Chapter 2.4. Isomorphous replacement and anomalous scattering

Two crystals are said to be isomorphous if (a) both have the same space group and unit-cell dimensions and (b) the types and the positions of atoms in both are the same except for a replacement of one or more atoms in one structure with different types of atoms in the other (isomorphous replacement) or the presence of one or more additional atoms in one of them (isomorphous addition). Consider two crystal structures with identical space groups and unit-cell dimensions, one containing N atoms and the other M atoms. The N atoms in the first structure contain subsets P and Q whereas the M atoms in the second structure contain subsets P, Q′ and R. The subset P is common to both structures in terms of atomic positions and atom types. The atomic positions are identical in subsets Q and Q′, but at any given atomic position the atom type is different in Q and Q′. The subset R exists only in the second structure. If and denote the structure factors of the two structures for a given reflection, and where the quantities on the right-hand side represent contributions from different subsets. From (2.4.2.1) and (2.4.2.2) we have The above equations are illustrated in the Argand diagram shown in Fig. 2.4.2.1. and would be collinear if all the atoms in Q were of the same type and those in Q′ of another single type, as in the replacement of chlorine atoms in a structure by bromine atoms.

Figure 2.4.2.1 | top | pdf | Vector relationship between and .

We have a case of `isomorphous replacement' if and a case of `isomorphous addition' if . Once is known, in addition to the magnitudes of and , which can be obtained experimentally, the two cases can be treated in an equivalent manner in reciprocal space. In deference to common practice, the term `isomorphous replacement' will be used to cover both cases. Also, in as much as is the vector sum of and , and will be used synonymously. Thus

The number of replaceable (or `added') atoms is usually small and they generally have high atomic numbers. Their positions are often determined by a Patterson synthesis of one type or another (see Chapter 2.3 ). It will therefore be assumed in the following discussion that is known. Then it can be readily seen by referring to Fig. 2.4.2.2 that when is derived from its cosine function, it could obviously be positive or negative. Hence, there are two possible solutions for . These two solutions are distributed symmetrically about . One of these would correspond to the correct value of . Therefore, in general, the phase angle cannot be unambiguously determined using a pair of isomorphous crystals.

Figure 2.4.2.2 | top | pdf | Relationship between and .

The twofold ambiguity in phase angle vanishes when the structures are centrosymmetric. and are all real in centric data and the corresponding phase angles are 0 or . From (2.4.2.4) The sign of is already known and the signs of and can be readily determined from (2.4.2.6) (Robertson & Woodward, 1937).

When the data are acentric, the best one can do is to use both the possible phase angles simultaneously in a Fourier synthesis (Bokhoven et al., 1951). This double-phased synthesis, which is equivalent to the isomorphous synthesis of Ramachandran & Raman (1959), contains the structure and its inverse when the replaceable atoms have a centrosymmetric distribution (Ramachandran & Srinivasan, 1970). When the distribution is noncentrosymmetric, however, the synthesis contains peaks corresponding to the structure and general background. Fourier syntheses computed using the single isomorphous replacement method of Blow & Rossmann (1961) and Kartha (1961) have the same properties. In this method, the phase angle is taken to be the average of the two possible solutions of , which is always or . Also, the Fourier coefficients are multiplied by cos , following arguments based on the Blow & Crick (1959) formulation of phase evaluation (see Section 2.4.4.4). Although Blow & Rossmann (1961) have shown that this method could yield interpretable protein Fourier maps, it is rarely used as such in protein crystallography as the Fourier maps computed using it usually have unacceptable background levels (Blundell & Johnson, 1976).

The ambiguity in in a noncentrosymmetric crystal can be resolved only if at least two crystals isomorphous to it are available (Bokhoven et al., 1951). We then have two equations of the type (2.4.2.5), namely, where subscripts 1 and 2 refer to isomorphous crystals 1 and 2, respectively. This is demonstrated graphically in Fig. 2.4.2.3 with the aid of the Harker (1956) construction. A circle is drawn with as radius and the origin of the vector diagram as the centre. Two more circles are drawn with and as radii and the ends of vectors and , respectively as centres. Each of these circles intersects the circle at two points corresponding to the two possible solutions. One of the points of intersection is common and this point defines the correct value of . With the assumption of perfect isomorphism and if errors are neglected, the phase circles corresponding to all the crystals would intersect at a common point if a number of isomorphous crystals were used for phase determination.

Figure 2.4.2.3 | top | pdf | Harker construction when two heavy-atom derivatives are available.

Atomic scattering factors are normally calculated on the assumption that the binding energy of the electrons in an atom is negligible compared to the energy of the incident X-rays and the distribution of electrons is spherically symmetric. The transition frequencies within the atom are then negligibly small compared to the frequency of the radiation used and the scattering power of each electron in the atom is close to that of a free electron. When this assumption is valid, the atomic scattering factor is a real positive number and its value decreases as the scattering angle increases because of the finite size of the atom. When the binding energy of the electrons is appreciable, the atomic scattering factor at any given angle is given by where is a real positive number and corresponds to the atomic scattering factor for a spherically symmetric collection of free electrons in the atom. The second and third terms are, respectively, referred to as the real and the imaginary components of the `dispersion correction' (IT IV, 1974). f′ is usually negative whereas f″ is positive. For any given atom, f″ is obviously ahead of the real part of the scattering factor given by

The variation of f′ and f″ as a function of atomic number for two typical radiations is given in Fig. 2.4.3.1 (Srinivasan, 1972; Cromer, 1965). The dispersion effects are pronounced when an absorption edge of the atom concerned is in the neighbourhood of the wavelength of the incident radiation. Atoms with high atomic numbers have several absorption edges and the dispersion-correction terms in their scattering factors always have appreciable values. The values of f′ and f″ do not vary appreciably with the angle of scattering as they are caused by core electrons confined to a very small volume around the nucleus. An atom is usually referred to as an anomalous scatterer if the dispersion-correction terms in its scattering factor have appreciable values. The effects on the structure factors or intensities of Bragg reflections resulting from dispersion corrections are referred to as anomalous-dispersion effects or anomalous-scattering effects.

