Theory of Coupled Electron and Proton Transfer...

Theory of Coupled Electron and Proton Transfer Reactions

Sharon Hammes-Schiffer*,† and Alexei A. Stuchebrukhov*,‡

Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, United States, and Department of Chemistry,University of California, Davis, California 95616, United States

Received May 14, 2010

Contents

1. Introduction 69392. Concerted PCET Reactions 6940

2.1. Theoretical Framework 69402.2. Rate Constants 69422.3. Vibronic Coupling: Electronically Adiabatic and

Nonadiabatic Proton Transfer6943

2.4. Hydrogen Atom Transfer 69442.5. Kinetic Isotope Effects 6944

3. Sequential Electron and Proton Transfer. FourState Models

6944

3.1. Rate of a Sequential Coupled Reaction 69443.2. pH and Isotope Effects 69463.3. PT vs ET: Which is Faster? 69473.4. Statistics of Transitions in a Single Molecule 69473.5. ET Induced by PT 69483.6. Concerted vs Sequential Proton Transport 6949

4. Concerted vs Sequential PCET 69495. Kinetics of Electron Transfer Reactions Coupled to

Proton Translocation6950

5.1. Coordinates and Energetics 69505.2. Reaction Dynamics 69515.3. Rate Constants in Limiting Cases 6951

5.3.1. Fast PT, Slow ET 69515.3.2. Fast ET, Slow PT: Effective Proton

Potential6952

5.3.3. Fast ET, Slow PT: Reaction Rate Constant 69525.3.4. Proton Induced Electron Transfer 6953

6. Applications 69536.1. HAT vs PCET Mechanisms 69536.2. Concerted PCET in Solution and Enzymes 69556.3. Cytochrome c Oxidase 6955

6.3.1. Proton Pumping Mechanism 69566.3.2. Coupled Electron and Proton Transfer

Reactions6957

7. Conclusions 69578. Acknowledgments 69589. References 6958

1. IntroductionCoupled electron and proton transfer reactions play a key

role in the mechanisms of biological energy transduction.1-3

Such reactions are also fundamental for artificial energy-related systems such as fuel cells, chemical sensors, and otherelectrochemical devices. Biological examples include, among

others, cytochrome c oxidase,4,5 bc1 complex,6,7 and photo-synthetic reaction centers.8,9 In such systems, electrons tunnelbetween redox cofactors of an enzyme, while the coupledprotons are transferred either across a single hydrogen bondor between protonatable groups along special proton-conducting channels.

In this review general theories and models of coupledelectron transfer and proton transfer (ET and PT) reactionsare discussed. Pure electron transfer reactions in proteins havebeen thoroughly studied in the past, both experimentally10-17

and theoretically.18-25 The coupled reactions are relativelynew and currently are gaining attention in the field.6,8,26-43

Two types of coupled reactions can be distinguished. Inconcerted electron and proton transfer reactions, both the ETand PT transitions occur in one step. These concertedreactions have been denoted proton-coupled electron transfer(PCET) in the literature,29,30,43-45 but the term PCET is usedmore generally in this review. Such concerted processesoccur in reactions in which proton transfer is typically limited

† Pennsylvania State University.‡ University of California, Davis.

Sharon Hammes-Schiffer received her B.A. in 1988 from PrincetonUniversity and her Ph.D. in Chemistry at Stanford University in 1993.She was a postdoc at AT&T Bell Laboratories and was appointed theClare Boothe Luce Assistant Professor of Chemistry and Biochemistry atthe University of Notre Dame in 1995. She has been at PennsylvaniaState University since 2000, when she was appointed the Shaffer AssociateProfessor of Chemistry and subsequently appointed the Eberly Professorof Biotechnology in 2006. She is a Senior Editor for The Journal of PhysicalChemistry and The Journal of Physical Chemistry Letters. Dr. Hammes-Schiffer’s current research centers on the investigation of proton, electron,and proton-coupled electron transfer reactions in chemical, biological, andinterfacial processes. Her work encompasses the development of analyticaltheories and computational methods, as well as applications to a widerange of experimentally relevant systems. She has been a recipient ofthe National Science Foundation CAREER Award (1996), an Alfred P.Sloan Research Fellowship (1998), the Camille Dreyfus Teacher-ScholarAward (1999), the Iota Sigma Pi Agnes Fay Morgan Research Award(2005), the International Academy of Quantum Molecular Science Medal(2005), and the American Chemical Society Akron Section Award (2008).

Chem. Rev. 2010, 110, 6939–6960 6939

10.1021/cr1001436 2010 American Chemical SocietyPublished on Web 11/04/2010

to one hydrogen bond; however, examples with multiplehydrogen bond rearrangements are also known.46 In sequen-tial reactions, the transitions occur in two steps: ET followedby PT or PT followed by ET. Typically, each individual stepis uphill in energy, while the coupled reaction is downhill.

A sequential reaction can proceed along two parallel channels:ET then PT (ET/PT) or PT then ET (PT/ET). In each channelthe reaction involves two sequential steps: uphill activation andthen downhill reaction to the final product state. The lifetimeof the activated complex is limited by the back reaction. Thegeneral formula for the rate of such reactions can be easilydeveloped. In the context of bioenergetics issues, however, itis interesting to analyze all of the possible cases separatelybecause each corresponds to a different mechanism: forexample, an electron can go first and pull out a proton;alternatively, a proton can go first and pull out an electron; oran electron can jump back and forth between donor and acceptorand gradually pull out a proton. In enzymes involving coupledproton and electron transport, the exact mechanism of thereaction is of prime interest.

The initial part of this review will cover general theoreticalconcepts of coupled electron and proton transfer reactions.First we will consider a simple four-state model of reactionswhere the proton moves across a single hydrogen bond; bothconcerted and sequential reactions will be treated. Then wewill consider models for long-distance proton transfer, alsodenoted proton transport or proton translocation. Typically,electron transfer coupled to proton translocation in proteinsinvolves an electron tunneling over a long distance betweentwo redox cofactors, coupled to a proton moving along aproton conducting channel in a classical, diffusion-likerandom walk fashion. Again, separately the electron andproton transfer reactions are typically uphill, while thecoupled reaction is downhill in energy. The schematics of

this process is shown in Figure 1. The kinetics of suchreactions can be much different from those involving protontransfer across a single hydrogen bond. In this paper, wewill discuss the specifics of such long-distance proton-coupled reactions.

Following the review of theoretical concepts, a fewapplications will be discussed. First the phenoxyl/phenol andbenzyl/toluene self-exchange reactions will be examined. Thephenoxyl/phenol reaction involves electronically nonadiabaticproton transfer and corresponds to a concerted PCETmechanism, whereas the benzyl/toluene reaction involveselectronically adiabatic proton transfer and corresponds to ahydrogen atom transfer (HAT) mechanism. Comparison ofthese two systems provides insight into fundamental aspectsof electron-proton interactions in these types of systems.Next a series of theoretical calculations on experimentallystudied PCET reactions in solution and enzymes will besummarized, along with general predictions concerning thedependence of rates and kinetic isotope effects (the ratio ofthe rate constants for hydrogen and deuterium transfer) onsystem properties such as temperature and driving force. Thefinal application that will be discussed is cytochrome coxidase (CcO). CcO is the terminal component of the electrontransport chain of the respiratory system in mitochondria andis one of the key enzymes responsible for energy generationin cells. The intricate correlation between the electron andproton transport via electrostatic interactions, as well as thekinetics of the coupled transitions, appears to be the basisof the pumping mechanism in this enzyme.

2. Concerted PCET Reactions

2.1. Theoretical FrameworkIf one electron and one proton are transferred, the reaction

can be described in terms of four diabatic electronic states,as depicted in Figure 2:47

Alexei Stuchebrukhov received a Ph.D. degree in theoretical physics fromMoscow Physical and Technical Institute in 1985, in the theory group ofAcademician Vitaly L. Ginzburg, and was a research fellow in the Instituteof Laser Spectroscopy and the Research Center for Laser Technology ofthe Russian Academy of Sciences until 1990. During this time he workedin a Laboratory directed by V. S. Letokhov. In 1990 he joined the groupof Rudy Marcus at California Institute of Technology, with whom he wasa research fellow until 1994. Until this time his research interests weremainly in the area of vibrational dynamics and spectroscopy of polyatomicmolecules. Since 1994 he has been at University of California, Davis,where he is currently a Professor of Chemistry and Biophysics. His currentinterests are in the general area of chemical kinetics and biological electronand proton transfer reactions; his main current efforts are focused onunderstanding molecular aspects of the electron transport chain of aerobiccells and biological energy transduction. He has been an Alfred P. SloanFoundation Fellow (1996) and a recipient of an Arnold and Mabel BeckmanFoundation Young Investigator Award (1997).

Figure 1. Schematics of the electron transfer reaction coupled to protontranslocation. In the reaction, an electron is tunneling over a long distancebetween two redox cofactors, O and R, and a coupled proton is transferredover a proton conducting channel. The initial and final states of the protonare two protonatable groups of the protein, U and P. Separately, bothelectron and proton transfer are uphill in energy; however, the coupledreaction is downhill. Figure reprinted with permission from ref 42.Copyright 2003 World Scientific.

6940 Chemical Reviews, 2010, Vol. 110, No. 12 Hammes-Schiffer and Stuchebrukhov

where the O/R (oxidized/reduced) and U/P (unprotonated/protonated) symbols are used to denote the state of theacceptor group. In concerted mechanisms, the electron andproton transfer simultaneously (i.e., OU f RP), along thediagonal in Figure 2. In sequential mechanisms, the protontransfers prior to the electron (i.e., OU f OP f RP) or theelectron transfers prior to the proton (i.e., OUf RUf RP),along the edges of the scheme in Figure 2. This model iseasily extended to proton-coupled electron transport ortranslocation processes involving multiple proton and elec-tron transfers by including additional diabatic electronicstates.47

In general, a concerted PCET mechanism is defined as aPCET reaction that does not involve a stable intermediatearising from single electron or proton transfer. ConcertedPCET reactions are often described in terms of reactant andproduct states corresponding to the electron being localizedon the donor or acceptor, respectively.48 In this case, thereactant state is dominated by OU, and the product state isdominated by RP. The proton vibrational states can becalculated for the reactant and product electronic states,leading to two sets of electron-proton vibronic states.Typically concerted PCET reactions can be described interms of nonadiabatic transitions between the reactant andproduct electron-proton vibronic states.

Analogous to Marcus theory for electron transfer, PCETreactions can be described in terms of reorganization of thesolvent environment. Figure 3 depicts a slice of the freeenergy surfaces along a collective solvent coordinate. Alsoshown are the proton potential energy curves and associatedproton vibrational wave functions. Typically the proton donorwell is lower in energy when the electron is localized on itsdonor, whereas the proton acceptor well is lower in energywhen the electron is localized on its acceptor. Thus, theproton vibrational ground state wave function is localizednear its donor in the reactant state and near its acceptor inthe product state. Note that this general description encom-passes the cases in which the electron and proton aretransferred in the same or in different directions. The shapesof the proton potential energy curves are usually notsignificantly influenced by the solvent coordinate in the

region of interest because the asymmetry is dominated bythe electrostatic interaction between the proton and the soluteelectronic charge distribution. The relative energies of theproton potential energy curves, however, are stronglyinfluenced by the solvent coordinate.

The basic mechanism for concerted PCET may beanalyzed in the context of Figure 3. Initially, the system isin thermal equilibrium in the reactant state, and both theelectron and proton are localized near their donors. Fluctua-tions of the solvent environment cause the system to evolveto the intersection between the two curves, where a nona-diabatic transition from the reactant to the product vibronicstate occurs with a probability proportional to the square ofthe vibronic coupling. After this nonadiabatic transition, theelectron and proton are localized near their acceptors, andthe system relaxes to thermal equilibrium in the product state.Often excited vibronic states and the proton donor-acceptorvibrational motion also play important roles in PCETreactions.31 Figure 4 depicts slices of the free energy surfacesand the corresponding proton vibrational wave functions fortwo reactant and four product vibronic states for a concertedPCET reaction.

Figure 2. Four-state model for coupled electron and proton transferreactions. In the initial state OU, the electron is on the donor siteO (oxidized acceptor), and the proton is on the U site (unprotonatedacceptor). In the final state RP, the electron is on the acceptor siteR (reduced acceptor), and the proton is on the P site (protonatedacceptor). Intermediate states OP and RU correspond to activatedstates for sequential PT/ET and ET/PT reactions. The concertedreaction corresponds to an OU to RP transition. Figure reprintedwith permission from ref 42. Copyright 2003 World Scientific.

