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provided by Elsevier - Publisher Connector Biophysical Journal Volume 79 September 2000 1237–1252 1237

Self-Regulation Phenomena in Bacterial Reaction Centers. I. General Theory

Alexander O. Goushcha,*† Valery N. Kharkyanen,‡ Gary W. Scott,† and Alfred R. Holzwarth* *Max-Planck-Institut fu¨r Strahlenchemie, Ruhr 45470, Germany; †Department of Chemistry, University of California at Riverside, Riverside, California 92521 USA; and ‡Institute for Physics, National Academy of Science–Ukraine, Kyiv 252028, Ukraine

ABSTRACT A model for light-induced charge separation in a donor-acceptor system of the reaction center of photosyn- thetic bacteria is described. This description is predicated on a self-regulation of the flow of photo-activated electrons due to self-consistent, slow structural rearrangements of the macromolecule. Effects of the interaction between the separated charges and the slow structural modes of the biomolecule may accumulate during multiple, sequential charge transfer events. This accumulation produces non-linear dynamic effects on system function, providing a regulation of the charge separation efficiency. For a biomolecule with a finite number of different charge-transfer states, the quasi-stationary populations of these states with a localized electron on different cofactors may deviate from a Lagmuir law dependence with actinic light intensity. Such deviations are predicted by the model to be due to light-induced structural changes. The theory of self-regulation developed here assumes that light-induced changes in the effective adiabatic potential occur along a slow structural coordinate. In this model, a “light-adapted” conformational state appears when bifurcation produces a new minimum in the adiabatic potential. In this state, the lifetime of the charge-separated state may be quite different from that of the “dark- adapted” conformation. The results predicted by this theory agree with previously obtained experimental results on photo- synthetic reaction centers.

INTRODUCTION Biological energy conversion and storage take place bacteriorhodopsins indicate slow (tens of seconds) struc- through elementary events of charge transport in biomol- tural motions induced by a proton flux (Nagel et al., 1998; ecules. Transient, localized charges interact with ionized, Sass et al., 1998). Long-lived structural modes are also polarizable, or dipolar structural elements of the macromol- important for the function of cytochrome oxidases (Einars- ecule to perturb cofactor and/or protein structural modes. dottir et al., 1993) and ATPases (Noji et al., 1997). For such These interactions couple localized electron states to nuclear modes, transient, localized charges interact with structural degrees of freedom that may be reduced to a single (general- elements of the biomolecule, and the effects of these inter- ized) coordinate (see, e.g., Agmon and Hopfield, 1983). This actions accumulate during successive events. Accumulated coordinate may be either collective or localized, corresponding structural changes produce feedback on the charge transfer in the latter case to motion of specific structural groups. Char- rate. Thus, slow conformational modes function as control acteristic relaxation times of structural motion may vary modes to determine long-time biomolecule behavior widely, facilitating either an adiabatic or a non-adiabatic ele- (Haken, 1983). The action of a charged particle flux on slow mentary charge transfer event in the biomolecule (see Hoff and conformational modes and structural feedback on charge Deisenhofer, 1997, for a review). The present work focuses on transfer rate constants produce non-linear, self-regulation effects that occur during multiple, sequential charge transfer effects (Chinarov et al., 1992; Tributsch and Pohlmann, events when structural relaxation is significantly slower than 1998; Goushcha et al., 1997a; Gushcha et al., 1994). These the charge transfer rate itself. self-regulation processes should be quite important for the The relevance of slow structural dynamics to the function function of charge transfer biomolecules, modulating the of biological charge transfer system function has been dem- charge-transfer rate. To describe these effects, we propose a onstrated many times. Photosynthetic reaction centers self-consistent, adiabatic theory of charge transfer and struc- (RCs) exhibit a long-lived, structural relaxation for minutes tural motion. This theory develops a correlation of structural after completion of electron transfer (ET) (Puchenkov et al., dynamics and electron transfer, ensuring a correct statistical 1995; Kalman and Maroti, 1997). Numerous studies of description of electron-conformational dynamics in macro- molecules by considering system diffusion along an effec- tive adiabatic potential. This approach generalizes an adia- Received for publication 22 October 1998 and in final form 24 May 2000. batic theory for a single ET event to the case of multiple, Address reprint requests to Dr. Alexander O. Goushcha or Prof. Gary W. successive ET events, each of which induces small but Scott, Dept. of Chemistry, University of California at Riverside, Riverside, long-lived structural changes that accumulate to influence CA 92521. Tel.: 909-787-4732; Fax: 909-787-4713; E-mail: goushcha@ ucrac1.ucr.edu or [email protected]. subsequent events. A preliminary account of these results was presented at the Third Sympo- We develop this theory for photosynthetic reaction cen- sium on Biological Physics, Santa Fe, New Mexico, September 1998. ters, but most of our results can be readily generalized to © 2000 by the Biophysical Society other macromolecular charge transfer systems. For an intact 0006-3495/00/09/1237/16 $2.00 photosynthetic system, charge separation efficiency is de- 1238 Goushcha et al.

termined by the quantum yield of the primary charge sep- time. The solution of Eq. 2 is: aration event and the lifetime of the charge-separated state. ␳͑t, D͒ ϭ 1 Ϫ ␳͑t, A͒ The term “efficiency of charge separation” emphasizes that (3) the average survival time of this state, as in isolated RCs, is ϭ ␳ ͑ϱ ͒ ϩ ͓␳͑ ͒ Ϫ ␳ ͑ϱ ͔͒ ͑Ϫ␬ ͒ I , D 0, D I , D exp t ; the determining factor in intact systems, in which the quan- tum yield of the primary charge separation event is Ϸ1. (see in which Wraight and Clayton, 1973). ␬ ϭ I ϩ k ; (4) In this paper, we proceed as follows: 1) In the next (first) rec section, we develop a theory for a two-state charge-transfer krec system and for the more general case of a finite number of ␳ ͑ϱ, D͒ ϵ lim ␳͑t, D͒ ϭ ; (5) I I ϩ k charge-transfer states. We show that the survival time of the t3ϱ rec charge-separated state reflects the light-induced structural and changes of the system. 2) In the second section, we develop a general, kinetic description of the electron-conformational I ␳ ͑ϱ ͒ ϵ Ϫ ␳ ͑ϱ ͒ ϭ I , A 1 I , D ϩ . interaction in macromolecular systems. We show that slow, I krec structural dynamics determine self-regulation effects in bi- omolecules. 3) In the third section, we analyze the depen- For a more general system with an arbitrary number of dence of stationary-state structural variable values with light charge-transfer states, but only a single photodonor D, the ␴ ϭ Ϫ ␳ intensity. 4) Finally, in the fourth section we apply the quantity (t) 1 (t, D) defines the probability of charge theory to photosynthetic RC recombination kinetics and separation at t. For a fixed, constant actinic light intensity I, ␴ ϱ ϵ Ϫ ␳ ϱ quasi-stationary-state, light-induced effects. I( ) 1 I( , D). For the case of a simple donor- acceptor pair (Eq. 1), we obtain: I ␴ ͑ϱ͒ ϭ , (6) SURVIVAL TIME OF THE CHARGE-SEPARATED I I ϩ k STATE: DEPENDENCE UPON rec MACROMOLECULAR STRUCTURE corresponding to a Langmuir dissociation isotherm with a half-saturation intensity, k . Consider first the average lifetime of the charge-separated rec The efficiency of charge separation under stationary-state state for a simple system consisting of a photodonor D and illumination with a single photoactivated electron that trans- an acceptor A, both inserted into a suitable matrix. The fers between a finite number of localized electron states is scheme of electron transfers in this system may be described defined as the ratio of the stationary state probability of by charge separation to the number of charge separation events

kIϭ␩I per unit time -|0 ϩ Ϫ DA D A , (1) ␴ ͑ϱ͒ krec ␶ ϭ I d ␳ ͑ϱ ͒ (7) I , D I in which k ϭ ␩I is the first-order rate constant for photo- I ␶ induced electron transfer from the light-absorbing photodo- Here d gives the average lifetime or “survival time” (Ag- nor D to the acceptor A. The rate of this process is propor- mon and Hopfield, 1983) of separated charges relative to tional to the intensity of absorbed actinic light I, with a recombination. For the two-state system under consider- ␩ ␶ ϭ Ϫ1 proportionality coefficient ; krec is the first-order rate con- ation, d (krec) . We show in the Appendix that, for the stant for charge recombination. general case of a system with a finite number of localized ␶ Let ␳(t, D) and ␳(t, A) be the normalized populations of electron states and a fixed structure, the value of d, given the states, DA and DϩAϪ, respectively, at time t. Then these by Eq. 7, depends only on structural organization and not ␶ quantities satisfy simple coupled differential rate equations upon the actinic light intensity. Moreover, d can be mea- for a fixed structure of the system: sured by the system response ␴(t) to a short, saturating actinic flash or upon ceasation of continuous photoexcita- Ѩ␳͑t, D͒ tion: ϭ Ϫ␩ ␳͑ ͒ ϩ ␳͑ ͒ Ѩ I t, D krec t, A ; t ϱ (2) ␶ ϭ ͵ ␴͑t͒dt. (8) Ѩ␳͑t, A͒ d ϭ ␩I␳͑t, D͒ Ϫ k ␳͑t, A͒; 0 Ѩt rec ␶ Thus, d equals the area under the recombination probability in which we will take ␩ ϭ 1. This substitution specifies the function. In the case of multiphasic relaxation this param- units of I as photoinduced charge separation events per unit eter is identical to the average lifetime of the charge-sepa-

