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Electron transfer via a midway molecule as seen in primary processes in photosynthesis: Superexchange or sequential, or unified?
ELSEVIER
Journalof Electwanalytic,al Chemistry438 (1997) 11-20
Electron transfer via a midway molecule as seen in primary processes in photosynthesis: Su~rexchange or sequential, or unified? ~ Hitoshi Sumi Institute of Materials Science. University of Tsulmba. TsukaOa. Ibaraki J~)5.
Received20 Septemlx~1996;receivedin revisedform2 Janumy 1997
Abstract A typical e~ample of electron transfer (ET) mediated by a midway molecule is the initial El" in bacterial photosymhesis fiem the excited special pair (P" ) to the bacteriopheophytin (BPh) mediated by the accessory, chlompbyil monomer (BChl) stationed in between. It has been argued intensively whether this El' is sequential (where the state Ira) of P+-BChl--BPh exists as a teal chemical imennedi~) or superexchange (where it is passed as a quantum-mechanical ~rtual slate). This ET is isomorphic to second-tmkr optical pmcessas (SOOP) where the initial state composed of an incoming photon and a matter is connected to the final state composed of an ealgoing photon and a matter with phonons left excited tlmmgh an intermediate state composed of a matter cleclnmicallyexcited withom a pht~on. SOOP is a single process reducing to Raman scattering or an a b s o ~ - l u m i n e s c e a c e sequence in mutually opposite limits. Correspondingly this ET is also a single process, not composed of two coexisting parallel channels by superexchange and suluemiat ETs. It reduces to them only in mutually opposite limits determined by competition between the lifetime of aa electnm and the r e ~ time of the medium at ira). The rate constant of this ET can be formulated by extending the fonn,!a6on for SOOP. It can describe satisfactorily the initial El" in photosynthesis observed. We can predict, ~ v e r , that s u p e r e x ~ El" should begin to manifest itself in a low-temperature region when Ira) is raised by several l,undmt cm - t from its native position, with its validity range toward higher temperatures as [m) is raised further. © 1997 Elsevier Scienc q.A. © 1997 Elsevier Science S.A. Keywords: Electron transfer, b~idwaymolecule:Photosymhesis:AccessoryChlorophyll
1. Introduction Recently, a new stage seems to have emerged in the investigation of electron transfer (ET), being devoted to ET mediated by a midway molecule in condensed matter, rather than the direct ET between two molecules investigated so far. The dawn of this stage was the first clarification in 1984 o~ the three-dimensional structure of the protein-pigment complex of the photosynthetic reactieq center of purple bacteria [1]` The beautiful arrangement of pigments clarified has n ~ only disclosed a pathway for ET, but also stimulated imagination about its development in the evolution of photosynthesis, initiating many researches (for a review see Ref. [2]). In the reaction center, El" (to he mine exact, charge separation) starts from the excited bacteriochlorophyl! dimer ( P " ) to the bacteriopbeophyfin (BPl0, and then an electron is further trmmfened front the reduced bacteriopbeophytin (BPh-) to the quinone (Q). As a result, an electron is transferred from the inner side ot" the bacterial membrane to the outer side counter to the membrane potential, and the energy of sol=" radiation exciting the bacteriochlorophyll dimer (P) is converted into the electrostatic energy across the membrane. Here, P and BPh are so far apart from each other (by a ~ t 17 A center to center and about 10A edge to edge) that direct coupling for El" from P* to BPh is too weak to account for the observed reaction time which is as small as about 3ps [2]. To mediate the El', a bacteriochlorophyll monomer (BChl) is stationed between P and BPh. Then it has been argued intensively [2] whether the ET mediated by BChl is sequential or superexehange. In the sequential ET, the i n t e m ~ i a t e state of P + - B C h l - - B P h exi~As as a chemical reality between the initial state P * - B C h I - B P h and the final state P + - B C h I - B P h - , while in the
t This paper was presentedat the InternationalSymposiumon EtectronTzansferin Proteinand SuWanmle~ar Assembiiesat Interfacesheld in Shomm Village, Kanagewa,Japan on 17 to 20 March 1996. 0022-0728/97/$17.00 © .'997 ElsevierScienceS.A. All fights resecved. Pil S0022-0728(97)00023-5
12
H. Swni/Journal of ElectroanalyticalChemistry438 (1997) 11-20
superexchange El', it is passed as a quantum-mechani~cal virtual state. It has been argued Rat ET from BPh- to Q is also mediated by a midway molecule. They are about 14A apart center to center and about IUA apart edge to edge [3], and it has been argued that the F'l" is mediated by a tryptophan residue of the protein matrix [4]. ET via a midway molecule can also be regarded as an example of long-range ET in protein matrix, which has been attracting much attention recently [3,5]. It is also incorporated in scanning tnnncling microscopy (STM) processes mediated by a redox state in an adsorbete molecule on a substrate [6j. This .;s important as an electrochemical application of STM for obtaining an atomic-scale image of biopolymers in liquids. Let us express ET via a midway molecule as occurring from a donor state [d) to an acceptor state la) mediated by an intermediate ~ Ira). This ET is isomorphic to second-order optical processes in which an initial state li) composed of an incoming photon (of energy E) and the matter is converted into a final state I f ) composed of an oul~oing photon (of energy smaller by A than E) and the matter with a phonon (of energy A) left excited inside. Since there exists no direct coupling between these states, they are mediated by an intermediate state Ira) which is composed only of matter electronically excited (at energy E,,) without photon. When E , is much higher than E, the intermediate state is pas~d only as a quantum-mechatdcal virtual state since energy cannot be conserved there. This situation is described as Ranlan scattering of light by matter, which is a concept corresponding to the superexchange ET in our original problem. When E,, i~. comparable to E on the seale of w~dths of ]/) and If), the situation of resonance Raman scattering emerges. It is known [7,8] that two quantities play an important role in this situation. One is the thermalization (or dephasing) time ~-,, of the matter at Ira), while the other is the lifetime l,, of an electron at Ira). When ~-,,:~ Ira, Raman scattering is applicable since ~'m is so large that the quantum-mechanical coh~reuce between Ii) and I f ) is not lost through the mediation at Ira). When ~-,,,~ l,n, on the other hand, the whole process can be divided into light absorption by a matter (from li) to Ira>) and subsequent luminescence from it (from Ira) to If>) since ~'m is so short that the quantum-mechani~:al coherence is quickly disrupted at Ira). This case of consecutive absorption and luminescence processes corresponds to the sequential ET in our original problem. In general, neither Raman scattering nor the absorption-luminescence sequence are appropriate. They can be applied only in the mutually-opposite limits of a unified single process, new in itself, called the resonance Raman scattering. This knowledge on the resonance Raman scattering can be applied directly to ET via a midway molecule. In this case, ~'m should more ~,pt~riateIy be called the reorganization time of the medium surrounding the midway molecule at its reduced state, since thermalizafion occurs in association with reorganization of the medium at a single localized state. Since both concepts of superexchange and sequential ETs are applicable only in the mutually opposite limits (respectively of ~'m:~ l,, and ~'m"~ l,~), especially, they cannot coexist as two parallel channels. Unfortunately. however, the initial El" in photosynthesis has often been a,alyzed by an assm~tption of i~,, coex~tlng chaJ,tels by superexchange and sequential ETs, and the branching ratio between them has been discussed [9-13]. In general, ET as seen there should be described as neither superexchange nor sequential in terms of old concepts for direct ET between two molecules. We are looking at a unified single process, new in itself, called ET mediated by a midway molecule in parallel to the resonance Raman scattering in second-order optical processes. An analytical formula for the rate constant of the ET can be obtained by extending the formulation for resonauce Raman scattering, with the procedure sketched in Ref. [ 14]. The purpose of the present work is to show how the mutually opposite limits of superexchange and sequential ETs are bridged by this formula for the ET as a unified single process, with an example of an explicit calculation taken from the initial El" in photosynthesis.
2. Rate constant for ET via a midway molecule as a unified single process Let us write as kad the rate constant for ET via a midway molecule as a single process. In the mutually opposite limits, kad should reduce to rate constants for superexchange and sequential ETs, denoted respectively as j.(sx) "ad and k (sQ) ad , whose expressions are well known as written in section 2 of Ref. [14]. Sin~e k (sx)oJis fourth order in electronic coupling while k~dQ) is second order, it is in principle impossible to derive a formula fi)r kad by a simple perturbational expansion in electronic coupling. It was first noted in Ref. [14] that the formulation ,:ould be realized by a renormalization appfied to the perturbational aplm3ach which was used for resonance Ramari scattering. Its explicit reduction will be given in a fortlgoming paper, but the final form of kad is sketched in Appendix A. It was noted by Kharkats et al. [15] that ~" ~.(sx) ad contains a c-mponent which describes a real transition of an electron from ]d) to Ira) followed by that from Ira) to la). In this component, the second transition takes place during vibrational relaxation at [m) triggered by the first transition, and its role was intensively discussed by them. Since k~x) is fourth order in electronic coupling, however, the component never reduces to k(asQ). Anyhow, this phenomenon is among the so-called hot transitions which were first noted by Hizhnyakov and Tehver [16,17] in solid-state physics. The rate constant kad for the unified single process covers this phenomenon naturally too. Ref. [14] starts from the divergent fourth-order perturbational expansion of the Liouville operator. The divergence was suppressed by renormalization in Ref. [14]. Hu and Mukamel [18],
H. . Y ~ / Jomml qf ~
CkeJm'y 438 (199"/)11-20
!~
however, suppressed it by taking into account distribution of electn~ic energies at Id), Ira) an~ ~a). Thus, the magnitude of the rate constant obtained by them depends essentially on the width of the ~ b u t i o n , remaining fourth order in etecaemie coupling. It was shown in Ref. [14] and also in Appendix A that in ET via a midway molecule as a unified single laxgess its degree of sequentiality can be d e t ¢ ~ by a quantity ( < 1) PSQ m
[kSQ)/k.d]X (survival probability of an electron at Ira) until ~-,)
{1)
Since the superexchange and the sequential ETs do not coexist as mentioned earlier, 1 - PsQ does not in general mexn the degree of superexelumgeness except when Psq "~ I or 1 - PsQ "~ 1. This PSQ describes conectly various situatiom in the ET. For example, when Ira) is much higher than both Id) and la), it should be t ~ d e d as superexchange. This ~ is described as PSQ ~ 1 for k(SQ)/kad / ad < 1 in Eq. (1) since it is vet). difficult for an electron to make a n:al ttmmitkm from [d) to Ira). Even if Ira) is located near Id), the ET should be regarded as supe~xchange in the large 1"., limit 11m situation is described as PSQ ' ~ 1 since the second factor on the fight-hand side of Eq. (1) becomes mneh smaller thaa onity although the first factor there is not much smaller than unity.