De--Dp-H+······Ap-Ae (OU)

De--Dp······

+H-Ap-Ae (OP)

De-Dp-H+······Ap-Ae- (RU)

De-Dp······+H-Ap-Ae

- (RP)

(2.1)

Figure 3. Slices of the free energy surfaces for the ground reactant(I) and product (II) vibronic states along a collective solventcoordinate. The proton potential energy curves along the protoncoordinate and the corresponding ground state proton vibrationalwave functions are depicted for the reactant minimum, the crossingpoint, and the product minimum of the free energy curves. Theenergies of these proton vibrational states correspond to the opencircles on the free energy curves. The proton potential energy curvesassociated with the crossing point are shifted higher in energy forclarity. Figure and caption reprinted with permission from ref 31.Copyright 2008 American Chemical Society.

Figure 4. Calculated free energy curves for the PCET reaction ina rhenium-tyrosine complex. In the center frame are slices of thefree energy surfaces along a collective solvent coordinate. In theleft/right frames are the reactant/product proton potential energycurves and the corresponding proton vibrational wave functionsalong the proton coordinate. Figure reprinted with permission fromref 137. Copyright 2007 American Chemical Society.

Theory of Coupled Electron and Proton Transfer Reactions Chemical Reviews, 2010, Vol. 110, No. 12 6941

2.2. Rate ConstantsA series of rate constant expressions for vibronically

nonadiabatic concerted PCET have been derived49,50 invarious well-defined limits using Fermi’s Golden ruleformalism and linear response theory. For fixed protondonor-acceptor distance R, the rate constant is49

where the summations are over reactant and product vibronicstates, Pµ is the Boltzmann population for the reactant stateµ, Vµν is the vibronic coupling between the reactant andproduct vibronic states µ and ν, λµν is the solvent reorganiza-tion energy for states µ and ν, and ∆Gµν° is the free energyof reaction for states µ and ν. All of these quantities dependon the fixed proton donor-acceptor distance R. As discussedin section 2.3, the vibronic coupling is the product of theelectronic coupling and the overlap between the reactant andproduct proton vibrational wave functions in the electroni-cally nonadiabatic regime.

Rate constant expressions including the dynamical effectsof the R coordinate and the solvent have also been derived.50

In these derivations, the vibronic coupling is assumed todepend exponentially on R:

where Rjµ is the equilibrium value of R for the reactant stateµ and Vµν

(0) is the vibronic coupling between states µ and ν atdistance Rjµ. This form of the coupling is a reasonableapproximation in the region of R near its equilibrium value,as illustrated by expanding ln[Vµν/Vµν

(0)] in a Taylor seriesaround R ) Rjµ and retaining only the linear terms.51 Typicallythe Condon approximation, in which the electron-protonvibronic coupling is assumed to be independent of the nuclearconfiguration, is invoked for PCET reactions, with theimportant exception of the R-mode.

In the dynamical formulation,50 the rate constant isrepresented by the time integral of a time-dependent prob-ability flux correlation function, which is expressed in termsof the vibronic coupling and the time correlation functionsof the R coordinate, the energy gap, and the derivative ofthe energy gap with respect to the R coordinate. These timecorrelation functions can be calculated from classical mo-lecular dynamics simulations on the reactant surface. Thisformulation can be used with any potential energy surfaceand includes the dynamical effects of the solvent and R mode,but it has a complicated form and requires numericalintegration over time.

Using the short-time, high-temperature approximation forthe solvent modes and representing the R-mode time cor-relation function by that of a quantum mechanical harmonicoscillator, the rate constant can be expressed as50

with the dimensionless parameters defined as

Here ) 1/kBT, λµν(R) is the coupling reorganization energy

defined as λµν(R) ) p2Rµν

2 /2M, and λR is the R-mode reorganiza-tion energy defined as λR ) MΩ2δR2/2, where M and Ω arethe R-mode effective mass and frequency, respectively, andδR ) Rjν - Rjµ. Here δR and the solvent reorganization energyλs are assumed to be the same for all pairs of states, althoughin general they could be allowed to vary for different pairsof states. The short-time, high-temperature approximationfor the solvent is valid when the dynamics of the solventfluctuations are fast on the time scale of the coherentnonadiabatic transitions. Note that this rate constant stillrequires integration over time. Further simplified expressionshave been derived in limiting regimes pertaining to theR-mode frequency.

In the high-temperature (low-frequency) limit for theR-mode (pΩ , kBT), the rate constant simplifies to52

where the total reorganization energy is defined as Λµν ) λs

+ λR + λµν(R). This rate constant can be further simplified by

assuming that δR ) 0 (i.e., the equilibrium R value is thesame for the reactant and product vibronic states) and λµν

(R)

, λs, which is equivalent to the replacement of the R-modetime correlation function with its value at zero time. Theresulting rate constant expression is

Related expressions have been derived for vibrationallynonadiabatic proton transfer reactions and for electrontransfer reactions.53-57

In the low-temperature (high-frequency) limit for theR-mode (pΩ . kBT), the rate constant simplifies to50

This analytical expression was derived using the stationaryphase method and is valid only in the strong solvation regime(i.e., λs > |∆Gµν° | for all relevant pairs of states). In this limit,the R-mode remains predominantly in its ground state, andthe vibronic coupling is averaged over the ground statevibrational wave function of the R-mode. In principle, otherhigh-frequency solute modes could also be included in therate constant expression.

k ) ∑µ

Pµ ∑ν

|Vµν|2

p πλµνkBT

exp[- (∆Gµν° + λµν)2

4λµνkBT ](2.2)

Vµν ) Vµν(0) exp[-Rµν(R - Rjµ)] (2.3)

k ) ∑µ

Pµ ∑ν

|Vµν(0)|2

p2Ωexp[2λµν

(R)pΩ ] ×

∫-∞∞

dτ exp[-12

τ2 + a(cos τ - 1) + i(b sin τ + θτ)](2.4)

) coth(12pΩ); )2λs

p2Ω2; θ )

∆Gµν° + λs

pΩ

a ) λR + λµν

(R)

pΩ+ Rµν δR; b )

λR + λµν(R)

pΩ+ Rµν δR

(2.5)

k ) ∑µ

Pµ ∑ν

|Vµν(0)|2

pexp[2kBTRµν

2

MΩ2 ] πΛµνkBT

×

exp[- (∆Gµν° + Λµν + 2RµνδRkBT)2

4ΛµνkBT ] (2.6)

k ) ∑µ

Pµ ∑ν

|Vµν(0)|2

pexp[2kBTRµν

2

MΩ2 ] πλskBT

×

exp[- (∆Gµν° + λs)2

4λskBT ] (2.7)

k ) ∑µ

Pµ ∑ν

|Vµν(0)|2

p πλskBT

exp[λµν(R) - λR

pΩ- Rµν δR] ×

exp[- (∆Gµν° + λs)2

4λskBT ] (2.8)


The effects of intramolecular solute modes (i.e., inner-sphere reorganization) have been incorporated within thistheoretical framework49 in various well-defined limits.58-60

In the high-temperature approximation for the uncoupledsolute modes, the rate constant expressions given above aremodified by adding the inner-sphere reorganization energyto the solvent reorganization energy.

In some cases, the assumption that the vibronic couplingdecreases exponentially with the proton donor-acceptordistance R is not valid. Typically this assumption is validonly near the equilibrium R value and will break down forlower proton donor-acceptor vibrational frequencies thatenable sampling of a wider range of R values.61 In addition,this assumption will not be valid when the proton vibrationalwave functions change character in the relevant range of Rvalues. Specifically, the shapes of the proton potentials maychange with R (i.e., the barrier of an asymmetric double wellpotential energy curve could become lower, possibly becom-ing only a shoulder, as R decreases). In this case, a givenproton vibrational state could shift from being localized onone side to the other, or possibly being delocalized, as Rchanges. The assumption in eq 2.3 will break down in thesesituations, and the rate constant expressions based on thisassumption are no longer valid.

For the general form of the vibronic coupling, the effectsof the R-mode can be included with the expression

where k(R) is the rate constant in eq 2.2 evaluated at a givenR value and P(R) is the normalized probability distributionfunction for R at a specified temperature. In practice, P(R)is often chosen to be a classical or quantum mechanicalharmonic oscillator probability distribution function.62 Ingeneral, it could be a more complicated probability distribu-tion function reflecting the Boltzmann probabilities for therelevant R values. The R value corresponding to themaximum of the integrand in eq 2.9 is determined by abalance between the probability distribution function, whichtends to be greatest at the equilibrium R value, and thevibronic coupling, which tends to favor shorter R valuesbecause of the larger overlap between the reactant andproduct proton vibrational wave functions.63 Thus, typicallythe dominant contribution to the rate constant arises froman R value that is shorter than the equilibrium value.

Recently PCET rate constant expressions that include theeffects of solvent dynamics and interpolate between thegolden rule and solvent-controlled limits were derived.64

The golden rule limit is defined in terms of weak vibroniccoupling and fast solvent relaxation. As shown above, therate constant is proportional to the square of the vibroniccoupling and is independent of the solvent relaxation timein this limit. In contrast, the rate constant is independent ofthe vibronic coupling and increases as the solvent relaxationtime decreases in the solvent-controlled limit. The intercon-version between the solvent-controlled and golden rule limitscan be induced by altering the proton donor-acceptor modefrequency, the vibronic coupling, or the solvent relaxationtime. The kinetic isotope effect behaves differently in thesolvent-controlled and golden rule limits and thus providesa unique probe for characterizing the nature of PCETprocesses.64

These theoretical formulations have been extended to elec-trochemical PCET at metal-solution interfaces.52,64,65 In addi-

tion, a theory has been developed for studying the ultrafastdynamics of both homogeneous and interfacial photoinducedPCET reactions.66,67 These directions are important for thedevelopment of solar cells and other energy conversion devices.

2.3. Vibronic Coupling: Electronically Adiabaticand Nonadiabatic Proton Transfer

All of the rate constant expressions given above dependon the vibronic coupling Vµν, which is defined as theHamiltonian matrix element between the reactant and productelectron-proton vibronic wave functions. Concerted PCETreactions are usually vibronically nonadiabatic, i.e., Vµν ,kBT, and the quantum subsystem comprised of the electronsand transferring proton does not respond instantaneously tothe solvent motions. The rate constant expressions given inSection 2.2 are applicable in this vibronically nonadiabaticlimit. In the vibronically adiabatic limit, where Vµν . kBTand the electron-proton quantum subsystem responds instan-taneously to the solvent motions, the system moves on asingle electron-proton vibronic surface (i.e., the ground state),and the rate constant has the general form obtained withmultidimensional transition state theory.

Even for vibronically nonadiabatic reactions, however, theproton transfer can be electronically adiabatic, electronicallynonadiabatic, or in the intermediate regime. These regimes aredefined in terms of the relative time scales of the rearrangingelectrons and the transferring proton. In the electronicallyadiabatic limit, the electrons respond instantaneously to theproton motion, but in the electronically nonadiabatic limit, theelectronic response is slower than the proton tunneling.

A semiclassical expression for the vibronic coupling thatspans both of these limits has been derived.43 In thisformulation, the general vibronic coupling Vµν

(sc) is given by

where Vµν(ad) is the adiabatic vibronic coupling and the factor

κ is defined as

Here Γ(x) is the gamma-function and p is the protonadiabaticity parameter, defined as

where Vel is the electronic coupling between the diabaticelectronic states, Vt is the tunneling velocity of the proton atthe crossing point of the two proton potential energy curves,and |∆F| is the difference between the slopes of the protonpotential energy curves at the crossing point. The tunnelingvelocity Vt can be expressed in terms of the energy Vc atwhich the potential energy curves cross, the tunneling energyE, and the mass m of the proton: Vt ) [2(Vc - E)/m]1/2.

