PhotosynthesisResearch 43: 149-154, 1995. © 1995 KluwerAcademicPublishers. Printedin the Netherlands, Regular paper
Description of energy migration and trapping in Photosystem I by a model with two distance scaling parameters L e o n a s V a l k u n a s 1, V l a d a s L i u o l i a 1, Jan P. D e k k e r z & R i e n k v a n G r o n d e l l e : , *
Ilnstitute of Physics, A. Gostauto 12, Vilnius, 2600, Lithuania; 2Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, the Netherlands; *Author for correspondence Received 17 June 1994;acceptedin revisedform20 January 1995
Key words: Photosystem I, energy transfer, trapping, low-energy chlorophyll
Abstract The energy transfer and trapping kinetics in the core antenna of Photosystem I are described in a new model in which the distance between the core antenna chlorophylls and P700 is proposed to be considerably longer than the distance between the chlorophylls within the antenna. Structurally, the model describes the Photosystem I core antenna as a regular sphere around P700, while energetically it consists of three levels representing the bulk antenna, P700 and the red-shifted antenna pigments absorbing at longer wavelength than P700, respectively. It is shown that the model explains experimental results obtained from the Photosystem I complex of the cyanobacterium Synechococcus sp. (A.R. Holzwarth, G. Schatz, H Brock, and E. Bittersman (1993) Biophys. J. 64: 1813-1826) quite well, and that no unrealistic charge separation rate and organization of the long-wavelength pigments has to be assumed. We suggest that excitation energy transfer and trapping in Photosystem I should be described as a 'transfer-to-the-trap'-limited process.
Abbreviations: C h l - chlorophyll; PS - photosystem; R C - reaction center Introduction In the reaction center core complex of Photosystem I (PS I), light energy is captured by one of the 50-100 antenna chlorophyll (Chl) molecules and efficiently trapped in a charge separated state within some tens of picoseconds (see van Grondelle et al. 1994, for a recent review). This fast rate is especially remarkable in view of the presence of chlorophyll molecules that absorb at considerably longer wavelength than the primary electron donor P700. In the P S I core antenna from the cyanobacterium Synechocystis PCC 6803 two Chl a molecules determine an inhomogeneously broadened absorption band around 708 nm (Gobets et al. 1994), whereas in the thermophilic cyanobacterium Synechococcus sp. four Chl a absorbing at 712 nm and one Chl a at 724 nm have been proposed (Holzwarth et al. 1993).
The basic energy transfer step in the bulk P S I antenna most likely occurs with a time constant of a few hundreds fs (Du et al. 1993; Lin et al. 1992). Time-resolved fluorescence experiments have shown an equilibration phase between the bulk PS I pigments and the far red population with a typical time constant of 12 ps, while trapping in a charge-separated state occurs within some tens of picoseconds (Holzwarth et al. 1993; Turconi et al. 1993; Hastings et al. 1994). A number of authors have proposed models which describe excitation energy transfer and trapping in PS I and which make predictions on the role of the far red pigments (Shipman et al. 1980; Jean et al. 1989; Beauregard et al. 1991; Werst et al. 1992; Trinkunas and Holzwarth 1994; Laible et al. 1994). Most of these models have in common that the P S I antenna is described by a regular two-dimensional lattice, in which P700 occupies one of the lattice positions. Laible et al. (1994) model the PSI antenna by an ellip-
150 soidal structure in which most of the pigments (apart from the two redmost) are randomly distributed. In all models the pairwise energy transfer times between the different sites (including P700) are at most a few ps. Most of the experiments and model calculations have suggested that the process of excitation trapping at room temperature is trap-limited. Within the context of this model the excitation distribution in the PS I antenna reaches an equilibrium (transfer equilibrium in the terminology of Laible et al. 1994) within at most 10-20 ps, after which the relative probability to find the excitation on a specific lattice position is no longer dependent on time (Jean et al. 1989; Beauregard et al. 1991; Laible et al. 1994). The observed trapping rate is then only determined by the relative probability to find the excitation on P700 multiplied by the intrinsic rate of charge separation. To explain the temperature and wavelength dependence of the fluorescence lifetime in PSI cores, Werst et al. (1992) concluded that a random distribution of the pigments around P700 in combination with the localization of the red most pigments next to P700 explained their results. A similar arrangement was chosen by Laible et al. (1994). On the other hand, Trinkunas and Holzwarth (1994) placed the five red-most pigments on one of the corners of the two-dimensional lattice and separated P700 from the most red pigment by two lattice constants. The latter model has one major shortcoming: to obtain a good fit for the observed kinetics the time-constant for charge separation had to be chosen unrealistically short (of the order of 0.5 ps). In addition, the specific location of P700 and the longwavelength pigments suggest that P700 as a 'goalkeeper' protects excitations to reach the far red pigments, which seems the arrangement to reduce the role of the far red pigments as much as possible. We note that in this model the excitation density on the red pigments did not reach equilibrium before trapping. In this paper we propose an alternative description of energy transfer and trapping in PS I by a model with two distance scaling parameters. This model is based on the organization of the bacterial system, in which the distance between the core (LH 1) antenna pigments and the primary electron donor is considerably longer than the average distance between the individual pigments in the antenna. Recently, we explained the excitation trapping kinetics in the bacterial RC-LH 1 complex as a function of temperature and as a function of the rate constant of charge separation by models (Valkunas et al. 1992; Beekman et al. 1994; Somsen et al. 1994) in which the pairwise energy transfer in the bulk
antenna is ultrafast (< 1 ps) and in which the excitation equilibration with the primary donor is slow (20-30 ps). It was argued that this huge difference in rate constants is due to the presence of two different distance scaling parameters: one for the interpigment (or interdimer) distance in the antenna and one for the distance between P and the neighboring antenna pigments. Thus, we propose also for the situation in P S I a model in which the distance between P700 and the bulk antenna pigments is much larger than the average distance between the antenna chlorophylls. Moreover, we assume that the coordination number of P700 is large, i.e. that many antenna sites are connected to P700 through relatively 'slow' excitation transfer. This contrasts the thusfar applied lattice models, in which these distances were assumed to be (almost) the same and in which the coordination number of P700 was assumed to be small. It should be stressed that our model is, just as the previously applied regular lattice models, purely hypothetical, because the structural data by Xray crystallography (Krauss et al. 1993) are, as yet, not conclusive enough to exclude any of the possibilities. The tendency seems to emerge, however, that the heart of the P S I complex contains a structure that at least remotely resembles the purple bacterial reaction center (Fromme et al. 1994). If the similarity extends to all transmembrane a-helices of the LM proteins a relatively pigment-poor direct environment of P700 is expected, due to which the application of a model with two distance scaling parameters becomes reasonable. In this contribution we present calculations using a model with two different distance scaling parameters. The results explain the available data quite well, and lead to realistic predictions of the time-constant for charge separation. In addition, no 'special' organization of the red-most pigments has to be assumed in order to get appropriate fits.
