Revised manuscript (June 10, 2005)

Non-Linear Enhancement of Oxygen Evolution in Thylakoid Membranes:

Modelling the Effect of Light Intensity and

ββββ

-Cyclodextrin Concentration

Mário Fragata,* and Subhan Dudekula1

Université du Québec à Trois-Rivières, Département de Chimie-Biologie, Section de Chimie et Biochimie, Trois-Rivières, Québec, G9A 5H7, Canada

*Corresponding author. Phone: 819-3765011. Fax: 819-3765057. E-mail: [email protected].

1

Present address: Center for Cellular and Molecular Biology, Habsiguda, Hyderabad, Andhrapradesh, 500007 India.

Abstract. Electron transport through photosystem II, measured as oxygen evolution (OE), was investigated in isolated thylakoid membranes treated with β-cyclodextrin (β-CD, a cyclic oligosaccharide constituted of 7 α-D-glucose residues linked by α-1,4 glycosidic bonds) and irradiated with white light of variable intensity. First, we found that the light-response curves of oxygen evolution are well fitted with a hyperbolic function, the shape of which is not affected by the β-CD concentration. Secondly, we showed that in conditions of irradiation with white light of saturating intensity (~5000 µmol photons.m-2.s-1) β-CD enhances the oxygen evolution in the thylakoid membranes according to a sigmoid function displaying a sharp inflection point, or transition. Unexpectedely, this β-CD effect is not observed at irradiances of less than ~300 µmol photons.m-2.s-1. We attempted the theoretical analysis of the combined effect of irradiance and β-CD concentration on oxygen evolution (OE_th). For this purpose, the effect of irradiance (I) was modelled with a hyperbola (i), and the β-CD concentration (C) contribution with a Hill equation, i.e., a sigmoid function (ii). The mathematical simulations generated the general expressions

OE_th = [OE_max(0) G₁(C)] I / [L_½(0) G₂(C) + I] (i) and G_i(C) = 1 + p[Cn/ (K_½n + Cn)] (ii) where OE_max(0) is the OE maximum (OE_max) in the absence of β-CD, L_½(0) the photon flux density giving OE_max/2 in the absence of β-CD, and G₁(C) or G₂(C) are obtained from G_i(C) where i is 1 or 2, n the Hill coefficient, p a parameter to account for the β-CD-mediated maximum OE increase, and K_½ the β-CD concentration giving half-maximal OE activity. The results of the calculations yielded the expression

OE_th = 151[1 + 3.3C4.8/(13.14.8 + C4.8)] I

/

{97.5[1 + 5.2C7.8/(14.87.8 + C7.8)] + I} (iii) which agrees well with the experimental data for a broad range of I and C. Note that for C=0, eq (iii) reverts to the light-response curve of oxygen evolution in the absence of β-CD. We conclude that eq (iii) is a good approximation of the combined effect of irradiance and β-CD concentration, meaning that the model has a significant value for predicting the outcome of associated photochemical and biochemical reactions.

I. Introduction

The cyclodextrins (CD), a group of cyclic oligosaccharide constituted of various units of D-glucose linked by α-1,4 bonds,1,2 are currently used in various fundamental and applied aspects of the chemical and biomedical sciences.1-4 The CDs have a truncated cone geometry with a narrower- (nR) and a wider-rim (wR) where the primary 6-hydroxyl groups are located on the nR side and the secondary 2- and 3-hydroxyls on the wR side.1,2 This molecular arrangement renders the external surface of the CDs hydrophylic, whereas the internal cavity is less polar or hydrophobic. The low polarity of the cyclodextrins interior favours their interaction with the hydrophobic moities of neutral or ionic molecules of small size, therefore facilitating the formation of a wide variety of inclusion complexes which are the framework of their mode of action.

In the past few years, the cyclodextrins were applied successfully to the study of the photosynthetic activity in the thylakoid membrane of plant chloroplasts.5-10 It was shown that β-CD, a cyclodextrin containing seven D-glucose units, affects the spectroscopic characteristics and the electron transport properties of isolated thylakoid membranes.7-10 In short, β-CD induces a red-shift from 681 to 683 nm in the absorption and second derivative spectra of the thylakoids7 which is assigned to perturbations affecting the Q_Y(0,0) electronic transition in the plane of the chlorophyll (Chl) a molecule (see refs 11 and 12). However, the molecular interactions underlying this spectral shift are still not completely understood.

A plausible explanation is to attribute the spectral shift to cyclodextrin interactions with the π-electron system of the Chl tetrapyrrole macrocycle in the pigment-proteins of photosystem II (PSII). But a recent study using a combination of UV/vis absorption, circular dichroism, NMR and steady-state and time resolved fluorescence measurements, does not seem to support this argument since it showed that heptakis(2,3,6-tri-O-methyl)-β-CD and hydroxypropyl-β-CD in aqueous solution with Chl a form respectively 1:1 and 1:2 inclusion complexes with the phytyl chain of the pigment.13 In the thylakoid membrane, a more

effective role of β-CD is a molecular or structural membrane rearrangement causing new pigment-pigment interactions that would give rise to the Q_Y(0,0) electronic transition changes at the origin of the spectral shift discussed above.

