A model of bacterial DNA segregation based upon helical geometry

A new mechanism to segregate daughter genomes in bacterial cells is suggested that is based upon the rules of geometry governing the helix clock (Mendelson, 1982a). The reorientation of cell surface string arrays used as a timing reference in the helix clock is capable of drawing apart the initial products of DNA replication. Physically linking the sister DNA replication origins to the ends of the initial cell surface string inserted into the cell surface at the start of a helix clock cycle, and linking the DNA terminus to a point along the length of the same string provides a means to mark the locations to which the genomes will segregate as well as the place where cell division will occur. The parallel packing of additional cell surface strings into an array which includes the string to which DNA is attached provides the necessary spatial rearrangements. The helical segregation model can account for the precise registration of cell divisions with the completion of replication forks in a multifork replication system, provides a basis for determining the relationship of sister cell sizes at division, and can also accommodate the asymmetrical divisions associated with minicell production and sporulation. Examination of the helical segregation theory under multifork DNA replication conditions moreover reveals that adjacent helical clocks are physically linked to one another although totally independent in terms of their progression through the clock cycle. A relationship between the initiation of DNA replication forks and the insertion of the first cell surface string associated with the start of a helix clock cycle is predicted by the model.     

Geometrical principles of homomeric β‐barrels and β‐helices: Application to modeling amyloid protofilaments

Examples of homomeric β‐helices and β‐barrels have recently emerged. Here we generalize the theory for the shear number in β‐barrels to encompass β‐helices and homomeric structures. We introduce the concept of the “β‐strip,” the set of parallel or antiparallel neighboring strands, from which the whole helix can be generated giving it n‐fold rotational symmetry. In this context, the shear number is interpreted as the sum around the helix of the fixed register shift between neighboring identical β‐strips. Using this approach, we have derived relationships between helical width, pitch, angle between strand direction and helical axis, mass per length, register shift, and number of strands. The validity and unifying power of the method is demonstrated with known structures including α‐hemolysin, T4 phage spike, cylindrin, and the HET‐s(218‐289) prion. From reported dimensions measured by X‐ray fiber diffraction on amyloid fibrils, the relationships can be used to predict the register shift and the number of strands within amyloid protofilaments. This was used to construct models of transthyretin and Alzheimer β(40) amyloid protofilaments that comprise a single strip of in‐register β‐strands folded into a “β‐strip helix.” Results suggest both stabilization of an individual β‐strip helix and growth by addition of further β‐strip helices can involve the same pair of sequence segments associating with β‐sheet hydrogen bonding at the same register shift. This process would be aided by a repeat sequence. Hence, understanding how the register shift (as the distance between repeat sequences) relates to helical dimensions will be useful for nanotube design.     

Parallel stranded DNA under scanning tunnelling microscope: the main characteristics of the double helix.

Using the scanning tunnelling microscopy we have directly observed the parallel stranded DNA of 43 bp in length, containing alternating AT-stretches. The double helix is right-handed and has the same width of each grooves equal to 17.4 A. The average pitch of the helical turn is about 34 A. The parallel double helix possesses no more than 8.6 bases per one turn. The diameter of the parallel stranded DNA molecule is 17-18 A. We conclude that in parallel DNA double helix the angle between N-glycoside bounds in trans-Crick-Watson base pairs is close to 180 degrees.     

Spontaneous formation of helical structures from phospholipid-nucleoside conjugates.