Results and discussion
The regular sphere model For antennae which are characterized by fast (< 1 ps) energy migration among the majority of the pigments and slow transfer (20-30 ps) from the antenna pigments to the special pair of the reaction center (i.e. antennae containing two distance scaling parameters), a variation in the symmetry of the antenna pigment arrangement has only a minor effect on the overall excitation
151 energy trapping kinetics (Somsen et al. 1994). Therefore, the most simple structural model of an antenna/RC system can be used. This is a model in which the antenna pigments are situated on a regular sphere surrounding P700. For clarity, we note that strictly speaking the organization of the antenna as a regular sphere around the primary electron donor is not realistic, because electrons have to go in and out. A more realistic representation was discussed recently by van Grondelle et al. (1994). However, the mathematical features of this representation are not principally different from those of a regular sphere model (Somsen et al. 1994), due to which we will only consider the regular sphere model in the following. A feature of the regular sphere model is that the distance from the antenna pigments to P700 differs from the interpigment distance in the antenna. If, for instance, 100 Chl per P700 are assumed (Holzwarth et al. 1993, Trinkunas and Holzwarth 1994), a pigmentpigment distance of 1 nm on the sphere corresponds to an average antenna-P700 distance of about 3 nm. In such a system the transfer time between two antenna pigments will be a factor of 500 faster than that of excitation transfer from individual antenna sites to P700. Therefore, if the energy transfer within the bulk P S I antenna proceeds in the few 100 fs timerange (Duet al. 1993), an average transfer time from each antenna pigment to P700 of about 50 ps is not unexpected. We note that these estimates are very rough, because of variations in spectral properties of the pigments, orientation parameters and distances. We can therefore not exclude that faster and/or slower rates exist (they most likely do!), but as we will argue, these will not affect the major behavior of this model. Within the context of the regular sphere model, the major factor in the experimentally observed 25-50 ps trapping times in cores of PS I arises from the slow rate of excitation energy delivery from the majority of the pigments (including the red pigments) to P700. A consequence of this representation is that, in contrast to all other recent models, the kinetics can not be described as trap-limited, but as 'transfer-to-the-traplimited'.
Description of energy transfer and trapping By taking into account the intrinsic symmetry of our model and, for the moment, ignoring the presence of red pigments, the excitation trapping kinetics is determined by a two level scheme, in which the first level determines the excitation population on the antenna
and the second level describes the excitation on P700. The dynamics of such a two-level system is described by the following kinetic equations: z
daOdt -
('Y + r ° l + zWc) ao + ~ - - ~ W a a l
da, dt -
(
Z ~
w
(1)
) a q- 7-0-1
al q- zWcao
where ao and al represent the probabilities for the excitation to be situated on PT00 and on the surrounding antenna, respectively, 7 is the rate of charge separation by P700, To is the excitation lifetime due to other excitation decay channels except trapping, Wa and Wc are the excitation transfer rates from the antenna to P700 and back, respectively, z is the coordination number (i.e. the number of antenna molecules in actual contact with P700 through the rate constants Wa and We) and N is the number of antenna pigments per PT00. It is evident that in this extremely simple model the rate of excitation transfer between pigments within the antenna cancels because of the symmetrization procedure (Valkunas et al. 1984). The decay kinetics of the excitation in the antenna is double-exponential and the amplitude of each exponential component is determined by the initial conditions. However, for isotropic excitation the amplitude of one of the exponentials is close to zero and the decay is essentially monoexponential with the eigenvalue given by :A+rol
=--~
1--
1
4")'ZWa "~
(2)
where z {9 = 7 q- ZWc q- ~ - ~ T W a
(3)
In the case of fast charge separation, i.e. when 7 >~> Wa, Wc
(4)
then A + r o - l = - z / ( N - 1 ) Wa and, consequently, the observed excitation lifetime is directly determined by Wa. We are fully aware of the fact that in a 'real' system with a 'real' structure and a 'real' spectral and spatial distribution of the pigments the situation is more complex, but an analysis of such a model is not relevant for illustrating the basic idea behind the hypothesis. We note that the very simple expressions (1-3) were used to understand the dynamics of excitation trapping in the core antenna of photosynthetic purple bacteria (Beekman et al. 1994). There it was shown that a 56 fold slowing down of the rate of charge separation
152
Wa Wl
W2 3
2
_
Fig. I. Schematicenergeticmodelof the reactioncentercore complex of PSI. Level 1 representsthe bulk antenna chlorophylls,level 2 the long-wavelengthchlorophylls and level 3 the photochemical trap P700. The energy transfer rate constants between the various levels are denoted Wn, while photochemicalcharge separation is representedby the rate constant %
(induced by site-directed mutagenesis of the residue M210 in the M chain of the bacterial RC) only slowed down the trapping kinetics by 30%, demonstrating that the rate of delivery of excitations to P is the rate limiting step in this system.
