ARTICLE pubs.acs.org/JPCB
Excitation Migration, Quenching, and Regulation of Photosynthetic Light Harvesting in Photosystem II Leonas Valkunas,*,†,‡ Jevgenij Chmeliov,†,‡ Gediminas Trinkunas,†,‡ Christopher D. P. Duffy,§ Rienk van Grondelle,|| and Alexander V. Ruban§ †
Institute of Physics, Center for Physical Sciences and Technology, Savanoriu Ave 231, LT-02300 Vilnius, Lithuania Department of Theoretical Physics, Faculty of Physics, Vilnius University, Sauletekio Ave 9, build. 3, LT-10222 Vilnius, Lithuania § School of Biological and Chemical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom Department of Physics and Astronomy, Faculty of Sciences, VU Universiteit Amsterdam, De Boelelaan 1081, NL-1081 HV Amsterdam, The Netherlands
)
‡
ABSTRACT: Excitation energy transfer and quenching in LHCII aggregates is considered in terms of a coarse-grained model. The model assumes that the excitation energy transfer within a pigmentprotein complex is much faster than the intercomplex excitation energy transfer, whereas the quenching ability is attributed to a specific pigmentprotein complex responsible for the nonphotochemical quenching (NPQ). It is demonstrated that the pumpprobe experimental data obtained at low excitation intensities for LHCII aggregates under NPQ conditions can be equally well explained at two limiting cases, either describing the excitation kinetics in the migration-limited or in the trap-limited regime. Thus, it is concluded that low excitation conditions do not allow one to unambiguously define the relationship between the mean times of excitation migration and trapping. However, this could be achieved by using high excitation conditions when excitonexciton annihilation is dominant. In this case it was found that in the trap-limited regime the excitation kinetics in the aggregate should be almost insensitive to the excitation density, meaning that singletsinglet annihilation has little effect on the NPQ decay kinetics, whereas in the migration-limited case there is a clear intensity dependence. In order to account for the random distribution of the NPQ-traps within the LHCII aggregates, excitation diffusion in a continuous medium with random static traps was considered. This description demonstrates a very good correspondence to the experimental fluorescence kinetics assuming a lamellar (quasi-3D) structure of the antenna characterized by the dimension d = 2.4 and therefore justifying the diffusionlimited approach on which the model is based. Using the coarse-grained model to describe the aggregate we estimate one NPQ-trap per 100 monomeric LHCII complexes. Finally we discuss the origin of the traps responsible for excitation quenching under NPQ conditions.
’ INTRODUCTION The excitation lifetime in the light-harvesting antenna and its trapping efficiency by reaction centers (RC) rely on three characteristic parameters: the time needed for the excitation migration in antenna (τmig), for excitation delivery to (τdel) and trapping in the RC (τtrap).1 The overall excitation lifetime attributed to photochemical quenching (PQ) by the RC then equals to the sum of these characteristic times τ ¼ τmig + τdel + τtrap
ð1Þ
The first term is the so-called “first passage time”, which is the time, required for the excitation to reach the RC,24 and it dominates the overall excitation lifetime when the charge separation in the RC is very fast. Evidently, the excitation trapping process can be considered as migration-limited in this case. The last term, τtrap, dominates the expression in the case of very slow charge separation, and then the process can be considered as r 2011 American Chemical Society
trap-limited. And finally, τdel determines the excitation transfer to the RC from its nearest surrounding. The value of τdel strongly depends on the geometrical organization of the system. For instance, this characteristic time dominates in the case of excitation transfer from light-harvesting complex 1 (LH1) to the RC in photosynthetic purple bacteria,5 where the distance between pigments in LH1 is much smaller than the relevant distance to the special pair of the RC. In this case the process should be considered as transfer-to-trap limited.1 In order to control the excitation density in oxygen evolving photosystem II (PSII), a physiologically important strategy to regulate photosynthetic light harvesting has evolved by plants resulting in so-called nonphotochemical quenching (NPQ) of the excitations.6 This rapid regulation of the excitation density is Received: February 14, 2011 Revised: June 14, 2011 Published: June 15, 2011 9252
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additional experiments. Nonlinear excitonexciton annihilation is the process of choice to experimentally define the migration term separately.1,15 In this work, by combining all of these results, we define the characteristic parameters responsible for NPQ. An appropriate analysis of the excitation kinetics in the LHCII aggregates under NPQ conditions will be presented here. Special emphasis will be put on the possibility to discriminate between the two limiting cases: migration-limited vs trap-limited. On the basis of this analysis a possible specific mechanism responsible for the quenching process taking place in NPQ is discussed.
