Chemical Science

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aDepartment of Chemistry, Yeshiva Univer

10033, USAbCenter for Biological Physics, Arizona Sta

Arizona 85287, USA. E-mail: dmitrym@asu.cThe Biodesign Institute, and Department

State University, 1001 McAllister Ave., Temp

† Electronic supplementary informationexperimental kinetic data, and calculat10.1039/c3sc51327k

Cite this: DOI: 10.1039/c3sc51327k

Received 13th May 2013Accepted 12th August 2013

DOI: 10.1039/c3sc51327k

www.rsc.org/chemicalscience

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Protein dynamics to optimize and control bacterialphotosynthesis†

David N. LeBard,a Daniel R. Martin,b Su Lin,c Neal W. Woodburyc

and Dmitry V. Matyushov*b

Proteins function by sampling conformational sub-states within a given fold. How this configurational

flexibility and the associated protein dynamics affect the rates of chemical reactions are open questions.

The difficulty in exploring this issue arises in part from the need to identify the relevant nuclear modes

affecting the reaction rate for each characteristic time-scale of the reaction. Proteins as reaction media

display a hierarchy of such nuclear modes, of increasingly collective character, that produce both a

broad spectrum of static fluctuations and a broad spectrum of relaxation times. In order to understand

the effect of protein dynamics on reaction rates, we have chosen to study a sub-nanosecond electron

transfer reaction between the bacteriopheophytin and primary quinone cofactors of the photosynthetic

bacterial reaction center. We show that dynamics affects the activation barrier of the reaction through a

dynamical restriction of the configurational space sampled by the protein–water solvent on the reaction

time-scale. The modes which become dynamically arrested on the reaction time-scale of hundreds of

picoseconds are related to elastic motions of the protein that are strongly coupled to the hydration

layer of water. Several mechanistic consequences for protein electron transfer emerge from this picture.

Importantly, energy parameters used to define the activation barrier of electron transfer reactions lose

their direct connection to equilibrium thermodynamics and become dependent in a very direct way on

the relative magnitudes of the reaction and nuclear reorganization time-scales. As a result, the

energetics of protein electron transfer need to be defined on each specific reaction time-scale. This

perspective offers a mechanism to optimize protein electron transfer by tuning the reaction rate to the

relaxation spectrum of the reaction coordinate.

Introduction

Due to the explosion of protein structures that have beendetermined in the past 30 years, there is now a large body ofinformation concerning the static or equilibrium conforma-tions of a very wide variety of protein systems. The structure–function relationships that have been formulated using thisstructural database form the core of our mechanistic under-standing of biochemistry. The next frontier, currently activelyexplored,1–8 is how to move from a static picture offered bystructural studies to a conceptual and theoretical frameworkdirectly incorporating protein dynamics into protein-mediatedchemistry.9 Protein dynamics occur over a broad range of time-

sity, 500 West 185th St., New York, NY

te University, PO Box 871504, Tempe,

edu

of Chemistry and Biochemistry, Arizona

e, Arizona 85287, USA

(ESI) available: Simulation protocol,ions of the reaction rates. See DOI:

Chemistry 2013

scales and can potentially couple to many chemical processesranging from typical enzymatic reactions on the millisecondtime-scale8 to the ultra-fast reactions of photosynthesis,10

exciton energy transfer,11 and photoinduced isomerization12 onthe picosecond time-scale.

The ability of proteins to explore many alternative congu-rational states, nominally belonging to the same native struc-ture, is well established.4,5,13–16 What is not well understood ishow the dynamics of moving between these sub-states affectbiochemical reactions at a structurally based, mechanisticlevel.17–19 The relevant question here is whether the largecongurational space of a protein and its hydration layer14,20

merely affects the equilibrium free energy barrier of the reac-tion, a thermodynamic quantity, or whether protein dynamics,i.e., the time evolution of the nuclear coordinates, contribute tothe non-equilibrium reaction energetics during the time-courseof the reaction. This is a nontrivial question of fundamentalsignicance: standard theories of condensed-phase chemicalkinetics offer little room for efficient control of reaction rates bynuclear dynamics. Indeed, the standard formulation of transi-tion-state theory3,21,22 prescribes that most of the mediumeffects on the chemical rates are static and enter the exponential

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factor of the rate as the free energy barrier, a thermodynamicproperty. Modications of transition-state theory, such asKramers' diffusional dynamics along the reaction coordinate,23

introduce dissipation and characteristic relaxation times intothe pre-exponential factor of the rate, but do not affect theArrhenius activation term given by the Boltzmann occupation ofthe transition state.

