J. Phys. Chem. 1994,98, 5251-5264
5257
Effects of Heterogeneity on Relaxation Dynamics and Electron-Transfer Rates in Photosynthetic Reaction Centers Ewa Gudowska-Nowakt Department of Applied Science, Brookhaven National Laboratory, Upton, New York 11973 Received: June 15, 1993; In Final Form: March 21, 1994’
The excitation and relaxation of the nuclear medium modes which couple to the electronic states can significantly modulate the dynamics of electron transfer (ET) when the relaxation process is sufficiently slow. Recent molecular dynamics (MD) studies of reaction centers (RC) show that the dynamical response of the proteic environment has noticeable nonlinear behavior, and the observed slow decay of the time correlation functions is suggestive of glassy dynamics in the system. In this context, we consider two possible models for the R C relaxation times. The first model of a “static disorder” discusses relaxation dynamics in the presence of a “rough” free energy potential. A rough potential has a smooth background on which randomly fluctuating local perturbations are superimposed. Distribution of the perturbations determines the relaxation dynamics and may be responsible for the dynamical, temperature-dependent phase change in the medium. In particular, the mean sojourn time in such a free energy potential well is no longer exponentially distributed, and this effect may explain the complex decay dynamics of the primary donor in RCs. A similar behavior can be found in a complementary model of a “dynamic disorder” where spontaneous conformational isomerization of chromophores causes the transition from one configurational state to another and back. The importance of such a transition, induced by a polar group isomerization in the neighboring amino acid side chains of bacteriochlorophylls, has been suggested in the MD studies of the time-variations of the electrostatic energy at the chromophores in RCs. With the assumption that the time the system spends in a particular conformation state is Poisson distributed, the decay time is evaluated by a stochastic analysis of the isomerization process and compared with kinetics predicted by other models of the primary ET in RCs.
I. Introduction The excitation received from the light-harvesting complex promotes the primary donor in the reaction center, a special pair P of bacteriochlorophyll molecules, into an excited state from which the electron is ejected into the nearby bacteriopheophytin (BPhe) molecule. This first photochemical step takes about 3 ps. The edge-to-edge distance between the ?r systems of the donor special pair to the edge of the a system of the acceptor, BPhe, is about 10 A. The reaction proceeds from above room temperature to 2 K with approximately unit quantum yield. From basic electron-transfer theory, it follows that the rate is too fast to be accounted for by a direct nonaided charge transfer. All the mechanisms proposed to describe the primary charge separation attribute a central role to the accessory monomeric Bchl situated in between the primary donor and the primary acceptor, BPhe. The question arises of how this intermediate functionsand whether it is part of a two-step ET reaction with the state P+(Bchl)-(BPhe) as a discrete intermediate or whether it is a superexchange site in a single-stepreaction. The controversy in discussing theoretical models for the primary process is complicated by increasing experimental evidence of a nonexponential decay of the optical transient states associated with the initial event.’ Various explanations for the relaxation behavior of P* observed in the experiments have been discussed, pointing out possible incomplete vibrational relaxation after excitation? heterogeneity of the sample,’ intrinsic biphasic kinetics of the primary step with a chemical intermediary carrier B ~ h l or , ~ combinations of all of these factor^.^ In particular, it has been shown596 that a Gaussian distribution of vertical energy gaps46gives rise to a nonexponential behavior in the simulation of P* decay with a superexchange mechanism. As a supportive argument for the inhomogeneity, Du et a1.l have t On leave of absence from the Institute of Physics, Jagellonian University, Krakow, Poland. 0 Abstract published in Advance ACS Abstracts, May 1, 1994.
