Photosynthesis Research

, Volume 97, Issue 3, pp 215–222

Energy migration as related to the mutual position and orientation of donor and acceptor molecules in LH1 and LH2 antenna complexes of purple bacteria

Authors

    • A.N. Belozersky Institute of Physico-Chemical BiologyM.V. Lomonosov Moscow State University
  • A. V. Rybina
    • Faculty of Bioengineering and BioinformaticsM.V. Lomonosov Moscow State University
Regular Paper

DOI: 10.1007/s11120-008-9318-x

Cite this article as:
Borisov, A.Y. & Rybina, A.V. Photosynth Res (2008) 97: 215. doi:10.1007/s11120-008-9318-x

Abstract

Many approaches to discovering the interaction energy of molecular transition dipoles use the well-known coefficient ξ(φ, ψ1ψ2) = (cos φ − 3 cos ψ1 cos ψ2)2, where φ, Ψ1, and Ψ2 are inter-dipole angles. Unfortunately, this formula often yields rather approximate results, in particular, when it is applied to closely positioned molecules. This problem is of great importance when dealing with energy migration in photosynthetic organisms, because the major part of excitation transfers in their chlorophyllous antenna proceed between closely positioned molecules. In this paper, the authors introduce corrected values of the orientation factor for several types of mutual orientation of molecules exchanging with electronic excitations for realistic ratios of dipole lengths and spacing. The corrected magnitudes of interaction energies of neighboring bacteriochlorophyll molecules in LH2 and LH1 light-absorbing complexes are calculated for the class of photosynthetic purple bacteria. Some advantageous factors are revealed in their mutual positions and orientations in vivo.

Keywords

Bacterial photosynthesisEnergy migrationPrecision of theory

Abbreviations

SEE

Singlet electronic excitation

/B/Chl

/Bacterio/chlorophyll

RC

Reaction center

B800, B850, B875

Light-harvesting BChl fractions, having absorption maxima at about 800, 850, and 875 nm, respectively

Introduction

Excitation migration often takes place in physics and photochemistry, but, most importantly, in various biological particles and tissues. In particular, in photosynthetic organisms, the migration of light-induced singlet electronic excitations (SEEs) take place between /bacterio/chlorophyll (/B/Chl) molecules in thin phospholipid–protein membranes, which incorporate so-called bulk or “antenna” /B/Chl and carotenoid ensembles and specialized reaction centers (RCs). The spatial arrangement of these pigments is provided by transmembrane polypeptide subunits (Brunisholz and Zuber 1992). The primary photophysical processes in membranes include the absorption of a wide spectral range of visible and near-infrared solar radiation by these pigments, the migration of SEEs thus formed down the energy gradient to the long-wavelength /B/Chl fractions, and from them to RCs. RCs trap incoming SEEs and convert them into the electrochemical energy of molecular redox pairs separated across the membrane (van Grondelle et al. 1994; Robert et al. 2003). In various photosynthetic organisms, one RC serves tens to hundreds of Chls and carotenoids. Therefore, it becomes a task of high importance to reveal detailed molecular mechanism(s) which is(are) responsible for efficient SEE delivery from antenna pigments to energy-converting RCs.

It was shown in a vast series of papers (Duysens 1951; Clayton 1966 were among the first) that, in photosynthesis, SEE migration proceeds via the lowest singlet excited states (S1*) of photosynthetic pigments. Many authors advanced arguments that SEE migration proceeds via a so-called “slow inductive resonance mechanism,” which was developed by Förster (1948) (see also the monograph of Agranovich and Galanin 1982). Later, it was proved that SEEs often form excitons delocalized over several closely positioned molecules (see the reviews of van Grondelle et al. 1994; Robert et al. 2003). However, so-called “bottlenecks” were discovered in plant and bacterial pigment complexes on SEE paths from antenna Chls and bacteriochlorophylls (BChls) to RCs, with the interchromophore spacing of up to 20–40 Å. This range of spacing is amenable to Förster’s migration mechanism. Bearing in mind the very sharp, R−6-dependence of migration rate constants on the inter-chromophore distance (Förster 1948; Agranovich and Galanin 1982), one can see that these bottlenecks are mostly responsible for the net quantum efficiency of SEE delivery from vast antenna pigments to RCs. For example, in BChl spectral fractions B850 and B875 of Rhodopseudomonas acidophila and Rhodospirillum molishianum, single SEE jumps between neighboring BChls proceed in about 0.1 ps, while between these fractions and from the latter to its RCs, it proceeds in about 5 and 50 ps, respectively (van Grondelle et al. 1994; Robert et al. 2003).

