Computer modeling of electron and proton transport in chloroplasts.

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BIO 3481 1–21

BioSystems xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

BioSystems journal homepage: www.elsevier.com/locate/biosystems

Review Article

Computer modeling of electron and proton transport in chloroplasts

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Alexander N. Tikhonov ∗ , Alexey V. Vershubskii Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow 119991, Russia

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a r t i c l e

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Article history: Received 16 March 2014 Received in revised form 27 April 2014 Accepted 28 April 2014 Available online xxx

Keywords: Photosynthesis Chloroplasts Electron and proton transport Regulation Mathematical modeling

Contents

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1. 2.

Photosynthesis is one of the most important biological processes in biosphere, which provides production of organic substances from atmospheric CO2 and water at expense of solar energy. In this review, we contemplate computer models of oxygenic photosynthesis in the context of feedback regulation of photosynthetic electron transport in chloroplasts, the energy-transducing organelles of plant cell. We start with a brief overview of electron and proton transport processes in chloroplasts coupled to ATP synthesis and consider basic regulatory mechanisms of oxygenic photosynthesis. General approaches to computer simulation of photosynthetic processes are considered, including the random walk models of plastoquinone diffusion in thylakoid membranes and deterministic approach to modeling electron transport in chloroplasts based on the mass action law. Then we focus on a kinetic model of oxygenic photosynthesis that includes key stages of the linear electron transport, alternative pathways of electron transfer around photosystem I (PSI), transmembrane proton transport and ATP synthesis in chloroplasts. This model includes different regulatory processes: pH-dependent control of the intersystem electron transport, down-regulation of photosystem II (PSII) activity (non-photochemical quenching), the light-induced activation of the Bassham–Benson–Calvin (BBC) cycle. The model correctly describes pH-dependent feedback control of electron transport in chloroplasts and adequately reproduces a variety of experimental data on induction events observed under different experimental conditions in intact chloroplasts (variations of CO2 and O2 concentrations in atmosphere), including a complex kinetics of P700 (primary electron donor in PSI) photooxidation, CO2 consumption in the BBC cycle, and photorespiration. Finally, we describe diffusion-controlled photosynthetic processes in chloroplasts within the framework of the model that takes into account complex architecture of chloroplasts and lateral heterogeneity of lamellar system of thylakoids. The lateral proﬁles of pH in the thylakoid lumen and in the narrow gap between grana thylakoids have been calculated under different metabolic conditions. Analyzing topological aspects of diffusion-controlled stages of electron and proton transport in chloroplasts, we conclude that along with the NPQ mechanism of attenuation of PSII activity and deceleration of PQH2 oxidation by the cytochrome b6 f complex, the intersystem electron transport may be down-regulated due to light-induced alkalization of the narrow partition between adjacent thylakoids of grana. The computer models of electron and proton transport described in this article may be integrated as appropriate modules into a comprehensive model of oxygenic photosynthesis. © 2014 Published by Elsevier Ireland Ltd.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pathways of electron transport and its regulation in chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The intersystem electron transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abbreviations: PSI and PSII, photosystem I and photosystem II, respectively; P700 , primary electron donor of PSI; BBC, Bassham–Benson–Calvin cycle; b6 f, plastoquinone–plastocyanin–oxidoreductase (b6 f complex); CEF, cyclic electron ﬂow; ETC, electron transport chain; Fd, ferredoxin; LHCII, light-harvesting complex II; LEF, linear electron ﬂux; NPQ, non-photochemical quenching; PQ, plastoquinone; PQH2 , plastoquinol; Pc, plastocyanin; WWC, water–water cycle (pseudocyclic electron transport); WOC, water-oxidizing complex. Q2 ∗ Corresponding author. Tel.: +7 495 9392973. E-mail addresses: an [emailprotected] (A.N. Tikhonov), [emailprotected] (A.V. Vershubskii). http://dx.doi.org/10.1016/j.biosystems.2014.04.007 0303-2647/© 2014 Published by Elsevier Ireland Ltd.

Please cite this article in press as: Tikhonov, A.N., Vershubskii, A.V., Computer modeling of electron and proton transport in chloroplasts. BioSystems (2014), http://dx.doi.org/10.1016/j.biosystems.2014.04.007

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2.2. Alternative pathways of electron transport on the acceptor side of photosystem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Activation of the BBC cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Light energy partitioning between PSI and PSII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. pH-dependent regulation of the intersystem electron transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General aspects of computer simulation of electron transport in chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Intersystem electron transport: random walk models of plastoquinone and plastocyanin diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Deterministic approach to modeling electron transport processes in chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Electron transport in multi-enzyme complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A comprehensive mathematical model of photosynthetic electron and proton transport in chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Transmembrane proton ﬂuxes and ATP synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. pH-dependent regulation of the intersystem electron transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Induction events in oxygenic photosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Redox transients of P700 in dark-adapted leaves and cyanobacteria cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Alternative pathways of electron transport on the acceptor side of photosystem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Effects of CO2 and O2 on photosynthetic electron transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion-controlled processes in heterogeneous lamellar system of chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Lateral heterogeneity of the chloroplast lamellar system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Lateral proﬁles of proton potential in chloroplasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. On the peculiarities of thermodynamic description of small systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Energetic and regulatory aspects of non-uniform lateral distribution of proton potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Photosynthesis is one of the main biological processes in biosphere, which provides production of organic substances from atmospheric CO2 and water. The primary processes of photosynthesis are initiated by light absorption in the light-harvesting antenna, followed by migration of energy and charge separation in photoreaction centers. Photosynthetic organisms of oxygenic type (higher plants, algae, cyanobacteria) have two multisubunit pigment–protein complexes, photosystem I (PSI) and photosystem II (PSII) (Nelson and Yocum, 2006). PSI and PSII are interconnected via the membrane-bound cytochrome b6 f complex and mobile electron carriers, plastoquinone (PQ) and plastocyanin (Pc) (see cartoon in Fig. 1). The pigment–protein complex of PSII contains the photoreaction center and water-oxidizing complex (WOC). Photoexcitation of PSII leads to the extraction of electrons from water, producing molecular oxygen by WOC, and the reduction of plastoquinone (PQ) to plastoquinol (PQH2 ) (see for review Müh et al., 2012). The cytochrome b6 f complex mediates electron transfer between PSII and PSI by oxidizing PQH2 and reducing Pc (see for review Cramer et al., 2006; Hasan et al., 2013; Tikhonov, 2014). The two-electron oxidation of PQH2 occurs at the quinone-binding center Qo on the electropositive (lumenal) side of the thylakoid membrane. This reaction is accompanied by dissociation of two protons into the bulk phase of the thylakoid lumen. Photoexcitation of PSI provides the oxidation of Pc (or cytochrome c6 in cyanobacteria) and reduction of ferredoxin (Fd), a mobile electron carrier on the stromal side of the membrane. Ferredoxin reduces NADP+ to NADPH via the ferredoxin-NADP+ -oxidoreductase (FNR) (Benz et al., 2010). Thus, acting in tandem, PSII and PSI provide electron transfer along the photosynthetic electron transport chain (ETC) from water to NADP+ , the terminal electron acceptor of PSI (H2 O → PSII → PQ → b6 f → Pc → PSI → NADP+ ). In chloroplasts (the energy-transducing organelles of plant cell), the multisubunit electron transport complexes are embedded into lamellar membranes of thylakoids, closed vesicles situated under the chloroplast envelope. Electron transport is coupled to translocation of hydrogen ions from stroma to thylakoid lumen, thus generating the transthylakoid difference in electrochemical

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potentials of protons, ˜ H+ , which serves as the proton motive force (pmf) used to drive ATP synthesis from ADP and Pi (ADP + Pi → ATP) by the CF0 –CF1 ATP synthase (see for review Mitchell, 1966; Boyer, 1993, 1997; Kramer et al., 1999, 2003; von Ballmoos et al., 2009; Romanovsky and Tikhonov, 2010; Tikhonov, 2013). The products of the light-induced stages of photosynthesis, ATP and NADPH, are used in reductive biosynthetic reactions in the Bassham–Benson–Calvin (BBC) cycle (Edwards and Walker, 1983). The pmf value is determined by two components: the pH difference (pH = pHout − pHin ) and the difference in electric potentials, = in − out . Both components of ˜ H+ are competent as the sources of energy for operation of ATP synthases (Gräber, 1982; Junesche and Gräber, 1991). In chloroplasts, under steady state conditions, pH is a major component of pmf (Johnson and Ruban, 2014), although under certain conditions one cannot ignore the contribution of (Cruz et al., 2001, 2005). Elucidation of the mechanisms of regulation of electron transport and adaptation of the photosynthetic apparatus to varying environmental conditions is a challenging task of biochemistry and biophysics of photosynthesis (Buchanan, 1980, 1991; Noctor and Foyer, 2000; Kramer et al., 2004; Eberhard et al., 2008). Among the diversity of regulatory processes that serve to optimize electron transport in oxygenic photosynthesis, noteworthy are the feedbacks associated with the light-induced changes in the lumen and stroma pH. The light-induced acidiﬁcation of the lumen (pHi ↓) markedly slows down PQH2 oxidation by the b6 f complex, on the one hand, and attenuates the activity of PSII due to the enhancement of energy dissipation in the PSII light-harvesting antenna (see for review Jahns and Holzwarth, 2012; Järvi et al., 2013; Ruban et al., 2012; Tikhonov, 2012, 2013). The light-induced alkalization of stroma (pHo ↑) activates the BBC cycle, thus stimulating the consumption of NADPH and ATP (Werdan et al., 1975; Mott and Berry, 1986; Andersson, 2008). Redox-regulation of photosynthetic electron transport associated with optimal distribution of absorbed light energy between PSI and PSII and redistribution of electron ﬂuxes between alternative pathways (noncyclic, cyclic, and pseudocyclic electron ﬂow) are another means of the feedback regulation of electron transport and energy transduction in chloroplasts (Bendall and Manasse, 1995; Backhausen et al., 2000; Allen,

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Fig. 1. A scheme of electron transfer and proton transport pathways in chloroplasts and arrangement of protein complexes (photosystem I, photosystem II, the cytochrome b6 f and ATP synthase complexes) in grana thylakoids and stroma lamellae membrane. Stacked grana thylakoids are enriched with the PSII and light-harvesting complex II (LHCII); most of PSI and ATP synthase complexes are localized in unstacked stroma-exposed thylakoids, grana margins, and grana end membranes. The cytochrome b6 f complexes are distributed uniformly throughout all the domains of the chloroplast lamellae. The amount of PQ molecules is about 10 times higher than that of PSI or PSII. Q5 Blue arrows show electron transfer reactions, red arrows depict proton transport pathways. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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2003a,b; Breyton et al., 2006; Joliot and Joliot, 2006; Rochaix, 2011; Michelet et al., 2013). A deep insight into the regulatory network of oxygenic photosynthesis is complicated by a large number of interacting components, entangled with numerous regulatory feedbacks that act over vastly different time scales. Computational modeling is increasingly recognized as an efﬁcient research tool to understand the functional organization of photosynthetic systems. Mathematical models of photosynthetic processes has long been used for analyzing the regulatory mechanisms of oxygenic photosynthesis (see, e.g., Kukushkin et al., 1973; Rubin and Shinkarev, 1984; Laisk and Walker, 1986; Karavaev and Kukushkin, 1993; Berry and Rumberg, 2000; Laisk et al., 2006; Laverne, 2009; Nedbal et al., 2007; Rubin and Riznichenko, 2009; von Caemmerer et al., 2009; Igamberdiev and Kleczkowski, 2011; Xin et al., 2013; Zaks et al., 2012; Antal et al., 2013; Zhu et al., 2013; and references therein). The photosynthetic system of the plant cell represents a highly apt object for mathematical modeling of metabolic processes. It is largely separated from other subsystems of the plant cell, which may be readily seen from the widely disseminated metabolic maps. Mathematical modeling of oxygenic photosynthesis became one of efﬁcient instruments for scrutinizing the regulatory processes in the complex network of photosynthetic processes (see, for instance, the state-of-the-art in the recent book “Photosynthesis in silico. Understanding Complexity from Molecules to Ecosystems” (Laisk et al., 2009) and Special issue of the BioSystems journal (Igamberdiev and Kleczkowski, 2011)). These publications provide the comprehensive overview of the problem, demonstrating how mathematical and computational approaches can be used for deep understanding all the variety of complex photosynthetic processes. It should be noted that the complexity of mathematical description of photosynthetic processes is not a mere consequence of a large number of components, but it often results from non-uniform distribution of main electron transport and ATP synthase complexes between

spatially segregated domains of laterally heterogeneous membrane lamellae. In this review, we focus on mathematical description of oxygenic photosynthesis in the context of light-induced regulation of photosynthetic electron and proton transport. The structure of the article is following: (1) we start with a brief overview of electron transport in chloroplasts and its regulation (Section 2), and consider general approaches to mathematical description of photosynthetic processes (Section 3); (2) then we describe results of numerical experiments on regulation of photosynthetic processes performed within the frameworks of our comprehensive model of electron and proton transport coupled to ATP synthesis in chloroplasts (Section 4); and, ﬁnally, (3) in view of complex architecture of chloroplasts and lateral heterogeneity of thylakoid membranes, we analyze topological aspects of diffusion-controlled stages of electron and proton transport in chloroplasts (Section 5). 2. Pathways of electron transport and its regulation in chloroplasts 2.1. The intersystem electron transport Fig. 1 schematically depicts a chloroplast ETC, which provides the light-induced electron transfer from the water molecule oxidized by PSII to NADP+ and generation of pmf. Electron ﬂow from PSI to NADP+ (linear electron ﬂow, LEF) provides NADPH formation at the expense of electrons donated by PSII to PSI via the intersystem ETC (H2 O → PSII → PQ → b6 f → Pc → PSI → NADP+ ). There are two diffusion-controlled stages in the chain of long-range electron transfer between PSII and PSI: (i) electron transfer from PSII to the cytochrome b6 f complex, which is mediated by PQH2 molecules diffusing in the thylakoid membrane, and (ii) electron transfer from the b6 f complex to PSI as mediated by Pc diffusing in the thylakoid lumen. The peculiarities of the chloroplast architecture raise the