presented results of measured width of the P absorption band which increases with increasing amplitude of the long-time component in the fluorescence decay. A model of sample heterogeneity has also been proposed by Kirmaier and Holten? whose results of the near-infrared absorption spectroscopy conducted at both room and low temperatures have been analyzed in terms of a rate of ET that varies as a function of the wavelength at which the photochemistry is monitored. These findings have led to the suggestion that the RC may exist in more than one form with the possibility of rapid (at the millisecond time scale of the meas~rements)~ interconversion at room temperatures. Such a postulate would imply that the relevant potential energy surfaces contain barriers which make the passage between the substates of the distribution slower than the primary electron-transfer reactions. These barriers may reflect restricted pigment and protein motions of various types, such as motions involving hydrogen bonds, porphyrin ring puckering, or low-frequency torsions. Lowering the temperature would result in a collapse of the distribution to a form with the fastest inherent rates of ET.*-10 It has been speculateds that a major contribution to the process may be a general contraction of the proteinl’J2 at low temperatures. Protein relaxation by itself may be a source of intrinsically nonexponential ET dynamics. Chromophores and protein relaxation on the electron-transfer time scale may lead to a timedependent fluorescence spectrum,’ which would give nonexponential time dependence at a particular emission wavelength. Guided by spectral hole-burning data, Small et ~ 1 . 1 6have derived a model of a dispersive kinetics for the initial phase of charge separation in RCs. These authors have related a significant site inhomoheneous line broadening (rl) observed in Rb. sphaeroides and Rps. viridis to the “glasslike” structural heterogeneity of proteins. Nonexponentiality of the decay of P* could then be understood as the effect of dispersive kinetics stemming from a distribution of the energy gaps relevant to the primary charge-separation process. The existance of rl is indicative of random fluctuations in the RC structure for a given
0 1994 American Chemical Society 0022-3654/94/209~-5251~04.50/0
5258 The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 complex and may be attributed to the distributions of the energy gaps. In this paper we analyze the dynamics of the ET in R C in terms of (1) motion in the presence of random free energy barriers imposed on the smooth “background” structure of the potential (a “static disorder” model) and (2) conformational isomerization of a chromophores taking place on the time scale of the E T kinetics (a model of “dynamic disorder”). Model 1 is discussed in section I1 where the medium relaxation effects on ET dynamics are analyzed. In particular, it is shown that, due to enhancement of the effective solvent longitudinal relaxation time in the presence of a “rough potential”, it becomes considerably easier for the solvent dynamics to influence the nonadiabatic ET. Section 111presents a formalism to derive the kinetic rate for the relaxation of the excited state P* after a model of isomeric conformation changes of the chromophores is adopted. Conformational readjustments are presented in the form a dichotomous non-Gaussian Markov process which leads to nonexponential decay of the excited state. We briefly discuss the concept of possible Gaussianity of the underlying kinetics (section IV) * A particular aspect of the biopolymers is the usual presence of the hydration shell, a monolayer or two of bound water molecules on the surface of the protein. In heme proteins, the hydration shell displays a broad liquid-glass-like transition,Iz which in turn can be characterized by general features displayed by this class of transitions studied in amorphous materials. The most significant points are the divergence of the transport or inverse transport properties (such as viscosity, inverse diffusion constant, and relaxation times) and the extremely broad relaxation phenomena of the stress and modulus.13 It has been well recognized that the relaxation time follows an unusual law in physics, described by the Vogel-Fulcher LawI4J5 and can be correlated with some characteristic (not necessarily Gaussian) distribution in kinetic parameters.l5-I7 Provided the characteristics of glasses and spin-glasses have correspondences in proteins,i8-M efforts to improve M D free energy calculations in RCs should be attempted together with the experimental search for the relevance of the glass transition in the systems.
Gudowska-Nowak This combination of both steric and electronc factors requires a full analysis in theoretical and experimental studies of the primary charge separation in RCs. The theoretical exploration of the E T in these systems rests on conventional nonadiabatic ET theory. Implementation of the theory faces many problems which remain to be resolved on a quantitative level, such as energetic parameters controlling the transport, evaluation of the electronic matrix elements responsible for the effective coupling, the solvent contribution to the activation energy, dynamical aspects of the glassy protein response, and its projection on the detailed structure of the solvent. Experimental studies point toward a correlation27-28 between the rate constant of a fast electron-transfer reaction and the reciprocal of the longitudinal dielectric relaxation time of the solvent. In thenonadiabaticlimit, the E T process is usually weakly correlated with the dynamics, provided the adiabaticity parameterZ7
