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ARTICLE IN PRESS Journal of Theoretical Biology 233 (2005) 287–300 www.elsevier.com/locate/yjtbi A theoretical study on effect of the initial redox state of cytochrome b559 on maximal chlorophyll fluorescence level (FM): implications for photoinhibition of photosystem II Dušan Lazára,, Petr Ilı́ka, Jerzy Krukb, Kazimierz Strza"kab, Jan Nauša a Laboratory of Biophysics, Department of Experimental Physics, Faculty of Science, Palacký University, tř. Svobody 26, 771 46 Olomouc, Czech Republic b Department of Plant Physiology and Biochemistry, Faculty of Biotechnology, Jagiellonian University, Gronostajowa 7, 30-387 Kraków, Poland Received 18 August 2004; received in revised form 6 October 2004; accepted 8 October 2004 Available online 1 December 2004 Abstract In this work, we extended the reversible radical pair model which describes energy utilization and electron transfer up to the first quinone electron acceptor (QA) in photosystem II (PSII), by redox reactions involving cytochrome (cyt) b559. In the model, cyt b559 accepts electrons from the reduced primary electron acceptor in PSII, pheophytin, and donates electrons to the oxidized primary electron donor in PSII (P680+). Theoretical simulations of chlorophyll fluorescence rise based on the model show that the maximal fluorescence, FM, increases with an increasing amount of initially reduced cyt b559. In this work we applied, the first to our knowledge, metabolic control analysis (MCA) to a model of reactions in PSII. The MCA was used to determine to what extent the reactions occurring in the model control the FM level and how this control depends on the initial redox state of cyt b559. The simulations also revealed that increasing the amount of initially reduced cyt b559 could protect PSII against photoinhibition. Also experimental data, which might be used to validate our theory, are presented and discussed. r 2004 Elsevier Ltd. All rights reserved. Keywords: Chlorophyll fluorescence induction; Cytochrome b559; DCMU; Photoinhibition; Reversible radical pair model 1. Introduction Although cytochrome (cyt) b559 is an essential part of the functional photosystem II (PSII) reaction centre, the Abbreviations: Cyt, cytochrome; DCMU, 3-(30 , 40 -dichlorophenyl)1, 1-dimethylurea; F0, minimal fluorescence; FM, maximal fluorescence; FV/FM, maximal quantum yield of photosystem II photochemistry; FLR, fluorescence rise; HP, high potential; LP, low potential; MCA, metabolic control analysis; OEC, oxygen evolving complex; P680, primary electron donor in photosystem II; Pheo, (pheophytin) primary electron acceptor in photosystem II; PQ, plastoquinone; PSII, photosystem II; QA, the first quinone electron acceptor in photosystem II; QB, the second quinone electron acceptor in photosystem II; RCII, reaction centre of photosystem II; RRP, reversible radical pair; 3P680, triplet state of primary electron donor in photosystem II Corresponding author. Tel.: +420 58 5225737; fax: +420 58 5634153. E-mail address: [email protected] (D. Lazár). 0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2004.10.015 function of cyt b559 is not yet fully understood (see the latest reviews on cyt b559 by Cramer et al. (1993); Shuvalov (1994); Whitmarsh et al. (1994); Whitmarsh and Pakrasi (1996); Stewart and Brudvig (1998)). Different results on the cyt b559 function are based on the fact that cyt b559 was identified in four redox forms in dependence on the measurement conditions and sample preparation: the extra low potential (LP) form (Em=–45 mV), the LP form (Em=+30 to +80 mV), the intermediary potential form (Em=+150 mV), and the high potential (HP) form (Em=+350 mV). The function of cyt b559 is mainly ascribed to its protective role against photoinhibition of PSII as it arises from numerous in vitro studies. In donor side photoinhibition, when oxygen evolving complex (OEC) cannot effectively pass through its S-states and does not thereby effectively donate electrons to primary electron donor in photo- ARTICLE IN PRESS 288 D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 system II (P680+), cyt b559 substitutes for the role of OEC and donates electrons to P680+. In such a way cyt b559 prevents P680+ accumulation which could initiate oxidation of nearby chromophores or residues leading to PSII degradation (Thompson and Brudvig, 1988; Blubaugh et al., 1991; Buser et al., 1992; van der Bolt and Vermaas, 1992). Under acceptor side photoinhibition, when electron transfer from Q A (reduced the first quinone electron acceptor in PSII) to QB (the second quinone electron acceptor in PSII) is disabled, cyt b559 can accept electrons from the primary reduced electron acceptor in PSII, pheophytin (Pheo), directly (Nedbal et al., 1992; Barber and De Las Rivas, 1993; Poulson et al., 1995) or via an additional plastoquinone (PQ) molecule in PSII (Kruk and Strzalka, 2001) to prevent a recombination of Pheo with P680+. This recombination can result in the formation of a triplet state of P680 (3P680) which could initiate the formation of reactive singlet oxygen causing PSII degradation once again (Styring et al., 1990; Vass et al., 1992). In addition to the acceptor and donor side photoinhibitions, the unifying model for the photoinhibition of PSII occurring in vivo in a steady state was proposed which depends on the generation and maintenance of an increased concentration of P680+Pheo radical pair and where the damage of PSII was suggested to be caused by P680+ rather than by singlet oxygen (Anderson et al., 1998). The role of cyt b559 in the unifying model for the photoinhibition of PSII has not been described as yet. In order to access the cyt b559 function, one can use the measurement of chlorophyll (chl) a fluorescence rise (FLR) as cyt b559 is a part of PSII and FLR is assumed to come exclusively from PSII (see the latest reviews on the FLR by Krause and Weis (1991); Govindjee (1995); Strasser et al. (1995); and Lazár (1999)). An increase in the chl fluorescence level reflects accumulation of the reduced primary quinone electron acceptor in PSII (Q A) as was originally suggested by Duysens and Sweers (1963). When the FLR is measured at high intensity of exciting light with an untreated sample, it is characterized by typical O2J2I2P transient. The J step is explained by transient accumulation of only reduced QA and hence the O–J transient is called the photochemical phase of the FLR (Neubauer and Schreiber, 1987; Strasser et al., 1995; Tomek et al., 2001; Lazár, 2003). The J–I–P transient is called the thermal phase and, in addition to QA reduction, probably reflects reduction of secondary quinone electron acceptor in PSII (QB) and PQ pool (Neubauer and Schreiber, 1987; Lazár et al., 1997; Stirbet et al., 1998; Tomek et al., 2001). Other processes can also be involved in this phase of FLR (Schreiber et al., 1989; Schreiber and