Chapter 18.4. Refinement at atomic resolution

Ab initio methods of phase calculation normally depend on the assumption of positivity and atomicity of the electron density. Such methods rely largely on the availability of atomic resolution data. In addition, approaches such as solvent flattening and automated map interpretation benefit enormously from such data. The fact that current ab initio methods in the absence of heavy atoms are only effective when meaningful data extend beyond 1.2 Å reinforces the idea that this is a reasonable working criterion for atomic resolution.

As can be seen from equation (18.4.1.2), the measure of information about all or part of the crystal contents depends strongly on the quality and quantity of the data. Of course, before the experiment is carried out some questions should be answered. Firstly, what is the aim of the experiment? Secondly, what is the cost of the experiment and what are the available resources? With modern techniques, if synchrotron radiation (SR) is used with an efficient detector, the cost of the experiment for different resolutions does not vary greatly (provided that a suitable quality crystal is available). In practice, the apparent increase in cost to attain high-resolution data will generally provide a saving in terms of the time spent by the investigator, since the interpretation of the resulting electron density is much easier and faster. In general, to answer the same question is much easier and cheaper if high-resolution data are available. In addition, high-resolution data mean that answers to some of the questions which may arise during analysis of the experiment will already be addressable. In contrast, low-resolution data not only make it difficult to answer the question currently being asked, but may also necessitate further experiments to address other problems that arise.

While the information content of the data appears to depend quantitatively on the nominal resolution, in fact it is dependent on the data quality throughout the resolution range, and both high- and low-resolution completeness and their statistical significance affect the information content of the data and derived model. High-intensity low-resolution terms remain important for refinement at atomic resolution, as they define the contrast in the density maps between solvent and protein, and because their omission biases the refinement, especially that of parameters such as the ADPs. The rejection of low-intensity observations will have a similar biasing effect. In particular, all the maps calculated for visual or computer inspection by Fourier transformation are diminished in quality by omission of any terms, but are especially affected by omission of strong low-resolution data. This is particularly true in the early stages of structure solution, where low-resolution data can be vital. Although most phase-improvement algorithms rely on relations between all reflections, terms involving low-resolution reflections will be large, will be involved in many relations and will play a dominant role. Hence, omission of these terms will severely degrade the power of these methods, which may indeed converge to solutions that have nothing whatsoever to do with the real structure.

The intensity data from a crystal may display anisotropy, i.e., the intensity fall-off with resolution will vary with direction, and may be much higher along one crystal axis than along another. If the structure is to be refined with an isotropic atomic model (either because there are insufficient data or the programs used cannot handle anisotropic parameters), then the fall-off of the calculated values will, of necessity, also be isotropic. In this situation, an improved agreement between observed and calculated values can be obtained either by using anisotropic scaling during data reduction to the expected Wilson distribution of intensities, or by including a maximum of six overall anisotropic parameters during refinement. This will result in an isotropic set of values. For crystals with a high degree of anisotropy in the experimental data, this can lead to a substantial drop of several per cent in R and R_free (Sheriff & Hendrickson, 1987; Murshudov et al., 1998).

This ambiguity effectively disappears with use of an anisotropic atomic model. The individual ADPs, including contributions from both static and thermal disorder, take up relative individual displacements, but also the overall anisotropy of the experimental values. The significance of the overall anisotropy is a point of some contention, and its physical meaning is not clear. It may represent asymmetric crystal imperfection or anisotropic overall displacement of molecules in the lattice related to TLS parameters. Refinement of TLS parameters, which can be performed using, for example, RESTRAIN (Driessen et al., 1989), removes the overall crystal contribution to the ADP.

The principles of the least-squares method of minimization are described in IT C (2004). Least squares involves the construction of a matrix of order N × N, where N is the number of parameters, representing a system of least-squares equations, whose solution provides estimates of adjustments to the current atomic parameters. The problem is nonlinear and the matrix construction and solution must be iterated until convergence is achieved. In addition, inversion of the matrix at convergence provides an approximation to standard uncertainties for each individual parameter refined. Indeed, this is the only method available so far that gives such estimates properly.