Another line of evidence favouring the structural aspect of the β-CD effect comes from a study of the fluorescence induction in isolated thylakoid membranes.8 It was shown that β -CD enhances the transfer of electrons between the excited state of the reaction center Chl of PSII (P680*) and the oxidized pheophytin (Phe), then from Phe- to the primary quinone Q_A. An explanation of these observations is the increase of light absorption by P680 and the re-activation of closed photochemical centres upon treatment of the thylakoid membranes with the cyclodextrin.8 We emphasize, in this respect, that the oxygen evolution yield is dependent on the number of open photochemical centers, or traps, in the thylakoid membrane, and is functionally related to the photochemical efficiency of chlorophyll absorption of excitation photons.14

A novel finding is the observation that the electron transport through photosystem II, measured as oxygen evolution, varies with the β-CD concentration according to a S-shaped, or sigmoid function displaying a sharp inflection point, or transition.9 The non-linearity of the β-CD concentration-response curves has been justified on the ground of an augmentation of the structural and functional cooperativity between PSII units. It is important to note, however, that the experiments reported in ref 9 were performed with isolated thylakoid membranes irradiated with white light of saturating intensity, i.e., about 5000 µmol photons.m-2.s-1. What is more, it is argued in previous works that the linear and non-linear effects observed in photosynthesis are brought about mainly by differences in the light intensity levels used to irradiate the plant material.15-17 To examine further this question, we recall that the relationship between electron transport through PSII measured as the rate of oxygen evolution (νO₂) and the quantum yield of PSII (Φ_PSII) is expressed as18-20

where I is the rate of photon absorption per PSII unit, [RC] the concentration of reaction centers, i.e., the density of PSII units, and Φ_PSII the (F_m – F_o)/F_m ratio obtained from Chl a fluorescence induction (FI) measurements. In FI experiments, F_o is the minimal level of chlorophyll fluorescence when all PSII centres are open after dark adaptation of the plant material, and F_m the maximal level of fluorescence when all PSII centres are closed after a saturating light flash.20,21

Generally, νO₂ in eq 1 ought to be a linear function of any of the independent variables in the right hand side of the equation. Nevertheless, eq 1 was shown to be linear at high irradiance, but not in low light intensity conditions where significant deviations from linearity are observed.15-20,22 This is an intriguing question that so far has not been solved satisfactorily. In this perspective, we examine hereunder the effect of β-CD on the oxygen evolution in thylakoid membranes irradiated with a broad range of light intensities, and put special emphasis on non-saturating low irradiance conditions.

In the first part of this work (section III), we investigate whether the non-linear effect of the β-CD concentration on the oxygen evolution in thylakoid membranes irradiated with white light of saturating intensity9 is also seen upon irradiating the membranes with low light intensities. In the second part of the work (section IV), we modelled the combined effect of light intensity and β-CD concentration on oxygen evolution with a set of mathematical expressions where the effect of irradiance is represented with a hyperbolic function, and the β-CD effect is fitted with an equation of the Hill type for the hypothesis of an allosteric transition with many binding sites (see discussions in ref 23). We show that the results of the calculations agree well with the experimental data, therefore confirming the quality of the model. The article ends with some remarks on the phenomenological significance of the mathematical conclusions (section V).

II. Materials and Methods

Chemicals. β-Cyclodextrin was purchased from Fluka-Chemie (Buchs, Switzerland) and 2,6-dichloro-p-benzoquinone (DCBQ) was obtained from Pfaltz and Bauer (Waterbury, CT). All other chemicals were from Fisher Scientific Company (Fair Lawn, NJ).

Isolation of thylakoid membranes. Primary leaves from 6-8 day old barley seedlings were used to isolate thylakoids from chloroplasts according to procedures described before.24,25 Briefly, the leaves were homogenized in a buffer containing 50 mM Tricine-NaOH (N-tris[hydroxymethyl]-methylglycine-NaOH) (pH 7.8), 400 mM sorbitol, 10 mM NaCl and 5 mM MgCl₂ (buffer A) at 273 K. The resultant slurry was filtered through eight layers of cheesecloth. The filtrate was centrifuged at 1000 g for 5 min at 277 K to precipitate the chloroplasts which were centrifuged again upon suspension in buffer A. This chloroplast preparation was collected in a buffer containing 50 mM Tricine-NaOH (pH 7.8), 10 mM NaCl and 5 mM MgCl₂ (buffer B), and centrifuged immediately at 1000 g for 5 min at 277 K. The pellet contained thethylakoid membranes which were dispersed in a buffer containing 20 mM MES-NaOH (2-[N-morpholino]ethanesulfonic acid-NaOH) (pH 6.5), 400 mM sucrose, 15 mM NaCl and 5 mM MgCl₂ (buffer C), and centrifuged at 1000 g for 5 min at 277 K. The final pellet was diluted in buffer C to give a final chlorophyll concentration of 2 mg/mL, and stored at 143 K. The chlorophyll concentration in the thylakoid preparations was measured in 80 % (v/v) acetone.26

The polypeptide composition of the isolated thylakoid membranes was analysed by sodium dodecylsulfate-polyacrylamide gel electrophoresis.27 The standard proteins and the thylakoid polypeptides were resolved on a linear gradient gel as described in ref 28 that gives also the procedures for staining and destaining the electrophoresis gels. To estimate the molecular mass of the proteins a set of markers was used (RPN 800 kit from Amersham International plc, Buckinghamshire, England).

membranes used in oxygen evolution experiments were treated with β-CD as reported in refs 7 and 8. Briefly, β-CD was solubilized in an incubation buffer (pH 6.5) containing 20 mM MES-NaOH and 400 mM sorbitol to yield concentrations of 2 to 16 mg/mL upon addition of the thylakoid membrane sample (50 µg Chl/mL). These suspensions were incubated at 273 K for 10 min in darkness followed immediately by a centrifugation at 8000 g for 5 min at 277 K to precipitate the thylakoid membranes, thus eliminating unbound β-CD. The control untreated thylakoid membranes and the β-CD-treated samples were resuspended in a measurement buffer containing 20 mM MES-NaOH, 400 mM sucrose, 15 mM NaCl and 10 mM MgCl₂ (pH 6.5) to perform oxygen evolution determinations.

Electron transport measurements and irradiation of the thylakoid membranes.