Figure 1. (a) Coarse-grained model for the LHCII aggregate with a NPQ-trap in one of LHCII monomers. Green circles represent Chl a molecules with very fast excitation equilibration between them within the same LHCII monomer, reddish hexagon is the NPQ-trap itself; green bars depict possible excitation transfer paths between different LHCII monomers with a rate constant kh, and the reddish bars represent paths for the excitation delivery to or detrapping from the monomer containing NPQ-trap (with the rate constants WT and WD, respectively). (b) Schematic energy diagram of part (a) illustrating eq 1: the total excitation lifetime is determined by the excitation migration through the antenna term τmig (depending on kh and the size of the aggregate N), excitation delivery to the NPQ site term τdel (depending on WT, WD, kh, and N), and the excitation relaxation in the NPQ-trap term τtrap (depending on the excitation quenching rate kQ, WT, WD, and N). (c) Simplified model describing excitation dynamics in LHCII aggregate at high excitation conditions when excitonexciton annihilation has to be taken into account; γ determines the rate constant of excitonexciton annihilation, τA = τmig + τdel is the excitation lifetime in the antenna, and τdis1 is the rate of the intrinsic dissipation processes.
performed by generating additional trapping centers in the lightharvesting system of PSII under high excitation conditions.6,7 Despite recent extensive discussions about the origin of the NPQ-traps,812 a common consensus has not been achieved so far. Evidently, shortening of the excitation lifetime due to the presence of the NPQ-traps can also be described in a way similar to the PQ kinetics as defined by eq 1.13 To have a measurable effect, the mechanism responsible for NPQ should be competitive with the excitation trapping by an open RC. Therefore, the corresponding trapping terms τtrap in both cases of PQ and NPQ should be comparable, possible differences in these values might be compensated by the delivery term, τdel. However, as demonstrated recently,14 the relative contributions of the trapping time, τtrap, and the migration time, τmig, cannot be established by analyzing the excitation dynamics due to the presence of the NPQ-traps in oligomers of the light-harvesting complexes from PSII (LHCII). The results could be explained reasonably well with both models: trap-limited and migration-limited. Each of the terms τtrap and τdel could be estimated from calculations based on the available structural data or by performing
2. MODEL OF LHCII AGGREGATES No direct evidence about the structural organization of the LHCII aggregates is available. However, since the main building block of LHCII aggregates is the LHCII trimer, the model already developed to describe the excitation kinetics in BBY particles can be used to determine the excitation migration time, τmig. As follows from the structural organization and pigment distribution in PSII, at least two scaling parameters must be proposed to evaluate the excitation energy transfer. The first determines the mean value characteristic for the excitation migration and/or exciton relaxation within a pigmentprotein complex (occurring on a time scale of less than 1 ps), and the second defines the mean time corresponding to the intercomplex energy transfer (on a time scale of about 10 ps). Since both characteristic rates of the excitation energy transfer differ at least by 1 order of magnitude,16 the coarse-grained model schematically shown in Figure 1a can be applied to describe the excitation energy transfer. A similar coarse-grained model has been already used to evaluate the fluorescence kinetics in BBY particles17,18 and to model NPQ in PSII.13 Within the frame of such a coarse-grained approach the excitation migration time is determined by the excitation hopping time, τh (or excitation hopping rate kh = 