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Kinetics of exciton trapping by monocoordinate reaction centers
LUMINESCENCE
Journal of Luminescence 51(1992)139—147
JOURNAL OF
Kinetics of exciton trapping by monocoordinate reaction centers Robert M. Pearistein Physics Department, Indiana-Purdue University, Indianapolis, IN 46205, USA
A model of exciton migration and trapping in photosynthetic membranes is presented that differs from earlier models of hopping excitons on lattices. In the new model, the trap, which is the special pair of chlorophyll (ChI) molecules within the reaction center (RC), is assumed monocoordinate, i.e. to have but one nearest-neighboring antenna lattice site; in earlier models, the trap is assumed isocoordinate with the antenna sites. In this monocoordinate-RC model, which it is argued is more realistic at least for some of the photosynthetic bacteria, antenna ChI excitation energy decays more slowly than in the isocoordinate-RC models, for equal values of parameters. The principal exponential decay lifetime is predicted to increase typically by a factor of 2—3. However, in the analysis of an experiment on the picosecond kinetics of excitation decay in Rps. viridis [1], where the exponential lifetime is fixed by the measurement, one is forced to conclude the opposite, i.e. unknown Förster parameters must be larger than previously thought, in fact approaching theoretical maximum values. As a side benefit of the relatively simpler calculation for the monocoordinate-RC model, coupling of the exciton trapping to the first two steps of electron transport in the RC has been readily included in the equations. This yields a hi-exponential decay in which the second exponential component, describable as a long-time tail of small amplitude (—. 0.2%), can nonetheless be responsible for a large fraction of the total quantum yield, at least in a system with pre-reduced RCs. The utility of this result for the analysis of delayed fluorescence experiments is discussed.
1. Introduction The shortened fluorescence lifetime of chlorophyll (Chi) in vivo was first explained theoretically to be a consequence of the migration of hopping excitons into traps almost three decades ago [2,31. Since then, the theory has been extended and reviewed a number of times [4—101.It has been applied to the analysis of experiments on energy transfer kinetics in both green plants and photosynthetic bacteria [1,11]. Recent arguments [12] regarding the relative placement of reaction centers (RCs) and antenna protein complexes in photosynthetic membranes suggest that one of the basic tenets of the theory may not hold, at least in some of the photosynthetic bacteria. This principle is that, in a lattice model of the arrangement of the Chls in the membrane, every lattice site including that of the trap has the same number of nearest neighbors [41.The purpose of this paper is to develop a simple model to treat another situation, hopefully more realistic, in which the trap has but one neighbor while, in general, other sites have many. 0022-2313/92/$05.00 © 1992
—
The model is described and justified in section 2. Section 3 provides details of the mathematical methods used in the calculations, the results of which are presented in section 4. In section 5, the results are used to analyze the very nice kinetic data on membranes from Rhodopseudomonas viridis obtained by Zhang et al. [11,whose paper provided the major stimulus for the work presented here. The last section considers puzzling points, mentions possible generalizations, and emphasizes certain predictive features of the theory, especially in the analysis of prompt versus delayed fluorescence yields.
2. Description of model A standard lattice model of exciton migration and trapping in a photosynthetic membrane is shown schematically in fig. 1(a). The trap, designated by the circle with the “T”, is isocoordinate with the antenna sites, designated by the open circles. While still schematic, fig. 1(b) gives a
Elsevier Science Publishers B.V. All rights reserved
140
R.M. Pearistein
~
ffi
/ Kinetics of exciton trapping by monocoordinate reaction centers
~
(b)
Fig. 1. Lattice models of exciton migration and trapping in a photosynthetic membrane. (a) “Standard” or isocoordinate trap model (b) Monocoordinate trap model that includes vacancies in the antenna lattice due to the presence of the reaction center. (c) Simplified monocoordinate trap model used for calculations; see text for details.
somewhat more realistic view of the lattice geometry in the vicinity of a reaction center (RC). The RC not only provides a trap, the special pair of bacteriochlorophyll (BChl) molecules denoted “P”, but also creates a large set of connected vacancies in the otherwise regular lattice of antenna BChls. The result is that the trap cannot be simultaneously isocoordinate and kinetically well-connected to the antenna sites, because an isocoordinate position would place the trap quite far from any of those sites. For example, if the antenna lattice constant is 20 A and the trap is isocoordinate but 30 A from each of its nearestneighboring antenna sites, the relative Förster rate constant for transfer from an antenna neighbor to the trap is (30/20)6 = 11 times smaller than between nearest antenna neighbors; this does not take into account a possibly less favorable spectral overlap for the antenna-trap transfer, which would further diminish that rate constant.
If the trap is placed at a distance of one lattice constant from a single antenna site, so that it is monocoordinate as in fig. 1(b), the rate constant for energy transfer from nearest-neighboring lattice sites to the trap is diminished by a factor equal to the lattice coordination number (number of nearest neighbors of each site in a perfect lattice of the same type) [2,41.For a two-dimensional lattice that number is at most 6, so that in the foregoing example there is still a net gain of nearly a factor of two in that energy-transfer rate constant. It is therefore assumed here that the trap is strongly coupled kinetically to a single antenna lattice site. It may be noted from fig. 1(b) that the presence of the large RC also reduces the coordination of those antenna sites near its perimeter. However, in general, the reduction is not so large, i.e. none of the antenna sites becomes monocoordinate. Moreover, arguments based on anisotropic exciton hopping theory [4] suggest that as long as the intra-antenna transfer is not restricted to one-dimensional domains, the kinetics of that transfer differs little from that on a perfect lattice. For simplicity, the vacancies in the antenna lattice are ignored here. Figure 1(c) illustrates the monocoordinate-RC model of this paper. The two-dimensional antenna lattice is perfect. The monocoordinate trap is “nondisruptive” [4]; the dashed line in fig. 1(c) that connects the trap to its sole nearest-neighboring antenna site may be interpreted as passing through the third dimension. That the kinetic connection to the trap may be accurately represented in this unphysical way is another assumption of the model. Figure 2 contrasts the kinetic relationships of the two models. In each case, the box distinguishes kinetic processes that occur after the exciton arrives at the “central” site the trap itself in the isocoordinate-RC model, the sole nearest-neighbor of the trap in the monocoordinate-RC model. Other than as described above, standard lattice-model assumptions are made [4]. This ineludes the assumption that the RCs occur periodically within the overall lattice. However, as has been shown, this is not a terribly significant re—
R.M. Pearistein
/ Kinetics of exciton trapping by monocoordinate reaction
centers
141
one lattice site to any one of its nearest neighbors, and i dp/dt. The quantity [211, is the rate constant, in units of FA, for the transfer from site I to site 1’. The presence of lattice imperfections, such as a trap, requires either extra terms to be added to the right side of eq. (1), or separate equations to be written for those lattice sites that are either themselves imperfections or that transfer energy directly to an imperfect site. For the simplified monocoordinate-RC model (section 2) it is only
(a)
=
PC (b)
A
0~PC
necessary to consider a single imperfect site and a single regular site that transfers energy to it. The
Fig. 2. Kinetic relationships in the lattice models. (a) Isocoordinate trap. (b) Simplified monocoordinate trap. Each “F” is a Fbrster rate constant, each “k” a rate constant for electron transfer in the reaction center. States of reaction center components are indicated in the usual way. In (b), A~3denotes the nearest-neighbor antenna site (to Pin the reaction0A’ center) the primary in its excited state, A0 in its quinone. ground state; “Q” is
latter is designated by I 0, and the former given the subscript ~ Additional equations can be written for states of the RC kinetically connected to P”. Here, only two such additional states, P~IQ and P~IQ~,are considered. model, be Thus, writtenforinthis three parts:the master equation can
striction. In any case, it probably holds, at least
,i,(t)
=
‘FA
~ [2
—p,(t)}
11~[p,~(t)
approximately, in most photosynthetic bacteria (if not green plants) [12]. If, as argued by some for certain organisms [13], the RCs are paired at each RC-site, the trap (P) of one RC is almost certainly physically far enough from that of the other so as not to modify the trapping kinetics, Note that neither this nor the isocoordinate-RC models applies if the core antenna (surrounding the RC) contains special long-wavelength components that pre-trap the excitation prior to its arrival at the RC [14]. This case may be monocoordinate-like, but includes at least one extra step in the energy transfer sequence.
