View metadata, citation and similar papers at core.ac.uk brought to you by CORE 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.