Photosynthesis Research 48: 255-262, 1996. (~) 1996 KluwerAcademic Publishers. Printedin the Netherlands. Regular paper

Competition between annihilation and trapping leads to strongly reduced yields of photochemistry under ps-flash excitation Karsten Wulf & Hans-Wilhelm Trissl Abteilung Biophysik, Fachbereich Biologie/Chemie, Universit~it Osnabriick, Barbarastrafle 11, D-49069 Osnabrfick, Germany Received 8 August 1995; acceptedin revised form 11 June 1996

Key words: photochemical yield, photosynthesis, Photosystem I, Photosystem II, Rps. viridis, Rs. rubrum

Abstract

Excitation of photosynthetic systems with short intense flashes is known to lead to exciton-exciton annihilation processes. Here we quantify the effect of competition between annihilation and trapping for Photosystem II, Photosystem I (thylakoids from peas and membranes from the cyanobacterium Synechocystis sp.), as well as for the purple bacterium Rhodospirillum rubrum. In none of the cases it was possible to reach complete product saturation (i.e. closure of reaction centers) even with an excitation energy exceeding 10 hits per photosynthetic unit. The parameter a introduced by Deprez et al. ((1990) Biochim. Biophys. Acta 1015: 295-303) describing the competition between exciton-exciton annihilation and trapping was calculated to range between ~ 4.5 (PS II) and 6 (Rs. rubrum). The rate constants for bimolecular exciton-exciton annihilation ranged between (42 p s ) - l and (2.5 ps)- l for PS II and PS I-membranes of Synechocystis, respectively. The data are interpreted in terms of hopping times (i.e. mean residence time of the excited state on a chromophore) according to random walk in isoenergetic antenna.

Abbreviations: DCMU-3-(3,4-dichlorophenyl)- 1,1-dimethylurea; LHC II-light harvesting complex II; P-primary donor; PS I-Photosystem I; PS II-Photosystem II; PSU-photosynthetic unit; RC-reaction center Introduction

Short laser flashes with pico- and femtosecond duration are a common excitation source to study the primary reactions of photosynthesis. At higher flash energies, when more than one exciton is delivered to a photosynthetic unit (PSU), exciton-exciton annihilation becomes prominent. Annihilation studies have been used to learn about the antenna organization and exciton migration dynamics. In particular, the number of PSUs over which the exciton can diffuse (domain size) and the mean exciton transfer time between antenna pigments (hopping time) could be estimated. The various aspects connected with annihilation effects have been reviewed (van Grondelle 1985; Geacintov and Breton 1987).

The extent of losses by exciton-exciton annihilation may be very pronounced. It has been realized already in 1978 that in purple bacteria not all reaction centers (RC) could be closed by a single ps-flash no matter how intense the flash was (Moskowitz and Malley 1978). Obviously, in this case losses through annihilation competed efficiently with trap closure (Bakker et al. 1983). Comparable results for the photosynthetic systems of higher plants are not available. Theoretical approaches that account for this competition have been formulated (Paillotin et al. 1979; Den Hollander et al. 1983; Deprez et al. 1990). A central role for quantification is played by the PSU (Emerson and Arnold 1932), defined as the stoichiometric ratio of antenna pigments to RCs. This ratio is termed antenna size, N. The collision of two singlet excitons ($1) on one pigment leads to a higher excit-

256 ed state which rapidly relaxes back to the $1. For two colliding excitons one may write for this annihilation process: S l -I- Sl

ka) S0-1- S1,

where ka is the overall biexcitation rate constant for a pair of excitons. The process leads to the loss of one exciton which is then no longer available for the photochemical utilization. Apart from annihilation, three other quenching processes and two quenching states of the RC have to be considered: (i) losses of excitons in the antenna system itself, kt, (ii) photochemical conversion in the open RC, i.e. trapping, ko, (iii) and quenching at RCs with oxidized donor (closed RC), kc. The present study is concerned with the relative efficiency of these paths. We restrict ourselves to a model of the functional antenna organization of perfectly connected PSUs (lake model). This model can be described by a system of differential equations (Deprez et al. 1990; Wulf and Trissl 1995). The time-dependence of the concentration of excitons is expressed as photons per PSU, z(O, whereas 5(t) denotes the number of excitons created by the flash. The time-dependent concentration of the product is Y(t) and its final yield is Y0. Exciton-exciton annihilation is accounted for by a quadratic term.