is small, where T L is the longitudinal medium relaxation time, V is the electronic coupling, and X stands for the reorganization energy. Warshe129 reported a preliminary computer simulation of solvent dynamics in an E T reaction. More recent studies7J1 have established a relationship between the autocorrelation function of the time-dependent vertical energy gap and the rate of E T in solution. In macroscopic models of the solvent, T L is equivalent to the autocorrelation time of the energy gap7-29s30and has been simulated in MD calculations of R C S , yielding ~ values in the range of 0.1-1.0 ps. On the basis of the simple phenomenological model presented below, we infer the possibility of longer effective relaxation times in a spatially disordered medium, the result being consistent with the recent findings on MD dynamics of proteinsZwhich have used the method of umbrella sampling to determine the probability distribution of proteic substates. The general nonadiabatic system consists of a two-level system coupled to a reaction c o ~ r d i n a t e . ~Time-evolution ~-~~ of the reactants’ population can be viewed in terms of time-dependent properties of the diagonal elements of the density matrix p relevant 11. Solvent Dynamics in a Rough Potential for that p r ~ b l e m . ~ ~Adopting J - ~ ~ the coarse-grained kinetics Recent molecular dynamics studies by Marchi et ~ 1 . show ~ 3 ~ ~ for the diagonal elements of p introduced by ZusmanZ7 gives the that the dynamical response of the proteic environment has evolution equations for the system: noticeable nonlinear behavior. Proteins at room temperature are known to exhibit many thermally accessible energy minima in the neighborhood of the native structure. The existence of states and substates implies two types of motion in proteins-equilibrium fluctuations and functionally important motions which are nonequilibrium processes driving the system from one functional state to the another. Interpretations of the flash photolysis experiments on proteins led some authors20J2 to the conclusion that substates and fluctuations in proteins possess a hierarchical character which results in nonexponential relaxation. Similar conclusions can be drawn from the theoretical models of photosynthetic chromophores.lS6 The MD calculations perwhere pR(x,f) stands for the probability distribution function for formed on photosynthetic conformers and their models7923-24 the reaction c0ordinate4~x value at time t belonging to the suggest that they possess high structural flexibility with relatively “reactants” and “products” wells, respectively, H R represents ~ low steric energy differences between various conformational the electronic coupling between the two diabatic surfaces, and forms. L R ,is~ a Liouvillan operator for the classical motion within the Crystallographic data on deliberately crowded porphinoids have harmonic well. If the dynamic properties of the medium described established that considerable puckering or deformation of the by L R ,can ~ be approximated by the ~ v e r d a m p e dDebye ~ ~ solvent skeletons can be induced with significant consequences to the LRPtakes the form of the Smoluchowski operator electronic properties of these systems.25-26 Clearly, a combination of axial ligation, hydrogen bonding, and nearby residues can define a structural scaffolding that determines the conformations of the m o l e c ~ l e sand ~ ~ *the ~ ~orientations of their substituents that, in turn, control their photophysical and photochemical character(3) istics.
Effects of Heterogeneity in Photosynthetic RCs (X2), = 2XkBt
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5259 9 one obtains
where X represents the reorganization energy of the solvent and T L stands for a characteristic longitudinal relaxation time of the medium, related to the friction coefficient E:
where D being a diffusion constant, mL representing a solvent polarization mass (an effective mass associated with the fluctuations 6&p(t), and WL standing for the longitudinal solvent frequency.49 For a non-Debye solvent, the Liouvillan will be explicitly timedependent, but with the assumption of the overdamped motion within the well, solvent relaxation can be characterized in terms of the finite number of exponentials, so that one can choose an equivalent “effective”, time-independent Liouvillan to describe rate-limiting diffusive dynamics of the ~ o l v e n t . ~ ~ . ~Equation *J3 2 can be solved by use of the Laplace-transform technique with suitably chosen initial conditions (the usual assumption is that at time t = 0, the system resides in either one of two diabatic states with the equilibrium distribution pq). The time evolution of variations
is given by
and in the overdamped limit, after the Debye representation of the solvent is assumed, leads to the Kramers expre~sion27333.3~ for the rate krxn:50
so that eventually the free energy function can be expressed as
A similar analysis can be repeated in the case of the diffusive well dynamics described by the Liouvillan operator in eq 2. In this coarse-grained approach, our knowledge of particular types of hamiltonians governing the evolution of “solvent” and “imperfections” dynamics is limited; we are assuming that the free energy potential can be described as a sum
F ( x ) = F,(x)
+ .fix)
(12)
whereflx) stands for random, local variations of the potential Fo(x). To calculate the population within the well, we need to estimate the integral (eq 13) which, evaluated over small “distances” dx,9 can be approximated by
s
d x e+F(X)
s
d x e+F~(X)(e+flX))
(13)
where brackets denote the spatial average (smoothing of the potential). Equation 13 suggests a description of the diffusive dynamics in terms of the effective potential
(14) From the general form of the Smoluchowski equation,