The precise microstructures of many antenna complexes were recently obtained with very high atomic resolution (Karrasch et al. 1995; Roszak et al. 2003; McDermott et al. 2002; Koepke et al. 1996; Papiz et al. 2003; Fenna et al. 1974; Kühlbrandt et al. 1994; Schubert et al. 1997; Liu et al. 2004; Jordan et al. 2001; Ben-Shem et al. 2003). In particular, mutual pigment positions were established in them with precision better than 1 Å. The positions of their transition dipoles were also precisely determined. Generally, quantum mechanics is based on the Coulomb interaction of individual charges. But we are concerned with the phenomenon of the intermolecular migration of SEEs, many theories, starting from classical Förster’ theory and several modern ones such as Redfield or incoherent exciton theories, often simplify this task: they suggest using Hamiltonian operators for S1* → S0 and S0 → S1* singlet transition dipoles of interacting molecules. The error thus involved depends on several factors, which were discussed by Chang (1977); Scholes (1996); Krueger et al. (1998), but it depends to a substantial extent on the ratio of the efficient lengths of donor and acceptor dipoles involved (pD and pA) to the distance between their centers (RD,A). Yet, in some monographs, the lengths of dye transition dipoles are accepted to be on the order of 1 Å (for example, in Chla and BChla, the transition dipole p ≈ 5 Debye was associated with the oscillating charge of one electron on the base of 1 Å). However, the quantum-mechanical solutions have proved that eigenfunctions of excited electrons of typical dye molecules expand over their π-electronic networks. Therefore, in chlorophylls, the efficient length of their S0 → S1* transition dipoles may be approximately related to the opposite tetrapyrol rings and recognized as 4–5 Å. Related to this, the spacing between neighboring BChl chromophores of LH1 and LH2 complexes of about 9–10 Å is too short for the precise use of Förster’s theory.

However, the posterior analysis has proved (Scholes et al. 2001; Kenkre and Knox 1974; Rahman et al. 1979) that, even in time intervals comparable with that of the Frank-Condon intervals, the application of Förster’s theory is justified, and more so that the spectra of “hot” fluorescence are now available for many molecules and they are not very blue-shifted relative to the quasi-equilibrated intervals. The minimal values of SEE jump times which fit Förster’s theory were estimated by Kenkre and Knox to be about 10−13 s for Chla and possibly slightly shorter for BChla (Kenkre and Knox 1974).

So, one may conclude that the limiting inter-molecular distance should be about 10 Å for BChls in vivo, having long wavelength absorption borders at about 900 nm. It is important to note that, even in RCs of purple bacteria, the Dexter-type of orbit overlapping contribution between closely positioned H870 cules was estimated by Jordanides et al. (2001) and some other researchers to be small as that related to their Coulomb coupling. An inter-BChl spacing of about 10 Å is often present in photosynthetic complexes, in particular, in antenna systems and RCs of purple bacteria. It is almost on the same order as the dimensions of π-electronic clouds of BChl chromophores. However, the precision of Förster’s theory suffers substantially if it is applied to ensembles with closely positioned dye molecules. This situation was analyzed in several publications (Chang 1977; Scholes 1996; Krueger et al. 1998; Beenken and Pullerits 2004; Madjet et al. 2006). In Chang (1977), the author suggested the introduction of a correction to Förster’s approach: to proceed to the monopole model in the calculation of the interaction between the transition dipoles of donor and acceptor molecules. In more precise approaches developed by Scholes (1996) and Krueger et al. (1998), the transition-density cube method was suggested, as well as the generalization of Coulomb interactions by including the contribution of multipole interactions. However, Scholes (1996) demonstrated that even nearly forbidden singlet excited states may contribute to intermolecular excitation transfers. Later, these theories were further generalized by Beenken and Pullerits (2004) and Madjet et al. (2006). Yet, the most important problems in this structural division are how to account for the local micro dielectric parameters, especially in cases with some mobile atoms and atomic groups having non-compensated charges, and whether the crystallized structures of bioparticles do not differ from their in vivo structures.

On the background of these circumstantial theoretical developments, our task pursued rather particular goals. It is well documented for many photosynthetic organisms that the transition dipoles of Chls and BChls form nearly planar arrays in their corresponding membranes (Koepke et al. 1996; Verméglio and Joliot 2002; Hu et al. 1998). For these reasons, our analysis is confined to the 2D dipole models. In order to make this analysis more vivid and pictorial to experimentalists, we used a simple monopole approximation, which apparently enables one to calculate the main part of the correction needed for the Förster-type approach to energy migration in/between the antenna complexes of purple bacteria. Then, we analyzed the precision of excitation migration at very short inter-molecular distances in two-dimensional dye complexes, in particular, in LH2 light-harvesting bacteriochlorophyll complexes of purple bacteria.