1/τh), between monomeric LHCII complexes and, thus, τdel = 0 if NPQ originates from the trapping ability of a single LHCII complex.1 It is clear that the excitation hopping time together with the structural arrangement of the LHCII complexes determines the excitation migration time τmig in the LHCII aggregate.13 Similar to the excitation quenching by the RC (PQ), we can relate NPQ to a quenching process characterized by a rate kQ in a particular LHCII monomer, where the additional excitation quenching pathway has emerged due to specific external/internal conditions. The PQ kinetics in the BBY particles is single exponential if the excitation trapping by the RC is characterized by a single parameter, the trapping rate.13 Deviation from single exponential decay is typically obtained by assuming a more complex model for the excitation relaxation in the RC, for instance, by assuming the presence of at least two charge separated states between which equilibration occurs.17,18 This type of deviation becomes even stronger if the whole process is trap-limited, resulting in the so-called excitonradical pair model, which was developed to describe the multiphase excited state kinetics in PSII.1921 2.1. Kinetics Modeling. Let us consider the excitation decay kinetics in LHCII aggregates measured at low excitation conditions, when nonlinear annihilation processes are insignificant.14 As follows from transient absorption measurements, the excitation decay kinetics is multiexponential with decay components in the time range of hundreds of fs and some distribution of times around 4 and 200 ps. The fastest component evidently should 9253
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Figure 2. Schematic representation of the slow/fast (S/F) (a) and fast/ slow (F/S) (b) oversimplified models, where the whole light-harvesting antenna is characterized by a single excited state, connected to the second excited state corresponding to the NPQ-trap. The numbers next to the arrows denote corresponding excitation transfer/relaxation rates (in ns1) taken from ref 14.
not be taken into consideration in the description of the excitation kinetics within the frame of the coarse-grained model; it reflects the excitation equilibration within a monomer, which is a constituent element of the structure. Thus, only the two slow components, 4 and 200 ps, will be considered. In the simplest version of this model some structural arrangement of N LHCII monomeric complexes is assumed, which are connected via energy transfer, characterized by the hopping time, τh. One of the LHCII monomers is assumed to be the NPQ-trap with a characteristic quenching rate, kQ. For the two- or three-dimensional aggregates containing a substantial amount of LHCII complexes (N . 10) the trap plays the role of a local perturbation and the excitation kinetics is single exponential described by eq 1 (see Figure 1b for a schematic representation).1 In the case of smaller aggregates the perturbation for the excitation migration by the presence of the trap is not local anymore and a biexponential kinetics is expected, as follows from the perturbed two-level model.1,22 According to this model, the excitation lifetimes are defined as 1 ðN 1ÞW T W D + gd ðNÞτh τ( ¼ λ( ðW D λ( Þ2 + W T W D where λ( ¼
kQ + W T + W D (
2 ð3Þ zWT N 1
W D ¼ zWD
dnA 2 + 3:6 nA ¼ 1000 dt dnNPQ 3:6 220 nA nNPQ ¼ 1000 1000 dt
ð4Þ ð5Þ
z is the coordination number of the NPQ-trap, N is the number of the LHCII monomers in the aggregate, WT is the excitation transfer rate from the LHCII monomers to the NPQ-trap, WD is the detrapping rate of the excitation from the monomer containing NPQ-trap characterized by the excitation quenching rate kQ, and gd(N) is the function reflecting the structural organization of the aggregate (d is the dimensionality of the
ð6Þ
where rates are expressed in ps1. For the initial conditions nA(0) = 1 and nNPQ(0) = 0 the solution is given by nNPQ
