Photosynth Res (2013) 117:289–320 DOI 10.1007/s11120-013-9895-1
REGULAR PAPER
The energy flux theory 35 years later: formulations and applications Merope Tsimilli-Michael • Reto J. Strasser
Received: 5 March 2013 / Accepted: 11 July 2013 / Published online: 17 September 2013 Springer Science+Business Media Dordrecht 2013
Abstract Several models have been proposed for the energetic behavior of the photosynthetic apparatus and a variety of experimental techniques are nowadays available to determine parameters that can quantify this behavior. The Energy Flux Theory (EFT) developed by Strasser 35 years ago provides a straightforward way to formulate any possible energetic communication between any complex arrangement of interconnected pigment systems and any energy transduction by these systems. We here revisit the EFT, starting from the basic general definitions and equations and presenting applications in formulating the energy distribution in photosystem (PS) II units with variable connectivity, as originally derived, where certain simplifications were adopted. We then proceed to the derivation of equations for a PSII model of higher complexity, which corresponds, from the formalistic point of view, to the later formulated and now broadly accepted exciton–radical-pair model. We also compare
the formulations derived with the EFT with those obtained, by different approaches, in the classic papers on energetic connectivity. Moreover, we apply the EFT for the evaluation of the excitation energy distribution between PSII and PSI and the distinction between state transitions and PSII to PSI excitation energy migration. Our analysis demonstrates that the EFT is a powerful approach for the formulation of any possible model, at any complexity level, even of models that may be proposed in the future, with the advantage that any possible energetic communication or energy transduction can be easily formulated mathematically by trivial algebraic equations. Moreover, the biophysical parameters introduced by the EFT and applicable for any possible model can be linked with obtainable experimental signals, provided that the theoretical resolution of the model does not go beyond the experimental resolution. Keywords Energetic connectivity (grouping) Energy distribution Energy flux Grouping probability Photosynthetic unit Photosystem II models
M. Tsimilli-Michael (&) Ath. Phylactou str., 3, 1100 Nicosia, Cyprus e-mail:
[email protected] R. J. Strasser Rte du Petit Lullier, 23, 1254 Jussy, Switzerland e-mail:
[email protected] R. J. Strasser Nanjing Agricultural University, Nanjing, China R. J. Strasser North-West University, Potchefstroom, South Africa R. J. Strasser University of Geneva, Geneva, Switzerland
Introduction All events in a living organism are related with transformations of energy. All life on earth depends on the energy transductions of the photosynthetic processes, from absorbed light-energy to excitation energy, to oxidoreduction energy (electron transport) and, ultimately, to energy stored in produced biomass and ordered structures (entropy decrease). The energetic behavior of the photosynthetic apparatus is fundamental in photosynthesis research. Several models
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have been proposed and a variety of experimental techniques are nowadays available to determine parameters that can quantify this behavior. An energetic concept and an applicable theory are needed to construct and formulate any proposed model. Thereafter, the behavior of the system according to the model can be predicted and the predictions can be then compared with the experimentally measured functions of the photosynthetic system. A broader challenge would be the development of a theoretical approach that would not be restricted to the case of the photosynthetic apparatus, but would be applicable for the function of any system in any kind of biomembranes where complex energy transductions take place. Such a theoretical approach is the Energy Flux Theory (EFT) developed by Strasser 35 years ago (Strasser 1978), which provides a straightforward way to formulate any possible energetic communication between any complex arrangement of interconnected pigment systems and any energy transduction by these systems. All terms often used in the analysis of energy distribution and transduction in any system (energy transfer, yield, probabilities and rate constants of energy transfer, exciton density, unit–unit energy transfer, energy distribution in the two photosystems, excitation energy transfer between the photosystems, etc.) are rigorously defined. The basis of the EFT is that each of the pigment systems under consideration is supplied by a continuous energy input, which it converts to energy outputs of different forms. As will be shown, it leads to systems of algebraic equations that are trivially solved. We will here revisit the EFT, starting from the basic general definitions and equations and presenting the applications in formulating the energy distribution in the photosynthetic apparatus, as derived in the original papers, where certain simplifications were adopted (Strasser 1980, 1981). We will then proceed to the derivation of equations for the model of higher complexity, which corresponds, from the formalistic point of view, to the later formulated (see e.g., Lavergne and Trissl 1995) and now broadly accepted exciton–radical-pair model. We will also show that the formulations derived with the EFT are in agreement with those obtained, by different approaches, in the classic papers of Joliot and Joliot (1964) and Paillotin (1976a, b, 1977), with the added advantage that they can describe models of higher and any complexity.
The EFT: variables and equations According to the EFT, any possible energetic communication between any complex arrangement of interconnected pigment systems and any energy transduction by
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each of them can be formulated by simple algebraic equations, using the below defined five magnitudes: The energy influx Ehi from a pigment system h to the pigment system i. The total energy influx Ei(in), otherwise the excitation rate of the pigment system i (in number of excitation events per time and area), is then given as the sum of all energy influxes, i.e. of the light energy flux Ji directly absorbed by pigment system i and of all the energy fluxes Ehi transferred to i from the surrounding pigment systems h (h = 1, 2,…, n): n X Ehi ð1Þ EiðinÞ ¼ Ji þ h¼1
The energy outflux Eij from the excited pigment system i to a destination site j; j denotes another pigment system or stands for another form of energy, i.e. photochemistry, heat dissipation, fluorescence emission, and energy transfer to a quencher. The total energy outflux Ei(out), otherwise the deexcitation rate of the pigment system i, is then given by the sum of all energy oufluxes to all possible destinations j (j = 1, 2, …, m). m X EiðoutÞ ¼ Eij ð2Þ j¼1
The probability pij that the de-excitation of the pigment system i is realized by the energy outflux Eij from i to j. This probability is, by definition, equal to the ratio of the energy transfer flux Eij to the total energy outflux EiðoutÞ : pij ¼
Eij EiðoutÞ
ð3Þ
As influxes and outfluxes are rapidly equilibrated in a pigment system, the total outflux is equal to the total influx, and they can be both substituted by Ei: EiðoutÞ ¼ EiðinÞ ¼ Ei
ð4Þ
Thereafter, Eq. 3 gives: Eij ¼ Ei pij
ð5Þ
Combining Eqs. 1, 4 and 5, we get Ei ¼ J i þ
n X h¼1
Ehi ¼ Ji þ
n X
ðEh phi Þ
ð6Þ
h¼1
The variables above defined for any pigment system i can be easily related with the other, commonly used parameters, i.e. the exciton density, the different deexcitation rate constants and the life time of the pigment system, as follows: Each individual de-excitation from i to j is characterized by a rate constant kij. The de-excitation flux Eij can thus be expressed, assuming that the de-excitation is of first order, as the product of the total energy content Pi (equivalently, the
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exciton density of the pigment system i) and the corresponding de-excitation rate constant: Eij ¼ Pi kij ð7Þ Note In the case of a second order de-excitation reaction, the outflux Eij is given by the product of the exciton density, the rate constant and the concentration of the energy acceptor, e.g. a quencher. Summing up for all the partial outfluxes Eij, Eq. 7 gives: m m X X Eij ¼ Pi kij ð8Þ Ei ¼ j¼1
j¼1
We recall that the lifetime si of the excited state of any pigment system i is equal to the reciprocal of the sum of all the rate constants that characterize the de-excitation outfluxes from this system: " # 1 m X si ¼ kij ð9Þ j¼1
Hence, Eq. 8 gives the following equation which expresses the energy content of the pigment system i as the product of its total excitation energy influx Ei and the lifetime si of its excited state: Pi ¼ E i s i ð10Þ Utilizing Eq. 5 and dividing Eq. 8 by Eq. 7, we get the relation of the de-excitation probability pij with the corresponding de-excitation rate constant kij and the sum of all de-excitation rate constants or, using Eq. 10, with the lifetime of the excited state of the pigment system i: Pi kij Eij kij pij ¼ ¼ ¼P ð11Þ m m P Ei ½Pi kij kij j¼1
j¼1
pij ¼ kij si
ð12Þ
In summary, all types of de-excitation fluxes Eij from a pigment system i to any destination j can be equivalently expressed as: Eij ¼ Ei pij ¼ Ei si kij ¼ ðEi si Þkij ¼ Pi kij ð13Þ
Formulation by EFT of energy distribution and transduction in the photosynthetic apparatus Chlorophyll (Chl) a fluorescence, though corresponding to a very small fraction of the dissipated energy from the photosynthetic apparatus is widely accepted to provide access to the understanding of its structure and function, hence characterized as ‘‘a signature of photosynthesis’’ (Papageorgiou and Govindjee 2004). There is a general
agreement that, at room temperature, Chl a fluorescence of plants, algae and cyanobacteria, in the 680–740 nm spectral region, originates almost entirely from photosystem (PS) II (see e.g. Paillotin 1976a) and it can therefore serve as an intrinsic probe of the fate of its excitation energy. This means that, though PSI fluorescence is not zero, it can be well assumed as being negligible; we here adopt this assumption, as in the vast majority of publications presenting analysis and utilization of the experimental fluorescence kinetics at room temperature. The fluorescence yield of PSII is determined by several structural and conformational parameters. We call ‘‘structure’’ the chemical composition and architecture of pigment assemblies, and we attribute the term ‘‘conformation’’ to the constellation of de-excitation rate constants (Strasser et al. 2004). For a given constellation of those parameters, since fluorescence and photochemistry are alternative fates of Chl a excitation energy in PSII, the fluorescence yield depends on the state, ‘‘open’’ (conducting photochemistry) or ‘‘closed’’ (not conducting photochemistry) of the PSII reaction centers (RCs); hence the closure of the RCs upon illumination is reflected in the fluorescence rise from F0 (all RCs open) to FM (all RCs closed). For many decades, one of the fundamental aims in photosynthesis research has been to formulate this dependence. Such formulations require modeling of the structure and conformation of the PSII unit (PSUII). According to the concept of photosynthetic unit (Emerson and Arnold 1932a, b; Gaffron and Wohl 1936), the units were considered as separate structural entities, each consisting of an antenna (pigments’ assembly) channeling excitation energy to the (one) associated RC (‘‘separate units’’, or ‘‘separate package’’, or ‘‘puddle’’ model), i.e. excitation energy transfer occurs only within the unit (see e.g. formulations in Kitajima and Butler 1975a). On the other hand, studies in photosynthetic bacteria led to the proposition of the ‘‘lake’’ model (or ‘‘matrix’’ model), where the RCs are embedded in a common antenna system of infinite size (pure lake model), or of definite size (domains), hence the excitation energy transfer in the according pigment bed is unrestricted (Vredenberg and Duysens 1963; Clayton 1966, 1967; Kitajima and Butler 1975a; for the terms ‘‘lake’’ vs. ‘‘puddle’’, see Robinson 1967). A model, intermediate between the pure lake model and the separate package model, was introduced by Joliot and Joliot (1964), according to which PSUIIs are structurally distinct but energetically connected in the sense that an exciton reaching a closed RC of a certain PSUII has a probability (\1) to move to a neighbor PSUII, where its fate is similarly determined by the state of the RC. This approach was the basis of the model and formulations proposed by Paillotin (1976b) and, also, from Butler’s group (see e.g. Butler and Strasser 1977; Butler 1980).
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In 1978 Strasser proposed the EFT (see also Strasser 1981), with which any model can be formulated, and applied it for models of higher complexity, in which he extended the concept of energetically connected units to include a probability for energy output to neighbor units also from open PSUIIs, both from the core antenna and the light-harvesting complex, with the latter being considered as a component of PSUII according to the tripartite model (as e.g. in Butler and Strasser 1977). Moreover, Strasser (1978) generalized the concept of forward and backward energy transfer within each PSUII, which was introduced in his previous work with Butler, to include also core antenna $ RC energy transfer both in open and closed units, as well as from PSUII components to PSUI. Below we present Strasser’ s model(s) and their formulation, along with the concomitant formulations of the dependence of fluorescence yield and photochemical yield on the fraction of open and closed RCs, which were deduced by applying the EFT. Note In this article, ‘‘energetic connectivity’’ (equivalently, ‘‘grouping’’) means restricted connectivity, in contrast to the non-restricted that refers to the lake model.
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THE GROUPING CONCEPT 3 2
2 13)
b
2
p32 = 0
b
2
2
14)
1)
b
b
b
b
pbb = 0 2
p32 = 0 p32 = 0 3
3
2
2
3 pbb = 0
b
b
3
p22 = 0
2
2
pbb = 0
b
p33 = 0
p32 = 0
3
3
2
2
b
2
b
b
p22 = 0
2 6)
b
b
p32 = 0 3 2
3 pbb = 0
3
2
11)
3
p22 = 0
pbb = 0
b
2
9)
b
2 12)
b
3
3)
b
b
2
p33 = 0
2
b
8)
b
3
p22 = 0
3
2
2)
2 7)
2 5)
b
b
b
3
2
2
p33 = 0
10)
b
b
p33 = 0
pbb = 0 3
3
2
2 4)
b
Models of different complexity levels according to the Grouping Concept of Strasser (1978) Figure 1 is reproduced from Strasser’s publication (1978), where the EFT and the accordingly derived formulations of energy distribution and transduction in photosystem II units were first proposed and analyzed (for the notations, see next section). In that publication the term ‘‘grouping’’ was introduced for the energetic connectivity among PSUIIs and, as a consequence of the EFT conceptual approach, the ‘‘overall grouping probability’’ among units was formulated by also considering, and hence comprising, the probabilities that govern all types of energetic communication within each unit, i.e. between the pigment pools of each unit. Accordingly, the concept (and the figure) was given the title ‘‘The Grouping Concept’’. Figure 1 presents schematically 14 models (numbered in italics) of different complexity levels and further demonstrates with which sequential simplifications each of them is derived from the model with the highest complexity level (model 10). Each line joining two pigment sites, both within and between PSUIIs, indicates forward and backward energy transfers, with probabilities pij and pji (where 0 pij \1 and 0 pji \1Þ: Note that, each model of the outer circle is derived from a corresponding model in the inner circle by the simplification that there is no energetic connectivity between RCs (pbb ¼ 0Þ:
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b
Fig. 1 The ‘‘Grouping Concept of the Photosynthetic Apparatus’’, as first presented 35 years ago (Strasser 1978). Only PSII components (sites) are drawn, with b standing for RCII, ‘‘2’’ for core antenna and ‘‘3’’ for LHCII. Each line between the pigments symbolizes energy transfer forth and back, i.e. energy cycling
Model of highest complexity level A detailed scheme of the model with highest complexity level (however, with pbb ¼ 0Þ is presented in panels A and B of Fig. 2 (based on a scheme in Strasser et al. 2004), which provide details for the paths of energy transduction indicated in the model 4 of Fig. 1 and for the paths for energy dissipation. Panel A indicates the energy fluxes in general and panel B the rate constants governing the energy fluxes in open and closed PSUIIs. The models adopt the dogma that the state of the RC and, accordingly of the PSUII in which it belongs, is defined by the redox state of the PSII primary quinone electron acceptor QA (Duysens and Sweers 1963), i.e. open RCs correspond to QA and closed RCs to Q A , though the basic equations that we will present for the formulation of the models have a wider validity. The model in panel A shows the two photosystems, PSI and PSII (at the open state), linked with the intersystem electron transport chain, which is a series of electron carriers from the primary reduced product of PSII
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(A) J3
PSI (B)
PSII
J1
3
E33 E3D E32 E23 E22 E22
J2
k32 k23
E21
2
E2D E2b Eb2
b
1
C
EbD
k2N T
a
EbP intersystem e− transport chain
k op bN
k3N
k32 k23 k22
2
k op 2b
3
k33
2
k22 k bop2
b k op bP (or
k3N k op bN = 0
J2 k clbN = 0 cl k op 2b = k 2b k 2b
k2N
k cl2b k clb 2
b k bP )
closed (cl)
J3
J3 k33
3
PSII open (op)
closed (cl)
J3 k3N
E31
(C)
PSII open (op)
M
G E33
J2
293
k bop2
=0
J2
3
J3 k33 k33
k32 k23
2
k2N
k3N
k32 k23 k22 k22
2
J2
k2N
k 2b k clb 2
k 2b
b
k clbN
3
b
k bP
P A P + A−
P A P + A−
Fig. 2 A Tripartite model of high complexity level. Boxes and circles present pigment pools and RCs, respectively. Arrows indicate energy fluxes Eij (from site i to site/destination j) governed by first order rate constants kij (not shown). The absorbed light energy influxes (doubleline arrows) are denoted as Ji. The model assumes two antenna pigment pools of PSII, i.e. LHCII labeled as ‘‘3’’ and core antenna labeled as ‘‘2’’, supplying with excitation energy the PSII reaction centre (RCII), labeled as b, while the PSI antenna pigment pools channeling excitation energy to RCI (labeled as a), are regarded as one site (labeled as ‘‘1’’). The model assumes energy dissipation EiD from each of the PS II sites, as heat, fluorescence emission, and energy flux to any quencher (Q), i.e. EiD ¼ EiH þ EiF þ EiQ , governed by the respective rate constants, energetic communication between core antenna and RCII within a PSUII (E2b and Eb2; expressed by T—see text), between core antenna and LHCII within a PSUII (E23 and E32; expressed by C—see text), between like antenna sites of neighbor PSUII (connectivity or grouping: E33 and E22; denoted by G) and from PSII antenna sites to PSI antenna site (migration or spill-over: E31 and E21; denoted by M). The energy
outflux from RCII that is transformed to photochemical energy (destination P) is also indicated (EbP). Concerning PSI, the schemes shows only the arrows presenting the energetic communication between antenna and reaction centre and the energy outflux from RCI that is transformed to photochemical energy. The wide grey arrow represents the intersystem electron transport chain, from the primary reduced product of PSII photochemistry to the primary oxidized product of PSI photochemistry. B The same model, showing only PSII and indicating for each arrow (energy flux) the corresponding rate constant. The arrows presenting energy outfluxes for dissipation (EiD) and spill-over (Ei1) are replaced by one and, accordingly, the sum of the respective rate constants is indicated, kiN ¼ ki1 þ kiD ¼ ki1 þ kiH þ kiF þ kiQ (kiQ of pseudo-first order), where subscript N stands for any destination site that does not belong to PSII. The scheme presents one PSUII with open and one PSUII with closed RC. The rate constants for the energy fluxes to and from the RC depend on its state and are accordingly distinguished (superscripts ‘‘op’’ and ‘‘cl’’). C The model used for the first application of the EFT (Strasser 1978), derived upon the indicated simplifications from model (B)
photochemistry to the primary oxidized product of PSI photochemistry. Each photosystem consists of antenna pigment complexes, which supply with excitation energy a special chlorophyll (Chl) a complex that can transform it to photochemical energy and is, hence, called reaction centre (RC). The model is of high complexity level, both in respect to pigment assemblies and in respect to the dynamics of energetic communication:
complexes (CP29, CP26, and CP24) and the major Chl a/b-binding protein complex (with subunits). The light energy influxes Ji (presented by double-line arrows) absorbed by the antenna pigment complexes are accordingly denoted as J3, J2, and J1, while the light energy fluxes absorbed by the RCs are regarded as negligible due to the small amount of RCs relatively to the amount of antenna pigments. The energy fluxes Eij from any site i to any site/destination j, are presented in the scheme of panel A by arrows.
