Thermodynamic Factors Controlling Electron Transfer amongst the Terminal Electron Acceptors of Photosystem I: Insight from Kinetic Modelling

1. Introduction

Photosystem I (PSI) is a fundamental component of the oxygenic photosynthetic electron transport chain. It catalyses the light-dependent oxidation of soluble electron shuttles, most commonly either the cupper-binding protein plastocyanin or the small cytochrome c₆, at its so-called donor side and the reduction of ferredoxin, which is also a soluble protein, at its acceptor side. Photosystem I is a very large membrane-embedded cofactor-protein supercomplex, composed of more than 10 protein subunits (e.g. [1,2]), the exact number of which depends on the species. Under an operational perspective it can be considered as being composed, analogously to the other photosystems, of two functional moieties: i) a core complex which has the role of harvesting sunlight and to perform primary charge separation and successive charge stabilisation through electron transfer (ET) reactions involving a series of redox-active cofactors, and ii) a peripheral antenna, which has light harvesting function only, and which composition in terms of proteins and chromophores, varies largely amongst different organisms.

The characteristics of the subunits composing the core complex of PSI appear instead to be generally well-conserved throughout evolutionary divergent species, particularly considering the two largest subunits PsaA and PsaB which form an heterodimer and coordinate, together with the core-antenna pigments, the majority of the redox-active cofactors. Also well conserved is the PsaC subunit which binds two 4Fe-4S clusters, referred to as $F_{\mathrm{A}}$ and $F_{\mathrm{B}}$, acting as the terminal electron acceptors within the photosystem and being, in turn, responsible for ferredoxin reduction. Structural studies [3,4,5] have shown that, in analogy to all other known photosynthetic reaction centres, the PsaA- and PsaB-coordinated cofactors, identified as participating in ET reactions, are organised in a highly symmetric fashion. The redox-active cofactors display a mirror symmetry arrangement with respect to the reference axis being putatively perpendicular to the membrane plane (Figure 1A). The symmetric organisation of the cofactors results in the presence of two putative ET chains, which are however not completely independent, sharing the (hetero)dimer of Chlorophyll (Chl) a/Chl a’ (the latter being an epimer [3]), assigned to the long-lived, terminal, electron donor, $P_{700}^{(+)}$ and the 4Fe-4S cluster $F_{\mathrm{X}}$, which is the last PsaA/PsaB-bound acceptor before the PsaC-bound $F_{\mathrm{A}}$ and $F_{\mathrm{B}}$ centres. Both $P_{700}^{ }$, where the number indicates the maximal absorption bleaching upon oxidation [2,6], and $F_{\mathrm{X}}$ are coordinated at the interface of PsaA and PsaB.

A general consensus has been reached over the years that in PSI the two parallel cofactor chains are both functional ([2,7,8,9,10,11,12]) and references therein). This represents a major functional difference with respect to Photosystem II, and the purple bacteria Reaction Centre (RC), in which primary photochemistry and ET reactions involve exclusively one cofactor chain, with the exclusion of the terminal electron acceptor, the quinone $Q_{\mathrm{B}}$, which is structurally part of the “inactive” chain, but accepts electrons from the “photochemically functional chain” bound quinone, $Q_{\mathrm{A}}$, instead. The need to control inter-quinone ET and the full-reduction of $Q_{\mathrm{B}}$ to quinole, by two consecutive charge separation events, is thought to be one of the main reasons for having an asymmetric, also referred to as mono-directional, ET in Type II photosynthetic RC. In PSI, and generally in Type I RC to whom it belongs, ET reactions involve only one-electron exchange, so that the need to control the accumulation of either oxidative or reducing equivalent at any level of the ET chain is not stringent.

Nonetheless, the probability by which electrons are transferred by each ET chain of PSI remains somewhat debated, with figures ranging from almost equal utilisation [12,13,14,15,16,17,18,19], to very asymmetric proportions in favour of the PsaA-coordinated branch [20,21,22,23,24]. Further functional differences are also apparent at the level of intermediate steps, particularly in the lifetime of oxidation of the reduced phylloquinone, $A_1^{ }$, by next acceptor in the chain, $F_{\mathrm{X}}$. This reaction shows a complex kinetic behaviour, which in its simplest description is accounted by two main phases characterised by lifetimes of 5-20 ns and 200-300 ns [6,7,8,9,10,25,26]. Studies of mutants at the level of the phylloquinone binding site have led to the assignment of the 200-300 ns phase to the oxidation of $A_{1\mathrm{A}}^{-}$ [6,7,8,9,10,27,28,29,30,31] (the subscript indicates the subunit which primarily coordinates the cofactor) and the 5-20 ns phase to the oxidation of $A_{1\mathrm{B}}^{-}$ [6,7,8,9,10,27,31]. These phases also show significantly different temperature dependences, with the slowest one displaying a larger activation energy (65-130 meV) with respect to the fast one (6-20 meV) [32,33,34]. Since $F_{\mathrm{X}}$ is a common cofactor to both reactions, and because of the similarity in the co-ordination geometry of the phylloquinones, the one-order-of-magnitude difference in the $A_{1\mathrm{A}}^{-}$ and $A_{1\mathrm{B}}^{-}$ oxidation kinetics was attributed, in the framework of ET-theory (e.g. [35,36,37] and references therein), to an asymmetry in the driving force for these reactions, with $A_{1\mathrm{B}}^{-}$ oxidation being more thermodynamically favourable (e.g. $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$<$\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$) [7,11,38,39]. Even though there is a general consensus concerning this interpretation, different estimates for the free energies have been advanced, also depending on the approach utilised for their evaluation ([12] and reference therein). Because of the very negative value of the phylloquinone potential direct redox titration is, at least, cumbersome [40]. Contradictory results were also obtained for the titration of $F_{\mathrm{X}}$, although its potential ($E_{F_{\mathrm{X}}}^0$) can be safely regarded as less than –680 mV ([41,42,43] and see discussion in [6]). Moreover, even direct titration can suffer of the bias linked to progressive accumulation of charges on the acceptor side (i.e. on $F_{\mathrm{A}}$, $F_{\mathrm{B}}$, $F_{\mathrm{X}}$ and so on). The direct electrochemically determined midpoint potentials can therefore differ, also significantly, from the “operational” ones, when the nearby cofactors are oxidised instead (see Ptsushenko et al. [44] for further discussion). Thus, most of the estimated $A_{1\mathrm{A}}^{ }$ and $A_{1\mathrm{B}}^{ }$ redox potentials were derived either from ET-theory-based modelling of the oxidation kinetic [7,11,38,39] or from structure-based computational methods [44,45,46,47]. None of these approaches is fully water-proof, since both require some specific simplifications and assumptions to address the problem. A detailed description of these issues is beyond the scope of the present paper. Yet, currently only two energetic pictures were proven to account for the oxidation kinetics at room temperature, the temperature dependence of the reactions [48] and the effect of mutations in the phylloquinone binding site, at least, in the latter case, concerning the $A_{1\mathrm{A}}^{-}$ to $F_{\mathrm{X}}$ reaction [49], for which information are more abundant. These are the so-called “weak driving force” and “large driving force” scenarios. In the former, $A_{1\mathrm{A}}^{-}$ oxidation is associated with a free energy difference of ~ – 30 meV <$\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$< + 30 meV whereas $A_{1\mathrm{B}}^{-}$ oxidation with $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$< – 50 meV (note that the values are broad, and rounded, because they depend on the exact parameters set describing the reaction rate constants) [7,11,12,48,49]. It is worth noticing that in the lower driving force limit $A_{1\mathrm{A}}^{-}$ oxidation becomes thermodynamically unfavourable. Within the weak-driving force configuration, experimental observable are semi-quantitatively reproduced considering a total reorganisation energy (${\lambda }_{tot}$) associated to phylloquinone oxidation, as well as $F_{\mathrm{A}}$ reduction by $F_{\mathrm{X}}$, of about 0.7 eV [7,11,12,48,49]. The “large driving force” scenario invokes that both $\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$ and $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$ < – 50/75 meV and that $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$<<$\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$, with the former corresponding to driving forces even larger than ~150 meV [39,44]. This energetic configuration, however, required to consider much larger reorganisation energies, in the order of 1 eV [48,49], to describe the experimental phylloquinone oxidation kinetics. When assuming an homogenous value of ${\lambda }_{tot}$ for all the ET reactions, the difference in the values adopted to simulate $A_1^{-}$ oxidation (i.e. 1 eV with respect to 0.7 eV) resulted in pronounced differences in the simulations of $F_X^{ }$ oxidation for the two energetic configurations described above even when keeping all other relevant simulation parameters, including $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$ which was set at ~ –150 meV [6,7], equal. In this framework, the electron transfer from $F_X^{ }$ to the terminal Fe-S clusters was predicted to be about 5 folds slower for the large $A_1^{-}$ oxidation driving force than for the weak $A_1^{-}$ oxidation driving force enegetics ([48] and vide infra for further detail).