Electron transport through photosystem II estimated as oxygen evolution was measured with a Hansatech Oxygen Electrode (Hansatech Instruments Ltd., Norfolk, UK) connected to a temperature controlled water circulator at 298 K. The assay mixtures contained untreated or β-CD-treated samples of thylakoid membranes (12.5 µg Chl/mL) in a oxygen evolution measurement buffer (pH 6.5) constituted of 20 mM MES-NaOH, 400 mM sucrose, 15 mM NaCl, 5 mM MgCl₂, and 350 µM 2,6-dichloro-p-benzoquinone which accepts electrons at the Q_A site, i.e., the primary quinone acceptor of PSII.29

Irradiation of the thylakoid membrane suspensions was performed with white light from a Fiber-Lite High Intensity Illuminator, model 180, from Dolan-Jenner Industries Inc. (Lawrence, MA). Incident photon flux densities were measured with a Quantum Photometer, model LI-185B, from LI-COR, Inc. (Lincoln, NE) which was equipped with a LI-190SB quantum sensor. Photon flux densities of 60 to 1100 µmol photons.m-2.s-1 were obtained with a series of calibrated optical neutral-density filters (Tiffen Optical Co., Roslyn Heights, LI, NY). The uniformity of the spectral characteristics of the neutral density filters was checked by registering the absorbance of chlorophyll a solutions in diethyl ether.

Data analysis. The theoretical curves displayed in sections III and IV are fits of the experimental data with mathematical expressions discussed in refs 14, 23, 30 and 31. The

software programs used are Origin, version 5, from Microcal Software, Inc. (Northampton, MA), Maple V, release 5.1, from Waterloo Maple Inc. (Waterloo, ON, Canada), and Mathematica, version 4.0.1, from Wolfram Research (Champaign, IL). For several other calculations, we used the QuickBasic programming language, version 4.0, from Microsoft Corporation.

III. Results and Discussion

A. Light-Response Curves of Oxygen Evolution in the Thylakoid Membrane. Figure 1 displays the oxygen evolution (OE) observed in thylakoid membranes irradiated with white light of photon flux densities (I) between 60 and 1000 µmol.m-2.s-1. We note first that the variation of OE with I is characterized by a sharp increase at low light intensities followed by a quite lower dOE/dI rate at high irradiance. This trend has been usually reported in the literature (see, e.g., refs 18 and 32). The first question addressed here is to investigate which mathematical model is the best representation (or mathematical solution) of the experimental OE vs I data. This is specially important for the study of the combined effect of light intensity and β-CD concentration on the oxygen evolution in the thylakoid membrane undertaken in section IV.

Hereunder, we use two mathematical models, i.e., an exponential function (Model I) or a hyperbola (Model II), which have relevant phenomenological significance. That is, in Model I we determine the best mathematical fit with the hypothesis of the cumulative one-hit Poisson probability distribution,14 and in Model II we use the steady-state approximation for the case of alternating slow and fast reactions.33

Model I. The experimental data displayed in Figure 1 is fitted with an exponential expression for OE_th, i.e., the theoretical oxygen evolution, as a function of I, the photon flux density,

OE_th = OE_th(max) (1 - e-kI) (2)

where OE_th(max), the maximum oxygen evolution, is given in µmol oxygen evolution.(mg Chl.h)-1, I in µmol photons.m-2.s-1, and k (= cross section for absorption of a photon x duration of illumination) in m2.(µmol photons)-1.s. The best fit of OE_th vs I is the theoretical light-response curve shown in Figure 1. The mathematical simulations performed with the curve fitting tool of the Origin software (see Materials and Methods) yielded the expression

OE_th = 132.9 (± 3.6) (1 - e-0.0082 (± 0.0007) I) (3)

that clearly cannot represent adequately the experimental oxygen evolution data obtained with light intensities between 60 and 1000 µmol photons.m-2.s-1.

It is interesting that eq 2 is similar to the mathematical expression for the cumulative one-hit Poisson probability distribution developed by Mauzerall and Greenbaum,14 that is,

Y(z) = Y_o(z) (1 – e-σ(z)E) (4)

where Y(z) is the yield of a photoproduct, Y_o(z) the yield of the photoproduct per hit or closing of a reaction center, σ(z) the optical cross section for absorption of a photon by a unit forming z, E the fluence, i.e., the photons per unit area, and the term e-σ(z)E gives the fraction of targets which were not hit. A simple transformation of eq 2 to change -kI into -k_o(z)It where the term It, with t = duration of irradiation, is the fluence (or E in eq 4), shows that k_o(z) is the equivalent of the optical cross section for absorption of a photon by a unit forming z, or σ(z) (eq 4).

The above discussions and the theoretical result displayed in Figure 1 are a good indication that the cumulative one-hit Poisson probability distribution14 does not yield a reliable representation of the experimental data. This conclusion is further illustrated on comparing

the results in Figure 1 (untreated thylakoid membranes) with data obtained with β-CD-treated thylakoid membranes (Figure 2 and Table 1). Figure 2 shows the light-response curves of oxygen evolution in thylakoid membranes treated with β-CD concentrations from 10 to 14 mM. For comparison purposes, the figure contains also the results obtained with control (untreated) preparations.

Figure 2 shows that all β-CD concentrations used enhance considerably the oxygen evolution. Furthermore, the dOE/dI rate varies according to a trend which is similar to the one observed in control (untreated) thylakoid membranes. The results of the calculations are summarized in Table 1. The table collects (i) the oxygen evolution data obtained experimentally, OE_obs, on irradiating untreated and β-CD-treated thylakoids thylakoid membranes with white light of saturating intensity, i.e., ~5000 µmol photons.m-2.s-1, (ii) the theoretical oxygen evolution maxima, OE_th(max), computed for β-CD concentrations between 10 and 14 mM, and (iii) the values of the parameter k (see eq 2).

