Photosynthesis: basics, history and modelling

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Abstract

Background

With limited agricultural land and increasing human population, it is essential to enhance overall photosynthesis and thus productivity. Oxygenic photosynthesis begins with light absorption, followed by excitation energy transfer to the reaction centres, primary photochemistry, electron and proton transport, NADPH and ATP synthesis, and then CO₂ fixation (Calvin–Benson cycle, as well as Hatch–Slack cycle). Here we cover some of the discoveries related to this process, such as the existence of two light reactions and two photosystems connected by an electron transport ‘chain’ (the Z-scheme), chemiosmotic hypothesis for ATP synthesis, water oxidation clock for oxygen evolution, steps for carbon fixation, and finally the diverse mechanisms of regulatory processes, such as ‘state transitions’ and ‘non-photochemical quenching’ of the excited state of chlorophyll a.

Scope

In this review, we emphasize that mathematical modelling is a highly valuable tool in understanding and making predictions regarding photosynthesis. Different mathematical models have been used to examine current theories on diverse photosynthetic processes; these have been validated through simulation(s) of available experimental data, such as chlorophyll a fluorescence induction, measured with fluorometers using continuous (or modulated) exciting light, and absorbance changes at 820 nm (ΔA₈₂₀) related to redox changes in P700, the reaction centre of photosystem I.

Conclusions

We highlight here the important role of modelling in deciphering and untangling complex photosynthesis processes taking place simultaneously, as well as in predicting possible ways to obtain higher biomass and productivity in plants, algae and cyanobacteria.

Keywords: Calvin–Benson cycle, chlorophyll a fluorescence induction, non-photochemical quenching (of the excited state of chlorophyll a), modelling, discoveries in photosynthesis, photosynthetic electron transport, state transitions

‘Complexity is the prodigy of the world. Simplicity is the sensation of the universe. Behind complexity, there is always simplicity to be revealed. Inside simplicity, there is always complexity to be discovered.’

Gang Yu

INTRODUCTION

With limited agricultural land and increasing human population, it is essential to enhance photosynthetic activities. Oxygenic photosynthesis is a very important process, not only because it is the source of our food, fibre and many useful substances, but also because almost all life on the Earth depends on it, either directly or indirectly. Plants, algae and cyanobacteria are oxygenic photosynthetizers that use light energy to generate organic molecules [e.g. glucose (C₆H₁₂O₆), sugars, starch] from carbon dioxide (CO₂) and water (H₂O), and release molecular oxygen (O₂) into the atmosphere (for a background on photosynthesis see, Eaton-Rye et al., 2012; Blankenship, 2014; Shevela et al., 2019):

Note that the above global equation of photosynthesis emphasizes that the oxygen molecules released into the atmosphere originate from water oxidation, not from carbon dioxide, as established using 18 O-labelled water (Ruben et al., 1941).

This process starts in the thylakoid membrane (TM) with two light reactions taking place simultaneously at photosystem (PS) II and PSI reaction centres (RCs; for PSII and PSI, see the review by Nelson and Junge, 2015). The light energy absorbed by pigment–protein antenna complexes of the PSs is converted, with very high efficiency, into redox chemical energy; a small part is, however, dissipated as heat (internal conversion), and as chlorophyll (Chl) fluorescence (2–10 %, Latimer et al., 1956). Furthermore, water is oxidized to oxygen, and NADP + is reduced to NADPH, and, in addition, ATP is produced (Rabinowitch and Govindjee, 1969; Blankenship, 2014; Shevela et al., 2019). Both NADPH and ATP are then used for CO₂ assimilation in the stroma (for a historical background of the Calvin–Benson cycle, see, Bassham, 2005; Benson, 2005); here, Rubisco (ribulose 1,5-bisphosphate carboxylase/oxygenase) is a key enzyme, which catalyses the fixation of CO₂ on a five-carbon compound, RuBP (ribulose 1,5-bisphosphate). A diagram of the photosynthetic apparatus and the electron transport (ET) reactions is shown in Fig. 1 .

Diagram of the photosynthetic apparatus and electron transport (ET) pathways in plants and algae. Four major protein complexes in the thylakoid membrane (TM) participate in the production of ATP and nicotinamide adenine dinucleotide phosphate in reduced form (NADPH), needed for the Calvin–Benson cycle to fix CO₂ to produce sugars: two photosystems (PSII and PSI) connected in series via the cytochrome (Cyt) b₆/f, and the ATP synthase. Light is absorbed simultaneously by pigments in the light harvesting complexes of PSI and PSII (LHCI and LHCII); excitation energy is transferred to reaction centre (RC) P700 (in PSI) and P680 (in PSII), where primary charge separation takes place, initiating a chain of redox reactions. PSII functions as a water/PQ (photo)-oxidoreductase, which has a manganese complex [Mn₄O₅Ca], and a tyrosine-161 (Y_Z), located on D1 protein on the electron donor side, as well as pheophytin (Pheo), plastoquinones Q_A and Q_B, and a non-haem (heme) iron binding a bicarbonate ion (HCO₃ ‒ ) on the electron acceptor side. By contrast, PSI is a plastocyanin (PC)/ferredoxin (Fd) (photo)-oxidoreductase; it uses reduced PC as an electron donor, and a particular Chl a molecule (A₀), vitamin K₁ (A₁), and three non-haem iron–sulfur centres (shown in the figure as Fe-S) are on the acceptor side of PSI. The Cyt b₆/f complex includes a Cyt f, a Rieske iron–sulfur protein (Fe-S), two cytochromes b (Cyt b_p and Cyt b_n) that participate in the oxidation and reduction of PQH₂ and PQ: PQH₂ is oxidized at the Q_p-site by Cyt b_p, while PQ is reduced at the Q_n-site by Cyt b_n. The Q_p- and Q_n-sides are also called Q_o- and Q_i-sides, respectively. Besides the linear ET flow from water to NADP + , there are several pathways leading to electron donation to alternative electron acceptors: cyclic electron flow (CEF) around PSI mediated by Fd (involving Fd-NADP + -reductase, FNR, and a proton gradient regulator, PGR5), or NADPH (via NADPH dehydrogenase, NDH); water–water cycle (WWC); chlororespiration (through the plastid terminal oxidase, PTOX); and the malate valve (through malate dehydrogenase, MDH). The proton motive force (pmf) [consisting of the proton concentration difference (ΔpH) and the electric potential (ΔΨ) across TM] is used by ATP synthase to produce ATP from ADP and phosphate (P_i); in the pmf formula, R is the gas constant, F is the Faraday constant, and T is the absolute temperature (in K). Modified from Alric (2010).

The availability of high-performance computers and detailed knowledge of the various steps of photosynthesis have provided new opportunities to use mathematical modelling to better understand the dynamics of this process (see reviews by Lazár and Schansker, 2009; Jablonsky et al., 2011; Stirbet et al., 2014). In addition, several studies (Zhu et al., 2010; Long et al., 2006, 2015; Ort et al., 2015; South et al., 2018; Simkin et al., 2019) strongly support the idea that the photosynthetic processes can be improved through genetic engineering to increase the yield potential of various crops (see also Rosenthal et al., 2011; Simkin et al., 2015, 2017; Kromdijk et al., 2016; McGrath and Long, 2016). Furthermore, mathematical modelling can be used to predict opportunities for specific genetic modifications and devise optimized engineering designs to improve photosynthesis (Zhu et al., 2007).

In this review, we first provide a background of oxygenic photosynthesis that forms the basis of its modelling. We then discuss a few selected studies on mathematical models describing photosynthetic processes. Partial reactions of photosynthesis have been often modelled separately, such as: (1) the primary photochemical reactions (e.g. Schatz et al., 1988; Roelofs et al., 1992); (2) water ‘splitting’ reactions (e.g. Kok et al., 1970; Mar and Govindjee, 1972; Jablonsky and Lazár, 2008; Shen, 2015); (3) reduction of Q_B, the secondary plastoquinone (PQ) acceptor of PSII (e.g. Velthuys and Amesz, 1974; Petrouleas and Crofts, 2005); and (4) the redox reactions of the PQ pool at the Cyt b₆/f complex (which may include the Q-cycle; see e.g. Mitchell, 1975; Cramer et al., 2011). However, in this review we mainly discuss larger models, which include several steps, providing information on complex photosynthetic processes.

PHOTOSYNTHESIS IN PLANTS, ALGAE AND CYANOBACTERIA: SOME BASICS

Early discoveries

Not much was known about photosynthesis before the 20th century; for earlier discoveries in photosynthesis see chapter 2 in Rabinowitch (1945) and the timeline in Govindjee and Krogmann (2004). The key discoveries were as follows (see chapter 1 in Rabinowitch and Govindjee, 1969): Jan van Helmont (1648) showed that plant growth was mainly from the water that plants had absorbed; it was only later that Nicolas Théodore de Saussure (1804) clearly showed that water was an essential reactant of photosynthesis. Joseph Priestley (1776) showed, in elegant experiments, that plants produced ‘oxygen’ (then called de-phlogisticated air) needed by a mouse to live, whereas Jan Ingen-Housz (1773) convincingly established that light was necessary for photosynthesis. The role of CO₂ in photosynthesis was shown by Jean Senebier (1782), whereas the synthesis of starch was shown by Julius von Sachs (1862, 1864). However, the involvement of chlorophyll (Chl) in this process has a long history. For some of the earliest concepts, we must remember to mention Pierre Joseph Pelletier and Joseph Bienaimé Caventou (1817, 1818), and René Joachim Henri Dutrochet (1837). However, Theodor Engelmann (1882) provided the first action spectrum of photosynthesis, showing that red and blue light, absorbed by Chl, produce oxygen (see figure 1.1 and its description in Shevela et al., 2019).

Physiological and biochemical advances

An understanding of how photosynthesis functions began only after 1900, but by 1960 a basic model at the molecular level, including generation of NADPH and ATP as well as the steps leading to the assimilation of CO₂ to produce carbohydrates, was established (see Govindjee and Krogmann, 2004; Govindjee et al., 2005; Nickelsen, 2016).

By measuring photosynthesis as a function of light intensity, Frederick Frost Blackman (1905) suggested that photosynthesis consists of two separate phases: a light-dependent phase (i.e. so-called ‘light’ reactions), and a temperature-dependent biochemical phase (so-called ‘dark’ reactions, or ‘Blackman reaction’; see Warburg and Uyesugi, 1924). However, because CO₂ fixation uses NADPH and ATP, formed in the light phase, these so-called ‘dark’ reactions are also light-dependent. Moreover, many enzymes, involved in CO₂ assimilation reactions, function only when they are ‘light-activated’, being controlled through the ferredoxin:thioredoxin reductase (FTR) system (see reviews by Buchanan et al., 2002; Nikkanen and Rintamäki, 2019). Therefore, the term ‘dark phase’ is inappropriate; Buchanan (2016) has proposed the use of ‘carbon reactions’ for ‘dark reactions’. Furthermore, the true ‘light reactions’ end after the primary charge separation steps in the RCs; both the electron transfer and the proton transfer reactions, in principle, can occur in darkness.

Cornelis B. van Niel (1931, 1941) showed that certain photosynthetic bacteria use H₂S instead of H₂O as an electron donor, producing sulfur instead of oxygen, and the global reaction of photosynthesis is:

where A is sulfur in sulfur bacteria and oxygen in plants, algae and cyanobacteria. By analogy with photosynthetic bacteria, van Niel suggested that O₂ released by plants is derived from H₂O rather than CO₂. This was confirmed by Sam Ruben, Merle Randall, Martin Kamen and James Logan Hyde (see Ruben et al., 1941), based on results using 18 O-labelled water.

