From Map to Model

Having obtained initial phases as described in the previous section, an electron density map can be calculated by Fourier transformation. It is not always (!) possible to recognize features in a first electron density map, however (Figure 10a). In order to improve the map (phases), it is necessary to use information about protein crystals and their electron densities. One of the most universally applied forms of initial phase improvement is solvent flattening. Protein crystals contain large amounts of solvent; this will in general be disordered, and so will not contribute to the crystal diffraction.

By knowing the protein content of the crystal (from calculating the Matthews coefficient VM; see above), it is therefore possible to estimate the threshold density below which is noise; points with density below the threshold are set to a suitable average value. This method is particularly useful for locating molecular boundaries (Figure 10). If the asymmetric unit contains more than one molecule, then real space averaging can lead to a dramatic improvement in the map. Provided that the transformation between individual molecules can be identified (e.g., by the coordinates of heavy atoms or anomalous scatterers, or through location of a noncrystallographic symmetry axis), then the densities of the various copies can be equivalenced to yield an averaged density.

These density modification procedures (which also include so-called histogram matching and map skeletonization) become particularly powerful when carried out in a cyclic process (Figure 10). Repeated density modification, phase calculation and combination, and map calculation until convergence results in most cases in an electron density map of superior quality, and can even be used for phase extension, i.e., determination of phases to a higher resolution than those at which the experimental phases are reliable. Once again, the FoM can be used to ascertain the quality of the phases.

With reasonable phases and sufficiently high resolution, the initial interpretation of the electron density map should pose few problems; at resolutions better than about 2A (Figure 11), modern programs can even build a model completely automatically. In many cases, however, it will be necessary to build at least part of the model manually. The general procedure is (1) identification of secondary structure elements, (2) interpretation of possible side chains, assigning them as large or small, interior or exterior, and (3) matching of the protein sequence to the density. The ease with which this process can be carried out is dependent on the quality of the phases and the resolution of the data (Figure 11). It is important to remember that the connectivity of the electron density (i.e., the continuity of the density for connected atoms in the protein) can be disrupted by errors in the phases, mobility of the residues involved

Phase combination

New phases and amplitudes Fpca;c, ^pca;c

Phases and amplitudes fp, ^p

Fourier transformation

Inverse Fourier transformation

Pmod(r)

Modified density map

Pmod(r)

Modified density map

Electron p(r) density map

Map modification

Figure 10 Density modification. (a) The initial electron density map from a SeMet MAD experiment was of insufficient quality for model building (yellow spheres reveal the positions of the Se atoms used for phasing). (b) Repeated cycles of solvent flattening and phase recombination resulted in clear differentiation between protein and solvent and allowed interpretation of secondary structure (b-strands are clearly visible). Interestingly, although there are two molecules in the asymmetric unit, electron density averaging failed to converge; subsequent completion of the structure revealed significant conformational differences between the two monomers. Data taken from the structure determination of surfactin thioesterase domain SrfTE.63

(leading to reduced or missing density), alternative conformations or ambiguous density - one example is the presence of disulfide bridges, which can cause problems at lower resolutions. Clues as to the possible interpretation of individual side chains can come from neighboring heavy atom positions used in the phase determination (in particular, SeMet phasing reveals the positions of methionines at an early stage of the interpretation), or to already interpreted and well-defined surrounding atoms (such as charged groups, where charge compensation is usually necessary).

In nearly all cases, the first building session will result in a partial model with errors. To this end, the model must be refined - the xyz positions of the atoms of the model are allowed to move (and their B-factors allowed to vary) according to a modified force field that resembles a molecular dynamics simulation, with a 'pseudoenergy' related to the Rfac described in eqn [7]. Unless working at atomic resolution (more the exception than the rule for biological macromolecules), refinement of the structure requires that geometrical parameters such as bond lengths, bond angles, torsion angles etc. are restrained to minimize deviation from ideal values. Simulated annealing,41 in which the model is heated in silico to high temperatures and allowed to cool down slowly, provides a method to overcome local energy barriers and is often valuable in the early stages of refinement. At low resolution, the parameter to observable ratio can be improved by applying constraints, where geometrical parameters are fixed to ideal values. In place of free refinement of the x, y, and z coordinates for each atom, torsion angle dynamics can be particularly useful in reducing the number of variables.

