International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 19.5, pp. 583-592   | 1 | 2 |
https://doi.org/10.1107/97809553602060000871

Chapter 19.5. Fibre diffraction

R. Chandrasekarana* and G. Stubbsb

aWhistler Center for Carbohydrate Research, Purdue University, West Lafayette, IN 47907, USA, and  bDepartment of Molecular Biology, Vanderbilt University, Nashville, TN 37235, USA
Correspondence e-mail:  chandra@purdue.edu

Fibre diffraction, diffraction from a specimen in which the diffracting units are randomly oriented about an axis (the fibre axis), is a powerful technique for determining the structural details of helical polymers, often at atomic resolution. Fibre specimens may be noncrystalline or polycrystalline. Expressions are given for diffraction from both types of fibre. Diffraction patterns are recorded photographically or using imaging plates or charge-coupled devices (CCDs). Data processing includes correction for experimental effects, background subtraction and integration of reflections. Initial models, constructed so as to be consistent with the helical parameters or determined directly, are refined against diffraction data by various methods. Difference Fourier–Bessel syntheses are used to locate missing fragments of the structure, including ions and water molecules. Developments in theory and practice and the availability of fast computers have led to descriptions of the molecular architecture of a wide variety of biopolymers, ranging from simple polypeptides, polynucleotides and polysaccharides to complex filamentous viruses, cytoskeletal filaments and other large macromolecular assemblies.

19.5.1. Introduction

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Many biopolymers are long helical structures and have a natural tendency to form fibres. This tendency severely impedes the growth of single crystals from these polymers, and even if crystals can be grown, the molecular interactions in the crystals rarely correspond to the biologically significant interactions in the fibres. Conventional macromolecular crystallography is therefore often not applicable to these systems. Fibre diffraction, however, is a powerful technique for determining the structural details of such polymers. It has been used to study a wide variety of bio­polymers, ranging from simple polypeptides, polynucleotides and polysaccharides to complex filamentous viruses and cytoskeletal filaments.

Fibres can have relatively high degrees of order, although falling short of true three-dimensional crystallinity. The key difference between fibres and crystals, however, is that in fibres the fundamental structural aggregates, although parallel to each other, are randomly oriented about the fibre axis. Consequently, the diffraction pattern is cylindrically averaged. This cylindrical averaging is the defining characteristic of fibre diffraction.

On the basis of this definition, fibre diffraction may also be considered to include diffraction from many biological mem­­brane specimens, and much of fibre-diffraction theory also applies to membrane diffraction. In general, however, the diffracting units in fibres have helical symmetry, whereas those of membranes do not.

In addition to the loss of information due to cylindrical averaging, fibre-diffraction patterns reflect a generally limited degree of order and rarely extend beyond 3 Å resolution. Consequently, the number of data obtainable from a fibre is considerably less than that from a single crystal having a similar size of asymmetric unit. The use of stereochemical information to supplement the diffraction data is therefore essential. For polymers with small asymmetric units, such as polynucleotides, structural chemical information can be used to construct models consistent with the helical parameters and molecular dimensions obtained from the diffraction data. For the larger asymmetric units found in aggregates, such as viruses, initial models must be constructed in other ways. However, in all cases the combination of diffraction data and stereochemistry can be used to refine both molecular structures and packing parameters. Refinement in this way is very similar to that used in macromolecular crystallography, but because of the limited number of experimental data, stereochemical restraints are particularly important in fibre diffraction. As in crystallography, difference-electron-density maps are used in conjunction with refinement to identify missing portions and determine the correctness of the models and, in favourable cases, to locate ions and solvent molecules associated with the polymers.

19.5.2. Types of fibres

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Fibres fall into essentially two classes with respect to the degree of ordering of the polymer chains. Within each class, there are varying degrees of disorder; furthermore, many fibres exhibit properties intermediate between those of the two ideal classes.

In noncrystalline fibres, the polymers are parallel to each other, but their positions and orientations are otherwise uncorrelated. Diffraction patterns from these fibres are confined to layer lines (Fig. 19.5.2.1a[link]) because of the repeating nature of the polymer helix, but are otherwise continuous and correspond to the cylindrical average of the Fourier transform of a single particle.

[Figure 19.5.2.1]

Figure 19.5.2.1 | top | pdf |

X-ray diffraction patterns showing (a) continuous intensity on layer lines from an oriented nucleic acid fibre and (b) Bragg reflections from an oriented and polycrystalline polysaccharide fibre.

In polycrystalline fibres, the polymers form fully ordered microcrystallites, and each fibre consists of many such microcrystallites, randomly oriented about the fibre axis. In diffraction patterns from polycrystalline fibres, the layer lines are sampled to form discrete reflections (Fig. 19.5.2.1b[link]); the diffraction pattern is the cylindrical average of a single-crystal diffraction pattern and is, in fact, equivalent to the diffraction pattern that would be obtained from a rotating single crystal.

Polycrystalline fibres may be disordered in various ways. For example, the helical polymers may be subject to rotational or translational disorder, and this disorder may be partial (a small number of alternative packings for each particle) or complete (for example, completely random rotational particle orientations). Rotational disorder may be coupled to translational disorder (screw disorder). The resulting diffraction patterns may contain both discrete reflections and continuous diffraction along layer lines; depending upon the type of disorder, the discrete reflections may be confined to the equator (layer line zero) or the low-resolution part of the pattern, or they may be dispersed throughout the pattern. Variations in diffraction effects due to different types of disorder have been discussed by Arnott (1980[link]) and Stroud & Millane (1995[link]).