Chlorophyll a fluorescence

As mentioned earlier, in addition to primary photochemistry, photosynthetic organisms lose some energy as heat (internal conversion) and as light (fluorescence). Fluorescence is radiative deactivation of (usually) the first singlet excited state of a molecule to the ground state. Kautsky and Hirsch (1931) discovered what others later called the ‘Kautsky effect’, which is Chl a fluorescence induction (ChlFI; see Govindjee, 1995). Kautsky and Hirsch observed (visually) transitory variations in Chl a fluorescence (ChlF) emitted by samples that were illuminated after a period of darkness; this ChlF has an increasing phase (peak, ~1 s) followed by a slower (5–10 min) decreasing phase. McAlister and Myers (1940) made an important observation by showing an inverse relationship between ChlF emission and CO₂ uptake. These ChlF transients were then studied, among other places, in the Photosynthesis Laboratory at the University of Illinois, Urbana-Champaign (beginning in the 1950s; see Govindjee and Papageorgiou, 1971; Papageorgiou, 1975; Govindjee and Satoh, 1986; Papageorgiou et al., 2007). Because ChlF has been shown to be directly or indirectly affected by complex physical and biochemical processes taking place during photosynthesis, analysis of ChlFI curves is of importance in photosynthesis research (see reviews by Krause and Weis, 1991; Lazár, 1999, 2015; Strasser et al., 2004; Stirbet and Govindjee, 2011; Stirbet et al., 2018).

Photosynthetic unit (antenna and reaction centres): excitation energy transfer

An essential concept related to the light phase of photosynthesis is ‘photosynthetic unit’. It was developed based on the crucial discovery by Emerson and Arnold (1932a, b) that ~2400 Chl molecules cooperate to evolve one molecule of O₂, while the minimum quantum requirement for the evolution of one O₂ molecule was 8–10 (Emerson, 1958; for the history of this discovery, see Nickelsen and Govindjee, 2011; Nickelsen, 2016). Gaffron and Wohl (1936) suggested the existence of ‘photosynthetic units’, where light energy absorbed by any antenna molecule is transferred as excitation energy among the pigment molecules, until finally it is trapped with high efficiency by a limiting enzyme (a ‘photoenzyme’, as implied by Emerson and Arnold, 1932b), which is equivalent to what we now call reaction centre (RC), a term introduced by Duysens (1952). Here, the primary charge separation (i.e. photochemistry) takes place (see e.g. Myers, 1994; Govindjee and Krogmann, 2004). Experimental evidence for excitation energy transfer (EET) between photosynthetic pigments was initially obtained by comparing action spectra of photosynthesis and of sensitized ChlF in green, brown and red algae (see chapters 10–12 in Rabinowitch and Govindjee, 1969). We now have much more detailed knowledge on the molecular mechanisms of electronic EET in antenna, as well as on exciton trapping by the RCs (e.g. Croce and van Amerongen, 2013; van Amerongen and Croce, 2013; Roden et al., 2016; Mirkovic et al., 2017; Chan et al., 2018).

Taking things apart

Robert Hill (1937) found that the ‘light phase’ of photosynthesis can operate independently from the ‘dark phase’ (the carbon reaction phase), since isolated chloroplasts can evolve O₂ in the presence of artificial electron acceptors [this reaction is called the ‘Hill-reaction’ in honor of Robert (Robin) Hill], even in the absence of CO₂. This concept led to a ‘modularization’ in the study of photosynthesis (Nickelsen, 2016), since even if these two partial processes are interrelated, the tendency after 1940 was to investigate them separately. Note that Mehler (1951) had found that molecular oxygen is also a Hill electron acceptor, and this reaction, called the ‘Mehler reaction’, has been shown to play an important role in photoprotection of photosynthetic organisms (Miyake, 2010).

The carbon reactions

The long-lived form of radioactive carbon, 14 C, was discovered by Samuel Ruben and Martin Kamen (1941). This radioactive isotope was used to decipher the major pathway of CO₂ reduction by photosynthetic organisms, by Andrew Benson (who did most of the early pioneering work, using 14 C), Melvin Calvin, James A. Bassham and co-workers (see Calvin et al., 1950; Calvin, 1989; Bassham, 2005; Benson, 2005). For example, they found that ribulose 1,5-bisphosphate (RuBP; a 5-C sugar) was the acceptor of CO₂; the first stable product of CO₂ reduction was 3-phosphoglyceraldehyde (G3P; a triose phosphate); and that there was a cycle to regenerate the RuBP. Melvin Calvin received the Nobel Prize in Chemistry in 1961 for these discoveries; we are of the opinion that Andrew Benson should have been a co-recepient.

Photophosphorylation

Daniel Arnon et al. (1954a, b) showed that isolated chloroplasts can produce ATP in light; in addition, they showed that intact isolated chloroplasts can even perform complete photosynthesis (i.e. CO₂ fixation). Furthermore, Allen et al. (1958) found that photophosphorylation can be ‘cyclical’ (i.e. ATP is produced when there is a cyclic ET, which was shown to involve cyclic electron flow around PSI via Cyt b₆/f, CEF-PSI), or when there is ‘non-cyclic’ [i.e. during linear electron flow (LEF) from PSII to PSI) (see also Arnon, 1984; Tagawa et al., 1963). A third pathway, labelled as ‘pseudo-cyclic photophosphorylation’, was also established, in which molecular oxygen plays the role of a terminal electron acceptor (i.e. the Mehler reaction; Mehler, 1951; Heber, 2002). Furthermore, a coupling mechanism between ATP synthesis and the ET, also in chloroplasts, was demonstrated by Dave Krogmann, Mordhay Avron and André Jagendorf (see Krogmann et al., 1959). Note that the chloroplast coupling factor (CF1) for photophosphorylation, today known as ATP synthase, was discovered by Avron (1963).

The two-light reaction and the two-pigment system concept

The idea of two light reactions and two types of PSs had its beginning in the 1943 experiments of Robert Emerson and Charleton Lewis on the ‘red drop’ in the action spectrum of the quantum yield of photosynthesis (Emerson and Lewis, 1943) and in the 1957 ‘Emerson enhancement’ effect, that is when the rate of photosynthesis in two lights given together was higher than the sum of the rates of photosynthesis measured when the two lights were given separately (Emerson et al., 1957; also see: Govindjee and Rabinowitch, 1960); this discovery led to the well-known ‘Z’-scheme of photosynthesis (Hill and Bendall, 1960; for the evolution of the Z-scheme, see Govindjee et al., 2017). The very first Chl electron donors in the two PSs are P700 for PSI (identified also by an absorbance change around 705 nm; see Kok, 1956; Govindjee and Renger, 1993), and P680 in PSII, first suggested by Krey and Govindjee (1964) and shown to exist by Döring et al. (1969). Key experiments proving the Z-scheme were provided by Duysens et al. (1961) on the red alga Porphyridium cruentum, who showed the antagonistic effect of light 1 and light 2 on the redox state of cytochrome (Cyt). (Here, light absorbed by PSI was ~680 nm, and that absorbed by PSII was ~562 nm.) Furthermore, based on flashing light experiments, Witt et al. (1961a, b) provided evidence for the kinetics and on the existence of other intermediate steps in the Z-scheme; details of the ET components involved in the photosynthetic electron transport chain (PETC) are given in Fig. 1 . However, of course, the physical confirmation for the existence and organization of the two PSs was the isolation and characterization via X-ray crystallography of the high-resolution spatial structure of PSII (e.g. Zouni et al., 2001) and PSI (e.g. Jordan et al., 2001).

Evidence from Chl a fluorescence measurements

Additional evidence for the two-pigment-system/two-light-reaction scheme in oxygenic photosynthesis was obtained by Govindjee et al. (1960) on Chlorella cells, using ChlF measurements. They showed an antagonistic effect of light 1 (i.e. predominantly absorbed by PSI) and light 2 (i.e. predominantly absorbed by PSII) on ChlF: addition of far-red light (light 1) to a shorter wavelength light (light 2) caused a decline (rather than an enhancement) of ChlF yield, compared to that produced by the two beams separately. As an explanation of this effect, Duysens and Sweers (1963) proposed that light 2 reduces a quencher Q, while light 1 oxidizes Q ‒ back to Q. The quencher theory of Duysens and Sweers was based not only on ChlF data published by Govindjee et al. (1960), but also by Butler (1962), who showed that variable fluorescence is mostly from PSII, and far-red light, absorbed by PSI, gives a smaller amount of PSI fluorescence. The quencher Q (named X-320, but now labelled Q_A) was identified using single turnover flashes, and has an absorption spectrum with maximal spectral changes in the UV, at 270 and 320 nm (Stiehl and Witt, 1968). In several experimental studies (Stiehl and Witt, 1969; van Gorkom, 1974; see also Witt, 2004), plastoquinone difference spectra in the near UV (300–350 nm) were similar to light-minus-dark spectra of the first plastoquinone acceptor of PSII (i.e. Q_A −• − Q_A). According to Duysens and Sweers (1963), ChlF is proportional to the fraction of the reduced quencher ([Q_A − ]/[Q_A]_total; see a discussion in Stirbet and Govindjee, 2012; for other views see, Schansker et al., 2011, 2014; Magyar et al., 2018). Later, it was shown that several non-photochemical quenching (NPQ) processes take place in parallel with the photochemical quenching (i.e. by Q_A) during the so-called slow (~10 min) phase of the ChlF transient, and the proportionality of the fluorescence yield with [Q_A − ]/[Q_A]_total, observed during the initial (et al., 1986).

Vredenberg and Duysens (1963) observed that closure of RCs is accompanied by an increase in fluorescence yield of bacteriochlorophyll in Rhodospirillum rubrum, a purple anoxygenic photosynthetic bacterium, and concluded that several RCs share the same antenna. In an oxygenic photosynthesizer, the green alga Chlorella, Anne and Pierre Joliot (Joliot and Joliot, 1964) measured the rate of steady-state oxygen evolution, and correlated it with the fraction of active PSIIs (see also Joliot and Joliot, 2003). Joliot and Joliot (1964) observed that both the oxygen yield and the fluorescence yield are related, in a hyperbolic manner, to the fraction of closed PSII centres; this suggested that there is an energetic connectivity within PSIIs, that is an excitation visiting a closed PSII (i.e. with Q_A reduced) is redirected to another PSII. In this manner, the trapping cross-section of the open PSIIs increases as their neighbouring PSIIs become closed (see a review on PSII excitonic connectivity by Stirbet, 2013). Joliot and Joliot (1964) also derived theoretical equations describing the dependence of the ChlF yield (Φ _F) and the photochemical yield (Φ _P) on the fraction of open PSIIs, which included a connectivity parameter (p) for the probability of excitation energy transfer from a closed PSII to a neighbouring PSII (either closed or open). This was followed by publication of detailed papers on PSII excitonic connectivity by Paillotin (1976), Strasser (1978) and Butler (1980), the last two describing the process, using bipartite and tripartite PSII models of Butler and co-workers (Butler and Kitajima, 1975; Butler and Strasser, 1977). Later, Lavergne and Trissl (1995) and Trissl and Lavergne (1995) extended the concept of PSII excitonic connectivity, using an exciton–radical pair equilibrium model. The latter is equivalent to the reversible radical pair (RRP) model of Schatz et al. (1988); it assumes rapid exciton equilibration between all PSII pigments, including P680, and describes primary photochemistry (charge separation, recombination and stabilization) leading to closed PSII RCs. The major feature of the RRP model is equilibrium, i.e. reversibility of charge separation, meaning fast charge separation followed by fast charge recombination, in both the open and the closed PSII centres (see Fig. 2 ).

Scheme showing the RRP (reversible radical pair) model and related reactions. The original RRP model is represented by the reactions on lines I and II, which are reactions occurring in an open PSII RC (when Q_A is initially oxidized) and a closed PSII RC (when Q_A is initially reduced), respectively. (L–P680)* denotes Chls in the light harvesting antenna of PSII (L) plus P680, which are in ultrafast excitation kinetic equilibrium, the asterisk (*) indicating the excited state. The rates constants are: k_L, overall rate constant of antenna excitation; k₃, overall rate constant of the excited state deactivation through heat dissipation and ChlF emission; k₁ o and k₁ c , rate constants of the primary charge separation in open and closed PSIIs, respectively; k_-1 o and k_-1 c , rate constants of the radiative (i.e. to the excited state) charge recombination between P680 + and Pheo − in open and closed PSIIs, respectively; k₂ o , rate constant of charge stabilization in an open PSII, i.e. the ET from Pheo ‒ to Q_A; k₂ c , rate constant of non-radiative (i.e. to the ground state) charge recombination between P680 + and Pheo ‒ in a closed PSII. The scheme presented here also includes excitation energy transfer (the energetic connectivity) between open and closed PSIIs (rate constant k_UU) and reversible reduction of P680 + by Y_Z (rate constants k_Pred and k_Pox), as well as the reduction of Y_Z + by the manganese cluster of the oxygen-evolving complex (OEC; rate constant k_Yred), which produces an S-state transition from S_i to S_i+1, where S_i and S_i+1 represent particular S-states. Modified from Lazár and Schansker (2009).

