Analysis of structure-function correlations in light-harvesting photosynthetic antenna: Structure optimization parameters

J. theor. Biol. (1985) 112, 41-75

Analysis of Structure-function Correlations in Light-harvesting Photosynthetic Antenna: Structure Optimization Parameters Z. G. FeTISOVA, A. Yu. BORmOV

A. N. Belozersky Laboratory of Molecular Biology and Bioorganic Chemistry, Moscow State University, Moscow 119899 AND

M. V. F o K

P. N. Lebedev Institute of Physics, U.S.S.R. Academy of Sciences, Moscow 117333, U.S.S.R. (Received 15 September 1983, and in revised form 18 April 1984) Our previous theoretical analysis of the consistency between the results of experimental studies of two most important stages in the primary conversion of light energy in photosynthesis--the excitation energy transfer within a light-harvesting antenna and the stabilization of this energy in reaction centres--showed that the photosynthetic unit structure is to be strongly optimized in oioo to operate with a 90% quantum yield of primary charge separation in reaction centres, which means that a macroscopic photosynthetic unit is neither uniform nor isotropic. The principles of the structural organization of a light-harvesting antenna of a model photosynthetic unit which allow optimization of excitation energy transfer from antenna molecules to reaction centres are studied here. The time of excitation energy trapping by reaction centres was computed for two-dimensional systems simulating a photosynthetic unit of purple bacteria and the longest-wavelength part of higher plant Photosystem-I photosynthetic unit. The calculations assume a F~rster inductive resonance mechanism for energy transfer within antenna and pairwise dipolar interactions. It was shown that the co-operative effect of several optimizing factors, i.e. (1) anisotropy of intermolecular distances in an antenna; (2) its spectral heterogeneity; (3) mutual orientation of transition moment vectors; (4) availability of the "focusing zone" around reaction centres; (5) definite spatial clustering of reaction centres in macroscopic antenna, can decrease 300-fold the time required for a high probability of excitation trapping to be obtained compared to a similar array of undifferentiated antenna molecules. As the analysis that we offeris of more or less general nature, it is to be hoped that our major conclusions will be valid for any photosynthetic organisms. 41

0022-5193/85/010041 +35 $03.00/0

© 1985 Academic Press Inc. (London) Ltd

42

Z. G. F E T 1 S O V A

ET AL.

The basic principles of the structural organization of an optimal artificial light-converting system are formulated. 1. Introduction

Primary processes of photosynthesis have been studied in various organisms different in the size, N, of the light-harvesting antenna of a photosynthetic unit (PSU), N being the number of antenna molecules per each reaction centre (RC). These studies have shown that the time of excitation energy transfer from an antanna to an RC is quite short regardless of the antenna size. Namely, in non-saturated light photosynthesis these times equal 20100 psec, for both small antennae with N --- 30 and large ones with N = 1000, while the quantum yield of primary charge separation in an RC exceeds 90%. In our previous works (Fok & Fetisova, 1983; Fetisova & Fok, 1984) an analysis is given for the consistency between the two bodies of evidence for the major stages of the primary conversion of light energy in photosynthesis (the excitation energy transfer within a light-harvesting antenna of PSUs and the stabilization of this energy in reaction centres) that have so far been studied unrelatedly. We have shown that even in the case of extremely fast excitation energy transfer within the large PSU antenna, a 90% quantum yield of primary charge separation in an RC can be achieved only in systems strictly optimized over a number of parameters which define the structural and functional organization of a PSU. In particular, it has been shown that the coupling between excitation energy transfer within a light-harvesting antenna and stabilization of this energy in an RC with such a high quantum yield imposes strong restrictions on the antenna structure; it is neither uniform nor isotropic. In the analysis performed, the rate of energy transfer is assumed to be a non-limiting factor. Lower rates of excitation energy transfer impose even more rigorous requirements for optimization (Fok & Fetisova, 1984). Here we proceed from the same standpoint and use mathematical simulation to study the principal features of a PSU structure which make possible the optimization of energy transfer from an antenna to an RC. This problem is equally important for both the search for fundamental principles of the PSU structural organization which provide large and highly efficient PSUs in vivo and the designing of efficient ariificial systems for light energy conversion. Optimization of energy transfer to an RC is ensured by the following basic properties of structural organization of a PSU: (1) structure of a molecular lattice; (2) spectral heterogeneity of an antenna; (3) mutual orientation of transition moment vectors of antenna molecules and an RC.

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As the analysis that we offer is of more or less general nature, it is to be hoped that our major conclusions will be valid for any photosynthetic organisms. 2. Methods

We assume that the exciton motion is predominantly diffusive in vivo. Therefore, we calculated the time of excitation energy trapping by an RC for the case of inductive-resonance energy transfer within an antenna only (Ffrster, 1948). However, we suppose that qualitative conclusions derived here should not fail in the cases of other mechanisms of energy transfer in photosynthesis. We assume that at any given moment the excitation energy is localized in one o f the antenna molecules, and after a At time interval it can be found either in another molecule within the same antenna, or in an RC, or it can remain in the former molecule. According to F6rster's theory, the probability for excitation energy to be transferred from a ruth to a nth chlorophyll molecule during a At time interval equals

P,..At=

---~ k,..

F"(v) A.(~)--~

At,

(1)

where C = const, and R,.. is the distance between the centres of macrocycles m and n; the orientation factor, k"., is equal to k,.. = cos ot - 3 cos

tim COS ft.,

(2)

a being the angle between the two dipoles, and/3,, a n d / 3 , are the angles made by each of the dipoles with the vector drawn from the centre o f macrocycle m to the centre of macrocycle n. The integral in equation (1) is given by the product o f a normalized fluorescence spectrum, F,.(v), of the d o n o r molecule and absorption spectrum, A . ( v ) , of the acceptor molecule. The probability of the excitation to be found at the same mth molecule during a small At time interval is given by the sum of Pro. over all n # m: P". = 1-

E P " . At.

(3)

rl#"

To simulate energy transfer processes, we used the probability matrix approach. It can be easily seen that the P,.. values comprise a square matrix of the N t h order, N being the number o f molecules in the aggregate. Each element of this matrix gives the probability that the excitation energy localized in the ruth molecule at time t should be found in the nth molecule

44

Z.G.

FETISOVA

ET AL.

at time t+At. The RC was considered as one of the molecules of the same aggregate, with a number r. Only low light levels, far from saturating, are considered. The probabilities of energy transfer to an RC were calculated according to equation (1), while the probabilities of reverse energy transfer were taken to be zero, that is, in contrast to all the antenna molecules we assumed the following for an RC: Pr. --- 0;

Pr~ = 1.

(4)

To find the probability distribution of excitation in the aggregate molecules, (Q.), at time to+ At, to = 0, one should multiply the P,.. matrix by the linear matrix of the N t h order, (Q.(0)) = 1/N, which describes the initial distribution of the excitation: (Q.(At)) = (Pro.) (Q.(0)).

(5)

To find ( Q . ( 2 . A t ) ) , the procedure should be repeated: (Q.(2. At)) = (p,..)(Q.(At)),

(6)

and so on. At each step the Q, value gives the probability that the excitation is trapped by an RC, N

Qr=l-

~ Q.,

(7)

n=l rl~r

and the number of cycles of multiplication equals, on a certain representative scale, the time that elapsed after excitation of an antenna. It is the number of cycles necessary for Or to attain a certain value that characterizes the efficiency of energy transfer to an RC, or the time of excitation energy trapping by an RC, since the excitation may be deactivated by trivial mechanisms during all the time o f its migration within an antenna. We computed the number of these cycles necessary for the sum in equation (7) to decrease to e -1, e -2, and 0.1, that is for the probability Q, to reach (1- e - ' ) , ( 1 - e-2), and 0.9, respectively. The trapping times which corresponded to these cycle numbers were denoted tl, t2, and t3, respectively. The calculated t values are normalized so that the time scale is the same for all the cases considered. We assumed in calculations that P.,~ = 0 unless the ruth and nth molecules were the nearest neighbours, which is valid in the first appr~oximation since the probability of energy transfer decreases dramatically,t~s th~ distance increases and is diminished by at least an order of magnitude/f~r//the transfer to farther neighbours. This assumption can fail, however, in some ordered systems (see section 3(c)). Therefore, in calculations of the time of excitation trapping by an RC in ordered systems,

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

45

interactions with eight nearest neighbours (in a square lattice) were taken into account. Boundaries of aggregates were considered totally reflective, which was equivalent to an assumption of translational symmetry of a PSU, or in other words, multicentral PSUs were studied. Energy transfer in spectrally heterogeneous antennae were exemplified by the absorption spectra of several abundant purple photosynthetic bacteria with maxima in the 800-890 nm region. The absorption band of each spectral form was approximated with Gaussian curve, with 300 cm -~ half-width at half height and 100 cm -~ Stokes shift was assumed. For the PSU of higher plants the absorption bands were approximated with Gaussian curve, with 200 cm -I half-width at half height. Reaction centres were believed to be localized in the longest-wavelength spectral form of a light-harvesting antenna. As large N limits the computation rate, we have confined ourselves to small PSUs. But the principal conclusions should hold for large PSUs (Fetisova, 1975) as well as for PSUs of any photosynthetic organisms. Let us analyze each of the above factors which favour optimization of the light-harvest by an antenna.