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1239

Ϫ␥ ␴ ␴ ϭ͚ it rated pair. Approximating (t)as (t) i Aie , in which transfer in chemical and biological systems was given by ␥ { i} are a set of relaxation rate constants, with correspond- Levich and Dogonadze, 1959; Marcus, 1956; Marcus and ␶ ϭ͚ ␥ ing weights, Ai, then it follows from Eq. 8 that d i (Ai/ i). Sutin, 1985; and Jortner, 1976. See also the review by Hoff ␶ Thus, d is the time constant for some effective single-expo- and Deisenhofer, 1997. It was shown that, in the high ␻ nential relaxation process that gives the area under the relax- temperature limit, the rate constant ij of ET between the ith ation kinetics curve equal to that of the real process. As a and jth cofactors depends exponentially on both the donor- ␭ ϩ⌬ ° 2 ␭ consequence, for the general case, using Eqs. 5 and 7, we acceptor distance Rij and the value of ( ij Gij) / ij,in ␭ ⌬ ° obtain which ij is the nuclear reorganization energy and Gij is the standard Gibbs free energy difference between the donor I ␴ ͑ϱ͒ ϭ and acceptor levels (Marcus, 1956; Marcus and Sutin, I ϩ ␶Ϫ1 . (9) I d 1985). This means that either a change in the distance between cofactors on the order of ϳ1 Å or a change in the Thus, the stationary state probability of photo-separated ␭ ϩ⌬ ° 2 ␭ macromolecule structure such that ( ij Gij) / ij changes charges depends upon the actinic light intensity strictly in by more than k T may cause a significant difference in ␻ . ␶ Ϫ1 B ij accordance with the Langmuir law, with a value ( d) for The generalization of the Marcus expression for the rate the half-saturation intensity. This value is fixed at a fixed constant of non-adiabatic ET in continuous media to the ␴ ϱ structure. Thus, any deviation of the experimental I( ) case of any solvent model shows that the rate constant for from a Langmuir curve implies that light-induced structural charge transfer may be expressed in terms of a function of ␶ rearrangements occur and that d depends on I. the free energy difference between electron-localized donor What physical mechanisms may correlate macromolecu- and acceptor sites produced by a fluctuating polar medium lar structural dynamics with photoactivated charge transfer (Tachiya, 1993). This approach leads to a Gaussian-like along a cofactor chain? The following facts are relevant: dependence of the charge transfer rate constant on the local 1. Electric fields produced by photo-induced separated electrostatic potential of the medium. Many current theories charges at angstrom distances are calculated to be on the of charge transfer reactions in proteins are based on a order of 107–108 V/cm, much higher than those that exist similar evaluation of the probability distribution for a free ⌬ across biomembranes in vivo. energy difference Vij between product and reactant states 2. Protein subunits of biomacromolecules contain charged (Warshel, 1982; Parson et al., 1998; Tachiya, 1993; Ban- or polar groups with redox properties that depend upon dyopadhyay et al., 1999; Warshel and Parson, 1991; Web- the surrounding media. ster et al., 1994). Such an approach not only provides for a 3. Experiments show that characteristic time constants for correct molecular dynamic calculation of the potential sur- structural relaxation in proteins range from nanoseconds faces of the reactant and product states, but also enables to minutes. prediction of the influence of adiabatic structural motions. 4. Slight perturbations in the equilibrium positions of mac- Molecular dynamic simulations show that photoinduced romolecule structural elements may dramatically change charge separation in photosynthetic reaction centers occurs the rates of electron transfer between cofactors. in much shorter times than those required for the system to approach conformational equilibrium after the charge trans- Statements 1–3 require no additional discussion. Support fer step (Parson et al., 1998). Recent molecular dynamics for statement 4, although previously discussed, is now am- studies also show that, for long-range electron transfer in plified. For a system with a finite number of localized proteins, cooperation between vibrational modes of the in- electron states but with no interactions with its surround- tervening medium and the transferring electron (inelastic ings, electron motion should be completely coherent and ET) may significantly facilitate the ET reaction, even mak- may be described in terms of a non-equilibrium density ing it an activationless process (Daizadeh et al., 1997; matrix as periodic oscillations of the electronic populations Medvedev and Stuchebrukhov, 1997). The resulting analyt- of these states (Landau and Lifshitz, 1965). However, ab- ical, modified Marcus expression for the ET rate constant, solute coherence of photoelectron motion is destroyed by using a diabatic model of electron tunneling in fluctuating interaction with thermal oscillations of the nuclei with re- medium, shows that the activation energy may be signifi- laxation times of Ϸ10Ϫ13–10Ϫ11 s. This means that non- cantly reduced due to inelastic interaction with phonons. diagonal elements of the density matrix may be neglected This description is similar to the idea of adiabatic self- for slow steps of charge separation. The non-adiabatic de- organized ET in an active medium (Tributsch and Pohl- scription given by Fermi’s Golden Rule is appropriate for mann, 1998; Gushcha et al., 1994). this type of donor-acceptor transition (Landau and Lifshitz, Adiabatic theories of particle transfer over a potential 1965). The theory of non-adiabatic transitions has been barrier lead to a Kramers-type dependence of the reaction well-developed in solid state physics by Fo¨rster, Dexter, and rate constant, one that depends exponentially on the barrier

Galanin (Fo¨rster, 1949; Dexter, 1953; Galanin, 1951). An height Eb (Kramers, 1940). Kramers’ theory came from an appropriate description of elementary steps of electron assumption of a linear interaction between the particle and

Biophysical Journal 79(3) 1237–1252 1240 Goushcha et al.

its environment. Recent studies by Tributsch show that a laxation times longer than a second, describe the global nonlinearity with energy of frictional force dependence may dynamics of macromolecular structural rearrangement. result in a greatly increased probability of escape from a Let us assume that there exist long-lived, light-induced potential well (Tributsch and Pohlmann, 1998). In this de- structural rearrangements of the macromolecule. Because of ␻ scription, the rate constant ij depends exponentially on their long relaxation times, these rearrangements can pro- ␣ Ϫ 2 ␣ ( /2)(Eb ELC) , with being a coupling coefficient with duce effects that accumulate from one single electron-trans- the medium and ELC denoting the mean energy of exchange fer step to the next. Under stationary-state illumination between the particle and medium during oscillation. This conditions, accumulated structural changes produce a new, idea has been substantiated analytically for the problem of quasi-stable structure. The extent of structural changes de- particle escape over a potential barrier in the case of strong pends only upon the illumination intensity. In particular, in interactions between the particle and structural modes of the the case of a large electron-conformational interaction, a surroundings (Cˇ apek and Tributsch, 1999). The authors “dark-adapted” conformational state may convert to a com- gave a description of uphill particle transfer for the simplest pletely new conformational state under high-intensity illu- case. In this case, the coupling of the transferred particle to mination. Furthermore, this new “light-adapted” conforma- its surroundings was assumed to be mediated by only one tional state may coexist with the “dark-adapted” one over an specific mode, localized in the vicinity of the transferred intermediate range of illumination intensity. This result particle. Exact calculations for more realistic cases of many means that there is a photo-induced bistability of the mac- interacting modes were not performed, but expected results romolecular structure. Necessary conditions for realization for such calculations should be qualitatively the same, pro- of such an effect are a strong charge-conformational inter- viding support for the phenomenological result obtained action and a long structural relaxation time relative to lo- earlier (Tributsch and Pohlmann, 1998). calized electron relaxation. The slow structural modes, rep- In this work we use a phenomenological description for resented in this theory as “slow, generalized coordinates” modeling non-equilibrium structural effects that occur dur- function as “control modes.” These modes lead to a self- ing sequential charge transfer through a protein. We take regulation of the photoexcited electron flux through the ␻ into account dependence of the rate constants ij on biomol- macromolecule, as recently demonstrated for photosyn- ecule structure by coupling to different structural motion. In thetic RCs (Gushcha et al., 1994, Goushcha et al., 1997a,b). both the adiabatic and non-adiabatic descriptions, small Below we develop a self-consistent, statistical theory of ⌬ ° ␭ changes in the values of parameters such as Gij, ij, Eb, electronic-conformational transitions to describe such ef- ⌬ ELC, Rij, Vij, which might be caused by electron transfer fects. between cofactors, significantly affect the kinetics of the ET. Therefore, we assume that ET rate constants should be expressed as exponential functions of a structural parameter A KINETIC DESCRIPTION OF THE ELECTRON- ϭ X X(x1, x2, x3,..., xi,...) that in turn depends on a CONFORMATIONAL INTERACTION IN A ␻ ϰ complete set of structural variables {x1,..., xN}, ij CHARGE-TRANSFER MACROMOLECULE exp(ϪX). The structural factor X is defined by either the For the theoretical treatment of light-induced structural adiabatic E , E , ⌬V ,...,ornon-adiabatic ⌬G° , ␭ , R , b LC ij ij ij ij changes in a macromolecule undergoing photoinduced ⌬V ,..., parameters of the system. Thus, statement 4 ij charge transfer and separation, we use a Langevin equation above indicates that charge-conformational interaction, by with two random forces to describe the mechanical motion which ET is coupled to structural dynamics, may signifi- of a flexible structure (Chandrasekhar, 1943): cantly affect the main reaction rate. For the simplest two-

