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* Environmental Biology Group, Research School of Biological Sciences, Australian National University, Canberra, Australia; and Institut für Botanik, Westfälische Wilhelms-Universität Münster, Münster, Germany
Correspondence: Address reprint requests to Ira Tremmel, E-mail: ira{at}germanynet.de.
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ABSTRACT |
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INTRODUCTION |
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; Kirchhoff et al., 2002; Tremmel et al., 2003) plastoquinone diffusion may be severely restricted, depending on the organization of the integral proteins. However, the exact nature of the protein organization is still not understood. Lavergne and co-workers (1992) assume a random organization. Kirchhoff and co-workers (2000) suggest that the protein distribution is more ordered because of protein-protein interactions.
In an earlier work (Tremmel et al., 2003), we investigated PQ diffusion in thylakoids using a Monte Carlo simulation. Assuming a random protein distribution it was found that a considerable proportion of binding sites for electron transfer is obstructed by proteins. Furthermore, the area occupied by integral proteins is close to the percolation threshold beyond which the obstacles form closed domains within which PQ can diffuse freely but cannot leave—a situation that could be wasteful if PQ has no access to its binding sites. Slight changes in the protein arrangement lead to pronounced changes in diffusion behavior under such conditions. Therefore, factors were investigated that may influence the protein arrangement and, hence, PQ diffusion. It was found that boundary lipids and the mobility of the integral proteins may play an important role in PQ migration. However, the model did not include protein-protein interactions. Therefore, in our current work the model was extended to account for protein-protein interactions, and the impact of such interactions on the arrangement of proteins was investigated. We started our investigations with general considerations related to interacting and noninteracting spheres and extended them to the situation in thylakoids.
Protein-protein interactions and supercomplexes
During the last two decades a number of experiments have shown that multisubunit proteins (e.g., LHC II, PS I, PS II, cyt bf) can associate to produce higher-aggregation forms. These constitute homo- or heterooligomeric assemblies (often referred to as "supercomplexes"). Examples for homooligomers are the trimeric form of LHC II ((LHC II)3) (Janson, 1994; Kühlbrandt and Wang, 1991) and the cyt bf dimer (Hope, 1993; Huang et al., 1994; Cramer et al., 1996). Heterooligomers exist for the majority of PS II (PS II
, (PS II-LHC II)2) (Hankammer et al., 1997) and PS I (PS I-(LHC I)8) (Janson, 1994; Boekema et al., 1994). Although many proteins tend to form oligomers, others do not: PS IIß and the ATPase do not form higher aggregation states.
Recent evidence supports the existence of an even higher level of protein assemblage, located mainly in the grana region. First, heptameric LHC II-trimers (Dekker et al., 1999) as well as complexes formed by (LHC II)3 and (PS II-(LHC II)3)2 supercomplexes (Boekema et al., 1999, 2000; Yakushevska et al., 2001) have been isolated by fast solubilization of grana stacks, followed by single-particle analysis using electron microscopy. The authors conclude that these aggregates also exist in the native membrane. Second, the analysis of light-induced changes in circular dichroism of thylakoids indicates the existence of a long-range chiral order. These signals are interpreted as indicating a macrodomain structure in grana stacks (Istokovics et al., 1997; Garab and Mustárdy, 1999).
The segregation and aggregation of protein assemblies is of functional significance. Phosphorylation of the LHC II proteins results in a segregation from PS II and lateral migration of the complexes, which is regarded as an essential mechanism for the regulation of the energy distribution between the two photosystems in plants (Allen et al., 1981, 1992a,b). The modification of LHC II by the negatively charged phosphate group is likely to decrease protein-protein interactions. On the other hand, the most likely explanation for the well-established phenomenon of energy transfer between PS II units (cooperativity) (Krause and Weis, 1991) is that several PS II centers share a common antenna bed mediated by LHC II complexes. To facilitate energy transfer, close contact between several of these complexes must occur. Furthermore, using a functional analysis of electron transport reactions, a microdomain hypothesis was developed by Joliot, Lavergne, and co-workers (Joliot et al., 1992; Lavergne et al., 1992) and extended by others (Blackwell et al., 1994; Kirchhoff et al., 2000). According to the extended microdomain hypothesis, (PS II-(LHC II)3)2 and (LHC II)3 build up a networklike arrangement, within which plastoquinone molecules are temporarily trapped. The participation of cyt bf complexes in these networks is unclear. In contrast to supercomplexes, it is expected that microdomains are less stable structures with a finite lifetime (Kirchhoff et al., 2000; Joliot and Joliot, 1992).