Fibres are also subject to orientational disorder. The polymer helices in noncrystalline fibres and the microcrystallites in crystalline fibres are not perfectly aligned to the fibre axis; the deviation from parallelism is called the disorientation of the fibre. Disorientation causes the reflections from crystalline fibres and the diffracted intensity from noncrystalline fibres to be spread into arcs (Debye–Scherrer arcs).

19.5.3. Diffraction by helical molecules

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19.5.3.1. Fibre diffraction patterns

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As noted above, the diffraction pattern from a fibre is confined to layer lines because of the repeating nature of the polymer helix. The layer lines in reciprocal space are perpendicular to the fibre axis in real space. The layer line passing through the origin in reciprocal space is called the equator or zero layer line. The line in a diffraction pattern normal to the equator and passing through the origin is called the meridian. If the fibre axis is perpendicular to the incident X-ray beam, the recorded diffraction pattern is symmetric about both the equator and the meridian. If the fibre is not normal to the incident beam, the pattern is symmetric only about the meridian.

19.5.3.2. Helical symmetry

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It is convenient to use cylindrical coordinates to describe helical molecules. In real space we use coordinates (r, ϕ, z); in reciprocal space (R, ψ, Z). By convention, the z axis is the helix axis and the line [(\varphi = 0, z = 0)] corresponds to the x axis in Cartesian coordinates. The repeat distance along the z axis is c. Within this distance, there are u repeating units in t turns of the helix. If the coordinates of a point in the first repeating unit are (r, ϕ, z), then applying the helical symmetry gives the coordinates of the corresponding point in the (k + 1)th repeating unit as [(r, \varphi + 2\pi kt/u, z + kc/u)].

19.5.3.3. Structure factors

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Cochran et al. (1952)[link] showed that the structure factor on layer line l of a helix made up of repeating subunits is[{\hbox{{\bf F}}(R,\! \psi,\! Z) = \!\textstyle\sum\limits_{j}\! \textstyle\sum\limits_{n}\displaystyle f_{j}\, J_{n} (2\pi Rr_{j}) \exp \{i\,[n(\psi + \pi/2) - n\varphi_{j}\! + \!2\pi lz_{j}/c]\}.} \eqno(19.5.3.1)]Diffraction occurs only for [Z = l/c.\ r_{j}, \varphi_{j}] and [z_{j}] are the real-space coordinates of atom j in the repeating unit of the helix; [f_{j}] is the atomic scattering factor of that atom. [J_{n}] is the Bessel function of the first kind of order n. The summation over n includes only those values of n that satisfy the selection rule [l = tn + um, \eqno(19.5.3.2)]where m is any integer. In practice, the summation may be limited to values of [|n|] less than [2\pi r_{\max} R + 2], where [r_{\max}] is the radius of the outermost atom in the polymer, because the value of a Bessel function [J_{n}(x)] is negligible for n greater than about x + 2. For low-order Bessel functions or applications requiring greater accuracy, slight variations of this limitation are used.

The structure factor F is a complex number with an amplitude and phase, and is fully equivalent to that derived using the trigonometric functions in crystallography. The expression for intensity [I = \hbox{{\bf FF}}^{*} = |\hbox{{\bf F}}|^{2}] holds good.

Equation (19.5.3.1[link]) can be rewritten (Klug et al., 1958[link]) as [\hbox{{\bf F}}(R, \psi, l/c) = \textstyle\sum\limits_{n}\displaystyle \hbox{{\bf G}}_{n, \, l} (R) \exp [in(\psi + \pi /2)], \eqno(19.5.3.3)]where the Fourier–Bessel structure factor [\hbox{{\bf G}}_{n, \, l} (R)] is independent of ψ and is given by [\hbox{{\bf G}}_{n, \, l} (R) = \textstyle\sum\limits_{j}\displaystyle f_{j} \,J_{n} (2\pi Rr_{j}) \exp [i(-n\varphi_{j} + 2\pi lz_{j}/c)]. \eqno(19.5.3.4)][J_{n} (0)] is 1 when [n = 0] and 0 otherwise. For this reason, the struc­ture factors on the meridian [(R = 0)] are nonzero only on layer lines for which l is an integral multiple of u. Hence, a visual inspection of the diffraction pattern often helps to determine u.

19.5.3.4. Fourier–Bessel syntheses

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Electron densities may be calculated for crystalline fibres, as they are in crystallography, using Fourier syntheses with coefficients determined for the crystalline reflections.

For noncrystalline fibres, it is more convenient to use Fourier–Bessel syntheses: the electron density ρ at point [(r, \varphi, z)] is [{\rho (r, \varphi, z) = (1/c) \textstyle\sum\limits_{l}\displaystyle \textstyle\sum\limits_{n}\displaystyle \hbox{{\bf g}}_{n, \, l} (r) \exp [i(n\varphi - 2\pi lz/c)],}\hfill\!\!\! \eqno(19.5.3.5)]where [\hbox{{\bf g}}_{n, \, l} (r) = \textstyle\int\limits_{0}^{\infty}\displaystyle \hbox{{\bf G}}_{n, \, l} (R) \,J_{n} (2\pi Rr)2\pi R \kern3pt\hbox{d}R. \eqno(19.5.3.6)]

19.5.3.5. Diffracted intensities: noncrystalline fibres

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The intensity in the diffraction pattern of a noncrystalline fibre is the cylindrical average of the square of the Fourier transform (Franklin & Klug, 1955[link]): [\eqalignno{ I (R, l) &= \langle|\hbox{{\bf F}}(R, \psi, l/c)|^{2}\rangle_{\psi} &\cr &= \textstyle\sum\limits_{n}\displaystyle \hbox{{\bf G}}_{n, \, l} (R) {\hbox{{\bf G}}^{*}_{n, \, l}} (R) &\cr &= \textstyle\sum\limits_{n}\displaystyle |\hbox{{\bf G}}_{n, \, l} (R)|^{2}. &(19.5.3.7)\cr}]The intensity varies continuously as a function of R along each layer line (Fig. 19.5.2.1a[link]).