state system (Eq. 1) the only kinetic parameter that depends d ѨT͑x˙͒ ѨVϩ͑x͒ ѨR͑x˙͒ ϭ Ϫ Ϫ ϩ ͱ ␽ ͑ ͒ ϩ ͑ x͒ upon macromolecular structure is the recombination rate Ѩ Ѩ Ѩ 2Di i t Fi t, , dt x˙i xi x˙i constant, krec. Thus we write, (11) ͑ ͒ ϭ 0 ͑Ϫ ͒ x ϵ x˙ ϵ krec X krec exp X . (10) in which {xi}, {x˙i} are the sets of structural variables (degrees of freedom; i ϭ 1, 2, . . .) with rates of In this equation, the structural factor X is dimensionless, structural rearrangements; T(x˙) and Vϩ (x) are the kinetic normalized with a scaling factor that depends on details of and potential energies for structural modes of the photoac- the particular system. tivated macromolecule, respectively; and R(x˙) is a dissipa-

A complete set of structural variables {x1,...,xN} may tive function of structural motion. be selected to span the coordinate space in many ways. Here The last two terms on the right-hand side of Eq. 11 we take the structural variables as a set of variables that are represent random forces. The first is a random force due to

each distinguished by different relaxation times. The fastest thermal motion. This force acts on the structural variable xi, variables (␶ Յ 10Ϫ13 s) describe fast motion of single atoms while the quantity ␽ (t) describes ␦-correlated random pro- i ͌ and small groups, whereas the slowest variables, with re- cesses with amplitudes 2Di to model thermal fluctuations

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1241

x ␶ ϽϽ ␶ of the structural variables. The last term in Eq. 11 is a tively. Those variables fast, for which x el, belong to x random force corresponding to interaction of a photoacti- the first group. For the second group ( equal) the time ␶ ϳ ␶ vated electron with the macromolecular structure. Thus, constants are of the same order: x el. The third group is x x ␶ Fi(t, ) describes a discrete random process: characterized by slow structural motions ( slow) for which x ϾϾ ␶ . The two types of variables, x and x , should ͑ x͒ ʦ ͓ i ͑x͔͒ ϭ el slow equal Fi t, fn , n D, A1, A2, . . . (12) be explicitly retained in a description of self-regulation i x effects for a system involving photoexcited electron transfer in which f n ( ) is a force describing the interaction of a photoactivated electron, localized on cofactor n, with struc- within a flexible structure. However, in the present treat- x x tural mode i. The probability of each component f i (x)is ment we retain only slow, and ignore equal. The fast vari- n x determined by the probability of electron localization on ables, fast, are not important for self-regulation effects, cofactor n at a fixed x. These probabilities (␳(t, n͉x)) can be because these variables relax on much shorter time scales. determined from the system of differential rate equations, We can integrate over these variables, using the substitution ͑x͒ Ѩ␳͑t, n͉x͒ Vn ϭ ͸ ͕Ϫ␻ ͑x͒␳ ͑ ͉x͒ ϩ ␻ ͑x͒␳ ͑ ͉x͖͒ expͩ Ϫ ͪ nm n t, n mn m t, m . k T Ѩt ͑ x͒ ϭ ͑ x ͒ B m P t; n, P˜ t; n, slow fast , Zn (13) (17) ͑x͒ These are the master equations for a random process (Eq. Vn ␻ x Zfast ϭ ͵ expͩ Ϫ ͪ dx , 12) (Horsthemke and Lefever, 1984). The quantity ( ) n fast nm kBT in Eq. 13 defines the rate constants of non-adiabatic transi- tions between the n and m cofactors at a fixed structure. We in which P˜ (t; n, x ) is the distribution function for elec- x ϭ slow assume that the variables {xi} represent overdamped tron and slow structural variables. conformational motions, a valid description for flexible x Putting Eq. 17 into Eq. 14 and integrating over fast,we structures like proteins. Although this assumption may be obtain equations identical to Eq. 14 that are valid for time incorrect for high-frequency oscillations, these variables are ϾϾ ␶ intervals t fast. They may also be derived from Eq. 14 thermally equilibrated and excluded from detailed consid- with a simple substitution, eration. From Eq. 11 and in accord with the results of x 3 x 3 ␻ 3 ␻ 3 Horsthemke and Lefever, 1984, for a coupled random pro- slow; P P˜ ; nm ˜ nm; Vn V˜ n. (18) cess (Christophorov, 1995) we obtain the fundamental ki- netic equation for the distribution function of both electron For ET rate constants between cofactors n and m, after such and structural variables of the macromolecule, P(t; n, x), substitutions, we obtain: ͑x͒ ѨP͑t; n, x͒ Vn ϭ ͑x͒ ͑ x͒ expͩ Ϫ ͪ Dˆ n P t; n, Ѩt kBT (14) ␻˜ ͑x ͒ ϭ ͵ ␻ ͑x͒ ⅐ dx . (19) nm slow nm Zfast fast ϩ ͸ ͕Ϫ␻ ͑x͒ ͑ x͒ ϩ ␻ ͑x͒ ͑ x͖͒ n nm P t; n, mn P t; m, , m x These transitions are non-adiabatic with respect to fast, but x in which adiabatic with respect to slow. The potential energy expression corresponding to slow Ѩ 1 ѨV ͑x͒ Ѩ structural variables can be easily obtained after substitution ˆ ͑x͒ ϭ ͸ ͫ n ϩ ͬ Dn Di , (15) x Ѩx k T Ѩx Ѩx of Eq. 17 into Eq. 14 and integrating over fast, i i B i i V ͑x͒ ѨV ͑x͒ ѨVϩ͑x͒ n n i V˜ ͑x ͒ ϭ Ϫk T ln͵ expͩ Ϫ ͪdx . (20) Ϫ ϭϪ ϩ f ͑x͒, (16) n slow B fast Ѩ Ѩ n kBT xi xi ˜ x ϵ x and Di is a diffusion constant corresponding to motion of It is obvious that the quantity Vn ( slow) Gn ( slow), in x the structural variables {xi} along the conformational po- which Gn( slow) is a so-called quasi-free energy for an x tential surface Vn( ) for electron localization on binding site electronic state n, depends parametrically on the slow struc- n. This equation is general, but we further simplify it to tural variables (Stratonovich, 1992; 1994). This means that ˜ x reveal the role of control modes on macromolecule struc- Vn( slow) represents the standard free energy for the electron x tural dynamics. state n with respect to the fast structural variables fast, but To simplify, we separate the variables {xi} into three it corresponds to the potential energy of electron state n with x groups, depending upon the relative magnitudes of the re- respect to the slow variables slow. ␶ ␶ laxation time constants x and el of the distribution function We further proceed from Eq. 14, taking into account the P(t; n, x) over the structural and electron variables, respec- substitutions (Eq. 18) and the actual role of the slow vari-

Biophysical Journal 79(3) 1237–1252 1242 Goushcha et al.

