Báo cáo khoa học: Kinetics of the quinone binding reaction at the QB site of reaction centers from the purple bacteria Rhodobacter sphaeroides reconstituted in liposomes docx

The results obtained for the quinone release kinetic constant are com-parable to the rate of the charge recombination reaction from the state D+QA–.. Some considerations regarding the ex

Kinetics of the quinone binding reaction at the QB site of reaction

reconstituted in liposomes

Francesco Milano1, Angela Agostiano1,2, Fabio Mavelli2and Massimo Trotta1

1

CNR, Istituto per i Processi Chimico-Fisici – Sezione di Bari and2Dipartimento di Chimica, Universita´ di Bari, Italy

Transmembrane proton translocation in the photosynthetic

membranes of the purple bacterium Rhodobacter

sphaero-idesis driven by light and performed by two

transmem-brane complexes; the photosynthetic reaction center and the

ubiquinol–cytochrome c oxidoreductase complex, coupled

by two mobile electron carriers; the cytochrome and the

quinone This paper focuses on the kinetics and

thermo-dynamics of the interaction between the lipophylic electron

carrier ubiquinone-10 and the photosynthetic enzyme

reconstituted in liposomes The collected data were

simula-ted with an existing recognized kinetic scheme [Shinkarev,

V.P & Wraight, C.A (1993) In The Photosynthetic

Reac-tion Center (Deisenhofer, J & Norris, J.R., eds.), pp 193–

255 Academic Press, San Diego, CA, USA] and the kinetic constants of the uptake (7.2· 107M )1Æs)1) and release (40 s)1) processes of the ligand were inferred The results obtained for the quinone release kinetic constant are com-parable to the rate of the charge recombination reaction from the state D+QA– Values for the kinetic constants are discussed as part of the overall photocycle, suggesting that its bottleneckmay not be the quinone uptake reaction in agreement with a previous report (Gerencser, L., Laczko, G

& Maro´ti, P (1999) Biochemistry 38, 16866–16875) Keywords: reaction center; quinone binding; liposomes; photosynthesis

4The photosynthetic apparatus of the nonsulfur purple

bacterium Rhodobacter sphaeroides sits primarily in

dedica-ted portions of the cell membrane called intracytoplasmatic

membranes (ICM) [1,2] The key enzymes involved in the

build-up of the transmembrane proton gradient [3,4] that

eventually trigger ATP synthesis [5] are located in the ICM

The increase in the photosynthetic transmembrane proton

gradient occurs following absorption of solar

electromag-netic radiation, which is performed by light harvesting

complexes (LHCs) [6,7] The LHCs channel excitons to the

reaction center (RC), a transmembrane enzyme, where they

generate a cascade of electron transfer reactions that results

in the double reduction

carrier, ubiquinone-10 Following reduction the ubiquinone

takes up two protons from the cytoplasm, exits the RC and

migrates towards the ubiquinol–cytochrome c

oxidoreduc-tase (bc1), a second transmembrane complex In the bc1

complex the electrons are utilized to attract two more

protons and reduce the cytochrome c2, a water soluble

electron carrier that will eventually donate electrons to an

oxidized quinone sitting in the RC, thereby concluding the cyclic electron transport driven by the solar radiation [8] The net result of the entire photocycle is the light-sustained translocation of a proton through the membrane, therefore

it is not surprising that a great effort has been made to characterize the mechanism by which the excitons that are absorbed by the RC, excite and shuttle electrons across the enzyme The large amounts of spectroscopic and structural information that have been gathered have enabled a relatively clear description of the electron transfer chain reaction, which is initiated by the absorption of a photon or

an exciton The excited electron is transferred from the primary electron donor excited state D* (a dimer of bacteriochlorophyll a)

6 , to a chain of electron acceptors located inside the protein at increasing distances from D [9] Due to the spatial organization and the relative energies of the cofactor redox couples, the forward electron transfer reactions occur faster than the recombination reactions and therefore, within hundreds of picoseconds, the electron reaches the primary electron acceptor, ubiquinone-10, sitting

in the QA

7 pocket In the absence of exogenous electron

donors (i.e cytochrome) the charge separated state D+QA–

has a lifetime

8 of 100 ms unless a loosely bound

ubiquinone-10 molecule is present in the

it acts as secondary electron acceptor The state D+QB–is more stable, with a lifetime of one or two seconds In the presence of cytochrome, the secondary quinone can allocate

a second electron yielded from the absorption of a new photon, thereby functioning as a two-electron gate [3,10] During transfer of the second electron from the primary to the secondary quinone, protons reach the interior of the protein [11] Finally the quinol leaves the RC and is replaced

by the oxidized quinone sitting in the membrane pool [12]

Correspondence to M Trotta, Istituto per i Processi Chimico-Fisici –

Sezione di Bari, Via Orabona 4-I 70126 BARI, Italy.

Fax: + 39 080 5442029, Tel.: + 39 080 5442027,

E-mail: m.trotta@area.ba.cnr.it

Abbreviations: bc 1 , ubiquinol-cytochrome c oxidoreductase; ICM,

intracytoplasmatic membranes; LDAO, lauryl dimethyl amino

N-oxide; LHC, light harvesting complex; RC, reaction center.

Dedication: Dedicated to the memory of Professor Mario Della

Monica.

(Received 18 September 2002, revised 12 September 2003,

accepted 22 September 2003)

Under saturating illumination, the photocycle time scale

is in the order of milliseconds A key role in the photocycle is

played by the exchange of the two redox forms of the

quinone, between the protein interior and the bilayer Some

considerations regarding the exchange reaction for the

oxidized quinone are made in this paper, based on

investigations into the charge recombination reactions that

take place in purified RCs reconstituted in proteoliposomes,

and in the absence of exogenous electron donors

Proteo-liposomes were selected because they can be considered a

good mimicking system for the photosynthetic membrane,

in which the relative amounts of enzyme and quinone can

be altered easily, in contrast to the isolated ICM, called

chromatophores, where changing quinone concentration is

a laborious task[13] Moreover, in the ICM the presence of

the entire and active electron transport chain would require

the use of decouplers in order to focus the RC–quinone

interaction A final consideration for using liposomes is that

the solubilizing environment may play a role, particularly

when the QBpocket is under investigation [14,15]

