Theory and Electrochemistry of Cytochrome C

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Theory and Electrochemistry of Cytochrome C

Journal: The Journal of Physical Chemistry

Manuscript ID jp-2017-009174.R1

Manuscript Type: Article

Date Submitted by the Author: n/a

Complete List of Authors: Seyedi, Salman; Arizona State University, Physics M. Waskasi, Morteza; Arizona State University, Chemistry and Biochemistry Matyushov, Dmitry; Arizona State University, Chemistry and Biochemistry

ACS Paragon Plus Environment

The Journal of Physical Chemistry

Theory and Electrochemistry of Cytochrome c

Salman S. Seyedi,† Morteza M. Waskasi,‡ and Dmitry V. Matyushov†,‡ ∗,

†Department of Physics, PO Box 871504, Tempe, AZ 85287-1504‡School of Molecular Sciences, Arizona State University, PO Box 871604, Tempe, AZ

85287-1604

E-mail: [email protected]

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Abstract

Extensive simulations of cytochrome c in solution are performed to address the appar-ent contradiction between large reorganization energies of protein electron transfer typicallyreported by atomistic simulations and much smaller values produced by protein electrochem-istry. The two sets of data are reconciled by deriving the activation barrier for electrochem-ical reaction in terms of an effective reorganization energy composed of half the Stokes shift(characterizing the medium polarization in response to electron transfer) and the variancereorganization energy (characterizing the breadth of electrostatic fluctuations). This effectivereorganization energy is much smaller than each of the two components contributing to it andis fully consistent with electrochemical measurements. Calculations in the range of temper-atures between 280 and 360 K combine long classical molecular dynamics simulations withquantum calculation of the protein active site. The results agree with the Arrhenius plots forthe reaction rates and with cyclic voltammetry of cytochrome c immobilized on self-assembledmonolayers. Small effective reorganization energy, and the resulting small activation barrier,is a general phenomenology of protein electron transfer allowing fast electron transport withinbiological energy chains.

Introduction

Redox proteins participate in metabolic redoxreactions of biology and in photosynthetic andrespiratory energy chains responsible for thecross-membrane electron transport. The con-ditions of operation of redox proteins withinenergy chains require some design trade-offs toaccommodate both the small reaction free ener-gies and the need to place bulky cofactors in thechain at sufficiently long distances. The over-all rate of protein electron transfer is the prod-uct of the Boltzmann factor describing the ac-tivation barrier and the electron coupling (tun-neling probability) decaying exponentially withthe distance between the donor and acceptor.1–3

The Marcus theory of electron transfer4 de-fines the free energy barrier for electron trans-fer, ∆G†, in terms of the reorganization energyλ and the reaction free energy ∆G0

∆G† =(λ+∆G0)2

4λ(1)

For reactions involving small values of ∆G0,typical for biology,5 the reorganization energybecomes the most important factor determiningthe reaction barrier.It is often assumed that proteins provide

a nonpolar environment for electron transfer,blocking the access of highly polar water tothe active sites. This view is supported by

the low dielectric constant of protein powders(∼ 2 − 56,7) and the low magnitude of thescreening factor (effective dielectric constant8)required in the Coulomb law to screen the in-teraction between the charges. However, hy-dration of the protein causes ionization of thesurface groups and their increased mobility.9 Asa result, any active site of a hydrated protein issurrounded by a nearly uniform density of sur-face charges10 maintaining the stability of thefolded protein in solution and allowing its solu-bility in water.11 While these charges mostly donot affect the electrostatic screening inside theprotein, their motions, caused by thermally ac-tivated elastic deformations of the protein, pro-duce a significant electrostatic noise at the pro-tein active site.12 It is this electrostatic noisethat affects the electronic energy levels of thedonor and acceptor, bringing them into reso-nance for electron tunneling.4

The reorganization energy in the Marcusequation (eq 1) is both the measure of thechange in the distribution of charges (polariza-tion) in the medium caused by transferring theelectron (nominator in eq 1) and the measure ofthe breadth of electrostatic fluctuations causedby thermal agitation (denominator in eq 1).7,12

The Boltzmann factor exp[−∆G†/(kBT )] thenbecomes a Gaussian distribution of the vari-able ∆G0 with the mean −λ and the variance2kBTλ.

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D

A

X = !ω

aE

!ω

em. abs.2λ

St b

Gi(X)c

∆G0

X

Pi(!ω)

⟨X⟩1

⟨X⟩1

⟨X⟩2

⟨X⟩2

λ

σX

0

Figure 1: (a) Reaction coordinate X = !ωfor solution electron transfer between the donor(D) and acceptor (A).13 (b) Probability densi-ties for absorbing (abs.) and emitting (em.) aphoton in a charge-transfer optical transition;⟨X⟩i stand for the average transition energies.The separation between the peaks of opticaltransitions represents the Stokes shift and thecorresponding reorganization energy λSt. (c)The free energy surfaces of electron transferGi(X) = Gi

0−kBT ln[Pi(X)] following from theoptical transition probabilities Pi(X). The re-organization energy λ defines the curvature ofthe free energy surface near the bottom (shownby the double arrow). It also provides the mea-sure of inhomogeneous broadening of the op-tical charge-transfer band14 (σ2

X = ⟨(δX)2⟩ =2kBTλ in (b) and in eq 3). The filled dots in(b) and (c) indicate, respectively, P2(0) and thecrossing point of Gi(X) representing the tran-sition state, X = 0, of the electron-transfer re-action.

