INTRODUCTION

TOP ABSTRACT INTRODUCTION SURVIVAL TIME OF THE... A KINETIC DESCRIPTION OF... STABLE STATES OF THE... APPLICATION TO BACTERIAL... CONCLUSION APPENDIX REFERENCES

Biological energy conversion and storage take place through elementary events of charge transport in biomolecules. Transient, localized charges interact with ionized, polarizable, or dipolar structural elements of the macromolecule to perturb cofactor and/or protein structural modes. These interactions couple localized electron states to nuclear degrees of freedom that may be reduced to a single (generalized) coordinate (see, e.g., Agmon and Hopfield, 1983). This coordinate may be either collective or localized, corresponding in the latter case to motion of specific structural groups. Characteristic relaxation times of structural motion may vary widely, facilitating either an adiabatic or a non-adiabatic elementary charge transfer event in the biomolecule (see Hoff and Deisenhofer, 1997, for a review). The present work focuses on effects that occur during multiple, sequential charge transfer events when structural relaxation is significantly slower than the charge transfer rate itself.

The relevance of slow structural dynamics to the function of biological charge transfer system function has been demonstrated many times. Photosynthetic reaction centers (RCs) exhibit a long-lived, structural relaxation for minutes after completion of electron transfer (ET) (Puchenkov et al., 1995; Kalman and Maroti, 1997). Numerous studies of bacteriorhodopsins indicate slow (tens of seconds) structural motions induced by a proton flux (Nagel et al., 1998; Sass et al., 1998). Long-lived structural modes are also important for the function of cytochrome oxidases (Einarsdottir et al., 1993) and ATPases (Noji et al., 1997). For such modes, transient, localized charges interact with structural elements of the biomolecule, and the effects of these interactions accumulate during successive events. Accumulated structural changes produce feedback on the charge transfer rate. Thus, slow conformational modes function as control modes to determine long-time biomolecule behavior (Haken, 1983). The action of a charged particle flux on slow conformational modes and structural feedback on charge transfer rate constants produce non-linear, self-regulation effects (Chinarov et al., 1992; Tributsch and Pohlmann, 1998; Goushcha et al., 1997a; Gushcha et al., 1994). These self-regulation processes should be quite important for the function of charge transfer biomolecules, modulating the charge-transfer rate. To describe these effects, we propose a self-consistent, adiabatic theory of charge transfer and structural motion. This theory develops a correlation of structural dynamics and electron transfer, ensuring a correct statistical description of electron-conformational dynamics in macromolecules by considering system diffusion along an effective adiabatic potential. This approach generalizes an adiabatic theory for a single ET event to the case of multiple, successive ET events, each of which induces small but long-lived structural changes that accumulate to influence subsequent events.

We develop this theory for photosynthetic reaction centers, but most of our results can be readily generalized to other macromolecular charge transfer systems. For an intact photosynthetic system, charge separation efficiency is determined by the quantum yield of the primary charge separation event and the lifetime of the charge-separated state. The term "efficiency of charge separation" emphasizes that the average survival time of this state, as in isolated RCs, is the determining factor in intact systems, in which the quantum yield of the primary charge separation event is 1. (see Wraight and Clayton, 1973).

In this paper, we proceed as follows: 1) In the next (first) section, we develop a theory for a two-state charge-transfer system and for the more general case of a finite number of charge-transfer states. We show that the survival time of the charge-separated state reflects the light-induced structural changes of the system. 2) In the second section, we develop a general, kinetic description of the electron-conformational interaction in macromolecular systems. We show that slow, structural dynamics determine self-regulation effects in biomolecules. 3) In the third section, we analyze the dependence of stationary-state structural variable values with light intensity. 4) Finally, in the fourth section we apply the theory to photosynthetic RC recombination kinetics and quasi-stationary-state, light-induced effects.