In this work, RCs were reconstituted in

phosphatidyl-choline liposomes, which are recognized for producing the

best results in the formation of small unilamellar vesicles

The kinetics and equilibrium of the exchange between the

QB pocket and the quinone pool were estimated The

collected data were simulated with the well-known kinetic

scheme of Shinkarev & Wraight [16], and the kinetic

constants of the ligand uptake (kin) and release (kout)

processes were inferred The single species time evolution

involved in the kinetic scheme was extracted from the

output of the numerical simulation Recombination

reac-tions were also compared to different solubilizing

environ-ments such as reverse and direct micelles

Materials and methods

Isolation of reaction centers andQBsite depletion

Reaction centers were isolated from Rhodobacter

sphaero-ides strain R-26.1 following the procedure illustrated by

Isaacson et al [17] Protein purity was established using the

ratio of absorbance at 280 and 802 nm (A280/A802), which

was kept below 1.3, and the ratio of absorbance at 760 and

865 nm (A760/A865), which was equal to or lower than 1 The

average quinone content was 1.8 when defined by (Q/RC)

Depletion of the QB site was accomplished using the

procedure of Okamura et al [18], with the final

prepara-tions exhibiting a quinone content (Q/RC)¼ 1.05 ± 0.05

as determined by the charge recombination decay No

changes to the photobleaching amplitude were observed

upon addition of quinone

Charge recombination kinetics were recorded at 865 nm

using a kinetic spectrophotometer implemented with an

Hamamatsu R928 photomultiplier (Hamamatsu

Photo-nics K.K., Hamamatsu City, Japan), and a Nd-Yag Laser

(Quanta System, Milan, Italy) which was used for RC

photoexcitation Data were collected onto a Digital

Oscilloscope (Tektronix, Inc., TKS3052, Beaverton, OR,

USA) and trace deconvolution was performed using

software developed in-house The decay traces were

recorded until complete recovery occurred following

photobleaching Absorbance changes were measured

taking the baseline recorded before the flash as the starting value Even at high quinone concentrations, the trace deconvolution was obtained with a high correlation coefficient (r2) using bi-exponential functions A drift of less than 1.5% was observed in samples illuminated by the sole measuring beam in the time range of the experiments Each point in the data shown below is the average of three different liposome preparations

Reaction center reconstitution in proteoliposomes

RC reconstitution in liposomes was accomplished following the procedure outlined in [19–21] One to eight milligrams of 1,2-diacyl-sn-glycero-3-phosphocholine (used at 48% pur-ity, Sigma) were dissolved in 500 lL of chloroform to which, when needed, aliquots of a 1 mM ubiquinone-10 (Sigma) solution were added The resulting solution was carefully dried under a stream of nitrogen in an Eppendorf tube, to form an evenly distributed film of lipids Five hundred microlitres of a 4% (w/v) sodium cholate solution (Sigma) in phosphate buffer, pH 6.8, 100 mM KCl were added to the lipid film Lipids were solubilized by 10–20 repeated one-second sonications (Sonifier Mod 250, Bran-son UltraBran-sonic Corporation, Danbury, CT, USA) to form a homogenous solution This solution was added to the

QBsite-depleted RC (90 lM), shaken vigorously and stored for 15 min at 4C Finally, the solution was loaded onto a

15 cm Sephadex G-50 Superfine column (Pharmacia) previously equilibrated with the phosphate buffer The band containing RC incorporating liposomes elutes rapidly, and optical measurements were carried out Proteolipo-somes were prepared with different quinone/RC (Q/RC) ratios while still maintaining a constant enzyme concentra-tion The RC orientation in the liposome bilayer was inferred from the decrease in the total amount of photo-bleaching at 865 nm before and after the addition of reduced cytochrome c (Sigma) The two possible orienta-tions of RCs were found to be equally distributed Dynamic light scattering measurements The hydro-dynamic diameter of liposomes was determined by means of dynamic light scattering using a Brookhaven Instruments Corporation goniometer (BI-200SM) (New York, USA) equipped with a helium/neon laser source (wavelength 632.8 nm) Samples were contained in cylindrical optical cells with a diameter of 1 cm while an external thermostat maintained the temperature at 20.0 ± 0.1C All dynamic light scattering determinations were made at a scattering angle of 90 Data were acquired within the 1–104 ns decay time range that is necessary to determine the signal from particles

The diffusion coefficient D,

measured autocorrelation function by a cumulants method [22,23] usingBI-PCSW SIMPLE CUMULANTSsoftware (Brook-haven Instruments Corporation, New York, USA)

In this method, the logarithm of the correlation function, g(s),

31 fits to a power series of the correlation time (s):

ln gðsÞf g ¼ A þ Bs þ Cs2þ :::

where A is a constant that depends on the instrument setting and

B¼ 
C¼ DQ2

Q¼ [4p · n · sen(Q/2)/k], with Q being the modulus of

the scattering vector, n being the refraction index of the

solution, k being the wavelength and Q/2 being the

scattering angle); and C is equal to

33

1 2

Z1 0

ðC 
CÞ2CðCÞdC

2 4

3 5

where, C and C(C) are the decay velocity and the decay

velocity distribution, respectively) The ratio C/B2

rep-resents the size polydispersity distribution

In the hypothesis that particles behave like hard spheres

the average hydrodynamic radius (R) was calculated from

D using the Stokes–Einstein equation,

R¼ kBT=6pgD where g is the water viscosity, kB is the Boltzmann

constant and T is the absolute temperature

The geometry of the liposomes is in agreement with that

obtained by Palazzo et al [24] for liposomes prepared in the

same way Combining the parameters obtained for the

preparation of liposomes as summarized in Table 1, it is

possible to estimate a RC/liposome ratio of 500 ± 150

depending on the lipid/protein ratio used to prepare the

liposomes (see below) These values correspond to an RC

surface concentration ranging from 2.7 to 20.0 nmolÆm2

The lower concentration is in agreement with 3.0 nmolÆm2

calculated for chromatophores assigned a radius of 50 nm

[25,26] and using the 50–60 RC/chromatophore ratio as

found by Saphon et al [27]

It is well known that the radius of liposomes is influenced

by the molar ratio of lipid/detergent in the mixed micelles

36

starting solution, and in our preparations this ratio was

always below the critical value of 1.33 at which the

transition between the extended bilayer sheet and the

micelle takes place Each of the above described experiments

exhibits no significant variation in the diameter of the

liposomes with varying lipid/detergent molar ratio

Due to dispersion of the data for the same sample we

conclude that an average value of 110 ± 25 nm can be

assumed as a reasonable

The measurements made on both liposomes containing the

RC (proteoliposomes), and pure liposomes (not containing

protein), gave substantially the same results

Reconstitution of the protein was confirmed by

prepar-ing liposomes in the presence of a fluorescent lipid

(1-palmitoyl-2-[12-[(7-nitro-2-1,3-benzoxadiazol-4-yl)

amino]-dodecanoyl]-sn-glycero-3-phosphocholine

Avanti Polar Lipids Inc., Alabaster, AL, USA), and

recording the visible spectra and fluorescence of the solution

eluted from the column

39 [19] The RC elutes in a single sharp

band that coincides with the lipid elution, indicating that the proteins are completely reconstituted into liposomes