+[FDpp*(t) —F~pj(t)f6p), ~~~(t)
(2a)
=FTpO(t) —(F0+k~~p~*(t)
+ k~~~pp÷1_(t), and 1i~+1_(t) k~~pp*(t) (k_c, + =
(2b)
—
(2c) The additional rate constants in eq. (2) are as defined in fig. 2; ~ is the Kronecker delta. The initial conditions are taken to be pp+i-(O) 0, p,(O) [1 p~~(0)]/N,and 0 pp*(O) 1; N is the number of regular antenna sites per trap. As in ref. [4], one defines a dimensionless Laplace transform by =
=
—
3. Mathematical methods l5,(s)~FAJ exp(—sFAt)pI(t)
In a perfect isotropic lattice the kinetics of hopping excitons is described by the master equation [41, 1i1(t)
=
FA ~ Q11~[p1(t) —p,(t)]
,
(1)
dt.
(3)
o Applying this transform to eq. (2), one obtains (s
+ =
[1 —p~*(0)]/N+ ~
l’~1
Here, I and I’ are dimensionless lattice vectors, p,(t) is the probability at time t that the exciton resides at the site whose lattice vector is I, FA is the rate constant for Förster energy transfer from
1’5~
+F~[F~,5~~(s) —FT,5l(s)]~Io, + k~1),5~*(s) 5o(s) + k_~~i5~+ pp*(O) + FT, 1_(s),
(4a)
(sFA + FD
(4b)
142
R.M. Pearlstein
/ Kinetics of cxciton trapping
and (sEA
+k_~,+k2~,)/5p+,~(s) =k~,p~~(s). (4c)
In eq. (4a), q, = ~ If the transform is applied to eq. (1), then formally ~,
by monocoordinate reaction Centers
gives the mean time for the excitation decay due to exciton trapping (and subsequent steps in the RC). The second equality in eq. (8) follows from an elementary property of Laplace transforms. The prefactor of the integral in eq. (8) does not appear in earlier studies because in those P was considered part of the antenna lattice (i.e. “P” 0 as in fig. 1(a)), whereas here P is considered =
(s +q,)/5,(s)
= p,(O) +
~ (211~j5,~(s).
(5)
l’*l
For the “artificial” condition p,(O) = ~ where I” may~be any issite, the the solution to eq. (5), Green designated called perfect lattice function, which has the following useful properties [4]: (i) ~,g 11~~(s) ~ = 1/s. (ii) g,,~~(s) depends only on s and n Il—I’ ~ ~° may be written ga(s). (iii) For small s, one may expand ga(s) = —(NsY’ —a~+ 0(s), where a,, >0; the abbreviation a a0 is used. In terms of the Green function one has —
=
j5,(s)
=
—
—
Lg,,~(s){p,~~(0)
+ F~’ [ED =
[1
—
/5 p
*
( s)
—
FT/So(s)]
~
—FD/5p*(s)]g,(s),
(6) where properties (I) and (ii) of the Green function have been used to obtain the second equality; here / = I I. Equation (6) for 1 = 0, together with eqs. (4b) and (4c), may be solved simultaneously for /50(s), j5~~(s), and /5~+~-(s). The probability that the antenna remains in the excited state at time t is given by the function ~i(t)
~p,(t), —
[1
pp*(0)I~JcP(t) dt 0
=
Having previously solved for ~
and
pp*(S),
one finds c~(0)= lim~_,~(s)from eq. (9), and thus M0 from eq. (8). (In actually taking this limit, use is made of property (iii) of the Green function.) One may also straightforwardly calculate the exponential lifetime of the longest-lived exciton diffusive mode (“zero mode”). This quantity is given by [4] t0 = —(FA5OY where s~is that pole of ~P(s) closest to the origin in the complex s-plane. The value of s~may be found to a very good approximation again by taking advantage of property (iii) of the Green function: Expand the function g0(s) that appears in the denominator of ~(s), keep terms that are linear in s, and equate the resulting expression to zero. If M0 and t0 differ significantly, it is worth calculating the fractional contribution of the zero mode, given by C0=
[I —p~~(0)}’Res~,.[I~(s)J,
where the residue of ct(s) at the pole s given by Res,ç[~(s)}
=
lim [(s -s0)~P(s)j.
(10) = S0 iS
(11)
(7)
where cP(0) = 1 pp*(O). The zeroth moment of cf with respect to t, M0
=
~,
pp*(0)](Ns)
+F,~[FT,5o(s)
separately. Taking the Laplace transforms of both sides (7), and using eq. (6), one obtains 5(s)of eq. 1 —p~,(O) scl +F~[FDf5I,*(s) —FT/5~)(S)]. (9)
[1 —p~~(0)] F,~*J)(0),
(8)
4. Results of calculations In this section, to simplify the calculations, only the initial condition pp*(O)O is considered. This is often an easily achievable condition experimentally, at least approximately (see seetion 5).
R.M. Pearistein
/ Kinetics of exciton
Using eq. (6) for / = 0, and eqs. (4b) and (4c), one obtains 5FA+kC,+k2C,’ k~~,5~~(s)
=
(12a)
FTh(s)15o(s),
pp*(s)
(12b)
and =
(Ns)~{1
—
(F~/F~)g0(s)
x[1 _FDh(s)]}_i,
(12c)
where h(s)
(sEA +
k_c,
+
k2~,)
With the aid of property (iii) of the Green function (section 3), one finds for small s that +
(FT/FA)(l
+
Nas)
x [1 FDh(s)], (14) and from eq. (9), with the aid of eq. (12b), that —
s~D(s)= 1
(FT/FA),50(s)[FDh(s) —1].
+
(15)
After some algebraic manipulation, the last expression may be written in the form Nau(s)
-
(1
+
+
N(FA/FT)v(s)
Nets) u(s)
+ N( FA/FT)
~ ( s)
5FA( k~,+
k_~~ + k2~,)+ k~,k2~,,
(19)
In eq. (19), use is made of the equality [7] FD/FT = ID/IT’ where ‘D and ‘T are the Förster overlap integrals. The expansion for M0 given by eq. (19) may be compared with that for the isocoordinate-RC model. For the same initial conditions, i.e. p,(O) = 1/N, pp*(O)0, and neglecting terms of order 1/N, one may write eq. (13) of ref. [7] in the form qA)
+(a_q~)F~t~.
5FA(FD + + FD(
of antenna and trap coordination numbers is made explicit. (In the monocoordinate-RC model, to which eq. (19) applies, the trap coordination number, q~,is unity, while q~,upon which a depends implicitly, remains general, and may be 3, 4, or 6 in a simple two-dimensional lattice.) Note that in ref. [7] kr,. is denoted k0, and in eq. (13) of ref. [7] ID/IT is written as ED/FT. Also, in eq. (20) it is assumed that N ‘D/’T>> 1, a condition that usually prevails at least in purple
(17a)
sions term-by-term. The coefficient of FK’ can be much greater in eq. (19) than in eq. (20). For example, if q~= 6, a = 0.0459 ln N + 0.0392; for
k~,+ k_~~ + k2~,)
k_~~ + k2~~) + k~~k2~,;
(17b)
2) in u(s) and u(s) have been neglected 0(s time for excitation decay, from eq. The mean (8), is obtained from =
Na X
+ N( EA/FT)
[i
+ (FD/kCl)(1 +
(20)
In eq. (20), the antenna lattice coordination number is denoted q~(q in ref. [7]), and the equality
the typical value N 25, a = 0.1869, while a = 0.0203, more than 9 times smaller. However, this change, while appearing dramatic, is likely to have little practical effect (see section 5). Much more important is the absence of the factor q~’from the term in Ff’ in eq. (19). This term is thus q~times larger in the current model than in that of ref. [7]. Also of considerable interest are =
u(s)
+
photosynthetic bacteria. With the foregoing provisos, one may now examine the differences between the two expres-
and
~(0)
(ID/IT)k~’(1
+aF~I.
