The membrane environment modulates self-association of the human GpA TM domain—Implications for membrane protein folding and transmembrane signaling

The influence of lipid bilayer properties on a defined and sequence-specific transmembrane helix-helix interaction is not well characterized yet. To study the potential impact of changing bilayer properties on a sequence-specific transmembrane helix-helix interaction, we have traced the association of fluorescent-labeled glycophorin A transmembrane peptides by fluorescence spectroscopy in model membranes with varying lipid compositions. The observed changes of the glycophorin A dimerization propensities in different lipid bilayers suggest that the lipid bilayer thickness severely influences the monomer-dimer equilibrium of this transmembrane domain, and dimerization was most efficient under hydrophobic matching conditions. Moreover, cholesterol considerably promotes self-association of transmembrane helices in model membranes by affecting the lipid acyl chain ordering. In general, the order of the lipid acyl chains appears to be an important factor involved in determining the strength and stability of transmembrane helix-helix interactions. As discussed, the described influences of membrane properties on transmembrane helix-helix interactions are highly important for understanding the mechanism of transmembrane protein folding and functioning as well as for gaining a deeper insight into the regulation of signal transduction via membrane integral proteins by bilayer properties.     

Comparison of helix interactions in membrane and soluble alpha-bundle proteins

Helix-helix interactions are important for the folding, stability, and function of membrane proteins. Here, two independent and complementary methods are used to investigate the nature and distribution of amino acids that mediate helix-helix interactions in membrane and soluble alpha-bundle proteins. The first method characterizes the packing density of individual amino acids in helical proteins based on the van der Waals surface area occluded by surrounding atoms. We have recently used this method to show that transmembrane helices pack more tightly, on average, than helices in soluble proteins. These studies are extended here to characterize the packing of interfacial and noninterfacial amino acids and the packing of amino acids in the interfaces of helices that have either right- or left-handed crossing angles, and either parallel or antiparallel orientations. We show that the most abundant tightly packed interfacial residues in membrane proteins are Gly, Ala, and Ser, and that helices with left-handed crossing angles are more tightly packed on average than helices with right-handed crossing angles. The second method used to characterize helix-helix interactions involves the use of helix contact plots. We find that helices in membrane proteins exhibit a broader distribution of interhelical contacts than helices in soluble proteins. Both helical membrane and soluble proteins make use of a general motif for helix interactions that relies mainly on four residues (Leu, Ala, Ile, Val) to mediate helix interactions in a fashion characteristic of left-handed helical coiled coils. However, a second motif for mediating helix interactions is revealed by the high occurrence and high average packing values of small and polar residues (Ala, Gly, Ser, Thr) in the helix interfaces of membrane proteins. Finally, we show that there is a strong linear correlation between the occurrence of residues in helix-helix interfaces and their packing values, and discuss these results with respect to membrane protein structure prediction and membrane protein stability.     

Design and Characterization of a Membrane Protein Unfolding Platform in Lipid Bilayers

Accurate measurement of membrane protein stability—and particularly how it may vary as a result of disease-phenotypic mutations—ideally requires a denaturant that can unfold a membrane-embedded structure while leaving the solubilizing environment unaffected. The steric trap method fulfills this requirement by using monovalent streptavidin (mSA) molecules to unfold membrane proteins engineered with two spatially close biotin tags. Here we adapted this method to an 87-residue helix-loop-helix (hairpin) construct derived from helices 3 and 4 in the transmembrane domain of the human cystic fibrosis transmembrane conductance regulator (CFTR), wherein helix-helix tertiary interactions are anticipated to confer a portion of construct stability. The wild type CFTR TM3/4 hairpin construct was modified with two accessible biotin tags for mSA-induced unfolding, along with two helix-terminal pyrene labels to monitor loss of inter-helical contacts by pyrene excimer fluorescence. A series of eight constructs with biotin tags at varying distances from the helix-terminal pyrene labels were expressed, purified and labeled appropriately; all constructs exhibited largely helical circular dichroism spectra. We found that addition of mSA to an optimized construct in lipid vesicles led to a complete and reversible loss in pyrene excimer fluorescence and mSA binding, and hence hairpin unfolding—results further supported by SDS-PAGE visualization of mSA bound and unbound species. While some dimeric/oligomeric populations persist that may affect quantitation of the unfolding step, our characterization of the design yields a promising prototype of a future platform for the systematic study of membrane protein folding in a lipid bilayer environment.     