In this theoretical study the role of protein-protein interactions in the organization of proteins in a membrane is investigated, starting with interacting spheres corresponding to the size of LHC II trimers. Different energies for the interactions are considered. The arrangement of the particles is then examined by a nearest-neighbor analysis and pair-correlation analysis. Investigations are carried out for different particle densities to get a basic understanding of the influence of protein-protein interactions on pattern formation. In the second part, the influence of noninteracting ("disturbing") spheres on the organization of the particles is investigated. This part relates to the cytochrome bf complexes in the thylakoids, for which there are no reports about interactions with other proteins. Finally, proteins with shapes corresponding to photosynthetic proteins are considered. LHC II particles are assumed to interact with each other, cyt bf is considered to be noninteracting, and PS II is assumed to contain integral, tightly bound LHC II complexes that interact in the same manner as free LHC II.
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SIMULATION |
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. Periodic boundary conditions were assumed, i.e., the actual shape of the lattice corresponded to a torus. Consequently, a particle that was positioned at one edge of the lattice was continued at the opposite side. A 1-nm lattice spacing was chosen. This corresponds to the typical size of lipids.
LHC II
The number of attributes of the LHC II was extended beyond those discussed in the model presented in Tremmel et al. (2003) by an additional layer of grid points surrounding them. These grid points reflected the binding sites. If another LHC II came to lie on these grid points (from here on referred to as binding sites), each occupied point was counted as a bound site (see Fig. 1). The number of bound sites times the interaction energy was considered to constitute the binding energy of the particle.
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PS II
PS II usually forms oligomers consisting of two PS II monomers and two LHC II trimers (see Fig. 2). It was assumed that the LHC II trimers that are tightly bound to PS II interact with free LHC II and other LHC II tightly bound to PS II. Accordingly, those parts of the (PS II-(LHC II)3)2 (from now on referred to as PS II) where the LHC II is bound should interact with other LHC II complexes.
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1 and 2, see Fig. 3). Between these angles all nearest and next nearest neighboring lattice sites of a PS II were considered as binding sites.
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The binding energy of a particle (e.g., PS II or LHC II) was considered to be the sum of bound sites with which the particle was involved (see Fig. 1) times the assumed energy (E, in units of kT). When two particles collided they stuck together until one particle detached itself. Because of its thermal energy a particle could unbind from its neighbors with the probability e–
E/kT. T is the temperature and E is the energy change due to the unbinding. It was assumed that E = nE, where n is the number of bound sites of a particle. At each step a randomly chosen particle was tested for whether it was unbinding. To test for unbinding a random number between zero and 1 was chosen. If e–nE/kT was larger than this random number, the transition was accepted; otherwise, it was rejected. This is similar to the procedure used by Shih and co-workers (1987). Correspondingly, the higher the binding energy (E), the lower the probability for unbinding. When the procedure was repeated as often as there were particles on the grid, this corresponded to one Monte Carlo step.
Movement of a particle
A particle that underwent an unbinding transition was moved to a random nearest-neighbor site on the lattice. If the site was occupied another neighboring site was tried. Particles bound to others were considered to be stationary because of their larger mass. This corresponds to a form of the Multicenter Diffusion Limited Aggregation (standard aggregation models like Multicenter Diffusion Limited Aggregation are summarized in, e.g., Saxton (1992, 1993), with a high concentration of immobile seeds. The assumption of immobility of clusters is a simplification. However, cross-membrane interactions between proteins in adjacent grana discs are expected to restrict the mobility of clusters much more than that of single particles.
For nonspherical particles, e.g., particles with the shape of photosynthetic proteins, the exact calculation of their translation and rotation in a membrane is very difficult, and therefore some simplifications were needed. Accordingly, it was assumed that translation in all dimensions was equally probable. Furthermore, it was assumed that at each time step each particle rotated 
±10°. That is, forces tangential to the body surface (viscosity of the matrix molecules) and forces normal to the surface (pressure forces caused by momentum transfer between the particles and the matrix molecules) were not addressed directly but were subsumed into the ratio of rotational energy/translational energy. This is somewhat arbitrary, but the exact degree of rotation should not influence the general outcome.