	SURVIVAL TIME OF THE CHARGE-SEPARATED STATE: DEPENDENCE UPON MACROMOLECULAR STRUCTURE

TOP ABSTRACT INTRODUCTION SURVIVAL TIME OF THE... A KINETIC DESCRIPTION OF... STABLE STATES OF THE... APPLICATION TO BACTERIAL... CONCLUSION APPENDIX REFERENCES

Consider first the average lifetime of the charge-separated state for a simple system consisting of a photodonor D and an acceptor A, both inserted into a suitable matrix. The scheme of electron transfers in this system may be described by

(1)

in which k_I = I is the first-order rate constant for photoinduced electron transfer from the light-absorbing photodonor D to the acceptor A. The rate of this process is proportional to the intensity of absorbed actinic light I, with a proportionality coefficient ; k_rec is the first-order rate constant for charge recombination.

Let (t, D) and (t, A) be the normalized populations of the states, DA and D⁺A, respectively, at time t. Then these quantities satisfy simple coupled differential rate equations for a fixed structure of the system:

(2)

in which we will take  = 1. This substitution specifies the units of I as photoinduced charge separation events per unit time. The solution of Eq. 2 is:

(3)

in which

(4)

(5)

and
For a more general system with an arbitrary number of charge-transfer states, but only a single photodonor D, the quantity (t) = 1  (t, D) defines the probability of charge separation at t. For a fixed, constant actinic light intensity I, _I()  1  _I(, D). For the case of a simple donor-acceptor pair (Eq. 1), we obtain:

(6)

corresponding to a Langmuir dissociation isotherm with a half-saturation intensity, k_rec.

The efficiency of charge separation under stationary-state illumination with a single photoactivated electron that transfers between a finite number of localized electron states is defined as the ratio of the stationary state probability of charge separation to the number of charge separation events per unit time

(7)

Here _d gives the average lifetime or "survival time" (Agmon and Hopfield, 1983) of separated charges relative to recombination. For the two-state system under consideration, _d = (k_rec)¹. We show in the Appendix that, for the general case of a system with a finite number of localized electron states and a fixed structure, the value of _d, given by Eq. 7, depends only on structural organization and not upon the actinic light intensity. Moreover, _d can be measured by the system response (t) to a short, saturating actinic flash or upon ceasation of continuous photoexcitation:

(8)

Thus, _d equals the area under the recombination probability function. In the case of multiphasic relaxation this parameter is identical to the average lifetime of the charge-separated pair. Approximating (t) as (t) = _i A_ie^_it, in which {_i} are a set of relaxation rate constants, with corresponding weights, A_i, then it follows from Eq. 8 that _d = _i (A_i/_i). Thus, _d is the time constant for some effective single-exponential relaxation process that gives the area under the relaxation kinetics curve equal to that of the real process. As a consequence, for the general case, using Eqs. 5 and 7, we obtain

(9)

Thus, the stationary state probability of photo-separated charges depends upon the actinic light intensity strictly in accordance with the Langmuir law, with a value (_d)¹ for the half-saturation intensity. This value is fixed at a fixed structure. Thus, any deviation of the experimental _I() from a Langmuir curve implies that light-induced structural rearrangements occur and that _d depends on I.

What physical mechanisms may correlate macromolecular structural dynamics with photoactivated charge transfer along a cofactor chain? The following facts are relevant:

1.	Electric fields produced by photo-induced separated charges at angstrom distances are calculated to be on the order of 107-108 V/cm, much higher than those that exist across biomembranes in vivo.
2.	Protein subunits of biomacromolecules contain charged or polar groups with redox properties that depend upon the surrounding media.
3.	Experiments show that characteristic time constants for structural relaxation in proteins range from nanoseconds to minutes.
4.	Slight perturbations in the equilibrium positions of macromolecule structural elements may dramatically change the rates of electron transfer between cofactors.