Fluorescence analysis of the lipid binding-induced conformational change of apolipoprotein e4

Apolipoprotein (apo) E is thought to undergo conformational changes in the N-terminal helix bundle domain upon lipid binding, modulating its receptor binding activity. In this study, site-specific fluorescence labeling of the N-terminal (S94) and C-terminal (W264 or S290) helices in apoE4 by pyrene maleimide or acrylodan was employed to probe the conformational organization and lipid binding behavior of the N- and C-terminal domains. Guanidine denaturation experiments monitored by acrylodan fluorescence demonstrated the less organized, more solvent-exposed structure of the C-terminal helices compared to the N-terminal helix bundle. Pyrene excimer fluorescence together with gel filtration chromatography indicated that there are extensive intermolecular helix-helix contacts through the C-terminal helices of apoE4. Comparison of increases in pyrene fluorescence upon binding of pyrene-labeled apoE4 to egg phosphatidylcholine small unilamellar vesicles suggests a two-step lipid-binding process; apoE4 initially binds to a lipid surface through the C-terminal helices followed by the slower conformational reorganization of the N-terminal helix bundle domain. Consistent with this, fluorescence resonance energy transfer measurements from Trp residues to acrylodan attached at position 94 demonstrated that upon binding to the lipid surface, opening of the N-terminal helix bundle occurs at the same rate as the increase in pyrene fluorescence of the N-terminal domain. Such a two-step mechanism of lipid binding of apoE4 is likely to apply to mostly phospholipid-covered lipoproteins such as VLDL. However, monitoring pyrene fluorescence upon binding to HDL(3) suggests that not only apoE-lipid interactions but also protein-protein interactions are important for apoE4 binding to HDL(3).     

Identifying interactions between transmembrane helices from the adenosine A2A receptor

The human adenosine A(2A) receptor (A(2A)R) is an integral membrane protein and a member of the G-protein-coupled receptor (GPCR) superfamily, characterized by seven transmembrane (TM) helices. Although helix-helix association in the lipid bilayer is known to be an essential step in the folding of GPCRs, the determinants of their structures, folding, and assembly in the cell membrane are poorly understood. Previous studies in our group showed that while peptides corresponding to all seven TM domains of A(2A)R form stable helical structures in detergent micelles and lipid vesicles, they display significant variability in their helical propensity. This finding suggested to us that some TM domains might need to interact with other domains to properly insert and fold in hydrophobic environments. In this study, we assessed the ability of TM peptides to interact in pairwise combinations. We analyzed peptide interactions in hydrophobic milieus using circular dichroism spectroscopy and F?rster resonance energy transfer. We find that specific interactions between TM helices occur, leading to additional helical content, especially in weakly helical TM domains, suggesting that some TM domains need a partner for proper folding in the membrane. The approach developed in this study will enable complete analysis of the TM domain interactions and the modeling of a folding pathway for A(2A)R.     

Helix-helix interactions in lipid bilayers.

Using a continuum model, we calculated the electrostatic interaction free energy between two alpha-helices in three environments: the aqueous phase, a low dielectric alkane phase, and a simple representation of a lipid bilayer. As was found in previous work, helix-helix interactions in the aqueous phase are quite weak, because of solvent screening, and slightly repulsive, because of desolvation effects that accompany helix assembly. In contrast, the interactions can be quite strong in a hypothetical alkane phase because desolvation effects are essentially nonexistent and because helix-helix interactions are not well screened. In this type of environment, the antiparallel helix orientation is strongly favored over the parallel orientation. In previous work we found that the free energy penalty associated with burying helix termini in a bilayer is quite high, which is why the termini tend to protrude into the solvent. Under these conditions the electrostatic interaction is strongly screened by solvent; indeed, it is sufficient for the termini to protrude a few angstroms from the two surfaces of the bilayer for their interaction to diminish almost completely. The effect is consistent with the classical model of the helix dipole in which the dipole moment is represented by point charges located at either terminus. Our results suggest, in agreement with previous models, that there is no significant nonspecific driving force for helix aggregation and, hence, that membrane protein folding must be driven by specific interactions such as close packing and salt-bridge and hydrogen bond formation.     

