- Email: [email protected]
Exciton transport in the photosynthetic unit
.I. theor. Biol. (1978) 72, 91-116
Exciton Transport in the Photosynthetic Unit T. MARKVART~ Department
of Physics,
University
of Keele, England
(Received 25 October 1977) In this paper we present a theoretical study of the exciton transport in the photosynthetic unit and attempt to correlate the structural and organizational properties of the unit with the exciton transport efficiency. The transport mechanism is assumed to be of the Forster type. This mechanism is shown to be equivalent to the thermally assisted exciton hopping. The transport equations are solved by the Green’s function technique in a more general way than has been presented previously. We thus find it possible to study the effect of the proper Rm6 coupling and random orientations of the pigment molecules. The equations are also solved for a low concentration of reaction centres when the conventional random walk picture is not valid. Various models of chlorophyll a and b mixtures are studied and by comparing their different exciton transport properties, an explanation is proposed for several observed features of the primary photochemical process.
1. Introduction Jt it well established that the light energy required for photosynthesis is collected by the aggregate of accessory pigments in the photosynthetic unit and subsequently, this energy migrates to a special molecule (or molecules) of chlorophyll where the chemistry begins. Although other mechanisms for the energy transfer have been suggested in the past it is generally accepted now that the transport agents are excitons (Duysens, 1964; Clayton, 1965). In the primary chemical reaction an electron is transferred from the reaction centre (R.C.) to the primary electron acceptor A: (R.C.)*
A + (R.C.)+A-.
(1)
The fluorescence of excitons in the phototsynthetic unit depends on the state of Al, and sometimes also on the state of the primary electron donor D. The minimum fluorescence is usually called the constant fluorescence and the remainder variable fluorescence. By the nature of this paper we shall be mainly interested in the yield of the constant fluorescence which is closely t Present address: Department
of Mathematics,
University of Southampton.
91
OO22-5193/78/0508-0491 %02.00/O
10 1978 Academic Press Inc. (London)
Ltd.
92
T.
MARKVART
related to the maximum photochemical yield and to the efficiency of the exciton transport. The most important accessory pigments in higher plants are the chlorophylls. Because of their structure their position is usually deduced to be on the interface between the hydrophobic and hydrophyllic parts of the photosynthetic membrane in a two-dimensional layer. Due to larger intermolecular distances the interaction energy responsible for the exciton transfer in the photosynthetic unit is smaller than in the crystalline state of the pigments. Consequently, the excitons are more susceptible to scattering and the simple wave-like picture usually employed to describe the exciton transport in molecular crystals (Markvart, 1976) is probably not applicable here. In the opposite “incoherent” limit one can use the Forster mechanism for the excitation transfer (Forster, 1948). If one assumes that the absorption and fluorescence bands are Gaussian and have the same widths the transfer rate between identical molecules can be written as: W=$E$(g)aexp
[-51
assuming the dipole-dipole approximation. The quantity rf if the natural fluorescence lifetime, A is the width of the spectral bands, a is the transition wavelength divided by 27c, Es is the energy of the Stokes shift, Anat is the natural width of the band defined by Anat = h/rf and R is the position vector connecting the charge centres of the two molecules. The orientational factor K is given by: d
d 1.2
-3
(a_,W&.R)
(3)
R2
where d, and d, are the transition dipole vectors. For a “random” orientation of the molecules assuming all orientations equally likely, K' = 4 and if we define the usual transfer distance parameter: 9
the “average” transition
(4)
rate is:
(5) Care must be taken in interpreting the words “random” and “average”. Strictly speaking, they can only be applied in special situations when each molecule experiences an average environment. In our problem averaging of
THE
PHOTOSYNTHETIC
UNIT
93
the type above may not be the correct procedure. This will become apparent later in this paper. ‘The excitation transfer in Forster’s very weak coupling limit can be pictured as thermally activated exciton hopping. A simple model (Appendix A) gives for the transition rate:
where the activation
energy E, is given by:
Et,= bEs, or: E =!.!? a
The resonance-interaction
4 k,O’
matrix element is :
is Boltzmann’s constant and 8 is the absolute temperature. For the majority of chlorophyll in the photosynthetic unit the absorption wavelength is between 650 and 700 nm, z/ = 15 ns (Brody & Rabinowitch, 1957) and we estimate E, = 50-100 cm -I. Equation (4) then gives R, between 9 and 10 nm which is at the upper end of the experimentally quoted values (Knox, 1975). The molecular distances are usually estimated at about 1.7 nm (Duysens, k,
1964).
In this paper we shall adopt, as a starting point in our study of the problem, a homogeneous model in which the transfer between individual molecules is described by the Forster very weak coupling formalism. As the data above indicate this should not be far wrong for average molecular distances in the unit. We shall develop a general formalism for this model but will be forced by the complexity of the problem to introduce some even more restricting assumptions later. However, by treating the inherent difficulties in turn we shall look at the exciton transport from several angles and thus hope to offer a farily comprehensive view of the problem. We hope to present refinements to this “homogeneous” model in a future publication. 2. The Transport Formalism The photosynthetic units associated with different reaction centres are probably not independent and there is a possibility of exciton transport between different units. Joliot & co-workers (1964, 1969) reported exciton migration between traps in photo-system (PS) II; transfer between PS I
94
T.
MARKVART
units and between units of different photosystems has also been suggested. Only independent units will be studied in this paper. This assumption is not as restrictive as it may seem. Any infinite connected model with active traps distributed at the points of a two-dimensional periodic lattice and in which the distribution of accessory pigments shows the same periodicity, can be reduced to the independent unit problem, supplemented perhaps by a different set of boundary conditions. This problem is treated more fully in another publication by the author (Markvart, 1977); a similar problem in the random walk setting is discussed by Chandraskhar (1943). In a similar fashion, a number of finite connected models with open traps can be divided into sets of independent units by using symmetry considerations. The maximum photochemical yield and constant fluorescence are therefore seen to be affected only little by whether the units are connected or not. Another situation when the methods of this paper can be applied arises in a large connected model, when most traps are inactive and the excitations created near one active trap have a negligible probability of reaching another. Each active trap may in this case be considered as belonging to a large and independent “superunit”. The intermediate problem of non-periodically distributed traps of appreciable concentration is treated elsewhere (Markvart, 1977). We shall be concerned with exciton transport following a brief flash of illumination. In view of the assumption of “incoherent” exciton hopping the transport equations are:
with initial conditions p,(O) = pi”), where p,(t) is the probablility that the exciton is at the nth molecule at time t; n = no corresponds to the trap. W,, is the exciton transfer from molecule m to II (m # n), the diagonal elements are defined as:
K” = - mfn c Km. The exciton “loss rate” is given by:
where r,, is the lifetime for non-radiative de-excitation, and k,, is the rate of the primary chemical reaction (1). The photochemical yield is: x = k, r p,,(t) dt, 0
THE
PHOTOSYNTHETIC
95
UNIT
and the fluorescence yield of the nth molecule:
Equations (6) are most conveniently solved by the method of Laplace transforms : this brings them into a set of algebraic equations : ; i(s + k + QL,kL
- w,,)p”,(s) = do’,
where p,, is the Laplace transform of pn. Introducing by :
the Green’s function
c W,, - ~mJG,(s) = 4,w one readily obtains : ; C%,(S + k,) + kp@m,(s f O%tt(s + k,) -
p”“(S)= with :
There are two limiting cases to be considered. The first arises when the exciton transport in the photosynthetic unit is very fast and the rate of the complete primary photochemical process : (A.P.)* (A.P.) (9) D (R.C.) A + D (R.C.)’ A-, where (A.P.) denotes the accessory pigment aggregate, is limited by the rate of the reaction (1). In this situation both constant and variable fluorescence are emitted by excitons in thermal equilibrium (Paillotin, 1976) and accordingly, the constant and variable fluorescence have identical static properties (spectrum, polarization etc.). This, however, seems to be in contradiction with the experimental evidence : see, for example, Duysens (1964) and the reviews by Fork & Amesz (1969) and Goedheer (1972). In the remaining part of this paper we shall therefore be concerned with the other limiting case when the primary process described by equation (9) is limited by the exciton transport, and the rate of reaction (1) is very fast. In this case expressions (7) and (8) give:
c &m,(Wnm(kJ - G,,,(kl>G,,,(kl>}p,“’ (10) %,n,W ’
96
T.