3. Optimization Factors (A) S T R U C T U R E O F M O L E C U L A R L A T T I C E O F A N A N T E N N A

To mimic photosynthetic units, we considered two-dimensional aggregates of 24 or 25 molecules each, which comprised infinite square lattices with translational symmetry. The approach of trapping time computation applied here can be extended to other antenna models, e.g. triangle or hexagonal two-dimensional lattices, plane or non-plane, and also of onedimensional or three-dimensional antennae. For a globular antenna model this approach can be used directly if the spatial positions of chlorophyll molecules in a globule and the position of globules with respect to each other are given. On the other hand, the approach can be generalized assuming that energy transfer within each globule is extremely fast, and its time can be neglected if compared to the times of energy migration between globules. Then N denotes the number of globules, and R equals the distance between the nearest chlorophyll molecules which belong to the adjacent globules. In such a generalization, it should be taken into account that excitation is distributed all over the chlorophyll molecules in a globule, and the relative probability that the excitation is localized in each molecule should be introduced in one way or another. Then this approach can be used to study much larger PSUs during the same computation time. The

46

z.G.

FETISOVA

ET

AL.

same approach can be used to antennae made of dimers. In this particular case, all the results reported here should be valid for aggregates of N = 47 or N = 4 9 . We shall not dwell on all the specificities of antenna lattice structure, since, on the one hand, these problems have been partially analysed elsewhere (Pearlstein, 1967; Knox, 1968; 1977; Fetisova, 1984), and, on the other hand, the aim of our work is to study the possibilities of optimization of energy transfer from an antenna to an RC, which can be done with any chosen type of an antenna lattice without any loss in generality. Therefore, sophistications introduced by complexed lattice structures would be superfluous. Here we deal with several models shown in Figs 1-3 and different in spectral heterogeneity, mutual position of RCs, and other features. In particular, effects of clustering of RCs (see models (a) and (c) of Figs 1 and 2) are interesting to study in terms of designing artificial light-converting systems. Figure 4 shows the decay kinetics of excitation in these antennae due to its trapping by RCs. Table l comprises excitation decay times, tl, t2, and t3 (which correspond to decay to e -t, e -2, and 0.1 levels, resepctively) for PSU models with a square lattice and random orientation of transition dipole moment vectors in three-dimensional space (k2= 2) shown in Figs I-3. As seen in Fig. 4, the efficiency of light-harvest by a homogeneous antenna with a square lattice is decreased if the anisotropy of distribution of RCs in macroscopic PSU is increased. Consequently, from the four studied types of homogeneous PSU models, the highest efficiency is found in isotropic model (g) (Fig. 3), the lowest, in model (c) (Fig. 2). Thus clustering of RCs in homogeneous antennae with a square lattice with constant N decreases its light-harvesting efficiency. The anisotropy of the distances between antenna molecules can favour both acceleration of energy transfer and direction of excitation transfer from an antenna to an RC. Really, chlorophyll molecules resemble flat disks 16/~ in diameter, which may form ordered stack, i.e. with the planes of the disks been parallel and the centres forming a common aggregate axis perpendicular to these planes. In a light-harvesting antenna made of such close-packed stacks of disks with parallel axes, the intermolecular distances in the plane of the disks should be equal to 16 ~ , while the distances along the axis of each stack can be reduced at most five-fold (Netzel, 1982). Since the energy transfer rate depends dramatically on intermolecular distances, the rate of excitation transfer along the stacks would be one to two orders of magnitude higher than that across the stacks. Hence, the very shape of chlorophyll molecules is in principle capable of providing selective increase

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48

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ET AL.

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LIGHT-CONVERTING

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49

TABLE 1

Excitation decay in model photosynthetic units shown in Figs 1-3. Decay times tj, t2, and t3 correspond to decay from 1 to e -I, e -2, and 0.1 respectively; 77her is the operation efficiency of heterogeneous PSUs: ~7he' = t2ho~/t2he,• Orientation factor is k 2=2. tt

t2

t3

~het

(a) hom. het. (b) hom. het. (c) hom. het. (d) hom. het. (g) horn. het.

266 197 235 186 536 368 235 174 204 113

558 371 509 389 1223 745 509 329 420 204

646 423 592 452 1431 855 592 372 486 230

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50

Z. G. F E T I S O V A

I-W

ET AL.

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0-2

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1500

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FIG. 4. Kinetics of excitation decay (due to excitation trapping by reaction centres) in homogeneous (broken line) and heterogeneous (solid line) PSU antennae shown in Figs 1-3. W is the fraction of excitation energy trapped by reaction centres, t is the time measured in relative units. Intercepts o f t h e f ( t ) = 1 - W curves with straight lines, 1 - W = e -~, I - W = e -2, and 1 - W = 0.1 yield the t~, t2, and t 3 times for each model. These times correspond to ~ 6 3 % , ~86% and 90% of the excitation energy trapped by reaction centres. Initial parts of kinetic curves are shown on an enlarged time scale.

in the energy transfer rate is required direction. Let us consider the extent of acceleration for energy transfer towards an RC. It is reasonable to start with spectrally homogeneous models shown in Figs 1-3. Let us change intermolecular distances along a certain direction, for example, along OO' shown by arrows in Figs 1-3. In other words, we shall stretch and compress the model aggregate lattices along the OO' direction and compare the t2 values obtained (Fig. 5). From Fig. 5 one can see that: (1) for each model the t2 time increases sharply upon stretching which means an increase of the lattice period in the OO' direction, while in compressed aggregates the t2 time decreases and tends asymptotically to minimal limiting value; (2) a similar behaviour of t2 values is observed in aggregates compressed along any of the two axes (compare models (b) and (d)), however, the minimal t2 value depends on the mutual position of RCs in the macroantenna and on the direction of compressions; (3) for elongated aggregates, maximal acceleration of energy transfer from an antenna to an RC is attained by a decrease of intermolecular distances along the longest axis of an aggregate (compare models (b) and

(d));

L I G H T - C O N V E R T I N G SYSTEM O P T I M I Z A T I O N

51

900 800 700 60C

0

~0'

.~ 50C 40C '

,b

50C

I o ° !_

• .........

