Helical structures in complex plasma II: Collective interaction

The conditions for the formation and stability of quasi-crystal helical structures formed of charged dust grains of equal size in a complex plasma are investigated. A study is made of both the transverse confinement of such structures by means of an external parabolic potential well and the possibility of their self-confinement or confinement by the field of the smaller background grains. It is shown that, in the presence of dissipation, any initial random distribution of dust grains in a cylindrical external potential evolves into an equilibrium helical structure with a constant pitch angle. It is found that, when the external control parameter is smoothly varied, the pitch angle of the helix changes abruptly and the structure of the quasi-crystal undergoes bifurcations, during which the number of interwoven helices changes. The smaller the confinement parameter and the weaker the attractive forces between the grains, the larger the number of bifurcations. In addition, it is found that, because of the attraction (both noncollective and collective) between the dust grains, stable helical structures may exist not only at zero value of the external confinement parameter, but also at a negative one, and that the collective interaction between the grains not only increases the number of bifurcations but also changes their character.     

Spectral properties of an N-pole helical structure consisting of like-charged equal-size dust grains

The oscillations and stability of an N-pole helical structure that consists of like-charged equal-size dust grains and is confined in a plasma in an axisymmetric potential well are studied theoretically. Self-confining structures, as well as their linear collective modes corresponding to three coupled types of grain displacements (change in the radius of the structure, change in the distance between neighboring lattice planes of the structure, and angular displacements in the lattice planes), are found. On the whole, the coupled oscillations have the form of wormlike perturbations. Dispersion relations for the oscillation modes of helical structures composed of N interwoven helices are derived and solved numerically.     

Simulation studies on bacteriorhodopsin bundle of transmembrane α segments

Bacteriorhodopsin (BR) is a membrane protein which pumps protons through the plasma membrane. Seven transmembrane BR helical segments are subjected to simulation studies in order to investigate the packing process of transmembrane helices. A Monte Carlo simulated annealing protocol is employed to optimize the helix bundle system. Helix packing is optimized according to a semi-empirical potential mainly composed of six components: a bilayer potential, a crossing angle potential, a helix dipole potential, a helix-helix distance potential, a helix orientation potential and a helix-helix distance restraint potential (a loop potential). Necessary parameters are derived from theoretical studies and statistical analysis of experimentally determined protein structures. The structures from the simulations are compared with the experimentally determined structures in terms of geometry. The structures generated show similar shapes to the experimentally suggested structure even without the helix-helix distance restraint potential. However, the relative locations of individual helices were reproduced only when the helix-helix distance restraint potential was used with restraint conditions. Our results suggest that transmembrane helix bundles resembling those observed experimentally may be generated by simulations using simple potentials. Received: 19 April 1999 / Revised version: 6 September 1999 / Accepted: 17 September 1999     

QHELIX: A Computational Tool for the Improved Measurement of Inter-Helical Angles in Proteins

Knowledge about the assembled structures of the secondary elements in proteins is essential to understanding protein folding and functionality. In particular, the analysis of helix geometry is required to study helix packing with the rest of the protein and formation of super secondary structures, such as, coiled coils and helix bundles, formed by packing of two or more helices. Here we present an improved computational method, QHELIX, for the calculation of the orientation angles between helices. Since a large number of helices are known to be in curved shapes, an appropriate definition of helical axes is a prerequisite for calculating the orientation angle between helices. The present method provides a quantitative measure on the irregularity of helical shape, resulting in discriminating irregular-shaped helices from helices with an ideal geometry in a large-scale analysis of helix geometry. It is also capable of straightforwardly assigning the direction of orientation angles in a consistent way. These improvements will find applications in finding a new insight on the assembly of protein secondary structure. Electronic Supplementary Material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Simulation studies on bacteriorhodopsin α-helices

Bacteriorhodopsin (BR) is a membrane protein which pumps protons through the plasma membrane. Transmembrane BR helical segments are subjected to simulation studies in order to investigate the effect of bilayer environment in various simulation conditions. A bilayer potential is introduced to the system to mimic the lipid membrane. The structures from the simulations are compared with the experimentally determined structures in terms of geometrical properties. Electrostatic contribution to the helix packing is also investigated. The simulation results show that the packing geometry of the transmembrane helices is highly affected by the bilayer potential. The results obtained from the simulations may be used for further simulation studies and analysis in investigating transmembrane helix packing. Received: 5 July 1999 / Revised version: 5 October 1999 / Accepted: 15 October 1999     