Due to the long computing time required for all conditions only 2–3 different runs were averaged for each condition. However, a large lattice (200 x 200 nm) was used with many particles (at least 303). Therefore the variability of different runs was not very pronounced. To illustrate the variability, the single results of the particle pair correlation function for the conditions showing the most variability are shown in Fig. 4. This illustrates that the variability was reasonably small compared to the effects observed.
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RESULTS |
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E/kT. Particles initially placed randomly on a lattice thus rearranged until a steady state was reached where the frequency of binding equaled that of unbinding (i.e., after a number of steps well within the plateau in Fig. 12). This binding and unbinding led to different patterns of arrangements according to the interaction energy, E. In Fig. 6, particle arrangements are shown that result from the same initial distribution but using different interaction energies. From Fig. 6 it can be seen that higher interaction energies led to more elongated patterns whereas lower interaction energies resulted in more clustered patterns. This is in agreement with the results of Shih and co-workers (1987) (see also Saxton, 1992, and references therein).
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). The NNDA was computed for a given sample by determining the distance between a particle and its nearest neighbor for each of the particles. From the list of nearest-neighbor distances the mean (accumulated) probability (Pacc(r)) of finding the nearest neighbor within the distance r was calculated and plotted versus r. Aggregated particles (attractive interaction forces) show a relative excess of small nearest-neighbor distances, whereas repelled particles with the same average particle density show a relative deficiency of small nearest-neighbor distances. The NNDA is of course dependent on the particle density. Fig. 7 shows that the higher the particle density the smaller the distance to the nearest neighbor. The diameter of a sphere is 6.25 nm, and particles cannot be closer than that.
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0, of particles in the vicinity of one particle. The PCA was computed for a given sample of particles by determining the mean number of particles (n(r)) found in a shell of radii r – dr and r + dr around an average particle. In other words, the resulting function 0(r) describes the deviation of the local particle density from the average density. 0 was calculated in the following way (similar to the function described in McQuarrie, 1976):
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= particle density and dr = 0.5 nm (lattice spacing = 1.0 nm). Fig. 9 (see also Fig. 11) shows the PCA for different particle densities and varied interaction energies.
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) (local minimum). This effect can also be seen in Fig. 6. Interestingly, the pronounced maximum at 6 nm for 1 kT was shifted to 7 nm for 5 kT. In Fig. 10, three spheres are shown illustrating two possible smallest distances to the nearest neighbor. In the linear position (upper left sphere to lower sphere) the distance is 7 nm, due to the resolution of the lattice (diameter, 6.25 nm; lattice spacing, 1 nm). In the angular arrangement (upper left sphere to upper right sphere), the minimal distance is 6.08 nm. Thus the shift in the maximum indicates that for higher energies particles were arranged in rather linear chains, whereas for lower energies particles were arranged at oblique angles.
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Next, the density dependence of the PCA of randomly distributed particles (E = 0 kT) was investigated. The results are shown in Fig. 11, left. It can be seen that a high density without interaction energy may also have the effect of cluster formation. This reflects the "organizing effect of entropy": entropy may lead to higher levels of organization (Onsager, 1949; Dinsmore et al., 1996; Chow, 1999). For an area of occupation of 0.75, due to the high particle density, there are more possibilities to arrange the spheres in the lattice when they are ordered. For 0.75 occupied area fraction, oscillation around the mean density occurred. This reflects the tendency of the spheres to be arranged in an ordered way, due to the high particle density. The most compact packing of spheres (hexagonal closest packing) in a continuous area is expected to result in an oscillating PCA. For larger distances, r, the oscillations will become broader and less extreme. This is because more combinations of possibilities occur for a particle to occupy sites oblique to the considered particle, leading to slightly different distances. On the other hand, due to the square characteristics, at least two different positions of a nearest neighbor are possible (6.08 nm and 7 nm, see also Fig. 10). In the continuum, the closest distance to a particle is simply its diameter (here 6.25 nm). Therefore the square nature of the lattice leads to broader peaks at small r. With increasing r, the grid nature of the simulation will have less effect. However, comparing Fig. 9 with Fig. 11 shows that without interaction the peaks are relatively broad, whereas interaction energy leads to clearly distinct maxima.
To illustrate the effect of the particle density, together with the interaction energy, the PCA for different densities and 1 kT interaction energy is shown in Fig. 11, middle, whereas Fig. 11, left, shows the same for 5 kT. For both interaction energies (1 kT and 5 kT, Fig. 11, middle and right), the clustering was most pronounced for an occupied area fraction of 0.25. For 0.75 area fraction there was still clustering but it equals that for noninteracting spheres of the same density.