nA ¼ et=178:57 ¼ 0:017et=178:57 0:017et=4:55
ð7Þ
On the other hand, for the fast/slow (F/S) model shown in Figure 2b, we obtain dnA 2 + 205 50 nA + nNPQ ¼ 1000 1000 dt dnNPQ 205 50 + 6:5 nA nNPQ ¼ 1000 1000 dt
ð8Þ
and for the same initial conditions nA(0) = 1 and nNPQ(0) = 0 the solution can be defined accordingly nA ¼ 0:202et=178:41 + 0:798et=3:878 nNPQ ¼ 0:813et=178:41 0:813et=3:878
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðkQ + W T + W D Þ2 4kQ W T
WT ¼
system). Evidently, the second term in eq 2 defines the excitation migration. According to this model, partitioning of the excitation lifetime into constituents τmig, τtrap, and τdel defined in eq 1 can be obtained in some limiting cases of the rate parameters. For instance, in the case of fast trapping, i.e., when kQ . W D + W T, λ describes the excitation delivery to the trap, λ ≈ W T, and λ + is defined by the excitation quenching rate by the NPQ-trap, λ + ≈ kQ. Evidently, in the opposite case, when W T . kQ, we will also get the same two values: λ ≈ kQ and λ + ≈ W T. In order to connect the perturbed 2-level model to the experimentally observed kinetics, we will first evaluate the experimentally obtained excitation kinetics, as has been originally suggested in ref 14, using the oversimplified compartment model shown in Figure 2. In the case of the so-called slow/fast (S/F) conditions presented in Figure 2a, the population kinetics on the antenna state (nA) and the NPQ state (nNPQ) follows the linear kinetic equations given by
ð9Þ
As follows from the solutions given by eq 7, in the case of the S/F model the decay of the excited state population in the antenna is single-exponential, and the two-exponential kinetics only relates to the NPQ-trap population. However, the amplitudes of the population decay of the NPQ-trap are very small and even assuming that the bleaching signal in the chlorophyll a band (at 680 nm)14 reflects the sum of populations of both states, i.e. by attributing the signal to (nA + nNPQ), the bleaching signal will still be very close to a single exponential. In the other, F/S, case the nonexponentiality of the antenna signal (nA) is well pronounced. However, by assuming that also in this case the bleaching signal corresponds to (nA + nNPQ), the resulting signal again becomes almost single exponential. Some changes in the kinetics can be achieved by changing the initial conditions. For the S/F case the kinetics of the antenna signal (nA) remains single exponential independently of variation of the initial conditions, while the fast exponent becomes more evident in the total signal (nA + nNPQ) when the larger portion of excitation is initially generated in the NPQ-trap. For the F/S 9254
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Table 1. Parameters of the Coarse-Grained Model (Figure 1) for the LHCII Aggregates of Different Size N Containing a Trap with a Coordination Number z Defined by Fitting the Experimental Kinetics Assuming Either Slow (τh = 5.8 ps) or Fast (τh = 0.1 ps) Hopping Ratea slow/fast model
fast/slow model
N
z
WT1 (ps)
WD1 (ps)
WT1 (ps)
WD1 (ps)
3
2
191.8
289
4.86
39.9
6 9
3 4
114.5 95.1
818 2381
2.88 2.39
59.2 78.4
12
5
86.3
5591
2.16
97.6
15
5
67.7
2088
1.70
97.9
18
5
55.2
3823
1.39
97.0
21
5
46.8
3697
1.18
96.9
The quenching time kQ1 was fitted to be equal to 3.2 and 156 ps independently of N in slow/fast and fast/slow model approaches, respectively. a
Figure 3. Excitation decay kinetics calculated according to the S/F (a) and F/S (b) model approaches by analyzing the experimental data. Squares, circles and stars correspond to the simple 2-state model (Figure 2) while the solid lines represent the kinetics calculated according to the coarse-grained model using any set of parameters from Table 1 (in this case nA(t) corresponds to the sum of excitation population of all LHCII complexes).
