Deceleration of the electron transfer reaction in the photosynthetic reaction centre as a manifestation of its structure fluctuations

Aim. To extract information on the nature of protein structural relaxation from the kinetics of electron transfer reaction in the photosynthetic reaction centre (RC). Methods. The kinetic curves obtained by absorption spectroscopy are processed by a maximum entropy method to get the spectrum of relaxation times. Results. A series of distinctive peaks of this spectrum in the interval from 0.1 s to hundreds of seconds is revealed. With the time of exposure of the sample to actinic light increasing, the positions of the peak maxima grow linearly. Conclusions. Theoretical analysis of these results reveals the formation of several structural states of the RC protein. Remarkably, in each of these states the slow reaction kinetics follow the same fractional power law that reflects the glass-like properties of the protein.

Introduction.Biochemical reactions with the participation of biological macromolecules (proteins mostly) are usually known to demonstrate "deviations from simple behaviour" [1].
Even "simple" reactions of monomolecular type, for instance, binding of ligands or one-electron oxidation/reduction, are featured by complicated and evidently non-exponential kinetics.Besides, thermal behaviour of the reaction rate constants is non-Arrhenius.Both factors testify that description of these reactions in the framework of standard chemical kinetics is insufficient.
In recent decades the deviations are generally explained as a result of direct impact of structural movements of protein and its conformational fluctuations on the reaction.The fluctuations change more or less the characteristics of an active centre, promote the evolution of a reaction barrier, etc.The time spectrum of fluctuations can be extremely wideup to 10 orders -which causes high dispersion of the reaction rate "constants" (characteristic times).
A clas sic ex am ple (which is still one of the main reac tions while in ves ti gat ing a reg u la tory role of slow struc tural mo tions of pro tein) is the re verse bind ing of photodetached CO ligand to macromolecule of myoglobin, the ki net ics of which is reg is tered in the inter val from submicroseconds to sec onds and lon ger [2,3].A sim i lar sit u a tion is also no ta ble for the re ac tions of elec tron trans fer in the pig ment-pro tein com plex of the photosynthetic reaction centre (RC) [4,5].
Nat u rally, such sig nif i cant de cel er a tion of el e mentary bio chem i cal re ac tions com pli cates their sim u lation, as it al most elim i nates the pos si bil ity of com puter sim u la tions of MD-type even in case of well-de termined static struc ture of pro tein.It is also un real to mon i tor thor oughly its dy nam ics dur ing the ex per iment.The lat ter is usu ally per formed to reg is ter only the ki net ics of the main re ac tion (i.e. the state of ligand, electron or any other "substrate" of the reaction).
Therefore, all suggested mechanisms of structural regulation are of hidden nature and so far cannot be directly proven.The only criterion of adequacy of models is their capability of reproducing observed fine details of the respective reaction, and, more seldom, their predicting capability for independent experiments.It should be noted that the quality of both experimental data and methods of their analysis becomes critical.
The re sults of nu mer ous works, de voted to the re actions in com pli cated re lax ing en vi ron ment [6,7], includ ing those of pro teins, may be sum ma rized as follows.First, the mo tion along conformational de grees of free dom is dif fu sive, while the de scrip tion of re ac tions in the frame work of two-three dis crete con for ma tions ("ac tive-in ac tive" type, etc.) is in suf fi cient (see e.g.[8,9]).Sec ond, this dif fu sion is of non-stan dard char ac ter, re lated to hi er ar chi cal "tier" struc ture of the pro tein energy land scape [10] and to pe cu liar hi er ar chy of freedom de grees, when faster ones "limit" mo tion along slower [11,12].Usu ally, it co mes]to a time-de pend ent dif fu sion co ef fi cient for the vari able de ter min ing the re ac tion bar rier.In its turn, this causes the ob served re -lax ation dependences of "stretched ex po nen tial" type (exp(-t/t 0 ) b , where b < 1) or power de crease (t -a , a > 0), which is typ i cal for glass-like materials similar in this sense to proteins [7,[13][14][15].