This formulation provides simplified expressions in theelectronically adiabatic and nonadiabatic limits. In theelectronically adiabatic limit, p . 1, κ ) 1, and the vibroniccoupling becomes Vµν

(ad), which is defined as half of thetunneling splitting on the electronically adiabatic groundstate. In the electronically nonadiabatic limit, p , 1, κ )(2πp)1/2, and the vibronic coupling becomes Vµν

(na) ) VelSµν,where Sµν is the overlap between the reactant and product

k ) ∫0

∞k(R) P(R) dR (2.9)

Vµν(sc) ) κVµν

(ad) (2.10)

κ ) √2πpep ln p-p

Γ(p + 1)(2.11)

p ) |Vel|2

p|∆F|Vt(2.12)


proton vibrational wave functions. These limits can beunderstood in terms of the relative time scales for the protontunneling and the electronic transition. The adiabaticityparameter is the ratio of the proton tunneling time τp andthe electronic transition time τe: p ) τp/τe. The protontunneling time is defined as the time spent by the tunnelingproton in the crossing region, τp ∼ Vel/|∆F|νt, and theelectronic transition time is defined as the time required tochange the electronic state, τe ∼ p/Vel. When the protontunneling time is much longer than the electronic transitiontime, the electronic states have enough time to mix com-pletely, and the proton transfer occurs on the electronicallyadiabatic ground state surface (i.e., the proton transfer iselectronically adiabatic). When the proton tunneling time ismuch less than the electronic transition time, the protontransfer is electronically nonadiabatic because the electronicstates no longer have enough time to mix completely duringthe proton tunneling process.

2.4. Hydrogen Atom TransferHydrogen atom transfer (HAT) may be viewed as a special

case of concerted PCET. In HAT, the electron and protontransfer between the same donor and acceptor (i.e., De ≡ Dp

and Ae ≡ Ap). Such reactions do not involve substantialcharge redistribution, and the solvent reorganization energyis small. Thus, these reactions may require an explicitdynamical treatment of intramolecular solute modes ratherthan the treatment described above in terms of collectivesolvent coordinates. In addition, the probability flux correla-tion function formalism may require special treatment suchas inclusion of coupling between the proton donor-acceptorvibrational mode and the solvent modes to avoid divergentintegrals.68

The criteria for distinguishing between HAT and concertedPCET have been debated in the literature. The distinctionbased on the same proton and electron donors and acceptorsis not rigorous because of the quantum mechanical behaviorand associated delocalization of the electron and proton. Thedistinction based on molecular orbital analysis32 is alsoproblematic due to the dependence on the level of theoryand representation. The degree of electronic nonadiabaticityfor the proton transfer reaction described in the previoussubsection provides a quantitative diagnostic for differentiat-ing between HAT and PCET.69 The PCET mechanismcorresponds to the electronically nonadiabatic limit (i.e., p, 1), and the HAT mechanism corresponds to the electroni-cally adiabatic limit (i.e., p . 1). This distinction betweenHAT and PCET has been illustrated by a comparisonbetween the phenoxyl/phenol and benzyl/toluene self-exchange reactions.69 Based on this type of analysis, thephenoxyl/phenol reaction, which involves electronicallynonadiabatic proton transfer, corresponds to PCET, whilethe benzyl/toluene reaction, which involves electronicallyadiabatic proton transfer, corresponds to HAT. This analysisis discussed in more detail below in section 6.1. The relationbetween PCET and HAT has also been explored in thecontext of the photosystem II oxygen-evolving complex.36

2.5. Kinetic Isotope EffectsA hallmark of a concerted PCET reaction is the observa-

tion of a kinetic isotope effect (KIE). Typically the magnitudeof the KIE is strongly influenced by the vibronic coupling.As discussed above, in the electronically nonadiabatic

regime, the vibronic coupling is the product of the electroniccoupling and the overlap between the reactant and productproton vibrational wave functions. Thus, for a given pair ofvibronic states, the rate constant is proportional to SH

2, thesquare of the hydrogen vibrational overlap for that pair ofstates at the equilibrium proton donor-acceptor distance, andthe KIE is proportional to the ratio SH

2/SD2. Due to the smaller

mass of hydrogen, the hydrogen overlap is typically sub-stantially larger than the deuterium overlap, leading to asignificant KIE. Moreover, the deuterium overlap SD de-creases faster than the hydrogen overlap SH as the protondonor-acceptor distance increases, so RD > RH. Thisparameter, which represents the distance dependence of thevibronic coupling defined in eq 2.3, influences the temper-ature dependence of the KIE in certain regimes.

The dependence of the KIE on system properties can beanalyzed for each specific rate constant expression givenabove. Neglecting the isotopic dependence of all quantitiesexcept the vibronic coupling and including only the groundreactant and product vibronic states, eq 2.2 for fixed protondonor-acceptor distance predicts that the KIE will beSH

2 /SD2 , which is independent of temperature. This ratio will

increase as the equilibrium proton donor-acceptor distanceincreases. In the low-frequency limit for the protondonor-acceptor mode (eq 2.7), the same approximations leadto the following expression:

indicating that the KIE will decrease with increasing tem-perature. In this regime, the magnitude of the KIE isdetermined mainly by the ratio of the squares of the hydrogenand deuterium overlaps, the proton donor-acceptor modefrequency, and the distance dependence of the vibroniccoupling. The temperature dependence of the KIE dependsstrongly on the proton donor-acceptor mode frequency inthis regime.61

When excited vibronic states contribute significantly tothe total reaction rate, these simple expressions for the KIEare not valid. The relative weightings of the contributionsfrom each pair of reactant and product vibronic states aredetermined by several competing factors, including temper-ature, and are usually different for hydrogen and deuterium.Moreover, altering only a single parameter without influenc-ing the other parameters is not always experimentallypossible. For example, increasing the proton donor-acceptorequilibrium distance often decreases the associated frequency.Thus, the dependence of the KIE on system properties, suchas proton donor-acceptor distance or temperature, is oftennot straightforward and may not be intuitive.61 In the high-frequency limit for the proton donor-acceptor mode, the KIEcould even increase with increasing temperature, as observedfor a quinol oxidation reaction.70

3. Sequential Electron and Proton Transfer. FourState Models

The reactions discussed in this section occur along theedges of the scheme shown in Figure 2.

3.1. Rate of a Sequential Coupled ReactionIn the literature the rate constant of a sequential electron and

proton transfer reaction is often written in the following form:

KIE ≈|SH|2

|SD|2exp -2kBT

MΩ2(RD

2 - RH2) (2.13)


where ke and kp are the individual electron and proton transferreaction rate constants. This expression assumes that thereaction occurs in sequence: first electron (proton) transferand then proton (electron) transfer. The reaction is completewhen both an electron and a proton are in their final states.The overall average time to complete two transfers is thesum of those for each of the consecutive steps, as stated inthe above formula. The overall rate is limited by the sloweststep.

A tacit assumption made in eq 3.1 is that the first electron/proton step in the reaction is irreversible; that is, if a protonjumps first to the final state, it stays there long enough foran electron to make its own transfer. (The coupling betweenan electron and a proton is such that electron transfer willmost likely occur when the proton is in the final state.) Thiscan be the case when, for example, the first step of thereaction is downhill in free energy or if the second step isfast enough to beat the reverse reaction of the first step.

This is not the most common situation in enzymaticreactions.8,27,28 Typically, each of the separate ET and PTreactions is uphill in energy. As a result, the lifetime of theintermediate excited state after the first transfer may not belong enough for the second reaction to be complete. Such isthe case, for example, in the photosynthetic reaction center(PRC) for the quinone reduction reaction.8,26

The simplest type of such reactions can be described interms of the scheme shown in Figure 2. The states aredenoted according to the state of the acceptor group:(oxidized/reduced) and (unprotonated/protonated). Thus, theinitial state is OU and the final state is RP. There are twointermediate states, RU and OP. Electron transfer can occurbefore or after protonation, with corresponding rate constantske

U and keP. Likewise, the proton transfer can occur before or

after reduction, with rate constants kpO and kp

R. The reversereaction for each of these steps will be denoted as ke- andkp- with the appropriate superscript. The rate constant ofsuch a reaction is

where

and

The reaction can proceed along two channels: ET first andPT second (ET/PT), with rate constant kep, or PT first andET second (PT/ET), with rate constant kpe, as shown in Figure5. The overall rate constant is the sum of the rate constantsof the two channels. Each rate constant is a product of therate constant for the formation of the activated complex (forET without PT, the rate constant is ke

U; for PT without ET,the rate constant is kp

O), and the probability that the reactionwill proceed from the intermediate activated state in thedirection of the final state RP, instead of the initial state viaback reaction.

Consider the ET/PT channel. If the proton is “fast”, sothat the proton transfer reaction can be completed while theelectron is in the activated state, kp

R . ke-U , the reaction rate

constant is given by the rate constant for formation of theactivated complex,

If the proton is “slow” (kpR , ke-

U ), then the lifetime of theactivated complex, τe- ) 1/ke-

U , is shorter than the time scaleof the PT reaction, τp ) 1/kp

R, and several electron transfer“attempts” are needed to complete the PT reaction. Thenumber of such attempts is τp/τe-; hence, the overall timeneeded to complete the reaction is τep ) τe(τp/τe-), where τe

) 1/keU. Here we assume τe . τe-. Then the rate constant is

given by

where Ke is the equilibrium constant for electron transfer(without proton transfer):

where ∆GeU is the free energy of the electron transfer reaction

without proton transfer. It will be assumed that ET is uphill,so ∆Ge

U > 0 and -∆GeU is a negative driving force. The

equilibrium constant Ke can be interpreted as the populationof the activated (reduced) state, from which the protontransfer occurs with rate constant kp

R. The overall rate constantin this limit is given by eq 3.6.

Similarly, for the PT/ET channel, if the electron is “fast,”the rate constant is

If the electron is “slow,” the rate constant is

where

and -∆GpO is the driving force of the proton transfer reaction

without electron transfer. The equilibrium constant Kp canbe interpreted as the population of the activated (protonated)state, from which electron transfer occurs with rate ke

P. Theoverall rate constant in this limit is given by eq 3.9.

In general, either the ET/PT or PT/ET channel candominate, depending on the rate of the individual reactions.For example, when the electron is slow/proton fast

1/k ) 1/ke + 1/kp (3.1)

k ) kep + kpe (3.2)

kep ) keU kp

R

kpR + ke-

U(3.3)

kpe ) kpO ke

P

keP + kp-

O(3.4)

Figure 5. Free energies of sequential ET/PT and PT/ET reactionsshown in Figure 2. Figure reprinted with permission from ref 42.Copyright 2003 World Scientific.

kep ) keU (3.5)

kep ) kpRKe (3.6)

Ke ) keU/ke-

U ) τe-/τe ) exp(-∆GeU/kBT) (3.7)

kpe ) kpO (3.8)

kpe ) kePKp (3.9)

Kp ) kpO/kp-

O ) τp-/τp ) exp(-∆GpO/kBT)

(3.10)


and in the opposite case of a slow proton/fast electron

When both the electron and the proton are fast

Consider the slow electron case, eq 3.11. Although it isexpected that ke

U , keP, for an uphill proton reaction, Kp ,

1, and hence, the relative magnitude of the two terms willbe determined by the energetics of the reaction (or how fastthe proton is). If the first term dominates, one can say thatan electron jumps first and “pulls out” a proton. If the secondterm dominates, a proton goes first and, by frequent but shortvisits of the acceptor site, it eventually pulls out an electronfrom its donor state. A similar interpretation can be givenfor the slow proton case described by eq 3.12. In the thirdcase, eq 3.13, the dominant channel is defined by the relativerates of PT (first term) and ET (second term) uphill reactions.A more detailed classification of all possible cases isdescribed in ref 42.

The differences in the rate constants for ET to protonatedand unprotonated acceptor, ke

P and keU, and in the rate constants

for PT to oxidized and reduced acceptor, kpO and kp

R, arerelated to the interaction energy between an electron and aproton in the final state. For reactions in which the electronand proton transfer in the same direction, the positive drivingforce for the overall reaction is due to the favorableelectrostatic interaction between an electron and a proton inthe final state. Symbolically, this stabilizing energy will bedescribed as

where R0 is the “effective distance” between an electron anda proton in the final state. This qualitative description is validonly when the electron and proton transfer in the samedirection. The free energy of the coupled reaction is then

The driving force for electron transfer with a protonatedacceptor is

Similarly, the driving force for proton transfer with a reducedacceptor is

The variations in ∆Gp (via pKa) and ∆Ge can be used inexperimental studies to probe the nature of the coupledreaction, as in refs 8, 26, and 27. Note that the change inpKa value of the proton acceptor site, which alters the valueof ∆Gp, does not affect the driving force of electron transfer,unless the change in pKa is electrostatic in nature, in whichcase it directly affects the reduction potential of the electronacceptor site. (The independence of the electron transferdriving force with respect to the pKa value of the protonacceptor may also break down when both electron and proton

acceptors are on the same molecule, as, for example, inquinone. The magnitude of such variations, or even their sign,is difficult to predict without detailed ab initio calculations.)