40

Results and discussion

The kinetic scheme and data analysis The reaction scheme outlined in Fig 1 shows the kinetic constants for the final electron acceptor reactions The reactions take place in the neutral state (lower row), and in the charge separated state that is generated in the RC following the absorption of a photon in the absence of an exogenous electron donor (upper row) Several descriptions

of the scheme are available, the most detailed of which was given by Shinkarev & Wraight [16]

In the darkthe RCs undergo a binding equilibrium in which the loosely bound quinones are taken up and released from the QBsite [12] After a short light pulse, the RCs undergo a charge separation process, where an electron is transferred from D to a primary quinone acceptor located in the QAbinding site For proteins in which the QBpocket is empty, a charge recombination occurs with a phenomen-ological monoexponential decay constant [9] kF¼ kAD

41which is 8 s)1(kFis the phenomenological delay constant

of the fast phase and kAD is the backelectron transfer constant from the D+QA–and D+QA–QBstates) In RCs which have the QB pocket occupied, the electron rapidly equilibrates between the two final acceptors with an equilib-rium constant (LAB) that can be expressed as

kAB/kBA(kABbeing the forward electron transfer constant from D+QA–QBto D+QAQB–and kBAbeing the backward electron transfer from D+QAQB–to D+QA )QB) When the

QB pockets are fully occupied, the charge recombination reaction is also monoexponential, with a phenomenological rate constant, ks:

Table 1 Parameters of proteliposomes preparations RC area assumes

a horizontal section as an ellipse [9] of 0.3 · 0.4 nm 2

Liposome radius derived from experimental data.

RC area

(nm2)

Liposome

radius (nm)

Liposome area (nm2)

(Liposome area/RC area)

10 110 ± 25 (1.5 ± 0.6) · 10 5 1.5 · 10 4

Fig 1 The kinetic scheme for reaction centers in the presence of quinone association and dissociation (quinone exchange), both in the dark and in the charge separated state The constants in the scheme are defined as follows: k AD ¼ backelectron transfer constant from the D + Q A–and

D + Q A–Q B states, assuming that the charge recombination process from

Q A –

is not affected by the functional occupancy of the Q B site;

k in ¼ quinone uptake constant; k out ¼ quinone release kinetic con-stant; k AB ¼ forward electron transfer constant from D + Q A–Q B to

D+Q A Q B –

; k BA ¼ backward electron transfer from D +

Q A Q B –

to

D+Q A–Q B The direct recombination route from D+Q A Q B– is not shown as its constant is negligible compared to the others k in and k out

are assumed to be independent of the redox state of Q A (see text for discussion).

ðkADþ kBDLABÞ 1

1þ LAB

kAD

1

1þ LAB

Eqnð1Þ This approximation holds because the direct recombination

reaction from the D+QAQB–state has a negligible kinetic

constant (kBD< 0.1 s)1) [28,29] In the presence of a

subsaturating quinone concentration, only a fraction of the

QBsites can be filled and the decay can be fitted with the

sum of two exponential decays:

DAðtÞ ¼ DA0

fastexpðkFtÞ þ DA0

slowexpðkStÞ Eqn ð2Þ where t is the time, DA(t) represents the amplitude at any

instant t, and DAfast0 and DA0slow represent the amplitudes of

the fast and slow phase respectively

Proteoliposomes were prepared using QBdepleted

reac-tion centers in the presence of increasing amounts of

ubiquinone-10,

45 the naturally occurring quinone in the QB

site Charge recombination kinetics were recorded and time

evolution traces of absorbance changes were fitted

(r2> 0.995) using Eqn (2) where kFand kSrepresent the

phenomenological decay constants of the fast and slow

phase, respectively In this workthe kFconstant is assumed

equivalent to the kinetic constant kAD(8.3 s)1) of the decay

from the QA– containing states (Fig 1) Indeed, upon

addition of inhibitors of QBfunctionality, the decay of the

charge separated state is monoexponential, with a constant

slightly faster than the kF ( 10 s)1), indicating that the

secondary quinone is displaced from its binding site as

observed in detergent

46

In contrast, kS results from more than one clear-cut

process as discussed below As the quinone/RC ratio

47increases, a rise

in the decay constant are observed Figure 2 shows the

dependence of the slow phase relative amplitude (titration

curve) on the increase of Q/RC Similarly the dependence of

the decay constant is shown in Fig 3

Under these conditions the binding reaction has a role in

the slow component of the charge recombination The slow

decay constant depends both on the rate ratio between the quinone exchange and the charge recombination from the states D+QA–or D+QA–QB, in addition to Q/RC The quinone release rate kout[D+QA )QB] can be normal-ized to the backelectron transfer rate from the appropriate state; kout[D+QA )QB]/kAD[D+QA )QB], and the ratio can be used to describe the quinone exchange regime For instance,

if (kout/kAD) > 1 the exchange is defined as fast, whereas for (kout/kAD) < 1 the exchange is defined as slow The different kinetic behaviour of the protein when solubilized in different environments (e.g direct micelles, reverse micelles, and proteoliposomes) comes from the influence played by the surroundings on kin, koutand LAB For instance, in direct lauryl dimethyl amino N-oxide (LDAO)

49 micelles a decay sum of two exponential is observed [18] with a subsaturating quinone concentration [i.e a fast phase with a decay constant (kF¼ kAD) and a slow phase with a decay constant (kS) given by Eqn (1)], that can be explained only by considering a slow exchange The quinone uptake and release can be neglected during the charge recombination reaction, hence the relative amplitude

of the slow phase is proportional to the QBsite occupancy

On the other hand, in direct Triton X-100 micelles a decay sum of two exponentials is observed [30] with a subsaturat-ing quinone concentration in which the kSdepends on the concentration of added quinone, ranging from 1.1 s)1to 2.7 s)1, showing a fast exchange at the QBsite Agostiano

et al.[31] found that the charge separated state of RCs solubilized in phospholipid reverse micelles will decay as the sum of two exponentials The reverse micelles are dissolved

in hexane where the unbound quinone is highly soluble The decay has a kF¼ kADand a slow phase with a constant kS decreasing from 3 s)1 to 1 s)1, and a relative amplitude increasing to 1.0 for 400£ Q/RC £ 7000 Such behaviour was explained in terms of fast quinone exchange

Assuming quinone molecules uniformly distributed among vesicles of different sizes, Palazzo et al [24] studied the influence of the spread of the local solute concentration

on the phenomenological kinetic constants

In the present workthe Q/RC ratio ranged from 0.02 to

4, and full QBreconstitution was obtained for values higher than 3 The long chain exogenous quinone was confined to

