Structure factor computing from model

When low-temperature studies are performed, the maximum resolution is imposed by data collection geometry and fall-off of the scattered intensities below the noise level, rather than by negligible high-resolution structure factor amplitudes. Use of Ag Ka radiation would for example allow measurement of diffracted intensities up to 0.35 A for amino-acid crystals below 30 K [40]. Similarly, model calculations show that noise-free structure factors computed from atomic core electrons would be still non-zero up to O.lA. 

There are therefore four adjustable parameters per atom in the refinement (xy, yy, Zj, By). In the computer experiments we have carried out to test the assumptions of the nucleic acid refinement model we have generated sets of observed structure factors F (Q), from the Z-DNA molecular dynamics trajectories. The thermal averaging implicit in Equation III.3 is accomplished by averaging the atomic structure factors obtained from coordinate sets sampled along the molecular dynamics trajectories at each temperature ... 

To check this prediction, a number of MaxEnt charge density calculations have been performed with the computer program BUSTER [42] on a set of synthetic structure factors, obtained from a reference model density for a crystal of L-alanine at 23 K. The set of 1500 synthetic structure factors, complete up to a resolution of 0.555 A [45], was calculated from a multipolar expansion of the density, with the computer program VALRAY[ 46],... 

Equation (5.15) describes one structure factor in terms of diffractive contributions from all atoms in the unit cell. Equation (5.16) describes one structure factor in terms of diffractive contributions from all volume elements of electron density in the unit cell. These equations suggest that we can calculate all of the structure factors either from an atomic model of the protein or from an electron density function. In short, if we know the structure, we can calculate the diffraction pattern, including the phases of all reflections. This computation, of course, appears to go in just the opposite direction that the crystallographer desires. It turns out, however, that computing structure factors from a model of the unit cell (back-transforming the model) is an essential part of crystallography, for several reasons. 

In words, for each reflection, we compute the difference between the observed structure-factor amplitude from the native data set IFobsl and the calculated amplitude from the model in its current trial location IFcalcl and take the absolute value, giving the magnitude of the difference. We add these magnitudes for all reflections. Then we divide by the sum of the observed structure-factor amplitudes (the reflection intensities). 

As the model is built, the viewer sees the model within the map, as shown in Plate 2 b. As the model is constructed or adjusted, the program stores current atom locations in the form of three-dimensional coordinates. The crystallogra-pher, while building a model interactively on the computer screen, is actually building a list of atoms, each with a set of coordinates (x,y,z) to specify its location. Coordinates are automatically updated whenever the model is adjusted. This list of coordinates is the output file from the map-fitting program and the input file for calculation of new structure factors. When the model is correct and complete, this file becomes the means by which the model is shared with the community of scientists who study proteins (see Section VII). 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

In this section we briefly discuss an approximate formalism that allows incorporation of the experimental error variances in the constrained maximisation of the Bayesian score. The problem addressed here is the derivation of a likelihood function that not only gives the distribution of a structure factor amplitude as computed from the current structural model, but also takes into account the variance due to the experimental error. 

A test of the computational strategy outlined in the previous paragraph has been performed on a set of synthetic noisy structure factor amplitudes. The diffraction data were computed from the same model density for L-alanine at 23 K as the one used for the noise-free calculations described in Section 3.1. 

Gaussian noise has been added onto the structure factor amplitudes squared as computed from the L-alanine model density for each datum, the amount of noise added was proportional to the experimental esd for the corresponding intensity measurement ... 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

In order to learn more about the Rouse model and its limits a detailed quantitative comparison was recently performed of molecular dynamics (MD) computer simulations on a 100 C-atom PE chain with NSE experiments on PE chains of similar molecular weight [52]. Both the experiment and the simulation were carried out at T=509 K. Simulations were imdertaken,both for an explicit (EA) as well as for an united (l/A) atom model. In the latter the H-atoms are not explicitly taken into account but reinserted when calculating the dynamic structure factor. The potential parameters for the MD-simulation were either based on quantum chemical calculations or taken from literature. No adjusting... 

I will discuss the iterative improvement of phases and electron-density maps in Chapter 7. For now just take note that obtaining the final structure entails both calculating p(x,y,z) from structure factors and calculating structure factors from some preliminary form of p(x,y,z). Note further that when we compute structure factors from a known or assumed model, the results include the phases. In other words, the computed results give all the information needed for a "full-color" diffraction pattern, like that shown in Plate 3d, whereas experimentally obtained diffraction patterns lack the phases and are merely black and white, like Plate 3e. 

Displaying an electron-density map and adjusting the models to improve its fit to the map (see Plate 21 and the cover of this book). SPV can display maps of several types (CCP4, X-PLOR, DN6). I am aware of no programs currently available for computation of maps from structure factors on personal computers, but I am sure this will soon change. 

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