Interactions of alpha-helices with lipid bilayers: a review of simulation studies

Membrane proteins, of which the majority seem to contain one or more alpha-helix, constitute approx. 30% of most genomes. A complete understanding of the nature of helix/bilayer interactions is necessary for an understanding of the structural principles underlying membrane proteins. This review describes computer simulation studies of helix/bilayer interactions. Key experimental studies of the interactions of alpha-helices and lipid bilayers are briefly reviewed. Surface associated helices are found in some membrane-bound enzymes (e.g. prostaglandin synthase), and as stages in the mechanisms of antimicrobial peptides and of pore-forming bacterial toxins. Transmembrane alpha-helices are found in most integral membrane proteins, and also in channels formed by amphipathic peptides or by bacterial toxins. Mean field simulations, in which the lipid bilayer is approximated as a hydrophobic continuum, have been used in studies of membrane-active peptides (e.g. alamethicin, melittin, magainin and dermaseptin) and of simple membrane proteins (e.g. phage Pf1 coat protein). All atom molecular dynamics simulations of fully solvated bilayers with transmembrane helices have been applied to: the constituent helices of bacteriorhodopsin; peptide-16 (a simple model TM helix); and a number of pore-lining helices from ion channels. Surface associated helices (e.g. melittin and dermaseptin) have been simulated, as have alpha-helical bundles such as bacteriorhodopsin and alamethicin. From comparison of the results from the two classes of simulation, it emerges that a major theoretical challenge is to exploit the results of all atom simulations in order to improve the mean field approach.     

The effect of interactions involving ionizable residues flanking membrane-inserted hydrophobic helices upon helix-helix interaction

We examined the effect of ionizable residues at positions flanking the hydrophobic core of helix-forming polyLeu peptides upon helix-helix interactions within model membrane vesicles composed of dioleoylphosphatidylcholine. The peptides studied were flanked on both the N and C termini either by two Lys (K(2)-flanked peptide), one Lys plus one Asp (DK-flanked peptide), or one Lys plus three Asp (KD(3)-flanked peptide). The fluorescence of a Trp residue positioned at the center of the hydrophobic sequence was used to evaluate peptide behavior. As judged by the concentration dependence of the maximum wavelength of Trp emission, there was significant oligomerization of the KD(3)- and DK-flanked peptides, but not the K(2)-flanked peptide, at neutral pH. At neutral pH mixtures of K(2)- and KD(3)-flanked peptides associated with each other, but mixtures of the K(2)- and DK-flanked peptides did not. Oligomerization by the DK- and KD(3)-flanked peptides decreased under low pH conditions in which the Asp residues were protonated. Additional experiments showed that at neutral pH the KD(3)-flanked peptide showed an increased tendency to oligomerize when as little as 10-15 mol % of an anionic lipid, phosphatidylglycerol, was present. The behavior of the other peptides was not strongly influenced by phosphatidylglycerol. These results can largely be explained by modulation of helix-helix interactions via electrostatic interactions involving the helix-flanking ionizable residues. Such interactions may influence membrane protein folding. The self-association of anionic KD(3)-flanked peptides suggests that additional interactions involving charged residues also can modulate helix-helix association.     

Discovery of a significant,nontopological preference for antiparallel alignment of helices with parallel regions in sheets

To help elucidate the role of secondary structure packing preferences in protein folding, here we present an analysis of the packing geometry observed between alpha-helices and between alpha-helices and beta-sheets in 1316 diverse, nonredundant protein structures. Finite-length vectors were fit to the alpha-carbon atoms in each of the helices and strands, and the packing angle between the vectors, Omega, was determined at the closest point of approach within each helix-helix or helix-sheet pair. Helix-sheet interactions were found in 391 of the proteins, and the distributions of Omega values were calculated for all the helix-sheet and helix-helix interactions. The packing angle preferences for helix-helix interactions are similar to those previously observed. However, analysis of helix-strand packing preferences uncovered a remarkable tendency for helices to align antiparallel to parallel regions of beta-sheets, independent of the topological constraints or prevalence of beta-alpha-beta motifs in the proteins. This packing angle preference is significantly diminished in helix interactions involving mixed and antiparallel beta-sheets, suggesting a role for helix-sheet dipole alignment in guiding supersecondary structure formation in protein folding. This knowledge of preferred packing angles can be used to guide the engineering of stable protein modules.     