In traditional formalisms, the pre-exponential factor in therate equation carries the dynamical information, a concept thathas been well established in solvent controlled electron-transferreactions.24–28 This pre-exponential term does not signicantlychange among different condensed media and thus does notsignicantly modify the reaction rate. This is particularly true inapplication to protein-mediated chemistry because of thelimited range of thermodynamic conditions that permit astable, native state of the protein. Therefore, if the congura-tional dynamics of a protein affect the rate, this effect, in thetraditional viewpoint, must be accounted for primarily by theactivation entropy.18

While the dynamical effect on chemical reactions can beviewed as the crossing of the activation barrier affected by timeevolution (dynamics) of the nuclear modes and thus accom-modated in the transmission coefficient of transition-statetheory,29,30 the notion of the “dynamical effect” in enzymatickinetics carries a somewhat different connotation. It has beensuggested that motions of the protein can act as a “gate” suchthat some conformations have much lower energy barriers thanothers.31,32 This does not constitute the dynamical effect refer-enced to above since lowering of the free energy barrier byexploring a multi-dimensional conguration space is consistentwith a purely statistical description of reaction kinetics. A moreconsistent way to describe the dynamical effect of the protein–water thermal bath is to require the involvement of a particularnuclear mode in the activation process in a way that goesbeyond the statistical ensemble description, i.e., cannot bedescribed by a Gibbs energy barrier.3 This is the meaning of thedynamical effect that we pursue here and apply to sub-nano-second intra-protein electron transfer33–35 in photosyntheticbacterial reaction centers. We have chosen the bacterialphotosynthetic reaction because the reactions are initiated bylight,11 making detailed kinetic measurements possible,10 andbecause the structure is well known and easily modiedthrough genetic engineering.36

We advocate a view of dynamical control of protein-mediatedchemistry which emphasizes the effect of dynamics on theactivation barrier, i.e., on the Arrhenius factor in the rate,instead of the pre-exponential factor described by diffusionalKramers' kinetics.23–28 While this general mechanism is notlimited to proteins,37 we suggest that there are specic proper-ties of proteins that make it particularly important to biologicalreactions occurring on sub-nanosecond (and potentially longer)time-scales. A broad distribution of relaxation times, a propertydistinguishing proteins from many redox pairs employed insynthetic chemistry, provides control over chemical reactions inways not anticipated by standard models of chemical kineticsoperating in terms of equilibrium, thermodynamic reactionbarriers.38 The dynamical control of the reaction rates is

Chem. Sci.

achieved through the dependence of the activation barrier (theexponential term) on the relaxation time(s) of the medium, asoriginally suggested by van der Zwan and Hynes39 and Sumi andMarcus,25 and comes in addition to the dependence of thereaction rate pre-exponential factor on the relaxation timealready anticipated by Kramers-type models of chemicalkinetics in general23 and solvent-dynamic models of electrontransfer in particular.24–28 In our description, the dynamics enterthe activation barrier through the nonergodic cutoff of the slowportion of the relaxation spectrum by the time-scale of thereaction. We explain in detail below how this dynamical controlis achieved for electron transfer in bacterial photosynthesis,but, in general terms, we propose a novel picture of thedynamical effect of the protein–water solvent on the reactionactivation energy through nonergodic restriction of the cong-urational space available to the reaction.

Proteins provide both a broad statistical spectrum of uc-tuations affecting the energy of the active site3,40 and a broadspectrum of relaxation times.41–45 The latter can be used tomodify the distribution of the statistical uctuations asrequired for the optimization of reaction efficiency.38 Proteinsas dynamic media thus allow dynamic tuning of the reactionrates, in contrast to the commonly assumed thermodynamictuning achieved by adjusting the reaction free energy (redoxpotentials). These general ideas are applied here to the inter-mediate range of sub-nanosecond time-scales,6 focusing onelectron transfer between bacteriopheophytin and quinonecofactors in the bacterial reaction center. We offer both exper-imental and theoretical perspectives on the problem, the latterachieved by MD simulations and kinetic analysis of the simu-lation data.

The bacterial reaction center

The bacterial photosynthetic reaction center (RC) is almostuniquely suited to serve as an initial biochemical system inwhich to develop the emerging eld of structure–dynamics–function relationships in biochemistry. There are multiplephotoinduced electron transfer reactions that occur acrosstime-scales from picoseconds, to hundreds of picoseconds, tonanoseconds, to microseconds, to milliseconds andseconds.10,48 There are also many very well studied and oenstructurally characterized mutants and cofactor substitutionsthat alter the energetics of these reactions.49 Finally, recentdevelopments in protein50 and lipid51 force-eld parameteriza-tion and scalability of numerical simulations52 have providedatomistic detail of primary charge separation (�3 ps) in bacte-rial reaction centers.40,53–56 This progress, which has beenlimited to relatively short reaction times so far, has neverthelessopened the door for systematic studies of bacterial photosyn-thesis combining time-resolved spectroscopy49 with atomisticdetail from molecular dynamics (MD) simulations.38

Fig. 1A shows the structure of the RC protein of Rhodobacter(Rb.) sphaeroides and its electron transfer pathway and kinetics.Light is absorbed by a bacteriochlorophyll dimer (P), and anelectron is transferred via a monomer bacteriochlorophyll (BA)to a bacteriopheophytin (HA) in a few picoseconds. Primary

