X-ray diffraction patterns from oriented polycrystalline fibers of some biopolymers show that the molecules are disordered within the microcrystallites. Quantifying the disorder in such specimens is a necessary step for the use of their diffraction patterns for accurate structure determination. Theory and algorithms for calculating diffraction patterns from such fibers have recently been described [Stroud & Millane (1995). Acta Cryst. A51, 000-000]. Here the application of these methods to determining the kind and degree of disorder in two polynucleotide fibers is described. The more ordered system shows random screw disorder accompanied by small lattice distortions, and the more disordered system shows larger lattice distortions and significant rotational disorder. These results show the potential of these methods for determining disorder in polycrystalline fibers; uniqueness of the solutions and implications for structure determination are discussed.

An expression is derived for the intensity diffracted by a non-crystalline fiber made up of aggregates of helical molecules. This expression is useful for the efficient calculation of diffraction from such specimens and provides insight into the effects of aggregation on diffraction patterns. Example calculations show a number of implications for structure determination.

X-ray diffraction patterns from some polycrystalline fibers show that the constituent microcrystallites are disordered. The relationship between the crystal structure and the diffracted intensities is then quite complicated and depends on the precise kind and degree of disorder present. The effects of disorder on diffracted intensities must be included in structure determinations using diffraction data from such specimens. Theory and algorithms are developed here that allow the full diffraction pattern to be calculated for a disordered polycrystalline fiber made up of helical molecules. The model accommodates various kinds of disorder and includes the effects of finite crystallite size and cylindrical averaging of the diffracted intensities from a fiber. Simulations using these methods show how different kinds, or components, of disorder produce particular diffraction effects. General properties of disordered arrays of helical molecules and their effects on diffraction patterns are described. Implications for structure determination are discussed.

The largest likely R-factor is useful for evaluating the significance of R-factors obtained in structure determinations. Numerical expressions have been derived previously for calculating largest likely R-factors in fiber diffraction analyses. Analytical approximations to largest likely R-factors (R) in fiber diffraction are derived here that show the dependence on resolution (), helix symmetry (u_o) and molecular radius (). The simplest approximation is R ∝ (u/)^1/2 which represents the overall behavior of R-factors reasonably well. More accurate approximations are also derived. These are applied to various structures and the dependence on different structural parameters is examined. These results provide insight into the behavior of R-factors in fiber diffraction and may be useful in further analysis.

An expression is obtained for the largest likely R factor for data that are normally distributed. For zero-mean data, the largest likely R factor is 2^1/2 and, for positive data (μ >> σ.), it is equal to 2σ/(μπ^1/2). These results are applied to fiber diffraction and other possible applications in crystallography are discussed.

It is shown that by considering the cylindrically averaged intensity diffracted by a non-crystalline fiber as being due to the atoms in one c repeat, rather than, as is usual, one helix repeat, some relationships in fiber diffraction can be easily derived. These include the helix selection rule, a new expression for the diffracted intensity, and the cylindrically averaged Patterson. This approach may have other applications as well.

The probability distributions of X-ray intensities in fiber diffraction are different from those for single crystals (Wilson statistics) because of the cylindrical averaging of the diffraction data. Stubbs [Acta Cryst. (1989), A45, 254-258] has recently determined the intensity distributions on a fiber diffraction pattern for a fixed number of overlapping Fourier-Bessel terms. Some properties of the amplitude and intensity distributions are derived here. It is shown that the amplitudes and intensities are approximately normally distributed (the distributions being asymptotically normal with increasing number of Fourier-Bessel terms). Improved approximations using an Edgeworth series are derived. Other statistical properties and some asymptotic expansions are also derived, and normalization of fiber diffraction amplitudes is discussed. The accuracies of the normal approximations are illustrated for particular fiber structures, and possible applications of intensity statistics in fiber diffraction are discussed.

The largest likely R factor is useful for evaluating the significance of R factors obtained in structure determinations, and is smaller in fiber diffraction than in traditional crystallography. Very simple approximations to functions used to calculate the largest likely R factor in fiber diffraction are derived. For example, the largest R factor (R_m) for m overlapping terms is very well approximated by R_m ≃ (2/π)^1/2 m ^-1/2. These are a useful alternative to the exact, but quite complicated, expressions derived previously. More significantly, they provide insight into the behavior of R factors in fiber diffraction and may be useful in further analysis.

The largest likely R factor (that for a structure uncorrelated with the correct structure) is smaller for an X-ray fiber diffraction analysis than for a traditional single-crystal analysis. For example, the largest likely R factor for tobacco mosaic virus determined by fiber diffraction at 3 Å resolution is 0.31, compared to 0.59 for a single-crystal analysis. Earlier treatments of largest likely R factors in fiber diffraction for a fixed number of overlapping Fourier-Bessel structure factors are extended to general fiber diffraction patterns. The theory is illustrated with applications to particular structures thereby elucidating some general features of fiber diffraction R factors. These results are useful for interpreting the reliability of structure determinations, and may also be useful for further developments of fiber diffraction theory in general.

Simple expressions are obtained for the largest likely R factor in X-ray fiber diffraction recently derived by Stubbs [Acta Cryst. (1989), A45, 254-258]. These generalize the largest likely R factors obtained by Wilson [Acta Cryst. (1950), 3, 397-399] for centric and acentric crystals. Expressions are obtained in terms of special functions and as finite series that simplify the calculation of R factors. These may be useful for further analysis and understanding of the effects of particle diameter and symmetry and diffraction data resolution on the reliability of structure determinations.

A method of constructing closed rings of specified pucker is described. A ring is constructed such that a set of puckering parameters as well as the ring bond lengths and bond angles are optimized towards desired values. A computer program is described which generates N-membered rings in this way and is illustrated with examples.

Background can be a major source of error in measurement of diffracted intensities in fiber diffraction patterns. Errors can be large when poorly oriented less-crystalline specimens give diffraction patterns with little uncontaminated background. A method for estimating and removing a general global background in such cases is described and illustrated with an example.