Protein folding: evaluation of some simple rules for the assembly of helices into tertiary structures with myoglobin as an example

A computer program designed to fold a peptide chain consisting solely of helical segments and connecting links of known length is described and evaluated. This study is a detailed extension of certain aspects of the earlier work of Ptitsyn &; Rashin (1975). Possible interaction sites on the helices are sequence dependent and are calculated as described by Richmond &; Richards (1978) using probable changes in solvent contact area as a guide. The helices are then paired according to the list of potential sites, with each helix being paired at least once. The lists of pairings are then examined geometrically, each site having a defined dihedral helix axis angle, a specified inter-helix axis distance, and defined rotations, when required, about each helix axis. Two simplified filters are used: (1) lengths of connecting links must be equal to or greater than the end-to-end distances of the helices; and (2) non-paired helices must not collide. With myoglobin as a test example and only six of the eight helices being considered, a conformation space consisting of more than 3 × 10⁸ structures was surveyed. The two filters reduced the acceptable structure list to 121. Slight readjustment of the parameters in the filters would have reduced this to 20 structures. Of these 20, one closely resembles the actual distribution of helices in myoglobin. The possible utility and pitfalls of this approach as part of an overall protein folding program are discussed.     

HELANAL: a program to characterize helix geometry in proteins

A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C alpha atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.     

Structures of the transmembrane helices of the G-protein coupled receptor, rhodopsin.

An hypothesis is tested that individual peptides corresponding to the transmembrane helices of the membrane protein, rhodopsin, would form helices in solution similar to those in the native protein. Peptides containing the sequences of helices 1, 4 and 5 of rhodopsin were synthesized. Two peptides, with overlapping sequences at their termini, were synthesized to cover each of the helices. The peptides from helix 1 and helix 4 were helical throughout most of their length. The N- and C-termini of all the peptides were disordered and proline caused opening of the helical structure in both helix 1 and helix 4. The peptides from helix 5 were helical in the middle segment of each peptide, with larger disordered regions in the N- and C-termini than for helices 1 and 4. These observations show that there is a strong helical propensity in the amino acid sequences corresponding to the transmembrane domain of this G-protein coupled receptor. In the case of the peptides from helix 4, it was possible to superimpose the structures of the overlapping sequences to produce a construct covering the whole of the sequence of helix 4 of rhodopsin. As similar superposition for the peptides from helix 1 also produced a construct, but somewhat less successfully because of the disordering in the region of sequence overlap. This latter problem was more severe for helix 5 and therefore a single peptide was synthesized for the entire sequence of this helix, and its structure determined. It proved to be helical throughout. Comparison of all these structures with the recent crystal structure of rhodopsin revealed that the peptide structures mimicked the structures seen in the whole protein. Thus similar studies of peptides may provide useful information on the secondary structure of other transmembrane proteins built around helical bundles.     

HELANAL: A Program to Characterize Helix Geometry in Proteins

Abstract

A detailed analysis of structural and position dependent characteristic features of helices will give a better understanding of the secondary structure formation in globular proteins. Here we describe an algorithm that quantifies the geometry of helices in proteins on the basis of their C^α atoms alone. The Fortran program HELANAL can extract the helices from the PDB files and then characterises the overall geometry of each helix as being linear, curved or kinked, in terms of its local structural features, viz. local helical twist and rise, virtual torsion angle, local helix origins and bending angles between successive local helix axes. Even helices with large radius of curvature are unambiguously identified as being linear or curved. The program can also be used to differentiate a kinked helix and other motifs, such as helix-loop-helix or a helix-turn-helix (with a single residue linker) with the help of local bending angles. In addition to these, the program can also be used to characterise the helix start and end as well as other types of secondary structures.     

Differentiation of lipid-associating helices by use of three-dimensional molecular hydrophobicity potential calculations.