3.2. pH and Isotope EffectsAbove we assumed that a proton is available in the donor

state. In fact, protonation of the donor site in a protein willdepend on the pH of the medium. The coupled electron andproton transfer reaction will take place only if the protondonor site is protonated. Hence all rate constants discussedabove will be proportional to the protonation fraction:

where pKa refers to the proton donor site. For an effectiveelectron/proton coupling, the pKa’s of the protonation sitesin the enzyme should depend on the redox state. In the rateconstant expressions above we assumed that the pH in thesystem is such that the pKa’s of the donor (don) and acceptor(acc) sites satisfy

In other words, initially (in the OU state, Figure 2) a protonis available at the proton donor site and the proton acceptorsite is empty. We do not consider here the exchange kineticsbetween the medium and the proton donor and acceptor sites,which may introduce additional complications into thekinetics of the coupled reaction. For a discussion of suchcases, see ref 71.

The pKa values of those protonation sites that depend onthe redox state of the enzyme are said to be redox coupled.We assume, for example, that when an electron is placed onits acceptor site, the proton equilibrium is shifted toward itsfinal state. In this case, the pKa value of the proton acceptorsite in the reduced form is higher than that of the donor site:

After electron transfer, the pKa value of the proton donorsite, pKdon

R , is different from that before electron transfer,pKdon

O . The relative magnitude of the former with respect tothe pH of the medium determines whether an electrontransfer reaction will induce an uptake of a proton from themedium or not. If pKdon

R > pH, then one additional protonwill be uptaken (i.e., after ET, both proton donor and protonacceptor sites are protonated). If pKdon

R < pH, no extra protonswill be taken from the medium upon electron transfer, andthe proton transfer is intraprotein. In general, the pKa valuesof the protonation sites in a protein not only are redoxdependent but also depend on the protonation states of othersites due to their Coulomb interactions. Thus, the kineticsof proton redistribution associated with a change in the redoxstate of the enzyme can be very complicated (see discussionof the pumping mechanisms in section 6.3).

Generally, reactions involving protons have either large,or small but observable kinetic isotope effects.30,31,72-74 Thisis not always the case with the coupled reactions. It shouldbe noticed that in both the ET/PT and PT/ET channels, whenthe electron is slow and the proton is fast, the rate-limitingstep is electron transfer. The reaction in this case does nothave a significant KIE, as seen from eqs 3.5 and 3.9.

k ) keU + ke

PKp (3.11)

k ) kpO + kp

RKe (3.12)

k ) kpO + ke

U (3.13)

∆Eep ) e2/R0 (3.14)

∆Gep ) ∆GeU + ∆Gp

O - e2/R0 (3.15)

∆GeP ) ∆Ge

U - e2/R0 ) ∆Gep - ∆GpO (3.16)

∆GpR ) ∆Gp

O - e2/R0 ) ∆Gep - ∆GeU (3.17)

R ) 1

1 + 10pH-pKa(3.18)

pKaccO < pH < pKdon

O (3.19)

pKaccR > pKdon

R (3.20)


For the quinone reaction in the PRC, for example,according to our classification, we have the case of a slowelectron, in which a fast proton goes first and makes severaltransitions to the activated intermediate state, establishesquasi-equilibrium with this state, and gradually pulls out anelectron. The rate constant is then Kpke

P. There should be nostrong KIE, other than the relatively small equilibriumisotope effect, and the overall observed rate constant is ordersof magnitude (by a factor of Kp) slower than that of pureelectron transfer to a protonated acceptor, ke

P.To measure ke

P, one can achieve the protonated state ofthe acceptor site by lowering the pH. However, one needsto remember that in a real protein at low pH several othersites may become protonated as well. The driving force forelectron transfer in this case may not be the same as thatassumed in ke

P. Alternatively, at fixed pH one can increaseprotonation of the acceptor site by increasing its pKa value.Again, one needs to make sure that the driving force for ke

P

(i.e., reduction potential of the electron acceptor) will notbe affected by the changes in pKa.

3.3. PT vs ET: Which is Faster?Typically, electron transfer reactions in proteins are

nonadiabatic. Therefore, for similar activation energies, thePT rate constant is expected to be much larger than the ETrate constant. In the simplest case, the rate constant of aproton transfer reaction would be given by

where Ea is the activation (free) energy and ν0 is a typicalfrequency of nuclear motions that result in attempts toovercome the activation barrier. This frequency is in therange 1012 to 1013 s-1. (Here PT is assumed to be adiabatic.For a discussion of nonadiabatic PT, see ref 43 and section2.3.) For nonadiabatic electron transfer, the rate constant isroughly

where PLZ is the Landau-Zener (LZ) parameter for electrontransfer, which for nonadiabatic reactions is less than unity.The LZ parameter for an ET reaction is

where λ is the reorganization energy, which is typically ofthe order 0.5 eV, VDA is the electronic coupling, which istypically less than 1 cm-1 in proteins,10,11,75 and ε is the rateof energy fluctuations. Hence, typically

On the basis of the above estimates, and assuming thesame activation energies, one could conclude that the fastproton/slow electron case would be the most likely one. Forexample, such is the case for the quinone reduction reactionin PRC.8,26-28 However, proton transfer reactions in proteinsare generally complex and are likely to include manyintermediates. In this case, the free energy of activation forPT may be significantly higher than that of ET, and theopposite case of slow proton/fast electron will be realized.For example, in cytochrome oxidase, the ET reactions occur

on the time scale of 10 µs or faster (see section 5.3), and PTis believed to be on the time scale of 100 µs or slower.76 Inaddition, the proton transfer is often electronically nonadia-batic, as discussed in section 2.3.

3.4. Statistics of Transitions in a Single MoleculeIn a single protein molecule, the transition between the

initial and the final state of each of the individual reactionsis a random event (i.e., follows a Poisson process). For atypical two-state reaction, the system would jump betweeninitial and final states at random, as shown in Figure 6.

The average times that the system will spend in the initialand final states are related as

where ∆Gx is the free energy of the electron or protontransfer reaction. For an uphill reaction, the system stayssome long time in the initial state and then at random makesa quick transition to the activated state, stays there a shortperiod of time, and goes back. Such random jumps repeatin time. The transition time itself between the initial and finalstates is considered here to be much smaller than both τx

and τx-. In this case, the four state model described in eq2.1 is applicable.

The probability (per unit time) that such a randomtransition will occur at time t is

The average lifetime, as seen from above, is τ ) k-1.In the ET/PT channel, when an electron is in the activated

state, and the proton is fast enough, the electrostatic attractioncan pull out the proton from the donor. The proton transferwill occur as a random jump within time τp < τe-.Alternatively, when the proton is slow, several electron jumpattempts are required to complete the reaction; that is, timeis required to find a coincidence that the proton will makeits jump when the electron is already in the final state. Thesame picture is applicable to the PT/ET channel.

In proteins, both electrons and protons are typicallytransferred over long distances (tens of angstroms). Themechanisms of transfer, however, are completely different.Electrons are light and can tunnel over long distances inproteins. (In addition, hopping mechanisms of long-distanceelectron transport are also possible.) Protons are heavy andcan tunnel only over short distances, less than one angstrom,

kp ) k0 ) ν0 exp(-Ea/kBT) (3.21)

ke ) k0PLZ (3.22)

PLZ ) 2πp

VDA2

ε) 2πp

VDA2

ν0√2λkBT(3.23)

PLZ ) 10-3 to 10-4 (3.24)

Figure 6. Random transitions of charge between donor andacceptor states in an ET or PT reaction of a single molecule assumedin the four-state model. Figure reprinted with permission from ref42. Copyright 2003 World Scientific.

τx-/τx ) kx/kx- ) Kx ) exp(-∆Gx/kBT)(3.25)

P(t) ) ke-kt (3.26)


and many steps are required for the net transport of a protonover a long distance. The carrier water molecules are alsoneeded to pass protons along the chain via the Grotthussmechanism, as depicted in Figure 7. If water molecules arealready assembled in a hydrogen bonded chain (proton wire),the transfer could be as fast as in liquid water, i.e., transferbetween two water molecules in 1 ps, or significantly slower,depending on the state of the proton conducting wire.77-83

While for an electron transfer the time of a single tunnelingjump is in the subfemtosecond range,43,84 and the picture ofinstantaneous transition between the two states is typicallyvalid, for protons such an idealization may not always becorrect. The protons are moving via many intermediate steps,and a more accurate picture should explicitly include arandom walk along the proton conducting channel.71

3.5. ET Induced by PTHere we consider one interesting possibility that occurs

in the PT/ET channel. Namely, when electrons are slow, keP

, kp-O , several proton transitions (order of kp-

O /keP) to the

activated state OP are required to pull out an electron fromthe donor state. As the proton moves along a proton-conducting channel, such as the one depicted in Figure 7,the energy of the electron acceptor state continuously changesand at some point crosses that of the donor state. At thispoint, the usual Landau-Zener electronic transition mayoccur. Thus, every time the proton moves back and forthalong the channel, the energy levels cross, and with somenonzero probability, electron transfer induced by the protonmay occur. (In a similar way, motion of other charges ofthe protein medium may induce the usual electron transfer.)The question is whether this additional channel of coupledelectron and proton transfer can be competitive with thesequential channel that we considered so far, for which therate constant is ke

PKp.In this case, the proton motion can be considered as part

of the reorganization of the medium for electron transfer.The proton coordinate, however, is quite different from thoseof other fluctuating charges in the system. The latter aredescribed as a set of harmonic oscillators in the usual modelof electron transfer.85 This description is obviously notapplicable to a proton moving along the proton conductingchannel. The complete model for such a system will bediscussed in section 5; here we consider a simple case ofnonadiabatic transitions.

As the two energy levels cross, the Landau-Zenerprobability of an electronic transition is given by eq 3.23.The velocity of the level crossing, ε, depends on the protonvelocity at the moment when the LZ level crossing occurs.In this case, the overall rate constant of the reaction is given

by the LZ probability times the number of instances, persecond, that such a crossing will occur. Thus, the rateconstant for proton-induced ET is

The factor of 2 accounts for the two level crossings as theproton moves up and down the channel during a singleactivation event. This rate is to be compared with

To estimate the ratio of the two rate constants,

we write the rate constant of the electron transfer reactionin the standard form:

where λ is the reorganization energy and EaeP ) (∆Ge

P + λ)2/4λ. The rate constant of the proton back transfer reaction(from the oxidized acceptor) is written in the form

where Eap-O is the activation free energy for the reaction and

τ0-1 is a characteristic frequency with which the proton moves.

(It is difficult to ascribe a precise value to the prefactor;roughly, τ0 is the characteristic time for a proton to transferfrom one water molecule to another at the transition state,which is in the range 10-12 to 10-13 s.)

On the other hand, the velocity of the crossing of electroniclevels is related to τ0 as follows:

where ∆ε is the shift of electronic energy levels as the protonjumps from one water molecule to another in the channel.For reactions in which the electron and proton transfer inthe same direction, the shift of energy has an electrostaticnature and can be estimated as

Figure 7. Qualitative scheme of proton transfer via a chain of hydrogen bonds in a proton conducting channel. The proton transfer canoccur either as a random walk of a localized charge along the chain or as a single transition of a delocalized soliton-like object, in whichseveral protons along the chain are transferred simultaneously. Figure reprinted with permission from ref 42. Copyright 2003 World Scientific.

kPIET ) 2kpOPLZ (3.27)

kpe ) kePKp ) kp

O keP

kp-O

(3.28)

kPIET/kpe ) 2PLZkp-O /ke

P (3.29)

keP ) 2π

p

|VDA|2

√4πλkBTe-Eae

P /kBT (3.30)

kp-O ) τ0

-1 e-Eap-O /kBT (3.31)

ε ) ∆ε/τ0 (3.32)

∆ε ∼ e2

R0

∆RR0

(3.33)


where R0 is the typical distance that characterizes thestabilizing electrostatic interaction between an electron anda proton, and ∆R is the distance that a proton travels in asingle jump, i.e., on the order of one angstrom. (Here we donot consider protein dielectric effects, assuming instead thatthey are accounted for in the effective distance.) Realisticnumbers for the above order of magnitude estimate wouldbe e2/R0 ∼ 0.5 eV, ∆R/R0 ∼ 0.1, and, therefore, ∆ε ∼ 500cm-1. This value should be compared with (4πλkBT)1/2.Typically, in proteins λ ∼ 0.5 eV and, therefore, ∆ε ∼(4πλkBT)1/2. We conclude that

i.e., the ratio depends on the relative magnitudes of theactivation free energy barriers for the proton back-reactionin the oxidized state (rate constant kp-

O ) and for electrontransfer in the protonated state (rate constant ke

P). Bothreactions are downhill, but here we are assuming that thereare reaction barriers.

