Fluctuational tunnel effects and vibrational dispersion in primary electron transfer processes of bacterial photosynthesis

Spectrochimica Acta Part A 54 (1998) 1201 – 1209

Fluctuational tunnel effects and vibrational dispersion in primary electron transfer processes of bacterial photosynthesis Aleksandr M. Kuznetsov a, Jens Ulstrup b,* a

The A.N. Frumkin Institute of Electrochemistry of the Russian Academy of Sciences, Leninskij Prospect 31, Moscow 117071, Russia b Department of Chemistry, Technical Uni6ersity of Denmark, DK-2800 Lyngby, Denmark

Abstract We have investigated forward electron transfer (ET) from reduced bacteriopheophytin (bph − ) to the primary quinone (QA), and the recombination ET from QA− to the oxidized special pair (bch)2+ in the photosynthetic reaction centre of Rh. sphaeroides. The same reactant, QA, is here engaged in two ET reactions of widely different distances. The processes can also be followed over broad temperature ranges. Focus is on two elements not commonly considered in ET theory. One is continuous environmental dielectric dispersion which is both formally and physically important. The other one is environmental fluctuational modulation of the tunnel factor which is important when the donor and acceptor are separated by environmental matter. The analysis shows, first that a broad dielectric dispersion is in line with the ET data up to :180°K. Fluctuational modulation of the tunnel factor is, secondly, unimportant ˚ centre-to-centre distance) but could be important in the much in shorter-range forward ET from bph − to QA (13 A ˚ ). © 1998 Elsevier Science B.V. All rights reserved. longer-range recombination ET from QA− to (bch)2+ (28 A Keywords: Tunnel effects; Vibrational dispersion; Bacterial photosynthesis

1. Introduction The electron transfer (ET) processes in the bacterial photosynthetic reaction centre have come to offer exciting probes for the controlling factors of biological charge transport. Early focus was on low-temperature ET and nuclear tunnelling where photosynthetic ET constitutes one of the few cases for detailed study [1 – 3]. Other, more recent issues are driving force effects as investigated by external electric field variation [4 – 7], site-directed * Corresponding author. Tel.: +45 45 252525; fax: + 45 45 883136; e-mail: [email protected]

mutagenesis [8,9], ubiquinone substitution by other quinones [10,11], and auxiliary chlorophyll replacement by other metal analogues [12]. Such efforts have been crucial in disentangling electronic–vibrational coupling [8,10,11,13], the function of auxiliary bacteriochlorophyll in initial superexchange or vibrationally coherent ET [12,14–16], and in assessment of the relative importance of the L- and M-branch of the organized pigments [12,17–19]. The present communication addresses the electronic tunnel factor. Broad interest has been aroused by observations regarding the distance dependence of this factor. A large number of

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activation Gibbs free energy corrected rate constants appear, first, to follow a broad exponential distance dependence over many orders of magnitude [20] ˚ −1 WDA 8 exp(− bR); b : 1.4 A

(1)

WDA is the rate constant (s − 1), R the geometric donor–acceptor distance, and b an approximately system independent decay factor. This observation is intriguing as theoretical work would, secondly, testify that the rate constant decay depends strongly on the properties of the intermediate medium [21–26]. Such directional specificity is, thirdly, supported by data for intramolecular ET in modified metalloproteins [27,28] which are better in line with coupling to intermediate residues not aligned along the geometrically shortest route. The two kinds of observation are not in real conflict. Firstly, there is often a broad correlation between the geometric distance and the directional order of best overlap. There is, secondly, notable scatter—an order of magnitude or so — in the geometric exponential correlation [20]. This prompts focus on the accuracy with which for example the activation Gibbs free energy and the three-dimensional structures are known [29]. Thirdly, the approximately exponential correlation should be approached with the understanding that it applies to some (many) systems but other systems equally unambiguously follow the directional distance correlation. The latter is supported by quantum chemical calculations which in principle include the whole protein [23,30]. In addition to dominating ET routes there would, finally, often be (many) close van der Waals’ contacting side groups of potentially efficient electronic overlap, providing additional competitive tunnel routes. Soft environmental material, in strongly exothermic ET adds still another feature. The electronic transition occurs at vibrationally highly excited nuclear configurations strongly distorted from equilibrium. The donor – acceptor overlap can therefore be quite different from its value at equilibrium [31–33]. At the same time nuclear vibrational dispersion is a better conceptual and physical representation of the environment than