Given that a hydrated protein is a softmedium possessing a large density of chargeat the protein-water interface, it is hardly asurprise that atomistic computer simulationsconsistently show large reorganization energiesfor electron transfer, λ ≃ 1 − 2 eV,15–18 oreven higher19,20 when the simulation trajecto-ries are sufficiently long. More surprising wasthe realization that the reorganization energycharacterizing the fluctuations of the energylevels (thermal agitation) was distinct fromthe reorganization energy characterizing theirshift upon electron transfer (medium polariza-tion).19 We denote the former, which we call thevariance reorganization energy, with the stan-dard Marcus notation λ and the latter withthe “Stokes-shift” reorganization energy21 λSt.The reason for the latter assignment is thatλSt, characterizing the shift in average energies,is half of the Stokes shift separating the max-ima of absorption and emission charge-transferbands22,23 (Figure 1).The typical phenomenology of electrostatic

fluctuations at active sites of proteins as cal-culated from atomistic simulations is the in-equality12,19 λ ≫ λSt. The reason for this re-sult can be traced to the non-Boltzmann (non-ergodic) sampling of the phase-space availableto the protein on the reaction time-scale,21 po-larizability of the active site,24,25 or the com-bination of both these factors and/or someother reasons not yet identified. This phe-nomenology, as well as some analytic mod-els allowing non-Gaussian fluctuations affectingelectron transfer,24,26,27 provides an extensionof the standard Gaussian picture of the Mar-cus model,28 which stipulates29 λSt = λ. Atleast some of these extensions24 require non-parabolic free energy surfaces. Since our sim-ulations do not provide sufficient sampling todistinguish such features, the phenomenologyof equal-curvature parabolas21 is used here.Specifically, the variance reorganization ener-gies in the oxidized and reduced states areconsidered to be equal (given by λ) and dis-tinct from λSt. This phenomenological ap-proach allows us to accommodate both the non-Boltzmann (non-equilibrium) sampling and po-larizability effects (requiring non-parabolic free

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energy surfaces24) in terms of only two reorga-nization parameters, λ and λSt.Large values of the reorganization energies

appearing in simulations come in direct contra-diction to often small, in the range 0.3 − 0.6eV30–36 (or even smaller, < 0.25 eV37,38), valuesof the reorganization energy reported by elec-trochemistry of redox proteins. Since electro-statics is not much sensitive to the details offorce fields employed by atomistic simulations,the problem cannot be simply related to stillexisting deficiencies of the atomistic force-fieldmodels. Here we argue that the reorganizationenergy reported by electrochemistry of proteinsis an effective “reaction” (superscript “r”) re-organization energy combining two reorganiza-tion energies typically reported by simulations

λr =(λSt)2

λ(2)

The notion that proteins are characterized bythe condition λ ≫ λSt explains why relativelysmall values of λr are reported by electrochem-ical measurements. We derive the equation forthe activation barrier, from which the expres-sion for λr follows, in eqs 5–8 below. Sufficeit to say here that λr in the form of eq 2 isa direct consequence of the parabolic shape ofthe free energy surfaces of electron transfer.One comes back to the standard Marcus pic-ture with λr = λ = λSt when λSt = λ.In order to show the consistency of our

theoretical model with experimental data, wehave performed extensive simulations of a muchstudied39,40 heme protein cytochrome c (Cyt-c,wild type from horse). We show that the tem-perature dependence of the reaction reorgani-zation energy λr(T ) is consistent with the Ar-rhenius plots for electrochemical rates obtainedfrom cyclic voltammetry.33 We also show thatthe distribution of the energy levels (density ofstates) of the oxidized heme, caused by ther-mal agitation of the bath, is consistent withthe corresponding distribution obtained by tak-ing the derivative of the cathodic current withthe overpotential, dic/dη. The cathodic cur-rent ic is obtained from cyclic voltammetry af-ter correction for mass transport.30,31,41,42 We

report an overall good agreement between ex-periment30,33,34,43 and the combined applicationof the analytic theory and computer simula-tions.21,25

Producing reliable values of reorganizationparameters of Cyt-c has required the combina-tion of long trajectories of classical moleculardynamics (MD) simulations with quantum cal-culations of the heme’s active site. Since longsimulation times are required for the conver-gence of the reorganization energies, our quan-tum calculations are based on Warshel’s empiri-cal valence-bond method,44,45 which involves di-agonalizing the quantum Hamiltonian, affectedby fluctuating electrostatics, along the MDtrajectory.16,25 This specific form of a generalQM/MM methodology46 allows one to combinelong trajectories required for sufficient samplingof electrostatic fluctuations with a large num-ber, M ≃ 100, of excited quantum states of theactive site. These excited states are coupledto the fluctuating electrostatic field through aset of transition dipoles and thus allow us toaccount for the polarizability of the active siteand the corresponding deformation of the elec-tronic density in response to the medium fluc-tuations.16,24,25 This part of the calculation for-malism turns out to be very essential for achiev-ing low values of λr consistent with experiment.

