Thermodynamically accurate modeling of the catalytic cycle of photosynthetic oxygen evolution: A mathematical solution to asymmetric Markov chains

Forty-three years ago, Kok and coworkers introduced a phenomenological model describing period-four oscillations in O₂ flash yields during photosynthetic water oxidation (WOC), which had been first reported by Joliot and coworkers. The original two-parameter Kok model was subsequently extended in its level of complexity to better simulate diverse data sets, including intact cells and isolated PSII-WOCs, but at the expense of introducing physically unrealistic assumptions necessary to enable numerical solutions. To date, analytical solutions have been found only for symmetric Kok models (inefficiencies are equally probable for all intermediates, called “S-states”). However, it is widely accepted that S-state reaction steps are not identical and some are not reversible (by thermodynamic restraints) thereby causing asymmetric cycles. We have developed a mathematically more rigorous foundation that eliminates unphysical assumptions known to be in conflict with experiments and adopts a new experimental constraint on solutions. This new algorithm termed STEAMM for S-state Transition Eigenvalues of Asymmetric Markov Models enables solutions to models having fewer adjustable parameters and uses automated fitting to experimental data sets, yielding higher accuracy and precision than the classic Kok or extended Kok models. This new tool provides a general mathematical framework for analyzing damped oscillations arising from any cycle period using any appropriate Markov model, regardless of symmetry. We illustrate applications of STEAMM that better describe the intrinsic inefficiencies for photon-to-charge conversion within PSII-WOCs that are responsible for damped period-four and period-two oscillations of flash O₂ yields across diverse species, while using simpler Markov models free from unrealistic assumptions.     

Earlier researches on the mechanism of oxygen evolution: A personal account

A brief overview is given of the research which led to the discovery of the period-4 oscillations of the flash-induced oxygen production and which is the basis of the generally accepted Kok's model for water splitting and oxygen evolution.In this paper I discuss the earlier work of the groups of Robert Emerson, James Franck, C. P. Whittingham, and myself in relation to the development of new techniques for the detection of photosynthetic oxygen evolution. Also discussed are various hypotheses and speculations related to the concept of a priming photoreaction which is required for oxygen evolution. Finally, I discuss my long scientific collaboration with Bessel Kok which led to the elaboration by Kok of the classical model in which the formation of oxygen requires the sequential accumulation of four positive charges on the same photochemical center.Written at the invitation of Govindjee and Gernot Renger.     

An investigation of water splitting in the Kok scheme of photosynthetic oxygen evolution

The involvement of OH bond breaking in the 4 dark reactions of the Kok scheme of photosynthetic oxygen evolution was investigated using Chlorella and spinach chloroplasts. When the photosynthetic material was suspended in a ²H₂O based medium, the reaction rates in all 4 cases were only slightly reduced as compared to the rates observed in an H₂O based medium. This was evidence that these rate processes were probably not limited by the breaking of an OH bond. Observations were also made on the yields of O₂ from dark adapted Chlorella subjected to a sequence of brief saturating light flashes. The oscillating flash yield sequence observed with algae suspended in ²H₂O showed greater damping of the oscillations than when the algae were suspended in H₂O. A computer fit of the Kok model to these results revealed a slightly higher proportion of misses, (i.e. absorbed quanta that do not drive photochemistry) in the ²H₂O case.     

How fast can Photosystem II split water? Kinetic performance at high and low frequencies

Molecular oxygen evolution from water is a universal signature of oxygenic photosynthesis. Detection of the presence, speed and efficiency of the enzymatic machinery that catalyzes this process in vivo has been limited. We describe a laser-based fast repetition rate fluorometer (FRRF) that allows highly accurate and rapid measurements of these properties via the kinetics of Chl-a variable fluorescence yield (Fv) in living cells and leaves at repetition rates up to 10 kHz. Application to the detection of quenching of Fv is described and compared to flash-induced O₂ yield data. Period-four oscillations in both Fv and O₂, caused by stimulation of primary charge recombination by the O₂ evolving complex (WOC) within Photosystem II (PS II), are directly compared. The first quantitative calculations of the enzymatic parameters of the Kok model (α – miss; β – double hit; S-state populations) are reported from Fv data over a 5 kHz range of flash frequencies that is 100-fold wider than previously examined. Comparison of a few examples of cyanobacteria, green algae and spinach reveals that Arthrospira m., a cyanobacterium that thrives in alkaline carbonate lakes, exhibits the fastest water-splitting rates ever observed thus farin vivo. In all oxygenic phototrophs examined thus far, an unprecedented large increase in the Kok α and β parameters occur at both high and low flash frequencies, which together with their strong correlation, indicates that PS II-WOC centers split water at remarkably lower efficiencies and possibly by different mechanisms at these extreme flash frequencies. Revisions to the classic Kok model are anticipated.     