19.5.3.6. Diffracted intensities: polycrystalline fibres

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The intensity in the diffraction pattern of a polycrystalline fibre consists of Bragg reflections on layer lines (Fig. 19.5.2.1b[link]). On each layer line, owing to the lattice sampling that arises from the lateral organization of the polymers, intensities are observed at discrete R values defined by the reciprocal-lattice points. In the case of monoclinic (with c as the unique axis), orthorhombic and hexagonal systems, the reflection positions are determined by equations (19.5.3.8[link]), (19.5.3.9[link]) and (19.5.3.10[link]), respectively. [\eqalignno{ R^{2}_{hk} &= h^{2}a^{*2} + k^{2}b^{*2} + 2hka^{*}b^{*} \cos \gamma^{*}, &(19.5.3.8)\cr R^{2}_{hk} &= h^{2}a^{*2} + k^{2}b^{*2}, &(19.5.3.9)\cr R^{2}_{hk} &= (h^{2} + k^{2} + hk) a^{*2}. &(19.5.3.10)} %fd(19.5.3.10)]Consequently, on each layer line, superposition occurs between reciprocal-lattice points (hkl) and ([\bar{h}\bar{k}l]) for monoclinic; (hkl), ([\bar{h}kl]), ([h\bar{k}l]) and ([\bar{h}\bar{k}l]) for orthorhombic; and (hkl), ([\bar{h}\bar{k}l]), (khl), ([\bar{k}\bar{h}l]), (kil), ([\overline{ki}l]), (ikl), ([\overline{ik}l]), (ihl), ([\overline{ih}l]), (hil) and ([\overline{hi}l]), where [i = -(h + k)], for hexagonal systems. Depending upon the unit-cell dimensions, other reflections having the same R value may also be superposed to give a single intensity, and those having R values close to each other may be difficult to resolve. All superposed reflections must be considered individually when calculating such composite intensities.

19.5.4. Fibre preparation

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Natural fibre specimens may require only the selection of fibres or regions of fibres in which the polymers are well oriented, but many other fibres must be made in the laboratory, and orientation can often be improved by a wide variety of procedures.

Orientation often requires controlled conditions of relative humidity and temperature during the preparation of the fibre; in many cases, these conditions must be maintained during data collection. In some cases, tension must be applied to the fibres; in an increasing number of cases, magnetic fields have been found to improve orientation. Fibres may be drawn directly from concentrated polymer solutions or made by stretching gels, using weights to stretch strips of polymer films cast on Teflon blocks or applying radial heating while forming polymer films (Arnott, Guss et al., 1974[link]; Chandrasekaran, Radha, Lee & Zhang, 1994[link]). A drop of concentrated polymer solution may simply be dried while suspended between two supports (McDonald et al., 2008[link]). Magnetic fields have dramatically improved the orientation in dried fibres of polymers having significant dipole moments (Torbet, 1987[link]).

Oriented sols, generally enclosed in glass capillaries, are usually made using shearing forces, either by moving the sol in the capillary (Gregory & Holmes, 1965[link]; Kendall & Stubbs, 2006[link]) or by centrifugation (Cohen et al., 1971[link]). Again, magnetic fields can greatly improve orientation, sometimes in combination with centrifugation (Yamashita, Suzuki & Namba, 1998[link]).

Any of these stretching or orienting processes might facilitate the growth of long microcrystallites along the fibre axis. Crystallization in general and lateral organization in particular are achieved primarily by careful choice of solution conditions, including solvent, pH, additives, relative humidity and temperature. In both crystalline and noncrystalline specimens, annealing processes are often important to both crystallization and orientation.

19.5.5. Data collection

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Fibre-diffraction data were historically collected using laboratory X-ray sources and photographic film, but synchrotron sources (Shotton et al., 1998[link]) are now generally used for final data collection (Janaswamy & Chandrasekaran, 2005[link], 2006[link]; Tsuruta & Irving, 2008[link]), taking advantage of reduced exposure time, the potential for time-resolved studies and the fact that many fibres (or the well oriented regions of fibres) are too small for laboratory data collection. Charge-coupled device (CCD) detectors have largely replaced film. Pinhole cameras, mirror-monochromator optics and double-mirror optics are used in different applications. Diffraction by most fibres is inherently weak, and very long repeat distances often require long distances between the specimen and the detector, so fibre cameras are often flushed with helium or kept under vacuum to reduce air scatter. Constant relative humidity is often required, and is achieved by bubbling a helium stream through a saturated salt solution followed by a salt trap or by isolating the sample in equilibrium with a saturated salt solution (McDonald et al., 2008[link]).

The X-ray beam commonly strikes a stationary fibre perpendicular to the fibre axis. Because of the cylindrical averaging of the data, this procedure allows most of the diffraction pattern to be collected in a single exposure. There is, however, a `blind region' around the meridian, where the Ewald sphere does not intersect the diffraction pattern (Fraser et al., 1976[link]). Data in this region are collected by tilting the fibre.