conditions the total signal is weakly dependent on variation of the initial conditions. Thus, this oversimplified scheme allows to demonstrate the excitation trapping process in the limiting cases, when the excitation migration is slow and the τmig term cannot be neglected (S/F model), and the opposite case of very fast excitation migration (F/S model). The latter can be well understood in terms of the perturbed two-level model. Indeed, for the limiting cases mentioned above, i.e., when either kQ . W D + W T or W T . kQ is fulfilled, two exponents defined by eq 2 correspond to the following characteristic times: τA = τmig + τdel and τQ = 1/kQ. It is noteworthy that the kinetics of the total signal in the oversimplified scheme corresponding to the initial conditions, when the excitation is generated only in the antenna, is the same for both S/F and F/S cases (see Figure 3). The experimentally observed excitation decay kinetics in LHCII aggregates14 can be also well fitted numerically by means of the coarse-grained model shown in Figure 1a by assuming either slow excitation migration in the light-harvesting antenna (similar to the S/F model), or by assuming fast energy transfer between the LHCII monomers (analogous to the F/S model). If we assume the excitation hopping time between LHCII monomers, τh, fixed, the fitting parameters are related to the NPQ-trap only. Since the size of the LHCII aggregate, N, is also not defined, the
energy hopping times τh, WT, and WD are sensitive to variation of the N value. Such variability is well demonstrated by fitting the kinetics as follows from the oversimplified model calculations (see eqs 7 and 9) using the coarse-grained model for both limiting cases (see Table 1) as shown in Figure 3. 2.2. Slow/Fast vs Fast/Slow. As demonstrated in Figure 3, the coarse-grained model approach is applicable for the description of the excitation kinetics in LHCII aggregates in the presence of NPQ-traps. Some further simplification in the description of the excitation trapping leads to the S/F simplified model. According to this model, the excitation trapping by the NPQ-trap from the antenna state is defined by a single value, τmig, while the trapping time is defined by the characteristics of the NPQ-trap. Thus, by using rates estimated from the experimental data,14 the excitation transfer to the NPQ-trap equals to τmig = 278 ps (1/τmig = 3.6 ns1, see Figure 2a). This value of the excitation migration time can be obtained for an aggregate containing more than 60 LHCII complexes arranged in a so-called supercomplex (in three layers) and assuming that τh = 5.8 ps.13 Moreover, the excitation trapping time by the NPQ-trap should then be equal to τQ = 4.5 ps (kQ = 220 ns1) as follows from the analysis of the NPQ trapping efficiency in vivo.13 In the opposite case of the F/S model the perturbed two-level model can be combined with the coarse-grained model. In this case fast equilibration of the excitation should be assumed taking place in the whole aggregate. As a result, the parameter determining the excitation transfer from the nearest LHCII monomers to the NPQ-trap equals to τdel = 5 ps (1/τdel = 205 ns1, see Figure 2b). Within the frame of the perturbed two-level model the excitation transfer from the nearest LHCII monomeric complex to the NPQ-trap should be slower than the rate of excitation transfer between LHCII monomers in the aggregate. This transfer value is defined as W T (see eq 4). For two- or three-dimensional aggregates, z is on the order of 5; thus, by assuming that (N 1) equals to 50, we will get that 1/WT is on the order of 0.5 ps, which is obviously faster than the excitation transfer time between monomers in the aggregate and comparable with the excitation relaxation times in the LHCII monomer. In order to fulfill the demands for the perturbed twolevel model, the WT value has to be slower than τh1, and consequently we have to assume that the aggregate is small, 9255