Krieger, 1996; Koblı́žek et al., 2001; Pospı́šil and Dau, 2002; Bukhov et al., 2003; Hsu and Leu, 2003; Heredia and De Las Rivas, 2003; Schansker et al., 2003; Vredenberg and Bulychev, 2003). However, even if significant progress in understanding the FLR has occurred, the O–J–I–P transient is still not fully understood at present. Therefore, an inhibitor of electron transfer between QA and QB (Oettmeier and Soll, 1983; Trebst and Draber, 1986; Trebst, 1987; Shigematsu et al., 1989), 3-(30 ,40 -dichlorophenyl)-1,1-dimethylurea (DCMU), is often used to make the reactions in PSII simpler. When a sample is treated with DCMU, the FLR measured at high intensity of exciting light is characterized by a steep fluorescence increase reaching the maximal saturation level at approximately the position of the J step measured with the sample without DCMU (Strasser et al., 1995; Lazár et al., 1998; Lazár and Pospı́šil, 1999). Mathematical modelling is at present an important tool in better understanding the explored processes. This is also valid in understanding plant metabolic pathways (for reviews see Giersch, 2000; Morgan and Rhodes, 2002) as well as the FLR (for review see Lazár, 1999). Excitation energy utilization and photochemical reactions in the reaction centre of PSII (RCII), which is also the case when whole PSII (thylakoid membranes) is treated by DCMU, can be well described by the reversible radical pair (RRP) model, also called the exciton radical pair equilibrium model and first proposed by Breton (1983) (see also van Grondelle, 1985; Schatz et al., 1988; Leibl et al., 1989; Roelofs et al., 1992; Dau, 1994). The RRP model and its derivatives have already been used for simulation or fitting of the FLR measured both with and without DCMU (Baake and Schlöder, 1992; Trissl et al., 1993; Trissl and Lavergne, 1995; Lavergne and Trissl, 1995; Lazár and Pospı́šil, 1999; Lazár, 2003). Although a modified RRP model was used for studying PSII photoinhibition occurring in DCMU-treated thylakoid membranes (Vavilin et al., 1998), cyt b559 has not yet been included in the RRP model. The contribution of cyt b559 to the FLR and the protective function of cyt b559 against PSII photoinhibition have also not yet been studied with the help of theoretical simulations. With respect to known electron transport pathways in which cyt b559 is involved (see above) and the formulation of the RRP model, cyt b559 can be included into the RRP model as the electron acceptor from Pheo and/or electron donor to P680+. Each of these electron pathways separately, or considered together as one pathway enabling a cycle around PSII (Heber et al., 1979; Falkowski et al., 1986; Arnon and Tang, 1988; Prasil et al., 1996), should protect PSII against photoinhibition. Moreover, addition of a new component (cyt b559) to the RRP model should lead to a change in fluorescence properties of the RRP model. A powerful tool in theoretical studying of explored phenomenon is the metabolic control analysis (MCA; for reviews see Fell, 1992; Visser and Heijnen, 2002). In MCA, in addition to other coefficients, the control coefficient C is calculated which exactly quantifies to ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 what extent particular model parameter (concentration or activity or rate constant) controls selected variable of the model (concentration or flux). MCA has already been used for exploration of several plant metabolic pathways, e.g. Calvin cycle (Poolman et al., 2000) or the malate valve (Fridlyand et al., 1998) but it has never been used for exploration of the FLR. In this work, we extended the RRP model by redox reactions involving cyt b559. Firstly, using this model, we focused on the effect of the initial redox state of cyt b559 on the maximal fluorescence level FM in the presence of DCMU. Further, we performed an analysis of fluxes in our model (MCA) in order to discover to what extent the reactions occurring in the model control the FM level and how this control depends on the initial redox state of cyt b559. Additionally, we used theoretical simulations to explore whether changes in the initial redox state of cyt b559 can protect PSII against photoinhibition. Finally, we presented and discussed experimental results, which might be used to validate our theory. 2. Material and methods 2.1. Plant material and preparation of thylakoid membranes Pea seedlings (Pissum sativum) were cultivated in perlite for 14 days at 22 1C. The seedlings were watered regularly and exposed to a periodic light–dark cycle of 16 h of white light (90 mmol photons of photosynthetically active radiation m2 s1) and 8 h of dark. Detached pea leaves were used for the thylakoid membrane preparation. Thylakoid membranes were isolated according to Dau et al. (1995) with a slight modification. The pea leaves were mixed with a chilled grinding buffer (0.4 M sucrose, 0.4 M NaCl, 35 mM HEPES (pH 7.5), 4 mM MgCl2, 1 mM EDTA, 5 mM ascorbic acid and 2 g l1 BSA) and homogenized three times for 5 s by T25 Basic homogenizer (IKA Labortechnik, Staufen, Germany). The homogenate was filtered through miracloth and centrifuged for 6 min at 5000g. The chloroplast pellet was resuspended in a buffer containing 25 mM HEPES (pH 7.5), 150 mM NaCl, 8 mM MgCl2 and then centrifuged for 10 min at 5000g. The resulting pellet consisting of intact thylakoid membranes was resuspended in a small amount of the buffer containing 50 mM MES (pH 6.0), 15 mM NaCl, and 10 mM MgCl2. All steps were performed under dim green light at 4 1C. The chl content was determined spectrophotometrically in 80% acetone according to Lichtenthaler (1987). Thylakoid membranes were diluted in the last mentioned buffer to the chl concentration of 50 mg ml1 prior to the fluorescence or spectroscopic measurements. 289 2.2. Measurement of FLR curves FLR curves were measured with 300 ml of thylakoid membranes in a standard 0.5 0.5 cm glass cuvette using a Plant Efficiency Analyser (Hansatech, Norfolk, United Kingdom). LEDs producing strong red light (l ¼ 650 nm) of an intensity of 12,000 mmol photons m2 s1 were used for excitation and fluorescence emission was detected by PIN diode above 700 nm. The FRs were recorded for up to 1 s starting at 10 ms after the onset of irradiation. A fluorescence signal at 50 ms was considered as minimal fluorescence F0 (Strasser et al., 1995). Before each measurement, thylakoid membranes were treated with DCMU (20 mM, 6 min in dark) or additionally with FeCy (20 mM, 6 min in dark) or Asc (5 mM, 6 min in dark). A higher volume of thylakoid membrane suspension with a chl concentration of 50 mg ml1 was always prepared. 