For cryo-EM maps with resolutions of 15 A or worse, molecular volume is a rather unreliable guide to the selection of density threshold, since the limited resolution leads both to an infilling of interior channels and crevices, and a truncation of thin components sticking out into the solvent. This becomes quite evident when comparing a structure displayed at different resolutions (see figure 6.6, where X-ray structure coordinates are used to generate density maps of the ribosome at different resolutions). The volume is a quantity with fractal properties (see section 5.3.2), and hence very sensitive to resolution.
The easiest way to obtain the density associated with a desired molecular volume is by computing a cumulative histogram, which expresses the number of voxels above a given density dth as a function of dth. It is related to the density histogram described above by integration. The threshold dth can be immediately looked up as that density for which the associated contour encloses a given molecular volume.
An example for the use of volume constraint is contained in the work of de Haas and coworkers (1996c, 1997), who used it to determine the thresholds for the display of, respectively, Riftia pachyptila hemoglobin and Lumbricus terrestris extracellular hemoglobin, as obtained by single-particle reconstruction. In the first of these papers, the authors go into a comparison of the volume obtained from an examination of the density histogram with the volume estimated from molecular mass, and find a large discrepancy (a factor of 1.7 in favor of the volume estimated from the histogram). It is quite likely that at resolutions as low as 36 A, used in the study, volume deduced from the density map simply ceases to be useful as a guide.
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