On the helical arrangements of protein molecules

Abstract

Helical structures are prevalent in biology. In the PDB, there are many examples where protein molecules are helically arranged, not only according to strict crystallographic screw axes but also according to approximate noncrystallographic screws. The preponderance of such screws is rather striking as helical arrangements in crystals must preserve an integer number of subunits per turn, while intuition and simple packing arguments would seem to favor fractional helices. The article provides insights into such questions, based on stereochemistry, trigonometry, and topology, and illustrates the findings with concrete PDB structures. Updated statistics of Sohncke space groups in the PDB are also presented.

Keywords: DNA double helix; Protein Data Bank (PDB); crystal packing; helical virus capsids; noncrystallographic symmetry; screw axes; space group frequency; α-helix.

Figure 1

Examples of noncrystallographic helices in the PDB protein crystal structures, selected from Table 1, shown in A–D down (left) and across (right) the helix axis (marked); and the corresponding plots of the self‐rotation Patterson functions (E–H) calculated for these structures. In the stereographic projections of the self‐rotation function, ω (inclination, from 0° in the center to 90° at the perimeter) and ϕ (azimuth, marked on the perimeter) define the orientation of a rotation axis with unit rotation κ. The structures, identified by their PDB codes, are as follows: (A, E) 2PNL, with 5₃ screw axis; (B, F) 5V50 with 7₅ screw axis; (C, G) 3KBX with 10₉ screw axis within 10₉22 symmetry; (D, H) 5HM2 with 15₂ screw axis. The structural figures (A–D) were prepared with PyMol (version 1.6, Schrodinger LLC, http://www.pymol.org). The stereographic plots (E–H) of the self‐rotation Patterson functions for the same structures were prepared with XPREP (Sheldrick, 2008).

Figure 2

A fragment of a discrete helix (polygonal chain) with the external (r, φ, h) and internal (d, α, τ) parameters indicated.

Figure 3

Interdependence of the internal angles α and τ (°) for helices with the same number of unites per one complete turn, with u = 360°/φ. The curves correspond to integer values of u from 2 to 11, except for the dashed black curve for u = 3.6, which corresponds to the α‐helix. See text for a detailed description.

Figure 4

Illustration of two noncrystallographic helical screw axes: (A) 5₂ and (B) 9₃. These screws have multiple threads marked in different colors. The 9₃ screw contains in addition the normal threefold axis, marked in orange.

Figure 5

The capsid of the TMV virus (PDB ID 4UFT) viewed down the helix axis (seen as a black dot). The figure, prepared with Rasmol (http://www.rasmol.org), shows the Cα traces of the protein subunits forming the helical capsid. The molecules are colored successively along the helix from blue to red.