& Caleulaflen ef the rate comtaut ef ET in mmive ranetim eeatens In the real system, the thermalization time vm at ]m) has been determined as a quantity of the erda" of ~te inverse of the width of the frequency distribution of the phonon spectrum in the medium around an electron at Ira) [7,8,16,17]. In the case of bacterial photosynthesis, the medium is a protein around BChl-. It has been obseaved by hale-burning specuoscepy [19] that the phonon spectrum in proteins in reaction centers spreads broadly around a central energy of almut 30cm -t. Then, it is reasonable to estimate the width of frequency distribution of the spectnnn to be about 30cm -t divided by the Planck constant, that is, about 1THz. Therefore, it is reasonable to estimate %, to be about I ps. Free energie~ at ira) and ia) seen from Id) are respectively written as AG,, and AGo. The values of these energies were contmvenial for a long time, but it seems that they are converging to AGm ~ --500cm -t [13,20] and AG,-- --2000cm -t [21] on the basis of receat measurements. These values are adopted ~ the present cakulation. It has usually been believed that biological lax~esses have been adjusted so as to be most efficient in an eavinxauem in which they work, as a result of evolution over a ioag time. Based on this beficf, tbe initial charge scpmatkm tram P*-BChl-BPh to P+-BChI-BPh- in photosynthesis should have been adjusted so as to be fastest. Without the fastest charge separation, an excitation in P" would have been annihilated by fiucceaceuce. To accomplish the ~ the total reorganization energy associated with it must coincide with the coaesponding Gibbs energy ditfenmfe, givea by IAGm[~ 2000cm- t, since in this situation a banier'~,~s ET is realized at the boondmy between the normal and the inverted region from the standpoint of the direct ET. This situation expected from the belief is also adopted, as in many other articles. Since P, BChi and BPh are sufficiently apart mutually, let us consider as the first aplwoxinmion ~utt the which takes place when an electron is located at Id), Ira) or la) is mutually independent. In this case., the total reorganization energy on the initial charge separation is the sum of remganizatm energies at Id) and la). It is n~oaable to assume the reorganization anergy at [d) at a value comparable to 25otto-t, which was obml--uedas that UlPortexcilafiee of P to P" [19]. R is assumed to be 250cm -1, and the reorganization energy at la) is taken at 1750 ( 2 0 0 0 - 250)¢m-'. Since the former is considerably smaller than the typ/cal energy quantum (of the order of 1000¢m - i ) of iatmmolecu~ vibrations in chlorophyll species, it is reasonable to consider that it originates only from reorganizatm of the lamein medium, which corresponds to onter-snhere reo~aniT~_~ in tem~ of ~l~,ml theories for ET, being writtan as Ad. It is considered, on the other hano, tltat the latter is composed not only of the reorganization energy of the l a e ~ n medium (written as A.) but also of the inner-sphere reorganjT~i~ energy in BPh-. h is written as S.A % with the eueggy quantum A % of an intramolecolar vibration interacting with a transferred electron and the dimensionless coupling constant S. (called the Huang-Rhys factor). Considering half-and-half division, let us assume Aa ~ 850cm-' and SaA % ~, 900 cm - I , and moreover Sa a, 1.0 as a typical value. Without a clue for estimating the r e o r g a n i ~ energy at Ira), it is put as a trial at about 450cm -t, as composed of only the reorganization energy Am of the ~ medium. The remaining parameters which we need for numerical calculation are the slreagth of electronic coupling relarSonted by transfer integrals J.~ and Jam between Id) and Ira). and Ira) and la) respectively. The J . . value is the most impmtem. Unfortunately, however, it scatters from 9 to 60cm -t in the INDO calculation [22.23] and at 40cm -t in the extended Hllekel calculation [24] for Rhodopseudomonas viridis. Therefore. it is taken to be the average of them at J , da, 35¢m" ', close to the value obtained by the extended Hikkei calculation. Since it gives 20cm -t for J , , [24], it is taken at J,,, ,~ 15cm -~ in parallel to 35cm -I taken for J,,d" The transfer integral J decreases, rapidly with t h e ~ r of ET. It has been known empirically [3] that j2 decays as e x p ( - f l r ) with /]-7 1.4A -t in the region of 5A_
14
H. Sumi / Journal of Electroanalytical Chemistry 438 (1997) 1i -20 2.0
t
TO~
~m
I
,
I
= ° "~O cm'l
,
I
--. ~ef~J~
Q ~"
0.5 ! 0.5. 0.0
o
,~o
~;o
~o
00
Temperature / K
Fig. 1. Temperature dependence of the rate constant of ET from the donor Idt to tire acceptor lat state mediated by the intermediate lint state; the solid line with the ET treated correctly as a unified single process, the dashed line with the ET approximated as sequential, while the dotted line with the ET approxiramed as superexchange. The degree of sequentiality of the El" is given by the dash-dot line plotted with a scale on the right-hand side. The calculation was performed with parameters of AG,n = -500 and AGo= -2000cm -I as the free energy respectivelyat lint and lot seen from Id). A d = 2 5 0 , A m ~ 450 and Aa = 850cm- t as the reorganization energy of the protein medium respectivelyat Id), lint and [at, S,~he%= 900cm- i as the reorganization energy at la) by an intramolecularphonon with energy quantum h o~a = 900cm-~ and So = 1.0 as the coupling constant, Jm,ew 35 and Jan, = 15cm-I as electroniccoupling represented by the transfer integral respectivelybetween ]d) and [mt and between [mr and ]at, and ~'~,= 1.0ps as the re,Jrganizationtime of the protein medium at lint. means that j 2 becomes half when r increases by only 0.5,~. Unfortunately, however, it is at the 2.3,~ resolution that the structure of the protein-pigment complex has been detern'~ned [1]. Therefore, these values of Jm,t and Join are subject to change with the future achievement of higher resolution structure determination a n d / o r the development of calculation teclmiques. These values are also important in determining how fast an electron decays from Ira) to ]d) or la). With these Yal~es set above, therefore, we have determined the lifetime I m of an electron at [,,~ whose magnitude in comparison witi, ~-,, is important in determining the degree of sequentiality in ET. The rate constant koa calculated with the formula in Appendix A is shown by a solid line in Fig. 1 as a function of temperature below 350 K. This nicely reproduces the rate constant [25] observed in R h o d o p s e u d o m o n a s viridis with 7,espect to both its magnitude and its temperature dependence. The dotted line in Fig. ! shows k ~ x) justified in the superexchange limit, while the dashed line therein shows '~od"(sQ)justified in the opposite sequential !Emit. Neither reproduces the observed rate constant.. The degree of sequentiality PsQ of Eq. (1) is plotted as the dash-dot line with a scale on the right-hand side. It is about 0.8 over the whole temperature range. This is consistent with recent tendencies towards an agreement that the initial ET in photosynthesis in purple bacteria is understandable in terms of sequential ET [26]. By analyzing the content of the calculation, we can see that after transfer of an electron from Id) to Ira>, the electron is transferred to la) before ~',n in
2..0
~:)
, Z~m=
1.5- - -
I , - 500 cm" "",
I
, ~
I ~
AG. = - 2000 cm "I
1.0
1.0. ~
0.5
..................................................................... 0.0
o
o
1~o
~o
,
~o
0.0
Temperature / K
Fig. 2. Samequantifiesas shown in Fig. !, calculated with ~rm = 0.5 ps and other lxaramcterskept unchanged.
H. Sumi/Journal of Electroanal)vical Cl~misu'y 438 fl997) ! 1-20 ,
'1.6
~. ~.4 i
-,
I
,
I
~
~
= -500~
~.
= . ~OO
I
~
-
c m "~
1.2 1.0
0.8 O.a ~
0.4 0.2 0.0 209. Tempma!ufe/K
Fig. 3. Temperaturedependenceof the rate constantof the ET in Fig. I as a unifiedsingleprocess,calculatedwith J,a = 25cm-t and other peramete~ kept unchanged,shown in cc:nparisonwi~ ~ one for 3,,a = 35cm-t in Fig. 1.
the course of reorganization at Im) (that is, not af~t completion of reorganization) at a probability of about 0.6. Therefore, the second ET takes place as a hot transfer, whose role was intensively discussed fwst in Ref. [15] as mentioned in Section 2. In the calculation for Fig. 1, the phonon frequency distribution in the protein medium was assumed to have a width of about 30cm - I , giving 1"m = I ps. Let us change it to T,, = 0.5 ps, assuming the distribution to have a width of abont 65cm -! centered at about 65 cm -~. The distribution detected in the hole-burning spectrum is so broad that this does not seem unacceptable either. Numerical calculation for ~', ffi 0.5 ps with other parameters kept unchanged gives Fig. 2. The rate constant k~d shown by the solid line therein is nearly unchanged from that in Fig. 1 for ~-,,= !.0 ps in both magnitude and temperature depet~dence. The distance between P and BChl is shorter by about 0.5 h, on average in Rhodobactor qd~aero/des than in Rhodopseudomonas viridis in the crystalline state structural determination [27], although that between BChl and BPh is unchanged. If this difference is taken fiterally, J , d should be about 1/~/2 times as small in the former as in the latter, from the empirical rule on the distance dependence of the transfer integral in the protein medium mentioned earlier. Since 3,d in the latter was taken at 35cm - t in the calculation in Fig. 1, it is interesting to calculate the rate constant k~,~ for J , d = 25cm - I ( = 35/~/2cm - I ) with other parameters kept unchanged from those used in Fig. 1. The result is shown as a function of temperature in Fig. 3 in comparison with that shown in Fig. 1. We see that the rate constant calculated with JMd~ 25cm - I reproduces nicely that [25] observed in Rhodobactor sphaemides with respect to b o ~ magnitude and temperature dependence. This agreement seems surprising, however, since the free energy AG~ at Ira) relative to Id) is also very important in determining the rate constant as shown in the next section.