d~(t) = 2(t) -- [kt + ko(1 - Y ( t ) ) + k~Y(t)]z(t) - ½kaz(t) 2

(1)

and

d Y ( t ) _ ko(1 - Y ( t ) ) z ( t ) . (2) dt For &function flashes these differential equations have an analytical solution of implicit form for the dependence of the yield of closed RCs after the flash, Y0, on the excitation energy, zo, defined as z0 = f o 2 ( r ) d r (Deprez et al. 1990). Introducing the abbreviations a = ka/2ko and ~ = ko/kc the solution is: +

(1 - y0)

:

(31

+

It is important to note that the shapes of such saturation curves (Y0 versus z0) are not superimposable for different values of a and/3. Thus, the shape of a particular saturation curve allows the numerical evaluation of both parameters. The parameter a is called competition parameter because it describes the ratio of rate constants for annihilation and photochemical utilization, while/3 is the

ratio of rate constants for losses at the reduced and oxidized primary donor, respectively. Equation (3) converges in the absence of annihilation (a = 0) and for the case that the quenching powers of RCs in the redoxstates P and P+ are equal (ko = kc or fl = 1) to an exponential saturation law (Deprez et al. 1990):

Y(zo)= 1-exp(

ko

k t + k o z°)

(4)

In this study we have used fluorescence yield and absorbance change measurements to assay the product yield produced by flashes of different duration (30 ps and 15 #s) to investigate annihilation losses in the photosynthetic systems of PS II, PS I, and Rs. rubrum. In all cases studied, losses by annihilation prevented a 100% closure of the reaction centers. The fit of the experimental data with the above theory yields the annihilation rate constant and this in turn allows the estimation of hopping times. In all systems studied hopping times were on the order of several 100 fs.

Materials and methods

We used 30 ps-flashes (FWHM) from a frequencydoubled Nd-YAG laser or 15/zs flashes from a Xenon bulb. Homogeneous illumination was attempted to achieve by placing a diffusor plate in front of the cuvette. The samples were optically thin at the excitation wavelength of 532 nm (OD < 0.1). Kinetic absorption difference measurements were carried out as described by Junge (1976). Care was taken that the measuring light was attenuated to a negligible actinic level. Thylakoid membranes from peas were prepared as described by Polle and Junge (1986), BBYmembranes as by Berthold et al. (1981) with modifications (Ono and Inoue 1985), PS I-membranes from Synechocystis sp. PCC 6803 as by Bottin and S6tif ( 1991), and chromatophores from Rs. rubrum S 1 as by Vos et al. (1986). The data were fitted to a saturation curve with essentially two parameters, a scaling factor to obtain the abscissa in excitons per PSU and the competition parameter, a. Note, that the ordinate scaling had not to be adjusted, since it was in all cases a directly accessible quantity. The rate constants involved in our analysis (kt, ko, and kc in Equations (1)-(4)) for the photosynthetic systems studied were taken from work cited in van Grondelle (1985), Holzwarth (1991), van Grondelle et al. (1994).

257

1.0I ot=~._? ....

| >.°

o.o

in Figure 1) yields an erroneous competition parameter of cz = 2.2.

expollen~l satum~on;Eq. 4

/ /'-=:o:)-1 iI

"

.

