Incoherent Oscillations Accompanying Charge Separation in Photosynthetic Reaction Centers

doi:10.4236/ojapps.2013.33B001

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Open Journal of Applied Sciences, 2013, 3, 1-5

http://dx.doi.org/10.4236/ojapps.2013.33B001 Published Online July 2013 (http://www.scirp.org/journal/ojapps)

Incoherent Oscillations Accompanying Charge Separation

in Photosynthetic Re ac tion Cent er s

A. G. Yakovlev1, V. A. Shuvalov1,2

1A.N. Belozersky Institute of Physico-Chemical Biology, Moscow State University, Mos c ow , Rus si a

2Institute of Basic Biological Problems, Russian Academy of Sciences, Moscow, Russia

Email: ya [email protected], [email protected]

Received May 2013

ABSTRACT

Early events of charge separation in reaction centers (RCs) of bacterial photosynthesis are modeled by kinetic equations

with time-dependent rate constants. An illustrative case of regular motion along a “slow” coordinates leading to oscilla-

tions in the kinetics is examined. Different schemes of charge separation are investigated. A good fitting of experimen-

tal kinetics of native Rba. sphaeroides RCs is achieved in the fiv e states model P*1BAHA ↔ P*2BAHA ↔ I ↔ P+

−

↔ P+BA

−

with two excited states

BAHA and

BAHA and three charge separated states I, P+

−

HA and P+BA

−

(P is a primary electron donor, bacteriochlorophyll dimer, BA and HA are an electron acceptor, monomeric bacterioch-

lorophyll and bacteriopheophytin in active A-branch, respectively). In the model only the first excited state is directly

populated by optical excitation. The emission of the two excited states is assumed to be at 905 and 940 nm, respectively.

The intermediate state I is assumed to absorb at 1020 nm as well as the P+

−

HA state. The model explains the deep

oscillations in the kinetics of the

1,2

stimulated emission and of the

−

absorption. In the simpler schemes without

the I state or with only one excited state the accordance with the experiment is achieved at unreal parameter values. A

possible nature of the I and

BAHA states and a possible incoherent nature of the oscillations are discussed.

Keywords: Photosynthesis; Charge Separation; Reaction Center; Electron Transfer

1. Introduction

In reaction centers (RCs ) of purple bacteria Rhodobacter

(Rba.) sphaeroides primary charge separation consisted

in electron is transferred from excited dimer of bacteri-

ochlorophyll P* to monomeric bacteriochlorophyll BA

within ~3 psec and from

−

to bacteriopheophytin HA

within ~1 psec at room temperature (for reviews see

[1,2]). Then electron is transferred to primary quinone

QA within ~200 psec. At cryogenic temperatures these

reactions are accelerated by 2 - 3 times. Excitation of

RCs by broadband femtosecond light pulses leads to os-

cillations in the kinetics of P* emission [3] and of

−

absorption [4] with a frequencies at 10 - 400 cm−1 (peak

at 130 - 150 cm−1). These oscillations are observed in

various native and mutant RCs in a wide range of tempe-

ratures. According to modern ideas, a vibrational or elec-

tronic (or bo th) coherence is a possible explanation of the

oscillatory phenomena in the kinetics of the excited and

charge separated states. A periodical motion of a vibra-

tional wavepacket [5] or a quantum beats between the

levels with close energies [6] can produce the damped

oscillations in the populations of the RC states. Analysis

of the coherent electron transfer was performed on the

base of Redfield theory in [7] and in the dispersed pola-

ron model [8]. The oscillations in the P* stimulated

emission band were theoretically studied by an approach

of single electronic transition coupled to one or two vi-

brational modes [9-11]. At room temperature the quan-

tum coherence should be destroyed very quickly due to

thermal motions [12]. On the other hand, the molecular

dynamics calculations revealed a number of classical

vibrational modes in RCs at 20 - 200 cm–1 [13]. These

modes can reflect the nuclear motions in protein matrix

as well as inside the RC pigments. A classical stochastic

Langevin equation was used to calculate the oscillatory

dynamics of the P* ↔ P+

−

Reaction controlled by

protein relaxation [14] in modified RCs with blocked

electron transfers to HA.

