Anomalous scattering method 83

Diffraction, under normal conditions, results when the electrons absorb energy from the incident radiation, are set to vibrate and re-emit radiation of the same frequency. This is called coherent scattering and can occur only when electrons can vibrate freely inside atoms. Under such conditions, reflections related by inversion through the origin (Friedel pair), have the same amplitude but opposite phase, known as Friedel’s law (Equation 3.30). However, the electrons absorb differently as the wavelength of the incident radiation varies. At certain wavelengths, characteristic of the type of atom and the quantum level of the electron (K, L, M, etc.), the absorption decreases abruptly and the corresponding region of wavelength is called the absorption edge. At wavelengths just short of the absorption edge of an element, which correspond to atomic transitions, the energy of the X-rays is sufficient to eject electrons out of their shell (transition energy). Once ejected, the electrons do not behave as free electrons any more and scatter radiation with a frequency and phase different from the incident wave (incoherent scattering). Under these abnormal or anomalous conditions, Friedel’s law does not hold and the phenomenon is known as anomalous scattering. It was first shown by Bijvoet that these differences in the intensities of Friedel pairs, arising due to the presence of anomalous scatterers, could be used to estimate the phase of the diffracted waves.

Fhkl = F!h!k!l (3.29)

At X-ray wavelengths generally used for diffraction experiments, lighter atoms like carbon, oxygen or nitrogen do not show anomalous scattering but elements like sulphur or those heavier do. These atoms are known as heavy atoms and one of the elements widely used for anomalous scattering experiments is selenium. Selenium can be incorporated into a protein by growing E. coli cells expressing the target protein in a minimal medium containing selenomethionine instead of normal methionine. Anomalous data can be collected by adjusting the wavelength of the X-rays according to the heavy atom used, which can easily be achieved in the synchrotron sources.

When the heavy atoms present in a protein absorb X-rays with a wavelength near its absorption edge, they emit radiation with an altered phase. The scattering factor, under these conditions, is composed of an anomalous component apart from the normal component (the scattering factor at wavelengths far apart from the edge). Furthermore, the anomalous component of the altered atomic scattering factor is made of a real part and an imaginary part that is perpendicular to the real part. This can be represented as in Equation 3.30 and is depicted in Figure 3-7.

f_anom = f_N + δf ’ + f ” (3.30)

Replacing the real components (f_{N + δf ’) by f ’, Equation 3.30 can be written as:}

f_anom = f ’ + f ” (3.31)

f

f

δf’

f”

f

Figure 3-7: Vector representation of anomalous scattering

fN is the normal atomic scattering factor at wavelengths far apart from the absorption edge. The

anomalous component (fH) consists of a real part (δf ’) and an imaginary part (f ”), perpendicular to each

other. The total anomalous scattering is denoted by fanom.

3.6.3.1 Relation between Friedel mates during anomalous scattering

As was mentioned earlier, as a result of anomalous scattering, Friedel’s law is broken and thus, the Friedel mates have different intensities and are no longer related by opposite phases to each other. The Friedel pair of reflections that differ from each other due to anomalous scattering are known as Bijvoet pairs. This has been depicted in Figure 3-8, where the FHP+ and FHP- represent the anomalous scattering factors of a

Bijvoet pair and are oriented at different angles from the horizontal axis. The real contributions to the anomalous scattering, FH(r)+ and FH(r)- are still the reflections of each other along the X-axis and are thus, related by opposite phases. However, the imaginary parts, FH(i)+ and FH(i)- are inverted reflections of each other (rather than ordinary reflections) and this difference is responsible for the differences in the intensity as well as the phase of Friedel mates under conditions of anomalous scattering.

F_N- F_HP- F_N+ F_HP+ F_{H (i)}- F_{H (i)}+ F_{H (r)}+ F_{H (r)}-

Figure 3-8: Violation of Friedel’s law

Friedel’s law does not relate the scattering factors FHP+ and FHP- as they are not the mirror images of each

other, unlike FN+ and FN-. The corresponding pair of reflections, under the conditions of anomalous

scattering is known as a Bijvoet pair.

3.6.3.2 Calculation of phases from the anomalous data

From Figure 3-8 and Equation 3.30,

where FHP is the anomalous scattering factor, FN is the non-anomalous or normal scattering factor and FH(r) and FH(i) are the real and imaginary parts of the anomalous scattering. All the terms in Equation 3.32 involve the reflection with a positive phase and the same equation holds for its Friedel mate (with a negative phase). Solving Equation 3.32 for only FN+ results in a phase ambiguity, as in the case of single isomorphous replacement. However, the actual phase can be obtained when the anomalous scattering component of the Friedel mate (FN- _{in this case) is taken into} account. This explains the importance of anomalous scattering in calculating the phases of the diffraction data.

Equation 3.33 can be rearranged as (without specifying any Friedel mate):

FN= FHP- FH(r) - FH(i) (3.33)

The magnitudes of FH(r) and FH(i) are constants for a given element (e.g., selenium) while their phases can be calculated from the positions of the heavy atoms (calculated by Patterson methods). Thus, FNcan be obtained as all the terms on the right hand side of Equation 3.33 can be calculated with the help of the heavy atom anomalous data. Solution of the phase problem by MAD or multiwavelength anomalous diffraction method involves collection of anomalous data at different wavelengths. The data are generally collected at wavelengths corresponding to – (a) the peak, where the anomalous signal is maximum; (b) the absorption edge, also known as the inflection, and (c) the remote, which is far from the peak wavelength. The real and imaginary parts of the anomalous scattering, FH(r) and FH(i) vary considerably at different wavelengths and this difference is used for precise calculation of the phases. Besides, the intensities of the reflections also vary at different wavelengths and these differences can also be harnessed to aid in determination of the phase.