MARKVART
and
(11) 3. Units with One Bind of Accessory Pigment An approximation often employed in the theories of exciton transport in the photosynthetic unit is that all accessory pigment molecules are identical. From the discussion of the transfer rate it follows that this will be permissible if the spread in energies of the relevant energy levels (the first singlet) is small compared to k,B. To some approximation we can therefore regard the different moieties of chlorophyll a at room temperature as identical, but not chlorophyll a and b. Even though this approximation is rather restrictive it simplifies the problem considerably and allows an insight into a number of interesting features of the exciton transport. The usual random walk approximation (Pearlstein, 1964, 1966, Montroll, 1969, Bedeaux et al., 1977) is equivalent to expanding the product sG(s) in (positive) powers of s. We write the Green’s function in the spectral representation as :
where the 1’s are the eigenvalues of the transfer matrix W and t(A) are the corresponding eigenvectors. It is clear from equation (12) that the random walk approximation is valid provided: s c & = min (In]; 1# 01. (13) In this paper we shall keep only the lowest order terms in the expansion and put:
G,z,(4= ks (1+w,,), where : c km,(4 a ’ 1#0 and N is the number of molecules in the unit. We denote 9 nm =
-N
T, = grin
l--,(m)
(15)
Wa)
(16~)
= g?. ”
THE
PHOTOSYNTHETIC
97
UNIT
It will be seen that T, is the mean first passage time of the excitation through the point n, assuming the starting points are distributed throughout the unit with equal probability. The time Tis the average of T, over all positions in the unit. Clearly, (17) Condition (13) gives an upper bound for s which is a decreasing function of N and which decays to zero as N + co. It is seen therefore that the random walk approximation may not always be valid. In particular, it will not be valid for very large units and for units with slow excitation transfer rates. We put P”(‘I = l/N and using the random walk approximation we obtain from equations (10) and (11): 1 l+k,T,,
‘= 0,
= -$
lf 1 (#)w ” = __Nk,z,
I1-ly;k[T%
\ =N’ (1 -rn,(n))q+)
1 no 1 +kTJ& 1 +k,T,,
)
(18)
1
= N {1+ k,T,,l-,,(n)}iD(“)
where the total fluorescence yields are:
The superscripts c and v refer to the constant and variable fluorescence, respectively. We shall also consider briefly another approximation for the Green’s function. When : c-e1 W-4 Ao < k, we may approximate The photochemical
4
the sum in equation (12) by an integral. yield is in this case:
Wb)
98
T.
MARKVART
where :
and Nn(L)di
is the number of eigenvalues of Win the interval J. --f 1+dL.
(A)
UNIFORM
UNITS
WITH
ISOTROPIC
COUPLING
A convenient starting point in the study of units with one kind of accessory pigment are models where the transfer rates depend only on the relative positions of the molecules concerned. It is also convenient to discard the orientational dependence of the coupling at this stage and consider a transfer matrix of the form: (22) Km = WIRn-R,I) for all m # ~1, where Ri is the position vector of molecule i. Coupling of the type (22), with matrix elements non-zero only between nearest neighbours, was considered by Pearlstein (1966); an analytic solution was given by Montroll (1969) In this section we shall consider a more general coupling of the type (22) and assume only that the sum:
;co ]Rn\2+e W(Rn(h converges for some E > 0. The subscript co indicates that the summation extends over the entire infinite lattice. We denote: to = {“FOX3 Ww-‘, 4&J
= to WRA
o2 = cm Rn2tiR,), n
PO (23bj (23~)
In the random walk terminology (Spiker, 1964) w(r) is the transition function of the random walk (the probability of hopping distance r in one step) and to is the average time between jumps. We shall assume that the pigment molecules form a two-dimensional square lattice and the co-ordinates of the molecules are given by: R,i = mid,
i = 1,2; mi = 1,2,. . ., M.
(24)
The method, however, can be applied without major modifications to other two-dimensional periodic structures. The eigenvalues and eigenvectors of the
THE
PHOTOSYNTHETIC
UNIT
99
matrix W are, for N 9 ts2,
where : i = 1, 2;ji = 0, 1,. . .) M-l
ki = s;,
(26)
A,, = &
if ki = 0
J2’
= 1
otherwise,
provided molecule n is not too near the unit boundary. In the special case of transitions between nearest neighbours only formulae (25) are exact for any N > 1 and any position of molecule 12. We note that: (27) for small jkl. Let us denote the trapping time T, when the trap is at the centre of the unit. For large N there is a simple relationship between T, and T: T(N) = 4~(N/4) _ LN t e 3a2 O’
where we explicitely depicted the dependence of the trapping times on the size of the unit. To evaluate T we write:
In the first sum we substitute for k from equation (26) and the resulting sum is evaluated directly (Appendix B). The second sum is replaced by an integral. Hence, T = ntO,
W4 where n, the number of steps the exciton will take to reach the trap for the first time, is given by: Nd2 nSC)= 2 7-m
In N+2C+rc+rcl-
E + b 71 >
Wb)
100
T.
MARKVART
where C is Euler’s constant, G is Catalan’s constant, = O-0076
and (31)
For the nearest-neighbour form :
coupling integral (31) can be obtained in a closed
I,-, = Sf - AIn !Jf = 1.0994. TT
7c
(32)
8
Substitution of equation (32) into (30b), noting that cr = d for the nearest neighbour coupling, and using equation (28) gives the first two terms of Montroll’s result (Montroll, 1969). Of particular interest is the coupling: (33)
where
which has a Forster-like dependence on the molecular separation. case I and a2 have to be calculated numerically with the result: IF = O-5696, c2 = 1.2936d2,
In this
(34)
The effect of the trap position on the trapping time is shown in Fig. 1. The behaviour of the function r with the trap at the centre of the unit is shown in Fig. 2. The trapping time also depends on the shape of the unit. For example, for a diamond-shaped unit one obtains along similar lines as above. Nd2 ndiam
=
2
In (2N)+2C+7r+7rI
- ff_G+ b . n
>
(35)
Let us consider integral (21) for the transfer matrix of the form (22). We can choose periodic boundary conditions for the eigenvectors
THE
PHOTOSYNTHETIC
UNIT
101
FIG. 1. Distribution of the trapping times with different positions of the trap. The edges of the unit are represented by double lines, the eentre of the unit is denoted by c. Only a Part of the unit is shown, the remainder can be obtained by symmetry.
to obtain: ‘r dk, ‘[ dk2 t k ipckj
J == kg
= !$$
x
0 1 nld d dkl f dk2
x
nld t,k, ;‘.‘Fi’
+ j dkr
’
n/d
s dkz x
’
[
tends to (rr/cr)2Z and the fist
In the limit krt, -+ 0 the second integral integral is evaluated directly. Thus: tok,d2 J = 2 -ln(k,to)+21n ?c(T We neglected terms of the order (klt0)2.
($
+nZ-F}.
(36)
102
T.
MARKVART
FIG. 2. Three dimensional “graph” of the transfer function r. The edges of the unit are represented by double lines, the centre of the unit is denoted by c. Only a part of the unit is shown, the remainder can be obtained by symmetry.
(B)
RANDOM
SYSTEMS
: ORIENTATIONAL
DISORDER
In this section we shall study numerically the model of the photosynthetic unit in the extreme case of complete orientational disorder of the pigment molecules. The trapping time and transfer function r were chosen for investigation, rather than the fluorescence and photochemical yields, because of their simple interpretation and a more universal character. At each point of the lattice [equations (24)J an orientation of the dipole moment vector is generated by the computer so that: (i) All orientations are equally likely and (ii) orientations of dipole moment vectors at different sites are independent. The elements of the transfer matrix W were calculated from:
wnn= -c mWm. The normalization
was chosen so that:=-1
for molecules far from the boundary
of the unit.
THE
PHOTOSYNTHETIC
UNIT
103
The eigenvalues and eigenvectors of W were found by numerical methods. The trapping time and the transfer function I’ were subsequently determined using formulae (15), (16a, c) and (17). This procedure was repeated 100 times which was found to guarantee reasonable accuracy of the average values. A similar program was also run for the nearest-neighbour transition -rates. The results for the average trapping time [equation (17)] using Forster’s coupling are summarized in Table 1. The values for the orientationallydisordered model are seen to differ by a factor of 1.6-l -7 from those TABLE
1.
Comparison of results for the average trapping time using averaging at difirent stages of the calculation. Column 1 shows the average trapping time calculated numerically from equation (17), the averaging performed correctly at the end of the calculation. Column 2 contains the corresponding figures using the averaged transition rates (5) or (33). N is the number of molecues in the unit. N
1
2
49 100 121 196
192 403 492 814
109 240 296 502
obtained for the uniform model described by equations (22) and (33) with the corresponding averaged transition rates. For the nearest-neighbour coupling the difference is even more marked: we found that in this case the results differed from the corresponding results of section 3A by a factor of 2.5-2.7. The spatial variation of the ratio T,/T and of the function r, on the other hand, are almost unaffected by the randomness and were found to agree (within the small statistical error) with the corresponding values of section 3A.
4. Units Containing Chlorophyll A and B Two kinds of models will be considered in this section. The units in models of the first kind will consist of an ordered inhomogeneous arrangement of the two pigments, chlorophyll a forming the inside of the unit, surrounded by chlorophyll b. Models of this type were studied by Seely (1973) and Paillotin (1976). The units of the second kind will contain a
104
T.
MARKVART
homogeneous mixture of chlorophyll u and b throughout the unit. This problem was considered by Pearlstein (1964) and Swenberg et al. (1976). Because of the position of the spectral bands exciton transfer from chlorophyll a to chlorophyll b is inefficient and will be in this paper neglected. (A)
MODELS
OF THE
FIRST
KIND
Two models of the first kind are shown in Fig. 3. Characteristically, the average distance travelled by the excitons absorbed in chlorophyll b is appreciably larger than by those absorbed in chlorophyll a. This is reflected in the transport times and consequently should be observable as an inhomogeneity in the fluorescence and photochemical yields. As compared with unit of the same size composed of only one pigment, however, the exciton transport in this arrangement of pigments is clearly more efficient.
(a)
iti
FIG. 3. Two models for the PS II units of the first kind.