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question whether the lateral diffusion of mobile electron carriers, PQ and Pc, limits the intersystem electron transfer. In chloroplasts, PSI, PSII, and b6 f complexes are distributed non-uniformly between spatially separated thylakoids of grana and stroma (Andersson and Anderson, 1980; Murphy, 1986; Albertsson, 2001; Danielsson et al., 2004; Dekker and Boekema, 2005). Stacked grana thylakoids are enriched with PSII and light-harvesting complex II (LHCII); whereas most of PSI and ATP synthase complexes are localized in unstacked stroma-exposed thylakoids, grana margins and grana end membranes. The cytochrome b6 f complexes are distributed throughout all the domains of the chloroplast lamellae (Anderson, 1982). The rate of the intersystem electron transport is determined mainly by PQ turnover as a shuttle connecting PSII and b6 f complexes (Haehnel, 1984). PQ turnover is determined by several events: PQ reduction to PQH2 (PQB + 2e− + 2H+ out → PQB H2 ), dissociation of PQH2 from PSII and its diffusion toward the b6 f complex, and oxidation of PQH2 at the Qo -site of the b6 f complex. The question arises: which stage of PQ turnover determines the rate of the intersystem electron transport – lateral diffusion of PQH2 from PSII to the b6 f complex or PQH2 oxidation after PQH2 binding to the b6 f complex? The issue remains open. On the one hand, there are certain indications that inefﬁcient lateral diffusion of mobile electron carriers may limit electron transfer between PSII and PSI (Lavergne and Joliot, 1991; Lavergne et al., 1992; Kirchhoff, 2008, 2014; Kirchhoff et al., 2011). In particular, the long-range lateral diffusion of PQH2 from PSII to the b6 f complexes may be restricted by slow percolation of PQH2 through the lipid domains of over-crowded thylakoid membranes. Restricted diffusion of plastocyanine inside the narrow thylakoid lumen may restrict communication between the b6 f and PSI complexes. On the other hand, there is conclusive experimental evidence that the lateral diffusion of PQH2 in the thylakoid membrane and Pc movement within the lumen do not limit the overall rate of electron transfer from PSII to PSI, which appears to be true at least under certain experimental conditions. According to earlier studies on electron transfer from PSII to PSI in isolated chloroplasts (Haehnel, 1976; Tikhonov et al., 1984), PQH2 formation and its diffusion toward the b6 f complex do not limit the overall rate of the intersystem electron transport. Within a wide range of experimental conditions (pH, ionic strength, and temperature), the light-induced reduction of PQ to PQH2 (PQB + 2e− + 2H+ out → PQB H2 ), its dissociation from PSII (PQB H2 → PQH2 ), and PQH2 diffusion toward to the b6 f complex occur much more rapidly than PQH2 oxidation after its binding to the b6 f complex (see for review Tikhonov, 2013, 2014). This indicates that the rate of PQH2 turnover is determined mainly by the processes occurring after PQH2 binding to the b6 f complex. The conclusion that the major rate-limiting step in the intersystem electron transport is not diffusion of PQH2 , but its oxidation by the b6 f complex ﬁnds support in recent work by Laisk et al. (2014), who investigated temporal kinetics of electron transport in intact leaves. Characteristic time of PQH2 oxidation at the Qo -site of the b6 f complex and electron transfer the cytochrome f is about 10–20 ms (Stiehl and Witt, 1969; Haehnel, 1984). Further , occurs rapidly as compared to electron transfer, cyt f → Pc → P+ 700 PQH2 oxidation, with time constants ∼35–350 ␮s and ∼10–20 ␮s, respectively (Hope, 2000; Kirchhoff et al., 2004a,b). 2.2. Alternative pathways of electron transport on the acceptor side of photosystem I The feedback regulation of electron transport on the acceptor side of PSI is associated with redox-dependent redistribution of electron ﬂuxes, thus providing optimal operation of photosynthetic apparatus and its efﬁcient interaction with other metabolic systems. There are diverse pathways of electron transport on the acceptor side of PSI, when electrons donated by PSI can be delivered to different metabolic channels. Three main mechanisms are

proposed to account for balancing of the ATP/NADPH output ratio: (1) cyclic electron ﬂux around PSI (CEF) (Bendall and Manasse, 1995; Allen, 2003a); (2) the water–water cycle, in which electrons from PSI reduce O2 to H2 O (the Mehler reaction) (Mehler, 1951; Asada, 1999); (3) the malate shunt, in which electrons from LEF are shuttled to oxidative phosphorylation in mitochondria (Scheibe, 2004). Proton pumping into the thylakoid lumen, associated with the alternative pathways of electron transfer, drives ATP synthesis without net reduction of NADPH, thereby increasing the ATP/NADPH output ratio and initiating photoprotection by acidiﬁcation of the lumen (Cruz et al., 2007). As noted above, apart from the mainstream “linear” electron ﬂow from PSI to NADP+ (LEF), the electron ﬂux may be diverted to cyclic routes of electron ﬂow around PSI. According to commonly accepted point of view, CEF helps to sustain required ratio between ATP and NADPH (ATP/NADPH = 3/2) used for CO2 accumulation in the BBC cycle (see for review Kramer et al., 2004; Cruz et al., 2007; Michelet et al., 2013). In the CEF around PSI electrons are returned into the ETC between PSII and PSI via illusive ferredoxin-quinone reductase (FQR) (see for review Bendall and Manasse, 1995; Iwai et al., 2010); this way may be called as a “short” CEF. In principle, there may exist an alternative, “long” CEF around PSI, where electrons from NADPH return to ETC via the NAD(P) oxidoreductase (NDH) (Shikanai, 2007; Johnson, 2011). Electron ﬂow from PSI to molecular oxygen (Mehler, 1951) represents additional channel of electron drain from PSI (Asada, 1999; Badger et al., 2000; Heber, 2002). When O2 rather than NADP+ acts as a terminal electron acceptor in PSI (the Mehler reaction), it is eventually reduced to water during operation of pseudocyclic electron transport (“water–water cycle”: H2 O → PSII → PSI → O2 → H2 O). Taken together, alternative pathways of electron transfer on the acceptor side of PSI (CEF and WWC) are accompanied by proton translocation into the lumen, thus supporting the ATP formation without the reduction of NADP+ , providing an optimal stoichiometry of NADPH and ATP used in the BBC cycle (Kramer et al., 2004; Cruz et al., 2007). 2.3. Activation of the BBC cycle In dark-adapted chloroplasts, activity of the BBC cycle is low (Buchanan, 1980; Edwards and Walker, 1983). Therefore, immediately after the onset of the actinic light, a low rate of NADPH consumption in the BBC cycle will cause the over-reduction of electron carriers on the acceptor side of PSI. This would divert electrons from PSI to thioredoxin, which, in turn, will activate other enzymes, including those of the BBC cycle (Buchanan, 1991; Motohashi and Hisabori, 2006; Dietz and Pfannschmidt, 2011; Michelet et al., 2013). Activation of the BBC cycle will stimulate regeneration of NADP+ , thereby accelerating the outﬂow of electrons from PSI. Activation of the BBC cycle enzymes is associated with the redox- and pH-dependent modulation of electron transport on the acceptor side of PSI. Redox-dependent modulation of activities of photosynthetic enzymes occurs via the thioredoxin/thioredoxin reductase system. During the induction phase (the initial period of illumination of dark-adapted chloroplasts), when the BBC cycle is inactive, the electron outﬂow from PSI to NADP+ is slow. This is because the acceptor side of PSI becomes easily over-reduced due to limited consumption of NADPH. At the excess of reductants on the acceptor side of PSI, the outﬂow of electron from PSI is diverted from ferredoxin to thioredoxin (Tr). Reduced Tr activates other photosynthetic enzymes, including those of the BBC cycle. As a result of the BBC cycle activation, consumption of NADPH accelerates, providing a rapid regeneration of NADP+ . Therefore, the outﬂow of electrons from PSI to NADP+ would not limit linear (noncyclic) electron transport in chloroplasts, thereby promoting photooxidation of P700 (Ryzhikov and Tikhonov, 1988; Trubitsin et al., 2003, 2005).

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The light-induced alkalization of the stroma may be another factor of the BBC cycle activation. The light-induced increase in stromal pHo (the rise of pHo from 7.0–7.2 to 7.8–8.0 (Heldt et al., 1973; Robinson, 1985)) is caused by proton consumption upon the protonation of reduced plastoquinone (PQ + 2e– + 2H+ → PQH2 ) and NADP+ reduction (NADP+ + 2e– + H+ → NADPH). The alkalization of stroma is accompanied by an increase in Mg2+ concentration (Barber, 1976), which is necessary for activation of Rubisco, the key enzyme of the BBC cycle (Gardemann et al., 1986). Thus, the lightinduced activation of the BBC cycle will stimulate consumption of NADPH and ATP, increasing the outﬂow of electrons from PSI.

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One of the mechanisms of electron transport regulation in chloroplasts is associated with redistribution of light energy between PSI and PSII (state transitions, see for review (Allen, 2003b; Lemeille and Rochaix, 2010; Minagawa, 2011; Tikkanen et al., 2011; Tikkanen and Aro, 2012, 2014)). The over-reduction of the PQ pool induces activation of a protein kinase that catalyses phosphorylation of the light-harvesting antenna of PSII (collectively abbreviated as LHCII), thereby initiating structural changes in photosynthetic apparatus of plants and enhancing the light energy delivery to PSI at expense of PSII (so-called State I → State II transition). This process is triggered by PQH2 binding to the Qo -site of the b6 f complex. According to traditional (classical) point of view (Allen, 2003a; Lemeille and Rochaix, 2010; Minagawa, 2011), during State I → State II transition phosphorylated LHCII complexes dissociate from the PSII–LHCII supercomplex and migrates to associate with PSI, thus increasing the light harvesting capacity of PSI. There are biochemical and structural evidences (Kouril et al., 2005; Zhang and Scheller, 2004; Galka et al., 2012; Hofmann, 2012) that the loosely bound form of phosphorylated LHCII dissociates from PSII and migrates to PSI. Novel insights into LHCII phosphorylation and “state transitions” in higher plants have been suggested by Tikkanen et al. (2011) and Tikkanen and Aro (2012, 2014).