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Fig. 1 (A) Arrangement of the cofactors in the active (A, right) and inactive (B, left) sides of the bacterial photosynthetic reaction center from Rb. sphaeroides. Theprimary electron donor (P), bacteriopheophytin (HA), and quinone molecule (QA) are colored in green; the corresponding cofactors of the inactive branch, themonomer bacteriochlorophyll (BA), and the non-heme iron are shown in grey. (B) Free energies of electronic states and time constants of bacterial electron transport.Arrows indicate the reaction pathways, the bold arrow highlights the HA to QA electron transfer investigated here. (C) Cartoon of the simulation setup showing thereaction center in the POPC bilipid membrane: the water is represented as a transparent surface, the lipid membrane is rendered in a van der Waals representation, theprotein is shown in purple, and the cofactors are rendered as green licorice bonds. (D) A zoomed-in image of the electron transfer cofactors in two conformations of theM214LG mutant studied here.46 Shown are the BA (upper) and HA (lower) cofactors and space-filling rendering of the mutation site. The latter shows how the changein the residue volume allows for more room to flip the tail. WT and M214LG(R) are shown in green, M214LG(L) is in blue. (E) Normalized kinetic traces recorded for RCsamples from wild type, and the M214LG mutant at probe wavelengths 837 nm (inset) and 675 nm upon excitation at 865 nm, reflecting the kinetics of P+ and HA�,respectively.47 (F) Nonergodic reorganization energy of HA to QA electron transfer calculated from the Stokes-shift dynamics fromMD simulations. The dashed verticalline indicates the experimental rate of HA to QA electron transfer with the time constant of 200 ps.

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electron transfer (P* / HA) is very robust: the overall rate andyield of this reaction depend only weakly on driving force andtemperature.10,57,58 Primary charge separation is followed by a�200 ps electron transfer to the rst of two quinones (QA) and,nally, to a second quinone (QB) in 200 ms. The back reactionsof each step are 2–4 orders of magnitude slower than theforward reactions (Fig. 1B), resulting in a quantum yield nearunity.10,48,59 The time-scale of the HA� / QA reaction is in therange of time-scales that have been shown both experimentallyand through simulation as being important for the onset ofcollective motions in proteins.14,44,60 It is therefore a nearly idealreaction to study the impact of collective protein dynamics onbiochemical reactivity, which is the objective of this work.

Primary charge separation in the reaction center has beenstudied by atomistic MD simulations previously.40,53–56 Earlystudies53,54 assumed direct, superexchange electron transferbetween P* and HA.61 It later became clear from experiment thatstep-wise electron transfer, including the intermediate BAcofactor (Fig. 1A and B), was involved.62 Subsequent reactions ofbacterial electron transfer turned out to be more challenging toaddress by computations, mostly due to the prohibitivelyexpensive simulation length of hundreds of nanosecondsrequired to reliably sample the statistics and dynamics of thedonor–acceptor energy gap. Here we report the rst step in this

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direction by studying HA� / QA electron transfer by atomistic,large scale simulations of the membrane-bound bacterial RC(Fig. 1C) on time-scales sufficiently long to characterize theelectron transfer reactions that follow the primary chargeseparation.

Recently, Beatty and colleagues created a series of RCmutants that substituted amino acids with successively smallervolumes for the leucine at position M214, adjacent to the activeside HA (Fig. 1D).46 Using these mutants, it was determined thatwhile electron transfer to HA from P* was essentially unper-turbed, the rate of electron transfer from HA� to QA becameboth slower and very heterogeneous, with part of the reactionapparently taking place on the nanosecond time scale instead ofthe usual few hundred picosecond time scale (Fig. 1E and S2 inthe ESI†).47

The most extreme of these mutants, M214LG has beencrystallized and the structure solved.46 The decrease in aminoacid side chain volume apparently made it possible for the RC totake on at least two distinct conformations, with the majordifference being the position of the phytyl tail of the A-sidebacteriochlorophyll, BA (Fig. 1A and D). If, as suggested above,the protein represents a bath with many relaxation times,classical approaches to determining thermodynamic parame-ters (e.g., equilibrium redox measurements) will not provide

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particularly useful values in modeling reaction kinetics andmechanistic function. Instead, structure-motivated models ofprotein dynamics will be required and the parameters ofinterest will need to be calculated specically over the timeprogression of the reaction under study. Here, such a model isdescribed and applied to understanding the functional rami-cations of the M214LG mutant.8

ResultsMD simulations of the membrane-bound reaction center

This study reports the rst MD simulation of a membrane-bound RC of Rb. sphaeroides (Fig. 1C). The simulation protocoland the details of the trajectory analysis are given in theMethods section and in the ESI.† Both the initial, HA�–QA andnal, HA–QA� states of the reaction center were studied. Thefollowing nomenclature was adopted in order to distinguishbetween two conformational states of the mutant RCs. Thephytyl tail at the BA site adopts either “right” (labeled asM214LG(R)) or “le” (labeled as M214LG(L)) orientation. Theright orientation is identical to that in the WT RC (Fig. 1D).Correspondingly, two conformational states of the M214LGmutant RC, for which experimental structural data are avail-able,46 have been simulated in each oxidation state. Altogether,eight redox states were simulated, corresponding to two elec-tronic states of the HA–QA pair and two conformational statesof the WT and mutant RCs. The charge distributions of thecofactors (e.g., HA� and HA) are the same in different confor-mations, based on the spectroscopic evidence showing noalteration of absorbance spectra of the cofactors between WTand the mutants (ESI†).47

The phytyl tail of BA was ipped by using a computationaltechnique, generally known as metadynamics, to align it withthe X-ray crystal structure46 (see ESI†). A large (30 kcal mol�1)free energy barrier has been crossed, which brought the BA tailwithin a small average deviation (0.6–1.3 A) of the congurationfound in the X-ray structure. These simulations have alsoindicated a much larger barrier for the tail ipping in the WTRC compared to the M214LG mutant. It appears that thismutation, opening more free volume, facilitates conformationalexibility near BA. Given a larger conformational barrier, wepresent here the data only for the right conformation of WT RC(denoted for brevity as WT). The simulation data for the leconformation of WT RC are presented in the ESI.† Given thehigh barrier for conformational transitions, the phytyl tailremained either in one conformational state or the other duringthe simulation runs.

