Fig 1.
The Ramachandran plot is an important way of describing protein secondary structure.
(a) The state of a residue within a peptide (top) and a peptoid (bottom) can be largely specified by the two dihedral angles ϕ and ψ. (b) Regular protein secondary structures, such as α-helices and β-sheets, correspond to single diffuse regions on a plot drawn in terms of ϕ and ψ, called a Ramachandran plot (see Methods). (c) Peptoid Σ-sheets [16] harbor a secondary-structure motif in which backbone residues alternate between two regions on the Ramachandran plot. In order to describe each region in terms of a single number, so that the state of each residue in a backbone can be compactly indicated, we describe in this paper the development and properties of a structurally meaningful combination of ϕ and ψ that we call the Ramachandran number, . [Panel (a) was adapted from an image found on Wikimedia Commons (link) by Dcrjsr (CC BY 3.0 (link)). The contours in (b) and (c) represent regions within which 70% of a secondary structure resides; see Section 4.1.].
Fig 2.
Physical trends within the Ramachandran plot suggest a way of describing regions of it with a single number.
(a) First, the sense of residue twist changes from right-handed (‘D’) to left-handed (‘L’) as one moves from the bottom left of the Ramachandran plot to the top right. Second, contours (colored) of end-to-end polymer distance Re (here calculated for a 20-residue glycine) have a negative slope, resulting in the general trend shown in panel (b). Panel (c) indicates one method of indexing the Ramachandran plot so as to move from the region of right-handed twist to the region of left-handed twist with Re changing as slowly as possible. This method provides the basis for the construction of the Ramachandran number, .
Fig 3.
Potential pathologies of are avoided by the sparse occupancy of the Ramachandran plot.
(a) We construct by slicing across the Ramachandran plot, which can cause points distant in dihedral angle space to be grouped together, the more so as we approach the negative-sloping diagonal (near
= 0.5). This grouping can be inferred by superposing the standard deviation (error bars) in polymer end-to-end distance on top of the mean value (smooth line) for hypothetical structures built from the relevant part of the Ramachandran diagram. (b) However, many structures distant in dihedral angle space but close in
do not arise in proteins; the Ramachandran diagram is in general relatively sparsely occupied. Consequently,
can resolve the major types of protein secondary structure, which can be inferred from the fact that lines parallel to the negative-sloping diagonal (marked), along which
varies only slowly, can touch each region of known secondary structure (colored) individually. This sensitivity allows
to function as an order parameter for protein geometry. [Data in (a) were calculated for a 5-residue peptoid;
values are shown at discrete intervals of 0.01.].
Fig 4.
The indexing system defined by Eqs (6) and (1) collapses the Ramachandran plot into a single line, the Ramachandran number .
This number can act as an order parameter to distinguish secondary structures of different geometry, as shown (the overlap between distributions exists in the original Ramachandran plot representation; see Fig 3(b)). Top: interpolates between regions of right-handed and left-handed twist, with polymer extension Re varying smoothly throughout.
Fig 5.
Four ways of looking at secondary structure: a) Molecular configurations; b) Ramachandran plot; c) Histogram (-code) of Ramachandran numbers; and (d)
as a function of residue number (for the Σ-sheet we have chosen a single polymer).
Panel (c) provides a compact assay-by-geometry of the residues within molecular structures, while panel (d) shows that one can use to identify the spatial connectivity of domains of secondary structure within a polymer.
Fig 6.
(a) -codes for the SCOP protein dataset reveal at a glance several geometric properties of the set. Each column represents a histogram of the indicated protein class, normalized so that the largest value is unity. A feature common to all classes is the prominence of α-helices (
≈0.36). Another common feature is the presence of loops that connect ordered secondary structure (
≈0.62). Moreover, α-helical regions are prominently visible in ‘all-β’ proteins. (b) The
-code for a peptoid nanosheet shows two dominant rotational states, which coexist within a single secondary structure (see Figs 5 and 7).
Fig 7.
(a) Molecular dynamics simulations of the peptoid nanosheet [16, 23] show the existence of the Σ-strand secondary structure motif, within which residues possess two distinct rotational states (colored red and blue in the bottom-right-hand cutaway). (b) A time series of the -code of the bilayer shows the emergence (to the right of the vertical dotted line) of the Σ-strand motif within molecular dynamics simulations. Polymers in these simulations were initially fully extended, and adopted the Σ-strand motif upon relaxation of their backbone constraints. (c) Geometric state of each residue in one peptoid as a function of time, revealing the emergence of the Σ-strand structure and the subsequent fluctuations of individual residues on a nanosecond timescale.
Fig 8.
(a) Coordinate transformation applied to the Ramachandran plot in order to compute the Ramachandran number . A rotation, shift, and rescaling of ϕ and ψ results in a representation in which horizontal cuts run roughly along contours of polymer extension; see Fig 2. (b) The indexing system defined by Eq (1), where bold numbers are those that fall on the original Ramachandran plot.
Fig 9.
Dihedral angles converted to Ramachandran numbers can be recovered only approximately, but the error incurred during this back-mapping can be made much smaller than the standard error (typically 1Å) associated with structures in the protein databank.
Here we show the root-mean-squared-deviation (RMSD) in dihedral angles (a) and in protein α-carbon spatial coordinates (b) generated upon taking 8560 protein structures obtained from SCOP [36], converting their dihedral angles to Ramachandran numbers, and recovering approximately those dihedral angles using Eqs (7) and (8). The parameter σ indicates the grid resolution used to calculate R; see Eq (1).