Fig. 2. The sketches illustrating the distribution of proton ﬂuxes across the thylakoid membrane in metabolic states 3 and 4 (modiﬁed Fig. 13 from Tikhonov, 2014). The feedback regulation of the intersystem electron transport is governed by the lightinduced changes in the lumen pHin . Depending on the ADP/ATP ratio, the CF1 -CF0 complex functions either in the ATP synthase mode (ATP formation) or in the ATPase mode (ATP hydrolysis). In metabolic state 3 (at the surplus of ADP and Pi ), the efﬁcient operation of CF1 –CF0 complexes in the ATP synthase mode is accompanied by stoichiometric drain of protons from the lumen to stroma, which precludes too strong acidiﬁcation of the lumen (pHin ≥ 6). In state 3, chloroplasts retain a high rate of electron transport, which is comparable with accelerated electron ﬂow in uncoupled chloroplasts. Metabolic state 4 (the state of “photosynthetic control”) is attained after acute shortage of the ATP synthesis substrates (ADP and/or Pi ) and accumulation of surplus amounts of ATP. In state 4, the ATP formation and the ATP hydrolysis reactions are balanced; the overall ﬂux of protons through the CF1 –CF0 and production of ATP become negligible and more signiﬁcant acidiﬁcation of the lumen occurs (pHin < 6). Owing to greater acidiﬁcation of the lumen, the intersystem electron transport decelerates in state 4; the light-induced injection of protons into the lumen is balanced by an equal passive drain of protons to stroma via the thylakoid membrane.

2.5. pH-dependent regulation of the intersystem electron transport There are two basic mechanisms of pH-dependent regulation of electron ﬂow from PSII to PSI. Since pH is a dominant component of pmf, the intersystem electron transport is governed mainly by the light-induced changes in the lumen pHin (see for illustration cartoon in Fig. 2). One of the regulatory mechanisms is realized at the level of PQH2 oxidation by the b6 f complex. The light-induced decrease in pHin retards PQH2 oxidation, thereby decelerating electron ﬂow between PSII and PSI (Stiehl and Witt, 1969; Witt, 1979; Haehnel, 1984). Another mechanism of downregulation of electron transport is associated with attenuation of PSII activity due to thermal dissipation of excess light energy (“nonphotochemical quenching”, NPQ) in the light-harvesting antenna of PSII. An increase in NPQ is caused by conformational changes in photosynthetic apparatus induced by energization of thylakoid membranes, which create a quenching channel for dissipation of excess energy in LHCII. NPQ plays a protective role, precluding too strong acidiﬁcation of the lumen and decreasing the probability of damage to the photosynthetic apparatus under the solar stress conditions. The rate of the intersystem electron transport correlates with the metabolic state of chloroplasts characterized by the phosphate potential, P = [ATP]/([ADP] × [Pi ]), where [ATP], [ADP], and [Pi ] stand for concentrations of ATP, ADP, and Pi , respectively. According to the convention in bioenergetics coined by Chance and Williams (1956), the metabolic state established during intensive ATP

synthesis (the surplus of ADP and Pi , low P) is termed as metabolic “state 3”. A state of photosynthetic control (exhausted pool of ADP or Pi , and signiﬁcant surplus of ATP, high P), when the overall production of ATP is virtually zero, is usually termed as metabolic “state 4”. The chloroplast ATP synthase is a reversible molecular machine, which can operate in two modes. Depending on the ADP/ATP ratio, CF1 –CF0 functions either in the ATP synthase mode (H+ → H+ out , ATP formation) or in the ATPase mode (ATP hydrolin → H+ ysis, H+ out ). At the surplus of ADP and Pi (metabolic “state in 3”), the efﬁcient operation of CF1 –CF0 in the ATP synthase mode is accompanied by stoichiometric drain of protons from the lumen to stroma, which precludes too strong acidiﬁcation of the lumen (pHin ≈ 6–6.2, see for review (Tikhonov, 2012, 2013)). In this case, chloroplasts retain a high rate of electron transport, which is comparable with accelerated electron ﬂow in uncoupled chloroplasts (pHin ≈ pHout ). More signiﬁcant acidiﬁcation of the lumen occurs in the state 4, when the overall ﬂux of protons through the CF1 –CF0 complexes tends to zero. After acute shortage of the ATP synthesis substrates (ADP and/or Pi ), the overall production of ATP becomes negligible, and the ATP formation and the ATP hydrolysis catalyzed by chloroplast ATP synthases are balanced. In state 4, the proton ﬂux through the CF1 –CF0 complexes is virtually zero, and, therefore, the lumen becomes more acidic (pHin < 6) than in state 3, and the intersystem electron transport decelerates more signiﬁcantly.

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3. General aspects of computer simulation of electron transport in chloroplasts In this section, we brieﬂy outline mathematical approaches to description of electron transport processes in chloroplasts. 3.1. Intersystem electron transport: random walk models of plastoquinone and plastocyanin diffusion The Monte Carlo method is one of the efﬁcient tools for analysis of diffusion-controlled processes in biological systems (Saxton, 1987, 1989; Blackwell and Whitmarsh, 1990; Drepper et al., 1993; Blackwell et al., 1994; Kirchhoff et al., 2000, 2002, 2008) and metabolite diffusion in complex supramolecular structures (Aliev and Tikhonov, 2004, 2011; Shorten and Sneyd, 2009; Riznichenko et al., 2010). There are two important advantages of the Monte Carlo method for modeling diffusion-controlled processes in chloroplasts. First, a computer simulation of random walks allows overcoming mathematical problems posed by complex architecture of chloroplasts. Second, the Monte Carlo method is an efﬁcient tool for modeling the over-crowding effects (the inﬂuence of impermeable obstacles on the mobility of PQ and Pc). As mentioned above, lateral electron transfer between electron transport complexes is mediated by two mobile electron carriers: PQ and Pc. Hydrophobic PQ molecules diffuse in the thylakoid membrane, providing electron transfer between PSII and the b6 f complexes. Hydrophylic Pc molecules diffuse in the intrathylakoid space, mediating electron ﬂow from the b6 f complex to PSI. PQ turnover is the rate-limiting step in the intersystem electron transport. Efﬁcient turnover of PQ as a mediator of the intersystem electron transport implies sufﬁciently high lateral mobility of PQ in over-crowded thylakoid membranes. According to Albertsson (2001), Dekker and Boekema (2005), and Kirchhoff et al. (2002, 2004, 2008), protein complexes occupy at least 70% of the total membrane area of thylakoids. Collisions of PQ molecules with impermeable protein obstacles restrict their lateral trafﬁc in the membrane. The coefﬁcient of obstructed PQ diffusion in photosynthetic membranes was found to be at least two or three orders of magnitude less than in pure lipid membranes free of proteins (Blackwell et al., 1994). The question arises whether the restricted diffusion of PQ the thylakoid membrane may provide an efﬁcient long-range communication between PSII and b6 f complexes? Monte Carlo calculations of PQ random walks in percolated systems help to avoid ambiguity in understanding experimental data on electron transport and shed a new light on the mechanisms of long-range communications between segregated electron transport complexes. According to percolation theory (Saxton, 1987, 1989; Kirchhoff, 2008, 2014), an apparent diffusion coefﬁcient of low-molecular species (e.g., PQ or Pc) signiﬁcantly declines in the membrane crowded by impermeable obstacles. If an occupation of the thylakoid membrane by integral protein complexes is higher than critical value cP (“percolation threshold”), macromolecular obstacles will prevent the long-range diffusion of PQ, restricting PQ random walks within enclosed diffusion domains. The two-dimentional percolation threshold is deﬁned as the surface fraction above which the long-range diffusion of small objects is restricted. In thylakoid membranes, integral proteins may occupy an area close to 70%, which is close to the percolation threshold (cP ≈ 0.7–0.75). This suggests that rapid diffusion of PQ may be restricted to relatively small areas of the thylakoid membrane (“microdomains”). According to the microdomain concept of structural organization of thylakoid membranes (Lavergne and Joliot, 1991; Lavergne et al., 1992; Kirchhoff et al., 2000; Kirchhoff, 2014), rapid diffusion of PQ occurs within small lipid areas near active PSIIs surrounded by other protein complexes. The average distance between PSII and cyt b6 f complexes within the microdomains of

grana thylakoids has been evaluated as r ∼ 15–20 nm (Tremmel et al., 2003). Diffusion of PQH2 and PQ within each microdomain does not limit electron transport between PSII and the b6 f complex. Typical diameter of grana falls in the range of 300–600 nm (Dekker and Boekema, 2005; Kirchhoff, 2014). It should be noted, however, that PQ diffusion in the lipid areas is very fast. Therefore, if PQH2 is not trapped inside the microdomain, it could migrate over a large area within PQ turnover time (≈20 ms). Is it possible that PQH2 molecules may be efﬁcient in the long-range trafﬁc between the grana thylakoid domains and stroma lamellae? Lateral diffusion coefﬁcients for PQ-9, decyl PQ and PQ-2 in soybean phosphatidylcholine liposomes and in spinach thylakoid membranes have been estimated in the range of (0.1–3) × 10−9 cm2 /s (Blackwell et al., 1994). PQ turnover is determined by PQH2 oxidation, which is characterized by the half-time t1/2 ≈ 20 ms (Stiehl and Witt, 1969; Haehnel, 1984). For random walk motions in two-dimensional systems, the mean square displacement of particles may be evaluated using the Einstein equation: r2 =4D × t, where r2 is the mean square of the particle displacement during the time interval t, and D is the diffusion coefﬁcient. If DPQ ∼ (0.1–3) × 10−9 cm2 /s (Blackwell et al., 1994), PQH2 would traverse a distance about ∼30–150 nm within the time interval t = 20 ms. It is important to note, however, that numerical experiments performed by Tremmel et al. (2003) have demonstrated that mobility of macromolecular obstacles in the membrane increases the percolation threshold cP. In this case, the apparent diffusion coefﬁcient for PQ should increase, thereby allowing the long-range diffusion of PQ within the over-crowded membrane. According to simulations by Tremmel et al. (2003), the apparent coefﬁcient of the long-range diffusion of PQ in the thylakoid membrane approaches to DPQ ≈ 2.1 × 10−8 cm2 /s. This means that PQ could travel farther than 400 nm in 20 ms, suggesting that PQ migration in the thylakoid membrane from PSII to the b6 f complex would not limit electron transfer between PSII and PSI. Consider Pc turnover and its diffusion inside the thylakoid lumen. Electron transfer from the cyt f heme of the b6 f complex to Pc and further from Pc− to P+ occurs rapidly. Characteristic times 700 of these processes (∼35–350 ␮s, and ∼10–20 ␮s, respectively) are signiﬁcantly shorter than PQ turnover time (∼20 ms), suggesting that Pc diffusion between the b6 f and PSI complexes occurs rapidly and, therefore, should not limit the intersystem electron transport. How the latter statement could reconcile with a relatively small volume for Pc diffusion within the over-crowded granal lumen? The face-to-face distance between the inner surfaces of the opposite thylakoid membranes is relatively short, li ≤ 10–20 nm (Dekker and Boekema, 2005; Kirchhoff, 2014). The WOC domain of PSII protrudes from the membrane to the thylakoid lumen, thereby limiting the space available for Pc diffusion inside the lumen. Pc is the 10.5-kDa protein of 4 × 3 × 3 nm sizes (Guss et al., 1986). Therefore, the gap between adjacent WOCs should not be smaller than 3 nm to allow efﬁcient lateral diffusion of Pc inside the granal lumen. According to Kirchhoff et al. (2011), diffusion of Pc within the granal thylakoid lumen of dark-adapted thylakoids is highly restricted because of relatively narrow interthylakoid gap. In the meantime, demonstrated that the grana lumen undergoes signiﬁcant (≈100%) expansion in the light, thereby considerably increasing the space available for Pc diffusion and providing conditions for facilitated lateral trafﬁc of Pc (Kirchhoff et al., 2011). Takano et al. (1982) evaluated the coefﬁcient of Pc diffusion in the thylakoid lumen as DPc ≈ 2 × 10−9 cm2 /s. This yields the Pc displacement r ≈ 130 nm during t = 20 ms. Taking together, Pc movement inside the thylakoid lumen, as well as the lateral diffusion of PQH2 in the thylakoid membrane, can inﬂuence the intersystem electron transport. However, under a wide range of experimental conditions, the diffusion-dependent steps of electron transport mediated by PQ and Pc may not be the