x ␵ ables, slow, only. Thus we can use an adiabatic approach has its minimum at x0. fn( ) has the same meaning as the that enables us to make the following simplification in Eq. force introduced in Eq. 12 for i ϭ 1. Calculation of C(I) will 14. [Here we restrict consideration to the simpler case of be discussed elsewhere. I ordinary slaving (Haken, 1983) in which fluctuations of the The quantity Vad(x) serves as the effective adiabatic po- fast variable influence the evolution of the slow one only on tential for the slow structural mode. This potential deter- average, without causing any noticeable fluctuations of the mines the average value of x over the electron distribution latter. The softer types of slaving will be discussed in a function. This potential is of a statistical nature, depending separate paper.] upon a stationary-state distribution of localized electron populations at a fixed structure. This structure is determined ͑ x͒ ϭ ␳͑ ͉x͒ ͑ x͒ ͸ ␳͑ ͉x͒ ϭ P t; n, t, n P t, , t, n 1, (21) and controlled by the illumination intensity, I. For times t Ͼ n ␶ xslow, stationary-state conditions are reached. The corre- in which ␳(t, n͉x) are the relative probabilities to find an sponding stationary-state distribution function can be writ- electron localized on cofactor n at fixed x, as determined ten as from Eq. 13 using Eq. 18, and P(t,x) is a distribution VI ͑x͒ function for the slow structural variables. This function is ͑ϱ ͒ ϭ Ϫ1 ͩ Ϫ ad ͪ PI , x Z exp ; defined by the expression kBT (27) ͑ x͒ ϭ ͸ ͑ x͒ I ͑ ͒ P t, P t; n, . (22) Vad x Z ϭ ͵ dx ⅐ expͩ Ϫ ͪ. n kBT Putting Eq. 21 into Eq. 14, we obtain an equation that describes the time evolution of this function, The minima and maxima of this function, xext, define the stationary states of the macromolecule at a fixed I. Thus, the ѨP͑t, x͒ Ѩ 1 Ѩ effective adiabatic potential VI (x) for the open non-equi- ϭ ͸ ͫ Ϫ I ͑x͒ ϩ ͬ ͑ x͒ ad Ѩ Di Ѩ FadϪi Ѩ P t, , (23) t xi kBT xi librium system described by Eq. 1 is the analog of a stan- i dard Gibbs free energy, GЊ, which determines the probabil- I x in which FadϪi( ) is a statistical quantity with dimensions of ity to find a closed system in a particular equilibrium state a force. This quantity describes the adiabatic action of the with given free energy. The stationary states defined by xext electron transfer upon the ith slow structural degree of in this open system are the analog of the equilibrium states freedom under conditions of slowly varying illumination in a closed system, and the values of xext can be determined intensity, I, ensuring that electronic relaxation processes are from Ѩ complete: ͉ ln I(t)͉ ϾϾ ␶ , and Ѩt el ѨVI ͑x͒ ѨV ͑x͒ ad ϭ ͸ ␳ ͑ϱ ͉ ͒ n ϭ Ѩ ͑x͒ ͯ I ; n x ͯ 0. (28) Vn Ѩ Ѩ I x xϭx x ϭ F ⅐ ͑x͒ ϭϪ͸ ␳ ͑ϱ, n͉x͒ . (24) ext n x xext ad i I Ѩx n i Those states that correspond to potential minima define the Equations 13, 23, and 24 provide the basis for self- conformational coordinates of the system. The functions regulation of a photoactivated electron flux by slow struc- ␳ ͉ ␳ ϱ ͉ I(t, n x), P(t, x) as well as their stationary values I( , n x), tural variables of a macromolecule. ϱ PI( , x), and the xext(I) depend on the structure of the Assume that the system can be characterized by a single system and determine each experimentally measured quan- slow structural degree of freedom: the generalized config- tity q(n, x) by averaging. Averaging over the electron vari- urational coordinate x. Then Eq. 23 may be rewritten as: ables, Ѩ ͑ ͒ Ѩ Ѩ I ͑ ͒ Ѩ P t, x 1 Vad x ៮͑ ͒ ϵ ͗ ͑ ͒͘ ϭ ͸ ␳͑ ͉ ͒ ⅐ ͑ ͒ ϭ ϩ ⅐ ͑ ͒ q t q n, x el t, n x q n, x . (29) Ѩ D Ѩ ͫ Ѩ Ѩ ͬ P t, x , (25) t x kBT x x n

I in which the adiabatic potential of the system, Vad(x), at When averaged over both the electron variables and gener- fixed light intensity I is determined from Eqs. 16 and 24 alized configurational coordinates, with an uncertainty C(I)

x qញ͑t͒ ϵ ͗͗q͑n, x͒͘͘ ϭ ͵ dx ⅐ P͑t, x͒q៮͑t, x͒. (30) el, x I V ͑x͒ ϭ Vϩ͑x͒ Ϫ ͸͵ f ͑␵͒␳I͑ϱ, n͉␵͒d␵ ϩ C͑I͒. (26) ad n n x 0 Before proceeding to the next section we should comment I Note that the subscript “ad” in the expression Vad means on our use of a single structural variable (Eq. 25). In “adiabatic,” not to be confused with the free energy differ- practice, the application of any theory to a particular bio- ⌬ ence VAD between the donor and acceptor levels. Vϩ(x) molecular system often requires a decrease in the system

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1243

dimension to a few generalized structural coordinates or STABLE STATES OF THE TWO-STATE SYSTEM: even to a single coordinate. Of course the system configu- A CONFORMATIONAL APPROACH ration is determined by tens of thousands of physical vari- To determine the light intensity dependence of macromol- ables, and the configuration of one system is a point in the ecule stationary states from Eq. 28 we select, as an example, multi-dimensional configurational space. As shown above, a harmonic potential with effective elastic constant ␹, Vϩ the fast structural variables relax to quasi-equilibrium val- (x) ϭ ␹(x2/2), and, hence an effective adiabatic potential, ues and fluctuate about them. These quasi-equilibrium val- VI (x). Such a potential represents Gaussian fluctuations of ues themselves continue adiabatically to follow changes in ad x around equilibrium (see, e.g., Zusman, 1980). The quan- the slow variables until at long-time the system dynamics tities f (x)(n ϭ D, A) are defined as additional stochastic may be determined by a small number of slow variables or n forces that act on the configurational coordinate when an even a single slowest variable. The dynamics of such slow electron is localized on cofactors D and A, respectively. We variables is determined by the potential profile of the quasi- assume that these forces are constant, but not equal to each ء free energy (see Eq. 20). Such a hierarchy is characteristic ϭ ϭ ␹ other. That is, fD(x) fD xD is a force acting on the for the relaxation of complex systems with a large number structure when an electron is localized on donor D; and ء of variables. It has been called “the slaving principle” ϭ ϭ ␹ fA(x) fA xA is a force acting on the structure when (Haken, 1983). Thus it is reasonable to assume that at long  an electron is localized on acceptor A. In general, xA xD. times, the slow structural rearrangement of a biomolecule ␰ ϭ ͉ Ϫ ͉ The quantity xA xD characterizes the electron- may be described by a small number of variables. For a conformational interaction of the system. It is proportional description of phase transitions in systems with a large to the additional force acting on the configurational coor- number of variables, consideration is often restricted to a dinate in a charge-separated donor-acceptor pair. We as- single “control mode” or “order parameter.” Similarly, the sume here that the force constant ␹ is the same for the description of chemical reactions in complex molecular charge-neutral and charge-separated states, although this systems may also be described with a single “reaction ␹  ␹ need not necessarily be true (i.e., in general D A). coordinate.” For example, long-time relaxation processes in Furthermore, we showed in our recent paper (Goushcha et biopolymers are often described by bounded diffusion of al., 1999) that for photosynthetic bacterial reaction centers initial multi-dimensional distribution function along a par- the probe potential Vϩ (x) is probably not harmonic with

ticular trajectory of the potential surface (see, e.g., Agmon different curvatures in the charge-neutral (PQAQB) and in ϩ Ϫ and Hopfield, 1983; Rubin et al., 1990; Gudowska-Nowak, the charge-separated (P QAQB ) states. Parson and co- 1994; Frauenfelder et al., 1991, 1999). workers arrived at a similar conclusion in their molecular Thus, we restrict present considerations to a one-dimen- dynamic studies of the ET reaction P*BH 3 PϩBϪH. They sional model of slow structural rearrangements induced by showed that the force constant for the PϩBϪH state is larger a photoinduced charge separation in biomolecules. We treat than that it is for the P*BH state (Parson et al., 1998). In the x as a global configurational coordinate that describes slow current work, we explore qualitatively the conditions for photoinduced structural changes. In this case, the structural non-equilibrium structural transitions and the emergence of factor X(x) introduced above may also be identified as a new conformational states. For this treatment, the fact that ␹  ␹ ␹ ϭ ␹ ϵ ␹ configurational coordinate because as it is well known that D A is not essential, and we assume that D A the dimension of generalized variables does not affect the to obtain analytical solutions. calculation of trajectories and free energies (Goldstein, Using expressions for light-dependent, stationary-state 1980). Consequently, we modify Eq. 10 making the substi- electron populations (Eq. 5), the recombination rate con- tution x 3 X, assuming that the configurational coordinate stant dependence on x (Eq. 31), and stochastic forces (Eqs. x is monotonic in configurational space because system 12, 16, and fA,D(x)), the equation defining the stationary energy decreases during relaxation, and slow system diffu- states of the system (Eq. 28) becomes sion is determined by the trajectory. Note that x does not I describe arbitrary structural rearrangements, but only those ϭ ϩ ͑ Ϫ ͒ xext xD xA xD ϩ 0 ͑Ϫ ͒. (32) responsible for slow structural relaxation to a new potential I krecexp xext minimum in configurational space. Finally, we obtain: This equation is valid for macromolecular ET systems that can be accurately described as two-state systems (Eq. 1), k ͑x͒ ϭ k0 exp͑Ϫx͒, (31) rec rec explicitly indicating the dependence of xext on I. For this specific example, xext(I) (Eq. 32) at specific an expression that will be used in subsequent sections. It is values of ␰ was determined in our recent work (Goushcha et important to note that the adiabatic potential for the struc- al., 1999). A small, monotonic, light-induced increase in ϭ ␰ ϭ tural factor X has its minimum at Xmin X(xmin); therefore xext(I) was obtained for the case of a weak interaction, Ϫ Յ the proposed substitution of variables leads to the equivalent (xA xD) 4. The concomitant increase in the lifetime of ␶ consideration in our phenomenological model. the charge separated state, d, obtained using Eq. 32 is

Biophysical Journal 79(3) 1237–1252 1244 Goushcha et al.