However, even for small molecules there may be some disordered regions that will require the imposition of restraints, as is the case for macromolecules (see below), and the presence of such restraints means that the error estimates no longer reflect the information from the X-ray data alone. If the problem of how restraints affect the error estimates could be resolved, then inversion of the matrix corresponding to the second derivative of the posterior distribution would provide standard uncertainties incorporating both the prior knowledge, such as the restraints and the experimental data. Equation (18.4.1.2) for information measure could then be applied. For small structures, the speed and memory of modern computers have reduced the requirements for such calculations to the level of seconds, and the computational requirements form a trivial part of the structure analysis.

The size of the computational problem increases dramatically with the size of the unit cell, as the number of terms in the matrix increases with the square of the number of parameters. Furthermore, construction of each element depends on the number of reflections. For macromolecular structures, computation of a full matrix is at present prohibitively expensive in terms of CPU time and memory. A variety of simplifying approaches have been developed, but all suffer from a poorer estimate of the standard uncertainty and from a more limited range and speed of convergence.

The first approach is the block-matrix approximation, where instead of the full matrix, only square blocks along the matrix diagonal are constructed, involving groups of parameters that are expected to be correlated. The correlation between parameters belonging to different blocks is therefore neglected completely. In this way, the whole least-squares minimization is split into a set of smaller independent units. In principle this leads to the same solution, but more slowly and with less precise error estimates. Nevertheless, block-matrix approaches remain essential for tractable matrix inversion for macromolecular structures.

A further simplification involves the conjugate-gradient method or the diagonal approximation to the normal matrix (the second derivative of minus the log of the likelihood function in the case of maximum likelihood), which essentially ignores all off-diagonal terms of the least-squares matrix. For the conjugate-gradient approach, all diagonal terms of the matrix are equal. However, the range and speed of convergence are substantially reduced, and standard uncertainties can no longer be estimated directly by matrix inversion.

Conventional least-squares programs use the structure-factor equation and associated derivatives, with the summation extending over all atoms and all reflections. This is immensely slow in computational terms for large structures, but it has the advantage of providing precise values.

An alternative procedure, where the computer time is reduced from being proportional to N² to N log N, involves the use of fast Fourier algorithms for the computation of structure factors and derivatives (Ten Eyck, 1973, 1977; Agarwal, 1978). This can involve some interpolation and the limitation of the volume of electron-density maps to which individual atoms contribute. Such algorithms have been exploited extensively in macromolecular refinement programs, such as PROLSQ (Konnert & Hendrickson, 1980), XPLOR (Brünger, 1992b), TNT (Tronrud, 1997), RESTRAIN (Driessen et al., 1989), REFMAC (Murshudov et al., 1997) and CNS (Brünger et al., 1998), but have been largely restricted to the diagonal approximation. XPLOR and CNS use the conjugate-gradient method that relies only on the first derivatives, ignoring the second derivatives. In all other programs, the diagonal approximation is used for the second-derivative matrix.

This provides a more statistically sound alternative to least squares, especially in the early stages of refinement when the model lies far from the minimum. This approach increases the radius of convergence, takes into account experimental uncertainties, and in the final stages gives results similar to least squares, but with improved weights (Murshudov et al., 1997; Bricogne, 1997). The maximum-likelihood approach has been extended to allow refinement of a full atomic anisotropic model, while retaining the use of fast Fourier algorithms (Murshudov et al., 1999). A remaining limitation is the use of the diagonal approximation, which prevents the computation of standard uncertainties of individual parameters. Algorithms that will alleviate this limitation can be foreseen, and they are expected to be implemented in the near future.