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limiting factors. The rate of electron transfer between PSII and PSI is determined predominantly by the events associated with PQH2 oxidation after its binding to the cytochrome b6 f complex. 3.2. Deterministic approach to modeling electron transport processes in chloroplasts Deterministic approach to modeling photosynthetic processes is based on the use of the mass action law and Michaelis–Menten kinetics for mathematical description of molecular interactions. In the vast majority of traditional deterministic models of photosynthetic electron transport, concentrations of electron carriers are usually considered as the variables independent of spatial coordinates (see, for instance, mathematical models of photosynthetic processes described in Kukushkin et al. (1973), Karavaev and Kukushkin (1993), Rovers and Giersch (1995), Laisk and Edwards (2000), Laisk et al. (2006, 2009), Nedbal et al. (2007), Belyaeva et al. (2011), Vershubskii et al. (2011, 2014), von Caemmerer et al. (2009), Zaks et al. (2012), Zhu et al. (2013) and references therein). The question arises: is it correct to describe the interaction between distant electron transport complexes using deterministic approximation based on the mass action law for diffusion-controlled processes? The application of the mass action law can be approved if the pools of mobile electron carriers provide rapid interconnections between electron transport complexes (so-called “lake model”). This may be particularly true for chloroplasts where PQ and Pc connect PSII, b6 f, and PSI complexes. Actually, titration of chloroplasts by DCMU (an inhibitor which selectively blocks PSII activity due to occupation of the quinone-binding site) demonstrated that even ∼10% of active (DCMU-untreated) PSII complexes were able to donate electrons to almost all b6 f and PSI complexes (Siggel et al., 1972). This demonstrates that spatially segregated PSII, b6 f, and PSI complexes are interconnected via the common pool of mobile electron carriers, PQ and Pc. This means that the PQ molecules behave as a pool of freely diffusing molecules dissolved in the lipid bilayer, which connect the PSII and b6 f complexes (at least within the microdomains, where different PQH2 molecules are channeled from PSII to the b6 f complexes (Kirchhoff et al., 2008)). In the meantime, the b6 f complexes, which are dispersed over the granal and stromal domains of the thylakoid membranes, can freely communicate with PSI due to Pc mobility inside the lumen. The pool type behavior of PQ and Pc molecules sets the basis for the widely accepted random diffusion model of electron transport processes in chloroplasts. This model tacitly implies that integral protein complexes, including the basic electron transport complexes and ATP synthases, are randomly dispersed in the ﬂuid domains (or microdomains) of the thylakoid membrane. The interactions between spatially segregated complexes are mediated by mobile electron carriers, PQ and Pc. Rapid PQ diffusion does not limit electron transport between PSII and the b6 f complex within the microdomains. Bearing in mind high mobility of PQ and Pc within the microdomain, one may use (as a ﬁrst approximation) a conventional approach of chemical kinetics based on the mass action law, when the reaction rate Jik is assumed to be proportional to the product of concentrations of the reacting components, ci (t) and cj (t): Jik (t)∼kij × ci (t) × cj (t).

(1)

Here, kij is the apparent rate constant. Concentrations of reagents, ci (t) and cj (t), should be considered as local concentrations of interacting reagents. In the vast majority of kinetic models of photosynthetic electron transport, concentrations of electron carriers are usually considered as the variables independent of spatial coordinates. This approach is valid, as a ﬁrst approximation, if we assume that all the processes considered are independent of the reaction vessel geometry.

However, for correct description of diffusion-dependent processes in chloroplasts one cannot ignore the spatial heterogeneity of the lamellar membranes. The long-range lateral diffusion of PQ through the over-crowded areas of the granal and stromal membranes will be retarded. As noted above, integral protein complexes present the obstacles that can cause signiﬁcant decrease in the apparent coefﬁcient of PQ diffusion (Kirchhoff, 2008, 2014). Lateral diffusion of protons inside the narrow compartments conﬁned by the thylakoid membranes may also be decelerated. In this case, one have to take into account that electron and proton processes should be described using differential equations with partial derivatives: ∂ci (r , t) = Di × ∇ 2 ci (r , t) + Fi , ∂t

(2)

where ci (r , t) stands for the local concentration of a species “i” (in the space with coordinate r ), and Fi is a function, which describes interactions of “i” with other components of the system located in the vicinity of “i”. In order to illustrate the application of this approach to study of photosynthetic systems, we consider below results of modeling diffusion-controlled processes of proton transport within the framework of our model designed for description of heterogeneous thylakoid system (Section 5). 3.3. Electron transport in multi-enzyme complexes

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Spatially separated multiprotein electron transport complexes can interact via mobile electron carriers. However, electron transport processes within each individual complex occur independently of the states of other complexes. In the latter case, one cannot use the conventional approach based on the mass action law. Modeling electron transfer processes inside the multiprotein complexes (PSI, PSII, cyt b6 f) is usually based on the “master equation” method, when the catalytic cycle is described as a sequence of transitions between different states of the multiprotein complex (Shinkarev and Venedictov, 1977; Rubin and Shinkarev, 1984; Shinkarev, 1998). A complete number of states, N, is determined by the number of electron carriers in the complex and possible states of each electron carrier (reduced or oxidized, excited or in the ground state, protonated or deprotonated). Let pi is the probability to ﬁnd the complex in state “i” {i = 1, 2, . . ., N}. The transitions between the states are determined by the set of ordinary differential equations:

dpi = (kji pj − kij pi ). dt

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Here, kij is the rate constant of the transition from the state “i” to state “j”. The “master equation” approach has been widely used to the study electron transport and related processes in photosynthetic electron transport complexes (see for references Shinkarev, 1998; Rubin and Riznichenko, 2009). A master equation theory of ﬂuorescence induction, photochemical yield, and singlet–triplet exciton quenching in photosynthetic systems was developed by Paillotin et al. (1983). This approach has been successfully employed for simulation of the light-induced changes in the yield of chlorophyll ﬂuorescence in PSII (Belyaeva et al., 2011). Energy relaxation in the exciton transfer in photosynthetic antenna complexes have been studied theoretically using various models formulated in terms of generalized master equations (Nalbach et al., 2011; Singh and Brumer, 2012). 4. A comprehensive mathematical model of photosynthetic electron and proton transport in chloroplasts Is this section, we consider our comprehensive model developed for quantitative analysis of regulatory processes in oxygenic

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Fig. 3. A scheme of electron transport processes considered in the model. Two electrons extracted from the water molecule in PSII are used to reduce PQ to PQH2 . Electrons from PQH2 are transfered through the cytochrome b6 f complex to reduce plastocyanin (Pc). PSI oxidizes plastocyanin on the lumenal side of the thylakoid membrane and reduce a mobile electron carrier ferredoxin (Fd) on the stromal side of the membrane. Reduced Fd molecules donate electrons to NADP+ . Reduced protonated NADPH molecules are consumed in the BBC cycle. Along with the linear electron ﬂow from H2 O to NADP+ , the model takes into account alternative electron transport routes around PSI, Fd-mediated return of electrons to the PQ pool and O2 reduction. Abbreviations: FNR – ferredoxin-NAD(P)-reductase; FQR – ferredoxinquinone-reductase; NDH – NAD(P)-dehydrogenase.

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photosynthesis. The model takes into account key stages of photosynthetic electron and proton transport coupled to ATP synthesis, as well as the up- and down-regulation feedbacks in chloroplasts and cyanobacteria (Vershubskii et al., 2001, 2003, 2004a,b, 2006, 2007, 2011, 2014; Kuvykin et al., 2008, 2009a,b; Vershubskii and Tikhonov, 2013). As noted above, the kinetic behavior of PQ and Pc provides the basis for a random diffusion model of electron transfer in chloroplasts. At ﬁrst approximation, we assume that electron transport complexes and ATP synthases are randomly dispersed in the thylakoid membrane, neglecting the spatial heterogeneity of the chloroplast lamellar system. It turns out that this model can effectively mimic the typical kinetics of photosynthetic electron transport, the light-induced generation of transthylakoid pH difference (pH), and CO2 uptake in intact chloroplasts. Peculiar effects of lateral heterogeneity of thylakoid membranes on electron and proton transport processes are considered separately in Section 5.

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A scheme of the processes considered in our generalized model is depicted in Fig. 3. For numerical description of electron transport processes, we consider the following variables: [P+ ] and 700 ], concentrations of oxidized photoreaction centers P700 and [P+ 680 P680 (electron donors in PSI and PSII, respectively); [Pc], concentra(Pc in chloroplasts tion of oxidized carriers electron donors for P+ 700 and/or cyt c6 in cyanobacteria); [PQ], concentration of oxidized plastoquinone; [Fd], concentration of oxidized ferredoxin; [N+ ] and [NH], concentrations of NADP+ and NADPH; [O2 ], concentration of O2 . Variable [ATP] is the concentration of ATP; [H+ ] and [H+ o] i are the concentrations of hydrogen ions in the lumen and in the stroma, respectively. In general, the variables [PQ], [Pc], [Fd], [N+ ], [NH], [ATP], [H+ ], and [H+ o ], which describe the local concentrations i of mobile components of the system, should be considered as the functions of time t and the local coordinate r . However, using the rapid mixture approximation, one can assume that all the variable are independent of r . The intensities of the light exciting photoreaction centers of PSI and PSII are described by the model parameters L1 and L2 . The model comprises the set of 11 ordinary differential equations

Fig. 4. Schemes illustrating the modeled compartmentalization of hydrogen ions and transmembrane proton ﬂuxes (panel A) and mechanisms of proton transfer through the ATP synthase (ﬂux JATP ) and passive proton ﬂow (ﬂux Jpas ) (panel B). See text and Supplementary materials for explanations.

described by Vershubskii et al. (2011). Apparent rate constants for partial electron transport reactions have been found by ﬁtting calculated kinetic curves to relevant experimental data available from the literature, as described in Vershubskii et al. (2001, 2004, 2011). For detailed description of the set of equations comprising the core of the model see Supplementary materials. 4.1.1. Transmembrane proton ﬂuxes and ATP synthesis For description of proton transport processes we consider three compartments with different pH (Fig. 4A): the intrathylakoid lumen (pHi ), the stromal volume (pHo ), and cytoplasm (pHcyt ). The transmembrane ﬂuxes of protons, JATP (the proton ﬂux through CF0 –CF1 coupled to ATP synthesis), Jpas (passive drain of protons from the lumen to stroma), and JCyt (exchange of protons between stroma and cytoplasm), were calculated as algebraic functions of variables [H+ ] and [H+ o ]. Equations for JATP , Jpas and JCyt were derived from the i kinetic model, which describes the transmembrane proton transfer as a set of proton-exchange processes occurring with the participation of membrane acid-base groups (Dubinskii and Tikhonov, 1995; Vershubskii et al., 2001, 2004a,b; see also Supplementary materials). Buffer capacities of the lumen and stroma compartments are taken into account by consideration of proton-accepting species Bi and Bo , respectively. Dynamics of ATP changes is determined by its formation due to operation of the ATP synthase complexes and ATP consumption in the BBC cycle (Eq. (8) in (Vershubskii et al., 2011), and Eq. (8) in Supplementary materials). We assume that the rate of ATP synthesis is tightly coupled to the proton ﬂux JATP from the lumen to stroma

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through active CF0 –CF1 complexes, as depicted in Fig. 4B. The proton ﬂux JATP through the active ATP synthases was calculated as: JATP = kATP × ([A]0 − [ATP]) ×

] · [10pH − 1] [H+ i pH ˛ + [H+ + ˇ] o ] · [10

(4)