° I

0

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•

t

~

2

R±/R= FIG. 5. Dependence of the 12 time of excitation energy trapping by reaction centres for homogeneous PSU models shown in Figs I-3 on the degree of compression or stretching of their lattices along the OO' direction indicated by arrows in Figs I-3. Upon deformation a square lattice is deformed into a rectangular one. The lattice constant in the direction perpendicular to OO', Rl, is kept constant, while the lattice constant along the OO' direction, R=, is varied. At (R±/R=)= 1 the t2(R±/R=)values correspond to those for PSU models with square lattices. The t2 time corresponds to excitation decay from 1 to e -2 level due to excitation trapping by reaction centres. (4) d e v i a t i o n o f h f r o m its l i m i t v a l u e is n e g l i g i b l y s m a l l in a g g r e g a t e s c o m p r e s s e d o n l y t w o to t h r e e times, w h i c h e x a c t l y c o r r e s p o n d s to the ratio o f i n t e r m o l e c u l a r d i s t a n c e s b e t w e e n c h l o r o p h y l l m o l e c u l e s l o c a t e d in the s a m e s t a c k a n d in a d j a c e n t stacks, as d i s c u s s e d a b o v e ; (5) c l u s t e r i n g o f R C s w h i c h a l l o w s a s i g n i f i c a n t d e c r e a s e in t h e d i s t a n c e s b e t w e e n R C s in a m a c r o - a n t e n n a a l o n g the s h o r t e s t axis o f an e l e m e n t a r y a g g r e g a t e (Fig. 2), c o n v e r t s the least effective m o d e l into t h e m o s t effective one u p o n c o m p r e s s i o n o f the a g g r e g a t e a l o n g its l o n g e s t axis as c l e a r l y d e m o n s t r a t e d b y m o d e l (c) (Fig. 5). Thus, any influence which changes intermolecular distances may strongly affect t h e efficiency o f e n e r g y t r a n s f e r to a n R C . T h e m o s t d e m o n s t r a t i v e e x p e r i m e n t w h i c h s h o w e d t h a t the c h a n g e in the e n e r g y m i g r a t i o n rate r e s u l t e d f r o m the c h a n g e o f i n t e r m o l e c u l a r d i s t a n c e s w a s p e r f o r m e d b y

52

Z. G. FETISOVA

ET AL.

II'ina, Kotova & Borisov (1981). Variations of a detergent concentration led to a seven-fold change in the fluorescence lifetime from 0.5 to 3.5 nsec(!) in pigment-protein complexes of Photosystem-I from higher plants. As far as artificial antennae are concerned, it was shown (Zenkevich et al., 1978; Losev, 1982) that the maximal singlet-singlet energy transfer rate attained in model systems (2xl0~3sec-~) was observed in quasi-crystal systems assembled similarly to the stacks described above, namely, in cylinders formed by disks of chlorophyll molecules associated in a binary solvent mixture (dioxane-water) and packed with their planes parallel to each other, at intermolecular distances along the cylinder axis equal to approximately 6/~. Such chlorophyll stacks were not found in the PSUs in vivo. However, the very principle of the antenna organization with the above-mentioned specific anisotropy of intermolecular distances (not only for the limiting denseness of the molecular packing) seems attractive for the systems in vivo and perhaps is realized in vivo at least partially (for example, in threedimensional antennae), as it permits not only to decrease the intermolecular distances in selected direction, but (as necessary consequence) to increase significantly the degree of ordering of mutual orientation of antenna molecules. This essentially accelerates the energy trapping by RCs (see section 3(C). (B) S P E C T R A L

HETEROGENEITY

OF AN ANTENNA

It is beyond doubt now that in most photosynthetic organisms lightharvesting antennae are made of spectrally heterogeneous spatially distinct pigment-protein complexes (Clayton, 1980). In each of the organisms the spectral forms may be represented either by one or by several types of chlorophyll pigments. The expedience of spectral heterogeneity of a PSU antenna is evident since it not only ensured a wider spectral region of solar radiation absorbed by photosynthetic organisms, but also more energy absorbed in each spectral interval per an RC, provided that a natural "energy funnel" for delivery of energy to RCs is formed due to a proper (nonrandom !) mutual arrangement of short- and long-wavelength spectral forms (Borisov & Fetisova, 1971; Seely, 1973). Various spectral forms may comprise a united macro-antenna by one of two ways: (i) each spectral form is assembled into an unbroken homogeneous layer, the layers made by different forms lie close together but are totally distinct spatially; the multicentrality of such PSUs is realized due to all spectral forms (Figs 1 and 2, models (a), (c), (d)); (ii) otherwise, an antenna can be "discrete" in its short-wavelength spectral forms. Such

LIGHT-CONVERTING

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OPTIMIZATION

53

PSUs act as multicentral only due to the long-wavelength spectral form where RCs are localized (Figs 1 and 3, models (b), (g)). Obviously, any type of packing should ensure the effective flow of energy to RCs. It is noteworthy that both ways do not contradict to the structure of an antenna made of pigment-protein globules. We assume that all three spectral forms are spatially separated and equidistant. This assumption is not as restrictive as it may seem, as the qualitative conclusions are not affected. Figure 4 and Table 1 show the possible contribution of antenna spectral heterogeneity to acceleration of energy transfer to an Rc. As one can see from Fig. 4, a heterogeneous antenna has a pseudo-exponential kinetics of excitation decay. Data concerning the times, tj, t2, and t3, of excitation decay (due to its trapping by an RC) to the e -j, e -2, and 0.1 levels are also given in Table 1, together with the operation efficiency for heterogeneous PSUs, r/h~t: ,r/het ___

thorn/_t2het.

In the heterogeneous models studied here, the 77her value is greater than unity, in some cases by several tens per cent, in other cases two-fold, the difference caused by variations in structural organization of antennae. It shows the importance of the detailed structure of an antenna, especially of the position in it of the long-wavelength spectral form. This is most clearly seen from comparison of the t2 values computed for various models. In model (g), for example, short chains of the longest-wavelength form molecules reach RC from four different sides (Fig. 3). Each of these chains touches short-wavelength form molecules from two sides. Therefore the possibility of energy transfer to the longest-wavelength form is maximal here, and at the same time the possibilities of useless energy migration between longest-wavelength form molecules are minimized (an effect of "channels" leading to traps (Borisov & Fetisova, 1971)). As a result, the r/bet value found here was the highest among all studied models. Comparison between results derived here and those reported in the earlier work (Borisov & Fetisova, 1971) shows that the operation efficiency for heterogeneous PSUs, 77h~t, is considerably higher if various spectral forms in an antenna are assembled in concentric layers around each RC, rather than in lamellar heterogeneous structures. However, the heterogeneous macro-antenna structure comprised of concentric layers surrounding each RC is incompatible with the multicentrality of a two-dimensional PSU and, consequently, can hardly be found in vivo. For each spectral interval of light absorption, one can find an optimal number of spectral forms and their mutual arrangement, as we have reported previously (Borisov & Fetisova, 1971). Moreover, one can determine the

54

z.G.

FETISOVA

ET AL.

optimal positions of their absorption maxima. Let us show it for heterogeneous models (d) and (g) (Figs 2 and 3). We shall keep the absorption maxima for the shortest- and the longest-wavelength spectral forms at ~.~ = 800 nm and A3= 890 nm, respectively, and vary the absorption maximum wavelength of an intermediate form, A2, in the range of 800 to 890 nm. In model (d), the optimal spectral position of the absorption maximum, A~pt, was found in the 860-870 nm interval for which t2 = 285 + 4. At other ~t2 wavelengths, the t2 values increased to t2 = 327 at A2= 850 nm and t2 = 305 at ;t2 = 880 nm. In model (g), the ~pt wavelength fell into the 810-860 nm interval, i.e. this interval is much wider. This is explained by a high etticiency of the energy flow to the long-wavelength form A3 = 890 nm. Thus, the r/her value depends on the following structural features of a heterogeneous antenna complex: (1) the number and relative concentration of various spectral forms; (2) their mutual arrangement, and (3) the positions of their absorption maxima. This problem has been partly studied before (Borisov & Fetisova, 1971; Seely, 1973); therefore we shall not dwell on its details. However, it is noteworthy that the essence of the spectral heterogeneity effect lies in limitation of regions of homogeneous energy migration due to the optimal ratio between the overlap integrals which govern uphill and downhill excitation energy transfer. This can be illustrated by the following example. The probability of energy transfer among molecules of the same spectral form, for instance, for the 850~ 850 nm energy transfer, is twice higher than that for the 850~890 nm energy transfer. Consequently, the excitation localized in a molecule which belongs to the 850 nm spectral form (for example, in model (a)) can be transferred to an 890 nm molecule with a 20% probability only, while the probability of homogeneous transfer to another 850 nm molecule is almost 80%. If, however, the excitation has been transferred to the 890 nm form, it will be localized therein with a probability exceeding 95%. It results not only in a reduction of number of jumps necessary for excitation energy trapping by an RC (Fetisova, Fok & Borisov, 1983), but also in a decrease of the time of excitation trapping by an RC (Table 1). It appears that large antennae can only be spectrally heterogeneous (Fok & Fetisova, 1983) and have significant spectral shifts between their forms. Considerable increase in the energy migration rate within an antenna, per se, does not provide an equally high increase in the trapping time by RCs, as factors k 2 and R contribute equally to the rates of both downhill and uphill energy transfer (that is, they equally accelerate the migration towards and from an RC). Therefore, non-random spatial distribution of various

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

55

spectral forms in a macroscopic PSU should reduce the size of the regions of a PSU where the excitation can migrate randomly. The operation efficiency of a heterogeneous antenna can also be increased if intermolecular distances are reduced along one direction of a square lattice. In this case the excitation trapping rates can become equal in a heterogeneous antenna and in a corresponding homogeneous one if the model aggregates of these PSUs are similarly compressed two to three times along the axis perpendicular to the direction of heterogeneous layers (Fig. 6). However, this effect is observed only for the particular case of a lamellar structure of heterogeneous layers in a macro-antenna, and only upon its compression in the mentioned direction (Fig. 7). This interesting fact should apparently be taken into account in designing of artificial light-harvesting antennae. 1

I

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900 800 700 600

.~

l/r l /

500

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~ . \"

400 3O0

°,.