There are no longer any restrictions on the full-matrix refinement of small-molecule crystal structures. However, the large size of the matrix, which increases as , where N is the number of parameters, means that for macromolecules, which contain thousands of independent atoms, this approach is intractable with the computing resources normally available to the crystallographer. By extrapolating the progress in computing power experienced in recent years, it can be envisaged that the limitations will disappear during the next decade, as those for small structures have disappeared since the 1960s. Indeed, the advances in the speed of CPUs, computer memory and disk capacity continue to transform the field, which makes it hard to predict the optimal strategies for atomic resolution refinement, even over the next ten years.

The X-ray experiment provides two-dimensional diffraction images. These are transformed to integrated but unscaled data, which are transformed to Bragg reflection intensities that are subsequently transformed to structure-factor amplitudes. At each transformation some assumptions are used, and the results will depend on their validity. Invalid assumptions will introduce bias toward these assumptions into the resulting data. Ideally, refinement (or estimation of parameters) should be against data that are as close as possible to the experimental observations, eliminating at least some of the invalid assumptions. Extrapolating this to the extreme, refinement should use the images as observable data, but this poses several severe problems, depending on data quantity and the lack of an appropriate statistical model.

Alternatively, the transformation of data can be improved by revising the assumptions. The intensities are closer to the real experiment than are the structure-factor amplitudes, and use of intensities would reduce the bias. However, there are some difficulties in the implementation of intensity-based likelihood refinement (Pannu & Read, 1996).

Gaussian approximation to intensity-based likelihood (Murshudov et al., 1997) would avoid these difficulties, since a Gaussian distribution of error can be assumed in the intensities but not the amplitudes. However, errors in intensities may not only be the result of counting statistics, but may have additional contributions from factors such as crystal disorder and motion of the molecules in the lattice during data collection.

Nevertheless, the problem of how to treat weak reflections remains. Some of the measured intensities will be negative, as a result of statistical errors of observation, and the proportion of such measurements will be relatively large for weakly diffracting macromolecular structures, especially at atomic resolution. For intensity-based likelihood, this is less important than for the amplitude-based approach. French & Wilson (1978) have given a Bayesian approach for the derivation of structure-factor amplitudes from intensities using Wilson's distribution (Wilson, 1942) as a prior, but there is room for improvement in this approach. Firstly, the assumed Wilson distribution could be upgraded using the scaling techniques suggested by Cowtan & Main (1998) and Blessing (1997), and secondly, information about effects such as pseudosymmetry could be exploited.

Another argument for the use of intensities rather than amplitudes is relevant to least squares where the derivative for amplitude-based refinement with respect to when is equal to zero is singular (Schwarzenbach et al., 1995). This is not the case for intensity-based least squares. In applying maximum likelihood, this problem does not arise (Pannu & Read, 1996; Murshudov et al., 1997).

Finally, while there may be some advantages in refining against F², Fourier syntheses always require structure-factor amplitudes.

Even for small-molecule structures, disordered regions of the unit cell require the imposition of stereochemical restraints or constraints if the chemical integrity is to be preserved and the ADPs are to be realistic. The restraints are comparable to those used for proteins at lower resolution and this makes sense, since the poorly ordered regions with high ADPs in effect do not contribute to the high-angle diffraction terms, and as a result their parameters are only defined by the lower-angle amplitudes.

Thus, even for a macromolecule for which the crystals diffract to atomic resolution, there will be regions possessing substantial thermal or static disorder, and restraints on the positional parameters and ADPs are essential for these parts. Their effect on the ordered regions will be minimal, as the X-ray terms will dominate the refinement, provided the relative weighting of X-ray and geometric contributions is appropriate.