Here, kATP is the normalizing coefﬁcient, [A]0 is the total concentration of ATP and ADP, and pH is the transthylakoid pH difference (pH = pHo − pHi ). Eq. (4) implies that the turnover rate of ATP synthase depends on the ADP concentration, ([A]0 − [ATP]). The last factor in Eq. (4) describes how the proton ﬂux through the membrane part of the ATP synthase (CF0 ) depends on pH, which serves as the driving force for operation of the ATP synthase. The translocation of protons through CF0 is associated with the protonation/deprotonation of subunits c of the CF0 fragment. Hydrophobic subunits c form the ring-like cluster, cm -ring, consisting of m = 13–15 hydrophobic subunits c (Seelert et al., 2000; Pogoryelov et al., 2007; Vollmar et al., 2009). The cm -ring can rotate directionally in the membrane driven by pH generated across the thylakoid membrane. Rotating in the membrane, the cm -ring provides the coupling of the proton transfer through CF0 to ATP synthesis from ADP and Pi in the catalytic centers of CF1 (see for review Junge et al., 2009; Vollmar et al., 2009; von Ballmoos et al., 2009; Romanovsky and Tikhonov, 2010; Futai et al., 2012; Iino and Noji, 2012). Our model is based on the well-known fact that the carboxyl group of subunit c is directly involved into the proton transfer through the CF0 –CF1 complex. Being exposed to acidic reservoir (lumen) through “semi-channel” A, this group → COOH). Then, after the rotabecomes protonated ( COO− + H+ in tion of the cm -ring, when the carboxyl group comes to contact with stroma-exposed “semi-channel” B, the proton dissociates from the carboxyl group ( COOH → COO− + H+ out ) (Fig. 4B). The lightinduced generation of pH will promote the protonation of the carboxyl groups exposed to lumen and deprotonation of the carboxyl groups exposed to stroma, thus providing the conditions for directional rotation of the cm -ring. The last term in Eq. (4) is proportional to a steady-state rate of the proton efﬂux from the lumen to stroma that can be obtained from the kinetic model of protonation/deprotonation events depicted in Fig. 4B. Eq. (4), which describes the pH-dependence of JATP , contains the model parameters ˛ = 10−pKa (1 + k2 /k1 ) and ˇ = k2 /k1 . These parameters are determined by pKA of the carboxyl group A− of the subunit c and efﬁcient rate constants k1 and k2 , which characterize proton transport to A− from the lumen and stroma, respectively (for more details see Vershubskii et al. (2004a, 2011) and Supplementary materials). The value pKA = 7.1 was chosen from the literature data (Assadi-Porter and Fillingame, 1995; Vollmar et al., 2009). Passive proton ﬂuxes Jpas and JCyt through the thylakoid membrane (lumen ↔ stroma, Jpas ) and across the outer chloroplast membrane (stroma ↔ cytoplasm, JCyt ) were described within the framework of a simple kinetic model suggested by Dubinskii and Tikhonov (1995). This model is based on the assumption that the proton from the lumen ﬁrst binds to the acid–base group M k 1

k 2

(H+ + M− −→MH) and then dissociates to the stroma (MH−→M− + i H+ ) o (Fig. 4B). This group is characterized by the equilibrium con , where k and k , k and k stant K M = k1 /k−1 = k2 − k−2 are 1 −1 2 −2 the forward and backward reaction constants for proton transfer from the lumen to M and proton dissociation from MH to stroma. Mathematical expression for passive proton ﬂux Jpas is presented in Supplementary materials. The proton exchange between the stroma and cytoplasm (JCyt ) was described within the framework of the same model (Vershubskii et al., 2004a). The model parameters used for calculation of ﬂuxes JATP , Jpas and JCyt were ﬁtted using experimental data on steady state values of pHo in stroma (Heldt et al., 1973; Robinson, 1985; Laisk et al., 1989a) and pHi in the lumen established in metabolic states 3 and 4 (Tikhonov and Timoshin,

1985; Trubitsin and Tikhonov, 2003; Tikhonov et al., 2008). The description of constants used for calculation of ﬂuxes JATP , Jpas and JCyt , as well as veriﬁcation of the models for numerical description of transmembrane proton ﬂuxes, may be found in our earlier works (Vershubskii et al., 2004a, 2011).

4.1.2. pH-dependent regulation of the intersystem electron transport In our model, the inﬂuence of the lumen pHi on the intersystem electron transport has been taken into account at two steps of electron transfer: (i) PQ reduction to PQH2 in PSII, and (ii) PQH2 oxidation (Fig. 3). A key role in the description of pHi -dependent oxidation of PQH2 belongs to the function kQ ([Q], [Pc], [H+ ]) = i 1/Q , where Q is a characteristic time of PQH2 oxidation and further electron transfer to Pc from the b6 f complex (see for details (Dubinsky and Tikhonov, 1997; Vershubskii et al., 2001, 2011)). For simulation of non-photochemical losses of energy in PSII, we consider that the model parameter L2 , which speciﬁes a number of light quanta exciting P680 centers per time unit, decreases with acidiﬁcation of the lumen (see for details (Kuvykin et al., 2009a,b)). The value of pKNPQ = 6.0 was taken on the basis of experimental data (Pfündel and Dilley, 1993).

4.2. Induction events in oxygenic photosynthesis Regulatory events in dark-adapted photosynthetic systems, which reveal themselves after a dark-to-light transition, are often called by a general term “induction events” (Edwards and Walker, 1983). The Kautsky effect (non-monotonous changes in chlorophyll ﬂuorescence) is one of examples of induction events in photosynthesis (Govindjee, 1995). Induction events are the results of gradual activation of different enzymes and generation of pmf. The light-induced activation of the BBC cycle is another example of induction effects in oxygenic photosynthesis. Illumination of intact chloroplasts activates key photosynthetic enzymes of the reductive pentose phosphate cycle prior to achieve high rates of CO2 assimilation (Buchanan, 1980, 1991; Edwards and Walker, 1983; Woodrow and Berry, 1988; Andersson, 2008; Michelet et al., 2013). During the induction phase (the initial period of illumination of dark-adapted chloroplasts), when the BBC cycle is inactive, the electron outﬂow from PSI to NADP+ is slow. This is because the acceptor side of PSI becomes easily over-reduced due to limited consumption of NADPH. The light-induced activation of BBC cycle is associated with the redox- and pH-dependent modulation of electron transport on the acceptor side of PSI (Gardemann et al., 1986; Mott and Berry, 1986; Buchanan, 1991; Hertle et al., 2013; Michelet et al., 2013). At the excess of reductants on the acceptor side of PSI, the outﬂow of electron from PSI is diverted from ferredoxin to thioredoxin (Tr). Reduced Tr activates other photosynthetic enzymes, including those of the BBC cycle (Michelet et al., 2013). A crucial role in activation of the BBC cycle may belong to the light-induced alkalization of stroma (Heldt et al., 1973; Robinson, 1985) and concomitant accumulation of Mg2+ ions in stroma in exchange for proton uptake (Barber, 1976). An increase in stromal Mg2+ is one of the factors of activation of Rubisco, the key enzyme of the BBC cycle (Andersson, 2008). The stromal pHo increases due to consumption of protons upon the NADP+ reduction and protonation of reduced plastoquinone. As a result of the light-induced activation of the BBC cycle, consumption of NADPH accelerates, providing a rapid regeneration of NADP+ , thereby promoting the outﬂow of electrons from PSI to NADP+ (Ryzhikov and Tikhonov, 1988; Trubitsin et al., 2003, 2005; Joliot and Joliot, 2005, 2006). Below we illustrate the applicability of our model for simulation of induction events in intact chloroplasts.

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Fig. 5. Comparison of computed time-courses of P700 photooxidation with experimental kinetic curves obtained for dark-adapted (10 min) Hibiscus rosa-sinensis leaves (panel A, from Kuvykin et al., 2011) and cyanobacteria Synechosystis sp. PCC 6803 (panel B, from Trubitsin et al., 2005). Initial conditions for computed curves in panel A: curve 1 – PQ pool 60% initially oxidized, curve 2 – PQ pool 90% initially oxidized. PSII/PSI = 1.5.

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4.2.1. Redox transients of P700 in dark-adapted leaves and cyanobacteria cells Induction events reveal itself as a complex kinetics of the light-induced redox transients of P700 observed in dark-adapted leaves, algae, and cyanobacteria cells (Tikhonov and Ruuge, 1975; Maxwell and Biggins, 1977; Trubitsin et al., 2003, 2005; Joliot and Joliot, 2005, 2006). In order to illustrate this point, let us compare experimental and theoretical kinetics of P700 photooxidation in dark-adapted leaves and cyanobacteria. Fig. 5A shows that photooxidation of P700 in dark-adapted leaves of Hibiscus rosa-sinensis reveals a certain lag-phase. Immediately after switching on illumination, a concentration of oxidized centers P+ rises to the 700 intermediate level A, and then declines to the level B. An amplitude of the spike A depends on the initial conditions (see inset in Fig. 5A). A relatively low level of P+ at the initial phase of the 700 induction curve (phase A–B) may be explained by several factors: (i) slow outﬂow of electrons from PSI to the BBC cycle that causes the over-reduction of the pool of electron carriers on the acceptor side of PSI, (ii) recycling of electrons from PSI side to the intersystem ETC (CEF around PSI), and (iii) accelerated electron transfer from PSII to P+ because of moderate acidiﬁcation of the lumen at 700 the initial phase of the induction curve. During the lag-period of the induction curve (phase A–B), the acceptor side of PSI becomes over-reduced because the efﬂux from PSI to the BBC cycle is limited owing to slow consumption of NADPH in the BBC cycle. The subsequent growth of [P+ ] to a steady state level C is associated with 700 the BBC cycle activation caused by alkalization of stroma (pHo ↑) and deceleration of electron ﬂux from PSII to PSI caused by the lumen acidiﬁcation (pHi ↓) (for detail see Kuvykin et al., 2011). Acceleration of electron efﬂux from PSI to the BBC cycle is accompanied by a decrease in CEF around PSI (Section 4.2). There are other factors that inﬂuence the kinetics of P700 photooxidation. For instance, a pattern of the kinetic curve is sensitive to PSII/PSI stoichiometry. Varying the stoichiometry ratio PSII/PSI, we could simulate the kinetics of P700 photooxidation in leaves and cycnobacteria. In Fig. 5A and B we compare the time-courses of P700 photooxidation in dark-adapted leaves Hibiscus rosa-sinensis (Kuvykin et al., 2011) and in cyanobacteria Synechosystic sp. PCC 6803 (Trubitsin et al., 2005). In general, the patterns of P700 photooxidation in leaves and cyanobacteria are similar. Similarly to leaves, cyanobacterial cells show a multiphase kinetics, although the peculiarities of kinetic curves in leaves and cyanobacteria are somewhat different. We can explain the difference, in particular,

Fig. 6. Time-courses of the redox state of the speciﬁed electron carriers (panel A) and pH values in the lumen (pHi ) and in the stroma (pHo ) (panel B) calculated for the same conditions as in Fig. 5.

by different stoichiometry of PSI and PSII complexes in leaves and cyanobacteria. It is well-known that in chloroplasts of higher plants the ratio PSII/PSI ∼ 1–1.5 (Danielsson et al., 2004). Stoichiometric ratio PSII/PSI = 1.5, assayed by various methods, has been reported for market spinach (Chow et al., 2012). In cyanobacteria, a content of PSII is much less than that of PSI (Schmetterer, 1994). As evident from Fig. 5A, the model parameter PSII/PSI = 1.5 is suitable for adequate simulation of P700 photooxidation in leaves. In the case of cyanobacteria (Fig. 5B), the best ﬁt to experimental data is obtained at stoichiometric ratio PSII/PSI = 0.5, which is consistent with the literature data (Schmetterer, 1994). Note that in cyanobacteria the attenuation of electron ﬂux from PSII to PSI at low PSII/PSI ratio is compensated by additional inﬂux of electrons into the intersystem ETC (at the PQ segment of the intersystem ETC) from endogeneous electron donors. Fig. 6A shows the kinetics of light-induced redox transitions of mobile electron carriers (PQ, Fd, NADPH) simulated for chloroplasts at atmospheric [CO2 ] and [O2 ]. Temporal changes of relative concentrations of PQ, Fd, and NADPH were found to follow nonmonotonic kinetic patterns. Immediately after switching the light on, NADP+ rapidly reduces, the initially oxidized ferredoxin pool and partly oxidized PQ pool also become reduced. The lightinduced activation of the BBC cycle stimulates the efﬂux of electrons from PSI. Therefore, after the lag-phase, when the NADPH uptake in the BBC cycle accelerates, the NADP+ /NADPH ratio increases. Acceleration of LEF leads to reoxidation of the plastoquinone and ferredoxin pools. Non-monotonous behavior of variables [PQ], [Fd] and [NADPH] correlate with the light-induced changes in the lumen and stromal pH (Fig. 6B). Acidiﬁcation of the thylakoid lumen (pHi ↓) and alkalization of stroma (pHo ↑) are essential factors of electron transport control in chloroplasts. Electron ﬂow from PSII to PSI

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Fig. 7. Computed time-courses of electron ﬂuxes on the acceptor side of PSI: JLEF is the electron ﬂux from ferredoxin to NADP+ ; JCEF is the electron ﬂux from ferredoxin to quinone, and J O2 is the electron ﬂux from ferredoxin to oxygen (the Mehler reaction); JPSII is the overall ﬂux through PSI. Model parameters are the same as in Fig. 5.