Statements 1-3 require no additional discussion. Support for statement 4, although previously discussed, is now amplified. For a system with a finite number of localized electron states but with no interactions with its surroundings, electron motion should be completely coherent and may be described in terms of a non-equilibrium density matrix as periodic oscillations of the electronic populations of these states (Landau and Lifshitz, 1965). However, absolute coherence of photoelectron motion is destroyed by interaction with thermal oscillations of the nuclei with relaxation times of 10¹³-10¹¹ s. This means that non-diagonal elements of the density matrix may be neglected for slow steps of charge separation. The non-adiabatic description given by Fermi's Golden Rule is appropriate for this type of donor-acceptor transition (Landau and Lifshitz, 1965). The theory of non-adiabatic transitions has been well-developed in solid state physics by Förster, Dexter, and Galanin (Förster, 1949; Dexter, 1953; Galanin, 1951). An appropriate description of elementary steps of electron transfer in chemical and biological systems was given by Levich and Dogonadze, 1959; Marcus, 1956; Marcus and Sutin, 1985; and Jortner, 1976. See also the review by Hoff and Deisenhofer, 1997. It was shown that, in the high temperature limit, the rate constant _ij of ET between the ith and jth cofactors depends exponentially on both the donor-acceptor distance R_ij and the value of (_ij + G_ij^°)²/_ij, in which _ij is the nuclear reorganization energy and G_ij^° is the standard Gibbs free energy difference between the donor and acceptor levels (Marcus, 1956; Marcus and Sutin, 1985). This means that either a change in the distance between cofactors on the order of ~1 Å or a change in the macromolecule structure such that (_ij + G_ij^°)²/_ij changes by more than k_BT may cause a significant difference in _ij.

The generalization of the Marcus expression for the rate constant of non-adiabatic ET in continuous media to the case of any solvent model shows that the rate constant for charge transfer may be expressed in terms of a function of the free energy difference between electron-localized donor and acceptor sites produced by a fluctuating polar medium (Tachiya, 1993). This approach leads to a Gaussian-like dependence of the charge transfer rate constant on the local electrostatic potential of the medium. Many current theories of charge transfer reactions in proteins are based on a similar evaluation of the probability distribution for a free energy difference V_ij between product and reactant states (Warshel, 1982; Parson et al., 1998; Tachiya, 1993; Bandyopadhyay et al., 1999; Warshel and Parson, 1991; Webster et al., 1994). Such an approach not only provides for a correct molecular dynamic calculation of the potential surfaces of the reactant and product states, but also enables prediction of the influence of adiabatic structural motions. Molecular dynamic simulations show that photoinduced charge separation in photosynthetic reaction centers occurs in much shorter times than those required for the system to approach conformational equilibrium after the charge transfer step (Parson et al., 1998). Recent molecular dynamics studies also show that, for long-range electron transfer in proteins, cooperation between vibrational modes of the intervening medium and the transferring electron (inelastic ET) may significantly facilitate the ET reaction, even making it an activationless process (Daizadeh et al., 1997; Medvedev and Stuchebrukhov, 1997). The resulting analytical, modified Marcus expression for the ET rate constant, using a diabatic model of electron tunneling in fluctuating medium, shows that the activation energy may be significantly reduced due to inelastic interaction with phonons. This description is similar to the idea of adiabatic self-organized ET in an active medium (Tributsch and Pohlmann, 1998; Gushcha et al., 1994).

Adiabatic theories of particle transfer over a potential barrier lead to a Kramers-type dependence of the reaction rate constant, one that depends exponentially on the barrier height E_b (Kramers, 1940). Kramers' theory came from an assumption of a linear interaction between the particle and its environment. Recent studies by Tributsch show that a nonlinearity with energy of frictional force dependence may result in a greatly increased probability of escape from a potential well (Tributsch and Pohlmann, 1998). In this description, the rate constant _ij depends exponentially on (/2)(E_b  E_LC)², with  being a coupling coefficient with the medium and E_LC denoting the mean energy of exchange between the particle and medium during oscillation. This idea has been substantiated analytically for the problem of particle escape over a potential barrier in the case of strong interactions between the particle and structural modes of the surroundings (apek and Tributsch, 1999). The authors gave a description of uphill particle transfer for the simplest case. In this case, the coupling of the transferred particle to its surroundings was assumed to be mediated by only one specific mode, localized in the vicinity of the transferred particle. Exact calculations for more realistic cases of many interacting modes were not performed, but expected results for such calculations should be qualitatively the same, providing support for the phenomenological result obtained earlier (Tributsch and Pohlmann, 1998).