Another justification for use of restraints is that refinement can be considered a Bayesian estimation. From this point of view, all available and usable prior knowledge should be exploited, as it should not harm the parameter estimation during refinement. Bayesian estimation shows asymptotic behaviour (Box & Tiao, 1973), i.e., when the number of observations becomes large, the experimental data override the prior knowledge. In this sense, the purpose of the experiment is to enhance our knowledge about the molecule, and the procedure should be cumulative, i.e., the result of the old experiment should serve as prior knowledge for the design and treatment of new experiments (Box & Tiao, 1973; Stuart et al., 1999; O'Hagan, 1994). However, there are problems in using restraints. For example, the probability distribution reflecting the degree of belief in the restraints is not good enough. Use of a Gaussian approximation to distributions of distances, angles and other geometric properties has not been justified. Firstly, the distribution of geometric parameters depends strongly on ADPs, and secondly, different geometric parameters are correlated. This problem should be the subject of further investigation.

It may be necessary to refine one additional parameter, the occupancy factor of an atomic site, for structures possessing regions that are spatially or temporally disordered, with some atoms lying in more than one discrete site. The sum of the occupancies for alternative individual sites of a protein atom must be 1.0.

For macromolecules, the occupancy factor is important in several situations, including the following:

Unfortunately, the occupancy parameter is highly correlated with the ADP, and it is difficult to model these two parameters at resolutions less than atomic. Even at atomic resolution, it can prove difficult to refine the occupancy satisfactorily with statistical certainty.

The introduction of additional parameters into the model always results in a reduction in the least-squares or maximum-likelihood residual – in crystallographic terms, the R factor. However, the statistical significance of this reduction is not always clear, since this simultaneously reduces the observation-to-parameter ratio. It is therefore important to validate the significance of the introduction of further parameters into the model on a statistical basis. Early attempts to derive such an objective tool were made by Hamilton (1965). Unfortunately, they proved to be cumbersome in practice for large structures and did not provide the required objectivity.

Direct application of the Hamilton test is especially problematical for macromolecules because of the use of restraints. Attempts have been made to overcome these problems, using a direct extension of the Hamilton test itself (Bacchi et al., 1996) or with a combination of self and cross validation (Tickle et al., 1998).

Brünger (1992a) introduced the concept of statistical cross validation to evaluate the significance of introducing extra features into the atomic model. For this, a small and randomly distributed subset of the experimental observations is excluded from the refinement procedure, and the residual against this subset of reflections is termed . It is generally sufficient to include about 1000 reflections in the subset; further increase in this number provides little, if any, statistical advantage but diminishes the power of the minimization procedure. For atomic resolution structures, cross validation is important in establishing whether the introduction of an additional type of feature to the model (with its associated increase in parameters) is justified. There are two limitations to this. Firstly, if shows zero or minimal decrease compared to that in the R factor, the significance remains unclear. Secondly, the introduction of individual features, for example the partial occupancy of five water molecules, can provide only a very small change in , which will be impossible to substantiate. To recapitulate, at atomic resolution the prime use of cross validation is in establishing protocols with regard to extended sets of parameter types. The sets thus defined will depend on the quality of the data.

In the final analysis, validation of individual features depends on the electron density, and Fourier maps must be judiciously inspected. Nevertheless, this remains a somewhat subjective approach and is in practice intractable for extensive sets of parameters, such as the occupancies and ADPs of all solvent sites. For the latter, automated procedures, which are presently being developed, are an absolute necessity, but they may not be optimal in the final stages of structure analysis, and visual inspection of the model and density is often needed.

The problems of limited data and reparameterization of the model remain. At high resolution, reparameterization means having the same number of atoms, but changing the number of parameters to increase their statistical significance, for example switching from an anisotropic to an isotropic atomic model or vice versa. In contrast, when reparameterization is applied at low resolution, this usually involves reduction in the number of atoms, but this is not an ideal procedure, as real chemical entities of the model are sacrificed. Reducing the number of atoms will inevitably result in disagreement between the experiment and model, which in turn will affect the precision of other parameters. It would be more appropriate to reduce the number of parameters without sacrificing the number of atoms, for example by describing the model in torsion-angle space. Water poses a particular problem, as at low as well as at high resolution not all water molecules cannot be described as discrete atoms. Algorithms are needed to describe them as a continuous model with only a few parameters. In the simplest model, the solvent can be described as a constant electron density.