Methods

Theoretical formalism. Modern theoriesof electron transfer assign the energy gap Xbetween the donor and acceptor energy lev-els to the electron-transfer reaction coordi-nate.13,24,28,29 For electron transfer in solution,one considers the one-electron states of thedonor and acceptor and the instantaneous (fluc-tuating) energy gap X between them as the re-action coordinate. Thermal fluctuations reducethis gap to zero in the activated state of anelectron transfer reaction (radiationless transi-tion). The same energy gap comes in resonancewith the radiation photon in spectroscopy ofcharge-transfer transitions (Figure 1a).47 Oneobserves charge-transfer absorption or emissionbands with the maxima corresponding to the

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average excitation energies ⟨X⟩i. The separa-tion between the maxima is the spectroscopicStokes shift,14,22 which can be used to quantifythe reorganization energy labelled as λSt (Fig-ure 1b).If Pi(!ω) is the probability of absorbing (i =

1) or emitting (i = 2) a photon with the energyX = !ω, the free energy surfaces of electronare constructed to compliment this picture interms of the free energy (reversible work) re-quired to achieve a given value of X : Gi(X) =Gi

0−kBT ln[Pi(X)] (Figure 1c), where Gi0 is the

free energy at the minimum. The separationbetween the minima of the free-energy surfacesthen becomes equal to 2λSt. One additionallycan define the reorganization energy from thecurvature of the free energy surface at the min-imum, ∂2Gi(X)/∂X2|X0i

, which can be relatedto the variance of the reaction coordinate X ac-cording to the standard rules of statistical me-chanics7,12,29

λ = ⟨(δX)2⟩/(2kBT ) (3)

Returning to the picture of optical transitions,the variance reorganization energy λ determinesthe Gaussian width of the energy-gap fluctua-tions or the inhomogenoues width of a singlevibronic optical line.48 As mentioned above, inthe Marcus picture one has λSt = λ, whichis a specific case of a general result known asthe fluctuation-dissipation theorem.49 As men-tioned above, this phenomenology changes forprotein electron transfer in solution, where onefinds12,21 λ ≫ λSt.Electrode electron transfer. In the case of

electrode electron transfer, the energy gap in-volving one-electron states is between the fluc-tuating energy level of the oxidized reactant insolution ϵOx and the energy level in the metalϵ (cathodic process, Figure 2).50 Correspond-ingly, we replace i = {1, 2} for solution electrontransfer with i = {Ox,Red} for electrode reac-tions. While full description of the problem interms of finite-temperature distribution of theelectrons in the metal is possible,51–55 we firstsimplify the discussion by considering electrontransfer to a single level corresponding to thechemical potential of the electrons in the metal

µ = ϵF − eη. It is modified by the overpotentialη (e is the elementary charge) from the Fermienergy ϵF consistent with the equilibrium po-tential at the electrode (Figure 2).

−eη

ϵF

ϵOx

ϵ

e−

∆

σ2= 2kBTλ

Figure 2: Schematics of cathode electron trans-fer from the Fermi energy level ϵF , correspond-ing to the equilibrium electrode potential, to anoxidized reactant with the average energy ϵOx.Electron transfer predominantly occurs from ϵF toa nonequilibrium energy level in resonance with.The electrode-reactant electronic coupling ∆ char-acterizes the tunneling probability (eq 10). Thenonequilibrium energy level is a part of a Gaussianmanifold with the variance σ2 = 2kBTλ specifyingthe reorganization energy λ (eq 3). The overpo-tential η shifts the electrode chemical potential asµ = ϵF − eη.

In this picture, the initial state of the systemwith the electron in the metal is EOx

g +µ, whereEOx

g is the ground state of the oxidized state ofthe reactant. The final state, before any relax-ation of the nuclear subsystem has occurred,is the ground quantum state in the reducedstate ERed

g . Both energies refer to the same nu-clear configuration of the thermal bath. Theelectron-transfer reaction coordinate, monitor-ing the transition to the activation state X = 0,is the energy gap between the initial and finalstates13,29

X = EOxg −ERed

g + µ (4)

A generic Gaussian distribution of the reac-tion coordinate X results in the parabolic freeenergy surface12,28,56

Gi(X) = Gi0 +

(X − ⟨X⟩i)2

4λ(5)

where i = Ox,Red.We can apply the condition of crossing at zero

energy gap, GRed(0) = GOx(0), to obtain the

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average values

⟨X⟩Ox = −λSt − (λ/λSt)∆G0

⟨X⟩Red = λSt − (λ/λSt)∆G0

(6)

where ∆G0 = GRed0 − GOx

0 is the reaction freeenergy. The Stokes-shift reorganization energyfrom these equations is half of the separationbetween the minima of the crossing parabolas

λSt = 12 |⟨X⟩Red − ⟨X⟩Ox| (7)

The activation barrier for the cathodic pro-cess is the free energy difference between theactivated state, GOx(0), and the free energy atthe minimum, GOx