Flash-yield pattern for photosynthetic oxygen evolution in Chlorella and chloroplasts as a function of excitation intensity.

A study was carried out of the changes in flash-yield pattern for oxygen evolution at various light intensities. Oxygen measurements were made on algal and chloroplast samples, using a Joliot-type polarographic electrode. A mathematical model to describe the behavior of oxygen-evolving systems at various flash intensities was developed based on the binomial distribution. This model is capable of accurately predicting the oxygen flash-yield pattern over two orders of magnitude change in light intensity. The observed oxygen flash yields are accounted for at all flash intensities without changing the probability for misses and double hits by oxygen-evolving systems. It is concluded that intrinsic misses and double hits are observed, which are nearly independent of flash intensity. Also, the apparent optical cross section for oxygen evolution is found to increase as flash intensity decreases. It is suggested that inhomogeneity exists in the size of antenna-pigment aggregates associated with photosystem II reaction centers.     

Improvement of four sigma analysis for the investigation of oxygen evolution by Photosystem II

We present here an improvement to the analysis of oxygen evolution with four sigma coefficients (4-S) by computing z, the sum of the S-state probabilities, which was introduced earlier (Delrieu and Rosengard 1987, Biochim Biophys Acta 892: 163–171). We demonstrate that z is equal to the ratio of two consecutive Mean Y (the estimation of the steady state oxygen production based on local properties) found by three sigma analysis. The quantity z is useful for computing double-hits, and for showing the inactivation/activation processes of PS II complexes. Three sigma analysis assumes z=1 exactly; since this is not verified, it is argued that four sigma analysis is closer to the real workings of the water oxidizing complex. Oxygen evolution can then be interpreted in the frame of a modified Kok's model where the sum of the probabilities equals z. We therefore suggest that the closer fitting of four sigma analysis to oxygen production data is not simply due to an extra, unnecessary variable, but to the fact that PS II complexes can be inactivated and reactivated under flashing light. Finally, in order to facilitate the use of four sigma analysis, a computer program is made available upon request.     

Period-four oscillations of the flash-induced oxygen formation in photosynthesis*

In this minireview, the earlier researches that led to the discovery of the period-four oscillations of the flash-induced oxygen formation are presented. It also includes the background of the classical model proposed by Bessel Kok, in which the formation of oxygen requires the sequential accumulation of four positive charges on the donor side of the same reaction center. This revised version was published online in August 2006 with corrections to the Cover Date.     

The primary structure of D1 near the QB pocket influences oxygen evolution

Photosystem II (PS II) is the site of oxygen evolution. Activation of dark adapted samples by a train of saturating flashes produces oxygen with a yield per flash which oscillates with a periodicity of four. Damping of the oxygen oscillations is accounted for by misses and double hits. The mechanisms hidden behind these parameters are not yet fully understood. The components which participate in charge transfer and storage in PS II are believed to be anchored to the heterodimer formed by the D1 and D2 proteins. The secondary plastoquinone acceptor Q_B binds on D1 in a loop connecting the fourth and fifth helices (the Q_B pocket). Several D1 mutants, mutated in the Q_B binding region, have been studied over the past ten years.In the present report, our results on nine D1 mutants of Synechocystis PCC 6714 and 6803 are analyzed. When oxygen evolution is modified, it can be due to a change in the electron transfer kinetics at the level of the acceptor side of PS II and also in some specific mutants to a long ranging effect on the donor side of PS II. The different properties of the mutants enable us to propose a classification in three categories. Our results can fit in a model in which misses are substantially determined by the fraction of centers which have Q_A ^- before each flash due to the reversibility of the electron transfer reactions. This idea is not new but was more thoroughly studied in a recent paper by Shinkarev and Wraight (1993). However, we will show in the discussion that some doubts remain as to the true origin of misses and double hits.Abbreviations BQ p-benzoquinone - Chl chlorophyll - D1 and D2 proteins of the core of PS II - DCMU 3-(3,4-dichlorophenyl)-1,1 dimethyl urea - OEC oxygen evolving complex - P₆₈₀ chlorophyll center of PS II acting as the primary donor - PS II Photosystem II - Q_A and Q_B primary and secondary quinone electron acceptor - TL thermoluminescence     