It is not reasonable to give absolute rules for refinement of atomic resolution structures at this time, as the field is rather new and is developing rapidly. Pioneering work has been carried out by Teeter et al. (1993) on crambin, based on data recorded on this small and highly stable protein using a conventional diffractometer. Studies on perhaps more representative proteins are those on ribonuclease Sa at 1.1 Å (Sevcik et al., 1996) and triclinic lysozyme at 0.9 Å resolution (Walsh et al., 1998). These studies used data from a synchrotron source with an imaging-plate detector at room temperature for the ribonuclease and at 100 K for the lysozyme. The strategy involved the application of conventional restrained least squares or maximum-likelihood techniques in the early stages of refinement, followed by a switch over to SHELXL to introduce a full anisotropic model. A series of other papers have appeared in the literature following similar protocols, reflecting the fact that, until recently, only SHELXL was generally available for refining macromolecular structures with anisotropic models and appropriate stereochemical restraints. Programs such as REFMAC have now been extended to allow anisotropic models. As they use fast Fourier transforms for the structure-factor calculations, the speed advantage will mean that REFMAC or comparable programs are likely to be used extensively in this area in the future, even if SHELXL is used in the final step to extract error estimates.

Hydrogen atoms possess only a single electron and therefore have low electron density and are relatively poorly defined in X-ray studies. They play central roles in the function of proteins, but at the traditional resolution limits of macromolecular structure analyses their positions can only be inferred rather than defined from the experimental data. Indeed, even at a resolution of 2.5 Å, hydrogen atoms should be included in the refined model, as their exclusion biases the position of the heavier atoms, but with their `riding' positions fixed by those of the parent atoms.

As for small structures, independent refinement of hydrogen-atom positions and anisotropic parameters (see below) is not always warranted, even by atomic resolution data, and hydrogen atoms are rather attached as riding rigidly on the positions of the parent atoms. Nevertheless, atomic resolution data allow the experimental confirmation of the positions of many of the hydrogen atoms in the electron-density maps, as they account for one-sixth of the diffracting power of a carbon atom. Inspection of the maps can in principle allow the identification of (1) the presence or absence of hydrogen atoms on key residues, such as histidine, aspartate and glutamate or on ligands, and (2) the correct location of hydrogen atoms, where more than one position is possible, such as in the hydroxyl groups of serine, threonine or tyrosine.

The correct placement of hydrogen atoms riding on their parent atoms involves computation of the appropriate position after each cycle of refinement. This is done automatically by programs such as SHELXL (Sheldrick & Schneider, 1997) or HGEN from the CCP4 suite (Collaborative Computational Project, Number 4, 1994). For rigid groups such as the NH amide, aromatic rings, —CH₂— or =CH—, the position is accurately defined by the bonding scheme. For groups such as methyl CH₃ or OH, the position is not absolutely defined, and the software is required to make judgmental decisions. For example, SHELXL offers the opportunity to inspect the maximum density on a circular Fourier synthesis for optimal positioning. The bond length is fixed according to results from a small-molecule database. The location of hydrogen atoms on polar atoms can be assisted by software that analyses the local hydrogen-bonding networks; this involves maximization of the hydrogen-bonding potential of the relevant groups.

Refinement of an isotropic model involves four independent parameters per atom, three positional and one isotropic ADP. In contrast, an anisotropic model requires nine parameters, with the anisotropic atomic displacement described by an ellipsoid represented by six parameters. At 1 Å resolution, the data certainly justify an anisotropic atomic model. Extension of the model from isotropic to anisotropic should generally result in a reduction in the R factor of the order of 5–6% and a comparable drop in . As a consequence of the diminution of the observable-to-parameter ratio, the R factor at all resolutions will drop by a similar amount; however, will not. Experience shows that at 2 Å or less there is no drop in , and an anisotropic model is totally unsupported by the data. At intermediate resolutions, the result depends on the data quality and completeness. At lower resolution, to account for anisotropy of the atoms, the overall motion of molecules or domains can be refined using translation/libration/screw (TLS) parameters (Schomaker & Trueblood, 1968).