In gen eral, this prob lem is far from trust wor thy con clu sions on the mech a nisms of struc tural reg u lation.There fore, the non-triv ial ev i dences of the im pact of conformational fluc tu a tions on the ki net ics of electron trans fer be tween the co factors of re ac tion cen tres, an a lyzed in this work, may be quite informative.

Materials and Methods
The photosynthetic RC of bacteria and reactions of electron transfer in it are one of the most studied biophysical systems (for instance, see [16]).The scheme of primary oxidation-reduction reactions in RC, isolated from bacteria Rb. sphaeroides, may be presented as follows: where P, Q A , Q B are cofactors, built in RC protein (Pprimary donor of the electron, presented by a dimer of bacteriochlorophyll; Q A , Q B -primary and secondary acceptors of quinone nature, respectively).The light-activated photodonor at first very quickly (in ~100 ps) transfers the electron to the primary quinone, and then the charge separation is stabilized by electron transfer from Q A to Q B .If there is no further photochemical channel the electronin response to RC pulse excitation returns to the oxidized photodonor in ~1 s at physiological conditions; this very reaction will be a subject offurther analysis as it is the most liable to the impact of structural fluctuations [17].Since the intermediate state P + QA -QB of scheme (1) is negligibly populated, we considered a simplified scheme: It should be taken into account that the return of the electron via Q A depends on the reaction barrier determined by the difference in the electron free energies at Q A and Q B [18].The rate of direct reaction is proportional to the intensity of actinic light.It is noteworthy that contrary to the majority of studies on RC reactions in response to impulse excitation, we study the consequences of continuous excitation of specific duration and intensity, as only in this case the effects of RC conformational rearrangements are especially significant (for details see [19][20][21] ).
In the scheme (2) the kinetics is mostly registered by the methods of differential absorption spectroscopy, as the system has a stable optic marker -an absorption band at 865 nm, which bleaches at the donor P photooxidation of .The absorption changes in this band, taken with the opposite sign and properly normalized, are a quantitative indicator of the charge separation (i.e. they present the population of P + Q B state).The hardware -software complex for registration of these changes in time and the method of obtaining preparations of isolated RC are described in [22].Finally, the analysis of kinetics of recombination ( distribution of relaxation times) was performed using a version of the maximum entropy method (MEM) developed by us (for details see [22]).
Re sults Fig. 1 pres ents a typ i cal se ries of ex per imen tal ki netic curves, re flect ing the pro cess of photoseparation of charges and sub se quent re com bi nation ac cord ing to the scheme (2).In this case the in tensity of ex cit ing light was the same for all the curves, but the ex po sure time t exp was in the range from 10 to 100 s.The fol low ing dis tinc tive fea tures of the pro cess are clearly ev i dent.In the be gin ning of photoactivation there is fast ox i da tion of the do nor (re duc tion of the accep tor), ac com pa nied by ad di tional and rel a tively slow charge sep a ra tion (un til the mo ment of switch ing off the ex cit ing light).The lat ter is re lated to struc tural adjust ment of RC pro tein (in Q B en vi ron ment mainly) to a new charge state [19].The same is the rea son of ev ident de cel er a tion of the elec tron re turn af ter switch ing off the light and in crease of t exp .
For further detailed analysis of the curves, their relaxation (decreasing) part was expanded in a spectrum of relaxation times using MEM which is much more reliable method than approximation with a small number of exponents [22]: where n(t) in this case is the population of state with the electron transferred to Q B .A typical result of this procedure is presented in Fig. 2 for exposure t exp = 60 s, where five peaks g i (t) are well separated with the maxima at t » 0.1 s, 1 s, 5 s, 50 s, and 450 s, the area for which is proportional to the number of RCs with the photoexcited electron recombining with the time characteristic for the given peak.The relaxation parts of all the curves presented in Fig. 1 were subjected to the same expansion.As a result, the recombination curves after photoexcitation of intensity I during t exp can be presented as a sum of contributions from each of obtained peaks: where g i (t; t exp ) -separate peaks in the distribution of relaxation times.