4. Calculation of the rate constant of ET for modified reaction centers In the calculation in the previous section, the total reorganization energy given by Ad + Ao + S ~ ~ set so as t.o coincide with the free energy decrease IAGo[from Jd) to la) according to the belief mentionec, in the secou~ pm'agraph therein. It is in,cresting to check that belief. To this end, the position AG~ of ]a) relative to ~,d) was first lowered by 1000~m -I from its original value of - 2 0 0 0 c m - i to - 3 0 0 0 c m - ! with other parameters (including the total reorganizatioa energy) kept unchanged. The rate constant obtained was almost the same as that in Fig. I with respect to both magulmde and temperature dependence, although it is not explicitly slaown here. Next, ~G~ was raised by 1000cm -~ from its original value to - 1000cm -~, and we got the rate constant shown by the sofid fine in Fig. 4. It is still nearly unchanged from the rate constant shown in Fig. 1, although the rate constant decreases to about two thirds of that in Fig. 1 at room temperature. This is consistent with experimental finding that the rate constant decreases to about one half as much at room temperature when AG° was raised by about 1500cm - j , by replacing BPh in pheophytin in Rhodobactorsphaeroides [28]. Anyhow, the rate constant is nearly unchanged by the change in AGa. Therefore, the belief which has been widely e~cepted so far does not seem to be taken into account in riving organisms. With a change in the energy of Ira), on the other hand, the rate constant becomes subject to marked change..as ~m) is raised, especially, the ET should become describable as superexchange. It is interesting to investigate where it becomes so.
16
tt. Sumi /Journal of Electroanclytical Chemistry 438 (1997) 11-20
7m o
2.0
,
"
1.5-
i
I
,
I
,
AGm=" 500cm" " , : : ~ ",
1.o- ~ " 0.5 ,..
I
AGa = - 1000 CITI"1
"
1.0 0.5i
........................................................... 0.0
0
,
,
,
100 200 300 Temperature /K
0.0
Fig. 4. Same quantities as shown in Fig. 1, calculated with A Ga = - 1 0 0 0 c m - ~ and other parameters kept unchanged.
Let us first raise the free energy AG, of Ira) relative to [d), only a little, by 200cm -z from its original value to - 3 0 0 c m -m with other parameters kept unchanged. The rate constant calculated with &Gin= --300cm -I is shown by the solid line in Fig. 5 as a function of temperature. We see therein that the rate constant in the lowest temperature region decreases to about one thirtieth of that in Fig. 1 for AG, = - 5 o o c m -z, although the rate constant at room temperature decreases by a factor of only about one half. Especially, the temperature dependence becomes opposite to that in Fig. 1 below about 100 K, since the rate constant decreases with a decrease in temperature there, although the dependence remains unchanged from that in Fig. 1 above about 100K. The degree of sequentiality remains still about 0.8, similarly to Fig. 1. When AG, is raised by 5oocm -I from its original value to 0cm -i, the temperature dependence of the rate constant changes into that shown by the solid line in Fig. 6. It tends to d(:crease with a decrease in temperature in the entire temperature region. It should be noted, moreover, that the degree of sequentiality of this El', shown by the dash-dot line, becomes much smaller than unity below about 50 K although it d e c k s still to about 0.5 to 0.6 above about 150K. This means that an electron must be transferred only by means of supep;xchange ET below about 50 K since a real transition from [d) to ira> by thermal activation becomes difficult at low lemperatures due to the raise in AGm from its value estimated for native bacteria. (Note here that although AG~ vanishes in Fig. 6, there exists a non-vanishing thermal activation energy for real transitions from [d> to [m> because of a non-vanishing reorganization energy (700 cm - I ) associated with this El'.) In the temperature region above about 50 K, ET becomes describable as neither superexchange nor sequential since the degree of sequentiality is close to neither zero nor unity at 0.5 to 0.6. Let us raise AGm a little more by 1oo0cm -I from its original value to 5oocm -I. The temperature dependence of the rate constant obtained becomes that shown by the solid line in Fig. 7. The superexchange ET manifesting itself below about 50K in Fig. 6 extends its validity region to below about 100K in Fig. 7 since the degree of sequentiality shown by the ,
I
,
I
,
I
~m = .300~"I :_'_~" o.0 //'",
i'
~13t = . 2 0 0 0 I ~ "1
l
0.6-
0.4-
j
0.5
,
0.2-
0.0
0
1~o
~o
Temperature/K
'
~0
0.0
Fig. 5. Samequantitiesas shownin Fig. 1, calculatedwith AG,,= - 300cm-i and otherparameterskeptunchanged.
17 1.0
i
I
'
I
,
I
,
I
,.o| o.,
/"" . . . . . .
O0 ~
,
0
tO0
t
x
'
t
0.0
3OO
2O0
Tempeftum/K
Fig. 6. Samequantitiesas shownin Fig. I, cakulatedwith AG, - 0era-n aad odmr Imramem~kept ,mchMl~L dash-dot fine becomes much smaller than unity there. Below about 100K, in fact, the rate constant calculated as a miffed single process merges '~ad ~(sx) for the superexchange ET shown by the dotted line. Below about 100K, ~ , the rate constant becomes nearly independent of tempemaue. Above about looK, on ,he other hand, the rate ceamat decreases with a decrease in temperature. This temperature dependence does not originate from the (tlznmay activated) seqemial El" since the degree of sequentiality reaches at most about 0.4 at room tempmanue. Above about looK, the rate coastmt does not deviate so much from k~,~) for the sequential El" shown by the dashed line. This seems, however, ~ The temperature dependence in this region originates in the fact that it is going to appmuch the timmally activated one for real transitions from Id) to Ira), whose activation energy has reached 514cm -l due to the rise in AG,., as the degree of
sequentiality increases with increasing temperature. As AG, is raised f~a-therfrom 10Cgcm "~ in Fig. 7, the superexchange ET manifesting itself below about 100K in Fig. 7 extends its validity region further to higher temlmatures. In Fig. 8, AG= is raised by 2500cm - i from its original value to 20OOcm-L The degree of sequentiality shown by the dash-dot line is much smaller than unity below almut mere temperature, and the superexchange ET becomes prevailing in this tempemme ~0on- Below about 300 K, in fact, the rate constant calculated as a unified single Im3cess rapidly approaches k ~x) shown by the dotted line, and mmq~ cempletely into it below about 2OOK. The rate constant shown by the solid line is nearly uideimtdent of temperature in the n:gioa of IOO to 3OOK, lint it increases with a decrease in temperature below about IOOK. This tempermam depeadeuce reflects the situation of barrierless El', where reactant populations at the transition state increase at low temperaane~ It has been pre-installed for most speeding up the superexchange El', where the total ~ ~ energy on El" from Id) to la) coincides with the flee energy decrease on the El', adjusted according to the belief explained in ~ second pm'agral~ of Section 3. The superexchange ET expected to exist in the initial El' in photosynthesis [9] manifests itself at room 2.0
n
I
,
I
v
- - ~
~. ,.s. a~, = soom"
1.0
/
//
j ,o
o.e
o.o.
.
o.o
I
,
!
io.o
Temperaae / K Fig. 7. Samequanti~icsas shown in Fig. !. cakulm~ with AG,, - 500cm -n aadotherlxiramemsL-~ ~
H. Sumi / Journal of Electroanalytical Chemistry 438 ( ! 997) 11-20 I
,
5
3"
~4 %
..... ~
x ~ 3¸
I
,
~
I.
i, i i ~Gm = 2000 ~-1 / •1.,/ =.
g
0.8
| 0.6 o
0.4 ~ 0.2 ~
1-
0
,
, , 100 200 Temperature/ K
0
0.0 300
Fig. 8. Samequantitiesas shownin Fig. I, calculatedwith AGm= 2000cm-I and other parameterskept unchanged.
temperature for the first time when Im> is raised from the native position to about 2000cm -~ above Id>. In this superexchange ET disclosed, however, the rate constant is about 2 X 107 s -~ at room temperature with magnitude about 10 -4 times as small as that observed in bacteria. Jia et al. [29,30] replaced amino-acid residues in the protein matrix with different ones in the reaction center of Rhodobactor capsulatus by means of mutagenesis. They considered that this replacement mainly influenced the P / P + redox potential. Since P+ appears ; both P + - B C h l - - B P h (represented as Ira)) and P + - B C h l - B P h - (represented as la)) but in P ' - B C h I - B P h (represented as Id)), raising (or lowering) the P / P + redox potential by a certain amount has the same effect on the ET as lowering (or raising) the free energy of Id) by the same amount with those of Ira) and ]a) kept unchanged. Shown by the solid line in Fig. 9 is the rate constant at room temperatureplotted as a functionof the change in the free energy of Id) with those of [rn) and la) and other parameters kept unchanged. We see therein that the rate constant reaches a maximum when Id) is raised by about 5 0 0 c m - L This is consistent with their observation that the rate constant reached a maximum when the P / P + redox potential was lowered by about 300 to 500cm - t . The position of the rate constant maximum in Fig. 9 is mainly determined by the relative free energy difference between Id) and Ira) since the degree of sequentiality PsQ shown by the dash-dot line gives 1 - PSQ "~ 1 there and the ET can be regarded as sequential there. This can also be seen from the dashed line which represents the rate constant k (so) ad justifiable only in the sequential limit. The dashed line in Fig. 9 approaches very closely the solid line around its maximum and merges into it as we pass the maximum, raising Id) further.
