Simulations and model calculations

Glatter, O.

doi:10.1107/97809553602060000581

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 103-104

Section 2.6.1.7. Simulations and model calculations

O. Glatter^a

2.6.1.7. Simulations and model calculations

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2.6.1.7.1. Simulations

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Simulations can help to find the limits of the method and to estimate the systematic errors introduced by the data-evaluation procedure. Simulations are performed with exactly known model systems (test functions). These systems should be similar to the structures of interest. The model data are transformed according to the special experimental situation (collimation profiles and wavelength distribution) starting from the theoretical PDDF (or scattering function). `Experimental data points' are generated by sampling in a limited h range and adding statistical noise from a random-number generator. If necessary, a certain amount of background scattering can also be added. This simulated data set is subjected to the data-evaluation procedure and the result is compared with the starting function. Such simulation can reveal the influence of each approximation applied in the various evaluation routines.

On the other hand, simulations can also be used for the optimization of the experimental design for a special application. The experiment situation is characterized by several contradictory effects: a large width for the functions P(t), Q(x), and W(λ′) leads to a high statistical accuracy but considerable smearing effects. The quality of the results of the desmearing procedure is increased by high statistical accuracy, but decreased by large smearing effects. Simulations can help to find the optimum for a special application.

2.6.1.7.2. Model calculation

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In the section on data evaluation and interpretation, we have seen that we obtain a rough estimate for the structure of the particles under investigation directly from the experimental data. For further refinement, we have to compare our results with scattering functions or PDDF's from models.

2.6.1.7.3. Calculation of scattering intensities

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The scattering curves can be calculated semi-analytically for simple triaxial bodies and for models composed of some of these bodies. The scattering amplitude for regular bodies like ellipsoids, parallelepipeds, and cylinders can be calculated analytically for any orientation. The spatial averaging has to be performed numerically. Such calculations have been performed for a large number of different models by Porod (1948), Mittelbach & Porod (1961a,b, 1962), and by Mittelbach (1964). More complicated structures can be described by models composed of several such triaxial bodies, but the computing time necessary for such calculations can be hours on a mainframe computer.

Models composed only of spherical subunits can be evaluated with the Debye formula (Debye, 1915): $[I(h)=i_{\rm el} (h) \sum^N_{i=1} \sum^N_{k=1} \rho_iV_i\rho_kV_k\Phi_i(h)\Phi_k(h) {\textstyle \sin (hd_{ik})\over \textstyle hd_{ik}},\eqno (2.6.1.67)]$ where the spatial average is carried out analytically. Another possibility would be to use spherical harmonics as discussed in the previous section but the problem is how to find the expansion coefficients for a certain given geometrical structure.

2.6.1.7.4. Method of finite elements

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Models of arbitrary shape can be approximated by a large number of very small homogeneous elements of variable electron density. These elements have to be smaller than the smallest structural detail of interest.

Sphere method . In this method, the elements consist of spheres of equal size. The diameter of these spheres must be chosen independently of the distance between nearest neighbours, in such a way that the total volume of the model is represented correctly by the sum of all volume elements (which corresponds to a slight formal overlap between adjacent spheres). The scattering intensity is calculated using the Debye formula (2.6.1.67), with $[\Phi_i(h)=\Phi_k(h)=\Phi(h)]$ .

The computing time is mainly controlled by the number of mutual distances between the elements. The computing time can be lowered drastically by the use of approximate $[d_{ik}]$ values in (2.6.1.67). Negligible errors in I(h) result if $[d_{ik}]$ values are quantized to D_max/10000 (Glatter, 1980c). For the practical application (input operation), it is important that a certain number of elements can be combined to form so-called sub-structures that can be used in different positions with arbitrary weights and orientations to build the model.

The sphere method can also be used for the computation of scattering curves for macromolecules from a known crystal structure. The weights of the atoms are given by the effective number of electrons $[Z_{\rm eff}=Z-\rho _0V_{\rm eff},\eqno (2.6.1.68)]$ where $[V_{\rm eff}]$ is the apparent volume of the atom given by Langridge, Marvin, Seeds, Wilson, Cooper, Wilkins & Hamilton (1960).

Cube method . This method has been developed independently by Fedorov, Ptitsyn & Voronin (1972, 1974a,b) and by Ninio & Luzzati (1972) mainly for the computation of scattered intensities for macromolecules in solution whose crystal structure is known. In the cube method, the macromolecule is mentally placed in a parallelepiped, which is subdivided into small cubes (with edge lengths of 0.5–1.5 Å. Each cube is examined in order to decide whether it belongs to the molecule or to the solvent. Adjacent cubes in the z direction are joined to form parallelepipeds. The total scattering amplitude is the sum over the amplitudes from the parallelepipeds with different positions and lengths. The mathematical background is described by Fedorov, Ptitsyn & Voronin (1974a,b). The modified cube method of Fedorov & Denesyuk (1978) takes into account the possible penetration of the molecule by water molecules.

2.6.1.7.5. Calculation of distance-distribution functions

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The PDDF can be calculated analytically only for a few simple models (Porod, 1948; Goodisman, 1980); in all other cases, we have to use a finite element method with spheres. It is possible to define an analogous equation to the Debye formula (2.6.1.67) in real space (Glatter, 1980c). The PDDF can be expressed as $[\eqalignno {p(r)={}&\textstyle\sum\limits^N_{i=1}\rho_i^2p_0(r,R_i)\cr &+2\textstyle\sum\limits ^{N-1}_{i=1} \sum\limits^N_{k=i+1} \rho _i\rho_k p(r,d_{ik},R_i,R_k). &(2.6.1.69)}]$ [p_0(r,R_i] ) is the PDDF of a sphere with radius [R_i] and electron density equal to unity, $[p(r,d_{ik},R_i,R_k)]$ is the cross-term distance distribution between the ith and kth spheres (radii [R_i] and [R_k] ) with a mutual distance $[d_{ik}]$ .

Equation (2.6.1.69) [and (2.6.1.67)] can be used in two different ways for the calculation of model functions. Sometimes, it is possible to approximate a macromolecule as an aggregate of some spheres of well defined size representing different globular subunits (Pilz, Glatter, Kratky & Moring-Claesson, 1972). The form factors of the subunits are in such cases real parameters of the model. However, in most cases we have no such possibility and we have to use the method of finite elements, i.e. we fit our model with a large number of sufficiently small spheres of equal size, and, if necessary, different weight. The form factor of the small spheres is now not a real model parameter and introduces a limit of resolution.

Fourier transformation [equation (2.6.1.10)] can be used for the computation of the PDDF of any arbitrary model if the scattering function of the model is known over a sufficiently large range of h values.

References

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 103-104