Refinement of protein dynamic structure: normal mode refinement.

Abstract

An x-ray crystallographic refinement method, referred to as the normal mode refinement, is proposed. The Debye-Waller factor is expanded in terms of the effective normal modes whose amplitudes and eigenvectors are experimentally determined by the crystallographic refinement. In contrast to the conventional method, the atomic motions are treated generally as anisotropic and concerted. This method is assessed by using the simulated x-ray data given by a Monte Carlo simulation of human lysozyme. In this article, we refine the dynamic structure by fixing the average static structure to exact coordinates. It is found that the normal mode refinement, using a smaller number of variables, gives a better R factor and more information on the dynamics (anisotropy and collectivity in the motion).X-ray crystallography is recognized as one of the most powerful techniques for elucidating not only the static but also the dynamic structure ofproteins (1, 2). X-ray diffraction data from a protein crystal provide information about the protein dynamics in the form of the isotropic temperature factor By for a nonhydrogen atom j, which is related to the mean-square fluctuation (Arj2) from the average coordinates by By = (8ir2/3)(Arj2).[1]The temperature factor is determined by the least-squares refinement (3) of the x-ray structure, which minimizes a weighted sum of the residual[2] h where h = (h, k, I) refers to reciprocal-lattice points of the crystal and IFobsI and IFcalI denote the observed and calculated structure factor amplitudes, respectively. By assuming no correlations in the atomic motions belonging to different unit cells of the crystal, Fcal has the expression Fcal(h) = Xf/h)exp(2nrth.(rj))(exp(27rtihArj)), [3] where fj is the atomic structure factor usually given by four Gaussian functions and (rj) is the average coordinate. The dynamic part of the structure factor (exp(2mh-Arj) is referred to as the Debye-Waller factor and is usually approximated by the isotropic temperature factor in the form of (exp(2'1TihArj)[4]The approximation of Eq. 4 is based on the assumption that the atomic motion is (a) harmonic and (b) isotropic. In the refinement procedure, isotropic temperature factors Bj are treated as independent variables.A This means that the atomic motions are (c) independent of each other. When describing the dynamic structure of real proteins, however, these approximations, especially that of isotropic motion, could be a serious limitation in improving the refinement (4). The number of observable experimental data IFobsI is not large enough for adopting the anisotropic temperature factor, which requires six times more parameters in Fcg than the number required in the isotropic case.In the conventional refinement methods for determining the average static structure, a model of the atomic structure such as stereochemical knowledge (3) or conformational energy (5) has supplemented the limited diffraction data. In the same way, a model of dynamics should be used as supplemental information in the refinement of t...

Use of TLS parameters to model anisotropic displacements in macromolecular refinement

“…Pseudo-rigid-body variables, described by TLS parameters, are a basic example of this and are a necessary precursor to other methods. Future work will investigate the use of other collective variables, such as normal modes (Diamond, 1990;Kidera & Go, 1990. In situations where individual anisotropic re®nement is not viable, there are likely to be many other levels of modelling which may be considered.…”

“…Hence, derivatives of the residual with respect to atomic coordinates also include terms involving the derivative of the residual with respect to the elements of U, re¯ecting the fact that shifts in the atomic positions change the contribution of the TLS parameters to the calculated structure factor. Equation (5) relating the individual ADPs to the TLS parameters can be rewritten in terms of a 6 Â 6 variance± covariance matrix R for the translation and libration (Kidera & Go, 1992). Although this matrix should be positive de®nite, this is not guaranteed by the minimization procedure.…”

“…However, interatomic correlations cannot be determined directly from Bragg intensities [but see Kidera & Go (1992) for a discussion of the indirect inference of interatomic correlations], and one is left with atomic displacements only, each represented by a oneparameter isotropic model or a six-parameter anisotropic model. These atomic displacement parameters are usually individually determined in structure re®nement.…”

“…They concluded that these displacements must have an intramolecular component superimposed on a rigid-body motion or breathing motion. Diamond (1990) and Kidera & Go (1990 re®ned the amplitudes of normal modes describing the internal motion of the protein, together with TLS parameters for the overall motion of each molecule. These studies showed that while account of the internal motion was necessary for a complete description of the protein and is perhaps the most biologically informative aspect, the TLS parameters for the overall motion make the largest contribution to atomic displacements and are a necessary precursor for separating out the internal motion.…”

Normal mode refinement of anisotropic thermal parameters for a supramolecular complex at 3.42-Å crystallographic resolution

“…4 and 5). The concept of using low-frequency normal modes as x-ray crystallographic refinement parameters to model thermal factors dates back to the early 1990s (6)(7)(8)(9)(10)(11). In subsequent years, several attempts have been made to use normal modes in combination with x-ray data for molecular replacement (12,13), positional refinement (14), and protein dynamics (15)(16)(17).…”