Accuracy of the final model

The result of all the work described in the previous chapters will be a set of coordinates and other data suitable for deposit in the Protein Data Bank. You or I may use these coordinates, and we need to have some insight into their accuracy and reliability. In the previous chapters, indicators have been described, which may suggest aspects of the data or interpretation procedures that might lead to problems. But as the determination of protein crystal structures becomes more routine, many of these indicators are omitted from publications. Fortunately, crystallographic procedures are self-checking to a large extent. It is rare for a major error of interpretation to lead right through to a published refined structure. A high Rfree factor is a warning, especially if coupled with departures from the requirements of correct bond lengths, angles, and acceptable dihedral angles. On the other hand, there will always be a desire to squeeze more results from the data. All interpretations are subject to error; nearly all protein crystals have regions that are less ordered, where accurate interpretation is less feasible; and the structure may be overrefined, using too many variables for the data. If the majority of the molecule is correctly interpreted, a reasonable R factor may be obtained even though some small regions are completely wrong. During refinement it is usual to restrain the bond lengths and bond angles to be near their theoretical values, as described in Chapter 12. The extent to which bond lengths and bond angles depart from these values is often quoted as an indicator of accuracy. These departures are, however, difficult to interpret because they depend on how tightly the restraints have been applied. The same applies to the restraint of certain coordinates to lie in a plane. This difficulty illustrates a general problem. Designers of refinement procedures are understandably anxious to improve their procedures to lead directly to a well-refined structure. Every aspect of structure that can be recognized as having a regularity could, in principle, be expressed as a restraint which enforces it during refinement.

Structural refinement

At this stage, we have derived a model from an electron-density map and have interpreted it as closely as we can in terms of molecular structure. Provided the job has been done well enough, the next task of improving that interpretation can be left to computational procedures known as structural refinement. If further uninterpreted features of the structure are revealed, it will be necessary to go back to the methods of Chapter 11 to improve the interpretation. The purpose of structural refinement is to adjust a structure to give the best possible fit to the crystallographic observations. The intensities of the Bragg reflections constitute the observations, and the various quantities that define the structure are adjusted to give the best fit. Box 12.1 gives an outline of what is meant by refinement of quantities to fit observations. In structural refinement, a measure of the discrepancies between the calculated X-ray scattering by the model structure and the observed intensities is defined: this is called the refinement parameter. The purpose of the refinement procedure is to alter the model to give the lowest possible refinement parameter. Box 12.1 uses a simple example to bring out some important general points: 1. My model will be specified by a number of variables. In a diffraction experiment, they are usually the coordinates and B factor of every atom. If the number of observed quantities is less than the number of variables, the results can have no validity. 2. If the number of observations equals the number of variables, a perfect fit can be obtained, irrespective of the accuracy of the observations or of the model. (This is true of so-called linear problems, and approximately so in non-linear cases.) 3. If the number of observations exceeds the number of variables by only a small quantity, the estimate of the reliability of the model is questionable. In practice, refinement procedures can only work when there is a sufficient number of observations which are sufficiently accurate. Also, the model must already be good enough to make the refinement procedure meaningful.

Electron-density maps

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Intensity measurements

Once a suitable crystal has been obtained, a molecular structure investigation requires measurement of the intensities of as many Bragg reflections as possible. In this chapter, some of the options that must be decided by the experimenter will be considered, and some of the criteria used to assess the accuracy and completeness of the data will be presented. The experimenter has to make a number of strategic decisions in collecting the crystal intensity data. These include: • What X-ray source should be used? • What X-ray detector should be used? • Under what conditions should the crystal be maintained? • How long should each crystal be exposed? • What data collection technique will be used? • What resolution limit should be applied? The choice of source and detector will depend largely on what is available, but the major decision is whether to use facilities in the home laboratory or whether to use a synchrotron at a central facility. The energy released by absorption of X-rays in a crystal inevitably damages it. The process of radiation damage increases crystal disorder and reduces the intensity of scattering. The experimenter will ultimately have to abandon data collection from the damaged and disordered crystal. Under ideal experimental conditions, all the useful diffraction data can be obtained from a crystal long before radiation damage takes its toll, and radiation damage does not create a practical problem. At the other end of the scale, it may be necessary to combine the measurements from many crystals in order to obtain a complete set of diffracted intensities. There is no definite criterion to decide when a crystal is so badly damaged that it must be discarded. But if the measurements are going to be of highest quality, any observable change is bad news. The most serious effects occur in the part of the diffraction pattern at the highest observed resolution, where the observed intensities of the Bragg reflections will be altered most rapidly. The first observable effect of radiation damage is usually a reduction of high angle intensities due to increased disorder.