Fig 2 Fraction of slow phase obtained by fitting Eqn (1) to the

experimental traces d, phosphatidylcholine proteoliposomes prepared

with lipid/protein molar ratio of 1000 : 1; [Q]/[RC], concentration of

the species in the mixed micelles, where [RC] ¼ 8.3 lM; s, 2.1 l M RC

made up in 0.025% LDAO in 20 mM tris buffer pH 8, where the

quinone is solubilized in

Fig 3 Slow phase decay constant as a function of quinone/RC molar ratio d, liposomes; s, detergent.

the liposome bilayers Additionally, as a direct consequence

of our liposome preparation method, a solute molecule

distribution weighted by the bilayer vesicle volume was

considered (i.e larger vesicles will contain larger numbers of

solute molecules) As shown in the Appendix, under this

assumption the average local volume concentration of

quinones is the same for aggregates of all sizes and the

polydispersity can be neglected at high overall quinone

concentration [Q] In the investigated [Q] concentration

range this condition is not fulfilled for the first two values,

where the decay from the D+QA–state is predominant

Analogously to the previous case, the decay of the charge

separated state is fitted by the sum of two exponentials with

kF¼ kADand a slow phase kSdecreasing from 1.5 s)1to

0.5 s)1 Using the asymptotic kS value in the equilibrium

constant, LABis found to be 15.6 It should be noted that

when the condition (kout/kAD) > 1 occurs, the quinone

uptake and release take place during the charge

recombi-nation reaction The exchange regime and the value of some

constants for the three solubilizing environments are

summarized in Table 2

Numerical simulations

The set of differential equations [Eqn (3)] required for the

kinetic scheme shown in Fig 1 was numerically solved by a

fourth order Runge–Kutta method Using this approach a

value for the quinone uptake and release kinetic constants

and therefore the quinone binding constant (KB¼ kin/kout)

can be determined The symbols used in Eqn (3) are the

same as those used in [16] Numerical simulations have been

carried out for the lipid/protein molar ratio 1000 : 1 by

using the values listed in Table 3

dx/dt¼ ðkADþ kinqÞx þ kouty dy/dt¼ kinqxþ kBAz ðkADþ kABþ koutÞy dw/dt¼ koutuþ kADx kinqw

dz/dt¼ kABy ðkBAþ kBDÞz du/dt¼ kinqwþ kADyþ kBDz koutu dq/dt¼ kinqqðw þ xÞ þ kinðy þ uÞq

8

>

>

>

<

>

>

>

:

Eqnð3Þ

where x¼ ½DþQA=½RC; y¼ ½DþQAQB=½RC; z¼

½DþQAQB=½RC; w ¼ ½DQA=½RC; u ¼ ½DQAQB=½RC;

q¼ ½Qfree=½Qtotaland q¼½RC=½Qtotal Immediately after the flash, at time zero, the electron is found only in the charge separated states involving the primary electron acceptor, i.e D+QAand D+QA–QB, while

QBhas not yet been reached The D+QA–QBstate rapidly disappears, with constant kAB generating the state

D+QAQB–, until the equilibrium is attained within few milliseconds Simultaneously, the charge separated states start to decay and the different contributions cannot be resolved by monitoring the D+ decay The free quinone concentration ([Q]free) drops from its equilibrium dark value and is driven to the QBsite by the presence of the electron A typical time-evolution obtained by solving Eqn (3) is shown in Fig 4

The quinone binding constant KBwas varied in the range

1· 105) 1 · 107

M )1 and the quinone release constant

koutwas varied between 0.25 and 2500 s)1, spanning from a slow to a fast exchange regime; this is shown in Fig 5 where the charged species decay is simulated for seven kout

values at constant KB The slow decay constant ksis weakly dependent on kout for large and small kout/kAD values, whereas the dependence increases when this ratio is close

to 1 The overall dependencies of simulated ksand DA0slow upon KBand kout/kADare illustrated in Fig 6

The kin and koutvalues that minimize the square-root difference between the simulated and experimental traces were obtained by using the simple search method [32] with

a tolerance of 10)4 giving kin¼ 7.2 · 107M)1Æs)1 and

kout¼ 40 s)1 From the best fit values, KB¼ 1.8 · 106M )1

and kout/kAD¼ 4.8 were obtained

The agreement between the experimental and the simu-lated data for the reconstitution of QBsite experiments in proteoliposomes (Fig 7) is very satisfying

Discussion

An important issue arising from the above experimental and simulated data is the different behaviour of the quinone exchange when passing from direct micelles to

Table 2 Constants and exchange domains for three different solubilizing environments.

RC solubilizing environment k AD (s)1) L AB K B ( M )1 ) k out /k AD

Triton X-100 direct micelles [30] 8.3 6–8 a

5 · 10 7 >1

Phosphatidylcholine proteliposomesb 8.3 15 1.8 · 10 6

4.8

65 a Calculated from the equation (k AD /k S ) ) 1 using the values k AD ¼ 8.3 s)1and k s ¼ 0.8 s)1[16] b This work.

Table 3 Numerical value for the constants employed in the simulation of

D+decay L AB is taken from Table 2; k AB and k BD from [28] [29]; k BA

is obtained from the assumption that the forward electron transfer

constant in proteoliposomes remains unaltered Recently, Taly et al.

[52] measured, with 10% uncertainty, k AB ¼ 8700 s)1for the wild type

(Rb sphaeroides 2.4.1) in dimyristoylphosphatidylcholine liposomes.

The numerical simulation has also been tested for different k AB values

and it was found to be insensitive for rates in the range

5000 s)1) 15000 s )1

s)1

proteoliposomes The main difference between these two

solubilizing environments is their organization with the

enzyme RC–LDAO complexes have been characterized by

small angle neutron scattering [33,34] The complex is

formed by a toroidally shaped group of micelles

surround-ing the most hydrophobic part of the protein In these

complexes the detergent around the protein is organized

with the chain perpendicular to the protein surface and with

the terminal region sticking into the protein This reduces

the hydrophobic portion of the detergent in which free

quinone can diffuse ( 1500 A˚3) [33] Crystallographic data

[14] shows that the detergent itself is located in the channel

into which the quinone isoprenoid chain sits in the enzyme

This explains the slow exchange process of the quinone at its

binding site Conversely, the dimensions of Triton X-100

micelles [35] are larger than those formed by LDAO,

thereby allowing a larger quinone pool size as well as higher ligand mobility

The proteoliposomes are topologically similar to the detergent–RC

50 complexes, i.e they are disconnected solubi-lizing environments, but they differ because proteolipo-somes can allocate a large number of proteins, in the order

of hundreds of RCs per vesicle As a consequence, the number of quinones per liposome ranges from tens to hundreds, and fluctuations in the local concentration can be neglected Liposomes can therefore be used for drawing general conclusions on quinone binding at the QBsite The lipophylic environment represented by proteoliposomes has several advantages in describing the exchange of quinone in photosynthetic membranes compared to the RC–detergent complexes: (a) the quinone is arranged in the bilayer in a similar manner to chromatophores, where quinone can freely diffuse towards and away from the enzyme, and the large volume of the bilayer allows the accommodation a large number of ligands; (b) the arrangement of the lipid molecules around the RC is not known, but it can be reliably assumed that they will not attach with their chains into the protein No direct interaction with the QBsite is expected and the channel will always be accessible for the quinone exchange; (c) The absolute value of ksmeasured in detergent is larger than the one obtained in saturating conditions in liposomes, indicating a relative stabilization of