Mechanistic properties of protein electron transfer

Broken ergodicity (nonergodicity) necessarily occurs when thetime-scale of a chemical reaction becomes shorter than therelaxation time(s) of the thermal bath coupled to the reactioncoordinate.38 A reaction with the characteristic reaction time-scale sreac that falls inside the spectrum of relaxation times ofthe medium nds itself in out-of-equilibrium conditions. Forproteins, nonergodicity is encountered for nearly all electron

Chem. Sci.

transfer reactions because of the enormous breadth of therelaxation times supplied by the protein environment.41–45 Thedirect consequence of this physical reality is that propertiesmeasured by equilibrium experimental techniques, referring tomuch longer observation times seq [ sreac, such as electro-chemistry, are not relevant when the reaction rates are con-cerned. Our calculations and comparison to experiment belowclearly show this difficulty of relating kinetics to thermody-namics. Before proceeding to the details of the calculations, westart by presenting the basic ingredients of the nonergodictheory of electron transfer38 and its distinction from the tradi-tional Marcus formulation.33

The classical Marcus theory of electron transfer33 assumesthat the time of the electron transfer reaction sreac far exceedsthe relaxation times si of the nuclear modes coupled to thereaction coordinate, sreac[ si. The reaction is therefore ergodicwith respect to the thermal bath, and the activation barrierdepends on thermodynamic parameters. Two free energies arerequired in Marcus theory: the (free) energy of reorganizationand the reaction Gibbs energy; the latter is accessible fromredox potential measurements. Bacterial electron transfer forthe most part does not satisfy the equilibration criterion sincesreac falls inside the spectrum of relaxation times of the protein.The system loses ergodicity and the rules of nonergodic kineticsapply.40

The extension beyond Marcus theory is performed in twosteps. One rst needs to realize that electron-transfer events areuniquely determined by the energy-gap reaction coordinate X ¼DE(Q), equal to the instantaneous gap between the energies ofthe acceptor and donor electronic levels that depend on thenuclear coordinates Q.34,63,64 Since many degrees of freedom areinvolved, the statistics of X are mostly Gaussian, even for asystem out of equilibrium. The next step is to realize that theonly modes of the thermal bath that can potentially contributeto the reaction activation are those that are faster than sreac. Theensemble average can therefore be performed only on those fastmodes, producing partial free energies that depend on thereaction rate k ¼ sreac

�1, in place of equilibrium free energies.The reaction rate thus restricts the range of frequencies, andcorresponding nuclear modes, accessed by the system on thereaction time-scale.

The rst consequence of this perspective is the realizationthat nonergodic reorganization energy l(k) needs to replaceequilibrium Marcus reorganization energy l for nonergodicelectron transfer.34 The main qualitative difference betweenthe two is that the former depends on the reaction rate, or theobservation window of the experiment, and the latter is athermodynamic quantity, referring to an innite observationtime or effectively zero reaction rate. This physical reality oflimited access to the congurational space can be mathe-matically expressed by changing the rules of statisticalaverage from those supplied by the equilibrium Gibbsensemble to those determined by ensembles characterized bydynamically restricted phase space.65 The direct consequenceof this alteration of the ensemble of allowed statistical statesis the appearance of the low-frequency cutoff in the integraldening the nonergodic reorganization energy l(k) from the

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Stokes-shi loss functionc c00X(u) (spectral density,66 see

Fig. 2).

l(k) ¼ ÐNk c

00X(u)du/(pu) (1)

The Stokes-shi loss function c00X(u) species the rate of

energy dissipation into the bath when uctuations of thedonor–acceptor energy gap are induced. It is related to theFourier transform of the Stokes-shi correlation func-tion45,60,67,68 by the standard rules of the linear responseapproximation66,69 (see ESI for more detail†). Briey, one startswith the time correlation function of the reaction coordinateCX(t) ¼ hdX(t)dX(0)i, dX(t) ¼ X(t) � hXi. The frequency Fouriertransform CX(u) of CX(t) leads to the loss function as:

c00X(u) ¼ (bu/2)CX(u) (2)

where b ¼ 1/(kBT) is the inverse temperature.Since the reaction rate restricts the congurational space

available to the system, higher rates mean integration over asmaller portion of the loss spectrum in eqn (1) (Fig. 1C and 2A),leading to a smaller reorganization energy. Alternatively, forvery slow rates, l(k) / l at k / 0 and the standard Marcuspicture of ergodic, transition-state reaction kinetics is recov-ered. Even though l(k) appears from the formalism of dynam-ically restricted statistical ensembles,65,70 it is in fact anobservable quantity measured by optical spectroscopy insolvents approaching their glass transition.71–74 The algorithmof connecting l(k) to equilibrated Stokes-shi dynamics has