The point we want to make now is that, given the local fluctuation in steric effects of the potential which can be represented in the form of Gaussian-distributed “ripples” imposed on the otherwise smooth, averaged potential surface, the effective relaxation time for the medium becomes exponentially enhanced. Let us start our description by using the thermodynamic concept of the amorphous state. An amorphous state is a random but frozen state. Suppose that the total energy for the system is
we can infers2 that an effective diffusion coefficient changes to the value (the result is straightforward if the amplitude of fluctuations infix) does not depend on the coordinate x; for the discussion of more general cases, see ref 35)
(
De, = (eBflx))L(eiP/(x)) , D = (j3muLTL)-‘ D
(16)
which together with eq 14 defines a classical Liouvillan averaged over random contributions to the potential, A x ) where HOstands for the background “solvent” energy with the random structure of “impurities”. Interactions between the “background” and the impurities and among the impurities themselves yield an extra term in the energy (eq 8) of the system. The total thermodynamic potential (free energy function) for the system can be calculated by use of the partition function:
where x stands for collective coordinates in the system. From eq
The formal expression for the diffusive rate constant within the harmonic well with randomly distributed “roughness” follows a derivation based on the mean first passage time appr0ach,~6with a reflecting boundary condition at x = -m and an absorbing boundary at the top of the barrier, x = x*:
With the assumption that the fluctuations A x ) superimposed
Gudowska-Nowak
5260 The Journal of Physical Chemistry, Vol, 98, No. 20, 1994
over the mean potential Fo(x) are Gaussian distributed with cf(x)) = 0
u"(.,)
= u2
(19)
direct evaluation of eqs 14, 16, and 18 with the relaxation rate in the mean potential well predicted by eq 7 yields
so that duration of the process becomes exponentially enhanced. Note that the results can be interpreted as the effective enhancement of the longitudinal relaxation time of the medium, which can be now expressed as
where we have identified cfz) = u2 = k i T i . Such forms of the relaxation times have been discussed in temperature-dependent dynamics of spin-gla~ses.l2-~8~3~ For a Poisson distribution of f ( x ) , the same analysis leads to
properties are time-independent (as in the models of amorphous materials mentioned in the previous section) or they change on a time scale much shorter than the characteristic time evolution of the particle moving through the medium. In the latter case the medium acts on the system through the noises of "white spectrum" which can be treated by the Langevin equation or equivalent methods. Media with finite correlation times (finite memory), implying "colored noises", have been addressed with the aid of the generalized Langevin equation.39 Finite memory effects of media can influence rate processes when specific molecular environment configurations become coupled to a molecular reaction coordinate. In modeling nonexponential decay of P* within the fluctuating medium, we assume here that the relaxation process occurs with a competitive process of isomerization ("dynamic disorder") in which the complex undergoes conformational variations on the time scale of the decay. Spontaneous isomerization of chromophores causing the transition from one configurational state to another one and back has been suggested7 on the basis of visible variations of the electrostatic energy of chromophores studied in MD models of RCs. Let us assume that the time Ti the special pair exists in a given configuration i, before it flips to another one, follows an exponential law: Probability(Ti)t} = e-"',
predicting a "phase change" at a specific temperature at which r,ffdiverges. The result differs from the evaluation of the average escape time in a system of random energy barriers, as reported by Vilgis,16 where no underlying structure of the smooth free energy (a potential well) was used in the presentation of the model. A direct observation which follows from the derivation of formula 18 is that the distribution of first passage times is no longer exponential (cf. refs 16 and 35). The moments of first passage time in the system described by the Smoluchowski equation (eq 15) with the effective potential (eq 14) and the effective diffusion coefficient (eq 16) are given by the equation36
M , = k;l!ff
(23)
and, due to the dispersion caused by the randomness of&), do not follow the law of the exponential distribution
X i ) O , i = 1, 2, ..., N
(25)
At the macroscopic level, time variations of the population of the state P* are governed by phenomenological kinetic equations:
dx = -kx dt
- Z{X,
Z, = (Ai}, i = 1, 2, ...,N
(26)
where Zr describes the process of stochastic changes among various conformations and x ( t ) stands for the population of the primary donor. Since the pair process ( x ( t ) ,ZJ is Markovian, it is possible to derive a "Smoluchowski-like" evolution equation for the probability density p(x,Zt,t). Averaging over I, yields an equation for p(x,t) alone but also results in an explicit memory kernel in the evolution equation for that quantity.38 A particularily simple form of the evolution equation is derived if the conformational changes can be parametrized by two states with a symmetric stationary probability of occupation of a given ~ t a t e : ~ ~ , ~ ~
--a t - daxk x p ( x , t ) + A2&x J;-dt'exp(-
[ r - a , ka x ] ( f - t ' ) ) $ p ( x , t )
(27)
where
In this context, diffusive decay of the population within the harmonic well with a superimposed random roughness would result in nonexponential dynamics. From eqs 21 and 22, it becomes apparent that at high temperatures, compared to the intensity of fluctuations u,diffusion within the free energy well is essentially unaffected by the existence of randomly distributed barriers. At lower temperatures, effective relaxation time will strongly depend on the form of distribution off(x1.