(16)
‘
where u(s)
M0 = N[F.j~ +
=N[(qAFT)~+(ID/IT)k~ (13)
Ns
143
hence,
M0(qT =
x{(sFA+FD+kCt)(5FA+k_C(+k2Ct)
=
trapping by monocoordinate reaction centers
k_~,k~)I,(18)
the terms in ~
—
If k~,~ k
2~1, which holds for systems with open RCs, the two terms are identical. However, if secondary electron transfer is blocked, e.g., for a system in which the RCs are
144
R.M. Pearistein
/ Kinetics of exciton
trapping by monocoordinate reaction centers
in the state PIQA before excitation, one may expect the term in k~ to increase substantially. Although charge recombination is not included explicitly in the isocoordinate-RC model of ref. [7], it has been included in an extension of that model [151 that gives numerical simulations of ~1~(t)with parameters appropriate to green plant photosynthesis. Note that ref. [151does not make use of the analytic “slow back transfer” technique, nor gives numerical estimates of the relative quantum yields of long-time tails (see below). If ~17(t)is monoexponential, M0=Tex, the cxcited-state lifetime as limited by exciton trapping. It has been shown [16] for the isocoordinate-trap (ref. [7]) model that conditions necessary to observe any deviation from such exponentiality may be very difficult to achieve experimentally. However, in that case, the exciton trapping was not followed beyond the state P” in the RC. Here, the situation is more like the case of “slow back transfer” [4], for which it is known that the zeromode (section 3)splits into two modes that are jointly dominant, i.e. the remaining, higher, modes contribute negligibly. In this case, it is necessary to retain terms of order ~2 in the secular equation, which is obtained by equating to zero the denominator of i(s) in eq. (9). In evaluating terms, one takes advantage of the fact that k(>> k_c, and k~,>> k2~,.Noting that terms in the Green function of 3order would only contribute weres retained in the secular if terms ofone order s use the denominator of the equation, may expression in eq. (16) for 1(s), and thus to a very good approximation obtain for the two poles of smallest absolute magnitude, s~= k 2c1 E~ and = —(t~F~)~ where t =N[aF_i + (I II \k_i +F~’l (21~ —
A
\ D/
T/
T
ci
1
‘
‘
Consequently, I(t) may be written in the biexponential form, cP(t)
To evaluate C0, first note that in eqs. (16) and (17), since k_c, and k2,., are negligible compared to kr,, it follows that u(s0) = 0 and ti(s0) = FDk_(~.Therefore, .
C0 =
-
[(S
~
s0)~P(s)
(23)
=N(FA/FT)EDk_,/K,
where K=FAk. +NFA(ID/IT)(k “
+k2. ~‘
)
U
(24)
+N(FA/FT)kC,k2(,.
In eq. (24) a term equal to aNk k has been 2u neglected, since aNk2 <
Ci
Nk_Cf(ID/IT)
C0
=
k~,(1+ Nk2~,F~’) + N(1D/IT)k2C(
(25)
If the terms in k2~,are negligible, this simplifies to C~N(ID/IT)(k_(,/kC~). (26) If, using eqs. (22) et seq., one calculates f~Ji(t)dt, and neglects higher order terms, one obtains cxactly the result given by eq. (19), as expected.
5. Comparison with experiment As noted, the calculations of this paper were inspired by the experiments of Zhang et al. on Rps. uiridis membranes [1]. Because this organism contains no peripheral antenna complex, and no evidence is found for spectral heterogeneity of the core antenna (i.e. no evidence for special long-wavelength components), it is straightforward to analyze their results in terms of the monocoordinate-RC model. In addition, since the r’iridis antenna absorbs at longer wavelengths than
= C 0
exp( —k2~,t)+ (1
—
C0)
exp( —t/t1), (22~ ‘
/
where the zero-mode coefficient C0 is obtained from eqs. (10) and (11) with pp*(O) = 0. Since k2~,~ ~ the first exponential constitutes a long-time tail of cP(t), whose relative de-excitation yield is C0/[C0 + t1k2~,(1 C0)]. —
the RC, Zhang et al. could cleanly excite just the antenna with their laser pulse, i.e. the initial condition pp*(O) = 0 is clearly satisfied. For membranes with open RCs, i.e. all in state ~‘°A before excitation, Zhang et al. measure an exponential lifetime of 60 ps. This may be taken as the value of M0 to be used in eq. (19). The right-hand side of that equation must now be
R.M. Pearlstein
/
Kinetics of exciton trapping by monocoordinate reaction centers
evaluated. As noted, k_~~/k2~1 is negligible for open RCs. Zhang et al. argue, on the basis of anisotropy measurements, that FA> 10i2 s~’. For such a large value the term in F~ is negligible, at least in the isocoordinate RC model, to which eq. (20) applies. For a 6-coordinate (triangular) lattice, assumed applicable to the viridis antenna, the prefactor of F~ in eq. (20) is 0.2 inin the eq. (19). be 0.02, seenbut thatis even latterHowever, case, if itFAwill is indeed that large, its term is still at least an order of magnitude smaller than either of the other two terms, and thus the argument of Zhang et al. remains sound. Some issues regarding the magnitude of FA are considered further below. The value of k~ is lu~iownfrom experiments on isolated RCs to be 2.8 ps [17]. Data to calculate the overlap integral ratio are apparently not yet available, but one may accept the estimate of Zhang et al. that ID/IT 1. (It is certainly not reasonable for ID/IT < 1 in Rps. viridis, with its “uphill” energy transfer from antenna to RC —
[18].) It is known that in viridis there are 24 antenna BChls per RC [12], and therefore one might suppose that N = 24. If that is the case, then with ID/IT = 1, N(ID/IT)k~’= 67 ps. This might be accepted as giving reasonable agreement with the measured lifetime of Zhang et al., provided F.1’ is negligible. However, with N = 24, even Fi1’ = 0.5 Ps 15 not negligible, so that in this alternative one must have FT ~ FA. Unless the relative orientations of nearest-neighboring transition dipoles in the antenna are extremely unfavorable (a very unlikely circumstance), the last inequality cannot hold, and thus one must reject this argument. (Another possibility here is that FT and FA are each >> 10i2 s~,but this is also quite unlikely; see below.) It is proposed here that N = 12 is a more reasonable value. The argument is based on the fact that in any hopping-exciton model N is the number of lattice sites. If, as some now believe [19], the BChls of the core antenna occur as closely coupled pairs resembling the special-pair of the RC, such a pair may reasonably be expected to act as a single entity with regard to hopping transfer from one pair to the next. Under this supposition, each lattice site is occupied
145
by one pair of BChls, thus N = 12. (Note that this argument is in conflict with Knox’s conjecture [6] regarding the insensitivity of hopping exciton kinetics to any bunching of molecules on the lattice only if all pairwise interactions are described by Förster’s formula.) With N = 12 a value of F.j 2 Ps produces good agreement with the 60quite Ps lifetime of Zhang et al.geometry This value 1 is reasonable for the of of theFf monocoordinate-RC model (section 2). The rate constant FA is expressed by the Förster formula (see, e.g., ref. [5]) as FA =
r~( R
(27)
6. 0/R~0)
Here, the radiative lifetime of BCh1 is Trad 20 ns [20]. If each lattice site is occupied by a BCh1 dimer, nearest-neighboring sites are separated by ~ 20 A [19]. (R~~ may be > 20 A, but this would only strengthen the point being made.) With these values, if FA = 2 (ps)~1, then the Förster critical distance R 0 = 117 A, a rather large result [5]. It may be possible to justify on the grounds that it is the absorption of almost two BChls (for in-line, nearly parallel dipoles), not one, that contributes to the overlap integral to which R~ is proportional. Clearly, however, much larger values of FA, or FT, are unreasonable. To justify treating the exponential lifetime measured by Zhang et al. for membranes with open RCs as M0, one must verify that C0 is small. Using the parameter values discussed above, including FT = (2.2 ps) the value established by measurements [211 on isolated RCs of k2~1= (200 ps) ‘, and an estimate (based on this value of k2~,and the near-unit quantum yield of charge separation) of k_~1= (20 ns)~,one finds from eq. (25) that C0 = 0.0013. This means that for open RCs the long-time tail of 1(t) is predicted to have a relative yield of only — 0.4%, and thus would be difficult to observe. However, for pre-reduced RCs (initial RC state PIQ~),the relative yield of the tail may be much larger. In this case, one must reconsider the meaning of the rate constants (with primes to denote those for pre-reduced RCs): One knows from experiment [1] that t = 90 Ps, so that from eq. (21), assuming ~,