0 : ∆G† = GOx(0)−GOx0 . One

gets from eqs 5 and 6

∆G† =(λr +∆G0)2

4λr(8)

where λr is the effective reorganization energygiven by eq 2. Since the reduction and oxida-tion rates are equal at η = 0, eη = ∆G0 for theelectrochemical discharge. One therefore getsfor the barrier of electrochemical electron trans-fer

∆G† =(λr + eη)2

4λr(9)

The significant result of this derivation is thataccepting two equal-curvature parabolas, evenwith the variance reorganization energy distinctfrom the Stokes reorganization energy, does notalter the basic Marcus result4 for the depen-dence of the activation barrier on the reactionfree energy. The two reorganization energies,λ and λSt, combine into an effective reorganiza-tion energy λr (eq 2), which is the only parame-ter that can be reported from experiments alter-ing either the reaction free energy (solution re-actions) or the electrode overpotential (electro-chemical kinetics). In contrast, spectroscopy ofcharge-transfer bands allows one to distinguishbetween λSt and λ.22,23 The former parameterdetermines the spectroscopic Stokes shift, whilethe latter yields the inhomogeneous broadeningof the spectral lines (Figure 1b).14,48,57

The arguments presented here can be ex-tended to the calculation of the rate of non-adiabatic electron transfer, which involves sum-

mation of the Golden Rule transitions to all en-ergy levels of the metal below the chemical po-tential µ. The resulting cathodic rate is51,58–60

kc(η) =∆

!erfc

(

λr + eη√4kBTλr

)

(10)

where erfc(x) is the complimentary error func-tion and ∆ = πρFV 2 is the electronic couplingbetween the redox species and the electrode. Itis given in terms of the coupling V between thereactant and the individual energy state in themetal and the density of states ρF of the con-duction electrons at the Fermi level.52,54,61

The derivative of the rate over the overpoten-tial, dkc/dη, is thus proportional to the “den-sity of states” of the oxidized energy level in themedium

Pc(η) ∝ exp

[

−(λr + eη)2

4kBTλr

]

(11)

This distribution is distinct from the corre-sponding distribution along the reaction coor-dinate POx(X) ∝ exp[−βGOx(X)] (Figure S2 inSI). The function Pc(η) is a measure of the prob-ability POx(0) to reach the activated state X =0 when the average ⟨X⟩Ox = −λSt−(eη)(λ/λSt)is varied by applying the overpotential. Thedistribution function Pc(η) is directly accessi-ble from cyclic voltamemtry upon correction formass transport.41

Solvent dynamics can potentially affect thepreexponential factor of the rate constant.62–65

The rate constant of electron transfer betweenthe electrode and an adsorbed reactant, not af-fected by diffusion, is given by the following re-lation66–68

ksc(η) = (1 + g)−1kc(η) (12)

with the nonadiabatic rate constant kc(η) ac-cording eq 10. The factor in front of it, correct-ing for the solvent dynamics, is given by therelation

g =∆⟨τ⟩!

4kBTλr

(λr + eη)2(13)

The theory leading to eq 12 is the result ofapplying the Sumi-Marcus63 formalism to elec-trode kinetics.68 The analytical expressions in

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eqs 12 and 13 are obtained under the assump-tion of a sufficiently low overpotential such thatλr + eη ≫ kBT .68 Further, ⟨τ⟩ is the character-istic time of the Stokes-shift dynamics of theenergy gap X specified through the energy gapautocorrelation function69

CX(t) = ⟨δX(t)δX(0)⟩ (14)

where δX(t) = X(t)−⟨X⟩ and ⟨τ⟩ is defined asthe integral of the normalized time correlationfunction (average solvation time70)

⟨τ⟩ =∫ ∞

0

dtCX(t)/CX(0) (15)

The relaxation times in the range ⟨τ⟩ ≃ 300 −900 ps71–74 were determined from MD simula-tions (Figure S4 and Table S3 in SI). TheseStokes-shift relaxation times were used in eq 13to estimate the effect of the solvent dynamicson ks

c(0). It was found to be negligible for theexperimental data considered below.Simulations and data analysis. The NMR

solution structure of horse heart cytochrome c(PDB 1GIW) was adopted as the starting con-figuration for the classical MD simulations. Thesimulations were done with NAMD softwaresuite,75 with the trajectory length of ≥ 250 nsfor each temperature and oxidation state (over-all ≥ 4 µs of MD simulations). The classi-cal MD simulations were followed by empiricalvalence-bond calculations44,45 performed for thequantum center including the heme, histidine,methionine, and two cysteine amino acids (Fig-ure 3 and SI) following our protocol developedin the past.25 The electrostatic potential of thebath φ(r) acting on the quantum center wasexpanded around the potential φFe at the hemeiron up to the dipolar operator. This expansionleads to a set of transition dipoles µi

jk in thematrix of the quantum center Hamiltonian16,25

H ijk =

(

Eij +QiφFe

)

δjk − µijk · Eb (16)

Here, Eij is the energy of jth state in either

i = Ox or i = Red states and Qi is the to-tal charge of the quantum center. The excitedstates j and k are coupled through the electricfield of the thermal bath Eb multiplying transi-

tion dipoles µijk in eq 16. Physically, this term

in the Hamiltonian represents the polarizationof the heme by the medium field through a non-zero polarizability αi

j of state j, which is givenin terms of the transition dipoles as

αij = 2

∑

k =j

∣

∣µijk

∣

∣

2/∆Ejk (17)

where ∆Ejk = Ek − Ej .