␶ ␰ ϭ Ϫ shown in Fig. 1. The smooth, light-induced increase of d and xA xD may be determined from experiment as for curves 1 and 2 is due to deformation of the “dark” conformational state. This effect may be described as “self- ␶ ͑I 3 ϱ͒ ␰ ϭ d regulation” of the photoactivated electron transfer rate due ln ␶ ͑ 3 ͒ . (35) d I 0 to a slow structural deformation, reflecting macromolecule structural changes from an electron-conformational interac- The appearance of a new, light-induced stable structural tion modulated by the photoinduced electron flow. state for ␰ Ͼ 4 and the coexistence of this state with the More complex behavior of xext(I) occurs in the case of a ␰ Ͼ initial stable state represents a non-equilibrium phase tran- strong charge-conformational interaction 4. For illumi- sition of the “monostability-bistability” type (Haken, 1983, cr Ͼ Ͼ cr nation intensities, III I II , in which Stratonovich, 1994). The problem of thermodynamic stability of stationary xcr Ϫ x cr ϭ 0 I, II D Ϫ cr states at coordinates x(1) (I) and x(3) (I) and the related II, II krec Ϫ cr exp( xI, II) (33) ext ext xA xI, II problem of thermal fluctuations in the configurational co- ordinate around stationary-state values can be solved using and a distribution function over x of the form P(t, x) ϭ P (t, x) ϩ P (t, x) (see Eq. 22). An evolution equation for x ϩ x x Ϫ x 2 D A cr ϭ A D Ϯ ͱͩ A Dͪ Ϫ ͑ Ϫ ͒ this function is described by Eq. 23, where using Eq. 26, the xI, II xA xD , 2 2 I statistical potential Vad(x) (Goushcha et al., 1997a) is given (i) ϭ by three values of the extrema xext (I); i 1, 2, 3 are obtained. Two branches, x(1)(I) and x(3)(I), give minima adiabatic ext ext ␹ ͑ ͒ ϩ 0 Iexp x krec potential, corresponding to stable structural states of the VI ͑x͒ ϭ ͫ͑x Ϫ x ͒2 Ϫ 2͑x Ϫ x ͒ͩln ͪ (2) ad 2 D A D I exp͑x ͒ ϩ k0 system, while a third, xext(I), gives a maximum for an D rec 0 unstable state. The parameter krec slightly perturbs these ͑ Ϫ ͒ 2 (36) I xA xD dependencies, shifting them to a higher light intensity with ϩ ͩ ͪ ͬ. 0 I ϩ k0 exp͑Ϫx ͒ an increase in krec. Experimentally, one can only observe rec D stable branches of these dependencies. This means that experimentally measured system parameters may reveal dis- Previously, we analyzed this expression for many values continuities at particular illumination intensities. The sur- of the electron-conformational interaction parameter, ␰ ␶ vival time, d, of a charge-separated state, in the case of a (Goushcha et al., 1997a 1999). A second, light-induced (1) ␰ Ͼ strong charge-conformational interaction, for the branch xext potential minimum may appear for 4, a case corre- I ϱ (I) is significantly shorter than this time for one belonging to sponding to a distribution function Peq( , x) with two max- (3) ␰ Ͻ the branch xext (I). (Compare curves 3 or 4 with curve 1, Fig. ima (Fig. 2 A). For 4, the light-induced deformation of 1). In fact, following the discussion in the first section, the adiabatic potential causes a deformation of the distribu- tion function with only a shift in the distribution maximum ␶ ͓ (3)͑ ͔͒ d x I toward larger values of the conformational coordinate (Fig. ext ϭ ͑ (3)͑ ͒ Ϫ (1)͑ ͒͒ ␶ ͓ (1)͑ ͔͒ exp xext I xext I , (34) d xext I 2 B). The evolution of the distribution function with light intensity for the case of a weak interaction has been de- ϩ Ϫ 3 scribed in the literature. See, e.g., studies of the P QA PQA reaction in photosynthetic bacterial RCs (Shaitan et al., 1991; Uporov and Shaitan, 1990). I The abscissas of the potential Vad(x) extrema determine stationary values xext of the slow configurational coordinate ␰ (1) (3) as a function of (see Eq. 35). The values xext(I) and xext(I) (2) correspond to adiabatic potential minima, whereas xext(I) corresponds to a maximum. The minima determine the conformational states of the macromolecule at steady-state illumination intensity I. The thermodynamic stability of these conformational states is determined by both the depth of the potential minima and the height of the barrier be- tween them. FIGURE 1 Dependence of the survival time ␶ of the charge-separated d Often the ensemble properties of macromolecules may be state on illumination intensity I for various values of the electron-confor- ␰ ϭ Ϫ ␰ ϭ ␰ ϭ ␰ ϭ satisfactorily described by the most probable behavior of mational interaction parameter xA xD:1) 2; 2) 4; 3) 5.2; 4) ␰ ϭ 6. The curves were obtained for the following set of parameters: these macromolecules near potential minima. For this de- ϭ 0 ϭ xD 2; krec 10. scription, we use a conformational approach and introduce

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1245

the non-equilibrium distribution function for x, correspond- ing to the evolution Eq. 25, may be written as

␯ ͑ ͒ Ͻ (2)͑ ͒ dPd t, x , when x xext I ; P͑t, x͒ ϭ ͭ (39) ␯ ͑ ͒ Ͼ (2)͑ ͒ lPl t, x , when x xext I ; ␯ ␯ ϭ Ϫ ␯ in which d and l ( 1 d) are the integrated populations of the “dark” and “light” conformational states, respectively (see Eq. 37). These quantities depend on the illumination intensity and its prior variation. Calculation of these quan- tities was previously discussed (Goushcha et al., 1997a) and

is not repeated here. The distribution functions, Pd(t, x) and ⎣Ϫϱ (2) ⎦ Pl(t, x), equal P(t, x) for the intervals, , xext(I) and ⎣ (2) ϱ⎦ xext(I), , respectively, and are determined by Eq. 25. The theoretical expression for an experimentally mea- sured quantity, q(t), at steady-state illumination intensity I is ញ͑ϱ͒ ϭ ␯ ͗ ͑ ͒͘ ϩ ␯ ͗ ͑ ͒͘ q d q x d l q x l. (40) In Eq. 40, averaging is done over all values of x in the “dark” state (subscript “d”) and in the “light” state (sub- FIGURE 2 Quasi-equilibrium distribution functions calculated for ␰ ϭ 2 script “l”), respectively. This expression is valid outside the (A) and ␰ ϭ 5.3 (B) for the following values of I. a, curve 1: I ϭ 0; curve bistability domain if one recognizes that ␯ ϭ 1, ␯ ϭ 0, and ϭ ϭ ϭ ϭ d l 2: I 0.1; curve 3: I 0.5; curve 4: I 2.0; b, curve 1: I 0; curve 2: x(2) ϭϱat I Ͻ Icr; whereas ␯ ϭ 0, ␯ ϭ 1, and x(2) ϭϪϱ I ϭ 0.07; curve 3: I ϭ 0.1; curve 4: I ϭ 0.13; x ϭ 2; k0 ϭ 10. ext I d l ext D rec Ͼ cr at I III. In the conformational approach Eq. 40 can be simplified to the more convenient form, the function ញ͑ϱ͒ ϭ ␯ ⅐ ͑ (1)͑ ͒͒ ϩ ␯ ⅐ ͑ (3)͑ ͒͒ q d q xext I l q xext I . (41) In the final section of this paper, we apply Eqs. 40 and 41 ␯␣͑t͒ ϭ ͵ dx ⅐ P͑t, x Ϫ x␣͒, (37) to a brief analysis of illumination-dependent absorbance Ga changes in photosynthetic RCs. We also present other recent in which we integrate over all configurations of the coor- experimental observations that provide support for the idea ʦ ␣ of self-regulation phenomena in photosynthetic RCs. dinate x Ga near a potential minimum at the point x␣ (van Kampen, 1992). This function, ␯␣, determines the population at the minimum ␣ of the adiabatic potential. ʦ APPLICATION TO BACTERIAL States with an x such that x Ga are treated as the same conformational state, ␣. The probability of realizing differ- REACTION CENTERS ent conformational states is determined by the forward and Photosynthetic reaction centers have been among the most reverse transition rates over potential barrier (Kramers, comprehensively studied biological systems over the last 30 1940; van Kampen, 1992). These probabilities are obtained years (Hoff and Deisenhofer, 1997, and references therein). from Eq. 25 using Eq. 37. Using this conformational ap- Many researchers have discussed light-induced conforma- proach, the average value of an observable q(t) can be tional transitions in RCs (see, e.g., Graige et al., 1998; written as McMahon et al., 1998; Kalman and Maroti, 1997; Kleinfeld et al., 1984b; Shaitan et al., 1991). The charge recombina- qញ͑t͒ ϭ ͸ ␯␣͑t͒q៮͑t, x␣͒, (38) ␣ tion rate constant in RCs depends on structural coordinates such as the donor-acceptor distance (Kleinfeld et al., 1984b; in which q៮(t, x␣) is the average value over electron variables Shaitan et al., 1991). These authors experimentally deter- at t in conformational state ␣. mined, in effect, the distribution function for the generalized For a system undergoing photoinduced charge separation conformational coordinate both in the dark and under illu- and described by a double-minimum adiabatic potential, we mination by quenching structural relaxation at cryogenic introduce two distinct conformational states, “light” and temperatures. They obtained a light-induced increase in the “dark,” denoted as l and d, respectively. Providing that the donor-acceptor distance of ϳ1 Å. Recent EPR studies of barrier height between the two minima is sufficiently high, RCs show that light-induced conformational changes are then equilibration near the minima may occur without ther- not simple relative translations of the donor and primary mally activated transitions between the minima. In this case, quinone acceptor (QA) molecules, but that they are more