19.5.6. Data processing

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19.5.6.1. Coordinate transformation

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Data must be transformed from detector space into reciprocal space (Fraser et al., 1976[link]). Transformation of coordinates requires determination of the origin of the diffraction pattern in detector space, the fibre tilt angle β, the twist angle (often called in-plane tilt, the inclination of the projection of Z along the beam to the detector coordinate system) and the specimen-to-detector distance. It may also require determination of the detector mis-setting angles (the deviation of the normal to the detector plane from the beam). All of these parameters can be determined by comparing equivalent reflections in the diffraction pattern.

Most data-processing programs determine the transformation parameters by some form of minimization of the deviation from equivalence in the positions of well resolved equivalent reflections. The tilt was traditionally determined by comparing the apparent Z values of equivalent reflections, but the apparent value of R for near-meridional reflections is much more sensitive to tilt. The minimization set should therefore include some near-meridional reflections if the tilt value is to be determined accurately. The helical repeat distance and, for polycrystalline fibres, the unit-cell parameters must also be determined at this time, but helical repeat distance and specimen-to-detector distance are so highly correlated that it is not often practical to refine both. Data may be visualized and in many cases transformed and processed using a number of software applications, including the programs WCEN (Bian et al., 2006[link]), FibreFix (Rajkumar et al., 2005[link]) and FIT2D (http://www.esrf.eu/computing/scientific/FIT2D).

19.5.6.2. Intensity correction

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Data intensities must be corrected for geometric and polarization effects (Fraser et al., 1976[link]; Millane & Arnott, 1986[link]). The geometric correction has two components: a factor due to the geometry of the intersection between the diffraction pattern in reciprocal space and the sphere of reflection, and a factor due to the angle of incidence of the diffracted beam on the detector. The first factor is analogous to the Lorentz factor in crystallography, which arises because of the time taken for a reflection from a moving sample to pass through the Ewald sphere. The geometric correction can be applied to each data point as a single correction (Fraser et al., 1976[link]). For both crystalline and noncrystalline fibres, it has often been convenient to integrate reflections or arcs before applying Lorentz and polarization corrections, or to apply some corrections before integration and some after (Millane & Arnott, 1986[link]), but this practice is decreasing as computational power increases. The Lorentz correction for crystalline fibres is[{1/L = 2 \pi \sin \theta\, [\cos^{2}\theta \cos^{2}\beta - (\cos \sigma - \sin \theta \sin \beta)^{2}]^{1/2},}\hfill\!\!\!\! \eqno(19.5.6.1)]where θ is the Bragg angle and [\tan \sigma = R/Z] (Millane & Arnott, 1986[link]). The polarization correction for an unpolarized X-ray beam is[p = (1 + \cos^{2} 2\theta)/2. \eqno(19.5.6.2)]For a polarized beam, such as a synchrotron beam, the expression for p is more complicated (Azároff, 1955[link]; Kahn et al., 1982[link]; Bian et al., 2006[link]). Intensities should be divided by Lp. Intensities may also be corrected for nonlinearity of detector response and for absorption by the specimen and by detector components.

19.5.6.3. Background subtraction

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The background can be very high in fibre-diffraction data because of long exposure times and scattering from amorphous material. Because of specimen disorientation, fibre-diffraction data often contain large regions where there is no space between layer lines, so local-background-fitting methods are rarely useful. The background may be determined by fitting an analytical function to intensities at points between reflections (Millane & Arnott, 1985[link]; Lorenz & Holmes, 1993[link]), or by fitting a function that includes both signal and background components to the reflection data. This type of profile fitting has been described for individual reflections (Fraser et al., 1976[link]), for data in concentric rings about the centre of the diffraction pattern (Makowski, 1978[link]) and for entire data sets (Yamashita et al., 1995[link]; Ivanova & Makowski, 1998[link]).

19.5.6.4. Integration of crystalline fibre data

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The variation of reflection shape in detector space can be determined using a few sharp reflections and taking into account parameters related to crystallite size and disorientation in the specimen (Millane & Arnott, 1986[link]). This allows the integration boundary of a reflection to be determined. Sometimes, the boundary encompasses two or more reflections too close to separate; such reflections are considered to constitute a composite reflection.

19.5.6.5. Integration of continuous data

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In diffraction from noncrystalline fibres, intensity is a function of R on each layer line. Angular deconvolution (Makowski, 1978[link]; Namba & Stubbs, 1985[link]; Yamashita et al., 1995[link]) or profile fitting (Millane & Arnott, 1986[link]) corrects for disorientation and overlap between adjacent layer lines and may also incorporate background subtraction. The intensity determined in this way should be corrected for geometric and other effects if this has not been done previously (Section 19.5.6.2[link]; Namba & Stubbs, 1985[link]; Millane & Arnott, 1986[link]).

19.5.7. Determination of structures

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If the amplitude and phase of each diffracted wave are known, structure determination is, in principle, straightforward (Section 19.5.3.4[link]). In practice, however, the phase problem for fibres is more acute than for single crystals because of the limited resolution of the data, and because the diffracted intensities overlap as a result of disorientation and cylindrical averaging. Patterson methods (MacGillavry & Bruins, 1948[link]; Stubbs, 1987[link]) have some­times been useful, but the cylindrically averaged Patterson function is usually too complicated for detailed interpretation. Phasing by heavy-atom methods is not practical for polymers with small unit cells because of the difficulties in incorporating heavy atoms into the structures. Structures having small unit cells are instead determined by constructing initial models based on chemical information and the observed helical parameters. Extensions of the isomorphous-replacement method (Namba & Stubbs, 1985[link]) have been useful in determining structures, such as those of helical viruses, in which the unit cells are much larger. In all cases, refinement and evaluation of the model structures are essential. A flow chart of the sequential steps in the determination and refinement of fibre structures with small unit cells is shown in Fig. 19.5.7.1[link].