Table 1 shows that the OE_th(max) calculated according to Model I are quite different from OE_obs. For example, in thylakoid membranes treated with 12 mM β-CD one gets OE_th(max) = 278.2 (± 7.2) µmol O₂ evolution.(mg Chl.h)-1 which is about 24 % smaller than OE_obs at the same β-CD concentration, i.e., 364.9 (± 15.2) µmol O₂ evolution.(mg Chl.h)-1 (cf. Table 2). In brief, one sees in Table 2 that the estimated errors in the calculations of OE_th(max) with eq 2 vary between about 15 and 24 %.

Model II. We show in Figure 1 that the experimental data obtained with light intensities from 60 to 1100 µmol photons.m-2.s-1 are well fitted with the hyperbolic expression

OE_th = OE_th(max) I / (L_½ + I) (5)

where OE_th(max) is the theoretical oxygen evolution maximum that would be observed at very high irradiance (i.e., when the maximum possible number of PSII reaction centers is open), L_½ the irradiance giving OE_th(max)/2, and I is defined above. OE_th(max) and L_½ for

control thylakoid membranes (i.e., not treated with β-CD) were computed with eq 5 using the curve fitting tool of the Origin software (see Materials and Methods).

First, Table 1 shows that calculation of the data obtained with control (not treated) thylakoid membranes yields OE_th(max) = 151.0 (± 3.8) µmol O₂ evolution.(mg Chl.h)-1 and L_½ = 97.5 (± 9.0) µmol photons.m-2.s-1. We note that the OE_th(max) value is quite close to OE_obs which is about the maximum oxygen evolution obtained in experimental conditions of high irradiance. Secondly, Table 1 collects the result of calculations of data obtained with thylakoid membranes treated with 10-14 mM β-CD. In addition, Table 2 reveals that the estimated error in the determination of OE_th(max) with eq 5 is between 0.7 and 12 % which is significantly smaller than the error observed with the exponential function of eq 2 (Model I), i.e., between 14 and 24 % (see discussion above).

We conclude therefore that, in the conditions of our experiments, the steady-state approximation33 expressed by the hyperbolic function of eq 5 is a better representation of the effect of light intensity on the electron transport through PSII in isolated thylakoid membranes than the exponential model of eq 2 for the cumulative one-hit Poisson probability distribution.14 Furthermore, we note that none of the β-CD concentrations used in the experiments reported here change the hyperbolic shape of the light-response curves displayed in Figure 2. This corroborates the argument that Model II is a valid representation of the variation of oxygen evolution with the light intensity. Finally, it is interesting to remark that some experimental data published in the literature can as well be correctly fitted with an hyperbolic function (see note 34).

Toward a Phenomenological Interpretation of Model II. The OE_th dependence on I seen in Figure 1 is explained hereunder in terms of the steady-state approximation33 of the electron transfer kinetics which is consistent with the conditions of our experiments. An attractive aspect of the steady-state approximation is that it permits the mathematical representation of a series of coupled sequences of reactions where slow and fast kinetics (or vice versa) alternate. This concept is applied in Scheme I to describe the energy transfer from

an excited chlorophyll from the PSII antenna, i.e., Chl

a

_II*, to the PSII reaction center Chl (i.e., P680) followed by the transfer of an electron from P680* to an oxidized peophytin (Phe) (reaction 1), then from Phe- to the primary quinone Q_A (reaction 2). For the sake of a more comprehensible photochemical context, we have also represented in Scheme I the pathway of P680+ reduction by electrons originating in the H₂O photolysis and transferred to the oxidized Mn cluster (MnC+), then from MnC to the tyrosine Y_Z (Tyr161) in the D₁ protein (see, e.g., ref 35).

SCHEME I. Electron transfer through photosystem II

Reaction lifetime Ref ________________ _____

Reaction 2:

Reaction 1:

Q_A → Q_A k₂ ↑ e- Phe → Phe- → Phe k₁↑e- P680 → P680* → P680+ → P680 ↑ hν ↑ e- Chl

a

_II* Y_Z← Y_Z- ← Y_Z < 400 ps 6 to 10 ns (fast) (slow) 36,37 36,37 ↑ e ΜnC+→ MnC → MnC+ ↑ e 2H₂O → O₂ + 4H+

In Scheme 1, k₁ = [P680+][Phe-] / [P680*] [Phe] (reaction 1) and k₂ = [Phe][Q_A-] / [Phe-] [Q_A] (reaction 2). One sees that reaction 1 is slow (6 to 10 ns) comparatively to reaction 2 which takes place in less than 400 ps. This means that Phe- never attains a significant concentration during the course of the reaction since, as it is formed, it reacts rapidly with the primary quinone Q_A to produce the reduced form Q_A-. Hence, the velocity of reaction 1 is about identical to the velocity of reaction 2, that is,

k₁[P680*][Phe] ≅ k₂[Phe-][Q_A] (6) According to the steady-state approximation, equilibria such as the one described by eq 6 have mathematical solutions which are usually represented by hyperbolic functions.33 This is the case of the hyperbola of eq. 5 where consequently OE_th(max) and L_½ contain specific functional and structural information. A direct interpretation is to take (i) OE_th(max) as a measure of the maximum number of PSII reaction centers open for electron transport through PSII, and (ii) L_½ as the k₂/k₁ ratio, i.e., the equilibrium between the electron transfer from Phe- to the primary quinone Q_A and the formation of the reduced Phe molecule in the PSII reaction center (see also note 38).