The result, which we would like to highlight in this work, is presented in Fig. 4, a, where deceleration of the relaxation at increasing t exp is illustrated by the shift of the fifth peak towards even higher values, this rise being rather well linearly approximated.A similar effect is observed for the 3 rd , 4 th peaks and even for "immovable" first and second ones, though with very little n* 1,2 .The latter is natural as it is well known that the first peak corresponds to the RC fraction, inevitably present in the samples, in which the secondary acceptor is absent or inactive, and therefore, reflects fast recombination from Q A [16].The second one is related to the RC portion, which has not undergone structural changes under the charge photoseparation, and reflects a relaxation response of RC to pulse excitation (see also "Discussion").
Let us make an important assumption that even at the relaxation stage, i.e. after switching off the light, the "instant" recombination time t r (i) (t; t exp ) also follows linear law = - where the time dependence of the rate "constant" K i (t; t exp ) º 1/t r (i) (t; t exp ) reflects all effects of "static" and "dynamic" disorder [6] (i.e. an impact of conformational substates and transitions between them), then the formula (6) means hyperbolic decrease of this ["]constant["] in time.The integration of the equation (7) The kinetics of this type (transforming into the exponential one only in the limit n i ® 0) in protein reactions was observed as early as in [23,24]   two most popular systems -RC and MbCO (see also "Discussion").
The analytical distribution of relaxation times, corresponding to the kinetics of (8) type, is easy to find as according to (4), (1/t 2 )g i (t; t ex p) is nothing but Laplace transform h i (t; t exp ).The reverse transformation gives [25]: where G is gamma function.It should be noted that distribution (9) has one maximum at t i max / (n i + 1), close to t i max if n i << 1.Therefore, the distribution of relaxation times, obtained from the whole relaxation curve n(t;t exp ), corresponds to: Let us apply the theoretical dependence (10) to the experimental results, analyzed by MEM (shown in Fig. 2 by a solid curve).It is evident that [the] theory is in very good agreement with the experiment.The table presents the obtained parameters of all five peaks of distribution g(t; 60 s).
Good agreement is obtained also for the separate components.For instance, Fig. 4, b presents the kinetics of decay of the fifth component which is restored, on the one hand, by the last peak with maximum of ~450 s, and on the other -according to the formula (9) with corresponding parameters from the table.Finally, Fig. 5 contrary to Fig. 3 presents the comparison of relaxation curve n(t; 60 s), obtained experimentally, with the curve restored by the formula (3) with theoretical distribution (10).The observed evident coincidence confirms the validity of linear (hyperbolic) law of increase (decrease) of the instant recombination time (reaction rate constant) with time (6).
Similar results were obtained for all other exposures.
Finally, let us make another important remark.Under normal conditions the recombination of the electron from the secondary quinone acceptor is mainly determined by its thermoactivated transfer to the primary quinone acceptor [16], therefore, the following equation is valid where a is a constant; X(t) -difference between free energies of the electron on primary and secondary quinone acceptors, the changes in which reflect the evolution of reaction barrier.From this it follows directly that the hyperbolic dependence leads to the logarithmic law of change in the reaction barrier value 12) which was also observed in some models of structural diffusion [26].
Discussion Use of kinetic curves to receive the information about structural reorganizations of RC, caused by photoexcitation and recombination of the electron, is a non-trivial task and requires adequate methods of experimental data analysis.
The maximum entropy method (MEM), used in this work, permits to isolate specific relaxation components out of the kinetic curve of the electron recombination.As it was mentioned above, the observed component of recombination h 1 (t; t exp ) with characteristic time t 1 max » 0.1 s corresponds to recombination of the electron in RC without the secondary quinone acceptor.The rest of recombination components could be easily related to the initial differences in RC structure, unless the experimentally revealed RC redistributions between the components are taken into account (for details, see [22]).Supposing that these components correspond to initially identical RC, a significant difference in their kinetics can be explained by faster structural relaxation, which occurs right after localization of the photo-mobilized electron on the secondary quinone acceptor.This relaxation should result in the formation of three RC fractions, with different structural deformations showing up as corresponding components of h i (t; t exp ), i = 3, 4, 5, which have characteristic times t 3 max » 5 s, t 4 max » 50 s, t 5 max » 450 s (see Fig. 2), differing by orders of magnitude under exposure of 60 s.