\\

200, I OC I

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~>j_/% FIG. 6. Dependence of the t2 time of excitation energy trapping by reaction centres in heterogeneous models of photosynthetic units shown in Figs 1-3 (thick line) on the degree of compression or stretching of their lattices along the OO' direction indicated by arrows in Figs I-3. Upon deformation the square lattice is converted into a rectangular one. The lattice constant in the direction perpendicular to OO', R±, is constant, and only the lattice constant along the OO' direction, R=, is varied. To facilitate comparison between the effects observed in heterogeneous and homogeneous antennae, the same dependence for corresponding homogeneous antennae is shown here by thin line for the (R±/R=) > l values (see also Fig. 5).

56

Z. G. F E T I S O V A E T A L .

2 500~ --~'''°'"~

600

u',

'~

I

'°2V

, 2

,.

Rj/R=

R/R± FIG. 7. Dependence of the tz time of excitation energy trapping by reaction centres in heterogeneous (thick line) and homogeneous (thin line) models (d) on the degree of compression of their lattices along the OO' direction (full line) and perpendicular to the OO' direction (broken line). The OO' direction is indicated by the arrow in Fig. 2. Upon compression along OO', the abscissa values correspond to the R ± / R = ratio (R= and R~ are the lattice constants in the OO' direction and perpendicular to it, respectively). For aggregates compressed perpendicular to the OO' direction, the abscissa values correspond to the R = / R ± ratio. The t2 values correspond to the time necessary for excitation decay from 1 to e -2 due to excitation trapping by reaction centres.

(C) MUTUAL ORIENTATION OF TRANSITION DIPOLE MOMENT VECTORS OF CHLOROPHYLL MOLECULES

The rates of intermolecular energy transfer governed by the inductiveresonance mechanism are strongly dependent on the mutual orientation of transition dipole moment vectors which correspond to lowest-energy singletsinglet electron transitions in donor and acceptor molecules. For a dipoledipole interaction, these rates are proportional to the orientation factor, k 2, which varies from 0 to 4. Figure 8 shows the k 2 values for energy transfer in a pair of antenna molecules localized in a square lattice, with various orientations o f transition dipole moment vectors. For dipoles randomly oriented in three-dimensional space, k s = 2. Figure 9 shows the dependence of the k2(ot). 1 / R 6 factor from equation (l) for energy transfer from any molecule o f an antenna with a square lattice to eight nearest molecules, for the case of parallel vectors of their transition dipole moments. Let us compare the t2 times for models (i) with aligned transition dipoles which belong to antenna molecules and an RC, and (ii) with random orientation of these dipoles. In oriented systems, all vectors are collinear and make an

LIGHT-CONVERTING

SYSTEM

k 2 = ( cos

8,z -

OPTIMIZATION

57

3 cos e~. c o s Os )2

FIG. 8. k 2 orientation factors calculated for various mutual orientations of transition dipole m o m e n t vectors which belong to a pair of molecules in an a n t e n n a with a square lattice. @, a n t e n n a molecules involved in the energy transfer; I1, transition dipole m o m e n t vectors of these molecules; r, the radius-vector drawn from one molecule to the other; 8, the angles between It a n d r vectors s h o w n in the diagram. I

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1... 11

~ ! ~.....-T= /~ \ ~+"~""° / \ ..L."-::C'<-/ -

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30

60

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a,grod.

FIG. 9. D e p e n d e n c e o f probability of energy transfer from any a n t e n n a molecule to its eight nearest neighbours in a h o m o g e n e o u s a n t e n n a with a square lattice on the direction of transition dipole m o m e n t vectors (the a angle). Transition dipole m o m e n t vectors of a n t e n n a molecules are collinear; intermolecular distances are taken into account. Probabilities o f energy transfer in pairs o f molecules are s h o w n by broken line for the O --, I a n d O --, 5 transfer, by dot-and-dash line for the O ~ 3 a n d 0--*7 transfer, by full line for the O ~ 4 and O ~ 8 transfer, by dotted line for the 0 - * 2 and O ~ 6 transfer.

58

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60

Z.

G.

FETISOVA

ET

AL.

a a n g l e with the O O ' d i r e c t i o n i n d i c a t e d b y a r r o w s in Figs 1-3. It can b e easily seen f r o m Fig. 9 that if o n l y the n e a r e s t n e i g h b o u r i n t e r a c t i o n s are t a k e n into a c c o u n t (i.e., i n t e r a c t i o n o f e a c h m o l e c u l e with f o u r n e i g h b o u r s ) then k E ( a ) ~ 0 at a ~ 3 5 . 3 ° a n d at a ~ 5 4 - 7 °. T h e r e f o r e , all the s t u d i e d m o d e l s reveal e x t r e m e l y long t r a p p i n g t i m e s : t 2 ( a ) ~ oo at t h e s e a v a l u e s (Fig. 10). If, h o w e v e r , we r e g a r d i n t e r a c t i o n s o f e a c h m o l e c u l e with its eight n e i g h b o u r s r a t h e r t h a n with f o u r o n e s , the t 2 ( a ) d e p e n d e n c e w o u l d be d r a s t i c a l l y c h a n g e d , yet n e a r l y all q u a l i t a t i v e c o n c l u s i o n s will survive. F i g u r e 11 s h o w s the t2(~) d e p e n d e n c e s for t h e m o d e l s d e p i c t e d in Figs 1 a n d 2, p r o v i d e d t h a t i n t e r a c t i o n s with eight n e a r e s t n e i g h b o u r s are t a k e n into a c c o u n t . F i g u r e 12 s h o w s the s a m e d e p e n d e n c e s for m o d e l s (g) (see Fig. 3) c a l c u l a t e d in the b o t h cases: for i n t e r a c t i o n s with f o u r n e i g h b o u r s a n d

[

I

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i

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1500

'

\,/ i

,ooo

i -

t~

500-

/

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30

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60

1

-

90

a, grad

FIG. 12. Dependence of the t2 time of excitation energy trapping by reaction centres in homogeneous (full line) and heterogeneous (broken line) models (g) (shown in Fig. 3) on the angle between the OO' direction (indicated by the arrow in Fig. 3) and collinear transition dipole moment vectors of aggregate molecules (it). Horizontal lines show the t 2 values for the same PSU models with random orientation of antenna molecules in three-dimensional space. Calculations of the t2 times were made for the two cases where interactions between a given molecule and its four nearest neighbours were considered (thin line) and where interactions with eight nearest neighbours were taken into account (thick line).