Q 

B This difference in the semiquinone stability might be associated with small detrimental changes in the QBpocket, induced by the detergent hydrophobic chains that are absent

in the case of liposomes The absence of detrimental effects

in liposomes is also confirmed by the D þelectron nuclear double resonance spectra as recently reported by our group [19]

Some considerations on the absolute value of the quinone exchange constants in proteoliposomes can be useful in order to understand the same process in photosynthetic membranes For a bimolecular reaction of an enzyme with a small ligand, a reasonable approximation of the frequency

Fig 4 Numerical simulation of time evolution following light pulses of D+Q A

–

, D+Q A –

Q B ,

D + Q A Q B–and D+Q free Q/RC ¼ 0.74; [RC] ¼ 8.3 l M ; K B ¼ 10 6

M )1 ; k out ¼ 25 The initial ten milliseconds of the time-course are shown in the insert.

Fig 5 Simulated decay of D + obtained for Q/RC = 0.37 and a

binding constant of K B = 106M )1 Different decays were obtained

with different k out The noisy line represents the recorded trace in

the experimental conditions used for the simulation.

of collision (fC) in the diffusion controlled regime can be

obtained by simple considerations on the mobility of the

two species [36]:

fC¼ 4pr0ðDRCþ DQ 3NA 4pr0DQ 3NA

Eqnð4Þ

r0 is the minimum approaching distance in cm, assumed

to be equal to the radius of the protein; NA is the Avogadro Number; DRC and DQ represent the diffusion coefficients of the RC and quinone, respectively The approximation in Eqn (4) is based on the large difference

in the dimension of the colliding molecules For mito-chondrial cytochrome bc1, a diffusion constant D¼ 4.0 ·

10)11cm2Æs)1was measured [37,38] and a similar order of magnitude can be expected for the RC, as both are large membrane proteins

Several techniques have been used to measure the ubiquinone-10 diffusion coefficient DQ Using fluorescent quenching [39–42] the diffusion coefficient was found to span the range 1· 10)7) 5 · 10)6cm2Æs)1 With the fluorescent recovery after photobleaching technique, a value

in the range 1· 10)8) 5 · 10)8cm2Æs)1 was obtained [37,43,44] Electrochemical methods were also used, and a value of (2.0 ± 0.4)· 10)8cm2Æs)1 was obtained [45] According to Blackwell and coworkers [41,42] the DQ obtained using the fluorescence quenching method can be disregarded because it overestimates the actual value Therefore, using DQ¼ 2.0 · 10)8cm2Æs)1 in Eqn (4), an

fCvalue equal to 5.0· 107M )1Æs)1is obtained It should be noted that although the collision frequency is slightly overestimated because Eqn (4) is valid for three-dimen-sional systems, this value remains within the accuracy of these considerations

Assuming the surface of the QBchannel entrance to be the area of the protein where a successful collision can take place [46], an estimate of the association constant can be made Assuming 30 A˚2 and 5000 A˚2 for the quinone moiety, and L and M subunit surfaces in contact with the lipids respectively, a correction factor of 0.006 is obtained

As a consequence kincan be estimated to be 3· 105

M )1Æs)1 which is very close to the result of the best-fit procedure (kin¼ 7.2 · 107), corrected by the factor [L]v¢tail (see Appendix) giving kin[L]v¢tail¼ 3.6 · 105M )1Æs)1 This sug-gests that the rate limiting step for the association of the RC

Fig 6 Three dimensional representation of (A) k s and (B) DA0slowdependence on K B and k out /k AD For k out /k AD < 1 (slow exchange), the fraction of slow phase coincides with the fraction of occupied Q B sites in the darkadapted state The k s obtained under such conditions is independent, as expected, of the concentration of quinone in the solubilizing environment, matching the value from Eqn (1) For k out /k AD > 1 (fast exchange), the fraction of slow phase does not coincide with, and moreover, over-estimates the fraction of Q B sites occupied in the darkadapted state.

Fig 7 Comparison between simulated (s) and experimental values (d)

for k s (A) and DA0slow (B) as functions of Q/RC.

and quinone is the diffusion of the latter through the

proteoliposomes, implying that the ligand in the binding

channel, either taken up or released, moves at least as fast as

in the bilayer; this can be expressed as (DQ)channel‡ DQ

Assuming a random transfer for quinone, it will cover the

binding channel of length
X ( 50 A˚) in an average time of

(1/kdiff)£ (
X2/2DQ) 2 · 10)5s

The average time for quinone release, obtained from the

simulations of 1/kout¼ 25 ms, accounts for both the

residence time in the channel (1/kdiff) and for the time

required to unbind from the pocket (1/kP):

1

kout

¼ 1

kdiff

þ 1

kP

1

kP

As a consequence, the bottleneckin the quinone release

process is represented by the unbinding of the ligand from

its pocket The value of 25 ms obtained from Eqn (5), is

comparable with that of 2 ms obtained by NMR

measurements for systems kept in the dark in the presence

of ubiquinone-10 [47] The results differ by one order of

magnitude, and the discrepancy can be attributed to the

assumption that the charge separated and neutral RCs

exchange quinones with the same kinetics, regardless of the

redox state of QA(Fig 1), as we assume that the presence of

the hydrophobic tail has no influence on kP This suggests

that in the charge separated state the quinone release is

slower than in the dark This can be explained by invoking

the gated propeller twist imposed on QBby the presence of a

negative charge on QA[48] that buries the quinone head in

the inner part of the QB pocket, thereby increasing the

interaction energy between the ligand and the binding site

In a forthcoming workthe exchange kinetic dependence on

the QAredox state will be addressed

The results obtained in this paper can be related to the

RC photocycle, when the photochemistry takes place in the

presence of an exogenous electron donor able to doubly

reduce D+ Gerencser et al [49] have measured the

steady-state rate of cytochrome c turnover in detergent,

demon-strating that at low ionic strength the reaction of

cytochrome c3+unbinding from the RC is the rate limiting

step of the photocycle (1000 s)1< koff< 2000 s)1) By

employing the simulated kin¼ 7.2 · 107

M )1Æs)1value, it is possible to estimate the [Q]min at which quinone uptake

is not the rate limiting step: kin· [Q]min> 1000 s)1

[Q]min> 14 lm, which agrees with the value of 25 lMused

in Gerencser’s work Such [Q]min can easily be obtained

in our preparation and would give a quinone pool of

Q/RC 3, which is smaller than the average dimension of

the quinone pool in chromatophores [50]