In the present work the incoherent charge separation

dynamics of native Rba. sphaeroides RCs is modeled by

the kinetic equations with time-dependent rate constants.

This approach is based on the Marcus theory [15]. An

aim of the work was to study a possibility of incoherent

origin of the oscillatory phenomena observed at early

A. G. YAKOVLEV, V. A. SHUVAL OV

times of charge separation.

2. Model

According to Marcus theory [15], electron or energy

transfer reaction occurs at the intersection of the potential

energy surfaces of the initial and final states of the sys-

tem. In harmonic approximation these surfaces are para-

boloids shifted one from another: Uin = 1/2mω2x2 +

1/2MΩ2X2; Ufin = 1/2mω2 (x – x0)2 + 1/2MΩ2(X - X0)2 +

ΔG. Here x and X are effective coordinates of fast (ther-

mal) and s low (relaxation) motion, respe ctively; m and M

are effective masses, ω and Ω are self-frequencies, x0 and

X0 are the potential surface displacements along the x and

X coordinates, ΔG is the free energy change of the reac-

tion. In the high-temperature limit, the rate constants of

the forward and backward reaction can be written as fo l-

lows:

( )( )

f0 bf

exp; exp.

KKEkT KKGkT=−=

Here the activation energy E = 1/4λ(1 + G/λ)2; the

energy gap G = ΔG + λ1 − 2 λ1X/X0; the fast reorganiza-

tion energy λ = 1/2m

; the slow reorganization energy

λ1 = 1/2MΩ2

; kB is the Boltzmann constant; T is a

temperature; K0 is a constant. After an averaging over the

fast coordinate x the dynamics of the energy gap G(t) and

of the activation energy E(t) is determined by the dy-

namics of the slow coordinate X(t). Thus, the rate con-

stants Kf,b are time-dependent in this approach. In our

model we examine a simplest illustrative case of damped

cosine dependence of X on time. We studied the schemes

of three (P*BAHA ↔ P+

−

HA ↔ P+BA

−

), four (

BAHA ↔

BAHA ↔ P+

−

HA ↔ P+BA

−

) and five

(

BAHA ↔

BAHA ↔ I ↔ P

−

HA ↔ P+BA

−

)

states. The coordinates of each reaction were assumed to

be independent. The incoherent dynamics of relative

populations of these states was described by kinetic equ-

ations.

3. Results and Discussion

3.1. The Modeled Kinetics

The results of the modeling of the five states scheme (Fi-

gure 1) are shown in Figure 2 for native RCs of Rba.

sphaeroides. The calculation parameters are as follows:

temperature Т = 90 K, energy difference

= 35 cm–1,

= 85 cm–1,

= −400 cm–1,

= −900 cm–1;

reorganization energy λ12 = 80 cm–1, λ23 = 100 cm–1, λ34 =

450 cm–1, λ45 = 1500 cm–1; reactio n rate

= 26 psec–1,

K = 34 psec–1, 0

K = 12 psec–1,

= 40 psec–1;

G12(t) = 5 0 cm–1 exp(–3.5t)(sin(30.5t − 1.2) + sin(25. 5 t −

1.2)); G23(t) = 60 cm–1exp(−2t)(sin(22.5t − 0.1) +

sin(27.5t − 0.1)) + 20 cm–1 exp(−9t); G34(t) = –600

cm−1exp(−5t) + 320 cm–1exp(−0.5t)sin(12t − 2.4); G45(t)

Figure 1. An illustrative scheme of the energy levels of the

states at zero oscillations.

Figure 2. The modeled kinetics of the relative populations of

the

BAHA,

BAHA, I, I + P+

B−

HA and P+BAA

−

states

of the native Rba. sphaeroides RCs. For details see the text.

= 0; time t is given in psec. The calculated kinetics of the

states are similar with the experimental ones at t > 150

fsec [2-4,16]. A decay of the P* stimulated emission at

905 and 940 nm (the

BAHA and

BAHA states in

the model, respectively) and a bleaching of the HA ab-

sorption band at 760 nm (the P+BA

−

state) occur

within ~1.5 psec in native RCs at 90 K. These processes

are accompanied by the formation of the

−

absorption

band at 1020 nm. The modeled sum of the I and P+

−

populations is close to the experimental kinetics of the

−

absorption band. This means that the I state absorb

at 1020 nm. The pronounced damped oscillations with

the ~220 fsec period are observed in the kinetics of the

P* stimulated emission and of the

−

absorption. The

main contribution to the

−

oscillations is provided by

the I state. The oscillatio ns are completely damped with-

in ~600 fsec. The P* oscillation s at 905 and 940 nm have

the opposite phases, while the

−

absorption oscillates

in phase with the P* oscillations at 940 nm.