Fig. 2 (A) Loss spectra of the Stokes-shift dynamics of HA� to QA charge shift inWT and mutated RC vs. n ¼ u/(2p). The vertical dashed lines indicate the value ofthe experimental rate constants for HA� to QA reaction (kexp/(2p)) and of primarycharge separation P* to HA (kCS/(2p)). The shaded area shows the part of theStokes-shift spectrum contributing to the nonergodic reorganization energyl(kexp). (B) Separation of the total loss function into the protein (solid lines) andwater (dashed lines) components. The total loss function includes, in addition tothe individual components, the cross protein–water term, which significantlydiminishes the sensitivity of the individual water and protein components to themutations.

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also performed well in our previous modeling of primary chargeseparation in bacterial photosynthesis.40,75 The fast rate of thisreaction, kCS� 0.3 ps�1, cuts offmost of the slow nuclear modesof the protein, leaving only fast phonons and ballistic watermotions to inuence the rate (Fig. 2A).40 The result is the non-ergodic reorganization energy l(kCS) x 0.4 eV, signicantlylower than its value l x 2.5 eV reported by MD on the simu-lation time-scale of �15 ns.

Here we extend the formalism of nonergodic electrontransfer from the previously considered �3 ps time-scale to aconsiderably slower electron transfer from HA� to QA. Thereaction time-window, kexp

�1 � 200 ps, allows a broader spec-trum of nuclear modes to contribute to l(k) (Fig. 2A), which istherefore larger (Table 1). The immediate consequence is alarger reaction driving force (negative of the vertical separationbetween the free energy minima) corresponding to a nearlyactivationless reaction, as discussed in more detail below.

Fig. 2B shows the loss spectra of the Stokes shi dynamicsreferring to the water and protein components of the overallresponse for the M214LG(L) and M214LG(R) mutants. Both theprotein and water components are very sensitive to mutation.Further, the loss spectra of the protein and water are stronglycorrelated, suggesting a strong coupling between the twosubsystems.44,76–78 The most striking observation is the appear-ance of a very slow relaxation component in the water lossspectrum. For instance, the lowest-frequency water peak for theM214LG(L) system has the exponential decay time of x43 ns,which takes about 41% of the decay amplitude of the Stokes-shi time correlation function. This relaxation componentfollows, with some delay, the corresponding component of theprotein loss function. It can therefore be assigned to the proteinmoving domains of surface waters, polarized by the surfaceresidues76,79 and in some cases strongly bound to them.

The strong coupling between the protein and water subsys-tems results in a compensatory effect.76,79 As is seen fromcomparing Fig. 2A and B, the effect of mutations is signicantlydiminished in the total loss spectrum compared to the indi-vidual protein and water components. Correspondingly, totalreorganization energy l of the protein–water solvent includes across protein–water term, in addition to the direct contributionsfrom the protein, lp, and water, lw, parts. This cross term isnegative, such that l < lp + lw (Table 1). The physical reason for

Table 1 Rates of charge shift from HA� to QA and corresponding nonergodic(on the reaction time-scale) and equilibrium (obtained on the whole length ofsimulation) energetic parameters (eV); lp and lw refer to the protein and watercomponents of the reorganization energy, respectively

Systemk,109 s�1 l(kexp) DG hXi lp lw l

WT 4.5a 0.98 �0.55 �0.21 1.5 1.1 1.6M214LG(R) 4.7 0.90 �0.49 �0.21 1.0 1.0 1.3M214LG(L) 1.4 1.09 �0.84 �0.75 1.2 1.4 1.6

a This is the experimental rate constant kexp used to calculate theelectron transfer matrix element V and the vacuum component of theenergy gap X0.

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the negative cross protein–water correlation is assigned to thepolarization of the water domains at the protein surface by itscharged residues.80 For instance, a positive surface chargeorients the water dipoles with their negative ends pointing tothe charge. This preferential orientation creates an effectivelynegative surface charge density of the water dielectricsurrounding the positively charged residue. The opposite insign surface charge density is obviously created around anegative surface charge of the protein. The opposite signs of theionized residue and the dielectric surface charge induced by itare thus responsible for the negative sign of the cross-correla-tion term in the reorganization energy.

The qualitative picture emerging from the Stokes-shi lossfunctions is a broad distribution of relaxation times, withsome of the longer processes probably still unresolved on thetime-scale of simulations.43 Given that low frequenciescontribute strongly to the reorganization energy (note u�1

scaling of c00X(u) in eqn (1)), these dispersive dynamics project

into a substantial dependence of the nonergodic reorganiza-tion energy l(k) on the reaction rate k (Fig. 1F); the equilibriumreorganization energy corresponds to k ¼ 0, l(0) ¼ l. In otherwords, the effect is not only important conceptually, but is alsosignicant numerically. Accordingly, the values of l(k)obtained at the rate specic to the HA� to QA charge shi aresignicantly lower than “equilibrium” reorganization energiesreferring to the entire length of the simulation trajectory(Table 1). The real equilibrium l is still unknown to us, but itcan only exceed the values calculated so far. It is clear thatequilibrium Marcus reorganization (free) energy is a poormeasure of nuclear reorganization required to transfer anelectron across the membrane.40 We show below that the sameassessment applies to the reaction free energy deduced fromexperimental redox potentials.