300 ml of the suspension was chemically treated as described above and used for the measurements and another 300 ml of the suspension was not treated at all and used for measurement of the control (O2J2I2P) FLR. FM level of the control FRs subsistent to the DCMU, DCMU+FeCy, and DCMU+Asc FRs were normalized to the same level and appropriate DCMU, DCMU+FeCy, and DCMU+Asc FRs were then multiplied by the same normalization factors as the subsistent control FRs. This procedure was used in order to be able to present the FLR curves as they are shown in Fig. 5 and to avoid differences in the fluorescence signal caused by a different chl concentration in the measuring cuvette due to possibly incorrect pipetting. Each measurement (DCMU, DCMU+FeCy, DCMU+Asc) was performed four times with a new thylakoid membrane preparation each time with the mean value7standard deviation from all the measurements presented in the text. The light intensity was measured with a Quantum Radiometer LI-189 (LI-COR, Lincoln, USA). 2.3. Determination of the cyt b559 redox state The amount of particular redox forms of cyt b559 in the DCMU-treated thylakoid membranes was determined with a AMINCO DW-2000 UV/VIS spectrophotometer (SLM Instruments, Urbana, USA) using a split beam mode. Thylakoid membranes were treated with DCMU (20 mM, 6 min in dark), FeCy (20 mM, 6 min in dark), hydroquinone (HQ; 4 mM, 6 min in dark), and Asc (5 mM, 6 min in dark), step by step, and followed by the measurement of absorption spectrum in an interval of 520–590 nm. Relative amounts of particular redox forms of cyt b559 in thylakoid membranes were determined from absorbances at 559 nm related to the background absorbances at 540 nm in difference spectra obtained as follows: reduced HP cyt b559 ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 290 (spectrum 2–spectrum 1), oxidized HP cyt b559 (spectrum 3–spectrum 2), and oxidized LP cyt b559 (spectrum 4–spectrum 3). The spectra 1–4 were obtained as a difference of spectra of the sample and the reference (both with thylakoid membranes) treated in the following way: DCMU in both the sample and reference (spectrum 1), DCMU in the sample and DCMU and FeCy in the reference (spectrum 2), DCMU and HQ in the sample and DCMU and FeCy in the reference (spectrum 3), and DCMU, HQ, and Asc in the sample and DCMU and FeCy in the reference (spectrum 4). The contribution of cyt f to the absorbance at 559 nm was negligible. The amount of cyt b559 was measured four times with new thylakoid membrane preparation each time and with a mean value7standard deviation from all the measurements presented in the text. 2.4. Theoretical simulations The FLR curves were simulated using Gepasi software version 3.21 (P. Mendes, The University of Wales, Aberystwyth, United Kingdom) that was designed for simulations of chemical and biochemical kinetics (Mendes, 1993, 1997). Gepasi uses routine LSODA (Livermore Solver of Ordinary Differential Equations) for solution of a set of differential equations. LSODA is a sophisticated algorithm that measures the stiffness of equations and switches the integration method dynamically according to the measure of stiffness. As we made use of Gepasi software for our simulations, the expression of differential equations describing the time derivations of particular states in the model is not necessary. The model is entered into the software directly as chemical reactions as presented in Scheme 1. 2.5. Metabolic control analysis (MCA) In MCA, in addition to other parameters, the control coefficient is defined as a relative change in steady-state variable Xj upon a relative change in the steady-state rate vi, J CX vi ¼ qX j =X j q ln X j ¼ : qvi =vi q ln vi In our case, the steady-state variable Xj is the FM level and the steady-state rate vi is the rate of selected reaction in the theoretical model. As the fluorescence signal in the FLR is defined in our model as a deactivation of excited states via the rate constant of fluorescence emission (it is a product of the rate constant of fluorescence emission and the concentration of all excited states; see Theory), fluorescence emission amounts to flux. Therefore, the control coefficients calculated by us amount to the flux control coefficients. The Gepasi software (see above) calculated the control coefficients. For this purpose, any reversible reaction in the model was entered into the software as two irreversible reactions in order to obtain the flux control coefficients of the reaction in both directions. If a particular reaction occurred more than once in the model (e.g. a charge separation in open RCII; see Scheme 1), the overall flux control coefficient of this reaction was calculated as the sum of all the particular flux control coefficients. The correctness of the MCA results was checked by the calculation of the sum of all the flux control coefficients determined for the given amount of initially reduced cyt b559. According to the theory of MCA, the sum of all flux control coefficients is equal to unity (for reviews on MCA see Fell, 1992; Visser and Heijnen, 2002). As the simulated fluorescence is in our case the sum of the four fluxes (see Theory), the sum of all the flux control coefficients is 4 and not unity. This was verified as true for all the amounts of initially reduced cyt b559 used in our calculations. 3. Theory While the original RRP model describes charge separation, recombination and stabilization when QA is both oxidized and reduced (Breton, 1983; van Grondelle, 1985; Schatz et al., 1988; Leibl et al., 1989; Roelofs et al., 1992; for a review see Dau, 1994), we previously extended it (Lazár and Pospı́šil, 1999) by addition of the reduction of P680+ by electrons from OEC in the donor side of PSII. The model employed here for simulations of FLR curves in the presence of DCMU is based on the extended RRP model, which was further extended in this work by the addition of redox reactions involving cyt b559. We considered in our model the cyt b559, which accepts electrons from Pheo and donates electrons to P680+ (see Scheme 1). This cyt b559 electron pathway is only hypothetical but it is consistent with facts known from literature (see Introduction) and resulting theoretical simulations are in agreement with our experimental results (see Discussion). The initial conditions and values of the rate constants used in the simulations are summarized in the legend to Scheme 1 and Table 1, respectively. All the reactions in the model are assumed to be of the first order with respect to one component and follow the mass action theory. Fluorescence signal at time t in the FLR is defined to be proportional to the sum of the forms with excited chl a multiplied by the rate constant of chl a fluorescence, i.e. as F ðtÞ ¼ kF fððL PÞ IAÞðtÞ þ ððL PÞ I AÞðtÞ þ ððL PÞ IA ÞðtÞ þ ððL PÞ I A ÞðtÞg: See Scheme 1 for the meaning of the symbols. We had to estimate the value of the rate constant of cyt b559 reduction by Pheo (kCred; see Scheme 1), while ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 291 Scheme 1. A scheme for all the reactions in our model used for simulations of the FLR curves in the presence of DCMU. Part