Structure refinement

A more systematic way of improving on the trial structure is to use the least-square method. In this method the parameters of the system (such as the atomic coordinates and Debye-Waller factors) are altered by a numerical algorithm in the direction toward minimizing the sum 

In the simple least-squares method in two dimensions, the aim is to find a function y =f(x) that fits a series of observations (x y, (x2,y2).(xi-,yJ.), where each observation is a data point, a measured value of the independent variable x at some selected value y. (For example, y might be the temperature of a gas, and x might be its measured pressure.) The solution to the problem is a function fix) for which the sum of the squares of distances between the data points and the function itself is as small as possible. In other words,/(x) is the function that minimizes D, the sum of the squared differences between observed (yf.) and calculated [f(x )] values, as follows 

The differences are squared to make them all positive otherwise, for a large number of random differences, D simply equals zero. The term wt is an optional weighting factor that reflects the reliability of observation i, thus giving greater influence to the most reliable data. According to principles of statistics, wt should be 1/( t )2, where cr. is the standard deviation computed from multiple measurements of the same data point (x , v ). 

In the simplest case,/(x) is a straight line, for which the general equation is f(x) = mx + b, where m is the slope of the line and b is the intercept of the line on thef(x)-axis. Solving this problem entails finding the proper values of the parameters m and b. If we substitute (mx. + b) for eachf(x.) in Eq. (7.61, take the partial derivative of the right-hand side with respect to m and set it equal to zero, and then take the partial derivative with respect to b and set it equal to zero, the result is a set of simultaneous equations in m and b. Because all the squared differences are to be minimized simultaneously, the number of equations equals the number of observations, and there must be at least two 

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A molecular dynamics simulation nsnally starts with a molecular structure refined by geometry optimization, but wnthont atomic velocities. To completely describe the dynamics of a classical system con lain in g X atom s, yon m nsl define 6N variables. These correspond to ilX geometric coordinates (x, y, and /) and iSX variables for the velocities of each atom in the x, y, and /. directions. 

The greatest value of molecular dynamic simulations is that they complement and help to explain existing data for designing new experiments. The simulations are increasingly useful for structural refinement of models generated from NMR, distance geometry, and X-ray data. 

Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques. 

The first step for any structure elucidation is the assignment of the frequencies (chemical shifts) of the protons and other NMR-active nuclei ( C, N). Although the frequencies of the nuclei in the magnetic field depend on the local electronic environment produced by the three-dimensional structure, a direct correlation to structure is very complicated. The application of chemical shift in structure calculation has been limited to final structure refinements, using empirical relations [14,15] for proton and chemical shifts and ab initio calculation for chemical shifts of certain residues [16]. 

Depending on experimental parameters, NOE intensities will be affected by spin diffusion (Eig. 8). Magnetization can be transferred between two protons via third protons such that the NOE between the two protons is increased and may be observed even when the distance between the two protons is above the usual experimental limit. This is a consequence of the distance dependence of the NOE. Depending on the conformation, it can be more efficient to move magnetization over intennediate protons than directly. The treatment of spin diffusion during structure refinement is reviewed in more detail in Refs. 31, and 71-73. 

D Case. New directions m NMR spectral simulation and structure refinement. In WF van Gunsteren, PK Weiner, AJ WiUcmson, eds. Computer Simulation of Biomolecular Systems Theoretical and Experimental Applications, Vol 2. Leiden ESCOM, 1993, pp 382-406. 

Krishnaswamy, S., Rossmann, M.G. Structural refinement and analysis of mengo virus. /. Mol. Biol. 

Atomic structure refinements or determinations and residual stress measurements, all in bulk materials... 

In this chapter, the structure refinement program will be used to determine the structural parameters of graphitic carbons as shown in section 3. 

The structure refinement program for disordered carbons, which was recently developed by Shi et al [14,15] is ideally suited to studies of the powder diffraction patterns of graphitic carbons. By performing a least squares fit between the measured diffraction pattern and a theoretical calculation, parameters of the model structure are optimized. For graphitic carbon, the structure is well described by the two-layer model which was carefully described in section 2.1.3. 

Figure 6 shows the changes which occur in the diffraction patterns of the heated MCMB samples, and the excellent description of these patterns by the structure refinement program. The structural parameters, P, P, a, doo2> Lc the carbon samples are listed in Table 1 [6]. 