~ ( x l t= ) ~ ( x , A l , t >+ ~ ( x , A z , t > A, = - lA,l = A, (Zr) = 0, (ZtZr+J = A2eqr
Transition rates in a stochastic medium have been recently s t ~ d i e d ~ in ~ Jthe ~ Jcontext ~ of relaxation and transport in timedependent disordered systems. The stochastic medium is often represented as an array of randomly distributed, mobile objects (particles, traps, reacting sites, etc.). Two cases are usually considered: either their
(29)
and
x, = x,
= 712
(30)
The moments of the normalized populationX(t) = x ( t ) / x ( O ) can be calculated by use of the characteristic function for the process leading to the formula53
Z,,3E941
111. Relaxation Rate in a Stochastic Medium
(28)
1 r n= -(rZ + 4n2A2)'I2 2
where positivity of the rate requires A ( k .
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5261
Effects of Heterogeneity in Photosynthetic RCs
1
0.995
oa
0.99
3 'f0.985
0.6
8a
0.4
0.98 0.975 0.97
0.2
0.965 0.96
0
0
3.2
6.4
9.6
12.8
0
16
0.2
0.4
0.6
0.8
timc [pal
I Y
large values of y, eq 31 yields 0.8
(35) with time-dependent dispersion:
0.6
0.4
From the normalized decay curve of ( X ( t ) )one can calculate an average relaxation time
0.2
0 0
3.2
6.4
9.6
12.8
16
Ipal
Figure 2. Survival of the average normalized population (x(r)) (see the text) as a function of 7: (a) A = y = 0, k = 0.37 ps-'; (b) y = 2.0; (c) y = 0.5; (d) y = 0.1. Curves b-d are parametrized by k = 0.37 ps-Land
which can be used to estimate an effective quantum yield 6:
A = 0.3.
As can be easily deduced from eq 31 (cf. also Figures 1 and 2 ) decay of X(t) is no longer governed by a single exponential (which would be a limiting case if the effect of the fluctuating medium could be ignored). An effective relaxation rate for the decay of P*can be defined in terms of the linear relaxation time for a nonlinear process
(xftl):
where 7npcstands for the inverse rate of the nonphotochemical process which competes with the ET.55 Figure 3 displays 6 as a function of interconversion frequency y for different values of the noise intensity A. Within the range of chosen parameters y and A (Le. those which ensure the positivity of theoverall relaxation rate of the process), quantum yield decays with the higher values of noise intensities and remains close to one for increasing values 7.
IV. Concluding Remarks which, together with eq 31, yields y cosh(rt)
k,, = k + - r y sinh(Ft)
+ 2 r sinh(Ft) + 2 r cosh(rt)
r = rl In the limit of short times, t value, kcff = k, whereas for
(33)
--
0, ken tends to a "deterministic"
t
Ultrafast kinetic studies have shownz8that charge separation and charge recombination between donor/acceptor sites in naturally occurring systems can be strongly influenced by environment. The rates of charge-transfer processes may be limited by solvent motion owing to strong dielectric coupling between the charge-separated state of the chromophore and solvent dipoles. The precise impact of these solvent effects is knownz7 to vary with the degree of reaction adiabaticity. The overall rate constant describing the competition between E T and medium dielectric relaxation can be recast in the form27
k=: k
+
-i(y2
+ 4A2)'I2< k
(34)
As it can be also seen from Figures 1 and 2, stochastic dynamics underlying decay of the photoexcited state P* decreases the rate of P*relaxation compared with a single exponential deterministic54 rate k. The difference becomes small for larger values of y. For
= kNA
+ kdl!ff
(39)
where kNA stands for the conventional nonadiabatic rate constant and kdiffis a diffusive rate constant controlling the motion within the well (kdin = k,,,; cf. eqs 5-7). Dynamic medium effects are most evident for ET that is activationles~~~ or nearly activationless, as in this case the E T rate directly reflects the time scale of the solvent motion.28
5262
Gudowska-Nowak
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994
For a Debye solvent, a simple continuum description of the solvent predicts the rate kdiff to be inversely proportional to the Recent theolongitudinal relaxation time of the retical research has gone beyond the simple Debye model33 and introduced some aspects of the microscopic solute/solvent interactions. Many studies of solvent dynami~s*8-30~~~~~9 reveal that the solvent relaxation process is not as simple as the dielectric continuum model predicts. In most cases the solvent relaxation is not well described by a single time constant but rather by a distribution of times (see e.g. Rips and J ~ r t n e r ~Contemporary ~). theories which go beyond the dielectric continuum model also indicate that the relaxation process occurs with multiple time constants. On the basis of existing theories of ET, we have presented two models for the influence of medium dynamics on ET rates in RCs. Section I1 discusses effects of a static but random medium influencing relaxation