In Fig. 12, the number of bound sites (as described in Fig. 10) is plotted versus the number of Monte Carlo steps. All investigated particle densities are shown (0.25, 0.50, and 0.75). In accordance with the NNDA analysis and the PCA analysis, Fig. 12 shows that for an occupied area fraction of 0.75 the arrangement was rather similar, independent of the interaction energy. For an occupied area fraction of 0.50 and 0.25, Fig. 12 shows that for higher interaction energies (2 kT, 5 kT, and 10 kT) fewer bound sites were formed than for E = 1 kT. In Fig. 13, the number of bound sites in the equilibrium relative to that for E = 0 kT is plotted versus the interaction energy leading to the respective arrangement. In this figure, it can be seen clearly that the importance of interaction decreased with increasing particle density. Further, the introduction of particle interaction with 1 kT resulted in many more binding sites, but further increase of the interaction energy led to a decreased effect.
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As discussed above, for lower interaction energies the interacting particles formed denser clusters. More compact clustering of the interacting spheres in turn led to more free space for the noninteracting spheres. This may explain why the noninteracting particles showed less tendency to be close together and the PCA was shifted toward higher r. The effects of interacting forces were less pronounced for higher interaction energies.
Influence of protein-protein interactions on plastoquinone diffusion
In this section the influence of protein-protein interactions on particles exhibiting the shape of photosynthetic proteins is examined. The outlines of the photosynthetic proteins were taken from Hankammer et al. (1997) for PS II with tightly bound LHC II trimers, Breyton (2000) for cyt bf, and Kühlbrandt and Wang (1991) for LHC II. An occupied area fraction of 0.65 was chosen. This corresponds to the area fraction in thylakoids determined by Kirchhoff and co-workers (2002) and is slightly below the value of 0.70–0.77 for grana thylakoids estimated by Tremmel and co-workers (2003). However, 0.65 was chosen because it is expected to be very close to, but below, the percolation threshold for restricted long-range PQ diffusion if immobile obstacles of the shape of photosynthetic proteins without protein-protein interactions are assumed (see also Tremmel et al., 2003).
A random (E = 0 kT) arrangement of the particles is shown in Fig. 16 together with the steady-state arrangement resulting from protein-protein interactions with E = 1 kT. Despite the high protein density, significant reorganization of the particles was found. In agreement with the results reported above (see also Fig. 15) the noninteracting cyt bf seemed to have more open space surrounding it when located between interacting particles (Fig. 16, right) compared to a purely random distribution (Fig. 16, left). Particles other than cyt bf (i.e., PS II and LHC II) were lying close together due to their interaction. Consequently, diffusion of PQ was expected to be more hindered in an arrangement of interacting photosynthetic proteins than in a random distribution.
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). The results are shown in Fig. 18. It can be seen that the distance-dependent diffusion coefficient of PQ was significantly lower when interactions were introduced. For a distance of 15 nm (the mean distance between PS II and the nearest cyt bf in a random distribution (Tremmel et al., 2003)) the normalized diffusion coefficient in a random distribution (0 kT) is 0.135, whereas the diffusion coefficient between interacting proteins (1 kT) is 0.105, i.e., 
lower. As a consequence, PQ needs a much longer time to bridge a certain distance.
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DISCUSSION |
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suggest that protein-protein interactions may lead to the formation of diffusion domains for plastoquinone, preventing rapid redox equilibration across the membrane between PQ and cyt b. Therefore, in our work the arrangement of interacting particles on a square lattice was studied with the help of a Monte Carlo simulation assuming a simple hypothetical interaction potential. The NNDA and the PCA were examined. First, the effects of different interaction energies and particle densities on the arrangement of interacting spheres are discussed. Second, the influence of noninteracting particles disturbing the organization of interacting particles is considered. Finally, we examined how LHC II-LHC II, (PS II-(LHC II)3)2-LHC II, and (PS II-(LHC II)3)2-(PS II-(LHC II)3)2 interactions may influence plastoquinone diffusion and thus electron transport.
The results show that the arrangement of interacting particles was dependent on the interaction energy (see Figs. 6, 8–9, 12, and 13). Lower interaction energies (1 kT and 2 kT) led to a more clustered particle distribution, whereas higher energies (5 kT and 10 kT) result in ramified chains. This is in accordance with the results of Shih and co-workers (1987).