We conclude that the proton random motions along theproton conducting channel can induce electron transfer.When Eap-

O < EaeP , this type of transition can be the primary

mechanism for the coupled reaction. The opposite case of ausual sequential reaction occurs when Eap-

O > EaeP .

3.6. Concerted vs Sequential Proton TransportProton transport over long distances is a complicated

process, which involves the dynamics of the watermolecules, along which the protons move, and the protonsthemselves.77,78,81,86-93 Moreover, protein and membranesurfaces can significantly modify proton diffusion mech-anisms in biological systems.94-97 In contrast to electrons,which in principle can tunnel in proteins in any directionover distances up to 20-30 Å, proton transfer requiresspecial “wiring” of donor and acceptor sites by chains ofhydrogen bonds.98 This wiring occurs in proton conductingchannels, which require special structural organization of theprotein. A typical conducting channel consists of a chain ofwater molecules and possibly a few intermediate protonatableresidues. The intermediate protonatable sites would beconnected by a few (typically one to five) water molecules.Three to five water molecules can provide coupling over adistance of 10 Å, as depicted in Figure 7. If the state ofhydrogen bonding in the channel were the same as that inliquid water, protons would randomly jump between watermolecules at a rate of one jump per picosecond, which couldgive diffusion coefficients as high as 10-4 cm2/s. In proteinchannels, the dynamics of water molecules is different fromthat in the liquid state, and the corresponding rates of protontransport can be much slower than in the bulk.

Depending on the strength of hydrogen bonds along theconducting wire, proton transport can either occur as adelocalized soliton or as a localized (to a single hydrogenbond) charge.83 In the latter case, the transfer occurs as arandom walk, or through diffusion of a localized charge(positive or negative) along the wire, a process whichinvolves many activated steps. In the former case, thetransition can be viewed as an activated single step processin which several protons shift coherently along the wire.Strictly speaking, the concerted PCET rate constant expres-sions presented in section 2 are applicable only to the first

type of proton transfer, although the general theoreticalformulation is easily extended to the second type of protontransfer using additional diabatic states. The diffusion typeprocess will be discussed in section 5.

If the proton transport occurs via a delocalized soliton,then the wire should be formed first. The formation of thewire is itself an activated process.99 There is also a finitelifetime associated with the wire. (If the wire were athermodynamically stable structure, there would be noreorientations of water molecules required for Grotthusstransfer.77,86,88,89,92) Proton transfer along the channel istherefore a “gated” sequential reaction. The rate constant ofproton transfer along a proton conducting channel can thenbe written as83

where kw+ is the rate constant for formation of the wire, τw is

its lifetime, and kPT is the rate constant of proton transferalong the assembled wire. The latter itself is a complexactivated process. If the PT transition occurs via a delocalizedsoliton, then the transition itself is a fast process, on the orderof one period of nuclear vibration (in which all protons alongthe wire shift in a concerted way), while most of the reactiontime involves the system “waiting” until the necessaryreorganization of the medium and the wire itself occurs. Theclassical (or adiabatic) rate constant of such a process is givenby the generic expression

where τ0 ∼ 10-12 to 10-13 s and Eas is the activation free

energy for proton transfer.On the other hand, if proton transport along the wire occurs

as a random walk of a localized charge, then the aboveformula for kp is not applicable when proton diffusion is tooslow, and kPT < 1/τw. In this case, the proton will never reachthe acceptor during the lifetime of a connected wire.Moreover, in the case of slow diffusion of a localized charge,the formation of a continuous wire along the whole channelis not necessary; thus, the above formalism is not applicable.In this case, proton transport can be described simply as arandom walk along the channel, with some effective diffusioncoefficient, and an energy profile along the channel, specificfor a given redox state of the enzyme.

On the basis of the structures of PRC, bc1 complex, andcytochrome oxidase, we cannot identify a unique organiza-tion of a proton conducting channel, and different scenariosof proton transport seem to be possible. It appears that themost likely type of long-distance proton transfer (or trans-location), however, is a random walk along a chain ofintermediate protonatable sites, with quick delocalized transi-tions between them. Each individual transition would bedescribed by eq 3.35. The transitions between intermediatesites should be quick because it is energetically costly tohave a charge on a water molecule (H3O+ or OH-) in a lowdielectric protein medium.

4. Concerted vs Sequential PCETAs discussed above, sequential and concerted PCET

reactions, respectively, can be defined in terms of thepresence or lack of a stable intermediate arising from single

kPIET/kpe ∼ exp(EaeP - Eap-

O

kBT ) (3.34)

kp ) kw+ kPT

kPT + (1/τw)(3.35)

kPTs ) 1

τ0e-Ea

s/kBT (3.36)


electron or proton transfer. From the experimental perspec-tive, a stable intermediate would be defined in terms of aspecified lifetime, but the ability to detect such an intermedi-ate species may depend on the experimental apparatus. Inthe context of computing potential energy surfaces, a stableintermediate would be characterized by a minimum on thepotential energy surface. In the context of Marcus theoryand the four diabatic states defined in eq 2.1, a concertedmechanism corresponds to OUf RP, along the diagonal ofthe scheme in Figure 2, while sequential mechanismscorrespond to OU f OP f RP or OU f RU f RP, alongthe edges of the scheme in Figure 2. In this framework, themechanism is determined mainly by the relative energies ofthe four diabatic states and the couplings between them. APCET reaction is typically concerted when the energies ofthe intermediates OP and RU are significantly higher thanthe energies of the PCET reactant and product OU and RP.

While the detection of a stable intermediate provides proofof a sequential PCET mechanism, proving a concerted PCETmechanism is more challenging. The most convincingevidence can be obtained from the relative energetics of thecharge transfer reactions. In some cases, the experimentalmeasurement of redox potentials and pKa values indicatesthat the single electron and single proton transfer reactionsare thermodynamically unfavorable, with reaction free ener-gies greater than ∼30 kcal/mol, whereas the combinedelectron-proton transfer is more thermodynamically favor-able and possibly even exoergic.34,63,70 The measurement ofan unusually large KIE (i.e., greater than ∼10) is consistentwith a concerted PCET mechanism, but not all concertedPCET reactions exhibit such large KIEs. Thus, a moderateKIE does not distinguish between the sequential andconcerted mechanisms.

In some situations, the distinction between concerted andsequential mechanisms is not well-defined. One particularexample is the ultrafast dynamics in photoinduced PCETreactions. The experimental detection of intermediates onthe femtosecond time scale is challenging. Typically suchintermediates will not be thermally equilibrated, since thesephotoinduced reactions are inherently nonequilibrium pro-cesses. From the perspective of propagating nonadiabaticmolecular dynamics trajectories on the excited state vibronicsurfaces, many different pathways that encompass multiplemechanisms may be explored.100 Thus, the terminology ofconcerted and sequential mechanisms may not be applicableto these types of processes.

5. Kinetics of Electron Transfer ReactionsCoupled to Proton Translocation

The four-state model discussed above does not considerintermediate proton states in a proton-conducting channel.This approximation can be justified based on the fact thatthe proton spends most of the time on either the donor orthe acceptor site, and very little time in the channel itself.While for a tunneling electron the two-state representationis typically adequate, for a proton the neglect of intermediatestates is not always possible. Here we discuss a model thatincludes intermediate proton states in the channel explicitly.42

5.1. Coordinates and EnergeticsLet X be a one-dimensional proton coordinate, representing

the center of positive charge along the channel, that variesfrom 0 to L. Let R represent the rest of the nuclear

coordinates of the protein medium relevant to electrontransfer. (Note that this R is not to be confused with theproton donor-acceptor distance R in section 2.) In thesemiclassical description, the reaction coordinate for electrontransfer is the difference between the energies of the acceptorand donor electronic states, ∆da, which depends on thecoordinates of the medium. In our case,

The dynamics of both X and R results in variations in ∆da.Whenever energy levels cross, ∆da ) 0, the system canundergo an electronic transition with some nonzero prob-ability. To a good approximation, the coordinates X and Rcan be treated as independent; therefore, our model is a two-dimensional generalization of the usual electron transfertheory. This two-dimensional model is reminiscent of thatof Sumi and Marcus for solvent-controlled ET reactions.101,102

Various models for Ea(R,X) and Ed(R,X) can be developed.Given Ea(R,X) and Ed(R,X), the free energies of the proton

in the donor and acceptor electronic states are

Here R represents a set of appropriately chosen dimensionlesscoordinates. (Note that the functional dependence of G(X)would not depend on units of R, except for an unimportantconstant.) These free energies determine the equilibriumprobability distributions of X in both electronic states,

Figure 8. Qualitative free energy profiles along the protonconducting channel. The initial and final states of the proton (i.e.,the centers of positive charge for these two states) are at X ) 0and X ) L, respectively. The two curves Gp

d(X) and Gpa(X)

correspond to electron oxidized (O) and reduced (R) states,respectively. ∆Ge° and ∆Gp° are the free energies of electron andproton (uphill) transfer, respectively, in the uncoupled reaction. Forthe slow proton diffusion case, the effective potential is the lowerof the two curves for a given X. ∆Gp

† is the activation free energyfor such a reaction. Figure reprinted with permission from ref 42.Copyright 2003 World Scientific.

∆da(R,X) ) Ea(R,X) - Ed(R,X) (5.1)

Gpd(X) ) -kBT ln(∫ exp(-Ed(R,X)/kBT) dR)

(5.2)

Gpa(X) ) -kBT ln(∫ exp(-Ea(R,X)/kBT) dR)

(5.3)


where Zpd/a is the partition function for the proton. Qualita-

tively, the proton free energy profiles along the channel inthe two electronic states would typically resemble theschematic diagram in Figure 8.

5.2. Reaction DynamicsIn the course of thermal dynamics, the coordinates R and

X and the corresponding energy difference ∆da(R,X) undergorandom variations. The randomness of R is due to its complexnature: R represents many nuclear coordinates, point charges,dipoles, and so forth, in the protein medium. The randomnessof X is due to a diffusion-like, random walk motion of theprotons. Whenever the variables X and R assume values suchthat ∆da(R,X) ) 0, the energy levels of the donor and acceptorelectronic states cross, and a transition between these statescan occur. The probability of the transition is given by theLandau-Zener formula.

For simplicity, we will assume that the energy mismatchbetween the two electonic states, ∆da(R,X), is such that thecontribution of the proton to the energy shift is additive:

Thus for a given position of the proton X, the R-coordinate’sfluctuations can cause the Landau-Zener transition, andhence induce electron transfer; the rate of such a reactionwill depend on the position of the proton, X. Alternatively,for a given configuration of R, the proton motion in thechannel can cause the electron transfer.

If one assumes that the electronic coupling TDA issufficiently small (which is the case for long-distance ET),the reaction can be viewed as a usual electron transferreaction whose driving force, ∆Ge°, depends on the positionof the proton in the proton conduction channel, X. The rateconstant for such a reaction is

Various simple models of ∆Ge°(X) can be developed,42 andthe overall rate constant of such reactions will be determinedby the distribution function of the proton position X in theproton conducting channel. This distribution should beconsidered as changing in a self-consistent manner togetherwith the progress of the reaction, as in the Sumi-Marcustype models of electron transfer; see, for example, refs 103and 104. Below we consider a few simple limiting cases andshow how the approach extends some known results for therate constants.

5.3. Rate Constants in Limiting CasesIn the initial state of the system, the electron is on the

donor (d), and the proton is in the lower end of the channel(X ) 0). In this state, the electron acceptor is oxidized (O)and the proton acceptor site is unprotonated (U), as illustratedby Figures 1 and 2. In the final state, the electron is on theacceptor (a) and the proton is in the upper part of the channel(X ) L). In this state, the electron acceptor is reduced (R),and the proton acceptor site is protonated (P). The population

of the initial state is described by Ppd(X) for X in the region

close to X ) 0, where the proton donor site is located; thepopulation of the final state is described by Pp

a(X) with X inthe region X ) L, where the proton acceptor is located.

The kinetics of both Ppd(X) and Pp

a(X) is due to twocontributions: diffusion along the channel and electronictransitions. These two contributions correspond to the twosequential processes PT and ET in the four-state modeldepicted in Figure 2. Obviously, multiple transitions suchas ET/PT-PT/ET-ET/PT-... are now possible. We nowconsider the limiting cases that correspond to the k ) kp

O, k) Kekp

R, k ) keU, and k ) Kpke

P cases in the four-state model.