B. Non-Linearity of the ββββ-CD Concentration Curves of Oxygen Evolution. We show in Figure 3 that the effect of the β-CD concentration on oxygen evolution in isolated thylakoid membranes irradiated with white light of saturation intensity (~5000 µmol photons.m-2.s-1) is the enhancement of oxygen evolution according to a sigmoid curve displaying a sharp inflexion point, or transition. The theoretical representation of the non-linear enhancement of oxygen evolution brought about by increasing β-CD concentration was attempted previously9 with a mathematical expression for the hypothesis of an allosteric transition with many binding sites (see discussions in refs 23, 30 and 31). The theoretical dose-response curve is the Hill equation

OE_th = OE_th(max) [Cn/ (K_½n + Cn)] (7)

where OE_th is the steady-state rate of oxygen evolution at various β-CD concentrations, OE_th(max) the maximum oxygen-evolving activity, C the cyclodextrin concentration, K_½ the concentration of cyclodextrin that resulted in half-maximal oxygen-evolving activity, that is, OE_th(max)/2, and n the Hill coefficient, i.e., the number of cyclodextrin binding sites. In eq 7, the term [Cn/ (K_½n + Cn)] is the fraction of active cyclodextrin receptors in the thylakoid

membrane. It is noted that the theoretical upper limit of oxygen evolution in Figure 3, i.e., OE_th(max) ≅ 453 µmol O₂ evolutiom.(mg Chl.h)-1, can only be attained at β-CD concentrations higher than 20 mM which are out of the solubility range of the cyclodextrins. However, the oxygen evolution observed in thylakoid membranes treated with 14 mM β-CD, i.e., 427.2 µmol O₂ evolutiom.(mg Chl.h)-1, is quite close to the theoretical maximum.

To determine the characteristics of the β-CD effect at irradiances far from the saturating light conditions of Figure 3 we performed experiments in low light intensity conditions and at irradiances up to 1100 µmol photons.m-2.s-1. The three-dimensional graph in Figure 4 presents the combined effect of irradiance (60 to 1100 µmol photons.m-2.s-1) and β-CD concentration (6 to 14 mM) on oxygen evolution in isolated thylakoid membranes. We remark first that all light-response curves of oxygen evolution in Figure 4 were shown to be well fitted with hyperbolic functions (cf. Table 1). Unexpectedly, however, Figure 4 reveals that a neat S-shaped (sigmoid) curve of oxygen evolution enhancement with increasing β-CD concentration of the type seen in Figure 3 is observed only at irradiances higher than approximately 300 µmol photons.m-2.s-1.

At photon flux densities lower than about 300 µmol photons.m-2.s-1 the oxygen evolution variation with β-CD concentration is small. This is best seen in Figure 5 which displays the oxygen evolution observed in β-CD-treated thylakoid membranes irradiated with light intensities of 66, 100 and 1000 µmol photons.m-2.s-1. We note at first that the shape of the low light intensity curves is quite different from the sigmoid shape seen at high light intensity. In short, the oxygen evolution increases to a maximum at β-CD concentrations between 10 and 12 mM, then decreases to a lower steady level.

IV. Modelling the Combined Effect of Irradiance and ββββ-CD Concentration

We examine in this section the constraints to be introduced in the light-response curves of oxygen evolution (Figure 2) in order to include the sigmoid character of the β-CD concentration effect as is seen in the `OE evolution vs. β-CD concentration` curve displayed in Figure 3. The aim is to deduce a mathematical expression more general than the hyperbolic function of eq 5 to include the sigmoid aspect of the β-CD concentration effect on oxygen evolution. In other words, the modified eq 5 might describe completely the oxygen evolution data seen in the three-dimensional graph of Figure 4 for a broad range of irradiances and β-CD concentrations.

Α. β Α. β Α. β

Α. β-CD Effect on OE_max and L_½ in Light-Response Curves of Oxygen Evolution.

The purpose here is to re-write eq 5 in order to correct the values of OE_max and L_½ observed in control thylakoid membranes (in the absence of β-CD), i.e., OE_max(0) and L_½(0). To this end, OE_max(0) and L_½(0) are multiplied by scale functions dependent on the β -CD-concentration (C), i.e., G₁(C) and G₂(C), to transform eq 5 into eq 8

OE_th = [OE_max(0) G₁(C)] I / [L_½(0) G₂(C) + I] (8)

Determination of G₁(C) and G₂(C): step 1. Analysis of the data in Table 1 reveals that OE_max and L_½ vary according to sigmoid functions of the type given by eq 7 according to the general expression

G_i(C) = 1 + p[C n/ (K_½n + Cn)] (9)

where G_i(C) is G₁(C) or G₂(C) in eq 8, n the Hill coefficient (cf. eq 7), p a parameter to take into account the β-CD-mediated maximum increase of oxygen evolution, and K_½ is defined

above (cf. eq 7). It is emphasized that in the absence of β-CD, i.e., for C = 0 in eq 9, G_i(C) is equal to 1 and eq 8 reverts to the expression given by eq 5.

Determination of G₁(C) and G₂(C): step 2. The results of the mathematical simulations are the following two expressions

G₁(C) = 1 + 3.3C4.8 / (13.14.8 + C4.8) (10) G₂(C) = 1 + 5.2C7.8 / (14.87.8 + C7.8) (11)

Eqs 10 and 11 show that the non-linear variation of G₁(C) and G₂(C) is sigmoidal as was expected from analysis of the OE_max(0) and L_½(0) data in Table 1. However, the sigmoicity is not identical as is seen in the relative magnitudes of p, n and K_½ which, therefore, shall influence the data yielded by eq 8.

Surface of occupancy. The computed values of the Hill coefficients, n, in eqs 10 and 11 are largely different, i.e., 4.8 in G₁(C) and 7.8 in G₂(C). We recall that the size of n is a mesure of the number of cyclodextrin binding sites in the thylakoid membrane.9 Therefore, the magnitude of n in eqs 10 and 11 indicates the presence in the thylakoid membrane surface of at least two specific molecular targets, or binding sites, for β-CD which have quite different sizes. Taking into account that the cross-section of a β-CD molecule is about 1.84 nm2 (cf. refs 1 and 2), a simple calculation shows that for n = 4.8, or say 5 β-CD molecules, one has a thylakoid membrane surface of occupancy of about 9.20 nm2, whereas for n = 7.8, or ~8, the calculated surface of occupancy is approximately 14.72 nm2. These size differences of the molecular surface of docking of β-CD in the thylakoid membrane is likely at the origin of distinct structural and functional changes.