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

61

for interactions with eight neighbours. Figures 11 and 12 reveal that: (1) For each of the models the t2(a) function has two deep minima, near = 0 ° and a = 90 °, where its values are significantly lower than those for randomly oriented systems (in the models studied, up to five-fold lower). Sometimes t2(a) has a third minimum near o/=45 ° (heterogeneous model (g), Fig. 12); however, such oriented systems are of low efficiency since the t2(a) values near a = 45 ° are always higher than those for randomly oriented systems because the k2(t~) values are small near o/=45 ° (see also Fig. 9). (2) Comparison between the t2(0 °) and t2(90 °) values for various models with two axes of different lengths shows that the highest efficiency of light-harvest (which corresponds to minimal t2 values) is provided by the orientation with a zero fl angle between the longest axis of a model aggregate and the direction of collinear transition dipoles. For example, the optimal o/ angle is a °pt= 90 ° for models (a) and (b), while it equals o/°pt= 0 ° for models (c) and (d) (Fig. 13). Therefore, the orientation of the transition dipole moment vectors in optimal PSUs should provide maximal rates of energy transfer (which correspond to k 2= 4) along "limiting" directions in the macro-antenna independently of position of RCs therein (compare the

[3AO E]AO

DAo ODAAAAO0 DDAAAAO0 DOAAAAO0

E]&O ID&0

[3AO t3AO

Longoxisdirecti~ a ~t = 0 o

a(~pt = 9 0 °

/3 = 0 °

=O °

t Opfimol/~

direction

FIG. 13. Optimization of orientation of transition dipole moment vectors for molecules of model PSUs with two axes of different length.

62

Z. G. FETISOVA

ET AL.

(a) and (c) models). It is noteworthy that this principle is also inherent in model aggregates with only slightly elongated heterogeneous domains in a PSU antenna. For example, the homogeneous (g) model (Fig. 3) has two equal axes and therefore possesses two equally optimal s values: s = 0 ° and s = 9 0 ° (t2= 144). However, in heterogeneous model (g), the spectral forms with absorption maxima at 800 and 850nm comprise elongated (perpendicularly to the OO' direction), although short, domains (Fig. 3). Therefore, heterogenous model (g) has the maximal light-harvesting efficiency (or the minimal t2 value) at a°pt=90° (t2=87), while at s = 0 ° the efficiency is lower, t2=91 (Fig. 12). (3) It can be seen from Figs l I and I2 that both minima are broad; in other words, near s = 0 ° and s = 90 ° the t2 values vary by less than 20% within t~ deviations, As = ±(11-15°). Therefore, molecular thermal vibrations cannot disturb the system and drive it out of these minima. It means that the system is stable at these s values. Here the greater As value, at which t2 does not appreciably change, corresponds just to s °pt. Let us consider the following example for homogeneous model (d): s°Pt=0°, t~ in= 127at s - - 0 °, t2 grows by 20% at A s = ± 1 5 °. At s = 9 0 ° t2=247, t2 grows by 20% at As = +12 °. Usually a ±20 ° As deviation from the optimal s value does yield the trapping time which is at least twice smaller than that in systems randomly oriented in three-dimensional space. Experimental evidence for well-oriented PSU macro-antennae in vivo is still lacking; yet for many photosynthetic organisms it has been shown that transition dipoles of their antenna molecules and o f the RC special pair are approximately parallel to the membrane plane both in higher plant PSUs (Junge & Eckhof, 1973; Gagliano, Geacintov & Breton, 1977; Haworth et al., 1982) and in bacterial PSUs (Morita & Miyazaki, 1978; Rafferty & Clayton, 1979; Paillotin, Vermeglio & Breton, 1979; Abdourakhmanov et al., 1979; Abdourakhmanov, 1980). However, even in the case of random orientation within the membrane plane, the excitation transfer rate for a two-dimensional random orientation is higher than that for three-dimensional random ones since the corresponding k 2 values equal 5 and 2, respectively (see the Appendix). The following example is given for a t 2 computation in homogeneous model (g): the t2 trapping time equals t2=355 in the case of three-dimensional random orientation, while in the case of two-dimensional random orientation it is equal to t2 = 189, i.e. twice lower (Fetisova, 1984). Thus operation efficiency of PSUs is tolerant to heat motion o f PSU molecules both in the case of the optimal orientation o f collinear p. vectors of PSU molecules and in the case of two-dimensional random orientation o f the vectors in the PSU plane (see the Appendix).

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

63

It is not excluded that transition dipoles of macro-antenna molecules can be also optimally arranged in the membrane plane, yet a specific character of this arrangement may in principle lead to low dichroic values. 4. The Role of Optimization Factors in Creating of a Molecular Focusing Zone of a Reaction Centre

If all the structure parameters considered above are optimized not for an antenna as a whole but only for its part, an acceleration of energy transfer from an antenna to an RC would nevertheless occur although its operation efficiency will be lower than in the case of optimization of the parameters for a total antenna. In particular, optimization of the structure parameters for the part of an antenna nearest to an RC may form a certain "focusing zone" of an RC which favours reduction of the trapping time due to an increase in the "effective radius" of excitation energy trapping by an RC. This interesting effect of appreciable acceleration of energy trapping was found in our earlier theoretical study (Borisov & Fetisova, 1971). There it occurred if a homogeneous antenna was supplemented with a small number of molecules adjacent to an RC which had an absorption spectrum similar to that of RCs while the remaining part of antenna molecules had a shorter wavelength of absorption maximum. Let us consider the ways of creating of an RC "focusing zone" (Fetisova, 1982). A "focusing zone" of an RC may be formed by a small number of antenna molecules nearest to the RC (Fig. 14). A focusing effect, or reduction of the excitation energy trapping time, occurs if at least one of the following requirements is fulfilled: (1) transition dipole moment vectors (p.) of these molecules and of the RC are collinear (focusing by it, f~); (2) distances from these molecules to the RC are shorter than intermolecular distances in the antenna, for example, due to a greater size of an RC (focusing by R, JR); (3) these molecules comprise a minor long-wavelength spectral form of antenna pigments (focusing by A, fx). Let us demonstrate the focusing effect yielded by these three factors (f~, fR, and fA) in the case of homogeneous model d where a "focusing zone" is formed by four antenna molecules adjacent to the RC (Fig. 14). As the Table given to Fig. 14 shows, the effect of a "focusing zone" in this model results in over a two-fold decrease of the t2 value. From the same table one can find an optimal relation between spectral positions of the absorption maxima which belong to the major part of the antenna and to the "focusing zone" in the case of the fA focusing. In the given example we have chosen

64

Z . (3. F E T I S O V A

ET AL.

f2 rel.units 0 0 0 0 0 0 0 0 O 0 0 S O 0 0 0

509

0 0 0 0 0 0 0 0 0 0 0 ~ 0 0 0 0 285

o o ~ o o o 0 0 0 ~ 0 0 0 0

AR,~, 0

0

O0 ~

0

f~ 0 0 ~

0

0

0

I

0

0

0

3

286

0

0

0

5

232

0

0

0

0

0

0

0

0

0 0

0 0

® • @ 0 0 ® 0 0

0 0

0 0

408

Xo,nm ® 0

Xf = 885 rim

8.50 860 870 880 885

3:34 520 345 423 483

FIG. 14. Effect of the "'focusing zone" around a reaction centre in a PSU with a square lattice on the t2 time of excitation energy trapping: fo, no focusing of excitation energy (a homogeneous PSU with undifferentiated antenna molecules). All molecules of the PSU are randomly oriented in three-dimensional space (k 2 = 2/3); fro,, focusing by collinear orientation of transition dipole moment vectors (ix) which belong to the RC (~,) and to its four nearest neighbour antenna molecules (,~). All the rest of the antenna molecules have randomly oriented It vectors (k 2= 2/3); fR, focusing by a decrease of distances between the RC (@) and its four nearest neighbour antenna molecules (O). The distances (in the antenna with R = 15 A) are reduced by AR for these four molecules; t z are calculated for AR = 1, 3, and 5 A. All antenna molecules of the PSU are randomly oriented (k 2= 2/3). In all the above cases, f0, f~, and fR, the absorption maximum wavelengths of a reaction center, ARc, and of antenna molecules, :to, are equal to 890 nm. fA, focusing by a minor long-wavelength spectral form. Reaction centres (@) absorb at ARc = 890 nm, their four nearest neighbour antenna molecules (®) comprise the minor long-wavelength spectral form with an absorption maximum at hf = 885 nm; all the remaining antenna molecules (O) absorb at h,. Calculations of t2 are for A, = 850, 860, 870, 880, and 885 nm provided that all PSU molecules are randomly oriented (k 2 = 2/3). t2, time necessary for excitation decay from 1 to e -2 level due to excitation trapping by reaction centres.