Rate of the charge shi from HA� to QA

The above arguments, and the Stokes shi loss functions shownin Fig. 2, suggest that the parameters affecting the rate losedirect connection to their equilibrium thermodynamic cousinsand, if accessed separately, should be measured on the reactiontime-scale. These statements might seem trivial at rst glance.However, this reality has not been sufficiently appreciated bythe standard thermodynamic formulations of the transition-state theory of chemical rates, therefore the special label of“nonergodic kinetics” used to distinguish our formulation. Weshow next that these new concepts are important in anymechanistic understanding of photosynthetic electron transferkinetics and that some modication in the way these rates arecalculated is also required. The main modication is that therate constant must be calculated from a self-consistent equa-tion, typically solved by iteration.40

The standard transition-state rate constant is a product of afrequency factor un and the Boltzmann factor given in terms ofthe activation free energy Ea and the bath temperature T.21 Thefrequency factor un represents the frequency of attempts to crossthe barrier, which can be affected by the rate of energy dissipa-tion along the reaction coordinate and by the corresponding

Chem. Sci.

relaxation time(s). This is the picture advocated by diffusionalmodels of chemical activation going back to Kramers.23

However, these models follow the traditional transition-statetheory that demands the reaction barrier Ea, a thermodynamicproperty, not be affected by energy dissipation. In this case, thedynamics of biomolecules do not affect the height of the acti-vation barrier.19

This perspective changes when nonergodicity is introduced.The energy parameters affecting the barrier height, includingthe reorganization energy for electron transfer reactions,become functions of the rate itself, replacing the thermody-namic barrier Ea with its nonergodic counterpart Ea(k). Howmuch Ea(k) changes with the rate depends on the number ofrelaxation processes coupled to the reaction coordinate and onthe position of k within the spectrum of relaxation times si(Fig. 2A). It is at this point when the unique nature of proteins asthermal reservoirs enters the picture. A large number of oenoverlapping relaxation processes ensures a continuous, andsignicant, change of the barrier when the rate is tuned tooptimize biological function. The direct mathematical conse-quence of this picture is that the transition-state rate expressionis replaced by a self-consistent equation for the rate constant k,which needs to be solved iteratively.38,40

k ¼ un exp[�Ea(k)/(kBT)] (3)

The details of the rate calculations based on the MD inputfor the Stokes-shi dynamics are given in the ESI.†Wementionhere that we have used the standard form for the rate constantof non-adiabatic electron transfer,33–35 when the pre-exponentialfactor, un f V2, is not affected by the medium dynamics, but ismostly determined by the electron-transfer overlap V. Thisparameter and the vacuum component of the average donor–acceptor energy gap X0, which is dictated by the chemicalidentities of the donor and acceptor, were obtained by ttingthe experimental rate of HA� to QA electron transfer inWT RC,81

kexp ¼ 4.5 � 109 s�1. The result of this t is X0 ¼ 0.38 eV and V¼5.3 cm�1. The sum of X0 and the solvent-induced average energygap Xs, available from MD, makes the average Franck–Condonenergy gap hXi ¼ X0 + Xs listed in Table 1. Given the presence oftwo adjustable parameters in our rate calculations, we nowpresent arguments that, once these parameters are xed, ourcalculations are consistent with experimental evidence.

Fig. 3A shows the free energy surfaces against the reactioncoordinate X: F1(X) for HA� to QA electron transfer and F2(X) forthe backward reaction, QA� to HA, also studied here by MDsimulations. The green lines show parabolas obtained for WTRC with nonergodic energy parameters referring to the experi-mental rate kexp (see ESI†). The purple lines indicate the equi-librium Marcus parabolas referring to the entire length of thesimulation trajectory. The mean reorganization energy, �l¼ (l1 +l2)/2, between the forward and backward reactions was used inthe analysis. We note, however, that there is a consistentasymmetry, observed in all simulations, between the forwardand backward reactions originating from the water componentof the overall solvent reorganization and leading to l2 > l1 (seeESI†).

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Fig. 3 (A) Free energy surfaces Fi(X) for the forward electron shift from HA� toQA (i ¼ 1) and the backward transfer from QA� to HA (i ¼ 2). Shown are thenonergodic surfaces (obtained on the experimental reaction time of �200 ps,green) and near-equilibrium surfaces (obtained on the entire length of MDsimulations, �100 ns, purple). The free energy surface of the HA�–QA electrontransfer for the M214LG(L) mutant (Fig. 1D) is colored blue. The solid arrowsindicate the transformation of Fi(X) from nonergodic (200 ps) to near-equilibrium(100 ns) surfaces when the protein matrix is allowed to relax on the entire time ofMD simulation. The vertical separation between the minima of the curveschanges from �0.98 eV in the nonergodic description to �0.56 eV in the near-equilibrium limit. (B) Rate of HA� to quinone charge shift (eqn (3)) vs. the ther-modynamic reaction Gibbs energy DG (from equilibrium redox potentials).81 Thetheoretical solid line is compared to experimental kinetic data for WT reactioncenters depleted of the native ubiquinone and reconstructed with a number ofquinones81 (points): (1) anthraquinone, (2) 2-ethylanthraquinone, (3) 2-aminoanthraquinone.