1012
l , , , , I , , , , I , , , , l l l , B l l , , ~ l
300 K
,""'''"",
4. ~
,,"
1.o
{~ !011
j~ 0.5
101°
i
I ....
i ....
I ....
0.0
-IOO0 0 I000 Free Energy Change of the Donor State/crn-I Fig. 9. Same quantitiesas shown in Fig. I. calculatedat 300K by changingthe free energyat Id) seen from ]rn) and la) with other parameters kept unchanged.
H. S.mi/Jo.a~ o
y
~
C h e m ~ 438 t1~) H-2o
& Discussion and conclusien We see thus that the initial ET in bacterial pho~+,synthesis is c~ntrolled mainly by the free energy ~ between the donor Id) and the intermediate Ira) state, not by that between the donor Id) and the a c c e l ~ la) state, in a ~ a d i ~ t ~ i e a to the general belief. Although the ET can roughly be regarded as sequentbJ in native reaction centegs,the ET manifests itself as the free energy of Ira) is raised, extending the ~ region for its validity temperatures with raising Ira). When Ira) is raised to about 2000cm -~ above Id), ET becomes ~ as superexchange below about room temperature although its rate constant decreases to very small values. The wesent calculation of the rate constant was performed under the condition that all the ~ me sing~ valued, In native reaction centers, however, they ate considered to have a distribution with a n o n - ~ ~ It has known, especially, that the free energy d i f f ~ between Id) and la) has a distribution with half-width of abeut 400cm -~ above and below its average value [21]. This distribution does not play an intpommt role in determining the ~ ef the rate constant of ET between them. On the other hand, the distribution in the free er~'gy difference between Id) Ira), if appt~iable, should do since the rate constant is mainly controlled by this free e n e r ~ difference. Its width, however, has not been determined yet. Also when Jm) is raised sufficiently from its native position in modified nugtiea centers, latter distribution should play an impommt role. In this case, the sequential El" becomes gradually diWgult, but the distribution of the free energy difference with a non-vanishing width provides low-energy intermediate states tlmmgh which the sequential ET can take place. The initial ET in photosynthesis gives a typical example of ET mediated by a midway molecule. In general, it is describable as neither superexchange nor sequential, but should be described as a unified single process new in itself. The initial El" in photosynthesis gives a typical example in this respect too. Both the s u p e r e x d m ~ and the sequential El" ale concepts justified only in the mutually opposite limits of the unified single process, and they cannot coexist as two parallel channels -, often assumed [9-13]. An analytical formula for the rate constant of the El" via a midway molecule can be obtained by a f ~ akmg the lines sketched in Ref. [14]. The present work gives examples of an actual calculation by this formula. It was shown hem that this formula is very successful in reproducing the rate constant of the initial ET in hactefial I~homsyntla~ It is natural to expect that this formula has general applicabifity to other systems too. Acknowledgements The author wishes to thang Professor T. Kakitani of Nagoya University for valuable discussions in the initial stage of the present work. Appendix A When the inner-sphere (intra-pigment) reorganization is token into account only at the ncceptor state la), the rate constant of ET mediated by a midway molecule is finally given as a form of integration in time ~" spent virtually at the intermediate state ImP, as 2
~
'r
r
where C.O') represents the decay-rate constant of an electron at Ira) by transitions to la) or to the donor state Id) at time I". Function f(l") in the integrand in Eq. (AI) is given by f ( 1 " ) - t(D~
+D~)( 2 D,~+D2 2 1 )-
Xexp(_Sa_y2_ 1
. _0,/2 n~o ® l
D.(I")
[ E(a~(~') + E'a]2
9~[erf(x + iy)]
2 D~ + D$ + 2D~ - 2D.(~) ~
with 2
2
2
2
2
4 -;/~"
)
(A2)
20
H. Sumi/ Journal of Electroanalytical Chemistry 438 (1997) 11-20
and y=
I[
+
- Om( )2]
_ [ O: +
- Om( )2]
{ 2 [ ( O 2 ÷ Om2)(O 2 ÷ 02 ) - Om("r)4] [ 02a ÷ 02 ÷ 202m -- Din( T)2]/I/2 where e r g x + i y ) represents the error function for a complex argument x + i y and 9t represents the real part. Energies Ema and E(~(~F) are defmed by Emd = A~m ÷ ~m ÷ ~d
(A3)
~anm)("l*) = AGa -- AGm ÷ Aa ÷ Jim -- 2Am('r) ÷ nlit°a
(A4)
and
where Am(~') represents the energy of outer-sphere (protein) reorganization at [m) at time ~" with Am(0)~-0 and Am(QO)= Am. The variance of fluctuations in the outer-sphere reorganization energy has been represented by D d, D m and 1:). respectively at Id), Im) and la) in thermal equilibrium, and DIn(T) represents its value at Ira) at time 1" with Din(0)** 0 and Dm(oo) *ffiD m. In reality, ~-= o0 means that ~" is much larger than the reorganization (i.e. thermalization) time ~'m of the protein medium around an elecmJn at Im). Following Ref. [14], the degree of sequentiality is defined by P S Q - - f ~ g ( ~ ' ) d ~ ' / ~ ® g ( ~ ' ) d~" with g ( ~ ' ) - - - - f ( ~ ' ) e x p [ - f 0 r C m ( ' l J ) d'r ']
(A5)
considering that thermalization takes place in the protein medium as time z exceeds r m.
[1] J. Deisenhofer, H. Michel, Ann. Rev. Biophys. Biophys. Chem. 20 (1991) 247. [2] R.A. Friesner, Y. Won, Biochim. Biophys. Acta 977 (1989) 99. [3] C.C. Moser, J.M. Keske, IL Wamcke, R.S. Farid, P.L. Dutton, Na~:~ 355 (1992) 796. [4] M. Plato, M.E. MicheI-Beyerle,M. Bixon, J. Jortuer, FEBS Left. 249 (1989) 70. [5] P.L. Dutten, H.B. Gray, J.R. Winkler, in press. [6] A.M. Kuznetsov,P. Soramer-Larsen,J. Ulstrup, Surf. Sci. 275 (1992) 52. [7] Y. Toyozawa, J. Phys. Soc. Jpn. 41 0976) 400. [8] Y. Toyozawa, A. Kotani, A. Suml, J. Phys. Soc. Jpn. 42 (1977) 1495. [9] M. Bixon, J. Jottuer, M.E. Mivhei-Beyerle,Biochim. Biophys. Acta 1056 (1991) 301. [10] C.-K. Chart, TJ. DiMagno, L.X.-Q. Chen. LR. Norris, G.R. Fleming, Biophys. 88 (1991) 11202. [11] J.S. Joseph, W. Bialek, J. Phys. Chem. 97 (1993) 3245. [12] V. Nagarajan, W.W. Parson, D. Davis, C.C. Schenck, Biochem. 32 (1993) 12324. [13] M. Bixon, J. Jortuer, M.E. Michel-Beyerle,Chem. Phys. 197 (1995) 389. [14] H. Sumi, T. Kakitani, Chem. Phys. Lett. 252 (1996) 85. [15] Y.L Kharkats, A.M. Kuzaetsov,J. Ulstmp, J. Phys. Chem. 99 (1995) 13545. [16] V. Hizlmyakov,I. Tehver, Phys. Status Solidi 21 (1967) 755; 39 (1970) 67. [17] H. Sumi, Phys. Rev. Lett. 50 (1983) 1709; J. Phys. C 17 (1984) 6071. [18] Y. Hu, S. Mukamel, J. Chem. Phys. 91 (1989) 6973. [19] P.A. Lyle, S.V. Kolaczkowski,GJ. Small, J. Phys. Chem. 97 (1993) 6924. [20] S. Schraidt, T. Arlt, P. Harem, H. Huher, T. N'dgele,J. Wachtveiti,M. Meyer, H. Scheer, W. Zinth, Chem. Phys. Lett. 223 (1994) 116. [21] A. Ogrodnik, W. Keupp, M. Volk, G. Aumeier,b,.E. bllchei-Beyerle,J. Phys. Chem. 98 (1994) 3432. [22] P.OJ. Scherer, S.F. Hscher, Chem. Phys. 1M (1989) 115. [23] P.O.J. Scherer,C. Schamagl, S.F. Fischer, Chem. Phys. 197 (1995) 333. [24] H. Nakagawa, T. Okada, Y. Koyama, Biospectroscopy 1 0995) 169. [25] G.IL Fleming, J.-L. Martin, J. Breton, Nature 333 (1988) 190. [26] T. Arlt, S. Schmidt, ~,V.Kaiser, C. Lauterwasser, M. Meyer, H. Scheer, W. Zinth, Proc. Natl. Acad. Sci. USA 90 (1993) 11757. [27] J.P. Allen, G. Feller,1".O.Yeates, H. Komiya, D.C. Rees, Proc. Natl. Acad. Sci. USA 84 (1987) 5730. [28] H. Huher, M. Meyer, T. N'~igele,I. Hard, H. Scheer, W. Zinth, J. Wachtveitl,Chem. Phys. 197 (1995) 297. [29] Y. Jia, TJ. DiMaguo, C.IL Chart, Z. Wang, M. Du, D.K. Hanson, M. Schiffer,J.R. Norris, G.R. Fleming, M.S. Popov, J. Phys. Chem. 97 (1993) 13180. [30] J.R. Noms, in press.