B. Theoretical Representation of the Effect of Light Intensity and ββββ-CD Concentration. Substitution of the expressions given by eqs 10 and 11 for G₁(C) and G₂(C) in eq 8 results in the following general expression (eq 12) for oxygen evolution in β -CD-treated thylakoid membranes irradiated with low and high photon flux densities

OE_th = 151[1 + 3.3C4.8/(13.14.8 + C4.8)] I

/

{97.5[1 + 5.2C7.8/(14.87.8 + C7.8)] + I} (12) The three-dimensional representation displayed in Figure 6 was obtained with OE_max(0) = 151.0 µmol O₂ evolution.(mg Chl.h)-1 and L_½(0) = 97.5 µmol photons.m-2.s-1 as is indicated in eq 12 and Table 1.

The theoretical representation of the oxygen evolution (OE_th) as a function of I and C (Figure 6) was tested against the experimental data displayed in Figure 2 (OE_obs). For this purpose, we used software written with the QuickBasic programming language, version 4.0, from Microsoft Corporation. The calculated percentage error (%OEr) is represented in Figure 7. One sees that the average %OEr is between 2 and 9 %, and in most cases not greater than 13 %. Furthermore, the predictive value of the model was assayed for several sets of observed and theoretical data. For example, eq 12 yields OE_th = 198.4 µmol O₂ evolution.(mg Chl.h)-1 for I = 400 µmol photons.m-2.s-1and C = 10 mM, whereas for these same I and C the observed oxygen evolution is 209 µmol O₂ evolution.(mg Chl.h)-1 (cf. Figure 2). That is to say, an error of approximately 5.3 % which is small enough to confirm the quality of the model. In general, the calculations performed with eq 12 show a notable convergence between theory and experiment.

V. Concluding Remarks

First, we showed that the light-response of oxygen evolution in isolated thylakoid membranes irradiated with white light in steady-state conditions is well represented by a hyperbolic function (Figure 1 and Table 1), the shape of which is not affected by the β-CD concentration (Figure 2). We acknowledge, however, that this is at variance with the exponential functions, such as eq 4 (section III), which are used to describe the effect of light

flashes of high intensity and very short duration on whole leaves and other plant materials. On the one hand, a Poisson statistics is mandatory for the interpretation of data obtained with optically thin samples irradiated with single turnover flashes of monochromatic light.14 On the other hand, the steady-state approximation applied to the kinetics of electron transport through PSII justifies the use of a hyperbolic representation instead of an exponential function. These apparent discrepancies indicate the need for further investigations.

Secondly, we found that the sigmoid trend of the β-CD concentration effect on the oxygen evolution in thylakoid membranes irradiated with white light of saturating intensity, i.e., about 5000 µmol photons.m-2.s-1 (Figure 3), is not seen in low light intensity conditions, e.g., less than ~300 µmol photons.m-2.s-1 (Figure 4). This dual aspect of the β-CD effect has some connection to the photosynthetic activity differences observed previously in plant materials irradiated with low and high light intensities.15-17 In this respect, it is particularly interesting that the differences between low and high irradiance are ascribed in ref 17 to dissimilarities in molecular organization, that is, to the presence in the photosynthetic membrane of at least two populations of PSII centers with distinct properties. This conclusion gives support to our previous suggestion that the cyclodextrins cause the cooperative association of PSII units.9

A major issue in this work is the demonstration that the combined effect of light intensity and β-CD concentration on oxygen evolution (Figures 4 and 6) is well represented by eq 12 in which the irradiance effect on oxygen evolution is fitted with a hyperbolic function and the β -CD concentration effect is described by a mathematical expression for the hypothesis of an allosteric transition with many binding sites (see discussions in refs 23, 30 and 31). In the latter case, the theoretical β-CD dose-response curve is eq 7, that is, a Hill-type equation. We conclude that the theoretical result yielded by eq 12 is a reliable approximation of the combined effect of light intensity and β-CD concentration on oxygen evolution in isolated thylakoid membranes, therefore indicating that the model has a significant value for predicting the outcome of associated photochemical and biochemical reactions.

Acknowledgments. This work was supported by grants to M.F. from the Natural Sciences and Engineering Research Council of Canada. We thank Dr. S. Govindachary, Mr. P.-O. Hébert-Mercier and Mr. E. Richard for very helpful discussions, and the reviewers for having called our attention to a few ambiguities in the text thereby helping to improve the quality of the paper. M.F. dedicates this work to Luís de Camões, a great poet and humanist.

References and Notes

(1) Li, S.; Purdy, W. C. Chem. Rev. 1992, 92, 1457.

(2) D`Souza, V.-T.; Lipkowitz, K. B. Chem. Rev. 1998, 98, 1741. (3) Szejtli, J. J. Cyclodextrin Technology; Springer: Berlin, 1988

(4) Mikami, B.; Hehre, E. J.; Sato, M.; Katsube, Y.; Hirose, M.; Morita, Y.; Sacchettini, J.C. Biochemistry 1993, 32, 6836.

(5) Rawyler. A.; Siegenthaler, P.-A. Biochim. Biophys. Acta 1996, 1278, 89. (6) Duchêne, S.; Siegenthaler, P.-A. Lipids 2000, 37, 201.

(7) Sridharan, G.; Gaudreau, S.; Dalstein, L.; Huiban, C.; Lejeune, A.; Fragata, M. Z. Naturforsch. 2001, 56c, 792.

(8) Sridharan, G.; Daneau, E.; Fragata, M. Can. J. Bot. 2002, 80, 741. (9) Dudekula, S.; Sridharan, G.; Fragata, M. Can. J. Bot. 2005, 83, 320.