The problem studied was whether a simplified approach on the basis of Coulomb’s law may yield deviations from the classical formulae (Eq. 2) not far from those derived from more precise models (Scholes 1996; Krueger et al. 1998).

Some corrections specific to 2-D dipole–dipole interactions

In the case of SEE migration between closely positioned molecules, one should estimate the error thus induced, but a better method is to introduce correction (Chang 1977; Scholes 1996; Krueger et al. 1998), which may be precise, provided that one knows the real spacing and mutual position of donor and acceptor dipoles. According to general theory, if two dipoles fixed at a distance RD,A much greater than the dipole lengths (RD,A ≫ d), their interaction energy is equal to (Förster 1948; Agranovich and Galanin 1982):
$$ W_{{\operatorname{int} }} \approx \frac{{{\left( {{\mathbf{p}}_{{\text{D}}} \cdot {\mathbf{p}}_{{\text{A}}} } \right)}}} {{{\mathbf{R}}^{3}_{{{\text{D}},{\text{A}}}} }} - \frac{{3{\left( {{\mathbf{p}}_{{\text{D}}} \cdot {\mathbf{R}}_{{{\text{D}},{\text{A}}}} } \right)}{\left( {{\mathbf{p}}_{{\text{A}}} \cdot {\mathbf{R}}_{{{\text{D}},{\text{A}}}} } \right)}}} {{{\mathbf{R}}^{5}_{{{\text{D}},{\text{A}}}} }} = \frac{{{\mathbf{p}}^{2} {\mathbf{\xi }}^{2} }} {{{\mathbf{R}}^{3}_{\text{D},\text{A}} }} $$
(1)
where pD = pA = p is the vector magnitude of the transition dipole, RD,A is the vector which ranges between the centers of the donor and acceptor dipoles, ξ2 is the factor of mutual orientation of these dipoles. The value of ξ was derived as (Förster 1948; Agranovich and Galanin 1982):
$$ \xi {\left( {\varphi ,\;\psi _{1} ,\;\psi _{2} } \right)} = \cos \varphi - 3\cos \psi _{1} \cos \psi _{2} $$
(2)
where φ is the angle between interacting dipoles and ψ1 and ψ2 are the angles between these dipoles and the line connecting their centers.

In many dye ensembles, the condition (RD,A ≫ d) is not fulfilled. This problem is especially important for various chlorophyll and bacteriochlorophyll complexes of photosynthetic organisms.

Corrections for interactions of planar dipoles

It was well documented for many photosynthetic organisms that the transition dipoles of most Chls and BChls form nearly planar arrays in their corresponding membranes (Koepke et al. 1996; Madjet et al. 2006; Abdourakhmanov et al. 1979). For simplicity, we operate with similar dipoles, such that transition dipoles in various spectral fractions of Chls and BChls are indeed very alike.

It is evident that most of the migration acts take place between dye molecules having their dipoles close to each other and in advantageous mutual orientations. Figure 1a shows three extreme positions of two dipoles: symmetrical parallel (SPD) with ξ(φ, ψ1ψ2) = ±1.0, coaxial dipoles (CD) with ξ(φ, ψ1ψ2) = ±2, and orthogonal dipoles (OD) with ξ(φ, ψ1ψ2) = 0. It may seem from these examples that parallel dipoles are the most advantageous for molecular interaction and, hence, for SEE migration. However, the situation is not so simple. Below, several modes of planar dipoles are analyzed.
https://static-content.springer.com/image/art%3A10.1007%2Fs11120-008-9318-x/MediaObjects/11120_2008_9318_Fig1_HTML.gif
Fig. 1(a, b)

Mutual positions of transition dipoles in a plane. (a) Three extreme positions of two interacting dipoles: symmetrical parallel dipoles (SPD), coaxial dipoles (CD) and orthogonal dipoles (OD). (b) Donor dipole (dD) is positioned in the center of a circle. It makes an angle ψ with the line connecting its and the acceptor dipole’s (dA) centers. Angle φ was scanned from 0° to 90°. The acceptor dipole was shifted parallel to itself along the circle trajectory so that the distance R = RD,A between the centers of the two dipoles remained constant. Both dipoles have equal lengths and equal charges (±q) fixed in points 1, 2 and 3, 4 at the dipole ends, respectively

Parallel dipoles (φ = 0°, Fig. 1b)