3.2. The Parameters of the Calculations

The parameter values used in the model are in accor-

dance with the extensive theoretical and experimental

studies reviewed in [1,2,5,14]. According to different

estimations, in Rba. sphaeroides RCs the Р+A

B−HA and

P+ВА

−

energy levels are placed below the P*BAHA

energy level by 300 - 700 and 900 - 2000 cm–1, respec-

tively. In our model the energies of the

BAHA,

P*2BAHA and I states are different by less than 100 cm–1.

( 5)

( 4)

( 3)

( 2)

( 1)

0,0 0,4 0,8 1,2

0,0

0,4

0,8

Ι + P

Population, rel. un.

Time, ps

A. G. YAKOVLEV, V. A. SHUVAL OV

The rates of the forw ard and backward reactions between

these states oscillate with approximately opposite phases

(Figu r e 3 ). Such correlation between the dynamics of the

forward and backward reactions leads to the deep oscilla-

tions in the populations.

The molecular dynamics calculations estimate the re-

organization energy of the P*BA → Р+ВА– and Р*HA →

Р+HА– reactions by ~700 cm–1 [17] and ~2000 cm–1 [18],

respectively. In our model the fast reorganization energy

of the I → P+

−

HA (450 cm–1) and P+

−

HA →

P+BAHA– (1500 cm–1) reactions is much greater than that

of the

BAHA →

BAHA (80 cm–1) and

BAHA → I

(100 cm–1) reactions. This means the small displacement

between the potential surfaces of the

BAHA,

BAHA

and I states.

If we assume X/X0 = 1 for convenience, then the slow

reorganization energy λ1 = 25, 40, 460, 0 cm–1 for the

BAHA ↔

BAHA,

BAHA ↔ I, I ↔ P+A

B−HA and P+

−

HA ↔ P+BA

−

reactions, respectively. Thus, the

main part of the slow reorganization is corresponded to

the I ↔ P+A

B−HA reaction, and the noticeable part of this

reorganization (160 cm–1) is corresponded to the aperi-

odic motion.

In the model the values K0 = 26, 34, 12 and 40 cm–1

were used for the

BAHA ↔ P*2BAHA,

BAHA ↔ I, I

→ P+

−

HA and P+BA–HA → P+BA

−

reactions, respec-

tively. In the adiabatic approximation these K0 values are

corresponded to the effective frequency υ = 400 - 1300

cm–1. In the nonadiabatic approximation in high tempera-

ture limit these K0 values together with the λ values give

the electron coupling energy for the above mentioned

reactions V = 74, 89, 77 and 190 cm–1. The estimations of

the V value for the P*B ↔ P+

−

and P+

−

↔ P+

−

reactions are varied from 5.4 cm–1 [19] to 80 cm–1 [20]

and from 15 cm –1 [21] to 480 cm–1 [20], respectively.

The qualitative agreement of our model with the expe-

rimental data can be achieved in the wide range of the

parameters that indicates the stability of the model against

the parameter fluctuations. The two-fold change of the

Figure 3. The calculated dependences of the forward K12,

K23 and backward K21, K32 reaction rates (1–

BAHA, 2–

BAHA, 3–I) on time for the native Rba. sphaeroides RCs. For

details see the text.

energy differences, reaction rates or reorganization ener-

gies causes minor quantitative changes in the kinetics

shown in Figure 2.