Journalof Electwanalytic,al Chemistry438 (1997) 11-20
Electron transfer via a midway molecule as seen in primary processes in photosynthesis: Su~rexchange or sequential, or unified? ~ Hitoshi Sumi Institute of Materials Science. University of Tsulmba. TsukaOa. Ibaraki J~)5.
Received20 Septemlx~1996;receivedin revisedform2 Janumy 1997
Abstract A typical e~ample of electron transfer (ET) mediated by a midway molecule is the initial El" in bacterial photosymhesis fiem the excited special pair (P" ) to the bacteriopheophytin (BPh) mediated by the accessory, chlompbyil monomer (BChl) stationed in between. It has been argued intensively whether this El' is sequential (where the state Ira) of P+-BChl--BPh exists as a teal chemical imennedi~) or superexchange (where it is passed as a quantum-mechanical ~rtual slate). This ET is isomorphic to second-tmkr optical pmcessas (SOOP) where the initial state composed of an incoming photon and a matter is connected to the final state composed of an ealgoing photon and a matter with phonons left excited tlmmgh an intermediate state composed of a matter cleclnmicallyexcited withom a pht~on. SOOP is a single process reducing to Raman scattering or an a b s o ~ - l u m i n e s c e a c e sequence in mutually opposite limits. Correspondingly this ET is also a single process, not composed of two coexisting parallel channels by superexchange and suluemiat ETs. It reduces to them only in mutually opposite limits determined by competition between the lifetime of aa electnm and the r e ~ time of the medium at ira). The rate constant of this ET can be formulated by extending the fonn,!a6on for SOOP. It can describe satisfactorily the initial El" in photosynthesis observed. We can predict, ~ v e r , that s u p e r e x ~ El" should begin to manifest itself in a low-temperature region when Ira) is raised by several l,undmt cm - t from its native position, with its validity range toward higher temperatures as [m) is raised further. © 1997 Elsevier Scienc q.A. © 1997 Elsevier Science S.A. Keywords: Electron transfer, b~idwaymolecule:Photosymhesis:AccessoryChlorophyll
1. Introduction Recently, a new stage seems to have emerged in the investigation of electron transfer (ET), being devoted to ET mediated by a midway molecule in condensed matter, rather than the direct ET between two molecules investigated so far. The dawn of this stage was the first clarification in 1984 o~ the three-dimensional structure of the protein-pigment complex of the photosynthetic reactieq center of purple bacteria [1]` The beautiful arrangement of pigments clarified has n ~ only disclosed a pathway for ET, but also stimulated imagination about its development in the evolution of photosynthesis, initiating many researches (for a review see Ref. [2]). In the reaction center, El" (to he mine exact, charge separation) starts from the excited bacteriochlorophyl! dimer ( P " ) to the bacteriopbeophyfin (BPl0, and then an electron is further trmmfened front the reduced bacteriopbeophytin (BPh-) to the quinone (Q). As a result, an electron is transferred from the inner side ot" the bacterial membrane to the outer side counter to the membrane potential, and the energy of sol=" radiation exciting the bacteriochlorophyll dimer (P) is converted into the electrostatic energy across the membrane. Here, P and BPh are so far apart from each other (by a ~ t 17 A center to center and about 10A edge to edge) that direct coupling for El" from P* to BPh is too weak to account for the observed reaction time which is as small as about 3ps [2]. To mediate the El', a bacteriochlorophyll monomer (BChl) is stationed between P and BPh. Then it has been argued intensively [2] whether the ET mediated by BChl is sequential or superexehange. In the sequential ET, the i n t e m ~ i a t e state of P + - B C h l - - B P h exi~As as a chemical reality between the initial state P * - B C h I - B P h and the final state P + - B C h I - B P h - , while in the
t This paper was presentedat the InternationalSymposiumon EtectronTzansferin Proteinand SuWanmle~ar Assembiiesat Interfacesheld in Shomm Village, Kanagewa,Japan on 17 to 20 March 1996. 0022-0728/97/$17.00 © .'997 ElsevierScienceS.A. All fights resecved. Pil S0022-0728(97)00023-5
12
H. Swni/Journal of ElectroanalyticalChemistry438 (1997) 11-20
superexchange El', it is passed as a quantum-mechani~cal virtual state. It has been argued Rat ET from BPh- to Q is also mediated by a midway molecule. They are about 14A apart center to center and about IUA apart edge to edge [3], and it has been argued that the F'l" is mediated by a tryptophan residue of the protein matrix [4]. ET via a midway molecule can also be regarded as an example of long-range ET in protein matrix, which has been attracting much attention recently [3,5]. It is also incorporated in scanning tnnncling microscopy (STM) processes mediated by a redox state in an adsorbete molecule on a substrate [6j. This .;s important as an electrochemical application of STM for obtaining an atomic-scale image of biopolymers in liquids. Let us express ET via a midway molecule as occurring from a donor state [d) to an acceptor state la) mediated by an intermediate ~ Ira). This ET is isomorphic to second-order optical processes in which an initial state li) composed of an incoming photon (of energy E) and the matter is converted into a final state I f ) composed of an oul~oing photon (of energy smaller by A than E) and the matter with a phonon (of energy A) left excited inside. Since there exists no direct coupling between these states, they are mediated by an intermediate state Ira) which is composed only of matter electronically excited (at energy E,,) without photon. When E , is much higher than E, the intermediate state is pas~d only as a quantum-mechatdcal virtual state since energy cannot be conserved there. This situation is described as Ranlan scattering of light by matter, which is a concept corresponding to the superexchange ET in our original problem. When E,, i~. comparable to E on the seale of w~dths of ]/) and If), the situation of resonance Raman scattering emerges. It is known [7,8] that two quantities play an important role in this situation. One is the thermalization (or dephasing) time ~-,, of the matter at Ira), while the other is the lifetime l,, of an electron at Ira). When ~-,,:~ Ira, Raman scattering is applicable since ~'m is so large that the quantum-mechanical coh~reuce between Ii) and I f ) is not lost through the mediation at Ira). When ~-,,,~ l,n, on the other hand, the whole process can be divided into light absorption by a matter (from li) to Ira>) and subsequent luminescence from it (from Ira) to If>) since ~'m is so short that the quantum-mechani~:al coherence is quickly disrupted at Ira). This case of consecutive absorption and luminescence processes corresponds to the sequential ET in our original problem. In general, neither Raman scattering nor the absorption-luminescence sequence are appropriate. They can be applied only in the mutually-opposite limits of a unified single process, new in itself, called the resonance Raman scattering. This knowledge on the resonance Raman scattering can be applied directly to ET via a midway molecule. In this case, ~'m should more ~,pt~riateIy be called the reorganization time of the medium surrounding the midway molecule at its reduced state, since thermalizafion occurs in association with reorganization of the medium at a single localized state. Since both concepts of superexchange and sequential ETs are applicable only in the mutually opposite limits (respectively of ~'m:~ l,, and ~'m"~ l,~), especially, they cannot coexist as two parallel channels. Unfortunately. however, the initial El" in photosynthesis has often been a,alyzed by an assm~tption of i~,, coex~tlng chaJ,tels by superexchange and sequential ETs, and the branching ratio between them has been discussed [9-13]. In general, ET as seen there should be described as neither superexchange nor sequential in terms of old concepts for direct ET between two molecules. We are looking at a unified single process, new in itself, called ET mediated by a midway molecule in parallel to the resonance Raman scattering in second-order optical processes. An analytical formula for the rate constant of the ET can be obtained by extending the formulation for resonauce Raman scattering, with the procedure sketched in Ref. [ 14]. The purpose of the present work is to show how the mutually opposite limits of superexchange and sequential ETs are bridged by this formula for the ET as a unified single process, with an example of an explicit calculation taken from the initial El" in photosynthesis.
2. Rate constant for ET via a midway molecule as a unified single process Let us write as kad the rate constant for ET via a midway molecule as a single process. In the mutually opposite limits, kad should reduce to rate constants for superexchange and sequential ETs, denoted respectively as j.(sx) "ad and k (sQ) ad , whose expressions are well known as written in section 2 of Ref. [14]. Sin~e k (sx)oJis fourth order in electronic coupling while k~dQ) is second order, it is in principle impossible to derive a formula fi)r kad by a simple perturbational expansion in electronic coupling. It was first noted in Ref. [14] that the formulation ,:ould be realized by a renormalization appfied to the perturbational aplm3ach which was used for resonance Ramari scattering. Its explicit reduction will be given in a fortlgoming paper, but the final form of kad is sketched in Appendix A. It was noted by Kharkats et al. [15] that ~" ~.(sx) ad contains a c-mponent which describes a real transition of an electron from ]d) to Ira) followed by that from Ira) to la). In this component, the second transition takes place during vibrational relaxation at [m) triggered by the first transition, and its role was intensively discussed by them. Since k~x) is fourth order in electronic coupling, however, the component never reduces to k(asQ). Anyhow, this phenomenon is among the so-called hot transitions which were first noted by Hizhnyakov and Tehver [16,17] in solid-state physics. The rate constant kad for the unified single process covers this phenomenon naturally too. Ref. [14] starts from the divergent fourth-order perturbational expansion of the Liouville operator. The divergence was suppressed by renormalization in Ref. [14]. Hu and Mukamel [18],
H. . Y ~ / Jomml qf ~
CkeJm'y 438 (199"/)11-20
!~
however, suppressed it by taking into account distribution of electn~ic energies at Id), Ira) an~ ~a). Thus, the magnitude of the rate constant obtained by them depends essentially on the width of the ~ b u t i o n , remaining fourth order in etecaemie coupling. It was shown in Ref. [14] and also in Appendix A that in ET via a midway molecule as a unified single laxgess its degree of sequentiality can be d e t ¢ ~ by a quantity ( < 1) PSQ m
[kSQ)/k.d]X (survival probability of an electron at Ira) until ~-,)
{1)
Since the superexchange and the sequential ETs do not coexist as mentioned earlier, 1 - PsQ does not in general mexn the degree of superexelumgeness except when Psq "~ I or 1 - PsQ "~ 1. This PSQ describes conectly various situatiom in the ET. For example, when Ira) is much higher than both Id) and la), it should be t ~ d e d as superexchange. This ~ is described as PSQ ~ 1 for k(SQ)/kad / ad < 1 in Eq. (1) since it is vet). difficult for an electron to make a n:al ttmmitkm from [d) to Ira). Even if Ira) is located near Id), the ET should be regarded as supe~xchange in the large 1"., limit 11m situation is described as PSQ ' ~ 1 since the second factor on the fight-hand side of Eq. (1) becomes mneh smaller thaa onity although the first factor there is not much smaller than unity.