146
R.M. Pear/stein
/ Kinetics of exciton trapping
other parameters unchanged, one infers that k,’.~ = (53 p~)i• Also from experiment [211, it is 1,where ~ is known that ~ +~ = (12 ns)~ the sum of rate constants for all decay channels of the state p + I Q~other than the channel that regenerates P * IQ,~.While it has been argued [22] that ~ ~z ~ it has also been noted [15] that if ~ is too small there can be no delayed fluorescence resulting from charge recombination. In the absence of precise information, consider as an example the case ~ ~ (24 nsY’. With these values, eq. (26) gives C0 = 0.00265, which is still very small; but the relative yield of the tail is now [1 + (tk~~,/C0)]~ = 41%. Thus, by the use of sensitive techniques such as are used to observe delayed fluorescence, one may be able to measure the parameters of the tail, and thus obtain information about k_c, and other parameters. In their experiments, which make use of the pump-probe technique, Zhang et al. observe long-time tails in the multi-nanosecond region, but, as they note, these are clearly due to absorbance by p + in the RC. The latter absorbance masks the long-time tails predicted here, which may only be observable in fluorescence. As noted, Zhang et al. observe a 50% increase of t1 for membranes with pre-reduced RCs compared to those with open RCs. The current model is silent regarding their reasonable inference (used above) that this results from a reduced magnitude of kr,, possibly due to the negative charge on QA’
6. Discussion It is interesting that, in the analysis of the results of Zhang et al. [1] by the monocoordinateRC model, one is virtually forced to assign the largest plausible value to each Förster constant. (Note that this condition would not be quite so stringent if the isocoordinate-RC model were applicable.) The interpretation that this represents some sort of optimal evolution on the part of the ancestors of Rps. viridis suggests itself. However, this leaves unexplained the relative sizes of the three terms in the expression for t~,eq. (21). If “optimal evolution” includes equality of the
by monocoordinate reaction centers
terms, so that no one kinetic step is a bottleneck, why is the term in FA so much smaller than the other two, which are nearly equal in the analysis? Another possibility is that the mechanism for the energy homotransfer in the antenna is simply different, e.g., at least partially coherent exciton transport [23]. In fast, fully coherent, transport, one expects [24,25] no term involving purely intra-antenna kinetic factors in the expression for M 0 (or t1). As more details of the antenna structure emerge, it should become possible to model this type of transport reliably, with the aim of finding observable differences between the predictions of the different models. A potentially useful feature of the model as presented here is its prediction of a long-time tail of the antenna excitation, which may be able to account quantitatively for some observations of delayed fluorescence [261. Its ability to predict relative quantum yields of prompt and delayed fluorescence may be especially significant. With some work, the theory can be extended to include any number of subsequent electron transport steps (from which back transfer is possible), thereby allowing the calculation of extremely long-time tails. Other extensions of the theory as presented here may be of value. For example, it is straightforward to generalize the calculations to include any value of pp*(O) such that 0 pp*(O) 1. This has the potential to provide more information from the analysis of experiments in which excitation wavelength can be varied. The basic premise of the monocoordinate-RC model, that there is a fast energy-transfer pathway to P from only one antenna site, ultimately may require an atomic-resolution structure of coupled antenna and RC (or at least of the antenna) for its verification. In the meantime, it is possible to test the assumptions specifically of the simplified model (fig. 1(c)), used in the calculations described here, by including the lattice “vacancies” created by the RC in either numerical or analytical calculations. However, such an effort may not be worthwhile, because including the vacancies is likely only to affect the term in EA (in M~1or t1). Unless that term is increased significantly, which seems unlikely, the arguments
R.M Pearistein
/ Kinetics of exciton trapping by monocoordinate reaction centers
of section 5 suggest there would be no observable consequence of including the vacancies.
Acknowledgement I am grateful to F.G. Zhang, T. Gillbro, R. van Grondelle, and V. Sundström for approving my use of their manuscript, which I originally received as one of its referees, prior to its publication.
References [1] F.G. Zhang, T. Gillbro, R. van Grondelle and V. Sundstrom, Biophys. J., in press. [2] Z. Bay and R.M. Pearlstein, Proc. NatI. Acad. Sci. USA 50 (1963) 1071. [3] R.M. Pearlstein, Brookhaven Symposia in Biology 19 (1966) 8. [4] R.P. Hemenger, R.M. Pearlstein and K. Lakatos-Lindenberg, J. Math. Phys. 13 (1972) 1056. [5] R.S. Knox, in: Bioenergetics of Photosynthesis, ed. Govindjee (Academic Press, New York, 1975) p. 183. [6] R.S. Knox, in: Primary Processes of Photosynthesis, ed. J. Barber (Elsevier/North-Holland Biomedical Press, New York, 1979) p. 55. [7] R.M. Pearlstein, Photochem. Photobiol. 35 (1982) 835. [8] R.M. Pearistein, in: Photosynthesis: Energy Conversion by Plants and Bacteria, ed. Govindjee (Academic Press, New York, 1982) p. 293. [9] R. van Grondelle, Biochim. Biophys. Acta 811 (1985) 147.
147
[10] R. van Grondelle and J. Amesz, in: Light Emission by Plants and Bacteria, eds. Govindjee, J. Amesz and D.C. Fork (Academic Press, New York, 1986) p. 191. [11] T.G. Owens, S.P. Webb, L. Mets, R.S. Alberte and G.R. Fleming, Proc. NatI. Acad. Sci. USA 84 (1987) 1532. [12] H. Zuber and R.A. Brunisholz, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 627. [13] P. Joliot, A. Vermeglio and A. Joliot, Biochemistry 29 [14] V. Sundström and R. van Grondelle, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 1097. [15] B. Källebring and O. Hansson, Chem. Phys. 149 (1991) 361. [16] R.M. Pearlstein, in: Advances in Photosynthesis Research, Vol. I, Part 1, ed. C. Sybesma (Martinus Nijhoff/Dr. W. Junk Publishers, The Hague, 1984) p. 13. [17] J. Breton, J.-L. Martin, A. Migus, H. Antonetti and A. Orszag, Proc. NatI. Acad. Sci. USA 83 (1986) 5121. [18] K.L. Zankel and R.K. Clayton, Photochem. Photobiol. 9 (1969) 7. [19] R.M. Pearlstein, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 1047. 1201 K.L. Zankel, D.W. Reed and R.K. Clayton, Proc. NatI. Acad. Sci. USA 61(1968) p. 1243. [21] W.W. Parson and B. Ke, in: Photosynthesis: Energy Conversion by Plants and Bacteria, ed. Govindjee (Academic Press, New York, 1982) p. 331. [22] C. Kirmaier and D. Holten, Photosynth. Res. 13 (1987) 225.
[23] R. Silbey, Ann. Rev. Phys. Chem. 27 (1976) 203. [24] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431. [25] R.P. Hemenger and R.M. Pearlstein, Chem. Phys. 2 (1973) 424. [26] P. Jursinic, in: Light Emission by Plants and Bacteria, eds. Govindjee, J. Amesz and D.C. Fork (Academic Press, New York, 1986) p. 291.