Interestingly the structure of the QB pocket and the

quinone in the illuminated crystals [48] shows a strong

interaction between the protein residues and the quinoid

moiety of the ligand, based on the formation of hydrogen

bonds These bonds will, of course, disappear following

the double reduction of the RC photocycle and

protona-tion of the quinone; in some way driving the release of the

quinol It is quite tempting to conclude that the release of

the quinol from the binding pocket would be faster than

the quinone release because of the weaker interaction

between the QBpocket and the reduced ligand Presently

however, the results only permit the setting of a lower

limit on the release rate of quinol, which will lead to a result larger than the same rate for the oxidized form:

53

(kout)QH2‡ (kout)Q Conversely, from the hypothesis that cytochrome turnover would be unchanged in both the vesicle and in detergent, i.e that the unbinding of oxidized cytochrome would remain the slowest step of the photo-cycle, the upper limit for the quinol release can be set to (kout)QH2£ 1000 s)1

Conclusions

By studying the charge recombination kinetics of reaction centers incorporated into liposomes, thermodynamic and kinetic parameters have been inferred which regulate the photosynthetic turnover of this important protein These values, although obtained in a simpler environment, can be reasonably taken as a fair approximation to the ones actually working in the natural ICMs

The high value found for the quinone equilibrium binding constant KB¼ (kin/kout)¼ 1.8 · 106

M )1, makes it possible for the reaction centers to efficiently workwith a small quinone pool: we found that with a quinone/protein molar ratio as small as three, the QBsite was fully occupied When the electron reaches the QAsite in a reaction center without the quinone in the QBpocket, only the charge recombina-tion reacrecombina-tion can occur, which results in a loss of excitarecombina-tion energy However, in physiological conditions, where the quinone pool size has been estimated to be 10 or larger, this

is very unlikely to happen It would be interesting to investigate similar reconstituted systems prepared with

RC mutants with smaller charge recombination rate (kAD) constants which fulfil the slow exchange regime in liposomes

54

Acknowledgements

The authors are grateful to Professor E Caponnetti and Dr Lucia Pedone of Dipartimento di Chimica Fisica – Universita´ di Palermo for performing the dynamic light scattering measurements Thanks also to La´szlo´ Nagy and Pe´ter Maro´ti for helpful discussions This workwas made possible thanks to the financial support of the Grants: Meccanismi Molecolari della Fotosintesi (FIRB-MIUR) and Cofin – MIUR 2002.

References

1 Collins, M.L.P & Remsen, C.C (1991) The purple phototrophic bacteria In Structure of Phototrophic Prokaryotes (Stolz, J.F., ed.), pp 49–77 CRC Press, Boca Raton, FL, USA.

2 Oelze, J & Drews, G (1972) Membranes of photosynthetic bac-teria Biochim Biophys Acta 265, 209–239.

3 Verme´glio, A., Joliot, P & Joliot, A (1995) Organization of electron transfer components and supercomplexes In Anoxygenic Photosynthetic Bacteria (Blankenship, R.E., Madigan, T.M & Bauer, C.E., eds), pp 279–295 Kluwer Academic Publisher, Dordrecht, Boston, London.

4 Zuber, H & Cogdell, R.J (1995) Structure and organization of purple bacterial reaction antennae complexes In Anoxygenic Photosynthetic Bacteria (Blankenship, R.E., Madigan, T.M & Bauer, C.E., eds), pp 315–348 Kluwer Academic Publisher, Dordrecht, Boston, London.

5 Gromet-Elhanan, Z

55 (1995) The proton-translocating F 0 F 1 ATP synthase–ATPase complex In Anoxygenic Photosynthetic Bacteria 52

(Blankenship, R.E., Madigan, T.M., & Bauer, C.E., eds),

pp 807–820 Kluwer Academic Publisher, Dordrecht, Boston,

London.

6 Freiberg, A (1995) Coupling of antennaes to reaction centers In

Anoxygenic Photosynthetic Bacteria (Blankenship, R.E., Madigan,

T.M., & Bauer, C.E., eds), pp 385–398 Kluwer Academic

Pub-lisher, Dordrecht, Boston, London.

7 Mauzerall, D.C & Greenbaum, N.L (1989) The absolute size of a

photosynthetic unit Biochim Biophys Acta 974, 119–140.

8 de Grooth, B.G., van Grondelle, R., Romijn, J.C & Pulles, M.P.

(1978) The mechanism of reduction of the ubiquinone pool in

photosynthetic bacteria at different redox potentials Biochim.

Biophys Acta 503, 480–490.

9 Feher, G., Allen, J.P., Okamura, M.Y & Rees, D.C (1989)

Structure and function of bacterial photosynthetic reaction

cen-tres Nature 339, 111–116.

10 Wraight, C.A (1977) Electron acceptors of photosynthetic

bac-terial reaction centers Direct observation of oscillatory behaviour

suggesting two closely equivalent ubiquinones Biochim Biophys.

Acta 459, 525–531.

11 Okamura, M.Y & Feher, G (1995) Proton-coupled electron

transfer reactions of QB in reaction centers from photosynthetic

bacteria In Anoxygenic Photosynthetic Bacteria (Blankenship,

R.E., Madigan, T.M & Bauer, C.E., eds), pp 577–594 Kluwer

Academic Publisher, Dordrecht, Boston, London.

12 McPherson, P.H., Okamura, M.Y & Feher, G (1990) Electron

transfer from the reaction centres of Rb sphaeroides to the

qui-none pool: doubly reduced Q B leaves the reaction centres Biochim.

Biophys Acta 1016, 289–292.

13 Venturoli, G., Fernandez-Velasco, J.G., Crofts, A.R & Melandri,

B.A (1986) Demonstration of a collisional interaction of

ubiqui-nol with the ubiquiubiqui-nol-cytochrome c2 oxidoreductase complex in

chromatophores from Rhodobacter sphaeroides Biochim Biophys.