The magnitudes of dipole interaction energies, PDΣWint in the monopole approach are presented in Fig. 2a as the function of angle ψ for realistic d/R ratios equal to 0.2, 0.3, 0.4, and 0.5. They are presented in the scale in which 1.0 correspond to Wint = q2R, where ε is the coefficient of dielectric permeability of the host media. Its extreme magnitudes for ψ = 90° and ψ = 0° apparently correspond to SPD and CD dipole mutual positions, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs11120-008-9318-x/MediaObjects/11120_2008_9318_Fig2_HTML.gif
Fig. 2(a, b)

The values of interaction energies, Wint of the two equal dipoles. The numbers on the curves stand for the d/R ratios. Interaction energies are expressed in ±Wint units (see text). (a) Two parallel dipoles and (b) Two orthogonal dipoles, both of which are in the same plane

Assume that, in Fig. 1b, the acceptor dipole was moved from infinity to its present position. Then, the work needed is equal to that for overcoming the repulsion between charges 1 and 3, 2 and 4 minus the attraction between charges 1 and 4 and 2 and 3. The interaction energy between two point charges is known to be ±Wint = q2R. Thus, it will be:
$$ {}^{{{\text{PD}}}}{\sum {W_{{\operatorname{int} }} } } = {q^{2} } \mathord{\left/ {\vphantom {{q^{2} } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon {\left( { - 1 \mathord{\left/ {\vphantom {1 {R_{{1,\;3}} }}} \right. \kern-\nulldelimiterspace} {R_{{1,\;3}} } - 1 \mathord{\left/ {\vphantom {1 {R_{{2,\;4}} }}} \right. \kern-\nulldelimiterspace} {R_{{2,\;4}} } + 1 \mathord{\left/ {\vphantom {1 {R_{{1,\;4}} + 1 \mathord{\left/ {\vphantom {1 {R_{{2,\;3}} }}} \right. \kern-\nulldelimiterspace} {R_{{2,\;3}} }}}} \right. \kern-\nulldelimiterspace} {R_{{1,\;4}} + 1 \mathord{\left/ {\vphantom {1 {R_{{2,\;3}} }}} \right. \kern-\nulldelimiterspace} {R_{{2,\;3}} }}} \right)} $$
(3)
It should be noted that the equations below were obtained under the assumption that the host media has homogeneous dielectric properties and that no energetic disorder is present. In the case of parallel dipoles, four inter-charge distances may be expressed by the following expressions:
$$ L_{{1,\;3}} = L_{{2,\;4}} = R $$
$$ L_{{1,\;4}} \approx {\left( {R^{2} + d^{2} + 2Rd\cos \psi } \right)}^{{1/2}} = R{\left( {1 + {d^{2} }/ {R^{2} } + {2d}/{R\cos \psi }} \right)}^{{1/2}} $$
$$ L_{{2,\;3}} \approx {\left( {R^{2} + d^{2} - 2Rd\cos \psi } \right)}^{{1 / 2}} = R{\left( {1 + {d^{2} } / {R^{2} } - {2d}/ {R\cos \psi }} \right)}^{{1 /2}} $$

Let us determine the values of ΣWint for several particular cases.

Coaxial dipoles (φ = ψ = 0°)

Obtain by means of elementary algebraic transformations:
$$ \begin{aligned} {}^{{{\text{CD}}}}{\sum {W_{{\operatorname{int} }} } } & = {q^{2} } \mathord{\left/ {\vphantom {{q^{2} } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon {\left( {1 \mathord{\left/ {\vphantom {1 {L_{{1,\;3}} }}} \right. \kern-\nulldelimiterspace} {L_{{1,\;3}} } + 1 \mathord{\left/ {\vphantom {1 {L_{{2,\;4}} }}} \right. \kern-\nulldelimiterspace} {L_{{2,\;4}} } - 1 \mathord{\left/ {\vphantom {1 {L_{{1,\;4}} }}} \right. \kern-\nulldelimiterspace} {L_{{1,\;4}} } - 1 \mathord{\left/ {\vphantom {1 {L_{{2,\;3}} }}} \right. \kern-\nulldelimiterspace} {L_{{2,\;3}} }} \right)}\\ & = {q^{2} } \mathord{\left/ {\vphantom {{q^{2} } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon {\left[ {{\left( {R - d} \right)}^{{ - 1}} + {\left( {R + d} \right)}^{{ - 1}} - 2 \mathord{\left/ {\vphantom {2 R}} \right. \kern-\nulldelimiterspace} R} \right]} = {{\left( {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {\varepsilon R}}} \right. \kern-\nulldelimiterspace} {\varepsilon R}} \right)}} \mathord{\left/ {\vphantom {{{\left( {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {\varepsilon R}}} \right. \kern-\nulldelimiterspace} {\varepsilon R}} \right)}} {{\left[ {{2d^{2} } \mathord{\left/ {\vphantom {{2d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }{\left( {1 - {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }} \right)}} \right]}}}} \right. \kern-\nulldelimiterspace} {{\left[ {{2d^{2} } \mathord{\left/ {\vphantom {{2d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }{\left( {1 - {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }} \right)}} \right]}} \\ & = {}^{ \pm }{\sum {W_{{\operatorname{int} }} {2d^{2} } \mathord{\left/ {\vphantom {{2d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }{\left( {1 - {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }} \right)}} } \\ \end{aligned} $$
(4a)
where ±Wint = q2R is the interaction energy of +q and −q point charges at distance R. If d/R ≪ 1.0, the dipole interaction aspires to the generally used formulae,
$$ {}^{{{\text{CD}}}}{\sum {W_{{\operatorname{int} }} } } = {}^{ \pm }W_{{\operatorname{int} }} {2d^{2} } \mathord{\left/ {\vphantom {{2d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} } $$
(4b)
in which ξ = 2. The multiplier 1/(1 − d2/R2) in Eq. 4a is >1.0 and increases as d/R increases.