[Figure 19.5.7.1]

Figure 19.5.7.1 | top | pdf |

Flow chart of the principal steps in the determination and refinement of fibre structures with small unit cells.

19.5.7.1. Initial models: small unit cells

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For many biopolymers, especially polypeptides, polynucleotides and polysaccharides, the repeating unit is a monomer or a small oligomer and the unit-cell dimensions are in the range 10 to 50 Å. Such unit cells can accommodate one or more polymer helices, packed in an organized fashion.

An initial model is constructed from the primary structure of the repeating unit, using bond lengths, bond angles and some conformation angles derived from surveys of accurate single-crystal analyses. The model must satisfy the observed helical parameters and have reasonable intra- and inter-chain non-bonded, hydrogen-bonded and polar interactions.

This preliminary model provides an approximate solution to the phase problem and a starting point for refinement. Since there is no assurance that the refined model represents the true structure, however, stereochemically plausible alternatives must be carefully considered, refined and objectively adjudicated. Alternatives can include both right- and left-handed helices, single helices, and multistranded helices with parallel and antiparallel strands. The next stage involves the packing arrangement in the unit cell. If two or more helices are present, their positions, orientations and relative polarities must be varied in refinement.

19.5.7.2. Refinement: small unit cells

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The widely used linked-atom least-squares (LALS) technique (Arnott & Wonacott, 1966[link]; Smith & Arnott, 1978[link]) and the variable virtual bond (PS79) method (Zugenmaier & Sarko, 1980[link]) were developed for fibre structures. They are similar in principle to the least-squares refinement procedure for crystalline proteins (Hendrickson, 1985[link]), although bond lengths and bond angles are usually kept fixed in the fibre refinements. The function minimized by the LALS program is of the form [\Omega = \textstyle\sum\limits_{m}\displaystyle w_{m}\Delta F_{m}^{2} + \textstyle\sum\limits_{i}\displaystyle e_{i}\Delta \theta_{i}^{2} + \textstyle\sum\limits_{j}\displaystyle k_{j}\Delta c_{j}^{2} + \textstyle\sum\limits_{n}\displaystyle \lambda_{n}G_{n}. \eqno(19.5.7.1)] The first term on the right-hand side is the weighted sum of the squares of the differences, [\Delta F_{m}], between observed and calculated X-ray structure amplitudes of Bragg reflections or continuous diffraction. Either or both types of data can be used as necessary. The weights, [w_{m}], are inversely proportional to the estimated variance of the data. The second term minimizes the differences, [\Delta\theta_{i}], between the expected (standard) values of conformation and bond angles and those in the model; the weights, [e_{i}], are based on empirically determined variances. The third term is designed to take care of non-bonded interactions and thus keep the model free from steric compression. It includes the deviations from target values of both intra- and inter-chain hydrogen bonds and the differences between acceptable and calculated non-bonded distances for those contacts that are smaller than the acceptable limiting values. The weights, [k_{j}], are based on the Buckingham energy function for non-bonded contacts and empirical variances for hydrogen bonds. Finally, the fourth term imposes constraints ([G_{h}], with Lagrange multipliers [\lambda_{h}]) for helix connectivity and ring closure, as in a furanose or pyranose, and it vanishes when all such constraints are satisfied. During the refinement, the structure factors are calculated with either the conventional atomic scattering factor f or with a solvent-corrected atomic scattering factor [f_{w}] (Fraser et al., 1978[link]; Chandrasekaran & Radha, 1992[link]) given by the function [f_{w}(D) = f(D) - v \sigma_{s} \exp (-\pi v^{2/3} D^{2}), \eqno(19.5.7.2)]where [D = (R^{2} + Z^{2})^{1/2}] , [\sigma_{s}] is the electron density of the solvent and v is the excluded volume of the atom. If the van der Waals radius of water is taken as 2 Å, [\sigma_{s}] for water is 0.2984 e Å−3. Equation (19.5.7.2[link]) allows for the solvent contribution to the diffracted intensity and is particularly useful in studying hydrated fibres in which structured and amorphous water can account for up to 50% of the total mass.

19.5.7.3. Data-to-parameter ratio

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The total number of data used in this refinement process is M + I + J, where M, I and J are, respectively, the number of observations in the first three terms of equation (19.5.7.1[link]). If P is the number of parameters refined and H is the number of independent constraints in the last term, then the number of degrees of freedom of the system is [P - H]. The effective number of data is given by [D = (M + I + J) - (P - H)]. The data-to-parameter ratio (D/P), a measure of the dependability of the final results, must be greater than one for meaningful refinement. D/P is typically in the range 3 to 11 in the analysis of polynucleotide and polysaccharide structures. This ratio is comparable to those commonly reported for single-crystal structures, confirming that fibre-diffraction analysis of polymers, despite the limited number of X-ray data, can yield reliable results.

19.5.7.4. Initial models: large unit cells

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For large macromolecular aggregates, such as viruses and cytoskeletal filaments, initial models cannot usually be devised using the primary structure of the molecule alone. The largely α-helical filamentous bacteriophages form a rare class of exceptions (Makowski et al., 1980[link]). Molecular-replacement methods, in which initial models are constructed from single-crystal structure determinations of the separated components of the aggregate or from known related structures, can be useful, but because of the limited number of data in a fibre pattern such models can sometimes be difficult to refine.