& Caleulaflen ef the rate comtaut ef ET in mmive ranetim eeatens In the real system, the thermalization time vm at ]m) has been determined as a quantity of the erda" of ~te inverse of the width of the frequency distribution of the phonon spectrum in the medium around an electron at Ira) [7,8,16,17]. In the case of bacterial photosynthesis, the medium is a protein around BChl-. It has been obseaved by hale-burning specuoscepy [19] that the phonon spectrum in proteins in reaction centers spreads broadly around a central energy of almut 30cm -t. Then, it is reasonable to estimate the width of frequency distribution of the spectnnn to be about 30cm -t divided by the Planck constant, that is, about 1THz. Therefore, it is reasonable to estimate %, to be about I ps. Free energie~ at ira) and ia) seen from Id) are respectively written as AG,, and AGo. The values of these energies were contmvenial for a long time, but it seems that they are converging to AGm ~ --500cm -t [13,20] and AG,-- --2000cm -t [21] on the basis of receat measurements. These values are adopted ~ the present cakulation. It has usually been believed that biological lax~esses have been adjusted so as to be most efficient in an eavinxauem in which they work, as a result of evolution over a ioag time. Based on this beficf, tbe initial charge scpmatkm tram P*-BChl-BPh to P+-BChI-BPh- in photosynthesis should have been adjusted so as to be fastest. Without the fastest charge separation, an excitation in P" would have been annihilated by fiucceaceuce. To accomplish the ~ the total reorganization energy associated with it must coincide with the coaesponding Gibbs energy ditfenmfe, givea by IAGm[~ 2000cm- t, since in this situation a banier'~,~s ET is realized at the boondmy between the normal and the inverted region from the standpoint of the direct ET. This situation expected from the belief is also adopted, as in many other articles. Since P, BChi and BPh are sufficiently apart mutually, let us consider as the first aplwoxinmion ~utt the which takes place when an electron is located at Id), Ira) or la) is mutually independent. In this case., the total reorganization energy on the initial charge separation is the sum of remganizatm energies at Id) and la). It is n~oaable to assume the reorganization anergy at [d) at a value comparable to 25otto-t, which was obml--uedas that UlPortexcilafiee of P to P" [19]. R is assumed to be 250cm -1, and the reorganization energy at la) is taken at 1750 ( 2 0 0 0 - 250)¢m-'. Since the former is considerably smaller than the typ/cal energy quantum (of the order of 1000¢m - i ) of iatmmolecu~ vibrations in chlorophyll species, it is reasonable to consider that it originates only from reorganizatm of the lamein medium, which corresponds to onter-snhere reo~aniT~_~ in tem~ of ~l~,ml theories for ET, being writtan as Ad. It is considered, on the other hano, tltat the latter is composed not only of the reorganization energy of the l a e ~ n medium (written as A.) but also of the inner-sphere reorganjT~i~ energy in BPh-. h is written as S.A % with the eueggy quantum A % of an intramolecolar vibration interacting with a transferred electron and the dimensionless coupling constant S. (called the Huang-Rhys factor). Considering half-and-half division, let us assume Aa ~ 850cm-' and SaA % ~, 900 cm - I , and moreover Sa a, 1.0 as a typical value. Without a clue for estimating the r e o r g a n i ~ energy at Ira), it is put as a trial at about 450cm -t, as composed of only the reorganization energy Am of the ~ medium. The remaining parameters which we need for numerical calculation are the slreagth of electronic coupling relarSonted by transfer integrals J.~ and Jam between Id) and Ira). and Ira) and la) respectively. The J . . value is the most impmtem. Unfortunately, however, it scatters from 9 to 60cm -t in the INDO calculation [22.23] and at 40cm -t in the extended Hllekel calculation [24] for Rhodopseudomonas viridis. Therefore. it is taken to be the average of them at J , da, 35¢m" ', close to the value obtained by the extended Hikkei calculation. Since it gives 20cm -t for J , , [24], it is taken at J,,, ,~ 15cm -~ in parallel to 35cm -I taken for J,,d" The transfer integral J decreases, rapidly with t h e ~ r of ET. It has been known empirically [3] that j2 decays as e x p ( - f l r ) with /]-7 1.4A -t in the region of 5A_
14
H. Sumi / Journal of Electroanalytical Chemistry 438 (1997) 1i -20 2.0
t
TO~
~m
I
,
I
= ° "~O cm'l
,
I
--. ~ef~J~
Q ~"
0.5 ! 0.5. 0.0
o
,~o
~;o
~o
00
Temperature / K
Fig. 1. Temperature dependence of the rate constant of ET from the donor Idt to tire acceptor lat state mediated by the intermediate lint state; the solid line with the ET treated correctly as a unified single process, the dashed line with the ET approximated as sequential, while the dotted line with the ET approxiramed as superexchange. The degree of sequentiality of the El" is given by the dash-dot line plotted with a scale on the right-hand side. The calculation was performed with parameters of AG,n = -500 and AGo= -2000cm -I as the free energy respectivelyat lint and lot seen from Id). A d = 2 5 0 , A m ~ 450 and Aa = 850cm- t as the reorganization energy of the protein medium respectivelyat Id), lint and [at, S,~he%= 900cm- i as the reorganization energy at la) by an intramolecularphonon with energy quantum h o~a = 900cm-~ and So = 1.0 as the coupling constant, Jm,ew 35 and Jan, = 15cm-I as electroniccoupling represented by the transfer integral respectivelybetween ]d) and [mt and between [mr and ]at, and ~'~,= 1.0ps as the re,Jrganizationtime of the protein medium at lint. means that j 2 becomes half when r increases by only 0.5,~. Unfortunately, however, it is at the 2.3,~ resolution that the structure of the protein-pigment complex has been detern'~ned [1]. Therefore, these values of Jm,t and Join are subject to change with the future achievement of higher resolution structure determination a n d / o r the development of calculation teclmiques. These values are also important in determining how fast an electron decays from Ira) to ]d) or la). With these Yal~es set above, therefore, we have determined the lifetime I m of an electron at [,,~ whose magnitude in comparison witi, ~-,, is important in determining the degree of sequentiality in ET. The rate constant koa calculated with the formula in Appendix A is shown by a solid line in Fig. 1 as a function of temperature below 350 K. This nicely reproduces the rate constant [25] observed in R h o d o p s e u d o m o n a s viridis with 7,espect to both its magnitude and its temperature dependence. The dotted line in Fig. ! shows k ~ x) justified in the superexchange limit, while the dashed line therein shows '~od"(sQ)justified in the opposite sequential !Emit. Neither reproduces the observed rate constant.. The degree of sequentiality PsQ of Eq. (1) is plotted as the dash-dot line with a scale on the right-hand side. It is about 0.8 over the whole temperature range. This is consistent with recent tendencies towards an agreement that the initial ET in photosynthesis in purple bacteria is understandable in terms of sequential ET [26]. By analyzing the content of the calculation, we can see that after transfer of an electron from Id) to Ira>, the electron is transferred to la) before ~',n in
2..0
~:)
, Z~m=
1.5- - -
I , - 500 cm" "",
I
, ~
I ~
AG. = - 2000 cm "I
1.0
1.0. ~
0.5
..................................................................... 0.0
o
o
1~o
~o
,
~o
0.0
Temperature / K
Fig. 2. Samequantifiesas shown in Fig. !, calculated with ~rm = 0.5 ps and other lxaramcterskept unchanged.
H. Sumi/Journal of Electroanal)vical Cl~misu'y 438 fl997) ! 1-20 ,
'1.6
~. ~.4 i
-,
I
,
I
~
~
= -500~
~.
= . ~OO
I
~
-
c m "~
1.2 1.0
0.8 O.a ~
0.4 0.2 0.0 209. Tempma!ufe/K
Fig. 3. Temperaturedependenceof the rate constantof the ET in Fig. I as a unifiedsingleprocess,calculatedwith J,a = 25cm-t and other peramete~ kept unchanged,shown in cc:nparisonwi~ ~ one for 3,,a = 35cm-t in Fig. 1.
the course of reorganization at Im) (that is, not af~t completion of reorganization) at a probability of about 0.6. Therefore, the second ET takes place as a hot transfer, whose role was intensively discussed fwst in Ref. [15] as mentioned in Section 2. In the calculation for Fig. 1, the phonon frequency distribution in the protein medium was assumed to have a width of about 30cm - I , giving 1"m = I ps. Let us change it to T,, = 0.5 ps, assuming the distribution to have a width of abont 65cm -! centered at about 65 cm -~. The distribution detected in the hole-burning spectrum is so broad that this does not seem unacceptable either. Numerical calculation for ~', ffi 0.5 ps with other parameters kept unchanged gives Fig. 2. The rate constant k~d shown by the solid line therein is nearly unchanged from that in Fig. 1 for ~-,,= !.0 ps in both magnitude and temperature depet~dence. The distance between P and BChl is shorter by about 0.5 h, on average in Rhodobactor qd~aero/des than in Rhodopseudomonas viridis in the crystalline state structural determination [27], although that between BChl and BPh is unchanged. If this difference is taken fiterally, J , d should be about 1/~/2 times as small in the former as in the latter, from the empirical rule on the distance dependence of the transfer integral in the protein medium mentioned earlier. Since 3,d in the latter was taken at 35cm - t in the calculation in Fig. 1, it is interesting to calculate the rate constant k~,~ for J , d = 25cm - I ( = 35/~/2cm - I ) with other parameters kept unchanged from those used in Fig. 1. The result is shown as a function of temperature in Fig. 3 in comparison with that shown in Fig. 1. We see that the rate constant calculated with JMd~ 25cm - I reproduces nicely that [25] observed in Rhodobactor sphaemides with respect to b o ~ magnitude and temperature dependence. This agreement seems surprising, however, since the free energy AG~ at Ira) relative to Id) is also very important in determining the rate constant as shown in the next section.