The redox potential of the iron-sulphur clusters $F_{\mathrm{A}}$ an $F_{\mathrm{B}}$ is, together with the one of $P_{700}^{+}$, the more accurately determined, even in place of some variability in the estimations [6,50,51,52,53]. Knowledge of these values shall increase the margin of accuracy of kinetic modelling predictions. Nonetheless, the dynamics of ET involving these redox centres are not unambiguously understood yet. This is mainly due to the optical properties of the Fe-S cluster, which possess a broad, relatively unstructured spectrum that abundantly overlaps with those of other PSI-bound chromophores [54,55]. Moreover, ET involving species having close-to-the-same absorption spectra imply a small differential extinction coefficient, making their direct detection by optical methods extremely challenging. Most of the kinetic information relies then on alternative methods, exploiting changes in the electric field through the protein milieu caused by the electron displacement along the ET chain. These include direct methods, such as photovoltage [56,57,58,59,60], and indirect optical probes, like ET-dependent local-stark effect on chlorophylls and carotenoids [25]. From these methods a relatively broad range of values for the $F_{\mathrm{X}}$ to $F_{\mathrm{A}}$/$F_{\mathrm{B}}$ transfer time have been proposed, ranging from less than 25 ns to ~300 ns ([26,61] and references therein). A kinetic phase characterised by lifetimes of 160-180 ns, and attributed to ET between the iron-sulphur clusters, was also resolved in mutants in which the $A_{1\mathrm{A}}^{-}$ oxidation phase, to which it is otherwise overlapped, was slowed-down more than 3 times [30,62,63]. A component with similar lifetime is also resolvable in wild-type photosystems in temperature dependence studies, when it appears to have a lower activation, and becomes therefore discernible from the slow phase of $A_{1(\mathrm{A})}^{-}$ oxidation upon cooling [33,34]. Electron transfer between $F_{\mathrm{A}}$ and $F_{\mathrm{B}}$ is even less characterised, as the positioning of these cofactors leads to a vectorial ET almost parallel to the membrane plane (perpendicular to the symmetry axis) which is unfavourable for both local-stark and photo-voltage detection. Hence, either only the collective transfer to $F_{\mathrm{A}}$/$F_{\mathrm{B}}$ is generally detected or, alternatively, transfer to $F_{\mathrm{A}}$ alone after $F_{\mathrm{B}}$ is chemically inactivated/destabilised [26,61]. It is commonly assumed that the transfer time across all of the Fe-S shall be faster than 500 ns – 1 μs to account for the rapid reduction of pre-bound ferredoxin, that takes place in about 1 to 3 μs, although even faster sub-microsecond kinetics have been reported [64].

In order to gain further insight into the dynamics of ET reactions involving the Fe-S of PSI, kinetic simulations are here presented in which the effect of the most crucial thermodynamic parameters, i.e. the standard free energy difference and the reorganisation energy, are explored. The ET reactions involving the Fe-S clusters are contextualised within the “weak” and “large” driving force energetic scenarios for $A_1^{-}$ oxidation. The simulations allow to put forward predictions which could be, in principle, tested experimentally to get a better understanding of these ET events.

2. Results and Discussion

In order to gain some additional information concerning the ET reactions involving the 4Fe-4S centres in PSI, i.e. the reduction of $F_{\mathrm{A}}$ by $F_{\mathrm{X}}^{-}$ (hereafter the sign indicates the reduced state of the FeS centre, not its net charge) and the successive reduction of $F_{\mathrm{B}}$ by $F_{\mathrm{A}}^{-}$, kinetics simulations were performed starting from the two energetic scenarios which provide a satisfactory, semi-quantitative, description for the wild-type PSI. Successively, perturbations of parameters that control the rate constants of ET between the FeS clusters, mainly the standard free energies ($\Delta G_{DA}^0$) and the reorganisation energy (${\lambda }_{tot}$), are introduced.

2.1 Weak Driving Force Scenario for $A_1^{-}$ Oxidation

Figure 2 shows the simulated population evolutions of the reduced ET cofactors in PSI, after initial population of phylloquinones $A_{1\mathrm{A}}$ and $A_{1\mathrm{B}}$, within the so-called “weak-driving” force framework. The parameters used in the calculations are reported in Table 1. In brief, the standard free energy difference for $A_{1\mathrm{A}}^{-}$ and $A_{1\mathrm{B}}^{-}$ oxidation were taken as $\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$= +10 meV and $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$= –50 meV, and those for the 4Fe-4S centres oxidation were $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$= –150 meV and $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$= +25 meV. The two latter fall within the range derivable from direct redox titrations of the FeS cofactors [50,51,52,53]. Setting the midpoint potential of $F_{\mathrm{B}}$ at –555 mV, it then results that $E_{F_{\mathrm{A}}}^0$= –530 mV, $E_{F_{\mathrm{X}}}^0$= –680 mV $E_{A_{1\mathrm{A}}}^0$= –670 mV and $E_{A_{1\mathrm{B}}}^0$= –730 mV. Because of the relative spread of experimentally retrieved $E_{ }^0$ values and considering the effect of piling-up reducing equivalents upon titrations which likely result in a bias in the determined FeS clusters redox potentials, the free energy differences calculated from the experimental titrations and those employed in the simulations might and shall not match exactly. Those used in the simulations rather reflect the so-called “operational” potentials. Even with these approximations, the general consideration that the energy gap between $F_{\mathrm{X}}^{ }$ and $F_{\mathrm{A}}$ is relatively large, whereas the one between $F_{\mathrm{A}}$ and $F_{\mathrm{B}}$ is shallow, with the two PsaC-coordinated centres being almost iso-potential, remains valid. For the parameters described above, and in general for the “weak driving” force model, it is possible to consider a common value of the reorganisation energy for the $A_{1\mathrm{A}}^{-}$, $A_{1\mathrm{B}}^{-}$ and $F_{\mathrm{X}}^{-}$ oxidation equal to 0.7 eV whereas a larger value of 0.9 eV was considered for $F_{\mathrm{A}}^{-}$ oxidation. This is justified since the outer shell reconfiguration is expected to be larger for reactions involving metal centres only with respect to organic cofactors.

Figure 2. Panel A: reference energetic scheme for the “weak driving force” scenario for $A_1^{-}$ oxidation; standard redox potentials ($E^0$) are shown relative to $F_{\mathrm{X}}$. Panel B: simulated population evolution of the individual redox active cofactors, $A_{1\mathrm{A}}^{-}(t)$: dashed-dot red line, $A_{1\mathrm{B}}^{-}(t)$ dashed-dot blue line, $F_{\mathrm{X}}^{-}(t)$ dot golden line, $F_{\mathrm{A}}^{-}(t)$ dot orange line, $F_{\mathrm{B}}^{-}(t)$ dot burgundy line. Also shown are the total population evolutions of $A_{1,tot}^{-}(t)=A_{1\mathrm{A}}^{-}(t)+A_{1\mathrm{B}}^{-}(t)$ (black line) and $Fe{S}_{tot}^{-}(t)=F_{\mathrm{X}}^{-}(t)+F_{\mathrm{A}}^{-}(t)+F_{\mathrm{B}}^{-}(t)$ (purple line). The inset (C) shows the recombination kinetics simulated in absence of an exit from the system ($k_{out}$= 0). Temperature: 290 K.

Figure 2. Panel A: reference energetic scheme for the “weak driving force” scenario for $A_1^{-}$ oxidation; standard redox potentials ($E^0$) are shown relative to $F_{\mathrm{X}}$. Panel B: simulated population evolution of the individual redox active cofactors, $A_{1\mathrm{A}}^{-}(t)$: dashed-dot red line, $A_{1\mathrm{B}}^{-}(t)$ dashed-dot blue line, $F_{\mathrm{X}}^{-}(t)$ dot golden line, $F_{\mathrm{A}}^{-}(t)$ dot orange line, $F_{\mathrm{B}}^{-}(t)$ dot burgundy line. Also shown are the total population evolutions of $A_{1,tot}^{-}(t)=A_{1\mathrm{A}}^{-}(t)+A_{1\mathrm{B}}^{-}(t)$ (black line) and $Fe{S}_{tot}^{-}(t)=F_{\mathrm{X}}^{-}(t)+F_{\mathrm{A}}^{-}(t)+F_{\mathrm{B}}^{-}(t)$ (purple line). The inset (C) shows the recombination kinetics simulated in absence of an exit from the system ($k_{out}$= 0). Temperature: 290 K.