Journal of Luminescence 51(1992)139—147
JOURNAL OF
Kinetics of exciton trapping by monocoordinate reaction centers Robert M. Pearistein Physics Department, Indiana-Purdue University, Indianapolis, IN 46205, USA
A model of exciton migration and trapping in photosynthetic membranes is presented that differs from earlier models of hopping excitons on lattices. In the new model, the trap, which is the special pair of chlorophyll (ChI) molecules within the reaction center (RC), is assumed monocoordinate, i.e. to have but one nearest-neighboring antenna lattice site; in earlier models, the trap is assumed isocoordinate with the antenna sites. In this monocoordinate-RC model, which it is argued is more realistic at least for some of the photosynthetic bacteria, antenna ChI excitation energy decays more slowly than in the isocoordinate-RC models, for equal values of parameters. The principal exponential decay lifetime is predicted to increase typically by a factor of 2—3. However, in the analysis of an experiment on the picosecond kinetics of excitation decay in Rps. viridis [1], where the exponential lifetime is fixed by the measurement, one is forced to conclude the opposite, i.e. unknown Förster parameters must be larger than previously thought, in fact approaching theoretical maximum values. As a side benefit of the relatively simpler calculation for the monocoordinate-RC model, coupling of the exciton trapping to the first two steps of electron transport in the RC has been readily included in the equations. This yields a hi-exponential decay in which the second exponential component, describable as a long-time tail of small amplitude (—. 0.2%), can nonetheless be responsible for a large fraction of the total quantum yield, at least in a system with pre-reduced RCs. The utility of this result for the analysis of delayed fluorescence experiments is discussed.
1. Introduction The shortened fluorescence lifetime of chlorophyll (Chi) in vivo was first explained theoretically to be a consequence of the migration of hopping excitons into traps almost three decades ago [2,31. Since then, the theory has been extended and reviewed a number of times [4—101.It has been applied to the analysis of experiments on energy transfer kinetics in both green plants and photosynthetic bacteria [1,11]. Recent arguments [12] regarding the relative placement of reaction centers (RCs) and antenna protein complexes in photosynthetic membranes suggest that one of the basic tenets of the theory may not hold, at least in some of the photosynthetic bacteria. This principle is that, in a lattice model of the arrangement of the Chls in the membrane, every lattice site including that of the trap has the same number of nearest neighbors [41.The purpose of this paper is to develop a simple model to treat another situation, hopefully more realistic, in which the trap has but one neighbor while, in general, other sites have many. 0022-2313/92/$05.00 © 1992
—
The model is described and justified in section 2. Section 3 provides details of the mathematical methods used in the calculations, the results of which are presented in section 4. In section 5, the results are used to analyze the very nice kinetic data on membranes from Rhodopseudomonas viridis obtained by Zhang et al. [11,whose paper provided the major stimulus for the work presented here. The last section considers puzzling points, mentions possible generalizations, and emphasizes certain predictive features of the theory, especially in the analysis of prompt versus delayed fluorescence yields.
2. Description of model A standard lattice model of exciton migration and trapping in a photosynthetic membrane is shown schematically in fig. 1(a). The trap, designated by the circle with the “T”, is isocoordinate with the antenna sites, designated by the open circles. While still schematic, fig. 1(b) gives a
Elsevier Science Publishers B.V. All rights reserved
140
R.M. Pearistein
~
ffi
/ Kinetics of exciton trapping by monocoordinate reaction centers
~
(b)
Fig. 1. Lattice models of exciton migration and trapping in a photosynthetic membrane. (a) “Standard” or isocoordinate trap model (b) Monocoordinate trap model that includes vacancies in the antenna lattice due to the presence of the reaction center. (c) Simplified monocoordinate trap model used for calculations; see text for details.
somewhat more realistic view of the lattice geometry in the vicinity of a reaction center (RC). The RC not only provides a trap, the special pair of bacteriochlorophyll (BChl) molecules denoted “P”, but also creates a large set of connected vacancies in the otherwise regular lattice of antenna BChls. The result is that the trap cannot be simultaneously isocoordinate and kinetically well-connected to the antenna sites, because an isocoordinate position would place the trap quite far from any of those sites. For example, if the antenna lattice constant is 20 A and the trap is isocoordinate but 30 A from each of its nearestneighboring antenna sites, the relative Förster rate constant for transfer from an antenna neighbor to the trap is (30/20)6 = 11 times smaller than between nearest antenna neighbors; this does not take into account a possibly less favorable spectral overlap for the antenna-trap transfer, which would further diminish that rate constant.
If the trap is placed at a distance of one lattice constant from a single antenna site, so that it is monocoordinate as in fig. 1(b), the rate constant for energy transfer from nearest-neighboring lattice sites to the trap is diminished by a factor equal to the lattice coordination number (number of nearest neighbors of each site in a perfect lattice of the same type) [2,41.For a two-dimensional lattice that number is at most 6, so that in the foregoing example there is still a net gain of nearly a factor of two in that energy-transfer rate constant. It is therefore assumed here that the trap is strongly coupled kinetically to a single antenna lattice site. It may be noted from fig. 1(b) that the presence of the large RC also reduces the coordination of those antenna sites near its perimeter. However, in general, the reduction is not so large, i.e. none of the antenna sites becomes monocoordinate. Moreover, arguments based on anisotropic exciton hopping theory [4] suggest that as long as the intra-antenna transfer is not restricted to one-dimensional domains, the kinetics of that transfer differs little from that on a perfect lattice. For simplicity, the vacancies in the antenna lattice are ignored here. Figure 1(c) illustrates the monocoordinate-RC model of this paper. The two-dimensional antenna lattice is perfect. The monocoordinate trap is “nondisruptive” [4]; the dashed line in fig. 1(c) that connects the trap to its sole nearest-neighboring antenna site may be interpreted as passing through the third dimension. That the kinetic connection to the trap may be accurately represented in this unphysical way is another assumption of the model. Figure 2 contrasts the kinetic relationships of the two models. In each case, the box distinguishes kinetic processes that occur after the exciton arrives at the “central” site the trap itself in the isocoordinate-RC model, the sole nearest-neighbor of the trap in the monocoordinate-RC model. Other than as described above, standard lattice-model assumptions are made [4]. This ineludes the assumption that the RCs occur periodically within the overall lattice. However, as has been shown, this is not a terribly significant re—
R.M. Pearistein
/ Kinetics of exciton trapping by monocoordinate reaction
centers
141
one lattice site to any one of its nearest neighbors, and i dp/dt. The quantity [211, is the rate constant, in units of FA, for the transfer from site I to site 1’. The presence of lattice imperfections, such as a trap, requires either extra terms to be added to the right side of eq. (1), or separate equations to be written for those lattice sites that are either themselves imperfections or that transfer energy directly to an imperfect site. For the simplified monocoordinate-RC model (section 2) it is only
(a)
=
PC (b)
A
0~PC
necessary to consider a single imperfect site and a single regular site that transfers energy to it. The
Fig. 2. Kinetic relationships in the lattice models. (a) Isocoordinate trap. (b) Simplified monocoordinate trap. Each “F” is a Fbrster rate constant, each “k” a rate constant for electron transfer in the reaction center. States of reaction center components are indicated in the usual way. In (b), A~3denotes the nearest-neighbor antenna site (to Pin the reaction0A’ center) the primary in its excited state, A0 in its quinone. ground state; “Q” is
latter is designated by I 0, and the former given the subscript ~ Additional equations can be written for states of the RC kinetically connected to P”. Here, only two such additional states, P~IQ and P~IQ~,are considered. model, be Thus, writtenforinthis three parts:the master equation can
striction. In any case, it probably holds, at least
,i,(t)
=
‘FA
~ [2
—p,(t)}
11~[p,~(t)
approximately, in most photosynthetic bacteria (if not green plants) [12]. If, as argued by some for certain organisms [13], the RCs are paired at each RC-site, the trap (P) of one RC is almost certainly physically far enough from that of the other so as not to modify the trapping kinetics, Note that neither this nor the isocoordinate-RC models applies if the core antenna (surrounding the RC) contains special long-wavelength components that pre-trap the excitation prior to its arrival at the RC [14]. This case may be monocoordinate-like, but includes at least one extra step in the energy transfer sequence.