The Kok Effect in Chlamydomonas reinhardi

A Haxo-Blinks rate-measuring oxygen electrode together with a modulated light source gave an average current signal (change in net O₂ exchange) and a modulated current signal (photosynthetic O₂ evolution). Using this apparatus, net O₂ exchange and photosynthetic O₂ evolution at low intensities have been studied in the green alga, Chlamydomonas reinhardi. At both 645 nm and 695 nm, the curves of net O₂ exchange as a function of light intensity were steeper at lowest intensities than about compensation, indicative of the Kok effect. The effect was greater at 695 nm than at 645 nm. The corresponding curves of photosynthetic O₂ evolution, on the other hand, showed no Kok effect; here, the slope was lowest at lowest intensity. The absence of the Kok effect in O₂ evolution, together with its sensitivity to monofluoroacetic acid, show that it is due to an interaction of photosynthesis and respiration. The effect was exaggerated by 3-(3,4-dichlorophenyl)-1,1-dimethylurea. In the presence of concentrations of this inhibitor sufficient to inhibit O₂ evolution completely, a light-induced change in net O₂ exchange remained. This was interpreted as a system I dependent depression of respiratory O₂ uptake. The Kok effect remained undiminished in concentrations of carbonyl cyanide m-chlorophenylhydrazone and 2,4-dinitrophenol which partially uncoupled either oxidative phosphorylation alone or both oxidative and photosynthetic phosphorylations. The above results can be explained within a model of the Kok effect in which O₂ uptake is depressed by diversion of reductant away from respiratory electron transport and into photosystem I. The same photodepression of O₂ uptake also appears to account for a transient in net O₂ exchange seen in several algae upon turning off the light.     

Anomalies in the kinetics of photosynthetic oxygen emission in sequences of flashes revealed by matrix analysis. Effects of carbonyl cyanide m-chlorophenylhydrazone and variation in time parameters

The model of Kok et al. (Kok, B., Forbush, B. and McGloin, M. (1970) Photochem. Photobiol. 11, 457–475) is considered the best kinetic explanation of the damped oscillations of O₂ evolution induced in higher plants by a sequence of brief saturating flashes. Matrix analysis applied to this model shows that the parameters involved (distribution of S states at zero time, probabilities of transition between states induced by a flash) cannot be completely known from the O₂ yield sequence, Y_n. However, four quantities, with limited content of information, are readily derived from data, without additional assumptions. They are σ₁, σ₂ and σ₃, three quasi-symmetrical functions of the transition coefficients, and $\text{Y}$, a weighed average of four consecutive Y_n values. The extent of misses and double hits and their variations can be qualitatively ascertained by inspection of the relative values of σ₁, σ₂ and σ₃. In a regular sequence (strictly obeying Kok's model), all four quantities should be constant along the time axis.It is shown that actual sequences are seldom regular, in particular in the following conditions: (1) variable flashing frequency, (2) addition of carbonylcyanide m-chlorophenylhydrazone, (3) incomplete deactivation, (4) change of flashing frequency at steady state.In order to account for these anomalies, it is proposed to modify Kok's model by introducing, in parallel to the four state storage entity (S states), a side carrier C, which can reversibly exchange a positive charge with it. In the new model, the transition coefficients are essentially time varying, thus producing a nonregular behaviour of Y_n sequences.     

Control of misses in oxygen evolution by the oxido-reduction state of plastoquinone in Dunaliella tertiolecta

The way misses happen in oxygen evolution is subject to debate (Govindjee et al. 1985). We recently observed a linear lowering of the miss probability with the flash number (Meunier and Popovic 1989). Therefore, we investigated in Dunaliella tertiolecta the link between the average miss probability and the redox state of plastoquinone after n flashes. The effect of flashes was to oxidize the plastoquinone pool; we found that the oxidation of plastoquinone highly correlated (linear regression: R ²=0.996) with the lowering of the miss probability. The flash frequency was found to affect both the miss probability and the redox state of plastoquinone. When pre-flashes were given using a high flash frequency (10 Hz), the plastoquinone pool was oxidized and misses were low; however, if long dark intervals between flashes were used, the oxidizing effect of flashes was lost and the misses were high. We could not explain our results by assuming equal misses over all S-states; but unequal misses, caused by deactivations, were coherent with our results. We deduced that chlororespiration was responsible for the reduction of plastoquinone in the dark interval between flashes. We compared oxygen evolution with and without benzoquinone, using a low flash frequency (0.5 Hz) for maximum misses. Benzoquinone lowered the misses from 34% to 3%, and raised the amplitude of oxygen evolution by more than a factor of two (2). From this we deduced that the charge carrier C postulated to explain misses (Lavorel and Maison-Peteri 1983) did not account for more than 3% of miss probability in Dunaliella tertiolecta. These results indicate that the misses in oxygen evolution are controlled by the redox state of plastoquinone, through deactivations.     