Acta 851, 340–352.

14 Lancaster, C.R & Michel, H (1997) The coupling of light-induced

electron transfer and proton uptake as derived from crystal

structures of reaction centres from Rhodopseudomonas viridis

modified at the binding site of the secondary quinine, QB

Struc-ture 5, 1339–1359.

15 Nagy, L., Fodor, E., Farkas, T., Gedey, S & Kecskes, A (1998)

Lipids effect the charge stabilisation in wild type and mutant

reaction centres of photosynthetic bacteria Rhodobacter

sphaer-oides In Photosynthesis: Mechanisms and Effects.

the 11th International Congress on Photosynthesis (Garab, G., ed.),

pp 893–896 Kluwer Academic Publishers, Dordrecht, the

Nether-lands.

16 Shinkarev, V.P & Wraight, C.A (1993) Electron and proton

transfer in the acceptor quinone complex of reaction centres of

phototrophic bacteria In The Photosynthetic Reaction Center

(Deisenhofer, J & Norris, J.R., eds), pp 193–255 Academic Press,

San Diego, CA, USA.

17 Isaacson, R.A., Lendzian, F., Abresch, E.C., Lubitz, W & Feher,

G (1995) Electronic structure of Q-A in reaction centers from

Rhodobacter sphaeroides I Electron paramagnetic resonance in

single crystals Biophys J 69, 311–322.

18 Okamura, M.Y., Isaacson, R.A & Feher, G (1975) The primary

acceptor in bacterial photosynthesis: the obligatory role of

ubi-quinone in photoactive reaction centres of Rp Sphaeroides Proc.

Natl Acad Sci USA 72, 3491–3495.

19 Trotta, M., Milano, F., Nagy, L & Agostiano, A (2002)

Response of membrane protein to the environment: the case of

photosynthetic reaction centre Material Science and Engineering:

C 22, 263–267.

20 Venturoli, G., Trotta, M., Feick, R., Melandri, B.A & Zannoni,

D (1991) Temperature dependence of charge recombination from

the P + QA – and P + QB – states in photosynthetic reaction centres

isolated from thermophilic bacterium Chloroflexus aurantiacus Eur J Biochem 202, 625–634.

21 Kutuzov, M.A., Mamedov, M.D., Semenov, A.Yu., Shinkarev, V.P., Verkhovsky, M.I., Abdulaev, N.G., Drachev, L.A (1991) Functioning of quinone acceptors in the reaction center of the green photosynthetic bacterium Chloroflexus aurantiacus FEBS Lett 289, 179–182.

22 Ford, N.C Jr (1985)

58 Light Scattering Apparatus In Dynamic Light Scattering, Applications of Photon Correlation Spectroscopy (Pecora, R., ed), pp 7–34 Plenum Press, New York, NY, USA.

23 Schurtenberger, P & Hauser, H (1984) Characterisation of the size distribution of unilamellar vesicles by gel filtration, quasi-elastic light scattering and electron microscopy Biochim Biophys.

59 Acta 778, 470–480.

24 Palazzo, G., Mallardi, A., Giustini, M., Berti, D & Venturoli, G (2000) Cumulant analysis of charge recombination kinetics in bacterial reaction centres reconstituted in liposomes Biophys J.

79, 1171–1179.

25 Drews, G & Golecki, J.R (1995) Structure, molecular organisa-tion, and biosynthesis of membranes of purple bacteria In Anoxygenic Photosynthetic Bacteria (Blankenship, R.E., Madigan, T.M., & Bauer, C.E., eds), pp 231–257 Kluwer Academic Pub-lisher, Dordrecht, Boston, London.

26 Golecki, J.R & Oelze, J (1980) Differences in the architecture of cytoplasmic and intracytoplasmic membranes of three chemo-trophically and photochemo-trophically grown species of the Rhodos-pirillaceae J Bacteriol 144, 781–788.

27 Saphon, S., Jackson, J.B., Lerbs, V & Witt, H.T (1975) The functional unit of electrical events and phosphorylation in chro-matophores from Rhodopseudomonas sphaeroides Biochim Bio-phys Acta 408, 58–66.

28 Kleinfeld, D., Okamura, M.Y & Feher, G (1984) Electron transfer in reaction centres of Rhodopseudomonas sphaeroides I Determination of the charge recombination pathway of D+Q A Q–B

and free energy and kinetic relations between Q–A Q B and Q A Q–B Biochim Biophys Acta 766, 126–140.

29 Paddock, M.L., Graige, M.S., Feher, G & Okamura, M.Y (1999) Identification of the proton pathway in bacterial reaction centers: inhibition of proton transfer by binding of Zn2+Cd2+ Proc Natl Acad Sci USA 96, 6183–6188.

30 Wraight, C.A & Stein, R.R (1983) Bacterial reaction center as a model for PSII Turnover of the secondary acceptor quinone.

In Oxygen Evolving System of Photosynthesis (Inoue, Y., Crofts, A.R., Govindjee, Murata, N., Renger, G & Satoh, K., eds), pp 383–392 Academic Press, Tokyo, Japan

31 Agostiano, A., Catucci, L., Colafemmina, G., Della Monica, M., Palazzo, G., Giustini, M & Mallardi, A (1995) Charge recom-bination of photosynthetic reaction centres in different membrane models Gazz Chim Ital 125, 615–622.

32 Lagarias, J.C., Reeds, J.A., Wright, M.H & Wright, P.E (1998) Convergence properties of the Nelder-Mead simplex method in low dimensions SIAM J Optimization 9, 112–147.

33 Roth, M., Lewitt-Bentley, A., Michel, H., Deisenhofer, J., Huber, R & Oesterhelt, D (1989) Detergent structure in crystals

of a bacterial photosynthetic reaction centre Nature 340, 656–662.

34 Roth, M., Arnoux, B., Ducruix, A & Reiss-Husson, F (1991) Structure of the detergent phase and protein–detergent inter-actions in crystals of the wild-type (strain Y) Rhodobacter sph-aeroides Biochemistry 30, 9403–9413.

35 Gast, P., Hemelrijk, P.W., Van Gorkom, H.J & Hoff, A.J (1996) The association of different detergents with the photosynthetic reaction centre protein from Rhodobacter sphaeroides R26 and the effects on its photochemistry Eur J Biochem 239, 805–809.

36 Cantor, C.R & Schimmel, P.R (1980) Biophysical Chemistry: Techniques for the Studyof Biological Structure and Function.

Freeman & Co., San Francisco, CA, USA

37 Gupte, S., Wu, E.S., Hoechli, L., Hoechli, M., Jacobson, K.,

Sowers, A.E & Hackenbrock, C.R (1984) Relationship between

lateral diffusion, collision frequency, and electron transfer of

mitochondrial inner membrane oxidation-reduction components,

Proc Natl Acad Sci USA 81, 2606–2610.