Results

Photosystem H

0.0

0

5

10

15

zo Figure 1. Product yields (points) calculated by means of Equations (l) and (2) for 30 ps-flashes and the following set of parameters: ko = (60 ps) - l , ke = (180 ps) - 1 , kz = (1.0 ns) - t , ka = (7.5 ps) - l ,

a = 4.0, and fl = 3.0. Fit of the theoretical data by Equation (3) with a = 2.2 (solid line). For comparison two theoretical saturation curves without annihilation are shown, an exponential one (Equation (4); dashed line) and another one for which closed centers quench 3-times less than open centers (Equation (3); solid line).

In order to account for the finite width of the laser flash, instead of using the analytical solution of Equations (1) and (2), we applied a numerical procedure described by Wulf and Trissl (1995). The accuracy of this algorithm was tested by comparing theoretical saturation curves calculated numerically for 1 ps flashes (as an approximation for 5-function excitation) with those resulting from Equation (3) (exact (Lfunction) yielding differences of less than 3%. However, numerically calculated saturation curves for 1 psflashes and 30 ps-flashes deviated considerably, leading to an under-estimation of tx as noted by others (McRae and Kasha 1958). To illustrate the significance of the flash duration we calculated the product yields, Y0, for 30 ps-flashes by solving numerically Equations (1) and (2) with parameters published for Rs. rubrum (points in Figure 1). A trapping time in the low energy limit of 60 ps was assumed (van Grondelle 1985; van Grondelle et al. 1994) and a 3-fold higher fluorescence yield resulting from P-870 oxidation (Kingma et al. 1983) and this work) yielding kc = (180 ps) - l . A competition parameter of cz = 4 has been reported by Deprez et al. (1990). The fit of the 30 ps-points with Equation (3) (solid line

For this photosystem we define the product yield through the reduction of the first quinone acceptor, QA, and quantify the fraction of closed RCs by means of fluorescence yield measurements. A similar assay has been used before (Bakker et al. 1983). A fuorescence induction curve recorded in the presence of DCMU served as reference. In this case the redox transition is due to an one electron transfer for which the following relation between the variable fluorescence, Fv, and the fraction of closed RCs, Y0, holds (Lavergne and Trissl 1995):

Fv (Yo) -

-

Fm

Yo -

1 + J(1 - 110)

(5)

Here Fvm is the maximal variable fluorescence defined as the difference between the fluorescence yield of all closed and all open RCs, F m - - F o. A fit of the fluorescence induction curve from BBY-membranes with this equation yielded the sigmoidicity parameter J = 1.3 (data not shown). The annihilation experiment with BBY-membranes in the presence of DCMU was performed by a probe-pump-probe-pump-probe experiment. The probe-flashes originated from a Xenon-flashlamp (15 #s) and the pump-flash from the Nd-YAG laser. The first probe-flash served to quantify Ft. After 30 ms the actinic ps-flash was given and the resulting product yield assayed with the second probe-flash (delayed by l0 ms) that detected Fo + Fv. After a further delay of 20 ms all RCs were closed with dc-light (for 40 ms) and 20 ms later the maximal fluorescence yield, FE, was determined. The 15 #s-flashes and the ps-flash were passed through a bifurcated light-pipe. This prolonged the duration of the ps-flash to approximately 70 ps. Figure 2 shows the fraction of closed RCs as calculated according Equation (5) (dots) versus the normalized excitation energy of the ps-flash. The photochemical yield in the low energy limit was assumed to be Cp = 0.77 in accordance with available data from the literature (Leibl et al. 1989; Roelofs et al. 1992). The solid line represents the best fit of Equations (l) and (2) for 70 ps flashes to the ps-data with the two parameters ka = (42 ps) - l and fl = 1.0, yielding a = 4.6. The same

258

1.ot //"

1.0

/ ~ t r - - - 4

o.a

io.,

o..,p. IS. " "" "-

I

I

I

I

"

°."

~

0

0.2 l~l

!

'~o'

0

.° ..........

0.1

0.2

0.0

/P

I

5

. . . .

a

. . . .

10

I

15

. . . .

I

20

. . . .