properties of the system. The main ideology of this model stems from fairly recent attempts to characterize by exploring their similarity protein dynamics and to glasslike materials. Structural flexibility of proteins results in their multistate conformational energy landscape21942 with a broad distribution of relaxation frequencies. At high temperatures, compared with the intensity of freeenergy perturbations u, relaxation kinetics within the “effective potential well” (cf. eqs 12-20) is unaffected-the kinetic process essentially follows dynamics in a smooth “background potential”. Only for temperatures which are low compared with u would the diffusion process, as discussed in section 11, be influenced by distributions of “barriers” imposed on a smooth (Fo)structureof the potential. This picture is essentially consistent with the model of a statistical distribution of thermodynamic driving forces, as introduced by Fleming and Norris and their co-workersl~5to explain nonexponential decay of P*. In fact, by assuming that the “reactants” and “products” free energy potentials (Le., free energy potentials before and after the ET process, respectively) are slighly modified by random distribution of “ripples” (reflecting the existence of conformational substates in the proteic medium) around their parabolic envelope and do not change their curvature due to the ET process, one obtains (cf. eqs 12 and 14) X’
F , = ky ( x - a)’ F2 = k2 02
+ uJ1(x) + . J ~ ( x ) - AF
2x =k
where the “effective” free energy gap between the reactants’ and products’ states (defined as the energy difference between the bottoms of the parabolas) becomes a random variable. Our analysis shows that when applied to the theory of ET in proteic media, this feature would be responsible for an effective enhancement of a longitudinal relaxation time of the medium, which in turn may affect the nonadiabatic character of the processs6 (cf. eq 1 ) . On the other hand, the breakdown of the nonadiabatic limit in the presence of a relatively long relaxation time of the medium raises the question of a direct applicability of a conventional kinetic rate approach.57 Long medium dynamics of relaxation would not permit direct averaging of the FranckCondon factor, the effect which should be taken into account in contemporary theoretical models and MD evaluations of the overall kinetic rate in the system. In section 111 we have discussed a complementary model of dynamic configurational fluctuations in the system whose occurrence is possible on the same time scale as the time variations of the process of primary interest. To some extent, the model is
similar to the analysis of non-Markovian vibrational energy relaxation following the process of excitation in a quantum threelevel system studied by Bagchi and O ~ t o b y Our . ~ ~ “classical” p r e s e n t a t i ~ nwhich , ~ ~ differs entirely from the one used in ref 44, has mainly focused on derivation of a nonexponential decay rate of the excited state whose relaxation follows the path with possible configuration states. The time scaleof configurational variations has been assumed to be comparable with the time scale of the deexcitation leading to non-Markovian (memory-type) dynamics. As it stands, the model can easily be translated into a model of a fluctuating barrier (cf. Stein et al., ref 20) by rewriting eq 26 in the form
= -k,(cosh B - [ ( t ) sinh B)x
k, = Ke-A
- -
where x ( t ) stands for a symmetric dichotomous noise. In the limit of y m, A m, and A z / y = constant, the dichotomous process defined by eqs 28 and 29 tends to a Gaussian white n ~ i s e , ~producing ~,~’ effective Gaussian statistics of the fluctuating barrier. Such an interpretation is also close to the stochastic surface hopping model analyzed recently by Marchi et On the basis of their former MD studies,’ the authors proposed slow components of the energy gap fluctuations to be responsible for the observed nonexponential kinetics of the primary electron transfer in reaction centers. It has been discussed45that fluctuations of the slow energy gap components in a fairly linearly responding model can be viewed equivalently in terms of a Gaussian distribution of driving forces for the faster fluctuations. An additional comment should be added concerning types of probability distribution functions used in the models. Direct analysis of eq 6 displays interrelation between population-decay function and distribution of rate values (as studied, e.g., in section 11). A form of the kinetic rate distribution function can be deduced from the inverse Laplace transform of p(x,t). Obviously, a discrete distribution of k‘s would result in single- or multiexponential decay of P* (see also refs 42 and 33). More complex forms of the time decay can be inferred from such an analysis (a straightforward observation is that a power law distribution of p ( x , t ) a constant& would follow a uniform distribution of k ’ ~ l 3 - ~ ~ . 