The relaxation processes, causing the occurrence of these states, may be considered [11,12] as the relaxation of higher level and related to local rearrangements of RC structure close to the places of localization of separated charges.It is just this slow structural relaxation of a lower level seems to be observed in the experiment for the formed RC fractions.The logarithmic dependence of the reaction barrier of electron recombination, corresponding to the revealed linear dependences of the reverse recombination rate, would be logically attributed to slow non-specific relaxation of peripheral parts of macromolecule globule [7,26], notable for the relaxation processes in glass-like matrices.
The afore said may be pre sented as a scheme (Fig. 6), where each of the four two-level (elec tron on do nor or on ac cep tor) elec tronic schemes cor re sponds to its struc tural state.The struc tural state 2 cor re sponds to RC or gani sa tion with the elec tron on do nor (i.e. it is "dark-adapted" state); af ter the photoexcited elec tron gets on the ac cep tor, struc tural changes are ini ti ated result ing in RC tran si tion to one of the struc tural states 3-5.The RC in state 2 is re plen ished by RC from these frac tions af ter re com bi na tion of the photoexcited electron.Thus, the "dark-adapted" state 2 is of dy namic nature (see also [24]).The dis tri bu tion of re com bi na tion times for this struc ture does not de pend on the photoexcitation ex po sure, its max i mum is at t Parameters of distribution g i (t; 60 s) of type (9) for each of the peaks of total spectrum g(t; 60 s) obtained using MEM from kinetics of recombination its shape is well de scribed by Eq. ( 9).In de pend ence of the po si tion of peak of the "dark-adapted" state on the photoexcitation time may be ex plained by the fact that due to elec tron fast re com bi na tion, the struc ture remains in the ini tial state, corresponding to the position of the electron on the donor.These assumptions can be partially referred to the structural state 3, which t 3 max weakly depends on the exposure time.The structural states 4, 5, and partially 3, formed as a result of relaxation of higher level, continue to relax in accordance with the law (12), specific for the systems in glass-like matrix.The time for the structure to return to the "dark-adapted" state is likely to depend on "deformation depth" which results in strong dependence of t i max on the exposure time for the states 4, and especially 5, but weak -for 3.In other words, on the average, after recombination of the electron in states 4, 5 RC do not relax into the "dark-adapted" state before the repeated photoexcitation.The state 3 is characterized by much less recombination time and the dependence is weakly expressed.It is noteworthy that similar assumptions were previously used by us in the simulation of "light-" and "dark-adapted" RC under prolonged photoactivation [19][20][21].
Conclusions The experimental data obtained according to the proposed scheme of electron-conformational transitions in the RC permit to define the following stages in the photoexcitation process studied: 1. Electron transfer from the donor to the final quinone acceptor, the rate of which K i (0, t exp ) at any moment of photoexcitation t exp for each RC depends on its structural state i.
2. Relatively fast process of RC relaxation with the electron on the acceptor from the "dark" (dynamic) structural state 2 into one of the structural states of higher level 3, 4, 5, which is supposed to reach minimum free energy of the system in times not registered in our experiment.
3. Slow processes of further relaxation of the RC with the electron on the acceptor and in i-th structural state occur in accordance with the law (12), characteristic for the systems in glass-like matrix, which is likely to correspond to relaxation of peripheral parts of the RC protein globule, accompanying a faster (local) relaxation of higher level.
It is note wor thy that the scheme, sug gested in this work, is sim pli fied and does not take into ac count a detailed char ac ter of RC tran si tions be tween struc tural states of higher level, which may be re vealed at dif ferent in ten sity of photoexcitation.Thus, at con sid er ably lower in ten sity of photoexcitation we ob served the occur rence of a new struc tural state of the RC as a re sult of bi fur ca tion [22], which may be re lated to the in ter action of re lax ation processes of different levels.
The work is partially performed in the framework of the project "Fundamental properties of physical systems in extreme conditions" of the Department of Physics and Astronomy of NAS of Ukraine.

Fig. 4 .
Fig. 4. Dependence of the time of recombination at its initial moment t 5 max exp ( ) t on the exposure time (a) and kinetics of decay of the fifth components, restored by the fifth peak, obtained using MEM (triangles) and by formula (9) (solid line) (b)