4. Calculation of the rate constant of ET for modified reaction centers In the calculation in the previous section, the total reorganization energy given by Ad + Ao + S ~ ~ set so as t.o coincide with the free energy decrease IAGo[from Jd) to la) according to the belief mentionec, in the secou~ pm'agraph therein. It is in,cresting to check that belief. To this end, the position AG~ of ]a) relative to ~,d) was first lowered by 1000~m -I from its original value of - 2 0 0 0 c m - i to - 3 0 0 0 c m - ! with other parameters (including the total reorganizatioa energy) kept unchanged. The rate constant obtained was almost the same as that in Fig. I with respect to both magulmde and temperature dependence, although it is not explicitly slaown here. Next, ~G~ was raised by 1000cm -~ from its original value to - 1000cm -~, and we got the rate constant shown by the sofid fine in Fig. 4. It is still nearly unchanged from the rate constant shown in Fig. 1, although the rate constant decreases to about two thirds of that in Fig. 1 at room temperature. This is consistent with experimental finding that the rate constant decreases to about one half as much at room temperature when AG° was raised by about 1500cm - j , by replacing BPh in pheophytin in Rhodobactorsphaeroides [28]. Anyhow, the rate constant is nearly unchanged by the change in AGa. Therefore, the belief which has been widely e~cepted so far does not seem to be taken into account in riving organisms. With a change in the energy of Ira), on the other hand, the rate constant becomes subject to marked change..as ~m) is raised, especially, the ET should become describable as superexchange. It is interesting to investigate where it becomes so.
16
tt. Sumi /Journal of Electroanclytical Chemistry 438 (1997) 11-20
7m o
2.0
,
"
1.5-
i
I
,
I
,
AGm=" 500cm" " , : : ~ ",
1.o- ~ " 0.5 ,..
I
AGa = - 1000 CITI"1
"
1.0 0.5i
........................................................... 0.0
0
,
,
,
100 200 300 Temperature /K
0.0
Fig. 4. Same quantities as shown in Fig. 1, calculated with A Ga = - 1 0 0 0 c m - ~ and other parameters kept unchanged.
Let us first raise the free energy AG, of Ira) relative to [d), only a little, by 200cm -z from its original value to - 3 0 0 c m -m with other parameters kept unchanged. The rate constant calculated with &Gin= --300cm -I is shown by the solid line in Fig. 5 as a function of temperature. We see therein that the rate constant in the lowest temperature region decreases to about one thirtieth of that in Fig. 1 for AG, = - 5 o o c m -z, although the rate constant at room temperature decreases by a factor of only about one half. Especially, the temperature dependence becomes opposite to that in Fig. 1 below about 100 K, since the rate constant decreases with a decrease in temperature there, although the dependence remains unchanged from that in Fig. 1 above about 100K. The degree of sequentiality remains still about 0.8, similarly to Fig. 1. When AG, is raised by 5oocm -I from its original value to 0cm -i, the temperature dependence of the rate constant changes into that shown by the solid line in Fig. 6. It tends to d(:crease with a decrease in temperature in the entire temperature region. It should be noted, moreover, that the degree of sequentiality of this El', shown by the dash-dot line, becomes much smaller than unity below about 50 K although it d e c k s still to about 0.5 to 0.6 above about 150K. This means that an electron must be transferred only by means of supep;xchange ET below about 50 K since a real transition from [d) to ira> by thermal activation becomes difficult at low lemperatures due to the raise in AGm from its value estimated for native bacteria. (Note here that although AG~ vanishes in Fig. 6, there exists a non-vanishing thermal activation energy for real transitions from [d> to [m> because of a non-vanishing reorganization energy (700 cm - I ) associated with this El'.) In the temperature region above about 50 K, ET becomes describable as neither superexchange nor sequential since the degree of sequentiality is close to neither zero nor unity at 0.5 to 0.6. Let us raise AGm a little more by 1oo0cm -I from its original value to 5oocm -I. The temperature dependence of the rate constant obtained becomes that shown by the solid line in Fig. 7. The superexchange ET manifesting itself below about 50K in Fig. 6 extends its validity region to below about 100K in Fig. 7 since the degree of sequentiality shown by the ,
I
,
I
,
I
~m = .300~"I :_'_~" o.0 //'",
i'
~13t = . 2 0 0 0 I ~ "1
l
0.6-
0.4-
j
0.5
,
0.2-
0.0
0
1~o
~o
Temperature/K
'
~0
0.0
Fig. 5. Samequantitiesas shownin Fig. 1, calculatedwith AG,,= - 300cm-i and otherparameterskeptunchanged.
17 1.0
i
I
'
I
,
I
,
I
,.o| o.,
/"" . . . . . .
O0 ~
,
0
tO0
t
x
'
t
0.0
3OO
2O0
Tempeftum/K
Fig. 6. Samequantitiesas shownin Fig. I, cakulatedwith AG, - 0era-n aad odmr Imramem~kept ,mchMl~L dash-dot fine becomes much smaller than unity there. Below about 100K, in fact, the rate constant calculated as a miffed single process merges '~ad ~(sx) for the superexchange ET shown by the dotted line. Below about 100K, ~ , the rate constant becomes nearly independent of tempemaue. Above about looK, on ,he other hand, the rate ceamat decreases with a decrease in temperature. This temperature dependence does not originate from the (tlznmay activated) seqemial El" since the degree of sequentiality reaches at most about 0.4 at room tempmanue. Above about looK, the rate coastmt does not deviate so much from k~,~) for the sequential El" shown by the dashed line. This seems, however, ~ The temperature dependence in this region originates in the fact that it is going to appmuch the timmally activated one for real transitions from Id) to Ira), whose activation energy has reached 514cm -l due to the rise in AG,., as the degree of
sequentiality increases with increasing temperature. As AG, is raised f~a-therfrom 10Cgcm "~ in Fig. 7, the superexchange ET manifesting itself below about 100K in Fig. 7 extends its validity region further to higher temlmatures. In Fig. 8, AG= is raised by 2500cm - i from its original value to 20OOcm-L The degree of sequentiality shown by the dash-dot line is much smaller than unity below almut mere temperature, and the superexchange ET becomes prevailing in this tempemme ~0on- Below about 300 K, in fact, the rate constant calculated as a unified single Im3cess rapidly approaches k ~x) shown by the dotted line, and mmq~ cempletely into it below about 2OOK. The rate constant shown by the solid line is nearly uideimtdent of temperature in the n:gioa of IOO to 3OOK, lint it increases with a decrease in temperature below about IOOK. This tempermam depeadeuce reflects the situation of barrierless El', where reactant populations at the transition state increase at low temperaane~ It has been pre-installed for most speeding up the superexchange El', where the total ~ ~ energy on El" from Id) to la) coincides with the flee energy decrease on the El', adjusted according to the belief explained in ~ second pm'agral~ of Section 3. The superexchange ET expected to exist in the initial El' in photosynthesis [9] manifests itself at room 2.0
n
I
,
I
v
- - ~
~. ,.s. a~, = soom"
1.0
/
//
j ,o
o.e
o.o.
.
o.o
I
,
!
io.o
Temperaae / K Fig. 7. Samequanti~icsas shown in Fig. !. cakulm~ with AG,, - 500cm -n aadotherlxiramemsL-~ ~
H. Sumi / Journal of Electroanalytical Chemistry 438 ( ! 997) 11-20 I
,
5
3"
~4 %
..... ~
x ~ 3¸
I
,
~
I.
i, i i ~Gm = 2000 ~-1 / •1.,/ =.
g
0.8
| 0.6 o
0.4 ~ 0.2 ~
1-
0
,
, , 100 200 Temperature/ K
0
0.0 300
Fig. 8. Samequantitiesas shownin Fig. I, calculatedwith AGm= 2000cm-I and other parameterskept unchanged.
temperature for the first time when Im> is raised from the native position to about 2000cm -~ above Id>. In this superexchange ET disclosed, however, the rate constant is about 2 X 107 s -~ at room temperature with magnitude about 10 -4 times as small as that observed in bacteria. Jia et al. [29,30] replaced amino-acid residues in the protein matrix with different ones in the reaction center of Rhodobactor capsulatus by means of mutagenesis. They considered that this replacement mainly influenced the P / P + redox potential. Since P+ appears ; both P + - B C h l - - B P h (represented as Ira)) and P + - B C h l - B P h - (represented as la)) but in P ' - B C h I - B P h (represented as Id)), raising (or lowering) the P / P + redox potential by a certain amount has the same effect on the ET as lowering (or raising) the free energy of Id) by the same amount with those of Ira) and ]a) kept unchanged. Shown by the solid line in Fig. 9 is the rate constant at room temperatureplotted as a functionof the change in the free energy of Id) with those of [rn) and la) and other parameters kept unchanged. We see therein that the rate constant reaches a maximum when Id) is raised by about 5 0 0 c m - L This is consistent with their observation that the rate constant reached a maximum when the P / P + redox potential was lowered by about 300 to 500cm - t . The position of the rate constant maximum in Fig. 9 is mainly determined by the relative free energy difference between Id) and Ira) since the degree of sequentiality PsQ shown by the dash-dot line gives 1 - PSQ "~ 1 there and the ET can be regarded as sequential there. This can also be seen from the dashed line which represents the rate constant k (so) ad justifiable only in the sequential limit. The dashed line in Fig. 9 approaches very closely the solid line around its maximum and merges into it as we pass the maximum, raising Id) further.