Several types of lipid-associating helices exist: transmembrane helices such as in receptor proteins, pore-forming helices in ion channel proteins, fusion-inducing peptides in viral proteins, and amphipathic helices such as in plasma apolipoproteins. In order to propose a classification of these helices according to their molecular properties, we introduce the concept of molecular hydrophobicity potential for such helical segments. The calculation of this parameter for alpha-helices enables the visualization of the hydrophobic and hydrophilic envelopes around the peptide and their three-dimensional representation by molecular graphics. We have used this parameter to differentiate between pore-forming helices with a hydrophobic envelope larger than the hydrophilic component, membrane-spanning helices surrounded almost entirely by an hydrophobic envelope, fusiogenic peptides with an hydrophobicity gradient both around the helix and along the axis, and finally, amphipathic helices with a predominantly hydrophilic envelope. The structure of the lipid-protein complexes is determined by a number of different interactions: the hydrophobic interaction of the apolar faces of the helices with lipids, the polar interaction of the hydrophilic sides of different helices with each other, and the interaction of hydrophilic residues with the aqueous solvent. The relative magnitude of the hydrophobic and hydrophilic envelopes accounts for the differences in the structure of the lipid-protein complexes. Purely hydrophobic interactions stabilize transmembrane helical segments, while hydrophobic interactions with the lipid phase and with each other are involved in the stabilization of the pore-forming helices. In contrast, both hydrophobic interactions with the lipids and hydrophilic interactions with the aqueous phase contribute to the arrangement of amphipathic helices around the edges of the discoidal lipid-apoprotein complexes.     

Sequential unfolding of individual helices of bacterioopsin observed in molecular dynamics simulations of extraction from the purple membrane

Multiple molecular dynamics simulations of bacterioopsin pulling from its C-terminus show that its alpha-helices unfold individually. In the first metastable state observed in the simulations, helix G is unfolded at its C-terminal segment while the rest of helix G (residues 200-216) is folded and opposes resistance because of a salt-bridge network consisting of Asp-212 and Lys-216 on helix G and Arg-82 and Asp-85 on helix C. Helix G unfolds inside the bundle because the external force is applied to its C-terminal end in a direction perpendicular to the surface of the membrane. Inversely, helix F has to flip by 180 degrees to exit from the membrane because the applied force and the helical N-C axis point in opposite directions. At the highest peak of the force, which cannot be interpreted in single-molecule force spectroscopy experiments, helix F has a pronounced kink at Pro-186. Mutation of Pro-186 and/or the charged side chains mentioned above, which are involved in very favorable electrostatic interactions in the low-dielectric region of the membrane, are expected to reduce the highest peak of the force. Helices E and D unfold in a similar way to helices G and F, respectively. Hence, the force-distance profile and sequence of events during forced unfolding of bacterioopsin are influenced by the up-and-down topology of the seven-helix bundle. The sequential extraction of individual helices from the membrane suggests that the spontaneous (un)folding of bacterioopsin proceeds through metastable bundles of fewer than seven helices. The metastable states observed in the simulations provide atomic level evidence that corroborates the interpretation of very recent force spectroscopy experiments of bacteriorhodopsin refolding.     

The sequence TGAAKAVALVL from glyceraldehyde-3-phosphate dehydrogenase displays structural ambivalence and interconverts between alpha-helical and beta-hairpin conformations mediated by collapsed conformational states.

The peptide TGAAKAVALVL from glyceraldehyde-3-phosphate dehydrogenase adopts a helical conformation in the crystal structure and is a site for two hydrated helical segments, which are thought to be helical folding intermediates. Overlapping sequences of four to five residues from the peptide, sample both helical and strand conformations in known protein structures, which are dissimilar to glyceraldehyde-3-phosphate dehydrogenase suggesting that the peptide may have a structural ambivalence. Molecular dynamics simulations of the peptide sequence performed for a total simulation time of 1.2 micros, starting from the various initial conformations using GROMOS96 force field under NVT conditions, show that the peptide samples a large number of conformational forms with transitions from alpha-helix to beta-hairpin and vice versa. The peptide, therefore, displays a structural ambivalence. The mechanism from alpha-helix to beta-hairpin transition and vice versa reveals that the compact bends and turns conformational forms mediate such conformational transitions. These compact structures including helices and hairpins have similar hydrophobic radius of gyration (Rgh) values suggesting that similar hydrophobic interactions govern these conformational forms. The distribution of conformational energies is Gaussian with helix sampling lowest energy followed by the hairpins and coil. The lowest potential energy of the full helix may enable the peptide to take up helical conformation in the crystal structure of the glyceraldehyde-3-phosphate dehydrogenase, even though the peptide has a preference for hairpin too. The relevance of folding and unfolding events observed in our simulations to hydrophobic collapse model of protein folding are discussed.     