single- or discrete-mode models [34,35]. This applies particularly to strongly exothermic processes and at low temperatures, where nuclear coupling and electronic tunnel features have been obtained [13,20]. In this report we assess these particular effects, illustrated by the ET reactions of quinone A (QA) in the reaction centre of Rh. sphaeroides.

2. ET rates for vibrationally dispersive media The diabatic ET rate constant for a linear medium with arbitrary dispersion is, in the limit of strong electronic–vibrational coupling [26,31– 36] WDA = (b/')(TDA)2{(2p) − 1 F¦(u*) } − 1/2 exp[− bu*DG° −F(u*)]

(2)

where b= (kBT) − 1, kB is Boltzmann’s constant, T the temperature, 2p' Planck’s constant, DG° the driving force, and TDA the electron exchange factor which couples the donor (D) and acceptor (A) electronic wave functions. F(u*) contains all information about dispersion and electronic–vibrational coupling and can be brought to cover, e.g. Bardeen–Schockley pseudopotential interactions, acoustic modes, or dielectric interactions. In the latter case [35] F(u*)=

1 '

&

0

dv j (v) v s

sh(1/2 ('vu*))sh(1/2 ('v(1− u*))) sh(1/2( 'v))

(3)

where js(v) is the environmental reorganization free energy density at the frequency v, and is related to the total reorganization free energy, Es, and dielectric permittivity o(v) by js(v)= E=

2Es Imo(v) 1 −1 ; c= o − 0 −os ; pvc o(v) 2

&



dvjs(v)

(4)

0

where oo is the optical and os the static dielectric constant. The parameter u* coincides with the transfer coefficient, u°= − kBT d ln WDA/dDG° and is determined by

DG°=

2Es pc

A. M. Kuznetso6, J. Ulstrup / Spectrochimica Acta Part A 54 (1998) 1201–1209

&

0

dv Imo(v) sh(1/2(b'v(2u* − 1))) . v o(v) 2 sh(1/2( b'v)) (5)

The derivation of Eqs. (2) – (5) is shown elsewhere [35,36]. Presently we note: (a) Eqs. (2) – (5) represent a precise procedure for calculation of the rate constant; (b) the equations can be combined with local, harmonic and anharmonic nuclear modes; (c) the equations apply at all temperatures and provide simple limiting forms. In the high-T limit Eqs. (2) and (5) reduce, for example to the well known form WDA =





p kBT' 2Es



1/2

(TDA)2 exp −

(Es +DG°)2 4EskBT

1 DG° u* = + . 2 2Es

n

(6)

The following activationless form is appropriate to several photosynthetic ET steps

! &

1 'Es 2 W act1 DA = (TDA) ' pc

0



dv Imo(v) 1 cth b'v v o(v)2 2

"

−1

u* :0.

(7)

Other forms are available on expansion of the rate constant to higher order terms in u* .. For example close to the activationless region u* : 2(Es +DG°)/bD2;



D= 2 ('Es/pc)

&



dv 0



Imo 1 cth b'v o(v) 2 2

n

1/2

; (8)

(d) the dielectric susceptibility is available for water and some aqueous solutions [37 – 40]. The water spectrum has recently been used to investigate o(v) in simple ET reactions [41]. We shall use instead the Debye and resonance forms [42] oD(v)= oo +

os +oo 1− ivtD



(9)

n

1 1 − ivRtR oR(v)=oo + (os −oo) 2 1 − i(v + vR)tR +

1+ ivRtR 1+i(v+ vR)tR

(10)

where vD and vR are the Debye and resonance 1 frequency, respectively, and t − the damping coR

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efficient of the resonance mode; (e) a crucial feature of the Debye and resonance distributions is, finally that they are continuous even for the smallest frequencies. This is characteristic for disordered media and ensures process irreversibility at low temperatures. The condition is that a= lim v “ 0(p/')js(v); aD = 2Es/'vD; aR = 4Es/tR('vR)2

(11)

is finite, where aD and aR are the Debye and resonance forms of a, respectively.