During the molecular dynamics run, the pseudoenergy term is used to decrease the Rfac while maintaining the standard geometries of amino acids and proteins. If the model is for the most part correct, then the reduction in the Rfac will reflect an improvement in the phases, and a new map calculated from the ensuing phases should indicate clearer electron density for previously uninterpretable regions. At early stages of the structure solution, it is wise to combine the model phases with any experimental phase information at hand (from, e.g., MIR measurements) to avoid model bias. The latter is a very real problem for x-ray crystallography, in particular for molecular replacement - the phases tend to be dominated by the model, so that subsequent electron densities have a tendency of reproducing

Figure 11 Information content of 2Fobs — Fcalc maps at increasing resolution; each map contoured at 1.0 a (light blue), 2.5 a (magenta), and 3.7 a (yellow). Density shown for a Trp-Gly dipeptide from trypsin phased at the highest resolution.64 Given adequate phases, the overall shape and Ca trace can be evaluated at resolutions between 5 and 3.5 A. Side chains can be discerned at 3 A, carbonyl oyxgens at 2.8 A, while side chains and peptide bond planes are well resolved at resolutions better than 2.5 A. Holes in the aromatic rings of Phe and Tyr at resolutions of 1.5 A are indicators of correct phases, and individual atoms can be observed at 1.2A; at high resolution, hydrogen atoms can even be visualized.

Figure 11 Information content of 2Fobs — Fcalc maps at increasing resolution; each map contoured at 1.0 a (light blue), 2.5 a (magenta), and 3.7 a (yellow). Density shown for a Trp-Gly dipeptide from trypsin phased at the highest resolution.64 Given adequate phases, the overall shape and Ca trace can be evaluated at resolutions between 5 and 3.5 A. Side chains can be discerned at 3 A, carbonyl oyxgens at 2.8 A, while side chains and peptide bond planes are well resolved at resolutions better than 2.5 A. Holes in the aromatic rings of Phe and Tyr at resolutions of 1.5 A are indicators of correct phases, and individual atoms can be observed at 1.2A; at high resolution, hydrogen atoms can even be visualized.

model density, even when this is incorrect. For this reason it is essential during density interpretation to only build model fragments where there is a degree of certainty (this of course is relative, but a conservative interpretation is generally advisable). In particular, solvent and substrate molecules should only be built into the density at late stages of the refinement.

To avoid model bias (or to be able to recognise when it occurs), the free R-factor Rfree is generally monitored during refinement.42 Rfree is calculated in exactly the same way as Rfac (eqn [5]), but for only a fraction (between 5% and 10%) of the reflections (the test set). The essential difference is that reflections in the test set are not used for refinement (for this, the remaining 90-95% reflections of the working set are used) and are therefore independent of the model. The Rfree calculated in this way should decrease with refinement as the Rfac (Rfree is generally 2-5% higher than Rfac) if the model is correct; an increase in Rfree, however, indicates that something is wrong with the model and it is necessary to backtrack.

Recalling eqn [4], if we had perfect phases, then pobs (xyz) = ShkiFobs (hkl) expfifcalc (hkl)} exp{ — 2pi(hx + ky + lz)} [8]

The model phases are of course imperfect; if they are close to the correct solution, however, then a difference Fourier map calculated according to

DPobs(xyz) = Shki {Fobs (hkl)—Fcalc (hkl)} exp{i'fcalc(hkl)} exp{—2ra(hx + ky + lz)} [9]

should reveal errors or omissions in the model. Positive difference density Dpobs indicates that atoms are missing from the model, while negative difference density shows that an atom should not be at this position, or that its temperature factor is too low. Thus a so-called (Fobs — Fcalc) map (corresponding to Dpobs) is particularly useful in completing the structure. As pobs suffers from model bias, it is customary to work with maps relating to (pobs + Dpobs) or (2Fobs — Fcalc) maps: electron density for missing atoms is upweighted, while that for wrongly placed atoms is downweighted. Model bias can be reduced in part by calculating so-called omit maps, where a portion of the structure is removed from the refinement and phasing calculations prior to difference map synthesis. Electron density maps are commonly scaled according to their standard deviations a; it is usual to contour (2Fobs — Fcalc) maps at 0.8-1.0 a and (Fobs — Fcalc) maps at 2.5-3.0 a (Figure 11).

Most structure determinations involve multiple cycles of building and refinement before the Rfac (and Rfree) converge to reasonable values (around or below 20%), and so patience is required. The problem of model bias should always be borne in mind, as should the fact that the phases of all reflections contribute to the synthesis of an electron density map, and that all atoms of the model contribute to the phases of all reflections. This latter point is often neglected; a consequence thereof, however, is that a poorly or wrongly interpreted portion of the molecule remote from the point of interest (e.g., the binding of a ligand at the active site of an enzyme) may yet influence the electron density at this region. Particular care should be taken in the placement of solvent molecules, which are not constrained and laborious to insert and check by hand.

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