3If7 )experimentally , observed, some critical behavior in the decay analysis (eq 22) could be explained in terms of a specific (Poisson-like) distribution of the free energy fluctuations (Gaussian distributions yield a smooth temperature dependence of the rate; cf. eq 21). The implicit assumption for the existing ET models describing kinetics of primary charge separation in RCS“~is that configurational relaxation of the medium occurs on the subpicosecond time scale which, in the light of recent findings based on MD studies of polypeptides2’ and experimental observations of an ultrafast ET rate in RCs reported by Holzapfel et ~ 1 . :may ~ be an incorrect argument. Also, the protein medium by no means satisfies a condition of microscopic homogeneity. In the series of experiments,12,22referring to the short-time dynamics of proteins, 1 / T I electro-spin relaxation rates of the low-spin ferric ion in a number of heme and iron-sulfur proteins have been studied. Analysis of these experiments led to the conclusion that the structural excitations diffuse predominantly along pathways either defined by or identical with the backbone which is topologically one dimensional. The nature of the diffusional response, or more precisely the frequency dependence of the proton NMR relaxation rate l / T l , has been calculated12p22for a wide range of frequencies and temperatures. It has been shown that T I follows a power law distribution rather than a typical
Effects of Heterogeneity in Photosynthetic RCs exponential decay, ( T I ) ,= WC, with a crossover in the exponent between the high and low temperatures. The nonexponential time dependence of ligand binding and the low-temperature specific heat data12J9,20,22 have led to the suggestion that proteins are similar to glasses. An experimental Kohlrausch-Williams-Watts law has been found in studies of the broadening of relaxation processes in disordered materials.12-16 Its peculiarity lies in the special form of the response relaxation function, which in the case of dielectric response would result in a larger half-width in the imaginary part of the dielectric constant compared to the Debye process: p
We believe that studies of a similar type, especially the ones based on the observations of dielectric response of proteic media in RCs, would elucidate thecharacter and timescales of relaxation properties of the system. Our models show that the sequential protein relaxation can be considered within the framework of existing ET theories. However, complementary studies of conformational substates and their effects on calculated (in MD analysis) macromolecular free energy would be necessary to receive information on the precise nature of the process. In this context, one of the primary tasks of understanding the E T transport in RCs would rely on the analysis of basic features and underlying mechanisms of relaxation dynamics in proteins. The investigations of the electronic effects on the nuclear degrees of freedom and the dielectric response of the protein should be carried out in more detail. Acknowledgment. I acknowledge valuable discussions with J. Fajer, who brought my attention to electron-transfer problems in the reaction centers. I thank M. D. Newton for communicating the results of his paper prior to publication. This work was supported by the Division of Chemical Sciences, U.S.Department of Energy, under Contract DE-AC02-76CH00016 and by Grant 2.0387.91.01 from the Polish Government Project KBN. References and Notes (1) Vos, M. H.; Lambry, J.-C.; Robles, S.J.; Youvan, D. C.; Breton, J.; Martin, J. L.Proc. Narl. Acad. Sci. U S A . 1991,88,8885; 89,613. Du, M.; Rosentha1,S.J.;Xie,X.; DiMagno,T. J.;Schmidt, M.;Hansen,D,K.;Schiffer, M.; Norris, J. R.; Fleming, G. R. Proc. Narl. Acad. Sci. U.S.A. 1992, 89, 8517. Wang, Z.; Pearlstein, R. M.; Jia, Y.; Fleming, G. R.; Norris, J. R. Chem. Phys. 1993, 176,421. (21 Jean. J.: Friesner. R. A.: Fleming. G. R. Ber. Bunsen-Ges. Phvs. Chem.'1991,95,253. Jean: J.; Friesner, R.L.; Fleming, G. R.J. Chem. Pkys. 1992, 96, 5827. (3) Bixon, M.; Jortner, J.; Michel-Beyerle,M. E. Biochim. Biophys. Acta 1991, 1056, 301. (4) Norris, J. R. Isr. J . Chem. 1992, 32(4), 418. Friesner, R. A,; Won, Y. Biochim. Bioohvs. Acta 1989. 977, 99. ( 5 ) . Jia, Y.;'DiMagno, T. J.; Chan, C.; Wang, Z.; Du, M.; Hanson, D. K.; Schiffer, M.; Norris, J. R.; Fleming, G. R.; Popov, M. S. J. Phys. Chem. 1993, 97, 13180. (6) Small, G. J.; Hayes, J. M.; Silbey, R. J. J . Phys. Chem. 1992, 96, 7499. (7) Warshel, A.; Parson, W. W. Anu. Rev. Phys. Chem. 1991,42,279. Schulten, K.; Tesch, M. Chem. Phys. 1991,158,421. Nonella, M.; Schulten, K. J . Phys. Chem. 1991, 95, 2059. Treutlein, H.; Schulten, K.; Brunger, A.
