Refinement of X-ray structure

Reference electrode 300 Refinement of X-ray structure 136 Refolding of proteins 82 Regulation... 

Since atomic fluctuations are the basic elements of the dynamics of proteins (see Chapt. VI.A), it is important to have experimental tests of the accuracy of the simulation results. For the magnitudes of the motions, the most detailed data are provided, in principle, by an analysis of Debye-Waller or temperature factors obtained in crystallographic refinements of X-ray structure. 

A particularly important application of molecular dynamics, often in conjunction with the simulated annealing method, is in the refinement of X-ray and NMR data to determine the three-dimensional structures of large biological molecules such as proteins. The aim of such refinement is to determine the conformation (or conformations) that best explain the experimental data. A modified form of molecular dynamics called restrained moleculai dynarrdcs is usually used in which additional terms, called penalty functions, are added tc the potential energy function. These extra terms have the effect of penalising conformations... 

Energy minimization methods that exploit information about the second derivative of the potential are quite effective in the structural refinement of proteins. That is, in the process of X-ray structural determination one sometimes obtains bad steric interactions that can easily be relaxed by a small number of energy minimization cycles. The type of relaxation that can be obtained by energy minimization procedures is illustrated in Fig. 4.4. In fact, one can combine the potential U r) with the function which is usually optimized in X-ray structure determination (the R factor ) and minimize the sum of these functions (Ref. 4) by a conjugated gradient method, thus satisfying both the X-ray electron density constraints and steric constraint dictated by the molecular potential surface. 

Fluorapatite (FA) corresponds to the chemical formula Caio(P04)eF2 and crystallises in the hexagonal space group PGs/m, with Z = 1 and unit-cell parameters a = b = 9.367 A and c = 6.884 A [1] (Fig. 2). From a structural viewpoint, fluorapatite is often considered as a crystalline model for other apatites and is seen as a reference apatitic array [2]. It is one of the very first apatite structures to have been solved. It has been thoroughly studied since the 1930s [3] and is well documented in the literature. In particular, Sudarsanan et al. [1] reported the single crystal refinement of X-ray diffraction (XRD) data, and the detailed description of atomic positions and local symmetry is fully available [4,5],... 

The absolute square in Eq. (3.30.4) implies that the diffraction intensity Ihkii ) does not have an explicit phase and therefore masks the atom positions (x, /, zj),j = 1,2,..., n], the main goal of X-ray structure determination. This "phase problem" frustrated crystallographers for decennia. However, when one compares the experimental data (thousands of different diffraction intensities f a), with the goal (a few hundred atomic position and their thermal ellipsoid parameters B), one sees that this is a mathematically overdetermined problem. Therefore, first guessing the relative phases of some most intense low-order reflections, one can systematically exploit mutual relationships between intensities that share certain Miller indices, to build a list of many more, statistically likely mutual phases. Finally, a likely and chemically reasonable trial structure is obtained, whose correctness is proven by least-squares refinement. This has made large-angle X-ray structure determination easy for maybe 90% of the data sets collected. 

The SADMA option for the accommodation of small geometry variations is advantageous in studies of small macromolecular distortions, of protein folding processes, and potentially in the structure refinement process of x-ray structure determination. 

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

The preparation and structure of magnesium polyphosphide, MgP4, have been described. The compound was prepared by the reaction of gaseous phosphorus with the phosphide MgaP2 at 600 °C in a sealed silica tube. Evidence for a primitive monoclinic cell was obtained from electron microdiffraction. Refinement of X-ray powder diffraction data showed that the compound is isostructural with... 

At the onset of X-ray structural analysis, the structures of crystalline amino acids were studied to obtain information about bond lengths, valence and torsion angles of the primary building blocks of proteins. These data were used in refining the first crystal structures of proteins, as diffraction data then collected for biopolymers suffered from poor resolution [1],... 

Table 1. Examples of proposed ordering in rock-forming, K-white mica determined by means of X-ray structure refinements.

Jack and Levitt introduced molecular modelling techniques into the refinement in the form of an energy minimisation step (using a force field function) that was performed alternately with the least-squares refinement [Jack and Levitt 1978]. This approach was shown to give convergence to better structures. More recently, restrained molecular dynamics methods were introduced by Brunger, Kuriyan and Karplus [Brunger et al. 1987]. These methods have had a dramatic impact on the refinement of X-ray and NMR structure of proteins. 