5.3.1. Fast PT, Slow ET

Suppose that, in both electronic states, the proton transferin the channel is very fast, so that the proton quasi-equilibrium distributions are quickly established and main-tained during the reaction. Initially, Pp

a(X) ) 0. The rateconstant is given by

There is a point along the proton coordinate X where theintegrand in the above expression is maximum. This pointgives the maximum contribution to the rate constant and canbe considered a kind of transition state along the coordinateX (in the sense that one can consider the reaction occurringonly at that position of the proton along X).

First, we will explicitly show that both sequential andconcerted ET and PT are present in the above expression.For example, let the initial proton distribution in the channelbe such that the corresponding distribution in energymismatch between the two electronic states (∆p, see eq 5.5)is a generic Gaussian function written as

This form of distribution would be obtained if the protondynamics in the channel were similar to those of othernuclear degrees of freedom. The proton in this case couldbe treated as one of the medium degrees of freedom R. Interms of Pp, the rate constant is written as

where we assume ∆Ge° is evaluated at X ) 0, and the notationindicates that ke(X) is evaluated by replacing ∆Ge°(X) with∆Ge° + ∆p. Integration in the above formula results in astandard ET rate constant expression with λ ) λe + λp and∆G° ) ∆Ge° + ∆Gp

R ) ∆Gep (cf. section 3.1). Therefore,the proton motion in this case induces electron transfer inthe same way as the other coordinates of the medium. Thetransfer of the electron occurs in a concerted manner withthe motion of the proton.

Suppose now that the proton in the channel has two stablestates: X ) 0 (proton donor) and X ) L (proton acceptor).The energies of these states are different by ∆Gp°, as depictedin Figure 8. In this case, the equilibrium distribution for theproton is approximately

Ppd/a(X) ) exp(-Gp

d/a(X)/kBT)/Zpd/a (5.4)

∆da(R, X) ) ∆e(R) + ∆p(X) (5.5)

ke(X) ) 2πp

|TDA|2

√4πλekBTexp[-(∆Ge°(X) + λe)

2/4λekBT]

(5.6)

k ) ∫ ke(X) Ppd(X) dX (5.7)

Pp(∆p) )1

√4πλpkBTexp[-(∆Gp

R + λp - ∆p)2/4λpkBT]

(5.8)

k ) ∫ ke(∆Ge° + ∆p)Pp(∆p) d∆p (5.9)

Ppd(X) ) δ(X) + Kpδ(X - L) (5.10)


where Kp is the proton equilibrium constant. Then eq 5.7 gives

The first term is the rate constant for ET to unprotonatedacceptor, and the second term is proportional to the rateconstant for ET to protonated acceptor. Depending on theenergetics of the system, either the first or the second termcan dominate. This is identical to what we had in section 3for the four-state model. The use of a distribution functionis more realistic for the proton and can give more accurateresults than those obtained within the four-state model.

5.3.2. Fast ET, Slow PT: Effective Proton Potential

We now consider the opposite case of fast ET and slowPT. In this case, the rate-limiting process is proton diffusionalong the channel. In the four-state model, the slow protontransfer can occur in two ways. The proton can jump uphillto its final state first, and while it stays there, the fast electronis quickly transferred downhill to its protonated acceptor.The overall rate for this process is kp

O. In the secondmechanism, the electron makes several uphill jumps to itsunprotonated acceptor and via several attempts pulls out theproton. The overall rate here is Kekp

R. In the channel model,the transfer process is qualitatively similar; however, thedetails are different.

In the case of two electronic states, the free energy profilesalong the channel are Gp

d(X) and Gpa(X), as depicted in Figure

8. Note that the free energy along X will be referred to asthe proton potential, since the change in entropy along X isnegligible. In the course of the reaction, when the electronjumps back and forth between the donor and acceptor sites,the proton is moving in a time-dependent potential, whichswitches between Gp

d(X) and Gpa(X). When electron transfer

is fast, for every position of the proton in the channel,electronic equilibrium is quickly established. The averagetime that the electron remains on the donor, during whichthe proton potential is Gp

d(X), is τd ) 1/ke(X). The acceptortime, during which the potential is Gp

a(X), is τa ) τdKe(X),where Ke is the local equilibrium constant for electrontransfer,

Since the donor and acceptor times are not equal, theeffective proton potential is not simply the average of Gp

d

and Gpa. The potential and dynamics in the channel can be

described in terms of the effective potential Gp(X) for theproton in the channel:

Qualitatively,

as shown in Figure 8.The overall coupled reaction occurs as follows. If electron

transfer is fast, or proton diffusion is slow, for every positionof the proton in the channel, electronic equilibrium isestablished. Initially, for small X, Gp

d(X) < Gpa(X), the electron

equilibrium is shifted toward the donor state, and the potential

on which the diffusion occurs is Gpd(X). This diffusion is

uphill. When the proton reaches the transition point, X†, atwhich roughly Gp

d(X†) ) Gpa(X†), the electron equilibrium

shifts toward the acceptor state, and further diffusion for X> X† occurs on the Gp

a(X) potential. For X > X†, the protondiffusion is downhill. Since the diffusion is much sloweruphill than downhill, the overall proton diffusion time alongthe channel is equal to that of reaching the transition pointX†. The latter is the rate-limiting step for the overall ET/PTreaction. The following calculation gives an estimate for therate constant of such a reaction.

5.3.3. Fast ET, Slow PT: Reaction Rate Constant

The reaction occurs via one-dimensional diffusion over abarrier. In this case, the rate constant k is

where D† is the proton diffusion coefficient at the transitionstate point X†, ∆Gp

† is the activation free energy, and L0

and L† are partition functions (defined as integrals ofexp(-Gp(X)/kBT) over X and having units of length) ofthe proton donor site and the barrier site, respectively.The two lengths L0 and L† characterize the thermal widthsat the donor site and at the barrier, respectively. The pre-exponential factor in the rate constant expression, there-fore, is the inverse time for proton diffusion over acharacteristic length L ) (L0L†)1/2.

The simplified picture considered above assumes classicaldiffusion of a proton along the proton-conducting channel.It is clear, however, that the form of the expression willremain the same with a more detailed quantum descriptionof proton diffusion. By dimensional analysis, each partitionfunctionsthe characteristic lengths L0 and L†swill bereduced to a distance between the diffusion sites, which isroughly the distance a between water molecules in thechannel. Since the diffusion constant D ) a2/τp, with somecharacteristic time τp, the rate constant expression will havethe expected form

In the above expression, the pre-exponential factor shouldhave a weak isotope dependence, characteristic of protondiffusion, and the activation barrier should depend on thedriving force for electron transfer, ∆Ge°, as depicted in Figure8. The proton conducting channels in proteins are veryinhomogeneous, so D is expected to depend strongly on X.Then both the diffusion coefficient D(X) and the free energyprofile Gp(X) will be important in defining the position ofthe transition point X† in the channel.42

The coupled PT/ET reaction occurs when the proton passesthe critical point X†. To reach this point, the proton firstmoves uphill in energy along the channel, while the electronremains mainly on the donor site. Once the critical point ispassed, the electron is quickly transferred to its acceptor site,and the proton completes the reaction by moving downhillin energy to its acceptor site in the channel.

The described process resembles one in the four-statemodel, in which the proton jumps first onto the oxidizedacceptor and a quick downhill electron transfer follows. Theoverall rate constant for the reaction is kp

O. A similar

k ) ke(X ) 0) + Kpke(X ) L) ) keU + Kpke

P

(5.11)

Ke(X) ) exp(-∆Ge°(X)/kBT) (5.12)

Gp(X) ) -kBT ln(e-Gpd(X)/kBT + e-Gp

a(X)/kBT)(5.13)

Gp(X) ) min[Gpd(X),Gp

a(X)] (5.14)

k ) D†

L0L†e-∆Gp

†/kBT (5.15)

k ) 1τp

e-∆Gp†/kBT (5.16)


expression is obtained above. However, in the channel model,the proton does not need to move all the way up to theacceptor site before the electron makes the transition.

As in the four-state model, the above rate constant for thePT/ET process should be compared to the rate constant forthe ET/PT process. While the former rate constant corre-sponds to kp

O, the latter corresponds to KekpR in the four-state

model. In the ET/PT process, the proton transfer occurs inthe reduced state. In the present model, kp

R will be given by

where ∆Gp†a is the activation free energy barrier for proton

transfer along the channel in the electron acceptor state. Overall, itis a downhill reaction. Given that Ke ) exp(-∆Ge

U/kBT), the totalactivation free energy barrier for such a reaction is ∆Ge

U + ∆Gp†a.

This value should be compared with ∆Gp†d, the activation

free energy barrier for proton transfer along the channel inthe electron donor state, in order to determine which channel,PT/ET or ET/PT, is dominant in the coupled reaction.

5.3.4. Proton Induced Electron Transfer

Similarly, one can consider a reaction in which the protonmotion along the channel induces electron transfer. Themotion of the proton causes variations in the energydifference of the electron donor and acceptor states, ∆da(X).There can be such a point Xc in the channel at which theelectronic energy levels cross where an electronic transitioncan occur. The condition for level crossing is ∆da(R,X) ) 0.So far, we assumed that the coordinate R was very fast, sothat an equilibrium distribution in R was established for eachX. We now assume the opposite; that is, the coordinate R isslow, or equivalently, the change of ∆da due to variations ofR is small. In other words we assume that the main variationof ∆da is due to the motion of the proton along the channel.We assume there is an Xc such that ∆da(Xc) ) 0.

The rate constant in this case is

where Ppd(X) is the stationary distribution function of the

proton along the proton conducting channel in the electrondonor state. The unknown Pp

d(Xc) in the above equation canitself be expressed in terms of k in a self-consistent way.For simplicity, assume the diffusion coefficient D along theproton-conducting channel to be constant. Further analysisshows that the rate constant in this case can be written as42

where Zpd is the proton partition function in the donor state.

The above equation gives the rate constant for both theslow and fast proton diffusion cases. For slow diffusion, weobtain a diffusion-controlled reaction (k ∼ D), and for fastdiffusion, we obtain the usual nonadiabatic reaction (k ∼|TDA|2). In both cases, the activation free energy, ∆Gp(Xc) )Gp

d(Xc) - Gpd(0) depends on the free energy profile along the

channel. The free energy curves here are not the usualMarcus parabolas, however; therefore, the activation energy

dependence on the driving force, for example, is differentfrom the usual one. For linear dependence of free energieson X, we have (see Figure 8)

Then the activation free energy of the coupled reaction is

With the general formula in eq 5.19, different cases ofenergy profiles along the channel can be investigated. Boththe nonadiabatic and adiabatic diffusion controlled cases areincluded in one expression. These results are analogous tothose obtained by Zusman,105 Burshtein,106 and other workers,e.g.,101,107-111 for solvent-controlled ET reactions.

6. Applications

6.1. HAT vs PCET MechanismsIn general, HAT reactions can be viewed as a subset of

the concerted PCET reactions described in section 2.Nevertheless, to aid in discussions of chemical reactions, itis often useful to distinguish between these two types ofmechanisms. A textbook example of the fundamental dif-ferences between the PCET and HAT mechanisms isprovided by the comparison between the phenoxyl/phenoland benzyl/toluene self-exchange reactions. These systemshave been analyzed in terms of the singly occupied molecularorbitals (SOMOs) obtained from density functional theorycalculations of the transition state structures (i.e., the first-order saddle points).32 As depicted in Figure 9, the SOMOis dominated by 2p orbitals perpendicular to the protondonor-acceptor axis for the phenoxyl/phenol system, butthe SOMO is dominated by atomic orbitals oriented alongthe proton donor-acceptor axis for the benzyl/toluenesystem. These results were interpreted to signify that theelectron and proton are transferred between different sets oforbitals for the former system but between the same sets oforbitals for the latter system. This interpretation led to theidentification of the phenoxyl/phenol reaction as PCET andthe benzyl/toluene reaction as HAT.