Symmetric parallel dipoles (φ = 0°, ψ = 90°)

The corresponding formula was also derived algebraically as:
$$ {\sum {W_{{\operatorname{int} }} } } = \frac{{q^{2} }} {\varepsilon }{\left( {\frac{2} {R} - \frac{2} {{{\left( {R^{2} + d^{2} } \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right)} = \frac{{2q^{2} }} {{\varepsilon R}}{\left( {\frac{{{\left( {1 + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - 1}} {{{\left( {1 + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {R^{2} }}} \right. \kern-\nulldelimiterspace} {R^{2} }} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right)} $$
and after factorizing (1 + d2/R2)1/2 into the polinomial (1 + ax + bx2 + cx3 + dx4 + ···), we obtain:
$$ {\sum {W_{{\operatorname{int} }} } } = \frac{{q^{2} d^{2} }} {{\varepsilon R^{3} }} = \frac{{1 - {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {4R^{2} }}} \right. \kern-\nulldelimiterspace} {4R^{2} } - {d^{4} } \mathord{\left/ {\vphantom {{d^{4} } {8R^{4} }}} \right. \kern-\nulldelimiterspace} {8R^{4} }}} {{1 + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } {2R^{2} }}} \right. \kern-\nulldelimiterspace} {2R^{2} } - {d^{4} } \mathord{\left/ {\vphantom {{d^{4} } {8R^{4} }}} \right. \kern-\nulldelimiterspace} {8R^{4} }}} $$
(5a)
For d/R ≪ 1.0, this approximates to:
$$ {}^{{{\text{SPD}}}}{\sum {W_{{\operatorname{int} }} } } \approx {{}^{ \pm }W_{{\operatorname{int} }} d^{2} } \mathord{\left/ {\vphantom {{{}^{ \pm }W_{{\operatorname{int} }} d^{2} } {\varepsilon R^{2} }}} \right. \kern-\nulldelimiterspace} {\varepsilon R^{2} } $$
(5b)
in which ξ = 1. Contrary to Eq. 4a for coaxial dipoles, the multiplier which accompanies the core function ±Wintd2R3 in Eq. 5a is smaller than 1.0 and decreases as d/R increases.
It is seen from Eqs. 4a and 5a that, if the condition R ≫ d is fulfilled, the interaction energy of two ±q charges exceeds greatly of that between two dipoles. For example, for R = 10d, we obtain:
$$ \begin{aligned} & {}^{{{\text{CD}}}}{\sum {W_{{\operatorname{int} }} } } \approx {{}^{ \pm }W_{{\operatorname{int} }} } \mathord{\left/ {\vphantom {{{}^{ \pm }W_{{\operatorname{int} }} } {50}}} \right. \kern-\nulldelimiterspace} {50}\\ & {\text{and}} \\ & {}^{{{\text{SPD}}}}{\sum {W_{{\operatorname{int} }} } } \approx {{}^{ \pm }W_{{\operatorname{int} }} } \mathord{\left/ {\vphantom {{{}^{ \pm }W_{{\operatorname{int} }} } {100}}} \right. \kern-\nulldelimiterspace} {100} \\ \end{aligned} $$

Orthogonal dipoles (φ = 90°, ψ1 = 90° − ψ2 in Fig. 1a)