Wiener MC, White SH Structure of a fluid dioleoylphosphatidylcholine bilayer determined by joint refinement of X-ray and neutron diffraction data. III. Complete structure. Biophys. J. 1992, 61 434-447. 

For a correct setting of the bet-unit cell, a = 7.842 (a = o/V )-( ) ThSi2-type of structure (14,/amd) was claimed by Mayer and Felner (1973) from a refinement of X-ray powder data. The reUabihty value /t — 0.29 was in favor of an ordered Ni/Si distribution however, agreement was rather poor, thus the actual structure type might be better represented by the LaPtSi-type (ThSi2-derivative structure, 14, md see LaPtSi). 

Figure 9 Comparison of the X-ray structure of Yb +-CLaNP-5-tagged catal3ftic domain of MMP-1 (blue) with the structure of the same tagged protein domain (red) obtained from refining the X-ray structure of the imtagged protein domain (pdb code 3SHI) using the experimental PCS and RDC data obtained for the Ln +-CLaNP-5-tagged catal)ftic domain (Ln = Yb, Tm, and Tb) ...

In principle, it should now be possible to construct a map of electron density within the cell from a set of X-ray structure factors - but there is a major stumbling block. The structure factors are not just numbers but are complex quantities corresponding to sums of wave motions, and therefore have both amplitudes and phases (Eqs 10.8-10.10). All detectors measure intensities integrated over a period of time, so aU we can obtain are the moduli of the structure factors, Ffj i. The so-called crystallographic phase problem is therefore to deduce the phases of the structure factors, as well as the amplitudes. If we manage to do this, we have solved the structure. Note that the term structure solution is used specifically to describe the initial approximate identification of the atom positions within the unit cell, and is distinct from the subsequent refinement of those positions. 

The presence of p r -bonded phenylene rings in the main-chain construction increases the complexity of the intrachain structure. Extended n conjugation is best achieved if the polymer chains adopt a fully planar conformation. However, the steric repulsion between, in the case of PPV, adjacent vinylene and phenylene ring hydrogen atoms favors a nonplanar construction. In PPP these effects become even more pronounced. To further quantify the degree of nonplanarity, structure factor refinements of X-ray and neutron scattering data have been used. In PPV [43] and PPP [ 10] the room temperature mean ring deviation from planarity, defined by H in Fig. 25.5, is found to be approximately 5° and 10°, respectively. 

Two groups have independently developed a method for incorporating a normal mode description of protein dynamics into the refinement of X-ray data. The conventional approach to the refinement of protein structures involves proposing a structural model for the protein which includes thermal parameters for non-hydrogen atoms corresponding to the mean square fluctuation about the equilibrium position. The problem then becomes a least-squares refinement of the crystallographic... 

A.K. Soper, Joint structure refinement of X-ray and neutron diffraction data on disordered materials application to liquid water. J. Phys. Condens. Matter 19, 335206 (2007)... 

In the case of Pj o the assignment of experimental isotropic proton hyperfine couplings (hfc s) to nuclear positions rested mainly on the comparison with theoretical isotropic hfc s. The latter were derived from hydrogen s-spin densities obtained by MO calculations on the basis of X-ray structural data The same procedure was applied later to Pg65 using X-ray structural data of Rb. sphaeroides R-26 in an early refinement stage (March 1990). 

Computer simulations have been applied to studies of the structure of molten salts along two lines one is the fi ee standing application of the computer simulation to obtain the partial pair correlation functions, the other is the refining of x-ray and neutron diffraction and EXAFS measurements by means of a suitable model. In both cases a suitable potential function for the interactions of the ions must be employed, as discussed in Sect. 3.2.4. Such potential functirms are employed in both the Monte Carlo (MC) and the molecular dynamics (MD) simulation methods. A further aspect that has been considered in the case of molten salts is the long range coulombic interaction that exceeds the limits of the periodic simulation boxes usually involved (for 1000 ions altogether), requiring the Ewald summation that is expensive in computation time and is prone to truncation errors if not applied carefully. 

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