International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by M. G. Rossmann and E. Arnold © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. F. ch. 18.4, pp. 393402
https://doi.org/10.1107/97809553602060000696 Chapter 18.4. Refinement at atomic resolution^{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 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)]. Once the chemical identity of the atom is established, this parameter is fixed.
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 are used to describe the distribution of the atoms in different unit cells (Fig. 18.4.1.1). Atomic displacement parameters (ADPs) reflect both the thermal vibration of atoms about the mean position as a function of time (dynamic disorder) and the variation of positions between different unit cells of the crystal arising from its imperfection (static disorder). Contributors to the apparent ADP () can be thought of as follows (Murshudov et al., 1999): where represents the fact that a crystal itself is generally an anisotropic field that will result in the intensity falling off in an anisotropic manner, represents a translation/libration/screw (TLS), i.e. the overall motion of molecules or domains (Schomaker & Trueblood, 1968), is the oscillation along torsion angles and is the oscillation along and across bonds. In principle, all these contributors are highly correlated and it is difficult to separate them from one another. Nevertheless, an understanding of how is a sum of these different components makes it possible to apply atomic anisotropy parameters at different resolutions in a different manner. For example, can be applied at any resolution, as their refinement increases the number of parameters by at most five for and twenty per independent moiety for . In contrast, refinement of the third contributor does pose a problem, as there is a strong correlation between different torsion angles. As an alternative, ADPs along the internal degrees of freedom could in principle be refined. The fourth and final contributor, , can only be refined at very high resolution. In real applications, and are separated for convenient description of the system, but in practice their effect is indistinguishable.
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 introduces a tenth variable for each atom. In many cases, the fractional occupancies are not all independent, but are constant for sets of covalently or hydrogenbonded atoms or for those in nonoverlapping solvent networks. This would apply, for example, to partially occupied ligands or side chains with two conformations.
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), or more recently by maximumlikelihood procedures (Bricogne & Irwin, 1996; Pannu & Read, 1996; Murshudov et al., 1997).
Atomicity is the great simplifying feature of crystallography in terms of structure solution and refinement. If atomic resolution is achieved, there are sufficient accurately measured observables to refine a full atomic model for the ordered part of the structure, but this condition can only be defined somewhat subjectively. A pragmatic approach has been that data extending to 1.2 Å or better with at least 50% of the intensities in the outer shell being higher than 2σ is the acceptable limit (Sheldrick, 1990; Sheldrick & Schneider, 1997). In practice, this means the statistical problem of refinement is overdetermined. For smallmolecule structures, accurate amplitude data are normally available to around 0.8 Å, giving an observationtoparameter ratio of about seven, allowing positional parameters to be determined with an accuracy of around 0.001 Å. This reflects the high degree of order of such crystals, in which the molecules in the lattice are in a closely packed array.
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 Table 9.1.1.1 in Chapter 9.1). Secondly, the intrinsic disorder of the crystals further reduces the intensities at high Bragg angles and may lead to a resolution cutoff much less than atomic. Thirdly, the large solvent content leads to substantial decay of crystal quality under exposure to the Xray beam, especially at room temperature. The upper resolution limit of the data affects all stages of a crystallographic analysis, but especially restricts the features of the model that can be independently refined (Table 18.4.1.2). Solutions to the problem of refining macromolecular structures with a paucity of experimental data evolved during the 1970s and 1980s with the use of either constraints or restraints on the stereochemistry, based on that of known small molecules. With constraints, the structure is simplified as a set of rigid chemical units (Diamond, 1971; Herzberg & Sussman, 1983), whereas using restraints, the observationtoparameter ratio is increased by introduction of prior chemical knowledge of bond lengths and angles (Konnert & Hendrickson, 1980).

As expected, atoms with different ADPs contribute differently to the diffraction intensities, as discussed by Cruickshank (1999a,b). The relative contribution of the different atoms to a given reflection depends on the difference between their ADPs where . Clearly, if the average ADP of a molecule is small, then the spread will also be narrow, and most atoms will contribute to diffraction over the whole range of resolution. When the mean ADP is large, then the spread of the ADPs will be wide, and fewer atoms will contribute to the highresolution intensities (Fig. 18.4.1.2).
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 8 , 7 and 10 , respectively). These advances mean that there is no longer a sharp division between small and macromolecular crystallography, but a continuum from small through mediumsized structures, such as cyclodextrins and other supramolecules, to proteins. The inherent disorder in the crystal generally increases with the size of the structure, due in part to the increasing solvent content. However, it is now tractable to refine a significant number of proteins at atomic resolution with a full anisotropic model (Dauter, Lamzin & Wilson, 1997). This work of course benefits tremendously from the experience and algorithms of smallmolecule crystallography, but it does pose special problems of its own. The techniques of solving and refining macromolecular structures thus also overlap with those conventionally used for small molecules; a prime example is the use of SHELXL (Sheldrick & Schneider, 1997), which was developed for small structures and has now been extended to treat macromolecules.