Phospholipid-nucleoside conjugates containing two myristoyl groups and a nucleotidyl group, collectively designated as dimyristoyl-5'-phosphatidylnucleosides, were enzymatically synthesized and their self-organization, morphology, and physicochemical properties investigated. The dimyristoyl-5'-phosphatidylnucleosides spontaneously assembled to form various types of helical strands. Neutral and alkaline solutions of dimyristoyl-5'-phosphatidyladenosine (DMPA) produced multihelical strands. The multihelical strand consisted of several single helical strands of approximately 50 A in diameter and helical pitch approximately 100 A. DMPA produced cigar-like scrolls (tubular structures) in acidic solution, which consisted of many double-helical strands aligned parallel to each other. Diacyl-5'-phosphatidyladenosine with a shorter chain length as long as an alkyl group, dilauroyl-5'-phosphatidyladenosine (DLPA), didecanoyl-5'-phosphatidyladenosine (DDPA), and dioctanoyl-5'-phosphatidyladenosine (DOPA) formed extended tape structures having double-helical strands aligned parallel. Dimyristoyl-5'-phosphatidylcytidine (DMPC) produced network structures at an early stage, which were slowly transformed into multihelical strands. The multihelical strands contained some single-helical strands of approximately 55 A in diameter and helical pitch approximately 150 A. DMPA produced no definite helical structure in acidic solution but rather large lamellar structures. Dimyristoyl-5'-phosphatidyluridine (DMPU) produced crystalline platelet structures of approximately 1000 A in width in both alkaline and acidic solution. A 1:1 mixture of DMPA and DMPU formed a new hybrid helical strand having a wide and thick ribbon structure of approximately 300 A in diameter and helical pitch approximately 2000 A. The formation of different helical strands and effects of chain lengths of alkyl groups and a nucleotidyl group in phospholipid-nucleoside conjugates on that of helical strands in aqueous solution are discussed.     

Helical structures in complex plasma I: Noncollective interaction

The conditions for the formation and stability of helical quasi-crystals in a complex plasma containing dust grains of equal size are investigated. A study is made of both the confinement of such helical structures in a direction transverse to the cylinder axis by means of an external parabolic potential well and the possibility of their self-confinement. Computer simulations of the helical dust structures were carried for two cases: for a structure of infinite length along the symmetry axis (or a closed structure in toroidal geometry) and for a structure of finite length. The dust grains were assumed to interact through a potential that is a superposition of the non-Debye nonlinear screened potential and the nonscreened noncollective attractive potential (the Lesage effect). Molecular dynamics simulations showed that, in the presence of dissipation, any initial random distribution of the dust grains interacting through such a potential in cylindrical geometry evolves to an equilibrium helical structure. When the external control parameter was varied smoothly, the pitch angle of the helix was observed to bifurcate (i.e., to undergo sharp jumps). The structure of the helix was also observed to bifurcate when the external parameter was varied: a helix changed into two interwoven helices, which then changed into three interwoven helices, etc. The smaller the confinement parameter (and, accordingly, the larger the radius of the helical structures) and the stronger the attractive forces between the grains, the larger the number of bifurcations. The results of analytical calculations of the parameters of the equilibrium structures and of their energies are in complete agreement with numerical results. It is also shown that noncollective attraction between dust grains makes it probable that helical structures will exists when the external confinement parameter is zero or even when it is negative. Bifurcations in such systems may provide the possibility of creating new types of memory elements.     

Escherichia coli DnaA forms helical structures along the longitudinal cell axis distinct from MreB filaments

DnaA initiates chromosomal replication in Escherichia coli at a well-regulated time in the cell cycle. To determine how the spatial distribution of DnaA is related to the location of chromosomal replication and other cell cycle events, the localization of DnaA in living cells was visualized by confocal fluorescence microscopy. The gfp gene was randomly inserted into a dnaA -bearing plasmid via in vitro transposition to create a library that included internally GFP-tagged DnaA proteins. The library was screened for the ability to rescue dnaA ^ts mutants, and a candidate gfp–dnaA was used to replace the dnaA gene of wild-type cells. The resulting cells produce close to physiological levels of GFP–DnaA from the endogenous promoter as their only source of DnaA and somewhat under-initiate replication with moderate asynchrony. Visualization of GFP-tagged DnaA in living cells revealed that DnaA adopts a helical pattern that spirals along the long axis of the cell, a pattern also seen in wild-type cells by immunofluorescence with affinity purified anti-DnaA antibody. Although the DnaA helices closely resemble the helices of the actin analogue MreB, co-visualization of GFP-tagged DnaA and RFP-tagged MreB demonstrates that DnaA and MreB adopt discrete helical structures along the length of the longitudinal cell axis.     

The hand of the helix of deoxyhemoglobin S fibers.