ATP synthesis

Peter Mitchell (1961a, b) proposed a chemiosmotic theory for phosphorylation, which suggests that a ‘proton motive force’ (pmf), i.e. the electrochemical potential of protons, couples the ET reactions with ATP synthesis (from ADP and inorganic phosphate, P_i). Mitchell received the Nobel Prize in Chemistry in 1978 for this hypothesis. Later, Paul Boyer and John E. Walker received the Nobel Prize in Chemistry in 1997 for their work on the structure of F1 mitochondrial ATPase and the mechanism of ATP synthesis (see e.g. Boyer, 2002). Hind and Jagendorf (1963) (see also Jagendorf and Uribe, 1966) showed how photosynthetic cells convert light energy into free energy stored in the ATP molecule on the basis of the chemiosmotic theory, particularly the ΔpH component. The pmf has two components, one due to the trans-thylakoid electric potential difference (i.e. the membrane potential, ΔΨ), and the other due to the trans-thylakoid difference in proton concentration (ΔpH), which builds up during water splitting reactions on the lumen side of PSII, and the translocation of stroma protons to the lumen during PQ pool reduction by PSII, and by Cyt b₆/f (including the Q-cycle; Mitchell, 1975) in relation to both the linear and the cyclic photosynthetic ET (see Fig. 1 , and a historical review by Jagendorf, 2002). We remind the readers that just as André Jagendorf’s work proved the importance of the ΔpH component (of pmf) for ATP synthesis, Wolfgang Junge’s work proved the importance of ΔΨ in making ATP (see mini-review by Junge, 2004). However, a high ∆Ψ component of the pmf was also shown to affect the equilibrium of redox reactions within PSII, and has been linked to higher rates of PSII charge recombination in vivo, and subsequent photodamage due to increased production of singlet oxygen (Davis et al., 2016). On the other hand, low pH has been shown to inactivate oxygen evolution (Schlodder and Meyer, 1987); furthermore, release of Ca 2+ from the oxygen evolving complex (OEC) has also been suggested to be the cause of this inactivation (Ono and Inoue, 1988; Krieger and Weis, 1993). For recent research (and reviews) on ΔΨ and ΔpH across the TM see, Strand and Kramer (2014), Kaňa and Govindjee (2016), and Lyu and Lazár (2017a, b).

Oxygen evolution

The key experiments that preceded the discovery of the water splitting mechanism, leading to O₂ evolution and P680 + reduction in PSII, were done by Pierre Joliot and co-workers (Joliot, 1965; Joliot et al., 1969). Joliot et al. (1969) discovered period 4 oscillations in oxygen evolution in algal suspensions when they were exposed to a sequence of single turnover (ST) saturating light flashes. These results were explained by Bessel Kok et al. (1970), who proposed a model (now known as Kok’s oxygen clock model, or the Kok–Joliot model to many), in which the formation of oxygen requires sequential accumulation of four positive charges on the OEC, which cycles through five redox states, labelled as S₀, S₁, S₂, S₃ and S₄ (see Fig. 3 ). For the history of this discovery, see Renger and Govindjee (1993) and Joliot (2003). The first evidence for the participation of Mn in the S-states was obtained by Chuck Dismukes and Yona Siderer (1980), who obtained electron paramagnetic resonance (EPR) signals for the same. For a review on the functioning of the OEC, see Najafpour et al. (2012). For a recent review on oxygen evolution, see Lubitz et al. (2019).

Highly simplified scheme of Kok’s oxygen clock model; misses and double hits are not shown. S_i (i = 0, 1, 2, 3, 4) represent the particular S-states of the manganese cluster of OEC. The S₄-state is assumed to be kinetically indistinguishable from the S₀-state. During an S-state transition, Y_Z + (formed through PSII reactions) is reduced (with rate constants k₀₁, k₁₂, k₂₃ and k₃₀). Modified from Lazár and Schansker (2009). For a review, including the involvement of manganese, see Najafpour et al. (2012).

Mechanistic models for early events in photosynthesis

Bay and Pearlstein (1963) provided one of the first mathematical models of the exciton kinetics and trapping in a photosynthetic system; it was based on electronic excitation transfer, FRET (Förster resonance energy transfer; see Förster 1946, 1948; also see a historical review by Clegg, 2006). According to this model, the electronic excitation energy moves in a so-called ‘random walk’, hopping from one Chl to another Chl in the antenna, until it is trapped by an RC, or is dissipated as heat or fluorescence (also see: Govindjee, 2004). Starting from FRET, other more complex and elegant theories have now been developed to characterize the exciton dynamics in antenna (e.g. Engel et al., 2007; Ishizaki and Fleming, 2009; Clegg et al., 2010; Fassioli et al., 2014).

On ‘state transition’ for regulation of balanced excitation in the two photosystems

State transition, a light-adaptive phenomenon that optimizes photosynthesis by synchronizing the turnover rates of PSII RCs and of PSI RCs, when there is an excitation imbalance between their antenna, was discovered by Cecilia Bonaventura and Jack Myers (1969) in Chlorella and, independently, by Norio Murata (1969a, b) in the red alga Porphyridium cruentum and spinach chloroplasts. The equilibration of PSII and PSI activities takes place through adjustment of the relative size of their antenna: During a transition from ‘state 1’ to ‘state 2’, the absorption cross-section (CS) of PSII antenna (which provides information on the PSII-specific rates of light absorption and represents an ‘apparent’ measure of PSII antenna size in situ, in units of Å 2 per PSII centre; see Osmond et al., 2017) decreases and that of PSI antenna increases, while the opposite occurs during transition from ‘state 2’ to ‘state 1’. The result is: the overall ChlF yield decreases in ‘state 2’ and increases in ‘state 1’, because, at room temperature, PSI has a much lower ChlF yield than PSII (Butler, 1962). State transitions have been shown by John Allen and collaborators to be regulated by the redox state of the PQ pool (Allen et al., 1981; see Allen, 2002): the transition from ‘state 1’ to ‘state 2’ is triggered by the reduction of the PQ pool, and the transition from ‘state 2’ to ‘state 1’ is triggered by the oxidation of the PQ pool. In plants and algae, the controlling events take place at the Qp site of Cyt b₆/f (i.e. the binding site of PQH₂; see Zito et al., 1999), where the PQ redox-state is sensed, which triggers the activation or inactivation of a protein kinase (Allen et al., 1981): PQ pool reduction activates the protein kinase, and thus induces phosphorylation of mobile light harvesting complex (LHC) II, followed by its attachment to PSI antenna, while PQ pool oxidation inhibits the protein kinase, followed by dephosphorylation of the mobile LHCIIs by a phosphatase, and their re-attachment to PSII antenna (see Fig. 4 and reviews by Papageorgiou and Govindjee, 2011; Rochaix, 2014). For background on PSII, see Wydrzynski and Satoh (2005), on PSI, see Golbeck (2006), and on the Cyt b6f complex, see Cramer and Kallas (2016). Note that extensive dynamic changes in the organization and structure of the TMs are associated with state transitions, which include PSII antenna dissociation after LHCII phosphorylation by Stt7/STN7 kinases, or association with PSII after dephosphorylation by PPHI/TAP38 phosphatases (see above, and Iwai et al., 2010). However, new research suggests that these protein kinases and phosphatases can also affect the likelihood of cyclic ET around PSI (see Wood et al., 2019). On the other hand, Pribil et al. (2018) have shown that the changes in the shape of grana stacks are mediated by the CURVATURE THYLAKOID1 (CURT1) protein complexes, which were shown to facilitate adjustments in membrane curvature at the grana margins in a dose-dependent manner.

Diagram of the mechanism of state transitions in plants and algae. In the diagram, the system is shown to be initially in ‘state 1’, with the absorption cross section (CS) of photosystem (PS) II being larger than that of PSI (it will have high Chl fluorescence yield because Chl in PSII is much more fluorescent than in PSI). During illumination, the plastoquinone (PQ) pool will be reduced by PSII because of higher absorption there. This is sensed by the Cyt b₆/f (via its PQH₂-oxidizing site, Q_p), and leads to activation of a kinase (Stt7/STN7) and phosphorylation of the mobile light harvesting complexes of PSII (LHCII), which then associate with the PSI antenna. The reverse occurs when the system is in ‘state 2’ initially, with the absorption CS of PSI being larger than that of PSII. Here, oxidation of the PQ pool by PSI during illumination will be sensed by the Cyt b₆/f, which leads to the inactivation of kinases, followed by de-phosphorylation of the mobile LHCIIs (by the phosphatases Pph1/TAB38) and their relocation to PSII. Abbreviations: A₀ and A₁, a particular Chl a molecule and a vitamin K1 molecule, respectively; Fe-S, three non-haem (heme) iron–sulfur centres; Fd, ferredoxin; Q_A and Q_B, plastoquinone electron acceptors of PSII; NADP + and NADPH, nicotinamide adenine dinucleotide phosphate in oxidized and reduced state; P680 and P700, reaction centre chlorophylls/primary electron donors of PSII and PSI; PC, plastocyanin. Figure modified from Allen (2003) and Rochaix (2014).

Two-electron gate on the electron acceptor side of PSII, and the requirement of bicarbonate

Bernadette Bouges-Bocquet (1973) and Bruno Velthuys and Jan Amesz (1974) independently discovered the two-electron gate (TEG) mechanism on the electron acceptor side of PSII in plants; it describes ET from Q_A to Q_B (see also Robinson and Crofts, 1983). As mentioned above, both Q_A and Q_B are PQs, but Q_A is a one-electron acceptor, and is permanently bound to the D2 protein of PSII. By contrast, Q_B is a two-electron acceptor that is bound to the D1 protein of PSII; it is strongly bound only when it is in Q_B − -state, but is weakly bound in its fully oxidized state (Q_B), and very weakly bound when in the fully reduced state (Q_BH₂). Following the primary charge separation: (1) Q_A is reduced to Q_A − (via pheophytin, Pheo; discovered by Vyacheslav Klimov et al., 1977); (2) Q_A − then reduces Q_B to Q_B − , and the latter remains tightly bound to D1; (3) after another light reaction, Q_B − is then further reduced by Q_A − , becoming fully reduced to Q_BH₂ (PQH₂), after the addition of two protons; and finally (4) because Q_BH₂ is loosely bound to D1, it is released in the membrane and replaced by another PQ molecule from the PQ pool (see Fig. 5 , and reviews discussing light-induced PQ pool reduction by PSII by Cardona et al., 2012; Müh et al., 2012). A bicarbonate ion has been shown to have a very important role in the functioning of the TEG and Q_BH₂ formation (Wydrzynski and Govindjee, 1975; see reviews by Govindjee and van Rensen, 1978; van Rensen, 2002; Shevela et al., 2012). A similar TEG was also discovered in bacteria, independently by Colin Wraight and André Vermeglio (see Vermeglio, 2002), but there is no bicarbonate effect there (see Wang et al., 1992, and references therein). Note that the TEG model, the Kok model and the RRP model are important partial models that are used in more complex (or complete) models describing the photosynthetic ET (e.g. Nedbal et al., 2009).

Scheme of the two-electron gate (TEG) model and related reactions. The two-electron gate mechanism, by which electrons are transferred from Q_A to Q_B, is represented by the reactions on line II. The rate constants are: k_L, overall rate constant of Q_A reduction; k_AB1 and k_AB2, rate constants of ET from the reduced Q_A to Q_B and Q_B ‒ , respectively; k_BA1 and k_BA2, rate constants of backward ET from Q_B ‒ and Q_B 2‒ to Q_A, respectively. The reactions above and below line II describe the reversible exchange of doubly reduced Q_B (after its double protonation, which is implicitly assumed) with a PQ molecule from the PQ pool (rate constants k_(B/PQ)exch and k_(PQ/B)exch); the reversible oxidation of the plastoquinol (rate constants k_ox and k_red) is implicitly assumed to be the result of chlororespiration and cyclical electron flow around PSI. Modified from Lazár and Schansker (2009).