In this work we use a phenomenological description for modeling non-equilibrium structural effects that occur during sequential charge transfer through a protein. We take into account dependence of the rate constants _ij on biomolecule structure by coupling to different structural motion. In both the adiabatic and non-adiabatic descriptions, small changes in the values of parameters such as G_ij^°, _ij, E_b, E_LC, R_ij, V_ij, which might be caused by electron transfer 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 X = X(x₁, x₂, x₃, ... , x_i, ...) that in turn depends on a complete set of structural variables {x₁, ... , x_N}, _ij  exp(X). The structural factor X is defined by either the adiabatic E_b, E_LC, V_ij, ... , or non-adiabatic G_ij^°, _ij, R_ij, V_ij, ... , parameters of the system. Thus, statement 4 above indicates that charge-conformational interaction, by which ET is coupled to structural dynamics, may significantly affect the main reaction rate. For the simplest two-state system (Eq. 1) the only kinetic parameter that depends upon macromolecular structure is the recombination rate constant, k_rec. Thus we write,

(10)

In this equation, the structural factor X is dimensionless, normalized with a scaling factor that depends on details of the particular system.

A complete set of structural variables {x₁, ... , x_N} may be selected to span the coordinate space in many ways. Here we take the structural variables as a set of variables that are each distinguished by different relaxation times. The fastest variables (  10¹³ s) describe fast motion of single atoms and small groups, whereas the slowest variables, with relaxation times longer than a second, describe the global dynamics of macromolecular structural rearrangement.

Let us assume that there exist long-lived, light-induced structural rearrangements of the macromolecule. Because of their long relaxation times, these rearrangements can produce effects that accumulate from one single electron-transfer step to the next. Under stationary-state illumination conditions, accumulated structural changes produce a new, quasi-stable structure. The extent of structural changes depends only upon the illumination intensity. In particular, in the case of a large electron-conformational interaction, a "dark-adapted" conformational state may convert to a completely new conformational state under high-intensity illumination. Furthermore, this new "light-adapted" conformational state may coexist with the "dark-adapted" one over an intermediate range of illumination intensity. This result means that there is a photo-induced bistability of the macromolecular structure. Necessary conditions for realization of such an effect are a strong charge-conformational interaction and a long structural relaxation time relative to localized electron relaxation. The slow structural modes, represented in this theory as "slow, generalized coordinates" function as "control modes." These modes lead to a self-regulation of the photoexcited electron flux through the macromolecule, as recently demonstrated for photosynthetic RCs (Gushcha et al., 1994, Goushcha et al., 1997a,b). Below we develop a self-consistent, statistical theory of electronic-conformational transitions to describe such effects.

	A KINETIC DESCRIPTION OF THE ELECTRON-CONFORMATIONAL INTERACTION IN A CHARGE-TRANSFER MACROMOLECULE

TOP ABSTRACT INTRODUCTION SURVIVAL TIME OF THE... A KINETIC DESCRIPTION OF... STABLE STATES OF THE... APPLICATION TO BACTERIAL... CONCLUSION APPENDIX REFERENCES

For the theoretical treatment of light-induced structural changes in a macromolecule undergoing photoinduced charge transfer and separation, we use a Langevin equation with two random forces to describe the mechanical motion of a flexible structure (Chandrasekhar, 1943):

(11)

in which x  {x_i},  {_i} are the sets of structural variables (degrees of freedom; i = 1, 2, ...) with rates of structural rearrangements; T() and V₊ (x) are the kinetic and potential energies for structural modes of the photoactivated macromolecule, respectively; and R() is a dissipative function of structural motion.