Until recently, anisotropic ADPs have only been handled by programs originally developed for small-molecule analysis, which use conventional algebraic computations of the calculated structure-factor amplitudes, SHELXL being a prime example. A limitation of this approach is the substantial computation time required. The use of fast-Fourier-transform algorithms for the structure-factor calculation leads to a significant saving in time (Murshudov et al., 1999). Anisotropic modelling of the individual ADPs is essential if the thermal vibration is to be analysed in terms of coordinated motion of the whole molecule or of domains (Schomaker & Trueblood, 1968).

Proteins are not rigid units with a single allowed conformation. In vivo they spontaneously fold from a linear sequence of amino acids to provide a three-dimensional phenotype that may exhibit substantial flexibility, which can play a central role in biological function, for example in the induced fit of an enzyme by a substrate or in allosteric conformational changes. Flexibility is reflected in the nature of the protein crystals, in particular the presence of regions of disordered solvent between neighbouring macromolecules in the lattice (see below).

The structure tends to be highly ordered at the core of the protein, or more properly, at the core of the individual domains. Atoms in these regions in the most ordered protein crystals have ADP values comparable to those of small molecules, reflecting the fact that they are in essence closely packed by surrounding protein. In general, as one moves towards the surface of the protein, the situation becomes increasingly fluid. Side chains and even limited stretches of the main chain may show two (or multiple) conformations. These may be significant for the biological function of the protein.

The ability to model the alternative conformations is highly resolution dependent. At atomic resolution, the occupancy of two alternative but well defined conformations can be refined to an accuracy of about 5%, thus second conformations can be seen, provided that their occupancy is about 10% or higher. The limited number of proteins for which atomic resolution structures are available suggest that up to 20% of the `ordered residues' show multiple conformations. This confers even further complexity on the description of the protein model. A constraint can be imposed on residues with multiple conformations: namely that the sum of all the alternatives must be 1.0. Protein regions, be they side- or main-chain, with alternative conformations and partial occupancy can form clusters in the unit cell with complementary occupancy. This often coincides with alternative sets of solvent sites, which should also be refined with complementary occupancies.

The atoms in two alternative conformations occupy independent and discrete sites in the lattice, about which each vibrates. However, if the spacing between two sites is small and the vibration of each is large, then it becomes impossible to differentiate a single site with high anisotropy from two separate sites. There is no absolute rule for such cases: programs such as SHELXL place an upper limit on the anisotropy and then suggest splitting the atom over two sites. Some regions can show even higher levels of disorder, with no electron density being visible for their constituent atoms. Such fully disordered regions do not contribute to the diffraction at high resolution, and the definition of their location will not be improved with atomic resolution data.

A protein crystal typically contains some 50% aqueous solvent. This is roughly divided into two separate zones. The first is a set of highly ordered sites close to the surface of the protein. The second, lying remote from the protein surface, is essentially composed of fluid water, with no order between different unit cells.

At room temperature, the solvent sites around the surface are assumed to be in dynamic equilibrium with the surrounding fluid, as for a protein in solution. Nevertheless, the observation of apparently ordered solvent sites on the surface indicates that these are occupied most of the time. The waters are organized in hydrogen-bonded networks, both to the protein and with one another. The most ordered water sites lie in the first solvent shell, where at least one contact is made directly to the protein. For the second and subsequent shells, the degree of order diminishes: such shells form an intermediate grey level between the ordered protein and the totally disordered fluid. Indeed, the flexible residues on the surface form part of the continuum between a solid and liquid phase.