Specific frequencies of the mean coupled nuclear modes were considered for $A_1^{-}$ oxidation, corresponding to $\hslash {\overline{\omega }}_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}$= 22 meV (175 cm^-1) and $\hslash {\overline{\omega }}_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}$= 46 meV (375 cm^-1). These values were obtained from the analysis of the phylloquinone oxidation kinetics as a function of the temperature [66] and resulted in an improvement of the simulations of the temperature dependence of the kinetics when explicitly considered. A coupling $\hslash {\overline{\omega }}_{DA}$ = 34 meV (275 cm^-1), which is the mean of the values employed for the quinone oxidation, was used for all other reactions. In the kinetic scheme shown above (Figure 1B) also the recombination reactions between all reduced cofactors and $P_{700}^{+}$ were included. In the simulations of the recombination reactions $E_{P_{700}}^0$ was +450 mV, in accordance with direct estimations from redox titrations [6], and the reorganisation energy was 0.9 eV for all reactions. The recombination reactions have virtually no influence on the simulations of forward ET kinetics because these rates are several order of magnitude slower than productive reactions, due to large distances between the cofactor pairs even in place of large favourable driving forces for all of these processes (see Appendix S1 of the Supplementary Information). However, their inclusion allows to compare the simulated and experimentally determined recombination kinetics, as shown in Figure 2C, simply by suppressing the exit rate from the system (i.e. $F_{\mathrm{B}}^{-}$ oxidation by external electron acceptors).

For the parameters discussed above it is obtained that, at room temperature (290 K), the system is characterised by five lifetimes (the number corresponding to the states present in the kinetics scheme) of 5.2, 22.5, 136, 243 and 3808 ns, which are common to the description of the population evolution of all cofactors. The 5.2, 22.5 and 243 ns lifetimes are those also retrieved from a three states model that does not explicitly include $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ (e.g. [7,11,12,48,49] and see Appendix S1 Figure S1 and Table S1), so that the remaining lifetimes of 136 and 3808 ns can be related to electron transfer reactions involving the terminal iron-sulphur cluster directly. The longest-lived lifetime describes the exit from the system, when this is considered ($F_{\mathrm{B}}\to Out$). This reaction is not modelled according to ET theory, a phenomenological rate of 1 x 10^-3 ns^-1 is considered instead and it is set to zero, when necessary, to simulate the recombination reactions.

Table 1. Rate-defining Parameters and Simulated ET kinetic for the weak driving force energetic scenario.

	Rate Determining Parameters						Electron Transfer Simulated Parameters
							Amplitudes
Reaction	$X_{DA}$ (Å)	$\Delta G_{DA}^0$ (eV)	${\lambda }_{t,DA}$ (eV)	${\overline{\omega }}_{DA}$ (eV/cm-1)	$k_{D\to A}$ (ns-1)	${\tau }_{ }$ (ns)	$p_{A_{1\mathrm{A}}}$	$p_{A_{1\mathrm{B}}}$	$p_{F_{\mathrm{X}}}$	$p_{F_{\mathrm{A}}}$	$p_{F_{\mathrm{B}}}$	$p_{A_{1\mathrm{tot}}}$	$p_{Fe{S}_{\mathrm{tot}}}$
$A_{1A}^{-}\to F_X$	9.1	0.01	0.700	0.022 (175)	0.0181	5.26	0.0754	0.382	-0.489	0.033	0.000	0.457	-0.457
$A_{1B}^{-}\to F_X$	9.0	-0.05	0.700	0.046 (375)	0.145	22.5	-0.173	0.051	0.172	-0.051	0.003	-0.123	0.123
$F_X^{-}\to F_A$	11.6	-0.15	0.700	0.034 (275)	0.0126	136	0.0035	0.000	0.001	-0.295	0.336	0.004	0.042
$F_A^{-}\to F_B$	9.5	0.025	0.900	0.034 (275)	0.0019	243	0.587	0.067	0.312	-0.489	-0.628	0.654	-0.806
$F_B^{-}\to$	–	–	–	–	0.00100	3808	0.0068	0.001	0.005	0.803	0.290	0.008	1.098

Table 1. Rate-defining Parameters and Simulated ET kinetic for the weak driving force energetic scenario.

	Rate Determining Parameters						Electron Transfer Simulated Parameters
							Amplitudes
Reaction	$X_{DA}$ (Å)	$\Delta G_{DA}^0$ (eV)	${\lambda }_{t,DA}$ (eV)	${\overline{\omega }}_{DA}$ (eV/cm-1)	$k_{D\to A}$ (ns-1)	${\tau }_{ }$ (ns)	$p_{A_{1\mathrm{A}}}$	$p_{A_{1\mathrm{B}}}$	$p_{F_{\mathrm{X}}}$	$p_{F_{\mathrm{A}}}$	$p_{F_{\mathrm{B}}}$	$p_{A_{1\mathrm{tot}}}$	$p_{Fe{S}_{\mathrm{tot}}}$
$A_{1A}^{-}\to F_X$	9.1	0.01	0.700	0.022 (175)	0.0181	5.26	0.0754	0.382	-0.489	0.033	0.000	0.457	-0.457
$A_{1B}^{-}\to F_X$	9.0	-0.05	0.700	0.046 (375)	0.145	22.5	-0.173	0.051	0.172	-0.051	0.003	-0.123	0.123
$F_X^{-}\to F_A$	11.6	-0.15	0.700	0.034 (275)	0.0126	136	0.0035	0.000	0.001	-0.295	0.336	0.004	0.042
$F_A^{-}\to F_B$	9.5	0.025	0.900	0.034 (275)	0.0019	243	0.587	0.067	0.312	-0.489	-0.628	0.654	-0.806
$F_B^{-}\to$	–	–	–	–	0.00100	3808	0.0068	0.001	0.005	0.803	0.290	0.008	1.098

Summary of the parameters utilised for the simulation of forward electron transfer in reference PSI within the weak-driving force scenario for $A_1^{-}$ oxidation. The rate-defining parameters which are common to all forward electron reactions are the electronic coupling matrix element, ${\left|H_0\right|}^2$=1.3 10^-3 eV², the barrier camping factor$\beta$=1.34 Å^-1 and the temperature T=290 K, which are defined in Equation 1 of the main text.

The table also lista the lifetimes (${\tau }_i$) and associated amplitudes ($p_i$) describing the population evolution of each of the cofactors described in the model. The total amplitude associated with the electron transfer kinetics of phylloquinones ($p_{A_{1\mathrm{tot}}}$) and iron-sulphur centres ($p_{Fe{S}_{\mathrm{tot}}}$) are also presented in the table. The initial conditions for the amplitude simulations were $A_{1\mathrm{A}}^{-}(0)=A_{1\mathrm{B}}^{-}(0)=0.5$ and zero on all other cofactors considered.

Within this energetic scheme, the oxidation of $A_{1\mathrm{B}}^{-}$ is dominated by the two fastest components (~5 and 22.5 ns), resulting in an average decay lifetime of 46 ns, whereas the oxidation of $A_{1\mathrm{A}}^{-}$ is dominated by the 243 ns component, and the resulting average decay is 331 ns. Despite the assumption of equal initial population at time zero of $A_{1\mathrm{A}}^{-}$ and $A_{1\mathrm{B}}^{-}$ the predicted ratio of fast (5 ns and 22.5 ns) to slow (all remaining ones) oxidation phases is 0.33:0.67, which was interpreted as a transient inter-quinone population transfer [67]. Since the detail of the model predictions concerning the phylloquinone ET reactions has already been discussed previously (e.g. [7,11,12,48,49]), more attention will be here dedicated to the transfer involving the 4Fe-4S clusters. Within the above mentioned kinetic/energetic scheme, the average reduction time of $F_{\mathrm{X}}^{ }$ is predicted to be relatively fast, and described uniquely by the 5.2 ns component, whereas the successive oxidation is multiphasic, dominated by the 22.5 and 243 ns lifetimes, giving rise to an average decay lifetime (${\tau }_{av}$) of 199 ns, which is only slightly slower than the value simulated for the total quinone oxidation ($A_{1,tot}^{-}$=$A_{1\mathrm{A}}^{-}$+$A_{1\mathrm{B}}^{-}$) being 160 ns. Similarly close values are obtained when comparing the mean population lifetime (i.e. the first moment of the population temporal evolution, ${\tau }_m$), that is estimated as being 900 ns for $F_{\mathrm{X}}^{-}$ with respect to 805 ns for $A_{1,tot}^{-}$ oxidations. The ${\tau }_m$ are larger than the ${\tau }_{av}$ values because of the increased weight of the slowest component in the parameter estimation, which is at least one order of magnitude slower than all other lifetimes. When the latter is omitted from the calculations, the mean lifetimes for $F_{\mathrm{X}}^{-}$ and $A_{1,tot}^{-}$ become 240 and 243 ns, respectively, that are then close to ${\tau }_{av}$. Yet, irrespectively of the parameters considered, these cofactors relax to their neutral state almost simultaneously. This would explain the difficulties in detecting an electrogenic phase specifically associated with the $F_{\mathrm{X}}^{ }$ oxidation/reduction reactions, since this would broadly overlap kinetically with $A_1^{-}$ oxidation. The successive redox acceptors $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ are reduced with average lifetimes of 192 and 243 ns, respectively, that is, almost co-ordinately with the relaxation of $A_{1,tot}^{-}$ and $F_{\mathrm{X}}^{-}$, and are oxidised with average lifetimes of 3.6 μs and 1.9 μs, respectively, clearly limited by the rate of exit from the system. The simulated mean lifetimes of $F_{\mathrm{A}}^{-}$ and $F_{\mathrm{B}}^{-}$ population evolutions are 4.0 μs and 4.2 μs, respectively, again indicating that the terminal acceptors are oxidised almost simultaneously, which is due to the weak, slightly endergonic, driving force associated to $F_{\mathrm{A}}^{-}$ oxidation, leading to the partitioning of reducing equivalents in favour of this cofactor, with respect to the terminal acceptor $F_{\mathrm{B}}^{ }$.