Diffraction by crystals

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

Diffraction

Diffraction refers to the effects observed when light is scattered into directions other than the original direction of the light, without change of wavelength. An X-ray photon may interact with an electron and set the electron oscillating with the X-ray frequency. The oscillating electron may radiate an X-ray photon of the same wavelength, in a random direction, when it returns to its unexcited state. Other processes may also occur, akin to fluorescence, which emit X-rays of longer wavelengths, but these processes do not give diffraction effects. Just as we see a red card because red light is scattered off the card into our eyes, objects are observed with X-rays because an illuminating X-ray beam is scattered into the X-ray detector. Our eye can analyse details of the card because its lens forms an image on the retina. Since no X-ray lens is available, the scattered X-ray beam cannot be converted directly into an image. Indirect computational procedures have to be used instead. X-rays are penetrating radiation, and can be scattered from electrons throughout the whole scattering object, while light only shows the external shape of an opaque object like a red card. This allows X-rays to provide a truly three-dimensional image. When X-rays pass near an atom, only a tiny fraction of them is scattered: most of the X-rays pass further into the object, and usually most of them come straight out the other side of the whole object. In forming an image, these ‘straight through’ X-rays tell us nothing about the structure, and they are usually captured by a beam stop and ignored. This chapter begins by explaining that the diffraction of light or X-rays can provide a precise physical realization of Fourier’s method of analysing a regularly repeating function. This method may be used to study regularly repeating distributions of scattering material. Beginning in one dimension, examples will be used to bring out some fundamental features of diffraction analysis. Graphic examples of two-dimensional diffraction provide further demonstrations. Although the analysis in three dimensions depends on exactly the same principles, diffraction by a three-dimensional crystal raises additional complications.

Density-modification procedures: improving a calculated map before interpretation

Procedures to determine the phases of the structure factors, by isomorphous replacement, by anomalous scattering, or by molecular replacement, were described in the Chapters 7–9. Using one or more of these methods, phases are generated which allow an electron-density map to be calculated, at a resolution to which the phases are thought to be reliable. In many cases this electron density can be confidently interpreted in terms of atomic positions. But this is not always the case. Quite often, the procedures so far described offer a tantalizing puzzle map, with some features which I think I can interpret, but raising many questions. Before devoting effort to interpreting an unsatisfactory electron-density map, a number of procedures are available, which might make a striking improvement. Perhaps the most important strategy is to seek out more isomorphous and anomalous scattering derivatives. Before doing that, there are other possibilities which may improve an electron-density map without any more experimental data. These methods are known collectively as density modification. The first group of methods exploits features of the electron density which result from the packing of molecules into a crystal. Macromolecular crystals composed of rigid molecules have voids between the molecules filled with disordered solvent, often including the precipitants used in the crystallization process. These solvent regions present featureless density between the structured density of the macromolecules. A high-quality electron-density map will show these featureless regions clearly. In a map of poorer quality, the voids between molecules may be clearly defined, but far from featureless. This provides a method to improve the map. Although some solvent molecules are immobilized on the surface of the macromolecule, those further from the surface are in a disordered liquid-like state which presents a uniform density. Except in very small proteins, the majority of solvent is disordered. If such uniform solvent regions can be recognized, they allow surfaces to be defined which separate solvent regions from protein regions. Two procedures are described below. It has become almost a matter of routine to use one or both of these methods.