Transmembrane helix-helix interactions are modulated by the sequence context and by lipid bilayer properties

Folding of polytopic transmembrane proteins involves interactions of individual transmembrane helices, and multiple TM helix-helix interactions need to be controlled and aligned to result in the final TM protein structure. While defined interaction motifs, such as the GxxxG motif, might be critically involved in transmembrane helix-helix interactions, the sequence context as well as lipid bilayer properties significantly modulate the strength of a sequence specific transmembrane helix-helix interaction. Structures of 11 transmembrane helix dimers have been described today, and the influence of the sequence context as well as of the detergent and lipid environment on a sequence specific dimerization is discussed in light of the available structural information. This article is part of a Special Issue entitled: Protein Folding in Membranes.     

Lipid-protein interactions in biological membranes: a structural perspective

Lipid molecules bound to membrane proteins are resolved in some high-resolution structures of membrane proteins. An analysis of these structures provides a framework within which to analyse the nature of lipid-protein interactions within membranes. Membrane proteins are surrounded by a shell or annulus of lipid molecules, equivalent to the solvent layer surrounding a water-soluble protein. The lipid bilayer extends right up to the membrane protein, with a uniform thickness around the protein. The surface of a membrane protein contains many shallow grooves and protrusions to which the fatty acyl chains of the surrounding lipids conform to provide tight packing into the membrane. An individual lipid molecule will remain in the annular shell around a protein for only a short period of time. Binding to the annular shell shows relatively little structural specificity. As well as the annular lipid, there is evidence for other lipid molecules bound between the transmembrane α-helices of the protein; these lipids are referred to as non-annular lipids. The average thickness of the hydrophobic domain of a membrane protein is about 29 Å, with a few proteins having significantly smaller or greater thicknesses than the average. Hydrophobic mismatch between a membrane protein and the surrounding lipid bilayer generally leads to only small changes in membrane thickness. Possible adaptations in the protein to minimise mismatch include tilting of the helices and rotation of side chains at the ends of the helices. Packing of transmembrane α-helices is dependent on the chain length of the surrounding phospholipids. The function of membrane proteins is dependent on the thickness of the surrounding lipid bilayer, sometimes on the presence of specific, usually anionic, phospholipids, and sometimes on the phase of the phospholipid.     

Helical packing patterns in membrane and soluble proteins

This article presents the results of a detailed analysis of helix-helix interactions in membrane and soluble proteins. A data set of interacting pairs of helices in membrane proteins of known structure was constructed and a structure alignment algorithm was used to identify pairs of helices in soluble proteins that superimpose well with pairs of helices in the membrane-protein data set. Most helix pairs in membrane proteins are found to have a significant number of structural homologs in soluble proteins, although in some cases, primarily involving irregular helices, no close homologs exist. An analysis of geometric relationships between interacting helices in the two sets of proteins identifies some differences in the distributions of helix length, interfacial area, packing angle, and distance between the polypeptide backbones. However, a subset of soluble-protein helix pairs that are close structural homologs to membrane-protein helix pairs exhibits distributions that mirror those observed in membrane proteins. The larger average interface size and smaller distance of closest approach seen for helices in membrane proteins appears due in part to a relative enrichment of alanines and glycines, particularly as components of the AxxxA and GxxxG motifs. It is argued that membrane helices are not on average more tightly packed than helices in soluble proteins; they are simply able to approach each other more closely. This enables them to interact over longer distances, which may in turn facilitate their remaining in contact over much of the width of the lipid bilayer. The close structural similarity seen between some pairs of helices in membrane and soluble proteins suggests that packing patterns observed in soluble proteins may be useful in the modeling of membrane proteins. Moreover, there do not appear to be fundamental differences between the magnitude of the forces that drive helix packing in membrane and soluble proteins, suggesting that strategies to make membrane proteins more soluble by mutating surface residues are likely to encounter success, at least in some cases.     