Figure 2 also shows the predicted recombination, in the absence of an exit from the system, where the oxidation of the terminal 4Fe-S cluster is dominated by a lifetime of 19 ms, which replaces the longest-lived ~4μs lifetime predicted from the simulations in the presence of an acceptor. The simulated recombination lifetime falls well within the spread of values reported in the literature, that most commonly are in the range of 10 – 200 ms (see [6,26,68,69,70,71] and reference therein). It is also worth mentioning that the observed lifetime, however, does not reflect the (inverse) direct recombination rate from the terminal 4Fe-4S clusters, that are simulated as 2.1 x 10^-17 ns^-1 and 4.8 x 10^-22 ns^-1 for the reactions involving $F_{\mathrm{A}}^{-}$ and $F_{\mathrm{B}}^{-}$ respectively. Rather the recombination rate/lifetime is determined by the energetically up-hill repopulation of the upstream cofactors, chiefly $F_{\mathrm{X}}^{-}$. The predicted recombination kinetics are virtually unaffected, in the energetic framework discussed above, by arbitrarily setting to zero the values of all the recombination to $P_{700}^{+}$, excluding the direct recombination from the phylloquinones (Appendix S1 Figure S2).

2.2. Large Driving Force Scenario for $A_1^{-}$ Oxidation

Figure 3 shows the simulations obtained within the “large driving force” scenario for $A_1^{-}$ oxidation. In this case, we considered values of $\Delta G_{A_{1\mathrm{A}}\to F_{\mathrm{X}}}^0$= –50 meV and $\Delta G_{A_{1\mathrm{B}}\to F_{\mathrm{X}}}^0$= –220 meV as suggested by Milanovsky et al. [39], whereas the free energies for the other reactions were the same as discussed above, $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$= –150 meV and $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$= +25 meV. Setting the midpoint potential of $F_{\mathrm{B}}$ at –555 mV, it then results that $E_{F_{\mathrm{A}}}^0$= –530 mV, $E_{F_{\mathrm{X}}}^0$= –680 mV $E_{A_{1\mathrm{A}}}^0$= –730 mV and $E_{A_{1\mathrm{B}}}^0$= –900 mV. The parameters set employed in the calculations is listed in Table 2. In order to facilitate the comparison between the two energetic scenarios considered, whenever feasible, the same parameters were adopted in the simulations. The main difference resides in the reorganisation energy value, because within the large driving force scheme, a value of ${\lambda }_{tot}$= 0.7 eV associated to the quinone reactions resulted in simulated lifetimes which are too fast with respect to the measured ones, particularly for the slowest phase of $A_1^{-}$ oxidation which would be simulated by a lifetime of ~100 ns, that is at least two times faster than the value retrieved from the experiments, i.e. ~250-360 ns (see Appendix S3 of the Supplementary Information Figure S4). It was therefore necessary, as also discussed previously [48], to use a larger value for ${\lambda }_{tot}$ of 1 eV for these ET steps, that was extended to the simulation of all other, forward and recombination, reactions. Within this energetic/kinetic scenario, the simulated lifetimes, common to all reactions are: 8.2, 295, 307, 1411 and 4415 ns. The 8.2, 307 and 1411 ns lifetimes are also retrieved for a kinetic scheme that not explicitly considers the terminal iron-sulphur centres (Appendix S1 Figure S1 and Table S1), therefore the remaining 295 and 4415 ns are associated mainly to ET processes directly involving $F_A^{ }$ and $F_{\mathrm{B}}^{ }$.

Figure 3. Panel A: reference energetic scheme for the “large driving force” scenario for $A_1^{-}$ oxidation; standard redox potentials ($E^0$) are shown relative to $F_{\mathrm{X}}$. Panel B: simulated population evolution of the individual redox active cofactors, $A_{1\mathrm{A}}^{-}(t)$: dashed-dot red line, $A_{1\mathrm{B}}^{-}(t)$ dashed-dot blue line, $F_{\mathrm{X}}^{-}(t)$ dot golden line, $F_{\mathrm{A}}^{-}(t)$ dot orange line, $F_{\mathrm{B}}^{-}(t)$ dot burgundy line. Also shown are the total population evolutions of $A_{1,tot}^{-}(t)=A_{1\mathrm{A}}^{-}(t)+A_{1\mathrm{B}}^{-}(t)$ (black line) and $Fe{S}_{tot}^{-}(t)=F_{\mathrm{X}}^{-}(t)+F_{\mathrm{A}}^{-}(t)+F_{\mathrm{B}}^{-}(t)$ (purple line). The inset (C) shows the recombination kinetics simulated in absence of an exit from the system ($k_{out}$= 0). Temperature: 290 K.

Figure 3. Panel A: reference energetic scheme for the “large driving force” scenario for $A_1^{-}$ oxidation; standard redox potentials ($E^0$) are shown relative to $F_{\mathrm{X}}$. Panel B: simulated population evolution of the individual redox active cofactors, $A_{1\mathrm{A}}^{-}(t)$: dashed-dot red line, $A_{1\mathrm{B}}^{-}(t)$ dashed-dot blue line, $F_{\mathrm{X}}^{-}(t)$ dot golden line, $F_{\mathrm{A}}^{-}(t)$ dot orange line, $F_{\mathrm{B}}^{-}(t)$ dot burgundy line. Also shown are the total population evolutions of $A_{1,tot}^{-}(t)=A_{1\mathrm{A}}^{-}(t)+A_{1\mathrm{B}}^{-}(t)$ (black line) and $Fe{S}_{tot}^{-}(t)=F_{\mathrm{X}}^{-}(t)+F_{\mathrm{A}}^{-}(t)+F_{\mathrm{B}}^{-}(t)$ (purple line). The inset (C) shows the recombination kinetics simulated in absence of an exit from the system ($k_{out}$= 0). Temperature: 290 K.

The oxidation of $A_{1\mathrm{B}}^{-}$ is dominated by the 8.2 ns lifetime and that of $A_{1\mathrm{A}}^{-}$ by the 307 ns one, with some contribution from the 1,4 μs component. The proportion of fast:slow $A_{1,tot}^{-}$ oxidation phases are simulated as 0.5:0.5, corresponding to the initial populations of these cofactors. Different oxidation phase partitions could be straightforwardly simulated by setting the boundary conditions accordingly. This in turn implies an asymmetric charge separation/stabilisation between the two active ET chains of PSI (e.g. [20,21,22,23,24]). Discussion of this issue is however beyond the scope of this work. For equal initial population of $A_{1\mathrm{B}}^{-}$ and $A_{1\mathrm{A}}^{-}$, the resulting average lifetimes are 10.5 and 733 ns, respectively, and 371 ns for $A_{1,tot}^{-}$. The predicted oxidation of $A_{1\mathrm{A}}^{-}$ is somewhat slower, within this energetic scheme and for the conditions described, due to the significant contribution of the 1.4 μs phase, which exact value is in part linked to ET between the Fe-S clusters. Within this model, the reduction of $F_{\mathrm{X}}^{ }$ is multiphasic, with contributions from all the sub-microsecond components, i.e. 8.2, 295, 307 ns (Table 2), giving rise to an average rise lifetime of 143 ns, whereas $F_{\mathrm{X}}^{-}$ oxidation is largely dominated by the 1.4 μs phase, corresponding closely to the average oxidation (${\tau }_{av}$) time and the mean lifetimes of the population evolution (${\tau }_m$), being both 1.6 μs, and hence slower that 1.1 μs estimated for $A_{1,tot}^{-}$. As apparent form the inspection of the population evolutions (Figure 3B), in this energetic scenario, the oxidation of $F_{\mathrm{X}}^{-}$ clearly follows that of $A_{1,tot}^{-}$,whereas the two processes largely overlapped in the simulations performed within the small driving force energetic scheme (Figure 2B). This is associated to the slower rate of $F_{\mathrm{A}}^{ }$ reduction from $F_{\mathrm{X}}^{-}$ due to the larger reorganisation energy (1 eV rather than 0.7 eV). Although not necessarily impossible, it would be unusual that the reorganisation for an ET event between 4Fe-4S cluster were to be lower than the one associated to ET from phylloquinones to an iron-sulphur centre. For this reason, as well as to minimise the number of variable parameters, ${\lambda }_{tot}$ was set to the same value. Simulations for different values of the reorganisation energy are however explored in Appendix S4 of the Supplementary Information where the effect of this parameter can be better appreciated.