Influence of the far-red pigments The red pigments that are prominently present in the absorption spectra of the core antenna of P S I seriously perturb our model, even at room temperature, by destroying the symmetry of the system. Now the excitation migrates over a definite area of the antenna, with size ND, and the excitation lifetime in the bulk antenna is determined by two competing relaxation channels: the transfer to P700 and the trapping on one of the red pigments. For this case we approximate the energy transfer and trapping process by a 'pool' or 'level' model. Within this approximation the migration in the antenna, the transfer of excitation energy from one of the bulk P S I pigments to P700 and the transfer between the red pigments and P700 are described as kinetic processes occurring between three discrete levels. The corresponding three level scheme is represented in Fig. 1, in which level 1 symbolizes the excitations situated in the bulk of the P S I antenna, level 2 symbolizes the excitations on the red pigments and level 3 describes the excitation of P700. The rate constants depicted in Fig. 1 represent mean or effective rates of the corresponding processes. Within the sphere model Wa reflects the slow excitation delivery from the bulk P S I core to P700, while WI is an effective rate representing the search for the red pigments in the antenna. Wr is a single step energy transfer from the red pigments to the RC, while rates We, Wb and W2 are the reverse ofWa, Wr and W1, respectively.
As stated above, W1 is determined by the multistep search by an excitation generated in the bulk antenna of P S I for the red pigments, i.e. W1=~'FVr-~ , where ~'Fvr is the 'first passage time', which is the time that an excitation requires to visit a specific pigment on the lattice, assuming that initially the excitation is homogeneously distributed. The value of ~'Fr,r can be calculated from (Valkunas et al. 1986; Somsen et al. 1994): 1
TFPT ---- ~NDfp(ND)Vhop
(5)
where ND is the number of bulk P S I pigments per red pigment, fp(ND) is the structural function dependent on the structural dimension p and on ND, Thop is the excitation hopping time between 'nearest neighbors' in the bulk antenna. Diagonalization of the corresponding set of kinetic equations results in a three exponential kinetics for each level represented by: ai(t) = Ail exp(Alt) + A~ exp(A2t) + A~ exp(A3t)
(6) where ai(t) determines the excitation occupation on the i-th energy level. In the case of a very fast charge separation the eigenvalue )~3 characterizes the charge separation process. Similarly, as for the two-level system, the amplitude corresponding to the fastest exponent is negligibly small with isotropic initial excitation conditions (Valkunas et al. 1986) and this three level system will show biexponential kinetics, with all the intrinsic excitation equilibration and trapping dynamics concentrated in the parameters/~1 and ~2.
Analysis of experimental results Using this simple three level system we will analyze fluorescence data for cores ofPS I from Synechococcus obtained by Holzwarth et al. (1993). The exponential decay time-constants extracted from a global analysis of the time-resolved fluorescence experiment are: - ~ 1 - 1 = 12 ps and - ~ 2 -1 = 34 ps. The first time constant largely reflects the excitation equilibration between the bulk antenna, the far red states and P700, while the second time constant mainly reflects the excitation decay from the whole antenna due to trapping by P700. At 690 nm, largely reflecting the bulk antenna decay (level 1), the experimentally obtained ratio of amplitudes of these two phases in the fluorescence decay is A~l/A21 = 1.8 upon initial excitation of the bulk P S I antenna. Following the analysis of Holzwarth et al. (1993), we first assume that level 2 corresponds to 724 nm and
153 Table 1. Rate constants calculated from data reported by Holzwarth et al. (1993) as described in the text. See Fig. 1 for a description of the several rate constants. The kinetic parameters W l - 1 and W2-1 were assumed to be 22 ps and 50 ps, respectively ,.,),--1, ps
Wt,/Wr (E2)
Wb -- 1,ps
Wa-- l, ps
0.1 1.0
W c , - 1, ps
I0 (724 rim)
3.0
40
20
10 (724 nm)
2.0
36
20
1.5 0.I
10 (724 nm) 7 (720 nm)
1.6 3.4
30 44
20 20
1.0
7 (720 nm)
2.5
36
20
2.0
7 (720 nm)
1.5
30
20
take level 3 (ERc) to correspond to 700 nm. Thus at room temperature exp(AE/kT) = 10, where AE = E3 E2, k is the Boltzmann constant and T is temperature. Using these relations for the energy levels, the parameters A1 and A2 and the relative amplitudes of the bulk fluorescence we can solve the kinetic scheme for various choices of the rate constant 3' (see Table 1). Note that for the choice "y-1 = 0.1 ps, which is identical to the value obtained by Trinkunas and Holzwarth (1994) using a lattice model, we obtain a satisfactory solution w i t h W r - 1 = 30 ps and Wa- l = 40 ps. Taking a much more realistic value for % hardly affects the choices for the other rates and yields W r - 1 = 20 ps and Wa- 1 = 36 ps. Also in this model there is a limit to the choice of "y; for the current parameter set this is about "t' - l = 1.5 ps. On the other hand, it seems not very realistic to take E2 as low as 724 nm. It is more appropriate to consider the set of redmost pigments in the core of PS I of Synechococcus as reflecting an inhomogeneously broadened absorption band effectively peaking at 720 nm. Also for this choice of level 2 we have solved the kinetic equations for a variety of values for '7 and the results are given in Table 1. We note that now we find, for instance for 7 -1 = 2 ps, Wr -1 = 10.5 ps, Wa - I = 30 ps, again rather similar to the earlier values.