A shows the main reaction scheme where numbers 1, 2, and 3 next to the arrows mean P+ reduction, C+ reduction, and excited states deactivation, respectively, as described in detail in part B. The reactions described in part B occur at any time when a redox state of form(s) in the top left part of each particular reaction scheme is formed and unrelated to, but conserving, the redox state of other forms. L, P, I, A, C, YZ, and Si mean all chlorophylls from PSII light harvesting antennae, P680, pheophytin (Pheo), QA, cyt b559, tyrosine Z, and the S-states of OEC, respectively. The asterisk indicates an excited P state. The rate constants mean: kL—excited states formation; kLHA HD —overall non-radiative loss of excited states in light harvesting antennae; kHD— (non-radiative) heat dissipation of excited states by P680+ (the fluorescence quenching by P680+); kPQ HD—(non-radiative) heat dissipation of excited states by PQ molecules (fluorescence quenching by PQ molecules); kUU—excited states transfer between RCIIs; kF—fluorescence; koðcÞ 1 —electron + transfer from P680 to Pheo (the charge separation) in the open (closed) RCIIs; koðcÞ 1 —backward electron transfer from Pheo to P680 (the charge recombination) in open (closed) RCIIs leading to formation of excited-state P680*; ko2 —electron transfer from Pheo to QA (the charge stabilization) in open RCIIs; kc2 —backward electron transfer from Pheo to P680+ (non-radiative charge recombination) in closed RCIIs leading to the ground * + reduction by YZ; state of P680; kPArec —charge recombination between P680+ and Q A leading to formation of excited-state P680 ; kPred —P680 kPox—backward electron transfers from P680 to Y þ in the previous reactions; k —electron donation from the S state of OEC to Y þ 12 1 Z Z during the S1 ! S2 transition; kCox—electron donation from cyt b559 to P680+; and kCred—electron donation from Pheo to cyt b559. The values of the rate constants used in all the simulations are compiled in Table 1 and the initial conditions were as follows: [(LP)IA]0=1 (i.e. there is no excitation formed, P680 is reduced and Pheo and QA are oxidized in the dark); [YZ]0=1 (i.e. Y Z is reduced in the dark); ½S1 0 ¼ 1 (i.e. all OEC is initially in the S1 state; Messinger and Renger, 1993); ½PQ0 ¼ 7 (i.e. there are 7 PQ molecules in the PQ pool per PSII and all are oxidized in the dark; McCauley and Melis, 1986); ½C0 and ½C þ 0 were changed from 0 to 1 (½C0 þ ½C þ 0 ¼ 1) to simulate the effect of the initial redox state of cyt b559 on the FLR curve; all other forms in the model have initially a zero relative concentration in the simulations. the values of all the other rate constants in the model were known from literature (see Table 1). The time constant of this cyt b559 reduction was roughly estimated by Satoh et al. (1990) to be in the submicrosecond range. As discussed by Whitmarsh et al. (1994), the rate of electron flow from Pheo to cyt b559 should be very high. In order to obtain an approximate value of this rate constant, we simulated the FLR curves in the presence of DCMU and with initially fully oxidized cyt b559 for the rate constant in the range of 107–1010 s1. We discovered that only when kCred was about the same or higher than the rate constant of electron transfer from Pheo to QA (k02; 2.3 109 s1), the theoretical FLR curves resembled the experimentally measured FLR curves in the presence of DCMU (data not shown). For lower kCred the simulated FRs had a local maximum at the position of the J step (see Introduction) thus differing from the experimental FLR measured with DCMU-treated samples (see Fig. 5). Therefore, we used in all our simulations of FLR curves in the presence of DCMU kCred equalled to the value of k02 (see Table 1). In addition, only when kCred was above 109 s1 and cyt b559 fully initially oxidized in our model, a significantly lower accumulation of stable Pheo (P680PheoQ A state) occurred (to protect against the acceptor side photoinhibition of PSII; see Nedbal et al., 1992) in comparison with the case at which cyt b559 is not included in the model (data not shown). See also discussion on kCred in the Discussion. ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 292 Table 1 Values of the rate constants used in the simulations Rate constant Used value (s1) Value in literature (s1) kL kF kLHA HD kPHD 4000 6.7 107 5 108 1 109 5.6 107 4000–5500a 6.7 107b 3–10 108c 1 109d 5.6 107e 1 109 3 109 3 108 2.3 109 4.8 108 3.4 108 1 109 10000 2.9 107 5.8 106 10000 667 2.3 109 1–6 109f 1.9–9.3 109c 3–20 108c 2–2.3 109c 4.8–15 108c 3.4–30 108c 3–10 108c 5000–10000g 2–5 107h 1.1–8.3 106i 7150–25000j 500–667k See the theory kPQ HD kUU ko1 ko1 ko2 kc1 K c1 K c2 kPArec kPred kPox K 12 kCox kCred For the meaning of the rate constants see the legend to Scheme 1. a For the light intensity used in our measurement, see Lazár and Pospı́šil (1999); Tomek et al. (2001) and Lazár (2003). b Rabinovich and Govindjee (1969). c Reviewed by Dau (1994). d Bruce et al. (1997). e Lazár (2003). f Lavergne and Trissl (1995) and Trissl and Lavergne (1995). g Haveman and Mathis (1976) and Renger and Wolff (1976). h Brettel et al. (1984) and Meyer et al. (1989). i For the equilibrium constant K ¼ kPred =kPox 3; see Brettel et al. (1984). j Summarized by Razeghifard et al. (1997). k Satoh et al. (1990) and Havaux (1998). 4. Results 4.1. Theoretical FLR curves for different amounts of initially reduced cyt b559 In our theoretical model, we hypothesize that cyt b559 can accept electrons from Pheo and donate them to P680+. We first tested how the initial redox state of such cyt b559 can affect and control the FLR. In addition, as the function of cyt b559 in PSII is mainly connected with its protective role against photoinhibition of PSII (see Introduction), we also tested if changes in the initial redox state of cyt b559 can protect PSII against its photoinhibition. The FLR curves in the presence of DCMU simulated for different amounts of initially reduced cyt b559 by our model are shown in Fig. 1. These curves indicate that both the F0 and the FM increase with an increasing amount of initially reduced cyt b559. The FM/F0 ratio is about 2.1–2.7 in the simulations which is close to about 2.5–3.1 found for the FM/F0 ratio for experimental FLR Fig. 1. FLR curves simulated on the basis of our model for different amounts of initially reduced cyt b559 as indicated above the simulated curves. The vertical dotted line at 50 ms indicates the time at which the F0 level is determined in the experimental FRs (see Fig. 5). Inset A shows the difference in simulated F0 levels obtained for different amounts of initially reduced cyt b559 in more detail. Inset B shows the dependence of FM of the simulated FLR curves (FM of the 100% curve is set as 100%) on the initial redox state of cyt b559. curves (see Fig. 5) but it is lower than 4.5–5 found for the FM/F0 ratio of FLR curves measured