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The vertical separation between theminima of the nonergodiccurves cannot be identied with the equilibrium reaction Gibbsenergy DG. Indeed, the minima of the nonergodic curves areseparated by a larger energy gap x�1 eV, ensuring an activa-tionless forward electron transfer.81 When the protein matrix isallowed to relax to the nal conguration with the electronlocalized at QA, the vertical gap actually gets smaller, reducing toDG ¼ �0.56 eV. This relaxed, near-equilibrium energy gap is inexcellent agreement with the experimentally reported value of�0.55 eV calculated from equilibrium redox potentials.81 A verysimilar situation was previously found by us for primary chargeseparation: only when the protein matrix was allowed to relaxcould the vertical separation of the free energy surfaces approachthe experimental82 reaction free energy of DG x �0.25 eV.40

Overall, these calculations clearly indicate that the equilib-rium thermodynamic potentials are not the appropriateparameters to use when reactions faster than protein relaxationare concerned. This statement applies to both the reorganiza-tion (free) energy and the reaction free energy. A thermody-namic link between the two clearly exists since the sum of themmakes the average Franck–Condon transition energy hXi. Thelatter is devoid of an entropy component, which cannot develop

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on the fast time of the electronic transition; the entropy parts ofl and DG must cancel out in hXi. We now show that the tran-sition-state theory based on equilibrium thermodynamicpotentials is also not appropriate for the calculation of electrontransfer rates.

The rates calculated by iterating k in eqn (3) are listed inTable 1. We have tested the ability of our model to recover theexperimentally observed dependence of the reaction rate onquinone substitution in the WT RC.81 The rates of electrontransfer from HA to each of several quinones were calculated byattributing the changes in the equilibrium reaction free energy(from redox potentials) to changes in the electron affinity of thespecic quinone, and assigning this to the energy gap X0. Thevariation of the reaction rate, calculated by iteration of thenonergodic expression for the rate in eqn (3) (solid line inFig. 3B), is consistent with the previously published kinetics ofthe quinone-substituted reaction centers (points in Fig. 3B).81

Effect of the M214LG mutation on the reaction kinetics

We found no signicant change in the rate of HA� to QA elec-tron transfer, compared to WT RC, when MD data for theM214LG(R) mutant were used in our nonergodic kineticsformalism. The equilibrium reaction driving force for themutant was calculated from the vertical separation of theequilibrium free energy surfaces constructed from the MD dataand is also listed in Table 1. In contrast to the M214LG(R)mutant, a ip of the phytyl tail in M214LG(L) produces a majorchange to the energetics of the HA� to QA charge shi. Theenergy of the HA� electronic state remains unaltered for bothconformations. However, the average energy gap hXi becomesmore negative by �0.54 eV due to an increased Coulombstabilization of QA by the protein charges in the M214LG(L)mutant (Table 1). It is conceivable that the protein matrix ismore exible around QA than it is around HA�. This is also seenfrom a larger spread of site energies of QA compared to HA�

(Fig. S1 in ESI†).The shi of hXi is the main factor altering the rate of HA� to

QA electron transfer in the M214LG(L) mutant. This change inthe average energy gap is projected into a x0.3 eV more nega-tive equilibrium reaction free energy (Table 1). The rateconstant calculated from MD data while keeping the rest of thekinetic parameters intact (V and X0) is about 3 times lower thanthe rate in theWT RC. This result is consistent with the decreasein the reaction rate recorded experimentally (Fig. 1E). Theexperimentally recorded kinetics are, however, heterogeneous,and one possible origin for such kinetic complexity is a mixtureof conformationally distinct RCs. This complication prevents amore direct theory–experiment comparison at the moment. Theprevious application of the same formalism to the kinetics ofprimary charge separation in a number of homogeneouslyprepared RC mutants49 was, however, very successful.40

Discussion

The key nding of this work is that the broadly dispersivedynamics of proteins cause the activation parameters of

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electron transfer and, presumably, of other protein-mediatedreactions to depend in a very direct way on the relative magni-tudes of the reaction and nuclear reorganization time-scales. Thisgeneral statement has been extended here from previouslystudied �3 ps ultrafast time-scales40 to much slower �100 pstime-scales. Importantly, this is different from the more familiarconcept of protein relaxation as a function of time, and the ideathat collective motions of the protein directly contribute to thecatalytic power of an enzyme.83 We suggest that the effect of theprotein dynamics on the activation barrier is both more subtleand more universal, and is achieved by dynamical arrest oenencountered in studies of glass-formers.84 Once the reaction rateexceeds the relaxation rate of a nuclear mode coupled to thereaction coordinate, dynamical arrest restricts the congura-tional space accessible to the reacting system on the reactiontime-window and makes the activation barrier a function of therate. The most dramatic change to the traditional mechanisticviews of electron transfer is the replacement of the equilibriumfree energies (reorganization and reaction free energies) withtheir nonergodic counterparts. The main consequence of thisperspective is that it provides a new mechanism to tune the ratesof protein-mediated reactions.

