Fig 1.
The averaging process for helix model computation.
In this example the first helical curve H14 is computed using the first quadruple of backbone atoms {a1,a2,a3,a4}, the second curve H25 the next quadruple of atoms {a2,a3,a4,a5} and so on. For a pair of two consecutive interior atoms up to three slightly different curves could be computed. The final model curve for the segment between a pair of consecutive atoms is their average (Eq 3).
Fig 2.
The spatial differences between the backbone atoms and their closest points on the helix ribbon diagram generated by the model are barely visible (indicated by the arrows) for either (a) an ultra-high resolution protein structure (pdbid 1EJG) or (b) a medium-resolution structure (pdbid 2RH1). The backbones of the helices are shown in stick-and-ball with the diameter of the ball to be the same as the thickness of the ribbon. The Cα atoms are colored in cyan. A detachment occurs when a Cα atom is not positioned inside the ribbon diagram. The larger the difference is between a Cα atom and its closest point on the model, the larger its detachment from the diagram. The protein helix diagrams in both the main paper and Supporting Information (SI) are colored as follows according to residue’s helix score (Eq 2): 0.0–20.0 in green, 20.0–50.0 in celeste, 50.0–100.0 in yellow, 100.0–200.0 in magenta, > 200.0 in red. Except for Fig 1, all the figures in both the main paper and SI are prepared using our own molecular visualization program written in C++/openGL/Qt.
Fig 3.
A DNA ribbon diagram and its local distortions.
In (a) the differences between the DNA ribbon diagram generated by our helix model and the backbone P atoms (colored in maize) are visible for some nucleotides (indicated by an arrow) in a leucine zipper protein (pdbid 1A02). The protein ribbon diagram is colored according to residue’s helix score as in Fig 2. The residues with high helix scores are concentrated in the protein-DNA interface. For clarity neither protein loops nor β-strands are displayed in (a). In (b) the local distortions in the DNA ribbon diagram at the protein-DNA interface are indicated by the arrows. Here a local distortion means a twist away from an ideal helix ribbon diagram generated by a genuine helical curve (S4f, S4j and S5 Figs in the SI for additional examples of DNA ribbon diagrams). The protein atoms in (b) are colored as follows: H in gray, C in green, N in blue and S in yellow.
Table 1.
Helix score vs solvent accessible area (SAA) for proteins.
The data are obtained on all the dssp [22] assigned helices on a set of 3,446 x-ray structures with a resolution between 1.0Å–2.0Å and with less than 70% sequence identity. There are 650,167 helix residues in total. In each column these residues are divided into two subsets, S≤ hT and S> hT, according to a helix score threshold hT with the residues in S≤ hT having hi ≤ hT while those in S> hT having hi > hT. The parameter λ is computed by fitting a SAA histogram to an exponential function, y = Aexp λt + b, where A and b are parameters and t and y are the variables. The parameter μ is the average of the SAAs for all the residues in S≤ hT or S> hT. The unit of μ is Å2. As shown in the table μ increases but λ decreases with the helix score, and thus the higher the helix score the higher probability of the residue being on a protein surface.
Fig 4.
The deviations from the standard protein helix of the residues in the binding sites of three GPCR structures.
In (a) the three GPCR structures (pdbid 1U19, 2RH1 and 3EML) are overlayed based on their sequence similarity [26]. The helix ribbon diagrams are colored as in Fig 2 except that the residues in 1U19, 2RH1 and 3EML with a helix score < 20.0 are colored respectively in green, classic rose and bright azure. The ligands and the helix center polylines in 1U19, 2RH1 and 3EML are similarly colored. In (b) each bend in a polyline indicates a deviation from the helix center polyline for a genuine helical curve, the latter is a straight line. Such bends are concentrated at the ligand binding sites.
Fig 5.
The helix ribbon diagrams generated by the model.
The two figures illustrate the differences between the helix ribbon diagrams generated by the model and the helix ribbon diagrams generated using a series of cubic Hermite splines that pass through every backbone atom. The latter is colored in purple and drawn in sausage-shape and overlayed upon the former. As is clear by the comparison, the diagrams generated by Hermite splines are choppy while those by our model are much more smooth. The helix diagrams are colored according to the residue’s helix score as in Fig 2.
Fig 6.
The comparisons of the helix ribbon diagrams by our model and two previous programs.
The protein structure is 1TIM (pdbid) for which Prof. Jane Richardson had hand-drawn an iconic ribbon diagram [7]. The helices are oriented as close as possible to their orientations in her diagram (see S4 Fig). Shown in (a) is the diagram generated by our model, in (b) the diagram by the program PyMOL [18]. The diagrams in both (c) and (d) are drawn by the program UCSF Chimera [16]. The series of splines in (c) pass exactly through the Cα atoms and are computed using a series of quadruples of Cα atoms. Additional steps have been applied to smooth out the choppiness in (c). The splines in (d) are generated using a series of a quintuple of backbone Cα atoms. The side-chains in (d) could become detached. The choppiness in the PyMOL’s diagrams that pass exactly through the backbone atoms is less pronounced than those by other programs (see S3 Fig) but still visible upon a careful examination. It is likely that the original Hermite splines have been smoothed out to some extent in PyMOL.