chem. 2004, 565, 29−35. (15) Evans, D. H. One-Electron and Two-Electron Transfers in Electrochemistry and Homogeneous Solution Reactions. Chem. Rev. 2008, 108, 2113−2144. (16) Honeychurch, M. J.; Bond, A. M. Numerical simulation of Fourier transform alternating current linear sweep voltammetry of surface bound molecules. J. Electroanal. Chem. 2002, 529, 3−11. (17) Guo, S.; Zhang, J.; Elton, D.; Bond, A. M. Fourier transform large-amplitude alternating current cyclic voltammetry of surfacebound azurin. Anal. Chem. 2004, 76, 166−177. (18) Zhang, J.; Bond, A. M. Theoretical studies of large amplitude alternating current voltammetry for a reversible surface-confined electron transfer process coupled to a pseudo first-order electrocatalytic process. J. Electroanal. Chem. 2007, 600, 23−34. (19) Fleming, B. D.; Zhang, J.; Elton, D.; Bond, A. M. Detailed analysis of the electron-transfer properties of azurin adsorbed on graphite electrodes using dc and large-amplitude Fourier transformed ac voltammetry. Anal. Chem. 2007, 79, 6515−6526. (20) Lee, C.-Y.; Stevenson, G. P.; Parkin, A.; Roessler, M. M.; Baker, R. E.; Gillow, K.; Gavaghan, D. J.; Armstrong, F. A.; Bond, A. M. Theoretical and Experimental Investigation of Surface-Confined Twocenter Metalloproteins by Large-Amplitude Fourier Transformed ac Voltammetry. J. Electroanal. Chem. 2011, 656, 293−303. (21) Laviron, E. Voltammetric methods for the study of adsorbed species. In Electroanalytical Chemistry; Bard, A. J., Ed.; CRC Press: Boca Raton, FL, 1982; Vol. 12, p 53. (22) Bond, A. M. Broadening Electrochemical Horizons: principles and illustration of voltammetric and related techniques; Oxford University Press: New York, 2002. (23) English, A. M.; Laberge, M.; Walsh, M. Rapid Procedure for the Isolation of Cytochrome c Peroxidase. Inorg. Chim. Acta 1986, 123, 113−116. (24) Bond, A. M.; Duffy, N. W.; Guo, S.; Zhang, J.; Elton, D. Changing the look of voltammetry. Can FT revolutionize voltammetric techniques as it did for NMR? Anal. Chem. 2005, 77, 186A− 195A. (25) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, 2nd ed.; Wiley: New York, 2001. (26) Plichon, V.; Laviron, E. Theoretical study of a two-step reversible electrochemical reaction associated with irreversible chemical reactions in thin layer linear potential sweep voltammetry. J. Electroanal. Chem. 1976, 71, 143−156. (27) Gavaghan, D. J.; Bond, A. M. A Complete numerical simulation of the techniques of alternating current linear sweep and cyclic voltammetry: analysis of a reversible process by conventional and fast Fourier transform methods. J. Electroanal. Chem. 2000, 480, 133−149.

The simulations show that for surface-confined systems, singleelectron transfer, simultaneous two-electron transfer and cooperative two-electron transfer are readily distinguished by analysis of the higher harmonic components available in large amplitude ac voltammetry, but limitations are still encountered by the presence of thermodynamic and kinetic dispersion and other factors. Experimentally, we confirm that electron transfer steps for CcP are slow, so a relatively small ac frequency ( f = 3.88 Hz) and large ac amplitude (ΔE = 200 mV) are required to access the higher harmonics. The theory−experiment comparisons based on the fifth to the eighth harmonics of ac voltammetry confirm the cooperative two-electron transfer of CcP, with k0 in the range of 3 to 10 s−1 and E0 = 0.745 ± 0.010 V for both processes. This reconciles the electrochemical observation of cooperative two-electron transfer with the long-reported observation that Compound II is a fully isolatable intermediate state.9,10

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ASSOCIATED CONTENT

S Supporting Information *

Simulations of dc (Figures S1 and S2) and FT-ac (Figures S3 and Figures S4) cyclic voltammograms for two-electron transfer reactions under designated conditions. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Australian Research Council for the award of a Discovery Project Grant that made this study possible. G.P.S. also thanks the EPSRC for financial support. Research in the FAA group was supported by grants from the EPSRC (Supergen) and BBSRC (Grant H003878-1). A.P. thanks Merton College for a Junior Research Fellowship.