1012
l , , , , I , , , , I , , , , l l l , B l l , , ~ l
300 K
,""'''"",
4. ~
,,"
1.o
{~ !011
j~ 0.5
101°
i
I ....
i ....
I ....
0.0
-IOO0 0 I000 Free Energy Change of the Donor State/crn-I Fig. 9. Same quantitiesas shown in Fig. I. calculatedat 300K by changingthe free energyat Id) seen from ]rn) and la) with other parameters kept unchanged.
H. S.mi/Jo.a~ o
y
~
C h e m ~ 438 t1~) H-2o
& Discussion and conclusien We see thus that the initial ET in bacterial pho~+,synthesis is c~ntrolled mainly by the free energy ~ between the donor Id) and the intermediate Ira) state, not by that between the donor Id) and the a c c e l ~ la) state, in a ~ a d i ~ t ~ i e a to the general belief. Although the ET can roughly be regarded as sequentbJ in native reaction centegs,the ET manifests itself as the free energy of Ira) is raised, extending the ~ region for its validity temperatures with raising Ira). When Ira) is raised to about 2000cm -~ above Id), ET becomes ~ as superexchange below about room temperature although its rate constant decreases to very small values. The wesent calculation of the rate constant was performed under the condition that all the ~ me sing~ valued, In native reaction centers, however, they ate considered to have a distribution with a n o n - ~ ~ It has known, especially, that the free energy d i f f ~ between Id) and la) has a distribution with half-width of abeut 400cm -~ above and below its average value [21]. This distribution does not play an intpommt role in determining the ~ ef the rate constant of ET between them. On the other hand, the distribution in the free er~'gy difference between Id) Ira), if appt~iable, should do since the rate constant is mainly controlled by this free e n e r ~ difference. Its width, however, has not been determined yet. Also when Jm) is raised sufficiently from its native position in modified nugtiea centers, latter distribution should play an impommt role. In this case, the sequential El" becomes gradually diWgult, but the distribution of the free energy difference with a non-vanishing width provides low-energy intermediate states tlmmgh which the sequential ET can take place. The initial ET in photosynthesis gives a typical example of ET mediated by a midway molecule. In general, it is describable as neither superexchange nor sequential, but should be described as a unified single process new in itself. The initial El" in photosynthesis gives a typical example in this respect too. Both the s u p e r e x d m ~ and the sequential El" ale concepts justified only in the mutually opposite limits of the unified single process, and they cannot coexist as two parallel channels -, often assumed [9-13]. An analytical formula for the rate constant of the El" via a midway molecule can be obtained by a f ~ akmg the lines sketched in Ref. [14]. The present work gives examples of an actual calculation by this formula. It was shown hem that this formula is very successful in reproducing the rate constant of the initial ET in hactefial I~homsyntla~ It is natural to expect that this formula has general applicabifity to other systems too. Acknowledgements The author wishes to thang Professor T. Kakitani of Nagoya University for valuable discussions in the initial stage of the present work. Appendix A When the inner-sphere (intra-pigment) reorganization is token into account only at the ncceptor state la), the rate constant of ET mediated by a midway molecule is finally given as a form of integration in time ~" spent virtually at the intermediate state ImP, as 2
~
'r
r
where C.O') represents the decay-rate constant of an electron at Ira) by transitions to la) or to the donor state Id) at time I". Function f(l") in the integrand in Eq. (AI) is given by f ( 1 " ) - t(D~
+D~)( 2 D,~+D2 2 1 )-
Xexp(_Sa_y2_ 1
. _0,/2 n~o ® l
D.(I")
[ E(a~(~') + E'a]2
9~[erf(x + iy)]
2 D~ + D$ + 2D~ - 2D.(~) ~
with 2
2
2
2
2
4 -;/~"
)
(A2)
20
H. Sumi/ Journal of Electroanalytical Chemistry 438 (1997) 11-20
and y=
I[
+
- Om( )2]
_ [ O: +
- Om( )2]
{ 2 [ ( O 2 ÷ Om2)(O 2 ÷ 02 ) - Om("r)4] [ 02a ÷ 02 ÷ 202m -- Din( T)2]/I/2 where e r g x + i y ) represents the error function for a complex argument x + i y and 9t represents the real part. Energies Ema and E(~(~F) are defmed by Emd = A~m ÷ ~m ÷ ~d
(A3)
~anm)("l*) = AGa -- AGm ÷ Aa ÷ Jim -- 2Am('r) ÷ nlit°a
(A4)
and
where Am(~') represents the energy of outer-sphere (protein) reorganization at [m) at time ~" with Am(0)~-0 and Am(QO)= Am. The variance of fluctuations in the outer-sphere reorganization energy has been represented by D d, D m and 1:). respectively at Id), Im) and la) in thermal equilibrium, and DIn(T) represents its value at Ira) at time 1" with Din(0)** 0 and Dm(oo) *ffiD m. In reality, ~-= o0 means that ~" is much larger than the reorganization (i.e. thermalization) time ~'m of the protein medium around an elecmJn at Im). Following Ref. [14], the degree of sequentiality is defined by P S Q - - f ~ g ( ~ ' ) d ~ ' / ~ ® g ( ~ ' ) d~" with g ( ~ ' ) - - - - f ( ~ ' ) e x p [ - f 0 r C m ( ' l J ) d'r ']
(A5)
considering that thermalization takes place in the protein medium as time z exceeds r m.
[1] J. Deisenhofer, H. Michel, Ann. Rev. Biophys. Biophys. Chem. 20 (1991) 247. [2] R.A. Friesner, Y. Won, Biochim. Biophys. Acta 977 (1989) 99. [3] C.C. Moser, J.M. Keske, IL Wamcke, R.S. Farid, P.L. Dutton, Na~:~ 355 (1992) 796. [4] M. Plato, M.E. MicheI-Beyerle,M. Bixon, J. Jortuer, FEBS Left. 249 (1989) 70. [5] P.L. Dutten, H.B. Gray, J.R. Winkler, in press. [6] A.M. Kuznetsov,P. Soramer-Larsen,J. Ulstrup, Surf. Sci. 275 (1992) 52. [7] Y. Toyozawa, J. Phys. Soc. Jpn. 41 0976) 400. [8] Y. Toyozawa, A. Kotani, A. Suml, J. Phys. Soc. Jpn. 42 (1977) 1495. [9] M. Bixon, J. Jottuer, M.E. Mivhei-Beyerle,Biochim. Biophys. Acta 1056 (1991) 301. [10] C.-K. Chart, TJ. DiMagno, L.X.-Q. Chen. LR. Norris, G.R. Fleming, Biophys. 88 (1991) 11202. [11] J.S. Joseph, W. Bialek, J. Phys. Chem. 97 (1993) 3245. [12] V. Nagarajan, W.W. Parson, D. Davis, C.C. Schenck, Biochem. 32 (1993) 12324. [13] M. Bixon, J. Jortuer, M.E. Michel-Beyerle,Chem. Phys. 197 (1995) 389. [14] H. Sumi, T. Kakitani, Chem. Phys. Lett. 252 (1996) 85. [15] Y.L Kharkats, A.M. Kuzaetsov,J. Ulstmp, J. Phys. Chem. 99 (1995) 13545. [16] V. Hizlmyakov,I. Tehver, Phys. Status Solidi 21 (1967) 755; 39 (1970) 67. [17] H. Sumi, Phys. Rev. Lett. 50 (1983) 1709; J. Phys. C 17 (1984) 6071. [18] Y. Hu, S. Mukamel, J. Chem. Phys. 91 (1989) 6973. [19] P.A. Lyle, S.V. Kolaczkowski,GJ. Small, J. Phys. Chem. 97 (1993) 6924. [20] S. Schraidt, T. Arlt, P. Harem, H. Huher, T. N'dgele,J. Wachtveiti,M. Meyer, H. Scheer, W. Zinth, Chem. Phys. Lett. 223 (1994) 116. [21] A. Ogrodnik, W. Keupp, M. Volk, G. Aumeier,b,.E. bllchei-Beyerle,J. Phys. Chem. 98 (1994) 3432. [22] P.OJ. Scherer, S.F. Hscher, Chem. Phys. 1M (1989) 115. [23] P.O.J. Scherer,C. Schamagl, S.F. Fischer, Chem. Phys. 197 (1995) 333. [24] H. Nakagawa, T. Okada, Y. Koyama, Biospectroscopy 1 0995) 169. [25] G.IL Fleming, J.-L. Martin, J. Breton, Nature 333 (1988) 190. [26] T. Arlt, S. Schmidt, ~,V.Kaiser, C. Lauterwasser, M. Meyer, H. Scheer, W. Zinth, Proc. Natl. Acad. Sci. USA 90 (1993) 11757. [27] J.P. Allen, G. Feller,1".O.Yeates, H. Komiya, D.C. Rees, Proc. Natl. Acad. Sci. USA 84 (1987) 5730. [28] H. Huher, M. Meyer, T. N'~igele,I. Hard, H. Scheer, W. Zinth, J. Wachtveitl,Chem. Phys. 197 (1995) 297. [29] Y. Jia, TJ. DiMaguo, C.IL Chart, Z. Wang, M. Du, D.K. Hanson, M. Schiffer,J.R. Norris, G.R. Fleming, M.S. Popov, J. Phys. Chem. 97 (1993) 13180. [30] J.R. Noms, in press.