Electron micrographs of deoxyhemoglobin S fiber cross sections provide an end-on view of the fiber whose appearance is sensitive to small changes in orientation. We have developed a procedure to exploit this sensitivity in order to determine the hand of these particles. In a sickle hemoglobin fiber the hemoglobin molecules form long pitch helical strands which twist about the particle axis with a pitch of about 3000 A. Tilting a 400-A-thick cross section by a few degrees aligns one of the long pitch helices so that it is nearly parallel to the direction of view. When a strand of hemoglobin molecules in a fiber is aligned in this manner it appears as a strongly contrasted bright spot. It is this spot, rather than the fiber axis, which appears to be the apparent center of rotation of the cross section. The direction of the displacement of the spot from the particle axis depends upon the particle hand and tilt direction. We have used this property to determine that sickle hemoglobin fibers are right-handed particles. This method may be applicable to other particles with long pitch helices as well.     

Analysis of superhelical structures of nucleic acid-lipid conjugates by image processing

Nucleoside-phospholipid conjugates containing a nucleotidyl residue and two long alkyl chains have been synthesized and their self-organization and morphology have been investigated. In particular, 5'-phosphatidylcytidine spontaneously assembled to form linear and circular strands. Image processing analysis of the electron micrograph of the strands confirmed that they are indeed double helix reminiscent of the double-helical structure of nucleic acids. The linear and circular strands from 5'-phosphatidylcytidine had grooves of approximately 100 A in diameter and right-handed helical pitch of approximately 240 A.     

The cell envelope of Thermoproteus tenax: three-dimensional structure of the surface layer and its role in shape maintenance

The sulphur-dependent archaebacterium Thermoproteus tenax has a cylindrical cell shape variable in length, but constant in diameter. Its whole surface is covered by a regular protein layer (S-layer). The lattice has p6 symmetry and a lattice constant of 32.8 nm. The three-dimensional reconstruction from a tilt series of isolated and negatively stained S-layer shows a complex mass distribution of the protein: a prominent, pillar-shaped protrusion is located at the 6-fold crystallographic axis with radiating arms connecting neighbouring hexamers in the vicinity of the 3-fold axis. The base vectors of the S-layer lattice have a preferred orientation with respect to the longitudinal axis of the cell. The layer can be seen as a helical structure consisting of a right-handed, two-stranded helix, with the individual chains running parallel. Supposing that new S-layer protein is inserted at lattice faults (wedge disclinations) near the poles, growing of the layer would then proceed by moving a disclination at the end of the helix. The constant shape of the cell, as well as the particular structure of the layer, strongly suggest that this S-layer has a shape-maintaining function.     

A structural model for the polyadenylic acid single helix.

In the crystal, the poly(A) fragment ApApA assumes a conformation with the 5′-terminal and middle adenosines in a single helical arrangement. From the atomic co-ordinates of these two nucleotides the structure of the poly(A) single helix was derived mathematically. The helix has a pitch height of 25.4 Å, nine nucleotides per turn and the normals to the adenine bases form an angle of 66 ° with the helix axis.     

Molecular dynamics simulations predict a tilted orientation for the helical region of dynorphin A(1-17) in dimyristoylphosphatidylcholine bilayers

The structural properties of the endogenous opioid peptide dynorphin A(1-17) (DynA), a potential analgesic, were studied with molecular dynamics simulations in dimyristoylphosphatidylcholine bilayers. Starting with the known NMR structure of the peptide in dodecylphosphocholine micelles, the N-terminal helical segment of DynA (encompassing residues 1-10) was initially inserted in the bilayer in a perpendicular orientation with respect to the membrane plane. Parallel simulations were carried out from two starting structures, systems A and B, that differ by 4 A in the vertical positioning of the peptide helix. The complex consisted of approximately 26,400 atoms (dynorphin + 86 lipids + approximately 5300 waters). After >2 ns of simulation, which included >1 ns of equilibration, the orientation of the helical segment of DynA had undergone a transition from parallel to tilted with respect to the bilayer normal in both the A and B systems. When the helix axis achieved a approximately 50 degrees angle with the bilayer normal, it remained stable for the next 1 ns of simulation. The two simulations with different starting points converged to the same final structure, with the helix inserted in the bilayer throughout the simulations. Analysis shows that the tilted orientation adopted by the N-terminal helix is due to specific interactions of residues in the DynA sequence with phospholipid headgroups, water, and the hydrocarbon chains. Key elements are the "snorkel model"-type interactions of arginine side chains, the stabilization of the N-terminal hydrophobic sequence in the lipid environment, and the specific interactions of the first residue, Tyr. Water penetration within the bilayer is facilitated by the immersed DynA, but it is not uniform around the surface of the helix. Many water molecules surround the arginine side chains, while water penetration near the helical surface formed by hydrophobic residues is negligible. A mechanism of receptor interaction is proposed for DynA, involving the tilted orientation observed from these simulations of the peptide in the lipid bilayer.     