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decreases with a decrease in pHi , whereas the outﬂow of electrons from PSI to NADP+ accelerates with a rise of stromal pHo (see for review (Tikhonov, 2012, 2013)).

thereby further reducing electron ﬂux form PSII to PSI. All these regulatory feedbacks are taken into account in our model. Note that the overall electron ﬂux through PSII (JPSII ) is markedly higher than the partial ﬂuxes JLEF , JCEF , and J O2 . A substantial redistribution of electron ﬂuxes occurs in the induction period. On the initial stage, electron ﬂux to NADP+ (JLEF ) is small, while alternative ﬂuxes are rather strong. CEF around the PSI (JCEF ) is signiﬁcant from the very beginning of illumination, when the linear electron ﬂux from PSI to the BBC cycle is small owing to over-reduction of NADP+ at low activity of the BBC cycle enzymes. On the initial stage of the induction phase, there is also noticeable contribution of the Mehler reaction to electron efﬂux from PSI (J O2 ). Electron ﬂux to O2 , which acts as an alternative electron acceptor in PSI, is best seen when the electron ﬂux to NADP+ (JLEF ) is relatively small. This is consistent with the current notion that the Mehler reaction provides a bypass to avoid “over-reduction” on the acceptor side of PSI. After a certain lag, when the BBC cycle activates and the linear electron ﬂux JLEF increases, both alternative ﬂuxes, JCEF and J O2 , gradually decline to their steady state levels. Taken together, our kinetic model, which takes into account several factors of the feedback control of photosynthetic electron transport, can portray some general features of induction events in photosynthesis. In particular, it describes the appearance of the lagphase in the kinetics of P700 photooxidation and redistribution of electron ﬂuxes between alternative pathways (noncyclic, cyclic and pseudocyclic electron ﬂuxes) in the course of induction transition. 4.3. Effects of CO2 and O2 on photosynthetic electron transport

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4.2.2. Alternative pathways of electron transport on the acceptor side of photosystem I Alternative pathways of electron outﬂow from PSI helps to optimize the operation of the photosynthetic apparatus (see review Bendall and Manasse, 1995; Asada, 1999; Allen, 2003a; Johnson, 2005; Munekage et al., 2002, 2004; Johnson, 2005, 2011; Joliot and Joliot, 2005, 2006; Breyton et al., 2006; Shikanai, 2007; Miyake, 2010). As noted above, CEF around PSI may be essential for balancing the ATP/NADPH output ratio; too much CEF activity will result in depletion of ADP, while too little will result in overreduction of the electron transfer chain (Kramer et al., 2004). When molecular oxygen rather than NADP+ acts as a terminal electron acceptor in PSI, it is eventually reduced to water during operation of pseudocyclic electron transport (water–water cycle: H2 O → PSII → PSI → O2 → H2 O) (Asada, 1999; Heber, 2002). Although electron ﬂows through cyclic and pseudocyclic chains do not reduce NADP+ , they generate the transmembrane electrochemical gradient for protons needed for operation of the ATP synthase complexes. Thus, the stoichiometric ATP/NADPH ratio of 3:2 required for functioning of the BBC cycle is attained. The mechanisms of CEF regulation may be realized by variations of stromal ADP or ATP levels (Joliot and Joliot, 2006), the redox state of NADPH/NADP+ (Munekage et al., 2004), the availability of PSI electron acceptors (Breyton et al., 2006), and/or by means of redox modulation of proteins involved into CEF (Michelet et al., 2013). Consider now dynamics of electron ﬂuxes during the induction period modeled within the framework of our model. Fig. 7 compares the patterns of light-induced changes in electron ﬂux through PSII (JPSII ) and electron ﬂuxes on the acceptor side of PSI (JLEF , JCEF , and J O2 ) simulated for chloroplasts. As one can see, the model predicts that after signiﬁcant initial jump the overall electron ﬂux from PSII to the PQ pool (JPSII ) decays non-monotonically to a steady state level. A decrease in JPSII can be explained by downregulation of PSII activity due to enhanced NPQ in the result of acidiﬁcation of the lumen (pHi ↓). The light-induced decrease in pHi also causes deceleration of PQH2 oxidation by the b6 f complex. To add, the light-induced alkalization of stroma (pHo ↑) may hinder protonation of PQ reduced by PSII (PQ + 2e− + 2H+ out → PQH2 ),

kBBC = fgas ([CO2 ], [O2 ]) × fBBC (pHo ) [ATP] × [NH] . k1s + k2s × [ATP] + k3s × [NH] + k4s [ATP] × [NH]

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The inﬂuence of atmospheric gases on photosynthetic electron transport is associated mainly with the interaction of CO2 and O2 with Rubisco, the key enzyme of the BBC cycle. The most general description of the BBC cycle reactions includes the stages of carboxylation, reduction and regeneration (Chernavskaya and Chernavskii, 1961; Hahn, 1984; Pettersson and Ryde-Pettersson, 1988; Karavaev and Kukushkin, 1993; Fridlyand and Scheibe, 1999; Igamberdiev and Lea, 2006; Laisk et al., 1989b, 2006, 2009; von Caemmerer et al., 2009; Dubinsky et al., 2010a,b; Igamberdiev and Kleczkowski, 2011), which can be divided into more elementary processes, down to the level of single reactions (Farquhar et al., 1980, 2001; Farazdaghi, 2011). A comprehensive critical review of the existing mathematical models of the BBC cycle has been presented in recent work by Arnold and Nikoloski (2011). They scrutinized 15 models and identiﬁed those models that provided quantitatively accurate predictions for the levels of BBC cycle intermediates and could be used in metabolic engineering and in the design of synthetic metabolic pathways for improved carbon ﬁxation, growth and yield. In our works on modeling electron and proton transport processes in oxygenic photosynthesis (Kuvykin et al., 2009a,b; Vershubskii et al., 2011, 2014; Vershubskii and Tikhonov, 2013), we described the consumption of NADPH and ATP in the BBC cycle using the following generalized semiempirical function kBBC :

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(5)

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In Eq. (5), the ﬁrst factor, fgas ([CO2 ], [O2 ]), takes into account an experimental dependence of CO2 assimilation on the concentrations of CO2 and O2 in the atmosphere: fgas ([CO2 ], [O2 ]) =

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(6)

Formula (6) describes the consumption of CO2 in the BBC cycle (photosynthesis) and the uptake of O2 due to oxygenase activity of

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Fig. 8. CO2 consumption (photosynthesis) (panel A) and O2 uptake (photorespiration) (panel B) as a function of CO2 concentration. Results of numerical experiments (open circles) are compared with experimental data obtained on wheat plant (André, 2011).

Fig. 9. Effects of atmosphere gas composition on the steady-state level of P+ . Panel 700 ] vs. concentration of CO2 in model experiments calculated A: dependences of [P+ 700 for aerated (21% O2 ) and de-aerated (2% O2 ) leaves (Kuvykin et al., 2011). Panel B: comparison of theoretical (Kuvykin et al., 2011) and experimental (Munekage et al., 2002) data.

Rubisco (photorespiration). Parameter corresponds to CO2 concentration at the compensation point ( = 35 ppm), KC and KO are the model parameters (KC = 245 ppm, KO = 0.8) taken from (Sharkey et al., 2007). Formula (5) describes the competition between CO2 and O2 for Rubisco and their inﬂuence on the ratio between the BBC cycle reactions and photorespiration. The second factor, fBBC (pHo ), describes the activation of the BBC cycle upon the light-induced alkalization of stroma with a phenomenological sigmoid function of the Boltzman-type (see for details Frolov and Tikhonov, 2007; Vershubskii et al., 2011). The third term in Eq. (5) represents the sigmoid function, which describes the consumption of NADPH and ATP in the BBC cycle (Vershubskii et al., 2011). Formula (6) provides a reasonably good numerical description of CO2 consumption (photosynthesis) and O2 uptake (photorespiration) obtained in numerical experiments performed within the framework of our model. In Fig. 8 we compare experimental data on apparent photosynthesis (Fig. 8A) and photorespiration (Fig. 8B) as a function of CO2 concentration, reported by André (2011) for wheat leaves at 21% and 2% O2 , with the results of our calculations. As one can see, theoretical curves ﬁt experimental data with fair accuracy. Consumption of CO2 increases with the rise of CO2 concentration in the leaf surroundings, saturating at high levels of CO2 . Photorespiration decreases with the rise of CO2 level. Owing to photorespiration, at atmospheric concentration of O2 (21%) the yield of photosynthesis was lower than under the oxygen-deﬁcient conditions (2% O2 ). Thus, the model adequately describes the Warburg effect (Turner and Brittain, 1962): de-aeration increases photosynthesis and decreases the oxygen uptake by chloroplasts (Fig. 7). Our model allowed us to simulate the effects of CO2 and O2 on electron transport processes in chloroplasts. Fig. 9A shows effects

of CO2 on a steady state level of P+ simulated for chloroplasts 700 under aerated (21% O2 ) or oxygen-deﬁcient conditions (2% O2 ). In “aerated chloroplasts”, variation of CO2 has very insigniﬁcant . In the meantime, “deaerated chloroeffect on the level of P+ 700 plasts” reveal a marked dependence of P+ on CO2 concentration: 700 . depletion of CO2 caused a decrease in the steady state level of P+ 700 All these results are is a good agreement with experimental data on the measurements of P700 photooxidation in the leaves of Arabidopsis (Munekage et al., 2002) and Hibiscus rosa-sinensis (Kuvykin et al., 2011). These data indicate that at low levels of CO2 , when the Rubisco turnover limits the rate of electron efﬂux from PSI, molecular oxygen plays the role of electron acceptor supporting the operation of PSI. In support of this point see also Fig. 9B, where we compare the results of modeling with experimental data on photooxidation of P700 in Arabidopsis leaves obtained by Munekage et al. (2002). Fig. 10 compares the effects of CO2 and O2 on the overall electron ﬂow through PSII as determined from the chlorophyll ﬂuorescence in Hibiscus rosa-sinensis leaves and calculated within the frames of our model (Kuvykin et al., 2011). As one could expect, electron ﬂux through PSII (JPSII ) increases with the rise of atmospheric CO2 , reaching a steady-state level at [CO2 ] ≥ 0.15%. At sub-saturating concentrations of CO2 , just as in experiment, depletion of oxygen causes a marked decrease in JPSII . In the meantime, the inhibitory effect of oxygen depletion (2% O2 ) disappears at saturating concentrations of CO2 . These results agree with experimental data (Munekage et al., 2002), suggesting that the electron outﬂow from PSI to oxygen increases the overall rate of electron transfer through PSII. Taken together, the model considered adequately describes the effects of CO2 and O2 on photosynthetic electron transport in higher

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Fig. 10. Effect of atmospheric CO2 on steady-state electron ﬂux through PSII (JPSII ). Concentration dependences (JPSII vs. [CO2 ]) were modeled for two concentrations of atmospheric dioxygen, 21% O2 (closed circles) and 2% O2 (open circles). Open diamonds show experimental points obtained for Hibiscus rosa-sinensis leaves exposed to air atmosphere with different concentrations of CO2. Modiﬁed Fig. 13 from Kuvykin et al. (2011).

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plant leaves. Down-regulation of LEF with lowering atmospheric CO2 and depletion of oxygen can be explained by several reasons: (i) impediments to electron transfer on the acceptor side of PSI, (ii) deceleration of PQH2 oxidation caused by the lumen acidiﬁcation, and (iii) pH-dependent enhancement of energy losses in PSII (NPQ). Results of numerical experiments described by Kuvykin et al. (2011) support the mechanism of pH-dependent up- and downregulation of LEF. The model predicts that the thylakoid lumen becomes more acidic with decreasing the partial pressure of CO2 . At ambient or low levels of CO2 , the depletion of oxygen causes additional decrease in pHi , which manifests itself in experiment as an increase in NPQ (Kuvykin et al., 2011).