(10) Fragata, M.; Dudekula, S. In Photosynthesis: Fundamental Aspects to Global Perspectives; van der Est, A., Bruce, D., Eds.; ACG Publishing, Lawrence, KS, 2005.

(11) Fragata, M.; Nordén, B.; Kurucsev, T. Photochem. Photobiol. 1988, 47, 133. (12) Nordén, B.; Fragata, M.; Kurucsev, T. Austral. J. Chem. 1992, 45, 1559.

(13) Dentuto, P. L.; Catucci, L.; Cosma, P.; Fini, P.; Agostiano, A.; D'Acolti, L.; Trevithick-Sutton, C. C.; Foote, C. S. J. Phys. Chem. 2005, 109, 1313.

(14) Mauzerall, D.; Greenbaum, N. L. Biochim. Biophys. Acta 1989, 974, 119. (15) van Wijk, K. J.; van Hasselt, P. R. Photosynth. Res. 1990, 25, 233.

(16) Edwards, G. E.; Baker, N. R. Photosynth. Res. 1993, 37, 89.

(17) Hormann, H.; Neubauer, C.; Schreiber, U. Photosynth. Res. 1994, 40, 93. (18) Genty, B.; Briantais, J.-M.; Baker, N. R. Biochim. Biophys. Acta 1989, 990, 87.

(19) Lavergne, J.; Trissl, H.-W. Biophys. J. 1995, 68, 2474.

(20) Lavergne, J.; Briantais, J.-M. In Oxygenic Photosynthesis: The Light Reactions; Ort, D. R., Yocum, C. F., Eds.; Kluwer Academic Publishers: Dordrecht, The Nederlands, 1996; p 265.

(21) Lazár, D. Biochim. Biophys. Acta 1999, 1412, 1.

(22) Falkowski, P. G.; Wyman, K.; Ley, A.C.; Mauzerall, D. C. Biochim. Biophys. Acta

1986, 849, 183.

(23) Wyman, J.; Gill, S. J. Binding and Linkage. Functional Chemistry of Biological Macromolecules; University Science Books: Mill Valley, 1990.

(24) Berthold, D. A.; Babcock, G. T.; Yocum, C. F. FEBS Lett. 1981, 134, 231. (25) Nénonéné, E. K.; Fragata, M. J. Plant Physiol. 1990, 136, 615.

(26) Arnon, D. I. Plant Physiol. 1949, 14, 552. (27) Chua, N. H. Meth. Enzymol. 1980, 69, 434.

(28) Nénonéné, E. K.; Méthot, M; Fragata, M. Z. Naturforsch. 1998, 53c, 39. (29) Mulo, P.; Laakso, S.; Maenpaa, P.; Aro, E.-M. Plant Physiol. 1998, 117, 483.

(30) Krause, R. M.; Buisson, B.; Bertrand, S.; Corringer, P.-J.; Galzi, J.-L.; Changeux, J.-P.; Bertrand, D. Mol. Pharmacol. 1998, 53, 283.

(31) Kimmel, J. L.; Reinhart, G. D. Proc. Natl. Acad. Sci. USA. 2000, 97, 3844. (32) Krömer, S.; Malmberg, G.; Gardestrom, P. Plant Physiol. 1993, 102, 947.

(33) Tinoco Jr., I.; Sauer, K.; Wang, J. C. Physical Chemistry. Principles and Applications in Biological Sciences, third ed.; Prentice Hall: Englewood Cliffs, New Jersey, 1995.

(34) The relationship between electron transport in photosynthetic membranes and the photon flux density has been reported often. For example, in Figure 2 of ref 18 experimental data are presented on the relation of the CO₂ assimilation rate (µmol.m-2.s-1) in leaves of wild-type barley and the photon flux density (µmol quanta.m-2.s-1). We performed the mathematical modelling of the empirical graphical representations given in ref 18 and showed that the data cannot be fitted with none of the several exponential functions studied. The best fit of the CO₂ assimilation rate (y) was found to be the expression y = 36.2 (1 - e-0.003 I) where I is the photon flux density. This equation is nevertheless clearly inadequate. On the contrary, a good representation of the experimental data in ref 18 is given by the hyperbolic function y = 33.6 I / (266 + I).

(35) Hankamer, B.; Barber, J.; Boekema, E. J. Annu. Rev. Plant Physiol. Plant Mol. Biol.

1997, 48, 641.

(36) Klimov, V. V.; Krasnovsky, A. A. Photosynthetica 1981, 15, 592.

(37) Jursinic, P. A. In Light Emission by Plants and Bacteria; Govindjee, Amesz, I., Fork, D.C., Eds.; Academic Press: New York, U.S.A., 1986; p 291.

(38) The application of the 'steady state approximation' to the reactions in Scheme 1 requires a set of boundary conditions which are simply delineated in the following generally accepted molecular equilibria (see, e.g., Figure 3 in ref 37) with rates k₁ and k₂:

k₁ k₂

P680*.Phe.Q_A → P680+.Phe-.Q_A → P680+.Phe.Q_A

-In this approximation, the sequence of photochemical and biochemical events is limited to the instant of the absorption of a photon by P680 and the subsequent electron transfer up to the oxidized form of Q_A. Interestingly, the final result of the mathematical deductions is invariably a hyperbolic expression of the type given by eq 5 with L_½ = k₂/k₁. A detailed account of this matter shall be presented in another paper.

Legends for the figures

Figure 1. Effect of the light intensity (µmol photons.m-2.s-1) on the oxygen evolution in isolated thylakoid membranes. The theoretical curves were obtained with Origin 5.0 (see Material and Methods). Chl, chlorophyll; I, photon flux density.

Figure 2. Effect of the light intensity (µmol photons.m-2.s-1) and the β-cyclodextrin (β-CD) concentration on the oxygen evolution in isolated thylakoid membranes. The theoretical curves were obtained with Origin 5.0 (see Material and Methods). Chl, chlorophyll.