such a position of the absorption maximum of "focusing zone" molecules, Ay = 885 n m , t h a t t h e o v e r l a p i n t e g r a l in e q u a t i o n (1) w a s m a x i m a l f o r e n e r g y t r a n s f e r f r o m t h e " f o c u s i n g z o n e " to t h e R C w h i c h a b s o r b s at 890 n m . T h e fA f o c u s i n g effect w a s t h e h i g h e s t f o r t h e a n t e n n a w i t h t h e a b s o r p t i o n m a x i m u m at A, = 860 n m . I f m o r e t h a n o n e f o c u s i n g effects o c c u r s i m u l t a n e o u s l y in t h e " f o c u s i n g z o n e " , t h e r e s u l t i n g f o c u s i n g e f f e c t is e v i d e n t l y i n c r e a s e d . F o r e x a m p l e , i f f o r t h e s a m e m o d e l u n d e r s t u d y ( F i g . 14) t h e f~

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

65

focusing (Ao = 870 nm and AI= 885 nm which yield t2 343 in Fig. 14) is accompanied by the f~, focusing (in a way similar to that shown in Fig. 14), then for the resulting (fA +fg) effect one obtains further decrease of the t2 trapping time: t2 235. If additionalfR focusing is introduced, for AR = 5 A, the total effect (fx +f~+fR) yields a still shorter trapping time: t2--- 200. The assumption used here, that a small fraction of antenna molecules with absorption maximum at slightly shorter wavelengths than that of RCs exists in vivo, is compatible with experimental analysis of absorption spectra of various photosynthetic organisms provided by differential spectrophotometry (Guljaev & Litvin, 1967; Wiessner & French, 1970; Shubin, 1975). However, experimental evidence for the role of these molecules in excitation energy focusing towards RCs is lacking so far. It is noteworthy that a longer-wavelength shift of the absorption maximum of these antenna molecules adjacent to an RC can result from either of the two following factors or from both of them: i.e. a specific orientation of their dipoles, and localization at shorter distances from an RC compared with intermolecular distances in an antenna, as shown in Fig. 14. In this case a co-operated focusing effect occurs, either f~ +f~, or fA +fu, or f~ + f , +fu. There are, however, experimental results which showed the existence of a small fraction of molecules adjacent to an RC which have a common feature in PSUs of various organisms, namely, their absorption maxima are shifted by 5-25 nm to longer wavelengths compared with the RC absorption in the same PSUs. Besides, it was shown that the functions of these molecule fractions are different from those of antenna molecules. The results are obtained just as for PSUs of higher plants (the specialized form "C705") as for bacterial PSUs (Ogawa & Vernon, 1970; Butler & Kitajiama, 1975; Shubin, 1975; Gomez et al., 1982; Borisov, 1983). There is as yet no unanimous opinion about their role. It is not excluded that all or some of these fractions are antenna fractions and couple the antenna with an RC. On the other hand, it is possible that these molecules act as a safety valve which lets out excess energy from an RC at high light intensities and operates only in case of emergency. On the contrary, it is possible that all or some of these molecule fractions are components of an RC and participate in charge separation processes. Whatever their function in vivo, it is important to find out whether the Chl forms which are lower in energy than RCs can play any role in the focusing of energy transfer towards an RC. In accordance with experimental results, we assumed that the transition dipole moment vectors of antenna molecules and of the RC special pair lie in the same plane (membrane plane). Then, according to the fluid-mosaic model of membrane structure formulated by Singer & Nicolson (1972), these vectors may be considered as randomly oriented in membrane plane, i.e. randomly =

=

z . G . FETISOVA ET AL.

66

oriented in two-dimensional space. Therefore, in t2 trapping time calculations (Tables 2-4) the orientation factor k 2 = 5 was used (see the Appendix). Let us consider three different cases: (1) Xa < ARc, (2) Aa = Aac, (3))ta > h a c , here Aa denotes the absorption maximum wavelength of the major part of an antenna, Aac is that o f an RC. The first case has been found in PSUs o f Photosystem-I in higher plants: )ta = 6 9 2 n m (the longest-wavelength spectral form of the antenna part adjacent to P700), Aac = 700 nm. The calculations below are made for a model PSU structure shown in Fig. 14 where the focusing zone is comprised by four molecules adjacent to an RC. Let us vary their absorption maximum from 692 to 720 nm. The t2 values calculated for this model are given in Table 2. The t 2 value at A: = ha = 692 nm corresponds to the lack of the TABLE 2

Effect o f a "focusing zone" around a reaction centre in a model photosynthetic unit shown in Fig. 14 on the excitation trapping time, t2, in the case Aa < ARC : ARC = 700 nm; Aa = 692 nm is the wavelength of the absorption maximum o f antenna molecules; Af is the absorption maximum wavelength of "focusing zone" molecules. Orientation factor is k 2 = 2. Focusing factors,L , f , . and fR, are as shown in Fig. 14(AR = 5 A ; R = 15/~). The t2 time corresponds to excitation decay from l to e -2 level due to excitation trapping by reaction centres. The underlined t values correspond to the trapping time of excitation in PSUs with a homogeneous antenna (by -- ha, i.e. focusingf~ is absent). Trapping time (t2) nm

A

A +f~

fa +fR

692

254

182

123

705 710 715 720

203 234 295 438

156 182 228 326

119 134 157 191

LIGHT-CONVERTING

SYSTEM OPTIMIZATION

67

focusing (fo) and equals h(fo) -- 254. From the same Table one can see that the h values at AI=705 nm (t2(f~)= 203) and at A:= 710 nm(t2(f~)=234) are appreciably lower than h(f0), i.e. the focusing effect is realized. As we have noted, the minor longest-wavelength spectral form itself can originate from a specific orientation o f transition dipoles o f its molecules (ordered dipoles in contrast to random dipoles in the rest of the antenna) or from smaller distances between these molecules and the RC, compared to intermolecular distances in the total antenna as shown in Fig. 14. Consequently, it is interesting to trace the co-operative effects o f these focusing factors on the efficiency o f energy transfer to an RC. These data are also given in Table 2 which shows that co-operative factors, f~ +f~,, or fx ÷fR, yield a more pronounced focusing effect, or the decrease in t 2 compared with t2(fo). However, the maximal focusing effect is achieved only due to the minor spectral form with Ay = 705 nm. Therefore, a C-705 chlorophyll spectral form can in principle comprise a focusing zone around a P700 reaction centre in vivo. The second and the third cases have been found in bacterial PSUs. Let us calculate the t2 values for the second case (Ao = ARC) for the same PSU model shown in Fig. 14, with A~ = ARC = 870 nm and with the absorption maximum wavelength Af o f the four molecules adjacent to the RC varied from 865 to 900 nm. Table 3 shows that the focusing effect does exist, with

TABLE 3

Effect o f a "focusing zone" around a reaction centre in a model photosynthetic unit shown in Fig. 14 on the excitation trapping time, t2, in the case Aa = ARC" Aa = 870 rim, ARC = 870 nm. All other values are as in Table 2 Trapping time

(/2)

xl nm

A

A +f~

A +fR

865 870

302 272

210 190

135 125

875 880 885 890 895

257 260 277 315 375

181 181 191 214 249

118 117 119 127 139

900

452

3O9

156

68

Z. G. F E T I S O V A

ET AL.

the fA focusing occurring owing to minor antenna forms with absorption maxima, Ay, shifted from the maximum of the major antenna absorption, Aa, to longer wavelengths, i.e. in this case the focusing effect can only occur if A: > Aa. The co-operative factors, fA +f~ and fA +fR, yield a more pronounced focusing effect, the maximal value of this effect (the maximal value of the decrease in t2 compared with t2(fo)= 272) being achieved when A:> Aa(Af-Aa can be as high as 15 nm). For the third case (Aa > ARC) the t2 values are calculated with Aa = 880 nm, ARC = 870 nm, and Af varied from 865 to 900 nm. Calculations are made for the same model in Fig. 14, provided that A: belongs to four molecules adjacent to the RC. Table 4 shows that the focusing effect occurs only in

TABLE 4

Effect o f a "focusing zone" around a reaction centre in a model photosynthetic unit shown in Fig. 14 on the excitation trapping time, t2, in the case Aa > ARC : Aa = 880 rim, ARc = 870 nm. All other values are as in Table 2 T r a p p i n g time (/2) A/ nm

A

A +f~

A +fu

A +f~+fR

865 870 875 880

464 380 350 342

318 259 235 225

191 156 139 132

179 145 129 121

885 890 895 900

370 410 476 617

237 257 290 364

126 128 137 152

116 116 120 131

the case of simultaneous contribution of several factors, f~ +f~, or fx +fR, or f A + f , + f R . The maximal focusing effect (the maximal value of the decrease in t2 compared with t2(fo)= 342) is achieved only if hy I> ha, and h : - h a can be as high as 10nm. In this case h / > ARC, and h : - - h R c can be as high as 25 nm! Thus, at ha/> ARC in PSU models considered the maximal focusing effect is achieved if Aye>ha; h : - ; t ~ can be as high as 15 nm and h:--hRC can be as high as 25 nm! At '~a
LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

69

only at small differences between the absorption maxima AI and ARC: lay - ARCl~ 5 rim.