On NPQ of the excited state of Chl

In general, NPQ processes can be defined as processes that decrease ChlF through mechanisms other than photochemical quenching (i.e. Q_A quenching; e.g. Müller et al., 2001; for a time line, see Papageorgiou and Govindjee, 2014). In this sense, the avoidance movement of chloroplasts in the leaf under high light conditions (i.e. qM; Dall’Osto et al., 2014; Baránková et al., 2016; Semer et al., 2018, 2019), the state 1 to state 2 transition (qT₁₂; see above), as well as the photoinhibition (qI), initiated by the photodamage of PSII (Tyystjärvi et al., 2005; Murata et al., 2012; Tyystjärvi, 2013), would all be considered to be NPQ processes. However, according to Papageorgiou and Govindjee (2014), it is preferable to consider as NPQ processes only those in which the excess energy accumulated as singlet excited Chl a ( 1 Chl a*) in PSII antenna is dissipated as heat (see Kitajima and Butler, 1975), such as the quickly reversible ‘high-energy non-photochemical quenching’ (qE), which develops in a few seconds and relaxes in 1–2 min (see Jahns and Holzwarth, 2012; and chapters in Demmig-Adams et al., 2014), or other less clearly elucidated sustained forms of ChlF quenching processes (such as qH; Malnoë, 2018). This type of NPQ is induced by low lumen pH, being fully activated only after the pmf is established across the TM, when the TM is in a ‘high-energy’ state; it regulates the utilization of the light energy in PSII antenna in order to reduce photo-oxidative events that can damage the RCs. The exact relationship between lumen pH and NPQ is not fully understood; however, see discussions by Johnson (2011) and Zaks et al. (2013). There are three main requirements for qE activation: (1) a trans-thylakoid ΔpH formed in light (Wraight and Crofts, 1970; Briantais et al., 1979); (2) the xanthophyll (VAZ) cycle, particularly the conversion of the carotenoid violaxanthin (V) to antheraxanthin (A) and zeaxanthin (Z) (Yamamoto et al., 1962; Yamamoto and Higashi, 1978); and (3) the PSII protein subunit S (PsbS) (Li et al., 2000; Brooks et al., 2014). Barbara Demmig-Adams et al. (1989) (see also a historical review by Demmig-Adams, 2003) were the first to demonstrate that the extent of qE is proportional to the Z content of leaves; Demmig et al. (1987) further showed a correlation between Z and a form of qI manifested as a dark-sustained NPQ. Thus, they proposed that Z, which is derived from V in the xanthophyll cycle, is the link between the high energy state of the membrane and the heat dissipation of excess excitation energy of Chl a (see also Rees et al., 1989, 1992). In the xanthophyll cycle, the content of V decreases during illumination and is restored in darkness: Light↝V⇄A⇄Z⇐Dark. Violaxanthin deepoxidase (VDE) has a higher affinity for A than for V (Yamamoto and Higashi 1978), and binds on the lumen side of the membrane, at pH ≈ 5.0 (Hager and Holocher, 1994), which induces qE. Also, the NPQ kinetics was shown to depend on [Z], its induction being faster and its relaxation being slower when Z is present (see Johnson et al., 2008). Adam Gilmore made an important contribution to the field, which included a successful collaboration with one of us (G) on the effects of intrathylakoid pH and VAZ cycle pigments on Chl a lifetime distributions and intensity in thylakoids (Gilmore et al., 1995, 1998; Gilmore, 1997). On the other hand, the role of PsbS protein in qE is that of a pH sensor and quenching amplifier, as its amount in plant modulates the maximal qE level, but the underlying event is not yet fully understood (Horton et al., 2008; Holzwarth et al., 2009; Brooks et al., 2014). However, there is also evidence that qE can be induced in the absence of PsbS (Johnson et al., 2011), or even xanthophylls (Johnson et al., 2012), if the lumenal pH is sufficiently low (i.e. lower than the value assumed by the ‘moderate lumen pH paradigm’; see Kramer et al., 1999). Finally, qE in algae is much more species-dependent than in plants. In unicellular green algae, or other algal groups (e.g. diatoms), the qE extent depends on the Light-Harvesting Complex Stress-Related (LHCSR) proteins (Peers et al., 2009). In most organisms, the LHCSR level is strongly light-dependent, and in some species, such as Chlamydomonas reinhardtii, acclimation to low light leads to very low NPQ levels (Peers et al., 2009).

Recently, Schreiber et al. (2019) have described a rapidly induced NPQ process during a pulse of high-light intensity in a dilute suspension of Chlorella vulgaris; they called this process HIQ [high (light) intensity quenching]. The amplitude of the HIQ increases linearly with the effective rate of quantum absorption by PSII, reaching ~8 % of F_M (i.e. the maximum Chl fluorescence measured in dark-adapted samples). This quenching rapidly relaxed after the pulse, and was shown to be caused by annihilation of 1 Chl* a by 3 Car* (excited state of a carotenoid in triplet state).

MODELLING CHL FLUORESCENCE INDUCTION IN PLANTS, ALGAE AND CYANOBACTERIA

ChlF emitted by plants and algae has little involvement in the process of photosynthesis, being one of the pathways in which excess excitation energy is dissipated by photosynthetic organisms. However, ChlFI kinetics is well recognized to have an intricate connection with many processes taking place during the conversion of light energy into a stable chemical form. Because it is a non-destructive measurement, although indirect, the ChlFI has numerous applications in the study of photosynthesis (see chapters in Papageorgiou and Govindjee, 2004), while its modelling is a straightforward way to verify various theories regarding different photosynthetic processes. Note that ChlFI in cyanobacteria is in part affected in different ways by the activity of the photosynthetic apparatus than in plants and algae, and this is due to their structural differences (see Stirbet et al., 2019), but its modelling is not described in this review.

Chlorophyll a fluorescence induction curves measured in leaves of 10-d-old barley (Hordeum vulgare L.) plants kept in darkness for 20 min before the measurement, shown on a logarithmic time scale (A), and on a linear time scale (B); a.u., arbitrary units. The O, J, I, P, S, M and T steps marked in the figure represent: O, the origin (minimum fluorescence F_O); J and I, intermediary fluorescence levels at 2 and 30 ms (F_J and F_I); P, the peak (F_P); S, a semi-steady state level; M, a maximum; and T, the terminal steady state. Measurements were made under continuous red (650 nm) light of 2500 μmol photons m –2 s –1 with a Plant Efficiency Analyser (Hansatech, UK). Modified from Stirbet et al. (2018).

The fast phase was labelled OIDP (Munday and Govindjee, 1969), as OI₁I₂P (Schreiber, 1986) and then OJIP (Strasser and Govindjee, 1991); the O-J-I-P curves are measured only under a high intensity of excitation light. At low light the J step is missing, so that only an O-I-P curve is observed (Strasser et al., 1995; Tomek et al., 2001). Below, we briefly discuss several models for the O-J-I-P fluorescence rise, as well as for the entire O-J-I-P-S-(M)-T transient or just the slow P-S-(M)-T phase (see also reviews by Lazár, 1999; Lazár and Schansker, 2009; Stirbet et al., 2014).

Modelling strategy, definition of the Chl fluorescence signal, and some selected partial models of PSII

Mathematical modelling is an essential part of modern biology and can have several purposes. In any experimental study, the measured data provide information about how the explored system works, and based on these, we formulate hypotheses about how the explored system is functioning. By converting the hypotheses into a mathematical model, running the model and comparing the calculated results with experimental data, we can judge if the model describes the data well or not. In this case, the structure of the model (i.e. the hypotheses as such) and also the values of model parameters can cause agreement/disagreement between the results obtained with the model and the measured data. Regarding the values of the model parameters, we can run the model with fixed parameter values, taken from the literature, or we can fit the values to get the best agreement between the model results and experimental data. However, in the latter case, we may find a perfect agreement, but only by using unrealistic values of the model parameters (based on the literature), which usually rules out the correctness of the model. On the other hand, when values of system variables are not known from the litrrature and/or are not directly accessible from experiments, the fitting can provide this information, assuming the model structure is correct.

Furthermore, a so-called metabolic control analysis (MCA) can be performed, which quantifies the extent to which a given process (hypothesis) affects a given result (for a review see Visser and Heijnen, 2002). Sometimes, this quantification can be made easy only by using modelling rather than by doing experiments, because it is not always possible to infer the desired (initial) state of the experimental system, or to experimentally modify the parameters of the system, as needed to perform MCA.

Finally, if we have a robust model that describes well the various measured data, we can modify the model parameters and track the results, or in other words, we can perform ‘experiments’ without measuring anything – i.e. biological experiments in silico. These in silico experiments are very useful in making predictions that allow us to determine the role of model parameters, or to design experiments to prove or refute certain predictions. Concerning the modelling of ChlFI discussed below, it is important to keep in mind that a qualitative agreement between experiment and theory is a useful goal. The ChlFI is a manifestation of a very complex biological system, and therefore describing it correctly and comprehensively is difficult – this is quite different from modelling technical systems, which can be described correctly, and where a quantitative agreement between experiments and theory is strictly required.

Several approaches have been used for the formulation of a fast ChlF rise model, or for the entire ChlFI. The variable ChlF is emitted mostly from PSII (reviewed by Krause and Weis, 1991; Dau, 1994; Govindjee, 1995; Lazár, 1999, 2006; Stirbet and Govindjee, 2011, 2012). The basic strategy for modelling the fast ChlF rise has been to first use a model of the ET reactions occurring only in PSII, but then later add ET reactions beyond PSII, especially for the modelling of the entire ChlFI. The formulation of a ChlFI model also depends on the specific ET components considered, and then, on the way, the variable ChlF emitted during the transient is defined. The basic approach in the definition of the variable ChlF is based on the early work of Duysens and Sweers (1963) and the quencher theory defined there, later identified to be due to Q_A (see above the subsection Evidence from Chl a fluorescence measurements). According to this theory, if Q_A is oxidized, ChlF is low and if Q_A is reduced, ChlF is high, and the variable ChlF is proportional to the fraction of Q_A − . Moreover, the energetic PSII connectivity (mentioned earlier) can be also considered in modelling the variable ChlF.

Taken together, the most basic approach used to model the fast ChlF rise has been to define a PSII model that describes the redox changes of Q_A during reduction of the PQ pool. These redox changes are modulated by Q_B, the second PQ electron acceptor of PSII, which unlike Q_A is a two-electron PQ acceptor of the PSII RC; originally, it came from the PQ pool, transiently binding to the Q_B-site. The reduction of Q_B to plastoquinol is described by the TEG model (Bouges-Bocquet, 1973; Velthuys and Amesz, 1974), which is the fundamental partial model used in ChlFI modelling (see discussion earlier, and Fig. 5 ). Thus, one group of models describing the fast ChlF rise, including the first ever models (see below the subsection Modelling the fast Chl fluorescence rise by using only models of PSII reactions), are based on the TEG model. The charge stabilization on Q_A (i.e. the reduction of Q_A by Pheo − ) means that the PSII RC is closed and thus the ChlF is high. However, this charge stabilization is preceded by the formation of P680 + Pheo − (see Fig. 2 ). Thus, when either P680 + and/or Pheo − are present, the PSII RC is closed, but the ChlF decreases in their presence, as both P680 + and Pheo − are quenchers of ChlF (for P680 + , see Okayama and Butler, 1972; Shinkarev and Govindjee, 1993; Steffen et al., 2001, 2005; for Pheo − , see Klimov et al., 1977). Quenching of Chl fluorescence by P680 + accumulation has been considered in several models of the fast ChlF rise (e.g. Lazár, 2003; Laisk and Oja, 2018). Accumulation of reduced Pheo was shown to take place only under illumination at 200–220 K (Klimov et al., 1980; Breton, 1982). Nonetheless, Vredenberg (2000, 2008, 2011) has assumed, in his O-J-I-P model, not only that Pheo ‒ accumulates at room temperature, but also that ChlF is higher when both Q_A and Pheo are reduced than when only Q_A is reduced. Strasser and Stirbet (2001) have also simulated and fitted a fast ChlF rise with a simple TEG-based model, but considering three different PSII redox states that contribute to the fluorescence signal: (1) with Q_A ‒ ; (2) with Pheo ‒ ; and (3) with PheoQ_A ‒ and Pheo ‒ Q_A ‒ ; ChlF in the presence of Pheo ‒ Q_A ‒ was considered to be two-fold larger than that when PheoQ_A ‒ was present. The experimental O-J-I-P curve was fitted quite well by all three models, but the parameters of the models and the kinetics of the PSII redox states were different in each case. Thus, overparametrized models cannot be validated by fitting one experimental curve, and other approaches must be also used to reach firm conclusions. These can be, for example, measurements of the kinetics of the redox states of PSII during the ChlF transient, as well as through in silico experiments, in which the basic parameters of the model are kept constant.