Figure 2.4.3.1 | top | pdf | Variation of (a) and (b) as a function of atomic number for Cu and Mo radiations. Adapted from Fig. 3 of Srinivasan (1972).

Consider a structure containing N atoms of which P are normal atoms and the remaining Q anomalous scatterers. Let denote the contribution of the P atoms to the structure, and and the real and imaginary components of the contribution of the Q atoms. The relation between the different contributions to a reflection h and its Friedel equivalent is illustrated in Fig. 2.4.3.2. For simplicity we assume here that all Q atoms are of the same type. The phase angle of is then exactly ahead of that of . The structure factors of h and are denoted in the figure by and , respectively. In the absence of anomalous scattering, or when the imaginary component of the dispersion correction is zero, the magnitudes of the two structure factors are equal and Friedel's law is obeyed; the phase angles have equal magnitudes, but opposite signs. As can be seen from Fig. 2.4.3.2, this is no longer true when has a nonzero value. Friedel's law is then violated. A composite view of the vector relationship for h and can be obtained, as in Fig. 2.4.3.3, by reflecting the vectors corresponding to about the real axis of the vector diagram. and corresponding to the two reflections superpose exactly, but do not. and then have different magnitudes and phases.

Figure 2.4.3.2 | top | pdf | Vector diagram illustrating the violation of Friedel's law when .

Figure 2.4.3.3 | top | pdf | A composite view of the vector relationship between and .

It is easily seen that Friedel's law is obeyed in centric data even when anomalous scatterers are present. and are then parallel to the real axis and perpendicular to it. The vector sum of the three components is the same for h and . It may, however, be noted that the phase angle of the structure factor is then no longer 0 or . Even when the structure is noncentrosymmetric, the effect of anomalous scattering in terms of intensity differences between Friedel equivalents varies from reflection to reflection. The difference between and is zero when or . The difference tends to the maximum possible value when .

Intensity differences between Friedel equivalents depend also on the ratio (in terms of number and scattering power) between anomalous and normal scatterers. Differences obviously do not occur when all the atoms are normal scatterers. On the other hand, a structure containing only anomalous scatterers of the same type also does not give rise to intensity differences. Expressions for intensity differences between Friedel equivalents have been derived by Zachariasen (1965) for the most general case of a structure containing normal as well as different types of anomalous scatterers. Statistical distributions of such differences under various conditions have also been derived (Parthasarathy & Srinivasan, 1964; Parthasarathy, 1967). It turns out that, with a single type of anomalous scatterer in the structure, the ratio has a maximum mean value when the scattering powers of the anomalous scatterers and the normal scatterers are nearly the same (Srinivasan, 1972). Also, for a given ratio between the scattering powers, the smaller the number of anomalous scatterers, the higher is the mean ratio.

The discussion so far has been concerned essentially with crystals belonging to space groups P1 and . In the centrosymmetric space group, the crystal and the diffraction pattern have the same symmetry, namely, an inversion centre. In P1, however, the crystals are noncentrosymmetric while the diffraction pattern has an inversion centre, in the absence of anomalous scattering. When anomalous scatterers are present in the structure , Friedel's law breaks down and the diffraction pattern no longer has an inversion centre. Thus the diffraction pattern displays the same symmetry as that of the crystal in the presence of anomalous scattering. The same is true with higher-symmetry space groups also. For example, consider a crystal with space group P222, containing anomalous scatterers. The magnitudes of are the same for all equivalent reflections; so are those of and . Their phase angles, however, differ from one equivalent to another, as can be seen from Table 2.4.3.1. When , the magnitudes of the vector sum of and are the same for all the equivalent reflections. The intensity pattern thus has point-group symmetry . When , the equivalent reflections can be grouped into two sets in terms of their intensities: , , and ; and , , and . The equivalents belonging to the first group have the same intensity; so have the equivalents in the second group. But the two intensities are different. Thus the symmetry of the pattern is 222, the same as that of the crystal.

Table 2.4.3.1 | top | pdf | Phase angles of different components of the structure factor in space group Reflection Phase angle of

In general, under conditions of anomalous scattering, equivalent reflections generated by the symmetry elements in the crystal have intensities different from those of equivalent reflections generated by the introduction of an additional inversion centre in normal scattering. There have been suggestions that a reflection from the first group and another from the second group should be referred to as a `Bijvoet pair' instead of a `Friedel pair', when the two reflections are not inversely related. Most often, however, the terms are used synonymously. The same practice will be followed in this article.

The determination of the absolute configuration of chiral molecules has been among the most important applications of anomalous scattering. Indeed, anomalous scattering is the only effective method for this purpose and the method, first used in the early 1950s (Peerdeman et al., 1951), has been extensively employed in structural crystallography (Ramaseshan, 1963; Vos, 1975).

Many molecules, particularly biologically important ones, are chiral in that the molecular structure is not superimposable on its mirror image. Chirality (handedness) arises primarily on account of the presence of asymmetric carbon atoms in the molecule. A tetravalent carbon is asymmetric when the four atoms (or groups) bonded to it are all different from one another. The substituents can then have two distinct arrangements which are mirror images of (or related by inversion to) each other. These optical isomers or enantiomers have the same chemical and physical properties except that they rotate the plane of polarization in opposite directions when polarized light passes through them. It is not, however, possible to calculate the sign of optical rotation, given the exact spatial arrangement or the `absolute configuration' of the molecule. Therefore, one cannot distinguish between the possible enantiomorphic configurations of a given asymmetric molecule from measurements of optical rotation. This is also true of molecules with chiralities generated by overall asymmetric geometry instead of the presence of asymmetric carbon atoms in them.