•
The model distinguishes two antenna pigment pools of PSII, i.e. the light-harvesting complex (LHCII) and the core antenna, labeled here as ‘‘3’’ and ‘‘2’’, respectively, while the antenna pigment pools of PSI are regarded as one site (labeled as ‘‘1’’); hence, it is characterized as a tripartite model (Butler and Strasser 1977). Antenna pigment pools are presented by boxes and RCs by circles. The RCII and RCI are labeled as b and a, respectively. Note According to the knowledge and terminology of today (see e.g. Ke 2001), the pigment pool ‘‘2’’ stands for the assembly of CP43 and CP47 core antennas, and the pigment pool ‘‘3’’ for the assembly of the minor Chl a/b-binding protein
•
•
•
The model assumes variable energetic communication (all possible energy fluxes Eij) between pigment sites (thin-line arrows), namely: –
– –
Forward and backward energetic communication between core antenna and RCII within a PSII unit (T), i.e. E2b and Eb2, Energetic communication between core antenna and LHCII within a PSII unit (C), i.e. E23 and E32, Energetic communication between like antenna sites of neighbor PSII units, i.e. core–core (E22) or
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Photosynth Res (2013) 117:289–320
–
•
•
LHCII–LHCII (E33), called energetic connectivity or grouping (G), Energy transfer from PSII to PSI sites within a PSU, called migration or spill-over (M), i.e. E31 and E21.
Note The product of probabilities referring to both directions of energy transfer between i and j expresses energy cycling. In this sense, Strasser (1978) denoted the product [p2b pb2] as ‘‘Trapping Product—T’’ and the product [p23 p32 ] as ‘‘Coupling Product—C’’. op The model denotes as EbP (standing for EbP ; thickline arrow) the energy outflux from an open RCII that is used for primary photochemistry (subscript ‘‘P’’), i.e. for QA reduction. For each PSII site, the model pools together energy dissipation EiD (dashed-line arrows) as the sum of heat dissipation, fluorescence emission, and energy flux to any acceptor other than photosynthetic pigments (hence considered as quencher, QÞ : EiD ¼ EiH þ EiF þ EiQ .
The equivalent scheme of panel B (showing only PSII) indicates for each arrow, instead of the energy flux Eij , the corresponding rate constant kij . The arrows in panel A presenting energy outfluxes for dissipation (EiD ) and spill-over (Ei1 ) are replaced by one arrow (thin-line arrow) in panel B and, accordingly, the sum of the respective rate constants is indicated, kiN ¼ ki1 þ kiD ¼ ki1 þ kiH þ kiF þ kiQ , where subscript ‘‘N’’ stands for any destination site that does not belong to PSII (for consistency of labeling, kbD is written as kbN ). All rate constants are of first order except for kiQ , which is of pseudo-first order, since it stands for the product of a firstorder rate constant and the concentration of the quencher. The scheme in panel B presents one open and one closed PSUII. The rate constants for energy dissipation from the two antenna pigment pools and the energetic communication between them and with PSI are characteristic of their structure, hence independent of the RC state, while those for the energy flux to the RC (k2b ) and from the RC (kb2 and kbN ) are regarded as depending on its state and accordingly distinguished (superscripts ‘‘op’’ and ‘‘cl’’). The rate constant for energy conservation in primary photochemistry PA ! Pþ A (P standing for P680, i.e. the PSII RC, and A for the primary electron acceptor of PSII), can be written as op kbP (instead of kbP Þ since it is meaningful only for open PSUIIs. Applying Eq. 1 for the two antenna pigment pools (i ¼ 2 and i ¼ 3) and the RC of PS II (i ¼ b), and denoting by B the fraction of closed RCs, B ¼ Q A =ðQA þ QA Þ ¼ QA =QA;total
ð14Þ
we get the following system of equations for the total influxes or excitation rates:
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E3op ¼ J3 þ ð1 BÞE3op p33 þ BE3cl p33 þ E2op pop 23 E3cl ¼ J3 þ ð1
BÞE3op p33
E2op
BÞE2op pop 22
¼ J2 þ ð1
ð15Þ
þ BE3cl p33 þ E2cl pcl 23 þ
BE2cl pcl 22
þ
Ebop pop b2
ð16Þ þ
E3op p32 ð17Þ
cl cl cl cl cl E2cl ¼ J2 þ ð1 BÞE2op pop 22 þ BE2 p22 þ Eb pb2 þ E3 p32
ð18Þ Ebop
¼
E2op pop 2b
Ebcl ¼ E2cl pcl 2b
ð19Þ ð20Þ
Each probability is determined by the appropriate rate constants, according to Eq. 11. We note that only p33 and p32 are independent of the state of the RC, while each of the others depends on rate constant(s), different in closed than in open RCs. Note Model 10 of Fig. 1 (and, concomitantly, all models of the inner circle derived from model 10 with the indicated simplifications) can be similarly formulated be considering energetic connectivity between RCs (pbb 6¼ 0), hence by modifying Eqs. 19 and 20 as op op cl cl Ebop ¼ E2op pop 2b þ ð1 BÞEb pbb þ BEb pbb
ð190 Þ
op op cl cl Ebcl ¼ E2cl pcl 2b þ ð1 BÞEb pbb þ BEb pbb
ð200 Þ
It is worth emphasizing that the EFT incorporated the concept of forward (E2b) and backward energy transfer (Eb2), both for open and closed RCs. In this sense, the model in Fig. 2B (Strasser 1978) can well be considered as an early ancestor of the exciton–radical-pair (ERP) model (see e.g. Lavergne and Trissl 1995), shown in Fig. 3. Despite the conceptual difference, i.e. despite the fact that the forward and backward energy transfer refers to the primary charge separation in the ERP model and to the trapping by the RC in Strasser’s original model, the formalism of the two models and, concomitantly, the derived equations (which aim to describe the phenomenology), are identical, with full formalistic correspondence of the deexcitation rate constants, as indicated in Fig. 3.
Formulation of the model of Fig. 2C (according to Strasser 1978, 1981) Though the model with full complexity (panels A, B in Fig. 2) was conceived and presented 35 years ago (Strasser 1978), the model that was formulated at that time was the one presented by the scheme of panel C, derived from the model in panel B by employing certain simplifications, as indicated in Fig. 2. Here we will first re-derive those formulations and further proceed to a much more thorough analysis of them than that presented in the original papers, as well as of
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constants that define each of them, according to Eq. 11, are considered (by the simplification adopted) to be the same for open and closed RCs. The same applies for the probabilities for energy dissipation (including fluorescence emission) and migration.
Fig. 3 The exciton–radical-pair (ERP) model for the trapping in a photosynthetic unit (U). The unit, composed of a core antenna and one reaction center, exchanges energy with the radical pair (R), whose status, oxidized (superscript ‘‘ox’’) or reduced (superscript ‘‘red’’; indicated in brackets), defines also the status of U. The rate constants are: k2ox (or, simply k2 since it refers only to Rox) for charge stabilization, kdox and kdred for energy losses from Rox and Rred, ktox ox red and ktred for energy transfer from U to R, k1 and k1 for backward energy transfer, i.e. from R to U, and k1U for energy losses (including fluorescence emission) from U. The correspondences between the rate constants of the EPR model and the model of Fig. 2B are also shown
issues not raised at that time; so far, only part of this extended analysis has been published (in Strasser et al. 2004). We will then revert to the formulation of the model of Fig. 2B (utilizing the system of Eqs. 15–20), which is here presented for the first time. With the simplifications adopted in the model of Fig. 2C, the system of Eqs. 15–20 is accordingly simplified as: E3op ¼ J3 þ ð1 BÞE3op p33 þ BE3cl p33 þ E2op p23 E3cl
¼ J3 þ ð1
BÞE3op p33
þ
BE3cl p33
þ
E2cl p23
E2op ¼ J2 þ ð1 BÞE2op p22 þ BE2cl p22 þ E3op p32
ð22Þ ð23Þ ð24Þ
Ebcl
Energy influxes for the extreme cases when all PSUIIs are open or closed op cl We denote by E2;0 and E2;M the total influxes when all RCs are open (B = 0; all QA oxidized, i.e. QA ¼ QA;total Þ or all are closed (B = 1; all QA reduced, i.e. Q A ¼ QA;total Þ respectively, where the subscripts ‘‘0’’ and ‘‘M’’ are taken as a loan from the labeling of the corresponding fluorescence signals F0 and FM. Solving the system of Eqs. 21– 26, we get: op E2;0 ¼
J2 þ J3 p32 =ð1 p33 Þ J ¼ 1 p22 C=ð1 p33 Þ 1 p22 C=ð1 p33 Þ
ð21Þ
E2cl ¼ J2 þ ð1 BÞE2op p22 þ BE2cl p22 þ Ebcl pb2 þ E3cl p32 Ebop
Solving the system of Eqs. 21–26, which is a trivial algebraic system, we get the expressions for the total energy influxes in each of the three PSII sites (‘‘3’’, ‘‘2’’ and b), both at the open and the closed state, in terms of the absorbed light energy fluxes, the probabilities of energy transfer and the fraction B of closed RCs. Once these expressions are derived, all partial energy fluxes within PSII, as well as the outflux from PSII to PSI (migration), can be derived in terms of the respective probabilities or deexcitation rate constants, according to Eq. 13.
¼
E2op p2b
ð25Þ
¼
E2cl p2b
ð26Þ
It is worth explaining the differences of this system of equations from that of Eqs. 15–20: (i) Since pop b2 ¼ 0 (because, upon the simplification, op kb2 ¼ 0), pcl b2 is the only probability of back energy flux to core antenna and was, hence written for simplicity as pb2 ; moreover, this probability is equal to 1 (kept, however, as pb2 in Eq. 24), since no dissipation from a closed RC is considered. (ii) Not only p33 and p32 , but also p22 , p23 , and p2b are independent of the state of the RC, since the rate
ð27Þ J2 þ J3 p32 =ð1 p33 Þ 1 p22 T C=ð1 p33 Þ J ¼ 1 p22 T C=ð1 p33 Þ
cl E2;M ¼
ð28Þ
where J is the influx to antenna pigment pool ‘‘2’’ that is solely due to absorbed energy (directly by ‘‘2’’ and also via ‘‘3’’): J ¼ J2 þ J3 p32 =ð1 p33 Þ ¼ J2 þ J3 k32 =ðk32 þ k31 þ k3D Þ ð29Þ We remind that C ¼ p23 p32 and T ¼ p2b pb2 . We also note that, according to the simplifications employed, cl Top = zero (since pop b2 ¼ 0) and T stands for T , which is cl equal to p2b (since pb2 ¼ 1). The corresponding expressions for the energy influxes in the pigment sites ‘‘3’’ and b can be deduced by substituting Eqs. 27 and 28 into Eqs. 21 and 22, or into Eqs. 25 and 26, respectively.