A database of 32 DNA triplets to study triple helices by molecular mechanics and dynamics

We present here a database of 32 deoxyribonucleotide triplets, that can be used as building blocks of triple helix forming deoxyribonucleotides on a computer. This database is made of all the pairing schemes of the triplets ATT, GCC⁺, ATA and GCG where the third base forms two hydrogen bonds with the purine of the first two Watson-Crick strands. The essential features of the known triple helices were preserved in the resulting structures. A triple helix can be easily built from any combination of these basic triplets. Four homogeneous and alternate triple helices thus obtained were studied by molecular mechanics and dynamics in vacuo. The results are in agreement with known experimental observations for ATT and suggest a possible structure for the GCG triple helix. In order to characterize the geometry of the structures obtained, the definitions of nucleic acid structure parameters (R.E. Dickerson et al., EMBO J. 8 (1989) 1–4) have been extended to triple helical polynucleotides.     

Packing of α-helices: Geometrical constraints and contact areas

The helical packing in sperm whale myoglobin has been examined. Using cylindrical co-ordinates based on each helix axis in turn, the overlap of the side-chain atoms of a helix with the surrounding atoms from other parts of the structure was 2.3 Å, but the distribution was not at all uniform and severe overlap occurred in at least one location for each helix. Simple axial translations or rotations of any helix in the native structure are not permitted motions. Translation perpendicular to the helix axis in at least one direction is not restricted by interlocking side-chains.The approach of two helices along the contact normal connecting their axes produces solvent exclusion effects at a distance of about 6 Å from the final position. The solvent-excluded area found in such interaction sites is equivalent to a large hydrophobic contribution to the free energy of association. The six principal sites correspond by themselves to 40% of the total area change in going from the extended sausage model to the native structure. The mean atom-packing densities for these sites and the standard deviations of these values are similar and are equal to that found for the protein as a whole.Helices of close-packed spheres form useful approximations to actual peptide helices. The helix of index number four corresponds closely to an α-helix. The required sphere size corresponds in volume to residues such as leucine or methionine. The predicted packing scheme for such helices corresponds to the three general classes of interactions actually seen.Making use of the geometry implied by the close-packed sphere helix, an algorithm is proposed for picking potentially strong helix-helix interaction sites in peptide chains of known sequence. When combined with preliminary secondary structure predictions, it is suggested that this algorithm might usefully restrict the search for these specific types of contact in the docking portion of a general folding program.     

Collagens as multidomain proteins

The number of proteins known to contain collagen-like triple helical domains is rapidly increasing. The functions of these domains are to provide molecular rods that separate spatially non-triple helical domains with varied properties and structures and to permit lateral interactions between molecules. Two-thirds of the amino acids of the triple helical domains have their side-chains at the surface of the protein. The triple helix is also a structure that is easily predictable from the primary structure. The structure of several recently discovered collagens are discussed in terms of domains and functions. The triple helical domains have sizes varying from 33 to over 1,000 amino acid residues. The longest uninterrupted triple helices are involved in the formation of the classical quarter-staggered fibrils. Other triple helical domains permit varied molecular aggregates. A very broad spectrum of non-triple helical or globular domains are interspersed by triple helices. Only those located at the extremities of the molecules are large in size, sometimes several hundred kDa, while the domains separating 2 triple helices are small (less than 50 amino acids) and provide the molecules with hinges, proteolytic cleavage sites or other specialized functions like a glycosaminoglycan attachment site. If the assembly of the 3 chains required for the triple helix formation can be controlled in vitro, collagen-like molecules offer an as yet unexploited potential for protein engineering.     

Structure stability of lytic peptides during their interactions with lipid bilayers.

In this work, molecular dynamics simulations were used to examine the consequences of a variety of analogs of cecropin A on lipid bilayers. Analog sequences were constructed by replacing either the N- or C-terminal helix with the other helix in native or reverse sequence order, by making palindromic peptides based on both the N- and C-terminal helices, and by deleting the hinge region. The structure of the peptides was monitored throughout the simulation. The hinge region appeared not to assist in maintaining helical structure but help in motion flexibility. In general, the N-terminal helix of peptides was less stable than the C-terminal one during the interaction with anionic lipid bilayers. Sequences with hydrophobic helices tended to regain helical structure after an initial loss while sequences with amphipathic helices were less able to do this. The results suggests that hydrophobic design peptides have a high structural stability in an anionic membrane and are the candidates for experimental investigation.     