5. Principles for Designing Optimal Artificial Photosynthetic Units Analysis of the efficiency provided by various optimization factors considered here is useful from the viewpoint of creation of artificial lightconverting systems since it allows one to formulate principles for'designing optimal artificial PSUs. Now we can summarize the basic principles for two-dimensional systems, which do not claim at providing final generalization. (I) The effects of optimizing factors studied here should be co-operated in optimal systems. (2) The most powerful factor that governs the rate of excitation energy trapping by a reaction centre (or by another trap) is the anisotropy of intermolecular distances which can easily be attained due to the fiat shape of a chlorophyll molecule (or of other dye molecules). As follows from section 3(A), an optimal antenna should have a strongly elongated shape and should be specifically anisotropic (a square lattice compressed along the longest axis of the elementary aggregate). In other words, for a given N number, an optimal antenna should be comprised of the minimal number of "cylinder" stacks made of close-packed chlorophyll molecules. Each stack should contain the maximal number of chlorophyll molecules arranged so that their planes are parallel to each other and perpendicular to the axis of the stack. Figure 6 shows clearly that the limiting t2 values are smaller for the aggregates with minimal length of the smaller axis. Therefore, for any given N number, an optimal ratio should be calculated for the longer and the shorter axes of an elementary model aggregate. (3) In an optimal macroscopic antenna, mutual position of traps for excitation energy should be also optimized. It is clear from Fig. 6 that in systems with anisotropy of energy transfer rates (provided by the method described in (2)), the most effective traps will be those arranged in clusters, so that the "chains" of traps should form continuous barriers across the longest axes of elementary aggregates in the macroscopic antenna (compare the (c) and (d) models in Fig. 6). Moreover, to increase the light-harvesting efficiency of an antenna (regardless of the position of the traps therein), it is necessary to create a focusing zone around each trap. The clusters should be organized so that each trap in a cluster should be coupled with the antenna via all the possible adjacent molecules (as in the case of the solitary traps in model (d)) or via the four nearest neighbour antenna molecules

70

Z. G. F E T I S O V A

ET AL.

(as in model (c) shown in Fig. 2). This, in turn, favours creation of focusing zones by antenna molecules nearest to each trap (that is, by four molecules in the case of a macro-antenna with a square lattice). These zones, together with the traps, form domains containing many clusters of reaction centers and the adjacent parts of the macro-antenna. Such domains spread in the form of unbroken wide bands ("channels" to traps) over the macroscopic PSU perpendicular to the longest axes of its elementary aggregates (Fig. 15(a)). From these "channels" the excitation is delivered to the traps without violation of the PSU multicentrality. The evidence for the clustering of RCs in vivo is obtained recently by Meyer, Snozzi & Bachofen, (1981). (4) In systems built as described above (in (2) and (3)), the rate of energy transfer along the longest axis of an elementary aggregate is about 200-fold higher than that across the longest axis, provided that the molecules are randomly oriented in three-dimensional space. Hence, any further increase of the transfer rate along the longest axis would decrease the trapping time

LOOAAADD

i

IO[]AAAOO [a~OAAAOD O[]AAAO~OAAAD[]

0

~O'

(b)

FIG. 15. Optimization of the (c) homogeneous model structure. (a) The structure of model (c) after its transformation into a spectral heterogeneous one and a 2-5-fold compression of its lattice along the direction OO' shown by the arrow in Fig. 2. IS], A, and O, the antenna molecules which correspond to the spectral forms with absorption maxima at 800, 850 and 875 nm; Q, the molecules of the focusing zones with an absorption maximum at 885 nm; O, the RCs with an absorption maximum at 890 rim. (b) The direction of transition dipole moment vectors in the PSU plane.

LIGHT-CONVERTING

SYSTEM

OPTIMIZATION

71

but insignificantly. Therefore, further optimization of the model aggregate structure can be favoured by ordered orientation of transition dipole moment vectors in a single way, namely, the direction of collinear dipoles should coincide with the aggregate axis along which the energy transfer is the slowest. Thus, transition dipole moment vectors in such system should lie in the aggregate plane and be oriented perpendicular to the longest axes of elementary aggregates. In other words, the factors of intermolecular distance anisotropy and of transition dipole orientation should be used to minimize the trapping time in mutually perpendicular directions in a planar macroscopic PSU. (5) The spectral heterogeneity of an antenna should be used to limit regions of homogeneous energy transfer, especially in large antennae. Without attempting at constructing a PSU model with the maximal 7/ value at given N, we shall show the ways of optimizing of the least effective model PSU, homogeneous model (c) in Fig. 2. An improved structure of this PSU model in which all the optimization factors studied here were used according to the principles listed above is shown in Fig. 15. In this macroscopic model PSU the focusing zones are made of four molecules which are the nearest neighbours of RCs, the reaction centres are arranged in clusters, and these clusters together with their focusing zones comprise domains, or local zones within the macro-antenna, where the excitation is captured and can be transferred to any RC so that the requirement of PSU multicentrality is fulfilled. Each domain spreads in the form of unbroken band over the macroscopic PSU perpendicular to the longest axes of its elementary aggregates (Fig. 15). The homogeneous isotropic model (c) in Fig. 2 with which we have started has the trapping time t2 = 1223 (see Table 1). When a spectral heterogeneity was introduced with the absorption maxima of antenna molecules at 800, 850, and 875 nm, and that of reaction centres at 890 nm, the trapping time was reduced to t2 = 339. Compression of the PSU lattice 2.5 times along the OO' direction (that is along the longest axes of elementary aggregates), together with orientation of transition dipoles within the PSU plane perpendicular to the OO' direction (Fig. 15(b)), and formation of focusing zones with AI= 885 nm yield a drastic decrease of trapping time to tz = 5. The additional fR focusing (as in Fig. 14) with AR = 2/~ at R = 15 A for the initial square lattice results in further reduction of the trapping time to t2 = 4. In other words, the t2 trapping time in the optimized model was reduced almost 310 times compared with the corresponding homogeneous isotropic PSU! We have so far assumed that a reaction centre acts as a perfect trap, i.e. its efficiency of excitation trapping (qb,) amounts to 100%. However, the calculations show that the conclusions drawn here are also valid in the case

Z. G. FETISOVA E T A L .

72

o f lower t r a p p i n g efficiency. It can be easily seen f r o m Table 5 calculated for m o d e l (d) d e p i c t e d in Fig. 2. Really, a decrease o f ~ , f r o m 100% to 60% c h a n g e d the "0 values by not m o r e than 2 5 % , a n d the r/ values are mostly increased. It means that in a realistic case at qb, < 100%, or even at • ,<< 100%, the optimization o f a PSU structure is even m o r e significant than in the cases studied here. TABLE 5

Effects o f optimizing factors for a model photosynthetic unit shown in Fig. 2 (homogeneous ( d) model) on the excitation trapping time, t2, when qbt = 100% and qb, = 6 0 % , qb, being the efficiency o f excitation trapping by reaction centres. ,op,, where t o is is the operation efficiency of an optimizing factor: 77 = ,O/ ,2/,2 the t: trapping time for a photosynthetic unit with an isotropic uniform antenna, t~p' is the t2 time for the same photosynthetic unit after optimization of its antenna structure. The focusing factors, f , and fR (AR = 3/~, R = 15/~,), are as shown in Fig. 14. Trapping efficiency 100% Optimizing factors None Spectral heterogeneity 2.2-fold lattice compression along OO' direction Collinear i~ orientation along OO' direction Focusing zone fix around RCs fa

60%

tz

r/

t2

509 329

1.5

71 I 395

1.8

50

10-2

56

12-7

130 285 286

3-9 1-8 1-8

183 341 339

3"9 2.1 2- i

It a p p e a r s that o p t i m i z a t i o n o f all the factors listed a b o v e occurs in vivo as well, especially in large and highly efficient PSUs. The same principles s h o u l d be fulfilled in the construction o f artificial p h o t o s y n t h e t i c units. The m e t h o d described a b o v e allows a theoretical analysis for expected efficiency o f various artificial light-converting systems. O n e should, however, bear in mind that the effects o f various optimization factors are not additive, and their co-operative contribution s h o u l d be calculated in every case.