Figure 3: Quantum center of Cyt-c used in thecalculations to compute the Hamiltonian matrix ineq 16. It includes the heme (gray, with Fe coloredred), histidine (blue), methionine (green), and twocysteine (orange) amino acids.

The quantum states j = 0, . . . ,M include theground state of the quantum center, j = 0, anda number of its excited states produced hereby ZINDO/S calculations for the oxidized (Ox,Q = −1) and reduced (Red, Q = −2) states.The number of states M = 100 was chosen toconverge the polarizability of the quantum cen-ter. Decreasing the number of states M makesthe quantum center less polarizable and even-tually brings the system back to the Marcusformulation with λSt ≃ λ.25 Additional detailsof the simulation protocol and of the quantumcalculations are given in the SI.A polarizable quantum center carrying the

polarizability αi gives rise to the polarizationfree energy −(1/2)Eb · αi · Eb. On the otherhand, the free energy (reversible work) investedin creating a fluctuation in the medium scalesquadratically with the field,24,76,77 (χ/2)E2

b .The sum of this term and the polarization free

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energy lowers the force constant for the mediumfluctuation from χ to ≃ (χ−αi), αi = 1

3Tr[αi].

When projected on the reaction coordinate ofelectron transfer X , less free energy investedin an electrostatic medium fluctuation implieslower curvature of the corresponding parabolaGi(X) and a higher reorganization energy λ.24

This physical picture is consistent with our sim-ulations.The Hamiltonian matrix in eq 16 is diago-

nalized at each instantaneous value of the po-tential φFe and the electric field Eb along thesimulation trajectory to produce the minimumeigenvalues EOx/Red

g corresponding to the quan-tum ground state in either oxidized or reducedstates of the active site. They are used in eq4 to produce the trajectory of the fluctuatingvariable X . The probability distributions of Xcalculated in the oxidized and reduced statesyield the free energy surfaces of the half reac-tion Gi(X) (Figure S2 in SI).The reorganization energies λSt and λ are ob-

tained from, correspondingly, the first and sec-ond moments of the variable X . The formeris defined in terms of the average energy gap⟨X⟩i in the Red and Ox states according to eq7. The latter is given through the variance,λi = ⟨(δX)2⟩i/(2kBT ), i = Ox,Red. Signifi-cantly longer simulations are required to con-verge λi compared to λSt (Figure S1 in the SI).We find λOx and λRed slightly different even af-ter 250 ns of simulations (Figure 4 and TableS1 in the SI). The values of λ used for the ki-netic analysis were therefore obtained by takingthe mean of the values in the corresponding re-dox states, λ = (λOx + λRed)/2 (black circles inFigure 4).A separate issue is the potential impact of

the polarizability of the protein-water solventon the reorganization energies calculated fromsimulations. Dielectric continuum models pre-dict that the reorganization energy is propor-tional to the Pekar factor c0 = n−2

D −ϵ−1s ,4 which

implies a drop by a factor of about n−2D in going

from a non-polarizable solvent to a polarizablesolvent with the refractive index nD (assuminga high static dielectric constant ϵs). This per-spective would suggest that the reorganizationenergies obtained by computer simulations in

non-polarizable solvents (TIP3P water in thisstudy) would need to be scaled down to accountfor the polarizability effects. We have recentlyaddress this problem by computer simulationsand liquid-state theories.78,79

It turns out that microscopic solvation mod-els do not support re-scaling of the reorgani-zation energy according to the rules stipulatedby dielectric continuum models. In contrastto those predictions, λ stays nearly constantwith increasing n2

D, or even slightly increases(for polarizable water models), instead of thepredicted drop. Given these new results andprevious simulations and calculations of the ef-fects of the solvent polarizability on electrontransfer,80,81 it is reasonable to suggest thatthe reorganization parameters obtained fromthe present simulations do not need further re-scaling. A good agreement with experimentalresults demonstrated below is another indica-tion that our calculation formalism is robust.We also note that electron transfer in re-

dox proteins is typically accompanied by smallstructural changes of the active site82 and, cor-respondingly, low reorganization energy of ac-tive site vibrations. Estimated values rangefrom 0.05−0.09 eV for Fe-porphins83 to 0.10−0.14 eV for Zn-porphyrins84 to ≃ 0.1 eV inazurins.85 This internal reorganization energyis generally split between quantum and classi-cal vibrations. The reorganization energy re-lated to quantum vibrations affects the rates inthe Marcus inverted region of electron trans-fer,86 which is not typically reached in eitherthe electrochemical experiment or at the typ-ical conditions of redox reactions in biologicalenergy chains. Therefore, only the classical partof the internal reorganization energy can poten-tially affect these reactions. While the splittingof the reorganization energy between the classi-cal and quantum modes is not known for Cyt-c,the classical part of the internal reorganizationenergy, remaining after subtracting the quan-tum component, is expected to be small, withinthe simulation uncertainties. We therefore donot include the internal reorganization energyin our calculations of the electrode kinetics.

