International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 4.5, pp. 478479

At low enough resolution, only one Fourier–Bessel structure factor contributes on each layer line of a fibre diffraction pattern, so that only the phase needs to be determined and the situation is no different to that in protein crystallography. If heavyatomderivative specimens can be prepared, the usual method of multiple isomorphous replacement (MIR) (Drenth, 1994) can be applied, which in principle requires only two heavyatom derivatives. At higher resolution, however, more than one Fourier–Bessel structure factor contributes on each layer line. A generalized form of isomorphous replacement which involves using diffraction data from several heavyatom derivatives to determine the real and imaginary components of each contributing is referred to as multidimensional isomorphous replacement (MDIR) (Namba & Stubbs, 1985). MDIR was first described and used to determine the structure of TMV at 6.7 Å resolution (Stubbs & Diamond, 1975; Holmes et al., 1975), and has since been used to extend the resolution to 2.9 Å (Namba, Pattanayek & Stubbs, 1989). A consequence of cylindrical averaging is that large numbers of heavyatom derivatives are required: at least two for each Bessel term to be separated. The theory of MDIR is outlined here.
The first step in MDIR is location of the heavy atoms in the derivative structures. The radial coordinate of a heavy atom can be determined by analysis of the intensity distribution in the lowresolution region of the equator where only the Bessel term contributes. Since is real, and can be measured continuously in R, inspection of the positions of the minima and maxima in the lowresolution region of the equator generally allows the sign of to be assigned to , i.e. can be determined from . If the sign is determined for both the native and a heavyatom derivative, referring to equation (4.5.2.13) shows that where is the value derived from the derivative data, o denotes the occupancy and the subscript h denotes values for the heavy atom. The parameters and on the righthand side of equation (4.5.2.68) can be searched in a trialanderror fashion to obtain the best agreement with the lefthand side (calculated from the data) to determine the radial coordinate of the heavy atom (Mandelkow & Holmes, 1974). Lobert et al. (1987) applied the same method to cucumber green mottle mosaic virus (CGMMV), except that the sign of was taken from that of TMV.
Two approaches have been used to determine the angular and axial coordinates of the heavy atom. Mandelkow & Holmes (1974) and Holmes et al. (1975) used a search procedure in which the quantity is varied and used to calculate the intensity of the Fourier–Bessel structure factor for the heavy atom alone. This is compared to on each layer line, where only one Bessel order contributes, and Φ chosen to minimize the meansquare difference. The values of Φ found for each layer line can then be combined to determine and . In the case of CGMMV, Lobert et al. (1987) used the phases and Besselorder separations from TMV to calculate Fourier–Bessel difference maps between the native and derivative data to determine the heavyatom coordinates .
Consider a set of isomorphous heavyatom derivatives indexed by j. Since the analysis is applied at any point on the fibre diffraction pattern, the symbol will be used for where no confusion arises. Denote by the value of for the jth derivative, so that where denotes the Fourier–Bessel structure factor of a structure containing the heavy atom only. Denote by and the real and imaginary parts, respectively, of (for the native structure), and by and the real and imaginary parts of , i.e. for the jth heavyatom structure alone. Equation (4.5.2.17) can then be written as for the native and for the jth derivative. If intensity data are available from J heavyatom derivatives, and can be calculated from the heavyatom positions, and equations (4.5.2.70) and (4.5.2.71) represent a system of secondorder equations for the m unknowns and . If , then the system of equations is overdetermined and can be solved for the and . The solution of this nonlinear system can be eased by deriving a system of linear equations by substituting from (4.5.2.70) into (4.5.2.71), giving Equation (4.5.2.72) is a system of linear equations for the unknowns and , the solution being subject to the constraint equation (4.5.2.70). However, since the original problem is secondorder, there may be up to m local minima. Stubbs & Diamond (1975) describe a numerical procedure for locating all the local minima and selecting the best of these based on `continuity' of the . This method was used to determine the structure of TMV at 6.7 Å resolution (Holmes et al., 1975) and 4 Å resolution (Stubbs et al., 1977). In current applications of MDIR a more direct solution technique is used in which the phasedetermining equations (4.5.2.70) and (4.5.2.71) are solved by first solving the linear equations (4.5.2.72) by linear least squares to obtain an approximate solution, which is then refined by solving the quadratic equations (4.5.2.70) and (4.5.2.71) directly using nonlinear least squares (Namba & Stubbs, 1985).