I

25

0.0

0

tion with 70 ps-flashes. The solid curve represents the best fit oftbe numerical solution of Equations (1) and (2) to the data points with ko = (386 ps)- 1, kt = (1.3 ns)- i, fl = 1.0, and the fit parameters ka = (42 ps)- l (c~= 4.6). The dotted lines represent theoretical curves for ka = (50 ps) - I (a = 3.8) and ka = (34 ps) -1 (a = 5.7). For comparison an exponential saturation law (Equation (4)) is also shown (dashed line).

q u e n c h i n g p o w e r o f P-680 +, kc, and open RCs, ko, (i.e. fl = 1.0) has b e e n e x p e r i m e n t a l l y established with a p s - p u m p - p r o b e flash technique (Deprez et al. 1983). In a second analysis o f the above data we took into a c c o u n t an c~-,fl-heterogeneity of PS II which resulted from a m o r e precise fit o f the fluorescence i n d u c t i o n curve. However, this i m p r o v e d analysis resulted in an o n l y slightly different value of ka = (40 p s ) - l ( a = 4.8).

Photosystem I of higher plants The p r o d u c t yield o f P-700 + o f P S I in peathylakoids was m e a s u r e d by m e a n s o f time-resolved flash absorption-difference-spectroscopy at 700 nm. Two saturation curves were measured, one with X e n o n - f l a s h e s without annihilation and a second one with 30 ps-laser flashes with annihilation (Figure 3). The data are c o m p a r a b l e to c o r r e s p o n d i n g photovoltage data (Trissl et al. 1987; F i g u r e 7 therein). As before, the solid curve represents the best fit of Equations (1) and (2) for 30 ps flashes to the ps-data with the fit parameters ka = (11 ps) -1 a n d / ~ = 1.0, yielding a = 4.6. For P S I it is well established that P-700 and P-

1 V ....... a ........

0.1

I 2

i 4

i II

i II

1

I 10

10

i

12

zo

Zo

Figure2. Product-saturation curve of BBY-membranes upon excita-

. ~.a

Figure3. Saturation curve for P-700 + in thylakoid-membranes upon excitation with 30 ps-flashes (dots) and with 15 #s-flasbes (squares). The solid curve represents the best fit of the numerical solution of Equations (1) and (2) to the data points with ko = (90 ps) - I , kt = (3 ns) -I,/~ = 1.0, and the fit parameters/ca = (11 ps) -1 (a = 4.1). The dotted lines represent theoretical curves for ka = (12 ps) -1 (a = 3.8) and ka = (10 ps)- l (a = 4.5). For comparison an exponential saturation law (Equation (4)) is also shown (dashed line). 700 + q u e n c h equally strong (Delepelaire and B e n n o u n 1978).

Photosystem I membranes from cyanobacteria The product yield of P-700 + in a P S I m e m b r a n e preparation from Synechocystis was measured as before. The solid curve in Figure 4 represents the best fit o f Equations (1) and (2) for 30 ps flashes to the ps-data with the fit parameters ka = (2.5 ps) -1 and/3 = 1.0, yielding a = 4.8. Also in this preparation P-700 and P-700 + q u e n c h equally strongly (Hecks et al. 1994).

Rhodospirillum rubrum The product yield P-870 + of Rs. rubrum chromatophores was d e t e r m i n e d by m e a s u r i n g the absorbance change of the oxidized primary d o n o r at 604 nm. The solid curve in F i g u r e 5 represents the best fit of Equations (1) and (2) for 30 ps-flashes to the ps-data with the fit parameters ka = (6 p s ) - I and ~3 = 3.2, yielding a = 5.0. Our value for the annihilation rate constant compares well with ka = (5.6 ps) -1 reported by Vos et al. (1986) and also by B a k k e r et al. (1983). A similar saturation curve has been published