without any treatment (Lazár et al., 1998; Lazár, 2003). The low FM/ F0 ratio, in case when DCMU is present (Fig. 5) or appropriate FLR curves are simulated (Fig. 1), is due to an increase of the F0 level and a decrease of the FM level. This is caused by backward electron flow from Q B to QA preceding DCMV binding and by the oxidized PQ pool quenching, respectively. As described and discussed in detail in Lazár (2003), F0 in the FLR simulated by the RRP-based model is established at about 3 ns after the beginning of irradiation and lasts to about 1 ms. This simulated F0 level reflects accumulation of chl excited states when all electron acceptors in the acceptor side of PSII are oxidized, which means the accumulation of P680*PheoQA state in our case. An increased accumulation of this state is the only reason (data not shown) why the F0 level with an increasing amount of initially reduced cyt b559 increases (inset A in Fig. 1). This increase is caused by a shift of equilibrium among all occurring reactions in the model in favour of accumulation of the P680*PheoQA state when cyt b559 is initially reduced to a greater extent. Inset B in Fig. 1 clearly shows that the FM level in the simulated FLRs increases with an increasing amount of initially reduced cyt b559. Simulated time courses of particular excited states in the model revealed (data not shown) that the main reason of the increase of FM is in an increase of the maximal (i.e. steady state) concentration of P680*PheoQ A state when cyt b559 is initially reduced to a greater extent. This increase is caused, as in ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 case of P680*PheoQA described above, by a shift of equilibrium among all occurring reactions. According to the original suggestion by Duysens and Sweers (1963), an increase in the FM level should reflect an increased accumulation of Q A. This is also valid in our case, as shown in Fig. 2, where the time courses of the sum of all the reduced QA in the model are simulated in dependence on the initial amount of reduced cyt b559. Fig. 2 also shows that for only 100% of cyt b559 initially reduced, the maximal accumulation of Q A reaches 1, in other words, only in this case the FM level actually has the meaning of fluorescence intensity coming from fully closed RCIIs. For lower amounts of initially reduced cyt b559 not all RCIIs are closed and the corresponding FM level does not have its definition meaning. This is surprising as the value of the rate constant of excited states formation (kL ; see Scheme 1) was calculated for high (=saturating) intensity of exciting light (see Lazár and Pospı́šil, 1999 for the calculation) which is used for measurements of the experimental FRs (see Fig. 5). Therefore, the presence of a certain amount of initially oxidized cyt b559 can prevent, even at very high intensity of exciting light, the reaching of the real FM level (in the sense of a definition) thus leading to the incorrect determination of the maximal quantum yield of PSII photochemistry which is expressed as the FV/FM (=(FMF0)/FM) ratio (Kitajama and Butler, 1975). Thus, a different amount of initially reduced cyt b559 reflecting the use of different buffers during preparation of chloroplasts or thylakoid membranes could be the reason why the measured values of fluorescence parameters (F0, FM, FV/FM) are different. Special attention should be paid to this fact particularly when the results 293 with one sample preparation are compared with the results obtained with another sample preparation. As electrons can be transferred from Pheo not only to QA but also to cyt b559 in our model, the steady-state level of reduced cyt b559 should reflect changes in maximal Q A accumulation and changes in FM level as described above. The inset in Fig. 2, however, reveals that the FM level (and also the maximal accumulation of Q A; data not shown) linearly increases with the steadystate amount of reduced cyt b559 only up to about 80% of cyt b559 initially reduced. When more than about 80% of cyt b559 is initially reduced, more than 99.9% of cyt b559 is reduced in the steady state and a further increase in the amount of initially reduced cyt b559 does not lead to a proportional increase in the steady-state amount of reduced cyt b559. In the latter case however, the FM level (and also the maximal accumulation of Q A) continually increases. This indicates that when cyt b559 is initially more reduced than a certain amount (80% in our case), changes in the FM level do not reflect changes in the steady-state amount of reduced cyt b559. A question arises as to what is a mechanistic explanation for the changes in FM level and maximal steady-state accumulations of reduced cyt b559 and QA in dependence on the initial redox state of cyt b559 as shown in Figs. 1 and 2. The changes can be easily explained by the fact that there is one more electron in PSII when cyt b559 is initially reduced when compared to the case when cyt b559 is initially oxidized. The presence or absence of the ‘‘additional’’ electron either stored or not initially in cyt b559 results in different equilibrium in steady state: PSII is more reduced (more reduced cyt b559 and QA in the steady state leading to high FM level) when cyt b559 is initially reduced and, vice versa, PSII is less reduced (less reduced cyt b559 and QA in the steady state leading to low FM level) when cyt b559 is initially oxidized. 4.2. Control of the FM level Fig. 2. Time courses of the sum of all Q A (all states in the model which have Q A) simulated on the basis of our model for different amounts of initially reduced cyt b559 (0%, 20%, 40%, 60%, 80%, and 100%). The inset shows the dependence of the simulated FM level (data from Fig. 1) on the maximal (i.e. steady-state) accumulation of reduced cyt b559 as resulted from the simulations for different amounts of initially reduced cyt b559 (indicated in the inset). In order to understand the control of the FM level in more detail, we applied the so-called MCA to our model. MCA is a sophisticated tool which substitutes an analysis of a model made using the time courses of reactant concentrations at different initial conditions. To the best of our knowledge, we are the first to apply MCA to a model describing electron transfer reactions X in PSII. In MCA, the control coefficient C vi j ; describing to what extent a particular reaction in a model (characterized by its steady-state rate vi) controls a selected steady-state variable Xj of the model (the FM level in our case), is defined (see Material and methods). The positive C FviM indicates that an increase in the steady-state rate vi results in an increase in the FM level (i.e. stimulation of fluorescence emission) and, on the other hand, the negative C FviM means that an increase in ARTICLE IN PRESS 294 D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 the steady-state rate vi