REFERENCES ABDOURAKHMANOV,I. A. (1980). Dissertation, Institute of Photosynthesis U.S.S.R. Academy of Sciences.

LIGHT-CONVERTING

SYSTEM OPTIMIZATION

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ABDOURAKHMANOV, I. A., GANAGO, A. O., EROKHIN, YU. E., SOLOV'EV, A. A. & CHUGUNOV, V. A. (1979). Biochim. biophys. Acta 546, 183. BORISOV, A. Yu. & FETISOVA, Z. G. (1971). Molekularnaya Biologia 5, 509 (in Russian). BORISOV, A. Yu. (1983). Proceedings VIth Int. Congr. Photosyn., Brussels (in press). BUTLER, W. L. & KITAJIMA, M. (1975). Biochim. biophys. Acta 396, 72. CLAYTON, R. K. (1980). Photosynthesis, (Hutchinson, F., Fuller, W., Mullins, L. J. eds). Cambridge, New York, Melbourn: Cambridge Univeristy Press. FETISOVA, Z. G. (1975). Dissertation, Moscow State University. FETISOVA, Z. G. (1982). In: I Y lnt. Syrup. on Photosynthetic Prokaryotes, Abstr. C 12. Bombannes, France. FETISOVA, Z. G. (1984). In: 8th Int. Biophysics Congress, Abstracts, p. 150. Bristol, England. FETISOVA, Z. G. & FOK, M. V. (1984). Molekularnaya Biologia, 18, 1650 (in Russian). FETISOVA, Z. G., FOK, M. V. & BORISOV, A. Yu. (1983). Molekularnaya Biologia, 17, 437 (in Russian). FOK, M. V. & FETISOVA, Z. G. (1983). Photobiochem. Photobiophys. 6, 127. FOK, M. V. & FETISOVA, Z. G. (1984). In: Abstrcts of the 5th Int. Conf. on Photochemical Conversion and Storage of Solar Energy. p. 118, Osaka, Japan. F(SRSTER, TH. (1948). Ann. Physik 2, 55. GAGLIANO, A. G., GEACINTOV, N. E. & BRETON, J. (1977). Biochim. biophys. Acta 461, 460. GOMEZ, I., SIEIRO, C., RAMIREZ, J. M., GOMEZ-AMORES, S. & DEE CAMPO, F. F. (1982). FEBS Lett. 144, 117. GULJAEV, B. A. & LITVIN, F. F. (1967). Biofizika 12, 845. HAWORTH, P., TAPIE, P., ARNTZEN, C. J. & BRETON, J. (1982). Biochim. biophys. Acta 682, 152; 504. IL'INA, M. D., KOTOVA, E. A. & BORISOV, A. YU. (1981). Biochim. biophys. Acta 636, 193. JUNGE, W. & ECKHOF, A. (1973). FEBS Left. 36, 207. KNOX, R. S. (1968). 3". theor. Biol. 21, 244. KNOX, R. S. (1977). In: Topics in Photosynthesis (Barber, J. ed), Vol. 2, p. 55. Amsterdam: Elsevier. LOSEV, A. P. (1982). Dissertation, Institute of Physics, B.S.S.R. Academy of Sciences, Minsk. MORITA, S. & MIYAZAKI, (1978). J. Biochem. 83, 1715. MEYER, R., SNOZZI, M. & BACHOFEN, R. (1981). Arch. Microbiol. 130, 125. NETZEL, T. L. (1982). In: Biological Events Probed by Ultrafast Laser Spectroscopy, (Alfano, R. R. ed.), p. 79. New York: Academic Press. OGAWA, T. & VERNON, L. P. (1970). Biochim. biophys. Acta 197, 332. PAILLOTIN, G., VERMEGLIO, A. & BRETON, J. (1979). Biochim. biophys. Acta 545, 249. PEARLSTEIN, R. M. (1967). Brookhaven Syrup. Biol. 19, 8. RAFFERTY, C. N. & CLAYTON, R. K. (1979). Biochim. biophys. Acta, 546, 189. SEELY, G. R. (1973). J. theor. Biol. 40, 173. SINGER, S. J. & NICOLSON, G. L. (1972). Science 175, 720 SHUBIN, V. V. (1975). Dissertation. Moscow State University. WIESSNER, W. & FRENCH, C. S. (1970). Planta 94, 78. ZENKEVITCH, E. I., KOCHUBEEV, G. A., LOSEV, A. P. & GURINOVITCH, G. P. (1978). Molekularnaya Biologia 12, 1002 (in Russian).

APPENDIX C a l c u l a t i o n o f k 2 factor L e t us a d o p t s u c h C a r t e s i a n c o o r d i n a t e s y s t e m t h a t t h e y - a x i s is p a r a l l e l to t h e l i n e d r a w n t h r o u g h m a c r o c y c l e c e n t r e s o f t w o c h l o r o p h y l l m o l e c u l e s between which the energy transfer occurs. In this system the unit vectors,

74

z.G.

ET

FETISOVA

AL.

M, and M2, which are parallel to transition dipole moment vectors of these molecules, are equal to: { sin 0' cos ~o') M~= sin 0' sin ~o'?; cos 0' J

[sin 0" cos ~") M2=~sin0"sin~"~ ( c o s 0" J

where 0' and 0" are the angles between these vectors and the z-axis, ~' and ~" are the angles between the x-axis and the projections of these vectors on the xy-plane. The unit vector, r, which is parallel to the y-axis, is equal to

Then, k 2 - [cos (M,, M2) - 3 cos (M,, r).cos (M2, r)] 2 = (sin O' sin

O"cos ~' cos ~o"- 2 sin

O' sin 0" sin ~o' sin ~o"

+ COS 0 ' C O S 0 " ) 2

where 0', 0", ~p', and ~o" change independently from each other, with - ,n-<~(p' ~< 7r,

- ,a- <~tp" <~ ,n-.

If all transition dipole moment vectors are in the same plane and the x-axis is also in this plane, then 77"

0 ' = 0"=--. 2

If these vectors depart from this plane in the range of angles A0, then 9"/"

~

77"

~-A0~-< 0 ~<--+ ; 2 A0

7rA0<~20 " < ~ _ +A0

By definition, d,' k 2 _

--rr

d~p"| --~r

dO'

a(~r/2)--a0

,_2~r#, . . . . . ~ , 0 , 0 " ) sin 0' sin 0" dO" ¢(~r/2)-ae

I f f d , f(~/2)+a°dO,f("/2)+a°sin O, sin O,,d8. •/ ( ' r r / 2 ) - - A 0

d (~'/2)--~.0

Let us perform the integration. Then k 2 = ~i ( 5 - - Tlos m • 2 AS+sin 4Ao).

L I G H T - C O N V E R T I N G SYSTEM O P T I M I Z A T I O N

75

For a random orientation of transition dipole mom__ent vectors in threedimensional space (i.e. A0 = zr/2), the well-known k 2 =2 can be obtained. For a random orientation of these vectors in two-dimensional space (i.e. A0:0),

k 2--5 --4"

It should be mentioned that if these randomly oriented vectors lie in the same plane but may depart from the plane in the range of angles A0 < 25 °, then the k 2 value is changed no more than by 10%.