3. Dispersive nuclear configurational fluctuation effects on the electronic tunnel factor The electronic tunnel factor in long-range ET is exposed to fluctuations in the environmental configuration, leading the latter to be mostly quite different in the transition state and in the reactants’ or products’ equilibrium states. This feature is not commonly addressed in ET analysis. The modulation of the electronic overlap is important already within the Condon approximation but self-consistent interaction between fluctuationally modulated wave functions and the environment also leads to potential surface distortion and nonCondon effects [31–33,43]. This was noted by Teichler [44] and by Ivanov and Kozhusner [45] in the context of solid state light particle transfer. Self-consistent schemes for electron density confined in a cavity surrounded by a dielectric was introduced by Basilievskij and Chudinov [46] and recently by Newton and associates [47]. The physical origin of these effects are different from those investigated recently by Stuchebrukhov and associates [48]. They addressed inelastic electronic–vibrational interactions similar to inelastic tunnel spectroscopy in solid-state tunnel configurations [49]. These analogies have also been discussed much earlier [50–52]. In several reports we have investigated the effects of continuous, vibrationally dispersive and nondispersive medium fluctuations on the electronic tunnel factor [26,31–33,43]. Variational schemes and analytical trial wave functions were used in the Condon limit and in a full self-consistent scheme. The

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effects are small in models where the donor and acceptor are confined in the same solvent-free cavity but much larger if they are separated by solvent. The effects are also attenuated by the space occupied by the reactants but overall strong if the reactants are separated by solvent, giving an approximately exponential dependence of the overlap factor on the driving force and distance. An attractive observation is that the fluctuational tunnel corrections reduce to simple analytical forms when the ET process is strongly exothermic and the excess electron localization notably weaker on one reactant than on the other. This might apply to the fast charge separation and recombination reactions of bacterial photosynthesis. We first summarize a few of these observations and then assess the effects in relation to the forward and recombination ET reactions of QA. The most important features are well represented by the Condon approximation and the single-electron donor and acceptor wave functions

wardly, to anisotropic wave functions such as pand d-orbitals, and to linear combinations of such orbitals [43]. The analytical form of the variational equations are then lost, and the magnitude and directional features of the modulation effects composite but the fundamental modulation phenomenon and self-consistently determined nonCondon effects [43] would remain. The variational calculus schemes are given elsewhere [31–33,43]. We consider here the simplest case where the ET process is approximately activationless and the electron significantly delocalized on one reactant, say lD B (B)lA. The overlap is then dominated by the smaller exponent lD. Expansion in the transfer coefficient gives

cD =(l 3D/p)1/2 exp(− lD r );

j=

cA =(l 3A/p)1/2 exp(−lA r −R )

(12)

where r is the space coordinate counted from the donor centre, and R the donor – acceptor distance. Fluctuational modulation is reflected in the orbital exponents lD and lA. These are obtained by variational calculus of the Gibbs free energy functional of the electronic densities represented by Eq. (12) in the polarization field P *(r) (space point r ), at the moment of ET. P*(r) is in turn determined by the equilibrium inertial polarization in the reactants’ and products’ states, and by the vibrationally dispersive dielectric permittivity. Eq. (12) appears crude but has broader perspectives than implied by its simple 1s-form. First, approximately exponential forms in fact emerge from detailed quantum chemical calculations [25,26] and from experimental data (Ref. [20], and above) when the donor and acceptor are well separated. Environmental and other effects are then inherent in the orbital exponents lD and lA. The variational scheme exploited below [31 – 33,43] can, secondly straightaway incorporate pseudopotential, cavity, and other local effects. The scheme also applies, in principle straightfor-

lD(u*): lD0(1−jyu*R)