Summary of the parameters utilised for the simulations of forward electron transfer in reference PSI within the large-driving-force scenario for $A_1^{-}$ oxidation. The rate-defining parameters which are common to all forward electron reactions are the electronic coupling matrix element, ${\left|H_0\right|}^2$=1.3 10^-3 eV², the barrier camping factor$\beta$=1.34 Å^-1 and the temperature T=290 K, which are defined in Equation 1 of the main text.

The tables also list the lifetimes (${\tau }_i$) and associated amplitudes ($p_i$) describing the population evolution of each of the cofactors described in the model. The total amplitude associated with the electron transfer kinetics of phylloquinones ($p_{A_{1\mathrm{tot}}}$) and iron-sulphur centres ($p_{Fe{S}_{\mathrm{tot}}}$) are also presented in the table. The initial conditions for the amplitude simulations were $A_{1\mathrm{A}}^{-}(0)=A_{1\mathrm{B}}^{-}(0)=0.5$ and zero on all other cofactors considered.

The terminal electron acceptors $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ are reduced with average lifetimes of 1.2 μs and 808 ns, respectively, and oxidised with average lifetimes of 3.4 and 1.8 μs respectively. The terminal acceptors oxidation is, basically, determined by the rate of exit from the system. The slower simulated dynamics of $F_{\mathrm{A}}^{-}$ oxidation are, as in the “weak driving force” scenario, due to considering a slightly endergonic reduction of $F_{\mathrm{B}}^{ }$. Nonetheless, the mean lifetimes for $F_{\mathrm{A}}^{-}$ and $F_{\mathrm{B}}^{-}$ are 5.8 and 6.2 μs indicating that, as also discussed above, the two centres are predicted to be oxidised almost concertedly by diffusible acceptors.

When the rate of electron donation from $F_{\mathrm{B}}^{-}$ is suppressed in order to simulate the recombination reactions, these are characterised by a lifetime of ~570 ms, which falls rather outside the spread of values reported in the literature in which the slowest ones are in the order of 100 – 200 ms [6,26,68,69,70,71]. Yet, no specific effort has been made here to reach a good agreement between the measured and simulated recombination rates. An improvement in the match between modelled (between 174 and 89 ms) and experimental values is obtained by increasing $\hslash {\overline{\omega }}_{DA}$ to 55 meV (450 cm^-1) (Appendix S2, Figure S3), which is the average recommended consensus value from a survey in redox-active proteins [37]. Variation of the values of ${\lambda }_{tot}$ (Appendix S3, Figure S6) or application of the parameter sets employed in the “weak driving force” model (Appendix S4, Figure S8), as discussed above, do not appear to improve the description of the recombination kinetics within the “large driving force” energetic scheme. Nonetheless, as in the “weak driving force” case, the recombination rate is limited by the uphill repopulation of $F_{\mathrm{X}}^{-}$ and, especially, $A_{1(\mathrm{A},\mathrm{B})}^{-}$. As discussed by Cherepanov and coworkers [72], a more detailed description of the Franck-Condon function, involving multiple nuclear modes and specific electron-phonon coupling terms, as well as explicit consideration of $A_{0(\mathrm{A},\mathrm{B})}^{-}$ (the up-stream electron donor) might be required to describe the recombination reactions in detail, at least within the large driving force energetic picture.

2.3. Effect of Changing the Driving Force of $F_{\mathrm{X}}^{-}$ Oxidation ($\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$)

The most straightforward change in PSI energetics that affects the ET between the iron-sulphur cluster directly, involves the tuning of the redox potential of the terminal electron acceptors $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$. Mutations in the PsaC subunit that modify the coordination, and hence the redox properties, of $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ have already been reported [73,74,75,76,77,78,79,80,81,82,83,84]. However, the effect of these mutations on the $A_{1\mathrm{A},\mathrm{B}}^{-}$ oxidation kinetics has not been investigated in details, since the main target of these studies was to identify the specific nature and coordination site of the terminal acceptor, which has been a matter of controversy before being definitely elucidated by the availability of high resolution structural models. Hence, these studies targeted primarily the residues directly involved in the binding of the FeS clusters [73,74,75,76,77,78,79,80,81,82,83,84], and the kinetic information reported related almost exclusively on the recombination rather than the forward ET reactions [73,74,75,76,77,78,79,80,81,82]. Similarly, mutants affecting either the direct coordination of $F_{\mathrm{X}}^{ }$ [85,86,87,88] and, in some cases, of nearby residues altering the cluster binding niche [89,90,91,92,93,94,95], have also been produced. Actually, these mutants pioneered the application of site-directed mutagenesis to the study of PSI [85,86,87]. The main target of these studies was also the identification of the ligands and the interaction of the $F_{\mathrm{X}}^{ }$-binding domain with the PSI acceptor side subunits. Most of these mutants resulted however in a loss or very low level of PSI accumulation, in centres lacking or having a heavily modified $F_{\mathrm{X}}^{ }$ centre [85,86,87] as well as in altered binding of PsaC and the neighbouring subunits [86,87,88,89,90,91,92,93,94,95]. This, in turn, restricted the possibility of studying the effect of the specific mutations on forward electron transfer, so that the characterisation of these modified reaction centres was then relatively limited [91,92,93,94,95]. Although mutants leading to the perturbations of the $F_{\mathrm{X}}^{ }$ properties are certainly interesting, these would lead to the simultaneous alteration of factors governing three ET events: the reduction of this cluster by both phylloquinones and its oxidation by$F_{\mathrm{A}}^{ }$. Here, then, the discussion will be initially focused on changes in the redox potential of $F_{\mathrm{A}}^{ }$ ($E_{F_{\mathrm{A}}}^0$), since the $F_{\mathrm{X}}^{ }\to F_{\mathrm{A}}$ reaction follows immediately the phylloquinones in the ET cascade, but is not expected to modify the uphill reaction rates directly. Yet, any alteration in the $F_A^{ }$ redox potential, leads to changes to both $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$ and $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$, albeit in the opposite direction, as the driving force for one reaction increase, the other decreases.

Figure 4 shows the simulated kinetics for all redox cofactors described in the energetic/kinetic model in which $E_{F_{\mathrm{A}}}^0$ was shifted by –25 mV (Figure 4A/B), –50 mV (Figure 4C/D) and –100 mV (Figure 4E/F), within the “weak driving force” scenario for phyllosemiquinone oxidation. For the –25 mV shift, $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ are isoenergetic, while for larger potential shifts the driving force for $F_{\mathrm{B}}^{ }$ reduction (by $F_{\mathrm{A}}^{ }$) increases progressively, whereas that of $F_{\mathrm{A}}^{ }$ reduction by $F_{\mathrm{X}}^{-}$ decreases accordingly, but always remains favourable from a thermodynamic point of view.

Figure 5 shows simulations for the same alterations of $E_{F_{\mathrm{A}}}^0$, but within the “large driving force” phyllosemiquinone oxidation scheme.

The simulations of Figure 4 and Figure 5 show some significant differences in the ET kinetics when equal perturbations are applied to the driving forces of reactions which are otherwise described by, basically, the same parameters (Figure 2 and 3).

In the “weak driving force” scheme, together with the modification of the population evolutions of $F_{\mathrm{X},\mathrm{A},\mathrm{B}}^{-}$, which are expected since the rate constants associated with these reactions are directly affected by the $F_{\mathrm{A}}^{ }$ redox potential perturbations, also significant alterations of the $A_{1,tot}^{-}$ oxidation kinetics are simulated (${\tau }_{av,A_1^{-}}$ was 258, 360 and 840 ns vs. ~180 ns in the reference energetic scenario). This is particularly clear for the simulated $A_{1\mathrm{A}}^{-}$ kinetics, due to the increase in the value of the lifetime that dominates its oxidation, that is simulated as 340 and 530 ns and 1 μs for the $E_{F_{\mathrm{A}}}^0$ perturbations discussed above vs 284 ns in the initial scenario. The kinetics of $A_{1,tot}^{-}$ oxidation became progressively slower as the rate of $F_{\mathrm{X}}^{-}\to F_{\mathrm{A}}$ became slower (7.2 x 10^-3, 5.9 x 10^-3 and 2.1 x 10^-3 ns^-1 vs. 1 x 10^-2 ns^-1 in the starting scenario), because of the decrease in driving force, $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$. However, irrespectively of the exact shift of the $F_{\mathrm{A}}^{ }$ redox potential, $F_{\mathrm{X}}^{-}$ reduction remained concerted with that of $A_{1,tot}^{-}$ and, on average, slowed down by the same extent (${\tau }_{av,F_{\mathrm{X}}^{-}}^d$ 326, ns 413 ns, 1 μs vs 199 ns in the reference system). On the other hand, $F_{\mathrm{A}}^{-}$ oxidation became progressively faster (as the potential of the cofactor became more reductive), and so did the oxidation of $F_{\mathrm{B}}^{-}$, upon shifting from endergonic (Figure 2) to exergonic conditions (Figure 4). The maximal population of these cofactors decreased (${[F_{(\mathrm{X}+\mathrm{A}+\mathrm{B})}^{-}]}^{\max }$ 0.86, 0.70, 0.53 vs. 0.86 in the reference) with increased free energy for the last inter-protein ET step. The effect is clearly more pronounced for the maximal $F_{\mathrm{A}}^{-}$ reduction level (${[F_{\mathrm{A}}^{-}]}^{\max }$ 0.45, 0.26 and 0.06 vs. 0.64 in the reference system).