The last two terms on the right-hand side of Eq. 11 represent random forces. The first is a random force due to thermal motion. This force acts on the structural variable x_i, while the quantity _i(t) describes -correlated random processes with amplitudes to model thermal fluctuations of the structural variables. The last term in Eq. 11 is a random force corresponding to interaction of a photoactivated electron with the macromolecular structure. Thus, F_i(t, x) describes a discrete random process:

(12)

in which f_nⁱ (x) is a force describing the interaction of a photoactivated electron, localized on cofactor n, with structural mode i. The probability of each component f_nⁱ (x) is determined by the probability of electron localization on cofactor n at a fixed x. These probabilities ((t, n|x)) can be determined from the system of differential rate equations,

(13)

These are the master equations for a random process (Eq. 12) (Horsthemke and Lefever, 1984). The quantity _nm(x) in Eq. 13 defines the rate constants of non-adiabatic transitions between the n and m cofactors at a fixed structure. We assume that the variables x = {x_i} represent overdamped conformational motions, a valid description for flexible structures like proteins. Although this assumption may be incorrect for high-frequency oscillations, these variables are thermally equilibrated and excluded from detailed consideration. From Eq. 11 and in accord with the results of Horsthemke and Lefever, 1984, for a coupled random process (Christophorov, 1995) we obtain the fundamental kinetic equation for the distribution function of both electron and structural variables of the macromolecule, P(t; n, x),

(14)

in which

(15)

(16)

and D_i is a diffusion constant corresponding to motion of the structural variables {x_i} along the conformational potential surface V_n(x) for electron localization on binding site n. This equation is general, but we further simplify it to reveal the role of control modes on macromolecule structural dynamics.

To simplify, we separate the variables {x_i} into three groups, depending upon the relative magnitudes of the relaxation time constants _x and _el of the distribution function P(t; n, x) over the structural and electron variables, respectively. Those variables x_fast, for which _{x el}, belong to the first group. For the second group (x_equal) the time constants are of the same order: _x ~ _el. The third group is characterized by slow structural motions (x_slow) for which _{x el}. The two types of variables, x_slow and x_equal, should be explicitly retained in a description of self-regulation effects for a system involving photoexcited electron transfer within a flexible structure. However, in the present treatment we retain only x_slow, and ignore x_equal. The fast variables, x_fast, are not important for self-regulation effects, because these variables relax on much shorter time scales. We can integrate over these variables, using the substitution

(17)

in which (t; n, x_slow) is the distribution function for electron and slow structural variables.

Putting Eq. 17 into Eq. 14 and integrating over x_fast, we obtain equations identical to Eq. 14 that are valid for time intervals t _fast. They may also be derived from Eq. 14 with a simple substitution,

(18)

For ET rate constants between cofactors n and m, after such substitutions, we obtain:

(19)

These transitions are non-adiabatic with respect to x_fast, but adiabatic with respect to x_slow.

The potential energy expression corresponding to slow structural variables can be easily obtained after substitution of Eq. 17 into Eq. 14 and integrating over x_fast,

(20)

It is obvious that the quantity _n (x_slow)  G_n (x_slow), in which G_n(x_slow) is a so-called quasi-free energy for an electronic state n, depends parametrically on the slow structural variables (Stratonovich, 1992; 1994). This means that _n(x_slow) represents the standard free energy for the electron state n with respect to the fast structural variables x_fast, but it corresponds to the potential energy of electron state n with respect to the slow variables x_slow.

We further proceed from Eq. 14, taking into account the substitutions (Eq. 18) and the actual role of the slow variables, x_slow, only. Thus we can use an adiabatic approach that enables us to make the following simplification in Eq. 14. [Here we restrict consideration to the simpler case of ordinary slaving (Haken, 1983) in which fluctuations of the fast variable influence the evolution of the slow one only on average, without causing any noticeable fluctuations of the latter. The softer types of slaving will be discussed in a separate paper.]

(21)

in which (t, n|x) are the relative probabilities to find an electron localized on cofactor n at fixed x, as determined from Eq. 13 using Eq. 18, and P(t,x) is a distribution function for the slow structural variables. This function is defined by the expression

(22)

Putting Eq. 21 into Eq. 14, we obtain an equation that describes the time evolution of this function,

(23)

in which F_adi^I(x) is a statistical quantity with dimensions of a force. This quantity describes the adiabatic action of the electron transfer upon the ith slow structural degree of freedom under conditions of slowly varying illumination intensity, I, ensuring that electronic relaxation processes are complete: | ln I(t)| _el, and

(24)

Equations 13, 23, and 24 provide the basis for self-regulation of a photoactivated electron flux by slow structural variables of a macromolecule.