Modulation of glycophorin A transmembrane helix interactions by lipid bilayers: molecular dynamics calculations

Starting from the glycophorin A dimer structure determined by NMR, we performed simulations of both dimer and monomer forms in explicit lipid bilayers with constant normal pressure, lateral area, and temperature using the CHARMM potential. Analysis of the trajectories in four different lipids reveals how lipid chain length and saturation modulate the structural and energetic properties of transmembrane helices. Helix tilt, helix-helix crossing angle, and helix accessible volume depend on lipid type in a manner consistent with hydrophobic matching concepts: the most relevant lipid property appears to be the bilayer thickness. Although the net helix-helix interaction enthalpy is strongly attractive, analysis of residue-residue interactions reveals significant unfavorable electrostatic repulsion between interfacial glycine residues previously shown to be critical for dimerization. Peptide volume is nearly conserved upon dimerization in all lipid types, indicating that the monomeric helices pack equally well with lipid as dimer helices do with one another. Enthalpy calculations indicate that the helix-environment interaction energy is lower in the dimer than in the monomer form, when solvated by unsaturated lipids. In all lipid environments there is a marked preference for lipids to interact with peptide predominantly through one rather than both acyl chains. Although our trajectories are not long enough to allow a full thermodynamic treatment, these results demonstrate that molecular dynamics simulations are a powerful method for investigating the protein-protein, protein-lipid, and lipid-lipid interactions that determine the structure, stability and dynamics of transmembrane alpha-helices in membranes.     

Assembly of the M2 Tetramer Is Strongly Modulated by Lipid Chain Length

The influenza virus matrix protein 2 (M2) assembles into a tetramer in the host membrane during viral uncoating and maturation. It has been used as a model system to understand the relative contributions of protein-lipid and protein-protein interactions to membrane protein structure and association. Here we investigate the effect of lipid chain length on the association of the M2 transmembrane domain into tetramers using Förster resonance energy transfer. We observe that the interactions between the M2 helices are much stronger in 1,2-dilauroyl-sn-glycero-3-phosphocholine than in 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine bilayers. Thus, lipid chain length and bilayer thickness not only modulate peptide interactions, but could also be a major determinant of the association of transmembrane helices into functional membrane protein oligomers.     

Control of the transmembrane orientation and interhelical interactions within membranes by hydrophobic helix length

We examined the effect of the length of the hydrophobic core of Lys-flanked poly(Leu) peptides on their behavior when inserted into model membranes. Peptide structure and membrane location were assessed by the fluorescence emission lambdamax of a Trp residue in the center of the peptide sequence, the quenching of Trp fluorescence by nitroxide-labeled lipids (parallax analysis), and circular dichroism. Peptides in which the hydrophobic core varied in length from 11 to 23 residues were found to be largely alpha-helical when inserted into the bilayer. In dioleoylphosphatidylcholine (diC18:1PC) bilayers, a peptide with a 19-residue hydrophobic core exhibited highly blue-shifted fluorescence, an indication of Trp location in a nonpolar environment, and quenching localized the Trp to the bilayer center, an indication of transmembrane structure. A peptide with an 11-residue hydrophobic core exhibited emission that was red-shifted, suggesting a more polar Trp environment, and quenching showed the Trp was significantly displaced from the bilayer center, indicating that this peptide formed a nontransmembranous structure. A peptide with a 23-residue hydrophobic core gave somewhat red-shifted fluorescence, but quenching demonstrated the Trp was still close to the bilayer center, consistent with a transmembrane structure. Analogous behavior was observed when the behavior of individual peptides was examined in model membranes with various bilayer widths. Other experiments demonstrated that in diC18:1PC bilayers the dilution of the membrane concentration of the peptide with a 23-residue hydrophobic core resulted in a blue shift of fluorescence, suggesting the red-shifted fluorescence at higher peptide concentrations was due to helix oligomerization. The intermolecular self-quenching of rhodamine observed when the peptide was rhodamine-labeled, and the concentration dependence of self-quenching, supported this conclusion. These studies indicate that the mismatch between helix length and bilayer width can control membrane location, orientation, and helix-helix interactions, and thus may mismatch control both membrane protein folding and the interactions between membrane proteins.     

How protein transmembrane segments sense the lipid environment

Integral membrane proteins have central roles in a vast number of vital cellular processes. A structural feature that most membrane proteins have in common is the presence of one or more alpha-helices with which they traverse the lipid bilayer. Because of the interaction with the surrounding lipids, the organization of these transmembrane helices will be sensitive to lipid properties like lateral packing, hydrophobic thickness, and headgroup charge. The helices may adapt to the lipids in different ways, which in turn can influence the structure and function of the intact membrane protein. In this review, we will focus on how the lipid environment influences two specific properties of transmembrane segments: their lateral association and their tilt with respect to the bilayer normal.     