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The Journal of Physical Chemistry B otherwise there is no basis for applying the F/S simplified model. By using the rate values determined from fitting the experimental data,14 the rate of the excitation detrapping from the NPQ-trap, defined accordingly as W D = zWD (see eq 5) equals to zWD = 50 ns1, see Figure 2b. With z = 5, we obtain 1/WD = 100 ps. Similarly we get τQ = 150 ps (kQ = 6.5 ns1, see Figure 2b). According to this F/S scheme, the excitation predominantly rests on the NPQ-trap state. If more than one NPQ trap per aggregate is assumed to be present, the WT value defined by fitting the experimental data will vary. Indeed, the N value describing the amount of complexes per single NPQ-trap decreases with increasing the average number of the NPQ-traps in the aggregate. In the limiting case when either kQ . W D + W T or W T . kQ is satisfied, the two exponents defined by λ( are approximately equal to kQ and W T in both cases. Thus, by decreasing N, a proportional increase of the WT value must be assumed in order to obtain both exponents unchanged. It means that the excitation delivery rate must be assumed even larger in the F/S scheme to keep the exponents unchanged. It is clear that this simplified F/S model meets serious restrictions: (i) the aggregate has to be small enough and (ii) the excitation equilibration in the aggregate has to be fast in comparison with the excitation delivery time to the NPQ-trap. However, the excitation transfer time between monomers in the aggregate is of the order of 5 ps, as follows from excitonexciton annihilation data.16,23 The excited state kinetics determined by using the simplified model approaches is either single exponential (in the case of the S/F model approach) or almost single exponential (in the opposite case of the F/S model) for the antenna state. Thus, it is not possible to discriminate between these two limiting cases by analyzing the pumpprobe experimental data in the chlorophyll a spectral region, which clearly displays nonexponentiality. Therefore, other experimental conditions should be considered for this purpose. 2.3. Kinetics at High Excitation Conditions. To discriminate between the S/F and F/S approaches by describing the NPQ kinetics in the LHCII aggregates, high excitation conditions when excitonexciton annihilation becomes dominant should be considered. We note that the process of excitonexciton annihilation is migration-limited resulting in changes of the excitation lifetime.1,24 Obviously, the intensity dependence of the excitation kinetics should be sensitive to the model approach: S/F vs F/S. In order to take excitonexciton annihilation into account we will consider the simplified S/F and F/S models presented in Figure 2 by substituting the corresponding rate parameters and including the additional channel of nonlinear annihilation as shown in Figure 1c. Thus, the kinetic equations are as follows: dnA 1 1 1 ¼ + nA γnA 2 + zWD nNPQ τA τdis 2 dt ð10Þ dnNPQ nA ¼ ðzWD + kQ ÞnNPQ dt τA where τdis is the intrinsic excitation decay time in the LHCII monomer with the value of 2 ns, γ describes the rate of excitonexciton annihilation, and τA = τmig + τdel is the excitation lifetime in the antenna. The term proportional to zWD is taken into account to describe the detrapping effect for the F/S conditions, when the population of the NPQ-trap is higher than
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Figure 4. Normalized annihilation kinetics in the slow/fast (a) and fast/ slow (b) models calculated with different values of initial excitation. The upper curves correspond to nA(0) = 0.1, the lower ones, to nA(0) = 1, and the ΔnA(0) between two adjacent curves is 0.1. The inset in (b) shows the initial part of the kinetics.
the antenna population. In the case of the S/F conditions this term should be discarded (WD = 0). The resulting kinetics at different excitation conditions (by assuming different values of the initial conditions for nA(0) while keeping nNPQ(0) = 0) are shown in Figure 4 for the F/S and S/F model approaches. The excitation intensity dependencies evidently allow us to discriminate between the two limiting models: the S/F model displays excited state kinetics dependent on the excitation intensity while the F/S model is almost insensitive to the excitation intensity. From experiments14,16,17 it followed that the excitation decay kinetics is sensitive to the excitation intensity. This difference clearly suggests that the simplified S/F model approach should be used to analyze the excitation decay kinetics in LHCII aggregates.