Normal X-ray scattering does not distinguish between enantiomers. Two structures and obviously produce the same diffraction pattern on account of Friedel's law. The situation is, however, different when anomalous scatterers are present in the structure. The intensity difference between reflections h and , or that between members of any Bijvoet pair, has the same magnitude, but opposite sign for structures A and B. If the atomic coordinates are known, the intensities of Bijvoet pairs can be readily calculated. The absolute configuration can then be determined, i.e. one can distinguish between A and B by comparing the calculated intensities with the observed ones for a few Bijvoet pairs with pronounced anomalous effects.

An important application of anomalous scattering is in the determination of phase angles using Bijvoet differences (Ramachandran & Raman, 1956; Peerdeman & Bijvoet, 1956). From Figs. 2.4.3.2 and 2.4.3.3, we have and Then In the above equations may be approximated to . Then can be evaluated from (2.4.3.5) except for the ambiguity in its sign. Therefore, from Fig. 2.4.3.2, we have The phase angle thus has two possible values symmetrically distributed about . Anomalous scatterers are usually heavy atoms and their positions can most often be determined by Patterson methods. can then be calculated and the two possible values of for each reflection evaluated using (2.4.3.6).

In practice, the twofold ambiguity in phase angles can often be resolved in a relatively straightforward manner. As indicated earlier, anomalous scatterers usually have relatively high atomic numbers. The `heavy-atom' phases calculated from their positions therefore contain useful information. For any given reflection, that phase angle which is closer to the heavy-atom phase, from the two phases calculated using (2.4.3.6), may be taken as the correct phase angle. This method has been successfully used in several structure determinations including that of a derivative of vitamin B₁₂ (Dale et al., 1963). The same method was also employed in a probabilistic fashion in the structure solution of a small protein (Hendrickson & Teeter, 1981). A method for obtaining a unique, but approximate, solution for phase angles from (2.4.3.6) has also been suggested (Srinivasan & Chacko, 1970). An accurate unique solution for phase angles can be obtained if one collects two sets of intensity data using two different wavelengths which have different dispersion-correction terms for the anomalous scatterers in the structure. Two equations of the type (2.4.3.6) are then available for each reflection and the solution common to both is obviously the correct phase angle. Different types of Patterson and Fourier syntheses can also be employed for structure solution using intensity differences between Bijvoet equivalents (Srinivasan, 1972; Okaya & Pepinsky, 1956; Pepinsky et al., 1957).

An interesting situation occurs when the replaceable atoms in a pair of isomorphous structures are anomalous scatterers. The phase angles can then be uniquely determined by combining isomorphous replacement and anomalous-scattering methods. Such situations normally occur in protein crystallography and are discussed in Section 2.4.4.5.

The phase determination, and hence the structure solution, outlined above relies on the imaginary component of the dispersion correction. Variation in the real component can also be used in structure analysis. In early applications of anomalous scattering, the real component of the dispersion correction was made use of to distinguish between atoms of nearly the same atomic numbers (Mark & Szillard, 1925; Bradley & Rodgers, 1934). For example, copper and manganese, with atomic numbers 29 and 25, respectively, are not easily distinguishable under normal X-ray scattering. However, the real components of the dispersion correction for the two elements are −1.129 and −3.367, respectively, when Fe radiation is used (IT IV, 1974). Therefore, the difference between the scattering factors of the two elements is accentuated when this radiation is used. The difference is more pronounced at high angles as the normal scattering factor falls off comparatively rapidly with increasing scattering angle whereas the dispersion-correction term does not.

The structure determination of KMnO₄ provides a typical example for the use of anomalous scattering without phase change in the determination of a centrosymmetric structure (Ramaseshan et al., 1957; Ramaseshan & Venkatesan, 1957). f′ and f″ for manganese for Cu radiation are and 2.808, respectively. The corresponding values for Fe radiation are and 0.481, respectively (IT IV, 1974). The data sets collected using the two radiations can now be treated as those arising from two perfectly isomorphous crystals. The intensity differences between a reflection in one set and the corresponding reflection in the other are obviously caused by the differences in the dispersion-correction terms. They can, however, be considered formally as intensity differences involving data from two perfectly isomorphous crystals. They can be used, as indeed they were, to determine the position of the manganese ion through an appropriate Patterson synthesis (see Section 2.4.4.2) and then to evaluate the signs of structure factors using (2.4.2.6) when the structure is centrosymmetric. When the structure is noncentrosymmetric, a twofold ambiguity exists in the phase angles in a manner analogous to that in the isomorphous replacement method. This ambiguity can be removed if the structure contains two different subsets of atoms Q1 and Q2 which, respectively, scatter radiations and anomalously. Data sets can then be collected with , which is scattered normally by all atoms, and . The three sets can be formally treated as those from three perfectly isomorphous structures and the phase determination effected using (2.4.2.7) (Ramaseshan, 1963).

The effect of anomalous scattering needs to be taken into account in the refinement of structures containing anomalous scatterers, if accurate atomic parameters are required. The effect of the real part of the dispersion correction is largely confined to the thermal parameters of anomalous scatterers. This effect can be eliminated by simply adding f′ to the normal scattering factor of the anomalous scatterers.

The effects of the imaginary component of the dispersion correction are, however, more complex. These effects could lead to serious errors in positional parameters when the space group is polar, if data in the entire diffraction sphere are not used (Ueki et al., 1966; Cruickshank & McDonald, 1967). For example, accessible data in a hemisphere are normally used for X-ray analysis when the space group is P1. If the hemisphere has say h positive, the x coordinates of all the atoms would be in error when the structure contains anomalous scatterers. The situation in other polar space groups has been discussed by Cruickshank & McDonald (1967). In general, in the presence of anomalous scattering, it is desirable to collect data for the complete sphere, if accurate structural parameters are required (Srinivasan, 1972).