Biophysical Journal 79(3) 1237–1252 1246 Goushcha et al.

likely rearrangements of the protein structure (Zech et al., The experimental dependence of absorbance changes at 1997). More recent x-ray studies of RCs from Rb. spha- 865 nm with steady-state illumination intensity is shown in eroides showed a large, ϳ5 Å, light-induced translation of Fig. 3. In this figure, curve 1 corresponds to RCs from Rb.

the secondary quinone acceptor from its location in the sphaeroides with inhibited ET from the primary QA to the dark-adapted system accompanied by a 180° rotation about secondary QB quinone. We call these RCs “QB-lacking,” the isoprene axis (Stowell et al., 1997). These authors also whereas RCs with allowed ET between quinone acceptors

reported light-induced changes in the protein structure that are called “QB-containing” RCs. [The isolation procedure affect the protonation of amino acid residues. Recent mo- for these RCs and the details of the experimental setup are lecular dynamic calculations demonstrated the existence of described elsewhere (Zakharova et al., 1981; Mueller et al., two distinctly different binding sites for the neutral second- 1991; Goushcha et al., 1997b)]. The value ␷ ϭ 0.9 was Ϫ ary quinone QB and semiquinone anion QB (Grafton and estimated as the ratio of the amplitude of absorbance change Wheeler, 1999). The authors showed for the first time that at saturating light intensity to the steady-state absorbance at ϭ ␦Ј the protonation of ASP L213 should occur prior to occupa- I 0. One expects a linear relationship of I vs. I for a Ϫ ϳ tion by QB of its stable (quasi-equilibrium) site, 5Å system with fixed structure obeying a Langmuir law. The distant from the site which is at equilibrium for neutral slope of this type of plot yields the survival time of the Ϫ quinone in the dark. Such motion of an ubi-semiquinone QB charge-separated state for the RCs. For QB-containing RCs, from the non-equilibrium position that is characteristic for a simple plot deviates from a Langmuir law (curve 2 in Fig. ␶ the dark-adapted structure to a quasi-equilibrium position 3). As Eq. 44 indicates, this deviation occurs because d that is stable for the light-adapted structure indicates the depends on I due to light-induced structural rearrangements importance of non-equilibrium structural transitions in RCs. of the RCs. These observations explain previously reported light-in- The conclusions drawn above are valid for a system in duced changes in the transient absorption spectrum of Rb. which both 1) a single photoactivated electron transfers sphaeroides RCs (Kleinfeld et al., 1984b), but the physical between a finite number of electronic states, and 2) charge phenomena responsible for these new conformations re- trapping by exogenous acceptors is negligible. The first mained unexplained. Using the theoretical approach de- condition is verified for the RCs by the absence of cyto- scribed above, we now elucidate reasonable mechanisms chrome c or any other exogenous electron donors that might that lead to these structural changes in RCs. rapidly reduce the bacteriochlorophyll dimer Pϩ after initial photoactivation in double-flash experiments. Thus there are The relationship between light-induced no states with doubly reduced quinone acceptors in these absorbance changes and the average survival experiments. The second condition was verified in recent time of the charge-separated state studies, which showed that large changes in the ET and charge recombination kinetics of RCs upon prolonged illu- Upon photoexcitation of RCs from purple bacteria mination are not related to the loss of photochemical activ- Rhodobacter (Rb.) sphaeroides, a large absorbance decrease ity of the RCs, but rather are due to the formation of new is observed in the 865-nm absorption band for the primary donor bacteriochlorophyll dimer (See Clayton, 1965). The

absorbance A865(t) depends on the illumination intensity through ␴(t), ͑ ͒ ϭ ͑ Ϫ ␷͒ ͑ ͒ ϩ ␷ ͑ ͒͑ Ϫ ␴͑ ͒͒ A865 t 1 A865 0 A865 0 1 t (42) in which ␷ Ͻ 1 is the fraction of total absorbance at 865 nm that is due to reduced (not photooxidized) photoelectron

donor P and A865(0) is the steady-state absorbance in the absence of photoactivation. From Eq. 42 one can easily determine the relationship ␴ between the theoretical quantity, I(t) (see first section) and ␦ ϭϪ Ϫ experimentally measured values of (t) {[A865(t) A865(0)]/[A865(0)]}. For stationary-state conditions, ␦Ј FIGURE 3 Experimentally measured optical absorbance changes, I,as a function of illumination intensity for Q -lacking (curve 1, open circles) I B ␦ ϵ lim ␦͑t͒ ϭ ␯␴ ͑ϱ͒ ϭ ␷ Ϫ . (43) and QB-containing (curve 2, solid squares) RCs from Rb. sphaeroides, wt. I I ϩ ␶ 1 ϳ ␮ ϭ Ϯ t3ϱ I d The RC concentration was 1 M. A280/A802 1.3 0.5. Buffer conditions: 10 mM HCl-Tris (pH ϭ 8.0), ambient temperature (T ϭ 20ЊC), For steady-state illumination, a more convenient form is 0.025%. LDAO. The samples where thoroughly degassed before experi- Ϫ ␦ ͑ ͒ Ϫ ͑ ͒ ments by multiple freeze-thaw-pump cycles at 77 K and the pressure 10 6 I A865 I A865 0 ␮ ␦Јϭ ϭϪ ϵ ␶ torr. QB-lacking RCs were prepared by addition of a 20 Mofo-phenan- I ␷ Ϫ ␦ ͑ ͒ ϩ ͑ Ϫ ␯͒ ͑ ͒ dI. (44) ␮ I A865 I 1 A865 0 throline solution to a 1 M solution of RCs.

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1247

“light-induced” conformations of the RCs (Kalman and in which kAB and kBA are the forward and reverse electron Maroti, 1997). transfer rate constants between the primary and the second- ␦ ⌬ The experimentally measured quantity (t) can be calcu- ary quinone acceptors, respectively, and GAB is the free ϩ Ϫ lated with the formalism developed above for the distribu- energy difference between the states P QA៮ QB and ϩ Ϫ ⌬ tion function. Consider a homogeneous sample of isolated P QAQB . Note that GAB for the non-equilibrium case RCs at concentration C and optical pathlength l that absorbs described in this work is the difference in the quasi-free energies G ϩ (x) and G ϩ Ϫ (x). This difference equals , ␦ ءwith an “ideal” theoretical absorbance AЈ ϭ ␧Ј lC 865 865 DA P QAQB P QA QB ␦ ϭ ϭ ϭ ⌬ ° in which DA 1 for A D and 0, otherwise. Taking an the conventional standard free energy difference GAB ensemble average over the distribution of RC electronic and when RCs reach their equilibrium stable state in the dark ϵ conformational states yields the following result: (when x xD). ⌬ The parameters kAP and GAB/kBT may both be impor- tant in modeling the correlation of photoelectron transfer Aញ ͑t͒ ϭ ͗͗AЈ ͑t, x͒͘͘ ϭ␧ lC͵ dxP͑t, x͒␳͑t; D͉x͒. 865 865 el,x 865 with RC structure. As discussed in the first section, both (45) parameters determine the survival time of the charge-sepa- rated state Hence, the calculated experimental quantity ␦(t) is given by ͑␶ ͒Ϫ1 ϭ d kAP for QB-lacking RCs; (48)

␦͑t͒ ϭ ͵dx␴͑t, x͒P͑t, x͒, (46) and ⌬ GAB ͑␶ ͒Ϫ1 ϭ k expͩ ͪ for Q -containing RCs. in which, for the simplest case of a reduced, two-level RC d AP k T B model, the quantity ␴(t, x) ϭ 1 Ϫ ␳(t, D͉x), and ␳(t, D͉x)is B defined by Eq. 3. Our experiments demonstrate that one may need only con- Equation 46 may be used to model both transient absorp- sider one of these two parameters. tion kinetics and quasi-stationary state effects for RCs. The Fig. 4 A shows the primary donor absorbance recovery calculated, light-induced, stationary-state absorption kinetics, measured at ␭ ϭ 865 nm, for RCs from Rb. Ϫ 3 changes of RCs from Eq. 46 reveal nonlinear behavior of sphaeroides (wild type, wt) with inhibited QA QB elec- ␦(ϱ) and were described theoretically elsewhere (Gushcha tron transfer. In this case, the recombination kinetics are et al., 1994; Goushcha et al., 1997a). The parameters xD and determined by kAP. Trace 1 of Fig. 4 A was obtained after xA, discussed above, have a straightforward physical inter- flash-activation of a dark-adapted sample, while trace 2 was pretation for RCs: they are generalized configurational co- measured after switching off 2 min of background illumi- ordinates for permanent localization of the photoelectron nation that pre-illuminated the RCs (␭ ϭ 700–900 nm and exp ϭ 2 Ϫ1 either on the donor D, as in thoroughly dark-adapted RCs, I 2 mW/cm ). The values kAP observed were 13 s Ϫ1 or on a quinone acceptor, QB or QA, for thoroughly light- and9s for the dark-adapted sample and the pre-illumi- adapted RCs, respectively. nated sample, respectively. Such small but detectable dif-