Differently, in the “large driving force” scheme, the modifications of the $F_{\mathrm{A}}^{-}$ oxidation potential, together with the resulting changes in driving forces associated to the reactions in which this cofactor participates, appear to affect almost exclusively inter-Fe-S electron transfer reactions, whereas the kinetics of $A_{1,tot}^{-}$ oxidation remain almost unaffected (Figure 5). The only significant effect on $A_1^{-}$ oxidation concerned the amplitude of a minor component displaying a long lifetime (~1.5 μs), which was also simulated for the reference system (Figure 3), but which value increased significantly upon $E_{F_{\mathrm{A}}}^0$ perturbation ($\tau$ 3.7, 5.2 and 14 μs, respectively, vs. 1.4 μs in the reference system). The large value of this lifetime, even when having a small associated amplitude (<10% of total), has a marked impact on the estimation of ${\tau }_{av,A_1^{-}}$ (which was 680 ns, 875 ns and 2.2 μs in the $F_{\mathrm{A}}^{ }$-perturbed vs. ~370 ns in the reference energetic system). It shall be recalled that this slow phase has never been detected experimentally, to the best of our knowledge. It is clearly connected to the explicit consideration of $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ reaction, as it is not simulated otherwise ([48,63,67] and Appendix S1, Table S1). Kinetic coupling in this scenario becomes significant since the rate of $F_{\mathrm{X}}^{-}\to F_{\mathrm{A}}$ transfer (8.8 10^-4 ns^-1) is almost the same as that of $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ (7.7 10^-4 ns^-1) and, especially, of the reverse, $F_{\mathrm{B}}^{-}\to F_{\mathrm{A}}$, reaction (1.5 10^-4 ns^-1). This is not the case for the “weak driving force” scenario by virtue of the lower reorganisation energy (0.7 vs. 1 eV) required to describe the $A_1^{-}\to F_{\mathrm{X}}$ and $F_{\mathrm{X}}^{-}\to F_{\mathrm{A}}$ reactions. Thus, when the μs-phase(s) is omitted from the calculations of ${\tau }_{av,A_1^{-}}$ in the “large driving force” simulations, the values of this parameter was almost unaffected by changes of $E_{F_{\mathrm{A}}}^0$, varying from 122 ns in the reference system to 139 ns for the largest potential shift of –100 mV considered in the simulations of Figure 5.

Comparing Figure 4 and Figure 5, it can be appreciated that the effect of rendering $F_{\mathrm{A}}$ potential more reductive, is instead far more pronounced on the ET kinetics involving all iron-sulphur clusters in the case of the large driving force scenario for $A_1^{-}$ oxidation. The cofactor showing the largest change in ET kinetic is $F_{\mathrm{X}}^{-}$, which oxidation/reduction dynamics became progressively slower with average decay lifetimes of 3.8, 5.2 and 13 μs with respect to 2.7 μs in the reference system. When decreasing the driving force for $F_{\mathrm{X}}^{-}$ oxidation, its depopulation became progressively overlapped with that of $F_{\mathrm{A},\mathrm{B}}^{-}$ and its maximal population increases from 0.59 in the reference system to 0.69, 0.72 and 0.79 as $F_{\mathrm{A}}^{ }$ becomes more reductive. The maximal cumulative population of the Fe-S clusters did not change significantly, indicating that the increase in transient $F_{\mathrm{X}}^{-}$ population level is accompanied by a parallel decrease in that of $F_{\mathrm{A}+\mathrm{B}}^{-}$ (Figure 5).

To obtain a more comprehensive picture of the effect resulting from perturbations of the $F_{\mathrm{A}}^{ }$ redox potential, and in turn the $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$ and $\Delta G_{F_A\to F_B}^0$ free energies, the value of $E_{F_{\mathrm{A}}}^0$ was changed in ±100 mV interval with respect to the reference systems. The effect of tuning the $E_{F_{\mathrm{A}}}^0$ redox potential on the simulated lifetimes is presented in Figure 6 starting from the initial reference being either the “weak” (Figure 6A, C and E) or “large” driving force (Figure 6B, D, F) energetics for $A_1^{-}$ oxidation. In Figure 6, lifetimes values were separated in classes for ease of presentation and comparison, so that Figure 6A and B show $\tau$<~50 ns, Figure 6C and Figure DFigure showFigure ~50Figure ns<$\tau$<~1 μs and Figure 6E and F, $\tau$>~1 μs. Since the lifetimes change upon $F_{\mathrm{A}}^{ }$ perturbation, this classification is not strict, but refers to the dominant behaviour of a given lifetime component. The values of $\tau$<50 are not greatly affected by any perturbation of $E_{F_{\mathrm{A}}}^0$ in the ±100 mV interval considered. These lifetimes, ${\tau }_1$ and ${\tau }_2$ in the reference weak-driving force and ${\tau }_1$ only in the reference large-driving force energetic scheme, reflect principally $A_{1\mathrm{B}}^{-}$ oxidation, that is exergonic in both. The simulated kinetics of Figure 4 and 5 show the prediction of little alterations in this kinetic phase when modifying successive ET reactions.

For lifetime values in the 50 ns<$\tau$<~1 μs range (Figure 6C and D), starting-model-dependent variations are predicted instead. In the starting weak-driving force case, $A_{1A}^{-}$ oxidation is dominated by the value of ${\tau }_4$, which simulated values increase with the increase of $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$, i.e. with the lowering of the reaction driving force (Figure 6C ${\tau }_3$ lifetime in this energetic configuration, relates principally to ET reactions amongst the Fe-S clusters and its reference value of ~160 ns is an agreement with experimental estimation [30,33,34]. Its dependence from the variation of $F_{\mathrm{A}}^{ }$ redox potential is not monotonic, consistent with it being determined by the kinetic coupling of reactions influenced by both $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$ and $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$.

In the starting large-driving force energetic scheme, $A_{1\mathrm{A}}^{-}$ oxidation is dominated by the value of ${\tau }_3$ instead, which is close to independent from the redox potential of $F_{\mathrm{A}}^{ }$. As discussed for the simulations of Figure 5, the two main phases of $A_1^{-}$ oxidation remain largely unaffected by perturbing $E_{F_{\mathrm{A}}}^0$, even in the large ± 100 mV range, within this reference scenario (Figure 6D), consistently with the phylloquinone oxidation being (mainly) kinetically decoupled from the successive ET steps. For this initial energetic configuration, ${\tau }_2$ displays a dependence to $E_{F_{\mathrm{A}}}^0$ variation similar to the one simulated for ${\tau }_3$ in the weak-driving force model, albeit its starting value is somewhat larger (~250 ns, this is because of the slightly larger value of ${\lambda }_{tot}$, i.e. 1 eV vs. 0.9 eV). This component shall then also be taken as reflecting principally transfer between Fe-S clusters. Since it is not simulated when not considering $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ explicitly ([48] and Appendix S1), it most likely reflects ET transfer between the terminal acceptors. This simulated lifetime is in general agreement with the inter 4Fe-4S clusters electron transfer estimated by Nuclear Magnetic Resonance methods in the ferredoxin of Chromatium vinosum (~300 ns) which is structurally analogous to the PsaC subunit [96]. The value of ${\tau }_4$ is also larger in the reference large driving scheme with respect to the weak-driving force configuration. Its initial value of ~1 μs is the same as the one simulated when $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ are not explicitly taken into account ([48] and Appendix S1) and therefore reflects mainly $F_{\mathrm{X}}^{-}$ oxidation. This lifetime shows a non monotonic dependence on $E_{F_{\mathrm{A}}}^0$, with an apparent discontinuity point at –40 mV potential shift (Figure 6D), which is for a decrease of $F_{\mathrm{X}}^{-}$ oxidation driving force (less exergonic) and an increase in the driving force (more exergonic) for $F_{\mathrm{A}}^{-}$ oxidation by $F_{\mathrm{B}}^{ }$.

Different behaviour with respect to shifts in the $E_{F_{\mathrm{A}}}^0$ value are also simulated for the slowest lifetime (${\tau }_5$). In the reference weak-driving force $A_1^{-}$ oxidation, ${\tau }_5$ shows a quasi-monotonic dependence, becoming faster as the driving force for $F_{\mathrm{A}}^{-}$ oxidation by $F_{\mathrm{B}}^{ }$ increases. For the largest favourable energetic configuration the value of this lifetime approaches the inverse of the out-put from the system $k_{out}^{-1}$~1 μs (Figure 6E). In the large-driving force $A_1^{-}$ oxidation framework, ${\tau }_5$ has a more complex dependence, with the lifetime also becoming faster for increased $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ oxidation driving forces. However, similarly to ${\tau }_4$, which is also re-plotted in Figure 6F for direct comparison a trend discontinuity point at ~–40 mV potential shift is simulated. Under these energetic conditions, the decrease in driving force for $F_{\mathrm{X}}^{-}$ oxidation dominates over the increased driving force for the successive ET steps.