38 Hackenbrock, C.R., Gupte, S., Wu, E.S & Jacobson, K (1984)

Lateral diffusion, collision and efficiency of oxidation-reduction

components in mitochondrial electron transport Biochem Soc.

Trans 12, 402–403.

39 Fato, R., Battino, M., Degli-Esposti, M., Parenti-Castelli, G &

Lenaz, G (1986) Determination of partition and lateral diffusion

coefficients of ubiquinones by fluorescence quenching of

N-(9-anthroyloxy) stearic acids in phospholipid vesicles and

mitochondrial membranes Biochemistry 25, 3378–3390.

40 Lenaz, G., Samorı`, B., Fato, R., Battino, M., Parenti-Castelli, G.

& Domini, I (1992) Localization and preferred orientations of

ubiquinone homologs in model bilayers Biochem Cell Biol 70,

504–514.

41 Blackwell, M.F & Whitmarsh, J (1990) Effect of integral

membrane proteins on the lateral mobility of plastoquinone

of phosphatidylcholine proteoliposomes Biophys J 58, 1259–

1271.

42 Blackwell, M.F., Gounaris, K., Zara, S.J & Barber, J (1987) A

method for estimating lateral diffusion coefficients in membranes

from steady-state fluorescence quenching studies Biophys J 51,

735–744.

43 Rajarathnam, K., Hochman, J., Schindler, M & Ferguson-Miller,

S (1989) Synthesis, location, and lateral mobility of fluorescently

labeled ubiquinone 10 in mitochondrial and artificial membranes.

Biochemistry 28, 3166–3176.

44 Chazotte, B & Hackenbrock, C.R (1991) Lateral diffusion of

redox components in the mitochondrial inner membrane is

unaffected by inner membrane folding and matrix density J Biol.

Chem 266, 5973–5979.

45 Marchal, D., Boireau, W., Laval, J.M., Moiroux, J & Bourdillon,

C (1998) Electrochemical measurement of lateral diffusion

coef-ficients of ubiquinones and plastoquinones of various isoprenoid

chain lengths incorporated in model bilayers Biophys J 74,

1937–1948.

46 Agostiano, A., Milano, F & Trotta, M (1999) Investigation on

the detergent role in the function of secondary quinone in bacterial

Reaction Centers Eur J Biochem 262, 358–364.

47 Trotta, M., Feher, G & Okamura, M.Y (1992) NMR studies of

quinone binding to reaction centres from Rb Phaeroides Biophys.

J 61, A101

48 Stowell, M.H., McPhillips, T.M., Rees, D.C., Soltis, S.M.,

Abresch, E.C & Feher, G (1997) Light-induced structural

chan-ges in photosynthetic reaction center: implications for mechanism

of electron-proton transfer Science 276, 812–816.

49 Gerencser, L., Laczko, G & Maroti, P (1999) Unbinding of

oxidized cytochrome c from photosynthetic reaction center of

Rhodobacter sphaeroides is the bottleneckof fast turnover

Bio-chemistry 38, 16866–16875.

50 Hauska, G & Hurt, E (1982) Pool function behaviour and

mobility of isoprenoid quinones In Function of Quinones in Energy

Conserving Systems (Trumpower, B.L., ed), pp 87–110 Academic

Press, New York, NY, USA.

51 Shinkarev, V.P & Wraight, C.A (1997) The interaction of

qui-none and detergent with reaction centres of purple bacteria I.

Slow quinone exchange between reaction centre micelles and pure

detergent micelles Biophys J 72, 2304–2319.

52 Taly, A., Baciou, L & Sebban, P (2002) The DMPC lipid phase

transition influences differently the first and the second electron

transfer reactions in bacterial reaction centers FEBS Lett 532,

91–96.

Appendix

In this section the distribution of a highly hydrophobic solute among vesicles of different sizes will be addressed, assuming vesicles to be spherical compartments with bi-layered boundaries of negligible thickness Moreover, the existence

of a density probability function P(R) should also be defined

as follows: P(R) dR equals the probability to find a VRvesicle with radius between R and R + dR At this level of approximation, the overall concentration of vesicles [V] can

be calculated in terms of the lipid concentration [L] by the surface area conservation law per unit volume:

½La ¼ ½V

Z 8pR2PðRÞdR ¼ ½V8phR2i Eqnð6Þ

a being the lipid head area Additionally, the bilayer volume can be also estimated according to Palazzo et al [24], as the product of the lipid number on the vesicle bilayer surface (8pR2/a) multiplied by the lipid tail volume vtail:

mðRÞ ¼ ð8pR2=aÞ  mtail Eqnð7Þ The distribution of solute molecules S, among spherical vesicles V, of different radius R, can be then described with the following density function:

Pðn; RÞ ¼ Pðn j RÞPðRÞ Eqnð8Þ where P(n, R) dR is the probability to find n solute molecules inside a VRvesicle (i.e an aggregate of size between R and

R+ dR), and it equals the products of the probability P(R) dR to find a VRvesicle multiplied by the conditional probability P(n|R) to find n solute molecules in this aggregate

As a consequence of Eqn (8) the average number of solute moleculesÆNæ among vesicles of any size is obtained

by the summation over all possible solute molecule numbers and the integration over all vesicle size ranges:

hNi ¼Z X

n

nPðn j RÞPðRÞdR ¼

Z hNðRÞiPðRÞdR

Eqnð9Þ

As shown by the previous equation,ÆNæ can also be expressed in terms of the average numbers of solute molecules among VR compartments: ÆN(R)æ ¼

SnnP(n|R) The termÆN(R)æ can also estimate, in terms

of macroscopic concentration, the ratio between the bulkconcentration of S molecules contained in VR vesicles ([SR]), divided by the bulkconcentration of these aggregates [VR]:

hNðRÞi ¼½SR

[VR] is directly linked to the overall vesicle bulk concentration [V] by the equation [VR]¼ [V]P(R) dR, whereas different hypotheses can be found on the relationship between [SR] and [S], depending on the experimental preparation method Herein two main assumptions will be considered: (a) the solute molecule distribution is independent of the vesicle radius, (b) the

protona-tion of the quinone; in some way driving the release of the

quinol It is quite tempting to conclude that the release of

the quinol from the. .. smaller than the average dimension of

the quinone pool in chromatophores [50]

Interestingly the structure of the QB pocket and the

quinone in the illuminated crystals... part of the QB pocket, thereby increasing the

interaction energy between the ligand and the binding site

In a forthcoming workthe exchange kinetic dependence on

the