These systems have also been analyzed with the semiclas-sical formalism described in section 2.3.69 The electronicallydiabatic potential energy curves corresponding to the reactantand product PCET states defined in section 2.1 were obtainedby fitting to the CASSCF (complete active space self-consistent-field) electronically adiabatic ground- and excited-state potential energy curves. Mixing the two diabatic stateswith the appropriate electronic coupling leads to the CASSCFelectronically adiabatic curves, as depicted in Figure 10. Theparameters comprising the electronic transition time τe andthe proton tunneling time τp defined in section 2.3, as wellas the proton adiabaticity parameter p ) τp/τe (eq 2.12), wereobtained from these diabatic potential energy curves. Asdiscussed in section 2.4, the degree of electronic nonadia-baticity for the proton transfer reaction provides a quantitativediagnostic for distinguishing between HAT and PCET.Specifically, the PCET mechanism corresponds to the

kpR ) 1

τpe-∆Gp

†a/kBT (5.17)

k ) 2πp

|TDA|2|∂∆da(X)/∂X|X)Xc

-1 Ppd(Xc) (5.18)

k ) ( kBT

D|dGpd(Xc)/dXc|

+|d∆da(X)/dX|X)Xc

2π|TDA|2/p )-1

×

e-∆Gp(Xc)/kBT/Zpd (5.19)

Gpd(X) ) (X/L)∆Gp° (5.20)

Gpa(X) ) ∆Ge° + (X/L)(∆Gep - ∆Ge°) (5.21)

∆Gp† )

∆Gp°∆Ge°∆Gp° + ∆Ge° - ∆Gep

(5.22)


electronically nonadiabatic limit (p , 1), and the HATmechanism corresponds to the electronically adiabatic limit(p . 1).

These calculations revealed that the phenoxyl/phenolreaction involves electronically nonadiabatic proton transferand corresponds to PCET, whereas the benzyl/toluenereaction involves electronically adiabatic proton transfer andcorresponds to HAT.69 For the phenoxyl/phenol system, theadiabaticity parameter p ) 0.013, which is in the electroni-cally nonadiabatic limit with τe ≈ 80τp. In this case, theelectronic transition time is significantly greater than the

proton tunneling time, so the electrons are not able torearrange fast enough for the proton to remain on theelectronically adiabatic ground state surface. For the benzyl/toluene system, the adiabaticity parameter p ) 3.45, whichis found to be in the electronically adiabatic limit with τe ≈0.29τp. In this case, the electronic transition time is less thanthe proton tunneling time, so the electrons can respondinstantaneously to the proton motion, allowing the protonto remain on the electronically adiabatic ground state surface.This characterization of the phenoxyl/phenol system as PCETand the benzyl/toluene system as HAT is consistent with thequalitative analysis based on the SOMOs.

Further analysis of the semiclassical calculations providesinsight into the fundamental differences between these twosystems. The electronic coupling Vel was estimated to be halfthe splitting between the two electronically adiabaticCASSCF potential energy curves at the midpoint betweenthe proton donor and acceptor atoms. From Figure 10, thiselectronic coupling was found to be 700 cm-1 for thephenoxyl/phenol system and 14,300 cm-1 for the benzyl/toluene system. Note also that the diabatic states exhibitgreater slopes at the crossing point for the former systemthan for the latter system, resulting in a larger |∆F| for thephenoxyl/phenol system. The smaller Vel and larger |∆F| leadto a smaller value of the proton adiabaticity parameter (eq2.12) for the phenoxyl/phenol system. As a result of thesedifferences, the vibronic coupling has different forms forthese two systems: it is the product of the electronic couplingand the overlap of the reactant and product proton vibrationalwave functions for the phenoxyl/phenol system, but it is halfthe tunneling splitting on the electronically adiabatic groundstate for the benzyl/toluene system.

We point out that the splittings between the ground andexcited electronic states are significantly larger than thethermal energy kBT at room temperature for both systems.The significance of electron-proton nonadiabatic effects isnot determined by comparing this splitting to the thermalenergy but rather by comparing the electronic transition timeto the proton tunneling time. In contrast, the overall vibroniccouplings for both systems were found to be much smallerthan the thermal energy, mainly due to contributions fromthe proton vibrational wave functions, indicating that bothreactions are vibronically nonadiabatic with respect to asolvent or protein environment. Thus, the types of rateconstant expressions given in section 2 are applicable to thesesystems.

Figure 9. The two highest-energy occupied molecular orbitals for (a) the phenoxyl/phenol and (b) the benzyl/toluene systems for thetransition state structures. Figure reprinted with permission from ref 69. Copyright 2006 American Chemical Society.

Figure 10. State-averaged CASSCF ground and excited stateelectronically adiabatic potential energy curves along the transfer-ring hydrogen coordinate for (a) the phenoxyl/phenol and (b) thebenzyl/toluene systems. The coordinates of all nuclei except thetransferring hydrogen correspond to the transition state geometry.The CASSCF results are depicted as open circles that are blue forthe ground state and red for the excited state. The black dashedlines represent the diabatic potential energy curves correspondingto the two localized diabatic states I and II. The mixing of thesetwo diabatic states with the electronic coupling Vel leads to theCASSCF ground and excited state electronically adiabatic curvesdepicted with solid colored lines following the colored open circles.For the phenoxyl/phenol system, the solid colored lines and theblack dashed lines are nearly indistinguishable because the adiabaticand diabatic potential energy curves are virtually identical exceptin the transition state region. Figure and caption reprinted withpermission from ref 69. Copyright 2006 American ChemicalSociety.


6.2. Concerted PCET in Solution and EnzymesThe parameters in the rate constant expressions given in

section 2 can be determined using various experimental andtheoretical methods. The driving force can be estimated fromexperimentally measured redox potentials and pKa values.34,63,70

The solvent reorganization energy of a PCET reaction is oftensimilar to that of the corresponding ET reaction because thesolvent reorganization energy for PT is typically muchsmaller than that for ET (i.e., the proton transfers a muchshorter distance than the electron). Thus, the solvent reor-ganization energy can be estimated from the experimentalmeasurement of the solvent reorganization energy for thecorresponding ET reaction. From the theoretical standpoint,the PCET solvent reorganization energy can be calculatedwith dielectric continuum models112-114 or molecular dynam-ics simulations,115 analogous to the methods used for ET. Inprinciple, the equilibrium proton donor-acceptor distancecould be obtained with experimental methods for structuredetermination, and the associated frequency could be deter-mined spectroscopically, although the separation of this modefrom other modes is challenging. In practice, the equilibriumproton donor-acceptor distance and frequency, as well asthe proton potential energy curves, are usually obtained fromelectronic structure calculations. Alternatively, moleculardynamics simulations may be used to estimate this distanceand frequency.115 The reactant and product proton vibrationalwave functions and the associated overlaps can be calculatedfor the proton potential energy curves using Fourier gridmethods.116,117 The electronic coupling can be calculated withthe same electronic structure methods that are used forcalculating this parameter in electron transfer theory.118-120

Note that the magnitude of the electronic coupling is usuallysimilar for related PCET and ET reactions, but the PCETelectron-proton vibronic coupling is substantially smallerdue to inclusion of the overlap between the reactant andproduct proton vibrational wave functions. Currently, meth-ods for calculating the combined electron-proton vibroniccoupling in terms of mixed nuclear-electronic wave func-tions are being developed.121-123

The formulation described in section 2 has been used topredict the dependence of the concerted PCET rate constantsand KIEs on system properties, including temperature anddrivingforce, aswell as theequilibriumprotondonor-acceptordistance and frequency.61 From the experimental standpoint,altering only a single parameter without influencing the otherparameters is often not possible. For example, an increasein the equilibrium proton donor-acceptor distance is oftenassociated with a decrease in the frequency. In addition,changing the driving force for proton transfer often impactsthe equilibrium proton donor-acceptor distance.

The driving force dependence of the rate constant isparticularly interesting in the context of the Marcus invertedregion behavior, where the rate constant decreases as thedriving force increases (i.e., as the reaction becomes moreexoergic). Inverted region behavior has been observedexperimentally for electron transfer reactions.124,125 Invertedregion behavior is predicted to be experimentally inaccessiblefor concerted PCET reactions because of the availability ofexcited electron-proton vibronic product states with greatervibronic coupling.61,126 As illustrated for model systems,however, apparent inverted region behavior could be ob-served if varying the driving force also impacts otherproperties of the system, such as the proton donor-acceptordistance.126

The theory described in section 2 has been applied to avariety of experimentally studied concerted PCET reactionsin solution and enzymes.31,127,128 These applications includePCET in amidinium-carboxylate salt bridges,114,129,130 ironbi-imidazoline complexes,131,132 ruthenium-polypyridyl com-plexes,133-135 ruthenium-polypyridyl-tyrosine systems,33,136

rhenium-polypyridyl-tyrosine systems,137-139 thymine-acryl-amide complexes,140,141 and ruthenium-polypyridyl-quinolsystems.70,142 This theory has also been applied to PCET inthe enzyme soybean lipoxygenase,63,115,143 as well as to aseries of mutant forms of this enzyme.144,145 Furthermore,the extension of this theory to electrochemical systems41,52,65

has been applied to osmium complexes attached to a self-assembled monolayer on a gold electrode.146,147 Three ofthese systems are depicted in Figure 11. All of thesecalculations have reproduced the experimentally observedtrends in the rates and KIEs, as well as the temperature andpH dependences in some cases. In addition, these studieshave elucidated the fundamental principles underlying PCETreactions, assisted in the interpretation of experimental data,and provided experimentally testable predictions.

6.3. Cytochrome c OxidaseA marvelous illustration of many aspects of coupled

electron and proton transfer reactions in biology is providedby cytochrome c oxidase. This enzyme is a redox-drivenproton pump that utilizes the energy of oxygen reduction topump protons across the membrane.148-151 The A-typeoxidases have two proton-conducting input channels (D- andK-channels) to transfer protons required for oxygen reductionand for proton pumping, with the D-channel transporting allpumped protons. Electrons are transferred along a chain ofmetal cofactors: from cyt c to CuA, then to heme a, andfinally to the Fea3-CuB binuclear catalytic center of theenzyme. The electron transport is coupled to proton trans-location in such an intricate way that as electrons flow tothe catalytic center of the enzyme to accomplish oxygenreduction (and produce water), the protons are pumped acrossthe membrane against the electrochemical proton gradient.

Figure 11. PCET systems studied theoretically with the frameworkdescribed in section 2. Figures reprinted with permission from refs70, 137, and 146. Copyright 2009, 2007, and 2010 AmericanChemical Society.


6.3.1. Proton Pumping Mechanism

Since the discovery of the proton pumping by CcO in 1977by Wikstrom,152 there were many proposals for the mecha-nism of the enzyme; see, for example, refs 153-157. Somecomputational studies158-161 point to the mechanism ofpumping shown in Figure 12. Recent experiments162 and theirmodeling163 support the mechanism in its general form;however, some details (i.e., the identity of the so-calledproton loading site; see below) remain experimentallyuntested and may in fact be different (see, e.g., discussionsin refs 162-165). Schematically, the proposed mechanismis as follows.

During the cycle, the stable state of the catalytic center,before an additional electron is supplied to the system, issuch that one of the metal centers is formally oxidized, e.g.,Fe3+H2O or Cu2+H2O. This state is established in a previousstep of the cycle, when a “chemical” proton is accepted byone of the hydroxy ligands of the binuclear center. (Hereand below for clarity, we distinguish protons that participatein oxygen reduction, which we denote “chemical” protons,from the protons that are pumped through the membrane,which we denote “pumped” protons.) In this state, His291,the so-called proton loading site (PLS), is deprotonated, andGlu242 is protonated. The following steps are involved:

(1 and 2) An electron is supplied to the system via cyt cand CuA, which is transferred to heme a and then to the hemea3-CuB binuclear center. One of the metal ions is reduced,and the overall charge of the binuclear center (BNC) becomesone charge unit more negative. The driving force of pureelectron transfer is about 20 meV.158

(3) In response to the increased negative charge of thebinuclear center, the proton from Glu242 now has a drivingforce to move closer to the binuclear center. There are twopathways leading from Glu242 to two possible sites:166 oneis the BNC itself, and the other is His291sthe PLS. Theassumption (gating) is made that the rate of proton transferto His291 is much higher than that to the binuclear center.Therefore, the protonation of His291 occurs before that ofthe binuclear center. The fast proton transfer from Glu242to His291 occurs by the Grotthuss mechanism via Arg438and PRDa3. The driving force for this transfer is about 100meV.158 With this transition, His291 becomes protonated.

(4) In this step Glu 242 is reprotonated.

(5) Now the second, chemical proton is transferred to thebinuclear center, using the second path connecting Glu242and the catalytic center of CcO. The driving force for thistransition is about 250 meV.158 A significant driving force,despite the presence of the proton on a nearby His291, isdue to formation of a water molecule in the BNCsthis isthe main source of energy in the process. This transitionoccurs after the first proton has moved to the His291 protonloading site.