It is convenient to introduce two transfer matrices, IV(‘) and FVcb),for description of the exciton transport in the bulk of chlorophyll a and b, respectively, without the possibility of transfer between the two pigments. It may then be shown by lengthy but straightforward manipulation of the appropriate Green’s functions that, keeping only the lowest order terms in the transport times,
(37)
THE
PHOTOSYNTHETIC
where xn (xb) is the photochemical cl (chlorophyll b), Ax
UNIT
105
yield for light absorbed by chlorophyll =
x,
--Xb,
0%’ is the constant fluorescence yield of chlorophyll CLfor light absorbed by chlorophyll b, and similarly for @g,’ and @ii’. The time T, is the mean trapping time for the unit with chlorophyll b removed and: (38) Tb = -Nb
%%‘b)
c
--j-7
Lb+0
b
(39)
where :
The 5’s and I’s are the eigenvectors and eigenvalues of IV(‘) or WCb), N,(N,) is the number of molecules of chlorophyll a (chlorophyll b), and the sum in equation (40) is to be taken over the boundary between the pigments of length L. The interpretation of formulae (37) is evident since (To+ T,‘) is the average time to reach the trap from the boundary between chlorophyll a and b, and Tb is the mean time to reach this boundary from chlorophyll b. If the pigments are arranged in a square lattice and the coupling is of the form (22) we have from equations (28) and (30). (41) The largest contribution to the sums in (38) and (39) comes from terms with small 1. General consideration using equation (27) gives: T; = ; N,tb”) Tb = 4 4 $’ ab
0
Nbtbb’, *
where c( is a constant independent of N, or Nb and C$is usually a linear function of the ratio NJN,,. In the present situation (Fig. 3): a = Q, and
with B equal to + and O-18 for the models of Figs 3(a) and (b) respectively.
106
T. (B)
MODELS
MARKVART OF THE SECOND
KIND
The exciton relaxation times in a unit of the second kind are very short (of the same order of magnitude as the hopping time to), and therefore the greatest part of the transport will take place among the energetically lower lying chlorophyll a molecules. The presence of chlorophyll b in the unit has two effects on the structure of the chlorophyll a aggregate: it increases the molecular separation and reduces the number of chlorophyll a molecules in the unitt. These two structural changes have opposite effects on the exciton transport: the former increases the hopping time t,,, the latter decrease the number of steps for the exciton to reach the trap. It will be seen that it is the increase in the hopping time that will predominate, making the exciton transport less efficient. We shall in this paper neglect
o chio x
chi b
l
trap
FIG. 4. A simple model of the secondkind.
the possible dielectric effects of chlorophyll b on the transition rates between molecules of chlorophyll a. This would slow down the exciton transport still further. The simplest model of the second kind is shown in Fig. 4. Molecules of the two pigments are arranged alternately in the square lattice [equation (24)]. Molecules of chlorophyll u clearly form a square lattice in a diamondshaped unit. We shall denote T,, tL2’ and n, the mean trapping time, the hopping time and the mean number of steps to reach the trap, respectively, for the exciton transport among the chlorophyll a molecules in the unit of Fig. 4, and T,, t$,” and n, the corresponding quantities in a unit obtained from the unit in figure 4 by replacing the molecules of chlorophyll b by chlorophyll a. Applying formulae equations (30) and (35) we obtain: n2 = 2nl,
and since: d, = d,f2, t We assume that the total amount of chlorophyll
in the unit is constant.
THE PHOTOSYNTHETIC
UNIT
where d, is the lattice constant of the chlorophyll using equations (2) and (23a), tb”’ = 8@‘. Therefore :
107
a lattice in Fig. 4, we have,
T2 = 4T,.
From this result one would intuitively chlorophyll b equal to c,
expect for the concentration
of
2 N (l-c)-“. In the next model the molecules of chlorophyll a and b are intermixed at the lattice points [equation (24)] at random. The coupling was taken in the form (33) and the average trapping time was computed using equation (17). The results for the ratio Ti2)/Ti1’ are given in Fig. 5. Because of the fast relaxation times in units of the second kind,
FIG. 5. The results for the average trapping time (17) as a function of chlorophyll b concentration c in PS II units consisting of a homogeneous mixture of chlorophyll a and b. The full line are the results of a numerical calculation for a random homogeneous mixture, the broken line is the graph of (l-~)-~.
108
T. MARKVART
5. Discussion In this paper we have presented new theoretical results for exciton transport in the photosynthetic unit, and examined some approximations often employed in previous analyses. The rigorous condition for validity of the random walk approximation [equation (13)] t oge th er with the data given in the Introduction shows, as expected, that this approximation is valid in this situation. From the results of section 3 it is seen, however, that other two common approximationstransitions between nearest neighbours only and the a priori use of the averaged transition rates-have only a qualitative character. Formula (30) with numerical values [equation (34)] show that using the proper Rv6 dependence of the coupling decreases the trapping time by the factor of about l-5? as compared with the result obtained using transitions between nearest neighbours only. In the case of complete orientational disorder among the pigment molecules we have found that the proper treatment by averaging at the end of the calculation gives results higher by a factor of about l-7 than when using the already averaged transition rates (33). Formulae (18) show that due to the finite exciton transport time across the photosynthetic unit, some of the constant and variable fluorescence is emitted by different parts of the unit. From the shape of the function r (Fig. 2) it is seen in particular that more variable fluorescence originates from the molecules in a relatively small neighbourhood of the trap. Some correlation between orientations of pigment molecules over distances short compared with the size of the unit should therefore result in the variable fluorescence being more polarized than the constant fluorescence. Some experimental evidence in support of this is given in the work of Whitmarsh & Levine (1974) who observed a slight increase of fluorescence polarization upon blocking the electron transport. It is well known that the majority of cholorophyll fluorescence is emitted by PS II. It is tempting to associate this with the relatively large content of chlorophyll b in this photosystem. Let us assume for the sake of the argument that the units of PS II contain chlorophyll a and b in equal amounts, units of PS I contain only chlorophyll a, and that the total amount of chlorophyll is evenly distributed between the two photosystems. If we take a model of the second kind for the units of PS II we obtain for the ratio of the constant fluorescence yields:
Q(') T ",A=4 Qf' - Tl
'
t This includes the factor /3/4 = 1.16 which reduces the hopping time lo.
THE
PHOTOSYNTHETIC
109
UNIT
The decay of fluorescence as a function of time is complicated in the present theory. At long times, however, the decay should approach exponential with the lifetime approximately equal to the appropriate trapping time. The ratio of the apparent fluorescence lifetimes of the two photosystems is therefore: (42)
Yu et al. (1975, 1977) obtained values 50-60 psc and 200-220 psc for the lifetimes of PS I and PS II fluorescence, respectively, by direct picosecond studies. Similar values (80 psc and 300 psc) were obtained by Paschenko et al. (1975), in excellent agreement with formulae (42). It would therefore appear that the units of PS II are best described as a homogeneous mixture of chlorophyll a and b. The models of the first kind (section 4A), on the other hand, represent the most likely possibility if the PS II units are inhomogeneous and consist of large blocks composed of preferentially one kind of pigment. This hypothesis can be tested experimentally since the transport times in equations (38, 39) are-though somewhat smaller-of the same order of magnitude as the trapping time [equation (41)] and this should manifest itself through inhomogeneities in the action spectra as a function of excitation wavelength. The model studied at the end of section 3A describes a situation when the random walk approximation is not valid. Let us consider a connected model consisting of many traps associated with a large number of identical accessory pigment molecules. By “poisoning” a large proportion of the traps a situation will arise when the remaining “open” traps may be considered as independent, that is, when the excitationscreated near one open trap have a negligible probability of reaching another. It will be seen that conditions (19) are satisfied for such a system and equation (20) may therefore be used to analyse the linear portion of the dependence of X on c, as c -+ 0, in the study of connected models. A preliminary calculation shows that the values of parameters consistent with those used in the study of the trapping time can be used in expression (36) to obtain a reasonable agreement with the data of Joliot et al. (1964, 1969). The author wishes to thank Dr E. Cane1 for suggesting this problem, and for valuable comments and discussions. The financial support provided by a Science Research Council grant is gratefully acknowledged. REFERENCES ALISTIN, I. G. & MOTT, N. F. (1969). Adv. Php. l&41. BEDEAUX, D., LAKATOS-LINDENBERG, K. & SHULER, K. E. (1971). BRODY, S. S. & RABINOWTCH, E. (1957). Science 125, 555. CWANDRASEKHAR, S.(1943). Rev. mod. Phys. 15, I.
J. math.
Phys.
12, 2116.
110
T.