3. RANDOM DISTRIBUTION OF THE NPQ-TRAPS The model description used so far for the excitation kinetics is based on the assumption of a single NPQ-trap in the aggregate. Evidently the excitation lifetime would decrease in the case when more than one NPQ-trap per aggregate is generated.13 The model described above also assumes that the aggregates (in average) are of the same size. This approach is an evident simplification and the possible distribution of both these parameters, the size of the aggregate and the distribution of the NPQ-traps, should be taken into account. For this purpose let us consider the excitation kinetics in an extended system (continuous medium) with 9256
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randomly distributed traps.25 Due to the randomness of their distribution the mean distance between the traps defined by the concentration of the traps is a single characteristic value. Thus, assuming that traps are very efficient we propose that for some particular random distribution of the traps the excitation is completely quenched at the distance R from the point of initial excitation. In other words, we can assume that the random distribution of the traps creates a (hypothetic) d-dimensional spherical surface of radius R representing a perfect trap for the excitation. The fate of an excitation initially being produced in the center of such a sphere is then defined by the excitation migration (diffusion) according to the following diffusion equation for the excitation distribution nA(rB,t|R) in the aggregate, where B r determines the excitation position ∂ 2 D∇ nA ð B r , tjRÞ ¼ 0 ð11Þ ∂t where D is the excitation diffusion constant. By separating both variables, coordinate and time, the solution of eq 11 can be written as r , tjRÞ ¼ nA ð B
∑n fn ð Br jRÞ expðεntÞ
ð12Þ
where εn are the eigenvalues of the corresponding stationary equation for the spatial excitation distribution, fn(rB|R) ðD∇2 + εn Þfn ð B rÞ ¼ 0
ð13Þ
with the boundary condition fn(rB|R)|r=R = 0 corresponding to the case of a perfect trapping surface. By solving eq 13 for the continuous systems of different dimensionality d, we obtain 8 rffiffiffiffiffi ! > εn > > r , d¼1 An cos > > D > > > > rffiffiffiffiffi ! > < εn r , d¼2 ð14Þ r jRÞ ¼ An J0 fn ð B D > > > > rffiffiffiffiffi ! > > 1 εn > > > An sin r , d¼3 > : r D where J0(ξ) is the zero-order Bessel function, An is the amplitude, and εn is now defined directly by the boundary conditions, which are typically assumed to reflect perfect trapping on the surface of radius R. At long times the excitation decay kinetics is defined by the dominant term in eq 12 containing the smallest eigenvalue ε0, thus, giving Dt nA ðtjRÞ µ expðε0 tÞ ¼ exp kd 2 ð15Þ R where
8 1 > > π2 , < 4 kd ¼ ξ 2 , 0 > > : π2 ,
d ¼1 d ¼2 d ¼3
ð16Þ
Here, ξ0 ≈ 2.4048 is the first zero-crossing of J0(ξ). The probability to find k NPQ-traps within the volume V is then given by the Poisson distribution ðkÞ
PNPQ ðV Þ ¼
N̅ k ̅N e k!
Figure 5. (a) Experimental fluorescence kinetics (squares) fitted by eq 23 for the systems of different dimensionality. (b) The average number of the LHCII monomers per NPQ-trap vs hopping time for the systems with different coordination number z.
where the average number of traps is given by N = cNPQV and cNPQ stands for the concentration of the NPQ-traps in the system. Thus the probability to find no additional NPQ-traps inside the sphere of radius R is given by ð0Þ
PNPQ ðRÞ ¼ exp½ cNPQ Vd ðRÞ where
8 > 2R, > < 2 2πd=2 d R ¼ πR , Vd ðRÞ ¼ > dΓðd=2Þ > 4πR 3 , : 3
d¼1 d¼2
ð18Þ
ð19Þ
d¼3
is the volume of the sphere in the d-dimensional system, Γ(d/2) is the gamma-function. The average concentration of the excitation surviving in the system at a fixed time t then can be defined as follows: Z ∞ ð0Þ dVd ðRÞPNPQ ðRÞnA ðtjRÞ ð20Þ n̅ A ðtÞ µ 0
After applying the saddle-point method to calculate the integral in eq 20 at longer times we get
ð17Þ
2=ðd + 2Þ
n̅ A ðtÞ µ ðDtÞd=½2ðd + 2Þ exp½ηd cNPQ 9257
ðDtÞd=ðd + 2Þ
ð21Þ
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where ηd ¼
d=ðd + 2Þ kd