Fig. 6. The X-ray diffraction patterns and calculated best fits from the structure refinement program for the samples MCMB2300, iVICMB2600 and iVfCMB2800.

Contributions in this section are important because they provide structural information (geometries, dipole moments, and rotational constants) of individual tautomers in the gas phase. The molecular structure and tautomer equilibrium of 1,2,3-triazole (20) has been determined by MW spectroscopy [88ACSA(A)500].This case is paradigmatic since it illustrates one of the limitations of this technique the sensitivity depends on the dipole moment and compounds without a permanent dipole are invisible for MW. In the case of 1,2,3-triazole, the dipole moments are 4.38 and 0.218 D for 20b and 20a, respectively. Hence the signals for 20a are very weak. Nevertheless, the relative abundance of the tautomers, estimated from intensity measurements, is 20b/20a 1 1000 at room temperature. The structural refinement of 20a was carried out based upon the electron diffraction data (Section V,D,4). 

In all these examples, the importance of good simulation and modeling cannot be stressed enough. A variety of methods have been used in this field to simulate the data in the cases studies described above. Blander et al. [4], for example, used a semi-empirical molecular orbital method, MNDO, to calculate the geometries of the free haloaluminate ions and used these as a basis for the modeling of the data by the RPSU model [12]. Badyal et al. [6] used reverse Monte Carlo simulations, whereas Bowron et al. [11] simulated the neutron data from [MMIM]C1 with the Empirical Potential Structure Refinement (EPSR) model [13]. 

More detailed aspects of protein function can be obtained also by force-field based approaches. Whereas protein function requires protein dynamics, no experimental technique can observe it directly on an atomic scale, and motions have to be simulated by molecular dynamics (MD) simulations. Also free energy differences (e.g. between binding energies of different protein ligands) can be characterised by MD simulations. Molecular mechanics or molecular dynamics based approaches are also necessary for homology modelling and for structure refinement in X-ray crystallography and NMR structure determination. 

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. 

Cluster 2 appears to be unique among Fe-S-containing proteins whose structures have so far been determined, emd it has been termed the hybrid cluster (6) because of its diverse chemical nature. Figure 14 is a schematic drawing of the cluster as interpreted from the final electron density synthesis and the structure refinement. The cluster contains both oxygen and sulfur bridges, and X represents a site whose precise nature has not been determined, but which may contain a partially occupied and/or disordered substrate molecule (see Section III,B,3,b). The environments of the four iron atoms can be described as follows. 

A review is given of the application of Molecular Dynamics (MD) computer simulation to complex molecular systems. Three topics are treated in particular the computation of free energy from simulations, applied to the prediction of the binding constant of an inhibitor to the enzyme dihydrofolate reductase the use of MD simulations in structural refinements based on two-dimensional high-resolution nuclear magnetic resonance data, applied to the lac repressor headpiece the simulation of a hydrated lipid bilayer in atomic detail. The latter shows a rather diffuse structure of the hydrophilic head group layer with considerable local compensation of charge density. 

Perrakis A, Morris R, Lamzin VS. Automated protein model building combined with iterative structure refinement. Nat Struct Biol 1998 6 458-63. 

Even if the main intent of the Rietveld analysis is the structure refinement in material science, sometimes the information relative to the structure is not the heart of the matter. 

The first step of the structure refinement is the appHcation of distance geometry (DG) calculations which do not use an energy function but only experimentally derived distances and restraints which follow directly from the constitution, the so-caUed holonomic constraints. Those constraints are, for example, distances between geminal protons, which normally are in the range between 1.7 and 1.8 A, or the distance between vicinal protons, which can not exceed 3.1 A when protons are in anti-periplanar orientation. 

Pintacuda, G., Moshref, A., Leonchiks, A., Sharipo, A., Otting, G. Site-specific labeling with a metal chelator for protein-structure refinement. /. Biomol. NMR 2004, 29, 351-361. 

Torda, A. E., Brunne, R. M., Huber, T., Kessler, H., Van Gunsteren, W. F. Structure refinement using time-averaged /-coupling constant resttaints. /. Biomol. NMR 1993, 3, 55-66. 

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