3.3. Different Rea ct ion Schemes

Calculation shows that the simplest scheme of three

states P*BAHA ↔ P+

−

HA ↔ P+BA

−

produces more

smooth oscillations than it is in experiment. In this

scheme very large amplitude of the energetics changes

~1000 cm–1 is necessary to obtain the oscillation ampli-

tude comparable with experimental one. It is clear that

this scheme does not explain the out-of-phase oscillations

of the P* stimulated emission at 905 and 940 nm. These

out-of-phase oscillations can be explained by the scheme

of four states

BAHA ↔

BAHA ↔ P+A

B−HA ↔

P+BA

−

. In this scheme the oscillatory behavior of the

P+

−

HA state can be explained if to assume that the

energetics of the

BAHA ↔ P+A

B−HA reaction oscil-

lates with unreal amplitude ~1 400 cm–1 or that the P+

−

HA energy level is higher than the

BAHA and

BAHA levels. The last assumption contradicts to the

number of theoretical and experimental works [5,17,22,

23]. An insertion of the intermediate state I between the

BAHA and P+A

B−HA states helps to explain the oscil-

latory kinetics of the

−

absorption if to suppose that

the I energy is close to the

BAHA energy and that the I

state has spectral properties of the A

B−.

3.4. A Possible Na t ure of the

BAHA and I

States

In our model the

BAHA state is presumably associated

with the P* stimulated emission band at 940 nm. This

band is spectrally and temporally differ from the initially

excited P* emission band at ~905 nm (

in the model)

and is clearly observed in native and mutant RCs at room

and low temperatures [3,24]. In the experimental ΔA

spectra the 940-nm emission band forms a long-wave-

length tail of the broadband negative signal mainly con-

sisted of the P absorption band bleaching at ~870 nm.

The nature of 940-nm emission band is not well unders-

tood. The visible and IR transient spectroscopy indicates

that this band may be ascribed to the state with partial

charge separation inside the dimer P [24-26]. From the

other hand, the hole burning experiments at low temper-

atures show an absence of the P* conversion into another

state on a time scale much shorter than 1 psec [27,28].

The possible charge transfer character of the P* state

may be a result of the electron-spin density shift from PA

to PB in P* calculated by quantum-mechanical methods

[29]. The appearance of the delayed 940-nm feature in

the P* emission spectra may originates from vibrational

relaxation or electronic relaxation (or both) or fro m exci-

tation energy redistr ibution over vibrational modes in the

0,0 0,2 0,4 0,6

K, 1/psec

Time, psec

A. G. YAKOVLEV, V. A. SHUVAL OV

P* state. It is clear that further studies are need in this

question.

As it was mentioned above, the I state was introduced

into the model for better fitting of the experimental

−

absorption kinetics at 1020 nm. The population dynamics

of the I state is close to the oscillatory component of the

experimental

−

absorption band dynamics. The cal-

culated sum of the I and P+

−

HA populations is close to

the experimental A

B− kinetics. These results indicate

that the I state may contain

−

or ВА

δ–. One may spe-

culate that I is the state with charge transfer character

between P* and BA. To distinguish the I and P+A

B−HA

states in experiment is very difficult because of their

spectral and temporal closeness. No clear experimental

evidence of the I state existence is available now.

In our opinion, an advantage of the schemes with the

BAHA and I intermediates consists in the high rates of

the reactions in which these states are involved. The

BAHA and I states act as fast mediators between the in-

itially excited (

BAHA) and charge separated states. A

stabilization of electron occurs lately in the P+

H−

state

and partially in the P+

−

state.

3.5. A Nature of the Oscillations

In our model the oscillations in the populations are

caused by external modulation of the reaction energetics.

A nature of this modulation is beyond the scope of the

model. The nuclear motion (not only inside P) can be a

possible source of this modulation. Shortly after the ex-

citation of P by broadband femtosecond pulse the nuclear

motion has a coherent character. When the coherence is

rapidly damped this motion may continue incoherently

due to the nuclear inertia. One can speculate that the in-

coherent motion may exist a longer timescale than the

coherent one, but the latter produces the greater ampli-

tude of the oscillations. The observed oscillations may

reflect both the coherent and incoherent motion. The

presented model shows th at at t > 150 fsec the oscillatory

features observed in the kinetics of the excited and

charge separated states can be explained by incoherent

modula t ion of the reaction energetics.

4. Acknowledgements

We are grateful to L.G. Vasilieva and A.Ya. Shkuropatov

for preparing RC samples. We acknowledge the partial

financial support of the Russian Foundation for Basic

Research.

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