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K. A,; Oesterhelt, D.; Feick, R.; Scheer, H.; Zinth, W. Biochim. Biophys. Acta 1993, 1142, 99. (44) Bagchi, B.; Oxtoby, D. J . Phys. Chem. 1982, 86, 2197. (45) Gehlen, J. N.; Marchi, M.; Chandler, D. Science 1994, 263, 499. (46) The energy gap function AE(1) (Le. the energy required for a virtual transfer of the electron) is a generalized microscopic “reaction coordinate” relevant for the electron-medium transfer coupling. In typical molecular dynamics studies,’ random occurrences of AE(t) follow Brownian dynamics. (47) In the original Zusman’s approach,27the reaction coordinate is defined as the time-dependent energy gap between the levels of a two-state system, cf. eq 2. This approach yields p as an explicit function of energy. Conversion to any generally chosen “reaction coordinate” y would require scaling of p according to p(x,t) dx = p ( y , t ) dy, which yields H R , = ~ 2nV/h(l/fyo) X [pR(y,r) - pp(y,t)], wheref = m ~ w *= 2X/yi, yo being the minimum of the reactants’ parabola. (48) Assumption of the overdamped dynamics is essential for the purpose of using the particular form of the kinetic rate (effective Kramers limit). The general variational state theory,” which is a suitable approach in the case of the cusplike potentials, predicts deviations from the rate (eq 7) beyond the strong-damping limit. (49) Solvent polarization massis related tothecurvatureofthewell through thespringconstantoL2 = k/mL = ((.k(t))2r2(r)), and thediffusioncoefficient
Gudowska-Nowak can be defined by use of the correlation function A(r) = (x2) -1(xx(t)), D = -(x*}&l(t)A(t) (cf. Marcus and Sutin3l and Carter andlynesm). (50) The formula (eq 7) is within 10% exact for barrier height @F(x*)2 5, whereF(x*) stands for the parabolic potentialvalueat the topof the barrier. For lower barriers, k, has to be corrected with a multiplying factor,)4 qqZ) &1 + erf x)21-1, = BF(~*), . . = (1 + erf L)[ze-~*pmdx (51) Note a general Eharacterbf x ( t ) which,.so far, has been defined as the reaction coordinate for the ET process; cf. eqs 2 and 3. (52) For a one-dimensional diffusion problem with random perturbations of the potential, the effective forms of the diffusion and potential have been postulated by Zwan~ig,’~ who studied the effect of “smoothing” over the randomness on the mean first passage time. All of the moments, generated ~ by the equation adjoint to eq 15, can be shown to be determined by D c and Fcffas defined above (cf. also a general formula (eq 23)). (53) We are reporting here results based on the formula for the characteristic function for the random telegraph signal as derived by Morita.“ More general derivations for various families of dichotomous processes can be found in papers by Fulinski,20 Kac,4I and Gaveau et ~ 1 . ~ ~ ~ 4 1 (54) In the calculations for Figures 1-3, k has been chosen as the roomtemperature ET rate of the wild type Rb. Capsulatus; cf. Wang et a/.’ (55) T , , ~has been chosen to be about 200 ps (cf. Wang et ~ 1 . 1 ) . (56) The criterion of adiabaticity requires that the energy uncertainty of the system in the mixing region (close to a transition point) is small compared to the splitting of energy levels within the region. For a diffusive motion along the reaction coordinate, the adiabaticity parameter can be estimated from the residence time in the transition region.28.3’ (57) A typical assignment for use of the rate theory is the assumption of excitations and relaxations of nuclear medium modes following a time scale faster than that of the electronic ET transfer. This situation calls for a perfect time scale separation in the system, the limit of which does not need to be achieved in slowly relaxing media. (58) The quantum version of our model would also lead to oscillatory relaxation of an excited state, as discussed in ref 44.