Methods have been derived to correct for dispersion effects in observed data from centrosymmetric and noncentrosymmetric crystals (Patterson, 1963). The methods are empirical and depend upon the refined parameters at the stage at which corrections are applied. This is obviously an unsatisfactory situation and it has been suggested that the measured structure factors of Bijvoet equivalents should instead be treated as independent observations in structure refinement (Ibers & Hamilton, 1964). The effect of dispersion corrections needs to be taken into account to arrive at the correct scale and temperature factors also (Wilson, 1975; Gilli & Cruickshank, 1973).

Perhaps the most spectacular applications of isomorphous replacement and anomalous-scattering methods have been in the structure solution of large biological macromolecules, primarily proteins. Since its first successful application on myoglobin and haemoglobin, the isomorphous replacement method, which is often used in conjunction with the anomalous-scattering method, has been employed in the solution of scores of proteins. The application of this method involves the preparation of protein heavy-atom derivatives, i.e. the attachment of heavy atoms like mercury, uranium and lead, or chemical groups containing them, to protein crystals in a coherent manner without changing the conformation of the molecules and their crystal packing. This is only rarely possible in ordinary crystals as the molecules in them are closely packed. Protein crystals, however, contain large solvent regions and isomorphous derivatives can be prepared by replacing the disordered solvent molecules by heavy-atom-containing groups without disturbing the original arrangement of protein molecules.

For any given reflection, the structure factor of the native protein crystal , that of a heavy-atom derivative , and the contribution of the heavy atoms in that derivative are related by the equation The value of depends not only on the positional and thermal parameters of the heavy atoms, but also on their occupancy factors, because, at a given position, the heavy atom may not often be present in all the unit cells. For example, if the heavy atom is present at a given position in only half the unit cells in the crystal, then the occupancy factor of the site is said to be 0.5.

For the successful determination of the heavy-atom parameters, as also for the subsequent phase determination, the data sets from the native and the derivative crystals should have the same relative scale. The different data sets should also have the same overall temperature factor. Different scaling procedures have been suggested (Blundell & Johnson, 1976) and, among them, the following procedure, based on Wilson's (1942) statistics, appears to be the most feasible in the early stages of structure analysis.

Assuming that the data from the native and the derivative crystals obey Wilson's statistics, we have, for any range of , and where and refer to the atomic scattering factors of protein atoms and heavy atoms, respectively. and are the scale factors to be applied to the intensities from the native and the derivative crystals, respectively, and and the temperature factors of the respective structure factors. Normally one would be able to derive the absolute scale factor and the temperature factor for both the data sets from (2.4.4.2) and (2.4.4.3) using the well known Wilson plot. The data from protein crystals, however, do not follow Wilson's statistics as protein molecules contain highly non-random features. Therefore, in practice, it is difficult to fit a straight line through the points in a Wilson plot, thus rendering the parameters derived from it unreliable. (2.4.4.2) and (2.4.4.3) can, however, be used in a different way. From the two equations we obtain The effects of structural non-randomness in the crystals obviously cancel out in (2.4.4.4). When the left-hand side of (2.4.4.4) is plotted against , it is called a comparison or difference Wilson plot. Such plots yield the ratio between the scales of the derivative and the native data, and the additional temperature factor of the derivative data. Initially, the number and the occupancy factors of heavy-atom sites are unknown, and are roughly estimated from intensity differences to evaluate . These estimates usually undergo considerable revision in the course of the determination and the refinement of heavy-atom parameters.

At first, heavy-atom positions are most often determined by Patterson syntheses of one type or another. Such syntheses are discussed in some detail elsewhere in Chapter 2.3 . They are therefore discussed here only briefly.

Equation (2.4.2.6) holds when the data are centric. is usually small compared to and , and the minus sign is then relevant on the left-hand side of (2.4.2.6). Thus the difference between the magnitudes of and , which can be obtained experimentally, normally gives a correct estimate of the magnitude of for most reflections. Then a Patterson synthesis with as coefficients corresponds to the distribution of vectors between heavy atoms, when the data are centric. But proteins are made up of L-amino acids and hence cannot crystallize in centrosymmetric space groups. However, many proteins crystallize in space groups with centrosymmetric projections. The centric data corresponding to these projections can then be used for determining heavy-atom positions through a Patterson synthesis of the type outlined above.

The situation is more complex for three-dimensional acentric data. It has been shown (Rossmann, 1961) that when is small compared to and . Patterson synthesis with as coefficients would, therefore, give an approximation to the heavy-atom vector distribution. An isomorphous difference Patterson synthesis of this type has been used extensively in protein crystallography to determine heavy-atom positions. The properties of this synthesis have been extensively studied (Ramachandran & Srinivasan, 1970; Rossmann, 1960; Phillips, 1966; Dodson & Vijayan, 1971) and it has been shown that this Patterson synthesis would provide a good approximation to the heavy-atom vector distribution even when is large compared to (Dodson & Vijayan, 1971).

As indicated earlier (see Section 2.4.3.1), heavy atoms are always anomalous scatterers, and the structure factors of any given reflection and its Friedel equivalent from a heavy-atom derivative have unequal magnitudes. If these structure factors are denoted by and and the real component of the heavy-atom contributions (including the real component of the dispersion correction) by , then it can be shown (Kartha & Parthasarathy, 1965) that where . Here it has been assumed that all the anomalous scatterers are of the same type with atomic scattering factor and dispersion-correction terms and . A Patterson synthesis with the left-hand side of (2.4.4.6) as coefficients would also yield the vector distribution corresponding to the heavy-atom positions (Rossmann, 1961; Kartha & Parthasarathy, 1965). However, is a small difference between two large quantities and is liable to be in considerable error. Patterson syntheses of this type are therefore rarely used to determine heavy-atom positions.