Isomorphous replacement

In this chapter, we shall see how this historic observation created the first method for determining the atomic structure of proteins, a method which is still amongst the most important. As discussed in Chapters 1 and 3, detectors of X-rays measure the energy of light entering them. They are insensitive to the phase (Fig. 7.1). The energy of light in a diffracted beam is proportional to the square of its amplitude: all we can detect is the intensity of each diffracted wave. This applies to all available detectors of short-wavelength electromagnetic waves. In order to reconstruct the electron-density distribution ρ(x) from the observed scattering it is necessary to know the phase angle of the scattering associated with each Bragg reflection. The difficulty in determining the phase of a scattered wave is referred to as the phase problem, and the next three chapters present methods that are used to overcome it for macromolecular structure analysis. There is one situation where the phase problem is much simplified. If a structure has a centre of symmetry at the origin, all the waves that represent its Fourier transform are cosine-like (with maxima or minima at the origin), and the phases are 0° or 180°. This was presented in Chapter 4 for the one-dimensional case (see Figs 4.6 and 4.11), and applies in just the same way in two or three dimensions. All crystallographic structure determinations before 1940 dealt with centrosymmetric structures. This does not help us directly, because biological macromolecules are never centrosymmetric, and all phase angles are possible. Having said this, there are cases where crystals of biological materials have some reflections where phase is limited to 0° or 180°. These are usually reflections with at least one index zero, depending on the symmetry: for monoclinic crystals the h0l reflections are always in this category. More precise conditions are defined in Box 7.1. Figure 7.2 shows how a projection of a non-centrosymmetric structure along a 2-fold axis gives a centrosymmetric projection. Such a centrosymmetric projection requires a group of reflections to have phases of 0 or π.

Anomalous scattering

X-rays set the electronic charges around atoms vibrating, and these vibrations generate radiation of the same frequency, which is propagated in all directions. This is the coherent scattering that gives rise to diffraction effects. Normally, the electrons vibrate in step with the incident beam. If, however, the incident photons have an energy close to a transition energy which can bring the atom to an excited state, the electronic vibration gets out of step. Instead of re-radiating in phase with the incident beam, the radiated energy has a different phase. Also, the intensity of coherent scattering is reduced, because some energy is absorbed to bring about the transition. This effect is called anomalous scattering. More detail is given in Box 8.1. Figure 8.1 shows how X-ray absorption varies with wavelength near the transition energy. X-rays become rapidly less penetrating as wavelength increases, but this trend is interrupted by a sharp ‘edge’ at a wavelength that corresponds to an electronic transition. In practice, at the wavelengths of X-rays convenient for diffraction experiments (less than 1.6 Å) atoms lighter than phosphorus or sulphur behave as normal scatterers because they have no transitions of corresponding energy. At wavelengths very near the energy of an electronic transition, anomalous scattering can become a significant fraction of the total scattering. In this chapter the atoms which are scattering anomalously are referred to as heavy atoms, H. Useful anomalous scatterers can be far lighter than the heavy elements needed for macromolecular isomorphous replacement. From iron (Z=26) to palladium (Z = 46) the K absorption edges are at convenient wavelengths. Much heavier atoms (the lanthanides and beyond) give strong anomalous effects from the L edges, at useful wavelengths. Remember that the X-ray scattering power of an atom is measured by comparison with the scattering of an electron. At low scattering angle and ‘normal’ wavelengths, an atom scatters according to the number of electrons it contains (Box 5.2). At wavelengths close to the absorption edge, the atomic scattering factor includes an ‘anomalous’ component, shown in Fig. 8.2, and presented in algebraic form in Box 8.2.

Waves

In this very short chapter, some basic facts about waves are presented. Attached to this short chapter are several boxes which give the fundamental mathematical basis for understanding waves in a more quantitative fashion. There are many physical examples of waves. Waves in water are perhaps the most familiar example. A water wave is created by a disturbance in the height of the water surface. The amount by which the height of the water is disturbed by the wave is called its amplitude. Another important form of wave is sound, which is a variation of pressure in a gas or liquid (or of stress in a solid). But for our purposes the most important waves are electromagnetic waves, specifically X-rays, with wavelengths of an Ångström or so. Electromagnetic waves create a disturbance in both the electric field and the magnetic field: usually the wave is represented by its electric component. All these waves carry energy. The rate of energy transfer is called the intensity, and at a given wavelength the intensity is proportional to the square of the amplitude. Detectors of X-rays, discussed in Chapter 1, respond to the quantity of energy delivered by the beam, which is also proportional to the number of photons. The amplitude of any wave is thus proportional to the square root of its intensity. The most simple form of wave is a sinusoidal disturbance which moves forward at a fixed velocity. ‘Sinusoidal’ means shaped like a sine wave (Fig. 3.1). For reasons that will emerge, we will work more often with a cosine function, which is just the same shape as a sine wave, but which has its origin at a maximum point of the wave. A sinusoidal wave can be described by several properties: • the wavelength, which is the distance from one peak to the next; • the amplitude, which is the height of the wave peak above its mean level; • the phase, which specifies where the peak of the wave is, relative to an origin of measurement at the position x=0, and the time t =0; • the wave velocity, which is the velocity at which the wave advances along the propagation direction.