Fourier series electron density

The Fourier theorem states that any periodic function may be resolved into cosine and sine terms involving known constants. Since a crystal has a periodically repeating internal structure, this can be represented, in a mathematically useful way, by a three-dimensional Fourier series, to give a three-dimensional Fourier or electron density map. In X-ray diffraction studies the magnitudes of the coefficients may be derived from... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

We wish to obtain an image of the scattering elements in three dimensions (the electron density). To do this, we perform a 3-D Fourier synthesis (summation). Fourier series are used because they can be applied to a regular periodic function crystals are regular periodic distributions of atoms. The Fourier synthesis is given in O Eq. 22.2 ... 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

Due to termination of the series, however, p(r) is severely affected by ripples. In addition, especially in the case of non-centrosymmetric crystals, the phase of vector F(S) is not known with precision and this affects a correct reconstruction of the density. Therefore, Fourier summation cannot be used for precise and accurate mapping of electron density. On the other hand, a model is necessary to overcome these limitations and to produce a function that is sufficiently close to the real, quantum mechanical />(r) in all regions of the crystal. 

The three-dimensional periodic electron-density distribution in a single crystal can be represented by a three-dimensional Fourier series with the so-called structure factors Fhkl as Fourier coefficients ... 

The electron density distribution within a crystal can be expressed in a similar way as a three-dimensional Fourier series ... 

There is another important point. If a Fourier series is cut off sharply when the terms are still appreciable, false detail will appear in the electron density map. To avoid this, for crystals giving strong reflections at large angles, an artificial temperature factor may be applied to the intensities, to make the F s fade off gradually instead of... 

If the absolute intensities of the X-ray reflections are not available— but only relative intensities—the value of the constant term (the equivalent of 000 in the equations given previously) in relation to the other terms of the Fourier series (which are in this case in arbitrary units) is not known the figures for the electron density obtained by calculation, omitting the constant term, will all be wrong by this amount but for the purpose of locating atomic centres, this is of no consequence the image formed by the electron density contours is of precisely the same form. 

If we are only interested in the variation of electron density in the direction normal to the plane of the film, a quantity defined in Section 2.3 as /p, we can treat / as a constant and expand p(x) as a Fourier series. 

Because the electron density we seek is a complicated periodic function, it can be described as a Fourier series. Do the many structure-factor equations, each a sum of wave equations describing one reflection in the diffraction pattern, have any connection with the Fourier series that describes the electron density As mentioned earlier, each structure-factor equation can be written as a sum in which each term describes diffraction from one atom in the unit cell. But this is only one of many ways to write a structure-factor equation. Another way is to imagine dividing the electron density in the unit cell into many small volume elements by inserting planes parallel to the cell edges (Fig. 2.16). 

In Bragg s way of looking at diffraction as reflection from sets of planes in the crystal, each set of parallel planes described here (as well as each additional set of planes interleaved between these sets) is treated as an independent diffractor and produces a single reflection. This model is useful for determining the geometry of data collection. Later, when I discuss structure determination, I will consider another model in which each atom or each small volume element of electron density is treated as an independent diffractor, represented by one term in a Fourier series that describes each reflection. Bragg s model tells us where to look for the data. The Fourier series model tells us what the data has to say about molecular structure. 

It is not coincidental that I use the variable names h, k, and I for both the indices of planes in the crystal and the indices of reflections in the diffraction pattern (Chapter 2, Section V). I will show later that in fact the set of planes (hkl) produces the reflection hkl of the diffraction pattern. In the terms used in Chapter 2, each set of parallel planes in the crystal produces one reflection, or one term in the Fourier series that describes the electron density within the unit cell. The intensity of that reflection depends upon the electron distribution and density along the planes that produce the reflection. 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

First I will discuss Fourier series and the Fourier transform in general terms. I will emphasize the form of these equations and the information they contain, in the hope of helping you to interpret the equations — that is, to translate the equations into words and visual images. Then I will present the specific types of Fourier series that represent structure factors and electron density and show how the Fourier transform inter con verts them. 

The Fourier series that the crystallographer seeks is p(x,y,z), the three-dimensional electron density of the molecules under study. This function is a wave equation or periodic function because it repeats itself in every unit cell. The waves described in the preceeding equations are one-dimensional they represent a numerical value/(x) that varies in one direction, along the x-axis. How do we write the equations of two-dimensional and three-dimensional waves First, what do the graphs of such waves look like ... 