It is interesting to note (Kartha & Parthasarathy, 1965) that addition of (2.4.4.5) and (2.4.4.6) readily leads to Thus, the magnitude of the heavy-atom contribution can be estimated if intensities of Friedel equivalents have been measured from the derivative crystal. is then not readily available, but to a good approximation A different and more accurate expression for estimating from isomorphous and anomalous differences was derived by Matthews (1966). According to a still more accurate expression derived by Singh & Ramaseshan (1966), The lower estimate in (2.4.4.9) is relevant when and the upper estimate is relevant when . The lower and the upper estimates may be referred to as and , respectively. It can be readily shown (Dodson & Vijayan, 1971) that the lower estimate would represent the correct value of for a vast majority of reflections. Thus, a Patterson synthesis with as coefficients would yield the vector distribution of heavy atoms in the derivative. Such a synthesis would normally be superior to those with the left-hand sides of (2.4.4.5) and (2.4.4.6) as coefficients. However, when the level of heavy-atom substitution is low, the anomalous differences are also low and susceptible to large percentage errors. In such a situation, a synthesis with as coefficients is likely to yield better results than that with as coefficients (Vijayan, 1981).

Direct methods employing different methodologies have also been used successfully for the determination of heavy-atom positions (Navia & Sigler, 1974). These methods, developed primarily for the analysis of smaller structures, have not yet been successful in a priori analysis of protein structures. The very size of protein structures makes the probability relations used in these methods weak. In addition, data from protein crystals do not normally extend to high enough angles to permit resolution of individual atoms in the structure and the feasibility of using many of the currently popular direct-method procedures in such a situation has been a topic of much discussion. The heavy atoms in protein derivative crystals, however, are small in number and are normally situated far apart from one another. They are thus expected to be resolved even when low-resolution X-ray data are used. In most applications, the magnitudes of the differences between and are formally considered as the `observed structure factors' of the heavy-atom distribution and conventional direct-method procedures are then applied to them.

Once the heavy-atom parameters in one or more derivatives have been determined, approximate protein phase angles, 's, can be derived using methods described later. These phase angles can then be readily used to determine the heavy-atom parameters in a new derivative employing a difference Fourier synthesis with coefficients Such syntheses are also used to confirm and to improve upon the information on heavy-atom parameters obtained through Patterson or direct methods. They are obviously very powerful when centric data corresponding to centrosymmetric projections are used. The synthesis yields satisfactory results even when the data are acentric although the difference Fourier technique becomes progressively less powerful as the level of heavy-atom substitution increases (Dodson & Vijayan, 1971).

While the positional parameters of heavy atoms can be determined with a reasonable degree of confidence using the above-mentioned methods, the corresponding temperature and occupancy factors cannot. Rough estimates of the latter are usually made from the strength and the size of appropriate peaks in difference syntheses. The estimated values are then refined, along with the positional parameters, using the techniques outlined below.

The least-squares method with different types of minimization functions is used for refining the heavy-atom parameters, including the occupancy factors. The most widely used method (Dickerson et al., 1961; Muirhead et al., 1967; Dickerson et al., 1968) involves the minimization of the function where the summation is over all the reflections and w is the weight factor associated with each reflection. Here is the observed magnitude of the structure factor for the particular derivative and is the calculated structure factor. The latter obviously depends upon the protein phase angle , and the magnitude and the phase angle of which are in turn dependent on the heavy-atom parameters. Let us assume that we have three derivatives A, B and C, and that we have already determined the heavy-atom parameters , and . Then, A set of approximate protein phase angles is first calculated, employing methods described later, making use of the unrefined heavy-atom parameters. These phase angles are used to construct for each derivative. (2.4.4.11) is then minimized, separately for each derivative, by varying for derivative A, for derivative B, and for derivative C. The refined values of , and are subsequently used to calculate a new set of protein phase angles. Alternate cycles of parameter refinement and phase-angle calculation are carried out until convergence is reached. The progress of refinement may be monitored by computing an R factor defined as (Kraut et al., 1962)

The above method has been successfully used for the refinement of heavy-atom parameters in the X-ray analysis of many proteins. However, it has one major drawback in that the refined parameters in one derivative are dependent on those in other derivatives through the calculation of protein phase angles. Therefore, it is important to ensure that the derivative, the heavy-atom parameters of which are being refined, is omitted from the phase-angle calculation (Blow & Matthews, 1973). Even when this is done, serious problems might arise when different derivatives are related by common sites. In practice, the occupancy factors of the common sites tend to be overestimated compared to those of the others (Vijayan, 1981; Dodson & Vijayan, 1971). Yet another factor which affects the occupancy factors is the accuracy of the phase angles. The inclusion of poorly phased reflections tends to result in the underestimation of occupancy factors. It is therefore advisable to omit from refinement cycles reflections with figures of merit less than a minimum threshold value or to assign a weight proportional to the figure of merit (as defined later) to each term in the minimization function (Dodson & Vijayan, 1971; Blow & Matthews, 1973).

If anomalous-scattering data from derivative crystals are available, the values of can be estimated using (2.4.4.7) or (2.4.4.9) and these can be used as the `observed' magnitudes of the heavy-atom contributions for the refinement of heavy-atom parameters, as has been done by many workers (Watenpaugh et al., 1975; Vijayan, 1981; Kartha, 1965). If (2.4.4.9) is used for estimating , the minimization function has the form The progress of refinement may be monitored using a reliability index defined as

The major advantage of using 's in refinement is that the heavy-atom parameters in each derivative can now be refined independently of all other derivatives. Care should, however, be taken to omit from calculations all reflections for which is likely to be the correct estimate of . This can be achieved in practice by excluding from least-squares calculations all reflections for which has a value less than the maximum expected value of for the given derivative (Vijayan, 1981; Dodson & Vijayan, 1971).

