^{a}National Cancer Institute, Brookhaven National Laboratory, Building 725AX9, Upton, NY 11973, USA,^{b}Structural Biology Laboratory, Department of Chemistry, University of York, York YO10 5DD, England, and CLRC, Daresbury Laboratory, Daresbury, Warrington, WA4 4AD, England, and ^{c}Structural Biology Laboratory, Department of Chemistry, University of York, York YO10 5DD, England
Correspondence email: dauter@bnl.gov
The first part of this chapter gives a definition of atomic resolution. This is followed by a discussion of data quality and anisotropic scaling of data. Computational algorithms and strategies are then covered as are computational options and tactics. Those features of the refined model that are especially enhanced in an atomic resolution analysis are described. Finally, the biological issues that are addressed by analysis of macromolecular structures at atomic resolution are discussed.
Keywords: ab initio phasing; alternative conformations; anisotropic scaling; atomic displacement parameters; atomic resolution; automatic location of water sites; bulk solvent modelling; constraints; deformation density; fast Fourier transform; leastsquares methods; ligands at atomic resolution; maximum likelihood; metal ions at atomic resolution; partial occupancy; phasing; refinement; restraints; scaling; solvent.
Xrays are diffracted by the electrons that are distributed around the atomic nuclei, and the result of an Xray crystallographic study is the derived threedimensional electrondensity distribution in the unit cell of the crystal. The elegant simplicity and power of Xray crystallography arise from the fact that molecular structures are composed of discrete atoms that can be treated as spherically symmetric in the usual approximation. This property places such strong restraints on the Fourier transform of the crystal structures of small molecules that the phase problem can be solved by knowledge of the amplitudes alone.
Each atom or ion can be described by up to eleven parameters (Table 18.4.1.1)
The first parameter is the scatteringfactor amplitude for the chemical nature of the atom in question, computed and tabulated for all atom types [International Tables for Crystallography, Volume C
(2004)
The next three parameters relate to the positional coordinates of the atom with respect to the origin of the unit cell.
At atomic resolution, six anisotropic atomic displacement parameters
In the special case when the tensor is isotropic, i.e., all nondiagonal elements are equal to zero and all diagonal terms are equal to each other, then the atom itself appears to be isotropic and its ADP can be described using only one parameter, .
Thus for a full description of a crystal structure in which all atoms only occupy a single site, nine parameters must be determined: three positional parameters and six anisotropic ADPs. This assumes that the sphericalatom approximation applies and ignores the socalled deformation density resulting from the nonspherical nature of the outer atomic and molecular orbitals involved in the chemistry of the atom (Coppens, 1997
For disordered regions or features, where atoms can be distributed over two or more identifiable sites, the occupancy
Thus, at atomic resolution, minimization of the discrepancy between the experimentally determined amplitudes or intensities of the Bragg reflections and those calculated from the atomic model requires refinement of, at most, ten (usually nine) independent parameters per atom. This has been achieved classically by least squares, as described in IT C
(2004)
Atomicity
Crystals of macromolecules deviate substantially from this ideal. Firstly, the large unitcell volume leads to an enormous number of reflections for which the average intensity is weak compared to those for small molecules (see
As expected, atoms with different ADPs contribute differently to the diffraction intensities, as discussed by Cruickshank (1999a
Three advances in experimental techniques have combined effectively to overcome these problems for an increasing number of well ordered macromolecular crystals, namely the use of highintensity synchrotron radiation, efficient twodimensional detectors and cryogenic freezing (discussed in Parts
An alternative and probably better approach to the definition of atomic resolution would be to employ a measure of the information content
Equation (18.4.1.2)
However, there are problems in applying equation (18.4.1.2)
This alternative definition of 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.
The quality of the refined model relies finally on that of the available experimental data. Data collection has been covered extensively in Chapter
As can be seen from equation (18.4.1.2)
While the information content of the data
The intensity data from a crystal may display anisotropy, i.e., the intensity falloff 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 falloff 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
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
The principles of the leastsquares method of minimization are described in IT C
(2004)
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 Xray data alone. If the problem of how restraints
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 blockmatrix approximation
A further simplification involves the conjugategradient method
Conventional leastsquares programs use the structurefactor 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 N^{2} to N log N, involves the use of fast Fourier algorithms for the computation of structure factors and derivatives (Ten Eyck, 1973
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
There are no longer any restrictions on the fullmatrix refinement of smallmolecule 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 Xray experiment provides twodimensional diffraction images. These are transformed to integrated but unscaled data, which are transformed to Bragg reflection intensities that are subsequently transformed to structurefactor 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 structurefactor amplitudes, and use of intensities would reduce the bias. However, there are some difficulties in the implementation of intensitybased likelihood refinement
Gaussian approximation to intensitybased likelihood
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 intensitybased likelihood, this is less important than for the amplitudebased approach. French & Wilson (1978)
Another argument for the use of intensities rather than amplitudes is relevant to least squares where the derivative for amplitudebased refinement with respect to when is equal to zero is singular (Schwarzenbach et al., 1995
Finally, while there may be some advantages in refining against F^{2}, Fourier syntheses always require structurefactor amplitudes.
Even for smallmolecule 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 highangle diffraction terms, and as a result their parameters are only defined by the lowerangle 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 Xray terms will dominate the refinement, provided the relative weighting of Xray 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
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 leastsquares or maximumlikelihood residual – in crystallographic terms, the R factor. However, the statistical significance of this reduction is not always clear, since this simultaneously reduces the observationtoparameter 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)
Direct application of the Hamilton test
Brünger (1992a)
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 torsionangle 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)
All features of the refined model are more accurately defined if the data extend to higher resolution (Fig. 18.4.5.1