In the ordered region, the solvent structure can be modelled by discrete sites whose positional parameters and ADPs can be refined. For sites with low ADPs, the refinement is stable and their behaviour well defined. As the ADPs increase, or more likely the associated occupancy in a particular site falls, the behaviour deteriorates, until finally the existence of the site becomes dubious. There is no hard cutoff for the reality of a weak solvent site. However, the number and significance of solvent sites are increased by atomic resolution data. Despite the fact that the waters contribute only weakly to the high-resolution terms, the improved accuracy of the rest of the structure means that their positions become better defined.

Indeed, the occupancy of some solvent sites can be refined if the resolution is sufficient, or at least their fractional occupancy can be estimated and kept fixed (Walsh et al., 1998). This leads to the possibility of defining overlapping water networks with alternative hydrogen-bonding schemes. This can be a most time consuming step in atomic resolution refinement, and a trade-off finally has to be made between the relevance of any improvement in the model and the time spent.

The protein itself has a clearly defined chemical structure, and the number of atoms to be positioned and how they are bonded to one another are known at the start of model building. The solvent region is in marked contrast to this, as the number of ordered water sites is not known a priori, and the distances between them are less well defined, their occupancy is uncertain, and there may be overlapping networks of partially occupied solvent sites. Those of low occupancy lie at the level of significance of the Fourier maps.

Selection of partially occupied solvent sites poses a most cumbersome problem in the modelling over and above that of the macromolecule itself, and can be highly subjective and very time consuming. Improved resolution of the data reveals additional weak or partially occupied solvent sites, which generally do not behave well during refinement. Water atoms modelled into relatively weak peaks in electron density tend to drift out of the density during refinement due to the weak gradients that define their positions.

Given the huge number of water sites in question, automatic and at least semi-objective protocols are required. Several procedures have been developed for the automated identification of water sites during refinement [inter alia ARP (Lamzin & Wilson, 1997) and SHELXL (Sheldrick & Schneider, 1997)] and others allow selective inspection of such sites using graphics [O (Jones et al., 1991) and Quanta (Molecular Simulations Inc., San Diego)]. These depend on a combination of peak height in the density map and geometric considerations.

As stated in the preceding section and first reviewed by Matthews (1968) and more recently by Andersson & Hovmöller (1998), macromolecular crystals contain substantial regions of totally disordered, or bulk, aqueous solvent, in addition to those solvent molecules bound to the surface. The average electron density of the crystal volume occupied by protein is 1.35 g cm⁻³ (according to Matthews) or 1.22 g cm⁻³ (according to Andersson & Hovmöller), while that of water is 1.0 g cm⁻³. This is because the atoms are more closely packed within the protein, as they are connected by covalent bonds, while in solvent regions they form sets of hydrogen-bonded networks.

To model both solvent and protein regions of the crystal appropriately, it is necessary to have a satisfactory representation of the bulk solvent. The high R factors generally observed for most proteins for the low-resolution shells are partly symptomatic of the poor modelling of this feature or of systematic errors in the recording of the intensities of the low-angle reflections. For atomic resolution structures, the R factor can fall to values as low as 6–7% around 3–5 Å resolution. However, in lower-resolution shells it then rises steadily, often reaching values in the range of 20–40% below 10 Å. These observations indicate serious deficiencies in our current models or data.

The poorest approach is to ignore bulk solvent and assign zero electron density to those regions where there are no discrete atomic sites, as this leads to a severe discontinuum. An improved approach is to assign a constant value of the electron density to all points of the Fourier transform that are not covered by the discrete, ordered sites. This provides substantial reduction in the R factor for low-resolution shells of the order of 10% and requires the introduction of only one extra parameter to the least-squares minimization. An improvement of this simplistic model is the introduction of a second parameter, , described by where and are the scale factors for the protein, and and are the equivalent parameters for the bulk solvent (Tronrud, 1997). In effect, this provides a resolution-dependent smoothing of the interface contribution, rather than an overall term applied equally to all data. The physical basis of this is discussed by Tronrud and implemented in several programs, for example SHELXL (Sheldrick & Schneider, 1997) and REFMAC (Murshudov et al., 1997) (Fig. 18.4.5.2).