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Energy influxes in open and closed PSUIIs for any mixture of open and closed PSUIIs Solving the system of Eqs. 20–25 for a mixture of open and closed PSUIIs (any value of B between 0 and 1), we get the expressions for the energy influxes E2op and E2cl and, conop ¼ comitantly, for the total influxes in all open units, E2;tot cl ð1 BÞE2op and in all closed units, E2;tot ¼ BE2cl :
and ‘‘3’’ ? ‘‘2’’. Therefore, the second term was denoted as the ‘‘overall grouping probability pG’’ (Strasser 1978, 1981): 1 p33 p22 þ C pG ¼ ð33Þ 1C 1 p33 Hence, Eq. 32 can be equivalently written as " # cl E2;M CHYP ¼ op 1 pG E2;0
ð320 Þ
op E2;tot ¼
Jð1 T CÞð1 BÞ ð1 BÞð1 T CÞ½1 p22 C=ð1 p33 Þ þ Bð1 CÞ½1 T p22 C=ð1 p33 Þ
ð30Þ
cl E2;tot ¼
Jð1 CÞB ð1 BÞð1 T CÞ½1 p22 C=ð1 p33 Þ þ Bð1 CÞ½1 T p22 C=ð1 p33 Þ
ð31Þ
Note The above equations are of course valid for any value of B, including the extremes B = 0 and B = 1, as can be easily found by the according substitutions. Equations 30 and 31 are equivalently written in terms of op cl E2;0 and E2;M (from Eqs. 27 and 28; derivation steps not shown), as: ð1 þ CHYP Þ op op E2;tot ¼ E2;0 ð1 BÞ 1 þ CHYP ð1 BÞ ð300 Þ 1 þ CHYP op op E2 ¼ E2;0 1 þ CHYP ð1 BÞ 1 cl cl E2;tot ¼ E2;M B 1 þ CHYP ð1 BÞ ð310 Þ 1 cl cl E2 ¼ E2;M 1 þ CHYP ð1 BÞ
Setting as equal to zero any of the probabilities comprising pG leads to models of lower complexity (as shown in Fig. 1), with pG transformed accordingly. For example, in the extreme case of no grouping between the PSUIIs units (p22 ¼ 0 and p33 ¼ 0), i.e. if the model of separate units is considered, CHYP becomes equal to zero and the hyperbolic dependence of the total influxes on the fraction B of closed RCs (Eqs. 300 and 310 ) degenerates to a op cl and E2;M linear dependence, with the expressions of E2;0 (from Eqs. 27 and 28) also simplified (superscript ‘‘un’’ standing for ‘‘ungrouped’’):
which are hyperbolic functions of the fraction B of closed RCs, with CHYP the curvature constant (C for constant and subscript ‘‘HYP’’ for hyperbola; Strasser 1978, 1981), equal to: " # cl E2;M 1 p33 CHYP ¼ p22 þ C ð32Þ op 1 1C 1 p33 E2;0
From energy fluxes to PSII fluorescence signals
The expressions for the energy influxes in the antenna pigment pool ‘‘3’’ or the RC—b can be deduced by substituting Eqs. 300 and 310 into Eqs. 21 and 22, or into Eqs. 25 and 26, respectively. It is worth noting that Eq. 32 formulates CHYP as the product of two terms, the first one defined by the energy influxes at the extremes (B = 1 and B = 0) and the second one comprising the probabilities of all types of energetic communication among the PSII pigment pools (within a PSUII and among PSUIIs), i.e. ‘‘2’’ ? ‘‘2’’, ‘‘2’’ ? ‘‘3’’, ‘‘3’’ ? ‘‘3’’
123
op;un op E2;tot ¼ ð1 BÞE2;0 ¼ ð1 BÞðJ2 þ J3 p32 Þ
ð34Þ
cl;un cl ¼ BE2;M ¼ BðJ2 þ J3 p32 Þ E2;tot
ð35Þ
The expressions for the energy outfluxes as fluorescence emission from the open and the closed PSUIIs (core antenna and LHC II) in a mixture of open and closed units, op op cl cl i.e. E2F;tot , E3F;tot , E2F;tot and E3F;tot , are derived by applying the general Eq. 13 (Eij ¼ Ei pij ) EiF ¼ Ei piF ). In the following, we assume (as in the original article; see also the section ‘‘What about the fluorescence from the LHCII?’’ below) that the experimental fluorescence signals emerge only from antenna pigment pool ‘‘2’’; therefore we will write simply F instead of F2. Though only a fraction of the fluorescence outfluxes reaches the light detector, the fractional (geometrical) factor can be taken as equal to 1 since the measured fluorescence intensities are in arbitrary units. Hence, the fluorescence signals (intensities) from all the open (F op ) and all the closed (F cl ) units are written as:
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op op F op ¼ E2F;tot ¼ E2;tot p2F
ð36Þ
cl cl ¼ E2;tot p2F F cl ¼ E2F;tot
ð37Þ
Accordingly, the fluorescence signals at the extremes (for B = 0 and B = 1) are written as: op F0 ¼ E2;0 p2F
ð38Þ
cl p2F FM ¼ E2;M
ð39Þ
It is worth showing here the transformation of Eqs. 38 and 39 to expressions in terms of de-excitation rate constants, for which Eqs. 27 and 28 and the general Eq. 11 were utilized: F0 ¼
FM ¼
J k2F k2b þ k21 þ k2D þ k23 ðk3D þ k31 Þ=ðk32 þ k31 þ k3D Þ ð40Þ J k2F k21 þ k2D þ k23 ðk3D þ k31 Þ=ðk32 þ k31 þ k3D Þ
ð41Þ
Rearrangement of the denominator of Eq. 40 as k3D k2b þ k2D þ k23 k32 þ k31 þ k3D k31 þ k21 þ k23 ; k32 þ k31 þ k3D
FM ¼ J
kF kN
F op ¼ F0 F cl ¼ FM
ð1 BÞð1 þ CHYP Þ 1 þ CHYP ð1 BÞ B 1 þ CHYP ð1 BÞ
F0 ð1 þ CHYP Þð1 BÞ þ FM B 1 þ CHYP ð1 BÞ B ¼ F0 þ ðFM F0 Þ 1 þ CHYP ð1 BÞ
ð42aÞ ð42bÞ
F ¼ F op þ F cl ¼
expresses it as the sum of three terms: The first is the rate constant k2b; since pop bP ¼ 1, k2b determines the primary photochemistry of PSII and can be hence written as kP. The second term determines dissipation from ‘‘2’’ both directly and via transfer to ‘‘3’’. The third term determines migration from ‘‘2’’ to ‘‘1’’ both directly and via transfer to ‘‘3’’. Since the last two terms refer to outfluxes not used for PSII photochemistry, their sum can be replaced by kN (subscript ‘‘N’’ stands for any destination site that does not belong to PSII; as also used for defining k2N and k3N in Fig. 2). Accordingly, the denominator of Eq. 41 is equal to kN. Hence, Eqs. 40 and 41 can be equivalently written as: kF F0 ¼ J kP þ kN
can not affect their exciton density since the exchanged energy between any two of them is the same in both directions. (ii) Though the model of Fig. 2C assumes energy cycling between RC and core antenna, the exchanged energy between them is the same in both directions due to the adopted simplification that there is no energy dissipation from a closed RC; hence, in the final equation, the net result is the same as if there was no energetic communication between ‘‘2’’ and b in closed units. Multiplying both sides of Eqs. 300 and 310 with p2F and using Eqs. 36–39, we get the relation with F0 and FM of the fluorescence intensities Fop and Fcl in a mixture of open and closed PSUIIs and, concomitantly of the total fluorescence F of the whole illuminated sample, i.e. of the experimentally measured fluorescence intensity (emitted by PSII):
ð400 Þ
ð43Þ
The above equation show that the fluorescence signals F and Fcl and, concomitantly their sum, F, are hyperbolic functions of B, which, during the fluorescence rise from F0 to FM, is a function of time t. Using the definitions of the maximum variable florescence, FV ¼ FM F0 , and the actual (for any value of B, i.e. at any time t during the fluorescence induction) variable fluorescence, Ft ¼ F F0 , Eq. 43 gives op
Ft F F0 ¼ FV
B 1 þ CHYP ð1 BÞ
ð44Þ
Hence, the relative variable fluorescence V, defined as V Ft =FV
ð45Þ
is expressed as a function of B by ð410 Þ
We see that Eqs. 400 and 410 are the same with those formulated in the early 1960s’ for the simple model of separate packages, which further assumed that energy transfer to the RC is blocked in closed photosynthetic units (see Kitajima and Butler 1975a). The degeneration of the model of Fig. 2C to the simplest, concerning F0 and FM, of all models, is due to: (i) At each of the extremes F0 and FM, all PSUIIs are at the same state (all open or all closed) and have, hence, the same exciton density; therefore, grouping
V¼
B 1 þ CHYP ð1 BÞ
ð46Þ
which is a hyperbola with vertical asymptote. Curves of V = f(B), calculated by numerical substitutions in Eq. 46 are presented in Fig. 4 for selected values of CHYP, as indicated. The case of separate packages (CHYP = 0) is also presented (straight line). Moreover, the case of the lake model is depicted (CHYP = 4; to be discussed in the next section). The difference of each of the curves from the curve referring to separate packages, which
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CHYP ¼ Gain
Relative variable fluorescence (V)
4 2
0.8
Hence, Eqs. 47 and 460 are equivalently written as:
0.4
V¼
1 0.4
0.6
0
1
B
0
0
0.4
separate packages
0.4 1
2
4
CHYP
Gain
0.2
lake model
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Fraction of closed RCs (B) Fig. 4 Curves of V = f(B) calculated by numerical substitutions in Eq. 46, for selected values of CHYP (as indicated). The case of ungrouped PSUIIs (pG ¼ 0 ) CHYP ¼ 0) is also presented (straight line). Moreover, the case of lake model is depicted, for which CHYP = FV/F0 = kP/kN, here set as equal to 4, as for all presented curves. The difference of each of the curves from the straight line, equal to B–V and denoted as gain, is presented vs. B in the insert of Fig. 3
is equal to B–V, is presented vs. B in the insert of Fig. 4; this difference is the Gain (see next sections). Note The derivation and formulation presented here comes from Strasser (1978, 1981) and Strasser and Greppin (1981). However, as already mentioned, Joliot and Joliot (1964) were the first to propose PSUIIs energetic connectivity and formulate the hyperbolic relation between fluorescence and fraction of open RCs (1 - B; according to our notation). The equivalence of the apparently different formulations of the two approaches will be shown later in this article. op cl Since F0 =FM ¼ E2;0 =E2;M (see Eqs. 38 and 39), the curvature constant, CHYP, given in Eq. 320 can be equivalently written as: FM FV CHYP ¼ 1 pG ¼ pG ð47Þ F0 F0 Hence, Eq. 46 writes as: V¼
B B ¼ 1 þ CHYP ð1 BÞ 1 þ ½ðFV =F0 ÞpG ð1 BÞ
ð460 Þ
Recalling Eqs. 400 and 410 , we get the link (equality) between the experimentally determined FV/F0 ratio and the biophysical ratio kP/kN: FV FM kP þ kN kP ¼ 1¼ 1¼ F0 F0 kN kN
123
ð48Þ
kP pG kN
B 1 þ ½pG ðkP =kN Þ ð1 BÞ
ð470 Þ ð4600 Þ
The factors affecting the curvature constant will be discussed below, in the section ‘‘Changes of CHYP: What information can we obtain?’’ Note For all curves in Fig. 4 we considered FV/F0 = kP/ kN = 4. Hence, according to Eq. 47, for CHYP = 0, 0.4, 1 and 2, the corresponding pG values were 0, 0.1, 0.25, and 0.5; strictly speaking, no pG value can be attributed to the lake model, for which CHYP = 4 (see next section). Using Eq. 46, we can now rewrite Eqs. 300 –310 and 42– 43 as: op op E2;tot ¼ E2;0 ð1 VÞ
ð49Þ
cl cl ¼ E2;M V E2;tot
ð50Þ
F op ¼ F0 ð1 VÞ
ð51Þ
cl
F ¼ FM V
ð52Þ
F ¼ F0 ð1 VÞ þ FM V ¼ F0 þ VðFM F0 Þ
ð53Þ
Notes (i) Eq. 53 is an identity, as it expresses the definition of V (: Ft/FV; see Eq. 45). (ii) Equations 51 and 52 are very useful for the deconvolution of the fluorescence kinetics into the fluorescence kinetics of open and of closed units (Strasser 1978, 1981), as will be shown later. Quantum yield of PSII primary photochemistry On the basis of the Eqs. 13 and 19, applied for the model of highest complexity (Fig. 2A or B) and for any mixture of open and closed PSUIIS, the energy outflux for PSII primary photochemistry, denoted according to our nomenop clature as EbP;tot (or simply as EbP;tot since it refers only to open RCs) is formulated (using Eqs. 25 and 26) as: op op op op EbP;tot ¼ Eb;tot pop bP ¼ E2;tot p2b pbP
ð54Þ
With the simplifications adopted in the original paper (Strasser 1978) that led to the model of Fig. 2C, pop bP is equal to 1, since for an open RC no backwards energy flux to core antenna and no energy dissipation are considered. Moreover, as already pointed out when discussing Eqs. 21– 0 26, pop 2b is simply written as p2b . Hence, and using Eqs. 30 and 49, Eq. 54 writes: op op EbP;tot ¼ Eb;tot ¼ E2;tot p2b ð1 þ CHYP Þ op ¼ E2;0 p2b ð1 BÞ 1 þ CHYP ð1 BÞ op ¼ E2;0 p2b ð1 VÞ
ð55Þ
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By definition, the quantum yield for primary photochemistry, for which the notation uP is used here (as in all our previous publications), is defined as the rate of primary photochemistry (here the energy outflux EbP;tot divided by the absorbed light intensity (flux), J. Hence, uPt ¼
op E2;0 p2b ð1 Vt Þ J
ð56Þ
where both uPt and Vt refer to the certain (but any) mixture of open and closed RCs, at a certain (but any) time during illumination of the photosynthetic sample (hence the subscript t). When all RCs are open (V = 0), EbP;tot and uPt are denoted as EbP;0 and uPo, respectively and, accordingly, written as: op EbP;0 ¼ E2;0 p2b
uPo ¼
op E2;0 p2b J
ð550 Þ ð560 Þ
From Eqs. 56 and 560 we get uPt ¼ uPo ð1 Vt Þ
ð57Þ
which degenerates to uun Pt ¼ uPo ð1 Bt Þ
ð570 Þ
for the case of ungrouped PSUIIs (superscript ‘‘un’’), i.e. of separate packages. We note that, according to our formulations, the absorbed light energy flux J is given by Eq. 29. However, since J cancels in the derivation of Eq. 57, the latter is valid for any expression of J. Below we will examine and discuss Eqs. 56, 560 , and 57: Like for the derivation of Eqs. 40 and 400 , Eq. 560 can be transformed to equivalent expression in terms of deexcitation rate constants, as k2b k2b þ k21 þ k2D þ k23 ðk3D þ k31 Þ=ðk32 þ k31 þ k3D Þ kP ð58Þ ¼ kp þ kN
uPo ¼
and to equivalent expressions in terms of fluorescence signals (using Eqs. 400 and 410 ), as uPo ¼ 1
F0 F M F 0 FV ¼ ¼ FM FM FM
Concomitantly, Eq. 57 writes: FM Ft upt ¼ uPo ð1 Vt Þ ¼ uPo FM F0 FV FM Ft ðFM Ft Þ Ft ¼ ¼1 ¼ FM FM FV FM
ð59Þ
ð60Þ
It is worth reminding in brief the ‘‘history’’ of the above equations. Equation 59 is the well-known Butler’s
formula (see e.g. Kitajima and Butler 1975a), derived both for the case of separate units, as well as for the lake model (called in their paper as ‘‘matrix’’; for this model see the below section). Paillotin (1976b), following a different approach, derived, like in Eq. 60, the equation uPt = uPo[(FM – Ft)/(FM – F0)] (written here with the symbols that we use, to facilitate the comparison). Applying the EFT, Strasser (1978) derived the equations, as here presented. It should be emphasized that Eq. 60 is the general equation expressing the actual quantum yield of photochemistry at any time t and at any state of the photosynthetic sample. For example, if we consider that the symbols as used in Eq. 60 refer to a dark-adapted sample, we can similarly write the following equation for any time t during light-adaptation (marking the parameters, for distinction, with the prime symbol): 0 0 0 0 FM Ft ðF F Þ 0 0 0 0 uPt ¼ uPo ð1 Vt Þ ¼ uPo 0 ¼ M 0 t 0 FM F0 FM 0 Ft ¼1 0 ð600 Þ FM Moreover, applied for the light-adapted steady-state ‘‘S’’ in the Kautsky transient (Kautsky and Hirsh 1931), Eq. 600 writes: 0 0 0 0 FM FS ðFM FS Þ 0 0 0 0 uPS ¼ uPo ð1 VS Þ ¼ uPo 0 ¼ 0 0 FM F0 FM 0
¼1
FS 0 FM
ð600 Þ
0 0 0 0 Note that the term ðFM FS Þ=ðFM F0 Þ , equal to 0 ð1 VS Þ according to the general equation (Eq. 60), is identical with the so-called photochemical quenching, qQ or qP, later introduced and commonly used for studies at the light-adapted steady-state S. However, in those studies, Eq. 6000 is known as Genty’s formula (Genty et al. 1989), introduced to express the quantum yield of electron transport (uPSII or ue) and equivalently written as uPSII (or ue) = DF/ FM (where DF = FM – FS). It needs therefore to be clarified that, fundamentally, Eq. 6000 expresses, as a special case of Eq. 60, the actual quantum yield of primary photochemistry 0 uPS at the light-adapted steady-state S and, consequently the quantum yield of electron transport uPSII, only because at the steady state the flux for electron transport is equal to the flux for primary photochemistry (hence, the yields are equal: 0 uPSII = uPS ). Note Obviously, at any other phase of the Kautsky transient, during which fluorescence intensity is increasing (closure of RCs) or decreasing (reopening of 0 closed RCs), the actual quantum yield of photochemistry uPS is bigger or smaller than uPSII, respectively; this is why the
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Genty’s formula is applicable only for the steady state (plateau) of the fluorescence kinetics. The big importance of Eq. 57 and consequently of Eq. 60, is that they link, independently of the model used (also for the lake model; see the relevant note below), the actual quantum yield of primary photochemistry with fluorescence experimental signals. Moreover, Eq. 57, which expresses the relation between the actual (uPt ) and the maximum (uPo ) quantum yield of primary photochemistry, also summarizes the essence of energetic connectivity for the ‘‘economy’’ of photosynthesis, showing clearly that, for the same B, the actual quantum yield of primary photosynthesis when the PSUIIs are grouped, ugPt ¼ uPo ð1 Vt Þ is bigger than when they are ungrouped, uun Pt ¼ uPo ð1 Bt Þ: For the comparison of the two cases (both with the same otherwise structure that defines uPo) the term gain factor was introduced by Strasser (1981) as /Pt =/un Pt ¼ ð1 Vt Þ=ð1 Bt Þ ¼ ð1 þ CHYP Þ=½1 þ CHYP ð1 Bt Þ. The gain factor, which by definition is equal to unity for the case of ungrouped units, attains bigger values as CHYP becomes bigger. For a given value of CHYP, the gain factor increases with the increase of the fraction B, getting values from 1 when B is minimal (B = 0) up to 1 þ CHYP when B is maximal (B = 1). In other words, as the fraction of closed RCs increases, the total energy influx in the each of the remaining open PSUIIs increases. However, it should be taken into account that the gain factor is expressed per open RC, hence when it reaches its maximal value there are no more open RCs to benefit. Therefore, for the realistic visualization of the benefit, we used instead, and plotted in the insert of Fig. 4, the dif ference (Bt – Vt), which, being equal to uPt uun Pt =uPo is the real Gain: it expresses the fraction of uPo that, for a certain Bt, is gained for photochemistry because of the grouping of the PSUIIs.