259 1.0

I--"11" /

1of

11"

f

0.8

. .° . ~ ° ° , ° . . . , . o ° ° ° . . o °

>e /

°°o o°°.°°*

.~

. 0.4 y

/ ~

0.6

0.4

0.2

0.2 0.1 0.0

/ /

0

I

l

2

*

I 4

*

1

I 6

I

8

Zo

0.0

0

I

l

1

2

, 0.1

t

3

J

[

4

zo

Figure 4. Saturation curve for P-700 + in PS I-membranes from Synechocystis upon excitation with 30 ps-flashes (dots) and with 15

Figure 5. Saturation curve for P-870 + measured at 604 nm in chromatophores of Rs. rubrum upon excitation with 30 ps-flashes (dots)

/zs-flashes (squares). The solid curve represents the best fit of the numerical solution of Equations (1) and (2) to the data points with ko = (24 p s ) - l, kt = (3 n s ) - 1, fl = 1.0, and the fit parameters/Ca = (2.5 ps)-t ( a = 4.8). The dotted lines represent theoretical curves for ka = (3.0 ps) - 1 ( a = 4.0) and/Ca = (2.0 ps) -1 ( a = 6.0). For comparison an exponential saturation law (Equation (4)) is also shown (dashed line).

and with 15/.ts-flashes (squares). The solid curve represents the best fit of the numerical solution of Equations (1) and (2) to the ps-data with/Co = (60 ps) -1 , kt = (1.1 ns) - l , / ~ = 3.2, and the fit parameters ka = (5 ps)-1 ( a = 6.0). The dotted lines represent theoretical curves for ka = (6.0 p s ) - I ( a = 5) and ka = (4.0 p s ) - t ( a = 7.5). The #s-data were fired with Equation (3) taking a = 0 and/3 = 3.2 (dashed line). The trapping time was assumed to be 60 ps.

by Bakker et al. (1983), even though the latter data failed to reach the real low energy limit and display a rather large scatter. In parallel experiments we have measured a 3.2 times higher fluorescence yield when the RCs were converted from the all open state (15 min darkadaption) to the all closed state (20 mM ferricyanide). This value agrees with the one of 3.3 -4- 0.1 reported by Kingma et al. (1983).

tal uncertainties accumulate into the final parameters. Alternatively, the shape of the fluorescence quenching curve may be taken to adjust the energy scale to units of hits per PSU. This procedure assumes that the theoretical model used accounts for all experimental realities, representing another source for uncertainties. The concept of comparing product saturation curves with and without annihilation circumvents the uncertainties connected with the z0-scaling in a cramping way. Two aspects connected with scaling are essential: (i) The ps-data in the low energy limit can be shifted either to an experimental saturation curve obtained under non-annihilation conditions or to a theoretical saturation curve with a = 0, since the inital slopes of a pair of saturation curves must be identical. (ii) The knowledge of the maximal product yield, an experimentally easily determinable quantity, yields the intermediary product yields scaled in absolute numbers. In the present procedure the annihilation rate constant follows from the fit parameter (a) that accounts for the diminished product formation. It can be determined with high precision as well as with high reliability, since the second parameter involved (8) can be measured independently and is known for many

Discussion With the exception of some photovoltage measurements, rate constants for exciton-exciton annihilation were mainly obtained from measurements of the decrease of the fluorescence yield with increasing energy of ps-excitation flashes as reviewed (Breton and Geacintov 1980; van Grondelle 1985; Geacintov and Breton 1987). The analysis of such data requires the determination of the excitation energy in hits per PSU and, hence, the knowledge of the molar absorption coefficients of the antenna pigments at the excitation wavelength. This is no trivial problem and experimen-

260 Table 1. List of the parameters used in this work. The competition parameter, cz, follows from the rate constants for excitonexciton annihilation, ka, as determined in this work and overall trapping, ko, taken from the literature (van Grondelle 1985; van Grondelle et al. 1994). The antenna sizes, N, are taken from van Grondelle (1985) kol/ps

/3

t~

kal/ps

N

(fit param.) PS II

.

.

.

.

.

.