A major problem associated with this refinement method is concerned with the effect of experimental errors on refined parameters. The values of are often comparable to the experimental errors associated with and . In such a situation, even random errors in and tend to increase systematically the observed difference between them (Dodson & Vijayan, 1971). In (2.4.4.7) and (2.4.4.9), this difference is multiplied by k or , a quantity much greater than unity, and then squared. This could lead to the systematic overestimation of 's and the consequent overestimation of occupancy factors. The situation can be improved by employing empirical values of k, evaluated using the relation (Kartha & Parthasarathy, 1965; Matthews, 1966) for estimating or by judiciously choosing the weighting factors in (2.4.4.14) (Dodson & Vijayan, 1971). The use of a modified form of , arrived at through statistical considerations, along with appropriate weighting factors, has also been advocated (Dodson et al., 1975).

When the data are centric, (2.4.4.9) reduces to Here, again, the lower estimate most often corresponds to the correct value of . (2.4.4.17) does not involve which, as indicated earlier, is prone to substantial error. Therefore, 's estimated using centric data are more reliable than those estimated using acentric data. Consequently, centric reflections, when available, are extensively used for the refinement of heavy-atom parameters. It may also be noted that in conditions under which corresponds to the correct estimate of , minimization functions (2.4.4.11) and (2.4.4.14) are identical for centric data.

A Patterson function correlation method with a minimization function of the type was among the earliest procedures suggested for heavy-atom-parameter refinement (Rossmann, 1960). This procedure would obviously work well when centric reflections are used. A modified version of this procedure, in which the origins of the Patterson functions are removed from the correlation, and centric and acentric data are treated separately, has been proposed (Terwilliger & Eisenberg, 1983).

As shown in Section 2.4.2.3, ideally, protein phase angles can be evaluated if two isomorphous heavy-atom derivatives are available. However, in practice, conditions are far from ideal on account of several factors such as imperfect isomorphism, errors in the estimation of heavy-atom parameters, and the experimental errors in the measurement of intensity from the native and the derivative crystals. It is therefore desirable to use as many derivatives as are available for phase determination. When isomorphism is imperfect and errors exist in data and heavy-atom parameters, all the circles in a Harker diagram would not intersect at a single point; instead, there would be a distribution of intersections, such as that illustrated in Fig. 2.4.4.1. Consequently, a unique solution for the phase angle cannot be deduced.

Figure 2.4.4.1 | top | pdf | Distribution of intersections in the Harker construction under non-ideal conditions.

The statistical procedure for computing protein phase angles using multiple isomorphous replacement (MIR) was derived by Blow & Crick (1959). In their treatment, Blow and Crick assume, for mathematical convenience, that all errors, including those arising from imperfect isomorphism, could be considered as residing in the magnitudes of the derivative structure factors only. They further assume that these errors could be described by a Gaussian distribution. With these simplifying assumptions, the statistical procedure for phase determination could be derived in the following manner.

Consider the vector diagram, shown in Fig. 2.4.4.2, for a reflection from the ith derivative for an arbitrary value for the protein phase angle. Then, If corresponds to the true protein phase angle , then coincides with . The amount by which differs from , namely, is a measure of the departure of from . is called the lack of closure. The probability for being the correct protein phase angle could now be defined as where is the normalization constant and is the estimated r.m.s. error. The methods for estimating will be outlined later. When several derivatives are used for phase determination, the total probability of the phase angle being the protein phase angle would be where the summation is over all the derivatives. A typical distribution of plotted around a circle of unit radius is shown in Fig. 2.4.4.3. The phase angle corresponding to the highest value of would obviously be the most probable protein phase, , of the given reflection. The most probable electron-density distribution is obtained if each is associated with the corresponding in a Fourier synthesis.

Figure 2.4.4.2 | top | pdf | Vector diagram indicating the calculated structure factor, , of the ith heavy-atom derivative for an arbitrary value for the phase angle of the structure factor of the native protein.

Figure 2.4.4.3 | top | pdf | The probability distribution of the protein phase angle. The point P is the centroid of the distribution.

Blow and Crick suggested a different way of using the probability distribution. In Fig. 2.4.4.3, the centroid of the probability distribution is denoted by P. The polar coordinates of P are m and , where m, a fractional positive number with a maximum value of unity, and are referred to as the `figure of merit' and the `best phase', respectively. One can then compute a `best Fourier' with coefficients The best Fourier is expected to provide an electron-density distribution with the lowest r.m.s. error. The figure of merit and the best phase are usually calculated using the equations where are calculated, say, at intervals (Dickerson et al., 1961). The figure of merit is statistically interpreted as the cosine of the expected error in the calculated phase angle and it is obviously a measure of the precision of phase determination. In general, m is high when and are close to each other and low when they are far apart.

When anomalous-scattering data have been collected from derivative crystals, and can be formally treated as arising from two independent derivatives. The corresponding Harker diagram is shown in Fig. 2.4.4.4. Thus, in principle, protein phase angles can be determined using a single derivative when anomalous-scattering effects are also made use of. It is interesting to note that the information obtained from isomorphous differences, , and that obtained from anomalous differences, , are complementary. The isomorphous difference for any given reflection is a maximum when and are parallel or antiparallel. The anomalous difference is then zero, if all the anomalous scatterers are of the same type, and is determined uniquely on the basis of the isomorphous difference. The isomorphous difference decreases and the anomalous difference increases as the inclination between and increases. The isomorphous difference tends to be small and the anomalous difference tends to have the maximum possible value when and are perpendicular to each other. The anomalous difference then has the predominant influence in determining the phase angle.

Figure 2.4.4.4 | top | pdf | Harker construction using anomalous-scattering data from a single derivative.