Analysis of the stability of hemoglobin S double strands.

The deoxyhemoglobin S (deoxy-HbS) double strand is the fundamental building block of both the crystals of deoxy-HbS and the physiologically relevant fibers present within sickle cells. To use the atomic-resolution detail of the hemoglobin-hemoglobin interaction known from the crystallography of HbS as a basis for understanding the interactions in the fibers, it is necessary to define precisely the relationship between the straight double strands in the crystal and the twisted, helical double strands in the fibers. The intermolecular contact conferring the stability of the double strand in both crystal and fiber is between the beta6 valine on one HbS molecule and residues near the EF corner of an adjacent molecule. Models for the helical double strands were constructed by a geometric transformation from crystal to fiber that preserves this critical interaction, minimizes distortion, and makes the transformation as smooth as possible. From these models, the energy of association was calculated over the range of all possible helical twists of the double strands and all possible distances of the double strands from the fiber axis. The calculated association energies reflect the fact that the axial interactions decrease as the distance between the double strand and the fiber axis increases, because of the increased length of the helical path taken by the double strand. The lateral interactions between HbS molecules in a double strand change relatively little between the crystal and possible helical double strands. If the twist of the fiber or the distance between the double strand and the fiber axis is too great, the lateral interaction is broken by intermolecular contacts in the region around the beta6 valine. Consequently, the geometry of the beta6 valine interaction and the residues surrounding it severely restricts the possible helical twist, radius, and handedness of helical aggregates constructed from the double strands. The limitations defined by this analysis establish the structural basis for the right-handed twist observed in HbS fibers and demonstrates that for a subunit twist of 8 degrees, the fiber diameter cannot be more than approximately 300 A, consistent with electron microscope observations. The energy of interaction among HbS molecules in a double strand is very slowly varying with helical pitch, explaining the variable pitch observed in HbS fibers. The analysis results in a model for the HbS double strand, for use in the analysis of interactions between double strands and for refinement of models of the HbS fibers against x-ray diffraction data.     

The bacterial cytoskeleton: in vivo dynamics of the actin-like protein Mbl of Bacillus subtilis

Mbl is a bacterial actin homolog that controls cell morphogenesis in Bacillus subtilis. A functional GFP-Mbl fusion protein was used to examine the behavior of the helical cables formed by Mbl protein in live B. subtilis cells. The cables undergo dynamic changes during cell cycle progression. They are stable but not rigid while elongating in parallel with cell growth, and they require septum formation to divide/cleave. Fluorescence recovery after photobleaching (FRAP) analysis showed that the cables are continuously remodeled during cell elongation. Turnover occurs along the length of the helical Mbl filaments, with no obvious polarity and a recovery half-time of about 8 min. These findings have important implications for the nature of bacterial cell wall architecture and synthesis.     

Polarization selectivity of interaction of DNA molecules with X-ray radiation

The optimum form of a long helical molecule, which DNA is, has been calculated in terms of the classical electromagnetic theory. Three different methods of classical electrodynamics are used: the theory of dipole radiation of electromagnetic waves, the energetic power approach, and a helical model of molecules of chiral medium. In all three cases, an identical result for the optimum geometrical form of a long spiral molecule has been obtained. The lead angle between the tangent to the helix and the plane normal to the axis of the helix should be equal to 24.5 degrees. This condition imposes restrictions on the radius and the pitch of the helical molecule. The experimentally measured geometrical characteristics of the DNA molecule satisfy the theoretically calculated condition precisely enough. Having the optimum geometrical form, the DNA molecule is not influenced by a circularly right-polarized electromagnetic wave in the soft X-ray range λ = 7–8 nm. This wave, for which the right-handed DNA molecule is “transparent,” should propagate orthogonally to the helix axis and form a right-handed screw in space. The wave radiated by the right-handed DNA molecule orthogonally to helix axis in the range of λ ≈ 7–8 nm has, accordingly, the left-handed circular polarization. The polarization selectivity of the DNA molecule by the action of X-ray radiation is exhibited strongly enough in the wavelength range of λ ≈ 1–35 nm. The results obtained are valid for any distribution of electric currents in DNA, i.e., for any sequence of nitrogenous bases in DNA.     