MARKVART
CLAYTON, R. K. (1965). Molecular Physics in Photosynthesis. New York: Blaisdell. DUYSENS, L. N. M. (1964). Prog. Biophys. mol. Biol. 14, 1. FORK, D. C. & AMESZ, J. (1969). A. Rev. PI. Physiol. 20, 505. FORSTER, T. (1948). Ann. Physik 2, 55. GOEDHEER, J. C. (1972). Ann. Rev. Pi. Physiol. 23, 87. GRAJXHTEYN, I. S. & RYZHIK, I. M. (I 965). Tables of Integrals, Series andProducts. London: Academic Press. HOLSTEIN, T. (1959). Ann. Phys. 8, 343. JOLIOT, A. & JOLIOT, P. (1964). Compt. Rend. 258,4622. JOLIOT, P., JOLIOT, A. & KOK, B. (1968). BBA 153, 653. KNOX, R. S. (1975). In Bioenergetics of Photosynthesis (Govindjee, ed.), p. 183. London: Academic Press. MARKVART, T. (1976). Phys. Stat. Solidi 73, 689; 74, 135; 76, 953. MARKVART, T. (1978). J. theor. Biol. 72, 117. MONTROLL, E. W. (1969). J. math. Phys. 10, 753. PAILLOTIN, G. (1976). J. theor. Biol. 58, 219. PASCHENKO, V. Z., PROTASOV, S., RUBIN, A. B., TIMOFEEV, K. N., ZAMAZOVA, L. M. & RUBIN, L. B. (1975). BBA 408,413. PEARLSTEIN, R. M. (1964). Proc. Natn. Acad. Sci. U.S.A. 52, 824. PEARLSTEIN, R. M. (1966). Brookhaven Synzp. Biol. No. 19, p.8. ROBINSON, G. W. (1966). Brookhaven Symp. Biol. No. 19, p.16. SEELY, G. R. (1973). J. theor. Biol. 40, 173, 189. SPITZER, F. (1964). Principles of Random Walk. Van Nostrand. SWENBERG, C. E., DOMINIJANNI, R. & GEACINTOV, N. E. (1976). Photochem. Photobiol. 24, 601. WHITMARSH, J. & LEVINE, R. P. (1974). BBA 368, 199. Yu, W., Ho, P., ALFANO, R. R. & SEIFERT, M. (1975) BBA 387, 159. Yu, W., PELLEGRINO, F. & ALFANO, R. R. BBA 460, 171. ZENNER, C. (1932). Proc. R. Sot. A137, 696.
APPENDIX (A)
PHONON-ASSISTED
A EXCITON
HOPPING
The model on which the calculations will be based consists of two identical molecules, each with one exicted state and one vibrational co-ordinate. In the Born-Oppenheimer approximation the solution of the Schrodinger equation for a single molecule has the form:
H(Q, r)q%,Q) = E(‘)(Qh”k
Q)3
where Q is the vibrational co-ordinate, r is the collection of electron coordinate vectors, H (Q, r) is the single molecule Hamiltonian and qCi) (r, Q) are the electronic wavefunctions. The superscript denotes the electronic state (1 the ground state, 2 the excited state). As is well known the eigenvalue E(‘) (Q) acts as the potential energy for the vibrational motion. We expand E(‘)(Q)
= EC’) f$m&)Z(Q
- Q”92
+
Or-Q
_
Q(i))31
THE
PHOTOSYNTHETIC
UNIT
111
The following assumptions will now be made: N(i)We neglect the terms 0 [(Q - Q$j))3] (ii) Set cJi) = o, same in both electronic states. These assumptions correspond, respectively, to the harmonic approximation and to the assumption of the exciton-phonon interaction linear in lattice co-ordinates. We shall choose Q so that Qb” vanishes and denote Qb”’ = Q,. Considering both molecules the total electronic wavefunctions can be written as : @ = ~,(QI, Qdd"(~i~'+
a2(Q1,
+A(Q,, Q2k~‘l~+&‘+ B(Q,, Q~~P(~+P(~).
Q2M2)d1’
(Al)
The indexes refer to the two molecules. We shall omit A and B since they are unimportant in our problem. Using representation (Al) the total Hamiltonian is given by: $a,= 9~2
{WQI, Q2) - m~2QoQ~>u~+ % = { WQ1, Q,>-m~2Q,Q,>~, +V*q,
WI
where :
u(Q1, Q2) =
+3m02(Q; +Q: +Qf)
E2-E,
and V is the usual resonance-interaction In the dipole-dipole approximation :
matrix element (5a).
where : di = -e J (p$“(r,.)(C r;)cp!“*(r;) dr,
i = 1, 2,
are the transition dipole moment vectors and R is the vector connecting the charge centres of the two molecules. Equations (A2) are formally the same as those considered by Holstein (1959). Below we briefly outline his approach to obtain the transition rate. First, the new potential surfaces are calculated: they are the eigenvalues of the Hamiltonian (A2):
EJQ,, Qd = WQI,
Q2)
- m+
(Ql +Qz) k f {FF)2
(Ql-Q2)2+lV12}’
(A3a)
112
T. MARKVART
and
a;" -=-
a$”
- mo2Qo 2 -1 mw2Qo(Qz-Q~) + [(-opt) ca,-e,,'+lVl']ij. 2 -
Wb)
V
The nature of the solutions (A3) is shown in Fig. 6. It is seen that the exciton moving along one of the surfaces will hop to the other molecule if it passes through the “bottleneck” Q, = Q, and remains on the same surface. Hence the exciton transfer rate is:
t mw%&tP~~ 2
E-Ut
FIG. 6. Adiabatic Holstein, 1959).
energy surfaces for the phonon-assisted
where P(V) is the probability with velocity
exciton hopping
that the exciton passing through v=
Q, = Q2 C-45)
l&Q,l~
remains on the same potential surface. This probability Zener (1932) to be (in our notation): 27-c pq’ P(v) = 1 - exp - - ____ h mw2Qov > ’
(after
was determined
by
646)
THE
PHOTOSYNTHETIC
113
UNIT
,f(v) is the encounter frequency of Q, = Q2 subject to (A5). Consideration based on elementary statistical mechanics gives : 2v j exp - Ek+U~~z~~}6(e,-~2-v)i5(Ql-Q2) B F(v) = -
dQ dQ ---
=k$vexp(-kg)exp(-a+), where : In the limit when the argument of the exponential integral (A4) gives:
in equation (A6) is small
where : E, = Srnw’Q& It is usual to express the transition rate W in spectroscopic terms. In our model it is readily observed that both absorption and fluorescence bands are Gaussian of width: A = 2(E,k,8)*, c48) and the energy of the Stoke’s shift is: E, = 4E,.
(A9) Substitution of parameters (A8) and (A9) into Forster’s result for the transition rate yields (A7) as expected. For two dissimilar molecules equations (A8) and (19) give the usual relation :
provided both molecules have equal Stoke’s shifts. E, and E, are the absorption energies of the two molecules. (B)
LIMITS
OF APPLICATION
The validity of expression (A7) is limited by two factors. In approximating expression (A6) to evaluate integral (A4) it was assumed that: 2
PI’ .g 1, h mwZQ,,vO
(All)
114
T. MARKVART
where uOis a typical value of o at Q, = Q2. Evaluating uOreduces expression (All) to: w 4 w(t). G412) In view of the use of equilibrium distribution function for the lattice co-ordinates, however, equation (A12) should be replaced by the more stringent: d$: where r, is the phonon relaxation time. The other assumption involved in the derivation of formulae (A7) is that the vibrations can be treated classically. Holstein (1969) shows that this can be done provided: E, $- ‘2
4’
and
0413)
Although conditions (A13) are necessary for validity of the formula (A7) thermally assisted hopping will occur in a much wider range of circumstances. Briefly, one may summarize the conditions for the “incoherent” exciton hopping as follows: (i) Distortion of lattice due to exciton creation is confined to a single molecule. (ii) The width of the band associatedwith the “coherent”exciton states is smaller than the uncertainty in energy corresponding to the thermally assisted hopping. Regime described by (i) is clearly equivalent to the union of Forster’s weak and very weak coupling regions where the vibrational states of the aggregate are closely related to those of individual molecules. This will be true if either E, < ho or 1Vl c Ea. In the former regime the exciton-phonon interaction is weak and the exciton will propagate in a band with occasional scattering. It is the latter case that is of interest here. When the resonance interaction can be treated by pertubation theory ($) the exciton bandwidth is approximately given by: a(O) = co exp t This is different from Holstein’s criterion (Holstein, 1959, equation 103). The reason is that Holstein used the average of u over the whole period of vibration to obtain v,,. $ This does not cover the whole of the region described by the latter inequality. The reader is referred to literature (Holstein, 1959; Austin & Mott, 1969) for a discussion of the so-called adiabatic regime.
THE
PHOTOSYNTHETIC
UNIT
115
where oO is the bandwidth in the absence of the exciton-phonon interaction. lt is, seen that o(O) is a rapidly decreasing function of temperature. By application of condition (ii) above incoherent site hopping will be dominant at temperatures higher than some critical temperature 6, defined by: o(&) = hW. In Forster’s terminology this equation describes the transition to very weak coupling. Holstein, (1959) estimates : where On is the Debye temperature
of the phonons concerned.
APPENDIX We shall evaluate the sum
in equation (29). We put:
where
We write:
The first sum readily gives:
The second sum we write in the form
where
B
from weak
116
T.
MARKVART
For large M the sum (B2) can be replaced by an integral: l cotan-’ dz = G &=J 9 1 1 2 0 Z where G is Catalan’s constant (see, for example, Gradshteyn 1965). The sum (Bl) is evaluated as follows: M-l z TCM- l coth (nn) - 1 +;“g’hA = .zl 2n coth (in) - ,$ = z 3 _____ n n 1 n 1 > B-tidz,ydq 0
-iyg:+2=$b
+g(C+InM)--g+O(M-‘),
where C is Euler’s constant and:
b=4
2 -i-
n= 1 n(eznn - 1)’
Hence, collecting all terms, s=~lnN+~b+qc+%_c+o(~-~).