.

.

Rps. viridis

.

.

.

.

.

1.0

34-50

3.8-5.7

220

-

883-1 300

1.0

10-12

3.8-4.5

180

-

317-381

1.0

2.0-3.0

4.0-6.0

90

-

127-190

60 .

.

.

.

(cubic lattice)

90 24

( Syneclmcystis ) Rs. rubrum

~-h l fs

(linear chain)

386

P S I (peas) PS I

Th / fs

.

60

3.2 .

.

.

.

.

.

.

.

.

-

.

5.0-7.5 .

.

.

.

.

.

.

.

.

4.0-6.0 .

.

9.7

photosynthetic systems. The comparison of saturation curves under non-annihilation and annihilation conditions makes possible a narrow confinement of the annihilation rate constant and, hence, the competition parameter, c~, since the low energy trapping rate constants, ko, are currently well determined. Table 1 summarizes the parameters used in the present study. To draw more general final conclusions we have added data reported for Rps. viridis (Deinum 1991). The theoretical approach applied here is based on concentration of states, implying that at all times the excitation distribution is random, and on the assumption of a lake model. It represents one possibility to treat annihilation. Other approaches have been elaborated. These comprise the concept of domains, defined by an integer number of excitonically well connected PSUs with the domains being isolated against each other, and the concept of random walk on lattices (van Grondelle 1985). However, the latter require much more computational power than the present simulations. Whatever approach is taken the final parameters are not expected to depend very much on the model chosen and/or on the domain size as discussed for instance by Deinum et al. (1989), Deinum (1991), and Wulf and Trissl (1995). Some values of the annihilation rate constant, ka, and the competition parameter c~reported in the present work agree with, whereas others are at variance with, values published before. For PS II an annihilation rate constant k~ = (74 ps) -1 and a competition parameter c~ ~ 2-3 have been reported (Leibl et al. 1989). For PS I in thylakoid membranes no direct literature values are available. The photovoltage data by Trissl et al. (1987) are in approximate agreement with the present ones. For Rs. rubrum the competition parameter reported in the literature is c~ = 4.0 applying the theory of Deprez

.

.

.

.

.

.

.

3.1

.

.

.

32 .

.

.

.

.

24

60-90 .

.

.

.

.

.

.

400

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

et al. (1990) to the data of Bakker et al. (1983). This value is at the lower range given in Table 1 which might be due to the unjustified application of Equation (3) to 30 ps-flash data. However, the value of c~ = 4.7 reported by Deinum (1991) is in excellent agreement with ours. In the case of Rps. viridis the competition parameters reported in the literature are a()~ = 532 nm) = 1.2 (Trissl et al. 1990) and c~(,~ = 532 nm) = 3.1 (Deinum 1991). The discrepancy could be due to negligence of the pulse length in Trissl (1990) and its consideration in Deinum (1991). A reanalysis of the data of Trissl et al. (1990) with Equations (1) and (2) for 30 ps-flashes yields a = 2.3 which reduced the discrepancy considerably. The present analysis neglects ground state bleaching, excited state absorption, and stimulated emission which become relevant at very high excition energy (z0 > 0.1 N). At the relative low maximal energies (zo < 10) used for data analysis these effects can be considered to be of second order, although they could be responsible for small systematic deviations between data points and theoretical curves (Figures 2, 3, 4, and 5). Finally, it is worth mentioning that inhomogeneous excitation originating from interference patterns of the coherent laser beam ('hot spots') - that exist even after the passage of diffusor plates - may account for different results obtained in different laboratories. The rate constants of the overall singlet-singlet annihilation, ka, (Table 1) refer formally to a PSU comprising N (bacterio-)chlorophyll antenna pigments and one RC. The probability of two excitons to meet each other in a PSU depends on the number of pigments that can be visited during its lifetime. In the Chl a and Chl b containing systems this number may be given by the Chl a pigments only, due to the extremely