Comparison of various models Our calculations demonstrate that within the framework of the regular sphere model reasonable values of energy transfer rates between bulk antenna (level 1), far-red pigments (level 2) and P700 (level 3) are calculated for a variety of charge separation rates (3,). This contrasts the calculations by Trinkunas and Holzwarth (1994) who on the basis of a regular lattice model had to take into account an unreasonably fast charge separation rate. According to Eq. (5), we can estimate with our model the mean time of a single jump of the excita-
tion in the main bulk of the antenna (Wl = f r e t - 1 ). If we assume that ND = 100 (Holzwarth et al. 1993) and fp(No) < 0.5 (Somsen et al. 1994) the corresponding excitation hopping time rhop is >0.88 ps. This number is of the same order as that obtained in other photosynthetic structures (Owens et al. 1987; Valkunas et al. 1992; Somsen et al. 1994) and in view of the crudeness of the model we consider this as very reasonable. According to our model, the fluxes of excitations going to P700 either via the 'red' pigments (level 2) or directly from the main bulk (level 1) are more or less similar. Experimental evidence in favor of this situation has been provided by Van der Lee et al. (1993) on the basis of the contribution of the red-most pigments in the 77 K fluorescence excitation spectrum. We note, however, that in the limiting cases (W1 = W2 = 0 - the 'red' pigments are connected with the main bulk via P700 only, and Wa = Wc = 0 - the main bulk of the antenna is connected with P700 via the 'red' pigments only) we also can reproduce the observed experimental values without any restrictions on the value of 7. Thus, our model does not seem to give any restrictions on the specific organization of the long-wavelength pigments, in contrast to some of the other models (Werst et al. 1992; Trinkunas and Holzwarth, 1994; Laible et al. 1994). It is, however, difficult to imagine a structure for the antenna in which the red pigments or P700 are completely disconnected from the main bulk of the antenna; a structure in which direct energy transfer to P700 competes with energy transfer through a population of red pigments seems more realistic (van Grondelle et al. 1994). The temperature dependent experiments by Mukerji and Sauer (1989) clearly support such a model. In previous models (Werst et al. 1992; Trinkunas and Holzwarth 1994; Laible et al. 1994) different arrangements of the long-wavelength pigments have been postulated. In our opinion, the different arrangements may in part have originated from the different P S I material that has been analyzed in these studies. Holzwarth et al. (1993) and Trinkunas and Holzwarth (1994) analyzed complexes from the thermophilic cyanobacterium Synechococcus sp., which most likely contain more red-shifted pigments than the particles used by the other groups (see also van Grondelle et al. 1994), and which cause the room temperature emission spectrum to shift to the red. In the latter case the increased amounts of red-shifted pigments has put a large constraint on their specific organization. However, from our analysis it follows that the different detailed arrangements that have been proposed are strongly related to the type of model being used.
154 I f our m o d e l is correct, then it implicates that excitation energy transfer and trapping in P S I should not be described as a trap-limited process (see, e.g., H o l z w a r t h et al. 1993; Laible et al. 1994), but as a 'transfer-to-the-trap'-limited process. This situation occurs w h e n the distance b e t w e e n the bulk antenna pigments and the p h o t o c h e m i c a l trap is (considerably) larger than the distances b e t w e e n the pigments within the antenna. In purple bacteria the structural e v i d e n c e for the need to apply two distance scaling parameters is quite strong, and also the experimental evidence to describe energy transfer and trapping by such a 'transfer-to-the-trap'-limited process is straightforward ( B e e k m a n et al. 1994). Since all types of reaction centers f r o m all types o f photosynthetic organisms s e e m to h a v e c o m m o n m o t i v e s in the general arrangem e n t o f the c h r o m o p h o r e s and the a - h e l i c e s o f the protein backbones, it seems w o r t h w h i l e to consider the 'transfer-to-the-trap'-limited process as a general m o d e l to describe the energy transfer and trapping kinetics in photosynthetic systems.
Acknowledgements T h e authors a c k n o w l e d g e their lively discussions on the subject with Drs G. Trinkunas and A.R. Holzwarth. This research was supported in part by the E C contract E R B - C I P A - C T - 9 2 - 0 4 1 1 (fellowship of V.L.), the D u t c h R o y a l A c a d e m y (J.P.D.) and a grant to L.V. f r o m the Institute for C o n d e n s e d Matter Physics and Spectroscopy of the Vrije Universiteit A m s t e r d a m .