(6) The previously formed state has two additional protonspresent in two closely located sites (the PLS (His291) andthe BNC) but only one additional electron residing on oneof the metal ions of the BNC. This state is metastable becauseof significant proton repulsion. The state is stabilized,therefore, by the expulsion of the proton from the His291PLS site. The additional energy of stabilization gained isabout 250 meV.158 The expulsion of the proton from His291from a state in which one of the metal centers is formallyoxidized is predicted by electrostatic calculations, whichshow that, in this redox state, His291 has to be deprotonated.

(7) Glu 242 is reprotonated again, and one turnover ofthe cycle is complete. The formed state is stable until thenext electron is passed through the system. This last step ofreprotonation of Glu242 may be correlated with the expulsionfrom His291 and/or subsequent rereduction of heme a in thenext turnover in the cycle.

Each time a proton is pumped, in a single turnover of thepump, a stable state is formed until the next electron isinjected into the system. Thus, for each electron passingthrough the chain, there is one pumped proton. For eachoxygen molecule, four electrons are required to form twowater molecules; therefore, a maximum of four protons canbe pumped.

The above model is based on the energetics of the systemdescribed originally in ref 158 (and subsequent studies160,161),and a key kinetic assumption is that, upon reduction of theBNC, the first (pumping) proton is transferred to the PLS ofthe pump, and later the second, chemical proton, arrives atthe BNC. The first proton transfer is fast; however, it leadsto a state (proton on the PLS) that is not most favorableenergetically. The most energetically favorable state (protonon OH- in the BNC) is achieved by the second, slow protontransfer to the BNC. Since the PLS and BNC sites are closelylocated, due to electrostatic repulsion, the two protons cannotcoexist, and the first proton is expelled for the sake ofachieving an energetically more stable state. Since bothchemical and pumping protons are derived from the samesource (protonated Glu242; one or two chemical protons canpossibly come via the K-channel167,168) and due to the specialarrangement of the two channels leading to the PLS and tothe BNC, the transfer of the second chemical proton blocksthe return (back transfer) of the first proton to Glu242.

The difference in rates of proton transfer along the “fast”pumping channel leading from Glu242 to His291 and the“slow” chemical channel leading from Glu242 to the BNCpresumably is based on the structural arrangements of thekey groups. The protonated Arg438 is located in theimmediate vicinity of His291, so that when the proton isneeded on His291, Arg438H+ quickly donates the proton tothis residue. This proton transfer is achieved via a watermolecule Wa3. Immediately after that, the chain of watermolecules connecting the PRDa3 and Arg438 site withGlu242 provides a proton from Glu242 to reprotonateArg438.159 The net result of these transitions is that a proton

Figure 12. (A) Key structural elements of the proposed pumpingmechanism of CcO and the sequence of transitions during onepumping cycle. Two protonation sites (the PLS and a site in theBNC) are shown as H-circles. PT and ET steps are shown by blueand red arrows, respectively. The residue notation is for bovineenzyme. (B) Schematic depiction of the model. The key assumptionof the model is that, upon ET between the hemes (step 2), the protontransfer to the proton loading site (step 3) occurs before the protontransfer to the BNC (step 5). Figure reprinted with permission fromref 159. Copyright 2004 Elsevier.


from Glu242 is quickly transferred to His291. The exactmechanism of gating, however, is still unknown. Recently,Wikstrom and co-workers proposed that water dynamics inthe catalytic cavity might in fact be responsible for protongating.169

For proper function of the pump (i.e., to pump protons inthe right direction, against the membrane electrochemicalgradient), another key requirement should be fulfilled,namely, that the protonation of the PLS occurs by a protonfrom the negative side of the membrane with low chemicalpotential and not from the opposite side with high chemicalpotential. Otherwise, the protons would flow in the wrongdirection. Closely related to this requirement is the one thatis usually assumed in the form of a mechanical gate thatwould prevent the leak of protons through the pump betweenthe pumping events.169

The two-step pumping mechanism, in which the pumpedproton is first loaded to the PLS from the N-side of themembrane and then expelled by the chemical proton arrivingto the BNC on the P-side of the membrane, is part of manyrecently discussed models. The main difference between themodels is the nature of the PLS site, whose identity is stillnot known; however, recent experimental and theoreticalstudies point to one of the residues in the group His291,propionate A or D of heme a3, or a group (including a watermolecule) nearby.162,163,170,171

6.3.2. Coupled Electron and Proton Transfer Reactions

The individual steps of the mechanism illustrate variousaspects of proton-coupled electron transfer discussed in thispaper. We will now describe qualitatively how these analyti-cal models can be utilized in the analysis of specific reactionsin cytochrome c oxidase.

Step 1 of ET from CuA to CuB is rather fast (on the orderof 10 µs) and does not show a KIE. Sometimes the absenceof a KIE (and also pH dependence) is taken as an indicationof the absence of the coupled proton, even internal to theenzyme. We have seen that this is not always the case; thereare several examples of coupled reactions where a KIE isnot present at all. In these cases, the actual rate constant forthe coupled reaction is often a product of the pure electrontransfer rate constant and the proton equilibrium constantKp. Obviously when the proton transfer is uphill, the overallrate constant is a factor of Kp , 1 smaller than the pureelectron transfer rate constant. In this specific case, however,the distance between redox cofactors is such that the 10 µstime scale is the maximum that one typically expects fromET in proteins. If indeed a proton were coupled, and 10 µs-1

is the rate constant of the coupled reaction, that would meanthat the rate of ET alone is much faster (by a factor 1/Kp),which is unlikely.

If there is no coupling, however, the role of heme a is notclear, except that in the next step the pure electron transferis much faster (on the order of nanoseconds172). In fact, thismay be the reason for heme a: the key electron transfer tothe BNC should be fast, because only in this case does acoupled reaction have a chance to be in the physiologicalrange of (order of magnitude) 0.1-1 ms.173 In addition, thereis an interesting proposal for electrostatic regulation of waterchains and proton gating in the region between heme a andheme a3, which would also explain the need for heme a.157

The most intriguing and most fundamental part of themechanism of CcO is the second step shown in Figure 12:the ET between heme a and heme a3. This electron transfer

appears to be driving all proton translocations in the enzyme;this was clearly demonstrated in recent potentiometricexperiments,162 where the injection of the electron into thesystem (10 µs phase) was observed to generate a sequenceof much slower, and much higher in amplitude kineticphases, which apparently refer to individual proton transferreactions coupled to the injected electron.163 There are severalinteresting aspects of the coupling in this transition.

(A) There is a proton loading reaction coupled to ET instep 2. The coupled proton is transferred from Glu242 alongthe chain of water molecules in the catalytic cavity; however,the dynamic character of the water chain connecting Glu242and PropD is not entirely clear. Moreover, the exact numberof water molecules in the catalytic cavity is not known andmost likely is even ill-defined, in the sense that watermolecules constantly move in and out of the cavity; thisprocess is likely to define the overall kinetics of the coupledreaction (2) + (3) (see discussion in section 3.6). There is agating of the proton: upon step 2 ET, the proton from Glu242does not go to the BNC but instead makes a “mistake” andmoves to the PLS.

The driving force of electron transfer (2) was estimatedto be about 20 meV.158 This means that an electron is initiallyequilibrated between the two hemes, and the transfer isincomplete. However, this electron transfer is directlycoupled with a proton transfer to the PLS located near theBNC. The proton transfer to the PLS significantly increasesthe redox potential of the BNC, thereby stabilizing theelectron at the BNC, which in turn further increases thedriving force for proton transfer to the PLS. Therefore, onecan say that the electron and the proton drive each other atthis step to the more stable (intermediate) state of the enzyme,where they occupy the BNC and the PLS, respectively. Thistype of reaction was discussed earlier in this review.

(B) After the loading is complete, and Glu242 getsreprotonated, the second proton is transferred from Glu242,this time to the BNC. In this case the reaction is completelydifferent, because the electron is already in the final state,but the electrostatic driving force for the chemical proton isdiminished by the presence of the first proton at the PLS.Yet, obviously there is nonzero driving force, as bothcalculations showed160,161 and as is evident from the kineticsof the potentiometry, which indicate that the second protontransfer (corresponding to the 800 µs kinetic phase in theexperimental measurement of the membrane potential)162 iswell separated from the first loading phase (150 µs). In thiscase it can be considered as pure proton transfer; its rate isdetermined by both the driving force and the kinetics of waterchain formation in the catalytic cavity connecting Glu242and the BNC. It was predicted earlier166 that there is a stablechain of water molecules; however, as mentioned earlier,the exact number of water molecules in the BNC cavity isnot known.174

(C) The expulsion of the proton from the PLS also appearsto be a pure proton transfer reaction, which is governed bythe proton-proton interactions, rather than electron-protoninteractions. The proton transfer pathway between the PLSand the outside of the protein is not known with certainty,but theory and simulations provide some clues about thisprocess.175,176

7. ConclusionsAs is evident from this review, the main characteristic of

proton-coupled electron transfer reactions is their great


diversity. There is no one magic formula for the reactionrate constant as in the pure ET case. As has been shown,however, most cases involve a reaction bottleneck typicallydescribed by a rate constant that can be cast in the form ofa product of a statistical (Arrhenius) exponential factor anda dynamic prefactor, or as a sum of such terms. In addition,the quantum mechanical effects of the electrons and thetransferring protons play an important role in these processes.As a result, PCET processes can exhibit large kinetic isotopeeffects and interesting dependence of the rate constants andkinetic isotope effects on temperature, pH, and driving force.Many cases of ET reactions coupled to proton translocation,where proton dynamics can be described as a random walk,bear close resemblance to common models of ET, where adistinction is made between “fast” vibrational modes and“slow” solvent or protein dynamics modes. Typically suchreactions are described in terms of the Sumi-Marcus model101

or related models developed in the field.177-180 Although thephysics and mechanics of these models are very similar, theactual realization is quite different.

The great variety of theoretical cases provides manypossibilities in the way that electrons can be coupled toprotons. The result of such coupling is that, in cases whereelectron or proton transfer is not possible separately (i.e., isenergetically unfavorable), a coupled reaction of bothelectron and proton transfer is possible. The main reasonfor such reactions is the electrostatic stabilization; as a result,either electrons can be driven to species where they areneeded, or protons can be delivered to a site where they areneeded with the help of collateral proton/electron transfer.In most cases, the fundamental reason for this coupling isobvious: positive protons and negative electrons are attractedto each other, thereby electrons can “pull” protons, or viceversa, protons can pull the electrons to sites where they areneeded. There are some nontrivial cases such as CcO,however, where negative electrons are somehow “pushing”positive protons against the external field gradient. Such casesunderscore the wonderful richness of the field of electronand proton coupled transport.

The future prospects in the theoretical study of PCET areexciting and challenging. Based on the significance ofdesigning solar cells, which often involve PCET at afundamental level, the development of methods to studyphotoinduced PCET reactions is critical. These reactions areoften inherently nonequilibrium processes, so the standardMarcus theory expressions may not be applicable. A promis-ing approach for studying the ultrafast dynamics of suchsystems is the use of nonadiabatic molecular dynamicssimulations on electron-proton vibronic surfaces.100 Anotherimportant direction is the further development of methodsto study proton-coupled electron transport involving multipleelectron and proton transfers.46,114 These extensions are vitalfor the investigation of the complex biological processes ofrespiration and photosynthesis, as well as the design ofcatalysts for various energy conversion processes.

Biological reactions involving electrons and protons areparticularly challenging to describe theoretically. Despite avariety of models that are fundamentally possible, the realchallenge is to achieve a quantitative level of description ofspecific biological systems, such as cytochrome c oxidase.The principal difficulty of studying proton translocationdriven by redox chemistry is that the proton motions in theprotein are extremely difficult to monitor experimentally. Incontrast to electrons, which can be detected using optical

spectroscopy of redox centers, protons are not as easy to“see” in proteins. In this case, computer simulations becomeabsolutely indispensable. Development of accurate methodsof simulation for biological systems that reflect the richnessof the field represents a major challenge for computationalchemists.

8. AcknowledgmentsThis work was supported in part by NSF Grant PHY

0646273, NIH Grant GM54052 (A.A.S.), NSF Grant CHE-07-49646, NSF Grant CHE-08-02907 for POWERING THEPLANET: A Chemical Bonding Center in the DirectConversion of Sunlight into Chemical Fuel, and NIH GrantGM56207 (S.H.-S.). We acknowledge helpful discussionswith Marshall Newton, Jim Mayer, and Leif Hammarstrom.

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