Assume that the system can be characterized by a single slow structural degree of freedom: the generalized configurational coordinate x. Then Eq. 23 may be rewritten as:

(25)

in which the adiabatic potential of the system, V_ad^I(x), at fixed light intensity I is determined from Eqs. 16 and 24 with an uncertainty C(I)

(26)

Note that the subscript "ad" in the expression V_ad^I means "adiabatic," not to be confused with the free energy difference V_AD between the donor and acceptor levels. V₊(x) has its minimum at x₀. f_n() has the same meaning as the force introduced in Eq. 12 for i = 1. Calculation of C(I) will be discussed elsewhere.

The quantity V_ad^I(x) serves as the effective adiabatic potential for the slow structural mode. This potential determines the average value of x over the electron distribution function. This potential is of a statistical nature, depending upon a stationary-state distribution of localized electron populations at a fixed structure. This structure is determined and controlled by the illumination intensity, I. For times t > _xslow, stationary-state conditions are reached. The corresponding stationary-state distribution function can be written as

(27)

The minima and maxima of this function, x_ext, define the stationary states of the macromolecule at a fixed I. Thus, the effective adiabatic potential V_ad^I(x) for the open non-equilibrium system described by Eq. 1 is the analog of a standard Gibbs free energy, G°, which determines the probability to find a closed system in a particular equilibrium state with given free energy. The stationary states defined by x_ext in this open system are the analog of the equilibrium states in a closed system, and the values of x_ext can be determined from

(28)

Those states that correspond to potential minima define the conformational coordinates of the system. The functions _I(t, n|x), P(t, x) as well as their stationary values _I(, n|x), P_I(, x), and the x_ext(I) depend on the structure of the system and determine each experimentally measured quantity q(n, x) by averaging. Averaging over the electron variables,

(29)

When averaged over both the electron variables and generalized configurational coordinates,

(30)

Before proceeding to the next section we should comment on our use of a single structural variable (Eq. 25). In practice, the application of any theory to a particular biomolecular system often requires a decrease in the system dimension to a few generalized structural coordinates or even to a single coordinate. Of course the system configuration is determined by tens of thousands of physical variables, and the configuration of one system is a point in the multi-dimensional configurational space. As shown above, the fast structural variables relax to quasi-equilibrium values and fluctuate about them. These quasi-equilibrium values themselves continue adiabatically to follow changes in the slow variables until at long-time the system dynamics may be determined by a small number of slow variables or even a single slowest variable. The dynamics of such slow variables is determined by the potential profile of the quasi-free energy (see Eq. 20). Such a hierarchy is characteristic for the relaxation of complex systems with a large number of variables. It has been called "the slaving principle" (Haken, 1983). Thus it is reasonable to assume that at long times, the slow structural rearrangement of a biomolecule may be described by a small number of variables. For a description of phase transitions in systems with a large number of variables, consideration is often restricted to a single "control mode" or "order parameter." Similarly, the description of chemical reactions in complex molecular systems may also be described with a single "reaction coordinate." For example, long-time relaxation processes in biopolymers are often described by bounded diffusion of initial multi-dimensional distribution function along a particular trajectory of the potential surface (see, e.g., Agmon and Hopfield, 1983; Rubin et al., 1990; Gudowska-Nowak, 1994; Frauenfelder et al., 1991, 1999).