Intramolecular interference effects in dynamic light scattering: rigid double spirals and superhelical DNAs

A theory is developed for dynamic light scattering (DLS) from rigid double spirals by treating an invisible rigid cylinder with two helical scattering stripes on opposite sides of its cylindrical surface. The exact initial, or first cumulant, diffusion coefficient Dapp (K) is obtained in terms of the translational diffusion coefficients (D parallel and D perpendicular) parallel and perpendicular to the symmetry axis, the rotational diffusion coefficients (DR parallel and DR perpendicular) around the symmetry and transverse axes, the length (L) and radius (b) of the cylindrical surface bearing the stripes, and the pitch (p). Interference effects, namely geometrical antiresonances, between strands, produce deep minima in the static structure factor S (K) and corresponding prominent peaks in Dapp (K). These peaks in Dapp (K) depend sensitively on the rotational dynamics around the symmetry axis, and nearly vanish when DR parallel = 0. Some results for single spirals are also presented. A simpler model in which scattering points are attached at opposite ends of an otherwise invisible thin rigid rod is also treated, and shown to exhibit modest minima in S (K) and corresponding maxima in Dapp (K). Confining this rod to a plane containing K enhances the amplitudes of the oscillations in S (K) and Dapp (K), as expected. Rigid double spirals are employed as crude models for interwound supercoiled DNAs in order to assess the possible occurrence of interference effects. Although native supercoiled DNAs exhibit a cylinder diameter that is much too small to exhibit geometrical antiresonances in the presently accessible range of K2, nearly relaxed supercoiled DNAs are predicted to exhibit their first maximum in Dapp (K) just inside this range. Previously reported data for the effect of Escherichia coli single-strand binding (ssb) protein on the DLS of supercoiled pBR322 DNA cannot be mimicked by a gradual homogeneous reduction of superhelix density with increasing ssb, but instead can be mimicked by inhomogeneous all-or-none binding in which uncomplexed native DNAs and nearly relaxed saturated ssb/DNA complexes coexist in varying proportions. Experimental Dapp (K) and S (K) data for a sample of relaxed pUC8 dimers display, respectively, a broad maximum and a corresponding minimum, in qualitative agreement with rough theoretical predictions.     

MICROTUBULES IN THE SPERMATIDS OF THE DOMESTIC FOWL

Spermiogenesis in chicken has been examined in order to see whether the radical changes observed in cell shape can be related to the presence of cytoplasmic microtubules. A highly ordered array of tubules has been found which surrounds the nucleus as it elongates from a sphere to a slender cylinder. The structure of the array has been determined by following the tubules through 12–14 adjacent serial sections, and it is a left-handed double helix. Faint cross-bridges connect consecutive turns of the two helices. After the change in nuclear shape is complete, the helical system of microtubules disappears and is replaced by a set of almost straight tubules which run parallel to the long axis of the nucleus. These tubules remain while the spermatid nucleus condenses isotropically to its final size. We suggest that the helix is the agent which effects nuclear elongation and that the subsequent system of paraxial tubules determines the curvature of the final sperm head. Evidence for these suggestions is found in the form of spermatids which have failed to develop properly. In an appendix we consider the kinematics of single and multiple helix systems and discuss the revelance of these models to the morphogenesis of chicken spermatids.     

Helix‐sheet packing in proteins

The three‐dimensional structure of a protein is organized around the packing of its secondary structure elements. Although much is known about the packing geometry observed between α‐helices and between β‐sheets, there has been little progress on characterizing helix–sheet interactions. We present an analysis of the conformation of αβ₂ motifs in proteins, corresponding to all occurrences of helices in contact with two strands that are hydrogen bonded. The geometry of the αβ₂ motif is characterized by the azimuthal angle θ between the helix axis and an average vector representing the two strands, the elevation angle ψ between the helix axis and the plane containing the two strands, and the distance D between the helix and the strands. We observe that the helix tends to align to the two strands, with a preference for an antiparallel orientation if the two strands are parallel; this preference is diminished for other topologies of the β‐sheet. Side‐chain packing at the interface between the helix and the strands is mostly hydrophobic, with a preference for aliphatic amino acids in the strand and aromatic amino acids in the helix. From the knowledge of the geometry and amino acid propensities of αβ₂ motifs in proteins, we have derived different statistical potentials that are shown to be efficient in picking native‐like conformations among a set of non‐native conformations in well‐known decoy datasets. The information on the geometry of αβ₂ motifs as well as the related statistical potentials have applications in the field of protein structure prediction. Proteins 2010. © 2010 Wiley‐Liss, Inc.     