2πd=2 dΓðd=2Þ
8 3 > ð2π2 Þ1=3 , > > < 2 pffiffiffi ¼ 2ξ0 π, > 5 > > : 41=5 π8=5 , 3
!2=ðd + 2Þ " 2=ðd + 2Þ # 2 d=ðd + 2Þ d + d 2
d ¼1 d ¼2
ð22Þ
d ¼3
This result can be even further generalized by performing quadratic interpolation of kd in eq 16 to fractal dimension d and then using eq 22 to obtain ηd with noninteger d. Thus the obtained kinetics defined by eq 21 is nonexponential and contains two fitting parameters d and b, since n̅ A ðtÞ µ t d=½2ðd + 2Þ exp½bt d=ðd + 2Þ
ð23Þ
In this description the nonexponentiality of the kinetics is different from that represented by simplified approach displayed in Figure 2 used to describe the pumpprobe experimental data as shown in Figure 3. As follows from the analysis of the experimental data,14 the two exponentials are clearly distributed, indicating a stronger nonexponentiality of the kinetics in agreement with what follows from the pumpprobe measurements. This conclusion is furthermore supported by fluorescence kinetics recorded for LHCII aggregates (see ref 26 and the supporting material in ref 14). Excitation decay traces corresponding to different dimensions d of the system together with the fluorescence kinetics of the LHCII aggregates at 680 nm taken from the supporting material in ref 14 are presented in Figure 5a. The experimental kinetics was normalized at t = 0 and was fitted for t > 20 ps since eq 23 does not describe the initial part of the kinetics. It is evident that kinetics corresponding to two-dimensional (d = 2) and three-dimensional (d = 3) cases give a satisfactory correspondence on a time scale extending up to nanoseconds. However, the best fits of the experimental kinetics was obtained by assuming d = 2.4, supporting the fractal arrangement of the aggregate, similar to the supercomplex organization as postulated by analyzing NPQ in vivo.13 These results support the migration-limited S/F model conditions. From fitting the experimental data by using eq 21 we are able to determine the concentration of the NPQ-traps if the excitation diffusion constant, which is directly related to the τh value as follows from the coarse-grained approach, is defined. Since the fitting parameter b is fixed, the concentration of the NPQ-traps is larger, the larger value of τh is assumed (see Figure 5b). Thus, for the chosen value τh = 5.8 ps one NPQ-trap (assuming it to be perfect) per 100 monomers has to be assumed.
4. POSSIBLE IDENTITY OF THE NPQ TRAP The NPQ phenomenon is usually attributed to some activated quenching species (a single molecule or the cooperative association of several) where the excitation undergoes nonradiative decay. The exact location of the quencher within the antenna and its precise nature are still matters of ongoing debate, with both chlorophylls and carotenoids being put forward as essential components of the quenching mechanism. The most promising candidates for the quenching mechanisms (in no order of preference) are (i) quenching by either charge transfer (CT) states14 or excimeric states6 of a
strongly coupled chlorophyll dimer; (ii) by a CT state between chlorophyll and xanthophyll resulting in generation of a localized cation radical state of the xanthophyll;8 (iii) via increased coupling of chlorophyll to a short-lived xanthophyll excited state;12 and (iv) by the direct transfer of energy from the chlorophyll pool to a particular carotenoid, thought to be lutein in the wild type.10 For the S/F regime additional discrimination between the various proposed quenching mechanisms is achieved. As follows from the structural data of the LHCII complex, it is natural to assume that excitation energy from the Qy states of the chlorophyll a (Chla) terminal emitter of LHCII (the exit point for excitation energy as it migrates to the PSII core), Chla610 Chla611Chla612, is transferred to the 2Ag (or S1) state of lutein 1.10,23 Since the energy relaxation in the structural arrangement of closely positioned pigment molecules in the single LHCII complex is fast (of the order of 100 fs27) and the lifetime of the 2Ag state of the lutein molecule is short (