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Description of energy migration and trapping in Photosystem I by a model with two distance scaling parameters L e o n a s V a l k u n a s 1, V l a d a s L i u o l i a 1, Jan P. D e k k e r z & R i e n k v a n G r o n d e l l e : , *
Ilnstitute of Physics, A. Gostauto 12, Vilnius, 2600, Lithuania; 2Department of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, the Netherlands; *Author for correspondence Received 17 June 1994;acceptedin revisedform20 January 1995
Key words: Photosystem I, energy transfer, trapping, low-energy chlorophyll
Abstract The energy transfer and trapping kinetics in the core antenna of Photosystem I are described in a new model in which the distance between the core antenna chlorophylls and P700 is proposed to be considerably longer than the distance between the chlorophylls within the antenna. Structurally, the model describes the Photosystem I core antenna as a regular sphere around P700, while energetically it consists of three levels representing the bulk antenna, P700 and the red-shifted antenna pigments absorbing at longer wavelength than P700, respectively. It is shown that the model explains experimental results obtained from the Photosystem I complex of the cyanobacterium Synechococcus sp. (A.R. Holzwarth, G. Schatz, H Brock, and E. Bittersman (1993) Biophys. J. 64: 1813-1826) quite well, and that no unrealistic charge separation rate and organization of the long-wavelength pigments has to be assumed. We suggest that excitation energy transfer and trapping in Photosystem I should be described as a 'transfer-to-the-trap'-limited process.
Abbreviations: C h l - chlorophyll; PS - photosystem; R C - reaction center Introduction In the reaction center core complex of Photosystem I (PS I), light energy is captured by one of the 50-100 antenna chlorophyll (Chl) molecules and efficiently trapped in a charge separated state within some tens of picoseconds (see van Grondelle et al. 1994, for a recent review). This fast rate is especially remarkable in view of the presence of chlorophyll molecules that absorb at considerably longer wavelength than the primary electron donor P700. In the P S I core antenna from the cyanobacterium Synechocystis PCC 6803 two Chl a molecules determine an inhomogeneously broadened absorption band around 708 nm (Gobets et al. 1994), whereas in the thermophilic cyanobacterium Synechococcus sp. four Chl a absorbing at 712 nm and one Chl a at 724 nm have been proposed (Holzwarth et al. 1993).
The basic energy transfer step in the bulk P S I antenna most likely occurs with a time constant of a few hundreds fs (Du et al. 1993; Lin et al. 1992). Time-resolved fluorescence experiments have shown an equilibration phase between the bulk PS I pigments and the far red population with a typical time constant of 12 ps, while trapping in a charge-separated state occurs within some tens of picoseconds (Holzwarth et al. 1993; Turconi et al. 1993; Hastings et al. 1994). A number of authors have proposed models which describe excitation energy transfer and trapping in PS I and which make predictions on the role of the far red pigments (Shipman et al. 1980; Jean et al. 1989; Beauregard et al. 1991; Werst et al. 1992; Trinkunas and Holzwarth 1994; Laible et al. 1994). Most of these models have in common that the P S I antenna is described by a regular two-dimensional lattice, in which P700 occupies one of the lattice positions. Laible et al. (1994) model the PSI antenna by an ellip-
150 soidal structure in which most of the pigments (apart from the two redmost) are randomly distributed. In all models the pairwise energy transfer times between the different sites (including P700) are at most a few ps. Most of the experiments and model calculations have suggested that the process of excitation trapping at room temperature is trap-limited. Within the context of this model the excitation distribution in the PS I antenna reaches an equilibrium (transfer equilibrium in the terminology of Laible et al. 1994) within at most 10-20 ps, after which the relative probability to find the excitation on a specific lattice position is no longer dependent on time (Jean et al. 1989; Beauregard et al. 1991; Laible et al. 1994). The observed trapping rate is then only determined by the relative probability to find the excitation on P700 multiplied by the intrinsic rate of charge separation. To explain the temperature and wavelength dependence of the fluorescence lifetime in PSI cores, Werst et al. (1992) concluded that a random distribution of the pigments around P700 in combination with the localization of the red most pigments next to P700 explained their results. A similar arrangement was chosen by Laible et al. (1994). On the other hand, Trinkunas and Holzwarth (1994) placed the five red-most pigments on one of the corners of the two-dimensional lattice and separated P700 from the most red pigment by two lattice constants. The latter model has one major shortcoming: to obtain a good fit for the observed kinetics the time-constant for charge separation had to be chosen unrealistically short (of the order of 0.5 ps). In addition, the specific location of P700 and the longwavelength pigments suggest that P700 as a 'goalkeeper' protects excitations to reach the far red pigments, which seems the arrangement to reduce the role of the far red pigments as much as possible. We note that in this model the excitation density on the red pigments did not reach equilibrium before trapping. In this paper we propose an alternative description of energy transfer and trapping in PS I by a model with two distance scaling parameters. This model is based on the organization of the bacterial system, in which the distance between the core (LH 1) antenna pigments and the primary electron donor is considerably longer than the average distance between the individual pigments in the antenna. Recently, we explained the excitation trapping kinetics in the bacterial RC-LH 1 complex as a function of temperature and as a function of the rate constant of charge separation by models (Valkunas et al. 1992; Beekman et al. 1994; Somsen et al. 1994) in which the pairwise energy transfer in the bulk
antenna is ultrafast (< 1 ps) and in which the excitation equilibration with the primary donor is slow (20-30 ps). It was argued that this huge difference in rate constants is due to the presence of two different distance scaling parameters: one for the interpigment (or interdimer) distance in the antenna and one for the distance between P and the neighboring antenna pigments. Thus, we propose also for the situation in P S I a model in which the distance between P700 and the bulk antenna pigments is much larger than the average distance between the antenna chlorophylls. Moreover, we assume that the coordination number of P700 is large, i.e. that many antenna sites are connected to P700 through relatively 'slow' excitation transfer. This contrasts the thusfar applied lattice models, in which these distances were assumed to be (almost) the same and in which the coordination number of P700 was assumed to be small. It should be stressed that our model is, just as the previously applied regular lattice models, purely hypothetical, because the structural data by Xray crystallography (Krauss et al. 1993) are, as yet, not conclusive enough to exclude any of the possibilities. The tendency seems to emerge, however, that the heart of the P S I complex contains a structure that at least remotely resembles the purple bacterial reaction center (Fromme et al. 1994). If the similarity extends to all transmembrane a-helices of the LM proteins a relatively pigment-poor direct environment of P700 is expected, due to which the application of a model with two distance scaling parameters becomes reasonable. In this contribution we present calculations using a model with two different distance scaling parameters. The results explain the available data quite well, and lead to realistic predictions of the time-constant for charge separation. In addition, no 'special' organization of the red-most pigments has to be assumed in order to get appropriate fits.