On the other hand, ChlF yield during ChlFI has also been defined by using ratios of the rate constants related to fluorescence emission, heat dissipation and photochemistry (Goltsev and Yordanov, 1997; Laisk et al., 2006; Ebenhöh et al., 2014; Stirbet and Govindjee, 2016). A better estimation of the ChlF signal, in models used to simulate the ChlFI, is obtained by considering fluorescence as a radiative deactivation of the singlet excited state of Chl (i.e. 1 Chl*); this was used in the modelling of the fast ChlF rise by Baake and Schlöder (1992) (see also Lebedeva et al., 2002; Lazár, 2003; Belyaeva, 2004). If the ChlF signal is defined by the redox states of Q_A or by the concentration of 1 Chl*, the model must include these entities. The reactions among the excited states of Chl a in PSII antenna that include P680 and Pheo, besides Q_A, have been described by the RRP model of Schatz et al. (1988); it was based on measurements of ChlF decay in the picosecond range after excitation by a short laser pulse. In the RRP model, charge separation between P680 and Pheo is reversible and is followed by charge stabilization (ET from Pheo − to Q_A) in the open PSII RCs, and by non-radiative charge recombination (to the ground state) in closed PSII RCs (see Fig. 2 ). Thus, the RRP model is the second fundamental partial model, in addition to the TEG model, which must be considered in modelling the ChlFI.

If the formation of P680 + is considered in a model, then the reduction of P680 + must be also included, i.e. reactions on the donor side of PSII, as well as the recombination reactions between P680 + and Pheo ‒ or Q_A ‒ . The P680 + is reduced by tyrosine 161 (i.e. Y_Z; Debus et al., 1988), which is, in turn, reduced by OEC. Electrons are donated to Y_Z + , by OEC, as it undergoes the S-state cycle (Kok et al., 1970; Fig. 3 ). Kok’s model of OEC is the third fundamental partial model for the description of PSII function. This model also includes parameters called ‘misses’ (when the light flash used does not lead to an S-state advancement) and ‘double hits’ (when the flash leads to an advancement by two S-states). Kok’s model has been modified by Jablonsky and Lazar (2008) by including the so-called intermediate S-states, which enable omission of the misses and double hits in the model.

Modelling of the fast Chl fluorescence rise measured after treatment with a herbicide

Because many photosynthetic processes affect ChlFI, herbicides that interrupt the ET from Q_A to Q_B have been used to simplify the observed curves. Note that many different herbicides are employed to kill weeds, and this can be achieved by using different substances that operate through various other mechanisms, but here we discuss only those that block the Q_B-pocket of PSII. DCMU (3-(3′,4′-dichlorophenyl)-1,1-dimethylurea) is a herbicide that has been frequently used in such studies; it binds to the Q_B-pocket, blocking ET beyond PSII (e.g. Oettmeier et al., 1980), which leads to a faster closure of PSII RCs during illumination and to a faster accumulation of Q_A ‒ . Binding of DCMU at the Q_B-pocket results in a faster sigmoidal ChlF rise to its maximal value (F_M), which is reached approximately at the J step (~2 ms) of the ChlF rise, measured (under saturating light) with an untreated sample. The gradual binding of DCMU to the Q_B-pocket of PSII, and thus the gradual closure of PSII, as reflected in changes in the O-J-I-P transient, was modelled by Lazár et al. (1998). Here, the diffusion of DCMU was described using Fick’s laws, and the reaction of DCMU at the Q_B-binding site of PSII, by second-order kinetics. From this work, Lazár et al. (1998) provided values of the diffusion coefficient of DCMU, and the second-order rate constant of DCMU binding to the Q_B-pocket of PSII.

The sigmoidal shape of the fast ChlF rise measured with DCMU has been suggested to reflect energetic connectivity (p) between the PSII units (Joliot and Joliot, 1964; also see above for discussion). This concept is tightly connected with a type of PSII heterogeneity, namely PSII α/β antenna heterogeneity (Melis and Homann, 1975). The PSIIα units, the main PSIIs, have a large and energetically connected light-harvesting antenna. The size of the antenna is reflected in the rate constant of the fast ChlF rise, measured with DCMU, and PSII connectivity is reflected in the value of the parameter p; the PSIIβ units have smaller antenna and a lower energetic connectivity. Several different procedures have been used to obtain quantitative information on this PSII heterogeneity (see Hsu et al., 1989). To increase the reliability and accuracy in the determination of PSII antenna heterogeneity, Lazár et al. (2001) have fitted the values of rate constants, the parameter p and the fractions of particular PSII types to several curves of fast ChlF rise in the presence of DCMU, measured at different light intensities, by using just one fitting procedure; results from this work were in good agreement with those in the literature.

The fast ChlF rise measured with DCMU has also been explored using the RRP model by Trissl et al. (1993), Lavergne and Trissl (1995), and Trissl and Lavergne (1995), with PSII energetic connectivity included. The RRP model has been further improved by Lazár and Pospíšil (1999) by the addition of P680 + reduction step(s) on the (electron) donor side of PSII; for this, they had used the fast ChlF rise in the presence of DCMU measured at high temperatures. Decreases in PSII energetic connectivity and in the rate of P680 + reduction by Y_Z were suggested to occur in the photosynthetic samples kept at high temperatures (e.g. 47 °C for 5 min; Guissé et al., 1995; Srivastava et al., 1997), but these conclusions were based on results on samples, without DCMU. By contrast, Lazár and Pospíšil (1999) have simulated the fast ChlF rise, in the presence of DCMU, at high temperatures by omitting PSII energetic connectivity, and by decreasing the rate constants related to the electron donation to P680 + .

To study photoinhibition in DCMU-treated samples, Vavilin et al. (1998) and Lazár et al. (2005) have simulated fast ChlF rise curves by using the RRP model. Lazár et al. (2005) further extended the RRP model by considering a possible protective function of Cyt b₅₅₉ against photoinhibition, as proposed by Thompson and Brudvig (1988) and by Nedbal et al. (1992). Cyt b₅₅₉ is indeed reduced by Pheo − , which then donates electrons to P680 + , involving a CEF around PSII. However, an argument against such an ET may be in the crystal structure of PSII (e.g. Zouni et al., 2001; Kamiya and Shen, 2003), which shows that the distance from the Pheo in the active D1 branch of PSII and the Cyt b₅₅₉ is too long (~45 Å) to allow an ET between them. However, the distance between Pheo in the inactive D2 branch of PSII and the Cyt b₅₅₉ is shorter (22 Å), and ET by tunnelling has been reported for such distances (Page et al., 1999). Thus, the Pheo in the model of Lazár et al. (2005) could be Pheo in the D2 branch of PSII.

Modelling the fast Chl fluorescence rise by using only models of PSII reactions

Mathematical analyses of the fast ChlF rise were published in the 1960s (Malkin and Kok, 1966; Malkin, 1966; Munday and Govindjee 1969). Munday and Govindjee (1969) measured the O-I-D-P (where D is for a dip) ChlF rise curve in Chlorella pyrenoidosa and related it successfully to variations in the fraction of reduced Q_A. In their paper, the dip was analysed by studying the transient oxidation of Q_A − by PSI.

In all likelihood, the first ‘real’ model of the fast ChlF rise [i.e. a scheme of ET reactions and a related set of coupled ordinary differential equations (ODEs)] was that of Holzapfel and Bauer (1975). This model was rather complex: it described the complete ET chain in the TM, including the formation of NADPH and ATP. On the other hand, some details of the photosynthetic ET were not included in the model, due to limited knowledge of the photosynthesis process at that time. In this model, the ChlF was assumed to be proportional to the amount of Q_A − . Holzapfel and Bauer (1975) were able to qualitatively simulate the rate of oxygen evolution at different light intensities, the fast ChlF rise of control samples, and of those treated with DCMU and/or 2,5-dibromo-3-methyl-6-isopropyl-p-benzoquinone (DBMIB, which blocks the electron flow between PQ and Cyt b₆/f; cf. Trebst and Reimer, 1973), as well as of samples that were dark-adapted under anaerobic conditions. This model was further used by Holzapfel (1978), where the effect of ΔΨ across the TM was included. It is unclear why these models were missed by others. However, several models on the fast O-I-P ChlF rise, measured using light intensities lower than 1200 µmol photons m −2 s −1 , are available (Renger and Schulze, 1985; Hsu, 1992a, b; Goltsev and Yordanov, 1997; Tomek et al., 2003); these models were based on the TEG model, where ChlF signal was assumed to be proportional to the amount of reduced Q_A (for an exception, see Goltsev and Yotdanov, 1997). Tomek et al. (2003) have further used the amplitude of the I step to estimate the fraction of ‘Q_B-non-reducing centres’ (i.e. PSIIs which cannot reduce Q_B).

Different TEG models, and PSII redox states with reduced Q_A to calculate the ChlF signal, were also used in modelling the O-J-I-P ChlF rise measured under saturating light (~3000 µmol photons m −2 s −1 ; Stirbet and Strasser, 1995, 1996; Lazár et al., 1997; Stirbet et al., 1998, 2001; Strasser and Stirbet, 2001; Tomek et al., 2001; Sušila et al., 2004). In these studies, the authors mainly showed how selected parameters of the models (e.g. initial concentrations and values of the rate constants) affect the shape of the O-J-I-P curves. However, Stirbet and Strasser (1996) showed that consideration of second-order kinetics for the reactions between Q_A and Q_B in the TEG model gives different simulated O-J-I-P curves compared to those obtained in the simulation where first-order kinetics is used. Strasser and Stirbet (1998) have also simulated O-J-I-P ChlF transients with a TEG model, by taking into account the heterogeneity of the PSII population in relation to PSII antenna, PSII energetic connectivity, and the ability of PSII to reduce Q_B (‘Q_B-reducing’ vs. ‘Q_B-non-reducing’ RCs).

Sušila et al. (2004) considered a hypothetical sample divided into ten layers of the same thickness, and calculated the light intensity in each layer, based on the Lambert–Beer attenuation law, in order to determine the light gradient inside the sample. They then simulated the fast ChlF rise curve for each layer, by using the same model as in Lazár et al. (1997) and Tomek et al. (2001), and summed the ChlF signal from all the layers to obtain the total ChlF signal. Their results showed that the light gradient inside a sample can significantly affect the shape of the fast ChlF transient. We note that in all the above models for the O-J-I-P ChlF rise, with the exception of those used by Stirbet et al. (1998, 2001) and Strasser and Stirbet (1998, 2001), the presence of an unknown component X that accepts electrons from the Q_B ‒ was assumed to exist.

Guo and Tan (2011) have extended the TEG model by taking in account the existence of a light-harvesting antenna system. Later, Feng et al. (2018) extended the above model by including the pH-dependent NPQ process, which allows the fitting of the decrease of the ChlF signal from the peak ‘P’ to ‘S’ and/or the ‘T’ level. To fit the O-J-I-P ChlF curves measured at different temperatures (20, 25, 30 °C), the rate constants in the model of Guo and Tan (2011) were assumed to be dependent on the temperature according to the Arrhenius law (Xia et al., 2018). Because the formation of 1 Chl* during illumination was included in the models used in all three studies above, the ChlF signal was defined as radiative deactivation of 1 Chl* in the PSII antenna.