An alternative and probably better approach to the definition of atomic resolution would be to employ a measure of the information content of the data. There are a variety of definitions of the information in the data about the postulated model (see, for example, O'Hagan, 1994). A suitable one is the Bayesian definition for quadratic information measure: where is the quadratic information measure, p is the vector of parameters, F is the experimental data, var(p) is the variance matrix corresponding to prior knowledge, var(p, F) is the variance matrix corresponding to the posterior distribution (which includes prior knowledge and likelihood), E is the expectation, tr is the trace operator (i.e. the sum of the diagonal terms of the matrix) and A is the matrix through which the relative importance of different parameters or combinations of parameters is introduced. For example, if A is the identity matrix, then the information measure is unitary and all parameters are assigned the same weight. If A is the identity matrix for positional parameters and zero for ADPs, then only the information about positional parameters is included. The appropriate choice of A allows the estimation of information on selected key features, such as the active site.
Equation (18.4.1.2) shows how much the experiment reduces the uncertainty in given parameters. Prior knowledge is usually taken to be information about bond lengths, bond angles and other chemical features of the molecule, known before the experiment has been carried out. In the case of an experiment designed to provide information about the ligated protein or mutant, when information about differences between two (or more) separate states is needed, the prior knowledge can be considered instead as knowledge about the native protein.
However, there are problems in applying equation (18.4.1.2). Firstly, careful analysis of the prior knowledge and its variance is essential. The target values used at present, or more properly the distributions for these values, need to be reevaluated. Another problem concerns the integration required to compute the expectation value (E). Nevertheless, the equation gives some idea about how much information about a postulated model can be extracted from a given experiment.
This alternative definition of atomic resolution assumes that the second term of equation (18.4.1.2) for positional parameters is sufficiently close to zero for most atoms to be resolved from all their neighbours. Defining atomic resolution using this information measure reflects the importance of both the quality and quantity of the data [through the posterior var(p, F)]. In addition, data may come from more than one crystal, in which case the information will be correspondingly increased. There may be additional data from mutant and/or complexed protein crystals, where, again, the information measure will be increased and, moreover, the differences between different states can be analysed. The effect of redundancy of crystal forms is to reduce the limit of data necessary for achieving atomic resolution, which is equivalent to the advantage of noncrystallographic averaging.
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 9.1 and will not be discussed here.
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 highresolution 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 highresolution data are available. In addition, highresolution data mean that answers to some of the questions which may arise during analysis of the experiment will already be addressable. In contrast, lowresolution 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 lowresolution completeness and their statistical significance affect the information content of the data and derived model. Highintensity lowresolution 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 lowintensity 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 lowresolution data. This is particularly true in the early stages of structure solution, where lowresolution data can be vital. Although most phaseimprovement algorithms rely on relations between all reflections, terms involving lowresolution 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 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; 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 leastsquares 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 leastsquares 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 Xray 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 blockmatrix 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 leastsquares 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, blockmatrix approaches remain essential for tractable matrix inversion for macromolecular structures.
A further simplification involves the conjugategradient 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 offdiagonal terms of the leastsquares matrix. For the conjugategradient 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 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, 1977; Agarwal, 1978). This can involve some interpolation and the limitation of the volume of electrondensity 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 conjugategradient method that relies only on the first derivatives, ignoring the second derivatives. In all other programs, the diagonal approximation is used for the secondderivative 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 maximumlikelihood 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 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 (Pannu & Read, 1996).
Gaussian approximation to intensitybased 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 intensitybased likelihood, this is less important than for the amplitudebased approach. French & Wilson (1978) have given a Bayesian approach for the derivation of structurefactor 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 amplitudebased refinement with respect to when is equal to zero is singular (Schwarzenbach et al., 1995). This is not the case for intensitybased 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 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), 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 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). 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 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) 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 imagingplate 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 maximumlikelihood 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 structurefactor 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.
All features of the refined model are more accurately defined if the data extend to higher resolution (Fig. 18.4.5.1). In this section, those features that are especially enhanced in an atomic resolution analysis are described. Introduction of an additional feature to the model should be assessed by the use of cross or selfvalidation tools: only then can the significance of the parameters added to the model be substantiated.
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) or HGEN from the CCP4 suite (Collaborative Computational Project, Number 4, 1994). For rigid groups such as the NH amide, aromatic rings, —CH_{2}— or =CH—, the position is accurately defined by the bonding scheme. For groups such as methyl CH_{3} 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 smallmolecule database. The location of hydrogen atoms on polar atoms can be assisted by software that analyses the local hydrogenbonding networks; this involves maximization of the hydrogenbonding 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 observabletoparameter 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 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). 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 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). This leads to the possibility of defining overlapping water networks with alternative hydrogenbonding schemes. This can be a most time consuming step in atomic resolution refinement, and a tradeoff 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 semiobjective 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 hydrogenbonded 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 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). In effect, this provides a resolutiondependent 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).
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)]. 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 nonwater 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 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 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}.
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) are derived from either the Cambridge Structural Database (Allen et al., 1979) or from protein structures in the Protein Data Bank (PDB; Bernstein et al., 1977). The availability of atomic resolution structures provides more objective data for the construction of target libraries. Stereochemical parameters, such as conformational angles φ, ψ, should ideally not be restrained, as they allow independent validation of the model. Analysis of eight structures determined at atomic resolution (EU 3D Validation Network, 1998) indicates that they follow the expected rules of chemistry more closely than those of lowerresolution analyses in the PDB, confirming that atomic resolution indeed provides more precise coordinates.
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.
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