Thus, we restrict present considerations to a one-dimensional model of slow structural rearrangements induced by a photoinduced charge separation in biomolecules. We treat x as a global configurational coordinate that describes slow photoinduced structural changes. In this case, the structural factor X(x) introduced above may also be identified as a configurational coordinate because as it is well known that the dimension of generalized variables does not affect the calculation of trajectories and free energies (Goldstein, 1980). Consequently, we modify Eq. 10 making the substitution x  X, assuming that the configurational coordinate x is monotonic in configurational space because system energy decreases during relaxation, and slow system diffusion is determined by the trajectory. Note that x does not describe arbitrary structural rearrangements, but only those responsible for slow structural relaxation to a new potential minimum in configurational space. Finally, we obtain:

(31)

an expression that will be used in subsequent sections. It is important to note that the adiabatic potential for the structural factor X has its minimum at X_min = X(x_min); therefore the proposed substitution of variables leads to the equivalent consideration in our phenomenological model.

	STABLE STATES OF THE TWO-STATE SYSTEM: A CONFORMATIONAL APPROACH

TOP ABSTRACT INTRODUCTION SURVIVAL TIME OF THE... A KINETIC DESCRIPTION OF... STABLE STATES OF THE... APPLICATION TO BACTERIAL... CONCLUSION APPENDIX REFERENCES

To determine the light intensity dependence of macromolecule stationary states from Eq. 28 we select, as an example, a harmonic potential with effective elastic constant , V₊ (x) = (x²/2), and, hence an effective adiabatic potential, V_ad^I(x). Such a potential represents Gaussian fluctuations of x around equilibrium (see, e.g., Zusman, 1980). The quantities f_n(x) (n = D, A) are defined as additional stochastic forces that act on the configurational coordinate when an electron is localized on cofactors D and A, respectively. We assume that these forces are constant, but not equal to each other. That is, f_D(x) = f_D =  * x_D is a force acting on the structure when an electron is localized on donor D; and f_A(x) = f_A =  * x_A is a force acting on the structure when an electron is localized on acceptor A. In general, x_A  x_D. The quantity  = |x_A  x_D| characterizes the electron-conformational interaction of the system. It is proportional to the additional force acting on the configurational coordinate in a charge-separated donor-acceptor pair. We assume here that the force constant  is the same for the charge-neutral and charge-separated states, although this need not necessarily be true (i.e., in general _D  A). Furthermore, we showed in our recent paper (Goushcha et al., 1999) that for photosynthetic bacterial reaction centers the probe potential V₊ (x) is probably not harmonic with different curvatures in the charge-neutral (PQ_AQ_B) and in the charge-separated (P⁺Q_AQ_B) states. Parson and co-workers arrived at a similar conclusion in their molecular dynamic studies of the ET reaction P*BH  P⁺BH. They showed that the force constant for the P⁺BH state is larger than that it is for the P*BH state (Parson et al., 1998). In the current work, we explore qualitatively the conditions for non-equilibrium structural transitions and the emergence of new conformational states. For this treatment, the fact that _D  A is not essential, and we assume that _D = _A   to obtain analytical solutions.

Using expressions for light-dependent, stationary-state electron populations (Eq. 5), the recombination rate constant dependence on x (Eq. 31), and stochastic forces (Eqs. 12, 16, and f_A,D(x)), the equation defining the stationary states of the system (Eq. 28) becomes

(32)

This equation is valid for macromolecular ET systems that can be accurately described as two-state systems (Eq. 1), explicitly indicating the dependence of x_ext on I.

For this specific example, x_ext(I) (Eq. 32) at specific values of  was determined in our recent work (Goushcha et al., 1999). A small, monotonic, light-induced increase in x_ext(I) was obtained for the case of a weak interaction,  = (x_A  x_D)  4. The concomitant increase in the lifetime of the charge separated state, _d, obtained using Eq. 32 is shown in Fig. 1. The smooth, light-induced increase of _d for curves 1 and 2 is due to deformation of the "dark" conformational state. This effect may be described as "self-regulation" of the photoactivated electron transfer rate due to a slow structural deformation, reflecting macromolecule structural changes from an electron-conformational interaction modulated by the photoinduced electron flow.

View larger version (17K): [in this window] [in a new window]   FIGURE 1   Dependence of the survival time d of the charge-separated state on illumination intensity I for various values of the electron-conformational interaction parameter  = xA  xD: 1)  = 2; 2)  = 4; 3)  = 5.2; 4)  = 6. The curves were obtained for the following set of parameters: xD = 2; krec0 = 10.