In both cases, the slowest values simulated, especially for conditions in which $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ is significantly endergonic, indicates that the equilibration between the terminal electron acceptors can significantly slow down ET to the diffusible acceptors carries (e.g. ferredoxin). A slow $F_{\mathrm{X}}^{-}\to F_{\mathrm{A}}$ ET reaction would have similar consequences, as shown by the ${\tau }_5$ dependence of Figure 6F as well as the simulations ofFigure 4. The latter effect, to our knowledge has not previously been explicitly considered, whereas it might have important implications for the photosystem functionality.

Figure 7 shows the dependence on the simulated recombination reactions, i.e. when $k_{out}^{ }$= 0. In both energetic scenarios, the lifetime for charge recombination displays a positive correlation with the increase in driving force for the $F_{\mathrm{X}}^{-}\to F_{\mathrm{A}}$ reaction. For values of $E_{F_{\mathrm{A}}}^0$ that yields $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$<0 ( i.e. ~< –25 mV shift) the recombination lifetime depends only weakly on the exact value of $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$. Yet, as the $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ transfer enters the endergonic regime ($\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$>>0), the recombination lifetimes display almost an exponential response with respect to the decrease of $\Delta G_{F_{\mathrm{X}}\to F_{\mathrm{A}}}^0$, i.e. an increase in driving force for the forward and a decrease of backward reactions, in agreement with back-population of $F_{\mathrm{X}}^{-}$ imposing important kinetic limitation to these recombination reactions.

2.4. Effect of Changing the Driving Force of $F_{\mathrm{A}}^{-}$ Oxidation ($\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$)

The driving force for $F_{\mathrm{A}}^{-}$ oxidation can be modulated, without directly affecting the energetics of $F_{\mathrm{X}}^{-}$, by modifying the redox potential of $F_{\mathrm{B}}^{ }$ ($E_{F_{\mathrm{B}}}^0$).

Figure 8 (“weak driving force” energetics) and Figure 9 (“large driving force” energetics) show the simulated kinetics of all redox cofactors considered in the models, upon shifting the $F_B^{ }$ potential by +25 mV (Figure 8/9B), –25 mV (Figure 8/9D) and –50 mV (Figure 8/9F). In this case, for the +25 mV shift, $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}^{ }$ becomes more endergonic ($\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$= +50 meV), for the –25 mV $E_{F_{\mathrm{B}}}^0$ shift $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ are isoenergetic ($\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$= 0 meV), and for the –50 mV shift $F_{\mathrm{A}}^{-}$ oxidation is exergonic ($\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$= –25 meV). The impact of exploring a ±100 mV variation of $E_{F_{\mathrm{B}}}^0$ on the lifetimes describing ET in PSI is shown in Figure 10.

Comparison of the simulations presented in Figure 8 and Figure 9 indicates that, irrespectively of the reference energetic model utilised to describe $A_1^{-}$ oxidation, alterations affecting selectively $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$ do not impact on the phyllosemiquinone oxidation kinetics, as they are, in both cases, basically indistinguishable from the respective reference system. The same hold true for wider changes in $E_{F_{\mathrm{B}}}^0$ of ± 100 mV, since the values of ${\tau }_1$, ${\tau }_2$, and ${\tau }_4$ (Figure 10A and C), which dominates $A_1^{-}$ oxidation in the “weak” driving force scenario, are barely affected by changes in $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$. The same is observed for ${\tau }_1$ and ${\tau }_3$ that dominates $A_1^{-}$ oxidation in the large-driving force $A_1^{-}$ oxidation framework (Figure 10B and D). The value of the ${\tau }_4$ which is also associated with a small-amplitude-μs $A_1^{-}$ oxidation phase in this energetic configuration, barely depends on $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$. On the other hand, as expected, changing $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$ affects the kinetics of electron transfer between the Fe-S clusters, especially the temporal evolutions of $F_{\mathrm{A}}^{-}$ and $F_{\mathrm{B}}^{-}$, whereas that of $F_{\mathrm{X}}^{-}$ is only slightly perturbed. In brief, the overall reduced population of $F_{\mathrm{A}}^{ }$ increased with the decrease in driving force for its oxidation by $F_{\mathrm{B}}^{ }$, and vice versa, so that with the decrease of $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$ the temporal evolution of the reduced state of this cofactors became less concerted and, basically, sequential for large driving forces.

Interestingly, the values of ${\tau }_3$ (“weak driving force” reference energetics, Figure 10C${\tau }_2$ (“large driving force” reference energetics, Figure 10D) show a non-monotonous dependence on variation of $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$, with the slowest values simulated, in both cases, upon $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ becoming close to iso-energetic. These behaviours are similar to those simulated for variations in the $E_{F_{\mathrm{A}}}^0$, which also affected $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$.

The lifetime showing the greatest dependence on $E_{F_{\mathrm{B}}}^0$ is the one representing the effective exit from the system (${\tau }_5$). An almost monotonous decrease (decay deceleration) is predicted upon progressive decreasing $\Delta G_{F_A\to F_B}^0$, for values of $E_{F_{\mathrm{B}}}^0$ under which the $F_{\mathrm{A}}^{-}\to F_{\mathrm{B}}$ reaction falls in the endergonic regime. A softer dependence is simulated instead for the large exergonic regimes, i.e. for $F_{\mathrm{B}}^{ }$ potential shifts >~60–80 mV (Figure 10 E/F). Under the latter conditions, the value of ${\tau }_5$ (at least in the weak-driving force scenario) approaches that of $k_{out}^{-1}$ (~1 μs).

Figure 11 shows the simulated recombination in the absence of an exit from the system. Irrespectively of the energetics associated to $A_1^{-}$ oxidation, the kinetics of charge recombination displayed a weak dependence on $\Delta G_{F_A\to F_B}^0$, maintaining the value simulated for the reference scenario, for shifts in $E_{F_{\mathrm{B}}}^0$ of less than about ~60 mV. A significant increase is however predicted when $F_A^{-}\to F_B$ enters the large endergonic regime, $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$> 60 mV. The overall response of charge recombination kinetics is similar, but less pronounced, than the one predicted upon perturbations of$E_{F_{\mathrm{A}}}^0$, that affects both $\Delta G_{F_{\mathrm{A}}\to F_{\mathrm{B}}}^0$ and $\Delta G_{F_X\to F_A}^0$ (Figure 7), albeit in contrasting fashion. This indicates that, although the repopulation of $F_{\mathrm{X}}^{-}$ imposes the largest kinetic limitations to charge recombination, back-population of $F_{\mathrm{A}}^{-}$ will also slow this reaction when its oxidation by $F_{\mathrm{B}}^{ }$ becomes largely favourable.

3. Materials and Methods

Kinetic simulations were performed as described previously [12,48,49]. In brief the population evolutions of the redox cofactors considered in the kinetic model (as schematically shown in Figure 1) are obtained from solving a system of ordinary differential equations, which in compact matrix form, is defined by $\dot{P}(t)=K_i\cdot P(t)$. $P(t)$and $\dot{P}(t)$ are the vectors describing the temporal evolution of the redox cofactors populations and its first derivative, respectively, and $K_i$ is a square matrix, frequently referred to as the rate matrix, that has as its elements the rate constants between pairs of electron donors and acceptors. The index i corresponds to the number of cofactors (states) considered in the modelling. The system of linear differential equation has a general solution of the form: $P(t)={\displaystyle \sum _{J=1}^i{c}_j{V}_j{e}^{\zeta {}_{j}t}}$, where ${\zeta }_j$ and $V_j$ are, the eigenvalues and the eigenvectors, respectively, that diagonalise the matrix $K_i$, and $c_j$ are a set of scalars that need to be determined according to some given initial conditions, that were here considered to be: $A_{1\mathrm{A}}^{-}(0)$=$A_{1\mathrm{B}}^{-}(0)$= 0.5 and $F_{\mathrm{X}}(0)$=$F_{\mathrm{A}}(0)$=$F_{\mathrm{B}}(0)$= 0. The general solution of the system corresponds to the empirical linear combination of weighted exponentials frequently employed to describe the experimental kinetics. Thus, the eigenvalues, that are univocally determined by the eigen-decomposition of $K_i$ reflect in the experimentally observed lifetimes (${\tau }_{j,obs}^{ }$) through the relation ${\tau }_{j,obs}^{ }=-{\zeta }_j^{-1}$. The product $c_j{V}_j$ relates only indirectly to the experimental pre-exponential amplitudes. The modelled amplitudes represent “weighted” molar fractions, and describe only the changes in the relative cofactor concentration over time, whereas the experimental values are also determined by the monitored properties of the cofactors (e.g. the wavelength-dependent extinction coefficient, fluorescence yield/spectra, and so on).