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λ

λ

λ

λ

Figure 4: Temperature dependence of reorganiza-tion energies from QM/MD simulations. Shownseparately are the variance reorganization energiesλi (eq 3) in the reduced (red diamonds) and ox-idized (blue triangles) states of Cyt-c and theirmean values (black circles). Squares refer to theStokes-shift reorganization energy λSt (eq 7) andblack diamonds refer to the reaction reorganiza-tion energy λr (eq 2). The dashed lines are linearregressions through the simulation points (the up-per dashed line is a linear regression through themean values λ(T )).

Results

The results of QM/MD simulations for the reor-ganization energies as functions of temperatureare shown in Figure 4. The corresponding val-ues at T = 300 K, estimated from linear regres-sions of the simulation data, are listed in Table1. As expected, both reorganization energies,λSt and λ, are fairly large and consistent witha large density of charge and polar groups sur-rounding the active site of a redox protein. Arelatively small value of the reaction reorganiza-tion energy λr (eq 2) is achieved due to λ ≫ λSt.As we already pointed out, this inequality inthe case of Cyt-c is the consequence of a highpolarizability of the active site allowing its elec-tronic density to deform in response to the fluc-tuations of the thermal bath. Reducing the po-larizability by either using fixed partial atomiccharges (zero polarizability) or a small numberof quantum states when diagonalizing the quan-tum Hamiltonian produce λSt ≃ λ ≃ 1.3 − 1.6eV consistent with the standard Marcus pictureof a single reorganization energy characterizingelectron transfer.25 However, these values of the

reorganization energy are too high to describethe experimental electrochemical data, as weshow below.

×

Figure 5: Rate constant k0 = kc(0) (eq 10) atη = 0 for horse Cyt-c on the gold electrode mod-ified with PyC11/C10 self-assembled monolayer33

(points, Exp.). The solid line shows the calcula-tions based on eqs 10 and 12, which give identi-cal results. The electron coupling ∆ = 2 × 10−9

eV is used to reproduce the experimental data.The temperature-dependent reorganization energyλr(T ) from Figure 4 was used in eqs 10 and 12.The dotted line shows the result of neglecting thetemperature dependence λr(T ) and putting λr =λr(300 K).

Not only the reorganization energy itself, butalso its temperature dependence is reduced forλr compared to λSt and λ. We list in Table 1the entropies of reorganization

Sλ = − (∂λ/∂T )V (18)

at constant volume consistent with the NVT en-semble used in the simulations (see SI). All reor-ganization energies are decaying functions withincreasing temperature, as is expected fromstudies of electron-transfer reactions in polarliquids.87,88

The decay of the reorganization energy withincreasing temperature is related to struc-tural fluctuations in polar liquids producingchanges in both orientations of the liquiddipoles and their positions (density fluctua-tions). While changes in orientations are mostlydriven by redistributing the thermal energy (en-ergy driven), the density rearrangements re-quire local repacking of the liquid against repul-sive molecular cores (entropy driven). The dif-

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ference in the character of fluctuations, energy-driven for rotations and entropy-driven fortranslations, projects on different effects of tem-perature on the corresponding components inthe reorganization energy. The reorganiza-tion energy arising from molecular rotations isnearly temperature-independent, while the re-organization energy arising from density fluc-tuations decays approximately hyperbolicallywith increasing temperature. The overall de-pendence of the reorganization energy on tem-perature is hyperbolic,87 as proven experimen-tally88,89 for systems with λSt ≃ λ. Figure 4shows that the general rule of the reorganiza-tion energy decaying with increasing tempera-ture extends to redox proteins with λ ≫ λSt.The largest entropy Sλ is observed for

the variance reorganization energy λ, withTSλ/λ ≃ 0.5 consistent with typical values ob-served for electron transfer in polar molecularliquids.87 In contrast, the temperature varia-tion of λr is significantly reduced, by an orderof magnitude, due to the mutual cancellationof the corresponding temperature effects on λSt

and λ. This cancellation achieves a significantrobustness of operation and insensitivity of theenzyme to the variations of thermodynamicconditions.1 Consistently, a very small reactionentropy was recently reported for electrochem-istry of immobilized myoglobin.90

Table 1: Reorganization parameters ofCyt-c at T = 300 K (eV).a

Parameter λ λSt λr

λ 2.85 1.26 0.56TSλ 1.34 0.45 0.14

aThe parameters in the Table are calculated fromlinear interpolations of the MD data in the range

of temperatures from 280 K to 360 K (Figure 4).