More complex behavior of x_ext(I) occurs in the case of a strong charge-conformational interaction  > 4. For illumination intensities, I_II^cr > I > I_I^cr, in which

(33)

and
three values of the extrema x_ext⁽ⁱ⁾ (I); i = 1, 2, 3 are obtained. Two branches, x_ext⁽¹⁾(I) and x_ext⁽³⁾(I), give minima adiabatic potential, corresponding to stable structural states of the system, while a third, x_ext⁽²⁾(I), gives a maximum for an unstable state. The parameter k_rec⁰ slightly perturbs these dependencies, shifting them to a higher light intensity with an increase in k_rec⁰. Experimentally, one can only observe stable branches of these dependencies. This means that experimentally measured system parameters may reveal discontinuities at particular illumination intensities. The survival time, _d, of a charge-separated state, in the case of a strong charge-conformational interaction, for the branch x_ext⁽¹⁾ (I) is significantly shorter than this time for one belonging to the branch x_ext⁽³⁾ (I). (Compare curves 3 or 4 with curve 1, Fig. 1). In fact, following the discussion in the first section,

(34)

and  = x_A  x_D may be determined from experiment as

(35)

The appearance of a new, light-induced stable structural state for  > 4 and the coexistence of this state with the initial stable state represents a non-equilibrium phase transition of the "monostability-bistability" type (Haken, 1983, Stratonovich, 1994).

The problem of thermodynamic stability of stationary states at coordinates x_ext⁽¹⁾ (I) and x_ext⁽³⁾ (I) and the related problem of thermal fluctuations in the configurational coordinate around stationary-state values can be solved using a distribution function over x of the form P(t, x) = P_D(t, x) + P_A(t, x) (see Eq. 22). An evolution equation for this function is described by Eq. 23, where using Eq. 26, the statistical potential V_ad^I(x) (Goushcha et al., 1997a) is given by

(36)

Previously, we analyzed this expression for many values of the electron-conformational interaction parameter,  (Goushcha et al., 1997a 1999). A second, light-induced potential minimum may appear for  > 4, a case corresponding to a distribution function P_eq^I(, x) with two maxima (Fig. 2 A). For  < 4, the light-induced deformation of the adiabatic potential causes a deformation of the distribution function with only a shift in the distribution maximum toward larger values of the conformational coordinate (Fig. 2 B). The evolution of the distribution function with light intensity for the case of a weak interaction has been described in the literature. See, e.g., studies of the P⁺Q_A  PQ_A reaction in photosynthetic bacterial RCs (Shaitan et al., 1991; Uporov and Shaitan, 1990).

View larger version (18K): [in this window] [in a new window]   FIGURE 2   Quasi-equilibrium distribution functions calculated for  = 2 (A) and  = 5.3 (B) for the following values of I. a, curve 1: I = 0; curve 2: I = 0.1; curve 3: I = 0.5; curve 4: I = 2.0; b, curve 1: I = 0; curve 2: I = 0.07; curve 3: I = 0.1; curve 4: I = 0.13; xD = 2; krec0 = 10.

The abscissas of the potential V_ad^I(x) extrema determine stationary values x_ext of the slow configurational coordinate as a function of  (see Eq. 35). The values x_ext⁽¹⁾(I) and x_ext⁽³⁾(I) correspond to adiabatic potential minima, whereas x_ext⁽²⁾(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 between them.

Often the ensemble properties of macromolecules may be satisfactorily described by the most probable behavior of these macromolecules near potential minima. For this description, we use a conformational approach and introduce the function

(37)

in which we integrate over all configurations of the coordinate x  G_a near a potential minimum  at the point x (van Kampen, 1992). This function, , determines the population at the minimum  of the adiabatic potential.

States with an x such that x  G_a are treated as the same conformational state, . The probability of realizing different conformational states is determined by the forward and reverse transition rates over potential barrier (Kramers, 1940; van Kampen, 1992). These probabilities are obtained from Eq. 25 using Eq. 37. Using this conformational approach, the average value of an observable q(t) can be written as