& Ryzhik,
Exciton Transport in the Photosynthetic Unit T. MARKVART~ Department
of Physics,
University
of Keele, England
(Received 25 October 1977) In this paper we present a theoretical study of the exciton transport in the photosynthetic unit and attempt to correlate the structural and organizational properties of the unit with the exciton transport efficiency. The transport mechanism is assumed to be of the Forster type. This mechanism is shown to be equivalent to the thermally assisted exciton hopping. The transport equations are solved by the Green’s function technique in a more general way than has been presented previously. We thus find it possible to study the effect of the proper Rm6 coupling and random orientations of the pigment molecules. The equations are also solved for a low concentration of reaction centres when the conventional random walk picture is not valid. Various models of chlorophyll a and b mixtures are studied and by comparing their different exciton transport properties, an explanation is proposed for several observed features of the primary photochemical process.
1. Introduction Jt it well established that the light energy required for photosynthesis is collected by the aggregate of accessory pigments in the photosynthetic unit and subsequently, this energy migrates to a special molecule (or molecules) of chlorophyll where the chemistry begins. Although other mechanisms for the energy transfer have been suggested in the past it is generally accepted now that the transport agents are excitons (Duysens, 1964; Clayton, 1965). In the primary chemical reaction an electron is transferred from the reaction centre (R.C.) to the primary electron acceptor A: (R.C.)*
A + (R.C.)+A-.
(1)
The fluorescence of excitons in the phototsynthetic unit depends on the state of Al, and sometimes also on the state of the primary electron donor D. The minimum fluorescence is usually called the constant fluorescence and the remainder variable fluorescence. By the nature of this paper we shall be mainly interested in the yield of the constant fluorescence which is closely t Present address: Department
of Mathematics,
University of Southampton.
91
OO22-5193/78/0508-0491 %02.00/O
10 1978 Academic Press Inc. (London)
Ltd.
92
T.
MARKVART
related to the maximum photochemical yield and to the efficiency of the exciton transport. The most important accessory pigments in higher plants are the chlorophylls. Because of their structure their position is usually deduced to be on the interface between the hydrophobic and hydrophyllic parts of the photosynthetic membrane in a two-dimensional layer. Due to larger intermolecular distances the interaction energy responsible for the exciton transfer in the photosynthetic unit is smaller than in the crystalline state of the pigments. Consequently, the excitons are more susceptible to scattering and the simple wave-like picture usually employed to describe the exciton transport in molecular crystals (Markvart, 1976) is probably not applicable here. In the opposite “incoherent” limit one can use the Forster mechanism for the excitation transfer (Forster, 1948). If one assumes that the absorption and fluorescence bands are Gaussian and have the same widths the transfer rate between identical molecules can be written as: W=$E$(g)aexp
[-51
assuming the dipole-dipole approximation. The quantity rf if the natural fluorescence lifetime, A is the width of the spectral bands, a is the transition wavelength divided by 27c, Es is the energy of the Stokes shift, Anat is the natural width of the band defined by Anat = h/rf and R is the position vector connecting the charge centres of the two molecules. The orientational factor K is given by: d
d 1.2
-3
(a_,W&.R)
(3)
R2
where d, and d, are the transition dipole vectors. For a “random” orientation of the molecules assuming all orientations equally likely, K' = 4 and if we define the usual transfer distance parameter: 9
the “average” transition
(4)
rate is:
(5) Care must be taken in interpreting the words “random” and “average”. Strictly speaking, they can only be applied in special situations when each molecule experiences an average environment. In our problem averaging of
THE
PHOTOSYNTHETIC
UNIT
93
the type above may not be the correct procedure. This will become apparent later in this paper. ‘The excitation transfer in Forster’s very weak coupling limit can be pictured as thermally activated exciton hopping. A simple model (Appendix A) gives for the transition rate:
where the activation
energy E, is given by:
Et,= bEs, or: E =!.!? a
The resonance-interaction
4 k,O’
matrix element is :
is Boltzmann’s constant and 8 is the absolute temperature. For the majority of chlorophyll in the photosynthetic unit the absorption wavelength is between 650 and 700 nm, z/ = 15 ns (Brody & Rabinowitch, 1957) and we estimate E, = 50-100 cm -I. Equation (4) then gives R, between 9 and 10 nm which is at the upper end of the experimentally quoted values (Knox, 1975). The molecular distances are usually estimated at about 1.7 nm (Duysens, k,
1964).
In this paper we shall adopt, as a starting point in our study of the problem, a homogeneous model in which the transfer between individual molecules is described by the Forster very weak coupling formalism. As the data above indicate this should not be far wrong for average molecular distances in the unit. We shall develop a general formalism for this model but will be forced by the complexity of the problem to introduce some even more restricting assumptions later. However, by treating the inherent difficulties in turn we shall look at the exciton transport from several angles and thus hope to offer a farily comprehensive view of the problem. We hope to present refinements to this “homogeneous” model in a future publication. 2. The Transport Formalism The photosynthetic units associated with different reaction centres are probably not independent and there is a possibility of exciton transport between different units. Joliot & co-workers (1964, 1969) reported exciton migration between traps in photo-system (PS) II; transfer between PS I
94
T.
MARKVART
units and between units of different photosystems has also been suggested. Only independent units will be studied in this paper. This assumption is not as restrictive as it may seem. Any infinite connected model with active traps distributed at the points of a two-dimensional periodic lattice and in which the distribution of accessory pigments shows the same periodicity, can be reduced to the independent unit problem, supplemented perhaps by a different set of boundary conditions. This problem is treated more fully in another publication by the author (Markvart, 1977); a similar problem in the random walk setting is discussed by Chandraskhar (1943). In a similar fashion, a number of finite connected models with open traps can be divided into sets of independent units by using symmetry considerations. The maximum photochemical yield and constant fluorescence are therefore seen to be affected only little by whether the units are connected or not. Another situation when the methods of this paper can be applied arises in a large connected model, when most traps are inactive and the excitations created near one active trap have a negligible probability of reaching another. Each active trap may in this case be considered as belonging to a large and independent “superunit”. The intermediate problem of non-periodically distributed traps of appreciable concentration is treated elsewhere (Markvart, 1977). We shall be concerned with exciton transport following a brief flash of illumination. In view of the assumption of “incoherent” exciton hopping the transport equations are:
with initial conditions p,(O) = pi”), where p,(t) is the probablility that the exciton is at the nth molecule at time t; n = no corresponds to the trap. W,, is the exciton transfer from molecule m to II (m # n), the diagonal elements are defined as:
K” = - mfn c Km. The exciton “loss rate” is given by:
where r,, is the lifetime for non-radiative de-excitation, and k,, is the rate of the primary chemical reaction (1). The photochemical yield is: x = k, r p,,(t) dt, 0
THE
PHOTOSYNTHETIC
95
UNIT
and the fluorescence yield of the nth molecule:
Equations (6) are most conveniently solved by the method of Laplace transforms : this brings them into a set of algebraic equations : ; i(s + k + QL,kL
- w,,)p”,(s) = do’,
where p,, is the Laplace transform of pn. Introducing by :
the Green’s function
c W,, - ~mJG,(s) = 4,w one readily obtains : ; C%,(S + k,) + kp@m,(s f O%tt(s + k,) -
p”“(S)= with :
There are two limiting cases to be considered. The first arises when the exciton transport in the photosynthetic unit is very fast and the rate of the complete primary photochemical process : (A.P.)* (A.P.) (9) D (R.C.) A + D (R.C.)’ A-, where (A.P.) denotes the accessory pigment aggregate, is limited by the rate of the reaction (1). In this situation both constant and variable fluorescence are emitted by excitons in thermal equilibrium (Paillotin, 1976) and accordingly, the constant and variable fluorescence have identical static properties (spectrum, polarization etc.). This, however, seems to be in contradiction with the experimental evidence : see, for example, Duysens (1964) and the reviews by Fork & Amesz (1969) and Goedheer (1972). In the remaining part of this paper we shall therefore be concerned with the other limiting case when the primary process described by equation (9) is limited by the exciton transport, and the rate of reaction (1) is very fast. In this case expressions (7) and (8) give:
c &m,(Wnm(kJ - G,,,(kl>G,,,(kl>}p,“’ (10) %,n,W ’
96
T.