results in a decrease in the FM level (i.e. fluorescence quenching). A higher absolute value for the C FviM indicates a higher control of the FM level by the reaction characterized by its steady-state rate vi. The results of the MCA for our model are presented in Fig. 3. Some reactions in the model only control the FM level in a minor way (Fig. 3A) whereas other reactions have a high control on the FM level (Fig. 3B). As for the 100% initial reduction of cyt b559, the control coefficients for reactions involving redox changes of cyt b559, i.e. its oxidation by P680+ (vCox) and of cyt b559 reduction by Pheo (vCred), are almost zero (0.009) and zero, respectively, the values of the other control coefficients provide information as to how particular reactions in the model control the FM level in the case when cyt b559 is not considered in the model. This case is worth mentioning as the simulations of the FLR in the presence of DCMU was reported several times in literature but always without MCA (Trissl et al., 1993; Lavergne and Trissl, 1995; Trissl and Lavergne, 1995; Vavilin et al., 1998; Lazár and Pospı́šil, 1999; Lazár, 2003). In our case, in addition to excitation formation Fig. 3. The dependence of the flux control coefficients C FviM , by which the rate vi (i.e. flux) of the particular reaction in our model controls the FM level of the simulated FLR curves, on the amount of initially reduced cyt b559 in the simulations. The subscripts and superscripts in the denotation of the rates have the same meaning as those in the denotation of the rate constants used in our model shown in Scheme 1. (vL) and fluorescence emission (vF), the radiative charge recombination between P680+ and Q A (vPArec) and the charge separation in the closed RCII (vc1 ) contribute the most to fluorescence emission in the FM level. On the other hand, overall non-radiative loss of excited states in light harvesting antennae (vLHA HD ), non-radiative charge recombination between Pheo and P680+ in the closed RCIIs (vc2 ), the fluorescence quenching by oxidized PQ molecules (vPQ HD), the charge separation between P680 and Pheo in the open RCIIs (vo1 ), and the charge stabilization between Pheo and QA (vo2 ) contribute the most to the quenching of the FM level. As the nonradiative charge recombination between Pheo and P680+ in the closed RCIIs (vc2 ) and overall non-radiative loss of excited states in light harvesting antennae (vLHA HD ) quench the FM level by about the same extent (for the values of the rate constants used in our simulations), MCA cannot finally judge if the quenching of the FM level is mainly located in the RCIIs or in the light harvesting antennae. A particularly interesting result of the MCA is that the initial redox state of cyt b559 strongly affects the extent in which particular reactions in the model control the FM level. While the control coefficients of certain reactions are almost unaffected by the initial redox state of cyt b559 (Fig. 3; excited states transfer between RCIIs (vUU), electron donation from the S1 state of the OEC to Yþ Z during the S1-S2 transition (v12 ), vF ), some reactions lose their control on the FM level when cyt b559 is initially more reduced (Fig. 3; electron donation from Pheo to cyt b559 (vCred), the fluorescence quenching by P680+ (vPHD), vo2 ) whereas some reactions first significantly increase and then decrease their control on the FM level when cyt b559 is initially reduced more (Fig. 3; backward electron transfer from Pheo to P680+ in the closed RCIIs (vc1 ), vo1 ; vL ; electron transfer from P680 to Pheo in the closed RCIIs (vc1 ), P680+ reduction by YZ (vPred), backward electron transfers from P680 to + Yþ Z (vPox), electron donation from cyt b559 to P680 LHA c (vCox), vHD , v2 ). In the latter case, maximum control of the FM level by the reactions is reached at about 77% of the initially reduced cyt b559. Therefore, the FM level is extremely sensitive to changes in the rates of the abovementioned reactions at around 77% initial reduction of cyt b559. Furthermore, the 77% of initially reduced cyt b559 corresponds to the roughly determined 80% initial reduction of cyt b559 representing the break point in linear dependence between the steady-state amount of reduced cyt b559 and the FM level (see the inset in Fig. 2). The ‘‘critical’’ amount of 77% of initially reduced cyt b559 for the control of the FM level need not be exactly 77% as found here but can vary in dependence on the values of the rate constants used in the model. It is also worth noting that for up to about 95% of initially reduced cyt b559, the non-radiative charge recombination between Pheo and P680+ in the closed ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 295 RCIIs (vc2 ), i.e. fluorescence quenching located in the RCII, mainly controls the lowering of the FM level. On the other hand when more than about 95% of cyt b559 is initially reduced, the FM level is mainly quenched by the overall non-radiative dissipation of excited states in light harvesting antennae (vLHA HD ), i.e. by fluorescence quenching located in the light harvesting antennae (Fig. 3). Although in O2 evolving samples (leaves, chloroplasts, thylakoid membranes, PSII membranes) most of cyt b559 is initially reduced (Hiller et al., 1971; Heber et al., 1976; Ortega et al., 1988; Thompson et al., 1989; Shinohara et al., 1992; Mizusawa et al., 1995; McNamara and Gounaris, 1995), cyt b559 has never been reported in literature to be fully reduced initially. Therefore, the fluorescence quenching by a non-radiative charge recombination between P680+ and Pheo in closed RCII seems to be the main process in PSII, which quenches the FM level. 4.3. Consequences for PSII photoinhibition With respect to the previous theoretical results, one can summarize that the initial redox state of cyt b559 can control the FM level and fine-tune the contribution of particular reactions in PSII to this control. Therefore and because the function of cyt b559 is mainly connected to the protection of PSII against its photoinhibition (see Introduction), we further explored, through theoretical simulations whether the initial redox state of cyt b559 can control and protect PSII against photoinhibition. Here it is important to note that, when DCMU is applied to a sample or an appropriate theoretical model is used for simulations as it is in our case (see also Theory and Introduction), electrons cannot be transferred from Q A to QB in the acceptor side of PSII and the acceptor side PSII photoinhibition can be triggered (see Introduction). When DCMU is applied, only two electrons can be stored in the electron acceptors of PSII (one on Pheo and one on QA) resulting in the impossibility of OEC to continuously pass through its S-states. This situation leads to the donor side photoinhibition of PSII (see Introduction). Therefore, the FRs measured with a DCMU-treated