The magnitude of λr, and its temperaturedependence, are fully consistent with the ex-perimental data. Figure 5 shows the tempera-ture variation of k0 = kc(0) measured for horseCyt-c33 immobilized on a self-assembled mono-layer (SAM) on a metal electrode.51,91,92 Im-mobilization is achieved by linking the hemeof Cyt-C to terminal pyridine group of thePyC11/C10 monolayer. The application of the

Figure 6: Normalized probability density Pc(η)(eq 11) obtained from experiment with tuna (filledcircles30) and horse (diamonds31) Cyt-c and fromMD simulations (solid line). The experimen-tal results were collected at T = 273 K fromvoltammograms with the electrode coated with theOH− (CH2)11SH ω-hydroxyalkenthiol. The MDvalue of the reorganization energy λr = 0.57 eVwas obtained by extrapolating the results shown inFigure 4 to 273 K. The dashed line is the Gaussianfit through the filled circles.

non-adiabatic electron-transfer rate requiresthe unknown parameter ∆ in eq 10, which doesnot affect the slope of the Arrhenius plot (ln k0vs 1/T ), but causes its vertical shift. The valueof ∆ was adjusted to fit the experimental data(points in Figure 5). A good agreement of theArrhenius slope with experiment suggests thatλr, and its temperature dependence, are reli-ably reproduced by the simulations. Neglect-ing the temperature dependence of λr resultsin a lower slope (dotted line), in accord withthe positive sign of Sλ in Table 1 affecting theenthalpy of activation according to the relation

∆H† ≃λ+ TSλ

4(19)

The estimate of the solvent dynamic ef-fect,62,64,65 with the relaxation time ⟨τ⟩ ≃300−900 ps obtained from simulations (see SI),shows that the term in the denominator in eq12, containing g ∝ ⟨τ⟩ (eq 13), can be neglectedfor this reaction (g ≃ 4× 10−4).An independent test of our results is provided

by voltammetry of horse Cyt-c performed30 onan electrode coated with ω-hydroxyalkenthiolSAM of a thickness comparable to the one

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used to produce data shown in Figure 5.33

In these experiments, the derivative of thediffusion-corrected41 electrode current dic/dηwas recorded (points in Figure 6). As men-tioned above, this derivative is proportional tothe probability density Pc(η) along the overpo-tential coordinate (eq 11). The probability den-sity based on our MD simulations (solid linein Figure 6) is in good agreement with experi-ment without any additional fitting. The valueof λr ≃ 0.57 eV used in the analysis is consis-tent with λr ≃ 0.58 ± 0.04 eV reported previ-ously31,32 (native rat Cyt-c in Ref. 32).

∆

Figure 7: Shift of the cathodic peak potential ∆Ep

vs the scan rate log10(v∗), v∗ = ev/(kBTkc(0)).

The points are experimental data33 and the solidline is the calculations93,94 performed with λr =0.56 eV and ∆ = 2 × 10−9 eV at T = 298 K. Thedashed line indicates Laviron’s95 irreversible reac-tion limit with the slope 2.3kBT/(αe) and with thetransfer coefficient α = 0.5 (Butler-Volmer kinet-ics50).

Our results are also consistent with the re-ported cyclic voltammograms33 without addi-tional fitting (Figure 7). The calculations wereperformed by applying the rate constant asgiven by eq 10 to kinetic equations describingredox adsorbates (see SI).93,94 The reorganiza-tion energy λr for the analysis is taken from ourMD data and ∆ = 2 × 10−9 eV is the fittingparameter from the Arrhenius plot shown inFigure 5. The calculations are consistent withthe expected limiting behavior for fully irre-versible reactions where Laviron’s solution95 forthe Butler-Volmer kinetics predicts the linearplot with the slope 2.3kBT/(αe) (dashed line inFigure 7 for the transfer coefficient α = 0.5).

Discussion & Conclusions

Effective (reaction) reorganization energy λr

(eq 2), combining the Stokes-shift and vari-ance reorganization energies, controls the ac-tivation barrier of electrode electron transfer.The resulting low activation barrier is con-sistent with the values obtained from cyclicvoltammograms. The low value of λr is specificto metalloproteins in solution. Here, we havenot directly simulated the protein attached tothe electrode and instead applied the resultsfor the protein solution to interfacial electrontransfer.The effect of temperature on the activation

barrier is significantly reduced compared to thestandard models due to the compensation of in-dividual temperature dependencies of λ and λSt

in λr. This result implies a robust operation ofthe redox enzyme, little affected by the alter-ation of temperature.Redox enzymes act to lower the activation

barrier for electron transfer. When the reac-tion free energy is low, lower barrier impliesminimizing the reorganization energy (eq 8). Ithas been long anticipated that redox proteinsshould have evolved mechanisms to achieve thisgoal. However, traditional thinking in terms ofa non-polar environment provided by the pro-tein and producing low solvation energy turnedout, with the help of atomistic simulations, tobe inconsistent with a wet and covered withcharges and polar groups environment of a typ-ical redox site. The mechanisms which evolvedin such a heterogenous environment appear tobe different from the standard thinking of theMarcus model considering polarization of a ho-mogeneous solvent in response to re-localizingthe electron.The heterogeneous protein-water thermal

bath is capable of producing the spectrum offluctuations deviating from the rules of theGibbs ensemble by the fact of being trappedin non-equilibrium states on the time-scale ofthe reaction.21 While this mechanism operatesfor a number of proteins, Cyt-c appears to bemore stable and rigid than many other proteins,thus disallowing a large number of trap states.Polarizability of the active site is involved in

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this case. It achieves the same result of anintense electrostatic noise effectively lowing thebarrier for electron transfer. It appears that dif-ferent mechanisms are involved with differentproteins, all reaching the same goal of minimiz-ing the reaction reorganization energy throughlarge-amplitude interfacial noise. The principleof a noisy protein-water interface, also washingout small differences in thermodynamic con-ditions and effects of insignificant mutations,might be a general principle by which energychains of biology achieve low activation barriersfor physiological electron transport.

