Fourier terms

In this representative dihedral potential, V is the dihedral force constant, n is the periodicity of the Fourier term, (jtg is the phase angle, and (]) is the dihedral angle. 

It is shown by empirical tests that the radial distribution function given by a sum of Fourier terms corresponding to the rings observed on an electron diffraction photograph of gas molecules... 

Interatomic vectors. Although, in the absence of knowledge of the signs of the Fourier terms, it is not possible to deduce directly the actual positions of the atoms in the cell, it is theoretically possible to deduce interatomic vectors, that is, the lengths and directions of lines joining atomic centres. Patterson (1934,1935 a) showed that aFourier synthesis employing values of F2 (which are of course all positive) yields this information. The Patterson function... 

Figure 2.14 Beginning of a Fourier series to approximate a target function, in this case, a step function or square wave./0 = 1 /, = cos 2 tt(x) /2 = (—1/3) cos 271. (3x) /3 = Vs) cos 2tt(5x). In the left column are the target and terms/, through f3. In the right column are/0 and the succeeding sums as each term is added to/0. Notice that the approximaton improves (i.e. each successive sum looks more like the target) as the number of Fourier terms in the sum increase. In the last graph, terms f4,f5 and f6 are added (but not shown separately) to show further improvement in the approximation.

Are all three of these parameters accessible in the data on our films We will see in Chapter 5 that the measurable intensity Ihkl of one reflection gives the amplitude of one Fourier term in the series that describes p(x,y,z), and that the position hkl specifies the frequency for that term. But the phase a of each reflection is not recorded on the film. In Chapter 6, we will see how to obtain the phase of each reflection, completing the information we need to calculate p (x,y,z). 

I also stated in Chapter 2 that any wave, no matter how complicated, can be described as the sum of simple waves. This sum is called a Fourier series and each simple wave equation in the series is called a Fourier term. Either Eq. (5.1) or (5.2) could be used as a single Fourier term. For example, we can write a Fourier series of n terms using Eq. (5.1) as follows... 

In words, this series is the sum of n simple Fourier terms, one for each integral value of h beginning with zero and ending with n. Each term is a simple wave with its own amplitude Fh, its own frequency h, and (implicitly) its own phase a. 

Equation (5.18) tells us, at last, how to obtain p(pc,y,z). We need merely to construct a Fourier series from the structure factors. The structure factors describe diffracted rays that produce the measured reflections. A full description of a diffracted ray, like any description of a wave, must include three parameters amplitude, frequency, and phase. In discussing data collection, however, I mentioned only two measurements the indices of each reflection and its intensity. Looking again at Eq. (5.18), you see that the indices of a reflection play the role of the three frequencies in one Fourier term. The only measurable variable remaining in the equation is Fhkf Does the measured intensity of a reflection, the only measurement we can make in addition to the indices, completely define Fhkp Unfortunately, the answer is "no."... 

The indices hkl of the reflection give the three frequencies necessary to describe the Fourier term as a simple wave in three dimensions. Recall from Chapter 2, Section VI.B, that any periodic function can be approximated by a Fourier series, and that the approximation improves as more terms are added to the series (see Fig. 2.14). The low-frequency terms in Eq. (5.18) determine gross features of the periodic function p(x,y,z), whereas the high-frequency terms improve the approximation by filling in fine details. You can also see in Eq. (5.18) that the low-frequency terms in the Fourier series that describes our desired function p(x,y,z) are given by reflections with low indices, that is, by reflections near the center of the diffraction pattern (Fig. 5.2). 

Just as the auto mechanic sometimes has parts left over, electron-density maps occasionally show clear, empty density after all known contents of the crystal have been located. Apparent density can appear as an artifact of missing Fourier terms, but this density disappears when a more complete set of data is obtained. Among the possible explanations for density that is not artifactual are ions like phosphate and sulfate from the mother liquor reagents like mercaptoethanol, dithiothreitol, or detergents used in purification or crystallization or cofactors, inhibitors, allosteric effectors, or other small molecules that survived the protein purification. Later discovery of previously unknown but important ligands has sometimes resulted in subsequent interpretation of empty density. 

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... 

The program and force field SHAPES, developed for transition metal complexes and tested for square planar geometries, uses a single Fourier term (Eq. 2.19), which is similar to the torsional angle term in many molecular mechanics programs (see Section 2.2.3 periodicity = m, phase shift = Fourier force constant kg is related to that of the harmonic potential, kg (Eq. 2.7) by Eq. 2.20. 

Table 2.3. Examples of potential energy functions for common rotors, using a single Fourier term (Eq. 2.8).

Barriers with low symmetry are usually much more complicated than other types. Several Fourier coefficients are normally necessary. Sufficient high-quality experimental data to derive the necessary Fourier terms are often very difficult to obtain. Despite these severe obstacles, several such barriers have been determined in recent years44). Some few of these potentials are presumably of high quality and represent, indeed, great experimental and intellectual achievements. 

Table 3. Some torsional dependent observed quantities for molecules with symmetric end-groups. Central distance (R) and u-values in pm, torsional amplitude (a ) in °, Fourier term (Vn) in kJmol"1...

The program and force field SHAPES, developed for transition metal complexes and tested for square-planar geometries, uses a single Fourier term (Eq. 

To accommodate better the presence of multiple minima and low energy barriers, both CHARMM and MM3 use Fourier expansions in the torsional angle. In fact, virtually all molecular mechanics force fields use Fourier terms for the torsional angle potential energies. 

Banerjee, K. Determinations of the Fourier terms in complete crystal analysis Proc. Roy. Soc. (London) A141, 188-193 (1933). 

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