Self-consistent quasineutral dust structures

Distributions of the dusty plasma parameters (electron, ion, and dust densities; dust grain charge; and ion drift velocity) in quasineutral dust structures whose dimensions are much greater than the mean free path of the ions in their interactions with neutral particles are calculated numerically under conditions such that ionization sources are located outside the structures. Planar, cylindrical, and spherical structures are investigated. It is shown that static equilibrium structures are governed by a single (basic) parameter: the electrostatic potential drop between the center of the structure and its boundary. It is found that the maximum value of the basic parameter (in energy units) does not exceed the electron temperature. The basic parameter also determines the total number of dust grains in the structure and the power of external ionization sources that are necessary to sustain this structure. The fact that the basic parameter varies within a limited range allows one to consider all the possible structures with a given dimensionality (planar, cylindrical, and spherical).     

The energy of formation of internal loops in triple-helical collagen polypeptides

The sequence-dependent local destabilization in the interior of the collagen triple helix has been evaluated by means of conformational energy computations. Using a model poly(Gly-Pro-Pro) triple helix as the reference state, a method was developed for generating local loops, i.e., internal deformations, and analyzing their conformations. A seven-residue Gly-Pro-Pro-Gly-Pro-Pro-Gly fragment was replaced by the Gly-Pro-Ala-Gly-Ala-Ala-Gly sequence in one, two, or all three of the strands of the loop region. A set of loop conformations was generated in which the ends of the loop were initially fixed in the triple-helical structure. The potential energy of the entire deformed triple helix was then minimized, resulting in a variety of structures that contained deformed loops. The conformations of the triple helices at the two ends of the loops remained essentially unchanged in many of the low-energy conformations. In numerous high-energy conformations, however, the triple-helical segments were also partially or totally disrupted. The minimum-energy conformations of the whole structures were compared in terms of rms deviations of atomic coordinates with respect to the original triple helix, and of the shapes of the loops (using a distance function derived from differential geometry). Three new geometrical parameters—stretch S, kink K, and unwinding U—were defined to describe the changes in the overall orientation of the triple helices at the two ends of the loop. It is shown that, when the number of Pro residues in a short fragment is reduced, the triple helical structure can accomodate internal loops (i.e., distortions) within a 5 kcal/mol cutoff from the essentially unperturbed triple helical structure. For structures with a Gly-Pro-Ala-Gly-Ala-Ala-Gly sequence in all three strands, the probability of finding conformations with internal loops is small, i.e., 0.06. Internal loops affect the overall orientation of these structures, as measured by the helix-distortion parameters S, K, and U. © 1995 John Wiley & Sons, Inc.     

Simulation of the packing of idealized transmembrane α-helix bundles

The aim of this study is to investigate if the packing motifs of native transmembrane helices can be produced by simulations with simple potentials and to develop a method for the rapid generation of initial candidate models for integral membrane proteins composed of bundles of transmembrane helices. Constituent residues are mapped along the helix axis in order to maintain the amino acid sequence-dependent properties of the helix. Helix packing is optimized according to a semi-empirical potential mainly composed of four components: a bilayer potential, a crossing angle potential, a helix dipole potential and a helix-helix distance potential. A Monte Carlo simulated annealing protocol is employed to optimize the helix bundle system. Necessary parameters are derived from theoretical studies and statistical analysis of experimentally determined protein structures. Preliminary testing of the method has been conducted with idealized seven Ala₂₀ helix bundles. The structures generated show a high degree of compactness. It was observed that both bacteriorhodopsin-like and δ-endotoxin-like structures are generated in seven-helix bundle simulations, within which the composition varies dependent upon the cooling rate. The simulation method has also been employed to explore the packing of N = 4 and N = 12 transmembrane helix bundles. The results suggest that seven and 12 transmembrane helix bundles resembling those observed experimentally (e.g., bacteriorhodopsin, rhodopsin and cytochrome c oxidase subunit I) may be generated by simulations using simple potentials. Received: 16 November 1998 / Revised version: 26 March 1999 / Accepted: 8 April 1999