Although isomorphous and anomalous differences have a complementary role in phase determination, their magnitudes are obviously unequal. Therefore, when and are treated as arising from two derivatives, the effect of anomalous differences on phase determination would be only marginal as, for any given reflection, is usually much smaller than . However, the magnitude of the error in the anomalous difference would normally be much smaller than that in the corresponding isomorphous difference. Firstly, the former is obviously free from the effects of imperfect isomorphism. Secondly, and are expected to have the same systematic errors as they are measured from the same crystal. These errors are eliminated in the difference between the two quantities. Therefore, as pointed out by North (1965), the r.m.s. error used for anomalous differences should be much smaller than that used for isomorphous differences. Denoting the r.m.s. error in anomalous differences by E′, the new expression for the probability distribution of protein phase angle may be written as where and Here is the phase angle of [see (2.4.4.19) and Fig. 2.4.4.2]. is the anomalous difference calculated for the assumed protein phase angle . may be taken as the average of and for calculating using (2.4.4.20).

Perhaps the most important parameters that control the reliability of phase evaluation using the Blow and Crick formulation are the isomorphous r.m.s. error and the anomalous r.m.s. error . For a given derivative, the sharpness of the peak in the phase probability distribution obviously depends upon the value of E and that of E′ when anomalous-scattering data have also been used. When several derivatives are used, an overall underestimation of r.m.s. errors leads to artifically sharper peaks, the movement of towards , and deceptively high figures of merit. Opposite effects result when E's are overestimated. Underestimation or overestimation of the r.m.s. error in the data from a particular derivative leads to distortions in the relative contribution of that derivative to the overall phase probability distributions. It is therefore important that the r.m.s. error in each derivative is correctly estimated.

Centric reflections, when present, obviously provide the best means for evaluating E using the expression As suggested by Blow & Crick (1959), values of E thus estimated can be used for acentric reflections as well. Once a set of approximate protein phase angles is available, can be calculated as the r.m.s. lack of closure corresponding to [i.e. in (2.4.4.20)] (Kartha, 1976). can be similarly evaluated as the r.m.s. difference between the observed anomalous difference and the anomalous difference calculated for [see (2.4.4.24)]. Normally, the value of is about a third of that of (North, 1965).

A different method, outlined below, can also be used to evaluate E and E′ when anomalous scattering is present (Vijayan, 1981; Adams, 1968). From Fig. 2.4.2.2, we have and where . Using arguments similar to those used in deriving (2.4.3.5), we obtain If is considered to be equal to , we obtain from (2.4.4.28) We obtain what may be called if the magnitude of is determined from (2.4.4.26) and the quadrant from (2.4.4.28). Similarly, we obtain if the magnitude of is determined from (2.4.4.28) and the quadrant from (2.4.4.26). Ideally, and should have the same value and the difference between them is a measure of the errors in the data. obtained from (2.4.4.27) using may be considered as its calculated value . Then, assuming all errors to lie in , we may write Similarly, the calculated anomalous difference may be evaluated from (2.4.4.29) using . Then If all errors are assumed to reside in , E can be evaluated in yet another way using the expression

Modifications to the Blow and Crick procedure of phase evaluation have been suggested by several workers, although none represent a fundamental departure from the essential features of their formulation. In one of the modifications (Cullis et al., 1961a; Ashida, 1976), all 's are assumed to be the same, but the lack-of-closure error for the ith derivative is measured as the distance from the mean of all intersections between phase circles to the point of intersection of the phase circle of that derivative with the phase circle of the native protein. Alternatively, individual values of are retained, but the lack of closure is measured from the weighted mean of all intersections (Ashida, 1976). This is obviously designed to undo the effects of the unduly high weight given to in the Blow and Crick formulation. In another modification (Raiz & Andreeva, 1970; Einstein, 1977), suggested for the same purpose, the and circles are treated as circular bands, the width of each band being related to the error in the appropriate structure factor. A comprehensive set of modifications suggested by Green (1979) treats different types of errors separately. In particular, errors arising from imperfect isomorphism are treated in a comprehensive manner.

Although the isomorphous replacement method still remains the method of choice for the ab initio determination of protein structures, additional items of phase information from other sources are increasingly being used to replace, supplement, or extend the information obtained through the application of the isomorphous replacement. Methods have been developed for the routine refinement of protein structures (Watenpaugh et al., 1973; Huber et al., 1974; Sussman et al., 1977; Jack & Levitt, 1978; Isaacs & Agarwal, 1978; Hendrickson & Konnert, 1980) and they provide a rich source of phase information. However, the nature of the problem and the inherent limitations of the Fourier technique are such that the possibility of refinement yielding misleading results exists (Vijayan, 1980a,b). It is therefore sometimes desirable to combine the phases obtained during refinement with the original isomorphous replacement phases. The other sources of phase information include molecular replacement (see Chapter 2.3 ), direct methods (Hendrickson & Karle, 1973; Sayre, 1974; de Rango et al., 1975; see also Chapter 2.2 ) and different types of electron-density modifications (Hoppe & Gassmann, 1968; Collins, 1975; Schevitz et al., 1981; Bhat & Blow, 1982; Agard & Stroud, 1982; Cannillo et al., 1983; Raghavan & Tulinsky, 1979; Wang, 1985).

The problem of combining isomorphous replacement phases with those obtained by other methods was first addressed by Rossmann & Blow (1961). The problem was subsequently examined by Hendrickson & Lattman (1970) and their method, which involves a modification of the Blow and Crick formulation, is perhaps the most widely used for combining phase information from different sources.

The Blow and Crick procedure is based on an assumed Gaussian `lumped' error in which leads to a lack of closure, , in defined by (2.4.4.20). Hendrickson and Lattman make an equally legitimate assumption that the lumped error, again assumed to be Gaussian, is associated with . Then, as in (2.4.4.20), we have where is the lack of closure associated with for an assumed protein phase angle . Then the probability for being the correct phase angle can be expressed as where is the r.m.s. error in , which can be evaluated using methods similar to those employed for evaluating . Hendrickson and Lattman have shown that the exponent in the probability expression (2.4.4.34) can be readily expressed as a linear combination of five terms in the following manner. where and are constants dependent on and . Thus, five constants are enough to store the complete probability distribution of any reflection. Expressions for the five constants have been derived for phase information from anomalous scattering, tangent formula, partial structure and molecular replacement. The combination of the phase information from all sources can then be achieved by simply taking the total value of each constant. Thus, the total probability of the protein phase angle being is given by where etc. are the constants appropriate for the sth source and N is the normalization constant.