(13)

where lD0 is the value of lD at configurational equilibrium. j and y are given by the donor core charge zD and the dielectric frequency dispersion 5/11 ; 1+ (16/11)(zD/cos)

y=(2/pc)

&

0

dv Imo(v) 1/2( b'v) . v o(v) 2 tan(1/2( b'v))

(14)

In the high-T limit y= 1, and lD reduces to lD(u*)lD0 : [1−(5/11)u*R] for a neutral core. By Eqs. (12)–(15) the electron exchange factor is TDA(u*; T): T 0DA exp[− l 0D(1−jy*R)]

T 0,eq DA exp(lD0 jyu*R); u*“ 0. (15) In combination with the nuclear factor the rate constant is WDA : W 0DA exp[2lD0 ju* −bEs(u*)2] 0 DA

(16)

where W is the pre-exponential factor in the absence of environmental modulation. Eqs. (13)–(16) call for some observations. Firstly, the wave functions expand on polarization fluctuation, i.e. TDA (u*\ 0)\T 0DA. These effects can be significant. By Eq. (15) the tunnel factor rises 20-fold when u* increases from 0 to 0.3 (zD = 0), corresponding to a DG°-change from 1 ˚ − 1 and R=15 A ˚ . The to 0.6× Es, if lD0 = 0.7 A corresponding drop in the nuclear factor is 2-fold

A. M. Kuznetso6, J. Ulstrup / Spectrochimica Acta Part A 54 (1998) 1201–1209

if Es =0.5 eV and 36-fold if Es =1 eV. Opposite effects, i.e. wave function contraction appears in the inverted region. Secondly the effect increases with increasing ET distance. The correction factor in Eq. (15), for u* =0.3 thus rises from : 7 ˚ . Thirdly to 45 when R increases from 10 to 20 A the modulation depends on T but only far from the activationless region where the simple form in Eq. (15) no longer applies. u* (Eq. (8)) thus increases whereas y (Eq. (14)) decreases with increasing T, the product u*y being independent of T. Fourthly the u*-variation of the tunnel factor gives a red-shift of the Gibbs free energy relation in the high-T range with maximum at u*m = (kBT/Es)lD0jR\ 0. u*m = 0 when only the nuclear factor is considered. The shift is from −DG° = Es to − DG°= (1 −2u*m )Es =(Es − 2kBTlD0jR)B Es. By Eq. (15) it amounts to ˚ − l and R =10 – 20 A ˚. 0.15 –0.3 eV for lD0 =0.7 A This effect can be addressed experimentally if free energy relations can be extended to broad ranges of temperature. A recent report on the back ET reaction from the primary quinone to the special pair in several mutant Rh. sphaeroides reaction centres in fact exhibits a shift in the approximately quadratic Gibbs free energy relation [53]. The observed effect is, however, in the opposite direction and dominated by the T-dependence of Es or DG° (freezing of environmental dipoles at low T) which is then even stronger than concluded in Ref. [53]. Fig. 1 shows the full DG°-dependence of lD and lA for a broad resonance dispersion (Eq. (10)). The exponents drop to about half their values at equilibrium in the u*-range 0 – 0.5 (DG°= − (Es −0). Fig. 2 shows the corresponding T-dependence of u* and y at given DG° and Es. Core, and other local charges have been disregarded in this calculation but are implicitly inherent in the equilibrium orbital exponents lD0 and lA0 [54]. Positive and negative donor core charges would attenuate and enhance the modulation effect, respectively, whereas the effects from the acceptor centre are opposite. Other local charges can be incorporated and would give a multifarious pattern depending on their location. The dielectric constant, os to be used when

1205

Fig. 1. DG°-dependence of lD/lD0 and lA/lA0. o(v) given by ˚ ; T= 298°K ( — ); Eq. (10). vR =tR− 1 =150 cm − 1; R =10 A T=37°K ( – – – ). No core charges.

local charges are incorporated would then warrant a note of observation. The intrinsic value of os inside a protein is small, say, : 5 due to the absence of orientational polarization. The actual or ‘effective’ value is, however, always much larger (10–20), as the dielectric screening is partly through the external aqueous medium in a finite-size protein medium. In this sense the dielectric constant holds, of course, characteristic distance and orientation dependences such as investigated numerically by molecular models [55] and analytically by finite-size globular dielectric models [56].