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5. Diffusion-controlled processes in heterogeneous lamellar system of chloroplasts

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5.1. Lateral heterogeneity of the chloroplast lamellar system

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PSI, PSII, and ATP synthase complexes are distributed nonuniformly between the grana stacks and stroma lamellae of chloroplasts (Murphy, 1986; Albertsson, 2001; Dekker and Boekema, 2005; Kirchhoff, 2008). Closely stacked thylakoids of grana are enriched with PSII and light-harvesting complex II (LHCII); whereas most of PSI and ATP synthase complexes are localized in unstacked stroma-exposed thylakoids, grana margins, and grana edge membranes (see cartoon in Fig. 1). Thus, signiﬁcant amounts of PSI and PSII complexes are laterally segregated. In contrast to PSI and PSII, the cytochrome b6 f complexes are distributed almost uniformly along the granal and stromal lamellae. According to Albertsson (2001), about 55% of the b6 f complexes are localized in appressed membranes of grana. Other b6 f complexes are distributed over the stromal lamellae, in the margins and grana end membranes. The amount of PQ molecules, which connects PSII with the b6 f complexes, is about 10 times higher than that of PSI or PSII (Stiehl and Witt, 1969; Witt, 1979; Haehnel, 1984). Above mentioned peculiarities of structural organization of photosynthetic apparatus raise some awkward questions when we consider diffusion-controlled reactions of electron and proton transport: (i)

how the restrictions to lateral diffusion of PQH2 in the thylakoid membrane would limit the intersystem electron transport, and (ii) how the slowing down of protons diffusion in small volumes of the lumen and the narrow gap between grana thylakoids will inﬂuence the operation of the chloroplast ETC? The formation of PQH2 requires two electrons donated by PSII and two protons, which enter from the stroma to PSII (PQB + 2e− + 2H+ out → PQB H2 ). How the light-induced changes in pH outside the thylakoids could inﬂuence the PQ turnover? Under the normal physiological conditions, the light-induced raise of pHs in the bulk of stroma does not exceed pHs ≈ 7.8–8.0 (Heldt et al., 1973; Robinson, 1985). Such a moderate elevation of pHs suggests that the light-induced alkalization of stroma will not inﬂuence the rate of PQH2 formation. It should be noted, however, that PQH2 formation may be affected by more signiﬁcant rise of pH in the local vicinity of PSII complexes (Vershubskii et al., 2004a). Most of PSII complexes are spatially separated from the stroma, their outer parts protrude from the membrane into the partition between appressed thylakoids of grana. The gap between the adjacent thylakoids is ∼2.5–3.5 nm; the protein molecules protruding from oppositely located membranes are in close contact with each other. The interthylakoid gap contains a lot of buffering groups, mostly of the large proteins which cover ∼75–80% of the thylakoid membrane area (Murphy, 1986; Kirchhoff, 2008; Kirchhoff et al., 2008, 2011). The protons percolating through the interthylakoid gap from stroma to PSII will interact with the buffering groups. It is not surprising that an efﬁcient coefﬁcient of proton diffusion inside the gap (from stroma to PSII) may be signiﬁcantly smaller than in the bulk water (Polle and Junge, 1986; Junge and Polle, 1986; Junge and McLaughlin, 1987; Polle and Junge, 1989). Therefore, the proton consumption upon the PQ reduction by PSII may cause signiﬁcant alkalization of the gap, because the slowing down of proton diffusion will preclude to rapid leveling of pH in the interthylakoid gap and in the stroma (pHgap > pHstroma ). The light-induced alkalization of the interthylakoid partition may turn to disadvantage in the formation of fully reduced form of plastoquinol, PQH2 . Deceleration of PQH2 formation would occur if pH in the gap will be higher than (or comparable with) pKa values of the acid-base groups involved in protonation of the secondary quinol (PQB + 2e− + 2H+ gap → PQB H2 ). The crystal structure of PSII (Umena et al., 2011) suggests that four residues of the subunit D1 may be involved in protonation of PQB . According to (Müh et al., 2012; Müh and Zouni, 2013; Saito et al., 2013), His252 and Ser264 provide the transfer of the ﬁrst proton to quinone binding site of PSII: His252 accepts the proton from the solvent, while Ser264 transfers the proton to the distal carbonyl oxygen •− •− + − of PQ− B (PQA PQB + HI → PQA PQB H ). According to computations by Ishikita and Knapp (2005), protonation of His252 signiﬁcantly changes in response to PQ− B formation. They concluded that the interruption of the proton transfer to PQB should hamper electron transfer from PQ− to QB . The second proton passes from the solA vent to the proximal carbonyl group of PQB H− (PQA PQB H− + H+ II → PQA PQB H2 ) via Tyr246, bicarbonate and His215 (Müh and Zouni, 2013). A question about pKa values of acid-base groups involved in protonation of reduced PQB is yet open. There are indications that the pKa value for protonation of PQ− B or of a neighboring group in PSII to be about 7.6–7.9 (Vermaas et al., 1984). The pKa for the − + second step of PQ2− B protonation (PQB H + H → PQB H2 ) might be − expected to be similar to that for PQH protonation in aqueous solution, pKa ∼ 10.8 (Rich, 1984). However, real pKa values of acid-base groups participating in protonation of PQH− in the quinone-binding site of PSII are likely to be signiﬁcantly lower (Saito et al., 2013). In bacterial reaction centers, the proton binding groups involved in protonation of ubiquinol have pKa ≈ 8.5–9.0 (McPherson et al., 1988, 1993; Wraight, 1989; Cherepanov et al., 2000). By analogy

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Fig. 11. Cartoon illustrating the geometry of a model system taking into account the peculiarities of structural organization of photosynthetic apparatus, associated with the non-uniform distribution of electron transport and ATP synthase complexes in thylakoid membrane lamellae.

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with bacterial reaction centers, we could suppose that pKa values of the acid-base groups involved in PQH2 formation in PSII (PQB + 2e− + 2H+ → PQB H2 ) are about 8.5–9.0. According to Knaff (1975), at pH ≥ 8.9 the reduced acceptor of PSII is predominantly unprotonated, reduction of the oxidized acceptor would not result in proton uptake. Taken together, the above estimates of pKa suggest that sufﬁciently strong alkalization of the gap (pHgap ≥ pKa ) would hinder to PQH2 formation in PSII.

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5.2. Lateral proﬁles of proton potential in chloroplasts

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The existence of non-uniform lateral proﬁles of pH has been discussed in the literature for a long time (Haraux and de Kouchkovsky, 1982; Haraux et al., 1983; Tikhonov and Blumenfeld, 1985; Blumenfeld and Tikhonov, 1994) in the context of alternative mechanisms for proton transfer to the ATP synthase («local» and «non-local» mechanisms energy coupling, Williams, 1978; Kell, 1979; Westerhoff et al., 1984; Dilley, 1991; Blumenfeld and Tikhonov, 1994). Mathematical analysis of the problem predicts that obstructed lateral diffusion of protons in the lumen (Dubinsky and Tikhonov, 1997) or in the invagin*tions of mitochondrial membranes (Kara-Ivanov, 1983) might be one of the reasons for non-uniform lateral proﬁles of pmf. Direct measurement of pH proﬁles in small compartments is a very arduous experimental task. In the absence of reliable experimental data about the lateral proﬁles of pHgap and pHi , mathematical modeling becomes a useful tool for quantitative analysis of pH distribution along the chloroplast lamellae. Mathematical models developed in our previous works (Dubinsky and Tikhonov, 1997; Vershubskii et al., 2001, 2004a,b, 2006, 2011) take into account the lateral heterogeneity of thylakoids. These models predict, in particular, signiﬁcant alkalization of the narrow interthylakoid gap, suggesting that this might one of the factors of down-regulation of electron transport in chloroplasts. In order to illustrate this point, let us consider the results of computer simulation of proton transport performed within the framework of our model, which takes into account non-uniform partitioning of PSI and PSII in chloroplast lamellae (Vershubskii et al., 2001, 2004a,b, 2006, 2011). Fig. 11 depicts the spatial arrangement of granal and stromal thylakoids considered in the model. The grana thylakoid is simulated by a ﬂattened cylinder of radius a. The stroma-exposed thylakoid is modeled as a wider outer cylinder of radius b, which protrudes from the grana thylakoid. The outer cylinder incorporates grana thylakoid, which gradually transforms into

the stroma-exposed thylakoid. The distances between the inner surfaces of the thylakoid, li , and the partition between neighboring thylakoids of grana, lo , are geometrical parameters of the model. PSI complexes are localized exclusively in the stromal domain, PSII complexes are localized in granal regions of the thylakoid lamellae. The cytochrome b6 f complexes are distributed almost uniformly along the membrane. ATP synthase complexes (CF0 –CF1 ) are localized in the stroma-exposed membranes. Mobile electron carriers, PQ and Pc, can diffuse along the membrane (PQ) and inside the thylakoid lumen (Pc), respectively. Since the lateral mobility of large electron transport complexes in the membrane is three orders of magnitude lower than that of plastoquinone (Kirchhoff, 2008; ˇ 2013), we assume that PSI, PSII, b6 f, Kirchhoff et al., 2008; Kana, and CF0 –CF1 complexes have ﬁxed positions in the thylakoid membrane. The local concentrations of protons in the lumen and in the )], as well as the local coninterthylakoid gap, [H+ (t, r )] and [H+ o (t, r i centrations of the mobile electron carriers, [PQ(t, r )] and [Pc(t, r )], are considered as the functions of time t and the spatial coordinate r . Other variables are independent of the spatial coordinate r . Let us now consider how restrictions to proton diffusion rate inside the narrow compartments of chloroplasts (the intrathylakoid lumen and partitions between grana thylakoids) could )]. It inﬂuence the lateral proﬁles of variables [H+ (t, r )] and [H+ o (t, r i is well known that an apparent coefﬁcient of proton diffusion near the thylakoid membrane surface may be signiﬁcantly lower than in the aqueous bulk phase (Nesbitt and Berg, 1982; Junge and Polle, 1986; Polle and Junge, 1986a, 1989; Trubitsin and Tikhonov, 2003). Experimental evidence for slow percolation of protons through the gap between appressed thylakoids of grana was obtained by Junge and Polle (1986), who demonstrated that stacking/unstacking of granal thylakoids dramatically inﬂuenced on the proton diffusion from the bulk water phase to PSII. After unstacking of granal thylakoids the rate of proton uptake increased dramatically (Polle and Junge, 1986). These data were used for ﬁtting the apparent coefﬁcient of proton diffusion, D, by comparison of calculated and experimental proton ﬂux Jdiff (Fig. 12A) through the interthylakoid gap. Fig. 12B shows typical patterns of the proton uptake kinetics in response to a short light pulse (1 ms), as simulated for two values of the model parameter, D = 0.02D0 and D = 0.1D0 . Here, the dimensionless coefﬁcient D0 = 1 corresponds to diffusion coefﬁcient DH+ = 10−5 cm2 /s. A reasonable agreement between theoretical and experimental rates of the proton uptake (t1/2 ∼ 60–100 ms) can been obtained for D = 0.02D0 , which provides deceleration of proton diffusion by a factor of ∼500 as compared to proton mobility in the aqueous bulk phase (for more details see Vershubskii et al., 2004b). Reduced mobility of protons may be one of the reasons for non-uniform proﬁles of pH established along the interthylakoid gap and inside the thylakoid lumen. Fig. 13 compares the lateral proﬁles pHi (r) and pHo (r) calculated for metabolic states 3 and 4 and the model parameter D = 0.02D0 . The rate of proton diffusion along the gap increases with the expansion of the partition between the opposite membranes. In Fig. 13, curves 1–3 correspond to different values of the model parameter lo /li (li = const). In state 4 (when the overall proton ﬂux through CF0 –CF1 is virtually zero), pHi decreases more signiﬁcantly than in state 3 (when protons can escape from the lumen via active CF0 –CF1 complexes). In state 3, there may establish non-uniform proﬁle of pHi (r), which shape depends on the model parameter lo /li . In the case of narrow gap (stacked thylakoids, lo /li = 0.1), in the center of granum the lumen becomes more acidic than in periphery enriched with CF0 –CF1 complexes. Destacking of granal thylakoids, as modeled by increasing the ratio lo /li , inﬂuences the pHi proﬁle: the level of pHi decreases with widening the gap (Fig. 13). This result can be explained by acceleration of overall electron transport and more intensive proton pumping into the lumen caused by rapid operation

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of PSII in unstacked thylakoids. The model predicts that acidiﬁcation of the lumen also depends on the proton mobility: pHi decreases with an increase of D (data not shown, for details see Vershubskii et al., 2004b). In state 3, at relatively low rate of proton diffusion (D = 0.02D0 ), the pH difference across the stromal thylakoids is less than in grana thylakoids. As the proton diffusion occurs more rapidly (D = 0.1D0 ), pHi proﬁle inside the thylakoids becomes leveled, and pHi reaches a lower level regardless of the interthylakoid gap width. One of the most interesting predictions of our model concerns the light-induced alkalization of the interthylakoid gap caused by the consumption of protons upon PQ reduction to PQH2 (pHgap ≥ 9.5–10, depending on geometry characteristics of the system, Fig. 13). Obstructed diffusion of protons through the narrow + channel (H+ stroma → Hgap ) cannot compensate a signiﬁcant rise of pH in the partition between the thylakoids of grana. Alkalization of the gap would hinder to protonation of double-reduced form + of plastoquinone (PQ2− B + 2Hgap → PQB H2 ), thus slowing down the operation of PSII. Acceleration of proton diffusion inside the gap, either due to destacking of granal thylakoids (as modeled by the rise of l0 ,) or due to an increase the diffusion coefﬁcient D (Fig. 12B and C), accelerates electron transport from PSII to PSI, thereby magnifying the proton pumping into the lumen. 5.3. On the peculiarities of thermodynamic description of small systems

Fig. 12. Modeled pathways of proton transfer (panel A), kinetics of proton uptake induced by the short light pulse (1 ms) calculated for two values of dimensionless diffusion coefﬁcient for protons, D = 0.02D0 and D = 0.1D0 (panel B), and the inﬂuence of geometrical parameter l0 (li = const) on the PSII activity calculated for D = 0.02D0 and D = 0.1D0 (panel C). Panels B and C are modiﬁed Figs. 2 and 5 from Vershubskii et al. (2004b).