Figure 18.4.5.2| top | pdf | Schematic representation of the bulk-solvent models described in the text. (a) No bulk-solvent correction, i.e. solvent density set to zero. (b) Constant level of solvent outside the macromolecule and ordered water envelope. Here, sharp edge effects remain. (c) The model as in (b), but smoothed at the edge of a macromolecule, equivalent to the application of a B value to the solvent model.

Nevertheless, there remain severe problems in the modelling of the interface. The border between the two regions is not abrupt, as there is a smooth and continuous change from the region with fully occupied, discrete sites to one which is truly fluid, but this passes through a volume with an increasing level of dynamic disorder and associated partial occupancy. Modelling of this region poses major problems, as described above, and the definition of disordered sites with low occupancy remains difficult even at atomic resolution. At which stage the occupancy and associated ADP can be defined with confidence is not yet an objective decision. In addition, refinement and modelling at this level of detail is very time consuming in terms of human intervention.

In general, proteins are crystallized from aqueous solutions which contain various additives, such as anions or cations (especially metals), organic solvents, including those used as cryoprotectants, and other ligands. Some of these may bind in specific or indeed non-specific sites in the ordered solvent shell, in addition to any functional binding sites of the protein. To identify such entities at limited resolution is often impossible, as the range of expected ADPs is large and there is very poor discrimination in the appearance of such sites and of water in the electron density. Atomic resolution assists in resolving ambiguities, as all the interatomic distances, ADPs and occupancies are better defined.

For metal ions, two additional criteria can be invoked. Firstly, the coordination geometry, with well defined bond lengths and angles, provides an indication of the identity of the ion, as different metals have different preferred ligand environments [see, for example, Nayal & Di Cera (1996)]. In addition, the value of the refined ADP and/or occupancy is helpful. Secondly, the anomalous signal in the data should reveal the presence of metal and some other non-water sites in the solvent by computation of the anomalous difference synthesis (Dauter & Dauter, 1999). While these approaches can be applied at lower resolution, they both become much more powerful at atomic resolution.

The presence of bound organic ligands has become especially relevant since the advent of cryogenic freezing. Compounds such as ethylene glycol and glycerol possess a number of functional hydrogen-bonding groups that can attach to sites on the protein in a defined way. Indeed, these may often bind in the active sites of enzymes such as glycosyl hydrolases, where they mimic the hydroxyl groups of the sugar substrate. It is most important to identify such moieties properly, particularly if substrate studies are to be planned successfully.

X-ray structures are generally modelled using the spherical-atom approximation for the scattering, which ignores the deviation from sphericity of the outer bonding and lone-pair electrons. Extensive studies over a long period have confirmed that the so-called deformation density, representing deviation from this spherical model, can be determined experimentally using data to very high resolution, usually from 0.8 to 0.5 Å. An excellent recent review of this field is provided by Coppens (1997). The observed deviations can be compared with those expected from the available theories of chemical bonding and the densities derived therefrom. Such studies have been applied to peptides and related molecules (Souhassou et al., 1992; Jelsch et al., 1998).

The application of atomic resolution analysis to proteins has allowed the first steps towards observation of the deformation density in macromolecules (Lamzin et al., 1999). Data for two proteins were analysed: crambin (molecular weight 6 kDa) at 0.67 Å resolution and a subtilisin (molecular weight 30 kDa) at 0.9 Å. Significant and interpretable deformation density could not be observed for the individual residues. However, on averaging the density over 40 peptide units for crambin and more than 250 for the subtilisin, the deformation density within the peptide unit was clearly visible and could be related to the expected bonding features in these units. This shows the real power of atomic resolution crystallography, which can reveal features containing no more than 0.2 e Å⁻³.