Multi-dimensional isomorphous replacement (MDIR), an extension of the isomorphous-replacement method of protein crystallography, has been useful in studying helical viruses (Stubbs & Diamond, 1975[link]; Namba & Stubbs, 1985[link]). The dimensions are the real and imaginary parts of the various overlapping structure factors at a given point in the diffraction pattern. Information about both the phases of the structure factors and the relative magnitudes of the overlapping structure factors is obtained from heavy-atom derivatives of the virus; at least twice as many heavy-atom derivatives as the number of significant G terms in equation (19.5.3.7[link]) are required. If the structure of a related aggregate is known, MDIR can be combined with molecular replacement (Namba & Stubbs, 1987a[link]; Wang & Stubbs, 1994[link]); in this case, fewer derivatives are required.

Layer-line splitting (Franklin & Klug, 1955[link]) arises when the helical symmetry of the scattering particles is close to, but not exactly, integral. For example, tobacco mosaic virus (TMV) has 49.02 subunits in three turns of the viral helix. In this case, the G terms in each layer line do not fall at exactly the same Z values in the diffraction pattern. The resulting shifts in the positions of the layer lines can be measured for the native aggregate and, in favourable cases, for heavy-atom derivatives, and used to provide additional phase information (Stubbs & Makowski, 1982[link]). Information from electron microscopy (Beese et al., 1987[link]) and neutron scattering (Nambudripad et al., 1991[link]) has also been used.

19.5.7.5. Refinement: large unit cells

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Refinement of fibre structures having large unit cells has many parallels to refinement in protein crystallography. Refinement in real space, especially the solvent-flattening approach, has been widely used to improve electron-density maps and is particularly valuable in structure determination of noncrystalline fibres. Since helical aggregates have finite radii, g terms [equation (19.5.3.6[link])] can be set to zero outside a maximum radius and back-transformed to obtain refined estimates of the phases of the G terms. More detailed solvent-flattening algorithms can also be used (Namba & Stubbs, 1985[link]).

Molecular models can be refined by methods conceptually related to those of LALS. The principal difference is that bond lengths and angles are not kept fixed, but are restrained to remain close to standard values. The restrained least-squares method (Hendrickson, 1985[link]), widely used in protein crystallography, has been adapted (Stubbs et al., 1986[link]) for fibre diffraction and used to refine a number of filamentous virus structures (Namba et al., 1989[link]; Nambudripad et al., 1991[link]). Although effective, the radius of convergence of this method is less than desired, probably because of the limited number of data available from fibre diffraction (Wang & Stubbs, 1993[link]).

Molecular-dynamics methods have been used to increase the radius of convergence of refinement (Wang & Stubbs, 1993[link]). The program X-PLOR (Brünger et al., 1987[link]) has been adapted for fibre diffraction and can handle data from both crystalline and noncrystalline fibres. A potential-energy function of the form [\Omega = E + S \textstyle\sum\limits_{l} \textstyle\sum\limits_{i} w_{li} \{[I_{o}(R_{i})]^{1/2} - [I_{c}(R_{i})]^{1/2}\}^{2} \eqno(19.5.7.3)]is minimized. The first term, E, is an empirical energy function that accounts for distortions in bond lengths, bond angles and conformation angles, and for non-bonded, electrostatic and hydrogen-bonding interactions. The second term accounts for the differences between the observed and calculated X-ray intensities at specific values of [R_{i}] on every layer line l; [w_{li}] is the weight for each observation and S is a normalizing factor. In the most effective use of this method, simulated annealing, the process of heating the structure to a temperature of 3000 to 4000 K is simulated, and then the structure is cooled (`annealed') in small increments. At high temperatures, energy barriers between the starting model and structures of lower potential can be over­come; in this way, the radius of convergence of the refinement is increased.

19.5.7.6. Difference Fourier methods

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As in crystallography, difference maps are used during refinement to correct errors and to identify missing fragments of the model and, in the final stages of refinement, to identify solvent molecules and associated ions.

In crystalline fibre diffraction, the most common difference maps use calculated phases with amplitudes of either [F_{o} - F_{c}] or [2F_{o} - F_{c}]. In both cases, weighting the coefficients on the basis of the observed and calculated structure amplitudes has been used to minimize the root-mean-square error in the electron-density maps. Reflections superposed by cylindrical averaging do, how­ever, present problems. One solution is to divide the observed intensity equally among the superposed reflections. This is a reasonable approach in the initial stages of structure analysis, when the reliability of the model is uncertain, and has the advantage of minimizing bias toward the model. Alternatively, the observed intensity may be split in the same ratio as the calculated intensity. This approach, although biased, is more effective for locating solvent molecules and ions in an otherwise well determined structure. Difference Fourier maps have played a significant role in determining the molecular structures and packing arrangements in unit cells mediated by water molecules and cations of several polynucleotide (Chandrasekaran et al., 1995[link], 1997[link]) and polysaccharide helices (Winter et al., 1975[link]; Chandrasekaran et al., 1988[link], 1998[link]; Chandrasekaran, Radha & Lee, 1994[link]; Janaswamy & Chandrasekaran, 2001[link], 2002[link], 2008[link]).