Assume that the acceptor dipole is orthogonal to the donor dipole. Then, we obtain:
$$ L{\left( {1,\;3} \right)} = {\left[ {{\left( {R\cos \psi _{1} - d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} + {\left( {R\sin \psi _{1} - d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} } \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$
$$ L{\left( {2,\;4} \right)} = {\left[ {{\left( {R\cos \psi _{1} + d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} + {\left( {R\sin \psi _{1} + d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} } \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$
$$ L{\left( {1,\;4} \right)} = {\left[ {{\left( {R\cos \psi _{1} - d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} + {\left( {R\sin \psi _{1} + d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} } \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$
$$ L{\left( {2,\;3} \right)} = {\left[ {{\left( {R\cos \psi _{1} + d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} + {\left( {R\sin \psi _{1} - d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{2} } \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$
$$ \begin{aligned} {}^{{{\text{OD}}}}W_{{\operatorname{int} }} & = {\left( {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon } \right)}{\left( {1 \mathord{\left/ {\vphantom {1 {L_{{1,\;3}} }}} \right. \kern-\nulldelimiterspace} {L_{{1,\;3}} } + 1 \mathord{\left/ {\vphantom {1 {L_{{2,\;4}} }}} \right. \kern-\nulldelimiterspace} {L_{{2,\;4}} } - 1 \mathord{\left/ {\vphantom {1 {L_{{1,\;4}} }}} \right. \kern-\nulldelimiterspace} {L_{{1,\;4}} } - 1 \mathord{\left/ {\vphantom {1 {L_{{2,\;3}} }}} \right. \kern-\nulldelimiterspace} {L_{{2,\;3}} }} \right)} \\ & = {\left( {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } \varepsilon }} \right. \kern-\nulldelimiterspace} \varepsilon } \right)} [{\left( {R^{2} + Rd{\left( {\cos \psi - \sin \psi } \right)} + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ & \quad + {\left( {R^{2} - Rd{\left( {\cos \psi - \sin \psi } \right)} + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ & \quad - {\left( {R^{2} + Rd{\left( {\cos \psi + \sin \psi } \right)} + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ & \quad - {\left( {R^{2} - Rd{\left( {\cos \psi + \sin \psi } \right)} + {d^{2} } \mathord{\left/ {\vphantom {{d^{2} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)}^{{ - 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}] \\ \end{aligned} $$
(6)

The modules of ODWint magnitude expressed in ±Wint units are shown in Fig. 2b as the function of angle ψ for four d/R ratios.

Examination of the data in Fig. 2a, b leads to some particular conclusions:
  1. 1.

    In the case of parallel dipoles, their interaction energy PDΣWint is small in the vicinity of ψ ≈ 35°, where it reaches zero. Thus, often, parallel dipoles are not optimal for excitation migration. At the angles 0° ≤ ψ < 34° 40′, the corrected values of PDΣWint are smaller, while at the angles 34° 40′ < ψ ≤ 90°, they exceed those calculated with the widely used formula in Eq. 2.

     
  2. 2.

    Within 25° < ψ < 60°, the interaction energies of orthogonal dipoles (Fig. 2b) exceed those of parallel dipoles.

     
  3. 3.

    In many real cases with d/R ≥ 0.4, the dipole interaction energy is not negligible as compared with that for +q and −q interacting point charges.

     
Table 1 demonstrates the corrections for ξ(φ, ψ1ψ2) values.
Table 1

Comparison of the modulus of the dipole interaction energies |±Wint|, calculated with the aid of the classical formula (Eq. 2; upper lines) and those calculated with the aid of the correcting formula (Eqs. 4a, 5a, and 6; lower lines)

 

Type of mutual orientation

Coaxial dipoles

Symmetric parallel dipoles

Orthogonal dipoles

d/R

Angle φ

90°

 

Angle ψ

90°

45°

0.3

Widely used (2)

0.18

0.09

0.135

Corrected (6–8)

0.198 (+10%)

0.084 (−5.55%)

0.138 (+2.2%)

0.4

Widely used (2)

0.32

0.16

0.240

Corrected (6–8)

0.38 (+18.7%)

0.143 (−9.0%)

0.249 (+3.75%)

0.5

Widely used (2)

0.50a

0.25a

0.375

Corrected (6–8)

0.667 (+33.3%)

0.212 (−15.5%)

0.400 (+6.7%)

Wint are expressed in the units of interaction energy ±Wint of two ± unit point charges at the distance of 1 Å. The upper lines were calculated with the widely used formula (Eq. 2) and the lower lines with the corrected formula. Their differences are given in brackets in percentages

aThese two entries correspond to ξ(φ, ψ) factors 2.0 and 1.0, respectively

Note, d/R ≈ 0.5 is very typical for neighboring BChls and Chls in light-harvesting complexes of photosynthetic organisms. For d/R = 0.5, the interaction energy for the OD-type of dipole mutual position is equal to 0.375, i.e., 1.5-times greater than that for SPD position (0.25), if calculated with the widely used formula (Eq. 2). But their ratio increases by up to 1.9-fold, provided one proceeds to the corrected values in Fig. 2 (0.40 and 0.211). The maximal value of corrected Wint for coaxial dipoles (0.667) exceeds more than three times that for symmetrical parallel dipoles (0.212).

