Algorithm for ribbon models of proteins.

A method is presented to draw smooth, 3D ribbon models of proteins. The procedure calculates closely-spaced guide coordinates based on the peptide plane and passes regular, nearly parallel B-spline curves through them. This becomes a simple process with a graphics device having built-in B-spline generating capabilities such as the Evans and Sutherland PS300. Examination of ribbons such as these provides a useful toolfor the crystallographer. Any irregularity in the ribbon is a strong visual cue, suggestive of potential problem areas during the refinement process.

Keywords: proteins, 3D ribbon models, B-spline curves, ubiquitin, purine nucleoside phosphorylase, calmodulin

received 25 November 1985, accepted 6 January 1986

The 'ribbon drawing' of Richardson(1) provides an elegant method enabling the vizualization of folding and secondary structure of proteins. Computer programs for the generation of drawings of this type have been written by several authors. Lesk and Hardman(2) plot the ribbon approximated as a series of linked trapezoids, one segment per residue. Pique(3) produces a series of short line segments perpendicular to a smooth curve fit to the protein backbone. The method presented below models the protein ribbon in much the same manner as the ribbons one uses to wrap presents, i.e., composed of many approximately parallel smooth threads running along the length of the ribbon. The key point in relation to protein chemistry is that the peptide plane is used as the basis for the geometrical construction. The key point relating to graphics is that B-splines(4) are used to produce the smooth and regular curves that model the ribbon.

ALGORITHM:

Defining peptide planes

The minimal required input is a list of the protein's alpha carbon and carbonyl oxygen coordinates ordered by sequence. The following procedures are carried out to define the peptide plane:

• Get vector A = Ca(i+l) - Ca(i)
• Get vector B = O(i) - Ca(i)
• Form vector C = A x B
• Form vector D = C x A
• Normalize C and D

Vector C is normal to the peptide plane approximated by atoms Ca(i), O(i), and Ca(i+1). The 'handedness' chosen here has C pointing away from the helix axis for righthanded alpha-helices. Vector D lies parallel to the peptide plane and is perpendicular to vector A.

Generating guide coordinates

The required inputs are the number of threads necessary to approximate the ribbon (nine is convenient), an assignment for each residue of secondary structural class (helix, sheet, coil or turn), and the desired ribbon widths (for example, 3.0 A for secondary structure, 1.0 A for turns and random coils).

The following procedures are carried out to define the B-spline guide points:

Form point P as the midpoint of Ca(i) and Ca(i+1)

If the peptide plane is part of a helix, translate in the direction of vector C, away from the helix axis (this step is elaborated on in the discussion)

Scale vector D by one half the desired ribbon width

Form points P(-) = P - D and P(+) = P + D

After these procedures have been carried out the line segment defined by P(-) --> P(+) is of the desired ribbon width and parallel to the peptide plane. The guide coordinates are evenly spaced along this line segment. A set of guide points is then generated from the next peptide plane in the sequence, and the test below is performed.

Determining the connectivity of the guide coordinates

It is desired to have a number of threads that make up the roughly parallel long fibres of the ribbon. A complication arises when the direction of the carbonyl oxygen flips, as is always the case between adjacent residues of beta sheets. This can be monitored by taking the vector product of the D vectors of successive peptide planes. If the angle between D(i-1), and D(i) is greater than 90 degrees, a flip is noted. This information is used to determine if the line segment defined in the previous step will have its first point, P-, or its last point, P+, assigned to the first thread of the ribbon.

.....

And the figures were all beautiful line-drawings complements of the E&S vector graphics, and photographed directly from the screen. Colour plate 1 has been scanned from a reprint.

Colour Plate 1 - Ribbon construction with the protein Ubiquitin.
A white 13-thread smooth ribbon is overlaid with a two-thread ribbon in which the flip of the carbonyl is not monitored. The red thread approximately traces the positions of the carbonyl oxygen atoms, and the blue thread the positions of the amide hydrogen atoms.