Figure 3. Effect of the concentration of β-cyclodextrin (β-CD) on the oxygen evolution (OE) in isolated thylakoid membranes irradiated with white light of saturating intensity (~5000 µmol photons.m-2.s-1). OE is 155.4 ± 3.3 µmol O₂ (mg Chl.hr)-1 in the absence of β-CD. The experimental data are given as ‘mean ± SE’. The theoretical curve was obtained from mathematical simulations performed according to the Hill equation with Origin 5.0 (see text and Materials and Methods). Chl, chlorophyll; SE, standard error.

Figure 4. Three-dimensional representation of the effect of the light intensity (µmol photons.m-2.s-1) and the β-cyclodextrin (β-CD) concentration on the oxygen evolution in isolated thylakoid membranes. Chl, chlorophyll.

Figure 5. Comparison of theoretical and experimental data on the effect of β-cyclodextrin concentration and irradiance (66, 102 and 1000 µmol photons.m-2.s-1) on oxygen evolution in isolated thylakoid membranes. The theoretical curves were obtained from mathematical simulations performed with Origin 5.0 (see Materials and Methods and eq 12). Chl, chlorophyll.

Figure 6. Theoretical three-dimensional representation of the effect of the light intensity (µmol photons.m-2.s-1) and the β-cyclodextrin (β-CD) concentration on the oxygen evolution in isolated thylakoid membranes. The mathematical simulations were performed with eq 12 (see text) using Origin 5.0 (see Materials and Methods). Chl, chlorophyll.

Figure 7. Error (%OEr) between theoretical and experimental data observed in the study of the effect of the light intensity (µmol photons.m-2.s-1) and the β-cyclodextrin (β-CD) concentration on the oxygen evolution in isolated thylakoid membranes. The theoretical data were obtained with eq 12 (see text and Figure 6) and the experimental data are those given in Figure 2.

TABLE 1: Comparison of oxygen evolution observed in isolated thylakoid membranes (OEobs)a,bwith data obtained from theoretical simulations (OEth)c,d

______________________________________________________________________________ β-CD mΜ Oxygen evolution µmol O₂.(mg Chl.h)-1 L_½ µmol photons.m-2.s-1 k m2.(µmol photons)-1.s ______ _____________________________ __________________ _____________________

OE_obse OEth(max) SE Cvarf L½ SE Cvar k SE _Cvar _______ _________ ____ _____ _______ ____ ____ _______ _______ ____ 0 155.4 3.3 2.1 132.9c 3.6 2.7 0.00820 0.00070 8.5 151.0d 3.8 2.5 97.5 9.0 9.2 10 282.9 9.5 3.4 218.1c 4.9 2.2 0.00736 0.00051 6.9 255.0d 7.4 2.9 118.7 11.5 9.7 12 364.9 15.2 4.2 278.2c 7.2 2.6 0.00475 0.00037 7.8 342.9d 8.4 2.4 209.1 15.3 7.3 14 427.2 16.5 3.9 349.4c 4.9 1.4 0.00373 0.00016 4.3 449.4d 11.4 2.5 295.5 20.6 7.0 a

Abbreviations: β-CD, β-cyclodextrin; I, light intensity in µmol photons.m-2.s-1; k, cross section for absorption of a photon x duration of illumination (see Model I below); L_½, light intensity giving OE_th(max)/2 (see Model II below); OE_th(max), theoretical oxygen evolution maximum; Sd, standard deviation; SE, standard error.

b

The isolated thylakoid membranes were irradiated with white light of saturating intensity (~5000 µmol photons.m-2.s-1).

c

Model I (see text, section III): OE_th = OE_th(max) (1 - e-kI). d

Model II (see text, section III): OE_th = OE_th(max) I / (L_½ + I). e

Experimental data of Figure 3. f

TABLE 2: Estimated error in the determination of oxygen evolution with Model I and Model II in relation to the observed oxygen evolution in thylakoid membranes untreated and treated with various ββββ-cyclodextrin concentrationsa

______________________________________________________________________________

Model µmol O₂ evolution.(mg Chl.h)-1 %∆OE_th(5K)b or %∆OE_th(max₎c ______________________________ ______________________________ β-CD, mM 0 10 12 14 0 10 12 14 ______ ______ ______ ______ ______ ______ ______ ______ OE_obsd 155.4 282.9 364.9 427.2 OE_th(5K) Ie 132.9 218.1 278.2 349.4 14.5 22.9 23.8 18.2 IIf 148.1 249.1 329.1 424.3 4.7 12.0 9.8 0.7 OE_th(max) I 132.9 218.1 278.2 349.4 14.5 22.9 23.8 18.2 II 151.0 255.0 342.9 449.4 2.8 9.9 6.0 -5.2 a

Abbreviations: β-CD, β-cyclodextrin; I, light intensity in µmol photons.m-2.s-1; k, cross section for absorption of a photon x duration of illumination (see Model I below); L_½, light intensity giving OE_th(max)/2 (see Model II below); OE_obs, oxygen evolution observed; OE_th(max), theoretical oxygen evolution maximum; OE_th(5K), theoretical oxygen evolution computed for an irradiance of 5000 µmol photons.m-2.s-1.

b

%∆OE_th(5K) = [OE_obs - OE_th(5K)] x 100 / OE_th(5K). c

%∆OE_th(max₎ = [OE_obs - OE_th(max)] x 100 / OE_th(max). d

The thylakoid membranes were irradiated with white light of saturating intensity (~5000 µmol photons.m-2.s-1).

e

Model I (see text, section III): OE_th = OE_th(max) (1 - e-kI). f

Figure 1, Fragata and Dudekula (2005)

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Figure 5, Fragata and Dudekula (2005)

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