Results and discussion
The regular sphere model For antennae which are characterized by fast (< 1 ps) energy migration among the majority of the pigments and slow transfer (20-30 ps) from the antenna pigments to the special pair of the reaction center (i.e. antennae containing two distance scaling parameters), a variation in the symmetry of the antenna pigment arrangement has only a minor effect on the overall excitation
151 energy trapping kinetics (Somsen et al. 1994). Therefore, the most simple structural model of an antenna/RC system can be used. This is a model in which the antenna pigments are situated on a regular sphere surrounding P700. For clarity, we note that strictly speaking the organization of the antenna as a regular sphere around the primary electron donor is not realistic, because electrons have to go in and out. A more realistic representation was discussed recently by van Grondelle et al. (1994). However, the mathematical features of this representation are not principally different from those of a regular sphere model (Somsen et al. 1994), due to which we will only consider the regular sphere model in the following. A feature of the regular sphere model is that the distance from the antenna pigments to P700 differs from the interpigment distance in the antenna. If, for instance, 100 Chl per P700 are assumed (Holzwarth et al. 1993, Trinkunas and Holzwarth 1994), a pigmentpigment distance of 1 nm on the sphere corresponds to an average antenna-P700 distance of about 3 nm. In such a system the transfer time between two antenna pigments will be a factor of 500 faster than that of excitation transfer from individual antenna sites to P700. Therefore, if the energy transfer within the bulk P S I antenna proceeds in the few 100 fs timerange (Duet al. 1993), an average transfer time from each antenna pigment to P700 of about 50 ps is not unexpected. We note that these estimates are very rough, because of variations in spectral properties of the pigments, orientation parameters and distances. We can therefore not exclude that faster and/or slower rates exist (they most likely do!), but as we will argue, these will not affect the major behavior of this model. Within the context of the regular sphere model, the major factor in the experimentally observed 25-50 ps trapping times in cores of PS I arises from the slow rate of excitation energy delivery from the majority of the pigments (including the red pigments) to P700. A consequence of this representation is that, in contrast to all other recent models, the kinetics can not be described as trap-limited, but as 'transfer-to-the-traplimited'.
Description of energy transfer and trapping By taking into account the intrinsic symmetry of our model and, for the moment, ignoring the presence of red pigments, the excitation trapping kinetics is determined by a two level scheme, in which the first level determines the excitation population on the antenna
and the second level describes the excitation on P700. The dynamics of such a two-level system is described by the following kinetic equations: z
daOdt -
('Y + r ° l + zWc) ao + ~ - - ~ W a a l
da, dt -
(
Z ~
w
(1)
) a q- 7-0-1
al q- zWcao
where ao and al represent the probabilities for the excitation to be situated on PT00 and on the surrounding antenna, respectively, 7 is the rate of charge separation by P700, To is the excitation lifetime due to other excitation decay channels except trapping, Wa and Wc are the excitation transfer rates from the antenna to P700 and back, respectively, z is the coordination number (i.e. the number of antenna molecules in actual contact with P700 through the rate constants Wa and We) and N is the number of antenna pigments per PT00. It is evident that in this extremely simple model the rate of excitation transfer between pigments within the antenna cancels because of the symmetrization procedure (Valkunas et al. 1984). The decay kinetics of the excitation in the antenna is double-exponential and the amplitude of each exponential component is determined by the initial conditions. However, for isotropic excitation the amplitude of one of the exponentials is close to zero and the decay is essentially monoexponential with the eigenvalue given by :A+rol
=--~
1--
1
4")'ZWa "~
(2)
where z {9 = 7 q- ZWc q- ~ - ~ T W a
(3)
In the case of fast charge separation, i.e. when 7 >~> Wa, Wc
(4)
then A + r o - l = - z / ( N - 1 ) Wa and, consequently, the observed excitation lifetime is directly determined by Wa. We are fully aware of the fact that in a 'real' system with a 'real' structure and a 'real' spectral and spatial distribution of the pigments the situation is more complex, but an analysis of such a model is not relevant for illustrating the basic idea behind the hypothesis. We note that the very simple expressions (1-3) were used to understand the dynamics of excitation trapping in the core antenna of photosynthetic purple bacteria (Beekman et al. 1994). There it was shown that a 56 fold slowing down of the rate of charge separation
152
Wa Wl
W2 3
2
_
Fig. I. Schematicenergeticmodelof the reactioncentercore complex of PSI. Level 1 representsthe bulk antenna chlorophylls,level 2 the long-wavelengthchlorophylls and level 3 the photochemical trap P700. The energy transfer rate constants between the various levels are denoted Wn, while photochemicalcharge separation is representedby the rate constant %
(induced by site-directed mutagenesis of the residue M210 in the M chain of the bacterial RC) only slowed down the trapping kinetics by 30%, demonstrating that the rate of delivery of excitations to P is the rate limiting step in this system.