ferences in kAP may well be due to light-induced structural changes in the RCs (Kleinfeld et al., 1984b; Shaitan et al., Recombination rate constants as exponential 1991). functions of conformational coordinates Fig. 4 B shows the primary donor recovery kinetics for Ϫ 3 Another remarkable property of RC dynamics is the depen- RCs from Rb. sphaeroides (wt) that are active in QA QB dence of primary donor recombination kinetics upon illu- electron transfer. We used isolated RCs with ϳ75% occu- mination conditions (Kleinfeld et al., 1984b; Shaitan et al., pation of the QB site with native ubiquinone, without any 1991; Kalman and Maroti, 1997). For Rb. sphaeroides RCs reconstitution of QB activity. The sample was thoroughly that lack a secondary quinone acceptor QB, the charge degassed before experiments by multiple freeze-thaw-pump Ϫ6 recombination rate constant krec (see Eq. 1) equals the cycles down to 77 K and 10 torr. This procedure allowed ϭ Ϫ1 recombination rate constant kAP 10 s for the radical us to avoid photo-oxidation of semiquinones by oxygen or ϩ Ϫ pair D QA (Parson and Ke, 1982; Kleinfeld et al., 1984b). other oxidants during illumination. Trace 1 in Fig. 4 B For QB-containing Rb. sphaeroides RCs, electron transfer corresponds to flash-activated kinetics of a thoroughly dark- may be described by a two-level electronic scheme under adapted sample. Trace 2 was measured after switching off 2 physiological conditions (Kleinfeld et al., 1984a; Labahn et min of pre-illumination of the sample with saturating actinic al., 1994). This situation is also exactly like the one of Eq. light Iexp ϭ 2 mW/cm2 at ␭ ϭ 700–900 nm. Curve 1 in Fig. 1 with an effective recombination rate constant, 4 B may be modeled with a recombination half-lifetime of ϳ1 s, whereas curve 2 reveals a decay half-lifetime of ϳ200 k ⌬G Ϸ BA ϭ ͩ ABͪ s. Such a drastic increase in the average survival time of the krec kAP kAP exp , (47) kAB kBT charge-separated state for a pre-illuminated sample cannot

Biophysical Journal 79(3) 1237–1252 1248 Goushcha et al.

(trace 3 in Fig. 4 B). This means that a significant fraction

(15–20% for QB-containing RCs) of the RCs remain trapped in a second, light-induced conformation, one with a relax- Ն ation time 10 min for QB-containing RCs. A similar observation was reported by Kalman and Maroti (1997) who applied much higher intensities of actinic illumination Ϫ to QB-lacking RCs to saturate the electronic state QA. This difference in experimental conditions may explain why our

results for QB-lacking RCs differ from those of Kalman and Maroti, as well as why our results for QB-containing RCs are similar to theirs, even though they worked with QB- lacking RCs. As noted above, the effects reported here cannot be ϩ explained by formation of states like P QAQB with the electron having been captured by exogenous acceptors. Nevertheless, we do not completely exclude formation of such states and their influence on the dynamics of the system. This very interesting problem requires special at- tention, but was not studied in the present work. Addition- ally, sample preparation protocols involving treatment with detergents play a role in the effects observed. These results FIGURE 4 Primary donor recovery kinetics for RCs from Rb. spha- are not reported here either, but we plan to report them in eroides, wt under different illumination conditions. (A) QB-lacking RCs: subsequent work. curve 1 gives the absorbance change kinetics for a sample that was dark From the comparison of Eq. 47 with Eq. 31 we may adapted and then flash-activated for 2 ms. Curve 2 gives these kinetics following the cessation of 2 min of saturating intensity illumination. (B) assume that for the case of bacterial photosynthetic reaction ͉⌬ ͉ QB-containing RCs: curves 1 and 2 are the kinetics for the same conditions centers the parameter GAB /kBT may be identified with the as in (A). Curve 3 was obtained as the response to a short, saturating flash slow generalized configurational coordinate x. This coordi- (2 ms) applied 12 min after turning off 2 min of saturating intensity nate reflects slow, light-induced structural rearrangements illumination to the sample. The buffer conditions were the same as used for in QB-containing RCs during system relaxation to its quasi- the experiments shown in Fig. 3. ϩ Ϫ equilibrium, charge-separated state P QAQB . During re- laxation, the system moves along a single trajectory in be explained by a light-induced decrease in the rate constant configurational space with a concomitant decrease in free kAP, which changes only slightly for similar pre-illumina- energy, which remains a single-valued function of x.As tion conditions (see Fig. 4 A). Consequently, this increase in already noted, this free energy should be considered as a the recombination time constant for a QB-containing sample quasi-free energy with respect to slow relaxation of medium ͉⌬ ͉ ⌬ may reflect a light-induced increase in the value GAB / polarization (Stratonovich, 1992, 1994). The quantity GAB kBT. is a single-valued function of the configurational coordinate In these experiments we assume a closed system with a during system relaxation along a chosen trajectory in con- fixed number of localized electron states and an absence of figurational space. It should be considered as a quasi-free exogenous donors or acceptors. These assumptions are sup- energy difference for the system, which does not yet reach ported by the following observations. Several minutes after thermodynamic equilibrium and is still subject to illumina- cessation of prolonged illumination, the RCs had undergone tion. Hence we define complete electron relaxation. Then we applied a saturating ͉⌬ ͉ GAB illumination flash. In cases of both QB-lacking and QB- ϭ x, (49) containing RCs, the bleaching amplitude caused by this kBT flash was the same as for a dark-adapted sample. Thus, the emphasizing thus the dependence of the quasi-free energy native activity of the pre-illuminated RCs was restored ⌬ difference GAB on the slow structural variables that gov- following charge recombination. However, the recovery ern system relaxation toward equilibrium in a charge-sepa- kinetics of the primary donor, in response to this illumina- rated or a charge-neutral state. Assuming neglect of the tion flash, were quite different from those of a dark-adapted relatively small, light-induced changes in the rate constant sample. These kinetics (not shown in Fig. 4 A) were nearly kAP we estimate from Eq. 35 the light-induced changes in x the same as those obtained for a 2-min, pre-illuminated as sample in the case of QB-lacking RCs. For QB-containing RCs, the kinetics were intermediate between those of the 200 ⌬x ϭ ln Ϸ 5.3. (50) thoroughly dark-adapted and of the pre-illuminated sample 1

Biophysical Journal 79(3) 1237–1252 General Self-Regulation Theory in RCs 1249

͉⌬ ͉ ␰ ϭ ͉ Ϫ ͉ Ͻ Therefore, the free energy difference GAB increases by interaction, xA xD 4, and results in a continuous Ն 130 meV as a result of structural changes caused by shift of the adiabatic potential minimum from xA to xD multiple successive turnovers of the RC. Note that one may following an increase in the illumination intensity. For the expect even larger light-induced changes in the standard case of a strong electron-conformational interaction, ␰ ϭ ⌬⌬ ° ͉ Ϫ ͉ Ն free energy difference, GAB, because the illumination xA xD 4, the initial “dark” conformational state shifts conditions used in our experiments are not necessarily suf- only slightly with an I increase until the light intensity cr ficiently intense or prolonged to satisfy RC equilibration reaches a critical value II . At this illumination level a new near the potential minimum xA. Thus, the generalized con- “light-adapted” conformational state of biomolecule ap- figurational coordinate, analogous to the perpendicular co- pears. The minimum for this new conformational state also ordinate in the Agmon-Hopfield model, Eq. 49, is suitable does not shift significantly in x with a further increase in I. for analysis of the correlation of structural dynamics and However, the relative free energies of the two conforma- electron states in QB-containing RCs. Several recent papers tional states and the shape of their potential profiles change have shown that slow structural dynamics play an important considerably with I variation. Such a behavior is an example role in ultrafast charge separation steps in RCs, providing of a first-order non-equilibrium phase transition (Haken, evidence for their adiabatic character and an illumination- 1983) in biomolecular systems. controlled drop of the acceptor free energy (Holzwarth and The second, “light-adapted” conformational state has a Mueller, 1996; Lin et al., 1996). An analysis shows that the much longer survival time for the charge-separated state structural changes in response to ET in QB-lacking RCs relative to the “dark-adapted” conformational state. This cause a decrease in the free energy gap between donor situation was shown experimentally for the case of photo- ϩ Ϫ (DQA) and acceptor (D QA) states of 120 meV (McMahon synthetic RCs. Functionally important, light-induced struc- et al., 1998), comparable to the above result. Such a de- tural rearrangements in RCs were discussed originally in crease in the free energy may be common in many biolog- 1984 by Kleinfeld et al. Because slow structural motion ical charge transfer systems. provides a structural “memory” effect, the “light-adapted” conformational state may remain unrelaxed for a long time, even during a significant decrease in the illumination CONCLUSION intensity. In the current paper we have developed a formalism for The theory of non-equilibrium charged particle flow describing the evolution of open, non-equilibrium biomo- through a flexible macromolecular structure developed in lecular systems with photoinduced charge separation. Upon this work may be relevant for a description of many non- steady-state illumination, such a system may have several linear effects that have been observed in other charge trans- stationary states, the number and relative thermodynamic fer, biomolecular systems. A strong correlation between the stability of which depends on the illumination intensity. The proton flux (concentration) and macromolecular conforma- thermodynamic stability of such an open system is deter- tion in bacteriorhodopsin may be explained within nonlinear mined by Eq. 27, in which the system effective adiabatic dynamic theory (Hong, 1999; Brown et al., 1997; Sass et al., I Њ potential Vad is used instead of the standard free energy G 1998). If the macromolecular system can be characterized to describe equilibrium. Note that the fundamental Gibbs by two generalized conformational coordinates that function Њ ϭ Ϫ1 Ϫ Њ relationship P(G ) Z exp( G /kBT) that defines the as control parameters, we predict that not only mono and probability of finding the system with a particular free bistable functioning regimes exist, as described here, but energy GЊ is valid only for a system in thermodynamic sustained or damped oscillations may additionally occur. equilibrium with an absence of external forces, which is not The period of these oscillations could be much longer than true in this case. Therefore, Eq. 27, which defines the the characteristic time for a single charge-transfer step. One probability of finding a non-equilibrium system in config- example might be the oscillations of dye luminescence and uration x, can be considered as a generalization of the Gibbs pH observed recently in bacteriorhodopsins for different relationship for the case of non-equilibrium biomolecular levels of optical excitation (Tributsch and Bogomolni, systems with photoinduced charge separation. 1994; Birge, 1994). For a system that includes three or more The general kinetic formalism developed in this work conformational coordinates or control modes, more compli- describes non-equilibrium changes in macromolecular cated, non-periodic regimes, such as dynamic chaos, might structure caused by light-induced redistribution of charge occur. density among redox co-factors, both under steady-state and We have described properties of charge-conformational non-stationary-state illumination intensities. Such structural systems of biomolecules that are responsible for their func- changes may have relaxation times of up to minutes due to tional behavior. Electron transfer enzymes, such as cyto- slow structural modes to provide adaptation of the macro- chrome c oxidase and cytochrome c peroxidase, proton molecule for prolonged illumination. This adaptation results pumping systems like bacteriorhodopsin and ATPases, and in a continuous deformation of the initial “dark” conforma- the ion channels of biomembranes, for which efficiency of tional state, for the case of weak electron-conformational function is likely determined by many structural control