In noncrystalline fibre diffraction, the superposition of intensities due to cylindrical averaging is more serious and must be taken into account. Namba & Stubbs (1987b[link]) have shown that the coefficients yielding the most accurate electron-density maps of the full structure have amplitudes of [NG_{o} - (N - 1)G_{c}], where N is the number of significant terms in equation (19.5.3.7[link]) (the number of superposed intensities), and the observed intensity is divided in the ratio of the calculated intensity. For filamentous viruses at moderate resolution, N is typically in the range four to six. As in crystallography and crystalline fibre diffraction, maps calculated from amplitudes of [F_{o} - F_{c}] have low noise levels and are most useful for checking the accuracy of final models and for locating solvent molecules.

19.5.7.7. Evaluation

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As in crystallography, fibre structures are evaluated by statistical measures, such as R values, and by the examination of difference maps. Fibre-diffraction R values are inherently lower than those expected in crystallography, particularly when large numbers of intensities have been superposed by cylindrical averaging (Stubbs, 1989[link]). The largest likely R value for non­crystalline TMV at 3 Å resolution is about 0.31 and for polycrystalline DNA at 3 Å resolution it is about 0.41, both significantly less than the value of 0.59 to be expected from noncentric single-crystal analyses (Millane, 1989[link]). Free R values, widely used in crystallography, have been used to a limited extent in fibre diffraction (Welsh et al., 2000[link]), sometimes with modifications to deal with the effect of overlapping Bessel orders (Welsh et al., 1998[link]). Some authors prefer to use correlation coefficients to evaluate models.

Comparison of R values alone is not necessarily a reliable way to discriminate between competing models. Such discrimination is often required for structures with small unit cells, for which alternative models are routinely refined (Sections 19.5.7.1[link] and 19.5.7.2[link]). The relative merits of any pair of competing models can be assessed on the basis of several types of statistics (Arnott, 1980[link]) using Hamilton's significance test (Hamilton, 1965[link]), which considers not only residuals but also numbers of degrees of freedom (Section 19.5.7.3[link]). Such a test is essential. There are many examples in the literature where R values have been lowered by the simple process of increasing the number of degrees of freedom; a decreased R value obtained in this way may or may not have any significance.

Difference Fourier maps have been used to evaluate crystalline fibre diffraction analyses for many years, for example, to reject the controversial Hoogsteen base pairing in double-stranded DNA (Arnott et al., 1965[link]), and later to discriminate between 10- and 11-fold double helices of RNA (Arnott et al., 1967[link]). Difference maps have been essential in the refinement of fibre structures with large unit cells (Namba et al., 1989[link]; Wang & Stubbs, 1994[link]), both to identify errors in early models and to confirm that the final structures contained no major errors or omissions.

19.5.8. Structures determined by X-ray fibre diffraction

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The α-helix of several synthetic polypeptides (Pauling & Corey, 1951[link]), the double helix of DNA (Franklin & Gosling, 1953a[link],b[link]; Watson & Crick, 1953[link]), the ribbon structure of cellulose (Meyer & Misch, 1937[link]) and the low-resolution structure of tobacco mosaic virus (Barrett et al., 1971[link]) were early examples of structures determined by fibre diffraction. Early workers also examined a number of fibrous proteins (Bailey et al., 1943[link]). In the past 50 years, developments in theory and practice and the availability of fast computers have made it possible to determine and refine about 200 biological polymer structures of varying complexities. The largest repeating units in polypeptides, polynucleotides and polysaccharides solved to date correspond to a tripeptide, a tetranucleotide and a heptasaccharide, respectively.

19.5.8.1. Polypeptides

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The structural details of the α-helix and β-sheet, the principal secondary-structure elements of proteins, have emerged from the analysis of synthetic polypeptides (Pauling & Corey, 1951[link], 1953[link]). Analysis of noncrystalline fibre-diffraction patterns led to the triple-helical coiled-coil model of collagen (Ramachandran & Kartha, 1955[link]; Rich & Crick, 1955[link]). There have been many recent studies on amyloid fibrils, associated with a large number of diseases including Alzheimer's and Parkinson's diseases, and including peptide fragments of prions, infectious amyloids associated with diseases such as bovine spongiform encephalopathy (`mad cow disease') (Inouye et al., 1993[link], 2000[link]; Malinchik et al., 1998[link]; Kishimoto et al., 2004[link]; Makin & Serpell, 2005[link]).

19.5.8.2. Polynucleotides

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The molecular structures of a series of DNA and RNA helices have been determined and refined using data from polycrystalline fibres (Arnott et al., 1969[link]; Chandrasekaran & Arnott, 1989[link]). These include the canonical A, B and C forms of DNA, corresponding, respectively, to 11-, 10- and 9.3-fold right-handed antiparallel Watson–Crick base-paired helices. Structural differences between the three have been attributed to changes in furanose puckerings and helical parameters: the A form has C3-endo, but B and C have C2-endo or analogous C3-exo puckers. All RNA duplexes are members of the A family. Later important structures included the sixfold single helix of poly (C) (Arnott et al., 1976[link]), a compact eightfold double helix for poly d(AT) and poly d(IC) (Arnott et al., 1983[link]), and the left-handed Z-DNA for poly d(GC) (Arnott et al., 1980[link]). Difference Fourier syntheses were instrumental in locating a spine of water molecules in the minor groove and a series of sodium ions and water molecules that bridge the phosphate groups of adjacent DNA molecules in the tenfold helices of poly (dA)·poly (dT) (Chandrasekaran et al., 1995[link]), poly (dA)·poly (dU) and poly d(AI)·poly d(CT) (Chandrasekaran et al., 1997[link]). Neutron fibre diffraction has been particularly useful for locating water molecules in polynucleotides (Shotton et al., 1997[link], 1998[link]). Data from noncrystalline fibres have been used to determine, among others, the structures of DNA·RNA hybrid duplexes (Arnott et al., 1986[link]), a DNA triple-stranded helix (Chandrasekaran et al., 2000a[link]) and two RNA triple-stranded helices (Chandrasekaran et al., 2000b[link],c[link]).