sample or simulated with an appropriate theoretical model represent a suitable system for studying the protective role of cyt b559 against photoinhibition of PSII. In the donor or acceptor side photoinhibition of PSII, accumulation of P680+ or 3P680, respectively, is the starting process occurring in PSII leading to its damage (see Introduction). As our model does not include the formation of the 3P680, we assumed in agreement with literature (Booth et al., 1990; van Mieghem et al., 1995; Pospı́šil, 1997) that the amount of formed 3P680 is proportional to the amount of forms with charged P680 and Pheo in both the open and closed RCII, i.e. to the sum of P680+PheoQA and P680+PheoQ A forms in Fig. 4. Time courses of the sum of all P680+ (all states in the model which have P680+; part A) and of the sum of all P680+Pheo (all states in the model which have P680+Pheo; part B) simulated on the basis of our model for different amounts of initially reduced cyt b559 (0%, 20%, 40%, 60%, 80%, and 100%). our model. The time courses of all P680+ and P680+Pheo accumulations, simulated for different amounts of initially reduced cyt b559 in our model, are shown in panels A and B, respectively, of Fig. 4. The simulated curves clearly show that there is a decrease in steady-state concentrations of both P680+ and P680+Pheo states with an increasing amount of initially reduced cyt b559. This means that an increase in the amount of initially reduced cyt b559 can protect PSII against its photodamage with respect to both the donor and acceptor side photoinhibitions of PSII. PSII photoinhibition in vivo was found to be a light dosage event and single hit process (Park et al., 1995, 1996, 1997; Sinclair et al., 1996; Anderson et al., 1998) implying, in the sense of our model concept, that after absorption of a certain light dose by a photosynthetic apparatus, its electron transfer chain is saturated (i.e. clogged) and PSII behaves like that treated with DCMU. Absorption of the last quantum of light then triggers the photodamage of PSII. We can also, therefore, use our model to test whether the initial redox state of cyt b559 can control and protect against PSII photoinhibition with respect to the unifying model of PSII photoinhibition in vivo in the steady state, which depends on the generation and maintenance of P680+Pheo radical pair (Anderson et al., 1998). In this case, we can use once again the simulations shown ARTICLE IN PRESS 296 D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 in panel B of Fig. 4 where now we take the accumulation of all P680+Pheo states in the model, not as a measure of 3P680 accumulation, but as a measure of the radical pair accumulation per se. The simulations in panel B of Fig. 4, showing that the increasing amount of initially reduced cyt b559 leads to a decrease of P680+Pheo states at steady-state level, demonstrate that the initial reduction of cyt b559 can protect PSII against its photodamage also with respect to the unifying model of PSII photoinhibition. In agreement with the protective role of reduced cyt b559 against PSII photoinhibition in vivo is also the fact that in O2 evolving samples most of cyt b559 is initially reduced (Hiller et al., 1971; Heber et al., 1976; Ortega et al., 1988; Thompson et al., 1989; Shinohara et al., 1992; Mizusawa et al., 1995; McNamara and Gounaris, 1995). It is also important to note that according to Fig. 4, the steady-state maximal accumulation of all P680+ (panel A) is by 4–5 orders of magnitude higher than of all P680+Pheo (panel B). The accumulation of P680+ is mainly caused by accumulation of P680+PheoQ A and P680+PheoQA states in the model and not by + P680+PheoQ A and P680 Pheo QA states (data not shown). Therefore, if we take accumulation of all P680+Pheo states in the model again as a measure of 3 P680 accumulation (see above), the results in Fig. 4 suggest that photoinhibition of PSII is more likely to be caused by accumulation of P680+ than by accumulation of 3P680. This finding is in agreement with the statement by Anderson et al. (1998) who suggested in the framework of their unifying model of PSII photoinhibition that damage of PSII in vivo occurs rather via generation of P680+ than via generation of singlet oxygen caused by the 3P680 formation. 5. Discussion Although theoretical simulations presented above are based on a model with hypothetical function of cyt b559, we present below experimental data, which might be used as a support for the correctness of our model. Fig. 5 shows typical FLR curves measured with pea thylakoid membranes at a high intensity of exciting red light (12,000 mmol photons m2 s1) in the presence of DCMU (20 mM) alone, DCMU with Asc (5 mM), and DCMU with FeCy (20 mM). Considering redox changes of electron carriers in PSII, the addition of Asc or FeCy in the used concentrations before the FLR measurements should lead to reduction or oxidation, respectively, of cyt b559 only. Therefore, the experimental DCMU-FLR curves can be directly compared with theoretical FRs simulated for different initial redox state of cyt b559 as shown in Fig. 1. When the FM level determined from the experimental DCMU-FRs is plotted in dependence on the amount of initially reduced cyt b559 (the inset in Fig. 5. Typical FLR curves measured at room temperature upon red light (l ¼ 650 nm) excitation of an intensity of 12,000 mmol photons m2 s1 with pea thylakoid membranes in the presence of DCMU. The thylakoid membranes were also treated with Asc (the DCMU+Asc curve) or with FeCy (the DCMU+FeCy curve) before the measurement to initially reduce or oxidize cyt b559, respectively. The inset shows the dependence of FM of the measured FLR curves (FM of the DCMU+Asc curve is set as 100%) on the initial redox state of cyt b559 determined spectroscopically (the bars indicate standard deviations). Fig. 5) and compared with the same plot determined from theoretical FRs (the inset B in Fig. 1), there is an almost numerical agreement between the simulated and experimental changes in the FM levels. However, it can be argued that the Asc and FeCy treatment could also lead to processes, which are different to changes in the initial redox state of cyt b559. For example in the case of the FM level, the Asc and FeCy driven changes in the redox state of the PQ pool leading to different fluorescence quenching by the oxidized PQ pool (Vernotte et al., 1979; Hsu and Lee, 1995; Kramer et al., 1995; Kurreck et al., 2000; Pospı́šil and Dau, 2000; Koblı́žek et al., 2001) could be mentioned. In this case, however, the strong exciting light employed in our experiments also excites photosystem I, which in DCMU-treated thylakoids should effectively oxidize the PQ pool. Therefore, the changes in the FM level caused by Asc or FeCy treatment are probably not caused by a different PQ-based fluorescence quenching. Further, it can be argued that the FM level could be under