MARKVART
and
(11) 3. Units with One Bind of Accessory Pigment An approximation often employed in the theories of exciton transport in the photosynthetic unit is that all accessory pigment molecules are identical. From the discussion of the transfer rate it follows that this will be permissible if the spread in energies of the relevant energy levels (the first singlet) is small compared to k,B. To some approximation we can therefore regard the different moieties of chlorophyll a at room temperature as identical, but not chlorophyll a and b. Even though this approximation is rather restrictive it simplifies the problem considerably and allows an insight into a number of interesting features of the exciton transport. The usual random walk approximation (Pearlstein, 1964, 1966, Montroll, 1969, Bedeaux et al., 1977) is equivalent to expanding the product sG(s) in (positive) powers of s. We write the Green’s function in the spectral representation as :
where the 1’s are the eigenvalues of the transfer matrix W and t(A) are the corresponding eigenvectors. It is clear from equation (12) that the random walk approximation is valid provided: s c & = min (In]; 1# 01. (13) In this paper we shall keep only the lowest order terms in the expansion and put:
G,z,(4= ks (1+w,,), where : c km,(4 a ’ 1#0 and N is the number of molecules in the unit. We denote 9 nm =
-N
T, = grin
l--,(m)
(15)
Wa)
(16~)
= g?. ”
THE
PHOTOSYNTHETIC
97
UNIT
It will be seen that T, is the mean first passage time of the excitation through the point n, assuming the starting points are distributed throughout the unit with equal probability. The time Tis the average of T, over all positions in the unit. Clearly, (17) Condition (13) gives an upper bound for s which is a decreasing function of N and which decays to zero as N + co. It is seen therefore that the random walk approximation may not always be valid. In particular, it will not be valid for very large units and for units with slow excitation transfer rates. We put P”(‘I = l/N and using the random walk approximation we obtain from equations (10) and (11): 1 l+k,T,,
‘= 0,
= -$
lf 1 (#)w ” = __Nk,z,
I1-ly;k[T%
\ =N’ (1 -rn,(n))q+)
1 no 1 +kTJ& 1 +k,T,,
)
(18)
1
= N {1+ k,T,,l-,,(n)}iD(“)
where the total fluorescence yields are:
The superscripts c and v refer to the constant and variable fluorescence, respectively. We shall also consider briefly another approximation for the Green’s function. When : c-e1 W-4 Ao < k, we may approximate The photochemical
4
the sum in equation (12) by an integral. yield is in this case:
Wb)
98
T.
MARKVART
where :
and Nn(L)di
is the number of eigenvalues of Win the interval J. --f 1+dL.
(A)
UNIFORM
UNITS
WITH
ISOTROPIC
COUPLING
A convenient starting point in the study of units with one kind of accessory pigment are models where the transfer rates depend only on the relative positions of the molecules concerned. It is also convenient to discard the orientational dependence of the coupling at this stage and consider a transfer matrix of the form: (22) Km = WIRn-R,I) for all m # ~1, where Ri is the position vector of molecule i. Coupling of the type (22), with matrix elements non-zero only between nearest neighbours, was considered by Pearlstein (1966); an analytic solution was given by Montroll (1969) In this section we shall consider a more general coupling of the type (22) and assume only that the sum:
;co ]Rn\2+e W(Rn(h converges for some E > 0. The subscript co indicates that the summation extends over the entire infinite lattice. We denote: to = {“FOX3 Ww-‘, 4&J
= to WRA
o2 = cm Rn2tiR,), n
PO (23bj (23~)
In the random walk terminology (Spiker, 1964) w(r) is the transition function of the random walk (the probability of hopping distance r in one step) and to is the average time between jumps. We shall assume that the pigment molecules form a two-dimensional square lattice and the co-ordinates of the molecules are given by: R,i = mid,
i = 1,2; mi = 1,2,. . ., M.
(24)
The method, however, can be applied without major modifications to other two-dimensional periodic structures. The eigenvalues and eigenvectors of the
THE
PHOTOSYNTHETIC
UNIT
99
matrix W are, for N 9 ts2,
where : i = 1, 2;ji = 0, 1,. . .) M-l
ki = s;,
(26)
A,, = &
if ki = 0
J2’
= 1
otherwise,
provided molecule n is not too near the unit boundary. In the special case of transitions between nearest neighbours only formulae (25) are exact for any N > 1 and any position of molecule 12. We note that: (27) for small jkl. Let us denote the trapping time T, when the trap is at the centre of the unit. For large N there is a simple relationship between T, and T: T(N) = 4~(N/4) _ LN t e 3a2 O’
where we explicitely depicted the dependence of the trapping times on the size of the unit. To evaluate T we write:
In the first sum we substitute for k from equation (26) and the resulting sum is evaluated directly (Appendix B). The second sum is replaced by an integral. Hence, T = ntO,
W4 where n, the number of steps the exciton will take to reach the trap for the first time, is given by: Nd2 nSC)= 2 7-m
In N+2C+rc+rcl-
E + b 71 >
Wb)
100
T.
MARKVART
where C is Euler’s constant, G is Catalan’s constant, = O-0076
and (31)
For the nearest-neighbour form :
coupling integral (31) can be obtained in a closed
I,-, = Sf - AIn !Jf = 1.0994. TT
7c
(32)
8
Substitution of equation (32) into (30b), noting that cr = d for the nearest neighbour coupling, and using equation (28) gives the first two terms of Montroll’s result (Montroll, 1969). Of particular interest is the coupling: (33)
where
which has a Forster-like dependence on the molecular separation. case I and a2 have to be calculated numerically with the result: IF = O-5696, c2 = 1.2936d2,
In this
(34)
The effect of the trap position on the trapping time is shown in Fig. 1. The behaviour of the function r with the trap at the centre of the unit is shown in Fig. 2. The trapping time also depends on the shape of the unit. For example, for a diamond-shaped unit one obtains along similar lines as above. Nd2 ndiam
=
2
In (2N)+2C+7r+7rI
- ff_G+ b . n
>
(35)
Let us consider integral (21) for the transfer matrix of the form (22). We can choose periodic boundary conditions for the eigenvectors
THE
PHOTOSYNTHETIC
UNIT
101
FIG. 1. Distribution of the trapping times with different positions of the trap. The edges of the unit are represented by double lines, the eentre of the unit is denoted by c. Only a Part of the unit is shown, the remainder can be obtained by symmetry.
to obtain: ‘r dk, ‘[ dk2 t k ipckj
J == kg
= !$$
x
0 1 nld d dkl f dk2
x
nld t,k, ;‘.‘Fi’
+ j dkr
’
n/d
s dkz x
’
[
tends to (rr/cr)2Z and the fist
In the limit krt, -+ 0 the second integral integral is evaluated directly. Thus: tok,d2 J = 2 -ln(k,to)+21n ?c(T We neglected terms of the order (klt0)2.
($
+nZ-F}.
(36)
102
T.
MARKVART
FIG. 2. Three dimensional “graph” of the transfer function r. The edges of the unit are represented by double lines, the centre of the unit is denoted by c. Only a part of the unit is shown, the remainder can be obtained by symmetry.
(B)
RANDOM
SYSTEMS
: ORIENTATIONAL
DISORDER
In this section we shall study numerically the model of the photosynthetic unit in the extreme case of complete orientational disorder of the pigment molecules. The trapping time and transfer function r were chosen for investigation, rather than the fluorescence and photochemical yields, because of their simple interpretation and a more universal character. At each point of the lattice [equations (24)J an orientation of the dipole moment vector is generated by the computer so that: (i) All orientations are equally likely and (ii) orientations of dipole moment vectors at different sites are independent. The elements of the transfer matrix W were calculated from:
wnn= -c mWm. The normalization
was chosen so that:
for molecules far from the boundary
of the unit.
THE
PHOTOSYNTHETIC
UNIT
103
The eigenvalues and eigenvectors of W were found by numerical methods. The trapping time and the transfer function I’ were subsequently determined using formulae (15), (16a, c) and (17). This procedure was repeated 100 times which was found to guarantee reasonable accuracy of the average values. A similar program was also run for the nearest-neighbour transition -rates. The results for the average trapping time [equation (17)] using Forster’s coupling are summarized in Table 1. The values for the orientationallydisordered model are seen to differ by a factor of 1.6-l -7 from those TABLE
1.
Comparison of results for the average trapping time using averaging at difirent stages of the calculation. Column 1 shows the average trapping time calculated numerically from equation (17), the averaging performed correctly at the end of the calculation. Column 2 contains the corresponding figures using the averaged transition rates (5) or (33). N is the number of molecues in the unit. N
1
2
49 100 121 196
192 403 492 814
109 240 296 502
obtained for the uniform model described by equations (22) and (33) with the corresponding averaged transition rates. For the nearest-neighbour coupling the difference is even more marked: we found that in this case the results differed from the corresponding results of section 3A by a factor of 2.5-2.7. The spatial variation of the ratio T,/T and of the function r, on the other hand, are almost unaffected by the randomness and were found to agree (within the small statistical error) with the corresponding values of section 3A.
4. Units Containing Chlorophyll A and B Two kinds of models will be considered in this section. The units in models of the first kind will consist of an ordered inhomogeneous arrangement of the two pigments, chlorophyll a forming the inside of the unit, surrounded by chlorophyll b. Models of this type were studied by Seely (1973) and Paillotin (1976). The units of the second kind will contain a
104
T.
MARKVART
homogeneous mixture of chlorophyll u and b throughout the unit. This problem was considered by Pearlstein (1964) and Swenberg et al. (1976). Because of the position of the spectral bands exciton transfer from chlorophyll a to chlorophyll b is inefficient and will be in this paper neglected. (A)
MODELS
OF THE
FIRST
KIND
Two models of the first kind are shown in Fig. 3. Characteristically, the average distance travelled by the excitons absorbed in chlorophyll b is appreciably larger than by those absorbed in chlorophyll a. This is reflected in the transport times and consequently should be observable as an inhomogeneity in the fluorescence and photochemical yields. As compared with unit of the same size composed of only one pigment, however, the exciton transport in this arrangement of pigments is clearly more efficient.
(a)
iti
FIG. 3. Two models for the PS II units of the first kind.