In some of the models just mentioned, the function of the PSII donor side was implicitly included. By contrast, in the models of Stirbet et al. (1998, 2001), Chernev et al. (2006), Lazár and Jablonský (2009), and Laisk and Oja (2018), the function of the PSII donor side was included explicitly, and that too in combination with the TEG model. Stirbet et al. (1998, 2001) not only included the S-states of OEC, but also the PSII energetic connectivity, and the quenching of the ChlF signal by P680 + and by the oxidized PQ molecules. Stirbet et al. (1998, 2001) then simulated (or fitted) the O-J-I-P ChlF transient by defining the ChlF signal to be proportional to the amount of reduced Q_A, and by considering different initial fractions of Q_B and Q_B ‒ , or of the S₁ and S₀ states of OEC. In the model of Lazár and Jablonský (2009), all the S-state transitions of OEC were taken into account, as well as the redox states of P680 + that were explicitly considered in combination with the TEG model, which was then used for simulation of the O-J-I-P ChlF transient. In their study, the effect on the simulated fast ChlF curve was described by using (1) first- or second-order reaction kinetics for electron donation from the OEC to P680 + ; (2) one second-order reaction or two subsequent reactions for the Q_B 2‒ /PQ exchange; and (3) all possible reactions between the ET components, or of fewer ‘logical’ reactions.

Other models used for simulation of the fast ChlF rise are those that include, in addition to the TEG model, the description of the fast events in the PSII RC (i.e. charge separation, recombination and stabilization) described by the RRP model. Models by Baake and Schlőder (1992) and Belyaeva et al. (2011) belong to this group, where reduction of P680 + by Y_Z (via OEC) was implicitly included. Other authors (Lazár, 2003; Zhu et al., 2005; Matsuoka et al., 2015) have explicitly included Y_Z and the S-state transitions of OEC.

Lazár (2003) provided a detailed analysis of how values of particular rate constants and initial conditions affect the simulated fast O-J-I-P ChlF curves. An important aspect of the ChlFI curves analysed by simulations in this work is the origin of the minimal ChlF level (F_O) which is the initial ChlF, when all PSII RCs have all Q_A in the oxidized state; F_O originates from the radiative deactivation of the excited PSII state [(antenna-P680)*PheoQ_AQ_B; see Fig. 7 ]. Interestingly, although the model of Lazár (2003) is one of the most detailed models of PSII reactions (consisting of a set of 44 coupled ODEs), yet it was not able to simulate typical O-J-I-P ChlF transients, as the ChlF signal increased from the J step to a maximum, which was reached at the I step position in the experimental curves ( Fig. 7 ).

Simulations of the O-J-I(=P) ChlF rise (see text) and of the model forms of photosystem (PS) II in the excited state, which mainly contribute to the (chlorophyll a) fluorescence transient, are shown on a logarithmic time scale. Abbreviations: (L-P)*, the excited state of the PSII antenna, which is equilibrated among all light harvesting Chls, including P680; Ph, pheophytin; A and B, the first and second plastoquinone acceptors of PSII (Q_A and Q_B). The time course of the PSII model form (L-P)*PhAB at the beginning of the transient, which represents excited open PSII RCs (i.e. with oxidized Q_A), is at the origin of the minimal ChlF, F_O. Modified from Lazár (2003).

The inability to simulate the proper time-dependence of the ChlF signal by the detailed model based only on PSII redox states is one of the arguments that a proper model for the O-J-I-P ChlF rise should also describe ET reactions occurring beyond the PQ pool, as already inferred by Munday and Govindjee (1969) and later confirmed in other studies (i.e. Schreiber et al., 1989; Schansker et al., 2003, 2005).

Modelling the fast Chl a fluorescence rise with models that consider electron transport in and around the TM

The last group of models used in simulation of the O-J-I-P ChlF transients are those that include ET reactions occurring in and around the TM (Lebedeva et al., 2002; Kroon and Thoms, 2006; Lazár, 2009; Makarov et al., 2012; Belyaeva et al., 2016, 2019), or even the metabolic reactions in the stroma (e.g. the Calvin–Benson cycle; see Laisk et al., 2006, 2009; Zhu et al., 2013). Given the all-inclusive nature of these models, some of them were also used for modelling of the the entire ChlFI (see below). A diagram of the reactions considered in the model proposed by Lazár (2009) is shown in Fig. 8 . This model consists of a set of 42 coupled ODEs, and the ChlF emission is defined as being proportional to the amount of reduced Q_A. In addition, the ΔA₈₂₀ signal, describing redox changes of P700 and plastocyanin (PC), was also modelled. To show that the ET reactions beyond the PQ pool affect the shape of the simulated fast ChlF transients, Lazár (2009) also analysed in silico the effects of DBMIB and MV [methylviologen, which accepts electrons from both the iron–sulfur cluster of PSI and ferredoxin (Fd); Sétif, 2015]. The shapes of the simulated fast ChlF transients and of ΔA₈₂₀ signal were qualitatively in agreement with the experimental curves (see Fig. 8 ). This model is also a part of e-photosynthesis.org (Šafránek et al., 2011), which is a web-based platform for modelling complex photosynthetic processes.

Diagram of the ET reactions used in the model of Lazár (2009) (A), the O-J-I-P ChlF transients measured on control (= untreated) leaves, as well as on leaves treated with 2,5-dibromo-3-methyl-6-isopropyl-p-benzoquinone (DBMIB, which inhibits ET in the cytochrome b₆/f, see A) or with methyl-viologen (MV, which accepts electrons from the iron–sulfur cluster of PSI and ferredoxin, Fd, see A) (B), and the respective curves simulated with the model (C), the ΔA₈₂₀ curves measured under the same conditions (D), and the respective curves simulated by the model (E). The curves are shown on a logarithmic time scale. Abbreviations: S_i, the S-states of the oxygen-evolving complex (OEC); f, b_L, b_H … c, cytochrome f, low/potential cytochrome b₆, and high-potential cytochrome b₆ in kinetic equilibrium with the haem c of cytochrome b₆/f complex; PC, plastocyanin. Modified from Lazár (2009).

In all the models mentioned above, the variable ChlF signal was assumed to originate from the PSII antenna. The problem with direct measurement of the variable ChlF from PSI in vivo (not from isolated PSI complexes) is that it overlaps spectrally with the PSII ChlF. However, some experimental results, presented in the literature (see Lazár, 2013), show the existence of a variable ChlF originating in PSI, at least under certain conditions. Lazár (2013) presented a very detailed model of the ET reactions in PSI (i.e. a set of 106 coupled ODEs), and simulated fast ChlF transients originating only from PSI. The ChlF signal was defined as the radiative deactivation of 1 Chl*. PSI was further shown to emit variable ChlF, and its contribution to the total maximal variable ChlF signal from the two PSI and PSII was ~8–17 % (Lazár, 2013). Future studies are needed to quantitatively assess these findings.

Rule-based modelling of the fast Chl fluorescence rise

All the models of the fast ChlF rise discussed thus far have described the photosynthetic processes by using sets of coupled ODEs. Each ODE was used to describe the time-change of a particular PSII redox form (i.e. state variable) of the model. This approach is deterministic, because in any run of the model, the same solution is obtained.

If too many state variables (coupled ODEs) are considered in a model, it becomes difficult to obtain model results, due to high requirements of computational time and hardware; this is because all ODEs must be solved simultaneously at each time of system evolution. While there are ways (specific for each model) to decrease the number of equations, this problem can be better overcome by employing a rule-based modelling approach, where rules are defined that are equivalent to the particular ET reactions. Furthermore, random numbers are generated, and these determine (using internal decision process) which rules should be considered in each particular step of the model run, i.e. in each ‘evolvement’ of the system in time. Thus, a time course of the system behaviour would be described by a sequence of particular rules, which are slightly different in different model runs, i.e. small differences between solutions are obtained after different runs of the model. Thus, this approach would be stochastic (i.e. random). The rule-based stochastic approach by means of kinetic Monte Carlo simulations has been applied for modelling of the O-J-I-P ChlF transient by Xin et al. (2013), Guo and Tan (2014), Maslakov et al. (2016) and Antal et al. (2018). However, in all these cases, the shapes of simulated ChlFI curves were the same (except for the noise) as when using the deterministic approach. Similarly, the O-J-I-P curve was also simulated using stochastic π-calculus (Tokarčík, 2012) and rule-based language-simplified Kappa (Nižnan, 2014). Much further work is needed to obtain conclusive results from this approach.

Modelling the slow PS(M)T phase of the Chl a fluorescence induction curve

The nomenclature of P-S-(M)-T for the slow phase of the ChlFI was first used by Papageorgiou and Govindjee (1968a, b). Compared with the fast ChlF rise, this phase is much more complex and less well understood, as the fluorescence yield is modulated by an increasing number of processes triggered during this phase, besides the photochemical quenching by Q_A (see above), such as: (1) the NPQ of excited singlet 1 Chl* a in PSII antenna, induced by low pH in the lumen (i.e. the high-energy NPQ qE; Horton et al., 1996; Rochaix, 2014); (2) state transitions (i.e. qT₁₂ or qT₂₁) that regulate the absorption CS of PSI and PSII (with ‘state 1’ being more fluorescent than ‘state 2’; see Papageorgiou and Govindjee, 2011, 2014); (3) photoinactivation processes (qI) due to the photodamage of PSII (e.g. Tyystjärvi, 2013); (4) cyclic electron flow around PSI (e.g. Miyake, 2010; Buchert et al., 2018), chlororespiration (Bennoun, 1982) and electron flow to molecular oxygen (Mehler, 1951; Asada, 1999); as well as (5) activation of the Calvin–Benson cycle. Therefore, besides the partial models necessary for modelling the fast ChlF rise discussed in the previous section (e.g. RRP, Kok’s oxygen clock, TEG, the Q-cycle at the Cyt b₆/f complex), the processes listed above are fundamental for modelling the whole ChlFI; however, qT and qI, with a few exceptions, have been usually neglected by most authors.

Laisk et al. (1997) were the first to model the qE process, which they used later to simulate successfully the slow PS(M)T phase of ChlFI (Laisk et al., 2006). This qE model was later adapted by Zhu et al. (2013) for C₃ photosynthesis, but the descending M-T phase is missing in their simulated ChlFI curve. Note that these two papers were centred on the detailed description of metabolic reactions.

The transmembrane pmf, i.e. both ΔpH and ΔΨ, was modelled by Lebedeva et al. (2002), which predicts that a sufficiently large transmembrane electric potential (positive inside) would slow the rate of PQH₂ oxidation by the Cyt b₆/f (the so-called backpressure effect; see van Kooten et al., 1986), and consequently the ET rate from PSII to PSI (see also comments in Stirbet et al., 2014). This pmf model was further used by, for example, Rubin et al. (2009) and Belyaeva et al. (2016, 2019) to model the complete ChlFI curve, with a TM model that describes the electron/proton transfer reactions between membrane protein complexes: PSII, PSI, Cyt b₆/f, mobile PQ pool in the TM, PC in lumen and Fd in stroma, CEF-PSI, and reduction of NADP + via Fd-NADP + -oxidoreductase (FNR) (see Fig. 1 ). Belyaeva et al. (2016) used the TM model to fit both ChlFI data and P700 redox changes (ΔA₈₁₀), measured in pea leaves, from 20 μs to 20 s. Belyaeva et al. (2019) added to their earlier TM model partial models for the light-induced activation of FNR and qE, with the goal to simulate the ChlFI and ΔA₈₁₀ kinetics on the time scale from 40 μs to 30 s. Their results showed that the time-dependent rate constants changed substantially upon the release of ET on the (electron) acceptor side of PSI and during qE induction. Belyaeva et al. (2019) also discussed differences between the parameters related to FNR activation and qE induction evaluated for dark-adapted and pre-illuminated pea leaves, and also analysed the transition between CEF-PSI and LEF modes.