The lake model We deemed that, since we often refer to the lake model, it would be useful for the reader to remind here the according equations for the fluorescence intensity and the quantum yield of primary photochemistry, as originally deduced (see e.g. Kitajima and Butler 1975a): kF kP ð1 BÞ þ kN
ð61Þ
kP ð1 BÞ kP ð1 BÞ þ kN
ð62Þ
Ft ¼ J uPt ¼
It should be clarified that Eqs. 61 and 62 assume energy exchange between core antenna and RC similar to those in the simplified model of Fig. 2C and, accordingly, kP and kN
123
have the same meaning as discussed for Eq. 400 . We will later derive the equations for a lake model with full complexity in respect to the energy exchange between core antenna and RC (like in the model of Fig. 2B). For B = 0 and B = 1, Eq. 61 coincides with Eqs. 400 and 410 , respectively and, for B = 0, Eq. 62 coincides with Eq. 59, as expected since the extremes do not depend on the extent of energetic connectivity. Though the lake model is conceptually different than that of Fig. 2C, hence the differing expressions for Ft and uPt (for 0 \ B \ 1), the relation of uPt with uPo is again expressed by Eq. 57. On the basis of Eqs. 61, 400 and 410 , the equation for the relative variable fluorescence is derived as: Vlake ¼
B B ¼ 1 þ ðFV =F0 Þð1 BÞ 1 þ ðkP =kN Þð1 BÞ
ð63Þ
The form of Eq. 63 is the same as that of Eqs. 46 and 46 , with the difference that CHYP ¼ FV =F0 ¼ kP =kN . So, it is as if pG equals 1; we emphasize the ‘‘as if’’, because the apparent equality is technical but not conceptually correct. It is obvious that the CHYP corresponding to the lake model attains the maximal possible value. We here remind that for all different values of CHYP used in constructing the theoretical curves V = f(B) in Fig. 4 we assumed FV =F0 ¼ kP =kN ¼ 4; hence, the curve for the lake model is characterized by CHYP = 4 (as indicated in Fig. 4). 0
Yields, probabilities, and efficiencies An important magnitude in studies of any energy transformation is the yield. In photosynthesis we deal mainly with two yields, commonly symbolized by ‘‘u’’ (or ‘‘U’’): the photochemical yield referring to the energy utilized for photochemistry and the fluorescence yield referring to the energy emitted as fluorescence. In both cases, it is defined as the ratio of the energy outflux in consideration to the absorbed light energy; hence, it is (or should be) more correctly called quantum yield. The quantum yield is often considered as synonymous with efficiency, probably due to an extension from Mechanics, where the efficiency of a machine is defined on the basis of the total energy input. However, care should be taken because the two terms are not necessarily equivalent. The key point is the necessity to distinguish the macro from the micro: The quantum yield is synonymous with the efficiency of the whole photosynthetic machine (macro), because the absorbed light energy flux is the only energy influx to the machine (the whole system communicates energetically only with the environment). On the other hand, the efficiency of a certain component of this machine (micro), i.e. of a pigment system, is the ratio of the energy outflux in consideration to the total energy influx in the certain component. The EFT postulates (and formulates)
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that the absorbed light energy flux is not necessarily the only component of the total energy influx to a pigment system, because of the other pigment assemblies with which the system in consideration is energetically communicating (further than communicating with the environment). Hence, the efficiency of a pigment system for a certain energy transduction is synonymous to the probability that the energy outflux follows the certain transduction path and not to the quantum yield. The distinction between quantum yield and probability is clearly reflected in Eq. 560 , which shows that the two magnitudes are not identical.
op p2b dB=dt ¼ ð1 BÞE2op p2b ¼ E2;tot ð1 þ CHYP Þð1 BÞ ¼ JuPo 1 þ CHYP ð1 BÞ
Using Eqs. 46 and 65a, we can also deduce the dV/dt as a function of V: dV=dt ¼
The redox state of QA is determined by its photochemical reduction due to PSII activity and its re-oxidation by the electron transport driven by PSI activity. In order to reduce the complexity of the in vivo system and facilitate the investigation of PSII properties, the utilization of 3-(3,4dichlorophenyl)-1,1-dimethylurea (DCMU), a post-QA inhibitor of electron transport, has been widely employed. In the presence of DCMU at room temperature, the Chl a fluorescence rise kinetics (fluorescence transients or induction curves) F = f(t) reflects pure photochemical events leading to the complete reduction of QA (full closure of RCs). Hence, the maximal recorded fluorescence FP is equal to FM. Under normal conditions QA is assumed to be completely (re)oxidized in the dark, i.e. all RCs (re)open, and the fluorescence signal at the onset of illumination is F0. Due to energetic connectivity of PSUIIs (including the case of the lake model), the Chl a fluorescence transients of DCMU-treated samples have a sigmoidal shape instead of the exponential expected for separate units. Joliot and Joliot (1964) were the first to interpret in this way the experimentally observed sigmoidicity, followed then by Paillotin (1976b) and Strasser (1978, 1981). We will here re-derive B = f(t) and, there from, V = f(t) on the basis of our above formulations for the Fig. 2C model. The rate of QA reduction is written (taking also in consideration pop bP ¼ 1Þ as: op op op op op d½Q A =dt ¼ ½QA Eb pbP ¼ ½QA E2 p2b pbP ¼ ½QA E2 p2b
ð64Þ Dividing both sides by the total concentration of the primary quinone acceptor, ½Qtotal A and using Eqs. 55–57 (or Eqs. 300 and 46), we get op p2b ¼ JuP ¼ JuPo ð1 VÞ dB=dt ¼ ð1 BÞE2op p2b ¼ E2;tot
ð65aÞ
JuPo ð1 VÞð1 þ CHYP VÞ2 ð1 þ CHYP Þ
ð66Þ
Equation 65b gives, by integration, the t = f(B) equation (note that the B = f(t) can not be derived explicitly): t¼
The equation of the fluorescence transient in the presence of DCMU
ð65bÞ
1 f lnð1 BÞ þ CHYP Bg J uPo ðCHYP þ 1Þ
ð67Þ
Substitution of B from Eq. 46, or integration of Eq. 66, leads to the t = f(V):
1 1V CHYP ðCHYP þ 1ÞV t¼ ln þ JuPo ðCHYP þ 1Þ 1 þ VCHYP 1 þ VCHYP ð68Þ For separate units, Eqs. 67 and 68 become identical (V = B), written as t¼
1 1 f lnð1 BÞg ¼ f lnð1 VÞg JuPo JuPo
ð69Þ
that can be now explicitly written as V = B = f(t), which is an exponential function, V ¼ B ¼ 1 eJuPo t
ð690 Þ
with Eq. 66 degenerating to dV=dt ¼ JuPo ð1 VÞ
ð660 Þ
In Fig. 5, the V = f(t) kinetics, as well as the B = f(t) kinetics (insert), were calculated by numerical substitutions in Eqs. 69 and 68, respectively, for the same CHYP values as in Fig. 4. The value of uPo was taken as 0.8, so that it corresponds to FV/F0 = 4 that was set for the calculations of the curves in Fig. 4. It should be pointed out that sigmoidicity is a feature only of V = f(t), while the B = f(t) kinetics proceeds faster than the exponential rise (separate package), with the rate of B increase being bigger for higher CHYP values, as expected because of the bigger gain (Fig. 4), i.e. of the bigger utilization of the absorbed light flux. It is worth pointing out that for CHYP smaller than a threshold value (and bigger than zero), the transient would not show the typical inflection characterizing a sigmoidicity, though it would still deviate from the exponential shape (characteristic of the separate packages). The threshold value of CHYP is therefore the minimum value that permits dV/dt, i.e. the first derivative of V = f(t), to exhibit a maximum or, equivalently, the second derivative,
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Relative variable fluorescence (V)
1.0
0.8
1.0 4
0.8
0.6
0
CHYP
0.6
B 0.4
0.4 0
4
CHYP 0.2 0.0
0.2
0.0
0.2
0.4 t 0.6
0.8
1.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Time (t) Fig. 5 Kinetics of the relative variable fluorescence, V = f(t), and of the fraction of closed RCs, B = f(t), in the insert, which were calculated for the same values of CHYP as in Fig. 4 (i.e. 0, 0.4, 1, 2, and 4) by numerical substitutions in Eqs. 68 and 67, respectively (derived for DCMU-treated samples). In accordance with FV/F0 = 4 (see legend of Fig. 4), the value of uPo used in the equations was 0.8
d2V/dt2, to become equal to zero, at a realistic value of V (let it be written as Vinfl). This means that the following equation should have a solution: d2 V d dV dV d dV ¼ ¼ 2 dt dt dt dV
dt dth i dV JuPo d ð1 VÞð1 þ CHYP VÞ2 ¼ dt ð1 þ CHYP Þ dt ¼
dV JuPo f2CHYP 3CHYP V 1Þð1 þ CHYP VÞg dt ð1 þ CHYP Þ
¼0
ð70Þ
The solution of Eq. 70 (excluding V = 1 that diminishes dV/dt, and which, anyway, is not of interest) is: 2CHYP 3CHYP Vinfl 1 ¼ 0 , CHYP ¼ 1=ð2 3Vinfl Þ , Vinfl ¼ ð2CHYP 1Þ=3CHYP ð71Þ which shows that only if CHYP [ 0.5 the fluorescence transient does exhibit an inflection point (and the dV/ dt does exhibit a maximum). As an example, we included in Fig. 5 the curve for CHYP = 0.4, which is, indeed, not sigmoidal. We will discuss below, in the section ‘‘Comparison with the formulations of Paillotin (1976b, 1977)’’, the equivalency of Eq. 71 with the prerequisite for sigmoidal shape as formulated by Paillotin. From Eq. 71 we get also the information that, the bigger the CHYP the bigger is the Vinfl, which nevertheless cannot exceed, from the mathematical point of view, the value of 2/3.
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However, the physiological threshold of Vinfl is even lower: since the maximum possible CHYP is equal to FV/F0 (lake model) we get (from Eq. 71) that Vinfl ½2ðFV =F0 Þ 1=ð3FV =F0 Þ, i.e. Vinfl ½2=3 ðF0 =3FV Þ. A proposition for the determination of CHYP by utilizing Eq. 71 will be presented in a following section. As we mentioned above, referring to cases that CHYP \ 0:5 (and = 0, so that Eq. 70 applies), the fluorescence kinetics, though not being sigmoidal, they still deviate from the exponential shape. This is an issue that needs to be emphasized because, in several publications, any fluorescence transient (of DCMU-treated photosynthetic material) that does not appear sigmoidal was characterized as exponential. It should be taken in consideration that an exponential transient fulfills certain criteria; according to Eqs. 660 (or 69 or 690 ), the following plots are straight lines: (i) the V vs: B : V ¼ B; (ii) the lnð1 VÞ vs. t : lnð1 VÞ ¼ JuPo t; (iii) the dV=dt vs. V : dV=dt ¼ ð1 VÞJuPo . Care should also be taken when deducing whether a fluorescence kinetics has inflection point, as it may need more than a visual examination of the transient. E.g., the kinetics for CHYP = 1 in Fig. 5 does not ‘‘look’’ to be sigmoidal but, as shown in Fig. 4, the V vs. B for this case is a hyperbola; moreover, as can be calculated from Eq. 71, this kinetics has indeed an inflection point (Vinfl = 1/3).
Changes of CHYP: what information can we obtain? Equations 47 and 470 show that the curvature of the hyperbola V = f(B) is defined both by the overall grouping probability pG and by the FV/F0 ratio, or equivalently, by the ratio kP/kN of de-excitation rate constants that do not depend on grouping. E.g., under conditions that cause or increase nonphotochemical quenching on the antenna pigment pools (increase of k2D and/or k3D), or spill-over enhancement (increase of k21 and/or k31), kN also increases (concomitantly, kP/kN decreases) since, by definition (see legend of Fig. 2 and relevant text), kiN ki1 þ kiD ¼ ki1 þ ðkiH þ kiF þ kiQ Þ. In such cases, pG will also be affected, not because of a change in the connectivity rate constants k22 and/or k33, but because it is comprised of probabilities (Eq. 33) that are also determined by the other de-excitation rate constants k2j and/or k3j (Eq. 11). Hence, the conclusion found in several publications that an increase of CHYP indicates that the photosynthetic units come physically closer to one another (increase of k22 or k33) is an overestimation of the information that the experimental signals can provide, if not a completely wrong interpretation of the signals. The case of inactivation of a fraction of RCs and the impact on CHYP needs a different analysis. It is worth mentioning here that this issue has been tackled also by
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303
Sorokin (1985), who proposed a correction of Paillotin’s formula in order to make it applicable for the case of a partial RCs’ inactivation; in the section ‘‘Comparison with the formulations of Paillotin (1976b, 1977)’’ we will discuss Sorokin’s correction in comparison with our analysis, presented below.
1.0
V' = f(B')
0.8
V , V'
(Bi' , Vi' )
0.6
Inactivation of RCs and the impact on CHYP
(x,Vx)
0.4
As we will present in the section ‘‘Inactive or silent centers (non-QA-reducing)’’, based on results that we obtained in studies of the effect of certain stresses on PSII behavior by analyzing the OJIP transient with the JIP-test, we deduced and proposed that those stresses caused the transformation of a fraction of RCs to ‘‘inactive’’ centers (which we have also termed as ‘‘silent’’), in the sense that they are still acting as efficient exciton traps but are dissipating, instead, the whole of the energy outflux that would be otherwise used for photochemistry; i.e., they become non-QA-reducing (see e.g., the review by Strasser et al. 2004). This definition means that the conformation of an inactive centre compared to that of open RC given in the models of Fig. 2 (panels B, C) is the substitution of kbP by a dissipation rate constant kbQ of equal magnitude. Concomitantly, the inactive centers, throughout the fluorescence induction, behave in respect to their fluorescence yield as open RCs but cannot get closed. The proposition for the existence of such inactive centers came originally from Sorokin (1985)—who called them ‘‘quenching centers’’—and from Cleland et al. (1986)—who called them ‘‘photodestructed’’ or ‘‘photoinhibited’’—and was integrated in the work of Krause on photoinhibition, where the formation of ‘‘heat sinks’’ (as they were called there) was considered as a possible protective mechanism (see e.g. Krause et al. 1990). It is worth adding here that the inactivation of RCs contributes also to the release of closed PSUIIs from excess excitation energy, since, via grouping, they supply energy to PSUIIs with inactive centers that, hence, dissipate it. Let us denote by (1 - x) the fraction of inactivated RCs, hence by x the fraction of the still active. Let us also plot (Fig. 6; in the B–V axes) the hyperbola V = f(B) that refers to the non-quenched state (all RCs active) and was hence numerically constructed according to Eq. 46 (i.e., similarly to those of Fig. 4 and here with CHYP = 3). Since the fraction of active RCs can not exceed x, the range of the realistic B and V values is limited between the point (0,0) and the point (marked by a closed circle) with coordinates (x,Vx); i.e., the graph is within the shaded rectangular. According to Eq. 46 x ð72Þ Vx ¼ 1 þ CHYP ð1 xÞ
(Bi , Vi )
0.2
V = f(B) 0.0 0.0
0.2
0.4
0.6
0.8
1.0
B , B' Fig. 6 A demonstration of how the inactivation of a fraction (1 - x) of RCs transforms the V vs. B to the V0 vs. B0 hyperbolic curve, with the latter characterized by a lower curvature than the former. For details, see text
For any point in the realistic part of the hyperbola, let it be the point marked by an open circle with coordinates (Bi,Vi), Eq. 46 writes as Vi ¼
Bi 1 þ CHYP ð1 Bi Þ
ð73Þ
If we now ‘‘blow up’’ the shaded rectangular in Fig. 6, so that the (x,Vx) point ‘‘moves’’ to the point marked by the closed square, with coordinates (1,1) in the B0 –V0 axes, the realistic part of the V = f(B) curve is transformed to the curve V0 = f(B0 ), with a smaller curvature than V = f(B), 0 0 and the (Bi, Vi) point ‘‘moves’’ to the (Bi , Vi ) point (marked by the open square). In other words, Bi has been normalized on x and Vi on Vx: 0
0
Bi ¼ Bi =x and Vi ¼ Vi =Vx
ð74Þ
Using Eqs. 72 and 73, we get Bi B0i 1 þ C HYP ð1 Bi Þ Vi0 ¼ ¼ x 1 þ CHYP ð1 xB0i Þ 1 þ CHYP ð1 xÞ 1 þ CHYP ð1 xÞ B0i ¼ xCHYP ð1 B0i Þ 1þ 1 þ CHYP ð1 xÞ
ð75Þ
Equation 75 has the same form as Eq. 72 (or Eq. 46), 0 however, with a different curvature constant, CHYP : 0
CHYP ¼
xCHYP 1 þ CHYP ð1 xÞ
ð76Þ
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ð760 Þ
Since it has been assumed (by definition) that F0 is not affected by the inactivation and reminding that Vx ¼ Ft;x =FV , where Ft,x is equal to the maximal variable fluorescence observed after inactivation (i.e. to FV0 after normalization), Eq. 760 is written as:
FV FV Ft;x 0 CHYP ¼ Vx p G pG F0 F0 FV
0 Ft;x FV ¼ ð7600 Þ pG ¼ pG F0 F0 Hence, Eq. 75 writes, for any B0 (\x): B
0
V ¼
0
0
1 þ CHYP ð1 B0 Þ
0
¼
B 0 1 þ ½pG ðFV =F0 Þ ð1 B0 Þ ð77Þ
This means that the basic formula (Eq. 460 ) is valid also after inactivation of a fraction of RCs and, concomitantly, pG can be determined as before inactivation (see ‘‘Experimental determination of the overall grouping probability pG’’ section). It also permits us to use Eq. 460 for the case of untreated samples, without questioning whether all RCs are active, and to simply consider it as the reference case (with x = 1). Moreover, the result of the above analysis has a general importance, as it proves that Eq. 460 is applicable and pG can be determined even if all RCs are active but the true FM (all RCs closed) is not reached experimentally: We can choose any F value in the fluorescence rise, calculate the 0 Ft = F - F0, denote it as FV and apply Eq. 77. A relevant question is how RCs’ inactivation affect the appearance of inflection point in the fluorescence induction kinetics, the prerequisite for which is CHYP [ 0.5 (Eq. 71). Accordingly, the prerequisite for the V0 = f(t) recorded after a treatment that caused inactivation of a fraction of 0 RCs’, is: CHYP [ 0:5. Let us take the case that the V ¼ f ðtÞ of the untreated photosynthetic material was indeed sigmoidal. Since the decrease of CHYP after inactivation does not affect pG but only the FV/F0 term, which depends on x, the question is how small x can be, or how big (1 - x) 0 can be, so that CHYP would remain bigger than 0.5. Using Eq. 76, the prerequisite is written as 0
xCHYP [ 0:5 ) 1 þ CHYP ð1 xÞ xCHYP [ 0:5 ½1 þ CHYP xCHYP ) 1:5ðxCHYP Þ [ 0:5ð1 þ CHYP Þ ) x [ ð1 þ CHYP Þ=3CHYP
CHYP ¼
123
As examples, let us take two cases of untreated samples, both with FV/F0 = 4 and both fulfilling CHYP [ 0.5: a case where the lake model applies (hence, CHYP = 4) and a case of connected units with pG = 0.2 (hence, CHYP = 0.8). Applying the above derived inequality, the fraction of active RCs (x) should be bigger than 5/12 and 3/4, respectively; i.e., the inactivated fraction should be less than *58 and *25 %, respectively.