It is often useful to have a Fourier representation of only the anomalous scatterers in a protein. The imaginary component of the electron-density distribution obviously provides such a representation. When the structure is known and and have been experimentally determined, Chacko & Srinivasan (1970) have shown that this representation is obtained in a Fourier synthesis with as coefficients, where , whose magnitude is , is the complex conjugate of . They also indicated a method for calculating the phase angles of and . It has been shown (Hendrickson & Sheriff, 1987) that the Bijvoet-difference Fourier synthesis proposed earlier by Kraut (1968) is an approximation of the true imaginary component of the electron density. The imaginary synthesis can be useful in identifying minor anomalous-scattering centres when the major centres are known and also in providing an independent check on the locations of anomalous scatterers and in distinguishing between anomalous scatterers with nearly equal atomic numbers (Sheriff & Hendrickson, 1987; Kitagawa et al., 1987).

Apart from the limitations introduced by experimental factors, such as the need for large crystals and the comparatively low flux of neutron beams, there are two fundamental reasons why neutrons are less suitable than X-rays for the ab initio determination of crystal structures. First, the neutron scattering lengths of different nuclei have comparable magnitudes whereas the atomic form factors for X-rays vary by two orders of magnitude. Therefore, Patterson techniques and the related heavy-atom method are much less suitable for use with neutron diffraction data than with X-ray data. Secondly, neutron scattering lengths could be positive or negative and hence, in general, the positivity criterion (Karle & Hauptman, 1950) or the squarability criterion (Sayre, 1952) does not hold good for nuclear density. Therefore, the direct methods based on these criteria are not strictly applicable to structure analysis using neutron data, although it has been demonstrated that these methods could be successfully used in favourable situations in neutron crystallography (Sikka, 1969). The anomalous-scattering method is, however, in principle more powerful in the neutron case than in the X-ray case for ab initio structure determination.

Thermal neutrons are scattered anomalously at appropriate wavelengths by several nuclei. In a manner analogous to (2.4.3.1), the neutron scattering length of these nuclei can be written as The correction terms b′ and b″ are strongly wavelength-dependent. In favourable cases, and can be of the order of 10 whereas they are small fractions in X-ray anomalous scattering. In view of this pronounced anomalous effect in neutron scattering, Ramaseshan (1966) suggested that it could be used for structure solution. Subsequently, Singh & Ramaseshan (1968) proposed a two-wavelength method for unique structure analysis using neutron diffraction. The first part of the method is the determination of the positions of the anomalous scatterers from the estimated values of . The method employed for estimating is analogous to that using (2.4.4.9) except that data collected at two appropriate wavelengths are used instead of those from two isomorphous crystals. The second stage of the two-wavelength method involves phase evaluation. Referring to Fig. 2.4.3.2 and in a manner analogous to (2.4.3.5), we have where and subscript 1 refers to data collected at wavelength . Singh and Ramaseshan showed that cos can also be determined when data are available at wavelength and . We may define and we have from (2.4.3.3), (2.4.3.4) and (2.4.5.3) Then, where x is the magnitude of the temperature-corrected geometrical part of . and hence can be calculated using (2.4.5.2) and (2.4.5.5). can also be obtained in a similar manner.

During the decade that followed Ramaseshan's suggestion, neutron anomalous scattering was used to solve half a dozen crystal structures, employing the multiple-wavelength methods as well as the methods developed for structure determination using X-ray anomalous scattering (Koetzle & Hamilton, 1975; Sikka & Rajagopal, 1975; Flook et al., 1977). It has also been demonstrated that measurable Bijvoet differences could be obtained, in favourable situations, in neutron diffraction patterns from protein crystals (Schoenborn, 1975). However, despite the early promise held by neutron anomalous scattering, the method has not been as successful as might have been hoped. In addition to the need for large crystals, the main problem with using this method appears to be the time and expense involved in data collection (Koetzle & Hamilton, 1975).

The most significant development in recent years in relation to anomalous scattering of X-rays has been the advent of synchrotron radiation (Helliwell, 1984). The advantage of using synchrotron radiation for making anomalous-scattering measurements essentially arises out of the tunability of the wavelength. Unlike the characteristic radiation from conventional X-ray sources, synchrotron radiation has a smooth spectrum and the wavelength to be used can be finely selected. Accurate measurements have shown that values in the neighbourhood of 30 electrons could be obtained in favourable cases for f′ and f″ (Templeton, Templeton, Phillips & Hodgson, 1980; Templeton, Templeton & Phizackerley, 1980; Templeton et al., 1982). Schemes for the optimization of the wavelengths to be used have also been suggested (Narayan & Ramaseshan, 1981). Interestingly, the anomalous differences obtainable using synchrotron radiation are comparable in magnitude to the isomorphous differences normally encountered in protein crystallography. Thus, the use of anomalous scattering at several wavelengths would obviously eliminate the need for employing many heavy-atom derivatives. The application of anomalous scattering of synchrotron radiation for macromolecular structure analysis began to yield encouraging results in the 1980s (Helliwell, 1985). Intensity measurements from macromolecular X-ray diffraction patterns using synchrotron radiation at first relied primarily upon oscillation photography (Arndt & Wonacott, 1977). This method is not particularly suitable for accurately evaluating anomalous differences. Much higher levels of accuracy began to be achieved with the use of position-sensitive detectors (Arndt, 1986). Anomalous scattering, in combination with such detectors, has developed into a major tool in macromolecular crystallography (see IT F , 2001).