I have stated that both structure factors and electron density can be expressed as Fourier series. A structure factor describes one diffracted X-ray, which produces one reflection received at the detector. A structure factor Fhkl can be written as a Fourier series in which each term gives the contribution of one atom to the reflection hkl [see Fig. 2.15 and Eq. (2.3)]. Here is a single term, called an atomic structure factor fhkl, in such a series, representing the contribution of the single atom j to reflection hkl ... 

In words, the structure factor that describes reflection hkl is a Fourier series in which each term is the contribution of one atom, treated as a simple sphere of electron density. So the contribution of each atom j to Fhkl depends on (1) what element it is. which determines jf, the amplitude of the contribution, and (2) its position in the unit cell (Xj, yj, z-)> which establishes the phase of its contribution. 

When we describe structure factors and electron density as Fourier series, we find that they are intimately related. The electron density is the Fourier transform of the structure factors, which means that we can convert the crystallographic data into an image of the unit cell and its contents. One necessary piece of information is, however, missing for each structure factor. We can measure only the intensity Ihkl of each reflection, not the complete structure factor Fhkl. What is the relationship between them It can be shown that the amplitude of structure factor Fhkl is proportional to the square root of... 

The molecular image that the crystallographer seeks is a contour map of the electron density p(x,y,z) throughout the unit cell. The electron density, like all periodic functions, can be represented by a Fourier series. The representation that connects p(x,y,z) to the diffraction pattern is... 

This equation gives the desired electron density as a function of the known amplitudes IFI and the unknown phases ot hkl of each reflection. Recall that this equation represents p(x,y,z) in a now-familiar form, as a Fourier series, but this time with the phase of each structure factor expressed explicitly. Each term in the series is a three-dimensional wave of amplitude IF I, phase (x hkl, and frequencies h along the x-axis, k along the y-axis, and 1 along the z-axis. 

As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of the heavy-atom derivative, and then solving a simpler "structure," that of one heavy atom (or a few) in the unit cell of the protein. The most powerful tool in determining the heavy-atom coordinates is a Fourier series called the Pattersonfunction P(u,v,w), a variation on the Fourier series used to compute p(x,y,z) from structure factors. The coordinates (u,v,w) locate a point in a Patterson map, in the same way that coordinates (x,y,z) locate a point in an electron-density map. The Patterson function or Patterson synthesis is a Fourier series without phases. The amplitude of each term is the square of one structure factor, which is proportional to the measured reflection intensity. Thus we can construct this series from intensity measurements, even though we have no phase information. Here is the Patterson function in general form... 

Here is the Fourier series that gives the first electron-density map... 

In words, the desired electron-density function is a Fourier series in which term hkl has amplitude IFobsl, which equals (7/, /)1/2, the square root of the measured intensity Ihkl from the native data set. The phase ot hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as described in Chapter 6. The term is weighted by the factor whU, which will be near 1.0 if ct hkl is among the most highly reliable phases, or smaller if the phase is questionable. This Fourier series is called an Fobs or Fo synthesis (and the map an Fo map) because the amplitude of each term hkl is iFobsl for reflection hkl. 

To compare apo- and holo-forms of proteins after both structures have been determined independently, crystallographers often compute difference Fourier syntheses (Chapter 7, Section IV.B), in which each Fourier term contains the structure-factor difference FAc>/c(-—F 0. A contour map of this Fourier series is called a difference map, and it shows only the differences between the holo-and apo- forms. Like the FQ — Fc map, the FAoio—F map contains both positive and negative density. Positive density occurs where the electron density of the holo-form is greater than that of the apo-form, so the ligand shows up clearly in positive density. In addition, conformational differences between holo- and apo-forms result in positive density where holo-protein atoms occupy regions that are unoccupied in the apo-form, and negative... 

Plate 12 Electron-density maps at increasing resolution. Maps were calculated using final phases, and Fourier series were truncated at the resolution limits indicated (a) 6.0 A (b) 4.5 A (c) 3.0 A (d) 1.6 A. (For discussion, see Chapter 7.) (Continues)... 

The Laue and the Bragg condition give us information about the angular distribution of the diffraction peaks. To calculate the peak intensities, we have to know more about the scattering properties of the atoms or molecules in the crystal. In the case of X-rays and electrons the scattering probability is proportional to the electron density ne(r) within the crystal. Since n,(r) has to have the same periodicity as the crystal lattice, we can write it as a three-dimensional Fourier series (using the notation eikx = cos kx + i sin kx) ... 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

The structure of the unit cell of a crystal may be described in terms of the electron density distribution in the cell. The x,y,z coordinates of the maxima of the electron density function correspond to the positions of the atoms. The electron density distribution function p (x,y,z) may be represented by a three-dimensional Fourier series... 

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