Deconvolution of the fluorescence transient From the fluorescence experimental values F0 and FM, the relative variable fluorescence Vt ¼ ðFt F0 Þ=ðFM F0 Þ is calculated for any Ft during the course of the fluorescence transient. The fluorescence transient can thus be deconvoluted into the fluorescence kinetics of open RCs, Ftop ¼ F0 ð1 Vt Þ, and that of closed RCs, Ftcl ¼ FM Vt (Eqs. 51 and 52), as shown in Fig. 7. What is very important is that the deconvolution is permitted (a) independently of whether the PSUIIs are energetically separated or connected and, (b) not only in DCMU-treated samples (as presented in Fig. 7), but under any physiological condition (see later in this article), provided that the extremes F0 and FM can be experimentally determined.
FM Fluorescence intensity ( F, Fop, Fcl )
which, using Eqs. 72 and 47, is written as:
FV FV 0 CHYP ¼ CHYP Vx ¼ pG Vx ¼ Vx p G F0 F0
Fcl
FV
F F F0
Fop Time (t) Fig. 7 Deconvolution of a fluorescence transient (referring to a DCMU-treated sample; like in Fig. 4) into the fluorescence kinetics of open RCs, Fop, and that of closed RCs, Fcl. The variable fluorescence Ft ¼ F F0 and the maximal variable fluorescence FV ¼ FM F0 are indicated
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305
Experimental determination of the overall grouping probability pG
Rearrangement of Eq. 46 as Bt/Vt vs. Bt gives a straight line, Bt =Vt ¼ ð1 þ CHYP Þ CHYP Bt
0
0
0
Rt
The integral 0 ð1 Vt Þdt corresponds to the area between the fluorescence transient plotted as V = f(t), the horizontal line at V = 1 and the vertical line at time t, called the complementary area and denoted here as st. Theoretically for t ? ? and practically for t ¼ tFM , where tFM is the time when FM is achieved and, hence, V (as well as B) become equal to 1, the integral corresponds to the total complementary area, denoted here as smax. Therefore, writing Eq. 78 for t ¼ tFM , we get: ZtFM 1 ¼ JuPo ð1 Vt Þdt ð780 Þ 0
Combination of Eqs. 78 and 780 gives: 8 t 9,8 t 9
0
ð79Þ where, St denotes the complementary area normalized on the total. The equality Bt = St is valid independently of the extent of energetic connectivity between the PSUIIs, from separate packages to the lake model. From the experimental fluorescence signals we first calculate the relative variable florescence V and plot the V = f(t) kinetics. Then we determine, with algorithmic calculations the st (for every t) and the smax and, consequently St, hence Bt. Then the V = f(B) plot is constructed, like for the theoretically derived curves in Fig. 4. The bigger the extent of grouping, the bigger would be the deviation of the curves from the straight line (separate packages). However, this is semi-quantitative information. Concerning the V = f(t) transients from DCMU-treated samples with different extent of grouping, the higher the value of the curvature constant CHYP, the more pronounced the sigmoidal shape of V = f(t) would be (see Fig. 5). Still, this is again semi-quantitative information. In order to derive the full quantitative information, simulation of the transient must be made (see e.g. Stirbet et al. 1998). However, Strasser (1981) proposed the following simple way to calculate the overall grouping probability in DCMU-treated samples:
which can be plotted from the experimentally determined Vt and Bt (=St; Eq. 79). So, the value of CHYP can be calculated either from the slope or the intercept of the Bt/Vt vs. Bt plot. Concomitantly, pG is calculated according to Eq. 47, since FV/F0 is already experimentally available. A new method for the determination of CHYP, based on our above arguments in respect to the inflection point, is here proposed. From the experimental V ¼ f ðtÞ; we calculate the first derivative, dV/dt, which we then plot vs. V. The curve will show a maximum, ðdV=dtÞmax , at that value of relative variable fluorescence for which the V ¼ f ðtÞ shows the inflection, i.e., at Vinfl. Hence, CHYP and, thereafter pG, are calculated using Eq. 71. Figure 8 demonstrates, as examples, the dV=dt ¼ f ðV Þ derived from the V ¼ f ðtÞ kinetics depicted in Fig. 5. For the case of separate units (CHYP = 0) the dV=dt ¼ f ðV Þ is a straight line (grey line; Eq. 660 ), while the dV=dt ¼ f ðV Þ for the case of CHYP = 0.4 (small open circles) does not exhibit a maximum, as predicted from Eq. 71. For each of the other three cases (black lines), the corresponding value of Vinfl (at the maximum of dV/dt) is indicated. In addition, the ðdV=dtÞmax (indicated by *) permits, by means of Eq. 66, the determination of J, since uPo can be experimentally determined. Note For the case of separate units the plot of dV=dt ¼ f ðV Þ, given by Eq. 660 as dV=dt ¼ JuPo ð1 VÞ, is a straight line (grey line in Fig. 8), with intercept equal
5
CHYP
* : (dV/dt)max
dV/dt
By integrating Eq. 65a (we remind that it refers to DCMUtreated photosynthetic samples), we get the fraction Bt of closed RCs at any time t in the course of the experimental fluorescence transient as follows: Zt ZB Zt Bt ¼ dB ¼ ðdB=dtÞ dt ¼ JuPo ð1 Vt Þdt ð78Þ
ð80Þ
4
*
4
3
* *
2
2
1
1
0.4 0 Vinfl :
0.33
0.50 0.58
0 0.0
0.2
0.4
0.6
0.8
1.0
Relative variable fluorescence (V)
Fig. 8 The time derivative of relative variable fluorescence dV/dt vs. V, constructed from the kinetics V vs. t of Fig. 5. Each of the three curves with CHYP = 1, 2 and 4 exhibits a maximum, ðdV=dtÞmax , at the inflection point, Vinfl, of the corresponding V vs. t of Fig. 5; these curves are presented by black lines (to be distinguished from the other two) and, for each of them, the value of Vinfl is indicated and the value of ðdV=dtÞmax is also marked (with asterisk). The grey line presents the curve for CHYP = 0 (straight line), while the curve with small open circles presents the case of CHYP = 0.4
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306
to JuPo and slope equal to 1 - JuPo. Hence, the determination of J is permitted. However, Vt vs. Bt usually deviates, in practice, from the pure hyperbolic function. Figure 9 presents, as examples, the experimental results from two cases: The main plot is from recent experiments with whole leaves (conducted with a HandyPEA fluorimeter), while the insert depicts results obtained by Strasser (1978) with chloroplasts that were under high (?) or low (-) salt conditions (±MgCl2), found to cause high or low grouping, respectively (Strasser 1978, 1981; Hipkins 1978); hence, under low salt conditions, the curve would be expected to be closer to a straight line (CHYP % 0). It is interesting to notice that both experimental curves obtained from chloroplasts deviate from the theoretically expected (i.e., from the vertical hyperbola at high salt and the straight line at low salt) in a similar way; namely, they both exhibit a bending ‘‘tail’’ at higher B values. The same is true for the curve obtained from whole leaves (main plot) with a fluorimeter of high-time-resolution and much better accuracy for F0 determination. As deduced by Strasser (1978), the deviation is due to the presence of a heterogeneous mixture of different types of PSUIIs, i.e., of big (with LHCII) grouped units together with big and small (deprived of LHCII) separate units, which is reflected both in the V = f(t) kinetics (superposition of one sigmoidal and two exponential functions of time) and in the V = f(B) curve (superposition of one
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vertical hyperbola and two straight lines). It was further shown (Strasser 1978, 1981) how an appropriate deconvolution of the experimentally measured V = f(t) into the three components and the plotting of V vs. B, for each component separately, enables not only to distinguish but also to estimate the fractions of the different types in the mixture and to determine, as above explained, the pG of the grouped units. The same method was applied for the simpler case of fluorescence kinetics deprived of the sigmoidal component (obtained from photosynthetic material (spinach) grown in a phytotron under conditions that did not permit the formation of grouped PSUIIs). In that case, the method led to the deconvolution of V = f(t) in two exponential curves and the concomitant determination of the fraction of big ungrouped and small ungrouped PSUIIs. We here remind a similar analysis by Melis and Homann (1976), in that case of the area growth, i.e. of B = f(t), which was considered as having a biphasic exponential shape, though the fluorescence induction did appear sigmoidal. That analysis yielded a fast a-component and a relatively slow b-component, both appearing to be exponential and, only after subtracting the contribution of the slow component from the fast one, it was recognized that the a-component deviates from exponentiality; still, no further deconvolution was attempted. On the other hand, Joliot and Joliot (1964) did obtain experimentally the pure hyperbolic function, however, in Chlorella pyrenoidosa cells’ suspension; a possible speculation would be to attribute the different behavior to the different photosynthetic material, assuming that in the certain cultures used by Joliot and Joliot (for which no specific information is available), Chlorella chloroplasts were highly homogeneous in respect to the type of their PSUIIs.
What about the fluorescence from the LHCII? As clarified in section ‘‘From energy fluxes to PSII fluorescence signals’’, though the model of Fig. 2C leads to the formulation of the fluorescence emissions (energy outop fluxes) both from core antenna and LHCII (i.e., E2F;tot ,
Fig. 9 Main plot The relative variable fluorescence (V) vs. the fraction of closed RCs (B), constructed from the fluorescence kinetics of whole leaves; B was calculated from the complementary area growth according to Eq. 79. Insert V vs. the complementary area growth, constructed from the fluorescence kinetics of broken chloroplasts (redrawn from Strasser 1978). The chloroplasts were under low (-) or high (?) salt (Mg2?) conditions. Chloroplasts and leaves were treated with DCMU
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op cl cl E3F;tot , E2F;tot , and E3F;tot ), the link of biophysical magnitudes (EiF; energy outfluxes) with experimental signals (F; fluorescence) considered only, as in the original paper (Strasser 1978), the fluorescence emitted by the core antenna (or pigment pool ‘‘2’’). We will here examine whether the derivations would hold if the fluorescence from the LHCII (pigment pool ‘‘3’’) is also considered. Using Eqs. 21 and 22, we get the expression of the total energy influx in the pigment pool ‘‘3’’, E3;tot , where ‘‘total’’ means in both open and closed PSUIIs:
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307
op cl E3;tot ¼ E3;tot þ Ecl ¼ ð1 BÞE3op þ BE3;tot 3;tot op cl J3 þ ð1 BÞE2 þ BE2 p23 J3 þ E2;tot p23 ¼ ¼ 1 p33 1 p33
ð81Þ Hence, E3F;tot , the total energy outflux emitted as fluorescence from pigment pool ‘‘3’’, is related with E2F;tot the total energy outflux emitted as fluorescence from pigment pool ‘‘2’’, as follows: E3F;tot ¼
ðJ3 þ E2F;tot p23 =p2F Þp3F 1 p33
ð82aÞ
Accordingly, the experimental signal F3 is related with the experimental signal F2 (which is the F used in all previous equations), as: F3 ¼ Kk
ðJ3 þ F2 p23 =p2F Þp3F 1 p33
ð82bÞ
Note In the section ‘‘From energy fluxes to PSII fluorescence signals’’, we explained that, since the experimentally measured intensity F (or F2) is in arbitrary units, we could take it as equal to the energy outflux E2F;tot . This is not applicable for the derivation of Eq. 82b from Eq. 82a, since F3, though in arbitrary units as well, is linked to F2. Therefore, we must take in consideration the relative contribution of E3F;tot and E2F;tot in the experimentally measured fluorescence intensity, which depends on the detection wavelength k; hence the introduction of Kk in Eq. 82b. The fluorescence emitted from the whole PSUII, which we denote as FII, is then given by Kk p23 p3F Kk J3 p3F FII ¼ F2 þ F3 ¼ F2 1 þ ð83Þ þ p2F ð1 p33 Þ 1 p33 or, in terms of rate constants, Kk k23 k3F Kk J3 k3F FII ¼ F2 1 þ þ ðk3N þ k32 Þk2F k3N þ k32
ð830 Þ
The above equations show that F3 and, concomitantly FII are linearly, but not proportionally, related with F2, as already reported by Butler and Strasser (1977). Applied for the extremes, Eq. 83 gives: Kk p23 p3F Kk J3 p3F FII;0 ¼ F2;0 1 þ ð84aÞ þ p2F ð1 p33 Þ 1 p33 Kk p23 p3F Kk J3 p3F ð84bÞ FII;M ¼ F2;M 1 þ þ p2F ð1 p33 Þ 1 p33 From the set of the above equations, we deduce VII ¼
FII FII;0 F2 F2;0 ¼ ¼ V2 FII;M FII;0 F2;M F2;0
ð85Þ
which means that the experimentally determined relative variable fluorescence is not affected by the consideration or not of the fluorescence emitted by the pigment pool ‘‘3’’; this results from the elimination, by the subtractions employed in Eq. 85, of the F2-independent component of Eq. 83. The consequence of the non-proportionality between FII and F2 is that Eqs. 400 and 410 , which relate F0 and FM (i.e. F2,0 and F2,M) with the de-excitation rate constants kP and kN, do not hold if the extremes stand for FII,0 and FII,M; neither does Eq. 59, which relates, through kP and kN, the maximum quantum yield of primary photochemistry with F0 and FM. This conclusion is independent of the theoretical approach leading to the certain equations, but intrinsic in a tripartite model. This means that the equations hold only if fluorescence emission from ‘‘3’’ is taken as negligible, or if the pigment pools ‘‘2’’ and ‘‘3’’ in a PSUII are considered as one pigment bed within which exciton transfer is unrestricted. The second option is only a convenient, but not justified, simplification, by which the tripartite model would degenerate to a ‘‘bipartite model’’ (one pigment pool in PSII and one in PSI; see e.g., Kitajima and Butler 1975a; Strasser 1986). Therefore, Eqs. 400 , 410 , and 59, can only be used if the experimental setup satisfies the first option, i.e., by choosing a detection wavelength at which the contribution of LHCII florescence is indeed negligible. In our knowledge, this issue has not been raised so far.
Formulations according to the model of full complexity: scheme in Fig. 2B We will now proceed to derive the formulations for the model of full complexity shown in the scheme of Fig. 2B, using accordingly the system of Eqs. 15–20. Since the probabilities for energy outfluxes from pigment pool ‘‘2’’ and the RC (b) depend on the state of the RC, we distinguish op cl cl cl the Trapping Product as T op ¼ pop 2b pb2 and T ¼ p2b pb2 and op the Coupling Product as C op ¼ p23 p32 and C cl ¼ pcl 23 p32 .