+[FDpp*(t) —F~pj(t)f6p), ~~~(t)
(2a)
=FTpO(t) —(F0+k~~p~*(t)
+ k~~~pp÷1_(t), and 1i~+1_(t) k~~pp*(t) (k_c, + =
(2b)
—
(2c) The additional rate constants in eq. (2) are as defined in fig. 2; ~ is the Kronecker delta. The initial conditions are taken to be pp+i-(O) 0, p,(O) [1 p~~(0)]/N,and 0 pp*(O) 1; N is the number of regular antenna sites per trap. As in ref. [4], one defines a dimensionless Laplace transform by =
=
—
3. Mathematical methods l5,(s)~FAJ exp(—sFAt)pI(t)
In a perfect isotropic lattice the kinetics of hopping excitons is described by the master equation [41, 1i1(t)
=
FA ~ Q11~[p1(t) —p,(t)]
,
(1)
dt.
(3)
o Applying this transform to eq. (2), one obtains (s
+ =
[1 —p~*(0)]/N+ ~
l’~1
Here, I and I’ are dimensionless lattice vectors, p,(t) is the probability at time t that the exciton resides at the site whose lattice vector is I, FA is the rate constant for Förster energy transfer from
1’5~
+F~[F~,5~~(s) —FT,5l(s)]~Io, + k~1),5~*(s) 5o(s) + k_~~i5~+ pp*(O) + FT, 1_(s),
(4a)
(sFA + FD
(4b)
142
R.M. Pearlstein
/ Kinetics of cxciton trapping
and (sEA
+k_~,+k2~,)/5p+,~(s) =k~,p~~(s). (4c)
In eq. (4a), q, = ~ If the transform is applied to eq. (1), then formally ~,
by monocoordinate reaction Centers
gives the mean time for the excitation decay due to exciton trapping (and subsequent steps in the RC). The second equality in eq. (8) follows from an elementary property of Laplace transforms. The prefactor of the integral in eq. (8) does not appear in earlier studies because in those P was considered part of the antenna lattice (i.e. “P” 0 as in fig. 1(a)), whereas here P is considered =
(s +q,)/5,(s)
= p,(O) +
~ (211~j5,~(s).
(5)
l’*l
For the “artificial” condition p,(O) = ~ where I” may~be any issite, the the solution to eq. (5), Green designated called perfect lattice function, which has the following useful properties [4]: (i) ~,g 11~~(s) ~ = 1/s. (ii) g,,~~(s) depends only on s and n Il—I’ ~ ~° may be written ga(s). (iii) For small s, one may expand ga(s) = —(NsY’ —a~+ 0(s), where a,, >0; the abbreviation a a0 is used. In terms of the Green function one has —
=
j5,(s)
=
—
—
Lg,,~(s){p,~~(0)
+ F~’ [ED =
[1
—
/5 p
*
( s)
—
FT/So(s)]
~
—FD/5p*(s)]g,(s),
(6) where properties (I) and (ii) of the Green function have been used to obtain the second equality; here / = I I. Equation (6) for 1 = 0, together with eqs. (4b) and (4c), may be solved simultaneously for /50(s), j5~~(s), and /5~+~-(s). The probability that the antenna remains in the excited state at time t is given by the function ~i(t)
~p,(t), —
[1
pp*(0)I~JcP(t) dt 0
=
Having previously solved for ~
and
pp*(S),
one finds c~(0)= lim~_,~(s)from eq. (9), and thus M0 from eq. (8). (In actually taking this limit, use is made of property (iii) of the Green function.) One may also straightforwardly calculate the exponential lifetime of the longest-lived exciton diffusive mode (“zero mode”). This quantity is given by [4] t0 = —(FA5OY where s~is that pole of ~P(s) closest to the origin in the complex s-plane. The value of s~may be found to a very good approximation again by taking advantage of property (iii) of the Green function: Expand the function g0(s) that appears in the denominator of ~(s), keep terms that are linear in s, and equate the resulting expression to zero. If M0 and t0 differ significantly, it is worth calculating the fractional contribution of the zero mode, given by C0=
[I —p~~(0)}’Res~,.[I~(s)J,
where the residue of ct(s) at the pole s given by Res,ç[~(s)}
=
lim [(s -s0)~P(s)j.
(10) = S0 iS
(11)
(7)
where cP(0) = 1 pp*(O). The zeroth moment of cf with respect to t, M0
=
~,
pp*(0)](Ns)
+F,~[FT,5o(s)
separately. Taking the Laplace transforms of both sides (7), and using eq. (6), one obtains 5(s)of eq. 1 —p~,(O) scl +F~[FDf5I,*(s) —FT/5~)(S)]. (9)
[1 —p~~(0)] F,~*J)(0),
(8)
4. Results of calculations In this section, to simplify the calculations, only the initial condition pp*(O)O is considered. This is often an easily achievable condition experimentally, at least approximately (see seetion 5).
R.M. Pearistein
/ Kinetics of exciton
Using eq. (6) for / = 0, and eqs. (4b) and (4c), one obtains 5FA+kC,+k2C,’ k~~,5~~(s)
=
(12a)
FTh(s)15o(s),
pp*(s)
(12b)
and =
(Ns)~{1
—
(F~/F~)g0(s)
x[1 _FDh(s)]}_i,
(12c)
where h(s)
(sEA +
k_c,
+
k2~,)
With the aid of property (iii) of the Green function (section 3), one finds for small s that +
(FT/FA)(l
+
Nas)
x [1 FDh(s)], (14) and from eq. (9), with the aid of eq. (12b), that —
s~D(s)= 1
(FT/FA),50(s)[FDh(s) —1].
+
(15)
After some algebraic manipulation, the last expression may be written in the form Nau(s)
-
(1
+
+
N(FA/FT)v(s)
Nets) u(s)
+ N( FA/FT)
~ ( s)
5FA( k~,+
k_~~ + k2~,)+ k~,k2~,,
(19)
In eq. (19), use is made of the equality [7] FD/FT = ID/IT’ where ‘D and ‘T are the Förster overlap integrals. The expansion for M0 given by eq. (19) may be compared with that for the isocoordinate-RC model. For the same initial conditions, i.e. p,(O) = 1/N, pp*(O)0, and neglecting terms of order 1/N, one may write eq. (13) of ref. [7] in the form qA)
+(a_q~)F~t~.
5FA(FD + + FD(
of antenna and trap coordination numbers is made explicit. (In the monocoordinate-RC model, to which eq. (19) applies, the trap coordination number, q~,is unity, while q~,upon which a depends implicitly, remains general, and may be 3, 4, or 6 in a simple two-dimensional lattice.) Note that in ref. [7] kr,. is denoted k0, and in eq. (13) of ref. [7] ID/IT is written as ED/FT. Also, in eq. (20) it is assumed that N ‘D/’T>> 1, a condition that usually prevails at least in purple
(17a)
sions term-by-term. The coefficient of FK’ can be much greater in eq. (19) than in eq. (20). For example, if q~= 6, a = 0.0459 ln N + 0.0392; for
k~,+ k_~~ + k2~,)
k_~~ + k2~~) + k~~k2~,;
(17b)
2) in u(s) and u(s) have been neglected 0(s time for excitation decay, from eq. The mean (8), is obtained from =
Na X
+ N( EA/FT)
[i
+ (FD/kCl)(1 +
(20)
In eq. (20), the antenna lattice coordination number is denoted q~(q in ref. [7]), and the equality
the typical value N 25, a = 0.1869, while a = 0.0203, more than 9 times smaller. However, this change, while appearing dramatic, is likely to have little practical effect (see section 5). Much more important is the absence of the factor q~’from the term in Ff’ in eq. (19). This term is thus q~times larger in the current model than in that of ref. [7]. Also of considerable interest are =
u(s)
+
photosynthetic bacteria. With the foregoing provisos, one may now examine the differences between the two expres-
and
~(0)
(ID/IT)k~’(1
+aF~I.