the direct control of QA/Asc and QA/FeCy redox pairs during the FLR measurement. This suggestion also seems to be improbable as we used in all the experiments a very high intensity of exciting red light (12,000 mmol photons m2 s1; l ¼ 650 nm) which is thought to be saturating for Q A reduction even for samples untreated with DCMU. Therefore, the different FM levels presented in Fig. 5 are probably caused by a different Q A accumulation which is driven by Asc- or FeCy-induced difference in the initial redox state of cyt b559 as it also implies from our theoretical simulations shown in Fig. 2. ARTICLE IN PRESS D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 The question arises as to which form of cyt b559 can accept electrons from Pheo and donate them to P680+ as it is assumed in our model. Here we can use again the experimental results of Fig. 5 but also facts know from the literature for a help. As we found that in our thylakoid membranes treated with DCMU, both the HP and LP forms of cyt b559 were present (60.679.8% of reduced HP form, 24.673.1% of oxidized HP form, and 14.778.9% of oxidized LP form; mean7standard deviation), both the forms could be involved in the observed changes in the FM level. As the LP form, however, of cyt b559 was present only in the oxidized state, it should not be involved in the change of the FM level caused by FeCy treatment. On the other hand, both the HP and LP forms of cyt b559 could be involved in the changes in the FM level caused by Asc treatment. Therefore, from the experiments, cyt b559, which can accept electrons from Pheo and donate them to P680+ seems to be the HP form. Further, to protect PSII against damage with respect to the unifying model of photoinhibition (see Introduction), cyt b559, which accepts electrons from Pheo and donates electrons to P680+ would provide the best protection. This statement is in line with the suggestion that cyt b559 participates in cyclic electron transfer around PSII (Heber et al., 1979; Falkowski et al., 1986; Arnon and Tang, 1988; Prasil et al., 1996). Moreover, cyt b559, which accepts electrons from Pheo and donates electrons to P680+ was suggested to have taken part in electron transfer reactions in PSII during photoactivation of PSII to protect it against photoinhibition (Magnuson et al., 1999). This cyt should be the HP form as the HP form is restored from the LP form during the photoactivation of PSII (Mizusawa et al., 1995, 1997). With respect to a high rate of electron transfer from Pheo to cyt b559 assumed in our model (kCred; see Table 1) describing the case when DCMU is present, one can argue that when DCMU is not present, assumed high rate of electron transfer from Pheo to cyt b559 can result in a decrease of quantum yield of PSII photochemistry and therefore in a decrease of overall rate of linear electron transport because electrons flow to cyt b559 instead of QA and further. But it seems, it is not true: using a model of Lazár (2003) that completely describes electron transfer reactions in PSII extended by the cyt b559 reactions described in this work, we found (data not shown) that only the height of J step decreases with increasing amount of initially oxidized cyt b559 but F0 and FM are always the same in all three cases when cyt b559 is either initially fully reduced or fully oxidized or not considered at all in the model. As F0 and FM are not changed, the maximal quantum yield of PSII photochemistry expressed by FV/FM ratio (=(FM–F0)/ FM; Kitajma and Butler, 1975) and therefore also the overall rate of linear electron transport are not decreased by electron flow to cyt b559. Therefore, high 297 rate of electron flow from Pheo to cyt b559 assumed in this work does not anyhow decrease functionality of PSII when DCMU is not present. Simulations also revealed that insensitivity of F0 and FM to no consideration of cyt b559 at all in the model of Lazár (2003) or to the initial redox state of cyt b559 in this model is caused (data not shown) by a small rate of electron transfer from reduced cyt b559 to P680+ (kCox; see Table 1). In other words, even if electrons are transferred fast from Pheo to cyt b559, the small rate of electron transfer from reduced cyt b559 to P680+ results in only a small electron flux through cyt b559 leading to no effect on electron flux from Pheo to QA and further in the case when DCMU is not present. Therefore, the fast electron transfer from Pheo to cyt b559 assumed in this work seems to have no effect on quantum yield of PSII photochemistry when DCMU is not present but it can protect PSII against photoinhibition in the case when PSII behaves like if DCMU is present (see also the Consequences for PSII photoinhibition section). 6. Conclusions We extended the RRP model by cyt b559, which accepts electrons from Pheo and donates electrons to P680+. Our theoretical simulations revealed that an increased amount of initially reduced cyt b559 leads to an increase in the FM level. MCA of our model revealed that the initial redox state of cyt b559 strongly affects the extent in which particular reactions in the model control the FM level. The simulations also demonstrated that increasing the amount of initially reduced cyt b559 can protect PSII against its degradation with respect to the donor and acceptor side photoinhibitions and also with respect to the unifying model of photoinhibition of PSII. Even if we assumed a high rate of electron transfer from Pheo to cyt b559, it seems, it has no effect on the quantum yield of PSII photochemistry of fully functional PSII (when electron transfer from Q A to further electron acceptors is not disabled, e.g. by long lasting high light exposure or DCMU application) but it can ensure protection of PSII against photoinhibition when electron transfer from Q A to further electron acceptors is disabled (i.e. when PSII behaves like when DCMU is present). Therefore, this work contributed to a better understanding of the ‘‘enigmatic’’ cyt b559 function and also showed the power of theoretical simulations and their benefits for further research. Acknowledgements This work was supported by the Ministry of Education of the Czech Republic (Grant No. MSM 153100010) and by a project cofinanced by a Centre of ARTICLE IN PRESS 298 D. Lazár et al. / Journal of Theoretical Biology 233 (2005) 287–300 Excellence grant from the European Union (contract no. BIER ICA1-CT-2000-70012) and a Polish Committee for Scientific Research Grant (No. 158/E-338/SPUB-M/ 5 PR UE/DZ 9/2001-2003). D.L. would like to thank the Grant Agency of the Czech Republic for their financial support (Grant No. 204/02/P071). References Anderson, J.M., Park, Y.-I., Chow, W.S., 1998. Unifying model for the photoinactivation of photosystem II in vivo under steady-state photosynthesis. Photosynth. Res. 56, 1–13. Arnon, D.I., Tang, G.M.-S., 1988. 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