Energy influxes in core antenna Solving the system of Eqs. 15–20 for a mixture of open and closed PSUIIs (any value of B between 0 and 1), we get the expressions for the energy influxes E2op and E2cl and, concomitantly, for the total influxes in all open units, op cl = ð1 BÞE2op and in all closed units, E2;tot ¼ BE2cl : E2;tot
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op E2;tot ¼
cl ¼ E2;tot
Jð1 T cl C cl Þð1 BÞ i h i C op op C op Þ 1 T cl pcl C cl ð1 BÞð1 T cl Ccl Þ 1 T op pop 22 22 1 p33 þ Bð1 T 1p33
ð86Þ
Jð1 T op Cop ÞB i h i C op op C op Þ 1 T cl pcl C cl ð1 BÞð1 T cl Ccl Þ 1 T op pop 22 22 1 p33 þ Bð1 T 1p33
ð87Þ
h
h
Applying Eqs. 86 and 87 for the extreme cases, i.e. op cl and E2;M , respectively: B = 0 and B = 1, we get the E2;0 op ¼ E2;0
cl ¼ E2;M
J C op =ð1 p33 Þ
ð860 Þ
J cl C cl =ð1 p Þ T 1 pcl 33 22
ð870 Þ
1
pop 22
T op
op op cl cl Writing E2;tot and E2;tot in terms of E2;0 and E2;M , respectively (derivation steps not shown), we get again Eqs. 300 and 310 (which we copy here to facilitate the reader): ð1 þ CHYP Þ op op E2;tot ¼ E2;0 ð1 BÞ ð300 Þ 1 þ CHYP ð1 BÞ 1 cl cl E2;tot ¼ E2;M B ð310 Þ 1 þ CHYP ð1 BÞ
However, it is obvious that the expression of each parameter in terms of probabilities is different than in the case of the simplified model of Fig. 2C (compare e.g. Eqs. 800 and 81 with Eqs. 27 and 28). Consequently, the expression of the curvature constant CHYP is different, because the expression of pG is different, as we will show below. Links of PSII fluorescence signals with energy influxes As explained and clarified above (in ‘‘What about the fluorescence from the LHCII?’’), we will consider only the experimental fluorescence signals emerging from antenna pigment pool ‘‘2’’. We follow again the approach described for the model of Fig. 2C, however, with the difference that the probabilities for energy outflux as fluorescence emission are now different in open than in closed PSUIIs. Hence, and using Eqs. 300 and 310 , we write: op op op op ð1 þ CHYP Þð1 BÞ op op F ¼ E2F;tot ¼ E2;tot p2F ¼ E2;0 p 1 þ CHYP ð1 BÞ 2F ð1 þ CHYP Þð1 BÞ ¼ F0 ð88Þ 1 þ CHYP ð1 BÞ B cl cl cl ¼ E2;tot pcl ¼ E pcl F cl ¼ E2F;tot 2F 2;M 1 þ CHYP ð1 BÞ 2F B ¼ FM ð89Þ 1 þ CHYP ð1 BÞ
123
As shown by Eqs. 88 and 89, the relation of F op with F0 and of F cl with FM are the same as in Eqs. 42a and 42b derived for the model of Fig. 2C; hence the total fluorescence F is given again by Eq. 43 and the relative variable fluorescence V by Eq. 46. Concomitantly, Eqs. 49 and 50, hence Eqs. 51 and 52 also, are valid and the deconvolution of F kinetics in kinetics of F op and F cl is therefore applicable. The derivations (not shown) led to the following expression for the curvature constant CHYP: op FV pop FV 22 þ C p33 =ð1 p33 Þ ½ pG ð90Þ CHYP ¼ ¼ op op ð1 T C Þ F0 F0 Hence, CHYP is equal to the product of FV/F0 with a term that corresponds to the overall grouping probability pG, like in Eq. 47. According to Eq. 90, the expression of pG for the model of Fig. 2B is different than for the model of Fig. 2C (Eq. 33); to facilitate comparison, we write it below in the form that Eq. 33 was written: 1 p33 op op pG ¼ p þC ð91Þ 1 T op C op 22 1 p33 Moreover, as will be shown below, FV/F0 is not any more equal to kP/kN as it was in Eq. 47. However, the differences are not affecting the general approach and neither the utilization of the routine experimental data. This is an example of a more general statement that the degree of the complexity of a model is meaningful only if the experimental signal has the corresponding resolution (Tsimilli-Michael and Strasser 2008). The fluorescence intensities at the extremes, F0 and FM, in terms of de-excitation rate constants We will here express the fluorescence intensities F0 and FM in terms of de-excitation rate constants, starting from Eqs. 86 and 87 and recalling (as in Eq. 29) that J stands for J2 þ J3 p32 =ð1 p33 Þ. op op F0 ¼ E2;0 p2F op J k2F =ðk2b þ k22 þ k2N þ k23 Þ op op k22 þ k2b pb2 þ k23 k32 =ðk32 þ k3N Þ 1 op ðk2b þ k22 þ k2N þ k23 Þ J k2F ¼ op op k2b ð1 pb2 Þ þ ½k23 k3N =ðk32 þ k3N Þ þ k2N
¼
ð92Þ
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309
We see that the denominator consists of two terms: the first refers to the net outflux from the core antenna to the RC (since pop b2 is the probability for backwards energy transfer). The second term (in brackets) is an overall de-excitation rate constant for outfluxes from ‘‘2’’ that are not directed to the RC and it corresponds to the term denoted by kN (after rearrangement of Eq. 40). On the other hand, the first term, unlike in Eq. 40, does not refer only to photochemistry since it also contains a component governing energy dissipation from open RCs. Hence, we can write: F0 ¼
op k2b ð1
J k2F pop b2 Þ þ kN
J k2F op op op op k2b kbN k2b kbP þ k þ N op op op op op op kbP þ kb2 þ kbN kbP þ kb2 þ kbN J k2F ¼ op op kP;ALL þ kN;ALL
We recall Eq. 54 (formulated for the model of full comop from Eq. 49 plexity), in which we further substitute E2;tot (which is valid for both models): op op op op op op op EbP;tot ¼ Eb;tot pop bP ¼ E2;tot p2b pbP ¼ E2;0 ð1 VÞp2b pbP
ð95Þ
The actual quantum yield of primary photochemistry (at any time t) is hence written as uPt ¼
¼
op op op E2;0 p2b pbP ð1 Vt Þ J
ð96Þ
Applied for the case that all RCs are open (V = 0), Eq. 96 is written as ð920 Þ
In Eq. 920 the denominator was rearranged to separate the ‘‘overall rate constant’’ governing the outflux from ‘‘2’’ used for PSII photochemistry (first term) from the ‘‘overall rate constant’’ that governs the total outflux from ‘‘2’’ that, not only directly and via ‘‘3’’, but also via RC, is finally not used for PSII photochemistry (second term). In other words the first and second term correspond to an overall kP and an op overall kN, respectively (accordingly denoted as kP;ALL and op 0 0 kN;ALL and used in Eq. 92 ); hence, Eq. 92 is principally the same with Eq. 400 , despite the different meaning of the corresponding rate constants. Similarly, and because of the symmetry of Eq. 87 with Eq. 86, we get J k2F J k2F J k2F FM ¼ cl ¼ cl cl ¼ cl cl k2b ð1 pb2 Þ þ kN kN;ALL k2b kbN þ kN cl þ kcl kb2 bN ð93Þ Equation 93 is principally the same with Eq. 410 , though cl 6¼ kN . We note moreover that, contrary to the case kN;ALL of Eqs. 400 and 410 , where kN is the same for open and closed units, Eqs. 920 and 93 contain two different terms, op cl and kN;ALL , respectively. i.e. kN;ALL As referred above, in respect to Eq. 90, FV/F0 is not equal to kP/kN when the model of Fig. 2B, instead of that in Fig. 2C, is considered. Using Eqs. 92, 920 , and 93, we can now get the expression of FV/F0 in terms of rate constants (or overall rate constants): FV FM kop ð1 pop Þ kcl ð1 pcl b2 Þ ¼ 1 ¼ 2b cl b2 cl 2b F0 F0 k2b ð1 pb2 Þ þ kN op op cl kP;ALL þ kN;ALL kN;ALL ¼ cl kN;ALL
Quantum yield of PSII primary photochemistry
ð94Þ
uPo
op op op E2;0 p2b pbP ¼ J
ð960 Þ
Hence, we get uPt ¼ uPo ð1 Vt Þ
ð57Þ
like for the model of Fig. 2C (hence we indicate it, again, as Eq. 57). We can now express Eq. 960 in terms of de-excitation rate constants: op kbP op kop þ kop þ kbN ¼ op bP opb2 k2b ð1 pb2 Þ þ kN op k2b
uPo
op kbP op op op kbP þ kb2 þ kbN op op kbP þ kbN op k2b op op op þ kN kbP þ kb2 þ kbN op k2b
¼
ð97Þ
Let us now examine the validity of the widely used formula uPo ¼ FV =FM ¼ 1 F0 =FM . From Eqs. 920 and 93 we get: FV F0 kcl ð1 pcl b2 Þ þ kN ¼1 ¼ 1 2b op FM FM k2b ð1 pop b2 Þ þ kN op op cl k þ k kbN op cl bP bN k2b op op op k2b cl cl kbP þ kb2 þ kbN kb2 þ kbN ¼ op op kbP þ kbN op k2b op op op þ kN kbP þ kb2 þ kbN
ð98Þ
Equations 97 and 98 give 9 8 op op k2b kbP > > > > > > op op op =F < kbP þ kb2 þ kbN V uPo ¼ op op op op cl cl > FM k2b kbN k2b kbP k2b kbN > > > > > : op op op þ op op op cl cl ; kbP þ kb2 þ kbN kbP þ kb2 þ kbN kb2 þ kbN ð99Þ
123
310
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or, using the expressions introduced for Eqs. 920 and 93, ( ) op kP;ALL FV uPo ¼ ð990 Þ op op cl kP;ALL þ ðkN;ALL kN;ALL Þ FM
Ebop ¼ E2 pop 2b
ð101Þ
Ebcl ¼ E2 pcl 2b
ð102Þ
According to Eq. 99, uPo is not equal, but only proportional to FV/FM, unless the term in brackets, i.e. the correction (or, proportionality) factor, would be equal to 1. Hence, uPo, currently calculated as equal to FV/FM, is underestimated or overestimated if the correction factor is bigger or smaller than 1, respectively and, in terms of rate cl cl cl cl constants, if k2b kbN =ðkb2 þ kbN Þ is bigger or smaller than op op op op op k2b kbN =ðkbP þ kb2 þ kbN Þ, respectively. It should be emphasized that this conclusion is independent of whether a tripartite or a bipartite model, with or without grouping, is assumed; as Eq. 99 reveals, the correction factor is only determined by rate constants governing the forward and backward energy fluxes between core antenna and RC and by the rate constant for energy dissipation from op cl the RC. For example, a preferential increase of kbN or kbN (i.e., the development of quenching on the closed or open cl cl cl cl kbN =ðkb2 þ kbN Þ or RCs, respectively) will increase k2b op op op op op k2b kbN =ðkbP þ kb2 þ kbN Þ, respectively and, concomitantly, alter the correction factor.
kop 2b and, op ð1BÞk2b þBkcl 2b þk 2N op k2j pop ¼ op 2j cl þ k ð1 BÞk2b þ Bk2b 2N pop 2b ¼
pcl 2b ¼ pcl 2j ¼
more generally;
kcl 2b and, op ð1BÞk2b þBkcl 2b þk 2N cl k2j op cl þ k ð1 BÞk2b þ Bk2b 2N
ð103Þ
more generally; ð104Þ
Substituting Eqs. 101 and 102 in Eq. 100, we get E2 ¼
J 1 ð1
op BÞpop 2b pb2
ð105Þ
cl Bpcl 2b pb2
Lake (or matrix) model
The lake model with full complexity in respect to core antenna $ RC energetic communication
J3
k3N
3 We will here apply the EFT to solve the case of a lake model, i.e. of a model with unrestricted excitation energy transfer in pigment pool ‘‘2’’, which further assumes full complexity in respect to the energetic communication between core antenna and RC, as in the model of Fig. 2B. The formulations of the model are independent of the energetic communication between ‘‘2’’ and ‘‘3’’ and of the structure of ‘‘3’’, except for the meaning of the term J (the energy influx to ‘‘2’’ that is only due to light absorption), which will be equal to J2 for a bipartite model (like model 14 in Fig. 1), or to J2 þ J3 p32 for a tripartite model with unrestricted excitation energy transfer in ‘‘3’’ or to J2 þ J3 p32 =ð1 p33 Þ for a tripartite with ‘‘3’’ $ ‘‘3’’ energetic connectivity. The three equivalent models are shown in Fig. 10. A new set of basic equations will be formulated, since the set of Eqs. 15–20 is not applicable. The main differences are that we will now deal with one E2 (not E2op and E2cl as in Eqs. 17–18) and that the probabilities p2j will depend not only on the relevant de-excitation rate constants, but also on the fraction B of closed RCs. The equations are: cl cl E2 ¼ J þ ð1 BÞEbop pop b2 þ BEb pb2
123
ð100Þ
k33 k33
k32 k23 J2 1–B
3
k32 k23
2
k2N B
cl cl op k op 2b k b 2 k 2b k b 2
k op bN
b k op bP
b
k clbN
+ − PA P A
Fig. 10 The lake (or matrix) model with full complexity in respect to energetic communication between core antenna and RC. The forward and backward energy transfer is indicated for one open and one closed RC, standing for the 1 – B and the B fractions of the total RCs, respectively. The model drawn with rigid lines is a bipartite lake model (like model 14 in Fig. 1). The model is transformed to a tripartite lake model (‘‘lake’’ refers only to the organization of pigment pool ‘‘2’’) when the components drawn with dotted lines are incorporated (pigment pool ‘‘3’’ and respective rate constants), however, without affecting the formulations, whatever the structure of pigment pool ‘‘3’’ would be; only J, the energy influx to ‘‘2’’ that is solely due to light absorption, will be different (see text). Symbols are as in Fig. 2B
Photosynth Res (2013) 117:289–320
311 cl cl cl cl E2cl ¼ J2 þ ð1 BÞE2op pop 22 þ BE2 p22 þ E3 p32 þ Eb pb2
Hence, Jpop 2F op cl cl 1 ð1 BÞpop 2b pb2 Bp2b pb2 Jk2F ¼ op op op cl cl pcl ð1 BÞk2b þ Bk2b þ k2N ð1 BÞk2b pb2 Bk2b b2 Jk2F ¼ ð106Þ op cl cl ð1 BÞk2b ð1 pop b2 Þ þ Bk2b ð1 pb2 Þ þ k2N
F¼
Applying Eq. 106 for B = 0 and B = 1, we get, respectively, F0 and FM: Jk2F op k2b ð1 pop b2 Þ þ k2N
ð107Þ
Jk2F FM ¼ cl k2b ð1 pcl b2 Þ þ k2N
ð108Þ
F0 ¼
ð112Þ Ebop
¼
E2op pop 2b
Ebcl ¼ E2cl pcl 2b
op ¼ E2;tot
Jð1 BÞ op cl cl ð1 BÞðpop 22 þ C Þ Bðp22 þ C Þ ð1 T op Þ 1 ð1 T op Þ ð1 T cl Þ ð115Þ
cl E2;tot ¼
JB op cl cl ð1 BÞðpop 22 þ C Þ Bðp22 þ C Þ ð1 T cl Þ 1 op ð1 T Þ ð1 T cl Þ
ð116Þ ð109Þ op E2;0 ¼
We note that, if we would set as equal to zero all magnitudes referring to LHCII in Eqs. 920 , 93, and 94 (and, accordingly. substitute kN by k2N) we would get Eqs. 107, 108, and 109, respectively. From Eqs. 106 and 108, we derive the same expression of relative variable fluorescence as in Eq. 63, i.e. Vlake ¼
ð114Þ
From the system of Eqs. 110–114, we get (intermediate derivation steps not shown), the following equations, where J ¼ J2 þ J3 p32 :
Hence, op cl cl FV k2b ð1 pop b2 Þ k2b ð1 pb2 Þ ¼ cl ð1 pcl Þ þ k F0 k2b 2N b2
ð113Þ
B 1 þ ðFV =F0 Þð1 BÞ
1
T op
J , F0 ¼ op pop 1 22 C
C op ð1150 Þ
cl E2;M ¼
J Jpcl 2F , F ¼ M cl cl 1 T cl pcl 1 T cl pcl 22 C 22 C ð1160 Þ
ð63Þ
however, with FV =F0 not being equal to kP =kN , but given by Eq. 109. The basic Eq. 57, uPt ¼ uPo ð1 Vt Þ, is also derived and, moreover, the expression of uPo in terms of rate constants (as in Eq. 97), as well as the relation of uPo with FV/FM (as in Eq. 99). The semi-matrix model with p22 = 0 and with full complexity in respect to core antenna $ RC energetic communication
Semi-matrix model with p22 ≠ 0
J3
k3N
ð110Þ
op op cl cl E2op ¼ J2 þ ð1 BÞE2op pop 22 þ BE2 p22 þ E3 p32 þ Eb pb2
ð111Þ
3
1–B k32 k23 J2 k2N
We will here consider the model shown in Fig. 11, which assumes a ‘‘lake’’ structure for the LHCII, like model 13 of Fig. 1 and called then as semi-matrix model (Strasser 1978, 1981), however, assuming now an energetic connectivity between core antennae of neighbor PSUIIs. The model assumes also full complexity in respect to the energetic communication between core antenna and RC, as in the model of Fig. 2B. The basic equations, formulated with EFT, are: cl cl E3 ¼ J3 þ ð1 BÞE2op pop 23 þ BE2 p23
Jpop 2F T op pop 22
2
B k32 k23
k22 k22
J2
2
k2N
cl cl op k op 2b k b 2 k 2b k b 2
k op bN
b k op bP
b
k clbN
+ − PA P A
Fig. 11 Semi-matrix model with p22 = 0 and with full complexity in respect to energetic communication between core antenna and RC. The forward and backward energy transfer is indicated for one open and one closed RC, standing for the 1 – B and the B fractions of the total RCs, respectively. Symbols are as in Fig. 2B
123
312
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op op ð1 þ CHYP Þð1 BÞ E2;tot ¼ E2;0 , 1 þ CHYP ð1 BÞ ð1 þ CHYP Þð1 BÞ F op ¼ F0 1 þ CHYP ð1 BÞ B cl cl E2;tot ¼ E2;M , 1 þ CHYP ð1 BÞ B F cl ¼ FM 1 þ CHYP ð1 BÞ
ð117Þ
Semi-matrix
CHYP = pG(FV/F0)
CHYP = pG(FV/F0)
Cop p 33 + 1 − p 33 pG = op 1 − T − Cop p op 22
ð118Þ
3
pG =
b
2
3
2
3
2
Cop + p op 22 1 − Top
2
b
2
b
3
4)
We note that Eqs. 117–118 are of the same form as Eqs. 88–89 and Eqs. 42a and 42b. The curvature constant is again found (not shown) equal to pG(FV/F0), however, with pG given by þ pop 22 T op
Limited connectivity
b b
3
2
b
3
2
b
b
2 p op 22 = 0
3)
3
p op 22 = 0
13)
2
b
op
pG ¼
C 1
ð119Þ
If the connectivity between core antennae of neighbour PSUIIs is not considered (p22 ¼ 0), the model is simplified to the originally (Butler and Strasser 1977) denoted as semi-matrix model (model 13 of Fig. 1) and pG is given as C op pG ¼ 1 T op
ð1190 Þ
If moreover the energetic communication between core antenna and RC is assumed as in the simplified model of Fig. 2C, the expression of pG simplifies to pG ¼ C op ¼ C. It is worth reminding that the expression deduced for the semi-matrix model in the original publications was, indeed, pG ¼ C (Strasser 1978, 1981). Summarizing the models Figure 12 depicts all the PSUII models analyzed above. The models are distinguished in three groups: in those with limited and variable connectivity (0 \p22 \ 1 and 0 \ p33 \ 1), which can degenerate to the separate packages ðp22 ¼ 0 and p33 ¼ 0Þ, the semi-matrix models (for p22 ¼ 0; or 0 \p22 \ 1) and the lake (or matrix) model (with or without pigment pool ‘‘3’’). For all of them, (i) the same expression of the vertical hyperbola V = f(B) applies, however, with CHYP equal to pG ðFV =F0 Þ for the models with limited connectivity and the semi-matrix models, zero for the separate package and FV =F0 for the lake model, (ii) the same important equation uPt ¼ uPo ð1 Vt Þ is valid and, (iii) the same relations of uPo with FV =FM (Eq. 99) and of FV =F0 with rate constants (Eq. 94) are valid (however, with the according simplifications if the Fig. 2C model is used instead of the Fig. 2B model). It is easy to see that, within each of the groups, the equation giving pG for any of them can be derived from the general equation (full complexity) by applying the adopted simplification.