Because the photosynthetic organisms are exposed continuously to fluctuations in the environmental conditions, the activity of their photosynthetic apparatus is dynamic, being feedback-regulated by several processes that reduce imbalances between the rate of energy trapping by the PSs and CO₂ assimilation. These serve to optimize the photosynthetic ET to, for example, light-induced pH changes in the lumen and in the stroma (see Tikhonov, 2013; Rochaix, 2014; Strand and Kramer, 2014), or changes in the PQ pool redox state, as modulated by variations in light irradiance, ATP/ADP ratio and the ambient CO₂ level (Rochaix, 2014, 2016). Light-induced acidification of the lumen slows down PQH₂ oxidation by the Cyt b₆/f (the backpressure effect), and also decreases PSII activity by inducing excitonic energy dissipation as heat in PSII antenna through qE (Jahns and Holzwarth, 2012; Rochaix, 2014, 2016). This reduces the excess of input energy in the system, and thus oxidative damage (Nishiyama et al., 2006), which occurs when singlet excited 1 Chl* forms triplet-state Chl ( 3 Chl) (Durrant et al., 1990) that interacts with ground state oxygen, generating ‘noxious’ reactive oxygen species (ROS), such as singlet oxygen ( 1 O₂). Furthermore, the alkalization of stroma activates the Calvin–Benson cycle, which stimulates the consumption of NADPH and ATP (Werdan et al., 1975; Noctor and Foyer, 2000). As shown earlier, state transitions re-equilibrate PSI and PSII activities through changes in their absorption CS, which are triggered by PQ pool redox state changes (for plants and algae, see reviews by Rochaix, 2014, 2016; Goldschmidt-Clermont and Bassi, 2015), and involve phosphorylation/dephosphorylation of the PSII mobile antenna by kinases and phosphatases (i.e. STN7/TAP38 in Arabidopsis thaliana, or Stt7/Pph1 in Chlamydomonas reinhardtii; Rochaix et al., 2012). Furthermore, during induction of the Calvin–Benson cycle, changes in illumination, or anaerobiosis, photosynthetic electron fluxes are optimally redistributed between the linear electron transport (LET) from water to NADP + , and alternative electron pathways, i.e. cyclic electron flows, pseudocyclic O₂-dependent electron flows and the malate valve (Backhausen et al., 2000; Miyake, 2010; Hemschemeier and Happe, 2011).

Modelling the state transition process

Ebenhöh et al. (2014) were the first to model state transitions in plants and algae based on a mechanism, described by Allen et al. (1981); they investigated the dynamics and regulation of state transitions by simulating experimental PAM-SP curves from Chlamydomonas reinhardtii cells, grown under dim light, and thus with little capacity for qE, having a low LHCSR3 content (Peers et al., 2009). Here, a simplified mathematical model (based on eight coupled ODEs) was used, where the most relevant ET routes, necessary for modelling state transitions in this green alga, were used: LEF, CEF-PSI, and chlororespiration through the plastid terminal oxidase PTOX (see Fig. 1 ; and Bennoun, 1982; McDonald et al., 2011). Individual reactions of the Calvin–Benson cycle were treated implicitly, using steady-state consumption of NADPH and ATP, and a quasi-steady state approximation for the dynamics of oxygen evolution and charge separation in PSII. For simplicity, in the partial model of state transitions, it was assumed that the PSII mobile antennas phosphorylated by the kinase Stt7 (activated by the PQ pool reduction) are relocated directly to PSI (i.e. state 1 to state 2 transition, qT₁₂); also, after the Stt7 inhibition (triggered by the PQ pool oxidation), the PSII mobile antennas are dephosphorylated by the phosphatase Pph1, and directly re-associate with PSII (i.e. state 2 to state 1 transition, qT₂₁) (see Fig. 4 ). Finally, the ChlF signal is defined by the ratios of rate constants related to fluorescence emission, heat dissipation and photochemistry, which include changes in the absorption CS of PSI and PSII (due to state transition). Ebenhöh et al. (2014) successfully simulated with their model the main features of the experimental fluorescence signal measured with a PAM instrument from dark-adapted wild-type Chlamydomonas cells illuminated for 10 min with low light (100 μmol photons m −2 s −1 ). The saturating F_M′ peaks during illumination reflect changes in the antenna CS of PSII (i.e. a partial state transition to ‘state 2’), which take place in parallel with the establishment of a stationary redox poise of the PQ pool.

State transitions were also modelled by Stirbet and Govindjee (2016), with the goal to simulate the slow PS(M)T phase of the ChlFI, in order to determine the origin of the S–M rise of Chlamydomonas reinhardtii cells (see Kodru et al., 2015; Zhou et al., 2015). Here, the photosynthesis model of Ebenhöh et al. (2014) was adapted for the simulation of ChlF data obtained by using a Plant Efficiency Analyser (PEA; Hansatech, UK). Stirbet and Govindjee (2016) confirmed that, under anaerobic conditions, in darkness, the PQ pool reduction through chlororespiration triggers a state 1 to state 2 transition (see Fig. 9A ), when the relative CS of PSII (CSII) is lower than that of PSI (see Bulté et al., 1990). Next, it was shown that, during the subsequent illumination, the hypothetical sample undergoes a transition from this ‘state 2’ to a ‘state 1’, which is the origin of the slow S-M fluorescence rise (see Fig. 9B ). However, if the dark-adaptation period is too short, and the transition to ‘state 2’ in darkness is not complete, the subsequent illumination triggers a state 1 to state 2 transition (see Fig. 9C ). We note, however, that the M-T fluorescence decline observed experimentally (Kodru et al., 2015; see also Fig. 6B ) is missing in the simulated curves, and, thus, further research is needed to determine its origin.

Simulated kinetics of the fraction of the oxidized plastoquinone (PQ) pool (i.e. PQ/PQ_tot) and the relative absorption cross-section of photosystem (PS) II (i.e. CSII) during dark adaptation under anoxic conditions of a hypothetical sample of Chlamydomonas reinhardtii cells (see A), as well as simulated time courses of PQ/PQ_tot, CSII and Chl fluorescence induction (F) during illumination in the presence of oxygen of the hypothetical sample after 600 s (see B) and 200 s (see C) anoxic dark adaptation. Note that a decrease in CSII reflects a state 1 to state 2 transition, while an increase reflects a state 2 to state 1 transition. Modified from Stirbet and Govindjee (2016).

Stirbet and Govindjee (2016) also examined in silico the influence of different factors on the amplitude of the S-M fluorescence rise under low light conditions (~100 to 300 μmol photons m −2 s −1 ). For example, they found that, under conditions that trigger a qT₂₁ during a dark-to-light transition (i.e. reduced PQ pool, and CSII < 0.5 at the beginning of illumination), an increase in the CEF-PSI rate leads to a lower CSII increase at the end of the state transition, and a smaller amplitude of the S-M fluorescence rise (see Fig. 10A ). This simulation also confirmed that, when the CEF-PSI is much more rapid, the ATP level increases, while the NADPH level decreases. When the light intensity is higher, the simulations also showed a decrease in the S-M fluorescence rise. This result is in agreement with the experimental ChlFI data on Chlorella published by Papageorgiou and Govindjee (1968a), who showed that the slow S-M fluorescence rise is larger at lower exciting light intensities. By contrast, under other conditions taken into account by Stirbet and Govindjee (2016), such as the increase in NADPH and ATP consumption by the Calvin–Benson cycle, or an increase in the rate of the Mehler reaction, the S-M amplitude increased, due to a larger increase in the PSII CS during the qT₂₁ (see Fig. 10B ). However, the increase in the S-M rise becomes saturated by further increasing these rate constants. The conclusion is that the factors reducing the PQ pool (e.g. higher light intensity, or more rapid CEF-PSI) decrease the S-M amplitude, and those that oxidize further the PQ pool (e.g. more rapid NADPH consumption or Mehler reaction) increase the S-M amplitude.

Simulated kinetics of the fraction of the oxidized plastoquinone (PQ) pool (i.e. PQ/PQ_tot), the relative absorption cross-section of photosystem (PS) II (i.e. CSII), and of chlorophyll (Chl) fluorescence induction (F) during illumination in the presence of oxygen of a hypothetical sample of Chlamydomonas reinharditii cells dark-adapted for 600 s under anoxic conditions, by considering that: (1) the illumination is equivalent to 100 μmol photons m −2 s −1 , and the rate constant of the cyclic electron flow (CEF) around PSI is k_Cyc = 1 or 5 s −1 (see A); and (2) the illumination is equivalent to 300 μmol photons m −2 s −1 , the rate constant of CEF-PSI is 1 s −1 , and that of the Mehler reaction (i.e. ET from ferredoxin to O₂) k_O2 = 0 or 11 s −1 (see B). Note that an increase in CSII reflects a state 2 to state 1 transition. These simulations show that the S-M fluorescence rise decreases when light intensity increases or when CEF-PSI is faster, but increases when the Mehler reaction is also functioning. Modified from Stirbet and Govindjee (2016).

Modelling the qE component of NPQ

Because NPQ in plants and algae is associated with LHCs of PSII (see Horton et al., 1996; Tian et al., 2017), models simulating qE usually include reactions around PSII, and focus on describing the ChlFI (see reviews by Zaks et al., 2013; Matuszyńska and Ebenhöh, 2015). Different photosynthesis models have been used to simulate either ChlFI curves measured with instruments using direct light (e.g. PEA), or with PAM-SP fluorometers (for a review see Stirbet et al., (2014). But, of course, the main phenomenon under analysis with either of these instruments is the same. Besides measurements of ChlF lifetime (e.g. Gilmore et al., 1995, 1998; Sylak-Glassman et al., 2016), measurements of Chl fluorescence yield with PAM-SP fluorometers are especially suitable for the study of NPQ processes (Müller et al., 2001). It is clear that models that simulate experimental PAM data are valuable tools to analyse the qE component of NPQ.

Several original qE models have been proposed by, for example, Ebenhöh et al. (2011) and Zaks et al. (2012); these have been used for the simulation of the dynamics of ChlF quenching, as measured by PAM-SP instruments (see review by Stirbet et al., 2014). Now, photosynthesis models that include qE are available (Ebenhöh et al., 2014; Matuszyńska et al., 2016, 2019; Snellenburg et al., 2017; Morales et al., 2018a, b).

The qE model of Ebenhöh et al. (2014) takes into account the induction of qE by low pH in the lumen (see above), but it is based on the simplifying assumption that the xanthophyll cycle is the only component involved in qE-dependent quenching. Thus, it was assumed that the decrease in lumenal pH leads directly to the formation of ‘Z’ through the xanthophyll cycle (i.e. the de-epoxidation of V via A), which then acts as a fluorescence quencher in the PSII antenna; the quencher acts by increasing the rate constant of the non-radiative deactivation of the 1 Chl*. Furthermore, the qE is reversed in darkness as Z is epoxidized to V by an active epoxidase. The results of the simulations, obtained with this qE model, showed that high light illumination leads to a plateau of the PQ pool redox state, which is relatively constant for a range of CSII values. Based on these theoretical results, Ebenhöh et al. (2014) concluded that, due to qE induction, the requirement to adjust the antenna CS through state transition under high light is much lower than under low light conditions. Indeed. Allorent et al. (2013) showed that the phosphorylation of LHCII antenna, mainly mediated by the STN7/Stt7 kinase in low light, is inhibited by high light, due either to a negative regulation of the kinase through the thioredoxin pathway under high light (see e.g. Lemeille and Rochaix, 2010), or to a conformational change in the PSII antenna (Vink et al., 2004).

To avoid the harmful effects of over-excitation, plants optimize their photosynthetic performance based on their illumination history through a process in which Z seems to play a key role (e.g. Ruban et al., 2012). Matuszyńska et al. (2016) used a combined experimental and theoretical approach in the study of qE, particularly designed to determine if plants have a ‘memory’ of their recent (minutes to hours) light exposure, similar to what occurs after really long (days, months) periods of stress (Demmig et al., 1987; Adams and Demmig-Adams, 2004). In these studies, fluorescence measurements were made on Epipremnum aureum (a shadow (shade)-tolerant, ornamental plant) by PAM-SP. Here, F_M′ was used instead of NPQ, as suggested by Holzwarth et al. (2013), to avoid mathematical distortion of the ChlF quenching kinetics. Additionally, the pigment composition was measured at the end of each phase of the experiment, in order to determine the contribution of Z to the ‘memory’ effect. These data confirmed the presence of a short-term ‘memory’ effect, which is influenced by both light intensity and the period of dark-relaxation between two light exposures. Matuszyńska et al. (2016) concluded that the ‘memory’ of recent light exposure related to qE can be assigned to dynamic changes in pigment composition, being due to a slower conversion of Z to V, as observed by, for example, Demmig et al. (1987) and Reinhold et al. (2008). By implementing a qE model based on the ‘4 state-2-site quenching’ system (Holzwarth and Jahns, 2014) in the photosynthesis model of Ebenhöh et al. (2014) (but without state transitions), Matuszyńska et al. (2016) were able to simulate successfully changes in the quantum yield of ChlF during the PAM-SP experiments, discussed above. In these simulations, the ChlF signal was also calculated using ratios of rate constants related to fluorescence emission, heat dissipation and photochemistry, where the rate constant of the heat dissipation was assumed to be modulated by the concentration of a quencher (Q), which was, in turn, calculated by taking into account the concentrations at any time of both Z and the protonated PsbS protein. [Note that Snellenburg et al. (2017) and Morales et al. (2018a, b) have used similar qE models, depending on the relative concentrations of Z and protonated PsbS.]