Fig. 13. Lateral proﬁles of pH in the thylakoid lumen (pHi ) and in the interthylakoid gap (pHo ) in steady states 3 and 4 as calculated for different values of the interthylakoid gap width lo . Curves 1, 2, and 3 correspond to the ratio lo /li = 0.1, 0.25; and 0.5 (li = const), respectively. Efﬁcient coefﬁcient of proton diffusion D = 0.02D0 . Stromal pH was assumed to be constant, pHs = 8. Modiﬁed Figs. 3 and 4 from Vershubskii et al. (2004b).

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In the context of thermodynamic description of bioenergetic systems in terms of the proton potential difference, pH, let us consider the question of the physical meaning of the hydrogen ion activity (concentration) inside small vesicles. Speciﬁc features of thermodynamics of small energy-transducing systems have been analyzed in (Tikhonov and Blumenfeld, 1985; Blumenfeld et al., 1991; Blumenfeld and Tikhonov, 1994). Is it correct to speak about pH inside small vesicles in conventional terms of physical chemistry when we consider, for instance, hydrogen ions inside a single thylakoid? The very essence of this question becomes clear when we calculate a number of hydrogen ions in the bulk phase of the vesicle. Let us evaluate a concentration of hydrogen ions inside a single thylakoid, using the formula [H+ ]in = nin /vin , where nin is the number of free (not bound to buffer groups) hydrogen ions located in the aqueous (osmotic) bulk phase of a thylakoid of volume vin . Under physiological conditions, the vin value can be estimated as vin = (1–6) × 109 (Å3 ) (Tikhonov and Blumenfeld, 1985). In this case, at moderate pHin a number of free protons nin inside a single thylakoid should be only a few. For instance, at pHin = 6.0 we obtain nin = vin × 10−pHin = 0.6–3.6. Is it correct to speak about pHin if a mean value of nin of free hydrogen ions is only a few or a probability of detecting even of one free (unbound) hydrogen ion is < 1 (nin < 1)? This question was scrutinized by Blumenfeld et al. (1991), who used conventional approaches of statistical thermodynamics to analyze a reaction of the type PQ ↔ P + Q (an analog of the dissociation reaction AH ↔ H+ + A– ). The use of statistical thermodynamics for modeling this reaction is reasonable because the total number of particles P inside small vesicles is high, NPQ + NP 1. In the meantime, a number NP of unbound (free) particles P inside a small vesicle can be very small (NP ∼ 1); ﬂuctuations of NP would sharply increase with a decrease in the vesicle volume vin . In the latter case, the law of mass action may be violated dramatically (Blumenfeld et al., 1991). It should be stressed, however, that the Nernst equation for the free energy change upon the particle P transfer from the vesicle to the external volume remains valid, regardless of a number of free particles inside the vesicle: G = −kB T ln( NP /Nout ).

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(7)

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Fig. 14. Cartoon illustrating non-uniform distribution on pH in different compartments of the chloroplast. Since most of CF0 –CF1 complexes are localized in stroma-exposed thylakoids, the efﬁcient drain of protons from the lumen via CF0 –CF1 may lead to the non-uniform proﬁle of the intrathylakoid pHi , when the transthylakoid pH difference across the stromal membrane is lower than pH grana grana in grana (pHi < pHstroma ). The model predicts signiﬁcant alkalization (pHo > i ) pHstroma o

of the interthylakoid partition caused by limitations in the proton diffusion along the narrow interthylakoid gap. Non-uniform distribution of protons may cause the lateral circulation of proton ﬂuxes: the “proton pressure” developed in the granal regions of the lumen will divert the protons to CF0 –CF1 , while the operation of PSII supports the inﬂux of protons from the stroma to the interthylakoid gap. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

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Here, NP is the mean number of free P particles inside a single vesicle, kB is the Boltzmann’s constant, and Nout is the number of free particles P located outside the vesicle. Formally, the NP

value may be virtually less then 1. In this case, NP reﬂects a probability of detecting a single particle inside the vesicle. Eq. (7) is valid regardless of the vesicle volume, of course, if we deal with the mean number of free particles, NP . In this case, the notion of chemical potential difference across the membrane is correct even for small vesicles, being rigorously approved by traditional methods of statistical thermodynamics (Blumenfeld et al., 1991). In chloroplasts, almost all the protons taken up by illuminated chloroplasts (∼99%) are bound to acid-base groups of the thylakoid membrane and buffer groups of the molecules dissolved in the lumen (Tikhonov and Timoshin, 1985; Tikhonov and Shevyakova, 1985). The presence of a large number of buffer groups has to damp ﬂuctuations of the number of unbound hydrogen ions inside the thylakoids. 5.4. Energetic and regulatory aspects of non-uniform lateral distribution of proton potentials Fig. 14 symbolically cartoons the scheme of proton partitioning based on experimental measurements of the intrathylakoid (Rumberg and Siggel, 1969; Tikhonov et al., 1981; Kramer et al., 1999; Trubitsin and Tikhonov, 2003; Tikhonov et al., 2008), stromal pH (Heldt et al., 1973; Robinson, 1985; Laisk et al., 1989a), and the results of modeling of electron and proton processes in chloroplasts. The light-induced acidiﬁcation of the thylakoid lumen generates pmf, which is the driving force for ATP synthase operation. Since most of CF0 –CF1 complexes are localized in stroma-exposed thylakoids, the efﬁcient drain of protons from the lumen via CF0 –CF1 in state 3 may lead to the non-uniform proﬁle of pHi , when pH difference across the stromal membrane is lower than pH in grana. Note that the steady state value of pH generated in stromal thylakoids (pH ≈ 2.5) is sufﬁcient to support the operation of CF0 –CF1 in the ATP synthesis mode (see for

review Tikhonov, 2013, 2014). It is also interesting to note that there may be lateral circulation of proton ﬂuxes: the “proton pressure” grana developed in the granal regions of the lumen (pHi < pHstroma ) i diverts the protons to CF0 –CF1 , while the operation of PSII supports the inﬂux of protons from the stroma to the interthylakoid grana gap (pHo > pHstroma ). o Apart from its energy role in ATP synthesis, pH plays important regulatory role. The light-induced decrease in the lumen pHi decelerates PQH2 oxidation by the b6 f complex and induces the attenuation of PSII activity due to the NPQ mechanism of energy dissipation in LHCII. It is interesting to note that our model predicts another mechanism of down-regulation of PSII activity, which is associated with alkalization of the interthylakoid gap. Obstructed diffusion of protons is unable to equilibrate pH values in the gap center (pHgap > 9) and in the bulk of stroma (pHout ∼ 7.8–8.0). Signiﬁcant alkalization of the gap will hinder in protonation of fully reduced plastoquinol, PQ2− B , thus slowing down the formation of protonated quinol and its dissociation from PSII. The model also predicts that unstacking of granal thylakoids, which releases the constraints to protons diffusion toward PSII, should stimulate the turnover of PSII (Fig. 12C). This prediction is in a good agreement with experimental data on the inﬂuence of osmotic conditions on electron transport in chloroplasts. Kirchhoff et al. (2000) reported that in destacked tobacco chloroplasts nearly all PQ molecules became reduced by a 120 ms light pulse (about 11 electrons were stored in the ETC, equivalent to 5–6 PQH2 molecules). Otherwise, in stacked thylakoids, a large fraction of PQ molecules remained oxidized after the light pulse (about 40% in tobacco and 50% in spinach thylakoids). The authors concluded, in the line with observations made by Lavergne et al. (1992), that their observations could be explained by inability of PQ to migrate rapidly throughout the membrane. Results of our calculations suggest alternative explanation of the stacking/destacking experiment data: in stacked membranes the light-induced formation of PQH2 may be limited by slow diffusion of protons throughout the narrow interthylakoid gap; unstacking of granal thylakoids stimulates diffusion of protons, thereby promoting the light-induced formation of PQH2 . This result is also in a good agreement with the observation that swelling of thylakoids due to osmotic effects stimulates the rate of electron transport and ATP synthesis in bean chloroplasts (Masarova and Tikhonov, 1989). Summing up, computer experiments suggest that along with the NPQ mechanism of attenuation of PSII activity and deceleration of PQH2 oxidation by the b6 f complex, the intersystem electron transport may be down-regulated due to the light-induced alkalization of the interthylakoid partition.

6. Concluding remarks Among a great variety of biological processes, photosynthesis is a unique set of molecular events that unite different kinds of energy transducing processes, from light capture, energy migration and primary photochemical processes in photoreactions centers to numerous biochemical reactions of organic substances synthesis from atmospheric CO2 and water. Mathematical modeling of photosynthesis provides a framework for in-depth analysis of feedbacks in the network of plant cell metabolism and creates a platform for investigation of the regulatory mechanisms that provide optimal functioning of photosynthetic apparatus and its plasticity in the processes of energy transduction and photoprotection of plants. As one of examples of computer modeling of oxygenic photosynthesis, we have described our model of electron and proton transport in chloroplasts. This model includes key stages of the linear electron transport, alternative pathways of electron transfer around PSI, transmembrane proton transport and ATP

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synthesis in chloroplasts. The model considers different regulatory processes: pH-dependent control of the intersystem electron transport, down-regulation of PSII due to the NPQ activity mechanism, and the light-induced activation of the BBC cycle. Results of numerical experiments performed within the framework of this model show that the model adequately reproduces a variety of experimental data on induction events observed under different experimental conditions in intact chloroplasts (variations of CO2 and O2 concentrations in atmosphere), including a complex kinetics of P700 (primary electron donor in PSI) photooxidation, CO2 consumption in the BBC cycle, and photorespiration. The model, therefore, may provide a module for a comprehensive model of oxygenic photosynthesis. Simulation of diffusion-controlled photosynthetic processes in laterally heterogeneous thylakoids has been used to calculate the lateral proﬁles of pH in the thylakoid lumen and in the narrow gap between grana thylakoids. Results of these calculations suggest that along with the NPQ mechanism of attenuation of PSII activity and deceleration of PQH2 oxidation by the cytochrome b6 f complex, the intersystem electron transport may be down-regulated due to light-induced alkalization of the narrow partition between adjacent thylakoids of grana. Further prospects for mathematical modeling of photosynthesis will be associated with two general trends of development emodels: ﬁrst, the expansion, and, second, the extension of e-models of photosynthesis. The ﬁrst trend in modeling metabolic processes in the plant cell is associated with integration of photosynthetic processes with other metabolic processes, e.g., interactions of chloroplasts with mitochondria via common metabolites, carbon and nitrogen metabolism, which are closely related to photophosphorylation and electron and proton transport in chloroplasts. Mathematical modeling of oxygenic photosynthesis helps to analyze the feedback and feedforward controls in the plant cell metabolic network. The reader can ﬁnd in the literature several examples of comprehensive computer models that consider a variety of photosynthetic processes, from light capture to sucrose synthesis (Laisk et al., 2006, 2009; Nedbal et al., 2007; Zaks et al., 2012; Zhu et al., 2013). Further development of comprehensive models of e-photosynthesis would provide reliable and workable platforms for profound analysis of photosynthetic energy conversion in nature, from seconds to seasons (Demmig-Adams et al., 2012). Another trend in theoretical study of energy transduction in associated with in-depth study of partial reactions of energy transducing procecces at the molecular level. Recent development of sophisticated computational methods, such as quantum chemical calculations (see for review Blomberg and Siegbahn, 2010) and molecular dynamics simulations (see, e.g., Mulkidjanian et al., 2005; Postila et al., 2013), made it possible to scrutinize energetics and reaction mechanisms of electron and proton transfer processes in photosynthetic and related energy transducing systems.

Uncited references Foyer et al. (2012), Kovalenko et al. (2011), Lazar (2003), Lazar and Schansker (2009), Leister and Shikanai (2013) and Mitchell et al. (1990).

Acknowledgements

This work was partly supported by grant 12-04-01267 from the Q4 Russian Foundation for Basic Researches. We thank V.I. Priklonskii 1480 and I.V. Kuvykin, who took part in our previous works on mathe1481 matical modeling of photosynthetic processes in chloroplasts. 1482 1479

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.biosystems.2014.04.007.

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Computer modeling of electron and proton transport in chloroplasts. - PDF Download Free (2024)