123
2
3
2
b b
2
For all: Vt =
b
3 2
pG = 0
2)
3
CHYP = (FV/F0)
p33 = 0
5)
3
Lake or matrix
b
3
14)
b
Bt , ϕ Pt = ϕ Po (1 − Vt ) 1 + CHYP (1 − B t )
⎫ ⎧t ⎫ ⎧t B t = ⎨∫ (1 − Vt )dt ⎬ ⎨ ∫ (1 − Vt )dt ⎬ ⎭ ⎩0 ⎩0 ⎭
(Eq. 79)
B t / Vt = (1 + C HYP ) − C HYP ⋅ B t
(Eq. 80)
FM
Fig. 12 A summary of all analyzed models, where full complexity in respect to the energetic communication between core antenna and RC, as in the model of Fig. 2B, was assumed. The models belong to three groups: in those with limited and variable connectivity (0\p22 \1 and 0\p33 \1), the semi-matrix models (for two cases, i.e. with p22 ¼ 0; or 0\p22 \1) and the lake (or matrix) model (with or without pigment pool ‘‘3’’); numbers in italics refer to the numbering of models in Fig. 1. The models with the extreme values of CHYP (for a given value of FV/F0) are distinguished by drawing the separate package model, i.e. the model with CHYP = 0 (minimum CHYP) with grey line, and by using a grey background for the lake model, i.e. the model with CHYP = FV/F0 (maximum possible CHYP); op op cl cl according to Eq. 94, FV/F0 = ðkP;ALL þ kN;ALL kN;ALL Þ=kN;ALL , an expression that degenerates to FV =F0 ¼ kP =kN (Eq. 48) after the simplifications employed in the model of Fig. 2C. Key equations for calculating Bt and CHYP from experimental signals are also presented
Comparison of formulations with those in the classic papers Since the first publication that introduced the concept of energetic connectivity (Joliot and Joliot 1964), a number of reports have produced or, mostly, reproduced equations formulating the concept. Whatever set of parameters can be chosen for the formulations and with whatever approach they
Photosynth Res (2013) 117:289–320
313
are derived, it is essential to recognize whether they are equivalent or not, even if they ‘‘look’’ different. We will show below the basic equivalence of our formulations with those obtained in the classic papers of Joliot and Joliot (1964) and Paillotin (1976b, 1977). The equivalence of the formulations later deduced (Lavergne and Trissl 1995) for the exciton–radical-pair model, where energetic connectivity was incorporated, with our formulations has been already discussed in relation to Fig. 3. Comparison with the formulations of Joliot and Joliot (1964) Joliot and Joliot (1964) approached the issue of energetic connectivity as summarized below. To facilitate comparison we also write (in brackets) the terms and symbols as in the present paper. Their hypotheses were: When excitation energy reaches a photochemically active complex (open RC), whose concentration was denoted as e (=1 - B), it is used only for photochemistry—other pathways were considered as negligible. When excitation energy reaches a photochemically inactive complex (closed RC), whose concentration was denoted as e0 (=B), it has a probability, p, to move to a neighbor complex (RC) and, there from, its fate is determined again by the state of the complex, photochemically active or inactive; hence, successive transfers may follow. The mathematical analysis of these hypotheses led to the following equations for 0 the rate of photochemical reaction v ¼ de =dt ð¼ dB=dtÞ and for the fluorescence intensity F, which, however, was taken as equal only to the variable fluorescence since at that time F0 was still considered as ‘‘dead fluorescence’’. Each original equation is rewritten with the symbols used in the present paper. 0 h i de kie 0 0 0 ¼ kie ð1 þ e p þ ðe pÞ2 þ þ ðe pÞn ¼ 1 e0 p dt dB kJð1 BÞ ¼ ð120Þ , dt 1 Bp
v¼
0
0
0
k ie k JB F¼ , , F F0 ¼ 1 Bp 1 e0 p 0 k JB V¼ ðFM F0 Þð1 BpÞ
ð121Þ
where i is the light intensity (J) and k and k0 constants. Note Equation 120 and, accordingly, Eq. 121 utilize that the sum of an infinite convergent geometric series, i.e. of a series of terms following a geometric progression with the common ratio R between any two successive terms smaller 0 than unity (here R ¼ e p\1), is equal to 1=ð1 RÞ. In the case that the thermal reactions are blocked (inhibited by DCMU) the kinetics of e vs. t, expressed as t = f(e), was derived by integration of Eq. 120 as:
p þ ð1 pÞ ln e þ ep ¼ kit , 1 ½Bp ð1 pÞ lnð1 BÞ ¼ t Jk
ð122Þ
Comparison of Eqs. 120, 121 and 122 with Eqs. 65b, 46 and 67, respectively, shows that they are equivalent, however with the parameters used in the two sets of equations different but keeping the following correspondence: p , CHYP =ð1þ 0 CHYP Þ; k , uPo ; k , ðFM F0 Þ=½J ð1 þ CHYP Þ: We should here remind that Strasser’ s model (1978) assumed energy outflux to neighbor PSUIIs both from open and closed units, while Joliot and Joliot (1964) assumed a probability for such transfer only from closed units. This difference does not affect the equivalence of the two approaches, as already proved above, since the net result of grouping in Strasser’ s model is the ‘‘feeding’’ with excitation energy of open units by the closed units (hence the gain; see Fig. 4). Note The formulations by Paillotin (1976b) and by the Butler’s group (see e.g. Butler and Strasser 1977) were also based, like those by Joliot and Joliot (1964), on the feature of converging series, while the EFT of Strasser (1978) employs a different approach that further permits the formulation of models with high complexity level.
Comparison with the formulations of Paillotin (1976b, 1977) It might be doubted whether the comparison of our approach with that of Paillotin (1976a, b, 1977) is legitimate since we take as unrestricted the number of PSUIIs that can be energetically connected, while Paillotin’s derivations refer to domains, for which he adopted the definition of Clayton (1976) that domain is an ensemble of identical units able to exchange excitation energy. However, the key point that permits the comparison is that, as Paillotin clarified, the PSII domains must contain at least four PSUIIs, since the fluorescence of domains with fewer units would exhibit high fluctuations among them (resulting from high fluctuations in the fraction of closed RCs) while experimental data proved that the fluctuation remained negligible throughout the fluorescence rise kinetics (\5 %; see Paillotin 1976a and references therein). On the basis of this precondition and clarification, Paillotin (1976a) concluded that ‘‘the microscopic and macroscopic relations between fluorescence yield and fraction of centers in state P are identical’’ (where ‘‘centers in state P’’ means closed RCs). It is worth adding that this is also the key point for the equivalence of Paillotin’s formulae with those of Joliots (1965), who also considered as unrestricted the number of PSUIIs that can be energetically connected. Paillotin’s formulations are presented in two publications (Paillotin 1976b, 1977). As a simplification of the domain
123
314
Photosynth Res (2013) 117:289–320
concept that he elaborated in a general form (Paillotin 1976a), where more than two states of RC were considered, he analyzed mainly the case (with which we compare our approach) of only two possible states for each RC, i.e. open and closed, denoting (Paillotin 1977) the according parameters with the subscripts ‘‘0’’ and ‘‘1’’, respectively. The symbols used in Paillotin 1977 were: q0 and q1 ¼ 1 q0 for the fraction of open and closed RCs; T for ‘‘the rate with which an excitation may leave a given unit’’ (to another one in a domain); U0 and U1 for the fluorescence yield of open and closed units; U for the fluorescence yield of a domain (with open and closed RCs); P0 and P1 for the yield of energy transfer from open and closed units; K0 and K1 for the ‘‘deactivation rate’’ of open and closed units when no energy transfer is considered. Paillotin derived the following equation: U¼
ð1 P0 Þq0 U0 þ ð1 P1 Þq1 U1 ð1 P0 Þq0 þ ð1 P1 Þq1
ð123Þ
where the parameters were defined as: U0 ¼ kF =K0
U1 ¼ kF =K1
ð124Þ
P0 ¼ T=ðK0 þ T Þ
P1 ¼ T=ðK1 þ T Þ
ð125Þ
Note We are not presenting the derivations of Eq. 123 (Paillotin 1976b) which, in our opinion, are quite complicated to be followed by readers without a strong mathematical background. From Eq. 123, the following equation was derived: U U0 q1 ð1 pÞ ¼ 1 pq1 U1 U0
ð126Þ
where p, denoted as the ‘‘connection parameter’’ was given as P 1 P0 U0 p¼ ¼ P1 1 1 P0 U1
ð127Þ
F F0 Bð1 pÞ ¼ 1 pB FM F0
ð1260 Þ
Equation 1260 ‘‘looks’’ different, and has been indeed taken as different in many publications, than our Eq. 460 , which we copy here: V¼
B 1 þ ½pG ðFV =F0 Þ ð1 BÞ
ð460 Þ
However, rearranging Eq. 1260 , we get: V¼
B B B ¼ ¼ 1 pB 1 p þ p pB 1 þ p ð1 BÞ ð1 pÞ ð1 pÞ ð1 pÞ
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The term ðU1 U0 Þ=U0 in the right part of Eq. 128, written as (FM - F0)/F0 in our notations, is equal to FV/F0. As for P0, it corresponds (by definition; Eq. 124) to our pop 22 . Hence, Paillotin’ s equation (Eq. 126) coincides with our general equation (Eq. 460 ) for the special case that the latter applies for a bi-partite model (Cop = 0, p33 = 0) with no backwards energy transfer from an open RC to the core antenna (Top = 0), since under those simplifications, pG (given in Eq. 91) degenerates to pop 22 : 1 p33 op op pG ¼ p þ C ) 1 T op C op 22 1 p33 pop 22 ) pG ¼ pop pG ¼ 22 1 T op Hence, the correspondence between the two approaches is concluded, with p=ð1 pÞ ¼ CHYP
Let us now write Eq. 126 in our notations: V
Comparison of Eq. 12600 with Eq. 460 shows that it differs only in respect to the curvature constant (our CHYP), i.e. the coefficient of (1 - B) in the denominator, which is equal to p/(1 - p) instead of pG ðFV =F0 Þ. Does it indeed differ? Let us transform p=ð1 pÞ on the basis of Eqs. 124, 125 and 127: U0 P1 1 p U1 ¼ U0 1p 1 P1 1 U1 T U0 T K0 K1 1 K1 þ T K1 þ T U1 K ¼ 0 ¼ T U0 T K0 K1 1 1 1 K1 þ T K1 þ T U1 K0 TðK0 K1 Þ ¼ K0 ðK1 þ TÞ TðK0 K1 Þ K0 K1 T U1 U0 ¼ ½ P0 ð128Þ ¼ K0 þ T K1 U0
ð12600 Þ
ð129Þ
Note The same correspondence is concluded by comparing the explicit equation t = f(V) presented by Paillotin (1976b) with our Eq. 68. The equation of Paillotin for the lake model (Paillotin 1976b) is again Eq. 126 (or, after rearrangement, Eq. 12600 ), however with p ¼ 1 U0 =U1 , which, in our notations is written as 1 F0 =FM ¼ FV =FM . Concomitantly, the curvature constant in Eq. 12600 , i.e. p=ð1 pÞ, is the same as that derived with our approach (Eq. 63): p=ð1 pÞ ¼ ðFV =FM Þ=½1 ðFV =FM Þ ¼ ðFV =FM Þ=ðF0 = FM Þ ¼ FV =F0 . Though whatever is deduced from equivalent formulae should obviously be equivalent, it is worth presenting the equivalence in respect to the criterion for the appearance of
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an inflection point in the fluorescence induction curve. In Paillotin 1976b, the criterion is given as ð3x 1Þ=3x [ 1=Rp where x was used instead of P1 (defined in Eq. 125) and Rp ¼ U1 =U0 (with U0 and U1 defined in Eq. 124). By rearrangement (and using Eq. 127), the criterion can be written as ½1 1=Rp [ 1 ½ð3x 1Þ=3x , ½1 1=Rp x [ 1=3 , p [ 1=3 which, by using Eq. 129, gives the same result as found (Eq. 71) by our approach: p=ð1 pÞ ¼ CHYP [ ð1=3Þ=ð1 1=3Þ , CHYP [ 0:5 Comparison with Sorokin’s correction (1985) of Paillotin’s formula Sorokin (1985) reported that light and some other factors (like ageing) cause a gradual reduction of the sigmoidal shape of induction curves and of FM without any perceptible change of F0 and he concluded for transformation of a fraction of RCs to ‘‘quenching centers’’ (see also section ‘‘Inactivation of RCs and the impact on CHYP’’). He further proposed a correction of Paillotin’s formula (Eq. 126) in order to make it applicable for such a case. In order to avoid confusion by using the symbols in Sorokin’s article (different than in Paillotin’s), we re-write his corrected equation in our notations (as we re-wrote Eq. 126 as Eq. 1260 ): 0
B ð1 pxÞ V ¼ 1 pxB0 0
ð130Þ
where, according to our notations in section ‘‘Changes of CHYP: What information can we obtain?’’, we wrote as V0 the relative variable fluorescence (based on the experi0 mental maximal fluorescence FM ), x the ‘‘fraction of normally operating photosynthetic centers’’ and B0 ‘‘the relative number of such centers closed’’ (excerpts from Sorokin 1985, where the symbols were written as DqR , H and a, respectively). Rearranging Eq. 130 (as we did for Eq. 1260 to give Eq. 12600 ), we get 0
0
V ¼
B px 0 ð1 B Þ 1þ ð1 pxÞ
ð131Þ
Hence, the curvature constant (as we termed it) is 0
CHYP ¼
px ð1 pxÞ
ð132Þ
Dividing Eq. 132 by Eq. 129, we get 0
CHYP px=ð1 pxÞ xð1 pÞ ¼ ¼ p=ð1 pÞ ð1 pxÞ CHYP
ð133Þ
According to Paillotin’s equation (Eq. 126, equivalently written as Eq. 1260 ), the right part of Eq. 133 is the relative variable fluorescence of the experimental maximal 0 fluorescence FM in respect to the true FM (all RCs active and closed); hence, it is our Vx (see section ‘‘Changes of CHYP: 0 What information can we obtain?’’). Hence, CHYP ¼ CHYP Vx , as we indeed concluded by Eq. 760 . However, we have there 0 0 deduced also (Eq. 7600 ) that CHYP ¼ ðFV =F0 ÞpG . We deem, therefore, that these findings, though obviously independent of the approach used in the derivation of the hyperbolic function V = f(B), have been facilitated by the way this function is written in our approach (Eq. 46): all factors that determine the sigmoidicity of V = f(t) comprise a single parameter, the curvature constant CHYP, which appears in Eq. 46 only in the denominator, as the coefficient of (1 - B).