Modelling alternative electron flows

Besides LEF, which provides the Calvin–Benson cycle with NADPH and ATP, other ET routes function during oxygenic photosynthesis (see Fig. 1 ; Alric and Johnson, 2017; Shikanai and Yamamoto, 2017): (1) CEF-PSI via ferredoxin-plastoquinone reductase, or NADPH dehydrogenase (NDH); and (2) ‘alternative’ non-cyclic pathways that involve reduction of electron acceptors such as O₂ [the water–water cycle (WWC); see a model by Valero et al. (2009)], or oxaloacetate [by malate dehydrogenase (MDH); see a model by Fridlyand et al. (1998)]. The main role of CEF-PSI is to increase the ATP/NADPH ratio, as ‘required’ by the metabolic reactions in stroma or other energy-dependent processes in the chloroplast; furthermore, the pH difference, which induces qE, protects PSI and PSII against photoinhibition (Strand et al, 2016, 2017). The electron pathway to molecular oxygen (Mehler reaction, WWC), besides contributing to the acidification of the lumen and to the reduction of the excitation pressure on PSs, is also important in chloroplast redox signalling during abiotic stress, and in the regulation of CEF-PSI (Miyake, 2010). The respective contributions of alternative electron pathways to the total ET is strictly regulated, depending on environmental conditions, but further research is needed to understand how these diverse pathways and their regulatory mechanisms function (see Yamori et al., 2016; Nawrocki et al., 2019).

Comprehensive dynamic C₃ photosynthesis models, such as those by Laisk et al. (2006, 2009) and Zhu et al. (2013), include light reactions, proton and electron transport, detailed carbon metabolism reactions, exchange of intermediates between cytosol and stroma, photorespiration, amino acid synthesis, and regulatory mechanisms. However, because these models involve a large number of model parameters, simplified photosynthesis models are much more suitable, and practical, for the study of dynamic responses of the photosynthetic apparatus to diverse changes of environmental factors. Indeed, a number of simplified photosynthesis models have been used in several studies to analyse PETC regulation in silico, through simulation of experimental data measured with a variety of methods (Ebenhöh et al., 2011; Zaks et al., 2013; Tikhonov and Vershubskii, 2014; Stirbet and Govindjee, 2016; Snellenburg et al., 2017; Morales et al., 2018a, b; Matuszyńska et al., 2016, 2019). According to Morales et al. (2018b), the term ‘regulation’ means: reaching simultaneously, during environmental fluctuations, a suitable redox state of PETC, dissipation of excess excitation energy and ATP/NADPH ratio through adjustments of NPQ processes, CEF-PSI and reduction of alternative electron acceptors (also including the reduction of NO₂ ‒ during NH₄ + assimilation, NiR), as well as pmf optimization through changes in ATP synthase activity.

We have reviewed above results obtained in studies of photosynthesis regulation through state transitions and qE, based on simulations of ChlFI data. By contrast, Morales et al. (2018b) used, for simulations, several types of experimental data on Arabidopsis thaliana, such as PAM-SP ChlF data (for effective quantum yield of PSII and NPQ), ΔA₈₂₀ (for the P700 redox state, which is related to LET and alternative ET pathways), the electrochromic shift in A₅₂₀ (for pmf and its components), and net CO₂ assimilation (A_n, for the Calvin–Benson cycle and CO₂ diffusion). The results of these simulations have shown that CEF-PSI and alternative ET pathways are strongly interacting, and, thus, changes in FQR- or NDH-dependent CEF-PSI kinetics indirectly influence WWC, NiR and MDH activities, due to changes in the redox state of Fd. It is also known that the steady-state pH in the lumen cannot be controlled only by CEF-PSI and alternative ET, because it is also greatly affected by the pH sensitivity of qE, Cyt b₆/f and ATP synthase. Additionally, Morales et al. (2018b) have examined the influence of the ADP/ATP ratio in stroma on the metabolic regulation of ATP synthesis, and their simulations showed that there is a coordination between the regulation of Rubisco, NPQ and PETC over a large range of light intensities and CO₂ concentrations. These are important observations for programming plants for better productivity.

MODELLING THE REGULATORY DEPENDENCE BETWEEN THE LIGHT REACTIONS AND THE CARBON REACTIONS

The slow part of the ChlFI induction also reflects changes due to the induction of the Calvin–Benson cycle during a dark to light transition. The activation and gradual increase in CO₂ assimilation during this phase leads to a parallel activation of ATP synthesis and an increase in the rate of LEF, which decreases the initial excitation pressure. As a result: (1) the level of Q_A reduction decreases and photochemical quenching increases; and (2) qE decreases, because, due to a faster synthesis of ATP, the ΔpH decreases. Therefore, only models that include the induction of the Calvin–Benson cycle are suitable for correctly modelling the slow part of the ChlFI induction (see Laisk et al., 2006, 2009; Zhu et al., 2013).

The Calvin–Benson cycle is one of the best-studied plant metabolic processes. Besides photosynthesis models, which include both the light and carbon reactions (e.g. Laisk et al., 2006, 2009; Zhu et al., 2013; Belassio, 2019; Matuszyńska et al., 2019), the carbon assimilation was often modelled separately, by considering a simplified relationship for NADPH and ATP supply (see review by Jablonsky et al., 2011). In these models, carbon metabolism was analysed either by taking into account the kinetic properties of the enzymes involved, i.e. dynamic modelling (Pettersson and Ryde-Pettersson, 1988; Zhu et al., 2007), or without the need to use these, i.e. stoichiometric modelling (Boyle and Morgan, 2009). In addition, a combination of both the above approaches has also been used (Fleming et al., 2010). In many models for the Calvin–Benson cycle, the steady-state behaviour of the photosynthetic apparatus has been analysed based on the equations of Farquhar et al. (1980). Here we briefly mention some recent results on (short-term) regulation of photosynthesis obtained with the photosynthesis models of Morales et al. (2018a), Belassio (2019) and Matuszyńska et al. (2019).

Fluctuating irradiances, which were shown to limit the performance of photosynthesis (Pearcy, 1990), can be due to transient sun exposure, penumbra effects, shading by clouds, gaps in the canopy that produce ‘light (sun) flecks’, or movements of the leaves in the wind. Morales et al. (2018a) used a simplified dynamic model of CO₂ assimilation in a leaf to analyse the effects of fluctuating irradiances. In this study, they extended the canonical steady-state model by adding original empirical (phenomenological) partial models for the effects of chloroplast movement (qM; Dall’Osto et al., 2014; Baránková et al., 2016; Semer et al., 2018, 2019), qE, qI, regulation of enzyme activity in the Calvin–Benson cycle, metabolite concentrations, and the dynamic CO₂ diffusion through different leaf compartments. Changes in qE were assumed to follow PsbS protonation and Z generation, as was the case with the approach used by Matuszyńska et al. (2016). With their model, Morales et al. (2018a) analysed potential improvements in CO₂ assimilation that may result after removing the kinetic limitation of different regulatory processes. Their simulations predicted that the most limiting steps in the carbon reactions are the activation rates of the Calvin–Benson cycle enzymes and stomatal opening (up to 17 % improvement), followed by the rate of qE relaxation and chloroplast movement (up to 10 % improvement), depending on the frequency of light fluctuations. However, up to 32 % improvement in CO₂ assimilation has been predicted, when all the kinetic limitations were simultaneously removed. Belassio (2019) has presented a dynamic photosynthesis model which also includes both light and carbon reactions, coupled to a mechanistic hydro-mechanical partial model for stomatal behaviour. This model successfully simulates responses to rapid changes in light intensity (light flecks), as well as in atmospheric CO₂ and O₂ concentrations. This model is freely available (as a supplement to the paper), and runs as a stand-alone workbook in Microsoft Excel.

Finally, Matuszyńska et al. (2019) have proposed a dynamic photosynthetic model describing the light reactions and the Calvin–Benson cycle in C₃ plants, for which they have used their earlier models [for light reactions: Ebenhöh et al. (2014) and Matuszyńska et al. (2016); for carbon reactions: Pettersson and Ryde-Pettersson (1988) and Poolman et al. (2000)]. This newly merged model is based on nine coupled ODEs for the PETC, and 15 coupled ODEs for the carbon reactions. Analysis of this model shows the need for a ‘stand-by’ mode of the Calvin–Benson cycle in darkness, so that it can be restarted after prolonged dark periods; in this sense, the oxidative pentose phosphate pathway can play this function. Matuszyńska et al. (2019) have also used MCA (e.g. Visser and Heijnen, 2002) and metabolic supply–demand analysis (Hofmeyr and Cornish-Bowden, 2000) to investigate the regulatory dependence between the PETC and the Calvin–Benson cycle, and to quantify the ‘control distribution’ of supply and demand under different light conditions and CO₂ assimilation rates. Th results obtained with MCA have indicated that, when CO₂ is saturating, the demand reactions control the flux under light-saturating conditions (with seduheptulose-1,7-bisphosphatase maintaining the highest overall flux control; see Poolman et al., 2000), while the supply reactions display higher overall flux control under light-limited conditions, with PSII and PSI activities sustaining the highest overall flux control.

CONCLUSIONS

In this review, we have shown the important role played by models in deciphering and untangling different less well-understood and complex processes of photosynthesis, emphasizing the necessity and importance of modelling in the analysis of hypotheses developed from experimental studies. One major example, used in this review, is the ChlFI, which is simultaneously influenced by various photosynthetic processes affecting different segments of the fluorescence transient. As shown here, this process has been simulated by many modellers, who were focused either on understanding the dynamics of the redox states of different PETC components (see also reviews by Lazár, 1999; Lazár and Schansker, 2009; Stirbet et al., 2014), or that of more complex, regulatory mechanisms involved in processes such as state transitions and qE, or of the relative contributions of alternative ET pathways, as well as their relationship with the CO₂ assimilation (the Calvin–Benson cycle) (see also Stirbet et al., 2014). From the examples discussed in this review, it is evident that correctly simplified but complete dynamic models of photosynthesis are well suited to obtaining information about how the photosynthetic organisms cope with variable environmental conditions (see also Matuszyńska and Ebenhöh, 2015). Indeed, modelling is a very efficient method to identify important morphological and physiological parameters of a biological system and to find their optimal values. In addition, by using a larger variety of experimental data to verify such models, the simulations can lead to much more meaningful information about the organizational principles of the photosynthetic apparatus, which can also reveal original ways and means to improve the photosynthetic efficiency of plant crops (Zhu et al., 2007; Rosenthal et al., 2011; Kromdijk et al., 2016), besides being of theoretical interest. Moreover, multi-scale plant models (also known as plant system models), which quantitatively integrate physical, biochemical and physiological processes at different organizational levels (e.g. molecular, cell, organ, plant, population, or ecosystem), are able to predict physiological and growth properties of plants beyond photosynthetic metabolism, and they represent the future challenge in plant modelling (see Zhu et al., 2016; Marshall-Colón et al., 2017; Chang et al., 2019; Marshall-Colón and Kliebenstein, 2019).

FUNDING

D.L. was supported by European Regional Development Fund project ‘Plants as a tool for sustainable global development’ [No. CZ.02.1.01/0.0/0.0/16_019/0000827].

ACKNOWLEDGEMENTS

Govindjee is grateful for IT support provided by the UIUC Life Sciences Office of Information Technology (Andrew Debevec, Karl Schlipf, Thomas Uebele, Jeffrey Haas), and the staff of the Department of Plant Biology and of the Department of Biochemistry, University of Illinois at Urbana-Champaign; he encourages all readers to visit his web site (http://www.life.illinois.edu/govindjee/) to download available educational material on photosynthesis for personal use.

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