The number of heavyatom derivatives required can be quite demanding experimentally, although phasing with fewer heavyatom derivatives is possible, particularly if additional information is available, such as from a related structure. The different Bessel terms may be assumed to contribute the same amplitude each, or, if the structure of a related molecule is known, the ratios of the amplitudes can be taken as being the same as those for the related molecule. Using the amplitude estimates derived using either of these two approaches, applied to both native and derivative data, the phases of the Bessel terms can be estimated using conventional MIR and data from at least two heavyatom derivatives, allowing an initial electrondensity map to be calculated. If only one heavyatom derivative is available then two phase solutions are obtained, but the method of conventional single isomorphous replacement (SIR) (Drenth, 1994) can be used to obtain an estimate of the electron density. The electron density obtained by MIR, and particularly by SIR, in this way tends to be noisy and low contrast as a result of inaccurate division of the intensities, as well as the usual sources of errors in MIR. The electron density can, however, be improved using solvent levelling. If no heavyatom derivatives are available, both the relative amplitudes and the phases can be based on those of a related structure. Model bias can, however, be more serious than in conventional crystallography since both the phases and the relative amplitudes are based on the model.
The feasibility of structure determination with a limited number of heavyatom derivatives was first demonstrated by Namba & Stubbs (1987b) using data from TMV at 4 Å resolution. The structure of CGMMV has been determined at 5 Å resolution using data from two heavyatom derivatives and the techniques described above (Lobert et al., 1987; Lobert & Stubbs, 1990). Structure determination at this resolution using MDIR would theoretically require six heavyatom derivatives. Initial separation of the Besselterm amplitudes was based on the equalamplitude assumption and also on the relative amplitudes for (homologous) TMV.
In general, the equalamplitude assumption appears to produce reliable electrondensity maps where only two or three Bessel terms contribute. The corresponding resolution depends on the helix symmetry and the molecular diameter, but can be relatively high for molecules with high helix symmetry. At higher resolution where more Bessel terms contribute, use of related or partial structures can be used to calculate initial Besselterm amplitudes and can lead to successful phasing.
If the molecule has only approximate helix symmetry, then layerline splitting (Section 4.5.2.3.3) can provide additional information which reduces the number of heavyatom derivatives required. The degree of splitting is usually significantly less than the breadth of the layer lines so that the different Bessel terms within a (split) layer line overlap. The effect of splitting can be observed, however, since the centre of a layer line, at a particular value of R, is shifted towards the position of the stronger Bessel term contributing at that radius. The shift depends on the relative magnitudes of the contributing Bessel terms, and can be measured and used in phase determination as detailed by Stubbs & Makowski (1982). If P of the heavyatom derivatives (in addition to the native) give accurate splitting information, then an additional P linear equations [analogous to equation (4.5.2.72)] and one quadratic equation [analogous to equation (4.5.2.70)] are available for solution of the phase problem, and the number of heavyatom derivatives required is reduced by a factor of up to two. The value of layerline splitting was first demonstrated by recalculating an electrondensity map of TMV at 6.7 Å resolution using only two derivatives, rather than using six derivatives without the use of splitting data (Stubbs & Makowski, 1982). Layerline splitting was subsequently used in a structure determination of TMV at 3.6 Å resolution (Namba & Stubbs, 1985).
Macromolecular fibre structures that have been built into an electrondensity map have been refined using both restrained leastsquares (RLS) and moleculardynamics (MD) refinements. Restrained least squares has been used to refine the structure of TMV at 2.9 Å resolution (Namba, Pattanayek & Stubbs, 1989); however, Wang & Stubbs (1993) have shown that a larger radius of convergence is obtained using MD refinement (as in protein crystallography).
Moleculardynamics refinement in fibre diffraction has been implemented by adding a fibre diffraction option (Wang & Stubbs, 1993) to the XPLOR program (Brünger, 1992). This involves including the cylindrically averaged fibre diffraction intensities in the energy term and taking account of the interhelical subunit contacts and covalent connections in the same way as described above for RLS refinement. The effective potentialenergy function E used is where is the empirical energy function (which typically includes bondlength, bondangle and torsionangle distortions, van der Waals and electrostatic interactions, and other terms such as ring planarity), and are the observed and calculated, respectively, cylindrically averaged diffraction intensities sampled at , the are weights for the observed intensities and k is a scale factor between the calculated and observed data. The quantity S is a weight to make the gradients of the two terms in equation (4.5.2.73) comparable (Wang & Stubbs, 1993), and can be estimated using the method of Brünger (1992). Moleculardynamics refinement has been successfully used to refine the structure of CGMMV at 3.4 Å resolution (Wang & Stubbs, 1994). In the case of ribgrass mosaic virus (RMV), the close isomorphism with TMV (identical helix symmetry, similar repeat distance, significant sequence homology and similar diffraction pattern) allowed an initial model to be built based on the TMV structure, and a solution obtained at 2.9 Å by alternating moleculardynamics refinement with differencemap and omitmap calculations (Wang et al., 1997).
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