(16)
‘
where u(s)
M0 = N[F.j~ +
=N[(qAFT)~+(ID/IT)k~ (13)
Ns
143
hence,
M0(qT =
x{(sFA+FD+kCt)(5FA+k_C(+k2Ct)
=
trapping by monocoordinate reaction centers
k_~,k~)I,(18)
the terms in ~
—
If k~,~ k
2~1, which holds for systems with open RCs, the two terms are identical. However, if secondary electron transfer is blocked, e.g., for a system in which the RCs are
144
R.M. Pearistein
/ Kinetics of exciton
trapping by monocoordinate reaction centers
in the state PIQA before excitation, one may expect the term in k~ to increase substantially. Although charge recombination is not included explicitly in the isocoordinate-RC model of ref. [7], it has been included in an extension of that model [151 that gives numerical simulations of ~1~(t)with parameters appropriate to green plant photosynthesis. Note that ref. [151does not make use of the analytic “slow back transfer” technique, nor gives numerical estimates of the relative quantum yields of long-time tails (see below). If ~17(t)is monoexponential, M0=Tex, the cxcited-state lifetime as limited by exciton trapping. It has been shown [16] for the isocoordinate-trap (ref. [7]) model that conditions necessary to observe any deviation from such exponentiality may be very difficult to achieve experimentally. However, in that case, the exciton trapping was not followed beyond the state P” in the RC. Here, the situation is more like the case of “slow back transfer” [4], for which it is known that the zeromode (section 3)splits into two modes that are jointly dominant, i.e. the remaining, higher, modes contribute negligibly. In this case, it is necessary to retain terms of order ~2 in the secular equation, which is obtained by equating to zero the denominator of i(s) in eq. (9). In evaluating terms, one takes advantage of the fact that k(>> k_c, and k~,>> k2~,.Noting that terms in the Green function of 3order would only contribute weres retained in the secular if terms ofone order s use the denominator of the equation, may expression in eq. (16) for 1(s), and thus to a very good approximation obtain for the two poles of smallest absolute magnitude, s~= k 2c1 E~ and = —(t~F~)~ where t =N[aF_i + (I II \k_i +F~’l (21~ —
A
\ D/
T/
T
ci
1
‘
‘
Consequently, I(t) may be written in the biexponential form, cP(t)
To evaluate C0, first note that in eqs. (16) and (17), since k_c, and k2,., are negligible compared to kr,, it follows that u(s0) = 0 and ti(s0) = FDk_(~.Therefore, .
C0 =
-
[(S
~
s0)~P(s)
(23)
=N(FA/FT)EDk_,/K,
where K=FAk. +NFA(ID/IT)(k “
+k2. ~‘
)
U
(24)
+N(FA/FT)kC,k2(,.
In eq. (24) a term equal to aNk k has been 2u neglected, since aNk2 <
Ci
Nk_Cf(ID/IT)
C0
=
k~,(1+ Nk2~,F~’) + N(1D/IT)k2C(
(25)
If the terms in k2~,are negligible, this simplifies to C~N(ID/IT)(k_(,/kC~). (26) If, using eqs. (22) et seq., one calculates f~Ji(t)dt, and neglects higher order terms, one obtains cxactly the result given by eq. (19), as expected.
5. Comparison with experiment As noted, the calculations of this paper were inspired by the experiments of Zhang et al. on Rps. uiridis membranes [1]. Because this organism contains no peripheral antenna complex, and no evidence is found for spectral heterogeneity of the core antenna (i.e. no evidence for special long-wavelength components), it is straightforward to analyze their results in terms of the monocoordinate-RC model. In addition, since the r’iridis antenna absorbs at longer wavelengths than
= C 0
exp( —k2~,t)+ (1
—
C0)
exp( —t/t1), (22~ ‘
/
where the zero-mode coefficient C0 is obtained from eqs. (10) and (11) with pp*(O) = 0. Since k2~,~ ~ the first exponential constitutes a long-time tail of cP(t), whose relative de-excitation yield is C0/[C0 + t1k2~,(1 C0)]. —
the RC, Zhang et al. could cleanly excite just the antenna with their laser pulse, i.e. the initial condition pp*(O) = 0 is clearly satisfied. For membranes with open RCs, i.e. all in state ~‘°A before excitation, Zhang et al. measure an exponential lifetime of 60 ps. This may be taken as the value of M0 to be used in eq. (19). The right-hand side of that equation must now be
R.M. Pearlstein
/
Kinetics of exciton trapping by monocoordinate reaction centers
evaluated. As noted, k_~~/k2~1 is negligible for open RCs. Zhang et al. argue, on the basis of anisotropy measurements, that FA> 10i2 s~’. For such a large value the term in F~ is negligible, at least in the isocoordinate RC model, to which eq. (20) applies. For a 6-coordinate (triangular) lattice, assumed applicable to the viridis antenna, the prefactor of F~ in eq. (20) is 0.2 inin the eq. (19). be 0.02, seenbut thatis even latterHowever, case, if itFAwill is indeed that large, its term is still at least an order of magnitude smaller than either of the other two terms, and thus the argument of Zhang et al. remains sound. Some issues regarding the magnitude of FA are considered further below. The value of k~ is lu~iownfrom experiments on isolated RCs to be 2.8 ps [17]. Data to calculate the overlap integral ratio are apparently not yet available, but one may accept the estimate of Zhang et al. that ID/IT 1. (It is certainly not reasonable for ID/IT < 1 in Rps. viridis, with its “uphill” energy transfer from antenna to RC —
[18].) It is known that in viridis there are 24 antenna BChls per RC [12], and therefore one might suppose that N = 24. If that is the case, then with ID/IT = 1, N(ID/IT)k~’= 67 ps. This might be accepted as giving reasonable agreement with the measured lifetime of Zhang et al., provided F.1’ is negligible. However, with N = 24, even Fi1’ = 0.5 Ps 15 not negligible, so that in this alternative one must have FT ~ FA. Unless the relative orientations of nearest-neighboring transition dipoles in the antenna are extremely unfavorable (a very unlikely circumstance), the last inequality cannot hold, and thus one must reject this argument. (Another possibility here is that FT and FA are each >> 10i2 s~,but this is also quite unlikely; see below.) It is proposed here that N = 12 is a more reasonable value. The argument is based on the fact that in any hopping-exciton model N is the number of lattice sites. If, as some now believe [19], the BChls of the core antenna occur as closely coupled pairs resembling the special-pair of the RC, such a pair may reasonably be expected to act as a single entity with regard to hopping transfer from one pair to the next. Under this supposition, each lattice site is occupied
145
by one pair of BChls, thus N = 12. (Note that this argument is in conflict with Knox’s conjecture [6] regarding the insensitivity of hopping exciton kinetics to any bunching of molecules on the lattice only if all pairwise interactions are described by Förster’s formula.) With N = 12 a value of F.j 2 Ps produces good agreement with the 60quite Ps lifetime of Zhang et al.geometry This value 1 is reasonable for the of of theFf monocoordinate-RC model (section 2). The rate constant FA is expressed by the Förster formula (see, e.g., ref. [5]) as FA =
r~( R
(27)
6. 0/R~0)
Here, the radiative lifetime of BCh1 is Trad 20 ns [20]. If each lattice site is occupied by a BCh1 dimer, nearest-neighboring sites are separated by ~ 20 A [19]. (R~~ may be > 20 A, but this would only strengthen the point being made.) With these values, if FA = 2 (ps)~1, then the Förster critical distance R 0 = 117 A, a rather large result [5]. It may be possible to justify on the grounds that it is the absorption of almost two BChls (for in-line, nearly parallel dipoles), not one, that contributes to the overlap integral to which R~ is proportional. Clearly, however, much larger values of FA, or FT, are unreasonable. To justify treating the exponential lifetime measured by Zhang et al. for membranes with open RCs as M0, one must verify that C0 is small. Using the parameter values discussed above, including FT = (2.2 ps) the value established by measurements [211 on isolated RCs of k2~1= (200 ps) ‘, and an estimate (based on this value of k2~,and the near-unit quantum yield of charge separation) of k_~1= (20 ns)~,one finds from eq. (25) that C0 = 0.0013. This means that for open RCs the long-time tail of 1(t) is predicted to have a relative yield of only — 0.4%, and thus would be difficult to observe. However, for pre-reduced RCs (initial RC state PIQ~),the relative yield of the tail may be much larger. In this case, one must reconsider the meaning of the rate constants (with primes to denote those for pre-reduced RCs): One knows from experiment [1] that t = 90 Ps, so that from eq. (21), assuming ~,