Fig. 2. T-dependence of: u* ( —); y ( · · · · ); u*y( – – – ) for the same dielectric permittivity function as in Fig. 1. Es =0.5 eV; DG° = − 0.2 eV. Left scale:u and u*y. Right scale: y.

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Fig. 3. T-dependence of the activationless rate constants for the dielectric permittivity function in Fig. 1. Es = 0.6 eV: (a) forward ET from bph − to QA; (b) recombination ET from − to (bch)2− . Data points from Refs [6,49]. QA

4. Fluctuation and distance effects in photosynthetic charge separation and recombination Primary, ultrafast charge separation from the special pair, (bch)2 to bacteriopheophytin, bph is a case for coherent two-step ET via vibrationally unrelaxed reduced auxiliary chlorophyll. Longrange ET in the bacterial photosynthetic reaction centre is, however, rather associated with the forward and recombination reactions of the primary and secondary quinones QA and QB [6,8 – 11,57] bph − + QA “ bph + QA −

(17)

(bch)2+ + QA− “(bch)2 +QA

(18)

These two reactions constitute a case for distance, temperature, and electronic fluctuation effects for the same reactant engaged in two different biological long-range ET processes where ET is along approximately the same directions. Half-lives, t1/2, and driving forces used in the following are from Refs [6,58]. For Eq. (17): t1/2 :0.2 ns, DG° : − 0.6 eV. For Eq. (18): t1/2 :0.05 s, DG° : − 0.5 eV. The centre-to-centre distances are 13 and 28 ˚ for Eqs. (17) and (18), respectively [58]. The TA and DG°-dependence of the rates are also available [6,8,10,11,53,57]. The T-dependence of the rates (Fig. 3) shows, first that both reactions are independent of T up to :50°K but decrease weakly at higher temperatures. This suggests that the reactions are activationless but Eq. (7) only reproduces the data up to

: 180°K. Quantitative agreement over the whole T-range requires either that some environmental protein modes represent relative donor–acceptor group motion, major conformational reorganization, phase transitions etc., or that the kinetic parameters themselves depend on T. This conclusion also applies if local high-frequency modes are added. T-dependence of the kinetic parameters is, interestingly, supported by data for the recombination, which suggest that the reorganization Gibbs free energy is larger at room temperature (: 1.3 eV) than at cryogenic temperatures (0.6– 0.8 eV) [8,10,11,13]. This would transfer the process from the activationless to the normal free energy range as the temperature is raised, giving a faster rate drop in an intermediate T-range. The electron exchange factor obtained from Fig. 3 and Eq. (7) are 6×10 − 4 eV for forward ET and 5× 10 − 8 eV for the recombination. This can be correlated with the exponential distance dependence in Eq. (15). Use of T 0DA = 1–10 eV from electronic mobility of organic crystals [3] is presently adequate. Combination of Eq. (18) with the centre-to-centre distances then gives the same ˚ − 1 for both decay factor lD (u*)= 0.6− 0.7 A reactions. This value is also in line with the expo˚ − 1(Eq. (1)) in the broad distance nent of b=1.4 A relation reported [21]. The orbital exponent of the long-distance recombination reaction is little affected if edge-to-edge distances are used instead, ˚ − 1 while the value for giving lD(u*)= 0.7–0.9 A the forward reaction assumes the unphysical value ˚ − 1. These observations pave the way of 1.4–1.9 A for concluding that the centre-to-centre distance is a major controlling factor for the widely different TDA-values of the two reactions in Eqs. (17) and (18). We proceed next to assessment of environmental fluctuational effects. The recent data for the long-range ET between QA− and (bch)2+ in Rh. sphaeroides [53] indicated that these effects could here be significant although the observed effects could also be rooted in structural changes over the broad temperature range used (10–298°K). We invoke three constraints: (a) the equilibrium orbital exponents of the semiquinone should be the same in the two reactions since the ET directions are only slightly different; (b) the exponent