The link with theory is provided by using explicitly a physical description of the rate constants rather than setting their values as free simulation parameters. To this end, here we adopted the description between a pair of donor-acceptor molecules, originally derived by Hopfield [36,65], when considering a single (mean) nuclear mode coupled to the ET event:

$$\begin{array}{l}{k}_{D\to A}={\displaystyle \frac{2\pi }{\hslash }}{\left|H_{DA}\right|}^2{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-{\displaystyle \frac{{(\Delta G_{{ }^{DA}}^0+{\lambda }_{tot})}^2}{2{\sigma }^2}}}\\ {\sigma }^2={\lambda }_{tot}\hslash \overline{\omega }\coth {\displaystyle \frac{\hslash \overline{\omega }}{2k_{B}T}}\end{array}$$

(1)

Where ${\lambda }_{tot}$ is the total reorganisation energy, $\overline{\omega }$ is the angular frequency of the mean coupled nuclear mode, $\hslash$ is the Dirac constant, $k_B$ is the Boltzmann constant, T is the temperature and ${\left|H_{DA}\right|}^2$ is the electronic coupling term, which can be to a good level approximated as ${\left|H_{DA}\right|}^2={\left|H_0\right|}^2{e}^{-\beta (X_{DA}-3.6)}$ where ${\left|H_0\right|}^2$, is the maximal value of the electronic coupling at wavefunctions overlap; $\beta$, a damping term associated to the probability of tunnelling the potential barrier, X_DA, the edge-to-edge cofactor distance (in Å) and 3.6 represents a correction for the van der Waals radii. When $\hslash \overline{\omega }$<<$k_{B}T$, ${\sigma }_{ }^2=2\lambda k_{B}T$and the expression therefore simplifies, yielding the semi-classical relation derived by Marcus [35].

In the presented simulations values of ${\left|H_0\right|}^2$~1.3 10^-3 eV² (e.g. [48,49]) and $\beta$=1.38 Å^-1 [37] were considered. Specific values of $\Delta G_{DA}^0$, $\hslash \overline{\omega }$ and ${\lambda }_{tot}$ will be discussed below.

It is often convenient to compare the average (${\tau }_{av}^{ }$) and mean (${\tau }_m^{ }$) lifetime obtained from the simulations, rather than comparing the individual pre-exponential amplitudes ($p_i^{ }$) and lifetimes. These parameters are defined, for each of the level (cofactor) considered, as:

$${\tau }_{av}^{ }={\displaystyle \frac{{\displaystyle \sum _i{p}_i{\tau }_i}}{{\displaystyle \sum _i{p}_i}}}$$

(2)

Since the sum at the denominator is zero for all cofactors which have no initial population, two other derived parameter associated to it are considered for these cofactors, the average rise (${\tau }_{av}^r$) and depopulation (${\tau }_{av}^d$) lifetimes. These are obtained by performing the summations in Eqn. 2 considering only the pre-exponential having negative ($p_{i-}^{ }$) or positive ($p_{i+}^{ }$) amplitudes, respectively.

The mean lifetimes represent the first moment of the population evolution, that being described by a linear combination of exponential is defined for each cofactors as:

$${\tau }_m^{ }={\displaystyle \frac{{\displaystyle \sum _i{p}_i{\tau }_i^2}}{{\displaystyle \sum _i{p}_i{\tau }_i}}}$$

(3)

which is always defined irrespectively of the initial amplitudes of the cofactors.

4. Conclusions

In this work the effect of altering the redox potential of the terminal acceptors of PSI, the iron-sulphur clusters, $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$, has been explored employing theory-based kinetic model simulations. This approach allows to predict the electron transfer kinetics between the Fe-S clusters, which has proven cumbersome to address experimentally by spectroscopic methods because of the characteristics of these cofactors and because of the spectral crowding occurring in large chromophore-pigment super-complexes, like PSI. The effect of altering $F_{\mathrm{A}}^{ }$ and $F_{\mathrm{B}}^{ }$ redox reactivity was comparatively explored by considering two energetic scenarios for ET reactions involving the upstream redox cofactors, the iron-sulphur cluster $F_{\mathrm{X}}^{ }$ and the phylloquinones $A_{1\mathrm{A}}^{ }$ and $A_{1\mathrm{B}}^{ }$, as the latter are better characterised by time-resolved spectroscopic approaches. From the presented analysis it is possible to infer that:

i) tuning the $F_{\mathrm{A}}^{ }$ redox potential, together with an alteration of the ET kinetics directly involving this redox centre, also affects the phyllosemiquinones oxidation kinetics, particularly of the slow kinetic phase of this process that is dominated by the depopulation of $A_{1\mathrm{A}}^{-}$. Significant kinetic perturbations are predicted within the “weak driving force” scenario for $A_1^{-}$ oxidation by $F_{\mathrm{X}}^{ }$. Tuning $E_{F_{\mathrm{A}}}^0$ has instead a limited effect on $A_1^{-}$ oxidation kinetics when considering the “large driving force” energetic scheme for these cofactors oxidation reactions. Since both energetic schemes describe adequately several measured parameters, they can both be considered realistic. A more in-depth analysis of mutants affecting $F_{\mathrm{A}}^{ }$ coordination in the PsaC subunit, some of which have already been reported (e.g. [73,74,75,76,77,78,79,80,81,82,83,84]) but not studied with sufficient (ns) temporal resolution, is expected to shed further light on the energy gap between $A_{1(\mathrm{A}/\mathrm{B})}^{ }$ and $F_{\mathrm{X}}^{ }$, especially since the redox potential of the modified $F_{\mathrm{A}}^{ }$ cofactor shall be accessible by direct electrochemistry and henceforth correlated to the kinetic perturbation. Similar approaches have already been explored and proven feasible (e.g. [75,76,77]) yet, as already discussed, the effect of the mutations on the kinetics of forward ET reactions has not investigated in detail yet.

ii) tuning the $F_{\mathrm{B}}^{ }$ redox potential is predicted to affect almost exclusively ET involving the iron-sulphur clusters instead, especially the oxidation kinetics of $F_A^{-}$ and $F_B^{-}$, without significantly impacting on the $A_1^{-}$ oxidation kinetics, irrespectively of the energetic scenario employed to describe the latter reaction.

iii) alterations of the $F_{\mathrm{B}}^{ }$ redox potential, which are also been reported for mutants affecting its coordination site in the PsaC subunit (e.g. [73,74,75,76,77,78,79,80,81,82,83,84]), can however help in elucidating the actual energetics of ET between the terminal iron-sulphur clusters, which might be inferred from differences in both the recombination rates and forward ET to ferredoxin, in comparison with similar perturbation of the $F_{\mathrm{A}}^{ }$ redox potential.

iv) the ~180-ns kinetic phase experimentally retrieved in investigations of $A_1^{-}$ oxidation kinetics [30,33,34,62,63], appears to be associated principally with $F_{\mathrm{A}}^{-}$ oxidation by $F_{\mathrm{B}}^{ }$, whereas it was previously assigned to $F_{\mathrm{X}}^{-}$ oxidation by $F_{\mathrm{A}}^{ }$ [30]. The ~180-ns lifetime appears as a significant contribution also in the $F_{\mathrm{X}}^{-}$ oxidation kinetics, indicating that $F_{\mathrm{A}}^{-}$ oxidation by $F_{\mathrm{B}}^{ }$ is not fully kinetically decoupled from upstream reactions. This lifetime is correctly predicted in the “weak driving force” $A_1^{-}$ oxidation scenario, whereas a slight overestimation (~300 ns) is obtained in the “large driving force” energetic scheme.

v) the need to consider ${\lambda }_{tot}^{ }$~ 1 eV in the large driving force $A_1^{-}$ oxidation scenario led to simulate a relatively slow $F_{\mathrm{X}}^{-}$ oxidation, which couples to a small-amplitude $A_{1\mathrm{A}}^{-}$ oxidation phase in the μs windows. This phase has not been detected experimentally either because not present or not resolvable. Multiple pieces of evidence suggest that $F_{\mathrm{X}}^{-}$ oxidation shall proceed in the sub-μs scale. This can be taken as an indication that ${\lambda }_{tot}^{ }$~ 1 eV represents an upper limit/overestimation of this parameter. The calculated recombination rates, which are limited by the back-population of $F_{\mathrm{X}}^{-}$, argues in the same direction, as they are overestimated, in general, by about an order of magnitude within the “large driving” force scenario (see Appendixes S2 and S3 for further discussion and simulations).

vi) uphill ET from $F_{\mathrm{A}}^{-}$ to $F_{\mathrm{B}}^{ }$ can significantly slow down donation from the terminal electron acceptor(s) to diffusible carriers, at least when pre-bound complexes are formed, i.e. the speed of electron transfer to ferredoxin (or other acceptors) might not depend solely on the rate of transfer from $F_{\mathrm{B}}^{-}$ to the redox active centre in the acceptor molecules, particularly if the intrinsic rate constant for this reaction is as-fast-as, or comparable with, the electron transfer kinetics involving the terminal electron acceptors within the photosystem..