Crystal structure analysis of an A-DNA fragment at 1.8 A resolution: d(GCCCGGGC).

Single crystals of the self-complementary octadeoxyribonucleotide d(GCCCGGGC) have been analysed by X-ray diffraction methods at a resolution of 1.8 A. The tetragonal unit cell of space group P4(3)2(1)2 has dimensions of a = 43.25 A and c = 24.61 A and contains eight strands of the oligonucleotide. The structure was refined by standard crystallographic techniques to an R factor of 17.1% using 1359 3 sigma structure factor observations. Two strands of the oligonucleotide are related by the crystallographic dyad axis to form a DNA helix in the A conformation. The d(GCCCGGGC) helix is characterized by a wide open major groove, a near perpendicular orientation of base pairs to the helix axis and an unusually small average helix twist angle of 31.3 degrees indicating a slightly underwound helix with 11.5 base pairs per turn. Extensive cross-strand stacking between guanine bases at the central cytosine-guanine step is made possible by a number of local conformational adjustments including a fully extended sugar-phosphate backbone of the central guanosine nucleotide.     

Does Paramecium sense gravity?

In order to get an insight into the cellular mechanisms for the integration of the effects of gravity, we investigated the gravitactic behaviour in Paramecium. There are two main categories for the model of the mechanism of gravitaxis; one is derived on the basis of the mechanistic properties of the cell (physical model) and the other of the physiological properties including cellular gravireception (physiological model). In this review article, we criticized the physical models and introduced a new physiological model. Physical models postulated so far can be divided into two; one explaining the negative gravitactic orientation of the cell in terms of the static torque generated by the structural properties of the cell (gravity-buoyancy model by Verworn, 1889 and drag-gravity model by Roberts, 1970), and the other explaining it in terms of the dynamic torque generated by the helical swimming of the cell (propulsion-gravity model by Winet and Jahn, 1974 and lifting-force model by Nowakowska and Grebecki, 1977). Among those we excluded the possibility of dynamic-torque models because of their incorrect theoretical assumptions. According to the passive orientation of Ni(2+)-immobilized cells, the physical effect of the static torque should be inevitable for the gravitactic orientation. Downward orientation of the immobilized cells in the course of floating up in the hyper-density medium demonstrated the gravitactic orientation is not resulted by the nonuniform distribution of cellular mass (gravity-buoyancy model) but by the fore-aft asymmetry of the cell (drag-gravity model). A new model explaining the gravitactic behaviour is derived on the basis of the cellular gravity sensation through mechanoreceptor channels of the cell membrane. Paramecium is known to have depolarizing receptor channels in the anterior and hyperpolarizing receptors in the posterior of the cell. The uneven distribution of the receptor may lead to the bidirectional changes of the membrane potential by the selective deformation of the anterior and posterior cell membrane responding to the orientation of the cell in the gravity field; i.e. negative- and positive-going shift of the potential due to the upward and downward orientation, respectively. The orientation dependent changes in membrane potential with respect to gravity, in combination with the close coupling of the membrane potential and the ciliary locomotor activity, may allow the changes in swimming direction along with those in the helical nature of the swimming path; upward shift of axis of helix by decreasing the pitch angle due to hyperpolarization in the upward-orienting cell, and also the upward shift by increasing the pitch angle due to depolarization in the downward-orienting cell. Computer simulation of the model demonstrated that the cell can swim upward along the "super-helical" trajectory consisting of a small helix winding helically an axis parallel to the gravity vector, after which the model was named as "Super-helix model". Three-dimensional recording of the trajectories of the swimming cells demonstrated that about a quarter of the cell population drew super-helical trajectory under the unbounded, thermal convection-free conditions. In addition, quantitative analysis of the orientation rate of the swimming cell indicated that gravity-dependent orientation of the swimming trajectory could not be explained solely by the physical static torque but complementarily by the physiological mechanism as proposed in the super-helix model.