Our corrected values are in good accordance with those estimated with the same point dipole approach in Chang (1977) and are in semi-quantitative accordance with the corresponding magnitudes obtained with the more precise transition density cube method in the work by Krueger et al. (1998). Taking into account the dependence of Förster’s rate constants on ±Wint in the second power (Förster 1948; Agranovich and Galanin 1982), the above differences may develop to be much stronger. When d/R increases, the dipole–dipole interaction energy increases greater in cases of coaxial and orthogonal dipoles, as compared with its increase obtained with the aid of the widely used formula (Eq. 2). In the case of symmetric parallel dipoles, this increase is much weaker.

Bacteriochlorophyll Coulomb interactions in LH1 and LH2 light-absorbing complexes of photosynthetic purple bacteria

Application of an X-ray method to crystallized antennae microparticles enabled Karrasch et al. (1995), Roszak et al. (2003), McDermott et al. (2002), Koepke et al. (1996), and Papiz et al. (2003) to reveal the fine structures of LH1 and LH2 BChl-protein complexes of two purple bacteria. The spectral hierarchy of the BChl molecules that they contain furnishes a cascade-like system of excited states that funnels SEEs from outer LH2 through LH1 to the RC special pairs. The mutual positions of B875, B850, and B800 transition dipoles (875, 850, and 800 stands for the approximate positions of the absorption peaks of these BChls) are shown in Fig. 3.
https://static-content.springer.com/image/art%3A10.1007%2Fs11120-008-9318-x/MediaObjects/11120_2008_9318_Fig3_HTML.gif
Fig. 3

Simplified 2-D model showing the mutual orientations and positions of BChl-a transition dipoles in typical purple bacteria. Segments 1 and 2 stand for the transition moments of B875s and B850s (B800s are not shown). Segment 3 stands for the P870 special pair in RC. The black (4) and white (5) circles represent cross-sections of transmembrane α/β-protein helices which carry BChls

The co-ordinates of the atoms which govern the directions and mutual positions of the transition dipoles of three neighboring B850 molecules of LH2 are presented in Table 2. The data in this table enabled us to reveal: (1) the angles which govern factors of mutual orientation of the transition dipoles of three neighboring B850 molecules in the LH2 circle; (2) their interaction energies in monopole approximation (see Table 3); (3) that the Z-coordinates of all Mg atoms are the same within the error, thus, the B850 circle is, indeed, parallel to the membrane plane; (4) in the LH2 structure, the Mg-atoms of β-850 molecules stick out by 0.2 Ǻ from the centers of their tetrapyrol rings; (5) the transition dipoles of B850 molecules are inclined by 2.8 ± 0.2° to z-plane, thus, the 2D-approximation is quite reasonable here.
Table 2

Mutual positions of central magnesium (Mg) and four nitrogen (N-1, N-2, N-3, N-4) atoms of tetrapyrol rings of three subsequent B850 molecules of Rhodopseudomonas acidophila

Atom

β-B8501

A-Β8501

β-8502

X

Y

Z

X

Y

Z

X

Y

Z

Mg

0.61

27.15

18.20

9.35

24.13

18.16

17.92

20.41

18.20

N-1

−1.25

27.88

17.89

11.13

22.91

17.87

16.96

22.16

17.89

N-3

2.17

25.91

18.40

8.19

25.55

18.41

18.31

18.45

18.40

N-1 and N-3 stand for the nitrogen atom of the first and third pyrol rings of B850 molecules, respectively. They specify the directions of transition dipoles. All digits are in Ǻ. The Z-axis is perpendicular to the membrane plane. Digits under β-8501 and α-Β8501 are within the same α/β-pair. Digits under β-8502 are between two α/β-pairs. All co-ordinates were borrowed from the Brookhaven protein bank (http://www.rcsb.org/pdb/, identification number 1LGH

Table 3

B850s parameters calculated on the basis of the data presented in Table 2

Interacting B850 molecules

Spacing LD,A (Ǻ)

d/LD,A

ξ(φ, ψ1ψ2)