It is convenient to introduce two transfer matrices, IV(‘) and FVcb),for description of the exciton transport in the bulk of chlorophyll a and b, respectively, without the possibility of transfer between the two pigments. It may then be shown by lengthy but straightforward manipulation of the appropriate Green’s functions that, keeping only the lowest order terms in the transport times,
(37)
THE
PHOTOSYNTHETIC
where xn (xb) is the photochemical cl (chlorophyll b), Ax
UNIT
105
yield for light absorbed by chlorophyll =
x,
--Xb,
0%’ is the constant fluorescence yield of chlorophyll CLfor light absorbed by chlorophyll b, and similarly for @g,’ and @ii’. The time T, is the mean trapping time for the unit with chlorophyll b removed and: (38) Tb = -Nb
%%‘b)
c
--j-7
Lb+0
b
(39)
where :
The 5’s and I’s are the eigenvectors and eigenvalues of IV(‘) or WCb), N,(N,) is the number of molecules of chlorophyll a (chlorophyll b), and the sum in equation (40) is to be taken over the boundary between the pigments of length L. The interpretation of formulae (37) is evident since (To+ T,‘) is the average time to reach the trap from the boundary between chlorophyll a and b, and Tb is the mean time to reach this boundary from chlorophyll b. If the pigments are arranged in a square lattice and the coupling is of the form (22) we have from equations (28) and (30). (41) The largest contribution to the sums in (38) and (39) comes from terms with small 1. General consideration using equation (27) gives: T; = ; N,tb”) Tb = 4 4 $’ ab
0
Nbtbb’, *
where c( is a constant independent of N, or Nb and C$is usually a linear function of the ratio NJN,,. In the present situation (Fig. 3): a = Q, and
with B equal to + and O-18 for the models of Figs 3(a) and (b) respectively.
106
T. (B)
MODELS
MARKVART OF THE SECOND
KIND
The exciton relaxation times in a unit of the second kind are very short (of the same order of magnitude as the hopping time to), and therefore the greatest part of the transport will take place among the energetically lower lying chlorophyll a molecules. The presence of chlorophyll b in the unit has two effects on the structure of the chlorophyll a aggregate: it increases the molecular separation and reduces the number of chlorophyll a molecules in the unitt. These two structural changes have opposite effects on the exciton transport: the former increases the hopping time t,,, the latter decrease the number of steps for the exciton to reach the trap. It will be seen that it is the increase in the hopping time that will predominate, making the exciton transport less efficient. We shall in this paper neglect
o chio x
chi b
l
trap
FIG. 4. A simple model of the secondkind.
the possible dielectric effects of chlorophyll b on the transition rates between molecules of chlorophyll a. This would slow down the exciton transport still further. The simplest model of the second kind is shown in Fig. 4. Molecules of the two pigments are arranged alternately in the square lattice [equation (24)]. Molecules of chlorophyll u clearly form a square lattice in a diamondshaped unit. We shall denote T,, tL2’ and n, the mean trapping time, the hopping time and the mean number of steps to reach the trap, respectively, for the exciton transport among the chlorophyll a molecules in the unit of Fig. 4, and T,, t$,” and n, the corresponding quantities in a unit obtained from the unit in figure 4 by replacing the molecules of chlorophyll b by chlorophyll a. Applying formulae equations (30) and (35) we obtain: n2 = 2nl,
and since: d, = d,f2, t We assume that the total amount of chlorophyll
in the unit is constant.
THE PHOTOSYNTHETIC
UNIT
where d, is the lattice constant of the chlorophyll using equations (2) and (23a), tb”’ = 8@‘. Therefore :
107
a lattice in Fig. 4, we have,
T2 = 4T,.
From this result one would intuitively chlorophyll b equal to c,
expect for the concentration
of
2 N (l-c)-“. In the next model the molecules of chlorophyll a and b are intermixed at the lattice points [equation (24)] at random. The coupling was taken in the form (33) and the average trapping time was computed using equation (17). The results for the ratio Ti2)/Ti1’ are given in Fig. 5. Because of the fast relaxation times in units of the second kind,
FIG. 5. The results for the average trapping time (17) as a function of chlorophyll b concentration c in PS II units consisting of a homogeneous mixture of chlorophyll a and b. The full line are the results of a numerical calculation for a random homogeneous mixture, the broken line is the graph of (l-~)-~.
108
T. MARKVART
5. Discussion In this paper we have presented new theoretical results for exciton transport in the photosynthetic unit, and examined some approximations often employed in previous analyses. The rigorous condition for validity of the random walk approximation [equation (13)] t oge th er with the data given in the Introduction shows, as expected, that this approximation is valid in this situation. From the results of section 3 it is seen, however, that other two common approximationstransitions between nearest neighbours only and the a priori use of the averaged transition rates-have only a qualitative character. Formula (30) with numerical values [equation (34)] show that using the proper Rv6 dependence of the coupling decreases the trapping time by the factor of about l-5? as compared with the result obtained using transitions between nearest neighbours only. In the case of complete orientational disorder among the pigment molecules we have found that the proper treatment by averaging at the end of the calculation gives results higher by a factor of about l-7 than when using the already averaged transition rates (33). Formulae (18) show that due to the finite exciton transport time across the photosynthetic unit, some of the constant and variable fluorescence is emitted by different parts of the unit. From the shape of the function r (Fig. 2) it is seen in particular that more variable fluorescence originates from the molecules in a relatively small neighbourhood of the trap. Some correlation between orientations of pigment molecules over distances short compared with the size of the unit should therefore result in the variable fluorescence being more polarized than the constant fluorescence. Some experimental evidence in support of this is given in the work of Whitmarsh & Levine (1974) who observed a slight increase of fluorescence polarization upon blocking the electron transport. It is well known that the majority of cholorophyll fluorescence is emitted by PS II. It is tempting to associate this with the relatively large content of chlorophyll b in this photosystem. Let us assume for the sake of the argument that the units of PS II contain chlorophyll a and b in equal amounts, units of PS I contain only chlorophyll a, and that the total amount of chlorophyll is evenly distributed between the two photosystems. If we take a model of the second kind for the units of PS II we obtain for the ratio of the constant fluorescence yields:
Q(') T ",A=4 Qf' - Tl
'
t This includes the factor /3/4 = 1.16 which reduces the hopping time lo.
THE
PHOTOSYNTHETIC
109
UNIT
The decay of fluorescence as a function of time is complicated in the present theory. At long times, however, the decay should approach exponential with the lifetime approximately equal to the appropriate trapping time. The ratio of the apparent fluorescence lifetimes of the two photosystems is therefore: (42)
Yu et al. (1975, 1977) obtained values 50-60 psc and 200-220 psc for the lifetimes of PS I and PS II fluorescence, respectively, by direct picosecond studies. Similar values (80 psc and 300 psc) were obtained by Paschenko et al. (1975), in excellent agreement with formulae (42). It would therefore appear that the units of PS II are best described as a homogeneous mixture of chlorophyll a and b. The models of the first kind (section 4A), on the other hand, represent the most likely possibility if the PS II units are inhomogeneous and consist of large blocks composed of preferentially one kind of pigment. This hypothesis can be tested experimentally since the transport times in equations (38, 39) are-though somewhat smaller-of the same order of magnitude as the trapping time [equation (41)] and this should manifest itself through inhomogeneities in the action spectra as a function of excitation wavelength. The model studied at the end of section 3A describes a situation when the random walk approximation is not valid. Let us consider a connected model consisting of many traps associated with a large number of identical accessory pigment molecules. By “poisoning” a large proportion of the traps a situation will arise when the remaining “open” traps may be considered as independent, that is, when the excitationscreated near one open trap have a negligible probability of reaching another. It will be seen that conditions (19) are satisfied for such a system and equation (20) may therefore be used to analyse the linear portion of the dependence of X on c, as c -+ 0, in the study of connected models. A preliminary calculation shows that the values of parameters consistent with those used in the study of the trapping time can be used in expression (36) to obtain a reasonable agreement with the data of Joliot et al. (1964, 1969). The author wishes to thank Dr E. Cane1 for suggesting this problem, and for valuable comments and discussions. The financial support provided by a Science Research Council grant is gratefully acknowledged. REFERENCES ALISTIN, I. G. & MOTT, N. F. (1969). Adv. Php. l&41. BEDEAUX, D., LAKATOS-LINDENBERG, K. & SHULER, K. E. (1971). BRODY, S. S. & RABINOWTCH, E. (1957). Science 125, 555. CWANDRASEKHAR, S.(1943). Rev. mod. Phys. 15, I.
J. math.
Phys.
12, 2116.
110
T.