Influence of the far-red pigments The red pigments that are prominently present in the absorption spectra of the core antenna of P S I seriously perturb our model, even at room temperature, by destroying the symmetry of the system. Now the excitation migrates over a definite area of the antenna, with size ND, and the excitation lifetime in the bulk antenna is determined by two competing relaxation channels: the transfer to P700 and the trapping on one of the red pigments. For this case we approximate the energy transfer and trapping process by a 'pool' or 'level' model. Within this approximation the migration in the antenna, the transfer of excitation energy from one of the bulk P S I pigments to P700 and the transfer between the red pigments and P700 are described as kinetic processes occurring between three discrete levels. The corresponding three level scheme is represented in Fig. 1, in which level 1 symbolizes the excitations situated in the bulk of the P S I antenna, level 2 symbolizes the excitations on the red pigments and level 3 describes the excitation of P700. The rate constants depicted in Fig. 1 represent mean or effective rates of the corresponding processes. Within the sphere model Wa reflects the slow excitation delivery from the bulk P S I core to P700, while WI is an effective rate representing the search for the red pigments in the antenna. Wr is a single step energy transfer from the red pigments to the RC, while rates We, Wb and W2 are the reverse ofWa, Wr and W1, respectively.
As stated above, W1 is determined by the multistep search by an excitation generated in the bulk antenna of P S I for the red pigments, i.e. W1=~'FVr-~ , where ~'Fvr is the 'first passage time', which is the time that an excitation requires to visit a specific pigment on the lattice, assuming that initially the excitation is homogeneously distributed. The value of ~'Fr,r can be calculated from (Valkunas et al. 1986; Somsen et al. 1994): 1
TFPT ---- ~NDfp(ND)Vhop
(5)
where ND is the number of bulk P S I pigments per red pigment, fp(ND) is the structural function dependent on the structural dimension p and on ND, Thop is the excitation hopping time between 'nearest neighbors' in the bulk antenna. Diagonalization of the corresponding set of kinetic equations results in a three exponential kinetics for each level represented by: ai(t) = Ail exp(Alt) + A~ exp(A2t) + A~ exp(A3t)
(6) where ai(t) determines the excitation occupation on the i-th energy level. In the case of a very fast charge separation the eigenvalue )~3 characterizes the charge separation process. Similarly, as for the two-level system, the amplitude corresponding to the fastest exponent is negligibly small with isotropic initial excitation conditions (Valkunas et al. 1986) and this three level system will show biexponential kinetics, with all the intrinsic excitation equilibration and trapping dynamics concentrated in the parameters/~1 and ~2.
Analysis of experimental results Using this simple three level system we will analyze fluorescence data for cores ofPS I from Synechococcus obtained by Holzwarth et al. (1993). The exponential decay time-constants extracted from a global analysis of the time-resolved fluorescence experiment are: - ~ 1 - 1 = 12 ps and - ~ 2 -1 = 34 ps. The first time constant largely reflects the excitation equilibration between the bulk antenna, the far red states and P700, while the second time constant mainly reflects the excitation decay from the whole antenna due to trapping by P700. At 690 nm, largely reflecting the bulk antenna decay (level 1), the experimentally obtained ratio of amplitudes of these two phases in the fluorescence decay is A~l/A21 = 1.8 upon initial excitation of the bulk P S I antenna. Following the analysis of Holzwarth et al. (1993), we first assume that level 2 corresponds to 724 nm and
153 Table 1. Rate constants calculated from data reported by Holzwarth et al. (1993) as described in the text. See Fig. 1 for a description of the several rate constants. The kinetic parameters W l - 1 and W2-1 were assumed to be 22 ps and 50 ps, respectively ,.,),--1, ps
Wt,/Wr (E2)
Wb -- 1,ps
Wa-- l, ps
0.1 1.0
W c , - 1, ps
I0 (724 rim)
3.0
40
20
10 (724 nm)
2.0
36
20
1.5 0.I
10 (724 nm) 7 (720 nm)
1.6 3.4
30 44
20 20
1.0
7 (720 nm)
2.5
36
20
2.0
7 (720 nm)
1.5
30
20
take level 3 (ERc) to correspond to 700 nm. Thus at room temperature exp(AE/kT) = 10, where AE = E3 E2, k is the Boltzmann constant and T is temperature. Using these relations for the energy levels, the parameters A1 and A2 and the relative amplitudes of the bulk fluorescence we can solve the kinetic scheme for various choices of the rate constant 3' (see Table 1). Note that for the choice "y-1 = 0.1 ps, which is identical to the value obtained by Trinkunas and Holzwarth (1994) using a lattice model, we obtain a satisfactory solution w i t h W r - 1 = 30 ps and Wa- l = 40 ps. Taking a much more realistic value for % hardly affects the choices for the other rates and yields W r - 1 = 20 ps and Wa- 1 = 36 ps. Also in this model there is a limit to the choice of "y; for the current parameter set this is about "t' - l = 1.5 ps. On the other hand, it seems not very realistic to take E2 as low as 724 nm. It is more appropriate to consider the set of redmost pigments in the core of PS I of Synechococcus as reflecting an inhomogeneously broadened absorption band effectively peaking at 720 nm. Also for this choice of level 2 we have solved the kinetic equations for a variety of values for '7 and the results are given in Table 1. We note that now we find, for instance for 7 -1 = 2 ps, Wr -1 = 10.5 ps, Wa - I = 30 ps, again rather similar to the earlier values.