146
R.M. Pear/stein
/ Kinetics of exciton trapping
other parameters unchanged, one infers that k,’.~ = (53 p~)i• Also from experiment [211, it is 1,where ~ is known that ~ +~ = (12 ns)~ the sum of rate constants for all decay channels of the state p + I Q~other than the channel that regenerates P * IQ,~.While it has been argued [22] that ~ ~z ~ it has also been noted [15] that if ~ is too small there can be no delayed fluorescence resulting from charge recombination. In the absence of precise information, consider as an example the case ~ ~ (24 nsY’. With these values, eq. (26) gives C0 = 0.00265, which is still very small; but the relative yield of the tail is now [1 + (tk~~,/C0)]~ = 41%. Thus, by the use of sensitive techniques such as are used to observe delayed fluorescence, one may be able to measure the parameters of the tail, and thus obtain information about k_c, and other parameters. In their experiments, which make use of the pump-probe technique, Zhang et al. observe long-time tails in the multi-nanosecond region, but, as they note, these are clearly due to absorbance by p + in the RC. The latter absorbance masks the long-time tails predicted here, which may only be observable in fluorescence. As noted, Zhang et al. observe a 50% increase of t1 for membranes with pre-reduced RCs compared to those with open RCs. The current model is silent regarding their reasonable inference (used above) that this results from a reduced magnitude of kr,, possibly due to the negative charge on QA’
6. Discussion It is interesting that, in the analysis of the results of Zhang et al. [1] by the monocoordinateRC model, one is virtually forced to assign the largest plausible value to each Förster constant. (Note that this condition would not be quite so stringent if the isocoordinate-RC model were applicable.) The interpretation that this represents some sort of optimal evolution on the part of the ancestors of Rps. viridis suggests itself. However, this leaves unexplained the relative sizes of the three terms in the expression for t~,eq. (21). If “optimal evolution” includes equality of the
by monocoordinate reaction centers
terms, so that no one kinetic step is a bottleneck, why is the term in FA so much smaller than the other two, which are nearly equal in the analysis? Another possibility is that the mechanism for the energy homotransfer in the antenna is simply different, e.g., at least partially coherent exciton transport [23]. In fast, fully coherent, transport, one expects [24,25] no term involving purely intra-antenna kinetic factors in the expression for M 0 (or t1). As more details of the antenna structure emerge, it should become possible to model this type of transport reliably, with the aim of finding observable differences between the predictions of the different models. A potentially useful feature of the model as presented here is its prediction of a long-time tail of the antenna excitation, which may be able to account quantitatively for some observations of delayed fluorescence [261. Its ability to predict relative quantum yields of prompt and delayed fluorescence may be especially significant. With some work, the theory can be extended to include any number of subsequent electron transport steps (from which back transfer is possible), thereby allowing the calculation of extremely long-time tails. Other extensions of the theory as presented here may be of value. For example, it is straightforward to generalize the calculations to include any value of pp*(O) such that 0 pp*(O) 1. This has the potential to provide more information from the analysis of experiments in which excitation wavelength can be varied. The basic premise of the monocoordinate-RC model, that there is a fast energy-transfer pathway to P from only one antenna site, ultimately may require an atomic-resolution structure of coupled antenna and RC (or at least of the antenna) for its verification. In the meantime, it is possible to test the assumptions specifically of the simplified model (fig. 1(c)), used in the calculations described here, by including the lattice “vacancies” created by the RC in either numerical or analytical calculations. However, such an effort may not be worthwhile, because including the vacancies is likely only to affect the term in EA (in M~1or t1). Unless that term is increased significantly, which seems unlikely, the arguments
R.M Pearistein
/ Kinetics of exciton trapping by monocoordinate reaction centers
of section 5 suggest there would be no observable consequence of including the vacancies.
Acknowledgement I am grateful to F.G. Zhang, T. Gillbro, R. van Grondelle, and V. Sundström for approving my use of their manuscript, which I originally received as one of its referees, prior to its publication.
References [1] F.G. Zhang, T. Gillbro, R. van Grondelle and V. Sundstrom, Biophys. J., in press. [2] Z. Bay and R.M. Pearlstein, Proc. NatI. Acad. Sci. USA 50 (1963) 1071. [3] R.M. Pearlstein, Brookhaven Symposia in Biology 19 (1966) 8. [4] R.P. Hemenger, R.M. Pearlstein and K. Lakatos-Lindenberg, J. Math. Phys. 13 (1972) 1056. [5] R.S. Knox, in: Bioenergetics of Photosynthesis, ed. Govindjee (Academic Press, New York, 1975) p. 183. [6] R.S. Knox, in: Primary Processes of Photosynthesis, ed. J. Barber (Elsevier/North-Holland Biomedical Press, New York, 1979) p. 55. [7] R.M. Pearlstein, Photochem. Photobiol. 35 (1982) 835. [8] R.M. Pearistein, in: Photosynthesis: Energy Conversion by Plants and Bacteria, ed. Govindjee (Academic Press, New York, 1982) p. 293. [9] R. van Grondelle, Biochim. Biophys. Acta 811 (1985) 147.
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[10] R. van Grondelle and J. Amesz, in: Light Emission by Plants and Bacteria, eds. Govindjee, J. Amesz and D.C. Fork (Academic Press, New York, 1986) p. 191. [11] T.G. Owens, S.P. Webb, L. Mets, R.S. Alberte and G.R. Fleming, Proc. NatI. Acad. Sci. USA 84 (1987) 1532. [12] H. Zuber and R.A. Brunisholz, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 627. [13] P. Joliot, A. Vermeglio and A. Joliot, Biochemistry 29 [14] V. Sundström and R. van Grondelle, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 1097. [15] B. Källebring and O. Hansson, Chem. Phys. 149 (1991) 361. [16] R.M. Pearlstein, in: Advances in Photosynthesis Research, Vol. I, Part 1, ed. C. Sybesma (Martinus Nijhoff/Dr. W. Junk Publishers, The Hague, 1984) p. 13. [17] J. Breton, J.-L. Martin, A. Migus, H. Antonetti and A. Orszag, Proc. NatI. Acad. Sci. USA 83 (1986) 5121. [18] K.L. Zankel and R.K. Clayton, Photochem. Photobiol. 9 (1969) 7. [19] R.M. Pearlstein, in: Chlorophylls, ed. H. Scheer (CRC Press, Boca Raton, 1991) p. 1047. 1201 K.L. Zankel, D.W. Reed and R.K. Clayton, Proc. NatI. Acad. Sci. USA 61(1968) p. 1243. [21] W.W. Parson and B. Ke, in: Photosynthesis: Energy Conversion by Plants and Bacteria, ed. Govindjee (Academic Press, New York, 1982) p. 331. [22] C. Kirmaier and D. Holten, Photosynth. Res. 13 (1987) 225.
[23] R. Silbey, Ann. Rev. Phys. Chem. 27 (1976) 203. [24] R.M. Pearlstein, J. Chem. Phys. 56 (1972) 2431. [25] R.P. Hemenger and R.M. Pearlstein, Chem. Phys. 2 (1973) 424. [26] P. Jursinic, in: Light Emission by Plants and Bacteria, eds. Govindjee, J. Amesz and D.C. Fork (Academic Press, New York, 1986) p. 291.