FC mapping

Fig. 2 A 1. 8-A resolution sigma-weighted (IFg-Fc) map of the octapeptide, contoured at la level. B Stereo view of the superimposed positions of the peptide [80] and carbohydrate [82] ligands. Atom types are the same as in Fig. 1. The ordered solvent (water) molecules (S) associated with the peptide complex are also shown. The solvent molecules S2, S9 and Si and the residue Rha C are shown to occupy a similar area of the site, the deep pocket of the combining site groove (see also Fig. 1). Reproduced from [80]. 2003 by The National Academy of Sciences of the USA...

For every pair a/ , bRj and fixed genus g > 2, the number of strictly face-regular possibilities is, clearly, finite. But it is, certainly, extremely large. An interesting question would be to decide, what type of strict face-regularity can appear on surfaces of genus g > 2. Another direction is to study all weakly face-regular ( a, b), fc)-maps with k>A. 

The Fo Fc map emphasizes errors in the current model, but it lacks the familiar appearance of the molecular surface found in an FQ map. In addition, if the model still contains many errors, the F — Fc map is "noisy," full of small positive and negative peaks that are difficult to interpret. The FQ— Fc map is most useful near the end of the structure determination, when most of the model errors have been eliminated. The FQ— Fc map is a great aid in detecting subtle errors after most of the serious errors are corrected. 

A more easily interpreted and intuitively satisfying difference map, but one that still allows undue influence by the model to be detected, is the 2FQ — Fc map, calculated as follows... 

With each successive map, new molecular features are added as they can be discerned, and errors in the model, such as side-chain conformations that no longer fit the electron density, are corrected. As the structure nears completion, the crystallographer may use 2F0 Fc and FQ— Fc maps simultaneously to track down the most subtle disagreements between the model and the data. 

At the end of successful refinement, the 2F0 Fc map almost looks like a space-filling model of the protein. (Refer to Plate 2 b, which is the final model... 

Occupancies rij for atoms of the protein (but not necessarily its ligands, which may be present at lower occupancies) are usually constrained at 1.0 early in refinement, and in many refinements are never released, so that both thermal motion and disorder show their effects upon the final B values. In some cases, after refinement converges, a few B values fall far outside the average range for the model. This is sometimes an indication of disorder. Careful examination of 2Fq— Fc and FQ— Fc maps may give evidence for more than one conformation in such a troublesome region. If so, inclusion of multiple conformations followed by refinement of their occupancies may improve the R-factor and the map, revealing the nature of the disorder more clearly. 

In between rounds of computerized refinement, maps were computed using IFobsls from the ALBP data set and acalcs from the current model [taken from IFcalcls computed by Eq. (5.15)]. The model was corrected where the fit to maps was poor, or where the Ramachandran angles and P were forbidden. Notice that the use of 2F0 — Fc and Fc — Fc maps [Eqs. (7.4) and (7.5)] is as... 

To compare apo- and holo-forms of proteins after both structures have been determined independently, crystallographers often compute difference Fourier syntheses (Chapter 7, Section IV.B), in which each Fourier term contains the structure-factor difference FAc>/c(-—F 0. A contour map of this Fourier series is called a difference map, and it shows only the differences between the holo-and apo- forms. Like the FQ — Fc map, the FAoio—F map contains both positive and negative density. Positive density occurs where the electron density of the holo-form is greater than that of the apo-form, so the ligand shows up clearly in positive density. In addition, conformational differences between holo- and apo-forms result in positive density where holo-protein atoms occupy regions that are unoccupied in the apo-form, and negative... 

In order to discriminate this special case from dynamical mapping we introduce the term FC mapping. As an immediate consequence we note that ... 

The photodissociation of H2O via the AlB state (the first absorption band) provides a perfect example for FC mapping. Figure 10.4 depicts the corresponding PES as function of one of the O-H bonds and the HOH bending angle with the other O-H bond being fixed at the equilibrium value in H20(X). Because the center-of-mass of OH nearly coincides with the oxygen atom the bond coordinates (/ OHia) are almost equivalent to the Jacobi coordinates (R, 7). 

Fig. 18 The solution of epothilone bound to tubulin by electron crystallography, a 2Fo-Fc map and model of epothilone A bound to tubulin at la (ltvk) [3], b Energy optimized model derived from ltvk through MAID protocol used as template for analysis of SAR and design of new analogs...

Fig. 26.11 The rationale behind the 2Fq - Fc maps. Shown is a carbohydrate attachment site at Asn109 in human leukocyte elastase. The carbohydrate chain has not been added to the model. The Fq map in (a) mainly shows the electron density of the model, owing to the model bias. Some electron density is present for the carbohydrate, but it is very weak and not connected to the Asn residue. The Fo - Fc map in (b) shows only features which have not been accounted for in the model, in this case the carbohydrate moiety. The 2Fo - Fg map in (c) is a combination of the two and shows both density belonging to the model and strong connected density for the carbohydrate. From these figures, it is clear why crystallographers prefer a 2Fo - Fc map over an Fq map.

Water molecules are never ordered unless they are hydrogen bonded to another atona. In such a case, it would be best to remove the orphaned water molecule from the model and run another round of automated refinement. Examine the Fo—Fc map in this region after the refinement. Chances are that the density will have disappeared, confirming that the water was modeled incorrectly. If, however, the positive density is found, then one should broaden one s view and look for the possibility that the density arose from an alternate side chain conformation or a larger ligand. If the resolution of the data is worse than 2.7 A, there are usually little or no water molecules to be found, as is the case in the lja3 example. 

The function can also be mapped into a conformally scaled fC-map directly from the canonical expansion, this being essentially the same process as entering the minterms (or maxterms) from the truth table. The function may also be directly mapped from any PS or SP expansion form. Another means of obtaining the K-map is to formulate it as a function table, as illustrated for the multiplication off - and 2-b numbers in Fig. 1.48. 

Figure 9 Model for the difference electron density of CoilSerL16Pen using either (a) Cys or (b) Val as a model for the non-coded for amino acid penicillamine (Pen) in the interior of a three stranded coiled coil. Shown is a top down view bom the N-terminus of the three-stranded coiled coil (PDB 3H5F) during the building of the Pen side chain [64]. The 2Fa-Fc map (blue, contoured at 1.5a) and Fq-Fc map (green, contoiued at 3o) after one round of refinement, show that the density for the methyl groups (at the position of the P-methylene protons) is missing for the Cys model, as is the thiol group when Val is used. The other amino acid side chains have been omitted for clarity.

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