Determination of the overall grouping probability pG and deconvolution of fluorescence kinetics in physiological photosynthetic material With new instrumentation, new analytical tools have been developed for the utilization of the fluorescence transient emitted in vivo, i.e. by physiological (non-DCMU treated) photosynthetic material. Based on the EFT, the JIP-test was introduced (Strasser and Strasser 1995) and further elaborated (see e.g. Strasser et al. 2000, 2004, 2010), by which the fast fluorescence rise OJIP is translated to structural/conformational and functional parameters. Though it is out of the scope of the present paper to explore the JIP-test as application of EFT, we deemed that it would be of interest to refer in brief to two of the issues handled by the JIP-test, as they express extensions of topics that the present paper deals with. Determination of pG As shown by Strasser and Strasser (1995), the OJ phase (20 ls to 2 ms), when expressed as WOJ ¼ ðF F0 Þ= ðFJ F0 Þ ¼ f ðtÞ, i.e. when normalized between 0 and 1, coincides with the V ¼ f ðtÞ in DCMU-treated samples, indicating that it reflects QA reduction by single turnover events (photochemical phase). Hence, it can be processed for the determination of the overall grouping (or connectivity) probability pG in the same way as presented above for the full fluorescence induction (F0 to FM) in DCMUtreated samples (see e.g. Strasser et al. 2004). Taking in consideration Eq. 77, care should be taken to use Ft;J ¼ 0 ðFJ F0 Þ of the full transient as the FV .
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Applying the JIP-test for investigating the impact of stress on PSII behavior, we have witnessed, in many cases, a pronounced decrease of uPo (measured as FV/FM) and a stability of the initial slope of the normalized transient W0J ðF F0 Þ=ðFJ F0 Þ, while no absorption changes were detected (by reflectance measurements). We then concluded and proposed that this apparent controversy is due to the transformation of a fraction of RCs to ‘‘inactive’’ centers (which we have also termed as ‘‘silent’’), in the sense that they are still acting as efficient exciton traps but are dissipating, instead, the whole of the energy outflux that would be used for photochemistry if they were active, i.e. they become non-QA-reducing; we further derived the formula for the determination of this fraction (Strasser and Tsimilli-Michael 1998; TsimilliMichael et al. 1999; for reviews, see Strasser et al. 2000, 2004). According to our proposition, the conformation of an inactive centre compared to that of open RC given in the models of Fig. 2 (panels B and C) is the substitution of kbP by a dissipation rate constant kbQ of equal magnitude. Concomitantly, the inactive centers behave in respect to their fluorescence as open RCs throughout the fluorescence induction, as shown in the right panel of Fig. 13, where the experimental fluorescence transient OJIP is deconvoluted into the fluorescence kinetics of open RCs
(Fop), closed RCs (Fcl) and inactive centers (Fin), the latter being, by definition, a horizontal line. In the left panel of Fig. 13 the deconvolution into the transients of Fop and Fcl (like in Fig. 7) of the fluorescence transient emitted by the sample before exposed to stress (reference sample; assumed to have only active RCs) is depicted. Comparison of the plots in the two panels shows that they have the same F0 and different FM. This could be considered as a criterion for the detection of inactivation, since a lowering of FM would be associated with a lowering of F0 if quenching at the antenna would occur. However, this is not always an applicable criterion in practice, since the photosynthetic samples in vivo can exhibit different F0 because of heterogeneity in respect to their Chl a content. It is worth clarifying that the inactivation of RCs is not equivalent with an increase of the probability for energy dissipation from an active RC (quenching at the RC). The basic difference, holding both for the model of Fig. 2C and that of Fig. 2B, is that quenching decreases the probability for photochemistry in all RCs, while inactivation diminishes this probability but only in a fraction of RCs. Concomitantly, in the first case the rate of QA reduction (rate of RCs’ closure) decreases but, eventually, though with a delay, all QA are reduced and the true FM (all RCs close) is reached, while in the second case a fraction of reaction centers can not close and the apparent FM is hence lower than the true FM.
Fig. 13 Left panel The fluorescence transient OJIP obtained from a whole leaf of a non-stressed plant (reference sample) is deconvoluted into the fluorescence kinetics of open RCs, Fop (open circles) and closed RCs, Fcl (closed circles), as in Fig. 7. Right panel The OJIP obtained after the plant was exposed to stress is deconvoluted into the Fop kinetics (open circles), Fcl kinetics (closed circles) and the fluorescence kinetics (horizontal line; constant fraction) of inactive
centers Fin (grey closed circles). The fluorescence kinetics of only the active RCs, Fop ? Fcl, is also depicted (stars); upon addition of the constant Fin, the Fop ? Fcl kinetics is vertically shifted, as indicated by the grey vertical lines, to give the experimental Ft kinetics. The partial inactivation (reflected in the transformation of the left to the right panel) is assumed to be caused by a stress that did not affect other structural/conformational parameters
Inactive or silent centers (non-QA-reducing)
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Fluorescence measurements at 77 K address PSI energetics concerning state transitions and spill-over In any living photosynthetic system, regulatory changes of PSII and PSI energy influxes take place, which serve to balance their exciton density, hence their function, under the perpetually changing internal and external conditions (e.g. light regimes and metabolic demands) (see e.g. Allen 1992, 1995). The regulation in plants is realized by two possible mechanisms, state transitions and spill-over. The first refers to the movement, upon phosphorylation, of a ‘‘mobile’’ part of the LHCII to PSI (state 1 ? state 2 transition) and the backwards movement, upon de-phosphorylation (state 2 ? state 1) or, equivalently, to a change of the effective absorption cross sections of the two photosystems; the second refers to energy migration from PSII to PSI. Fluorescence measurements could be used as a tool for recognizing and evaluating such changes, provided that the distinction between fluorescence emitted from PSII and PSI would be permitted. However, as already stated at the beginning of this article, there is a general agreement that, at room temperature, Chl a fluorescence of plants, algae and cyanobacteria, in the 680-740 nm spectral region, is emitted mainly by PSII, as revealed by the fluorescence emission spectra that exhibit only one peak, since the quantum yield of PSI fluorescence is much lower than that of PSII fluorescence. Moreover, when changes in the PSI excitation rate take place, the whole spectrum changes at every wavelength in an almost similar way. Very precise measurements are needed in order to utilize such signals as sources of specific information, like e.g. those proposed by Lombard and Strasser (1984) or the more recent and highly elaborated global spectral-kinetic analysis (see e.g. Gilmore et al. 2000). In our analysis so far, the equations derived with the EFT for the models we presented were linked with fluorescence emitted at room temperature, for which we adopted the commonly used approximation that it emerges only from PSII. Though in our models the energy flux from PSII to PSI, E21 and E31, were taken in consideration (M - migration; Fig. 2A), the corresponding rate constants k21 and k31 had to be incorporated in k2N and k3N respectively, since the experimental resolution does not permit their distinction in the sum (subscript N) of all rate constants that govern energy outfluxes to destination sites not belonging to PSII. However, since we were dealing with the fluorescence induction, whose duration is too short to permit changes of spill-over or state transitions, we could indeed consider that k21 and k31, as well as the absorbed light energy fluxes by PSII and PSI, were remaining constant. On the other hand, the experimental resolution at room temperature does not permit the comparison of any two photosynthetic samples that might be at different states or differ concerning energy migration.
317
Quite different is the case of Chl a fluorescence emission at 77 K. The fluorescence emission spectra show three main emission bands, originating from distinct pigment complexes (for a review see Govindjee 1995). It is accepted that the long wavelength band, exhibited at 725 nm by young plants and leaves greened in flashing light, at 735 nm by mature chloroplasts of higher plants and at about 715 nm by many green algae, comes from PSI (core and peripheral antenna; pigment pool ‘‘1’’ in the model of Fig. 2A), except for a small fraction which is due to the long wavelength tail of the 685 nm and 695 nm emission bands. The 695 and 685 nm bands are accepted to originate from the core antenna of PSII (pigment pool ‘‘2’’ in our model) and, specifically, from the CP-47 and CP-43 Chl a protein complex, respectively. At 77 K, the fluorescence transients obtained at 685 nm, 695 nm or 735 nm (as well as at all emission wavelengths) were found to have the same non-sigmoidal shape (typical of 77 K), starting from an initial fluorescence F0 and leveling off at a maximum fluorescence FM, however, with different values of the F0/FM ratio (see e.g., Strasser and Butler 1976, 1977a; Strasser and Greppin 1981). An example is given in Fig. 14, which depicts the fluorescence induction kinetics measured simultaneously at 735 nm,
F1,M
F1
F1,V
M1(2) = p1F (k21/k2F)
F1,0 F1,α = I1(2)
F2
t 0
0 F2,0 F2,V F2,M
t
Fig. 14 Fluorescence induction kinetics measured simultaneously at 735 nm (F1) and 695 nm (F2) in spinach chloroplasts (or, similarly, in leaves) at 77 K. The plot of F1 vs. F2, derived from the two kinetics (as demonstrated by the thin dash-dot lines for one pair of values), is a straight line, from which the intercept I1(2), equal to F1(a), and the slope M1(2), equal to p1F (k21/k2F), are determined. The minimal (initial), maximal and maximal variable fluorescence at 735 and 695 nm are indicated, i.e. F0;1 ; FM;1 ; FV;1 ¼ FM;1 F0;1 ; and F0;2 ; FM;2 ; FV;2 ¼ FM;2 F0;2 , respectively (modified from Strasser 1986)
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F735 F1 ¼ f ðtÞ, and 695 nm, F695 ¼ f ðtÞ, in spinach chloroplasts at 77 K (Strasser 1986). It should be clarified that, at that time, the band at 685 nm was considered as originating from the LHCII, based on findings that is was missing in leaves greened in flashing light (deprived of LHCII) and that it appeared when transfer of these leaves under continuous light induced LHCII formation (Strasser and Butler 1976); hence, F695 was attributed to the whole core antenna (i.e. it was taken as equal to F2). Though this is not holding anymore, we kept the notations used in the original figures and concept, assuming that F695—which now reads as FCP47—is proportional to F2. On the basis of the above, Kitajima and Butler (1975b) proposed that F1 is comprised of two different components, F1(a) and F1(b), according to the energy source that generates E1, the excitation rate (energy influx) of PSI: F1(a) is the component that originates from the excitation rate E1(a) of PSI that is generated by the light energy directly absorbed by PSI, and F1(b) is the component that originates from the excitation rate E1(b) of PSI that is generated by the energy migration (spill-over) from PSII to PSI. Hence, according to our notations, E1(a) = J1 and E1(b) = E21. Using for simplicity a bi-partite model and based on previous findings (Kitajima and Butler 1975b) that the fluorescence emitted by PSI (F1) is independent of the redox state of PSI RCs, Strasser and Butler (1976, 1977a) formulated equations, which we here re-write with the notations used in EFT as: F1 ¼ F1ðaÞ þ F1ðbÞ ¼ J1 p1F þ E21 p1F ¼ J1 p1F þ E2 p21 p1F ¼ J1 p1F þ F2 ðp21 =p2F Þp1F ) F1 ¼ J1 p1F þ F2 ðk21 =k2F Þp1F
ð134Þ
According to Eq. 134 the plot of F1 vs. F2 can well be used as a criterion for the origin of the PSI variable fluorescence: If it is a straight line, as Eq. 134 predicts and as indeed found experimentally in most studied cases (Strasser and Butler 1976, 1977a), it means that the PSI variable fluorescence originates solely from PS II, governed by the state of PSII RCs, which defines the exciton density in PSII antenna pigment pools and, concomitantly, the migration energy flux from PSII to PSI. Figure 14 (modified from Strasser 1986) presents an example of such cases, demonstrating also how F1 vs. F2 is constructed from simultaneous measurements of the fluorescence induction kinetics F1 ¼ f ðtÞ and F2 ¼ f ðtÞ Any deviation from the straight line indicates that the state of PSI RCs (redox state of P700) is also involved (as for example in red algae; see Ley and Butler 1977). Moreover, from the F1 vs. F2 plot the intercept and the slope, denoted as I1(2) and M1(2) respectively, can be determined (Fig. 14). According to Eq. 134, I1ð2Þ ¼ F1ðaÞ and M1ð2Þ ¼ ðk21 =k2F Þp1F ; hence the F1 vs. F2 plot can
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Fig. 15 The three fluorescence components F1(a), F1(b) and F2 (see text) vs. both the emission and the excitation wavelength. The samples were flashed bean leaves measured at 77 K (taken from Strasser 1986)
provide specific and distinct information: A change of the slope, induced by a certain treatment of the photosynthetic material, will indicate a change in the rate constant k21 (since k2F and p1F can well be considered as remaining constant), while a change of the intercept (change of F1(a), hence of J1) will indicate that a change in the initial distribution of the absorbed light energy between the two photosystems has occurred. For example, under conditions of LHCII phosphorylation, the intercept was found to increase, reflecting the induced increase of PSI effective absorption cross section, while the slope remained unchanged indicating that state 1 ? state 2 transition was not affecting the spill-over rate constants (Tsala and Strasser 1984). On the other hand, it was found (Strasser and Butler 1976; Tsala and Strasser 1984) that addition of Mg2? in chloroplasts that were under low salt conditions resulted in a decrease of both the slope and the
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intercept, in agreement with previous findings that high salt conditions decrease the extent of spill-over (Murata 1969; Butler and Kitajima 1975) and, also, enhance the fraction of light energy absorbed by PSII (Butler and Kitajima 1975), i.e. they favor the establishment of state 1. The indications for the spill-over changes were found to be in accordance with FV/FM changes and also confirmed by simultaneous measurements of P700 photo-oxidation (Satoh et al. 1976; Strasser and Butler 1977b). Since F1(a) can be determined from a F1 vs. F2 plot, F1(b) is also determined (F1(b) = F1 - F1(a)). Thereafter, F1(a), F1(b) and F2 can be plotted vs. both the emission wavelength and the excitation wavelength. The 3-dimensional plots shown in Fig. 15 (Strasser 1986) clearly demonstrate the features of the three distinguished fluorescence fluxes: Both the excitation and the emission spectra of F1(a) are those characteristic of PSI, while F1(b) has the excitation spectrum characteristic of PSII, like F2, and the emission spectrum of PSI.
Concluding remarks Construction and analysis of conceptual models are essential to address and understand the complexity of structures and functions in nature. Models of any theoretical complexity level can be proposed; however, they are meaningful only if they can be experimentally validated. Our analysis demonstrates that the EFT, which does provide the links with obtainable experimental signals, is a powerful approach for the formulation of any possible model, at any complexity level, even of future models that will incorporate new discoveries related with further technological advancements. A core advantage of EFT is that any possible energetic communication, between any complex arrangement of interconnected pigment systems, and any energy transduction by these systems can be easily formulated mathematically with trivial algebraic equations. It should be also added that the EFT is a general theory that can be applied to formulate the energetic behavior of any system, in any kind of biomembranes, as it rigorously defines all the terms used for analyzing energy distribution and transduction (exciton density, energy transfer, yield, probabilities and rate constants of energy transfer/transduction). The EFT has been indeed applied for the study of protochlorophyll(ide) to chlorophyll(ide) photoreduction (Strasser 1984) and for energy absorption and transduction by retina rhodopsin and bacteriorhodopsin (Montal et al. 1978; Strasser 1980), whose presentation was not in the frame of this paper. Acknowledgments M T-M thanks Dr Pierre Haldimann for stimulating discussions, critical comments and valuable suggestions during the preparation of this manuscript.
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