Biophysical Journal 79(3) 1237–1252 1250 Goushcha et al.

parameters, may be additional examples. Finally, the me- Hence, in accord with the definition of the first section, chanics of slow structural changes that accumulate by the I ␴ ͑ϱ͒ ϵ Ϫ ␳ ͑ϱ ͒ ϭ action of multiple, elementary events likely play a signifi- I 1 I , D Ϫ1 . (A9) cant role in the function of a variety of enzymes, not ϱ I ϩ ͵ ␴A͑t͒dt ͩ 0 ͪ necessarily only those involving charge transfer. 0

Then, in agreement with Eq. 7 the survival time of the charge-separated APPENDIX state that characterizes the efficiency of stationary state charge separation in intact systems is ␳៮ ϭ͚ ␳ ͉ ͘ ͉ ͘ ϭ We denote (t) n (t; n) n , in which n , n 1,...N is a system of ͗ ͉ ͘ϭ␦ normalized orthogonal vectors satisfying the condition n m nm. Then ϱ ␳ Eq. 2 for (t; n), in the case of an arbitrary but finite number of electronic ␶ ϭ ͵ ␴A͑t͒dt, (A10) states, may be written as follows: d 0 Ѩ␳͑t͒ ϭ Lˆ␳៮͑t͒, (A1) and does not depend on the intensity of illumination. Ѩt

in which the operator Lˆ is We acknowledge Prof. K. Schaffner for his support of this work, Prof. M. Brodyn and Prof. I. Blonsky for valuable remarks and interest in the work, .␳͑ ͒ Dr. G. Trinkunas for fruitful discussion, and Dr. N. I. Zakharova and Mء͔͉ ͗͘ ͉ Ϫ ͉ ͗͘ ͉͓ء ϭ ˆ Ϫ ˆ L L0 I D D A D t, D , (A2) Reus for isolating RCs.

and ͉D͘ and ͉A͘ are normalized orthogonal vectors corresponding to states Partial financial support by Volkswagenstiftung, Hannover, Germany, with an electron on the donor P or on the acceptor A, respectively. We within the Schwerpunkt program “Intra and Intermolecular Electron Trans- -␳(t, D) in the last term of the right-hand side fer in Chemistry and Biology” is gratefully acknowledged. G.W.S. grateءintroduced terms containing I of Eq. A2. fully acknowledges the California Space Institute Research Grant Program ␳ for partial support of this research. Let 0 be a solution of the equation, Ѩ␳៮ ͑ ͒ 0 t ϭ Lˆ ␳៮ ͑t͒. (A3) REFERENCES Ѩt 0 0 Agmon, N., and J. J. Hopfield. 1983. Transient kinetics of chemical It is easy to show, after substitution, that the formal solution of Eq. A1 is reactions with bounded diffusion perpendicular to the reaction coordinate: intramolecular processes with slow conformational changes. J. Chem. Phys. 78:6947–6959. t Bandyopadhyay, T., A. Okada, and M. Tachiya. 1999. Two-electron trans- D A ␳៮͑t͒ ϭ ␳៮ ͑t͒ Ϫ I ͵ d␶͓␳៮ ͑t Ϫ ␶͒ Ϫ ␳៮ ͑tϪ␶͔͒␳͑␶, D͒, (A4) 0 0 0 fer reactions involving three paraboloidal potential surfaces in solvents 0 with multiple solvation time scales. J. Chem. Phys. 110:9630–9645. Birge, R. R. 1994. Bacteriorhodopsin: a nonlinear proton pump. Nature. 371:659–660. in which Brown, L., H. Kamikubo, L. Zimanyi, M. Kataoka, F. Tokunaga, P. ␳៮ D͑ ͒ ϵ͉ ͘ ␳៮ A͑ ͒ ϵ͉ ͘ Verdegem, J. Lugtenburg, and J. Lanyi. 1997. A local electrostatic 0 0 D , 0 0 A . (A5) change is the cause of the large-scale protein conformation shift in bacteriorhodopsin. Proc. Natl. Acad. Sci. USA. 94:5040–5044. ␳ ϭ͐ϱ Ϫ ␳ Using the Laplace transformation ˜(s) 0 exp( st) (t)dt on Eq. A4 and C˘ apek, V., and H. Tributsch. 1999. Particle (electron, proton) transfer and multiplying it from the left side by ͗D͉ we obtain self-organization in active thermodynamic reservoirs. J. Phys. Chem. 103:3711–3719. ␳͑ ͒ ϭ ␳ ͑ ͒ Ϫ ͓␳D͑ ͒ Ϫ ␳A͑ ͔͒␳͑ ͒ ˜ s; D ˜ 0 s; D I ˜ 0 s; D ˜ 0 s; D ˜ s; D , (A6) Chandrasekhar, S. 1943. Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15:1–89. from which we observe Chinarov, V. A., Yu. B. Gaididei, V. N. Kharkyanen, and S. P. Sit’ko. 1992. Ion pores in biological membranes as self-organized bistable systems. Phys. Rev. A. 46:5232–5241. ␳˜ ͑ ͒ ␳͑ ͒ ϭ 0 s; D Christophorov, L. N. 1995. Conformation-dependent charge transport: a ˜ s; D ϩ ͓␳D͑ ͒Ϫ␳A͑ ͔͒. (A7) 1 I ˜ 0 s; D ˜ 0 s; D new stochastic approach. Phys. Lett. A. 205:14–17. Clayton, R. K. 1965. Molecular Physics in Photosynthesis. Blaisdell Pub- ␳D ϭ lishing Co., New York. Obviously ˜0 (s; D) 1/s. Using the known correlation (Korn and Korn, ៮͑ ͒ ϭ ͑ ͒ 2 Ϫ Daizadeh, I., E. Medvedev, and A. Stuchebrukhov. 1997. Effects of protein 1968): lim 3 sf s lim 3ϱ f t and taking into account that (1/s ) s 0 t dynamics on biological electron transfer. Proc. Natl. Acad. Sci. USA. (␳˜A(s; D)/s) defines the Laplace-transform of ͐t [1 Ϫ ␳A (tЈ, D)]dtЈ, we find 0 0 0 94:3703–3708. from Eq. A7 that Dexter, D. L. 1953. A theory of sensitized luminescence in solids. J. Chem. Phys. 21:836–850. 1 ␳ ͑ϱ, D͒ ϭ . (A8) Einarsdottir, O., R. B. Dyer, D. D. Lemon, P. M. Killough, S. M. Hubig, I ϱ S. J. Atherton, J. J. Lopez-Garriga, G. Palmer, and W. H. Woodruff. 1 ϩ I ͵ ␴A͑t͒dt 1993. Photodissociation and recombination of Carbonmonoxycyto- 0 chrome oxidase: dynamics from picoseconds to kiloseconds. Biochem- 0 istry. 32:12013–12024.

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