Sequence specificity in the dimerization of transmembrane alpha-helices.

While several reports have suggested a role for helix-helix interactions in membrane protein oligomerization, there are few direct biochemical data bearing on this subject. Here, using mutational analysis, we show that dimerization of the transmembrane alpha-helix of glycophorin A in a detergent environment is spontaneous and highly specific. Very subtle changes in the side-chain structure at certain sensitive positions disrupt the helix-helix association. These sensitive positions occur at approximately every 3.9 residues along the helix, consistent with their comprising the interface of a closely fit transmembranous supercoil of alpha-helices. By contrast with other reported cases of interactions between transmembrane helices, the set of interfacial residues in this case contains no highly polar groups. Amino acids with aliphatic side chains define much of the interface, indicating that precise packing interactions between the helices may provide much of the energy for association. These data highlight the potential general importance of specific interactions between the hydrophobic anchors of integral membrane proteins.     

Evidence for assembly of small multidrug resistance proteins by a "two-faced" transmembrane helix

Clinically significant bacterial resistance to drugs and cytotoxic compounds can be conferred by the energy-dependent efflux of toxicants, catalyzed by proteins embedded in the bacterial cell membrane. One such group of proteins, the small multidrug resistance family, are drug/proton antiporters that must oligomerize to function, a process that requires the assembly of at least two inactive monomers by intermolecular association of their four transmembrane helices. Here, we have used peptides that correspond to each of the four wild type transmembrane helices of the Halobacterium salinarum protein Hsmr and a corresponding library of mutant peptides to determine the interactive surfaces that likely contribute to protein oligomerization. Hsmr peptides were examined for strong (sodium dodecyl sulfate-resistant) and weaker (perfluorooctanoate-resistant) helix-helix interactions, in conjunction with circular dichroism, fluorescence energy transfer measurements, and molecular modeling. The results are compatible with a scheme in which two faces of helix four permit self-assembly via a higher affinity asymmetric pairing and a lower affinity symmetric interaction, resulting in a discrete tetramer. Our finding that two surfaces of helix four can contribute to the stability of small multidrug resistance protein assembly provides a molecular basis for the design of therapeutics that target this antibiotic resistance mechanism.     

Transmembrane helix-helix interactions: comparative simulations of the glycophorin a dimer

The glycophorin helix dimer is a paradigm for the exploration of helix-helix interactions in integral membrane proteins. Two NMR structures of the dimer are known, one in a detergent micelle and one in a lipid bilayer. Multiple (4 x 50 ns) molecular dynamics simulations starting from each of the two NMR structures, with each structure in either a dodecyl phosphocholine (DPC) micelle or a dimyristoyl phosphatidylcholine (DMPC) bilayer, have been used to explore the conformational dynamics of the helix dimer. Analysis of the helix-helix interaction, mediated by the GxxxG sequence motif, suggests convergence of the simulations to a common model. This is closer to the NMR structure determined in a bilayer than to micelle structure. The stable dimer interface in the final simulation model is characterized by (i) Gly/Gly packing and (ii) Thr/Thr interhelix H-bonds. These results demonstrate the ability of extended molecular dynamics simulations in a lipid bilayer environment to refine membrane protein structures or models derived from experimental data obtained in protein/detergent micelles.     

Mapping the energy surface of transmembrane helix-helix interactions

Transmembrane helices are no longer believed to be just hydrophobic segments that exist solely to anchor proteins to a lipid bilayer, but rather they appear to have the capacity to specify function and structure. Specific interactions take place between hydrophobic segments within the lipid bilayer whereby subtle mutations that normally would be considered innocuous can result in dramatic structural differences. That such specificity takes place within the lipid bilayer implies that it may be possible to identify the most favorable interaction surface of transmembrane alpha-helices based on computational methods alone, as shown in this study. Herein, an attempt is made to map the energy surface of several transmembrane helix-helix interactions for several homo-oligomerizing proteins, where experimental data regarding their structure exist (glycophorin A, phospholamban, Influenza virus A M2, Influenza virus C CM2, and HIV vpu). It is shown that due to symmetry constraints in homo-oligomers the computational problem can be simplified. The results obtained are mostly consistent with known structural data and may additionally provide a view of possible alternate and intermediate configurations.