Improved 5-step modeling of the Photosystem II S-state mechanism in cyanobacteria

We present a model of the S-state mechanism, as well as an improved eigenvalue analysis, that integrate into a coherent ensemble several features found since the S-state model was initially developed. These features include the presence of S_–1, deactivations in the dark interval between flashes, and the change in the number of active PS II centers by photoinhibition or photoactivation. A new feature is the capacity to predict the steady-state distribution of S-states under conditions of steady photoinhibition or photoactivation. The improved eigenvalue analysis allowed the calculation of the initial S-state distribution. In addition, the model resolved true photochemical misses from apparent misses due to deactivations in the dark interval between flashes. The model suggested that most of the misses that are commonly reported are due to deactivations, and not to an intrinsic inefficiency of the photochemical mechanism of PS II. Because models that allow double-hits encompassing the S₂ to S₃ transition often predict negative initial quantities of S₂ in cyanobacteria, our proposed model specifically prohibited them. The model accounts for inhomogeneous misses and a steady-state distribution of the type (S₂)(S₁)>(S₃)(S₀). This 5-step model uses only 4 probabilities, and is therefore easy to handle. The use of this model is critical for the analysis of several cyanobacterial strains, as well as for any species that show non-negligible deactivations in the dark interval between flashes.     

Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading-unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah-Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.     

A mathematical model of oxygen transport in intact muscle with imposed surface oscillations

A one-dimensional (1D) reaction-diffusion equation is presented to model oxygen delivery by the microcirculation and oxygen diffusion and consumption in intact muscle. This model is motivated by in vivo experiments in which oscillatory boundary conditions are used to study the mechanisms of local blood flow regulation in response to changes in the tissue oxygen environment. An exact periodic solution is presented for the 1D 'in vivo' model and shown to agree with experimental data for the case where the blood flow regulation system is not activated. Approximate low- and high-frequency solutions are presented, and the latter is shown to agree with the pure diffusion solution in the absence of sources or sinks. For the low frequencies considered experimentally, the 1D in vivo model shows that as depth increases: (i) the mean of tissue O(2) oscillations changes exponentially, (ii) the amplitude of oscillations decreases very rapidly, and (iii) the phase of oscillations remains nearly the same as that of the imposed surface oscillations. The 1D in vivo model also shows that the dependence on depth of the mean, amplitude, and phase of tissue O(2) oscillations is nearly the same for all stimulation periods >30s, implying that experimentally varying the forcing period in this range will not change the spatial distribution of the O(2) stimulation.     

The influence of anions on oxygen evolution by isolated spinach chloroplasts

A study has been made of the onset of chloride deprivation on the oxygen-evolving characteristics of isolated spinach chloroplasts. Using a modulated oxygen electrode it is found that the type of inhibition depends on the anion replacing chloride in the bathing medium. With nitrate a large increase in phase lag accompanies a relatively small inhibition which can be shown to be consistent with a decrease in the rate constant of the reaction which limits the rate of electron transport between water and Photosystem II. With sulphate there is a very small phase change but a larger inhibition which suggests that replacing chloride with sulphate in an electron-transport chain shuts off that chain. With acetate there is a moderate increase in phase lag and the largest inhibitory effect. The phase-lag increase suggests that acetate is affecting the same chloride-sensitive site as nitrate. However, the inhibition cannot be explained by this effect alone and points to the existence of a second chloride-sensitive site. Of the four forward reactions associated with the Kok model of oxygen evolution (Kok, B., Forbush, B. and McGloin, M. (1970) Photochem. Photobiol. 11, 457–475) only S1₃ → S₀ is slowed down when chloride is replaced by nitrate. This reaction is not slowed down by replacing chloride with sulphate.     

Determination of carbon dioxide evolution rate using on-line gas analysis during dynamic biodegradation experiments

Respirometry is a precious tool for determining the activity of microbial populations. The measurement of oxygen uptake rate is commonly used but cannot be applied in anoxic or anaerobic conditions or for insoluble substrate. Carbon dioxide production can be measured accurately by gas balance techniques, especially with an on-line infrared analyzer. Unfortunately, in dynamic systems, and hence in the case of short-term batch experiments, chemical and physical transfer limitations for carbon dioxide can be sufficient to make the observed carbon dioxide evolution rate (OCER) deduced from direct gas analysis very different from the biological carbon dioxide evolution rate (CER).To take these transfer phenomena into account and calculate the real CER, a mathematical model based on mass balance equations is proposed. In this work, the chemical equilibrium involving carbon dioxide and the measured pH evolution of the liquid medium are considered. The mass transfer from the liquid to the gas phase is described, and the response time of the analysis system is evaluated.Global mass transfer coefficients (K(L)a) for carbon dioxide and oxygen are determined and compared to one another, improving the choice of hydrodynamic hypotheses. The equations presented are found to give good predictions of the disturbance of gaseous responses during pH changes.Finally, the mathematical model developed associated with a laboratory-scale reactor, is used successfully to determine the CER in nonstationary conditions, during batch experiments performed with microorganisms coming from an activated sludge system. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 53: 243-252, 1997.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999