MARKVART
CLAYTON, R. K. (1965). Molecular Physics in Photosynthesis. New York: Blaisdell. DUYSENS, L. N. M. (1964). Prog. Biophys. mol. Biol. 14, 1. FORK, D. C. & AMESZ, J. (1969). A. Rev. PI. Physiol. 20, 505. FORSTER, T. (1948). Ann. Physik 2, 55. GOEDHEER, J. C. (1972). Ann. Rev. Pi. Physiol. 23, 87. GRAJXHTEYN, I. S. & RYZHIK, I. M. (I 965). Tables of Integrals, Series andProducts. London: Academic Press. HOLSTEIN, T. (1959). Ann. Phys. 8, 343. JOLIOT, A. & JOLIOT, P. (1964). Compt. Rend. 258,4622. JOLIOT, P., JOLIOT, A. & KOK, B. (1968). BBA 153, 653. KNOX, R. S. (1975). In Bioenergetics of Photosynthesis (Govindjee, ed.), p. 183. London: Academic Press. MARKVART, T. (1976). Phys. Stat. Solidi 73, 689; 74, 135; 76, 953. MARKVART, T. (1978). J. theor. Biol. 72, 117. MONTROLL, E. W. (1969). J. math. Phys. 10, 753. PAILLOTIN, G. (1976). J. theor. Biol. 58, 219. PASCHENKO, V. Z., PROTASOV, S., RUBIN, A. B., TIMOFEEV, K. N., ZAMAZOVA, L. M. & RUBIN, L. B. (1975). BBA 408,413. PEARLSTEIN, R. M. (1964). Proc. Natn. Acad. Sci. U.S.A. 52, 824. PEARLSTEIN, R. M. (1966). Brookhaven Synzp. Biol. No. 19, p.8. ROBINSON, G. W. (1966). Brookhaven Symp. Biol. No. 19, p.16. SEELY, G. R. (1973). J. theor. Biol. 40, 173, 189. SPITZER, F. (1964). Principles of Random Walk. Van Nostrand. SWENBERG, C. E., DOMINIJANNI, R. & GEACINTOV, N. E. (1976). Photochem. Photobiol. 24, 601. WHITMARSH, J. & LEVINE, R. P. (1974). BBA 368, 199. Yu, W., Ho, P., ALFANO, R. R. & SEIFERT, M. (1975) BBA 387, 159. Yu, W., PELLEGRINO, F. & ALFANO, R. R. BBA 460, 171. ZENNER, C. (1932). Proc. R. Sot. A137, 696.
APPENDIX (A)
PHONON-ASSISTED
A EXCITON
HOPPING
The model on which the calculations will be based consists of two identical molecules, each with one exicted state and one vibrational co-ordinate. In the Born-Oppenheimer approximation the solution of the Schrodinger equation for a single molecule has the form:
H(Q, r)q%,Q) = E(‘)(Qh”k
Q)3
where Q is the vibrational co-ordinate, r is the collection of electron coordinate vectors, H (Q, r) is the single molecule Hamiltonian and qCi) (r, Q) are the electronic wavefunctions. The superscript denotes the electronic state (1 the ground state, 2 the excited state). As is well known the eigenvalue E(‘) (Q) acts as the potential energy for the vibrational motion. We expand E(‘)(Q)
= EC’) f$m&)Z(Q
- Q”92
+
Or-Q
_
Q(i))31
THE
PHOTOSYNTHETIC
UNIT
111
The following assumptions will now be made: N(i)We neglect the terms 0 [(Q - Q$j))3] (ii) Set cJi) = o, same in both electronic states. These assumptions correspond, respectively, to the harmonic approximation and to the assumption of the exciton-phonon interaction linear in lattice co-ordinates. We shall choose Q so that Qb” vanishes and denote Qb”’ = Q,. Considering both molecules the total electronic wavefunctions can be written as : @ = ~,(QI, Qdd"(~i~'+
a2(Q1,
+A(Q,, Q2k~‘l~+&‘+ B(Q,, Q~~P(~+P(~).
Q2M2)d1’
(Al)
The indexes refer to the two molecules. We shall omit A and B since they are unimportant in our problem. Using representation (Al) the total Hamiltonian is given by: $a,= 9~2
{WQI, Q2) - m~2QoQ~>u~+ % = { WQ1, Q,>-m~2Q,Q,>~, +V*q,
WI
where :
u(Q1, Q2) =
+3m02(Q; +Q: +Qf)
E2-E,
and V is the usual resonance-interaction In the dipole-dipole approximation :
matrix element (5a).
where : di = -e J (p$“(r,.)(C r;)cp!“*(r;) dr,
i = 1, 2,
are the transition dipole moment vectors and R is the vector connecting the charge centres of the two molecules. Equations (A2) are formally the same as those considered by Holstein (1959). Below we briefly outline his approach to obtain the transition rate. First, the new potential surfaces are calculated: they are the eigenvalues of the Hamiltonian (A2):
EJQ,, Qd = WQI,
Q2)
- m+
(Ql +Qz) k f {FF)2
(Ql-Q2)2+lV12}’
(A3a)
112
T. MARKVART
and
a;" -=-
a$”
- mo2Qo 2 -1 mw2Qo(Qz-Q~) + [(-opt) ca,-e,,'+lVl']ij. 2 -
Wb)
V
The nature of the solutions (A3) is shown in Fig. 6. It is seen that the exciton moving along one of the surfaces will hop to the other molecule if it passes through the “bottleneck” Q, = Q, and remains on the same surface. Hence the exciton transfer rate is:
t mw%&tP~~ 2
E-Ut
FIG. 6. Adiabatic Holstein, 1959).
energy surfaces for the phonon-assisted
where P(V) is the probability with velocity
exciton hopping
that the exciton passing through v=
Q, = Q2 C-45)
l&Q,l~
remains on the same potential surface. This probability Zener (1932) to be (in our notation): 27-c pq’ P(v) = 1 - exp - - ____ h mw2Qov > ’
(after
was determined
by
646)
THE
PHOTOSYNTHETIC
113
UNIT
,f(v) is the encounter frequency of Q, = Q2 subject to (A5). Consideration based on elementary statistical mechanics gives : 2v j exp - Ek+U~~z~~}6(e,-~2-v)i5(Ql-Q2) B F(v) = -
dQ dQ ---
=k$vexp(-kg)exp(-a+), where : In the limit when the argument of the exponential integral (A4) gives:
in equation (A6) is small
where : E, = Srnw’Q& It is usual to express the transition rate W in spectroscopic terms. In our model it is readily observed that both absorption and fluorescence bands are Gaussian of width: A = 2(E,k,8)*, c48) and the energy of the Stoke’s shift is: E, = 4E,.
(A9) Substitution of parameters (A8) and (A9) into Forster’s result for the transition rate yields (A7) as expected. For two dissimilar molecules equations (A8) and (19) give the usual relation :
provided both molecules have equal Stoke’s shifts. E, and E, are the absorption energies of the two molecules. (B)
LIMITS
OF APPLICATION
The validity of expression (A7) is limited by two factors. In approximating expression (A6) to evaluate integral (A4) it was assumed that: 2
PI’ .g 1, h mwZQ,,vO
(All)
114
T. MARKVART
where uOis a typical value of o at Q, = Q2. Evaluating uOreduces expression (All) to: w 4 w(t). G412) In view of the use of equilibrium distribution function for the lattice co-ordinates, however, equation (A12) should be replaced by the more stringent: d$: where r, is the phonon relaxation time. The other assumption involved in the derivation of formulae (A7) is that the vibrations can be treated classically. Holstein (1969) shows that this can be done provided: E, $- ‘2
4’
and
0413)
Although conditions (A13) are necessary for validity of the formula (A7) thermally assisted hopping will occur in a much wider range of circumstances. Briefly, one may summarize the conditions for the “incoherent” exciton hopping as follows: (i) Distortion of lattice due to exciton creation is confined to a single molecule. (ii) The width of the band associatedwith the “coherent”exciton states is smaller than the uncertainty in energy corresponding to the thermally assisted hopping. Regime described by (i) is clearly equivalent to the union of Forster’s weak and very weak coupling regions where the vibrational states of the aggregate are closely related to those of individual molecules. This will be true if either E, < ho or 1Vl c Ea. In the former regime the exciton-phonon interaction is weak and the exciton will propagate in a band with occasional scattering. It is the latter case that is of interest here. When the resonance interaction can be treated by pertubation theory ($) the exciton bandwidth is approximately given by: a(O) = co exp t This is different from Holstein’s criterion (Holstein, 1959, equation 103). The reason is that Holstein used the average of u over the whole period of vibration to obtain v,,. $ This does not cover the whole of the region described by the latter inequality. The reader is referred to literature (Holstein, 1959; Austin & Mott, 1969) for a discussion of the so-called adiabatic regime.
THE
PHOTOSYNTHETIC
UNIT
115
where oO is the bandwidth in the absence of the exciton-phonon interaction. lt is, seen that o(O) is a rapidly decreasing function of temperature. By application of condition (ii) above incoherent site hopping will be dominant at temperatures higher than some critical temperature 6, defined by: o(&) = hW. In Forster’s terminology this equation describes the transition to very weak coupling. Holstein, (1959) estimates : where On is the Debye temperature
of the phonons concerned.
APPENDIX We shall evaluate the sum
in equation (29). We put:
where
We write:
The first sum readily gives:
The second sum we write in the form
where
B
from weak
116
T.
MARKVART
For large M the sum (B2) can be replaced by an integral: l cotan-’ dz = G &=J 9 1 1 2 0 Z where G is Catalan’s constant (see, for example, Gradshteyn 1965). The sum (Bl) is evaluated as follows: M-l z TCM- l coth (nn) - 1 +;“g’hA = .zl 2n coth (in) - ,$ = z 3 _____ n n 1 n 1 > B-tidz,ydq 0
-iyg:+2=$b
+g(C+InM)--g+O(M-‘),
where C is Euler’s constant and:
b=4
2 -i-
n= 1 n(eznn - 1)’
Hence, collecting all terms, s=~lnN+~b+qc+%_c+o(~-~).
& Ryzhik,