This general mechanism can potentially provide a resolutionto the conundrum between spectroscopic studies of proteins,clearly showing transitions between sub-states of the proteinnative state,4,15,85 and theories of chemical kinetics, lacking aformalism to incorporate the dynamics of these transitions andtheir corresponding relaxation times into the reaction activa-tion barrier. [Note that theories considering the effect ofdynamics on the probability of crossing the activated barrier arewell established.3,22,30] Though speculative, it is hard to avoid theparallel between our present study of the protein dynamics inelectron transfer and the larger problem of protein dynamics inenzymatic reactions.3–8 We show that the strongest effect ofnonergodicity on the activation barrier occurs when the reac-tion rate is near one of the peaks in the loss spectrum of thereaction coordinate (c

00X(u) for electron transfer). There is

nothing that limits this concept to electron transfer.65 If it isapplied to enzyme reactions, then the millisecond conforma-tional dynamics recorded by NMR4,5 should affect the activationbarrier to the greatest extent when (millisecond) rates ofconformational transitions are close to (millisecond) rates ofcatalytic reactions. In order for the effect to be signicant, thecorresponding nuclear mode should produce a sufficiently highpeak in the loss spectrum of the reaction coordinate. Eventhough we cannot explore millisecond relaxation times by MDsimulation, the formalism used here has been successfullyapplied to this time-scale when the experimentally recordedStokes shi of phosphorescence shows ergodicity breaking nearthe glass transition of the solvent.74

Applying these general ideas to electron transfer, the greatestalteration of the rate, or, in other words, greatest tunability, isachieved when the rate constant is close in magnitude to one ofthe relaxation times of the Stokes-shi dynamics, i.e., k x si

�1

falls close to a peak in the corresponding loss spectrum (Fig. 2).The rates of photosynthetic electron transfer for both primarycharge separation and the charge shi from HA� to QA seem to

Chem. Sci.

avoid those regions, instead falling into the gap between the twomajor peaks in the loss spectrum (kCS and kexp in Fig. 2A). Thehigher-frequency peak is the projection of the phonon spectrumof the protein on the electron transfer reaction coordinate,while the lower frequency peak arises from shape-alteringelastic motions of the entire protein modulating the electro-static potential inside the protein.86 The fact that reaction ratesavoid both regions might reect the need for a robust perfor-mance, with low sensitivity to external conditions. Whetherslower rates of photosynthetic charge transfer fall in otherdynamic windows and whether, conversely, some reactions areoptimized for a greater sensitivity by tuning their rates to peaksof the lost spectrum is currently not known.

The ability to tune the reaction rate by the protein “thermalbath” comes not in the least part from a broad breadth ofelectrostatic uctuations produced by the protein matrix at theactive site.18,38 For electron transfer, this broad spectrum ofuctuations is reected by large values of the equilibriumreorganization energy calculated from the variance of thedonor–acceptor energy gap.79,87,88 The origin of the broad uc-tuation spectrum is the abundance of ionized and polar groupsat the surface of the protein pushed to the surface by folding.These polar/ionized residues are moved by so, low-frequencymotions altering the shape of the protein and thus creatinglarge-amplitude electrostatic uctuations.38,40,86 The combina-tion of a broad static spectrum of uctuations with the largenumber of dynamical processes that contribute to it makes theprotein a unique thermal bath in the context of activating long-distance tunneling of electrons. Nonergodic cutoff of the uc-tuation spectrum comes into this picture as an efficient tool toregulate rates of redox reactions affected by electrostatics.

Methods

The MD simulations were performed on the membrane-boundRC of the Rb. sphaeroides (Fig. 1C) as described in the ESI.†Production runs consisted of a 90 ns simulation with a savingfrequency of 1 frame per picosecond, followed by another 10 nswith a saving frequency of 20 frames per picosecond. Thisprotocol provided a total 100 ns of production data at the lowerframe density, and ensured convergence of data collection forthe high frequency motions. All simulations were performed inthe NPT ensemble using three-dimensional periodic boundaryconditions. A 1.2 nm cutoff, with a smoothing function at 1.0nm, was applied to Lennard-Jones interactions. The long-ranged electrostatics were calculated with the smooth particlemesh Ewald method. Weak coupling to a respective Langevinbarostat and thermostat at 1 atm and 300 K was used for allsimulations. All simulations and minimizations were per-formed using the NAMD 2.8 soware package.52 Eight systemswere run in total, and required an aggregate production simu-lation time of 0.8 ms.

Acknowledgements

This research was supported by the National Science Founda-tion (MCB-1157788) and by start-up funding from Yeshiva

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University (DNL). CPU time was provided by the NationalScience Foundation through XSEDE resources(TGMCB080116N). The authors thank Dr Beatty, Dr Murphy andcollaborators for access to the coordinates of the M214LGreaction center mutant46 prior to publication.

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