However, the arrangement of the particles was not only dependent on the interaction energy but also on the particle density (see Figs. 5, 7, and 11). The ordering effect of high particle densities was very similar to that of interaction energies. Both led to a steeper increase in the NNDA and higher PCA at low r, i.e., generally a higher probability for one particle to be close to another. The higher the particle density, the lower was the effect of interaction.
For occupied-area fractions of 0.75 the arrangement of the spheres was independent of the interaction energy. This is interesting, because in thylakoids the area occupied by proteins is 0.7. Therefore, interaction energies have presumably only a minor impact on the thylakoid architecture. However, using only homogeneous interacting spheres with binding sites all around the surface is a very crude simplifying assumption for the modeling of thylakoids, and further refinement is needed. Nevertheless, interacting spheres used in the simulation resemble LHC II, which is rather cylindrical and of similar size to the spheres used in the simulation (compare Kühlbrandt and Wang, 1991). Consequently these simulations are suited to describing, e.g., LHC II reconstituted in liposomes or BBY preparations.
In thylakoids not all photosynthetic proteins may be appropriately modeled by interacting spheres of the same size. Not all photosynthetic proteins may interact with other proteins. For example, for cytochrome bf nothing is known about interactions with other proteins. Furthermore, the largest complexes, here referred to as PS II, are in fact PS II with tightly bound LHC II (((PS II)-(LHC II)3)2). Therefore, they are expected to interact with free LHC II or other LHC II tightly bound to PS II. However, ((PS II)–(LHC II)3)2 probably does not interact on its whole surface but rather where the LHC II is located. In addition, PS II (((PS II)-(LHC II)3)2) is very large compared to LHC II. Therefore, using the same area fraction, more space may be left for LHC II to take a tortuous path between noninteracting proteins until it binds to an interacting protein.
Therefore, the simulation was refined step by step. First the simulation was extended to account for noninteracting particles. The results are shown in Figs. 14 and 15. The noninteracting spheres were assumed to be of the same size as the interacting spheres. This is because the sizes of LHC II (interacting) and cyt bf (probably noninteracting) are in the same range. It should be noted that the ratio of interacting/noninteracting spheres used in the simulations (i.e., 4:1) does not reflect the stoichiometries of the photosynthetic proteins. The principal effects were investigated and the relatively large proportion of noninteracting spheres was chosen because it was expected to lead to clearer effects. The noninteracting spheres did not alter qualitatively the behavior of interacting spheres. In contrast noninteracting spheres were influenced by interacting spheres. The relative density of noninteracting particles became higher at larger distances from each other (larger r). This shows that whereas interacting spheres tended to form clusters, the noninteracting spheres tended to be kept apart by the interacting spheres. In thylakoids this could lead to a more homogeneous distribution of the noninteracting cyt bf . This may be of importance because cyt bf is involved in the rate-limiting step of photosynthesis, the PQH2 oxidation. In thylakoids densely packed with proteins, PQH2 diffusion may be severely restricted (Blackwell et al., 1994; Lavergne and Joliot, 1991; Joliot et al., 1992; Lavergne et al., 1992; Joliot and Joliot, 1992; Kirchhoff et al., 2000; Tremmel et al., 2003). Therefore, the distribution of cyt bf in the thylakoids may be of importance.
As a second step, the influence of the interaction between LHC II and parts of PS II was examined, taking into account the realistic shapes of the proteins. Despite the high density of proteins in the thylakoids the introduction of protein-protein interactions did exert a strong effect on the arrangement of the proteins. Furthermore, PQ diffusion was influenced by the interactions: compared to the diffusion in randomly arranged proteins, the diffusion coefficient of PQ was significantly decreased when protein-protein interactions were introduced. On the other hand, in the case of interacting LHC II and PS II there seemed to be much more free space around the noninteracting cyt bf complexes. Further, since cyt bf is not likely to interact with other proteins, it is expected to be much more mobile than interacting proteins. Accordingly the probability for a binding site on cyt bf to be obstructed permanently is low. PQH2 oxidation at cyt bf is considered to be the rate-limiting step in electron transport. Taking that into account, interaction energies resulting in a more homogeneous distribution of cyt bf and an increased accessibility of the binding sites may play an important role in electron flux. This holds particularly if PQ diffusion is restricted. On the other hand, protein-protein interactions may indeed increase the retarding effect of the high protein density in thylakoids.
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FOOTNOTES |
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Submitted on May 11, 2004; accepted for publication January 4, 2005.
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