Wcorrect

Β-8501–α-B8501

9.55

0.482

1.58

0.405

Α-8501–β-B8502

9.15

0.502

1.20

0.303

Β-8501–β-8502

18.6

0.247

1.13

0.069

The dipole length d was assumed to be 4.6 Ǻ

Wcorrect = corrected values of interaction energy expressed in the units ±Wint; d/LD,A = the ratios in first two lines are close to 0.5, therefore, their Wcorrectvalues are in the same scale as Wint in Table 1 and in Fig. 2

The magnitudes of the dipole–dipole interaction energies presented in Table 3 are close to those determined with a more precise transition density method in the work of Krueger et al. (1998). Thus, the simple approach of point charge dipoles enables the experimentalist to reveal semi-quantitatively the sign and magnitude of the difference between the real value of the dipole–dipole interaction energy and its value calculated with the widely used formula (Eq. 2).

Discussion

It may seem strange that mutual orientations of neighboring molecules B850 in LH2 are more advantageous for SEE migration along their circles than between closest B850 and B875 molecules. In fact, the value of SEE jump-time between closest B850 and B875 molecules is about 0.05 ps (Koolhaus et al. 1998; Jimenez et al. 1997; Monshouwer et al. 1998), while between B850s and B875s, it was about 3–5 ps (Pullerits and Sundström 1996; Yang et al. 2001). The latter is evidently the narrow site in this part of the SEE migration route. Assume temporarily that B850s and B875s transition dipoles are turned by about 80° to the radial directions in their rings. Then, the closest B850 (in the LH2 complex) and B875 (in the LH1 complex) dipoles fall within the CD type of mutual arrangement. The distance between the closest B850 and B875 dipoles in vivo is about 25 Å (Chachisvilis et al. 1997). So, the interaction energy in this imaginable CD configuration should be about twice as high as that in the native configuration, provided that one calculates the interactions in a monomer presentation.

But such a conclusion is not correct. In real mutual orientations of dipoles in the LH2 and LH1 complexes, one should take into consideration the effect of excitons delocalized over several BChls. It stimulates greatly SEE migration in the above cited narrow sites (see Hu et al. 1998; Koolhaus et al. 1998; Jimenez et al. 1997; Monshouwer et al. 1998). The interaction energies between neighboring B850s of about 280–370 cm−1 (Krueger et al. 1998; Pullerits and Sundström 1996) cause SEE delocalization in 4–6 BChl molecules (Hu et al. 1998; Koolhaus et al. 1998; Jimenez et al. 1997; Monshouwer et al. 1998; Pullerits and Sundström 1996). The efficient dipole strength of these delocalized excitons in B850 and B875 absorption bands does not differ greatly from the vector sum of dipoles of separate BChls involved in such an interaction. Thus, one may consider a group of several excitonically coupled molecules as a single super-molecule with greater dipole moment. For two to three α/β-pairs of B850s (or B875s), this sum may comprise about 3.7–5.2 times the value of the dipole moment of BChl monomer. According to general theory, the rate constant of inter-molecular SEE migration is proportional to the operator of the product of the dipole strengths of the donor and acceptor molecules. Thus, for excitation migration between B850 and B875 molecules, the factor of energy coupling turns out to be greater than that estimated for the monomer model with BChls arranged in the even more advantageous CD configuration.

This is indeed an impressive achievement of natural photosynthesis. Let us compare it with SEE migration in the preceding B800* → B850 stage in the same purple bacterium. Bearing in mind the rather long B800–B850 interdipole distances (≥17.6 Ǻ), the use of the classic formula (Eq. 2) for ξ(ψ, θ1, θ2)2 looks reasonable. The mean value of the factors of mutual orientation of B800 dipole and those of the three closest B850 molecules was calculated in Bahatyreva et al. (2004) to be about 0.58, i.e., it is even smaller than the averaged value of 2/3 for chaotic molecular ensembles in Förster’s theory. It apparently proves that, contrary to the B850* → B875 stage, there was no tendency for improvement of the migration factors.

Thus, one may conclude that, in the course of the evolution of purple bacteria, nature has developed a progressive microstructure which ensures higher rates of excitation migration and, hence, higher values of quantum efficiency for energy delivery from vast B800–B850 light-harvesting antenna to the main B875 molecules. This exciton effect is also at work in the terminal migration stage between B875 molecules and the special pairs of energy-converting reaction centers.

Acknowledgment

AYB is grateful to the Russian Government Foundation for Leading Scientific Schools, grant 1710.2005.04, for the financial support.

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© Springer Science+Business Media B.V. 2008