A. M. Kuznetso6, J. Ulstrup / Spectrochimica Acta Part A 54 (1998) 1201–1209

for the confined (bch)2+ ground state acceptor orbital is significantly larger than for the anion radical states bph − and QA− ; (c) the electron exchange factor is dominated by the wave function with the lower orbital exponent. We can then distinguish two limits: (a) modulation can be insignificant. The ‘observed’ constant decay factor ˚ − 1 then suggests that the equilibrium of :0.7 A orbital exponents follow approximately l bph Do : sp.p l quin B(B )l where the superscripts refer to Do Do the pigments (bacteriopheophytin (bph); quinone (quin); special pair (sp.p)). If, for example l bph D0 B quin ˚− ( B)l quin , then we would have l \( \ )0.7 A D0 D0 quin 1 in conflict with the value of l D0 in the back reaction; (b) two implications follow when modulation is significant. Recombination accords, first sp.p sp.p with l quin D0 B (B)l D0 (u* :0) Bl D0 . This implies −1 quin ˚ that l D0 :0.7 A while, from Fig. 1 l sp.p D0 :1.2 −1 bph bph ˚ A . l D (u*): l D0 .in the forward reaction must ˚ − 1, as l bph ˚ −1 then also be :0.7 A D0 B( B)0.7 A would give a correspondingly smaller ‘observed’ value. If fluctuation effects are important in the recombination reaction, they cannot therefore be important in the forward reaction. If the opposite inequality should apply in the recombination resp.p action, i.e. l quin D0 \(\ )l D0 (u : 0), then the decay − 1 ˚ factor of : 0.7 A represents l sp.p D ,and would quin ˚ −1 imply that l D0 \(\ )0.7 and l sp.p D :1.2 A (Fig. 1). This condition is only compatible with the data for the forward reaction if also l quin D0 : quin ˚ − 1. Otherwise l bph 1.2 A \l (u* : 0) in forD0 D ward ET and a smaller lD for this reaction should ˚ again be observed. l quin D0 :1.2 A is, however, far too large for a weakly bound anion radical state in a medium. The conclusion is therefore, first that environmental fluctuation effects on the tunnel factor is unimportant in the forward, shorter-range ET reaction of QA but could be important in the much longer-range ET of recombination if the equilibrium decay factor of the confined (bch)2+ ˚ − 1. acceptor wave function :1.2 A

1207

general affect the electronic tunnel factor in longrange ET when the donor and acceptor centres are separated by intermediate solvent or protein. A particularly interesting effect is the enhanced ET in the normal or activationless free energy range. The QA-reactions in the bacterial photosynthetic reaction centres hold perspectives for illuminating these effects. The effect seems unimportant in forward ET from bph to QA, probably ˚ centre-to-cendue to the short ET distance (13 A ˚ edge-to-edge) between the reaction tre, :5 A partners. On the other hand, the data indicate that the effect could be important in the longrange recombination ET from QA− to (bch)2+ (28 ˚ centre-to-centre, 22 A ˚ edge-to-edge). The A present data analysis is of course not definite as to the actual importance of the effects in the systems considered. The conclusion is rather that ET theory and suitable models point to the order and physical features of the effects, while the data are in line with the prevalence of such effects but also with different parametric features in the absence of the effects. The analysis also points to further steps. One would be quantum chemical calculations of the wave functions of the reaction partners, bph − , QA− and (bch)2+ which would provide distinction between the alternatives noted. The other one would be to extend broad Gibbs free energy relations of modified metalloproteins and other biological long-range ET systems to wide temperature ranges, such as in the recent data by Mathis and associates [53]. We note finally that still another fluctuation effect can assist. Spatially randomly fluctuating tunnel barriers facilitate tunnelling compared with the average barrier [59]. This could in principle be given a more specific substantiation in terms of the structural coordinates of amino acid fragments in the bacterial photosynthetic reaction center.

Acknowledgements 5. Concluding remarks Nuclear configurational fluctuations must in

Financial support from the Danish Natural Science Research Council and the EU programme INTAS is acknowledged.

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