Comparison of various models Our calculations demonstrate that within the framework of the regular sphere model reasonable values of energy transfer rates between bulk antenna (level 1), far-red pigments (level 2) and P700 (level 3) are calculated for a variety of charge separation rates (3,). This contrasts the calculations by Trinkunas and Holzwarth (1994) who on the basis of a regular lattice model had to take into account an unreasonably fast charge separation rate. According to Eq. (5), we can estimate with our model the mean time of a single jump of the excita-
tion in the main bulk of the antenna (Wl = f r e t - 1 ). If we assume that ND = 100 (Holzwarth et al. 1993) and fp(No) < 0.5 (Somsen et al. 1994) the corresponding excitation hopping time rhop is >0.88 ps. This number is of the same order as that obtained in other photosynthetic structures (Owens et al. 1987; Valkunas et al. 1992; Somsen et al. 1994) and in view of the crudeness of the model we consider this as very reasonable. According to our model, the fluxes of excitations going to P700 either via the 'red' pigments (level 2) or directly from the main bulk (level 1) are more or less similar. Experimental evidence in favor of this situation has been provided by Van der Lee et al. (1993) on the basis of the contribution of the red-most pigments in the 77 K fluorescence excitation spectrum. We note, however, that in the limiting cases (W1 = W2 = 0 - the 'red' pigments are connected with the main bulk via P700 only, and Wa = Wc = 0 - the main bulk of the antenna is connected with P700 via the 'red' pigments only) we also can reproduce the observed experimental values without any restrictions on the value of 7. Thus, our model does not seem to give any restrictions on the specific organization of the long-wavelength pigments, in contrast to some of the other models (Werst et al. 1992; Trinkunas and Holzwarth, 1994; Laible et al. 1994). It is, however, difficult to imagine a structure for the antenna in which the red pigments or P700 are completely disconnected from the main bulk of the antenna; a structure in which direct energy transfer to P700 competes with energy transfer through a population of red pigments seems more realistic (van Grondelle et al. 1994). The temperature dependent experiments by Mukerji and Sauer (1989) clearly support such a model. In previous models (Werst et al. 1992; Trinkunas and Holzwarth 1994; Laible et al. 1994) different arrangements of the long-wavelength pigments have been postulated. In our opinion, the different arrangements may in part have originated from the different P S I material that has been analyzed in these studies. Holzwarth et al. (1993) and Trinkunas and Holzwarth (1994) analyzed complexes from the thermophilic cyanobacterium Synechococcus sp., which most likely contain more red-shifted pigments than the particles used by the other groups (see also van Grondelle et al. 1994), and which cause the room temperature emission spectrum to shift to the red. In the latter case the increased amounts of red-shifted pigments has put a large constraint on their specific organization. However, from our analysis it follows that the different detailed arrangements that have been proposed are strongly related to the type of model being used.
154 I f our m o d e l is correct, then it implicates that excitation energy transfer and trapping in P S I should not be described as a trap-limited process (see, e.g., H o l z w a r t h et al. 1993; Laible et al. 1994), but as a 'transfer-to-the-trap'-limited process. This situation occurs w h e n the distance b e t w e e n the bulk antenna pigments and the p h o t o c h e m i c a l trap is (considerably) larger than the distances b e t w e e n the pigments within the antenna. In purple bacteria the structural e v i d e n c e for the need to apply two distance scaling parameters is quite strong, and also the experimental evidence to describe energy transfer and trapping by such a 'transfer-to-the-trap'-limited process is straightforward ( B e e k m a n et al. 1994). Since all types of reaction centers f r o m all types o f photosynthetic organisms s e e m to h a v e c o m m o n m o t i v e s in the general arrangem e n t o f the c h r o m o p h o r e s and the a - h e l i c e s o f the protein backbones, it seems w o r t h w h i l e to consider the 'transfer-to-the-trap'-limited process as a general m o d e l to describe the energy transfer and trapping kinetics in photosynthetic systems.
Acknowledgements T h e authors a c k n o w l e d g e their lively discussions on the subject with Drs G. Trinkunas and A.R. Holzwarth. This research was supported in part by the E C contract E R B - C I P A - C T - 9 2 - 0 4 1 1 (fellowship of V.L.), the D u t c h R o y a l A c a d e m y (J.P.D.) and a grant to L.V. f r o m the Institute for C o n d e n s e d Matter Physics and Spectroscopy of the Vrije Universiteit A m s t e r d a m .
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