Supporting Information Available: Sim-ulation protocols, additional data analysis, andderivation of equations used in the text. Thismaterial is available free of charge via the In-ternet at http://pubs.acs.org/.

Author Information:Corresponding author: ∗[email protected]: The authors declare no competing fi-nancial interest.

Acknowledgement The authors acknowledgeHadi Dinpajooh and Daniel Martin for their help

in setting up the simulations. David Waldeck pro-vided us with experimental kinetic data. Thisresearch was supported by the National Science

Foundation (CHE-1464810). CPU time was pro-vided by the National Science Foundation through

XSEDE resources (TG-MCB080071).

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Figure Legends

Figure 1 (a) Reaction coordinate X = !ωfor solution electron transfer between thedonor (D) and acceptor (A).13 (b) Proba-bility densities for absorbing (abs.) andemitting (em.) a photon in a charge-transfer optical transition; ⟨X⟩i stand forthe average transition energies. The sep-aration between the peaks of optical tran-sitions represents the Stokes shift and thecorresponding reorganization energy λSt.(c) The free energy surfaces of electrontransfer Gi(X) = Gi

0 − kBT ln[Pi(X)] fol-lowing from the optical transition prob-abilities Pi(X). The reorganization en-ergy λ defines the curvature of the freeenergy surface near the bottom (shownby the double arrow). It also providesthe measure of inhomogeneous broaden-ing of the optical charge-transfer band14

(σ2X = ⟨(δX)2⟩ = 2kBTλ in (b) and in

eq 3). The filled dots in (b) and (c) indi-cate, respectively, P2(0) and the crossingpoint of Gi(X) representing the transitionstate, X = 0, of the electron-transfer re-action.

Figure 2 Schematics of cathode electrontransfer from the Fermi energy level ϵF ,corresponding to the equilibrium elec-trode potential, to an oxidized reactantwith the average energy ϵOx. Electrontransfer predominantly occurs from ϵFto a nonequilibrium energy level in reso-nance with it with the electrode-reactantelectronic coupling ∆ (eq 10). Thenonequilibrium energy level is a part ofa Gaussian manifold with the varianceσ2 = 2kBTλ specifying the reorganiza-tion energy λ (eq 3). The overpotential ηshifts the electrode chemical potential asµ = ϵF − eη.

Figure 3 Quantum center of Cyt-c used inthe calculations to compute the Hamil-tonian matrix in eq 16. It includes theheme (gray, with Fe colored red), histi-dine (blue), methionine (green), and twocysteine (orange) amino acids.

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Figure 4 Temperature dependence of reorga-nization energies from QM/MD simula-tions. Shown separately are the variancereorganization energies λi (eq 3) in the re-duced (red diamonds) and oxidized (bluetriangles) states of Cyt-c and their meanvalues (black circles). Squares refer tothe Stokes-shift reorganization energy λSt

(eq 7) and black diamonds refer to thereaction reorganization energy λr (eq 2).The dashed lines are linear regressionsthrough the simulation points (the upperdashed line is a linear regression throughthe mean values λ(T )).

Figure 5 Rate constant k0 = kc(0) (eq 10)at η = 0 for horse Cyt-c on the goldelectrode modified with PyC11/C10 self-assembled monolayer33 (points, Exp.).The solid line shows the calculationsbased on eqs 10 and 12, which give identi-cal results. The electron coupling∆ = 2×10−9 eV is used to reproduce the experi-mental data. The temperature-dependentreorganization energy λr(T ) from Figure4 was used in eqs 10 and 12. The dot-ted line shows the result of neglectingthe temperature dependence λr(T ) andputting λr = λr(300 K).

Figure 6 Normalized probability densityPc(η) (eq 11) obtained from experimentwith tuna (filled circles30) and horse(diamonds31) Cyt-c and from MD sim-ulations (solid line). The experimen-tal results were collected at T = 273K from voltammograms with the elec-trode coated with the OH− (CH2)11SHω-hydroxyalkenthiol. The MD value ofthe reorganization energy λr = 0.57 eVwas obtained by extrapolating the resultsshown in Figure 4 to 273 K. The dashedline is the Gaussian fit through the filledcircles.

Figure 7 Shift of the cathodic peak potential∆Ep vs the scan rate log10(v

∗), v∗ =ev/(kBTkc(0)). The points are experi-mental data33 and the solid line is the cal-culations93,94 performed with λr = 0.56

eV and ∆ = 2 × 10−9 eV at T = 298K. The dashed line indicates Laviron’s95

irreversible reaction limit with the slope2.3kBT/(αe) and with the transfer coeffi-cient α = 0.5 (Butler-Volmer kinetics50).

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Graphical TOC Entry

Cyt-c

electrode

e

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Theory and Electrochemistry of Cytochrome C