Hydrogen atoms possess only a single electron and therefore have low electron density and are relatively poorly defined in Xray 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 hydrogenatom 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 electrondensity maps, as they account for onesixth 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
Refinement of an isotropic model involves four independent parameters per atom, three positional and one isotropic ADP. In contrast, an anisotropic model
Until recently, anisotropic ADPs have only been handled by programs originally developed for smallmolecule analysis, which use conventional algebraic computations of the calculated structurefactor amplitudes, SHELXL being a prime example. A limitation of this approach is the substantial computation time required. The use of fastFouriertransform algorithms for the structurefactor calculation leads to a significant saving in time (Murshudov et al., 1999
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 threedimensional 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 mainchain, 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 hydrogenbonded 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 highresolution 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
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
Given the huge number of water sites in question, automatic and at least semiobjective protocols are required. Several procedures have been developed for the automated identification of water sites during refinement [inter alia ARP (Lamzin & Wilson, 1997
As stated in the preceding section and first reviewed by Matthews (1968)
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 lowresolution shells are partly symptomatic of the poor modelling of this feature or of systematic errors in the recording of the intensities of the lowangle reflections. For atomic resolution structures, the R factor can fall to values as low as 6–7% around 3–5 Å resolution. However, in lowerresolution 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 lowresolution shells of the order of 10% and requires the introduction of only one extra parameter to the leastsquares 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
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 nonspecific 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)
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 hydrogenbonding 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.
Xray structures are generally modelled using the sphericalatom approximation for the scattering, which ignores the deviation from sphericity of the outer bonding and lonepair electrons. Extensive studies over a long period have confirmed that the socalled 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 application of atomic resolution analysis to proteins has allowed the first steps towards observation of the deformation density in macromolecules (Lamzin et al., 1999
The refinement of proteins at resolution lower than atomic depends upon the use of restraints on the geometry and ADPs. Most target libraries for refinement and validation of structures (e.g. Engh & Huber, 1991
A question arises as to what biological issues are addressed by analysis of macromolecular structures at atomic resolution. For any protein, the overall structure of its fold, and hence its homology with other proteins, can already be provided by analyses at low to medium resolution. However, proteins are the active entities of cells and carry out recognition of other macromolecules, ligand binding and catalytic roles that depend upon subtle details of chemistry, for which accurate positioning of the atoms is required. Even at atomic resolution, the accuracy of structural definition is less than what would ideally be required for the changes observed during a chemical reaction. At lower resolutions, structure–function relations require yet further extrapolation of the experimental data.
To understand the function of many macromolecules, such as enzymes, it is not sufficient to determine the structure of a single state. Alongside the native structure, those of various complexes will also be required. The differences between the states provide additional information on the functionality. For an understanding of the chemistry involved, atomic resolution has tremendous advantages in terms of accuracy, as reliable judgments can be based on the experimental data alone.
Advantages of atomic resolution include the following:
Almost all atomic resolution analyses require data recorded from cryogenically frozen crystals. This does pose some problems of biological relevance, as proteins in vivo have adapted to operate at ambient cellular temperatures. The required structure is that of the protein and surrounding solvent at the corresponding temperature. The tradeoff is that cryogenic structures may be better defined, but only because of the increased order of protein and solvent at low temperature. This has to be weighed against the lack of fine detail in a mediumresolution analysis at room temperature.
A question often raised with regard to the worth of atomic resolution data concerns the effort required in refining a protein at such resolution. To define all details, such as alternative conformations, hydrogenatom positions and solvent, is certainly time consuming, especially if an anisotropic model is adopted. However, the advantages outweigh the disadvantages, as even if a full anisotropic model is not refined to exhaustion, nevertheless all density maps will be clearer if the resolution is better, resulting in an improved definition of the features of interest.
The thermalellipsoid model used to represent anisotropic atomic displacement, with major axes indicated. The ellipsoid is drawn with a specified probability of finding an atom inside its contour. Six parameters are necessary to describe the ellipsoid: three represent the dimensions of the major axes and three the orientation of these axes. These six parameters are expressed in terms of a symmetric U tensor and contribute to atomic scattering through the term
The thermalellipsoid model used to represent anisotropic atomic displacement, with major axes indicated
Histograms of B values for a protein structure, Micrococcus lysodecticus catalase (Murshudov et al., 1999
Histograms of B values for a protein structure, Micrococcus lysodecticus catalase (Murshudov et al
(a), (b) Representative electrondensity maps for the refinement of Clostridium acidurici ferredoxin at 0.94 Å resolution (Dauter, Wilson et al., 1997
(a), (b) Representative electrondensity maps for the refinement of Clostridium acidurici ferredoxin at 0.94 Å resolution (Dauter, Wilson et al
Schematic representation of the bulksolvent models described in the text. (a) No bulksolvent 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.
Schematic representation of the bulksolvent models described in the text



Figure 18.4.1.1
The thermalellipsoid model used to represent anisotropic atomic displacement, with major axes indicated. The ellipsoid is drawn with a specified probability of finding an atom inside its contour. Six parameters are necessary to describe the ellipsoid: three represent the dimensions of the major axes and three the orientation of these axes. These six parameters are expressed in terms of a symmetric U tensor and contribute to atomic scattering through the term 

Figure 18.4.1.2
Histograms of B values for a protein structure, Micrococcus lysodecticus catalase (Murshudov et al., 1999 

Figure 18.4.5.1
(a), (b) Representative electrondensity maps for the refinement of Clostridium acidurici ferredoxin at 0.94 Å resolution (Dauter, Wilson et al., 1997 

Figure 18.4.5.2
Schematic representation of the bulksolvent models described in the text. (a) No bulksolvent 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. 