261 fast energy transfer from Chl b to Chl a (van Grondelle et al. 1994). In a most simple approximation (k~) -1 may be identified with the time required for two excitons to meet each other on the same pigment (first encounter time, Tfet) if the annihilation is assumed to be instantaneous and the transfer probability to a pigment in the excited state is similar to one in the ground-state (Den Hollander et al. 1983). Then one can estimate an upper limit for the single-step transfer times (hopping times), 7"h. The estimation may be carried out by means of a random walk treatment of two excitons either for a cubic lattice (Equation (6a)) or a linear chain (Equation (6b)) using equations derived for the first passage time, ~-fpt (one mobile exciton and one fixed lattice point) (Montroll 1969; Pearlstein 1982a,b): kal

:

Tfet :

1

2 Tfpt - -

rh 1 . 4 N , 2"Cn

(6a)

and 1 k a 1 ~_ Tfet ~-. 2 Tfpt __

7"h N 2 - 1

2. Cn

6

(6b)

Here cn is the coordination number) Equation (6a) (cubic lattice) is applied to the Chl systems (PSI and PS II) assuming a coordination number of four. Equation (6b) (linear chain) is applied to the BChl systems (Rs. rubrum and Rps. viridis), since the ring-like grouping of the antenna pigments (Karrasch et al. 1995) resembles very much the linear chain case. Estimated values for N and the resulting 7"h for the two types of lattices are given in Table 1. The upper limit hopping times of the Chl systems do not depend very much on the coordination number of the cubic lattice. In the two photosystems I and II they differ significantly. The approximately 3 times longer hopping times in PS II compared to P S I could indicate subtle differences in the arrangement of the antenna pigments, e.g. applying the linear chain case, Equation 6b, would result in a hopping time of 60 fs. The real arrangement in PS II may then be anywhere in between: Substructures present in the antenna system might require an analysis with a composite of lattice arrangements with reduced coordination numbers. However, it could as well be that the difference between the PS II and P S I data is due to an experimental imperfection: the ps-excitation of PS II occurred through fiber optics - which causes a homogeneous illumination without detectable spreckle patterns - whereas the excitation of P S I occurred via a diffusor plate, which still causes spreckle patterns as judged by eye.

With the exception of PS II, our estimates of hopping times agree reasonably well with single step hopping times estimated by other methods (van Grondelle 1985; van Grondelle et al. 1994). In particular, the estimate for Rs. rubrum gives the right order of magnitude when compared with recent sub-picosecond transient absorption measurements that yield single-site lifetimes of 50-70 fs (Giirtner and Towner 1995). In summary, • a proper quantification of the annihilation effect requires the consideration of the actual flash duration, • excitation of photosynthetic systems with psflashes (or fs-flashes) leads to significantly reduced yields, compared to Xenon-flash-excitation, • significant annihilation sets in already below O. 1 hits per PSU, • exciton-exciton annihilation competes efficiently with trapping and prevents complete closure of RCs by single fs/ps-flashes even at very high energies.

Acknowledgements

The authors thank Prof. G. Drews for a Rs. rubrum culture, Dipl. Biol. O. B6gershausen for flash spectroscopic measurements, Dr. J. Lavergne and Dr. W. Leibl for helpful discussions, and Prof. W. Junge for laboratory facilities. Financial support by the Deutsche Forschungsgenmeinschaft (SFB 171, TP-A1 and TR129/4-1) is acknowledged.

Note 1. The factor of 1/2 between the first passage time and the first encounter time results from the following consideration: A hopping step of exciton B while exciton A is fixed can be replaced by a hopping step of A with fixed B (yielding the first passage time). If A and B move the problem may be described with respect to a fixed reference lattice. Assuming that both A and B perform a hop during a clock beat, this can be replaced by a first half beat with A hopping and B fixed, then a second half beat with A fixed and B hopping. This is equivalent to keeping B fixed and having A performing random walk at double hopping frequency (yielding for the first encounter time half the first passage time).

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262

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