19.5.8.3. Polysaccharides

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Among the three-dimensional structures determined for industrially useful and biologically important polysaccharides are the gel-forming calcium i-carrageenan (Arnott, Scott et al., 1974[link]; Janaswamy & Chandrasekaran, 2001[link], 2002[link], 2008[link]), sodium pectate (Walkinshaw & Arnott, 1981[link]), gellan (Chandrasekaran et al., 1988[link]) and welan (Chandrasekaran, Radha & Lee, 1994[link]), and a series of distinct helical forms of the glycosaminoglycan hyaluronan (Arnott & Mitra, 1984[link]). The conformations of these molecules are delicately controlled by ions, such as sodium, potassium and calcium. The repeating units range from a simple monosaccharide to a branched pentasaccharide.

Specific interactions among the polysaccharides and their associated small molecules can be correlated with their observed properties. A number of neutral polysaccharides, such as cellulose, chitin and mannan, are twofold ribbon-like helices, which aggregate and are hence water insoluble. The A and B forms of amylose, the main constituents of starch granules, are sixfold left-handed parallel double helices. Derivatization of amylose leads to the formation of single helices (Chandrasekaran, 1997[link]). The water-soluble galactomannan derives its high viscosity in aqueous solution from intermolecular side-chain interactions (Chandrasekaran et al., 1998[link]). Gels of calcium iota-carrageenan are stronger owing to more intimate sulfate-group interactions between double helices than in the sodium form (Janaswamy & Chandrasekaran, 2001[link], 2002[link], 2008[link]).

19.5.8.4. Helical viruses and bacteriophages

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The largest repeating units in structures determined by fibre diffraction are those of several members of the tobamovirus family, including tobacco mosaic virus (Namba et al., 1989[link]; Kendall et al., 2007[link]), cucumber green mottle mosaic virus (Wang & Stubbs, 1994[link]) and ribgrass mosaic virus (Wang et al., 1997[link]). These viruses are rod-shaped, 3000 Å long and about 180 Å in diameter. Oriented sols yield exceptionally good diffraction patterns (Fig. 19.5.8.1[link]). The asymmetric unit consists of a protein subunit of approximate molecular weight 18 000 Da and three nucleotides of RNA. The coat proteins are folded like globular proteins and are about 40% α-helical, with small regions of β-sheet. All of the amino acids, all three nucleotides, and in some cases water molecules and calcium ions, are seen in the electron-density maps. The TMV structure was determined by MDIR; the remaining structures were determined by molecular replacement from TMV or by a combination of molecular replacement and isomorphous replacement. All of the structures were refined by restrained least-squares or molecular-dynamics methods to R values of less than 0.10 at resolutions between 2.9 and 3.5 Å. Fibre diffraction data from a number of flexible filamentous viruses, including members of the Flexiviridae and Potyviridae families, have been used in conjunction with data from electron cryomicroscopy to determine symmetries and low-resolution structures (Kendall et al., 2008[link]).

[Figure 19.5.8.1]

Figure 19.5.8.1 | top | pdf |

X-ray diffraction pattern from an oriented sol of the U2 strain of tobacco mosaic virus.

Several filamentous bacteriophage structures, including fd, Pf1 and related strains, have been determined and refined. Filamentous bacteriophages are flexible viruses, about 60 Å in diameter and 10 000 to 20 000 Å in length. Several thousand copies of a coat protein of about 50 residues wrap around a central single-stranded circular DNA. The DNA does not appear to be sufficiently ordered to appear in electron-density maps. The coat-protein molecules have an unusually simple structure, being almost entirely α-helical (Marvin et al., 1974[link]). Model-building approaches have therefore been used, sometimes supplemented by isomorphous replacement (Bryan et al., 1983[link]). Neutron scattering from bacteriophages with selectively deuterated amino-acid residues has also been used to assist model building (Nambudripad et al., 1991[link]). Both restrained least-squares (Nam­budripad et al., 1991[link]) and molecular-dynamics (Gonzalez et al., 1995[link]) refinement methods have been used. The coat protein forms two α-helical layers (Welsh et al., 1998[link]).

19.5.8.5. Other large assemblies

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Low-resolution X-ray fibre-diffraction data have been successfully used to model the structural details of a number of complex assemblies. For example, the structure of the F-actin helix at 8 Å resolution has been described by combining the single-crystal structure of the G-actin monomer with fibre-diffraction data (Holmes et al., 1990[link]; Oda et al., 2009[link]). This structure, in turn, has been used to model the muscle thin filament, composed of F-actin monomers and tropomyosin, at about 25 Å resolution, both in the resting and activated states, and hence to understand the movement of tropomyosin in muscle function (Squire et al., 1993[link]). The structure of the microtubule has been determined at 18 Å resolution using information from electron microscopy and fibre diffraction (Beese et al., 1987[link]). A similar but more sophisticated approach was used for bacterial flagellar filaments at 9 Å resolution (Yamashita, Hasegawa et al., 1998[link]); the diffraction patterns obtained from these filaments are of such high quality that prospects for a complete molecular structure are excellent.

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