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REFERENCES

(1) Eddowes, M. J.; Hill, H. A. O. Novel Method for the Investigation of the Electrochemistry of Metalloproteins: cytochrome c. J. Chem. Soc. Chem. Comm. 1977, 21, 771b−772. (2) Eddowes, M. J.; Hill, H. A. O. Electrochemistry of Horse Heart Cytochrome c. J. Am. Chem. Soc. 1979, 101, 4461−4464. (3) Yeh, P.; Kuwana, T. Reversible electrode reaction of cytochrome c. Chem. Lett. 1977, 6, 1145−1148. (4) Cass, A. E. G.; Eddowes, M. J.; Hill, H. A. O.; Uosaki, K.; Hammond, R. C.; Higgins, I. J.; Plotkin, E. Electrochemical, photoelectrochemical, electrocatalytic and catalytic reduction of redox proteins. Nature 1980, 285, 673−674. (5) Eddowes, M. J.; Hill, H. A. O.; Uosaki, K. The Electrochemistry of Cytochrome c. Investigation of the Mechanism of the 4,4′-Bipyridyl Surface Modified Gold Electrode. J. Electroanal. Chem. 1980, 116, 527−537. (6) Armstrong, F. A.; Heering, H. A.; Hirst, J. Reaction of complex metalloproteins studied by protein-film voltammetry. Chem. Soc. Rev. 1997, 26, 169−179. (7) Léger, C.; Bertrand, P. Direct electrochemistry of redox enzymes as a tool for mechanistic studies. Chem. Rev. 2008, 108, 2379−2438. (8) Armstrong, F. A.; Hill, H. A. O.; Oliver, B. N.; Walton, N. Direct Electrochemistry of Redox Proteins at Pyrolytic Graphite Electrodes. J. Am. Chem. Soc. 1984, 106, 921−923. 9876


Langmuir

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(28) Gavaghan, D. J.; Elton, D.; Bond, A. M. A comparison of sinusoidal, square wave, sawtooth, and staircase forms of transient ramped voltammetry when a reversible process is analysed in the frequency domain. J. Electroanal. Chem. 2001, 513, 73−86. (29) Gavaghan, D. J.; Elton, D.; Oldham, K. B.; Bond, A. M. Analysis of ramped square-wave voltammetry in the frequency domain. J. Electroanal. Chem. 2001, 512, 1−15. (30) Gavaghan, D. J.; Bond, A. M. Numerical simulation of the effects of experimental error on the higher harmonic components of Fourier transformed ac voltammetry. Electroanalysis 2006, 18, 333−344. (31) Sher, A. A.; Bond, A. M.; Gavaghan, D. J.; Harriman, K.; Feldberg, S. W.; Duffy, N. W. Resistance, capacitance, and electrode kinetic effects in fourier-transformed large-amplitude sinusoidal voltammetry: emergence of powerful and intuitively obvious tools for recognition of patterns of behaviour. Anal. Chem. 2004, 76, 6214. (32) Mashkina, E.; Bond, A. M. Implementation of a Statistically Supported Heuristic Approach to Alternating Current Voltammetric Harmonic Component Analysis: Re-evaluation of the Macrodisk Glassy Carbon Electrode Kinetics for Oxidation of Ferrocene in Acetonitrile. Anal. Chem. 2010, 83, 1791−1799. (33) Lee, C.-Y.; Bullock, J. P.; Kennedy, G. F.; Bond, A. M. Effects of coupled homogeneous chemical reactions on the response of largeamplitude AC voltammetry: extraction of kinetic and mechanistic information by Fourier transform analysis of higher harmonic data. J. Phys. Chem. A 2010, 114, 10122−10134. (34) Lee, C.-Y.; Bond, A. M. Evaluation of Levels of Defect Sites Present in Highly Ordered Pyrolytic Graphite Electrodes Using Capacitive and Faradaic Current Components Derived Simultaneously from Large-Amplitude Fourier Transformed ac Voltammetric Experiments. Anal. Chem. 2009, 81, 584−594. (35) Heering, H. A.; Mondal, M. S.; Armstrong, F. A. Using the Pulsed Nature of Staircase Cyclic Voltammetry To Determine Interfacial Electron-Transfer Rates of Adsorbed Species. Anal. Chem. 1999, 71, 174−182. (36) Heering, H. A.; Hirst, J.; Armstrong, F. A. Interpreting the Catalytic Voltammetry of Electroactive Enzymes Adsorbed on Electrodes. J. Phys. Chem. B 1998, 102, 6889−6902. (37) Lee, C.-Y.; Bond, A. M. A Comparison of the Higher Order Harmonic Components Derived from Large-Amplitude Fourier Transformed ac Voltammetry of Myoglobin and Heme in DDAB Films at a Pyrolytic Graphite Electrode. Langmuir 2010, 5243−5253. (38) Davis, J. J.; Burgess, H.; Zauner, G.; Kuznetsova, S.; Salverda, J.; Aartsma, T.; Canters, G. W. Monitoring Interfacial Bioelectrochemistry Using a FRET Switch. J. Phys. Chem. B 2006, 110, 20649−20654. (39) Leger, C.; Jones, A. K.; Albracht, S. P. J.; Armstrong, F. A. Effect of a Dispersion of Interfacial Electron-transfer Rates on Steady-state Catalytic Electron Transport in [NiFe]-Hydrogenase and Other Enzymes. J. Phys. Chem. B 2002, 106, 13058−13063.

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Theoretical Analysis of the Two-Electron Transfer Reaction and

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